LMFDB - Embedded newform 1620.4.a.h.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:	$4$
Coefficient field:	4.4.438516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 48 $ x^4 - x^3 - 20*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.75789$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +17.9035 q^{7} -39.5614 q^{11} -23.4839 q^{13} +16.2456 q^{17} -28.3443 q^{19} +105.531 q^{23} +25.0000 q^{25} +99.8442 q^{29} -321.899 q^{31} +89.5174 q^{35} -205.760 q^{37} -327.644 q^{41} +375.061 q^{43} -88.9336 q^{47} -22.4655 q^{49} -85.0314 q^{53} -197.807 q^{55} -796.806 q^{59} -317.936 q^{61} -117.420 q^{65} -525.490 q^{67} +760.854 q^{71} -101.076 q^{73} -708.287 q^{77} -225.633 q^{79} +503.051 q^{83} +81.2278 q^{85} -868.775 q^{89} -420.444 q^{91} -141.722 q^{95} +1749.44 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{5} - 13 q^{7} - 57 q^{11} + 14 q^{13} - 3 q^{17} - 31 q^{19} + 69 q^{23} + 100 q^{25} + 69 q^{29} - 58 q^{31} - 65 q^{35} - 388 q^{37} - 396 q^{41} + 371 q^{43} - 129 q^{47} + 111 q^{49} - 1356 q^{53}+ \cdots - 1495 q^{97}+O(q^{100}) $ 4 * q + 20 * q^5 - 13 * q^7 - 57 * q^11 + 14 * q^13 - 3 * q^17 - 31 * q^19 + 69 * q^23 + 100 * q^25 + 69 * q^29 - 58 * q^31 - 65 * q^35 - 388 * q^37 - 396 * q^41 + 371 * q^43 - 129 * q^47 + 111 * q^49 - 1356 * q^53 - 285 * q^55 - 15 * q^59 - 1441 * q^61 + 70 * q^65 + 368 * q^67 - 168 * q^71 - 955 * q^73 - 342 * q^77 - 1408 * q^79 + 789 * q^83 - 15 * q^85 - 1617 * q^89 - 1406 * q^91 - 155 * q^95 - 1495 * 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
$7$	17.9035	0.966697	0.483348	−	0.875428i	$-0.339420\pi$
$7$	17.9035	0.966697	0.483348	+	0.875428i	$0.339420\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−39.5614	−1.08438	−0.542191	−	0.840255i	$-0.682405\pi$
$11$	−39.5614	−1.08438	−0.542191	+	0.840255i	$0.682405\pi$
$12$	0	0
$13$	−23.4839	−0.501020	−0.250510	−	0.968114i	$-0.580598\pi$
$13$	−23.4839	−0.501020	−0.250510	+	0.968114i	$0.580598\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16.2456	0.231772	0.115886	−	0.993263i	$-0.463029\pi$
$17$	16.2456	0.231772	0.115886	+	0.993263i	$0.463029\pi$
$18$	0	0
$19$	−28.3443	−0.342244	−0.171122	−	0.985250i	$-0.554739\pi$
$19$	−28.3443	−0.342244	−0.171122	+	0.985250i	$0.554739\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	105.531	0.956728	0.478364	−	0.878162i	$-0.341230\pi$
$23$	105.531	0.956728	0.478364	+	0.878162i	$0.341230\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	99.8442	0.639331	0.319666	−	0.947530i	$-0.396429\pi$
$29$	99.8442	0.639331	0.319666	+	0.947530i	$0.396429\pi$
$30$	0	0
$31$	−321.899	−1.86499	−0.932496	−	0.361180i	$-0.882374\pi$
$31$	−321.899	−1.86499	−0.932496	+	0.361180i	$0.882374\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	89.5174	0.432320
$36$	0	0
$37$	−205.760	−0.914237	−0.457119	−	0.889406i	$-0.651118\pi$
$37$	−205.760	−0.914237	−0.457119	+	0.889406i	$0.651118\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−327.644	−1.24803	−0.624017	−	0.781411i	$-0.714500\pi$
$41$	−327.644	−1.24803	−0.624017	+	0.781411i	$0.714500\pi$
$42$	0	0
$43$	375.061	1.33015	0.665073	−	0.746778i	$-0.268400\pi$
$43$	375.061	1.33015	0.665073	+	0.746778i	$0.268400\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−88.9336	−0.276006	−0.138003	−	0.990432i	$-0.544068\pi$
$47$	−88.9336	−0.276006	−0.138003	+	0.990432i	$0.544068\pi$
$48$	0	0
$49$	−22.4655	−0.0654972
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−85.0314	−0.220377	−0.110188	−	0.993911i	$-0.535145\pi$
$53$	−85.0314	−0.220377	−0.110188	+	0.993911i	$0.535145\pi$
$54$	0	0
$55$	−197.807	−0.484951
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−796.806	−1.75823	−0.879113	−	0.476614i	$-0.841864\pi$
$59$	−796.806	−1.75823	−0.879113	+	0.476614i	$0.841864\pi$
$60$	0	0
$61$	−317.936	−0.667337	−0.333669	−	0.942690i	$-0.608287\pi$
$61$	−317.936	−0.667337	−0.333669	+	0.942690i	$0.608287\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−117.420	−0.224063
$66$	0	0
$67$	−525.490	−0.958192	−0.479096	−	0.877763i	$-0.659035\pi$
$67$	−525.490	−0.958192	−0.479096	+	0.877763i	$0.659035\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	760.854	1.27179	0.635893	−	0.771778i	$-0.280632\pi$
$71$	760.854	1.27179	0.635893	+	0.771778i	$0.280632\pi$
$72$	0	0
$73$	−101.076	−0.162056	−0.0810279	−	0.996712i	$-0.525820\pi$
$73$	−101.076	−0.162056	−0.0810279	+	0.996712i	$0.525820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−708.287	−1.04827
$78$	0	0
$79$	−225.633	−0.321338	−0.160669	−	0.987008i	$-0.551365\pi$
$79$	−225.633	−0.321338	−0.160669	+	0.987008i	$0.551365\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	503.051	0.665266	0.332633	−	0.943056i	$-0.392063\pi$
$83$	503.051	0.665266	0.332633	+	0.943056i	$0.392063\pi$
$84$	0	0
$85$	81.2278	0.103652
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−868.775	−1.03472	−0.517359	−	0.855768i	$-0.673085\pi$
$89$	−868.775	−1.03472	−0.517359	+	0.855768i	$0.673085\pi$
$90$	0	0
$91$	−420.444	−0.484335
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−141.722	−0.153056
$96$	0	0
$97$	1749.44	1.83123	0.915613	−	0.402060i	$-0.131706\pi$
$97$	1749.44	1.83123	0.915613	+	0.402060i	$0.131706\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1294.16	1.27498	0.637492	−	0.770457i	$-0.279972\pi$
$101$	1294.16	1.27498	0.637492	+	0.770457i	$0.279972\pi$
$102$	0	0
$103$	191.807	0.183488	0.0917441	−	0.995783i	$-0.470756\pi$
$103$	191.807	0.183488	0.0917441	+	0.995783i	$0.470756\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1417.70	1.28089	0.640443	−	0.768006i	$-0.278751\pi$
$107$	1417.70	1.28089	0.640443	+	0.768006i	$0.278751\pi$
$108$	0	0
$109$	129.747	0.114014	0.0570071	−	0.998374i	$-0.481844\pi$
$109$	129.747	0.114014	0.0570071	+	0.998374i	$0.481844\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−830.855	−0.691684	−0.345842	−	0.938293i	$-0.612407\pi$
$113$	−830.855	−0.691684	−0.345842	+	0.938293i	$0.612407\pi$
$114$	0	0
$115$	527.655	0.427862
$116$	0	0
$117$	0	0
$118$	0	0
$119$	290.852	0.224053
$120$	0	0
$121$	234.104	0.175886
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2385.67	−1.66688	−0.833441	−	0.552608i	$-0.813633\pi$
$127$	−2385.67	−1.66688	−0.833441	+	0.552608i	$0.813633\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	904.316	0.603133	0.301567	−	0.953445i	$-0.402490\pi$
$131$	904.316	0.603133	0.301567	+	0.953445i	$0.402490\pi$
$132$	0	0
$133$	−507.462	−0.330846
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−808.364	−0.504111	−0.252056	−	0.967713i	$-0.581107\pi$
$137$	−808.364	−0.504111	−0.252056	+	0.967713i	$0.581107\pi$
$138$	0	0
$139$	−2864.72	−1.74807	−0.874037	−	0.485859i	$-0.838507\pi$
$139$	−2864.72	−1.74807	−0.874037	+	0.485859i	$0.838507\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	929.056	0.543298
$144$	0	0
$145$	499.221	0.285918
$146$	0	0
$147$	0	0
$148$	0	0
$149$	679.411	0.373554	0.186777	−	0.982402i	$-0.440196\pi$
$149$	679.411	0.373554	0.186777	+	0.982402i	$0.440196\pi$
$150$	0	0
$151$	−2386.23	−1.28602	−0.643008	−	0.765859i	$-0.722314\pi$
$151$	−2386.23	−1.28602	−0.643008	+	0.765859i	$0.722314\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1609.49	−0.834050
$156$	0	0
$157$	−2698.35	−1.37167	−0.685833	−	0.727759i	$-0.740562\pi$
$157$	−2698.35	−1.37167	−0.685833	+	0.727759i	$0.740562\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1889.37	0.924866
$162$	0	0
$163$	1500.47	0.721017	0.360509	−	0.932756i	$-0.382603\pi$
$163$	1500.47	0.721017	0.360509	+	0.932756i	$0.382603\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3429.43	1.58909	0.794543	−	0.607208i	$-0.207710\pi$
$167$	3429.43	1.58909	0.794543	+	0.607208i	$0.207710\pi$
$168$	0	0
$169$	−1645.51	−0.748979
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2007.94	0.882435	0.441217	−	0.897400i	$-0.354547\pi$
$173$	2007.94	0.882435	0.441217	+	0.897400i	$0.354547\pi$
$174$	0	0
$175$	447.587	0.193339
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3836.95	−1.60216	−0.801082	−	0.598555i	$-0.795742\pi$
$179$	−3836.95	−1.60216	−0.801082	+	0.598555i	$0.795742\pi$
$180$	0	0
$181$	−1822.10	−0.748263	−0.374131	−	0.927376i	$-0.622059\pi$
$181$	−1822.10	−0.748263	−0.374131	+	0.927376i	$0.622059\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1028.80	−0.408859
$186$	0	0
$187$	−642.697	−0.251330
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4195.97	−1.58958	−0.794790	−	0.606885i	$-0.792419\pi$
$191$	−4195.97	−1.58958	−0.794790	+	0.606885i	$0.792419\pi$
$192$	0	0
$193$	−1810.36	−0.675194	−0.337597	−	0.941291i	$-0.609614\pi$
$193$	−1810.36	−0.675194	−0.337597	+	0.941291i	$0.609614\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3827.21	−1.38415	−0.692074	−	0.721826i	$-0.743303\pi$
$197$	−3827.21	−1.38415	−0.692074	+	0.721826i	$0.743303\pi$
$198$	0	0
$199$	−477.853	−0.170222	−0.0851108	−	0.996371i	$-0.527124\pi$
$199$	−477.853	−0.170222	−0.0851108	+	0.996371i	$0.527124\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1787.56	0.618039
$204$	0	0
$205$	−1638.22	−0.558137
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1121.34	0.371123
$210$	0	0
$211$	−216.265	−0.0705607	−0.0352803	−	0.999377i	$-0.511232\pi$
$211$	−216.265	−0.0705607	−0.0352803	+	0.999377i	$0.511232\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1875.31	0.594860
$216$	0	0
$217$	−5763.11	−1.80288
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−381.509	−0.116123
$222$	0	0
$223$	−5380.77	−1.61580	−0.807899	−	0.589321i	$-0.799395\pi$
$223$	−5380.77	−1.61580	−0.807899	+	0.589321i	$0.799395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	554.590	0.162156	0.0810780	−	0.996708i	$-0.474164\pi$
$227$	554.590	0.162156	0.0810780	+	0.996708i	$0.474164\pi$
$228$	0	0
$229$	3410.67	0.984206	0.492103	−	0.870537i	$-0.336228\pi$
$229$	3410.67	0.984206	0.492103	+	0.870537i	$0.336228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2640.60	−0.742452	−0.371226	−	0.928543i	$-0.621062\pi$
$233$	−2640.60	−0.742452	−0.371226	+	0.928543i	$0.621062\pi$
$234$	0	0
$235$	−444.668	−0.123434
$236$	0	0
$237$	0	0
$238$	0	0
$239$	182.951	0.0495151	0.0247575	−	0.999693i	$-0.492119\pi$
$239$	182.951	0.0495151	0.0247575	+	0.999693i	$0.492119\pi$
$240$	0	0
$241$	−7433.77	−1.98693	−0.993467	−	0.114117i	$-0.963596\pi$
$241$	−7433.77	−1.98693	−0.993467	+	0.114117i	$0.963596\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−112.328	−0.0292912
$246$	0	0
$247$	665.635	0.171471
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2170.36	0.545785	0.272893	−	0.962045i	$-0.412020\pi$
$251$	2170.36	0.545785	0.272893	+	0.962045i	$0.412020\pi$
$252$	0	0
$253$	−4174.96	−1.03746
$254$	0	0
$255$	0	0
$256$	0	0
$257$	78.3663	0.0190209	0.00951043	−	0.999955i	$-0.496973\pi$
$257$	78.3663	0.0190209	0.00951043	+	0.999955i	$0.496973\pi$
$258$	0	0
$259$	−3683.82	−0.883790
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1216.81	−0.285291	−0.142646	−	0.989774i	$-0.545561\pi$
$263$	−1216.81	−0.285291	−0.142646	+	0.989774i	$0.545561\pi$
$264$	0	0
$265$	−425.157	−0.0985555
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1249.44	−0.283196	−0.141598	−	0.989924i	$-0.545224\pi$
$269$	−1249.44	−0.283196	−0.141598	+	0.989924i	$0.545224\pi$
$270$	0	0
$271$	7675.60	1.72051	0.860257	−	0.509860i	$-0.170303\pi$
$271$	7675.60	1.72051	0.860257	+	0.509860i	$0.170303\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−989.035	−0.216877
$276$	0	0
$277$	731.747	0.158723	0.0793617	−	0.996846i	$-0.474712\pi$
$277$	731.747	0.158723	0.0793617	+	0.996846i	$0.474712\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1129.95	−0.239883	−0.119942	−	0.992781i	$-0.538271\pi$
$281$	−1129.95	−0.239883	−0.119942	+	0.992781i	$0.538271\pi$
$282$	0	0
$283$	−2451.26	−0.514883	−0.257442	−	0.966294i	$-0.582880\pi$
$283$	−2451.26	−0.514883	−0.257442	+	0.966294i	$0.582880\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5865.96	−1.20647
$288$	0	0
$289$	−4649.08	−0.946282
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6245.86	1.24535	0.622674	−	0.782481i	$-0.286046\pi$
$293$	6245.86	1.24535	0.622674	+	0.782481i	$0.286046\pi$
$294$	0	0
$295$	−3984.03	−0.786302
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2478.28	−0.479340
$300$	0	0
$301$	6714.90	1.28585
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1589.68	−0.298442
$306$	0	0
$307$	10638.4	1.97774	0.988870	−	0.148782i	$-0.0475354\pi$
$307$	10638.4	1.97774	0.988870	+	0.148782i	$0.0475354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	37.2693	0.00679533	0.00339767	−	0.999994i	$-0.498918\pi$
$311$	37.2693	0.00679533	0.00339767	+	0.999994i	$0.498918\pi$
$312$	0	0
$313$	947.572	0.171118	0.0855590	−	0.996333i	$-0.472732\pi$
$313$	947.572	0.171118	0.0855590	+	0.996333i	$0.472732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10576.6	1.87395	0.936976	−	0.349394i	$-0.113612\pi$
$317$	10576.6	1.87395	0.936976	+	0.349394i	$0.113612\pi$
$318$	0	0
$319$	−3949.98	−0.693279
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−460.469	−0.0793225
$324$	0	0
$325$	−587.098	−0.100204
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1592.22	−0.266814
$330$	0	0
$331$	−8719.36	−1.44791	−0.723957	−	0.689845i	$-0.757678\pi$
$331$	−8719.36	−1.44791	−0.723957	+	0.689845i	$0.757678\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2627.45	−0.428516
$336$	0	0
$337$	−7773.62	−1.25655	−0.628273	−	0.777993i	$-0.716238\pi$
$337$	−7773.62	−1.25655	−0.628273	+	0.777993i	$0.716238\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12734.8	2.02236
$342$	0	0
$343$	−6543.10	−1.03001
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7089.94	1.09685	0.548427	−	0.836199i	$-0.315227\pi$
$347$	7089.94	1.09685	0.548427	+	0.836199i	$0.315227\pi$
$348$	0	0
$349$	12085.5	1.85364	0.926819	−	0.375509i	$-0.122532\pi$
$349$	12085.5	1.85364	0.926819	+	0.375509i	$0.122532\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8976.95	−1.35353	−0.676763	−	0.736201i	$-0.736618\pi$
$353$	−8976.95	−1.35353	−0.676763	+	0.736201i	$0.736618\pi$
$354$	0	0
$355$	3804.27	0.568760
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2038.36	0.299667	0.149834	−	0.988711i	$-0.452126\pi$
$359$	2038.36	0.299667	0.149834	+	0.988711i	$0.452126\pi$
$360$	0	0
$361$	−6055.60	−0.882869
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−505.381	−0.0724736
$366$	0	0
$367$	7047.82	1.00243	0.501217	−	0.865322i	$-0.332886\pi$
$367$	7047.82	1.00243	0.501217	+	0.865322i	$0.332886\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1522.36	−0.213037
$372$	0	0
$373$	−4285.50	−0.594892	−0.297446	−	0.954739i	$-0.596135\pi$
$373$	−4285.50	−0.594892	−0.297446	+	0.954739i	$0.596135\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2344.73	−0.320318
$378$	0	0
$379$	−1366.86	−0.185253	−0.0926267	−	0.995701i	$-0.529526\pi$
$379$	−1366.86	−0.185253	−0.0926267	+	0.995701i	$0.529526\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12454.8	−1.66164	−0.830822	−	0.556537i	$-0.812130\pi$
$383$	−12454.8	−1.66164	−0.830822	+	0.556537i	$0.812130\pi$
$384$	0	0
$385$	−3541.43	−0.468800
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−436.562	−0.0569012	−0.0284506	−	0.999595i	$-0.509057\pi$
$389$	−436.562	−0.0569012	−0.0284506	+	0.999595i	$0.509057\pi$
$390$	0	0
$391$	1714.41	0.221743
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1128.17	−0.143707
$396$	0	0
$397$	−7117.74	−0.899822	−0.449911	−	0.893073i	$-0.648544\pi$
$397$	−7117.74	−0.899822	−0.449911	+	0.893073i	$0.648544\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7769.47	−0.967553	−0.483777	−	0.875192i	$-0.660735\pi$
$401$	−7769.47	−0.967553	−0.483777	+	0.875192i	$0.660735\pi$
$402$	0	0
$403$	7559.45	0.934399
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8140.16	0.991383
$408$	0	0
$409$	−12832.6	−1.55142	−0.775711	−	0.631089i	$-0.782608\pi$
$409$	−12832.6	−1.55142	−0.775711	+	0.631089i	$0.782608\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14265.6	−1.69967
$414$	0	0
$415$	2515.26	0.297516
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14903.2	1.73764	0.868818	−	0.495132i	$-0.164880\pi$
$419$	14903.2	1.73764	0.868818	+	0.495132i	$0.164880\pi$
$420$	0	0
$421$	−1218.77	−0.141091	−0.0705456	−	0.997509i	$-0.522474\pi$
$421$	−1218.77	−0.141091	−0.0705456	+	0.997509i	$0.522474\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	406.139	0.0463544
$426$	0	0
$427$	−5692.17	−0.645113
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9989.48	1.11642	0.558209	−	0.829700i	$-0.311489\pi$
$431$	9989.48	1.11642	0.558209	+	0.829700i	$0.311489\pi$
$432$	0	0
$433$	−6010.34	−0.667063	−0.333532	−	0.942739i	$-0.608240\pi$
$433$	−6010.34	−0.667063	−0.333532	+	0.942739i	$0.608240\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2991.20	−0.327434
$438$	0	0
$439$	10234.7	1.11270	0.556348	−	0.830949i	$-0.312202\pi$
$439$	10234.7	1.11270	0.556348	+	0.830949i	$0.312202\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4958.56	−0.531802	−0.265901	−	0.964000i	$-0.585669\pi$
$443$	−4958.56	−0.531802	−0.265901	+	0.964000i	$0.585669\pi$
$444$	0	0
$445$	−4343.88	−0.462740
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7715.87	0.810989	0.405495	−	0.914097i	$-0.367099\pi$
$449$	7715.87	0.810989	0.405495	+	0.914097i	$0.367099\pi$
$450$	0	0
$451$	12962.0	1.35335
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2102.22	−0.216601
$456$	0	0
$457$	−491.734	−0.0503333	−0.0251667	−	0.999683i	$-0.508012\pi$
$457$	−491.734	−0.0503333	−0.0251667	+	0.999683i	$0.508012\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3213.49	−0.324658	−0.162329	−	0.986737i	$-0.551901\pi$
$461$	−3213.49	−0.324658	−0.162329	+	0.986737i	$0.551901\pi$
$462$	0	0
$463$	3324.70	0.333719	0.166860	−	0.985981i	$-0.446637\pi$
$463$	3324.70	0.333719	0.166860	+	0.985981i	$0.446637\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11322.6	−1.12194	−0.560972	−	0.827835i	$-0.689572\pi$
$467$	−11322.6	−1.12194	−0.560972	+	0.827835i	$0.689572\pi$
$468$	0	0
$469$	−9408.10	−0.926281
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14837.9	−1.44239
$474$	0	0
$475$	−708.608	−0.0684487
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7182.32	0.685112	0.342556	−	0.939497i	$-0.388707\pi$
$479$	7182.32	0.685112	0.342556	+	0.939497i	$0.388707\pi$
$480$	0	0
$481$	4832.05	0.458051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8747.21	0.818949
$486$	0	0
$487$	−242.899	−0.0226012	−0.0113006	−	0.999936i	$-0.503597\pi$
$487$	−242.899	−0.0226012	−0.0113006	+	0.999936i	$0.503597\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3491.36	−0.320902	−0.160451	−	0.987044i	$-0.551295\pi$
$491$	−3491.36	−0.320902	−0.160451	+	0.987044i	$0.551295\pi$
$492$	0	0
$493$	1622.02	0.148179
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13621.9	1.22943
$498$	0	0
$499$	−714.219	−0.0640738	−0.0320369	−	0.999487i	$-0.510199\pi$
$499$	−714.219	−0.0640738	−0.0320369	+	0.999487i	$0.510199\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18948.7	1.67968	0.839842	−	0.542832i	$-0.182648\pi$
$503$	18948.7	1.67968	0.839842	+	0.542832i	$0.182648\pi$
$504$	0	0
$505$	6470.78	0.570190
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11841.3	−1.03115	−0.515574	−	0.856845i	$-0.672421\pi$
$509$	−11841.3	−1.03115	−0.515574	+	0.856845i	$0.672421\pi$
$510$	0	0
$511$	−1809.62	−0.156659
$512$	0	0
$513$	0	0
$514$	0	0
$515$	959.034	0.0820584
$516$	0	0
$517$	3518.34	0.299296
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21836.5	1.83623	0.918114	−	0.396317i	$-0.129712\pi$
$521$	21836.5	1.83623	0.918114	+	0.396317i	$0.129712\pi$
$522$	0	0
$523$	6595.18	0.551409	0.275705	−	0.961242i	$-0.411089\pi$
$523$	6595.18	0.551409	0.275705	+	0.961242i	$0.411089\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5229.43	−0.432253
$528$	0	0
$529$	−1030.19	−0.0846708
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7694.36	0.625290
$534$	0	0
$535$	7088.52	0.572829
$536$	0	0
$537$	0	0
$538$	0	0
$539$	888.768	0.0710240
$540$	0	0
$541$	−7877.56	−0.626031	−0.313015	−	0.949748i	$-0.601339\pi$
$541$	−7877.56	−0.626031	−0.313015	+	0.949748i	$0.601339\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	648.737	0.0509887
$546$	0	0
$547$	15421.0	1.20540	0.602702	−	0.797966i	$-0.294091\pi$
$547$	15421.0	1.20540	0.602702	+	0.797966i	$0.294091\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2830.01	−0.218807
$552$	0	0
$553$	−4039.62	−0.310637
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17146.1	−1.30431	−0.652156	−	0.758085i	$-0.726135\pi$
$557$	−17146.1	−1.30431	−0.652156	+	0.758085i	$0.726135\pi$
$558$	0	0
$559$	−8807.90	−0.666430
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13330.5	−0.997892	−0.498946	−	0.866633i	$-0.666279\pi$
$563$	−13330.5	−0.997892	−0.498946	+	0.866633i	$0.666279\pi$
$564$	0	0
$565$	−4154.27	−0.309330
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18582.1	1.36907	0.684536	−	0.728979i	$-0.260005\pi$
$569$	18582.1	1.36907	0.684536	+	0.728979i	$0.260005\pi$
$570$	0	0
$571$	5572.92	0.408440	0.204220	−	0.978925i	$-0.434534\pi$
$571$	5572.92	0.408440	0.204220	+	0.978925i	$0.434534\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2638.28	0.191346
$576$	0	0
$577$	−11047.6	−0.797083	−0.398542	−	0.917150i	$-0.630484\pi$
$577$	−11047.6	−0.797083	−0.398542	+	0.917150i	$0.630484\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9006.37	0.643111
$582$	0	0
$583$	3363.96	0.238973
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.2551	−0.00142422	−0.000712110	−	1.00000i	$-0.500227\pi$
$587$	−20.2551	−0.00142422	−0.000712110		1.00000i	$0.500227\pi$
$588$	0	0
$589$	9124.00	0.638282
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21868.4	−1.51438	−0.757190	−	0.653194i	$-0.773428\pi$
$593$	−21868.4	−1.51438	−0.757190	+	0.653194i	$0.773428\pi$
$594$	0	0
$595$	1454.26	0.100200
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9359.66	0.638439	0.319220	−	0.947681i	$-0.396579\pi$
$599$	9359.66	0.638439	0.319220	+	0.947681i	$0.396579\pi$
$600$	0	0
$601$	1007.30	0.0683668	0.0341834	−	0.999416i	$-0.489117\pi$
$601$	1007.30	0.0683668	0.0341834	+	0.999416i	$0.489117\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1170.52	0.0786585
$606$	0	0
$607$	−6975.60	−0.466443	−0.233222	−	0.972424i	$-0.574927\pi$
$607$	−6975.60	−0.466443	−0.233222	+	0.972424i	$0.574927\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2088.51	0.138285
$612$	0	0
$613$	25416.8	1.67468	0.837338	−	0.546686i	$-0.184111\pi$
$613$	25416.8	1.67468	0.837338	+	0.546686i	$0.184111\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9850.85	0.642755	0.321378	−	0.946951i	$-0.395854\pi$
$617$	9850.85	0.642755	0.321378	+	0.946951i	$0.395854\pi$
$618$	0	0
$619$	17017.0	1.10496	0.552481	−	0.833526i	$-0.313681\pi$
$619$	17017.0	1.10496	0.552481	+	0.833526i	$0.313681\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15554.1	−1.00026
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3342.69	−0.211895
$630$	0	0
$631$	6738.60	0.425134	0.212567	−	0.977147i	$-0.431818\pi$
$631$	6738.60	0.425134	0.212567	+	0.977147i	$0.431818\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11928.4	−0.745452
$636$	0	0
$637$	527.579	0.0328154
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6290.15	0.387591	0.193796	−	0.981042i	$-0.437920\pi$
$641$	6290.15	0.387591	0.193796	+	0.981042i	$0.437920\pi$
$642$	0	0
$643$	8883.66	0.544848	0.272424	−	0.962177i	$-0.412175\pi$
$643$	8883.66	0.544848	0.272424	+	0.962177i	$0.412175\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13436.4	0.816444	0.408222	−	0.912883i	$-0.366149\pi$
$647$	13436.4	0.816444	0.408222	+	0.912883i	$0.366149\pi$
$648$	0	0
$649$	31522.8	1.90659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17166.0	1.02873	0.514363	−	0.857572i	$-0.328028\pi$
$653$	17166.0	1.02873	0.514363	+	0.857572i	$0.328028\pi$
$654$	0	0
$655$	4521.58	0.269729
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20418.8	−1.20699	−0.603493	−	0.797369i	$-0.706225\pi$
$659$	−20418.8	−1.20699	−0.603493	+	0.797369i	$0.706225\pi$
$660$	0	0
$661$	5244.03	0.308577	0.154288	−	0.988026i	$-0.450692\pi$
$661$	5244.03	0.308577	0.154288	+	0.988026i	$0.450692\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2537.31	−0.147959
$666$	0	0
$667$	10536.7	0.611666
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12578.0	0.723649
$672$	0	0
$673$	14954.5	0.856542	0.428271	−	0.903650i	$-0.359123\pi$
$673$	14954.5	0.856542	0.428271	+	0.903650i	$0.359123\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27051.4	−1.53570	−0.767850	−	0.640629i	$-0.778674\pi$
$677$	−27051.4	−1.53570	−0.767850	+	0.640629i	$0.778674\pi$
$678$	0	0
$679$	31321.1	1.77024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26054.5	−1.45966	−0.729829	−	0.683630i	$-0.760400\pi$
$683$	−26054.5	−1.45966	−0.729829	+	0.683630i	$0.760400\pi$
$684$	0	0
$685$	−4041.82	−0.225445
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1996.87	0.110413
$690$	0	0
$691$	−6295.60	−0.346593	−0.173297	−	0.984870i	$-0.555442\pi$
$691$	−6295.60	−0.346593	−0.173297	+	0.984870i	$0.555442\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14323.6	−0.781763
$696$	0	0
$697$	−5322.76	−0.289259
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4014.82	−0.216316	−0.108158	−	0.994134i	$-0.534495\pi$
$701$	−4014.82	−0.216316	−0.108158	+	0.994134i	$0.534495\pi$
$702$	0	0
$703$	5832.13	0.312892
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23169.9	1.23252
$708$	0	0
$709$	−22598.7	−1.19706	−0.598528	−	0.801102i	$-0.704248\pi$
$709$	−22598.7	−1.19706	−0.598528	+	0.801102i	$0.704248\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−33970.3	−1.78429
$714$	0	0
$715$	4645.28	0.242970
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6579.83	−0.341288	−0.170644	−	0.985333i	$-0.554585\pi$
$719$	−6579.83	−0.341288	−0.170644	+	0.985333i	$0.554585\pi$
$720$	0	0
$721$	3434.01	0.177378
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2496.10	0.127866
$726$	0	0
$727$	33217.2	1.69458	0.847290	−	0.531131i	$-0.178233\pi$
$727$	33217.2	1.69458	0.847290	+	0.531131i	$0.178233\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6093.08	0.308291
$732$	0	0
$733$	16051.2	0.808821	0.404411	−	0.914578i	$-0.367477\pi$
$733$	16051.2	0.808821	0.404411	+	0.914578i	$0.367477\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20789.1	1.03905
$738$	0	0
$739$	14855.1	0.739452	0.369726	−	0.929141i	$-0.379452\pi$
$739$	14855.1	0.739452	0.369726	+	0.929141i	$0.379452\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23494.9	−1.16009	−0.580044	−	0.814585i	$-0.696965\pi$
$743$	−23494.9	−1.16009	−0.580044	+	0.814585i	$0.696965\pi$
$744$	0	0
$745$	3397.05	0.167058
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25381.8	1.23823
$750$	0	0
$751$	8301.68	0.403372	0.201686	−	0.979450i	$-0.435358\pi$
$751$	8301.68	0.403372	0.201686	+	0.979450i	$0.435358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11931.1	−0.575124
$756$	0	0
$757$	7018.98	0.337000	0.168500	−	0.985702i	$-0.446108\pi$
$757$	7018.98	0.337000	0.168500	+	0.985702i	$0.446108\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31808.3	1.51518	0.757589	−	0.652732i	$-0.226377\pi$
$761$	31808.3	1.51518	0.757589	+	0.652732i	$0.226377\pi$
$762$	0	0
$763$	2322.93	0.110217
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18712.1	0.880907
$768$	0	0
$769$	−21932.9	−1.02851	−0.514253	−	0.857639i	$-0.671931\pi$
$769$	−21932.9	−1.02851	−0.514253	+	0.857639i	$0.671931\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25629.2	−1.19252	−0.596260	−	0.802791i	$-0.703347\pi$
$773$	−25629.2	−1.19252	−0.596260	+	0.802791i	$0.703347\pi$
$774$	0	0
$775$	−8047.47	−0.372998
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9286.83	0.427131
$780$	0	0
$781$	−30100.4	−1.37910
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13491.7	−0.613428
$786$	0	0
$787$	29789.3	1.34927	0.674633	−	0.738153i	$-0.264302\pi$
$787$	29789.3	1.34927	0.674633	+	0.738153i	$0.264302\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14875.2	−0.668648
$792$	0	0
$793$	7466.39	0.334350
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13115.4	−0.582901	−0.291450	−	0.956586i	$-0.594138\pi$
$797$	−13115.4	−0.582901	−0.291450	+	0.956586i	$0.594138\pi$
$798$	0	0
$799$	−1444.78	−0.0639706
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3998.72	0.175731
$804$	0	0
$805$	9446.87	0.413613
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9743.58	−0.423444	−0.211722	−	0.977330i	$-0.567907\pi$
$809$	−9743.58	−0.423444	−0.211722	+	0.977330i	$0.567907\pi$
$810$	0	0
$811$	13466.5	0.583075	0.291538	−	0.956559i	$-0.405833\pi$
$811$	13466.5	0.583075	0.291538	+	0.956559i	$0.405833\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7502.35	0.322449
$816$	0	0
$817$	−10630.8	−0.455234
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25128.2	1.06818	0.534092	−	0.845426i	$-0.320654\pi$
$821$	25128.2	1.06818	0.534092	+	0.845426i	$0.320654\pi$
$822$	0	0
$823$	37032.4	1.56849	0.784245	−	0.620452i	$-0.213051\pi$
$823$	37032.4	1.56849	0.784245	+	0.620452i	$0.213051\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33611.4	1.41328	0.706640	−	0.707574i	$-0.250210\pi$
$827$	33611.4	1.41328	0.706640	+	0.707574i	$0.250210\pi$
$828$	0	0
$829$	17321.9	0.725710	0.362855	−	0.931846i	$-0.381802\pi$
$829$	17321.9	0.725710	0.362855	+	0.931846i	$0.381802\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−364.965	−0.0151804
$834$	0	0
$835$	17147.2	0.710661
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13870.7	0.570761	0.285381	−	0.958414i	$-0.407880\pi$
$839$	13870.7	0.570761	0.285381	+	0.958414i	$0.407880\pi$
$840$	0	0
$841$	−14420.1	−0.591256
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8227.53	−0.334953
$846$	0	0
$847$	4191.27	0.170028
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21714.1	−0.874676
$852$	0	0
$853$	−18237.0	−0.732030	−0.366015	−	0.930609i	$-0.619278\pi$
$853$	−18237.0	−0.732030	−0.366015	+	0.930609i	$0.619278\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37468.9	−1.49348	−0.746741	−	0.665115i	$-0.768383\pi$
$857$	−37468.9	−1.49348	−0.746741	+	0.665115i	$0.768383\pi$
$858$	0	0
$859$	8485.07	0.337028	0.168514	−	0.985699i	$-0.446103\pi$
$859$	8485.07	0.337028	0.168514	+	0.985699i	$0.446103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20829.5	−0.821603	−0.410801	−	0.911725i	$-0.634751\pi$
$863$	−20829.5	−0.821603	−0.410801	+	0.911725i	$0.634751\pi$
$864$	0	0
$865$	10039.7	0.394637
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8926.37	0.348454
$870$	0	0
$871$	12340.6	0.480074
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2237.93	0.0864640
$876$	0	0
$877$	41591.2	1.60141	0.800704	−	0.599061i	$-0.204459\pi$
$877$	41591.2	1.60141	0.800704	+	0.599061i	$0.204459\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47222.3	1.80586	0.902928	−	0.429792i	$-0.141413\pi$
$881$	47222.3	1.80586	0.902928	+	0.429792i	$0.141413\pi$
$882$	0	0
$883$	−3201.36	−0.122010	−0.0610048	−	0.998137i	$-0.519430\pi$
$883$	−3201.36	−0.122010	−0.0610048	+	0.998137i	$0.519430\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29115.9	1.10216	0.551080	−	0.834453i	$-0.314216\pi$
$887$	29115.9	1.10216	0.551080	+	0.834453i	$0.314216\pi$
$888$	0	0
$889$	−42711.8	−1.61137
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2520.76	0.0944614
$894$	0	0
$895$	−19184.8	−0.716509
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−32139.7	−1.19235
$900$	0	0
$901$	−1381.38	−0.0510772
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9110.50	−0.334633
$906$	0	0
$907$	−39255.4	−1.43710	−0.718552	−	0.695473i	$-0.755195\pi$
$907$	−39255.4	−1.43710	−0.718552	+	0.695473i	$0.755195\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36666.8	1.33351	0.666754	−	0.745278i	$-0.267683\pi$
$911$	36666.8	1.33351	0.666754	+	0.745278i	$0.267683\pi$
$912$	0	0
$913$	−19901.4	−0.721403
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16190.4	0.583047
$918$	0	0
$919$	−11786.5	−0.423070	−0.211535	−	0.977370i	$-0.567846\pi$
$919$	−11786.5	−0.423070	−0.211535	+	0.977370i	$0.567846\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17867.8	−0.637190
$924$	0	0
$925$	−5144.01	−0.182847
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40009.2	−1.41298	−0.706490	−	0.707723i	$-0.749723\pi$
$929$	−40009.2	−1.41298	−0.706490	+	0.707723i	$0.749723\pi$
$930$	0	0
$931$	636.770	0.0224160
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3213.48	−0.112398
$936$	0	0
$937$	−14272.7	−0.497620	−0.248810	−	0.968552i	$-0.580039\pi$
$937$	−14272.7	−0.497620	−0.248810	+	0.968552i	$0.580039\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26221.6	0.908395	0.454197	−	0.890901i	$-0.349926\pi$
$941$	26221.6	0.908395	0.454197	+	0.890901i	$0.349926\pi$
$942$	0	0
$943$	−34576.6	−1.19403
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41671.9	1.42994	0.714971	−	0.699155i	$-0.246440\pi$
$947$	41671.9	1.42994	0.714971	+	0.699155i	$0.246440\pi$
$948$	0	0
$949$	2373.67	0.0811933
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24374.3	0.828500	0.414250	−	0.910163i	$-0.364044\pi$
$953$	24374.3	0.828500	0.414250	+	0.910163i	$0.364044\pi$
$954$	0	0
$955$	−20979.8	−0.710882
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14472.5	−0.487323
$960$	0	0
$961$	73827.9	2.47820
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9051.79	−0.301956
$966$	0	0
$967$	−18670.4	−0.620888	−0.310444	−	0.950592i	$-0.600478\pi$
$967$	−18670.4	−0.620888	−0.310444	+	0.950592i	$0.600478\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−139.756	−0.00461893	−0.00230946	−	0.999997i	$-0.500735\pi$
$971$	−139.756	−0.00461893	−0.00230946	+	0.999997i	$0.500735\pi$
$972$	0	0
$973$	−51288.4	−1.68986
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11504.5	0.376726	0.188363	−	0.982099i	$-0.439682\pi$
$977$	11504.5	0.376726	0.188363	+	0.982099i	$0.439682\pi$
$978$	0	0
$979$	34370.0	1.12203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1414.94	−0.0459101	−0.0229550	−	0.999736i	$-0.507307\pi$
$983$	−1414.94	−0.0459101	−0.0229550	+	0.999736i	$0.507307\pi$
$984$	0	0
$985$	−19136.0	−0.619010
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39580.6	1.27259
$990$	0	0
$991$	12991.4	0.416434	0.208217	−	0.978083i	$-0.433234\pi$
$991$	12991.4	0.416434	0.208217	+	0.978083i	$0.433234\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2389.27	−0.0761254
$996$	0	0
$997$	21151.2	0.671880	0.335940	−	0.941883i	$-0.390946\pi$
$997$	21151.2	0.671880	0.335940	+	0.941883i	$0.390946\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.h.1.4		4
3.2	odd	2		1620.4.a.g.1.4		4
9.2	odd	6		180.4.i.b.121.3	yes	8
9.4	even	3		540.4.i.b.181.1		8
9.5	odd	6		180.4.i.b.61.3	✓	8
9.7	even	3		540.4.i.b.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.b.61.3	✓	8	9.5	odd	6
180.4.i.b.121.3	yes	8	9.2	odd	6
540.4.i.b.181.1		8	9.4	even	3
540.4.i.b.361.1		8	9.7	even	3
1620.4.a.g.1.4		4	3.2	odd	2
1620.4.a.h.1.4		4	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)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +17.9035 q^{7} -39.5614 q^{11} -23.4839 q^{13} +16.2456 q^{17} -28.3443 q^{19} +105.531 q^{23} +25.0000 q^{25} +99.8442 q^{29} -321.899 q^{31} +89.5174 q^{35} -205.760 q^{37} -327.644 q^{41} +375.061 q^{43} -88.9336 q^{47} -22.4655 q^{49} -85.0314 q^{53} -197.807 q^{55} -796.806 q^{59} -317.936 q^{61} -117.420 q^{65} -525.490 q^{67} +760.854 q^{71} -101.076 q^{73} -708.287 q^{77} -225.633 q^{79} +503.051 q^{83} +81.2278 q^{85} -868.775 q^{89} -420.444 q^{91} -141.722 q^{95} +1749.44 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{5} - 13 q^{7} - 57 q^{11} + 14 q^{13} - 3 q^{17} - 31 q^{19} + 69 q^{23} + 100 q^{25} + 69 q^{29} - 58 q^{31} - 65 q^{35} - 388 q^{37} - 396 q^{41} + 371 q^{43} - 129 q^{47} + 111 q^{49} - 1356 q^{53}+ \cdots - 1495 q^{97}+O(q^{100}) \) 4 * q + 20 * q^5 - 13 * q^7 - 57 * q^11 + 14 * q^13 - 3 * q^17 - 31 * q^19 + 69 * q^23 + 100 * q^25 + 69 * q^29 - 58 * q^31 - 65 * q^35 - 388 * q^37 - 396 * q^41 + 371 * q^43 - 129 * q^47 + 111 * q^49 - 1356 * q^53 - 285 * q^55 - 15 * q^59 - 1441 * q^61 + 70 * q^65 + 368 * q^67 - 168 * q^71 - 955 * q^73 - 342 * q^77 - 1408 * q^79 + 789 * q^83 - 15 * q^85 - 1617 * q^89 - 1406 * q^91 - 155 * q^95 - 1495 * q^97

Introduction

L-functions

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