LMFDB - Embedded newform 10000.2.a.bo.1.3

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 $ x^12 - x^11 - 20x^10 + 11x^9 + 144x^8 - 29x^7 - 440x^6 + 4x^5 + 556x^4 + 15x^3 - 285*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.571452$ 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)

$f(q)$	$=$	$q-2.94508 q^{3} +3.90152 q^{7} +5.67351 q^{9} +O(q^{10})$	$q-2.94508 q^{3} +3.90152 q^{7} +5.67351 q^{9} +2.05881 q^{11} -4.35223 q^{13} +5.68781 q^{17} -4.55839 q^{19} -11.4903 q^{21} +4.68450 q^{23} -7.87369 q^{27} +6.49095 q^{29} -7.63015 q^{31} -6.06335 q^{33} -4.18794 q^{37} +12.8177 q^{39} +10.5349 q^{41} -7.38455 q^{43} -9.29573 q^{47} +8.22187 q^{49} -16.7511 q^{51} -3.50752 q^{53} +13.4248 q^{57} +2.15720 q^{59} -0.531840 q^{61} +22.1353 q^{63} -15.2658 q^{67} -13.7962 q^{69} -10.0316 q^{71} -9.49559 q^{73} +8.03248 q^{77} +2.87144 q^{79} +6.16815 q^{81} -7.90239 q^{83} -19.1164 q^{87} +6.81774 q^{89} -16.9803 q^{91} +22.4714 q^{93} -14.1754 q^{97} +11.6806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 26 * q^9	$ 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * 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
$3$	−2.94508	−1.70034	−0.850172	−	0.526505i	$-0.823502\pi$
$3$	−2.94508	−1.70034	−0.850172	+	0.526505i	$0.823502\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.90152	1.47464	0.737318	−	0.675546i	$-0.236092\pi$
$7$	3.90152	1.47464	0.737318	+	0.675546i	$0.236092\pi$
$8$	0	0
$9$	5.67351	1.89117
$10$	0	0
$11$	2.05881	0.620753	0.310377	−	0.950614i	$-0.399545\pi$
$11$	2.05881	0.620753	0.310377	+	0.950614i	$0.399545\pi$
$12$	0	0
$13$	−4.35223	−1.20709	−0.603546	−	0.797328i	$-0.706246\pi$
$13$	−4.35223	−1.20709	−0.603546	+	0.797328i	$0.706246\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.68781	1.37950	0.689748	−	0.724049i	$-0.257721\pi$
$17$	5.68781	1.37950	0.689748	+	0.724049i	$0.257721\pi$
$18$	0	0
$19$	−4.55839	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$19$	−4.55839	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$20$	0	0
$21$	−11.4903	−2.50739
$22$	0	0
$23$	4.68450	0.976786	0.488393	−	0.872624i	$-0.337583\pi$
$23$	4.68450	0.976786	0.488393	+	0.872624i	$0.337583\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−7.87369	−1.51529
$28$	0	0
$29$	6.49095	1.20534	0.602670	−	0.797991i	$-0.294104\pi$
$29$	6.49095	1.20534	0.602670	+	0.797991i	$0.294104\pi$
$30$	0	0
$31$	−7.63015	−1.37042	−0.685208	−	0.728348i	$-0.740289\pi$
$31$	−7.63015	−1.37042	−0.685208	+	0.728348i	$0.740289\pi$
$32$	0	0
$33$	−6.06335	−1.05549
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.18794	−0.688493	−0.344246	−	0.938879i	$-0.611866\pi$
$37$	−4.18794	−0.688493	−0.344246	+	0.938879i	$0.611866\pi$
$38$	0	0
$39$	12.8177	2.05247
$40$	0	0
$41$	10.5349	1.64527	0.822636	−	0.568569i	$-0.192503\pi$
$41$	10.5349	1.64527	0.822636	+	0.568569i	$0.192503\pi$
$42$	0	0
$43$	−7.38455	−1.12613	−0.563067	−	0.826411i	$-0.690379\pi$
$43$	−7.38455	−1.12613	−0.563067	+	0.826411i	$0.690379\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.29573	−1.35592	−0.677961	−	0.735098i	$-0.737136\pi$
$47$	−9.29573	−1.35592	−0.677961	+	0.735098i	$0.737136\pi$
$48$	0	0
$49$	8.22187	1.17455
$50$	0	0
$51$	−16.7511	−2.34562
$52$	0	0
$53$	−3.50752	−0.481795	−0.240897	−	0.970551i	$-0.577442\pi$
$53$	−3.50752	−0.481795	−0.240897	+	0.970551i	$0.577442\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.4248	1.77816
$58$	0	0
$59$	2.15720	0.280844	0.140422	−	0.990092i	$-0.455154\pi$
$59$	2.15720	0.280844	0.140422	+	0.990092i	$0.455154\pi$
$60$	0	0
$61$	−0.531840	−0.0680951	−0.0340475	−	0.999420i	$-0.510840\pi$
$61$	−0.531840	−0.0680951	−0.0340475	+	0.999420i	$0.510840\pi$
$62$	0	0
$63$	22.1353	2.78879
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.2658	−1.86502	−0.932510	−	0.361144i	$-0.882386\pi$
$67$	−15.2658	−1.86502	−0.932510	+	0.361144i	$0.882386\pi$
$68$	0	0
$69$	−13.7962	−1.66087
$70$	0	0
$71$	−10.0316	−1.19053	−0.595267	−	0.803528i	$-0.702954\pi$
$71$	−10.0316	−1.19053	−0.595267	+	0.803528i	$0.702954\pi$
$72$	0	0
$73$	−9.49559	−1.11137	−0.555687	−	0.831391i	$-0.687545\pi$
$73$	−9.49559	−1.11137	−0.555687	+	0.831391i	$0.687545\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.03248	0.915386
$78$	0	0
$79$	2.87144	0.323062	0.161531	−	0.986868i	$-0.448357\pi$
$79$	2.87144	0.323062	0.161531	+	0.986868i	$0.448357\pi$
$80$	0	0
$81$	6.16815	0.685350
$82$	0	0
$83$	−7.90239	−0.867401	−0.433700	−	0.901057i	$-0.642792\pi$
$83$	−7.90239	−0.867401	−0.433700	+	0.901057i	$0.642792\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19.1164	−2.04949
$88$	0	0
$89$	6.81774	0.722679	0.361340	−	0.932434i	$-0.382319\pi$
$89$	6.81774	0.722679	0.361340	+	0.932434i	$0.382319\pi$
$90$	0	0
$91$	−16.9803	−1.78002
$92$	0	0
$93$	22.4714	2.33018
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.1754	−1.43929	−0.719647	−	0.694340i	$-0.755696\pi$
$97$	−14.1754	−1.43929	−0.719647	+	0.694340i	$0.755696\pi$
$98$	0	0
$99$	11.6806	1.17395
$100$	0	0
$101$	−6.35704	−0.632549	−0.316275	−	0.948668i	$-0.602432\pi$
$101$	−6.35704	−0.632549	−0.316275	+	0.948668i	$0.602432\pi$
$102$	0	0
$103$	−15.2415	−1.50179	−0.750893	−	0.660424i	$-0.770377\pi$
$103$	−15.2415	−1.50179	−0.750893	+	0.660424i	$0.770377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.23376	−0.795987	−0.397994	−	0.917388i	$-0.630293\pi$
$107$	−8.23376	−0.795987	−0.397994	+	0.917388i	$0.630293\pi$
$108$	0	0
$109$	6.73656	0.645245	0.322623	−	0.946528i	$-0.395436\pi$
$109$	6.73656	0.645245	0.322623	+	0.946528i	$0.395436\pi$
$110$	0	0
$111$	12.3338	1.17067
$112$	0	0
$113$	−11.6882	−1.09953	−0.549765	−	0.835319i	$-0.685283\pi$
$113$	−11.6882	−1.09953	−0.549765	+	0.835319i	$0.685283\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−24.6924	−2.28282
$118$	0	0
$119$	22.1911	2.03426
$120$	0	0
$121$	−6.76132	−0.614665
$122$	0	0
$123$	−31.0261	−2.79753
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.8996	1.05592	0.527958	−	0.849271i	$-0.322958\pi$
$127$	11.8996	1.05592	0.527958	+	0.849271i	$0.322958\pi$
$128$	0	0
$129$	21.7481	1.91481
$130$	0	0
$131$	13.1320	1.14735	0.573675	−	0.819083i	$-0.305517\pi$
$131$	13.1320	1.14735	0.573675	+	0.819083i	$0.305517\pi$
$132$	0	0
$133$	−17.7846	−1.54212
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.6530	1.33733	0.668664	−	0.743565i	$-0.266867\pi$
$137$	15.6530	1.33733	0.668664	+	0.743565i	$0.266867\pi$
$138$	0	0
$139$	2.14856	0.182239	0.0911193	−	0.995840i	$-0.470956\pi$
$139$	2.14856	0.182239	0.0911193	+	0.995840i	$0.470956\pi$
$140$	0	0
$141$	27.3767	2.30553
$142$	0	0
$143$	−8.96041	−0.749307
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−24.2141	−1.99714
$148$	0	0
$149$	−4.62324	−0.378751	−0.189375	−	0.981905i	$-0.560646\pi$
$149$	−4.62324	−0.378751	−0.189375	+	0.981905i	$0.560646\pi$
$150$	0	0
$151$	9.12433	0.742527	0.371264	−	0.928527i	$-0.378925\pi$
$151$	9.12433	0.742527	0.371264	+	0.928527i	$0.378925\pi$
$152$	0	0
$153$	32.2698	2.60886
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.62507	0.768165	0.384082	−	0.923299i	$-0.374518\pi$
$157$	9.62507	0.768165	0.384082	+	0.923299i	$0.374518\pi$
$158$	0	0
$159$	10.3299	0.819216
$160$	0	0
$161$	18.2767	1.44040
$162$	0	0
$163$	11.5015	0.900863	0.450432	−	0.892811i	$-0.351270\pi$
$163$	11.5015	0.900863	0.450432	+	0.892811i	$0.351270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.96441	−0.616305	−0.308152	−	0.951337i	$-0.599711\pi$
$167$	−7.96441	−0.616305	−0.308152	+	0.951337i	$0.599711\pi$
$168$	0	0
$169$	5.94195	0.457073
$170$	0	0
$171$	−25.8620	−1.97772
$172$	0	0
$173$	17.2428	1.31095	0.655474	−	0.755218i	$-0.272469\pi$
$173$	17.2428	1.31095	0.655474	+	0.755218i	$0.272469\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.35313	−0.477531
$178$	0	0
$179$	−7.55281	−0.564523	−0.282262	−	0.959337i	$-0.591085\pi$
$179$	−7.55281	−0.564523	−0.282262	+	0.959337i	$0.591085\pi$
$180$	0	0
$181$	2.69133	0.200045	0.100022	−	0.994985i	$-0.468109\pi$
$181$	2.69133	0.200045	0.100022	+	0.994985i	$0.468109\pi$
$182$	0	0
$183$	1.56631	0.115785
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.7101	0.856327
$188$	0	0
$189$	−30.7194	−2.23451
$190$	0	0
$191$	6.93649	0.501907	0.250954	−	0.967999i	$-0.419256\pi$
$191$	6.93649	0.501907	0.250954	+	0.967999i	$0.419256\pi$
$192$	0	0
$193$	−17.0161	−1.22484	−0.612421	−	0.790532i	$-0.709804\pi$
$193$	−17.0161	−1.22484	−0.612421	+	0.790532i	$0.709804\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.18477	0.369399	0.184700	−	0.982795i	$-0.440869\pi$
$197$	5.18477	0.369399	0.184700	+	0.982795i	$0.440869\pi$
$198$	0	0
$199$	−14.2933	−1.01322	−0.506611	−	0.862175i	$-0.669102\pi$
$199$	−14.2933	−1.01322	−0.506611	+	0.862175i	$0.669102\pi$
$200$	0	0
$201$	44.9592	3.17117
$202$	0	0
$203$	25.3246	1.77744
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.5775	1.84727
$208$	0	0
$209$	−9.38484	−0.649163
$210$	0	0
$211$	−2.64462	−0.182063	−0.0910317	−	0.995848i	$-0.529016\pi$
$211$	−2.64462	−0.182063	−0.0910317	+	0.995848i	$0.529016\pi$
$212$	0	0
$213$	29.5439	2.02432
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−29.7692	−2.02086
$218$	0	0
$219$	27.9653	1.88972
$220$	0	0
$221$	−24.7547	−1.66518
$222$	0	0
$223$	17.6067	1.17903	0.589516	−	0.807756i	$-0.299318\pi$
$223$	17.6067	1.17903	0.589516	+	0.807756i	$0.299318\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.7437	0.713083	0.356541	−	0.934280i	$-0.383956\pi$
$227$	10.7437	0.713083	0.356541	+	0.934280i	$0.383956\pi$
$228$	0	0
$229$	21.6719	1.43212	0.716058	−	0.698041i	$-0.245944\pi$
$229$	21.6719	1.43212	0.716058	+	0.698041i	$0.245944\pi$
$230$	0	0
$231$	−23.6563	−1.55647
$232$	0	0
$233$	−5.77486	−0.378324	−0.189162	−	0.981946i	$-0.560577\pi$
$233$	−5.77486	−0.378324	−0.189162	+	0.981946i	$0.560577\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.45663	−0.549317
$238$	0	0
$239$	−2.97793	−0.192626	−0.0963132	−	0.995351i	$-0.530705\pi$
$239$	−2.97793	−0.192626	−0.0963132	+	0.995351i	$0.530705\pi$
$240$	0	0
$241$	−13.0194	−0.838655	−0.419328	−	0.907835i	$-0.637734\pi$
$241$	−13.0194	−0.838655	−0.419328	+	0.907835i	$0.637734\pi$
$242$	0	0
$243$	5.45537	0.349962
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.8392	1.26234
$248$	0	0
$249$	23.2732	1.47488
$250$	0	0
$251$	−8.23470	−0.519770	−0.259885	−	0.965640i	$-0.583685\pi$
$251$	−8.23470	−0.519770	−0.259885	+	0.965640i	$0.583685\pi$
$252$	0	0
$253$	9.64448	0.606343
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.14453	−0.508042	−0.254021	−	0.967199i	$-0.581753\pi$
$257$	−8.14453	−0.508042	−0.254021	+	0.967199i	$0.581753\pi$
$258$	0	0
$259$	−16.3393	−1.01528
$260$	0	0
$261$	36.8264	2.27950
$262$	0	0
$263$	8.64826	0.533275	0.266637	−	0.963797i	$-0.414087\pi$
$263$	8.64826	0.533275	0.266637	+	0.963797i	$0.414087\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−20.0788	−1.22880
$268$	0	0
$269$	−6.38246	−0.389146	−0.194573	−	0.980888i	$-0.562332\pi$
$269$	−6.38246	−0.389146	−0.194573	+	0.980888i	$0.562332\pi$
$270$	0	0
$271$	−29.6292	−1.79985	−0.899924	−	0.436048i	$-0.856378\pi$
$271$	−29.6292	−1.79985	−0.899924	+	0.436048i	$0.856378\pi$
$272$	0	0
$273$	50.0085	3.02665
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.58529	0.455756	0.227878	−	0.973690i	$-0.426821\pi$
$277$	7.58529	0.455756	0.227878	+	0.973690i	$0.426821\pi$
$278$	0	0
$279$	−43.2897	−2.59169
$280$	0	0
$281$	18.9216	1.12877	0.564385	−	0.825512i	$-0.309113\pi$
$281$	18.9216	1.12877	0.564385	+	0.825512i	$0.309113\pi$
$282$	0	0
$283$	15.0440	0.894272	0.447136	−	0.894466i	$-0.352444\pi$
$283$	15.0440	0.894272	0.447136	+	0.894466i	$0.352444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.1020	2.42618
$288$	0	0
$289$	15.3512	0.903011
$290$	0	0
$291$	41.7477	2.44729
$292$	0	0
$293$	−30.7869	−1.79859	−0.899294	−	0.437345i	$-0.855919\pi$
$293$	−30.7869	−1.79859	−0.899294	+	0.437345i	$0.855919\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.2104	−0.940623
$298$	0	0
$299$	−20.3880	−1.17907
$300$	0	0
$301$	−28.8110	−1.66064
$302$	0	0
$303$	18.7220	1.07555
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.9552	−0.796465	−0.398233	−	0.917284i	$-0.630376\pi$
$307$	−13.9552	−0.796465	−0.398233	+	0.917284i	$0.630376\pi$
$308$	0	0
$309$	44.8874	2.55355
$310$	0	0
$311$	−1.18583	−0.0672421	−0.0336211	−	0.999435i	$-0.510704\pi$
$311$	−1.18583	−0.0672421	−0.0336211	+	0.999435i	$0.510704\pi$
$312$	0	0
$313$	−22.6459	−1.28002	−0.640012	−	0.768365i	$-0.721071\pi$
$313$	−22.6459	−1.28002	−0.640012	+	0.768365i	$0.721071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.9997	1.12329	0.561647	−	0.827377i	$-0.310168\pi$
$317$	19.9997	1.12329	0.561647	+	0.827377i	$0.310168\pi$
$318$	0	0
$319$	13.3636	0.748218
$320$	0	0
$321$	24.2491	1.35345
$322$	0	0
$323$	−25.9272	−1.44263
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19.8397	−1.09714
$328$	0	0
$329$	−36.2675	−1.99949
$330$	0	0
$331$	−24.6827	−1.35668	−0.678342	−	0.734746i	$-0.737301\pi$
$331$	−24.6827	−1.35668	−0.678342	+	0.734746i	$0.737301\pi$
$332$	0	0
$333$	−23.7603	−1.30206
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.4739	−1.49660	−0.748298	−	0.663363i	$-0.769129\pi$
$337$	−27.4739	−1.49660	−0.748298	+	0.663363i	$0.769129\pi$
$338$	0	0
$339$	34.4226	1.86958
$340$	0	0
$341$	−15.7090	−0.850690
$342$	0	0
$343$	4.76714	0.257402
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.5579	−0.942557	−0.471278	−	0.881985i	$-0.656207\pi$
$347$	−17.5579	−0.942557	−0.471278	+	0.881985i	$0.656207\pi$
$348$	0	0
$349$	−19.9386	−1.06729	−0.533645	−	0.845709i	$-0.679178\pi$
$349$	−19.9386	−1.06729	−0.533645	+	0.845709i	$0.679178\pi$
$350$	0	0
$351$	34.2682	1.82910
$352$	0	0
$353$	26.3391	1.40189	0.700944	−	0.713216i	$-0.252762\pi$
$353$	26.3391	1.40189	0.700944	+	0.713216i	$0.252762\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−65.3546	−3.45893
$358$	0	0
$359$	−0.0441951	−0.00233253	−0.00116626	−	0.999999i	$-0.500371\pi$
$359$	−0.0441951	−0.00233253	−0.00116626	+	0.999999i	$0.500371\pi$
$360$	0	0
$361$	1.77890	0.0936264
$362$	0	0
$363$	19.9126	1.04514
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.427626	−0.0223219	−0.0111610	−	0.999938i	$-0.503553\pi$
$367$	−0.427626	−0.0223219	−0.0111610	+	0.999938i	$0.503553\pi$
$368$	0	0
$369$	59.7697	3.11149
$370$	0	0
$371$	−13.6847	−0.710472
$372$	0	0
$373$	−8.82769	−0.457080	−0.228540	−	0.973534i	$-0.573395\pi$
$373$	−8.82769	−0.457080	−0.228540	+	0.973534i	$0.573395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−28.2501	−1.45496
$378$	0	0
$379$	18.2014	0.934941	0.467470	−	0.884009i	$-0.345165\pi$
$379$	18.2014	0.934941	0.467470	+	0.884009i	$0.345165\pi$
$380$	0	0
$381$	−35.0452	−1.79542
$382$	0	0
$383$	13.3746	0.683411	0.341706	−	0.939807i	$-0.388995\pi$
$383$	13.3746	0.683411	0.341706	+	0.939807i	$0.388995\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−41.8963	−2.12971
$388$	0	0
$389$	−5.31345	−0.269403	−0.134701	−	0.990886i	$-0.543008\pi$
$389$	−5.31345	−0.269403	−0.134701	+	0.990886i	$0.543008\pi$
$390$	0	0
$391$	26.6445	1.34747
$392$	0	0
$393$	−38.6749	−1.95089
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.1142	−1.05969	−0.529844	−	0.848095i	$-0.677750\pi$
$397$	−21.1142	−1.05969	−0.529844	+	0.848095i	$0.677750\pi$
$398$	0	0
$399$	52.3772	2.62214
$400$	0	0
$401$	17.8040	0.889088	0.444544	−	0.895757i	$-0.353366\pi$
$401$	17.8040	0.889088	0.444544	+	0.895757i	$0.353366\pi$
$402$	0	0
$403$	33.2082	1.65422
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.62215	−0.427384
$408$	0	0
$409$	−13.2234	−0.653854	−0.326927	−	0.945050i	$-0.606013\pi$
$409$	−13.2234	−0.653854	−0.326927	+	0.945050i	$0.606013\pi$
$410$	0	0
$411$	−46.0994	−2.27392
$412$	0	0
$413$	8.41636	0.414142
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.32769	−0.309868
$418$	0	0
$419$	2.18602	0.106794	0.0533970	−	0.998573i	$-0.482995\pi$
$419$	2.18602	0.106794	0.0533970	+	0.998573i	$0.482995\pi$
$420$	0	0
$421$	19.1418	0.932917	0.466458	−	0.884543i	$-0.345530\pi$
$421$	19.1418	0.932917	0.466458	+	0.884543i	$0.345530\pi$
$422$	0	0
$423$	−52.7394	−2.56428
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.07498	−0.100415
$428$	0	0
$429$	26.3891	1.27408
$430$	0	0
$431$	20.8065	1.00221	0.501106	−	0.865386i	$-0.332927\pi$
$431$	20.8065	1.00221	0.501106	+	0.865386i	$0.332927\pi$
$432$	0	0
$433$	−10.3653	−0.498122	−0.249061	−	0.968488i	$-0.580122\pi$
$433$	−10.3653	−0.498122	−0.249061	+	0.968488i	$0.580122\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.3538	−1.02149
$438$	0	0
$439$	−1.50131	−0.0716535	−0.0358267	−	0.999358i	$-0.511406\pi$
$439$	−1.50131	−0.0716535	−0.0358267	+	0.999358i	$0.511406\pi$
$440$	0	0
$441$	46.6468	2.22128
$442$	0	0
$443$	7.15150	0.339778	0.169889	−	0.985463i	$-0.445659\pi$
$443$	7.15150	0.339778	0.169889	+	0.985463i	$0.445659\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.6158	0.644006
$448$	0	0
$449$	10.0455	0.474075	0.237038	−	0.971500i	$-0.423824\pi$
$449$	10.0455	0.474075	0.237038	+	0.971500i	$0.423824\pi$
$450$	0	0
$451$	21.6893	1.02131
$452$	0	0
$453$	−26.8719	−1.26255
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.4504	0.535625	0.267813	−	0.963471i	$-0.413699\pi$
$457$	11.4504	0.535625	0.267813	+	0.963471i	$0.413699\pi$
$458$	0	0
$459$	−44.7841	−2.09034
$460$	0	0
$461$	−18.2719	−0.851006	−0.425503	−	0.904957i	$-0.639903\pi$
$461$	−18.2719	−0.851006	−0.425503	+	0.904957i	$0.639903\pi$
$462$	0	0
$463$	−19.5202	−0.907178	−0.453589	−	0.891211i	$-0.649857\pi$
$463$	−19.5202	−0.907178	−0.453589	+	0.891211i	$0.649857\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.2567	1.16874	0.584370	−	0.811487i	$-0.301342\pi$
$467$	25.2567	1.16874	0.584370	+	0.811487i	$0.301342\pi$
$468$	0	0
$469$	−59.5600	−2.75023
$470$	0	0
$471$	−28.3466	−1.30614
$472$	0	0
$473$	−15.2034	−0.699051
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−19.8999	−0.911155
$478$	0	0
$479$	−6.18475	−0.282588	−0.141294	−	0.989968i	$-0.545126\pi$
$479$	−6.18475	−0.282588	−0.141294	+	0.989968i	$0.545126\pi$
$480$	0	0
$481$	18.2269	0.831075
$482$	0	0
$483$	−53.8263	−2.44918
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.3047	−0.965408	−0.482704	−	0.875784i	$-0.660345\pi$
$487$	−21.3047	−0.965408	−0.482704	+	0.875784i	$0.660345\pi$
$488$	0	0
$489$	−33.8727	−1.53178
$490$	0	0
$491$	−0.623077	−0.0281191	−0.0140595	−	0.999901i	$-0.504475\pi$
$491$	−0.623077	−0.0281191	−0.0140595	+	0.999901i	$0.504475\pi$
$492$	0	0
$493$	36.9193	1.66276
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.1386	−1.75561
$498$	0	0
$499$	−30.9543	−1.38570	−0.692852	−	0.721080i	$-0.743646\pi$
$499$	−30.9543	−1.38570	−0.692852	+	0.721080i	$0.743646\pi$
$500$	0	0
$501$	23.4558	1.04793
$502$	0	0
$503$	11.1646	0.497803	0.248902	−	0.968529i	$-0.419930\pi$
$503$	11.1646	0.497803	0.248902	+	0.968529i	$0.419930\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.4995	−0.777181
$508$	0	0
$509$	−26.5186	−1.17542	−0.587708	−	0.809073i	$-0.699970\pi$
$509$	−26.5186	−1.17542	−0.587708	+	0.809073i	$0.699970\pi$
$510$	0	0
$511$	−37.0473	−1.63887
$512$	0	0
$513$	35.8914	1.58464
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.1381	−0.841693
$518$	0	0
$519$	−50.7815	−2.22906
$520$	0	0
$521$	24.2969	1.06447	0.532233	−	0.846598i	$-0.321353\pi$
$521$	24.2969	1.06447	0.532233	+	0.846598i	$0.321353\pi$
$522$	0	0
$523$	5.54700	0.242554	0.121277	−	0.992619i	$-0.461301\pi$
$523$	5.54700	0.242554	0.121277	+	0.992619i	$0.461301\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.3989	−1.89048
$528$	0	0
$529$	−1.05547	−0.0458899
$530$	0	0
$531$	12.2389	0.531122
$532$	0	0
$533$	−45.8503	−1.98600
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.2436	0.959883
$538$	0	0
$539$	16.9272	0.729107
$540$	0	0
$541$	6.43545	0.276682	0.138341	−	0.990385i	$-0.455823\pi$
$541$	6.43545	0.276682	0.138341	+	0.990385i	$0.455823\pi$
$542$	0	0
$543$	−7.92619	−0.340145
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.5605	1.64873	0.824364	−	0.566060i	$-0.191533\pi$
$547$	38.5605	1.64873	0.824364	+	0.566060i	$0.191533\pi$
$548$	0	0
$549$	−3.01740	−0.128779
$550$	0	0
$551$	−29.5883	−1.26050
$552$	0	0
$553$	11.2030	0.476400
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.4414	1.12036	0.560179	−	0.828372i	$-0.310732\pi$
$557$	26.4414	1.12036	0.560179	+	0.828372i	$0.310732\pi$
$558$	0	0
$559$	32.1393	1.35935
$560$	0	0
$561$	−34.4872	−1.45605
$562$	0	0
$563$	−34.4034	−1.44993	−0.724964	−	0.688786i	$-0.758144\pi$
$563$	−34.4034	−1.44993	−0.724964	+	0.688786i	$0.758144\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	24.0652	1.01064
$568$	0	0
$569$	−38.8282	−1.62776	−0.813881	−	0.581031i	$-0.802649\pi$
$569$	−38.8282	−1.62776	−0.813881	+	0.581031i	$0.802649\pi$
$570$	0	0
$571$	37.4325	1.56650	0.783250	−	0.621707i	$-0.213561\pi$
$571$	37.4325	1.56650	0.783250	+	0.621707i	$0.213561\pi$
$572$	0	0
$573$	−20.4285	−0.853415
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21.6087	−0.899583	−0.449792	−	0.893134i	$-0.648502\pi$
$577$	−21.6087	−0.899583	−0.449792	+	0.893134i	$0.648502\pi$
$578$	0	0
$579$	50.1137	2.08265
$580$	0	0
$581$	−30.8314	−1.27910
$582$	0	0
$583$	−7.22130	−0.299076
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.2180	0.463018	0.231509	−	0.972833i	$-0.425634\pi$
$587$	11.2180	0.463018	0.231509	+	0.972833i	$0.425634\pi$
$588$	0	0
$589$	34.7812	1.43313
$590$	0	0
$591$	−15.2696	−0.628106
$592$	0	0
$593$	−37.5291	−1.54114	−0.770568	−	0.637357i	$-0.780028\pi$
$593$	−37.5291	−1.54114	−0.770568	+	0.637357i	$0.780028\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	42.0948	1.72283
$598$	0	0
$599$	0.0240427	0.000982357 0	0.000491179	−	1.00000i	$-0.499844\pi$
$599$	0.0240427	0.000982357 0	0.000491179		1.00000i	$0.499844\pi$
$600$	0	0
$601$	12.9683	0.528987	0.264493	−	0.964388i	$-0.414795\pi$
$601$	12.9683	0.528987	0.264493	+	0.964388i	$0.414795\pi$
$602$	0	0
$603$	−86.6109	−3.52707
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.9496	−1.05326	−0.526630	−	0.850094i	$-0.676545\pi$
$607$	−25.9496	−1.05326	−0.526630	+	0.850094i	$0.676545\pi$
$608$	0	0
$609$	−74.5830	−3.02225
$610$	0	0
$611$	40.4572	1.63672
$612$	0	0
$613$	19.5369	0.789089	0.394545	−	0.918877i	$-0.370902\pi$
$613$	19.5369	0.789089	0.394545	+	0.918877i	$0.370902\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.3862	1.58563	0.792815	−	0.609462i	$-0.208615\pi$
$617$	39.3862	1.58563	0.792815	+	0.609462i	$0.208615\pi$
$618$	0	0
$619$	−0.568306	−0.0228422	−0.0114211	−	0.999935i	$-0.503636\pi$
$619$	−0.568306	−0.0228422	−0.0114211	+	0.999935i	$0.503636\pi$
$620$	0	0
$621$	−36.8843	−1.48012
$622$	0	0
$623$	26.5996	1.06569
$624$	0	0
$625$	0	0
$626$	0	0
$627$	27.6391	1.10380
$628$	0	0
$629$	−23.8202	−0.949773
$630$	0	0
$631$	−12.0545	−0.479880	−0.239940	−	0.970788i	$-0.577128\pi$
$631$	−12.0545	−0.479880	−0.239940	+	0.970788i	$0.577128\pi$
$632$	0	0
$633$	7.78863	0.309570
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−35.7835	−1.41779
$638$	0	0
$639$	−56.9145	−2.25150
$640$	0	0
$641$	2.86993	0.113355	0.0566777	−	0.998393i	$-0.481949\pi$
$641$	2.86993	0.113355	0.0566777	+	0.998393i	$0.481949\pi$
$642$	0	0
$643$	−14.3101	−0.564335	−0.282168	−	0.959365i	$-0.591053\pi$
$643$	−14.3101	−0.564335	−0.282168	+	0.959365i	$0.591053\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.4459	−1.00038	−0.500190	−	0.865916i	$-0.666736\pi$
$647$	−25.4459	−1.00038	−0.500190	+	0.865916i	$0.666736\pi$
$648$	0	0
$649$	4.44126	0.174335
$650$	0	0
$651$	87.6727	3.43616
$652$	0	0
$653$	−26.6527	−1.04300	−0.521501	−	0.853251i	$-0.674628\pi$
$653$	−26.6527	−1.04300	−0.521501	+	0.853251i	$0.674628\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−53.8733	−2.10180
$658$	0	0
$659$	13.2299	0.515362	0.257681	−	0.966230i	$-0.417042\pi$
$659$	13.2299	0.515362	0.257681	+	0.966230i	$0.417042\pi$
$660$	0	0
$661$	−30.4140	−1.18297	−0.591483	−	0.806317i	$-0.701457\pi$
$661$	−30.4140	−1.18297	−0.591483	+	0.806317i	$0.701457\pi$
$662$	0	0
$663$	72.9046	2.83138
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.4069	1.17736
$668$	0	0
$669$	−51.8532	−2.00476
$670$	0	0
$671$	−1.09495	−0.0422703
$672$	0	0
$673$	5.71047	0.220122	0.110061	−	0.993925i	$-0.464895\pi$
$673$	5.71047	0.220122	0.110061	+	0.993925i	$0.464895\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.1510	−1.19723	−0.598616	−	0.801036i	$-0.704282\pi$
$677$	−31.1510	−1.19723	−0.598616	+	0.801036i	$0.704282\pi$
$678$	0	0
$679$	−55.3056	−2.12243
$680$	0	0
$681$	−31.6410	−1.21249
$682$	0	0
$683$	−18.5614	−0.710234	−0.355117	−	0.934822i	$-0.615559\pi$
$683$	−18.5614	−0.710234	−0.355117	+	0.934822i	$0.615559\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−63.8254	−2.43509
$688$	0	0
$689$	15.2655	0.581571
$690$	0	0
$691$	−10.6396	−0.404751	−0.202376	−	0.979308i	$-0.564866\pi$
$691$	−10.6396	−0.404751	−0.202376	+	0.979308i	$0.564866\pi$
$692$	0	0
$693$	45.5723	1.73115
$694$	0	0
$695$	0	0
$696$	0	0
$697$	59.9204	2.26965
$698$	0	0
$699$	17.0074	0.643280
$700$	0	0
$701$	33.8576	1.27878	0.639392	−	0.768881i	$-0.279186\pi$
$701$	33.8576	1.27878	0.639392	+	0.768881i	$0.279186\pi$
$702$	0	0
$703$	19.0902	0.720002
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.8021	−0.932780
$708$	0	0
$709$	34.4933	1.29542	0.647711	−	0.761886i	$-0.275727\pi$
$709$	34.4933	1.29542	0.647711	+	0.761886i	$0.275727\pi$
$710$	0	0
$711$	16.2911	0.610966
$712$	0	0
$713$	−35.7434	−1.33860
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.77025	0.327531
$718$	0	0
$719$	−18.9423	−0.706430	−0.353215	−	0.935542i	$-0.614912\pi$
$719$	−18.9423	−0.706430	−0.353215	+	0.935542i	$0.614912\pi$
$720$	0	0
$721$	−59.4649	−2.21459
$722$	0	0
$723$	38.3433	1.42600
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21.7904	−0.808160	−0.404080	−	0.914724i	$-0.632408\pi$
$727$	−21.7904	−0.808160	−0.404080	+	0.914724i	$0.632408\pi$
$728$	0	0
$729$	−34.5710	−1.28041
$730$	0	0
$731$	−42.0019	−1.55350
$732$	0	0
$733$	20.7988	0.768222	0.384111	−	0.923287i	$-0.374508\pi$
$733$	20.7988	0.768222	0.384111	+	0.923287i	$0.374508\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.4294	−1.15772
$738$	0	0
$739$	−39.5326	−1.45423	−0.727115	−	0.686515i	$-0.759140\pi$
$739$	−39.5326	−1.45423	−0.727115	+	0.686515i	$0.759140\pi$
$740$	0	0
$741$	−58.4280	−2.14641
$742$	0	0
$743$	−9.86832	−0.362033	−0.181017	−	0.983480i	$-0.557939\pi$
$743$	−9.86832	−0.362033	−0.181017	+	0.983480i	$0.557939\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−44.8343	−1.64040
$748$	0	0
$749$	−32.1242	−1.17379
$750$	0	0
$751$	29.0218	1.05902	0.529510	−	0.848304i	$-0.322376\pi$
$751$	29.0218	1.05902	0.529510	+	0.848304i	$0.322376\pi$
$752$	0	0
$753$	24.2519	0.883787
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.3704	1.75805	0.879027	−	0.476772i	$-0.158193\pi$
$757$	48.3704	1.75805	0.879027	+	0.476772i	$0.158193\pi$
$758$	0	0
$759$	−28.4038	−1.03099
$760$	0	0
$761$	0.0910199	0.00329947	0.00164974	−	0.999999i	$-0.499475\pi$
$761$	0.0910199	0.00329947	0.00164974	+	0.999999i	$0.499475\pi$
$762$	0	0
$763$	26.2828	0.951502
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.38864	−0.339004
$768$	0	0
$769$	40.9229	1.47572	0.737859	−	0.674955i	$-0.235837\pi$
$769$	40.9229	1.47572	0.737859	+	0.674955i	$0.235837\pi$
$770$	0	0
$771$	23.9863	0.863846
$772$	0	0
$773$	−25.8947	−0.931369	−0.465685	−	0.884951i	$-0.654192\pi$
$773$	−25.8947	−0.931369	−0.465685	+	0.884951i	$0.654192\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	48.1207	1.72632
$778$	0	0
$779$	−48.0221	−1.72057
$780$	0	0
$781$	−20.6532	−0.739028
$782$	0	0
$783$	−51.1078	−1.82644
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.0083	0.784512	0.392256	−	0.919856i	$-0.371695\pi$
$787$	22.0083	0.784512	0.392256	+	0.919856i	$0.371695\pi$
$788$	0	0
$789$	−25.4698	−0.906750
$790$	0	0
$791$	−45.6016	−1.62141
$792$	0	0
$793$	2.31469	0.0821971
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.1943	−0.361101	−0.180551	−	0.983566i	$-0.557788\pi$
$797$	−10.1943	−0.361101	−0.180551	+	0.983566i	$0.557788\pi$
$798$	0	0
$799$	−52.8723	−1.87049
$800$	0	0
$801$	38.6805	1.36671
$802$	0	0
$803$	−19.5496	−0.689890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.7969	0.661681
$808$	0	0
$809$	52.4286	1.84329	0.921645	−	0.388034i	$-0.126846\pi$
$809$	52.4286	1.84329	0.921645	+	0.388034i	$0.126846\pi$
$810$	0	0
$811$	−44.9021	−1.57672	−0.788362	−	0.615211i	$-0.789071\pi$
$811$	−44.9021	−1.57672	−0.788362	+	0.615211i	$0.789071\pi$
$812$	0	0
$813$	87.2605	3.06036
$814$	0	0
$815$	0	0
$816$	0	0
$817$	33.6617	1.17767
$818$	0	0
$819$	−96.3380	−3.36632
$820$	0	0
$821$	−43.2651	−1.50996	−0.754981	−	0.655747i	$-0.772354\pi$
$821$	−43.2651	−1.50996	−0.754981	+	0.655747i	$0.772354\pi$
$822$	0	0
$823$	5.85120	0.203960	0.101980	−	0.994786i	$-0.467482\pi$
$823$	5.85120	0.203960	0.101980	+	0.994786i	$0.467482\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.9128	0.727210	0.363605	−	0.931553i	$-0.381546\pi$
$827$	20.9128	0.727210	0.363605	+	0.931553i	$0.381546\pi$
$828$	0	0
$829$	17.4679	0.606685	0.303342	−	0.952882i	$-0.401897\pi$
$829$	17.4679	0.606685	0.303342	+	0.952882i	$0.401897\pi$
$830$	0	0
$831$	−22.3393	−0.774942
$832$	0	0
$833$	46.7644	1.62029
$834$	0	0
$835$	0	0
$836$	0	0
$837$	60.0775	2.07658
$838$	0	0
$839$	−56.4386	−1.94848	−0.974238	−	0.225523i	$-0.927591\pi$
$839$	−56.4386	−1.94848	−0.974238	+	0.225523i	$0.927591\pi$
$840$	0	0
$841$	13.1324	0.452843
$842$	0	0
$843$	−55.7258	−1.91930
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.3794	−0.906408
$848$	0	0
$849$	−44.3058	−1.52057
$850$	0	0
$851$	−19.6184	−0.672510
$852$	0	0
$853$	−9.85998	−0.337599	−0.168800	−	0.985650i	$-0.553989\pi$
$853$	−9.85998	−0.337599	−0.168800	+	0.985650i	$0.553989\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.4622	−1.62128	−0.810639	−	0.585546i	$-0.800880\pi$
$857$	−47.4622	−1.62128	−0.810639	+	0.585546i	$0.800880\pi$
$858$	0	0
$859$	28.0225	0.956114	0.478057	−	0.878329i	$-0.341341\pi$
$859$	28.0225	0.956114	0.478057	+	0.878329i	$0.341341\pi$
$860$	0	0
$861$	−121.049	−4.12534
$862$	0	0
$863$	−4.32252	−0.147140	−0.0735701	−	0.997290i	$-0.523439\pi$
$863$	−4.32252	−0.147140	−0.0735701	+	0.997290i	$0.523439\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−45.2105	−1.53543
$868$	0	0
$869$	5.91174	0.200542
$870$	0	0
$871$	66.4405	2.25125
$872$	0	0
$873$	−80.4242	−2.72195
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.5746	0.627221	0.313611	−	0.949552i	$-0.398461\pi$
$877$	18.5746	0.627221	0.313611	+	0.949552i	$0.398461\pi$
$878$	0	0
$879$	90.6698	3.05822
$880$	0	0
$881$	2.95222	0.0994629	0.0497315	−	0.998763i	$-0.484163\pi$
$881$	2.95222	0.0994629	0.0497315	+	0.998763i	$0.484163\pi$
$882$	0	0
$883$	−28.9252	−0.973412	−0.486706	−	0.873566i	$-0.661802\pi$
$883$	−28.9252	−0.973412	−0.486706	+	0.873566i	$0.661802\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−57.7976	−1.94065	−0.970326	−	0.241799i	$-0.922262\pi$
$887$	−57.7976	−1.94065	−0.970326	+	0.241799i	$0.922262\pi$
$888$	0	0
$889$	46.4264	1.55709
$890$	0	0
$891$	12.6990	0.425434
$892$	0	0
$893$	42.3735	1.41798
$894$	0	0
$895$	0	0
$896$	0	0
$897$	60.0444	2.00483
$898$	0	0
$899$	−49.5269	−1.65182
$900$	0	0
$901$	−19.9501	−0.664634
$902$	0	0
$903$	84.8507	2.82365
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−38.8757	−1.29085	−0.645423	−	0.763825i	$-0.723319\pi$
$907$	−38.8757	−1.29085	−0.645423	+	0.763825i	$0.723319\pi$
$908$	0	0
$909$	−36.0667	−1.19626
$910$	0	0
$911$	−35.8296	−1.18709	−0.593543	−	0.804802i	$-0.702271\pi$
$911$	−35.8296	−1.18709	−0.593543	+	0.804802i	$0.702271\pi$
$912$	0	0
$913$	−16.2695	−0.538442
$914$	0	0
$915$	0	0
$916$	0	0
$917$	51.2348	1.69192
$918$	0	0
$919$	14.2496	0.470051	0.235025	−	0.971989i	$-0.424483\pi$
$919$	14.2496	0.470051	0.235025	+	0.971989i	$0.424483\pi$
$920$	0	0
$921$	41.0992	1.35426
$922$	0	0
$923$	43.6600	1.43709
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−86.4726	−2.84013
$928$	0	0
$929$	−38.8326	−1.27406	−0.637028	−	0.770841i	$-0.719836\pi$
$929$	−38.8326	−1.27406	−0.637028	+	0.770841i	$0.719836\pi$
$930$	0	0
$931$	−37.4785	−1.22831
$932$	0	0
$933$	3.49236	0.114335
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.4285	1.22273	0.611367	−	0.791347i	$-0.290620\pi$
$937$	37.4285	1.22273	0.611367	+	0.791347i	$0.290620\pi$
$938$	0	0
$939$	66.6941	2.17648
$940$	0	0
$941$	−15.2256	−0.496339	−0.248170	−	0.968717i	$-0.579829\pi$
$941$	−15.2256	−0.496339	−0.248170	+	0.968717i	$0.579829\pi$
$942$	0	0
$943$	49.3506	1.60708
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.7983	1.39076	0.695380	−	0.718643i	$-0.255236\pi$
$947$	42.7983	1.39076	0.695380	+	0.718643i	$0.255236\pi$
$948$	0	0
$949$	41.3270	1.34153
$950$	0	0
$951$	−58.9007	−1.90998
$952$	0	0
$953$	26.2492	0.850296	0.425148	−	0.905124i	$-0.360222\pi$
$953$	26.2492	0.850296	0.425148	+	0.905124i	$0.360222\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−39.3569	−1.27223
$958$	0	0
$959$	61.0706	1.97207
$960$	0	0
$961$	27.2192	0.878039
$962$	0	0
$963$	−46.7143	−1.50535
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−7.32965	−0.235706	−0.117853	−	0.993031i	$-0.537601\pi$
$967$	−7.32965	−0.235706	−0.117853	+	0.993031i	$0.537601\pi$
$968$	0	0
$969$	76.3579	2.45297
$970$	0	0
$971$	−28.0952	−0.901618	−0.450809	−	0.892620i	$-0.648864\pi$
$971$	−28.0952	−0.901618	−0.450809	+	0.892620i	$0.648864\pi$
$972$	0	0
$973$	8.38266	0.268736
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.28601	0.297086	0.148543	−	0.988906i	$-0.452542\pi$
$977$	9.28601	0.297086	0.148543	+	0.988906i	$0.452542\pi$
$978$	0	0
$979$	14.0364	0.448606
$980$	0	0
$981$	38.2199	1.22027
$982$	0	0
$983$	29.9990	0.956819	0.478410	−	0.878137i	$-0.341213\pi$
$983$	29.9990	0.956819	0.478410	+	0.878137i	$0.341213\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	106.811	3.39982
$988$	0	0
$989$	−34.5929	−1.09999
$990$	0	0
$991$	−0.819483	−0.0260317	−0.0130159	−	0.999915i	$-0.504143\pi$
$991$	−0.819483	−0.0260317	−0.0130159	+	0.999915i	$0.504143\pi$
$992$	0	0
$993$	72.6926	2.30683
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.52531	0.0799773	0.0399886	−	0.999200i	$-0.487268\pi$
$997$	2.52531	0.0799773	0.0399886	+	0.999200i	$0.487268\pi$
$998$	0	0
$999$	32.9745	1.04327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bo.1.3		12
4.3	odd	2		5000.2.a.p.1.10	yes	12
5.4	even	2		10000.2.a.bp.1.10		12
20.19	odd	2		5000.2.a.o.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.o.1.3	✓	12	20.19	odd	2
5000.2.a.p.1.10	yes	12	4.3	odd	2
10000.2.a.bo.1.3		12	1.1	even	1	trivial
10000.2.a.bp.1.10		12	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)

\(f(q)\)	\(=\)	\(q-2.94508 q^{3} +3.90152 q^{7} +5.67351 q^{9} +O(q^{10})\)	\(q-2.94508 q^{3} +3.90152 q^{7} +5.67351 q^{9} +2.05881 q^{11} -4.35223 q^{13} +5.68781 q^{17} -4.55839 q^{19} -11.4903 q^{21} +4.68450 q^{23} -7.87369 q^{27} +6.49095 q^{29} -7.63015 q^{31} -6.06335 q^{33} -4.18794 q^{37} +12.8177 q^{39} +10.5349 q^{41} -7.38455 q^{43} -9.29573 q^{47} +8.22187 q^{49} -16.7511 q^{51} -3.50752 q^{53} +13.4248 q^{57} +2.15720 q^{59} -0.531840 q^{61} +22.1353 q^{63} -15.2658 q^{67} -13.7962 q^{69} -10.0316 q^{71} -9.49559 q^{73} +8.03248 q^{77} +2.87144 q^{79} +6.16815 q^{81} -7.90239 q^{83} -19.1164 q^{87} +6.81774 q^{89} -16.9803 q^{91} +22.4714 q^{93} -14.1754 q^{97} +11.6806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 26 * q^9	\( 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

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Database

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