LMFDB - Embedded newform 648.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.39417$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.33932 q^{5} +19.1946 q^{7} +O(q^{10})$	$q-6.33932 q^{5} +19.1946 q^{7} -41.2460 q^{11} +39.2655 q^{13} -59.9385 q^{17} -41.2583 q^{19} +53.8079 q^{23} -84.8130 q^{25} +65.2595 q^{29} +64.7493 q^{31} -121.681 q^{35} +293.515 q^{37} -207.272 q^{41} -377.997 q^{43} -323.783 q^{47} +25.4314 q^{49} +340.588 q^{53} +261.471 q^{55} -406.962 q^{59} -757.235 q^{61} -248.917 q^{65} +844.344 q^{67} -859.728 q^{71} -789.079 q^{73} -791.699 q^{77} -217.909 q^{79} -101.178 q^{83} +379.969 q^{85} -1247.68 q^{89} +753.685 q^{91} +261.549 q^{95} -1747.01 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5}+O(q^{10}) $ 4 * q - 8 * q^5	$ 4 q - 8 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{17} + 80 q^{19} - 200 q^{23} + 8 q^{25} - 216 q^{29} + 80 q^{31} - 408 q^{35} - 276 q^{37} - 384 q^{41} + 160 q^{43} - 768 q^{47} - 268 q^{49} - 944 q^{53} + 304 q^{55} - 992 q^{59} - 548 q^{61} - 1328 q^{65} + 464 q^{67} - 1720 q^{71} - 764 q^{73} - 1728 q^{77} + 688 q^{79} - 2128 q^{83} - 1324 q^{85} - 2112 q^{89} + 1776 q^{91} - 2056 q^{95} - 1816 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^17 + 80 * q^19 - 200 * q^23 + 8 * q^25 - 216 * q^29 + 80 * q^31 - 408 * q^35 - 276 * q^37 - 384 * q^41 + 160 * q^43 - 768 * q^47 - 268 * q^49 - 944 * q^53 + 304 * q^55 - 992 * q^59 - 548 * q^61 - 1328 * q^65 + 464 * q^67 - 1720 * q^71 - 764 * q^73 - 1728 * q^77 + 688 * q^79 - 2128 * q^83 - 1324 * q^85 - 2112 * q^89 + 1776 * q^91 - 2056 * q^95 - 1816 * 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
$5$	−6.33932	−0.567006	−0.283503	−	0.958971i	$-0.591497\pi$
$5$	−6.33932	−0.567006	−0.283503	+	0.958971i	$0.591497\pi$
$6$	0	0
$7$	19.1946	1.03641	0.518205	−	0.855257i	$-0.326601\pi$
$7$	19.1946	1.03641	0.518205	+	0.855257i	$0.326601\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−41.2460	−1.13056	−0.565279	−	0.824900i	$-0.691231\pi$
$11$	−41.2460	−1.13056	−0.565279	+	0.824900i	$0.691231\pi$
$12$	0	0
$13$	39.2655	0.837715	0.418857	−	0.908052i	$-0.362431\pi$
$13$	39.2655	0.837715	0.418857	+	0.908052i	$0.362431\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−59.9385	−0.855131	−0.427565	−	0.903984i	$-0.640629\pi$
$17$	−59.9385	−0.855131	−0.427565	+	0.903984i	$0.640629\pi$
$18$	0	0
$19$	−41.2583	−0.498173	−0.249087	−	0.968481i	$-0.580130\pi$
$19$	−41.2583	−0.498173	−0.249087	+	0.968481i	$0.580130\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	53.8079	0.487814	0.243907	−	0.969799i	$-0.421571\pi$
$23$	53.8079	0.487814	0.243907	+	0.969799i	$0.421571\pi$
$24$	0	0
$25$	−84.8130	−0.678504
$26$	0	0
$27$	0	0
$28$	0	0
$29$	65.2595	0.417875	0.208938	−	0.977929i	$-0.432999\pi$
$29$	65.2595	0.417875	0.208938	+	0.977929i	$0.432999\pi$
$30$	0	0
$31$	64.7493	0.375139	0.187570	−	0.982251i	$-0.439939\pi$
$31$	64.7493	0.375139	0.187570	+	0.982251i	$0.439939\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−121.681	−0.587650
$36$	0	0
$37$	293.515	1.30415	0.652076	−	0.758154i	$-0.273898\pi$
$37$	293.515	1.30415	0.652076	+	0.758154i	$0.273898\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−207.272	−0.789524	−0.394762	−	0.918783i	$-0.629173\pi$
$41$	−207.272	−0.789524	−0.394762	+	0.918783i	$0.629173\pi$
$42$	0	0
$43$	−377.997	−1.34056	−0.670280	−	0.742108i	$-0.733826\pi$
$43$	−377.997	−1.34056	−0.670280	+	0.742108i	$0.733826\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−323.783	−1.00486	−0.502432	−	0.864617i	$-0.667561\pi$
$47$	−323.783	−1.00486	−0.502432	+	0.864617i	$0.667561\pi$
$48$	0	0
$49$	25.4314	0.0741442
$50$	0	0
$51$	0	0
$52$	0	0
$53$	340.588	0.882706	0.441353	−	0.897334i	$-0.354499\pi$
$53$	340.588	0.882706	0.441353	+	0.897334i	$0.354499\pi$
$54$	0	0
$55$	261.471	0.641033
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−406.962	−0.897999	−0.448999	−	0.893532i	$-0.648219\pi$
$59$	−406.962	−0.897999	−0.448999	+	0.893532i	$0.648219\pi$
$60$	0	0
$61$	−757.235	−1.58941	−0.794705	−	0.606996i	$-0.792374\pi$
$61$	−757.235	−1.58941	−0.794705	+	0.606996i	$0.792374\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−248.917	−0.474990
$66$	0	0
$67$	844.344	1.53960	0.769799	−	0.638287i	$-0.220357\pi$
$67$	844.344	1.53960	0.769799	+	0.638287i	$0.220357\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−859.728	−1.43706	−0.718528	−	0.695499i	$-0.755184\pi$
$71$	−859.728	−1.43706	−0.718528	+	0.695499i	$0.755184\pi$
$72$	0	0
$73$	−789.079	−1.26513	−0.632566	−	0.774506i	$-0.717998\pi$
$73$	−789.079	−1.26513	−0.632566	+	0.774506i	$0.717998\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−791.699	−1.17172
$78$	0	0
$79$	−217.909	−0.310338	−0.155169	−	0.987888i	$-0.549592\pi$
$79$	−217.909	−0.310338	−0.155169	+	0.987888i	$0.549592\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−101.178	−0.133804	−0.0669018	−	0.997760i	$-0.521311\pi$
$83$	−101.178	−0.133804	−0.0669018	+	0.997760i	$0.521311\pi$
$84$	0	0
$85$	379.969	0.484864
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1247.68	−1.48600	−0.742998	−	0.669294i	$-0.766597\pi$
$89$	−1247.68	−1.48600	−0.742998	+	0.669294i	$0.766597\pi$
$90$	0	0
$91$	753.685	0.868216
$92$	0	0
$93$	0	0
$94$	0	0
$95$	261.549	0.282467
$96$	0	0
$97$	−1747.01	−1.82868	−0.914340	−	0.404948i	$-0.867290\pi$
$97$	−1747.01	−1.82868	−0.914340	+	0.404948i	$0.867290\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1410.54	−1.38964	−0.694821	−	0.719183i	$-0.744516\pi$
$101$	−1410.54	−1.38964	−0.694821	+	0.719183i	$0.744516\pi$
$102$	0	0
$103$	879.881	0.841721	0.420861	−	0.907125i	$-0.361728\pi$
$103$	879.881	0.841721	0.420861	+	0.907125i	$0.361728\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−504.967	−0.456234	−0.228117	−	0.973634i	$-0.573257\pi$
$107$	−504.967	−0.456234	−0.228117	+	0.973634i	$0.573257\pi$
$108$	0	0
$109$	726.560	0.638457	0.319229	−	0.947678i	$-0.396576\pi$
$109$	726.560	0.638457	0.319229	+	0.947678i	$0.396576\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1518.76	−1.26436	−0.632182	−	0.774820i	$-0.717841\pi$
$113$	−1518.76	−1.26436	−0.632182	+	0.774820i	$0.717841\pi$
$114$	0	0
$115$	−341.106	−0.276594
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1150.49	−0.886265
$120$	0	0
$121$	370.230	0.278159
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1330.07	0.951722
$126$	0	0
$127$	2119.86	1.48116	0.740578	−	0.671970i	$-0.234552\pi$
$127$	2119.86	1.48116	0.740578	+	0.671970i	$0.234552\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2499.74	1.66720	0.833599	−	0.552369i	$-0.186276\pi$
$131$	2499.74	1.66720	0.833599	+	0.552369i	$0.186276\pi$
$132$	0	0
$133$	−791.935	−0.516312
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1373.11	0.856299	0.428149	−	0.903708i	$-0.359166\pi$
$137$	1373.11	0.856299	0.428149	+	0.903708i	$0.359166\pi$
$138$	0	0
$139$	1598.13	0.975194	0.487597	−	0.873069i	$-0.337874\pi$
$139$	1598.13	0.975194	0.487597	+	0.873069i	$0.337874\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1619.54	−0.947085
$144$	0	0
$145$	−413.701	−0.236938
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2495.97	−1.37233	−0.686167	−	0.727444i	$-0.740708\pi$
$149$	−2495.97	−1.37233	−0.686167	+	0.727444i	$0.740708\pi$
$150$	0	0
$151$	−1977.95	−1.06598	−0.532992	−	0.846120i	$-0.678932\pi$
$151$	−1977.95	−1.06598	−0.532992	+	0.846120i	$0.678932\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−410.467	−0.212706
$156$	0	0
$157$	153.278	0.0779165	0.0389583	−	0.999241i	$-0.487596\pi$
$157$	153.278	0.0779165	0.0389583	+	0.999241i	$0.487596\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1032.82	0.505575
$162$	0	0
$163$	55.9069	0.0268648	0.0134324	−	0.999910i	$-0.495724\pi$
$163$	55.9069	0.0268648	0.0134324	+	0.999910i	$0.495724\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	403.234	0.186845	0.0934226	−	0.995627i	$-0.470219\pi$
$167$	403.234	0.186845	0.0934226	+	0.995627i	$0.470219\pi$
$168$	0	0
$169$	−655.219	−0.298234
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1865.74	0.819940	0.409970	−	0.912099i	$-0.365539\pi$
$173$	1865.74	0.819940	0.409970	+	0.912099i	$0.365539\pi$
$174$	0	0
$175$	−1627.95	−0.703208
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2223.69	−0.928527	−0.464263	−	0.885697i	$-0.653681\pi$
$179$	−2223.69	−0.928527	−0.464263	+	0.885697i	$0.653681\pi$
$180$	0	0
$181$	−3287.28	−1.34995	−0.674976	−	0.737840i	$-0.735846\pi$
$181$	−3287.28	−1.34995	−0.674976	+	0.737840i	$0.735846\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1860.69	−0.739462
$186$	0	0
$187$	2472.22	0.966774
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4100.14	−1.55328	−0.776638	−	0.629947i	$-0.783077\pi$
$191$	−4100.14	−1.55328	−0.776638	+	0.629947i	$0.783077\pi$
$192$	0	0
$193$	−3941.53	−1.47004	−0.735020	−	0.678046i	$-0.762827\pi$
$193$	−3941.53	−1.47004	−0.735020	+	0.678046i	$0.762827\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	466.195	0.168604	0.0843020	−	0.996440i	$-0.473134\pi$
$197$	466.195	0.168604	0.0843020	+	0.996440i	$0.473134\pi$
$198$	0	0
$199$	−2136.33	−0.761007	−0.380503	−	0.924780i	$-0.624249\pi$
$199$	−2136.33	−0.761007	−0.380503	+	0.924780i	$0.624249\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1252.63	0.433090
$204$	0	0
$205$	1313.96	0.447665
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1701.74	0.563214
$210$	0	0
$211$	4145.34	1.35250	0.676249	−	0.736673i	$-0.263604\pi$
$211$	4145.34	1.35250	0.676249	+	0.736673i	$0.263604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2396.25	0.760106
$216$	0	0
$217$	1242.83	0.388798
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2353.52	−0.716356
$222$	0	0
$223$	6511.56	1.95536	0.977682	−	0.210089i	$-0.0673752\pi$
$223$	6511.56	1.95536	0.977682	+	0.210089i	$0.0673752\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3573.92	1.04498	0.522488	−	0.852647i	$-0.325004\pi$
$227$	3573.92	1.04498	0.522488	+	0.852647i	$0.325004\pi$
$228$	0	0
$229$	−1849.26	−0.533637	−0.266818	−	0.963747i	$-0.585972\pi$
$229$	−1849.26	−0.533637	−0.266818	+	0.963747i	$0.585972\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5359.31	1.50687	0.753434	−	0.657524i	$-0.228396\pi$
$233$	5359.31	1.50687	0.753434	+	0.657524i	$0.228396\pi$
$234$	0	0
$235$	2052.56	0.569764
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5334.48	1.44376	0.721881	−	0.692017i	$-0.243278\pi$
$239$	5334.48	1.44376	0.721881	+	0.692017i	$0.243278\pi$
$240$	0	0
$241$	−2327.39	−0.622075	−0.311038	−	0.950398i	$-0.600677\pi$
$241$	−2327.39	−0.622075	−0.311038	+	0.950398i	$0.600677\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−161.218	−0.0420402
$246$	0	0
$247$	−1620.03	−0.417327
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−938.870	−0.236099	−0.118050	−	0.993008i	$-0.537664\pi$
$251$	−938.870	−0.236099	−0.118050	+	0.993008i	$0.537664\pi$
$252$	0	0
$253$	−2219.36	−0.551502
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4383.44	1.06393	0.531967	−	0.846765i	$-0.321453\pi$
$257$	4383.44	1.06393	0.531967	+	0.846765i	$0.321453\pi$
$258$	0	0
$259$	5633.90	1.35163
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4379.32	1.02677	0.513384	−	0.858159i	$-0.328392\pi$
$263$	4379.32	1.02677	0.513384	+	0.858159i	$0.328392\pi$
$264$	0	0
$265$	−2159.10	−0.500500
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1981.97	0.449231	0.224615	−	0.974447i	$-0.427887\pi$
$269$	1981.97	0.449231	0.224615	+	0.974447i	$0.427887\pi$
$270$	0	0
$271$	806.801	0.180847	0.0904237	−	0.995903i	$-0.471178\pi$
$271$	806.801	0.180847	0.0904237	+	0.995903i	$0.471178\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3498.19	0.767087
$276$	0	0
$277$	2.55558	0.000554333 0	0.000277166	−	1.00000i	$-0.499912\pi$
$277$	2.55558	0.000554333 0	0.000277166		1.00000i	$0.499912\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	82.6189	0.0175396	0.00876981	−	0.999962i	$-0.497208\pi$
$281$	82.6189	0.0175396	0.00876981	+	0.999962i	$0.497208\pi$
$282$	0	0
$283$	−4448.26	−0.934351	−0.467176	−	0.884165i	$-0.654728\pi$
$283$	−4448.26	−0.934351	−0.467176	+	0.884165i	$0.654728\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3978.50	−0.818270
$288$	0	0
$289$	−1320.38	−0.268752
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3413.56	0.680623	0.340311	−	0.940313i	$-0.389468\pi$
$293$	3413.56	0.680623	0.340311	+	0.940313i	$0.389468\pi$
$294$	0	0
$295$	2579.86	0.509171
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2112.80	0.408649
$300$	0	0
$301$	−7255.50	−1.38937
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4800.35	0.901205
$306$	0	0
$307$	2102.23	0.390817	0.195409	−	0.980722i	$-0.437397\pi$
$307$	2102.23	0.390817	0.195409	+	0.980722i	$0.437397\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7092.19	1.29312	0.646562	−	0.762862i	$-0.276206\pi$
$311$	7092.19	1.29312	0.646562	+	0.762862i	$0.276206\pi$
$312$	0	0
$313$	10485.4	1.89352	0.946759	−	0.321944i	$-0.104336\pi$
$313$	10485.4	1.89352	0.946759	+	0.321944i	$0.104336\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7517.49	1.33194	0.665969	−	0.745980i	$-0.268018\pi$
$317$	7517.49	1.33194	0.665969	+	0.745980i	$0.268018\pi$
$318$	0	0
$319$	−2691.69	−0.472432
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2472.96	0.426003
$324$	0	0
$325$	−3330.23	−0.568393
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6214.87	−1.04145
$330$	0	0
$331$	2227.14	0.369832	0.184916	−	0.982754i	$-0.440799\pi$
$331$	2227.14	0.369832	0.184916	+	0.982754i	$0.440799\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5352.57	−0.872961
$336$	0	0
$337$	3923.52	0.634208	0.317104	−	0.948391i	$-0.397290\pi$
$337$	3923.52	0.634208	0.317104	+	0.948391i	$0.397290\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2670.65	−0.424116
$342$	0	0
$343$	−6095.59	−0.959566
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12108.5	−1.87325	−0.936623	−	0.350340i	$-0.886066\pi$
$347$	−12108.5	−1.87325	−0.936623	+	0.350340i	$0.886066\pi$
$348$	0	0
$349$	5587.78	0.857040	0.428520	−	0.903532i	$-0.359035\pi$
$349$	5587.78	0.857040	0.428520	+	0.903532i	$0.359035\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3914.17	−0.590170	−0.295085	−	0.955471i	$-0.595348\pi$
$353$	−3914.17	−0.590170	−0.295085	+	0.955471i	$0.595348\pi$
$354$	0	0
$355$	5450.09	0.814819
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9037.04	1.32857	0.664285	−	0.747479i	$-0.268736\pi$
$359$	9037.04	1.32857	0.664285	+	0.747479i	$0.268736\pi$
$360$	0	0
$361$	−5156.76	−0.751823
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5002.22	0.717338
$366$	0	0
$367$	−4282.73	−0.609146	−0.304573	−	0.952489i	$-0.598514\pi$
$367$	−4282.73	−0.609146	−0.304573	+	0.952489i	$0.598514\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6537.45	0.914844
$372$	0	0
$373$	−2907.11	−0.403551	−0.201775	−	0.979432i	$-0.564671\pi$
$373$	−2907.11	−0.403551	−0.201775	+	0.979432i	$0.564671\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2562.45	0.350060
$378$	0	0
$379$	−2096.88	−0.284194	−0.142097	−	0.989853i	$-0.545385\pi$
$379$	−2096.88	−0.284194	−0.142097	+	0.989853i	$0.545385\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7174.92	0.957236	0.478618	−	0.878023i	$-0.341138\pi$
$383$	7174.92	0.957236	0.478618	+	0.878023i	$0.341138\pi$
$384$	0	0
$385$	5018.83	0.664372
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2737.71	0.356831	0.178415	−	0.983955i	$-0.442903\pi$
$389$	2737.71	0.356831	0.178415	+	0.983955i	$0.442903\pi$
$390$	0	0
$391$	−3225.17	−0.417145
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1381.40	0.175964
$396$	0	0
$397$	1561.87	0.197451	0.0987256	−	0.995115i	$-0.468523\pi$
$397$	1561.87	0.197451	0.0987256	+	0.995115i	$0.468523\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4853.86	−0.604465	−0.302232	−	0.953234i	$-0.597732\pi$
$401$	−4853.86	−0.604465	−0.302232	+	0.953234i	$0.597732\pi$
$402$	0	0
$403$	2542.41	0.314260
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12106.3	−1.47442
$408$	0	0
$409$	1143.74	0.138274	0.0691370	−	0.997607i	$-0.477975\pi$
$409$	1143.74	0.138274	0.0691370	+	0.997607i	$0.477975\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7811.46	−0.930694
$414$	0	0
$415$	641.398	0.0758675
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1320.74	0.153992	0.0769958	−	0.997031i	$-0.475467\pi$
$419$	1320.74	0.153992	0.0769958	+	0.997031i	$0.475467\pi$
$420$	0	0
$421$	−11282.7	−1.30614	−0.653069	−	0.757299i	$-0.726519\pi$
$421$	−11282.7	−1.30614	−0.653069	+	0.757299i	$0.726519\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5083.56	0.580210
$426$	0	0
$427$	−14534.8	−1.64728
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5443.17	0.608325	0.304163	−	0.952620i	$-0.401623\pi$
$431$	5443.17	0.608325	0.304163	+	0.952620i	$0.401623\pi$
$432$	0	0
$433$	4235.84	0.470119	0.235060	−	0.971981i	$-0.424471\pi$
$433$	4235.84	0.470119	0.235060	+	0.971981i	$0.424471\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2220.02	−0.243016
$438$	0	0
$439$	−14160.7	−1.53953	−0.769765	−	0.638328i	$-0.779627\pi$
$439$	−14160.7	−1.53953	−0.769765	+	0.638328i	$0.779627\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15671.0	1.68070	0.840352	−	0.542042i	$-0.182349\pi$
$443$	15671.0	1.68070	0.840352	+	0.542042i	$0.182349\pi$
$444$	0	0
$445$	7909.43	0.842569
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14045.2	−1.47624	−0.738122	−	0.674667i	$-0.764287\pi$
$449$	−14045.2	−1.47624	−0.738122	+	0.674667i	$0.764287\pi$
$450$	0	0
$451$	8549.14	0.892602
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4777.85	−0.492284
$456$	0	0
$457$	−890.871	−0.0911886	−0.0455943	−	0.998960i	$-0.514518\pi$
$457$	−890.871	−0.0911886	−0.0455943	+	0.998960i	$0.514518\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7981.86	−0.806404	−0.403202	−	0.915111i	$-0.632103\pi$
$461$	−7981.86	−0.806404	−0.403202	+	0.915111i	$0.632103\pi$
$462$	0	0
$463$	3300.75	0.331315	0.165658	−	0.986183i	$-0.447025\pi$
$463$	3300.75	0.331315	0.165658	+	0.986183i	$0.447025\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11181.4	−1.10795	−0.553977	−	0.832532i	$-0.686891\pi$
$467$	−11181.4	−1.10795	−0.553977	+	0.832532i	$0.686891\pi$
$468$	0	0
$469$	16206.8	1.59565
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15590.9	1.51558
$474$	0	0
$475$	3499.24	0.338013
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2889.55	−0.275630	−0.137815	−	0.990458i	$-0.544008\pi$
$479$	−2889.55	−0.275630	−0.137815	+	0.990458i	$0.544008\pi$
$480$	0	0
$481$	11525.0	1.09251
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11074.9	1.03687
$486$	0	0
$487$	−13486.6	−1.25490	−0.627448	−	0.778658i	$-0.715901\pi$
$487$	−13486.6	−1.25490	−0.627448	+	0.778658i	$0.715901\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21088.0	−1.93827	−0.969134	−	0.246535i	$-0.920708\pi$
$491$	−21088.0	−1.93827	−0.969134	+	0.246535i	$0.920708\pi$
$492$	0	0
$493$	−3911.56	−0.357338
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16502.1	−1.48938
$498$	0	0
$499$	9870.15	0.885469	0.442734	−	0.896653i	$-0.354009\pi$
$499$	9870.15	0.885469	0.442734	+	0.896653i	$0.354009\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3945.75	0.349766	0.174883	−	0.984589i	$-0.444045\pi$
$503$	3945.75	0.349766	0.174883	+	0.984589i	$0.444045\pi$
$504$	0	0
$505$	8941.86	0.787935
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13604.1	1.18466	0.592331	−	0.805695i	$-0.298208\pi$
$509$	13604.1	1.18466	0.592331	+	0.805695i	$0.298208\pi$
$510$	0	0
$511$	−15146.0	−1.31119
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5577.85	−0.477261
$516$	0	0
$517$	13354.7	1.13606
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8044.57	−0.676467	−0.338233	−	0.941062i	$-0.609829\pi$
$521$	−8044.57	−0.676467	−0.338233	+	0.941062i	$0.609829\pi$
$522$	0	0
$523$	7785.42	0.650923	0.325462	−	0.945555i	$-0.394480\pi$
$523$	7785.42	0.650923	0.325462	+	0.945555i	$0.394480\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3880.98	−0.320793
$528$	0	0
$529$	−9271.71	−0.762037
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8138.65	−0.661396
$534$	0	0
$535$	3201.15	0.258687
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1048.94	−0.0838242
$540$	0	0
$541$	−2520.37	−0.200294	−0.100147	−	0.994973i	$-0.531931\pi$
$541$	−2520.37	−0.200294	−0.100147	+	0.994973i	$0.531931\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4605.90	−0.362009
$546$	0	0
$547$	−1243.27	−0.0971819	−0.0485910	−	0.998819i	$-0.515473\pi$
$547$	−1243.27	−0.0971819	−0.0485910	+	0.998819i	$0.515473\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2692.49	−0.208174
$552$	0	0
$553$	−4182.68	−0.321637
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14215.4	−1.08137	−0.540687	−	0.841224i	$-0.681836\pi$
$557$	−14215.4	−1.08137	−0.540687	+	0.841224i	$0.681836\pi$
$558$	0	0
$559$	−14842.3	−1.12301
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21612.4	−1.61786	−0.808928	−	0.587907i	$-0.799952\pi$
$563$	−21612.4	−1.61786	−0.808928	+	0.587907i	$0.799952\pi$
$564$	0	0
$565$	9627.93	0.716903
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11385.6	−0.838857	−0.419428	−	0.907788i	$-0.637769\pi$
$569$	−11385.6	−0.838857	−0.419428	+	0.907788i	$0.637769\pi$
$570$	0	0
$571$	23265.3	1.70512	0.852560	−	0.522630i	$-0.175049\pi$
$571$	23265.3	1.70512	0.852560	+	0.522630i	$0.175049\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4563.61	−0.330984
$576$	0	0
$577$	−5871.82	−0.423652	−0.211826	−	0.977307i	$-0.567941\pi$
$577$	−5871.82	−0.423652	−0.211826	+	0.977307i	$0.567941\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1942.06	−0.138675
$582$	0	0
$583$	−14047.9	−0.997949
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9743.60	0.685113	0.342557	−	0.939497i	$-0.388707\pi$
$587$	9743.60	0.685113	0.342557	+	0.939497i	$0.388707\pi$
$588$	0	0
$589$	−2671.44	−0.186884
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11058.0	−0.765764	−0.382882	−	0.923797i	$-0.625068\pi$
$593$	−11058.0	−0.765764	−0.382882	+	0.923797i	$0.625068\pi$
$594$	0	0
$595$	7293.35	0.502518
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26348.2	−1.79726	−0.898629	−	0.438709i	$-0.855436\pi$
$599$	−26348.2	−1.79726	−0.898629	+	0.438709i	$0.855436\pi$
$600$	0	0
$601$	−15110.3	−1.02556	−0.512782	−	0.858519i	$-0.671385\pi$
$601$	−15110.3	−1.02556	−0.512782	+	0.858519i	$0.671385\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2347.01	−0.157718
$606$	0	0
$607$	−13482.7	−0.901561	−0.450780	−	0.892635i	$-0.648854\pi$
$607$	−13482.7	−0.901561	−0.450780	+	0.892635i	$0.648854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12713.5	−0.841789
$612$	0	0
$613$	12821.6	0.844798	0.422399	−	0.906410i	$-0.361188\pi$
$613$	12821.6	0.844798	0.422399	+	0.906410i	$0.361188\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18125.0	1.18263	0.591317	−	0.806439i	$-0.298608\pi$
$617$	18125.0	1.18263	0.591317	+	0.806439i	$0.298608\pi$
$618$	0	0
$619$	−23499.8	−1.52591	−0.762954	−	0.646453i	$-0.776252\pi$
$619$	−23499.8	−1.52591	−0.762954	+	0.646453i	$0.776252\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−23948.6	−1.54010
$624$	0	0
$625$	2169.87	0.138872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17592.9	−1.11522
$630$	0	0
$631$	−4893.60	−0.308734	−0.154367	−	0.988014i	$-0.549334\pi$
$631$	−4893.60	−0.308734	−0.154367	+	0.988014i	$0.549334\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13438.5	−0.839825
$636$	0	0
$637$	998.579	0.0621117
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28784.0	1.77363	0.886816	−	0.462122i	$-0.152912\pi$
$641$	28784.0	1.77363	0.886816	+	0.462122i	$0.152912\pi$
$642$	0	0
$643$	−896.295	−0.0549711	−0.0274855	−	0.999622i	$-0.508750\pi$
$643$	−896.295	−0.0549711	−0.0274855	+	0.999622i	$0.508750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30780.9	1.87036	0.935179	−	0.354174i	$-0.115238\pi$
$647$	30780.9	1.87036	0.935179	+	0.354174i	$0.115238\pi$
$648$	0	0
$649$	16785.5	1.01524
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13734.1	−0.823059	−0.411529	−	0.911396i	$-0.635005\pi$
$653$	−13734.1	−0.823059	−0.411529	+	0.911396i	$0.635005\pi$
$654$	0	0
$655$	−15846.6	−0.945312
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16587.2	0.980494	0.490247	−	0.871583i	$-0.336906\pi$
$659$	16587.2	0.980494	0.490247	+	0.871583i	$0.336906\pi$
$660$	0	0
$661$	28246.1	1.66209	0.831047	−	0.556202i	$-0.187742\pi$
$661$	28246.1	1.66209	0.831047	+	0.556202i	$0.187742\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5020.33	0.292752
$666$	0	0
$667$	3511.48	0.203846
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31232.9	1.79692
$672$	0	0
$673$	−12562.7	−0.719551	−0.359775	−	0.933039i	$-0.617147\pi$
$673$	−12562.7	−0.719551	−0.359775	+	0.933039i	$0.617147\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26532.4	−1.50624	−0.753119	−	0.657885i	$-0.771451\pi$
$677$	−26532.4	−1.50624	−0.753119	+	0.657885i	$0.771451\pi$
$678$	0	0
$679$	−33533.1	−1.89526
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2528.33	0.141645	0.0708226	−	0.997489i	$-0.477438\pi$
$683$	2528.33	0.141645	0.0708226	+	0.997489i	$0.477438\pi$
$684$	0	0
$685$	−8704.60	−0.485527
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13373.4	0.739456
$690$	0	0
$691$	18836.2	1.03699	0.518496	−	0.855080i	$-0.326492\pi$
$691$	18836.2	1.03699	0.518496	+	0.855080i	$0.326492\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10131.1	−0.552941
$696$	0	0
$697$	12423.6	0.675146
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26573.0	1.43174	0.715869	−	0.698234i	$-0.246031\pi$
$701$	26573.0	1.43174	0.715869	+	0.698234i	$0.246031\pi$
$702$	0	0
$703$	−12109.9	−0.649694
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27074.7	−1.44024
$708$	0	0
$709$	6220.82	0.329517	0.164759	−	0.986334i	$-0.447315\pi$
$709$	6220.82	0.329517	0.164759	+	0.986334i	$0.447315\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3484.03	0.182998
$714$	0	0
$715$	10266.8	0.537003
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13175.7	0.683407	0.341704	−	0.939808i	$-0.388996\pi$
$719$	13175.7	0.683407	0.341704	+	0.939808i	$0.388996\pi$
$720$	0	0
$721$	16888.9	0.872368
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5534.85	−0.283530
$726$	0	0
$727$	−20675.1	−1.05474	−0.527370	−	0.849636i	$-0.676822\pi$
$727$	−20675.1	−1.05474	−0.527370	+	0.849636i	$0.676822\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22656.6	1.14635
$732$	0	0
$733$	5609.02	0.282638	0.141319	−	0.989964i	$-0.454866\pi$
$733$	5609.02	0.282638	0.141319	+	0.989964i	$0.454866\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34825.8	−1.74060
$738$	0	0
$739$	14434.8	0.718526	0.359263	−	0.933236i	$-0.383028\pi$
$739$	14434.8	0.718526	0.359263	+	0.933236i	$0.383028\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24769.3	1.22301	0.611506	−	0.791240i	$-0.290564\pi$
$743$	24769.3	1.22301	0.611506	+	0.791240i	$0.290564\pi$
$744$	0	0
$745$	15822.7	0.778121
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9692.63	−0.472845
$750$	0	0
$751$	2381.46	0.115713	0.0578566	−	0.998325i	$-0.481573\pi$
$751$	2381.46	0.115713	0.0578566	+	0.998325i	$0.481573\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12538.9	0.604419
$756$	0	0
$757$	14556.5	0.698897	0.349449	−	0.936956i	$-0.386369\pi$
$757$	14556.5	0.698897	0.349449	+	0.936956i	$0.386369\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10105.5	−0.481370	−0.240685	−	0.970603i	$-0.577372\pi$
$761$	−10105.5	−0.481370	−0.240685	+	0.970603i	$0.577372\pi$
$762$	0	0
$763$	13946.0	0.661703
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15979.6	−0.752267
$768$	0	0
$769$	−11631.1	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$769$	−11631.1	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16482.4	−0.766922	−0.383461	−	0.923557i	$-0.625268\pi$
$773$	−16482.4	−0.766922	−0.383461	+	0.923557i	$0.625268\pi$
$774$	0	0
$775$	−5491.58	−0.254534
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8551.69	0.393320
$780$	0	0
$781$	35460.3	1.62467
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−971.677	−0.0441792
$786$	0	0
$787$	−14600.7	−0.661320	−0.330660	−	0.943750i	$-0.607271\pi$
$787$	−14600.7	−0.661320	−0.330660	+	0.943750i	$0.607271\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29152.0	−1.31040
$792$	0	0
$793$	−29733.2	−1.33147
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21830.2	0.970222	0.485111	−	0.874453i	$-0.338779\pi$
$797$	21830.2	0.970222	0.485111	+	0.874453i	$0.338779\pi$
$798$	0	0
$799$	19407.1	0.859290
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32546.3	1.43030
$804$	0	0
$805$	−6547.38	−0.286664
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14798.7	0.643133	0.321567	−	0.946887i	$-0.395791\pi$
$809$	14798.7	0.643133	0.321567	+	0.946887i	$0.395791\pi$
$810$	0	0
$811$	−23276.2	−1.00782	−0.503908	−	0.863757i	$-0.668105\pi$
$811$	−23276.2	−1.00782	−0.503908	+	0.863757i	$0.668105\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−354.412	−0.0152325
$816$	0	0
$817$	15595.5	0.667831
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27340.9	1.16225	0.581124	−	0.813815i	$-0.302613\pi$
$821$	27340.9	1.16225	0.581124	+	0.813815i	$0.302613\pi$
$822$	0	0
$823$	−14930.1	−0.632357	−0.316179	−	0.948700i	$-0.602400\pi$
$823$	−14930.1	−0.632357	−0.316179	+	0.948700i	$0.602400\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10707.3	−0.450219	−0.225109	−	0.974334i	$-0.572274\pi$
$827$	−10707.3	−0.450219	−0.225109	+	0.974334i	$0.572274\pi$
$828$	0	0
$829$	−28515.9	−1.19469	−0.597345	−	0.801984i	$-0.703778\pi$
$829$	−28515.9	−1.19469	−0.597345	+	0.801984i	$0.703778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1524.32	−0.0634029
$834$	0	0
$835$	−2556.23	−0.105942
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38422.0	−1.58102	−0.790510	−	0.612449i	$-0.790185\pi$
$839$	−38422.0	−1.58102	−0.790510	+	0.612449i	$0.790185\pi$
$840$	0	0
$841$	−20130.2	−0.825380
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4153.65	0.169100
$846$	0	0
$847$	7106.40	0.288287
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15793.4	0.636184
$852$	0	0
$853$	−16696.6	−0.670201	−0.335100	−	0.942182i	$-0.608770\pi$
$853$	−16696.6	−0.670201	−0.335100	+	0.942182i	$0.608770\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9336.14	−0.372131	−0.186066	−	0.982537i	$-0.559574\pi$
$857$	−9336.14	−0.372131	−0.186066	+	0.982537i	$0.559574\pi$
$858$	0	0
$859$	22903.3	0.909721	0.454861	−	0.890563i	$-0.349689\pi$
$859$	22903.3	0.909721	0.454861	+	0.890563i	$0.349689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31426.0	1.23958	0.619789	−	0.784769i	$-0.287218\pi$
$863$	31426.0	1.23958	0.619789	+	0.784769i	$0.287218\pi$
$864$	0	0
$865$	−11827.5	−0.464911
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8987.88	0.350855
$870$	0	0
$871$	33153.6	1.28974
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25530.2	0.986374
$876$	0	0
$877$	−39180.1	−1.50857	−0.754286	−	0.656546i	$-0.772017\pi$
$877$	−39180.1	−1.50857	−0.754286	+	0.656546i	$0.772017\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23946.2	0.915743	0.457871	−	0.889018i	$-0.348612\pi$
$881$	23946.2	0.915743	0.457871	+	0.889018i	$0.348612\pi$
$882$	0	0
$883$	3969.15	0.151271	0.0756357	−	0.997136i	$-0.475901\pi$
$883$	3969.15	0.151271	0.0756357	+	0.997136i	$0.475901\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9450.30	0.357734	0.178867	−	0.983873i	$-0.442757\pi$
$887$	9450.30	0.357734	0.178867	+	0.983873i	$0.442757\pi$
$888$	0	0
$889$	40689.7	1.53508
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13358.7	0.500597
$894$	0	0
$895$	14096.7	0.526480
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4225.51	0.156761
$900$	0	0
$901$	−20414.4	−0.754829
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20839.1	0.765431
$906$	0	0
$907$	40371.4	1.47796	0.738980	−	0.673728i	$-0.235308\pi$
$907$	40371.4	1.47796	0.738980	+	0.673728i	$0.235308\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16280.7	0.592099	0.296050	−	0.955173i	$-0.404331\pi$
$911$	16280.7	0.592099	0.296050	+	0.955173i	$0.404331\pi$
$912$	0	0
$913$	4173.17	0.151273
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47981.4	1.72790
$918$	0	0
$919$	17967.5	0.644934	0.322467	−	0.946581i	$-0.395488\pi$
$919$	17967.5	0.644934	0.322467	+	0.946581i	$0.395488\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−33757.7	−1.20384
$924$	0	0
$925$	−24893.9	−0.884872
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17416.6	0.615090	0.307545	−	0.951533i	$-0.400492\pi$
$929$	17416.6	0.615090	0.307545	+	0.951533i	$0.400492\pi$
$930$	0	0
$931$	−1049.26	−0.0369367
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15672.2	−0.548167
$936$	0	0
$937$	12121.1	0.422604	0.211302	−	0.977421i	$-0.432230\pi$
$937$	12121.1	0.422604	0.211302	+	0.977421i	$0.432230\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15403.5	0.533622	0.266811	−	0.963749i	$-0.414030\pi$
$941$	15403.5	0.533622	0.266811	+	0.963749i	$0.414030\pi$
$942$	0	0
$943$	−11152.9	−0.385141
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31886.5	1.09416	0.547082	−	0.837079i	$-0.315739\pi$
$947$	31886.5	1.09416	0.547082	+	0.837079i	$0.315739\pi$
$948$	0	0
$949$	−30983.6	−1.05982
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38339.2	1.30318	0.651589	−	0.758572i	$-0.274103\pi$
$953$	38339.2	1.30318	0.651589	+	0.758572i	$0.274103\pi$
$954$	0	0
$955$	25992.1	0.880717
$956$	0	0
$957$	0	0
$958$	0	0
$959$	26356.3	0.887476
$960$	0	0
$961$	−25598.5	−0.859271
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24986.6	0.833521
$966$	0	0
$967$	−56307.0	−1.87251	−0.936253	−	0.351328i	$-0.885730\pi$
$967$	−56307.0	−1.87251	−0.936253	+	0.351328i	$0.885730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35994.8	1.18963	0.594814	−	0.803864i	$-0.297226\pi$
$971$	35994.8	1.18963	0.594814	+	0.803864i	$0.297226\pi$
$972$	0	0
$973$	30675.5	1.01070
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30144.1	−0.987099	−0.493549	−	0.869718i	$-0.664301\pi$
$977$	−30144.1	−0.987099	−0.493549	+	0.869718i	$0.664301\pi$
$978$	0	0
$979$	51461.7	1.68000
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38865.8	−1.26106	−0.630532	−	0.776163i	$-0.717163\pi$
$983$	−38865.8	−1.26106	−0.630532	+	0.776163i	$0.717163\pi$
$984$	0	0
$985$	−2955.36	−0.0955995
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20339.3	−0.653944
$990$	0	0
$991$	−43210.3	−1.38509	−0.692543	−	0.721377i	$-0.743510\pi$
$991$	−43210.3	−1.38509	−0.692543	+	0.721377i	$0.743510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13542.9	0.431495
$996$	0	0
$997$	31814.6	1.01061	0.505306	−	0.862940i	$-0.331380\pi$
$997$	31814.6	1.01061	0.505306	+	0.862940i	$0.331380\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.g.1.2	✓	4
3.2	odd	2		648.4.a.j.1.3	yes	4
4.3	odd	2		1296.4.a.x.1.2		4
9.2	odd	6		648.4.i.u.433.2		8
9.4	even	3		648.4.i.v.217.3		8
9.5	odd	6		648.4.i.u.217.2		8
9.7	even	3		648.4.i.v.433.3		8
12.11	even	2		1296.4.a.bb.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.2	✓	4	1.1	even	1	trivial
648.4.a.j.1.3	yes	4	3.2	odd	2
648.4.i.u.217.2		8	9.5	odd	6
648.4.i.u.433.2		8	9.2	odd	6
648.4.i.v.217.3		8	9.4	even	3
648.4.i.v.433.3		8	9.7	even	3
1296.4.a.x.1.2		4	4.3	odd	2
1296.4.a.bb.1.3		4	12.11	even	2

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q-6.33932 q^{5} +19.1946 q^{7} +O(q^{10})\)	\(q-6.33932 q^{5} +19.1946 q^{7} -41.2460 q^{11} +39.2655 q^{13} -59.9385 q^{17} -41.2583 q^{19} +53.8079 q^{23} -84.8130 q^{25} +65.2595 q^{29} +64.7493 q^{31} -121.681 q^{35} +293.515 q^{37} -207.272 q^{41} -377.997 q^{43} -323.783 q^{47} +25.4314 q^{49} +340.588 q^{53} +261.471 q^{55} -406.962 q^{59} -757.235 q^{61} -248.917 q^{65} +844.344 q^{67} -859.728 q^{71} -789.079 q^{73} -791.699 q^{77} -217.909 q^{79} -101.178 q^{83} +379.969 q^{85} -1247.68 q^{89} +753.685 q^{91} +261.549 q^{95} -1747.01 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5}+O(q^{10}) \) 4 * q - 8 * q^5	\( 4 q - 8 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{17} + 80 q^{19} - 200 q^{23} + 8 q^{25} - 216 q^{29} + 80 q^{31} - 408 q^{35} - 276 q^{37} - 384 q^{41} + 160 q^{43} - 768 q^{47} - 268 q^{49} - 944 q^{53} + 304 q^{55} - 992 q^{59} - 548 q^{61} - 1328 q^{65} + 464 q^{67} - 1720 q^{71} - 764 q^{73} - 1728 q^{77} + 688 q^{79} - 2128 q^{83} - 1324 q^{85} - 2112 q^{89} + 1776 q^{91} - 2056 q^{95} - 1816 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^17 + 80 * q^19 - 200 * q^23 + 8 * q^25 - 216 * q^29 + 80 * q^31 - 408 * q^35 - 276 * q^37 - 384 * q^41 + 160 * q^43 - 768 * q^47 - 268 * q^49 - 944 * q^53 + 304 * q^55 - 992 * q^59 - 548 * q^61 - 1328 * q^65 + 464 * q^67 - 1720 * q^71 - 764 * q^73 - 1728 * q^77 + 688 * q^79 - 2128 * q^83 - 1324 * q^85 - 2112 * q^89 + 1776 * q^91 - 2056 * q^95 - 1816 * q^97

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