LMFDB - Embedded newform 9280.2.a.bo.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,3,0,-3,0,2,0,-6,0,1,0,-1,0,5,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47283 q^{3} +1.00000 q^{5} +1.11491 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.47283 q^{13} +1.47283 q^{15} -4.06058 q^{17} -5.51396 q^{19} +1.64207 q^{21} +1.24302 q^{23} +1.00000 q^{25} -5.64207 q^{27} +1.00000 q^{29} +1.83076 q^{31} +3.28415 q^{33} +1.11491 q^{35} -1.05433 q^{37} -2.16924 q^{39} -4.22982 q^{41} -7.83076 q^{43} -0.830760 q^{45} +2.71585 q^{47} -5.75698 q^{49} -5.98055 q^{51} -9.34472 q^{53} +2.22982 q^{55} -8.12115 q^{57} -0.904539 q^{59} -13.2166 q^{61} -0.926221 q^{63} -1.47283 q^{65} -1.43171 q^{67} +1.83076 q^{69} +6.56829 q^{71} +11.7221 q^{73} +1.47283 q^{75} +2.48604 q^{77} +8.98680 q^{79} -5.81756 q^{81} -6.94567 q^{83} -4.06058 q^{85} +1.47283 q^{87} -3.05433 q^{89} -1.64207 q^{91} +2.69641 q^{93} -5.51396 q^{95} -12.6483 q^{97} -1.85244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} + q^{13} - q^{15} + 5 q^{17} - 2 q^{19} + 4 q^{21} + 11 q^{23} + 3 q^{25} - 16 q^{27} + 3 q^{29} + q^{31} + 8 q^{33} - 3 q^{35} - 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 + q^13 - q^15 + 5 * q^17 - 2 * q^19 + 4 * q^21 + 11 * q^23 + 3 * q^25 - 16 * q^27 + 3 * q^29 + q^31 + 8 * q^33 - 3 * q^35 - 14 * q^37 - 11 * q^39 - 19 * q^43 + 2 * q^45 + 10 * q^47 - 10 * q^49 - 26 * q^51 - 9 * q^53 - 6 * q^55 + 10 * q^57 - q^59 - 7 * q^61 + q^65 - 8 * q^67 + q^69 + 16 * q^71 + 9 * q^73 - q^75 + 22 * q^77 + 7 * q^79 + 7 * q^81 - 10 * q^83 + 5 * q^85 - q^87 - 20 * q^89 - 4 * q^91 + 18 * q^93 - 2 * q^95 - 9 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.47283	0.850341	0.425171	−	0.905113i	$-0.360214\pi$
$3$	1.47283	0.850341	0.425171	+	0.905113i	$0.360214\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.11491	0.421395	0.210698	−	0.977551i	$-0.432426\pi$
$7$	1.11491	0.421395	0.210698	+	0.977551i	$0.432426\pi$
$8$	0	0
$9$	−0.830760	−0.276920
$10$	0	0
$11$	2.22982	0.672315	0.336157	−	0.941806i	$-0.390873\pi$
$11$	2.22982	0.672315	0.336157	+	0.941806i	$0.390873\pi$
$12$	0	0
$13$	−1.47283	−0.408491	−0.204245	−	0.978920i	$-0.565474\pi$
$13$	−1.47283	−0.408491	−0.204245	+	0.978920i	$0.565474\pi$
$14$	0	0
$15$	1.47283	0.380284
$16$	0	0
$17$	−4.06058	−0.984834	−0.492417	−	0.870359i	$-0.663887\pi$
$17$	−4.06058	−0.984834	−0.492417	+	0.870359i	$0.663887\pi$
$18$	0	0
$19$	−5.51396	−1.26499	−0.632495	−	0.774565i	$-0.717969\pi$
$19$	−5.51396	−1.26499	−0.632495	+	0.774565i	$0.717969\pi$
$20$	0	0
$21$	1.64207	0.358330
$22$	0	0
$23$	1.24302	0.259187	0.129594	−	0.991567i	$-0.458633\pi$
$23$	1.24302	0.259187	0.129594	+	0.991567i	$0.458633\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.64207	−1.08582
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.83076	0.328814	0.164407	−	0.986393i	$-0.447429\pi$
$31$	1.83076	0.328814	0.164407	+	0.986393i	$0.447429\pi$
$32$	0	0
$33$	3.28415	0.571697
$34$	0	0
$35$	1.11491	0.188454
$36$	0	0
$37$	−1.05433	−0.173331	−0.0866656	−	0.996237i	$-0.527621\pi$
$37$	−1.05433	−0.173331	−0.0866656	+	0.996237i	$0.527621\pi$
$38$	0	0
$39$	−2.16924	−0.347356
$40$	0	0
$41$	−4.22982	−0.660586	−0.330293	−	0.943878i	$-0.607148\pi$
$41$	−4.22982	−0.660586	−0.330293	+	0.943878i	$0.607148\pi$
$42$	0	0
$43$	−7.83076	−1.19418	−0.597090	−	0.802174i	$-0.703676\pi$
$43$	−7.83076	−1.19418	−0.597090	+	0.802174i	$0.703676\pi$
$44$	0	0
$45$	−0.830760	−0.123842
$46$	0	0
$47$	2.71585	0.396148	0.198074	−	0.980187i	$-0.436531\pi$
$47$	2.71585	0.396148	0.198074	+	0.980187i	$0.436531\pi$
$48$	0	0
$49$	−5.75698	−0.822426
$50$	0	0
$51$	−5.98055	−0.837445
$52$	0	0
$53$	−9.34472	−1.28360	−0.641798	−	0.766874i	$-0.721811\pi$
$53$	−9.34472	−1.28360	−0.641798	+	0.766874i	$0.721811\pi$
$54$	0	0
$55$	2.22982	0.300668
$56$	0	0
$57$	−8.12115	−1.07567
$58$	0	0
$59$	−0.904539	−0.117761	−0.0588805	−	0.998265i	$-0.518753\pi$
$59$	−0.904539	−0.117761	−0.0588805	+	0.998265i	$0.518753\pi$
$60$	0	0
$61$	−13.2166	−1.69221	−0.846107	−	0.533013i	$-0.821060\pi$
$61$	−13.2166	−1.69221	−0.846107	+	0.533013i	$0.821060\pi$
$62$	0	0
$63$	−0.926221	−0.116693
$64$	0	0
$65$	−1.47283	−0.182683
$66$	0	0
$67$	−1.43171	−0.174911	−0.0874553	−	0.996168i	$-0.527874\pi$
$67$	−1.43171	−0.174911	−0.0874553	+	0.996168i	$0.527874\pi$
$68$	0	0
$69$	1.83076	0.220398
$70$	0	0
$71$	6.56829	0.779513	0.389757	−	0.920918i	$-0.372559\pi$
$71$	6.56829	0.779513	0.389757	+	0.920918i	$0.372559\pi$
$72$	0	0
$73$	11.7221	1.37197	0.685984	−	0.727617i	$-0.259372\pi$
$73$	11.7221	1.37197	0.685984	+	0.727617i	$0.259372\pi$
$74$	0	0
$75$	1.47283	0.170068
$76$	0	0
$77$	2.48604	0.283310
$78$	0	0
$79$	8.98680	1.01109	0.505547	−	0.862799i	$-0.331291\pi$
$79$	8.98680	1.01109	0.505547	+	0.862799i	$0.331291\pi$
$80$	0	0
$81$	−5.81756	−0.646395
$82$	0	0
$83$	−6.94567	−0.762386	−0.381193	−	0.924495i	$-0.624487\pi$
$83$	−6.94567	−0.762386	−0.381193	+	0.924495i	$0.624487\pi$
$84$	0	0
$85$	−4.06058	−0.440431
$86$	0	0
$87$	1.47283	0.157904
$88$	0	0
$89$	−3.05433	−0.323759	−0.161879	−	0.986811i	$-0.551756\pi$
$89$	−3.05433	−0.323759	−0.161879	+	0.986811i	$0.551756\pi$
$90$	0	0
$91$	−1.64207	−0.172136
$92$	0	0
$93$	2.69641	0.279604
$94$	0	0
$95$	−5.51396	−0.565721
$96$	0	0
$97$	−12.6483	−1.28424	−0.642121	−	0.766603i	$-0.721945\pi$
$97$	−12.6483	−1.28424	−0.642121	+	0.766603i	$0.721945\pi$
$98$	0	0
$99$	−1.85244	−0.186177
$100$	0	0
$101$	4.29735	0.427602	0.213801	−	0.976877i	$-0.431416\pi$
$101$	4.29735	0.427602	0.213801	+	0.976877i	$0.431416\pi$
$102$	0	0
$103$	16.9193	1.66710	0.833552	−	0.552441i	$-0.186303\pi$
$103$	16.9193	1.66710	0.833552	+	0.552441i	$0.186303\pi$
$104$	0	0
$105$	1.64207	0.160250
$106$	0	0
$107$	−19.4053	−1.87598	−0.937990	−	0.346661i	$-0.887315\pi$
$107$	−19.4053	−1.87598	−0.937990	+	0.346661i	$0.887315\pi$
$108$	0	0
$109$	2.79811	0.268010	0.134005	−	0.990981i	$-0.457216\pi$
$109$	2.79811	0.268010	0.134005	+	0.990981i	$0.457216\pi$
$110$	0	0
$111$	−1.55286	−0.147391
$112$	0	0
$113$	7.59398	0.714382	0.357191	−	0.934031i	$-0.383735\pi$
$113$	7.59398	0.714382	0.357191	+	0.934031i	$0.383735\pi$
$114$	0	0
$115$	1.24302	0.115912
$116$	0	0
$117$	1.22357	0.113119
$118$	0	0
$119$	−4.52717	−0.415005
$120$	0	0
$121$	−6.02792	−0.547993
$122$	0	0
$123$	−6.22982	−0.561724
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.74378	−0.864620	−0.432310	−	0.901725i	$-0.642301\pi$
$127$	−9.74378	−0.864620	−0.432310	+	0.901725i	$0.642301\pi$
$128$	0	0
$129$	−11.5334	−1.01546
$130$	0	0
$131$	−6.60719	−0.577273	−0.288636	−	0.957439i	$-0.593202\pi$
$131$	−6.60719	−0.577273	−0.288636	+	0.957439i	$0.593202\pi$
$132$	0	0
$133$	−6.14756	−0.533061
$134$	0	0
$135$	−5.64207	−0.485592
$136$	0	0
$137$	5.70265	0.487210	0.243605	−	0.969875i	$-0.421670\pi$
$137$	5.70265	0.487210	0.243605	+	0.969875i	$0.421670\pi$
$138$	0	0
$139$	11.0885	0.940514	0.470257	−	0.882529i	$-0.344161\pi$
$139$	11.0885	0.940514	0.470257	+	0.882529i	$0.344161\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−3.28415	−0.274634
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−8.47908	−0.699342
$148$	0	0
$149$	14.0823	1.15366	0.576832	−	0.816863i	$-0.304289\pi$
$149$	14.0823	1.15366	0.576832	+	0.816863i	$0.304289\pi$
$150$	0	0
$151$	−17.2966	−1.40758	−0.703790	−	0.710408i	$-0.748510\pi$
$151$	−17.2966	−1.40758	−0.703790	+	0.710408i	$0.748510\pi$
$152$	0	0
$153$	3.37336	0.272720
$154$	0	0
$155$	1.83076	0.147050
$156$	0	0
$157$	−18.3121	−1.46146	−0.730731	−	0.682665i	$-0.760821\pi$
$157$	−18.3121	−1.46146	−0.730731	+	0.682665i	$0.760821\pi$
$158$	0	0
$159$	−13.7632	−1.09149
$160$	0	0
$161$	1.38585	0.109220
$162$	0	0
$163$	−5.28415	−0.413886	−0.206943	−	0.978353i	$-0.566352\pi$
$163$	−5.28415	−0.413886	−0.206943	+	0.978353i	$0.566352\pi$
$164$	0	0
$165$	3.28415	0.255671
$166$	0	0
$167$	−18.6678	−1.44455	−0.722277	−	0.691603i	$-0.756905\pi$
$167$	−18.6678	−1.44455	−0.722277	+	0.691603i	$0.756905\pi$
$168$	0	0
$169$	−10.8308	−0.833135
$170$	0	0
$171$	4.58078	0.350301
$172$	0	0
$173$	9.93246	0.755151	0.377576	−	0.925979i	$-0.376758\pi$
$173$	9.93246	0.755151	0.377576	+	0.925979i	$0.376758\pi$
$174$	0	0
$175$	1.11491	0.0842791
$176$	0	0
$177$	−1.33224	−0.100137
$178$	0	0
$179$	0.904539	0.0676084	0.0338042	−	0.999428i	$-0.489238\pi$
$179$	0.904539	0.0676084	0.0338042	+	0.999428i	$0.489238\pi$
$180$	0	0
$181$	1.66152	0.123500	0.0617499	−	0.998092i	$-0.480332\pi$
$181$	1.66152	0.123500	0.0617499	+	0.998092i	$0.480332\pi$
$182$	0	0
$183$	−19.4659	−1.43896
$184$	0	0
$185$	−1.05433	−0.0775160
$186$	0	0
$187$	−9.05433	−0.662118
$188$	0	0
$189$	−6.29039	−0.457559
$190$	0	0
$191$	0.737534	0.0533661	0.0266831	−	0.999644i	$-0.491506\pi$
$191$	0.737534	0.0533661	0.0266831	+	0.999644i	$0.491506\pi$
$192$	0	0
$193$	2.75698	0.198452	0.0992259	−	0.995065i	$-0.468363\pi$
$193$	2.75698	0.198452	0.0992259	+	0.995065i	$0.468363\pi$
$194$	0	0
$195$	−2.16924	−0.155342
$196$	0	0
$197$	1.22357	0.0871759	0.0435879	−	0.999050i	$-0.486121\pi$
$197$	1.22357	0.0871759	0.0435879	+	0.999050i	$0.486121\pi$
$198$	0	0
$199$	−8.12115	−0.575693	−0.287847	−	0.957677i	$-0.592939\pi$
$199$	−8.12115	−0.575693	−0.287847	+	0.957677i	$0.592939\pi$
$200$	0	0
$201$	−2.10866	−0.148734
$202$	0	0
$203$	1.11491	0.0782512
$204$	0	0
$205$	−4.22982	−0.295423
$206$	0	0
$207$	−1.03265	−0.0717742
$208$	0	0
$209$	−12.2951	−0.850471
$210$	0	0
$211$	−1.31055	−0.0902223	−0.0451112	−	0.998982i	$-0.514364\pi$
$211$	−1.31055	−0.0902223	−0.0451112	+	0.998982i	$0.514364\pi$
$212$	0	0
$213$	9.67401	0.662852
$214$	0	0
$215$	−7.83076	−0.534053
$216$	0	0
$217$	2.04113	0.138561
$218$	0	0
$219$	17.2647	1.16664
$220$	0	0
$221$	5.98055	0.402296
$222$	0	0
$223$	−8.86565	−0.593688	−0.296844	−	0.954926i	$-0.595934\pi$
$223$	−8.86565	−0.593688	−0.296844	+	0.954926i	$0.595934\pi$
$224$	0	0
$225$	−0.830760	−0.0553840
$226$	0	0
$227$	−2.82452	−0.187470	−0.0937349	−	0.995597i	$-0.529881\pi$
$227$	−2.82452	−0.187470	−0.0937349	+	0.995597i	$0.529881\pi$
$228$	0	0
$229$	−10.5202	−0.695195	−0.347597	−	0.937644i	$-0.613002\pi$
$229$	−10.5202	−0.695195	−0.347597	+	0.937644i	$0.613002\pi$
$230$	0	0
$231$	3.66152	0.240910
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	2.71585	0.177163
$236$	0	0
$237$	13.2361	0.859774
$238$	0	0
$239$	1.17548	0.0760357	0.0380178	−	0.999277i	$-0.487896\pi$
$239$	1.17548	0.0760357	0.0380178	+	0.999277i	$0.487896\pi$
$240$	0	0
$241$	17.2166	1.10902	0.554510	−	0.832177i	$-0.312906\pi$
$241$	17.2166	1.10902	0.554510	+	0.832177i	$0.312906\pi$
$242$	0	0
$243$	8.35793	0.536161
$244$	0	0
$245$	−5.75698	−0.367800
$246$	0	0
$247$	8.12115	0.516736
$248$	0	0
$249$	−10.2298	−0.648288
$250$	0	0
$251$	9.05433	0.571504	0.285752	−	0.958304i	$-0.407757\pi$
$251$	9.05433	0.571504	0.285752	+	0.958304i	$0.407757\pi$
$252$	0	0
$253$	2.77170	0.174255
$254$	0	0
$255$	−5.98055	−0.374517
$256$	0	0
$257$	−1.87885	−0.117199	−0.0585997	−	0.998282i	$-0.518664\pi$
$257$	−1.87885	−0.117199	−0.0585997	+	0.998282i	$0.518664\pi$
$258$	0	0
$259$	−1.17548	−0.0730410
$260$	0	0
$261$	−0.830760	−0.0514228
$262$	0	0
$263$	13.7438	0.847478	0.423739	−	0.905784i	$-0.360717\pi$
$263$	13.7438	0.847478	0.423739	+	0.905784i	$0.360717\pi$
$264$	0	0
$265$	−9.34472	−0.574042
$266$	0	0
$267$	−4.49852	−0.275305
$268$	0	0
$269$	16.9673	1.03452	0.517259	−	0.855829i	$-0.326952\pi$
$269$	16.9673	1.03452	0.517259	+	0.855829i	$0.326952\pi$
$270$	0	0
$271$	10.8106	0.656697	0.328348	−	0.944557i	$-0.393508\pi$
$271$	10.8106	0.656697	0.328348	+	0.944557i	$0.393508\pi$
$272$	0	0
$273$	−2.41850	−0.146374
$274$	0	0
$275$	2.22982	0.134463
$276$	0	0
$277$	21.4876	1.29106	0.645531	−	0.763734i	$-0.276636\pi$
$277$	21.4876	1.29106	0.645531	+	0.763734i	$0.276636\pi$
$278$	0	0
$279$	−1.52092	−0.0910553
$280$	0	0
$281$	14.1817	0.846011	0.423005	−	0.906127i	$-0.360975\pi$
$281$	14.1817	0.846011	0.423005	+	0.906127i	$0.360975\pi$
$282$	0	0
$283$	6.94567	0.412877	0.206439	−	0.978460i	$-0.433813\pi$
$283$	6.94567	0.412877	0.206439	+	0.978460i	$0.433813\pi$
$284$	0	0
$285$	−8.12115	−0.481055
$286$	0	0
$287$	−4.71585	−0.278368
$288$	0	0
$289$	−0.511728	−0.0301016
$290$	0	0
$291$	−18.6289	−1.09204
$292$	0	0
$293$	−12.3774	−0.723094	−0.361547	−	0.932354i	$-0.617751\pi$
$293$	−12.3774	−0.723094	−0.361547	+	0.932354i	$0.617751\pi$
$294$	0	0
$295$	−0.904539	−0.0526643
$296$	0	0
$297$	−12.5808	−0.730011
$298$	0	0
$299$	−1.83076	−0.105876
$300$	0	0
$301$	−8.73057	−0.503222
$302$	0	0
$303$	6.32928	0.363608
$304$	0	0
$305$	−13.2166	−0.756781
$306$	0	0
$307$	4.14756	0.236714	0.118357	−	0.992971i	$-0.462237\pi$
$307$	4.14756	0.236714	0.118357	+	0.992971i	$0.462237\pi$
$308$	0	0
$309$	24.9193	1.41761
$310$	0	0
$311$	−7.55509	−0.428410	−0.214205	−	0.976789i	$-0.568716\pi$
$311$	−7.55509	−0.428410	−0.214205	+	0.976789i	$0.568716\pi$
$312$	0	0
$313$	22.9193	1.29547	0.647737	−	0.761864i	$-0.275716\pi$
$313$	22.9193	1.29547	0.647737	+	0.761864i	$0.275716\pi$
$314$	0	0
$315$	−0.926221	−0.0521866
$316$	0	0
$317$	8.12115	0.456129	0.228065	−	0.973646i	$-0.426760\pi$
$317$	8.12115	0.456129	0.228065	+	0.973646i	$0.426760\pi$
$318$	0	0
$319$	2.22982	0.124846
$320$	0	0
$321$	−28.5808	−1.59522
$322$	0	0
$323$	22.3899	1.24581
$324$	0	0
$325$	−1.47283	−0.0816981
$326$	0	0
$327$	4.12115	0.227900
$328$	0	0
$329$	3.02792	0.166935
$330$	0	0
$331$	−21.5529	−1.18465	−0.592326	−	0.805698i	$-0.701790\pi$
$331$	−21.5529	−1.18465	−0.592326	+	0.805698i	$0.701790\pi$
$332$	0	0
$333$	0.875897	0.0479989
$334$	0	0
$335$	−1.43171	−0.0782224
$336$	0	0
$337$	−2.75002	−0.149803	−0.0749016	−	0.997191i	$-0.523864\pi$
$337$	−2.75002	−0.149803	−0.0749016	+	0.997191i	$0.523864\pi$
$338$	0	0
$339$	11.1847	0.607468
$340$	0	0
$341$	4.08226	0.221067
$342$	0	0
$343$	−14.2229	−0.767962
$344$	0	0
$345$	1.83076	0.0985648
$346$	0	0
$347$	−8.59470	−0.461388	−0.230694	−	0.973026i	$-0.574100\pi$
$347$	−8.59470	−0.461388	−0.230694	+	0.973026i	$0.574100\pi$
$348$	0	0
$349$	−2.42074	−0.129579	−0.0647895	−	0.997899i	$-0.520638\pi$
$349$	−2.42074	−0.129579	−0.0647895	+	0.997899i	$0.520638\pi$
$350$	0	0
$351$	8.30984	0.443546
$352$	0	0
$353$	−10.8804	−0.579103	−0.289552	−	0.957162i	$-0.593506\pi$
$353$	−10.8804	−0.579103	−0.289552	+	0.957162i	$0.593506\pi$
$354$	0	0
$355$	6.56829	0.348609
$356$	0	0
$357$	−6.66776	−0.352895
$358$	0	0
$359$	−4.06754	−0.214676	−0.107338	−	0.994223i	$-0.534233\pi$
$359$	−4.06754	−0.214676	−0.107338	+	0.994223i	$0.534233\pi$
$360$	0	0
$361$	11.4038	0.600199
$362$	0	0
$363$	−8.87813	−0.465981
$364$	0	0
$365$	11.7221	0.613563
$366$	0	0
$367$	−20.0125	−1.04464	−0.522322	−	0.852749i	$-0.674934\pi$
$367$	−20.0125	−1.04464	−0.522322	+	0.852749i	$0.674934\pi$
$368$	0	0
$369$	3.51396	0.182930
$370$	0	0
$371$	−10.4185	−0.540902
$372$	0	0
$373$	−18.4813	−0.956926	−0.478463	−	0.878108i	$-0.658806\pi$
$373$	−18.4813	−0.956926	−0.478463	+	0.878108i	$0.658806\pi$
$374$	0	0
$375$	1.47283	0.0760568
$376$	0	0
$377$	−1.47283	−0.0758548
$378$	0	0
$379$	−11.4442	−0.587849	−0.293924	−	0.955829i	$-0.594961\pi$
$379$	−11.4442	−0.587849	−0.293924	+	0.955829i	$0.594961\pi$
$380$	0	0
$381$	−14.3510	−0.735222
$382$	0	0
$383$	−22.6678	−1.15827	−0.579134	−	0.815232i	$-0.696609\pi$
$383$	−22.6678	−1.15827	−0.579134	+	0.815232i	$0.696609\pi$
$384$	0	0
$385$	2.48604	0.126700
$386$	0	0
$387$	6.50548	0.330692
$388$	0	0
$389$	−24.3510	−1.23464	−0.617321	−	0.786711i	$-0.711782\pi$
$389$	−24.3510	−1.23464	−0.617321	+	0.786711i	$0.711782\pi$
$390$	0	0
$391$	−5.04737	−0.255257
$392$	0	0
$393$	−9.73129	−0.490879
$394$	0	0
$395$	8.98680	0.452175
$396$	0	0
$397$	6.48131	0.325288	0.162644	−	0.986685i	$-0.447998\pi$
$397$	6.48131	0.325288	0.162644	+	0.986685i	$0.447998\pi$
$398$	0	0
$399$	−9.05433	−0.453284
$400$	0	0
$401$	−13.3517	−0.666751	−0.333376	−	0.942794i	$-0.608188\pi$
$401$	−13.3517	−0.666751	−0.333376	+	0.942794i	$0.608188\pi$
$402$	0	0
$403$	−2.69641	−0.134318
$404$	0	0
$405$	−5.81756	−0.289077
$406$	0	0
$407$	−2.35097	−0.116533
$408$	0	0
$409$	30.9582	1.53078	0.765391	−	0.643565i	$-0.222546\pi$
$409$	30.9582	1.53078	0.765391	+	0.643565i	$0.222546\pi$
$410$	0	0
$411$	8.39905	0.414295
$412$	0	0
$413$	−1.00848	−0.0496239
$414$	0	0
$415$	−6.94567	−0.340949
$416$	0	0
$417$	16.3315	0.799758
$418$	0	0
$419$	−15.3058	−0.747739	−0.373869	−	0.927481i	$-0.621969\pi$
$419$	−15.3058	−0.747739	−0.373869	+	0.927481i	$0.621969\pi$
$420$	0	0
$421$	−18.2423	−0.889075	−0.444538	−	0.895760i	$-0.646632\pi$
$421$	−18.2423	−0.889075	−0.444538	+	0.895760i	$0.646632\pi$
$422$	0	0
$423$	−2.25622	−0.109701
$424$	0	0
$425$	−4.06058	−0.196967
$426$	0	0
$427$	−14.7353	−0.713091
$428$	0	0
$429$	−4.83700	−0.233533
$430$	0	0
$431$	23.1880	1.11693	0.558463	−	0.829530i	$-0.311391\pi$
$431$	23.1880	1.11693	0.558463	+	0.829530i	$0.311391\pi$
$432$	0	0
$433$	12.7159	0.611085	0.305542	−	0.952178i	$-0.401162\pi$
$433$	12.7159	0.611085	0.305542	+	0.952178i	$0.401162\pi$
$434$	0	0
$435$	1.47283	0.0706170
$436$	0	0
$437$	−6.85396	−0.327869
$438$	0	0
$439$	−29.1227	−1.38995	−0.694975	−	0.719034i	$-0.744584\pi$
$439$	−29.1227	−1.38995	−0.694975	+	0.719034i	$0.744584\pi$
$440$	0	0
$441$	4.78267	0.227746
$442$	0	0
$443$	−21.4200	−1.01770	−0.508848	−	0.860856i	$-0.669928\pi$
$443$	−21.4200	−1.01770	−0.508848	+	0.860856i	$0.669928\pi$
$444$	0	0
$445$	−3.05433	−0.144789
$446$	0	0
$447$	20.7408	0.981007
$448$	0	0
$449$	−30.3635	−1.43294	−0.716470	−	0.697618i	$-0.754243\pi$
$449$	−30.3635	−1.43294	−0.716470	+	0.697618i	$0.754243\pi$
$450$	0	0
$451$	−9.43171	−0.444122
$452$	0	0
$453$	−25.4751	−1.19692
$454$	0	0
$455$	−1.64207	−0.0769816
$456$	0	0
$457$	39.3789	1.84207	0.921033	−	0.389484i	$-0.127347\pi$
$457$	39.3789	1.84207	0.921033	+	0.389484i	$0.127347\pi$
$458$	0	0
$459$	22.9101	1.06935
$460$	0	0
$461$	35.5606	1.65622	0.828112	−	0.560563i	$-0.189415\pi$
$461$	35.5606	1.65622	0.828112	+	0.560563i	$0.189415\pi$
$462$	0	0
$463$	24.7019	1.14800	0.573998	−	0.818857i	$-0.305392\pi$
$463$	24.7019	1.14800	0.573998	+	0.818857i	$0.305392\pi$
$464$	0	0
$465$	2.69641	0.125043
$466$	0	0
$467$	−11.8308	−0.547462	−0.273731	−	0.961806i	$-0.588258\pi$
$467$	−11.8308	−0.547462	−0.273731	+	0.961806i	$0.588258\pi$
$468$	0	0
$469$	−1.59622	−0.0737066
$470$	0	0
$471$	−26.9706	−1.24274
$472$	0	0
$473$	−17.4611	−0.802864
$474$	0	0
$475$	−5.51396	−0.252998
$476$	0	0
$477$	7.76322	0.355454
$478$	0	0
$479$	31.8432	1.45496	0.727478	−	0.686131i	$-0.240693\pi$
$479$	31.8432	1.45496	0.727478	+	0.686131i	$0.240693\pi$
$480$	0	0
$481$	1.55286	0.0708041
$482$	0	0
$483$	2.04113	0.0928746
$484$	0	0
$485$	−12.6483	−0.574330
$486$	0	0
$487$	15.2166	0.689530	0.344765	−	0.938689i	$-0.387959\pi$
$487$	15.2166	0.689530	0.344765	+	0.938689i	$0.387959\pi$
$488$	0	0
$489$	−7.78267	−0.351945
$490$	0	0
$491$	−27.4876	−1.24050	−0.620248	−	0.784406i	$-0.712968\pi$
$491$	−27.4876	−1.24050	−0.620248	+	0.784406i	$0.712968\pi$
$492$	0	0
$493$	−4.06058	−0.182879
$494$	0	0
$495$	−1.85244	−0.0832611
$496$	0	0
$497$	7.32304	0.328483
$498$	0	0
$499$	34.5785	1.54795	0.773974	−	0.633217i	$-0.218266\pi$
$499$	34.5785	1.54795	0.773974	+	0.633217i	$0.218266\pi$
$500$	0	0
$501$	−27.4945	−1.22836
$502$	0	0
$503$	−37.2313	−1.66006	−0.830032	−	0.557716i	$-0.811678\pi$
$503$	−37.2313	−1.66006	−0.830032	+	0.557716i	$0.811678\pi$
$504$	0	0
$505$	4.29735	0.191230
$506$	0	0
$507$	−15.9519	−0.708449
$508$	0	0
$509$	−0.984562	−0.0436399	−0.0218200	−	0.999762i	$-0.506946\pi$
$509$	−0.984562	−0.0436399	−0.0218200	+	0.999762i	$0.506946\pi$
$510$	0	0
$511$	13.0691	0.578141
$512$	0	0
$513$	31.1102	1.37355
$514$	0	0
$515$	16.9193	0.745552
$516$	0	0
$517$	6.05585	0.266336
$518$	0	0
$519$	14.6289	0.642136
$520$	0	0
$521$	−9.38362	−0.411104	−0.205552	−	0.978646i	$-0.565899\pi$
$521$	−9.38362	−0.411104	−0.205552	+	0.978646i	$0.565899\pi$
$522$	0	0
$523$	14.2159	0.621618	0.310809	−	0.950472i	$-0.399400\pi$
$523$	14.2159	0.621618	0.310809	+	0.950472i	$0.399400\pi$
$524$	0	0
$525$	1.64207	0.0716660
$526$	0	0
$527$	−7.43394	−0.323828
$528$	0	0
$529$	−21.4549	−0.932822
$530$	0	0
$531$	0.751455	0.0326104
$532$	0	0
$533$	6.22982	0.269843
$534$	0	0
$535$	−19.4053	−0.838964
$536$	0	0
$537$	1.33224	0.0574902
$538$	0	0
$539$	−12.8370	−0.552929
$540$	0	0
$541$	35.5606	1.52887	0.764435	−	0.644701i	$-0.223018\pi$
$541$	35.5606	1.52887	0.764435	+	0.644701i	$0.223018\pi$
$542$	0	0
$543$	2.44714	0.105017
$544$	0	0
$545$	2.79811	0.119858
$546$	0	0
$547$	13.0015	0.555905	0.277952	−	0.960595i	$-0.410344\pi$
$547$	13.0015	0.555905	0.277952	+	0.960595i	$0.410344\pi$
$548$	0	0
$549$	10.9798	0.468608
$550$	0	0
$551$	−5.51396	−0.234903
$552$	0	0
$553$	10.0194	0.426070
$554$	0	0
$555$	−1.55286	−0.0659151
$556$	0	0
$557$	−7.44643	−0.315515	−0.157758	−	0.987478i	$-0.550426\pi$
$557$	−7.44643	−0.315515	−0.157758	+	0.987478i	$0.550426\pi$
$558$	0	0
$559$	11.5334	0.487811
$560$	0	0
$561$	−13.3355	−0.563026
$562$	0	0
$563$	21.4200	0.902746	0.451373	−	0.892335i	$-0.350934\pi$
$563$	21.4200	0.902746	0.451373	+	0.892335i	$0.350934\pi$
$564$	0	0
$565$	7.59398	0.319481
$566$	0	0
$567$	−6.48604	−0.272388
$568$	0	0
$569$	17.5793	0.736961	0.368481	−	0.929635i	$-0.379878\pi$
$569$	17.5793	0.736961	0.368481	+	0.929635i	$0.379878\pi$
$570$	0	0
$571$	−43.0426	−1.80128	−0.900639	−	0.434567i	$-0.856901\pi$
$571$	−43.0426	−1.80128	−0.900639	+	0.434567i	$0.856901\pi$
$572$	0	0
$573$	1.08627	0.0453794
$574$	0	0
$575$	1.24302	0.0518375
$576$	0	0
$577$	−5.70961	−0.237694	−0.118847	−	0.992913i	$-0.537920\pi$
$577$	−5.70961	−0.237694	−0.118847	+	0.992913i	$0.537920\pi$
$578$	0	0
$579$	4.06058	0.168752
$580$	0	0
$581$	−7.74378	−0.321266
$582$	0	0
$583$	−20.8370	−0.862981
$584$	0	0
$585$	1.22357	0.0505885
$586$	0	0
$587$	−12.9332	−0.533810	−0.266905	−	0.963723i	$-0.586001\pi$
$587$	−12.9332	−0.533810	−0.266905	+	0.963723i	$0.586001\pi$
$588$	0	0
$589$	−10.0947	−0.415947
$590$	0	0
$591$	1.80212	0.0741292
$592$	0	0
$593$	45.6476	1.87452	0.937261	−	0.348628i	$-0.113352\pi$
$593$	45.6476	1.87452	0.937261	+	0.348628i	$0.113352\pi$
$594$	0	0
$595$	−4.52717	−0.185596
$596$	0	0
$597$	−11.9611	−0.489535
$598$	0	0
$599$	−19.3836	−0.791993	−0.395997	−	0.918252i	$-0.629601\pi$
$599$	−19.3836	−0.791993	−0.395997	+	0.918252i	$0.629601\pi$
$600$	0	0
$601$	45.3525	1.84997	0.924983	−	0.380008i	$-0.124079\pi$
$601$	45.3525	1.84997	0.924983	+	0.380008i	$0.124079\pi$
$602$	0	0
$603$	1.18940	0.0484363
$604$	0	0
$605$	−6.02792	−0.245070
$606$	0	0
$607$	−31.6351	−1.28403	−0.642015	−	0.766692i	$-0.721901\pi$
$607$	−31.6351	−1.28403	−0.642015	+	0.766692i	$0.721901\pi$
$608$	0	0
$609$	1.64207	0.0665402
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−21.0496	−0.850186	−0.425093	−	0.905150i	$-0.639759\pi$
$613$	−21.0496	−0.850186	−0.425093	+	0.905150i	$0.639759\pi$
$614$	0	0
$615$	−6.22982	−0.251210
$616$	0	0
$617$	20.9798	0.844616	0.422308	−	0.906452i	$-0.361220\pi$
$617$	20.9798	0.844616	0.422308	+	0.906452i	$0.361220\pi$
$618$	0	0
$619$	−19.9861	−0.803308	−0.401654	−	0.915791i	$-0.631565\pi$
$619$	−19.9861	−0.803308	−0.401654	+	0.915791i	$0.631565\pi$
$620$	0	0
$621$	−7.01320	−0.281430
$622$	0	0
$623$	−3.40530	−0.136430
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.1087	−0.723190
$628$	0	0
$629$	4.28120	0.170702
$630$	0	0
$631$	16.2562	0.647150	0.323575	−	0.946203i	$-0.395115\pi$
$631$	16.2562	0.647150	0.323575	+	0.946203i	$0.395115\pi$
$632$	0	0
$633$	−1.93023	−0.0767197
$634$	0	0
$635$	−9.74378	−0.386670
$636$	0	0
$637$	8.47908	0.335953
$638$	0	0
$639$	−5.45668	−0.215863
$640$	0	0
$641$	6.87590	0.271582	0.135791	−	0.990738i	$-0.456642\pi$
$641$	6.87590	0.271582	0.135791	+	0.990738i	$0.456642\pi$
$642$	0	0
$643$	32.8759	1.29650	0.648249	−	0.761428i	$-0.275501\pi$
$643$	32.8759	1.29650	0.648249	+	0.761428i	$0.275501\pi$
$644$	0	0
$645$	−11.5334	−0.454128
$646$	0	0
$647$	−12.9721	−0.509985	−0.254992	−	0.966943i	$-0.582073\pi$
$647$	−12.9721	−0.509985	−0.254992	+	0.966943i	$0.582073\pi$
$648$	0	0
$649$	−2.01696	−0.0791724
$650$	0	0
$651$	3.00624	0.117824
$652$	0	0
$653$	−9.31055	−0.364350	−0.182175	−	0.983266i	$-0.558314\pi$
$653$	−9.31055	−0.364350	−0.182175	+	0.983266i	$0.558314\pi$
$654$	0	0
$655$	−6.60719	−0.258164
$656$	0	0
$657$	−9.73825	−0.379925
$658$	0	0
$659$	−16.1212	−0.627991	−0.313995	−	0.949424i	$-0.601668\pi$
$659$	−16.1212	−0.627991	−0.313995	+	0.949424i	$0.601668\pi$
$660$	0	0
$661$	15.2966	0.594970	0.297485	−	0.954726i	$-0.403852\pi$
$661$	15.2966	0.594970	0.297485	+	0.954726i	$0.403852\pi$
$662$	0	0
$663$	8.80836	0.342088
$664$	0	0
$665$	−6.14756	−0.238392
$666$	0	0
$667$	1.24302	0.0481299
$668$	0	0
$669$	−13.0576	−0.504837
$670$	0	0
$671$	−29.4706	−1.13770
$672$	0	0
$673$	−14.3385	−0.552708	−0.276354	−	0.961056i	$-0.589126\pi$
$673$	−14.3385	−0.552708	−0.276354	+	0.961056i	$0.589126\pi$
$674$	0	0
$675$	−5.64207	−0.217164
$676$	0	0
$677$	40.1895	1.54461	0.772304	−	0.635253i	$-0.219104\pi$
$677$	40.1895	1.54461	0.772304	+	0.635253i	$0.219104\pi$
$678$	0	0
$679$	−14.1017	−0.541174
$680$	0	0
$681$	−4.16004	−0.159413
$682$	0	0
$683$	−19.1880	−0.734207	−0.367104	−	0.930180i	$-0.619651\pi$
$683$	−19.1880	−0.734207	−0.367104	+	0.930180i	$0.619651\pi$
$684$	0	0
$685$	5.70265	0.217887
$686$	0	0
$687$	−15.4945	−0.591153
$688$	0	0
$689$	13.7632	0.524337
$690$	0	0
$691$	−24.8587	−0.945669	−0.472834	−	0.881151i	$-0.656769\pi$
$691$	−24.8587	−0.945669	−0.472834	+	0.881151i	$0.656769\pi$
$692$	0	0
$693$	−2.06530	−0.0784543
$694$	0	0
$695$	11.0885	0.420611
$696$	0	0
$697$	17.1755	0.650568
$698$	0	0
$699$	26.5110	1.00274
$700$	0	0
$701$	35.5918	1.34428	0.672141	−	0.740423i	$-0.265375\pi$
$701$	35.5918	1.34428	0.672141	+	0.740423i	$0.265375\pi$
$702$	0	0
$703$	5.81355	0.219262
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	4.79115	0.180190
$708$	0	0
$709$	−45.9427	−1.72541	−0.862707	−	0.505703i	$-0.831233\pi$
$709$	−45.9427	−1.72541	−0.862707	+	0.505703i	$0.831233\pi$
$710$	0	0
$711$	−7.46587	−0.279992
$712$	0	0
$713$	2.27567	0.0852245
$714$	0	0
$715$	−3.28415	−0.122820
$716$	0	0
$717$	1.73129	0.0646563
$718$	0	0
$719$	25.2966	0.943405	0.471703	−	0.881758i	$-0.343640\pi$
$719$	25.2966	0.943405	0.471703	+	0.881758i	$0.343640\pi$
$720$	0	0
$721$	18.8634	0.702510
$722$	0	0
$723$	25.3572	0.943045
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−11.5140	−0.427029	−0.213515	−	0.976940i	$-0.568491\pi$
$727$	−11.5140	−0.427029	−0.213515	+	0.976940i	$0.568491\pi$
$728$	0	0
$729$	29.7625	1.10232
$730$	0	0
$731$	31.7974	1.17607
$732$	0	0
$733$	17.3789	0.641904	0.320952	−	0.947095i	$-0.395997\pi$
$733$	17.3789	0.641904	0.320952	+	0.947095i	$0.395997\pi$
$734$	0	0
$735$	−8.47908	−0.312755
$736$	0	0
$737$	−3.19244	−0.117595
$738$	0	0
$739$	−13.6490	−0.502088	−0.251044	−	0.967976i	$-0.580774\pi$
$739$	−13.6490	−0.502088	−0.251044	+	0.967976i	$0.580774\pi$
$740$	0	0
$741$	11.9611	0.439402
$742$	0	0
$743$	−4.14756	−0.152159	−0.0760796	−	0.997102i	$-0.524240\pi$
$743$	−4.14756	−0.152159	−0.0760796	+	0.997102i	$0.524240\pi$
$744$	0	0
$745$	14.0823	0.515934
$746$	0	0
$747$	5.77018	0.211120
$748$	0	0
$749$	−21.6351	−0.790530
$750$	0	0
$751$	37.2144	1.35797	0.678986	−	0.734151i	$-0.262420\pi$
$751$	37.2144	1.35797	0.678986	+	0.734151i	$0.262420\pi$
$752$	0	0
$753$	13.3355	0.485974
$754$	0	0
$755$	−17.2966	−0.629489
$756$	0	0
$757$	−47.8649	−1.73968	−0.869840	−	0.493334i	$-0.835778\pi$
$757$	−47.8649	−1.73968	−0.869840	+	0.493334i	$0.835778\pi$
$758$	0	0
$759$	4.08226	0.148177
$760$	0	0
$761$	−19.1538	−0.694325	−0.347162	−	0.937805i	$-0.612855\pi$
$761$	−19.1538	−0.694325	−0.347162	+	0.937805i	$0.612855\pi$
$762$	0	0
$763$	3.11963	0.112938
$764$	0	0
$765$	3.37336	0.121964
$766$	0	0
$767$	1.33224	0.0481043
$768$	0	0
$769$	−22.8370	−0.823523	−0.411762	−	0.911292i	$-0.635086\pi$
$769$	−22.8370	−0.823523	−0.411762	+	0.911292i	$0.635086\pi$
$770$	0	0
$771$	−2.76723	−0.0996595
$772$	0	0
$773$	25.8524	0.929848	0.464924	−	0.885351i	$-0.346082\pi$
$773$	25.8524	0.929848	0.464924	+	0.885351i	$0.346082\pi$
$774$	0	0
$775$	1.83076	0.0657629
$776$	0	0
$777$	−1.73129	−0.0621097
$778$	0	0
$779$	23.3230	0.835635
$780$	0	0
$781$	14.6461	0.524078
$782$	0	0
$783$	−5.64207	−0.201631
$784$	0	0
$785$	−18.3121	−0.653586
$786$	0	0
$787$	23.4831	0.837082	0.418541	−	0.908198i	$-0.362542\pi$
$787$	23.4831	0.837082	0.418541	+	0.908198i	$0.362542\pi$
$788$	0	0
$789$	20.2423	0.720645
$790$	0	0
$791$	8.46659	0.301037
$792$	0	0
$793$	19.4659	0.691253
$794$	0	0
$795$	−13.7632	−0.488131
$796$	0	0
$797$	−9.43171	−0.334088	−0.167044	−	0.985949i	$-0.553422\pi$
$797$	−9.43171	−0.334088	−0.167044	+	0.985949i	$0.553422\pi$
$798$	0	0
$799$	−11.0279	−0.390140
$800$	0	0
$801$	2.53742	0.0896553
$802$	0	0
$803$	26.1381	0.922394
$804$	0	0
$805$	1.38585	0.0488448
$806$	0	0
$807$	24.9901	0.879693
$808$	0	0
$809$	17.2841	0.607678	0.303839	−	0.952723i	$-0.401731\pi$
$809$	17.2841	0.607678	0.303839	+	0.952723i	$0.401731\pi$
$810$	0	0
$811$	23.4270	0.822633	0.411316	−	0.911493i	$-0.365069\pi$
$811$	23.4270	0.822633	0.411316	+	0.911493i	$0.365069\pi$
$812$	0	0
$813$	15.9222	0.558416
$814$	0	0
$815$	−5.28415	−0.185096
$816$	0	0
$817$	43.1785	1.51063
$818$	0	0
$819$	1.36417	0.0476679
$820$	0	0
$821$	10.3774	0.362173	0.181086	−	0.983467i	$-0.442039\pi$
$821$	10.3774	0.362173	0.181086	+	0.983467i	$0.442039\pi$
$822$	0	0
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	0	0
$825$	3.28415	0.114339
$826$	0	0
$827$	−50.0078	−1.73894	−0.869470	−	0.493986i	$-0.835540\pi$
$827$	−50.0078	−1.73894	−0.869470	+	0.493986i	$0.835540\pi$
$828$	0	0
$829$	−38.0078	−1.32006	−0.660032	−	0.751237i	$-0.729457\pi$
$829$	−38.0078	−1.32006	−0.660032	+	0.751237i	$0.729457\pi$
$830$	0	0
$831$	31.6476	1.09784
$832$	0	0
$833$	23.3767	0.809953
$834$	0	0
$835$	−18.6678	−0.646025
$836$	0	0
$837$	−10.3293	−0.357032
$838$	0	0
$839$	36.0250	1.24372	0.621860	−	0.783128i	$-0.286377\pi$
$839$	36.0250	1.24372	0.621860	+	0.783128i	$0.286377\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	20.8873	0.719398
$844$	0	0
$845$	−10.8308	−0.372589
$846$	0	0
$847$	−6.72058	−0.230922
$848$	0	0
$849$	10.2298	0.351086
$850$	0	0
$851$	−1.31055	−0.0449252
$852$	0	0
$853$	−17.5140	−0.599667	−0.299833	−	0.953992i	$-0.596931\pi$
$853$	−17.5140	−0.599667	−0.299833	+	0.953992i	$0.596931\pi$
$854$	0	0
$855$	4.58078	0.156659
$856$	0	0
$857$	−38.1600	−1.30352	−0.651761	−	0.758424i	$-0.725970\pi$
$857$	−38.1600	−1.30352	−0.651761	+	0.758424i	$0.725970\pi$
$858$	0	0
$859$	−48.4023	−1.65147	−0.825733	−	0.564061i	$-0.809238\pi$
$859$	−48.4023	−1.65147	−0.825733	+	0.564061i	$0.809238\pi$
$860$	0	0
$861$	−6.94567	−0.236708
$862$	0	0
$863$	9.78491	0.333082	0.166541	−	0.986035i	$-0.446740\pi$
$863$	9.78491	0.333082	0.166541	+	0.986035i	$0.446740\pi$
$864$	0	0
$865$	9.93246	0.337714
$866$	0	0
$867$	−0.753690	−0.0255967
$868$	0	0
$869$	20.0389	0.679773
$870$	0	0
$871$	2.10866	0.0714494
$872$	0	0
$873$	10.5077	0.355632
$874$	0	0
$875$	1.11491	0.0376908
$876$	0	0
$877$	−0.800344	−0.0270257	−0.0135129	−	0.999909i	$-0.504301\pi$
$877$	−0.800344	−0.0270257	−0.0135129	+	0.999909i	$0.504301\pi$
$878$	0	0
$879$	−18.2298	−0.614876
$880$	0	0
$881$	−31.7702	−1.07036	−0.535182	−	0.844737i	$-0.679757\pi$
$881$	−31.7702	−1.07036	−0.535182	+	0.844737i	$0.679757\pi$
$882$	0	0
$883$	−41.5918	−1.39967	−0.699837	−	0.714303i	$-0.746744\pi$
$883$	−41.5918	−1.39967	−0.699837	+	0.714303i	$0.746744\pi$
$884$	0	0
$885$	−1.33224	−0.0447826
$886$	0	0
$887$	−45.7688	−1.53676	−0.768382	−	0.639991i	$-0.778938\pi$
$887$	−45.7688	−1.53676	−0.768382	+	0.639991i	$0.778938\pi$
$888$	0	0
$889$	−10.8634	−0.364347
$890$	0	0
$891$	−12.9721	−0.434581
$892$	0	0
$893$	−14.9751	−0.501123
$894$	0	0
$895$	0.904539	0.0302354
$896$	0	0
$897$	−2.69641	−0.0900304
$898$	0	0
$899$	1.83076	0.0610593
$900$	0	0
$901$	37.9450	1.26413
$902$	0	0
$903$	−12.8587	−0.427910
$904$	0	0
$905$	1.66152	0.0552308
$906$	0	0
$907$	7.83076	0.260016	0.130008	−	0.991513i	$-0.458500\pi$
$907$	7.83076	0.260016	0.130008	+	0.991513i	$0.458500\pi$
$908$	0	0
$909$	−3.57007	−0.118412
$910$	0	0
$911$	−31.0426	−1.02849	−0.514244	−	0.857644i	$-0.671928\pi$
$911$	−31.0426	−1.02849	−0.514244	+	0.857644i	$0.671928\pi$
$912$	0	0
$913$	−15.4876	−0.512563
$914$	0	0
$915$	−19.4659	−0.643522
$916$	0	0
$917$	−7.36640	−0.243260
$918$	0	0
$919$	2.78562	0.0918892	0.0459446	−	0.998944i	$-0.485370\pi$
$919$	2.78562	0.0918892	0.0459446	+	0.998944i	$0.485370\pi$
$920$	0	0
$921$	6.10866	0.201287
$922$	0	0
$923$	−9.67401	−0.318424
$924$	0	0
$925$	−1.05433	−0.0346662
$926$	0	0
$927$	−14.0558	−0.461655
$928$	0	0
$929$	−27.1010	−0.889155	−0.444577	−	0.895741i	$-0.646646\pi$
$929$	−27.1010	−0.889155	−0.444577	+	0.895741i	$0.646646\pi$
$930$	0	0
$931$	31.7438	1.04036
$932$	0	0
$933$	−11.1274	−0.364295
$934$	0	0
$935$	−9.05433	−0.296108
$936$	0	0
$937$	−10.4163	−0.340285	−0.170142	−	0.985420i	$-0.554423\pi$
$937$	−10.4163	−0.340285	−0.170142	+	0.985420i	$0.554423\pi$
$938$	0	0
$939$	33.7563	1.10159
$940$	0	0
$941$	37.2841	1.21543	0.607714	−	0.794156i	$-0.292087\pi$
$941$	37.2841	1.21543	0.607714	+	0.794156i	$0.292087\pi$
$942$	0	0
$943$	−5.25774	−0.171216
$944$	0	0
$945$	−6.29039	−0.204626
$946$	0	0
$947$	−13.0955	−0.425545	−0.212773	−	0.977102i	$-0.568249\pi$
$947$	−13.0955	−0.425545	−0.212773	+	0.977102i	$0.568249\pi$
$948$	0	0
$949$	−17.2647	−0.560436
$950$	0	0
$951$	11.9611	0.387865
$952$	0	0
$953$	−21.2453	−0.688201	−0.344101	−	0.938933i	$-0.611816\pi$
$953$	−21.2453	−0.688201	−0.344101	+	0.938933i	$0.611816\pi$
$954$	0	0
$955$	0.737534	0.0238660
$956$	0	0
$957$	3.28415	0.106161
$958$	0	0
$959$	6.35793	0.205308
$960$	0	0
$961$	−27.6483	−0.891881
$962$	0	0
$963$	16.1212	0.519497
$964$	0	0
$965$	2.75698	0.0887504
$966$	0	0
$967$	−25.2408	−0.811689	−0.405844	−	0.913942i	$-0.633023\pi$
$967$	−25.2408	−0.811689	−0.405844	+	0.913942i	$0.633023\pi$
$968$	0	0
$969$	32.9765	1.05936
$970$	0	0
$971$	−38.2159	−1.22641	−0.613203	−	0.789925i	$-0.710119\pi$
$971$	−38.2159	−1.22641	−0.613203	+	0.789925i	$0.710119\pi$
$972$	0	0
$973$	12.3627	0.396328
$974$	0	0
$975$	−2.16924	−0.0694713
$976$	0	0
$977$	−2.13507	−0.0683070	−0.0341535	−	0.999417i	$-0.510874\pi$
$977$	−2.13507	−0.0683070	−0.0341535	+	0.999417i	$0.510874\pi$
$978$	0	0
$979$	−6.81060	−0.217668
$980$	0	0
$981$	−2.32456	−0.0742174
$982$	0	0
$983$	29.1924	0.931094	0.465547	−	0.885023i	$-0.345858\pi$
$983$	29.1924	0.931094	0.465547	+	0.885023i	$0.345858\pi$
$984$	0	0
$985$	1.22357	0.0389862
$986$	0	0
$987$	4.45963	0.141952
$988$	0	0
$989$	−9.73378	−0.309516
$990$	0	0
$991$	−3.58373	−0.113841	−0.0569205	−	0.998379i	$-0.518128\pi$
$991$	−3.58373	−0.113841	−0.0569205	+	0.998379i	$0.518128\pi$
$992$	0	0
$993$	−31.7438	−1.00736
$994$	0	0
$995$	−8.12115	−0.257458
$996$	0	0
$997$	27.8260	0.881259	0.440630	−	0.897689i	$-0.354755\pi$
$997$	27.8260	0.881259	0.440630	+	0.897689i	$0.354755\pi$
$998$	0	0
$999$	5.94862	0.188206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.bo.1.3		3
4.3	odd	2		9280.2.a.bq.1.1		3
8.3	odd	2		2320.2.a.p.1.3		3
8.5	even	2		1160.2.a.g.1.1	✓	3
40.29	even	2		5800.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.g.1.1	✓	3	8.5	even	2
2320.2.a.p.1.3		3	8.3	odd	2
5800.2.a.q.1.3		3	40.29	even	2
9280.2.a.bo.1.3		3	1.1	even	1	trivial
9280.2.a.bq.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.47283 q^{3} +1.00000 q^{5} +1.11491 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.47283 q^{13} +1.47283 q^{15} -4.06058 q^{17} -5.51396 q^{19} +1.64207 q^{21} +1.24302 q^{23} +1.00000 q^{25} -5.64207 q^{27} +1.00000 q^{29} +1.83076 q^{31} +3.28415 q^{33} +1.11491 q^{35} -1.05433 q^{37} -2.16924 q^{39} -4.22982 q^{41} -7.83076 q^{43} -0.830760 q^{45} +2.71585 q^{47} -5.75698 q^{49} -5.98055 q^{51} -9.34472 q^{53} +2.22982 q^{55} -8.12115 q^{57} -0.904539 q^{59} -13.2166 q^{61} -0.926221 q^{63} -1.47283 q^{65} -1.43171 q^{67} +1.83076 q^{69} +6.56829 q^{71} +11.7221 q^{73} +1.47283 q^{75} +2.48604 q^{77} +8.98680 q^{79} -5.81756 q^{81} -6.94567 q^{83} -4.06058 q^{85} +1.47283 q^{87} -3.05433 q^{89} -1.64207 q^{91} +2.69641 q^{93} -5.51396 q^{95} -12.6483 q^{97} -1.85244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} + q^{13} - q^{15} + 5 q^{17} - 2 q^{19} + 4 q^{21} + 11 q^{23} + 3 q^{25} - 16 q^{27} + 3 q^{29} + q^{31} + 8 q^{33} - 3 q^{35} - 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 + q^13 - q^15 + 5 * q^17 - 2 * q^19 + 4 * q^21 + 11 * q^23 + 3 * q^25 - 16 * q^27 + 3 * q^29 + q^31 + 8 * q^33 - 3 * q^35 - 14 * q^37 - 11 * q^39 - 19 * q^43 + 2 * q^45 + 10 * q^47 - 10 * q^49 - 26 * q^51 - 9 * q^53 - 6 * q^55 + 10 * q^57 - q^59 - 7 * q^61 + q^65 - 8 * q^67 + q^69 + 16 * q^71 + 9 * q^73 - q^75 + 22 * q^77 + 7 * q^79 + 7 * q^81 - 10 * q^83 + 5 * q^85 - q^87 - 20 * q^89 - 4 * q^91 + 18 * q^93 - 2 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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