LMFDB - Embedded newform 3520.2.p.c.351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(351,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3520.351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.p (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:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.4
Root			$-0.760198 + 1.19252i$ of defining polynomial
Character	$\chi$	$=$	3520.351
Dual form			3520.2.p.c.351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76156 q^{3} +1.00000i q^{5} +2.02561 q^{7} +4.62620 q^{9} +O(q^{10})$	$q-2.76156 q^{3} +1.00000i q^{5} +2.02561 q^{7} +4.62620 q^{9} +(-2.62620 + 2.02561i) q^{11} -5.31965 q^{13} -2.76156i q^{15} +4.32540i q^{17} -7.34525i q^{19} -5.59383 q^{21} -1.52311i q^{23} -1.00000 q^{25} -4.49084 q^{27} +5.04546 q^{29} -4.49084i q^{31} +(7.25240 - 5.59383i) q^{33} +2.02561i q^{35} +7.14931i q^{37} +14.6905 q^{39} -0.548369i q^{41} -9.91923i q^{43} +4.62620i q^{45} +5.52311i q^{47} -2.89692 q^{49} -11.9448i q^{51} -2.37380i q^{53} +(-2.02561 - 2.62620i) q^{55} +20.2843i q^{57} -12.9817 q^{59} +3.05697 q^{61} +9.37086 q^{63} -5.31965i q^{65} +11.2524 q^{67} +4.20617i q^{69} -5.74324i q^{71} -0.720062i q^{73} +2.76156 q^{75} +(-5.31965 + 4.10308i) q^{77} -17.2274 q^{79} -1.47689 q^{81} +9.91923i q^{83} -4.32540 q^{85} -13.9333 q^{87} +5.14931 q^{89} -10.7755 q^{91} +12.4017i q^{93} +7.34525 q^{95} +1.25240 q^{97} +(-12.1493 + 9.37086i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} + 20 q^{9}+O(q^{10}) $ 12 * q - 8 * q^3 + 20 * q^9	$ 12 q - 8 q^{3} + 20 q^{9} + 4 q^{11} - 12 q^{25} - 8 q^{27} + 16 q^{33} - 20 q^{49} - 64 q^{59} + 64 q^{67} + 8 q^{75} - 68 q^{81} - 24 q^{89} - 8 q^{91} - 56 q^{97} - 60 q^{99}+O(q^{100}) $ 12 * q - 8 * q^3 + 20 * q^9 + 4 * q^11 - 12 * q^25 - 8 * q^27 + 16 * q^33 - 20 * q^49 - 64 * q^59 + 64 * q^67 + 8 * q^75 - 68 * q^81 - 24 * q^89 - 8 * q^91 - 56 * q^97 - 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$.

$n$	$321$	$1541$	$2751$	$2817$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.76156			−1.59439			−0.797193	−	0.603725i	$-0.793683\pi$
$3$	−2.76156			−1.59439			−0.797193	+	0.603725i	$0.793683\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.02561			0.765607			0.382804	−	0.923830i	$-0.374959\pi$
$7$	2.02561			0.765607			0.382804	+	0.923830i	$0.374959\pi$
$8$		0			0
$9$	4.62620			1.54207
$10$		0			0
$11$	−2.62620	+	2.02561i	−0.791829	+	0.610743i
$11$	−2.62620	+	2.02561i	−0.791829	+	0.610743i
$12$		0			0
$13$	−5.31965			−1.47540			−0.737702	−	0.675126i	$-0.764089\pi$
$13$	−5.31965			−1.47540			−0.737702	+	0.675126i	$0.764089\pi$
$14$		0			0
$15$		−	2.76156i		−	0.713031i
$16$		0			0
$17$			4.32540i			1.04906i	0.851391	+	0.524532i	$0.175760\pi$
$17$			4.32540i			1.04906i	−0.851391	+	0.524532i	$0.824240\pi$
$18$		0			0
$19$		−	7.34525i		−	1.68512i	−0.538605	−	0.842558i	$-0.681049\pi$
$19$		−	7.34525i		−	1.68512i	0.538605	−	0.842558i	$-0.318951\pi$
$20$		0			0
$21$	−5.59383			−1.22067
$22$		0			0
$23$		−	1.52311i		−	0.317591i	−0.987311	−	0.158796i	$-0.949239\pi$
$23$		−	1.52311i		−	0.317591i	0.987311	−	0.158796i	$-0.0507611\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.49084			−0.864262
$28$		0			0
$29$	5.04546			0.936919			0.468459	−	0.883485i	$-0.344809\pi$
$29$	5.04546			0.936919			0.468459	+	0.883485i	$0.344809\pi$
$30$		0			0
$31$		−	4.49084i		−	0.806578i	−0.915073	−	0.403289i	$-0.867867\pi$
$31$		−	4.49084i		−	0.806578i	0.915073	−	0.403289i	$-0.132133\pi$
$32$		0			0
$33$	7.25240	−	5.59383i	1.26248	−	0.973761i
$34$		0			0
$35$			2.02561i			0.342390i
$36$		0			0
$37$			7.14931i			1.17534i	0.809101	+	0.587670i	$0.199955\pi$
$37$			7.14931i			1.17534i	−0.809101	+	0.587670i	$0.800045\pi$
$38$		0			0
$39$	14.6905			2.35236
$40$		0			0
$41$		−	0.548369i		−	0.0856408i	−0.999083	−	0.0428204i	$-0.986366\pi$
$41$		−	0.548369i		−	0.0856408i	0.999083	−	0.0428204i	$-0.0136343\pi$
$42$		0			0
$43$		−	9.91923i		−	1.51267i	−0.654186	−	0.756334i	$-0.726988\pi$
$43$		−	9.91923i		−	1.51267i	0.654186	−	0.756334i	$-0.273012\pi$
$44$		0			0
$45$			4.62620i			0.689633i
$46$		0			0
$47$			5.52311i			0.805629i	0.915282	+	0.402815i	$0.131968\pi$
$47$			5.52311i			0.805629i	−0.915282	+	0.402815i	$0.868032\pi$
$48$		0			0
$49$	−2.89692			−0.413845
$50$		0			0
$51$		−	11.9448i		−	1.67261i
$52$		0			0
$53$		−	2.37380i		−	0.326067i	−0.986621	−	0.163033i	$-0.947872\pi$
$53$		−	2.37380i		−	0.326067i	0.986621	−	0.163033i	$-0.0521278\pi$
$54$		0			0
$55$	−2.02561	−	2.62620i	−0.273133	−	0.354116i
$56$		0			0
$57$			20.2843i			2.68673i
$58$		0			0
$59$	−12.9817			−1.69007			−0.845035	−	0.534711i	$-0.820421\pi$
$59$	−12.9817			−1.69007			−0.845035	+	0.534711i	$0.820421\pi$
$60$		0			0
$61$	3.05697			0.391405			0.195702	−	0.980663i	$-0.437301\pi$
$61$	3.05697			0.391405			0.195702	+	0.980663i	$0.437301\pi$
$62$		0			0
$63$	9.37086			1.18062
$64$		0			0
$65$		−	5.31965i		−	0.659821i
$66$		0			0
$67$	11.2524			1.37470			0.687349	−	0.726327i	$-0.258774\pi$
$67$	11.2524			1.37470			0.687349	+	0.726327i	$0.258774\pi$
$68$		0			0
$69$			4.20617i			0.506363i
$70$		0			0
$71$		−	5.74324i		−	0.681597i	−0.940136	−	0.340798i	$-0.889303\pi$
$71$		−	5.74324i		−	0.681597i	0.940136	−	0.340798i	$-0.110697\pi$
$72$		0			0
$73$		−	0.720062i		−	0.0842769i	−0.999112	−	0.0421385i	$-0.986583\pi$
$73$		−	0.720062i		−	0.0842769i	0.999112	−	0.0421385i	$-0.0134171\pi$
$74$		0			0
$75$	2.76156			0.318877
$76$		0			0
$77$	−5.31965	+	4.10308i	−0.606230	+	0.467590i
$78$		0			0
$79$	−17.2274			−1.93823			−0.969115	−	0.246609i	$-0.920684\pi$
$79$	−17.2274			−1.93823			−0.969115	+	0.246609i	$0.920684\pi$
$80$		0			0
$81$	−1.47689			−0.164098
$82$		0			0
$83$			9.91923i			1.08878i	0.838833	+	0.544388i	$0.183238\pi$
$83$			9.91923i			1.08878i	−0.838833	+	0.544388i	$0.816762\pi$
$84$		0			0
$85$	−4.32540			−0.469155
$86$		0			0
$87$	−13.9333			−1.49381
$88$		0			0
$89$	5.14931			0.545826			0.272913	−	0.962039i	$-0.412013\pi$
$89$	5.14931			0.545826			0.272913	+	0.962039i	$0.412013\pi$
$90$		0			0
$91$	−10.7755			−1.12958
$92$		0			0
$93$			12.4017i			1.28600i
$94$		0			0
$95$	7.34525			0.753607
$96$		0			0
$97$	1.25240			0.127162			0.0635808	−	0.997977i	$-0.479748\pi$
$97$	1.25240			0.127162			0.0635808	+	0.997977i	$0.479748\pi$
$98$		0			0
$99$	−12.1493	+	9.37086i	−1.22105	+	0.941807i
$100$		0			0
$101$	6.58808			0.655538			0.327769	−	0.944758i	$-0.393703\pi$
$101$	6.58808			0.655538			0.327769	+	0.944758i	$0.393703\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$		−	5.59383i		−	0.545902i
$106$		0			0
$107$			4.77128i			0.461257i	0.973042	+	0.230628i	$0.0740782\pi$
$107$			4.77128i			0.461257i	−0.973042	+	0.230628i	$0.925922\pi$
$108$		0			0
$109$	14.6905			1.40710			0.703548	−	0.710648i	$-0.251598\pi$
$109$	14.6905			1.40710			0.703548	+	0.710648i	$0.251598\pi$
$110$		0			0
$111$		−	19.7432i		−	1.87394i
$112$		0			0
$113$	−2.54144			−0.239078			−0.119539	−	0.992829i	$-0.538142\pi$
$113$	−2.54144			−0.239078			−0.119539	+	0.992829i	$0.538142\pi$
$114$		0			0
$115$	1.52311			0.142031
$116$		0			0
$117$	−24.6097			−2.27517
$118$		0			0
$119$			8.76156i			0.803171i
$120$		0			0
$121$	2.79383	−	10.6393i	0.253985	−	0.967208i
$122$		0			0
$123$			1.51435i			0.136544i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−1.81680			−0.161215			−0.0806075	−	0.996746i	$-0.525686\pi$
$127$	−1.81680			−0.161215			−0.0806075	+	0.996746i	$0.525686\pi$
$128$		0			0
$129$			27.3925i			2.41178i
$130$		0			0
$131$			8.44199i			0.737580i	0.929513	+	0.368790i	$0.120228\pi$
$131$			8.44199i			0.737580i	−0.929513	+	0.368790i	$0.879772\pi$
$132$		0			0
$133$		−	14.8786i		−	1.29014i
$134$		0			0
$135$		−	4.49084i		−	0.386510i
$136$		0			0
$137$	16.7110			1.42771			0.713857	−	0.700292i	$-0.246947\pi$
$137$	16.7110			1.42771			0.713857	+	0.700292i	$0.246947\pi$
$138$		0			0
$139$		−	4.05121i		−	0.343619i	−0.985130	−	0.171810i	$-0.945039\pi$
$139$		−	4.05121i		−	0.343619i	0.985130	−	0.171810i	$-0.0549614\pi$
$140$		0			0
$141$		−	15.2524i		−	1.28448i
$142$		0			0
$143$	13.9704	−	10.7755i	1.16827	−	0.901093i
$144$		0			0
$145$			5.04546i			0.419003i
$146$		0			0
$147$	8.00000			0.659829
$148$		0			0
$149$	−0.994247			−0.0814519			−0.0407259	−	0.999170i	$-0.512967\pi$
$149$	−0.994247			−0.0814519			−0.0407259	+	0.999170i	$0.512967\pi$
$150$		0			0
$151$	19.8385			1.61443			0.807215	−	0.590257i	$-0.200974\pi$
$151$	19.8385			1.61443			0.807215	+	0.590257i	$0.200974\pi$
$152$		0			0
$153$			20.0101i			1.61772i
$154$		0			0
$155$	4.49084			0.360713
$156$		0			0
$157$		−	5.08476i		−	0.405808i	−0.979199	−	0.202904i	$-0.934962\pi$
$157$		−	5.08476i		−	0.405808i	0.979199	−	0.202904i	$-0.0650380\pi$
$158$		0			0
$159$			6.55539i			0.519876i
$160$		0			0
$161$		−	3.08523i		−	0.243150i
$162$		0			0
$163$	17.4725			1.36855			0.684277	−	0.729223i	$-0.260118\pi$
$163$	17.4725			1.36855			0.684277	+	0.729223i	$0.260118\pi$
$164$		0			0
$165$	5.59383	+	7.25240i	0.435479	+	0.564598i
$166$		0			0
$167$	16.1677			1.25110			0.625549	−	0.780185i	$-0.284875\pi$
$167$	16.1677			1.25110			0.625549	+	0.780185i	$0.284875\pi$
$168$		0			0
$169$	15.2986			1.17682
$170$		0			0
$171$		−	33.9806i		−	2.59856i
$172$		0			0
$173$	−22.0729			−1.67817			−0.839085	−	0.544001i	$-0.816909\pi$
$173$	−22.0729			−1.67817			−0.839085	+	0.544001i	$0.816909\pi$
$174$		0			0
$175$	−2.02561			−0.153121
$176$		0			0
$177$	35.8496			2.69462
$178$		0			0
$179$	17.2803			1.29159			0.645795	−	0.763511i	$-0.276526\pi$
$179$	17.2803			1.29159			0.645795	+	0.763511i	$0.276526\pi$
$180$		0			0
$181$			9.79383i			0.727970i	0.931405	+	0.363985i	$0.118584\pi$
$181$			9.79383i			0.727970i	−0.931405	+	0.363985i	$0.881416\pi$
$182$		0			0
$183$	−8.44199			−0.624050
$184$		0			0
$185$	−7.14931			−0.525628
$186$		0			0
$187$	−8.76156	−	11.3594i	−0.640709	−	0.830678i
$188$		0			0
$189$	−9.09667			−0.661686
$190$		0			0
$191$		−	24.8034i		−	1.79471i	−0.441307	−	0.897356i	$-0.645485\pi$
$191$		−	24.8034i		−	1.79471i	0.441307	−	0.897356i	$-0.354515\pi$
$192$		0			0
$193$			14.4163i			1.03771i	0.854862	+	0.518855i	$0.173641\pi$
$193$			14.4163i			1.03771i	−0.854862	+	0.518855i	$0.826359\pi$
$194$		0			0
$195$			14.6905i			1.05201i
$196$		0			0
$197$	14.5188			1.03442			0.517211	−	0.855858i	$-0.326970\pi$
$197$	14.5188			1.03442			0.517211	+	0.855858i	$0.326970\pi$
$198$		0			0
$199$		−	24.5833i		−	1.74266i	−0.490694	−	0.871332i	$-0.663257\pi$
$199$		−	24.5833i		−	1.74266i	0.490694	−	0.871332i	$-0.336743\pi$
$200$		0			0
$201$	−31.0741			−2.19180
$202$		0			0
$203$	10.2201			0.717312
$204$		0			0
$205$	0.548369			0.0382997
$206$		0			0
$207$		−	7.04623i		−	0.489747i
$208$		0			0
$209$	14.8786	+	19.2901i	1.02917	+	1.33432i
$210$		0			0
$211$			25.1210i			1.72940i	0.502289	+	0.864700i	$0.332492\pi$
$211$			25.1210i			1.72940i	−0.502289	+	0.864700i	$0.667508\pi$
$212$		0			0
$213$			15.8603i			1.08673i
$214$		0			0
$215$	9.91923			0.676486
$216$		0			0
$217$		−	9.09667i		−	0.617523i
$218$		0			0
$219$			1.98849i			0.134370i
$220$		0			0
$221$		−	23.0096i		−	1.54779i
$222$		0			0
$223$			7.58767i			0.508108i	0.967190	+	0.254054i	$0.0817640\pi$
$223$			7.58767i			0.508108i	−0.967190	+	0.254054i	$0.918236\pi$
$224$		0			0
$225$	−4.62620			−0.308413
$226$		0			0
$227$			20.5585i			1.36452i	0.731111	+	0.682258i	$0.239002\pi$
$227$			20.5585i			1.36452i	−0.731111	+	0.682258i	$0.760998\pi$
$228$		0			0
$229$		−	25.7572i		−	1.70208i	−0.525098	−	0.851041i	$-0.675972\pi$
$229$		−	25.7572i		−	1.70208i	0.525098	−	0.851041i	$-0.324028\pi$
$230$		0			0
$231$	14.6905	−	11.3309i	0.966564	−	0.745519i
$232$		0			0
$233$		−	2.26268i		−	0.148233i	−0.997250	−	0.0741165i	$-0.976386\pi$
$233$		−	2.26268i		−	0.148233i	0.997250	−	0.0741165i	$-0.0236137\pi$
$234$		0			0
$235$	−5.52311			−0.360288
$236$		0			0
$237$	47.5744			3.09029
$238$		0			0
$239$	2.95448			0.191109			0.0955546	−	0.995424i	$-0.469538\pi$
$239$	2.95448			0.191109			0.0955546	+	0.995424i	$0.469538\pi$
$240$		0			0
$241$			7.55406i			0.486600i	0.969951	+	0.243300i	$0.0782299\pi$
$241$			7.55406i			0.486600i	−0.969951	+	0.243300i	$0.921770\pi$
$242$		0			0
$243$	17.5510			1.12590
$244$		0			0
$245$		−	2.89692i		−	0.185077i
$246$		0			0
$247$			39.0741i			2.48623i
$248$		0			0
$249$		−	27.3925i		−	1.73593i
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$	3.08523	+	4.00000i	0.193967	+	0.251478i
$254$		0			0
$255$	11.9448			0.748015
$256$		0			0
$257$	19.9634			1.24528			0.622640	−	0.782508i	$-0.286060\pi$
$257$	19.9634			1.24528			0.622640	+	0.782508i	$0.286060\pi$
$258$		0			0
$259$			14.4817i			0.899849i
$260$		0			0
$261$	23.3413			1.44479
$262$		0			0
$263$	3.53996			0.218283			0.109142	−	0.994026i	$-0.465190\pi$
$263$	3.53996			0.218283			0.109142	+	0.994026i	$0.465190\pi$
$264$		0			0
$265$	2.37380			0.145821
$266$		0			0
$267$	−14.2201			−0.870257
$268$		0			0
$269$		−	11.0462i		−	0.673500i	−0.941594	−	0.336750i	$-0.890672\pi$
$269$		−	11.0462i		−	0.673500i	0.941594	−	0.336750i	$-0.109328\pi$
$270$		0			0
$271$	26.4265			1.60530			0.802649	−	0.596452i	$-0.203423\pi$
$271$	26.4265			1.60530			0.802649	+	0.596452i	$0.203423\pi$
$272$		0			0
$273$	29.7572			1.80099
$274$		0			0
$275$	2.62620	−	2.02561i	0.158366	−	0.122149i
$276$		0			0
$277$	5.86801			0.352575			0.176287	−	0.984339i	$-0.443591\pi$
$277$	5.86801			0.352575			0.176287	+	0.984339i	$0.443591\pi$
$278$		0			0
$279$		−	20.7755i		−	1.24380i
$280$		0			0
$281$		−	10.6393i		−	0.634687i	−0.948311	−	0.317343i	$-0.897209\pi$
$281$		−	10.6393i		−	0.634687i	0.948311	−	0.317343i	$-0.102791\pi$
$282$		0			0
$283$			7.38237i			0.438836i	0.975631	+	0.219418i	$0.0704159\pi$
$283$			7.38237i			0.438836i	−0.975631	+	0.219418i	$0.929584\pi$
$284$		0			0
$285$	−20.2843			−1.20154
$286$		0			0
$287$		−	1.11078i		−	0.0655672i
$288$		0			0
$289$	−1.70907			−0.100534
$290$		0			0
$291$	−3.45856			−0.202745
$292$		0			0
$293$	27.6950			1.61796			0.808979	−	0.587838i	$-0.200021\pi$
$293$	27.6950			1.61796			0.808979	+	0.587838i	$0.200021\pi$
$294$		0			0
$295$		−	12.9817i		−	0.755823i
$296$		0			0
$297$	11.7938	−	9.09667i	0.684348	−	0.527843i
$298$		0			0
$299$			8.10243i			0.468576i
$300$		0			0
$301$		−	20.0925i		−	1.15811i
$302$		0			0
$303$	−18.1933			−1.04518
$304$		0			0
$305$			3.05697i			0.175041i
$306$		0			0
$307$		−	6.28563i		−	0.358740i	−0.983782	−	0.179370i	$-0.942594\pi$
$307$		−	6.28563i		−	0.358740i	0.983782	−	0.179370i	$-0.0574059\pi$
$308$		0			0
$309$			11.0462i			0.628398i
$310$		0			0
$311$			4.34922i			0.246622i	0.992368	+	0.123311i	$0.0393512\pi$
$311$			4.34922i			0.246622i	−0.992368	+	0.123311i	$0.960649\pi$
$312$		0			0
$313$	−23.7572			−1.34284			−0.671418	−	0.741079i	$-0.734314\pi$
$313$	−23.7572			−1.34284			−0.671418	+	0.741079i	$0.734314\pi$
$314$		0			0
$315$			9.37086i			0.527988i
$316$		0			0
$317$		−	35.1772i		−	1.97575i	−0.155253	−	0.987875i	$-0.549619\pi$
$317$		−	35.1772i		−	1.97575i	0.155253	−	0.987875i	$-0.450381\pi$
$318$		0			0
$319$	−13.2504	+	10.2201i	−0.741879	+	0.572217i
$320$		0			0
$321$		−	13.1762i		−	0.735421i
$322$		0			0
$323$	31.7711			1.76779
$324$		0			0
$325$	5.31965			0.295081
$326$		0			0
$327$	−40.5687			−2.24345
$328$		0			0
$329$			11.1877i			0.616796i
$330$		0			0
$331$	−7.72928			−0.424840			−0.212420	−	0.977178i	$-0.568134\pi$
$331$	−7.72928			−0.424840			−0.212420	+	0.977178i	$0.568134\pi$
$332$		0			0
$333$			33.0741i			1.81245i
$334$		0			0
$335$			11.2524i			0.614784i
$336$		0			0
$337$			33.7064i			1.83610i	0.396459	+	0.918052i	$0.370239\pi$
$337$			33.7064i			1.83610i	−0.396459	+	0.918052i	$0.629761\pi$
$338$		0			0
$339$	7.01832			0.381183
$340$		0			0
$341$	9.09667	+	11.7938i	0.492613	+	0.638672i
$342$		0			0
$343$	−20.0473			−1.08245
$344$		0			0
$345$	−4.20617			−0.226452
$346$		0			0
$347$		−	8.89672i		−	0.477601i	−0.971069	−	0.238800i	$-0.923246\pi$
$347$		−	8.89672i		−	0.477601i	0.971069	−	0.238800i	$-0.0767542\pi$
$348$		0			0
$349$	5.14795			0.275564			0.137782	−	0.990463i	$-0.456003\pi$
$349$	5.14795			0.275564			0.137782	+	0.990463i	$0.456003\pi$
$350$		0			0
$351$	23.8897			1.27514
$352$		0			0
$353$	27.8863			1.48424			0.742119	−	0.670268i	$-0.233821\pi$
$353$	27.8863			1.48424			0.742119	+	0.670268i	$0.233821\pi$
$354$		0			0
$355$	5.74324			0.304819
$356$		0			0
$357$		−	24.1955i		−	1.28056i
$358$		0			0
$359$	−1.44012			−0.0760069			−0.0380034	−	0.999278i	$-0.512100\pi$
$359$	−1.44012			−0.0760069			−0.0380034	+	0.999278i	$0.512100\pi$
$360$		0			0
$361$	−34.9527			−1.83962
$362$		0			0
$363$	−7.71533	+	29.3810i	−0.404950	+	1.54210i
$364$		0			0
$365$	0.720062			0.0376898
$366$		0			0
$367$		−	1.49521i		−	0.0780492i	−0.999238	−	0.0390246i	$-0.987575\pi$
$367$		−	1.49521i		−	0.0780492i	0.999238	−	0.0390246i	$-0.0124251\pi$
$368$		0			0
$369$		−	2.53686i		−	0.132064i
$370$		0			0
$371$		−	4.80839i		−	0.249639i
$372$		0			0
$373$	−6.41638			−0.332228			−0.166114	−	0.986107i	$-0.553122\pi$
$373$	−6.41638			−0.332228			−0.166114	+	0.986107i	$0.553122\pi$
$374$		0			0
$375$			2.76156i			0.142606i
$376$		0			0
$377$	−26.8401			−1.38233
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$	5.01720			0.257039
$382$		0			0
$383$			12.5414i			0.640837i	0.947276	+	0.320419i	$0.103824\pi$
$383$			12.5414i			0.640837i	−0.947276	+	0.320419i	$0.896176\pi$
$384$		0			0
$385$	−4.10308	−	5.31965i	−0.209113	−	0.271114i
$386$		0			0
$387$		−	45.8883i		−	2.33263i
$388$		0			0
$389$		−	0.412335i		−	0.0209062i	−0.999945	−	0.0104531i	$-0.996673\pi$
$389$		−	0.412335i		−	0.0209062i	0.999945	−	0.0104531i	$-0.00332739\pi$
$390$		0			0
$391$	6.58808			0.333173
$392$		0			0
$393$		−	23.3130i		−	1.17599i
$394$		0			0
$395$		−	17.2274i		−	0.866803i
$396$		0			0
$397$		−	8.50479i		−	0.426843i	−0.976960	−	0.213422i	$-0.931539\pi$
$397$		−	8.50479i		−	0.426843i	0.976960	−	0.213422i	$-0.0684608\pi$
$398$		0			0
$399$			41.0881i			2.05698i
$400$		0			0
$401$	−18.4663			−0.922161			−0.461080	−	0.887358i	$-0.652538\pi$
$401$	−18.4663			−0.922161			−0.461080	+	0.887358i	$0.652538\pi$
$402$		0			0
$403$			23.8897i			1.19003i
$404$		0			0
$405$		−	1.47689i		−	0.0733870i
$406$		0			0
$407$	−14.4817	−	18.7755i	−0.717831	−	0.930668i
$408$		0			0
$409$		−	25.4040i		−	1.25615i	−0.778154	−	0.628074i	$-0.783844\pi$
$409$		−	25.4040i		−	1.25615i	0.778154	−	0.628074i	$-0.216156\pi$
$410$		0			0
$411$	−46.1483			−2.27633
$412$		0			0
$413$	−26.2958			−1.29393
$414$		0			0
$415$	−9.91923			−0.486916
$416$		0			0
$417$			11.1877i			0.547862i
$418$		0			0
$419$	−1.79383			−0.0876345			−0.0438172	−	0.999040i	$-0.513952\pi$
$419$	−1.79383			−0.0876345			−0.0438172	+	0.999040i	$0.513952\pi$
$420$		0			0
$421$		−	29.3082i		−	1.42839i	−0.699944	−	0.714197i	$-0.746792\pi$
$421$		−	29.3082i		−	1.42839i	0.699944	−	0.714197i	$-0.253208\pi$
$422$		0			0
$423$			25.5510i			1.24233i
$424$		0			0
$425$		−	4.32540i		−	0.209813i
$426$		0			0
$427$	6.19221			0.299662
$428$		0			0
$429$	−38.5802	+	29.7572i	−1.86267	+	1.43669i
$430$		0			0
$431$	13.1762			0.634673			0.317336	−	0.948313i	$-0.397212\pi$
$431$	13.1762			0.634673			0.317336	+	0.948313i	$0.397212\pi$
$432$		0			0
$433$	−10.9538			−0.526405			−0.263202	−	0.964741i	$-0.584779\pi$
$433$	−10.9538			−0.526405			−0.263202	+	0.964741i	$0.584779\pi$
$434$		0			0
$435$		−	13.9333i		−	0.668052i
$436$		0			0
$437$	−11.1877			−0.535178
$438$		0			0
$439$	−10.7135			−0.511328			−0.255664	−	0.966766i	$-0.582294\pi$
$439$	−10.7135			−0.511328			−0.255664	+	0.966766i	$0.582294\pi$
$440$		0			0
$441$	−13.4017			−0.638177
$442$		0			0
$443$	−5.70138			−0.270881			−0.135440	−	0.990786i	$-0.543245\pi$
$443$	−5.70138			−0.270881			−0.135440	+	0.990786i	$0.543245\pi$
$444$		0			0
$445$			5.14931i			0.244101i
$446$		0			0
$447$	2.74567			0.129866
$448$		0			0
$449$	28.8034			1.35932			0.679659	−	0.733529i	$-0.262128\pi$
$449$	28.8034			1.35932			0.679659	+	0.733529i	$0.262128\pi$
$450$		0			0
$451$	1.11078	+	1.44012i	0.0523046	+	0.0678128i
$452$		0			0
$453$	−54.7850			−2.57403
$454$		0			0
$455$		−	10.7755i		−	0.505164i
$456$		0			0
$457$			9.39912i			0.439672i	0.975537	+	0.219836i	$0.0705523\pi$
$457$			9.39912i			0.439672i	−0.975537	+	0.219836i	$0.929448\pi$
$458$		0			0
$459$		−	19.4247i		−	0.906666i
$460$		0			0
$461$	18.2958			0.852122			0.426061	−	0.904694i	$-0.359901\pi$
$461$	18.2958			0.852122			0.426061	+	0.904694i	$0.359901\pi$
$462$		0			0
$463$		−	31.6156i		−	1.46930i	−0.678446	−	0.734650i	$-0.737346\pi$
$463$		−	31.6156i		−	1.46930i	0.678446	−	0.734650i	$-0.262654\pi$
$464$		0			0
$465$	−12.4017			−0.575115
$466$		0			0
$467$	−35.1526			−1.62667			−0.813335	−	0.581796i	$-0.802350\pi$
$467$	−35.1526			−1.62667			−0.813335	+	0.581796i	$0.802350\pi$
$468$		0			0
$469$	22.7929			1.05248
$470$		0			0
$471$			14.0419i			0.647015i
$472$		0			0
$473$	20.0925	+	26.0499i	0.923852	+	1.19777i
$474$		0			0
$475$			7.34525i			0.337023i
$476$		0			0
$477$		−	10.9817i		−	0.502816i
$478$		0			0
$479$	−30.4035			−1.38917			−0.694586	−	0.719410i	$-0.744412\pi$
$479$	−30.4035			−1.38917			−0.694586	+	0.719410i	$0.744412\pi$
$480$		0			0
$481$		−	38.0318i		−	1.73410i
$482$		0			0
$483$			8.52004i			0.387675i
$484$		0			0
$485$			1.25240i			0.0568684i
$486$		0			0
$487$			19.6156i			0.888866i	0.895812	+	0.444433i	$0.146595\pi$
$487$			19.6156i			0.888866i	−0.895812	+	0.444433i	$0.853405\pi$
$488$		0			0
$489$	−48.2514			−2.18200
$490$		0			0
$491$		−	32.6750i		−	1.47460i	−0.675563	−	0.737302i	$-0.736099\pi$
$491$		−	32.6750i		−	1.47460i	0.675563	−	0.737302i	$-0.263901\pi$
$492$		0			0
$493$			21.8236i			0.982887i
$494$		0			0
$495$	−9.37086	−	12.1493i	−0.421189	−	0.546071i
$496$		0			0
$497$		−	11.6335i		−	0.521835i
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$		0			0
$501$	−44.6481			−1.99473
$502$		0			0
$503$	−19.4618			−0.867758			−0.433879	−	0.900971i	$-0.642855\pi$
$503$	−19.4618			−0.867758			−0.433879	+	0.900971i	$0.642855\pi$
$504$		0			0
$505$			6.58808i			0.293166i
$506$		0			0
$507$	−42.2480			−1.87630
$508$		0			0
$509$			26.0925i			1.15653i	0.815850	+	0.578264i	$0.196270\pi$
$509$			26.0925i			1.15653i	−0.815850	+	0.578264i	$0.803730\pi$
$510$		0			0
$511$		−	1.45856i		−	0.0645231i
$512$		0			0
$513$			32.9863i			1.45638i
$514$		0			0
$515$	4.00000			0.176261
$516$		0			0
$517$	−11.1877	−	14.5048i	−0.492033	−	0.637920i
$518$		0			0
$519$	60.9555			2.67565
$520$		0			0
$521$	15.3815			0.673876			0.336938	−	0.941527i	$-0.390609\pi$
$521$	15.3815			0.673876			0.336938	+	0.941527i	$0.390609\pi$
$522$		0			0
$523$			23.5130i			1.02815i	0.857745	+	0.514076i	$0.171865\pi$
$523$			23.5130i			1.02815i	−0.857745	+	0.514076i	$0.828135\pi$
$524$		0			0
$525$	5.59383			0.244135
$526$		0			0
$527$	19.4247			0.846152
$528$		0			0
$529$	20.6801			0.899136
$530$		0			0
$531$	−60.0558			−2.60620
$532$		0			0
$533$			2.91713i			0.126355i
$534$		0			0
$535$	−4.77128			−0.206280
$536$		0			0
$537$	−47.7205			−2.05929
$538$		0			0
$539$	7.60788	−	5.86801i	0.327694	−	0.252753i
$540$		0			0
$541$	43.0773			1.85204			0.926018	−	0.377479i	$-0.123209\pi$
$541$	43.0773			1.85204			0.926018	+	0.377479i	$0.123209\pi$
$542$		0			0
$543$		−	27.0462i		−	1.16066i
$544$		0			0
$545$			14.6905i			0.629272i
$546$		0			0
$547$			24.6097i			1.05224i	0.850412	+	0.526118i	$0.176353\pi$
$547$			24.6097i			1.05224i	−0.850412	+	0.526118i	$0.823647\pi$
$548$		0			0
$549$	14.1421			0.603572
$550$		0			0
$551$		−	37.0602i		−	1.57882i
$552$		0			0
$553$	−34.8959			−1.48392
$554$		0			0
$555$	19.7432			0.838054
$556$		0			0
$557$	9.44509			0.400201			0.200101	−	0.979775i	$-0.435873\pi$
$557$	9.44509			0.400201			0.200101	+	0.979775i	$0.435873\pi$
$558$		0			0
$559$			52.7668i			2.23180i
$560$		0			0
$561$	24.1955	+	31.3695i	1.02154	+	1.32442i
$562$		0			0
$563$			18.3650i			0.773994i	0.922081	+	0.386997i	$0.126488\pi$
$563$			18.3650i			0.773994i	−0.922081	+	0.386997i	$0.873512\pi$
$564$		0			0
$565$		−	2.54144i		−	0.106919i
$566$		0			0
$567$	−2.99159			−0.125635
$568$		0			0
$569$			25.4040i			1.06499i	0.846433	+	0.532496i	$0.178746\pi$
$569$			25.4040i			1.06499i	−0.846433	+	0.532496i	$0.821254\pi$
$570$		0			0
$571$		−	32.2009i		−	1.34757i	−0.738929	−	0.673783i	$-0.764668\pi$
$571$		−	32.2009i		−	1.34757i	0.738929	−	0.673783i	$-0.235332\pi$
$572$		0			0
$573$			68.4961i			2.86146i
$574$		0			0
$575$			1.52311i			0.0635183i
$576$		0			0
$577$	24.9171			1.03731			0.518657	−	0.854983i	$-0.326432\pi$
$577$	24.9171			1.03731			0.518657	+	0.854983i	$0.326432\pi$
$578$		0			0
$579$		−	39.8115i		−	1.65451i
$580$		0			0
$581$			20.0925i			0.833576i
$582$		0			0
$583$	4.80839	+	6.23407i	0.199143	+	0.258189i
$584$		0			0
$585$		−	24.6097i		−	1.01749i
$586$		0			0
$587$	13.9369			0.575237			0.287618	−	0.957745i	$-0.407136\pi$
$587$	13.9369			0.575237			0.287618	+	0.957745i	$0.407136\pi$
$588$		0			0
$589$	−32.9863			−1.35918
$590$		0			0
$591$	−40.0945			−1.64927
$592$		0			0
$593$		−	44.3740i		−	1.82222i	−0.412163	−	0.911110i	$-0.635227\pi$
$593$		−	44.3740i		−	1.82222i	0.412163	−	0.911110i	$-0.364773\pi$
$594$		0			0
$595$	−8.76156			−0.359189
$596$		0			0
$597$			67.8882i			2.77848i
$598$		0			0
$599$		−	15.2018i		−	0.621129i	−0.950552	−	0.310564i	$-0.899482\pi$
$599$		−	15.2018i		−	0.621129i	0.950552	−	0.310564i	$-0.100518\pi$
$600$		0			0
$601$			2.53686i			0.103481i	0.998661	+	0.0517404i	$0.0164768\pi$
$601$			2.53686i			0.103481i	−0.998661	+	0.0517404i	$0.983523\pi$
$602$		0			0
$603$	52.0558			2.11988
$604$		0			0
$605$	10.6393	+	2.79383i	0.432549	+	0.113585i
$606$		0			0
$607$	−41.4975			−1.68433			−0.842167	−	0.539218i	$-0.818720\pi$
$607$	−41.4975			−1.68433			−0.842167	+	0.539218i	$0.818720\pi$
$608$		0			0
$609$	−28.2234			−1.14367
$610$		0			0
$611$		−	29.3810i		−	1.18863i
$612$		0			0
$613$	23.1696			0.935812			0.467906	−	0.883778i	$-0.345009\pi$
$613$	23.1696			0.935812			0.467906	+	0.883778i	$0.345009\pi$
$614$		0			0
$615$	−1.51435			−0.0610646
$616$		0			0
$617$	20.2986			0.817192			0.408596	−	0.912715i	$-0.366019\pi$
$617$	20.2986			0.817192			0.408596	+	0.912715i	$0.366019\pi$
$618$		0			0
$619$	−19.7572			−0.794108			−0.397054	−	0.917795i	$-0.629968\pi$
$619$	−19.7572			−0.794108			−0.397054	+	0.917795i	$0.629968\pi$
$620$		0			0
$621$			6.84006i			0.274482i
$622$		0			0
$623$	10.4305			0.417888
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−41.0881	−	53.2707i	−1.64090	−	2.12743i
$628$		0			0
$629$	−30.9236			−1.23301
$630$		0			0
$631$			8.63246i			0.343653i	0.985127	+	0.171826i	$0.0549668\pi$
$631$			8.63246i			0.343653i	−0.985127	+	0.171826i	$0.945033\pi$
$632$		0			0
$633$		−	69.3730i		−	2.75733i
$634$		0			0
$635$		−	1.81680i		−	0.0720975i
$636$		0			0
$637$	15.4106			0.610589
$638$		0			0
$639$		−	26.5693i		−	1.05107i
$640$		0			0
$641$	−17.6907			−0.698743			−0.349371	−	0.936984i	$-0.613605\pi$
$641$	−17.6907			−0.698743			−0.349371	+	0.936984i	$0.613605\pi$
$642$		0			0
$643$	−19.7153			−0.777497			−0.388748	−	0.921344i	$-0.627092\pi$
$643$	−19.7153			−0.777497			−0.388748	+	0.921344i	$0.627092\pi$
$644$		0			0
$645$	−27.3925			−1.07858
$646$		0			0
$647$		−	20.9817i		−	0.824875i	−0.910986	−	0.412437i	$-0.864678\pi$
$647$		−	20.9817i		−	0.824875i	0.910986	−	0.412437i	$-0.135322\pi$
$648$		0			0
$649$	34.0925	−	26.2958i	1.33825	−	1.03220i
$650$		0			0
$651$			25.1210i			0.984569i
$652$		0			0
$653$		−	0.822781i		−	0.0321979i	−0.999870	−	0.0160990i	$-0.994875\pi$
$653$		−	0.822781i		−	0.0321979i	0.999870	−	0.0160990i	$-0.00512468\pi$
$654$		0			0
$655$	−8.44199			−0.329856
$656$		0			0
$657$		−	3.33115i		−	0.129961i
$658$		0			0
$659$			15.9960i			0.623118i	0.950227	+	0.311559i	$0.100851\pi$
$659$			15.9960i			0.623118i	−0.950227	+	0.311559i	$0.899149\pi$
$660$		0			0
$661$			19.6647i			0.764869i	0.923983	+	0.382435i	$0.124914\pi$
$661$			19.6647i			0.764869i	−0.923983	+	0.382435i	$0.875086\pi$
$662$		0			0
$663$			63.5423i			2.46778i
$664$		0			0
$665$	14.8786			0.576967
$666$		0			0
$667$		−	7.68481i		−	0.297557i
$668$		0			0
$669$		−	20.9538i		−	0.810120i
$670$		0			0
$671$	−8.02820	+	6.19221i	−0.309925	+	0.239048i
$672$		0			0
$673$			30.7519i			1.18540i	0.805423	+	0.592700i	$0.201938\pi$
$673$			30.7519i			1.18540i	−0.805423	+	0.592700i	$0.798062\pi$
$674$		0			0
$675$	4.49084			0.172852
$676$		0			0
$677$	28.5867			1.09868			0.549338	−	0.835600i	$-0.314880\pi$
$677$	28.5867			1.09868			0.549338	+	0.835600i	$0.314880\pi$
$678$		0			0
$679$	2.53686			0.0973558
$680$		0			0
$681$		−	56.7735i		−	2.17557i
$682$		0			0
$683$	−4.79820			−0.183598			−0.0917990	−	0.995778i	$-0.529262\pi$
$683$	−4.79820			−0.183598			−0.0917990	+	0.995778i	$0.529262\pi$
$684$		0			0
$685$			16.7110i			0.638493i
$686$		0			0
$687$			71.1299i			2.71378i
$688$		0			0
$689$			12.6278i			0.481080i
$690$		0			0
$691$	−11.8988			−0.452652			−0.226326	−	0.974052i	$-0.572671\pi$
$691$	−11.8988			−0.452652			−0.226326	+	0.974052i	$0.572671\pi$
$692$		0			0
$693$	−24.6097	+	18.9817i	−0.934846	+	0.721054i
$694$		0			0
$695$	4.05121			0.153671
$696$		0			0
$697$	2.37191			0.0898426
$698$		0			0
$699$			6.24851i			0.236341i
$700$		0			0
$701$	7.03395			0.265669			0.132834	−	0.991138i	$-0.457592\pi$
$701$	7.03395			0.265669			0.132834	+	0.991138i	$0.457592\pi$
$702$		0			0
$703$	52.5135			1.98058
$704$		0			0
$705$	15.2524			0.574438
$706$		0			0
$707$	13.3449			0.501885
$708$		0			0
$709$		−	19.7572i		−	0.741997i	−0.928634	−	0.370998i	$-0.879016\pi$
$709$		−	19.7572i		−	0.741997i	0.928634	−	0.370998i	$-0.120984\pi$
$710$		0			0
$711$	−79.6972			−2.98888
$712$		0			0
$713$	−6.84006			−0.256162
$714$		0			0
$715$	10.7755	+	13.9704i	0.402981	+	0.522465i
$716$		0			0
$717$	−8.15896			−0.304702
$718$		0			0
$719$			20.7249i			0.772909i	0.922309	+	0.386454i	$0.126300\pi$
$719$			20.7249i			0.772909i	−0.922309	+	0.386454i	$0.873700\pi$
$720$		0			0
$721$		−	8.10243i		−	0.301750i
$722$		0			0
$723$		−	20.8610i		−	0.775828i
$724$		0			0
$725$	−5.04546			−0.187384
$726$		0			0
$727$		−	4.12910i		−	0.153140i	−0.997064	−	0.0765700i	$-0.975603\pi$
$727$		−	4.12910i		−	0.153140i	0.997064	−	0.0765700i	$-0.0243969\pi$
$728$		0			0
$729$	−44.0375			−1.63102
$730$		0			0
$731$	42.9046			1.58688
$732$		0			0
$733$	−28.1868			−1.04110			−0.520552	−	0.853830i	$-0.674274\pi$
$733$	−28.1868			−1.04110			−0.520552	+	0.853830i	$0.674274\pi$
$734$		0			0
$735$			8.00000i			0.295084i
$736$		0			0
$737$	−29.5510	+	22.7929i	−1.08853	+	0.839588i
$738$		0			0
$739$		−	33.4322i		−	1.22982i	−0.788596	−	0.614912i	$-0.789191\pi$
$739$		−	33.4322i		−	1.22982i	0.788596	−	0.614912i	$-0.210809\pi$
$740$		0			0
$741$		−	107.905i		−	3.96401i
$742$		0			0
$743$	5.45423			0.200096			0.100048	−	0.994983i	$-0.468100\pi$
$743$	5.45423			0.200096			0.100048	+	0.994983i	$0.468100\pi$
$744$		0			0
$745$		−	0.994247i		−	0.0364264i
$746$		0			0
$747$			45.8883i			1.67897i
$748$		0			0
$749$			9.66473i			0.353142i
$750$		0			0
$751$			39.0881i			1.42634i	0.700989	+	0.713172i	$0.252742\pi$
$751$			39.0881i			1.42634i	−0.700989	+	0.713172i	$0.747258\pi$
$752$		0			0
$753$	−55.2311			−2.01273
$754$		0			0
$755$			19.8385i			0.721995i
$756$		0			0
$757$		−	4.99042i		−	0.181380i	−0.995879	−	0.0906899i	$-0.971093\pi$
$757$		−	4.99042i		−	0.181380i	0.995879	−	0.0906899i	$-0.0289072\pi$
$758$		0			0
$759$	−8.52004	−	11.0462i	−0.309258	−	0.400953i
$760$		0			0
$761$			24.3638i			0.883187i	0.897215	+	0.441594i	$0.145587\pi$
$761$			24.3638i			0.883187i	−0.897215	+	0.441594i	$0.854413\pi$
$762$		0			0
$763$	29.7572			1.07728
$764$		0			0
$765$	−20.0101			−0.723468
$766$		0			0
$767$	69.0579			2.49354
$768$		0			0
$769$		−	25.7474i		−	0.928475i	−0.885711	−	0.464238i	$-0.846328\pi$
$769$		−	25.7474i		−	0.928475i	0.885711	−	0.464238i	$-0.153672\pi$
$770$		0			0
$771$	−55.1299			−1.98546
$772$		0			0
$773$			23.3555i			0.840038i	0.907515	+	0.420019i	$0.137977\pi$
$773$			23.3555i			0.840038i	−0.907515	+	0.420019i	$0.862023\pi$
$774$		0			0
$775$			4.49084i			0.161316i
$776$		0			0
$777$		−	39.9920i		−	1.43471i
$778$		0			0
$779$	−4.02791			−0.144315
$780$		0			0
$781$	11.6335	+	15.0829i	0.416281	+	0.539708i
$782$		0			0
$783$	−22.6584			−0.809743
$784$		0			0
$785$	5.08476			0.181483
$786$		0			0
$787$		−	4.35366i		−	0.155191i	−0.996985	−	0.0775957i	$-0.975276\pi$
$787$		−	4.35366i		−	0.155191i	0.996985	−	0.0775957i	$-0.0247243\pi$
$788$		0			0
$789$	−9.77580			−0.348028
$790$		0			0
$791$	−5.14795			−0.183040
$792$		0			0
$793$	−16.2620			−0.577480
$794$		0			0
$795$	−6.55539			−0.232496
$796$		0			0
$797$		−	41.1020i		−	1.45591i	−0.685625	−	0.727955i	$-0.740471\pi$
$797$		−	41.1020i		−	1.45591i	0.685625	−	0.727955i	$-0.259529\pi$
$798$		0			0
$799$	−23.8897			−0.845156
$800$		0			0
$801$	23.8217			0.841700
$802$		0			0
$803$	1.45856	+	1.89103i	0.0514716	+	0.0667329i
$804$		0			0
$805$	3.08523			0.108740
$806$		0			0
$807$			30.5048i			1.07382i
$808$		0			0
$809$			40.0203i			1.40704i	0.710676	+	0.703519i	$0.248389\pi$
$809$			40.0203i			1.40704i	−0.710676	+	0.703519i	$0.751611\pi$
$810$		0			0
$811$			27.6579i			0.971198i	0.874182	+	0.485599i	$0.161399\pi$
$811$			27.6579i			0.971198i	−0.874182	+	0.485599i	$0.838601\pi$
$812$		0			0
$813$	−72.9784			−2.55946
$814$		0			0
$815$			17.4725i			0.612036i
$816$		0			0
$817$	−72.8592			−2.54902
$818$		0			0
$819$	−49.8496			−1.74189
$820$		0			0
$821$	−9.12494			−0.318463			−0.159231	−	0.987241i	$-0.550902\pi$
$821$	−9.12494			−0.318463			−0.159231	+	0.987241i	$0.550902\pi$
$822$		0			0
$823$			31.4865i			1.09755i	0.835970	+	0.548775i	$0.184906\pi$
$823$			31.4865i			1.09755i	−0.835970	+	0.548775i	$0.815094\pi$
$824$		0			0
$825$	−7.25240	+	5.59383i	−0.252496	+	0.194752i
$826$		0			0
$827$			2.91354i			0.101314i	0.998716	+	0.0506568i	$0.0161315\pi$
$827$			2.91354i			0.101314i	−0.998716	+	0.0506568i	$0.983869\pi$
$828$		0			0
$829$		−	11.7014i		−	0.406406i	−0.979137	−	0.203203i	$-0.934865\pi$
$829$		−	11.7014i		−	0.406406i	0.979137	−	0.203203i	$-0.0651351\pi$
$830$		0			0
$831$	−16.2049			−0.562140
$832$		0			0
$833$		−	12.5303i		−	0.434150i
$834$		0			0
$835$			16.1677i			0.559508i
$836$		0			0
$837$			20.1676i			0.697095i
$838$		0			0
$839$			23.6801i			0.817529i	0.912640	+	0.408764i	$0.134040\pi$
$839$			23.6801i			0.817529i	−0.912640	+	0.408764i	$0.865960\pi$
$840$		0			0
$841$	−3.54333			−0.122184
$842$		0			0
$843$			29.3810i			1.01194i
$844$		0			0
$845$			15.2986i			0.526289i
$846$		0			0
$847$	5.65921	−	21.5510i	0.194453	−	0.740502i
$848$		0			0
$849$		−	20.3868i		−	0.699674i
$850$		0			0
$851$	10.8892			0.373278
$852$		0			0
$853$	39.7744			1.36185			0.680924	−	0.732354i	$-0.261578\pi$
$853$	39.7744			1.36185			0.680924	+	0.732354i	$0.261578\pi$
$854$		0			0
$855$	33.9806			1.16211
$856$		0			0
$857$		−	21.5528i		−	0.736228i	−0.929781	−	0.368114i	$-0.880004\pi$
$857$		−	21.5528i		−	0.736228i	0.929781	−	0.368114i	$-0.119996\pi$
$858$		0			0
$859$	−33.2524			−1.13456			−0.567279	−	0.823526i	$-0.692004\pi$
$859$	−33.2524			−1.13456			−0.567279	+	0.823526i	$0.692004\pi$
$860$		0			0
$861$			3.06748i			0.104539i
$862$		0			0
$863$		−	35.8988i		−	1.22201i	−0.791627	−	0.611005i	$-0.790766\pi$
$863$		−	35.8988i		−	1.22201i	0.791627	−	0.611005i	$-0.209234\pi$
$864$		0			0
$865$		−	22.0729i		−	0.750500i
$866$		0			0
$867$	4.71970			0.160289
$868$		0			0
$869$	45.2425	−	34.8959i	1.53475	−	1.18376i
$870$		0			0
$871$	−59.8588			−2.02824
$872$		0			0
$873$	5.79383			0.196092
$874$		0			0
$875$		−	2.02561i		−	0.0684780i
$876$		0			0
$877$	−45.8883			−1.54954			−0.774769	−	0.632244i	$-0.782134\pi$
$877$	−45.8883			−1.54954			−0.774769	+	0.632244i	$0.782134\pi$
$878$		0			0
$879$	−76.4812			−2.57965
$880$		0			0
$881$	44.6897			1.50563			0.752817	−	0.658230i	$-0.228694\pi$
$881$	44.6897			1.50563			0.752817	+	0.658230i	$0.228694\pi$
$882$		0			0
$883$	11.0235			0.370972			0.185486	−	0.982647i	$-0.440614\pi$
$883$	11.0235			0.370972			0.185486	+	0.982647i	$0.440614\pi$
$884$		0			0
$885$			35.8496i			1.20507i
$886$		0			0
$887$	−11.3594			−0.381410			−0.190705	−	0.981647i	$-0.561077\pi$
$887$	−11.3594			−0.381410			−0.190705	+	0.981647i	$0.561077\pi$
$888$		0			0
$889$	−3.68012			−0.123427
$890$		0			0
$891$	3.87859	−	2.99159i	0.129938	−	0.100222i
$892$		0			0
$893$	40.5687			1.35758
$894$		0			0
$895$			17.2803i			0.577617i
$896$		0			0
$897$		−	22.3753i		−	0.747090i
$898$		0			0
$899$		−	22.6584i		−	0.755698i
$900$		0			0
$901$	10.2676			0.342065
$902$		0			0
$903$			55.4865i			1.84647i
$904$		0			0
$905$	−9.79383			−0.325558
$906$		0			0
$907$	21.1372			0.701851			0.350925	−	0.936403i	$-0.385867\pi$
$907$	21.1372			0.701851			0.350925	+	0.936403i	$0.385867\pi$
$908$		0			0
$909$	30.4777			1.01088
$910$		0			0
$911$		−	9.44461i		−	0.312914i	−0.987685	−	0.156457i	$-0.949993\pi$
$911$		−	9.44461i		−	0.312914i	0.987685	−	0.156457i	$-0.0500072\pi$
$912$		0			0
$913$	−20.0925	−	26.0499i	−0.664963	−	0.862124i
$914$		0			0
$915$		−	8.44199i		−	0.279084i
$916$		0			0
$917$			17.1002i			0.564697i
$918$		0			0
$919$	14.6905			0.484595			0.242298	−	0.970202i	$-0.422099\pi$
$919$	14.6905			0.484595			0.242298	+	0.970202i	$0.422099\pi$
$920$		0			0
$921$			17.3581i			0.571970i
$922$		0			0
$923$			30.5520i			1.00563i
$924$		0			0
$925$		−	7.14931i		−	0.235068i
$926$		0			0
$927$		−	18.5048i		−	0.607777i
$928$		0			0
$929$	−35.5616			−1.16674			−0.583370	−	0.812207i	$-0.698266\pi$
$929$	−35.5616			−1.16674			−0.583370	+	0.812207i	$0.698266\pi$
$930$		0			0
$931$			21.2786i			0.697377i
$932$		0			0
$933$		−	12.0106i		−	0.393210i
$934$		0			0
$935$	11.3594	−	8.76156i	0.371491	−	0.286534i
$936$		0			0
$937$		−	47.4027i		−	1.54858i	−0.632832	−	0.774289i	$-0.718108\pi$
$937$		−	47.4027i		−	1.54858i	0.632832	−	0.774289i	$-0.281892\pi$
$938$		0			0
$939$	65.6068			2.14100
$940$		0			0
$941$	−8.67906			−0.282929			−0.141465	−	0.989943i	$-0.545181\pi$
$941$	−8.67906			−0.282929			−0.141465	+	0.989943i	$0.545181\pi$
$942$		0			0
$943$	−0.835228			−0.0271988
$944$		0			0
$945$		−	9.09667i		−	0.295915i
$946$		0			0
$947$	−10.4263			−0.338809			−0.169404	−	0.985547i	$-0.554184\pi$
$947$	−10.4263			−0.338809			−0.169404	+	0.985547i	$0.554184\pi$
$948$		0			0
$949$			3.83048i			0.124343i
$950$		0			0
$951$			97.1439i			3.15011i
$952$		0			0
$953$			2.68029i			0.0868232i	0.999057	+	0.0434116i	$0.0138227\pi$
$953$			2.68029i			0.0868232i	−0.999057	+	0.0434116i	$0.986177\pi$
$954$		0			0
$955$	24.8034			0.802620
$956$		0			0
$957$	36.5917	−	28.2234i	1.18284	−	0.912334i
$958$		0			0
$959$	33.8498			1.09307
$960$		0			0
$961$	10.8324			0.349431
$962$		0			0
$963$			22.0729i			0.711288i
$964$		0			0
$965$	−14.4163			−0.464078
$966$		0			0
$967$	38.1997			1.22842			0.614209	−	0.789143i	$-0.289475\pi$
$967$	38.1997			1.22842			0.614209	+	0.789143i	$0.289475\pi$
$968$		0			0
$969$	−87.7378			−2.81855
$970$		0			0
$971$	−61.4373			−1.97162			−0.985809	−	0.167873i	$-0.946310\pi$
$971$	−61.4373			−1.97162			−0.985809	+	0.167873i	$0.946310\pi$
$972$		0			0
$973$		−	8.20617i		−	0.263078i
$974$		0			0
$975$	−14.6905			−0.470473
$976$		0			0
$977$	−31.7572			−1.01600			−0.508001	−	0.861356i	$-0.669615\pi$
$977$	−31.7572			−1.01600			−0.508001	+	0.861356i	$0.669615\pi$
$978$		0			0
$979$	−13.5231	+	10.4305i	−0.432201	+	0.333360i
$980$		0			0
$981$	67.9612			2.16983
$982$		0			0
$983$		−	46.0925i		−	1.47012i	−0.678001	−	0.735061i	$-0.737154\pi$
$983$		−	46.0925i		−	1.47012i	0.678001	−	0.735061i	$-0.262846\pi$
$984$		0			0
$985$			14.5188i			0.462608i
$986$		0			0
$987$		−	30.8954i		−	0.983410i
$988$		0			0
$989$	−15.1081			−0.480410
$990$		0			0
$991$			1.21575i			0.0386196i	0.999814	+	0.0193098i	$0.00614689\pi$
$991$			1.21575i			0.0386196i	−0.999814	+	0.0193098i	$0.993853\pi$
$992$		0			0
$993$	21.3449			0.677358
$994$		0			0
$995$	24.5833			0.779343
$996$		0			0
$997$	−27.0901			−0.857951			−0.428976	−	0.903316i	$-0.641125\pi$
$997$	−27.0901			−0.857951			−0.428976	+	0.903316i	$0.641125\pi$
$998$		0			0
$999$		−	32.1064i		−	1.01580i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.p.c.351.4	yes	12
4.3	odd	2		3520.2.p.d.351.11	yes	12
8.3	odd	2	inner	3520.2.p.c.351.1	✓	12
8.5	even	2		3520.2.p.d.351.10	yes	12
11.10	odd	2	inner	3520.2.p.c.351.3	yes	12
44.43	even	2		3520.2.p.d.351.12	yes	12
88.21	odd	2		3520.2.p.d.351.9	yes	12
88.43	even	2	inner	3520.2.p.c.351.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3520.2.p.c.351.1	✓	12	8.3	odd	2	inner
3520.2.p.c.351.2	yes	12	88.43	even	2	inner
3520.2.p.c.351.3	yes	12	11.10	odd	2	inner
3520.2.p.c.351.4	yes	12	1.1	even	1	trivial
3520.2.p.d.351.9	yes	12	88.21	odd	2
3520.2.p.d.351.10	yes	12	8.5	even	2
3520.2.p.d.351.11	yes	12	4.3	odd	2
3520.2.p.d.351.12	yes	12	44.43	even	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:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.76156 q^{3} +1.00000i q^{5} +2.02561 q^{7} +4.62620 q^{9} +O(q^{10})\)	\(q-2.76156 q^{3} +1.00000i q^{5} +2.02561 q^{7} +4.62620 q^{9} +(-2.62620 + 2.02561i) q^{11} -5.31965 q^{13} -2.76156i q^{15} +4.32540i q^{17} -7.34525i q^{19} -5.59383 q^{21} -1.52311i q^{23} -1.00000 q^{25} -4.49084 q^{27} +5.04546 q^{29} -4.49084i q^{31} +(7.25240 - 5.59383i) q^{33} +2.02561i q^{35} +7.14931i q^{37} +14.6905 q^{39} -0.548369i q^{41} -9.91923i q^{43} +4.62620i q^{45} +5.52311i q^{47} -2.89692 q^{49} -11.9448i q^{51} -2.37380i q^{53} +(-2.02561 - 2.62620i) q^{55} +20.2843i q^{57} -12.9817 q^{59} +3.05697 q^{61} +9.37086 q^{63} -5.31965i q^{65} +11.2524 q^{67} +4.20617i q^{69} -5.74324i q^{71} -0.720062i q^{73} +2.76156 q^{75} +(-5.31965 + 4.10308i) q^{77} -17.2274 q^{79} -1.47689 q^{81} +9.91923i q^{83} -4.32540 q^{85} -13.9333 q^{87} +5.14931 q^{89} -10.7755 q^{91} +12.4017i q^{93} +7.34525 q^{95} +1.25240 q^{97} +(-12.1493 + 9.37086i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} + 20 q^{9}+O(q^{10}) \) 12 * q - 8 * q^3 + 20 * q^9	\( 12 q - 8 q^{3} + 20 q^{9} + 4 q^{11} - 12 q^{25} - 8 q^{27} + 16 q^{33} - 20 q^{49} - 64 q^{59} + 64 q^{67} + 8 q^{75} - 68 q^{81} - 24 q^{89} - 8 q^{91} - 56 q^{97} - 60 q^{99}+O(q^{100}) \) 12 * q - 8 * q^3 + 20 * q^9 + 4 * q^11 - 12 * q^25 - 8 * q^27 + 16 * q^33 - 20 * q^49 - 64 * q^59 + 64 * q^67 + 8 * q^75 - 68 * q^81 - 24 * q^89 - 8 * q^91 - 56 * q^97 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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