LMFDB - Embedded newform 252.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, 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("252.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.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:	$14.8684813214$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.29150 q^{5} -7.00000 q^{7} +O(q^{10})$	$q+5.29150 q^{5} -7.00000 q^{7} -68.7895 q^{11} -30.0000 q^{13} +121.705 q^{17} -100.000 q^{19} +89.9555 q^{23} -97.0000 q^{25} -232.826 q^{29} -180.000 q^{31} -37.0405 q^{35} -118.000 q^{37} -15.8745 q^{41} -412.000 q^{43} +285.741 q^{47} +49.0000 q^{49} +285.741 q^{53} -364.000 q^{55} -836.057 q^{59} -378.000 q^{61} -158.745 q^{65} +244.000 q^{67} +439.195 q^{71} +670.000 q^{73} +481.527 q^{77} +216.000 q^{79} +804.308 q^{83} +644.000 q^{85} -968.345 q^{89} +210.000 q^{91} -529.150 q^{95} +574.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7}+O(q^{10}) $ 2 * q - 14 * q^7	$ 2 q - 14 q^{7} - 60 q^{13} - 200 q^{19} - 194 q^{25} - 360 q^{31} - 236 q^{37} - 824 q^{43} + 98 q^{49} - 728 q^{55} - 756 q^{61} + 488 q^{67} + 1340 q^{73} + 432 q^{79} + 1288 q^{85} + 420 q^{91} + 1148 q^{97}+O(q^{100}) $ 2 * q - 14 * q^7 - 60 * q^13 - 200 * q^19 - 194 * q^25 - 360 * q^31 - 236 * q^37 - 824 * q^43 + 98 * q^49 - 728 * q^55 - 756 * q^61 + 488 * q^67 + 1340 * q^73 + 432 * q^79 + 1288 * q^85 + 420 * q^91 + 1148 * 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
$5$	5.29150	0.473286	0.236643	−	0.971597i	$-0.423953\pi$
$5$	5.29150	0.473286	0.236643	+	0.971597i	$0.423953\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−68.7895	−1.88553	−0.942765	−	0.333459i	$-0.891784\pi$
$11$	−68.7895	−1.88553	−0.942765	+	0.333459i	$0.891784\pi$
$12$	0	0
$13$	−30.0000	−0.640039	−0.320019	−	0.947411i	$-0.603689\pi$
$13$	−30.0000	−0.640039	−0.320019	+	0.947411i	$0.603689\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	121.705	1.73633	0.868167	−	0.496271i	$-0.165298\pi$
$17$	121.705	1.73633	0.868167	+	0.496271i	$0.165298\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	89.9555	0.815523	0.407761	−	0.913088i	$-0.366309\pi$
$23$	89.9555	0.815523	0.407761	+	0.913088i	$0.366309\pi$
$24$	0	0
$25$	−97.0000	−0.776000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−232.826	−1.49085	−0.745426	−	0.666588i	$-0.767754\pi$
$29$	−232.826	−1.49085	−0.745426	+	0.666588i	$0.767754\pi$
$30$	0	0
$31$	−180.000	−1.04287	−0.521435	−	0.853291i	$-0.674603\pi$
$31$	−180.000	−1.04287	−0.521435	+	0.853291i	$0.674603\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−37.0405	−0.178885
$36$	0	0
$37$	−118.000	−0.524299	−0.262150	−	0.965027i	$-0.584431\pi$
$37$	−118.000	−0.524299	−0.262150	+	0.965027i	$0.584431\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−15.8745	−0.0604678	−0.0302339	−	0.999543i	$-0.509625\pi$
$41$	−15.8745	−0.0604678	−0.0302339	+	0.999543i	$0.509625\pi$
$42$	0	0
$43$	−412.000	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$43$	−412.000	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	285.741	0.886801	0.443400	−	0.896324i	$-0.353772\pi$
$47$	285.741	0.886801	0.443400	+	0.896324i	$0.353772\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	285.741	0.740558	0.370279	−	0.928921i	$-0.379262\pi$
$53$	285.741	0.740558	0.370279	+	0.928921i	$0.379262\pi$
$54$	0	0
$55$	−364.000	−0.892395
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−836.057	−1.84484	−0.922419	−	0.386191i	$-0.873790\pi$
$59$	−836.057	−1.84484	−0.922419	+	0.386191i	$0.873790\pi$
$60$	0	0
$61$	−378.000	−0.793409	−0.396704	−	0.917946i	$-0.629846\pi$
$61$	−378.000	−0.793409	−0.396704	+	0.917946i	$0.629846\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−158.745	−0.302922
$66$	0	0
$67$	244.000	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$67$	244.000	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	439.195	0.734124	0.367062	−	0.930196i	$-0.380364\pi$
$71$	439.195	0.734124	0.367062	+	0.930196i	$0.380364\pi$
$72$	0	0
$73$	670.000	1.07421	0.537107	−	0.843514i	$-0.319517\pi$
$73$	670.000	1.07421	0.537107	+	0.843514i	$0.319517\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	481.527	0.712663
$78$	0	0
$79$	216.000	0.307619	0.153809	−	0.988101i	$-0.450846\pi$
$79$	216.000	0.307619	0.153809	+	0.988101i	$0.450846\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	804.308	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$83$	804.308	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$84$	0	0
$85$	644.000	0.821784
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−968.345	−1.15331	−0.576654	−	0.816989i	$-0.695642\pi$
$89$	−968.345	−1.15331	−0.576654	+	0.816989i	$0.695642\pi$
$90$	0	0
$91$	210.000	0.241912
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−529.150	−0.571470
$96$	0	0
$97$	574.000	0.600834	0.300417	−	0.953808i	$-0.402874\pi$
$97$	574.000	0.600834	0.300417	+	0.953808i	$0.402874\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	672.021	0.662065	0.331033	−	0.943619i	$-0.392603\pi$
$101$	672.021	0.662065	0.331033	+	0.943619i	$0.392603\pi$
$102$	0	0
$103$	−964.000	−0.922192	−0.461096	−	0.887350i	$-0.652544\pi$
$103$	−964.000	−0.922192	−0.461096	+	0.887350i	$0.652544\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−428.612	−0.387247	−0.193624	−	0.981076i	$-0.562024\pi$
$107$	−428.612	−0.387247	−0.193624	+	0.981076i	$0.562024\pi$
$108$	0	0
$109$	2122.00	1.86469	0.932343	−	0.361575i	$-0.117761\pi$
$109$	2122.00	1.86469	0.932343	+	0.361575i	$0.117761\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−296.324	−0.246689	−0.123344	−	0.992364i	$-0.539362\pi$
$113$	−296.324	−0.246689	−0.123344	+	0.992364i	$0.539362\pi$
$114$	0	0
$115$	476.000	0.385976
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−851.932	−0.656273
$120$	0	0
$121$	3401.00	2.55522
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1174.71	−0.840557
$126$	0	0
$127$	−1464.00	−1.02291	−0.511453	−	0.859311i	$-0.670892\pi$
$127$	−1464.00	−1.02291	−0.511453	+	0.859311i	$0.670892\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1926.11	1.28462	0.642308	−	0.766446i	$-0.277977\pi$
$131$	1926.11	1.28462	0.642308	+	0.766446i	$0.277977\pi$
$132$	0	0
$133$	700.000	0.456374
$134$	0	0
$135$	0	0
$136$	0	0
$137$	52.9150	0.0329988	0.0164994	−	0.999864i	$-0.494748\pi$
$137$	52.9150	0.0329988	0.0164994	+	0.999864i	$0.494748\pi$
$138$	0	0
$139$	352.000	0.214793	0.107397	−	0.994216i	$-0.465749\pi$
$139$	352.000	0.214793	0.107397	+	0.994216i	$0.465749\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2063.69	1.20681
$144$	0	0
$145$	−1232.00	−0.705600
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2254.18	1.23939	0.619697	−	0.784841i	$-0.287256\pi$
$149$	2254.18	1.23939	0.619697	+	0.784841i	$0.287256\pi$
$150$	0	0
$151$	376.000	0.202639	0.101319	−	0.994854i	$-0.467694\pi$
$151$	376.000	0.202639	0.101319	+	0.994854i	$0.467694\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−952.470	−0.493576
$156$	0	0
$157$	2198.00	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	2198.00	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−629.689	−0.308239
$162$	0	0
$163$	−292.000	−0.140314	−0.0701571	−	0.997536i	$-0.522350\pi$
$163$	−292.000	−0.140314	−0.0701571	+	0.997536i	$0.522350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3439.48	−1.59374	−0.796870	−	0.604150i	$-0.793513\pi$
$167$	−3439.48	−1.59374	−0.796870	+	0.604150i	$0.793513\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2058.39	−0.904606	−0.452303	−	0.891864i	$-0.649397\pi$
$173$	−2058.39	−0.904606	−0.452303	+	0.891864i	$0.649397\pi$
$174$	0	0
$175$	679.000	0.293300
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1571.58	0.656230	0.328115	−	0.944638i	$-0.393587\pi$
$179$	1571.58	0.656230	0.328115	+	0.944638i	$0.393587\pi$
$180$	0	0
$181$	−3246.00	−1.33300	−0.666501	−	0.745504i	$-0.732209\pi$
$181$	−3246.00	−1.33300	−0.666501	+	0.745504i	$0.732209\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−624.397	−0.248144
$186$	0	0
$187$	−8372.00	−3.27391
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2206.56	−0.835921	−0.417960	−	0.908465i	$-0.637255\pi$
$191$	−2206.56	−0.835921	−0.417960	+	0.908465i	$0.637255\pi$
$192$	0	0
$193$	3998.00	1.49110	0.745550	−	0.666450i	$-0.232187\pi$
$193$	3998.00	1.49110	0.745550	+	0.666450i	$0.232187\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3502.97	1.26689	0.633443	−	0.773789i	$-0.281641\pi$
$197$	3502.97	1.26689	0.633443	+	0.773789i	$0.281641\pi$
$198$	0	0
$199$	224.000	0.0797937	0.0398968	−	0.999204i	$-0.487297\pi$
$199$	224.000	0.0797937	0.0398968	+	0.999204i	$0.487297\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1629.78	0.563489
$204$	0	0
$205$	−84.0000	−0.0286186
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6878.95	2.27668
$210$	0	0
$211$	−3460.00	−1.12889	−0.564446	−	0.825470i	$-0.690910\pi$
$211$	−3460.00	−1.12889	−0.564446	+	0.825470i	$0.690910\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2180.10	−0.691542
$216$	0	0
$217$	1260.00	0.394168
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3651.14	−1.11132
$222$	0	0
$223$	−4648.00	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$223$	−4648.00	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3841.63	−1.12325	−0.561626	−	0.827392i	$-0.689824\pi$
$227$	−3841.63	−1.12325	−0.561626	+	0.827392i	$0.689824\pi$
$228$	0	0
$229$	−4646.00	−1.34068	−0.670341	−	0.742053i	$-0.733852\pi$
$229$	−4646.00	−1.34068	−0.670341	+	0.742053i	$0.733852\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4878.77	−1.37175	−0.685877	−	0.727718i	$-0.740581\pi$
$233$	−4878.77	−1.37175	−0.685877	+	0.727718i	$0.740581\pi$
$234$	0	0
$235$	1512.00	0.419711
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2143.06	0.580012	0.290006	−	0.957025i	$-0.406343\pi$
$239$	2143.06	0.580012	0.290006	+	0.957025i	$0.406343\pi$
$240$	0	0
$241$	−826.000	−0.220777	−0.110389	−	0.993888i	$-0.535210\pi$
$241$	−826.000	−0.220777	−0.110389	+	0.993888i	$0.535210\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	259.284	0.0676123
$246$	0	0
$247$	3000.00	0.772816
$248$	0	0
$249$	0	0
$250$	0	0
$251$	857.223	0.215568	0.107784	−	0.994174i	$-0.465625\pi$
$251$	857.223	0.215568	0.107784	+	0.994174i	$0.465625\pi$
$252$	0	0
$253$	−6188.00	−1.53769
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3984.50	0.967107	0.483553	−	0.875315i	$-0.339346\pi$
$257$	3984.50	0.967107	0.483553	+	0.875315i	$0.339346\pi$
$258$	0	0
$259$	826.000	0.198167
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6206.93	1.45527	0.727635	−	0.685965i	$-0.240620\pi$
$263$	6206.93	1.45527	0.727635	+	0.685965i	$0.240620\pi$
$264$	0	0
$265$	1512.00	0.350496
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2344.14	−0.531318	−0.265659	−	0.964067i	$-0.585589\pi$
$269$	−2344.14	−0.531318	−0.265659	+	0.964067i	$0.585589\pi$
$270$	0	0
$271$	−7980.00	−1.78875	−0.894374	−	0.447321i	$-0.852378\pi$
$271$	−7980.00	−1.78875	−0.894374	+	0.447321i	$0.852378\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6672.58	1.46317
$276$	0	0
$277$	−1954.00	−0.423843	−0.211921	−	0.977287i	$-0.567972\pi$
$277$	−1954.00	−0.423843	−0.211921	+	0.977287i	$0.567972\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3989.79	−0.847015	−0.423507	−	0.905893i	$-0.639201\pi$
$281$	−3989.79	−0.847015	−0.423507	+	0.905893i	$0.639201\pi$
$282$	0	0
$283$	4492.00	0.943540	0.471770	−	0.881722i	$-0.343615\pi$
$283$	4492.00	0.943540	0.471770	+	0.881722i	$0.343615\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	111.122	0.0228547
$288$	0	0
$289$	9899.00	2.01486
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8111.87	−1.61741	−0.808704	−	0.588215i	$-0.799831\pi$
$293$	−8111.87	−1.61741	−0.808704	+	0.588215i	$0.799831\pi$
$294$	0	0
$295$	−4424.00	−0.873136
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2698.67	−0.521966
$300$	0	0
$301$	2884.00	0.552262
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2000.19	−0.375510
$306$	0	0
$307$	2788.00	0.518305	0.259152	−	0.965836i	$-0.416557\pi$
$307$	2788.00	0.518305	0.259152	+	0.965836i	$0.416557\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3799.30	0.692728	0.346364	−	0.938100i	$-0.387416\pi$
$311$	3799.30	0.692728	0.346364	+	0.938100i	$0.387416\pi$
$312$	0	0
$313$	−7166.00	−1.29408	−0.647039	−	0.762457i	$-0.723993\pi$
$313$	−7166.00	−1.29408	−0.647039	+	0.762457i	$0.723993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3143.15	0.556899	0.278450	−	0.960451i	$-0.410179\pi$
$317$	3143.15	0.556899	0.278450	+	0.960451i	$0.410179\pi$
$318$	0	0
$319$	16016.0	2.81105
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12170.5	−2.09654
$324$	0	0
$325$	2910.00	0.496670
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2000.19	−0.335179
$330$	0	0
$331$	2108.00	0.350049	0.175024	−	0.984564i	$-0.444000\pi$
$331$	2108.00	0.350049	0.175024	+	0.984564i	$0.444000\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1291.13	0.210572
$336$	0	0
$337$	8814.00	1.42472	0.712358	−	0.701816i	$-0.247627\pi$
$337$	8814.00	1.42472	0.712358	+	0.701816i	$0.247627\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12382.1	1.96636
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4905.22	−0.758865	−0.379433	−	0.925219i	$-0.623881\pi$
$347$	−4905.22	−0.758865	−0.379433	+	0.925219i	$0.623881\pi$
$348$	0	0
$349$	−8498.00	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$349$	−8498.00	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8207.12	1.23745	0.618727	−	0.785606i	$-0.287649\pi$
$353$	8207.12	1.23745	0.618727	+	0.785606i	$0.287649\pi$
$354$	0	0
$355$	2324.00	0.347451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2576.96	0.378849	0.189425	−	0.981895i	$-0.439338\pi$
$359$	2576.96	0.378849	0.189425	+	0.981895i	$0.439338\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3545.31	0.508411
$366$	0	0
$367$	6848.00	0.974013	0.487006	−	0.873398i	$-0.338089\pi$
$367$	6848.00	0.974013	0.487006	+	0.873398i	$0.338089\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2000.19	−0.279905
$372$	0	0
$373$	1966.00	0.272911	0.136455	−	0.990646i	$-0.456429\pi$
$373$	1966.00	0.272911	0.136455	+	0.990646i	$0.456429\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6984.78	0.954203
$378$	0	0
$379$	1324.00	0.179444	0.0897220	−	0.995967i	$-0.471402\pi$
$379$	1324.00	0.179444	0.0897220	+	0.995967i	$0.471402\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7132.95	−0.951636	−0.475818	−	0.879544i	$-0.657848\pi$
$383$	−7132.95	−0.951636	−0.475818	+	0.879544i	$0.657848\pi$
$384$	0	0
$385$	2548.00	0.337294
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7154.11	−0.932462	−0.466231	−	0.884663i	$-0.654389\pi$
$389$	−7154.11	−0.932462	−0.466231	+	0.884663i	$0.654389\pi$
$390$	0	0
$391$	10948.0	1.41602
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1142.96	0.145592
$396$	0	0
$397$	−7002.00	−0.885190	−0.442595	−	0.896722i	$-0.645942\pi$
$397$	−7002.00	−0.885190	−0.442595	+	0.896722i	$0.645942\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11397.9	−1.41941	−0.709705	−	0.704498i	$-0.751172\pi$
$401$	−11397.9	−1.41941	−0.709705	+	0.704498i	$0.751172\pi$
$402$	0	0
$403$	5400.00	0.667477
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8117.17	0.988582
$408$	0	0
$409$	7310.00	0.883756	0.441878	−	0.897075i	$-0.354312\pi$
$409$	7310.00	0.883756	0.441878	+	0.897075i	$0.354312\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5852.40	0.697283
$414$	0	0
$415$	4256.00	0.503419
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4032.12	0.470125	0.235062	−	0.971980i	$-0.424471\pi$
$419$	4032.12	0.470125	0.235062	+	0.971980i	$0.424471\pi$
$420$	0	0
$421$	−11586.0	−1.34125	−0.670626	−	0.741796i	$-0.733974\pi$
$421$	−11586.0	−1.34125	−0.670626	+	0.741796i	$0.733974\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11805.3	−1.34740
$426$	0	0
$427$	2646.00	0.299880
$428$	0	0
$429$	0	0
$430$	0	0
$431$	460.361	0.0514496	0.0257248	−	0.999669i	$-0.491811\pi$
$431$	460.361	0.0514496	0.0257248	+	0.999669i	$0.491811\pi$
$432$	0	0
$433$	−9718.00	−1.07856	−0.539281	−	0.842126i	$-0.681304\pi$
$433$	−9718.00	−1.07856	−0.539281	+	0.842126i	$0.681304\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8995.55	−0.984704
$438$	0	0
$439$	1680.00	0.182647	0.0913235	−	0.995821i	$-0.470890\pi$
$439$	1680.00	0.182647	0.0913235	+	0.995821i	$0.470890\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3201.36	0.343343	0.171672	−	0.985154i	$-0.445083\pi$
$443$	3201.36	0.343343	0.171672	+	0.985154i	$0.445083\pi$
$444$	0	0
$445$	−5124.00	−0.545845
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1037.13	0.109010	0.0545049	−	0.998514i	$-0.482642\pi$
$449$	1037.13	0.109010	0.0545049	+	0.998514i	$0.482642\pi$
$450$	0	0
$451$	1092.00	0.114014
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1111.22	0.114494
$456$	0	0
$457$	−2330.00	−0.238496	−0.119248	−	0.992864i	$-0.538048\pi$
$457$	−2330.00	−0.238496	−0.119248	+	0.992864i	$0.538048\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9614.66	−0.971365	−0.485683	−	0.874135i	$-0.661429\pi$
$461$	−9614.66	−0.971365	−0.485683	+	0.874135i	$0.661429\pi$
$462$	0	0
$463$	−4880.00	−0.489833	−0.244917	−	0.969544i	$-0.578761\pi$
$463$	−4880.00	−0.489833	−0.244917	+	0.969544i	$0.578761\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6720.21	0.665898	0.332949	−	0.942945i	$-0.391956\pi$
$467$	6720.21	0.665898	0.332949	+	0.942945i	$0.391956\pi$
$468$	0	0
$469$	−1708.00	−0.168162
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28341.3	2.75504
$474$	0	0
$475$	9700.00	0.936982
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16117.9	−1.53747	−0.768733	−	0.639570i	$-0.779113\pi$
$479$	−16117.9	−1.53747	−0.768733	+	0.639570i	$0.779113\pi$
$480$	0	0
$481$	3540.00	0.335572
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3037.32	0.284366
$486$	0	0
$487$	−8728.00	−0.812122	−0.406061	−	0.913846i	$-0.633098\pi$
$487$	−8728.00	−0.812122	−0.406061	+	0.913846i	$0.633098\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14525.2	1.33505	0.667527	−	0.744585i	$-0.267353\pi$
$491$	14525.2	1.33505	0.667527	+	0.744585i	$0.267353\pi$
$492$	0	0
$493$	−28336.0	−2.58862
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3074.36	−0.277473
$498$	0	0
$499$	−2204.00	−0.197725	−0.0988623	−	0.995101i	$-0.531520\pi$
$499$	−2204.00	−0.197725	−0.0988623	+	0.995101i	$0.531520\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7492.77	−0.664187	−0.332093	−	0.943246i	$-0.607755\pi$
$503$	−7492.77	−0.664187	−0.332093	+	0.943246i	$0.607755\pi$
$504$	0	0
$505$	3556.00	0.313346
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8461.11	−0.736802	−0.368401	−	0.929667i	$-0.620095\pi$
$509$	−8461.11	−0.736802	−0.368401	+	0.929667i	$0.620095\pi$
$510$	0	0
$511$	−4690.00	−0.406014
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5101.01	−0.436461
$516$	0	0
$517$	−19656.0	−1.67209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2460.55	−0.206907	−0.103454	−	0.994634i	$-0.532989\pi$
$521$	−2460.55	−0.206907	−0.103454	+	0.994634i	$0.532989\pi$
$522$	0	0
$523$	−9424.00	−0.787921	−0.393961	−	0.919127i	$-0.628895\pi$
$523$	−9424.00	−0.787921	−0.393961	+	0.919127i	$0.628895\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21906.8	−1.81077
$528$	0	0
$529$	−4075.00	−0.334922
$530$	0	0
$531$	0	0
$532$	0	0
$533$	476.235	0.0387018
$534$	0	0
$535$	−2268.00	−0.183279
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3370.69	−0.269361
$540$	0	0
$541$	−8730.00	−0.693775	−0.346887	−	0.937907i	$-0.612761\pi$
$541$	−8730.00	−0.693775	−0.346887	+	0.937907i	$0.612761\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11228.6	0.882530
$546$	0	0
$547$	1668.00	0.130381	0.0651906	−	0.997873i	$-0.479234\pi$
$547$	1668.00	0.130381	0.0651906	+	0.997873i	$0.479234\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	23282.6	1.80013
$552$	0	0
$553$	−1512.00	−0.116269
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−264.575	−0.0201264	−0.0100632	−	0.999949i	$-0.503203\pi$
$557$	−264.575	−0.0201264	−0.0100632	+	0.999949i	$0.503203\pi$
$558$	0	0
$559$	12360.0	0.935192
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21536.4	1.61217	0.806085	−	0.591800i	$-0.201582\pi$
$563$	21536.4	1.61217	0.806085	+	0.591800i	$0.201582\pi$
$564$	0	0
$565$	−1568.00	−0.116754
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25304.0	1.86432	0.932159	−	0.362048i	$-0.117922\pi$
$569$	25304.0	1.86432	0.932159	+	0.362048i	$0.117922\pi$
$570$	0	0
$571$	−4068.00	−0.298144	−0.149072	−	0.988826i	$-0.547629\pi$
$571$	−4068.00	−0.298144	−0.149072	+	0.988826i	$0.547629\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8725.69	−0.632846
$576$	0	0
$577$	7618.00	0.549639	0.274819	−	0.961496i	$-0.411382\pi$
$577$	7618.00	0.549639	0.274819	+	0.961496i	$0.411382\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5630.16	−0.402028
$582$	0	0
$583$	−19656.0	−1.39634
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4497.78	−0.316257	−0.158129	−	0.987419i	$-0.550546\pi$
$587$	−4497.78	−0.316257	−0.158129	+	0.987419i	$0.550546\pi$
$588$	0	0
$589$	18000.0	1.25921
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21890.9	−1.51594	−0.757971	−	0.652288i	$-0.773809\pi$
$593$	−21890.9	−1.51594	−0.757971	+	0.652288i	$0.773809\pi$
$594$	0	0
$595$	−4508.00	−0.310605
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18525.6	−1.26366	−0.631831	−	0.775106i	$-0.717696\pi$
$599$	−18525.6	−1.26366	−0.631831	+	0.775106i	$0.717696\pi$
$600$	0	0
$601$	−11590.0	−0.786632	−0.393316	−	0.919403i	$-0.628672\pi$
$601$	−11590.0	−0.786632	−0.393316	+	0.919403i	$0.628672\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17996.4	1.20935
$606$	0	0
$607$	9144.00	0.611439	0.305720	−	0.952122i	$-0.401103\pi$
$607$	9144.00	0.611439	0.305720	+	0.952122i	$0.401103\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8572.23	−0.567587
$612$	0	0
$613$	−29874.0	−1.96835	−0.984176	−	0.177195i	$-0.943298\pi$
$613$	−29874.0	−1.96835	−0.984176	+	0.177195i	$0.943298\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26722.1	−1.74358	−0.871792	−	0.489877i	$-0.837042\pi$
$617$	−26722.1	−1.74358	−0.871792	+	0.489877i	$0.837042\pi$
$618$	0	0
$619$	4744.00	0.308041	0.154021	−	0.988068i	$-0.450778\pi$
$619$	4744.00	0.308041	0.154021	+	0.988068i	$0.450778\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6778.41	0.435909
$624$	0	0
$625$	5909.00	0.378176
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14361.1	−0.910359
$630$	0	0
$631$	−13144.0	−0.829246	−0.414623	−	0.909993i	$-0.636087\pi$
$631$	−13144.0	−0.829246	−0.414623	+	0.909993i	$0.636087\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7746.76	−0.484127
$636$	0	0
$637$	−1470.00	−0.0914341
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23928.2	−1.47442	−0.737212	−	0.675661i	$-0.763858\pi$
$641$	−23928.2	−1.47442	−0.737212	+	0.675661i	$0.763858\pi$
$642$	0	0
$643$	20228.0	1.24061	0.620307	−	0.784359i	$-0.287008\pi$
$643$	20228.0	1.24061	0.620307	+	0.784359i	$0.287008\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20160.6	1.22503	0.612516	−	0.790458i	$-0.290157\pi$
$647$	20160.6	1.22503	0.612516	+	0.790458i	$0.290157\pi$
$648$	0	0
$649$	57512.0	3.47850
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16065.0	0.962744	0.481372	−	0.876516i	$-0.340139\pi$
$653$	16065.0	0.962744	0.481372	+	0.876516i	$0.340139\pi$
$654$	0	0
$655$	10192.0	0.607991
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16504.2	0.975588	0.487794	−	0.872959i	$-0.337802\pi$
$659$	16504.2	0.975588	0.487794	+	0.872959i	$0.337802\pi$
$660$	0	0
$661$	−27242.0	−1.60301	−0.801506	−	0.597987i	$-0.795968\pi$
$661$	−27242.0	−1.60301	−0.801506	+	0.597987i	$0.795968\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3704.05	0.215995
$666$	0	0
$667$	−20944.0	−1.21582
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26002.4	1.49600
$672$	0	0
$673$	17606.0	1.00841	0.504206	−	0.863583i	$-0.331785\pi$
$673$	17606.0	1.00841	0.504206	+	0.863583i	$0.331785\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15064.9	0.855231	0.427616	−	0.903961i	$-0.359354\pi$
$677$	15064.9	0.855231	0.427616	+	0.903961i	$0.359354\pi$
$678$	0	0
$679$	−4018.00	−0.227094
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2905.03	0.162750	0.0813749	−	0.996684i	$-0.474069\pi$
$683$	2905.03	0.162750	0.0813749	+	0.996684i	$0.474069\pi$
$684$	0	0
$685$	280.000	0.0156179
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8572.23	−0.473986
$690$	0	0
$691$	29720.0	1.63618	0.818091	−	0.575088i	$-0.195032\pi$
$691$	29720.0	1.63618	0.818091	+	0.575088i	$0.195032\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1862.61	0.101659
$696$	0	0
$697$	−1932.00	−0.104992
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14117.7	0.760655	0.380328	−	0.924852i	$-0.375811\pi$
$701$	14117.7	0.760655	0.380328	+	0.924852i	$0.375811\pi$
$702$	0	0
$703$	11800.0	0.633066
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4704.15	−0.250237
$708$	0	0
$709$	16658.0	0.882376	0.441188	−	0.897415i	$-0.354557\pi$
$709$	16658.0	0.882376	0.441188	+	0.897415i	$0.354557\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16192.0	−0.850484
$714$	0	0
$715$	10920.0	0.571168
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12699.6	0.658714	0.329357	−	0.944205i	$-0.393168\pi$
$719$	12699.6	0.658714	0.329357	+	0.944205i	$0.393168\pi$
$720$	0	0
$721$	6748.00	0.348556
$722$	0	0
$723$	0	0
$724$	0	0
$725$	22584.1	1.15690
$726$	0	0
$727$	24316.0	1.24048	0.620241	−	0.784411i	$-0.287035\pi$
$727$	24316.0	1.24048	0.620241	+	0.784411i	$0.287035\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−50142.3	−2.53704
$732$	0	0
$733$	12570.0	0.633402	0.316701	−	0.948525i	$-0.397425\pi$
$733$	12570.0	0.633402	0.316701	+	0.948525i	$0.397425\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16784.6	−0.838901
$738$	0	0
$739$	7356.00	0.366164	0.183082	−	0.983098i	$-0.441393\pi$
$739$	7356.00	0.366164	0.183082	+	0.983098i	$0.441393\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36559.0	−1.80514	−0.902570	−	0.430543i	$-0.858322\pi$
$743$	−36559.0	−1.80514	−0.902570	+	0.430543i	$0.858322\pi$
$744$	0	0
$745$	11928.0	0.586588
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3000.28	0.146366
$750$	0	0
$751$	−27768.0	−1.34923	−0.674613	−	0.738172i	$-0.735689\pi$
$751$	−27768.0	−1.34923	−0.674613	+	0.738172i	$0.735689\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1989.60	0.0959061
$756$	0	0
$757$	23826.0	1.14395	0.571975	−	0.820271i	$-0.306177\pi$
$757$	23826.0	1.14395	0.571975	+	0.820271i	$0.306177\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28727.6	1.36843	0.684214	−	0.729281i	$-0.260145\pi$
$761$	28727.6	1.36843	0.684214	+	0.729281i	$0.260145\pi$
$762$	0	0
$763$	−14854.0	−0.704785
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25081.7	1.18077
$768$	0	0
$769$	23770.0	1.11465	0.557326	−	0.830293i	$-0.311827\pi$
$769$	23770.0	1.11465	0.557326	+	0.830293i	$0.311827\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29362.5	−1.36623	−0.683116	−	0.730310i	$-0.739376\pi$
$773$	−29362.5	−1.36623	−0.683116	+	0.730310i	$0.739376\pi$
$774$	0	0
$775$	17460.0	0.809267
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1587.45	0.0730120
$780$	0	0
$781$	−30212.0	−1.38421
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11630.7	0.528813
$786$	0	0
$787$	−40656.0	−1.84146	−0.920731	−	0.390199i	$-0.872406\pi$
$787$	−40656.0	−1.84146	−0.920731	+	0.390199i	$0.872406\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2074.27	0.0932396
$792$	0	0
$793$	11340.0	0.507812
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30939.4	1.37507	0.687535	−	0.726152i	$-0.258693\pi$
$797$	30939.4	1.37507	0.687535	+	0.726152i	$0.258693\pi$
$798$	0	0
$799$	34776.0	1.53978
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−46089.0	−2.02546
$804$	0	0
$805$	−3332.00	−0.145885
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20277.0	0.881215	0.440607	−	0.897700i	$-0.354763\pi$
$809$	20277.0	0.881215	0.440607	+	0.897700i	$0.354763\pi$
$810$	0	0
$811$	−3248.00	−0.140632	−0.0703161	−	0.997525i	$-0.522401\pi$
$811$	−3248.00	−0.140632	−0.0703161	+	0.997525i	$0.522401\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1545.12	−0.0664088
$816$	0	0
$817$	41200.0	1.76427
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25134.6	−1.06846	−0.534230	−	0.845339i	$-0.679398\pi$
$821$	−25134.6	−1.06846	−0.534230	+	0.845339i	$0.679398\pi$
$822$	0	0
$823$	−31904.0	−1.35128	−0.675640	−	0.737232i	$-0.736133\pi$
$823$	−31904.0	−1.35128	−0.675640	+	0.737232i	$0.736133\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24589.6	1.03394	0.516968	−	0.856005i	$-0.327061\pi$
$827$	24589.6	1.03394	0.516968	+	0.856005i	$0.327061\pi$
$828$	0	0
$829$	25762.0	1.07931	0.539657	−	0.841885i	$-0.318554\pi$
$829$	25762.0	1.07931	0.539657	+	0.841885i	$0.318554\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5963.52	0.248048
$834$	0	0
$835$	−18200.0	−0.754296
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5450.25	−0.224271	−0.112136	−	0.993693i	$-0.535769\pi$
$839$	−5450.25	−0.224271	−0.112136	+	0.993693i	$0.535769\pi$
$840$	0	0
$841$	29819.0	1.22264
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6863.08	−0.279405
$846$	0	0
$847$	−23807.0	−0.965783
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10614.8	−0.427578
$852$	0	0
$853$	−17050.0	−0.684386	−0.342193	−	0.939630i	$-0.611170\pi$
$853$	−17050.0	−0.684386	−0.342193	+	0.939630i	$0.611170\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6968.91	0.277775	0.138888	−	0.990308i	$-0.455647\pi$
$857$	6968.91	0.277775	0.138888	+	0.990308i	$0.455647\pi$
$858$	0	0
$859$	−10948.0	−0.434856	−0.217428	−	0.976076i	$-0.569767\pi$
$859$	−10948.0	−0.434856	−0.217428	+	0.976076i	$0.569767\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34611.7	1.36523	0.682617	−	0.730777i	$-0.260842\pi$
$863$	34611.7	1.36523	0.682617	+	0.730777i	$0.260842\pi$
$864$	0	0
$865$	−10892.0	−0.428138
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14858.5	−0.580025
$870$	0	0
$871$	−7320.00	−0.284763
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8223.00	0.317701
$876$	0	0
$877$	−16154.0	−0.621986	−0.310993	−	0.950412i	$-0.600662\pi$
$877$	−16154.0	−0.621986	−0.310993	+	0.950412i	$0.600662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9318.34	−0.356348	−0.178174	−	0.983999i	$-0.557019\pi$
$881$	−9318.34	−0.356348	−0.178174	+	0.983999i	$0.557019\pi$
$882$	0	0
$883$	−45772.0	−1.74445	−0.872226	−	0.489104i	$-0.837324\pi$
$883$	−45772.0	−1.74445	−0.872226	+	0.489104i	$0.837324\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7461.02	−0.282431	−0.141216	−	0.989979i	$-0.545101\pi$
$887$	−7461.02	−0.282431	−0.141216	+	0.989979i	$0.545101\pi$
$888$	0	0
$889$	10248.0	0.386622
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28574.1	−1.07077
$894$	0	0
$895$	8316.00	0.310585
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41908.7	1.55476
$900$	0	0
$901$	34776.0	1.28586
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17176.2	−0.630891
$906$	0	0
$907$	3900.00	0.142775	0.0713877	−	0.997449i	$-0.477257\pi$
$907$	3900.00	0.142775	0.0713877	+	0.997449i	$0.477257\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11604.3	−0.422027	−0.211013	−	0.977483i	$-0.567676\pi$
$911$	−11604.3	−0.422027	−0.211013	+	0.977483i	$0.567676\pi$
$912$	0	0
$913$	−55328.0	−2.00557
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13482.7	−0.485539
$918$	0	0
$919$	−18416.0	−0.661031	−0.330516	−	0.943801i	$-0.607223\pi$
$919$	−18416.0	−0.661031	−0.330516	+	0.943801i	$0.607223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13175.8	−0.469868
$924$	0	0
$925$	11446.0	0.406856
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42273.8	1.49296	0.746479	−	0.665408i	$-0.231743\pi$
$929$	42273.8	1.49296	0.746479	+	0.665408i	$0.231743\pi$
$930$	0	0
$931$	−4900.00	−0.172493
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−44300.5	−1.54950
$936$	0	0
$937$	−30910.0	−1.07768	−0.538840	−	0.842408i	$-0.681137\pi$
$937$	−30910.0	−1.07768	−0.538840	+	0.842408i	$0.681137\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24706.0	0.855891	0.427945	−	0.903805i	$-0.359238\pi$
$941$	24706.0	0.855891	0.427945	+	0.903805i	$0.359238\pi$
$942$	0	0
$943$	−1428.00	−0.0493129
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35405.4	1.21491	0.607456	−	0.794353i	$-0.292190\pi$
$947$	35405.4	1.21491	0.607456	+	0.794353i	$0.292190\pi$
$948$	0	0
$949$	−20100.0	−0.687538
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32362.8	1.10004	0.550018	−	0.835153i	$-0.314621\pi$
$953$	32362.8	1.10004	0.550018	+	0.835153i	$0.314621\pi$
$954$	0	0
$955$	−11676.0	−0.395630
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−370.405	−0.0124724
$960$	0	0
$961$	2609.00	0.0875768
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21155.4	0.705717
$966$	0	0
$967$	−51136.0	−1.70054	−0.850270	−	0.526346i	$-0.823562\pi$
$967$	−51136.0	−1.70054	−0.850270	+	0.526346i	$0.823562\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10794.7	−0.356763	−0.178382	−	0.983961i	$-0.557086\pi$
$971$	−10794.7	−0.356763	−0.178382	+	0.983961i	$0.557086\pi$
$972$	0	0
$973$	−2464.00	−0.0811842
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45432.8	−1.48774	−0.743872	−	0.668322i	$-0.767013\pi$
$977$	−45432.8	−1.48774	−0.743872	+	0.668322i	$0.767013\pi$
$978$	0	0
$979$	66612.0	2.17460
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37146.3	1.20527	0.602637	−	0.798015i	$-0.294117\pi$
$983$	37146.3	1.20527	0.602637	+	0.798015i	$0.294117\pi$
$984$	0	0
$985$	18536.0	0.599600
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−37061.7	−1.19160
$990$	0	0
$991$	−49720.0	−1.59375	−0.796876	−	0.604143i	$-0.793516\pi$
$991$	−49720.0	−1.59375	−0.796876	+	0.604143i	$0.793516\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1185.30	0.0377653
$996$	0	0
$997$	−20554.0	−0.652910	−0.326455	−	0.945213i	$-0.605854\pi$
$997$	−20554.0	−0.652910	−0.326455	+	0.945213i	$0.605854\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.a.e.1.2	yes	2
3.2	odd	2	inner	252.4.a.e.1.1	✓	2
4.3	odd	2		1008.4.a.bd.1.2		2
7.2	even	3		1764.4.k.t.361.1		4
7.3	odd	6		1764.4.k.w.1549.2		4
7.4	even	3		1764.4.k.t.1549.1		4
7.5	odd	6		1764.4.k.w.361.2		4
7.6	odd	2		1764.4.a.t.1.1		2
12.11	even	2		1008.4.a.bd.1.1		2
21.2	odd	6		1764.4.k.t.361.2		4
21.5	even	6		1764.4.k.w.361.1		4
21.11	odd	6		1764.4.k.t.1549.2		4
21.17	even	6		1764.4.k.w.1549.1		4
21.20	even	2		1764.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.a.e.1.1	✓	2	3.2	odd	2	inner
252.4.a.e.1.2	yes	2	1.1	even	1	trivial
1008.4.a.bd.1.1		2	12.11	even	2
1008.4.a.bd.1.2		2	4.3	odd	2
1764.4.a.t.1.1		2	7.6	odd	2
1764.4.a.t.1.2		2	21.20	even	2
1764.4.k.t.361.1		4	7.2	even	3
1764.4.k.t.361.2		4	21.2	odd	6
1764.4.k.t.1549.1		4	7.4	even	3
1764.4.k.t.1549.2		4	21.11	odd	6
1764.4.k.w.361.1		4	21.5	even	6
1764.4.k.w.361.2		4	7.5	odd	6
1764.4.k.w.1549.1		4	21.17	even	6
1764.4.k.w.1549.2		4	7.3	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q+5.29150 q^{5} -7.00000 q^{7} +O(q^{10})\)	\(q+5.29150 q^{5} -7.00000 q^{7} -68.7895 q^{11} -30.0000 q^{13} +121.705 q^{17} -100.000 q^{19} +89.9555 q^{23} -97.0000 q^{25} -232.826 q^{29} -180.000 q^{31} -37.0405 q^{35} -118.000 q^{37} -15.8745 q^{41} -412.000 q^{43} +285.741 q^{47} +49.0000 q^{49} +285.741 q^{53} -364.000 q^{55} -836.057 q^{59} -378.000 q^{61} -158.745 q^{65} +244.000 q^{67} +439.195 q^{71} +670.000 q^{73} +481.527 q^{77} +216.000 q^{79} +804.308 q^{83} +644.000 q^{85} -968.345 q^{89} +210.000 q^{91} -529.150 q^{95} +574.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7}+O(q^{10}) \) 2 * q - 14 * q^7	\( 2 q - 14 q^{7} - 60 q^{13} - 200 q^{19} - 194 q^{25} - 360 q^{31} - 236 q^{37} - 824 q^{43} + 98 q^{49} - 728 q^{55} - 756 q^{61} + 488 q^{67} + 1340 q^{73} + 432 q^{79} + 1288 q^{85} + 420 q^{91} + 1148 q^{97}+O(q^{100}) \) 2 * q - 14 * q^7 - 60 * q^13 - 200 * q^19 - 194 * q^25 - 360 * q^31 - 236 * q^37 - 824 * q^43 + 98 * q^49 - 728 * q^55 - 756 * q^61 + 488 * q^67 + 1340 * q^73 + 432 * q^79 + 1288 * q^85 + 420 * q^91 + 1148 * q^97

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