LMFDB - Embedded newform 5488.2.a.h.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] 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("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-5,0,11,0,0,0,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.8219006293$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1279733.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 $ x^6 - 2x^5 - 6x^4 + 10x^3 + 10x^2 - 11*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0849355$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40003 q^{3} +3.29412 q^{5} -1.03992 q^{9} -1.30976 q^{11} +2.58875 q^{13} +4.61185 q^{15} +5.10591 q^{17} -6.41058 q^{19} +5.61669 q^{23} +5.85120 q^{25} -5.65600 q^{27} +5.85067 q^{29} +5.23043 q^{31} -1.83370 q^{33} +8.05400 q^{37} +3.62433 q^{39} -1.40563 q^{41} -3.50061 q^{43} -3.42562 q^{45} +4.81866 q^{47} +7.14842 q^{51} -10.2983 q^{53} -4.31450 q^{55} -8.97499 q^{57} -4.83173 q^{59} -12.2149 q^{61} +8.52766 q^{65} +8.93174 q^{67} +7.86353 q^{69} +7.60661 q^{71} +3.59628 q^{73} +8.19184 q^{75} +7.58967 q^{79} -4.79880 q^{81} -5.15263 q^{83} +16.8195 q^{85} +8.19111 q^{87} +3.39197 q^{89} +7.32275 q^{93} -21.1172 q^{95} +9.10089 q^{97} +1.36205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 11 q^{5} + 9 q^{9} + q^{11} + 7 q^{13} + 5 q^{15} + 26 q^{17} + 3 q^{19} + 9 q^{23} + 15 q^{25} - 8 q^{27} - 2 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{37} - 7 q^{39} + 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 + 11 * q^5 + 9 * q^9 + q^11 + 7 * q^13 + 5 * q^15 + 26 * q^17 + 3 * q^19 + 9 * q^23 + 15 * q^25 - 8 * q^27 - 2 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^37 - 7 * q^39 + 28 * q^41 - 5 * q^43 + 3 * q^45 - 18 * q^47 - 13 * q^51 - q^53 + 22 * q^55 - 26 * q^57 - 5 * q^59 - 17 * q^61 - 14 * q^65 + 22 * q^67 - 20 * q^69 + 2 * q^71 + 12 * q^73 + 27 * q^75 + 2 * q^79 - 30 * q^81 - 7 * q^83 + 37 * q^85 + 25 * q^87 + 39 * q^89 - 22 * q^93 + 11 * q^95 + 35 * q^97 - 12 * 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
$3$	1.40003	0.808307	0.404153	−	0.914691i	$-0.367566\pi$
$3$	1.40003	0.808307	0.404153	+	0.914691i	$0.367566\pi$
$4$	0	0
$5$	3.29412	1.47317	0.736587	−	0.676343i	$-0.236436\pi$
$5$	3.29412	1.47317	0.736587	+	0.676343i	$0.236436\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.03992	−0.346641
$10$	0	0
$11$	−1.30976	−0.394908	−0.197454	−	0.980312i	$-0.563267\pi$
$11$	−1.30976	−0.394908	−0.197454	+	0.980312i	$0.563267\pi$
$12$	0	0
$13$	2.58875	0.717991	0.358996	−	0.933339i	$-0.383119\pi$
$13$	2.58875	0.717991	0.358996	+	0.933339i	$0.383119\pi$
$14$	0	0
$15$	4.61185	1.19078
$16$	0	0
$17$	5.10591	1.23837	0.619183	−	0.785247i	$-0.287464\pi$
$17$	5.10591	1.23837	0.619183	+	0.785247i	$0.287464\pi$
$18$	0	0
$19$	−6.41058	−1.47069	−0.735344	−	0.677694i	$-0.762979\pi$
$19$	−6.41058	−1.47069	−0.735344	+	0.677694i	$0.762979\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.61669	1.17116	0.585581	−	0.810614i	$-0.300866\pi$
$23$	5.61669	1.17116	0.585581	+	0.810614i	$0.300866\pi$
$24$	0	0
$25$	5.85120	1.17024
$26$	0	0
$27$	−5.65600	−1.08850
$28$	0	0
$29$	5.85067	1.08644	0.543222	−	0.839589i	$-0.317204\pi$
$29$	5.85067	1.08644	0.543222	+	0.839589i	$0.317204\pi$
$30$	0	0
$31$	5.23043	0.939413	0.469706	−	0.882823i	$-0.344360\pi$
$31$	5.23043	0.939413	0.469706	+	0.882823i	$0.344360\pi$
$32$	0	0
$33$	−1.83370	−0.319206
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.05400	1.32407	0.662035	−	0.749473i	$-0.269693\pi$
$37$	8.05400	1.32407	0.662035	+	0.749473i	$0.269693\pi$
$38$	0	0
$39$	3.62433	0.580357
$40$	0	0
$41$	−1.40563	−0.219523	−0.109761	−	0.993958i	$-0.535009\pi$
$41$	−1.40563	−0.219523	−0.109761	+	0.993958i	$0.535009\pi$
$42$	0	0
$43$	−3.50061	−0.533838	−0.266919	−	0.963719i	$-0.586006\pi$
$43$	−3.50061	−0.533838	−0.266919	+	0.963719i	$0.586006\pi$
$44$	0	0
$45$	−3.42562	−0.510662
$46$	0	0
$47$	4.81866	0.702874	0.351437	−	0.936212i	$-0.385693\pi$
$47$	4.81866	0.702874	0.351437	+	0.936212i	$0.385693\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.14842	1.00098
$52$	0	0
$53$	−10.2983	−1.41458	−0.707290	−	0.706924i	$-0.750082\pi$
$53$	−10.2983	−1.41458	−0.707290	+	0.706924i	$0.750082\pi$
$54$	0	0
$55$	−4.31450	−0.581767
$56$	0	0
$57$	−8.97499	−1.18877
$58$	0	0
$59$	−4.83173	−0.629038	−0.314519	−	0.949251i	$-0.601843\pi$
$59$	−4.83173	−0.629038	−0.314519	+	0.949251i	$0.601843\pi$
$60$	0	0
$61$	−12.2149	−1.56395	−0.781977	−	0.623308i	$-0.785788\pi$
$61$	−12.2149	−1.56395	−0.781977	+	0.623308i	$0.785788\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.52766	1.05773
$66$	0	0
$67$	8.93174	1.09119	0.545593	−	0.838050i	$-0.316304\pi$
$67$	8.93174	1.09119	0.545593	+	0.838050i	$0.316304\pi$
$68$	0	0
$69$	7.86353	0.946658
$70$	0	0
$71$	7.60661	0.902739	0.451369	−	0.892337i	$-0.350936\pi$
$71$	7.60661	0.902739	0.451369	+	0.892337i	$0.350936\pi$
$72$	0	0
$73$	3.59628	0.420913	0.210456	−	0.977603i	$-0.432505\pi$
$73$	3.59628	0.420913	0.210456	+	0.977603i	$0.432505\pi$
$74$	0	0
$75$	8.19184	0.945912
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.58967	0.853905	0.426952	−	0.904274i	$-0.359587\pi$
$79$	7.58967	0.853905	0.426952	+	0.904274i	$0.359587\pi$
$80$	0	0
$81$	−4.79880	−0.533200
$82$	0	0
$83$	−5.15263	−0.565575	−0.282788	−	0.959183i	$-0.591259\pi$
$83$	−5.15263	−0.565575	−0.282788	+	0.959183i	$0.591259\pi$
$84$	0	0
$85$	16.8195	1.82433
$86$	0	0
$87$	8.19111	0.878179
$88$	0	0
$89$	3.39197	0.359548	0.179774	−	0.983708i	$-0.442463\pi$
$89$	3.39197	0.359548	0.179774	+	0.983708i	$0.442463\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.32275	0.759334
$94$	0	0
$95$	−21.1172	−2.16658
$96$	0	0
$97$	9.10089	0.924056	0.462028	−	0.886865i	$-0.347122\pi$
$97$	9.10089	0.924056	0.462028	+	0.886865i	$0.347122\pi$
$98$	0	0
$99$	1.36205	0.136891
$100$	0	0
$101$	18.4360	1.83446	0.917228	−	0.398363i	$-0.130422\pi$
$101$	18.4360	1.83446	0.917228	+	0.398363i	$0.130422\pi$
$102$	0	0
$103$	10.9284	1.07681	0.538403	−	0.842687i	$-0.319028\pi$
$103$	10.9284	1.07681	0.538403	+	0.842687i	$0.319028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9747	−1.06096	−0.530482	−	0.847696i	$-0.677989\pi$
$107$	−10.9747	−1.06096	−0.530482	+	0.847696i	$0.677989\pi$
$108$	0	0
$109$	9.80577	0.939222	0.469611	−	0.882873i	$-0.344394\pi$
$109$	9.80577	0.939222	0.469611	+	0.882873i	$0.344394\pi$
$110$	0	0
$111$	11.2758	1.07025
$112$	0	0
$113$	−3.12877	−0.294330	−0.147165	−	0.989112i	$-0.547015\pi$
$113$	−3.12877	−0.294330	−0.147165	+	0.989112i	$0.547015\pi$
$114$	0	0
$115$	18.5020	1.72532
$116$	0	0
$117$	−2.69210	−0.248885
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.28453	−0.844048
$122$	0	0
$123$	−1.96793	−0.177442
$124$	0	0
$125$	2.80393	0.250792
$126$	0	0
$127$	1.62243	0.143967	0.0719835	−	0.997406i	$-0.477067\pi$
$127$	1.62243	0.143967	0.0719835	+	0.997406i	$0.477067\pi$
$128$	0	0
$129$	−4.90095	−0.431505
$130$	0	0
$131$	−8.95123	−0.782072	−0.391036	−	0.920375i	$-0.627883\pi$
$131$	−8.95123	−0.782072	−0.391036	+	0.920375i	$0.627883\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−18.6315	−1.60355
$136$	0	0
$137$	2.37668	0.203053	0.101527	−	0.994833i	$-0.467627\pi$
$137$	2.37668	0.203053	0.101527	+	0.994833i	$0.467627\pi$
$138$	0	0
$139$	3.18945	0.270525	0.135263	−	0.990810i	$-0.456812\pi$
$139$	3.18945	0.270525	0.135263	+	0.990810i	$0.456812\pi$
$140$	0	0
$141$	6.74626	0.568138
$142$	0	0
$143$	−3.39065	−0.283540
$144$	0	0
$145$	19.2728	1.60052
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.0375	−1.06808	−0.534038	−	0.845460i	$-0.679326\pi$
$149$	−13.0375	−1.06808	−0.534038	+	0.845460i	$0.679326\pi$
$150$	0	0
$151$	2.56546	0.208774	0.104387	−	0.994537i	$-0.466712\pi$
$151$	2.56546	0.208774	0.104387	+	0.994537i	$0.466712\pi$
$152$	0	0
$153$	−5.30975	−0.429268
$154$	0	0
$155$	17.2296	1.38392
$156$	0	0
$157$	8.67435	0.692289	0.346144	−	0.938181i	$-0.387491\pi$
$157$	8.67435	0.692289	0.346144	+	0.938181i	$0.387491\pi$
$158$	0	0
$159$	−14.4179	−1.14341
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.06962	0.475409	0.237705	−	0.971337i	$-0.423605\pi$
$163$	6.06962	0.475409	0.237705	+	0.971337i	$0.423605\pi$
$164$	0	0
$165$	−6.04042	−0.470246
$166$	0	0
$167$	−15.3746	−1.18972	−0.594862	−	0.803828i	$-0.702793\pi$
$167$	−15.3746	−1.18972	−0.594862	+	0.803828i	$0.702793\pi$
$168$	0	0
$169$	−6.29835	−0.484488
$170$	0	0
$171$	6.66650	0.509800
$172$	0	0
$173$	8.31524	0.632196	0.316098	−	0.948726i	$-0.397627\pi$
$173$	8.31524	0.632196	0.316098	+	0.948726i	$0.397627\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.76456	−0.508455
$178$	0	0
$179$	15.0266	1.12314	0.561569	−	0.827430i	$-0.310198\pi$
$179$	15.0266	1.12314	0.561569	+	0.827430i	$0.310198\pi$
$180$	0	0
$181$	−23.4392	−1.74222	−0.871112	−	0.491084i	$-0.836601\pi$
$181$	−23.4392	−1.74222	−0.871112	+	0.491084i	$0.836601\pi$
$182$	0	0
$183$	−17.1012	−1.26415
$184$	0	0
$185$	26.5308	1.95058
$186$	0	0
$187$	−6.68752	−0.489040
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.82469	−0.493817	−0.246909	−	0.969039i	$-0.579415\pi$
$191$	−6.82469	−0.493817	−0.246909	+	0.969039i	$0.579415\pi$
$192$	0	0
$193$	6.10039	0.439116	0.219558	−	0.975599i	$-0.429539\pi$
$193$	6.10039	0.439116	0.219558	+	0.975599i	$0.429539\pi$
$194$	0	0
$195$	11.9390	0.854966
$196$	0	0
$197$	14.0840	1.00344	0.501722	−	0.865029i	$-0.332700\pi$
$197$	14.0840	1.00344	0.501722	+	0.865029i	$0.332700\pi$
$198$	0	0
$199$	20.4947	1.45283	0.726416	−	0.687255i	$-0.241185\pi$
$199$	20.4947	1.45283	0.726416	+	0.687255i	$0.241185\pi$
$200$	0	0
$201$	12.5047	0.882013
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.63032	−0.323395
$206$	0	0
$207$	−5.84092	−0.405972
$208$	0	0
$209$	8.39632	0.580786
$210$	0	0
$211$	−1.47045	−0.101230	−0.0506149	−	0.998718i	$-0.516118\pi$
$211$	−1.47045	−0.101230	−0.0506149	+	0.998718i	$0.516118\pi$
$212$	0	0
$213$	10.6495	0.729689
$214$	0	0
$215$	−11.5314	−0.786435
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.03490	0.340227
$220$	0	0
$221$	13.2180	0.889136
$222$	0	0
$223$	−5.47586	−0.366691	−0.183345	−	0.983049i	$-0.558693\pi$
$223$	−5.47586	−0.366691	−0.183345	+	0.983049i	$0.558693\pi$
$224$	0	0
$225$	−6.08478	−0.405652
$226$	0	0
$227$	5.03683	0.334306	0.167153	−	0.985931i	$-0.446543\pi$
$227$	5.03683	0.334306	0.167153	+	0.985931i	$0.446543\pi$
$228$	0	0
$229$	−21.6621	−1.43147	−0.715736	−	0.698370i	$-0.753909\pi$
$229$	−21.6621	−1.43147	−0.715736	+	0.698370i	$0.753909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.3274	−1.06964	−0.534821	−	0.844966i	$-0.679621\pi$
$233$	−16.3274	−1.06964	−0.534821	+	0.844966i	$0.679621\pi$
$234$	0	0
$235$	15.8732	1.03545
$236$	0	0
$237$	10.6258	0.690217
$238$	0	0
$239$	6.61138	0.427655	0.213827	−	0.976871i	$-0.431407\pi$
$239$	6.61138	0.427655	0.213827	+	0.976871i	$0.431407\pi$
$240$	0	0
$241$	−1.00382	−0.0646619	−0.0323310	−	0.999477i	$-0.510293\pi$
$241$	−1.00382	−0.0646619	−0.0323310	+	0.999477i	$0.510293\pi$
$242$	0	0
$243$	10.2496	0.657509
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.5954	−1.05594
$248$	0	0
$249$	−7.21383	−0.457158
$250$	0	0
$251$	27.6782	1.74703	0.873516	−	0.486796i	$-0.161835\pi$
$251$	27.6782	1.74703	0.873516	+	0.486796i	$0.161835\pi$
$252$	0	0
$253$	−7.35652	−0.462501
$254$	0	0
$255$	23.5477	1.47462
$256$	0	0
$257$	7.67171	0.478548	0.239274	−	0.970952i	$-0.423091\pi$
$257$	7.67171	0.478548	0.239274	+	0.970952i	$0.423091\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.08424	−0.376605
$262$	0	0
$263$	−10.6366	−0.655878	−0.327939	−	0.944699i	$-0.606354\pi$
$263$	−10.6366	−0.655878	−0.327939	+	0.944699i	$0.606354\pi$
$264$	0	0
$265$	−33.9238	−2.08392
$266$	0	0
$267$	4.74885	0.290625
$268$	0	0
$269$	2.92590	0.178395	0.0891975	−	0.996014i	$-0.471570\pi$
$269$	2.92590	0.178395	0.0891975	+	0.996014i	$0.471570\pi$
$270$	0	0
$271$	1.97491	0.119968	0.0599838	−	0.998199i	$-0.480895\pi$
$271$	1.97491	0.119968	0.0599838	+	0.998199i	$0.480895\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.66366	−0.462136
$276$	0	0
$277$	6.82618	0.410145	0.205073	−	0.978747i	$-0.434257\pi$
$277$	6.82618	0.410145	0.205073	+	0.978747i	$0.434257\pi$
$278$	0	0
$279$	−5.43924	−0.325639
$280$	0	0
$281$	−27.8219	−1.65971	−0.829857	−	0.557977i	$-0.811578\pi$
$281$	−27.8219	−1.65971	−0.829857	+	0.557977i	$0.811578\pi$
$282$	0	0
$283$	8.27801	0.492076	0.246038	−	0.969260i	$-0.420871\pi$
$283$	8.27801	0.492076	0.246038	+	0.969260i	$0.420871\pi$
$284$	0	0
$285$	−29.5647	−1.75126
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.07035	0.533550
$290$	0	0
$291$	12.7415	0.746920
$292$	0	0
$293$	−6.28753	−0.367321	−0.183661	−	0.982990i	$-0.558795\pi$
$293$	−6.28753	−0.367321	−0.183661	+	0.982990i	$0.558795\pi$
$294$	0	0
$295$	−15.9163	−0.926681
$296$	0	0
$297$	7.40801	0.429856
$298$	0	0
$299$	14.5402	0.840884
$300$	0	0
$301$	0	0
$302$	0	0
$303$	25.8110	1.48280
$304$	0	0
$305$	−40.2372	−2.30397
$306$	0	0
$307$	−13.8159	−0.788514	−0.394257	−	0.919000i	$-0.628998\pi$
$307$	−13.8159	−0.788514	−0.394257	+	0.919000i	$0.628998\pi$
$308$	0	0
$309$	15.3001	0.870390
$310$	0	0
$311$	−10.1060	−0.573060	−0.286530	−	0.958071i	$-0.592502\pi$
$311$	−10.1060	−0.573060	−0.286530	+	0.958071i	$0.592502\pi$
$312$	0	0
$313$	−11.9191	−0.673710	−0.336855	−	0.941557i	$-0.609363\pi$
$313$	−11.9191	−0.673710	−0.336855	+	0.941557i	$0.609363\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.2134	−1.47229	−0.736145	−	0.676823i	$-0.763356\pi$
$317$	−26.2134	−1.47229	−0.736145	+	0.676823i	$0.763356\pi$
$318$	0	0
$319$	−7.66298	−0.429045
$320$	0	0
$321$	−15.3649	−0.857584
$322$	0	0
$323$	−32.7319	−1.82125
$324$	0	0
$325$	15.1473	0.840221
$326$	0	0
$327$	13.7283	0.759179
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.2728	−0.894435	−0.447218	−	0.894425i	$-0.647585\pi$
$331$	−16.2728	−0.894435	−0.447218	+	0.894425i	$0.647585\pi$
$332$	0	0
$333$	−8.37553	−0.458976
$334$	0	0
$335$	29.4222	1.60751
$336$	0	0
$337$	−22.1009	−1.20391	−0.601955	−	0.798530i	$-0.705612\pi$
$337$	−22.1009	−1.20391	−0.601955	+	0.798530i	$0.705612\pi$
$338$	0	0
$339$	−4.38037	−0.237909
$340$	0	0
$341$	−6.85061	−0.370981
$342$	0	0
$343$	0	0
$344$	0	0
$345$	25.9034	1.39459
$346$	0	0
$347$	−20.2320	−1.08611	−0.543054	−	0.839698i	$-0.682732\pi$
$347$	−20.2320	−1.08611	−0.543054	+	0.839698i	$0.682732\pi$
$348$	0	0
$349$	20.3826	1.09105	0.545527	−	0.838093i	$-0.316330\pi$
$349$	20.3826	1.09105	0.545527	+	0.838093i	$0.316330\pi$
$350$	0	0
$351$	−14.6420	−0.781532
$352$	0	0
$353$	3.17871	0.169186	0.0845929	−	0.996416i	$-0.473041\pi$
$353$	3.17871	0.169186	0.0845929	+	0.996416i	$0.473041\pi$
$354$	0	0
$355$	25.0570	1.32989
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.33067	0.228564	0.114282	−	0.993448i	$-0.463543\pi$
$359$	4.33067	0.228564	0.114282	+	0.993448i	$0.463543\pi$
$360$	0	0
$361$	22.0955	1.16292
$362$	0	0
$363$	−12.9986	−0.682250
$364$	0	0
$365$	11.8466	0.620078
$366$	0	0
$367$	−10.7431	−0.560786	−0.280393	−	0.959885i	$-0.590465\pi$
$367$	−10.7431	−0.560786	−0.280393	+	0.959885i	$0.590465\pi$
$368$	0	0
$369$	1.46175	0.0760956
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.2207	0.632762	0.316381	−	0.948632i	$-0.397532\pi$
$373$	12.2207	0.632762	0.316381	+	0.948632i	$0.397532\pi$
$374$	0	0
$375$	3.92559	0.202716
$376$	0	0
$377$	15.1460	0.780057
$378$	0	0
$379$	11.3737	0.584227	0.292113	−	0.956384i	$-0.405642\pi$
$379$	11.3737	0.584227	0.292113	+	0.956384i	$0.405642\pi$
$380$	0	0
$381$	2.27144	0.116369
$382$	0	0
$383$	−24.5408	−1.25398	−0.626988	−	0.779029i	$-0.715713\pi$
$383$	−24.5408	−1.25398	−0.626988	+	0.779029i	$0.715713\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.64036	0.185050
$388$	0	0
$389$	14.3914	0.729674	0.364837	−	0.931071i	$-0.381125\pi$
$389$	14.3914	0.729674	0.364837	+	0.931071i	$0.381125\pi$
$390$	0	0
$391$	28.6784	1.45033
$392$	0	0
$393$	−12.5320	−0.632154
$394$	0	0
$395$	25.0013	1.25795
$396$	0	0
$397$	−2.74397	−0.137716	−0.0688579	−	0.997626i	$-0.521936\pi$
$397$	−2.74397	−0.137716	−0.0688579	+	0.997626i	$0.521936\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.86195	0.392607	0.196304	−	0.980543i	$-0.437106\pi$
$401$	7.86195	0.392607	0.196304	+	0.980543i	$0.437106\pi$
$402$	0	0
$403$	13.5403	0.674490
$404$	0	0
$405$	−15.8078	−0.785496
$406$	0	0
$407$	−10.5488	−0.522885
$408$	0	0
$409$	−13.2463	−0.654987	−0.327493	−	0.944853i	$-0.606204\pi$
$409$	−13.2463	−0.654987	−0.327493	+	0.944853i	$0.606204\pi$
$410$	0	0
$411$	3.32742	0.164129
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.9734	−0.833190
$416$	0	0
$417$	4.46531	0.218667
$418$	0	0
$419$	20.5917	1.00597	0.502985	−	0.864295i	$-0.332235\pi$
$419$	20.5917	1.00597	0.502985	+	0.864295i	$0.332235\pi$
$420$	0	0
$421$	−36.0621	−1.75756	−0.878780	−	0.477227i	$-0.841642\pi$
$421$	−36.0621	−1.75756	−0.878780	+	0.477227i	$0.841642\pi$
$422$	0	0
$423$	−5.01103	−0.243645
$424$	0	0
$425$	29.8757	1.44918
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.74700	−0.229187
$430$	0	0
$431$	−21.2703	−1.02455	−0.512277	−	0.858821i	$-0.671198\pi$
$431$	−21.2703	−1.02455	−0.512277	+	0.858821i	$0.671198\pi$
$432$	0	0
$433$	8.27108	0.397483	0.198741	−	0.980052i	$-0.436315\pi$
$433$	8.27108	0.397483	0.198741	+	0.980052i	$0.436315\pi$
$434$	0	0
$435$	26.9825	1.29371
$436$	0	0
$437$	−36.0063	−1.72241
$438$	0	0
$439$	1.94835	0.0929898	0.0464949	−	0.998919i	$-0.485195\pi$
$439$	1.94835	0.0929898	0.0464949	+	0.998919i	$0.485195\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.4707	−1.30517	−0.652586	−	0.757714i	$-0.726316\pi$
$443$	−27.4707	−1.30517	−0.652586	+	0.757714i	$0.726316\pi$
$444$	0	0
$445$	11.1735	0.529677
$446$	0	0
$447$	−18.2529	−0.863333
$448$	0	0
$449$	0.960250	0.0453170	0.0226585	−	0.999743i	$-0.492787\pi$
$449$	0.960250	0.0453170	0.0226585	+	0.999743i	$0.492787\pi$
$450$	0	0
$451$	1.84104	0.0866913
$452$	0	0
$453$	3.59172	0.168754
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.5614	−1.85060	−0.925302	−	0.379232i	$-0.876188\pi$
$457$	−39.5614	−1.85060	−0.925302	+	0.379232i	$0.876188\pi$
$458$	0	0
$459$	−28.8791	−1.34796
$460$	0	0
$461$	−24.5076	−1.14143	−0.570716	−	0.821148i	$-0.693334\pi$
$461$	−24.5076	−1.14143	−0.570716	+	0.821148i	$0.693334\pi$
$462$	0	0
$463$	4.23774	0.196944	0.0984722	−	0.995140i	$-0.468604\pi$
$463$	4.23774	0.196944	0.0984722	+	0.995140i	$0.468604\pi$
$464$	0	0
$465$	24.1220	1.11863
$466$	0	0
$467$	−23.8784	−1.10496	−0.552479	−	0.833527i	$-0.686318\pi$
$467$	−23.8784	−1.10496	−0.552479	+	0.833527i	$0.686318\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.1443	0.559581
$472$	0	0
$473$	4.58496	0.210817
$474$	0	0
$475$	−37.5096	−1.72106
$476$	0	0
$477$	10.7094	0.490350
$478$	0	0
$479$	34.0296	1.55485	0.777425	−	0.628975i	$-0.216525\pi$
$479$	34.0296	1.55485	0.777425	+	0.628975i	$0.216525\pi$
$480$	0	0
$481$	20.8498	0.950670
$482$	0	0
$483$	0	0
$484$	0	0
$485$	29.9794	1.36129
$486$	0	0
$487$	11.2696	0.510676	0.255338	−	0.966852i	$-0.417813\pi$
$487$	11.2696	0.510676	0.255338	+	0.966852i	$0.417813\pi$
$488$	0	0
$489$	8.49764	0.384277
$490$	0	0
$491$	−31.3811	−1.41621	−0.708104	−	0.706108i	$-0.750449\pi$
$491$	−31.3811	−1.41621	−0.708104	+	0.706108i	$0.750449\pi$
$492$	0	0
$493$	29.8730	1.34541
$494$	0	0
$495$	4.48674	0.201664
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.4626	0.602671	0.301335	−	0.953518i	$-0.402568\pi$
$499$	13.4626	0.602671	0.301335	+	0.953518i	$0.402568\pi$
$500$	0	0
$501$	−21.5249	−0.961661
$502$	0	0
$503$	4.47867	0.199694	0.0998471	−	0.995003i	$-0.468165\pi$
$503$	4.47867	0.199694	0.0998471	+	0.995003i	$0.468165\pi$
$504$	0	0
$505$	60.7305	2.70247
$506$	0	0
$507$	−8.81787	−0.391615
$508$	0	0
$509$	−10.8851	−0.482475	−0.241238	−	0.970466i	$-0.577553\pi$
$509$	−10.8851	−0.482475	−0.241238	+	0.970466i	$0.577553\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	36.2583	1.60084
$514$	0	0
$515$	35.9994	1.58632
$516$	0	0
$517$	−6.31129	−0.277570
$518$	0	0
$519$	11.6416	0.511008
$520$	0	0
$521$	21.2127	0.929346	0.464673	−	0.885482i	$-0.346172\pi$
$521$	21.2127	0.929346	0.464673	+	0.885482i	$0.346172\pi$
$522$	0	0
$523$	−35.3234	−1.54458	−0.772292	−	0.635268i	$-0.780890\pi$
$523$	−35.3234	−1.54458	−0.772292	+	0.635268i	$0.780890\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.7061	1.16334
$528$	0	0
$529$	8.54726	0.371620
$530$	0	0
$531$	5.02462	0.218050
$532$	0	0
$533$	−3.63884	−0.157616
$534$	0	0
$535$	−36.1519	−1.56298
$536$	0	0
$537$	21.0376	0.907840
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.34396	−0.143768	−0.0718841	−	0.997413i	$-0.522901\pi$
$541$	−3.34396	−0.143768	−0.0718841	+	0.997413i	$0.522901\pi$
$542$	0	0
$543$	−32.8156	−1.40825
$544$	0	0
$545$	32.3013	1.38364
$546$	0	0
$547$	−12.9906	−0.555440	−0.277720	−	0.960662i	$-0.589579\pi$
$547$	−12.9906	−0.555440	−0.277720	+	0.960662i	$0.589579\pi$
$548$	0	0
$549$	12.7025	0.542130
$550$	0	0
$551$	−37.5062	−1.59782
$552$	0	0
$553$	0	0
$554$	0	0
$555$	37.1439	1.57667
$556$	0	0
$557$	−21.8103	−0.924130	−0.462065	−	0.886846i	$-0.652891\pi$
$557$	−21.8103	−0.924130	−0.462065	+	0.886846i	$0.652891\pi$
$558$	0	0
$559$	−9.06222	−0.383291
$560$	0	0
$561$	−9.36272	−0.395294
$562$	0	0
$563$	31.2308	1.31622	0.658111	−	0.752921i	$-0.271356\pi$
$563$	31.2308	1.31622	0.658111	+	0.752921i	$0.271356\pi$
$564$	0	0
$565$	−10.3065	−0.433599
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.2729	1.85602	0.928009	−	0.372558i	$-0.121519\pi$
$569$	44.2729	1.85602	0.928009	+	0.372558i	$0.121519\pi$
$570$	0	0
$571$	33.2319	1.39071	0.695356	−	0.718665i	$-0.255247\pi$
$571$	33.2319	1.39071	0.695356	+	0.718665i	$0.255247\pi$
$572$	0	0
$573$	−9.55476	−0.399156
$574$	0	0
$575$	32.8644	1.37054
$576$	0	0
$577$	13.3294	0.554911	0.277455	−	0.960739i	$-0.410509\pi$
$577$	13.3294	0.554911	0.277455	+	0.960739i	$0.410509\pi$
$578$	0	0
$579$	8.54072	0.354940
$580$	0	0
$581$	0	0
$582$	0	0
$583$	13.4883	0.558628
$584$	0	0
$585$	−8.86809	−0.366651
$586$	0	0
$587$	−29.4138	−1.21404	−0.607018	−	0.794688i	$-0.707634\pi$
$587$	−29.4138	−1.21404	−0.607018	+	0.794688i	$0.707634\pi$
$588$	0	0
$589$	−33.5301	−1.38158
$590$	0	0
$591$	19.7180	0.811091
$592$	0	0
$593$	33.0462	1.35704	0.678522	−	0.734580i	$-0.262621\pi$
$593$	33.0462	1.35704	0.678522	+	0.734580i	$0.262621\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.6932	1.17433
$598$	0	0
$599$	41.2922	1.68716	0.843578	−	0.537007i	$-0.180445\pi$
$599$	41.2922	1.68716	0.843578	+	0.537007i	$0.180445\pi$
$600$	0	0
$601$	9.01739	0.367827	0.183914	−	0.982942i	$-0.441123\pi$
$601$	9.01739	0.367827	0.183914	+	0.982942i	$0.441123\pi$
$602$	0	0
$603$	−9.28831	−0.378249
$604$	0	0
$605$	−30.5843	−1.24343
$606$	0	0
$607$	23.7157	0.962589	0.481294	−	0.876559i	$-0.340167\pi$
$607$	23.7157	0.962589	0.481294	+	0.876559i	$0.340167\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.4743	0.504657
$612$	0	0
$613$	−8.39552	−0.339092	−0.169546	−	0.985522i	$-0.554230\pi$
$613$	−8.39552	−0.339092	−0.169546	+	0.985522i	$0.554230\pi$
$614$	0	0
$615$	−6.48257	−0.261403
$616$	0	0
$617$	−13.2575	−0.533727	−0.266863	−	0.963734i	$-0.585987\pi$
$617$	−13.2575	−0.533727	−0.266863	+	0.963734i	$0.585987\pi$
$618$	0	0
$619$	10.0839	0.405305	0.202653	−	0.979251i	$-0.435044\pi$
$619$	10.0839	0.405305	0.202653	+	0.979251i	$0.435044\pi$
$620$	0	0
$621$	−31.7680	−1.27481
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−20.0195	−0.800780
$626$	0	0
$627$	11.7551	0.469453
$628$	0	0
$629$	41.1230	1.63968
$630$	0	0
$631$	25.8656	1.02969	0.514847	−	0.857282i	$-0.327849\pi$
$631$	25.8656	1.02969	0.514847	+	0.857282i	$0.327849\pi$
$632$	0	0
$633$	−2.05867	−0.0818246
$634$	0	0
$635$	5.34446	0.212088
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.91028	−0.312926
$640$	0	0
$641$	14.6633	0.579166	0.289583	−	0.957153i	$-0.406483\pi$
$641$	14.6633	0.579166	0.289583	+	0.957153i	$0.406483\pi$
$642$	0	0
$643$	41.7512	1.64651	0.823253	−	0.567675i	$-0.192157\pi$
$643$	41.7512	1.64651	0.823253	+	0.567675i	$0.192157\pi$
$644$	0	0
$645$	−16.1443	−0.635681
$646$	0	0
$647$	6.36539	0.250249	0.125125	−	0.992141i	$-0.460067\pi$
$647$	6.36539	0.250249	0.125125	+	0.992141i	$0.460067\pi$
$648$	0	0
$649$	6.32841	0.248412
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.5686	−0.765778	−0.382889	−	0.923794i	$-0.625071\pi$
$653$	−19.5686	−0.765778	−0.382889	+	0.923794i	$0.625071\pi$
$654$	0	0
$655$	−29.4864	−1.15213
$656$	0	0
$657$	−3.73985	−0.145905
$658$	0	0
$659$	−32.3906	−1.26176	−0.630878	−	0.775882i	$-0.717305\pi$
$659$	−32.3906	−1.26176	−0.630878	+	0.775882i	$0.717305\pi$
$660$	0	0
$661$	−18.3198	−0.712558	−0.356279	−	0.934380i	$-0.615955\pi$
$661$	−18.3198	−0.712558	−0.356279	+	0.934380i	$0.615955\pi$
$662$	0	0
$663$	18.5055	0.718694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.8615	1.27240
$668$	0	0
$669$	−7.66636	−0.296398
$670$	0	0
$671$	15.9985	0.617617
$672$	0	0
$673$	15.7991	0.609010	0.304505	−	0.952511i	$-0.401509\pi$
$673$	15.7991	0.609010	0.304505	+	0.952511i	$0.401509\pi$
$674$	0	0
$675$	−33.0944	−1.27380
$676$	0	0
$677$	−29.7924	−1.14502	−0.572508	−	0.819899i	$-0.694030\pi$
$677$	−29.7924	−1.14502	−0.572508	+	0.819899i	$0.694030\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7.05171	0.270222
$682$	0	0
$683$	−31.9938	−1.22421	−0.612105	−	0.790776i	$-0.709677\pi$
$683$	−31.9938	−1.22421	−0.612105	+	0.790776i	$0.709677\pi$
$684$	0	0
$685$	7.82905	0.299133
$686$	0	0
$687$	−30.3276	−1.15707
$688$	0	0
$689$	−26.6597	−1.01566
$690$	0	0
$691$	16.9349	0.644233	0.322117	−	0.946700i	$-0.395606\pi$
$691$	16.9349	0.644233	0.322117	+	0.946700i	$0.395606\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.5064	0.398531
$696$	0	0
$697$	−7.17704	−0.271850
$698$	0	0
$699$	−22.8588	−0.864598
$700$	0	0
$701$	0.686127	0.0259147	0.0129573	−	0.999916i	$-0.495875\pi$
$701$	0.686127	0.0259147	0.0129573	+	0.999916i	$0.495875\pi$
$702$	0	0
$703$	−51.6308	−1.94729
$704$	0	0
$705$	22.2230	0.836965
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15.8562	0.595493	0.297746	−	0.954645i	$-0.403765\pi$
$709$	15.8562	0.595493	0.297746	+	0.954645i	$0.403765\pi$
$710$	0	0
$711$	−7.89266	−0.295998
$712$	0	0
$713$	29.3777	1.10020
$714$	0	0
$715$	−11.1692	−0.417704
$716$	0	0
$717$	9.25612	0.345676
$718$	0	0
$719$	45.9438	1.71341	0.856707	−	0.515804i	$-0.172507\pi$
$719$	45.9438	1.71341	0.856707	+	0.515804i	$0.172507\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.40538	−0.0522667
$724$	0	0
$725$	34.2334	1.27140
$726$	0	0
$727$	−39.4429	−1.46286	−0.731428	−	0.681918i	$-0.761146\pi$
$727$	−39.4429	−1.46286	−0.731428	+	0.681918i	$0.761146\pi$
$728$	0	0
$729$	28.7461	1.06467
$730$	0	0
$731$	−17.8738	−0.661086
$732$	0	0
$733$	−37.2055	−1.37422	−0.687108	−	0.726555i	$-0.741120\pi$
$733$	−37.2055	−1.37422	−0.687108	+	0.726555i	$0.741120\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.6984	−0.430918
$738$	0	0
$739$	24.5703	0.903831	0.451916	−	0.892061i	$-0.350741\pi$
$739$	24.5703	0.903831	0.451916	+	0.892061i	$0.350741\pi$
$740$	0	0
$741$	−23.2341	−0.853524
$742$	0	0
$743$	−1.84492	−0.0676834	−0.0338417	−	0.999427i	$-0.510774\pi$
$743$	−1.84492	−0.0676834	−0.0338417	+	0.999427i	$0.510774\pi$
$744$	0	0
$745$	−42.9471	−1.57346
$746$	0	0
$747$	5.35834	0.196051
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.7325	−0.683558	−0.341779	−	0.939780i	$-0.611029\pi$
$751$	−18.7325	−0.683558	−0.341779	+	0.939780i	$0.611029\pi$
$752$	0	0
$753$	38.7502	1.41214
$754$	0	0
$755$	8.45093	0.307561
$756$	0	0
$757$	−24.7836	−0.900774	−0.450387	−	0.892834i	$-0.648714\pi$
$757$	−24.7836	−0.900774	−0.450387	+	0.892834i	$0.648714\pi$
$758$	0	0
$759$	−10.2993	−0.373842
$760$	0	0
$761$	−0.595628	−0.0215915	−0.0107958	−	0.999942i	$-0.503436\pi$
$761$	−0.595628	−0.0215915	−0.0107958	+	0.999942i	$0.503436\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−17.4909	−0.632386
$766$	0	0
$767$	−12.5082	−0.451644
$768$	0	0
$769$	−38.4360	−1.38604	−0.693019	−	0.720920i	$-0.743720\pi$
$769$	−38.4360	−1.38604	−0.693019	+	0.720920i	$0.743720\pi$
$770$	0	0
$771$	10.7406	0.386814
$772$	0	0
$773$	−2.33290	−0.0839086	−0.0419543	−	0.999120i	$-0.513358\pi$
$773$	−2.33290	−0.0839086	−0.0419543	+	0.999120i	$0.513358\pi$
$774$	0	0
$775$	30.6043	1.09934
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.01092	0.322850
$780$	0	0
$781$	−9.96284	−0.356498
$782$	0	0
$783$	−33.0914	−1.18259
$784$	0	0
$785$	28.5743	1.01986
$786$	0	0
$787$	−35.2852	−1.25778	−0.628890	−	0.777494i	$-0.716490\pi$
$787$	−35.2852	−1.25778	−0.628890	+	0.777494i	$0.716490\pi$
$788$	0	0
$789$	−14.8915	−0.530150
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−31.6213	−1.12290
$794$	0	0
$795$	−47.4942	−1.68445
$796$	0	0
$797$	26.1684	0.926933	0.463467	−	0.886114i	$-0.346605\pi$
$797$	26.1684	0.926933	0.463467	+	0.886114i	$0.346605\pi$
$798$	0	0
$799$	24.6037	0.870415
$800$	0	0
$801$	−3.52738	−0.124634
$802$	0	0
$803$	−4.71027	−0.166222
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.09633	0.144198
$808$	0	0
$809$	8.19891	0.288258	0.144129	−	0.989559i	$-0.453962\pi$
$809$	8.19891	0.288258	0.144129	+	0.989559i	$0.453962\pi$
$810$	0	0
$811$	−31.6920	−1.11286	−0.556429	−	0.830895i	$-0.687829\pi$
$811$	−31.6920	−1.11286	−0.556429	+	0.830895i	$0.687829\pi$
$812$	0	0
$813$	2.76494	0.0969705
$814$	0	0
$815$	19.9940	0.700360
$816$	0	0
$817$	22.4409	0.785109
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.97247	0.243341	0.121670	−	0.992571i	$-0.461175\pi$
$821$	6.97247	0.243341	0.121670	+	0.992571i	$0.461175\pi$
$822$	0	0
$823$	49.0630	1.71023	0.855115	−	0.518438i	$-0.173486\pi$
$823$	49.0630	1.71023	0.855115	+	0.518438i	$0.173486\pi$
$824$	0	0
$825$	−10.7293	−0.373548
$826$	0	0
$827$	51.0207	1.77416	0.887081	−	0.461614i	$-0.152729\pi$
$827$	51.0207	1.77416	0.887081	+	0.461614i	$0.152729\pi$
$828$	0	0
$829$	6.24791	0.216999	0.108499	−	0.994097i	$-0.465395\pi$
$829$	6.24791	0.216999	0.108499	+	0.994097i	$0.465395\pi$
$830$	0	0
$831$	9.55684	0.331523
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−50.6458	−1.75267
$836$	0	0
$837$	−29.5833	−1.02255
$838$	0	0
$839$	−29.5930	−1.02166	−0.510831	−	0.859681i	$-0.670662\pi$
$839$	−29.5930	−1.02166	−0.510831	+	0.859681i	$0.670662\pi$
$840$	0	0
$841$	5.23040	0.180358
$842$	0	0
$843$	−38.9514	−1.34156
$844$	0	0
$845$	−20.7475	−0.713735
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.5894	0.397749
$850$	0	0
$851$	45.2368	1.55070
$852$	0	0
$853$	30.6985	1.05110	0.525548	−	0.850764i	$-0.323860\pi$
$853$	30.6985	1.05110	0.525548	+	0.850764i	$0.323860\pi$
$854$	0	0
$855$	21.9602	0.751024
$856$	0	0
$857$	−48.1238	−1.64388	−0.821939	−	0.569576i	$-0.807108\pi$
$857$	−48.1238	−1.64388	−0.821939	+	0.569576i	$0.807108\pi$
$858$	0	0
$859$	−18.9605	−0.646925	−0.323462	−	0.946241i	$-0.604847\pi$
$859$	−18.9605	−0.646925	−0.323462	+	0.946241i	$0.604847\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.5822	−1.00699	−0.503495	−	0.863998i	$-0.667953\pi$
$863$	−29.5822	−1.00699	−0.503495	+	0.863998i	$0.667953\pi$
$864$	0	0
$865$	27.3914	0.931335
$866$	0	0
$867$	12.6987	0.431272
$868$	0	0
$869$	−9.94065	−0.337213
$870$	0	0
$871$	23.1221	0.783462
$872$	0	0
$873$	−9.46422	−0.320315
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.8873	0.637780	0.318890	−	0.947792i	$-0.396690\pi$
$877$	18.8873	0.637780	0.318890	+	0.947792i	$0.396690\pi$
$878$	0	0
$879$	−8.80271	−0.296908
$880$	0	0
$881$	4.56470	0.153789	0.0768943	−	0.997039i	$-0.475500\pi$
$881$	4.56470	0.153789	0.0768943	+	0.997039i	$0.475500\pi$
$882$	0	0
$883$	25.3199	0.852081	0.426040	−	0.904704i	$-0.359908\pi$
$883$	25.3199	0.852081	0.426040	+	0.904704i	$0.359908\pi$
$884$	0	0
$885$	−22.2832	−0.749043
$886$	0	0
$887$	−29.3736	−0.986270	−0.493135	−	0.869953i	$-0.664149\pi$
$887$	−29.3736	−0.986270	−0.493135	+	0.869953i	$0.664149\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.28528	0.210565
$892$	0	0
$893$	−30.8904	−1.03371
$894$	0	0
$895$	49.4992	1.65458
$896$	0	0
$897$	20.3567	0.679692
$898$	0	0
$899$	30.6015	1.02062
$900$	0	0
$901$	−52.5822	−1.75177
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−77.2115	−2.56660
$906$	0	0
$907$	−46.0953	−1.53057	−0.765284	−	0.643692i	$-0.777402\pi$
$907$	−46.0953	−1.53057	−0.765284	+	0.643692i	$0.777402\pi$
$908$	0	0
$909$	−19.1720	−0.635897
$910$	0	0
$911$	−25.0040	−0.828420	−0.414210	−	0.910181i	$-0.635942\pi$
$911$	−25.0040	−0.828420	−0.414210	+	0.910181i	$0.635942\pi$
$912$	0	0
$913$	6.74872	0.223350
$914$	0	0
$915$	−56.3332	−1.86232
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−57.2666	−1.88905	−0.944526	−	0.328437i	$-0.893478\pi$
$919$	−57.2666	−1.88905	−0.944526	+	0.328437i	$0.893478\pi$
$920$	0	0
$921$	−19.3426	−0.637361
$922$	0	0
$923$	19.6916	0.648158
$924$	0	0
$925$	47.1255	1.54948
$926$	0	0
$927$	−11.3647	−0.373265
$928$	0	0
$929$	58.7295	1.92685	0.963427	−	0.267972i	$-0.0863536\pi$
$929$	58.7295	1.92685	0.963427	+	0.267972i	$0.0863536\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.1487	−0.463208
$934$	0	0
$935$	−22.0295	−0.720441
$936$	0	0
$937$	−2.21448	−0.0723437	−0.0361719	−	0.999346i	$-0.511516\pi$
$937$	−2.21448	−0.0723437	−0.0361719	+	0.999346i	$0.511516\pi$
$938$	0	0
$939$	−16.6871	−0.544564
$940$	0	0
$941$	15.5903	0.508228	0.254114	−	0.967174i	$-0.418216\pi$
$941$	15.5903	0.508228	0.254114	+	0.967174i	$0.418216\pi$
$942$	0	0
$943$	−7.89501	−0.257097
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.4310	−1.05386	−0.526932	−	0.849907i	$-0.676658\pi$
$947$	−32.4310	−1.05386	−0.526932	+	0.849907i	$0.676658\pi$
$948$	0	0
$949$	9.30989	0.302212
$950$	0	0
$951$	−36.6995	−1.19006
$952$	0	0
$953$	44.7409	1.44930	0.724650	−	0.689117i	$-0.242002\pi$
$953$	44.7409	1.44930	0.724650	+	0.689117i	$0.242002\pi$
$954$	0	0
$955$	−22.4813	−0.727479
$956$	0	0
$957$	−10.7284	−0.346800
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.64260	−0.117503
$962$	0	0
$963$	11.4128	0.367773
$964$	0	0
$965$	20.0954	0.646894
$966$	0	0
$967$	−10.6192	−0.341489	−0.170745	−	0.985315i	$-0.554617\pi$
$967$	−10.6192	−0.341489	−0.170745	+	0.985315i	$0.554617\pi$
$968$	0	0
$969$	−45.8255	−1.47213
$970$	0	0
$971$	−39.8498	−1.27884	−0.639420	−	0.768857i	$-0.720826\pi$
$971$	−39.8498	−1.27884	−0.639420	+	0.768857i	$0.720826\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	21.2067	0.679156
$976$	0	0
$977$	−39.0141	−1.24817	−0.624086	−	0.781355i	$-0.714529\pi$
$977$	−39.0141	−1.24817	−0.624086	+	0.781355i	$0.714529\pi$
$978$	0	0
$979$	−4.44267	−0.141988
$980$	0	0
$981$	−10.1972	−0.325572
$982$	0	0
$983$	−53.6270	−1.71043	−0.855217	−	0.518270i	$-0.826576\pi$
$983$	−53.6270	−1.71043	−0.855217	+	0.518270i	$0.826576\pi$
$984$	0	0
$985$	46.3944	1.47825
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.6618	−0.625210
$990$	0	0
$991$	−29.1681	−0.926555	−0.463277	−	0.886213i	$-0.653327\pi$
$991$	−29.1681	−0.926555	−0.463277	+	0.886213i	$0.653327\pi$
$992$	0	0
$993$	−22.7824	−0.722978
$994$	0	0
$995$	67.5120	2.14027
$996$	0	0
$997$	20.2783	0.642219	0.321110	−	0.947042i	$-0.395944\pi$
$997$	20.2783	0.642219	0.321110	+	0.947042i	$0.395944\pi$
$998$	0	0
$999$	−45.5534	−1.44125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.h.1.5		6
4.3	odd	2		343.2.a.d.1.4	yes	6
7.6	odd	2		5488.2.a.p.1.2		6
12.11	even	2		3087.2.a.j.1.3		6
20.19	odd	2		8575.2.a.n.1.3		6
28.3	even	6		343.2.c.e.324.3		12
28.11	odd	6		343.2.c.d.324.3		12
28.19	even	6		343.2.c.e.18.3		12
28.23	odd	6		343.2.c.d.18.3		12
28.27	even	2		343.2.a.c.1.4	✓	6
84.83	odd	2		3087.2.a.k.1.3		6
140.139	even	2		8575.2.a.o.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.c.1.4	✓	6	28.27	even	2
343.2.a.d.1.4	yes	6	4.3	odd	2
343.2.c.d.18.3		12	28.23	odd	6
343.2.c.d.324.3		12	28.11	odd	6
343.2.c.e.18.3		12	28.19	even	6
343.2.c.e.324.3		12	28.3	even	6
3087.2.a.j.1.3		6	12.11	even	2
3087.2.a.k.1.3		6	84.83	odd	2
5488.2.a.h.1.5		6	1.1	even	1	trivial
5488.2.a.p.1.2		6	7.6	odd	2
8575.2.a.n.1.3		6	20.19	odd	2
8575.2.a.o.1.3		6	140.139	even	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:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40003 q^{3} +3.29412 q^{5} -1.03992 q^{9} -1.30976 q^{11} +2.58875 q^{13} +4.61185 q^{15} +5.10591 q^{17} -6.41058 q^{19} +5.61669 q^{23} +5.85120 q^{25} -5.65600 q^{27} +5.85067 q^{29} +5.23043 q^{31} -1.83370 q^{33} +8.05400 q^{37} +3.62433 q^{39} -1.40563 q^{41} -3.50061 q^{43} -3.42562 q^{45} +4.81866 q^{47} +7.14842 q^{51} -10.2983 q^{53} -4.31450 q^{55} -8.97499 q^{57} -4.83173 q^{59} -12.2149 q^{61} +8.52766 q^{65} +8.93174 q^{67} +7.86353 q^{69} +7.60661 q^{71} +3.59628 q^{73} +8.19184 q^{75} +7.58967 q^{79} -4.79880 q^{81} -5.15263 q^{83} +16.8195 q^{85} +8.19111 q^{87} +3.39197 q^{89} +7.32275 q^{93} -21.1172 q^{95} +9.10089 q^{97} +1.36205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 11 q^{5} + 9 q^{9} + q^{11} + 7 q^{13} + 5 q^{15} + 26 q^{17} + 3 q^{19} + 9 q^{23} + 15 q^{25} - 8 q^{27} - 2 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{37} - 7 q^{39} + 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 + 11 * q^5 + 9 * q^9 + q^11 + 7 * q^13 + 5 * q^15 + 26 * q^17 + 3 * q^19 + 9 * q^23 + 15 * q^25 - 8 * q^27 - 2 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^37 - 7 * q^39 + 28 * q^41 - 5 * q^43 + 3 * q^45 - 18 * q^47 - 13 * q^51 - q^53 + 22 * q^55 - 26 * q^57 - 5 * q^59 - 17 * q^61 - 14 * q^65 + 22 * q^67 - 20 * q^69 + 2 * q^71 + 12 * q^73 + 27 * q^75 + 2 * q^79 - 30 * q^81 - 7 * q^83 + 37 * q^85 + 25 * q^87 + 39 * q^89 - 22 * q^93 + 11 * q^95 + 35 * q^97 - 12 * q^99

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