LMFDB - Embedded newform 10000.2.a.bl.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:	$0$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.67989$ 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)

$f(q)$	$=$	$q-0.470831 q^{3} -1.95931 q^{7} -2.77832 q^{9} +O(q^{10})$	$q-0.470831 q^{3} -1.95931 q^{7} -2.77832 q^{9} -5.25723 q^{11} -2.75128 q^{13} -2.64004 q^{17} +4.38492 q^{19} +0.922505 q^{21} -3.26121 q^{23} +2.72061 q^{27} -8.52457 q^{29} +0.781931 q^{31} +2.47527 q^{33} -3.45889 q^{37} +1.29539 q^{39} -9.03471 q^{41} +0.494035 q^{43} +8.14706 q^{47} -3.16110 q^{49} +1.24301 q^{51} -0.131204 q^{53} -2.06456 q^{57} -5.18375 q^{59} +1.39727 q^{61} +5.44359 q^{63} -15.2408 q^{67} +1.53548 q^{69} -5.90377 q^{71} -8.40117 q^{73} +10.3006 q^{77} -7.03607 q^{79} +7.05401 q^{81} +5.19942 q^{83} +4.01363 q^{87} -1.88392 q^{89} +5.39061 q^{91} -0.368157 q^{93} -18.9515 q^{97} +14.6063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * 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
$3$	−0.470831	−0.271834	−0.135917	−	0.990720i	$-0.543398\pi$
$3$	−0.470831	−0.271834	−0.135917	+	0.990720i	$0.543398\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.95931	−0.740550	−0.370275	−	0.928922i	$-0.620737\pi$
$7$	−1.95931	−0.740550	−0.370275	+	0.928922i	$0.620737\pi$
$8$	0	0
$9$	−2.77832	−0.926106
$10$	0	0
$11$	−5.25723	−1.58512	−0.792558	−	0.609797i	$-0.791251\pi$
$11$	−5.25723	−1.58512	−0.792558	+	0.609797i	$0.791251\pi$
$12$	0	0
$13$	−2.75128	−0.763067	−0.381533	−	0.924355i	$-0.624604\pi$
$13$	−2.75128	−0.763067	−0.381533	+	0.924355i	$0.624604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.64004	−0.640305	−0.320152	−	0.947366i	$-0.603734\pi$
$17$	−2.64004	−0.640305	−0.320152	+	0.947366i	$0.603734\pi$
$18$	0	0
$19$	4.38492	1.00597	0.502985	−	0.864295i	$-0.332235\pi$
$19$	4.38492	1.00597	0.502985	+	0.864295i	$0.332235\pi$
$20$	0	0
$21$	0.922505	0.201307
$22$	0	0
$23$	−3.26121	−0.680009	−0.340005	−	0.940424i	$-0.610429\pi$
$23$	−3.26121	−0.680009	−0.340005	+	0.940424i	$0.610429\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.72061	0.523582
$28$	0	0
$29$	−8.52457	−1.58297	−0.791487	−	0.611187i	$-0.790692\pi$
$29$	−8.52457	−1.58297	−0.791487	+	0.611187i	$0.790692\pi$
$30$	0	0
$31$	0.781931	0.140439	0.0702195	−	0.997532i	$-0.477630\pi$
$31$	0.781931	0.140439	0.0702195	+	0.997532i	$0.477630\pi$
$32$	0	0
$33$	2.47527	0.430889
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.45889	−0.568638	−0.284319	−	0.958730i	$-0.591767\pi$
$37$	−3.45889	−0.568638	−0.284319	+	0.958730i	$0.591767\pi$
$38$	0	0
$39$	1.29539	0.207428
$40$	0	0
$41$	−9.03471	−1.41098	−0.705492	−	0.708718i	$-0.749274\pi$
$41$	−9.03471	−1.41098	−0.705492	+	0.708718i	$0.749274\pi$
$42$	0	0
$43$	0.494035	0.0753397	0.0376699	−	0.999290i	$-0.488006\pi$
$43$	0.494035	0.0753397	0.0376699	+	0.999290i	$0.488006\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.14706	1.18837	0.594186	−	0.804328i	$-0.297474\pi$
$47$	8.14706	1.18837	0.594186	+	0.804328i	$0.297474\pi$
$48$	0	0
$49$	−3.16110	−0.451585
$50$	0	0
$51$	1.24301	0.174057
$52$	0	0
$53$	−0.131204	−0.0180223	−0.00901113	−	0.999959i	$-0.502868\pi$
$53$	−0.131204	−0.0180223	−0.00901113	+	0.999959i	$0.502868\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.06456	−0.273457
$58$	0	0
$59$	−5.18375	−0.674867	−0.337433	−	0.941349i	$-0.609559\pi$
$59$	−5.18375	−0.674867	−0.337433	+	0.941349i	$0.609559\pi$
$60$	0	0
$61$	1.39727	0.178903	0.0894513	−	0.995991i	$-0.471489\pi$
$61$	1.39727	0.178903	0.0894513	+	0.995991i	$0.471489\pi$
$62$	0	0
$63$	5.44359	0.685828
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.2408	−1.86195	−0.930977	−	0.365077i	$-0.881043\pi$
$67$	−15.2408	−1.86195	−0.930977	+	0.365077i	$0.881043\pi$
$68$	0	0
$69$	1.53548	0.184850
$70$	0	0
$71$	−5.90377	−0.700649	−0.350325	−	0.936628i	$-0.613929\pi$
$71$	−5.90377	−0.700649	−0.350325	+	0.936628i	$0.613929\pi$
$72$	0	0
$73$	−8.40117	−0.983283	−0.491641	−	0.870798i	$-0.663603\pi$
$73$	−8.40117	−0.983283	−0.491641	+	0.870798i	$0.663603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3006	1.17386
$78$	0	0
$79$	−7.03607	−0.791620	−0.395810	−	0.918332i	$-0.629536\pi$
$79$	−7.03607	−0.791620	−0.395810	+	0.918332i	$0.629536\pi$
$80$	0	0
$81$	7.05401	0.783779
$82$	0	0
$83$	5.19942	0.570711	0.285355	−	0.958422i	$-0.407888\pi$
$83$	5.19942	0.570711	0.285355	+	0.958422i	$0.407888\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.01363	0.430306
$88$	0	0
$89$	−1.88392	−0.199695	−0.0998474	−	0.995003i	$-0.531835\pi$
$89$	−1.88392	−0.199695	−0.0998474	+	0.995003i	$0.531835\pi$
$90$	0	0
$91$	5.39061	0.565089
$92$	0	0
$93$	−0.368157	−0.0381761
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.9515	−1.92423	−0.962115	−	0.272643i	$-0.912102\pi$
$97$	−18.9515	−1.92423	−0.962115	+	0.272643i	$0.912102\pi$
$98$	0	0
$99$	14.6063	1.46799
$100$	0	0
$101$	−16.4512	−1.63696	−0.818478	−	0.574538i	$-0.805182\pi$
$101$	−16.4512	−1.63696	−0.818478	+	0.574538i	$0.805182\pi$
$102$	0	0
$103$	4.29785	0.423479	0.211740	−	0.977326i	$-0.432087\pi$
$103$	4.29785	0.423479	0.211740	+	0.977326i	$0.432087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.18358	0.791137	0.395568	−	0.918437i	$-0.370548\pi$
$107$	8.18358	0.791137	0.395568	+	0.918437i	$0.370548\pi$
$108$	0	0
$109$	−10.4015	−0.996281	−0.498140	−	0.867096i	$-0.665984\pi$
$109$	−10.4015	−0.996281	−0.498140	+	0.867096i	$0.665984\pi$
$110$	0	0
$111$	1.62855	0.154575
$112$	0	0
$113$	14.5245	1.36635	0.683177	−	0.730253i	$-0.260598\pi$
$113$	14.5245	1.36635	0.683177	+	0.730253i	$0.260598\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.64392	0.706681
$118$	0	0
$119$	5.17267	0.474178
$120$	0	0
$121$	16.6385	1.51259
$122$	0	0
$123$	4.25382	0.383554
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.2564	−1.70873	−0.854366	−	0.519672i	$-0.826054\pi$
$127$	−19.2564	−1.70873	−0.854366	+	0.519672i	$0.826054\pi$
$128$	0	0
$129$	−0.232607	−0.0204799
$130$	0	0
$131$	7.09112	0.619554	0.309777	−	0.950809i	$-0.399746\pi$
$131$	7.09112	0.619554	0.309777	+	0.950809i	$0.399746\pi$
$132$	0	0
$133$	−8.59143	−0.744972
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.405646	0.0346567	0.0173283	−	0.999850i	$-0.494484\pi$
$137$	0.405646	0.0346567	0.0173283	+	0.999850i	$0.494484\pi$
$138$	0	0
$139$	5.28107	0.447935	0.223967	−	0.974597i	$-0.428099\pi$
$139$	5.28107	0.447935	0.223967	+	0.974597i	$0.428099\pi$
$140$	0	0
$141$	−3.83589	−0.323040
$142$	0	0
$143$	14.4641	1.20955
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.48834	0.122756
$148$	0	0
$149$	2.02769	0.166115	0.0830575	−	0.996545i	$-0.473531\pi$
$149$	2.02769	0.166115	0.0830575	+	0.996545i	$0.473531\pi$
$150$	0	0
$151$	5.24744	0.427031	0.213515	−	0.976940i	$-0.431509\pi$
$151$	5.24744	0.427031	0.213515	+	0.976940i	$0.431509\pi$
$152$	0	0
$153$	7.33488	0.592990
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.1326	1.68657	0.843283	−	0.537469i	$-0.180620\pi$
$157$	21.1326	1.68657	0.843283	+	0.537469i	$0.180620\pi$
$158$	0	0
$159$	0.0617749	0.00489907
$160$	0	0
$161$	6.38973	0.503581
$162$	0	0
$163$	−4.03632	−0.316149	−0.158074	−	0.987427i	$-0.550529\pi$
$163$	−4.03632	−0.316149	−0.158074	+	0.987427i	$0.550529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.6049	−1.59445	−0.797226	−	0.603681i	$-0.793700\pi$
$167$	−20.6049	−1.59445	−0.797226	+	0.603681i	$0.793700\pi$
$168$	0	0
$169$	−5.43048	−0.417729
$170$	0	0
$171$	−12.1827	−0.931635
$172$	0	0
$173$	24.4105	1.85590	0.927950	−	0.372706i	$-0.121570\pi$
$173$	24.4105	1.85590	0.927950	+	0.372706i	$0.121570\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.44067	0.183452
$178$	0	0
$179$	21.9577	1.64119	0.820597	−	0.571508i	$-0.193641\pi$
$179$	21.9577	1.64119	0.820597	+	0.571508i	$0.193641\pi$
$180$	0	0
$181$	−22.8503	−1.69845	−0.849224	−	0.528033i	$-0.822930\pi$
$181$	−22.8503	−1.69845	−0.849224	+	0.528033i	$0.822930\pi$
$182$	0	0
$183$	−0.657880	−0.0486319
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.8793	1.01496
$188$	0	0
$189$	−5.33053	−0.387739
$190$	0	0
$191$	20.8481	1.50851	0.754257	−	0.656579i	$-0.227997\pi$
$191$	20.8481	1.50851	0.754257	+	0.656579i	$0.227997\pi$
$192$	0	0
$193$	19.9665	1.43722	0.718609	−	0.695414i	$-0.244779\pi$
$193$	19.9665	1.43722	0.718609	+	0.695414i	$0.244779\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.3348	−1.16381	−0.581904	−	0.813257i	$-0.697692\pi$
$197$	−16.3348	−1.16381	−0.581904	+	0.813257i	$0.697692\pi$
$198$	0	0
$199$	11.0764	0.785186	0.392593	−	0.919712i	$-0.371578\pi$
$199$	11.0764	0.785186	0.392593	+	0.919712i	$0.371578\pi$
$200$	0	0
$201$	7.17582	0.506143
$202$	0	0
$203$	16.7023	1.17227
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.06068	0.629761
$208$	0	0
$209$	−23.0526	−1.59458
$210$	0	0
$211$	−20.1174	−1.38494	−0.692470	−	0.721447i	$-0.743478\pi$
$211$	−20.1174	−1.38494	−0.692470	+	0.721447i	$0.743478\pi$
$212$	0	0
$213$	2.77968	0.190460
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.53205	−0.104002
$218$	0	0
$219$	3.95553	0.267290
$220$	0	0
$221$	7.26349	0.488595
$222$	0	0
$223$	26.3150	1.76218	0.881092	−	0.472944i	$-0.156809\pi$
$223$	26.3150	1.76218	0.881092	+	0.472944i	$0.156809\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.8387	0.984881	0.492440	−	0.870346i	$-0.336105\pi$
$227$	14.8387	0.984881	0.492440	+	0.870346i	$0.336105\pi$
$228$	0	0
$229$	−12.7756	−0.844233	−0.422116	−	0.906542i	$-0.638713\pi$
$229$	−12.7756	−0.844233	−0.422116	+	0.906542i	$0.638713\pi$
$230$	0	0
$231$	−4.84982	−0.319095
$232$	0	0
$233$	23.3584	1.53026	0.765130	−	0.643875i	$-0.222675\pi$
$233$	23.3584	1.53026	0.765130	+	0.643875i	$0.222675\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.31280	0.215189
$238$	0	0
$239$	19.5390	1.26387	0.631937	−	0.775019i	$-0.282260\pi$
$239$	19.5390	1.26387	0.631937	+	0.775019i	$0.282260\pi$
$240$	0	0
$241$	−16.3952	−1.05611	−0.528054	−	0.849211i	$-0.677078\pi$
$241$	−16.3952	−1.05611	−0.528054	+	0.849211i	$0.677078\pi$
$242$	0	0
$243$	−11.4831	−0.736640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0641	−0.767622
$248$	0	0
$249$	−2.44805	−0.155139
$250$	0	0
$251$	3.62159	0.228593	0.114296	−	0.993447i	$-0.463539\pi$
$251$	3.62159	0.228593	0.114296	+	0.993447i	$0.463539\pi$
$252$	0	0
$253$	17.1449	1.07789
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.4730	1.08993	0.544967	−	0.838457i	$-0.316542\pi$
$257$	17.4730	1.08993	0.544967	+	0.838457i	$0.316542\pi$
$258$	0	0
$259$	6.77704	0.421105
$260$	0	0
$261$	23.6840	1.46600
$262$	0	0
$263$	17.4561	1.07639	0.538195	−	0.842820i	$-0.319106\pi$
$263$	17.4561	1.07639	0.538195	+	0.842820i	$0.319106\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.887006	0.0542839
$268$	0	0
$269$	−5.00311	−0.305045	−0.152523	−	0.988300i	$-0.548740\pi$
$269$	−5.00311	−0.305045	−0.152523	+	0.988300i	$0.548740\pi$
$270$	0	0
$271$	−18.7351	−1.13807	−0.569037	−	0.822312i	$-0.692684\pi$
$271$	−18.7351	−1.13807	−0.569037	+	0.822312i	$0.692684\pi$
$272$	0	0
$273$	−2.53806	−0.153611
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.9952	0.961059	0.480530	−	0.876979i	$-0.340444\pi$
$277$	15.9952	0.961059	0.480530	+	0.876979i	$0.340444\pi$
$278$	0	0
$279$	−2.17245	−0.130061
$280$	0	0
$281$	0.473737	0.0282608	0.0141304	−	0.999900i	$-0.495502\pi$
$281$	0.473737	0.0282608	0.0141304	+	0.999900i	$0.495502\pi$
$282$	0	0
$283$	1.52053	0.0903861	0.0451931	−	0.998978i	$-0.485610\pi$
$283$	1.52053	0.0903861	0.0451931	+	0.998978i	$0.485610\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.7018	1.04490
$288$	0	0
$289$	−10.0302	−0.590010
$290$	0	0
$291$	8.92294	0.523072
$292$	0	0
$293$	−2.40677	−0.140605	−0.0703025	−	0.997526i	$-0.522396\pi$
$293$	−2.40677	−0.140605	−0.0703025	+	0.997526i	$0.522396\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−14.3029	−0.829938
$298$	0	0
$299$	8.97249	0.518892
$300$	0	0
$301$	−0.967970	−0.0557928
$302$	0	0
$303$	7.74574	0.444981
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.7898	1.58605	0.793024	−	0.609191i	$-0.208506\pi$
$307$	27.7898	1.58605	0.793024	+	0.609191i	$0.208506\pi$
$308$	0	0
$309$	−2.02356	−0.115116
$310$	0	0
$311$	−1.00543	−0.0570125	−0.0285062	−	0.999594i	$-0.509075\pi$
$311$	−1.00543	−0.0570125	−0.0285062	+	0.999594i	$0.509075\pi$
$312$	0	0
$313$	−14.1804	−0.801524	−0.400762	−	0.916182i	$-0.631255\pi$
$313$	−14.1804	−0.801524	−0.400762	+	0.916182i	$0.631255\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.2589	0.913189	0.456595	−	0.889675i	$-0.349069\pi$
$317$	16.2589	0.913189	0.456595	+	0.889675i	$0.349069\pi$
$318$	0	0
$319$	44.8157	2.50920
$320$	0	0
$321$	−3.85308	−0.215058
$322$	0	0
$323$	−11.5764	−0.644127
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.89734	0.270823
$328$	0	0
$329$	−15.9626	−0.880049
$330$	0	0
$331$	−12.7917	−0.703097	−0.351548	−	0.936170i	$-0.614345\pi$
$331$	−12.7917	−0.703097	−0.351548	+	0.936170i	$0.614345\pi$
$332$	0	0
$333$	9.60989	0.526619
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.9111	−1.03015	−0.515076	−	0.857145i	$-0.672236\pi$
$337$	−18.9111	−1.03015	−0.515076	+	0.857145i	$0.672236\pi$
$338$	0	0
$339$	−6.83860	−0.371422
$340$	0	0
$341$	−4.11079	−0.222612
$342$	0	0
$343$	19.9088	1.07497
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.87103	0.529905	0.264952	−	0.964262i	$-0.414644\pi$
$347$	9.87103	0.529905	0.264952	+	0.964262i	$0.414644\pi$
$348$	0	0
$349$	28.5586	1.52871	0.764354	−	0.644796i	$-0.223058\pi$
$349$	28.5586	1.52871	0.764354	+	0.644796i	$0.223058\pi$
$350$	0	0
$351$	−7.48515	−0.399528
$352$	0	0
$353$	2.34375	0.124745	0.0623726	−	0.998053i	$-0.480133\pi$
$353$	2.34375	0.124745	0.0623726	+	0.998053i	$0.480133\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.43545	−0.128898
$358$	0	0
$359$	−21.6870	−1.14459	−0.572297	−	0.820046i	$-0.693948\pi$
$359$	−21.6870	−1.14459	−0.572297	+	0.820046i	$0.693948\pi$
$360$	0	0
$361$	0.227543	0.0119759
$362$	0	0
$363$	−7.83392	−0.411174
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.1833	−1.05356	−0.526779	−	0.850002i	$-0.676600\pi$
$367$	−20.1833	−1.05356	−0.526779	+	0.850002i	$0.676600\pi$
$368$	0	0
$369$	25.1013	1.30672
$370$	0	0
$371$	0.257070	0.0133464
$372$	0	0
$373$	11.1585	0.577767	0.288883	−	0.957364i	$-0.406716\pi$
$373$	11.1585	0.577767	0.288883	+	0.957364i	$0.406716\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.4534	1.20791
$378$	0	0
$379$	−26.7114	−1.37207	−0.686036	−	0.727567i	$-0.740651\pi$
$379$	−26.7114	−1.37207	−0.686036	+	0.727567i	$0.740651\pi$
$380$	0	0
$381$	9.06652	0.464492
$382$	0	0
$383$	15.3515	0.784427	0.392213	−	0.919874i	$-0.371709\pi$
$383$	15.3515	0.784427	0.392213	+	0.919874i	$0.371709\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.37259	−0.0697726
$388$	0	0
$389$	1.27141	0.0644632	0.0322316	−	0.999480i	$-0.489739\pi$
$389$	1.27141	0.0644632	0.0322316	+	0.999480i	$0.489739\pi$
$390$	0	0
$391$	8.60974	0.435413
$392$	0	0
$393$	−3.33872	−0.168416
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.5463	−1.23194	−0.615971	−	0.787769i	$-0.711236\pi$
$397$	−24.5463	−1.23194	−0.615971	+	0.787769i	$0.711236\pi$
$398$	0	0
$399$	4.04511	0.202509
$400$	0	0
$401$	−13.1320	−0.655781	−0.327890	−	0.944716i	$-0.606338\pi$
$401$	−13.1320	−0.655781	−0.327890	+	0.944716i	$0.606338\pi$
$402$	0	0
$403$	−2.15131	−0.107164
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.1842	0.901357
$408$	0	0
$409$	−22.3514	−1.10521	−0.552604	−	0.833444i	$-0.686366\pi$
$409$	−22.3514	−1.10521	−0.552604	+	0.833444i	$0.686366\pi$
$410$	0	0
$411$	−0.190991	−0.00942088
$412$	0	0
$413$	10.1566	0.499773
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.48649	−0.121764
$418$	0	0
$419$	15.7307	0.768496	0.384248	−	0.923230i	$-0.374461\pi$
$419$	15.7307	0.768496	0.384248	+	0.923230i	$0.374461\pi$
$420$	0	0
$421$	−4.99080	−0.243237	−0.121618	−	0.992577i	$-0.538808\pi$
$421$	−4.99080	−0.243237	−0.121618	+	0.992577i	$0.538808\pi$
$422$	0	0
$423$	−22.6351	−1.10056
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.73770	−0.132486
$428$	0	0
$429$	−6.81014	−0.328797
$430$	0	0
$431$	−11.3859	−0.548438	−0.274219	−	0.961667i	$-0.588419\pi$
$431$	−11.3859	−0.548438	−0.274219	+	0.961667i	$0.588419\pi$
$432$	0	0
$433$	−25.2351	−1.21272	−0.606361	−	0.795190i	$-0.707371\pi$
$433$	−25.2351	−1.21272	−0.606361	+	0.795190i	$0.707371\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.3002	−0.684069
$438$	0	0
$439$	−5.27404	−0.251716	−0.125858	−	0.992048i	$-0.540168\pi$
$439$	−5.27404	−0.251716	−0.125858	+	0.992048i	$0.540168\pi$
$440$	0	0
$441$	8.78253	0.418216
$442$	0	0
$443$	3.66444	0.174103	0.0870513	−	0.996204i	$-0.472256\pi$
$443$	3.66444	0.174103	0.0870513	+	0.996204i	$0.472256\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.954700	−0.0451557
$448$	0	0
$449$	−11.5593	−0.545517	−0.272759	−	0.962082i	$-0.587936\pi$
$449$	−11.5593	−0.545517	−0.272759	+	0.962082i	$0.587936\pi$
$450$	0	0
$451$	47.4976	2.23657
$452$	0	0
$453$	−2.47066	−0.116082
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.61173	−0.356062	−0.178031	−	0.984025i	$-0.556973\pi$
$457$	−7.61173	−0.356062	−0.178031	+	0.984025i	$0.556973\pi$
$458$	0	0
$459$	−7.18253	−0.335252
$460$	0	0
$461$	−7.98881	−0.372076	−0.186038	−	0.982543i	$-0.559565\pi$
$461$	−7.98881	−0.372076	−0.186038	+	0.982543i	$0.559565\pi$
$462$	0	0
$463$	13.4277	0.624036	0.312018	−	0.950076i	$-0.398995\pi$
$463$	13.4277	0.624036	0.312018	+	0.950076i	$0.398995\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.7333	−1.60726	−0.803632	−	0.595126i	$-0.797102\pi$
$467$	−34.7333	−1.60726	−0.803632	+	0.595126i	$0.797102\pi$
$468$	0	0
$469$	29.8614	1.37887
$470$	0	0
$471$	−9.94989	−0.458467
$472$	0	0
$473$	−2.59726	−0.119422
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.364527	0.0166905
$478$	0	0
$479$	−21.1304	−0.965470	−0.482735	−	0.875766i	$-0.660357\pi$
$479$	−21.1304	−0.965470	−0.482735	+	0.875766i	$0.660357\pi$
$480$	0	0
$481$	9.51635	0.433908
$482$	0	0
$483$	−3.00848	−0.136891
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.0976	−1.72637	−0.863183	−	0.504891i	$-0.831533\pi$
$487$	−38.0976	−1.72637	−0.863183	+	0.504891i	$0.831533\pi$
$488$	0	0
$489$	1.90042	0.0859401
$490$	0	0
$491$	−20.3388	−0.917878	−0.458939	−	0.888468i	$-0.651770\pi$
$491$	−20.3388	−0.917878	−0.458939	+	0.888468i	$0.651770\pi$
$492$	0	0
$493$	22.5052	1.01359
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.5673	0.518866
$498$	0	0
$499$	5.13297	0.229783	0.114892	−	0.993378i	$-0.463348\pi$
$499$	5.13297	0.229783	0.114892	+	0.993378i	$0.463348\pi$
$500$	0	0
$501$	9.70140	0.433427
$502$	0	0
$503$	−40.2594	−1.79508	−0.897539	−	0.440936i	$-0.854647\pi$
$503$	−40.2594	−1.79508	−0.897539	+	0.440936i	$0.854647\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.55684	0.113553
$508$	0	0
$509$	−5.61485	−0.248874	−0.124437	−	0.992228i	$-0.539712\pi$
$509$	−5.61485	−0.248874	−0.124437	+	0.992228i	$0.539712\pi$
$510$	0	0
$511$	16.4605	0.728170
$512$	0	0
$513$	11.9297	0.526708
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−42.8310	−1.88371
$518$	0	0
$519$	−11.4932	−0.504497
$520$	0	0
$521$	16.3340	0.715607	0.357803	−	0.933797i	$-0.383526\pi$
$521$	16.3340	0.715607	0.357803	+	0.933797i	$0.383526\pi$
$522$	0	0
$523$	30.3040	1.32510	0.662551	−	0.749017i	$-0.269474\pi$
$523$	30.3040	1.32510	0.662551	+	0.749017i	$0.269474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.06433	−0.0899237
$528$	0	0
$529$	−12.3645	−0.537587
$530$	0	0
$531$	14.4021	0.624998
$532$	0	0
$533$	24.8570	1.07667
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.3384	−0.446133
$538$	0	0
$539$	16.6186	0.715815
$540$	0	0
$541$	−25.3253	−1.08882	−0.544409	−	0.838820i	$-0.683246\pi$
$541$	−25.3253	−1.08882	−0.544409	+	0.838820i	$0.683246\pi$
$542$	0	0
$543$	10.7586	0.461697
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.38648	0.0592815	0.0296407	−	0.999561i	$-0.490564\pi$
$547$	1.38648	0.0592815	0.0296407	+	0.999561i	$0.490564\pi$
$548$	0	0
$549$	−3.88207	−0.165683
$550$	0	0
$551$	−37.3796	−1.59242
$552$	0	0
$553$	13.7859	0.586234
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.5415	0.828000	0.414000	−	0.910277i	$-0.364131\pi$
$557$	19.5415	0.828000	0.414000	+	0.910277i	$0.364131\pi$
$558$	0	0
$559$	−1.35923	−0.0574892
$560$	0	0
$561$	−6.53482	−0.275900
$562$	0	0
$563$	−1.64191	−0.0691981	−0.0345991	−	0.999401i	$-0.511015\pi$
$563$	−1.64191	−0.0691981	−0.0345991	+	0.999401i	$0.511015\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−13.8210	−0.580427
$568$	0	0
$569$	40.7240	1.70724	0.853619	−	0.520898i	$-0.174403\pi$
$569$	40.7240	1.70724	0.853619	+	0.520898i	$0.174403\pi$
$570$	0	0
$571$	42.3335	1.77160	0.885800	−	0.464067i	$-0.153610\pi$
$571$	42.3335	1.77160	0.885800	+	0.464067i	$0.153610\pi$
$572$	0	0
$573$	−9.81592	−0.410066
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.977188	−0.0406809	−0.0203404	−	0.999793i	$-0.506475\pi$
$577$	−0.977188	−0.0406809	−0.0203404	+	0.999793i	$0.506475\pi$
$578$	0	0
$579$	−9.40083	−0.390685
$580$	0	0
$581$	−10.1873	−0.422640
$582$	0	0
$583$	0.689770	0.0285674
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.3350	0.426569	0.213285	−	0.976990i	$-0.431584\pi$
$587$	10.3350	0.426569	0.213285	+	0.976990i	$0.431584\pi$
$588$	0	0
$589$	3.42871	0.141277
$590$	0	0
$591$	7.69094	0.316363
$592$	0	0
$593$	4.17978	0.171643	0.0858214	−	0.996311i	$-0.472649\pi$
$593$	4.17978	0.171643	0.0858214	+	0.996311i	$0.472649\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.21512	−0.213441
$598$	0	0
$599$	15.1195	0.617765	0.308882	−	0.951100i	$-0.400045\pi$
$599$	15.1195	0.617765	0.308882	+	0.951100i	$0.400045\pi$
$600$	0	0
$601$	−15.3961	−0.628021	−0.314010	−	0.949420i	$-0.601673\pi$
$601$	−15.3961	−0.628021	−0.314010	+	0.949420i	$0.601673\pi$
$602$	0	0
$603$	42.3437	1.72437
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.9258	−1.45818	−0.729091	−	0.684416i	$-0.760057\pi$
$607$	−35.9258	−1.45818	−0.729091	+	0.684416i	$0.760057\pi$
$608$	0	0
$609$	−7.86396	−0.318664
$610$	0	0
$611$	−22.4148	−0.906806
$612$	0	0
$613$	33.2867	1.34444	0.672219	−	0.740352i	$-0.265341\pi$
$613$	33.2867	1.34444	0.672219	+	0.740352i	$0.265341\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.6630	−1.27471	−0.637353	−	0.770572i	$-0.719971\pi$
$617$	−31.6630	−1.27471	−0.637353	+	0.770572i	$0.719971\pi$
$618$	0	0
$619$	31.6845	1.27351	0.636753	−	0.771067i	$-0.280277\pi$
$619$	31.6845	1.27351	0.636753	+	0.771067i	$0.280277\pi$
$620$	0	0
$621$	−8.87248	−0.356040
$622$	0	0
$623$	3.69118	0.147884
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.8539	0.433461
$628$	0	0
$629$	9.13162	0.364101
$630$	0	0
$631$	46.3726	1.84606	0.923032	−	0.384722i	$-0.125703\pi$
$631$	46.3726	1.84606	0.923032	+	0.384722i	$0.125703\pi$
$632$	0	0
$633$	9.47190	0.376474
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.69705	0.344590
$638$	0	0
$639$	16.4026	0.648875
$640$	0	0
$641$	18.5332	0.732017	0.366008	−	0.930612i	$-0.380724\pi$
$641$	18.5332	0.732017	0.366008	+	0.930612i	$0.380724\pi$
$642$	0	0
$643$	−20.3699	−0.803311	−0.401656	−	0.915791i	$-0.631565\pi$
$643$	−20.3699	−0.803311	−0.401656	+	0.915791i	$0.631565\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.0060	−1.76937	−0.884685	−	0.466189i	$-0.845627\pi$
$647$	−45.0060	−1.76937	−0.884685	+	0.466189i	$0.845627\pi$
$648$	0	0
$649$	27.2522	1.06974
$650$	0	0
$651$	0.721335	0.0282713
$652$	0	0
$653$	−36.5325	−1.42963	−0.714813	−	0.699315i	$-0.753488\pi$
$653$	−36.5325	−1.42963	−0.714813	+	0.699315i	$0.753488\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	23.3411	0.910624
$658$	0	0
$659$	−31.3819	−1.22247	−0.611233	−	0.791451i	$-0.709326\pi$
$659$	−31.3819	−1.22247	−0.611233	+	0.791451i	$0.709326\pi$
$660$	0	0
$661$	−25.6055	−0.995940	−0.497970	−	0.867194i	$-0.665921\pi$
$661$	−25.6055	−0.995940	−0.497970	+	0.867194i	$0.665921\pi$
$662$	0	0
$663$	−3.41987	−0.132817
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.8004	1.07644
$668$	0	0
$669$	−12.3899	−0.479022
$670$	0	0
$671$	−7.34580	−0.283581
$672$	0	0
$673$	28.5845	1.10185	0.550927	−	0.834554i	$-0.314274\pi$
$673$	28.5845	1.10185	0.550927	+	0.834554i	$0.314274\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.7033	−0.872557	−0.436279	−	0.899812i	$-0.643704\pi$
$677$	−22.7033	−0.872557	−0.436279	+	0.899812i	$0.643704\pi$
$678$	0	0
$679$	37.1318	1.42499
$680$	0	0
$681$	−6.98653	−0.267724
$682$	0	0
$683$	41.4547	1.58622	0.793110	−	0.609078i	$-0.208460\pi$
$683$	41.4547	1.58622	0.793110	+	0.609078i	$0.208460\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.01513	0.229491
$688$	0	0
$689$	0.360978	0.0137522
$690$	0	0
$691$	−15.9595	−0.607127	−0.303564	−	0.952811i	$-0.598177\pi$
$691$	−15.9595	−0.607127	−0.303564	+	0.952811i	$0.598177\pi$
$692$	0	0
$693$	−28.6182	−1.08712
$694$	0	0
$695$	0	0
$696$	0	0
$697$	23.8520	0.903460
$698$	0	0
$699$	−10.9979	−0.415977
$700$	0	0
$701$	44.9076	1.69614	0.848069	−	0.529886i	$-0.177765\pi$
$701$	44.9076	1.69614	0.848069	+	0.529886i	$0.177765\pi$
$702$	0	0
$703$	−15.1670	−0.572033
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.2330	1.21225
$708$	0	0
$709$	−19.9111	−0.747778	−0.373889	−	0.927473i	$-0.621976\pi$
$709$	−19.9111	−0.747778	−0.373889	+	0.927473i	$0.621976\pi$
$710$	0	0
$711$	19.5484	0.733124
$712$	0	0
$713$	−2.55004	−0.0954998
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.19958	−0.343565
$718$	0	0
$719$	−15.1002	−0.563141	−0.281570	−	0.959541i	$-0.590855\pi$
$719$	−15.1002	−0.563141	−0.281570	+	0.959541i	$0.590855\pi$
$720$	0	0
$721$	−8.42082	−0.313608
$722$	0	0
$723$	7.71937	0.287086
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.9039	0.997810	0.498905	−	0.866657i	$-0.333736\pi$
$727$	26.9039	0.997810	0.498905	+	0.866657i	$0.333736\pi$
$728$	0	0
$729$	−15.7554	−0.583535
$730$	0	0
$731$	−1.30428	−0.0482404
$732$	0	0
$733$	14.4571	0.533987	0.266993	−	0.963698i	$-0.413970\pi$
$733$	14.4571	0.533987	0.266993	+	0.963698i	$0.413970\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	80.1242	2.95141
$738$	0	0
$739$	−42.1335	−1.54991	−0.774954	−	0.632018i	$-0.782227\pi$
$739$	−42.1335	−1.54991	−0.774954	+	0.632018i	$0.782227\pi$
$740$	0	0
$741$	5.68016	0.208666
$742$	0	0
$743$	−3.04065	−0.111551	−0.0557753	−	0.998443i	$-0.517763\pi$
$743$	−3.04065	−0.111551	−0.0557753	+	0.998443i	$0.517763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.4456	−0.528539
$748$	0	0
$749$	−16.0342	−0.585877
$750$	0	0
$751$	−26.5617	−0.969251	−0.484626	−	0.874722i	$-0.661044\pi$
$751$	−26.5617	−0.969251	−0.484626	+	0.874722i	$0.661044\pi$
$752$	0	0
$753$	−1.70516	−0.0621394
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.12203	0.186163	0.0930817	−	0.995658i	$-0.470328\pi$
$757$	5.12203	0.186163	0.0930817	+	0.995658i	$0.470328\pi$
$758$	0	0
$759$	−8.07237	−0.293008
$760$	0	0
$761$	24.2272	0.878235	0.439118	−	0.898430i	$-0.355291\pi$
$761$	24.2272	0.878235	0.439118	+	0.898430i	$0.355291\pi$
$762$	0	0
$763$	20.3797	0.737796
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.2619	0.514968
$768$	0	0
$769$	−54.8738	−1.97880	−0.989399	−	0.145220i	$-0.953611\pi$
$769$	−54.8738	−1.97880	−0.989399	+	0.145220i	$0.953611\pi$
$770$	0	0
$771$	−8.22682	−0.296282
$772$	0	0
$773$	−32.1930	−1.15790	−0.578950	−	0.815363i	$-0.696537\pi$
$773$	−32.1930	−1.15790	−0.578950	+	0.815363i	$0.696537\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.19084	−0.114471
$778$	0	0
$779$	−39.6165	−1.41941
$780$	0	0
$781$	31.0375	1.11061
$782$	0	0
$783$	−23.1920	−0.828816
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.5252	0.838583	0.419292	−	0.907852i	$-0.362278\pi$
$787$	23.5252	0.838583	0.419292	+	0.907852i	$0.362278\pi$
$788$	0	0
$789$	−8.21888	−0.292600
$790$	0	0
$791$	−28.4581	−1.01185
$792$	0	0
$793$	−3.84429	−0.136515
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.75754	0.310208	0.155104	−	0.987898i	$-0.450429\pi$
$797$	8.75754	0.310208	0.155104	+	0.987898i	$0.450429\pi$
$798$	0	0
$799$	−21.5086	−0.760920
$800$	0	0
$801$	5.23412	0.184939
$802$	0	0
$803$	44.1669	1.55862
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.35562	0.0829217
$808$	0	0
$809$	1.49946	0.0527182	0.0263591	−	0.999653i	$-0.491609\pi$
$809$	1.49946	0.0527182	0.0263591	+	0.999653i	$0.491609\pi$
$810$	0	0
$811$	9.07983	0.318836	0.159418	−	0.987211i	$-0.449038\pi$
$811$	9.07983	0.318836	0.159418	+	0.987211i	$0.449038\pi$
$812$	0	0
$813$	8.82105	0.309368
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.16631	0.0757895
$818$	0	0
$819$	−14.9768	−0.523333
$820$	0	0
$821$	14.2992	0.499044	0.249522	−	0.968369i	$-0.419726\pi$
$821$	14.2992	0.499044	0.249522	+	0.968369i	$0.419726\pi$
$822$	0	0
$823$	15.8158	0.551305	0.275653	−	0.961257i	$-0.411106\pi$
$823$	15.8158	0.551305	0.275653	+	0.961257i	$0.411106\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.19161	0.250077	0.125038	−	0.992152i	$-0.460095\pi$
$827$	7.19161	0.250077	0.125038	+	0.992152i	$0.460095\pi$
$828$	0	0
$829$	11.5276	0.400370	0.200185	−	0.979758i	$-0.435846\pi$
$829$	11.5276	0.400370	0.200185	+	0.979758i	$0.435846\pi$
$830$	0	0
$831$	−7.53104	−0.261249
$832$	0	0
$833$	8.34543	0.289152
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.12733	0.0735313
$838$	0	0
$839$	−55.2612	−1.90783	−0.953914	−	0.300082i	$-0.902986\pi$
$839$	−55.2612	−1.90783	−0.953914	+	0.300082i	$0.902986\pi$
$840$	0	0
$841$	43.6683	1.50580
$842$	0	0
$843$	−0.223050	−0.00768225
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−32.6000	−1.12015
$848$	0	0
$849$	−0.715912	−0.0245700
$850$	0	0
$851$	11.2802	0.386679
$852$	0	0
$853$	55.1014	1.88664	0.943318	−	0.331891i	$-0.107687\pi$
$853$	55.1014	1.88664	0.943318	+	0.331891i	$0.107687\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.4488	0.561879	0.280940	−	0.959725i	$-0.409354\pi$
$857$	16.4488	0.561879	0.280940	+	0.959725i	$0.409354\pi$
$858$	0	0
$859$	−22.1277	−0.754987	−0.377494	−	0.926012i	$-0.623214\pi$
$859$	−22.1277	−0.754987	−0.377494	+	0.926012i	$0.623214\pi$
$860$	0	0
$861$	−8.33456	−0.284041
$862$	0	0
$863$	40.5565	1.38056	0.690279	−	0.723543i	$-0.257488\pi$
$863$	40.5565	1.38056	0.690279	+	0.723543i	$0.257488\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.72251	0.160385
$868$	0	0
$869$	36.9903	1.25481
$870$	0	0
$871$	41.9315	1.42080
$872$	0	0
$873$	52.6532	1.78204
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.0587	−0.609797	−0.304899	−	0.952385i	$-0.598623\pi$
$877$	−18.0587	−0.609797	−0.304899	+	0.952385i	$0.598623\pi$
$878$	0	0
$879$	1.13318	0.0382212
$880$	0	0
$881$	1.28285	0.0432204	0.0216102	−	0.999766i	$-0.493121\pi$
$881$	1.28285	0.0432204	0.0216102	+	0.999766i	$0.493121\pi$
$882$	0	0
$883$	35.1374	1.18247	0.591234	−	0.806500i	$-0.298641\pi$
$883$	35.1374	1.18247	0.591234	+	0.806500i	$0.298641\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.3320	−0.649104	−0.324552	−	0.945868i	$-0.605214\pi$
$887$	−19.3320	−0.649104	−0.324552	+	0.945868i	$0.605214\pi$
$888$	0	0
$889$	37.7294	1.26540
$890$	0	0
$891$	−37.0846	−1.24238
$892$	0	0
$893$	35.7242	1.19547
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.22452	−0.141053
$898$	0	0
$899$	−6.66562	−0.222311
$900$	0	0
$901$	0.346384	0.0115397
$902$	0	0
$903$	0.455750	0.0151664
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.2888	0.540861	0.270430	−	0.962740i	$-0.412834\pi$
$907$	16.2888	0.540861	0.270430	+	0.962740i	$0.412834\pi$
$908$	0	0
$909$	45.7067	1.51600
$910$	0	0
$911$	8.98753	0.297770	0.148885	−	0.988855i	$-0.452432\pi$
$911$	8.98753	0.297770	0.148885	+	0.988855i	$0.452432\pi$
$912$	0	0
$913$	−27.3346	−0.904642
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.8937	−0.458811
$918$	0	0
$919$	−11.3046	−0.372903	−0.186452	−	0.982464i	$-0.559699\pi$
$919$	−11.3046	−0.372903	−0.186452	+	0.982464i	$0.559699\pi$
$920$	0	0
$921$	−13.0843	−0.431142
$922$	0	0
$923$	16.2429	0.534642
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−11.9408	−0.392187
$928$	0	0
$929$	47.5518	1.56012	0.780062	−	0.625702i	$-0.215187\pi$
$929$	47.5518	1.56012	0.780062	+	0.625702i	$0.215187\pi$
$930$	0	0
$931$	−13.8612	−0.454281
$932$	0	0
$933$	0.473386	0.0154979
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.3175	0.565738	0.282869	−	0.959159i	$-0.408714\pi$
$937$	17.3175	0.565738	0.282869	+	0.959159i	$0.408714\pi$
$938$	0	0
$939$	6.67657	0.217882
$940$	0	0
$941$	47.5005	1.54847	0.774235	−	0.632898i	$-0.218135\pi$
$941$	47.5005	1.54847	0.774235	+	0.632898i	$0.218135\pi$
$942$	0	0
$943$	29.4641	0.959482
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.6343	0.508048	0.254024	−	0.967198i	$-0.418246\pi$
$947$	15.6343	0.508048	0.254024	+	0.967198i	$0.418246\pi$
$948$	0	0
$949$	23.1139	0.750310
$950$	0	0
$951$	−7.65518	−0.248236
$952$	0	0
$953$	−13.8976	−0.450188	−0.225094	−	0.974337i	$-0.572269\pi$
$953$	−13.8976	−0.450188	−0.225094	+	0.974337i	$0.572269\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−21.1006	−0.682085
$958$	0	0
$959$	−0.794787	−0.0256650
$960$	0	0
$961$	−30.3886	−0.980277
$962$	0	0
$963$	−22.7366	−0.732677
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.9592	−0.963424	−0.481712	−	0.876330i	$-0.659985\pi$
$967$	−29.9592	−0.963424	−0.481712	+	0.876330i	$0.659985\pi$
$968$	0	0
$969$	5.45052	0.175096
$970$	0	0
$971$	41.6487	1.33657	0.668285	−	0.743906i	$-0.267029\pi$
$971$	41.6487	1.33657	0.668285	+	0.743906i	$0.267029\pi$
$972$	0	0
$973$	−10.3473	−0.331718
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.0537	0.353639	0.176820	−	0.984243i	$-0.443419\pi$
$977$	11.0537	0.353639	0.176820	+	0.984243i	$0.443419\pi$
$978$	0	0
$979$	9.90419	0.316539
$980$	0	0
$981$	28.8986	0.922662
$982$	0	0
$983$	−47.1166	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$983$	−47.1166	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.51570	0.239227
$988$	0	0
$989$	−1.61115	−0.0512317
$990$	0	0
$991$	39.0047	1.23902	0.619512	−	0.784987i	$-0.287330\pi$
$991$	39.0047	1.23902	0.619512	+	0.784987i	$0.287330\pi$
$992$	0	0
$993$	6.02274	0.191126
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.27227	−0.103634	−0.0518169	−	0.998657i	$-0.516501\pi$
$997$	−3.27227	−0.103634	−0.0518169	+	0.998657i	$0.516501\pi$
$998$	0	0
$999$	−9.41029	−0.297728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bl.1.3		8
4.3	odd	2		5000.2.a.k.1.6	✓	8
5.4	even	2		10000.2.a.bg.1.6		8
20.19	odd	2		5000.2.a.n.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.6	✓	8	4.3	odd	2
5000.2.a.n.1.3	yes	8	20.19	odd	2
10000.2.a.bg.1.6		8	5.4	even	2
10000.2.a.bl.1.3		8	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.470831 q^{3} -1.95931 q^{7} -2.77832 q^{9} +O(q^{10})\)	\(q-0.470831 q^{3} -1.95931 q^{7} -2.77832 q^{9} -5.25723 q^{11} -2.75128 q^{13} -2.64004 q^{17} +4.38492 q^{19} +0.922505 q^{21} -3.26121 q^{23} +2.72061 q^{27} -8.52457 q^{29} +0.781931 q^{31} +2.47527 q^{33} -3.45889 q^{37} +1.29539 q^{39} -9.03471 q^{41} +0.494035 q^{43} +8.14706 q^{47} -3.16110 q^{49} +1.24301 q^{51} -0.131204 q^{53} -2.06456 q^{57} -5.18375 q^{59} +1.39727 q^{61} +5.44359 q^{63} -15.2408 q^{67} +1.53548 q^{69} -5.90377 q^{71} -8.40117 q^{73} +10.3006 q^{77} -7.03607 q^{79} +7.05401 q^{81} +5.19942 q^{83} +4.01363 q^{87} -1.88392 q^{89} +5.39061 q^{91} -0.368157 q^{93} -18.9515 q^{97} +14.6063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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