LMFDB - Embedded newform 1620.4.a.j.1.4

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 151x^{4} - 212x^{3} + 4412x^{2} + 8656x - 19148 $ x^6 - 2x^5 - 151x^4 - 212x^3 + 4412x^2 + 8656*x - 19148
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.35846$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +6.21414 q^{7} -28.3492 q^{11} -8.60548 q^{13} -90.1352 q^{17} +114.170 q^{19} -48.4106 q^{23} +25.0000 q^{25} +305.958 q^{29} +93.5957 q^{31} +31.0707 q^{35} -282.892 q^{37} -28.7685 q^{41} -354.443 q^{43} -522.998 q^{47} -304.384 q^{49} -66.9285 q^{53} -141.746 q^{55} -7.63223 q^{59} +9.41124 q^{61} -43.0274 q^{65} -494.391 q^{67} -560.709 q^{71} +1116.68 q^{73} -176.166 q^{77} +1041.87 q^{79} +45.5107 q^{83} -450.676 q^{85} -357.159 q^{89} -53.4757 q^{91} +570.849 q^{95} -120.006 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53}+ \cdots - 1188 q^{97}+O(q^{100}) $ 6 * q + 30 * q^5 + 12 * q^7 - 84 * q^13 - 12 * q^17 - 114 * q^19 - 30 * q^23 + 150 * q^25 - 168 * q^29 - 324 * q^31 + 60 * q^35 - 492 * q^37 - 312 * q^41 - 156 * q^43 - 462 * q^47 - 588 * q^49 - 1014 * q^53 - 1008 * q^59 + 36 * q^61 - 420 * q^65 + 144 * q^67 - 1212 * q^71 - 900 * q^73 - 672 * q^77 - 936 * q^79 - 288 * q^83 - 60 * q^85 + 120 * q^89 + 2286 * q^91 - 570 * q^95 - 1188 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	6.21414	0.335532	0.167766	−	0.985827i	$-0.446345\pi$
$7$	6.21414	0.335532	0.167766	+	0.985827i	$0.446345\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−28.3492	−0.777054	−0.388527	−	0.921437i	$-0.627016\pi$
$11$	−28.3492	−0.777054	−0.388527	+	0.921437i	$0.627016\pi$
$12$	0	0
$13$	−8.60548	−0.183595	−0.0917973	−	0.995778i	$-0.529261\pi$
$13$	−8.60548	−0.183595	−0.0917973	+	0.995778i	$0.529261\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−90.1352	−1.28594	−0.642970	−	0.765891i	$-0.722298\pi$
$17$	−90.1352	−1.28594	−0.642970	+	0.765891i	$0.722298\pi$
$18$	0	0
$19$	114.170	1.37854	0.689272	−	0.724503i	$-0.257931\pi$
$19$	114.170	1.37854	0.689272	+	0.724503i	$0.257931\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−48.4106	−0.438883	−0.219441	−	0.975626i	$-0.570423\pi$
$23$	−48.4106	−0.438883	−0.219441	+	0.975626i	$0.570423\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	305.958	1.95914	0.979568	−	0.201113i	$-0.0644557\pi$
$29$	305.958	1.95914	0.979568	+	0.201113i	$0.0644557\pi$
$30$	0	0
$31$	93.5957	0.542267	0.271134	−	0.962542i	$-0.412601\pi$
$31$	93.5957	0.542267	0.271134	+	0.962542i	$0.412601\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	31.0707	0.150055
$36$	0	0
$37$	−282.892	−1.25695	−0.628475	−	0.777830i	$-0.716320\pi$
$37$	−282.892	−1.25695	−0.628475	+	0.777830i	$0.716320\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−28.7685	−0.109583	−0.0547913	−	0.998498i	$-0.517449\pi$
$41$	−28.7685	−0.109583	−0.0547913	+	0.998498i	$0.517449\pi$
$42$	0	0
$43$	−354.443	−1.25703	−0.628513	−	0.777799i	$-0.716336\pi$
$43$	−354.443	−1.25703	−0.628513	+	0.777799i	$0.716336\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−522.998	−1.62313	−0.811565	−	0.584262i	$-0.801384\pi$
$47$	−522.998	−1.62313	−0.811565	+	0.584262i	$0.801384\pi$
$48$	0	0
$49$	−304.384	−0.887418
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−66.9285	−0.173459	−0.0867296	−	0.996232i	$-0.527642\pi$
$53$	−66.9285	−0.173459	−0.0867296	+	0.996232i	$0.527642\pi$
$54$	0	0
$55$	−141.746	−0.347509
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.63223	−0.0168412	−0.00842061	−	0.999965i	$-0.502680\pi$
$59$	−7.63223	−0.0168412	−0.00842061	+	0.999965i	$0.502680\pi$
$60$	0	0
$61$	9.41124	0.0197539	0.00987693	−	0.999951i	$-0.496856\pi$
$61$	9.41124	0.0197539	0.00987693	+	0.999951i	$0.496856\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−43.0274	−0.0821060
$66$	0	0
$67$	−494.391	−0.901485	−0.450743	−	0.892654i	$-0.648841\pi$
$67$	−494.391	−0.901485	−0.450743	+	0.892654i	$0.648841\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−560.709	−0.937239	−0.468619	−	0.883400i	$-0.655248\pi$
$71$	−560.709	−0.937239	−0.468619	+	0.883400i	$0.655248\pi$
$72$	0	0
$73$	1116.68	1.79038	0.895192	−	0.445680i	$-0.147038\pi$
$73$	1116.68	1.79038	0.895192	+	0.445680i	$0.147038\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−176.166	−0.260726
$78$	0	0
$79$	1041.87	1.48379	0.741896	−	0.670515i	$-0.233927\pi$
$79$	1041.87	1.48379	0.741896	+	0.670515i	$0.233927\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	45.5107	0.0601862	0.0300931	−	0.999547i	$-0.490420\pi$
$83$	45.5107	0.0601862	0.0300931	+	0.999547i	$0.490420\pi$
$84$	0	0
$85$	−450.676	−0.575090
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−357.159	−0.425379	−0.212690	−	0.977120i	$-0.568222\pi$
$89$	−357.159	−0.425379	−0.212690	+	0.977120i	$0.568222\pi$
$90$	0	0
$91$	−53.4757	−0.0616019
$92$	0	0
$93$	0	0
$94$	0	0
$95$	570.849	0.616503
$96$	0	0
$97$	−120.006	−0.125616	−0.0628079	−	0.998026i	$-0.520006\pi$
$97$	−120.006	−0.125616	−0.0628079	+	0.998026i	$0.520006\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−498.559	−0.491173	−0.245586	−	0.969375i	$-0.578980\pi$
$101$	−498.559	−0.491173	−0.245586	+	0.969375i	$0.578980\pi$
$102$	0	0
$103$	−254.686	−0.243641	−0.121820	−	0.992552i	$-0.538873\pi$
$103$	−254.686	−0.243641	−0.121820	+	0.992552i	$0.538873\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	311.077	0.281056	0.140528	−	0.990077i	$-0.455120\pi$
$107$	311.077	0.281056	0.140528	+	0.990077i	$0.455120\pi$
$108$	0	0
$109$	−661.173	−0.580999	−0.290500	−	0.956875i	$-0.593822\pi$
$109$	−661.173	−0.580999	−0.290500	+	0.956875i	$0.593822\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−565.537	−0.470807	−0.235404	−	0.971898i	$-0.575641\pi$
$113$	−565.537	−0.470807	−0.235404	+	0.971898i	$0.575641\pi$
$114$	0	0
$115$	−242.053	−0.196274
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−560.113	−0.431474
$120$	0	0
$121$	−527.326	−0.396188
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1809.60	1.26438	0.632190	−	0.774814i	$-0.282156\pi$
$127$	1809.60	1.26438	0.632190	+	0.774814i	$0.282156\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2029.83	−1.35379	−0.676897	−	0.736078i	$-0.736676\pi$
$131$	−2029.83	−1.35379	−0.676897	+	0.736078i	$0.736676\pi$
$132$	0	0
$133$	709.467	0.462546
$134$	0	0
$135$	0	0
$136$	0	0
$137$	545.350	0.340091	0.170045	−	0.985436i	$-0.445609\pi$
$137$	545.350	0.340091	0.170045	+	0.985436i	$0.445609\pi$
$138$	0	0
$139$	−1762.64	−1.07558	−0.537788	−	0.843080i	$-0.680740\pi$
$139$	−1762.64	−1.07558	−0.537788	+	0.843080i	$0.680740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	243.958	0.142663
$144$	0	0
$145$	1529.79	0.876152
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1223.53	−0.672722	−0.336361	−	0.941733i	$-0.609196\pi$
$149$	−1223.53	−0.672722	−0.336361	+	0.941733i	$0.609196\pi$
$150$	0	0
$151$	2849.03	1.53543	0.767716	−	0.640790i	$-0.221393\pi$
$151$	2849.03	1.53543	0.767716	+	0.640790i	$0.221393\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	467.978	0.242509
$156$	0	0
$157$	−942.831	−0.479275	−0.239637	−	0.970862i	$-0.577029\pi$
$157$	−942.831	−0.479275	−0.239637	+	0.970862i	$0.577029\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−300.830	−0.147259
$162$	0	0
$163$	417.774	0.200752	0.100376	−	0.994950i	$-0.467995\pi$
$163$	417.774	0.200752	0.100376	+	0.994950i	$0.467995\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1392.76	−0.645360	−0.322680	−	0.946508i	$-0.604584\pi$
$167$	−1392.76	−0.645360	−0.322680	+	0.946508i	$0.604584\pi$
$168$	0	0
$169$	−2122.95	−0.966293
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3671.35	−1.61345	−0.806727	−	0.590924i	$-0.798763\pi$
$173$	−3671.35	−1.61345	−0.806727	+	0.590924i	$0.798763\pi$
$174$	0	0
$175$	155.354	0.0671064
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2387.27	−0.996834	−0.498417	−	0.866938i	$-0.666085\pi$
$179$	−2387.27	−0.996834	−0.498417	+	0.866938i	$0.666085\pi$
$180$	0	0
$181$	2078.63	0.853608	0.426804	−	0.904344i	$-0.359639\pi$
$181$	2078.63	0.853608	0.426804	+	0.904344i	$0.359639\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1414.46	−0.562125
$186$	0	0
$187$	2555.26	0.999245
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2991.03	−1.13311	−0.566553	−	0.824026i	$-0.691723\pi$
$191$	−2991.03	−1.13311	−0.566553	+	0.824026i	$0.691723\pi$
$192$	0	0
$193$	−5055.79	−1.88561	−0.942807	−	0.333338i	$-0.891825\pi$
$193$	−5055.79	−1.88561	−0.942807	+	0.333338i	$0.891825\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3665.56	−1.32569	−0.662843	−	0.748758i	$-0.730651\pi$
$197$	−3665.56	−1.32569	−0.662843	+	0.748758i	$0.730651\pi$
$198$	0	0
$199$	3217.92	1.14629	0.573147	−	0.819452i	$-0.305722\pi$
$199$	3217.92	1.14629	0.573147	+	0.819452i	$0.305722\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1901.27	0.657353
$204$	0	0
$205$	−143.843	−0.0490068
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3236.61	−1.07120
$210$	0	0
$211$	−3909.15	−1.27544	−0.637718	−	0.770270i	$-0.720121\pi$
$211$	−3909.15	−1.27544	−0.637718	+	0.770270i	$0.720121\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1772.22	−0.562159
$216$	0	0
$217$	581.617	0.181948
$218$	0	0
$219$	0	0
$220$	0	0
$221$	775.657	0.236092
$222$	0	0
$223$	808.723	0.242852	0.121426	−	0.992600i	$-0.461253\pi$
$223$	808.723	0.242852	0.121426	+	0.992600i	$0.461253\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.7516	0.00548275	0.00274138	−	0.999996i	$-0.499127\pi$
$227$	18.7516	0.00548275	0.00274138	+	0.999996i	$0.499127\pi$
$228$	0	0
$229$	−1022.99	−0.295202	−0.147601	−	0.989047i	$-0.547155\pi$
$229$	−1022.99	−0.295202	−0.147601	+	0.989047i	$0.547155\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2881.43	0.810166	0.405083	−	0.914280i	$-0.367243\pi$
$233$	2881.43	0.810166	0.405083	+	0.914280i	$0.367243\pi$
$234$	0	0
$235$	−2614.99	−0.725886
$236$	0	0
$237$	0	0
$238$	0	0
$239$	114.828	0.0310779	0.0155390	−	0.999879i	$-0.495054\pi$
$239$	114.828	0.0310779	0.0155390	+	0.999879i	$0.495054\pi$
$240$	0	0
$241$	−1968.74	−0.526214	−0.263107	−	0.964767i	$-0.584747\pi$
$241$	−1968.74	−0.526214	−0.263107	+	0.964767i	$0.584747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1521.92	−0.396865
$246$	0	0
$247$	−982.485	−0.253093
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1706.47	−0.429129	−0.214564	−	0.976710i	$-0.568833\pi$
$251$	−1706.47	−0.429129	−0.214564	+	0.976710i	$0.568833\pi$
$252$	0	0
$253$	1372.40	0.341035
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6191.41	−1.50276	−0.751380	−	0.659869i	$-0.770612\pi$
$257$	−6191.41	−1.50276	−0.751380	+	0.659869i	$0.770612\pi$
$258$	0	0
$259$	−1757.93	−0.421747
$260$	0	0
$261$	0	0
$262$	0	0
$263$	138.736	0.0325278	0.0162639	−	0.999868i	$-0.494823\pi$
$263$	138.736	0.0325278	0.0162639	+	0.999868i	$0.494823\pi$
$264$	0	0
$265$	−334.643	−0.0775733
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3362.16	−0.762061	−0.381030	−	0.924562i	$-0.624431\pi$
$269$	−3362.16	−0.762061	−0.381030	+	0.924562i	$0.624431\pi$
$270$	0	0
$271$	3502.68	0.785140	0.392570	−	0.919722i	$-0.371586\pi$
$271$	3502.68	0.785140	0.392570	+	0.919722i	$0.371586\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−708.729	−0.155411
$276$	0	0
$277$	3853.36	0.835833	0.417917	−	0.908485i	$-0.362760\pi$
$277$	3853.36	0.835833	0.417917	+	0.908485i	$0.362760\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8259.91	1.75354	0.876770	−	0.480910i	$-0.159694\pi$
$281$	8259.91	1.75354	0.876770	+	0.480910i	$0.159694\pi$
$282$	0	0
$283$	−5171.06	−1.08618	−0.543088	−	0.839676i	$-0.682745\pi$
$283$	−5171.06	−1.08618	−0.543088	+	0.839676i	$0.682745\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−178.772	−0.0367685
$288$	0	0
$289$	3211.35	0.653644
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4190.68	−0.835571	−0.417786	−	0.908546i	$-0.637194\pi$
$293$	−4190.68	−0.835571	−0.417786	+	0.908546i	$0.637194\pi$
$294$	0	0
$295$	−38.1611	−0.00753162
$296$	0	0
$297$	0	0
$298$	0	0
$299$	416.596	0.0805765
$300$	0	0
$301$	−2202.56	−0.421772
$302$	0	0
$303$	0	0
$304$	0	0
$305$	47.0562	0.00883420
$306$	0	0
$307$	3745.74	0.696355	0.348177	−	0.937429i	$-0.386801\pi$
$307$	3745.74	0.696355	0.348177	+	0.937429i	$0.386801\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4854.78	−0.885175	−0.442588	−	0.896725i	$-0.645939\pi$
$311$	−4854.78	−0.885175	−0.442588	+	0.896725i	$0.645939\pi$
$312$	0	0
$313$	3171.06	0.572649	0.286324	−	0.958133i	$-0.407567\pi$
$313$	3171.06	0.572649	0.286324	+	0.958133i	$0.407567\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5837.07	−1.03420	−0.517101	−	0.855924i	$-0.672989\pi$
$317$	−5837.07	−1.03420	−0.517101	+	0.855924i	$0.672989\pi$
$318$	0	0
$319$	−8673.65	−1.52235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10290.7	−1.77273
$324$	0	0
$325$	−215.137	−0.0367189
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3249.99	−0.544612
$330$	0	0
$331$	5595.31	0.929142	0.464571	−	0.885536i	$-0.346209\pi$
$331$	5595.31	0.929142	0.464571	+	0.885536i	$0.346209\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2471.96	−0.403157
$336$	0	0
$337$	−8661.15	−1.40001	−0.700004	−	0.714139i	$-0.746819\pi$
$337$	−8661.15	−1.40001	−0.700004	+	0.714139i	$0.746819\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2653.36	−0.421371
$342$	0	0
$343$	−4022.94	−0.633289
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10555.3	1.63297	0.816483	−	0.577369i	$-0.195921\pi$
$347$	10555.3	1.63297	0.816483	+	0.577369i	$0.195921\pi$
$348$	0	0
$349$	6399.71	0.981572	0.490786	−	0.871280i	$-0.336710\pi$
$349$	6399.71	0.981572	0.490786	+	0.871280i	$0.336710\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9778.11	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$353$	−9778.11	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$354$	0	0
$355$	−2803.55	−0.419146
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2893.60	0.425399	0.212700	−	0.977118i	$-0.431774\pi$
$359$	2893.60	0.425399	0.212700	+	0.977118i	$0.431774\pi$
$360$	0	0
$361$	6175.72	0.900382
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5583.42	0.800684
$366$	0	0
$367$	−7307.82	−1.03941	−0.519707	−	0.854345i	$-0.673959\pi$
$367$	−7307.82	−1.03941	−0.519707	+	0.854345i	$0.673959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−415.903	−0.0582012
$372$	0	0
$373$	8272.24	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$373$	8272.24	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2632.91	−0.359687
$378$	0	0
$379$	−740.953	−0.100423	−0.0502113	−	0.998739i	$-0.515989\pi$
$379$	−740.953	−0.100423	−0.0502113	+	0.998739i	$0.515989\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1233.49	−0.164566	−0.0822828	−	0.996609i	$-0.526221\pi$
$383$	−1233.49	−0.164566	−0.0822828	+	0.996609i	$0.526221\pi$
$384$	0	0
$385$	−880.828	−0.116600
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6958.20	0.906927	0.453464	−	0.891275i	$-0.350188\pi$
$389$	6958.20	0.906927	0.453464	+	0.891275i	$0.350188\pi$
$390$	0	0
$391$	4363.50	0.564377
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5209.35	0.663572
$396$	0	0
$397$	5973.68	0.755190	0.377595	−	0.925971i	$-0.376751\pi$
$397$	5973.68	0.755190	0.377595	+	0.925971i	$0.376751\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4716.88	−0.587406	−0.293703	−	0.955897i	$-0.594888\pi$
$401$	−4716.88	−0.587406	−0.293703	+	0.955897i	$0.594888\pi$
$402$	0	0
$403$	−805.436	−0.0995574
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8019.74	0.976717
$408$	0	0
$409$	7815.88	0.944915	0.472458	−	0.881353i	$-0.343367\pi$
$409$	7815.88	0.944915	0.472458	+	0.881353i	$0.343367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−47.4277	−0.00565077
$414$	0	0
$415$	227.554	0.0269161
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2955.17	−0.344557	−0.172278	−	0.985048i	$-0.555113\pi$
$419$	−2955.17	−0.344557	−0.172278	+	0.985048i	$0.555113\pi$
$420$	0	0
$421$	−6770.08	−0.783737	−0.391869	−	0.920021i	$-0.628171\pi$
$421$	−6770.08	−0.783737	−0.391869	+	0.920021i	$0.628171\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2253.38	−0.257188
$426$	0	0
$427$	58.4828	0.00662805
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6082.40	0.679765	0.339883	−	0.940468i	$-0.389613\pi$
$431$	6082.40	0.679765	0.339883	+	0.940468i	$0.389613\pi$
$432$	0	0
$433$	−8993.04	−0.998102	−0.499051	−	0.866573i	$-0.666318\pi$
$433$	−8993.04	−0.998102	−0.499051	+	0.866573i	$0.666318\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5527.02	−0.605019
$438$	0	0
$439$	17180.5	1.86784	0.933919	−	0.357485i	$-0.116366\pi$
$439$	17180.5	1.86784	0.933919	+	0.357485i	$0.116366\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12231.3	−1.31180	−0.655901	−	0.754847i	$-0.727711\pi$
$443$	−12231.3	−1.31180	−0.655901	+	0.754847i	$0.727711\pi$
$444$	0	0
$445$	−1785.79	−0.190235
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1182.08	−0.124245	−0.0621224	−	0.998069i	$-0.519787\pi$
$449$	−1182.08	−0.124245	−0.0621224	+	0.998069i	$0.519787\pi$
$450$	0	0
$451$	815.563	0.0851515
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−267.378	−0.0275492
$456$	0	0
$457$	−7353.03	−0.752648	−0.376324	−	0.926488i	$-0.622812\pi$
$457$	−7353.03	−0.752648	−0.376324	+	0.926488i	$0.622812\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3706.94	−0.374510	−0.187255	−	0.982311i	$-0.559959\pi$
$461$	−3706.94	−0.374510	−0.187255	+	0.982311i	$0.559959\pi$
$462$	0	0
$463$	−14015.4	−1.40680	−0.703401	−	0.710794i	$-0.748336\pi$
$463$	−14015.4	−1.40680	−0.703401	+	0.710794i	$0.748336\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12187.1	1.20760	0.603801	−	0.797135i	$-0.293652\pi$
$467$	12187.1	1.20760	0.603801	+	0.797135i	$0.293652\pi$
$468$	0	0
$469$	−3072.22	−0.302477
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10048.2	0.976776
$474$	0	0
$475$	2854.24	0.275709
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−64.6375	−0.00616568	−0.00308284	−	0.999995i	$-0.500981\pi$
$479$	−64.6375	−0.00616568	−0.00308284	+	0.999995i	$0.500981\pi$
$480$	0	0
$481$	2434.42	0.230769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−600.029	−0.0561771
$486$	0	0
$487$	−3763.31	−0.350168	−0.175084	−	0.984553i	$-0.556020\pi$
$487$	−3763.31	−0.350168	−0.175084	+	0.984553i	$0.556020\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2106.94	−0.193655	−0.0968277	−	0.995301i	$-0.530870\pi$
$491$	−2106.94	−0.193655	−0.0968277	+	0.995301i	$0.530870\pi$
$492$	0	0
$493$	−27577.6	−2.51933
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3484.33	−0.314474
$498$	0	0
$499$	2130.97	0.191173	0.0955866	−	0.995421i	$-0.469527\pi$
$499$	2130.97	0.191173	0.0955866	+	0.995421i	$0.469527\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12920.0	1.14528	0.572639	−	0.819808i	$-0.305920\pi$
$503$	12920.0	1.14528	0.572639	+	0.819808i	$0.305920\pi$
$504$	0	0
$505$	−2492.79	−0.219659
$506$	0	0
$507$	0	0
$508$	0	0
$509$	685.808	0.0597209	0.0298604	−	0.999554i	$-0.490494\pi$
$509$	685.808	0.0597209	0.0298604	+	0.999554i	$0.490494\pi$
$510$	0	0
$511$	6939.24	0.600731
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1273.43	−0.108959
$516$	0	0
$517$	14826.6	1.26126
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21259.3	−1.78769	−0.893845	−	0.448377i	$-0.852002\pi$
$521$	−21259.3	−1.78769	−0.893845	+	0.448377i	$0.852002\pi$
$522$	0	0
$523$	6099.14	0.509937	0.254968	−	0.966949i	$-0.417935\pi$
$523$	6099.14	0.509937	0.254968	+	0.966949i	$0.417935\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8436.26	−0.697323
$528$	0	0
$529$	−9823.42	−0.807382
$530$	0	0
$531$	0	0
$532$	0	0
$533$	247.567	0.0201188
$534$	0	0
$535$	1555.39	0.125692
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8629.04	0.689572
$540$	0	0
$541$	9552.07	0.759104	0.379552	−	0.925170i	$-0.376078\pi$
$541$	9552.07	0.759104	0.379552	+	0.925170i	$0.376078\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3305.87	−0.259831
$546$	0	0
$547$	24878.0	1.94462	0.972309	−	0.233698i	$-0.0750828\pi$
$547$	24878.0	1.94462	0.972309	+	0.233698i	$0.0750828\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	34931.1	2.70075
$552$	0	0
$553$	6474.33	0.497860
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10625.7	−0.808306	−0.404153	−	0.914692i	$-0.632434\pi$
$557$	−10625.7	−0.808306	−0.404153	+	0.914692i	$0.632434\pi$
$558$	0	0
$559$	3050.16	0.230783
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16188.0	1.21180	0.605900	−	0.795541i	$-0.292813\pi$
$563$	16188.0	1.21180	0.605900	+	0.795541i	$0.292813\pi$
$564$	0	0
$565$	−2827.68	−0.210551
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7919.46	−0.583482	−0.291741	−	0.956497i	$-0.594234\pi$
$569$	−7919.46	−0.583482	−0.291741	+	0.956497i	$0.594234\pi$
$570$	0	0
$571$	13505.3	0.989808	0.494904	−	0.868948i	$-0.335203\pi$
$571$	13505.3	0.989808	0.494904	+	0.868948i	$0.335203\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1210.26	−0.0877765
$576$	0	0
$577$	−7873.07	−0.568042	−0.284021	−	0.958818i	$-0.591669\pi$
$577$	−7873.07	−0.568042	−0.284021	+	0.958818i	$0.591669\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	282.810	0.0201944
$582$	0	0
$583$	1897.37	0.134787
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20117.0	1.41451	0.707254	−	0.706959i	$-0.249934\pi$
$587$	20117.0	1.41451	0.707254	+	0.706959i	$0.249934\pi$
$588$	0	0
$589$	10685.8	0.747539
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4394.15	0.304294	0.152147	−	0.988358i	$-0.451381\pi$
$593$	4394.15	0.304294	0.152147	+	0.988358i	$0.451381\pi$
$594$	0	0
$595$	−2800.56	−0.192961
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14244.7	0.971660	0.485830	−	0.874053i	$-0.338517\pi$
$599$	14244.7	0.971660	0.485830	+	0.874053i	$0.338517\pi$
$600$	0	0
$601$	16435.6	1.11551	0.557755	−	0.830006i	$-0.311663\pi$
$601$	16435.6	1.11551	0.557755	+	0.830006i	$0.311663\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2636.63	−0.177180
$606$	0	0
$607$	27431.2	1.83426	0.917131	−	0.398586i	$-0.130499\pi$
$607$	27431.2	1.83426	0.917131	+	0.398586i	$0.130499\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4500.65	0.297998
$612$	0	0
$613$	−15161.9	−0.998995	−0.499497	−	0.866315i	$-0.666482\pi$
$613$	−15161.9	−0.998995	−0.499497	+	0.866315i	$0.666482\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6160.44	−0.401961	−0.200981	−	0.979595i	$-0.564413\pi$
$617$	−6160.44	−0.401961	−0.200981	+	0.979595i	$0.564413\pi$
$618$	0	0
$619$	−26799.2	−1.74015	−0.870074	−	0.492922i	$-0.835929\pi$
$619$	−26799.2	−1.74015	−0.870074	+	0.492922i	$0.835929\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2219.44	−0.142728
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25498.5	1.61636
$630$	0	0
$631$	−26100.0	−1.64663	−0.823315	−	0.567585i	$-0.807878\pi$
$631$	−26100.0	−1.64663	−0.823315	+	0.567585i	$0.807878\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9048.01	0.565448
$636$	0	0
$637$	2619.37	0.162925
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3235.84	−0.199389	−0.0996944	−	0.995018i	$-0.531787\pi$
$641$	−3235.84	−0.199389	−0.0996944	+	0.995018i	$0.531787\pi$
$642$	0	0
$643$	−18448.4	−1.13147	−0.565733	−	0.824589i	$-0.691407\pi$
$643$	−18448.4	−1.13147	−0.565733	+	0.824589i	$0.691407\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17173.6	−1.04353	−0.521765	−	0.853089i	$-0.674726\pi$
$647$	−17173.6	−1.04353	−0.521765	+	0.853089i	$0.674726\pi$
$648$	0	0
$649$	216.367	0.0130865
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26039.4	1.56049	0.780245	−	0.625474i	$-0.215094\pi$
$653$	26039.4	1.56049	0.780245	+	0.625474i	$0.215094\pi$
$654$	0	0
$655$	−10149.1	−0.605435
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2706.12	0.159963	0.0799815	−	0.996796i	$-0.474514\pi$
$659$	2706.12	0.159963	0.0799815	+	0.996796i	$0.474514\pi$
$660$	0	0
$661$	27271.4	1.60474	0.802371	−	0.596825i	$-0.203571\pi$
$661$	27271.4	1.60474	0.802371	+	0.596825i	$0.203571\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3547.33	0.206857
$666$	0	0
$667$	−14811.6	−0.859831
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−266.801	−0.0153498
$672$	0	0
$673$	−12793.0	−0.732738	−0.366369	−	0.930470i	$-0.619399\pi$
$673$	−12793.0	−0.732738	−0.366369	+	0.930470i	$0.619399\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27114.1	1.53926	0.769629	−	0.638491i	$-0.220441\pi$
$677$	27114.1	1.53926	0.769629	+	0.638491i	$0.220441\pi$
$678$	0	0
$679$	−745.733	−0.0421481
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33488.7	1.87615	0.938075	−	0.346433i	$-0.112607\pi$
$683$	33488.7	1.87615	0.938075	+	0.346433i	$0.112607\pi$
$684$	0	0
$685$	2726.75	0.152093
$686$	0	0
$687$	0	0
$688$	0	0
$689$	575.952	0.0318462
$690$	0	0
$691$	−1728.99	−0.0951864	−0.0475932	−	0.998867i	$-0.515155\pi$
$691$	−1728.99	−0.0951864	−0.0475932	+	0.998867i	$0.515155\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8813.19	−0.481012
$696$	0	0
$697$	2593.05	0.140917
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9451.65	0.509250	0.254625	−	0.967040i	$-0.418048\pi$
$701$	9451.65	0.509250	0.254625	+	0.967040i	$0.418048\pi$
$702$	0	0
$703$	−32297.7	−1.73276
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3098.11	−0.164804
$708$	0	0
$709$	−23271.7	−1.23271	−0.616353	−	0.787470i	$-0.711391\pi$
$709$	−23271.7	−1.23271	−0.616353	+	0.787470i	$0.711391\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4531.02	−0.237992
$714$	0	0
$715$	1219.79	0.0638008
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24974.3	−1.29539	−0.647695	−	0.761900i	$-0.724267\pi$
$719$	−24974.3	−1.29539	−0.647695	+	0.761900i	$0.724267\pi$
$720$	0	0
$721$	−1582.66	−0.0817493
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7648.95	0.391827
$726$	0	0
$727$	1666.52	0.0850177	0.0425089	−	0.999096i	$-0.486465\pi$
$727$	1666.52	0.0850177	0.0425089	+	0.999096i	$0.486465\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31947.8	1.61646
$732$	0	0
$733$	−8694.92	−0.438137	−0.219068	−	0.975710i	$-0.570302\pi$
$733$	−8694.92	−0.438137	−0.219068	+	0.975710i	$0.570302\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14015.6	0.700503
$738$	0	0
$739$	−19373.0	−0.964342	−0.482171	−	0.876077i	$-0.660152\pi$
$739$	−19373.0	−0.964342	−0.482171	+	0.876077i	$0.660152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27444.0	1.35508	0.677538	−	0.735488i	$-0.263047\pi$
$743$	27444.0	1.35508	0.677538	+	0.735488i	$0.263047\pi$
$744$	0	0
$745$	−6117.65	−0.300850
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1933.08	0.0943032
$750$	0	0
$751$	−3861.56	−0.187630	−0.0938152	−	0.995590i	$-0.529906\pi$
$751$	−3861.56	−0.187630	−0.0938152	+	0.995590i	$0.529906\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14245.1	0.686666
$756$	0	0
$757$	19846.9	0.952903	0.476452	−	0.879201i	$-0.341923\pi$
$757$	19846.9	0.952903	0.476452	+	0.879201i	$0.341923\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30259.4	−1.44140	−0.720698	−	0.693250i	$-0.756178\pi$
$761$	−30259.4	−1.44140	−0.720698	+	0.693250i	$0.756178\pi$
$762$	0	0
$763$	−4108.63	−0.194944
$764$	0	0
$765$	0	0
$766$	0	0
$767$	65.6790	0.00309196
$768$	0	0
$769$	29877.1	1.40104	0.700518	−	0.713634i	$-0.252952\pi$
$769$	29877.1	1.40104	0.700518	+	0.713634i	$0.252952\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9509.77	−0.442487	−0.221244	−	0.975219i	$-0.571012\pi$
$773$	−9509.77	−0.442487	−0.221244	+	0.975219i	$0.571012\pi$
$774$	0	0
$775$	2339.89	0.108453
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3284.49	−0.151064
$780$	0	0
$781$	15895.6	0.728285
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4714.16	−0.214338
$786$	0	0
$787$	−13809.0	−0.625461	−0.312730	−	0.949842i	$-0.601244\pi$
$787$	−13809.0	−0.625461	−0.312730	+	0.949842i	$0.601244\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3514.33	−0.157971
$792$	0	0
$793$	−80.9882	−0.00362670
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16821.9	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$797$	16821.9	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$798$	0	0
$799$	47140.5	2.08725
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31657.1	−1.39122
$804$	0	0
$805$	−1504.15	−0.0658563
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28306.2	−1.23015	−0.615075	−	0.788468i	$-0.710874\pi$
$809$	−28306.2	−1.23015	−0.615075	+	0.788468i	$0.710874\pi$
$810$	0	0
$811$	13617.1	0.589594	0.294797	−	0.955560i	$-0.404748\pi$
$811$	13617.1	0.589594	0.294797	+	0.955560i	$0.404748\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2088.87	0.0897790
$816$	0	0
$817$	−40466.7	−1.73286
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30903.3	1.31368	0.656841	−	0.754029i	$-0.271892\pi$
$821$	30903.3	1.31368	0.656841	+	0.754029i	$0.271892\pi$
$822$	0	0
$823$	20273.3	0.858665	0.429333	−	0.903146i	$-0.358749\pi$
$823$	20273.3	0.858665	0.429333	+	0.903146i	$0.358749\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32437.6	1.36393	0.681963	−	0.731387i	$-0.261127\pi$
$827$	32437.6	1.36393	0.681963	+	0.731387i	$0.261127\pi$
$828$	0	0
$829$	−7539.68	−0.315879	−0.157940	−	0.987449i	$-0.550485\pi$
$829$	−7539.68	−0.315879	−0.157940	+	0.987449i	$0.550485\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27435.7	1.14117
$834$	0	0
$835$	−6963.81	−0.288614
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20839.4	0.857518	0.428759	−	0.903419i	$-0.358951\pi$
$839$	20839.4	0.857518	0.428759	+	0.903419i	$0.358951\pi$
$840$	0	0
$841$	69221.2	2.83822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10614.7	−0.432139
$846$	0	0
$847$	−3276.88	−0.132934
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13695.0	0.551653
$852$	0	0
$853$	−36319.7	−1.45787	−0.728934	−	0.684584i	$-0.759984\pi$
$853$	−36319.7	−1.45787	−0.728934	+	0.684584i	$0.759984\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10265.3	0.409168	0.204584	−	0.978849i	$-0.434416\pi$
$857$	10265.3	0.409168	0.204584	+	0.978849i	$0.434416\pi$
$858$	0	0
$859$	38218.5	1.51804	0.759022	−	0.651065i	$-0.225677\pi$
$859$	38218.5	1.51804	0.759022	+	0.651065i	$0.225677\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27385.0	1.08018	0.540090	−	0.841608i	$-0.318390\pi$
$863$	27385.0	1.08018	0.540090	+	0.841608i	$0.318390\pi$
$864$	0	0
$865$	−18356.7	−0.721559
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29536.2	−1.15299
$870$	0	0
$871$	4254.48	0.165508
$872$	0	0
$873$	0	0
$874$	0	0
$875$	776.768	0.0300109
$876$	0	0
$877$	31319.6	1.20592	0.602958	−	0.797773i	$-0.293989\pi$
$877$	31319.6	1.20592	0.602958	+	0.797773i	$0.293989\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6309.71	0.241293	0.120647	−	0.992696i	$-0.461503\pi$
$881$	6309.71	0.241293	0.120647	+	0.992696i	$0.461503\pi$
$882$	0	0
$883$	−24665.0	−0.940026	−0.470013	−	0.882660i	$-0.655751\pi$
$883$	−24665.0	−0.940026	−0.470013	+	0.882660i	$0.655751\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27988.7	1.05949	0.529746	−	0.848156i	$-0.322287\pi$
$887$	27988.7	1.05949	0.529746	+	0.848156i	$0.322287\pi$
$888$	0	0
$889$	11245.1	0.424240
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−59710.6	−2.23756
$894$	0	0
$895$	−11936.4	−0.445798
$896$	0	0
$897$	0	0
$898$	0	0
$899$	28636.3	1.06238
$900$	0	0
$901$	6032.62	0.223058
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10393.1	0.381745
$906$	0	0
$907$	23263.2	0.851645	0.425822	−	0.904807i	$-0.359985\pi$
$907$	23263.2	0.851645	0.425822	+	0.904807i	$0.359985\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38345.8	1.39457	0.697286	−	0.716793i	$-0.254391\pi$
$911$	38345.8	1.39457	0.697286	+	0.716793i	$0.254391\pi$
$912$	0	0
$913$	−1290.19	−0.0467679
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12613.6	−0.454241
$918$	0	0
$919$	−11675.3	−0.419077	−0.209539	−	0.977800i	$-0.567196\pi$
$919$	−11675.3	−0.419077	−0.209539	+	0.977800i	$0.567196\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4825.17	0.172072
$924$	0	0
$925$	−7072.29	−0.251390
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34050.4	1.20254	0.601269	−	0.799047i	$-0.294662\pi$
$929$	34050.4	1.20254	0.601269	+	0.799047i	$0.294662\pi$
$930$	0	0
$931$	−34751.5	−1.22334
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12776.3	0.446876
$936$	0	0
$937$	15395.5	0.536766	0.268383	−	0.963312i	$-0.413511\pi$
$937$	15395.5	0.536766	0.268383	+	0.963312i	$0.413511\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25708.7	−0.890626	−0.445313	−	0.895375i	$-0.646908\pi$
$941$	−25708.7	−0.890626	−0.445313	+	0.895375i	$0.646908\pi$
$942$	0	0
$943$	1392.70	0.0480939
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7035.07	−0.241403	−0.120702	−	0.992689i	$-0.538514\pi$
$947$	−7035.07	−0.241403	−0.120702	+	0.992689i	$0.538514\pi$
$948$	0	0
$949$	−9609.61	−0.328705
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15233.0	0.517782	0.258891	−	0.965907i	$-0.416643\pi$
$953$	15233.0	0.517782	0.258891	+	0.965907i	$0.416643\pi$
$954$	0	0
$955$	−14955.1	−0.506740
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3388.88	0.114111
$960$	0	0
$961$	−21030.8	−0.705946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25278.9	−0.843273
$966$	0	0
$967$	18311.7	0.608959	0.304479	−	0.952519i	$-0.401518\pi$
$967$	18311.7	0.608959	0.304479	+	0.952519i	$0.401518\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37567.0	−1.24159	−0.620795	−	0.783973i	$-0.713190\pi$
$971$	−37567.0	−1.24159	−0.620795	+	0.783973i	$0.713190\pi$
$972$	0	0
$973$	−10953.3	−0.360890
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50040.0	−1.63861	−0.819305	−	0.573357i	$-0.805641\pi$
$977$	−50040.0	−1.63861	−0.819305	+	0.573357i	$0.805641\pi$
$978$	0	0
$979$	10125.2	0.330543
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3328.05	0.107984	0.0539921	−	0.998541i	$-0.482805\pi$
$983$	3328.05	0.107984	0.0539921	+	0.998541i	$0.482805\pi$
$984$	0	0
$985$	−18327.8	−0.592865
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17158.8	0.551687
$990$	0	0
$991$	47485.9	1.52214	0.761069	−	0.648671i	$-0.224675\pi$
$991$	47485.9	1.52214	0.761069	+	0.648671i	$0.224675\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16089.6	0.512639
$996$	0	0
$997$	101.392	0.00322076	0.00161038	−	0.999999i	$-0.499487\pi$
$997$	101.392	0.00322076	0.00161038	+	0.999999i	$0.499487\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.j.1.4	yes	6
3.2	odd	2		1620.4.a.i.1.4	✓	6
9.2	odd	6		1620.4.i.x.1081.3		12
9.4	even	3		1620.4.i.w.541.3		12
9.5	odd	6		1620.4.i.x.541.3		12
9.7	even	3		1620.4.i.w.1081.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.4	✓	6	3.2	odd	2
1620.4.a.j.1.4	yes	6	1.1	even	1	trivial
1620.4.i.w.541.3		12	9.4	even	3
1620.4.i.w.1081.3		12	9.7	even	3
1620.4.i.x.541.3		12	9.5	odd	6
1620.4.i.x.1081.3		12	9.2	odd	6

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)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +6.21414 q^{7} -28.3492 q^{11} -8.60548 q^{13} -90.1352 q^{17} +114.170 q^{19} -48.4106 q^{23} +25.0000 q^{25} +305.958 q^{29} +93.5957 q^{31} +31.0707 q^{35} -282.892 q^{37} -28.7685 q^{41} -354.443 q^{43} -522.998 q^{47} -304.384 q^{49} -66.9285 q^{53} -141.746 q^{55} -7.63223 q^{59} +9.41124 q^{61} -43.0274 q^{65} -494.391 q^{67} -560.709 q^{71} +1116.68 q^{73} -176.166 q^{77} +1041.87 q^{79} +45.5107 q^{83} -450.676 q^{85} -357.159 q^{89} -53.4757 q^{91} +570.849 q^{95} -120.006 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53}+ \cdots - 1188 q^{97}+O(q^{100}) \) 6 * q + 30 * q^5 + 12 * q^7 - 84 * q^13 - 12 * q^17 - 114 * q^19 - 30 * q^23 + 150 * q^25 - 168 * q^29 - 324 * q^31 + 60 * q^35 - 492 * q^37 - 312 * q^41 - 156 * q^43 - 462 * q^47 - 588 * q^49 - 1014 * q^53 - 1008 * q^59 + 36 * q^61 - 420 * q^65 + 144 * q^67 - 1212 * q^71 - 900 * q^73 - 672 * q^77 - 936 * q^79 - 288 * q^83 - 60 * q^85 + 120 * q^89 + 2286 * q^91 - 570 * q^95 - 1188 * q^97

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