LMFDB - Embedded newform 1620.4.a.l.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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.0415615$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} -13.7344 q^{7} +21.6430 q^{11} -79.6266 q^{13} -64.3628 q^{17} -144.503 q^{19} -26.9451 q^{23} +25.0000 q^{25} +297.006 q^{29} -96.6507 q^{31} -68.6720 q^{35} +401.580 q^{37} -11.6890 q^{41} +292.271 q^{43} +175.362 q^{47} -154.366 q^{49} +155.406 q^{53} +108.215 q^{55} +595.163 q^{59} -62.1070 q^{61} -398.133 q^{65} -160.664 q^{67} -959.723 q^{71} +763.424 q^{73} -297.254 q^{77} +63.9649 q^{79} +917.326 q^{83} -321.814 q^{85} -1318.88 q^{89} +1093.62 q^{91} -722.513 q^{95} -128.169 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 35 q^{5} + 8 q^{7} + 27 q^{11} + 32 q^{13} + 123 q^{17} - 67 q^{19} + 42 q^{23} + 175 q^{25} + 324 q^{29} + 98 q^{31} + 40 q^{35} + 356 q^{37} + 339 q^{41} + 119 q^{43} - 96 q^{47} + 813 q^{49} + 858 q^{53}+ \cdots + 1991 q^{97}+O(q^{100}) $ 7 * q + 35 * q^5 + 8 * q^7 + 27 * q^11 + 32 * q^13 + 123 * q^17 - 67 * q^19 + 42 * q^23 + 175 * q^25 + 324 * q^29 + 98 * q^31 + 40 * q^35 + 356 * q^37 + 339 * q^41 + 119 * q^43 - 96 * q^47 + 813 * q^49 + 858 * q^53 + 135 * q^55 + 549 * q^59 + 260 * q^61 + 160 * q^65 + 881 * q^67 + 342 * q^71 + 737 * q^73 - 456 * q^77 + 1886 * q^79 - 132 * q^83 + 615 * q^85 + 396 * q^89 + 1264 * q^91 - 335 * q^95 + 1991 * 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
$7$	−13.7344	−0.741588	−0.370794	−	0.928715i	$-0.620914\pi$
$7$	−13.7344	−0.741588	−0.370794	+	0.928715i	$0.620914\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21.6430	0.593237	0.296619	−	0.954996i	$-0.404141\pi$
$11$	21.6430	0.593237	0.296619	+	0.954996i	$0.404141\pi$
$12$	0	0
$13$	−79.6266	−1.69880	−0.849401	−	0.527747i	$-0.823037\pi$
$13$	−79.6266	−1.69880	−0.849401	+	0.527747i	$0.823037\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−64.3628	−0.918251	−0.459125	−	0.888371i	$-0.651837\pi$
$17$	−64.3628	−0.918251	−0.459125	+	0.888371i	$0.651837\pi$
$18$	0	0
$19$	−144.503	−1.74480	−0.872399	−	0.488795i	$-0.837437\pi$
$19$	−144.503	−1.74480	−0.872399	+	0.488795i	$0.837437\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−26.9451	−0.244280	−0.122140	−	0.992513i	$-0.538976\pi$
$23$	−26.9451	−0.244280	−0.122140	+	0.992513i	$0.538976\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	297.006	1.90182	0.950908	−	0.309472i	$-0.100152\pi$
$29$	297.006	1.90182	0.950908	+	0.309472i	$0.100152\pi$
$30$	0	0
$31$	−96.6507	−0.559967	−0.279984	−	0.960005i	$-0.590329\pi$
$31$	−96.6507	−0.559967	−0.279984	+	0.960005i	$0.590329\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−68.6720	−0.331648
$36$	0	0
$37$	401.580	1.78431	0.892153	−	0.451734i	$-0.149194\pi$
$37$	401.580	1.78431	0.892153	+	0.451734i	$0.149194\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.6890	−0.0445246	−0.0222623	−	0.999752i	$-0.507087\pi$
$41$	−11.6890	−0.0445246	−0.0222623	+	0.999752i	$0.507087\pi$
$42$	0	0
$43$	292.271	1.03653	0.518267	−	0.855219i	$-0.326577\pi$
$43$	292.271	1.03653	0.518267	+	0.855219i	$0.326577\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	175.362	0.544239	0.272119	−	0.962263i	$-0.412275\pi$
$47$	175.362	0.544239	0.272119	+	0.962263i	$0.412275\pi$
$48$	0	0
$49$	−154.366	−0.450048
$50$	0	0
$51$	0	0
$52$	0	0
$53$	155.406	0.402767	0.201383	−	0.979513i	$-0.435456\pi$
$53$	155.406	0.402767	0.201383	+	0.979513i	$0.435456\pi$
$54$	0	0
$55$	108.215	0.265304
$56$	0	0
$57$	0	0
$58$	0	0
$59$	595.163	1.31328	0.656641	−	0.754203i	$-0.271977\pi$
$59$	595.163	1.31328	0.656641	+	0.754203i	$0.271977\pi$
$60$	0	0
$61$	−62.1070	−0.130360	−0.0651802	−	0.997874i	$-0.520762\pi$
$61$	−62.1070	−0.130360	−0.0651802	+	0.997874i	$0.520762\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−398.133	−0.759728
$66$	0	0
$67$	−160.664	−0.292958	−0.146479	−	0.989214i	$-0.546794\pi$
$67$	−160.664	−0.292958	−0.146479	+	0.989214i	$0.546794\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−959.723	−1.60420	−0.802100	−	0.597190i	$-0.796284\pi$
$71$	−959.723	−1.60420	−0.802100	+	0.597190i	$0.796284\pi$
$72$	0	0
$73$	763.424	1.22400	0.612000	−	0.790858i	$-0.290365\pi$
$73$	763.424	1.22400	0.612000	+	0.790858i	$0.290365\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−297.254	−0.439938
$78$	0	0
$79$	63.9649	0.0910964	0.0455482	−	0.998962i	$-0.485497\pi$
$79$	63.9649	0.0910964	0.0455482	+	0.998962i	$0.485497\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	917.326	1.21313	0.606564	−	0.795035i	$-0.292548\pi$
$83$	917.326	1.21313	0.606564	+	0.795035i	$0.292548\pi$
$84$	0	0
$85$	−321.814	−0.410654
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1318.88	−1.57080	−0.785399	−	0.618990i	$-0.787542\pi$
$89$	−1318.88	−1.57080	−0.785399	+	0.618990i	$0.787542\pi$
$90$	0	0
$91$	1093.62	1.25981
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−722.513	−0.780297
$96$	0	0
$97$	−128.169	−0.134161	−0.0670804	−	0.997748i	$-0.521368\pi$
$97$	−128.169	−0.134161	−0.0670804	+	0.997748i	$0.521368\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−460.408	−0.453587	−0.226793	−	0.973943i	$-0.572824\pi$
$101$	−460.408	−0.453587	−0.226793	+	0.973943i	$0.572824\pi$
$102$	0	0
$103$	717.461	0.686345	0.343173	−	0.939272i	$-0.388498\pi$
$103$	717.461	0.686345	0.343173	+	0.939272i	$0.388498\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1361.76	1.23034	0.615171	−	0.788393i	$-0.289087\pi$
$107$	1361.76	1.23034	0.615171	+	0.788393i	$0.289087\pi$
$108$	0	0
$109$	225.862	0.198474	0.0992369	−	0.995064i	$-0.468360\pi$
$109$	225.862	0.198474	0.0992369	+	0.995064i	$0.468360\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1968.47	1.63874	0.819371	−	0.573263i	$-0.194323\pi$
$113$	1968.47	1.63874	0.819371	+	0.573263i	$0.194323\pi$
$114$	0	0
$115$	−134.725	−0.109245
$116$	0	0
$117$	0	0
$118$	0	0
$119$	883.984	0.680964
$120$	0	0
$121$	−862.580	−0.648069
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	751.204	0.524871	0.262435	−	0.964950i	$-0.415474\pi$
$127$	751.204	0.524871	0.262435	+	0.964950i	$0.415474\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−79.5797	−0.0530757	−0.0265378	−	0.999648i	$-0.508448\pi$
$131$	−79.5797	−0.0530757	−0.0265378	+	0.999648i	$0.508448\pi$
$132$	0	0
$133$	1984.66	1.29392
$134$	0	0
$135$	0	0
$136$	0	0
$137$	803.584	0.501130	0.250565	−	0.968100i	$-0.419384\pi$
$137$	803.584	0.501130	0.250565	+	0.968100i	$0.419384\pi$
$138$	0	0
$139$	1761.67	1.07498	0.537491	−	0.843270i	$-0.319372\pi$
$139$	1761.67	1.07498	0.537491	+	0.843270i	$0.319372\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1723.36	−1.00779
$144$	0	0
$145$	1485.03	0.850518
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2409.91	−1.32502	−0.662509	−	0.749054i	$-0.730508\pi$
$149$	−2409.91	−1.32502	−0.662509	+	0.749054i	$0.730508\pi$
$150$	0	0
$151$	1847.68	0.995774	0.497887	−	0.867242i	$-0.334109\pi$
$151$	1847.68	0.995774	0.497887	+	0.867242i	$0.334109\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−483.254	−0.250425
$156$	0	0
$157$	−86.1401	−0.0437881	−0.0218940	−	0.999760i	$-0.506970\pi$
$157$	−86.1401	−0.0437881	−0.0218940	+	0.999760i	$0.506970\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	370.074	0.181155
$162$	0	0
$163$	−1127.15	−0.541625	−0.270812	−	0.962632i	$-0.587292\pi$
$163$	−1127.15	−0.541625	−0.270812	+	0.962632i	$0.587292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−968.645	−0.448838	−0.224419	−	0.974493i	$-0.572048\pi$
$167$	−968.645	−0.448838	−0.224419	+	0.974493i	$0.572048\pi$
$168$	0	0
$169$	4143.39	1.88593
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1752.07	0.769987	0.384993	−	0.922919i	$-0.374204\pi$
$173$	1752.07	0.769987	0.384993	+	0.922919i	$0.374204\pi$
$174$	0	0
$175$	−343.360	−0.148318
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−880.297	−0.367578	−0.183789	−	0.982966i	$-0.558836\pi$
$179$	−880.297	−0.367578	−0.183789	+	0.982966i	$0.558836\pi$
$180$	0	0
$181$	335.846	0.137919	0.0689593	−	0.997619i	$-0.478032\pi$
$181$	335.846	0.137919	0.0689593	+	0.997619i	$0.478032\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2007.90	0.797966
$186$	0	0
$187$	−1393.00	−0.544741
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3977.29	−1.50674	−0.753368	−	0.657599i	$-0.771572\pi$
$191$	−3977.29	−1.50674	−0.753368	+	0.657599i	$0.771572\pi$
$192$	0	0
$193$	4159.11	1.55119	0.775593	−	0.631233i	$-0.217451\pi$
$193$	4159.11	1.55119	0.775593	+	0.631233i	$0.217451\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1581.17	0.571846	0.285923	−	0.958253i	$-0.407700\pi$
$197$	1581.17	0.571846	0.285923	+	0.958253i	$0.407700\pi$
$198$	0	0
$199$	−2139.37	−0.762090	−0.381045	−	0.924557i	$-0.624436\pi$
$199$	−2139.37	−0.762090	−0.381045	+	0.924557i	$0.624436\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4079.20	−1.41036
$204$	0	0
$205$	−58.4448	−0.0199120
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3127.47	−1.03508
$210$	0	0
$211$	−176.284	−0.0575162	−0.0287581	−	0.999586i	$-0.509155\pi$
$211$	−176.284	−0.0575162	−0.0287581	+	0.999586i	$0.509155\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1461.36	0.463552
$216$	0	0
$217$	1327.44	0.415265
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5124.99	1.55993
$222$	0	0
$223$	−3622.37	−1.08777	−0.543883	−	0.839161i	$-0.683046\pi$
$223$	−3622.37	−1.08777	−0.543883	+	0.839161i	$0.683046\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	776.343	0.226994	0.113497	−	0.993538i	$-0.463795\pi$
$227$	776.343	0.226994	0.113497	+	0.993538i	$0.463795\pi$
$228$	0	0
$229$	5086.09	1.46768	0.733838	−	0.679324i	$-0.237727\pi$
$229$	5086.09	1.46768	0.733838	+	0.679324i	$0.237727\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3896.93	1.09569	0.547846	−	0.836579i	$-0.315448\pi$
$233$	3896.93	1.09569	0.547846	+	0.836579i	$0.315448\pi$
$234$	0	0
$235$	876.812	0.243391
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1234.73	−0.334176	−0.167088	−	0.985942i	$-0.553436\pi$
$239$	−1234.73	−0.334176	−0.167088	+	0.985942i	$0.553436\pi$
$240$	0	0
$241$	1918.45	0.512772	0.256386	−	0.966574i	$-0.417468\pi$
$241$	1918.45	0.512772	0.256386	+	0.966574i	$0.417468\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−771.832	−0.201267
$246$	0	0
$247$	11506.2	2.96407
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7109.99	1.78796	0.893981	−	0.448105i	$-0.147901\pi$
$251$	7109.99	1.78796	0.893981	+	0.448105i	$0.147901\pi$
$252$	0	0
$253$	−583.172	−0.144916
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4182.28	−1.01511	−0.507555	−	0.861619i	$-0.669451\pi$
$257$	−4182.28	−1.01511	−0.507555	+	0.861619i	$0.669451\pi$
$258$	0	0
$259$	−5515.46	−1.32322
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4216.61	−0.988620	−0.494310	−	0.869286i	$-0.664579\pi$
$263$	−4216.61	−0.988620	−0.494310	+	0.869286i	$0.664579\pi$
$264$	0	0
$265$	777.029	0.180123
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3271.14	−0.741431	−0.370715	−	0.928747i	$-0.620887\pi$
$269$	−3271.14	−0.741431	−0.370715	+	0.928747i	$0.620887\pi$
$270$	0	0
$271$	6407.18	1.43619	0.718096	−	0.695944i	$-0.245014\pi$
$271$	6407.18	1.43619	0.718096	+	0.695944i	$0.245014\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	541.075	0.118647
$276$	0	0
$277$	6326.26	1.37223	0.686116	−	0.727492i	$-0.259314\pi$
$277$	6326.26	1.37223	0.686116	+	0.727492i	$0.259314\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3982.57	−0.845481	−0.422741	−	0.906251i	$-0.638932\pi$
$281$	−3982.57	−0.845481	−0.422741	+	0.906251i	$0.638932\pi$
$282$	0	0
$283$	−3679.68	−0.772913	−0.386456	−	0.922308i	$-0.626301\pi$
$283$	−3679.68	−0.772913	−0.386456	+	0.922308i	$0.626301\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	160.541	0.0330189
$288$	0	0
$289$	−770.434	−0.156815
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8946.28	1.78378	0.891890	−	0.452253i	$-0.149380\pi$
$293$	8946.28	1.78378	0.891890	+	0.452253i	$0.149380\pi$
$294$	0	0
$295$	2975.82	0.587318
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2145.54	0.414983
$300$	0	0
$301$	−4014.17	−0.768681
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−310.535	−0.0582989
$306$	0	0
$307$	−9746.88	−1.81200	−0.906000	−	0.423278i	$-0.860879\pi$
$307$	−9746.88	−1.81200	−0.906000	+	0.423278i	$0.860879\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5279.02	0.962527	0.481264	−	0.876576i	$-0.340178\pi$
$311$	5279.02	0.962527	0.481264	+	0.876576i	$0.340178\pi$
$312$	0	0
$313$	4534.39	0.818847	0.409424	−	0.912344i	$-0.365730\pi$
$313$	4534.39	0.818847	0.409424	+	0.912344i	$0.365730\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2234.77	0.395953	0.197977	−	0.980207i	$-0.436563\pi$
$317$	2234.77	0.395953	0.197977	+	0.980207i	$0.436563\pi$
$318$	0	0
$319$	6428.11	1.12823
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9300.58	1.60216
$324$	0	0
$325$	−1990.66	−0.339761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2408.50	−0.403601
$330$	0	0
$331$	−1841.53	−0.305799	−0.152899	−	0.988242i	$-0.548861\pi$
$331$	−1841.53	−0.305799	−0.152899	+	0.988242i	$0.548861\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−803.318	−0.131015
$336$	0	0
$337$	7034.18	1.13702	0.568511	−	0.822676i	$-0.307520\pi$
$337$	7034.18	1.13702	0.568511	+	0.822676i	$0.307520\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2091.81	−0.332193
$342$	0	0
$343$	6831.03	1.07534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11775.1	−1.82168	−0.910839	−	0.412761i	$-0.864564\pi$
$347$	−11775.1	−1.82168	−0.910839	+	0.412761i	$0.864564\pi$
$348$	0	0
$349$	7898.58	1.21147	0.605733	−	0.795668i	$-0.292880\pi$
$349$	7898.58	1.21147	0.605733	+	0.795668i	$0.292880\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8967.91	1.35216	0.676082	−	0.736826i	$-0.263677\pi$
$353$	8967.91	1.35216	0.676082	+	0.736826i	$0.263677\pi$
$354$	0	0
$355$	−4798.62	−0.717420
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8351.48	1.22778	0.613892	−	0.789390i	$-0.289603\pi$
$359$	8351.48	1.22778	0.613892	+	0.789390i	$0.289603\pi$
$360$	0	0
$361$	14022.0	2.04432
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3817.12	0.547390
$366$	0	0
$367$	110.518	0.0157193	0.00785964	−	0.999969i	$-0.497498\pi$
$367$	110.518	0.0157193	0.00785964	+	0.999969i	$0.497498\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2134.41	−0.298687
$372$	0	0
$373$	4895.94	0.679631	0.339816	−	0.940492i	$-0.389635\pi$
$373$	4895.94	0.679631	0.339816	+	0.940492i	$0.389635\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23649.6	−3.23081
$378$	0	0
$379$	10326.8	1.39962	0.699808	−	0.714331i	$-0.253269\pi$
$379$	10326.8	1.39962	0.699808	+	0.714331i	$0.253269\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8332.52	−1.11168	−0.555838	−	0.831291i	$-0.687602\pi$
$383$	−8332.52	−1.11168	−0.555838	+	0.831291i	$0.687602\pi$
$384$	0	0
$385$	−1486.27	−0.196746
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2071.50	0.269998	0.134999	−	0.990846i	$-0.456897\pi$
$389$	2071.50	0.269998	0.134999	+	0.990846i	$0.456897\pi$
$390$	0	0
$391$	1734.26	0.224310
$392$	0	0
$393$	0	0
$394$	0	0
$395$	319.825	0.0407396
$396$	0	0
$397$	−7891.46	−0.997635	−0.498818	−	0.866707i	$-0.666232\pi$
$397$	−7891.46	−0.997635	−0.498818	+	0.866707i	$0.666232\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1733.64	0.215894	0.107947	−	0.994157i	$-0.465572\pi$
$401$	1733.64	0.215894	0.107947	+	0.994157i	$0.465572\pi$
$402$	0	0
$403$	7695.97	0.951274
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8691.39	1.05852
$408$	0	0
$409$	−9412.60	−1.13795	−0.568977	−	0.822353i	$-0.692661\pi$
$409$	−9412.60	−1.13795	−0.568977	+	0.822353i	$0.692661\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8174.21	−0.973914
$414$	0	0
$415$	4586.63	0.542527
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3868.71	−0.451071	−0.225535	−	0.974235i	$-0.572413\pi$
$419$	−3868.71	−0.451071	−0.225535	+	0.974235i	$0.572413\pi$
$420$	0	0
$421$	12925.0	1.49626	0.748129	−	0.663554i	$-0.230953\pi$
$421$	12925.0	1.49626	0.748129	+	0.663554i	$0.230953\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1609.07	−0.183650
$426$	0	0
$427$	853.002	0.0966737
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13183.6	−1.47339	−0.736697	−	0.676223i	$-0.763616\pi$
$431$	−13183.6	−1.47339	−0.736697	+	0.676223i	$0.763616\pi$
$432$	0	0
$433$	−7584.46	−0.841769	−0.420884	−	0.907114i	$-0.638280\pi$
$433$	−7584.46	−0.841769	−0.420884	+	0.907114i	$0.638280\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3893.63	0.426219
$438$	0	0
$439$	−7546.46	−0.820439	−0.410220	−	0.911987i	$-0.634548\pi$
$439$	−7546.46	−0.820439	−0.410220	+	0.911987i	$0.634548\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2045.04	0.219330	0.109665	−	0.993969i	$-0.465022\pi$
$443$	2045.04	0.219330	0.109665	+	0.993969i	$0.465022\pi$
$444$	0	0
$445$	−6594.40	−0.702482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10723.2	1.12708	0.563541	−	0.826088i	$-0.309439\pi$
$449$	10723.2	1.12708	0.563541	+	0.826088i	$0.309439\pi$
$450$	0	0
$451$	−252.984	−0.0264137
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5468.11	0.563405
$456$	0	0
$457$	−4660.58	−0.477052	−0.238526	−	0.971136i	$-0.576664\pi$
$457$	−4660.58	−0.477052	−0.238526	+	0.971136i	$0.576664\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16640.4	−1.68117	−0.840584	−	0.541681i	$-0.817788\pi$
$461$	−16640.4	−1.68117	−0.840584	+	0.541681i	$0.817788\pi$
$462$	0	0
$463$	−3468.69	−0.348172	−0.174086	−	0.984730i	$-0.555697\pi$
$463$	−3468.69	−0.348172	−0.174086	+	0.984730i	$0.555697\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9920.55	−0.983016	−0.491508	−	0.870873i	$-0.663554\pi$
$467$	−9920.55	−0.983016	−0.491508	+	0.870873i	$0.663554\pi$
$468$	0	0
$469$	2206.62	0.217254
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6325.63	0.614911
$474$	0	0
$475$	−3612.56	−0.348959
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18264.7	1.74224	0.871120	−	0.491070i	$-0.163394\pi$
$479$	18264.7	1.74224	0.871120	+	0.491070i	$0.163394\pi$
$480$	0	0
$481$	−31976.4	−3.03118
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−640.845	−0.0599985
$486$	0	0
$487$	−3244.55	−0.301899	−0.150949	−	0.988542i	$-0.548233\pi$
$487$	−3244.55	−0.301899	−0.150949	+	0.988542i	$0.548233\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−591.093	−0.0543293	−0.0271646	−	0.999631i	$-0.508648\pi$
$491$	−591.093	−0.0543293	−0.0271646	+	0.999631i	$0.508648\pi$
$492$	0	0
$493$	−19116.2	−1.74635
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13181.2	1.18965
$498$	0	0
$499$	−9644.97	−0.865267	−0.432634	−	0.901570i	$-0.642416\pi$
$499$	−9644.97	−0.865267	−0.432634	+	0.901570i	$0.642416\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	723.099	0.0640982	0.0320491	−	0.999486i	$-0.489797\pi$
$503$	723.099	0.0640982	0.0320491	+	0.999486i	$0.489797\pi$
$504$	0	0
$505$	−2302.04	−0.202850
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2684.55	−0.233773	−0.116886	−	0.993145i	$-0.537291\pi$
$509$	−2684.55	−0.233773	−0.116886	+	0.993145i	$0.537291\pi$
$510$	0	0
$511$	−10485.2	−0.907704
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3587.31	0.306943
$516$	0	0
$517$	3795.37	0.322863
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4203.80	−0.353497	−0.176748	−	0.984256i	$-0.556558\pi$
$521$	−4203.80	−0.353497	−0.176748	+	0.984256i	$0.556558\pi$
$522$	0	0
$523$	10571.8	0.883883	0.441942	−	0.897044i	$-0.354290\pi$
$523$	10571.8	0.883883	0.441942	+	0.897044i	$0.354290\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6220.71	0.514190
$528$	0	0
$529$	−11441.0	−0.940327
$530$	0	0
$531$	0	0
$532$	0	0
$533$	930.752	0.0756386
$534$	0	0
$535$	6808.82	0.550226
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3340.95	−0.266985
$540$	0	0
$541$	15310.3	1.21671	0.608356	−	0.793664i	$-0.291829\pi$
$541$	15310.3	1.21671	0.608356	+	0.793664i	$0.291829\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1129.31	0.0887601
$546$	0	0
$547$	−894.345	−0.0699075	−0.0349538	−	0.999389i	$-0.511128\pi$
$547$	−894.345	−0.0699075	−0.0349538	+	0.999389i	$0.511128\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−42918.2	−3.31829
$552$	0	0
$553$	−878.520	−0.0675560
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25371.2	1.93000	0.965001	−	0.262245i	$-0.0844627\pi$
$557$	25371.2	1.93000	0.965001	+	0.262245i	$0.0844627\pi$
$558$	0	0
$559$	−23272.6	−1.76087
$560$	0	0
$561$	0	0
$562$	0	0
$563$	848.897	0.0635466	0.0317733	−	0.999495i	$-0.489885\pi$
$563$	848.897	0.0635466	0.0317733	+	0.999495i	$0.489885\pi$
$564$	0	0
$565$	9842.34	0.732868
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8636.66	−0.636323	−0.318161	−	0.948037i	$-0.603065\pi$
$569$	−8636.66	−0.636323	−0.318161	+	0.948037i	$0.603065\pi$
$570$	0	0
$571$	−7231.98	−0.530033	−0.265017	−	0.964244i	$-0.585377\pi$
$571$	−7231.98	−0.530033	−0.265017	+	0.964244i	$0.585377\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−673.627	−0.0488559
$576$	0	0
$577$	−5882.23	−0.424403	−0.212201	−	0.977226i	$-0.568063\pi$
$577$	−5882.23	−0.424403	−0.212201	+	0.977226i	$0.568063\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12598.9	−0.899640
$582$	0	0
$583$	3363.45	0.238936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19332.8	1.35937	0.679684	−	0.733505i	$-0.262117\pi$
$587$	19332.8	1.35937	0.679684	+	0.733505i	$0.262117\pi$
$588$	0	0
$589$	13966.3	0.977029
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4318.26	−0.299039	−0.149519	−	0.988759i	$-0.547773\pi$
$593$	−4318.26	−0.299039	−0.149519	+	0.988759i	$0.547773\pi$
$594$	0	0
$595$	4419.92	0.304536
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15381.6	1.04921	0.524605	−	0.851346i	$-0.324213\pi$
$599$	15381.6	1.04921	0.524605	+	0.851346i	$0.324213\pi$
$600$	0	0
$601$	4172.55	0.283198	0.141599	−	0.989924i	$-0.454776\pi$
$601$	4172.55	0.283198	0.141599	+	0.989924i	$0.454776\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4312.90	−0.289825
$606$	0	0
$607$	692.820	0.0463274	0.0231637	−	0.999732i	$-0.492626\pi$
$607$	692.820	0.0463274	0.0231637	+	0.999732i	$0.492626\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13963.5	−0.924555
$612$	0	0
$613$	11649.8	0.767584	0.383792	−	0.923419i	$-0.374618\pi$
$613$	11649.8	0.767584	0.383792	+	0.923419i	$0.374618\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21787.2	−1.42159	−0.710793	−	0.703402i	$-0.751664\pi$
$617$	−21787.2	−1.42159	−0.710793	+	0.703402i	$0.751664\pi$
$618$	0	0
$619$	−4970.00	−0.322716	−0.161358	−	0.986896i	$-0.551587\pi$
$619$	−4970.00	−0.322716	−0.161358	+	0.986896i	$0.551587\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18114.0	1.16488
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25846.8	−1.63844
$630$	0	0
$631$	−16907.2	−1.06666	−0.533331	−	0.845906i	$-0.679060\pi$
$631$	−16907.2	−1.06666	−0.533331	+	0.845906i	$0.679060\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3756.02	0.234729
$636$	0	0
$637$	12291.7	0.764542
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21469.9	1.32295	0.661474	−	0.749968i	$-0.269931\pi$
$641$	21469.9	1.32295	0.661474	+	0.749968i	$0.269931\pi$
$642$	0	0
$643$	−17712.4	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$643$	−17712.4	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12632.9	0.767619	0.383809	−	0.923412i	$-0.374612\pi$
$647$	12632.9	0.767619	0.383809	+	0.923412i	$0.374612\pi$
$648$	0	0
$649$	12881.1	0.779088
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16170.2	0.969047	0.484523	−	0.874778i	$-0.338993\pi$
$653$	16170.2	0.969047	0.484523	+	0.874778i	$0.338993\pi$
$654$	0	0
$655$	−397.899	−0.0237362
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7752.76	0.458277	0.229139	−	0.973394i	$-0.426409\pi$
$659$	7752.76	0.458277	0.229139	+	0.973394i	$0.426409\pi$
$660$	0	0
$661$	11693.5	0.688087	0.344044	−	0.938954i	$-0.388203\pi$
$661$	11693.5	0.688087	0.344044	+	0.938954i	$0.388203\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9923.28	0.578659
$666$	0	0
$667$	−8002.85	−0.464575
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1344.18	−0.0773347
$672$	0	0
$673$	−4118.91	−0.235917	−0.117959	−	0.993019i	$-0.537635\pi$
$673$	−4118.91	−0.235917	−0.117959	+	0.993019i	$0.537635\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13603.9	−0.772293	−0.386146	−	0.922438i	$-0.626194\pi$
$677$	−13603.9	−0.772293	−0.386146	+	0.922438i	$0.626194\pi$
$678$	0	0
$679$	1760.32	0.0994920
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28073.3	1.57276	0.786381	−	0.617742i	$-0.211952\pi$
$683$	28073.3	1.57276	0.786381	+	0.617742i	$0.211952\pi$
$684$	0	0
$685$	4017.92	0.224112
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12374.4	−0.684221
$690$	0	0
$691$	−15917.5	−0.876311	−0.438155	−	0.898899i	$-0.644368\pi$
$691$	−15917.5	−0.876311	−0.438155	+	0.898899i	$0.644368\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8808.33	0.480746
$696$	0	0
$697$	752.334	0.0408848
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17451.6	0.940280	0.470140	−	0.882592i	$-0.344203\pi$
$701$	17451.6	0.940280	0.470140	+	0.882592i	$0.344203\pi$
$702$	0	0
$703$	−58029.3	−3.11325
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6323.42	0.336375
$708$	0	0
$709$	30178.3	1.59855	0.799273	−	0.600968i	$-0.205218\pi$
$709$	30178.3	1.59855	0.799273	+	0.600968i	$0.205218\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2604.26	0.136789
$714$	0	0
$715$	−8616.79	−0.450699
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7009.26	−0.363562	−0.181781	−	0.983339i	$-0.558186\pi$
$719$	−7009.26	−0.363562	−0.181781	+	0.983339i	$0.558186\pi$
$720$	0	0
$721$	−9853.90	−0.508985
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7425.16	0.380363
$726$	0	0
$727$	−31166.0	−1.58994	−0.794968	−	0.606651i	$-0.792513\pi$
$727$	−31166.0	−1.58994	−0.794968	+	0.606651i	$0.792513\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18811.4	−0.951798
$732$	0	0
$733$	32076.1	1.61632	0.808158	−	0.588966i	$-0.200465\pi$
$733$	32076.1	1.61632	0.808158	+	0.588966i	$0.200465\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3477.24	−0.173794
$738$	0	0
$739$	−35798.6	−1.78197	−0.890984	−	0.454035i	$-0.849984\pi$
$739$	−35798.6	−1.78197	−0.890984	+	0.454035i	$0.849984\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1043.92	−0.0515449	−0.0257724	−	0.999668i	$-0.508205\pi$
$743$	−1043.92	−0.0515449	−0.0257724	+	0.999668i	$0.508205\pi$
$744$	0	0
$745$	−12049.6	−0.592566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18703.0	−0.912407
$750$	0	0
$751$	−40244.7	−1.95546	−0.977731	−	0.209864i	$-0.932698\pi$
$751$	−40244.7	−1.95546	−0.977731	+	0.209864i	$0.932698\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9238.39	0.445324
$756$	0	0
$757$	1971.31	0.0946480	0.0473240	−	0.998880i	$-0.484931\pi$
$757$	1971.31	0.0946480	0.0473240	+	0.998880i	$0.484931\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14876.5	−0.708635	−0.354317	−	0.935125i	$-0.615287\pi$
$761$	−14876.5	−0.708635	−0.354317	+	0.935125i	$0.615287\pi$
$762$	0	0
$763$	−3102.08	−0.147186
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−47390.8	−2.23101
$768$	0	0
$769$	2284.74	0.107139	0.0535695	−	0.998564i	$-0.482940\pi$
$769$	2284.74	0.107139	0.0535695	+	0.998564i	$0.482940\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15649.7	0.728177	0.364089	−	0.931364i	$-0.381380\pi$
$773$	15649.7	0.728177	0.364089	+	0.931364i	$0.381380\pi$
$774$	0	0
$775$	−2416.27	−0.111993
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1689.09	0.0776865
$780$	0	0
$781$	−20771.3	−0.951671
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−430.700	−0.0195826
$786$	0	0
$787$	−30420.7	−1.37786	−0.688932	−	0.724826i	$-0.741920\pi$
$787$	−30420.7	−1.37786	−0.688932	+	0.724826i	$0.741920\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27035.7	−1.21527
$792$	0	0
$793$	4945.37	0.221457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18616.0	−0.827367	−0.413683	−	0.910421i	$-0.635758\pi$
$797$	−18616.0	−0.827367	−0.413683	+	0.910421i	$0.635758\pi$
$798$	0	0
$799$	−11286.8	−0.499748
$800$	0	0
$801$	0	0
$802$	0	0
$803$	16522.8	0.726123
$804$	0	0
$805$	1850.37	0.0810149
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42989.2	1.86826	0.934129	−	0.356935i	$-0.116178\pi$
$809$	42989.2	1.86826	0.934129	+	0.356935i	$0.116178\pi$
$810$	0	0
$811$	−28303.4	−1.22548	−0.612741	−	0.790284i	$-0.709933\pi$
$811$	−28303.4	−1.22548	−0.612741	+	0.790284i	$0.709933\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5635.73	−0.242222
$816$	0	0
$817$	−42234.0	−1.80854
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10594.2	0.450353	0.225176	−	0.974318i	$-0.427704\pi$
$821$	10594.2	0.450353	0.225176	+	0.974318i	$0.427704\pi$
$822$	0	0
$823$	−24598.3	−1.04185	−0.520925	−	0.853602i	$-0.674413\pi$
$823$	−24598.3	−1.04185	−0.520925	+	0.853602i	$0.674413\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15104.9	−0.635126	−0.317563	−	0.948237i	$-0.602865\pi$
$827$	−15104.9	−0.635126	−0.317563	+	0.948237i	$0.602865\pi$
$828$	0	0
$829$	−33824.6	−1.41710	−0.708551	−	0.705660i	$-0.750651\pi$
$829$	−33824.6	−1.41710	−0.708551	+	0.705660i	$0.750651\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9935.45	0.413257
$834$	0	0
$835$	−4843.22	−0.200727
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5366.62	0.220830	0.110415	−	0.993886i	$-0.464782\pi$
$839$	5366.62	0.220830	0.110415	+	0.993886i	$0.464782\pi$
$840$	0	0
$841$	63823.8	2.61691
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20717.0	0.843414
$846$	0	0
$847$	11847.0	0.480600
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10820.6	−0.435870
$852$	0	0
$853$	16383.3	0.657623	0.328812	−	0.944396i	$-0.393352\pi$
$853$	16383.3	0.657623	0.328812	+	0.944396i	$0.393352\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28118.4	−1.12078	−0.560389	−	0.828230i	$-0.689348\pi$
$857$	−28118.4	−1.12078	−0.560389	+	0.828230i	$0.689348\pi$
$858$	0	0
$859$	32222.9	1.27990	0.639949	−	0.768417i	$-0.278955\pi$
$859$	32222.9	1.27990	0.639949	+	0.768417i	$0.278955\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37447.4	1.47709	0.738543	−	0.674206i	$-0.235514\pi$
$863$	37447.4	1.47709	0.738543	+	0.674206i	$0.235514\pi$
$864$	0	0
$865$	8760.37	0.344349
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1384.39	0.0540418
$870$	0	0
$871$	12793.1	0.497678
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1716.80	−0.0663296
$876$	0	0
$877$	−7833.73	−0.301626	−0.150813	−	0.988562i	$-0.548189\pi$
$877$	−7833.73	−0.301626	−0.150813	+	0.988562i	$0.548189\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3195.87	−0.122215	−0.0611076	−	0.998131i	$-0.519463\pi$
$881$	−3195.87	−0.122215	−0.0611076	+	0.998131i	$0.519463\pi$
$882$	0	0
$883$	6033.49	0.229947	0.114973	−	0.993369i	$-0.463322\pi$
$883$	6033.49	0.229947	0.114973	+	0.993369i	$0.463322\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3479.21	−0.131703	−0.0658515	−	0.997829i	$-0.520976\pi$
$887$	−3479.21	−0.131703	−0.0658515	+	0.997829i	$0.520976\pi$
$888$	0	0
$889$	−10317.3	−0.389238
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25340.3	−0.949587
$894$	0	0
$895$	−4401.49	−0.164386
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28705.9	−1.06496
$900$	0	0
$901$	−10002.3	−0.369841
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1679.23	0.0616791
$906$	0	0
$907$	4383.31	0.160469	0.0802345	−	0.996776i	$-0.474433\pi$
$907$	4383.31	0.160469	0.0802345	+	0.996776i	$0.474433\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4685.36	−0.170398	−0.0851992	−	0.996364i	$-0.527153\pi$
$911$	−4685.36	−0.170398	−0.0851992	+	0.996364i	$0.527153\pi$
$912$	0	0
$913$	19853.7	0.719672
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1092.98	0.0393603
$918$	0	0
$919$	36353.0	1.30487	0.652434	−	0.757846i	$-0.273748\pi$
$919$	36353.0	1.30487	0.652434	+	0.757846i	$0.273748\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	76419.5	2.72522
$924$	0	0
$925$	10039.5	0.356861
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8542.11	0.301677	0.150838	−	0.988558i	$-0.451803\pi$
$929$	8542.11	0.301677	0.150838	+	0.988558i	$0.451803\pi$
$930$	0	0
$931$	22306.3	0.785242
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6965.02	−0.243615
$936$	0	0
$937$	33521.5	1.16873	0.584364	−	0.811491i	$-0.301344\pi$
$937$	33521.5	1.16873	0.584364	+	0.811491i	$0.301344\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24074.0	0.833994	0.416997	−	0.908908i	$-0.363083\pi$
$941$	24074.0	0.833994	0.416997	+	0.908908i	$0.363083\pi$
$942$	0	0
$943$	314.960	0.0108765
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41080.3	−1.40964	−0.704820	−	0.709386i	$-0.748972\pi$
$947$	−41080.3	−1.40964	−0.704820	+	0.709386i	$0.748972\pi$
$948$	0	0
$949$	−60788.8	−2.07934
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28047.0	−0.953337	−0.476668	−	0.879083i	$-0.658156\pi$
$953$	−28047.0	−0.953337	−0.476668	+	0.879083i	$0.658156\pi$
$954$	0	0
$955$	−19886.5	−0.673833
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11036.7	−0.371632
$960$	0	0
$961$	−20449.6	−0.686437
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20795.5	0.693712
$966$	0	0
$967$	−5313.03	−0.176686	−0.0883432	−	0.996090i	$-0.528157\pi$
$967$	−5313.03	−0.176686	−0.0883432	+	0.996090i	$0.528157\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17611.7	0.582065	0.291032	−	0.956713i	$-0.406001\pi$
$971$	17611.7	0.582065	0.291032	+	0.956713i	$0.406001\pi$
$972$	0	0
$973$	−24195.4	−0.797193
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43273.1	−1.41702	−0.708511	−	0.705700i	$-0.750633\pi$
$977$	−43273.1	−1.41702	−0.708511	+	0.705700i	$0.750633\pi$
$978$	0	0
$979$	−28544.5	−0.931856
$980$	0	0
$981$	0	0
$982$	0	0
$983$	46719.7	1.51590	0.757948	−	0.652315i	$-0.226202\pi$
$983$	46719.7	1.51590	0.757948	+	0.652315i	$0.226202\pi$
$984$	0	0
$985$	7905.85	0.255737
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7875.27	−0.253204
$990$	0	0
$991$	−4961.78	−0.159047	−0.0795237	−	0.996833i	$-0.525340\pi$
$991$	−4961.78	−0.159047	−0.0795237	+	0.996833i	$0.525340\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10696.8	−0.340817
$996$	0	0
$997$	29340.1	0.932007	0.466004	−	0.884783i	$-0.345693\pi$
$997$	29340.1	0.932007	0.466004	+	0.884783i	$0.345693\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.l.1.3		7
3.2	odd	2		1620.4.a.k.1.3		7
9.2	odd	6		540.4.i.c.361.5		14
9.4	even	3		180.4.i.c.61.1	✓	14
9.5	odd	6		540.4.i.c.181.5		14
9.7	even	3		180.4.i.c.121.1	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.1	✓	14	9.4	even	3
180.4.i.c.121.1	yes	14	9.7	even	3
540.4.i.c.181.5		14	9.5	odd	6
540.4.i.c.361.5		14	9.2	odd	6
1620.4.a.k.1.3		7	3.2	odd	2
1620.4.a.l.1.3		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -13.7344 q^{7} +21.6430 q^{11} -79.6266 q^{13} -64.3628 q^{17} -144.503 q^{19} -26.9451 q^{23} +25.0000 q^{25} +297.006 q^{29} -96.6507 q^{31} -68.6720 q^{35} +401.580 q^{37} -11.6890 q^{41} +292.271 q^{43} +175.362 q^{47} -154.366 q^{49} +155.406 q^{53} +108.215 q^{55} +595.163 q^{59} -62.1070 q^{61} -398.133 q^{65} -160.664 q^{67} -959.723 q^{71} +763.424 q^{73} -297.254 q^{77} +63.9649 q^{79} +917.326 q^{83} -321.814 q^{85} -1318.88 q^{89} +1093.62 q^{91} -722.513 q^{95} -128.169 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 35 q^{5} + 8 q^{7} + 27 q^{11} + 32 q^{13} + 123 q^{17} - 67 q^{19} + 42 q^{23} + 175 q^{25} + 324 q^{29} + 98 q^{31} + 40 q^{35} + 356 q^{37} + 339 q^{41} + 119 q^{43} - 96 q^{47} + 813 q^{49} + 858 q^{53}+ \cdots + 1991 q^{97}+O(q^{100}) \) 7 * q + 35 * q^5 + 8 * q^7 + 27 * q^11 + 32 * q^13 + 123 * q^17 - 67 * q^19 + 42 * q^23 + 175 * q^25 + 324 * q^29 + 98 * q^31 + 40 * q^35 + 356 * q^37 + 339 * q^41 + 119 * q^43 - 96 * q^47 + 813 * q^49 + 858 * q^53 + 135 * q^55 + 549 * q^59 + 260 * q^61 + 160 * q^65 + 881 * q^67 + 342 * q^71 + 737 * q^73 - 456 * q^77 + 1886 * q^79 - 132 * q^83 + 615 * q^85 + 396 * q^89 + 1264 * q^91 - 335 * q^95 + 1991 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more