LMFDB - Embedded newform 1004.2.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1004,2,Mod(1,1004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1004 = 2^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$8.01698036294$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.164919$ of defining polynomial
Character	$\chi$	$=$	1004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.164919 q^{3} -1.38175 q^{5} +1.87004 q^{7} -2.97280 q^{9} +O(q^{10})$	$q+0.164919 q^{3} -1.38175 q^{5} +1.87004 q^{7} -2.97280 q^{9} +0.536413 q^{11} -3.43559 q^{13} -0.227876 q^{15} -5.29146 q^{17} +2.51612 q^{19} +0.308405 q^{21} +1.08295 q^{23} -3.09078 q^{25} -0.985029 q^{27} +1.95108 q^{29} -10.5408 q^{31} +0.0884647 q^{33} -2.58392 q^{35} -6.13316 q^{37} -0.566594 q^{39} +6.36274 q^{41} -0.652721 q^{43} +4.10765 q^{45} +10.2991 q^{47} -3.50295 q^{49} -0.872662 q^{51} -1.31866 q^{53} -0.741186 q^{55} +0.414956 q^{57} -13.8057 q^{59} -8.27363 q^{61} -5.55926 q^{63} +4.74711 q^{65} -12.4629 q^{67} +0.178598 q^{69} -8.24013 q^{71} +7.43258 q^{73} -0.509728 q^{75} +1.00311 q^{77} -7.33283 q^{79} +8.75596 q^{81} +7.83006 q^{83} +7.31145 q^{85} +0.321771 q^{87} +10.4182 q^{89} -6.42469 q^{91} -1.73838 q^{93} -3.47664 q^{95} -12.2648 q^{97} -1.59465 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.164919	0.0952160	0.0476080	−	0.998866i	$-0.484840\pi$
$3$	0.164919	0.0952160	0.0476080	+	0.998866i	$0.484840\pi$
$4$	0	0
$5$	−1.38175	−0.617935	−0.308968	−	0.951073i	$-0.599983\pi$
$5$	−1.38175	−0.617935	−0.308968	+	0.951073i	$0.599983\pi$
$6$	0	0
$7$	1.87004	0.706809	0.353405	−	0.935471i	$-0.385024\pi$
$7$	1.87004	0.706809	0.353405	+	0.935471i	$0.385024\pi$
$8$	0	0
$9$	−2.97280	−0.990934
$10$	0	0
$11$	0.536413	0.161735	0.0808673	−	0.996725i	$-0.474231\pi$
$11$	0.536413	0.161735	0.0808673	+	0.996725i	$0.474231\pi$
$12$	0	0
$13$	−3.43559	−0.952861	−0.476431	−	0.879212i	$-0.658070\pi$
$13$	−3.43559	−0.952861	−0.476431	+	0.879212i	$0.658070\pi$
$14$	0	0
$15$	−0.227876	−0.0588373
$16$	0	0
$17$	−5.29146	−1.28337	−0.641684	−	0.766970i	$-0.721764\pi$
$17$	−5.29146	−1.28337	−0.641684	+	0.766970i	$0.721764\pi$
$18$	0	0
$19$	2.51612	0.577237	0.288619	−	0.957444i	$-0.406804\pi$
$19$	2.51612	0.577237	0.288619	+	0.957444i	$0.406804\pi$
$20$	0	0
$21$	0.308405	0.0672996
$22$	0	0
$23$	1.08295	0.225810	0.112905	−	0.993606i	$-0.463984\pi$
$23$	1.08295	0.225810	0.112905	+	0.993606i	$0.463984\pi$
$24$	0	0
$25$	−3.09078	−0.618156
$26$	0	0
$27$	−0.985029	−0.189569
$28$	0	0
$29$	1.95108	0.362307	0.181154	−	0.983455i	$-0.442017\pi$
$29$	1.95108	0.362307	0.181154	+	0.983455i	$0.442017\pi$
$30$	0	0
$31$	−10.5408	−1.89319	−0.946594	−	0.322427i	$-0.895501\pi$
$31$	−10.5408	−1.89319	−0.946594	+	0.322427i	$0.895501\pi$
$32$	0	0
$33$	0.0884647	0.0153997
$34$	0	0
$35$	−2.58392	−0.436762
$36$	0	0
$37$	−6.13316	−1.00829	−0.504143	−	0.863620i	$-0.668191\pi$
$37$	−6.13316	−1.00829	−0.504143	+	0.863620i	$0.668191\pi$
$38$	0	0
$39$	−0.566594	−0.0907277
$40$	0	0
$41$	6.36274	0.993694	0.496847	−	0.867838i	$-0.334491\pi$
$41$	6.36274	0.993694	0.496847	+	0.867838i	$0.334491\pi$
$42$	0	0
$43$	−0.652721	−0.0995391	−0.0497695	−	0.998761i	$-0.515849\pi$
$43$	−0.652721	−0.0995391	−0.0497695	+	0.998761i	$0.515849\pi$
$44$	0	0
$45$	4.10765	0.612333
$46$	0	0
$47$	10.2991	1.50228	0.751140	−	0.660142i	$-0.229504\pi$
$47$	10.2991	1.50228	0.751140	+	0.660142i	$0.229504\pi$
$48$	0	0
$49$	−3.50295	−0.500421
$50$	0	0
$51$	−0.872662	−0.122197
$52$	0	0
$53$	−1.31866	−0.181131	−0.0905657	−	0.995890i	$-0.528868\pi$
$53$	−1.31866	−0.181131	−0.0905657	+	0.995890i	$0.528868\pi$
$54$	0	0
$55$	−0.741186	−0.0999415
$56$	0	0
$57$	0.414956	0.0549622
$58$	0	0
$59$	−13.8057	−1.79734	−0.898672	−	0.438622i	$-0.855467\pi$
$59$	−13.8057	−1.79734	−0.898672	+	0.438622i	$0.855467\pi$
$60$	0	0
$61$	−8.27363	−1.05933	−0.529665	−	0.848207i	$-0.677682\pi$
$61$	−8.27363	−1.05933	−0.529665	+	0.848207i	$0.677682\pi$
$62$	0	0
$63$	−5.55926	−0.700401
$64$	0	0
$65$	4.74711	0.588806
$66$	0	0
$67$	−12.4629	−1.52259	−0.761293	−	0.648408i	$-0.775435\pi$
$67$	−12.4629	−1.52259	−0.761293	+	0.648408i	$0.775435\pi$
$68$	0	0
$69$	0.178598	0.0215007
$70$	0	0
$71$	−8.24013	−0.977923	−0.488961	−	0.872305i	$-0.662624\pi$
$71$	−8.24013	−0.977923	−0.488961	+	0.872305i	$0.662624\pi$
$72$	0	0
$73$	7.43258	0.869918	0.434959	−	0.900450i	$-0.356763\pi$
$73$	7.43258	0.869918	0.434959	+	0.900450i	$0.356763\pi$
$74$	0	0
$75$	−0.509728	−0.0588584
$76$	0	0
$77$	1.00311	0.114316
$78$	0	0
$79$	−7.33283	−0.825008	−0.412504	−	0.910956i	$-0.635346\pi$
$79$	−7.33283	−0.825008	−0.412504	+	0.910956i	$0.635346\pi$
$80$	0	0
$81$	8.75596	0.972884
$82$	0	0
$83$	7.83006	0.859460	0.429730	−	0.902957i	$-0.358609\pi$
$83$	7.83006	0.859460	0.429730	+	0.902957i	$0.358609\pi$
$84$	0	0
$85$	7.31145	0.793038
$86$	0	0
$87$	0.321771	0.0344975
$88$	0	0
$89$	10.4182	1.10433	0.552165	−	0.833735i	$-0.313802\pi$
$89$	10.4182	1.10433	0.552165	+	0.833735i	$0.313802\pi$
$90$	0	0
$91$	−6.42469	−0.673491
$92$	0	0
$93$	−1.73838	−0.180262
$94$	0	0
$95$	−3.47664	−0.356695
$96$	0	0
$97$	−12.2648	−1.24530	−0.622648	−	0.782502i	$-0.713943\pi$
$97$	−12.2648	−1.24530	−0.622648	+	0.782502i	$0.713943\pi$
$98$	0	0
$99$	−1.59465	−0.160268
$100$	0	0
$101$	−0.782050	−0.0778169	−0.0389084	−	0.999243i	$-0.512388\pi$
$101$	−0.782050	−0.0778169	−0.0389084	+	0.999243i	$0.512388\pi$
$102$	0	0
$103$	5.94779	0.586053	0.293026	−	0.956104i	$-0.405338\pi$
$103$	5.94779	0.586053	0.293026	+	0.956104i	$0.405338\pi$
$104$	0	0
$105$	−0.426138	−0.0415868
$106$	0	0
$107$	16.2918	1.57499	0.787494	−	0.616322i	$-0.211378\pi$
$107$	16.2918	1.57499	0.787494	+	0.616322i	$0.211378\pi$
$108$	0	0
$109$	17.9055	1.71503	0.857517	−	0.514456i	$-0.172006\pi$
$109$	17.9055	1.71503	0.857517	+	0.514456i	$0.172006\pi$
$110$	0	0
$111$	−1.01148	−0.0960050
$112$	0	0
$113$	−18.2128	−1.71332	−0.856658	−	0.515885i	$-0.827463\pi$
$113$	−18.2128	−1.71332	−0.856658	+	0.515885i	$0.827463\pi$
$114$	0	0
$115$	−1.49635	−0.139536
$116$	0	0
$117$	10.2133	0.944222
$118$	0	0
$119$	−9.89524	−0.907095
$120$	0	0
$121$	−10.7123	−0.973842
$122$	0	0
$123$	1.04934	0.0946156
$124$	0	0
$125$	11.1794	0.999916
$126$	0	0
$127$	5.52940	0.490655	0.245327	−	0.969440i	$-0.421105\pi$
$127$	5.52940	0.490655	0.245327	+	0.969440i	$0.421105\pi$
$128$	0	0
$129$	−0.107646	−0.00947772
$130$	0	0
$131$	1.20519	0.105298	0.0526489	−	0.998613i	$-0.483234\pi$
$131$	1.20519	0.105298	0.0526489	+	0.998613i	$0.483234\pi$
$132$	0	0
$133$	4.70525	0.407997
$134$	0	0
$135$	1.36106	0.117141
$136$	0	0
$137$	−5.11105	−0.436667	−0.218333	−	0.975874i	$-0.570062\pi$
$137$	−5.11105	−0.436667	−0.218333	+	0.975874i	$0.570062\pi$
$138$	0	0
$139$	10.4966	0.890308	0.445154	−	0.895454i	$-0.353149\pi$
$139$	10.4966	0.890308	0.445154	+	0.895454i	$0.353149\pi$
$140$	0	0
$141$	1.69852	0.143041
$142$	0	0
$143$	−1.84290	−0.154111
$144$	0	0
$145$	−2.69590	−0.223882
$146$	0	0
$147$	−0.577703	−0.0476481
$148$	0	0
$149$	7.12252	0.583499	0.291750	−	0.956495i	$-0.405763\pi$
$149$	7.12252	0.583499	0.291750	+	0.956495i	$0.405763\pi$
$150$	0	0
$151$	−7.70467	−0.626997	−0.313499	−	0.949589i	$-0.601501\pi$
$151$	−7.70467	−0.626997	−0.313499	+	0.949589i	$0.601501\pi$
$152$	0	0
$153$	15.7305	1.27173
$154$	0	0
$155$	14.5647	1.16987
$156$	0	0
$157$	2.96694	0.236788	0.118394	−	0.992967i	$-0.462225\pi$
$157$	2.96694	0.236788	0.118394	+	0.992967i	$0.462225\pi$
$158$	0	0
$159$	−0.217472	−0.0172466
$160$	0	0
$161$	2.02515	0.159604
$162$	0	0
$163$	−9.73897	−0.762815	−0.381407	−	0.924407i	$-0.624561\pi$
$163$	−9.73897	−0.762815	−0.381407	+	0.924407i	$0.624561\pi$
$164$	0	0
$165$	−0.122236	−0.00951604
$166$	0	0
$167$	−21.8240	−1.68879	−0.844396	−	0.535719i	$-0.820041\pi$
$167$	−21.8240	−1.68879	−0.844396	+	0.535719i	$0.820041\pi$
$168$	0	0
$169$	−1.19672	−0.0920556
$170$	0	0
$171$	−7.47992	−0.572004
$172$	0	0
$173$	21.7976	1.65724	0.828619	−	0.559813i	$-0.189127\pi$
$173$	21.7976	1.65724	0.828619	+	0.559813i	$0.189127\pi$
$174$	0	0
$175$	−5.77989	−0.436918
$176$	0	0
$177$	−2.27681	−0.171136
$178$	0	0
$179$	4.63283	0.346274	0.173137	−	0.984898i	$-0.444610\pi$
$179$	4.63283	0.346274	0.173137	+	0.984898i	$0.444610\pi$
$180$	0	0
$181$	18.9389	1.40772	0.703860	−	0.710339i	$-0.251458\pi$
$181$	18.9389	1.40772	0.703860	+	0.710339i	$0.251458\pi$
$182$	0	0
$183$	−1.36448	−0.100865
$184$	0	0
$185$	8.47447	0.623055
$186$	0	0
$187$	−2.83841	−0.207565
$188$	0	0
$189$	−1.84204	−0.133989
$190$	0	0
$191$	9.66753	0.699518	0.349759	−	0.936840i	$-0.386263\pi$
$191$	9.66753	0.699518	0.349759	+	0.936840i	$0.386263\pi$
$192$	0	0
$193$	24.1503	1.73838	0.869190	−	0.494479i	$-0.164641\pi$
$193$	24.1503	1.73838	0.869190	+	0.494479i	$0.164641\pi$
$194$	0	0
$195$	0.782889	0.0560638
$196$	0	0
$197$	−19.0445	−1.35687	−0.678434	−	0.734662i	$-0.737341\pi$
$197$	−19.0445	−1.35687	−0.678434	+	0.734662i	$0.737341\pi$
$198$	0	0
$199$	14.6461	1.03823	0.519117	−	0.854703i	$-0.326261\pi$
$199$	14.6461	1.03823	0.519117	+	0.854703i	$0.326261\pi$
$200$	0	0
$201$	−2.05537	−0.144975
$202$	0	0
$203$	3.64861	0.256082
$204$	0	0
$205$	−8.79169	−0.614038
$206$	0	0
$207$	−3.21938	−0.223763
$208$	0	0
$209$	1.34968	0.0933593
$210$	0	0
$211$	−19.6827	−1.35501	−0.677507	−	0.735516i	$-0.736940\pi$
$211$	−19.6827	−1.35501	−0.677507	+	0.735516i	$0.736940\pi$
$212$	0	0
$213$	−1.35895	−0.0931140
$214$	0	0
$215$	0.901895	0.0615087
$216$	0	0
$217$	−19.7118	−1.33812
$218$	0	0
$219$	1.22577	0.0828301
$220$	0	0
$221$	18.1793	1.22287
$222$	0	0
$223$	−12.9059	−0.864246	−0.432123	−	0.901815i	$-0.642235\pi$
$223$	−12.9059	−0.864246	−0.432123	+	0.901815i	$0.642235\pi$
$224$	0	0
$225$	9.18828	0.612552
$226$	0	0
$227$	22.5143	1.49433	0.747165	−	0.664639i	$-0.231415\pi$
$227$	22.5143	1.49433	0.747165	+	0.664639i	$0.231415\pi$
$228$	0	0
$229$	6.72186	0.444193	0.222097	−	0.975025i	$-0.428710\pi$
$229$	6.72186	0.444193	0.222097	+	0.975025i	$0.428710\pi$
$230$	0	0
$231$	0.165433	0.0108847
$232$	0	0
$233$	17.6949	1.15923	0.579617	−	0.814889i	$-0.303202\pi$
$233$	17.6949	1.15923	0.579617	+	0.814889i	$0.303202\pi$
$234$	0	0
$235$	−14.2308	−0.928312
$236$	0	0
$237$	−1.20932	−0.0785540
$238$	0	0
$239$	13.2463	0.856834	0.428417	−	0.903581i	$-0.359071\pi$
$239$	13.2463	0.856834	0.428417	+	0.903581i	$0.359071\pi$
$240$	0	0
$241$	−21.0850	−1.35820	−0.679102	−	0.734044i	$-0.737631\pi$
$241$	−21.0850	−1.35820	−0.679102	+	0.734044i	$0.737631\pi$
$242$	0	0
$243$	4.39911	0.282203
$244$	0	0
$245$	4.84018	0.309228
$246$	0	0
$247$	−8.64435	−0.550027
$248$	0	0
$249$	1.29132	0.0818344
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.580906	0.0365213
$254$	0	0
$255$	1.20580	0.0755099
$256$	0	0
$257$	−19.5223	−1.21777	−0.608884	−	0.793259i	$-0.708383\pi$
$257$	−19.5223	−1.21777	−0.608884	+	0.793259i	$0.708383\pi$
$258$	0	0
$259$	−11.4693	−0.712665
$260$	0	0
$261$	−5.80019	−0.359022
$262$	0	0
$263$	−14.4220	−0.889300	−0.444650	−	0.895704i	$-0.646672\pi$
$263$	−14.4220	−0.889300	−0.444650	+	0.895704i	$0.646672\pi$
$264$	0	0
$265$	1.82205	0.111927
$266$	0	0
$267$	1.71817	0.105150
$268$	0	0
$269$	23.9613	1.46095	0.730473	−	0.682942i	$-0.239300\pi$
$269$	23.9613	1.46095	0.730473	+	0.682942i	$0.239300\pi$
$270$	0	0
$271$	18.3038	1.11188	0.555940	−	0.831223i	$-0.312359\pi$
$271$	18.3038	1.11188	0.555940	+	0.831223i	$0.312359\pi$
$272$	0	0
$273$	−1.05955	−0.0641271
$274$	0	0
$275$	−1.65794	−0.0999772
$276$	0	0
$277$	25.8177	1.55123	0.775617	−	0.631204i	$-0.217439\pi$
$277$	25.8177	1.55123	0.775617	+	0.631204i	$0.217439\pi$
$278$	0	0
$279$	31.3358	1.87602
$280$	0	0
$281$	−21.8894	−1.30581	−0.652905	−	0.757440i	$-0.726450\pi$
$281$	−21.8894	−1.30581	−0.652905	+	0.757440i	$0.726450\pi$
$282$	0	0
$283$	9.00380	0.535220	0.267610	−	0.963527i	$-0.413766\pi$
$283$	9.00380	0.535220	0.267610	+	0.963527i	$0.413766\pi$
$284$	0	0
$285$	−0.573363	−0.0339631
$286$	0	0
$287$	11.8986	0.702352
$288$	0	0
$289$	10.9995	0.647031
$290$	0	0
$291$	−2.02269	−0.118572
$292$	0	0
$293$	−10.3231	−0.603080	−0.301540	−	0.953454i	$-0.597501\pi$
$293$	−10.3231	−0.603080	−0.301540	+	0.953454i	$0.597501\pi$
$294$	0	0
$295$	19.0759	1.11064
$296$	0	0
$297$	−0.528382	−0.0306598
$298$	0	0
$299$	−3.72056	−0.215165
$300$	0	0
$301$	−1.22062	−0.0703551
$302$	0	0
$303$	−0.128975	−0.00740941
$304$	0	0
$305$	11.4321	0.654597
$306$	0	0
$307$	−7.06324	−0.403120	−0.201560	−	0.979476i	$-0.564601\pi$
$307$	−7.06324	−0.403120	−0.201560	+	0.979476i	$0.564601\pi$
$308$	0	0
$309$	0.980903	0.0558016
$310$	0	0
$311$	−5.97829	−0.338998	−0.169499	−	0.985530i	$-0.554215\pi$
$311$	−5.97829	−0.338998	−0.169499	+	0.985530i	$0.554215\pi$
$312$	0	0
$313$	−4.34574	−0.245635	−0.122818	−	0.992429i	$-0.539193\pi$
$313$	−4.34574	−0.245635	−0.122818	+	0.992429i	$0.539193\pi$
$314$	0	0
$315$	7.68148	0.432802
$316$	0	0
$317$	11.4975	0.645766	0.322883	−	0.946439i	$-0.395348\pi$
$317$	11.4975	0.645766	0.322883	+	0.946439i	$0.395348\pi$
$318$	0	0
$319$	1.04659	0.0585976
$320$	0	0
$321$	2.68683	0.149964
$322$	0	0
$323$	−13.3139	−0.740807
$324$	0	0
$325$	10.6187	0.589017
$326$	0	0
$327$	2.95295	0.163299
$328$	0	0
$329$	19.2598	1.06183
$330$	0	0
$331$	−0.857090	−0.0471099	−0.0235550	−	0.999723i	$-0.507498\pi$
$331$	−0.857090	−0.0471099	−0.0235550	+	0.999723i	$0.507498\pi$
$332$	0	0
$333$	18.2327	0.999144
$334$	0	0
$335$	17.2206	0.940859
$336$	0	0
$337$	−29.5645	−1.61048	−0.805239	−	0.592950i	$-0.797963\pi$
$337$	−29.5645	−1.61048	−0.805239	+	0.592950i	$0.797963\pi$
$338$	0	0
$339$	−3.00364	−0.163135
$340$	0	0
$341$	−5.65424	−0.306194
$342$	0	0
$343$	−19.6409	−1.06051
$344$	0	0
$345$	−0.246777	−0.0132860
$346$	0	0
$347$	0.552284	0.0296482	0.0148241	−	0.999890i	$-0.495281\pi$
$347$	0.552284	0.0296482	0.0148241	+	0.999890i	$0.495281\pi$
$348$	0	0
$349$	7.92529	0.424231	0.212115	−	0.977245i	$-0.431965\pi$
$349$	7.92529	0.424231	0.212115	+	0.977245i	$0.431965\pi$
$350$	0	0
$351$	3.38415	0.180633
$352$	0	0
$353$	17.0252	0.906157	0.453079	−	0.891471i	$-0.350326\pi$
$353$	17.0252	0.906157	0.453079	+	0.891471i	$0.350326\pi$
$354$	0	0
$355$	11.3858	0.604293
$356$	0	0
$357$	−1.63191	−0.0863700
$358$	0	0
$359$	−5.01161	−0.264503	−0.132251	−	0.991216i	$-0.542221\pi$
$359$	−5.01161	−0.264503	−0.132251	+	0.991216i	$0.542221\pi$
$360$	0	0
$361$	−12.6691	−0.666797
$362$	0	0
$363$	−1.76666	−0.0927254
$364$	0	0
$365$	−10.2699	−0.537553
$366$	0	0
$367$	15.4377	0.805844	0.402922	−	0.915234i	$-0.367995\pi$
$367$	15.4377	0.805844	0.402922	+	0.915234i	$0.367995\pi$
$368$	0	0
$369$	−18.9152	−0.984685
$370$	0	0
$371$	−2.46594	−0.128025
$372$	0	0
$373$	10.5612	0.546839	0.273419	−	0.961895i	$-0.411845\pi$
$373$	10.5612	0.546839	0.273419	+	0.961895i	$0.411845\pi$
$374$	0	0
$375$	1.84370	0.0952080
$376$	0	0
$377$	−6.70312	−0.345228
$378$	0	0
$379$	−34.3556	−1.76473	−0.882365	−	0.470565i	$-0.844050\pi$
$379$	−34.3556	−1.76473	−0.882365	+	0.470565i	$0.844050\pi$
$380$	0	0
$381$	0.911903	0.0467182
$382$	0	0
$383$	−17.1854	−0.878131	−0.439066	−	0.898455i	$-0.644690\pi$
$383$	−17.1854	−0.878131	−0.439066	+	0.898455i	$0.644690\pi$
$384$	0	0
$385$	−1.38605	−0.0706396
$386$	0	0
$387$	1.94041	0.0986367
$388$	0	0
$389$	−24.5501	−1.24474	−0.622371	−	0.782723i	$-0.713830\pi$
$389$	−24.5501	−1.24474	−0.622371	+	0.782723i	$0.713830\pi$
$390$	0	0
$391$	−5.73036	−0.289797
$392$	0	0
$393$	0.198759	0.0100260
$394$	0	0
$395$	10.1321	0.509801
$396$	0	0
$397$	−3.07180	−0.154169	−0.0770847	−	0.997025i	$-0.524561\pi$
$397$	−3.07180	−0.154169	−0.0770847	+	0.997025i	$0.524561\pi$
$398$	0	0
$399$	0.775984	0.0388478
$400$	0	0
$401$	5.49127	0.274221	0.137110	−	0.990556i	$-0.456218\pi$
$401$	5.49127	0.274221	0.137110	+	0.990556i	$0.456218\pi$
$402$	0	0
$403$	36.2140	1.80395
$404$	0	0
$405$	−12.0985	−0.601179
$406$	0	0
$407$	−3.28991	−0.163075
$408$	0	0
$409$	−5.66156	−0.279946	−0.139973	−	0.990155i	$-0.544702\pi$
$409$	−5.66156	−0.279946	−0.139973	+	0.990155i	$0.544702\pi$
$410$	0	0
$411$	−0.842910	−0.0415777
$412$	0	0
$413$	−25.8171	−1.27038
$414$	0	0
$415$	−10.8191	−0.531091
$416$	0	0
$417$	1.73109	0.0847716
$418$	0	0
$419$	−32.4938	−1.58742	−0.793712	−	0.608294i	$-0.791854\pi$
$419$	−32.4938	−1.58742	−0.793712	+	0.608294i	$0.791854\pi$
$420$	0	0
$421$	20.5313	1.00063	0.500317	−	0.865842i	$-0.333217\pi$
$421$	20.5313	1.00063	0.500317	+	0.865842i	$0.333217\pi$
$422$	0	0
$423$	−30.6172	−1.48866
$424$	0	0
$425$	16.3547	0.793321
$426$	0	0
$427$	−15.4720	−0.748744
$428$	0	0
$429$	−0.303928	−0.0146738
$430$	0	0
$431$	−17.9439	−0.864327	−0.432164	−	0.901795i	$-0.642250\pi$
$431$	−17.9439	−0.864327	−0.432164	+	0.901795i	$0.642250\pi$
$432$	0	0
$433$	−23.7738	−1.14249	−0.571247	−	0.820778i	$-0.693540\pi$
$433$	−23.7738	−1.14249	−0.571247	+	0.820778i	$0.693540\pi$
$434$	0	0
$435$	−0.444605	−0.0213172
$436$	0	0
$437$	2.72482	0.130346
$438$	0	0
$439$	−10.3008	−0.491632	−0.245816	−	0.969317i	$-0.579056\pi$
$439$	−10.3008	−0.491632	−0.245816	+	0.969317i	$0.579056\pi$
$440$	0	0
$441$	10.4136	0.495884
$442$	0	0
$443$	36.5923	1.73855	0.869276	−	0.494327i	$-0.164586\pi$
$443$	36.5923	1.73855	0.869276	+	0.494327i	$0.164586\pi$
$444$	0	0
$445$	−14.3953	−0.682405
$446$	0	0
$447$	1.17464	0.0555585
$448$	0	0
$449$	−24.1955	−1.14186	−0.570929	−	0.821000i	$-0.693417\pi$
$449$	−24.1955	−1.14186	−0.570929	+	0.821000i	$0.693417\pi$
$450$	0	0
$451$	3.41306	0.160715
$452$	0	0
$453$	−1.27065	−0.0597002
$454$	0	0
$455$	8.87729	0.416174
$456$	0	0
$457$	3.77405	0.176543	0.0882713	−	0.996096i	$-0.471866\pi$
$457$	3.77405	0.176543	0.0882713	+	0.996096i	$0.471866\pi$
$458$	0	0
$459$	5.21224	0.243286
$460$	0	0
$461$	−17.2665	−0.804183	−0.402091	−	0.915600i	$-0.631717\pi$
$461$	−17.2665	−0.804183	−0.402091	+	0.915600i	$0.631717\pi$
$462$	0	0
$463$	9.59954	0.446129	0.223064	−	0.974804i	$-0.428394\pi$
$463$	9.59954	0.446129	0.223064	+	0.974804i	$0.428394\pi$
$464$	0	0
$465$	2.40200	0.111390
$466$	0	0
$467$	−25.5113	−1.18052	−0.590261	−	0.807213i	$-0.700975\pi$
$467$	−25.5113	−1.18052	−0.590261	+	0.807213i	$0.700975\pi$
$468$	0	0
$469$	−23.3061	−1.07618
$470$	0	0
$471$	0.489305	0.0225460
$472$	0	0
$473$	−0.350128	−0.0160989
$474$	0	0
$475$	−7.77677	−0.356823
$476$	0	0
$477$	3.92010	0.179489
$478$	0	0
$479$	−12.2561	−0.559998	−0.279999	−	0.960000i	$-0.590334\pi$
$479$	−12.2561	−0.559998	−0.279999	+	0.960000i	$0.590334\pi$
$480$	0	0
$481$	21.0710	0.960756
$482$	0	0
$483$	0.333986	0.0151969
$484$	0	0
$485$	16.9468	0.769513
$486$	0	0
$487$	−38.3034	−1.73569	−0.867846	−	0.496833i	$-0.834496\pi$
$487$	−38.3034	−1.73569	−0.867846	+	0.496833i	$0.834496\pi$
$488$	0	0
$489$	−1.60614	−0.0726322
$490$	0	0
$491$	−21.1124	−0.952788	−0.476394	−	0.879232i	$-0.658056\pi$
$491$	−21.1124	−0.952788	−0.476394	+	0.879232i	$0.658056\pi$
$492$	0	0
$493$	−10.3241	−0.464973
$494$	0	0
$495$	2.20340	0.0990354
$496$	0	0
$497$	−15.4094	−0.691205
$498$	0	0
$499$	−35.2535	−1.57816	−0.789081	−	0.614289i	$-0.789443\pi$
$499$	−35.2535	−1.57816	−0.789081	+	0.614289i	$0.789443\pi$
$500$	0	0
$501$	−3.59919	−0.160800
$502$	0	0
$503$	4.44605	0.198240	0.0991198	−	0.995076i	$-0.468397\pi$
$503$	4.44605	0.198240	0.0991198	+	0.995076i	$0.468397\pi$
$504$	0	0
$505$	1.08059	0.0480858
$506$	0	0
$507$	−0.197362	−0.00876517
$508$	0	0
$509$	−0.579375	−0.0256804	−0.0128402	−	0.999918i	$-0.504087\pi$
$509$	−0.579375	−0.0256804	−0.0128402	+	0.999918i	$0.504087\pi$
$510$	0	0
$511$	13.8992	0.614866
$512$	0	0
$513$	−2.47845	−0.109426
$514$	0	0
$515$	−8.21833	−0.362143
$516$	0	0
$517$	5.52458	0.242971
$518$	0	0
$519$	3.59483	0.157796
$520$	0	0
$521$	−41.5438	−1.82006	−0.910032	−	0.414537i	$-0.863943\pi$
$521$	−41.5438	−1.82006	−0.910032	+	0.414537i	$0.863943\pi$
$522$	0	0
$523$	21.0263	0.919417	0.459708	−	0.888070i	$-0.347954\pi$
$523$	21.0263	0.919417	0.459708	+	0.888070i	$0.347954\pi$
$524$	0	0
$525$	−0.953213	−0.0416016
$526$	0	0
$527$	55.7763	2.42966
$528$	0	0
$529$	−21.8272	−0.949010
$530$	0	0
$531$	41.0415	1.78105
$532$	0	0
$533$	−21.8598	−0.946852
$534$	0	0
$535$	−22.5111	−0.973241
$536$	0	0
$537$	0.764041	0.0329708
$538$	0	0
$539$	−1.87903	−0.0809354
$540$	0	0
$541$	30.9743	1.33169	0.665844	−	0.746091i	$-0.268072\pi$
$541$	30.9743	1.33169	0.665844	+	0.746091i	$0.268072\pi$
$542$	0	0
$543$	3.12339	0.134038
$544$	0	0
$545$	−24.7408	−1.05978
$546$	0	0
$547$	−31.7028	−1.35551	−0.677757	−	0.735286i	$-0.737048\pi$
$547$	−31.7028	−1.35551	−0.677757	+	0.735286i	$0.737048\pi$
$548$	0	0
$549$	24.5959	1.04973
$550$	0	0
$551$	4.90916	0.209137
$552$	0	0
$553$	−13.7127	−0.583123
$554$	0	0
$555$	1.39760	0.0593248
$556$	0	0
$557$	11.4341	0.484480	0.242240	−	0.970216i	$-0.422118\pi$
$557$	11.4341	0.484480	0.242240	+	0.970216i	$0.422118\pi$
$558$	0	0
$559$	2.24248	0.0948469
$560$	0	0
$561$	−0.468107	−0.0197635
$562$	0	0
$563$	9.17811	0.386811	0.193406	−	0.981119i	$-0.438047\pi$
$563$	9.17811	0.386811	0.193406	+	0.981119i	$0.438047\pi$
$564$	0	0
$565$	25.1654	1.05872
$566$	0	0
$567$	16.3740	0.687643
$568$	0	0
$569$	1.61458	0.0676866	0.0338433	−	0.999427i	$-0.489225\pi$
$569$	1.61458	0.0676866	0.0338433	+	0.999427i	$0.489225\pi$
$570$	0	0
$571$	24.1569	1.01094	0.505468	−	0.862846i	$-0.331320\pi$
$571$	24.1569	1.01094	0.505468	+	0.862846i	$0.331320\pi$
$572$	0	0
$573$	1.59436	0.0666053
$574$	0	0
$575$	−3.34715	−0.139586
$576$	0	0
$577$	15.6838	0.652924	0.326462	−	0.945210i	$-0.394144\pi$
$577$	15.6838	0.652924	0.326462	+	0.945210i	$0.394144\pi$
$578$	0	0
$579$	3.98285	0.165522
$580$	0	0
$581$	14.6425	0.607474
$582$	0	0
$583$	−0.707345	−0.0292952
$584$	0	0
$585$	−14.1122	−0.583468
$586$	0	0
$587$	10.7368	0.443155	0.221578	−	0.975143i	$-0.428879\pi$
$587$	10.7368	0.443155	0.221578	+	0.975143i	$0.428879\pi$
$588$	0	0
$589$	−26.5220	−1.09282
$590$	0	0
$591$	−3.14081	−0.129196
$592$	0	0
$593$	22.6536	0.930273	0.465136	−	0.885239i	$-0.346005\pi$
$593$	22.6536	0.930273	0.465136	+	0.885239i	$0.346005\pi$
$594$	0	0
$595$	13.6727	0.560526
$596$	0	0
$597$	2.41542	0.0988565
$598$	0	0
$599$	6.97985	0.285189	0.142594	−	0.989781i	$-0.454456\pi$
$599$	6.97985	0.285189	0.142594	+	0.989781i	$0.454456\pi$
$600$	0	0
$601$	−36.7290	−1.49821	−0.749104	−	0.662453i	$-0.769516\pi$
$601$	−36.7290	−1.49821	−0.749104	+	0.662453i	$0.769516\pi$
$602$	0	0
$603$	37.0497	1.50878
$604$	0	0
$605$	14.8016	0.601771
$606$	0	0
$607$	−38.8007	−1.57487	−0.787435	−	0.616398i	$-0.788591\pi$
$607$	−38.8007	−1.57487	−0.787435	+	0.616398i	$0.788591\pi$
$608$	0	0
$609$	0.601725	0.0243831
$610$	0	0
$611$	−35.3835	−1.43147
$612$	0	0
$613$	10.2993	0.415983	0.207992	−	0.978131i	$-0.433307\pi$
$613$	10.2993	0.415983	0.207992	+	0.978131i	$0.433307\pi$
$614$	0	0
$615$	−1.44992	−0.0584663
$616$	0	0
$617$	−9.60079	−0.386513	−0.193257	−	0.981148i	$-0.561905\pi$
$617$	−9.60079	−0.386513	−0.193257	+	0.981148i	$0.561905\pi$
$618$	0	0
$619$	−27.4324	−1.10260	−0.551300	−	0.834307i	$-0.685868\pi$
$619$	−27.4324	−1.10260	−0.551300	+	0.834307i	$0.685868\pi$
$620$	0	0
$621$	−1.06673	−0.0428065
$622$	0	0
$623$	19.4825	0.780551
$624$	0	0
$625$	0.00682458	0.000272983 0
$626$	0	0
$627$	0.222588	0.00888930
$628$	0	0
$629$	32.4534	1.29400
$630$	0	0
$631$	−12.4909	−0.497254	−0.248627	−	0.968599i	$-0.579979\pi$
$631$	−12.4909	−0.497254	−0.248627	+	0.968599i	$0.579979\pi$
$632$	0	0
$633$	−3.24606	−0.129019
$634$	0	0
$635$	−7.64022	−0.303193
$636$	0	0
$637$	12.0347	0.476832
$638$	0	0
$639$	24.4963	0.969057
$640$	0	0
$641$	−4.41453	−0.174364	−0.0871818	−	0.996192i	$-0.527786\pi$
$641$	−4.41453	−0.174364	−0.0871818	+	0.996192i	$0.527786\pi$
$642$	0	0
$643$	−28.2482	−1.11400	−0.557001	−	0.830512i	$-0.688048\pi$
$643$	−28.2482	−1.11400	−0.557001	+	0.830512i	$0.688048\pi$
$644$	0	0
$645$	0.148740	0.00585662
$646$	0	0
$647$	47.4682	1.86617	0.933084	−	0.359660i	$-0.117107\pi$
$647$	47.4682	1.86617	0.933084	+	0.359660i	$0.117107\pi$
$648$	0	0
$649$	−7.40553	−0.290693
$650$	0	0
$651$	−3.25085	−0.127411
$652$	0	0
$653$	−30.4075	−1.18994	−0.594968	−	0.803749i	$-0.702835\pi$
$653$	−30.4075	−1.18994	−0.594968	+	0.803749i	$0.702835\pi$
$654$	0	0
$655$	−1.66526	−0.0650672
$656$	0	0
$657$	−22.0956	−0.862031
$658$	0	0
$659$	0.0569585	0.00221879	0.00110939	−	0.999999i	$-0.499647\pi$
$659$	0.0569585	0.00221879	0.00110939	+	0.999999i	$0.499647\pi$
$660$	0	0
$661$	−44.7208	−1.73944	−0.869718	−	0.493548i	$-0.835700\pi$
$661$	−44.7208	−1.73944	−0.869718	+	0.493548i	$0.835700\pi$
$662$	0	0
$663$	2.99811	0.116437
$664$	0	0
$665$	−6.50145	−0.252115
$666$	0	0
$667$	2.11292	0.0818125
$668$	0	0
$669$	−2.12843	−0.0822900
$670$	0	0
$671$	−4.43809	−0.171330
$672$	0	0
$673$	29.7172	1.14551	0.572757	−	0.819725i	$-0.305874\pi$
$673$	29.7172	1.14551	0.572757	+	0.819725i	$0.305874\pi$
$674$	0	0
$675$	3.04451	0.117183
$676$	0	0
$677$	−1.55948	−0.0599357	−0.0299679	−	0.999551i	$-0.509540\pi$
$677$	−1.55948	−0.0599357	−0.0299679	+	0.999551i	$0.509540\pi$
$678$	0	0
$679$	−22.9356	−0.880187
$680$	0	0
$681$	3.71304	0.142284
$682$	0	0
$683$	3.15386	0.120679	0.0603395	−	0.998178i	$-0.480782\pi$
$683$	3.15386	0.120679	0.0603395	+	0.998178i	$0.480782\pi$
$684$	0	0
$685$	7.06217	0.269832
$686$	0	0
$687$	1.10856	0.0422943
$688$	0	0
$689$	4.53036	0.172593
$690$	0	0
$691$	−41.3424	−1.57274	−0.786369	−	0.617756i	$-0.788042\pi$
$691$	−41.3424	−1.57274	−0.786369	+	0.617756i	$0.788042\pi$
$692$	0	0
$693$	−2.98206	−0.113279
$694$	0	0
$695$	−14.5036	−0.550153
$696$	0	0
$697$	−33.6682	−1.27527
$698$	0	0
$699$	2.91823	0.110378
$700$	0	0
$701$	−18.6405	−0.704041	−0.352020	−	0.935992i	$-0.614505\pi$
$701$	−18.6405	−0.704041	−0.352020	+	0.935992i	$0.614505\pi$
$702$	0	0
$703$	−15.4318	−0.582020
$704$	0	0
$705$	−2.34692	−0.0883902
$706$	0	0
$707$	−1.46246	−0.0550017
$708$	0	0
$709$	−24.4135	−0.916867	−0.458434	−	0.888729i	$-0.651589\pi$
$709$	−24.4135	−0.916867	−0.458434	+	0.888729i	$0.651589\pi$
$710$	0	0
$711$	21.7990	0.817528
$712$	0	0
$713$	−11.4151	−0.427500
$714$	0	0
$715$	2.54641	0.0952304
$716$	0	0
$717$	2.18457	0.0815844
$718$	0	0
$719$	3.04053	0.113393	0.0566964	−	0.998391i	$-0.481943\pi$
$719$	3.04053	0.113393	0.0566964	+	0.998391i	$0.481943\pi$
$720$	0	0
$721$	11.1226	0.414227
$722$	0	0
$723$	−3.47732	−0.129323
$724$	0	0
$725$	−6.03037	−0.223962
$726$	0	0
$727$	−1.94456	−0.0721197	−0.0360599	−	0.999350i	$-0.511481\pi$
$727$	−1.94456	−0.0721197	−0.0360599	+	0.999350i	$0.511481\pi$
$728$	0	0
$729$	−25.5424	−0.946014
$730$	0	0
$731$	3.45385	0.127745
$732$	0	0
$733$	42.7869	1.58037	0.790185	−	0.612868i	$-0.209984\pi$
$733$	42.7869	1.58037	0.790185	+	0.612868i	$0.209984\pi$
$734$	0	0
$735$	0.798238	0.0294434
$736$	0	0
$737$	−6.68526	−0.246255
$738$	0	0
$739$	−2.64693	−0.0973689	−0.0486845	−	0.998814i	$-0.515503\pi$
$739$	−2.64693	−0.0973689	−0.0486845	+	0.998814i	$0.515503\pi$
$740$	0	0
$741$	−1.42562	−0.0523714
$742$	0	0
$743$	35.6220	1.30685	0.653423	−	0.756993i	$-0.273332\pi$
$743$	35.6220	1.30685	0.653423	+	0.756993i	$0.273332\pi$
$744$	0	0
$745$	−9.84150	−0.360565
$746$	0	0
$747$	−23.2772	−0.851668
$748$	0	0
$749$	30.4663	1.11322
$750$	0	0
$751$	5.91359	0.215790	0.107895	−	0.994162i	$-0.465589\pi$
$751$	5.91359	0.215790	0.107895	+	0.994162i	$0.465589\pi$
$752$	0	0
$753$	0.164919	0.00600998
$754$	0	0
$755$	10.6459	0.387444
$756$	0	0
$757$	−2.55989	−0.0930408	−0.0465204	−	0.998917i	$-0.514813\pi$
$757$	−2.55989	−0.0930408	−0.0465204	+	0.998917i	$0.514813\pi$
$758$	0	0
$759$	0.0958025	0.00347741
$760$	0	0
$761$	14.3371	0.519721	0.259860	−	0.965646i	$-0.416323\pi$
$761$	14.3371	0.519721	0.259860	+	0.965646i	$0.416323\pi$
$762$	0	0
$763$	33.4840	1.21220
$764$	0	0
$765$	−21.7355	−0.785848
$766$	0	0
$767$	47.4306	1.71262
$768$	0	0
$769$	−28.6419	−1.03285	−0.516426	−	0.856332i	$-0.672738\pi$
$769$	−28.6419	−1.03285	−0.516426	+	0.856332i	$0.672738\pi$
$770$	0	0
$771$	−3.21960	−0.115951
$772$	0	0
$773$	38.9087	1.39945	0.699725	−	0.714412i	$-0.253306\pi$
$773$	38.9087	1.39945	0.699725	+	0.714412i	$0.253306\pi$
$774$	0	0
$775$	32.5794	1.17029
$776$	0	0
$777$	−1.89150	−0.0678572
$778$	0	0
$779$	16.0094	0.573597
$780$	0	0
$781$	−4.42011	−0.158164
$782$	0	0
$783$	−1.92187	−0.0686821
$784$	0	0
$785$	−4.09956	−0.146320
$786$	0	0
$787$	18.5441	0.661027	0.330514	−	0.943801i	$-0.392778\pi$
$787$	18.5441	0.661027	0.330514	+	0.943801i	$0.392778\pi$
$788$	0	0
$789$	−2.37847	−0.0846757
$790$	0	0
$791$	−34.0587	−1.21099
$792$	0	0
$793$	28.4248	1.00939
$794$	0	0
$795$	0.300490	0.0106573
$796$	0	0
$797$	2.45640	0.0870103	0.0435052	−	0.999053i	$-0.486148\pi$
$797$	2.45640	0.0870103	0.0435052	+	0.999053i	$0.486148\pi$
$798$	0	0
$799$	−54.4974	−1.92798
$800$	0	0
$801$	−30.9714	−1.09432
$802$	0	0
$803$	3.98693	0.140696
$804$	0	0
$805$	−2.79824	−0.0986252
$806$	0	0
$807$	3.95167	0.139105
$808$	0	0
$809$	10.2785	0.361372	0.180686	−	0.983541i	$-0.442168\pi$
$809$	10.2785	0.361372	0.180686	+	0.983541i	$0.442168\pi$
$810$	0	0
$811$	8.79705	0.308906	0.154453	−	0.988000i	$-0.450638\pi$
$811$	8.79705	0.308906	0.154453	+	0.988000i	$0.450638\pi$
$812$	0	0
$813$	3.01865	0.105869
$814$	0	0
$815$	13.4568	0.471370
$816$	0	0
$817$	−1.64232	−0.0574577
$818$	0	0
$819$	19.0993	0.667385
$820$	0	0
$821$	−22.2393	−0.776158	−0.388079	−	0.921626i	$-0.626861\pi$
$821$	−22.2393	−0.776158	−0.388079	+	0.921626i	$0.626861\pi$
$822$	0	0
$823$	−17.2681	−0.601927	−0.300963	−	0.953636i	$-0.597308\pi$
$823$	−17.2681	−0.601927	−0.300963	+	0.953636i	$0.597308\pi$
$824$	0	0
$825$	−0.273425	−0.00951944
$826$	0	0
$827$	36.4068	1.26599	0.632995	−	0.774156i	$-0.281825\pi$
$827$	36.4068	1.26599	0.632995	+	0.774156i	$0.281825\pi$
$828$	0	0
$829$	−35.2447	−1.22410	−0.612050	−	0.790819i	$-0.709655\pi$
$829$	−35.2447	−1.22410	−0.612050	+	0.790819i	$0.709655\pi$
$830$	0	0
$831$	4.25783	0.147702
$832$	0	0
$833$	18.5357	0.642224
$834$	0	0
$835$	30.1552	1.04356
$836$	0	0
$837$	10.3830	0.358890
$838$	0	0
$839$	54.4281	1.87907	0.939533	−	0.342459i	$-0.111260\pi$
$839$	54.4281	1.87907	0.939533	+	0.342459i	$0.111260\pi$
$840$	0	0
$841$	−25.1933	−0.868734
$842$	0	0
$843$	−3.60997	−0.124334
$844$	0	0
$845$	1.65357	0.0568844
$846$	0	0
$847$	−20.0324	−0.688320
$848$	0	0
$849$	1.48490	0.0509616
$850$	0	0
$851$	−6.64188	−0.227681
$852$	0	0
$853$	45.1884	1.54722	0.773611	−	0.633661i	$-0.218448\pi$
$853$	45.1884	1.54722	0.773611	+	0.633661i	$0.218448\pi$
$854$	0	0
$855$	10.3353	0.353461
$856$	0	0
$857$	16.3569	0.558740	0.279370	−	0.960183i	$-0.409874\pi$
$857$	16.3569	0.558740	0.279370	+	0.960183i	$0.409874\pi$
$858$	0	0
$859$	55.4909	1.89332	0.946662	−	0.322228i	$-0.104432\pi$
$859$	55.4909	1.89332	0.946662	+	0.322228i	$0.104432\pi$
$860$	0	0
$861$	1.96230	0.0668751
$862$	0	0
$863$	39.4103	1.34154	0.670771	−	0.741665i	$-0.265964\pi$
$863$	39.4103	1.34154	0.670771	+	0.741665i	$0.265964\pi$
$864$	0	0
$865$	−30.1187	−1.02407
$866$	0	0
$867$	1.81403	0.0616077
$868$	0	0
$869$	−3.93343	−0.133432
$870$	0	0
$871$	42.8174	1.45081
$872$	0	0
$873$	36.4607	1.23401
$874$	0	0
$875$	20.9059	0.706749
$876$	0	0
$877$	19.1323	0.646053	0.323027	−	0.946390i	$-0.395300\pi$
$877$	19.1323	0.646053	0.323027	+	0.946390i	$0.395300\pi$
$878$	0	0
$879$	−1.70247	−0.0574229
$880$	0	0
$881$	28.6268	0.964460	0.482230	−	0.876045i	$-0.339827\pi$
$881$	28.6268	0.964460	0.482230	+	0.876045i	$0.339827\pi$
$882$	0	0
$883$	−20.5608	−0.691926	−0.345963	−	0.938248i	$-0.612448\pi$
$883$	−20.5608	−0.691926	−0.345963	+	0.938248i	$0.612448\pi$
$884$	0	0
$885$	3.14598	0.105751
$886$	0	0
$887$	−14.0885	−0.473045	−0.236522	−	0.971626i	$-0.576008\pi$
$887$	−14.0885	−0.473045	−0.236522	+	0.971626i	$0.576008\pi$
$888$	0	0
$889$	10.3402	0.346799
$890$	0	0
$891$	4.69681	0.157349
$892$	0	0
$893$	25.9138	0.867173
$894$	0	0
$895$	−6.40139	−0.213975
$896$	0	0
$897$	−0.613591	−0.0204872
$898$	0	0
$899$	−20.5660	−0.685916
$900$	0	0
$901$	6.97761	0.232458
$902$	0	0
$903$	−0.201303	−0.00669894
$904$	0	0
$905$	−26.1688	−0.869880
$906$	0	0
$907$	20.0693	0.666390	0.333195	−	0.942858i	$-0.391873\pi$
$907$	20.0693	0.666390	0.333195	+	0.942858i	$0.391873\pi$
$908$	0	0
$909$	2.32488	0.0771114
$910$	0	0
$911$	−20.6039	−0.682637	−0.341319	−	0.939948i	$-0.610874\pi$
$911$	−20.6039	−0.682637	−0.341319	+	0.939948i	$0.610874\pi$
$912$	0	0
$913$	4.20014	0.139004
$914$	0	0
$915$	1.88536	0.0623282
$916$	0	0
$917$	2.25375	0.0744254
$918$	0	0
$919$	3.63168	0.119798	0.0598990	−	0.998204i	$-0.480922\pi$
$919$	3.63168	0.119798	0.0598990	+	0.998204i	$0.480922\pi$
$920$	0	0
$921$	−1.16486	−0.0383835
$922$	0	0
$923$	28.3097	0.931825
$924$	0	0
$925$	18.9563	0.623278
$926$	0	0
$927$	−17.6816	−0.580740
$928$	0	0
$929$	−51.1167	−1.67709	−0.838543	−	0.544836i	$-0.816592\pi$
$929$	−51.1167	−1.67709	−0.838543	+	0.544836i	$0.816592\pi$
$930$	0	0
$931$	−8.81383	−0.288862
$932$	0	0
$933$	−0.985934	−0.0322780
$934$	0	0
$935$	3.92196	0.128262
$936$	0	0
$937$	−13.6650	−0.446418	−0.223209	−	0.974771i	$-0.571653\pi$
$937$	−13.6650	−0.446418	−0.223209	+	0.974771i	$0.571653\pi$
$938$	0	0
$939$	−0.716694	−0.0233884
$940$	0	0
$941$	−54.4871	−1.77623	−0.888114	−	0.459623i	$-0.847985\pi$
$941$	−54.4871	−1.77623	−0.888114	+	0.459623i	$0.847985\pi$
$942$	0	0
$943$	6.89050	0.224386
$944$	0	0
$945$	2.54524	0.0827965
$946$	0	0
$947$	−15.5186	−0.504286	−0.252143	−	0.967690i	$-0.581135\pi$
$947$	−15.5186	−0.504286	−0.252143	+	0.967690i	$0.581135\pi$
$948$	0	0
$949$	−25.5353	−0.828911
$950$	0	0
$951$	1.89616	0.0614873
$952$	0	0
$953$	27.4414	0.888915	0.444458	−	0.895800i	$-0.353396\pi$
$953$	27.4414	0.888915	0.444458	+	0.895800i	$0.353396\pi$
$954$	0	0
$955$	−13.3581	−0.432257
$956$	0	0
$957$	0.172602	0.00557943
$958$	0	0
$959$	−9.55788	−0.308640
$960$	0	0
$961$	80.1091	2.58416
$962$	0	0
$963$	−48.4323	−1.56071
$964$	0	0
$965$	−33.3696	−1.07421
$966$	0	0
$967$	−52.5276	−1.68917	−0.844586	−	0.535419i	$-0.820154\pi$
$967$	−52.5276	−1.68917	−0.844586	+	0.535419i	$0.820154\pi$
$968$	0	0
$969$	−2.19572	−0.0705367
$970$	0	0
$971$	16.7966	0.539029	0.269514	−	0.962996i	$-0.413137\pi$
$971$	16.7966	0.539029	0.269514	+	0.962996i	$0.413137\pi$
$972$	0	0
$973$	19.6290	0.629278
$974$	0	0
$975$	1.75122	0.0560839
$976$	0	0
$977$	−11.9633	−0.382739	−0.191369	−	0.981518i	$-0.561293\pi$
$977$	−11.9633	−0.382739	−0.191369	+	0.981518i	$0.561293\pi$
$978$	0	0
$979$	5.58848	0.178609
$980$	0	0
$981$	−53.2294	−1.69948
$982$	0	0
$983$	20.4967	0.653743	0.326872	−	0.945069i	$-0.394005\pi$
$983$	20.4967	0.653743	0.326872	+	0.945069i	$0.394005\pi$
$984$	0	0
$985$	26.3147	0.838456
$986$	0	0
$987$	3.17630	0.101103
$988$	0	0
$989$	−0.706862	−0.0224769
$990$	0	0
$991$	−44.5827	−1.41622	−0.708109	−	0.706103i	$-0.750451\pi$
$991$	−44.5827	−1.41622	−0.708109	+	0.706103i	$0.750451\pi$
$992$	0	0
$993$	−0.141350	−0.00448562
$994$	0	0
$995$	−20.2372	−0.641561
$996$	0	0
$997$	5.99320	0.189807	0.0949033	−	0.995486i	$-0.469746\pi$
$997$	5.99320	0.189807	0.0949033	+	0.995486i	$0.469746\pi$
$998$	0	0
$999$	6.04134	0.191140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1004.2.a.a.1.4	✓	7
3.2	odd	2		9036.2.a.i.1.4		7
4.3	odd	2		4016.2.a.g.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.4	✓	7	1.1	even	1	trivial
4016.2.a.g.1.4		7	4.3	odd	2
9036.2.a.i.1.4		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1004 = 2^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.164919 q^{3} -1.38175 q^{5} +1.87004 q^{7} -2.97280 q^{9} +O(q^{10})\)	\(q+0.164919 q^{3} -1.38175 q^{5} +1.87004 q^{7} -2.97280 q^{9} +0.536413 q^{11} -3.43559 q^{13} -0.227876 q^{15} -5.29146 q^{17} +2.51612 q^{19} +0.308405 q^{21} +1.08295 q^{23} -3.09078 q^{25} -0.985029 q^{27} +1.95108 q^{29} -10.5408 q^{31} +0.0884647 q^{33} -2.58392 q^{35} -6.13316 q^{37} -0.566594 q^{39} +6.36274 q^{41} -0.652721 q^{43} +4.10765 q^{45} +10.2991 q^{47} -3.50295 q^{49} -0.872662 q^{51} -1.31866 q^{53} -0.741186 q^{55} +0.414956 q^{57} -13.8057 q^{59} -8.27363 q^{61} -5.55926 q^{63} +4.74711 q^{65} -12.4629 q^{67} +0.178598 q^{69} -8.24013 q^{71} +7.43258 q^{73} -0.509728 q^{75} +1.00311 q^{77} -7.33283 q^{79} +8.75596 q^{81} +7.83006 q^{83} +7.31145 q^{85} +0.321771 q^{87} +10.4182 q^{89} -6.42469 q^{91} -1.73838 q^{93} -3.47664 q^{95} -12.2648 q^{97} -1.59465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

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Database

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