LMFDB - Embedded newform 930.2.d.i.559.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(559,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("930.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.d (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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.11669056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58 $ x^6 - 2x^5 + 7x^4 + 8x^3 - x^2 + 54x + 58
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.3
Root			$1.60509 - 2.15264i$ of defining polynomial
Character	$\chi$	$=$	930.559
Dual form			930.2.d.i.559.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.605092 - 2.15264i) q^{5} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.605092 - 2.15264i) q^{5} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-2.15264 - 0.605092i) q^{10} +5.21018 q^{11} -1.00000i q^{12} +0.789816i q^{13} -2.00000 q^{14} +(2.15264 + 0.605092i) q^{15} +1.00000 q^{16} +0.115086i q^{17} +1.00000i q^{18} -0.115086 q^{19} +(-0.605092 + 2.15264i) q^{20} +2.00000 q^{21} -5.21018i q^{22} -4.42037i q^{23} -1.00000 q^{24} +(-4.26773 - 2.60509i) q^{25} +0.789816 q^{26} -1.00000i q^{27} +2.00000i q^{28} -4.42037 q^{29} +(0.605092 - 2.15264i) q^{30} -1.00000 q^{31} -1.00000i q^{32} +5.21018i q^{33} +0.115086 q^{34} +(-4.30528 - 1.21018i) q^{35} +1.00000 q^{36} -6.61056i q^{37} +0.115086i q^{38} -0.789816 q^{39} +(2.15264 + 0.605092i) q^{40} +2.00000 q^{41} -2.00000i q^{42} -8.61056i q^{43} -5.21018 q^{44} +(-0.605092 + 2.15264i) q^{45} -4.42037 q^{46} -0.115086i q^{47} +1.00000i q^{48} +3.00000 q^{49} +(-2.60509 + 4.26773i) q^{50} -0.115086 q^{51} -0.789816i q^{52} -0.190196i q^{53} -1.00000 q^{54} +(3.15264 - 11.2157i) q^{55} +2.00000 q^{56} -0.115086i q^{57} +4.42037i q^{58} +4.19020 q^{59} +(-2.15264 - 0.605092i) q^{60} +12.4955 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +(1.70019 + 0.477911i) q^{65} +5.21018 q^{66} -5.82075i q^{67} -0.115086i q^{68} +4.42037 q^{69} +(-1.21018 + 4.30528i) q^{70} -13.8207 q^{71} -1.00000i q^{72} -10.8407i q^{73} -6.61056 q^{74} +(2.60509 - 4.26773i) q^{75} +0.115086 q^{76} -10.4204i q^{77} +0.789816i q^{78} -2.30528 q^{79} +(0.605092 - 2.15264i) q^{80} +1.00000 q^{81} -2.00000i q^{82} +15.1460i q^{83} -2.00000 q^{84} +(0.247739 + 0.0696377i) q^{85} -8.61056 q^{86} -4.42037i q^{87} +5.21018i q^{88} +2.23017 q^{89} +(2.15264 + 0.605092i) q^{90} +1.57963 q^{91} +4.42037i q^{92} -1.00000i q^{93} -0.115086 q^{94} +(-0.0696377 + 0.247739i) q^{95} +1.00000 q^{96} +9.21018i q^{97} -3.00000i q^{98} -5.21018 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9	$ 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 + 16 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 + 4 * q^20 + 12 * q^21 - 6 * q^24 - 8 * q^25 + 20 * q^26 + 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^34 + 6 * q^36 - 20 * q^39 + 12 * q^41 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 4 * q^51 - 6 * q^54 + 6 * q^55 + 12 * q^56 + 4 * q^59 + 28 * q^61 - 6 * q^64 - 8 * q^65 + 16 * q^66 - 4 * q^69 + 8 * q^70 - 16 * q^71 + 12 * q^74 + 8 * q^75 - 4 * q^76 + 12 * q^79 - 4 * q^80 + 6 * q^81 - 12 * q^84 - 22 * q^85 + 4 * q^89 + 40 * q^91 + 4 * q^94 - 28 * q^95 + 6 * q^96 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−1.00000			−0.500000
$5$	0.605092	−	2.15264i	0.270605	−	0.962690i
$5$	0.605092	−	2.15264i	0.270605	−	0.962690i
$6$	1.00000			0.408248
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$			1.00000i			0.353553i
$9$	−1.00000			−0.333333
$10$	−2.15264	−	0.605092i	−0.680725	−	0.191347i
$11$	5.21018			1.57093			0.785465	−	0.618906i	$-0.212424\pi$
$11$	5.21018			1.57093			0.785465	+	0.618906i	$0.212424\pi$
$12$		−	1.00000i		−	0.288675i
$13$			0.789816i			0.219056i	0.993984	+	0.109528i	$0.0349339\pi$
$13$			0.789816i			0.219056i	−0.993984	+	0.109528i	$0.965066\pi$
$14$	−2.00000			−0.534522
$15$	2.15264	+	0.605092i	0.555810	+	0.156234i
$16$	1.00000			0.250000
$17$			0.115086i			0.0279125i	0.999903	+	0.0139562i	$0.00444255\pi$
$17$			0.115086i			0.0279125i	−0.999903	+	0.0139562i	$0.995557\pi$
$18$			1.00000i			0.235702i
$19$	−0.115086			−0.0264026			−0.0132013	−	0.999913i	$-0.504202\pi$
$19$	−0.115086			−0.0264026			−0.0132013	+	0.999913i	$0.504202\pi$
$20$	−0.605092	+	2.15264i	−0.135303	+	0.481345i
$21$	2.00000			0.436436
$22$		−	5.21018i		−	1.11081i
$23$		−	4.42037i		−	0.921710i	−0.887475	−	0.460855i	$-0.847543\pi$
$23$		−	4.42037i		−	0.921710i	0.887475	−	0.460855i	$-0.152457\pi$
$24$	−1.00000			−0.204124
$25$	−4.26773	−	2.60509i	−0.853545	−	0.521018i
$26$	0.789816			0.154896
$27$		−	1.00000i		−	0.192450i
$28$			2.00000i			0.377964i
$29$	−4.42037			−0.820842			−0.410421	−	0.911896i	$-0.634618\pi$
$29$	−4.42037			−0.820842			−0.410421	+	0.911896i	$0.634618\pi$
$30$	0.605092	−	2.15264i	0.110474	−	0.393017i
$31$	−1.00000			−0.179605
$31$	−1.00000			−0.179605
$32$		−	1.00000i		−	0.176777i
$33$			5.21018i			0.906977i
$34$	0.115086			0.0197371
$35$	−4.30528	−	1.21018i	−0.727726	−	0.204558i
$36$	1.00000			0.166667
$37$		−	6.61056i		−	1.08677i	−0.839484	−	0.543385i	$-0.817142\pi$
$37$		−	6.61056i		−	1.08677i	0.839484	−	0.543385i	$-0.182858\pi$
$38$			0.115086i			0.0186694i
$39$	−0.789816			−0.126472
$40$	2.15264	+	0.605092i	0.340362	+	0.0956735i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		−	2.00000i		−	0.308607i
$43$		−	8.61056i		−	1.31310i	−0.754283	−	0.656549i	$-0.772015\pi$
$43$		−	8.61056i		−	1.31310i	0.754283	−	0.656549i	$-0.227985\pi$
$44$	−5.21018			−0.785465
$45$	−0.605092	+	2.15264i	−0.0902018	+	0.320897i
$46$	−4.42037			−0.651748
$47$		−	0.115086i		−	0.0167870i	−0.999965	−	0.00839352i	$-0.997328\pi$
$47$		−	0.115086i		−	0.0167870i	0.999965	−	0.00839352i	$-0.00267177\pi$
$48$			1.00000i			0.144338i
$49$	3.00000			0.428571
$50$	−2.60509	+	4.26773i	−0.368416	+	0.603548i
$51$	−0.115086			−0.0161153
$52$		−	0.789816i		−	0.109528i
$53$		−	0.190196i		−	0.0261254i	−0.999915	−	0.0130627i	$-0.995842\pi$
$53$		−	0.190196i		−	0.0261254i	0.999915	−	0.0130627i	$-0.00415811\pi$
$54$	−1.00000			−0.136083
$55$	3.15264	−	11.2157i	0.425102	−	1.51232i
$56$	2.00000			0.267261
$57$		−	0.115086i		−	0.0152435i
$58$			4.42037i			0.580423i
$59$	4.19020			0.545517			0.272759	−	0.962083i	$-0.412064\pi$
$59$	4.19020			0.545517			0.272759	+	0.962083i	$0.412064\pi$
$60$	−2.15264	−	0.605092i	−0.277905	−	0.0781170i
$61$	12.4955			1.59988			0.799941	−	0.600079i	$-0.204864\pi$
$61$	12.4955			1.59988			0.799941	+	0.600079i	$0.204864\pi$
$62$			1.00000i			0.127000i
$63$			2.00000i			0.251976i
$64$	−1.00000			−0.125000
$65$	1.70019	+	0.477911i	0.210883	+	0.0592776i
$66$	5.21018			0.641329
$67$		−	5.82075i		−	0.711118i	−0.934654	−	0.355559i	$-0.884291\pi$
$67$		−	5.82075i		−	0.711118i	0.934654	−	0.355559i	$-0.115709\pi$
$68$		−	0.115086i		−	0.0139562i
$69$	4.42037			0.532150
$70$	−1.21018	+	4.30528i	−0.144645	+	0.514580i
$71$	−13.8207			−1.64022			−0.820111	−	0.572205i	$-0.806088\pi$
$71$	−13.8207			−1.64022			−0.820111	+	0.572205i	$0.806088\pi$
$72$		−	1.00000i		−	0.117851i
$73$		−	10.8407i		−	1.26881i	−0.773000	−	0.634406i	$-0.781245\pi$
$73$		−	10.8407i		−	1.26881i	0.773000	−	0.634406i	$-0.218755\pi$
$74$	−6.61056			−0.768462
$75$	2.60509	−	4.26773i	0.300810	−	0.492795i
$76$	0.115086			0.0132013
$77$		−	10.4204i		−	1.18751i
$78$			0.789816i			0.0894290i
$79$	−2.30528			−0.259365			−0.129682	−	0.991556i	$-0.541396\pi$
$79$	−2.30528			−0.259365			−0.129682	+	0.991556i	$0.541396\pi$
$80$	0.605092	−	2.15264i	0.0676513	−	0.240673i
$81$	1.00000			0.111111
$82$		−	2.00000i		−	0.220863i
$83$			15.1460i			1.66249i	0.555905	+	0.831246i	$0.312372\pi$
$83$			15.1460i			1.66249i	−0.555905	+	0.831246i	$0.687628\pi$
$84$	−2.00000			−0.218218
$85$	0.247739	+	0.0696377i	0.0268711	+	0.00755327i
$86$	−8.61056			−0.928501
$87$		−	4.42037i		−	0.473913i
$88$			5.21018i			0.555407i
$89$	2.23017			0.236398			0.118199	−	0.992990i	$-0.462288\pi$
$89$	2.23017			0.236398			0.118199	+	0.992990i	$0.462288\pi$
$90$	2.15264	+	0.605092i	0.226908	+	0.0637823i
$91$	1.57963			0.165590
$92$			4.42037i			0.460855i
$93$		−	1.00000i		−	0.103695i
$94$	−0.115086			−0.0118702
$95$	−0.0696377	+	0.247739i	−0.00714468	+	0.0254175i
$96$	1.00000			0.102062
$97$			9.21018i			0.935153i	0.883953	+	0.467576i	$0.154873\pi$
$97$			9.21018i			0.935153i	−0.883953	+	0.467576i	$0.845127\pi$
$98$		−	3.00000i		−	0.303046i
$99$	−5.21018			−0.523643
$100$	4.26773	+	2.60509i	0.426773	+	0.260509i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$			0.115086i			0.0113952i
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	−0.789816			−0.0774478
$105$	1.21018	−	4.30528i	0.118102	−	0.420153i
$106$	−0.190196			−0.0184735
$107$			3.03093i			0.293011i	0.989210	+	0.146506i	$0.0468027\pi$
$107$			3.03093i			0.293011i	−0.989210	+	0.146506i	$0.953197\pi$
$108$			1.00000i			0.0962250i
$109$	8.19020			0.784479			0.392239	−	0.919863i	$-0.371701\pi$
$109$	8.19020			0.784479			0.392239	+	0.919863i	$0.371701\pi$
$110$	−11.2157	−	3.15264i	−1.06937	−	0.300593i
$111$	6.61056			0.627447
$112$		−	2.00000i		−	0.188982i
$113$			10.3804i			0.976505i	0.872702	+	0.488253i	$0.162366\pi$
$113$			10.3804i			0.976505i	−0.872702	+	0.488253i	$0.837634\pi$
$114$	−0.115086			−0.0107788
$115$	−9.51547	−	2.67473i	−0.887322	−	0.249420i
$116$	4.42037			0.410421
$117$		−	0.789816i		−	0.0730185i
$118$		−	4.19020i		−	0.385739i
$119$	0.230172			0.0210999
$120$	−0.605092	+	2.15264i	−0.0552371	+	0.196508i
$121$	16.1460			1.46782
$122$		−	12.4955i		−	1.13129i
$123$			2.00000i			0.180334i
$124$	1.00000			0.0898027
$125$	−8.19020	+	7.61056i	−0.732553	+	0.680710i
$126$	2.00000			0.178174
$127$			10.4204i			0.924658i	0.886708	+	0.462329i	$0.152986\pi$
$127$			10.4204i			0.924658i	−0.886708	+	0.462329i	$0.847014\pi$
$128$			1.00000i			0.0883883i
$129$	8.61056			0.758118
$130$	0.477911	−	1.70019i	0.0419156	−	0.149117i
$131$	−0.190196			−0.0166175			−0.00830875	−	0.999965i	$-0.502645\pi$
$131$	−0.190196			−0.0166175			−0.00830875	+	0.999965i	$0.502645\pi$
$132$		−	5.21018i		−	0.453488i
$133$			0.230172i			0.0199585i
$134$	−5.82075			−0.502836
$135$	−2.15264	−	0.605092i	−0.185270	−	0.0520780i
$136$	−0.115086			−0.00986855
$137$			21.4513i			1.83271i	0.400369	+	0.916354i	$0.368882\pi$
$137$			21.4513i			1.83271i	−0.400369	+	0.916354i	$0.631118\pi$
$138$		−	4.42037i		−	0.376287i
$139$	−16.4204			−1.39276			−0.696379	−	0.717674i	$-0.745207\pi$
$139$	−16.4204			−1.39276			−0.696379	+	0.717674i	$0.745207\pi$
$140$	4.30528	+	1.21018i	0.363863	+	0.102279i
$141$	0.115086			0.00969200
$142$			13.8207i			1.15981i
$143$			4.11509i			0.344121i
$144$	−1.00000			−0.0833333
$145$	−2.67473	+	9.51547i	−0.222124	+	0.790216i
$146$	−10.8407			−0.897186
$147$			3.00000i			0.247436i
$148$			6.61056i			0.543385i
$149$	3.63055			0.297426			0.148713	−	0.988880i	$-0.452487\pi$
$149$	3.63055			0.297426			0.148713	+	0.988880i	$0.452487\pi$
$150$	−4.26773	−	2.60509i	−0.348458	−	0.212705i
$151$	−15.7566			−1.28225			−0.641126	−	0.767435i	$-0.721533\pi$
$151$	−15.7566			−1.28225			−0.641126	+	0.767435i	$0.721533\pi$
$152$		−	0.115086i		−	0.00933472i
$153$		−	0.115086i		−	0.00930416i
$154$	−10.4204			−0.839697
$155$	−0.605092	+	2.15264i	−0.0486022	+	0.172904i
$156$	0.789816			0.0632359
$157$			3.96002i			0.316044i	0.987436	+	0.158022i	$0.0505117\pi$
$157$			3.96002i			0.316044i	−0.987436	+	0.158022i	$0.949488\pi$
$158$			2.30528i			0.183398i
$159$	0.190196			0.0150835
$160$	−2.15264	−	0.605092i	−0.170181	−	0.0478367i
$161$	−8.84074			−0.696748
$162$		−	1.00000i		−	0.0785674i
$163$			9.82075i			0.769220i	0.923079	+	0.384610i	$0.125664\pi$
$163$			9.82075i			0.769220i	−0.923079	+	0.384610i	$0.874336\pi$
$164$	−2.00000			−0.156174
$165$	11.2157	+	3.15264i	0.873138	+	0.245433i
$166$	15.1460			1.17556
$167$		−	2.00000i		−	0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$			2.00000i			0.154303i
$169$	12.3762			0.952015
$170$	0.0696377	−	0.247739i	0.00534097	−	0.0190007i
$171$	0.115086			0.00880085
$172$			8.61056i			0.656549i
$173$		−	0.535454i		−	0.0407098i	−0.999793	−	0.0203549i	$-0.993520\pi$
$173$		−	0.535454i		−	0.0407098i	0.999793	−	0.0203549i	$-0.00647962\pi$
$174$	−4.42037			−0.335107
$175$	−5.21018	+	8.53545i	−0.393853	+	0.645220i
$176$	5.21018			0.392732
$177$			4.19020i			0.314954i
$178$		−	2.23017i		−	0.167158i
$179$	−10.4313			−0.779673			−0.389836	−	0.920884i	$-0.627468\pi$
$179$	−10.4313			−0.779673			−0.389836	+	0.920884i	$0.627468\pi$
$180$	0.605092	−	2.15264i	0.0451009	−	0.160448i
$181$	17.4513			1.29714			0.648572	−	0.761153i	$-0.275366\pi$
$181$	17.4513			1.29714			0.648572	+	0.761153i	$0.275366\pi$
$182$		−	1.57963i		−	0.117090i
$183$			12.4955i			0.923692i
$184$	4.42037			0.325874
$185$	−14.2302	−	4.00000i	−1.04622	−	0.294086i
$186$	−1.00000			−0.0733236
$187$			0.599620i			0.0438485i
$188$			0.115086i			0.00839352i
$189$	−2.00000			−0.145479
$190$	0.247739	+	0.0696377i	0.0179729	+	0.00505205i
$191$	19.2611			1.39368			0.696842	−	0.717224i	$-0.254588\pi$
$191$	19.2611			1.39368			0.696842	+	0.717224i	$0.254588\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			14.4313i			1.03879i	0.854535	+	0.519394i	$0.173842\pi$
$193$			14.4313i			1.03879i	−0.854535	+	0.519394i	$0.826158\pi$
$194$	9.21018			0.661253
$195$	−0.477911	+	1.70019i	−0.0342239	+	0.121753i
$196$	−3.00000			−0.214286
$197$		−	10.3804i		−	0.739572i	−0.929117	−	0.369786i	$-0.879431\pi$
$197$		−	10.3804i		−	0.739572i	0.929117	−	0.369786i	$-0.120569\pi$
$198$			5.21018i			0.370272i
$199$	−14.5355			−1.03039			−0.515196	−	0.857073i	$-0.672281\pi$
$199$	−14.5355			−1.03039			−0.515196	+	0.857073i	$0.672281\pi$
$200$	2.60509	−	4.26773i	0.184208	−	0.301774i
$201$	5.82075			0.410564
$202$		0			0
$203$			8.84074i			0.620498i
$204$	0.115086			0.00805764
$205$	1.21018	−	4.30528i	0.0845229	−	0.300694i
$206$	−6.00000			−0.418040
$207$			4.42037i			0.307237i
$208$			0.789816i			0.0547639i
$209$	−0.599620			−0.0414766
$210$	−4.30528	−	1.21018i	−0.297093	−	0.0835106i
$211$	24.8407			1.71011			0.855053	−	0.518540i	$-0.173524\pi$
$211$	24.8407			1.71011			0.855053	+	0.518540i	$0.173524\pi$
$212$			0.190196i			0.0130627i
$213$		−	13.8207i		−	0.946982i
$214$	3.03093			0.207190
$215$	−18.5355	−	5.21018i	−1.26411	−	0.355332i
$216$	1.00000			0.0680414
$217$			2.00000i			0.135769i
$218$		−	8.19020i		−	0.554710i
$219$	10.8407			0.732549
$220$	−3.15264	+	11.2157i	−0.212551	+	0.756159i
$221$	−0.0908968			−0.00611438
$222$		−	6.61056i		−	0.443672i
$223$		−	19.6306i		−	1.31456i	−0.753647	−	0.657280i	$-0.771707\pi$
$223$		−	19.6306i		−	1.31456i	0.753647	−	0.657280i	$-0.228293\pi$
$224$	−2.00000			−0.133631
$225$	4.26773	+	2.60509i	0.284515	+	0.173673i
$226$	10.3804			0.690493
$227$			6.42037i			0.426135i	0.977038	+	0.213067i	$0.0683454\pi$
$227$			6.42037i			0.426135i	−0.977038	+	0.213067i	$0.931655\pi$
$228$			0.115086i			0.00762176i
$229$	29.7166			1.96373			0.981864	−	0.189585i	$-0.0607142\pi$
$229$	29.7166			1.96373			0.981864	+	0.189585i	$0.0607142\pi$
$230$	−2.67473	+	9.51547i	−0.176366	+	0.627431i
$231$	10.4204			0.685610
$232$		−	4.42037i		−	0.290211i
$233$			5.38944i			0.353074i	0.984294	+	0.176537i	$0.0564895\pi$
$233$			5.38944i			0.353074i	−0.984294	+	0.176537i	$0.943511\pi$
$234$	−0.789816			−0.0516319
$235$	−0.247739	−	0.0696377i	−0.0161607	−	0.00454266i
$236$	−4.19020			−0.272759
$237$		−	2.30528i		−	0.149744i
$238$		−	0.230172i		−	0.0149198i
$239$	−7.03093			−0.454793			−0.227397	−	0.973802i	$-0.573021\pi$
$239$	−7.03093			−0.454793			−0.227397	+	0.973802i	$0.573021\pi$
$240$	2.15264	+	0.605092i	0.138952	+	0.0390585i
$241$	−0.190196			−0.0122516			−0.00612580	−	0.999981i	$-0.501950\pi$
$241$	−0.190196			−0.0122516			−0.00612580	+	0.999981i	$0.501950\pi$
$242$		−	16.1460i		−	1.03791i
$243$			1.00000i			0.0641500i
$244$	−12.4955			−0.799941
$245$	1.81528	−	6.45792i	0.115974	−	0.412582i
$246$	2.00000			0.127515
$247$		−	0.0908968i		−	0.00578363i
$248$		−	1.00000i		−	0.0635001i
$249$	−15.1460			−0.959840
$250$	7.61056	+	8.19020i	0.481334	+	0.517993i
$251$	14.4204			0.910206			0.455103	−	0.890439i	$-0.349603\pi$
$251$	14.4204			0.910206			0.455103	+	0.890439i	$0.349603\pi$
$252$		−	2.00000i		−	0.125988i
$253$		−	23.0309i		−	1.44794i
$254$	10.4204			0.653832
$255$	−0.0696377	+	0.247739i	−0.00436088	+	0.0155140i
$256$	1.00000			0.0625000
$257$			1.76983i			0.110399i	0.998475	+	0.0551994i	$0.0175795\pi$
$257$			1.76983i			0.110399i	−0.998475	+	0.0551994i	$0.982421\pi$
$258$		−	8.61056i		−	0.536070i
$259$	−13.2211			−0.821521
$260$	−1.70019	−	0.477911i	−0.105441	−	0.0296388i
$261$	4.42037			0.273614
$262$			0.190196i			0.0117504i
$263$			14.2302i			0.877470i	0.898616	+	0.438735i	$0.144573\pi$
$263$			14.2302i			0.877470i	−0.898616	+	0.438735i	$0.855427\pi$
$264$	−5.21018			−0.320665
$265$	−0.409424	−	0.115086i	−0.0251507	−	0.00706968i
$266$	0.230172			0.0141128
$267$			2.23017i			0.136484i
$268$			5.82075i			0.355559i
$269$	1.15926			0.0706815			0.0353408	−	0.999375i	$-0.488748\pi$
$269$	1.15926			0.0706815			0.0353408	+	0.999375i	$0.488748\pi$
$270$	−0.605092	+	2.15264i	−0.0368247	+	0.131006i
$271$	22.6857			1.37806			0.689028	−	0.724734i	$-0.258038\pi$
$271$	22.6857			1.37806			0.689028	+	0.724734i	$0.258038\pi$
$272$			0.115086i			0.00697812i
$273$			1.57963i			0.0956037i
$274$	21.4513			1.29592
$275$	−22.2356	−	13.5730i	−1.34086	−	0.818483i
$276$	−4.42037			−0.266075
$277$			9.63055i			0.578644i	0.957232	+	0.289322i	$0.0934298\pi$
$277$			9.63055i			0.578644i	−0.957232	+	0.289322i	$0.906570\pi$
$278$			16.4204i			0.984828i
$279$	1.00000			0.0598684
$280$	1.21018	−	4.30528i	0.0723223	−	0.257290i
$281$	10.8407			0.646704			0.323352	−	0.946279i	$-0.395190\pi$
$281$	10.8407			0.646704			0.323352	+	0.946279i	$0.395190\pi$
$282$		−	0.115086i		−	0.00685328i
$283$		−	19.7808i		−	1.17584i	−0.808917	−	0.587922i	$-0.799946\pi$
$283$		−	19.7808i		−	1.17584i	0.808917	−	0.587922i	$-0.200054\pi$
$284$	13.8207			0.820111
$285$	−0.247739	−	0.0696377i	−0.0146748	−	0.00412498i
$286$	4.11509			0.243330
$287$		−	4.00000i		−	0.236113i
$288$			1.00000i			0.0589256i
$289$	16.9868			0.999221
$290$	9.51547	+	2.67473i	0.558767	+	0.157066i
$291$	−9.21018			−0.539911
$292$			10.8407i			0.634406i
$293$			31.2211i			1.82396i	0.410237	+	0.911979i	$0.365446\pi$
$293$			31.2211i			1.82396i	−0.410237	+	0.911979i	$0.634554\pi$
$294$	3.00000			0.174964
$295$	2.53545	−	9.01999i	0.147620	−	0.525164i
$296$	6.61056			0.384231
$297$		−	5.21018i		−	0.302326i
$298$		−	3.63055i		−	0.210312i
$299$	3.49128			0.201906
$300$	−2.60509	+	4.26773i	−0.150405	+	0.246397i
$301$	−17.2211			−0.992609
$302$			15.7566i			0.906689i
$303$		0			0
$304$	−0.115086			−0.00660064
$305$	7.56091	−	26.8983i	0.432937	−	1.54019i
$306$	−0.115086			−0.00657904
$307$			13.5796i			0.775031i	0.921863	+	0.387515i	$0.126667\pi$
$307$			13.5796i			0.775031i	−0.921863	+	0.387515i	$0.873333\pi$
$308$			10.4204i			0.593756i
$309$	6.00000			0.341328
$310$	2.15264	+	0.605092i	0.122262	+	0.0343669i
$311$	18.2811			1.03663			0.518313	−	0.855191i	$-0.326560\pi$
$311$	18.2811			1.03663			0.518313	+	0.855191i	$0.326560\pi$
$312$		−	0.789816i		−	0.0447145i
$313$		−	14.4603i		−	0.817347i	−0.912681	−	0.408673i	$-0.865991\pi$
$313$		−	14.4603i		−	0.817347i	0.912681	−	0.408673i	$-0.134009\pi$
$314$	3.96002			0.223477
$315$	4.30528	+	1.21018i	0.242575	+	0.0681862i
$316$	2.30528			0.129682
$317$			3.46455i			0.194588i	0.995256	+	0.0972941i	$0.0310188\pi$
$317$			3.46455i			0.194588i	−0.995256	+	0.0972941i	$0.968981\pi$
$318$		−	0.190196i		−	0.0106657i
$319$	−23.0309			−1.28948
$320$	−0.605092	+	2.15264i	−0.0338257	+	0.120336i
$321$	−3.03093			−0.169170
$322$			8.84074i			0.492675i
$323$		−	0.0132448i		−	0.000736961i
$324$	−1.00000			−0.0555556
$325$	2.05754	−	3.37072i	0.114132	−	0.186974i
$326$	9.82075			0.543921
$327$			8.19020i			0.452919i
$328$			2.00000i			0.110432i
$329$	−0.230172			−0.0126898
$330$	3.15264	−	11.2157i	0.173547	−	0.617402i
$331$	−20.8008			−1.14331			−0.571657	−	0.820493i	$-0.693699\pi$
$331$	−20.8008			−1.14331			−0.571657	+	0.820493i	$0.693699\pi$
$332$		−	15.1460i		−	0.831246i
$333$			6.61056i			0.362257i
$334$	−2.00000			−0.109435
$335$	−12.5300	−	3.52209i	−0.684586	−	0.192432i
$336$	2.00000			0.109109
$337$			23.0709i			1.25675i	0.777910	+	0.628376i	$0.216280\pi$
$337$			23.0709i			1.25675i	−0.777910	+	0.628376i	$0.783720\pi$
$338$		−	12.3762i		−	0.673176i
$339$	−10.3804			−0.563786
$340$	−0.247739	−	0.0696377i	−0.0134355	−	0.00377663i
$341$	−5.21018			−0.282147
$342$		−	0.115086i		−	0.00622314i
$343$		−	20.0000i		−	1.07990i
$344$	8.61056			0.464251
$345$	2.67473	−	9.51547i	0.144003	−	0.512295i
$346$	−0.535454			−0.0287862
$347$			31.9868i			1.71714i	0.512697	+	0.858569i	$0.328646\pi$
$347$			31.9868i			1.71714i	−0.512697	+	0.858569i	$0.671354\pi$
$348$			4.42037i			0.236957i
$349$	−23.4513			−1.25532			−0.627660	−	0.778488i	$-0.715987\pi$
$349$	−23.4513			−1.25532			−0.627660	+	0.778488i	$0.715987\pi$
$350$	8.53545	+	5.21018i	0.456239	+	0.278496i
$351$	0.789816			0.0421573
$352$		−	5.21018i		−	0.277704i
$353$			3.88491i			0.206773i	0.994641	+	0.103387i	$0.0329679\pi$
$353$			3.88491i			0.206773i	−0.994641	+	0.103387i	$0.967032\pi$
$354$	4.19020			0.222706
$355$	−8.36283	+	29.7511i	−0.443853	+	1.57903i
$356$	−2.23017			−0.118199
$357$			0.230172i			0.0121820i
$358$			10.4313i			0.551312i
$359$	21.3604			1.12736			0.563680	−	0.825994i	$-0.309385\pi$
$359$	21.3604			1.12736			0.563680	+	0.825994i	$0.309385\pi$
$360$	−2.15264	−	0.605092i	−0.113454	−	0.0318912i
$361$	−18.9868			−0.999303
$362$		−	17.4513i		−	0.917220i
$363$			16.1460i			0.847446i
$364$	−1.57963			−0.0827952
$365$	−23.3362	−	6.55964i	−1.22147	−	0.343347i
$366$	12.4955			0.653149
$367$			20.4713i			1.06859i	0.845297	+	0.534296i	$0.179423\pi$
$367$			20.4713i			1.06859i	−0.845297	+	0.534296i	$0.820577\pi$
$368$		−	4.42037i		−	0.230428i
$369$	−2.00000			−0.104116
$370$	−4.00000	+	14.2302i	−0.207950	+	0.739791i
$371$	−0.380392			−0.0197490
$372$			1.00000i			0.0518476i
$373$		−	9.76983i		−	0.505863i	−0.967484	−	0.252931i	$-0.918605\pi$
$373$		−	9.76983i		−	0.505863i	0.967484	−	0.252931i	$-0.0813946\pi$
$374$	0.599620			0.0310056
$375$	−7.61056	−	8.19020i	−0.393008	−	0.422940i
$376$	0.115086			0.00593511
$377$		−	3.49128i		−	0.179810i
$378$			2.00000i			0.102869i
$379$	−3.65474			−0.187731			−0.0938657	−	0.995585i	$-0.529922\pi$
$379$	−3.65474			−0.187731			−0.0938657	+	0.995585i	$0.529922\pi$
$380$	0.0696377	−	0.247739i	0.00357234	−	0.0127087i
$381$	−10.4204			−0.533852
$382$		−	19.2611i		−	0.985484i
$383$		−	25.4913i		−	1.30254i	−0.758845	−	0.651272i	$-0.774236\pi$
$383$		−	25.4913i		−	1.30254i	0.758845	−	0.651272i	$-0.225764\pi$
$384$	−1.00000			−0.0510310
$385$	−22.4313	−	6.30528i	−1.14321	−	0.321347i
$386$	14.4313			0.734534
$387$			8.61056i			0.437700i
$388$		−	9.21018i		−	0.467576i
$389$	−4.65054			−0.235792			−0.117896	−	0.993026i	$-0.537615\pi$
$389$	−4.65054			−0.235792			−0.117896	+	0.993026i	$0.537615\pi$
$390$	1.70019	+	0.477911i	0.0860925	+	0.0242000i
$391$	0.508723			0.0257272
$392$			3.00000i			0.151523i
$393$		−	0.190196i		−	0.00959412i
$394$	−10.3804			−0.522957
$395$	−1.39491	+	4.96245i	−0.0701854	+	0.249688i
$396$	5.21018			0.261822
$397$		−	26.7124i		−	1.34066i	−0.742064	−	0.670329i	$-0.766153\pi$
$397$		−	26.7124i		−	1.34066i	0.742064	−	0.670329i	$-0.233847\pi$
$398$			14.5355i			0.728596i
$399$	−0.230172			−0.0115230
$400$	−4.26773	−	2.60509i	−0.213386	−	0.130255i
$401$	16.0509			0.801545			0.400772	−	0.916178i	$-0.368742\pi$
$401$	16.0509			0.801545			0.400772	+	0.916178i	$0.368742\pi$
$402$		−	5.82075i		−	0.290313i
$403$		−	0.789816i		−	0.0393435i
$404$		0			0
$405$	0.605092	−	2.15264i	0.0300673	−	0.106966i
$406$	8.84074			0.438758
$407$		−	34.4423i		−	1.70724i
$408$		−	0.115086i		−	0.00569761i
$409$	−9.87167			−0.488123			−0.244061	−	0.969760i	$-0.578480\pi$
$409$	−9.87167			−0.488123			−0.244061	+	0.969760i	$0.578480\pi$
$410$	−4.30528	−	1.21018i	−0.212623	−	0.0597667i
$411$	−21.4513			−1.05811
$412$			6.00000i			0.295599i
$413$		−	8.38039i		−	0.412372i
$414$	4.42037			0.217249
$415$	32.6039	+	9.16474i	1.60046	+	0.449879i
$416$	0.789816			0.0387239
$417$		−	16.4204i		−	0.804109i
$418$			0.599620i			0.0293284i
$419$	0.420368			0.0205363			0.0102682	−	0.999947i	$-0.496731\pi$
$419$	0.420368			0.0205363			0.0102682	+	0.999947i	$0.496731\pi$
$420$	−1.21018	+	4.30528i	−0.0590509	+	0.210076i
$421$	−39.2211			−1.91152			−0.955760	−	0.294146i	$-0.904965\pi$
$421$	−39.2211			−1.91152			−0.955760	+	0.294146i	$0.904965\pi$
$422$		−	24.8407i		−	1.20923i
$423$			0.115086i			0.00559568i
$424$	0.190196			0.00923674
$425$	0.299810	−	0.491156i	0.0145429	−	0.0238246i
$426$	−13.8207			−0.669617
$427$		−	24.9910i		−	1.20940i
$428$		−	3.03093i		−	0.146506i
$429$	−4.11509			−0.198678
$430$	−5.21018	+	18.5355i	−0.251257	+	0.893859i
$431$	−6.88071			−0.331432			−0.165716	−	0.986173i	$-0.552994\pi$
$431$	−6.88071			−0.331432			−0.165716	+	0.986173i	$0.552994\pi$
$432$		−	1.00000i		−	0.0481125i
$433$		−	1.38944i		−	0.0667720i	−0.999443	−	0.0333860i	$-0.989371\pi$
$433$		−	1.38944i		−	0.0667720i	0.999443	−	0.0333860i	$-0.0106291\pi$
$434$	2.00000			0.0960031
$435$	−9.51547	−	2.67473i	−0.456232	−	0.128243i
$436$	−8.19020			−0.392239
$437$			0.508723i			0.0243355i
$438$		−	10.8407i		−	0.517990i
$439$	−19.8717			−0.948423			−0.474212	−	0.880411i	$-0.657267\pi$
$439$	−19.8717			−0.948423			−0.474212	+	0.880411i	$0.657267\pi$
$440$	11.2157	+	3.15264i	0.534685	+	0.150296i
$441$	−3.00000			−0.142857
$442$			0.0908968i			0.00432352i
$443$			8.61056i			0.409100i	0.978856	+	0.204550i	$0.0655731\pi$
$443$			8.61056i			0.409100i	−0.978856	+	0.204550i	$0.934427\pi$
$444$	−6.61056			−0.313723
$445$	1.34946	−	4.80076i	0.0639705	−	0.227578i
$446$	−19.6306			−0.929534
$447$			3.63055i			0.171719i
$448$			2.00000i			0.0944911i
$449$	10.3694			0.489364			0.244682	−	0.969603i	$-0.421316\pi$
$449$	10.3694			0.489364			0.244682	+	0.969603i	$0.421316\pi$
$450$	2.60509	−	4.26773i	0.122805	−	0.201183i
$451$	10.4204			0.490676
$452$		−	10.3804i		−	0.488253i
$453$		−	15.7566i		−	0.740309i
$454$	6.42037			0.301323
$455$	0.955823	−	3.40038i	0.0448097	−	0.159412i
$456$	0.115086			0.00538940
$457$			21.4913i			1.00532i	0.864484	+	0.502660i	$0.167645\pi$
$457$			21.4913i			1.00532i	−0.864484	+	0.502660i	$0.832355\pi$
$458$		−	29.7166i		−	1.38857i
$459$	0.115086			0.00537176
$460$	9.51547	+	2.67473i	0.443661	+	0.124710i
$461$	3.34946			0.156000			0.0779999	−	0.996953i	$-0.475147\pi$
$461$	3.34946			0.156000			0.0779999	+	0.996953i	$0.475147\pi$
$462$		−	10.4204i		−	0.484799i
$463$		−	28.0109i		−	1.30178i	−0.759172	−	0.650889i	$-0.774396\pi$
$463$		−	28.0109i		−	1.30178i	0.759172	−	0.650889i	$-0.225604\pi$
$464$	−4.42037			−0.205210
$465$	−2.15264	−	0.605092i	−0.0998263	−	0.0280605i
$466$	5.38944			0.249661
$467$			37.6815i			1.74369i	0.489781	+	0.871845i	$0.337077\pi$
$467$			37.6815i			1.74369i	−0.489781	+	0.871845i	$0.662923\pi$
$468$			0.789816i			0.0365093i
$469$	−11.6415			−0.537554
$470$	−0.0696377	+	0.247739i	−0.00321215	+	0.0114274i
$471$	−3.96002			−0.182468
$472$			4.19020i			0.192869i
$473$		−	44.8626i		−	2.06279i
$474$	−2.30528			−0.105885
$475$	0.491156	+	0.299810i	0.0225358	+	0.0137562i
$476$	−0.230172			−0.0105499
$477$			0.190196i			0.00870848i
$478$			7.03093i			0.321587i
$479$	10.2811			0.469755			0.234878	−	0.972025i	$-0.424531\pi$
$479$	10.2811			0.469755			0.234878	+	0.972025i	$0.424531\pi$
$480$	0.605092	−	2.15264i	0.0276185	−	0.0982542i
$481$	5.22113			0.238063
$482$			0.190196i			0.00866319i
$483$		−	8.84074i		−	0.402267i
$484$	−16.1460			−0.733910
$485$	19.8262	+	5.57301i	0.900262	+	0.253057i
$486$	1.00000			0.0453609
$487$			32.9316i			1.49227i	0.665792	+	0.746137i	$0.268094\pi$
$487$			32.9316i			1.49227i	−0.665792	+	0.746137i	$0.731906\pi$
$488$			12.4955i			0.565644i
$489$	−9.82075			−0.444110
$490$	−6.45792	−	1.81528i	−0.291739	−	0.0820058i
$491$	−41.7034			−1.88205			−0.941023	−	0.338342i	$-0.890134\pi$
$491$	−41.7034			−1.88205			−0.941023	+	0.338342i	$0.890134\pi$
$492$		−	2.00000i		−	0.0901670i
$493$		−	0.508723i		−	0.0229117i
$494$	−0.0908968			−0.00408964
$495$	−3.15264	+	11.2157i	−0.141701	+	0.504106i
$496$	−1.00000			−0.0449013
$497$			27.6415i			1.23989i
$498$			15.1460i			0.678709i
$499$	−22.2302			−0.995159			−0.497580	−	0.867418i	$-0.665778\pi$
$499$	−22.2302			−0.995159			−0.497580	+	0.867418i	$0.665778\pi$
$500$	8.19020	−	7.61056i	0.366277	−	0.340355i
$501$	2.00000			0.0893534
$502$		−	14.4204i		−	0.643613i
$503$		−	12.3453i		−	0.550448i	−0.961380	−	0.275224i	$-0.911248\pi$
$503$		−	12.3453i		−	0.550448i	0.961380	−	0.275224i	$-0.0887520\pi$
$504$	−2.00000			−0.0890871
$505$		0			0
$506$	−23.0309			−1.02385
$507$			12.3762i			0.549646i
$508$		−	10.4204i		−	0.462329i
$509$	−30.9910			−1.37365			−0.686825	−	0.726823i	$-0.740996\pi$
$509$	−30.9910			−1.37365			−0.686825	+	0.726823i	$0.740996\pi$
$510$	0.247739	+	0.0696377i	0.0109701	+	0.00308361i
$511$	−21.6815			−0.959132
$512$		−	1.00000i		−	0.0441942i
$513$			0.115086i			0.00508118i
$514$	1.76983			0.0780638
$515$	−12.9158	−	3.63055i	−0.569140	−	0.159981i
$516$	−8.61056			−0.379059
$517$		−	0.599620i		−	0.0263713i
$518$			13.2211i			0.580903i
$519$	0.535454			0.0235038
$520$	−0.477911	+	1.70019i	−0.0209578	+	0.0745583i
$521$	28.8008			1.26178			0.630892	−	0.775871i	$-0.282689\pi$
$521$	28.8008			1.26178			0.630892	+	0.775871i	$0.282689\pi$
$522$		−	4.42037i		−	0.193474i
$523$		−	27.7698i		−	1.21429i	−0.794591	−	0.607145i	$-0.792315\pi$
$523$		−	27.7698i		−	1.21429i	0.794591	−	0.607145i	$-0.207685\pi$
$524$	0.190196			0.00830875
$525$	−8.53545	−	5.21018i	−0.372518	−	0.227391i
$526$	14.2302			0.620465
$527$		−	0.115086i		−	0.00501323i
$528$			5.21018i			0.226744i
$529$	3.46034			0.150450
$530$	−0.115086	+	0.409424i	−0.00499902	+	0.0177842i
$531$	−4.19020			−0.181839
$532$		−	0.230172i		−	0.00997923i
$533$			1.57963i			0.0684214i
$534$	2.23017			0.0965090
$535$	6.52451	+	1.83399i	0.282079	+	0.0792904i
$536$	5.82075			0.251418
$537$		−	10.4313i		−	0.450144i
$538$		−	1.15926i		−	0.0499794i
$539$	15.6306			0.673256
$540$	2.15264	+	0.605092i	0.0926349	+	0.0260390i
$541$	34.9426			1.50230			0.751149	−	0.660132i	$-0.229500\pi$
$541$	34.9426			1.50230			0.751149	+	0.660132i	$0.229500\pi$
$542$		−	22.6857i		−	0.974433i
$543$			17.4513i			0.748907i
$544$	0.115086			0.00493428
$545$	4.95582	−	17.6306i	0.212284	−	0.755210i
$546$	1.57963			0.0676020
$547$			23.6415i			1.01084i	0.862874	+	0.505419i	$0.168662\pi$
$547$			23.6415i			1.01084i	−0.862874	+	0.505419i	$0.831338\pi$
$548$		−	21.4513i		−	0.916354i
$549$	−12.4955			−0.533294
$550$	−13.5730	+	22.2356i	−0.578755	+	0.948131i
$551$	0.508723			0.0216723
$552$			4.42037i			0.188143i
$553$			4.61056i			0.196061i
$554$	9.63055			0.409163
$555$	4.00000	−	14.2302i	0.169791	−	0.604037i
$556$	16.4204			0.696379
$557$		−	22.9910i		−	0.974158i	−0.873358	−	0.487079i	$-0.838062\pi$
$557$		−	22.9910i		−	0.974158i	0.873358	−	0.487079i	$-0.161938\pi$
$558$		−	1.00000i		−	0.0423334i
$559$	6.80076			0.287642
$560$	−4.30528	−	1.21018i	−0.181931	−	0.0511396i
$561$	−0.599620			−0.0253160
$562$		−	10.8407i		−	0.457289i
$563$			7.87167i			0.331751i	0.986147	+	0.165876i	$0.0530450\pi$
$563$			7.87167i			0.331751i	−0.986147	+	0.165876i	$0.946955\pi$
$564$	−0.115086			−0.00484600
$565$	22.3453	+	6.28109i	0.940072	+	0.264248i
$566$	−19.7808			−0.831448
$567$		−	2.00000i		−	0.0839921i
$568$		−	13.8207i		−	0.579906i
$569$	40.6724			1.70508			0.852538	−	0.522664i	$-0.175062\pi$
$569$	40.6724			1.70508			0.852538	+	0.522664i	$0.175062\pi$
$570$	−0.0696377	+	0.247739i	−0.00291680	+	0.0103766i
$571$	14.2302			0.595514			0.297757	−	0.954642i	$-0.403761\pi$
$571$	14.2302			0.595514			0.297757	+	0.954642i	$0.403761\pi$
$572$		−	4.11509i		−	0.172060i
$573$			19.2611i			0.804644i
$574$	−4.00000			−0.166957
$575$	−11.5155	+	18.8649i	−0.480228	+	0.786722i
$576$	1.00000			0.0416667
$577$		−	7.63055i		−	0.317664i	−0.987306	−	0.158832i	$-0.949227\pi$
$577$		−	7.63055i		−	0.317664i	0.987306	−	0.158832i	$-0.0507728\pi$
$578$		−	16.9868i		−	0.706556i
$579$	−14.4313			−0.599745
$580$	2.67473	−	9.51547i	0.111062	−	0.395108i
$581$	30.2920			1.25673
$582$			9.21018i			0.381774i
$583$		−	0.990956i		−	0.0410412i
$584$	10.8407			0.448593
$585$	−1.70019	−	0.477911i	−0.0702942	−	0.0197592i
$586$	31.2211			1.28973
$587$			17.8449i			0.736539i	0.929719	+	0.368270i	$0.120050\pi$
$587$			17.8449i			0.736539i	−0.929719	+	0.368270i	$0.879950\pi$
$588$		−	3.00000i		−	0.123718i
$589$	0.115086			0.00474204
$590$	−9.01999	−	2.53545i	−0.371347	−	0.104383i
$591$	10.3804			0.426992
$592$		−	6.61056i		−	0.271693i
$593$		−	47.4732i		−	1.94949i	−0.223320	−	0.974745i	$-0.571690\pi$
$593$		−	47.4732i		−	1.94949i	0.223320	−	0.974745i	$-0.428310\pi$
$594$	−5.21018			−0.213776
$595$	0.139275	−	0.495478i	0.00570973	−	0.0203126i
$596$	−3.63055			−0.148713
$597$		−	14.5355i		−	0.594897i
$598$		−	3.49128i		−	0.142769i
$599$	−41.4622			−1.69410			−0.847051	−	0.531512i	$-0.821624\pi$
$599$	−41.4622			−1.69410			−0.847051	+	0.531512i	$0.821624\pi$
$600$	4.26773	+	2.60509i	0.174229	+	0.106352i
$601$	−8.77887			−0.358098			−0.179049	−	0.983840i	$-0.557302\pi$
$601$	−8.77887			−0.358098			−0.179049	+	0.983840i	$0.557302\pi$
$602$			17.2211i			0.701881i
$603$			5.82075i			0.237039i
$604$	15.7566			0.641126
$605$	9.76983	−	34.7566i	0.397200	−	1.41306i
$606$		0			0
$607$		−	32.0619i		−	1.30135i	−0.759356	−	0.650675i	$-0.774486\pi$
$607$		−	32.0619i		−	1.30135i	0.759356	−	0.650675i	$-0.225514\pi$
$608$			0.115086i			0.00466736i
$609$	−8.84074			−0.358245
$610$	−26.8983	−	7.56091i	−1.08908	−	0.306132i
$611$	0.0908968			0.00367729
$612$			0.115086i			0.00465208i
$613$		−	23.2321i		−	0.938335i	−0.883109	−	0.469167i	$-0.844554\pi$
$613$		−	23.2321i		−	0.938335i	0.883109	−	0.469167i	$-0.155446\pi$
$614$	13.5796			0.548029
$615$	4.30528	+	1.21018i	0.173606	+	0.0487993i
$616$	10.4204			0.419849
$617$			13.6196i			0.548305i	0.961686	+	0.274152i	$0.0883973\pi$
$617$			13.6196i			0.548305i	−0.961686	+	0.274152i	$0.911603\pi$
$618$		−	6.00000i		−	0.241355i
$619$	−25.3894			−1.02049			−0.510244	−	0.860030i	$-0.670445\pi$
$619$	−25.3894			−1.02049			−0.510244	+	0.860030i	$0.670445\pi$
$620$	0.605092	−	2.15264i	0.0243011	−	0.0864521i
$621$	−4.42037			−0.177383
$622$		−	18.2811i		−	0.733005i
$623$		−	4.46034i		−	0.178700i
$624$	−0.789816			−0.0316179
$625$	11.4270	+	22.2356i	0.457080	+	0.889426i
$626$	−14.4603			−0.577952
$627$		−	0.599620i		−	0.0239465i
$628$		−	3.96002i		−	0.158022i
$629$	0.760784			0.0303345
$630$	1.21018	−	4.30528i	0.0482149	−	0.171527i
$631$	−28.3804			−1.12981			−0.564903	−	0.825157i	$-0.691086\pi$
$631$	−28.3804			−1.12981			−0.564903	+	0.825157i	$0.691086\pi$
$632$		−	2.30528i		−	0.0916992i
$633$			24.8407i			0.987331i
$634$	3.46455			0.137595
$635$	22.4313	+	6.30528i	0.890159	+	0.250217i
$636$	−0.190196			−0.00754176
$637$			2.36945i			0.0938809i
$638$			23.0309i			0.911803i
$639$	13.8207			0.546740
$640$	2.15264	+	0.605092i	0.0850906	+	0.0239184i
$641$	3.59058			0.141819			0.0709096	−	0.997483i	$-0.477410\pi$
$641$	3.59058			0.141819			0.0709096	+	0.997483i	$0.477410\pi$
$642$			3.03093i			0.119621i
$643$		−	38.6505i		−	1.52423i	−0.647443	−	0.762114i	$-0.724161\pi$
$643$		−	38.6505i		−	1.52423i	0.647443	−	0.762114i	$-0.275839\pi$
$644$	8.84074			0.348374
$645$	5.21018	−	18.5355i	0.205151	−	0.729833i
$646$	−0.0132448			−0.000521110
$647$		−	32.9026i		−	1.29353i	−0.762687	−	0.646767i	$-0.776120\pi$
$647$		−	32.9026i		−	1.29353i	0.762687	−	0.646767i	$-0.223880\pi$
$648$			1.00000i			0.0392837i
$649$	21.8317			0.856969
$650$	−3.37072	−	2.05754i	−0.132210	−	0.0807035i
$651$	−2.00000			−0.0783862
$652$		−	9.82075i		−	0.384610i
$653$			25.5264i			0.998926i	0.866335	+	0.499463i	$0.166469\pi$
$653$			25.5264i			0.998926i	−0.866335	+	0.499463i	$0.833531\pi$
$654$	8.19020			0.320262
$655$	−0.115086	+	0.409424i	−0.00449679	+	0.0159975i
$656$	2.00000			0.0780869
$657$			10.8407i			0.422937i
$658$			0.230172i			0.00897305i
$659$	−6.84074			−0.266477			−0.133239	−	0.991084i	$-0.542538\pi$
$659$	−6.84074			−0.266477			−0.133239	+	0.991084i	$0.542538\pi$
$660$	−11.2157	−	3.15264i	−0.436569	−	0.122716i
$661$	−21.2611			−0.826961			−0.413481	−	0.910513i	$-0.635687\pi$
$661$	−21.2611			−0.826961			−0.413481	+	0.910513i	$0.635687\pi$
$662$			20.8008i			0.808445i
$663$		−	0.0908968i		−	0.00353014i
$664$	−15.1460			−0.587780
$665$	0.495478	+	0.139275i	0.0192138	+	0.00540087i
$666$	6.61056			0.256154
$667$			19.5397i			0.756578i
$668$			2.00000i			0.0773823i
$669$	19.6306			0.758961
$670$	−3.52209	+	12.5300i	−0.136070	+	0.484075i
$671$	65.1037			2.51330
$672$		−	2.00000i		−	0.0771517i
$673$			39.7034i			1.53045i	0.643762	+	0.765226i	$0.277373\pi$
$673$			39.7034i			1.53045i	−0.643762	+	0.765226i	$0.722627\pi$
$674$	23.0709			0.888658
$675$	−2.60509	+	4.26773i	−0.100270	+	0.164265i
$676$	−12.3762			−0.476007
$677$			3.45130i			0.132644i	0.997798	+	0.0663221i	$0.0211265\pi$
$677$			3.45130i			0.132644i	−0.997798	+	0.0663221i	$0.978874\pi$
$678$			10.3804i			0.398657i
$679$	18.4204			0.706909
$680$	−0.0696377	+	0.247739i	−0.00267048	+	0.00950036i
$681$	−6.42037			−0.246029
$682$			5.21018i			0.199508i
$683$		−	20.4603i		−	0.782893i	−0.920201	−	0.391447i	$-0.871975\pi$
$683$		−	20.4603i		−	0.782893i	0.920201	−	0.391447i	$-0.128025\pi$
$684$	−0.115086			−0.00440043
$685$	46.1770	+	12.9800i	1.76433	+	0.495941i
$686$	−20.0000			−0.763604
$687$			29.7166i			1.13376i
$688$		−	8.61056i		−	0.328275i
$689$	0.150220			0.00572292
$690$	−9.51547	−	2.67473i	−0.362248	−	0.101825i
$691$	−9.94678			−0.378393			−0.189197	−	0.981939i	$-0.560588\pi$
$691$	−9.94678			−0.378393			−0.189197	+	0.981939i	$0.560588\pi$
$692$			0.535454i			0.0203549i
$693$			10.4204i			0.395837i
$694$	31.9868			1.21420
$695$	−9.93583	+	35.3472i	−0.376888	+	1.34079i
$696$	4.42037			0.167554
$697$			0.230172i			0.00871839i
$698$			23.4513i			0.887645i
$699$	−5.38944			−0.203847
$700$	5.21018	−	8.53545i	0.196926	−	0.322610i
$701$	−14.7898			−0.558604			−0.279302	−	0.960203i	$-0.590103\pi$
$701$	−14.7898			−0.558604			−0.279302	+	0.960203i	$0.590103\pi$
$702$		−	0.789816i		−	0.0298097i
$703$			0.760784i			0.0286935i
$704$	−5.21018			−0.196366
$705$	0.0696377	−	0.247739i	0.00262271	−	0.00933039i
$706$	3.88491			0.146211
$707$		0			0
$708$		−	4.19020i		−	0.157477i
$709$	10.4336			0.391843			0.195921	−	0.980620i	$-0.437230\pi$
$709$	10.4336			0.391843			0.195921	+	0.980620i	$0.437230\pi$
$710$	29.7511	+	8.36283i	1.11654	+	0.313851i
$711$	2.30528			0.0864548
$712$			2.23017i			0.0835792i
$713$			4.42037i			0.165544i
$714$	0.230172			0.00861398
$715$	8.85830	+	2.49001i	0.331282	+	0.0931209i
$716$	10.4313			0.389836
$717$		−	7.03093i		−	0.262575i
$718$		−	21.3604i		−	0.797163i
$719$	20.3320			0.758256			0.379128	−	0.925344i	$-0.376224\pi$
$719$	20.3320			0.758256			0.379128	+	0.925344i	$0.376224\pi$
$720$	−0.605092	+	2.15264i	−0.0225504	+	0.0802242i
$721$	−12.0000			−0.446903
$722$			18.9868i			0.706614i
$723$		−	0.190196i		−	0.00707347i
$724$	−17.4513			−0.648572
$725$	18.8649	+	11.5155i	0.700626	+	0.427674i
$726$	16.1460			0.599235
$727$			37.5132i			1.39129i	0.718387	+	0.695643i	$0.244881\pi$
$727$			37.5132i			1.39129i	−0.718387	+	0.695643i	$0.755119\pi$
$728$			1.57963i			0.0585450i
$729$	−1.00000			−0.0370370
$730$	−6.55964	+	23.3362i	−0.242783	+	0.863712i
$731$	0.990956			0.0366518
$732$		−	12.4955i		−	0.461846i
$733$			28.6505i			1.05823i	0.848550	+	0.529116i	$0.177476\pi$
$733$			28.6505i			1.05823i	−0.848550	+	0.529116i	$0.822524\pi$
$734$	20.4713			0.755609
$735$	6.45792	+	1.81528i	0.238204	+	0.0669575i
$736$	−4.42037			−0.162937
$737$		−	30.3272i		−	1.11712i
$738$			2.00000i			0.0736210i
$739$	−21.6415			−0.796095			−0.398048	−	0.917365i	$-0.630312\pi$
$739$	−21.6415			−0.796095			−0.398048	+	0.917365i	$0.630312\pi$
$740$	14.2302	+	4.00000i	0.523112	+	0.147043i
$741$	0.0908968			0.00333918
$742$			0.380392i			0.0139646i
$743$		−	5.76983i		−	0.211674i	−0.994383	−	0.105837i	$-0.966248\pi$
$743$		−	5.76983i		−	0.211674i	0.994383	−	0.105837i	$-0.0337522\pi$
$744$	1.00000			0.0366618
$745$	2.19682	−	7.81528i	0.0804852	−	0.286330i
$746$	−9.76983			−0.357699
$747$		−	15.1460i		−	0.554164i
$748$		−	0.599620i		−	0.0219243i
$749$	6.06187			0.221496
$750$	−8.19020	+	7.61056i	−0.299064	+	0.277899i
$751$	−31.7698			−1.15930			−0.579649	−	0.814866i	$-0.696810\pi$
$751$	−31.7698			−1.15930			−0.579649	+	0.814866i	$0.696810\pi$
$752$		−	0.115086i		−	0.00419676i
$753$			14.4204i			0.525507i
$754$	−3.49128			−0.127145
$755$	−9.53418	+	33.9183i	−0.346984	+	1.23441i
$756$	2.00000			0.0727393
$757$		−	14.6905i		−	0.533936i	−0.963705	−	0.266968i	$-0.913978\pi$
$757$		−	14.6905i		−	0.533936i	0.963705	−	0.266968i	$-0.0860218\pi$
$758$			3.65474i			0.132746i
$759$	23.0309			0.835970
$760$	−0.247739	−	0.0696377i	−0.00898644	−	0.00252602i
$761$	44.1528			1.60054			0.800268	−	0.599642i	$-0.204690\pi$
$761$	44.1528			1.60054			0.800268	+	0.599642i	$0.204690\pi$
$762$			10.4204i			0.377490i
$763$		−	16.3804i		−	0.593010i
$764$	−19.2611			−0.696842
$765$	−0.247739	−	0.0696377i	−0.00895703	−	0.00251776i
$766$	−25.4913			−0.921037
$767$			3.30948i			0.119499i
$768$			1.00000i			0.0360844i
$769$	−2.07995			−0.0750050			−0.0375025	−	0.999297i	$-0.511940\pi$
$769$	−2.07995			−0.0750050			−0.0375025	+	0.999297i	$0.511940\pi$
$770$	−6.30528	+	22.4313i	−0.227227	+	0.808368i
$771$	−1.76983			−0.0637388
$772$		−	14.4313i		−	0.519394i
$773$			30.1502i			1.08443i	0.840240	+	0.542214i	$0.182414\pi$
$773$			30.1502i			1.08443i	−0.840240	+	0.542214i	$0.817586\pi$
$774$	8.61056			0.309500
$775$	4.26773	+	2.60509i	0.153301	+	0.0935777i
$776$	−9.21018			−0.330626
$777$		−	13.2211i		−	0.474305i
$778$			4.65054i			0.166730i
$779$	−0.230172			−0.00824678
$780$	0.477911	−	1.70019i	0.0171120	−	0.0608766i
$781$	−72.0086			−2.57667

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.605092 - 2.15264i) q^{5} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.605092 - 2.15264i) q^{5} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-2.15264 - 0.605092i) q^{10} +5.21018 q^{11} -1.00000i q^{12} +0.789816i q^{13} -2.00000 q^{14} +(2.15264 + 0.605092i) q^{15} +1.00000 q^{16} +0.115086i q^{17} +1.00000i q^{18} -0.115086 q^{19} +(-0.605092 + 2.15264i) q^{20} +2.00000 q^{21} -5.21018i q^{22} -4.42037i q^{23} -1.00000 q^{24} +(-4.26773 - 2.60509i) q^{25} +0.789816 q^{26} -1.00000i q^{27} +2.00000i q^{28} -4.42037 q^{29} +(0.605092 - 2.15264i) q^{30} -1.00000 q^{31} -1.00000i q^{32} +5.21018i q^{33} +0.115086 q^{34} +(-4.30528 - 1.21018i) q^{35} +1.00000 q^{36} -6.61056i q^{37} +0.115086i q^{38} -0.789816 q^{39} +(2.15264 + 0.605092i) q^{40} +2.00000 q^{41} -2.00000i q^{42} -8.61056i q^{43} -5.21018 q^{44} +(-0.605092 + 2.15264i) q^{45} -4.42037 q^{46} -0.115086i q^{47} +1.00000i q^{48} +3.00000 q^{49} +(-2.60509 + 4.26773i) q^{50} -0.115086 q^{51} -0.789816i q^{52} -0.190196i q^{53} -1.00000 q^{54} +(3.15264 - 11.2157i) q^{55} +2.00000 q^{56} -0.115086i q^{57} +4.42037i q^{58} +4.19020 q^{59} +(-2.15264 - 0.605092i) q^{60} +12.4955 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +(1.70019 + 0.477911i) q^{65} +5.21018 q^{66} -5.82075i q^{67} -0.115086i q^{68} +4.42037 q^{69} +(-1.21018 + 4.30528i) q^{70} -13.8207 q^{71} -1.00000i q^{72} -10.8407i q^{73} -6.61056 q^{74} +(2.60509 - 4.26773i) q^{75} +0.115086 q^{76} -10.4204i q^{77} +0.789816i q^{78} -2.30528 q^{79} +(0.605092 - 2.15264i) q^{80} +1.00000 q^{81} -2.00000i q^{82} +15.1460i q^{83} -2.00000 q^{84} +(0.247739 + 0.0696377i) q^{85} -8.61056 q^{86} -4.42037i q^{87} +5.21018i q^{88} +2.23017 q^{89} +(2.15264 + 0.605092i) q^{90} +1.57963 q^{91} +4.42037i q^{92} -1.00000i q^{93} -0.115086 q^{94} +(-0.0696377 + 0.247739i) q^{95} +1.00000 q^{96} +9.21018i q^{97} -3.00000i q^{98} -5.21018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9	\( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 + 16 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 + 4 * q^20 + 12 * q^21 - 6 * q^24 - 8 * q^25 + 20 * q^26 + 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^34 + 6 * q^36 - 20 * q^39 + 12 * q^41 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 4 * q^51 - 6 * q^54 + 6 * q^55 + 12 * q^56 + 4 * q^59 + 28 * q^61 - 6 * q^64 - 8 * q^65 + 16 * q^66 - 4 * q^69 + 8 * q^70 - 16 * q^71 + 12 * q^74 + 8 * q^75 - 4 * q^76 + 12 * q^79 - 4 * q^80 + 6 * q^81 - 12 * q^84 - 22 * q^85 + 4 * q^89 + 40 * q^91 + 4 * q^94 - 28 * q^95 + 6 * q^96 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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