LMFDB - Embedded newform 1840.2.e.d.369.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not 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:	$14.6924739719$
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:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.3
Root			$-0.199724 + 0.199724i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.d.369.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39945i q^{3} +(-1.19972 + 1.88697i) q^{5} +4.60747i q^{7} +1.04155 q^{9} +O(q^{10})$	$q-1.39945i q^{3} +(-1.19972 + 1.88697i) q^{5} +4.60747i q^{7} +1.04155 q^{9} -1.56592 q^{11} +2.60055i q^{13} +(2.64072 + 1.67895i) q^{15} -0.559006i q^{17} -1.16647 q^{19} +6.44791 q^{21} +1.00000i q^{23} +(-2.12133 - 4.52769i) q^{25} -5.65593i q^{27} +3.17339 q^{29} -10.0554 q^{31} +2.19143i q^{33} +(-8.69416 - 5.52769i) q^{35} -5.07341i q^{37} +3.63934 q^{39} -11.8127 q^{41} +2.76426i q^{43} +(-1.24957 + 1.96537i) q^{45} +9.32298i q^{47} -14.2288 q^{49} -0.782299 q^{51} -5.54789i q^{53} +(1.87867 - 2.95485i) q^{55} +1.63242i q^{57} +3.84044 q^{59} -4.29832 q^{61} +4.79889i q^{63} +(-4.90717 - 3.11994i) q^{65} +2.60747i q^{67} +1.39945 q^{69} -7.89582 q^{71} +9.90579i q^{73} +(-6.33626 + 2.96868i) q^{75} -7.21494i q^{77} +12.0485 q^{79} -4.79054 q^{81} -13.3856i q^{83} +(1.05483 + 0.670652i) q^{85} -4.44099i q^{87} +7.71551 q^{89} -11.9820 q^{91} +14.0720i q^{93} +(1.39945 - 2.20111i) q^{95} +2.62130i q^{97} -1.63098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 6 * q^5 - 8 * q^9	$ 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.39945i		−	0.807971i	−0.914765	−	0.403986i	$-0.867625\pi$
$3$		−	1.39945i		−	0.807971i	0.914765	−	0.403986i	$-0.132375\pi$
$4$		0			0
$5$	−1.19972	+	1.88697i	−0.536533	+	0.843879i
$5$	−1.19972	+	1.88697i	−0.536533	+	0.843879i
$6$		0			0
$7$			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			0
$9$	1.04155			0.347182
$10$		0			0
$11$	−1.56592			−0.472143			−0.236072	−	0.971736i	$-0.575860\pi$
$11$	−1.56592			−0.472143			−0.236072	+	0.971736i	$0.575860\pi$
$12$		0			0
$13$			2.60055i			0.721264i	0.932708	+	0.360632i	$0.117439\pi$
$13$			2.60055i			0.721264i	−0.932708	+	0.360632i	$0.882561\pi$
$14$		0			0
$15$	2.64072	+	1.67895i	0.681830	+	0.433503i
$16$		0			0
$17$		−	0.559006i		−	0.135579i	−0.997700	−	0.0677894i	$-0.978405\pi$
$17$		−	0.559006i		−	0.135579i	0.997700	−	0.0677894i	$-0.0215946\pi$
$18$		0			0
$19$	−1.16647			−0.267608			−0.133804	−	0.991008i	$-0.542719\pi$
$19$	−1.16647			−0.267608			−0.133804	+	0.991008i	$0.542719\pi$
$20$		0			0
$21$	6.44791			1.40705
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−2.12133	−	4.52769i	−0.424265	−	0.905538i
$26$		0			0
$27$		−	5.65593i		−	1.08848i
$28$		0			0
$29$	3.17339			0.589284			0.294642	−	0.955608i	$-0.404800\pi$
$29$	3.17339			0.589284			0.294642	+	0.955608i	$0.404800\pi$
$30$		0			0
$31$	−10.0554			−1.80600			−0.903000	−	0.429641i	$-0.858640\pi$
$31$	−10.0554			−1.80600			−0.903000	+	0.429641i	$0.858640\pi$
$32$		0			0
$33$			2.19143i			0.381478i
$34$		0			0
$35$	−8.69416	−	5.52769i	−1.46958	−	0.934350i
$36$		0			0
$37$		−	5.07341i		−	0.834064i	−0.908892	−	0.417032i	$-0.863070\pi$
$37$		−	5.07341i		−	0.834064i	0.908892	−	0.417032i	$-0.136930\pi$
$38$		0			0
$39$	3.63934			0.582760
$40$		0			0
$41$	−11.8127			−1.84484			−0.922419	−	0.386190i	$-0.873791\pi$
$41$	−11.8127			−1.84484			−0.922419	+	0.386190i	$0.873791\pi$
$42$		0			0
$43$			2.76426i			0.421546i	0.977535	+	0.210773i	$0.0675982\pi$
$43$			2.76426i			0.421546i	−0.977535	+	0.210773i	$0.932402\pi$
$44$		0			0
$45$	−1.24957	+	1.96537i	−0.186275	+	0.292980i
$46$		0			0
$47$			9.32298i			1.35990i	0.733260	+	0.679948i	$0.237998\pi$
$47$			9.32298i			1.35990i	−0.733260	+	0.679948i	$0.762002\pi$
$48$		0			0
$49$	−14.2288			−2.03268
$50$		0			0
$51$	−0.782299			−0.109544
$52$		0			0
$53$		−	5.54789i		−	0.762061i	−0.924562	−	0.381030i	$-0.875569\pi$
$53$		−	5.54789i		−	0.762061i	0.924562	−	0.381030i	$-0.124431\pi$
$54$		0			0
$55$	1.87867	−	2.95485i	0.253320	−	0.398432i
$56$		0			0
$57$			1.63242i			0.216219i
$58$		0			0
$59$	3.84044			0.499983			0.249991	−	0.968248i	$-0.419572\pi$
$59$	3.84044			0.499983			0.249991	+	0.968248i	$0.419572\pi$
$60$		0			0
$61$	−4.29832			−0.550343			−0.275172	−	0.961395i	$-0.588735\pi$
$61$	−4.29832			−0.550343			−0.275172	+	0.961395i	$0.588735\pi$
$62$		0			0
$63$			4.79889i			0.604604i
$64$		0			0
$65$	−4.90717	−	3.11994i	−0.608659	−	0.386981i
$66$		0			0
$67$			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			0
$69$	1.39945			0.168474
$70$		0			0
$71$	−7.89582			−0.937062			−0.468531	−	0.883447i	$-0.655217\pi$
$71$	−7.89582			−0.937062			−0.468531	+	0.883447i	$0.655217\pi$
$72$		0			0
$73$			9.90579i			1.15938i	0.814835	+	0.579692i	$0.196827\pi$
$73$			9.90579i			1.15938i	−0.814835	+	0.579692i	$0.803173\pi$
$74$		0			0
$75$	−6.33626	+	2.96868i	−0.731649	+	0.342794i
$76$		0			0
$77$		−	7.21494i		−	0.822219i
$78$		0			0
$79$	12.0485			1.35556			0.677779	−	0.735266i	$-0.262943\pi$
$79$	12.0485			1.35556			0.677779	+	0.735266i	$0.262943\pi$
$80$		0			0
$81$	−4.79054			−0.532282
$82$		0			0
$83$		−	13.3856i		−	1.46926i	−0.678470	−	0.734628i	$-0.737357\pi$
$83$		−	13.3856i		−	1.46926i	0.678470	−	0.734628i	$-0.262643\pi$
$84$		0			0
$85$	1.05483	+	0.670652i	0.114412	+	0.0727425i
$86$		0			0
$87$		−	4.44099i		−	0.476125i
$88$		0			0
$89$	7.71551			0.817843			0.408921	−	0.912570i	$-0.365905\pi$
$89$	7.71551			0.817843			0.408921	+	0.912570i	$0.365905\pi$
$90$		0			0
$91$	−11.9820			−1.25605
$92$		0			0
$93$			14.0720i			1.45920i
$94$		0			0
$95$	1.39945	−	2.20111i	0.143580	−	0.225829i
$96$		0			0
$97$			2.62130i			0.266153i	0.991106	+	0.133076i	$0.0424856\pi$
$97$			2.62130i			0.266153i	−0.991106	+	0.133076i	$0.957514\pi$
$98$		0			0
$99$	−1.63098			−0.163920
$100$		0			0
$101$	−8.88199			−0.883791			−0.441895	−	0.897067i	$-0.645694\pi$
$101$	−8.88199			−0.883791			−0.441895	+	0.897067i	$0.645694\pi$
$102$		0			0
$103$			8.65593i			0.852894i	0.904512	+	0.426447i	$0.140235\pi$
$103$			8.65593i			0.852894i	−0.904512	+	0.426447i	$0.859765\pi$
$104$		0			0
$105$	−7.73571	+	12.1670i	−0.754928	+	1.18738i
$106$		0			0
$107$		−	6.91662i		−	0.668655i	−0.942457	−	0.334327i	$-0.891491\pi$
$107$		−	6.91662i		−	0.668655i	0.942457	−	0.334327i	$-0.108509\pi$
$108$		0			0
$109$	6.84736			0.655858			0.327929	−	0.944702i	$-0.393649\pi$
$109$	6.84736			0.655858			0.327929	+	0.944702i	$0.393649\pi$
$110$		0			0
$111$	−7.09998			−0.673900
$112$		0			0
$113$		−	4.60747i		−	0.433434i	−0.976234	−	0.216717i	$-0.930465\pi$
$113$		−	4.60747i		−	0.433434i	0.976234	−	0.216717i	$-0.0695349\pi$
$114$		0			0
$115$	−1.88697	−	1.19972i	−0.175961	−	0.111875i
$116$		0			0
$117$			2.70860i			0.250410i
$118$		0			0
$119$	2.57560			0.236105
$120$		0			0
$121$	−8.54789			−0.777081
$122$		0			0
$123$			16.5313i			1.49058i
$124$		0			0
$125$	11.0886	+	1.42909i	0.991797	+	0.127822i
$126$		0			0
$127$			11.5324i			1.02334i	0.859182	+	0.511669i	$0.170973\pi$
$127$			11.5324i			1.02334i	−0.859182	+	0.511669i	$0.829027\pi$
$128$		0			0
$129$	3.86844			0.340597
$130$		0			0
$131$	12.9639			1.13266			0.566332	−	0.824177i	$-0.308362\pi$
$131$	12.9639			1.13266			0.566332	+	0.824177i	$0.308362\pi$
$132$		0			0
$133$		−	5.37450i		−	0.466028i
$134$		0			0
$135$	10.6726	+	6.78556i	0.918550	+	0.584008i
$136$		0			0
$137$			17.1457i			1.46485i	0.680846	+	0.732427i	$0.261612\pi$
$137$			17.1457i			1.46485i	−0.680846	+	0.732427i	$0.738388\pi$
$138$		0			0
$139$	−19.4798			−1.65225			−0.826127	−	0.563485i	$-0.809460\pi$
$139$	−19.4798			−1.65225			−0.826127	+	0.563485i	$0.809460\pi$
$140$		0			0
$141$	13.0470			1.09876
$142$		0			0
$143$		−	4.07226i		−	0.340540i
$144$		0			0
$145$	−3.80719	+	5.98810i	−0.316170	+	0.497285i
$146$		0			0
$147$			19.9124i			1.64235i
$148$		0			0
$149$	−13.2288			−1.08374			−0.541872	−	0.840461i	$-0.682284\pi$
$149$	−13.2288			−1.08374			−0.541872	+	0.840461i	$0.682284\pi$
$150$		0			0
$151$	2.50749			0.204057			0.102028	−	0.994781i	$-0.467467\pi$
$151$	2.50749			0.204057			0.102028	+	0.994781i	$0.467467\pi$
$152$		0			0
$153$		−	0.582231i		−	0.0470706i
$154$		0			0
$155$	12.0637	−	18.9742i	0.968978	−	1.52405i
$156$		0			0
$157$			8.52824i			0.680628i	0.940312	+	0.340314i	$0.110533\pi$
$157$			8.52824i			0.680628i	−0.940312	+	0.340314i	$0.889467\pi$
$158$		0			0
$159$	−7.76398			−0.615723
$160$		0			0
$161$	−4.60747			−0.363119
$162$		0			0
$163$			7.73240i			0.605648i	0.953046	+	0.302824i	$0.0979294\pi$
$163$			7.73240i			0.605648i	−0.953046	+	0.302824i	$0.902071\pi$
$164$		0			0
$165$	−4.13516	−	2.62911i	−0.321922	−	0.204676i
$166$		0			0
$167$		−	14.3064i		−	1.10706i	−0.832829	−	0.553531i	$-0.813280\pi$
$167$		−	14.3064i		−	1.10706i	0.832829	−	0.553531i	$-0.186720\pi$
$168$		0			0
$169$	6.23713			0.479779
$170$		0			0
$171$	−1.21494			−0.0929086
$172$		0			0
$173$			15.4714i			1.17627i	0.808763	+	0.588135i	$0.200138\pi$
$173$			15.4714i			1.17627i	−0.808763	+	0.588135i	$0.799862\pi$
$174$		0			0
$175$	20.8612	−	9.77394i	1.57696	−	0.738841i
$176$		0			0
$177$		−	5.37450i		−	0.403972i
$178$		0			0
$179$	5.98612			0.447424			0.223712	−	0.974655i	$-0.428183\pi$
$179$	5.98612			0.447424			0.223712	+	0.974655i	$0.428183\pi$
$180$		0			0
$181$	−5.69892			−0.423597			−0.211799	−	0.977313i	$-0.567932\pi$
$181$	−5.69892			−0.423597			−0.211799	+	0.977313i	$0.567932\pi$
$182$		0			0
$183$			6.01527i			0.444662i
$184$		0			0
$185$	9.57339	+	6.08670i	0.703850	+	0.447503i
$186$		0			0
$187$			0.875359i			0.0640126i
$188$		0			0
$189$	26.0595			1.89555
$190$		0			0
$191$	8.34402			0.603752			0.301876	−	0.953347i	$-0.402387\pi$
$191$	8.34402			0.603752			0.301876	+	0.953347i	$0.402387\pi$
$192$		0			0
$193$			16.0250i			1.15350i	0.816920	+	0.576751i	$0.195680\pi$
$193$			16.0250i			1.15350i	−0.816920	+	0.576751i	$0.804320\pi$
$194$		0			0
$195$	−4.36620	+	6.86733i	−0.312670	+	0.491779i
$196$		0			0
$197$		−	5.78893i		−	0.412444i	−0.978505	−	0.206222i	$-0.933883\pi$
$197$		−	5.78893i		−	0.412444i	0.978505	−	0.206222i	$-0.0661169\pi$
$198$		0			0
$199$	−18.8085			−1.33330			−0.666651	−	0.745370i	$-0.732273\pi$
$199$	−18.8085			−1.33330			−0.666651	+	0.745370i	$0.732273\pi$
$200$		0			0
$201$	3.64902			0.257382
$202$		0			0
$203$			14.6213i			1.02621i
$204$		0			0
$205$	14.1720	−	22.2903i	0.989816	−	1.55682i
$206$		0			0
$207$			1.04155i			0.0723925i
$208$		0			0
$209$	1.82661			0.126349
$210$		0			0
$211$	4.88199			0.336090			0.168045	−	0.985779i	$-0.446255\pi$
$211$	4.88199			0.336090			0.168045	+	0.985779i	$0.446255\pi$
$212$		0			0
$213$			11.0498i			0.757119i
$214$		0			0
$215$	−5.21609	−	3.31635i	−0.355734	−	0.226173i
$216$		0			0
$217$		−	46.3299i		−	3.14508i
$218$		0			0
$219$	13.8626			0.936750
$220$		0			0
$221$	1.45372			0.0977880
$222$		0			0
$223$			13.4786i			0.902596i	0.892373	+	0.451298i	$0.149039\pi$
$223$			13.4786i			0.902596i	−0.892373	+	0.451298i	$0.850961\pi$
$224$		0			0
$225$	−2.20946	−	4.71580i	−0.147297	−	0.314387i
$226$		0			0
$227$			13.6894i			0.908598i	0.890849	+	0.454299i	$0.150110\pi$
$227$			13.6894i			0.908598i	−0.890849	+	0.454299i	$0.849890\pi$
$228$		0			0
$229$	−27.3964			−1.81040			−0.905202	−	0.424981i	$-0.860281\pi$
$229$	−27.3964			−1.81040			−0.905202	+	0.424981i	$0.860281\pi$
$230$		0			0
$231$	−10.0969			−0.664329
$232$		0			0
$233$		−	0.0387841i		−	0.00254083i	−0.999999	−	0.00127041i	$-0.999596\pi$
$233$		−	0.0387841i		−	0.00254083i	0.999999	−	0.00127041i	$-0.000404386\pi$
$234$		0			0
$235$	−17.5922	−	11.1850i	−1.14759	−	0.729629i
$236$		0			0
$237$		−	16.8612i		−	1.09525i
$238$		0			0
$239$	−13.9030			−0.899312			−0.449656	−	0.893202i	$-0.648453\pi$
$239$	−13.9030			−0.899312			−0.449656	+	0.893202i	$0.648453\pi$
$240$		0			0
$241$	−11.9972			−0.772810			−0.386405	−	0.922329i	$-0.626283\pi$
$241$	−11.9972			−0.772810			−0.386405	+	0.922329i	$0.626283\pi$
$242$		0			0
$243$		−	10.2637i		−	0.658416i
$244$		0			0
$245$	17.0706	−	26.8493i	1.09060	−	1.71534i
$246$		0			0
$247$		−	3.03348i		−	0.193016i
$248$		0			0
$249$	−18.7324			−1.18712
$250$		0			0
$251$	17.7806			1.12230			0.561150	−	0.827714i	$-0.310359\pi$
$251$	17.7806			1.12230			0.561150	+	0.827714i	$0.310359\pi$
$252$		0			0
$253$		−	1.56592i		−	0.0984487i
$254$		0			0
$255$	0.938543	−	1.47618i	0.0587738	−	0.0924418i
$256$		0			0
$257$		−	10.2315i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$257$		−	10.2315i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$258$		0			0
$259$	23.3756			1.45249
$260$		0			0
$261$	3.30524			0.204589
$262$		0			0
$263$			3.99856i			0.246562i	0.992372	+	0.123281i	$0.0393416\pi$
$263$			3.99856i			0.246562i	−0.992372	+	0.123281i	$0.960658\pi$
$264$		0			0
$265$	10.4687	+	6.65593i	0.643088	+	0.408871i
$266$		0			0
$267$		−	10.7975i		−	0.660794i
$268$		0			0
$269$	2.88084			0.175648			0.0878239	−	0.996136i	$-0.472009\pi$
$269$	2.88084			0.175648			0.0878239	+	0.996136i	$0.472009\pi$
$270$		0			0
$271$	−14.1944			−0.862250			−0.431125	−	0.902292i	$-0.641883\pi$
$271$	−14.1944			−0.862250			−0.431125	+	0.902292i	$0.641883\pi$
$272$		0			0
$273$			16.7681i			1.01485i
$274$		0			0
$275$	3.32183	+	7.09001i	0.200314	+	0.427544i
$276$		0			0
$277$		−	9.13576i		−	0.548915i	−0.961599	−	0.274457i	$-0.911502\pi$
$277$		−	9.13576i		−	0.548915i	0.961599	−	0.274457i	$-0.0884982\pi$
$278$		0			0
$279$	−10.4731			−0.627011
$280$		0			0
$281$	14.8640			0.886709			0.443355	−	0.896346i	$-0.353788\pi$
$281$	14.8640			0.886709			0.443355	+	0.896346i	$0.353788\pi$
$282$		0			0
$283$		−	3.03878i		−	0.180637i	−0.995913	−	0.0903185i	$-0.971212\pi$
$283$		−	3.03878i		−	0.180637i	0.995913	−	0.0903185i	$-0.0287885\pi$
$284$		0			0
$285$	−3.08033	−	1.95845i	−0.182463	−	0.116009i
$286$		0			0
$287$		−	54.4268i		−	3.21271i
$288$		0			0
$289$	16.6875			0.981618
$290$		0			0
$291$	3.66837			0.215044
$292$		0			0
$293$		−	19.2288i		−	1.12336i	−0.827356	−	0.561678i	$-0.810156\pi$
$293$		−	19.2288i		−	1.12336i	0.827356	−	0.561678i	$-0.189844\pi$
$294$		0			0
$295$	−4.60747	+	7.24681i	−0.268257	+	0.421925i
$296$		0			0
$297$			8.85675i			0.513921i
$298$		0			0
$299$	−2.60055			−0.150394
$300$		0			0
$301$	−12.7363			−0.734106
$302$		0			0
$303$			12.4299i			0.714078i
$304$		0			0
$305$	5.15680	−	8.11081i	0.295277	−	0.464423i
$306$		0			0
$307$		−	20.5395i		−	1.17225i	−0.810220	−	0.586127i	$-0.800652\pi$
$307$		−	20.5395i		−	1.17225i	0.810220	−	0.586127i	$-0.199348\pi$
$308$		0			0
$309$	12.1135			0.689114
$310$		0			0
$311$	9.83929			0.557935			0.278967	−	0.960301i	$-0.410008\pi$
$311$	9.83929			0.557935			0.278967	+	0.960301i	$0.410008\pi$
$312$		0			0
$313$			22.4659i			1.26985i	0.772574	+	0.634925i	$0.218969\pi$
$313$			22.4659i			1.26985i	−0.772574	+	0.634925i	$0.781031\pi$
$314$		0			0
$315$	−9.05538	−	5.75735i	−0.510213	−	0.324390i
$316$		0			0
$317$			17.8404i			1.00202i	0.865442	+	0.501010i	$0.167038\pi$
$317$			17.8404i			1.00202i	−0.865442	+	0.501010i	$0.832962\pi$
$318$		0			0
$319$	−4.96928			−0.278226
$320$		0			0
$321$	−9.67944			−0.540254
$322$		0			0
$323$			0.652066i			0.0362819i
$324$		0			0
$325$	11.7745	−	5.51662i	0.653132	−	0.306007i
$326$		0			0
$327$		−	9.58252i		−	0.529914i
$328$		0			0
$329$	−42.9554			−2.36821
$330$		0			0
$331$	−13.3130			−0.731750			−0.365875	−	0.930664i	$-0.619230\pi$
$331$	−13.3130			−0.731750			−0.365875	+	0.930664i	$0.619230\pi$
$332$		0			0
$333$		−	5.28420i		−	0.289572i
$334$		0			0
$335$	−4.92022	−	3.12824i	−0.268820	−	0.170914i
$336$		0			0
$337$		−	3.08338i		−	0.167962i	−0.996467	−	0.0839812i	$-0.973236\pi$
$337$		−	3.08338i		−	0.167962i	0.996467	−	0.0839812i	$-0.0267636\pi$
$338$		0			0
$339$	−6.44791			−0.350202
$340$		0			0
$341$	15.7459			0.852691
$342$		0			0
$343$		−	33.3063i		−	1.79837i
$344$		0			0
$345$	−1.67895	+	2.64072i	−0.0903916	+	0.142171i
$346$		0			0
$347$			11.5617i			0.620666i	0.950628	+	0.310333i	$0.100441\pi$
$347$			11.5617i			0.620666i	−0.950628	+	0.310333i	$0.899559\pi$
$348$		0			0
$349$	16.4980			0.883117			0.441558	−	0.897232i	$-0.354426\pi$
$349$	16.4980			0.883117			0.441558	+	0.897232i	$0.354426\pi$
$350$		0			0
$351$	14.7085			0.785084
$352$		0			0
$353$			13.7252i			0.730518i	0.930906	+	0.365259i	$0.119020\pi$
$353$			13.7252i			0.730518i	−0.930906	+	0.365259i	$0.880980\pi$
$354$		0			0
$355$	9.47280	−	14.8992i	0.502764	−	0.790767i
$356$		0			0
$357$		−	3.60442i		−	0.190766i
$358$		0			0
$359$	15.1442			0.799282			0.399641	−	0.916672i	$-0.369135\pi$
$359$	15.1442			0.799282			0.399641	+	0.916672i	$0.369135\pi$
$360$		0			0
$361$	−17.6393			−0.928386
$362$		0			0
$363$			11.9623i			0.627859i
$364$		0			0
$365$	−18.6919	−	11.8842i	−0.978381	−	0.622048i
$366$		0			0
$367$			11.2719i			0.588390i	0.955745	+	0.294195i	$0.0950515\pi$
$367$			11.2719i			0.588390i	−0.955745	+	0.294195i	$0.904949\pi$
$368$		0			0
$369$	−12.3035			−0.640495
$370$		0			0
$371$	25.5617			1.32710
$372$		0			0
$373$		−	35.3296i		−	1.82930i	−0.404252	−	0.914648i	$-0.632468\pi$
$373$		−	35.3296i		−	1.82930i	0.404252	−	0.914648i	$-0.367532\pi$
$374$		0			0
$375$	1.99994	−	15.5180i	0.103277	−	0.801344i
$376$		0			0
$377$			8.25257i			0.425029i
$378$		0			0
$379$	−36.8598			−1.89336			−0.946679	−	0.322178i	$-0.895585\pi$
$379$	−36.8598			−1.89336			−0.946679	+	0.322178i	$0.895585\pi$
$380$		0			0
$381$	16.1390			0.826828
$382$		0			0
$383$			4.27065i			0.218220i	0.994030	+	0.109110i	$0.0348001\pi$
$383$			4.27065i			0.218220i	−0.994030	+	0.109110i	$0.965200\pi$
$384$		0			0
$385$	13.6144	+	8.65593i	0.693853	+	0.441147i
$386$		0			0
$387$			2.87911i			0.146353i
$388$		0			0
$389$	6.42988			0.326008			0.163004	−	0.986625i	$-0.447882\pi$
$389$	6.42988			0.326008			0.163004	+	0.986625i	$0.447882\pi$
$390$		0			0
$391$	0.559006			0.0282701
$392$		0			0
$393$		−	18.1423i		−	0.915160i
$394$		0			0
$395$	−14.4548	+	22.7351i	−0.727301	+	1.14393i
$396$		0			0
$397$			33.9539i			1.70410i	0.523462	+	0.852049i	$0.324640\pi$
$397$			33.9539i			1.70410i	−0.523462	+	0.852049i	$0.675360\pi$
$398$		0			0
$399$	−7.52133			−0.376537
$400$		0			0
$401$	27.6421			1.38038			0.690189	−	0.723629i	$-0.257527\pi$
$401$	27.6421			1.38038			0.690189	+	0.723629i	$0.257527\pi$
$402$		0			0
$403$		−	26.1495i		−	1.30260i
$404$		0			0
$405$	5.74732	−	9.03961i	0.285587	−	0.449182i
$406$		0			0
$407$			7.94457i			0.393798i
$408$		0			0
$409$	13.2161			0.653494			0.326747	−	0.945112i	$-0.394048\pi$
$409$	13.2161			0.653494			0.326747	+	0.945112i	$0.394048\pi$
$410$		0			0
$411$	23.9945			1.18356
$412$		0			0
$413$			17.6947i			0.870700i
$414$		0			0
$415$	25.2582	+	16.0590i	1.23988	+	0.788304i
$416$		0			0
$417$			27.2609i			1.33497i
$418$		0			0
$419$	23.4772			1.14694			0.573468	−	0.819228i	$-0.305598\pi$
$419$	23.4772			1.14694			0.573468	+	0.819228i	$0.305598\pi$
$420$		0			0
$421$	23.1138			1.12650			0.563249	−	0.826287i	$-0.309551\pi$
$421$	23.1138			1.12650			0.563249	+	0.826287i	$0.309551\pi$
$422$		0			0
$423$			9.71032i			0.472132i
$424$		0			0
$425$	−2.53100	+	1.18583i	−0.122772	+	0.0575214i
$426$		0			0
$427$		−	19.8044i		−	0.958401i
$428$		0			0
$429$	−5.69892			−0.275146
$430$		0			0
$431$	9.52709			0.458904			0.229452	−	0.973320i	$-0.426307\pi$
$431$	9.52709			0.458904			0.229452	+	0.973320i	$0.426307\pi$
$432$		0			0
$433$			2.79445i			0.134293i	0.997743	+	0.0671464i	$0.0213895\pi$
$433$			2.79445i			0.134293i	−0.997743	+	0.0671464i	$0.978611\pi$
$434$		0			0
$435$	8.38003	+	5.32797i	0.401792	+	0.255456i
$436$		0			0
$437$		−	1.16647i		−	0.0558001i
$438$		0			0
$439$	2.09578			0.100026			0.0500129	−	0.998749i	$-0.484074\pi$
$439$	2.09578			0.100026			0.0500129	+	0.998749i	$0.484074\pi$
$440$		0			0
$441$	−14.8199			−0.705711
$442$		0			0
$443$			27.0870i			1.28694i	0.765471	+	0.643470i	$0.222506\pi$
$443$			27.0870i			1.28694i	−0.765471	+	0.643470i	$0.777494\pi$
$444$		0			0
$445$	−9.25648	+	14.5590i	−0.438799	+	0.690161i
$446$		0			0
$447$			18.5130i			0.875633i
$448$		0			0
$449$	0.470272			0.0221935			0.0110967	−	0.999938i	$-0.496468\pi$
$449$	0.470272			0.0221935			0.0110967	+	0.999938i	$0.496468\pi$
$450$		0			0
$451$	18.4978			0.871028
$452$		0			0
$453$		−	3.50910i		−	0.164872i
$454$		0			0
$455$	14.3750	−	22.6096i	0.673913	−	1.05996i
$456$		0			0
$457$		−	5.83768i		−	0.273075i	−0.990635	−	0.136538i	$-0.956403\pi$
$457$		−	5.83768i		−	0.273075i	0.990635	−	0.136538i	$-0.0435974\pi$
$458$		0			0
$459$	−3.16170			−0.147575
$460$		0			0
$461$	10.8487			0.505277			0.252638	−	0.967561i	$-0.418702\pi$
$461$	10.8487			0.505277			0.252638	+	0.967561i	$0.418702\pi$
$462$		0			0
$463$		−	1.06258i		−	0.0493825i	−0.999695	−	0.0246912i	$-0.992140\pi$
$463$		−	1.06258i		−	0.0493825i	0.999695	−	0.0246912i	$-0.00786026\pi$
$464$		0			0
$465$	−26.5534	−	16.8825i	−1.23139	−	0.782906i
$466$		0			0
$467$		−	4.99885i		−	0.231319i	−0.993289	−	0.115660i	$-0.963102\pi$
$467$		−	4.99885i		−	0.231319i	0.993289	−	0.115660i	$-0.0368982\pi$
$468$		0			0
$469$	−12.0138			−0.554747
$470$		0			0
$471$	11.9348			0.549928
$472$		0			0
$473$		−	4.32862i		−	0.199030i
$474$		0			0
$475$	2.47447	+	5.28144i	0.113537	+	0.242329i
$476$		0			0
$477$		−	5.77838i		−	0.264574i
$478$		0			0
$479$	33.4283			1.52738			0.763688	−	0.645585i	$-0.223387\pi$
$479$	33.4283			1.52738			0.763688	+	0.645585i	$0.223387\pi$
$480$		0			0
$481$	13.1937			0.601580
$482$		0			0
$483$			6.44791i			0.293390i
$484$		0			0
$485$	−4.94632	−	3.14484i	−0.224601	−	0.142800i
$486$		0			0
$487$		−	10.7901i		−	0.488945i	−0.969656	−	0.244473i	$-0.921385\pi$
$487$		−	10.7901i		−	0.488945i	0.969656	−	0.244473i	$-0.0786148\pi$
$488$		0			0
$489$	10.8211			0.489346
$490$		0			0
$491$	−15.8127			−0.713618			−0.356809	−	0.934177i	$-0.616135\pi$
$491$	−15.8127			−0.713618			−0.356809	+	0.934177i	$0.616135\pi$
$492$		0			0
$493$		−	1.77394i		−	0.0798944i
$494$		0			0
$495$	1.95673	−	3.07762i	0.0879483	−	0.138329i
$496$		0			0
$497$		−	36.3798i		−	1.63185i
$498$		0			0
$499$	12.9711			0.580668			0.290334	−	0.956925i	$-0.406234\pi$
$499$	12.9711			0.580668			0.290334	+	0.956925i	$0.406234\pi$
$500$		0			0
$501$	−20.0210			−0.894474
$502$		0			0
$503$			0.998849i			0.0445365i	0.999752	+	0.0222682i	$0.00708878\pi$
$503$			0.998849i			0.0445365i	−0.999752	+	0.0222682i	$0.992911\pi$
$504$		0			0
$505$	10.6559	−	16.7601i	0.474183	−	0.745813i
$506$		0			0
$507$		−	8.72853i		−	0.387648i
$508$		0			0
$509$	20.6936			0.917228			0.458614	−	0.888636i	$-0.348346\pi$
$509$	20.6936			0.917228			0.458614	+	0.888636i	$0.348346\pi$
$510$		0			0
$511$	−45.6406			−2.01902
$512$		0			0
$513$			6.59750i			0.291287i
$514$		0			0
$515$	−16.3335	−	10.3847i	−0.719740	−	0.457606i
$516$		0			0
$517$		−	14.5991i		−	0.642066i
$518$		0			0
$519$	21.6514			0.950393
$520$		0			0
$521$	39.2620			1.72010			0.860049	−	0.510212i	$-0.170433\pi$
$521$	39.2620			1.72010			0.860049	+	0.510212i	$0.170433\pi$
$522$		0			0
$523$			1.74162i			0.0761556i	0.999275	+	0.0380778i	$0.0121235\pi$
$523$			1.74162i			0.0761556i	−0.999275	+	0.0380778i	$0.987877\pi$
$524$		0			0
$525$	−13.6781	−	29.1941i	−0.596962	−	1.27414i
$526$		0			0
$527$			5.62101i			0.244855i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	30.7196i		−	1.33061i
$534$		0			0
$535$	13.0515	+	8.29803i	0.564264	+	0.358755i
$536$		0			0
$537$		−	8.37726i		−	0.361505i
$538$		0			0
$539$	22.2811			0.959717
$540$		0			0
$541$	−8.18117			−0.351736			−0.175868	−	0.984414i	$-0.556273\pi$
$541$	−8.18117			−0.351736			−0.175868	+	0.984414i	$0.556273\pi$
$542$		0			0
$543$			7.97534i			0.342254i
$544$		0			0
$545$	−8.21494	+	12.9208i	−0.351889	+	0.553465i
$546$		0			0
$547$		−	24.2470i		−	1.03673i	−0.855160	−	0.518364i	$-0.826541\pi$
$547$		−	24.2470i		−	1.03673i	0.855160	−	0.518364i	$-0.173459\pi$
$548$		0			0
$549$	−4.47690			−0.191069
$550$		0			0
$551$	−3.70168			−0.157697
$552$		0			0
$553$			55.5129i			2.36065i
$554$		0			0
$555$	8.51801	−	13.3975i	0.361569	−	0.568690i
$556$		0			0
$557$			35.0966i			1.48709i	0.668685	+	0.743546i	$0.266858\pi$
$557$			35.0966i			1.48709i	−0.668685	+	0.743546i	$0.733142\pi$
$558$		0			0
$559$	−7.18862			−0.304046
$560$		0			0
$561$	1.22502			0.0517204
$562$		0			0
$563$			6.98179i			0.294247i	0.989118	+	0.147124i	$0.0470016\pi$
$563$			6.98179i			0.294247i	−0.989118	+	0.147124i	$0.952998\pi$
$564$		0			0
$565$	8.69416	+	5.52769i	0.365766	+	0.232552i
$566$		0			0
$567$		−	22.0723i		−	0.926948i
$568$		0			0
$569$	−21.7280			−0.910886			−0.455443	−	0.890265i	$-0.650519\pi$
$569$	−21.7280			−0.910886			−0.455443	+	0.890265i	$0.650519\pi$
$570$		0			0
$571$	4.71016			0.197114			0.0985570	−	0.995131i	$-0.468577\pi$
$571$	4.71016			0.197114			0.0985570	+	0.995131i	$0.468577\pi$
$572$		0			0
$573$		−	11.6770i		−	0.487814i
$574$		0			0
$575$	4.52769	−	2.12133i	0.188818	−	0.0884654i
$576$		0			0
$577$			9.49085i			0.395109i	0.980292	+	0.197555i	$0.0633000\pi$
$577$			9.49085i			0.395109i	−0.980292	+	0.197555i	$0.936700\pi$
$578$		0			0
$579$	22.4261			0.931996
$580$		0			0
$581$	61.6736			2.55865
$582$		0			0
$583$			8.68756i			0.359802i
$584$		0			0
$585$	−5.11105	−	3.24957i	−0.211316	−	0.134353i
$586$		0			0
$587$			10.4952i			0.433184i	0.976262	+	0.216592i	$0.0694942\pi$
$587$			10.4952i			0.433184i	−0.976262	+	0.216592i	$0.930506\pi$
$588$		0			0
$589$	11.7293			0.483299
$590$		0			0
$591$	−8.10130			−0.333243
$592$		0			0
$593$		−	18.2138i		−	0.747951i	−0.927439	−	0.373975i	$-0.877994\pi$
$593$		−	18.2138i		−	0.747951i	0.927439	−	0.373975i	$-0.122006\pi$
$594$		0			0
$595$	−3.09001	+	4.86009i	−0.126678	+	0.199244i
$596$		0			0
$597$			26.3215i			1.07727i
$598$		0			0
$599$	−20.8055			−0.850091			−0.425045	−	0.905172i	$-0.639742\pi$
$599$	−20.8055			−0.850091			−0.425045	+	0.905172i	$0.639742\pi$
$600$		0			0
$601$	−47.4004			−1.93350			−0.966752	−	0.255716i	$-0.917689\pi$
$601$	−47.4004			−1.93350			−0.966752	+	0.255716i	$0.917689\pi$
$602$		0			0
$603$			2.71580i			0.110596i
$604$		0			0
$605$	10.2551	−	16.1296i	0.416929	−	0.655762i
$606$		0			0
$607$		−	5.77118i		−	0.234245i	−0.993117	−	0.117123i	$-0.962633\pi$
$607$		−	5.77118i		−	0.234245i	0.993117	−	0.117123i	$-0.0373670\pi$
$608$		0			0
$609$	20.4617			0.829152
$610$		0			0
$611$	−24.2449			−0.980844
$612$		0			0
$613$			38.9329i			1.57248i	0.617919	+	0.786242i	$0.287976\pi$
$613$			38.9329i			1.57248i	−0.617919	+	0.786242i	$0.712024\pi$
$614$		0			0
$615$	−31.1941	−	19.8330i	−1.25787	−	0.799743i
$616$		0			0
$617$		−	10.1653i		−	0.409241i	−0.978841	−	0.204620i	$-0.934404\pi$
$617$		−	10.1653i		−	0.409241i	0.978841	−	0.204620i	$-0.0655959\pi$
$618$		0			0
$619$	2.09923			0.0843752			0.0421876	−	0.999110i	$-0.486567\pi$
$619$	2.09923			0.0843752			0.0421876	+	0.999110i	$0.486567\pi$
$620$		0			0
$621$	5.65593			0.226965
$622$		0			0
$623$			35.5490i			1.42424i
$624$		0			0
$625$	−16.0000	+	19.2094i	−0.639998	+	0.768377i
$626$		0			0
$627$		−	2.55624i		−	0.102087i
$628$		0			0
$629$	−2.83607			−0.113081
$630$		0			0
$631$	32.3066			1.28611			0.643053	−	0.765821i	$-0.277667\pi$
$631$	32.3066			1.28611			0.643053	+	0.765821i	$0.277667\pi$
$632$		0			0
$633$		−	6.83209i		−	0.271551i
$634$		0			0
$635$	−21.7614	−	13.8357i	−0.863575	−	0.549055i
$636$		0			0
$637$		−	37.0027i		−	1.46610i
$638$		0			0
$639$	−8.22387			−0.325331
$640$		0			0
$641$	14.5758			0.575711			0.287856	−	0.957674i	$-0.407058\pi$
$641$	14.5758			0.575711			0.287856	+	0.957674i	$0.407058\pi$
$642$		0			0
$643$			42.5090i			1.67639i	0.545369	+	0.838196i	$0.316389\pi$
$643$			42.5090i			1.67639i	−0.545369	+	0.838196i	$0.683611\pi$
$644$		0			0
$645$	−4.64106	+	7.29964i	−0.182742	+	0.287423i
$646$		0			0
$647$		−	3.04979i		−	0.119900i	−0.998201	−	0.0599498i	$-0.980906\pi$
$647$		−	3.04979i		−	0.119900i	0.998201	−	0.0599498i	$-0.0190940\pi$
$648$		0			0
$649$	−6.01383			−0.236064
$650$		0			0
$651$	−64.8362			−2.54113
$652$		0			0
$653$			17.9600i			0.702830i	0.936220	+	0.351415i	$0.114299\pi$
$653$			17.9600i			0.702830i	−0.936220	+	0.351415i	$0.885701\pi$
$654$		0			0
$655$	−15.5531	+	24.4626i	−0.607711	+	0.955832i
$656$		0			0
$657$			10.3173i			0.402518i
$658$		0			0
$659$	46.3411			1.80519			0.902597	−	0.430486i	$-0.141658\pi$
$659$	46.3411			1.80519			0.902597	+	0.430486i	$0.141658\pi$
$660$		0			0
$661$	1.23165			0.0479056			0.0239528	−	0.999713i	$-0.492375\pi$
$661$	1.23165			0.0479056			0.0239528	+	0.999713i	$0.492375\pi$
$662$		0			0
$663$		−	2.03441i		−	0.0790099i
$664$		0			0
$665$	10.1415	+	6.44791i	0.393271	+	0.250039i
$666$		0			0
$667$			3.17339i			0.122874i
$668$		0			0
$669$	18.8626			0.729271
$670$		0			0
$671$	6.73083			0.259841
$672$		0			0
$673$			42.4664i			1.63696i	0.574537	+	0.818479i	$0.305182\pi$
$673$			42.4664i			1.63696i	−0.574537	+	0.818479i	$0.694818\pi$
$674$		0			0
$675$	−25.6083	+	11.9981i	−0.985664	+	0.461806i
$676$		0			0
$677$		−	1.41777i		−	0.0544893i	−0.999629	−	0.0272447i	$-0.991327\pi$
$677$		−	1.41777i		−	0.0544893i	0.999629	−	0.0272447i	$-0.00867332\pi$
$678$		0			0
$679$	−12.0776			−0.463495
$680$		0			0
$681$	19.1576			0.734121
$682$		0			0
$683$		−	4.60886i		−	0.176353i	−0.996105	−	0.0881766i	$-0.971896\pi$
$683$		−	4.60886i		−	0.176353i	0.996105	−	0.0881766i	$-0.0281040\pi$
$684$		0			0
$685$	−32.3534	−	20.5701i	−1.23616	−	0.785942i
$686$		0			0
$687$			38.3398i			1.46276i
$688$		0			0
$689$	14.4276			0.549647
$690$		0			0
$691$	16.8903			0.642537			0.321269	−	0.946988i	$-0.395891\pi$
$691$	16.8903			0.642537			0.321269	+	0.946988i	$0.395891\pi$
$692$		0			0
$693$		−	7.51470i		−	0.285460i
$694$		0			0
$695$	23.3704	−	36.7578i	0.886488	−	1.39430i
$696$		0			0
$697$			6.60338i			0.250121i
$698$		0			0
$699$	−0.0542763			−0.00205292
$700$		0			0
$701$	−11.9543			−0.451508			−0.225754	−	0.974184i	$-0.572484\pi$
$701$	−11.9543			−0.451508			−0.225754	+	0.974184i	$0.572484\pi$
$702$		0			0
$703$			5.91801i			0.223202i
$704$		0			0
$705$	−15.6528	+	24.6194i	−0.589519	+	0.927219i
$706$		0			0
$707$		−	40.9235i		−	1.53909i
$708$		0			0
$709$	12.6755			0.476040			0.238020	−	0.971260i	$-0.423502\pi$
$709$	12.6755			0.476040			0.238020	+	0.971260i	$0.423502\pi$
$710$		0			0
$711$	12.5490			0.470626
$712$		0			0
$713$		−	10.0554i		−	0.376577i
$714$		0			0
$715$	7.68425	+	4.88559i	0.287375	+	0.182711i
$716$		0			0
$717$			19.4566i			0.726618i
$718$		0			0
$719$	−34.7351			−1.29540			−0.647700	−	0.761896i	$-0.724269\pi$
$719$	−34.7351			−1.29540			−0.647700	+	0.761896i	$0.724269\pi$
$720$		0			0
$721$	−39.8819			−1.48528
$722$		0			0
$723$			16.7895i			0.624408i
$724$		0			0
$725$	−6.73180	−	14.3681i	−0.250013	−	0.533619i
$726$		0			0
$727$			27.6631i			1.02597i	0.858398	+	0.512984i	$0.171460\pi$
$727$			27.6631i			1.02597i	−0.858398	+	0.512984i	$0.828540\pi$
$728$		0			0
$729$	−28.7351			−1.06426
$730$		0			0
$731$	1.54524			0.0571528
$732$		0			0
$733$		−	2.14239i		−	0.0791309i	−0.999217	−	0.0395654i	$-0.987403\pi$
$733$		−	2.14239i		−	0.0791309i	0.999217	−	0.0395654i	$-0.0125974\pi$
$734$		0			0
$735$	−37.5742	−	23.8894i	−1.38594	−	0.881174i
$736$		0			0
$737$		−	4.08309i		−	0.150403i
$738$		0			0
$739$	−42.6249			−1.56798			−0.783991	−	0.620772i	$-0.786819\pi$
$739$	−42.6249			−1.56798			−0.783991	+	0.620772i	$0.786819\pi$
$740$		0			0
$741$	−4.24519			−0.155951
$742$		0			0
$743$		−	23.4687i		−	0.860982i	−0.902595	−	0.430491i	$-0.858340\pi$
$743$		−	23.4687i		−	0.860982i	0.902595	−	0.430491i	$-0.141660\pi$
$744$		0			0
$745$	15.8709	−	24.9623i	0.581464	−	0.914549i
$746$		0			0
$747$		−	13.9417i		−	0.510100i
$748$		0			0
$749$	31.8681			1.16444
$750$		0			0
$751$	27.2579			0.994654			0.497327	−	0.867563i	$-0.334315\pi$
$751$	27.2579			0.994654			0.497327	+	0.867563i	$0.334315\pi$
$752$		0			0
$753$		−	24.8830i		−	0.906786i
$754$		0			0
$755$	−3.00830	+	4.73157i	−0.109483	+	0.172199i
$756$		0			0
$757$			12.7382i			0.462976i	0.972838	+	0.231488i	$0.0743595\pi$
$757$			12.7382i			0.462976i	−0.972838	+	0.231488i	$0.925641\pi$
$758$		0			0
$759$	−2.19143			−0.0795437
$760$		0			0
$761$	6.61715			0.239871			0.119936	−	0.992782i	$-0.461731\pi$
$761$	6.61715			0.239871			0.119936	+	0.992782i	$0.461731\pi$
$762$		0			0
$763$			31.5490i			1.14215i
$764$		0			0
$765$	1.09865	+	0.698516i	0.0397219	+	0.0252549i
$766$		0			0
$767$			9.98727i			0.360619i
$768$		0			0
$769$	27.2116			0.981275			0.490638	−	0.871364i	$-0.336764\pi$
$769$	27.2116			0.981275			0.490638	+	0.871364i	$0.336764\pi$
$770$		0			0
$771$	−14.3185			−0.515668
$772$		0			0
$773$			22.9249i			0.824552i	0.911059	+	0.412276i	$0.135266\pi$
$773$			22.9249i			0.824552i	−0.911059	+	0.412276i	$0.864734\pi$
$774$		0			0
$775$	21.3307	+	45.5276i	0.766223	+	1.63540i
$776$		0			0
$777$		−	32.7129i		−	1.17357i
$778$		0			0
$779$	13.7792			0.493693
$780$		0			0
$781$	12.3642			0.442427
$782$		0			0
$783$		−	17.9485i		−	0.641427i
$784$		0			0
$785$	−16.0926	−	10.2315i	−0.574368	−	0.365179i
$786$		0			0
$787$			13.0620i			0.465610i	0.972523	+	0.232805i	$0.0747904\pi$
$787$			13.0620i			0.465610i	−0.972523	+	0.232805i	$0.925210\pi$
$788$		0			0
$789$	5.59578			0.199215
$790$		0			0
$791$	21.2288			0.754808
$792$		0			0
$793$		−	11.1780i		−	0.396943i
$794$		0			0
$795$	9.31463	−	14.6504i	0.330356	−	0.519596i
$796$		0			0
$797$		−	20.9305i		−	0.741395i	−0.928754	−	0.370697i	$-0.879119\pi$
$797$		−	20.9305i		−	0.741395i	0.928754	−	0.370697i	$-0.120881\pi$
$798$		0			0
$799$	5.21160			0.184373
$800$		0			0
$801$	8.03607			0.283941
$802$		0			0
$803$		−	15.5117i		−	0.547396i
$804$		0			0
$805$	5.52769	−	8.69416i	0.194825	−	0.306429i
$806$		0			0
$807$		−	4.03158i		−	0.141918i
$808$		0			0
$809$	−18.0664			−0.635180			−0.317590	−	0.948228i	$-0.602874\pi$
$809$	−18.0664			−0.635180			−0.317590	+	0.948228i	$0.602874\pi$
$810$		0			0
$811$	−34.8658			−1.22430			−0.612152	−	0.790740i	$-0.709696\pi$
$811$	−34.8658			−1.22430			−0.612152	+	0.790740i	$0.709696\pi$
$812$		0			0
$813$			19.8644i			0.696673i
$814$		0			0
$815$	−14.5908	−	9.27674i	−0.511094	−	0.324950i
$816$		0			0
$817$		−	3.22444i		−	0.112809i
$818$		0			0
$819$	−12.4798			−0.436079
$820$		0			0
$821$	14.5989			0.509507			0.254753	−	0.967006i	$-0.418006\pi$
$821$	14.5989			0.509507			0.254753	+	0.967006i	$0.418006\pi$
$822$		0			0
$823$		−	25.8126i		−	0.899771i	−0.893086	−	0.449886i	$-0.851465\pi$
$823$		−	25.8126i		−	0.899771i	0.893086	−	0.449886i	$-0.148535\pi$
$824$		0			0
$825$	9.92210	−	4.64873i	0.345443	−	0.161848i
$826$		0			0
$827$			54.9097i			1.90940i	0.297577	+	0.954698i	$0.403822\pi$
$827$			54.9097i			1.90940i	−0.297577	+	0.954698i	$0.596178\pi$
$828$		0			0
$829$	27.4809			0.954452			0.477226	−	0.878781i	$-0.341642\pi$
$829$	27.4809			0.954452			0.477226	+	0.878781i	$0.341642\pi$
$830$		0			0
$831$	−12.7850			−0.443507
$832$		0			0
$833$			7.95396i			0.275589i
$834$		0			0
$835$	26.9958	+	17.1637i	0.934226	+	0.593975i
$836$		0			0
$837$			56.8725i			1.96580i
$838$		0			0
$839$	−1.46577			−0.0506041			−0.0253020	−	0.999680i	$-0.508055\pi$
$839$	−1.46577			−0.0506041			−0.0253020	+	0.999680i	$0.508055\pi$
$840$		0			0
$841$	−18.9296			−0.652744
$842$		0			0
$843$		−	20.8013i		−	0.716436i
$844$		0			0
$845$	−7.48283	+	11.7693i	−0.257417	+	0.404876i
$846$		0			0
$847$		−	39.3841i		−	1.35325i
$848$		0			0
$849$	−4.25262			−0.145949
$850$		0			0
$851$	5.07341			0.173914
$852$		0			0
$853$			35.8044i			1.22592i	0.790115	+	0.612959i	$0.210021\pi$
$853$			35.8044i			1.22592i	−0.790115	+	0.612959i	$0.789979\pi$
$854$		0			0
$855$	1.45759	−	2.29255i	0.0498485	−	0.0784037i
$856$		0			0
$857$		−	28.1663i		−	0.962141i	−0.876682	−	0.481070i	$-0.840248\pi$
$857$		−	28.1663i		−	0.962141i	0.876682	−	0.481070i	$-0.159752\pi$
$858$		0			0
$859$	20.3185			0.693258			0.346629	−	0.938002i	$-0.387326\pi$
$859$	20.3185			0.693258			0.346629	+	0.938002i	$0.387326\pi$
$860$		0			0
$861$	−76.1674			−2.59578
$862$		0			0
$863$			15.1357i			0.515224i	0.966248	+	0.257612i	$0.0829356\pi$
$863$			15.1357i			0.515224i	−0.966248	+	0.257612i	$0.917064\pi$
$864$		0			0
$865$	−29.1941	−	18.5614i	−0.992631	−	0.631108i
$866$		0			0
$867$		−	23.3533i		−	0.793120i
$868$		0			0
$869$	−18.8670			−0.640018
$870$		0			0
$871$	−6.78086			−0.229761
$872$		0			0
$873$			2.73021i			0.0924036i
$874$		0			0
$875$	−6.58451	+	51.0905i	−0.222597	+	1.72717i
$876$		0			0
$877$		−	12.9668i		−	0.437856i	−0.975741	−	0.218928i	$-0.929744\pi$
$877$		−	12.9668i		−	0.437856i	0.975741	−	0.218928i	$-0.0702560\pi$
$878$		0			0
$879$	−26.9097			−0.907640
$880$		0			0
$881$	−34.3573			−1.15753			−0.578763	−	0.815496i	$-0.696464\pi$
$881$	−34.3573			−1.15753			−0.578763	+	0.815496i	$0.696464\pi$
$882$		0			0
$883$			14.7201i			0.495372i	0.968840	+	0.247686i	$0.0796701\pi$
$883$			14.7201i			0.495372i	−0.968840	+	0.247686i	$0.920330\pi$
$884$		0			0
$885$	10.1415	+	6.44791i	0.340904	+	0.216744i
$886$		0			0
$887$			20.9401i			0.703101i	0.936169	+	0.351550i	$0.114345\pi$
$887$			20.9401i			0.703101i	−0.936169	+	0.351550i	$0.885655\pi$
$888$		0			0
$889$	−53.1354			−1.78210
$890$		0			0
$891$	7.50161			0.251313
$892$		0			0
$893$		−	10.8750i		−	0.363919i
$894$		0			0
$895$	−7.18169	+	11.2956i	−0.240057	+	0.377572i
$896$		0			0
$897$			3.63934i			0.121514i
$898$		0			0
$899$	−31.9097			−1.06425
$900$		0			0
$901$	−3.10130			−0.103319
$902$		0			0
$903$			17.8237i			0.593137i
$904$		0			0
$905$	6.83713	−	10.7537i	0.227274	−	0.357465i
$906$		0			0
$907$		−	40.2008i		−	1.33485i	−0.744679	−	0.667423i	$-0.767397\pi$
$907$		−	40.2008i		−	1.33485i	0.744679	−	0.667423i	$-0.232603\pi$
$908$		0			0
$909$	−9.25101			−0.306837
$910$		0			0
$911$	−54.6058			−1.80917			−0.904586	−	0.426292i	$-0.859820\pi$
$911$	−54.6058			−1.80917			−0.904586	+	0.426292i	$0.859820\pi$
$912$		0			0
$913$			20.9608i			0.693700i
$914$		0			0
$915$	−11.3506	−	7.21666i	−0.375241	−	0.238576i
$916$		0			0
$917$			59.7309i			1.97249i
$918$		0			0
$919$	13.8019			0.455284			0.227642	−	0.973745i	$-0.426898\pi$
$919$	13.8019			0.455284			0.227642	+	0.973745i	$0.426898\pi$
$920$		0			0
$921$	−28.7440			−0.947147
$922$		0			0
$923$		−	20.5335i		−	0.675868i
$924$		0			0
$925$	−22.9708	+	10.7624i	−0.755277	+	0.353864i
$926$		0			0
$927$			9.01556i			0.296110i
$928$		0			0
$929$	−19.2775			−0.632475			−0.316237	−	0.948680i	$-0.602420\pi$
$929$	−19.2775			−0.632475			−0.316237	+	0.948680i	$0.602420\pi$
$930$		0			0
$931$	16.5975			0.543961
$932$		0			0
$933$		−	13.7696i		−	0.450795i
$934$		0			0
$935$	−1.65178	−	1.05019i	−0.0540189	−	0.0343449i
$936$		0			0
$937$		−	48.0855i		−	1.57089i	−0.618934	−	0.785443i	$-0.712435\pi$
$937$		−	48.0855i		−	1.57089i	0.618934	−	0.785443i	$-0.287565\pi$
$938$		0			0
$939$	31.4399			1.02600
$940$		0			0
$941$	−30.6853			−1.00031			−0.500156	−	0.865935i	$-0.666724\pi$
$941$	−30.6853			−1.00031			−0.500156	+	0.865935i	$0.666724\pi$
$942$		0			0
$943$		−	11.8127i		−	0.384675i
$944$		0			0
$945$	−31.2642	+	49.1736i	−1.01703	+	1.59962i
$946$		0			0
$947$		−	12.9280i		−	0.420103i	−0.977690	−	0.210051i	$-0.932637\pi$
$947$		−	12.9280i		−	0.420103i	0.977690	−	0.210051i	$-0.0673631\pi$
$948$		0			0
$949$	−25.7605			−0.836222
$950$		0			0
$951$	24.9668			0.809603
$952$		0			0
$953$		−	6.38285i		−	0.206761i	−0.994642	−	0.103380i	$-0.967034\pi$
$953$		−	6.38285i		−	0.206761i	0.994642	−	0.103380i	$-0.0329659\pi$
$954$		0			0
$955$	−10.0105	+	15.7449i	−0.323933	+	0.509494i
$956$		0			0
$957$			6.95425i			0.224799i
$958$		0			0
$959$	−78.9982			−2.55098
$960$		0			0
$961$	70.1107			2.26163
$962$		0			0
$963$		−	7.20398i		−	0.232145i
$964$		0			0
$965$	−30.2386	−	19.2255i	−0.973416	−	0.618891i
$966$		0			0
$967$		−	38.6028i		−	1.24138i	−0.784056	−	0.620691i	$-0.786852\pi$
$967$		−	38.6028i		−	1.24138i	0.784056	−	0.620691i	$-0.213148\pi$
$968$		0			0
$969$	0.912532			0.0293148
$970$		0			0
$971$	24.0216			0.770889			0.385444	−	0.922731i	$-0.374048\pi$
$971$	24.0216			0.770889			0.385444	+	0.922731i	$0.374048\pi$
$972$		0			0
$973$		−	89.7525i		−	2.87733i
$974$		0			0
$975$	−7.72022	−	16.4778i	−0.247245	−	0.527712i
$976$		0			0
$977$		−	45.5558i		−	1.45746i	−0.684801	−	0.728730i	$-0.740111\pi$
$977$		−	45.5558i		−	1.45746i	0.684801	−	0.728730i	$-0.259889\pi$
$978$		0			0
$979$	−12.0819			−0.386139
$980$		0			0
$981$	7.13184			0.227702
$982$		0			0
$983$			29.1396i			0.929408i	0.885466	+	0.464704i	$0.153839\pi$
$983$			29.1396i			0.929408i	−0.885466	+	0.464704i	$0.846161\pi$
$984$		0			0
$985$	10.9235	+	6.94511i	0.348053	+	0.221290i
$986$		0			0
$987$			60.1138i			1.91344i
$988$		0			0
$989$	−2.76426			−0.0878985
$990$		0			0
$991$	20.0415			0.636639			0.318320	−	0.947983i	$-0.396881\pi$
$991$	20.0415			0.636639			0.318320	+	0.947983i	$0.396881\pi$
$992$		0			0
$993$			18.6309i			0.591233i
$994$		0			0
$995$	22.5650	−	35.4912i	0.715360	−	1.12515i
$996$		0			0
$997$		−	0.340153i		−	0.0107728i	−0.999985	−	0.00538638i	$-0.998285\pi$
$997$		−	0.340153i		−	0.0107728i	0.999985	−	0.00538638i	$-0.00171455\pi$
$998$		0			0
$999$	−28.6949			−0.907866

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.2	✓	8	20.19	odd	2
115.2.b.b.24.7	yes	8	4.3	odd	2
575.2.a.i.1.4		4	20.3	even	4
575.2.a.j.1.1		4	20.7	even	4
1035.2.b.e.829.2		8	12.11	even	2
1035.2.b.e.829.7		8	60.59	even	2
1840.2.e.d.369.3		8	1.1	even	1	trivial
1840.2.e.d.369.6		8	5.4	even	2	inner
5175.2.a.bv.1.1		4	60.23	odd	4
5175.2.a.bw.1.4		4	60.47	odd	4
9200.2.a.ck.1.2		4	5.2	odd	4
9200.2.a.cq.1.3		4	5.3	odd	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.39945i q^{3} +(-1.19972 + 1.88697i) q^{5} +4.60747i q^{7} +1.04155 q^{9} +O(q^{10})\)	\(q-1.39945i q^{3} +(-1.19972 + 1.88697i) q^{5} +4.60747i q^{7} +1.04155 q^{9} -1.56592 q^{11} +2.60055i q^{13} +(2.64072 + 1.67895i) q^{15} -0.559006i q^{17} -1.16647 q^{19} +6.44791 q^{21} +1.00000i q^{23} +(-2.12133 - 4.52769i) q^{25} -5.65593i q^{27} +3.17339 q^{29} -10.0554 q^{31} +2.19143i q^{33} +(-8.69416 - 5.52769i) q^{35} -5.07341i q^{37} +3.63934 q^{39} -11.8127 q^{41} +2.76426i q^{43} +(-1.24957 + 1.96537i) q^{45} +9.32298i q^{47} -14.2288 q^{49} -0.782299 q^{51} -5.54789i q^{53} +(1.87867 - 2.95485i) q^{55} +1.63242i q^{57} +3.84044 q^{59} -4.29832 q^{61} +4.79889i q^{63} +(-4.90717 - 3.11994i) q^{65} +2.60747i q^{67} +1.39945 q^{69} -7.89582 q^{71} +9.90579i q^{73} +(-6.33626 + 2.96868i) q^{75} -7.21494i q^{77} +12.0485 q^{79} -4.79054 q^{81} -13.3856i q^{83} +(1.05483 + 0.670652i) q^{85} -4.44099i q^{87} +7.71551 q^{89} -11.9820 q^{91} +14.0720i q^{93} +(1.39945 - 2.20111i) q^{95} +2.62130i q^{97} -1.63098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 6 * q^5 - 8 * q^9	\( 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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