LMFDB - Embedded newform 1620.4.a.i.1.6

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:	$0$
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			1.6
Root			$1.37492$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +22.4267 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +22.4267 q^{7} +56.5145 q^{11} +43.5678 q^{13} +34.9879 q^{17} +77.1406 q^{19} +122.475 q^{23} +25.0000 q^{25} +273.120 q^{29} -297.626 q^{31} -112.133 q^{35} -267.925 q^{37} -181.028 q^{41} +369.860 q^{43} -112.561 q^{47} +159.955 q^{49} -23.1415 q^{53} -282.573 q^{55} +279.315 q^{59} -392.869 q^{61} -217.839 q^{65} +394.580 q^{67} +973.017 q^{71} -760.770 q^{73} +1267.43 q^{77} -831.753 q^{79} -519.890 q^{83} -174.940 q^{85} -1189.22 q^{89} +977.081 q^{91} -385.703 q^{95} -839.717 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 30 q^{5} + 12 q^{7}+O(q^{10}) $ 6 * q - 30 * q^5 + 12 * q^7	$ 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}+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

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
$7$	22.4267	1.21093	0.605463	−	0.795873i	$-0.292988\pi$
$7$	22.4267	1.21093	0.605463	+	0.795873i	$0.292988\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	56.5145	1.54907	0.774535	−	0.632531i	$-0.217984\pi$
$11$	56.5145	1.54907	0.774535	+	0.632531i	$0.217984\pi$
$12$	0	0
$13$	43.5678	0.929504	0.464752	−	0.885441i	$-0.346144\pi$
$13$	43.5678	0.929504	0.464752	+	0.885441i	$0.346144\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	34.9879	0.499166	0.249583	−	0.968353i	$-0.419707\pi$
$17$	34.9879	0.499166	0.249583	+	0.968353i	$0.419707\pi$
$18$	0	0
$19$	77.1406	0.931436	0.465718	−	0.884933i	$-0.345796\pi$
$19$	77.1406	0.931436	0.465718	+	0.884933i	$0.345796\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	122.475	1.11034	0.555171	−	0.831737i	$-0.312653\pi$
$23$	122.475	1.11034	0.555171	+	0.831737i	$0.312653\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	273.120	1.74886	0.874432	−	0.485147i	$-0.161234\pi$
$29$	273.120	1.74886	0.874432	+	0.485147i	$0.161234\pi$
$30$	0	0
$31$	−297.626	−1.72436	−0.862180	−	0.506603i	$-0.830901\pi$
$31$	−297.626	−1.72436	−0.862180	+	0.506603i	$0.830901\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−112.133	−0.541543
$36$	0	0
$37$	−267.925	−1.19045	−0.595224	−	0.803560i	$-0.702937\pi$
$37$	−267.925	−1.19045	−0.595224	+	0.803560i	$0.702937\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−181.028	−0.689555	−0.344777	−	0.938684i	$-0.612046\pi$
$41$	−181.028	−0.689555	−0.344777	+	0.938684i	$0.612046\pi$
$42$	0	0
$43$	369.860	1.31170	0.655850	−	0.754891i	$-0.272310\pi$
$43$	369.860	1.31170	0.655850	+	0.754891i	$0.272310\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−112.561	−0.349333	−0.174666	−	0.984628i	$-0.555885\pi$
$47$	−112.561	−0.349333	−0.174666	+	0.984628i	$0.555885\pi$
$48$	0	0
$49$	159.955	0.466342
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−23.1415	−0.0599759	−0.0299880	−	0.999550i	$-0.509547\pi$
$53$	−23.1415	−0.0599759	−0.0299880	+	0.999550i	$0.509547\pi$
$54$	0	0
$55$	−282.573	−0.692765
$56$	0	0
$57$	0	0
$58$	0	0
$59$	279.315	0.616334	0.308167	−	0.951332i	$-0.400284\pi$
$59$	279.315	0.616334	0.308167	+	0.951332i	$0.400284\pi$
$60$	0	0
$61$	−392.869	−0.824617	−0.412309	−	0.911044i	$-0.635277\pi$
$61$	−392.869	−0.824617	−0.412309	+	0.911044i	$0.635277\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−217.839	−0.415687
$66$	0	0
$67$	394.580	0.719487	0.359744	−	0.933051i	$-0.382864\pi$
$67$	394.580	0.719487	0.359744	+	0.933051i	$0.382864\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	973.017	1.62642	0.813210	−	0.581970i	$-0.197718\pi$
$71$	973.017	1.62642	0.813210	+	0.581970i	$0.197718\pi$
$72$	0	0
$73$	−760.770	−1.21975	−0.609873	−	0.792500i	$-0.708779\pi$
$73$	−760.770	−1.21975	−0.609873	+	0.792500i	$0.708779\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1267.43	1.87581
$78$	0	0
$79$	−831.753	−1.18455	−0.592275	−	0.805736i	$-0.701770\pi$
$79$	−831.753	−1.18455	−0.592275	+	0.805736i	$0.701770\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−519.890	−0.687534	−0.343767	−	0.939055i	$-0.611703\pi$
$83$	−519.890	−0.687534	−0.343767	+	0.939055i	$0.611703\pi$
$84$	0	0
$85$	−174.940	−0.223234
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1189.22	−1.41637	−0.708183	−	0.706028i	$-0.750485\pi$
$89$	−1189.22	−1.41637	−0.708183	+	0.706028i	$0.750485\pi$
$90$	0	0
$91$	977.081	1.12556
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−385.703	−0.416551
$96$	0	0
$97$	−839.717	−0.878973	−0.439486	−	0.898249i	$-0.644840\pi$
$97$	−839.717	−0.878973	−0.439486	+	0.898249i	$0.644840\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1041.22	1.02579	0.512896	−	0.858451i	$-0.328573\pi$
$101$	1041.22	1.02579	0.512896	+	0.858451i	$0.328573\pi$
$102$	0	0
$103$	1868.14	1.78712	0.893558	−	0.448948i	$-0.148201\pi$
$103$	1868.14	1.78712	0.893558	+	0.448948i	$0.148201\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	371.156	0.335336	0.167668	−	0.985844i	$-0.446376\pi$
$107$	371.156	0.335336	0.167668	+	0.985844i	$0.446376\pi$
$108$	0	0
$109$	−1204.94	−1.05883	−0.529414	−	0.848364i	$-0.677588\pi$
$109$	−1204.94	−1.05883	−0.529414	+	0.848364i	$0.677588\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2203.50	−1.83441	−0.917205	−	0.398416i	$-0.869560\pi$
$113$	−2203.50	−1.83441	−0.917205	+	0.398416i	$0.869560\pi$
$114$	0	0
$115$	−612.376	−0.496560
$116$	0	0
$117$	0	0
$118$	0	0
$119$	784.662	0.604453
$120$	0	0
$121$	1862.89	1.39962
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2753.36	1.92379	0.961896	−	0.273417i	$-0.0881539\pi$
$127$	2753.36	1.92379	0.961896	+	0.273417i	$0.0881539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	931.592	0.621325	0.310663	−	0.950520i	$-0.399449\pi$
$131$	931.592	0.621325	0.310663	+	0.950520i	$0.399449\pi$
$132$	0	0
$133$	1730.01	1.12790
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1294.46	0.807248	0.403624	−	0.914925i	$-0.367750\pi$
$137$	1294.46	0.807248	0.403624	+	0.914925i	$0.367750\pi$
$138$	0	0
$139$	976.665	0.595969	0.297984	−	0.954571i	$-0.403686\pi$
$139$	976.665	0.595969	0.297984	+	0.954571i	$0.403686\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2462.22	1.43987
$144$	0	0
$145$	−1365.60	−0.782116
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1738.93	0.956100	0.478050	−	0.878333i	$-0.341344\pi$
$149$	1738.93	0.956100	0.478050	+	0.878333i	$0.341344\pi$
$150$	0	0
$151$	−1493.15	−0.804710	−0.402355	−	0.915484i	$-0.631808\pi$
$151$	−1493.15	−0.804710	−0.402355	+	0.915484i	$0.631808\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1488.13	0.771157
$156$	0	0
$157$	1660.16	0.843917	0.421958	−	0.906615i	$-0.361343\pi$
$157$	1660.16	0.843917	0.421958	+	0.906615i	$0.361343\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2746.71	1.34454
$162$	0	0
$163$	−3344.58	−1.60716	−0.803582	−	0.595195i	$-0.797075\pi$
$163$	−3344.58	−1.60716	−0.803582	+	0.595195i	$0.797075\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−441.117	−0.204399	−0.102200	−	0.994764i	$-0.532588\pi$
$167$	−441.117	−0.204399	−0.102200	+	0.994764i	$0.532588\pi$
$168$	0	0
$169$	−298.843	−0.136023
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3353.39	−1.47372	−0.736860	−	0.676045i	$-0.763693\pi$
$173$	−3353.39	−1.47372	−0.736860	+	0.676045i	$0.763693\pi$
$174$	0	0
$175$	560.667	0.242185
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−823.449	−0.343840	−0.171920	−	0.985111i	$-0.554997\pi$
$179$	−823.449	−0.343840	−0.171920	+	0.985111i	$0.554997\pi$
$180$	0	0
$181$	−1754.26	−0.720404	−0.360202	−	0.932874i	$-0.617292\pi$
$181$	−1754.26	−0.720404	−0.360202	+	0.932874i	$0.617292\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1339.63	0.532385
$186$	0	0
$187$	1977.32	0.773242
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1114.23	0.422109	0.211055	−	0.977474i	$-0.432310\pi$
$191$	1114.23	0.422109	0.211055	+	0.977474i	$0.432310\pi$
$192$	0	0
$193$	3083.32	1.14996	0.574980	−	0.818168i	$-0.305010\pi$
$193$	3083.32	1.14996	0.574980	+	0.818168i	$0.305010\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2968.69	1.07366	0.536828	−	0.843691i	$-0.319622\pi$
$197$	2968.69	1.07366	0.536828	+	0.843691i	$0.319622\pi$
$198$	0	0
$199$	−3366.56	−1.19924	−0.599621	−	0.800284i	$-0.704682\pi$
$199$	−3366.56	−1.19924	−0.599621	+	0.800284i	$0.704682\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6125.17	2.11775
$204$	0	0
$205$	905.138	0.308378
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4359.56	1.44286
$210$	0	0
$211$	2984.62	0.973789	0.486894	−	0.873461i	$-0.338130\pi$
$211$	2984.62	0.973789	0.486894	+	0.873461i	$0.338130\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1849.30	−0.586610
$216$	0	0
$217$	−6674.75	−2.08807
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1524.35	0.463976
$222$	0	0
$223$	−3552.96	−1.06692	−0.533461	−	0.845824i	$-0.679109\pi$
$223$	−3552.96	−1.06692	−0.533461	+	0.845824i	$0.679109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1758.13	0.514058	0.257029	−	0.966404i	$-0.417256\pi$
$227$	1758.13	0.514058	0.257029	+	0.966404i	$0.417256\pi$
$228$	0	0
$229$	−4244.69	−1.22488	−0.612439	−	0.790518i	$-0.709812\pi$
$229$	−4244.69	−1.22488	−0.612439	+	0.790518i	$0.709812\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4604.60	1.29467	0.647334	−	0.762207i	$-0.275884\pi$
$233$	4604.60	1.29467	0.647334	+	0.762207i	$0.275884\pi$
$234$	0	0
$235$	562.803	0.156226
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3985.34	1.07862	0.539309	−	0.842108i	$-0.318685\pi$
$239$	3985.34	1.07862	0.539309	+	0.842108i	$0.318685\pi$
$240$	0	0
$241$	2996.52	0.800923	0.400462	−	0.916314i	$-0.368850\pi$
$241$	2996.52	0.800923	0.400462	+	0.916314i	$0.368850\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−799.776	−0.208554
$246$	0	0
$247$	3360.85	0.865773
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1586.05	0.398847	0.199423	−	0.979913i	$-0.436093\pi$
$251$	1586.05	0.398847	0.199423	+	0.979913i	$0.436093\pi$
$252$	0	0
$253$	6921.63	1.72000
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2612.80	−0.634171	−0.317085	−	0.948397i	$-0.602704\pi$
$257$	−2612.80	−0.634171	−0.317085	+	0.948397i	$0.602704\pi$
$258$	0	0
$259$	−6008.66	−1.44155
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2187.76	0.512940	0.256470	−	0.966552i	$-0.417441\pi$
$263$	2187.76	0.512940	0.256470	+	0.966552i	$0.417441\pi$
$264$	0	0
$265$	115.707	0.0268220
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3368.46	0.763489	0.381745	−	0.924268i	$-0.375323\pi$
$269$	3368.46	0.763489	0.381745	+	0.924268i	$0.375323\pi$
$270$	0	0
$271$	−1111.68	−0.249187	−0.124594	−	0.992208i	$-0.539763\pi$
$271$	−1111.68	−0.249187	−0.124594	+	0.992208i	$0.539763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1412.86	0.309814
$276$	0	0
$277$	−3736.19	−0.810418	−0.405209	−	0.914224i	$-0.632801\pi$
$277$	−3736.19	−0.810418	−0.405209	+	0.914224i	$0.632801\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	408.603	0.0867446	0.0433723	−	0.999059i	$-0.486190\pi$
$281$	408.603	0.0867446	0.0433723	+	0.999059i	$0.486190\pi$
$282$	0	0
$283$	1326.90	0.278714	0.139357	−	0.990242i	$-0.455496\pi$
$283$	1326.90	0.278714	0.139357	+	0.990242i	$0.455496\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4059.84	−0.835000
$288$	0	0
$289$	−3688.85	−0.750834
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1345.46	0.268268	0.134134	−	0.990963i	$-0.457175\pi$
$293$	1345.46	0.268268	0.134134	+	0.990963i	$0.457175\pi$
$294$	0	0
$295$	−1396.58	−0.275633
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5335.98	1.03207
$300$	0	0
$301$	8294.72	1.58837
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1964.34	0.368780
$306$	0	0
$307$	−5947.29	−1.10563	−0.552817	−	0.833303i	$-0.686447\pi$
$307$	−5947.29	−1.10563	−0.552817	+	0.833303i	$0.686447\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4845.63	−0.883507	−0.441753	−	0.897137i	$-0.645643\pi$
$311$	−4845.63	−0.883507	−0.441753	+	0.897137i	$0.645643\pi$
$312$	0	0
$313$	−5224.88	−0.943539	−0.471770	−	0.881722i	$-0.656385\pi$
$313$	−5224.88	−0.943539	−0.471770	+	0.881722i	$0.656385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6763.11	1.19828	0.599139	−	0.800645i	$-0.295510\pi$
$317$	6763.11	1.19828	0.599139	+	0.800645i	$0.295510\pi$
$318$	0	0
$319$	15435.2	2.70911
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2698.99	0.464941
$324$	0	0
$325$	1089.20	0.185901
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2524.36	−0.423016
$330$	0	0
$331$	3842.32	0.638045	0.319023	−	0.947747i	$-0.396645\pi$
$331$	3842.32	0.638045	0.319023	+	0.947747i	$0.396645\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1972.90	−0.321764
$336$	0	0
$337$	8417.64	1.36065	0.680323	−	0.732912i	$-0.261839\pi$
$337$	8417.64	1.36065	0.680323	+	0.732912i	$0.261839\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16820.2	−2.67115
$342$	0	0
$343$	−4105.09	−0.646221
$344$	0	0
$345$	0	0
$346$	0	0
$347$	943.282	0.145931	0.0729655	−	0.997334i	$-0.476754\pi$
$347$	943.282	0.145931	0.0729655	+	0.997334i	$0.476754\pi$
$348$	0	0
$349$	2188.86	0.335722	0.167861	−	0.985811i	$-0.446314\pi$
$349$	2188.86	0.335722	0.167861	+	0.985811i	$0.446314\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8052.43	1.21413	0.607065	−	0.794652i	$-0.292347\pi$
$353$	8052.43	1.21413	0.607065	+	0.794652i	$0.292347\pi$
$354$	0	0
$355$	−4865.08	−0.727357
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5099.37	−0.749679	−0.374839	−	0.927090i	$-0.622302\pi$
$359$	−5099.37	−0.749679	−0.374839	+	0.927090i	$0.622302\pi$
$360$	0	0
$361$	−908.321	−0.132428
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3803.85	0.545487
$366$	0	0
$367$	−1846.74	−0.262668	−0.131334	−	0.991338i	$-0.541926\pi$
$367$	−1846.74	−0.262668	−0.131334	+	0.991338i	$0.541926\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−518.986	−0.0726264
$372$	0	0
$373$	−5169.18	−0.717561	−0.358780	−	0.933422i	$-0.616807\pi$
$373$	−5169.18	−0.717561	−0.358780	+	0.933422i	$0.616807\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11899.2	1.62558
$378$	0	0
$379$	1189.17	0.161170	0.0805851	−	0.996748i	$-0.474321\pi$
$379$	1189.17	0.161170	0.0805851	+	0.996748i	$0.474321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1778.37	0.237260	0.118630	−	0.992939i	$-0.462150\pi$
$383$	1778.37	0.237260	0.118630	+	0.992939i	$0.462150\pi$
$384$	0	0
$385$	−6337.16	−0.838887
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12303.3	−1.60360	−0.801801	−	0.597591i	$-0.796125\pi$
$389$	−12303.3	−1.60360	−0.801801	+	0.597591i	$0.796125\pi$
$390$	0	0
$391$	4285.15	0.554244
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4158.76	0.529747
$396$	0	0
$397$	4361.53	0.551383	0.275692	−	0.961246i	$-0.411093\pi$
$397$	4361.53	0.551383	0.275692	+	0.961246i	$0.411093\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6888.68	0.857866	0.428933	−	0.903336i	$-0.358890\pi$
$401$	6888.68	0.857866	0.428933	+	0.903336i	$0.358890\pi$
$402$	0	0
$403$	−12966.9	−1.60280
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15141.6	−1.84409
$408$	0	0
$409$	−4622.63	−0.558861	−0.279431	−	0.960166i	$-0.590146\pi$
$409$	−4622.63	−0.558861	−0.279431	+	0.960166i	$0.590146\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6264.11	0.746335
$414$	0	0
$415$	2599.45	0.307475
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10105.7	−1.17827	−0.589134	−	0.808036i	$-0.700531\pi$
$419$	−10105.7	−1.17827	−0.589134	+	0.808036i	$0.700531\pi$
$420$	0	0
$421$	9607.80	1.11225	0.556123	−	0.831100i	$-0.312288\pi$
$421$	9607.80	1.11225	0.556123	+	0.831100i	$0.312288\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	874.698	0.0998331
$426$	0	0
$427$	−8810.73	−0.998551
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6722.50	0.751303	0.375651	−	0.926761i	$-0.377419\pi$
$431$	6722.50	0.751303	0.375651	+	0.926761i	$0.377419\pi$
$432$	0	0
$433$	−10997.0	−1.22052	−0.610258	−	0.792202i	$-0.708934\pi$
$433$	−10997.0	−1.22052	−0.610258	+	0.792202i	$0.708934\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9447.82	1.03421
$438$	0	0
$439$	13143.7	1.42897	0.714484	−	0.699652i	$-0.246662\pi$
$439$	13143.7	1.42897	0.714484	+	0.699652i	$0.246662\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13844.5	−1.48481	−0.742404	−	0.669952i	$-0.766315\pi$
$443$	−13844.5	−1.48481	−0.742404	+	0.669952i	$0.766315\pi$
$444$	0	0
$445$	5946.08	0.633419
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13502.6	1.41921	0.709605	−	0.704600i	$-0.248874\pi$
$449$	13502.6	1.41921	0.709605	+	0.704600i	$0.248874\pi$
$450$	0	0
$451$	−10230.7	−1.06817
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4885.41	−0.503366
$456$	0	0
$457$	4665.18	0.477523	0.238762	−	0.971078i	$-0.423259\pi$
$457$	4665.18	0.477523	0.238762	+	0.971078i	$0.423259\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15220.2	1.53769	0.768847	−	0.639433i	$-0.220831\pi$
$461$	15220.2	1.53769	0.768847	+	0.639433i	$0.220831\pi$
$462$	0	0
$463$	−10689.1	−1.07293	−0.536463	−	0.843924i	$-0.680240\pi$
$463$	−10689.1	−1.07293	−0.536463	+	0.843924i	$0.680240\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10642.9	−1.05459	−0.527295	−	0.849682i	$-0.676794\pi$
$467$	−10642.9	−1.05459	−0.527295	+	0.849682i	$0.676794\pi$
$468$	0	0
$469$	8849.12	0.871246
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20902.5	2.03192
$474$	0	0
$475$	1928.52	0.186287
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5192.48	0.495304	0.247652	−	0.968849i	$-0.420341\pi$
$479$	5192.48	0.495304	0.247652	+	0.968849i	$0.420341\pi$
$480$	0	0
$481$	−11672.9	−1.10653
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4198.59	0.393089
$486$	0	0
$487$	13302.4	1.23776	0.618882	−	0.785484i	$-0.287586\pi$
$487$	13302.4	1.23776	0.618882	+	0.785484i	$0.287586\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17036.2	−1.56585	−0.782926	−	0.622114i	$-0.786274\pi$
$491$	−17036.2	−1.56585	−0.782926	+	0.622114i	$0.786274\pi$
$492$	0	0
$493$	9555.89	0.872973
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21821.5	1.96947
$498$	0	0
$499$	14850.7	1.33228	0.666139	−	0.745828i	$-0.267946\pi$
$499$	14850.7	1.33228	0.666139	+	0.745828i	$0.267946\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12662.6	−1.12246	−0.561231	−	0.827659i	$-0.689672\pi$
$503$	−12662.6	−1.12246	−0.561231	+	0.827659i	$0.689672\pi$
$504$	0	0
$505$	−5206.09	−0.458748
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−797.331	−0.0694324	−0.0347162	−	0.999397i	$-0.511053\pi$
$509$	−797.331	−0.0694324	−0.0347162	+	0.999397i	$0.511053\pi$
$510$	0	0
$511$	−17061.5	−1.47702
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9340.68	−0.799223
$516$	0	0
$517$	−6361.30	−0.541141
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8374.09	0.704176	0.352088	−	0.935967i	$-0.385472\pi$
$521$	8374.09	0.704176	0.352088	+	0.935967i	$0.385472\pi$
$522$	0	0
$523$	1576.14	0.131778	0.0658889	−	0.997827i	$-0.479012\pi$
$523$	1576.14	0.131778	0.0658889	+	0.997827i	$0.479012\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10413.3	−0.860741
$528$	0	0
$529$	2833.18	0.232857
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7886.98	−0.640944
$534$	0	0
$535$	−1855.78	−0.149967
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9039.79	0.722395
$540$	0	0
$541$	17191.1	1.36618	0.683089	−	0.730335i	$-0.260636\pi$
$541$	17191.1	1.36618	0.683089	+	0.730335i	$0.260636\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6024.69	0.473522
$546$	0	0
$547$	16012.5	1.25164	0.625818	−	0.779969i	$-0.284765\pi$
$547$	16012.5	1.25164	0.625818	+	0.779969i	$0.284765\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21068.6	1.62895
$552$	0	0
$553$	−18653.4	−1.43440
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18982.1	−1.44398	−0.721992	−	0.691902i	$-0.756773\pi$
$557$	−18982.1	−1.44398	−0.721992	+	0.691902i	$0.756773\pi$
$558$	0	0
$559$	16114.0	1.21923
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1525.53	0.114198	0.0570991	−	0.998369i	$-0.481815\pi$
$563$	1525.53	0.114198	0.0570991	+	0.998369i	$0.481815\pi$
$564$	0	0
$565$	11017.5	0.820373
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12811.7	−0.943926	−0.471963	−	0.881618i	$-0.656454\pi$
$569$	−12811.7	−0.943926	−0.471963	+	0.881618i	$0.656454\pi$
$570$	0	0
$571$	8372.85	0.613648	0.306824	−	0.951766i	$-0.400734\pi$
$571$	8372.85	0.613648	0.306824	+	0.951766i	$0.400734\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3061.88	0.222068
$576$	0	0
$577$	−7014.66	−0.506107	−0.253054	−	0.967452i	$-0.581435\pi$
$577$	−7014.66	−0.506107	−0.253054	+	0.967452i	$0.581435\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11659.4	−0.832553
$582$	0	0
$583$	−1307.83	−0.0929068
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3851.93	−0.270845	−0.135423	−	0.990788i	$-0.543239\pi$
$587$	−3851.93	−0.270845	−0.135423	+	0.990788i	$0.543239\pi$
$588$	0	0
$589$	−22959.0	−1.60613
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22196.9	−1.53713	−0.768563	−	0.639774i	$-0.779028\pi$
$593$	−22196.9	−1.53713	−0.768563	+	0.639774i	$0.779028\pi$
$594$	0	0
$595$	−3923.31	−0.270319
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6815.36	0.464889	0.232444	−	0.972610i	$-0.425328\pi$
$599$	6815.36	0.464889	0.232444	+	0.972610i	$0.425328\pi$
$600$	0	0
$601$	9904.50	0.672234	0.336117	−	0.941820i	$-0.390886\pi$
$601$	9904.50	0.672234	0.336117	+	0.941820i	$0.390886\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9314.44	−0.625927
$606$	0	0
$607$	12027.7	0.804269	0.402135	−	0.915581i	$-0.368268\pi$
$607$	12027.7	0.804269	0.402135	+	0.915581i	$0.368268\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4904.02	−0.324706
$612$	0	0
$613$	−21430.6	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$613$	−21430.6	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28191.1	−1.83944	−0.919718	−	0.392579i	$-0.871583\pi$
$617$	−28191.1	−1.83944	−0.919718	+	0.392579i	$0.871583\pi$
$618$	0	0
$619$	−3269.20	−0.212278	−0.106139	−	0.994351i	$-0.533849\pi$
$619$	−3269.20	−0.212278	−0.106139	+	0.994351i	$0.533849\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−26670.1	−1.71512
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9374.14	−0.594231
$630$	0	0
$631$	16952.0	1.06949	0.534745	−	0.845013i	$-0.320408\pi$
$631$	16952.0	1.06949	0.534745	+	0.845013i	$0.320408\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13766.8	−0.860346
$636$	0	0
$637$	6968.90	0.433466
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10268.8	0.632750	0.316375	−	0.948634i	$-0.397534\pi$
$641$	10268.8	0.632750	0.316375	+	0.948634i	$0.397534\pi$
$642$	0	0
$643$	4297.81	0.263591	0.131796	−	0.991277i	$-0.457926\pi$
$643$	4297.81	0.263591	0.131796	+	0.991277i	$0.457926\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11265.0	−0.684503	−0.342252	−	0.939608i	$-0.611190\pi$
$647$	−11265.0	−0.684503	−0.342252	+	0.939608i	$0.611190\pi$
$648$	0	0
$649$	15785.4	0.954745
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5691.65	−0.341089	−0.170545	−	0.985350i	$-0.554553\pi$
$653$	−5691.65	−0.341089	−0.170545	+	0.985350i	$0.554553\pi$
$654$	0	0
$655$	−4657.96	−0.277865
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22005.3	−1.30077	−0.650383	−	0.759606i	$-0.725392\pi$
$659$	−22005.3	−1.30077	−0.650383	+	0.759606i	$0.725392\pi$
$660$	0	0
$661$	−19421.9	−1.14285	−0.571426	−	0.820654i	$-0.693609\pi$
$661$	−19421.9	−1.14285	−0.571426	+	0.820654i	$0.693609\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8650.04	−0.504412
$666$	0	0
$667$	33450.4	1.94184
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22202.8	−1.27739
$672$	0	0
$673$	−15301.6	−0.876422	−0.438211	−	0.898872i	$-0.644388\pi$
$673$	−15301.6	−0.876422	−0.438211	+	0.898872i	$0.644388\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4432.21	0.251616	0.125808	−	0.992055i	$-0.459848\pi$
$677$	4432.21	0.251616	0.125808	+	0.992055i	$0.459848\pi$
$678$	0	0
$679$	−18832.1	−1.06437
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17375.9	−0.973455	−0.486727	−	0.873554i	$-0.661809\pi$
$683$	−17375.9	−0.973455	−0.486727	+	0.873554i	$0.661809\pi$
$684$	0	0
$685$	−6472.29	−0.361012
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1008.22	−0.0557478
$690$	0	0
$691$	−20907.6	−1.15103	−0.575515	−	0.817791i	$-0.695198\pi$
$691$	−20907.6	−1.15103	−0.575515	+	0.817791i	$0.695198\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4883.32	−0.266525
$696$	0	0
$697$	−6333.78	−0.344202
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28112.3	−1.51467	−0.757337	−	0.653024i	$-0.773500\pi$
$701$	−28112.3	−1.51467	−0.757337	+	0.653024i	$0.773500\pi$
$702$	0	0
$703$	−20667.9	−1.10883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23351.0	1.24216
$708$	0	0
$709$	7009.34	0.371285	0.185643	−	0.982617i	$-0.440563\pi$
$709$	7009.34	0.371285	0.185643	+	0.982617i	$0.440563\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36451.8	−1.91463
$714$	0	0
$715$	−12311.1	−0.643927
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15546.9	0.806401	0.403200	−	0.915112i	$-0.367898\pi$
$719$	15546.9	0.806401	0.403200	+	0.915112i	$0.367898\pi$
$720$	0	0
$721$	41896.1	2.16407
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6828.00	0.349773
$726$	0	0
$727$	−29671.5	−1.51369	−0.756846	−	0.653594i	$-0.773261\pi$
$727$	−29671.5	−1.51369	−0.756846	+	0.653594i	$0.773261\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12940.6	0.654756
$732$	0	0
$733$	−35084.8	−1.76792	−0.883962	−	0.467558i	$-0.845134\pi$
$733$	−35084.8	−1.76792	−0.883962	+	0.467558i	$0.845134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22299.5	1.11454
$738$	0	0
$739$	−13071.3	−0.650657	−0.325328	−	0.945601i	$-0.605475\pi$
$739$	−13071.3	−0.650657	−0.325328	+	0.945601i	$0.605475\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7933.81	−0.391741	−0.195870	−	0.980630i	$-0.562753\pi$
$743$	−7933.81	−0.391741	−0.195870	+	0.980630i	$0.562753\pi$
$744$	0	0
$745$	−8694.66	−0.427581
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8323.78	0.406067
$750$	0	0
$751$	−29260.8	−1.42176	−0.710880	−	0.703313i	$-0.751703\pi$
$751$	−29260.8	−1.42176	−0.710880	+	0.703313i	$0.751703\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7465.77	0.359877
$756$	0	0
$757$	1284.23	0.0616593	0.0308296	−	0.999525i	$-0.490185\pi$
$757$	1284.23	0.0616593	0.0308296	+	0.999525i	$0.490185\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25337.9	1.20696	0.603480	−	0.797378i	$-0.293780\pi$
$761$	25337.9	1.20696	0.603480	+	0.797378i	$0.293780\pi$
$762$	0	0
$763$	−27022.7	−1.28216
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12169.2	0.572885
$768$	0	0
$769$	19238.4	0.902152	0.451076	−	0.892486i	$-0.351040\pi$
$769$	19238.4	0.902152	0.451076	+	0.892486i	$0.351040\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33366.9	1.55255	0.776276	−	0.630393i	$-0.217106\pi$
$773$	33366.9	1.55255	0.776276	+	0.630393i	$0.217106\pi$
$774$	0	0
$775$	−7440.64	−0.344872
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13964.6	−0.642276
$780$	0	0
$781$	54989.5	2.51944
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8300.78	−0.377411
$786$	0	0
$787$	−9875.76	−0.447310	−0.223655	−	0.974668i	$-0.571799\pi$
$787$	−9875.76	−0.447310	−0.223655	+	0.974668i	$0.571799\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−49417.3	−2.22133
$792$	0	0
$793$	−17116.4	−0.766485
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2643.92	−0.117506	−0.0587530	−	0.998273i	$-0.518712\pi$
$797$	−2643.92	−0.117506	−0.0587530	+	0.998273i	$0.518712\pi$
$798$	0	0
$799$	−3938.26	−0.174375
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−42994.5	−1.88947
$804$	0	0
$805$	−13733.5	−0.601297
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23696.4	−1.02982	−0.514908	−	0.857246i	$-0.672174\pi$
$809$	−23696.4	−1.02982	−0.514908	+	0.857246i	$0.672174\pi$
$810$	0	0
$811$	30745.5	1.33122	0.665611	−	0.746299i	$-0.268171\pi$
$811$	30745.5	1.33122	0.665611	+	0.746299i	$0.268171\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16722.9	0.718745
$816$	0	0
$817$	28531.2	1.22176
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20044.7	−0.852090	−0.426045	−	0.904702i	$-0.640093\pi$
$821$	−20044.7	−0.852090	−0.426045	+	0.904702i	$0.640093\pi$
$822$	0	0
$823$	−18262.9	−0.773518	−0.386759	−	0.922181i	$-0.626405\pi$
$823$	−18262.9	−0.773518	−0.386759	+	0.922181i	$0.626405\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20442.0	−0.859538	−0.429769	−	0.902939i	$-0.641405\pi$
$827$	−20442.0	−0.859538	−0.429769	+	0.902939i	$0.641405\pi$
$828$	0	0
$829$	−8012.09	−0.335671	−0.167835	−	0.985815i	$-0.553678\pi$
$829$	−8012.09	−0.335671	−0.167835	+	0.985815i	$0.553678\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5596.50	0.232782
$834$	0	0
$835$	2205.58	0.0914101
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35570.7	1.46369	0.731847	−	0.681469i	$-0.238659\pi$
$839$	35570.7	1.46369	0.731847	+	0.681469i	$0.238659\pi$
$840$	0	0
$841$	50205.4	2.05853
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1494.21	0.0608313
$846$	0	0
$847$	41778.4	1.69483
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−32814.2	−1.32180
$852$	0	0
$853$	43053.2	1.72815	0.864076	−	0.503361i	$-0.167903\pi$
$853$	43053.2	1.72815	0.864076	+	0.503361i	$0.167903\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9024.86	0.359724	0.179862	−	0.983692i	$-0.442435\pi$
$857$	9024.86	0.359724	0.179862	+	0.983692i	$0.442435\pi$
$858$	0	0
$859$	−15688.4	−0.623145	−0.311572	−	0.950222i	$-0.600856\pi$
$859$	−15688.4	−0.623145	−0.311572	+	0.950222i	$0.600856\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46241.1	1.82394	0.911972	−	0.410252i	$-0.134559\pi$
$863$	46241.1	1.82394	0.911972	+	0.410252i	$0.134559\pi$
$864$	0	0
$865$	16767.0	0.659068
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−47006.1	−1.83495
$870$	0	0
$871$	17191.0	0.668766
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2803.33	−0.108309
$876$	0	0
$877$	−40136.5	−1.54540	−0.772699	−	0.634773i	$-0.781094\pi$
$877$	−40136.5	−1.54540	−0.772699	+	0.634773i	$0.781094\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38113.5	1.45752	0.728761	−	0.684768i	$-0.240096\pi$
$881$	38113.5	1.45752	0.728761	+	0.684768i	$0.240096\pi$
$882$	0	0
$883$	18171.9	0.692563	0.346281	−	0.938131i	$-0.387444\pi$
$883$	18171.9	0.692563	0.346281	+	0.938131i	$0.387444\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50265.3	−1.90275	−0.951377	−	0.308029i	$-0.900330\pi$
$887$	−50265.3	−1.90275	−0.951377	+	0.308029i	$0.900330\pi$
$888$	0	0
$889$	61748.7	2.32957
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8682.99	−0.325381
$894$	0	0
$895$	4117.24	0.153770
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−81287.5	−3.01567
$900$	0	0
$901$	−809.671	−0.0299379
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8771.30	0.322175
$906$	0	0
$907$	9105.98	0.333362	0.166681	−	0.986011i	$-0.446695\pi$
$907$	9105.98	0.333362	0.166681	+	0.986011i	$0.446695\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32170.0	1.16997	0.584983	−	0.811045i	$-0.301101\pi$
$911$	32170.0	1.16997	0.584983	+	0.811045i	$0.301101\pi$
$912$	0	0
$913$	−29381.3	−1.06504
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20892.5	0.752379
$918$	0	0
$919$	16150.1	0.579699	0.289849	−	0.957072i	$-0.406395\pi$
$919$	16150.1	0.579699	0.289849	+	0.957072i	$0.406395\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	42392.2	1.51176
$924$	0	0
$925$	−6698.13	−0.238090
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33227.5	1.17348	0.586738	−	0.809777i	$-0.300412\pi$
$929$	33227.5	1.17348	0.586738	+	0.809777i	$0.300412\pi$
$930$	0	0
$931$	12339.0	0.434367
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9886.62	−0.345804
$936$	0	0
$937$	−43603.7	−1.52025	−0.760123	−	0.649779i	$-0.774861\pi$
$937$	−43603.7	−1.52025	−0.760123	+	0.649779i	$0.774861\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31005.3	−1.07412	−0.537059	−	0.843545i	$-0.680465\pi$
$941$	−31005.3	−1.07412	−0.537059	+	0.843545i	$0.680465\pi$
$942$	0	0
$943$	−22171.4	−0.765641
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5456.55	−0.187237	−0.0936187	−	0.995608i	$-0.529843\pi$
$947$	−5456.55	−0.187237	−0.0936187	+	0.995608i	$0.529843\pi$
$948$	0	0
$949$	−33145.1	−1.13376
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7491.89	−0.254655	−0.127327	−	0.991861i	$-0.540640\pi$
$953$	−7491.89	−0.254655	−0.127327	+	0.991861i	$0.540640\pi$
$954$	0	0
$955$	−5571.15	−0.188773
$956$	0	0
$957$	0	0
$958$	0	0
$959$	29030.4	0.977518
$960$	0	0
$961$	58790.0	1.97341
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15416.6	−0.514277
$966$	0	0
$967$	39454.4	1.31207	0.656033	−	0.754732i	$-0.272233\pi$
$967$	39454.4	1.31207	0.656033	+	0.754732i	$0.272233\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9799.12	0.323861	0.161930	−	0.986802i	$-0.448228\pi$
$971$	9799.12	0.323861	0.161930	+	0.986802i	$0.448228\pi$
$972$	0	0
$973$	21903.3	0.721674
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4049.07	0.132591	0.0662953	−	0.997800i	$-0.478882\pi$
$977$	4049.07	0.132591	0.0662953	+	0.997800i	$0.478882\pi$
$978$	0	0
$979$	−67208.0	−2.19405
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36487.1	−1.18388	−0.591941	−	0.805981i	$-0.701638\pi$
$983$	−36487.1	−1.18388	−0.591941	+	0.805981i	$0.701638\pi$
$984$	0	0
$985$	−14843.4	−0.480154
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45298.7	1.45643
$990$	0	0
$991$	−20630.9	−0.661313	−0.330656	−	0.943751i	$-0.607270\pi$
$991$	−20630.9	−0.661313	−0.330656	+	0.943751i	$0.607270\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16832.8	0.536318
$996$	0	0
$997$	−13215.7	−0.419804	−0.209902	−	0.977722i	$-0.567314\pi$
$997$	−13215.7	−0.419804	−0.209902	+	0.977722i	$0.567314\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.i.1.6	✓	6
3.2	odd	2		1620.4.a.j.1.6	yes	6
9.2	odd	6		1620.4.i.w.1081.1		12
9.4	even	3		1620.4.i.x.541.1		12
9.5	odd	6		1620.4.i.w.541.1		12
9.7	even	3		1620.4.i.x.1081.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.6	✓	6	1.1	even	1	trivial
1620.4.a.j.1.6	yes	6	3.2	odd	2
1620.4.i.w.541.1		12	9.5	odd	6
1620.4.i.w.1081.1		12	9.2	odd	6
1620.4.i.x.541.1		12	9.4	even	3
1620.4.i.x.1081.1		12	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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +22.4267 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +22.4267 q^{7} +56.5145 q^{11} +43.5678 q^{13} +34.9879 q^{17} +77.1406 q^{19} +122.475 q^{23} +25.0000 q^{25} +273.120 q^{29} -297.626 q^{31} -112.133 q^{35} -267.925 q^{37} -181.028 q^{41} +369.860 q^{43} -112.561 q^{47} +159.955 q^{49} -23.1415 q^{53} -282.573 q^{55} +279.315 q^{59} -392.869 q^{61} -217.839 q^{65} +394.580 q^{67} +973.017 q^{71} -760.770 q^{73} +1267.43 q^{77} -831.753 q^{79} -519.890 q^{83} -174.940 q^{85} -1189.22 q^{89} +977.081 q^{91} -385.703 q^{95} -839.717 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 30 q^{5} + 12 q^{7}+O(q^{10}) \) 6 * q - 30 * q^5 + 12 * q^7	\( 6 q - 30 q^{5} + 12 q^{7} - 84 q^{13} + 12 q^{17} - 114 q^{19} + 30 q^{23} + 150 q^{25} + 168 q^{29} - 324 q^{31} - 60 q^{35} - 492 q^{37} + 312 q^{41} - 156 q^{43} + 462 q^{47} - 588 q^{49} + 1014 q^{53} + 1008 q^{59} + 36 q^{61} + 420 q^{65} + 144 q^{67} + 1212 q^{71} - 900 q^{73} + 672 q^{77} - 936 q^{79} + 288 q^{83} - 60 q^{85} - 120 q^{89} + 2286 q^{91} + 570 q^{95} - 1188 q^{97}+O(q^{100}) \) 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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