LMFDB - Embedded newform 416.2.b.c.209.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(209,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	416.209
Dual form			416.2.b.c.209.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24914i q^{3} +1.00000i q^{5} -3.24914 q^{7} -7.55691 q^{9} -1.05863i q^{11} -1.00000i q^{13} -3.24914 q^{15} +1.00000 q^{17} +4.00000i q^{19} -10.5569i q^{21} +2.94137 q^{23} +4.00000 q^{25} -14.8061i q^{27} +7.43965i q^{29} -5.05863 q^{31} +3.43965 q^{33} -3.24914i q^{35} +6.55691i q^{37} +3.24914 q^{39} -9.43965 q^{41} +0.307774i q^{43} -7.55691i q^{45} -6.80605 q^{47} +3.55691 q^{49} +3.24914i q^{51} -1.55691i q^{53} +1.05863 q^{55} -12.9966 q^{57} +5.67418i q^{59} +9.67418i q^{61} +24.5535 q^{63} +1.00000 q^{65} +1.50172i q^{67} +9.55691i q^{69} +5.36641 q^{71} +3.55691 q^{73} +12.9966i q^{75} +3.43965i q^{77} +6.73281 q^{79} +25.4362 q^{81} -2.49828i q^{83} +1.00000i q^{85} -24.1725 q^{87} +2.11727 q^{89} +3.24914i q^{91} -16.4362i q^{93} -4.00000 q^{95} +7.67418 q^{97} +8.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{7} - 12 q^{9} - 2 q^{15} + 6 q^{17} + 16 q^{23} + 24 q^{25} - 32 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{57} + 44 q^{63} + 6 q^{65} + 18 q^{71} - 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 6 * q - 2 * q^7 - 12 * q^9 - 2 * q^15 + 6 * q^17 + 16 * q^23 + 24 * q^25 - 32 * q^31 - 16 * q^33 + 2 * q^39 - 20 * q^41 + 10 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^57 + 44 * q^63 + 6 * q^65 + 18 * q^71 - 12 * q^73 + 12 * q^79 + 46 * q^81 - 80 * q^87 + 16 * q^89 - 24 * q^95 + 16 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.24914i	1.87589i	0.346781	+	0.937946i	$0.387275\pi$
$3$	3.24914i	1.87589i	−0.346781	+	0.937946i	$0.612725\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	−3.24914	−1.22806	−0.614030	−	0.789283i	$-0.710453\pi$
$7$	−3.24914	−1.22806	−0.614030	+	0.789283i	$0.710453\pi$
$8$	0	0
$9$	−7.55691	−2.51897
$10$	0	0
$11$	− 1.05863i	− 0.319190i	−0.987183	−	0.159595i	$-0.948981\pi$
$11$	− 1.05863i	− 0.319190i	0.987183	−	0.159595i	$-0.0510188\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−3.24914	−0.838924
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 10.5569i	− 2.30371i
$22$	0	0
$23$	2.94137	0.613317	0.306659	−	0.951820i	$-0.400789\pi$
$23$	2.94137	0.613317	0.306659	+	0.951820i	$0.400789\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 14.8061i	− 2.84943i
$28$	0	0
$29$	7.43965i	1.38151i	0.723090	+	0.690754i	$0.242721\pi$
$29$	7.43965i	1.38151i	−0.723090	+	0.690754i	$0.757279\pi$
$30$	0	0
$31$	−5.05863	−0.908557	−0.454279	−	0.890860i	$-0.650103\pi$
$31$	−5.05863	−0.908557	−0.454279	+	0.890860i	$0.650103\pi$
$32$	0	0
$33$	3.43965	0.598766
$34$	0	0
$35$	− 3.24914i	− 0.549205i
$36$	0	0
$37$	6.55691i	1.07795i	0.842322	+	0.538975i	$0.181188\pi$
$37$	6.55691i	1.07795i	−0.842322	+	0.538975i	$0.818812\pi$
$38$	0	0
$39$	3.24914	0.520279
$40$	0	0
$41$	−9.43965	−1.47423	−0.737113	−	0.675770i	$-0.763811\pi$
$41$	−9.43965	−1.47423	−0.737113	+	0.675770i	$0.763811\pi$
$42$	0	0
$43$	0.307774i	0.0469350i	0.999725	+	0.0234675i	$0.00747063\pi$
$43$	0.307774i	0.0469350i	−0.999725	+	0.0234675i	$0.992529\pi$
$44$	0	0
$45$	− 7.55691i	− 1.12652i
$46$	0	0
$47$	−6.80605	−0.992765	−0.496383	−	0.868104i	$-0.665339\pi$
$47$	−6.80605	−0.992765	−0.496383	+	0.868104i	$0.665339\pi$
$48$	0	0
$49$	3.55691	0.508131
$50$	0	0
$51$	3.24914i	0.454971i
$52$	0	0
$53$	− 1.55691i	− 0.213859i	−0.994267	−	0.106929i	$-0.965898\pi$
$53$	− 1.55691i	− 0.213859i	0.994267	−	0.106929i	$-0.0341019\pi$
$54$	0	0
$55$	1.05863	0.142746
$56$	0	0
$57$	−12.9966	−1.72144
$58$	0	0
$59$	5.67418i	0.738715i	0.929287	+	0.369358i	$0.120422\pi$
$59$	5.67418i	0.738715i	−0.929287	+	0.369358i	$0.879578\pi$
$60$	0	0
$61$	9.67418i	1.23865i	0.785134	+	0.619326i	$0.212594\pi$
$61$	9.67418i	1.23865i	−0.785134	+	0.619326i	$0.787406\pi$
$62$	0	0
$63$	24.5535	3.09345
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	1.50172i	0.183464i	0.995784	+	0.0917321i	$0.0292403\pi$
$67$	1.50172i	0.183464i	−0.995784	+	0.0917321i	$0.970760\pi$
$68$	0	0
$69$	9.55691i	1.15052i
$70$	0	0
$71$	5.36641	0.636875	0.318438	−	0.947944i	$-0.396842\pi$
$71$	5.36641	0.636875	0.318438	+	0.947944i	$0.396842\pi$
$72$	0	0
$73$	3.55691	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$73$	3.55691	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$74$	0	0
$75$	12.9966i	1.50071i
$76$	0	0
$77$	3.43965i	0.391984i
$78$	0	0
$79$	6.73281	0.757501	0.378750	−	0.925499i	$-0.376354\pi$
$79$	6.73281	0.757501	0.378750	+	0.925499i	$0.376354\pi$
$80$	0	0
$81$	25.4362	2.82625
$82$	0	0
$83$	− 2.49828i	− 0.274222i	−0.990556	−	0.137111i	$-0.956218\pi$
$83$	− 2.49828i	− 0.274222i	0.990556	−	0.137111i	$-0.0437817\pi$
$84$	0	0
$85$	1.00000i	0.108465i
$86$	0	0
$87$	−24.1725	−2.59156
$88$	0	0
$89$	2.11727	0.224430	0.112215	−	0.993684i	$-0.464205\pi$
$89$	2.11727	0.224430	0.112215	+	0.993684i	$0.464205\pi$
$90$	0	0
$91$	3.24914i	0.340602i
$92$	0	0
$93$	− 16.4362i	− 1.70436i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	7.67418	0.779195	0.389597	−	0.920985i	$-0.372614\pi$
$97$	7.67418	0.779195	0.389597	+	0.920985i	$0.372614\pi$
$98$	0	0
$99$	8.00000i	0.804030i
$100$	0	0
$101$	3.11383i	0.309838i	0.987927	+	0.154919i	$0.0495116\pi$
$101$	3.11383i	0.309838i	−0.987927	+	0.154919i	$0.950488\pi$
$102$	0	0
$103$	17.6121	1.73537	0.867686	−	0.497112i	$-0.165606\pi$
$103$	17.6121	1.73537	0.867686	+	0.497112i	$0.165606\pi$
$104$	0	0
$105$	10.5569	1.03025
$106$	0	0
$107$	− 10.9414i	− 1.05774i	−0.848702	−	0.528871i	$-0.822616\pi$
$107$	− 10.9414i	− 1.05774i	0.848702	−	0.528871i	$-0.177384\pi$
$108$	0	0
$109$	− 10.3224i	− 0.988705i	−0.869262	−	0.494352i	$-0.835405\pi$
$109$	− 10.3224i	− 0.988705i	0.869262	−	0.494352i	$-0.164595\pi$
$110$	0	0
$111$	−21.3043	−2.02212
$112$	0	0
$113$	−17.1138	−1.60993	−0.804967	−	0.593320i	$-0.797817\pi$
$113$	−17.1138	−1.60993	−0.804967	+	0.593320i	$0.797817\pi$
$114$	0	0
$115$	2.94137i	0.274284i
$116$	0	0
$117$	7.55691i	0.698637i
$118$	0	0
$119$	−3.24914	−0.297848
$120$	0	0
$121$	9.87930	0.898118
$122$	0	0
$123$	− 30.6707i	− 2.76549i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	15.9379	1.41426	0.707131	−	0.707082i	$-0.249989\pi$
$127$	15.9379	1.41426	0.707131	+	0.707082i	$0.249989\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	7.69223i	0.672073i	0.941849	+	0.336036i	$0.109087\pi$
$131$	7.69223i	0.672073i	−0.941849	+	0.336036i	$0.890913\pi$
$132$	0	0
$133$	− 12.9966i	− 1.12694i
$134$	0	0
$135$	14.8061	1.27430
$136$	0	0
$137$	5.32238	0.454722	0.227361	−	0.973811i	$-0.426990\pi$
$137$	5.32238	0.454722	0.227361	+	0.973811i	$0.426990\pi$
$138$	0	0
$139$	12.4802i	1.05856i	0.848447	+	0.529280i	$0.177538\pi$
$139$	12.4802i	1.05856i	−0.848447	+	0.529280i	$0.822462\pi$
$140$	0	0
$141$	− 22.1138i	− 1.86232i
$142$	0	0
$143$	−1.05863	−0.0885274
$144$	0	0
$145$	−7.43965	−0.617829
$146$	0	0
$147$	11.5569i	0.953198i
$148$	0	0
$149$	− 9.11383i	− 0.746634i	−0.927704	−	0.373317i	$-0.878220\pi$
$149$	− 9.11383i	− 0.746634i	0.927704	−	0.373317i	$-0.121780\pi$
$150$	0	0
$151$	15.4216	1.25499	0.627496	−	0.778620i	$-0.284080\pi$
$151$	15.4216	1.25499	0.627496	+	0.778620i	$0.284080\pi$
$152$	0	0
$153$	−7.55691	−0.610940
$154$	0	0
$155$	− 5.05863i	− 0.406319i
$156$	0	0
$157$	− 3.79145i	− 0.302590i	−0.988489	−	0.151295i	$-0.951656\pi$
$157$	− 3.79145i	− 0.302590i	0.988489	−	0.151295i	$-0.0483444\pi$
$158$	0	0
$159$	5.05863	0.401176
$160$	0	0
$161$	−9.55691	−0.753190
$162$	0	0
$163$	4.17246i	0.326812i	0.986559	+	0.163406i	$0.0522481\pi$
$163$	4.17246i	0.326812i	−0.986559	+	0.163406i	$0.947752\pi$
$164$	0	0
$165$	3.43965i	0.267776i
$166$	0	0
$167$	−14.2897	−1.10577	−0.552886	−	0.833257i	$-0.686474\pi$
$167$	−14.2897	−1.10577	−0.552886	+	0.833257i	$0.686474\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 30.2277i	− 2.31157i
$172$	0	0
$173$	6.99656i	0.531939i	0.963981	+	0.265969i	$0.0856920\pi$
$173$	6.99656i	0.531939i	−0.963981	+	0.265969i	$0.914308\pi$
$174$	0	0
$175$	−12.9966	−0.982448
$176$	0	0
$177$	−18.4362	−1.38575
$178$	0	0
$179$	17.7474i	1.32650i	0.748396	+	0.663252i	$0.230824\pi$
$179$	17.7474i	1.32650i	−0.748396	+	0.663252i	$0.769176\pi$
$180$	0	0
$181$	8.11727i	0.603352i	0.953411	+	0.301676i	$0.0975460\pi$
$181$	8.11727i	0.603352i	−0.953411	+	0.301676i	$0.902454\pi$
$182$	0	0
$183$	−31.4328	−2.32358
$184$	0	0
$185$	−6.55691	−0.482074
$186$	0	0
$187$	− 1.05863i	− 0.0774149i
$188$	0	0
$189$	48.1070i	3.49927i
$190$	0	0
$191$	19.9379	1.44266	0.721329	−	0.692593i	$-0.243532\pi$
$191$	19.9379	1.44266	0.721329	+	0.692593i	$0.243532\pi$
$192$	0	0
$193$	4.11727	0.296367	0.148184	−	0.988960i	$-0.452657\pi$
$193$	4.11727	0.296367	0.148184	+	0.988960i	$0.452657\pi$
$194$	0	0
$195$	3.24914i	0.232676i
$196$	0	0
$197$	− 10.7914i	− 0.768859i	−0.923154	−	0.384429i	$-0.874398\pi$
$197$	− 10.7914i	− 0.768859i	0.923154	−	0.384429i	$-0.125602\pi$
$198$	0	0
$199$	−0.615547	−0.0436350	−0.0218175	−	0.999762i	$-0.506945\pi$
$199$	−0.615547	−0.0436350	−0.0218175	+	0.999762i	$0.506945\pi$
$200$	0	0
$201$	−4.87930	−0.344159
$202$	0	0
$203$	− 24.1725i	− 1.69657i
$204$	0	0
$205$	− 9.43965i	− 0.659294i
$206$	0	0
$207$	−22.2277	−1.54493
$208$	0	0
$209$	4.23453	0.292909
$210$	0	0
$211$	1.80949i	0.124571i	0.998058	+	0.0622853i	$0.0198389\pi$
$211$	1.80949i	0.124571i	−0.998058	+	0.0622853i	$0.980161\pi$
$212$	0	0
$213$	17.4362i	1.19471i
$214$	0	0
$215$	−0.307774	−0.0209900
$216$	0	0
$217$	16.4362	1.11576
$218$	0	0
$219$	11.5569i	0.780944i
$220$	0	0
$221$	− 1.00000i	− 0.0672673i
$222$	0	0
$223$	6.80605	0.455767	0.227884	−	0.973688i	$-0.426819\pi$
$223$	6.80605	0.455767	0.227884	+	0.973688i	$0.426819\pi$
$224$	0	0
$225$	−30.2277	−2.01518
$226$	0	0
$227$	− 6.67074i	− 0.442753i	−0.975188	−	0.221376i	$-0.928945\pi$
$227$	− 6.67074i	− 0.442753i	0.975188	−	0.221376i	$-0.0710549\pi$
$228$	0	0
$229$	18.1138i	1.19700i	0.801124	+	0.598498i	$0.204235\pi$
$229$	18.1138i	1.19700i	−0.801124	+	0.598498i	$0.795765\pi$
$230$	0	0
$231$	−11.1759	−0.735320
$232$	0	0
$233$	11.4362	0.749211	0.374606	−	0.927184i	$-0.377778\pi$
$233$	11.4362	0.749211	0.374606	+	0.927184i	$0.377778\pi$
$234$	0	0
$235$	− 6.80605i	− 0.443978i
$236$	0	0
$237$	21.8759i	1.42099i
$238$	0	0
$239$	−16.8613	−1.09066	−0.545332	−	0.838220i	$-0.683596\pi$
$239$	−16.8613	−1.09066	−0.545332	+	0.838220i	$0.683596\pi$
$240$	0	0
$241$	−5.23109	−0.336964	−0.168482	−	0.985705i	$-0.553887\pi$
$241$	−5.23109	−0.336964	−0.168482	+	0.985705i	$0.553887\pi$
$242$	0	0
$243$	38.2277i	2.45231i
$244$	0	0
$245$	3.55691i	0.227243i
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	8.11727	0.514411
$250$	0	0
$251$	2.05520i	0.129723i	0.997894	+	0.0648614i	$0.0206605\pi$
$251$	2.05520i	0.129723i	−0.997894	+	0.0648614i	$0.979339\pi$
$252$	0	0
$253$	− 3.11383i	− 0.195765i
$254$	0	0
$255$	−3.24914	−0.203469
$256$	0	0
$257$	−21.2277	−1.32414	−0.662072	−	0.749440i	$-0.730323\pi$
$257$	−21.2277	−1.32414	−0.662072	+	0.749440i	$0.730323\pi$
$258$	0	0
$259$	− 21.3043i	− 1.32379i
$260$	0	0
$261$	− 56.2208i	− 3.47998i
$262$	0	0
$263$	−23.5569	−1.45258	−0.726291	−	0.687388i	$-0.758757\pi$
$263$	−23.5569	−1.45258	−0.726291	+	0.687388i	$0.758757\pi$
$264$	0	0
$265$	1.55691	0.0956405
$266$	0	0
$267$	6.87930i	0.421006i
$268$	0	0
$269$	− 6.32582i	− 0.385692i	−0.981229	−	0.192846i	$-0.938228\pi$
$269$	− 6.32582i	− 0.385692i	0.981229	−	0.192846i	$-0.0617718\pi$
$270$	0	0
$271$	−3.45769	−0.210040	−0.105020	−	0.994470i	$-0.533491\pi$
$271$	−3.45769	−0.210040	−0.105020	+	0.994470i	$0.533491\pi$
$272$	0	0
$273$	−10.5569	−0.638934
$274$	0	0
$275$	− 4.23453i	− 0.255352i
$276$	0	0
$277$	− 28.5535i	− 1.71561i	−0.513974	−	0.857806i	$-0.671827\pi$
$277$	− 28.5535i	− 1.71561i	0.513974	−	0.857806i	$-0.328173\pi$
$278$	0	0
$279$	38.2277	2.28863
$280$	0	0
$281$	24.9966	1.49117	0.745585	−	0.666411i	$-0.232170\pi$
$281$	24.9966	1.49117	0.745585	+	0.666411i	$0.232170\pi$
$282$	0	0
$283$	− 21.8207i	− 1.29710i	−0.761170	−	0.648552i	$-0.775375\pi$
$283$	− 21.8207i	− 1.29710i	0.761170	−	0.648552i	$-0.224625\pi$
$284$	0	0
$285$	− 12.9966i	− 0.769850i
$286$	0	0
$287$	30.6707	1.81044
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	24.9345i	1.46169i
$292$	0	0
$293$	− 27.4362i	− 1.60284i	−0.598102	−	0.801420i	$-0.704078\pi$
$293$	− 27.4362i	− 1.60284i	0.598102	−	0.801420i	$-0.295922\pi$
$294$	0	0
$295$	−5.67418	−0.330364
$296$	0	0
$297$	−15.6742	−0.909508
$298$	0	0
$299$	− 2.94137i	− 0.170104i
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	−10.1173	−0.581222
$304$	0	0
$305$	−9.67418	−0.553942
$306$	0	0
$307$	− 26.6707i	− 1.52218i	−0.648646	−	0.761090i	$-0.724665\pi$
$307$	− 26.6707i	− 1.52218i	0.648646	−	0.761090i	$-0.275335\pi$
$308$	0	0
$309$	57.2242i	3.25537i
$310$	0	0
$311$	−6.87930	−0.390089	−0.195045	−	0.980794i	$-0.562485\pi$
$311$	−6.87930	−0.390089	−0.195045	+	0.980794i	$0.562485\pi$
$312$	0	0
$313$	17.6707	0.998809	0.499405	−	0.866369i	$-0.333552\pi$
$313$	17.6707	0.998809	0.499405	+	0.866369i	$0.333552\pi$
$314$	0	0
$315$	24.5535i	1.38343i
$316$	0	0
$317$	− 1.11383i	− 0.0625588i	−0.999511	−	0.0312794i	$-0.990042\pi$
$317$	− 1.11383i	− 0.0625588i	0.999511	−	0.0312794i	$-0.00995817\pi$
$318$	0	0
$319$	7.87586	0.440963
$320$	0	0
$321$	35.5500	1.98421
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	33.5389	1.85470
$328$	0	0
$329$	22.1138	1.21917
$330$	0	0
$331$	18.4983i	1.01676i	0.861134	+	0.508379i	$0.169755\pi$
$331$	18.4983i	1.01676i	−0.861134	+	0.508379i	$0.830245\pi$
$332$	0	0
$333$	− 49.5500i	− 2.71533i
$334$	0	0
$335$	−1.50172	−0.0820477
$336$	0	0
$337$	−17.4362	−0.949811	−0.474905	−	0.880037i	$-0.657518\pi$
$337$	−17.4362	−0.949811	−0.474905	+	0.880037i	$0.657518\pi$
$338$	0	0
$339$	− 55.6052i	− 3.02006i
$340$	0	0
$341$	5.35524i	0.290002i
$342$	0	0
$343$	11.1871	0.604045
$344$	0	0
$345$	−9.55691	−0.514527
$346$	0	0
$347$	− 8.98539i	− 0.482361i	−0.970480	−	0.241181i	$-0.922465\pi$
$347$	− 8.98539i	− 0.482361i	0.970480	−	0.241181i	$-0.0775346\pi$
$348$	0	0
$349$	− 10.9931i	− 0.588448i	−0.955736	−	0.294224i	$-0.904939\pi$
$349$	− 10.9931i	− 0.588448i	0.955736	−	0.294224i	$-0.0950612\pi$
$350$	0	0
$351$	−14.8061	−0.790289
$352$	0	0
$353$	23.2311	1.23647	0.618233	−	0.785995i	$-0.287849\pi$
$353$	23.2311	1.23647	0.618233	+	0.785995i	$0.287849\pi$
$354$	0	0
$355$	5.36641i	0.284819i
$356$	0	0
$357$	− 10.5569i	− 0.558731i
$358$	0	0
$359$	11.2863	0.595668	0.297834	−	0.954618i	$-0.403736\pi$
$359$	11.2863	0.595668	0.297834	+	0.954618i	$0.403736\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	32.0992i	1.68477i
$364$	0	0
$365$	3.55691i	0.186177i
$366$	0	0
$367$	−13.6121	−0.710546	−0.355273	−	0.934763i	$-0.615612\pi$
$367$	−13.6121	−0.710546	−0.355273	+	0.934763i	$0.615612\pi$
$368$	0	0
$369$	71.3346	3.71353
$370$	0	0
$371$	5.05863i	0.262631i
$372$	0	0
$373$	33.1070i	1.71421i	0.515139	+	0.857107i	$0.327740\pi$
$373$	33.1070i	1.71421i	−0.515139	+	0.857107i	$0.672260\pi$
$374$	0	0
$375$	−29.2423	−1.51006
$376$	0	0
$377$	7.43965	0.383161
$378$	0	0
$379$	33.1690i	1.70378i	0.523722	+	0.851889i	$0.324543\pi$
$379$	33.1690i	1.70378i	−0.523722	+	0.851889i	$0.675457\pi$
$380$	0	0
$381$	51.7846i	2.65300i
$382$	0	0
$383$	−33.0698	−1.68979	−0.844894	−	0.534934i	$-0.820337\pi$
$383$	−33.0698	−1.68979	−0.844894	+	0.534934i	$0.820337\pi$
$384$	0	0
$385$	−3.43965	−0.175301
$386$	0	0
$387$	− 2.32582i	− 0.118228i
$388$	0	0
$389$	− 19.5569i	− 0.991575i	−0.868444	−	0.495787i	$-0.834880\pi$
$389$	− 19.5569i	− 0.991575i	0.868444	−	0.495787i	$-0.165120\pi$
$390$	0	0
$391$	2.94137	0.148751
$392$	0	0
$393$	−24.9931	−1.26074
$394$	0	0
$395$	6.73281i	0.338765i
$396$	0	0
$397$	34.2208i	1.71749i	0.512402	+	0.858746i	$0.328756\pi$
$397$	34.2208i	1.71749i	−0.512402	+	0.858746i	$0.671244\pi$
$398$	0	0
$399$	42.2277	2.11403
$400$	0	0
$401$	−32.6707	−1.63150	−0.815750	−	0.578405i	$-0.803675\pi$
$401$	−32.6707	−1.63150	−0.815750	+	0.578405i	$0.803675\pi$
$402$	0	0
$403$	5.05863i	0.251988i
$404$	0	0
$405$	25.4362i	1.26394i
$406$	0	0
$407$	6.94137	0.344071
$408$	0	0
$409$	−6.20855	−0.306993	−0.153497	−	0.988149i	$-0.549053\pi$
$409$	−6.20855	−0.306993	−0.153497	+	0.988149i	$0.549053\pi$
$410$	0	0
$411$	17.2932i	0.853009i
$412$	0	0
$413$	− 18.4362i	− 0.907187i
$414$	0	0
$415$	2.49828	0.122636
$416$	0	0
$417$	−40.5500	−1.98574
$418$	0	0
$419$	0.369845i	0.0180681i	0.999959	+	0.00903405i	$0.00287567\pi$
$419$	0.369845i	0.0180681i	−0.999959	+	0.00903405i	$0.997124\pi$
$420$	0	0
$421$	3.87930i	0.189065i	0.995522	+	0.0945327i	$0.0301357\pi$
$421$	3.87930i	0.189065i	−0.995522	+	0.0945327i	$0.969864\pi$
$422$	0	0
$423$	51.4328	2.50075
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	− 31.4328i	− 1.52114i
$428$	0	0
$429$	− 3.43965i	− 0.166068i
$430$	0	0
$431$	−0.516327	−0.0248706	−0.0124353	−	0.999923i	$-0.503958\pi$
$431$	−0.516327	−0.0248706	−0.0124353	+	0.999923i	$0.503958\pi$
$432$	0	0
$433$	30.7846	1.47941	0.739706	−	0.672930i	$-0.234965\pi$
$433$	30.7846	1.47941	0.739706	+	0.672930i	$0.234965\pi$
$434$	0	0
$435$	− 24.1725i	− 1.15898i
$436$	0	0
$437$	11.7655i	0.562819i
$438$	0	0
$439$	−30.4362	−1.45264	−0.726321	−	0.687356i	$-0.758771\pi$
$439$	−30.4362	−1.45264	−0.726321	+	0.687356i	$0.758771\pi$
$440$	0	0
$441$	−26.8793	−1.27997
$442$	0	0
$443$	3.45769i	0.164280i	0.996621	+	0.0821400i	$0.0261755\pi$
$443$	3.45769i	0.164280i	−0.996621	+	0.0821400i	$0.973825\pi$
$444$	0	0
$445$	2.11727i	0.100368i
$446$	0	0
$447$	29.6121	1.40060
$448$	0	0
$449$	14.7880	0.697889	0.348945	−	0.937143i	$-0.386540\pi$
$449$	14.7880	0.697889	0.348945	+	0.937143i	$0.386540\pi$
$450$	0	0
$451$	9.99312i	0.470558i
$452$	0	0
$453$	50.1070i	2.35423i
$454$	0	0
$455$	−3.24914	−0.152322
$456$	0	0
$457$	31.8759	1.49109	0.745545	−	0.666455i	$-0.232189\pi$
$457$	31.8759	1.49109	0.745545	+	0.666455i	$0.232189\pi$
$458$	0	0
$459$	− 14.8061i	− 0.691087i
$460$	0	0
$461$	− 18.5569i	− 0.864282i	−0.901806	−	0.432141i	$-0.857758\pi$
$461$	− 18.5569i	− 0.864282i	0.901806	−	0.432141i	$-0.142242\pi$
$462$	0	0
$463$	24.8241	1.15367	0.576837	−	0.816859i	$-0.304287\pi$
$463$	24.8241	1.15367	0.576837	+	0.816859i	$0.304287\pi$
$464$	0	0
$465$	16.4362	0.762211
$466$	0	0
$467$	40.2829i	1.86407i	0.362371	+	0.932034i	$0.381967\pi$
$467$	40.2829i	1.86407i	−0.362371	+	0.932034i	$0.618033\pi$
$468$	0	0
$469$	− 4.87930i	− 0.225305i
$470$	0	0
$471$	12.3189	0.567627
$472$	0	0
$473$	0.325819	0.0149812
$474$	0	0
$475$	16.0000i	0.734130i
$476$	0	0
$477$	11.7655i	0.538704i
$478$	0	0
$479$	4.07324	0.186111	0.0930556	−	0.995661i	$-0.470337\pi$
$479$	4.07324	0.186111	0.0930556	+	0.995661i	$0.470337\pi$
$480$	0	0
$481$	6.55691	0.298970
$482$	0	0
$483$	− 31.0518i	− 1.41290i
$484$	0	0
$485$	7.67418i	0.348467i
$486$	0	0
$487$	−15.0518	−0.682060	−0.341030	−	0.940052i	$-0.610776\pi$
$487$	−15.0518	−0.682060	−0.341030	+	0.940052i	$0.610776\pi$
$488$	0	0
$489$	−13.5569	−0.613065
$490$	0	0
$491$	− 3.92676i	− 0.177212i	−0.996067	−	0.0886061i	$-0.971759\pi$
$491$	− 3.92676i	− 0.177212i	0.996067	−	0.0886061i	$-0.0282412\pi$
$492$	0	0
$493$	7.43965i	0.335065i
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−17.4362	−0.782121
$498$	0	0
$499$	− 32.3189i	− 1.44679i	−0.690432	−	0.723397i	$-0.742580\pi$
$499$	− 32.3189i	− 1.44679i	0.690432	−	0.723397i	$-0.257420\pi$
$500$	0	0
$501$	− 46.4293i	− 2.07431i
$502$	0	0
$503$	16.1104	0.718327	0.359163	−	0.933275i	$-0.383062\pi$
$503$	16.1104	0.718327	0.359163	+	0.933275i	$0.383062\pi$
$504$	0	0
$505$	−3.11383	−0.138564
$506$	0	0
$507$	− 3.24914i	− 0.144299i
$508$	0	0
$509$	− 16.2277i	− 0.719278i	−0.933091	−	0.359639i	$-0.882900\pi$
$509$	− 16.2277i	− 0.719278i	0.933091	−	0.359639i	$-0.117100\pi$
$510$	0	0
$511$	−11.5569	−0.511248
$512$	0	0
$513$	59.2242	2.61481
$514$	0	0
$515$	17.6121i	0.776082i
$516$	0	0
$517$	7.20512i	0.316881i
$518$	0	0
$519$	−22.7328	−0.997860
$520$	0	0
$521$	−8.32238	−0.364610	−0.182305	−	0.983242i	$-0.558356\pi$
$521$	−8.32238	−0.364610	−0.182305	+	0.983242i	$0.558356\pi$
$522$	0	0
$523$	− 32.2829i	− 1.41163i	−0.708396	−	0.705815i	$-0.750581\pi$
$523$	− 32.2829i	− 1.41163i	0.708396	−	0.705815i	$-0.249419\pi$
$524$	0	0
$525$	− 42.2277i	− 1.84297i
$526$	0	0
$527$	−5.05863	−0.220358
$528$	0	0
$529$	−14.3484	−0.623842
$530$	0	0
$531$	− 42.8793i	− 1.86080i
$532$	0	0
$533$	9.43965i	0.408877i
$534$	0	0
$535$	10.9414	0.473037
$536$	0	0
$537$	−57.6639	−2.48838
$538$	0	0
$539$	− 3.76547i	− 0.162190i
$540$	0	0
$541$	2.32238i	0.0998470i	0.998753	+	0.0499235i	$0.0158977\pi$
$541$	2.32238i	0.0998470i	−0.998753	+	0.0499235i	$0.984102\pi$
$542$	0	0
$543$	−26.3741	−1.13182
$544$	0	0
$545$	10.3224	0.442162
$546$	0	0
$547$	9.89390i	0.423033i	0.977374	+	0.211516i	$0.0678402\pi$
$547$	9.89390i	0.423033i	−0.977374	+	0.211516i	$0.932160\pi$
$548$	0	0
$549$	− 73.1070i	− 3.12013i
$550$	0	0
$551$	−29.7586	−1.26776
$552$	0	0
$553$	−21.8759	−0.930256
$554$	0	0
$555$	− 21.3043i	− 0.904319i
$556$	0	0
$557$	− 30.3155i	− 1.28451i	−0.766491	−	0.642255i	$-0.777999\pi$
$557$	− 30.3155i	− 1.28451i	0.766491	−	0.642255i	$-0.222001\pi$
$558$	0	0
$559$	0.307774	0.0130174
$560$	0	0
$561$	3.43965	0.145222
$562$	0	0
$563$	38.7440i	1.63286i	0.577441	+	0.816432i	$0.304051\pi$
$563$	38.7440i	1.63286i	−0.577441	+	0.816432i	$0.695949\pi$
$564$	0	0
$565$	− 17.1138i	− 0.719984i
$566$	0	0
$567$	−82.6458	−3.47080
$568$	0	0
$569$	−19.2345	−0.806354	−0.403177	−	0.915122i	$-0.632094\pi$
$569$	−19.2345	−0.806354	−0.403177	+	0.915122i	$0.632094\pi$
$570$	0	0
$571$	− 10.8320i	− 0.453307i	−0.973976	−	0.226653i	$-0.927222\pi$
$571$	− 10.8320i	− 0.453307i	0.973976	−	0.226653i	$-0.0727784\pi$
$572$	0	0
$573$	64.7811i	2.70627i
$574$	0	0
$575$	11.7655	0.490654
$576$	0	0
$577$	−45.1329	−1.87891	−0.939454	−	0.342674i	$-0.888667\pi$
$577$	−45.1329	−1.87891	−0.939454	+	0.342674i	$0.888667\pi$
$578$	0	0
$579$	13.3776i	0.555953i
$580$	0	0
$581$	8.11727i	0.336761i
$582$	0	0
$583$	−1.64820	−0.0682615
$584$	0	0
$585$	−7.55691	−0.312440
$586$	0	0
$587$	5.05863i	0.208792i	0.994536	+	0.104396i	$0.0332910\pi$
$587$	5.05863i	0.208792i	−0.994536	+	0.104396i	$0.966709\pi$
$588$	0	0
$589$	− 20.2345i	− 0.833749i
$590$	0	0
$591$	35.0629	1.44230
$592$	0	0
$593$	29.4328	1.20866	0.604330	−	0.796734i	$-0.293441\pi$
$593$	29.4328	1.20866	0.604330	+	0.796734i	$0.293441\pi$
$594$	0	0
$595$	− 3.24914i	− 0.133202i
$596$	0	0
$597$	− 2.00000i	− 0.0818546i
$598$	0	0
$599$	40.4293	1.65190	0.825949	−	0.563745i	$-0.190640\pi$
$599$	40.4293	1.65190	0.825949	+	0.563745i	$0.190640\pi$
$600$	0	0
$601$	−22.9931	−0.937909	−0.468955	−	0.883222i	$-0.655369\pi$
$601$	−22.9931	−0.937909	−0.468955	+	0.883222i	$0.655369\pi$
$602$	0	0
$603$	− 11.3484i	− 0.462141i
$604$	0	0
$605$	9.87930i	0.401650i
$606$	0	0
$607$	−17.7846	−0.721853	−0.360927	−	0.932594i	$-0.617540\pi$
$607$	−17.7846	−0.721853	−0.360927	+	0.932594i	$0.617540\pi$
$608$	0	0
$609$	78.5397	3.18259
$610$	0	0
$611$	6.80605i	0.275344i
$612$	0	0
$613$	24.2277i	0.978546i	0.872131	+	0.489273i	$0.162738\pi$
$613$	24.2277i	0.978546i	−0.872131	+	0.489273i	$0.837262\pi$
$614$	0	0
$615$	30.6707	1.23676
$616$	0	0
$617$	−18.2277	−0.733818	−0.366909	−	0.930257i	$-0.619584\pi$
$617$	−18.2277	−0.733818	−0.366909	+	0.930257i	$0.619584\pi$
$618$	0	0
$619$	43.1982i	1.73628i	0.496316	+	0.868142i	$0.334686\pi$
$619$	43.1982i	1.73628i	−0.496316	+	0.868142i	$0.665314\pi$
$620$	0	0
$621$	− 43.5500i	− 1.74760i
$622$	0	0
$623$	−6.87930	−0.275613
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	13.7586i	0.549465i
$628$	0	0
$629$	6.55691i	0.261441i
$630$	0	0
$631$	7.07668	0.281718	0.140859	−	0.990030i	$-0.455014\pi$
$631$	7.07668	0.281718	0.140859	+	0.990030i	$0.455014\pi$
$632$	0	0
$633$	−5.87930	−0.233681
$634$	0	0
$635$	15.9379i	0.632477i
$636$	0	0
$637$	− 3.55691i	− 0.140930i
$638$	0	0
$639$	−40.5535	−1.60427
$640$	0	0
$641$	−28.2277	−1.11493	−0.557463	−	0.830202i	$-0.688225\pi$
$641$	−28.2277	−1.11493	−0.557463	+	0.830202i	$0.688225\pi$
$642$	0	0
$643$	2.73281i	0.107772i	0.998547	+	0.0538858i	$0.0171607\pi$
$643$	2.73281i	0.107772i	−0.998547	+	0.0538858i	$0.982839\pi$
$644$	0	0
$645$	− 1.00000i	− 0.0393750i
$646$	0	0
$647$	14.5604	0.572427	0.286213	−	0.958166i	$-0.407603\pi$
$647$	14.5604	0.572427	0.286213	+	0.958166i	$0.407603\pi$
$648$	0	0
$649$	6.00688	0.235790
$650$	0	0
$651$	53.4036i	2.09305i
$652$	0	0
$653$	29.3415i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$653$	29.3415i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$654$	0	0
$655$	−7.69223	−0.300560
$656$	0	0
$657$	−26.8793	−1.04866
$658$	0	0
$659$	− 32.2829i	− 1.25756i	−0.777583	−	0.628781i	$-0.783554\pi$
$659$	− 32.2829i	− 1.25756i	0.777583	−	0.628781i	$-0.216446\pi$
$660$	0	0
$661$	24.4102i	0.949448i	0.880135	+	0.474724i	$0.157452\pi$
$661$	24.4102i	0.949448i	−0.880135	+	0.474724i	$0.842548\pi$
$662$	0	0
$663$	3.24914	0.126186
$664$	0	0
$665$	12.9966	0.503985
$666$	0	0
$667$	21.8827i	0.847303i
$668$	0	0
$669$	22.1138i	0.854970i
$670$	0	0
$671$	10.2414	0.395365
$672$	0	0
$673$	11.8793	0.457913	0.228957	−	0.973437i	$-0.426469\pi$
$673$	11.8793	0.457913	0.228957	+	0.973437i	$0.426469\pi$
$674$	0	0
$675$	− 59.2242i	− 2.27954i
$676$	0	0
$677$	− 8.76891i	− 0.337016i	−0.985700	−	0.168508i	$-0.946105\pi$
$677$	− 8.76891i	− 0.337016i	0.985700	−	0.168508i	$-0.0538950\pi$
$678$	0	0
$679$	−24.9345	−0.956898
$680$	0	0
$681$	21.6742	0.830556
$682$	0	0
$683$	3.38445i	0.129502i	0.997901	+	0.0647512i	$0.0206254\pi$
$683$	3.38445i	0.129502i	−0.997901	+	0.0647512i	$0.979375\pi$
$684$	0	0
$685$	5.32238i	0.203358i
$686$	0	0
$687$	−58.8544	−2.24543
$688$	0	0
$689$	−1.55691	−0.0593137
$690$	0	0
$691$	11.2242i	0.426989i	0.976944	+	0.213495i	$0.0684846\pi$
$691$	11.2242i	0.426989i	−0.976944	+	0.213495i	$0.931515\pi$
$692$	0	0
$693$	− 25.9931i	− 0.987397i
$694$	0	0
$695$	−12.4802	−0.473402
$696$	0	0
$697$	−9.43965	−0.357552
$698$	0	0
$699$	37.1579i	1.40544i
$700$	0	0
$701$	48.6707i	1.83827i	0.393944	+	0.919134i	$0.371110\pi$
$701$	48.6707i	1.83827i	−0.393944	+	0.919134i	$0.628890\pi$
$702$	0	0
$703$	−26.2277	−0.989195
$704$	0	0
$705$	22.1138	0.832855
$706$	0	0
$707$	− 10.1173i	− 0.380499i
$708$	0	0
$709$	− 11.7586i	− 0.441603i	−0.975319	−	0.220802i	$-0.929133\pi$
$709$	− 11.7586i	− 0.441603i	0.975319	−	0.220802i	$-0.0708673\pi$
$710$	0	0
$711$	−50.8793	−1.90812
$712$	0	0
$713$	−14.8793	−0.557234
$714$	0	0
$715$	− 1.05863i	− 0.0395906i
$716$	0	0
$717$	− 54.7846i	− 2.04597i
$718$	0	0
$719$	38.7191	1.44398	0.721989	−	0.691905i	$-0.243228\pi$
$719$	38.7191	1.44398	0.721989	+	0.691905i	$0.243228\pi$
$720$	0	0
$721$	−57.2242	−2.13114
$722$	0	0
$723$	− 16.9966i	− 0.632109i
$724$	0	0
$725$	29.7586i	1.10521i
$726$	0	0
$727$	8.93449	0.331362	0.165681	−	0.986179i	$-0.447018\pi$
$727$	8.93449	0.331362	0.165681	+	0.986179i	$0.447018\pi$
$728$	0	0
$729$	−47.8984	−1.77401
$730$	0	0
$731$	0.307774i	0.0113834i
$732$	0	0
$733$	36.3484i	1.34256i	0.741205	+	0.671279i	$0.234255\pi$
$733$	36.3484i	1.34256i	−0.741205	+	0.671279i	$0.765745\pi$
$734$	0	0
$735$	−11.5569	−0.426283
$736$	0	0
$737$	1.58977	0.0585599
$738$	0	0
$739$	− 8.44309i	− 0.310584i	−0.987869	−	0.155292i	$-0.950368\pi$
$739$	− 8.44309i	− 0.310584i	0.987869	−	0.155292i	$-0.0496318\pi$
$740$	0	0
$741$	12.9966i	0.477441i
$742$	0	0
$743$	−22.5354	−0.826745	−0.413372	−	0.910562i	$-0.635649\pi$
$743$	−22.5354	−0.826745	−0.413372	+	0.910562i	$0.635649\pi$
$744$	0	0
$745$	9.11383	0.333905
$746$	0	0
$747$	18.8793i	0.690757i
$748$	0	0
$749$	35.5500i	1.29897i
$750$	0	0
$751$	−2.43621	−0.0888986	−0.0444493	−	0.999012i	$-0.514153\pi$
$751$	−2.43621	−0.0888986	−0.0444493	+	0.999012i	$0.514153\pi$
$752$	0	0
$753$	−6.67762	−0.243346
$754$	0	0
$755$	15.4216i	0.561250i
$756$	0	0
$757$	− 43.1329i	− 1.56769i	−0.620955	−	0.783847i	$-0.713255\pi$
$757$	− 43.1329i	− 1.56769i	0.620955	−	0.783847i	$-0.286745\pi$
$758$	0	0
$759$	10.1173	0.367234
$760$	0	0
$761$	30.0191	1.08819	0.544096	−	0.839023i	$-0.316873\pi$
$761$	30.0191	1.08819	0.544096	+	0.839023i	$0.316873\pi$
$762$	0	0
$763$	33.5389i	1.21419i
$764$	0	0
$765$	− 7.55691i	− 0.273221i
$766$	0	0
$767$	5.67418	0.204883
$768$	0	0
$769$	−35.3484	−1.27469	−0.637347	−	0.770577i	$-0.719968\pi$
$769$	−35.3484	−1.27469	−0.637347	+	0.770577i	$0.719968\pi$
$770$	0	0
$771$	− 68.9716i	− 2.48395i
$772$	0	0
$773$	47.2277i	1.69866i	0.527862	+	0.849330i	$0.322994\pi$
$773$	47.2277i	1.69866i	−0.527862	+	0.849330i	$0.677006\pi$
$774$	0	0
$775$	−20.2345	−0.726846
$776$	0	0
$777$	69.2208	2.48328
$778$	0	0
$779$	− 37.7586i	− 1.35284i
$780$	0	0
$781$	− 5.68106i	− 0.203284i
$782$	0	0
$783$	110.152	3.93651
$784$	0	0
$785$	3.79145	0.135323
$786$	0	0
$787$	− 24.7880i	− 0.883597i	−0.897114	−	0.441799i	$-0.854341\pi$
$787$	− 24.7880i	− 0.883597i	0.897114	−	0.441799i	$-0.145659\pi$
$788$	0	0
$789$	− 76.5397i	− 2.72489i
$790$	0	0
$791$	55.6052	1.97709
$792$	0	0
$793$	9.67418	0.343540
$794$	0	0
$795$	5.05863i	0.179411i
$796$	0	0
$797$	− 32.1104i	− 1.13741i	−0.822542	−	0.568704i	$-0.807445\pi$
$797$	− 32.1104i	− 1.13741i	0.822542	−	0.568704i	$-0.192555\pi$
$798$	0	0
$799$	−6.80605	−0.240781
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	− 3.76547i	− 0.132880i
$804$	0	0
$805$	− 9.55691i	− 0.336837i
$806$	0	0
$807$	20.5535	0.723517
$808$	0	0
$809$	4.55004	0.159971	0.0799854	−	0.996796i	$-0.474513\pi$
$809$	4.55004	0.159971	0.0799854	+	0.996796i	$0.474513\pi$
$810$	0	0
$811$	48.3449i	1.69762i	0.528698	+	0.848810i	$0.322680\pi$
$811$	48.3449i	1.69762i	−0.528698	+	0.848810i	$0.677320\pi$
$812$	0	0
$813$	− 11.2345i	− 0.394012i
$814$	0	0
$815$	−4.17246	−0.146155
$816$	0	0
$817$	−1.23109	−0.0430706
$818$	0	0
$819$	− 24.5535i	− 0.857968i
$820$	0	0
$821$	24.3484i	0.849764i	0.905249	+	0.424882i	$0.139684\pi$
$821$	24.3484i	0.849764i	−0.905249	+	0.424882i	$0.860316\pi$
$822$	0	0
$823$	28.4431	0.991464	0.495732	−	0.868476i	$-0.334900\pi$
$823$	28.4431	0.991464	0.495732	+	0.868476i	$0.334900\pi$
$824$	0	0
$825$	13.7586	0.479013
$826$	0	0
$827$	− 34.6448i	− 1.20472i	−0.798226	−	0.602358i	$-0.794228\pi$
$827$	− 34.6448i	− 1.20472i	0.798226	−	0.602358i	$-0.205772\pi$
$828$	0	0
$829$	− 34.2208i	− 1.18854i	−0.804267	−	0.594268i	$-0.797442\pi$
$829$	− 34.2208i	− 1.18854i	0.804267	−	0.594268i	$-0.202558\pi$
$830$	0	0
$831$	92.7743	3.21830
$832$	0	0
$833$	3.55691	0.123240
$834$	0	0
$835$	− 14.2897i	− 0.494516i
$836$	0	0
$837$	74.8984i	2.58887i
$838$	0	0
$839$	−12.9345	−0.446548	−0.223274	−	0.974756i	$-0.571675\pi$
$839$	−12.9345	−0.446548	−0.223274	+	0.974756i	$0.571675\pi$
$840$	0	0
$841$	−26.3484	−0.908564
$842$	0	0
$843$	81.2173i	2.79727i
$844$	0	0
$845$	− 1.00000i	− 0.0344010i
$846$	0	0
$847$	−32.0992	−1.10294
$848$	0	0
$849$	70.8984	2.43323
$850$	0	0
$851$	19.2863i	0.661126i
$852$	0	0
$853$	− 31.6448i	− 1.08350i	−0.840541	−	0.541748i	$-0.817763\pi$
$853$	− 31.6448i	− 1.08350i	0.840541	−	0.541748i	$-0.182237\pi$
$854$	0	0
$855$	30.2277	1.03376
$856$	0	0
$857$	38.8724	1.32786	0.663928	−	0.747796i	$-0.268888\pi$
$857$	38.8724	1.32786	0.663928	+	0.747796i	$0.268888\pi$
$858$	0	0
$859$	− 29.8207i	− 1.01747i	−0.860924	−	0.508734i	$-0.830114\pi$
$859$	− 29.8207i	− 1.01747i	0.860924	−	0.508734i	$-0.169886\pi$
$860$	0	0
$861$	99.6536i	3.39618i
$862$	0	0
$863$	−24.5163	−0.834545	−0.417273	−	0.908781i	$-0.637014\pi$
$863$	−24.5163	−0.834545	−0.417273	+	0.908781i	$0.637014\pi$
$864$	0	0
$865$	−6.99656	−0.237890
$866$	0	0
$867$	− 51.9862i	− 1.76555i
$868$	0	0
$869$	− 7.12758i	− 0.241787i
$870$	0	0
$871$	1.50172	0.0508838
$872$	0	0
$873$	−57.9931	−1.96277
$874$	0	0
$875$	− 29.2423i	− 0.988569i
$876$	0	0
$877$	4.34836i	0.146834i	0.997301	+	0.0734169i	$0.0233904\pi$
$877$	4.34836i	0.146834i	−0.997301	+	0.0734169i	$0.976610\pi$
$878$	0	0
$879$	89.1441	3.00676
$880$	0	0
$881$	27.6967	0.933126	0.466563	−	0.884488i	$-0.345492\pi$
$881$	27.6967	0.933126	0.466563	+	0.884488i	$0.345492\pi$
$882$	0	0
$883$	− 14.0440i	− 0.472619i	−0.971678	−	0.236310i	$-0.924062\pi$
$883$	− 14.0440i	− 0.472619i	0.971678	−	0.236310i	$-0.0759379\pi$
$884$	0	0
$885$	− 18.4362i	− 0.619726i
$886$	0	0
$887$	3.66730	0.123136	0.0615680	−	0.998103i	$-0.480390\pi$
$887$	3.66730	0.123136	0.0615680	+	0.998103i	$0.480390\pi$
$888$	0	0
$889$	−51.7846	−1.73680
$890$	0	0
$891$	− 26.9276i	− 0.902109i
$892$	0	0
$893$	− 27.2242i	− 0.911024i
$894$	0	0
$895$	−17.7474	−0.593231
$896$	0	0
$897$	9.55691	0.319096
$898$	0	0
$899$	− 37.6344i	− 1.25518i
$900$	0	0
$901$	− 1.55691i	− 0.0518683i
$902$	0	0
$903$	3.24914	0.108125
$904$	0	0
$905$	−8.11727	−0.269827
$906$	0	0
$907$	− 9.39239i	− 0.311869i	−0.987767	−	0.155935i	$-0.950161\pi$
$907$	− 9.39239i	− 0.311869i	0.987767	−	0.155935i	$-0.0498389\pi$
$908$	0	0
$909$	− 23.5309i	− 0.780472i
$910$	0	0
$911$	−3.64496	−0.120763	−0.0603815	−	0.998175i	$-0.519232\pi$
$911$	−3.64496	−0.120763	−0.0603815	+	0.998175i	$0.519232\pi$
$912$	0	0
$913$	−2.64476	−0.0875289
$914$	0	0
$915$	− 31.4328i	− 1.03914i
$916$	0	0
$917$	− 24.9931i	− 0.825346i
$918$	0	0
$919$	−12.6516	−0.417339	−0.208670	−	0.977986i	$-0.566913\pi$
$919$	−12.6516	−0.417339	−0.208670	+	0.977986i	$0.566913\pi$
$920$	0	0
$921$	86.6570	2.85544
$922$	0	0
$923$	− 5.36641i	− 0.176637i
$924$	0	0
$925$	26.2277i	0.862360i
$926$	0	0
$927$	−133.093	−4.37135
$928$	0	0
$929$	37.1070	1.21744	0.608720	−	0.793385i	$-0.291683\pi$
$929$	37.1070	1.21744	0.608720	+	0.793385i	$0.291683\pi$
$930$	0	0
$931$	14.2277i	0.466293i
$932$	0	0
$933$	− 22.3518i	− 0.731765i
$934$	0	0
$935$	1.05863	0.0346210
$936$	0	0
$937$	50.8724	1.66193	0.830965	−	0.556325i	$-0.187789\pi$
$937$	50.8724	1.66193	0.830965	+	0.556325i	$0.187789\pi$
$938$	0	0
$939$	57.4147i	1.87366i
$940$	0	0
$941$	− 21.4622i	− 0.699647i	−0.936816	−	0.349824i	$-0.886241\pi$
$941$	− 21.4622i	− 0.699647i	0.936816	−	0.349824i	$-0.113759\pi$
$942$	0	0
$943$	−27.7655	−0.904168
$944$	0	0
$945$	−48.1070	−1.56492
$946$	0	0
$947$	61.1690i	1.98773i	0.110617	+	0.993863i	$0.464717\pi$
$947$	61.1690i	1.98773i	−0.110617	+	0.993863i	$0.535283\pi$
$948$	0	0
$949$	− 3.55691i	− 0.115462i
$950$	0	0
$951$	3.61899	0.117354
$952$	0	0
$953$	13.2345	0.428709	0.214354	−	0.976756i	$-0.431235\pi$
$953$	13.2345	0.428709	0.214354	+	0.976756i	$0.431235\pi$
$954$	0	0
$955$	19.9379i	0.645176i
$956$	0	0
$957$	25.5898i	0.827200i
$958$	0	0
$959$	−17.2932	−0.558425
$960$	0	0
$961$	−5.41023	−0.174524
$962$	0	0
$963$	82.6830i	2.66442i
$964$	0	0
$965$	4.11727i	0.132539i
$966$	0	0
$967$	−48.4441	−1.55786	−0.778929	−	0.627112i	$-0.784237\pi$
$967$	−48.4441	−1.55786	−0.778929	+	0.627112i	$0.784237\pi$
$968$	0	0
$969$	−12.9966	−0.417510
$970$	0	0
$971$	5.69910i	0.182893i	0.995810	+	0.0914464i	$0.0291490\pi$
$971$	5.69910i	0.182893i	−0.995810	+	0.0914464i	$0.970851\pi$
$972$	0	0
$973$	− 40.5500i	− 1.29997i
$974$	0	0
$975$	12.9966	0.416223
$976$	0	0
$977$	2.65164	0.0848334	0.0424167	−	0.999100i	$-0.486494\pi$
$977$	2.65164	0.0848334	0.0424167	+	0.999100i	$0.486494\pi$
$978$	0	0
$979$	− 2.24141i	− 0.0716357i
$980$	0	0
$981$	78.0054i	2.49052i
$982$	0	0
$983$	−27.2491	−0.869113	−0.434556	−	0.900645i	$-0.643095\pi$
$983$	−27.2491	−0.869113	−0.434556	+	0.900645i	$0.643095\pi$
$984$	0	0
$985$	10.7914	0.343844
$986$	0	0
$987$	71.8509i	2.28704i
$988$	0	0
$989$	0.905275i	0.0287861i
$990$	0	0
$991$	−22.7552	−0.722841	−0.361421	−	0.932403i	$-0.617708\pi$
$991$	−22.7552	−0.722841	−0.361421	+	0.932403i	$0.617708\pi$
$992$	0	0
$993$	−60.1035	−1.90733
$994$	0	0
$995$	− 0.615547i	− 0.0195142i
$996$	0	0
$997$	4.79488i	0.151856i	0.997113	+	0.0759278i	$0.0241918\pi$
$997$	4.79488i	0.151856i	−0.997113	+	0.0759278i	$0.975808\pi$
$998$	0	0
$999$	97.0820	3.07154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.b.c.209.6		6
3.2	odd	2		3744.2.g.c.1873.1		6
4.3	odd	2		104.2.b.c.53.3	✓	6
8.3	odd	2		104.2.b.c.53.4	yes	6
8.5	even	2	inner	416.2.b.c.209.1		6
12.11	even	2		936.2.g.c.469.4		6
16.3	odd	4		3328.2.a.bh.1.3		3
16.5	even	4		3328.2.a.bg.1.3		3
16.11	odd	4		3328.2.a.be.1.1		3
16.13	even	4		3328.2.a.bf.1.1		3
24.5	odd	2		3744.2.g.c.1873.4		6
24.11	even	2		936.2.g.c.469.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.c.53.3	✓	6	4.3	odd	2
104.2.b.c.53.4	yes	6	8.3	odd	2
416.2.b.c.209.1		6	8.5	even	2	inner
416.2.b.c.209.6		6	1.1	even	1	trivial
936.2.g.c.469.3		6	24.11	even	2
936.2.g.c.469.4		6	12.11	even	2
3328.2.a.be.1.1		3	16.11	odd	4
3328.2.a.bf.1.1		3	16.13	even	4
3328.2.a.bg.1.3		3	16.5	even	4
3328.2.a.bh.1.3		3	16.3	odd	4
3744.2.g.c.1873.1		6	3.2	odd	2
3744.2.g.c.1873.4		6	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.24914i q^{3} +1.00000i q^{5} -3.24914 q^{7} -7.55691 q^{9} -1.05863i q^{11} -1.00000i q^{13} -3.24914 q^{15} +1.00000 q^{17} +4.00000i q^{19} -10.5569i q^{21} +2.94137 q^{23} +4.00000 q^{25} -14.8061i q^{27} +7.43965i q^{29} -5.05863 q^{31} +3.43965 q^{33} -3.24914i q^{35} +6.55691i q^{37} +3.24914 q^{39} -9.43965 q^{41} +0.307774i q^{43} -7.55691i q^{45} -6.80605 q^{47} +3.55691 q^{49} +3.24914i q^{51} -1.55691i q^{53} +1.05863 q^{55} -12.9966 q^{57} +5.67418i q^{59} +9.67418i q^{61} +24.5535 q^{63} +1.00000 q^{65} +1.50172i q^{67} +9.55691i q^{69} +5.36641 q^{71} +3.55691 q^{73} +12.9966i q^{75} +3.43965i q^{77} +6.73281 q^{79} +25.4362 q^{81} -2.49828i q^{83} +1.00000i q^{85} -24.1725 q^{87} +2.11727 q^{89} +3.24914i q^{91} -16.4362i q^{93} -4.00000 q^{95} +7.67418 q^{97} +8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{7} - 12 q^{9} - 2 q^{15} + 6 q^{17} + 16 q^{23} + 24 q^{25} - 32 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{57} + 44 q^{63} + 6 q^{65} + 18 q^{71} - 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 6 * q - 2 * q^7 - 12 * q^9 - 2 * q^15 + 6 * q^17 + 16 * q^23 + 24 * q^25 - 32 * q^31 - 16 * q^33 + 2 * q^39 - 20 * q^41 + 10 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^57 + 44 * q^63 + 6 * q^65 + 18 * q^71 - 12 * q^73 + 12 * q^79 + 46 * q^81 - 80 * q^87 + 16 * q^89 - 24 * q^95 + 16 * q^97

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