LMFDB - Embedded newform 1296.4.a.q.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{201}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 50 $ x^2 - x - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 648)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.58872$ 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)

$f(q)$	$=$	$q-10.1774 q^{5} +29.1774 q^{7} +O(q^{10})$	$q-10.1774 q^{5} +29.1774 q^{7} -19.5323 q^{11} +60.1774 q^{13} -102.420 q^{17} +113.887 q^{19} +14.8226 q^{23} -21.4196 q^{25} +68.1774 q^{29} +158.839 q^{31} -296.952 q^{35} -75.8226 q^{37} +364.355 q^{41} -291.081 q^{43} -454.355 q^{47} +508.323 q^{49} +560.839 q^{53} +198.789 q^{55} -359.065 q^{59} +383.017 q^{61} -612.453 q^{65} -736.952 q^{67} +360.468 q^{71} -1010.19 q^{73} -569.904 q^{77} +592.275 q^{79} -231.257 q^{83} +1042.37 q^{85} +1323.00 q^{89} +1755.82 q^{91} -1159.08 q^{95} -472.872 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{5} + 30 q^{7}+O(q^{10}) $ 2 * q + 8 * q^5 + 30 * q^7	$ 2 q + 8 q^{5} + 30 q^{7} + 46 q^{11} + 92 q^{13} + 22 q^{17} + 86 q^{19} + 58 q^{23} + 184 q^{25} + 108 q^{29} - 136 q^{31} - 282 q^{35} - 180 q^{37} + 672 q^{41} + 70 q^{43} - 852 q^{47} + 166 q^{49} + 668 q^{53} + 1390 q^{55} - 548 q^{59} + 284 q^{61} - 34 q^{65} - 1162 q^{67} + 806 q^{71} - 1510 q^{73} - 516 q^{77} + 22 q^{79} - 1540 q^{83} + 3304 q^{85} + 2646 q^{89} + 1782 q^{91} - 1666 q^{95} + 472 q^{97}+O(q^{100}) $ 2 * q + 8 * q^5 + 30 * q^7 + 46 * q^11 + 92 * q^13 + 22 * q^17 + 86 * q^19 + 58 * q^23 + 184 * q^25 + 108 * q^29 - 136 * q^31 - 282 * q^35 - 180 * q^37 + 672 * q^41 + 70 * q^43 - 852 * q^47 + 166 * q^49 + 668 * q^53 + 1390 * q^55 - 548 * q^59 + 284 * q^61 - 34 * q^65 - 1162 * q^67 + 806 * q^71 - 1510 * q^73 - 516 * q^77 + 22 * q^79 - 1540 * q^83 + 3304 * q^85 + 2646 * q^89 + 1782 * q^91 - 1666 * q^95 + 472 * 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
$5$	−10.1774	−0.910299	−0.455149	−	0.890415i	$-0.650414\pi$
$5$	−10.1774	−0.910299	−0.455149	+	0.890415i	$0.650414\pi$
$6$	0	0
$7$	29.1774	1.57543	0.787717	−	0.616037i	$-0.211263\pi$
$7$	29.1774	1.57543	0.787717	+	0.616037i	$0.211263\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−19.5323	−0.535384	−0.267692	−	0.963505i	$-0.586261\pi$
$11$	−19.5323	−0.535384	−0.267692	+	0.963505i	$0.586261\pi$
$12$	0	0
$13$	60.1774	1.28386	0.641932	−	0.766762i	$-0.278133\pi$
$13$	60.1774	1.28386	0.641932	+	0.766762i	$0.278133\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−102.420	−1.46120	−0.730600	−	0.682806i	$-0.760759\pi$
$17$	−102.420	−1.46120	−0.730600	+	0.682806i	$0.760759\pi$
$18$	0	0
$19$	113.887	1.37513	0.687566	−	0.726122i	$-0.258679\pi$
$19$	113.887	1.37513	0.687566	+	0.726122i	$0.258679\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	14.8226	0.134379	0.0671895	−	0.997740i	$-0.478597\pi$
$23$	14.8226	0.134379	0.0671895	+	0.997740i	$0.478597\pi$
$24$	0	0
$25$	−21.4196	−0.171357
$26$	0	0
$27$	0	0
$28$	0	0
$29$	68.1774	0.436560	0.218280	−	0.975886i	$-0.429955\pi$
$29$	68.1774	0.436560	0.218280	+	0.975886i	$0.429955\pi$
$30$	0	0
$31$	158.839	0.920269	0.460135	−	0.887849i	$-0.347801\pi$
$31$	158.839	0.920269	0.460135	+	0.887849i	$0.347801\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−296.952	−1.43412
$36$	0	0
$37$	−75.8226	−0.336896	−0.168448	−	0.985711i	$-0.553876\pi$
$37$	−75.8226	−0.336896	−0.168448	+	0.985711i	$0.553876\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	364.355	1.38787	0.693935	−	0.720038i	$-0.255875\pi$
$41$	364.355	1.38787	0.693935	+	0.720038i	$0.255875\pi$
$42$	0	0
$43$	−291.081	−1.03231	−0.516157	−	0.856494i	$-0.672638\pi$
$43$	−291.081	−1.03231	−0.516157	+	0.856494i	$0.672638\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−454.355	−1.41010	−0.705048	−	0.709160i	$-0.749074\pi$
$47$	−454.355	−1.41010	−0.705048	+	0.709160i	$0.749074\pi$
$48$	0	0
$49$	508.323	1.48199
$50$	0	0
$51$	0	0
$52$	0	0
$53$	560.839	1.45353	0.726766	−	0.686885i	$-0.241023\pi$
$53$	560.839	1.45353	0.726766	+	0.686885i	$0.241023\pi$
$54$	0	0
$55$	198.789	0.487359
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−359.065	−0.792309	−0.396155	−	0.918184i	$-0.629656\pi$
$59$	−359.065	−0.792309	−0.396155	+	0.918184i	$0.629656\pi$
$60$	0	0
$61$	383.017	0.803939	0.401969	−	0.915653i	$-0.368326\pi$
$61$	383.017	0.803939	0.401969	+	0.915653i	$0.368326\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−612.453	−1.16870
$66$	0	0
$67$	−736.952	−1.34378	−0.671888	−	0.740653i	$-0.734516\pi$
$67$	−736.952	−1.34378	−0.671888	+	0.740653i	$0.734516\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	360.468	0.602530	0.301265	−	0.953540i	$-0.402591\pi$
$71$	360.468	0.602530	0.301265	+	0.953540i	$0.402591\pi$
$72$	0	0
$73$	−1010.19	−1.61965	−0.809824	−	0.586673i	$-0.800437\pi$
$73$	−1010.19	−1.61965	−0.809824	+	0.586673i	$0.800437\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−569.904	−0.843462
$78$	0	0
$79$	592.275	0.843496	0.421748	−	0.906713i	$-0.361417\pi$
$79$	592.275	0.843496	0.421748	+	0.906713i	$0.361417\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−231.257	−0.305828	−0.152914	−	0.988239i	$-0.548866\pi$
$83$	−231.257	−0.305828	−0.152914	+	0.988239i	$0.548866\pi$
$84$	0	0
$85$	1042.37	1.33013
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1323.00	1.57570	0.787852	−	0.615864i	$-0.211193\pi$
$89$	1323.00	1.57570	0.787852	+	0.615864i	$0.211193\pi$
$90$	0	0
$91$	1755.82	2.02264
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1159.08	−1.25178
$96$	0	0
$97$	−472.872	−0.494978	−0.247489	−	0.968891i	$-0.579605\pi$
$97$	−472.872	−0.494978	−0.247489	+	0.968891i	$0.579605\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1494.10	1.47196	0.735982	−	0.677001i	$-0.236721\pi$
$101$	1494.10	1.47196	0.735982	+	0.677001i	$0.236721\pi$
$102$	0	0
$103$	−1364.55	−1.30537	−0.652685	−	0.757629i	$-0.726358\pi$
$103$	−1364.55	−1.30537	−0.652685	+	0.757629i	$0.726358\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1239.87	1.12022	0.560108	−	0.828419i	$-0.310760\pi$
$107$	1239.87	1.12022	0.560108	+	0.828419i	$0.310760\pi$
$108$	0	0
$109$	197.405	0.173467	0.0867337	−	0.996232i	$-0.472357\pi$
$109$	197.405	0.173467	0.0867337	+	0.996232i	$0.472357\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1887.97	1.57173	0.785863	−	0.618400i	$-0.212219\pi$
$113$	1887.97	1.57173	0.785863	+	0.618400i	$0.212219\pi$
$114$	0	0
$115$	−150.856	−0.122325
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2988.34	−2.30202
$120$	0	0
$121$	−949.488	−0.713364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1490.18	1.06628
$126$	0	0
$127$	1612.95	1.12698	0.563490	−	0.826123i	$-0.309459\pi$
$127$	1612.95	1.12698	0.563490	+	0.826123i	$0.309459\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1983.67	1.32300	0.661502	−	0.749943i	$-0.269919\pi$
$131$	1983.67	1.32300	0.661502	+	0.749943i	$0.269919\pi$
$132$	0	0
$133$	3322.94	2.16643
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1760.03	−1.09759	−0.548794	−	0.835957i	$-0.684913\pi$
$137$	−1760.03	−1.09759	−0.548794	+	0.835957i	$0.684913\pi$
$138$	0	0
$139$	2233.26	1.36275	0.681377	−	0.731933i	$-0.261382\pi$
$139$	2233.26	1.36275	0.681377	+	0.731933i	$0.261382\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1175.41	−0.687360
$144$	0	0
$145$	−693.872	−0.397400
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1723.76	0.947758	0.473879	−	0.880590i	$-0.342853\pi$
$149$	1723.76	0.947758	0.473879	+	0.880590i	$0.342853\pi$
$150$	0	0
$151$	1959.71	1.05615	0.528077	−	0.849196i	$-0.322913\pi$
$151$	1959.71	1.05615	0.528077	+	0.849196i	$0.322913\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1616.58	−0.837720
$156$	0	0
$157$	3180.08	1.61655	0.808275	−	0.588805i	$-0.200401\pi$
$157$	3180.08	1.61655	0.808275	+	0.588805i	$0.200401\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	432.484	0.211705
$162$	0	0
$163$	1654.55	0.795056	0.397528	−	0.917590i	$-0.369868\pi$
$163$	1654.55	0.795056	0.397528	+	0.917590i	$0.369868\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−451.891	−0.209391	−0.104696	−	0.994504i	$-0.533387\pi$
$167$	−451.891	−0.209391	−0.104696	+	0.994504i	$0.533387\pi$
$168$	0	0
$169$	1424.33	0.648305
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2949.63	−1.29628	−0.648140	−	0.761521i	$-0.724453\pi$
$173$	−2949.63	−1.29628	−0.648140	+	0.761521i	$0.724453\pi$
$174$	0	0
$175$	−624.969	−0.269961
$176$	0	0
$177$	0	0
$178$	0	0
$179$	865.161	0.361258	0.180629	−	0.983551i	$-0.442187\pi$
$179$	865.161	0.361258	0.180629	+	0.983551i	$0.442187\pi$
$180$	0	0
$181$	2669.03	1.09606	0.548032	−	0.836457i	$-0.315377\pi$
$181$	2669.03	1.09606	0.548032	+	0.836457i	$0.315377\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	771.680	0.306676
$186$	0	0
$187$	2000.49	0.782303
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2293.31	0.868786	0.434393	−	0.900723i	$-0.356963\pi$
$191$	2293.31	0.868786	0.434393	+	0.900723i	$0.356963\pi$
$192$	0	0
$193$	3942.46	1.47038	0.735192	−	0.677859i	$-0.237092\pi$
$193$	3942.46	1.47038	0.735192	+	0.677859i	$0.237092\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−27.6302	−0.00999275	−0.00499637	−	0.999988i	$-0.501590\pi$
$197$	−27.6302	−0.00999275	−0.00499637	+	0.999988i	$0.501590\pi$
$198$	0	0
$199$	−3076.20	−1.09581	−0.547904	−	0.836541i	$-0.684574\pi$
$199$	−3076.20	−1.09581	−0.547904	+	0.836541i	$0.684574\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1989.24	0.687771
$204$	0	0
$205$	−3708.20	−1.26338
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2224.48	−0.736224
$210$	0	0
$211$	2987.05	0.974582	0.487291	−	0.873240i	$-0.337985\pi$
$211$	2987.05	0.974582	0.487291	+	0.873240i	$0.337985\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2962.46	0.939713
$216$	0	0
$217$	4634.52	1.44982
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6163.35	−1.87598
$222$	0	0
$223$	−1419.67	−0.426313	−0.213157	−	0.977018i	$-0.568374\pi$
$223$	−1419.67	−0.426313	−0.213157	+	0.977018i	$0.568374\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2384.67	−0.697251	−0.348625	−	0.937262i	$-0.613351\pi$
$227$	−2384.67	−0.697251	−0.348625	+	0.937262i	$0.613351\pi$
$228$	0	0
$229$	−1946.34	−0.561649	−0.280824	−	0.959759i	$-0.590608\pi$
$229$	−1946.34	−0.561649	−0.280824	+	0.959759i	$0.590608\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2063.74	0.580259	0.290129	−	0.956987i	$-0.406302\pi$
$233$	2063.74	0.580259	0.290129	+	0.956987i	$0.406302\pi$
$234$	0	0
$235$	4624.17	1.28361
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2117.00	−0.572960	−0.286480	−	0.958086i	$-0.592485\pi$
$239$	−2117.00	−0.572960	−0.286480	+	0.958086i	$0.592485\pi$
$240$	0	0
$241$	2835.97	0.758012	0.379006	−	0.925394i	$-0.376266\pi$
$241$	2835.97	0.758012	0.379006	+	0.925394i	$0.376266\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5173.43	−1.34906
$246$	0	0
$247$	6853.44	1.76548
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4736.31	1.19105	0.595524	−	0.803337i	$-0.296944\pi$
$251$	4736.31	1.19105	0.595524	+	0.803337i	$0.296944\pi$
$252$	0	0
$253$	−289.519	−0.0719443
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2183.16	−0.529891	−0.264945	−	0.964263i	$-0.585354\pi$
$257$	−2183.16	−0.529891	−0.264945	+	0.964263i	$0.585354\pi$
$258$	0	0
$259$	−2212.31	−0.530757
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8181.33	−1.91818	−0.959092	−	0.283093i	$-0.908640\pi$
$263$	−8181.33	−1.91818	−0.959092	+	0.283093i	$0.908640\pi$
$264$	0	0
$265$	−5707.91	−1.32315
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2165.62	0.490856	0.245428	−	0.969415i	$-0.421071\pi$
$269$	2165.62	0.490856	0.245428	+	0.969415i	$0.421071\pi$
$270$	0	0
$271$	−8529.79	−1.91199	−0.955993	−	0.293390i	$-0.905217\pi$
$271$	−8529.79	−1.91199	−0.955993	+	0.293390i	$0.905217\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	418.374	0.0917416
$276$	0	0
$277$	−1891.03	−0.410185	−0.205093	−	0.978743i	$-0.565750\pi$
$277$	−1891.03	−0.410185	−0.205093	+	0.978743i	$0.565750\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1204.00	−0.255603	−0.127802	−	0.991800i	$-0.540792\pi$
$281$	−1204.00	−0.255603	−0.127802	+	0.991800i	$0.540792\pi$
$282$	0	0
$283$	3977.16	0.835398	0.417699	−	0.908585i	$-0.362837\pi$
$283$	3977.16	0.835398	0.417699	+	0.908585i	$0.362837\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10630.9	2.18650
$288$	0	0
$289$	5576.77	1.13510
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6302.86	1.25671	0.628357	−	0.777925i	$-0.283728\pi$
$293$	6302.86	1.25671	0.628357	+	0.777925i	$0.283728\pi$
$294$	0	0
$295$	3654.36	0.721238
$296$	0	0
$297$	0	0
$298$	0	0
$299$	891.983	0.172524
$300$	0	0
$301$	−8493.01	−1.62634
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3898.13	−0.731824
$306$	0	0
$307$	−7852.27	−1.45978	−0.729890	−	0.683565i	$-0.760429\pi$
$307$	−7852.27	−1.45978	−0.729890	+	0.683565i	$0.760429\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2607.88	0.475496	0.237748	−	0.971327i	$-0.423591\pi$
$311$	2607.88	0.475496	0.237748	+	0.971327i	$0.423591\pi$
$312$	0	0
$313$	−1656.48	−0.299137	−0.149569	−	0.988751i	$-0.547788\pi$
$313$	−1656.48	−0.299137	−0.149569	+	0.988751i	$0.547788\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6202.79	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$317$	−6202.79	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$318$	0	0
$319$	−1331.67	−0.233727
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11664.3	−2.00934
$324$	0	0
$325$	−1288.98	−0.219998
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13256.9	−2.22151
$330$	0	0
$331$	7622.41	1.26576	0.632879	−	0.774251i	$-0.281873\pi$
$331$	7622.41	1.26576	0.632879	+	0.774251i	$0.281873\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7500.29	1.22324
$336$	0	0
$337$	5270.62	0.851956	0.425978	−	0.904734i	$-0.359930\pi$
$337$	5270.62	0.851956	0.425978	+	0.904734i	$0.359930\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3102.50	−0.492697
$342$	0	0
$343$	4823.71	0.759347
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9537.83	1.47556	0.737778	−	0.675044i	$-0.235875\pi$
$347$	9537.83	1.47556	0.737778	+	0.675044i	$0.235875\pi$
$348$	0	0
$349$	2088.69	0.320358	0.160179	−	0.987088i	$-0.448793\pi$
$349$	2088.69	0.320358	0.160179	+	0.987088i	$0.448793\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6473.11	0.976003	0.488001	−	0.872843i	$-0.337726\pi$
$353$	6473.11	0.976003	0.488001	+	0.872843i	$0.337726\pi$
$354$	0	0
$355$	−3668.64	−0.548482
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11916.0	1.75182	0.875912	−	0.482470i	$-0.160260\pi$
$359$	11916.0	1.75182	0.875912	+	0.482470i	$0.160260\pi$
$360$	0	0
$361$	6111.30	0.890990
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10281.2	1.47436
$366$	0	0
$367$	−6510.14	−0.925958	−0.462979	−	0.886369i	$-0.653220\pi$
$367$	−6510.14	−0.925958	−0.462979	+	0.886369i	$0.653220\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16363.9	2.28994
$372$	0	0
$373$	4833.94	0.671024	0.335512	−	0.942036i	$-0.391091\pi$
$373$	4833.94	0.671024	0.335512	+	0.942036i	$0.391091\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4102.74	0.560483
$378$	0	0
$379$	1435.28	0.194527	0.0972633	−	0.995259i	$-0.468991\pi$
$379$	1435.28	0.194527	0.0972633	+	0.995259i	$0.468991\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2964.93	0.395563	0.197781	−	0.980246i	$-0.436626\pi$
$383$	2964.93	0.395563	0.197781	+	0.980246i	$0.436626\pi$
$384$	0	0
$385$	5800.17	0.767802
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−212.003	−0.0276324	−0.0138162	−	0.999905i	$-0.504398\pi$
$389$	−212.003	−0.0276324	−0.0138162	+	0.999905i	$0.504398\pi$
$390$	0	0
$391$	−1518.12	−0.196354
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6027.85	−0.767833
$396$	0	0
$397$	12508.1	1.58127	0.790635	−	0.612288i	$-0.209751\pi$
$397$	12508.1	1.58127	0.790635	+	0.612288i	$0.209751\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2006.52	0.249878	0.124939	−	0.992164i	$-0.460127\pi$
$401$	2006.52	0.249878	0.124939	+	0.992164i	$0.460127\pi$
$402$	0	0
$403$	9558.53	1.18150
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1480.99	0.180369
$408$	0	0
$409$	−640.449	−0.0774283	−0.0387142	−	0.999250i	$-0.512326\pi$
$409$	−640.449	−0.0774283	−0.0387142	+	0.999250i	$0.512326\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10476.6	−1.24823
$414$	0	0
$415$	2353.61	0.278395
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2326.26	−0.271230	−0.135615	−	0.990762i	$-0.543301\pi$
$419$	−2326.26	−0.271230	−0.135615	+	0.990762i	$0.543301\pi$
$420$	0	0
$421$	3426.61	0.396681	0.198341	−	0.980133i	$-0.436445\pi$
$421$	3426.61	0.396681	0.198341	+	0.980133i	$0.436445\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2193.78	0.250386
$426$	0	0
$427$	11175.4	1.26655
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13417.5	−1.49953	−0.749767	−	0.661701i	$-0.769835\pi$
$431$	−13417.5	−1.49953	−0.749767	+	0.661701i	$0.769835\pi$
$432$	0	0
$433$	−8862.95	−0.983663	−0.491832	−	0.870690i	$-0.663672\pi$
$433$	−8862.95	−0.983663	−0.491832	+	0.870690i	$0.663672\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1688.10	0.184789
$438$	0	0
$439$	−6109.45	−0.664210	−0.332105	−	0.943242i	$-0.607759\pi$
$439$	−6109.45	−0.664210	−0.332105	+	0.943242i	$0.607759\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15285.5	−1.63936	−0.819678	−	0.572824i	$-0.805848\pi$
$443$	−15285.5	−1.63936	−0.819678	+	0.572824i	$0.805848\pi$
$444$	0	0
$445$	−13464.8	−1.43436
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2622.04	0.275594	0.137797	−	0.990460i	$-0.455998\pi$
$449$	2622.04	0.275594	0.137797	+	0.990460i	$0.455998\pi$
$450$	0	0
$451$	−7116.70	−0.743043
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17869.8	−1.84121
$456$	0	0
$457$	−5127.24	−0.524819	−0.262410	−	0.964957i	$-0.584517\pi$
$457$	−5127.24	−0.524819	−0.262410	+	0.964957i	$0.584517\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9806.08	0.990704	0.495352	−	0.868692i	$-0.335039\pi$
$461$	9806.08	0.990704	0.495352	+	0.868692i	$0.335039\pi$
$462$	0	0
$463$	−18430.2	−1.84995	−0.924973	−	0.380032i	$-0.875913\pi$
$463$	−18430.2	−1.84995	−0.924973	+	0.380032i	$0.875913\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17350.2	−1.71921	−0.859604	−	0.510960i	$-0.829290\pi$
$467$	−17350.2	−1.71921	−0.859604	+	0.510960i	$0.829290\pi$
$468$	0	0
$469$	−21502.4	−2.11703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5685.50	0.552684
$474$	0	0
$475$	−2439.42	−0.235638
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6921.52	−0.660234	−0.330117	−	0.943940i	$-0.607088\pi$
$479$	−6921.52	−0.660234	−0.330117	+	0.943940i	$0.607088\pi$
$480$	0	0
$481$	−4562.81	−0.432528
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4812.63	0.450578
$486$	0	0
$487$	6396.86	0.595215	0.297607	−	0.954688i	$-0.403811\pi$
$487$	6396.86	0.595215	0.297607	+	0.954688i	$0.403811\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9739.78	0.895214	0.447607	−	0.894230i	$-0.352276\pi$
$491$	9739.78	0.895214	0.447607	+	0.894230i	$0.352276\pi$
$492$	0	0
$493$	−6982.71	−0.637901
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10517.5	0.949247
$498$	0	0
$499$	−3537.43	−0.317349	−0.158674	−	0.987331i	$-0.550722\pi$
$499$	−3537.43	−0.317349	−0.158674	+	0.987331i	$0.550722\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17616.0	1.56155	0.780775	−	0.624812i	$-0.214824\pi$
$503$	17616.0	1.56155	0.780775	+	0.624812i	$0.214824\pi$
$504$	0	0
$505$	−15206.1	−1.33993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12935.3	−1.12641	−0.563207	−	0.826316i	$-0.690433\pi$
$509$	−12935.3	−1.12641	−0.563207	+	0.826316i	$0.690433\pi$
$510$	0	0
$511$	−29474.9	−2.55165
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13887.6	1.18828
$516$	0	0
$517$	8874.61	0.754942
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22931.9	−1.92834	−0.964168	−	0.265291i	$-0.914532\pi$
$521$	−22931.9	−1.92834	−0.964168	+	0.265291i	$0.914532\pi$
$522$	0	0
$523$	7422.01	0.620539	0.310269	−	0.950649i	$-0.399581\pi$
$523$	7422.01	0.620539	0.310269	+	0.950649i	$0.399581\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16268.2	−1.34470
$528$	0	0
$529$	−11947.3	−0.981942
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21925.9	1.78184
$534$	0	0
$535$	−12618.8	−1.01973
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9928.75	−0.793435
$540$	0	0
$541$	−9659.86	−0.767670	−0.383835	−	0.923402i	$-0.625397\pi$
$541$	−9659.86	−0.767670	−0.383835	+	0.923402i	$0.625397\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2009.08	−0.157907
$546$	0	0
$547$	5780.17	0.451814	0.225907	−	0.974149i	$-0.427466\pi$
$547$	5780.17	0.451814	0.225907	+	0.974149i	$0.427466\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7764.54	0.600328
$552$	0	0
$553$	17281.1	1.32887
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8160.01	0.620737	0.310369	−	0.950616i	$-0.399548\pi$
$557$	8160.01	0.620737	0.310369	+	0.950616i	$0.399548\pi$
$558$	0	0
$559$	−17516.5	−1.32535
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5762.56	−0.431373	−0.215686	−	0.976463i	$-0.569199\pi$
$563$	−5762.56	−0.431373	−0.215686	+	0.976463i	$0.569199\pi$
$564$	0	0
$565$	−19214.7	−1.43074
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14025.2	−1.03333	−0.516666	−	0.856187i	$-0.672827\pi$
$569$	−14025.2	−1.03333	−0.516666	+	0.856187i	$0.672827\pi$
$570$	0	0
$571$	−13967.7	−1.02370	−0.511848	−	0.859076i	$-0.671039\pi$
$571$	−13967.7	−1.02370	−0.511848	+	0.859076i	$0.671039\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−317.493	−0.0230267
$576$	0	0
$577$	−13336.0	−0.962193	−0.481097	−	0.876668i	$-0.659761\pi$
$577$	−13336.0	−0.962193	−0.481097	+	0.876668i	$0.659761\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6747.49	−0.481812
$582$	0	0
$583$	−10954.5	−0.778197
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18348.6	−1.29016	−0.645082	−	0.764113i	$-0.723177\pi$
$587$	−18348.6	−1.29016	−0.645082	+	0.764113i	$0.723177\pi$
$588$	0	0
$589$	18089.8	1.26549
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20361.5	1.41003	0.705016	−	0.709192i	$-0.250940\pi$
$593$	20361.5	1.41003	0.705016	+	0.709192i	$0.250940\pi$
$594$	0	0
$595$	30413.7	2.09553
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7749.09	−0.528580	−0.264290	−	0.964443i	$-0.585138\pi$
$599$	−7749.09	−0.528580	−0.264290	+	0.964443i	$0.585138\pi$
$600$	0	0
$601$	−10531.8	−0.714811	−0.357406	−	0.933949i	$-0.616339\pi$
$601$	−10531.8	−0.714811	−0.357406	+	0.933949i	$0.616339\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9663.36	0.649374
$606$	0	0
$607$	−7708.85	−0.515473	−0.257737	−	0.966215i	$-0.582977\pi$
$607$	−7708.85	−0.515473	−0.257737	+	0.966215i	$0.582977\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27341.9	−1.81037
$612$	0	0
$613$	−13630.8	−0.898112	−0.449056	−	0.893504i	$-0.648240\pi$
$613$	−13630.8	−0.898112	−0.449056	+	0.893504i	$0.648240\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19613.6	1.27976	0.639882	−	0.768473i	$-0.278983\pi$
$617$	19613.6	1.27976	0.639882	+	0.768473i	$0.278983\pi$
$618$	0	0
$619$	16695.4	1.08408	0.542038	−	0.840354i	$-0.317653\pi$
$619$	16695.4	1.08408	0.542038	+	0.840354i	$0.317653\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	38601.8	2.48242
$624$	0	0
$625$	−12488.8	−0.799280
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7765.71	0.492272
$630$	0	0
$631$	−12302.0	−0.776124	−0.388062	−	0.921633i	$-0.626855\pi$
$631$	−12302.0	−0.776124	−0.388062	+	0.921633i	$0.626855\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16415.7	−1.02589
$636$	0	0
$637$	30589.6	1.90268
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27452.8	1.69160	0.845802	−	0.533497i	$-0.179122\pi$
$641$	27452.8	1.69160	0.845802	+	0.533497i	$0.179122\pi$
$642$	0	0
$643$	5711.12	0.350272	0.175136	−	0.984544i	$-0.443964\pi$
$643$	5711.12	0.350272	0.175136	+	0.984544i	$0.443964\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5973.11	−0.362948	−0.181474	−	0.983396i	$-0.558087\pi$
$647$	−5973.11	−0.362948	−0.181474	+	0.983396i	$0.558087\pi$
$648$	0	0
$649$	7013.37	0.424190
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15650.9	−0.937929	−0.468965	−	0.883217i	$-0.655373\pi$
$653$	−15650.9	−0.937929	−0.468965	+	0.883217i	$0.655373\pi$
$654$	0	0
$655$	−20188.6	−1.20433
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3957.24	0.233919	0.116959	−	0.993137i	$-0.462685\pi$
$659$	3957.24	0.233919	0.116959	+	0.993137i	$0.462685\pi$
$660$	0	0
$661$	12214.0	0.718715	0.359357	−	0.933200i	$-0.382996\pi$
$661$	12214.0	0.718715	0.359357	+	0.933200i	$0.382996\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33819.0	−1.97210
$666$	0	0
$667$	1010.56	0.0586644
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7481.21	−0.430416
$672$	0	0
$673$	23567.0	1.34984	0.674920	−	0.737891i	$-0.264178\pi$
$673$	23567.0	1.34984	0.674920	+	0.737891i	$0.264178\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4103.28	−0.232942	−0.116471	−	0.993194i	$-0.537158\pi$
$677$	−4103.28	−0.232942	−0.116471	+	0.993194i	$0.537158\pi$
$678$	0	0
$679$	−13797.2	−0.779806
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34200.5	−1.91602	−0.958012	−	0.286729i	$-0.907432\pi$
$683$	−34200.5	−1.91602	−0.958012	+	0.286729i	$0.907432\pi$
$684$	0	0
$685$	17912.6	0.999133
$686$	0	0
$687$	0	0
$688$	0	0
$689$	33749.9	1.86614
$690$	0	0
$691$	−15121.6	−0.832492	−0.416246	−	0.909252i	$-0.636655\pi$
$691$	−15121.6	−0.832492	−0.416246	+	0.909252i	$0.636655\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22728.9	−1.24051
$696$	0	0
$697$	−37317.1	−2.02796
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24421.2	−1.31580	−0.657899	−	0.753106i	$-0.728555\pi$
$701$	−24421.2	−1.31580	−0.657899	+	0.753106i	$0.728555\pi$
$702$	0	0
$703$	−8635.22	−0.463277
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43594.0	2.31898
$708$	0	0
$709$	−2516.78	−0.133314	−0.0666570	−	0.997776i	$-0.521233\pi$
$709$	−2516.78	−0.133314	−0.0666570	+	0.997776i	$0.521233\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2354.40	0.123665
$714$	0	0
$715$	11962.6	0.625702
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4416.18	0.229062	0.114531	−	0.993420i	$-0.463463\pi$
$719$	4416.18	0.229062	0.114531	+	0.993420i	$0.463463\pi$
$720$	0	0
$721$	−39814.1	−2.05653
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1460.33	−0.0748074
$726$	0	0
$727$	13463.3	0.686831	0.343415	−	0.939184i	$-0.388416\pi$
$727$	13463.3	0.686831	0.343415	+	0.939184i	$0.388416\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29812.4	1.50842
$732$	0	0
$733$	−20457.3	−1.03084	−0.515420	−	0.856938i	$-0.672364\pi$
$733$	−20457.3	−1.03084	−0.515420	+	0.856938i	$0.672364\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14394.4	0.719436
$738$	0	0
$739$	7781.39	0.387339	0.193669	−	0.981067i	$-0.437961\pi$
$739$	7781.39	0.387339	0.193669	+	0.981067i	$0.437961\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8738.82	−0.431489	−0.215744	−	0.976450i	$-0.569218\pi$
$743$	−8738.82	−0.431489	−0.215744	+	0.976450i	$0.569218\pi$
$744$	0	0
$745$	−17543.5	−0.862742
$746$	0	0
$747$	0	0
$748$	0	0
$749$	36176.4	1.76483
$750$	0	0
$751$	−19380.4	−0.941677	−0.470838	−	0.882219i	$-0.656048\pi$
$751$	−19380.4	−0.941677	−0.470838	+	0.882219i	$0.656048\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−19944.9	−0.961416
$756$	0	0
$757$	20151.9	0.967548	0.483774	−	0.875193i	$-0.339266\pi$
$757$	20151.9	0.967548	0.483774	+	0.875193i	$0.339266\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1384.63	0.0659565	0.0329782	−	0.999456i	$-0.489501\pi$
$761$	1384.63	0.0659565	0.0329782	+	0.999456i	$0.489501\pi$
$762$	0	0
$763$	5759.76	0.273286
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21607.6	−1.01722
$768$	0	0
$769$	18244.3	0.855536	0.427768	−	0.903888i	$-0.359300\pi$
$769$	18244.3	0.855536	0.427768	+	0.903888i	$0.359300\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18353.9	−0.854001	−0.427001	−	0.904251i	$-0.640430\pi$
$773$	−18353.9	−0.854001	−0.427001	+	0.904251i	$0.640430\pi$
$774$	0	0
$775$	−3402.27	−0.157694
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41495.4	1.90851
$780$	0	0
$781$	−7040.78	−0.322585
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−32365.1	−1.47154
$786$	0	0
$787$	−4826.88	−0.218627	−0.109314	−	0.994007i	$-0.534865\pi$
$787$	−4826.88	−0.218627	−0.109314	+	0.994007i	$0.534865\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	55086.1	2.47615
$792$	0	0
$793$	23049.0	1.03215
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16423.6	0.729931	0.364966	−	0.931021i	$-0.381081\pi$
$797$	16423.6	0.729931	0.364966	+	0.931021i	$0.381081\pi$
$798$	0	0
$799$	46534.8	2.06043
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19731.5	0.867133
$804$	0	0
$805$	−4401.59	−0.192715
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16215.6	0.704708	0.352354	−	0.935867i	$-0.385381\pi$
$809$	16215.6	0.704708	0.352354	+	0.935867i	$0.385381\pi$
$810$	0	0
$811$	17229.9	0.746023	0.373012	−	0.927827i	$-0.378325\pi$
$811$	17229.9	0.746023	0.373012	+	0.927827i	$0.378325\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16839.1	−0.723738
$816$	0	0
$817$	−33150.4	−1.41957
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1388.22	0.0590124	0.0295062	−	0.999565i	$-0.490607\pi$
$821$	1388.22	0.0590124	0.0295062	+	0.999565i	$0.490607\pi$
$822$	0	0
$823$	23255.0	0.984955	0.492477	−	0.870325i	$-0.336091\pi$
$823$	23255.0	0.984955	0.492477	+	0.870325i	$0.336091\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29307.4	1.23231	0.616154	−	0.787626i	$-0.288690\pi$
$827$	29307.4	1.23231	0.616154	+	0.787626i	$0.288690\pi$
$828$	0	0
$829$	18350.8	0.768818	0.384409	−	0.923163i	$-0.374405\pi$
$829$	18350.8	0.768818	0.384409	+	0.923163i	$0.374405\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−52062.3	−2.16549
$834$	0	0
$835$	4599.09	0.190609
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3063.27	−0.126050	−0.0630249	−	0.998012i	$-0.520075\pi$
$839$	−3063.27	−0.126050	−0.0630249	+	0.998012i	$0.520075\pi$
$840$	0	0
$841$	−19740.8	−0.809416
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14496.0	−0.590151
$846$	0	0
$847$	−27703.6	−1.12386
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1123.88	−0.0452717
$852$	0	0
$853$	3558.08	0.142821	0.0714105	−	0.997447i	$-0.477250\pi$
$853$	3558.08	0.142821	0.0714105	+	0.997447i	$0.477250\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6410.13	−0.255503	−0.127751	−	0.991806i	$-0.540776\pi$
$857$	−6410.13	−0.255503	−0.127751	+	0.991806i	$0.540776\pi$
$858$	0	0
$859$	23612.0	0.937873	0.468936	−	0.883232i	$-0.344637\pi$
$859$	23612.0	0.937873	0.468936	+	0.883232i	$0.344637\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33640.6	1.32693	0.663463	−	0.748209i	$-0.269086\pi$
$863$	33640.6	1.32693	0.663463	+	0.748209i	$0.269086\pi$
$864$	0	0
$865$	30019.7	1.18000
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11568.5	−0.451594
$870$	0	0
$871$	−44347.9	−1.72522
$872$	0	0
$873$	0	0
$874$	0	0
$875$	43479.6	1.67986
$876$	0	0
$877$	10587.4	0.407654	0.203827	−	0.979007i	$-0.434662\pi$
$877$	10587.4	0.407654	0.203827	+	0.979007i	$0.434662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12213.9	−0.467080	−0.233540	−	0.972347i	$-0.575031\pi$
$881$	−12213.9	−0.467080	−0.233540	+	0.972347i	$0.575031\pi$
$882$	0	0
$883$	−34545.1	−1.31657	−0.658287	−	0.752767i	$-0.728719\pi$
$883$	−34545.1	−1.31657	−0.658287	+	0.752767i	$0.728719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17946.2	−0.679340	−0.339670	−	0.940545i	$-0.610315\pi$
$887$	−17946.2	−0.679340	−0.339670	+	0.940545i	$0.610315\pi$
$888$	0	0
$889$	47061.8	1.77548
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−51745.2	−1.93907
$894$	0	0
$895$	−8805.13	−0.328853
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10829.2	0.401753
$900$	0	0
$901$	−57440.9	−2.12390
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27163.9	−0.997745
$906$	0	0
$907$	15206.0	0.556678	0.278339	−	0.960483i	$-0.410216\pi$
$907$	15206.0	0.556678	0.278339	+	0.960483i	$0.410216\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27342.8	−0.994410	−0.497205	−	0.867633i	$-0.665640\pi$
$911$	−27342.8	−0.994410	−0.497205	+	0.867633i	$0.665640\pi$
$912$	0	0
$913$	4516.99	0.163736
$914$	0	0
$915$	0	0
$916$	0	0
$917$	57878.3	2.08431
$918$	0	0
$919$	−31391.3	−1.12677	−0.563386	−	0.826194i	$-0.690501\pi$
$919$	−31391.3	−1.12677	−0.563386	+	0.826194i	$0.690501\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21692.0	0.773566
$924$	0	0
$925$	1624.09	0.0577293
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18230.3	0.643829	0.321914	−	0.946769i	$-0.395674\pi$
$929$	18230.3	0.643829	0.321914	+	0.946769i	$0.395674\pi$
$930$	0	0
$931$	57891.5	2.03794
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20359.9	−0.712129
$936$	0	0
$937$	19475.4	0.679012	0.339506	−	0.940604i	$-0.389740\pi$
$937$	19475.4	0.679012	0.339506	+	0.940604i	$0.389740\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7602.91	−0.263388	−0.131694	−	0.991290i	$-0.542042\pi$
$941$	−7602.91	−0.263388	−0.131694	+	0.991290i	$0.542042\pi$
$942$	0	0
$943$	5400.67	0.186501
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25591.1	−0.878142	−0.439071	−	0.898452i	$-0.644692\pi$
$947$	−25591.1	−0.878142	−0.439071	+	0.898452i	$0.644692\pi$
$948$	0	0
$949$	−60790.9	−2.07941
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6621.51	0.225070	0.112535	−	0.993648i	$-0.464103\pi$
$953$	6621.51	0.225070	0.112535	+	0.993648i	$0.464103\pi$
$954$	0	0
$955$	−23340.0	−0.790855
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−51353.2	−1.72918
$960$	0	0
$961$	−4561.12	−0.153104
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−40124.1	−1.33849
$966$	0	0
$967$	−15252.1	−0.507211	−0.253606	−	0.967308i	$-0.581617\pi$
$967$	−15252.1	−0.507211	−0.253606	+	0.967308i	$0.581617\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−55183.2	−1.82380	−0.911901	−	0.410410i	$-0.865386\pi$
$971$	−55183.2	−1.82380	−0.911901	+	0.410410i	$0.865386\pi$
$972$	0	0
$973$	65160.8	2.14693
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24293.0	−0.795497	−0.397749	−	0.917494i	$-0.630208\pi$
$977$	−24293.0	−0.795497	−0.397749	+	0.917494i	$0.630208\pi$
$978$	0	0
$979$	−25841.3	−0.843607
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1258.23	−0.0408253	−0.0204126	−	0.999792i	$-0.506498\pi$
$983$	−1258.23	−0.0408253	−0.0204126	+	0.999792i	$0.506498\pi$
$984$	0	0
$985$	281.205	0.00909638
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4314.57	−0.138721
$990$	0	0
$991$	−4198.63	−0.134585	−0.0672926	−	0.997733i	$-0.521436\pi$
$991$	−4198.63	−0.134585	−0.0672926	+	0.997733i	$0.521436\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31307.9	0.997513
$996$	0	0
$997$	−18058.2	−0.573629	−0.286814	−	0.957986i	$-0.592596\pi$
$997$	−18058.2	−0.573629	−0.286814	+	0.957986i	$0.592596\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.q.1.1		2
3.2	odd	2		1296.4.a.m.1.2		2
4.3	odd	2		648.4.a.f.1.1	yes	2
12.11	even	2		648.4.a.c.1.2	✓	2
36.7	odd	6		648.4.i.n.433.2		4
36.11	even	6		648.4.i.t.433.1		4
36.23	even	6		648.4.i.t.217.1		4
36.31	odd	6		648.4.i.n.217.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.c.1.2	✓	2	12.11	even	2
648.4.a.f.1.1	yes	2	4.3	odd	2
648.4.i.n.217.2		4	36.31	odd	6
648.4.i.n.433.2		4	36.7	odd	6
648.4.i.t.217.1		4	36.23	even	6
648.4.i.t.433.1		4	36.11	even	6
1296.4.a.m.1.2		2	3.2	odd	2
1296.4.a.q.1.1		2	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)

\(f(q)\)	\(=\)	\(q-10.1774 q^{5} +29.1774 q^{7} +O(q^{10})\)	\(q-10.1774 q^{5} +29.1774 q^{7} -19.5323 q^{11} +60.1774 q^{13} -102.420 q^{17} +113.887 q^{19} +14.8226 q^{23} -21.4196 q^{25} +68.1774 q^{29} +158.839 q^{31} -296.952 q^{35} -75.8226 q^{37} +364.355 q^{41} -291.081 q^{43} -454.355 q^{47} +508.323 q^{49} +560.839 q^{53} +198.789 q^{55} -359.065 q^{59} +383.017 q^{61} -612.453 q^{65} -736.952 q^{67} +360.468 q^{71} -1010.19 q^{73} -569.904 q^{77} +592.275 q^{79} -231.257 q^{83} +1042.37 q^{85} +1323.00 q^{89} +1755.82 q^{91} -1159.08 q^{95} -472.872 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{5} + 30 q^{7}+O(q^{10}) \) 2 * q + 8 * q^5 + 30 * q^7	\( 2 q + 8 q^{5} + 30 q^{7} + 46 q^{11} + 92 q^{13} + 22 q^{17} + 86 q^{19} + 58 q^{23} + 184 q^{25} + 108 q^{29} - 136 q^{31} - 282 q^{35} - 180 q^{37} + 672 q^{41} + 70 q^{43} - 852 q^{47} + 166 q^{49} + 668 q^{53} + 1390 q^{55} - 548 q^{59} + 284 q^{61} - 34 q^{65} - 1162 q^{67} + 806 q^{71} - 1510 q^{73} - 516 q^{77} + 22 q^{79} - 1540 q^{83} + 3304 q^{85} + 2646 q^{89} + 1782 q^{91} - 1666 q^{95} + 472 q^{97}+O(q^{100}) \) 2 * q + 8 * q^5 + 30 * q^7 + 46 * q^11 + 92 * q^13 + 22 * q^17 + 86 * q^19 + 58 * q^23 + 184 * q^25 + 108 * q^29 - 136 * q^31 - 282 * q^35 - 180 * q^37 + 672 * q^41 + 70 * q^43 - 852 * q^47 + 166 * q^49 + 668 * q^53 + 1390 * q^55 - 548 * q^59 + 284 * q^61 - 34 * q^65 - 1162 * q^67 + 806 * q^71 - 1510 * q^73 - 516 * q^77 + 22 * q^79 - 1540 * q^83 + 3304 * q^85 + 2646 * q^89 + 1782 * q^91 - 1666 * q^95 + 472 * q^97

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