LMFDB - Embedded newform 10000.2.a.bn.1.6

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:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.68341$ 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)

$f(q)$	$=$	$q+2.11675 q^{3} -0.973070 q^{7} +1.48063 q^{9} +O(q^{10})$	$q+2.11675 q^{3} -0.973070 q^{7} +1.48063 q^{9} +5.38225 q^{11} +1.99670 q^{13} +2.04301 q^{17} -6.20428 q^{19} -2.05975 q^{21} +1.93813 q^{23} -3.21612 q^{27} +4.81812 q^{29} +6.64014 q^{31} +11.3929 q^{33} +0.978913 q^{37} +4.22652 q^{39} +2.73319 q^{41} +3.99413 q^{43} +7.21339 q^{47} -6.05313 q^{49} +4.32454 q^{51} -13.2746 q^{53} -13.1329 q^{57} -6.54024 q^{59} +2.72672 q^{61} -1.44076 q^{63} +9.56957 q^{67} +4.10254 q^{69} +5.68520 q^{71} +9.35281 q^{73} -5.23731 q^{77} +3.18171 q^{79} -11.2496 q^{81} +6.11387 q^{83} +10.1988 q^{87} -3.00743 q^{89} -1.94293 q^{91} +14.0555 q^{93} -5.95526 q^{97} +7.96914 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9	$ 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9} - q^{11} - 10 q^{13} - 15 q^{17} + 10 q^{19} - 14 q^{21} + 30 q^{23} + 20 q^{27} + 10 q^{29} + 9 q^{31} - 5 q^{33} + 10 q^{37} - 8 q^{39} - 4 q^{41} + 30 q^{47} - 4 q^{49} + 14 q^{51} - 10 q^{53} + 10 q^{57} + 5 q^{59} + 6 q^{61} + 10 q^{67} + 3 q^{69} + 9 q^{71} - 5 q^{77} + 20 q^{79} + 8 q^{81} + 40 q^{83} + 40 q^{87} - 5 q^{89} - 6 q^{91} + 40 q^{93} + 22 q^{99}+O(q^{100}) $ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 - q^11 - 10 * q^13 - 15 * q^17 + 10 * q^19 - 14 * q^21 + 30 * q^23 + 20 * q^27 + 10 * q^29 + 9 * q^31 - 5 * q^33 + 10 * q^37 - 8 * q^39 - 4 * q^41 + 30 * q^47 - 4 * q^49 + 14 * q^51 - 10 * q^53 + 10 * q^57 + 5 * q^59 + 6 * q^61 + 10 * q^67 + 3 * q^69 + 9 * q^71 - 5 * q^77 + 20 * q^79 + 8 * q^81 + 40 * q^83 + 40 * q^87 - 5 * q^89 - 6 * q^91 + 40 * q^93 + 22 * 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
$3$	2.11675	1.22211	0.611053	−	0.791589i	$-0.290746\pi$
$3$	2.11675	1.22211	0.611053	+	0.791589i	$0.290746\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.973070	−0.367786	−0.183893	−	0.982946i	$-0.558870\pi$
$7$	−0.973070	−0.367786	−0.183893	+	0.982946i	$0.558870\pi$
$8$	0	0
$9$	1.48063	0.493544
$10$	0	0
$11$	5.38225	1.62281	0.811405	−	0.584484i	$-0.198703\pi$
$11$	5.38225	1.62281	0.811405	+	0.584484i	$0.198703\pi$
$12$	0	0
$13$	1.99670	0.553785	0.276892	−	0.960901i	$-0.410695\pi$
$13$	1.99670	0.553785	0.276892	+	0.960901i	$0.410695\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.04301	0.495502	0.247751	−	0.968824i	$-0.420308\pi$
$17$	2.04301	0.495502	0.247751	+	0.968824i	$0.420308\pi$
$18$	0	0
$19$	−6.20428	−1.42336	−0.711680	−	0.702504i	$-0.752065\pi$
$19$	−6.20428	−1.42336	−0.711680	+	0.702504i	$0.752065\pi$
$20$	0	0
$21$	−2.05975	−0.449474
$22$	0	0
$23$	1.93813	0.404128	0.202064	−	0.979372i	$-0.435235\pi$
$23$	1.93813	0.404128	0.202064	+	0.979372i	$0.435235\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.21612	−0.618943
$28$	0	0
$29$	4.81812	0.894702	0.447351	−	0.894359i	$-0.352368\pi$
$29$	4.81812	0.894702	0.447351	+	0.894359i	$0.352368\pi$
$30$	0	0
$31$	6.64014	1.19260	0.596302	−	0.802760i	$-0.296636\pi$
$31$	6.64014	1.19260	0.596302	+	0.802760i	$0.296636\pi$
$32$	0	0
$33$	11.3929	1.98325
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.978913	0.160932	0.0804662	−	0.996757i	$-0.474359\pi$
$37$	0.978913	0.160932	0.0804662	+	0.996757i	$0.474359\pi$
$38$	0	0
$39$	4.22652	0.676784
$40$	0	0
$41$	2.73319	0.426853	0.213427	−	0.976959i	$-0.431538\pi$
$41$	2.73319	0.426853	0.213427	+	0.976959i	$0.431538\pi$
$42$	0	0
$43$	3.99413	0.609100	0.304550	−	0.952496i	$-0.401494\pi$
$43$	3.99413	0.609100	0.304550	+	0.952496i	$0.401494\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.21339	1.05218	0.526091	−	0.850428i	$-0.323657\pi$
$47$	7.21339	1.05218	0.526091	+	0.850428i	$0.323657\pi$
$48$	0	0
$49$	−6.05313	−0.864734
$50$	0	0
$51$	4.32454	0.605557
$52$	0	0
$53$	−13.2746	−1.82340	−0.911700	−	0.410857i	$-0.865230\pi$
$53$	−13.2746	−1.82340	−0.911700	+	0.410857i	$0.865230\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.1329	−1.73950
$58$	0	0
$59$	−6.54024	−0.851467	−0.425733	−	0.904849i	$-0.639984\pi$
$59$	−6.54024	−0.851467	−0.425733	+	0.904849i	$0.639984\pi$
$60$	0	0
$61$	2.72672	0.349121	0.174560	−	0.984646i	$-0.444150\pi$
$61$	2.72672	0.349121	0.174560	+	0.984646i	$0.444150\pi$
$62$	0	0
$63$	−1.44076	−0.181519
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.56957	1.16911	0.584555	−	0.811354i	$-0.301269\pi$
$67$	9.56957	1.16911	0.584555	+	0.811354i	$0.301269\pi$
$68$	0	0
$69$	4.10254	0.493888
$70$	0	0
$71$	5.68520	0.674710	0.337355	−	0.941378i	$-0.390468\pi$
$71$	5.68520	0.674710	0.337355	+	0.941378i	$0.390468\pi$
$72$	0	0
$73$	9.35281	1.09466	0.547332	−	0.836916i	$-0.315644\pi$
$73$	9.35281	1.09466	0.547332	+	0.836916i	$0.315644\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.23731	−0.596847
$78$	0	0
$79$	3.18171	0.357970	0.178985	−	0.983852i	$-0.442719\pi$
$79$	3.18171	0.357970	0.178985	+	0.983852i	$0.442719\pi$
$80$	0	0
$81$	−11.2496	−1.24996
$82$	0	0
$83$	6.11387	0.671085	0.335542	−	0.942025i	$-0.391080\pi$
$83$	6.11387	0.671085	0.335542	+	0.942025i	$0.391080\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.1988	1.09342
$88$	0	0
$89$	−3.00743	−0.318787	−0.159393	−	0.987215i	$-0.550954\pi$
$89$	−3.00743	−0.318787	−0.159393	+	0.987215i	$0.550954\pi$
$90$	0	0
$91$	−1.94293	−0.203674
$92$	0	0
$93$	14.0555	1.45749
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.95526	−0.604666	−0.302333	−	0.953202i	$-0.597765\pi$
$97$	−5.95526	−0.604666	−0.302333	+	0.953202i	$0.597765\pi$
$98$	0	0
$99$	7.96914	0.800929
$100$	0	0
$101$	−7.77373	−0.773515	−0.386758	−	0.922181i	$-0.626405\pi$
$101$	−7.77373	−0.773515	−0.386758	+	0.922181i	$0.626405\pi$
$102$	0	0
$103$	4.95056	0.487793	0.243897	−	0.969801i	$-0.421574\pi$
$103$	4.95056	0.487793	0.243897	+	0.969801i	$0.421574\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.16798	0.112913	0.0564567	−	0.998405i	$-0.482020\pi$
$107$	1.16798	0.112913	0.0564567	+	0.998405i	$0.482020\pi$
$108$	0	0
$109$	−17.6879	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$109$	−17.6879	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$110$	0	0
$111$	2.07212	0.196676
$112$	0	0
$113$	−5.03643	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$113$	−5.03643	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.95638	0.273317
$118$	0	0
$119$	−1.98799	−0.182239
$120$	0	0
$121$	17.9687	1.63351
$122$	0	0
$123$	5.78549	0.521660
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.79457	0.691656	0.345828	−	0.938298i	$-0.387598\pi$
$127$	7.79457	0.691656	0.345828	+	0.938298i	$0.387598\pi$
$128$	0	0
$129$	8.45458	0.744385
$130$	0	0
$131$	9.79015	0.855369	0.427685	−	0.903928i	$-0.359329\pi$
$131$	9.79015	0.855369	0.427685	+	0.903928i	$0.359329\pi$
$132$	0	0
$133$	6.03720	0.523491
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.63326	0.310411	0.155205	−	0.987882i	$-0.450396\pi$
$137$	3.63326	0.310411	0.155205	+	0.987882i	$0.450396\pi$
$138$	0	0
$139$	1.86079	0.157830	0.0789149	−	0.996881i	$-0.474854\pi$
$139$	1.86079	0.157830	0.0789149	+	0.996881i	$0.474854\pi$
$140$	0	0
$141$	15.2690	1.28588
$142$	0	0
$143$	10.7467	0.898688
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.8130	−1.05680
$148$	0	0
$149$	16.8530	1.38066	0.690328	−	0.723497i	$-0.257466\pi$
$149$	16.8530	1.38066	0.690328	+	0.723497i	$0.257466\pi$
$150$	0	0
$151$	14.3201	1.16536	0.582678	−	0.812703i	$-0.302005\pi$
$151$	14.3201	1.16536	0.582678	+	0.812703i	$0.302005\pi$
$152$	0	0
$153$	3.02495	0.244552
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.6869	1.73080	0.865400	−	0.501081i	$-0.167064\pi$
$157$	21.6869	1.73080	0.865400	+	0.501081i	$0.167064\pi$
$158$	0	0
$159$	−28.0989	−2.22839
$160$	0	0
$161$	−1.88594	−0.148633
$162$	0	0
$163$	−19.7210	−1.54467	−0.772334	−	0.635217i	$-0.780911\pi$
$163$	−19.7210	−1.54467	−0.772334	+	0.635217i	$0.780911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.07947	0.470444	0.235222	−	0.971942i	$-0.424418\pi$
$167$	6.07947	0.470444	0.235222	+	0.971942i	$0.424418\pi$
$168$	0	0
$169$	−9.01319	−0.693322
$170$	0	0
$171$	−9.18626	−0.702491
$172$	0	0
$173$	−15.0309	−1.14278	−0.571388	−	0.820680i	$-0.693595\pi$
$173$	−15.0309	−1.14278	−0.571388	+	0.820680i	$0.693595\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.8441	−1.04058
$178$	0	0
$179$	20.0167	1.49612	0.748060	−	0.663631i	$-0.230986\pi$
$179$	20.0167	1.49612	0.748060	+	0.663631i	$0.230986\pi$
$180$	0	0
$181$	20.2993	1.50883	0.754417	−	0.656395i	$-0.227920\pi$
$181$	20.2993	1.50883	0.754417	+	0.656395i	$0.227920\pi$
$182$	0	0
$183$	5.77178	0.426663
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.9960	0.804106
$188$	0	0
$189$	3.12951	0.227638
$190$	0	0
$191$	9.47681	0.685718	0.342859	−	0.939387i	$-0.388605\pi$
$191$	9.47681	0.685718	0.342859	+	0.939387i	$0.388605\pi$
$192$	0	0
$193$	−2.72478	−0.196134	−0.0980671	−	0.995180i	$-0.531266\pi$
$193$	−2.72478	−0.196134	−0.0980671	+	0.995180i	$0.531266\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.780036	−0.0555753	−0.0277876	−	0.999614i	$-0.508846\pi$
$197$	−0.780036	−0.0555753	−0.0277876	+	0.999614i	$0.508846\pi$
$198$	0	0
$199$	7.42526	0.526363	0.263181	−	0.964746i	$-0.415228\pi$
$199$	7.42526	0.526363	0.263181	+	0.964746i	$0.415228\pi$
$200$	0	0
$201$	20.2564	1.42878
$202$	0	0
$203$	−4.68836	−0.329059
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.86966	0.199455
$208$	0	0
$209$	−33.3930	−2.30984
$210$	0	0
$211$	2.98128	0.205240	0.102620	−	0.994721i	$-0.467277\pi$
$211$	2.98128	0.205240	0.102620	+	0.994721i	$0.467277\pi$
$212$	0	0
$213$	12.0342	0.824567
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.46132	−0.438623
$218$	0	0
$219$	19.7976	1.33779
$220$	0	0
$221$	4.07927	0.274402
$222$	0	0
$223$	24.8720	1.66556	0.832778	−	0.553607i	$-0.186749\pi$
$223$	24.8720	1.66556	0.832778	+	0.553607i	$0.186749\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.5594	−1.16546	−0.582729	−	0.812667i	$-0.698015\pi$
$227$	−17.5594	−1.16546	−0.582729	+	0.812667i	$0.698015\pi$
$228$	0	0
$229$	7.01849	0.463795	0.231897	−	0.972740i	$-0.425507\pi$
$229$	7.01849	0.463795	0.231897	+	0.972740i	$0.425507\pi$
$230$	0	0
$231$	−11.0861	−0.729410
$232$	0	0
$233$	8.69126	0.569383	0.284692	−	0.958619i	$-0.408109\pi$
$233$	8.69126	0.569383	0.284692	+	0.958619i	$0.408109\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.73488	0.437477
$238$	0	0
$239$	−21.3430	−1.38057	−0.690283	−	0.723540i	$-0.742514\pi$
$239$	−21.3430	−1.38057	−0.690283	+	0.723540i	$0.742514\pi$
$240$	0	0
$241$	−21.3897	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$241$	−21.3897	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$242$	0	0
$243$	−14.1643	−0.908639
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.3881	−0.788235
$248$	0	0
$249$	12.9415	0.820137
$250$	0	0
$251$	−14.9016	−0.940580	−0.470290	−	0.882512i	$-0.655851\pi$
$251$	−14.9016	−0.940580	−0.470290	+	0.882512i	$0.655851\pi$
$252$	0	0
$253$	10.4315	0.655823
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.0689	−1.18948	−0.594741	−	0.803917i	$-0.702746\pi$
$257$	−19.0689	−1.18948	−0.594741	+	0.803917i	$0.702746\pi$
$258$	0	0
$259$	−0.952551	−0.0591887
$260$	0	0
$261$	7.13386	0.441575
$262$	0	0
$263$	−6.78341	−0.418283	−0.209142	−	0.977885i	$-0.567067\pi$
$263$	−6.78341	−0.418283	−0.209142	+	0.977885i	$0.567067\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.36597	−0.389591
$268$	0	0
$269$	0.494663	0.0301601	0.0150801	−	0.999886i	$-0.495200\pi$
$269$	0.494663	0.0301601	0.0150801	+	0.999886i	$0.495200\pi$
$270$	0	0
$271$	−9.82610	−0.596893	−0.298446	−	0.954426i	$-0.596468\pi$
$271$	−9.82610	−0.596893	−0.298446	+	0.954426i	$0.596468\pi$
$272$	0	0
$273$	−4.11270	−0.248912
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.1399	−1.51051	−0.755255	−	0.655431i	$-0.772487\pi$
$277$	−25.1399	−1.51051	−0.755255	+	0.655431i	$0.772487\pi$
$278$	0	0
$279$	9.83161	0.588603
$280$	0	0
$281$	−7.77050	−0.463549	−0.231775	−	0.972770i	$-0.574453\pi$
$281$	−7.77050	−0.463549	−0.231775	+	0.972770i	$0.574453\pi$
$282$	0	0
$283$	28.9617	1.72160	0.860798	−	0.508946i	$-0.169965\pi$
$283$	28.9617	1.72160	0.860798	+	0.508946i	$0.169965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.65959	−0.156991
$288$	0	0
$289$	−12.8261	−0.754477
$290$	0	0
$291$	−12.6058	−0.738966
$292$	0	0
$293$	0.154329	0.00901602	0.00450801	−	0.999990i	$-0.498565\pi$
$293$	0.154329	0.00901602	0.00450801	+	0.999990i	$0.498565\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−17.3100	−1.00443
$298$	0	0
$299$	3.86987	0.223800
$300$	0	0
$301$	−3.88657	−0.224018
$302$	0	0
$303$	−16.4551	−0.945318
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.4875	−0.826845	−0.413423	−	0.910539i	$-0.635667\pi$
$307$	−14.4875	−0.826845	−0.413423	+	0.910539i	$0.635667\pi$
$308$	0	0
$309$	10.4791	0.596135
$310$	0	0
$311$	−18.5385	−1.05122	−0.525610	−	0.850726i	$-0.676163\pi$
$311$	−18.5385	−1.05122	−0.525610	+	0.850726i	$0.676163\pi$
$312$	0	0
$313$	−6.85703	−0.387582	−0.193791	−	0.981043i	$-0.562078\pi$
$313$	−6.85703	−0.387582	−0.193791	+	0.981043i	$0.562078\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.2810	1.53225	0.766127	−	0.642689i	$-0.222181\pi$
$317$	27.2810	1.53225	0.766127	+	0.642689i	$0.222181\pi$
$318$	0	0
$319$	25.9323	1.45193
$320$	0	0
$321$	2.47233	0.137992
$322$	0	0
$323$	−12.6754	−0.705278
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−37.4410	−2.07049
$328$	0	0
$329$	−7.01914	−0.386978
$330$	0	0
$331$	13.0429	0.716901	0.358450	−	0.933549i	$-0.383305\pi$
$331$	13.0429	0.716901	0.358450	+	0.933549i	$0.383305\pi$
$332$	0	0
$333$	1.44941	0.0794273
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.6507	1.07044	0.535222	−	0.844712i	$-0.320228\pi$
$337$	19.6507	1.07044	0.535222	+	0.844712i	$0.320228\pi$
$338$	0	0
$339$	−10.6609	−0.579018
$340$	0	0
$341$	35.7389	1.93537
$342$	0	0
$343$	12.7016	0.685823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.38943	0.235637	0.117818	−	0.993035i	$-0.462410\pi$
$347$	4.38943	0.235637	0.117818	+	0.993035i	$0.462410\pi$
$348$	0	0
$349$	27.1955	1.45574	0.727872	−	0.685713i	$-0.240510\pi$
$349$	27.1955	1.45574	0.727872	+	0.685713i	$0.240510\pi$
$350$	0	0
$351$	−6.42163	−0.342761
$352$	0	0
$353$	15.9317	0.847960	0.423980	−	0.905672i	$-0.360633\pi$
$353$	15.9317	0.847960	0.423980	+	0.905672i	$0.360633\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.20808	−0.222715
$358$	0	0
$359$	5.01508	0.264686	0.132343	−	0.991204i	$-0.457750\pi$
$359$	5.01508	0.264686	0.132343	+	0.991204i	$0.457750\pi$
$360$	0	0
$361$	19.4931	1.02595
$362$	0	0
$363$	38.0352	1.99633
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.3669	1.48074	0.740370	−	0.672200i	$-0.234651\pi$
$367$	28.3669	1.48074	0.740370	+	0.672200i	$0.234651\pi$
$368$	0	0
$369$	4.04686	0.210671
$370$	0	0
$371$	12.9171	0.670621
$372$	0	0
$373$	32.7218	1.69427	0.847135	−	0.531377i	$-0.178325\pi$
$373$	32.7218	1.69427	0.847135	+	0.531377i	$0.178325\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.62033	0.495472
$378$	0	0
$379$	−4.48483	−0.230370	−0.115185	−	0.993344i	$-0.536746\pi$
$379$	−4.48483	−0.230370	−0.115185	+	0.993344i	$0.536746\pi$
$380$	0	0
$381$	16.4992	0.845278
$382$	0	0
$383$	16.2839	0.832069	0.416034	−	0.909349i	$-0.363420\pi$
$383$	16.2839	0.832069	0.416034	+	0.909349i	$0.363420\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.91385	0.300618
$388$	0	0
$389$	21.4899	1.08958	0.544791	−	0.838572i	$-0.316609\pi$
$389$	21.4899	1.08958	0.544791	+	0.838572i	$0.316609\pi$
$390$	0	0
$391$	3.95962	0.200246
$392$	0	0
$393$	20.7233	1.04535
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.8870	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$397$	20.8870	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$398$	0	0
$399$	12.7792	0.639762
$400$	0	0
$401$	−0.633423	−0.0316316	−0.0158158	−	0.999875i	$-0.505035\pi$
$401$	−0.633423	−0.0316316	−0.0158158	+	0.999875i	$0.505035\pi$
$402$	0	0
$403$	13.2584	0.660447
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.26876	0.261163
$408$	0	0
$409$	−4.29822	−0.212534	−0.106267	−	0.994338i	$-0.533890\pi$
$409$	−4.29822	−0.212534	−0.106267	+	0.994338i	$0.533890\pi$
$410$	0	0
$411$	7.69071	0.379355
$412$	0	0
$413$	6.36411	0.313158
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.93882	0.192885
$418$	0	0
$419$	−33.6215	−1.64252	−0.821258	−	0.570556i	$-0.806728\pi$
$419$	−33.6215	−1.64252	−0.821258	+	0.570556i	$0.806728\pi$
$420$	0	0
$421$	−21.3768	−1.04184	−0.520920	−	0.853605i	$-0.674411\pi$
$421$	−21.3768	−1.04184	−0.520920	+	0.853605i	$0.674411\pi$
$422$	0	0
$423$	10.6804	0.519298
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.65329	−0.128402
$428$	0	0
$429$	22.7482	1.09829
$430$	0	0
$431$	−26.0905	−1.25673	−0.628367	−	0.777917i	$-0.716277\pi$
$431$	−26.0905	−1.25673	−0.628367	+	0.777917i	$0.716277\pi$
$432$	0	0
$433$	−11.7676	−0.565513	−0.282757	−	0.959192i	$-0.591249\pi$
$433$	−11.7676	−0.565513	−0.282757	+	0.959192i	$0.591249\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.0247	−0.575220
$438$	0	0
$439$	−4.92572	−0.235092	−0.117546	−	0.993067i	$-0.537503\pi$
$439$	−4.92572	−0.235092	−0.117546	+	0.993067i	$0.537503\pi$
$440$	0	0
$441$	−8.96247	−0.426784
$442$	0	0
$443$	9.35961	0.444689	0.222344	−	0.974968i	$-0.428629\pi$
$443$	9.35961	0.444689	0.222344	+	0.974968i	$0.428629\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.6737	1.68731
$448$	0	0
$449$	7.41602	0.349983	0.174992	−	0.984570i	$-0.444010\pi$
$449$	7.41602	0.349983	0.174992	+	0.984570i	$0.444010\pi$
$450$	0	0
$451$	14.7107	0.692702
$452$	0	0
$453$	30.3121	1.42419
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.79714	−0.411513	−0.205756	−	0.978603i	$-0.565965\pi$
$457$	−8.79714	−0.411513	−0.205756	+	0.978603i	$0.565965\pi$
$458$	0	0
$459$	−6.57056	−0.306688
$460$	0	0
$461$	−42.1526	−1.96324	−0.981621	−	0.190838i	$-0.938879\pi$
$461$	−42.1526	−1.96324	−0.981621	+	0.190838i	$0.938879\pi$
$462$	0	0
$463$	23.0565	1.07153	0.535763	−	0.844368i	$-0.320024\pi$
$463$	23.0565	1.07153	0.535763	+	0.844368i	$0.320024\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.37964	0.202666	0.101333	−	0.994853i	$-0.467689\pi$
$467$	4.37964	0.202666	0.101333	+	0.994853i	$0.467689\pi$
$468$	0	0
$469$	−9.31186	−0.429982
$470$	0	0
$471$	45.9057	2.11522
$472$	0	0
$473$	21.4974	0.988453
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−19.6547	−0.899929
$478$	0	0
$479$	−14.1913	−0.648417	−0.324209	−	0.945986i	$-0.605098\pi$
$479$	−14.1913	−0.648417	−0.324209	+	0.945986i	$0.605098\pi$
$480$	0	0
$481$	1.95460	0.0891219
$482$	0	0
$483$	−3.99206	−0.181645
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.0694	1.00006	0.500029	−	0.866009i	$-0.333323\pi$
$487$	22.0694	1.00006	0.500029	+	0.866009i	$0.333323\pi$
$488$	0	0
$489$	−41.7444	−1.88775
$490$	0	0
$491$	7.95704	0.359096	0.179548	−	0.983749i	$-0.442536\pi$
$491$	7.95704	0.359096	0.179548	+	0.983749i	$0.442536\pi$
$492$	0	0
$493$	9.84345	0.443327
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.53210	−0.248149
$498$	0	0
$499$	4.09384	0.183266	0.0916328	−	0.995793i	$-0.470791\pi$
$499$	4.09384	0.183266	0.0916328	+	0.995793i	$0.470791\pi$
$500$	0	0
$501$	12.8687	0.574932
$502$	0	0
$503$	26.9554	1.20188	0.600941	−	0.799293i	$-0.294792\pi$
$503$	26.9554	1.20188	0.600941	+	0.799293i	$0.294792\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−19.0787	−0.847314
$508$	0	0
$509$	33.8837	1.50187	0.750935	−	0.660376i	$-0.229603\pi$
$509$	33.8837	1.50187	0.750935	+	0.660376i	$0.229603\pi$
$510$	0	0
$511$	−9.10093	−0.402602
$512$	0	0
$513$	19.9537	0.880978
$514$	0	0
$515$	0	0
$516$	0	0
$517$	38.8243	1.70749
$518$	0	0
$519$	−31.8166	−1.39659
$520$	0	0
$521$	−26.6469	−1.16742	−0.583711	−	0.811962i	$-0.698400\pi$
$521$	−26.6469	−1.16742	−0.583711	+	0.811962i	$0.698400\pi$
$522$	0	0
$523$	−1.80935	−0.0791172	−0.0395586	−	0.999217i	$-0.512595\pi$
$523$	−1.80935	−0.0791172	−0.0395586	+	0.999217i	$0.512595\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.5659	0.590938
$528$	0	0
$529$	−19.2437	−0.836680
$530$	0	0
$531$	−9.68370	−0.420237
$532$	0	0
$533$	5.45737	0.236385
$534$	0	0
$535$	0	0
$536$	0	0
$537$	42.3704	1.82842
$538$	0	0
$539$	−32.5795	−1.40330
$540$	0	0
$541$	−33.5233	−1.44128	−0.720640	−	0.693309i	$-0.756152\pi$
$541$	−33.5233	−1.44128	−0.720640	+	0.693309i	$0.756152\pi$
$542$	0	0
$543$	42.9685	1.84396
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−27.5146	−1.17644	−0.588219	−	0.808702i	$-0.700171\pi$
$547$	−27.5146	−1.17644	−0.588219	+	0.808702i	$0.700171\pi$
$548$	0	0
$549$	4.03727	0.172306
$550$	0	0
$551$	−29.8929	−1.27348
$552$	0	0
$553$	−3.09602	−0.131656
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.16761	−0.0918448	−0.0459224	−	0.998945i	$-0.514623\pi$
$557$	−2.16761	−0.0918448	−0.0459224	+	0.998945i	$0.514623\pi$
$558$	0	0
$559$	7.97509	0.337310
$560$	0	0
$561$	23.2758	0.982704
$562$	0	0
$563$	33.4836	1.41116	0.705582	−	0.708628i	$-0.250686\pi$
$563$	33.4836	1.41116	0.705582	+	0.708628i	$0.250686\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.9467	0.459717
$568$	0	0
$569$	17.2323	0.722415	0.361208	−	0.932485i	$-0.382365\pi$
$569$	17.2323	0.722415	0.361208	+	0.932485i	$0.382365\pi$
$570$	0	0
$571$	−42.7928	−1.79082	−0.895411	−	0.445240i	$-0.853118\pi$
$571$	−42.7928	−1.79082	−0.895411	+	0.445240i	$0.853118\pi$
$572$	0	0
$573$	20.0600	0.838020
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.8385	1.32546	0.662728	−	0.748860i	$-0.269399\pi$
$577$	31.8385	1.32546	0.662728	+	0.748860i	$0.269399\pi$
$578$	0	0
$579$	−5.76769	−0.239697
$580$	0	0
$581$	−5.94923	−0.246816
$582$	0	0
$583$	−71.4470	−2.95903
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.8027	−1.02372	−0.511858	−	0.859070i	$-0.671043\pi$
$587$	−24.8027	−1.02372	−0.511858	+	0.859070i	$0.671043\pi$
$588$	0	0
$589$	−41.1973	−1.69750
$590$	0	0
$591$	−1.65114	−0.0679189
$592$	0	0
$593$	−17.8431	−0.732728	−0.366364	−	0.930472i	$-0.619397\pi$
$593$	−17.8431	−0.732728	−0.366364	+	0.930472i	$0.619397\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.7174	0.643271
$598$	0	0
$599$	28.1028	1.14825	0.574126	−	0.818767i	$-0.305342\pi$
$599$	28.1028	1.14825	0.574126	+	0.818767i	$0.305342\pi$
$600$	0	0
$601$	−26.6296	−1.08624	−0.543121	−	0.839654i	$-0.682758\pi$
$601$	−26.6296	−1.08624	−0.543121	+	0.839654i	$0.682758\pi$
$602$	0	0
$603$	14.1690	0.577007
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.2757	−0.782375	−0.391187	−	0.920311i	$-0.627936\pi$
$607$	−19.2757	−0.782375	−0.391187	+	0.920311i	$0.627936\pi$
$608$	0	0
$609$	−9.92410	−0.402145
$610$	0	0
$611$	14.4030	0.582682
$612$	0	0
$613$	0.465680	0.0188086	0.00940432	−	0.999956i	$-0.497006\pi$
$613$	0.465680	0.0188086	0.00940432	+	0.999956i	$0.497006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	44.2765	1.78251	0.891253	−	0.453507i	$-0.149827\pi$
$617$	44.2765	1.78251	0.891253	+	0.453507i	$0.149827\pi$
$618$	0	0
$619$	36.1620	1.45347	0.726736	−	0.686917i	$-0.241036\pi$
$619$	36.1620	1.45347	0.726736	+	0.686917i	$0.241036\pi$
$620$	0	0
$621$	−6.23326	−0.250132
$622$	0	0
$623$	2.92644	0.117245
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−70.6847	−2.82287
$628$	0	0
$629$	1.99993	0.0797424
$630$	0	0
$631$	0.199431	0.00793921	0.00396961	−	0.999992i	$-0.498736\pi$
$631$	0.199431	0.00793921	0.00396961	+	0.999992i	$0.498736\pi$
$632$	0	0
$633$	6.31062	0.250825
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.0863	−0.478876
$638$	0	0
$639$	8.41770	0.332999
$640$	0	0
$641$	−30.9078	−1.22079	−0.610393	−	0.792099i	$-0.708988\pi$
$641$	−30.9078	−1.22079	−0.610393	+	0.792099i	$0.708988\pi$
$642$	0	0
$643$	−22.2489	−0.877412	−0.438706	−	0.898631i	$-0.644563\pi$
$643$	−22.2489	−0.877412	−0.438706	+	0.898631i	$0.644563\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.0696	−1.37873	−0.689364	−	0.724415i	$-0.742110\pi$
$647$	−35.0696	−1.37873	−0.689364	+	0.724415i	$0.742110\pi$
$648$	0	0
$649$	−35.2012	−1.38177
$650$	0	0
$651$	−13.6770	−0.536044
$652$	0	0
$653$	4.00012	0.156537	0.0782683	−	0.996932i	$-0.475061\pi$
$653$	4.00012	0.156537	0.0782683	+	0.996932i	$0.475061\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.8481	0.540265
$658$	0	0
$659$	−10.0349	−0.390904	−0.195452	−	0.980713i	$-0.562617\pi$
$659$	−10.0349	−0.390904	−0.195452	+	0.980713i	$0.562617\pi$
$660$	0	0
$661$	−48.5672	−1.88905	−0.944523	−	0.328445i	$-0.893475\pi$
$661$	−48.5672	−1.88905	−0.944523	+	0.328445i	$0.893475\pi$
$662$	0	0
$663$	8.63481	0.335348
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.33814	0.361574
$668$	0	0
$669$	52.6479	2.03549
$670$	0	0
$671$	14.6759	0.566557
$672$	0	0
$673$	−5.06306	−0.195167	−0.0975834	−	0.995227i	$-0.531111\pi$
$673$	−5.06306	−0.195167	−0.0975834	+	0.995227i	$0.531111\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.6669	−1.29393	−0.646963	−	0.762521i	$-0.723961\pi$
$677$	−33.6669	−1.29393	−0.646963	+	0.762521i	$0.723961\pi$
$678$	0	0
$679$	5.79489	0.222387
$680$	0	0
$681$	−37.1689	−1.42431
$682$	0	0
$683$	32.4523	1.24175	0.620877	−	0.783908i	$-0.286777\pi$
$683$	32.4523	1.24175	0.620877	+	0.783908i	$0.286777\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.8564	0.566807
$688$	0	0
$689$	−26.5053	−1.00977
$690$	0	0
$691$	30.7811	1.17097	0.585484	−	0.810684i	$-0.300905\pi$
$691$	30.7811	1.17097	0.585484	+	0.810684i	$0.300905\pi$
$692$	0	0
$693$	−7.75453	−0.294570
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.58394	0.211507
$698$	0	0
$699$	18.3972	0.695847
$700$	0	0
$701$	−13.1755	−0.497633	−0.248817	−	0.968551i	$-0.580042\pi$
$701$	−13.1755	−0.497633	−0.248817	+	0.968551i	$0.580042\pi$
$702$	0	0
$703$	−6.07345	−0.229065
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.56439	0.284488
$708$	0	0
$709$	−10.6307	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$709$	−10.6307	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$710$	0	0
$711$	4.71094	0.176674
$712$	0	0
$713$	12.8695	0.481965
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−45.1779	−1.68720
$718$	0	0
$719$	15.3464	0.572324	0.286162	−	0.958181i	$-0.407620\pi$
$719$	15.3464	0.572324	0.286162	+	0.958181i	$0.407620\pi$
$720$	0	0
$721$	−4.81724	−0.179404
$722$	0	0
$723$	−45.2767	−1.68386
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.8156	0.883272	0.441636	−	0.897194i	$-0.354398\pi$
$727$	23.8156	0.883272	0.441636	+	0.897194i	$0.354398\pi$
$728$	0	0
$729$	3.76661	0.139504
$730$	0	0
$731$	8.16005	0.301810
$732$	0	0
$733$	4.09180	0.151134	0.0755670	−	0.997141i	$-0.475923\pi$
$733$	4.09180	0.151134	0.0755670	+	0.997141i	$0.475923\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	51.5059	1.89724
$738$	0	0
$739$	−2.45909	−0.0904590	−0.0452295	−	0.998977i	$-0.514402\pi$
$739$	−2.45909	−0.0904590	−0.0452295	+	0.998977i	$0.514402\pi$
$740$	0	0
$741$	−26.2225	−0.963307
$742$	0	0
$743$	−11.1921	−0.410598	−0.205299	−	0.978699i	$-0.565817\pi$
$743$	−11.1921	−0.410598	−0.205299	+	0.978699i	$0.565817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.05240	0.331210
$748$	0	0
$749$	−1.13653	−0.0415279
$750$	0	0
$751$	10.3616	0.378101	0.189051	−	0.981967i	$-0.439459\pi$
$751$	10.3616	0.378101	0.189051	+	0.981967i	$0.439459\pi$
$752$	0	0
$753$	−31.5430	−1.14949
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.84722	0.212521	0.106260	−	0.994338i	$-0.466112\pi$
$757$	5.84722	0.212521	0.106260	+	0.994338i	$0.466112\pi$
$758$	0	0
$759$	22.0809	0.801486
$760$	0	0
$761$	41.6931	1.51137	0.755687	−	0.654932i	$-0.227303\pi$
$761$	41.6931	1.51137	0.755687	+	0.654932i	$0.227303\pi$
$762$	0	0
$763$	17.2116	0.623102
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.0589	−0.471530
$768$	0	0
$769$	−17.3076	−0.624129	−0.312065	−	0.950061i	$-0.601021\pi$
$769$	−17.3076	−0.624129	−0.312065	+	0.950061i	$0.601021\pi$
$770$	0	0
$771$	−40.3640	−1.45367
$772$	0	0
$773$	−17.8695	−0.642723	−0.321361	−	0.946957i	$-0.604140\pi$
$773$	−17.8695	−0.642723	−0.321361	+	0.946957i	$0.604140\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.01631	−0.0723348
$778$	0	0
$779$	−16.9575	−0.607565
$780$	0	0
$781$	30.5992	1.09493
$782$	0	0
$783$	−15.4956	−0.553769
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.91104	0.353290	0.176645	−	0.984275i	$-0.443475\pi$
$787$	9.91104	0.353290	0.176645	+	0.984275i	$0.443475\pi$
$788$	0	0
$789$	−14.3588	−0.511187
$790$	0	0
$791$	4.90080	0.174252
$792$	0	0
$793$	5.44444	0.193338
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.7590	−1.40834	−0.704168	−	0.710034i	$-0.748680\pi$
$797$	−39.7590	−1.40834	−0.704168	+	0.710034i	$0.748680\pi$
$798$	0	0
$799$	14.7370	0.521359
$800$	0	0
$801$	−4.45290	−0.157335
$802$	0	0
$803$	50.3392	1.77643
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.04708	0.0368589
$808$	0	0
$809$	−5.05110	−0.177587	−0.0887935	−	0.996050i	$-0.528301\pi$
$809$	−5.05110	−0.177587	−0.0887935	+	0.996050i	$0.528301\pi$
$810$	0	0
$811$	−11.7743	−0.413452	−0.206726	−	0.978399i	$-0.566281\pi$
$811$	−11.7743	−0.413452	−0.206726	+	0.978399i	$0.566281\pi$
$812$	0	0
$813$	−20.7994	−0.729467
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.7807	−0.866968
$818$	0	0
$819$	−2.87676	−0.100522
$820$	0	0
$821$	5.25401	0.183366	0.0916832	−	0.995788i	$-0.470775\pi$
$821$	5.25401	0.183366	0.0916832	+	0.995788i	$0.470775\pi$
$822$	0	0
$823$	−37.0042	−1.28989	−0.644943	−	0.764230i	$-0.723119\pi$
$823$	−37.0042	−1.28989	−0.644943	+	0.764230i	$0.723119\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.1147	0.977645	0.488822	−	0.872383i	$-0.337427\pi$
$827$	28.1147	0.977645	0.488822	+	0.872383i	$0.337427\pi$
$828$	0	0
$829$	−14.4532	−0.501979	−0.250990	−	0.967990i	$-0.580756\pi$
$829$	−14.4532	−0.501979	−0.250990	+	0.967990i	$0.580756\pi$
$830$	0	0
$831$	−53.2149	−1.84600
$832$	0	0
$833$	−12.3666	−0.428477
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−21.3555	−0.738154
$838$	0	0
$839$	−16.3694	−0.565134	−0.282567	−	0.959248i	$-0.591186\pi$
$839$	−16.3694	−0.565134	−0.282567	+	0.959248i	$0.591186\pi$
$840$	0	0
$841$	−5.78575	−0.199509
$842$	0	0
$843$	−16.4482	−0.566506
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.4848	−0.600784
$848$	0	0
$849$	61.3048	2.10397
$850$	0	0
$851$	1.89726	0.0650373
$852$	0	0
$853$	46.8216	1.60314	0.801571	−	0.597900i	$-0.203998\pi$
$853$	46.8216	1.60314	0.801571	+	0.597900i	$0.203998\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.9569	−0.989148	−0.494574	−	0.869136i	$-0.664676\pi$
$857$	−28.9569	−0.989148	−0.494574	+	0.869136i	$0.664676\pi$
$858$	0	0
$859$	−24.4996	−0.835917	−0.417959	−	0.908466i	$-0.637254\pi$
$859$	−24.4996	−0.835917	−0.417959	+	0.908466i	$0.637254\pi$
$860$	0	0
$861$	−5.62968	−0.191859
$862$	0	0
$863$	10.4696	0.356391	0.178195	−	0.983995i	$-0.442974\pi$
$863$	10.4696	0.356391	0.178195	+	0.983995i	$0.442974\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−27.1497	−0.922052
$868$	0	0
$869$	17.1247	0.580917
$870$	0	0
$871$	19.1076	0.647435
$872$	0	0
$873$	−8.81756	−0.298429
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.9121	−1.34774	−0.673868	−	0.738852i	$-0.735368\pi$
$877$	−39.9121	−1.34774	−0.673868	+	0.738852i	$0.735368\pi$
$878$	0	0
$879$	0.326677	0.0110185
$880$	0	0
$881$	−7.88965	−0.265809	−0.132904	−	0.991129i	$-0.542430\pi$
$881$	−7.88965	−0.265809	−0.132904	+	0.991129i	$0.542430\pi$
$882$	0	0
$883$	−15.1304	−0.509178	−0.254589	−	0.967049i	$-0.581940\pi$
$883$	−15.1304	−0.509178	−0.254589	+	0.967049i	$0.581940\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.6886	1.60123	0.800614	−	0.599180i	$-0.204507\pi$
$887$	47.6886	1.60123	0.800614	+	0.599180i	$0.204507\pi$
$888$	0	0
$889$	−7.58466	−0.254381
$890$	0	0
$891$	−60.5483	−2.02845
$892$	0	0
$893$	−44.7539	−1.49763
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.19154	0.273508
$898$	0	0
$899$	31.9930	1.06703
$900$	0	0
$901$	−27.1200	−0.903499
$902$	0	0
$903$	−8.22690	−0.273774
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−38.9321	−1.29272	−0.646359	−	0.763033i	$-0.723709\pi$
$907$	−38.9321	−1.29272	−0.646359	+	0.763033i	$0.723709\pi$
$908$	0	0
$909$	−11.5100	−0.381764
$910$	0	0
$911$	−37.6740	−1.24819	−0.624097	−	0.781347i	$-0.714533\pi$
$911$	−37.6740	−1.24819	−0.624097	+	0.781347i	$0.714533\pi$
$912$	0	0
$913$	32.9064	1.08904
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.52650	−0.314593
$918$	0	0
$919$	−14.4646	−0.477145	−0.238572	−	0.971125i	$-0.576679\pi$
$919$	−14.4646	−0.477145	−0.238572	+	0.971125i	$0.576679\pi$
$920$	0	0
$921$	−30.6664	−1.01049
$922$	0	0
$923$	11.3516	0.373644
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.32997	0.240748
$928$	0	0
$929$	−40.8990	−1.34185	−0.670927	−	0.741524i	$-0.734103\pi$
$929$	−40.8990	−1.34185	−0.670927	+	0.741524i	$0.734103\pi$
$930$	0	0
$931$	37.5553	1.23083
$932$	0	0
$933$	−39.2413	−1.28470
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33.2563	−1.08644	−0.543218	−	0.839592i	$-0.682794\pi$
$937$	−33.2563	−1.08644	−0.543218	+	0.839592i	$0.682794\pi$
$938$	0	0
$939$	−14.5146	−0.473666
$940$	0	0
$941$	39.1577	1.27650	0.638252	−	0.769827i	$-0.279658\pi$
$941$	39.1577	1.27650	0.638252	+	0.769827i	$0.279658\pi$
$942$	0	0
$943$	5.29728	0.172503
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.5840	−0.928854	−0.464427	−	0.885611i	$-0.653740\pi$
$947$	−28.5840	−0.928854	−0.464427	+	0.885611i	$0.653740\pi$
$948$	0	0
$949$	18.6747	0.606208
$950$	0	0
$951$	57.7471	1.87258
$952$	0	0
$953$	43.3568	1.40446	0.702232	−	0.711948i	$-0.252187\pi$
$953$	43.3568	1.40446	0.702232	+	0.711948i	$0.252187\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	54.8923	1.77442
$958$	0	0
$959$	−3.53542	−0.114165
$960$	0	0
$961$	13.0915	0.422306
$962$	0	0
$963$	1.72936	0.0557277
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.2551	1.03726	0.518628	−	0.855000i	$-0.326443\pi$
$967$	32.2551	1.03726	0.518628	+	0.855000i	$0.326443\pi$
$968$	0	0
$969$	−26.8306	−0.861925
$970$	0	0
$971$	−10.9579	−0.351654	−0.175827	−	0.984421i	$-0.556260\pi$
$971$	−10.9579	−0.351654	−0.175827	+	0.984421i	$0.556260\pi$
$972$	0	0
$973$	−1.81068	−0.0580476
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.1536	0.644770	0.322385	−	0.946609i	$-0.395516\pi$
$977$	20.1536	0.644770	0.322385	+	0.946609i	$0.395516\pi$
$978$	0	0
$979$	−16.1867	−0.517330
$980$	0	0
$981$	−26.1894	−0.836162
$982$	0	0
$983$	−16.0511	−0.511951	−0.255975	−	0.966683i	$-0.582397\pi$
$983$	−16.0511	−0.511951	−0.255975	+	0.966683i	$0.582397\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−14.8578	−0.472928
$988$	0	0
$989$	7.74115	0.246154
$990$	0	0
$991$	−20.5569	−0.653010	−0.326505	−	0.945195i	$-0.605871\pi$
$991$	−20.5569	−0.653010	−0.326505	+	0.945195i	$0.605871\pi$
$992$	0	0
$993$	27.6085	0.876129
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21.7296	−0.688183	−0.344092	−	0.938936i	$-0.611813\pi$
$997$	−21.7296	−0.688183	−0.344092	+	0.938936i	$0.611813\pi$
$998$	0	0
$999$	−3.14830	−0.0996079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bn.1.6		8
4.3	odd	2		625.2.a.e.1.2	✓	8
5.4	even	2		10000.2.a.be.1.3		8
12.11	even	2		5625.2.a.be.1.7		8
20.3	even	4		625.2.b.d.624.15		16
20.7	even	4		625.2.b.d.624.2		16
20.19	odd	2		625.2.a.g.1.7	yes	8
60.59	even	2		5625.2.a.s.1.2		8
100.3	even	20		625.2.e.j.249.8		32
100.11	odd	10		625.2.d.p.501.1		16
100.19	odd	10		625.2.d.m.251.1		16
100.23	even	20		625.2.e.k.124.8		32
100.27	even	20		625.2.e.k.124.1		32
100.31	odd	10		625.2.d.q.251.4		16
100.39	odd	10		625.2.d.n.501.4		16
100.47	even	20		625.2.e.j.249.1		32
100.59	odd	10		625.2.d.n.126.4		16
100.63	even	20		625.2.e.k.499.1		32
100.67	even	20		625.2.e.j.374.8		32
100.71	odd	10		625.2.d.q.376.4		16
100.79	odd	10		625.2.d.m.376.1		16
100.83	even	20		625.2.e.j.374.1		32
100.87	even	20		625.2.e.k.499.8		32
100.91	odd	10		625.2.d.p.126.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.2	✓	8	4.3	odd	2
625.2.a.g.1.7	yes	8	20.19	odd	2
625.2.b.d.624.2		16	20.7	even	4
625.2.b.d.624.15		16	20.3	even	4
625.2.d.m.251.1		16	100.19	odd	10
625.2.d.m.376.1		16	100.79	odd	10
625.2.d.n.126.4		16	100.59	odd	10
625.2.d.n.501.4		16	100.39	odd	10
625.2.d.p.126.1		16	100.91	odd	10
625.2.d.p.501.1		16	100.11	odd	10
625.2.d.q.251.4		16	100.31	odd	10
625.2.d.q.376.4		16	100.71	odd	10
625.2.e.j.249.1		32	100.47	even	20
625.2.e.j.249.8		32	100.3	even	20
625.2.e.j.374.1		32	100.83	even	20
625.2.e.j.374.8		32	100.67	even	20
625.2.e.k.124.1		32	100.27	even	20
625.2.e.k.124.8		32	100.23	even	20
625.2.e.k.499.1		32	100.63	even	20
625.2.e.k.499.8		32	100.87	even	20
5625.2.a.s.1.2		8	60.59	even	2
5625.2.a.be.1.7		8	12.11	even	2
10000.2.a.be.1.3		8	5.4	even	2
10000.2.a.bn.1.6		8	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)

\(f(q)\)	\(=\)	\(q+2.11675 q^{3} -0.973070 q^{7} +1.48063 q^{9} +O(q^{10})\)	\(q+2.11675 q^{3} -0.973070 q^{7} +1.48063 q^{9} +5.38225 q^{11} +1.99670 q^{13} +2.04301 q^{17} -6.20428 q^{19} -2.05975 q^{21} +1.93813 q^{23} -3.21612 q^{27} +4.81812 q^{29} +6.64014 q^{31} +11.3929 q^{33} +0.978913 q^{37} +4.22652 q^{39} +2.73319 q^{41} +3.99413 q^{43} +7.21339 q^{47} -6.05313 q^{49} +4.32454 q^{51} -13.2746 q^{53} -13.1329 q^{57} -6.54024 q^{59} +2.72672 q^{61} -1.44076 q^{63} +9.56957 q^{67} +4.10254 q^{69} +5.68520 q^{71} +9.35281 q^{73} -5.23731 q^{77} +3.18171 q^{79} -11.2496 q^{81} +6.11387 q^{83} +10.1988 q^{87} -3.00743 q^{89} -1.94293 q^{91} +14.0555 q^{93} -5.95526 q^{97} +7.96914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9	\( 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9} - q^{11} - 10 q^{13} - 15 q^{17} + 10 q^{19} - 14 q^{21} + 30 q^{23} + 20 q^{27} + 10 q^{29} + 9 q^{31} - 5 q^{33} + 10 q^{37} - 8 q^{39} - 4 q^{41} + 30 q^{47} - 4 q^{49} + 14 q^{51} - 10 q^{53} + 10 q^{57} + 5 q^{59} + 6 q^{61} + 10 q^{67} + 3 q^{69} + 9 q^{71} - 5 q^{77} + 20 q^{79} + 8 q^{81} + 40 q^{83} + 40 q^{87} - 5 q^{89} - 6 q^{91} + 40 q^{93} + 22 q^{99}+O(q^{100}) \) 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 - q^11 - 10 * q^13 - 15 * q^17 + 10 * q^19 - 14 * q^21 + 30 * q^23 + 20 * q^27 + 10 * q^29 + 9 * q^31 - 5 * q^33 + 10 * q^37 - 8 * q^39 - 4 * q^41 + 30 * q^47 - 4 * q^49 + 14 * q^51 - 10 * q^53 + 10 * q^57 + 5 * q^59 + 6 * q^61 + 10 * q^67 + 3 * q^69 + 9 * q^71 - 5 * q^77 + 20 * q^79 + 8 * q^81 + 40 * q^83 + 40 * q^87 - 5 * q^89 - 6 * q^91 + 40 * q^93 + 22 * q^99

Introduction

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