LMFDB - Embedded newform 3234.2.e.c.2155.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(2155,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3234.2155");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.e (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:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2155.7
Character	$\chi$	$=$	3234.2155
Dual form			3234.2.e.c.2155.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.313884i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.313884i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +0.313884 q^{10} +(-0.720050 + 3.23752i) q^{11} +1.00000i q^{12} +4.42216 q^{13} +0.313884 q^{15} +1.00000 q^{16} -3.88254 q^{17} +1.00000i q^{18} +1.19962 q^{19} -0.313884i q^{20} +(3.23752 + 0.720050i) q^{22} -7.80896 q^{23} +1.00000 q^{24} +4.90148 q^{25} -4.42216i q^{26} +1.00000i q^{27} -6.32404i q^{29} -0.313884i q^{30} -1.97170i q^{31} -1.00000i q^{32} +(3.23752 + 0.720050i) q^{33} +3.88254i q^{34} +1.00000 q^{36} -3.89552 q^{37} -1.19962i q^{38} -4.42216i q^{39} -0.313884 q^{40} -11.2575 q^{41} -4.06657i q^{43} +(0.720050 - 3.23752i) q^{44} -0.313884i q^{45} +7.80896i q^{46} +9.92277i q^{47} -1.00000i q^{48} -4.90148i q^{50} +3.88254i q^{51} -4.42216 q^{52} -11.7453 q^{53} +1.00000 q^{54} +(-1.01621 - 0.226012i) q^{55} -1.19962i q^{57} -6.32404 q^{58} +8.87397i q^{59} -0.313884 q^{60} -11.6955 q^{61} -1.97170 q^{62} -1.00000 q^{64} +1.38805i q^{65} +(0.720050 - 3.23752i) q^{66} -12.2415 q^{67} +3.88254 q^{68} +7.80896i q^{69} +10.0331 q^{71} -1.00000i q^{72} -4.85587 q^{73} +3.89552i q^{74} -4.90148i q^{75} -1.19962 q^{76} -4.42216 q^{78} -6.61990i q^{79} +0.313884i q^{80} +1.00000 q^{81} +11.2575i q^{82} +13.4283 q^{83} -1.21867i q^{85} -4.06657 q^{86} -6.32404 q^{87} +(-3.23752 - 0.720050i) q^{88} -11.7289i q^{89} -0.313884 q^{90} +7.80896 q^{92} -1.97170 q^{93} +9.92277 q^{94} +0.376541i q^{95} -1.00000 q^{96} +11.7519i q^{97} +(0.720050 - 3.23752i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^4 - 24 * q^6 - 24 * q^9	$ 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} + 24 q^{16} + 16 q^{17} - 32 q^{19} - 8 q^{22} + 24 q^{24} - 8 q^{25} - 8 q^{33} + 24 q^{36} + 16 q^{37} - 16 q^{41} + 24 q^{54} + 16 q^{55} - 16 q^{62} - 24 q^{64} - 64 q^{67} - 16 q^{68} + 64 q^{71} + 32 q^{76} + 24 q^{81} - 16 q^{83} + 8 q^{88} - 16 q^{93} + 64 q^{94} - 24 q^{96}+O(q^{100}) $ 24 * q - 24 * q^4 - 24 * q^6 - 24 * q^9 + 24 * q^16 + 16 * q^17 - 32 * q^19 - 8 * q^22 + 24 * q^24 - 8 * q^25 - 8 * q^33 + 24 * q^36 + 16 * q^37 - 16 * q^41 + 24 * q^54 + 16 * q^55 - 16 * q^62 - 24 * q^64 - 64 * q^67 - 16 * q^68 + 64 * q^71 + 32 * q^76 + 24 * q^81 - 16 * q^83 + 8 * q^88 - 16 * q^93 + 64 * q^94 - 24 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times$.

$n$	$199$	$1079$	$2059$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$			0.313884i			0.140373i	0.997534	+	0.0701866i	$0.0223595\pi$
$5$			0.313884i			0.140373i	−0.997534	+	0.0701866i	$0.977641\pi$
$6$	−1.00000			−0.408248
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$	−1.00000			−0.333333
$10$	0.313884			0.0992588
$11$	−0.720050	+	3.23752i	−0.217103	+	0.976149i
$11$	−0.720050	+	3.23752i	−0.217103	+	0.976149i
$12$			1.00000i			0.288675i
$13$	4.42216			1.22649			0.613244	−	0.789894i	$-0.289864\pi$
$13$	4.42216			1.22649			0.613244	+	0.789894i	$0.289864\pi$
$14$		0			0
$15$	0.313884			0.0810445
$16$	1.00000			0.250000
$17$	−3.88254			−0.941653			−0.470827	−	0.882226i	$-0.656044\pi$
$17$	−3.88254			−0.941653			−0.470827	+	0.882226i	$0.656044\pi$
$18$			1.00000i			0.235702i
$19$	1.19962			0.275211			0.137606	−	0.990487i	$-0.456059\pi$
$19$	1.19962			0.275211			0.137606	+	0.990487i	$0.456059\pi$
$20$		−	0.313884i		−	0.0701866i
$21$		0			0
$22$	3.23752	+	0.720050i	0.690241	+	0.153515i
$23$	−7.80896			−1.62828			−0.814140	−	0.580668i	$-0.802791\pi$
$23$	−7.80896			−1.62828			−0.814140	+	0.580668i	$0.802791\pi$
$24$	1.00000			0.204124
$25$	4.90148			0.980295
$26$		−	4.42216i		−	0.867258i
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	6.32404i		−	1.17435i	−0.809462	−	0.587173i	$-0.800241\pi$
$29$		−	6.32404i		−	1.17435i	0.809462	−	0.587173i	$-0.199759\pi$
$30$		−	0.313884i		−	0.0573071i
$31$		−	1.97170i		−	0.354127i	−0.984199	−	0.177064i	$-0.943340\pi$
$31$		−	1.97170i		−	0.354127i	0.984199	−	0.177064i	$-0.0566599\pi$
$32$		−	1.00000i		−	0.176777i
$33$	3.23752	+	0.720050i	0.563580	+	0.125345i
$34$			3.88254i			0.665849i
$35$		0			0
$36$	1.00000			0.166667
$37$	−3.89552			−0.640420			−0.320210	−	0.947347i	$-0.603754\pi$
$37$	−3.89552			−0.640420			−0.320210	+	0.947347i	$0.603754\pi$
$38$		−	1.19962i		−	0.194604i
$39$		−	4.42216i		−	0.708113i
$40$	−0.313884			−0.0496294
$41$	−11.2575			−1.75813			−0.879064	−	0.476705i	$-0.841831\pi$
$41$	−11.2575			−1.75813			−0.879064	+	0.476705i	$0.841831\pi$
$42$		0			0
$43$		−	4.06657i		−	0.620146i	−0.950713	−	0.310073i	$-0.899647\pi$
$43$		−	4.06657i		−	0.620146i	0.950713	−	0.310073i	$-0.100353\pi$
$44$	0.720050	−	3.23752i	0.108552	−	0.488074i
$45$		−	0.313884i		−	0.0467911i
$46$			7.80896i			1.15137i
$47$			9.92277i			1.44738i	0.690123	+	0.723692i	$0.257557\pi$
$47$			9.92277i			1.44738i	−0.690123	+	0.723692i	$0.742443\pi$
$48$		−	1.00000i		−	0.144338i
$49$		0			0
$50$		−	4.90148i		−	0.693174i
$51$			3.88254i			0.543664i
$52$	−4.42216			−0.613244
$53$	−11.7453			−1.61334			−0.806669	−	0.591004i	$-0.798732\pi$
$53$	−11.7453			−1.61334			−0.806669	+	0.591004i	$0.798732\pi$
$54$	1.00000			0.136083
$55$	−1.01621	−	0.226012i	−0.137025	−	0.0304755i
$56$		0			0
$57$		−	1.19962i		−	0.158893i
$58$	−6.32404			−0.830387
$59$			8.87397i			1.15529i	0.816287	+	0.577646i	$0.196029\pi$
$59$			8.87397i			1.15529i	−0.816287	+	0.577646i	$0.803971\pi$
$60$	−0.313884			−0.0405222
$61$	−11.6955			−1.49746			−0.748730	−	0.662875i	$-0.769336\pi$
$61$	−11.6955			−1.49746			−0.748730	+	0.662875i	$0.769336\pi$
$62$	−1.97170			−0.250406
$63$		0			0
$64$	−1.00000			−0.125000
$65$			1.38805i			0.172166i
$66$	0.720050	−	3.23752i	0.0886320	−	0.398511i
$67$	−12.2415			−1.49554			−0.747769	−	0.663959i	$-0.768875\pi$
$67$	−12.2415			−1.49554			−0.747769	+	0.663959i	$0.768875\pi$
$68$	3.88254			0.470827
$69$			7.80896i			0.940088i
$70$		0			0
$71$	10.0331			1.19071			0.595354	−	0.803464i	$-0.297012\pi$
$71$	10.0331			1.19071			0.595354	+	0.803464i	$0.297012\pi$
$72$		−	1.00000i		−	0.117851i
$73$	−4.85587			−0.568337			−0.284168	−	0.958774i	$-0.591717\pi$
$73$	−4.85587			−0.568337			−0.284168	+	0.958774i	$0.591717\pi$
$74$			3.89552i			0.452846i
$75$		−	4.90148i		−	0.565974i
$76$	−1.19962			−0.137606
$77$		0			0
$78$	−4.42216			−0.500712
$79$		−	6.61990i		−	0.744797i	−0.928073	−	0.372399i	$-0.878535\pi$
$79$		−	6.61990i		−	0.744797i	0.928073	−	0.372399i	$-0.121465\pi$
$80$			0.313884i			0.0350933i
$81$	1.00000			0.111111
$82$			11.2575i			1.24318i
$83$	13.4283			1.47395			0.736974	−	0.675921i	$-0.236254\pi$
$83$	13.4283			1.47395			0.736974	+	0.675921i	$0.236254\pi$
$84$		0			0
$85$		−	1.21867i		−	0.132183i
$86$	−4.06657			−0.438509
$87$	−6.32404			−0.678008
$88$	−3.23752	−	0.720050i	−0.345121	−	0.0767576i
$89$		−	11.7289i		−	1.24326i	−0.783309	−	0.621632i	$-0.786470\pi$
$89$		−	11.7289i		−	1.24326i	0.783309	−	0.621632i	$-0.213530\pi$
$90$	−0.313884			−0.0330863
$91$		0			0
$92$	7.80896			0.814140
$93$	−1.97170			−0.204456
$94$	9.92277			1.02346
$95$			0.376541i			0.0386323i
$96$	−1.00000			−0.102062
$97$			11.7519i			1.19322i	0.802530	+	0.596612i	$0.203487\pi$
$97$			11.7519i			1.19322i	−0.802530	+	0.596612i	$0.796513\pi$
$98$		0			0
$99$	0.720050	−	3.23752i	0.0723678	−	0.325383i
$100$	−4.90148			−0.490148
$101$	−8.27979			−0.823869			−0.411935	−	0.911213i	$-0.635147\pi$
$101$	−8.27979			−0.823869			−0.411935	+	0.911213i	$0.635147\pi$
$102$	3.88254			0.384428
$103$			14.5682i			1.43545i	0.696329	+	0.717723i	$0.254815\pi$
$103$			14.5682i			1.43545i	−0.696329	+	0.717723i	$0.745185\pi$
$104$			4.42216i			0.433629i
$105$		0			0
$106$			11.7453i			1.14080i
$107$		−	2.86420i		−	0.276892i	−0.990370	−	0.138446i	$-0.955789\pi$
$107$		−	2.86420i		−	0.276892i	0.990370	−	0.138446i	$-0.0442108\pi$
$108$		−	1.00000i		−	0.0962250i
$109$		−	2.95882i		−	0.283404i	−0.989909	−	0.141702i	$-0.954743\pi$
$109$		−	2.95882i		−	0.283404i	0.989909	−	0.141702i	$-0.0452574\pi$
$110$	−0.226012	+	1.01621i	−0.0215494	+	0.0968914i
$111$			3.89552i			0.369747i
$112$		0			0
$113$	−0.597983			−0.0562535			−0.0281268	−	0.999604i	$-0.508954\pi$
$113$	−0.597983			−0.0562535			−0.0281268	+	0.999604i	$0.508954\pi$
$114$	−1.19962			−0.112354
$115$		−	2.45111i		−	0.228567i
$116$			6.32404i			0.587173i
$117$	−4.42216			−0.408829
$118$	8.87397			0.816915
$119$		0			0
$120$			0.313884i			0.0286535i
$121$	−9.96306	−	4.66235i	−0.905732	−	0.423850i
$122$			11.6955i			1.05886i
$123$			11.2575i			1.01506i
$124$			1.97170i			0.177064i
$125$			3.10791i			0.277980i
$126$		0			0
$127$		−	8.86982i		−	0.787069i	−0.919310	−	0.393535i	$-0.871252\pi$
$127$		−	8.86982i		−	0.787069i	0.919310	−	0.393535i	$-0.128748\pi$
$128$			1.00000i			0.0883883i
$129$	−4.06657			−0.358041
$130$	1.38805			0.121740
$131$	−2.87234			−0.250957			−0.125479	−	0.992096i	$-0.540047\pi$
$131$	−2.87234			−0.250957			−0.125479	+	0.992096i	$0.540047\pi$
$132$	−3.23752	−	0.720050i	−0.281790	−	0.0626723i
$133$		0			0
$134$			12.2415i			1.05750i
$135$	−0.313884			−0.0270148
$136$		−	3.88254i		−	0.332925i
$137$	−5.10296			−0.435975			−0.217988	−	0.975952i	$-0.569949\pi$
$137$	−5.10296			−0.435975			−0.217988	+	0.975952i	$0.569949\pi$
$138$	7.80896			0.664743
$139$	1.71162			0.145177			0.0725887	−	0.997362i	$-0.476874\pi$
$139$	1.71162			0.145177			0.0725887	+	0.997362i	$0.476874\pi$
$140$		0			0
$141$	9.92277			0.835648
$142$		−	10.0331i		−	0.841957i
$143$	−3.18418	+	14.3168i	−0.266275	+	1.19723i
$144$	−1.00000			−0.0833333
$145$	1.98501			0.164847
$146$			4.85587i			0.401875i
$147$		0			0
$148$	3.89552			0.320210
$149$			19.5364i			1.60048i	0.599679	+	0.800241i	$0.295295\pi$
$149$			19.5364i			1.60048i	−0.599679	+	0.800241i	$0.704705\pi$
$150$	−4.90148			−0.400204
$151$		−	6.75468i		−	0.549688i	−0.961489	−	0.274844i	$-0.911374\pi$
$151$		−	6.75468i		−	0.549688i	0.961489	−	0.274844i	$-0.0886262\pi$
$152$			1.19962i			0.0973018i
$153$	3.88254			0.313884
$154$		0			0
$155$	0.618884			0.0497100
$156$			4.42216i			0.354057i
$157$			0.904021i			0.0721487i	0.999349	+	0.0360744i	$0.0114853\pi$
$157$			0.904021i			0.0721487i	−0.999349	+	0.0360744i	$0.988515\pi$
$158$	−6.61990			−0.526651
$159$			11.7453i			0.931461i
$160$	0.313884			0.0248147
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$	9.98569			0.782140			0.391070	−	0.920361i	$-0.372105\pi$
$163$	9.98569			0.782140			0.391070	+	0.920361i	$0.372105\pi$
$164$	11.2575			0.879064
$165$	−0.226012	+	1.01621i	−0.0175950	+	0.0791115i
$166$		−	13.4283i		−	1.04224i
$167$	−1.46451			−0.113327			−0.0566635	−	0.998393i	$-0.518046\pi$
$167$	−1.46451			−0.113327			−0.0566635	+	0.998393i	$0.518046\pi$
$168$		0			0
$169$	6.55554			0.504272
$170$	−1.21867			−0.0934674
$171$	−1.19962			−0.0917370
$172$			4.06657i			0.310073i
$173$	−2.11320			−0.160663			−0.0803317	−	0.996768i	$-0.525598\pi$
$173$	−2.11320			−0.160663			−0.0803317	+	0.996768i	$0.525598\pi$
$174$			6.32404i			0.479424i
$175$		0			0
$176$	−0.720050	+	3.23752i	−0.0542758	+	0.244037i
$177$	8.87397			0.667008
$178$	−11.7289			−0.879120
$179$	−25.0586			−1.87297			−0.936483	−	0.350714i	$-0.885939\pi$
$179$	−25.0586			−1.87297			−0.936483	+	0.350714i	$0.885939\pi$
$180$			0.313884i			0.0233955i
$181$			18.3609i			1.36475i	0.731000	+	0.682377i	$0.239054\pi$
$181$			18.3609i			1.36475i	−0.731000	+	0.682377i	$0.760946\pi$
$182$		0			0
$183$			11.6955i			0.864559i
$184$		−	7.80896i		−	0.575684i
$185$		−	1.22274i		−	0.0898978i
$186$			1.97170i			0.144572i
$187$	2.79562	−	12.5698i	0.204436	−	0.919193i
$188$		−	9.92277i		−	0.723692i
$189$		0			0
$190$	0.376541			0.0273171
$191$	1.90284			0.137684			0.0688422	−	0.997628i	$-0.478070\pi$
$191$	1.90284			0.137684			0.0688422	+	0.997628i	$0.478070\pi$
$192$			1.00000i			0.0721688i
$193$			0.0227424i			0.00163703i	1.00000		0.000818516i	$0.000260542\pi$
$193$			0.0227424i			0.00163703i	−1.00000		0.000818516i	$0.999739\pi$
$194$	11.7519			0.843737
$195$	1.38805			0.0994001
$196$		0			0
$197$			8.73950i			0.622663i	0.950301	+	0.311332i	$0.100775\pi$
$197$			8.73950i			0.622663i	−0.950301	+	0.311332i	$0.899225\pi$
$198$	−3.23752	−	0.720050i	−0.230080	−	0.0511717i
$199$		−	3.66906i		−	0.260093i	−0.991508	−	0.130046i	$-0.958487\pi$
$199$		−	3.66906i		−	0.260093i	0.991508	−	0.130046i	$-0.0415126\pi$
$200$			4.90148i			0.346587i
$201$			12.2415i			0.863449i
$202$			8.27979i			0.582564i
$203$		0			0
$204$		−	3.88254i		−	0.271832i
$205$		−	3.53355i		−	0.246794i
$206$	14.5682			1.01501
$207$	7.80896			0.542760
$208$	4.42216			0.306622
$209$	−0.863785	+	3.88378i	−0.0597492	+	0.268647i
$210$		0			0
$211$		−	20.2522i		−	1.39422i	−0.716966	−	0.697108i	$-0.754470\pi$
$211$		−	20.2522i		−	1.39422i	0.716966	−	0.697108i	$-0.245530\pi$
$212$	11.7453			0.806669
$213$		−	10.0331i		−	0.687455i
$214$	−2.86420			−0.195793
$215$	1.27643			0.0870518
$216$	−1.00000			−0.0680414
$217$		0			0
$218$	−2.95882			−0.200397
$219$			4.85587i			0.328129i
$220$	1.01621	+	0.226012i	0.0685125	+	0.0152377i
$221$	−17.1692			−1.15493
$222$	3.89552			0.261451
$223$			17.5626i			1.17608i	0.808833	+	0.588038i	$0.200100\pi$
$223$			17.5626i			1.17608i	−0.808833	+	0.588038i	$0.799900\pi$
$224$		0			0
$225$	−4.90148			−0.326765
$226$			0.597983i			0.0397773i
$227$	6.00479			0.398552			0.199276	−	0.979943i	$-0.436141\pi$
$227$	6.00479			0.398552			0.199276	+	0.979943i	$0.436141\pi$
$228$			1.19962i			0.0794466i
$229$			19.7840i			1.30736i	0.756769	+	0.653682i	$0.226777\pi$
$229$			19.7840i			1.30736i	−0.756769	+	0.653682i	$0.773223\pi$
$230$	−2.45111			−0.161621
$231$		0			0
$232$	6.32404			0.415194
$233$			5.93788i			0.389004i	0.980902	+	0.194502i	$0.0623090\pi$
$233$			5.93788i			0.389004i	−0.980902	+	0.194502i	$0.937691\pi$
$234$			4.42216i			0.289086i
$235$	−3.11460			−0.203174
$236$		−	8.87397i		−	0.577646i
$237$	−6.61990			−0.430009
$238$		0			0
$239$		−	30.7844i		−	1.99128i	−0.0932798	−	0.995640i	$-0.529735\pi$
$239$		−	30.7844i		−	1.99128i	0.0932798	−	0.995640i	$-0.470265\pi$
$240$	0.313884			0.0202611
$241$	26.6455			1.71639			0.858195	−	0.513324i	$-0.171586\pi$
$241$	26.6455			1.71639			0.858195	+	0.513324i	$0.171586\pi$
$242$	−4.66235	+	9.96306i	−0.299707	+	0.640449i
$243$		−	1.00000i		−	0.0641500i
$244$	11.6955			0.748730
$245$		0			0
$246$	11.2575			0.717752
$247$	5.30491			0.337543
$248$	1.97170			0.125203
$249$		−	13.4283i		−	0.850985i
$250$	3.10791			0.196562
$251$		−	0.276414i		−	0.0174471i	−0.999962	−	0.00872356i	$-0.997223\pi$
$251$		−	0.276414i		−	0.0174471i	0.999962	−	0.00872356i	$-0.00277683\pi$
$252$		0			0
$253$	5.62284	−	25.2816i	0.353505	−	1.58944i
$254$	−8.86982			−0.556542
$255$	−1.21867			−0.0763158
$256$	1.00000			0.0625000
$257$		−	20.9764i		−	1.30847i	−0.756290	−	0.654236i	$-0.772990\pi$
$257$		−	20.9764i		−	1.30847i	0.756290	−	0.654236i	$-0.227010\pi$
$258$			4.06657i			0.253173i
$259$		0			0
$260$		−	1.38805i		−	0.0860830i
$261$			6.32404i			0.391448i
$262$			2.87234i			0.177454i
$263$		−	3.53926i		−	0.218240i	−0.994029	−	0.109120i	$-0.965197\pi$
$263$		−	3.53926i		−	0.218240i	0.994029	−	0.109120i	$-0.0348033\pi$
$264$	−0.720050	+	3.23752i	−0.0443160	+	0.199256i
$265$		−	3.68665i		−	0.226469i
$266$		0			0
$267$	−11.7289			−0.717799
$268$	12.2415			0.747769
$269$			11.1093i			0.677347i	0.940904	+	0.338674i	$0.109978\pi$
$269$			11.1093i			0.677347i	−0.940904	+	0.338674i	$0.890022\pi$
$270$			0.313884i			0.0191024i
$271$	−14.7241			−0.894424			−0.447212	−	0.894428i	$-0.647583\pi$
$271$	−14.7241			−0.894424			−0.447212	+	0.894428i	$0.647583\pi$
$272$	−3.88254			−0.235413
$273$		0			0
$274$			5.10296i			0.308281i
$275$	−3.52931	+	15.8686i	−0.212825	+	0.956914i
$276$		−	7.80896i		−	0.470044i
$277$		−	25.2978i		−	1.52000i	−0.649925	−	0.759998i	$-0.725200\pi$
$277$		−	25.2978i		−	1.52000i	0.649925	−	0.759998i	$-0.274800\pi$
$278$		−	1.71162i		−	0.102656i
$279$			1.97170i			0.118042i
$280$		0			0
$281$			27.4309i			1.63639i	0.574939	+	0.818196i	$0.305026\pi$
$281$			27.4309i			1.63639i	−0.574939	+	0.818196i	$0.694974\pi$
$282$		−	9.92277i		−	0.590892i
$283$	6.31585			0.375438			0.187719	−	0.982223i	$-0.439891\pi$
$283$	6.31585			0.375438			0.187719	+	0.982223i	$0.439891\pi$
$284$	−10.0331			−0.595354
$285$	0.376541			0.0223043
$286$	14.3168	+	3.18418i	0.846573	+	0.188285i
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	−1.92592			−0.113289
$290$		−	1.98501i		−	0.116564i
$291$	11.7519			0.688908
$292$	4.85587			0.284168
$293$	−5.67806			−0.331716			−0.165858	−	0.986150i	$-0.553039\pi$
$293$	−5.67806			−0.331716			−0.165858	+	0.986150i	$0.553039\pi$
$294$		0			0
$295$	−2.78540			−0.162172
$296$		−	3.89552i		−	0.226423i
$297$	−3.23752	−	0.720050i	−0.187860	−	0.0417815i
$298$	19.5364			1.13171
$299$	−34.5325			−1.99707
$300$			4.90148i			0.282987i
$301$		0			0
$302$	−6.75468			−0.388688
$303$			8.27979i			0.475661i
$304$	1.19962			0.0688028
$305$		−	3.67104i		−	0.210203i
$306$		−	3.88254i		−	0.221950i
$307$	19.2444			1.09834			0.549169	−	0.835711i	$-0.314944\pi$
$307$	19.2444			1.09834			0.549169	+	0.835711i	$0.314944\pi$
$308$		0			0
$309$	14.5682			0.828755
$310$		−	0.618884i		−	0.0351503i
$311$			10.8759i			0.616717i	0.951270	+	0.308359i	$0.0997797\pi$
$311$			10.8759i			0.616717i	−0.951270	+	0.308359i	$0.900220\pi$
$312$	4.42216			0.250356
$313$			16.5392i			0.934851i	0.884033	+	0.467425i	$0.154818\pi$
$313$			16.5392i			0.934851i	−0.884033	+	0.467425i	$0.845182\pi$
$314$	0.904021			0.0510169
$315$		0			0
$316$			6.61990i			0.372399i
$317$	−16.6033			−0.932534			−0.466267	−	0.884644i	$-0.654401\pi$
$317$	−16.6033			−0.932534			−0.466267	+	0.884644i	$0.654401\pi$
$318$	11.7453			0.658642
$319$	20.4742	+	4.55363i	1.14634	+	0.254954i
$320$		−	0.313884i		−	0.0175466i
$321$	−2.86420			−0.159864
$322$		0			0
$323$	−4.65756			−0.259153
$324$	−1.00000			−0.0555556
$325$	21.6751			1.20232
$326$		−	9.98569i		−	0.553056i
$327$	−2.95882			−0.163623
$328$		−	11.2575i		−	0.621592i
$329$		0			0
$330$	1.01621	+	0.226012i	0.0559402	+	0.0124416i
$331$	−16.9166			−0.929822			−0.464911	−	0.885357i	$-0.653914\pi$
$331$	−16.9166			−0.929822			−0.464911	+	0.885357i	$0.653914\pi$
$332$	−13.4283			−0.736974
$333$	3.89552			0.213473
$334$			1.46451i			0.0801343i
$335$		−	3.84241i		−	0.209933i
$336$		0			0
$337$			16.4320i			0.895109i	0.894257	+	0.447554i	$0.147705\pi$
$337$			16.4320i			0.895109i	−0.894257	+	0.447554i	$0.852295\pi$
$338$		−	6.55554i		−	0.356574i
$339$			0.597983i			0.0324780i
$340$			1.21867i			0.0660914i
$341$	6.38341	+	1.41972i	0.345681	+	0.0768822i
$342$			1.19962i			0.0648679i
$343$		0			0
$344$	4.06657			0.219255
$345$	−2.45111			−0.131963
$346$			2.11320i			0.113606i
$347$		−	10.0447i		−	0.539227i	−0.962969	−	0.269614i	$-0.913104\pi$
$347$		−	10.0447i		−	0.539227i	0.962969	−	0.269614i	$-0.0868959\pi$
$348$	6.32404			0.339004
$349$	−27.7696			−1.48647			−0.743236	−	0.669029i	$-0.766710\pi$
$349$	−27.7696			−1.48647			−0.743236	+	0.669029i	$0.766710\pi$
$350$		0			0
$351$			4.42216i			0.236038i
$352$	3.23752	+	0.720050i	0.172560	+	0.0383788i
$353$			22.0619i			1.17424i	0.809501	+	0.587118i	$0.199738\pi$
$353$			22.0619i			1.17424i	−0.809501	+	0.587118i	$0.800262\pi$
$354$		−	8.87397i		−	0.471646i
$355$			3.14922i			0.167143i
$356$			11.7289i			0.621632i
$357$		0			0
$358$			25.0586i			1.32439i
$359$		−	1.20528i		−	0.0636125i	−0.999494	−	0.0318062i	$-0.989874\pi$
$359$		−	1.20528i		−	0.0636125i	0.999494	−	0.0318062i	$-0.0101259\pi$
$360$	0.313884			0.0165431
$361$	−17.5609			−0.924259
$362$	18.3609			0.965027
$363$	−4.66235	+	9.96306i	−0.244710	+	0.522925i
$364$		0			0
$365$		−	1.52418i		−	0.0797792i
$366$	11.6955			0.611335
$367$		−	18.3216i		−	0.956381i	−0.878256	−	0.478191i	$-0.841293\pi$
$367$		−	18.3216i		−	0.956381i	0.878256	−	0.478191i	$-0.158707\pi$
$368$	−7.80896			−0.407070
$369$	11.2575			0.586042
$370$	−1.22274			−0.0635674
$371$		0			0
$372$	1.97170			0.102228
$373$			20.8069i			1.07734i	0.842517	+	0.538670i	$0.181073\pi$
$373$			20.8069i			1.07734i	−0.842517	+	0.538670i	$0.818927\pi$
$374$	−12.5698	−	2.79562i	−0.649968	−	0.144558i
$375$	3.10791			0.160492
$376$	−9.92277			−0.511728
$377$		−	27.9660i		−	1.44032i
$378$		0			0
$379$	−12.8436			−0.659729			−0.329865	−	0.944028i	$-0.607003\pi$
$379$	−12.8436			−0.659729			−0.329865	+	0.944028i	$0.607003\pi$
$380$		−	0.376541i		−	0.0193161i
$381$	−8.86982			−0.454415
$382$		−	1.90284i		−	0.0973575i
$383$		−	32.1738i		−	1.64400i	−0.569486	−	0.822001i	$-0.692858\pi$
$383$		−	32.1738i		−	1.64400i	0.569486	−	0.822001i	$-0.307142\pi$
$384$	1.00000			0.0510310
$385$		0			0
$386$	0.0227424			0.00115756
$387$			4.06657i			0.206715i
$388$		−	11.7519i		−	0.596612i
$389$	0.425402			0.0215687			0.0107844	−	0.999942i	$-0.496567\pi$
$389$	0.425402			0.0215687			0.0107844	+	0.999942i	$0.496567\pi$
$390$		−	1.38805i		−	0.0702865i
$391$	30.3186			1.53328
$392$		0			0
$393$			2.87234i			0.144890i
$394$	8.73950			0.440289
$395$	2.07788			0.104550
$396$	−0.720050	+	3.23752i	−0.0361839	+	0.162691i
$397$		−	3.56911i		−	0.179128i	−0.995981	−	0.0895642i	$-0.971453\pi$
$397$		−	3.56911i		−	0.179128i	0.995981	−	0.0895642i	$-0.0285474\pi$
$398$	−3.66906			−0.183913
$399$		0			0
$400$	4.90148			0.245074
$401$	32.6415			1.63004			0.815019	−	0.579435i	$-0.196727\pi$
$401$	32.6415			1.63004			0.815019	+	0.579435i	$0.196727\pi$
$402$	12.2415			0.610551
$403$		−	8.71917i		−	0.434333i
$404$	8.27979			0.411935
$405$			0.313884i			0.0155970i
$406$		0			0
$407$	2.80497	−	12.6118i	0.139037	−	0.625145i
$408$	−3.88254			−0.192214
$409$	−20.9821			−1.03750			−0.518750	−	0.854926i	$-0.673602\pi$
$409$	−20.9821			−1.03750			−0.518750	+	0.854926i	$0.673602\pi$
$410$	−3.53355			−0.174510
$411$			5.10296i			0.251710i
$412$		−	14.5682i		−	0.717723i
$413$		0			0
$414$		−	7.80896i		−	0.383789i
$415$			4.21493i			0.206903i
$416$		−	4.42216i		−	0.216814i
$417$		−	1.71162i		−	0.0838182i
$418$	3.88378	+	0.863785i	0.189962	+	0.0422491i
$419$			3.85865i			0.188507i	0.995548	+	0.0942536i	$0.0300465\pi$
$419$			3.85865i			0.188507i	−0.995548	+	0.0942536i	$0.969954\pi$
$420$		0			0
$421$	−1.02513			−0.0499618			−0.0249809	−	0.999688i	$-0.507952\pi$
$421$	−1.02513			−0.0499618			−0.0249809	+	0.999688i	$0.507952\pi$
$422$	−20.2522			−0.985860
$423$		−	9.92277i		−	0.482462i
$424$		−	11.7453i		−	0.570401i
$425$	−19.0302			−0.923098
$426$	−10.0331			−0.486104
$427$		0			0
$428$			2.86420i			0.138446i
$429$	14.3168	+	3.18418i	0.691224	+	0.153734i
$430$		−	1.27643i		−	0.0615549i
$431$			38.0689i			1.83371i	0.399218	+	0.916856i	$0.369282\pi$
$431$			38.0689i			1.83371i	−0.399218	+	0.916856i	$0.630718\pi$
$432$			1.00000i			0.0481125i
$433$		−	13.6047i		−	0.653798i	−0.945059	−	0.326899i	$-0.893996\pi$
$433$		−	13.6047i		−	0.653798i	0.945059	−	0.326899i	$-0.106004\pi$
$434$		0			0
$435$		−	1.98501i		−	0.0951742i
$436$			2.95882i			0.141702i
$437$	−9.36776			−0.448121
$438$	4.85587			0.232023
$439$	8.72564			0.416452			0.208226	−	0.978081i	$-0.433231\pi$
$439$	8.72564			0.416452			0.208226	+	0.978081i	$0.433231\pi$
$440$	0.226012	−	1.01621i	0.0107747	−	0.0484457i
$441$		0			0
$442$			17.1692i			0.816656i
$443$	11.8030			0.560779			0.280390	−	0.959886i	$-0.409536\pi$
$443$	11.8030			0.560779			0.280390	+	0.959886i	$0.409536\pi$
$444$		−	3.89552i		−	0.184873i
$445$	3.68152			0.174521
$446$	17.5626			0.831612
$447$	19.5364			0.924038
$448$		0			0
$449$	−7.11503			−0.335779			−0.167889	−	0.985806i	$-0.553695\pi$
$449$	−7.11503			−0.335779			−0.167889	+	0.985806i	$0.553695\pi$
$450$			4.90148i			0.231058i
$451$	8.10597	−	36.4464i	0.381695	−	1.71619i
$452$	0.597983			0.0281268
$453$	−6.75468			−0.317363
$454$		−	6.00479i		−	0.281819i
$455$		0			0
$456$	1.19962			0.0561772
$457$			14.9142i			0.697655i	0.937187	+	0.348828i	$0.113420\pi$
$457$			14.9142i			0.697655i	−0.937187	+	0.348828i	$0.886580\pi$
$458$	19.7840			0.924446
$459$		−	3.88254i		−	0.181221i
$460$			2.45111i			0.114283i
$461$	−23.4773			−1.09345			−0.546723	−	0.837314i	$-0.684125\pi$
$461$	−23.4773			−1.09345			−0.546723	+	0.837314i	$0.684125\pi$
$462$		0			0
$463$	35.7950			1.66354			0.831768	−	0.555124i	$-0.187329\pi$
$463$	35.7950			1.66354			0.831768	+	0.555124i	$0.187329\pi$
$464$		−	6.32404i		−	0.293586i
$465$		−	0.618884i		−	0.0287001i
$466$	5.93788			0.275067
$467$		−	17.3569i		−	0.803180i	−0.915819	−	0.401590i	$-0.868458\pi$
$467$		−	17.3569i		−	0.803180i	0.915819	−	0.401590i	$-0.131542\pi$
$468$	4.42216			0.204415
$469$		0			0
$470$			3.11460i			0.143666i
$471$	0.904021			0.0416551
$472$	−8.87397			−0.408458
$473$	13.1656	+	2.92813i	0.605354	+	0.134636i
$474$			6.61990i			0.304062i
$475$	5.87990			0.269788
$476$		0			0
$477$	11.7453			0.537779
$478$	−30.7844			−1.40805
$479$	29.3682			1.34187			0.670933	−	0.741518i	$-0.265894\pi$
$479$	29.3682			1.34187			0.670933	+	0.741518i	$0.265894\pi$
$480$		−	0.313884i		−	0.0143268i
$481$	−17.2267			−0.785468
$482$		−	26.6455i		−	1.21367i
$483$		0			0
$484$	9.96306	+	4.66235i	0.452866	+	0.211925i
$485$	−3.68873			−0.167497
$486$	−1.00000			−0.0453609
$487$	11.7596			0.532877			0.266439	−	0.963852i	$-0.414153\pi$
$487$	11.7596			0.532877			0.266439	+	0.963852i	$0.414153\pi$
$488$		−	11.6955i		−	0.529432i
$489$		−	9.98569i		−	0.451569i
$490$		0			0
$491$			31.4910i			1.42117i	0.703612	+	0.710585i	$0.251570\pi$
$491$			31.4910i			1.42117i	−0.703612	+	0.710585i	$0.748430\pi$
$492$		−	11.2575i		−	0.507528i
$493$			24.5533i			1.10583i
$494$		−	5.30491i		−	0.238679i
$495$	1.01621	+	0.226012i	0.0456750	+	0.0101585i
$496$		−	1.97170i		−	0.0885319i
$497$		0			0
$498$	−13.4283			−0.601737
$499$	−2.16815			−0.0970597			−0.0485298	−	0.998822i	$-0.515454\pi$
$499$	−2.16815			−0.0970597			−0.0485298	+	0.998822i	$0.515454\pi$
$500$		−	3.10791i		−	0.138990i
$501$			1.46451i			0.0654294i
$502$	−0.276414			−0.0123370
$503$	7.07263			0.315353			0.157677	−	0.987491i	$-0.449600\pi$
$503$	7.07263			0.315353			0.157677	+	0.987491i	$0.449600\pi$
$504$		0			0
$505$		−	2.59889i		−	0.115649i
$506$	−25.2816	−	5.62284i	−1.12391	−	0.249966i
$507$		−	6.55554i		−	0.291142i
$508$			8.86982i			0.393535i
$509$			0.538666i			0.0238759i	0.999929	+	0.0119380i	$0.00380006\pi$
$509$			0.538666i			0.0238759i	−0.999929	+	0.0119380i	$0.996200\pi$
$510$			1.21867i			0.0539634i
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$			1.19962i			0.0529644i
$514$	−20.9764			−0.925230
$515$	−4.57272			−0.201498
$516$	4.06657			0.179021
$517$	−32.1252	−	7.14489i	−1.41286	−	0.314232i
$518$		0			0
$519$			2.11320i			0.0927590i
$520$	−1.38805			−0.0608699
$521$		−	26.0419i		−	1.14092i	−0.821326	−	0.570459i	$-0.806765\pi$
$521$		−	26.0419i		−	1.14092i	0.821326	−	0.570459i	$-0.193235\pi$
$522$	6.32404			0.276796
$523$	15.1764			0.663617			0.331809	−	0.943347i	$-0.392341\pi$
$523$	15.1764			0.663617			0.331809	+	0.943347i	$0.392341\pi$
$524$	2.87234			0.125479
$525$		0			0
$526$	−3.53926			−0.154319
$527$			7.65519i			0.333465i
$528$	3.23752	+	0.720050i	0.140895	+	0.0313362i
$529$	37.9798			1.65130
$530$	−3.68665			−0.160138
$531$		−	8.87397i		−	0.385097i
$532$		0			0
$533$	−49.7825			−2.15632
$534$			11.7289i			0.507560i
$535$	0.899026			0.0388683
$536$		−	12.2415i		−	0.528752i
$537$			25.0586i			1.08136i
$538$	11.1093			0.478957
$539$		0			0
$540$	0.313884			0.0135074
$541$		−	33.1237i		−	1.42410i	−0.702130	−	0.712049i	$-0.747767\pi$
$541$		−	33.1237i		−	1.42410i	0.702130	−	0.712049i	$-0.252233\pi$
$542$			14.7241i			0.632453i
$543$	18.3609			0.787941
$544$			3.88254i			0.166462i
$545$	0.928727			0.0397823
$546$		0			0
$547$		−	28.0871i		−	1.20092i	−0.799656	−	0.600459i	$-0.794985\pi$
$547$		−	28.0871i		−	1.20092i	0.799656	−	0.600459i	$-0.205015\pi$
$548$	5.10296			0.217988
$549$	11.6955			0.499153
$550$	15.8686	+	3.52931i	0.676640	+	0.150490i
$551$		−	7.58643i		−	0.323193i
$552$	−7.80896			−0.332371
$553$		0			0
$554$	−25.2978			−1.07480
$555$	−1.22274			−0.0519025
$556$	−1.71162			−0.0725887
$557$		−	7.15504i		−	0.303169i	−0.988444	−	0.151584i	$-0.951562\pi$
$557$		−	7.15504i		−	0.303169i	0.988444	−	0.151584i	$-0.0484375\pi$
$558$	1.97170			0.0834686
$559$		−	17.9830i		−	0.760601i
$560$		0			0
$561$	−12.5698	−	2.79562i	−0.530697	−	0.118031i
$562$	27.4309			1.15710
$563$	−11.4715			−0.483464			−0.241732	−	0.970343i	$-0.577716\pi$
$563$	−11.4715			−0.483464			−0.241732	+	0.970343i	$0.577716\pi$
$564$	−9.92277			−0.417824
$565$		−	0.187697i		−	0.00789649i
$566$		−	6.31585i		−	0.265475i
$567$		0			0
$568$			10.0331i			0.420979i
$569$		−	17.5534i		−	0.735878i	−0.929850	−	0.367939i	$-0.880064\pi$
$569$		−	17.5534i		−	0.735878i	0.929850	−	0.367939i	$-0.119936\pi$
$570$		−	0.376541i		−	0.0157716i
$571$		−	36.5396i		−	1.52914i	−0.644543	−	0.764568i	$-0.722953\pi$
$571$		−	36.5396i		−	1.52914i	0.644543	−	0.764568i	$-0.277047\pi$
$572$	3.18418	−	14.3168i	0.133137	−	0.598617i
$573$		−	1.90284i		−	0.0794921i
$574$		0			0
$575$	−38.2754			−1.59620
$576$	1.00000			0.0416667
$577$		−	3.01392i		−	0.125471i	−0.998030	−	0.0627355i	$-0.980018\pi$
$577$		−	3.01392i		−	0.125471i	0.998030	−	0.0627355i	$-0.0199824\pi$
$578$			1.92592i			0.0801077i
$579$	0.0227424			0.000945141
$580$	−1.98501			−0.0824233
$581$		0			0
$582$		−	11.7519i		−	0.487132i
$583$	8.45719	−	38.0256i	0.350261	−	1.57486i
$584$		−	4.85587i		−	0.200937i
$585$		−	1.38805i		−	0.0573887i
$586$			5.67806i			0.234558i
$587$		−	35.7499i		−	1.47556i	−0.675042	−	0.737779i	$-0.735875\pi$
$587$		−	35.7499i		−	1.47556i	0.675042	−	0.737779i	$-0.264125\pi$
$588$		0			0
$589$		−	2.36528i		−	0.0974598i
$590$			2.78540i			0.114673i
$591$	8.73950			0.359495
$592$	−3.89552			−0.160105
$593$	−27.4252			−1.12622			−0.563109	−	0.826383i	$-0.690395\pi$
$593$	−27.4252			−1.12622			−0.563109	+	0.826383i	$0.690395\pi$
$594$	−0.720050	+	3.23752i	−0.0295440	+	0.132837i
$595$		0			0
$596$		−	19.5364i		−	0.800241i
$597$	−3.66906			−0.150165
$598$			34.5325i			1.41214i
$599$	−44.4086			−1.81449			−0.907244	−	0.420605i	$-0.861818\pi$
$599$	−44.4086			−1.81449			−0.907244	+	0.420605i	$0.861818\pi$
$600$	4.90148			0.200102
$601$	−44.2914			−1.80668			−0.903342	−	0.428922i	$-0.858894\pi$
$601$	−44.2914			−1.80668			−0.903342	+	0.428922i	$0.858894\pi$
$602$		0			0
$603$	12.2415			0.498512
$604$			6.75468i			0.274844i
$605$	1.46344	−	3.12724i	0.0594972	−	0.127141i
$606$	8.27979			0.336343
$607$	−1.98763			−0.0806754			−0.0403377	−	0.999186i	$-0.512843\pi$
$607$	−1.98763			−0.0806754			−0.0403377	+	0.999186i	$0.512843\pi$
$608$		−	1.19962i		−	0.0486509i
$609$		0			0
$610$	−3.67104			−0.148636
$611$			43.8801i			1.77520i
$612$	−3.88254			−0.156942
$613$			10.1472i			0.409842i	0.978779	+	0.204921i	$0.0656937\pi$
$613$			10.1472i			0.409842i	−0.978779	+	0.204921i	$0.934306\pi$
$614$		−	19.2444i		−	0.776642i
$615$	−3.53355			−0.142486
$616$		0			0
$617$	6.71985			0.270531			0.135265	−	0.990809i	$-0.456811\pi$
$617$	6.71985			0.270531			0.135265	+	0.990809i	$0.456811\pi$
$618$		−	14.5682i		−	0.586018i
$619$			17.9109i			0.719899i	0.932972	+	0.359950i	$0.117206\pi$
$619$			17.9109i			0.719899i	−0.932972	+	0.359950i	$0.882794\pi$
$620$	−0.618884			−0.0248550
$621$		−	7.80896i		−	0.313363i
$622$	10.8759			0.436085
$623$		0			0
$624$		−	4.42216i		−	0.177028i
$625$	23.5319			0.941274
$626$	16.5392			0.661039
$627$	3.88378	+	0.863785i	0.155103	+	0.0344962i
$628$		−	0.904021i		−	0.0360744i
$629$	15.1245			0.603054
$630$		0			0
$631$	−35.8544			−1.42734			−0.713671	−	0.700481i	$-0.752969\pi$
$631$	−35.8544			−1.42734			−0.713671	+	0.700481i	$0.752969\pi$
$632$	6.61990			0.263326
$633$	−20.2522			−0.804951
$634$			16.6033i			0.659401i
$635$	2.78409			0.110483
$636$		−	11.7453i		−	0.465731i
$637$		0			0
$638$	4.55363	−	20.4742i	0.180280	−	0.810581i
$639$	−10.0331			−0.396902
$640$	−0.313884			−0.0124074
$641$	−41.4397			−1.63677			−0.818384	−	0.574671i	$-0.805130\pi$
$641$	−41.4397			−1.63677			−0.818384	+	0.574671i	$0.805130\pi$
$642$			2.86420i			0.113041i
$643$		−	19.6233i		−	0.773867i	−0.922108	−	0.386933i	$-0.873534\pi$
$643$		−	19.6233i		−	0.773867i	0.922108	−	0.386933i	$-0.126466\pi$
$644$		0			0
$645$		−	1.27643i		−	0.0502594i
$646$			4.65756i			0.183249i
$647$		−	2.65766i		−	0.104484i	−0.998634	−	0.0522418i	$-0.983363\pi$
$647$		−	2.65766i		−	0.104484i	0.998634	−	0.0522418i	$-0.0166367\pi$
$648$			1.00000i			0.0392837i
$649$	−28.7296	−	6.38970i	−1.12774	−	0.250818i
$650$		−	21.6751i		−	0.850169i
$651$		0			0
$652$	−9.98569			−0.391070
$653$	−22.7849			−0.891641			−0.445820	−	0.895122i	$-0.647088\pi$
$653$	−22.7849			−0.891641			−0.445820	+	0.895122i	$0.647088\pi$
$654$			2.95882i			0.115699i
$655$		−	0.901581i		−	0.0352277i
$656$	−11.2575			−0.439532
$657$	4.85587			0.189446
$658$		0			0
$659$			2.89032i			0.112591i	0.998414	+	0.0562955i	$0.0179289\pi$
$659$			2.89032i			0.112591i	−0.998414	+	0.0562955i	$0.982071\pi$
$660$	0.226012	−	1.01621i	0.00879751	−	0.0395557i
$661$		−	40.6647i		−	1.58167i	−0.612027	−	0.790837i	$-0.709646\pi$
$661$		−	40.6647i		−	1.58167i	0.612027	−	0.790837i	$-0.290354\pi$
$662$			16.9166i			0.657484i
$663$			17.1692i			0.666797i
$664$			13.4283i			0.521120i
$665$		0			0
$666$		−	3.89552i		−	0.150949i
$667$			49.3842i			1.91216i
$668$	1.46451			0.0566635
$669$	17.5626			0.679008
$670$	−3.84241			−0.148445
$671$	8.42137	−	37.8645i	0.325103	−	1.46174i
$672$		0			0
$673$			4.76054i			0.183505i	0.995782	+	0.0917526i	$0.0292469\pi$
$673$			4.76054i			0.183505i	−0.995782	+	0.0917526i	$0.970753\pi$
$674$	16.4320			0.632938
$675$			4.90148i			0.188658i
$676$	−6.55554			−0.252136
$677$	−9.85362			−0.378705			−0.189353	−	0.981909i	$-0.560639\pi$
$677$	−9.85362			−0.378705			−0.189353	+	0.981909i	$0.560639\pi$
$678$	0.597983			0.0229654
$679$		0			0
$680$	1.21867			0.0467337
$681$		−	6.00479i		−	0.230104i
$682$	1.41972	−	6.38341i	0.0543639	−	0.244433i
$683$	32.7366			1.25263			0.626315	−	0.779570i	$-0.284562\pi$
$683$	32.7366			1.25263			0.626315	+	0.779570i	$0.284562\pi$
$684$	1.19962			0.0458685
$685$		−	1.60174i		−	0.0611992i
$686$		0			0
$687$	19.7840			0.754807
$688$		−	4.06657i		−	0.155036i
$689$	−51.9395			−1.97874
$690$			2.45111i			0.0933120i
$691$			20.2918i			0.771938i	0.922512	+	0.385969i	$0.126133\pi$
$691$			20.2918i			0.771938i	−0.922512	+	0.385969i	$0.873867\pi$
$692$	2.11320			0.0803317
$693$		0			0
$694$	−10.0447			−0.381291
$695$			0.537249i			0.0203790i
$696$		−	6.32404i		−	0.239712i
$697$	43.7077			1.65555
$698$			27.7696i			1.05109i
$699$	5.93788			0.224591
$700$		0			0
$701$			23.3858i			0.883268i	0.897195	+	0.441634i	$0.145601\pi$
$701$			23.3858i			0.883268i	−0.897195	+	0.441634i	$0.854399\pi$
$702$	4.42216			0.166904
$703$	−4.67314			−0.176251
$704$	0.720050	−	3.23752i	0.0271379	−	0.122019i
$705$			3.11460i			0.117303i
$706$	22.0619			0.830310
$707$		0			0
$708$	−8.87397			−0.333504
$709$	47.0000			1.76512			0.882561	−	0.470198i	$-0.155817\pi$
$709$	47.0000			1.76512			0.882561	+	0.470198i	$0.155817\pi$
$710$	3.14922			0.118188
$711$			6.61990i			0.248266i
$712$	11.7289			0.439560
$713$			15.3969i			0.576619i
$714$		0			0
$715$	−4.49383	−	0.999463i	−0.168060	−	0.0373778i
$716$	25.0586			0.936483
$717$	−30.7844			−1.14967
$718$	−1.20528			−0.0449808
$719$			4.12031i			0.153662i	0.997044	+	0.0768309i	$0.0244802\pi$
$719$			4.12031i			0.153662i	−0.997044	+	0.0768309i	$0.975520\pi$
$720$		−	0.313884i		−	0.0116978i
$721$		0			0
$722$			17.5609i			0.653550i
$723$		−	26.6455i		−	0.990958i
$724$		−	18.3609i		−	0.682377i
$725$		−	30.9971i		−	1.15121i
$726$	9.96306	+	4.66235i	0.369764	+	0.173036i
$727$		−	6.30961i		−	0.234011i	−0.993131	−	0.117005i	$-0.962671\pi$
$727$		−	6.30961i		−	0.234011i	0.993131	−	0.117005i	$-0.0373294\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	−1.52418			−0.0564124
$731$			15.7886i			0.583962i
$732$		−	11.6955i		−	0.432279i
$733$	46.1627			1.70506			0.852528	−	0.522681i	$-0.175068\pi$
$733$	46.1627			1.70506			0.852528	+	0.522681i	$0.175068\pi$
$734$	−18.3216			−0.676264
$735$		0			0
$736$			7.80896i			0.287842i
$737$	8.81449	−	39.6321i	0.324686	−	1.45987i
$738$		−	11.2575i		−	0.414395i
$739$			47.1589i			1.73477i	0.497638	+	0.867385i	$0.334201\pi$
$739$			47.1589i			1.73477i	−0.497638	+	0.867385i	$0.665799\pi$
$740$			1.22274i			0.0449489i
$741$		−	5.30491i		−	0.194881i
$742$		0			0
$743$			1.18549i			0.0434915i	0.999764	+	0.0217458i	$0.00692244\pi$
$743$			1.18549i			0.0434915i	−0.999764	+	0.0217458i	$0.993078\pi$
$744$		−	1.97170i		−	0.0722860i
$745$	−6.13215			−0.224665
$746$	20.8069			0.761795
$747$	−13.4283			−0.491316
$748$	−2.79562	+	12.5698i	−0.102218	+	0.459597i
$749$		0			0
$750$		−	3.10791i		−	0.113485i
$751$	21.2874			0.776787			0.388393	−	0.921494i	$-0.373030\pi$
$751$	21.2874			0.776787			0.388393	+	0.921494i	$0.373030\pi$
$752$			9.92277i			0.361846i
$753$	−0.276414			−0.0100731
$754$	−27.9660			−1.01846
$755$	2.12019			0.0771614
$756$		0			0
$757$	−21.3152			−0.774715			−0.387357	−	0.921930i	$-0.626612\pi$
$757$	−21.3152			−0.774715			−0.387357	+	0.921930i	$0.626612\pi$
$758$			12.8436i			0.466499i
$759$	−25.2816	−	5.62284i	−0.917666	−	0.204096i
$760$	−0.376541			−0.0136586
$761$	34.3734			1.24604			0.623018	−	0.782207i	$-0.285906\pi$
$761$	34.3734			1.24604			0.623018	+	0.782207i	$0.285906\pi$
$762$			8.86982i			0.321320i
$763$		0			0
$764$	−1.90284			−0.0688422
$765$			1.21867i			0.0440609i
$766$	−32.1738			−1.16249
$767$			39.2422i			1.41695i
$768$		−	1.00000i		−	0.0360844i
$769$	−43.5831			−1.57165			−0.785823	−	0.618452i	$-0.787760\pi$
$769$	−43.5831			−1.57165			−0.785823	+	0.618452i	$0.787760\pi$
$770$		0			0
$771$	−20.9764			−0.755447
$772$		−	0.0227424i		−	0.000818516i
$773$			33.9112i			1.21970i	0.792517	+	0.609850i	$0.208770\pi$
$773$			33.9112i			1.21970i	−0.792517	+	0.609850i	$0.791230\pi$
$774$	4.06657			0.146170
$775$		−	9.66423i		−	0.347150i
$776$	−11.7519			−0.421868
$777$		0			0
$778$		−	0.425402i		−	0.0152514i
$779$	−13.5047			−0.483856
$780$	−1.38805			−0.0497000
$781$	−7.22432	+	32.4823i	−0.258506	+	1.16231i
$782$		−	30.3186i		−	1.08419i
$783$	6.32404			0.226003
$784$		0			0
$785$	−0.283758			−0.0101277
$786$	2.87234			0.102453
$787$	49.6233			1.76888			0.884440	−	0.466654i	$-0.154541\pi$
$787$	49.6233			1.76888			0.884440	+	0.466654i	$0.154541\pi$
$788$		−	8.73950i		−	0.311332i
$789$	−3.53926			−0.126001
$790$		−	2.07788i		−	0.0739277i
$791$		0			0
$792$	3.23752	+	0.720050i	0.115040	+	0.0255859i
$793$	−51.7196			−1.83662
$794$	−3.56911			−0.126663
$795$	−3.68665			−0.130752
$796$			3.66906i			0.130046i
$797$		−	11.7829i		−	0.417374i	−0.977983	−	0.208687i	$-0.933081\pi$
$797$		−	11.7829i		−	0.417374i	0.977983	−	0.208687i	$-0.0669189\pi$
$798$		0			0
$799$		−	38.5255i		−	1.36293i
$800$		−	4.90148i		−	0.173293i
$801$			11.7289i			0.414421i
$802$		−	32.6415i		−	1.15261i
$803$	3.49647	−	15.7210i	0.123388	−	0.554781i
$804$		−	12.2415i		−	0.431724i
$805$		0			0
$806$	−8.71917			−0.307120
$807$	11.1093			0.391067
$808$		−	8.27979i		−	0.291282i
$809$		−	43.8765i		−	1.54262i	−0.636462	−	0.771308i	$-0.719603\pi$
$809$		−	43.8765i		−	1.54262i	0.636462	−	0.771308i	$-0.280397\pi$
$810$	0.313884			0.0110288
$811$	−38.1514			−1.33968			−0.669839	−	0.742506i	$-0.733637\pi$
$811$	−38.1514			−1.33968			−0.669839	+	0.742506i	$0.733637\pi$
$812$		0			0
$813$			14.7241i			0.516396i
$814$	−12.6118	−	2.80497i	−0.442045	−	0.0983143i
$815$			3.13435i			0.109791i
$816$			3.88254i			0.135916i
$817$		−	4.87833i		−	0.170671i
$818$			20.9821i			0.733623i
$819$		0			0
$820$			3.53355i			0.123397i
$821$		−	39.1064i		−	1.36482i	−0.730969	−	0.682411i	$-0.760932\pi$
$821$		−	39.1064i		−	1.36482i	0.730969	−	0.682411i	$-0.239068\pi$
$822$	5.10296			0.177986
$823$	39.3506			1.37168			0.685839	−	0.727754i	$-0.259436\pi$
$823$	39.3506			1.37168			0.685839	+	0.727754i	$0.259436\pi$
$824$	−14.5682			−0.507507
$825$	15.8686	+	3.52931i	0.552475	+	0.122875i
$826$		0			0
$827$		−	1.88958i		−	0.0657070i	−0.999460	−	0.0328535i	$-0.989541\pi$
$827$		−	1.88958i		−	0.0657070i	0.999460	−	0.0328535i	$-0.0104595\pi$
$828$	−7.80896			−0.271380
$829$		−	11.3169i		−	0.393053i	−0.980499	−	0.196526i	$-0.937034\pi$
$829$		−	11.3169i		−	0.393053i	0.980499	−	0.196526i	$-0.0629661\pi$
$830$	4.21493			0.146302
$831$	−25.2978			−0.877570
$832$	−4.42216			−0.153311
$833$		0			0
$834$	−1.71162			−0.0592684
$835$		−	0.459686i		−	0.0159081i
$836$	0.863785	−	3.88378i	0.0298746	−	0.134323i
$837$	1.97170			0.0681519
$838$	3.85865			0.133295
$839$		−	36.6681i		−	1.26592i	−0.774183	−	0.632961i	$-0.781839\pi$
$839$		−	36.6681i		−	1.26592i	0.774183	−	0.632961i	$-0.218161\pi$
$840$		0			0
$841$	−10.9935			−0.379086
$842$			1.02513i			0.0353283i
$843$	27.4309			0.944771
$844$			20.2522i			0.697108i
$845$			2.05768i			0.0707863i
$846$	−9.92277			−0.341152
$847$		0			0
$848$	−11.7453			−0.403334
$849$		−	6.31585i		−	0.216759i
$850$			19.0302i			0.652729i
$851$	30.4200			1.04278
$852$			10.0331i			0.343728i
$853$	6.33290			0.216834			0.108417	−	0.994105i	$-0.465422\pi$
$853$	6.33290			0.216834			0.108417	+	0.994105i	$0.465422\pi$
$854$		0			0
$855$		−	0.376541i		−	0.0128774i
$856$	2.86420			0.0978963
$857$	44.7905			1.53001			0.765007	−	0.644022i	$-0.222735\pi$
$857$	44.7905			1.53001			0.765007	+	0.644022i	$0.222735\pi$
$858$	3.18418	−	14.3168i	0.108706	−	0.488769i
$859$		−	8.44407i		−	0.288108i	−0.989570	−	0.144054i	$-0.953986\pi$
$859$		−	8.44407i		−	0.288108i	0.989570	−	0.144054i	$-0.0460139\pi$
$860$	−1.27643			−0.0435259
$861$		0			0
$862$	38.0689			1.29663
$863$	5.86648			0.199697			0.0998487	−	0.995003i	$-0.468164\pi$
$863$	5.86648			0.199697			0.0998487	+	0.995003i	$0.468164\pi$
$864$	1.00000			0.0340207
$865$		−	0.663298i		−	0.0225528i
$866$	−13.6047			−0.462305
$867$			1.92592i			0.0654077i
$868$		0			0
$869$	21.4321	+	4.76666i	0.727033	+	0.161698i
$870$	−1.98501			−0.0672983
$871$	−54.1339			−1.83426
$872$	2.95882			0.100198
$873$		−	11.7519i		−	0.397741i
$874$			9.36776i			0.316869i
$875$		0			0
$876$		−	4.85587i		−	0.164065i
$877$		−	46.4196i		−	1.56748i	−0.621090	−	0.783739i	$-0.713310\pi$
$877$		−	46.4196i		−	1.56748i	0.621090	−	0.783739i	$-0.286690\pi$
$878$		−	8.72564i		−	0.294476i
$879$			5.67806i			0.191516i
$880$	−1.01621	−	0.226012i	−0.0342563	−	0.00761887i
$881$			10.4075i			0.350638i	0.984512	+	0.175319i	$0.0560957\pi$
$881$			10.4075i			0.350638i	−0.984512	+	0.175319i	$0.943904\pi$
$882$		0			0
$883$	−45.2346			−1.52227			−0.761133	−	0.648596i	$-0.775356\pi$
$883$	−45.2346			−1.52227			−0.761133	+	0.648596i	$0.775356\pi$
$884$	17.1692			0.577463
$885$			2.78540i			0.0936301i
$886$		−	11.8030i		−	0.396531i
$887$	−16.8922			−0.567183			−0.283591	−	0.958945i	$-0.591526\pi$
$887$	−16.8922			−0.567183			−0.283591	+	0.958945i	$0.591526\pi$
$888$	−3.89552			−0.130725
$889$		0			0
$890$		−	3.68152i		−	0.123405i
$891$	−0.720050	+	3.23752i	−0.0241226	+	0.108461i
$892$		−	17.5626i		−	0.588038i
$893$			11.9035i			0.398336i
$894$		−	19.5364i		−	0.653394i
$895$		−	7.86548i		−	0.262914i
$896$		0			0
$897$			34.5325i			1.15301i
$898$			7.11503i			0.237432i
$899$	−12.4691			−0.415868
$900$	4.90148			0.163383
$901$	45.6014			1.51920
$902$	−36.4464	−	8.10597i	−1.21353	−	0.269899i
$903$		0			0
$904$		−	0.597983i		−	0.0198886i
$905$	−5.76319			−0.191575
$906$			6.75468i			0.224409i
$907$	23.6944			0.786760			0.393380	−	0.919376i	$-0.371306\pi$
$907$	23.6944			0.786760			0.393380	+	0.919376i	$0.371306\pi$
$908$	−6.00479			−0.199276
$909$	8.27979			0.274623
$910$		0			0
$911$	−44.5110			−1.47472			−0.737358	−	0.675502i	$-0.763927\pi$
$911$	−44.5110			−1.47472			−0.737358	+	0.675502i	$0.763927\pi$
$912$		−	1.19962i		−	0.0397233i
$913$	−9.66906	+	43.4744i	−0.319999	+	1.43879i
$914$	14.9142			0.493317
$915$	−3.67104			−0.121361
$916$		−	19.7840i		−	0.653682i
$917$		0			0
$918$	−3.88254			−0.128143
$919$			9.49698i			0.313276i	0.987656	+	0.156638i	$0.0500657\pi$
$919$			9.49698i			0.313276i	−0.987656	+	0.156638i	$0.949934\pi$
$920$	2.45111			0.0808106
$921$		−	19.2444i		−	0.634125i
$922$			23.4773i			0.773183i
$923$	44.3679			1.46039
$924$		0			0
$925$	−19.0938			−0.627801
$926$		−	35.7950i		−	1.17630i
$927$		−	14.5682i		−	0.478482i
$928$	−6.32404			−0.207597
$929$			0.396529i			0.0130097i	0.999979	+	0.00650484i	$0.00207057\pi$
$929$			0.396529i			0.0130097i	−0.999979	+	0.00650484i	$0.997929\pi$
$930$	−0.618884			−0.0202940
$931$		0			0
$932$		−	5.93788i		−	0.194502i
$933$	10.8759			0.356062
$934$	−17.3569			−0.567934
$935$	3.94545	+	0.877500i	0.129030	+	0.0286973i
$936$		−	4.42216i		−	0.144543i
$937$	−29.2790			−0.956504			−0.478252	−	0.878223i	$-0.658729\pi$
$937$	−29.2790			−0.956504			−0.478252	+	0.878223i	$0.658729\pi$
$938$		0			0
$939$	16.5392			0.539736
$940$	3.11460			0.101587
$941$	27.5117			0.896854			0.448427	−	0.893819i	$-0.351984\pi$
$941$	27.5117			0.896854			0.448427	+	0.893819i	$0.351984\pi$
$942$		−	0.904021i		−	0.0294546i
$943$	87.9094			2.86272
$944$			8.87397i			0.288823i
$945$		0			0
$946$	2.92813	−	13.1656i	0.0952018	−	0.428050i
$947$	25.4599			0.827334			0.413667	−	0.910428i	$-0.364248\pi$
$947$	25.4599			0.827334			0.413667	+	0.910428i	$0.364248\pi$
$948$	6.61990			0.215004
$949$	−21.4735			−0.697058
$950$		−	5.87990i		−	0.190769i
$951$			16.6033i			0.538399i
$952$		0			0
$953$			4.82076i			0.156160i	0.996947	+	0.0780799i	$0.0248789\pi$
$953$			4.82076i			0.156160i	−0.996947	+	0.0780799i	$0.975121\pi$
$954$		−	11.7453i		−	0.380267i
$955$			0.597269i			0.0193272i
$956$			30.7844i			0.995640i
$957$	4.55363	−	20.4742i	0.147198	−	0.661837i
$958$		−	29.3682i		−	0.948842i
$959$		0			0
$960$	−0.313884			−0.0101306
$961$	27.1124			0.874594
$962$			17.2267i			0.555410i
$963$			2.86420i			0.0922975i
$964$	−26.6455			−0.858195
$965$	−0.00713847			−0.000229795
$966$		0			0
$967$		−	28.9375i		−	0.930567i	−0.885162	−	0.465284i	$-0.845952\pi$
$967$		−	28.9375i		−	0.930567i	0.885162	−	0.465284i	$-0.154048\pi$
$968$	4.66235	−	9.96306i	0.149854	−	0.320225i
$969$			4.65756i			0.149622i
$970$			3.68873i			0.118438i
$971$			27.0612i			0.868435i	0.900808	+	0.434218i	$0.142975\pi$
$971$			27.0612i			0.868435i	−0.900808	+	0.434218i	$0.857025\pi$
$972$			1.00000i			0.0320750i
$973$		0			0
$974$		−	11.7596i		−	0.376801i
$975$		−	21.6751i		−	0.694160i
$976$	−11.6955			−0.374365
$977$	−7.23749			−0.231548			−0.115774	−	0.993276i	$-0.536935\pi$
$977$	−7.23749			−0.231548			−0.115774	+	0.993276i	$0.536935\pi$
$978$	−9.98569			−0.319307
$979$	37.9726	+	8.44542i	1.21361	+	0.269917i
$980$		0			0
$981$			2.95882i			0.0944679i
$982$	31.4910			1.00492
$983$		−	11.3461i		−	0.361885i	−0.983494	−	0.180942i	$-0.942085\pi$
$983$		−	11.3461i		−	0.361885i	0.983494	−	0.180942i	$-0.0579147\pi$
$984$	−11.2575			−0.358876
$985$	−2.74319			−0.0874052
$986$	24.5533			0.781937
$987$		0			0
$988$	−5.30491			−0.168772
$989$			31.7557i			1.00977i
$990$	0.226012	−	1.01621i	0.00718314	−	0.0322971i
$991$	−25.6080			−0.813464			−0.406732	−	0.913547i	$-0.633332\pi$
$991$	−25.6080			−0.813464			−0.406732	+	0.913547i	$0.633332\pi$
$992$	−1.97170			−0.0626015
$993$			16.9166i			0.536833i
$994$		0			0
$995$	1.15166			0.0365100
$996$			13.4283i			0.425492i
$997$	43.7341			1.38507			0.692536	−	0.721383i	$-0.256493\pi$
$997$	43.7341			1.38507			0.692536	+	0.721383i	$0.256493\pi$
$998$			2.16815i			0.0686316i
$999$		−	3.89552i		−	0.123249i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.e.c.2155.7	✓	24
7.6	odd	2		3234.2.e.d.2155.6	yes	24
11.10	odd	2		3234.2.e.d.2155.19	yes	24
77.76	even	2	inner	3234.2.e.c.2155.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.e.c.2155.7	✓	24	1.1	even	1	trivial
3234.2.e.c.2155.18	yes	24	77.76	even	2	inner
3234.2.e.d.2155.6	yes	24	7.6	odd	2
3234.2.e.d.2155.19	yes	24	11.10	odd	2

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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.313884i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.313884i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +0.313884 q^{10} +(-0.720050 + 3.23752i) q^{11} +1.00000i q^{12} +4.42216 q^{13} +0.313884 q^{15} +1.00000 q^{16} -3.88254 q^{17} +1.00000i q^{18} +1.19962 q^{19} -0.313884i q^{20} +(3.23752 + 0.720050i) q^{22} -7.80896 q^{23} +1.00000 q^{24} +4.90148 q^{25} -4.42216i q^{26} +1.00000i q^{27} -6.32404i q^{29} -0.313884i q^{30} -1.97170i q^{31} -1.00000i q^{32} +(3.23752 + 0.720050i) q^{33} +3.88254i q^{34} +1.00000 q^{36} -3.89552 q^{37} -1.19962i q^{38} -4.42216i q^{39} -0.313884 q^{40} -11.2575 q^{41} -4.06657i q^{43} +(0.720050 - 3.23752i) q^{44} -0.313884i q^{45} +7.80896i q^{46} +9.92277i q^{47} -1.00000i q^{48} -4.90148i q^{50} +3.88254i q^{51} -4.42216 q^{52} -11.7453 q^{53} +1.00000 q^{54} +(-1.01621 - 0.226012i) q^{55} -1.19962i q^{57} -6.32404 q^{58} +8.87397i q^{59} -0.313884 q^{60} -11.6955 q^{61} -1.97170 q^{62} -1.00000 q^{64} +1.38805i q^{65} +(0.720050 - 3.23752i) q^{66} -12.2415 q^{67} +3.88254 q^{68} +7.80896i q^{69} +10.0331 q^{71} -1.00000i q^{72} -4.85587 q^{73} +3.89552i q^{74} -4.90148i q^{75} -1.19962 q^{76} -4.42216 q^{78} -6.61990i q^{79} +0.313884i q^{80} +1.00000 q^{81} +11.2575i q^{82} +13.4283 q^{83} -1.21867i q^{85} -4.06657 q^{86} -6.32404 q^{87} +(-3.23752 - 0.720050i) q^{88} -11.7289i q^{89} -0.313884 q^{90} +7.80896 q^{92} -1.97170 q^{93} +9.92277 q^{94} +0.376541i q^{95} -1.00000 q^{96} +11.7519i q^{97} +(0.720050 - 3.23752i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^4 - 24 * q^6 - 24 * q^9	\( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} + 24 q^{16} + 16 q^{17} - 32 q^{19} - 8 q^{22} + 24 q^{24} - 8 q^{25} - 8 q^{33} + 24 q^{36} + 16 q^{37} - 16 q^{41} + 24 q^{54} + 16 q^{55} - 16 q^{62} - 24 q^{64} - 64 q^{67} - 16 q^{68} + 64 q^{71} + 32 q^{76} + 24 q^{81} - 16 q^{83} + 8 q^{88} - 16 q^{93} + 64 q^{94} - 24 q^{96}+O(q^{100}) \) 24 * q - 24 * q^4 - 24 * q^6 - 24 * q^9 + 24 * q^16 + 16 * q^17 - 32 * q^19 - 8 * q^22 + 24 * q^24 - 8 * q^25 - 8 * q^33 + 24 * q^36 + 16 * q^37 - 16 * q^41 + 24 * q^54 + 16 * q^55 - 16 * q^62 - 24 * q^64 - 64 * q^67 - 16 * q^68 + 64 * q^71 + 32 * q^76 + 24 * q^81 - 16 * q^83 + 8 * q^88 - 16 * q^93 + 64 * q^94 - 24 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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