LMFDB - Embedded newform 10000.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.18625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} + 9x + 41 $ x^4 - x^3 - 14x^2 + 9x + 41
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1250)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.08634$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46831 q^{3} +0.710571 q^{7} +3.09254 q^{9} +O(q^{10})$	$q-2.46831 q^{3} +0.710571 q^{7} +3.09254 q^{9} -5.37577 q^{11} -6.61184 q^{13} +7.22987 q^{17} +2.14353 q^{19} -1.75391 q^{21} -3.32241 q^{23} -0.228409 q^{27} -3.56084 q^{29} -5.67522 q^{31} +13.2690 q^{33} -3.28943 q^{37} +16.3200 q^{39} -1.46831 q^{41} -0.681421 q^{43} +5.37577 q^{47} -6.49509 q^{49} -17.8455 q^{51} +0.624230 q^{53} -5.29089 q^{57} -9.90746 q^{59} -5.84791 q^{61} +2.19747 q^{63} +8.00383 q^{67} +8.20072 q^{69} -8.07395 q^{71} -10.8771 q^{73} -3.81986 q^{77} +7.66283 q^{79} -8.71383 q^{81} -2.82495 q^{83} +8.78925 q^{87} +2.02915 q^{89} -4.69818 q^{91} +14.0082 q^{93} -1.79308 q^{97} -16.6248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 3 q^{7} + 17 q^{9}+O(q^{10}) $ 4 * q - q^3 + 3 * q^7 + 17 * q^9	$ 4 q - q^{3} + 3 q^{7} + 17 q^{9} - 8 q^{11} - 4 q^{13} + 2 q^{17} - 5 q^{19} - 7 q^{21} + 9 q^{23} - 10 q^{27} - 10 q^{29} - 18 q^{31} + 22 q^{33} - 13 q^{37} + 16 q^{39} + 3 q^{41} - 16 q^{43} + 8 q^{47} + 23 q^{49} - 13 q^{51} + 16 q^{53} + 15 q^{57} - 35 q^{59} + 8 q^{61} + 54 q^{63} + 23 q^{67} + 19 q^{69} + 17 q^{71} + q^{73} + 29 q^{77} - 10 q^{79} + 24 q^{81} - 11 q^{83} + 40 q^{87} - 5 q^{89} + 17 q^{91} - 38 q^{93} - 3 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - q^3 + 3 * q^7 + 17 * q^9 - 8 * q^11 - 4 * q^13 + 2 * q^17 - 5 * q^19 - 7 * q^21 + 9 * q^23 - 10 * q^27 - 10 * q^29 - 18 * q^31 + 22 * q^33 - 13 * q^37 + 16 * q^39 + 3 * q^41 - 16 * q^43 + 8 * q^47 + 23 * q^49 - 13 * q^51 + 16 * q^53 + 15 * q^57 - 35 * q^59 + 8 * q^61 + 54 * q^63 + 23 * q^67 + 19 * q^69 + 17 * q^71 + q^73 + 29 * q^77 - 10 * q^79 + 24 * q^81 - 11 * q^83 + 40 * q^87 - 5 * q^89 + 17 * q^91 - 38 * q^93 - 3 * q^97 - 4 * 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
$3$	−2.46831	−1.42508	−0.712539	−	0.701633i	$-0.752455\pi$
$3$	−2.46831	−1.42508	−0.712539	+	0.701633i	$0.752455\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.710571	0.268570	0.134285	−	0.990943i	$-0.457126\pi$
$7$	0.710571	0.268570	0.134285	+	0.990943i	$0.457126\pi$
$8$	0	0
$9$	3.09254	1.03085
$10$	0	0
$11$	−5.37577	−1.62086	−0.810428	−	0.585839i	$-0.800765\pi$
$11$	−5.37577	−1.62086	−0.810428	+	0.585839i	$0.800765\pi$
$12$	0	0
$13$	−6.61184	−1.83379	−0.916897	−	0.399124i	$-0.869314\pi$
$13$	−6.61184	−1.83379	−0.916897	+	0.399124i	$0.869314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.22987	1.75350	0.876751	−	0.480945i	$-0.159706\pi$
$17$	7.22987	1.75350	0.876751	+	0.480945i	$0.159706\pi$
$18$	0	0
$19$	2.14353	0.491760	0.245880	−	0.969300i	$-0.420923\pi$
$19$	2.14353	0.491760	0.245880	+	0.969300i	$0.420923\pi$
$20$	0	0
$21$	−1.75391	−0.382734
$22$	0	0
$23$	−3.32241	−0.692770	−0.346385	−	0.938092i	$-0.612591\pi$
$23$	−3.32241	−0.692770	−0.346385	+	0.938092i	$0.612591\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.228409	−0.0439573
$28$	0	0
$29$	−3.56084	−0.661232	−0.330616	−	0.943765i	$-0.607256\pi$
$29$	−3.56084	−0.661232	−0.330616	+	0.943765i	$0.607256\pi$
$30$	0	0
$31$	−5.67522	−1.01930	−0.509650	−	0.860382i	$-0.670225\pi$
$31$	−5.67522	−1.01930	−0.509650	+	0.860382i	$0.670225\pi$
$32$	0	0
$33$	13.2690	2.30984
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.28943	−0.540779	−0.270389	−	0.962751i	$-0.587152\pi$
$37$	−3.28943	−0.540779	−0.270389	+	0.962751i	$0.587152\pi$
$38$	0	0
$39$	16.3200	2.61330
$40$	0	0
$41$	−1.46831	−0.229311	−0.114655	−	0.993405i	$-0.536576\pi$
$41$	−1.46831	−0.229311	−0.114655	+	0.993405i	$0.536576\pi$
$42$	0	0
$43$	−0.681421	−0.103916	−0.0519579	−	0.998649i	$-0.516546\pi$
$43$	−0.681421	−0.103916	−0.0519579	+	0.998649i	$0.516546\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.37577	0.784137	0.392068	−	0.919936i	$-0.371760\pi$
$47$	5.37577	0.784137	0.392068	+	0.919936i	$0.371760\pi$
$48$	0	0
$49$	−6.49509	−0.927870
$50$	0	0
$51$	−17.8455	−2.49888
$52$	0	0
$53$	0.624230	0.0857446	0.0428723	−	0.999081i	$-0.486349\pi$
$53$	0.624230	0.0857446	0.0428723	+	0.999081i	$0.486349\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.29089	−0.700796
$58$	0	0
$59$	−9.90746	−1.28984	−0.644921	−	0.764249i	$-0.723110\pi$
$59$	−9.90746	−1.28984	−0.644921	+	0.764249i	$0.723110\pi$
$60$	0	0
$61$	−5.84791	−0.748748	−0.374374	−	0.927278i	$-0.622142\pi$
$61$	−5.84791	−0.748748	−0.374374	+	0.927278i	$0.622142\pi$
$62$	0	0
$63$	2.19747	0.276855
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00383	0.977823	0.488912	−	0.872333i	$-0.337394\pi$
$67$	8.00383	0.977823	0.488912	+	0.872333i	$0.337394\pi$
$68$	0	0
$69$	8.20072	0.987251
$70$	0	0
$71$	−8.07395	−0.958201	−0.479101	−	0.877760i	$-0.659037\pi$
$71$	−8.07395	−0.958201	−0.479101	+	0.877760i	$0.659037\pi$
$72$	0	0
$73$	−10.8771	−1.27306	−0.636532	−	0.771251i	$-0.719631\pi$
$73$	−10.8771	−1.27306	−0.636532	+	0.771251i	$0.719631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.81986	−0.435314
$78$	0	0
$79$	7.66283	0.862136	0.431068	−	0.902319i	$-0.358137\pi$
$79$	7.66283	0.862136	0.431068	+	0.902319i	$0.358137\pi$
$80$	0	0
$81$	−8.71383	−0.968203
$82$	0	0
$83$	−2.82495	−0.310079	−0.155039	−	0.987908i	$-0.549550\pi$
$83$	−2.82495	−0.310079	−0.155039	+	0.987908i	$0.549550\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.78925	0.942307
$88$	0	0
$89$	2.02915	0.215089	0.107545	−	0.994200i	$-0.465701\pi$
$89$	2.02915	0.215089	0.107545	+	0.994200i	$0.465701\pi$
$90$	0	0
$91$	−4.69818	−0.492503
$92$	0	0
$93$	14.0082	1.45258
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.79308	−0.182060	−0.0910299	−	0.995848i	$-0.529016\pi$
$97$	−1.79308	−0.182060	−0.0910299	+	0.995848i	$0.529016\pi$
$98$	0	0
$99$	−16.6248	−1.67085
$100$	0	0
$101$	7.26905	0.723297	0.361649	−	0.932314i	$-0.382214\pi$
$101$	7.26905	0.723297	0.361649	+	0.932314i	$0.382214\pi$
$102$	0	0
$103$	11.8402	1.16665	0.583327	−	0.812237i	$-0.301751\pi$
$103$	11.8402	1.16665	0.583327	+	0.812237i	$0.301751\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.86413	−0.276886	−0.138443	−	0.990370i	$-0.544210\pi$
$107$	−2.86413	−0.276886	−0.138443	+	0.990370i	$0.544210\pi$
$108$	0	0
$109$	−7.73352	−0.740737	−0.370369	−	0.928885i	$-0.620769\pi$
$109$	−7.73352	−0.740737	−0.370369	+	0.928885i	$0.620769\pi$
$110$	0	0
$111$	8.11932	0.770652
$112$	0	0
$113$	−13.3772	−1.25842	−0.629212	−	0.777234i	$-0.716622\pi$
$113$	−13.3772	−1.25842	−0.629212	+	0.777234i	$0.716622\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−20.4474	−1.89036
$118$	0	0
$119$	5.13733	0.470939
$120$	0	0
$121$	17.8989	1.62717
$122$	0	0
$123$	3.62423	0.326786
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.4621	1.54951	0.774756	−	0.632261i	$-0.217873\pi$
$127$	17.4621	1.54951	0.774756	+	0.632261i	$0.217873\pi$
$128$	0	0
$129$	1.68196	0.148088
$130$	0	0
$131$	−11.8064	−1.03153	−0.515763	−	0.856731i	$-0.672492\pi$
$131$	−11.8064	−1.03153	−0.515763	+	0.856731i	$0.672492\pi$
$132$	0	0
$133$	1.52313	0.132072
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.77159	−0.578536	−0.289268	−	0.957248i	$-0.593412\pi$
$137$	−6.77159	−0.578536	−0.289268	+	0.957248i	$0.593412\pi$
$138$	0	0
$139$	−19.1641	−1.62548	−0.812740	−	0.582627i	$-0.802025\pi$
$139$	−19.1641	−1.62548	−0.812740	+	0.582627i	$0.802025\pi$
$140$	0	0
$141$	−13.2690	−1.11746
$142$	0	0
$143$	35.5437	2.97231
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.0319	1.32229
$148$	0	0
$149$	−19.4497	−1.59338	−0.796691	−	0.604386i	$-0.793418\pi$
$149$	−19.4497	−1.59338	−0.796691	+	0.604386i	$0.793418\pi$
$150$	0	0
$151$	11.6095	0.944765	0.472383	−	0.881394i	$-0.343394\pi$
$151$	11.6095	0.944765	0.472383	+	0.881394i	$0.343394\pi$
$152$	0	0
$153$	22.3586	1.80759
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.94664	−0.634211	−0.317105	−	0.948390i	$-0.602711\pi$
$157$	−7.94664	−0.634211	−0.317105	+	0.948390i	$0.602711\pi$
$158$	0	0
$159$	−1.54079	−0.122193
$160$	0	0
$161$	−2.36081	−0.186058
$162$	0	0
$163$	−1.89181	−0.148178	−0.0740892	−	0.997252i	$-0.523605\pi$
$163$	−1.89181	−0.148178	−0.0740892	+	0.997252i	$0.523605\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.14736	−0.475697	−0.237849	−	0.971302i	$-0.576442\pi$
$167$	−6.14736	−0.475697	−0.237849	+	0.971302i	$0.576442\pi$
$168$	0	0
$169$	30.7164	2.36280
$170$	0	0
$171$	6.62895	0.506928
$172$	0	0
$173$	7.74592	0.588911	0.294456	−	0.955665i	$-0.404862\pi$
$173$	7.74592	0.588911	0.294456	+	0.955665i	$0.404862\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.4547	1.83812
$178$	0	0
$179$	−0.810766	−0.0605995	−0.0302998	−	0.999541i	$-0.509646\pi$
$179$	−0.810766	−0.0605995	−0.0302998	+	0.999541i	$0.509646\pi$
$180$	0	0
$181$	20.6528	1.53511	0.767556	−	0.640982i	$-0.221473\pi$
$181$	20.6528	1.53511	0.767556	+	0.640982i	$0.221473\pi$
$182$	0	0
$183$	14.4344	1.06702
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−38.8661	−2.84217
$188$	0	0
$189$	−0.162301	−0.0118056
$190$	0	0
$191$	6.45481	0.467053	0.233527	−	0.972350i	$-0.424973\pi$
$191$	6.45481	0.467053	0.233527	+	0.972350i	$0.424973\pi$
$192$	0	0
$193$	−4.28323	−0.308314	−0.154157	−	0.988046i	$-0.549266\pi$
$193$	−4.28323	−0.308314	−0.154157	+	0.988046i	$0.549266\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.7344	1.76225	0.881127	−	0.472880i	$-0.156785\pi$
$197$	24.7344	1.76225	0.881127	+	0.472880i	$0.156785\pi$
$198$	0	0
$199$	−20.3044	−1.43934	−0.719670	−	0.694316i	$-0.755707\pi$
$199$	−20.3044	−1.43934	−0.719670	+	0.694316i	$0.755707\pi$
$200$	0	0
$201$	−19.7559	−1.39347
$202$	0	0
$203$	−2.53023	−0.177587
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.2747	−0.714139
$208$	0	0
$209$	−11.5231	−0.797072
$210$	0	0
$211$	7.48599	0.515357	0.257678	−	0.966231i	$-0.417042\pi$
$211$	7.48599	0.515357	0.257678	+	0.966231i	$0.417042\pi$
$212$	0	0
$213$	19.9290	1.36551
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.03265	−0.273754
$218$	0	0
$219$	26.8479	1.81421
$220$	0	0
$221$	−47.8027	−3.21556
$222$	0	0
$223$	−9.77449	−0.654548	−0.327274	−	0.944929i	$-0.606130\pi$
$223$	−9.77449	−0.654548	−0.327274	+	0.944929i	$0.606130\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.37161	−0.489271	−0.244635	−	0.969615i	$-0.578668\pi$
$227$	−7.37161	−0.489271	−0.244635	+	0.969615i	$0.578668\pi$
$228$	0	0
$229$	18.8379	1.24484	0.622421	−	0.782683i	$-0.286149\pi$
$229$	18.8379	1.24484	0.622421	+	0.782683i	$0.286149\pi$
$230$	0	0
$231$	9.42860	0.620356
$232$	0	0
$233$	−1.04920	−0.0687355	−0.0343677	−	0.999409i	$-0.510942\pi$
$233$	−1.04920	−0.0687355	−0.0343677	+	0.999409i	$0.510942\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−18.9142	−1.22861
$238$	0	0
$239$	−5.39636	−0.349061	−0.174531	−	0.984652i	$-0.555841\pi$
$239$	−5.39636	−0.349061	−0.174531	+	0.984652i	$0.555841\pi$
$240$	0	0
$241$	−17.2481	−1.11105	−0.555524	−	0.831501i	$-0.687482\pi$
$241$	−17.2481	−1.11105	−0.555524	+	0.831501i	$0.687482\pi$
$242$	0	0
$243$	22.1936	1.42372
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.1727	−0.901786
$248$	0	0
$249$	6.97285	0.441886
$250$	0	0
$251$	−0.283233	−0.0178775	−0.00893875	−	0.999960i	$-0.502845\pi$
$251$	−0.283233	−0.0178775	−0.00893875	+	0.999960i	$0.502845\pi$
$252$	0	0
$253$	17.8605	1.12288
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.05482	0.0657981	0.0328991	−	0.999459i	$-0.489526\pi$
$257$	1.05482	0.0657981	0.0328991	+	0.999459i	$0.489526\pi$
$258$	0	0
$259$	−2.33737	−0.145237
$260$	0	0
$261$	−11.0120	−0.681628
$262$	0	0
$263$	9.86829	0.608505	0.304252	−	0.952591i	$-0.401593\pi$
$263$	9.86829	0.608505	0.304252	+	0.952591i	$0.401593\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.00856	−0.306519
$268$	0	0
$269$	−18.6805	−1.13897	−0.569486	−	0.822001i	$-0.692858\pi$
$269$	−18.6805	−1.13897	−0.569486	+	0.822001i	$0.692858\pi$
$270$	0	0
$271$	−17.1497	−1.04177	−0.520886	−	0.853626i	$-0.674398\pi$
$271$	−17.1497	−1.04177	−0.520886	+	0.853626i	$0.674398\pi$
$272$	0	0
$273$	11.5965	0.701855
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.6144	−1.53902	−0.769510	−	0.638634i	$-0.779500\pi$
$277$	−25.6144	−1.53902	−0.769510	+	0.638634i	$0.779500\pi$
$278$	0	0
$279$	−17.5508	−1.05074
$280$	0	0
$281$	−7.53732	−0.449639	−0.224819	−	0.974400i	$-0.572179\pi$
$281$	−7.53732	−0.449639	−0.224819	+	0.974400i	$0.572179\pi$
$282$	0	0
$283$	6.89744	0.410010	0.205005	−	0.978761i	$-0.434279\pi$
$283$	6.89744	0.410010	0.205005	+	0.978761i	$0.434279\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.04334	−0.0615861
$288$	0	0
$289$	35.2710	2.07477
$290$	0	0
$291$	4.42587	0.259449
$292$	0	0
$293$	14.8455	0.867286	0.433643	−	0.901085i	$-0.357228\pi$
$293$	14.8455	0.867286	0.433643	+	0.901085i	$0.357228\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.22787	0.0712485
$298$	0	0
$299$	21.9672	1.27040
$300$	0	0
$301$	−0.484198	−0.0279087
$302$	0	0
$303$	−17.9422	−1.03075
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.0515679	−0.00294314	−0.00147157	−	0.999999i	$-0.500468\pi$
$307$	−0.0515679	−0.00294314	−0.00147157	+	0.999999i	$0.500468\pi$
$308$	0	0
$309$	−29.2254	−1.66257
$310$	0	0
$311$	−25.0512	−1.42052	−0.710262	−	0.703938i	$-0.751423\pi$
$311$	−25.0512	−1.42052	−0.710262	+	0.703938i	$0.751423\pi$
$312$	0	0
$313$	25.4659	1.43942	0.719710	−	0.694275i	$-0.244275\pi$
$313$	25.4659	1.43942	0.719710	+	0.694275i	$0.244275\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.0683	0.621659	0.310830	−	0.950466i	$-0.399393\pi$
$317$	11.0683	0.621659	0.310830	+	0.950466i	$0.399393\pi$
$318$	0	0
$319$	19.1423	1.07176
$320$	0	0
$321$	7.06954	0.394584
$322$	0	0
$323$	15.4975	0.862302
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.0887	1.05561
$328$	0	0
$329$	3.81986	0.210596
$330$	0	0
$331$	11.1768	0.614335	0.307167	−	0.951656i	$-0.400619\pi$
$331$	11.1768	0.614335	0.307167	+	0.951656i	$0.400619\pi$
$332$	0	0
$333$	−10.1727	−0.557459
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.9510	−0.705486	−0.352743	−	0.935720i	$-0.614751\pi$
$337$	−12.9510	−0.705486	−0.352743	+	0.935720i	$0.614751\pi$
$338$	0	0
$339$	33.0191	1.79335
$340$	0	0
$341$	30.5087	1.65214
$342$	0	0
$343$	−9.58922	−0.517769
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.4889	1.20727	0.603634	−	0.797262i	$-0.293719\pi$
$347$	22.4889	1.20727	0.603634	+	0.797262i	$0.293719\pi$
$348$	0	0
$349$	−8.57413	−0.458962	−0.229481	−	0.973313i	$-0.573703\pi$
$349$	−8.57413	−0.458962	−0.229481	+	0.973313i	$0.573703\pi$
$350$	0	0
$351$	1.51020	0.0806087
$352$	0	0
$353$	−19.2710	−1.02569	−0.512847	−	0.858480i	$-0.671409\pi$
$353$	−19.2710	−1.02569	−0.512847	+	0.858480i	$0.671409\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.6805	−0.671124
$358$	0	0
$359$	14.7188	0.776827	0.388413	−	0.921485i	$-0.373023\pi$
$359$	14.7188	0.776827	0.388413	+	0.921485i	$0.373023\pi$
$360$	0	0
$361$	−14.4053	−0.758172
$362$	0	0
$363$	−44.1800	−2.31885
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.7186	0.559504	0.279752	−	0.960072i	$-0.409748\pi$
$367$	10.7186	0.559504	0.279752	+	0.960072i	$0.409748\pi$
$368$	0	0
$369$	−4.54079	−0.236384
$370$	0	0
$371$	0.443560	0.0230285
$372$	0	0
$373$	10.6292	0.550357	0.275179	−	0.961393i	$-0.411263\pi$
$373$	10.6292	0.550357	0.275179	+	0.961393i	$0.411263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.5437	1.21256
$378$	0	0
$379$	−33.1335	−1.70195	−0.850977	−	0.525203i	$-0.823989\pi$
$379$	−33.1335	−1.70195	−0.850977	+	0.525203i	$0.823989\pi$
$380$	0	0
$381$	−43.1018	−2.20817
$382$	0	0
$383$	25.1780	1.28653	0.643267	−	0.765642i	$-0.277578\pi$
$383$	25.1780	1.28653	0.643267	+	0.765642i	$0.277578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.10732	−0.107121
$388$	0	0
$389$	19.5075	0.989069	0.494534	−	0.869158i	$-0.335339\pi$
$389$	19.5075	0.989069	0.494534	+	0.869158i	$0.335339\pi$
$390$	0	0
$391$	−24.0206	−1.21477
$392$	0	0
$393$	29.1417	1.47001
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.7206	0.788994	0.394497	−	0.918897i	$-0.370919\pi$
$397$	15.7206	0.788994	0.394497	+	0.918897i	$0.370919\pi$
$398$	0	0
$399$	−3.75955	−0.188213
$400$	0	0
$401$	3.03918	0.151769	0.0758846	−	0.997117i	$-0.475822\pi$
$401$	3.03918	0.151769	0.0758846	+	0.997117i	$0.475822\pi$
$402$	0	0
$403$	37.5237	1.86919
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.6832	0.876524
$408$	0	0
$409$	31.4623	1.55571	0.777856	−	0.628443i	$-0.216307\pi$
$409$	31.4623	1.55571	0.777856	+	0.628443i	$0.216307\pi$
$410$	0	0
$411$	16.7144	0.824459
$412$	0	0
$413$	−7.03995	−0.346413
$414$	0	0
$415$	0	0
$416$	0	0
$417$	47.3029	2.31643
$418$	0	0
$419$	−8.18213	−0.399723	−0.199862	−	0.979824i	$-0.564049\pi$
$419$	−8.18213	−0.399723	−0.199862	+	0.979824i	$0.564049\pi$
$420$	0	0
$421$	2.66960	0.130108	0.0650542	−	0.997882i	$-0.479278\pi$
$421$	2.66960	0.130108	0.0650542	+	0.997882i	$0.479278\pi$
$422$	0	0
$423$	16.6248	0.808324
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.15535	−0.201092
$428$	0	0
$429$	−87.7328	−4.23578
$430$	0	0
$431$	25.7871	1.24212	0.621060	−	0.783763i	$-0.286702\pi$
$431$	25.7871	1.24212	0.621060	+	0.783763i	$0.286702\pi$
$432$	0	0
$433$	22.6283	1.08744	0.543722	−	0.839265i	$-0.317014\pi$
$433$	22.6283	1.08744	0.543722	+	0.839265i	$0.317014\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.12169	−0.340676
$438$	0	0
$439$	23.3967	1.11666	0.558331	−	0.829618i	$-0.311442\pi$
$439$	23.3967	1.11666	0.558331	+	0.829618i	$0.311442\pi$
$440$	0	0
$441$	−20.0863	−0.956491
$442$	0	0
$443$	−6.41563	−0.304816	−0.152408	−	0.988318i	$-0.548703\pi$
$443$	−6.41563	−0.304816	−0.152408	+	0.988318i	$0.548703\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	48.0079	2.27069
$448$	0	0
$449$	−4.06192	−0.191694	−0.0958470	−	0.995396i	$-0.530556\pi$
$449$	−4.06192	−0.191694	−0.0958470	+	0.995396i	$0.530556\pi$
$450$	0	0
$451$	7.89328	0.371680
$452$	0	0
$453$	−28.6557	−1.34636
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.5022	−1.28650	−0.643249	−	0.765657i	$-0.722414\pi$
$457$	−27.5022	−1.28650	−0.643249	+	0.765657i	$0.722414\pi$
$458$	0	0
$459$	−1.65137	−0.0770792
$460$	0	0
$461$	8.47486	0.394713	0.197357	−	0.980332i	$-0.436764\pi$
$461$	8.47486	0.394713	0.197357	+	0.980332i	$0.436764\pi$
$462$	0	0
$463$	7.47450	0.347370	0.173685	−	0.984801i	$-0.444433\pi$
$463$	7.47450	0.347370	0.173685	+	0.984801i	$0.444433\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.1388	−0.746814	−0.373407	−	0.927668i	$-0.621811\pi$
$467$	−16.1388	−0.746814	−0.373407	+	0.927668i	$0.621811\pi$
$468$	0	0
$469$	5.68729	0.262615
$470$	0	0
$471$	19.6147	0.903799
$472$	0	0
$473$	3.66316	0.168432
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.93046	0.0883895
$478$	0	0
$479$	−30.0745	−1.37414	−0.687069	−	0.726592i	$-0.741103\pi$
$479$	−30.0745	−1.37414	−0.687069	+	0.726592i	$0.741103\pi$
$480$	0	0
$481$	21.7492	0.991677
$482$	0	0
$483$	5.82719	0.265146
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.1217	−0.685229	−0.342615	−	0.939476i	$-0.611312\pi$
$487$	−15.1217	−0.685229	−0.342615	+	0.939476i	$0.611312\pi$
$488$	0	0
$489$	4.66958	0.211166
$490$	0	0
$491$	38.1187	1.72027	0.860137	−	0.510062i	$-0.170378\pi$
$491$	38.1187	1.72027	0.860137	+	0.510062i	$0.170378\pi$
$492$	0	0
$493$	−25.7444	−1.15947
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.73711	−0.257345
$498$	0	0
$499$	−6.98088	−0.312507	−0.156253	−	0.987717i	$-0.549942\pi$
$499$	−6.98088	−0.312507	−0.156253	+	0.987717i	$0.549942\pi$
$500$	0	0
$501$	15.1736	0.677905
$502$	0	0
$503$	1.13680	0.0506874	0.0253437	−	0.999679i	$-0.491932\pi$
$503$	1.13680	0.0506874	0.0253437	+	0.999679i	$0.491932\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−75.8175	−3.36717
$508$	0	0
$509$	7.98435	0.353900	0.176950	−	0.984220i	$-0.443377\pi$
$509$	7.98435	0.353900	0.176950	+	0.984220i	$0.443377\pi$
$510$	0	0
$511$	−7.72892	−0.341907
$512$	0	0
$513$	−0.489602	−0.0216164
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28.8989	−1.27097
$518$	0	0
$519$	−19.1193	−0.839244
$520$	0	0
$521$	32.8997	1.44136	0.720681	−	0.693267i	$-0.243829\pi$
$521$	32.8997	1.44136	0.720681	+	0.693267i	$0.243829\pi$
$522$	0	0
$523$	−1.72859	−0.0755858	−0.0377929	−	0.999286i	$-0.512033\pi$
$523$	−1.72859	−0.0755858	−0.0377929	+	0.999286i	$0.512033\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.0311	−1.78734
$528$	0	0
$529$	−11.9616	−0.520070
$530$	0	0
$531$	−30.6392	−1.32963
$532$	0	0
$533$	9.70820	0.420509
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.00122	0.0863590
$538$	0	0
$539$	34.9161	1.50394
$540$	0	0
$541$	43.2902	1.86119	0.930595	−	0.366051i	$-0.119290\pi$
$541$	43.2902	1.86119	0.930595	+	0.366051i	$0.119290\pi$
$542$	0	0
$543$	−50.9775	−2.18765
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−37.5390	−1.60505	−0.802526	−	0.596618i	$-0.796511\pi$
$547$	−37.5390	−1.60505	−0.802526	+	0.596618i	$0.796511\pi$
$548$	0	0
$549$	−18.0849	−0.771843
$550$	0	0
$551$	−7.63278	−0.325167
$552$	0	0
$553$	5.44498	0.231544
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.0519	1.86654	0.933269	−	0.359178i	$-0.116943\pi$
$557$	44.0519	1.86654	0.933269	+	0.359178i	$0.116943\pi$
$558$	0	0
$559$	4.50545	0.190560
$560$	0	0
$561$	95.9335	4.05032
$562$	0	0
$563$	38.5528	1.62481	0.812404	−	0.583095i	$-0.198159\pi$
$563$	38.5528	1.62481	0.812404	+	0.583095i	$0.198159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6.19179	−0.260031
$568$	0	0
$569$	8.51164	0.356827	0.178413	−	0.983956i	$-0.442904\pi$
$569$	8.51164	0.356827	0.178413	+	0.983956i	$0.442904\pi$
$570$	0	0
$571$	12.3845	0.518277	0.259138	−	0.965840i	$-0.416561\pi$
$571$	12.3845	0.518277	0.259138	+	0.965840i	$0.416561\pi$
$572$	0	0
$573$	−15.9324	−0.665587
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.6271	0.650566	0.325283	−	0.945617i	$-0.394540\pi$
$577$	15.6271	0.650566	0.325283	+	0.945617i	$0.394540\pi$
$578$	0	0
$579$	10.5723	0.439371
$580$	0	0
$581$	−2.00733	−0.0832780
$582$	0	0
$583$	−3.35572	−0.138980
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.4271	1.13204	0.566019	−	0.824392i	$-0.308483\pi$
$587$	27.4271	1.13204	0.566019	+	0.824392i	$0.308483\pi$
$588$	0	0
$589$	−12.1650	−0.501251
$590$	0	0
$591$	−61.0521	−2.51135
$592$	0	0
$593$	−11.8754	−0.487663	−0.243832	−	0.969818i	$-0.578404\pi$
$593$	−11.8754	−0.487663	−0.243832	+	0.969818i	$0.578404\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	50.1175	2.05117
$598$	0	0
$599$	−6.75648	−0.276062	−0.138031	−	0.990428i	$-0.544077\pi$
$599$	−6.75648	−0.276062	−0.138031	+	0.990428i	$0.544077\pi$
$600$	0	0
$601$	−35.4612	−1.44649	−0.723246	−	0.690590i	$-0.757351\pi$
$601$	−35.4612	−1.44649	−0.723246	+	0.690590i	$0.757351\pi$
$602$	0	0
$603$	24.7521	1.00798
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.5868	−0.429707	−0.214853	−	0.976646i	$-0.568927\pi$
$607$	−10.5868	−0.429707	−0.214853	+	0.976646i	$0.568927\pi$
$608$	0	0
$609$	6.24539	0.253076
$610$	0	0
$611$	−35.5437	−1.43794
$612$	0	0
$613$	5.32513	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$613$	5.32513	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.37939	−0.216566	−0.108283	−	0.994120i	$-0.534535\pi$
$617$	−5.37939	−0.216566	−0.108283	+	0.994120i	$0.534535\pi$
$618$	0	0
$619$	−23.3512	−0.938565	−0.469283	−	0.883048i	$-0.655487\pi$
$619$	−23.3512	−0.938565	−0.469283	+	0.883048i	$0.655487\pi$
$620$	0	0
$621$	0.758868	0.0304523
$622$	0	0
$623$	1.44185	0.0577667
$624$	0	0
$625$	0	0
$626$	0	0
$627$	28.4426	1.13589
$628$	0	0
$629$	−23.7822	−0.948256
$630$	0	0
$631$	−19.0073	−0.756669	−0.378334	−	0.925669i	$-0.623503\pi$
$631$	−19.0073	−0.756669	−0.378334	+	0.925669i	$0.623503\pi$
$632$	0	0
$633$	−18.4777	−0.734423
$634$	0	0
$635$	0	0
$636$	0	0
$637$	42.9445	1.70152
$638$	0	0
$639$	−24.9690	−0.987758
$640$	0	0
$641$	12.7751	0.504585	0.252292	−	0.967651i	$-0.418816\pi$
$641$	12.7751	0.504585	0.252292	+	0.967651i	$0.418816\pi$
$642$	0	0
$643$	−34.9242	−1.37728	−0.688638	−	0.725105i	$-0.741791\pi$
$643$	−34.9242	−1.37728	−0.688638	+	0.725105i	$0.741791\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.1090	0.987135	0.493568	−	0.869707i	$-0.335693\pi$
$647$	25.1090	0.987135	0.493568	+	0.869707i	$0.335693\pi$
$648$	0	0
$649$	53.2602	2.09065
$650$	0	0
$651$	9.95381	0.390121
$652$	0	0
$653$	28.3657	1.11004	0.555019	−	0.831838i	$-0.312711\pi$
$653$	28.3657	1.11004	0.555019	+	0.831838i	$0.312711\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−33.6377	−1.31233
$658$	0	0
$659$	−13.5523	−0.527922	−0.263961	−	0.964533i	$-0.585029\pi$
$659$	−13.5523	−0.527922	−0.263961	+	0.964533i	$0.585029\pi$
$660$	0	0
$661$	−23.1326	−0.899754	−0.449877	−	0.893091i	$-0.648532\pi$
$661$	−23.1326	−0.899754	−0.449877	+	0.893091i	$0.648532\pi$
$662$	0	0
$663$	117.992	4.58242
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.8306	0.458082
$668$	0	0
$669$	24.1264	0.932782
$670$	0	0
$671$	31.4370	1.21361
$672$	0	0
$673$	42.0453	1.62073	0.810364	−	0.585926i	$-0.199269\pi$
$673$	42.0453	1.62073	0.810364	+	0.585926i	$0.199269\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.8428	1.30069	0.650343	−	0.759641i	$-0.274625\pi$
$677$	33.8428	1.30069	0.650343	+	0.759641i	$0.274625\pi$
$678$	0	0
$679$	−1.27411	−0.0488959
$680$	0	0
$681$	18.1954	0.697249
$682$	0	0
$683$	−44.0853	−1.68688	−0.843438	−	0.537226i	$-0.819472\pi$
$683$	−44.0853	−1.68688	−0.843438	+	0.537226i	$0.819472\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−46.4977	−1.77400
$688$	0	0
$689$	−4.12731	−0.157238
$690$	0	0
$691$	27.6884	1.05331	0.526657	−	0.850078i	$-0.323445\pi$
$691$	27.6884	1.05331	0.526657	+	0.850078i	$0.323445\pi$
$692$	0	0
$693$	−11.8131	−0.448741
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.6157	−0.402097
$698$	0	0
$699$	2.58975	0.0979533
$700$	0	0
$701$	32.2086	1.21650	0.608251	−	0.793745i	$-0.291872\pi$
$701$	32.2086	1.21650	0.608251	+	0.793745i	$0.291872\pi$
$702$	0	0
$703$	−7.05099	−0.265933
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.16517	0.194256
$708$	0	0
$709$	2.42150	0.0909412	0.0454706	−	0.998966i	$-0.485521\pi$
$709$	2.42150	0.0909412	0.0454706	+	0.998966i	$0.485521\pi$
$710$	0	0
$711$	23.6976	0.888729
$712$	0	0
$713$	18.8554	0.706141
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.3199	0.497439
$718$	0	0
$719$	43.2412	1.61263	0.806313	−	0.591490i	$-0.201460\pi$
$719$	43.2412	1.61263	0.806313	+	0.591490i	$0.201460\pi$
$720$	0	0
$721$	8.41333	0.313329
$722$	0	0
$723$	42.5736	1.58333
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.8936	0.700726	0.350363	−	0.936614i	$-0.386058\pi$
$727$	18.8936	0.700726	0.350363	+	0.936614i	$0.386058\pi$
$728$	0	0
$729$	−28.6392	−1.06071
$730$	0	0
$731$	−4.92659	−0.182216
$732$	0	0
$733$	17.0056	0.628117	0.314059	−	0.949404i	$-0.398311\pi$
$733$	17.0056	0.628117	0.314059	+	0.949404i	$0.398311\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.0267	−1.58491
$738$	0	0
$739$	31.4709	1.15767	0.578837	−	0.815443i	$-0.303507\pi$
$739$	31.4709	1.15767	0.578837	+	0.815443i	$0.303507\pi$
$740$	0	0
$741$	34.9825	1.28511
$742$	0	0
$743$	−4.62477	−0.169666	−0.0848331	−	0.996395i	$-0.527036\pi$
$743$	−4.62477	−0.169666	−0.0848331	+	0.996395i	$0.527036\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.73627	−0.319643
$748$	0	0
$749$	−2.03517	−0.0743633
$750$	0	0
$751$	49.7747	1.81630	0.908152	−	0.418640i	$-0.137493\pi$
$751$	49.7747	1.81630	0.908152	+	0.418640i	$0.137493\pi$
$752$	0	0
$753$	0.699106	0.0254768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.6657	0.969183	0.484591	−	0.874741i	$-0.338968\pi$
$757$	26.6657	0.969183	0.484591	+	0.874741i	$0.338968\pi$
$758$	0	0
$759$	−44.0852	−1.60019
$760$	0	0
$761$	−15.0242	−0.544627	−0.272314	−	0.962209i	$-0.587789\pi$
$761$	−15.0242	−0.544627	−0.272314	+	0.962209i	$0.587789\pi$
$762$	0	0
$763$	−5.49522	−0.198940
$764$	0	0
$765$	0	0
$766$	0	0
$767$	65.5065	2.36530
$768$	0	0
$769$	49.5198	1.78573	0.892865	−	0.450323i	$-0.148691\pi$
$769$	49.5198	1.78573	0.892865	+	0.450323i	$0.148691\pi$
$770$	0	0
$771$	−2.60363	−0.0937674
$772$	0	0
$773$	20.5493	0.739108	0.369554	−	0.929209i	$-0.379510\pi$
$773$	20.5493	0.739108	0.369554	+	0.929209i	$0.379510\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.76935	0.206974
$778$	0	0
$779$	−3.14736	−0.112766
$780$	0	0
$781$	43.4037	1.55311
$782$	0	0
$783$	0.813328	0.0290660
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36.9401	−1.31677	−0.658386	−	0.752681i	$-0.728760\pi$
$787$	−36.9401	−1.31677	−0.658386	+	0.752681i	$0.728760\pi$
$788$	0	0
$789$	−24.3580	−0.867166
$790$	0	0
$791$	−9.50547	−0.337976
$792$	0	0
$793$	38.6654	1.37305
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.3335	1.92459	0.962296	−	0.272005i	$-0.0876869\pi$
$797$	54.3335	1.92459	0.962296	+	0.272005i	$0.0876869\pi$
$798$	0	0
$799$	38.8661	1.37498
$800$	0	0
$801$	6.27522	0.221724
$802$	0	0
$803$	58.4725	2.06345
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.1092	1.62312
$808$	0	0
$809$	41.5457	1.46067	0.730335	−	0.683089i	$-0.239364\pi$
$809$	41.5457	1.46067	0.730335	+	0.683089i	$0.239364\pi$
$810$	0	0
$811$	29.4405	1.03379	0.516897	−	0.856047i	$-0.327087\pi$
$811$	29.4405	1.03379	0.516897	+	0.856047i	$0.327087\pi$
$812$	0	0
$813$	42.3308	1.48461
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.46065	−0.0511016
$818$	0	0
$819$	−14.5293	−0.507694
$820$	0	0
$821$	50.6634	1.76816	0.884082	−	0.467331i	$-0.154784\pi$
$821$	50.6634	1.76816	0.884082	+	0.467331i	$0.154784\pi$
$822$	0	0
$823$	−5.31220	−0.185172	−0.0925859	−	0.995705i	$-0.529513\pi$
$823$	−5.31220	−0.185172	−0.0925859	+	0.995705i	$0.529513\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6889	0.928065	0.464033	−	0.885818i	$-0.346402\pi$
$827$	26.6889	0.928065	0.464033	+	0.885818i	$0.346402\pi$
$828$	0	0
$829$	21.4543	0.745140	0.372570	−	0.928004i	$-0.378477\pi$
$829$	21.4543	0.745140	0.372570	+	0.928004i	$0.378477\pi$
$830$	0	0
$831$	63.2242	2.19322
$832$	0	0
$833$	−46.9587	−1.62702
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.29627	0.0448057
$838$	0	0
$839$	−0.730418	−0.0252168	−0.0126084	−	0.999921i	$-0.504013\pi$
$839$	−0.730418	−0.0252168	−0.0126084	+	0.999921i	$0.504013\pi$
$840$	0	0
$841$	−16.3204	−0.562772
$842$	0	0
$843$	18.6044	0.640770
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.7184	0.437011
$848$	0	0
$849$	−17.0250	−0.584296
$850$	0	0
$851$	10.9288	0.374635
$852$	0	0
$853$	−17.2457	−0.590483	−0.295241	−	0.955423i	$-0.595400\pi$
$853$	−17.2457	−0.590483	−0.295241	+	0.955423i	$0.595400\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.260256	0.00889017	0.00444509	−	0.999990i	$-0.498585\pi$
$857$	0.260256	0.00889017	0.00444509	+	0.999990i	$0.498585\pi$
$858$	0	0
$859$	44.0243	1.50209	0.751044	−	0.660252i	$-0.229550\pi$
$859$	44.0243	1.50209	0.751044	+	0.660252i	$0.229550\pi$
$860$	0	0
$861$	2.57527	0.0877650
$862$	0	0
$863$	−52.0417	−1.77152	−0.885760	−	0.464143i	$-0.846362\pi$
$863$	−52.0417	−1.77152	−0.885760	+	0.464143i	$0.846362\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−87.0597	−2.95670
$868$	0	0
$869$	−41.1936	−1.39740
$870$	0	0
$871$	−52.9200	−1.79313
$872$	0	0
$873$	−5.54517	−0.187676
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.0619	−1.25149	−0.625746	−	0.780027i	$-0.715205\pi$
$877$	−37.0619	−1.25149	−0.625746	+	0.780027i	$0.715205\pi$
$878$	0	0
$879$	−36.6433	−1.23595
$880$	0	0
$881$	26.0497	0.877638	0.438819	−	0.898575i	$-0.355397\pi$
$881$	26.0497	0.877638	0.438819	+	0.898575i	$0.355397\pi$
$882$	0	0
$883$	0.0774468	0.00260629	0.00130315	−	0.999999i	$-0.499585\pi$
$883$	0.0774468	0.00260629	0.00130315	+	0.999999i	$0.499585\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.03771	0.135573	0.0677866	−	0.997700i	$-0.478406\pi$
$887$	4.03771	0.135573	0.0677866	+	0.997700i	$0.478406\pi$
$888$	0	0
$889$	12.4081	0.416153
$890$	0	0
$891$	46.8435	1.56932
$892$	0	0
$893$	11.5231	0.385607
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−54.2218	−1.81041
$898$	0	0
$899$	20.2086	0.673994
$900$	0	0
$901$	4.51310	0.150353
$902$	0	0
$903$	1.19515	0.0397721
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.0596	0.466840	0.233420	−	0.972376i	$-0.425008\pi$
$907$	14.0596	0.466840	0.233420	+	0.972376i	$0.425008\pi$
$908$	0	0
$909$	22.4798	0.745608
$910$	0	0
$911$	11.8375	0.392195	0.196097	−	0.980584i	$-0.437173\pi$
$911$	11.8375	0.392195	0.196097	+	0.980584i	$0.437173\pi$
$912$	0	0
$913$	15.1863	0.502593
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.38926	−0.277038
$918$	0	0
$919$	−37.1661	−1.22600	−0.612999	−	0.790084i	$-0.710037\pi$
$919$	−37.1661	−1.22600	−0.612999	+	0.790084i	$0.710037\pi$
$920$	0	0
$921$	0.127285	0.00419420
$922$	0	0
$923$	53.3836	1.75714
$924$	0	0
$925$	0	0
$926$	0	0
$927$	36.6164	1.20264
$928$	0	0
$929$	31.1665	1.02254	0.511270	−	0.859420i	$-0.329175\pi$
$929$	31.1665	1.02254	0.511270	+	0.859420i	$0.329175\pi$
$930$	0	0
$931$	−13.9224	−0.456289
$932$	0	0
$933$	61.8340	2.02436
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.1725	−1.57373	−0.786864	−	0.617127i	$-0.788297\pi$
$937$	−48.1725	−1.57373	−0.786864	+	0.617127i	$0.788297\pi$
$938$	0	0
$939$	−62.8577	−2.05128
$940$	0	0
$941$	20.1763	0.657728	0.328864	−	0.944377i	$-0.393334\pi$
$941$	20.1763	0.657728	0.328864	+	0.944377i	$0.393334\pi$
$942$	0	0
$943$	4.87831	0.158860
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.3323	−1.57059	−0.785294	−	0.619123i	$-0.787488\pi$
$947$	−48.3323	−1.57059	−0.785294	+	0.619123i	$0.787488\pi$
$948$	0	0
$949$	71.9173	2.33454
$950$	0	0
$951$	−27.3200	−0.885912
$952$	0	0
$953$	−32.6612	−1.05800	−0.529001	−	0.848621i	$-0.677433\pi$
$953$	−32.6612	−1.05800	−0.529001	+	0.848621i	$0.677433\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−47.2490	−1.52734
$958$	0	0
$959$	−4.81169	−0.155378
$960$	0	0
$961$	1.20818	0.0389734
$962$	0	0
$963$	−8.85742	−0.285426
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.1632	0.584090	0.292045	−	0.956405i	$-0.405664\pi$
$967$	18.1632	0.584090	0.292045	+	0.956405i	$0.405664\pi$
$968$	0	0
$969$	−38.2525	−1.22885
$970$	0	0
$971$	−5.01643	−0.160985	−0.0804924	−	0.996755i	$-0.525649\pi$
$971$	−5.01643	−0.160985	−0.0804924	+	0.996755i	$0.525649\pi$
$972$	0	0
$973$	−13.6175	−0.436556
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.7701	0.440545	0.220273	−	0.975438i	$-0.429305\pi$
$977$	13.7701	0.440545	0.220273	+	0.975438i	$0.429305\pi$
$978$	0	0
$979$	−10.9082	−0.348629
$980$	0	0
$981$	−23.9162	−0.763586
$982$	0	0
$983$	0.853324	0.0272168	0.0136084	−	0.999907i	$-0.495668\pi$
$983$	0.853324	0.0272168	0.0136084	+	0.999907i	$0.495668\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9.42860	−0.300116
$988$	0	0
$989$	2.26396	0.0719897
$990$	0	0
$991$	29.3689	0.932932	0.466466	−	0.884539i	$-0.345527\pi$
$991$	29.3689	0.932932	0.466466	+	0.884539i	$0.345527\pi$
$992$	0	0
$993$	−27.5879	−0.875475
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.98761	−0.189629	−0.0948147	−	0.995495i	$-0.530226\pi$
$997$	−5.98761	−0.189629	−0.0948147	+	0.995495i	$0.530226\pi$
$998$	0	0
$999$	0.751335	0.0237712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.u.1.2		4
4.3	odd	2		1250.2.a.j.1.3	yes	4
5.4	even	2		10000.2.a.v.1.3		4
20.3	even	4		1250.2.b.d.1249.3		8
20.7	even	4		1250.2.b.d.1249.6		8
20.19	odd	2		1250.2.a.g.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.2.a.g.1.2	✓	4	20.19	odd	2
1250.2.a.j.1.3	yes	4	4.3	odd	2
1250.2.b.d.1249.3		8	20.3	even	4
1250.2.b.d.1249.6		8	20.7	even	4
10000.2.a.u.1.2		4	1.1	even	1	trivial
10000.2.a.v.1.3		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46831 q^{3} +0.710571 q^{7} +3.09254 q^{9} +O(q^{10})\)	\(q-2.46831 q^{3} +0.710571 q^{7} +3.09254 q^{9} -5.37577 q^{11} -6.61184 q^{13} +7.22987 q^{17} +2.14353 q^{19} -1.75391 q^{21} -3.32241 q^{23} -0.228409 q^{27} -3.56084 q^{29} -5.67522 q^{31} +13.2690 q^{33} -3.28943 q^{37} +16.3200 q^{39} -1.46831 q^{41} -0.681421 q^{43} +5.37577 q^{47} -6.49509 q^{49} -17.8455 q^{51} +0.624230 q^{53} -5.29089 q^{57} -9.90746 q^{59} -5.84791 q^{61} +2.19747 q^{63} +8.00383 q^{67} +8.20072 q^{69} -8.07395 q^{71} -10.8771 q^{73} -3.81986 q^{77} +7.66283 q^{79} -8.71383 q^{81} -2.82495 q^{83} +8.78925 q^{87} +2.02915 q^{89} -4.69818 q^{91} +14.0082 q^{93} -1.79308 q^{97} -16.6248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 3 q^{7} + 17 q^{9}+O(q^{10}) \) 4 * q - q^3 + 3 * q^7 + 17 * q^9	\( 4 q - q^{3} + 3 q^{7} + 17 q^{9} - 8 q^{11} - 4 q^{13} + 2 q^{17} - 5 q^{19} - 7 q^{21} + 9 q^{23} - 10 q^{27} - 10 q^{29} - 18 q^{31} + 22 q^{33} - 13 q^{37} + 16 q^{39} + 3 q^{41} - 16 q^{43} + 8 q^{47} + 23 q^{49} - 13 q^{51} + 16 q^{53} + 15 q^{57} - 35 q^{59} + 8 q^{61} + 54 q^{63} + 23 q^{67} + 19 q^{69} + 17 q^{71} + q^{73} + 29 q^{77} - 10 q^{79} + 24 q^{81} - 11 q^{83} + 40 q^{87} - 5 q^{89} + 17 q^{91} - 38 q^{93} - 3 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - q^3 + 3 * q^7 + 17 * q^9 - 8 * q^11 - 4 * q^13 + 2 * q^17 - 5 * q^19 - 7 * q^21 + 9 * q^23 - 10 * q^27 - 10 * q^29 - 18 * q^31 + 22 * q^33 - 13 * q^37 + 16 * q^39 + 3 * q^41 - 16 * q^43 + 8 * q^47 + 23 * q^49 - 13 * q^51 + 16 * q^53 + 15 * q^57 - 35 * q^59 + 8 * q^61 + 54 * q^63 + 23 * q^67 + 19 * q^69 + 17 * q^71 + q^73 + 29 * q^77 - 10 * q^79 + 24 * q^81 - 11 * q^83 + 40 * q^87 - 5 * q^89 + 17 * q^91 - 38 * q^93 - 3 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more