LMFDB - Embedded newform 8036.2.a.p.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.70664$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70664 q^{3} +0.469317 q^{5} +4.32591 q^{9} +O(q^{10})$	$q+2.70664 q^{3} +0.469317 q^{5} +4.32591 q^{9} +4.72750 q^{11} +1.70239 q^{13} +1.27027 q^{15} -3.62142 q^{17} +0.482699 q^{19} -0.0771055 q^{23} -4.77974 q^{25} +3.58875 q^{27} +0.891222 q^{29} +9.17820 q^{31} +12.7957 q^{33} +2.10312 q^{37} +4.60775 q^{39} +1.00000 q^{41} +6.61340 q^{43} +2.03022 q^{45} +4.76867 q^{47} -9.80188 q^{51} -12.1209 q^{53} +2.21870 q^{55} +1.30649 q^{57} +8.46582 q^{59} +1.21796 q^{61} +0.798959 q^{65} -3.96167 q^{67} -0.208697 q^{69} +5.12294 q^{71} -10.6052 q^{73} -12.9370 q^{75} +14.6429 q^{79} -3.26426 q^{81} +1.02509 q^{83} -1.69960 q^{85} +2.41222 q^{87} -3.74768 q^{89} +24.8421 q^{93} +0.226539 q^{95} +11.0675 q^{97} +20.4507 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

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
$3$	2.70664	1.56268	0.781340	−	0.624106i	$-0.214537\pi$
$3$	2.70664	1.56268	0.781340	+	0.624106i	$0.214537\pi$
$4$	0	0
$5$	0.469317	0.209885	0.104943	−	0.994478i	$-0.466534\pi$
$5$	0.469317	0.209885	0.104943	+	0.994478i	$0.466534\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.32591	1.44197
$10$	0	0
$11$	4.72750	1.42540	0.712698	−	0.701471i	$-0.247473\pi$
$11$	4.72750	1.42540	0.712698	+	0.701471i	$0.247473\pi$
$12$	0	0
$13$	1.70239	0.472157	0.236078	−	0.971734i	$-0.424138\pi$
$13$	1.70239	0.472157	0.236078	+	0.971734i	$0.424138\pi$
$14$	0	0
$15$	1.27027	0.327983
$16$	0	0
$17$	−3.62142	−0.878323	−0.439162	−	0.898408i	$-0.644724\pi$
$17$	−3.62142	−0.878323	−0.439162	+	0.898408i	$0.644724\pi$
$18$	0	0
$19$	0.482699	0.110739	0.0553694	−	0.998466i	$-0.482366\pi$
$19$	0.482699	0.110739	0.0553694	+	0.998466i	$0.482366\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.0771055	−0.0160776	−0.00803881	−	0.999968i	$-0.502559\pi$
$23$	−0.0771055	−0.0160776	−0.00803881	+	0.999968i	$0.502559\pi$
$24$	0	0
$25$	−4.77974	−0.955948
$26$	0	0
$27$	3.58875	0.690655
$28$	0	0
$29$	0.891222	0.165496	0.0827479	−	0.996571i	$-0.473630\pi$
$29$	0.891222	0.165496	0.0827479	+	0.996571i	$0.473630\pi$
$30$	0	0
$31$	9.17820	1.64845	0.824227	−	0.566259i	$-0.191610\pi$
$31$	9.17820	1.64845	0.824227	+	0.566259i	$0.191610\pi$
$32$	0	0
$33$	12.7957	2.22744
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.10312	0.345750	0.172875	−	0.984944i	$-0.444694\pi$
$37$	2.10312	0.345750	0.172875	+	0.984944i	$0.444694\pi$
$38$	0	0
$39$	4.60775	0.737830
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	6.61340	1.00853	0.504267	−	0.863548i	$-0.331763\pi$
$43$	6.61340	1.00853	0.504267	+	0.863548i	$0.331763\pi$
$44$	0	0
$45$	2.03022	0.302648
$46$	0	0
$47$	4.76867	0.695582	0.347791	−	0.937572i	$-0.386932\pi$
$47$	4.76867	0.695582	0.347791	+	0.937572i	$0.386932\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.80188	−1.37254
$52$	0	0
$53$	−12.1209	−1.66493	−0.832466	−	0.554076i	$-0.813072\pi$
$53$	−12.1209	−1.66493	−0.832466	+	0.554076i	$0.813072\pi$
$54$	0	0
$55$	2.21870	0.299169
$56$	0	0
$57$	1.30649	0.173049
$58$	0	0
$59$	8.46582	1.10216	0.551078	−	0.834454i	$-0.314217\pi$
$59$	8.46582	1.10216	0.551078	+	0.834454i	$0.314217\pi$
$60$	0	0
$61$	1.21796	0.155944	0.0779719	−	0.996956i	$-0.475156\pi$
$61$	1.21796	0.155944	0.0779719	+	0.996956i	$0.475156\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.798959	0.0990987
$66$	0	0
$67$	−3.96167	−0.483995	−0.241998	−	0.970277i	$-0.577803\pi$
$67$	−3.96167	−0.483995	−0.241998	+	0.970277i	$0.577803\pi$
$68$	0	0
$69$	−0.208697	−0.0251242
$70$	0	0
$71$	5.12294	0.607981	0.303991	−	0.952675i	$-0.401681\pi$
$71$	5.12294	0.607981	0.303991	+	0.952675i	$0.401681\pi$
$72$	0	0
$73$	−10.6052	−1.24124	−0.620622	−	0.784110i	$-0.713120\pi$
$73$	−10.6052	−1.24124	−0.620622	+	0.784110i	$0.713120\pi$
$74$	0	0
$75$	−12.9370	−1.49384
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.6429	1.64745	0.823725	−	0.566989i	$-0.191892\pi$
$79$	14.6429	1.64745	0.823725	+	0.566989i	$0.191892\pi$
$80$	0	0
$81$	−3.26426	−0.362695
$82$	0	0
$83$	1.02509	0.112519	0.0562594	−	0.998416i	$-0.482083\pi$
$83$	1.02509	0.112519	0.0562594	+	0.998416i	$0.482083\pi$
$84$	0	0
$85$	−1.69960	−0.184347
$86$	0	0
$87$	2.41222	0.258617
$88$	0	0
$89$	−3.74768	−0.397253	−0.198626	−	0.980075i	$-0.563648\pi$
$89$	−3.74768	−0.397253	−0.198626	+	0.980075i	$0.563648\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.8421	2.57601
$94$	0	0
$95$	0.226539	0.0232424
$96$	0	0
$97$	11.0675	1.12373	0.561867	−	0.827228i	$-0.310083\pi$
$97$	11.0675	1.12373	0.561867	+	0.827228i	$0.310083\pi$
$98$	0	0
$99$	20.4507	2.05538
$100$	0	0
$101$	10.9512	1.08969	0.544844	−	0.838537i	$-0.316589\pi$
$101$	10.9512	1.08969	0.544844	+	0.838537i	$0.316589\pi$
$102$	0	0
$103$	−12.2036	−1.20246	−0.601230	−	0.799076i	$-0.705323\pi$
$103$	−12.2036	−1.20246	−0.601230	+	0.799076i	$0.705323\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.4055	1.10262	0.551308	−	0.834302i	$-0.314129\pi$
$107$	11.4055	1.10262	0.551308	+	0.834302i	$0.314129\pi$
$108$	0	0
$109$	6.49074	0.621700	0.310850	−	0.950459i	$-0.399386\pi$
$109$	6.49074	0.621700	0.310850	+	0.950459i	$0.399386\pi$
$110$	0	0
$111$	5.69238	0.540297
$112$	0	0
$113$	11.5427	1.08584	0.542922	−	0.839783i	$-0.317318\pi$
$113$	11.5427	1.08584	0.542922	+	0.839783i	$0.317318\pi$
$114$	0	0
$115$	−0.0361870	−0.00337445
$116$	0	0
$117$	7.36436	0.680835
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.3493	1.03175
$122$	0	0
$123$	2.70664	0.244050
$124$	0	0
$125$	−4.58980	−0.410524
$126$	0	0
$127$	−1.91879	−0.170265	−0.0851326	−	0.996370i	$-0.527131\pi$
$127$	−1.91879	−0.170265	−0.0851326	+	0.996370i	$0.527131\pi$
$128$	0	0
$129$	17.9001	1.57602
$130$	0	0
$131$	9.65847	0.843864	0.421932	−	0.906627i	$-0.361352\pi$
$131$	9.65847	0.843864	0.421932	+	0.906627i	$0.361352\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.68426	0.144958
$136$	0	0
$137$	−20.8543	−1.78171	−0.890853	−	0.454291i	$-0.849893\pi$
$137$	−20.8543	−1.78171	−0.890853	+	0.454291i	$0.849893\pi$
$138$	0	0
$139$	6.00753	0.509552	0.254776	−	0.967000i	$-0.417998\pi$
$139$	6.00753	0.509552	0.254776	+	0.967000i	$0.417998\pi$
$140$	0	0
$141$	12.9071	1.08697
$142$	0	0
$143$	8.04803	0.673010
$144$	0	0
$145$	0.418266	0.0347351
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.5867	−1.52268	−0.761342	−	0.648350i	$-0.775459\pi$
$149$	−18.5867	−1.52268	−0.761342	+	0.648350i	$0.775459\pi$
$150$	0	0
$151$	−2.37742	−0.193472	−0.0967360	−	0.995310i	$-0.530840\pi$
$151$	−2.37742	−0.193472	−0.0967360	+	0.995310i	$0.530840\pi$
$152$	0	0
$153$	−15.6659	−1.26651
$154$	0	0
$155$	4.30749	0.345986
$156$	0	0
$157$	−11.4328	−0.912434	−0.456217	−	0.889869i	$-0.650796\pi$
$157$	−11.4328	−0.912434	−0.456217	+	0.889869i	$0.650796\pi$
$158$	0	0
$159$	−32.8069	−2.60176
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.28613	0.492368	0.246184	−	0.969223i	$-0.420823\pi$
$163$	6.28613	0.492368	0.246184	+	0.969223i	$0.420823\pi$
$164$	0	0
$165$	6.00522	0.467506
$166$	0	0
$167$	−11.3925	−0.881577	−0.440788	−	0.897611i	$-0.645301\pi$
$167$	−11.3925	−0.881577	−0.440788	+	0.897611i	$0.645301\pi$
$168$	0	0
$169$	−10.1019	−0.777068
$170$	0	0
$171$	2.08811	0.159682
$172$	0	0
$173$	−21.5221	−1.63629	−0.818147	−	0.575009i	$-0.804998\pi$
$173$	−21.5221	−1.63629	−0.818147	+	0.575009i	$0.804998\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	22.9139	1.72232
$178$	0	0
$179$	−6.43044	−0.480633	−0.240317	−	0.970695i	$-0.577251\pi$
$179$	−6.43044	−0.480633	−0.240317	+	0.970695i	$0.577251\pi$
$180$	0	0
$181$	17.3408	1.28893	0.644465	−	0.764634i	$-0.277080\pi$
$181$	17.3408	1.28893	0.644465	+	0.764634i	$0.277080\pi$
$182$	0	0
$183$	3.29658	0.243690
$184$	0	0
$185$	0.987030	0.0725679
$186$	0	0
$187$	−17.1203	−1.25196
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.106130	−0.00767931	−0.00383966	−	0.999993i	$-0.501222\pi$
$191$	−0.106130	−0.00767931	−0.00383966	+	0.999993i	$0.501222\pi$
$192$	0	0
$193$	2.28602	0.164552	0.0822758	−	0.996610i	$-0.473781\pi$
$193$	2.28602	0.164552	0.0822758	+	0.996610i	$0.473781\pi$
$194$	0	0
$195$	2.16250	0.154860
$196$	0	0
$197$	1.89180	0.134785	0.0673924	−	0.997727i	$-0.478532\pi$
$197$	1.89180	0.134785	0.0673924	+	0.997727i	$0.478532\pi$
$198$	0	0
$199$	−0.291932	−0.0206945	−0.0103472	−	0.999946i	$-0.503294\pi$
$199$	−0.291932	−0.0206945	−0.0103472	+	0.999946i	$0.503294\pi$
$200$	0	0
$201$	−10.7228	−0.756329
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.469317	0.0327786
$206$	0	0
$207$	−0.333551	−0.0231834
$208$	0	0
$209$	2.28196	0.157847
$210$	0	0
$211$	23.3493	1.60743	0.803717	−	0.595012i	$-0.202853\pi$
$211$	23.3493	1.60743	0.803717	+	0.595012i	$0.202853\pi$
$212$	0	0
$213$	13.8660	0.950080
$214$	0	0
$215$	3.10378	0.211676
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−28.7044	−1.93967
$220$	0	0
$221$	−6.16505	−0.414706
$222$	0	0
$223$	−13.1824	−0.882757	−0.441379	−	0.897321i	$-0.645510\pi$
$223$	−13.1824	−0.882757	−0.441379	+	0.897321i	$0.645510\pi$
$224$	0	0
$225$	−20.6767	−1.37845
$226$	0	0
$227$	10.7195	0.711479	0.355740	−	0.934585i	$-0.384229\pi$
$227$	10.7195	0.711479	0.355740	+	0.934585i	$0.384229\pi$
$228$	0	0
$229$	−12.8550	−0.849482	−0.424741	−	0.905315i	$-0.639635\pi$
$229$	−12.8550	−0.849482	−0.424741	+	0.905315i	$0.639635\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.1885	−0.798492	−0.399246	−	0.916844i	$-0.630728\pi$
$233$	−12.1885	−0.798492	−0.399246	+	0.916844i	$0.630728\pi$
$234$	0	0
$235$	2.23802	0.145992
$236$	0	0
$237$	39.6330	2.57444
$238$	0	0
$239$	4.28190	0.276973	0.138487	−	0.990364i	$-0.455776\pi$
$239$	4.28190	0.276973	0.138487	+	0.990364i	$0.455776\pi$
$240$	0	0
$241$	−1.35442	−0.0872456	−0.0436228	−	0.999048i	$-0.513890\pi$
$241$	−1.35442	−0.0872456	−0.0436228	+	0.999048i	$0.513890\pi$
$242$	0	0
$243$	−19.6014	−1.25743
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.821740	0.0522861
$248$	0	0
$249$	2.77456	0.175831
$250$	0	0
$251$	11.0524	0.697620	0.348810	−	0.937193i	$-0.386586\pi$
$251$	11.0524	0.697620	0.348810	+	0.937193i	$0.386586\pi$
$252$	0	0
$253$	−0.364517	−0.0229170
$254$	0	0
$255$	−4.60019	−0.288075
$256$	0	0
$257$	13.7831	0.859766	0.429883	−	0.902884i	$-0.358555\pi$
$257$	13.7831	0.859766	0.429883	+	0.902884i	$0.358555\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.85534	0.238640
$262$	0	0
$263$	−5.36247	−0.330664	−0.165332	−	0.986238i	$-0.552870\pi$
$263$	−5.36247	−0.330664	−0.165332	+	0.986238i	$0.552870\pi$
$264$	0	0
$265$	−5.68855	−0.349445
$266$	0	0
$267$	−10.1436	−0.620779
$268$	0	0
$269$	9.47122	0.577471	0.288735	−	0.957409i	$-0.406765\pi$
$269$	9.47122	0.577471	0.288735	+	0.957409i	$0.406765\pi$
$270$	0	0
$271$	−2.27313	−0.138083	−0.0690413	−	0.997614i	$-0.521994\pi$
$271$	−2.27313	−0.138083	−0.0690413	+	0.997614i	$0.521994\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.5962	−1.36260
$276$	0	0
$277$	−18.7734	−1.12798	−0.563991	−	0.825781i	$-0.690735\pi$
$277$	−18.7734	−1.12798	−0.563991	+	0.825781i	$0.690735\pi$
$278$	0	0
$279$	39.7040	2.37702
$280$	0	0
$281$	−5.35062	−0.319192	−0.159596	−	0.987182i	$-0.551019\pi$
$281$	−5.35062	−0.319192	−0.159596	+	0.987182i	$0.551019\pi$
$282$	0	0
$283$	10.6731	0.634452	0.317226	−	0.948350i	$-0.397249\pi$
$283$	10.6731	0.634452	0.317226	+	0.948350i	$0.397249\pi$
$284$	0	0
$285$	0.613160	0.0363205
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.88533	−0.228549
$290$	0	0
$291$	29.9557	1.75604
$292$	0	0
$293$	−11.9925	−0.700608	−0.350304	−	0.936636i	$-0.613922\pi$
$293$	−11.9925	−0.700608	−0.350304	+	0.936636i	$0.613922\pi$
$294$	0	0
$295$	3.97316	0.231326
$296$	0	0
$297$	16.9658	0.984457
$298$	0	0
$299$	−0.131263	−0.00759115
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.6411	1.70283
$304$	0	0
$305$	0.571610	0.0327303
$306$	0	0
$307$	9.58847	0.547243	0.273622	−	0.961837i	$-0.411778\pi$
$307$	9.58847	0.547243	0.273622	+	0.961837i	$0.411778\pi$
$308$	0	0
$309$	−33.0309	−1.87906
$310$	0	0
$311$	−8.06040	−0.457063	−0.228532	−	0.973536i	$-0.573392\pi$
$311$	−8.06040	−0.457063	−0.228532	+	0.973536i	$0.573392\pi$
$312$	0	0
$313$	15.2940	0.864467	0.432233	−	0.901762i	$-0.357726\pi$
$313$	15.2940	0.864467	0.432233	+	0.901762i	$0.357726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.16644	−0.290176	−0.145088	−	0.989419i	$-0.546347\pi$
$317$	−5.16644	−0.290176	−0.145088	+	0.989419i	$0.546347\pi$
$318$	0	0
$319$	4.21325	0.235897
$320$	0	0
$321$	30.8707	1.72303
$322$	0	0
$323$	−1.74806	−0.0972645
$324$	0	0
$325$	−8.13696	−0.451357
$326$	0	0
$327$	17.5681	0.971518
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.7547	−0.646097	−0.323049	−	0.946382i	$-0.604708\pi$
$331$	−11.7547	−0.646097	−0.323049	+	0.946382i	$0.604708\pi$
$332$	0	0
$333$	9.09789	0.498561
$334$	0	0
$335$	−1.85928	−0.101583
$336$	0	0
$337$	13.0144	0.708941	0.354471	−	0.935067i	$-0.384661\pi$
$337$	13.0144	0.708941	0.354471	+	0.935067i	$0.384661\pi$
$338$	0	0
$339$	31.2419	1.69683
$340$	0	0
$341$	43.3900	2.34970
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.0979451	−0.00527319
$346$	0	0
$347$	16.1209	0.865418	0.432709	−	0.901534i	$-0.357558\pi$
$347$	16.1209	0.865418	0.432709	+	0.901534i	$0.357558\pi$
$348$	0	0
$349$	9.39124	0.502701	0.251351	−	0.967896i	$-0.419125\pi$
$349$	9.39124	0.502701	0.251351	+	0.967896i	$0.419125\pi$
$350$	0	0
$351$	6.10943	0.326097
$352$	0	0
$353$	25.6538	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$353$	25.6538	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$354$	0	0
$355$	2.40429	0.127606
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.6006	−0.559476	−0.279738	−	0.960076i	$-0.590248\pi$
$359$	−10.6006	−0.559476	−0.279738	+	0.960076i	$0.590248\pi$
$360$	0	0
$361$	−18.7670	−0.987737
$362$	0	0
$363$	30.7184	1.61230
$364$	0	0
$365$	−4.97720	−0.260519
$366$	0	0
$367$	−25.0989	−1.31015	−0.655077	−	0.755562i	$-0.727364\pi$
$367$	−25.0989	−1.31015	−0.655077	+	0.755562i	$0.727364\pi$
$368$	0	0
$369$	4.32591	0.225198
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.1194	−1.09352	−0.546760	−	0.837290i	$-0.684139\pi$
$373$	−21.1194	−1.09352	−0.546760	+	0.837290i	$0.684139\pi$
$374$	0	0
$375$	−12.4229	−0.641518
$376$	0	0
$377$	1.51720	0.0781399
$378$	0	0
$379$	−23.5309	−1.20870	−0.604352	−	0.796718i	$-0.706568\pi$
$379$	−23.5309	−1.20870	−0.604352	+	0.796718i	$0.706568\pi$
$380$	0	0
$381$	−5.19348	−0.266070
$382$	0	0
$383$	−2.96252	−0.151378	−0.0756890	−	0.997131i	$-0.524116\pi$
$383$	−2.96252	−0.151378	−0.0756890	+	0.997131i	$0.524116\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.6089	1.45427
$388$	0	0
$389$	−4.62441	−0.234467	−0.117233	−	0.993104i	$-0.537403\pi$
$389$	−4.62441	−0.234467	−0.117233	+	0.993104i	$0.537403\pi$
$390$	0	0
$391$	0.279231	0.0141213
$392$	0	0
$393$	26.1420	1.31869
$394$	0	0
$395$	6.87215	0.345775
$396$	0	0
$397$	−9.06114	−0.454766	−0.227383	−	0.973805i	$-0.573017\pi$
$397$	−9.06114	−0.454766	−0.227383	+	0.973805i	$0.573017\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.7406	0.985797	0.492898	−	0.870087i	$-0.335937\pi$
$401$	19.7406	0.985797	0.492898	+	0.870087i	$0.335937\pi$
$402$	0	0
$403$	15.6248	0.778329
$404$	0	0
$405$	−1.53197	−0.0761244
$406$	0	0
$407$	9.94249	0.492831
$408$	0	0
$409$	35.6659	1.76356	0.881782	−	0.471658i	$-0.156344\pi$
$409$	35.6659	1.76356	0.881782	+	0.471658i	$0.156344\pi$
$410$	0	0
$411$	−56.4452	−2.78424
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.481095	0.0236160
$416$	0	0
$417$	16.2602	0.796267
$418$	0	0
$419$	18.8381	0.920303	0.460151	−	0.887840i	$-0.347795\pi$
$419$	18.8381	0.920303	0.460151	+	0.887840i	$0.347795\pi$
$420$	0	0
$421$	10.1778	0.496034	0.248017	−	0.968756i	$-0.420221\pi$
$421$	10.1778	0.496034	0.248017	+	0.968756i	$0.420221\pi$
$422$	0	0
$423$	20.6288	1.00301
$424$	0	0
$425$	17.3094	0.839631
$426$	0	0
$427$	0	0
$428$	0	0
$429$	21.7831	1.05170
$430$	0	0
$431$	14.1486	0.681515	0.340757	−	0.940151i	$-0.389317\pi$
$431$	14.1486	0.681515	0.340757	+	0.940151i	$0.389317\pi$
$432$	0	0
$433$	−0.0822900	−0.00395461	−0.00197730	−	0.999998i	$-0.500629\pi$
$433$	−0.0822900	−0.00395461	−0.00197730	+	0.999998i	$0.500629\pi$
$434$	0	0
$435$	1.13210	0.0542799
$436$	0	0
$437$	−0.0372188	−0.00178042
$438$	0	0
$439$	33.2966	1.58916	0.794580	−	0.607159i	$-0.207691\pi$
$439$	33.2966	1.58916	0.794580	+	0.607159i	$0.207691\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.3965	−0.684000	−0.342000	−	0.939700i	$-0.611104\pi$
$443$	−14.3965	−0.684000	−0.342000	+	0.939700i	$0.611104\pi$
$444$	0	0
$445$	−1.75885	−0.0833775
$446$	0	0
$447$	−50.3076	−2.37947
$448$	0	0
$449$	−2.30585	−0.108820	−0.0544100	−	0.998519i	$-0.517328\pi$
$449$	−2.30585	−0.108820	−0.0544100	+	0.998519i	$0.517328\pi$
$450$	0	0
$451$	4.72750	0.222609
$452$	0	0
$453$	−6.43483	−0.302335
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.7308	−0.876188	−0.438094	−	0.898929i	$-0.644346\pi$
$457$	−18.7308	−0.876188	−0.438094	+	0.898929i	$0.644346\pi$
$458$	0	0
$459$	−12.9964	−0.606618
$460$	0	0
$461$	−21.5879	−1.00545	−0.502725	−	0.864447i	$-0.667669\pi$
$461$	−21.5879	−1.00545	−0.502725	+	0.864447i	$0.667669\pi$
$462$	0	0
$463$	17.5466	0.815458	0.407729	−	0.913103i	$-0.366321\pi$
$463$	17.5466	0.815458	0.407729	+	0.913103i	$0.366321\pi$
$464$	0	0
$465$	11.6588	0.540665
$466$	0	0
$467$	3.70864	0.171615	0.0858076	−	0.996312i	$-0.472653\pi$
$467$	3.70864	0.171615	0.0858076	+	0.996312i	$0.472653\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−30.9444	−1.42584
$472$	0	0
$473$	31.2649	1.43756
$474$	0	0
$475$	−2.30718	−0.105861
$476$	0	0
$477$	−52.4338	−2.40078
$478$	0	0
$479$	2.20086	0.100560	0.0502798	−	0.998735i	$-0.483989\pi$
$479$	2.20086	0.100560	0.0502798	+	0.998735i	$0.483989\pi$
$480$	0	0
$481$	3.58032	0.163248
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.19417	0.235855
$486$	0	0
$487$	−5.11732	−0.231888	−0.115944	−	0.993256i	$-0.536989\pi$
$487$	−5.11732	−0.231888	−0.115944	+	0.993256i	$0.536989\pi$
$488$	0	0
$489$	17.0143	0.769413
$490$	0	0
$491$	−1.26077	−0.0568976	−0.0284488	−	0.999595i	$-0.509057\pi$
$491$	−1.26077	−0.0568976	−0.0284488	+	0.999595i	$0.509057\pi$
$492$	0	0
$493$	−3.22749	−0.145359
$494$	0	0
$495$	9.59788	0.431393
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−25.1914	−1.12772	−0.563862	−	0.825869i	$-0.690685\pi$
$499$	−25.1914	−1.12772	−0.563862	+	0.825869i	$0.690685\pi$
$500$	0	0
$501$	−30.8354	−1.37762
$502$	0	0
$503$	−29.0850	−1.29684	−0.648418	−	0.761285i	$-0.724569\pi$
$503$	−29.0850	−1.29684	−0.648418	+	0.761285i	$0.724569\pi$
$504$	0	0
$505$	5.13960	0.228709
$506$	0	0
$507$	−27.3422	−1.21431
$508$	0	0
$509$	−16.5493	−0.733535	−0.366767	−	0.930313i	$-0.619536\pi$
$509$	−16.5493	−0.733535	−0.366767	+	0.930313i	$0.619536\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.73229	0.0764823
$514$	0	0
$515$	−5.72738	−0.252379
$516$	0	0
$517$	22.5439	0.991480
$518$	0	0
$519$	−58.2526	−2.55700
$520$	0	0
$521$	43.9342	1.92479	0.962396	−	0.271649i	$-0.0875689\pi$
$521$	43.9342	1.92479	0.962396	+	0.271649i	$0.0875689\pi$
$522$	0	0
$523$	−2.18798	−0.0956736	−0.0478368	−	0.998855i	$-0.515233\pi$
$523$	−2.18798	−0.0956736	−0.0478368	+	0.998855i	$0.515233\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.2381	−1.44788
$528$	0	0
$529$	−22.9941	−0.999742
$530$	0	0
$531$	36.6223	1.58927
$532$	0	0
$533$	1.70239	0.0737385
$534$	0	0
$535$	5.35282	0.231423
$536$	0	0
$537$	−17.4049	−0.751076
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.5895	−1.83107	−0.915533	−	0.402243i	$-0.868231\pi$
$541$	−42.5895	−1.83107	−0.915533	+	0.402243i	$0.868231\pi$
$542$	0	0
$543$	46.9353	2.01419
$544$	0	0
$545$	3.04622	0.130486
$546$	0	0
$547$	−25.1538	−1.07550	−0.537749	−	0.843105i	$-0.680725\pi$
$547$	−25.1538	−1.07550	−0.537749	+	0.843105i	$0.680725\pi$
$548$	0	0
$549$	5.26878	0.224866
$550$	0	0
$551$	0.430192	0.0183268
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.67153	0.113400
$556$	0	0
$557$	−40.6947	−1.72429	−0.862145	−	0.506662i	$-0.830879\pi$
$557$	−40.6947	−1.72429	−0.862145	+	0.506662i	$0.830879\pi$
$558$	0	0
$559$	11.2586	0.476186
$560$	0	0
$561$	−46.3384	−1.95641
$562$	0	0
$563$	11.8745	0.500452	0.250226	−	0.968187i	$-0.419495\pi$
$563$	11.8745	0.500452	0.250226	+	0.968187i	$0.419495\pi$
$564$	0	0
$565$	5.41718	0.227902
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.3873	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$569$	−10.3873	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$570$	0	0
$571$	27.4888	1.15037	0.575184	−	0.818024i	$-0.304930\pi$
$571$	27.4888	1.15037	0.575184	+	0.818024i	$0.304930\pi$
$572$	0	0
$573$	−0.287256	−0.0120003
$574$	0	0
$575$	0.368544	0.0153694
$576$	0	0
$577$	31.5606	1.31388	0.656942	−	0.753941i	$-0.271849\pi$
$577$	31.5606	1.31388	0.656942	+	0.753941i	$0.271849\pi$
$578$	0	0
$579$	6.18745	0.257142
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−57.3015	−2.37319
$584$	0	0
$585$	3.45622	0.142897
$586$	0	0
$587$	23.0329	0.950670	0.475335	−	0.879805i	$-0.342327\pi$
$587$	23.0329	0.950670	0.475335	+	0.879805i	$0.342327\pi$
$588$	0	0
$589$	4.43031	0.182548
$590$	0	0
$591$	5.12041	0.210626
$592$	0	0
$593$	−4.97167	−0.204162	−0.102081	−	0.994776i	$-0.532550\pi$
$593$	−4.97167	−0.204162	−0.102081	+	0.994776i	$0.532550\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.790154	−0.0323389
$598$	0	0
$599$	39.7974	1.62608	0.813038	−	0.582211i	$-0.197812\pi$
$599$	39.7974	1.62608	0.813038	+	0.582211i	$0.197812\pi$
$600$	0	0
$601$	−37.1297	−1.51455	−0.757276	−	0.653094i	$-0.773470\pi$
$601$	−37.1297	−1.51455	−0.757276	+	0.653094i	$0.773470\pi$
$602$	0	0
$603$	−17.1378	−0.697906
$604$	0	0
$605$	5.32641	0.216549
$606$	0	0
$607$	−3.52274	−0.142984	−0.0714918	−	0.997441i	$-0.522776\pi$
$607$	−3.52274	−0.142984	−0.0714918	+	0.997441i	$0.522776\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.11811	0.328424
$612$	0	0
$613$	−33.7942	−1.36493	−0.682467	−	0.730916i	$-0.739093\pi$
$613$	−33.7942	−1.36493	−0.682467	+	0.730916i	$0.739093\pi$
$614$	0	0
$615$	1.27027	0.0512224
$616$	0	0
$617$	−14.1546	−0.569843	−0.284922	−	0.958551i	$-0.591968\pi$
$617$	−14.1546	−0.569843	−0.284922	+	0.958551i	$0.591968\pi$
$618$	0	0
$619$	−24.0838	−0.968011	−0.484006	−	0.875065i	$-0.660819\pi$
$619$	−24.0838	−0.968011	−0.484006	+	0.875065i	$0.660819\pi$
$620$	0	0
$621$	−0.276712	−0.0111041
$622$	0	0
$623$	0	0
$624$	0	0
$625$	21.7446	0.869785
$626$	0	0
$627$	6.17645	0.246664
$628$	0	0
$629$	−7.61627	−0.303680
$630$	0	0
$631$	40.8465	1.62607	0.813037	−	0.582213i	$-0.197813\pi$
$631$	40.8465	1.62607	0.813037	+	0.582213i	$0.197813\pi$
$632$	0	0
$633$	63.1983	2.51191
$634$	0	0
$635$	−0.900522	−0.0357361
$636$	0	0
$637$	0	0
$638$	0	0
$639$	22.1614	0.876690
$640$	0	0
$641$	31.7190	1.25283	0.626413	−	0.779491i	$-0.284522\pi$
$641$	31.7190	1.25283	0.626413	+	0.779491i	$0.284522\pi$
$642$	0	0
$643$	42.8966	1.69168	0.845838	−	0.533440i	$-0.179101\pi$
$643$	42.8966	1.69168	0.845838	+	0.533440i	$0.179101\pi$
$644$	0	0
$645$	8.40083	0.330782
$646$	0	0
$647$	−41.1500	−1.61777	−0.808886	−	0.587966i	$-0.799929\pi$
$647$	−41.1500	−1.61777	−0.808886	+	0.587966i	$0.799929\pi$
$648$	0	0
$649$	40.0222	1.57101
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.7638	0.460351	0.230176	−	0.973149i	$-0.426070\pi$
$653$	11.7638	0.460351	0.230176	+	0.973149i	$0.426070\pi$
$654$	0	0
$655$	4.53289	0.177115
$656$	0	0
$657$	−45.8770	−1.78983
$658$	0	0
$659$	−31.1634	−1.21396	−0.606978	−	0.794719i	$-0.707618\pi$
$659$	−31.1634	−1.21396	−0.606978	+	0.794719i	$0.707618\pi$
$660$	0	0
$661$	9.04814	0.351932	0.175966	−	0.984396i	$-0.443695\pi$
$661$	9.04814	0.351932	0.175966	+	0.984396i	$0.443695\pi$
$662$	0	0
$663$	−16.6866	−0.648053
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.0687182	−0.00266078
$668$	0	0
$669$	−35.6800	−1.37947
$670$	0	0
$671$	5.75791	0.222281
$672$	0	0
$673$	−30.4453	−1.17358	−0.586791	−	0.809739i	$-0.699609\pi$
$673$	−30.4453	−1.17358	−0.586791	+	0.809739i	$0.699609\pi$
$674$	0	0
$675$	−17.1533	−0.660231
$676$	0	0
$677$	−36.7599	−1.41280	−0.706399	−	0.707814i	$-0.749682\pi$
$677$	−36.7599	−1.41280	−0.706399	+	0.707814i	$0.749682\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	29.0139	1.11181
$682$	0	0
$683$	2.06099	0.0788615	0.0394308	−	0.999222i	$-0.487446\pi$
$683$	2.06099	0.0788615	0.0394308	+	0.999222i	$0.487446\pi$
$684$	0	0
$685$	−9.78731	−0.373954
$686$	0	0
$687$	−34.7938	−1.32747
$688$	0	0
$689$	−20.6344	−0.786109
$690$	0	0
$691$	17.4256	0.662901	0.331451	−	0.943473i	$-0.392462\pi$
$691$	17.4256	0.662901	0.331451	+	0.943473i	$0.392462\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.81944	0.106947
$696$	0	0
$697$	−3.62142	−0.137171
$698$	0	0
$699$	−32.9898	−1.24779
$700$	0	0
$701$	−36.8136	−1.39043	−0.695216	−	0.718801i	$-0.744691\pi$
$701$	−36.8136	−1.39043	−0.695216	+	0.718801i	$0.744691\pi$
$702$	0	0
$703$	1.01517	0.0382880
$704$	0	0
$705$	6.05752	0.228139
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.9160	−0.522628	−0.261314	−	0.965254i	$-0.584156\pi$
$709$	−13.9160	−0.522628	−0.261314	+	0.965254i	$0.584156\pi$
$710$	0	0
$711$	63.3436	2.37557
$712$	0	0
$713$	−0.707690	−0.0265032
$714$	0	0
$715$	3.77708	0.141255
$716$	0	0
$717$	11.5896	0.432821
$718$	0	0
$719$	−18.9441	−0.706497	−0.353248	−	0.935530i	$-0.614923\pi$
$719$	−18.9441	−0.706497	−0.353248	+	0.935530i	$0.614923\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.66592	−0.136337
$724$	0	0
$725$	−4.25981	−0.158205
$726$	0	0
$727$	24.0404	0.891608	0.445804	−	0.895131i	$-0.352918\pi$
$727$	24.0404	0.891608	0.445804	+	0.895131i	$0.352918\pi$
$728$	0	0
$729$	−43.2612	−1.60227
$730$	0	0
$731$	−23.9499	−0.885819
$732$	0	0
$733$	−26.0115	−0.960755	−0.480378	−	0.877062i	$-0.659500\pi$
$733$	−26.0115	−0.960755	−0.480378	+	0.877062i	$0.659500\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.7288	−0.689884
$738$	0	0
$739$	−32.4689	−1.19439	−0.597193	−	0.802097i	$-0.703718\pi$
$739$	−32.4689	−1.19439	−0.597193	+	0.802097i	$0.703718\pi$
$740$	0	0
$741$	2.22416	0.0817064
$742$	0	0
$743$	33.3797	1.22458	0.612291	−	0.790633i	$-0.290248\pi$
$743$	33.3797	1.22458	0.612291	+	0.790633i	$0.290248\pi$
$744$	0	0
$745$	−8.72308	−0.319589
$746$	0	0
$747$	4.43446	0.162249
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.44890	0.271814	0.135907	−	0.990722i	$-0.456605\pi$
$751$	7.44890	0.271814	0.135907	+	0.990722i	$0.456605\pi$
$752$	0	0
$753$	29.9148	1.09016
$754$	0	0
$755$	−1.11577	−0.0406069
$756$	0	0
$757$	23.2573	0.845301	0.422650	−	0.906293i	$-0.361100\pi$
$757$	23.2573	0.845301	0.422650	+	0.906293i	$0.361100\pi$
$758$	0	0
$759$	−0.986615	−0.0358119
$760$	0	0
$761$	−28.0263	−1.01595	−0.507976	−	0.861371i	$-0.669606\pi$
$761$	−28.0263	−1.01595	−0.507976	+	0.861371i	$0.669606\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−7.35229	−0.265823
$766$	0	0
$767$	14.4121	0.520390
$768$	0	0
$769$	25.2387	0.910131	0.455065	−	0.890458i	$-0.349616\pi$
$769$	25.2387	0.910131	0.455065	+	0.890458i	$0.349616\pi$
$770$	0	0
$771$	37.3059	1.34354
$772$	0	0
$773$	12.3719	0.444988	0.222494	−	0.974934i	$-0.428580\pi$
$773$	12.3719	0.444988	0.222494	+	0.974934i	$0.428580\pi$
$774$	0	0
$775$	−43.8694	−1.57584
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.482699	0.0172945
$780$	0	0
$781$	24.2187	0.866614
$782$	0	0
$783$	3.19837	0.114301
$784$	0	0
$785$	−5.36560	−0.191506
$786$	0	0
$787$	−17.7301	−0.632012	−0.316006	−	0.948757i	$-0.602342\pi$
$787$	−17.7301	−0.632012	−0.316006	+	0.948757i	$0.602342\pi$
$788$	0	0
$789$	−14.5143	−0.516722
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.07344	0.0736299
$794$	0	0
$795$	−15.3969	−0.546070
$796$	0	0
$797$	−27.8951	−0.988095	−0.494048	−	0.869435i	$-0.664483\pi$
$797$	−27.8951	−0.988095	−0.494048	+	0.869435i	$0.664483\pi$
$798$	0	0
$799$	−17.2694	−0.610946
$800$	0	0
$801$	−16.2121	−0.572826
$802$	0	0
$803$	−50.1360	−1.76926
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.6352	0.902402
$808$	0	0
$809$	−38.5032	−1.35370	−0.676850	−	0.736121i	$-0.736655\pi$
$809$	−38.5032	−1.35370	−0.676850	+	0.736121i	$0.736655\pi$
$810$	0	0
$811$	24.3236	0.854118	0.427059	−	0.904224i	$-0.359550\pi$
$811$	24.3236	0.854118	0.427059	+	0.904224i	$0.359550\pi$
$812$	0	0
$813$	−6.15254	−0.215779
$814$	0	0
$815$	2.95019	0.103341
$816$	0	0
$817$	3.19228	0.111684
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.0376	−1.53692	−0.768462	−	0.639895i	$-0.778978\pi$
$821$	−44.0376	−1.53692	−0.768462	+	0.639895i	$0.778978\pi$
$822$	0	0
$823$	41.5903	1.44975	0.724873	−	0.688883i	$-0.241899\pi$
$823$	41.5903	1.44975	0.724873	+	0.688883i	$0.241899\pi$
$824$	0	0
$825$	−61.1599	−2.12931
$826$	0	0
$827$	12.2346	0.425437	0.212719	−	0.977113i	$-0.431768\pi$
$827$	12.2346	0.425437	0.212719	+	0.977113i	$0.431768\pi$
$828$	0	0
$829$	33.6302	1.16803	0.584013	−	0.811744i	$-0.301482\pi$
$829$	33.6302	1.16803	0.584013	+	0.811744i	$0.301482\pi$
$830$	0	0
$831$	−50.8128	−1.76268
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.34669	−0.185030
$836$	0	0
$837$	32.9383	1.13851
$838$	0	0
$839$	12.3839	0.427541	0.213771	−	0.976884i	$-0.431426\pi$
$839$	12.3839	0.427541	0.213771	+	0.976884i	$0.431426\pi$
$840$	0	0
$841$	−28.2057	−0.972611
$842$	0	0
$843$	−14.4822	−0.498794
$844$	0	0
$845$	−4.74099	−0.163095
$846$	0	0
$847$	0	0
$848$	0	0
$849$	28.8883	0.991445
$850$	0	0
$851$	−0.162162	−0.00555884
$852$	0	0
$853$	11.8119	0.404432	0.202216	−	0.979341i	$-0.435186\pi$
$853$	11.8119	0.404432	0.202216	+	0.979341i	$0.435186\pi$
$854$	0	0
$855$	0.979987	0.0335149
$856$	0	0
$857$	−2.06114	−0.0704072	−0.0352036	−	0.999380i	$-0.511208\pi$
$857$	−2.06114	−0.0704072	−0.0352036	+	0.999380i	$0.511208\pi$
$858$	0	0
$859$	39.0418	1.33209	0.666045	−	0.745911i	$-0.267986\pi$
$859$	39.0418	1.33209	0.666045	+	0.745911i	$0.267986\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.5583	0.938095	0.469048	−	0.883173i	$-0.344597\pi$
$863$	27.5583	0.938095	0.469048	+	0.883173i	$0.344597\pi$
$864$	0	0
$865$	−10.1007	−0.343434
$866$	0	0
$867$	−10.5162	−0.357148
$868$	0	0
$869$	69.2242	2.34827
$870$	0	0
$871$	−6.74429	−0.228521
$872$	0	0
$873$	47.8770	1.62039
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−54.7549	−1.84894	−0.924471	−	0.381253i	$-0.875493\pi$
$877$	−54.7549	−1.84894	−0.924471	+	0.381253i	$0.875493\pi$
$878$	0	0
$879$	−32.4593	−1.09483
$880$	0	0
$881$	51.4637	1.73386	0.866929	−	0.498432i	$-0.166091\pi$
$881$	51.4637	1.73386	0.866929	+	0.498432i	$0.166091\pi$
$882$	0	0
$883$	−28.5358	−0.960305	−0.480152	−	0.877185i	$-0.659419\pi$
$883$	−28.5358	−0.960305	−0.480152	+	0.877185i	$0.659419\pi$
$884$	0	0
$885$	10.7539	0.361489
$886$	0	0
$887$	−21.7123	−0.729027	−0.364514	−	0.931198i	$-0.618765\pi$
$887$	−21.7123	−0.729027	−0.364514	+	0.931198i	$0.618765\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−15.4318	−0.516984
$892$	0	0
$893$	2.30183	0.0770279
$894$	0	0
$895$	−3.01792	−0.100878
$896$	0	0
$897$	−0.355283	−0.0118625
$898$	0	0
$899$	8.17982	0.272812
$900$	0	0
$901$	43.8948	1.46235
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.13833	0.270527
$906$	0	0
$907$	−41.4824	−1.37740	−0.688700	−	0.725046i	$-0.741818\pi$
$907$	−41.4824	−1.37740	−0.688700	+	0.725046i	$0.741818\pi$
$908$	0	0
$909$	47.3740	1.57130
$910$	0	0
$911$	34.7992	1.15295	0.576474	−	0.817115i	$-0.304428\pi$
$911$	34.7992	1.15295	0.576474	+	0.817115i	$0.304428\pi$
$912$	0	0
$913$	4.84614	0.160384
$914$	0	0
$915$	1.54714	0.0511469
$916$	0	0
$917$	0	0
$918$	0	0
$919$	15.9080	0.524755	0.262378	−	0.964965i	$-0.415493\pi$
$919$	15.9080	0.524755	0.262378	+	0.964965i	$0.415493\pi$
$920$	0	0
$921$	25.9526	0.855166
$922$	0	0
$923$	8.72122	0.287062
$924$	0	0
$925$	−10.0524	−0.330519
$926$	0	0
$927$	−52.7918	−1.73391
$928$	0	0
$929$	−54.5601	−1.79006	−0.895029	−	0.446008i	$-0.852845\pi$
$929$	−54.5601	−1.79006	−0.895029	+	0.446008i	$0.852845\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21.8166	−0.714243
$934$	0	0
$935$	−8.03484	−0.262767
$936$	0	0
$937$	−3.49463	−0.114164	−0.0570822	−	0.998369i	$-0.518180\pi$
$937$	−3.49463	−0.114164	−0.0570822	+	0.998369i	$0.518180\pi$
$938$	0	0
$939$	41.3953	1.35088
$940$	0	0
$941$	−18.2788	−0.595871	−0.297936	−	0.954586i	$-0.596298\pi$
$941$	−18.2788	−0.595871	−0.297936	+	0.954586i	$0.596298\pi$
$942$	0	0
$943$	−0.0771055	−0.00251090
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.6963	−0.445069	−0.222534	−	0.974925i	$-0.571433\pi$
$947$	−13.6963	−0.445069	−0.222534	+	0.974925i	$0.571433\pi$
$948$	0	0
$949$	−18.0541	−0.586061
$950$	0	0
$951$	−13.9837	−0.453452
$952$	0	0
$953$	−41.4865	−1.34388	−0.671940	−	0.740606i	$-0.734539\pi$
$953$	−41.4865	−1.34388	−0.671940	+	0.740606i	$0.734539\pi$
$954$	0	0
$955$	−0.0498088	−0.00161177
$956$	0	0
$957$	11.4038	0.368631
$958$	0	0
$959$	0	0
$960$	0	0
$961$	53.2394	1.71740
$962$	0	0
$963$	49.3393	1.58994
$964$	0	0
$965$	1.07287	0.0345369
$966$	0	0
$967$	−9.55841	−0.307378	−0.153689	−	0.988119i	$-0.549115\pi$
$967$	−9.55841	−0.307378	−0.153689	+	0.988119i	$0.549115\pi$
$968$	0	0
$969$	−4.73136	−0.151993
$970$	0	0
$971$	38.0415	1.22081	0.610405	−	0.792090i	$-0.291007\pi$
$971$	38.0415	1.22081	0.610405	+	0.792090i	$0.291007\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−22.0238	−0.705327
$976$	0	0
$977$	52.4259	1.67725	0.838626	−	0.544708i	$-0.183360\pi$
$977$	52.4259	1.67725	0.838626	+	0.544708i	$0.183360\pi$
$978$	0	0
$979$	−17.7171	−0.566242
$980$	0	0
$981$	28.0783	0.896471
$982$	0	0
$983$	−22.4487	−0.716002	−0.358001	−	0.933721i	$-0.616542\pi$
$983$	−22.4487	−0.716002	−0.358001	+	0.933721i	$0.616542\pi$
$984$	0	0
$985$	0.887853	0.0282893
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.509930	−0.0162148
$990$	0	0
$991$	−38.6350	−1.22728	−0.613641	−	0.789585i	$-0.710296\pi$
$991$	−38.6350	−1.22728	−0.613641	+	0.789585i	$0.710296\pi$
$992$	0	0
$993$	−31.8158	−1.00964
$994$	0	0
$995$	−0.137009	−0.00434347
$996$	0	0
$997$	−46.4862	−1.47223	−0.736116	−	0.676855i	$-0.763342\pi$
$997$	−46.4862	−1.47223	−0.736116	+	0.676855i	$0.763342\pi$
$998$	0	0
$999$	7.54756	0.238794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.9	yes	10
7.6	odd	2		8036.2.a.o.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.2	✓	10	7.6	odd	2
8036.2.a.p.1.9	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70664 q^{3} +0.469317 q^{5} +4.32591 q^{9} +O(q^{10})\)	\(q+2.70664 q^{3} +0.469317 q^{5} +4.32591 q^{9} +4.72750 q^{11} +1.70239 q^{13} +1.27027 q^{15} -3.62142 q^{17} +0.482699 q^{19} -0.0771055 q^{23} -4.77974 q^{25} +3.58875 q^{27} +0.891222 q^{29} +9.17820 q^{31} +12.7957 q^{33} +2.10312 q^{37} +4.60775 q^{39} +1.00000 q^{41} +6.61340 q^{43} +2.03022 q^{45} +4.76867 q^{47} -9.80188 q^{51} -12.1209 q^{53} +2.21870 q^{55} +1.30649 q^{57} +8.46582 q^{59} +1.21796 q^{61} +0.798959 q^{65} -3.96167 q^{67} -0.208697 q^{69} +5.12294 q^{71} -10.6052 q^{73} -12.9370 q^{75} +14.6429 q^{79} -3.26426 q^{81} +1.02509 q^{83} -1.69960 q^{85} +2.41222 q^{87} -3.74768 q^{89} +24.8421 q^{93} +0.226539 q^{95} +11.0675 q^{97} +20.4507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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