LMFDB - Embedded newform 160.8.d.a.81.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [160,8,Mod(81,160)] 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("160.81"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	160.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$49.9816040775$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.18
Character	$\chi$	$=$	160.81
Dual form			160.8.d.a.81.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+21.9337i q^{3} -125.000i q^{5} +1766.19 q^{7} +1705.91 q^{9} -4654.38i q^{11} -10214.0i q^{13} +2741.72 q^{15} -12616.9 q^{17} +8565.99i q^{19} +38739.2i q^{21} +7657.10 q^{23} -15625.0 q^{25} +85386.1i q^{27} -97383.8i q^{29} -282907. q^{31} +102088. q^{33} -220774. i q^{35} -468640. i q^{37} +224032. q^{39} +233023. q^{41} -134095. i q^{43} -213239. i q^{45} -892914. q^{47} +2.29589e6 q^{49} -276735. i q^{51} +979881. i q^{53} -581797. q^{55} -187884. q^{57} +921947. i q^{59} -1.96677e6i q^{61} +3.01297e6 q^{63} -1.27675e6 q^{65} -2.76945e6i q^{67} +167949. i q^{69} +4.08400e6 q^{71} -294889. q^{73} -342714. i q^{75} -8.22052e6i q^{77} +3.23939e6 q^{79} +1.85799e6 q^{81} -2.58186e6i q^{83} +1.57711e6i q^{85} +2.13599e6 q^{87} -2.22040e6 q^{89} -1.80400e7i q^{91} -6.20520e6i q^{93} +1.07075e6 q^{95} -1.81261e6 q^{97} -7.93995e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) $ 28 * q - 1372 * q^7 - 20412 * q^9 + 13500 * q^15 - 2588 * q^23 - 437500 * q^25 - 268024 * q^31 - 99016 * q^33 + 283944 * q^39 - 601208 * q^41 + 2076460 * q^47 + 4316268 * q^49 - 1331000 * q^55 + 3788536 * q^57 + 14839060 * q^63 + 1001656 * q^71 + 2534128 * q^73 + 22040096 * q^79 + 11166300 * q^81 - 34127184 * q^87 + 3807224 * q^89 + 13718000 * q^95 + 15198608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	21.9337i	0.469016i	0.972114	+	0.234508i	$0.0753480\pi$
$3$	21.9337i	0.469016i	−0.972114	+	0.234508i	$0.924652\pi$
$4$	0	0
$5$	− 125.000i	− 0.447214i
$5$	− 125.000i	− 0.447214i
$6$	0	0
$7$	1766.19	1.94623	0.973117	−	0.230312i	$-0.0739747\pi$
$7$	1766.19	1.94623	0.973117	+	0.230312i	$0.0739747\pi$
$8$	0	0
$9$	1705.91	0.780024
$10$	0	0
$11$	− 4654.38i	− 1.05436i	−0.849755	−	0.527178i	$-0.823250\pi$
$11$	− 4654.38i	− 1.05436i	0.849755	−	0.527178i	$-0.176750\pi$
$12$	0	0
$13$	− 10214.0i	− 1.28942i	−0.764426	−	0.644712i	$-0.776977\pi$
$13$	− 10214.0i	− 1.28942i	0.764426	−	0.644712i	$-0.223023\pi$
$14$	0	0
$15$	2741.72	0.209751
$16$	0	0
$17$	−12616.9	−0.622845	−0.311423	−	0.950271i	$-0.600805\pi$
$17$	−12616.9	−0.622845	−0.311423	+	0.950271i	$0.600805\pi$
$18$	0	0
$19$	8565.99i	0.286510i	0.989686	+	0.143255i	$0.0457569\pi$
$19$	8565.99i	0.286510i	−0.989686	+	0.143255i	$0.954243\pi$
$20$	0	0
$21$	38739.2i	0.912816i
$22$	0	0
$23$	7657.10	0.131225	0.0656126	−	0.997845i	$-0.479100\pi$
$23$	7657.10	0.131225	0.0656126	+	0.997845i	$0.479100\pi$
$24$	0	0
$25$	−15625.0	−0.200000
$26$	0	0
$27$	85386.1i	0.834860i
$28$	0	0
$29$	− 97383.8i	− 0.741470i	−0.928739	−	0.370735i	$-0.879106\pi$
$29$	− 97383.8i	− 0.741470i	0.928739	−	0.370735i	$-0.120894\pi$
$30$	0	0
$31$	−282907.	−1.70560	−0.852800	−	0.522237i	$-0.825098\pi$
$31$	−282907.	−1.70560	−0.852800	+	0.522237i	$0.825098\pi$
$32$	0	0
$33$	102088.	0.494510
$34$	0	0
$35$	− 220774.i	− 0.870382i
$36$	0	0
$37$	− 468640.i	− 1.52102i	−0.649329	−	0.760508i	$-0.724950\pi$
$37$	− 468640.i	− 1.52102i	0.649329	−	0.760508i	$-0.275050\pi$
$38$	0	0
$39$	224032.	0.604761
$40$	0	0
$41$	233023.	0.528026	0.264013	−	0.964519i	$-0.414954\pi$
$41$	233023.	0.528026	0.264013	+	0.964519i	$0.414954\pi$
$42$	0	0
$43$	− 134095.i	− 0.257201i	−0.991696	−	0.128601i	$-0.958951\pi$
$43$	− 134095.i	− 0.257201i	0.991696	−	0.128601i	$-0.0410485\pi$
$44$	0	0
$45$	− 213239.i	− 0.348837i
$46$	0	0
$47$	−892914.	−1.25449	−0.627245	−	0.778822i	$-0.715817\pi$
$47$	−892914.	−1.25449	−0.627245	+	0.778822i	$0.715817\pi$
$48$	0	0
$49$	2.29589e6	2.78783
$50$	0	0
$51$	− 276735.i	− 0.292125i
$52$	0	0
$53$	979881.i	0.904082i	0.891997	+	0.452041i	$0.149304\pi$
$53$	979881.i	0.904082i	−0.891997	+	0.452041i	$0.850696\pi$
$54$	0	0
$55$	−581797.	−0.471522
$56$	0	0
$57$	−187884.	−0.134378
$58$	0	0
$59$	921947.i	0.584418i	0.956354	+	0.292209i	$0.0943903\pi$
$59$	921947.i	0.584418i	−0.956354	+	0.292209i	$0.905610\pi$
$60$	0	0
$61$	− 1.96677e6i	− 1.10943i	−0.832041	−	0.554714i	$-0.812828\pi$
$61$	− 1.96677e6i	− 1.10943i	0.832041	−	0.554714i	$-0.187172\pi$
$62$	0	0
$63$	3.01297e6	1.51811
$64$	0	0
$65$	−1.27675e6	−0.576648
$66$	0	0
$67$	− 2.76945e6i	− 1.12495i	−0.826815	−	0.562474i	$-0.809850\pi$
$67$	− 2.76945e6i	− 1.12495i	0.826815	−	0.562474i	$-0.190150\pi$
$68$	0	0
$69$	167949.i	0.0615468i
$70$	0	0
$71$	4.08400e6	1.35420	0.677098	−	0.735893i	$-0.263237\pi$
$71$	4.08400e6	1.35420	0.677098	+	0.735893i	$0.263237\pi$
$72$	0	0
$73$	−294889.	−0.0887215	−0.0443608	−	0.999016i	$-0.514125\pi$
$73$	−294889.	−0.0887215	−0.0443608	+	0.999016i	$0.514125\pi$
$74$	0	0
$75$	− 342714.i	− 0.0938033i
$76$	0	0
$77$	− 8.22052e6i	− 2.05202i
$78$	0	0
$79$	3.23939e6	0.739212	0.369606	−	0.929189i	$-0.379493\pi$
$79$	3.23939e6	0.739212	0.369606	+	0.929189i	$0.379493\pi$
$80$	0	0
$81$	1.85799e6	0.388460
$82$	0	0
$83$	− 2.58186e6i	− 0.495631i	−0.968807	−	0.247816i	$-0.920287\pi$
$83$	− 2.58186e6i	− 0.495631i	0.968807	−	0.247816i	$-0.0797127\pi$
$84$	0	0
$85$	1.57711e6i	0.278545i
$86$	0	0
$87$	2.13599e6	0.347762
$88$	0	0
$89$	−2.22040e6	−0.333862	−0.166931	−	0.985969i	$-0.553386\pi$
$89$	−2.22040e6	−0.333862	−0.166931	+	0.985969i	$0.553386\pi$
$90$	0	0
$91$	− 1.80400e7i	− 2.50952i
$92$	0	0
$93$	− 6.20520e6i	− 0.799955i
$94$	0	0
$95$	1.07075e6	0.128131
$96$	0	0
$97$	−1.81261e6	−0.201653	−0.100826	−	0.994904i	$-0.532149\pi$
$97$	−1.81261e6	−0.201653	−0.100826	+	0.994904i	$0.532149\pi$
$98$	0	0
$99$	− 7.93995e6i	− 0.822422i
$100$	0	0
$101$	1.75572e7i	1.69562i	0.530296	+	0.847812i	$0.322081\pi$
$101$	1.75572e7i	1.69562i	−0.530296	+	0.847812i	$0.677919\pi$
$102$	0	0
$103$	−4.32525e6	−0.390015	−0.195007	−	0.980802i	$-0.562473\pi$
$103$	−4.32525e6	−0.390015	−0.195007	+	0.980802i	$0.562473\pi$
$104$	0	0
$105$	4.84240e6	0.408224
$106$	0	0
$107$	1.85044e7i	1.46027i	0.683304	+	0.730134i	$0.260542\pi$
$107$	1.85044e7i	1.46027i	−0.683304	+	0.730134i	$0.739458\pi$
$108$	0	0
$109$	− 1.47307e7i	− 1.08951i	−0.838596	−	0.544754i	$-0.816623\pi$
$109$	− 1.47307e7i	− 1.08951i	0.838596	−	0.544754i	$-0.183377\pi$
$110$	0	0
$111$	1.02790e7	0.713381
$112$	0	0
$113$	7.44542e6	0.485416	0.242708	−	0.970099i	$-0.421964\pi$
$113$	7.44542e6	0.485416	0.242708	+	0.970099i	$0.421964\pi$
$114$	0	0
$115$	− 957138.i	− 0.0586857i
$116$	0	0
$117$	− 1.74242e7i	− 1.00578i
$118$	0	0
$119$	−2.22838e7	−1.21220
$120$	0	0
$121$	−2.17604e6	−0.111665
$122$	0	0
$123$	5.11107e6i	0.247653i
$124$	0	0
$125$	1.95312e6i	0.0894427i
$126$	0	0
$127$	3.47873e7	1.50698	0.753491	−	0.657458i	$-0.228368\pi$
$127$	3.47873e7	1.50698	0.753491	+	0.657458i	$0.228368\pi$
$128$	0	0
$129$	2.94120e6	0.120632
$130$	0	0
$131$	− 5.24357e6i	− 0.203787i	−0.994795	−	0.101894i	$-0.967510\pi$
$131$	− 5.24357e6i	− 0.203787i	0.994795	−	0.101894i	$-0.0324902\pi$
$132$	0	0
$133$	1.51292e7i	0.557616i
$134$	0	0
$135$	1.06733e7	0.373361
$136$	0	0
$137$	3.16092e7	1.05025	0.525123	−	0.851026i	$-0.324019\pi$
$137$	3.16092e7	1.05025	0.525123	+	0.851026i	$0.324019\pi$
$138$	0	0
$139$	− 3.30038e7i	− 1.04235i	−0.853451	−	0.521174i	$-0.825494\pi$
$139$	− 3.30038e7i	− 1.04235i	0.853451	−	0.521174i	$-0.174506\pi$
$140$	0	0
$141$	− 1.95849e7i	− 0.588376i
$142$	0	0
$143$	−4.75400e7	−1.35951
$144$	0	0
$145$	−1.21730e7	−0.331596
$146$	0	0
$147$	5.03575e7i	1.30754i
$148$	0	0
$149$	2.88822e7i	0.715283i	0.933859	+	0.357641i	$0.116419\pi$
$149$	2.88822e7i	0.715283i	−0.933859	+	0.357641i	$0.883581\pi$
$150$	0	0
$151$	−2.47359e7	−0.584668	−0.292334	−	0.956316i	$-0.594432\pi$
$151$	−2.47359e7	−0.584668	−0.292334	+	0.956316i	$0.594432\pi$
$152$	0	0
$153$	−2.15233e7	−0.485834
$154$	0	0
$155$	3.53633e7i	0.762768i
$156$	0	0
$157$	1.13532e7i	0.234137i	0.993124	+	0.117068i	$0.0373497\pi$
$157$	1.13532e7i	0.234137i	−0.993124	+	0.117068i	$0.962650\pi$
$158$	0	0
$159$	−2.14924e7	−0.424029
$160$	0	0
$161$	1.35239e7	0.255395
$162$	0	0
$163$	6.59553e7i	1.19287i	0.802662	+	0.596435i	$0.203417\pi$
$163$	6.59553e7i	1.19287i	−0.802662	+	0.596435i	$0.796583\pi$
$164$	0	0
$165$	− 1.27610e7i	− 0.221152i
$166$	0	0
$167$	−3.84566e7	−0.638945	−0.319472	−	0.947596i	$-0.603506\pi$
$167$	−3.84566e7	−0.638945	−0.319472	+	0.947596i	$0.603506\pi$
$168$	0	0
$169$	−4.15780e7	−0.662613
$170$	0	0
$171$	1.46128e7i	0.223485i
$172$	0	0
$173$	1.18583e8i	1.74125i	0.491945	+	0.870626i	$0.336286\pi$
$173$	1.18583e8i	1.74125i	−0.491945	+	0.870626i	$0.663714\pi$
$174$	0	0
$175$	−2.75968e7	−0.389247
$176$	0	0
$177$	−2.02217e7	−0.274102
$178$	0	0
$179$	− 1.75108e7i	− 0.228202i	−0.993469	−	0.114101i	$-0.963601\pi$
$179$	− 1.75108e7i	− 0.228202i	0.993469	−	0.114101i	$-0.0363988\pi$
$180$	0	0
$181$	− 2.11922e7i	− 0.265645i	−0.991140	−	0.132823i	$-0.957596\pi$
$181$	− 2.11922e7i	− 0.265645i	0.991140	−	0.132823i	$-0.0424040\pi$
$182$	0	0
$183$	4.31386e7	0.520340
$184$	0	0
$185$	−5.85800e7	−0.680219
$186$	0	0
$187$	5.87236e7i	0.656700i
$188$	0	0
$189$	1.50808e8i	1.62483i
$190$	0	0
$191$	−3.98972e7	−0.414310	−0.207155	−	0.978308i	$-0.566420\pi$
$191$	−3.98972e7	−0.414310	−0.207155	+	0.978308i	$0.566420\pi$
$192$	0	0
$193$	3.94627e7	0.395127	0.197563	−	0.980290i	$-0.436697\pi$
$193$	3.94627e7	0.395127	0.197563	+	0.980290i	$0.436697\pi$
$194$	0	0
$195$	− 2.80040e7i	− 0.270457i
$196$	0	0
$197$	− 1.81451e8i	− 1.69094i	−0.534023	−	0.845470i	$-0.679321\pi$
$197$	− 1.81451e8i	− 1.69094i	0.534023	−	0.845470i	$-0.320679\pi$
$198$	0	0
$199$	−2.02144e7	−0.181834	−0.0909168	−	0.995858i	$-0.528980\pi$
$199$	−2.02144e7	−0.181834	−0.0909168	+	0.995858i	$0.528980\pi$
$200$	0	0
$201$	6.07445e7	0.527619
$202$	0	0
$203$	− 1.71999e8i	− 1.44307i
$204$	0	0
$205$	− 2.91279e7i	− 0.236141i
$206$	0	0
$207$	1.30623e7	0.102359
$208$	0	0
$209$	3.98693e7	0.302084
$210$	0	0
$211$	1.32774e8i	0.973024i	0.873674	+	0.486512i	$0.161731\pi$
$211$	1.32774e8i	0.973024i	−0.873674	+	0.486512i	$0.838269\pi$
$212$	0	0
$213$	8.95773e7i	0.635140i
$214$	0	0
$215$	−1.67619e7	−0.115024
$216$	0	0
$217$	−4.99668e8	−3.31950
$218$	0	0
$219$	− 6.46802e6i	− 0.0416119i
$220$	0	0
$221$	1.28869e8i	0.803111i
$222$	0	0
$223$	2.03764e8	1.23044	0.615221	−	0.788355i	$-0.289067\pi$
$223$	2.03764e8	1.23044	0.615221	+	0.788355i	$0.289067\pi$
$224$	0	0
$225$	−2.66549e7	−0.156005
$226$	0	0
$227$	− 2.61770e7i	− 0.148535i	−0.997238	−	0.0742675i	$-0.976338\pi$
$227$	− 2.61770e7i	− 0.148535i	0.997238	−	0.0742675i	$-0.0236619\pi$
$228$	0	0
$229$	1.61808e8i	0.890383i	0.895435	+	0.445191i	$0.146864\pi$
$229$	1.61808e8i	0.890383i	−0.895435	+	0.445191i	$0.853136\pi$
$230$	0	0
$231$	1.80307e8	0.962432
$232$	0	0
$233$	8.50027e7	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$233$	8.50027e7	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$234$	0	0
$235$	1.11614e8i	0.561025i
$236$	0	0
$237$	7.10519e7i	0.346702i
$238$	0	0
$239$	−1.52631e8	−0.723184	−0.361592	−	0.932336i	$-0.617767\pi$
$239$	−1.52631e8	−0.723184	−0.361592	+	0.932336i	$0.617767\pi$
$240$	0	0
$241$	−1.10727e8	−0.509557	−0.254778	−	0.966999i	$-0.582002\pi$
$241$	−1.10727e8	−0.509557	−0.254778	+	0.966999i	$0.582002\pi$
$242$	0	0
$243$	2.27492e8i	1.01705i
$244$	0	0
$245$	− 2.86987e8i	− 1.24675i
$246$	0	0
$247$	8.74933e7	0.369433
$248$	0	0
$249$	5.66297e7	0.232459
$250$	0	0
$251$	1.24960e8i	0.498784i	0.968403	+	0.249392i	$0.0802308\pi$
$251$	1.24960e8i	0.498784i	−0.968403	+	0.249392i	$0.919769\pi$
$252$	0	0
$253$	− 3.56390e7i	− 0.138358i
$254$	0	0
$255$	−3.45919e7	−0.130642
$256$	0	0
$257$	5.23524e8	1.92385	0.961925	−	0.273315i	$-0.0881200\pi$
$257$	5.23524e8	1.92385	0.961925	+	0.273315i	$0.0881200\pi$
$258$	0	0
$259$	− 8.27709e8i	− 2.96025i
$260$	0	0
$261$	− 1.66128e8i	− 0.578364i
$262$	0	0
$263$	−9.90641e6	−0.0335793	−0.0167896	−	0.999859i	$-0.505345\pi$
$263$	−9.90641e6	−0.0335793	−0.0167896	+	0.999859i	$0.505345\pi$
$264$	0	0
$265$	1.22485e8	0.404318
$266$	0	0
$267$	− 4.87017e7i	− 0.156587i
$268$	0	0
$269$	− 5.47899e8i	− 1.71620i	−0.513483	−	0.858100i	$-0.671645\pi$
$269$	− 5.47899e8i	− 1.71620i	0.513483	−	0.858100i	$-0.328355\pi$
$270$	0	0
$271$	1.19755e8	0.365511	0.182755	−	0.983158i	$-0.441498\pi$
$271$	1.19755e8	0.365511	0.182755	+	0.983158i	$0.441498\pi$
$272$	0	0
$273$	3.95683e8	1.17701
$274$	0	0
$275$	7.27246e7i	0.210871i
$276$	0	0
$277$	− 3.82549e7i	− 0.108145i	−0.998537	−	0.0540727i	$-0.982780\pi$
$277$	− 3.82549e7i	− 0.108145i	0.998537	−	0.0540727i	$-0.0172203\pi$
$278$	0	0
$279$	−4.82614e8	−1.33041
$280$	0	0
$281$	4.70011e8	1.26368	0.631838	−	0.775101i	$-0.282301\pi$
$281$	4.70011e8	1.26368	0.631838	+	0.775101i	$0.282301\pi$
$282$	0	0
$283$	− 2.27280e8i	− 0.596087i	−0.954552	−	0.298043i	$-0.903666\pi$
$283$	− 2.27280e8i	− 0.596087i	0.954552	−	0.298043i	$-0.0963340\pi$
$284$	0	0
$285$	2.34855e7i	0.0600957i
$286$	0	0
$287$	4.11564e8	1.02766
$288$	0	0
$289$	−2.51153e8	−0.612064
$290$	0	0
$291$	− 3.97574e7i	− 0.0945785i
$292$	0	0
$293$	8.24016e7i	0.191381i	0.995411	+	0.0956905i	$0.0305059\pi$
$293$	8.24016e7i	0.191381i	−0.995411	+	0.0956905i	$0.969494\pi$
$294$	0	0
$295$	1.15243e8	0.261360
$296$	0	0
$297$	3.97419e8	0.880239
$298$	0	0
$299$	− 7.82099e7i	− 0.169205i
$300$	0	0
$301$	− 2.36838e8i	− 0.500574i
$302$	0	0
$303$	−3.85094e8	−0.795276
$304$	0	0
$305$	−2.45846e8	−0.496151
$306$	0	0
$307$	7.43722e8i	1.46699i	0.679697	+	0.733493i	$0.262111\pi$
$307$	7.43722e8i	1.46699i	−0.679697	+	0.733493i	$0.737889\pi$
$308$	0	0
$309$	− 9.48689e7i	− 0.182923i
$310$	0	0
$311$	−4.54714e8	−0.857191	−0.428595	−	0.903497i	$-0.640991\pi$
$311$	−4.54714e8	−0.857191	−0.428595	+	0.903497i	$0.640991\pi$
$312$	0	0
$313$	5.08221e8	0.936802	0.468401	−	0.883516i	$-0.344830\pi$
$313$	5.08221e8	0.936802	0.468401	+	0.883516i	$0.344830\pi$
$314$	0	0
$315$	− 3.76621e8i	− 0.678919i
$316$	0	0
$317$	− 1.48942e8i	− 0.262610i	−0.991342	−	0.131305i	$-0.958083\pi$
$317$	− 1.48942e8i	− 0.262610i	0.991342	−	0.131305i	$-0.0419167\pi$
$318$	0	0
$319$	−4.53261e8	−0.781773
$320$	0	0
$321$	−4.05871e8	−0.684889
$322$	0	0
$323$	− 1.08076e8i	− 0.178452i
$324$	0	0
$325$	1.59594e8i	0.257885i
$326$	0	0
$327$	3.23099e8	0.510997
$328$	0	0
$329$	−1.57706e9	−2.44153
$330$	0	0
$331$	− 1.08603e9i	− 1.64605i	−0.568006	−	0.823024i	$-0.692285\pi$
$331$	− 1.08603e9i	− 1.64605i	0.568006	−	0.823024i	$-0.307715\pi$
$332$	0	0
$333$	− 7.99459e8i	− 1.18643i
$334$	0	0
$335$	−3.46182e8	−0.503092
$336$	0	0
$337$	1.35022e8	0.192177	0.0960885	−	0.995373i	$-0.469367\pi$
$337$	1.35022e8	0.192177	0.0960885	+	0.995373i	$0.469367\pi$
$338$	0	0
$339$	1.63306e8i	0.227668i
$340$	0	0
$341$	1.31675e9i	1.79831i
$342$	0	0
$343$	2.60046e9	3.47953
$344$	0	0
$345$	2.09936e7	0.0275245
$346$	0	0
$347$	1.28568e8i	0.165189i	0.996583	+	0.0825944i	$0.0263206\pi$
$347$	1.28568e8i	0.165189i	−0.996583	+	0.0825944i	$0.973679\pi$
$348$	0	0
$349$	− 1.00723e9i	− 1.26835i	−0.773190	−	0.634175i	$-0.781340\pi$
$349$	− 1.00723e9i	− 1.26835i	0.773190	−	0.634175i	$-0.218660\pi$
$350$	0	0
$351$	8.72136e8	1.07649
$352$	0	0
$353$	2.65853e8	0.321685	0.160843	−	0.986980i	$-0.448579\pi$
$353$	2.65853e8	0.321685	0.160843	+	0.986980i	$0.448579\pi$
$354$	0	0
$355$	− 5.10500e8i	− 0.605615i
$356$	0	0
$357$	− 4.88767e8i	− 0.568543i
$358$	0	0
$359$	−2.05582e8	−0.234507	−0.117253	−	0.993102i	$-0.537409\pi$
$359$	−2.05582e8	−0.234507	−0.117253	+	0.993102i	$0.537409\pi$
$360$	0	0
$361$	8.20496e8	0.917912
$362$	0	0
$363$	− 4.77287e7i	− 0.0523729i
$364$	0	0
$365$	3.68612e7i	0.0396775i
$366$	0	0
$367$	−6.18317e8	−0.652950	−0.326475	−	0.945206i	$-0.605861\pi$
$367$	−6.18317e8	−0.652950	−0.326475	+	0.945206i	$0.605861\pi$
$368$	0	0
$369$	3.97517e8	0.411873
$370$	0	0
$371$	1.73066e9i	1.75955i
$372$	0	0
$373$	1.61904e9i	1.61539i	0.589601	+	0.807695i	$0.299285\pi$
$373$	1.61904e9i	1.61539i	−0.589601	+	0.807695i	$0.700715\pi$
$374$	0	0
$375$	−4.28393e7	−0.0419501
$376$	0	0
$377$	−9.94681e8	−0.956069
$378$	0	0
$379$	− 3.04862e7i	− 0.0287651i	−0.999897	−	0.0143825i	$-0.995422\pi$
$379$	− 3.04862e7i	− 0.0287651i	0.999897	−	0.0143825i	$-0.00457826\pi$
$380$	0	0
$381$	7.63016e8i	0.706799i
$382$	0	0
$383$	3.11383e7	0.0283204	0.0141602	−	0.999900i	$-0.495493\pi$
$383$	3.11383e7	0.0283204	0.0141602	+	0.999900i	$0.495493\pi$
$384$	0	0
$385$	−1.02757e9	−0.917692
$386$	0	0
$387$	− 2.28754e8i	− 0.200623i
$388$	0	0
$389$	7.95666e7i	0.0685341i	0.999413	+	0.0342671i	$0.0109097\pi$
$389$	7.95666e7i	0.0685341i	−0.999413	+	0.0342671i	$0.989090\pi$
$390$	0	0
$391$	−9.66086e7	−0.0817330
$392$	0	0
$393$	1.15011e8	0.0955797
$394$	0	0
$395$	− 4.04924e8i	− 0.330585i
$396$	0	0
$397$	1.83383e9i	1.47093i	0.677561	+	0.735466i	$0.263037\pi$
$397$	1.83383e9i	1.47093i	−0.677561	+	0.735466i	$0.736963\pi$
$398$	0	0
$399$	−3.31840e8	−0.261531
$400$	0	0
$401$	−1.19601e9	−0.926252	−0.463126	−	0.886292i	$-0.653272\pi$
$401$	−1.19601e9	−0.926252	−0.463126	+	0.886292i	$0.653272\pi$
$402$	0	0
$403$	2.88962e9i	2.19924i
$404$	0	0
$405$	− 2.32249e8i	− 0.173725i
$406$	0	0
$407$	−2.18123e9	−1.60369
$408$	0	0
$409$	−1.57423e9	−1.13773	−0.568863	−	0.822432i	$-0.692617\pi$
$409$	−1.57423e9	−1.13773	−0.568863	+	0.822432i	$0.692617\pi$
$410$	0	0
$411$	6.93307e8i	0.492583i
$412$	0	0
$413$	1.62834e9i	1.13741i
$414$	0	0
$415$	−3.22732e8	−0.221653
$416$	0	0
$417$	7.23897e8	0.488878
$418$	0	0
$419$	8.82743e8i	0.586253i	0.956074	+	0.293127i	$0.0946958\pi$
$419$	8.82743e8i	0.586253i	−0.956074	+	0.293127i	$0.905304\pi$
$420$	0	0
$421$	− 4.45601e8i	− 0.291044i	−0.989355	−	0.145522i	$-0.953514\pi$
$421$	− 4.45601e8i	− 0.291044i	0.989355	−	0.145522i	$-0.0464861\pi$
$422$	0	0
$423$	−1.52323e9	−0.978532
$424$	0	0
$425$	1.97139e8	0.124569
$426$	0	0
$427$	− 3.47369e9i	− 2.15921i
$428$	0	0
$429$	− 1.04273e9i	− 0.637633i
$430$	0	0
$431$	1.83573e9	1.10443	0.552215	−	0.833701i	$-0.313783\pi$
$431$	1.83573e9	1.10443	0.552215	+	0.833701i	$0.313783\pi$
$432$	0	0
$433$	2.04642e9	1.21140	0.605701	−	0.795693i	$-0.292893\pi$
$433$	2.04642e9	1.21140	0.605701	+	0.795693i	$0.292893\pi$
$434$	0	0
$435$	− 2.66999e8i	− 0.155524i
$436$	0	0
$437$	6.55907e7i	0.0375973i
$438$	0	0
$439$	−1.20727e9	−0.681049	−0.340524	−	0.940236i	$-0.610605\pi$
$439$	−1.20727e9	−0.681049	−0.340524	+	0.940236i	$0.610605\pi$
$440$	0	0
$441$	3.91659e9	2.17457
$442$	0	0
$443$	3.22365e9i	1.76171i	0.473385	+	0.880856i	$0.343032\pi$
$443$	3.22365e9i	1.76171i	−0.473385	+	0.880856i	$0.656968\pi$
$444$	0	0
$445$	2.77550e8i	0.149308i
$446$	0	0
$447$	−6.33493e8	−0.335479
$448$	0	0
$449$	−1.67638e9	−0.873998	−0.436999	−	0.899462i	$-0.643959\pi$
$449$	−1.67638e9	−0.873998	−0.436999	+	0.899462i	$0.643959\pi$
$450$	0	0
$451$	− 1.08458e9i	− 0.556727i
$452$	0	0
$453$	− 5.42551e8i	− 0.274219i
$454$	0	0
$455$	−2.25499e9	−1.12229
$456$	0	0
$457$	−2.24910e9	−1.10230	−0.551152	−	0.834405i	$-0.685812\pi$
$457$	−2.24910e9	−1.10230	−0.551152	+	0.834405i	$0.685812\pi$
$458$	0	0
$459$	− 1.07730e9i	− 0.519989i
$460$	0	0
$461$	6.25433e8i	0.297322i	0.988888	+	0.148661i	$0.0474963\pi$
$461$	6.25433e8i	0.297322i	−0.988888	+	0.148661i	$0.952504\pi$
$462$	0	0
$463$	2.33057e9	1.09126	0.545632	−	0.838025i	$-0.316290\pi$
$463$	2.33057e9	1.09126	0.545632	+	0.838025i	$0.316290\pi$
$464$	0	0
$465$	−7.75650e8	−0.357751
$466$	0	0
$467$	1.74239e9i	0.791655i	0.918325	+	0.395828i	$0.129542\pi$
$467$	1.74239e9i	0.791655i	−0.918325	+	0.395828i	$0.870458\pi$
$468$	0	0
$469$	− 4.89139e9i	− 2.18941i
$470$	0	0
$471$	−2.49018e8	−0.109814
$472$	0	0
$473$	−6.24129e8	−0.271182
$474$	0	0
$475$	− 1.33844e8i	− 0.0573020i
$476$	0	0
$477$	1.67159e9i	0.705205i
$478$	0	0
$479$	−2.31629e9	−0.962984	−0.481492	−	0.876450i	$-0.659905\pi$
$479$	−2.31629e9	−0.962984	−0.481492	+	0.876450i	$0.659905\pi$
$480$	0	0
$481$	−4.78671e9	−1.96123
$482$	0	0
$483$	2.96630e8i	0.119784i
$484$	0	0
$485$	2.26577e8i	0.0901819i
$486$	0	0
$487$	2.19818e9	0.862404	0.431202	−	0.902255i	$-0.358090\pi$
$487$	2.19818e9	0.862404	0.431202	+	0.902255i	$0.358090\pi$
$488$	0	0
$489$	−1.44665e9	−0.559475
$490$	0	0
$491$	8.80140e8i	0.335557i	0.985825	+	0.167779i	$0.0536594\pi$
$491$	8.80140e8i	0.335557i	−0.985825	+	0.167779i	$0.946341\pi$
$492$	0	0
$493$	1.22868e9i	0.461821i
$494$	0	0
$495$	−9.92494e8	−0.367798
$496$	0	0
$497$	7.21313e9	2.63558
$498$	0	0
$499$	3.87072e9i	1.39457i	0.716795	+	0.697284i	$0.245608\pi$
$499$	3.87072e9i	1.39457i	−0.716795	+	0.697284i	$0.754392\pi$
$500$	0	0
$501$	− 8.43497e8i	− 0.299676i
$502$	0	0
$503$	2.27841e9	0.798258	0.399129	−	0.916895i	$-0.369313\pi$
$503$	2.27841e9	0.798258	0.399129	+	0.916895i	$0.369313\pi$
$504$	0	0
$505$	2.19465e9	0.758307
$506$	0	0
$507$	− 9.11960e8i	− 0.310776i
$508$	0	0
$509$	1.63663e9i	0.550096i	0.961431	+	0.275048i	$0.0886937\pi$
$509$	1.63663e9i	0.550096i	−0.961431	+	0.275048i	$0.911306\pi$
$510$	0	0
$511$	−5.20831e8	−0.172673
$512$	0	0
$513$	−7.31416e8	−0.239196
$514$	0	0
$515$	5.40657e8i	0.174420i
$516$	0	0
$517$	4.15596e9i	1.32268i
$518$	0	0
$519$	−2.60097e9	−0.816676
$520$	0	0
$521$	5.84018e9	1.80923	0.904616	−	0.426228i	$-0.140158\pi$
$521$	5.84018e9	1.80923	0.904616	+	0.426228i	$0.140158\pi$
$522$	0	0
$523$	− 2.35065e9i	− 0.718509i	−0.933240	−	0.359254i	$-0.883031\pi$
$523$	− 2.35065e9i	− 0.718509i	0.933240	−	0.359254i	$-0.116969\pi$
$524$	0	0
$525$	− 6.05300e8i	− 0.182563i
$526$	0	0
$527$	3.56940e9	1.06233
$528$	0	0
$529$	−3.34619e9	−0.982780
$530$	0	0
$531$	1.57276e9i	0.455860i
$532$	0	0
$533$	− 2.38011e9i	− 0.680850i
$534$	0	0
$535$	2.31305e9	0.653051
$536$	0	0
$537$	3.84077e8	0.107031
$538$	0	0
$539$	− 1.06860e10i	− 2.93936i
$540$	0	0
$541$	− 4.38266e9i	− 1.19000i	−0.803725	−	0.595001i	$-0.797152\pi$
$541$	− 4.38266e9i	− 1.19000i	0.803725	−	0.595001i	$-0.202848\pi$
$542$	0	0
$543$	4.64825e8	0.124592
$544$	0	0
$545$	−1.84134e9	−0.487243
$546$	0	0
$547$	− 5.06550e9i	− 1.32333i	−0.749801	−	0.661663i	$-0.769851\pi$
$547$	− 5.06550e9i	− 1.32333i	0.749801	−	0.661663i	$-0.230149\pi$
$548$	0	0
$549$	− 3.35513e9i	− 0.865380i
$550$	0	0
$551$	8.34189e8	0.212439
$552$	0	0
$553$	5.72139e9	1.43868
$554$	0	0
$555$	− 1.28488e9i	− 0.319034i
$556$	0	0
$557$	2.49888e9i	0.612706i	0.951918	+	0.306353i	$0.0991088\pi$
$557$	2.49888e9i	0.612706i	−0.951918	+	0.306353i	$0.900891\pi$
$558$	0	0
$559$	−1.36965e9	−0.331641
$560$	0	0
$561$	−1.28803e9	−0.308003
$562$	0	0
$563$	7.56075e7i	0.0178560i	0.999960	+	0.00892802i	$0.00284192\pi$
$563$	7.56075e7i	0.0178560i	−0.999960	+	0.00892802i	$0.997158\pi$
$564$	0	0
$565$	− 9.30677e8i	− 0.217085i
$566$	0	0
$567$	3.28158e9	0.756035
$568$	0	0
$569$	−3.55415e9	−0.808804	−0.404402	−	0.914581i	$-0.632520\pi$
$569$	−3.55415e9	−0.808804	−0.404402	+	0.914581i	$0.632520\pi$
$570$	0	0
$571$	− 1.63366e9i	− 0.367226i	−0.982999	−	0.183613i	$-0.941221\pi$
$571$	− 1.63366e9i	− 0.367226i	0.982999	−	0.183613i	$-0.0587794\pi$
$572$	0	0
$573$	− 8.75094e8i	− 0.194318i
$574$	0	0
$575$	−1.19642e8	−0.0262450
$576$	0	0
$577$	1.71765e9	0.372238	0.186119	−	0.982527i	$-0.440409\pi$
$577$	1.71765e9	0.372238	0.186119	+	0.982527i	$0.440409\pi$
$578$	0	0
$579$	8.65564e8i	0.185321i
$580$	0	0
$581$	− 4.56006e9i	− 0.964614i
$582$	0	0
$583$	4.56073e9	0.953223
$584$	0	0
$585$	−2.17803e9	−0.449799
$586$	0	0
$587$	1.59750e9i	0.325991i	0.986627	+	0.162996i	$0.0521157\pi$
$587$	1.59750e9i	0.325991i	−0.986627	+	0.162996i	$0.947884\pi$
$588$	0	0
$589$	− 2.42338e9i	− 0.488672i
$590$	0	0
$591$	3.97990e9	0.793078
$592$	0	0
$593$	−3.49911e9	−0.689075	−0.344537	−	0.938773i	$-0.611964\pi$
$593$	−3.49911e9	−0.689075	−0.344537	+	0.938773i	$0.611964\pi$
$594$	0	0
$595$	2.78548e9i	0.542113i
$596$	0	0
$597$	− 4.43376e8i	− 0.0852830i
$598$	0	0
$599$	3.77602e9	0.717862	0.358931	−	0.933364i	$-0.383141\pi$
$599$	3.77602e9	0.717862	0.358931	+	0.933364i	$0.383141\pi$
$600$	0	0
$601$	2.38785e9	0.448690	0.224345	−	0.974510i	$-0.427976\pi$
$601$	2.38785e9	0.448690	0.224345	+	0.974510i	$0.427976\pi$
$602$	0	0
$603$	− 4.72444e9i	− 0.877486i
$604$	0	0
$605$	2.72005e8i	0.0499382i
$606$	0	0
$607$	−1.78453e9	−0.323864	−0.161932	−	0.986802i	$-0.551773\pi$
$607$	−1.78453e9	−0.323864	−0.161932	+	0.986802i	$0.551773\pi$
$608$	0	0
$609$	3.77257e9	0.676826
$610$	0	0
$611$	9.12025e9i	1.61757i
$612$	0	0
$613$	1.05192e10i	1.84446i	0.386637	+	0.922232i	$0.373637\pi$
$613$	1.05192e10i	1.84446i	−0.386637	+	0.922232i	$0.626363\pi$
$614$	0	0
$615$	6.38883e8	0.110754
$616$	0	0
$617$	−9.65840e9	−1.65542	−0.827708	−	0.561160i	$-0.810355\pi$
$617$	−9.65840e9	−1.65542	−0.827708	+	0.561160i	$0.810355\pi$
$618$	0	0
$619$	− 4.79549e9i	− 0.812674i	−0.913723	−	0.406337i	$-0.866806\pi$
$619$	− 4.79549e9i	− 0.812674i	0.913723	−	0.406337i	$-0.133194\pi$
$620$	0	0
$621$	6.53810e8i	0.109555i
$622$	0	0
$623$	−3.92166e9	−0.649773
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	8.74483e8i	0.141682i
$628$	0	0
$629$	5.91277e9i	0.947357i
$630$	0	0
$631$	3.65513e8	0.0579162	0.0289581	−	0.999581i	$-0.490781\pi$
$631$	3.65513e8	0.0579162	0.0289581	+	0.999581i	$0.490781\pi$
$632$	0	0
$633$	−2.91222e9	−0.456364
$634$	0	0
$635$	− 4.34842e9i	− 0.673943i
$636$	0	0
$637$	− 2.34503e10i	− 3.59469i
$638$	0	0
$639$	6.96694e9	1.05630
$640$	0	0
$641$	1.01508e9	0.152229	0.0761145	−	0.997099i	$-0.475749\pi$
$641$	1.01508e9	0.152229	0.0761145	+	0.997099i	$0.475749\pi$
$642$	0	0
$643$	6.44588e9i	0.956188i	0.878309	+	0.478094i	$0.158672\pi$
$643$	6.44588e9i	0.956188i	−0.878309	+	0.478094i	$0.841328\pi$
$644$	0	0
$645$	− 3.67651e8i	− 0.0539481i
$646$	0	0
$647$	−1.14867e10	−1.66736	−0.833679	−	0.552250i	$-0.813769\pi$
$647$	−1.14867e10	−1.66736	−0.833679	+	0.552250i	$0.813769\pi$
$648$	0	0
$649$	4.29109e9	0.616185
$650$	0	0
$651$	− 1.09596e10i	− 1.55690i
$652$	0	0
$653$	− 2.25835e9i	− 0.317392i	−0.987328	−	0.158696i	$-0.949271\pi$
$653$	− 2.25835e9i	− 0.317392i	0.987328	−	0.158696i	$-0.0507289\pi$
$654$	0	0
$655$	−6.55446e8	−0.0911365
$656$	0	0
$657$	−5.03055e8	−0.0692049
$658$	0	0
$659$	6.56257e9i	0.893254i	0.894720	+	0.446627i	$0.147375\pi$
$659$	6.56257e9i	0.893254i	−0.894720	+	0.446627i	$0.852625\pi$
$660$	0	0
$661$	1.39055e10i	1.87275i	0.350997	+	0.936377i	$0.385843\pi$
$661$	1.39055e10i	1.87275i	−0.350997	+	0.936377i	$0.614157\pi$
$662$	0	0
$663$	−2.82658e9	−0.376672
$664$	0	0
$665$	1.89115e9	0.249373
$666$	0	0
$667$	− 7.45678e8i	− 0.0972995i
$668$	0	0
$669$	4.46931e9i	0.577097i
$670$	0	0
$671$	−9.15408e9	−1.16973
$672$	0	0
$673$	1.42405e10	1.80083	0.900414	−	0.435035i	$-0.143264\pi$
$673$	1.42405e10	1.80083	0.900414	+	0.435035i	$0.143264\pi$
$674$	0	0
$675$	− 1.33416e9i	− 0.166972i
$676$	0	0
$677$	4.39329e9i	0.544164i	0.962274	+	0.272082i	$0.0877122\pi$
$677$	4.39329e9i	0.544164i	−0.962274	+	0.272082i	$0.912288\pi$
$678$	0	0
$679$	−3.20143e9	−0.392463
$680$	0	0
$681$	5.74158e8	0.0696653
$682$	0	0
$683$	1.44759e9i	0.173849i	0.996215	+	0.0869246i	$0.0277039\pi$
$683$	1.44759e9i	0.173849i	−0.996215	+	0.0869246i	$0.972296\pi$
$684$	0	0
$685$	− 3.95115e9i	− 0.469684i
$686$	0	0
$687$	−3.54906e9	−0.417604
$688$	0	0
$689$	1.00085e10	1.16574
$690$	0	0
$691$	− 6.56593e9i	− 0.757047i	−0.925592	−	0.378523i	$-0.876432\pi$
$691$	− 6.56593e9i	− 0.757047i	0.925592	−	0.378523i	$-0.123568\pi$
$692$	0	0
$693$	− 1.40235e10i	− 1.60063i
$694$	0	0
$695$	−4.12548e9	−0.466152
$696$	0	0
$697$	−2.94002e9	−0.328879
$698$	0	0
$699$	1.86443e9i	0.206479i
$700$	0	0
$701$	1.52450e10i	1.67153i	0.549088	+	0.835764i	$0.314975\pi$
$701$	1.52450e10i	1.67153i	−0.549088	+	0.835764i	$0.685025\pi$
$702$	0	0
$703$	4.01437e9	0.435786
$704$	0	0
$705$	−2.44812e9	−0.263130
$706$	0	0
$707$	3.10094e10i	3.30008i
$708$	0	0
$709$	− 1.23751e9i	− 0.130403i	−0.997872	−	0.0652015i	$-0.979231\pi$
$709$	− 1.23751e9i	− 0.130403i	0.997872	−	0.0652015i	$-0.0207690\pi$
$710$	0	0
$711$	5.52612e9	0.576602
$712$	0	0
$713$	−2.16625e9	−0.223818
$714$	0	0
$715$	5.94249e9i	0.607992i
$716$	0	0
$717$	− 3.34776e9i	− 0.339185i
$718$	0	0
$719$	−1.15168e10	−1.15552	−0.577762	−	0.816205i	$-0.696074\pi$
$719$	−1.15168e10	−1.15552	−0.577762	+	0.816205i	$0.696074\pi$
$720$	0	0
$721$	−7.63923e9	−0.759060
$722$	0	0
$723$	− 2.42865e9i	− 0.238990i
$724$	0	0
$725$	1.52162e9i	0.148294i
$726$	0	0
$727$	7.83600e9	0.756352	0.378176	−	0.925734i	$-0.376551\pi$
$727$	7.83600e9	0.756352	0.378176	+	0.925734i	$0.376551\pi$
$728$	0	0
$729$	−9.26316e8	−0.0885549
$730$	0	0
$731$	1.69186e9i	0.160197i
$732$	0	0
$733$	− 1.71394e9i	− 0.160743i	−0.996765	−	0.0803713i	$-0.974389\pi$
$733$	− 1.71394e9i	− 0.160743i	0.996765	−	0.0803713i	$-0.0256106\pi$
$734$	0	0
$735$	6.29469e9	0.584748
$736$	0	0
$737$	−1.28901e10	−1.18609
$738$	0	0
$739$	1.15208e10i	1.05009i	0.851073	+	0.525047i	$0.175952\pi$
$739$	1.15208e10i	1.05009i	−0.851073	+	0.525047i	$0.824048\pi$
$740$	0	0
$741$	1.91906e9i	0.173270i
$742$	0	0
$743$	−1.76670e10	−1.58016	−0.790080	−	0.613004i	$-0.789961\pi$
$743$	−1.76670e10	−1.58016	−0.790080	+	0.613004i	$0.789961\pi$
$744$	0	0
$745$	3.61027e9	0.319884
$746$	0	0
$747$	− 4.40442e9i	− 0.386604i
$748$	0	0
$749$	3.26824e10i	2.84202i
$750$	0	0
$751$	−1.69635e10	−1.46142	−0.730711	−	0.682687i	$-0.760811\pi$
$751$	−1.69635e10	−1.46142	−0.730711	+	0.682687i	$0.760811\pi$
$752$	0	0
$753$	−2.74084e9	−0.233938
$754$	0	0
$755$	3.09199e9i	0.261471i
$756$	0	0
$757$	3.33112e8i	0.0279097i	0.999903	+	0.0139548i	$0.00444211\pi$
$757$	3.33112e8i	0.0279097i	−0.999903	+	0.0139548i	$0.995558\pi$
$758$	0	0
$759$	7.81697e8	0.0648922
$760$	0	0
$761$	−2.08615e10	−1.71593	−0.857963	−	0.513712i	$-0.828270\pi$
$761$	−2.08615e10	−1.71593	−0.857963	+	0.513712i	$0.828270\pi$
$762$	0	0
$763$	− 2.60172e10i	− 2.12044i
$764$	0	0
$765$	2.69041e9i	0.217272i
$766$	0	0
$767$	9.41680e9	0.753563
$768$	0	0
$769$	7.32167e9	0.580588	0.290294	−	0.956937i	$-0.406247\pi$
$769$	7.32167e9	0.580588	0.290294	+	0.956937i	$0.406247\pi$
$770$	0	0
$771$	1.14828e10i	0.902317i
$772$	0	0
$773$	1.19518e10i	0.930690i	0.885129	+	0.465345i	$0.154070\pi$
$773$	1.19518e10i	0.930690i	−0.885129	+	0.465345i	$0.845930\pi$
$774$	0	0
$775$	4.42042e9	0.341120
$776$	0	0
$777$	1.81547e10	1.38841
$778$	0	0
$779$	1.99607e9i	0.151285i
$780$	0	0
$781$	− 1.90085e10i	− 1.42780i
$782$	0	0
$783$	8.31522e9	0.619024
$784$	0	0
$785$	1.41915e9	0.104709
$786$	0	0
$787$	1.42890e10i	1.04494i	0.852658	+	0.522470i	$0.174989\pi$
$787$	1.42890e10i	1.04494i	−0.852658	+	0.522470i	$0.825011\pi$
$788$	0	0
$789$	− 2.17284e8i	− 0.0157492i
$790$	0	0
$791$	1.31500e10	0.944734
$792$	0	0
$793$	−2.00887e10	−1.43052
$794$	0	0
$795$	2.68655e9i	0.189632i
$796$	0	0
$797$	1.49311e10i	1.04469i	0.852733	+	0.522347i	$0.174943\pi$
$797$	1.49311e10i	1.04469i	−0.852733	+	0.522347i	$0.825057\pi$
$798$	0	0
$799$	1.12658e10	0.781353
$800$	0	0
$801$	−3.78781e9	−0.260420
$802$	0	0
$803$	1.37253e9i	0.0935440i
$804$	0	0
$805$	− 1.69049e9i	− 0.114216i
$806$	0	0
$807$	1.20175e10	0.804926
$808$	0	0
$809$	−1.06200e10	−0.705189	−0.352594	−	0.935776i	$-0.614700\pi$
$809$	−1.06200e10	−0.705189	−0.352594	+	0.935776i	$0.614700\pi$
$810$	0	0
$811$	− 1.29606e10i	− 0.853204i	−0.904439	−	0.426602i	$-0.859710\pi$
$811$	− 1.29606e10i	− 0.853204i	0.904439	−	0.426602i	$-0.140290\pi$
$812$	0	0
$813$	2.62667e9i	0.171431i
$814$	0	0
$815$	8.24441e9	0.533467
$816$	0	0
$817$	1.14866e9	0.0736908
$818$	0	0
$819$	− 3.07746e10i	− 1.95748i
$820$	0	0
$821$	− 8.63474e9i	− 0.544563i	−0.962218	−	0.272282i	$-0.912222\pi$
$821$	− 8.63474e9i	− 0.544563i	0.962218	−	0.272282i	$-0.0877782\pi$
$822$	0	0
$823$	−1.41772e10	−0.886528	−0.443264	−	0.896391i	$-0.646180\pi$
$823$	−1.41772e10	−0.886528	−0.443264	+	0.896391i	$0.646180\pi$
$824$	0	0
$825$	−1.59512e9	−0.0989020
$826$	0	0
$827$	− 6.22560e9i	− 0.382747i	−0.981517	−	0.191373i	$-0.938706\pi$
$827$	− 6.22560e9i	− 0.382747i	0.981517	−	0.191373i	$-0.0612942\pi$
$828$	0	0
$829$	− 1.14802e10i	− 0.699853i	−0.936777	−	0.349927i	$-0.886207\pi$
$829$	− 1.14802e10i	− 0.699853i	0.936777	−	0.349927i	$-0.113793\pi$
$830$	0	0
$831$	8.39072e8	0.0507219
$832$	0	0
$833$	−2.89670e10	−1.73638
$834$	0	0
$835$	4.80708e9i	0.285745i
$836$	0	0
$837$	− 2.41563e10i	− 1.42394i
$838$	0	0
$839$	1.32294e10	0.773344	0.386672	−	0.922217i	$-0.373624\pi$
$839$	1.32294e10	0.773344	0.386672	+	0.922217i	$0.373624\pi$
$840$	0	0
$841$	7.76627e9	0.450222
$842$	0	0
$843$	1.03091e10i	0.592685i
$844$	0	0
$845$	5.19725e9i	0.296330i
$846$	0	0
$847$	−3.84331e9	−0.217327
$848$	0	0
$849$	4.98511e9	0.279574
$850$	0	0
$851$	− 3.58843e9i	− 0.199595i
$852$	0	0
$853$	− 5.99775e9i	− 0.330877i	−0.986220	−	0.165439i	$-0.947096\pi$
$853$	− 5.99775e9i	− 0.330877i	0.986220	−	0.165439i	$-0.0529039\pi$
$854$	0	0
$855$	1.82660e9	0.0999454
$856$	0	0
$857$	−1.05936e10	−0.574926	−0.287463	−	0.957792i	$-0.592812\pi$
$857$	−1.05936e10	−0.574926	−0.287463	+	0.957792i	$0.592812\pi$
$858$	0	0
$859$	5.11747e9i	0.275473i	0.990469	+	0.137737i	$0.0439827\pi$
$859$	5.11747e9i	0.275473i	−0.990469	+	0.137737i	$0.956017\pi$
$860$	0	0
$861$	9.02713e9i	0.481991i
$862$	0	0
$863$	−2.02627e10	−1.07315	−0.536575	−	0.843853i	$-0.680282\pi$
$863$	−2.02627e10	−1.07315	−0.536575	+	0.843853i	$0.680282\pi$
$864$	0	0
$865$	1.48229e10	0.778712
$866$	0	0
$867$	− 5.50873e9i	− 0.287068i
$868$	0	0
$869$	− 1.50773e10i	− 0.779392i
$870$	0	0
$871$	−2.82873e10	−1.45053
$872$	0	0
$873$	−3.09216e9	−0.157294
$874$	0	0
$875$	3.44960e9i	0.174076i
$876$	0	0
$877$	1.26460e10i	0.633076i	0.948580	+	0.316538i	$0.102521\pi$
$877$	1.26460e10i	0.633076i	−0.948580	+	0.316538i	$0.897479\pi$
$878$	0	0
$879$	−1.80737e9	−0.0897609
$880$	0	0
$881$	2.84861e10	1.40352	0.701758	−	0.712415i	$-0.252399\pi$
$881$	2.84861e10	1.40352	0.701758	+	0.712415i	$0.252399\pi$
$882$	0	0
$883$	− 6.91021e9i	− 0.337776i	−0.985635	−	0.168888i	$-0.945982\pi$
$883$	− 6.91021e9i	− 0.337776i	0.985635	−	0.168888i	$-0.0540176\pi$
$884$	0	0
$885$	2.52772e9i	0.122582i
$886$	0	0
$887$	−2.27208e9	−0.109318	−0.0546590	−	0.998505i	$-0.517407\pi$
$887$	−2.27208e9	−0.109318	−0.0546590	+	0.998505i	$0.517407\pi$
$888$	0	0
$889$	6.14411e10	2.93294
$890$	0	0
$891$	− 8.64780e9i	− 0.409575i
$892$	0	0
$893$	− 7.64869e9i	− 0.359424i
$894$	0	0
$895$	−2.18885e9	−0.102055
$896$	0	0
$897$	1.71544e9	0.0793598
$898$	0	0
$899$	2.75505e10i	1.26465i
$900$	0	0
$901$	− 1.23630e10i	− 0.563103i
$902$	0	0
$903$	5.19473e9	0.234777
$904$	0	0
$905$	−2.64903e9	−0.118800
$906$	0	0
$907$	− 2.05338e10i	− 0.913787i	−0.889521	−	0.456893i	$-0.848962\pi$
$907$	− 2.05338e10i	− 0.913787i	0.889521	−	0.456893i	$-0.151038\pi$
$908$	0	0
$909$	2.99510e10i	1.32263i
$910$	0	0
$911$	5.14530e9	0.225474	0.112737	−	0.993625i	$-0.464038\pi$
$911$	5.14530e9	0.225474	0.112737	+	0.993625i	$0.464038\pi$
$912$	0	0
$913$	−1.20169e10	−0.522571
$914$	0	0
$915$	− 5.39232e9i	− 0.232703i
$916$	0	0
$917$	− 9.26115e9i	− 0.396618i
$918$	0	0
$919$	2.24863e10	0.955682	0.477841	−	0.878446i	$-0.341419\pi$
$919$	2.24863e10	0.955682	0.477841	+	0.878446i	$0.341419\pi$
$920$	0	0
$921$	−1.63126e10	−0.688041
$922$	0	0
$923$	− 4.17141e10i	− 1.74613i
$924$	0	0
$925$	7.32250e9i	0.304203i
$926$	0	0
$927$	−7.37850e9	−0.304221
$928$	0	0
$929$	−9.09332e9	−0.372107	−0.186053	−	0.982540i	$-0.559570\pi$
$929$	−9.09332e9	−0.372107	−0.186053	+	0.982540i	$0.559570\pi$
$930$	0	0
$931$	1.96666e10i	0.798740i
$932$	0	0
$933$	− 9.97358e9i	− 0.402036i
$934$	0	0
$935$	7.34045e9	0.293685
$936$	0	0
$937$	2.06278e10	0.819153	0.409576	−	0.912276i	$-0.365676\pi$
$937$	2.06278e10	0.819153	0.409576	+	0.912276i	$0.365676\pi$
$938$	0	0
$939$	1.11472e10i	0.439375i
$940$	0	0
$941$	− 4.71969e10i	− 1.84650i	−0.384195	−	0.923252i	$-0.625521\pi$
$941$	− 4.71969e10i	− 1.84650i	0.384195	−	0.923252i	$-0.374479\pi$
$942$	0	0
$943$	1.78428e9	0.0692903
$944$	0	0
$945$	1.88510e10	0.726648
$946$	0	0
$947$	− 3.96596e10i	− 1.51748i	−0.651393	−	0.758740i	$-0.725815\pi$
$947$	− 3.96596e10i	− 1.51748i	0.651393	−	0.758740i	$-0.274185\pi$
$948$	0	0
$949$	3.01201e9i	0.114400i
$950$	0	0
$951$	3.26686e9	0.123168
$952$	0	0
$953$	4.01775e10	1.50369	0.751845	−	0.659340i	$-0.229164\pi$
$953$	4.01775e10	1.50369	0.751845	+	0.659340i	$0.229164\pi$
$954$	0	0
$955$	4.98715e9i	0.185285i
$956$	0	0
$957$	− 9.94170e9i	− 0.366664i
$958$	0	0
$959$	5.58279e10	2.04402
$960$	0	0
$961$	5.25236e10	1.90907
$962$	0	0
$963$	3.15669e10i	1.13904i
$964$	0	0
$965$	− 4.93284e9i	− 0.176706i
$966$	0	0
$967$	−4.00487e10	−1.42428	−0.712141	−	0.702036i	$-0.752274\pi$
$967$	−4.00487e10	−1.42428	−0.712141	+	0.702036i	$0.752274\pi$
$968$	0	0
$969$	2.37051e9	0.0836967
$970$	0	0
$971$	2.91461e9i	0.102168i	0.998694	+	0.0510838i	$0.0162676\pi$
$971$	2.91461e9i	0.102168i	−0.998694	+	0.0510838i	$0.983732\pi$
$972$	0	0
$973$	− 5.82911e10i	− 2.02865i
$974$	0	0
$975$	−3.50050e9	−0.120952
$976$	0	0
$977$	−2.59846e10	−0.891424	−0.445712	−	0.895176i	$-0.647049\pi$
$977$	−2.59846e10	−0.891424	−0.445712	+	0.895176i	$0.647049\pi$
$978$	0	0
$979$	1.03346e10i	0.352009i
$980$	0	0
$981$	− 2.51293e10i	− 0.849842i
$982$	0	0
$983$	−1.21350e10	−0.407477	−0.203738	−	0.979025i	$-0.565309\pi$
$983$	−1.21350e10	−0.407477	−0.203738	+	0.979025i	$0.565309\pi$
$984$	0	0
$985$	−2.26814e10	−0.756211
$986$	0	0
$987$	− 3.45908e10i	− 1.14512i
$988$	0	0
$989$	− 1.02678e9i	− 0.0337513i
$990$	0	0
$991$	3.89139e10	1.27013	0.635063	−	0.772460i	$-0.280974\pi$
$991$	3.89139e10	1.27013	0.635063	+	0.772460i	$0.280974\pi$
$992$	0	0
$993$	2.38206e10	0.772024
$994$	0	0
$995$	2.52679e9i	0.0813185i
$996$	0	0
$997$	4.38562e10i	1.40152i	0.713399	+	0.700758i	$0.247155\pi$
$997$	4.38562e10i	1.40152i	−0.713399	+	0.700758i	$0.752845\pi$
$998$	0	0
$999$	4.00153e10	1.26984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.d.a.81.18		28
4.3	odd	2		40.8.d.a.21.25	✓	28
8.3	odd	2		40.8.d.a.21.26	yes	28
8.5	even	2	inner	160.8.d.a.81.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.d.a.21.25	✓	28	4.3	odd	2
40.8.d.a.21.26	yes	28	8.3	odd	2
160.8.d.a.81.11		28	8.5	even	2	inner
160.8.d.a.81.18		28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+21.9337i q^{3} -125.000i q^{5} +1766.19 q^{7} +1705.91 q^{9} -4654.38i q^{11} -10214.0i q^{13} +2741.72 q^{15} -12616.9 q^{17} +8565.99i q^{19} +38739.2i q^{21} +7657.10 q^{23} -15625.0 q^{25} +85386.1i q^{27} -97383.8i q^{29} -282907. q^{31} +102088. q^{33} -220774. i q^{35} -468640. i q^{37} +224032. q^{39} +233023. q^{41} -134095. i q^{43} -213239. i q^{45} -892914. q^{47} +2.29589e6 q^{49} -276735. i q^{51} +979881. i q^{53} -581797. q^{55} -187884. q^{57} +921947. i q^{59} -1.96677e6i q^{61} +3.01297e6 q^{63} -1.27675e6 q^{65} -2.76945e6i q^{67} +167949. i q^{69} +4.08400e6 q^{71} -294889. q^{73} -342714. i q^{75} -8.22052e6i q^{77} +3.23939e6 q^{79} +1.85799e6 q^{81} -2.58186e6i q^{83} +1.57711e6i q^{85} +2.13599e6 q^{87} -2.22040e6 q^{89} -1.80400e7i q^{91} -6.20520e6i q^{93} +1.07075e6 q^{95} -1.81261e6 q^{97} -7.93995e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) \) 28 * q - 1372 * q^7 - 20412 * q^9 + 13500 * q^15 - 2588 * q^23 - 437500 * q^25 - 268024 * q^31 - 99016 * q^33 + 283944 * q^39 - 601208 * q^41 + 2076460 * q^47 + 4316268 * q^49 - 1331000 * q^55 + 3788536 * q^57 + 14839060 * q^63 + 1001656 * q^71 + 2534128 * q^73 + 22040096 * q^79 + 11166300 * q^81 - 34127184 * q^87 + 3807224 * q^89 + 13718000 * q^95 + 15198608 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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