LMFDB - Embedded newform 3872.2.g.d.1935.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1935,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3872.1935");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.g (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:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1935.11
Character	$\chi$	$=$	3872.1935
Dual form			3872.2.g.d.1935.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24811 q^{3} +3.46137i q^{5} +1.43093 q^{7} -1.44222 q^{9} -6.59738 q^{13} -4.32018i q^{15} -0.752791i q^{17} +2.75168i q^{19} -1.78596 q^{21} -4.40738i q^{23} -6.98111 q^{25} +5.54438 q^{27} -0.125227 q^{29} -3.84067i q^{31} +4.95298i q^{35} -2.01363i q^{37} +8.23427 q^{39} +2.65007i q^{41} -4.68327i q^{43} -4.99205i q^{45} -4.61654i q^{47} -4.95244 q^{49} +0.939567i q^{51} -5.21442i q^{53} -3.43440i q^{57} -6.16277 q^{59} +2.93925 q^{61} -2.06371 q^{63} -22.8360i q^{65} +10.7156 q^{67} +5.50091i q^{69} -11.7635i q^{71} +6.25654i q^{73} +8.71321 q^{75} -1.63879 q^{79} -2.59336 q^{81} +12.2151i q^{83} +2.60569 q^{85} +0.156297 q^{87} +10.6955 q^{89} -9.44039 q^{91} +4.79358i q^{93} -9.52458 q^{95} +11.3372 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{3} - 44 q^{25} - 8 q^{27} + 52 q^{49} + 72 q^{59} + 28 q^{67} + 76 q^{75} - 112 q^{81} + 20 q^{89} - 48 q^{91} + 8 q^{97}+O(q^{100}) $ 32 * q - 8 * q^3 - 44 * q^25 - 8 * q^27 + 52 * q^49 + 72 * q^59 + 28 * q^67 + 76 * q^75 - 112 * q^81 + 20 * q^89 - 48 * q^91 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−1.24811	−0.720598	−0.360299	−	0.932837i	$-0.617325\pi$
$3$	−1.24811	−0.720598	−0.360299	+	0.932837i	$0.617325\pi$
$4$	0	0
$5$	3.46137i	1.54797i	0.633202	+	0.773987i	$0.281740\pi$
$5$	3.46137i	1.54797i	−0.633202	+	0.773987i	$0.718260\pi$
$6$	0	0
$7$	1.43093	0.540840	0.270420	−	0.962742i	$-0.412837\pi$
$7$	1.43093	0.540840	0.270420	+	0.962742i	$0.412837\pi$
$8$	0	0
$9$	−1.44222	−0.480739
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.59738	−1.82979	−0.914893	−	0.403697i	$-0.867725\pi$
$13$	−6.59738	−1.82979	−0.914893	+	0.403697i	$0.867725\pi$
$14$	0	0
$15$	− 4.32018i	− 1.11547i
$16$	0	0
$17$	− 0.752791i	− 0.182579i	−0.995824	−	0.0912893i	$-0.970901\pi$
$17$	− 0.752791i	− 0.182579i	0.995824	−	0.0912893i	$-0.0290988\pi$
$18$	0	0
$19$	2.75168i	0.631278i	0.948879	+	0.315639i	$0.102219\pi$
$19$	2.75168i	0.631278i	−0.948879	+	0.315639i	$0.897781\pi$
$20$	0	0
$21$	−1.78596	−0.389728
$22$	0	0
$23$	− 4.40738i	− 0.919003i	−0.888177	−	0.459502i	$-0.848028\pi$
$23$	− 4.40738i	− 0.919003i	0.888177	−	0.459502i	$-0.151972\pi$
$24$	0	0
$25$	−6.98111	−1.39622
$26$	0	0
$27$	5.54438	1.06702
$28$	0	0
$29$	−0.125227	−0.0232541	−0.0116270	−	0.999932i	$-0.503701\pi$
$29$	−0.125227	−0.0232541	−0.0116270	+	0.999932i	$0.503701\pi$
$30$	0	0
$31$	− 3.84067i	− 0.689804i	−0.938639	−	0.344902i	$-0.887912\pi$
$31$	− 3.84067i	− 0.689804i	0.938639	−	0.344902i	$-0.112088\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.95298i	0.837206i
$36$	0	0
$37$	− 2.01363i	− 0.331039i	−0.986207	−	0.165520i	$-0.947070\pi$
$37$	− 2.01363i	− 0.331039i	0.986207	−	0.165520i	$-0.0529301\pi$
$38$	0	0
$39$	8.23427	1.31854
$40$	0	0
$41$	2.65007i	0.413872i	0.978355	+	0.206936i	$0.0663491\pi$
$41$	2.65007i	0.413872i	−0.978355	+	0.206936i	$0.933651\pi$
$42$	0	0
$43$	− 4.68327i	− 0.714192i	−0.934068	−	0.357096i	$-0.883767\pi$
$43$	− 4.68327i	− 0.714192i	0.934068	−	0.357096i	$-0.116233\pi$
$44$	0	0
$45$	− 4.99205i	− 0.744171i
$46$	0	0
$47$	− 4.61654i	− 0.673391i	−0.941613	−	0.336696i	$-0.890691\pi$
$47$	− 4.61654i	− 0.673391i	0.941613	−	0.336696i	$-0.109309\pi$
$48$	0	0
$49$	−4.95244	−0.707492
$50$	0	0
$51$	0.939567i	0.131566i
$52$	0	0
$53$	− 5.21442i	− 0.716256i	−0.933672	−	0.358128i	$-0.883415\pi$
$53$	− 5.21442i	− 0.716256i	0.933672	−	0.358128i	$-0.116585\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.43440i	− 0.454897i
$58$	0	0
$59$	−6.16277	−0.802324	−0.401162	−	0.916007i	$-0.631394\pi$
$59$	−6.16277	−0.802324	−0.401162	+	0.916007i	$0.631394\pi$
$60$	0	0
$61$	2.93925	0.376333	0.188166	−	0.982137i	$-0.439746\pi$
$61$	2.93925	0.376333	0.188166	+	0.982137i	$0.439746\pi$
$62$	0	0
$63$	−2.06371	−0.260003
$64$	0	0
$65$	− 22.8360i	− 2.83246i
$66$	0	0
$67$	10.7156	1.30912	0.654559	−	0.756011i	$-0.272854\pi$
$67$	10.7156	1.30912	0.654559	+	0.756011i	$0.272854\pi$
$68$	0	0
$69$	5.50091i	0.662232i
$70$	0	0
$71$	− 11.7635i	− 1.39607i	−0.716064	−	0.698034i	$-0.754058\pi$
$71$	− 11.7635i	− 1.39607i	0.716064	−	0.698034i	$-0.245942\pi$
$72$	0	0
$73$	6.25654i	0.732273i	0.930561	+	0.366136i	$0.119320\pi$
$73$	6.25654i	0.732273i	−0.930561	+	0.366136i	$0.880680\pi$
$74$	0	0
$75$	8.71321	1.00611
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.63879	−0.184378	−0.0921892	−	0.995742i	$-0.529386\pi$
$79$	−1.63879	−0.184378	−0.0921892	+	0.995742i	$0.529386\pi$
$80$	0	0
$81$	−2.59336	−0.288151
$82$	0	0
$83$	12.2151i	1.34079i	0.742006	+	0.670393i	$0.233874\pi$
$83$	12.2151i	1.34079i	−0.742006	+	0.670393i	$0.766126\pi$
$84$	0	0
$85$	2.60569	0.282627
$86$	0	0
$87$	0.156297	0.0167568
$88$	0	0
$89$	10.6955	1.13372	0.566858	−	0.823815i	$-0.308159\pi$
$89$	10.6955	1.13372	0.566858	+	0.823815i	$0.308159\pi$
$90$	0	0
$91$	−9.44039	−0.989622
$92$	0	0
$93$	4.79358i	0.497071i
$94$	0	0
$95$	−9.52458	−0.977201
$96$	0	0
$97$	11.3372	1.15112	0.575560	−	0.817760i	$-0.304784\pi$
$97$	11.3372	1.15112	0.575560	+	0.817760i	$0.304784\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.83322	0.878938	0.439469	−	0.898258i	$-0.355167\pi$
$101$	8.83322	0.878938	0.439469	+	0.898258i	$0.355167\pi$
$102$	0	0
$103$	17.5871i	1.73291i	0.499253	+	0.866456i	$0.333608\pi$
$103$	17.5871i	1.73291i	−0.499253	+	0.866456i	$0.666392\pi$
$104$	0	0
$105$	− 6.18187i	− 0.603289i
$106$	0	0
$107$	4.63834i	0.448405i	0.974543	+	0.224203i	$0.0719777\pi$
$107$	4.63834i	0.448405i	−0.974543	+	0.224203i	$0.928022\pi$
$108$	0	0
$109$	−11.8543	−1.13544	−0.567718	−	0.823223i	$-0.692173\pi$
$109$	−11.8543	−1.13544	−0.567718	+	0.823223i	$0.692173\pi$
$110$	0	0
$111$	2.51324i	0.238546i
$112$	0	0
$113$	2.25464	0.212099	0.106049	−	0.994361i	$-0.466180\pi$
$113$	2.25464	0.212099	0.106049	+	0.994361i	$0.466180\pi$
$114$	0	0
$115$	15.2556	1.42259
$116$	0	0
$117$	9.51486	0.879649
$118$	0	0
$119$	− 1.07719i	− 0.0987458i
$120$	0	0
$121$	0	0
$122$	0	0
$123$	− 3.30758i	− 0.298235i
$124$	0	0
$125$	− 6.85737i	− 0.613342i
$126$	0	0
$127$	−13.8767	−1.23135	−0.615677	−	0.787999i	$-0.711117\pi$
$127$	−13.8767	−1.23135	−0.615677	+	0.787999i	$0.711117\pi$
$128$	0	0
$129$	5.84525i	0.514645i
$130$	0	0
$131$	− 8.50766i	− 0.743317i	−0.928369	−	0.371659i	$-0.878789\pi$
$131$	− 8.50766i	− 0.743317i	0.928369	−	0.371659i	$-0.121211\pi$
$132$	0	0
$133$	3.93745i	0.341420i
$134$	0	0
$135$	19.1912i	1.65171i
$136$	0	0
$137$	4.86400	0.415560	0.207780	−	0.978176i	$-0.433376\pi$
$137$	4.86400	0.415560	0.207780	+	0.978176i	$0.433376\pi$
$138$	0	0
$139$	− 16.0697i	− 1.36301i	−0.731812	−	0.681507i	$-0.761325\pi$
$139$	− 16.0697i	− 1.36301i	0.731812	−	0.681507i	$-0.238675\pi$
$140$	0	0
$141$	5.76196i	0.485244i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 0.433457i	− 0.0359967i
$146$	0	0
$147$	6.18120	0.509817
$148$	0	0
$149$	−21.2722	−1.74269	−0.871343	−	0.490675i	$-0.836750\pi$
$149$	−21.2722	−1.74269	−0.871343	+	0.490675i	$0.836750\pi$
$150$	0	0
$151$	15.9141	1.29507	0.647536	−	0.762035i	$-0.275800\pi$
$151$	15.9141	1.29507	0.647536	+	0.762035i	$0.275800\pi$
$152$	0	0
$153$	1.08569i	0.0877726i
$154$	0	0
$155$	13.2940	1.06780
$156$	0	0
$157$	2.08944i	0.166756i	0.996518	+	0.0833778i	$0.0265708\pi$
$157$	2.08944i	0.166756i	−0.996518	+	0.0833778i	$0.973429\pi$
$158$	0	0
$159$	6.50819i	0.516133i
$160$	0	0
$161$	− 6.30665i	− 0.497034i
$162$	0	0
$163$	10.9694	0.859191	0.429596	−	0.903021i	$-0.358656\pi$
$163$	10.9694	0.859191	0.429596	+	0.903021i	$0.358656\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.9846	1.00478	0.502391	−	0.864641i	$-0.332454\pi$
$167$	12.9846	1.00478	0.502391	+	0.864641i	$0.332454\pi$
$168$	0	0
$169$	30.5255	2.34811
$170$	0	0
$171$	− 3.96851i	− 0.303480i
$172$	0	0
$173$	−3.48265	−0.264781	−0.132390	−	0.991198i	$-0.542265\pi$
$173$	−3.48265	−0.264781	−0.132390	+	0.991198i	$0.542265\pi$
$174$	0	0
$175$	−9.98947	−0.755133
$176$	0	0
$177$	7.69182	0.578153
$178$	0	0
$179$	−2.70436	−0.202133	−0.101067	−	0.994880i	$-0.532226\pi$
$179$	−2.70436	−0.202133	−0.101067	+	0.994880i	$0.532226\pi$
$180$	0	0
$181$	− 11.0955i	− 0.824719i	−0.911021	−	0.412359i	$-0.864705\pi$
$181$	− 11.0955i	− 0.824719i	0.911021	−	0.412359i	$-0.135295\pi$
$182$	0	0
$183$	−3.66852	−0.271184
$184$	0	0
$185$	6.96993	0.512440
$186$	0	0
$187$	0	0
$188$	0	0
$189$	7.93362	0.577086
$190$	0	0
$191$	− 13.6009i	− 0.984127i	−0.870559	−	0.492064i	$-0.836243\pi$
$191$	− 13.6009i	− 0.984127i	0.870559	−	0.492064i	$-0.163757\pi$
$192$	0	0
$193$	5.39032i	0.388003i	0.981001	+	0.194002i	$0.0621467\pi$
$193$	5.39032i	0.388003i	−0.981001	+	0.194002i	$0.937853\pi$
$194$	0	0
$195$	28.5019i	2.04106i
$196$	0	0
$197$	−6.42744	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$197$	−6.42744	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$198$	0	0
$199$	− 3.62012i	− 0.256623i	−0.991734	−	0.128312i	$-0.959044\pi$
$199$	− 3.62012i	− 0.256623i	0.991734	−	0.128312i	$-0.0409558\pi$
$200$	0	0
$201$	−13.3743	−0.943348
$202$	0	0
$203$	−0.179191	−0.0125767
$204$	0	0
$205$	−9.17289	−0.640662
$206$	0	0
$207$	6.35640i	0.441801i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 20.9334i	− 1.44112i	−0.693394	−	0.720558i	$-0.743886\pi$
$211$	− 20.9334i	− 1.44112i	0.693394	−	0.720558i	$-0.256114\pi$
$212$	0	0
$213$	14.6821i	1.00600i
$214$	0	0
$215$	16.2106	1.10555
$216$	0	0
$217$	− 5.49572i	− 0.373074i
$218$	0	0
$219$	− 7.80886i	− 0.527674i
$220$	0	0
$221$	4.96645i	0.334080i
$222$	0	0
$223$	2.65371i	0.177706i	0.996045	+	0.0888528i	$0.0283201\pi$
$223$	2.65371i	0.177706i	−0.996045	+	0.0888528i	$0.971680\pi$
$224$	0	0
$225$	10.0683	0.671218
$226$	0	0
$227$	− 3.48209i	− 0.231115i	−0.993301	−	0.115557i	$-0.963135\pi$
$227$	− 3.48209i	− 0.231115i	0.993301	−	0.115557i	$-0.0368654\pi$
$228$	0	0
$229$	13.1778i	0.870816i	0.900233	+	0.435408i	$0.143396\pi$
$229$	13.1778i	0.870816i	−0.900233	+	0.435408i	$0.856604\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.97965i	0.457252i	0.973514	+	0.228626i	$0.0734233\pi$
$233$	6.97965i	0.457252i	−0.973514	+	0.228626i	$0.926577\pi$
$234$	0	0
$235$	15.9796	1.04239
$236$	0	0
$237$	2.04540	0.132863
$238$	0	0
$239$	2.38421	0.154221	0.0771107	−	0.997023i	$-0.475431\pi$
$239$	2.38421	0.154221	0.0771107	+	0.997023i	$0.475431\pi$
$240$	0	0
$241$	− 10.0044i	− 0.644440i	−0.946665	−	0.322220i	$-0.895571\pi$
$241$	− 10.0044i	− 0.644440i	0.946665	−	0.322220i	$-0.104429\pi$
$242$	0	0
$243$	−13.3963	−0.859376
$244$	0	0
$245$	− 17.1423i	− 1.09518i
$246$	0	0
$247$	− 18.1539i	− 1.15510i
$248$	0	0
$249$	− 15.2459i	− 0.966167i
$250$	0	0
$251$	17.4623	1.10221	0.551106	−	0.834435i	$-0.314206\pi$
$251$	17.4623	1.10221	0.551106	+	0.834435i	$0.314206\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.25219	−0.203660
$256$	0	0
$257$	−8.15614	−0.508766	−0.254383	−	0.967104i	$-0.581872\pi$
$257$	−8.15614	−0.508766	−0.254383	+	0.967104i	$0.581872\pi$
$258$	0	0
$259$	− 2.88136i	− 0.179039i
$260$	0	0
$261$	0.180604	0.0111791
$262$	0	0
$263$	21.0996	1.30106	0.650530	−	0.759481i	$-0.274547\pi$
$263$	21.0996	1.30106	0.650530	+	0.759481i	$0.274547\pi$
$264$	0	0
$265$	18.0491	1.10875
$266$	0	0
$267$	−13.3491	−0.816953
$268$	0	0
$269$	− 20.9725i	− 1.27872i	−0.768909	−	0.639358i	$-0.779200\pi$
$269$	− 20.9725i	− 1.27872i	0.768909	−	0.639358i	$-0.220800\pi$
$270$	0	0
$271$	19.9480	1.21175	0.605877	−	0.795559i	$-0.292822\pi$
$271$	19.9480	1.21175	0.605877	+	0.795559i	$0.292822\pi$
$272$	0	0
$273$	11.7827	0.713119
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.07111	0.0643568	0.0321784	−	0.999482i	$-0.489756\pi$
$277$	1.07111	0.0643568	0.0321784	+	0.999482i	$0.489756\pi$
$278$	0	0
$279$	5.53907i	0.331616i
$280$	0	0
$281$	3.48939i	0.208159i	0.994569	+	0.104080i	$0.0331897\pi$
$281$	3.48939i	0.208159i	−0.994569	+	0.104080i	$0.966810\pi$
$282$	0	0
$283$	− 20.4791i	− 1.21736i	−0.793418	−	0.608678i	$-0.791700\pi$
$283$	− 20.4791i	− 1.21736i	0.793418	−	0.608678i	$-0.208300\pi$
$284$	0	0
$285$	11.8877	0.704169
$286$	0	0
$287$	3.79206i	0.223838i
$288$	0	0
$289$	16.4333	0.966665
$290$	0	0
$291$	−14.1501	−0.829494
$292$	0	0
$293$	−25.9914	−1.51843	−0.759217	−	0.650837i	$-0.774418\pi$
$293$	−25.9914	−1.51843	−0.759217	+	0.650837i	$0.774418\pi$
$294$	0	0
$295$	− 21.3316i	− 1.24198i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.0772i	1.68158i
$300$	0	0
$301$	− 6.70143i	− 0.386264i
$302$	0	0
$303$	−11.0248	−0.633361
$304$	0	0
$305$	10.1739i	0.582553i
$306$	0	0
$307$	− 9.33420i	− 0.532731i	−0.963872	−	0.266366i	$-0.914177\pi$
$307$	− 9.33420i	− 0.532731i	0.963872	−	0.266366i	$-0.0858228\pi$
$308$	0	0
$309$	− 21.9507i	− 1.24873i
$310$	0	0
$311$	− 19.4661i	− 1.10382i	−0.833904	−	0.551909i	$-0.813899\pi$
$311$	− 19.4661i	− 1.10382i	0.833904	−	0.551909i	$-0.186101\pi$
$312$	0	0
$313$	14.7990	0.836490	0.418245	−	0.908334i	$-0.362645\pi$
$313$	14.7990	0.836490	0.418245	+	0.908334i	$0.362645\pi$
$314$	0	0
$315$	− 7.14327i	− 0.402478i
$316$	0	0
$317$	− 31.7286i	− 1.78205i	−0.453950	−	0.891027i	$-0.649985\pi$
$317$	− 31.7286i	− 1.78205i	0.453950	−	0.891027i	$-0.350015\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	− 5.78916i	− 0.323120i
$322$	0	0
$323$	2.07144	0.115258
$324$	0	0
$325$	46.0571	2.55479
$326$	0	0
$327$	14.7955	0.818192
$328$	0	0
$329$	− 6.60594i	− 0.364197i
$330$	0	0
$331$	0.462745	0.0254347	0.0127174	−	0.999919i	$-0.495952\pi$
$331$	0.462745	0.0254347	0.0127174	+	0.999919i	$0.495952\pi$
$332$	0	0
$333$	2.90409i	0.159143i
$334$	0	0
$335$	37.0907i	2.02648i
$336$	0	0
$337$	− 22.4702i	− 1.22403i	−0.790847	−	0.612014i	$-0.790360\pi$
$337$	− 22.4702i	− 1.22403i	0.790847	−	0.612014i	$-0.209640\pi$
$338$	0	0
$339$	−2.81404	−0.152838
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−17.1031	−0.923480
$344$	0	0
$345$	−19.0407	−1.02512
$346$	0	0
$347$	− 18.0797i	− 0.970570i	−0.874356	−	0.485285i	$-0.838716\pi$
$347$	− 18.0797i	− 0.970570i	0.874356	−	0.485285i	$-0.161284\pi$
$348$	0	0
$349$	0.318230	0.0170344	0.00851722	−	0.999964i	$-0.497289\pi$
$349$	0.318230	0.0170344	0.00851722	+	0.999964i	$0.497289\pi$
$350$	0	0
$351$	−36.5784	−1.95241
$352$	0	0
$353$	−35.8303	−1.90706	−0.953529	−	0.301303i	$-0.902579\pi$
$353$	−35.8303	−1.90706	−0.953529	+	0.301303i	$0.902579\pi$
$354$	0	0
$355$	40.7178	2.16108
$356$	0	0
$357$	1.34445i	0.0711560i
$358$	0	0
$359$	−23.2207	−1.22554	−0.612771	−	0.790261i	$-0.709945\pi$
$359$	−23.2207	−1.22554	−0.612771	+	0.790261i	$0.709945\pi$
$360$	0	0
$361$	11.4283	0.601489
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.6562	−1.13354
$366$	0	0
$367$	− 13.1173i	− 0.684716i	−0.939570	−	0.342358i	$-0.888775\pi$
$367$	− 13.1173i	− 0.684716i	0.939570	−	0.342358i	$-0.111225\pi$
$368$	0	0
$369$	− 3.82198i	− 0.198964i
$370$	0	0
$371$	− 7.46147i	− 0.387380i
$372$	0	0
$373$	11.1209	0.575820	0.287910	−	0.957657i	$-0.407040\pi$
$373$	11.1209	0.575820	0.287910	+	0.957657i	$0.407040\pi$
$374$	0	0
$375$	8.55876i	0.441973i
$376$	0	0
$377$	0.826170	0.0425499
$378$	0	0
$379$	−4.41034	−0.226544	−0.113272	−	0.993564i	$-0.536133\pi$
$379$	−4.41034	−0.226544	−0.113272	+	0.993564i	$0.536133\pi$
$380$	0	0
$381$	17.3196	0.887311
$382$	0	0
$383$	1.78696i	0.0913094i	0.998957	+	0.0456547i	$0.0145374\pi$
$383$	1.78696i	0.0913094i	−0.998957	+	0.0456547i	$0.985463\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.75429i	0.343340i
$388$	0	0
$389$	− 18.7996i	− 0.953177i	−0.879127	−	0.476588i	$-0.841873\pi$
$389$	− 18.7996i	− 0.953177i	0.879127	−	0.476588i	$-0.158127\pi$
$390$	0	0
$391$	−3.31784	−0.167790
$392$	0	0
$393$	10.6185i	0.535633i
$394$	0	0
$395$	− 5.67247i	− 0.285413i
$396$	0	0
$397$	− 16.9423i	− 0.850309i	−0.905121	−	0.425154i	$-0.860220\pi$
$397$	− 16.9423i	− 0.850309i	0.905121	−	0.425154i	$-0.139780\pi$
$398$	0	0
$399$	− 4.91438i	− 0.246027i
$400$	0	0
$401$	27.2395	1.36028	0.680138	−	0.733084i	$-0.261920\pi$
$401$	27.2395	1.36028	0.680138	+	0.733084i	$0.261920\pi$
$402$	0	0
$403$	25.3384i	1.26219i
$404$	0	0
$405$	− 8.97659i	− 0.446051i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.4878i	0.716376i	0.933650	+	0.358188i	$0.116605\pi$
$409$	14.4878i	0.716376i	−0.933650	+	0.358188i	$0.883395\pi$
$410$	0	0
$411$	−6.07082	−0.299452
$412$	0	0
$413$	−8.81848	−0.433929
$414$	0	0
$415$	−42.2812	−2.07550
$416$	0	0
$417$	20.0568i	0.982184i
$418$	0	0
$419$	29.5266	1.44247	0.721234	−	0.692691i	$-0.243575\pi$
$419$	29.5266	1.44247	0.721234	+	0.692691i	$0.243575\pi$
$420$	0	0
$421$	26.4920i	1.29114i	0.763702	+	0.645569i	$0.223380\pi$
$421$	26.4920i	1.29114i	−0.763702	+	0.645569i	$0.776620\pi$
$422$	0	0
$423$	6.65805i	0.323725i
$424$	0	0
$425$	5.25532i	0.254920i
$426$	0	0
$427$	4.20586	0.203536
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.3443	−0.931782	−0.465891	−	0.884842i	$-0.654266\pi$
$431$	−19.3443	−0.931782	−0.465891	+	0.884842i	$0.654266\pi$
$432$	0	0
$433$	6.50951	0.312827	0.156414	−	0.987692i	$-0.450007\pi$
$433$	6.50951	0.312827	0.156414	+	0.987692i	$0.450007\pi$
$434$	0	0
$435$	0.541003i	0.0259391i
$436$	0	0
$437$	12.1277	0.580146
$438$	0	0
$439$	3.89313	0.185809	0.0929044	−	0.995675i	$-0.470385\pi$
$439$	3.89313	0.185809	0.0929044	+	0.995675i	$0.470385\pi$
$440$	0	0
$441$	7.14250	0.340119
$442$	0	0
$443$	−29.4006	−1.39686	−0.698432	−	0.715677i	$-0.746118\pi$
$443$	−29.4006	−1.39686	−0.698432	+	0.715677i	$0.746118\pi$
$444$	0	0
$445$	37.0210i	1.75496i
$446$	0	0
$447$	26.5501	1.25578
$448$	0	0
$449$	−23.9418	−1.12989	−0.564943	−	0.825130i	$-0.691102\pi$
$449$	−23.9418	−1.12989	−0.564943	+	0.825130i	$0.691102\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−19.8626	−0.933225
$454$	0	0
$455$	− 32.6767i	− 1.53191i
$456$	0	0
$457$	33.7078i	1.57678i	0.615173	+	0.788392i	$0.289086\pi$
$457$	33.7078i	1.57678i	−0.615173	+	0.788392i	$0.710914\pi$
$458$	0	0
$459$	− 4.17376i	− 0.194814i
$460$	0	0
$461$	1.14908	0.0535180	0.0267590	−	0.999642i	$-0.491481\pi$
$461$	1.14908	0.0535180	0.0267590	+	0.999642i	$0.491481\pi$
$462$	0	0
$463$	− 35.5695i	− 1.65305i	−0.562897	−	0.826527i	$-0.690313\pi$
$463$	− 35.5695i	− 1.65305i	0.562897	−	0.826527i	$-0.309687\pi$
$464$	0	0
$465$	−16.5924	−0.769453
$466$	0	0
$467$	9.59802	0.444144	0.222072	−	0.975030i	$-0.428718\pi$
$467$	9.59802	0.444144	0.222072	+	0.975030i	$0.428718\pi$
$468$	0	0
$469$	15.3333	0.708024
$470$	0	0
$471$	− 2.60786i	− 0.120164i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 19.2098i	− 0.881404i
$476$	0	0
$477$	7.52033i	0.344332i
$478$	0	0
$479$	−13.1173	−0.599344	−0.299672	−	0.954042i	$-0.596877\pi$
$479$	−13.1173	−0.599344	−0.299672	+	0.954042i	$0.596877\pi$
$480$	0	0
$481$	13.2847i	0.605730i
$482$	0	0
$483$	7.87141i	0.358162i
$484$	0	0
$485$	39.2423i	1.78190i
$486$	0	0
$487$	− 13.8592i	− 0.628021i	−0.949420	−	0.314010i	$-0.898327\pi$
$487$	− 13.8592i	− 0.628021i	0.949420	−	0.314010i	$-0.101673\pi$
$488$	0	0
$489$	−13.6911	−0.619131
$490$	0	0
$491$	8.12590i	0.366717i	0.983046	+	0.183358i	$0.0586969\pi$
$491$	8.12590i	0.366717i	−0.983046	+	0.183358i	$0.941303\pi$
$492$	0	0
$493$	0.0942697i	0.00424569i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.8327i	− 0.755050i
$498$	0	0
$499$	6.57863	0.294500	0.147250	−	0.989099i	$-0.452958\pi$
$499$	6.57863	0.294500	0.147250	+	0.989099i	$0.452958\pi$
$500$	0	0
$501$	−16.2063	−0.724043
$502$	0	0
$503$	−36.1611	−1.61234	−0.806172	−	0.591682i	$-0.798464\pi$
$503$	−36.1611	−1.61234	−0.806172	+	0.591682i	$0.798464\pi$
$504$	0	0
$505$	30.5751i	1.36057i
$506$	0	0
$507$	−38.0992	−1.69205
$508$	0	0
$509$	− 18.8562i	− 0.835788i	−0.908496	−	0.417894i	$-0.862768\pi$
$509$	− 18.8562i	− 0.835788i	0.908496	−	0.417894i	$-0.137232\pi$
$510$	0	0
$511$	8.95266i	0.396043i
$512$	0	0
$513$	15.2563i	0.673584i
$514$	0	0
$515$	−60.8757	−2.68250
$516$	0	0
$517$	0	0
$518$	0	0
$519$	4.34673	0.190800
$520$	0	0
$521$	−26.9570	−1.18101	−0.590505	−	0.807034i	$-0.701071\pi$
$521$	−26.9570	−1.18101	−0.590505	+	0.807034i	$0.701071\pi$
$522$	0	0
$523$	− 26.3243i	− 1.15108i	−0.817774	−	0.575540i	$-0.804792\pi$
$523$	− 26.3243i	− 1.15108i	0.817774	−	0.575540i	$-0.195208\pi$
$524$	0	0
$525$	12.4680	0.544147
$526$	0	0
$527$	−2.89122	−0.125943
$528$	0	0
$529$	3.57496	0.155433
$530$	0	0
$531$	8.88805	0.385708
$532$	0	0
$533$	− 17.4835i	− 0.757296i
$534$	0	0
$535$	−16.0550	−0.694119
$536$	0	0
$537$	3.37534	0.145657
$538$	0	0
$539$	0	0
$540$	0	0
$541$	35.8266	1.54031	0.770153	−	0.637860i	$-0.220180\pi$
$541$	35.8266	1.54031	0.770153	+	0.637860i	$0.220180\pi$
$542$	0	0
$543$	13.8484i	0.594290i
$544$	0	0
$545$	− 41.0321i	− 1.75762i
$546$	0	0
$547$	2.59683i	0.111033i	0.998458	+	0.0555163i	$0.0176805\pi$
$547$	2.59683i	0.111033i	−0.998458	+	0.0555163i	$0.982320\pi$
$548$	0	0
$549$	−4.23904	−0.180918
$550$	0	0
$551$	− 0.344584i	− 0.0146798i
$552$	0	0
$553$	−2.34499	−0.0997193
$554$	0	0
$555$	−8.69926	−0.369263
$556$	0	0
$557$	−18.5778	−0.787166	−0.393583	−	0.919289i	$-0.628765\pi$
$557$	−18.5778	−0.787166	−0.393583	+	0.919289i	$0.628765\pi$
$558$	0	0
$559$	30.8973i	1.30682i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 11.7158i	− 0.493760i	−0.969046	−	0.246880i	$-0.920595\pi$
$563$	− 11.7158i	− 0.493760i	0.969046	−	0.246880i	$-0.0794054\pi$
$564$	0	0
$565$	7.80415i	0.328323i
$566$	0	0
$567$	−3.71092	−0.155844
$568$	0	0
$569$	0.0615051i	0.00257843i	0.999999	+	0.00128921i	$0.000410370\pi$
$569$	0.0615051i	0.00257843i	−0.999999	+	0.00128921i	$0.999590\pi$
$570$	0	0
$571$	− 13.5141i	− 0.565549i	−0.959186	−	0.282775i	$-0.908745\pi$
$571$	− 13.5141i	− 0.565549i	0.959186	−	0.282775i	$-0.0912549\pi$
$572$	0	0
$573$	16.9755i	0.709160i
$574$	0	0
$575$	30.7684i	1.28313i
$576$	0	0
$577$	−29.1720	−1.21445	−0.607224	−	0.794531i	$-0.707717\pi$
$577$	−29.1720	−1.21445	−0.607224	+	0.794531i	$0.707717\pi$
$578$	0	0
$579$	− 6.72772i	− 0.279594i
$580$	0	0
$581$	17.4790i	0.725151i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	32.9345i	1.36167i
$586$	0	0
$587$	16.7053	0.689503	0.344752	−	0.938694i	$-0.387963\pi$
$587$	16.7053	0.689503	0.344752	+	0.938694i	$0.387963\pi$
$588$	0	0
$589$	10.5683	0.435458
$590$	0	0
$591$	8.02216	0.329988
$592$	0	0
$593$	25.8884i	1.06311i	0.847024	+	0.531555i	$0.178392\pi$
$593$	25.8884i	1.06311i	−0.847024	+	0.531555i	$0.821608\pi$
$594$	0	0
$595$	3.72856	0.152856
$596$	0	0
$597$	4.51831i	0.184922i
$598$	0	0
$599$	− 34.0811i	− 1.39252i	−0.717792	−	0.696258i	$-0.754847\pi$
$599$	− 34.0811i	− 1.39252i	0.717792	−	0.696258i	$-0.245153\pi$
$600$	0	0
$601$	13.9718i	0.569920i	0.958539	+	0.284960i	$0.0919803\pi$
$601$	13.9718i	0.569920i	−0.958539	+	0.284960i	$0.908020\pi$
$602$	0	0
$603$	−15.4542	−0.629344
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.4069	0.747112	0.373556	−	0.927608i	$-0.378138\pi$
$607$	18.4069	0.747112	0.373556	+	0.927608i	$0.378138\pi$
$608$	0	0
$609$	0.223650	0.00906276
$610$	0	0
$611$	30.4571i	1.23216i
$612$	0	0
$613$	−6.17282	−0.249318	−0.124659	−	0.992200i	$-0.539784\pi$
$613$	−6.17282	−0.249318	−0.124659	+	0.992200i	$0.539784\pi$
$614$	0	0
$615$	11.4488	0.461660
$616$	0	0
$617$	1.39128	0.0560109	0.0280054	−	0.999608i	$-0.491084\pi$
$617$	1.39128	0.0560109	0.0280054	+	0.999608i	$0.491084\pi$
$618$	0	0
$619$	10.9762	0.441172	0.220586	−	0.975368i	$-0.429203\pi$
$619$	10.9762	0.441172	0.220586	+	0.975368i	$0.429203\pi$
$620$	0	0
$621$	− 24.4362i	− 0.980592i
$622$	0	0
$623$	15.3044	0.613159
$624$	0	0
$625$	−11.1696	−0.446786
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.51584	−0.0604406
$630$	0	0
$631$	2.03752i	0.0811124i	0.999177	+	0.0405562i	$0.0129130\pi$
$631$	2.03752i	0.0811124i	−0.999177	+	0.0405562i	$0.987087\pi$
$632$	0	0
$633$	26.1273i	1.03847i
$634$	0	0
$635$	− 48.0323i	− 1.90610i
$636$	0	0
$637$	32.6732	1.29456
$638$	0	0
$639$	16.9655i	0.671144i
$640$	0	0
$641$	−3.34806	−0.132241	−0.0661203	−	0.997812i	$-0.521062\pi$
$641$	−3.34806	−0.132241	−0.0661203	+	0.997812i	$0.521062\pi$
$642$	0	0
$643$	−42.0115	−1.65677	−0.828387	−	0.560157i	$-0.810741\pi$
$643$	−42.0115	−1.65677	−0.828387	+	0.560157i	$0.810741\pi$
$644$	0	0
$645$	−20.2326	−0.796657
$646$	0	0
$647$	11.5117i	0.452570i	0.974061	+	0.226285i	$0.0726581\pi$
$647$	11.5117i	0.452570i	−0.974061	+	0.226285i	$0.927342\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.85927i	0.268836i
$652$	0	0
$653$	41.9766i	1.64267i	0.570444	+	0.821336i	$0.306771\pi$
$653$	41.9766i	1.64267i	−0.570444	+	0.821336i	$0.693229\pi$
$654$	0	0
$655$	29.4482	1.15064
$656$	0	0
$657$	− 9.02329i	− 0.352032i
$658$	0	0
$659$	38.3263i	1.49298i	0.665396	+	0.746490i	$0.268263\pi$
$659$	38.3263i	1.49298i	−0.665396	+	0.746490i	$0.731737\pi$
$660$	0	0
$661$	− 12.9209i	− 0.502563i	−0.967914	−	0.251281i	$-0.919148\pi$
$661$	− 12.9209i	− 0.502563i	0.967914	−	0.251281i	$-0.0808520\pi$
$662$	0	0
$663$	− 6.19868i	− 0.240737i
$664$	0	0
$665$	−13.6290	−0.528510
$666$	0	0
$667$	0.551923i	0.0213706i
$668$	0	0
$669$	− 3.31213i	− 0.128054i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 42.2131i	− 1.62720i	−0.581428	−	0.813598i	$-0.697506\pi$
$673$	− 42.2131i	− 1.62720i	0.581428	−	0.813598i	$-0.302494\pi$
$674$	0	0
$675$	−38.7060	−1.48979
$676$	0	0
$677$	−8.58558	−0.329971	−0.164985	−	0.986296i	$-0.552758\pi$
$677$	−8.58558	−0.329971	−0.164985	+	0.986296i	$0.552758\pi$
$678$	0	0
$679$	16.2227	0.622572
$680$	0	0
$681$	4.34604i	0.166541i
$682$	0	0
$683$	−28.2213	−1.07986	−0.539928	−	0.841711i	$-0.681549\pi$
$683$	−28.2213	−1.07986	−0.539928	+	0.841711i	$0.681549\pi$
$684$	0	0
$685$	16.8361i	0.643276i
$686$	0	0
$687$	− 16.4474i	− 0.627508i
$688$	0	0
$689$	34.4016i	1.31060i
$690$	0	0
$691$	10.4254	0.396599	0.198300	−	0.980141i	$-0.436458\pi$
$691$	10.4254	0.396599	0.198300	+	0.980141i	$0.436458\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	55.6232	2.10991
$696$	0	0
$697$	1.99495	0.0755641
$698$	0	0
$699$	− 8.71139i	− 0.329495i
$700$	0	0
$701$	20.5951	0.777868	0.388934	−	0.921266i	$-0.372843\pi$
$701$	20.5951	0.777868	0.388934	+	0.921266i	$0.372843\pi$
$702$	0	0
$703$	5.54086	0.208978
$704$	0	0
$705$	−19.9443	−0.751146
$706$	0	0
$707$	12.6397	0.475365
$708$	0	0
$709$	17.7424i	0.666329i	0.942869	+	0.333164i	$0.108116\pi$
$709$	17.7424i	0.666329i	−0.942869	+	0.333164i	$0.891884\pi$
$710$	0	0
$711$	2.36349	0.0886379
$712$	0	0
$713$	−16.9273	−0.633932
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.97575	−0.111132
$718$	0	0
$719$	42.0348i	1.56763i	0.620992	+	0.783817i	$0.286730\pi$
$719$	42.0348i	1.56763i	−0.620992	+	0.783817i	$0.713270\pi$
$720$	0	0
$721$	25.1659i	0.937229i
$722$	0	0
$723$	12.4866i	0.464382i
$724$	0	0
$725$	0.874223	0.0324678
$726$	0	0
$727$	28.2673i	1.04838i	0.851602	+	0.524189i	$0.175631\pi$
$727$	28.2673i	1.04838i	−0.851602	+	0.524189i	$0.824369\pi$
$728$	0	0
$729$	24.5002	0.907416
$730$	0	0
$731$	−3.52552	−0.130396
$732$	0	0
$733$	32.5046	1.20058	0.600292	−	0.799781i	$-0.295051\pi$
$733$	32.5046	1.20058	0.600292	+	0.799781i	$0.295051\pi$
$734$	0	0
$735$	21.3955i	0.789183i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 23.6830i	− 0.871193i	−0.900142	−	0.435597i	$-0.856537\pi$
$739$	− 23.6830i	− 0.871193i	0.900142	−	0.435597i	$-0.143463\pi$
$740$	0	0
$741$	22.6580i	0.832364i
$742$	0	0
$743$	−12.4937	−0.458350	−0.229175	−	0.973385i	$-0.573603\pi$
$743$	−12.4937	−0.458350	−0.229175	+	0.973385i	$0.573603\pi$
$744$	0	0
$745$	− 73.6310i	− 2.69763i
$746$	0	0
$747$	− 17.6169i	− 0.644568i
$748$	0	0
$749$	6.63713i	0.242515i
$750$	0	0
$751$	− 4.36510i	− 0.159285i	−0.996823	−	0.0796425i	$-0.974622\pi$
$751$	− 4.36510i	− 0.159285i	0.996823	−	0.0796425i	$-0.0253779\pi$
$752$	0	0
$753$	−21.7950	−0.794252
$754$	0	0
$755$	55.0847i	2.00474i
$756$	0	0
$757$	− 19.6633i	− 0.714676i	−0.933975	−	0.357338i	$-0.883684\pi$
$757$	− 19.6633i	− 0.714676i	0.933975	−	0.357338i	$-0.116316\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.7437i	0.389457i	0.980857	+	0.194729i	$0.0623827\pi$
$761$	10.7437i	0.389457i	−0.980857	+	0.194729i	$0.937617\pi$
$762$	0	0
$763$	−16.9627	−0.614089
$764$	0	0
$765$	−3.75797	−0.135870
$766$	0	0
$767$	40.6582	1.46808
$768$	0	0
$769$	− 28.3787i	− 1.02336i	−0.859176	−	0.511680i	$-0.829023\pi$
$769$	− 28.3787i	− 1.02336i	0.859176	−	0.511680i	$-0.170977\pi$
$770$	0	0
$771$	10.1798	0.366616
$772$	0	0
$773$	− 13.4299i	− 0.483039i	−0.970396	−	0.241520i	$-0.922354\pi$
$773$	− 13.4299i	− 0.483039i	0.970396	−	0.241520i	$-0.0776458\pi$
$774$	0	0
$775$	26.8121i	0.963120i
$776$	0	0
$777$	3.59626i	0.129015i
$778$	0	0
$779$	−7.29213	−0.261268
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.694306	−0.0248125
$784$	0	0
$785$	−7.23234	−0.258133
$786$	0	0
$787$	53.3174i	1.90056i	0.311397	+	0.950280i	$0.399203\pi$
$787$	53.3174i	1.90056i	−0.311397	+	0.950280i	$0.600797\pi$
$788$	0	0
$789$	−26.3347	−0.937540
$790$	0	0
$791$	3.22623	0.114712
$792$	0	0
$793$	−19.3914	−0.688608
$794$	0	0
$795$	−22.5273	−0.798960
$796$	0	0
$797$	12.9017i	0.457001i	0.973544	+	0.228501i	$0.0733823\pi$
$797$	12.9017i	0.457001i	−0.973544	+	0.228501i	$0.926618\pi$
$798$	0	0
$799$	−3.47529	−0.122947
$800$	0	0
$801$	−15.4252	−0.545022
$802$	0	0
$803$	0	0
$804$	0	0
$805$	21.8297	0.769395
$806$	0	0
$807$	26.1760i	0.921440i
$808$	0	0
$809$	− 19.7127i	− 0.693063i	−0.938038	−	0.346532i	$-0.887359\pi$
$809$	− 19.7127i	− 0.693063i	0.938038	−	0.346532i	$-0.112641\pi$
$810$	0	0
$811$	− 3.21266i	− 0.112812i	−0.998408	−	0.0564059i	$-0.982036\pi$
$811$	− 3.21266i	− 0.112812i	0.998408	−	0.0564059i	$-0.0179641\pi$
$812$	0	0
$813$	−24.8973	−0.873187
$814$	0	0
$815$	37.9693i	1.33001i
$816$	0	0
$817$	12.8868	0.450853
$818$	0	0
$819$	13.6151	0.475750
$820$	0	0
$821$	34.1467	1.19173	0.595865	−	0.803085i	$-0.296809\pi$
$821$	34.1467	1.19173	0.595865	+	0.803085i	$0.296809\pi$
$822$	0	0
$823$	18.7711i	0.654318i	0.944969	+	0.327159i	$0.106091\pi$
$823$	18.7711i	0.654318i	−0.944969	+	0.327159i	$0.893909\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.2934i	1.67933i	0.543107	+	0.839663i	$0.317248\pi$
$827$	48.2934i	1.67933i	−0.543107	+	0.839663i	$0.682752\pi$
$828$	0	0
$829$	− 25.1260i	− 0.872662i	−0.899786	−	0.436331i	$-0.856278\pi$
$829$	− 25.1260i	− 0.872662i	0.899786	−	0.436331i	$-0.143722\pi$
$830$	0	0
$831$	−1.33687	−0.0463754
$832$	0	0
$833$	3.72815i	0.129173i
$834$	0	0
$835$	44.9447i	1.55537i
$836$	0	0
$837$	− 21.2941i	− 0.736033i
$838$	0	0
$839$	9.59287i	0.331183i	0.986194	+	0.165591i	$0.0529532\pi$
$839$	9.59287i	0.331183i	−0.986194	+	0.165591i	$0.947047\pi$
$840$	0	0
$841$	−28.9843	−0.999459
$842$	0	0
$843$	− 4.35514i	− 0.149999i
$844$	0	0
$845$	105.660i	3.63482i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25.5602i	0.877224i
$850$	0	0
$851$	−8.87485	−0.304226
$852$	0	0
$853$	−26.5509	−0.909086	−0.454543	−	0.890725i	$-0.650197\pi$
$853$	−26.5509	−0.909086	−0.454543	+	0.890725i	$0.650197\pi$
$854$	0	0
$855$	13.7365	0.469779
$856$	0	0
$857$	− 9.29115i	− 0.317380i	−0.987328	−	0.158690i	$-0.949273\pi$
$857$	− 9.29115i	− 0.317380i	0.987328	−	0.158690i	$-0.0507270\pi$
$858$	0	0
$859$	−0.473127	−0.0161429	−0.00807145	−	0.999967i	$-0.502569\pi$
$859$	−0.473127	−0.0161429	−0.00807145	+	0.999967i	$0.502569\pi$
$860$	0	0
$861$	− 4.73292i	− 0.161297i
$862$	0	0
$863$	− 6.23350i	− 0.212191i	−0.994356	−	0.106095i	$-0.966165\pi$
$863$	− 6.23350i	− 0.212191i	0.994356	−	0.106095i	$-0.0338349\pi$
$864$	0	0
$865$	− 12.0547i	− 0.409873i
$866$	0	0
$867$	−20.5106	−0.696577
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−70.6949	−2.39541
$872$	0	0
$873$	−16.3507	−0.553388
$874$	0	0
$875$	− 9.81241i	− 0.331720i
$876$	0	0
$877$	−58.1560	−1.96379	−0.981894	−	0.189431i	$-0.939336\pi$
$877$	−58.1560	−1.96379	−0.981894	+	0.189431i	$0.939336\pi$
$878$	0	0
$879$	32.4402	1.09418
$880$	0	0
$881$	−21.4583	−0.722949	−0.361475	−	0.932382i	$-0.617727\pi$
$881$	−21.4583	−0.722949	−0.361475	+	0.932382i	$0.617727\pi$
$882$	0	0
$883$	38.2306	1.28656	0.643281	−	0.765630i	$-0.277573\pi$
$883$	38.2306	1.28656	0.643281	+	0.765630i	$0.277573\pi$
$884$	0	0
$885$	26.6243i	0.894965i
$886$	0	0
$887$	38.3972	1.28925	0.644626	−	0.764498i	$-0.277013\pi$
$887$	38.3972	1.28925	0.644626	+	0.764498i	$0.277013\pi$
$888$	0	0
$889$	−19.8565	−0.665966
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.7032	0.425097
$894$	0	0
$895$	− 9.36079i	− 0.312897i
$896$	0	0
$897$	− 36.2916i	− 1.21174i
$898$	0	0
$899$	0.480955i	0.0160407i
$900$	0	0
$901$	−3.92537	−0.130773
$902$	0	0
$903$	8.36413i	0.278341i
$904$	0	0
$905$	38.4055	1.27664
$906$	0	0
$907$	−0.655162	−0.0217543	−0.0108771	−	0.999941i	$-0.503462\pi$
$907$	−0.655162	−0.0217543	−0.0108771	+	0.999941i	$0.503462\pi$
$908$	0	0
$909$	−12.7394	−0.422540
$910$	0	0
$911$	− 49.0780i	− 1.62603i	−0.582245	−	0.813013i	$-0.697826\pi$
$911$	− 49.0780i	− 1.62603i	0.582245	−	0.813013i	$-0.302174\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 12.6981i	− 0.419786i
$916$	0	0
$917$	− 12.1739i	− 0.402016i
$918$	0	0
$919$	−20.2852	−0.669146	−0.334573	−	0.942370i	$-0.608592\pi$
$919$	−20.2852	−0.669146	−0.334573	+	0.942370i	$0.608592\pi$
$920$	0	0
$921$	11.6501i	0.383885i
$922$	0	0
$923$	77.6082i	2.55451i
$924$	0	0
$925$	14.0574i	0.462204i
$926$	0	0
$927$	− 25.3645i	− 0.833078i
$928$	0	0
$929$	25.9159	0.850275	0.425137	−	0.905129i	$-0.360226\pi$
$929$	25.9159	0.850275	0.425137	+	0.905129i	$0.360226\pi$
$930$	0	0
$931$	− 13.6275i	− 0.446624i
$932$	0	0
$933$	24.2958i	0.795409i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.6496i	− 1.39330i	−0.717410	−	0.696651i	$-0.754673\pi$
$937$	− 42.6496i	− 1.39330i	0.717410	−	0.696651i	$-0.245327\pi$
$938$	0	0
$939$	−18.4708	−0.602773
$940$	0	0
$941$	3.61023	0.117690	0.0588451	−	0.998267i	$-0.481258\pi$
$941$	3.61023	0.117690	0.0588451	+	0.998267i	$0.481258\pi$
$942$	0	0
$943$	11.6799	0.380349
$944$	0	0
$945$	27.4612i	0.893314i
$946$	0	0
$947$	−24.5514	−0.797813	−0.398907	−	0.916992i	$-0.630610\pi$
$947$	−24.5514	−0.797813	−0.398907	+	0.916992i	$0.630610\pi$
$948$	0	0
$949$	− 41.2768i	− 1.33990i
$950$	0	0
$951$	39.6008i	1.28414i
$952$	0	0
$953$	5.04176i	0.163319i	0.996660	+	0.0816593i	$0.0260219\pi$
$953$	5.04176i	0.163319i	−0.996660	+	0.0816593i	$0.973978\pi$
$954$	0	0
$955$	47.0778	1.52340
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.96004	0.224752
$960$	0	0
$961$	16.2493	0.524170
$962$	0	0
$963$	− 6.68949i	− 0.215566i
$964$	0	0
$965$	−18.6579	−0.600619
$966$	0	0
$967$	−32.9415	−1.05933	−0.529664	−	0.848208i	$-0.677682\pi$
$967$	−32.9415	−1.05933	−0.529664	+	0.848208i	$0.677682\pi$
$968$	0	0
$969$	−2.58538	−0.0830545
$970$	0	0
$971$	−46.1030	−1.47952	−0.739758	−	0.672873i	$-0.765060\pi$
$971$	−46.1030	−1.47952	−0.739758	+	0.672873i	$0.765060\pi$
$972$	0	0
$973$	− 22.9946i	− 0.737172i
$974$	0	0
$975$	−57.4844	−1.84097
$976$	0	0
$977$	−32.6331	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$977$	−32.6331	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	17.0965	0.545848
$982$	0	0
$983$	− 8.45130i	− 0.269555i	−0.990876	−	0.134777i	$-0.956968\pi$
$983$	− 8.45130i	− 0.269555i	0.990876	−	0.134777i	$-0.0430319\pi$
$984$	0	0
$985$	− 22.2478i	− 0.708873i
$986$	0	0
$987$	8.24495i	0.262440i
$988$	0	0
$989$	−20.6410	−0.656345
$990$	0	0
$991$	− 29.6890i	− 0.943104i	−0.881838	−	0.471552i	$-0.843694\pi$
$991$	− 29.6890i	− 0.943104i	0.881838	−	0.471552i	$-0.156306\pi$
$992$	0	0
$993$	−0.577557	−0.0183282
$994$	0	0
$995$	12.5306	0.397246
$996$	0	0
$997$	−1.85056	−0.0586079	−0.0293039	−	0.999571i	$-0.509329\pi$
$997$	−1.85056	−0.0586079	−0.0293039	+	0.999571i	$0.509329\pi$
$998$	0	0
$999$	− 11.1643i	− 0.353224i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.g.d.1935.11		32
4.3	odd	2		968.2.g.e.483.17		32
8.3	odd	2	inner	3872.2.g.d.1935.9		32
8.5	even	2		968.2.g.e.483.15		32
11.6	odd	10		352.2.s.b.239.4		32
11.9	even	5		352.2.s.b.271.3		32
11.10	odd	2	inner	3872.2.g.d.1935.12		32
44.3	odd	10		968.2.k.e.475.2		32
44.7	even	10		968.2.k.e.699.1		32
44.15	odd	10		968.2.k.i.699.8		32
44.19	even	10		968.2.k.i.475.7		32
44.27	odd	10		968.2.k.h.723.4		32
44.31	odd	10		88.2.k.b.51.3	yes	32
44.35	even	10		968.2.k.h.403.6		32
44.39	even	10		88.2.k.b.19.5	yes	32
44.43	even	2		968.2.g.e.483.16		32
88.5	even	10		968.2.k.h.723.6		32
88.13	odd	10		968.2.k.h.403.4		32
88.21	odd	2		968.2.g.e.483.18		32
88.29	odd	10		968.2.k.e.699.2		32
88.37	even	10		968.2.k.i.699.7		32
88.43	even	2	inner	3872.2.g.d.1935.10		32
88.53	even	10		88.2.k.b.51.5	yes	32
88.61	odd	10		88.2.k.b.19.3	✓	32
88.69	even	10		968.2.k.e.475.1		32
88.75	odd	10		352.2.s.b.271.4		32
88.83	even	10		352.2.s.b.239.3		32
88.85	odd	10		968.2.k.i.475.8		32
132.83	odd	10		792.2.bp.b.19.4		32
132.119	even	10		792.2.bp.b.667.6		32
264.53	odd	10		792.2.bp.b.667.4		32
264.149	even	10		792.2.bp.b.19.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.k.b.19.3	✓	32	88.61	odd	10
88.2.k.b.19.5	yes	32	44.39	even	10
88.2.k.b.51.3	yes	32	44.31	odd	10
88.2.k.b.51.5	yes	32	88.53	even	10
352.2.s.b.239.3		32	88.83	even	10
352.2.s.b.239.4		32	11.6	odd	10
352.2.s.b.271.3		32	11.9	even	5
352.2.s.b.271.4		32	88.75	odd	10
792.2.bp.b.19.4		32	132.83	odd	10
792.2.bp.b.19.6		32	264.149	even	10
792.2.bp.b.667.4		32	264.53	odd	10
792.2.bp.b.667.6		32	132.119	even	10
968.2.g.e.483.15		32	8.5	even	2
968.2.g.e.483.16		32	44.43	even	2
968.2.g.e.483.17		32	4.3	odd	2
968.2.g.e.483.18		32	88.21	odd	2
968.2.k.e.475.1		32	88.69	even	10
968.2.k.e.475.2		32	44.3	odd	10
968.2.k.e.699.1		32	44.7	even	10
968.2.k.e.699.2		32	88.29	odd	10
968.2.k.h.403.4		32	88.13	odd	10
968.2.k.h.403.6		32	44.35	even	10
968.2.k.h.723.4		32	44.27	odd	10
968.2.k.h.723.6		32	88.5	even	10
968.2.k.i.475.7		32	44.19	even	10
968.2.k.i.475.8		32	88.85	odd	10
968.2.k.i.699.7		32	88.37	even	10
968.2.k.i.699.8		32	44.15	odd	10
3872.2.g.d.1935.9		32	8.3	odd	2	inner
3872.2.g.d.1935.10		32	88.43	even	2	inner
3872.2.g.d.1935.11		32	1.1	even	1	trivial
3872.2.g.d.1935.12		32	11.10	odd	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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.24811 q^{3} +3.46137i q^{5} +1.43093 q^{7} -1.44222 q^{9} -6.59738 q^{13} -4.32018i q^{15} -0.752791i q^{17} +2.75168i q^{19} -1.78596 q^{21} -4.40738i q^{23} -6.98111 q^{25} +5.54438 q^{27} -0.125227 q^{29} -3.84067i q^{31} +4.95298i q^{35} -2.01363i q^{37} +8.23427 q^{39} +2.65007i q^{41} -4.68327i q^{43} -4.99205i q^{45} -4.61654i q^{47} -4.95244 q^{49} +0.939567i q^{51} -5.21442i q^{53} -3.43440i q^{57} -6.16277 q^{59} +2.93925 q^{61} -2.06371 q^{63} -22.8360i q^{65} +10.7156 q^{67} +5.50091i q^{69} -11.7635i q^{71} +6.25654i q^{73} +8.71321 q^{75} -1.63879 q^{79} -2.59336 q^{81} +12.2151i q^{83} +2.60569 q^{85} +0.156297 q^{87} +10.6955 q^{89} -9.44039 q^{91} +4.79358i q^{93} -9.52458 q^{95} +11.3372 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{3} - 44 q^{25} - 8 q^{27} + 52 q^{49} + 72 q^{59} + 28 q^{67} + 76 q^{75} - 112 q^{81} + 20 q^{89} - 48 q^{91} + 8 q^{97}+O(q^{100}) \) 32 * q - 8 * q^3 - 44 * q^25 - 8 * q^27 + 52 * q^49 + 72 * q^59 + 28 * q^67 + 76 * q^75 - 112 * q^81 + 20 * q^89 - 48 * q^91 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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