LMFDB - Embedded newform 10000.2.a.bb.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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.17557$ 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)

$f(q)$	$=$	$q-0.175571 q^{3} +1.27346 q^{7} -2.96917 q^{9} +O(q^{10})$	$q-0.175571 q^{3} +1.27346 q^{7} -2.96917 q^{9} +4.36176 q^{11} -4.28408 q^{13} -0.0978870 q^{17} +0.333955 q^{19} -0.223582 q^{21} +6.31375 q^{23} +1.04801 q^{27} -4.67853 q^{29} -8.04029 q^{31} -0.765797 q^{33} -4.10851 q^{37} +0.752158 q^{39} +11.4317 q^{41} +10.2093 q^{43} -11.7052 q^{47} -5.37831 q^{49} +0.0171861 q^{51} -0.884927 q^{53} -0.0586326 q^{57} -7.60845 q^{59} -1.67667 q^{61} -3.78112 q^{63} -1.16901 q^{67} -1.10851 q^{69} +2.35706 q^{71} +1.60845 q^{73} +5.55452 q^{77} -8.13412 q^{79} +8.72353 q^{81} +7.42040 q^{83} +0.821412 q^{87} +1.70820 q^{89} -5.45559 q^{91} +1.41164 q^{93} -6.60741 q^{97} -12.9508 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9	$ 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 + 2 * q^11 - 14 * q^13 - 8 * q^17 - 2 * q^21 + 4 * q^23 + 10 * q^27 - 10 * q^29 - 8 * q^31 - 8 * q^33 - 18 * q^37 - 14 * q^39 - 12 * q^41 + 14 * q^43 + 8 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 2 * q^61 - 16 * q^63 - 2 * q^67 - 6 * q^69 - 8 * q^71 - 24 * q^73 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 14 * q^83 - 20 * q^87 - 20 * q^89 - 18 * q^91 - 8 * q^93 - 28 * q^97 - 14 * 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
$3$	−0.175571	−0.101366	−0.0506828	−	0.998715i	$-0.516140\pi$
$3$	−0.175571	−0.101366	−0.0506828	+	0.998715i	$0.516140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.27346	0.481322	0.240661	−	0.970609i	$-0.422636\pi$
$7$	1.27346	0.481322	0.240661	+	0.970609i	$0.422636\pi$
$8$	0	0
$9$	−2.96917	−0.989725
$10$	0	0
$11$	4.36176	1.31512	0.657560	−	0.753402i	$-0.271588\pi$
$11$	4.36176	1.31512	0.657560	+	0.753402i	$0.271588\pi$
$12$	0	0
$13$	−4.28408	−1.18819	−0.594095	−	0.804395i	$-0.702490\pi$
$13$	−4.28408	−1.18819	−0.594095	+	0.804395i	$0.702490\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.0978870	−0.0237411	−0.0118705	−	0.999930i	$-0.503779\pi$
$17$	−0.0978870	−0.0237411	−0.0118705	+	0.999930i	$0.503779\pi$
$18$	0	0
$19$	0.333955	0.0766145	0.0383073	−	0.999266i	$-0.487803\pi$
$19$	0.333955	0.0766145	0.0383073	+	0.999266i	$0.487803\pi$
$20$	0	0
$21$	−0.223582	−0.0487895
$22$	0	0
$23$	6.31375	1.31651	0.658254	−	0.752796i	$-0.271295\pi$
$23$	6.31375	1.31651	0.658254	+	0.752796i	$0.271295\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.04801	0.201690
$28$	0	0
$29$	−4.67853	−0.868781	−0.434391	−	0.900725i	$-0.643036\pi$
$29$	−4.67853	−0.868781	−0.434391	+	0.900725i	$0.643036\pi$
$30$	0	0
$31$	−8.04029	−1.44408	−0.722040	−	0.691852i	$-0.756795\pi$
$31$	−8.04029	−1.44408	−0.722040	+	0.691852i	$0.756795\pi$
$32$	0	0
$33$	−0.765797	−0.133308
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.10851	−0.675435	−0.337717	−	0.941248i	$-0.609655\pi$
$37$	−4.10851	−0.675435	−0.337717	+	0.941248i	$0.609655\pi$
$38$	0	0
$39$	0.752158	0.120442
$40$	0	0
$41$	11.4317	1.78534	0.892668	−	0.450715i	$-0.148831\pi$
$41$	11.4317	1.78534	0.892668	+	0.450715i	$0.148831\pi$
$42$	0	0
$43$	10.2093	1.55690	0.778452	−	0.627704i	$-0.216005\pi$
$43$	10.2093	1.55690	0.778452	+	0.627704i	$0.216005\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.7052	−1.70738	−0.853688	−	0.520784i	$-0.825640\pi$
$47$	−11.7052	−1.70738	−0.853688	+	0.520784i	$0.825640\pi$
$48$	0	0
$49$	−5.37831	−0.768329
$50$	0	0
$51$	0.0171861	0.00240653
$52$	0	0
$53$	−0.884927	−0.121554	−0.0607770	−	0.998151i	$-0.519358\pi$
$53$	−0.884927	−0.121554	−0.0607770	+	0.998151i	$0.519358\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.0586326	−0.00776608
$58$	0	0
$59$	−7.60845	−0.990536	−0.495268	−	0.868740i	$-0.664930\pi$
$59$	−7.60845	−0.990536	−0.495268	+	0.868740i	$0.664930\pi$
$60$	0	0
$61$	−1.67667	−0.214675	−0.107338	−	0.994223i	$-0.534233\pi$
$61$	−1.67667	−0.214675	−0.107338	+	0.994223i	$0.534233\pi$
$62$	0	0
$63$	−3.78112	−0.476376
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.16901	−0.142817	−0.0714084	−	0.997447i	$-0.522749\pi$
$67$	−1.16901	−0.142817	−0.0714084	+	0.997447i	$0.522749\pi$
$68$	0	0
$69$	−1.10851	−0.133449
$70$	0	0
$71$	2.35706	0.279732	0.139866	−	0.990170i	$-0.455333\pi$
$71$	2.35706	0.279732	0.139866	+	0.990170i	$0.455333\pi$
$72$	0	0
$73$	1.60845	0.188255	0.0941275	−	0.995560i	$-0.469994\pi$
$73$	1.60845	0.188255	0.0941275	+	0.995560i	$0.469994\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.55452	0.632996
$78$	0	0
$79$	−8.13412	−0.915160	−0.457580	−	0.889168i	$-0.651284\pi$
$79$	−8.13412	−0.915160	−0.457580	+	0.889168i	$0.651284\pi$
$80$	0	0
$81$	8.72353	0.969281
$82$	0	0
$83$	7.42040	0.814494	0.407247	−	0.913318i	$-0.366489\pi$
$83$	7.42040	0.814494	0.407247	+	0.913318i	$0.366489\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.821412	0.0880646
$88$	0	0
$89$	1.70820	0.181069	0.0905346	−	0.995893i	$-0.471142\pi$
$89$	1.70820	0.181069	0.0905346	+	0.995893i	$0.471142\pi$
$90$	0	0
$91$	−5.45559	−0.571901
$92$	0	0
$93$	1.41164	0.146380
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.60741	−0.670881	−0.335441	−	0.942061i	$-0.608885\pi$
$97$	−6.60741	−0.670881	−0.335441	+	0.942061i	$0.608885\pi$
$98$	0	0
$99$	−12.9508	−1.30161
$100$	0	0
$101$	−7.02311	−0.698825	−0.349413	−	0.936969i	$-0.613619\pi$
$101$	−7.02311	−0.698825	−0.349413	+	0.936969i	$0.613619\pi$
$102$	0	0
$103$	12.7466	1.25596	0.627982	−	0.778228i	$-0.283881\pi$
$103$	12.7466	1.25596	0.627982	+	0.778228i	$0.283881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.2022	1.17963	0.589817	−	0.807537i	$-0.299200\pi$
$107$	12.2022	1.17963	0.589817	+	0.807537i	$0.299200\pi$
$108$	0	0
$109$	−5.05092	−0.483790	−0.241895	−	0.970302i	$-0.577769\pi$
$109$	−5.05092	−0.483790	−0.241895	+	0.970302i	$0.577769\pi$
$110$	0	0
$111$	0.721333	0.0684659
$112$	0	0
$113$	−14.0351	−1.32031	−0.660155	−	0.751130i	$-0.729509\pi$
$113$	−14.0351	−1.32031	−0.660155	+	0.751130i	$0.729509\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.7202	1.17598
$118$	0	0
$119$	−0.124655	−0.0114271
$120$	0	0
$121$	8.02497	0.729543
$122$	0	0
$123$	−2.00707	−0.180972
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.31896	−0.117039	−0.0585194	−	0.998286i	$-0.518638\pi$
$127$	−1.31896	−0.117039	−0.0585194	+	0.998286i	$0.518638\pi$
$128$	0	0
$129$	−1.79245	−0.157817
$130$	0	0
$131$	1.78048	0.155561	0.0777804	−	0.996971i	$-0.475217\pi$
$131$	1.78048	0.155561	0.0777804	+	0.996971i	$0.475217\pi$
$132$	0	0
$133$	0.425277	0.0368762
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.56108	−0.816859	−0.408429	−	0.912790i	$-0.633923\pi$
$137$	−9.56108	−0.816859	−0.408429	+	0.912790i	$0.633923\pi$
$138$	0	0
$139$	4.59970	0.390141	0.195071	−	0.980789i	$-0.437506\pi$
$139$	4.59970	0.390141	0.195071	+	0.980789i	$0.437506\pi$
$140$	0	0
$141$	2.05509	0.173069
$142$	0	0
$143$	−18.6861	−1.56261
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.944272	0.0778822
$148$	0	0
$149$	2.03186	0.166457	0.0832284	−	0.996530i	$-0.473477\pi$
$149$	2.03186	0.166457	0.0832284	+	0.996530i	$0.473477\pi$
$150$	0	0
$151$	−18.9655	−1.54339	−0.771696	−	0.635992i	$-0.780591\pi$
$151$	−18.9655	−1.54339	−0.771696	+	0.635992i	$0.780591\pi$
$152$	0	0
$153$	0.290644	0.0234971
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.2430	−1.13672	−0.568359	−	0.822781i	$-0.692421\pi$
$157$	−14.2430	−1.13672	−0.568359	+	0.822781i	$0.692421\pi$
$158$	0	0
$159$	0.155367	0.0123214
$160$	0	0
$161$	8.04029	0.633664
$162$	0	0
$163$	−19.7147	−1.54417	−0.772086	−	0.635519i	$-0.780786\pi$
$163$	−19.7147	−1.54417	−0.772086	+	0.635519i	$0.780786\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.78183	0.447411	0.223706	−	0.974657i	$-0.428185\pi$
$167$	5.78183	0.447411	0.223706	+	0.974657i	$0.428185\pi$
$168$	0	0
$169$	5.35333	0.411795
$170$	0	0
$171$	−0.991571	−0.0758273
$172$	0	0
$173$	10.5278	0.800410	0.400205	−	0.916426i	$-0.368939\pi$
$173$	10.5278	0.800410	0.400205	+	0.916426i	$0.368939\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.33582	0.100406
$178$	0	0
$179$	−10.8589	−0.811630	−0.405815	−	0.913955i	$-0.633012\pi$
$179$	−10.8589	−0.811630	−0.405815	+	0.913955i	$0.633012\pi$
$180$	0	0
$181$	−10.2858	−0.764535	−0.382267	−	0.924052i	$-0.624857\pi$
$181$	−10.2858	−0.764535	−0.382267	+	0.924052i	$0.624857\pi$
$182$	0	0
$183$	0.294373	0.0217607
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.426960	−0.0312224
$188$	0	0
$189$	1.33460	0.0970777
$190$	0	0
$191$	9.64063	0.697572	0.348786	−	0.937202i	$-0.386594\pi$
$191$	9.64063	0.697572	0.348786	+	0.937202i	$0.386594\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7485	1.47827	0.739135	−	0.673558i	$-0.235235\pi$
$197$	20.7485	1.47827	0.739135	+	0.673558i	$0.235235\pi$
$198$	0	0
$199$	19.0629	1.35133	0.675666	−	0.737208i	$-0.263856\pi$
$199$	19.0629	1.35133	0.675666	+	0.737208i	$0.263856\pi$
$200$	0	0
$201$	0.205243	0.0144767
$202$	0	0
$203$	−5.95791	−0.418163
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.7466	−1.30298
$208$	0	0
$209$	1.45663	0.100757
$210$	0	0
$211$	−17.0196	−1.17168	−0.585838	−	0.810428i	$-0.699235\pi$
$211$	−17.0196	−1.17168	−0.585838	+	0.810428i	$0.699235\pi$
$212$	0	0
$213$	−0.413831	−0.0283552
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.2390	−0.695067
$218$	0	0
$219$	−0.282397	−0.0190826
$220$	0	0
$221$	0.419356	0.0282089
$222$	0	0
$223$	4.97793	0.333347	0.166673	−	0.986012i	$-0.446697\pi$
$223$	4.97793	0.333347	0.166673	+	0.986012i	$0.446697\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.02677	−0.0681490	−0.0340745	−	0.999419i	$-0.510848\pi$
$227$	−1.02677	−0.0681490	−0.0340745	+	0.999419i	$0.510848\pi$
$228$	0	0
$229$	−4.08507	−0.269949	−0.134975	−	0.990849i	$-0.543095\pi$
$229$	−4.08507	−0.269949	−0.134975	+	0.990849i	$0.543095\pi$
$230$	0	0
$231$	−0.975210	−0.0641641
$232$	0	0
$233$	4.41279	0.289092	0.144546	−	0.989498i	$-0.453828\pi$
$233$	4.41279	0.289092	0.144546	+	0.989498i	$0.453828\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.42811	0.0927659
$238$	0	0
$239$	−21.7522	−1.40703	−0.703515	−	0.710680i	$-0.748387\pi$
$239$	−21.7522	−1.40703	−0.703515	+	0.710680i	$0.748387\pi$
$240$	0	0
$241$	−15.8622	−1.02178	−0.510888	−	0.859648i	$-0.670683\pi$
$241$	−15.8622	−1.02178	−0.510888	+	0.859648i	$0.670683\pi$
$242$	0	0
$243$	−4.67563	−0.299942
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.43069	−0.0910326
$248$	0	0
$249$	−1.30280	−0.0825618
$250$	0	0
$251$	8.49087	0.535939	0.267970	−	0.963427i	$-0.413647\pi$
$251$	8.49087	0.535939	0.267970	+	0.963427i	$0.413647\pi$
$252$	0	0
$253$	27.5391	1.73137
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.98572	0.622892	0.311446	−	0.950264i	$-0.399187\pi$
$257$	9.98572	0.622892	0.311446	+	0.950264i	$0.399187\pi$
$258$	0	0
$259$	−5.23201	−0.325101
$260$	0	0
$261$	13.8914	0.859855
$262$	0	0
$263$	−7.73433	−0.476919	−0.238460	−	0.971152i	$-0.576642\pi$
$263$	−7.73433	−0.476919	−0.238460	+	0.971152i	$0.576642\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.299910	−0.0183542
$268$	0	0
$269$	−7.33865	−0.447446	−0.223723	−	0.974653i	$-0.571821\pi$
$269$	−7.33865	−0.447446	−0.223723	+	0.974653i	$0.571821\pi$
$270$	0	0
$271$	−13.2990	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$271$	−13.2990	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$272$	0	0
$273$	0.957841	0.0579712
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.5930	−1.17723	−0.588615	−	0.808413i	$-0.700327\pi$
$277$	−19.5930	−1.17723	−0.588615	+	0.808413i	$0.700327\pi$
$278$	0	0
$279$	23.8730	1.42924
$280$	0	0
$281$	−11.5065	−0.686421	−0.343210	−	0.939259i	$-0.611514\pi$
$281$	−11.5065	−0.686421	−0.343210	+	0.939259i	$0.611514\pi$
$282$	0	0
$283$	−19.6547	−1.16835	−0.584174	−	0.811628i	$-0.698582\pi$
$283$	−19.6547	−1.16835	−0.584174	+	0.811628i	$0.698582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.5578	0.859321
$288$	0	0
$289$	−16.9904	−0.999436
$290$	0	0
$291$	1.16007	0.0680043
$292$	0	0
$293$	−6.12203	−0.357653	−0.178827	−	0.983881i	$-0.557230\pi$
$293$	−6.12203	−0.357653	−0.178827	+	0.983881i	$0.557230\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.57118	0.265247
$298$	0	0
$299$	−27.0486	−1.56426
$300$	0	0
$301$	13.0011	0.749371
$302$	0	0
$303$	1.23305	0.0708369
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.1935	0.810064	0.405032	−	0.914302i	$-0.367260\pi$
$307$	14.1935	0.810064	0.405032	+	0.914302i	$0.367260\pi$
$308$	0	0
$309$	−2.23793	−0.127312
$310$	0	0
$311$	−7.18619	−0.407492	−0.203746	−	0.979024i	$-0.565312\pi$
$311$	−7.18619	−0.407492	−0.203746	+	0.979024i	$0.565312\pi$
$312$	0	0
$313$	−0.625638	−0.0353632	−0.0176816	−	0.999844i	$-0.505629\pi$
$313$	−0.625638	−0.0353632	−0.0176816	+	0.999844i	$0.505629\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.21006	0.404957	0.202479	−	0.979287i	$-0.435100\pi$
$317$	7.21006	0.404957	0.202479	+	0.979287i	$0.435100\pi$
$318$	0	0
$319$	−20.4066	−1.14255
$320$	0	0
$321$	−2.14235	−0.119574
$322$	0	0
$323$	−0.0326898	−0.00181891
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.886792	0.0490397
$328$	0	0
$329$	−14.9061	−0.821798
$330$	0	0
$331$	24.2144	1.33094	0.665471	−	0.746423i	$-0.268231\pi$
$331$	24.2144	1.33094	0.665471	+	0.746423i	$0.268231\pi$
$332$	0	0
$333$	12.1989	0.668495
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.2016	1.20940	0.604699	−	0.796454i	$-0.293294\pi$
$337$	22.2016	1.20940	0.604699	+	0.796454i	$0.293294\pi$
$338$	0	0
$339$	2.46415	0.133834
$340$	0	0
$341$	−35.0699	−1.89914
$342$	0	0
$343$	−15.7632	−0.851135
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.1583	1.35057	0.675285	−	0.737557i	$-0.264021\pi$
$347$	25.1583	1.35057	0.675285	+	0.737557i	$0.264021\pi$
$348$	0	0
$349$	−25.4708	−1.36342	−0.681710	−	0.731622i	$-0.738764\pi$
$349$	−25.4708	−1.36342	−0.681710	+	0.731622i	$0.738764\pi$
$350$	0	0
$351$	−4.48976	−0.239646
$352$	0	0
$353$	28.9214	1.53933	0.769665	−	0.638448i	$-0.220423\pi$
$353$	28.9214	1.53933	0.769665	+	0.638448i	$0.220423\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.0218857	0.00115832
$358$	0	0
$359$	8.43977	0.445434	0.222717	−	0.974883i	$-0.428507\pi$
$359$	8.43977	0.445434	0.222717	+	0.974883i	$0.428507\pi$
$360$	0	0
$361$	−18.8885	−0.994130
$362$	0	0
$363$	−1.40895	−0.0739506
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.99710	0.156447	0.0782236	−	0.996936i	$-0.475075\pi$
$367$	2.99710	0.156447	0.0782236	+	0.996936i	$0.475075\pi$
$368$	0	0
$369$	−33.9428	−1.76699
$370$	0	0
$371$	−1.12692	−0.0585066
$372$	0	0
$373$	17.5370	0.908032	0.454016	−	0.890994i	$-0.349991\pi$
$373$	17.5370	0.908032	0.454016	+	0.890994i	$0.349991\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.0432	1.03228
$378$	0	0
$379$	−31.7934	−1.63312	−0.816558	−	0.577263i	$-0.804121\pi$
$379$	−31.7934	−1.63312	−0.816558	+	0.577263i	$0.804121\pi$
$380$	0	0
$381$	0.231571	0.0118637
$382$	0	0
$383$	−19.6366	−1.00338	−0.501691	−	0.865047i	$-0.667289\pi$
$383$	−19.6366	−1.00338	−0.501691	+	0.865047i	$0.667289\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−30.3132	−1.54091
$388$	0	0
$389$	−29.5333	−1.49740	−0.748700	−	0.662909i	$-0.769322\pi$
$389$	−29.5333	−1.49740	−0.748700	+	0.662909i	$0.769322\pi$
$390$	0	0
$391$	−0.618034	−0.0312553
$392$	0	0
$393$	−0.312599	−0.0157685
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.2724	−1.11782	−0.558909	−	0.829229i	$-0.688780\pi$
$397$	−22.2724	−1.11782	−0.558909	+	0.829229i	$0.688780\pi$
$398$	0	0
$399$	−0.0746662	−0.00373798
$400$	0	0
$401$	7.37458	0.368269	0.184134	−	0.982901i	$-0.441052\pi$
$401$	7.37458	0.368269	0.184134	+	0.982901i	$0.441052\pi$
$402$	0	0
$403$	34.4453	1.71584
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.9203	−0.888278
$408$	0	0
$409$	−18.4318	−0.911396	−0.455698	−	0.890134i	$-0.650610\pi$
$409$	−18.4318	−0.911396	−0.455698	+	0.890134i	$0.650610\pi$
$410$	0	0
$411$	1.67864	0.0828014
$412$	0	0
$413$	−9.68904	−0.476767
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.807571	−0.0395469
$418$	0	0
$419$	−11.5953	−0.566469	−0.283234	−	0.959051i	$-0.591407\pi$
$419$	−11.5953	−0.566469	−0.283234	+	0.959051i	$0.591407\pi$
$420$	0	0
$421$	−2.91897	−0.142262	−0.0711310	−	0.997467i	$-0.522661\pi$
$421$	−2.91897	−0.142262	−0.0711310	+	0.997467i	$0.522661\pi$
$422$	0	0
$423$	34.7547	1.68983
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.13516	−0.103328
$428$	0	0
$429$	3.28073	0.158395
$430$	0	0
$431$	−7.57537	−0.364893	−0.182446	−	0.983216i	$-0.558402\pi$
$431$	−7.57537	−0.364893	−0.182446	+	0.983216i	$0.558402\pi$
$432$	0	0
$433$	−28.7822	−1.38318	−0.691591	−	0.722289i	$-0.743090\pi$
$433$	−28.7822	−1.38318	−0.691591	+	0.722289i	$0.743090\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.10851	0.100864
$438$	0	0
$439$	−17.7485	−0.847090	−0.423545	−	0.905875i	$-0.639214\pi$
$439$	−17.7485	−0.847090	−0.423545	+	0.905875i	$0.639214\pi$
$440$	0	0
$441$	15.9691	0.760435
$442$	0	0
$443$	19.3439	0.919058	0.459529	−	0.888163i	$-0.348018\pi$
$443$	19.3439	0.919058	0.459529	+	0.888163i	$0.348018\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.356736	−0.0168730
$448$	0	0
$449$	11.2932	0.532960	0.266480	−	0.963840i	$-0.414139\pi$
$449$	11.2932	0.532960	0.266480	+	0.963840i	$0.414139\pi$
$450$	0	0
$451$	49.8625	2.34793
$452$	0	0
$453$	3.32979	0.156447
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.60741	0.121970	0.0609848	−	0.998139i	$-0.480576\pi$
$457$	2.60741	0.121970	0.0609848	+	0.998139i	$0.480576\pi$
$458$	0	0
$459$	−0.102587	−0.00478833
$460$	0	0
$461$	−1.95751	−0.0911705	−0.0455853	−	0.998960i	$-0.514515\pi$
$461$	−1.95751	−0.0911705	−0.0455853	+	0.998960i	$0.514515\pi$
$462$	0	0
$463$	1.45797	0.0677575	0.0338787	−	0.999426i	$-0.489214\pi$
$463$	1.45797	0.0677575	0.0338787	+	0.999426i	$0.489214\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.5853	−1.18395	−0.591973	−	0.805958i	$-0.701651\pi$
$467$	−25.5853	−1.18395	−0.591973	+	0.805958i	$0.701651\pi$
$468$	0	0
$469$	−1.48868	−0.0687408
$470$	0	0
$471$	2.50066	0.115224
$472$	0	0
$473$	44.5305	2.04752
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.62750	0.120305
$478$	0	0
$479$	6.86723	0.313772	0.156886	−	0.987617i	$-0.449855\pi$
$479$	6.86723	0.313772	0.156886	+	0.987617i	$0.449855\pi$
$480$	0	0
$481$	17.6012	0.802545
$482$	0	0
$483$	−1.41164	−0.0642318
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.3233	1.91785	0.958926	−	0.283657i	$-0.0915479\pi$
$487$	42.3233	1.91785	0.958926	+	0.283657i	$0.0915479\pi$
$488$	0	0
$489$	3.46131	0.156526
$490$	0	0
$491$	−12.9499	−0.584423	−0.292211	−	0.956354i	$-0.594391\pi$
$491$	−12.9499	−0.584423	−0.292211	+	0.956354i	$0.594391\pi$
$492$	0	0
$493$	0.457967	0.0206258
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.00162	0.134641
$498$	0	0
$499$	12.9767	0.580915	0.290458	−	0.956888i	$-0.406192\pi$
$499$	12.9767	0.580915	0.290458	+	0.956888i	$0.406192\pi$
$500$	0	0
$501$	−1.01512	−0.0453522
$502$	0	0
$503$	21.2420	0.947134	0.473567	−	0.880758i	$-0.342966\pi$
$503$	21.2420	0.947134	0.473567	+	0.880758i	$0.342966\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.939887	−0.0417419
$508$	0	0
$509$	14.2523	0.631721	0.315861	−	0.948806i	$-0.397707\pi$
$509$	14.2523	0.631721	0.315861	+	0.948806i	$0.397707\pi$
$510$	0	0
$511$	2.04830	0.0906113
$512$	0	0
$513$	0.349988	0.0154524
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−51.0552	−2.24541
$518$	0	0
$519$	−1.84836	−0.0811341
$520$	0	0
$521$	−29.3687	−1.28667	−0.643334	−	0.765586i	$-0.722449\pi$
$521$	−29.3687	−1.28667	−0.643334	+	0.765586i	$0.722449\pi$
$522$	0	0
$523$	15.2398	0.666390	0.333195	−	0.942858i	$-0.391873\pi$
$523$	15.2398	0.666390	0.333195	+	0.942858i	$0.391873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.787040	0.0342840
$528$	0	0
$529$	16.8635	0.733194
$530$	0	0
$531$	22.5908	0.980358
$532$	0	0
$533$	−48.9744	−2.12132
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.90650	0.0822714
$538$	0	0
$539$	−23.4589	−1.01045
$540$	0	0
$541$	30.3322	1.30408	0.652041	−	0.758184i	$-0.273913\pi$
$541$	30.3322	1.30408	0.652041	+	0.758184i	$0.273913\pi$
$542$	0	0
$543$	1.80588	0.0774976
$544$	0	0
$545$	0	0
$546$	0	0
$547$	31.0484	1.32753	0.663767	−	0.747939i	$-0.268956\pi$
$547$	31.0484	1.32753	0.663767	+	0.747939i	$0.268956\pi$
$548$	0	0
$549$	4.97832	0.212469
$550$	0	0
$551$	−1.56242	−0.0665613
$552$	0	0
$553$	−10.3585	−0.440487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.0537	1.18867	0.594337	−	0.804216i	$-0.297415\pi$
$557$	28.0537	1.18867	0.594337	+	0.804216i	$0.297415\pi$
$558$	0	0
$559$	−43.7374	−1.84990
$560$	0	0
$561$	0.0749615	0.00316488
$562$	0	0
$563$	−23.1619	−0.976159	−0.488080	−	0.872799i	$-0.662302\pi$
$563$	−23.1619	−0.976159	−0.488080	+	0.872799i	$0.662302\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.1090	0.466536
$568$	0	0
$569$	−13.4815	−0.565175	−0.282588	−	0.959241i	$-0.591193\pi$
$569$	−13.4815	−0.565175	−0.282588	+	0.959241i	$0.591193\pi$
$570$	0	0
$571$	38.9587	1.63037	0.815186	−	0.579199i	$-0.196635\pi$
$571$	38.9587	1.63037	0.815186	+	0.579199i	$0.196635\pi$
$572$	0	0
$573$	−1.69261	−0.0707098
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−44.4264	−1.84950	−0.924748	−	0.380581i	$-0.875724\pi$
$577$	−44.4264	−1.84950	−0.924748	+	0.380581i	$0.875724\pi$
$578$	0	0
$579$	1.05342	0.0437788
$580$	0	0
$581$	9.44956	0.392034
$582$	0	0
$583$	−3.85984	−0.159858
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.2467	−1.61988	−0.809942	−	0.586510i	$-0.800501\pi$
$587$	−39.2467	−1.61988	−0.809942	+	0.586510i	$0.800501\pi$
$588$	0	0
$589$	−2.68510	−0.110637
$590$	0	0
$591$	−3.64282	−0.149846
$592$	0	0
$593$	−23.5637	−0.967644	−0.483822	−	0.875167i	$-0.660752\pi$
$593$	−23.5637	−0.967644	−0.483822	+	0.875167i	$0.660752\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.34688	−0.136979
$598$	0	0
$599$	−39.2513	−1.60376	−0.801882	−	0.597483i	$-0.796168\pi$
$599$	−39.2513	−1.60376	−0.801882	+	0.597483i	$0.796168\pi$
$600$	0	0
$601$	−13.1067	−0.534635	−0.267317	−	0.963609i	$-0.586137\pi$
$601$	−13.1067	−0.534635	−0.267317	+	0.963609i	$0.586137\pi$
$602$	0	0
$603$	3.47098	0.141349
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.14712	0.249504	0.124752	−	0.992188i	$-0.460186\pi$
$607$	6.14712	0.249504	0.124752	+	0.992188i	$0.460186\pi$
$608$	0	0
$609$	1.04603	0.0423874
$610$	0	0
$611$	50.1459	2.02869
$612$	0	0
$613$	−22.1506	−0.894654	−0.447327	−	0.894370i	$-0.647624\pi$
$613$	−22.1506	−0.894654	−0.447327	+	0.894370i	$0.647624\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.1247	0.488123	0.244062	−	0.969760i	$-0.421520\pi$
$617$	12.1247	0.488123	0.244062	+	0.969760i	$0.421520\pi$
$618$	0	0
$619$	−13.1696	−0.529331	−0.264666	−	0.964340i	$-0.585262\pi$
$619$	−13.1696	−0.529331	−0.264666	+	0.964340i	$0.585262\pi$
$620$	0	0
$621$	6.61688	0.265526
$622$	0	0
$623$	2.17533	0.0871526
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.255742	−0.0102133
$628$	0	0
$629$	0.402169	0.0160355
$630$	0	0
$631$	−32.3665	−1.28849	−0.644245	−	0.764819i	$-0.722828\pi$
$631$	−32.3665	−1.28849	−0.644245	+	0.764819i	$0.722828\pi$
$632$	0	0
$633$	2.98814	0.118768
$634$	0	0
$635$	0	0
$636$	0	0
$637$	23.0411	0.912921
$638$	0	0
$639$	−6.99853	−0.276858
$640$	0	0
$641$	−14.2278	−0.561966	−0.280983	−	0.959713i	$-0.590660\pi$
$641$	−14.2278	−0.561966	−0.280983	+	0.959713i	$0.590660\pi$
$642$	0	0
$643$	−18.7825	−0.740708	−0.370354	−	0.928891i	$-0.620764\pi$
$643$	−18.7825	−0.740708	−0.370354	+	0.928891i	$0.620764\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.0396	−0.591267	−0.295633	−	0.955301i	$-0.595531\pi$
$647$	−15.0396	−0.591267	−0.295633	+	0.955301i	$0.595531\pi$
$648$	0	0
$649$	−33.1863	−1.30267
$650$	0	0
$651$	1.79766	0.0704559
$652$	0	0
$653$	−23.7430	−0.929134	−0.464567	−	0.885538i	$-0.653790\pi$
$653$	−23.7430	−0.929134	−0.464567	+	0.885538i	$0.653790\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.77578	−0.186321
$658$	0	0
$659$	6.56255	0.255641	0.127820	−	0.991797i	$-0.459202\pi$
$659$	6.56255	0.255641	0.127820	+	0.991797i	$0.459202\pi$
$660$	0	0
$661$	−32.0324	−1.24592	−0.622958	−	0.782255i	$-0.714069\pi$
$661$	−32.0324	−1.24592	−0.622958	+	0.782255i	$0.714069\pi$
$662$	0	0
$663$	−0.0736265	−0.00285941
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−29.5391	−1.14376
$668$	0	0
$669$	−0.873978	−0.0337899
$670$	0	0
$671$	−7.31322	−0.282324
$672$	0	0
$673$	−49.7555	−1.91793	−0.958966	−	0.283520i	$-0.908498\pi$
$673$	−49.7555	−1.91793	−0.958966	+	0.283520i	$0.908498\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6822	1.10235	0.551174	−	0.834391i	$-0.314180\pi$
$677$	28.6822	1.10235	0.551174	+	0.834391i	$0.314180\pi$
$678$	0	0
$679$	−8.41426	−0.322910
$680$	0	0
$681$	0.180270	0.00690797
$682$	0	0
$683$	35.0788	1.34225	0.671126	−	0.741343i	$-0.265811\pi$
$683$	35.0788	1.34225	0.671126	+	0.741343i	$0.265811\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.717218	0.0273636
$688$	0	0
$689$	3.79110	0.144429
$690$	0	0
$691$	1.70164	0.0647334	0.0323667	−	0.999476i	$-0.489696\pi$
$691$	1.70164	0.0647334	0.0323667	+	0.999476i	$0.489696\pi$
$692$	0	0
$693$	−16.4923	−0.626492
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.11902	−0.0423858
$698$	0	0
$699$	−0.774756	−0.0293040
$700$	0	0
$701$	−5.26104	−0.198707	−0.0993534	−	0.995052i	$-0.531677\pi$
$701$	−5.26104	−0.198707	−0.0993534	+	0.995052i	$0.531677\pi$
$702$	0	0
$703$	−1.37206	−0.0517481
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.94363	−0.336360
$708$	0	0
$709$	−43.3189	−1.62688	−0.813438	−	0.581651i	$-0.802407\pi$
$709$	−43.3189	−1.62688	−0.813438	+	0.581651i	$0.802407\pi$
$710$	0	0
$711$	24.1516	0.905757
$712$	0	0
$713$	−50.7644	−1.90114
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.81904	0.142625
$718$	0	0
$719$	−24.6312	−0.918587	−0.459294	−	0.888284i	$-0.651897\pi$
$719$	−24.6312	−0.918587	−0.459294	+	0.888284i	$0.651897\pi$
$720$	0	0
$721$	16.2323	0.604522
$722$	0	0
$723$	2.78494	0.103573
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.0591	−0.521424	−0.260712	−	0.965417i	$-0.583957\pi$
$727$	−14.0591	−0.521424	−0.260712	+	0.965417i	$0.583957\pi$
$728$	0	0
$729$	−25.3497	−0.938877
$730$	0	0
$731$	−0.999357	−0.0369626
$732$	0	0
$733$	28.2337	1.04283	0.521417	−	0.853302i	$-0.325404\pi$
$733$	28.2337	1.04283	0.521417	+	0.853302i	$0.325404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.09893	−0.187821
$738$	0	0
$739$	34.7138	1.27697	0.638483	−	0.769636i	$-0.279562\pi$
$739$	34.7138	1.27697	0.638483	+	0.769636i	$0.279562\pi$
$740$	0	0
$741$	0.251187	0.00922758
$742$	0	0
$743$	−35.3024	−1.29512	−0.647559	−	0.762015i	$-0.724210\pi$
$743$	−35.3024	−1.29512	−0.647559	+	0.762015i	$0.724210\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.0325	−0.806125
$748$	0	0
$749$	15.5390	0.567783
$750$	0	0
$751$	−14.5513	−0.530983	−0.265491	−	0.964113i	$-0.585534\pi$
$751$	−14.5513	−0.530983	−0.265491	+	0.964113i	$0.585534\pi$
$752$	0	0
$753$	−1.49075	−0.0543258
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.9496	0.870462	0.435231	−	0.900319i	$-0.356667\pi$
$757$	23.9496	0.870462	0.435231	+	0.900319i	$0.356667\pi$
$758$	0	0
$759$	−4.83505	−0.175501
$760$	0	0
$761$	−21.6298	−0.784078	−0.392039	−	0.919949i	$-0.628230\pi$
$761$	−21.6298	−0.784078	−0.392039	+	0.919949i	$0.628230\pi$
$762$	0	0
$763$	−6.43213	−0.232859
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.5952	1.17694
$768$	0	0
$769$	44.3913	1.60079	0.800396	−	0.599472i	$-0.204623\pi$
$769$	44.3913	1.60079	0.800396	+	0.599472i	$0.204623\pi$
$770$	0	0
$771$	−1.75320	−0.0631399
$772$	0	0
$773$	−11.5903	−0.416874	−0.208437	−	0.978036i	$-0.566838\pi$
$773$	−11.5903	−0.416874	−0.208437	+	0.978036i	$0.566838\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.918587	0.0329541
$778$	0	0
$779$	3.81768	0.136783
$780$	0	0
$781$	10.2809	0.367881
$782$	0	0
$783$	−4.90315	−0.175224
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.75211	0.276333	0.138167	−	0.990409i	$-0.455879\pi$
$787$	7.75211	0.276333	0.138167	+	0.990409i	$0.455879\pi$
$788$	0	0
$789$	1.35792	0.0483432
$790$	0	0
$791$	−17.8731	−0.635494
$792$	0	0
$793$	7.18297	0.255075
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.4011	−1.43108	−0.715540	−	0.698571i	$-0.753819\pi$
$797$	−40.4011	−1.43108	−0.715540	+	0.698571i	$0.753819\pi$
$798$	0	0
$799$	1.14579	0.0405350
$800$	0	0
$801$	−5.07196	−0.179209
$802$	0	0
$803$	7.01569	0.247578
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.28845	0.0453556
$808$	0	0
$809$	9.88493	0.347536	0.173768	−	0.984787i	$-0.444406\pi$
$809$	9.88493	0.347536	0.173768	+	0.984787i	$0.444406\pi$
$810$	0	0
$811$	−5.64384	−0.198182	−0.0990911	−	0.995078i	$-0.531594\pi$
$811$	−5.64384	−0.198182	−0.0990911	+	0.995078i	$0.531594\pi$
$812$	0	0
$813$	2.33491	0.0818887
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.40945	0.119281
$818$	0	0
$819$	16.1986	0.566025
$820$	0	0
$821$	−8.32957	−0.290704	−0.145352	−	0.989380i	$-0.546431\pi$
$821$	−8.32957	−0.290704	−0.145352	+	0.989380i	$0.546431\pi$
$822$	0	0
$823$	−9.00735	−0.313976	−0.156988	−	0.987600i	$-0.550178\pi$
$823$	−9.00735	−0.313976	−0.156988	+	0.987600i	$0.550178\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.1702	1.64027	0.820135	−	0.572171i	$-0.193899\pi$
$827$	47.1702	1.64027	0.820135	+	0.572171i	$0.193899\pi$
$828$	0	0
$829$	−26.5456	−0.921968	−0.460984	−	0.887408i	$-0.652503\pi$
$829$	−26.5456	−0.921968	−0.460984	+	0.887408i	$0.652503\pi$
$830$	0	0
$831$	3.43996	0.119331
$832$	0	0
$833$	0.526466	0.0182410
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.42632	−0.291256
$838$	0	0
$839$	−0.293333	−0.0101270	−0.00506350	−	0.999987i	$-0.501612\pi$
$839$	−0.293333	−0.0101270	−0.00506350	+	0.999987i	$0.501612\pi$
$840$	0	0
$841$	−7.11134	−0.245219
$842$	0	0
$843$	2.02020	0.0695795
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.2195	0.351145
$848$	0	0
$849$	3.45078	0.118430
$850$	0	0
$851$	−25.9401	−0.889215
$852$	0	0
$853$	50.4369	1.72693	0.863463	−	0.504412i	$-0.168291\pi$
$853$	50.4369	1.72693	0.863463	+	0.504412i	$0.168291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.6554	−1.45708	−0.728540	−	0.685003i	$-0.759801\pi$
$857$	−42.6554	−1.45708	−0.728540	+	0.685003i	$0.759801\pi$
$858$	0	0
$859$	29.7833	1.01619	0.508097	−	0.861300i	$-0.330349\pi$
$859$	29.7833	1.01619	0.508097	+	0.861300i	$0.330349\pi$
$860$	0	0
$861$	−2.55592	−0.0871057
$862$	0	0
$863$	27.4151	0.933222	0.466611	−	0.884463i	$-0.345475\pi$
$863$	27.4151	0.933222	0.466611	+	0.884463i	$0.345475\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.98302	0.101309
$868$	0	0
$869$	−35.4791	−1.20355
$870$	0	0
$871$	5.00811	0.169693
$872$	0	0
$873$	19.6186	0.663988
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.33273	0.112538	0.0562692	−	0.998416i	$-0.482079\pi$
$877$	3.33273	0.112538	0.0562692	+	0.998416i	$0.482079\pi$
$878$	0	0
$879$	1.07485	0.0362537
$880$	0	0
$881$	−7.77695	−0.262012	−0.131006	−	0.991382i	$-0.541821\pi$
$881$	−7.77695	−0.262012	−0.131006	+	0.991382i	$0.541821\pi$
$882$	0	0
$883$	−5.63367	−0.189588	−0.0947941	−	0.995497i	$-0.530219\pi$
$883$	−5.63367	−0.189588	−0.0947941	+	0.995497i	$0.530219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.0411	−0.740068	−0.370034	−	0.929018i	$-0.620654\pi$
$887$	−22.0411	−0.740068	−0.370034	+	0.929018i	$0.620654\pi$
$888$	0	0
$889$	−1.67964	−0.0563333
$890$	0	0
$891$	38.0499	1.27472
$892$	0	0
$893$	−3.90900	−0.130810
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.74894	0.158562
$898$	0	0
$899$	37.6168	1.25459
$900$	0	0
$901$	0.0866228	0.00288582
$902$	0	0
$903$	−2.28261	−0.0759605
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−10.7033	−0.355398	−0.177699	−	0.984085i	$-0.556865\pi$
$907$	−10.7033	−0.355398	−0.177699	+	0.984085i	$0.556865\pi$
$908$	0	0
$909$	20.8528	0.691645
$910$	0	0
$911$	−54.6655	−1.81115	−0.905574	−	0.424188i	$-0.860560\pi$
$911$	−54.6655	−1.81115	−0.905574	+	0.424188i	$0.860560\pi$
$912$	0	0
$913$	32.3660	1.07116
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.26736	0.0748748
$918$	0	0
$919$	45.2037	1.49113	0.745566	−	0.666432i	$-0.232179\pi$
$919$	45.2037	1.49113	0.745566	+	0.666432i	$0.232179\pi$
$920$	0	0
$921$	−2.49195	−0.0821127
$922$	0	0
$923$	−10.0978	−0.332375
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−37.8470	−1.24306
$928$	0	0
$929$	37.0901	1.21689	0.608444	−	0.793597i	$-0.291794\pi$
$929$	37.0901	1.21689	0.608444	+	0.793597i	$0.291794\pi$
$930$	0	0
$931$	−1.79611	−0.0588652
$932$	0	0
$933$	1.26168	0.0413057
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.8727	−1.46593	−0.732964	−	0.680267i	$-0.761864\pi$
$937$	−44.8727	−1.46593	−0.732964	+	0.680267i	$0.761864\pi$
$938$	0	0
$939$	0.109844	0.00358461
$940$	0	0
$941$	46.7392	1.52365	0.761827	−	0.647780i	$-0.224303\pi$
$941$	46.7392	1.52365	0.761827	+	0.647780i	$0.224303\pi$
$942$	0	0
$943$	72.1771	2.35041
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.3671	−1.11678	−0.558391	−	0.829578i	$-0.688581\pi$
$947$	−34.3671	−1.11678	−0.558391	+	0.829578i	$0.688581\pi$
$948$	0	0
$949$	−6.89074	−0.223683
$950$	0	0
$951$	−1.26587	−0.0410487
$952$	0	0
$953$	15.1712	0.491443	0.245721	−	0.969340i	$-0.420975\pi$
$953$	15.1712	0.491443	0.245721	+	0.969340i	$0.420975\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.58280	0.115816
$958$	0	0
$959$	−12.1756	−0.393172
$960$	0	0
$961$	33.6463	1.08537
$962$	0	0
$963$	−36.2305	−1.16751
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53.4293	1.71817	0.859085	−	0.511832i	$-0.171033\pi$
$967$	53.4293	1.71817	0.859085	+	0.511832i	$0.171033\pi$
$968$	0	0
$969$	0.00573937	0.000184375 0
$970$	0	0
$971$	−27.1670	−0.871829	−0.435915	−	0.899988i	$-0.643575\pi$
$971$	−27.1670	−0.871829	−0.435915	+	0.899988i	$0.643575\pi$
$972$	0	0
$973$	5.85752	0.187783
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.3932	−0.364500	−0.182250	−	0.983252i	$-0.558338\pi$
$977$	−11.3932	−0.364500	−0.182250	+	0.983252i	$0.558338\pi$
$978$	0	0
$979$	7.45078	0.238128
$980$	0	0
$981$	14.9971	0.478819
$982$	0	0
$983$	9.10021	0.290252	0.145126	−	0.989413i	$-0.453641\pi$
$983$	9.10021	0.290252	0.145126	+	0.989413i	$0.453641\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.61706	0.0833021
$988$	0	0
$989$	64.4590	2.04968
$990$	0	0
$991$	−5.00007	−0.158832	−0.0794162	−	0.996842i	$-0.525306\pi$
$991$	−5.00007	−0.158832	−0.0794162	+	0.996842i	$0.525306\pi$
$992$	0	0
$993$	−4.25133	−0.134912
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49.3535	1.56304	0.781521	−	0.623879i	$-0.214444\pi$
$997$	49.3535	1.56304	0.781521	+	0.623879i	$0.214444\pi$
$998$	0	0
$999$	−4.30576	−0.136228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bb.1.2		4
4.3	odd	2		1250.2.a.i.1.3		4
5.4	even	2		10000.2.a.o.1.3		4
20.3	even	4		1250.2.b.c.1249.3		8
20.7	even	4		1250.2.b.c.1249.6		8
20.19	odd	2		1250.2.a.h.1.2		4
25.8	odd	20		400.2.y.a.289.2		8
25.22	odd	20		400.2.y.a.209.2		8
100.3	even	20		250.2.e.a.49.1		8
100.19	odd	10		250.2.d.c.51.1		8
100.31	odd	10		250.2.d.b.51.2		8
100.47	even	20		50.2.e.a.9.2	✓	8
100.67	even	20		250.2.e.a.199.1		8
100.71	odd	10		250.2.d.b.201.2		8
100.79	odd	10		250.2.d.c.201.1		8
100.83	even	20		50.2.e.a.39.2	yes	8
300.47	odd	20		450.2.l.b.109.1		8
300.83	odd	20		450.2.l.b.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.e.a.9.2	✓	8	100.47	even	20
50.2.e.a.39.2	yes	8	100.83	even	20
250.2.d.b.51.2		8	100.31	odd	10
250.2.d.b.201.2		8	100.71	odd	10
250.2.d.c.51.1		8	100.19	odd	10
250.2.d.c.201.1		8	100.79	odd	10
250.2.e.a.49.1		8	100.3	even	20
250.2.e.a.199.1		8	100.67	even	20
400.2.y.a.209.2		8	25.22	odd	20
400.2.y.a.289.2		8	25.8	odd	20
450.2.l.b.109.1		8	300.47	odd	20
450.2.l.b.289.1		8	300.83	odd	20
1250.2.a.h.1.2		4	20.19	odd	2
1250.2.a.i.1.3		4	4.3	odd	2
1250.2.b.c.1249.3		8	20.3	even	4
1250.2.b.c.1249.6		8	20.7	even	4
10000.2.a.o.1.3		4	5.4	even	2
10000.2.a.bb.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.175571 q^{3} +1.27346 q^{7} -2.96917 q^{9} +O(q^{10})\)	\(q-0.175571 q^{3} +1.27346 q^{7} -2.96917 q^{9} +4.36176 q^{11} -4.28408 q^{13} -0.0978870 q^{17} +0.333955 q^{19} -0.223582 q^{21} +6.31375 q^{23} +1.04801 q^{27} -4.67853 q^{29} -8.04029 q^{31} -0.765797 q^{33} -4.10851 q^{37} +0.752158 q^{39} +11.4317 q^{41} +10.2093 q^{43} -11.7052 q^{47} -5.37831 q^{49} +0.0171861 q^{51} -0.884927 q^{53} -0.0586326 q^{57} -7.60845 q^{59} -1.67667 q^{61} -3.78112 q^{63} -1.16901 q^{67} -1.10851 q^{69} +2.35706 q^{71} +1.60845 q^{73} +5.55452 q^{77} -8.13412 q^{79} +8.72353 q^{81} +7.42040 q^{83} +0.821412 q^{87} +1.70820 q^{89} -5.45559 q^{91} +1.41164 q^{93} -6.60741 q^{97} -12.9508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9	\( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 + 2 * q^11 - 14 * q^13 - 8 * q^17 - 2 * q^21 + 4 * q^23 + 10 * q^27 - 10 * q^29 - 8 * q^31 - 8 * q^33 - 18 * q^37 - 14 * q^39 - 12 * q^41 + 14 * q^43 + 8 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 2 * q^61 - 16 * q^63 - 2 * q^67 - 6 * q^69 - 8 * q^71 - 24 * q^73 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 14 * q^83 - 20 * q^87 - 20 * q^89 - 18 * q^91 - 8 * q^93 - 28 * q^97 - 14 * q^99

Introduction

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