LMFDB - Embedded newform 2912.2.i.a.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(337,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2912.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24325i q^{3} -4.24503 q^{5} +1.00000i q^{7} +1.45434 q^{9} -2.97219 q^{11} +(3.10071 + 1.84000i) q^{13} -5.27761i q^{15} +3.88084 q^{17} +3.68346 q^{19} -1.24325 q^{21} -4.72588 q^{23} +13.0203 q^{25} +5.53784i q^{27} +7.03148i q^{29} -2.28102i q^{31} -3.69517i q^{33} -4.24503i q^{35} +3.24019 q^{37} +(-2.28757 + 3.85494i) q^{39} +3.82209i q^{41} +8.00510i q^{43} -6.17371 q^{45} +0.318216i q^{47} -1.00000 q^{49} +4.82484i q^{51} +2.52928i q^{53} +12.6170 q^{55} +4.57944i q^{57} -10.7468 q^{59} -11.8802i q^{61} +1.45434i q^{63} +(-13.1626 - 7.81085i) q^{65} -8.65396 q^{67} -5.87543i q^{69} +5.29101i q^{71} -12.1985i q^{73} +16.1874i q^{75} -2.97219i q^{77} -2.81549 q^{79} -2.52188 q^{81} -2.11994 q^{83} -16.4743 q^{85} -8.74186 q^{87} +14.4314i q^{89} +(-1.84000 + 3.10071i) q^{91} +2.83587 q^{93} -15.6364 q^{95} -3.42799i q^{97} -4.32258 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.24325i			0.717788i	0.933378	+	0.358894i	$0.116846\pi$
$3$			1.24325i			0.717788i	−0.933378	+	0.358894i	$0.883154\pi$
$4$		0			0
$5$	−4.24503			−1.89843			−0.949217	−	0.314622i	$-0.898122\pi$
$5$	−4.24503			−1.89843			−0.949217	+	0.314622i	$0.898122\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.45434			0.484780
$10$		0			0
$11$	−2.97219			−0.896150			−0.448075	−	0.893996i	$-0.647890\pi$
$11$	−2.97219			−0.896150			−0.448075	+	0.893996i	$0.647890\pi$
$12$		0			0
$13$	3.10071	+	1.84000i	0.859982	+	0.510324i
$13$	3.10071	+	1.84000i	0.859982	+	0.510324i
$14$		0			0
$15$		−	5.27761i		−	1.36267i
$16$		0			0
$17$	3.88084			0.941243			0.470622	−	0.882335i	$-0.344030\pi$
$17$	3.88084			0.941243			0.470622	+	0.882335i	$0.344030\pi$
$18$		0			0
$19$	3.68346			0.845043			0.422521	−	0.906353i	$-0.361145\pi$
$19$	3.68346			0.845043			0.422521	+	0.906353i	$0.361145\pi$
$20$		0			0
$21$	−1.24325			−0.271298
$22$		0			0
$23$	−4.72588			−0.985414			−0.492707	−	0.870195i	$-0.663993\pi$
$23$	−4.72588			−0.985414			−0.492707	+	0.870195i	$0.663993\pi$
$24$		0			0
$25$	13.0203			2.60405
$26$		0			0
$27$			5.53784i			1.06576i
$28$		0			0
$29$			7.03148i			1.30571i	0.757481	+	0.652857i	$0.226430\pi$
$29$			7.03148i			1.30571i	−0.757481	+	0.652857i	$0.773570\pi$
$30$		0			0
$31$		−	2.28102i		−	0.409684i	−0.978795	−	0.204842i	$-0.934332\pi$
$31$		−	2.28102i		−	0.409684i	0.978795	−	0.204842i	$-0.0656680\pi$
$32$		0			0
$33$		−	3.69517i		−	0.643246i
$34$		0			0
$35$		−	4.24503i		−	0.717541i
$36$		0			0
$37$	3.24019			0.532684			0.266342	−	0.963879i	$-0.414185\pi$
$37$	3.24019			0.532684			0.266342	+	0.963879i	$0.414185\pi$
$38$		0			0
$39$	−2.28757	+	3.85494i	−0.366305	+	0.617285i
$40$		0			0
$41$			3.82209i			0.596910i	0.954424	+	0.298455i	$0.0964712\pi$
$41$			3.82209i			0.596910i	−0.954424	+	0.298455i	$0.903529\pi$
$42$		0			0
$43$			8.00510i			1.22077i	0.792106	+	0.610383i	$0.208985\pi$
$43$			8.00510i			1.22077i	−0.792106	+	0.610383i	$0.791015\pi$
$44$		0			0
$45$	−6.17371			−0.920323
$46$		0			0
$47$			0.318216i			0.0464166i	0.999731	+	0.0232083i	$0.00738809\pi$
$47$			0.318216i			0.0464166i	−0.999731	+	0.0232083i	$0.992612\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			4.82484i			0.675613i
$52$		0			0
$53$			2.52928i			0.347423i	0.984797	+	0.173712i	$0.0555761\pi$
$53$			2.52928i			0.347423i	−0.984797	+	0.173712i	$0.944424\pi$
$54$		0			0
$55$	12.6170			1.70128
$56$		0			0
$57$			4.57944i			0.606562i
$58$		0			0
$59$	−10.7468			−1.39911			−0.699556	−	0.714578i	$-0.746619\pi$
$59$	−10.7468			−1.39911			−0.699556	+	0.714578i	$0.746619\pi$
$60$		0			0
$61$		−	11.8802i		−	1.52111i	−0.649274	−	0.760555i	$-0.724927\pi$
$61$		−	11.8802i		−	1.52111i	0.649274	−	0.760555i	$-0.275073\pi$
$62$		0			0
$63$			1.45434i			0.183230i
$64$		0			0
$65$	−13.1626	−	7.81085i	−1.63262	−	0.968817i
$66$		0			0
$67$	−8.65396			−1.05725			−0.528625	−	0.848856i	$-0.677292\pi$
$67$	−8.65396			−1.05725			−0.528625	+	0.848856i	$0.677292\pi$
$68$		0			0
$69$		−	5.87543i		−	0.707319i
$70$		0			0
$71$			5.29101i			0.627927i	0.949435	+	0.313964i	$0.101657\pi$
$71$			5.29101i			0.627927i	−0.949435	+	0.313964i	$0.898343\pi$
$72$		0			0
$73$		−	12.1985i		−	1.42773i	−0.700284	−	0.713864i	$-0.746943\pi$
$73$		−	12.1985i		−	1.42773i	0.700284	−	0.713864i	$-0.253057\pi$
$74$		0			0
$75$			16.1874i			1.86916i
$76$		0			0
$77$		−	2.97219i		−	0.338713i
$78$		0			0
$79$	−2.81549			−0.316767			−0.158384	−	0.987378i	$-0.550628\pi$
$79$	−2.81549			−0.316767			−0.158384	+	0.987378i	$0.550628\pi$
$80$		0			0
$81$	−2.52188			−0.280208
$82$		0			0
$83$	−2.11994			−0.232693			−0.116347	−	0.993209i	$-0.537118\pi$
$83$	−2.11994			−0.232693			−0.116347	+	0.993209i	$0.537118\pi$
$84$		0			0
$85$	−16.4743			−1.78689
$86$		0			0
$87$	−8.74186			−0.937226
$88$		0			0
$89$			14.4314i			1.52972i	0.644194	+	0.764862i	$0.277193\pi$
$89$			14.4314i			1.52972i	−0.644194	+	0.764862i	$0.722807\pi$
$90$		0			0
$91$	−1.84000	+	3.10071i	−0.192884	+	0.325043i
$92$		0			0
$93$	2.83587			0.294066
$94$		0			0
$95$	−15.6364			−1.60426
$96$		0			0
$97$		−	3.42799i		−	0.348060i	−0.984740	−	0.174030i	$-0.944321\pi$
$97$		−	3.42799i		−	0.348060i	0.984740	−	0.174030i	$-0.0556789\pi$
$98$		0			0
$99$	−4.32258			−0.434436
$100$		0			0
$101$		−	4.85023i		−	0.482616i	−0.970449	−	0.241308i	$-0.922424\pi$
$101$		−	4.85023i		−	0.482616i	0.970449	−	0.241308i	$-0.0775764\pi$
$102$		0			0
$103$	10.9280			1.07677			0.538384	−	0.842700i	$-0.319035\pi$
$103$	10.9280			1.07677			0.538384	+	0.842700i	$0.319035\pi$
$104$		0			0
$105$	5.27761			0.515042
$106$		0			0
$107$		−	19.3059i		−	1.86637i	−0.359392	−	0.933187i	$-0.617016\pi$
$107$		−	19.3059i		−	1.86637i	0.359392	−	0.933187i	$-0.382984\pi$
$108$		0			0
$109$	−5.98308			−0.573075			−0.286538	−	0.958069i	$-0.592504\pi$
$109$	−5.98308			−0.573075			−0.286538	+	0.958069i	$0.592504\pi$
$110$		0			0
$111$			4.02835i			0.382354i
$112$		0			0
$113$	−11.8821			−1.11777			−0.558887	−	0.829244i	$-0.688772\pi$
$113$	−11.8821			−1.11777			−0.558887	+	0.829244i	$0.688772\pi$
$114$		0			0
$115$	20.0615			1.87074
$116$		0			0
$117$	4.50949	+	2.67598i	0.416902	+	0.247395i
$118$		0			0
$119$			3.88084i			0.355756i
$120$		0			0
$121$	−2.16606			−0.196915
$122$		0			0
$123$	−4.75179			−0.428455
$124$		0			0
$125$	−34.0462			−3.04519
$126$		0			0
$127$	−7.65675			−0.679426			−0.339713	−	0.940529i	$-0.610330\pi$
$127$	−7.65675			−0.679426			−0.339713	+	0.940529i	$0.610330\pi$
$128$		0			0
$129$	−9.95231			−0.876252
$130$		0			0
$131$		−	2.71789i		−	0.237463i	−0.992926	−	0.118732i	$-0.962117\pi$
$131$		−	2.71789i		−	0.237463i	0.992926	−	0.118732i	$-0.0378828\pi$
$132$		0			0
$133$			3.68346i			0.319396i
$134$		0			0
$135$		−	23.5083i		−	2.02327i
$136$		0			0
$137$			23.0466i			1.96901i	0.175363	+	0.984504i	$0.443890\pi$
$137$			23.0466i			1.96901i	−0.175363	+	0.984504i	$0.556110\pi$
$138$		0			0
$139$		−	6.61294i		−	0.560902i	−0.959868	−	0.280451i	$-0.909516\pi$
$139$		−	6.61294i		−	0.560902i	0.959868	−	0.280451i	$-0.0904840\pi$
$140$		0			0
$141$	−0.395621			−0.0333173
$142$		0			0
$143$	−9.21591	−	5.46884i	−0.770673	−	0.457327i
$144$		0			0
$145$		−	29.8488i		−	2.47881i
$146$		0			0
$147$		−	1.24325i		−	0.102541i
$148$		0			0
$149$	−3.51745			−0.288161			−0.144080	−	0.989566i	$-0.546022\pi$
$149$	−3.51745			−0.288161			−0.144080	+	0.989566i	$0.546022\pi$
$150$		0			0
$151$			10.1417i			0.825316i	0.910886	+	0.412658i	$0.135400\pi$
$151$			10.1417i			0.825316i	−0.910886	+	0.412658i	$0.864600\pi$
$152$		0			0
$153$	5.64407			0.456296
$154$		0			0
$155$			9.68301i			0.777758i
$156$		0			0
$157$			7.13358i			0.569322i	0.958628	+	0.284661i	$0.0918810\pi$
$157$			7.13358i			0.569322i	−0.958628	+	0.284661i	$0.908119\pi$
$158$		0			0
$159$	−3.14452			−0.249376
$160$		0			0
$161$		−	4.72588i		−	0.372452i
$162$		0			0
$163$	−20.6173			−1.61487			−0.807437	−	0.589954i	$-0.799146\pi$
$163$	−20.6173			−1.61487			−0.807437	+	0.589954i	$0.799146\pi$
$164$		0			0
$165$			15.6861i			1.22116i
$166$		0			0
$167$			6.39044i			0.494507i	0.968951	+	0.247253i	$0.0795280\pi$
$167$			6.39044i			0.494507i	−0.968951	+	0.247253i	$0.920472\pi$
$168$		0			0
$169$	6.22880	+	11.4106i	0.479139	+	0.877739i
$170$		0			0
$171$	5.35700			0.409660
$172$		0			0
$173$			3.69193i			0.280693i	0.990102	+	0.140346i	$0.0448216\pi$
$173$			3.69193i			0.280693i	−0.990102	+	0.140346i	$0.955178\pi$
$174$		0			0
$175$			13.0203i			0.984239i
$176$		0			0
$177$		−	13.3609i		−	1.00427i
$178$		0			0
$179$			20.4497i			1.52848i	0.644930	+	0.764242i	$0.276886\pi$
$179$			20.4497i			1.52848i	−0.644930	+	0.764242i	$0.723114\pi$
$180$		0			0
$181$			19.6160i			1.45805i	0.684488	+	0.729024i	$0.260026\pi$
$181$			19.6160i			1.45805i	−0.684488	+	0.729024i	$0.739974\pi$
$182$		0			0
$183$	14.7701			1.09183
$184$		0			0
$185$	−13.7547			−1.01127
$186$		0			0
$187$	−11.5346			−0.843495
$188$		0			0
$189$	−5.53784			−0.402819
$190$		0			0
$191$	6.04598			0.437472			0.218736	−	0.975784i	$-0.429807\pi$
$191$	6.04598			0.437472			0.218736	+	0.975784i	$0.429807\pi$
$192$		0			0
$193$		−	1.37318i		−	0.0988435i	−0.998778	−	0.0494217i	$-0.984262\pi$
$193$		−	1.37318i		−	0.0988435i	0.998778	−	0.0494217i	$-0.0157378\pi$
$194$		0			0
$195$	9.71080	−	16.3643i	0.695405	−	1.17188i
$196$		0			0
$197$	15.5202			1.10577			0.552884	−	0.833258i	$-0.313527\pi$
$197$	15.5202			1.10577			0.552884	+	0.833258i	$0.313527\pi$
$198$		0			0
$199$	−27.2936			−1.93479			−0.967395	−	0.253271i	$-0.918494\pi$
$199$	−27.2936			−1.93479			−0.967395	+	0.253271i	$0.918494\pi$
$200$		0			0
$201$		−	10.7590i		−	0.758881i
$202$		0			0
$203$	−7.03148			−0.493513
$204$		0			0
$205$		−	16.2249i		−	1.13319i
$206$		0			0
$207$	−6.87304			−0.477709
$208$		0			0
$209$	−10.9479			−0.757285
$210$		0			0
$211$			6.32498i			0.435429i	0.976012	+	0.217715i	$0.0698602\pi$
$211$			6.32498i			0.435429i	−0.976012	+	0.217715i	$0.930140\pi$
$212$		0			0
$213$	−6.57803			−0.450719
$214$		0			0
$215$		−	33.9819i		−	2.31754i
$216$		0			0
$217$	2.28102			0.154846
$218$		0			0
$219$	15.1658			1.02481
$220$		0			0
$221$	12.0334	+	7.14075i	0.809452	+	0.480339i
$222$		0			0
$223$		−	9.60586i		−	0.643256i	−0.946866	−	0.321628i	$-0.895770\pi$
$223$		−	9.60586i		−	0.643256i	0.946866	−	0.321628i	$-0.104230\pi$
$224$		0			0
$225$	18.9359			1.26239
$226$		0			0
$227$	14.4621			0.959882			0.479941	−	0.877301i	$-0.340658\pi$
$227$	14.4621			0.959882			0.479941	+	0.877301i	$0.340658\pi$
$228$		0			0
$229$	4.33134			0.286223			0.143112	−	0.989707i	$-0.454289\pi$
$229$	4.33134			0.286223			0.143112	+	0.989707i	$0.454289\pi$
$230$		0			0
$231$	3.69517			0.243124
$232$		0			0
$233$	−24.1730			−1.58362			−0.791812	−	0.610765i	$-0.790862\pi$
$233$	−24.1730			−1.58362			−0.791812	+	0.610765i	$0.790862\pi$
$234$		0			0
$235$		−	1.35084i		−	0.0881188i
$236$		0			0
$237$		−	3.50035i		−	0.227372i
$238$		0			0
$239$			13.4660i			0.871045i	0.900178	+	0.435523i	$0.143436\pi$
$239$			13.4660i			0.871045i	−0.900178	+	0.435523i	$0.856564\pi$
$240$		0			0
$241$		−	24.4534i		−	1.57518i	−0.616200	−	0.787590i	$-0.711329\pi$
$241$		−	24.4534i		−	1.57518i	0.616200	−	0.787590i	$-0.288671\pi$
$242$		0			0
$243$			13.4782i			0.864627i
$244$		0			0
$245$	4.24503			0.271205
$246$		0			0
$247$	11.4213	+	6.77756i	0.726722	+	0.431246i
$248$		0			0
$249$		−	2.63560i		−	0.167024i
$250$		0			0
$251$		−	20.9395i		−	1.32169i	−0.750522	−	0.660846i	$-0.770198\pi$
$251$		−	20.9395i		−	1.32169i	0.750522	−	0.660846i	$-0.229802\pi$
$252$		0			0
$253$	14.0462			0.883079
$254$		0			0
$255$		−	20.4816i		−	1.28261i
$256$		0			0
$257$	1.15045			0.0717633			0.0358816	−	0.999356i	$-0.488576\pi$
$257$	1.15045			0.0717633			0.0358816	+	0.999356i	$0.488576\pi$
$258$		0			0
$259$			3.24019i			0.201336i
$260$		0			0
$261$			10.2262i			0.632984i
$262$		0			0
$263$	4.03698			0.248931			0.124465	−	0.992224i	$-0.460278\pi$
$263$	4.03698			0.248931			0.124465	+	0.992224i	$0.460278\pi$
$264$		0			0
$265$		−	10.7369i		−	0.659560i
$266$		0			0
$267$	−17.9418			−1.09802
$268$		0			0
$269$		−	4.24698i		−	0.258943i	−0.991583	−	0.129472i	$-0.958672\pi$
$269$		−	4.24698i		−	0.258943i	0.991583	−	0.129472i	$-0.0413281\pi$
$270$		0			0
$271$			25.3475i			1.53975i	0.638194	+	0.769875i	$0.279682\pi$
$271$			25.3475i			1.53975i	−0.638194	+	0.769875i	$0.720318\pi$
$272$		0			0
$273$	−3.85494	−	2.28757i	−0.233312	−	0.138450i
$274$		0			0
$275$	−38.6987			−2.33362
$276$		0			0
$277$		−	6.89186i		−	0.414092i	−0.978331	−	0.207046i	$-0.933615\pi$
$277$		−	6.89186i		−	0.414092i	0.978331	−	0.207046i	$-0.0663849\pi$
$278$		0			0
$279$		−	3.31738i		−	0.198607i
$280$		0			0
$281$		−	1.99156i		−	0.118807i	−0.998234	−	0.0594033i	$-0.981080\pi$
$281$		−	1.99156i		−	0.118807i	0.998234	−	0.0594033i	$-0.0189198\pi$
$282$		0			0
$283$			5.18039i			0.307942i	0.988075	+	0.153971i	$0.0492062\pi$
$283$			5.18039i			0.307942i	−0.988075	+	0.153971i	$0.950794\pi$
$284$		0			0
$285$		−	19.4398i		−	1.15152i
$286$		0			0
$287$	−3.82209			−0.225611
$288$		0			0
$289$	−1.93904			−0.114061
$290$		0			0
$291$	4.26184			0.249833
$292$		0			0
$293$	−8.11470			−0.474066			−0.237033	−	0.971502i	$-0.576175\pi$
$293$	−8.11470			−0.474066			−0.237033	+	0.971502i	$0.576175\pi$
$294$		0			0
$295$	45.6204			2.65612
$296$		0			0
$297$		−	16.4595i		−	0.955079i
$298$		0			0
$299$	−14.6536	−	8.69562i	−0.847439	−	0.502881i
$300$		0			0
$301$	−8.00510			−0.461406
$302$		0			0
$303$	6.03003			0.346416
$304$		0			0
$305$			50.4320i			2.88773i
$306$		0			0
$307$	−22.6768			−1.29424			−0.647118	−	0.762390i	$-0.724026\pi$
$307$	−22.6768			−1.29424			−0.647118	+	0.762390i	$0.724026\pi$
$308$		0			0
$309$			13.5862i			0.772891i
$310$		0			0
$311$	−1.90309			−0.107914			−0.0539571	−	0.998543i	$-0.517183\pi$
$311$	−1.90309			−0.107914			−0.0539571	+	0.998543i	$0.517183\pi$
$312$		0			0
$313$	30.9361			1.74861			0.874306	−	0.485376i	$-0.161317\pi$
$313$	30.9361			1.74861			0.874306	+	0.485376i	$0.161317\pi$
$314$		0			0
$315$		−	6.17371i		−	0.347849i
$316$		0			0
$317$	10.9849			0.616975			0.308488	−	0.951228i	$-0.400177\pi$
$317$	10.9849			0.616975			0.308488	+	0.951228i	$0.400177\pi$
$318$		0			0
$319$		−	20.8989i		−	1.17012i
$320$		0			0
$321$	24.0020			1.33966
$322$		0			0
$323$	14.2949			0.795390
$324$		0			0
$325$	40.3721	+	23.9573i	2.23944	+	1.32891i
$326$		0			0
$327$		−	7.43844i		−	0.411347i
$328$		0			0
$329$	−0.318216			−0.0175438
$330$		0			0
$331$	21.1523			1.16264			0.581319	−	0.813676i	$-0.302537\pi$
$331$	21.1523			1.16264			0.581319	+	0.813676i	$0.302537\pi$
$332$		0			0
$333$	4.71234			0.258235
$334$		0			0
$335$	36.7363			2.00712
$336$		0			0
$337$	−22.7775			−1.24077			−0.620385	−	0.784297i	$-0.713024\pi$
$337$	−22.7775			−1.24077			−0.620385	+	0.784297i	$0.713024\pi$
$338$		0			0
$339$		−	14.7724i		−	0.802326i
$340$		0			0
$341$			6.77965i			0.367138i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			24.9414i			1.34280i
$346$		0			0
$347$			8.80009i			0.472414i	0.971703	+	0.236207i	$0.0759043\pi$
$347$			8.80009i			0.472414i	−0.971703	+	0.236207i	$0.924096\pi$
$348$		0			0
$349$	−7.29071			−0.390263			−0.195131	−	0.980777i	$-0.562513\pi$
$349$	−7.29071			−0.390263			−0.195131	+	0.980777i	$0.562513\pi$
$350$		0			0
$351$	−10.1896	+	17.1712i	−0.543882	+	0.916533i
$352$		0			0
$353$		−	16.0655i		−	0.855078i	−0.903997	−	0.427539i	$-0.859381\pi$
$353$		−	16.0655i		−	0.855078i	0.903997	−	0.427539i	$-0.140619\pi$
$354$		0			0
$355$		−	22.4605i		−	1.19208i
$356$		0			0
$357$	−4.82484			−0.255358
$358$		0			0
$359$		−	3.43324i		−	0.181200i	−0.995887	−	0.0905999i	$-0.971122\pi$
$359$		−	3.43324i		−	0.181200i	0.995887	−	0.0905999i	$-0.0288784\pi$
$360$		0			0
$361$	−5.43216			−0.285903
$362$		0			0
$363$		−	2.69295i		−	0.141343i
$364$		0			0
$365$			51.7830i			2.71045i
$366$		0			0
$367$	17.1507			0.895260			0.447630	−	0.894219i	$-0.352268\pi$
$367$	17.1507			0.895260			0.447630	+	0.894219i	$0.352268\pi$
$368$		0			0
$369$			5.55861i			0.289370i
$370$		0			0
$371$	−2.52928			−0.131314
$372$		0			0
$373$			9.16344i			0.474465i	0.971453	+	0.237233i	$0.0762403\pi$
$373$			9.16344i			0.474465i	−0.971453	+	0.237233i	$0.923760\pi$
$374$		0			0
$375$		−	42.3278i		−	2.18580i
$376$		0			0
$377$	−12.9379	+	21.8026i	−0.666337	+	1.12289i
$378$		0			0
$379$	30.7837			1.58125			0.790627	−	0.612298i	$-0.209755\pi$
$379$	30.7837			1.58125			0.790627	+	0.612298i	$0.209755\pi$
$380$		0			0
$381$		−	9.51922i		−	0.487684i
$382$		0			0
$383$		−	37.8476i		−	1.93392i	−0.254921	−	0.966962i	$-0.582049\pi$
$383$		−	37.8476i		−	1.93392i	0.254921	−	0.966962i	$-0.417951\pi$
$384$		0			0
$385$			12.6170i			0.643024i
$386$		0			0
$387$			11.6421i			0.591803i
$388$		0			0
$389$		−	32.2732i		−	1.63631i	−0.574994	−	0.818157i	$-0.694996\pi$
$389$		−	32.2732i		−	1.63631i	0.574994	−	0.818157i	$-0.305004\pi$
$390$		0			0
$391$	−18.3404			−0.927514
$392$		0			0
$393$	3.37901			0.170448
$394$		0			0
$395$	11.9518			0.601362
$396$		0			0
$397$	−6.45949			−0.324192			−0.162096	−	0.986775i	$-0.551825\pi$
$397$	−6.45949			−0.324192			−0.162096	+	0.986775i	$0.551825\pi$
$398$		0			0
$399$	−4.57944			−0.229259
$400$		0			0
$401$			2.05810i			0.102777i	0.998679	+	0.0513883i	$0.0163646\pi$
$401$			2.05810i			0.102777i	−0.998679	+	0.0513883i	$0.983635\pi$
$402$		0			0
$403$	4.19708	−	7.07279i	0.209072	−	0.352321i
$404$		0			0
$405$	10.7054			0.531957
$406$		0			0
$407$	−9.63048			−0.477365
$408$		0			0
$409$			25.4159i			1.25674i	0.777916	+	0.628368i	$0.216277\pi$
$409$			25.4159i			1.25674i	−0.777916	+	0.628368i	$0.783723\pi$
$410$		0			0
$411$	−28.6526			−1.41333
$412$		0			0
$413$		−	10.7468i		−	0.528815i
$414$		0			0
$415$	8.99918			0.441753
$416$		0			0
$417$	8.22151			0.402609
$418$		0			0
$419$			23.1356i			1.13025i	0.825006	+	0.565125i	$0.191172\pi$
$419$			23.1356i			1.13025i	−0.825006	+	0.565125i	$0.808828\pi$
$420$		0			0
$421$	7.32172			0.356839			0.178419	−	0.983955i	$-0.442902\pi$
$421$	7.32172			0.356839			0.178419	+	0.983955i	$0.442902\pi$
$422$		0			0
$423$			0.462794i			0.0225018i
$424$		0			0
$425$	50.5296			2.45105
$426$		0			0
$427$	11.8802			0.574926
$428$		0			0
$429$	6.79911	−	11.4576i	0.328264	−	0.553180i
$430$		0			0
$431$			29.5377i			1.42278i	0.702797	+	0.711390i	$0.251934\pi$
$431$			29.5377i			1.42278i	−0.702797	+	0.711390i	$0.748066\pi$
$432$		0			0
$433$	−27.4093			−1.31721			−0.658604	−	0.752489i	$-0.728853\pi$
$433$	−27.4093			−1.31721			−0.658604	+	0.752489i	$0.728853\pi$
$434$		0			0
$435$	37.1094			1.77926
$436$		0			0
$437$	−17.4076			−0.832717
$438$		0			0
$439$	−17.6695			−0.843320			−0.421660	−	0.906754i	$-0.638552\pi$
$439$	−17.6695			−0.843320			−0.421660	+	0.906754i	$0.638552\pi$
$440$		0			0
$441$	−1.45434			−0.0692543
$442$		0			0
$443$		−	1.74367i		−	0.0828444i	−0.999142	−	0.0414222i	$-0.986811\pi$
$443$		−	1.74367i		−	0.0828444i	0.999142	−	0.0414222i	$-0.0131889\pi$
$444$		0			0
$445$		−	61.2616i		−	2.90408i
$446$		0			0
$447$		−	4.37305i		−	0.206838i
$448$		0			0
$449$			2.54137i			0.119935i	0.998200	+	0.0599673i	$0.0190997\pi$
$449$			2.54137i			0.119935i	−0.998200	+	0.0599673i	$0.980900\pi$
$450$		0			0
$451$		−	11.3600i		−	0.534921i
$452$		0			0
$453$	−12.6086			−0.592402
$454$		0			0
$455$	7.81085	−	13.1626i	0.366178	−	0.617072i
$456$		0			0
$457$			39.5988i			1.85235i	0.377091	+	0.926176i	$0.376924\pi$
$457$			39.5988i			1.85235i	−0.377091	+	0.926176i	$0.623076\pi$
$458$		0			0
$459$			21.4915i			1.00314i
$460$		0			0
$461$	13.1410			0.612036			0.306018	−	0.952026i	$-0.401003\pi$
$461$	13.1410			0.612036			0.306018	+	0.952026i	$0.401003\pi$
$462$		0			0
$463$			26.9632i			1.25309i	0.779386	+	0.626544i	$0.215531\pi$
$463$			26.9632i			1.25309i	−0.779386	+	0.626544i	$0.784469\pi$
$464$		0			0
$465$	−12.0384			−0.558266
$466$		0			0
$467$		−	31.2099i		−	1.44422i	−0.691776	−	0.722112i	$-0.743171\pi$
$467$		−	31.2099i		−	1.44422i	0.691776	−	0.722112i	$-0.256829\pi$
$468$		0			0
$469$		−	8.65396i		−	0.399603i
$470$		0			0
$471$	−8.86879			−0.408652
$472$		0			0
$473$		−	23.7927i		−	1.09399i
$474$		0			0
$475$	47.9595			2.20053
$476$		0			0
$477$			3.67843i			0.168424i
$478$		0			0
$479$			7.12602i			0.325596i	0.986659	+	0.162798i	$0.0520519\pi$
$479$			7.12602i			0.325596i	−0.986659	+	0.162798i	$0.947948\pi$
$480$		0			0
$481$	10.0469	+	5.96195i	0.458099	+	0.271841i
$482$		0			0
$483$	5.87543			0.267341
$484$		0			0
$485$			14.5519i			0.660769i
$486$		0			0
$487$		−	1.00402i		−	0.0454964i	−0.999741	−	0.0227482i	$-0.992758\pi$
$487$		−	1.00402i		−	0.0454964i	0.999741	−	0.0227482i	$-0.00724160\pi$
$488$		0			0
$489$		−	25.6324i		−	1.15914i
$490$		0			0
$491$			9.80821i			0.442638i	0.975201	+	0.221319i	$0.0710362\pi$
$491$			9.80821i			0.442638i	−0.975201	+	0.221319i	$0.928964\pi$
$492$		0			0
$493$			27.2881i			1.22899i
$494$		0			0
$495$	18.3495			0.824748
$496$		0			0
$497$	−5.29101			−0.237334
$498$		0			0
$499$	−2.81067			−0.125823			−0.0629115	−	0.998019i	$-0.520039\pi$
$499$	−2.81067			−0.125823			−0.0629115	+	0.998019i	$0.520039\pi$
$500$		0			0
$501$	−7.94488			−0.354951
$502$		0			0
$503$	1.76210			0.0785682			0.0392841	−	0.999228i	$-0.487492\pi$
$503$	1.76210			0.0785682			0.0392841	+	0.999228i	$0.487492\pi$
$504$		0			0
$505$			20.5894i			0.916214i
$506$		0			0
$507$	−14.1862	+	7.74393i	−0.630031	+	0.343920i
$508$		0			0
$509$	−2.58451			−0.114556			−0.0572782	−	0.998358i	$-0.518242\pi$
$509$	−2.58451			−0.114556			−0.0572782	+	0.998358i	$0.518242\pi$
$510$		0			0
$511$	12.1985			0.539631
$512$		0			0
$513$			20.3984i			0.900611i
$514$		0			0
$515$	−46.3896			−2.04417
$516$		0			0
$517$		−	0.945800i		−	0.0415962i
$518$		0			0
$519$	−4.58998			−0.201478
$520$		0			0
$521$	−10.1321			−0.443896			−0.221948	−	0.975058i	$-0.571242\pi$
$521$	−10.1321			−0.443896			−0.221948	+	0.975058i	$0.571242\pi$
$522$		0			0
$523$		−	30.8439i		−	1.34871i	−0.738408	−	0.674355i	$-0.764422\pi$
$523$		−	30.8439i		−	1.34871i	0.738408	−	0.674355i	$-0.235578\pi$
$524$		0			0
$525$	−16.1874			−0.706475
$526$		0			0
$527$		−	8.85230i		−	0.385612i
$528$		0			0
$529$	−0.666049			−0.0289586
$530$		0			0
$531$	−15.6295			−0.678262
$532$		0			0
$533$	−7.03264	+	11.8512i	−0.304617	+	0.513332i
$534$		0			0
$535$			81.9542i			3.54319i
$536$		0			0
$537$	−25.4240			−1.09713
$538$		0			0
$539$	2.97219			0.128021
$540$		0			0
$541$	−39.8533			−1.71343			−0.856713	−	0.515793i	$-0.827497\pi$
$541$	−39.8533			−1.71343			−0.856713	+	0.515793i	$0.827497\pi$
$542$		0			0
$543$	−24.3876			−1.04657
$544$		0			0
$545$	25.3983			1.08795
$546$		0			0
$547$			33.2749i			1.42273i	0.702821	+	0.711366i	$0.251923\pi$
$547$			33.2749i			1.42273i	−0.702821	+	0.711366i	$0.748077\pi$
$548$		0			0
$549$		−	17.2779i		−	0.737404i
$550$		0			0
$551$			25.9002i			1.10338i
$552$		0			0
$553$		−	2.81549i		−	0.119727i
$554$		0			0
$555$		−	17.1005i		−	0.725875i
$556$		0			0
$557$	24.4137			1.03444			0.517222	−	0.855852i	$-0.326966\pi$
$557$	24.4137			1.03444			0.517222	+	0.855852i	$0.326966\pi$
$558$		0			0
$559$	−14.7294	+	24.8215i	−0.622986	+	1.04984i
$560$		0			0
$561$		−	14.3404i		−	0.605451i
$562$		0			0
$563$		−	11.8363i		−	0.498842i	−0.968395	−	0.249421i	$-0.919760\pi$
$563$		−	11.8363i		−	0.498842i	0.968395	−	0.249421i	$-0.0802403\pi$
$564$		0			0
$565$	50.4399			2.12202
$566$		0			0
$567$		−	2.52188i		−	0.105909i
$568$		0			0
$569$	−22.6270			−0.948573			−0.474286	−	0.880371i	$-0.657294\pi$
$569$	−22.6270			−0.948573			−0.474286	+	0.880371i	$0.657294\pi$
$570$		0			0
$571$		−	1.63950i		−	0.0686108i	−0.999411	−	0.0343054i	$-0.989078\pi$
$571$		−	1.63950i		−	0.0686108i	0.999411	−	0.0343054i	$-0.0109219\pi$
$572$		0			0
$573$			7.51664i			0.314012i
$574$		0			0
$575$	−61.5322			−2.56607
$576$		0			0
$577$			29.5910i			1.23189i	0.787789	+	0.615945i	$0.211226\pi$
$577$			29.5910i			1.23189i	−0.787789	+	0.615945i	$0.788774\pi$
$578$		0			0
$579$	1.70720			0.0709487
$580$		0			0
$581$		−	2.11994i		−	0.0879497i
$582$		0			0
$583$		−	7.51751i		−	0.311344i
$584$		0			0
$585$	−19.1429	−	11.3596i	−0.791461	−	0.469663i
$586$		0			0
$587$	−41.1847			−1.69988			−0.849938	−	0.526882i	$-0.823361\pi$
$587$	−41.1847			−1.69988			−0.849938	+	0.526882i	$0.823361\pi$
$588$		0			0
$589$		−	8.40205i		−	0.346200i
$590$		0			0
$591$			19.2954i			0.793707i
$592$		0			0
$593$			3.33933i			0.137130i	0.997647	+	0.0685650i	$0.0218421\pi$
$593$			3.33933i			0.137130i	−0.997647	+	0.0685650i	$0.978158\pi$
$594$		0			0
$595$		−	16.4743i		−	0.675380i
$596$		0			0
$597$		−	33.9326i		−	1.38877i
$598$		0			0
$599$	27.4683			1.12232			0.561162	−	0.827706i	$-0.310354\pi$
$599$	27.4683			1.12232			0.561162	+	0.827706i	$0.310354\pi$
$600$		0			0
$601$	−9.80041			−0.399767			−0.199884	−	0.979820i	$-0.564056\pi$
$601$	−9.80041			−0.399767			−0.199884	+	0.979820i	$0.564056\pi$
$602$		0			0
$603$	−12.5858			−0.512533
$604$		0			0
$605$	9.19499			0.373829
$606$		0			0
$607$	−24.7016			−1.00261			−0.501304	−	0.865271i	$-0.667146\pi$
$607$	−24.7016			−1.00261			−0.501304	+	0.865271i	$0.667146\pi$
$608$		0			0
$609$		−	8.74186i		−	0.354238i
$610$		0			0
$611$	−0.585517	+	0.986695i	−0.0236875	+	0.0399174i
$612$		0			0
$613$	33.1575			1.33922			0.669610	−	0.742713i	$-0.266461\pi$
$613$	33.1575			1.33922			0.669610	+	0.742713i	$0.266461\pi$
$614$		0			0
$615$	20.1715			0.813393
$616$		0			0
$617$		−	45.3381i		−	1.82524i	−0.408805	−	0.912622i	$-0.634054\pi$
$617$		−	45.3381i		−	1.82524i	0.408805	−	0.912622i	$-0.365946\pi$
$618$		0			0
$619$	22.2066			0.892560			0.446280	−	0.894893i	$-0.352749\pi$
$619$	22.2066			0.892560			0.446280	+	0.894893i	$0.352749\pi$
$620$		0			0
$621$		−	26.1712i		−	1.05021i
$622$		0			0
$623$	−14.4314			−0.578181
$624$		0			0
$625$	79.4259			3.17704
$626$		0			0
$627$		−	13.6110i		−	0.543570i
$628$		0			0
$629$	12.5747			0.501385
$630$		0			0
$631$		−	9.86072i		−	0.392549i	−0.980549	−	0.196275i	$-0.937116\pi$
$631$		−	9.86072i		−	0.392549i	0.980549	−	0.196275i	$-0.0628844\pi$
$632$		0			0
$633$	−7.86350			−0.312546
$634$		0			0
$635$	32.5031			1.28985
$636$		0			0
$637$	−3.10071	−	1.84000i	−0.122855	−	0.0729034i
$638$		0			0
$639$			7.69493i			0.304407i
$640$		0			0
$641$	−29.8909			−1.18062			−0.590311	−	0.807176i	$-0.700995\pi$
$641$	−29.8909			−1.18062			−0.590311	+	0.807176i	$0.700995\pi$
$642$		0			0
$643$	34.3349			1.35404			0.677019	−	0.735966i	$-0.263272\pi$
$643$	34.3349			1.35404			0.677019	+	0.735966i	$0.263272\pi$
$644$		0			0
$645$	42.2478			1.66351
$646$		0			0
$647$	9.52056			0.374292			0.187146	−	0.982332i	$-0.440076\pi$
$647$	9.52056			0.374292			0.187146	+	0.982332i	$0.440076\pi$
$648$		0			0
$649$	31.9415			1.25381
$650$		0			0
$651$			2.83587i			0.111147i
$652$		0			0
$653$		−	46.5326i		−	1.82096i	−0.413554	−	0.910480i	$-0.635713\pi$
$653$		−	46.5326i		−	1.82096i	0.413554	−	0.910480i	$-0.364287\pi$
$654$		0			0
$655$			11.5375i			0.450809i
$656$		0			0
$657$		−	17.7408i		−	0.692134i
$658$		0			0
$659$		−	24.5788i		−	0.957456i	−0.877963	−	0.478728i	$-0.841098\pi$
$659$		−	24.5788i		−	0.957456i	0.877963	−	0.478728i	$-0.158902\pi$
$660$		0			0
$661$	−19.4726			−0.757395			−0.378697	−	0.925521i	$-0.623628\pi$
$661$	−19.4726			−0.757395			−0.378697	+	0.925521i	$0.623628\pi$
$662$		0			0
$663$	−8.87771	+	14.9604i	−0.344782	+	0.581015i
$664$		0			0
$665$		−	15.6364i		−	0.606352i
$666$		0			0
$667$		−	33.2300i		−	1.28667i
$668$		0			0
$669$	11.9424			0.461722
$670$		0			0
$671$			35.3104i			1.36314i
$672$		0			0
$673$	−33.4160			−1.28809			−0.644046	−	0.764987i	$-0.722745\pi$
$673$	−33.4160			−1.28809			−0.644046	+	0.764987i	$0.722745\pi$
$674$		0			0
$675$			72.1041i			2.77529i
$676$		0			0
$677$			16.9070i			0.649790i	0.945750	+	0.324895i	$0.105329\pi$
$677$			16.9070i			0.649790i	−0.945750	+	0.324895i	$0.894671\pi$
$678$		0			0
$679$	3.42799			0.131554
$680$		0			0
$681$			17.9799i			0.688992i
$682$		0			0
$683$	−3.07903			−0.117816			−0.0589080	−	0.998263i	$-0.518762\pi$
$683$	−3.07903			−0.117816			−0.0589080	+	0.998263i	$0.518762\pi$
$684$		0			0
$685$		−	97.8336i		−	3.73803i
$686$		0			0
$687$			5.38492i			0.205448i
$688$		0			0
$689$	−4.65388	+	7.84257i	−0.177299	+	0.298778i
$690$		0			0
$691$	29.3592			1.11688			0.558439	−	0.829546i	$-0.311400\pi$
$691$	29.3592			1.11688			0.558439	+	0.829546i	$0.311400\pi$
$692$		0			0
$693$		−	4.32258i		−	0.164201i
$694$		0			0
$695$			28.0721i			1.06484i
$696$		0			0
$697$			14.8329i			0.561837i
$698$		0			0
$699$		−	30.0529i		−	1.13671i
$700$		0			0
$701$			23.1700i			0.875119i	0.899190	+	0.437559i	$0.144157\pi$
$701$			23.1700i			0.875119i	−0.899190	+	0.437559i	$0.855843\pi$
$702$		0			0
$703$	11.9351			0.450141
$704$		0			0
$705$	1.67942			0.0632506
$706$		0			0
$707$	4.85023			0.182412
$708$		0			0
$709$	−1.93611			−0.0727121			−0.0363561	−	0.999339i	$-0.511575\pi$
$709$	−1.93611			−0.0727121			−0.0363561	+	0.999339i	$0.511575\pi$
$710$		0			0
$711$	−4.09468			−0.153562
$712$		0			0
$713$			10.7798i			0.403708i
$714$		0			0
$715$	39.1218	+	23.2154i	1.46307	+	0.868205i
$716$		0			0
$717$	−16.7416			−0.625226
$718$		0			0
$719$	33.4382			1.24703			0.623517	−	0.781810i	$-0.285703\pi$
$719$	33.4382			1.24703			0.623517	+	0.781810i	$0.285703\pi$
$720$		0			0
$721$			10.9280i			0.406980i
$722$		0			0
$723$	30.4015			1.13065
$724$		0			0
$725$			91.5517i			3.40015i
$726$		0			0
$727$	47.6574			1.76752			0.883758	−	0.467943i	$-0.155005\pi$
$727$	47.6574			1.76752			0.883758	+	0.467943i	$0.155005\pi$
$728$		0			0
$729$	−24.3223			−0.900828
$730$		0			0
$731$			31.0666i			1.14904i
$732$		0			0
$733$	−10.5296			−0.388921			−0.194460	−	0.980910i	$-0.562296\pi$
$733$	−10.5296			−0.388921			−0.194460	+	0.980910i	$0.562296\pi$
$734$		0			0
$735$			5.27761i			0.194668i
$736$		0			0
$737$	25.7213			0.947455
$738$		0			0
$739$	37.2518			1.37033			0.685164	−	0.728388i	$-0.259730\pi$
$739$	37.2518			1.37033			0.685164	+	0.728388i	$0.259730\pi$
$740$		0			0
$741$	−8.42617	+	14.1995i	−0.309543	+	0.521632i
$742$		0			0
$743$		−	17.1129i		−	0.627810i	−0.949454	−	0.313905i	$-0.898363\pi$
$743$		−	17.1129i		−	0.627810i	0.949454	−	0.313905i	$-0.101637\pi$
$744$		0			0
$745$	14.9317			0.547054
$746$		0			0
$747$	−3.08311			−0.112805
$748$		0			0
$749$	19.3059			0.705423
$750$		0			0
$751$	−0.528361			−0.0192802			−0.00964009	−	0.999954i	$-0.503069\pi$
$751$	−0.528361			−0.0192802			−0.00964009	+	0.999954i	$0.503069\pi$
$752$		0			0
$753$	26.0330			0.948695
$754$		0			0
$755$		−	43.0516i		−	1.56681i
$756$		0			0
$757$			23.8685i			0.867515i	0.901030	+	0.433758i	$0.142813\pi$
$757$			23.8685i			0.867515i	−0.901030	+	0.433758i	$0.857187\pi$
$758$		0			0
$759$			17.4629i			0.633864i
$760$		0			0
$761$			24.1900i			0.876887i	0.898759	+	0.438444i	$0.144470\pi$
$761$			24.1900i			0.876887i	−0.898759	+	0.438444i	$0.855530\pi$
$762$		0			0
$763$		−	5.98308i		−	0.216602i
$764$		0			0
$765$	−23.9592			−0.866248
$766$		0			0
$767$	−33.3227	−	19.7741i	−1.20321	−	0.714001i
$768$		0			0
$769$		−	13.0043i		−	0.468947i	−0.972123	−	0.234474i	$-0.924663\pi$
$769$		−	13.0043i		−	0.468947i	0.972123	−	0.234474i	$-0.0753366\pi$
$770$		0			0
$771$			1.43030i			0.0515108i
$772$		0			0
$773$	3.76057			0.135258			0.0676292	−	0.997711i	$-0.478457\pi$
$773$	3.76057			0.135258			0.0676292	+	0.997711i	$0.478457\pi$
$774$		0			0
$775$		−	29.6995i		−	1.06684i
$776$		0			0
$777$	−4.02835			−0.144516
$778$		0			0
$779$			14.0785i			0.504414i
$780$		0			0
$781$		−	15.7259i		−	0.562717i
$782$		0			0
$783$	−38.9392			−1.39157
$784$		0			0
$785$		−	30.2822i		−	1.08082i
$786$		0			0
$787$	19.7264			0.703169			0.351584	−	0.936156i	$-0.385643\pi$
$787$	19.7264			0.703169			0.351584	+	0.936156i	$0.385643\pi$
$788$		0			0
$789$			5.01896i			0.178680i
$790$		0			0
$791$		−	11.8821i		−	0.422479i
$792$		0			0
$793$	21.8597	−	36.8372i	0.776259	−	1.30813i
$794$		0			0
$795$	13.3486			0.473425
$796$		0			0
$797$		−	13.5927i		−	0.481479i	−0.970590	−	0.240740i	$-0.922610\pi$
$797$		−	13.5927i		−	0.481479i	0.970590	−	0.240740i	$-0.0773900\pi$
$798$		0			0
$799$			1.23495i			0.0436893i
$800$		0			0
$801$			20.9881i			0.741579i
$802$		0			0
$803$			36.2564i			1.27946i
$804$		0			0
$805$			20.0615i			0.707075i
$806$		0			0
$807$	5.28004			0.185866
$808$		0			0
$809$	39.7054			1.39597			0.697983	−	0.716114i	$-0.254081\pi$
$809$	39.7054			1.39597			0.697983	+	0.716114i	$0.254081\pi$
$810$		0			0
$811$	−1.54998			−0.0544270			−0.0272135	−	0.999630i	$-0.508663\pi$
$811$	−1.54998			−0.0544270			−0.0272135	+	0.999630i	$0.508663\pi$
$812$		0			0
$813$	−31.5132			−1.10521
$814$		0			0
$815$	87.5211			3.06573
$816$		0			0
$817$			29.4864i			1.03160i
$818$		0			0
$819$	−2.67598	+	4.50949i	−0.0935065	+	0.157574i
$820$		0			0
$821$	43.4323			1.51580			0.757900	−	0.652371i	$-0.226226\pi$
$821$	43.4323			1.51580			0.757900	+	0.652371i	$0.226226\pi$
$822$		0			0
$823$	−56.1465			−1.95714			−0.978571	−	0.205908i	$-0.933985\pi$
$823$	−56.1465			−1.95714			−0.978571	+	0.205908i	$0.933985\pi$
$824$		0			0
$825$		−	48.1121i		−	1.67505i
$826$		0			0
$827$	0.504695			0.0175500			0.00877499	−	0.999961i	$-0.497207\pi$
$827$	0.504695			0.0175500			0.00877499	+	0.999961i	$0.497207\pi$
$828$		0			0
$829$		−	8.83750i		−	0.306939i	−0.988153	−	0.153470i	$-0.950955\pi$
$829$		−	8.83750i		−	0.306939i	0.988153	−	0.153470i	$-0.0490447\pi$
$830$		0			0
$831$	8.56827			0.297230
$832$		0			0
$833$	−3.88084			−0.134463
$834$		0			0
$835$		−	27.1276i		−	0.938789i
$836$		0			0
$837$	12.6319			0.436624
$838$		0			0
$839$			3.95785i			0.136640i	0.997663	+	0.0683202i	$0.0217639\pi$
$839$			3.95785i			0.136640i	−0.997663	+	0.0683202i	$0.978236\pi$
$840$		0			0
$841$	−20.4418			−0.704888
$842$		0			0
$843$	2.47600			0.0852780
$844$		0			0
$845$	−26.4414	−	48.4384i	−0.909613	−	1.66633i
$846$		0			0
$847$		−	2.16606i		−	0.0744267i
$848$		0			0
$849$	−6.44049			−0.221037
$850$		0			0
$851$	−15.3128			−0.524915
$852$		0			0
$853$	10.0094			0.342715			0.171358	−	0.985209i	$-0.445185\pi$
$853$	10.0094			0.342715			0.171358	+	0.985209i	$0.445185\pi$
$854$		0			0
$855$	−22.7406			−0.777712
$856$		0			0
$857$	16.3535			0.558624			0.279312	−	0.960200i	$-0.409894\pi$
$857$	16.3535			0.558624			0.279312	+	0.960200i	$0.409894\pi$
$858$		0			0
$859$			46.1150i			1.57342i	0.617322	+	0.786711i	$0.288218\pi$
$859$			46.1150i			1.57342i	−0.617322	+	0.786711i	$0.711782\pi$
$860$		0			0
$861$		−	4.75179i		−	0.161941i
$862$		0			0
$863$		−	48.4827i		−	1.65037i	−0.564864	−	0.825184i	$-0.691071\pi$
$863$		−	48.4827i		−	1.65037i	0.564864	−	0.825184i	$-0.308929\pi$
$864$		0			0
$865$		−	15.6724i		−	0.532876i
$866$		0			0
$867$		−	2.41071i		−	0.0818719i
$868$		0			0
$869$	8.36818			0.283871
$870$		0			0
$871$	−26.8334	−	15.9233i	−0.909216	−	0.539540i
$872$		0			0
$873$		−	4.98547i		−	0.168732i
$874$		0			0
$875$		−	34.0462i		−	1.15097i
$876$		0			0
$877$	−28.3417			−0.957033			−0.478516	−	0.878079i	$-0.658825\pi$
$877$	−28.3417			−0.957033			−0.478516	+	0.878079i	$0.658825\pi$
$878$		0			0
$879$		−	10.0886i		−	0.340279i
$880$		0			0
$881$	41.6725			1.40398			0.701991	−	0.712185i	$-0.252295\pi$
$881$	41.6725			1.40398			0.701991	+	0.712185i	$0.252295\pi$
$882$		0			0
$883$			13.9586i			0.469743i	0.972026	+	0.234872i	$0.0754670\pi$
$883$			13.9586i			0.469743i	−0.972026	+	0.234872i	$0.924533\pi$
$884$		0			0
$885$			56.7174i			1.90653i
$886$		0			0
$887$	26.0576			0.874929			0.437465	−	0.899236i	$-0.355876\pi$
$887$	26.0576			0.874929			0.437465	+	0.899236i	$0.355876\pi$
$888$		0			0
$889$		−	7.65675i		−	0.256799i
$890$		0			0
$891$	7.49550			0.251109
$892$		0			0
$893$			1.17213i			0.0392240i
$894$		0			0
$895$		−	86.8096i		−	2.90172i
$896$		0			0
$897$	10.8108	−	18.2180i	0.360962	−	0.608282i
$898$		0			0
$899$	16.0390			0.534930
$900$		0			0
$901$			9.81575i			0.327010i
$902$		0			0
$903$		−	9.95231i		−	0.331192i
$904$		0			0
$905$		−	83.2706i		−	2.76801i
$906$		0			0
$907$			5.70501i			0.189432i	0.995504	+	0.0947159i	$0.0301943\pi$
$907$			5.70501i			0.189432i	−0.995504	+	0.0947159i	$0.969806\pi$
$908$		0			0
$909$		−	7.05388i		−	0.233963i
$910$		0			0
$911$	37.8264			1.25325			0.626623	−	0.779323i	$-0.284437\pi$
$911$	37.8264			1.25325			0.626623	+	0.779323i	$0.284437\pi$
$912$		0			0
$913$	6.30086			0.208528
$914$		0			0
$915$	−62.6993			−2.07278
$916$		0			0
$917$	2.71789			0.0897527
$918$		0			0
$919$	−15.8594			−0.523152			−0.261576	−	0.965183i	$-0.584242\pi$
$919$	−15.8594			−0.523152			−0.261576	+	0.965183i	$0.584242\pi$
$920$		0			0
$921$		−	28.1929i		−	0.928987i
$922$		0			0
$923$	−9.73546	+	16.4059i	−0.320446	+	0.540006i
$924$		0			0
$925$	42.1881			1.38714
$926$		0			0
$927$	15.8930			0.521995
$928$		0			0
$929$			12.4095i			0.407141i	0.979060	+	0.203571i	$0.0652546\pi$
$929$			12.4095i			0.407141i	−0.979060	+	0.203571i	$0.934745\pi$
$930$		0			0
$931$	−3.68346			−0.120720
$932$		0			0
$933$		−	2.36600i		−	0.0774595i
$934$		0			0
$935$	48.9648			1.60132
$936$		0			0
$937$	27.8945			0.911275			0.455637	−	0.890165i	$-0.349411\pi$
$937$	27.8945			0.911275			0.455637	+	0.890165i	$0.349411\pi$
$938$		0			0
$939$			38.4612i			1.25513i
$940$		0			0
$941$	−33.5634			−1.09414			−0.547068	−	0.837088i	$-0.684256\pi$
$941$	−33.5634			−1.09414			−0.547068	+	0.837088i	$0.684256\pi$
$942$		0			0
$943$		−	18.0627i		−	0.588203i
$944$		0			0
$945$	23.5083			0.764724
$946$		0			0
$947$	−18.9092			−0.614466			−0.307233	−	0.951634i	$-0.599403\pi$
$947$	−18.9092			−0.614466			−0.307233	+	0.951634i	$0.599403\pi$
$948$		0			0
$949$	22.4453	−	37.8241i	0.728604	−	1.22782i
$950$		0			0
$951$			13.6570i			0.442858i
$952$		0			0
$953$	−21.7892			−0.705823			−0.352911	−	0.935657i	$-0.614808\pi$
$953$	−21.7892			−0.705823			−0.352911	+	0.935657i	$0.614808\pi$
$954$		0			0
$955$	−25.6654			−0.830512
$956$		0			0
$957$	25.9825			0.839895
$958$		0			0
$959$	−23.0466			−0.744215
$960$		0			0
$961$	25.7969			0.832159
$962$		0			0
$963$		−	28.0774i		−	0.904781i
$964$		0			0
$965$			5.82918i			0.187648i
$966$		0			0
$967$		−	42.4630i		−	1.36552i	−0.730643	−	0.682759i	$-0.760780\pi$
$967$		−	42.4630i		−	1.36552i	0.730643	−	0.682759i	$-0.239220\pi$
$968$		0			0
$969$			17.7721i			0.570922i
$970$		0			0
$971$		−	47.0765i		−	1.51076i	−0.655289	−	0.755378i	$-0.727453\pi$
$971$		−	47.0765i		−	1.51076i	0.655289	−	0.755378i	$-0.272547\pi$
$972$		0			0
$973$	6.61294			0.212001
$974$		0			0
$975$	−29.7848	+	50.1924i	−0.953876	+	1.60744i
$976$		0			0
$977$			23.5667i			0.753967i	0.926220	+	0.376984i	$0.123039\pi$
$977$			23.5667i			0.753967i	−0.926220	+	0.376984i	$0.876961\pi$
$978$		0			0
$979$		−	42.8929i		−	1.37086i
$980$		0			0
$981$	−8.70143			−0.277815
$982$		0			0
$983$			33.4979i			1.06842i	0.845353	+	0.534208i	$0.179390\pi$
$983$			33.4979i			1.06842i	−0.845353	+	0.534208i	$0.820610\pi$
$984$		0			0
$985$	−65.8836			−2.09923
$986$		0			0
$987$		−	0.395621i		−	0.0125927i
$988$		0			0
$989$		−	37.8312i		−	1.20296i
$990$		0			0
$991$	9.57436			0.304140			0.152070	−	0.988370i	$-0.451406\pi$
$991$	9.57436			0.304140			0.152070	+	0.988370i	$0.451406\pi$
$992$		0			0
$993$			26.2976i			0.834528i
$994$		0			0
$995$	115.862			3.67307
$996$		0			0
$997$			44.0185i			1.39408i	0.717033	+	0.697039i	$0.245500\pi$
$997$			44.0185i			1.39408i	−0.717033	+	0.697039i	$0.754500\pi$
$998$		0			0
$999$			17.9437i			0.567712i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.6		84
4.3	odd	2		728.2.i.a.701.52	yes	84
8.3	odd	2		728.2.i.a.701.34	yes	84
8.5	even	2	inner	2912.2.i.a.337.79		84
13.12	even	2	inner	2912.2.i.a.337.80		84
52.51	odd	2		728.2.i.a.701.33	✓	84
104.51	odd	2		728.2.i.a.701.51	yes	84
104.77	even	2	inner	2912.2.i.a.337.5		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.33	✓	84	52.51	odd	2
728.2.i.a.701.34	yes	84	8.3	odd	2
728.2.i.a.701.51	yes	84	104.51	odd	2
728.2.i.a.701.52	yes	84	4.3	odd	2
2912.2.i.a.337.5		84	104.77	even	2	inner
2912.2.i.a.337.6		84	1.1	even	1	trivial
2912.2.i.a.337.79		84	8.5	even	2	inner
2912.2.i.a.337.80		84	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.24325i q^{3} -4.24503 q^{5} +1.00000i q^{7} +1.45434 q^{9} -2.97219 q^{11} +(3.10071 + 1.84000i) q^{13} -5.27761i q^{15} +3.88084 q^{17} +3.68346 q^{19} -1.24325 q^{21} -4.72588 q^{23} +13.0203 q^{25} +5.53784i q^{27} +7.03148i q^{29} -2.28102i q^{31} -3.69517i q^{33} -4.24503i q^{35} +3.24019 q^{37} +(-2.28757 + 3.85494i) q^{39} +3.82209i q^{41} +8.00510i q^{43} -6.17371 q^{45} +0.318216i q^{47} -1.00000 q^{49} +4.82484i q^{51} +2.52928i q^{53} +12.6170 q^{55} +4.57944i q^{57} -10.7468 q^{59} -11.8802i q^{61} +1.45434i q^{63} +(-13.1626 - 7.81085i) q^{65} -8.65396 q^{67} -5.87543i q^{69} +5.29101i q^{71} -12.1985i q^{73} +16.1874i q^{75} -2.97219i q^{77} -2.81549 q^{79} -2.52188 q^{81} -2.11994 q^{83} -16.4743 q^{85} -8.74186 q^{87} +14.4314i q^{89} +(-1.84000 + 3.10071i) q^{91} +2.83587 q^{93} -15.6364 q^{95} -3.42799i q^{97} -4.32258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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