LMFDB - Embedded newform 1296.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, 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("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
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:	no (minimal twist has level 648)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39417$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-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	$-0.408503\pi$
$5$	6.33932	0.567006	0.283503	+	0.958971i	$0.408503\pi$
$6$	0	0
$7$	−19.1946	−1.03641	−0.518205	−	0.855257i	$-0.673399\pi$
$7$	−19.1946	−1.03641	−0.518205	+	0.855257i	$0.673399\pi$
$8$	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	$-0.359371\pi$
$17$	59.9385	0.855131	0.427565	+	0.903984i	$0.359371\pi$
$18$	0	0
$19$	41.2583	0.498173	0.249087	−	0.968481i	$-0.419870\pi$
$19$	41.2583	0.498173	0.249087	+	0.968481i	$0.419870\pi$
$20$	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	$-0.567001\pi$
$29$	−65.2595	−0.417875	−0.208938	+	0.977929i	$0.567001\pi$
$30$	0	0
$31$	−64.7493	−0.375139	−0.187570	−	0.982251i	$-0.560061\pi$
$31$	−64.7493	−0.375139	−0.187570	+	0.982251i	$0.560061\pi$
$32$	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	$-0.370827\pi$
$41$	207.272	0.789524	0.394762	+	0.918783i	$0.370827\pi$
$42$	0	0
$43$	377.997	1.34056	0.670280	−	0.742108i	$-0.266174\pi$
$43$	377.997	1.34056	0.670280	+	0.742108i	$0.266174\pi$
$44$	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	$-0.645501\pi$
$53$	−340.588	−0.882706	−0.441353	+	0.897334i	$0.645501\pi$
$54$	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	$-0.779643\pi$
$67$	−844.344	−1.53960	−0.769799	+	0.638287i	$0.779643\pi$
$68$	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	$-0.450408\pi$
$79$	217.909	0.310338	0.155169	+	0.987888i	$0.450408\pi$
$80$	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	$-0.233403\pi$
$89$	1247.68	1.48600	0.742998	+	0.669294i	$0.233403\pi$
$90$	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	$-0.255484\pi$
$101$	1410.54	1.38964	0.694821	+	0.719183i	$0.255484\pi$
$102$	0	0
$103$	−879.881	−0.841721	−0.420861	−	0.907125i	$-0.638272\pi$
$103$	−879.881	−0.841721	−0.420861	+	0.907125i	$0.638272\pi$
$104$	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	$-0.282159\pi$
$113$	1518.76	1.26436	0.632182	+	0.774820i	$0.282159\pi$
$114$	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	$-0.765448\pi$
$127$	−2119.86	−1.48116	−0.740578	+	0.671970i	$0.765448\pi$
$128$	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	$-0.640834\pi$
$137$	−1373.11	−0.856299	−0.428149	+	0.903708i	$0.640834\pi$
$138$	0	0
$139$	−1598.13	−0.975194	−0.487597	−	0.873069i	$-0.662126\pi$
$139$	−1598.13	−0.975194	−0.487597	+	0.873069i	$0.662126\pi$
$140$	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	$-0.259292\pi$
$149$	2495.97	1.37233	0.686167	+	0.727444i	$0.259292\pi$
$150$	0	0
$151$	1977.95	1.06598	0.532992	−	0.846120i	$-0.321068\pi$
$151$	1977.95	1.06598	0.532992	+	0.846120i	$0.321068\pi$
$152$	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	$-0.504276\pi$
$163$	−55.9069	−0.0268648	−0.0134324	+	0.999910i	$0.504276\pi$
$164$	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	$-0.634461\pi$
$173$	−1865.74	−0.819940	−0.409970	+	0.912099i	$0.634461\pi$
$174$	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	$-0.526866\pi$
$197$	−466.195	−0.168604	−0.0843020	+	0.996440i	$0.526866\pi$
$198$	0	0
$199$	2136.33	0.761007	0.380503	−	0.924780i	$-0.375751\pi$
$199$	2136.33	0.761007	0.380503	+	0.924780i	$0.375751\pi$
$200$	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	$-0.736396\pi$
$211$	−4145.34	−1.35250	−0.676249	+	0.736673i	$0.736396\pi$
$212$	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	$-0.932625\pi$
$223$	−6511.56	−1.95536	−0.977682	+	0.210089i	$0.932625\pi$
$224$	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	$-0.771604\pi$
$233$	−5359.31	−1.50687	−0.753434	+	0.657524i	$0.771604\pi$
$234$	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	$-0.678547\pi$
$257$	−4383.44	−1.06393	−0.531967	+	0.846765i	$0.678547\pi$
$258$	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	$-0.572113\pi$
$269$	−1981.97	−0.449231	−0.224615	+	0.974447i	$0.572113\pi$
$270$	0	0
$271$	−806.801	−0.180847	−0.0904237	−	0.995903i	$-0.528822\pi$
$271$	−806.801	−0.180847	−0.0904237	+	0.995903i	$0.528822\pi$
$272$	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	$-0.502792\pi$
$281$	−82.6189	−0.0175396	−0.00876981	+	0.999962i	$0.502792\pi$
$282$	0	0
$283$	4448.26	0.934351	0.467176	−	0.884165i	$-0.345272\pi$
$283$	4448.26	0.934351	0.467176	+	0.884165i	$0.345272\pi$
$284$	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	$-0.610532\pi$
$293$	−3413.56	−0.680623	−0.340311	+	0.940313i	$0.610532\pi$
$294$	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	$-0.562603\pi$
$307$	−2102.23	−0.390817	−0.195409	+	0.980722i	$0.562603\pi$
$308$	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	$-0.731982\pi$
$317$	−7517.49	−1.33194	−0.665969	+	0.745980i	$0.731982\pi$
$318$	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	$-0.559201\pi$
$331$	−2227.14	−0.369832	−0.184916	+	0.982754i	$0.559201\pi$
$332$	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	$-0.404652\pi$
$353$	3914.17	0.590170	0.295085	+	0.955471i	$0.404652\pi$
$354$	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	$-0.401486\pi$
$367$	4282.73	0.609146	0.304573	+	0.952489i	$0.401486\pi$
$368$	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	$-0.454615\pi$
$379$	2096.88	0.284194	0.142097	+	0.989853i	$0.454615\pi$
$380$	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	$-0.557097\pi$
$389$	−2737.71	−0.356831	−0.178415	+	0.983955i	$0.557097\pi$
$390$	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	$-0.402268\pi$
$401$	4853.86	0.604465	0.302232	+	0.953234i	$0.402268\pi$
$402$	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	$-0.220373\pi$
$439$	14160.7	1.53953	0.769765	+	0.638328i	$0.220373\pi$
$440$	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	$-0.235713\pi$
$449$	14045.2	1.47624	0.738122	+	0.674667i	$0.235713\pi$
$450$	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	$-0.367897\pi$
$461$	7981.86	0.806404	0.403202	+	0.915111i	$0.367897\pi$
$462$	0	0
$463$	−3300.75	−0.331315	−0.165658	−	0.986183i	$-0.552975\pi$
$463$	−3300.75	−0.331315	−0.165658	+	0.986183i	$0.552975\pi$
$464$	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	$-0.284099\pi$
$487$	13486.6	1.25490	0.627448	+	0.778658i	$0.284099\pi$
$488$	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	$-0.645991\pi$
$499$	−9870.15	−0.885469	−0.442734	+	0.896653i	$0.645991\pi$
$500$	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	$-0.701792\pi$
$509$	−13604.1	−1.18466	−0.592331	+	0.805695i	$0.701792\pi$
$510$	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	$-0.390171\pi$
$521$	8044.57	0.676467	0.338233	+	0.941062i	$0.390171\pi$
$522$	0	0
$523$	−7785.42	−0.650923	−0.325462	−	0.945555i	$-0.605520\pi$
$523$	−7785.42	−0.650923	−0.325462	+	0.945555i	$0.605520\pi$
$524$	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	$-0.484527\pi$
$547$	1243.27	0.0971819	0.0485910	+	0.998819i	$0.484527\pi$
$548$	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	$-0.318164\pi$
$557$	14215.4	1.08137	0.540687	+	0.841224i	$0.318164\pi$
$558$	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	$-0.362231\pi$
$569$	11385.6	0.838857	0.419428	+	0.907788i	$0.362231\pi$
$570$	0	0
$571$	−23265.3	−1.70512	−0.852560	−	0.522630i	$-0.824951\pi$
$571$	−23265.3	−1.70512	−0.852560	+	0.522630i	$0.824951\pi$
$572$	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	$-0.374932\pi$
$593$	11058.0	0.765764	0.382882	+	0.923797i	$0.374932\pi$
$594$	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	$-0.351146\pi$
$607$	13482.7	0.901561	0.450780	+	0.892635i	$0.351146\pi$
$608$	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	$-0.701392\pi$
$617$	−18125.0	−1.18263	−0.591317	+	0.806439i	$0.701392\pi$
$618$	0	0
$619$	23499.8	1.52591	0.762954	−	0.646453i	$-0.223748\pi$
$619$	23499.8	1.52591	0.762954	+	0.646453i	$0.223748\pi$
$620$	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	$-0.450666\pi$
$631$	4893.60	0.308734	0.154367	+	0.988014i	$0.450666\pi$
$632$	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	$-0.847088\pi$
$641$	−28784.0	−1.77363	−0.886816	+	0.462122i	$0.847088\pi$
$642$	0	0
$643$	896.295	0.0549711	0.0274855	−	0.999622i	$-0.491250\pi$
$643$	896.295	0.0549711	0.0274855	+	0.999622i	$0.491250\pi$
$644$	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	$-0.364995\pi$
$653$	13734.1	0.823059	0.411529	+	0.911396i	$0.364995\pi$
$654$	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	$-0.228549\pi$
$677$	26532.4	1.50624	0.753119	+	0.657885i	$0.228549\pi$
$678$	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	$-0.673508\pi$
$691$	−18836.2	−1.03699	−0.518496	+	0.855080i	$0.673508\pi$
$692$	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	$-0.753969\pi$
$701$	−26573.0	−1.43174	−0.715869	+	0.698234i	$0.753969\pi$
$702$	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	$-0.323178\pi$
$727$	20675.1	1.05474	0.527370	+	0.849636i	$0.323178\pi$
$728$	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	$-0.616972\pi$
$739$	−14434.8	−0.718526	−0.359263	+	0.933236i	$0.616972\pi$
$740$	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	$-0.518427\pi$
$751$	−2381.46	−0.115713	−0.0578566	+	0.998325i	$0.518427\pi$
$752$	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	$-0.422628\pi$
$761$	10105.5	0.481370	0.240685	+	0.970603i	$0.422628\pi$
$762$	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	$-0.374732\pi$
$773$	16482.4	0.766922	0.383461	+	0.923557i	$0.374732\pi$
$774$	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	$-0.392729\pi$
$787$	14600.7	0.661320	0.330660	+	0.943750i	$0.392729\pi$
$788$	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	$-0.661221\pi$
$797$	−21830.2	−0.970222	−0.485111	+	0.874453i	$0.661221\pi$
$798$	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	$-0.604209\pi$
$809$	−14798.7	−0.643133	−0.321567	+	0.946887i	$0.604209\pi$
$810$	0	0
$811$	23276.2	1.00782	0.503908	−	0.863757i	$-0.331895\pi$
$811$	23276.2	1.00782	0.503908	+	0.863757i	$0.331895\pi$
$812$	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	$-0.697387\pi$
$821$	−27340.9	−1.16225	−0.581124	+	0.813815i	$0.697387\pi$
$822$	0	0
$823$	14930.1	0.632357	0.316179	−	0.948700i	$-0.397600\pi$
$823$	14930.1	0.632357	0.316179	+	0.948700i	$0.397600\pi$
$824$	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	$-0.440426\pi$
$857$	9336.14	0.372131	0.186066	+	0.982537i	$0.440426\pi$
$858$	0	0
$859$	−22903.3	−0.909721	−0.454861	−	0.890563i	$-0.650311\pi$
$859$	−22903.3	−0.909721	−0.454861	+	0.890563i	$0.650311\pi$
$860$	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	$-0.651388\pi$
$881$	−23946.2	−0.915743	−0.457871	+	0.889018i	$0.651388\pi$
$882$	0	0
$883$	−3969.15	−0.151271	−0.0756357	−	0.997136i	$-0.524099\pi$
$883$	−3969.15	−0.151271	−0.0756357	+	0.997136i	$0.524099\pi$
$884$	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	$-0.764692\pi$
$907$	−40371.4	−1.47796	−0.738980	+	0.673728i	$0.764692\pi$
$908$	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	$-0.604512\pi$
$919$	−17967.5	−0.644934	−0.322467	+	0.946581i	$0.604512\pi$
$920$	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	$-0.599508\pi$
$929$	−17416.6	−0.615090	−0.307545	+	0.951533i	$0.599508\pi$
$930$	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	$-0.585970\pi$
$941$	−15403.5	−0.533622	−0.266811	+	0.963749i	$0.585970\pi$
$942$	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	$-0.725897\pi$
$953$	−38339.2	−1.30318	−0.651589	+	0.758572i	$0.725897\pi$
$954$	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	$-0.114270\pi$
$967$	56307.0	1.87251	0.936253	+	0.351328i	$0.114270\pi$
$968$	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	$-0.335699\pi$
$977$	30144.1	0.987099	0.493549	+	0.869718i	$0.335699\pi$
$978$	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	$-0.256490\pi$
$991$	43210.3	1.38509	0.692543	+	0.721377i	$0.256490\pi$
$992$	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	1296.4.a.bb.1.3		4
3.2	odd	2		1296.4.a.x.1.2		4
4.3	odd	2		648.4.a.j.1.3	yes	4
12.11	even	2		648.4.a.g.1.2	✓	4
36.7	odd	6		648.4.i.u.433.2		8
36.11	even	6		648.4.i.v.433.3		8
36.23	even	6		648.4.i.v.217.3		8
36.31	odd	6		648.4.i.u.217.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.2	✓	4	12.11	even	2
648.4.a.j.1.3	yes	4	4.3	odd	2
648.4.i.u.217.2		8	36.31	odd	6
648.4.i.u.433.2		8	36.7	odd	6
648.4.i.v.217.3		8	36.23	even	6
648.4.i.v.433.3		8	36.11	even	6
1296.4.a.x.1.2		4	3.2	odd	2
1296.4.a.bb.1.3		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.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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