LMFDB - Embedded newform 115.2.b.b.24.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(24,115)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.527896576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 2x^5 + 7x^4 - 10x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.2
Root			$-0.199724 + 0.199724i$ of defining polynomial
Character	$\chi$	$=$	115.24
Dual form			115.2.b.b.24.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.92022i q^{2} -1.39945i q^{3} -1.68725 q^{4} +(-1.19972 - 1.88697i) q^{5} -2.68725 q^{6} +4.60747i q^{7} -0.600553i q^{8} +1.04155 q^{9} +O(q^{10})$	\(q-1.92022i q^{2} -1.39945i q^{3} -1.68725 q^{4} +(-1.19972 - 1.88697i) q^{5} -2.68725 q^{6} +4.60747i q^{7} -0.600553i q^{8} +1.04155 q^{9} +(-3.62340 + 2.30373i) q^{10} +1.56592 q^{11} +2.36122i q^{12} -2.60055i q^{13} +8.84736 q^{14} +(-2.64072 + 1.67895i) q^{15} -4.52769 q^{16} +0.559006i q^{17} -2.00000i q^{18} +1.16647 q^{19} +(2.02423 + 3.18379i) q^{20} +6.44791 q^{21} -3.00692i q^{22} +1.00000i q^{23} -0.840442 q^{24} +(-2.12133 + 4.52769i) q^{25} -4.99364 q^{26} -5.65593i q^{27} -7.77394i q^{28} +3.17339 q^{29} +(3.22396 + 5.07076i) q^{30} +10.0554 q^{31} +7.49306i q^{32} -2.19143i q^{33} +1.07341 q^{34} +(8.69416 - 5.52769i) q^{35} -1.75735 q^{36} +5.07341i q^{37} -2.23989i q^{38} -3.63934 q^{39} +(-1.13323 + 0.720497i) q^{40} -11.8127 q^{41} -12.3814i q^{42} +2.76426i q^{43} -2.64210 q^{44} +(-1.24957 - 1.96537i) q^{45} +1.92022 q^{46} +9.32298i q^{47} +6.33626i q^{48} -14.2288 q^{49} +(8.69416 + 4.07341i) q^{50} +0.782299 q^{51} +4.38778i q^{52} +5.54789i q^{53} -10.8606 q^{54} +(-1.87867 - 2.95485i) q^{55} +2.76703 q^{56} -1.63242i q^{57} -6.09361i q^{58} -3.84044 q^{59} +(4.45555 - 2.83281i) q^{60} -4.29832 q^{61} -19.3086i q^{62} +4.79889i q^{63} +5.33295 q^{64} +(-4.90717 + 3.11994i) q^{65} -4.20802 q^{66} +2.60747i q^{67} -0.943181i q^{68} +1.39945 q^{69} +(-10.6144 - 16.6947i) q^{70} +7.89582 q^{71} -0.625504i q^{72} -9.90579i q^{73} +9.74208 q^{74} +(6.33626 + 2.96868i) q^{75} -1.96813 q^{76} +7.21494i q^{77} +6.98833i q^{78} -12.0485 q^{79} +(5.43198 + 8.54362i) q^{80} -4.79054 q^{81} +22.6830i q^{82} -13.3856i q^{83} -10.8792 q^{84} +(1.05483 - 0.670652i) q^{85} +5.30800 q^{86} -4.44099i q^{87} -0.940419i q^{88} +7.71551 q^{89} +(-3.77394 + 2.39945i) q^{90} +11.9820 q^{91} -1.68725i q^{92} -14.0720i q^{93} +17.9022 q^{94} +(-1.39945 - 2.20111i) q^{95} +10.4861 q^{96} -2.62130i q^{97} +27.3224i q^{98} +1.63098 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9	$ 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9} + 6 q^{10} + 4 q^{11} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{21} + 24 q^{24} - 16 q^{25} + 12 q^{26} - 8 q^{29} - 2 q^{30} - 28 q^{34} + 28 q^{35} - 16 q^{36} + 16 q^{39} - 10 q^{40} - 16 q^{41} - 12 q^{44} + 24 q^{45} + 28 q^{50} + 20 q^{51} - 44 q^{54} - 16 q^{55} + 28 q^{56} - 16 q^{60} - 16 q^{61} + 40 q^{64} - 14 q^{65} - 16 q^{66} + 4 q^{69} - 28 q^{70} - 48 q^{71} + 72 q^{74} - 36 q^{76} - 48 q^{79} - 2 q^{80} + 16 q^{81} - 4 q^{84} + 12 q^{85} + 28 q^{86} + 16 q^{89} - 4 q^{90} + 52 q^{91} + 84 q^{94} - 4 q^{95} + 60 q^{96} - 72 q^{99}+O(q^{100}) $ 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9 + 6 * q^10 + 4 * q^11 + 8 * q^14 + 6 * q^15 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 4 * q^21 + 24 * q^24 - 16 * q^25 + 12 * q^26 - 8 * q^29 - 2 * q^30 - 28 * q^34 + 28 * q^35 - 16 * q^36 + 16 * q^39 - 10 * q^40 - 16 * q^41 - 12 * q^44 + 24 * q^45 + 28 * q^50 + 20 * q^51 - 44 * q^54 - 16 * q^55 + 28 * q^56 - 16 * q^60 - 16 * q^61 + 40 * q^64 - 14 * q^65 - 16 * q^66 + 4 * q^69 - 28 * q^70 - 48 * q^71 + 72 * q^74 - 36 * q^76 - 48 * q^79 - 2 * q^80 + 16 * q^81 - 4 * q^84 + 12 * q^85 + 28 * q^86 + 16 * q^89 - 4 * q^90 + 52 * q^91 + 84 * q^94 - 4 * q^95 + 60 * q^96 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.92022i		−	1.35780i	−0.734230	−	0.678901i	$-0.762457\pi$
$2$		−	1.92022i		−	1.35780i	0.734230	−	0.678901i	$-0.237543\pi$
$3$		−	1.39945i		−	0.807971i	−0.914765	−	0.403986i	$-0.867625\pi$
$3$		−	1.39945i		−	0.807971i	0.914765	−	0.403986i	$-0.132375\pi$
$4$	−1.68725			−0.843624
$5$	−1.19972	−	1.88697i	−0.536533	−	0.843879i
$5$	−1.19972	−	1.88697i	−0.536533	−	0.843879i
$6$	−2.68725			−1.09706
$7$			4.60747i			1.74146i	0.491762	+	0.870730i	$0.336353\pi$
$7$			4.60747i			1.74146i	−0.491762	+	0.870730i	$0.663647\pi$
$8$		−	0.600553i		−	0.212327i
$9$	1.04155			0.347182
$10$	−3.62340	+	2.30373i	−1.14582	+	0.728505i
$11$	1.56592			0.472143			0.236072	−	0.971736i	$-0.424140\pi$
$11$	1.56592			0.472143			0.236072	+	0.971736i	$0.424140\pi$
$12$			2.36122i			0.681624i
$13$		−	2.60055i		−	0.721264i	−0.932708	−	0.360632i	$-0.882561\pi$
$13$		−	2.60055i		−	0.721264i	0.932708	−	0.360632i	$-0.117439\pi$
$14$	8.84736			2.36456
$15$	−2.64072	+	1.67895i	−0.681830	+	0.433503i
$16$	−4.52769			−1.13192
$17$			0.559006i			0.135579i	0.997700	+	0.0677894i	$0.0215946\pi$
$17$			0.559006i			0.135579i	−0.997700	+	0.0677894i	$0.978405\pi$
$18$		−	2.00000i		−	0.471405i
$19$	1.16647			0.267608			0.133804	−	0.991008i	$-0.457281\pi$
$19$	1.16647			0.267608			0.133804	+	0.991008i	$0.457281\pi$
$20$	2.02423	+	3.18379i	0.452632	+	0.711917i
$21$	6.44791			1.40705
$22$		−	3.00692i		−	0.641077i
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$	−0.840442			−0.171554
$25$	−2.12133	+	4.52769i	−0.424265	+	0.905538i
$26$	−4.99364			−0.979332
$27$		−	5.65593i		−	1.08848i
$28$		−	7.77394i		−	1.46914i
$29$	3.17339			0.589284			0.294642	−	0.955608i	$-0.404800\pi$
$29$	3.17339			0.589284			0.294642	+	0.955608i	$0.404800\pi$
$30$	3.22396	+	5.07076i	0.588611	+	0.925790i
$31$	10.0554			1.80600			0.903000	−	0.429641i	$-0.141360\pi$
$31$	10.0554			1.80600			0.903000	+	0.429641i	$0.141360\pi$
$32$			7.49306i			1.32460i
$33$		−	2.19143i		−	0.381478i
$34$	1.07341			0.184089
$35$	8.69416	−	5.52769i	1.46958	−	0.934350i
$36$	−1.75735			−0.292891
$37$			5.07341i			0.834064i	0.908892	+	0.417032i	$0.136930\pi$
$37$			5.07341i			0.834064i	−0.908892	+	0.417032i	$0.863070\pi$
$38$		−	2.23989i		−	0.363358i
$39$	−3.63934			−0.582760
$40$	−1.13323	+	0.720497i	−0.179179	+	0.113921i
$41$	−11.8127			−1.84484			−0.922419	−	0.386190i	$-0.873791\pi$
$41$	−11.8127			−1.84484			−0.922419	+	0.386190i	$0.873791\pi$
$42$		−	12.3814i		−	1.91049i
$43$			2.76426i			0.421546i	0.977535	+	0.210773i	$0.0675982\pi$
$43$			2.76426i			0.421546i	−0.977535	+	0.210773i	$0.932402\pi$
$44$	−2.64210			−0.398311
$45$	−1.24957	−	1.96537i	−0.186275	−	0.292980i
$46$	1.92022			0.283121
$47$			9.32298i			1.35990i	0.733260	+	0.679948i	$0.237998\pi$
$47$			9.32298i			1.35990i	−0.733260	+	0.679948i	$0.762002\pi$
$48$			6.33626i			0.914561i
$49$	−14.2288			−2.03268
$50$	8.69416	+	4.07341i	1.22954	+	0.576068i
$51$	0.782299			0.109544
$52$			4.38778i			0.608475i
$53$			5.54789i			0.762061i	0.924562	+	0.381030i	$0.124431\pi$
$53$			5.54789i			0.762061i	−0.924562	+	0.381030i	$0.875569\pi$
$54$	−10.8606			−1.47795
$55$	−1.87867	−	2.95485i	−0.253320	−	0.398432i
$56$	2.76703			0.369760
$57$		−	1.63242i		−	0.216219i
$58$		−	6.09361i		−	0.800130i
$59$	−3.84044			−0.499983			−0.249991	−	0.968248i	$-0.580428\pi$
$59$	−3.84044			−0.499983			−0.249991	+	0.968248i	$0.580428\pi$
$60$	4.45555	−	2.83281i	0.575209	−	0.365714i
$61$	−4.29832			−0.550343			−0.275172	−	0.961395i	$-0.588735\pi$
$61$	−4.29832			−0.550343			−0.275172	+	0.961395i	$0.588735\pi$
$62$		−	19.3086i		−	2.45219i
$63$			4.79889i			0.604604i
$64$	5.33295			0.666619
$65$	−4.90717	+	3.11994i	−0.608659	+	0.386981i
$66$	−4.20802			−0.517972
$67$			2.60747i			0.318553i	0.987234	+	0.159277i	$0.0509161\pi$
$67$			2.60747i			0.318553i	−0.987234	+	0.159277i	$0.949084\pi$
$68$		−	0.943181i		−	0.114378i
$69$	1.39945			0.168474
$70$	−10.6144	−	16.6947i	−1.26866	−	1.99540i
$71$	7.89582			0.937062			0.468531	−	0.883447i	$-0.344783\pi$
$71$	7.89582			0.937062			0.468531	+	0.883447i	$0.344783\pi$
$72$		−	0.625504i		−	0.0737163i
$73$		−	9.90579i		−	1.15938i	−0.814835	−	0.579692i	$-0.803173\pi$
$73$		−	9.90579i		−	1.15938i	0.814835	−	0.579692i	$-0.196827\pi$
$74$	9.74208			1.13249
$75$	6.33626	+	2.96868i	0.731649	+	0.342794i
$76$	−1.96813			−0.225760
$77$			7.21494i			0.822219i
$78$			6.98833i			0.791273i
$79$	−12.0485			−1.35556			−0.677779	−	0.735266i	$-0.737057\pi$
$79$	−12.0485			−1.35556			−0.677779	+	0.735266i	$0.737057\pi$
$80$	5.43198	+	8.54362i	0.607313	+	0.955206i
$81$	−4.79054			−0.532282
$82$			22.6830i			2.50492i
$83$		−	13.3856i		−	1.46926i	−0.678470	−	0.734628i	$-0.737357\pi$
$83$		−	13.3856i		−	1.46926i	0.678470	−	0.734628i	$-0.262643\pi$
$84$	−10.8792			−1.18702
$85$	1.05483	−	0.670652i	0.114412	−	0.0727425i
$86$	5.30800			0.572376
$87$		−	4.44099i		−	0.476125i
$88$		−	0.940419i		−	0.100249i
$89$	7.71551			0.817843			0.408921	−	0.912570i	$-0.365905\pi$
$89$	7.71551			0.817843			0.408921	+	0.912570i	$0.365905\pi$
$90$	−3.77394	+	2.39945i	−0.397809	+	0.252924i
$91$	11.9820			1.25605
$92$		−	1.68725i		−	0.175908i
$93$		−	14.0720i		−	1.45920i
$94$	17.9022			1.84647
$95$	−1.39945	−	2.20111i	−0.143580	−	0.225829i
$96$	10.4861			1.07024
$97$		−	2.62130i		−	0.266153i	−0.991106	−	0.133076i	$-0.957514\pi$
$97$		−	2.62130i		−	0.266153i	0.991106	−	0.133076i	$-0.0424856\pi$
$98$			27.3224i			2.75998i
$99$	1.63098			0.163920
$100$	3.57920	−	7.63934i	0.357920	−	0.763934i
$101$	−8.88199			−0.883791			−0.441895	−	0.897067i	$-0.645694\pi$
$101$	−8.88199			−0.883791			−0.441895	+	0.897067i	$0.645694\pi$
$102$		−	1.50219i		−	0.148739i
$103$			8.65593i			0.852894i	0.904512	+	0.426447i	$0.140235\pi$
$103$			8.65593i			0.852894i	−0.904512	+	0.426447i	$0.859765\pi$
$104$	−1.56177			−0.153144
$105$	−7.73571	−	12.1670i	−0.754928	−	1.18738i
$106$	10.6532			1.03473
$107$		−	6.91662i		−	0.668655i	−0.942457	−	0.334327i	$-0.891491\pi$
$107$		−	6.91662i		−	0.668655i	0.942457	−	0.334327i	$-0.108509\pi$
$108$			9.54296i			0.918272i
$109$	6.84736			0.655858			0.327929	−	0.944702i	$-0.393649\pi$
$109$	6.84736			0.655858			0.327929	+	0.944702i	$0.393649\pi$
$110$	−5.67397	+	3.60747i	−0.540992	+	0.343959i
$111$	7.09998			0.673900
$112$		−	20.8612i		−	1.97120i
$113$			4.60747i			0.433434i	0.976234	+	0.216717i	$0.0695349\pi$
$113$			4.60747i			0.433434i	−0.976234	+	0.216717i	$0.930465\pi$
$114$	−3.13461			−0.293583
$115$	1.88697	−	1.19972i	0.175961	−	0.111875i
$116$	−5.35430			−0.497134
$117$		−	2.70860i		−	0.250410i
$118$			7.37450i			0.678877i
$119$	−2.57560			−0.236105
$120$	1.00830	+	1.58589i	0.0920446	+	0.144771i
$121$	−8.54789			−0.777081
$122$			8.25372i			0.747257i
$123$			16.5313i			1.49058i
$124$	−16.9659			−1.52358
$125$	11.0886	−	1.42909i	0.991797	−	0.127822i
$126$	9.21494			0.820932
$127$			11.5324i			1.02334i	0.859182	+	0.511669i	$0.170973\pi$
$127$			11.5324i			1.02334i	−0.859182	+	0.511669i	$0.829027\pi$
$128$			4.74568i			0.419463i
$129$	3.86844			0.340597
$130$	5.99098	+	9.42285i	0.525444	+	0.826439i
$131$	−12.9639			−1.13266			−0.566332	−	0.824177i	$-0.691638\pi$
$131$	−12.9639			−1.13266			−0.566332	+	0.824177i	$0.691638\pi$
$132$			3.69748i			0.321824i
$133$			5.37450i			0.466028i
$134$	5.00692			0.432532
$135$	−10.6726	+	6.78556i	−0.918550	+	0.584008i
$136$	0.335712			0.0287871
$137$		−	17.1457i		−	1.46485i	−0.680846	−	0.732427i	$-0.738388\pi$
$137$		−	17.1457i		−	1.46485i	0.680846	−	0.732427i	$-0.261612\pi$
$138$		−	2.68725i		−	0.228754i
$139$	19.4798			1.65225			0.826127	−	0.563485i	$-0.190540\pi$
$139$	19.4798			1.65225			0.826127	+	0.563485i	$0.190540\pi$
$140$	−14.6692	+	9.32658i	−1.23977	+	0.788240i
$141$	13.0470			1.09876
$142$		−	15.1617i		−	1.27234i
$143$		−	4.07226i		−	0.340540i
$144$	−4.71580			−0.392983
$145$	−3.80719	−	5.98810i	−0.316170	−	0.497285i
$146$	−19.0213			−1.57421
$147$			19.9124i			1.64235i
$148$		−	8.56011i		−	0.703637i
$149$	−13.2288			−1.08374			−0.541872	−	0.840461i	$-0.682284\pi$
$149$	−13.2288			−1.08374			−0.541872	+	0.840461i	$0.682284\pi$
$150$	5.70053	−	12.1670i	0.465446	−	0.993434i
$151$	−2.50749			−0.204057			−0.102028	−	0.994781i	$-0.532533\pi$
$151$	−2.50749			−0.204057			−0.102028	+	0.994781i	$0.532533\pi$
$152$		−	0.700529i		−	0.0568204i
$153$			0.582231i			0.0470706i
$154$	13.8543			1.11641
$155$	−12.0637	−	18.9742i	−0.968978	−	1.52405i
$156$	6.14046			0.491631
$157$		−	8.52824i		−	0.680628i	−0.940312	−	0.340314i	$-0.889467\pi$
$157$		−	8.52824i		−	0.680628i	0.940312	−	0.340314i	$-0.110533\pi$
$158$			23.1357i			1.84058i
$159$	7.76398			0.615723
$160$	14.1392	−	8.98960i	1.11780	−	0.710690i
$161$	−4.60747			−0.363119
$162$			9.19889i			0.722733i
$163$			7.73240i			0.605648i	0.953046	+	0.302824i	$0.0979294\pi$
$163$			7.73240i			0.605648i	−0.953046	+	0.302824i	$0.902071\pi$
$164$	19.9310			1.55635
$165$	−4.13516	+	2.62911i	−0.321922	+	0.204676i
$166$	−25.7032			−1.99496
$167$		−	14.3064i		−	1.10706i	−0.832829	−	0.553531i	$-0.813280\pi$
$167$		−	14.3064i		−	1.10706i	0.832829	−	0.553531i	$-0.186720\pi$
$168$		−	3.87231i		−	0.298755i
$169$	6.23713			0.479779
$170$	−1.28780	−	2.02550i	−0.0987698	−	0.155349i
$171$	1.21494			0.0929086
$172$		−	4.66400i		−	0.355627i
$173$		−	15.4714i		−	1.17627i	−0.808763	−	0.588135i	$-0.799862\pi$
$173$		−	15.4714i		−	1.17627i	0.808763	−	0.588135i	$-0.200138\pi$
$174$	−8.52769			−0.646483
$175$	−20.8612	−	9.77394i	−1.57696	−	0.738841i
$176$	−7.09001			−0.534430
$177$			5.37450i			0.403972i
$178$		−	14.8155i		−	1.11047i
$179$	−5.98612			−0.447424			−0.223712	−	0.974655i	$-0.571817\pi$
$179$	−5.98612			−0.447424			−0.223712	+	0.974655i	$0.571817\pi$
$180$	2.10833	+	3.31607i	0.157146	+	0.247165i
$181$	−5.69892			−0.423597			−0.211799	−	0.977313i	$-0.567932\pi$
$181$	−5.69892			−0.423597			−0.211799	+	0.977313i	$0.567932\pi$
$182$		−	23.0080i		−	1.70547i
$183$			6.01527i			0.444662i
$184$	0.600553			0.0442733
$185$	9.57339	−	6.08670i	0.703850	−	0.447503i
$186$	−27.0213			−1.98130
$187$			0.875359i			0.0640126i
$188$		−	15.7302i		−	1.14724i
$189$	26.0595			1.89555
$190$	−4.22661	+	2.68725i	−0.306630	+	0.194953i
$191$	−8.34402			−0.603752			−0.301876	−	0.953347i	$-0.597613\pi$
$191$	−8.34402			−0.603752			−0.301876	+	0.953347i	$0.597613\pi$
$192$		−	7.46318i		−	0.538609i
$193$		−	16.0250i		−	1.15350i	−0.816920	−	0.576751i	$-0.804320\pi$
$193$		−	16.0250i		−	1.15350i	0.816920	−	0.576751i	$-0.195680\pi$
$194$	−5.03348			−0.361383
$195$	4.36620	+	6.86733i	0.312670	+	0.491779i
$196$	24.0075			1.71482
$197$			5.78893i			0.412444i	0.978505	+	0.206222i	$0.0661169\pi$
$197$			5.78893i			0.412444i	−0.978505	+	0.206222i	$0.933883\pi$
$198$		−	3.13184i		−	0.222570i
$199$	18.8085			1.33330			0.666651	−	0.745370i	$-0.267727\pi$
$199$	18.8085			1.33330			0.666651	+	0.745370i	$0.267727\pi$
$200$	2.71912	+	1.27397i	0.192271	+	0.0900831i
$201$	3.64902			0.257382
$202$			17.0554i			1.20001i
$203$			14.6213i			1.02621i
$204$	−1.31993			−0.0924138
$205$	14.1720	+	22.2903i	0.989816	+	1.55682i
$206$	16.6213			1.15806
$207$			1.04155i			0.0723925i
$208$			11.7745i			0.816414i
$209$	1.82661			0.126349
$210$	−23.3634	+	14.8543i	−1.61223	+	1.02504i
$211$	−4.88199			−0.336090			−0.168045	−	0.985779i	$-0.553745\pi$
$211$	−4.88199			−0.336090			−0.168045	+	0.985779i	$0.553745\pi$
$212$		−	9.36066i		−	0.642893i
$213$		−	11.0498i		−	0.757119i
$214$	−13.2814			−0.907900
$215$	5.21609	−	3.31635i	0.355734	−	0.226173i
$216$	−3.39668			−0.231115
$217$			46.3299i			3.14508i
$218$		−	13.1484i		−	0.890525i
$219$	−13.8626			−0.936750
$220$	3.16979	+	4.98557i	0.213707	+	0.336127i
$221$	1.45372			0.0977880
$222$		−	13.6335i		−	0.915022i
$223$			13.4786i			0.902596i	0.892373	+	0.451298i	$0.149039\pi$
$223$			13.4786i			0.902596i	−0.892373	+	0.451298i	$0.850961\pi$
$224$	−34.5240			−2.30673
$225$	−2.20946	+	4.71580i	−0.147297	+	0.314387i
$226$	8.84736			0.588518
$227$			13.6894i			0.908598i	0.890849	+	0.454299i	$0.150110\pi$
$227$			13.6894i			0.908598i	−0.890849	+	0.454299i	$0.849890\pi$
$228$			2.75430i			0.182408i
$229$	−27.3964			−1.81040			−0.905202	−	0.424981i	$-0.860281\pi$
$229$	−27.3964			−1.81040			−0.905202	+	0.424981i	$0.860281\pi$
$230$	−2.30373	−	3.62340i	−0.151904	−	0.238920i
$231$	10.0969			0.664329
$232$		−	1.90579i		−	0.125121i
$233$			0.0387841i			0.00254083i	0.999999	+	0.00127041i	$0.000404386\pi$
$233$			0.0387841i			0.00254083i	−0.999999	+	0.00127041i	$0.999596\pi$
$234$	−5.20111			−0.340007
$235$	17.5922	−	11.1850i	1.14759	−	0.729629i
$236$	6.47978			0.421798
$237$			16.8612i			1.09525i
$238$			4.94572i			0.320584i
$239$	13.9030			0.899312			0.449656	−	0.893202i	$-0.351547\pi$
$239$	13.9030			0.899312			0.449656	+	0.893202i	$0.351547\pi$
$240$	11.9564	−	7.60177i	0.771779	−	0.490692i
$241$	−11.9972			−0.772810			−0.386405	−	0.922329i	$-0.626283\pi$
$241$	−11.9972			−0.772810			−0.386405	+	0.922329i	$0.626283\pi$
$242$			16.4138i			1.05512i
$243$		−	10.2637i		−	0.658416i
$244$	7.25233			0.464283
$245$	17.0706	+	26.8493i	1.09060	+	1.71534i
$246$	31.7437			2.02391
$247$		−	3.03348i		−	0.193016i
$248$		−	6.03878i		−	0.383463i
$249$	−18.7324			−1.18712
$250$	−2.74418	−	21.2926i	−0.173557	−	1.34666i
$251$	−17.7806			−1.12230			−0.561150	−	0.827714i	$-0.689641\pi$
$251$	−17.7806			−1.12230			−0.561150	+	0.827714i	$0.689641\pi$
$252$		−	8.09693i		−	0.510058i
$253$			1.56592i			0.0984487i
$254$	22.1448			1.38949
$255$	−0.938543	−	1.47618i	−0.0587738	−	0.0924418i
$256$	19.7786			1.23617
$257$			10.2315i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$257$			10.2315i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$258$		−	7.42826i		−	0.462464i
$259$	−23.3756			−1.45249
$260$	8.27961	−	5.26412i	0.513480	−	0.326467i
$261$	3.30524			0.204589
$262$			24.8936i			1.53793i
$263$			3.99856i			0.246562i	0.992372	+	0.123281i	$0.0393416\pi$
$263$			3.99856i			0.246562i	−0.992372	+	0.123281i	$0.960658\pi$
$264$	−1.31607			−0.0809983
$265$	10.4687	−	6.65593i	0.643088	−	0.408871i
$266$	10.3202			0.632773
$267$		−	10.7975i		−	0.660794i
$268$		−	4.39945i		−	0.268739i
$269$	2.88084			0.175648			0.0878239	−	0.996136i	$-0.472009\pi$
$269$	2.88084			0.175648			0.0878239	+	0.996136i	$0.472009\pi$
$270$	13.0298	+	20.4937i	0.792966	+	1.24721i
$271$	14.1944			0.862250			0.431125	−	0.902292i	$-0.358117\pi$
$271$	14.1944			0.862250			0.431125	+	0.902292i	$0.358117\pi$
$272$		−	2.53100i		−	0.153465i
$273$		−	16.7681i		−	1.01485i
$274$	−32.9235			−1.98898
$275$	−3.32183	+	7.09001i	−0.200314	+	0.427544i
$276$	−2.36122			−0.142128
$277$			9.13576i			0.548915i	0.961599	+	0.274457i	$0.0884982\pi$
$277$			9.13576i			0.548915i	−0.961599	+	0.274457i	$0.911502\pi$
$278$		−	37.4055i		−	2.24343i
$279$	10.4731			0.627011
$280$	−3.31967	−	5.22130i	−0.198388	−	0.312033i
$281$	14.8640			0.886709			0.443355	−	0.896346i	$-0.353788\pi$
$281$	14.8640			0.886709			0.443355	+	0.896346i	$0.353788\pi$
$282$		−	25.0532i		−	1.49189i
$283$		−	3.03878i		−	0.180637i	−0.995913	−	0.0903185i	$-0.971212\pi$
$283$		−	3.03878i		−	0.180637i	0.995913	−	0.0903185i	$-0.0287885\pi$
$284$	−13.3222			−0.790528
$285$	−3.08033	+	1.95845i	−0.182463	+	0.116009i
$286$	−7.81964			−0.462385
$287$		−	54.4268i		−	3.21271i
$288$			7.80437i			0.459877i
$289$	16.6875			0.981618
$290$	−11.4985	+	7.31065i	−0.675214	+	0.429296i
$291$	−3.66837			−0.215044
$292$			16.7135i			0.978085i
$293$			19.2288i			1.12336i	0.827356	+	0.561678i	$0.189844\pi$
$293$			19.2288i			1.12336i	−0.827356	+	0.561678i	$0.810156\pi$
$294$	38.2362			2.22998
$295$	4.60747	+	7.24681i	0.268257	+	0.421925i
$296$	3.04685			0.177095
$297$		−	8.85675i		−	0.513921i
$298$			25.4022i			1.47151i
$299$	2.60055			0.150394
$300$	−10.6909	−	5.00891i	−0.617237	−	0.289189i
$301$	−12.7363			−0.734106
$302$			4.81494i			0.277069i
$303$			12.4299i			0.714078i
$304$	−5.28144			−0.302911
$305$	5.15680	+	8.11081i	0.295277	+	0.464423i
$306$	1.11801			0.0639125
$307$		−	20.5395i		−	1.17225i	−0.810220	−	0.586127i	$-0.800652\pi$
$307$		−	20.5395i		−	1.17225i	0.810220	−	0.586127i	$-0.199348\pi$
$308$		−	12.1734i		−	0.693643i
$309$	12.1135			0.689114
$310$	−36.4347	+	23.1649i	−2.06935	+	1.31568i
$311$	−9.83929			−0.557935			−0.278967	−	0.960301i	$-0.589992\pi$
$311$	−9.83929			−0.557935			−0.278967	+	0.960301i	$0.589992\pi$
$312$			2.18561i			0.123736i
$313$		−	22.4659i		−	1.26985i	−0.772574	−	0.634925i	$-0.781031\pi$
$313$		−	22.4659i		−	1.26985i	0.772574	−	0.634925i	$-0.218969\pi$
$314$	−16.3761			−0.924157
$315$	9.05538	−	5.75735i	0.510213	−	0.324390i
$316$	20.3287			1.14358
$317$		−	17.8404i		−	1.00202i	−0.865442	−	0.501010i	$-0.832962\pi$
$317$		−	17.8404i		−	1.00202i	0.865442	−	0.501010i	$-0.167038\pi$
$318$		−	14.9086i		−	0.836030i
$319$	4.96928			0.278226
$320$	−6.39807	−	10.0631i	−0.357663	−	0.562546i
$321$	−9.67944			−0.540254
$322$			8.84736i			0.493044i
$323$			0.652066i			0.0362819i
$324$	8.08283			0.449046
$325$	11.7745	+	5.51662i	0.653132	+	0.306007i
$326$	14.8479			0.822350
$327$		−	9.58252i		−	0.529914i
$328$			7.09416i			0.391710i
$329$	−42.9554			−2.36821
$330$	5.04846	+	7.94042i	0.277909	+	0.437106i
$331$	13.3130			0.731750			0.365875	−	0.930664i	$-0.380770\pi$
$331$	13.3130			0.731750			0.365875	+	0.930664i	$0.380770\pi$
$332$			22.5848i			1.23950i
$333$			5.28420i			0.289572i
$334$	−27.4714			−1.50317
$335$	4.92022	−	3.12824i	0.268820	−	0.170914i
$336$	−29.1941			−1.59267
$337$			3.08338i			0.167962i	0.996467	+	0.0839812i	$0.0267636\pi$
$337$			3.08338i			0.167962i	−0.996467	+	0.0839812i	$0.973236\pi$
$338$		−	11.9767i		−	0.651444i
$339$	6.44791			0.350202
$340$	−1.77976	+	1.13156i	−0.0965209	+	0.0613673i
$341$	15.7459			0.852691
$342$		−	2.33295i		−	0.126151i
$343$		−	33.3063i		−	1.79837i
$344$	1.66009			0.0895058
$345$	−1.67895	−	2.64072i	−0.0903916	−	0.142171i
$346$	−29.7085			−1.59714
$347$			11.5617i			0.620666i	0.950628	+	0.310333i	$0.100441\pi$
$347$			11.5617i			0.620666i	−0.950628	+	0.310333i	$0.899559\pi$
$348$			7.49306i			0.401670i
$349$	16.4980			0.883117			0.441558	−	0.897232i	$-0.354426\pi$
$349$	16.4980			0.883117			0.441558	+	0.897232i	$0.354426\pi$
$350$	−18.7681	+	40.0581i	−1.00320	+	2.14120i
$351$	−14.7085			−0.785084
$352$			11.7335i			0.625400i
$353$		−	13.7252i		−	0.730518i	−0.930906	−	0.365259i	$-0.880980\pi$
$353$		−	13.7252i		−	0.730518i	0.930906	−	0.365259i	$-0.119020\pi$
$354$	10.3202			0.548514
$355$	−9.47280	−	14.8992i	−0.502764	−	0.790767i
$356$	−13.0180			−0.689952
$357$			3.60442i			0.190766i
$358$			11.4947i			0.607512i
$359$	−15.1442			−0.799282			−0.399641	−	0.916672i	$-0.630865\pi$
$359$	−15.1442			−0.799282			−0.399641	+	0.916672i	$0.630865\pi$
$360$	−1.18031	+	0.750432i	−0.0622077	+	0.0395512i
$361$	−17.6393			−0.928386
$362$			10.9432i			0.575161i
$363$			11.9623i			0.627859i
$364$	−20.2165			−1.05964
$365$	−18.6919	+	11.8842i	−0.978381	+	0.622048i
$366$	11.5507			0.603762
$367$			11.2719i			0.588390i	0.955745	+	0.294195i	$0.0950515\pi$
$367$			11.2719i			0.588390i	−0.955745	+	0.294195i	$0.904949\pi$
$368$		−	4.52769i		−	0.236022i
$369$	−12.3035			−0.640495
$370$	−11.6878	−	18.3830i	−0.607620	−	0.955688i
$371$	−25.5617			−1.32710
$372$			23.7429i			1.23101i
$373$			35.3296i			1.82930i	0.404252	+	0.914648i	$0.367532\pi$
$373$			35.3296i			1.82930i	−0.404252	+	0.914648i	$0.632468\pi$
$374$	1.68088			0.0869164
$375$	−1.99994	−	15.5180i	−0.103277	−	0.801344i
$376$	5.59894			0.288743
$377$		−	8.25257i		−	0.425029i
$378$		−	50.0401i		−	2.57378i
$379$	36.8598			1.89336			0.946679	−	0.322178i	$-0.104415\pi$
$379$	36.8598			1.89336			0.946679	+	0.322178i	$0.104415\pi$
$380$	2.36122	+	3.71381i	0.121128	+	0.190514i
$381$	16.1390			0.826828
$382$			16.0224i			0.819775i
$383$			4.27065i			0.218220i	0.994030	+	0.109110i	$0.0348001\pi$
$383$			4.27065i			0.218220i	−0.994030	+	0.109110i	$0.965200\pi$
$384$	6.64133			0.338914
$385$	13.6144	−	8.65593i	0.693853	−	0.441147i
$386$	−30.7714			−1.56623
$387$			2.87911i			0.146353i
$388$			4.42279i			0.224533i
$389$	6.42988			0.326008			0.163004	−	0.986625i	$-0.447882\pi$
$389$	6.42988			0.326008			0.163004	+	0.986625i	$0.447882\pi$
$390$	13.1868	−	8.38407i	0.667739	−	0.424544i
$391$	−0.559006			−0.0282701
$392$			8.54512i			0.431594i
$393$			18.1423i			0.915160i
$394$	11.1160			0.560017
$395$	14.4548	+	22.7351i	0.727301	+	1.14393i
$396$	−2.75187			−0.138287
$397$		−	33.9539i		−	1.70410i	−0.523462	−	0.852049i	$-0.675360\pi$
$397$		−	33.9539i		−	1.70410i	0.523462	−	0.852049i	$-0.324640\pi$
$398$		−	36.1165i		−	1.81036i
$399$	7.52133			0.376537
$400$	9.60471	−	20.5000i	0.480235	−	1.02500i
$401$	27.6421			1.38038			0.690189	−	0.723629i	$-0.257527\pi$
$401$	27.6421			1.38038			0.690189	+	0.723629i	$0.257527\pi$
$402$		−	7.00692i		−	0.349473i
$403$		−	26.1495i		−	1.30260i
$404$	14.9861			0.745587
$405$	5.74732	+	9.03961i	0.285587	+	0.449182i
$406$	28.0761			1.39339
$407$			7.94457i			0.393798i
$408$		−	0.469812i		−	0.0232591i
$409$	13.2161			0.653494			0.326747	−	0.945112i	$-0.394048\pi$
$409$	13.2161			0.653494			0.326747	+	0.945112i	$0.394048\pi$
$410$	42.8023	−	27.2134i	2.11385	−	1.34397i
$411$	−23.9945			−1.18356
$412$		−	14.6047i		−	0.719522i
$413$		−	17.6947i		−	0.870700i
$414$	2.00000			0.0982946
$415$	−25.2582	+	16.0590i	−1.23988	+	0.788304i
$416$	19.4861			0.955384
$417$		−	27.2609i		−	1.33497i
$418$		−	3.50749i		−	0.171557i
$419$	−23.4772			−1.14694			−0.573468	−	0.819228i	$-0.694402\pi$
$419$	−23.4772			−1.14694			−0.573468	+	0.819228i	$0.694402\pi$
$420$	13.0521	+	20.5288i	0.636876	+	1.00170i
$421$	23.1138			1.12650			0.563249	−	0.826287i	$-0.309551\pi$
$421$	23.1138			1.12650			0.563249	+	0.826287i	$0.309551\pi$
$422$			9.37450i			0.456343i
$423$			9.71032i			0.472132i
$424$	3.33180			0.161806
$425$	−2.53100	−	1.18583i	−0.122772	−	0.0575214i
$426$	−21.2180			−1.02802
$427$		−	19.8044i		−	0.958401i
$428$			11.6701i			0.564093i
$429$	−5.69892			−0.275146
$430$	−6.36813	−	10.0160i	−0.307099	−	0.483017i
$431$	−9.52709			−0.458904			−0.229452	−	0.973320i	$-0.573693\pi$
$431$	−9.52709			−0.458904			−0.229452	+	0.973320i	$0.573693\pi$
$432$			25.6083i			1.23208i
$433$		−	2.79445i		−	0.134293i	−0.997743	−	0.0671464i	$-0.978611\pi$
$433$		−	2.79445i		−	0.134293i	0.997743	−	0.0671464i	$-0.0213895\pi$
$434$	88.9635			4.27039
$435$	−8.38003	+	5.32797i	−0.401792	+	0.255456i
$436$	−11.5532			−0.553298
$437$			1.16647i			0.0558001i
$438$			26.6193i			1.27192i
$439$	−2.09578			−0.100026			−0.0500129	−	0.998749i	$-0.515926\pi$
$439$	−2.09578			−0.100026			−0.0500129	+	0.998749i	$0.515926\pi$
$440$	−1.77454	+	1.12824i	−0.0845980	+	0.0537868i
$441$	−14.8199			−0.705711
$442$		−	2.79147i		−	0.132777i
$443$			27.0870i			1.28694i	0.765471	+	0.643470i	$0.222506\pi$
$443$			27.0870i			1.28694i	−0.765471	+	0.643470i	$0.777494\pi$
$444$	−11.9794			−0.568518
$445$	−9.25648	−	14.5590i	−0.438799	−	0.690161i
$446$	25.8819			1.22555
$447$			18.5130i			0.875633i
$448$			24.5714i			1.16089i
$449$	0.470272			0.0221935			0.0110967	−	0.999938i	$-0.496468\pi$
$449$	0.470272			0.0221935			0.0110967	+	0.999938i	$0.496468\pi$
$450$	9.05538	+	4.24265i	0.426875	+	0.200001i
$451$	−18.4978			−0.871028
$452$		−	7.77394i		−	0.365656i
$453$			3.50910i			0.164872i
$454$	26.2867			1.23370
$455$	−14.3750	−	22.6096i	−0.673913	−	1.05996i
$456$	−0.980354			−0.0459093
$457$			5.83768i			0.273075i	0.990635	+	0.136538i	$0.0435974\pi$
$457$			5.83768i			0.273075i	−0.990635	+	0.136538i	$0.956403\pi$
$458$			52.6071i			2.45817i
$459$	3.16170			0.147575
$460$	−3.18379	+	2.02423i	−0.148445	+	0.0943803i
$461$	10.8487			0.505277			0.252638	−	0.967561i	$-0.418702\pi$
$461$	10.8487			0.505277			0.252638	+	0.967561i	$0.418702\pi$
$462$		−	19.3883i		−	0.902027i
$463$		−	1.06258i		−	0.0493825i	−0.999695	−	0.0246912i	$-0.992140\pi$
$463$		−	1.06258i		−	0.0493825i	0.999695	−	0.0246912i	$-0.00786026\pi$
$464$	−14.3681			−0.667024
$465$	−26.5534	+	16.8825i	−1.23139	+	0.782906i
$466$	0.0744740			0.00344994
$467$		−	4.99885i		−	0.231319i	−0.993289	−	0.115660i	$-0.963102\pi$
$467$		−	4.99885i		−	0.231319i	0.993289	−	0.115660i	$-0.0368982\pi$
$468$			4.57008i			0.211252i
$469$	−12.0138			−0.554747
$470$	−21.4777	−	33.7809i	−0.990691	−	1.55820i
$471$	−11.9348			−0.549928
$472$			2.30639i			0.106160i
$473$			4.32862i			0.199030i
$474$	32.3772			1.48713
$475$	−2.47447	+	5.28144i	−0.113537	+	0.242329i
$476$	4.34568			0.199184
$477$			5.77838i			0.264574i
$478$		−	26.6969i		−	1.22109i
$479$	−33.4283			−1.52738			−0.763688	−	0.645585i	$-0.776613\pi$
$479$	−33.4283			−1.52738			−0.763688	+	0.645585i	$0.776613\pi$
$480$	−12.5805	−	19.7871i	−0.574217	−	0.903151i
$481$	13.1937			0.601580
$482$			23.0373i			1.04932i
$483$			6.44791i			0.293390i
$484$	14.4224			0.655564
$485$	−4.94632	+	3.14484i	−0.224601	+	0.142800i
$486$	−19.7085			−0.893998
$487$		−	10.7901i		−	0.488945i	−0.969656	−	0.244473i	$-0.921385\pi$
$487$		−	10.7901i		−	0.488945i	0.969656	−	0.244473i	$-0.0786148\pi$
$488$			2.58137i			0.116853i
$489$	10.8211			0.489346
$490$	51.5566	−	32.7793i	2.32909	−	1.48082i
$491$	15.8127			0.713618			0.356809	−	0.934177i	$-0.383865\pi$
$491$	15.8127			0.713618			0.356809	+	0.934177i	$0.383865\pi$
$492$		−	27.8924i		−	1.25749i
$493$			1.77394i			0.0798944i
$494$	−5.82495			−0.262077
$495$	−1.95673	−	3.07762i	−0.0879483	−	0.138329i
$496$	−45.5276			−2.04425
$497$			36.3798i			1.63185i
$498$			35.9703i			1.61187i
$499$	−12.9711			−0.580668			−0.290334	−	0.956925i	$-0.593766\pi$
$499$	−12.9711			−0.580668			−0.290334	+	0.956925i	$0.593766\pi$
$500$	−18.7093	+	2.41124i	−0.836704	+	0.107834i
$501$	−20.0210			−0.894474
$502$			34.1426i			1.52386i
$503$			0.998849i			0.0445365i	0.999752	+	0.0222682i	$0.00708878\pi$
$503$			0.998849i			0.0445365i	−0.999752	+	0.0222682i	$0.992911\pi$
$504$	2.88199			0.128374
$505$	10.6559	+	16.7601i	0.474183	+	0.745813i
$506$	3.00692			0.133674
$507$		−	8.72853i		−	0.387648i
$508$		−	19.4581i		−	0.863313i
$509$	20.6936			0.917228			0.458614	−	0.888636i	$-0.348346\pi$
$509$	20.6936			0.917228			0.458614	+	0.888636i	$0.348346\pi$
$510$	−2.83458	+	1.80221i	−0.125518	+	0.0798032i
$511$	45.6406			2.01902
$512$		−	28.4880i		−	1.25900i
$513$		−	6.59750i		−	0.291287i
$514$	19.6468			0.866583
$515$	16.3335	−	10.3847i	0.719740	−	0.457606i
$516$	−6.52702			−0.287336
$517$			14.5991i			0.642066i
$518$			44.8863i			1.97219i
$519$	−21.6514			−0.950393
$520$	1.87369	+	2.94701i	0.0821668	+	0.129235i
$521$	39.2620			1.72010			0.860049	−	0.510212i	$-0.170433\pi$
$521$	39.2620			1.72010			0.860049	+	0.510212i	$0.170433\pi$
$522$		−	6.34678i		−	0.277791i
$523$			1.74162i			0.0761556i	0.999275	+	0.0380778i	$0.0121235\pi$
$523$			1.74162i			0.0761556i	−0.999275	+	0.0380778i	$0.987877\pi$
$524$	21.8734			0.955543
$525$	−13.6781	+	29.1941i	−0.596962	+	1.27414i
$526$	7.67812			0.334782
$527$			5.62101i			0.244855i
$528$			9.92210i			0.431804i
$529$	−1.00000			−0.0434783
$530$	−12.7809	−	20.1022i	−0.555165	−	0.873185i
$531$	−4.00000			−0.173585
$532$		−	9.06811i		−	0.393152i
$533$			30.7196i			1.33061i
$534$	−20.7335			−0.897226
$535$	−13.0515	+	8.29803i	−0.564264	+	0.358755i
$536$	1.56592			0.0676375
$537$			8.37726i			0.361505i
$538$		−	5.53184i		−	0.238495i
$539$	−22.2811			−0.959717
$540$	18.0073	−	11.4489i	0.774911	−	0.492683i
$541$	−8.18117			−0.351736			−0.175868	−	0.984414i	$-0.556273\pi$
$541$	−8.18117			−0.351736			−0.175868	+	0.984414i	$0.556273\pi$
$542$		−	27.2564i		−	1.17076i
$543$			7.97534i			0.342254i
$544$	−4.18866			−0.179587
$545$	−8.21494	−	12.9208i	−0.351889	−	0.553465i
$546$	−32.1985			−1.37797
$547$		−	24.2470i		−	1.03673i	−0.855160	−	0.518364i	$-0.826541\pi$
$547$		−	24.2470i		−	1.03673i	0.855160	−	0.518364i	$-0.173459\pi$
$548$			28.9290i			1.23579i
$549$	−4.47690			−0.191069
$550$	13.6144	+	6.37865i	0.580519	+	0.271987i
$551$	3.70168			0.157697
$552$		−	0.840442i		−	0.0357716i
$553$		−	55.5129i		−	2.36065i
$554$	17.5427			0.745317
$555$	−8.51801	−	13.3975i	−0.361569	−	0.568690i
$556$	−32.8672			−1.39388
$557$		−	35.0966i		−	1.48709i	−0.668685	−	0.743546i	$-0.733142\pi$
$557$		−	35.0966i		−	1.48709i	0.668685	−	0.743546i	$-0.266858\pi$
$558$		−	20.1108i		−	0.851356i
$559$	7.18862			0.304046
$560$	−39.3645	+	25.0277i	−1.66345	+	1.05761i
$561$	1.22502			0.0517204
$562$		−	28.5421i		−	1.20397i
$563$			6.98179i			0.294247i	0.989118	+	0.147124i	$0.0470016\pi$
$563$			6.98179i			0.294247i	−0.989118	+	0.147124i	$0.952998\pi$
$564$	−22.0136			−0.926938
$565$	8.69416	−	5.52769i	0.365766	−	0.232552i
$566$	−5.83514			−0.245269
$567$		−	22.0723i		−	0.926948i
$568$		−	4.74186i		−	0.198964i
$569$	−21.7280			−0.910886			−0.455443	−	0.890265i	$-0.650519\pi$
$569$	−21.7280			−0.910886			−0.455443	+	0.890265i	$0.650519\pi$
$570$	3.76066	+	5.91492i	0.157517	+	0.247749i
$571$	−4.71016			−0.197114			−0.0985570	−	0.995131i	$-0.531423\pi$
$571$	−4.71016			−0.197114			−0.0985570	+	0.995131i	$0.531423\pi$
$572$			6.87092i			0.287288i
$573$			11.6770i			0.487814i
$574$	−104.511			−4.36222
$575$	−4.52769	−	2.12133i	−0.188818	−	0.0884654i
$576$	5.55452			0.231438
$577$		−	9.49085i		−	0.395109i	−0.980292	−	0.197555i	$-0.936700\pi$
$577$		−	9.49085i		−	0.395109i	0.980292	−	0.197555i	$-0.0633000\pi$
$578$		−	32.0437i		−	1.33284i
$579$	−22.4261			−0.931996
$580$	6.42368	+	10.1034i	0.266729	+	0.419521i
$581$	61.6736			2.55865
$582$			7.04409i			0.291987i
$583$			8.68756i			0.359802i
$584$	−5.94895			−0.246169
$585$	−5.11105	+	3.24957i	−0.211316	+	0.134353i
$586$	36.9235			1.52530
$587$			10.4952i			0.433184i	0.976262	+	0.216592i	$0.0694942\pi$
$587$			10.4952i			0.433184i	−0.976262	+	0.216592i	$0.930506\pi$
$588$		−	33.5972i		−	1.38552i
$589$	11.7293			0.483299
$590$	13.9155	−	8.84736i	0.572891	−	0.364240i
$591$	8.10130			0.333243
$592$		−	22.9708i		−	0.944096i
$593$			18.2138i			0.747951i	0.927439	+	0.373975i	$0.122006\pi$
$593$			18.2138i			0.747951i	−0.927439	+	0.373975i	$0.877994\pi$
$594$	−17.0069			−0.697802
$595$	3.09001	+	4.86009i	0.126678	+	0.199244i
$596$	22.3202			0.914272
$597$		−	26.3215i		−	1.07727i
$598$		−	4.99364i		−	0.204205i
$599$	20.8055			0.850091			0.425045	−	0.905172i	$-0.360258\pi$
$599$	20.8055			0.850091			0.425045	+	0.905172i	$0.360258\pi$
$600$	1.78285	−	3.80526i	0.0727846	−	0.155349i
$601$	−47.4004			−1.93350			−0.966752	−	0.255716i	$-0.917689\pi$
$601$	−47.4004			−1.93350			−0.966752	+	0.255716i	$0.917689\pi$
$602$			24.4564i			0.996770i
$603$			2.71580i			0.110596i
$604$	4.23076			0.172147
$605$	10.2551	+	16.1296i	0.416929	+	0.655762i
$606$	23.8681			0.969576
$607$		−	5.77118i		−	0.234245i	−0.993117	−	0.117123i	$-0.962633\pi$
$607$		−	5.77118i		−	0.234245i	0.993117	−	0.117123i	$-0.0373670\pi$
$608$			8.74046i			0.354473i
$609$	20.4617			0.829152
$610$	15.5745	−	9.90219i	0.630595	−	0.400928i
$611$	24.2449			0.980844
$612$		−	0.982368i		−	0.0397099i
$613$		−	38.9329i		−	1.57248i	−0.617919	−	0.786242i	$-0.712024\pi$
$613$		−	38.9329i		−	1.57248i	0.617919	−	0.786242i	$-0.287976\pi$
$614$	−39.4404			−1.59169
$615$	31.1941	−	19.8330i	1.25787	−	0.799743i
$616$	4.33295			0.174580
$617$			10.1653i			0.409241i	0.978841	+	0.204620i	$0.0655959\pi$
$617$			10.1653i			0.409241i	−0.978841	+	0.204620i	$0.934404\pi$
$618$		−	23.2606i		−	0.935680i
$619$	−2.09923			−0.0843752			−0.0421876	−	0.999110i	$-0.513433\pi$
$619$	−2.09923			−0.0843752			−0.0421876	+	0.999110i	$0.513433\pi$
$620$	20.3544	+	32.0142i	0.817453	+	1.28572i
$621$	5.65593			0.226965
$622$			18.8936i			0.757565i
$623$			35.5490i			1.42424i
$624$	16.4778			0.659639
$625$	−16.0000	−	19.2094i	−0.639998	−	0.768377i
$626$	−43.1396			−1.72420
$627$		−	2.55624i		−	0.102087i
$628$			14.3893i			0.574194i
$629$	−2.83607			−0.113081
$630$	−11.0554	−	17.3883i	−0.440457	−	0.692768i
$631$	−32.3066			−1.28611			−0.643053	−	0.765821i	$-0.722333\pi$
$631$	−32.3066			−1.28611			−0.643053	+	0.765821i	$0.722333\pi$
$632$			7.23574i			0.287822i
$633$			6.83209i			0.271551i
$634$	−34.2576			−1.36054
$635$	21.7614	−	13.8357i	0.863575	−	0.549055i
$636$	−13.0998			−0.519439
$637$			37.0027i			1.46610i
$638$		−	9.54212i		−	0.377776i
$639$	8.22387			0.325331
$640$	8.95496	−	5.69350i	0.353976	−	0.225055i
$641$	14.5758			0.575711			0.287856	−	0.957674i	$-0.407058\pi$
$641$	14.5758			0.575711			0.287856	+	0.957674i	$0.407058\pi$
$642$			18.5867i			0.733557i
$643$			42.5090i			1.67639i	0.545369	+	0.838196i	$0.316389\pi$
$643$			42.5090i			1.67639i	−0.545369	+	0.838196i	$0.683611\pi$
$644$	7.77394			0.306336
$645$	−4.64106	−	7.29964i	−0.182742	−	0.287423i
$646$	1.25211			0.0492636
$647$		−	3.04979i		−	0.119900i	−0.998201	−	0.0599498i	$-0.980906\pi$
$647$		−	3.04979i		−	0.119900i	0.998201	−	0.0599498i	$-0.0190940\pi$
$648$			2.87697i			0.113018i
$649$	−6.01383			−0.236064
$650$	10.5931	−	22.6096i	0.415497	−	0.886823i
$651$	64.8362			2.54113
$652$		−	13.0465i		−	0.510939i
$653$		−	17.9600i		−	0.702830i	−0.936220	−	0.351415i	$-0.885701\pi$
$653$		−	17.9600i		−	0.702830i	0.936220	−	0.351415i	$-0.114299\pi$
$654$	−18.4006			−0.719518
$655$	15.5531	+	24.4626i	0.607711	+	0.955832i
$656$	53.4844			2.08821
$657$		−	10.3173i		−	0.402518i
$658$			82.4838i			3.21555i
$659$	−46.3411			−1.80519			−0.902597	−	0.430486i	$-0.858342\pi$
$659$	−46.3411			−1.80519			−0.902597	+	0.430486i	$0.858342\pi$
$660$	6.97704	−	4.43595i	0.271581	−	0.172669i
$661$	1.23165			0.0479056			0.0239528	−	0.999713i	$-0.492375\pi$
$661$	1.23165			0.0479056			0.0239528	+	0.999713i	$0.492375\pi$
$662$		−	25.5639i		−	0.993570i
$663$		−	2.03441i		−	0.0790099i
$664$	−8.03874			−0.311963
$665$	10.1415	−	6.44791i	0.393271	−	0.250039i
$666$	10.1468			0.393182
$667$			3.17339i			0.122874i
$668$			24.1384i			0.933944i
$669$	18.8626			0.729271
$670$	−6.00692	−	9.44791i	−0.232067	−	0.365005i
$671$	−6.73083			−0.259841
$672$			48.3146i			1.86378i
$673$		−	42.4664i		−	1.63696i	−0.574537	−	0.818479i	$-0.694818\pi$
$673$		−	42.4664i		−	1.63696i	0.574537	−	0.818479i	$-0.305182\pi$
$674$	5.92077			0.228060
$675$	25.6083	+	11.9981i	0.985664	+	0.461806i
$676$	−10.5236			−0.404753
$677$			1.41777i			0.0544893i	0.999629	+	0.0272447i	$0.00867332\pi$
$677$			1.41777i			0.0544893i	−0.999629	+	0.0272447i	$0.991327\pi$
$678$		−	12.3814i		−	0.475505i
$679$	12.0776			0.463495
$680$	−0.402762	−	0.633480i	−0.0154452	−	0.0242928i
$681$	19.1576			0.734121
$682$		−	30.2357i		−	1.15778i
$683$		−	4.60886i		−	0.176353i	−0.996105	−	0.0881766i	$-0.971896\pi$
$683$		−	4.60886i		−	0.176353i	0.996105	−	0.0881766i	$-0.0281040\pi$
$684$	−2.04990			−0.0783800
$685$	−32.3534	+	20.5701i	−1.23616	+	0.785942i
$686$	−63.9555			−2.44183
$687$			38.3398i			1.46276i
$688$		−	12.5157i		−	0.477158i
$689$	14.4276			0.549647
$690$	−5.07076	+	3.22396i	−0.193041	+	0.122734i
$691$	−16.8903			−0.642537			−0.321269	−	0.946988i	$-0.604109\pi$
$691$	−16.8903			−0.642537			−0.321269	+	0.946988i	$0.604109\pi$
$692$			26.1041i			0.992330i
$693$			7.51470i			0.285460i
$694$	22.2011			0.842741
$695$	−23.3704	−	36.7578i	−0.886488	−	1.39430i
$696$	−2.66705			−0.101094
$697$		−	6.60338i		−	0.250121i
$698$		−	31.6798i		−	1.19910i
$699$	0.0542763			0.00205292
$700$	35.1980	+	16.4911i	1.33036	+	0.623304i
$701$	−11.9543			−0.451508			−0.225754	−	0.974184i	$-0.572484\pi$
$701$	−11.9543			−0.451508			−0.225754	+	0.974184i	$0.572484\pi$
$702$			28.2437i			1.06599i
$703$			5.91801i			0.223202i
$704$	8.35098			0.314740
$705$	−15.6528	−	24.6194i	−0.589519	−	0.927219i
$706$	−26.3554			−0.991899
$707$		−	40.9235i		−	1.53909i
$708$		−	9.06811i		−	0.340800i
$709$	12.6755			0.476040			0.238020	−	0.971260i	$-0.423502\pi$
$709$	12.6755			0.476040			0.238020	+	0.971260i	$0.423502\pi$
$710$	−28.6097	+	18.1899i	−1.07370	+	0.682654i
$711$	−12.5490			−0.470626
$712$		−	4.63357i		−	0.173650i
$713$			10.0554i			0.376577i
$714$	6.92128			0.259022
$715$	−7.68425	+	4.88559i	−0.287375	+	0.182711i
$716$	10.1001			0.377457
$717$		−	19.4566i		−	0.726618i
$718$			29.0803i			1.08527i
$719$	34.7351			1.29540			0.647700	−	0.761896i	$-0.275731\pi$
$719$	34.7351			1.29540			0.647700	+	0.761896i	$0.275731\pi$
$720$	5.65766	+	8.89858i	0.210848	+	0.331631i
$721$	−39.8819			−1.48528
$722$			33.8714i			1.26056i
$723$			16.7895i			0.624408i
$724$	9.61549			0.357357
$725$	−6.73180	+	14.3681i	−0.250013	+	0.533619i
$726$	22.9703			0.852508
$727$			27.6631i			1.02597i	0.858398	+	0.512984i	$0.171460\pi$
$727$			27.6631i			1.02597i	−0.858398	+	0.512984i	$0.828540\pi$
$728$		−	7.19580i		−	0.266694i
$729$	−28.7351			−1.06426
$730$	22.8203	+	35.8927i	0.844617	+	1.32845i
$731$	−1.54524			−0.0571528
$732$		−	10.1493i		−	0.375127i
$733$			2.14239i			0.0791309i	0.999217	+	0.0395654i	$0.0125974\pi$
$733$			2.14239i			0.0791309i	−0.999217	+	0.0395654i	$0.987403\pi$
$734$	21.6446			0.798917
$735$	37.5742	−	23.8894i	1.38594	−	0.881174i
$736$	−7.49306			−0.276198
$737$			4.08309i			0.150403i
$738$			23.6255i			0.869665i
$739$	42.6249			1.56798			0.783991	−	0.620772i	$-0.213181\pi$
$739$	42.6249			1.56798			0.783991	+	0.620772i	$0.213181\pi$
$740$	−16.1527	+	10.2698i	−0.593785	+	0.377524i
$741$	−4.24519			−0.155951
$742$			49.0841i			1.80194i
$743$		−	23.4687i		−	0.860982i	−0.902595	−	0.430491i	$-0.858340\pi$
$743$		−	23.4687i		−	0.860982i	0.902595	−	0.430491i	$-0.141660\pi$
$744$	−8.45096			−0.309827
$745$	15.8709	+	24.9623i	0.581464	+	0.914549i
$746$	67.8406			2.48382
$747$		−	13.9417i		−	0.510100i
$748$		−	1.47695i		−	0.0540026i
$749$	31.8681			1.16444
$750$	−29.7979	+	3.84033i	−1.08807	+	0.140229i
$751$	−27.2579			−0.994654			−0.497327	−	0.867563i	$-0.665685\pi$
$751$	−27.2579			−0.994654			−0.497327	+	0.867563i	$0.665685\pi$
$752$		−	42.2116i		−	1.53930i
$753$			24.8830i			0.906786i
$754$	−15.8468			−0.577105
$755$	3.00830	+	4.73157i	0.109483	+	0.172199i
$756$	−43.9689			−1.59913
$757$		−	12.7382i		−	0.462976i	−0.972838	−	0.231488i	$-0.925641\pi$
$757$		−	12.7382i		−	0.462976i	0.972838	−	0.231488i	$-0.0743595\pi$
$758$		−	70.7789i		−	2.57080i
$759$	2.19143			0.0795437
$760$	−1.32188	+	0.840442i	−0.0479496	+	0.0304860i
$761$	6.61715			0.239871			0.119936	−	0.992782i	$-0.461731\pi$
$761$	6.61715			0.239871			0.119936	+	0.992782i	$0.461731\pi$
$762$		−	30.9905i		−	1.12267i
$763$			31.5490i			1.14215i
$764$	14.0784			0.509340
$765$	1.09865	−	0.698516i	0.0397219	−	0.0252549i
$766$	8.20060			0.296300
$767$			9.98727i			0.360619i
$768$		−	27.6792i		−	0.998786i
$769$	27.2116			0.981275			0.490638	−	0.871364i	$-0.336764\pi$
$769$	27.2116			0.981275			0.490638	+	0.871364i	$0.336764\pi$
$770$	−16.6213	−	26.1426i	−0.598990	−	0.942115i
$771$	14.3185			0.515668
$772$			27.0381i			0.973121i
$773$		−	22.9249i		−	0.824552i	−0.911059	−	0.412276i	$-0.864734\pi$
$773$		−	22.9249i		−	0.824552i	0.911059	−	0.412276i	$-0.135266\pi$
$774$	5.52853			0.198719
$775$	−21.3307	+	45.5276i	−0.766223	+	1.63540i
$776$	−1.57423			−0.0565116
$777$			32.7129i			1.17357i
$778$		−	12.3468i		−	0.442654i
$779$	−13.7792			−0.493693
$780$	−7.36686	−	11.5869i	−0.263776	−	0.414877i
$781$	12.3642			0.442427
$782$			1.07341i			0.0383852i
$783$		−	17.9485i		−	0.641427i
$784$	64.4235			2.30084
$785$	−16.0926	+	10.2315i	−0.574368	+	0.365179i
$786$	34.8373			1.24261
$787$			13.0620i			0.465610i	0.972523	+	0.232805i	$0.0747904\pi$
$787$			13.0620i			0.465610i	−0.972523	+	0.232805i	$0.925210\pi$
$788$		−	9.76736i		−	0.347948i
$789$	5.59578			0.199215
$790$	43.6564	−	27.7565i	1.55323	−	0.987531i
$791$	−21.2288			−0.754808
$792$		−	0.979490i		−	0.0348047i
$793$			11.1780i			0.396943i
$794$	−65.1990			−2.31383
$795$	−9.31463	−	14.6504i	−0.330356	−	0.519596i
$796$	−31.7347			−1.12480
$797$			20.9305i			0.741395i	0.928754	+	0.370697i	$0.120881\pi$
$797$			20.9305i			0.741395i	−0.928754	+	0.370697i	$0.879119\pi$
$798$		−	14.4426i		−	0.511263i
$799$	−5.21160			−0.184373
$800$	−33.9263	−	15.8952i	−1.19947	−	0.561981i
$801$	8.03607			0.283941
$802$		−	53.0788i		−	1.87428i
$803$		−	15.5117i		−	0.547396i
$804$	−6.15680			−0.217133
$805$	5.52769	+	8.69416i	0.194825	+	0.306429i
$806$	−50.2129			−1.76867
$807$		−	4.03158i		−	0.141918i
$808$			5.33410i			0.187653i
$809$	−18.0664			−0.635180			−0.317590	−	0.948228i	$-0.602874\pi$
$809$	−18.0664			−0.635180			−0.317590	+	0.948228i	$0.602874\pi$
$810$	17.3581	−	11.0361i	0.609900	−	0.387770i
$811$	34.8658			1.22430			0.612152	−	0.790740i	$-0.290304\pi$
$811$	34.8658			1.22430			0.612152	+	0.790740i	$0.290304\pi$
$812$		−	24.6698i		−	0.865739i
$813$		−	19.8644i		−	0.696673i
$814$	15.2553			0.534699
$815$	14.5908	−	9.27674i	0.511094	−	0.324950i
$816$	−3.54201			−0.123995
$817$			3.22444i			0.112809i
$818$		−	25.3778i		−	0.887314i
$819$	12.4798			0.436079
$820$	−23.9117	−	37.6092i	−0.835033	−	1.31337i
$821$	14.5989			0.509507			0.254753	−	0.967006i	$-0.418006\pi$
$821$	14.5989			0.509507			0.254753	+	0.967006i	$0.418006\pi$
$822$			46.0747i			1.60704i
$823$		−	25.8126i		−	0.899771i	−0.893086	−	0.449886i	$-0.851465\pi$
$823$		−	25.8126i		−	0.899771i	0.893086	−	0.449886i	$-0.148535\pi$
$824$	5.19834			0.181093
$825$	9.92210	+	4.64873i	0.345443	+	0.161848i
$826$	−33.9778			−1.18224
$827$			54.9097i			1.90940i	0.297577	+	0.954698i	$0.403822\pi$
$827$			54.9097i			1.90940i	−0.297577	+	0.954698i	$0.596178\pi$
$828$		−	1.75735i		−	0.0610721i
$829$	27.4809			0.954452			0.477226	−	0.878781i	$-0.341642\pi$
$829$	27.4809			0.954452			0.477226	+	0.878781i	$0.341642\pi$
$830$	30.8368	+	48.5013i	1.07036	+	1.68350i
$831$	12.7850			0.443507
$832$		−	13.8686i		−	0.480808i
$833$		−	7.95396i		−	0.275589i
$834$	−52.3470			−1.81263
$835$	−26.9958	+	17.1637i	−0.934226	+	0.593975i
$836$	−3.08194			−0.106591
$837$		−	56.8725i		−	1.96580i
$838$			45.0814i			1.55731i
$839$	1.46577			0.0506041			0.0253020	−	0.999680i	$-0.491945\pi$
$839$	1.46577			0.0506041			0.0253020	+	0.999680i	$0.491945\pi$
$840$	−7.30694	+	4.64570i	−0.252113	+	0.160292i
$841$	−18.9296			−0.652744
$842$		−	44.3836i		−	1.52956i
$843$		−	20.8013i		−	0.716436i
$844$	8.23713			0.283534
$845$	−7.48283	−	11.7693i	−0.257417	−	0.404876i
$846$	18.6460			0.641062
$847$		−	39.3841i		−	1.35325i
$848$		−	25.1191i		−	0.862594i
$849$	−4.25262			−0.145949
$850$	−2.27706	+	4.86009i	−0.0781026	+	0.166700i
$851$	−5.07341			−0.173914
$852$			18.6437i			0.638724i
$853$		−	35.8044i		−	1.22592i	−0.790115	−	0.612959i	$-0.789979\pi$
$853$		−	35.8044i		−	1.22592i	0.790115	−	0.612959i	$-0.210021\pi$
$854$	−38.0288			−1.30132
$855$	−1.45759	−	2.29255i	−0.0498485	−	0.0784037i
$856$	−4.15379			−0.141974
$857$			28.1663i			0.962141i	0.876682	+	0.481070i	$0.159752\pi$
$857$			28.1663i			0.962141i	−0.876682	+	0.481070i	$0.840248\pi$
$858$			10.9432i			0.373594i
$859$	−20.3185			−0.693258			−0.346629	−	0.938002i	$-0.612674\pi$
$859$	−20.3185			−0.693258			−0.346629	+	0.938002i	$0.612674\pi$
$860$	−8.80084	+	5.59551i	−0.300106	+	0.190805i
$861$	−76.1674			−2.59578
$862$			18.2941i			0.623100i
$863$			15.1357i			0.515224i	0.966248	+	0.257612i	$0.0829356\pi$
$863$			15.1357i			0.515224i	−0.966248	+	0.257612i	$0.917064\pi$
$864$	42.3802			1.44180
$865$	−29.1941	+	18.5614i	−0.992631	+	0.631108i
$866$	−5.36597			−0.182343
$867$		−	23.3533i		−	0.793120i
$868$		−	78.1700i		−	2.65326i
$869$	−18.8670			−0.640018
$870$	10.2309	+	16.0915i	0.346859	+	0.545553i
$871$	6.78086			0.229761
$872$		−	4.11220i		−	0.139257i
$873$		−	2.73021i		−	0.0924036i
$874$	2.23989			0.0757654
$875$	6.58451	+	51.0905i	0.222597	+	1.72717i
$876$	23.3897			0.790265
$877$			12.9668i			0.437856i	0.975741	+	0.218928i	$0.0702560\pi$
$877$			12.9668i			0.437856i	−0.975741	+	0.218928i	$0.929744\pi$
$878$			4.02435i			0.135815i
$879$	26.9097			0.907640
$880$	8.50605	+	13.3787i	0.286739	+	0.450994i
$881$	−34.3573			−1.15753			−0.578763	−	0.815496i	$-0.696464\pi$
$881$	−34.3573			−1.15753			−0.578763	+	0.815496i	$0.696464\pi$
$882$			28.4575i			0.958215i
$883$			14.7201i			0.495372i	0.968840	+	0.247686i	$0.0796701\pi$
$883$			14.7201i			0.495372i	−0.968840	+	0.247686i	$0.920330\pi$
$884$	−2.45279			−0.0824963
$885$	10.1415	−	6.44791i	0.340904	−	0.216744i
$886$	52.0129			1.74741
$887$			20.9401i			0.703101i	0.936169	+	0.351550i	$0.114345\pi$
$887$			20.9401i			0.703101i	−0.936169	+	0.351550i	$0.885655\pi$
$888$		−	4.26391i		−	0.143087i
$889$	−53.1354			−1.78210
$890$	−27.9564	+	17.7745i	−0.937101	+	0.595802i
$891$	−7.50161			−0.251313
$892$		−	22.7418i		−	0.761451i
$893$			10.8750i			0.363919i
$894$	35.5490			1.18894
$895$	7.18169	+	11.2956i	0.240057	+	0.377572i
$896$	−21.8656			−0.730477
$897$		−	3.63934i		−	0.121514i
$898$		−	0.903025i		−	0.0301343i
$899$	31.9097			1.06425
$900$	3.72791	−	7.95673i	0.124264	−	0.265224i
$901$	−3.10130			−0.103319
$902$			35.5199i			1.18268i
$903$			17.8237i			0.593137i
$904$	2.76703			0.0920300
$905$	6.83713	+	10.7537i	0.227274	+	0.357465i
$906$	6.73825			0.223863
$907$		−	40.2008i		−	1.33485i	−0.744679	−	0.667423i	$-0.767397\pi$
$907$		−	40.2008i		−	1.33485i	0.744679	−	0.667423i	$-0.232603\pi$
$908$		−	23.0974i		−	0.766515i
$909$	−9.25101			−0.306837
$910$	−43.4155	+	27.6033i	−1.43921	+	0.915039i
$911$	54.6058			1.80917			0.904586	−	0.426292i	$-0.140180\pi$
$911$	54.6058			1.80917			0.904586	+	0.426292i	$0.140180\pi$
$912$			7.39109i			0.244744i
$913$		−	20.9608i		−	0.693700i
$914$	11.2096			0.370782
$915$	11.3506	−	7.21666i	0.375241	−	0.238576i
$916$	46.2245			1.52730
$917$		−	59.7309i		−	1.97249i
$918$		−	6.07116i		−	0.200378i
$919$	−13.8019			−0.455284			−0.227642	−	0.973745i	$-0.573102\pi$
$919$	−13.8019			−0.455284			−0.227642	+	0.973745i	$0.573102\pi$
$920$	−0.720497	−	1.13323i	−0.0237541	−	0.0373613i
$921$	−28.7440			−0.947147
$922$		−	20.8320i		−	0.686065i
$923$		−	20.5335i		−	0.675868i
$924$	−17.0360			−0.560444
$925$	−22.9708	−	10.7624i	−0.755277	−	0.353864i
$926$	−2.04040			−0.0670516
$927$			9.01556i			0.296110i
$928$			23.7784i			0.780565i
$929$	−19.2775			−0.632475			−0.316237	−	0.948680i	$-0.602420\pi$
$929$	−19.2775			−0.632475			−0.316237	+	0.948680i	$0.602420\pi$
$930$	32.4181	+	50.9884i	1.06303	+	1.67198i
$931$	−16.5975			−0.543961
$932$		−	0.0654384i		−	0.00214350i
$933$			13.7696i			0.450795i
$934$	−9.59889			−0.314085
$935$	1.65178	−	1.05019i	0.0540189	−	0.0343449i
$936$	−1.62666			−0.0531689
$937$			48.0855i			1.57089i	0.618934	+	0.785443i	$0.287565\pi$
$937$			48.0855i			1.57089i	−0.618934	+	0.785443i	$0.712435\pi$
$938$			23.0692i			0.753237i
$939$	−31.4399			−1.02600
$940$	−29.6824	+	18.8719i	−0.968134	+	0.615533i
$941$	−30.6853			−1.00031			−0.500156	−	0.865935i	$-0.666724\pi$
$941$	−30.6853			−1.00031			−0.500156	+	0.865935i	$0.666724\pi$
$942$			22.9175i			0.746693i
$943$		−	11.8127i		−	0.384675i
$944$	17.3883			0.565942
$945$	−31.2642	−	49.1736i	−1.01703	−	1.59962i
$946$	8.31191			0.270244
$947$		−	12.9280i		−	0.420103i	−0.977690	−	0.210051i	$-0.932637\pi$
$947$		−	12.9280i		−	0.420103i	0.977690	−	0.210051i	$-0.0673631\pi$
$948$		−	28.4490i		−	0.923981i
$949$	−25.7605			−0.836222
$950$	10.1415	+	4.75154i	0.329034	+	0.154160i
$951$	−24.9668			−0.809603
$952$			1.54678i			0.0501316i
$953$			6.38285i			0.206761i	0.994642	+	0.103380i	$0.0329659\pi$
$953$			6.38285i			0.206761i	−0.994642	+	0.103380i	$0.967034\pi$
$954$	11.0958			0.359239
$955$	10.0105	+	15.7449i	0.323933	+	0.509494i
$956$	−23.4579			−0.758681
$957$		−	6.95425i		−	0.224799i
$958$			64.1897i			2.07387i
$959$	78.9982			2.55098
$960$	−14.0828	+	8.95376i	−0.454521	+	0.288981i
$961$	70.1107			2.26163
$962$		−	25.3348i		−	0.816826i
$963$		−	7.20398i		−	0.232145i
$964$	20.2423			0.651961
$965$	−30.2386	+	19.2255i	−0.973416	+	0.618891i
$966$	12.3814			0.398365
$967$		−	38.6028i		−	1.24138i	−0.784056	−	0.620691i	$-0.786852\pi$
$967$		−	38.6028i		−	1.24138i	0.784056	−	0.620691i	$-0.213148\pi$
$968$			5.13346i			0.164996i
$969$	0.912532			0.0293148
$970$	6.03878	+	9.49803i	0.193894	+	0.304963i
$971$	−24.0216			−0.770889			−0.385444	−	0.922731i	$-0.625952\pi$
$971$	−24.0216			−0.770889			−0.385444	+	0.922731i	$0.625952\pi$
$972$			17.3174i			0.555456i
$973$			89.7525i			2.87733i
$974$	−20.7193			−0.663890
$975$	7.72022	−	16.4778i	0.247245	−	0.527712i
$976$	19.4615			0.622946
$977$			45.5558i			1.45746i	0.684801	+	0.728730i	$0.259889\pi$
$977$			45.5558i			1.45746i	−0.684801	+	0.728730i	$0.740111\pi$
$978$		−	20.7789i		−	0.664435i
$979$	12.0819			0.386139
$980$	−28.8023	−	45.3014i	−0.920057	−	1.44710i
$981$	7.13184			0.227702
$982$		−	30.3639i		−	0.968952i
$983$			29.1396i			0.929408i	0.885466	+	0.464704i	$0.153839\pi$
$983$			29.1396i			0.929408i	−0.885466	+	0.464704i	$0.846161\pi$
$984$	9.92791			0.316490
$985$	10.9235	−	6.94511i	0.348053	−	0.221290i
$986$	3.40636			0.108481
$987$			60.1138i			1.91344i
$988$			5.11823i			0.162833i
$989$	−2.76426			−0.0878985
$990$	−5.90970	+	3.75735i	−0.187823	+	0.119416i
$991$	−20.0415			−0.636639			−0.318320	−	0.947983i	$-0.603119\pi$
$991$	−20.0415			−0.636639			−0.318320	+	0.947983i	$0.603119\pi$
$992$			75.3456i			2.39222i
$993$		−	18.6309i		−	0.591233i
$994$	69.8572			2.21573
$995$	−22.5650	−	35.4912i	−0.715360	−	1.12515i
$996$	31.6062			1.00148
$997$			0.340153i			0.0107728i	0.999985	+	0.00538638i	$0.00171455\pi$
$997$			0.340153i			0.0107728i	−0.999985	+	0.00538638i	$0.998285\pi$
$998$			24.9074i			0.788431i
$999$	28.6949			0.907866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.b.b.24.2	✓	8
3.2	odd	2		1035.2.b.e.829.7		8
4.3	odd	2		1840.2.e.d.369.6		8
5.2	odd	4		575.2.a.i.1.4		4
5.3	odd	4		575.2.a.j.1.1		4
5.4	even	2	inner	115.2.b.b.24.7	yes	8
15.2	even	4		5175.2.a.bv.1.1		4
15.8	even	4		5175.2.a.bw.1.4		4
15.14	odd	2		1035.2.b.e.829.2		8
20.3	even	4		9200.2.a.ck.1.2		4
20.7	even	4		9200.2.a.cq.1.3		4
20.19	odd	2		1840.2.e.d.369.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.2	✓	8	1.1	even	1	trivial
115.2.b.b.24.7	yes	8	5.4	even	2	inner
575.2.a.i.1.4		4	5.2	odd	4
575.2.a.j.1.1		4	5.3	odd	4
1035.2.b.e.829.2		8	15.14	odd	2
1035.2.b.e.829.7		8	3.2	odd	2
1840.2.e.d.369.3		8	20.19	odd	2
1840.2.e.d.369.6		8	4.3	odd	2
5175.2.a.bv.1.1		4	15.2	even	4
5175.2.a.bw.1.4		4	15.8	even	4
9200.2.a.ck.1.2		4	20.3	even	4
9200.2.a.cq.1.3		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.92022i q^{2} -1.39945i q^{3} -1.68725 q^{4} +(-1.19972 - 1.88697i) q^{5} -2.68725 q^{6} +4.60747i q^{7} -0.600553i q^{8} +1.04155 q^{9} +O(q^{10})\)	\(q-1.92022i q^{2} -1.39945i q^{3} -1.68725 q^{4} +(-1.19972 - 1.88697i) q^{5} -2.68725 q^{6} +4.60747i q^{7} -0.600553i q^{8} +1.04155 q^{9} +(-3.62340 + 2.30373i) q^{10} +1.56592 q^{11} +2.36122i q^{12} -2.60055i q^{13} +8.84736 q^{14} +(-2.64072 + 1.67895i) q^{15} -4.52769 q^{16} +0.559006i q^{17} -2.00000i q^{18} +1.16647 q^{19} +(2.02423 + 3.18379i) q^{20} +6.44791 q^{21} -3.00692i q^{22} +1.00000i q^{23} -0.840442 q^{24} +(-2.12133 + 4.52769i) q^{25} -4.99364 q^{26} -5.65593i q^{27} -7.77394i q^{28} +3.17339 q^{29} +(3.22396 + 5.07076i) q^{30} +10.0554 q^{31} +7.49306i q^{32} -2.19143i q^{33} +1.07341 q^{34} +(8.69416 - 5.52769i) q^{35} -1.75735 q^{36} +5.07341i q^{37} -2.23989i q^{38} -3.63934 q^{39} +(-1.13323 + 0.720497i) q^{40} -11.8127 q^{41} -12.3814i q^{42} +2.76426i q^{43} -2.64210 q^{44} +(-1.24957 - 1.96537i) q^{45} +1.92022 q^{46} +9.32298i q^{47} +6.33626i q^{48} -14.2288 q^{49} +(8.69416 + 4.07341i) q^{50} +0.782299 q^{51} +4.38778i q^{52} +5.54789i q^{53} -10.8606 q^{54} +(-1.87867 - 2.95485i) q^{55} +2.76703 q^{56} -1.63242i q^{57} -6.09361i q^{58} -3.84044 q^{59} +(4.45555 - 2.83281i) q^{60} -4.29832 q^{61} -19.3086i q^{62} +4.79889i q^{63} +5.33295 q^{64} +(-4.90717 + 3.11994i) q^{65} -4.20802 q^{66} +2.60747i q^{67} -0.943181i q^{68} +1.39945 q^{69} +(-10.6144 - 16.6947i) q^{70} +7.89582 q^{71} -0.625504i q^{72} -9.90579i q^{73} +9.74208 q^{74} +(6.33626 + 2.96868i) q^{75} -1.96813 q^{76} +7.21494i q^{77} +6.98833i q^{78} -12.0485 q^{79} +(5.43198 + 8.54362i) q^{80} -4.79054 q^{81} +22.6830i q^{82} -13.3856i q^{83} -10.8792 q^{84} +(1.05483 - 0.670652i) q^{85} +5.30800 q^{86} -4.44099i q^{87} -0.940419i q^{88} +7.71551 q^{89} +(-3.77394 + 2.39945i) q^{90} +11.9820 q^{91} -1.68725i q^{92} -14.0720i q^{93} +17.9022 q^{94} +(-1.39945 - 2.20111i) q^{95} +10.4861 q^{96} -2.62130i q^{97} +27.3224i q^{98} +1.63098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9	\( 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9} + 6 q^{10} + 4 q^{11} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{21} + 24 q^{24} - 16 q^{25} + 12 q^{26} - 8 q^{29} - 2 q^{30} - 28 q^{34} + 28 q^{35} - 16 q^{36} + 16 q^{39} - 10 q^{40} - 16 q^{41} - 12 q^{44} + 24 q^{45} + 28 q^{50} + 20 q^{51} - 44 q^{54} - 16 q^{55} + 28 q^{56} - 16 q^{60} - 16 q^{61} + 40 q^{64} - 14 q^{65} - 16 q^{66} + 4 q^{69} - 28 q^{70} - 48 q^{71} + 72 q^{74} - 36 q^{76} - 48 q^{79} - 2 q^{80} + 16 q^{81} - 4 q^{84} + 12 q^{85} + 28 q^{86} + 16 q^{89} - 4 q^{90} + 52 q^{91} + 84 q^{94} - 4 q^{95} + 60 q^{96} - 72 q^{99}+O(q^{100}) \) 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9 + 6 * q^10 + 4 * q^11 + 8 * q^14 + 6 * q^15 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 4 * q^21 + 24 * q^24 - 16 * q^25 + 12 * q^26 - 8 * q^29 - 2 * q^30 - 28 * q^34 + 28 * q^35 - 16 * q^36 + 16 * q^39 - 10 * q^40 - 16 * q^41 - 12 * q^44 + 24 * q^45 + 28 * q^50 + 20 * q^51 - 44 * q^54 - 16 * q^55 + 28 * q^56 - 16 * q^60 - 16 * q^61 + 40 * q^64 - 14 * q^65 - 16 * q^66 + 4 * q^69 - 28 * q^70 - 48 * q^71 + 72 * q^74 - 36 * q^76 - 48 * q^79 - 2 * q^80 + 16 * q^81 - 4 * q^84 + 12 * q^85 + 28 * q^86 + 16 * q^89 - 4 * q^90 + 52 * q^91 + 84 * q^94 - 4 * q^95 + 60 * q^96 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more