LMFDB - Embedded newform 416.2.e.c.337.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(337,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4521217600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 $ x^8 + x^6 - 2x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$0.273147 + 1.38758i$ of defining polynomial
Character	$\chi$	$=$	416.337
Dual form			416.2.e.c.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51606i q^{3} -3.11473 q^{5} +2.77517i q^{7} +0.701562 q^{9} -2.56844 q^{11} +(0.546295 - 3.56393i) q^{13} -4.72212i q^{15} -5.70156 q^{17} -4.75362 q^{19} -4.20732 q^{21} -4.00000 q^{23} +4.70156 q^{25} +5.61179i q^{27} +7.12785i q^{29} -3.60338i q^{31} -3.89391i q^{33} -8.64391i q^{35} +4.20732 q^{37} +(5.40312 + 0.828216i) q^{39} +1.51606i q^{43} -2.18518 q^{45} +2.77517i q^{47} -0.701562 q^{49} -8.64391i q^{51} +6.06424i q^{53} +8.00000 q^{55} -7.20677i q^{57} +4.75362 q^{59} +1.94695i q^{63} +(-1.70156 + 11.1007i) q^{65} +12.0757 q^{67} -6.06424i q^{69} +6.66908i q^{71} +14.9946i q^{73} +7.12785i q^{75} -7.12785i q^{77} -14.8062 q^{79} -6.40312 q^{81} -9.89049 q^{83} +17.7588 q^{85} -10.8062 q^{87} -14.9946i q^{89} +(9.89049 + 1.51606i) q^{91} +5.46295 q^{93} +14.8062 q^{95} +3.89391i q^{97} -1.80192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{9} - 20 q^{17} - 32 q^{23} + 12 q^{25} - 8 q^{39} + 20 q^{49} + 64 q^{55} + 12 q^{65} - 16 q^{79} + 16 q^{87} + 16 q^{95}+O(q^{100}) $ 8 * q - 20 * q^9 - 20 * q^17 - 32 * q^23 + 12 * q^25 - 8 * q^39 + 20 * q^49 + 64 * q^55 + 12 * q^65 - 16 * q^79 + 16 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			1.51606i			0.875298i	0.899146	+	0.437649i	$0.144189\pi$
$3$			1.51606i			0.875298i	−0.899146	+	0.437649i	$0.855811\pi$
$4$		0			0
$5$	−3.11473			−1.39295			−0.696475	−	0.717581i	$-0.745250\pi$
$5$	−3.11473			−1.39295			−0.696475	+	0.717581i	$0.745250\pi$
$6$		0			0
$7$			2.77517i			1.04892i	0.851437	+	0.524458i	$0.175732\pi$
$7$			2.77517i			1.04892i	−0.851437	+	0.524458i	$0.824268\pi$
$8$		0			0
$9$	0.701562			0.233854
$10$		0			0
$11$	−2.56844			−0.774413			−0.387207	−	0.921993i	$-0.626560\pi$
$11$	−2.56844			−0.774413			−0.387207	+	0.921993i	$0.626560\pi$
$12$		0			0
$13$	0.546295	−	3.56393i	0.151515	−	0.988455i
$13$	0.546295	−	3.56393i	0.151515	−	0.988455i
$14$		0			0
$15$		−	4.72212i		−	1.21925i
$16$		0			0
$17$	−5.70156			−1.38283			−0.691416	−	0.722457i	$-0.743013\pi$
$17$	−5.70156			−1.38283			−0.691416	+	0.722457i	$0.743013\pi$
$18$		0			0
$19$	−4.75362			−1.09055			−0.545277	−	0.838256i	$-0.683576\pi$
$19$	−4.75362			−1.09055			−0.545277	+	0.838256i	$0.683576\pi$
$20$		0			0
$21$	−4.20732			−0.918113
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	4.70156			0.940312
$26$		0			0
$27$			5.61179i			1.07999i
$28$		0			0
$29$			7.12785i			1.32361i	0.749677	+	0.661804i	$0.230209\pi$
$29$			7.12785i			1.32361i	−0.749677	+	0.661804i	$0.769791\pi$
$30$		0			0
$31$		−	3.60338i		−	0.647187i	−0.946196	−	0.323593i	$-0.895109\pi$
$31$		−	3.60338i		−	0.647187i	0.946196	−	0.323593i	$-0.104891\pi$
$32$		0			0
$33$		−	3.89391i		−	0.677842i
$34$		0			0
$35$		−	8.64391i		−	1.46109i
$36$		0			0
$37$	4.20732			0.691680			0.345840	−	0.938294i	$-0.387594\pi$
$37$	4.20732			0.691680			0.345840	+	0.938294i	$0.387594\pi$
$38$		0			0
$39$	5.40312	+	0.828216i	0.865192	+	0.132621i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			1.51606i			0.231197i	0.993296	+	0.115598i	$0.0368786\pi$
$43$			1.51606i			0.231197i	−0.993296	+	0.115598i	$0.963121\pi$
$44$		0			0
$45$	−2.18518			−0.325747
$46$		0			0
$47$			2.77517i			0.404800i	0.979303	+	0.202400i	$0.0648741\pi$
$47$			2.77517i			0.404800i	−0.979303	+	0.202400i	$0.935126\pi$
$48$		0			0
$49$	−0.701562			−0.100223
$50$		0			0
$51$		−	8.64391i		−	1.21039i
$52$		0			0
$53$			6.06424i			0.832987i	0.909139	+	0.416494i	$0.136741\pi$
$53$			6.06424i			0.832987i	−0.909139	+	0.416494i	$0.863259\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		−	7.20677i		−	0.954560i
$58$		0			0
$59$	4.75362			0.618868			0.309434	−	0.950921i	$-0.399860\pi$
$59$	4.75362			0.618868			0.309434	+	0.950921i	$0.399860\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$			1.94695i			0.245293i
$64$		0			0
$65$	−1.70156	+	11.1007i	−0.211053	+	1.37687i
$66$		0			0
$67$	12.0757			1.47528			0.737639	−	0.675195i	$-0.235941\pi$
$67$	12.0757			1.47528			0.737639	+	0.675195i	$0.235941\pi$
$68$		0			0
$69$		−	6.06424i		−	0.730049i
$70$		0			0
$71$			6.66908i			0.791474i	0.918364	+	0.395737i	$0.129511\pi$
$71$			6.66908i			0.791474i	−0.918364	+	0.395737i	$0.870489\pi$
$72$		0			0
$73$			14.9946i			1.75498i	0.479592	+	0.877492i	$0.340785\pi$
$73$			14.9946i			1.75498i	−0.479592	+	0.877492i	$0.659215\pi$
$74$		0			0
$75$			7.12785i			0.823053i
$76$		0			0
$77$		−	7.12785i		−	0.812294i
$78$		0			0
$79$	−14.8062			−1.66583			−0.832917	−	0.553399i	$-0.813331\pi$
$79$	−14.8062			−1.66583			−0.832917	+	0.553399i	$0.813331\pi$
$80$		0			0
$81$	−6.40312			−0.711458
$82$		0			0
$83$	−9.89049			−1.08562			−0.542811	−	0.839855i	$-0.682640\pi$
$83$	−9.89049			−1.08562			−0.542811	+	0.839855i	$0.682640\pi$
$84$		0			0
$85$	17.7588			1.92622
$86$		0			0
$87$	−10.8062			−1.15855
$88$		0			0
$89$		−	14.9946i		−	1.58942i	−0.606988	−	0.794711i	$-0.707622\pi$
$89$		−	14.9946i		−	1.58942i	0.606988	−	0.794711i	$-0.292378\pi$
$90$		0			0
$91$	9.89049	+	1.51606i	1.03681	+	0.158926i
$92$		0			0
$93$	5.46295			0.566481
$94$		0			0
$95$	14.8062			1.51909
$96$		0			0
$97$			3.89391i			0.395366i	0.980266	+	0.197683i	$0.0633417\pi$
$97$			3.89391i			0.395366i	−0.980266	+	0.197683i	$0.936658\pi$
$98$		0			0
$99$	−1.80192			−0.181100
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	13.1047			1.27889
$106$		0			0
$107$			10.1600i			0.982201i	0.871103	+	0.491101i	$0.163405\pi$
$107$			10.1600i			0.982201i	−0.871103	+	0.491101i	$0.836595\pi$
$108$		0			0
$109$	2.02214			0.193686			0.0968431	−	0.995300i	$-0.469125\pi$
$109$	2.02214			0.193686			0.0968431	+	0.995300i	$0.469125\pi$
$110$		0			0
$111$			6.37855i			0.605425i
$112$		0			0
$113$	11.4031			1.07272			0.536358	−	0.843991i	$-0.319800\pi$
$113$	11.4031			1.07272			0.536358	+	0.843991i	$0.319800\pi$
$114$		0			0
$115$	12.4589			1.16180
$116$		0			0
$117$	0.383260	−	2.50031i	0.0354324	−	0.231154i
$118$		0			0
$119$		−	15.8228i		−	1.45047i
$120$		0			0
$121$	−4.40312			−0.400284
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.929554			0.0831419
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−2.29844			−0.202366
$130$		0			0
$131$		−	18.8039i		−	1.64290i	−0.570279	−	0.821451i	$-0.693165\pi$
$131$		−	18.8039i		−	1.64290i	0.570279	−	0.821451i	$-0.306835\pi$
$132$		0			0
$133$		−	13.1921i		−	1.14390i
$134$		0			0
$135$		−	17.4792i		−	1.50437i
$136$		0			0
$137$			11.1007i			0.948395i	0.880419	+	0.474197i	$0.157262\pi$
$137$			11.1007i			0.948395i	−0.880419	+	0.474197i	$0.842738\pi$
$138$		0			0
$139$			9.70752i			0.823381i	0.911324	+	0.411691i	$0.135062\pi$
$139$			9.70752i			0.823381i	−0.911324	+	0.411691i	$0.864938\pi$
$140$		0			0
$141$	−4.20732			−0.354320
$142$		0			0
$143$	−1.40312	+	9.15372i	−0.117335	+	0.765473i
$144$		0			0
$145$		−	22.2014i		−	1.84372i
$146$		0			0
$147$		−	1.06361i		−	0.0877251i
$148$		0			0
$149$	−13.5515			−1.11018			−0.555092	−	0.831789i	$-0.687317\pi$
$149$	−13.5515			−1.11018			−0.555092	+	0.831789i	$0.687317\pi$
$150$		0			0
$151$			13.8758i			1.12920i	0.825365	+	0.564600i	$0.190970\pi$
$151$			13.8758i			1.12920i	−0.825365	+	0.564600i	$0.809030\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$			11.2236i			0.901500i
$156$		0			0
$157$			1.06361i			0.0848853i	0.999099	+	0.0424427i	$0.0135140\pi$
$157$			1.06361i			0.0848853i	−0.999099	+	0.0424427i	$0.986486\pi$
$158$		0			0
$159$	−9.19375			−0.729112
$160$		0			0
$161$		−	11.1007i		−	0.874856i
$162$		0			0
$163$	−9.89049			−0.774683			−0.387342	−	0.921936i	$-0.626607\pi$
$163$	−9.89049			−0.774683			−0.387342	+	0.921936i	$0.626607\pi$
$164$		0			0
$165$			12.1285i			0.944201i
$166$		0			0
$167$		−	7.49729i		−	0.580158i	−0.957003	−	0.290079i	$-0.906318\pi$
$167$		−	7.49729i		−	0.580158i	0.957003	−	0.290079i	$-0.0936816\pi$
$168$		0			0
$169$	−12.4031	−	3.89391i	−0.954086	−	0.299531i
$170$		0			0
$171$	−3.33496			−0.255031
$172$		0			0
$173$			6.06424i			0.461056i	0.973066	+	0.230528i	$0.0740453\pi$
$173$			6.06424i			0.461056i	−0.973066	+	0.230528i	$0.925955\pi$
$174$		0			0
$175$			13.0476i			0.986308i
$176$		0			0
$177$			7.20677i			0.541694i
$178$		0			0
$179$		−	12.7396i		−	0.952205i	−0.879390	−	0.476103i	$-0.842049\pi$
$179$		−	12.7396i		−	0.952205i	0.879390	−	0.476103i	$-0.157951\pi$
$180$		0			0
$181$			13.1921i			0.980560i	0.871565	+	0.490280i	$0.163106\pi$
$181$			13.1921i			0.980560i	−0.871565	+	0.490280i	$0.836894\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.1047			−0.963476
$186$		0			0
$187$	14.6441			1.07088
$188$		0			0
$189$	−15.5737			−1.13282
$190$		0			0
$191$	2.80625			0.203053			0.101527	−	0.994833i	$-0.467627\pi$
$191$	2.80625			0.203053			0.101527	+	0.994833i	$0.467627\pi$
$192$		0			0
$193$			14.9946i			1.07933i	0.841879	+	0.539667i	$0.181450\pi$
$193$			14.9946i			1.07933i	−0.841879	+	0.539667i	$0.818550\pi$
$194$		0			0
$195$	−16.8293	−	2.57967i	−1.20517	−	0.184734i
$196$		0			0
$197$	4.20732			0.299759			0.149880	−	0.988704i	$-0.452111\pi$
$197$	4.20732			0.299759			0.149880	+	0.988704i	$0.452111\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			18.3074i			1.29131i
$202$		0			0
$203$	−19.7810			−1.38835
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.80625			−0.195048
$208$		0			0
$209$	12.2094			0.844540
$210$		0			0
$211$		−	18.8039i		−	1.29451i	−0.762272	−	0.647256i	$-0.775916\pi$
$211$		−	18.8039i		−	1.29451i	0.762272	−	0.647256i	$-0.224084\pi$
$212$		0			0
$213$	−10.1107			−0.692775
$214$		0			0
$215$		−	4.72212i		−	0.322046i
$216$		0			0
$217$	10.0000			0.678844
$218$		0			0
$219$	−22.7327			−1.53613
$220$		0			0
$221$	−3.11473	+	20.3199i	−0.209520	+	1.36687i
$222$		0			0
$223$		−	15.5323i		−	1.04012i	−0.854130	−	0.520059i	$-0.825910\pi$
$223$		−	15.5323i		−	1.04012i	0.854130	−	0.520059i	$-0.174090\pi$
$224$		0			0
$225$	3.29844			0.219896
$226$		0			0
$227$	−6.93880			−0.460544			−0.230272	−	0.973126i	$-0.573962\pi$
$227$	−6.93880			−0.460544			−0.230272	+	0.973126i	$0.573962\pi$
$228$		0			0
$229$	−7.48509			−0.494629			−0.247314	−	0.968935i	$-0.579548\pi$
$229$	−7.48509			−0.494629			−0.247314	+	0.968935i	$0.579548\pi$
$230$		0			0
$231$	10.8062			0.710999
$232$		0			0
$233$	−7.10469			−0.465443			−0.232722	−	0.972543i	$-0.574763\pi$
$233$	−7.10469			−0.465443			−0.232722	+	0.972543i	$0.574763\pi$
$234$		0			0
$235$		−	8.64391i		−	0.563867i
$236$		0			0
$237$		−	22.4472i		−	1.45810i
$238$		0			0
$239$		−	12.2194i		−	0.790408i	−0.918593	−	0.395204i	$-0.870674\pi$
$239$		−	12.2194i		−	0.790408i	0.918593	−	0.395204i	$-0.129326\pi$
$240$		0			0
$241$		−	7.20677i		−	0.464229i	−0.972688	−	0.232114i	$-0.925436\pi$
$241$		−	7.20677i		−	0.464229i	0.972688	−	0.232114i	$-0.0745644\pi$
$242$		0			0
$243$			7.12785i			0.457252i
$244$		0			0
$245$	2.18518			0.139606
$246$		0			0
$247$	−2.59688	+	16.9415i	−0.165235	+	1.07796i
$248$		0			0
$249$		−	14.9946i		−	0.950243i
$250$		0			0
$251$			24.4157i			1.54110i	0.637377	+	0.770552i	$0.280019\pi$
$251$			24.4157i			1.54110i	−0.637377	+	0.770552i	$0.719981\pi$
$252$		0			0
$253$	10.2738			0.645905
$254$		0			0
$255$			26.9235i			1.68601i
$256$		0			0
$257$	6.50781			0.405946			0.202973	−	0.979184i	$-0.434940\pi$
$257$	6.50781			0.405946			0.202973	+	0.979184i	$0.434940\pi$
$258$		0			0
$259$			11.6760i			0.725513i
$260$		0			0
$261$			5.00063i			0.309531i
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$		−	18.8885i		−	1.16031i
$266$		0			0
$267$	22.7327			1.39122
$268$		0			0
$269$		−	19.2563i		−	1.17408i	−0.809558	−	0.587040i	$-0.800293\pi$
$269$		−	19.2563i		−	1.17408i	0.809558	−	0.587040i	$-0.199707\pi$
$270$		0			0
$271$		−	27.2140i		−	1.65313i	−0.562839	−	0.826566i	$-0.690291\pi$
$271$		−	27.2140i		−	1.65313i	0.562839	−	0.826566i	$-0.309709\pi$
$272$		0			0
$273$	−2.29844	+	14.9946i	−0.139108	+	0.907513i
$274$		0			0
$275$	−12.0757			−0.728190
$276$		0			0
$277$		−	12.1285i		−	0.728730i	−0.931256	−	0.364365i	$-0.881286\pi$
$277$		−	12.1285i		−	0.728730i	0.931256	−	0.364365i	$-0.118714\pi$
$278$		0			0
$279$		−	2.52800i		−	0.151347i
$280$		0			0
$281$			7.20677i			0.429920i	0.976623	+	0.214960i	$0.0689621\pi$
$281$			7.20677i			0.429920i	−0.976623	+	0.214960i	$0.931038\pi$
$282$		0			0
$283$		−	4.09573i		−	0.243466i	−0.992563	−	0.121733i	$-0.961155\pi$
$283$		−	4.09573i		−	0.243466i	0.992563	−	0.121733i	$-0.0388451\pi$
$284$		0			0
$285$			22.4472i			1.32966i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.5078			0.912224
$290$		0			0
$291$	−5.90340			−0.346063
$292$		0			0
$293$	−29.4513			−1.72056			−0.860280	−	0.509821i	$-0.829712\pi$
$293$	−29.4513			−1.72056			−0.860280	+	0.509821i	$0.829712\pi$
$294$		0			0
$295$	−14.8062			−0.862053
$296$		0			0
$297$		−	14.4135i		−	0.836358i
$298$		0			0
$299$	−2.18518	+	14.2557i	−0.126372	+	0.824428i
$300$		0			0
$301$	−4.20732			−0.242506
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	29.6715			1.69344			0.846720	−	0.532038i	$-0.178574\pi$
$307$	29.6715			1.69344			0.846720	+	0.532038i	$0.178574\pi$
$308$		0			0
$309$		−	6.06424i		−	0.344983i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	−24.5078			−1.38526			−0.692632	−	0.721291i	$-0.743549\pi$
$313$	−24.5078			−1.38526			−0.692632	+	0.721291i	$0.743549\pi$
$314$		0			0
$315$		−	6.06424i		−	0.341681i
$316$		0			0
$317$	25.2439			1.41784			0.708920	−	0.705289i	$-0.249183\pi$
$317$	25.2439			1.41784			0.708920	+	0.705289i	$0.249183\pi$
$318$		0			0
$319$		−	18.3074i		−	1.02502i
$320$		0			0
$321$	−15.4031			−0.859719
$322$		0			0
$323$	27.1030			1.50805
$324$		0			0
$325$	2.56844	−	16.7560i	0.142471	−	0.929456i
$326$		0			0
$327$			3.06569i			0.169533i
$328$		0			0
$329$	−7.70156			−0.424601
$330$		0			0
$331$	5.52014			0.303414			0.151707	−	0.988425i	$-0.451523\pi$
$331$	5.52014			0.303414			0.151707	+	0.988425i	$0.451523\pi$
$332$		0			0
$333$	2.95170			0.161752
$334$		0			0
$335$	−37.6125			−2.05499
$336$		0			0
$337$	14.2984			0.778886			0.389443	−	0.921051i	$-0.372668\pi$
$337$	14.2984			0.778886			0.389443	+	0.921051i	$0.372668\pi$
$338$		0			0
$339$			17.2878i			0.938946i
$340$		0			0
$341$			9.25507i			0.501190i
$342$		0			0
$343$			17.4792i			0.943790i
$344$		0			0
$345$			18.8885i			1.01692i
$346$		0			0
$347$		−	6.67540i		−	0.358354i	−0.983817	−	0.179177i	$-0.942656\pi$
$347$		−	6.67540i		−	0.358354i	0.983817	−	0.179177i	$-0.0573435\pi$
$348$		0			0
$349$	3.44080			0.184182			0.0920910	−	0.995751i	$-0.470645\pi$
$349$	3.44080			0.184182			0.0920910	+	0.995751i	$0.470645\pi$
$350$		0			0
$351$	20.0000	+	3.06569i	1.06752	+	0.163634i
$352$		0			0
$353$		−	33.3020i		−	1.77249i	−0.463219	−	0.886244i	$-0.653306\pi$
$353$		−	33.3020i		−	1.77249i	0.463219	−	0.886244i	$-0.346694\pi$
$354$		0			0
$355$		−	20.7724i		−	1.10248i
$356$		0			0
$357$	23.9883			1.26960
$358$		0			0
$359$			18.5980i			0.981563i	0.871283	+	0.490782i	$0.163289\pi$
$359$			18.5980i			0.981563i	−0.871283	+	0.490782i	$0.836711\pi$
$360$		0			0
$361$	3.59688			0.189309
$362$		0			0
$363$		−	6.67540i		−	0.350368i
$364$		0			0
$365$		−	46.7041i		−	2.44461i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−16.8293			−0.873733
$372$		0			0
$373$			19.2563i			0.997055i	0.866874	+	0.498527i	$0.166126\pi$
$373$			19.2563i			0.997055i	−0.866874	+	0.498527i	$0.833874\pi$
$374$		0			0
$375$			1.40926i			0.0727739i
$376$		0			0
$377$	25.4031	+	3.89391i	1.30833	+	0.200546i
$378$		0			0
$379$	−4.75362			−0.244177			−0.122088	−	0.992519i	$-0.538959\pi$
$379$	−4.75362			−0.244177			−0.122088	+	0.992519i	$0.538959\pi$
$380$		0			0
$381$			18.1927i			0.932041i
$382$		0			0
$383$		−	1.11874i		−	0.0571648i	−0.999591	−	0.0285824i	$-0.990901\pi$
$383$		−	1.11874i		−	0.0571648i	0.999591	−	0.0285824i	$-0.00909930\pi$
$384$		0			0
$385$			22.2014i			1.13149i
$386$		0			0
$387$			1.06361i			0.0540663i
$388$		0			0
$389$		−	15.3193i		−	0.776720i	−0.921508	−	0.388360i	$-0.873042\pi$
$389$		−	15.3193i		−	0.776720i	0.921508	−	0.388360i	$-0.126958\pi$
$390$		0			0
$391$	22.8062			1.15336
$392$		0			0
$393$	28.5078			1.43803
$394$		0			0
$395$	46.1175			2.32042
$396$		0			0
$397$	6.22947			0.312648			0.156324	−	0.987706i	$-0.450036\pi$
$397$	6.22947			0.312648			0.156324	+	0.987706i	$0.450036\pi$
$398$		0			0
$399$	20.0000			1.00125
$400$		0			0
$401$			7.20677i			0.359889i	0.983677	+	0.179944i	$0.0575918\pi$
$401$			7.20677i			0.359889i	−0.983677	+	0.179944i	$0.942408\pi$
$402$		0			0
$403$	−12.8422	−	1.96851i	−0.639715	−	0.0980585i
$404$		0			0
$405$	19.9440			0.991026
$406$		0			0
$407$	−10.8062			−0.535646
$408$		0			0
$409$			33.3020i			1.64668i	0.567549	+	0.823340i	$0.307892\pi$
$409$			33.3020i			1.64668i	−0.567549	+	0.823340i	$0.692108\pi$
$410$		0			0
$411$	−16.8293			−0.830128
$412$		0			0
$413$			13.1921i			0.649140i
$414$		0			0
$415$	30.8062			1.51222
$416$		0			0
$417$	−14.7172			−0.720704
$418$		0			0
$419$			13.6445i			0.666579i	0.942824	+	0.333290i	$0.108159\pi$
$419$			13.6445i			0.666579i	−0.942824	+	0.333290i	$0.891841\pi$
$420$		0			0
$421$	36.4472			1.77633			0.888165	−	0.459525i	$-0.151980\pi$
$421$	36.4472			1.77633			0.888165	+	0.459525i	$0.151980\pi$
$422$		0			0
$423$			1.94695i			0.0946641i
$424$		0			0
$425$	−26.8062			−1.30029
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−13.8776	−	2.12722i	−0.670016	−	0.102703i
$430$		0			0
$431$		−	0.537693i		−	0.0258998i	−0.999916	−	0.0129499i	$-0.995878\pi$
$431$		−	0.537693i		−	0.0258998i	0.999916	−	0.0129499i	$-0.00412219\pi$
$432$		0			0
$433$	2.50781			0.120518			0.0602588	−	0.998183i	$-0.480807\pi$
$433$	2.50781			0.120518			0.0602588	+	0.998183i	$0.480807\pi$
$434$		0			0
$435$	33.6586			1.61381
$436$		0			0
$437$	19.0145			0.909585
$438$		0			0
$439$	14.8062			0.706664			0.353332	−	0.935498i	$-0.385049\pi$
$439$	14.8062			0.706664			0.353332	+	0.935498i	$0.385049\pi$
$440$		0			0
$441$	−0.492189			−0.0234376
$442$		0			0
$443$			1.51606i			0.0720302i	0.999351	+	0.0360151i	$0.0114664\pi$
$443$			1.51606i			0.0720302i	−0.999351	+	0.0360151i	$0.988534\pi$
$444$		0			0
$445$			46.7041i			2.21399i
$446$		0			0
$447$		−	20.5449i		−	0.971741i
$448$		0			0
$449$			11.6817i			0.551294i	0.961259	+	0.275647i	$0.0888922\pi$
$449$			11.6817i			0.551294i	−0.961259	+	0.275647i	$0.911108\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−21.0366			−0.988386
$454$		0			0
$455$	−30.8062	−	4.72212i	−1.44422	−	0.221377i
$456$		0			0
$457$		−	3.31286i		−	0.154969i	−0.996994	−	0.0774846i	$-0.975311\pi$
$457$		−	3.31286i		−	0.154969i	0.996994	−	0.0774846i	$-0.0246889\pi$
$458$		0			0
$459$		−	31.9960i		−	1.49344i
$460$		0			0
$461$	−8.25161			−0.384316			−0.192158	−	0.981364i	$-0.561549\pi$
$461$	−8.25161			−0.384316			−0.192158	+	0.981364i	$0.561549\pi$
$462$		0			0
$463$		−	11.3912i		−	0.529394i	−0.964332	−	0.264697i	$-0.914728\pi$
$463$		−	11.3912i		−	0.529394i	0.964332	−	0.264697i	$-0.0852719\pi$
$464$		0			0
$465$	−17.0156			−0.789081
$466$		0			0
$467$		−	16.2242i		−	0.750767i	−0.926870	−	0.375383i	$-0.877511\pi$
$467$		−	16.2242i		−	0.750767i	0.926870	−	0.375383i	$-0.122489\pi$
$468$		0			0
$469$			33.5120i			1.54744i
$470$		0			0
$471$	−1.61250			−0.0742999
$472$		0			0
$473$		−	3.89391i		−	0.179042i
$474$		0			0
$475$	−22.3494			−1.02546
$476$		0			0
$477$			4.25444i			0.194797i
$478$		0			0
$479$			28.8704i			1.31912i	0.751650	+	0.659562i	$0.229258\pi$
$479$			28.8704i			1.31912i	−0.751650	+	0.659562i	$0.770742\pi$
$480$		0			0
$481$	2.29844	−	14.9946i	0.104800	−	0.683694i
$482$		0			0
$483$	16.8293			0.765759
$484$		0			0
$485$		−	12.1285i		−	0.550726i
$486$		0			0
$487$			26.3858i			1.19565i	0.801625	+	0.597827i	$0.203969\pi$
$487$			26.3858i			1.19565i	−0.801625	+	0.597827i	$0.796031\pi$
$488$		0			0
$489$		−	14.9946i		−	0.678078i
$490$		0			0
$491$			3.64328i			0.164419i	0.996615	+	0.0822095i	$0.0261977\pi$
$491$			3.64328i			0.164419i	−0.996615	+	0.0822095i	$0.973802\pi$
$492$		0			0
$493$		−	40.6399i		−	1.83033i
$494$		0			0
$495$	5.61250			0.252263
$496$		0			0
$497$	−18.5078			−0.830189
$498$		0			0
$499$	−31.8567			−1.42610			−0.713050	−	0.701113i	$-0.752687\pi$
$499$	−31.8567			−1.42610			−0.713050	+	0.701113i	$0.752687\pi$
$500$		0			0
$501$	11.3663			0.507811
$502$		0			0
$503$	25.6125			1.14200			0.571002	−	0.820948i	$-0.306555\pi$
$503$	25.6125			1.14200			0.571002	+	0.820948i	$0.306555\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.90340	−	18.8039i	0.262179	−	0.835110i
$508$		0			0
$509$	−23.0588			−1.02206			−0.511031	−	0.859562i	$-0.670736\pi$
$509$	−23.0588			−1.02206			−0.511031	+	0.859562i	$0.670736\pi$
$510$		0			0
$511$	−41.6125			−1.84083
$512$		0			0
$513$		−	26.6763i		−	1.17779i
$514$		0			0
$515$	12.4589			0.549006
$516$		0			0
$517$		−	7.12785i		−	0.313482i
$518$		0			0
$519$	−9.19375			−0.403561
$520$		0			0
$521$	13.1047			0.574127			0.287063	−	0.957912i	$-0.407321\pi$
$521$	13.1047			0.574127			0.287063	+	0.957912i	$0.407321\pi$
$522$		0			0
$523$		−	4.09573i		−	0.179094i	−0.995983	−	0.0895469i	$-0.971458\pi$
$523$		−	4.09573i		−	0.179094i	0.995983	−	0.0895469i	$-0.0285419\pi$
$524$		0			0
$525$	−19.7810			−0.863313
$526$		0			0
$527$			20.5449i			0.894951i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	3.33496			0.144725
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	31.6456i		−	1.36816i
$536$		0			0
$537$	19.3141			0.833463
$538$		0			0
$539$	1.80192			0.0776141
$540$		0			0
$541$	−8.25161			−0.354764			−0.177382	−	0.984142i	$-0.556763\pi$
$541$	−8.25161			−0.354764			−0.177382	+	0.984142i	$0.556763\pi$
$542$		0			0
$543$	−20.0000			−0.858282
$544$		0			0
$545$	−6.29844			−0.269795
$546$		0			0
$547$			42.1559i			1.80246i	0.433343	+	0.901229i	$0.357334\pi$
$547$			42.1559i			1.80246i	−0.433343	+	0.901229i	$0.642666\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	33.8831i		−	1.44347i
$552$		0			0
$553$		−	41.0898i		−	1.74732i
$554$		0			0
$555$		−	19.8675i		−	0.843328i
$556$		0			0
$557$	12.2959			0.520994			0.260497	−	0.965475i	$-0.416114\pi$
$557$	12.2959			0.520994			0.260497	+	0.965475i	$0.416114\pi$
$558$		0			0
$559$	5.40312	+	0.828216i	0.228528	+	0.0350298i
$560$		0			0
$561$			22.2014i			0.937342i
$562$		0			0
$563$			27.9002i			1.17585i	0.808914	+	0.587927i	$0.200056\pi$
$563$			27.9002i			1.17585i	−0.808914	+	0.587927i	$0.799944\pi$
$564$		0			0
$565$	−35.5177			−1.49424
$566$		0			0
$567$		−	17.7698i		−	0.746259i
$568$		0			0
$569$	18.5078			0.775888			0.387944	−	0.921683i	$-0.373185\pi$
$569$	18.5078			0.775888			0.387944	+	0.921683i	$0.373185\pi$
$570$		0			0
$571$		−	18.8039i		−	0.786918i	−0.919342	−	0.393459i	$-0.871278\pi$
$571$		−	18.8039i		−	0.786918i	0.919342	−	0.393459i	$-0.128722\pi$
$572$		0			0
$573$			4.25444i			0.177732i
$574$		0			0
$575$	−18.8062			−0.784275
$576$		0			0
$577$			25.5142i			1.06217i	0.847318	+	0.531085i	$0.178216\pi$
$577$			25.5142i			1.06217i	−0.847318	+	0.531085i	$0.821784\pi$
$578$		0			0
$579$	−22.7327			−0.944738
$580$		0			0
$581$		−	27.4478i		−	1.13873i
$582$		0			0
$583$		−	15.5756i		−	0.645077i
$584$		0			0
$585$	−1.19375	+	7.78781i	−0.0493556	+	0.321986i
$586$		0			0
$587$	−15.0274			−0.620246			−0.310123	−	0.950696i	$-0.600370\pi$
$587$	−15.0274			−0.620246			−0.310123	+	0.950696i	$0.600370\pi$
$588$		0			0
$589$			17.1291i			0.705793i
$590$		0			0
$591$			6.37855i			0.262379i
$592$		0			0
$593$			7.78781i			0.319807i	0.987133	+	0.159904i	$0.0511183\pi$
$593$			7.78781i			0.319807i	−0.987133	+	0.159904i	$0.948882\pi$
$594$		0			0
$595$			49.2838i			2.02044i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$	−6.89531			−0.281266			−0.140633	−	0.990062i	$-0.544914\pi$
$601$	−6.89531			−0.281266			−0.140633	+	0.990062i	$0.544914\pi$
$602$		0			0
$603$	8.47183			0.345000
$604$		0			0
$605$	13.7146			0.557576
$606$		0			0
$607$	12.0000			0.487065			0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000			0.487065			0.243532	+	0.969893i	$0.421694\pi$
$608$		0			0
$609$		−	29.9892i		−	1.21522i
$610$		0			0
$611$	9.89049	+	1.51606i	0.400127	+	0.0613332i
$612$		0			0
$613$	−34.0990			−1.37725			−0.688623	−	0.725119i	$-0.741785\pi$
$613$	−34.0990			−1.37725			−0.688623	+	0.725119i	$0.741785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.5196i		−	0.423504i	−0.977323	−	0.211752i	$-0.932083\pi$
$617$		−	10.5196i		−	0.423504i	0.977323	−	0.211752i	$-0.0679170\pi$
$618$		0			0
$619$	−31.8567			−1.28043			−0.640214	−	0.768197i	$-0.721154\pi$
$619$	−31.8567			−1.28043			−0.640214	+	0.768197i	$0.721154\pi$
$620$		0			0
$621$		−	22.4472i		−	0.900774i
$622$		0			0
$623$	41.6125			1.66717
$624$		0			0
$625$	−26.4031			−1.05612
$626$		0			0
$627$			18.5101i			0.739224i
$628$		0			0
$629$	−23.9883			−0.956477
$630$		0			0
$631$		−	0.537693i		−	0.0214052i	−0.999943	−	0.0107026i	$-0.996593\pi$
$631$		−	0.537693i		−	0.0214052i	0.999943	−	0.0107026i	$-0.00340681\pi$
$632$		0			0
$633$	28.5078			1.13308
$634$		0			0
$635$	−37.3768			−1.48325
$636$		0			0
$637$	−0.383260	+	2.50031i	−0.0151853	+	0.0990661i
$638$		0			0
$639$			4.67877i			0.185089i
$640$		0			0
$641$	24.5969			0.971518			0.485759	−	0.874093i	$-0.338543\pi$
$641$	24.5969			0.971518			0.485759	+	0.874093i	$0.338543\pi$
$642$		0			0
$643$	−36.9935			−1.45888			−0.729441	−	0.684043i	$-0.760220\pi$
$643$	−36.9935			−1.45888			−0.729441	+	0.684043i	$0.760220\pi$
$644$		0			0
$645$	7.15902			0.281886
$646$		0			0
$647$	17.1938			0.675956			0.337978	−	0.941154i	$-0.390257\pi$
$647$	17.1938			0.675956			0.337978	+	0.941154i	$0.390257\pi$
$648$		0			0
$649$	−12.2094			−0.479260
$650$		0			0
$651$			15.1606i			0.594191i
$652$		0			0
$653$			19.2563i			0.753558i	0.926303	+	0.376779i	$0.122968\pi$
$653$			19.2563i			0.753558i	−0.926303	+	0.376779i	$0.877032\pi$
$654$		0			0
$655$			58.5691i			2.28848i
$656$		0			0
$657$			10.5196i			0.410410i
$658$		0			0
$659$			8.03275i			0.312912i	0.987685	+	0.156456i	$0.0500069\pi$
$659$			8.03275i			0.312912i	−0.987685	+	0.156456i	$0.949993\pi$
$660$		0			0
$661$	1.85911			0.0723109			0.0361555	−	0.999346i	$-0.488489\pi$
$661$	1.85911			0.0723109			0.0361555	+	0.999346i	$0.488489\pi$
$662$		0			0
$663$	−30.8062	−	4.72212i	−1.19642	−	0.183392i
$664$		0			0
$665$			41.0898i			1.59340i
$666$		0			0
$667$		−	28.5114i		−	1.10397i
$668$		0			0
$669$	23.5479			0.910413
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0890652			−0.00343321			−0.00171660	−	0.999999i	$-0.500546\pi$
$673$	−0.0890652			−0.00343321			−0.00171660	+	0.999999i	$0.500546\pi$
$674$		0			0
$675$			26.3842i			1.01553i
$676$		0			0
$677$			18.1927i			0.699203i	0.936898	+	0.349602i	$0.113683\pi$
$677$			18.1927i			0.699203i	−0.936898	+	0.349602i	$0.886317\pi$
$678$		0			0
$679$	−10.8062			−0.414706
$680$		0			0
$681$		−	10.5196i		−	0.403113i
$682$		0			0
$683$	15.7939			0.604336			0.302168	−	0.953255i	$-0.402290\pi$
$683$	15.7939			0.604336			0.302168	+	0.953255i	$0.402290\pi$
$684$		0			0
$685$		−	34.5756i		−	1.32107i
$686$		0			0
$687$		−	11.3478i		−	0.432947i
$688$		0			0
$689$	21.6125	+	3.31286i	0.823371	+	0.126210i
$690$		0			0
$691$	−2.56844			−0.0977080			−0.0488540	−	0.998806i	$-0.515557\pi$
$691$	−2.56844			−0.0977080			−0.0488540	+	0.998806i	$0.515557\pi$
$692$		0			0
$693$		−	5.00063i		−	0.189958i
$694$		0			0
$695$		−	30.2363i		−	1.14693i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	10.7711i		−	0.407402i
$700$		0			0
$701$		−	13.1921i		−	0.498258i	−0.968470	−	0.249129i	$-0.919856\pi$
$701$		−	13.1921i		−	0.498258i	0.968470	−	0.249129i	$-0.0801444\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$	13.1047			0.493551
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−39.2359			−1.47354			−0.736768	−	0.676146i	$-0.763649\pi$
$709$	−39.2359			−1.47354			−0.736768	+	0.676146i	$0.763649\pi$
$710$		0			0
$711$	−10.3875			−0.389562
$712$		0			0
$713$			14.4135i			0.539791i
$714$		0			0
$715$	4.37036	−	28.5114i	0.163442	−	1.06627i
$716$		0			0
$717$	18.5254			0.691842
$718$		0			0
$719$	5.19375			0.193694			0.0968471	−	0.995299i	$-0.469124\pi$
$719$	5.19375			0.193694			0.0968471	+	0.995299i	$0.469124\pi$
$720$		0			0
$721$		−	11.1007i		−	0.413411i
$722$		0			0
$723$	10.9259			0.406338
$724$		0			0
$725$			33.5120i			1.24461i
$726$		0			0
$727$	−42.8062			−1.58760			−0.793798	−	0.608182i	$-0.791899\pi$
$727$	−42.8062			−1.58760			−0.793798	+	0.608182i	$0.791899\pi$
$728$		0			0
$729$	−30.0156			−1.11169
$730$		0			0
$731$		−	8.64391i		−	0.319707i
$732$		0			0
$733$	−19.9440			−0.736649			−0.368325	−	0.929697i	$-0.620069\pi$
$733$	−19.9440			−0.736649			−0.368325	+	0.929697i	$0.620069\pi$
$734$		0			0
$735$			3.31286i			0.122197i
$736$		0			0
$737$	−31.0156			−1.14248
$738$		0			0
$739$	6.17228			0.227051			0.113525	−	0.993535i	$-0.463786\pi$
$739$	6.17228			0.227051			0.113525	+	0.993535i	$0.463786\pi$
$740$		0			0
$741$	−25.6844	−	3.93702i	−0.943539	−	0.144630i
$742$		0			0
$743$			39.9711i			1.46640i	0.680014	+	0.733199i	$0.261974\pi$
$743$			39.9711i			1.46640i	−0.680014	+	0.733199i	$0.738026\pi$
$744$		0			0
$745$	42.2094			1.54643
$746$		0			0
$747$	−6.93880			−0.253877
$748$		0			0
$749$	−28.1956			−1.03025
$750$		0			0
$751$	32.4187			1.18298			0.591488	−	0.806313i	$-0.298541\pi$
$751$	32.4187			1.18298			0.591488	+	0.806313i	$0.298541\pi$
$752$		0			0
$753$	−37.0156			−1.34892
$754$		0			0
$755$		−	43.2196i		−	1.57292i
$756$		0			0
$757$		−	34.5756i		−	1.25667i	−0.777942	−	0.628337i	$-0.783736\pi$
$757$		−	34.5756i		−	1.25667i	0.777942	−	0.628337i	$-0.216264\pi$
$758$		0			0
$759$			15.5756i			0.565359i
$760$		0			0
$761$		−	26.6763i		−	0.967015i	−0.875340	−	0.483508i	$-0.839363\pi$
$761$		−	26.6763i		−	0.967015i	0.875340	−	0.483508i	$-0.160637\pi$
$762$		0			0
$763$			5.61179i			0.203160i
$764$		0			0
$765$	12.4589			0.450454
$766$		0			0
$767$	2.59688	−	16.9415i	0.0937677	−	0.611723i
$768$		0			0
$769$			26.0953i			0.941019i	0.882395	+	0.470510i	$0.155930\pi$
$769$			26.0953i			0.941019i	−0.882395	+	0.470510i	$0.844070\pi$
$770$		0			0
$771$			9.86623i			0.355324i
$772$		0			0
$773$	42.3506			1.52325			0.761623	−	0.648020i	$-0.224403\pi$
$773$	42.3506			1.52325			0.761623	+	0.648020i	$0.224403\pi$
$774$		0			0
$775$		−	16.9415i		−	0.608558i
$776$		0			0
$777$	−17.7016			−0.635040
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	17.1291i		−	0.612928i
$782$		0			0
$783$	−40.0000			−1.42948
$784$		0			0
$785$		−	3.31286i		−	0.118241i
$786$		0			0
$787$	40.5974			1.44714			0.723570	−	0.690251i	$-0.242500\pi$
$787$	40.5974			1.44714			0.723570	+	0.690251i	$0.242500\pi$
$788$		0			0
$789$			24.2570i			0.863571i
$790$		0			0
$791$			31.6456i			1.12519i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	28.6361			1.01562
$796$		0			0
$797$			53.8320i			1.90683i	0.301668	+	0.953413i	$0.402457\pi$
$797$			53.8320i			1.90683i	−0.301668	+	0.953413i	$0.597543\pi$
$798$		0			0
$799$		−	15.8228i		−	0.559770i
$800$		0			0
$801$		−	10.5196i		−	0.371693i
$802$		0			0
$803$		−	38.5127i		−	1.35908i
$804$		0			0
$805$			34.5756i			1.21863i
$806$		0			0
$807$	29.1938			1.02767
$808$		0			0
$809$	−8.50781			−0.299119			−0.149559	−	0.988753i	$-0.547786\pi$
$809$	−8.50781			−0.299119			−0.149559	+	0.988753i	$0.547786\pi$
$810$		0			0
$811$	−31.0901			−1.09172			−0.545861	−	0.837876i	$-0.683797\pi$
$811$	−31.0901			−1.09172			−0.545861	+	0.837876i	$0.683797\pi$
$812$		0			0
$813$	41.2580			1.44698
$814$		0			0
$815$	30.8062			1.07910
$816$		0			0
$817$		−	7.20677i		−	0.252133i
$818$		0			0
$819$	6.93880	+	1.06361i	0.242461	+	0.0371656i
$820$		0			0
$821$	17.4328			0.608408			0.304204	−	0.952607i	$-0.401610\pi$
$821$	17.4328			0.608408			0.304204	+	0.952607i	$0.401610\pi$
$822$		0			0
$823$	25.6125			0.892796			0.446398	−	0.894835i	$-0.352707\pi$
$823$	25.6125			0.892796			0.446398	+	0.894835i	$0.352707\pi$
$824$		0			0
$825$		−	18.3074i		−	0.637383i
$826$		0			0
$827$	48.6860			1.69298			0.846488	−	0.532408i	$-0.178713\pi$
$827$	48.6860			1.69298			0.846488	+	0.532408i	$0.178713\pi$
$828$		0			0
$829$			16.3829i			0.569002i	0.958676	+	0.284501i	$0.0918280\pi$
$829$			16.3829i			0.569002i	−0.958676	+	0.284501i	$0.908172\pi$
$830$		0			0
$831$	18.3875			0.637855
$832$		0			0
$833$	4.00000			0.138592
$834$		0			0
$835$			23.3521i			0.808131i
$836$		0			0
$837$	20.2214			0.698955
$838$		0			0
$839$			18.5980i			0.642073i	0.947067	+	0.321037i	$0.104031\pi$
$839$			18.5980i			0.642073i	−0.947067	+	0.321037i	$0.895969\pi$
$840$		0			0
$841$	−21.8062			−0.751940
$842$		0			0
$843$	−10.9259			−0.376308
$844$		0			0
$845$	38.6324	+	12.1285i	1.32900	+	0.417232i
$846$		0			0
$847$		−	12.2194i		−	0.419864i
$848$		0			0
$849$	6.20937			0.213105
$850$		0			0
$851$	−16.8293			−0.576901
$852$		0			0
$853$	−28.0326			−0.959818			−0.479909	−	0.877318i	$-0.659330\pi$
$853$	−28.0326			−0.959818			−0.479909	+	0.877318i	$0.659330\pi$
$854$		0			0
$855$	10.3875			0.355245
$856$		0			0
$857$	12.8062			0.437453			0.218727	−	0.975786i	$-0.429810\pi$
$857$	12.8062			0.437453			0.218727	+	0.975786i	$0.429810\pi$
$858$		0			0
$859$		−	18.3514i		−	0.626143i	−0.949730	−	0.313071i	$-0.898642\pi$
$859$		−	18.3514i		−	0.626143i	0.949730	−	0.313071i	$-0.101358\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	22.7390i		−	0.774046i	−0.922070	−	0.387023i	$-0.873503\pi$
$863$		−	22.7390i		−	0.774046i	0.922070	−	0.387023i	$-0.126497\pi$
$864$		0			0
$865$		−	18.8885i		−	0.642228i
$866$		0			0
$867$			23.5108i			0.798468i
$868$		0			0
$869$	38.0289			1.29004
$870$		0			0
$871$	6.59688	−	43.0368i	0.223527	−	1.45825i
$872$		0			0
$873$			2.73182i			0.0924580i
$874$		0			0
$875$			2.57967i			0.0872088i
$876$		0			0
$877$	−43.3288			−1.46311			−0.731556	−	0.681782i	$-0.761205\pi$
$877$	−43.3288			−1.46311			−0.731556	+	0.681782i	$0.761205\pi$
$878$		0			0
$879$		−	44.6499i		−	1.50600i
$880$		0			0
$881$	35.7016			1.20282			0.601408	−	0.798942i	$-0.294607\pi$
$881$	35.7016			1.20282			0.601408	+	0.798942i	$0.294607\pi$
$882$		0			0
$883$			23.9632i			0.806427i	0.915106	+	0.403213i	$0.132107\pi$
$883$			23.9632i			0.806427i	−0.915106	+	0.403213i	$0.867893\pi$
$884$		0			0
$885$		−	22.4472i		−	0.754553i
$886$		0			0
$887$	51.2250			1.71997			0.859983	−	0.510322i	$-0.170474\pi$
$887$	51.2250			1.71997			0.859983	+	0.510322i	$0.170474\pi$
$888$		0			0
$889$			33.3020i			1.11691i
$890$		0			0
$891$	16.4460			0.550963
$892$		0			0
$893$		−	13.1921i		−	0.441456i
$894$		0			0
$895$			39.6806i			1.32638i
$896$		0			0
$897$	−21.6125	−	3.31286i	−0.721620	−	0.110613i
$898$		0			0
$899$	25.6844			0.856622
$900$		0			0
$901$		−	34.5756i		−	1.15188i
$902$		0			0
$903$		−	6.37855i		−	0.212265i
$904$		0			0
$905$		−	41.0898i		−	1.36587i
$906$		0			0
$907$		−	55.5067i		−	1.84307i	−0.388294	−	0.921536i	$-0.626935\pi$
$907$		−	55.5067i		−	1.84307i	0.388294	−	0.921536i	$-0.373065\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$	25.4031			0.840721
$914$		0			0
$915$		0			0
$916$		0			0
$917$	52.1839			1.72327
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$			44.9837i			1.48226i
$922$		0			0
$923$	23.7681	+	3.64328i	0.782336	+	0.119920i
$924$		0			0
$925$	19.7810			0.650395
$926$		0			0
$927$	−2.80625			−0.0921693
$928$		0			0
$929$			11.6817i			0.383265i	0.981467	+	0.191632i	$0.0613781\pi$
$929$			11.6817i			0.383265i	−0.981467	+	0.191632i	$0.938622\pi$
$930$		0			0
$931$	3.33496			0.109299
$932$		0			0
$933$		−	18.1927i		−	0.595603i
$934$		0			0
$935$	−45.6125			−1.49169
$936$		0			0
$937$	12.8062			0.418362			0.209181	−	0.977877i	$-0.432920\pi$
$937$	12.8062			0.418362			0.209181	+	0.977877i	$0.432920\pi$
$938$		0			0
$939$		−	37.1553i		−	1.21252i
$940$		0			0
$941$	−25.8474			−0.842602			−0.421301	−	0.906921i	$-0.638426\pi$
$941$	−25.8474			−0.842602			−0.421301	+	0.906921i	$0.638426\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	48.5078			1.57796
$946$		0			0
$947$	12.0757			0.392407			0.196203	−	0.980563i	$-0.437139\pi$
$947$	12.0757			0.392407			0.196203	+	0.980563i	$0.437139\pi$
$948$		0			0
$949$	53.4396	+	8.19146i	1.73472	+	0.265906i
$950$		0			0
$951$			38.2713i			1.24103i
$952$		0			0
$953$	−51.5234			−1.66901			−0.834504	−	0.551002i	$-0.814246\pi$
$953$	−51.5234			−1.66901			−0.834504	+	0.551002i	$0.814246\pi$
$954$		0			0
$955$	−8.74071			−0.282843
$956$		0			0
$957$	27.7552			0.897198
$958$		0			0
$959$	−30.8062			−0.994786
$960$		0			0
$961$	18.0156			0.581149
$962$		0			0
$963$			7.12785i			0.229692i
$964$		0			0
$965$		−	46.7041i		−	1.50346i
$966$		0			0
$967$			51.0718i			1.64236i	0.570671	+	0.821179i	$0.306683\pi$
$967$			51.0718i			1.64236i	−0.570671	+	0.821179i	$0.693317\pi$
$968$		0			0
$969$			41.0898i			1.32000i
$970$		0			0
$971$			30.0275i			0.963627i	0.876274	+	0.481814i	$0.160022\pi$
$971$			30.0275i			0.963627i	−0.876274	+	0.481814i	$0.839978\pi$
$972$		0			0
$973$	−26.9400			−0.863657
$974$		0			0
$975$	25.4031	+	3.89391i	0.813551	+	0.124705i
$976$		0			0
$977$			18.3074i			0.585707i	0.956157	+	0.292854i	$0.0946048\pi$
$977$			18.3074i			0.585707i	−0.956157	+	0.292854i	$0.905395\pi$
$978$		0			0
$979$			38.5127i			1.23087i
$980$		0			0
$981$	1.41866			0.0452943
$982$		0			0
$983$		−	29.9458i		−	0.955123i	−0.878598	−	0.477562i	$-0.841521\pi$
$983$		−	29.9458i		−	0.955123i	0.878598	−	0.477562i	$-0.158479\pi$
$984$		0			0
$985$	−13.1047			−0.417550
$986$		0			0
$987$		−	11.6760i		−	0.371652i
$988$		0			0
$989$		−	6.06424i		−	0.192832i
$990$		0			0
$991$	−41.6125			−1.32186			−0.660932	−	0.750446i	$-0.729839\pi$
$991$	−41.6125			−1.32186			−0.660932	+	0.750446i	$0.729839\pi$
$992$		0			0
$993$			8.36886i			0.265578i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			40.6399i			1.28708i	0.765413	+	0.643539i	$0.222535\pi$
$997$			40.6399i			1.28708i	−0.765413	+	0.643539i	$0.777465\pi$
$998$		0			0
$999$			23.6106i			0.747007i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.e.c.337.5		8
3.2	odd	2		3744.2.m.g.1585.8		8
4.3	odd	2		104.2.e.c.77.3	✓	8
8.3	odd	2		104.2.e.c.77.5	yes	8
8.5	even	2	inner	416.2.e.c.337.4		8
12.11	even	2		936.2.m.f.181.6		8
13.12	even	2	inner	416.2.e.c.337.6		8
24.5	odd	2		3744.2.m.g.1585.2		8
24.11	even	2		936.2.m.f.181.4		8
39.38	odd	2		3744.2.m.g.1585.1		8
52.51	odd	2		104.2.e.c.77.6	yes	8
104.51	odd	2		104.2.e.c.77.4	yes	8
104.77	even	2	inner	416.2.e.c.337.3		8
156.155	even	2		936.2.m.f.181.3		8
312.77	odd	2		3744.2.m.g.1585.7		8
312.155	even	2		936.2.m.f.181.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.e.c.77.3	✓	8	4.3	odd	2
104.2.e.c.77.4	yes	8	104.51	odd	2
104.2.e.c.77.5	yes	8	8.3	odd	2
104.2.e.c.77.6	yes	8	52.51	odd	2
416.2.e.c.337.3		8	104.77	even	2	inner
416.2.e.c.337.4		8	8.5	even	2	inner
416.2.e.c.337.5		8	1.1	even	1	trivial
416.2.e.c.337.6		8	13.12	even	2	inner
936.2.m.f.181.3		8	156.155	even	2
936.2.m.f.181.4		8	24.11	even	2
936.2.m.f.181.5		8	312.155	even	2
936.2.m.f.181.6		8	12.11	even	2
3744.2.m.g.1585.1		8	39.38	odd	2
3744.2.m.g.1585.2		8	24.5	odd	2
3744.2.m.g.1585.7		8	312.77	odd	2
3744.2.m.g.1585.8		8	3.2	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.51606i q^{3} -3.11473 q^{5} +2.77517i q^{7} +0.701562 q^{9} -2.56844 q^{11} +(0.546295 - 3.56393i) q^{13} -4.72212i q^{15} -5.70156 q^{17} -4.75362 q^{19} -4.20732 q^{21} -4.00000 q^{23} +4.70156 q^{25} +5.61179i q^{27} +7.12785i q^{29} -3.60338i q^{31} -3.89391i q^{33} -8.64391i q^{35} +4.20732 q^{37} +(5.40312 + 0.828216i) q^{39} +1.51606i q^{43} -2.18518 q^{45} +2.77517i q^{47} -0.701562 q^{49} -8.64391i q^{51} +6.06424i q^{53} +8.00000 q^{55} -7.20677i q^{57} +4.75362 q^{59} +1.94695i q^{63} +(-1.70156 + 11.1007i) q^{65} +12.0757 q^{67} -6.06424i q^{69} +6.66908i q^{71} +14.9946i q^{73} +7.12785i q^{75} -7.12785i q^{77} -14.8062 q^{79} -6.40312 q^{81} -9.89049 q^{83} +17.7588 q^{85} -10.8062 q^{87} -14.9946i q^{89} +(9.89049 + 1.51606i) q^{91} +5.46295 q^{93} +14.8062 q^{95} +3.89391i q^{97} -1.80192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{9} - 20 q^{17} - 32 q^{23} + 12 q^{25} - 8 q^{39} + 20 q^{49} + 64 q^{55} + 12 q^{65} - 16 q^{79} + 16 q^{87} + 16 q^{95}+O(q^{100}) \) 8 * q - 20 * q^9 - 20 * q^17 - 32 * q^23 + 12 * q^25 - 8 * q^39 + 20 * q^49 + 64 * q^55 + 12 * q^65 - 16 * q^79 + 16 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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