LMFDB - Embedded newform 9248.2.a.bz.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9248,2,Mod(1,9248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9248, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.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:	$73.8456517893$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 40 x^{18} + 620 x^{16} - 4784 x^{14} + 19585 x^{12} - 41912 x^{10} + 43536 x^{8} - 20328 x^{6} + \cdots + 8 $ x^20 - 40x^18 + 620x^16 - 4784x^14 + 19585x^12 - 41912x^10 + 43536x^8 - 20328x^6 + 4208x^4 - 352*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$0.646193$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23930 q^{3} +3.29169 q^{5} +2.75910 q^{7} +2.01445 q^{9} +1.05960 q^{11} +1.68352 q^{13} +7.37106 q^{15} +5.50343 q^{19} +6.17845 q^{21} +0.493141 q^{23} +5.83519 q^{25} -2.20694 q^{27} +9.81585 q^{29} +2.16148 q^{31} +2.37276 q^{33} +9.08209 q^{35} +3.43949 q^{37} +3.76990 q^{39} -7.08885 q^{41} +10.1052 q^{43} +6.63093 q^{45} -9.02539 q^{47} +0.612639 q^{49} -13.9767 q^{53} +3.48788 q^{55} +12.3238 q^{57} -11.2008 q^{59} +6.70511 q^{61} +5.55807 q^{63} +5.54162 q^{65} -4.89023 q^{67} +1.10429 q^{69} -5.80820 q^{71} -0.0130285 q^{73} +13.0667 q^{75} +2.92355 q^{77} -10.8521 q^{79} -10.9853 q^{81} -14.7087 q^{83} +21.9806 q^{87} -10.8032 q^{89} +4.64500 q^{91} +4.84020 q^{93} +18.1156 q^{95} -8.48329 q^{97} +2.13452 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) $ 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * q^89 - 32 * q^93

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
$3$	2.23930	1.29286	0.646429	−	0.762974i	$-0.276262\pi$
$3$	2.23930	1.29286	0.646429	+	0.762974i	$0.276262\pi$
$4$	0	0
$5$	3.29169	1.47209	0.736043	−	0.676935i	$-0.236692\pi$
$5$	3.29169	1.47209	0.736043	+	0.676935i	$0.236692\pi$
$6$	0	0
$7$	2.75910	1.04284	0.521421	−	0.853300i	$-0.325402\pi$
$7$	2.75910	1.04284	0.521421	+	0.853300i	$0.325402\pi$
$8$	0	0
$9$	2.01445	0.671483
$10$	0	0
$11$	1.05960	0.319482	0.159741	−	0.987159i	$-0.448934\pi$
$11$	1.05960	0.319482	0.159741	+	0.987159i	$0.448934\pi$
$12$	0	0
$13$	1.68352	0.466925	0.233462	−	0.972366i	$-0.424994\pi$
$13$	1.68352	0.466925	0.233462	+	0.972366i	$0.424994\pi$
$14$	0	0
$15$	7.37106	1.90320
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.50343	1.26257	0.631287	−	0.775549i	$-0.282527\pi$
$19$	5.50343	1.26257	0.631287	+	0.775549i	$0.282527\pi$
$20$	0	0
$21$	6.17845	1.34825
$22$	0	0
$23$	0.493141	0.102827	0.0514135	−	0.998677i	$-0.483627\pi$
$23$	0.493141	0.102827	0.0514135	+	0.998677i	$0.483627\pi$
$24$	0	0
$25$	5.83519	1.16704
$26$	0	0
$27$	−2.20694	−0.424726
$28$	0	0
$29$	9.81585	1.82276	0.911379	−	0.411568i	$-0.135019\pi$
$29$	9.81585	1.82276	0.911379	+	0.411568i	$0.135019\pi$
$30$	0	0
$31$	2.16148	0.388213	0.194107	−	0.980980i	$-0.437819\pi$
$31$	2.16148	0.388213	0.194107	+	0.980980i	$0.437819\pi$
$32$	0	0
$33$	2.37276	0.413045
$34$	0	0
$35$	9.08209	1.53515
$36$	0	0
$37$	3.43949	0.565449	0.282725	−	0.959201i	$-0.408762\pi$
$37$	3.43949	0.565449	0.282725	+	0.959201i	$0.408762\pi$
$38$	0	0
$39$	3.76990	0.603668
$40$	0	0
$41$	−7.08885	−1.10709	−0.553546	−	0.832818i	$-0.686726\pi$
$41$	−7.08885	−1.10709	−0.553546	+	0.832818i	$0.686726\pi$
$42$	0	0
$43$	10.1052	1.54103	0.770515	−	0.637422i	$-0.219999\pi$
$43$	10.1052	1.54103	0.770515	+	0.637422i	$0.219999\pi$
$44$	0	0
$45$	6.63093	0.988481
$46$	0	0
$47$	−9.02539	−1.31649	−0.658244	−	0.752805i	$-0.728700\pi$
$47$	−9.02539	−1.31649	−0.658244	+	0.752805i	$0.728700\pi$
$48$	0	0
$49$	0.612639	0.0875198
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.9767	−1.91985	−0.959927	−	0.280251i	$-0.909582\pi$
$53$	−13.9767	−1.91985	−0.959927	+	0.280251i	$0.909582\pi$
$54$	0	0
$55$	3.48788	0.470305
$56$	0	0
$57$	12.3238	1.63233
$58$	0	0
$59$	−11.2008	−1.45822	−0.729112	−	0.684394i	$-0.760067\pi$
$59$	−11.2008	−1.45822	−0.729112	+	0.684394i	$0.760067\pi$
$60$	0	0
$61$	6.70511	0.858501	0.429251	−	0.903185i	$-0.358778\pi$
$61$	6.70511	0.858501	0.429251	+	0.903185i	$0.358778\pi$
$62$	0	0
$63$	5.55807	0.700251
$64$	0	0
$65$	5.54162	0.687353
$66$	0	0
$67$	−4.89023	−0.597437	−0.298719	−	0.954341i	$-0.596559\pi$
$67$	−4.89023	−0.597437	−0.298719	+	0.954341i	$0.596559\pi$
$68$	0	0
$69$	1.10429	0.132941
$70$	0	0
$71$	−5.80820	−0.689307	−0.344654	−	0.938730i	$-0.612004\pi$
$71$	−5.80820	−0.689307	−0.344654	+	0.938730i	$0.612004\pi$
$72$	0	0
$73$	−0.0130285	−0.00152487	−0.000762435	−	1.00000i	$-0.500243\pi$
$73$	−0.0130285	−0.00152487	−0.000762435		1.00000i	$0.500243\pi$
$74$	0	0
$75$	13.0667	1.50882
$76$	0	0
$77$	2.92355	0.333169
$78$	0	0
$79$	−10.8521	−1.22096	−0.610481	−	0.792031i	$-0.709024\pi$
$79$	−10.8521	−1.22096	−0.610481	+	0.792031i	$0.709024\pi$
$80$	0	0
$81$	−10.9853	−1.22059
$82$	0	0
$83$	−14.7087	−1.61449	−0.807243	−	0.590220i	$-0.799041\pi$
$83$	−14.7087	−1.61449	−0.807243	+	0.590220i	$0.799041\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	21.9806	2.35657
$88$	0	0
$89$	−10.8032	−1.14514	−0.572568	−	0.819857i	$-0.694053\pi$
$89$	−10.8032	−1.14514	−0.572568	+	0.819857i	$0.694053\pi$
$90$	0	0
$91$	4.64500	0.486929
$92$	0	0
$93$	4.84020	0.501905
$94$	0	0
$95$	18.1156	1.85862
$96$	0	0
$97$	−8.48329	−0.861348	−0.430674	−	0.902508i	$-0.641724\pi$
$97$	−8.48329	−0.861348	−0.430674	+	0.902508i	$0.641724\pi$
$98$	0	0
$99$	2.13452	0.214527
$100$	0	0
$101$	−15.9634	−1.58841	−0.794207	−	0.607647i	$-0.792113\pi$
$101$	−15.9634	−1.58841	−0.794207	+	0.607647i	$0.792113\pi$
$102$	0	0
$103$	−1.74089	−0.171535	−0.0857676	−	0.996315i	$-0.527334\pi$
$103$	−1.74089	−0.171535	−0.0857676	+	0.996315i	$0.527334\pi$
$104$	0	0
$105$	20.3375	1.98474
$106$	0	0
$107$	−0.735099	−0.0710647	−0.0355324	−	0.999369i	$-0.511313\pi$
$107$	−0.735099	−0.0710647	−0.0355324	+	0.999369i	$0.511313\pi$
$108$	0	0
$109$	12.6147	1.20827	0.604134	−	0.796883i	$-0.293519\pi$
$109$	12.6147	1.20827	0.604134	+	0.796883i	$0.293519\pi$
$110$	0	0
$111$	7.70205	0.731046
$112$	0	0
$113$	−9.20738	−0.866158	−0.433079	−	0.901356i	$-0.642573\pi$
$113$	−9.20738	−0.866158	−0.433079	+	0.901356i	$0.642573\pi$
$114$	0	0
$115$	1.62326	0.151370
$116$	0	0
$117$	3.39137	0.313532
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.87724	−0.897931
$122$	0	0
$123$	−15.8740	−1.43131
$124$	0	0
$125$	2.74919	0.245895
$126$	0	0
$127$	2.07522	0.184146	0.0920732	−	0.995752i	$-0.470651\pi$
$127$	2.07522	0.184146	0.0920732	+	0.995752i	$0.470651\pi$
$128$	0	0
$129$	22.6286	1.99233
$130$	0	0
$131$	22.0829	1.92939	0.964695	−	0.263370i	$-0.0848339\pi$
$131$	22.0829	1.92939	0.964695	+	0.263370i	$0.0848339\pi$
$132$	0	0
$133$	15.1845	1.31667
$134$	0	0
$135$	−7.26455	−0.625233
$136$	0	0
$137$	2.94219	0.251368	0.125684	−	0.992070i	$-0.459887\pi$
$137$	2.94219	0.251368	0.125684	+	0.992070i	$0.459887\pi$
$138$	0	0
$139$	−8.25518	−0.700195	−0.350098	−	0.936713i	$-0.613852\pi$
$139$	−8.25518	−0.700195	−0.350098	+	0.936713i	$0.613852\pi$
$140$	0	0
$141$	−20.2105	−1.70203
$142$	0	0
$143$	1.78386	0.149174
$144$	0	0
$145$	32.3107	2.68326
$146$	0	0
$147$	1.37188	0.113151
$148$	0	0
$149$	1.87441	0.153558	0.0767788	−	0.997048i	$-0.475536\pi$
$149$	1.87441	0.153558	0.0767788	+	0.997048i	$0.475536\pi$
$150$	0	0
$151$	19.1894	1.56161	0.780804	−	0.624776i	$-0.214810\pi$
$151$	19.1894	1.56161	0.780804	+	0.624776i	$0.214810\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.11491	0.571484
$156$	0	0
$157$	−15.8204	−1.26261	−0.631303	−	0.775537i	$-0.717479\pi$
$157$	−15.8204	−1.26261	−0.631303	+	0.775537i	$0.717479\pi$
$158$	0	0
$159$	−31.2981	−2.48210
$160$	0	0
$161$	1.36062	0.107232
$162$	0	0
$163$	9.15180	0.716824	0.358412	−	0.933563i	$-0.383318\pi$
$163$	9.15180	0.716824	0.358412	+	0.933563i	$0.383318\pi$
$164$	0	0
$165$	7.81039	0.608038
$166$	0	0
$167$	−7.19323	−0.556629	−0.278314	−	0.960490i	$-0.589776\pi$
$167$	−7.19323	−0.556629	−0.278314	+	0.960490i	$0.589776\pi$
$168$	0	0
$169$	−10.1658	−0.781981
$170$	0	0
$171$	11.0864	0.847798
$172$	0	0
$173$	−2.66303	−0.202467	−0.101233	−	0.994863i	$-0.532279\pi$
$173$	−2.66303	−0.202467	−0.101233	+	0.994863i	$0.532279\pi$
$174$	0	0
$175$	16.0999	1.21704
$176$	0	0
$177$	−25.0820	−1.88528
$178$	0	0
$179$	14.6268	1.09326	0.546630	−	0.837374i	$-0.315910\pi$
$179$	14.6268	1.09326	0.546630	+	0.837374i	$0.315910\pi$
$180$	0	0
$181$	11.6738	0.867703	0.433852	−	0.900984i	$-0.357154\pi$
$181$	11.6738	0.867703	0.433852	+	0.900984i	$0.357154\pi$
$182$	0	0
$183$	15.0147	1.10992
$184$	0	0
$185$	11.3217	0.832390
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.08917	−0.442922
$190$	0	0
$191$	−18.0751	−1.30787	−0.653933	−	0.756552i	$-0.726882\pi$
$191$	−18.0751	−1.30787	−0.653933	+	0.756552i	$0.726882\pi$
$192$	0	0
$193$	1.41362	0.101754	0.0508772	−	0.998705i	$-0.483798\pi$
$193$	1.41362	0.101754	0.0508772	+	0.998705i	$0.483798\pi$
$194$	0	0
$195$	12.4093	0.888651
$196$	0	0
$197$	1.17020	0.0833732	0.0416866	−	0.999131i	$-0.486727\pi$
$197$	1.17020	0.0833732	0.0416866	+	0.999131i	$0.486727\pi$
$198$	0	0
$199$	−18.2507	−1.29376	−0.646879	−	0.762593i	$-0.723926\pi$
$199$	−18.2507	−1.29376	−0.646879	+	0.762593i	$0.723926\pi$
$200$	0	0
$201$	−10.9507	−0.772402
$202$	0	0
$203$	27.0829	1.90085
$204$	0	0
$205$	−23.3343	−1.62974
$206$	0	0
$207$	0.993407	0.0690466
$208$	0	0
$209$	5.83145	0.403370
$210$	0	0
$211$	23.8238	1.64010	0.820049	−	0.572293i	$-0.193946\pi$
$211$	23.8238	1.64010	0.820049	+	0.572293i	$0.193946\pi$
$212$	0	0
$213$	−13.0063	−0.891177
$214$	0	0
$215$	33.2632	2.26853
$216$	0	0
$217$	5.96374	0.404845
$218$	0	0
$219$	−0.0291747	−0.00197144
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.403945	−0.0270502	−0.0135251	−	0.999909i	$-0.504305\pi$
$223$	−0.403945	−0.0270502	−0.0135251	+	0.999909i	$0.504305\pi$
$224$	0	0
$225$	11.7547	0.783647
$226$	0	0
$227$	12.1563	0.806844	0.403422	−	0.915014i	$-0.367821\pi$
$227$	12.1563	0.806844	0.403422	+	0.915014i	$0.367821\pi$
$228$	0	0
$229$	12.9359	0.854830	0.427415	−	0.904056i	$-0.359424\pi$
$229$	12.9359	0.854830	0.427415	+	0.904056i	$0.359424\pi$
$230$	0	0
$231$	6.54670	0.430741
$232$	0	0
$233$	11.7240	0.768067	0.384034	−	0.923319i	$-0.374535\pi$
$233$	11.7240	0.768067	0.384034	+	0.923319i	$0.374535\pi$
$234$	0	0
$235$	−29.7087	−1.93798
$236$	0	0
$237$	−24.3012	−1.57853
$238$	0	0
$239$	−3.82714	−0.247557	−0.123779	−	0.992310i	$-0.539501\pi$
$239$	−3.82714	−0.247557	−0.123779	+	0.992310i	$0.539501\pi$
$240$	0	0
$241$	15.0221	0.967656	0.483828	−	0.875163i	$-0.339246\pi$
$241$	15.0221	0.967656	0.483828	+	0.875163i	$0.339246\pi$
$242$	0	0
$243$	−17.9786	−1.15333
$244$	0	0
$245$	2.01661	0.128837
$246$	0	0
$247$	9.26515	0.589527
$248$	0	0
$249$	−32.9371	−2.08730
$250$	0	0
$251$	17.9018	1.12995	0.564975	−	0.825108i	$-0.308886\pi$
$251$	17.9018	1.12995	0.564975	+	0.825108i	$0.308886\pi$
$252$	0	0
$253$	0.522533	0.0328514
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.52920	0.220145	0.110073	−	0.993924i	$-0.464892\pi$
$257$	3.52920	0.220145	0.110073	+	0.993924i	$0.464892\pi$
$258$	0	0
$259$	9.48991	0.589674
$260$	0	0
$261$	19.7735	1.22395
$262$	0	0
$263$	3.20443	0.197593	0.0987967	−	0.995108i	$-0.468501\pi$
$263$	3.20443	0.197593	0.0987967	+	0.995108i	$0.468501\pi$
$264$	0	0
$265$	−46.0070	−2.82619
$266$	0	0
$267$	−24.1915	−1.48050
$268$	0	0
$269$	−10.9062	−0.664963	−0.332482	−	0.943110i	$-0.607886\pi$
$269$	−10.9062	−0.664963	−0.332482	+	0.943110i	$0.607886\pi$
$270$	0	0
$271$	−31.2682	−1.89941	−0.949703	−	0.313153i	$-0.898615\pi$
$271$	−31.2682	−1.89941	−0.949703	+	0.313153i	$0.898615\pi$
$272$	0	0
$273$	10.4015	0.629530
$274$	0	0
$275$	6.18298	0.372848
$276$	0	0
$277$	12.6816	0.761963	0.380981	−	0.924583i	$-0.375586\pi$
$277$	12.6816	0.761963	0.380981	+	0.924583i	$0.375586\pi$
$278$	0	0
$279$	4.35419	0.260679
$280$	0	0
$281$	15.9496	0.951475	0.475737	−	0.879587i	$-0.342181\pi$
$281$	15.9496	0.951475	0.475737	+	0.879587i	$0.342181\pi$
$282$	0	0
$283$	7.06980	0.420256	0.210128	−	0.977674i	$-0.432612\pi$
$283$	7.06980	0.420256	0.210128	+	0.977674i	$0.432612\pi$
$284$	0	0
$285$	40.5661	2.40293
$286$	0	0
$287$	−19.5589	−1.15452
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−18.9966	−1.11360
$292$	0	0
$293$	6.41620	0.374838	0.187419	−	0.982280i	$-0.439988\pi$
$293$	6.41620	0.374838	0.187419	+	0.982280i	$0.439988\pi$
$294$	0	0
$295$	−36.8696	−2.14663
$296$	0	0
$297$	−2.33848	−0.135692
$298$	0	0
$299$	0.830213	0.0480124
$300$	0	0
$301$	27.8813	1.60705
$302$	0	0
$303$	−35.7467	−2.05359
$304$	0	0
$305$	22.0711	1.26379
$306$	0	0
$307$	30.6680	1.75031	0.875157	−	0.483839i	$-0.160758\pi$
$307$	30.6680	1.75031	0.875157	+	0.483839i	$0.160758\pi$
$308$	0	0
$309$	−3.89838	−0.221771
$310$	0	0
$311$	−15.1974	−0.861765	−0.430882	−	0.902408i	$-0.641798\pi$
$311$	−15.1974	−0.861765	−0.430882	+	0.902408i	$0.641798\pi$
$312$	0	0
$313$	2.59048	0.146422	0.0732112	−	0.997316i	$-0.476675\pi$
$313$	2.59048	0.146422	0.0732112	+	0.997316i	$0.476675\pi$
$314$	0	0
$315$	18.2954	1.03083
$316$	0	0
$317$	−14.1273	−0.793467	−0.396733	−	0.917934i	$-0.629856\pi$
$317$	−14.1273	−0.793467	−0.396733	+	0.917934i	$0.629856\pi$
$318$	0	0
$319$	10.4009	0.582339
$320$	0	0
$321$	−1.64611	−0.0918766
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9.82367	0.544919
$326$	0	0
$327$	28.2480	1.56212
$328$	0	0
$329$	−24.9020	−1.37289
$330$	0	0
$331$	20.4095	1.12181	0.560904	−	0.827881i	$-0.310454\pi$
$331$	20.4095	1.12181	0.560904	+	0.827881i	$0.310454\pi$
$332$	0	0
$333$	6.92869	0.379690
$334$	0	0
$335$	−16.0971	−0.879479
$336$	0	0
$337$	4.37565	0.238357	0.119178	−	0.992873i	$-0.461974\pi$
$337$	4.37565	0.238357	0.119178	+	0.992873i	$0.461974\pi$
$338$	0	0
$339$	−20.6181	−1.11982
$340$	0	0
$341$	2.29031	0.124027
$342$	0	0
$343$	−17.6234	−0.951573
$344$	0	0
$345$	3.63497	0.195700
$346$	0	0
$347$	12.2956	0.660064	0.330032	−	0.943970i	$-0.392940\pi$
$347$	12.2956	0.660064	0.330032	+	0.943970i	$0.392940\pi$
$348$	0	0
$349$	−4.57043	−0.244649	−0.122325	−	0.992490i	$-0.539035\pi$
$349$	−4.57043	−0.244649	−0.122325	+	0.992490i	$0.539035\pi$
$350$	0	0
$351$	−3.71543	−0.198315
$352$	0	0
$353$	−16.3984	−0.872797	−0.436398	−	0.899754i	$-0.643746\pi$
$353$	−16.3984	−0.872797	−0.436398	+	0.899754i	$0.643746\pi$
$354$	0	0
$355$	−19.1188	−1.01472
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.30795	0.174587	0.0872934	−	0.996183i	$-0.472178\pi$
$359$	3.30795	0.174587	0.0872934	+	0.996183i	$0.472178\pi$
$360$	0	0
$361$	11.2878	0.594094
$362$	0	0
$363$	−22.1181	−1.16090
$364$	0	0
$365$	−0.0428857	−0.00224474
$366$	0	0
$367$	18.9020	0.986674	0.493337	−	0.869838i	$-0.335777\pi$
$367$	18.9020	0.986674	0.493337	+	0.869838i	$0.335777\pi$
$368$	0	0
$369$	−14.2801	−0.743394
$370$	0	0
$371$	−38.5633	−2.00210
$372$	0	0
$373$	7.12337	0.368834	0.184417	−	0.982848i	$-0.440960\pi$
$373$	7.12337	0.368834	0.184417	+	0.982848i	$0.440960\pi$
$374$	0	0
$375$	6.15625	0.317907
$376$	0	0
$377$	16.5252	0.851091
$378$	0	0
$379$	18.0590	0.927626	0.463813	−	0.885933i	$-0.346481\pi$
$379$	18.0590	0.927626	0.463813	+	0.885933i	$0.346481\pi$
$380$	0	0
$381$	4.64704	0.238075
$382$	0	0
$383$	23.2334	1.18717	0.593587	−	0.804770i	$-0.297711\pi$
$383$	23.2334	1.18717	0.593587	+	0.804770i	$0.297711\pi$
$384$	0	0
$385$	9.62341	0.490454
$386$	0	0
$387$	20.3564	1.03478
$388$	0	0
$389$	−11.4442	−0.580244	−0.290122	−	0.956990i	$-0.593696\pi$
$389$	−11.4442	−0.580244	−0.290122	+	0.956990i	$0.593696\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	49.4501	2.49443
$394$	0	0
$395$	−35.7219	−1.79736
$396$	0	0
$397$	−17.2575	−0.866127	−0.433064	−	0.901363i	$-0.642567\pi$
$397$	−17.2575	−0.866127	−0.433064	+	0.901363i	$0.642567\pi$
$398$	0	0
$399$	34.0027	1.70226
$400$	0	0
$401$	4.10322	0.204905	0.102452	−	0.994738i	$-0.467331\pi$
$401$	4.10322	0.204905	0.102452	+	0.994738i	$0.467331\pi$
$402$	0	0
$403$	3.63890	0.181266
$404$	0	0
$405$	−36.1603	−1.79682
$406$	0	0
$407$	3.64450	0.180651
$408$	0	0
$409$	−7.26084	−0.359025	−0.179513	−	0.983756i	$-0.557452\pi$
$409$	−7.26084	−0.359025	−0.179513	+	0.983756i	$0.557452\pi$
$410$	0	0
$411$	6.58843	0.324983
$412$	0	0
$413$	−30.9042	−1.52070
$414$	0	0
$415$	−48.4163	−2.37666
$416$	0	0
$417$	−18.4858	−0.905254
$418$	0	0
$419$	11.5288	0.563217	0.281609	−	0.959529i	$-0.409132\pi$
$419$	11.5288	0.563217	0.281609	+	0.959529i	$0.409132\pi$
$420$	0	0
$421$	−19.0719	−0.929509	−0.464755	−	0.885439i	$-0.653857\pi$
$421$	−19.0719	−0.929509	−0.464755	+	0.885439i	$0.653857\pi$
$422$	0	0
$423$	−18.1812	−0.883999
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5001	0.895281
$428$	0	0
$429$	3.99460	0.192861
$430$	0	0
$431$	0.991174	0.0477432	0.0238716	−	0.999715i	$-0.492401\pi$
$431$	0.991174	0.0477432	0.0238716	+	0.999715i	$0.492401\pi$
$432$	0	0
$433$	1.72018	0.0826667	0.0413333	−	0.999145i	$-0.486839\pi$
$433$	1.72018	0.0826667	0.0413333	+	0.999145i	$0.486839\pi$
$434$	0	0
$435$	72.3532	3.46907
$436$	0	0
$437$	2.71397	0.129827
$438$	0	0
$439$	11.3457	0.541500	0.270750	−	0.962650i	$-0.412728\pi$
$439$	11.3457	0.541500	0.270750	+	0.962650i	$0.412728\pi$
$440$	0	0
$441$	1.23413	0.0587681
$442$	0	0
$443$	19.3737	0.920473	0.460236	−	0.887796i	$-0.347765\pi$
$443$	19.3737	0.920473	0.460236	+	0.887796i	$0.347765\pi$
$444$	0	0
$445$	−35.5607	−1.68574
$446$	0	0
$447$	4.19736	0.198528
$448$	0	0
$449$	34.4747	1.62696	0.813482	−	0.581590i	$-0.197569\pi$
$449$	34.4747	1.62696	0.813482	+	0.581590i	$0.197569\pi$
$450$	0	0
$451$	−7.51136	−0.353696
$452$	0	0
$453$	42.9707	2.01894
$454$	0	0
$455$	15.2899	0.716801
$456$	0	0
$457$	−27.9241	−1.30623	−0.653117	−	0.757257i	$-0.726539\pi$
$457$	−27.9241	−1.30623	−0.653117	+	0.757257i	$0.726539\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.14795	−0.146615	−0.0733073	−	0.997309i	$-0.523355\pi$
$461$	−3.14795	−0.146615	−0.0733073	+	0.997309i	$0.523355\pi$
$462$	0	0
$463$	8.05071	0.374148	0.187074	−	0.982346i	$-0.440100\pi$
$463$	8.05071	0.374148	0.187074	+	0.982346i	$0.440100\pi$
$464$	0	0
$465$	15.9324	0.738847
$466$	0	0
$467$	13.5785	0.628338	0.314169	−	0.949367i	$-0.398274\pi$
$467$	13.5785	0.628338	0.314169	+	0.949367i	$0.398274\pi$
$468$	0	0
$469$	−13.4926	−0.623033
$470$	0	0
$471$	−35.4266	−1.63237
$472$	0	0
$473$	10.7075	0.492331
$474$	0	0
$475$	32.1136	1.47347
$476$	0	0
$477$	−28.1554	−1.28915
$478$	0	0
$479$	25.4952	1.16491	0.582453	−	0.812864i	$-0.302093\pi$
$479$	25.4952	1.16491	0.582453	+	0.812864i	$0.302093\pi$
$480$	0	0
$481$	5.79046	0.264022
$482$	0	0
$483$	3.04684	0.138636
$484$	0	0
$485$	−27.9243	−1.26798
$486$	0	0
$487$	−12.2193	−0.553711	−0.276855	−	0.960912i	$-0.589292\pi$
$487$	−12.2193	−0.553711	−0.276855	+	0.960912i	$0.589292\pi$
$488$	0	0
$489$	20.4936	0.926752
$490$	0	0
$491$	−29.1011	−1.31331	−0.656657	−	0.754189i	$-0.728030\pi$
$491$	−29.1011	−1.31331	−0.656657	+	0.754189i	$0.728030\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	7.02615	0.315802
$496$	0	0
$497$	−16.0254	−0.718839
$498$	0	0
$499$	−38.6819	−1.73164	−0.865819	−	0.500357i	$-0.833202\pi$
$499$	−38.6819	−1.73164	−0.865819	+	0.500357i	$0.833202\pi$
$500$	0	0
$501$	−16.1078	−0.719642
$502$	0	0
$503$	19.9654	0.890215	0.445107	−	0.895477i	$-0.353165\pi$
$503$	19.9654	0.890215	0.445107	+	0.895477i	$0.353165\pi$
$504$	0	0
$505$	−52.5464	−2.33828
$506$	0	0
$507$	−22.7641	−1.01099
$508$	0	0
$509$	−6.53333	−0.289585	−0.144792	−	0.989462i	$-0.546251\pi$
$509$	−6.53333	−0.289585	−0.144792	+	0.989462i	$0.546251\pi$
$510$	0	0
$511$	−0.0359469	−0.00159020
$512$	0	0
$513$	−12.1457	−0.536248
$514$	0	0
$515$	−5.73047	−0.252515
$516$	0	0
$517$	−9.56332	−0.420594
$518$	0	0
$519$	−5.96332	−0.261761
$520$	0	0
$521$	−21.9798	−0.962951	−0.481476	−	0.876460i	$-0.659899\pi$
$521$	−21.9798	−0.962951	−0.481476	+	0.876460i	$0.659899\pi$
$522$	0	0
$523$	20.3613	0.890336	0.445168	−	0.895447i	$-0.353144\pi$
$523$	20.3613	0.890336	0.445168	+	0.895447i	$0.353144\pi$
$524$	0	0
$525$	36.0524	1.57346
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.7568	−0.989427
$530$	0	0
$531$	−22.5635	−0.979173
$532$	0	0
$533$	−11.9342	−0.516929
$534$	0	0
$535$	−2.41972	−0.104613
$536$	0	0
$537$	32.7538	1.41343
$538$	0	0
$539$	0.649154	0.0279610
$540$	0	0
$541$	18.4629	0.793781	0.396891	−	0.917866i	$-0.370089\pi$
$541$	18.4629	0.793781	0.396891	+	0.917866i	$0.370089\pi$
$542$	0	0
$543$	26.1410	1.12182
$544$	0	0
$545$	41.5236	1.77868
$546$	0	0
$547$	11.6164	0.496681	0.248341	−	0.968673i	$-0.420115\pi$
$547$	11.6164	0.496681	0.248341	+	0.968673i	$0.420115\pi$
$548$	0	0
$549$	13.5071	0.576469
$550$	0	0
$551$	54.0209	2.30137
$552$	0	0
$553$	−29.9422	−1.27327
$554$	0	0
$555$	25.3527	1.07616
$556$	0	0
$557$	−42.1765	−1.78708	−0.893539	−	0.448986i	$-0.851785\pi$
$557$	−42.1765	−1.78708	−0.893539	+	0.448986i	$0.851785\pi$
$558$	0	0
$559$	17.0123	0.719545
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.2212	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$563$	33.2212	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$564$	0	0
$565$	−30.3078	−1.27506
$566$	0	0
$567$	−30.3097	−1.27289
$568$	0	0
$569$	−43.6861	−1.83142	−0.915708	−	0.401845i	$-0.868369\pi$
$569$	−43.6861	−1.83142	−0.915708	+	0.401845i	$0.868369\pi$
$570$	0	0
$571$	−6.22856	−0.260657	−0.130328	−	0.991471i	$-0.541603\pi$
$571$	−6.22856	−0.260657	−0.130328	+	0.991471i	$0.541603\pi$
$572$	0	0
$573$	−40.4755	−1.69089
$574$	0	0
$575$	2.87757	0.120003
$576$	0	0
$577$	19.1580	0.797559	0.398779	−	0.917047i	$-0.369434\pi$
$577$	19.1580	0.797559	0.398779	+	0.917047i	$0.369434\pi$
$578$	0	0
$579$	3.16551	0.131554
$580$	0	0
$581$	−40.5827	−1.68365
$582$	0	0
$583$	−14.8098	−0.613359
$584$	0	0
$585$	11.1633	0.461546
$586$	0	0
$587$	11.2984	0.466334	0.233167	−	0.972437i	$-0.425091\pi$
$587$	11.2984	0.466334	0.233167	+	0.972437i	$0.425091\pi$
$588$	0	0
$589$	11.8956	0.490148
$590$	0	0
$591$	2.62042	0.107790
$592$	0	0
$593$	−9.01808	−0.370328	−0.185164	−	0.982708i	$-0.559282\pi$
$593$	−9.01808	−0.370328	−0.185164	+	0.982708i	$0.559282\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−40.8687	−1.67265
$598$	0	0
$599$	3.04853	0.124560	0.0622799	−	0.998059i	$-0.480163\pi$
$599$	3.04853	0.124560	0.0622799	+	0.998059i	$0.480163\pi$
$600$	0	0
$601$	−32.2052	−1.31368	−0.656839	−	0.754030i	$-0.728107\pi$
$601$	−32.2052	−1.31368	−0.656839	+	0.754030i	$0.728107\pi$
$602$	0	0
$603$	−9.85113	−0.401169
$604$	0	0
$605$	−32.5128	−1.32183
$606$	0	0
$607$	−10.2778	−0.417161	−0.208581	−	0.978005i	$-0.566884\pi$
$607$	−10.2778	−0.417161	−0.208581	+	0.978005i	$0.566884\pi$
$608$	0	0
$609$	60.6467	2.45753
$610$	0	0
$611$	−15.1944	−0.614701
$612$	0	0
$613$	11.9951	0.484479	0.242239	−	0.970216i	$-0.422118\pi$
$613$	11.9951	0.484479	0.242239	+	0.970216i	$0.422118\pi$
$614$	0	0
$615$	−52.2524	−2.10702
$616$	0	0
$617$	−0.362520	−0.0145945	−0.00729725	−	0.999973i	$-0.502323\pi$
$617$	−0.362520	−0.0145945	−0.00729725	+	0.999973i	$0.502323\pi$
$618$	0	0
$619$	−41.7327	−1.67738	−0.838689	−	0.544610i	$-0.816678\pi$
$619$	−41.7327	−1.67738	−0.838689	+	0.544610i	$0.816678\pi$
$620$	0	0
$621$	−1.08833	−0.0436732
$622$	0	0
$623$	−29.8071	−1.19420
$624$	0	0
$625$	−20.1265	−0.805060
$626$	0	0
$627$	13.0584	0.521500
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−12.8624	−0.512044	−0.256022	−	0.966671i	$-0.582412\pi$
$631$	−12.8624	−0.512044	−0.256022	+	0.966671i	$0.582412\pi$
$632$	0	0
$633$	53.3486	2.12042
$634$	0	0
$635$	6.83099	0.271079
$636$	0	0
$637$	1.03139	0.0408652
$638$	0	0
$639$	−11.7003	−0.462858
$640$	0	0
$641$	−25.2130	−0.995855	−0.497928	−	0.867219i	$-0.665905\pi$
$641$	−25.2130	−0.995855	−0.497928	+	0.867219i	$0.665905\pi$
$642$	0	0
$643$	36.5926	1.44307	0.721535	−	0.692378i	$-0.243437\pi$
$643$	36.5926	1.44307	0.721535	+	0.692378i	$0.243437\pi$
$644$	0	0
$645$	74.4861	2.93289
$646$	0	0
$647$	29.8075	1.17185	0.585926	−	0.810364i	$-0.300731\pi$
$647$	29.8075	1.17185	0.585926	+	0.810364i	$0.300731\pi$
$648$	0	0
$649$	−11.8684	−0.465876
$650$	0	0
$651$	13.3546	0.523408
$652$	0	0
$653$	−17.6393	−0.690280	−0.345140	−	0.938551i	$-0.612169\pi$
$653$	−17.6393	−0.690280	−0.345140	+	0.938551i	$0.612169\pi$
$654$	0	0
$655$	72.6899	2.84023
$656$	0	0
$657$	−0.0262453	−0.00102392
$658$	0	0
$659$	39.1712	1.52589	0.762946	−	0.646462i	$-0.223752\pi$
$659$	39.1712	1.52589	0.762946	+	0.646462i	$0.223752\pi$
$660$	0	0
$661$	−15.8114	−0.614991	−0.307495	−	0.951550i	$-0.599491\pi$
$661$	−15.8114	−0.614991	−0.307495	+	0.951550i	$0.599491\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	49.9827	1.93825
$666$	0	0
$667$	4.84060	0.187429
$668$	0	0
$669$	−0.904554	−0.0349721
$670$	0	0
$671$	7.10475	0.274276
$672$	0	0
$673$	25.1676	0.970138	0.485069	−	0.874476i	$-0.338794\pi$
$673$	25.1676	0.970138	0.485069	+	0.874476i	$0.338794\pi$
$674$	0	0
$675$	−12.8779	−0.495671
$676$	0	0
$677$	−41.3622	−1.58968	−0.794840	−	0.606820i	$-0.792445\pi$
$677$	−41.3622	−1.58968	−0.794840	+	0.606820i	$0.792445\pi$
$678$	0	0
$679$	−23.4063	−0.898250
$680$	0	0
$681$	27.2216	1.04314
$682$	0	0
$683$	21.3418	0.816622	0.408311	−	0.912843i	$-0.366118\pi$
$683$	21.3418	0.816622	0.408311	+	0.912843i	$0.366118\pi$
$684$	0	0
$685$	9.68476	0.370036
$686$	0	0
$687$	28.9674	1.10517
$688$	0	0
$689$	−23.5301	−0.896427
$690$	0	0
$691$	27.2046	1.03491	0.517456	−	0.855710i	$-0.326879\pi$
$691$	27.2046	1.03491	0.517456	+	0.855710i	$0.326879\pi$
$692$	0	0
$693$	5.88934	0.223718
$694$	0	0
$695$	−27.1735	−1.03075
$696$	0	0
$697$	0	0
$698$	0	0
$699$	26.2536	0.993003
$700$	0	0
$701$	−36.8975	−1.39360	−0.696800	−	0.717265i	$-0.745394\pi$
$701$	−36.8975	−1.39360	−0.696800	+	0.717265i	$0.745394\pi$
$702$	0	0
$703$	18.9290	0.713922
$704$	0	0
$705$	−66.5267	−2.50554
$706$	0	0
$707$	−44.0445	−1.65646
$708$	0	0
$709$	35.5912	1.33666	0.668328	−	0.743867i	$-0.267010\pi$
$709$	35.5912	1.33666	0.668328	+	0.743867i	$0.267010\pi$
$710$	0	0
$711$	−21.8611	−0.819856
$712$	0	0
$713$	1.06591	0.0399188
$714$	0	0
$715$	5.87191	0.219597
$716$	0	0
$717$	−8.57011	−0.320057
$718$	0	0
$719$	−25.7252	−0.959390	−0.479695	−	0.877435i	$-0.659253\pi$
$719$	−25.7252	−0.959390	−0.479695	+	0.877435i	$0.659253\pi$
$720$	0	0
$721$	−4.80330	−0.178884
$722$	0	0
$723$	33.6388	1.25104
$724$	0	0
$725$	57.2774	2.12723
$726$	0	0
$727$	15.3016	0.567505	0.283753	−	0.958897i	$-0.408421\pi$
$727$	15.3016	0.567505	0.283753	+	0.958897i	$0.408421\pi$
$728$	0	0
$729$	−7.30344	−0.270498
$730$	0	0
$731$	0	0
$732$	0	0
$733$	37.9771	1.40272	0.701359	−	0.712809i	$-0.252577\pi$
$733$	37.9771	1.40272	0.701359	+	0.712809i	$0.252577\pi$
$734$	0	0
$735$	4.51580	0.166568
$736$	0	0
$737$	−5.18170	−0.190870
$738$	0	0
$739$	−35.1581	−1.29331	−0.646655	−	0.762783i	$-0.723833\pi$
$739$	−35.1581	−1.29331	−0.646655	+	0.762783i	$0.723833\pi$
$740$	0	0
$741$	20.7474	0.762175
$742$	0	0
$743$	42.9372	1.57521	0.787606	−	0.616179i	$-0.211320\pi$
$743$	42.9372	1.57521	0.787606	+	0.616179i	$0.211320\pi$
$744$	0	0
$745$	6.16997	0.226050
$746$	0	0
$747$	−29.6299	−1.08410
$748$	0	0
$749$	−2.02821	−0.0741093
$750$	0	0
$751$	−19.8421	−0.724048	−0.362024	−	0.932169i	$-0.617914\pi$
$751$	−19.8421	−0.724048	−0.362024	+	0.932169i	$0.617914\pi$
$752$	0	0
$753$	40.0874	1.46087
$754$	0	0
$755$	63.1654	2.29882
$756$	0	0
$757$	26.6089	0.967117	0.483558	−	0.875312i	$-0.339344\pi$
$757$	26.6089	0.967117	0.483558	+	0.875312i	$0.339344\pi$
$758$	0	0
$759$	1.17011	0.0424722
$760$	0	0
$761$	24.9243	0.903505	0.451753	−	0.892143i	$-0.350799\pi$
$761$	24.9243	0.903505	0.451753	+	0.892143i	$0.350799\pi$
$762$	0	0
$763$	34.8052	1.26003
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.8568	−0.680881
$768$	0	0
$769$	32.9062	1.18663	0.593314	−	0.804971i	$-0.297819\pi$
$769$	32.9062	1.18663	0.593314	+	0.804971i	$0.297819\pi$
$770$	0	0
$771$	7.90292	0.284617
$772$	0	0
$773$	26.6766	0.959492	0.479746	−	0.877407i	$-0.340729\pi$
$773$	26.6766	0.959492	0.479746	+	0.877407i	$0.340729\pi$
$774$	0	0
$775$	12.6127	0.453060
$776$	0	0
$777$	21.2507	0.762365
$778$	0	0
$779$	−39.0130	−1.39779
$780$	0	0
$781$	−6.15439	−0.220221
$782$	0	0
$783$	−21.6630	−0.774172
$784$	0	0
$785$	−52.0758	−1.85866
$786$	0	0
$787$	45.6968	1.62891	0.814457	−	0.580224i	$-0.197035\pi$
$787$	45.6968	1.62891	0.814457	+	0.580224i	$0.197035\pi$
$788$	0	0
$789$	7.17566	0.255460
$790$	0	0
$791$	−25.4041	−0.903266
$792$	0	0
$793$	11.2882	0.400856
$794$	0	0
$795$	−103.023	−3.65386
$796$	0	0
$797$	13.5700	0.480675	0.240338	−	0.970689i	$-0.422742\pi$
$797$	13.5700	0.480675	0.240338	+	0.970689i	$0.422742\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−21.7625	−0.768939
$802$	0	0
$803$	−0.0138050	−0.000487169 0
$804$	0	0
$805$	4.47875	0.157855
$806$	0	0
$807$	−24.4222	−0.859703
$808$	0	0
$809$	−35.7755	−1.25780	−0.628899	−	0.777487i	$-0.716494\pi$
$809$	−35.7755	−1.25780	−0.628899	+	0.777487i	$0.716494\pi$
$810$	0	0
$811$	−3.33812	−0.117217	−0.0586087	−	0.998281i	$-0.518666\pi$
$811$	−3.33812	−0.117217	−0.0586087	+	0.998281i	$0.518666\pi$
$812$	0	0
$813$	−70.0187	−2.45566
$814$	0	0
$815$	30.1248	1.05523
$816$	0	0
$817$	55.6133	1.94566
$818$	0	0
$819$	9.35713	0.326965
$820$	0	0
$821$	13.5037	0.471281	0.235641	−	0.971840i	$-0.424281\pi$
$821$	13.5037	0.471281	0.235641	+	0.971840i	$0.424281\pi$
$822$	0	0
$823$	24.2284	0.844548	0.422274	−	0.906468i	$-0.361232\pi$
$823$	24.2284	0.844548	0.422274	+	0.906468i	$0.361232\pi$
$824$	0	0
$825$	13.8455	0.482040
$826$	0	0
$827$	−48.5460	−1.68811	−0.844055	−	0.536257i	$-0.819838\pi$
$827$	−48.5460	−1.68811	−0.844055	+	0.536257i	$0.819838\pi$
$828$	0	0
$829$	−12.5026	−0.434235	−0.217117	−	0.976145i	$-0.569665\pi$
$829$	−12.5026	−0.434235	−0.217117	+	0.976145i	$0.569665\pi$
$830$	0	0
$831$	28.3978	0.985110
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−23.6778	−0.819405
$836$	0	0
$837$	−4.77026	−0.164884
$838$	0	0
$839$	27.6954	0.956150	0.478075	−	0.878319i	$-0.341335\pi$
$839$	27.6954	0.956150	0.478075	+	0.878319i	$0.341335\pi$
$840$	0	0
$841$	67.3509	2.32245
$842$	0	0
$843$	35.7159	1.23012
$844$	0	0
$845$	−33.4625	−1.15114
$846$	0	0
$847$	−27.2523	−0.936401
$848$	0	0
$849$	15.8314	0.543332
$850$	0	0
$851$	1.69615	0.0581434
$852$	0	0
$853$	−38.4115	−1.31519	−0.657593	−	0.753373i	$-0.728425\pi$
$853$	−38.4115	−1.31519	−0.657593	+	0.753373i	$0.728425\pi$
$854$	0	0
$855$	36.4929	1.24803
$856$	0	0
$857$	−18.8117	−0.642595	−0.321298	−	0.946978i	$-0.604119\pi$
$857$	−18.8117	−0.642595	−0.321298	+	0.946978i	$0.604119\pi$
$858$	0	0
$859$	−13.5810	−0.463378	−0.231689	−	0.972790i	$-0.574425\pi$
$859$	−13.5810	−0.463378	−0.231689	+	0.972790i	$0.574425\pi$
$860$	0	0
$861$	−43.7981	−1.49264
$862$	0	0
$863$	−47.9451	−1.63207	−0.816035	−	0.578002i	$-0.803833\pi$
$863$	−47.9451	−1.63207	−0.816035	+	0.578002i	$0.803833\pi$
$864$	0	0
$865$	−8.76586	−0.298048
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.4990	−0.390076
$870$	0	0
$871$	−8.23281	−0.278958
$872$	0	0
$873$	−17.0892	−0.578381
$874$	0	0
$875$	7.58529	0.256430
$876$	0	0
$877$	−31.9005	−1.07720	−0.538602	−	0.842560i	$-0.681047\pi$
$877$	−31.9005	−1.07720	−0.538602	+	0.842560i	$0.681047\pi$
$878$	0	0
$879$	14.3678	0.484613
$880$	0	0
$881$	15.7524	0.530712	0.265356	−	0.964151i	$-0.414511\pi$
$881$	15.7524	0.530712	0.265356	+	0.964151i	$0.414511\pi$
$882$	0	0
$883$	−26.4467	−0.890001	−0.445000	−	0.895530i	$-0.646796\pi$
$883$	−26.4467	−0.890001	−0.445000	+	0.895530i	$0.646796\pi$
$884$	0	0
$885$	−82.5620	−2.77529
$886$	0	0
$887$	36.8779	1.23824	0.619119	−	0.785297i	$-0.287490\pi$
$887$	36.8779	1.23824	0.619119	+	0.785297i	$0.287490\pi$
$888$	0	0
$889$	5.72575	0.192036
$890$	0	0
$891$	−11.6401	−0.389958
$892$	0	0
$893$	−49.6706	−1.66216
$894$	0	0
$895$	48.1469	1.60937
$896$	0	0
$897$	1.85909	0.0620733
$898$	0	0
$899$	21.2168	0.707619
$900$	0	0
$901$	0	0
$902$	0	0
$903$	62.4345	2.07769
$904$	0	0
$905$	38.4263	1.27733
$906$	0	0
$907$	−32.2869	−1.07207	−0.536034	−	0.844196i	$-0.680078\pi$
$907$	−32.2869	−1.07207	−0.536034	+	0.844196i	$0.680078\pi$
$908$	0	0
$909$	−32.1574	−1.06659
$910$	0	0
$911$	3.33242	0.110408	0.0552039	−	0.998475i	$-0.482419\pi$
$911$	3.33242	0.110408	0.0552039	+	0.998475i	$0.482419\pi$
$912$	0	0
$913$	−15.5853	−0.515799
$914$	0	0
$915$	49.4238	1.63390
$916$	0	0
$917$	60.9289	2.01205
$918$	0	0
$919$	26.5711	0.876501	0.438250	−	0.898853i	$-0.355598\pi$
$919$	26.5711	0.876501	0.438250	+	0.898853i	$0.355598\pi$
$920$	0	0
$921$	68.6747	2.26291
$922$	0	0
$923$	−9.77823	−0.321855
$924$	0	0
$925$	20.0701	0.659901
$926$	0	0
$927$	−3.50694	−0.115183
$928$	0	0
$929$	35.6230	1.16875	0.584376	−	0.811483i	$-0.301339\pi$
$929$	35.6230	1.16875	0.584376	+	0.811483i	$0.301339\pi$
$930$	0	0
$931$	3.37162	0.110500
$932$	0	0
$933$	−34.0315	−1.11414
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.7918	1.72463	0.862317	−	0.506370i	$-0.169013\pi$
$937$	52.7918	1.72463	0.862317	+	0.506370i	$0.169013\pi$
$938$	0	0
$939$	5.80085	0.189304
$940$	0	0
$941$	18.8575	0.614736	0.307368	−	0.951591i	$-0.400552\pi$
$941$	18.8575	0.614736	0.307368	+	0.951591i	$0.400552\pi$
$942$	0	0
$943$	−3.49580	−0.113839
$944$	0	0
$945$	−20.0436	−0.652019
$946$	0	0
$947$	41.3325	1.34312	0.671562	−	0.740948i	$-0.265624\pi$
$947$	41.3325	1.34312	0.671562	+	0.740948i	$0.265624\pi$
$948$	0	0
$949$	−0.0219337	−0.000712000 0
$950$	0	0
$951$	−31.6351	−1.02584
$952$	0	0
$953$	12.3861	0.401227	0.200613	−	0.979671i	$-0.435707\pi$
$953$	12.3861	0.401227	0.200613	+	0.979671i	$0.435707\pi$
$954$	0	0
$955$	−59.4975	−1.92529
$956$	0	0
$957$	23.2907	0.752881
$958$	0	0
$959$	8.11780	0.262137
$960$	0	0
$961$	−26.3280	−0.849290
$962$	0	0
$963$	−1.48082	−0.0477188
$964$	0	0
$965$	4.65318	0.149791
$966$	0	0
$967$	1.76606	0.0567926	0.0283963	−	0.999597i	$-0.490960\pi$
$967$	1.76606	0.0567926	0.0283963	+	0.999597i	$0.490960\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.12081	0.260609	0.130305	−	0.991474i	$-0.458405\pi$
$971$	8.12081	0.260609	0.130305	+	0.991474i	$0.458405\pi$
$972$	0	0
$973$	−22.7769	−0.730193
$974$	0	0
$975$	21.9981	0.704503
$976$	0	0
$977$	26.7629	0.856221	0.428110	−	0.903726i	$-0.359179\pi$
$977$	26.7629	0.856221	0.428110	+	0.903726i	$0.359179\pi$
$978$	0	0
$979$	−11.4471	−0.365850
$980$	0	0
$981$	25.4117	0.811332
$982$	0	0
$983$	14.9517	0.476885	0.238442	−	0.971157i	$-0.423363\pi$
$983$	14.9517	0.476885	0.238442	+	0.971157i	$0.423363\pi$
$984$	0	0
$985$	3.85192	0.122733
$986$	0	0
$987$	−55.7629	−1.77495
$988$	0	0
$989$	4.98329	0.158459
$990$	0	0
$991$	10.1028	0.320925	0.160463	−	0.987042i	$-0.448701\pi$
$991$	10.1028	0.320925	0.160463	+	0.987042i	$0.448701\pi$
$992$	0	0
$993$	45.7030	1.45034
$994$	0	0
$995$	−60.0756	−1.90452
$996$	0	0
$997$	−47.0718	−1.49078	−0.745389	−	0.666630i	$-0.767736\pi$
$997$	−47.0718	−1.49078	−0.745389	+	0.666630i	$0.767736\pi$
$998$	0	0
$999$	−7.59075	−0.240161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bz.1.17		20
4.3	odd	2		9248.2.a.by.1.4		20
17.10	odd	16		544.2.bb.f.321.4	yes	20
17.12	odd	16		544.2.bb.f.161.4	yes	20
17.16	even	2	inner	9248.2.a.bz.1.4		20
68.27	even	16		544.2.bb.e.321.2	yes	20
68.63	even	16		544.2.bb.e.161.2	✓	20
68.67	odd	2		9248.2.a.by.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.bb.e.161.2	✓	20	68.63	even	16
544.2.bb.e.321.2	yes	20	68.27	even	16
544.2.bb.f.161.4	yes	20	17.12	odd	16
544.2.bb.f.321.4	yes	20	17.10	odd	16
9248.2.a.by.1.4		20	4.3	odd	2
9248.2.a.by.1.17		20	68.67	odd	2
9248.2.a.bz.1.4		20	17.16	even	2	inner
9248.2.a.bz.1.17		20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23930 q^{3} +3.29169 q^{5} +2.75910 q^{7} +2.01445 q^{9} +1.05960 q^{11} +1.68352 q^{13} +7.37106 q^{15} +5.50343 q^{19} +6.17845 q^{21} +0.493141 q^{23} +5.83519 q^{25} -2.20694 q^{27} +9.81585 q^{29} +2.16148 q^{31} +2.37276 q^{33} +9.08209 q^{35} +3.43949 q^{37} +3.76990 q^{39} -7.08885 q^{41} +10.1052 q^{43} +6.63093 q^{45} -9.02539 q^{47} +0.612639 q^{49} -13.9767 q^{53} +3.48788 q^{55} +12.3238 q^{57} -11.2008 q^{59} +6.70511 q^{61} +5.55807 q^{63} +5.54162 q^{65} -4.89023 q^{67} +1.10429 q^{69} -5.80820 q^{71} -0.0130285 q^{73} +13.0667 q^{75} +2.92355 q^{77} -10.8521 q^{79} -10.9853 q^{81} -14.7087 q^{83} +21.9806 q^{87} -10.8032 q^{89} +4.64500 q^{91} +4.84020 q^{93} +18.1156 q^{95} -8.48329 q^{97} +2.13452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) \) 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * q^89 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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