LMFDB - Embedded newform 9.22.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

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:	yes
Analytic conductor:	$25.1529609858$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+288.000 q^{2} -2.01421e6 q^{4} -2.16410e7 q^{5} -7.68079e8 q^{7} -1.18407e9 q^{8} +O(q^{10})$

\(q+288.000 q^{2} -2.01421e6 q^{4} -2.16410e7 q^{5} -7.68079e8 q^{7} -1.18407e9 q^{8} -6.23259e9 q^{10} +9.47249e10 q^{11} -8.06218e10 q^{13} -2.21207e11 q^{14} +3.88309e12 q^{16} -3.05228e12 q^{17} -7.92079e12 q^{19} +4.35894e13 q^{20} +2.72808e13 q^{22} +7.38454e13 q^{23} -8.50644e12 q^{25} -2.32191e13 q^{26} +1.54707e15 q^{28} +4.25303e15 q^{29} +1.90054e15 q^{31} +3.60151e15 q^{32} -8.79057e14 q^{34} +1.66220e16 q^{35} +2.21914e16 q^{37} -2.28119e15 q^{38} +2.56244e16 q^{40} +2.06228e16 q^{41} -1.93606e17 q^{43} -1.90796e17 q^{44} +2.12675e16 q^{46} -1.46961e17 q^{47} +3.13992e16 q^{49} -2.44986e15 q^{50} +1.62389e17 q^{52} -2.03827e18 q^{53} -2.04994e18 q^{55} +9.09460e17 q^{56} +1.22487e18 q^{58} +5.97588e18 q^{59} +6.19062e18 q^{61} +5.47356e17 q^{62} -7.10619e18 q^{64} +1.74473e18 q^{65} +1.69613e19 q^{67} +6.14793e18 q^{68} +4.78712e18 q^{70} +5.63276e18 q^{71} -4.32848e19 q^{73} +6.39113e18 q^{74} +1.59541e19 q^{76} -7.27562e19 q^{77} -5.12649e19 q^{79} -8.40337e19 q^{80} +5.93937e18 q^{82} -4.89119e19 q^{83} +6.60543e19 q^{85} -5.57585e19 q^{86} -1.12161e20 q^{88} +5.04303e20 q^{89} +6.19239e19 q^{91} -1.48740e20 q^{92} -4.23246e19 q^{94} +1.71413e20 q^{95} +8.08275e20 q^{97} +9.04297e18 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	288.000	0.198874	0.0994369	−	0.995044i	$-0.468296\pi$
$2$	288.000	0.198874	0.0994369	+	0.995044i	$0.468296\pi$
$3$	0	0
$3$	0	0
$4$	−2.01421e6	−0.960449
$5$	−2.16410e7	−0.991040	−0.495520	−	0.868596i	$-0.665023\pi$
$5$	−2.16410e7	−0.991040	−0.495520	+	0.868596i	$0.665023\pi$
$6$	0	0
$7$	−7.68079e8	−1.02772	−0.513862	−	0.857873i	$-0.671786\pi$
$7$	−7.68079e8	−1.02772	−0.513862	+	0.857873i	$0.671786\pi$
$8$	−1.18407e9	−0.389882
$9$	0	0
$10$	−6.23259e9	−0.197092
$11$	9.47249e10	1.10114	0.550568	−	0.834790i	$-0.314411\pi$
$11$	9.47249e10	1.10114	0.550568	+	0.834790i	$0.314411\pi$
$12$	0	0
$13$	−8.06218e10	−0.162199	−0.0810993	−	0.996706i	$-0.525843\pi$
$13$	−8.06218e10	−0.162199	−0.0810993	+	0.996706i	$0.525843\pi$
$14$	−2.21207e11	−0.204387
$15$	0	0
$16$	3.88309e12	0.882912
$17$	−3.05228e12	−0.367207	−0.183604	−	0.983000i	$-0.558776\pi$
$17$	−3.05228e12	−0.367207	−0.183604	+	0.983000i	$0.558776\pi$
$18$	0	0
$19$	−7.92079e12	−0.296385	−0.148192	−	0.988959i	$-0.547345\pi$
$19$	−7.92079e12	−0.296385	−0.148192	+	0.988959i	$0.547345\pi$
$20$	4.35894e13	0.951844
$21$	0	0
$22$	2.72808e13	0.218987
$23$	7.38454e13	0.371690	0.185845	−	0.982579i	$-0.440498\pi$
$23$	7.38454e13	0.371690	0.185845	+	0.982579i	$0.440498\pi$
$24$	0	0
$25$	−8.50644e12	−0.0178393
$26$	−2.32191e13	−0.0322571
$27$	0	0
$28$	1.54707e15	0.987076
$29$	4.25303e15	1.87724	0.938620	−	0.344954i	$-0.112105\pi$
$29$	4.25303e15	1.87724	0.938620	+	0.344954i	$0.112105\pi$
$30$	0	0
$31$	1.90054e15	0.416466	0.208233	−	0.978079i	$-0.433229\pi$
$31$	1.90054e15	0.416466	0.208233	+	0.978079i	$0.433229\pi$
$32$	3.60151e15	0.565470
$33$	0	0
$34$	−8.79057e14	−0.0730279
$35$	1.66220e16	1.01852
$36$	0	0
$37$	2.21914e16	0.758695	0.379347	−	0.925254i	$-0.376149\pi$
$37$	2.21914e16	0.758695	0.379347	+	0.925254i	$0.376149\pi$
$38$	−2.28119e15	−0.0589431
$39$	0	0
$40$	2.56244e16	0.386389
$41$	2.06228e16	0.239948	0.119974	−	0.992777i	$-0.461719\pi$
$41$	2.06228e16	0.239948	0.119974	+	0.992777i	$0.461719\pi$
$42$	0	0
$43$	−1.93606e17	−1.36615	−0.683077	−	0.730346i	$-0.739359\pi$
$43$	−1.93606e17	−1.36615	−0.683077	+	0.730346i	$0.739359\pi$
$44$	−1.90796e17	−1.05759
$45$	0	0
$46$	2.12675e16	0.0739195
$47$	−1.46961e17	−0.407543	−0.203771	−	0.979019i	$-0.565320\pi$
$47$	−1.46961e17	−0.407543	−0.203771	+	0.979019i	$0.565320\pi$
$48$	0	0
$49$	3.13992e16	0.0562160
$50$	−2.44986e15	−0.00354777
$51$	0	0
$52$	1.62389e17	0.155784
$53$	−2.03827e18	−1.60090	−0.800450	−	0.599399i	$-0.795406\pi$
$53$	−2.03827e18	−1.60090	−0.800450	+	0.599399i	$0.795406\pi$
$54$	0	0
$55$	−2.04994e18	−1.09127
$56$	9.09460e17	0.400691
$57$	0	0
$58$	1.22487e18	0.373334
$59$	5.97588e18	1.52214	0.761072	−	0.648667i	$-0.224673\pi$
$59$	5.97588e18	1.52214	0.761072	+	0.648667i	$0.224673\pi$
$60$	0	0
$61$	6.19062e18	1.11114	0.555572	−	0.831468i	$-0.312499\pi$
$61$	6.19062e18	1.11114	0.555572	+	0.831468i	$0.312499\pi$
$62$	5.47356e17	0.0828242
$63$	0	0
$64$	−7.10619e18	−0.770455
$65$	1.74473e18	0.160745
$66$	0	0
$67$	1.69613e19	1.13677	0.568387	−	0.822761i	$-0.307568\pi$
$67$	1.69613e19	1.13677	0.568387	+	0.822761i	$0.307568\pi$
$68$	6.14793e18	0.352684
$69$	0	0
$70$	4.78712e18	0.202556
$71$	5.63276e18	0.205357	0.102678	−	0.994715i	$-0.467259\pi$
$71$	5.63276e18	0.205357	0.102678	+	0.994715i	$0.467259\pi$
$72$	0	0
$73$	−4.32848e19	−1.17881	−0.589407	−	0.807837i	$-0.700638\pi$
$73$	−4.32848e19	−1.17881	−0.589407	+	0.807837i	$0.700638\pi$
$74$	6.39113e18	0.150885
$75$	0	0
$76$	1.59541e19	0.284662
$77$	−7.27562e19	−1.13166
$78$	0	0
$79$	−5.12649e19	−0.609166	−0.304583	−	0.952486i	$-0.598517\pi$
$79$	−5.12649e19	−0.609166	−0.304583	+	0.952486i	$0.598517\pi$
$80$	−8.40337e19	−0.875001
$81$	0	0
$82$	5.93937e18	0.0477194
$83$	−4.89119e19	−0.346014	−0.173007	−	0.984921i	$-0.555348\pi$
$83$	−4.89119e19	−0.346014	−0.173007	+	0.984921i	$0.555348\pi$
$84$	0	0
$85$	6.60543e19	0.363917
$86$	−5.57585e19	−0.271692
$87$	0	0
$88$	−1.12161e20	−0.429313
$89$	5.04303e20	1.71434	0.857170	−	0.515034i	$-0.172221\pi$
$89$	5.04303e20	1.71434	0.857170	+	0.515034i	$0.172221\pi$
$90$	0	0
$91$	6.19239e19	0.166695
$92$	−1.48740e20	−0.356990
$93$	0	0
$94$	−4.23246e19	−0.0810496
$95$	1.71413e20	0.293729
$96$	0	0
$97$	8.08275e20	1.11290	0.556450	−	0.830881i	$-0.312163\pi$
$97$	8.08275e20	1.11290	0.556450	+	0.830881i	$0.312163\pi$
$98$	9.04297e18	0.0111799
$99$	0	0
$100$	1.71337e19	0.0171337
$101$	1.00202e21	0.902612	0.451306	−	0.892369i	$-0.350958\pi$
$101$	1.00202e21	0.902612	0.451306	+	0.892369i	$0.350958\pi$
$102$	0	0
$103$	−5.89747e20	−0.432389	−0.216195	−	0.976350i	$-0.569365\pi$
$103$	−5.89747e20	−0.432389	−0.216195	+	0.976350i	$0.569365\pi$
$104$	9.54620e19	0.0632383
$105$	0	0
$106$	−5.87021e20	−0.318377
$107$	−1.12210e21	−0.551445	−0.275723	−	0.961237i	$-0.588917\pi$
$107$	−1.12210e21	−0.551445	−0.275723	+	0.961237i	$0.588917\pi$
$108$	0	0
$109$	1.72394e21	0.697499	0.348750	−	0.937216i	$-0.386606\pi$
$109$	1.72394e21	0.697499	0.348750	+	0.937216i	$0.386606\pi$
$110$	−5.90382e20	−0.217025
$111$	0	0
$112$	−2.98252e21	−0.907389
$113$	−4.95810e20	−0.137402	−0.0687008	−	0.997637i	$-0.521885\pi$
$113$	−4.95810e20	−0.137402	−0.0687008	+	0.997637i	$0.521885\pi$
$114$	0	0
$115$	−1.59809e21	−0.368360
$116$	−8.56649e21	−1.80299
$117$	0	0
$118$	1.72105e21	0.302715
$119$	2.34439e21	0.377387
$120$	0	0
$121$	1.57256e21	0.212501
$122$	1.78290e21	0.220978
$123$	0	0
$124$	−3.82809e21	−0.399994
$125$	1.05033e22	1.00872
$126$	0	0
$127$	1.63609e21	0.133005	0.0665027	−	0.997786i	$-0.478816\pi$
$127$	1.63609e21	0.133005	0.0665027	+	0.997786i	$0.478816\pi$
$128$	−9.59949e21	−0.718693
$129$	0	0
$130$	5.02483e20	0.0319680
$131$	1.38650e22	0.813898	0.406949	−	0.913451i	$-0.366593\pi$
$131$	1.38650e22	0.813898	0.406949	+	0.913451i	$0.366593\pi$
$132$	0	0
$133$	6.08379e21	0.304601
$134$	4.88486e21	0.226075
$135$	0	0
$136$	3.61412e21	0.143167
$137$	4.00789e22	1.47011	0.735055	−	0.678007i	$-0.237156\pi$
$137$	4.00789e22	1.47011	0.735055	+	0.678007i	$0.237156\pi$
$138$	0	0
$139$	4.47585e22	1.41000	0.705001	−	0.709206i	$-0.250946\pi$
$139$	4.47585e22	1.41000	0.705001	+	0.709206i	$0.250946\pi$
$140$	−3.34801e22	−0.978232
$141$	0	0
$142$	1.62223e21	0.0408400
$143$	−7.63689e21	−0.178603
$144$	0	0
$145$	−9.20396e22	−1.86042
$146$	−1.24660e22	−0.234435
$147$	0	0
$148$	−4.46982e22	−0.728688
$149$	−4.93289e22	−0.749283	−0.374641	−	0.927170i	$-0.622234\pi$
$149$	−4.93289e22	−0.749283	−0.374641	+	0.927170i	$0.622234\pi$
$150$	0	0
$151$	5.70415e22	0.753239	0.376620	−	0.926368i	$-0.377086\pi$
$151$	5.70415e22	0.753239	0.376620	+	0.926368i	$0.377086\pi$
$152$	9.37878e21	0.115555
$153$	0	0
$154$	−2.09538e22	−0.225058
$155$	−4.11295e22	−0.412735
$156$	0	0
$157$	6.35623e22	0.557511	0.278756	−	0.960362i	$-0.410078\pi$
$157$	6.35623e22	0.557511	0.278756	+	0.960362i	$0.410078\pi$
$158$	−1.47643e22	−0.121147
$159$	0	0
$160$	−7.79400e22	−0.560403
$161$	−5.67191e22	−0.381995
$162$	0	0
$163$	8.68484e22	0.513797	0.256899	−	0.966438i	$-0.417299\pi$
$163$	8.68484e22	0.513797	0.256899	+	0.966438i	$0.417299\pi$
$164$	−4.15386e22	−0.230458
$165$	0	0
$166$	−1.40866e22	−0.0688132
$167$	1.89411e23	0.868726	0.434363	−	0.900738i	$-0.356974\pi$
$167$	1.89411e23	0.868726	0.434363	+	0.900738i	$0.356974\pi$
$168$	0	0
$169$	−2.40565e23	−0.973692
$170$	1.90236e22	0.0723736
$171$	0	0
$172$	3.89962e23	1.31212
$173$	4.18508e23	1.32501	0.662506	−	0.749057i	$-0.269493\pi$
$173$	4.18508e23	1.32501	0.662506	+	0.749057i	$0.269493\pi$
$174$	0	0
$175$	6.53362e21	0.0183339
$176$	3.67825e23	0.972206
$177$	0	0
$178$	1.45239e23	0.340937
$179$	−4.76752e23	−1.05520	−0.527601	−	0.849493i	$-0.676908\pi$
$179$	−4.76752e23	−1.05520	−0.527601	+	0.849493i	$0.676908\pi$
$180$	0	0
$181$	−2.88627e22	−0.0568476	−0.0284238	−	0.999596i	$-0.509049\pi$
$181$	−2.88627e22	−0.0568476	−0.0284238	+	0.999596i	$0.509049\pi$
$182$	1.78341e22	0.0331513
$183$	0	0
$184$	−8.74383e22	−0.144915
$185$	−4.80244e23	−0.751897
$186$	0	0
$187$	−2.89127e23	−0.404345
$188$	2.96009e23	0.391424
$189$	0	0
$190$	4.93671e22	0.0584150
$191$	−8.86378e23	−0.992587	−0.496293	−	0.868155i	$-0.665306\pi$
$191$	−8.86378e23	−0.992587	−0.496293	+	0.868155i	$0.665306\pi$
$192$	0	0
$193$	8.63509e22	0.0866792	0.0433396	−	0.999060i	$-0.486200\pi$
$193$	8.63509e22	0.0866792	0.0433396	+	0.999060i	$0.486200\pi$
$194$	2.32783e23	0.221327
$195$	0	0
$196$	−6.32445e22	−0.0539926
$197$	6.99008e23	0.565701	0.282850	−	0.959164i	$-0.408720\pi$
$197$	6.99008e23	0.565701	0.282850	+	0.959164i	$0.408720\pi$
$198$	0	0
$199$	−1.24542e24	−0.906483	−0.453242	−	0.891388i	$-0.649732\pi$
$199$	−1.24542e24	−0.906483	−0.453242	+	0.891388i	$0.649732\pi$
$200$	1.00722e22	0.00695522
$201$	0	0
$202$	2.88581e23	0.179506
$203$	−3.26666e24	−1.92928
$204$	0	0
$205$	−4.46297e23	−0.237798
$206$	−1.69847e23	−0.0859909
$207$	0	0
$208$	−3.13061e23	−0.143207
$209$	−7.50296e23	−0.326360
$210$	0	0
$211$	3.50841e24	1.38085	0.690423	−	0.723406i	$-0.257425\pi$
$211$	3.50841e24	1.38085	0.690423	+	0.723406i	$0.257425\pi$
$212$	4.10549e24	1.53758
$213$	0	0
$214$	−3.23165e23	−0.109668
$215$	4.18981e24	1.35391
$216$	0	0
$217$	−1.45977e24	−0.428012
$218$	4.96495e23	0.138714
$219$	0	0
$220$	4.12900e24	1.04811
$221$	2.46081e23	0.0595605
$222$	0	0
$223$	−4.72350e24	−1.04007	−0.520035	−	0.854145i	$-0.674081\pi$
$223$	−4.72350e24	−1.04007	−0.520035	+	0.854145i	$0.674081\pi$
$224$	−2.76624e24	−0.581147
$225$	0	0
$226$	−1.42793e23	−0.0273256
$227$	5.44317e24	0.994444	0.497222	−	0.867623i	$-0.334353\pi$
$227$	5.44317e24	0.994444	0.497222	+	0.867623i	$0.334353\pi$
$228$	0	0
$229$	6.90677e24	1.15081	0.575403	−	0.817870i	$-0.304845\pi$
$229$	6.90677e24	1.15081	0.575403	+	0.817870i	$0.304845\pi$
$230$	−4.60249e23	−0.0732572
$231$	0	0
$232$	−5.03589e24	−0.731902
$233$	−4.53650e24	−0.630208	−0.315104	−	0.949057i	$-0.602039\pi$
$233$	−4.53650e24	−0.630208	−0.315104	+	0.949057i	$0.602039\pi$
$234$	0	0
$235$	3.18036e24	0.403891
$236$	−1.20367e25	−1.46194
$237$	0	0
$238$	6.75185e23	0.0750525
$239$	2.73493e24	0.290916	0.145458	−	0.989364i	$-0.453534\pi$
$239$	2.73493e24	0.290916	0.145458	+	0.989364i	$0.453534\pi$
$240$	0	0
$241$	−8.08907e24	−0.788351	−0.394175	−	0.919035i	$-0.628970\pi$
$241$	−8.08907e24	−0.788351	−0.394175	+	0.919035i	$0.628970\pi$
$242$	4.52898e23	0.0422609
$243$	0	0
$244$	−1.24692e25	−1.06720
$245$	−6.79508e23	−0.0557123
$246$	0	0
$247$	6.38588e23	0.0480732
$248$	−2.25038e24	−0.162373
$249$	0	0
$250$	3.02495e24	0.200608
$251$	−6.63927e24	−0.422227	−0.211113	−	0.977462i	$-0.567709\pi$
$251$	−6.63927e24	−0.422227	−0.211113	+	0.977462i	$0.567709\pi$
$252$	0	0
$253$	6.99500e24	0.409282
$254$	4.71195e23	0.0264513
$255$	0	0
$256$	1.21381e25	0.627526
$257$	−1.57278e24	−0.0780497	−0.0390249	−	0.999238i	$-0.512425\pi$
$257$	−1.57278e24	−0.0780497	−0.0390249	+	0.999238i	$0.512425\pi$
$258$	0	0
$259$	−1.70448e25	−0.779729
$260$	−3.51425e24	−0.154388
$261$	0	0
$262$	3.99311e24	0.161863
$263$	3.40077e25	1.32447	0.662235	−	0.749296i	$-0.269608\pi$
$263$	3.40077e25	1.32447	0.662235	+	0.749296i	$0.269608\pi$
$264$	0	0
$265$	4.41100e25	1.58656
$266$	1.75213e24	0.0605772
$267$	0	0
$268$	−3.41636e25	−1.09181
$269$	3.57975e25	1.10015	0.550077	−	0.835114i	$-0.314598\pi$
$269$	3.57975e25	1.10015	0.550077	+	0.835114i	$0.314598\pi$
$270$	0	0
$271$	2.46104e25	0.699746	0.349873	−	0.936797i	$-0.386225\pi$
$271$	2.46104e25	0.699746	0.349873	+	0.936797i	$0.386225\pi$
$272$	−1.18523e25	−0.324212
$273$	0	0
$274$	1.15427e25	0.292366
$275$	−8.05772e23	−0.0196435
$276$	0	0
$277$	−6.11679e25	−1.38193	−0.690965	−	0.722888i	$-0.742814\pi$
$277$	−6.11679e25	−1.38193	−0.690965	+	0.722888i	$0.742814\pi$
$278$	1.28904e25	0.280413
$279$	0	0
$280$	−1.96816e25	−0.397101
$281$	−1.73710e25	−0.337605	−0.168802	−	0.985650i	$-0.553990\pi$
$281$	−1.73710e25	−0.337605	−0.168802	+	0.985650i	$0.553990\pi$
$282$	0	0
$283$	7.57237e25	1.36607	0.683037	−	0.730383i	$-0.260659\pi$
$283$	7.57237e25	1.36607	0.683037	+	0.730383i	$0.260659\pi$
$284$	−1.13455e25	−0.197235
$285$	0	0
$286$	−2.19943e24	−0.0355194
$287$	−1.58399e25	−0.246600
$288$	0	0
$289$	−5.97755e25	−0.865159
$290$	−2.65074e25	−0.369989
$291$	0	0
$292$	8.71845e25	1.13219
$293$	−4.88684e25	−0.612235	−0.306118	−	0.951994i	$-0.599030\pi$
$293$	−4.88684e25	−0.612235	−0.306118	+	0.951994i	$0.599030\pi$
$294$	0	0
$295$	−1.29324e26	−1.50851
$296$	−2.62762e25	−0.295801
$297$	0	0
$298$	−1.42067e25	−0.149013
$299$	−5.95355e24	−0.0602877
$300$	0	0
$301$	1.48705e26	1.40403
$302$	1.64279e25	0.149800
$303$	0	0
$304$	−3.07571e25	−0.261681
$305$	−1.33971e26	−1.10119
$306$	0	0
$307$	−2.17987e26	−1.67293	−0.836466	−	0.548019i	$-0.815382\pi$
$307$	−2.17987e26	−1.67293	−0.836466	+	0.548019i	$0.815382\pi$
$308$	1.46546e26	1.08691
$309$	0	0
$310$	−1.18453e25	−0.0820821
$311$	4.04644e25	0.271075	0.135538	−	0.990772i	$-0.456724\pi$
$311$	4.04644e25	0.271075	0.135538	+	0.990772i	$0.456724\pi$
$312$	0	0
$313$	−8.74174e24	−0.0547498	−0.0273749	−	0.999625i	$-0.508715\pi$
$313$	−8.74174e24	−0.0547498	−0.0273749	+	0.999625i	$0.508715\pi$
$314$	1.83060e25	0.110874
$315$	0	0
$316$	1.03258e26	0.585073
$317$	3.19758e25	0.175267	0.0876334	−	0.996153i	$-0.472070\pi$
$317$	3.19758e25	0.175267	0.0876334	+	0.996153i	$0.472070\pi$
$318$	0	0
$319$	4.02868e26	2.06710
$320$	1.53785e26	0.763552
$321$	0	0
$322$	−1.63351e25	−0.0759688
$323$	2.41765e25	0.108835
$324$	0	0
$325$	6.85805e23	0.00289351
$326$	2.50123e25	0.102181
$327$	0	0
$328$	−2.44189e25	−0.0935514
$329$	1.12877e26	0.418841
$330$	0	0
$331$	−2.52215e26	−0.878166	−0.439083	−	0.898446i	$-0.644697\pi$
$331$	−2.52215e26	−0.878166	−0.439083	+	0.898446i	$0.644697\pi$
$332$	9.85186e25	0.332329
$333$	0	0
$334$	5.45504e25	0.172767
$335$	−3.67059e26	−1.12659
$336$	0	0
$337$	−1.53127e26	−0.441507	−0.220753	−	0.975330i	$-0.570852\pi$
$337$	−1.53127e26	−0.441507	−0.220753	+	0.975330i	$0.570852\pi$
$338$	−6.92826e25	−0.193642
$339$	0	0
$340$	−1.33047e26	−0.349524
$341$	1.80029e26	0.458586
$342$	0	0
$343$	4.04890e26	0.969949
$344$	2.29243e26	0.532639
$345$	0	0
$346$	1.20530e26	0.263510
$347$	−3.23436e26	−0.686008	−0.343004	−	0.939334i	$-0.611444\pi$
$347$	−3.23436e26	−0.686008	−0.343004	+	0.939334i	$0.611444\pi$
$348$	0	0
$349$	−6.77854e26	−1.35353	−0.676767	−	0.736197i	$-0.736620\pi$
$349$	−6.77854e26	−1.35353	−0.676767	+	0.736197i	$0.736620\pi$
$350$	1.88168e24	0.00364613
$351$	0	0
$352$	3.41153e26	0.622660
$353$	−6.84291e26	−1.21229	−0.606145	−	0.795354i	$-0.707285\pi$
$353$	−6.84291e26	−1.21229	−0.606145	+	0.795354i	$0.707285\pi$
$354$	0	0
$355$	−1.21898e26	−0.203517
$356$	−1.01577e27	−1.64654
$357$	0	0
$358$	−1.37304e26	−0.209852
$359$	6.85500e25	0.101746	0.0508728	−	0.998705i	$-0.483800\pi$
$359$	6.85500e25	0.101746	0.0508728	+	0.998705i	$0.483800\pi$
$360$	0	0
$361$	−6.51471e26	−0.912156
$362$	−8.31247e24	−0.0113055
$363$	0	0
$364$	−1.24728e26	−0.160102
$365$	9.36723e26	1.16825
$366$	0	0
$367$	1.13575e27	1.33749	0.668745	−	0.743492i	$-0.266832\pi$
$367$	1.13575e27	1.33749	0.668745	+	0.743492i	$0.266832\pi$
$368$	2.86748e26	0.328170
$369$	0	0
$370$	−1.38310e26	−0.149533
$371$	1.56555e27	1.64528
$372$	0	0
$373$	3.82975e26	0.380389	0.190195	−	0.981746i	$-0.439088\pi$
$373$	3.82975e26	0.380389	0.190195	+	0.981746i	$0.439088\pi$
$374$	−8.32687e25	−0.0804136
$375$	0	0
$376$	1.74012e26	0.158894
$377$	−3.42887e26	−0.304486
$378$	0	0
$379$	7.14767e25	0.0600417	0.0300208	−	0.999549i	$-0.490443\pi$
$379$	7.14767e25	0.0600417	0.0300208	+	0.999549i	$0.490443\pi$
$380$	−3.45262e26	−0.282112
$381$	0	0
$382$	−2.55277e26	−0.197400
$383$	1.38425e27	1.04142	0.520712	−	0.853732i	$-0.325666\pi$
$383$	1.38425e27	1.04142	0.520712	+	0.853732i	$0.325666\pi$
$384$	0	0
$385$	1.57451e27	1.12152
$386$	2.48690e25	0.0172382
$387$	0	0
$388$	−1.62803e27	−1.06888
$389$	−1.63213e27	−1.04300	−0.521500	−	0.853251i	$-0.674627\pi$
$389$	−1.63213e27	−1.04300	−0.521500	+	0.853251i	$0.674627\pi$
$390$	0	0
$391$	−2.25397e26	−0.136487
$392$	−3.71789e25	−0.0219176
$393$	0	0
$394$	2.01314e26	0.112503
$395$	1.10942e27	0.603708
$396$	0	0
$397$	1.21029e27	0.624581	0.312291	−	0.949987i	$-0.398904\pi$
$397$	1.21029e27	0.624581	0.312291	+	0.949987i	$0.398904\pi$
$398$	−3.58682e26	−0.180276
$399$	0	0
$400$	−3.30313e25	−0.0157505
$401$	6.51358e26	0.302555	0.151277	−	0.988491i	$-0.451661\pi$
$401$	6.51358e26	0.302555	0.151277	+	0.988491i	$0.451661\pi$
$402$	0	0
$403$	−1.53225e26	−0.0675502
$404$	−2.01827e27	−0.866913
$405$	0	0
$406$	−9.40799e26	−0.383684
$407$	2.10208e27	0.835427
$408$	0	0
$409$	−4.45251e27	−1.68078	−0.840389	−	0.541984i	$-0.817673\pi$
$409$	−4.45251e27	−1.68078	−0.840389	+	0.541984i	$0.817673\pi$
$410$	−1.28534e26	−0.0472918
$411$	0	0
$412$	1.18787e27	0.415288
$413$	−4.58995e27	−1.56434
$414$	0	0
$415$	1.05850e27	0.342914
$416$	−2.90360e26	−0.0917185
$417$	0	0
$418$	−2.16085e26	−0.0649044
$419$	−4.92912e27	−1.44385	−0.721925	−	0.691972i	$-0.756742\pi$
$419$	−4.92912e27	−1.44385	−0.721925	+	0.691972i	$0.756742\pi$
$420$	0	0
$421$	1.50145e27	0.418359	0.209180	−	0.977877i	$-0.432921\pi$
$421$	1.50145e27	0.418359	0.209180	+	0.977877i	$0.432921\pi$
$422$	1.01042e27	0.274614
$423$	0	0
$424$	2.41345e27	0.624162
$425$	2.59641e25	0.00655072
$426$	0	0
$427$	−4.75488e27	−1.14195
$428$	2.26014e27	0.529635
$429$	0	0
$430$	1.20667e27	0.269258
$431$	7.19221e27	1.56621	0.783107	−	0.621887i	$-0.213634\pi$
$431$	7.19221e27	1.56621	0.783107	+	0.621887i	$0.213634\pi$
$432$	0	0
$433$	4.23104e27	0.877656	0.438828	−	0.898571i	$-0.355394\pi$
$433$	4.23104e27	0.877656	0.438828	+	0.898571i	$0.355394\pi$
$434$	−4.20412e26	−0.0851203
$435$	0	0
$436$	−3.47237e27	−0.669913
$437$	−5.84914e26	−0.110163
$438$	0	0
$439$	4.53235e27	0.813665	0.406832	−	0.913503i	$-0.366633\pi$
$439$	4.53235e27	0.813665	0.406832	+	0.913503i	$0.366633\pi$
$440$	2.42727e27	0.425467
$441$	0	0
$442$	7.08712e25	0.0118450
$443$	4.61186e27	0.752726	0.376363	−	0.926472i	$-0.377174\pi$
$443$	4.61186e27	0.752726	0.376363	+	0.926472i	$0.377174\pi$
$444$	0	0
$445$	−1.09136e28	−1.69898
$446$	−1.36037e27	−0.206843
$447$	0	0
$448$	5.45811e27	0.791815
$449$	1.25692e26	0.0178123	0.00890615	−	0.999960i	$-0.497165\pi$
$449$	1.25692e26	0.0178123	0.00890615	+	0.999960i	$0.497165\pi$
$450$	0	0
$451$	1.95349e27	0.264216
$452$	9.98664e26	0.131967
$453$	0	0
$454$	1.56763e27	0.197769
$455$	−1.34009e27	−0.165202
$456$	0	0
$457$	1.15924e28	1.36475	0.682374	−	0.731003i	$-0.260948\pi$
$457$	1.15924e28	1.36475	0.682374	+	0.731003i	$0.260948\pi$
$458$	1.98915e27	0.228865
$459$	0	0
$460$	3.21888e27	0.353791
$461$	−1.02514e28	−1.10134	−0.550671	−	0.834723i	$-0.685628\pi$
$461$	−1.02514e28	−1.10134	−0.550671	+	0.834723i	$0.685628\pi$
$462$	0	0
$463$	−5.72801e27	−0.588035	−0.294017	−	0.955800i	$-0.594992\pi$
$463$	−5.72801e27	−0.588035	−0.294017	+	0.955800i	$0.594992\pi$
$464$	1.65149e28	1.65744
$465$	0	0
$466$	−1.30651e27	−0.125332
$467$	−1.12658e28	−1.05666	−0.528329	−	0.849040i	$-0.677181\pi$
$467$	−1.12658e28	−1.05666	−0.528329	+	0.849040i	$0.677181\pi$
$468$	0	0
$469$	−1.30276e28	−1.16829
$470$	9.15945e26	0.0803234
$471$	0	0
$472$	−7.07587e27	−0.593457
$473$	−1.83393e28	−1.50432
$474$	0	0
$475$	6.73777e25	0.00528729
$476$	−4.72210e27	−0.362462
$477$	0	0
$478$	7.87660e26	0.0578557
$479$	1.17373e28	0.843422	0.421711	−	0.906730i	$-0.361430\pi$
$479$	1.17373e28	0.843422	0.421711	+	0.906730i	$0.361430\pi$
$480$	0	0
$481$	−1.78911e27	−0.123059
$482$	−2.32965e27	−0.156782
$483$	0	0
$484$	−3.16747e27	−0.204097
$485$	−1.74918e28	−1.10293
$486$	0	0
$487$	4.75272e27	0.287004	0.143502	−	0.989650i	$-0.454164\pi$
$487$	4.75272e27	0.287004	0.143502	+	0.989650i	$0.454164\pi$
$488$	−7.33013e27	−0.433215
$489$	0	0
$490$	−1.95698e26	−0.0110797
$491$	2.59837e28	1.43994	0.719971	−	0.694004i	$-0.244155\pi$
$491$	2.59837e28	1.43994	0.719971	+	0.694004i	$0.244155\pi$
$492$	0	0
$493$	−1.29815e28	−0.689336
$494$	1.83913e26	0.00956049
$495$	0	0
$496$	7.37997e27	0.367703
$497$	−4.32640e27	−0.211050
$498$	0	0
$499$	3.84508e28	1.79825	0.899124	−	0.437695i	$-0.144205\pi$
$499$	3.84508e28	1.79825	0.899124	+	0.437695i	$0.144205\pi$
$500$	−2.11558e28	−0.968824
$501$	0	0
$502$	−1.91211e27	−0.0839699
$503$	3.27446e28	1.40824	0.704118	−	0.710083i	$-0.251343\pi$
$503$	3.27446e28	1.40824	0.704118	+	0.710083i	$0.251343\pi$
$504$	0	0
$505$	−2.16846e28	−0.894525
$506$	2.01456e27	0.0813954
$507$	0	0
$508$	−3.29543e27	−0.127745
$509$	2.02472e28	0.768826	0.384413	−	0.923161i	$-0.374404\pi$
$509$	2.02472e28	0.768826	0.384413	+	0.923161i	$0.374404\pi$
$510$	0	0
$511$	3.32461e28	1.21149
$512$	2.36274e28	0.843492
$513$	0	0
$514$	−4.52962e26	−0.0155220
$515$	1.27627e28	0.428515
$516$	0	0
$517$	−1.39208e28	−0.448760
$518$	−4.90889e27	−0.155068
$519$	0	0
$520$	−2.06589e27	−0.0626717
$521$	−4.75207e27	−0.141282	−0.0706411	−	0.997502i	$-0.522504\pi$
$521$	−4.75207e27	−0.141282	−0.0706411	+	0.997502i	$0.522504\pi$
$522$	0	0
$523$	−1.28180e26	−0.00366061	−0.00183030	−	0.999998i	$-0.500583\pi$
$523$	−1.28180e26	−0.00366061	−0.00183030	+	0.999998i	$0.500583\pi$
$524$	−2.79269e28	−0.781708
$525$	0	0
$526$	9.79422e27	0.263402
$527$	−5.80099e27	−0.152929
$528$	0	0
$529$	−3.40184e28	−0.861846
$530$	1.27037e28	0.315525
$531$	0	0
$532$	−1.22540e28	−0.292554
$533$	−1.66265e27	−0.0389192
$534$	0	0
$535$	2.42833e28	0.546504
$536$	−2.00834e28	−0.443208
$537$	0	0
$538$	1.03097e28	0.218792
$539$	2.97429e27	0.0619014
$540$	0	0
$541$	3.42747e28	0.686123	0.343061	−	0.939313i	$-0.388536\pi$
$541$	3.42747e28	0.686123	0.343061	+	0.939313i	$0.388536\pi$
$542$	7.08778e27	0.139161
$543$	0	0
$544$	−1.09928e28	−0.207645
$545$	−3.73077e28	−0.691250
$546$	0	0
$547$	−7.30329e28	−1.30212	−0.651061	−	0.759026i	$-0.725676\pi$
$547$	−7.30329e28	−1.30212	−0.651061	+	0.759026i	$0.725676\pi$
$548$	−8.07273e28	−1.41197
$549$	0	0
$550$	−2.32062e26	−0.00390658
$551$	−3.36874e28	−0.556385
$552$	0	0
$553$	3.93755e28	0.626055
$554$	−1.76164e28	−0.274830
$555$	0	0
$556$	−9.01529e28	−1.35424
$557$	3.32597e28	0.490274	0.245137	−	0.969488i	$-0.421167\pi$
$557$	3.32597e28	0.490274	0.245137	+	0.969488i	$0.421167\pi$
$558$	0	0
$559$	1.56089e28	0.221588
$560$	6.45445e28	0.899259
$561$	0	0
$562$	−5.00285e27	−0.0671407
$563$	8.28332e28	1.09110	0.545552	−	0.838077i	$-0.316320\pi$
$563$	8.28332e28	1.09110	0.545552	+	0.838077i	$0.316320\pi$
$564$	0	0
$565$	1.07298e28	0.136170
$566$	2.18084e28	0.271676
$567$	0	0
$568$	−6.66959e27	−0.0800648
$569$	−7.35414e28	−0.866669	−0.433335	−	0.901233i	$-0.642663\pi$
$569$	−7.35414e28	−0.866669	−0.433335	+	0.901233i	$0.642663\pi$
$570$	0	0
$571$	1.09131e29	1.23956	0.619780	−	0.784776i	$-0.287222\pi$
$571$	1.09131e29	1.23956	0.619780	+	0.784776i	$0.287222\pi$
$572$	1.53823e28	0.171539
$573$	0	0
$574$	−4.56190e27	−0.0490423
$575$	−6.28162e26	−0.00663070
$576$	0	0
$577$	1.30727e28	0.133051	0.0665257	−	0.997785i	$-0.478809\pi$
$577$	1.30727e28	0.133051	0.0665257	+	0.997785i	$0.478809\pi$
$578$	−1.72153e28	−0.172057
$579$	0	0
$580$	1.85387e29	1.78684
$581$	3.75682e28	0.355607
$582$	0	0
$583$	−1.93075e29	−1.76281
$584$	5.12523e28	0.459598
$585$	0	0
$586$	−1.40741e28	−0.121758
$587$	1.71459e29	1.45701	0.728503	−	0.685043i	$-0.240216\pi$
$587$	1.71459e29	1.45701	0.728503	+	0.685043i	$0.240216\pi$
$588$	0	0
$589$	−1.50538e28	−0.123434
$590$	−3.72452e28	−0.300002
$591$	0	0
$592$	8.61713e28	0.669861
$593$	−2.35277e29	−1.79682	−0.898412	−	0.439153i	$-0.855279\pi$
$593$	−2.35277e29	−1.79682	−0.898412	+	0.439153i	$0.855279\pi$
$594$	0	0
$595$	−5.07349e28	−0.374006
$596$	9.93587e28	0.719648
$597$	0	0
$598$	−1.71462e27	−0.0119896
$599$	1.33384e29	0.916481	0.458240	−	0.888828i	$-0.348480\pi$
$599$	1.33384e29	0.916481	0.458240	+	0.888828i	$0.348480\pi$
$600$	0	0
$601$	−3.64474e28	−0.241816	−0.120908	−	0.992664i	$-0.538581\pi$
$601$	−3.64474e28	−0.241816	−0.120908	+	0.992664i	$0.538581\pi$
$602$	4.28269e28	0.279225
$603$	0	0
$604$	−1.14893e29	−0.723448
$605$	−3.40317e28	−0.210597
$606$	0	0
$607$	2.71228e29	1.62126	0.810630	−	0.585559i	$-0.199125\pi$
$607$	2.71228e29	1.62126	0.810630	+	0.585559i	$0.199125\pi$
$608$	−2.85268e28	−0.167597
$609$	0	0
$610$	−3.85836e28	−0.218998
$611$	1.18482e28	0.0661029
$612$	0	0
$613$	−2.02699e29	−1.09274	−0.546370	−	0.837544i	$-0.683991\pi$
$613$	−2.02699e29	−1.09274	−0.546370	+	0.837544i	$0.683991\pi$
$614$	−6.27803e28	−0.332702
$615$	0	0
$616$	8.61486e28	0.441215
$617$	−3.04169e29	−1.53151	−0.765757	−	0.643130i	$-0.777636\pi$
$617$	−3.04169e29	−1.53151	−0.765757	+	0.643130i	$0.777636\pi$
$618$	0	0
$619$	1.21487e29	0.591262	0.295631	−	0.955302i	$-0.404470\pi$
$619$	1.21487e29	0.591262	0.295631	+	0.955302i	$0.404470\pi$
$620$	8.28434e28	0.396411
$621$	0	0
$622$	1.16538e28	0.0539097
$623$	−3.87345e29	−1.76187
$624$	0	0
$625$	−2.23245e29	−0.981842
$626$	−2.51762e27	−0.0108883
$627$	0	0
$628$	−1.28028e29	−0.535461
$629$	−6.77345e28	−0.278598
$630$	0	0
$631$	−1.79118e29	−0.712577	−0.356288	−	0.934376i	$-0.615958\pi$
$631$	−1.79118e29	−0.712577	−0.356288	+	0.934376i	$0.615958\pi$
$632$	6.07014e28	0.237503
$633$	0	0
$634$	9.20904e27	0.0348560
$635$	−3.54066e28	−0.131814
$636$	0	0
$637$	−2.53146e27	−0.00911815
$638$	1.16026e29	0.411091
$639$	0	0
$640$	2.07742e29	0.712254
$641$	3.41742e29	1.15263	0.576315	−	0.817228i	$-0.304490\pi$
$641$	3.41742e29	1.15263	0.576315	+	0.817228i	$0.304490\pi$
$642$	0	0
$643$	7.57582e28	0.247294	0.123647	−	0.992326i	$-0.460541\pi$
$643$	7.57582e28	0.247294	0.123647	+	0.992326i	$0.460541\pi$
$644$	1.14244e29	0.366887
$645$	0	0
$646$	6.96283e27	0.0216443
$647$	2.99088e29	0.914755	0.457377	−	0.889273i	$-0.348789\pi$
$647$	2.99088e29	0.914755	0.457377	+	0.889273i	$0.348789\pi$
$648$	0	0
$649$	5.66065e29	1.67609
$650$	1.97512e26	0.000575443 0
$651$	0	0
$652$	−1.74931e29	−0.493476
$653$	−5.40507e28	−0.150042	−0.0750210	−	0.997182i	$-0.523902\pi$
$653$	−5.40507e28	−0.150042	−0.0750210	+	0.997182i	$0.523902\pi$
$654$	0	0
$655$	−3.00051e29	−0.806606
$656$	8.00802e28	0.211853
$657$	0	0
$658$	3.25086e28	0.0832966
$659$	−2.26606e28	−0.0571446	−0.0285723	−	0.999592i	$-0.509096\pi$
$659$	−2.26606e28	−0.0571446	−0.0285723	+	0.999592i	$0.509096\pi$
$660$	0	0
$661$	−1.87483e29	−0.457980	−0.228990	−	0.973429i	$-0.573542\pi$
$661$	−1.87483e29	−0.457980	−0.228990	+	0.973429i	$0.573542\pi$
$662$	−7.26379e28	−0.174644
$663$	0	0
$664$	5.79151e28	0.134905
$665$	−1.31659e29	−0.301872
$666$	0	0
$667$	3.14067e29	0.697752
$668$	−3.81513e29	−0.834367
$669$	0	0
$670$	−1.05713e29	−0.224049
$671$	5.86406e29	1.22352
$672$	0	0
$673$	7.53001e29	1.52278	0.761390	−	0.648294i	$-0.224517\pi$
$673$	7.53001e29	1.52278	0.761390	+	0.648294i	$0.224517\pi$
$674$	−4.41005e28	−0.0878041
$675$	0	0
$676$	4.84547e29	0.935181
$677$	8.12052e29	1.54313	0.771565	−	0.636150i	$-0.219474\pi$
$677$	8.12052e29	1.54313	0.771565	+	0.636150i	$0.219474\pi$
$678$	0	0
$679$	−6.20819e29	−1.14375
$680$	−7.82130e28	−0.141885
$681$	0	0
$682$	5.18482e28	0.0912007
$683$	−4.69973e29	−0.814058	−0.407029	−	0.913415i	$-0.633435\pi$
$683$	−4.69973e29	−0.814058	−0.407029	+	0.913415i	$0.633435\pi$
$684$	0	0
$685$	−8.67346e29	−1.45694
$686$	1.16608e29	0.192897
$687$	0	0
$688$	−7.51789e29	−1.20619
$689$	1.64329e29	0.259664
$690$	0	0
$691$	3.58806e29	0.549971	0.274985	−	0.961448i	$-0.411327\pi$
$691$	3.58806e29	0.549971	0.274985	+	0.961448i	$0.411327\pi$
$692$	−8.42962e29	−1.27261
$693$	0	0
$694$	−9.31496e28	−0.136429
$695$	−9.68616e29	−1.39737
$696$	0	0
$697$	−6.29466e28	−0.0881106
$698$	−1.95222e29	−0.269183
$699$	0	0
$700$	−1.31601e28	−0.0176088
$701$	7.33110e29	0.966339	0.483170	−	0.875527i	$-0.339485\pi$
$701$	7.33110e29	0.966339	0.483170	+	0.875527i	$0.339485\pi$
$702$	0	0
$703$	−1.75774e29	−0.224865
$704$	−6.73133e29	−0.848376
$705$	0	0
$706$	−1.97076e29	−0.241093
$707$	−7.69629e29	−0.927636
$708$	0	0
$709$	1.20909e30	1.41473	0.707365	−	0.706848i	$-0.249884\pi$
$709$	1.20909e30	1.41473	0.707365	+	0.706848i	$0.249884\pi$
$710$	−3.51067e28	−0.0404741
$711$	0	0
$712$	−5.97131e29	−0.668390
$713$	1.40346e29	0.154796
$714$	0	0
$715$	1.65270e29	0.177003
$716$	9.60277e29	1.01347
$717$	0	0
$718$	1.97424e28	0.0202345
$719$	5.25189e29	0.530472	0.265236	−	0.964183i	$-0.414550\pi$
$719$	5.25189e29	0.530472	0.265236	+	0.964183i	$0.414550\pi$
$720$	0	0
$721$	4.52972e29	0.444377
$722$	−1.87624e29	−0.181404
$723$	0	0
$724$	5.81356e28	0.0545993
$725$	−3.61782e28	−0.0334886
$726$	0	0
$727$	7.23318e29	0.650456	0.325228	−	0.945636i	$-0.394559\pi$
$727$	7.23318e29	0.650456	0.325228	+	0.945636i	$0.394559\pi$
$728$	−7.33223e28	−0.0649915
$729$	0	0
$730$	2.69776e29	0.232335
$731$	5.90940e29	0.501662
$732$	0	0
$733$	−2.11409e30	−1.74394	−0.871969	−	0.489560i	$-0.837157\pi$
$733$	−2.11409e30	−1.74394	−0.871969	+	0.489560i	$0.837157\pi$
$734$	3.27097e29	0.265992
$735$	0	0
$736$	2.65955e29	0.210180
$737$	1.60666e30	1.25174
$738$	0	0
$739$	−7.42069e29	−0.561924	−0.280962	−	0.959719i	$-0.590653\pi$
$739$	−7.42069e29	−0.561924	−0.280962	+	0.959719i	$0.590653\pi$
$740$	9.67311e29	0.722159
$741$	0	0
$742$	4.50878e29	0.327204
$743$	−1.69968e30	−1.21614	−0.608071	−	0.793882i	$-0.708057\pi$
$743$	−1.69968e30	−1.21614	−0.608071	+	0.793882i	$0.708057\pi$
$744$	0	0
$745$	1.06752e30	0.742569
$746$	1.10297e29	0.0756494
$747$	0	0
$748$	5.82362e29	0.388353
$749$	8.61862e29	0.566733
$750$	0	0
$751$	5.90225e29	0.377397	0.188698	−	0.982035i	$-0.439573\pi$
$751$	5.90225e29	0.377397	0.188698	+	0.982035i	$0.439573\pi$
$752$	−5.70661e29	−0.359824
$753$	0	0
$754$	−9.87515e28	−0.0605542
$755$	−1.23443e30	−0.746490
$756$	0	0
$757$	1.33308e30	0.784058	0.392029	−	0.919953i	$-0.371773\pi$
$757$	1.33308e30	0.784058	0.392029	+	0.919953i	$0.371773\pi$
$758$	2.05853e28	0.0119407
$759$	0	0
$760$	−2.02966e29	−0.114520
$761$	7.86217e29	0.437526	0.218763	−	0.975778i	$-0.429798\pi$
$761$	7.86217e29	0.437526	0.218763	+	0.975778i	$0.429798\pi$
$762$	0	0
$763$	−1.32412e30	−0.716837
$764$	1.78535e30	0.953329
$765$	0	0
$766$	3.98665e29	0.207112
$767$	−4.81786e29	−0.246890
$768$	0	0
$769$	−1.65083e30	−0.823142	−0.411571	−	0.911378i	$-0.635020\pi$
$769$	−1.65083e30	−0.823142	−0.411571	+	0.911378i	$0.635020\pi$
$770$	4.53460e29	0.223042
$771$	0	0
$772$	−1.73929e29	−0.0832510
$773$	−3.77930e30	−1.78454	−0.892271	−	0.451499i	$-0.850889\pi$
$773$	−3.77930e30	−1.78454	−0.892271	+	0.451499i	$0.850889\pi$
$774$	0	0
$775$	−1.61668e28	−0.00742946
$776$	−9.57056e29	−0.433899
$777$	0	0
$778$	−4.70054e29	−0.207425
$779$	−1.63349e29	−0.0711169
$780$	0	0
$781$	5.33563e29	0.226126
$782$	−6.49144e28	−0.0271438
$783$	0	0
$784$	1.21926e29	0.0496337
$785$	−1.37555e30	−0.552516
$786$	0	0
$787$	−3.58219e30	−1.40092	−0.700461	−	0.713691i	$-0.747022\pi$
$787$	−3.58219e30	−1.40092	−0.700461	+	0.713691i	$0.747022\pi$
$788$	−1.40795e30	−0.543327
$789$	0	0
$790$	3.19514e29	0.120062
$791$	3.80821e29	0.141211
$792$	0	0
$793$	−4.99099e29	−0.180226
$794$	3.48564e29	0.124213
$795$	0	0
$796$	2.50854e30	0.870631
$797$	4.45901e30	1.52731	0.763654	−	0.645626i	$-0.223404\pi$
$797$	4.45901e30	1.52731	0.763654	+	0.645626i	$0.223404\pi$
$798$	0	0
$799$	4.48565e29	0.149653
$800$	−3.06360e28	−0.0100876
$801$	0	0
$802$	1.87591e29	0.0601702
$803$	−4.10015e30	−1.29803
$804$	0	0
$805$	1.22746e30	0.378572
$806$	−4.41288e28	−0.0134340
$807$	0	0
$808$	−1.18646e30	−0.351912
$809$	−3.09156e30	−0.905144	−0.452572	−	0.891728i	$-0.649493\pi$
$809$	−3.09156e30	−0.905144	−0.452572	+	0.891728i	$0.649493\pi$
$810$	0	0
$811$	−9.85711e29	−0.281210	−0.140605	−	0.990066i	$-0.544905\pi$
$811$	−9.85711e29	−0.281210	−0.140605	+	0.990066i	$0.544905\pi$
$812$	6.57974e30	1.85298
$813$	0	0
$814$	6.05400e29	0.166144
$815$	−1.87948e30	−0.509194
$816$	0	0
$817$	1.53351e30	0.404907
$818$	−1.28232e30	−0.334262
$819$	0	0
$820$	8.98935e29	0.228393
$821$	5.81930e30	1.45971	0.729856	−	0.683601i	$-0.239587\pi$
$821$	5.81930e30	1.45971	0.729856	+	0.683601i	$0.239587\pi$
$822$	0	0
$823$	1.82848e30	0.447086	0.223543	−	0.974694i	$-0.428238\pi$
$823$	1.82848e30	0.447086	0.223543	+	0.974694i	$0.428238\pi$
$824$	6.98303e29	0.168581
$825$	0	0
$826$	−1.32191e30	−0.311107
$827$	3.80997e30	0.885347	0.442673	−	0.896683i	$-0.354030\pi$
$827$	3.80997e30	0.885347	0.442673	+	0.896683i	$0.354030\pi$
$828$	0	0
$829$	9.24247e29	0.209395	0.104697	−	0.994504i	$-0.466613\pi$
$829$	9.24247e29	0.209395	0.104697	+	0.994504i	$0.466613\pi$
$830$	3.04848e29	0.0681966
$831$	0	0
$832$	5.72914e29	0.124967
$833$	−9.58392e28	−0.0206429
$834$	0	0
$835$	−4.09904e30	−0.860942
$836$	1.51125e30	0.313452
$837$	0	0
$838$	−1.41959e30	−0.287144
$839$	−2.18057e30	−0.435582	−0.217791	−	0.975995i	$-0.569885\pi$
$839$	−2.18057e30	−0.435582	−0.217791	+	0.975995i	$0.569885\pi$
$840$	0	0
$841$	1.29554e31	2.52403
$842$	4.32418e29	0.0832007
$843$	0	0
$844$	−7.06668e30	−1.32623
$845$	5.20605e30	0.964968
$846$	0	0
$847$	−1.20785e30	−0.218393
$848$	−7.91477e30	−1.41345
$849$	0	0
$850$	7.47765e27	0.00130277
$851$	1.63874e30	0.282000
$852$	0	0
$853$	−4.27754e30	−0.718173	−0.359087	−	0.933304i	$-0.616912\pi$
$853$	−4.27754e30	−0.718173	−0.359087	+	0.933304i	$0.616912\pi$
$854$	−1.36941e30	−0.227104
$855$	0	0
$856$	1.32865e30	0.214998
$857$	1.76322e29	0.0281843	0.0140922	−	0.999901i	$-0.495514\pi$
$857$	1.76322e29	0.0281843	0.0140922	+	0.999901i	$0.495514\pi$
$858$	0	0
$859$	4.74009e30	0.739365	0.369682	−	0.929158i	$-0.379467\pi$
$859$	4.74009e30	0.739365	0.369682	+	0.929158i	$0.379467\pi$
$860$	−8.43916e30	−1.30037
$861$	0	0
$862$	2.07136e30	0.311479
$863$	−8.10824e29	−0.120452	−0.0602259	−	0.998185i	$-0.519182\pi$
$863$	−8.10824e29	−0.120452	−0.0602259	+	0.998185i	$0.519182\pi$
$864$	0	0
$865$	−9.05691e30	−1.31314
$866$	1.21854e30	0.174543
$867$	0	0
$868$	2.94027e30	0.411084
$869$	−4.85607e30	−0.670775
$870$	0	0
$871$	−1.36745e30	−0.184383
$872$	−2.04127e30	−0.271942
$873$	0	0
$874$	−1.68455e29	−0.0219086
$875$	−8.06736e30	−1.03669
$876$	0	0
$877$	1.55255e30	0.194782	0.0973912	−	0.995246i	$-0.468950\pi$
$877$	1.55255e30	0.194782	0.0973912	+	0.995246i	$0.468950\pi$
$878$	1.30532e30	0.161817
$879$	0	0
$880$	−7.96009e30	−0.963496
$881$	5.75722e30	0.688598	0.344299	−	0.938860i	$-0.388117\pi$
$881$	5.75722e30	0.688598	0.344299	+	0.938860i	$0.388117\pi$
$882$	0	0
$883$	−1.29134e31	−1.50818	−0.754088	−	0.656774i	$-0.771921\pi$
$883$	−1.29134e31	−1.50818	−0.754088	+	0.656774i	$0.771921\pi$
$884$	−4.95657e29	−0.0572048
$885$	0	0
$886$	1.32822e30	0.149698
$887$	4.22752e30	0.470856	0.235428	−	0.971892i	$-0.424351\pi$
$887$	4.22752e30	0.470856	0.235428	+	0.971892i	$0.424351\pi$
$888$	0	0
$889$	−1.25665e30	−0.136693
$890$	−3.14312e30	−0.337883
$891$	0	0
$892$	9.51411e30	0.998935
$893$	1.16404e30	0.120789
$894$	0	0
$895$	1.03174e31	1.04575
$896$	7.37317e30	0.738618
$897$	0	0
$898$	3.61992e28	0.00354240
$899$	8.08306e30	0.781806
$900$	0	0
$901$	6.22137e30	0.587862
$902$	5.62606e29	0.0525455
$903$	0	0
$904$	5.87074e29	0.0535704
$905$	6.24617e29	0.0563383
$906$	0	0
$907$	1.34786e31	1.18787	0.593935	−	0.804513i	$-0.297574\pi$
$907$	1.34786e31	1.18787	0.593935	+	0.804513i	$0.297574\pi$
$908$	−1.09637e31	−0.955113
$909$	0	0
$910$	−3.85946e29	−0.0328543
$911$	1.28886e31	1.08458	0.542292	−	0.840190i	$-0.317557\pi$
$911$	1.28886e31	1.08458	0.542292	+	0.840190i	$0.317557\pi$
$912$	0	0
$913$	−4.63317e30	−0.381009
$914$	3.33860e30	0.271413
$915$	0	0
$916$	−1.39117e31	−1.10529
$917$	−1.06494e31	−0.836463
$918$	0	0
$919$	1.25018e30	0.0959753	0.0479876	−	0.998848i	$-0.484719\pi$
$919$	1.25018e30	0.0959753	0.0479876	+	0.998848i	$0.484719\pi$
$920$	1.89225e30	0.143617
$921$	0	0
$922$	−2.95239e30	−0.219028
$923$	−4.54123e29	−0.0333086
$924$	0	0
$925$	−1.88770e29	−0.0135346
$926$	−1.64967e30	−0.116945
$927$	0	0
$928$	1.53173e31	1.06152
$929$	1.41245e31	0.967850	0.483925	−	0.875109i	$-0.339211\pi$
$929$	1.41245e31	0.967850	0.483925	+	0.875109i	$0.339211\pi$
$930$	0	0
$931$	−2.48706e29	−0.0166615
$932$	9.13746e30	0.605283
$933$	0	0
$934$	−3.24455e30	−0.210142
$935$	6.25699e30	0.400722
$936$	0	0
$937$	−7.12851e30	−0.446409	−0.223204	−	0.974772i	$-0.571652\pi$
$937$	−7.12851e30	−0.446409	−0.223204	+	0.974772i	$0.571652\pi$
$938$	−3.75196e30	−0.232342
$939$	0	0
$940$	−6.40592e30	−0.387917
$941$	1.07254e29	0.00642275	0.00321138	−	0.999995i	$-0.498978\pi$
$941$	1.07254e29	0.00642275	0.00321138	+	0.999995i	$0.498978\pi$
$942$	0	0
$943$	1.52290e30	0.0891864
$944$	2.32049e31	1.34392
$945$	0	0
$946$	−5.28172e30	−0.299170
$947$	−2.92935e31	−1.64095	−0.820477	−	0.571680i	$-0.806292\pi$
$947$	−2.92935e31	−1.64095	−0.820477	+	0.571680i	$0.806292\pi$
$948$	0	0
$949$	3.48969e30	0.191202
$950$	1.94048e28	0.00105150
$951$	0	0
$952$	−2.77593e30	−0.147137
$953$	−3.04320e31	−1.59535	−0.797675	−	0.603088i	$-0.793937\pi$
$953$	−3.04320e31	−1.59535	−0.797675	+	0.603088i	$0.793937\pi$
$954$	0	0
$955$	1.91821e31	0.983694
$956$	−5.50872e30	−0.279410
$957$	0	0
$958$	3.38034e30	0.167735
$959$	−3.07838e31	−1.51087
$960$	0	0
$961$	−1.72134e31	−0.826556
$962$	−5.15264e29	−0.0244733
$963$	0	0
$964$	1.62931e31	0.757171
$965$	−1.86871e30	−0.0859026
$966$	0	0
$967$	−3.78485e31	−1.70244	−0.851218	−	0.524813i	$-0.824135\pi$
$967$	−3.78485e31	−1.70244	−0.851218	+	0.524813i	$0.824135\pi$
$968$	−1.86203e30	−0.0828504
$969$	0	0
$970$	−5.03765e30	−0.219344
$971$	1.62334e31	0.699212	0.349606	−	0.936897i	$-0.386316\pi$
$971$	1.62334e31	0.699212	0.349606	+	0.936897i	$0.386316\pi$
$972$	0	0
$973$	−3.43780e31	−1.44909
$974$	1.36878e30	0.0570776
$975$	0	0
$976$	2.40387e31	0.981043
$977$	−2.98858e30	−0.120662	−0.0603312	−	0.998178i	$-0.519216\pi$
$977$	−2.98858e30	−0.120662	−0.0603312	+	0.998178i	$0.519216\pi$
$978$	0	0
$979$	4.77701e31	1.88772
$980$	1.36867e30	0.0535088
$981$	0	0
$982$	7.48330e30	0.286367
$983$	2.41371e31	0.913846	0.456923	−	0.889506i	$-0.348951\pi$
$983$	2.41371e31	0.913846	0.456923	+	0.889506i	$0.348951\pi$
$984$	0	0
$985$	−1.51272e31	−0.560632
$986$	−3.73866e30	−0.137091
$987$	0	0
$988$	−1.28625e30	−0.0461718
$989$	−1.42969e31	−0.507786
$990$	0	0
$991$	−2.29436e31	−0.797790	−0.398895	−	0.916997i	$-0.630606\pi$
$991$	−2.29436e31	−0.797790	−0.398895	+	0.916997i	$0.630606\pi$
$992$	6.84481e30	0.235499
$993$	0	0
$994$	−1.24600e30	−0.0419723
$995$	2.69521e31	0.898361
$996$	0	0
$997$	3.55036e31	1.15871	0.579354	−	0.815076i	$-0.303305\pi$
$997$	3.55036e31	1.15871	0.579354	+	0.815076i	$0.303305\pi$
$998$	1.10738e31	0.357624
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.22.a.c.1.1		1
3.2	odd	2		1.22.a.a.1.1	✓	1
12.11	even	2		16.22.a.c.1.1		1
15.2	even	4		25.22.b.a.24.1		2
15.8	even	4		25.22.b.a.24.2		2
15.14	odd	2		25.22.a.a.1.1		1
21.20	even	2		49.22.a.a.1.1		1
24.5	odd	2		64.22.a.g.1.1		1
24.11	even	2		64.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.22.a.a.1.1	✓	1	3.2	odd	2
9.22.a.c.1.1		1	1.1	even	1	trivial
16.22.a.c.1.1		1	12.11	even	2
25.22.a.a.1.1		1	15.14	odd	2
25.22.b.a.24.1		2	15.2	even	4
25.22.b.a.24.2		2	15.8	even	4
49.22.a.a.1.1		1	21.20	even	2
64.22.a.a.1.1		1	24.11	even	2
64.22.a.g.1.1		1	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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