LMFDB - Embedded newform 9280.2.a.bz.1.4

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 = [4,0,-2,0,4,0,-2,0,8,0,2,0,-4,0,-2,0,-4,0,-6,0,20] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.552409$ 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)

$f(q)$	$=$	$q+2.87834 q^{3} +1.00000 q^{5} +2.87834 q^{7} +5.28487 q^{9} +1.77353 q^{11} +2.28487 q^{13} +2.87834 q^{15} -5.06185 q^{17} +5.16321 q^{19} +8.28487 q^{21} +8.05839 q^{23} +1.00000 q^{25} +6.57664 q^{27} -1.00000 q^{29} +9.16321 q^{31} +5.10482 q^{33} +2.87834 q^{35} +2.48521 q^{37} +6.57664 q^{39} -8.86151 q^{41} -6.87834 q^{43} +5.28487 q^{45} -3.69830 q^{47} +1.28487 q^{49} -14.5697 q^{51} +2.65187 q^{53} +1.77353 q^{55} +14.8615 q^{57} +10.9367 q^{59} -9.83192 q^{61} +15.2117 q^{63} +2.28487 q^{65} -7.90793 q^{67} +23.1948 q^{69} -7.75669 q^{71} -11.1034 q^{73} +2.87834 q^{75} +5.10482 q^{77} -3.61616 q^{79} +3.07523 q^{81} -4.66871 q^{83} -5.06185 q^{85} -2.87834 q^{87} -6.93674 q^{89} +6.57664 q^{91} +26.3749 q^{93} +5.16321 q^{95} -5.43266 q^{97} +9.37285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 8 q^{9} + 2 q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{17} - 6 q^{19} + 20 q^{21} + 14 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 10 q^{31} + 12 q^{33} - 2 q^{35} + 16 q^{37}+ \cdots - 6 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 8 * q^9 + 2 * q^11 - 4 * q^13 - 2 * q^15 - 4 * q^17 - 6 * q^19 + 20 * q^21 + 14 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 16 * q^37 + 4 * q^39 - 14 * q^43 + 8 * q^45 - 6 * q^47 - 8 * q^49 - 32 * q^51 - 8 * q^53 + 2 * q^55 + 24 * q^57 + 12 * q^59 - 16 * q^61 - 2 * q^63 - 4 * q^65 - 6 * q^67 + 8 * q^69 - 4 * q^71 + 12 * q^73 - 2 * q^75 + 12 * q^77 + 2 * q^79 + 16 * q^81 - 22 * q^83 - 4 * q^85 + 2 * q^87 + 4 * q^89 + 4 * q^91 + 16 * q^93 - 6 * q^95 - 8 * q^97 - 6 * q^99

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
$3$	2.87834	1.66181	0.830907	−	0.556412i	$-0.187822\pi$
$3$	2.87834	1.66181	0.830907	+	0.556412i	$0.187822\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.87834	1.08791	0.543956	−	0.839114i	$-0.316926\pi$
$7$	2.87834	1.08791	0.543956	+	0.839114i	$0.316926\pi$
$8$	0	0
$9$	5.28487	1.76162
$10$	0	0
$11$	1.77353	0.534738	0.267369	−	0.963594i	$-0.413846\pi$
$11$	1.77353	0.534738	0.267369	+	0.963594i	$0.413846\pi$
$12$	0	0
$13$	2.28487	0.633709	0.316854	−	0.948474i	$-0.397373\pi$
$13$	2.28487	0.633709	0.316854	+	0.948474i	$0.397373\pi$
$14$	0	0
$15$	2.87834	0.743185
$16$	0	0
$17$	−5.06185	−1.22768	−0.613839	−	0.789431i	$-0.710376\pi$
$17$	−5.06185	−1.22768	−0.613839	+	0.789431i	$0.710376\pi$
$18$	0	0
$19$	5.16321	1.18452	0.592261	−	0.805746i	$-0.298235\pi$
$19$	5.16321	1.18452	0.592261	+	0.805746i	$0.298235\pi$
$20$	0	0
$21$	8.28487	1.80791
$22$	0	0
$23$	8.05839	1.68029	0.840146	−	0.542361i	$-0.182469\pi$
$23$	8.05839	1.68029	0.840146	+	0.542361i	$0.182469\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	6.57664	1.26567
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.16321	1.64576	0.822881	−	0.568214i	$-0.192365\pi$
$31$	9.16321	1.64576	0.822881	+	0.568214i	$0.192365\pi$
$32$	0	0
$33$	5.10482	0.888635
$34$	0	0
$35$	2.87834	0.486529
$36$	0	0
$37$	2.48521	0.408565	0.204283	−	0.978912i	$-0.434514\pi$
$37$	2.48521	0.408565	0.204283	+	0.978912i	$0.434514\pi$
$38$	0	0
$39$	6.57664	1.05311
$40$	0	0
$41$	−8.86151	−1.38394	−0.691968	−	0.721929i	$-0.743256\pi$
$41$	−8.86151	−1.38394	−0.691968	+	0.721929i	$0.743256\pi$
$42$	0	0
$43$	−6.87834	−1.04894	−0.524469	−	0.851430i	$-0.675736\pi$
$43$	−6.87834	−1.04894	−0.524469	+	0.851430i	$0.675736\pi$
$44$	0	0
$45$	5.28487	0.787822
$46$	0	0
$47$	−3.69830	−0.539452	−0.269726	−	0.962937i	$-0.586933\pi$
$47$	−3.69830	−0.539452	−0.269726	+	0.962937i	$0.586933\pi$
$48$	0	0
$49$	1.28487	0.183553
$50$	0	0
$51$	−14.5697	−2.04017
$52$	0	0
$53$	2.65187	0.364262	0.182131	−	0.983274i	$-0.441700\pi$
$53$	2.65187	0.364262	0.182131	+	0.983274i	$0.441700\pi$
$54$	0	0
$55$	1.77353	0.239142
$56$	0	0
$57$	14.8615	1.96845
$58$	0	0
$59$	10.9367	1.42384	0.711921	−	0.702259i	$-0.247825\pi$
$59$	10.9367	1.42384	0.711921	+	0.702259i	$0.247825\pi$
$60$	0	0
$61$	−9.83192	−1.25885	−0.629424	−	0.777062i	$-0.716709\pi$
$61$	−9.83192	−1.25885	−0.629424	+	0.777062i	$0.716709\pi$
$62$	0	0
$63$	15.2117	1.91649
$64$	0	0
$65$	2.28487	0.283403
$66$	0	0
$67$	−7.90793	−0.966108	−0.483054	−	0.875591i	$-0.660472\pi$
$67$	−7.90793	−0.966108	−0.483054	+	0.875591i	$0.660472\pi$
$68$	0	0
$69$	23.1948	2.79233
$70$	0	0
$71$	−7.75669	−0.920550	−0.460275	−	0.887776i	$-0.652249\pi$
$71$	−7.75669	−0.920550	−0.460275	+	0.887776i	$0.652249\pi$
$72$	0	0
$73$	−11.1034	−1.29956	−0.649778	−	0.760124i	$-0.725138\pi$
$73$	−11.1034	−1.29956	−0.649778	+	0.760124i	$0.725138\pi$
$74$	0	0
$75$	2.87834	0.332363
$76$	0	0
$77$	5.10482	0.581748
$78$	0	0
$79$	−3.61616	−0.406850	−0.203425	−	0.979091i	$-0.565207\pi$
$79$	−3.61616	−0.406850	−0.203425	+	0.979091i	$0.565207\pi$
$80$	0	0
$81$	3.07523	0.341692
$82$	0	0
$83$	−4.66871	−0.512457	−0.256229	−	0.966616i	$-0.582480\pi$
$83$	−4.66871	−0.512457	−0.256229	+	0.966616i	$0.582480\pi$
$84$	0	0
$85$	−5.06185	−0.549034
$86$	0	0
$87$	−2.87834	−0.308591
$88$	0	0
$89$	−6.93674	−0.735293	−0.367646	−	0.929966i	$-0.619836\pi$
$89$	−6.93674	−0.735293	−0.367646	+	0.929966i	$0.619836\pi$
$90$	0	0
$91$	6.57664	0.689419
$92$	0	0
$93$	26.3749	2.73495
$94$	0	0
$95$	5.16321	0.529735
$96$	0	0
$97$	−5.43266	−0.551603	−0.275802	−	0.961215i	$-0.588943\pi$
$97$	−5.43266	−0.551603	−0.275802	+	0.961215i	$0.588943\pi$
$98$	0	0
$99$	9.37285	0.942007
$100$	0	0
$101$	−7.38969	−0.735301	−0.367651	−	0.929964i	$-0.619838\pi$
$101$	−7.38969	−0.735301	−0.367651	+	0.929964i	$0.619838\pi$
$102$	0	0
$103$	−5.81508	−0.572977	−0.286489	−	0.958084i	$-0.592488\pi$
$103$	−5.81508	−0.572977	−0.286489	+	0.958084i	$0.592488\pi$
$104$	0	0
$105$	8.28487	0.808520
$106$	0	0
$107$	−12.7210	−1.22978	−0.614892	−	0.788611i	$-0.710800\pi$
$107$	−12.7210	−1.22978	−0.614892	+	0.788611i	$0.710800\pi$
$108$	0	0
$109$	−16.4945	−1.57989	−0.789944	−	0.613180i	$-0.789890\pi$
$109$	−16.4945	−1.57989	−0.789944	+	0.613180i	$0.789890\pi$
$110$	0	0
$111$	7.15328	0.678959
$112$	0	0
$113$	4.56044	0.429010	0.214505	−	0.976723i	$-0.431186\pi$
$113$	4.56044	0.429010	0.214505	+	0.976723i	$0.431186\pi$
$114$	0	0
$115$	8.05839	0.751449
$116$	0	0
$117$	12.0752	1.11636
$118$	0	0
$119$	−14.5697	−1.33561
$120$	0	0
$121$	−7.85461	−0.714055
$122$	0	0
$123$	−25.5065	−2.29984
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.66871	0.769223	0.384612	−	0.923078i	$-0.374335\pi$
$127$	8.66871	0.769223	0.384612	+	0.923078i	$0.374335\pi$
$128$	0	0
$129$	−19.7982	−1.74314
$130$	0	0
$131$	20.1591	1.76131	0.880656	−	0.473757i	$-0.157102\pi$
$131$	20.1591	1.76131	0.880656	+	0.473757i	$0.157102\pi$
$132$	0	0
$133$	14.8615	1.28866
$134$	0	0
$135$	6.57664	0.566027
$136$	0	0
$137$	16.0914	1.37478	0.687392	−	0.726287i	$-0.258755\pi$
$137$	16.0914	1.37478	0.687392	+	0.726287i	$0.258755\pi$
$138$	0	0
$139$	−2.00690	−0.170223	−0.0851116	−	0.996371i	$-0.527125\pi$
$139$	−2.00690	−0.170223	−0.0851116	+	0.996371i	$0.527125\pi$
$140$	0	0
$141$	−10.6450	−0.896468
$142$	0	0
$143$	4.05227	0.338868
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	3.69830	0.305030
$148$	0	0
$149$	10.1168	0.828800	0.414400	−	0.910095i	$-0.363992\pi$
$149$	10.1168	0.828800	0.414400	+	0.910095i	$0.363992\pi$
$150$	0	0
$151$	−5.14638	−0.418806	−0.209403	−	0.977829i	$-0.567152\pi$
$151$	−5.14638	−0.418806	−0.209403	+	0.977829i	$0.567152\pi$
$152$	0	0
$153$	−26.7512	−2.16271
$154$	0	0
$155$	9.16321	0.736007
$156$	0	0
$157$	−10.3694	−0.827568	−0.413784	−	0.910375i	$-0.635793\pi$
$157$	−10.3694	−0.827568	−0.413784	+	0.910375i	$0.635793\pi$
$158$	0	0
$159$	7.63300	0.605336
$160$	0	0
$161$	23.1948	1.82801
$162$	0	0
$163$	16.3848	1.28336	0.641679	−	0.766973i	$-0.278238\pi$
$163$	16.3848	1.28336	0.641679	+	0.766973i	$0.278238\pi$
$164$	0	0
$165$	5.10482	0.397410
$166$	0	0
$167$	22.6350	1.75155	0.875776	−	0.482718i	$-0.160350\pi$
$167$	22.6350	1.75155	0.875776	+	0.482718i	$0.160350\pi$
$168$	0	0
$169$	−7.77938	−0.598413
$170$	0	0
$171$	27.2869	2.08668
$172$	0	0
$173$	5.26218	0.400076	0.200038	−	0.979788i	$-0.435893\pi$
$173$	5.26218	0.400076	0.200038	+	0.979788i	$0.435893\pi$
$174$	0	0
$175$	2.87834	0.217582
$176$	0	0
$177$	31.4797	2.36616
$178$	0	0
$179$	−2.61031	−0.195104	−0.0975520	−	0.995230i	$-0.531101\pi$
$179$	−2.61031	−0.195104	−0.0975520	+	0.995230i	$0.531101\pi$
$180$	0	0
$181$	−8.77556	−0.652282	−0.326141	−	0.945321i	$-0.605748\pi$
$181$	−8.77556	−0.652282	−0.326141	+	0.945321i	$0.605748\pi$
$182$	0	0
$183$	−28.2997	−2.09197
$184$	0	0
$185$	2.48521	0.182716
$186$	0	0
$187$	−8.97731	−0.656486
$188$	0	0
$189$	18.9298	1.37694
$190$	0	0
$191$	20.3165	1.47005	0.735025	−	0.678040i	$-0.237170\pi$
$191$	20.3165	1.47005	0.735025	+	0.678040i	$0.237170\pi$
$192$	0	0
$193$	−18.1490	−1.30640	−0.653198	−	0.757187i	$-0.726573\pi$
$193$	−18.1490	−1.30640	−0.653198	+	0.757187i	$0.726573\pi$
$194$	0	0
$195$	6.57664	0.470963
$196$	0	0
$197$	10.2165	0.727898	0.363949	−	0.931419i	$-0.381428\pi$
$197$	10.2165	0.727898	0.363949	+	0.931419i	$0.381428\pi$
$198$	0	0
$199$	17.3966	1.23321	0.616606	−	0.787272i	$-0.288507\pi$
$199$	17.3966	1.23321	0.616606	+	0.787272i	$0.288507\pi$
$200$	0	0
$201$	−22.7618	−1.60549
$202$	0	0
$203$	−2.87834	−0.202020
$204$	0	0
$205$	−8.86151	−0.618915
$206$	0	0
$207$	42.5876	2.96004
$208$	0	0
$209$	9.15709	0.633409
$210$	0	0
$211$	−26.7625	−1.84241	−0.921205	−	0.389079i	$-0.872793\pi$
$211$	−26.7625	−1.84241	−0.921205	+	0.389079i	$0.872793\pi$
$212$	0	0
$213$	−22.3264	−1.52978
$214$	0	0
$215$	−6.87834	−0.469099
$216$	0	0
$217$	26.3749	1.79044
$218$	0	0
$219$	−31.9594	−2.15962
$220$	0	0
$221$	−11.5657	−0.777990
$222$	0	0
$223$	23.2384	1.55616	0.778081	−	0.628164i	$-0.216193\pi$
$223$	23.2384	1.55616	0.778081	+	0.628164i	$0.216193\pi$
$224$	0	0
$225$	5.28487	0.352325
$226$	0	0
$227$	−16.0584	−1.06583	−0.532917	−	0.846168i	$-0.678904\pi$
$227$	−16.0584	−1.06583	−0.532917	+	0.846168i	$0.678904\pi$
$228$	0	0
$229$	15.1533	1.00136	0.500678	−	0.865633i	$-0.333084\pi$
$229$	15.1533	1.00136	0.500678	+	0.865633i	$0.333084\pi$
$230$	0	0
$231$	14.6934	0.966757
$232$	0	0
$233$	9.96633	0.652916	0.326458	−	0.945212i	$-0.394145\pi$
$233$	9.96633	0.652916	0.326458	+	0.945212i	$0.394145\pi$
$234$	0	0
$235$	−3.69830	−0.241250
$236$	0	0
$237$	−10.4086	−0.676109
$238$	0	0
$239$	9.14638	0.591630	0.295815	−	0.955245i	$-0.404409\pi$
$239$	9.14638	0.591630	0.295815	+	0.955245i	$0.404409\pi$
$240$	0	0
$241$	4.92984	0.317559	0.158779	−	0.987314i	$-0.449244\pi$
$241$	4.92984	0.317559	0.158779	+	0.987314i	$0.449244\pi$
$242$	0	0
$243$	−10.8783	−0.697846
$244$	0	0
$245$	1.28487	0.0820872
$246$	0	0
$247$	11.7973	0.750642
$248$	0	0
$249$	−13.4381	−0.851608
$250$	0	0
$251$	−5.99007	−0.378090	−0.189045	−	0.981968i	$-0.560539\pi$
$251$	−5.99007	−0.378090	−0.189045	+	0.981968i	$0.560539\pi$
$252$	0	0
$253$	14.2918	0.898516
$254$	0	0
$255$	−14.5697	−0.912392
$256$	0	0
$257$	1.41264	0.0881184	0.0440592	−	0.999029i	$-0.485971\pi$
$257$	1.41264	0.0881184	0.0440592	+	0.999029i	$0.485971\pi$
$258$	0	0
$259$	7.15328	0.444483
$260$	0	0
$261$	−5.28487	−0.327125
$262$	0	0
$263$	13.4412	0.828819	0.414409	−	0.910091i	$-0.363988\pi$
$263$	13.4412	0.828819	0.414409	+	0.910091i	$0.363988\pi$
$264$	0	0
$265$	2.65187	0.162903
$266$	0	0
$267$	−19.9663	−1.22192
$268$	0	0
$269$	−0.928854	−0.0566332	−0.0283166	−	0.999599i	$-0.509015\pi$
$269$	−0.928854	−0.0566332	−0.0283166	+	0.999599i	$0.509015\pi$
$270$	0	0
$271$	−24.5529	−1.49148	−0.745741	−	0.666236i	$-0.767904\pi$
$271$	−24.5529	−1.49148	−0.745741	+	0.666236i	$0.767904\pi$
$272$	0	0
$273$	18.9298	1.14569
$274$	0	0
$275$	1.77353	0.106948
$276$	0	0
$277$	−25.6786	−1.54288	−0.771440	−	0.636303i	$-0.780463\pi$
$277$	−25.6786	−1.54288	−0.771440	+	0.636303i	$0.780463\pi$
$278$	0	0
$279$	48.4264	2.89921
$280$	0	0
$281$	3.58861	0.214079	0.107039	−	0.994255i	$-0.465863\pi$
$281$	3.58861	0.214079	0.107039	+	0.994255i	$0.465863\pi$
$282$	0	0
$283$	−20.1752	−1.19929	−0.599645	−	0.800266i	$-0.704692\pi$
$283$	−20.1752	−1.19929	−0.599645	+	0.800266i	$0.704692\pi$
$284$	0	0
$285$	14.8615	0.880320
$286$	0	0
$287$	−25.5065	−1.50560
$288$	0	0
$289$	8.62228	0.507193
$290$	0	0
$291$	−15.6371	−0.916661
$292$	0	0
$293$	−2.98661	−0.174480	−0.0872400	−	0.996187i	$-0.527805\pi$
$293$	−2.98661	−0.174480	−0.0872400	+	0.996187i	$0.527805\pi$
$294$	0	0
$295$	10.9367	0.636762
$296$	0	0
$297$	11.6638	0.676805
$298$	0	0
$299$	18.4124	1.06482
$300$	0	0
$301$	−19.7982	−1.14115
$302$	0	0
$303$	−21.2701	−1.22193
$304$	0	0
$305$	−9.83192	−0.562974
$306$	0	0
$307$	−5.57460	−0.318159	−0.159080	−	0.987266i	$-0.550853\pi$
$307$	−5.57460	−0.318159	−0.159080	+	0.987266i	$0.550853\pi$
$308$	0	0
$309$	−16.7378	−0.952181
$310$	0	0
$311$	−1.21069	−0.0686520	−0.0343260	−	0.999411i	$-0.510928\pi$
$311$	−1.21069	−0.0686520	−0.0343260	+	0.999411i	$0.510928\pi$
$312$	0	0
$313$	27.8574	1.57459	0.787297	−	0.616574i	$-0.211480\pi$
$313$	27.8574	1.57459	0.787297	+	0.616574i	$0.211480\pi$
$314$	0	0
$315$	15.2117	0.857081
$316$	0	0
$317$	−26.7512	−1.50250	−0.751248	−	0.660020i	$-0.770548\pi$
$317$	−26.7512	−1.50250	−0.751248	+	0.660020i	$0.770548\pi$
$318$	0	0
$319$	−1.77353	−0.0992984
$320$	0	0
$321$	−36.6154	−2.04367
$322$	0	0
$323$	−26.1354	−1.45421
$324$	0	0
$325$	2.28487	0.126742
$326$	0	0
$327$	−47.4769	−2.62548
$328$	0	0
$329$	−10.6450	−0.586876
$330$	0	0
$331$	−1.89722	−0.104280	−0.0521402	−	0.998640i	$-0.516604\pi$
$331$	−1.89722	−0.104280	−0.0521402	+	0.998640i	$0.516604\pi$
$332$	0	0
$333$	13.1340	0.719738
$334$	0	0
$335$	−7.90793	−0.432057
$336$	0	0
$337$	34.5831	1.88386	0.941931	−	0.335806i	$-0.109009\pi$
$337$	34.5831	1.88386	0.941931	+	0.335806i	$0.109009\pi$
$338$	0	0
$339$	13.1265	0.712934
$340$	0	0
$341$	16.2512	0.880052
$342$	0	0
$343$	−16.4501	−0.888223
$344$	0	0
$345$	23.1948	1.24877
$346$	0	0
$347$	22.4777	1.20666	0.603332	−	0.797490i	$-0.293839\pi$
$347$	22.4777	1.20666	0.603332	+	0.797490i	$0.293839\pi$
$348$	0	0
$349$	−6.24738	−0.334415	−0.167207	−	0.985922i	$-0.553475\pi$
$349$	−6.24738	−0.334415	−0.167207	+	0.985922i	$0.553475\pi$
$350$	0	0
$351$	15.0268	0.802069
$352$	0	0
$353$	15.3859	0.818907	0.409454	−	0.912331i	$-0.365719\pi$
$353$	15.3859	0.818907	0.409454	+	0.912331i	$0.365719\pi$
$354$	0	0
$355$	−7.75669	−0.411682
$356$	0	0
$357$	−41.9367	−2.21953
$358$	0	0
$359$	−26.5006	−1.39865	−0.699325	−	0.714804i	$-0.746516\pi$
$359$	−26.5006	−1.39865	−0.699325	+	0.714804i	$0.746516\pi$
$360$	0	0
$361$	7.65877	0.403093
$362$	0	0
$363$	−22.6083	−1.18663
$364$	0	0
$365$	−11.1034	−0.581179
$366$	0	0
$367$	7.97935	0.416519	0.208259	−	0.978074i	$-0.433220\pi$
$367$	7.97935	0.416519	0.208259	+	0.978074i	$0.433220\pi$
$368$	0	0
$369$	−46.8319	−2.43797
$370$	0	0
$371$	7.63300	0.396286
$372$	0	0
$373$	4.44605	0.230208	0.115104	−	0.993353i	$-0.463280\pi$
$373$	4.44605	0.230208	0.115104	+	0.993353i	$0.463280\pi$
$374$	0	0
$375$	2.87834	0.148637
$376$	0	0
$377$	−2.28487	−0.117677
$378$	0	0
$379$	−15.4519	−0.793711	−0.396855	−	0.917881i	$-0.629899\pi$
$379$	−15.4519	−0.793711	−0.396855	+	0.917881i	$0.629899\pi$
$380$	0	0
$381$	24.9515	1.27831
$382$	0	0
$383$	10.5251	0.537810	0.268905	−	0.963167i	$-0.413338\pi$
$383$	10.5251	0.537810	0.268905	+	0.963167i	$0.413338\pi$
$384$	0	0
$385$	5.10482	0.260166
$386$	0	0
$387$	−36.3511	−1.84783
$388$	0	0
$389$	13.0120	0.659733	0.329867	−	0.944028i	$-0.392996\pi$
$389$	13.0120	0.659733	0.329867	+	0.944028i	$0.392996\pi$
$390$	0	0
$391$	−40.7904	−2.06286
$392$	0	0
$393$	58.0249	2.92697
$394$	0	0
$395$	−3.61616	−0.181949
$396$	0	0
$397$	−36.3236	−1.82303	−0.911515	−	0.411268i	$-0.865086\pi$
$397$	−36.3236	−1.82303	−0.911515	+	0.411268i	$0.865086\pi$
$398$	0	0
$399$	42.7765	2.14151
$400$	0	0
$401$	32.7082	1.63337	0.816685	−	0.577083i	$-0.195809\pi$
$401$	32.7082	1.63337	0.816685	+	0.577083i	$0.195809\pi$
$402$	0	0
$403$	20.9367	1.04293
$404$	0	0
$405$	3.07523	0.152809
$406$	0	0
$407$	4.40758	0.218475
$408$	0	0
$409$	4.35629	0.215405	0.107702	−	0.994183i	$-0.465651\pi$
$409$	4.35629	0.215405	0.107702	+	0.994183i	$0.465651\pi$
$410$	0	0
$411$	46.3167	2.28463
$412$	0	0
$413$	31.4797	1.54902
$414$	0	0
$415$	−4.66871	−0.229178
$416$	0	0
$417$	−5.77656	−0.282879
$418$	0	0
$419$	12.2742	0.599632	0.299816	−	0.953997i	$-0.403075\pi$
$419$	12.2742	0.599632	0.299816	+	0.953997i	$0.403075\pi$
$420$	0	0
$421$	−17.6478	−0.860100	−0.430050	−	0.902805i	$-0.641504\pi$
$421$	−17.6478	−0.860100	−0.430050	+	0.902805i	$0.641504\pi$
$422$	0	0
$423$	−19.5450	−0.950311
$424$	0	0
$425$	−5.06185	−0.245536
$426$	0	0
$427$	−28.2997	−1.36952
$428$	0	0
$429$	11.6638	0.563136
$430$	0	0
$431$	0.164267	0.00791244	0.00395622	−	0.999992i	$-0.498741\pi$
$431$	0.164267	0.00791244	0.00395622	+	0.999992i	$0.498741\pi$
$432$	0	0
$433$	−0.361514	−0.0173733	−0.00868664	−	0.999962i	$-0.502765\pi$
$433$	−0.361514	−0.0173733	−0.00868664	+	0.999962i	$0.502765\pi$
$434$	0	0
$435$	−2.87834	−0.138006
$436$	0	0
$437$	41.6072	1.99034
$438$	0	0
$439$	29.8329	1.42385	0.711923	−	0.702257i	$-0.247824\pi$
$439$	29.8329	1.42385	0.711923	+	0.702257i	$0.247824\pi$
$440$	0	0
$441$	6.79036	0.323351
$442$	0	0
$443$	10.3946	0.493860	0.246930	−	0.969033i	$-0.420578\pi$
$443$	10.3946	0.493860	0.246930	+	0.969033i	$0.420578\pi$
$444$	0	0
$445$	−6.93674	−0.328833
$446$	0	0
$447$	29.1196	1.37731
$448$	0	0
$449$	−11.3214	−0.534288	−0.267144	−	0.963657i	$-0.586080\pi$
$449$	−11.3214	−0.534288	−0.267144	+	0.963657i	$0.586080\pi$
$450$	0	0
$451$	−15.7161	−0.740043
$452$	0	0
$453$	−14.8130	−0.695978
$454$	0	0
$455$	6.57664	0.308318
$456$	0	0
$457$	2.06833	0.0967523	0.0483762	−	0.998829i	$-0.484595\pi$
$457$	2.06833	0.0967523	0.0483762	+	0.998829i	$0.484595\pi$
$458$	0	0
$459$	−33.2899	−1.55384
$460$	0	0
$461$	−6.31952	−0.294330	−0.147165	−	0.989112i	$-0.547015\pi$
$461$	−6.31952	−0.294330	−0.147165	+	0.989112i	$0.547015\pi$
$462$	0	0
$463$	5.36497	0.249331	0.124666	−	0.992199i	$-0.460214\pi$
$463$	5.36497	0.249331	0.124666	+	0.992199i	$0.460214\pi$
$464$	0	0
$465$	26.3749	1.22311
$466$	0	0
$467$	37.9574	1.75646	0.878229	−	0.478240i	$-0.158725\pi$
$467$	37.9574	1.75646	0.878229	+	0.478240i	$0.158725\pi$
$468$	0	0
$469$	−22.7618	−1.05104
$470$	0	0
$471$	−29.8467	−1.37526
$472$	0	0
$473$	−12.1989	−0.560907
$474$	0	0
$475$	5.16321	0.236904
$476$	0	0
$477$	14.0148	0.641693
$478$	0	0
$479$	22.3234	1.01998	0.509991	−	0.860180i	$-0.329649\pi$
$479$	22.3234	1.01998	0.509991	+	0.860180i	$0.329649\pi$
$480$	0	0
$481$	5.67837	0.258911
$482$	0	0
$483$	66.7627	3.03781
$484$	0	0
$485$	−5.43266	−0.246684
$486$	0	0
$487$	−34.2721	−1.55302	−0.776509	−	0.630107i	$-0.783011\pi$
$487$	−34.2721	−1.55302	−0.776509	+	0.630107i	$0.783011\pi$
$488$	0	0
$489$	47.1612	2.13270
$490$	0	0
$491$	−1.98316	−0.0894989	−0.0447495	−	0.998998i	$-0.514249\pi$
$491$	−1.98316	−0.0894989	−0.0447495	+	0.998998i	$0.514249\pi$
$492$	0	0
$493$	5.06185	0.227974
$494$	0	0
$495$	9.37285	0.421278
$496$	0	0
$497$	−22.3264	−1.00148
$498$	0	0
$499$	24.0831	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$499$	24.0831	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$500$	0	0
$501$	65.1514	2.91075
$502$	0	0
$503$	−42.9801	−1.91639	−0.958193	−	0.286122i	$-0.907634\pi$
$503$	−42.9801	−1.91639	−0.958193	+	0.286122i	$0.907634\pi$
$504$	0	0
$505$	−7.38969	−0.328837
$506$	0	0
$507$	−22.3917	−0.994451
$508$	0	0
$509$	35.2739	1.56349	0.781744	−	0.623600i	$-0.214331\pi$
$509$	35.2739	1.56349	0.781744	+	0.623600i	$0.214331\pi$
$510$	0	0
$511$	−31.9594	−1.41380
$512$	0	0
$513$	33.9566	1.49922
$514$	0	0
$515$	−5.81508	−0.256243
$516$	0	0
$517$	−6.55902	−0.288465
$518$	0	0
$519$	15.1464	0.664852
$520$	0	0
$521$	15.9248	0.697677	0.348838	−	0.937183i	$-0.386576\pi$
$521$	15.9248	0.697677	0.348838	+	0.937183i	$0.386576\pi$
$522$	0	0
$523$	−5.72914	−0.250518	−0.125259	−	0.992124i	$-0.539976\pi$
$523$	−5.72914	−0.250518	−0.125259	+	0.992124i	$0.539976\pi$
$524$	0	0
$525$	8.28487	0.361581
$526$	0	0
$527$	−46.3828	−2.02047
$528$	0	0
$529$	41.9377	1.82338
$530$	0	0
$531$	57.7992	2.50827
$532$	0	0
$533$	−20.2474	−0.877012
$534$	0	0
$535$	−12.7210	−0.549976
$536$	0	0
$537$	−7.51338	−0.324226
$538$	0	0
$539$	2.27875	0.0981526
$540$	0	0
$541$	38.0308	1.63507	0.817537	−	0.575876i	$-0.195339\pi$
$541$	38.0308	1.63507	0.817537	+	0.575876i	$0.195339\pi$
$542$	0	0
$543$	−25.2591	−1.08397
$544$	0	0
$545$	−16.4945	−0.706547
$546$	0	0
$547$	0.504440	0.0215683	0.0107842	−	0.999942i	$-0.496567\pi$
$547$	0.504440	0.0215683	0.0107842	+	0.999942i	$0.496567\pi$
$548$	0	0
$549$	−51.9604	−2.21762
$550$	0	0
$551$	−5.16321	−0.219960
$552$	0	0
$553$	−10.4086	−0.442617
$554$	0	0
$555$	7.15328	0.303640
$556$	0	0
$557$	−5.81587	−0.246426	−0.123213	−	0.992380i	$-0.539320\pi$
$557$	−5.81587	−0.246426	−0.123213	+	0.992380i	$0.539320\pi$
$558$	0	0
$559$	−15.7161	−0.664721
$560$	0	0
$561$	−25.8398	−1.09096
$562$	0	0
$563$	−36.4371	−1.53564	−0.767820	−	0.640665i	$-0.778659\pi$
$563$	−36.4371	−1.53564	−0.767820	+	0.640665i	$0.778659\pi$
$564$	0	0
$565$	4.56044	0.191859
$566$	0	0
$567$	8.85157	0.371731
$568$	0	0
$569$	7.91096	0.331645	0.165822	−	0.986156i	$-0.446972\pi$
$569$	7.91096	0.331645	0.165822	+	0.986156i	$0.446972\pi$
$570$	0	0
$571$	−41.6419	−1.74266	−0.871329	−	0.490699i	$-0.836741\pi$
$571$	−41.6419	−1.74266	−0.871329	+	0.490699i	$0.836741\pi$
$572$	0	0
$573$	58.4779	2.44295
$574$	0	0
$575$	8.05839	0.336058
$576$	0	0
$577$	11.8887	0.494933	0.247466	−	0.968896i	$-0.420402\pi$
$577$	11.8887	0.494933	0.247466	+	0.968896i	$0.420402\pi$
$578$	0	0
$579$	−52.2392	−2.17099
$580$	0	0
$581$	−13.4381	−0.557508
$582$	0	0
$583$	4.70316	0.194785
$584$	0	0
$585$	12.0752	0.499249
$586$	0	0
$587$	−10.3464	−0.427040	−0.213520	−	0.976939i	$-0.568493\pi$
$587$	−10.3464	−0.427040	−0.213520	+	0.976939i	$0.568493\pi$
$588$	0	0
$589$	47.3116	1.94944
$590$	0	0
$591$	29.4067	1.20963
$592$	0	0
$593$	−40.3412	−1.65662	−0.828308	−	0.560274i	$-0.810696\pi$
$593$	−40.3412	−1.65662	−0.828308	+	0.560274i	$0.810696\pi$
$594$	0	0
$595$	−14.5697	−0.597301
$596$	0	0
$597$	50.0734	2.04937
$598$	0	0
$599$	9.43737	0.385600	0.192800	−	0.981238i	$-0.438243\pi$
$599$	9.43737	0.385600	0.192800	+	0.981238i	$0.438243\pi$
$600$	0	0
$601$	35.2216	1.43672	0.718360	−	0.695672i	$-0.244893\pi$
$601$	35.2216	1.43672	0.718360	+	0.695672i	$0.244893\pi$
$602$	0	0
$603$	−41.7924	−1.70192
$604$	0	0
$605$	−7.85461	−0.319335
$606$	0	0
$607$	35.3285	1.43394	0.716969	−	0.697105i	$-0.245529\pi$
$607$	35.3285	1.43394	0.716969	+	0.697105i	$0.245529\pi$
$608$	0	0
$609$	−8.28487	−0.335720
$610$	0	0
$611$	−8.45012	−0.341855
$612$	0	0
$613$	−42.0831	−1.69972	−0.849861	−	0.527008i	$-0.823314\pi$
$613$	−42.0831	−1.69972	−0.849861	+	0.527008i	$0.823314\pi$
$614$	0	0
$615$	−25.5065	−1.02852
$616$	0	0
$617$	33.1963	1.33643	0.668215	−	0.743968i	$-0.267058\pi$
$617$	33.1963	1.33643	0.668215	+	0.743968i	$0.267058\pi$
$618$	0	0
$619$	−10.5123	−0.422526	−0.211263	−	0.977429i	$-0.567758\pi$
$619$	−10.5123	−0.422526	−0.211263	+	0.977429i	$0.567758\pi$
$620$	0	0
$621$	52.9972	2.12670
$622$	0	0
$623$	−19.9663	−0.799934
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	26.3573	1.05261
$628$	0	0
$629$	−12.5797	−0.501587
$630$	0	0
$631$	−21.1841	−0.843327	−0.421663	−	0.906752i	$-0.638554\pi$
$631$	−21.1841	−0.843327	−0.421663	+	0.906752i	$0.638554\pi$
$632$	0	0
$633$	−77.0318	−3.06174
$634$	0	0
$635$	8.66871	0.344007
$636$	0	0
$637$	2.93576	0.116319
$638$	0	0
$639$	−40.9931	−1.62166
$640$	0	0
$641$	−2.92068	−0.115360	−0.0576801	−	0.998335i	$-0.518370\pi$
$641$	−2.92068	−0.115360	−0.0576801	+	0.998335i	$0.518370\pi$
$642$	0	0
$643$	3.32439	0.131101	0.0655506	−	0.997849i	$-0.479120\pi$
$643$	3.32439	0.131101	0.0655506	+	0.997849i	$0.479120\pi$
$644$	0	0
$645$	−19.7982	−0.779555
$646$	0	0
$647$	−33.2693	−1.30795	−0.653975	−	0.756516i	$-0.726900\pi$
$647$	−33.2693	−1.30795	−0.653975	+	0.756516i	$0.726900\pi$
$648$	0	0
$649$	19.3966	0.761383
$650$	0	0
$651$	75.9160	2.97538
$652$	0	0
$653$	−28.6436	−1.12091	−0.560454	−	0.828185i	$-0.689374\pi$
$653$	−28.6436	−1.12091	−0.560454	+	0.828185i	$0.689374\pi$
$654$	0	0
$655$	20.1591	0.787682
$656$	0	0
$657$	−58.6800	−2.28933
$658$	0	0
$659$	−9.03262	−0.351861	−0.175930	−	0.984403i	$-0.556293\pi$
$659$	−9.03262	−0.351861	−0.175930	+	0.984403i	$0.556293\pi$
$660$	0	0
$661$	8.30881	0.323175	0.161588	−	0.986858i	$-0.448339\pi$
$661$	8.30881	0.323175	0.161588	+	0.986858i	$0.448339\pi$
$662$	0	0
$663$	−33.2899	−1.29287
$664$	0	0
$665$	14.8615	0.576305
$666$	0	0
$667$	−8.05839	−0.312022
$668$	0	0
$669$	66.8883	2.58605
$670$	0	0
$671$	−17.4372	−0.673154
$672$	0	0
$673$	−33.9635	−1.30920	−0.654598	−	0.755977i	$-0.727162\pi$
$673$	−33.9635	−1.30920	−0.654598	+	0.755977i	$0.727162\pi$
$674$	0	0
$675$	6.57664	0.253135
$676$	0	0
$677$	−23.2202	−0.892425	−0.446212	−	0.894927i	$-0.647227\pi$
$677$	−23.2202	−0.892425	−0.446212	+	0.894927i	$0.647227\pi$
$678$	0	0
$679$	−15.6371	−0.600096
$680$	0	0
$681$	−46.2216	−1.77122
$682$	0	0
$683$	20.7279	0.793130	0.396565	−	0.918007i	$-0.370202\pi$
$683$	20.7279	0.793130	0.396565	+	0.918007i	$0.370202\pi$
$684$	0	0
$685$	16.0914	0.614822
$686$	0	0
$687$	43.6164	1.66407
$688$	0	0
$689$	6.05918	0.230836
$690$	0	0
$691$	−29.1910	−1.11048	−0.555239	−	0.831691i	$-0.687373\pi$
$691$	−29.1910	−1.11048	−0.555239	+	0.831691i	$0.687373\pi$
$692$	0	0
$693$	26.9783	1.02482
$694$	0	0
$695$	−2.00690	−0.0761261
$696$	0	0
$697$	44.8556	1.69903
$698$	0	0
$699$	28.6865	1.08502
$700$	0	0
$701$	29.2285	1.10395	0.551973	−	0.833862i	$-0.313875\pi$
$701$	29.2285	1.10395	0.551973	+	0.833862i	$0.313875\pi$
$702$	0	0
$703$	12.8316	0.483955
$704$	0	0
$705$	−10.6450	−0.400913
$706$	0	0
$707$	−21.2701	−0.799943
$708$	0	0
$709$	−14.1206	−0.530310	−0.265155	−	0.964206i	$-0.585423\pi$
$709$	−14.1206	−0.530310	−0.265155	+	0.964206i	$0.585423\pi$
$710$	0	0
$711$	−19.1109	−0.716716
$712$	0	0
$713$	73.8408	2.76536
$714$	0	0
$715$	4.05227	0.151546
$716$	0	0
$717$	26.3264	0.983178
$718$	0	0
$719$	−21.8159	−0.813595	−0.406797	−	0.913518i	$-0.633354\pi$
$719$	−21.8159	−0.813595	−0.406797	+	0.913518i	$0.633354\pi$
$720$	0	0
$721$	−16.7378	−0.623349
$722$	0	0
$723$	14.1898	0.527723
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	37.6102	1.39489	0.697443	−	0.716640i	$-0.254321\pi$
$727$	37.6102	1.39489	0.697443	+	0.716640i	$0.254321\pi$
$728$	0	0
$729$	−40.5373	−1.50138
$730$	0	0
$731$	34.8171	1.28776
$732$	0	0
$733$	32.9731	1.21789	0.608944	−	0.793213i	$-0.291593\pi$
$733$	32.9731	1.21789	0.608944	+	0.793213i	$0.291593\pi$
$734$	0	0
$735$	3.69830	0.136414
$736$	0	0
$737$	−14.0249	−0.516615
$738$	0	0
$739$	31.2841	1.15080	0.575402	−	0.817871i	$-0.304846\pi$
$739$	31.2841	1.15080	0.575402	+	0.817871i	$0.304846\pi$
$740$	0	0
$741$	33.9566	1.24743
$742$	0	0
$743$	23.5787	0.865018	0.432509	−	0.901630i	$-0.357628\pi$
$743$	23.5787	0.865018	0.432509	+	0.901630i	$0.357628\pi$
$744$	0	0
$745$	10.1168	0.370650
$746$	0	0
$747$	−24.6735	−0.902756
$748$	0	0
$749$	−36.6154	−1.33790
$750$	0	0
$751$	14.6697	0.535305	0.267652	−	0.963516i	$-0.413752\pi$
$751$	14.6697	0.535305	0.267652	+	0.963516i	$0.413752\pi$
$752$	0	0
$753$	−17.2415	−0.628314
$754$	0	0
$755$	−5.14638	−0.187296
$756$	0	0
$757$	28.0430	1.01924	0.509620	−	0.860400i	$-0.329786\pi$
$757$	28.0430	1.01924	0.509620	+	0.860400i	$0.329786\pi$
$758$	0	0
$759$	41.1366	1.49317
$760$	0	0
$761$	53.4460	1.93742	0.968709	−	0.248201i	$-0.0798393\pi$
$761$	53.4460	1.93742	0.968709	+	0.248201i	$0.0798393\pi$
$762$	0	0
$763$	−47.4769	−1.71878
$764$	0	0
$765$	−26.7512	−0.967191
$766$	0	0
$767$	24.9890	0.902301
$768$	0	0
$769$	−34.7174	−1.25194	−0.625970	−	0.779847i	$-0.715297\pi$
$769$	−34.7174	−1.25194	−0.625970	+	0.779847i	$0.715297\pi$
$770$	0	0
$771$	4.06608	0.146436
$772$	0	0
$773$	−13.5485	−0.487304	−0.243652	−	0.969863i	$-0.578346\pi$
$773$	−13.5485	−0.487304	−0.243652	+	0.969863i	$0.578346\pi$
$774$	0	0
$775$	9.16321	0.329152
$776$	0	0
$777$	20.5896	0.738648
$778$	0	0
$779$	−45.7539	−1.63930
$780$	0	0
$781$	−13.7567	−0.492253
$782$	0	0
$783$	−6.57664	−0.235030
$784$	0	0
$785$	−10.3694	−0.370100
$786$	0	0
$787$	36.0316	1.28439	0.642194	−	0.766542i	$-0.278024\pi$
$787$	36.0316	1.28439	0.642194	+	0.766542i	$0.278024\pi$
$788$	0	0
$789$	38.6883	1.37734
$790$	0	0
$791$	13.1265	0.466725
$792$	0	0
$793$	−22.4646	−0.797743
$794$	0	0
$795$	7.63300	0.270715
$796$	0	0
$797$	−1.17765	−0.0417146	−0.0208573	−	0.999782i	$-0.506640\pi$
$797$	−1.17765	−0.0417146	−0.0208573	+	0.999782i	$0.506640\pi$
$798$	0	0
$799$	18.7202	0.662273
$800$	0	0
$801$	−36.6598	−1.29531
$802$	0	0
$803$	−19.6922	−0.694922
$804$	0	0
$805$	23.1948	0.817511
$806$	0	0
$807$	−2.67356	−0.0941138
$808$	0	0
$809$	−50.8020	−1.78610	−0.893052	−	0.449953i	$-0.851441\pi$
$809$	−50.8020	−1.78610	−0.893052	+	0.449953i	$0.851441\pi$
$810$	0	0
$811$	−40.7558	−1.43113	−0.715566	−	0.698546i	$-0.753831\pi$
$811$	−40.7558	−1.43113	−0.715566	+	0.698546i	$0.753831\pi$
$812$	0	0
$813$	−70.6717	−2.47857
$814$	0	0
$815$	16.3848	0.573935
$816$	0	0
$817$	−35.5144	−1.24249
$818$	0	0
$819$	34.7567	1.21450
$820$	0	0
$821$	3.77812	0.131857	0.0659287	−	0.997824i	$-0.478999\pi$
$821$	3.77812	0.131857	0.0659287	+	0.997824i	$0.478999\pi$
$822$	0	0
$823$	1.60828	0.0560610	0.0280305	−	0.999607i	$-0.491076\pi$
$823$	1.60828	0.0560610	0.0280305	+	0.999607i	$0.491076\pi$
$824$	0	0
$825$	5.10482	0.177727
$826$	0	0
$827$	−44.7320	−1.55548	−0.777741	−	0.628585i	$-0.783635\pi$
$827$	−44.7320	−1.55548	−0.777741	+	0.628585i	$0.783635\pi$
$828$	0	0
$829$	7.54396	0.262013	0.131006	−	0.991382i	$-0.458179\pi$
$829$	7.54396	0.262013	0.131006	+	0.991382i	$0.458179\pi$
$830$	0	0
$831$	−73.9119	−2.56398
$832$	0	0
$833$	−6.50381	−0.225344
$834$	0	0
$835$	22.6350	0.783318
$836$	0	0
$837$	60.2632	2.08300
$838$	0	0
$839$	−12.5343	−0.432732	−0.216366	−	0.976312i	$-0.569420\pi$
$839$	−12.5343	−0.432732	−0.216366	+	0.976312i	$0.569420\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	10.3293	0.355758
$844$	0	0
$845$	−7.77938	−0.267619
$846$	0	0
$847$	−22.6083	−0.776829
$848$	0	0
$849$	−58.0711	−1.99300
$850$	0	0
$851$	20.0268	0.686509
$852$	0	0
$853$	−13.6344	−0.466833	−0.233417	−	0.972377i	$-0.574991\pi$
$853$	−13.6344	−0.466833	−0.233417	+	0.972377i	$0.574991\pi$
$854$	0	0
$855$	27.2869	0.933192
$856$	0	0
$857$	35.3769	1.20845	0.604225	−	0.796813i	$-0.293483\pi$
$857$	35.3769	1.20845	0.604225	+	0.796813i	$0.293483\pi$
$858$	0	0
$859$	−5.35708	−0.182781	−0.0913906	−	0.995815i	$-0.529131\pi$
$859$	−5.35708	−0.182781	−0.0913906	+	0.995815i	$0.529131\pi$
$860$	0	0
$861$	−73.4164	−2.50203
$862$	0	0
$863$	0.452167	0.0153919	0.00769597	−	0.999970i	$-0.497550\pi$
$863$	0.452167	0.0153919	0.00769597	+	0.999970i	$0.497550\pi$
$864$	0	0
$865$	5.26218	0.178920
$866$	0	0
$867$	24.8179	0.842860
$868$	0	0
$869$	−6.41336	−0.217558
$870$	0	0
$871$	−18.0686	−0.612231
$872$	0	0
$873$	−28.7109	−0.971717
$874$	0	0
$875$	2.87834	0.0973058
$876$	0	0
$877$	31.7913	1.07352	0.536759	−	0.843736i	$-0.319649\pi$
$877$	31.7913	1.07352	0.536759	+	0.843736i	$0.319649\pi$
$878$	0	0
$879$	−8.59651	−0.289953
$880$	0	0
$881$	−24.0107	−0.808942	−0.404471	−	0.914551i	$-0.632544\pi$
$881$	−24.0107	−0.808942	−0.404471	+	0.914551i	$0.632544\pi$
$882$	0	0
$883$	−27.2976	−0.918638	−0.459319	−	0.888271i	$-0.651907\pi$
$883$	−27.2976	−0.918638	−0.459319	+	0.888271i	$0.651907\pi$
$884$	0	0
$885$	31.4797	1.05818
$886$	0	0
$887$	−22.3395	−0.750085	−0.375043	−	0.927008i	$-0.622372\pi$
$887$	−22.3395	−0.750085	−0.375043	+	0.927008i	$0.622372\pi$
$888$	0	0
$889$	24.9515	0.836847
$890$	0	0
$891$	5.45400	0.182716
$892$	0	0
$893$	−19.0951	−0.638993
$894$	0	0
$895$	−2.61031	−0.0872531
$896$	0	0
$897$	52.9972	1.76952
$898$	0	0
$899$	−9.16321	−0.305610
$900$	0	0
$901$	−13.4234	−0.447197
$902$	0	0
$903$	−56.9862	−1.89638
$904$	0	0
$905$	−8.77556	−0.291710
$906$	0	0
$907$	−29.5195	−0.980179	−0.490089	−	0.871672i	$-0.663036\pi$
$907$	−29.5195	−0.980179	−0.490089	+	0.871672i	$0.663036\pi$
$908$	0	0
$909$	−39.0535	−1.29532
$910$	0	0
$911$	−46.3137	−1.53444	−0.767220	−	0.641384i	$-0.778361\pi$
$911$	−46.3137	−1.53444	−0.767220	+	0.641384i	$0.778361\pi$
$912$	0	0
$913$	−8.28007	−0.274030
$914$	0	0
$915$	−28.2997	−0.935558
$916$	0	0
$917$	58.0249	1.91615
$918$	0	0
$919$	40.1869	1.32564	0.662822	−	0.748777i	$-0.269358\pi$
$919$	40.1869	1.32564	0.662822	+	0.748777i	$0.269358\pi$
$920$	0	0
$921$	−16.0456	−0.528722
$922$	0	0
$923$	−17.7230	−0.583360
$924$	0	0
$925$	2.48521	0.0817131
$926$	0	0
$927$	−30.7320	−1.00937
$928$	0	0
$929$	17.4797	0.573491	0.286745	−	0.958007i	$-0.407427\pi$
$929$	17.4797	0.573491	0.286745	+	0.958007i	$0.407427\pi$
$930$	0	0
$931$	6.63405	0.217422
$932$	0	0
$933$	−3.48479	−0.114087
$934$	0	0
$935$	−8.97731	−0.293590
$936$	0	0
$937$	35.8388	1.17080	0.585402	−	0.810743i	$-0.300937\pi$
$937$	35.8388	1.17080	0.585402	+	0.810743i	$0.300937\pi$
$938$	0	0
$939$	80.1833	2.61668
$940$	0	0
$941$	7.55875	0.246408	0.123204	−	0.992381i	$-0.460683\pi$
$941$	7.55875	0.246408	0.123204	+	0.992381i	$0.460683\pi$
$942$	0	0
$943$	−71.4095	−2.32541
$944$	0	0
$945$	18.9298	0.615788
$946$	0	0
$947$	27.3285	0.888056	0.444028	−	0.896013i	$-0.353549\pi$
$947$	27.3285	0.888056	0.444028	+	0.896013i	$0.353549\pi$
$948$	0	0
$949$	−25.3698	−0.823539
$950$	0	0
$951$	−76.9991	−2.49687
$952$	0	0
$953$	42.1586	1.36565	0.682826	−	0.730581i	$-0.260751\pi$
$953$	42.1586	1.36565	0.682826	+	0.730581i	$0.260751\pi$
$954$	0	0
$955$	20.3165	0.657426
$956$	0	0
$957$	−5.10482	−0.165015
$958$	0	0
$959$	46.3167	1.49564
$960$	0	0
$961$	52.9645	1.70853
$962$	0	0
$963$	−67.2287	−2.16641
$964$	0	0
$965$	−18.1490	−0.584238
$966$	0	0
$967$	22.8645	0.735274	0.367637	−	0.929969i	$-0.380167\pi$
$967$	22.8645	0.735274	0.367637	+	0.929969i	$0.380167\pi$
$968$	0	0
$969$	−75.2267	−2.41663
$970$	0	0
$971$	−32.2394	−1.03461	−0.517306	−	0.855800i	$-0.673065\pi$
$971$	−32.2394	−1.03461	−0.517306	+	0.855800i	$0.673065\pi$
$972$	0	0
$973$	−5.77656	−0.185188
$974$	0	0
$975$	6.57664	0.210621
$976$	0	0
$977$	−18.3879	−0.588280	−0.294140	−	0.955762i	$-0.595033\pi$
$977$	−18.3879	−0.588280	−0.294140	+	0.955762i	$0.595033\pi$
$978$	0	0
$979$	−12.3025	−0.393189
$980$	0	0
$981$	−87.1713	−2.78317
$982$	0	0
$983$	−19.2838	−0.615058	−0.307529	−	0.951539i	$-0.599502\pi$
$983$	−19.2838	−0.615058	−0.307529	+	0.951539i	$0.599502\pi$
$984$	0	0
$985$	10.2165	0.325526
$986$	0	0
$987$	−30.6399	−0.975278
$988$	0	0
$989$	−55.4284	−1.76252
$990$	0	0
$991$	54.1971	1.72163	0.860813	−	0.508921i	$-0.169955\pi$
$991$	54.1971	1.72163	0.860813	+	0.508921i	$0.169955\pi$
$992$	0	0
$993$	−5.46085	−0.173295
$994$	0	0
$995$	17.3966	0.551509
$996$	0	0
$997$	−36.3288	−1.15055	−0.575273	−	0.817962i	$-0.695104\pi$
$997$	−36.3288	−1.15055	−0.575273	+	0.817962i	$0.695104\pi$
$998$	0	0
$999$	16.3443	0.517111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.bz.1.4		4
4.3	odd	2		9280.2.a.cf.1.1		4
8.3	odd	2		4640.2.a.l.1.4	✓	4
8.5	even	2		4640.2.a.r.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.l.1.4	✓	4	8.3	odd	2
4640.2.a.r.1.1	yes	4	8.5	even	2
9280.2.a.bz.1.4		4	1.1	even	1	trivial
9280.2.a.cf.1.1		4	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)

\(f(q)\)	\(=\)	\(q+2.87834 q^{3} +1.00000 q^{5} +2.87834 q^{7} +5.28487 q^{9} +1.77353 q^{11} +2.28487 q^{13} +2.87834 q^{15} -5.06185 q^{17} +5.16321 q^{19} +8.28487 q^{21} +8.05839 q^{23} +1.00000 q^{25} +6.57664 q^{27} -1.00000 q^{29} +9.16321 q^{31} +5.10482 q^{33} +2.87834 q^{35} +2.48521 q^{37} +6.57664 q^{39} -8.86151 q^{41} -6.87834 q^{43} +5.28487 q^{45} -3.69830 q^{47} +1.28487 q^{49} -14.5697 q^{51} +2.65187 q^{53} +1.77353 q^{55} +14.8615 q^{57} +10.9367 q^{59} -9.83192 q^{61} +15.2117 q^{63} +2.28487 q^{65} -7.90793 q^{67} +23.1948 q^{69} -7.75669 q^{71} -11.1034 q^{73} +2.87834 q^{75} +5.10482 q^{77} -3.61616 q^{79} +3.07523 q^{81} -4.66871 q^{83} -5.06185 q^{85} -2.87834 q^{87} -6.93674 q^{89} +6.57664 q^{91} +26.3749 q^{93} +5.16321 q^{95} -5.43266 q^{97} +9.37285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 8 q^{9} + 2 q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{17} - 6 q^{19} + 20 q^{21} + 14 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 10 q^{31} + 12 q^{33} - 2 q^{35} + 16 q^{37}+ \cdots - 6 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 8 * q^9 + 2 * q^11 - 4 * q^13 - 2 * q^15 - 4 * q^17 - 6 * q^19 + 20 * q^21 + 14 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 16 * q^37 + 4 * q^39 - 14 * q^43 + 8 * q^45 - 6 * q^47 - 8 * q^49 - 32 * q^51 - 8 * q^53 + 2 * q^55 + 24 * q^57 + 12 * q^59 - 16 * q^61 - 2 * q^63 - 4 * q^65 - 6 * q^67 + 8 * q^69 - 4 * q^71 + 12 * q^73 - 2 * q^75 + 12 * q^77 + 2 * q^79 + 16 * q^81 - 22 * q^83 - 4 * q^85 + 2 * q^87 + 4 * q^89 + 4 * q^91 + 16 * q^93 - 6 * q^95 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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