LMFDB - Embedded newform 1620.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.244785.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 90x - 240 $ x^3 - x^2 - 90*x - 240
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.10730$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +22.6187 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +22.6187 q^{7} +10.9748 q^{11} -75.8057 q^{13} +15.9497 q^{17} +58.8812 q^{19} -56.6187 q^{23} +25.0000 q^{25} -5.61868 q^{29} -120.687 q^{31} -113.093 q^{35} +236.136 q^{37} -196.137 q^{41} +18.1873 q^{43} -311.791 q^{47} +168.604 q^{49} -33.3813 q^{53} -54.8742 q^{55} -520.981 q^{59} -150.259 q^{61} +379.029 q^{65} +506.122 q^{67} -961.593 q^{71} -251.877 q^{73} +248.236 q^{77} +834.834 q^{79} -691.323 q^{83} -79.7485 q^{85} +1006.10 q^{89} -1714.62 q^{91} -294.406 q^{95} +1285.65 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 - 3 * q^7	$ 3 q - 15 q^{5} - 3 q^{7} + 24 q^{11} + 3 q^{13} + 30 q^{17} - 27 q^{19} - 99 q^{23} + 75 q^{25} + 54 q^{29} + 72 q^{31} + 15 q^{35} - 18 q^{37} - 411 q^{41} + 444 q^{43} + 75 q^{47} + 204 q^{49} - 171 q^{53} - 120 q^{55} - 297 q^{59} + 684 q^{61} - 15 q^{65} + 12 q^{67} - 642 q^{71} - 66 q^{73} - 1044 q^{77} + 1122 q^{79} - 90 q^{83} - 150 q^{85} - 756 q^{89} - 1653 q^{91} + 135 q^{95} + 1536 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 - 3 * q^7 + 24 * q^11 + 3 * q^13 + 30 * q^17 - 27 * q^19 - 99 * q^23 + 75 * q^25 + 54 * q^29 + 72 * q^31 + 15 * q^35 - 18 * q^37 - 411 * q^41 + 444 * q^43 + 75 * q^47 + 204 * q^49 - 171 * q^53 - 120 * q^55 - 297 * q^59 + 684 * q^61 - 15 * q^65 + 12 * q^67 - 642 * q^71 - 66 * q^73 - 1044 * q^77 + 1122 * q^79 - 90 * q^83 - 150 * q^85 - 756 * q^89 - 1653 * q^91 + 135 * q^95 + 1536 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	22.6187	1.22129	0.610647	−	0.791903i	$-0.290910\pi$
$7$	22.6187	1.22129	0.610647	+	0.791903i	$0.290910\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.9748	0.300822	0.150411	−	0.988624i	$-0.451940\pi$
$11$	10.9748	0.300822	0.150411	+	0.988624i	$0.451940\pi$
$12$	0	0
$13$	−75.8057	−1.61729	−0.808643	−	0.588299i	$-0.799798\pi$
$13$	−75.8057	−1.61729	−0.808643	+	0.588299i	$0.799798\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	15.9497	0.227551	0.113776	−	0.993506i	$-0.463706\pi$
$17$	15.9497	0.227551	0.113776	+	0.993506i	$0.463706\pi$
$18$	0	0
$19$	58.8812	0.710962	0.355481	−	0.934684i	$-0.384317\pi$
$19$	58.8812	0.710962	0.355481	+	0.934684i	$0.384317\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−56.6187	−0.513296	−0.256648	−	0.966505i	$-0.582618\pi$
$23$	−56.6187	−0.513296	−0.256648	+	0.966505i	$0.582618\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.61868	−0.0359780	−0.0179890	−	0.999838i	$-0.505726\pi$
$29$	−5.61868	−0.0359780	−0.0179890	+	0.999838i	$0.505726\pi$
$30$	0	0
$31$	−120.687	−0.699226	−0.349613	−	0.936894i	$-0.613687\pi$
$31$	−120.687	−0.699226	−0.349613	+	0.936894i	$0.613687\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−113.093	−0.546179
$36$	0	0
$37$	236.136	1.04921	0.524603	−	0.851347i	$-0.324214\pi$
$37$	236.136	1.04921	0.524603	+	0.851347i	$0.324214\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−196.137	−0.747108	−0.373554	−	0.927609i	$-0.621861\pi$
$41$	−196.137	−0.747108	−0.373554	+	0.927609i	$0.621861\pi$
$42$	0	0
$43$	18.1873	0.0645010	0.0322505	−	0.999480i	$-0.489733\pi$
$43$	18.1873	0.0645010	0.0322505	+	0.999480i	$0.489733\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−311.791	−0.967647	−0.483824	−	0.875165i	$-0.660752\pi$
$47$	−311.791	−0.967647	−0.483824	+	0.875165i	$0.660752\pi$
$48$	0	0
$49$	168.604	0.491558
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−33.3813	−0.0865147	−0.0432573	−	0.999064i	$-0.513774\pi$
$53$	−33.3813	−0.0865147	−0.0432573	+	0.999064i	$0.513774\pi$
$54$	0	0
$55$	−54.8742	−0.134532
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−520.981	−1.14959	−0.574796	−	0.818296i	$-0.694919\pi$
$59$	−520.981	−1.14959	−0.574796	+	0.818296i	$0.694919\pi$
$60$	0	0
$61$	−150.259	−0.315388	−0.157694	−	0.987488i	$-0.550406\pi$
$61$	−150.259	−0.315388	−0.157694	+	0.987488i	$0.550406\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	379.029	0.723272
$66$	0	0
$67$	506.122	0.922875	0.461438	−	0.887173i	$-0.347334\pi$
$67$	506.122	0.922875	0.461438	+	0.887173i	$0.347334\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−961.593	−1.60733	−0.803663	−	0.595085i	$-0.797119\pi$
$71$	−961.593	−1.60733	−0.803663	+	0.595085i	$0.797119\pi$
$72$	0	0
$73$	−251.877	−0.403835	−0.201918	−	0.979403i	$-0.564717\pi$
$73$	−251.877	−0.403835	−0.201918	+	0.979403i	$0.564717\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	248.236	0.367392
$78$	0	0
$79$	834.834	1.18894	0.594470	−	0.804118i	$-0.297362\pi$
$79$	834.834	1.18894	0.594470	+	0.804118i	$0.297362\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−691.323	−0.914248	−0.457124	−	0.889403i	$-0.651120\pi$
$83$	−691.323	−0.914248	−0.457124	+	0.889403i	$0.651120\pi$
$84$	0	0
$85$	−79.7485	−0.101764
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1006.10	1.19827	0.599137	−	0.800647i	$-0.295511\pi$
$89$	1006.10	1.19827	0.599137	+	0.800647i	$0.295511\pi$
$90$	0	0
$91$	−1714.62	−1.97518
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−294.406	−0.317952
$96$	0	0
$97$	1285.65	1.34575	0.672874	−	0.739757i	$-0.265059\pi$
$97$	1285.65	1.34575	0.672874	+	0.739757i	$0.265059\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	834.136	0.821778	0.410889	−	0.911685i	$-0.365218\pi$
$101$	834.136	0.821778	0.410889	+	0.911685i	$0.365218\pi$
$102$	0	0
$103$	−532.950	−0.509837	−0.254918	−	0.966963i	$-0.582049\pi$
$103$	−532.950	−0.509837	−0.254918	+	0.966963i	$0.582049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	363.906	0.328786	0.164393	−	0.986395i	$-0.447433\pi$
$107$	363.906	0.328786	0.164393	+	0.986395i	$0.447433\pi$
$108$	0	0
$109$	−321.807	−0.282784	−0.141392	−	0.989954i	$-0.545158\pi$
$109$	−321.807	−0.282784	−0.141392	+	0.989954i	$0.545158\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1757.04	1.46273	0.731365	−	0.681986i	$-0.238884\pi$
$113$	1757.04	1.46273	0.731365	+	0.681986i	$0.238884\pi$
$114$	0	0
$115$	283.093	0.229553
$116$	0	0
$117$	0	0
$118$	0	0
$119$	360.761	0.277907
$120$	0	0
$121$	−1210.55	−0.909506
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1369.60	−0.956945	−0.478473	−	0.878102i	$-0.658809\pi$
$127$	−1369.60	−0.956945	−0.478473	+	0.878102i	$0.658809\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	127.908	0.0853083	0.0426541	−	0.999090i	$-0.486419\pi$
$131$	127.908	0.0853083	0.0426541	+	0.999090i	$0.486419\pi$
$132$	0	0
$133$	1331.81	0.868293
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2595.35	−1.61851	−0.809255	−	0.587457i	$-0.800129\pi$
$137$	−2595.35	−1.61851	−0.809255	+	0.587457i	$0.800129\pi$
$138$	0	0
$139$	1568.31	0.956995	0.478498	−	0.878089i	$-0.341182\pi$
$139$	1568.31	0.956995	0.478498	+	0.878089i	$0.341182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−831.956	−0.486515
$144$	0	0
$145$	28.0934	0.0160898
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3120.21	−1.71556	−0.857778	−	0.514020i	$-0.828156\pi$
$149$	−3120.21	−1.71556	−0.857778	+	0.514020i	$0.828156\pi$
$150$	0	0
$151$	−819.564	−0.441690	−0.220845	−	0.975309i	$-0.570881\pi$
$151$	−819.564	−0.441690	−0.220845	+	0.975309i	$0.570881\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	603.435	0.312703
$156$	0	0
$157$	−884.560	−0.449653	−0.224827	−	0.974399i	$-0.572182\pi$
$157$	−884.560	−0.449653	−0.224827	+	0.974399i	$0.572182\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1280.64	−0.626885
$162$	0	0
$163$	−3193.13	−1.53439	−0.767194	−	0.641415i	$-0.778348\pi$
$163$	−3193.13	−1.53439	−0.767194	+	0.641415i	$0.778348\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1707.49	−0.791194	−0.395597	−	0.918424i	$-0.629462\pi$
$167$	−1707.49	−0.791194	−0.395597	+	0.918424i	$0.629462\pi$
$168$	0	0
$169$	3549.51	1.61562
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3702.10	−1.62697	−0.813485	−	0.581586i	$-0.802432\pi$
$173$	−3702.10	−1.62697	−0.813485	+	0.581586i	$0.802432\pi$
$174$	0	0
$175$	565.467	0.244259
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4389.15	−1.83274	−0.916371	−	0.400331i	$-0.868895\pi$
$179$	−4389.15	−1.83274	−0.916371	+	0.400331i	$0.868895\pi$
$180$	0	0
$181$	−823.922	−0.338351	−0.169176	−	0.985586i	$-0.554111\pi$
$181$	−823.922	−0.338351	−0.169176	+	0.985586i	$0.554111\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1180.68	−0.469219
$186$	0	0
$187$	175.045	0.0684523
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4914.29	−1.86171	−0.930853	−	0.365394i	$-0.880934\pi$
$191$	−4914.29	−1.86171	−0.930853	+	0.365394i	$0.880934\pi$
$192$	0	0
$193$	−2501.87	−0.933103	−0.466551	−	0.884494i	$-0.654504\pi$
$193$	−2501.87	−0.933103	−0.466551	+	0.884494i	$0.654504\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1416.00	−0.512109	−0.256055	−	0.966662i	$-0.582423\pi$
$197$	−1416.00	−0.512109	−0.256055	+	0.966662i	$0.582423\pi$
$198$	0	0
$199$	1532.05	0.545751	0.272875	−	0.962049i	$-0.412025\pi$
$199$	1532.05	0.545751	0.272875	+	0.962049i	$0.412025\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−127.087	−0.0439397
$204$	0	0
$205$	980.684	0.334117
$206$	0	0
$207$	0	0
$208$	0	0
$209$	646.212	0.213873
$210$	0	0
$211$	−17.9805	−0.00586648	−0.00293324	−	0.999996i	$-0.500934\pi$
$211$	−17.9805	−0.00586648	−0.00293324	+	0.999996i	$0.500934\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−90.9367	−0.0288457
$216$	0	0
$217$	−2729.78	−0.853960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1209.08	−0.368015
$222$	0	0
$223$	961.340	0.288682	0.144341	−	0.989528i	$-0.453894\pi$
$223$	961.340	0.288682	0.144341	+	0.989528i	$0.453894\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1758.21	0.514082	0.257041	−	0.966401i	$-0.417253\pi$
$227$	1758.21	0.514082	0.257041	+	0.966401i	$0.417253\pi$
$228$	0	0
$229$	−1531.47	−0.441932	−0.220966	−	0.975282i	$-0.570921\pi$
$229$	−1531.47	−0.441932	−0.220966	+	0.975282i	$0.570921\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6668.03	−1.87484	−0.937419	−	0.348203i	$-0.886792\pi$
$233$	−6668.03	−1.87484	−0.937419	+	0.348203i	$0.886792\pi$
$234$	0	0
$235$	1558.96	0.432745
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2717.24	−0.735412	−0.367706	−	0.929942i	$-0.619857\pi$
$239$	−2717.24	−0.735412	−0.367706	+	0.929942i	$0.619857\pi$
$240$	0	0
$241$	6702.00	1.79134	0.895672	−	0.444716i	$-0.146695\pi$
$241$	6702.00	1.79134	0.895672	+	0.444716i	$0.146695\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−843.022	−0.219832
$246$	0	0
$247$	−4463.53	−1.14983
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5665.89	1.42481	0.712406	−	0.701768i	$-0.247606\pi$
$251$	5665.89	1.42481	0.712406	+	0.701768i	$0.247606\pi$
$252$	0	0
$253$	−621.381	−0.154411
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3968.72	−0.963276	−0.481638	−	0.876370i	$-0.659958\pi$
$257$	−3968.72	−0.963276	−0.481638	+	0.876370i	$0.659958\pi$
$258$	0	0
$259$	5341.09	1.28139
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1294.64	0.303541	0.151770	−	0.988416i	$-0.451503\pi$
$263$	1294.64	0.303541	0.151770	+	0.988416i	$0.451503\pi$
$264$	0	0
$265$	166.907	0.0386905
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2742.06	0.621511	0.310755	−	0.950490i	$-0.399418\pi$
$269$	2742.06	0.621511	0.310755	+	0.950490i	$0.399418\pi$
$270$	0	0
$271$	−2598.68	−0.582504	−0.291252	−	0.956646i	$-0.594072\pi$
$271$	−2598.68	−0.582504	−0.291252	+	0.956646i	$0.594072\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	274.371	0.0601644
$276$	0	0
$277$	2955.99	0.641184	0.320592	−	0.947217i	$-0.396118\pi$
$277$	2955.99	0.641184	0.320592	+	0.947217i	$0.396118\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5973.91	−1.26823	−0.634116	−	0.773238i	$-0.718636\pi$
$281$	−5973.91	−1.26823	−0.634116	+	0.773238i	$0.718636\pi$
$282$	0	0
$283$	3241.87	0.680950	0.340475	−	0.940254i	$-0.389412\pi$
$283$	3241.87	0.680950	0.340475	+	0.940254i	$0.389412\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4436.35	−0.912438
$288$	0	0
$289$	−4658.61	−0.948220
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3410.05	0.679923	0.339961	−	0.940439i	$-0.389586\pi$
$293$	3410.05	0.679923	0.339961	+	0.940439i	$0.389586\pi$
$294$	0	0
$295$	2604.91	0.514114
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4292.02	0.830147
$300$	0	0
$301$	411.373	0.0787746
$302$	0	0
$303$	0	0
$304$	0	0
$305$	751.295	0.141046
$306$	0	0
$307$	−7639.92	−1.42030	−0.710152	−	0.704048i	$-0.751374\pi$
$307$	−7639.92	−1.42030	−0.710152	+	0.704048i	$0.751374\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2883.95	−0.525832	−0.262916	−	0.964819i	$-0.584684\pi$
$311$	−2883.95	−0.525832	−0.262916	+	0.964819i	$0.584684\pi$
$312$	0	0
$313$	4088.79	0.738377	0.369188	−	0.929355i	$-0.379636\pi$
$313$	4088.79	0.738377	0.369188	+	0.929355i	$0.379636\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2481.64	0.439693	0.219846	−	0.975535i	$-0.429444\pi$
$317$	2481.64	0.439693	0.219846	+	0.975535i	$0.429444\pi$
$318$	0	0
$319$	−61.6641	−0.0108230
$320$	0	0
$321$	0	0
$322$	0	0
$323$	939.137	0.161780
$324$	0	0
$325$	−1895.14	−0.323457
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7052.31	−1.18178
$330$	0	0
$331$	6816.08	1.13186	0.565930	−	0.824453i	$-0.308517\pi$
$331$	6816.08	1.13186	0.565930	+	0.824453i	$0.308517\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2530.61	−0.412722
$336$	0	0
$337$	3249.23	0.525213	0.262607	−	0.964903i	$-0.415418\pi$
$337$	3249.23	0.525213	0.262607	+	0.964903i	$0.415418\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1324.52	−0.210342
$342$	0	0
$343$	−3944.60	−0.620957
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2549.51	0.394424	0.197212	−	0.980361i	$-0.436811\pi$
$347$	2549.51	0.394424	0.197212	+	0.980361i	$0.436811\pi$
$348$	0	0
$349$	−8644.39	−1.32586	−0.662928	−	0.748683i	$-0.730686\pi$
$349$	−8644.39	−1.32586	−0.662928	+	0.748683i	$0.730686\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1537.84	−0.231872	−0.115936	−	0.993257i	$-0.536987\pi$
$353$	−1537.84	−0.231872	−0.115936	+	0.993257i	$0.536987\pi$
$354$	0	0
$355$	4807.97	0.718818
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3830.53	−0.563141	−0.281571	−	0.959541i	$-0.590855\pi$
$359$	−3830.53	−0.563141	−0.281571	+	0.959541i	$0.590855\pi$
$360$	0	0
$361$	−3392.01	−0.494534
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1259.38	0.180601
$366$	0	0
$367$	−1273.49	−0.181133	−0.0905663	−	0.995890i	$-0.528868\pi$
$367$	−1273.49	−0.181133	−0.0905663	+	0.995890i	$0.528868\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−755.041	−0.105660
$372$	0	0
$373$	7287.41	1.01160	0.505801	−	0.862650i	$-0.331197\pi$
$373$	7287.41	1.01160	0.505801	+	0.862650i	$0.331197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	425.928	0.0581867
$378$	0	0
$379$	−11963.0	−1.62137	−0.810684	−	0.585484i	$-0.800905\pi$
$379$	−11963.0	−1.62137	−0.810684	+	0.585484i	$0.800905\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5316.91	−0.709350	−0.354675	−	0.934990i	$-0.615409\pi$
$383$	−5316.91	−0.709350	−0.354675	+	0.934990i	$0.615409\pi$
$384$	0	0
$385$	−1241.18	−0.164303
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10902.4	−1.42101	−0.710507	−	0.703690i	$-0.751534\pi$
$389$	−10902.4	−1.42101	−0.710507	+	0.703690i	$0.751534\pi$
$390$	0	0
$391$	−903.050	−0.116801
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4174.17	−0.531710
$396$	0	0
$397$	6551.82	0.828278	0.414139	−	0.910214i	$-0.364083\pi$
$397$	6551.82	0.828278	0.414139	+	0.910214i	$0.364083\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5928.47	−0.738289	−0.369145	−	0.929372i	$-0.620349\pi$
$401$	−5928.47	−0.738289	−0.369145	+	0.929372i	$0.620349\pi$
$402$	0	0
$403$	9148.76	1.13085
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2591.56	0.315624
$408$	0	0
$409$	−4119.31	−0.498012	−0.249006	−	0.968502i	$-0.580104\pi$
$409$	−4119.31	−0.498012	−0.249006	+	0.968502i	$0.580104\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11783.9	−1.40399
$414$	0	0
$415$	3456.61	0.408864
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15867.2	1.85003	0.925017	−	0.379926i	$-0.124050\pi$
$419$	15867.2	1.85003	0.925017	+	0.379926i	$0.124050\pi$
$420$	0	0
$421$	−13093.9	−1.51581	−0.757906	−	0.652364i	$-0.773777\pi$
$421$	−13093.9	−1.51581	−0.757906	+	0.652364i	$0.773777\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	398.742	0.0455102
$426$	0	0
$427$	−3398.66	−0.385182
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12996.6	1.45249	0.726247	−	0.687434i	$-0.241263\pi$
$431$	12996.6	1.45249	0.726247	+	0.687434i	$0.241263\pi$
$432$	0	0
$433$	14740.8	1.63602	0.818009	−	0.575205i	$-0.195078\pi$
$433$	14740.8	1.63602	0.818009	+	0.575205i	$0.195078\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3333.77	−0.364934
$438$	0	0
$439$	5985.20	0.650702	0.325351	−	0.945593i	$-0.394518\pi$
$439$	5985.20	0.650702	0.325351	+	0.945593i	$0.394518\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1663.75	0.178436	0.0892178	−	0.996012i	$-0.471563\pi$
$443$	1663.75	0.178436	0.0892178	+	0.996012i	$0.471563\pi$
$444$	0	0
$445$	−5030.50	−0.535884
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3707.44	0.389677	0.194838	−	0.980835i	$-0.437582\pi$
$449$	3707.44	0.389677	0.194838	+	0.980835i	$0.437582\pi$
$450$	0	0
$451$	−2152.57	−0.224746
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8573.12	0.883328
$456$	0	0
$457$	2757.23	0.282227	0.141113	−	0.989993i	$-0.454932\pi$
$457$	2757.23	0.282227	0.141113	+	0.989993i	$0.454932\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14975.3	1.51295	0.756475	−	0.654022i	$-0.226920\pi$
$461$	14975.3	1.51295	0.756475	+	0.654022i	$0.226920\pi$
$462$	0	0
$463$	−4783.63	−0.480161	−0.240080	−	0.970753i	$-0.577174\pi$
$463$	−4783.63	−0.480161	−0.240080	+	0.970753i	$0.577174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5341.87	0.529320	0.264660	−	0.964342i	$-0.414740\pi$
$467$	5341.87	0.529320	0.264660	+	0.964342i	$0.414740\pi$
$468$	0	0
$469$	11447.8	1.12710
$470$	0	0
$471$	0	0
$472$	0	0
$473$	199.603	0.0194033
$474$	0	0
$475$	1472.03	0.142192
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6659.28	−0.635220	−0.317610	−	0.948221i	$-0.602880\pi$
$479$	−6659.28	−0.635220	−0.317610	+	0.948221i	$0.602880\pi$
$480$	0	0
$481$	−17900.5	−1.69687
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6428.23	−0.601837
$486$	0	0
$487$	−1376.23	−0.128056	−0.0640278	−	0.997948i	$-0.520395\pi$
$487$	−1376.23	−0.128056	−0.0640278	+	0.997948i	$0.520395\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7898.69	−0.725993	−0.362997	−	0.931790i	$-0.618246\pi$
$491$	−7898.69	−0.725993	−0.362997	+	0.931790i	$0.618246\pi$
$492$	0	0
$493$	−89.6162	−0.00818683
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21750.0	−1.96302
$498$	0	0
$499$	−5196.44	−0.466182	−0.233091	−	0.972455i	$-0.574884\pi$
$499$	−5196.44	−0.466182	−0.233091	+	0.972455i	$0.574884\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1425.40	0.126353	0.0631763	−	0.998002i	$-0.479877\pi$
$503$	1425.40	0.126353	0.0631763	+	0.998002i	$0.479877\pi$
$504$	0	0
$505$	−4170.68	−0.367510
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8603.62	0.749212	0.374606	−	0.927184i	$-0.377778\pi$
$509$	8603.62	0.749212	0.374606	+	0.927184i	$0.377778\pi$
$510$	0	0
$511$	−5697.12	−0.493201
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2664.75	0.228006
$516$	0	0
$517$	−3421.86	−0.291090
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12519.4	−1.05276	−0.526378	−	0.850251i	$-0.676450\pi$
$521$	−12519.4	−1.05276	−0.526378	+	0.850251i	$0.676450\pi$
$522$	0	0
$523$	7569.86	0.632900	0.316450	−	0.948609i	$-0.397509\pi$
$523$	7569.86	0.632900	0.316450	+	0.948609i	$0.397509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1924.92	−0.159110
$528$	0	0
$529$	−8961.33	−0.736527
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14868.3	1.20829
$534$	0	0
$535$	−1819.53	−0.147038
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1850.41	0.147871
$540$	0	0
$541$	5700.08	0.452987	0.226493	−	0.974013i	$-0.427274\pi$
$541$	5700.08	0.452987	0.226493	+	0.974013i	$0.427274\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1609.03	0.126465
$546$	0	0
$547$	−2079.92	−0.162579	−0.0812897	−	0.996691i	$-0.525904\pi$
$547$	−2079.92	−0.162579	−0.0812897	+	0.996691i	$0.525904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−330.834	−0.0255790
$552$	0	0
$553$	18882.8	1.45204
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19507.2	1.48392	0.741961	−	0.670443i	$-0.233896\pi$
$557$	19507.2	1.48392	0.741961	+	0.670443i	$0.233896\pi$
$558$	0	0
$559$	−1378.70	−0.104317
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9826.58	0.735597	0.367798	−	0.929906i	$-0.380112\pi$
$563$	9826.58	0.735597	0.367798	+	0.929906i	$0.380112\pi$
$564$	0	0
$565$	−8785.21	−0.654153
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13064.7	−0.962569	−0.481284	−	0.876565i	$-0.659830\pi$
$569$	−13064.7	−0.962569	−0.481284	+	0.876565i	$0.659830\pi$
$570$	0	0
$571$	10580.6	0.775451	0.387726	−	0.921775i	$-0.373261\pi$
$571$	10580.6	0.775451	0.387726	+	0.921775i	$0.373261\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1415.47	−0.102659
$576$	0	0
$577$	9040.99	0.652307	0.326154	−	0.945317i	$-0.394247\pi$
$577$	9040.99	0.652307	0.326154	+	0.945317i	$0.394247\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15636.8	−1.11656
$582$	0	0
$583$	−366.355	−0.0260255
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6353.52	0.446742	0.223371	−	0.974733i	$-0.428294\pi$
$587$	6353.52	0.446742	0.223371	+	0.974733i	$0.428294\pi$
$588$	0	0
$589$	−7106.19	−0.497123
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14375.7	−0.995512	−0.497756	−	0.867317i	$-0.665842\pi$
$593$	−14375.7	−0.995512	−0.497756	+	0.867317i	$0.665842\pi$
$594$	0	0
$595$	−1803.80	−0.124284
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11848.3	0.808196	0.404098	−	0.914716i	$-0.367585\pi$
$599$	11848.3	0.808196	0.404098	+	0.914716i	$0.367585\pi$
$600$	0	0
$601$	−26705.5	−1.81255	−0.906273	−	0.422693i	$-0.861085\pi$
$601$	−26705.5	−1.81255	−0.906273	+	0.422693i	$0.861085\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6052.76	0.406744
$606$	0	0
$607$	16869.0	1.12799	0.563996	−	0.825777i	$-0.309263\pi$
$607$	16869.0	1.12799	0.563996	+	0.825777i	$0.309263\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23635.6	1.56496
$612$	0	0
$613$	26929.3	1.77433	0.887164	−	0.461454i	$-0.152672\pi$
$613$	26929.3	1.77433	0.887164	+	0.461454i	$0.152672\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14483.6	−0.945036	−0.472518	−	0.881321i	$-0.656655\pi$
$617$	−14483.6	−0.945036	−0.472518	+	0.881321i	$0.656655\pi$
$618$	0	0
$619$	−25375.0	−1.64767	−0.823836	−	0.566829i	$-0.808170\pi$
$619$	−25375.0	−1.64767	−0.823836	+	0.566829i	$0.808170\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	22756.6	1.46344
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3766.30	0.238748
$630$	0	0
$631$	−13091.2	−0.825915	−0.412958	−	0.910750i	$-0.635504\pi$
$631$	−13091.2	−0.825915	−0.412958	+	0.910750i	$0.635504\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6847.98	0.427959
$636$	0	0
$637$	−12781.2	−0.794991
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22880.0	1.40984	0.704920	−	0.709287i	$-0.250983\pi$
$641$	22880.0	1.40984	0.704920	+	0.709287i	$0.250983\pi$
$642$	0	0
$643$	27338.0	1.67668	0.838340	−	0.545148i	$-0.183526\pi$
$643$	27338.0	1.67668	0.838340	+	0.545148i	$0.183526\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27146.3	1.64951	0.824753	−	0.565493i	$-0.191314\pi$
$647$	27146.3	1.64951	0.824753	+	0.565493i	$0.191314\pi$
$648$	0	0
$649$	−5717.69	−0.345823
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24425.1	1.46375	0.731873	−	0.681441i	$-0.238646\pi$
$653$	24425.1	1.46375	0.731873	+	0.681441i	$0.238646\pi$
$654$	0	0
$655$	−639.541	−0.0381510
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10013.6	0.591922	0.295961	−	0.955200i	$-0.404360\pi$
$659$	10013.6	0.591922	0.295961	+	0.955200i	$0.404360\pi$
$660$	0	0
$661$	−7226.72	−0.425245	−0.212622	−	0.977134i	$-0.568200\pi$
$661$	−7226.72	−0.425245	−0.212622	+	0.977134i	$0.568200\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6659.07	−0.388312
$666$	0	0
$667$	318.122	0.0184674
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1649.07	−0.0948757
$672$	0	0
$673$	28080.4	1.60835	0.804176	−	0.594391i	$-0.202607\pi$
$673$	28080.4	1.60835	0.804176	+	0.594391i	$0.202607\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	435.394	0.0247172	0.0123586	−	0.999924i	$-0.496066\pi$
$677$	435.394	0.0247172	0.0123586	+	0.999924i	$0.496066\pi$
$678$	0	0
$679$	29079.6	1.64355
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13348.0	−0.747799	−0.373900	−	0.927469i	$-0.621980\pi$
$683$	−13348.0	−0.747799	−0.373900	+	0.927469i	$0.621980\pi$
$684$	0	0
$685$	12976.8	0.723820
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2530.50	0.139919
$690$	0	0
$691$	−4087.78	−0.225046	−0.112523	−	0.993649i	$-0.535893\pi$
$691$	−4087.78	−0.225046	−0.112523	+	0.993649i	$0.535893\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7841.55	−0.427981
$696$	0	0
$697$	−3128.32	−0.170005
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16568.9	0.892721	0.446360	−	0.894853i	$-0.352720\pi$
$701$	16568.9	0.892721	0.446360	+	0.894853i	$0.352720\pi$
$702$	0	0
$703$	13904.0	0.745945
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18867.0	1.00363
$708$	0	0
$709$	1974.55	0.104592	0.0522962	−	0.998632i	$-0.483346\pi$
$709$	1974.55	0.104592	0.0522962	+	0.998632i	$0.483346\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6833.13	0.358910
$714$	0	0
$715$	4159.78	0.217576
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16084.1	0.834265	0.417132	−	0.908846i	$-0.363035\pi$
$719$	16084.1	0.834265	0.417132	+	0.908846i	$0.363035\pi$
$720$	0	0
$721$	−12054.6	−0.622660
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−140.467	−0.00719560
$726$	0	0
$727$	−35993.9	−1.83623	−0.918116	−	0.396311i	$-0.870290\pi$
$727$	−35993.9	−1.83623	−0.918116	+	0.396311i	$0.870290\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	290.082	0.0146773
$732$	0	0
$733$	−18210.7	−0.917635	−0.458818	−	0.888530i	$-0.651727\pi$
$733$	−18210.7	−0.917635	−0.458818	+	0.888530i	$0.651727\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5554.61	0.277621
$738$	0	0
$739$	14620.2	0.727756	0.363878	−	0.931447i	$-0.381452\pi$
$739$	14620.2	0.727756	0.363878	+	0.931447i	$0.381452\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31638.5	1.56219	0.781093	−	0.624414i	$-0.214662\pi$
$743$	31638.5	1.56219	0.781093	+	0.624414i	$0.214662\pi$
$744$	0	0
$745$	15601.1	0.767220
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8231.08	0.401545
$750$	0	0
$751$	2865.85	0.139249	0.0696247	−	0.997573i	$-0.477820\pi$
$751$	2865.85	0.139249	0.0696247	+	0.997573i	$0.477820\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4097.82	0.197530
$756$	0	0
$757$	10442.1	0.501351	0.250676	−	0.968071i	$-0.419347\pi$
$757$	10442.1	0.501351	0.250676	+	0.968071i	$0.419347\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4251.49	−0.202518	−0.101259	−	0.994860i	$-0.532287\pi$
$761$	−4251.49	−0.202518	−0.101259	+	0.994860i	$0.532287\pi$
$762$	0	0
$763$	−7278.84	−0.345363
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39493.4	1.85922
$768$	0	0
$769$	−26445.9	−1.24013	−0.620066	−	0.784549i	$-0.712894\pi$
$769$	−26445.9	−1.24013	−0.620066	+	0.784549i	$0.712894\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.12943	0.000285201 0	0.000142601	−	1.00000i	$-0.499955\pi$
$773$	6.12943	0.000285201 0	0.000142601		1.00000i	$0.499955\pi$
$774$	0	0
$775$	−3017.17	−0.139845
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11548.8	−0.531165
$780$	0	0
$781$	−10553.3	−0.483519
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4422.80	0.201091
$786$	0	0
$787$	−16384.6	−0.742122	−0.371061	−	0.928609i	$-0.621006\pi$
$787$	−16384.6	−0.742122	−0.371061	+	0.928609i	$0.621006\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	39741.9	1.78642
$792$	0	0
$793$	11390.5	0.510073
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1511.64	0.0671832	0.0335916	−	0.999436i	$-0.489305\pi$
$797$	1511.64	0.0671832	0.0335916	+	0.999436i	$0.489305\pi$
$798$	0	0
$799$	−4972.97	−0.220189
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2764.31	−0.121482
$804$	0	0
$805$	6403.20	0.280352
$806$	0	0
$807$	0	0
$808$	0	0
$809$	961.644	0.0417918	0.0208959	−	0.999782i	$-0.493348\pi$
$809$	961.644	0.0417918	0.0208959	+	0.999782i	$0.493348\pi$
$810$	0	0
$811$	33187.3	1.43695	0.718474	−	0.695554i	$-0.244841\pi$
$811$	33187.3	1.43695	0.718474	+	0.695554i	$0.244841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15965.6	0.686199
$816$	0	0
$817$	1070.89	0.0458577
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29885.6	−1.27042	−0.635210	−	0.772340i	$-0.719086\pi$
$821$	−29885.6	−1.27042	−0.635210	+	0.772340i	$0.719086\pi$
$822$	0	0
$823$	37301.2	1.57988	0.789938	−	0.613186i	$-0.210113\pi$
$823$	37301.2	1.57988	0.789938	+	0.613186i	$0.210113\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21334.5	−0.897067	−0.448533	−	0.893766i	$-0.648053\pi$
$827$	−21334.5	−0.897067	−0.448533	+	0.893766i	$0.648053\pi$
$828$	0	0
$829$	43890.0	1.83880	0.919399	−	0.393326i	$-0.128676\pi$
$829$	43890.0	1.83880	0.919399	+	0.393326i	$0.128676\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2689.19	0.111855
$834$	0	0
$835$	8537.44	0.353833
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38729.5	1.59367	0.796835	−	0.604197i	$-0.206506\pi$
$839$	38729.5	1.59367	0.796835	+	0.604197i	$0.206506\pi$
$840$	0	0
$841$	−24357.4	−0.998706
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17747.5	−0.722525
$846$	0	0
$847$	−27381.1	−1.11077
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13369.7	−0.538553
$852$	0	0
$853$	−30658.7	−1.23064	−0.615318	−	0.788279i	$-0.710972\pi$
$853$	−30658.7	−1.23064	−0.615318	+	0.788279i	$0.710972\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3417.80	−0.136231	−0.0681153	−	0.997677i	$-0.521699\pi$
$857$	−3417.80	−0.136231	−0.0681153	+	0.997677i	$0.521699\pi$
$858$	0	0
$859$	−3226.67	−0.128164	−0.0640819	−	0.997945i	$-0.520412\pi$
$859$	−3226.67	−0.128164	−0.0640819	+	0.997945i	$0.520412\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30102.5	−1.18737	−0.593684	−	0.804698i	$-0.702327\pi$
$863$	−30102.5	−1.18737	−0.593684	+	0.804698i	$0.702327\pi$
$864$	0	0
$865$	18510.5	0.727603
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9162.18	0.357659
$870$	0	0
$871$	−38366.9	−1.49255
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2827.33	−0.109236
$876$	0	0
$877$	−28742.6	−1.10669	−0.553345	−	0.832952i	$-0.686649\pi$
$877$	−28742.6	−1.10669	−0.553345	+	0.832952i	$0.686649\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34320.7	1.31248	0.656239	−	0.754553i	$-0.272146\pi$
$881$	34320.7	1.31248	0.656239	+	0.754553i	$0.272146\pi$
$882$	0	0
$883$	11327.9	0.431728	0.215864	−	0.976423i	$-0.430743\pi$
$883$	11327.9	0.431728	0.215864	+	0.976423i	$0.430743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40694.5	−1.54046	−0.770229	−	0.637767i	$-0.779858\pi$
$887$	−40694.5	−1.54046	−0.770229	+	0.637767i	$0.779858\pi$
$888$	0	0
$889$	−30978.5	−1.16871
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−18358.6	−0.687960
$894$	0	0
$895$	21945.8	0.819627
$896$	0	0
$897$	0	0
$898$	0	0
$899$	678.101	0.0251568
$900$	0	0
$901$	−532.422	−0.0196865
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4119.61	0.151315
$906$	0	0
$907$	−33122.1	−1.21257	−0.606285	−	0.795247i	$-0.707341\pi$
$907$	−33122.1	−1.21257	−0.606285	+	0.795247i	$0.707341\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−52805.6	−1.92045	−0.960224	−	0.279230i	$-0.909921\pi$
$911$	−52805.6	−1.92045	−0.960224	+	0.279230i	$0.909921\pi$
$912$	0	0
$913$	−7587.16	−0.275026
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2893.11	0.104186
$918$	0	0
$919$	−43815.6	−1.57274	−0.786368	−	0.617758i	$-0.788041\pi$
$919$	−43815.6	−1.57274	−0.786368	+	0.617758i	$0.788041\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	72894.3	2.59951
$924$	0	0
$925$	5903.41	0.209841
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4331.65	−0.152978	−0.0764891	−	0.997070i	$-0.524371\pi$
$929$	−4331.65	−0.152978	−0.0764891	+	0.997070i	$0.524371\pi$
$930$	0	0
$931$	9927.63	0.349479
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−875.227	−0.0306128
$936$	0	0
$937$	39344.1	1.37173	0.685867	−	0.727727i	$-0.259423\pi$
$937$	39344.1	1.37173	0.685867	+	0.727727i	$0.259423\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29224.0	−1.01241	−0.506203	−	0.862414i	$-0.668951\pi$
$941$	−29224.0	−1.01241	−0.506203	+	0.862414i	$0.668951\pi$
$942$	0	0
$943$	11105.0	0.383487
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3091.33	−0.106077	−0.0530383	−	0.998592i	$-0.516891\pi$
$947$	−3091.33	−0.106077	−0.0530383	+	0.998592i	$0.516891\pi$
$948$	0	0
$949$	19093.7	0.653117
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−53176.7	−1.80751	−0.903757	−	0.428046i	$-0.859202\pi$
$953$	−53176.7	−1.80751	−0.903757	+	0.428046i	$0.859202\pi$
$954$	0	0
$955$	24571.5	0.832580
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−58703.4	−1.97668
$960$	0	0
$961$	−15225.7	−0.511083
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12509.4	0.417296
$966$	0	0
$967$	14200.2	0.472232	0.236116	−	0.971725i	$-0.424125\pi$
$967$	14200.2	0.472232	0.236116	+	0.971725i	$0.424125\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	56247.8	1.85899	0.929494	−	0.368838i	$-0.120244\pi$
$971$	56247.8	1.85899	0.929494	+	0.368838i	$0.120244\pi$
$972$	0	0
$973$	35473.1	1.16877
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29372.3	0.961824	0.480912	−	0.876769i	$-0.340306\pi$
$977$	29372.3	0.961824	0.480912	+	0.876769i	$0.340306\pi$
$978$	0	0
$979$	11041.8	0.360467
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3257.53	0.105696	0.0528479	−	0.998603i	$-0.483170\pi$
$983$	3257.53	0.105696	0.0528479	+	0.998603i	$0.483170\pi$
$984$	0	0
$985$	7079.98	0.229022
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1029.74	−0.0331081
$990$	0	0
$991$	−50330.0	−1.61330	−0.806652	−	0.591027i	$-0.798723\pi$
$991$	−50330.0	−1.61330	−0.806652	+	0.591027i	$0.798723\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7660.27	−0.244067
$996$	0	0
$997$	−15461.9	−0.491157	−0.245578	−	0.969377i	$-0.578978\pi$
$997$	−15461.9	−0.491157	−0.245578	+	0.969377i	$0.578978\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.c.1.3	✓	3
3.2	odd	2		1620.4.a.e.1.3	yes	3
9.2	odd	6		1620.4.i.t.1081.1		6
9.4	even	3		1620.4.i.v.541.1		6
9.5	odd	6		1620.4.i.t.541.1		6
9.7	even	3		1620.4.i.v.1081.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.c.1.3	✓	3	1.1	even	1	trivial
1620.4.a.e.1.3	yes	3	3.2	odd	2
1620.4.i.t.541.1		6	9.5	odd	6
1620.4.i.t.1081.1		6	9.2	odd	6
1620.4.i.v.541.1		6	9.4	even	3
1620.4.i.v.1081.1		6	9.7	even	3

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +22.6187 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +22.6187 q^{7} +10.9748 q^{11} -75.8057 q^{13} +15.9497 q^{17} +58.8812 q^{19} -56.6187 q^{23} +25.0000 q^{25} -5.61868 q^{29} -120.687 q^{31} -113.093 q^{35} +236.136 q^{37} -196.137 q^{41} +18.1873 q^{43} -311.791 q^{47} +168.604 q^{49} -33.3813 q^{53} -54.8742 q^{55} -520.981 q^{59} -150.259 q^{61} +379.029 q^{65} +506.122 q^{67} -961.593 q^{71} -251.877 q^{73} +248.236 q^{77} +834.834 q^{79} -691.323 q^{83} -79.7485 q^{85} +1006.10 q^{89} -1714.62 q^{91} -294.406 q^{95} +1285.65 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 - 3 * q^7	\( 3 q - 15 q^{5} - 3 q^{7} + 24 q^{11} + 3 q^{13} + 30 q^{17} - 27 q^{19} - 99 q^{23} + 75 q^{25} + 54 q^{29} + 72 q^{31} + 15 q^{35} - 18 q^{37} - 411 q^{41} + 444 q^{43} + 75 q^{47} + 204 q^{49} - 171 q^{53} - 120 q^{55} - 297 q^{59} + 684 q^{61} - 15 q^{65} + 12 q^{67} - 642 q^{71} - 66 q^{73} - 1044 q^{77} + 1122 q^{79} - 90 q^{83} - 150 q^{85} - 756 q^{89} - 1653 q^{91} + 135 q^{95} + 1536 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 - 3 * q^7 + 24 * q^11 + 3 * q^13 + 30 * q^17 - 27 * q^19 - 99 * q^23 + 75 * q^25 + 54 * q^29 + 72 * q^31 + 15 * q^35 - 18 * q^37 - 411 * q^41 + 444 * q^43 + 75 * q^47 + 204 * q^49 - 171 * q^53 - 120 * q^55 - 297 * q^59 + 684 * q^61 - 15 * q^65 + 12 * q^67 - 642 * q^71 - 66 * q^73 - 1044 * q^77 + 1122 * q^79 - 90 * q^83 - 150 * q^85 - 756 * q^89 - 1653 * q^91 + 135 * q^95 + 1536 * q^97

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