LMFDB - Embedded newform 10000.2.a.bb.1.4

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			1.4
Root			$1.90211$ 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)

$f(q)$	$=$	$q+2.90211 q^{3} -1.07768 q^{7} +5.42226 q^{9} +O(q^{10})$	$q+2.90211 q^{3} -1.07768 q^{7} +5.42226 q^{9} +2.06706 q^{11} -5.79360 q^{13} -0.824429 q^{17} -3.41164 q^{19} -3.12756 q^{21} -1.96261 q^{23} +7.02967 q^{27} -1.04801 q^{29} -2.11507 q^{31} +5.99885 q^{33} -8.69572 q^{37} -16.8137 q^{39} -6.57408 q^{41} +7.47684 q^{43} +8.65176 q^{47} -5.83860 q^{49} -2.39259 q^{51} -2.56816 q^{53} -9.90096 q^{57} -4.70228 q^{59} -9.28293 q^{61} -5.84348 q^{63} -4.36176 q^{67} -5.69572 q^{69} -4.90398 q^{71} -1.29772 q^{73} -2.22764 q^{77} +7.19566 q^{79} +4.13412 q^{81} +14.9680 q^{83} -3.04145 q^{87} -11.7082 q^{89} +6.24367 q^{91} -6.13818 q^{93} -0.510678 q^{97} +11.2081 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.90211	1.67554	0.837768	−	0.546027i	$-0.183860\pi$
$3$	2.90211	1.67554	0.837768	+	0.546027i	$0.183860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.07768	−0.407326	−0.203663	−	0.979041i	$-0.565285\pi$
$7$	−1.07768	−0.407326	−0.203663	+	0.979041i	$0.565285\pi$
$8$	0	0
$9$	5.42226	1.80742
$10$	0	0
$11$	2.06706	0.623243	0.311621	−	0.950206i	$-0.399128\pi$
$11$	2.06706	0.623243	0.311621	+	0.950206i	$0.399128\pi$
$12$	0	0
$13$	−5.79360	−1.60686	−0.803428	−	0.595401i	$-0.796993\pi$
$13$	−5.79360	−1.60686	−0.803428	+	0.595401i	$0.796993\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.824429	−0.199954	−0.0999768	−	0.994990i	$-0.531877\pi$
$17$	−0.824429	−0.199954	−0.0999768	+	0.994990i	$0.531877\pi$
$18$	0	0
$19$	−3.41164	−0.782684	−0.391342	−	0.920245i	$-0.627989\pi$
$19$	−3.41164	−0.782684	−0.391342	+	0.920245i	$0.627989\pi$
$20$	0	0
$21$	−3.12756	−0.682489
$22$	0	0
$23$	−1.96261	−0.409233	−0.204616	−	0.978842i	$-0.565595\pi$
$23$	−1.96261	−0.409233	−0.204616	+	0.978842i	$0.565595\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	7.02967	1.35286
$28$	0	0
$29$	−1.04801	−0.194611	−0.0973054	−	0.995255i	$-0.531022\pi$
$29$	−1.04801	−0.194611	−0.0973054	+	0.995255i	$0.531022\pi$
$30$	0	0
$31$	−2.11507	−0.379878	−0.189939	−	0.981796i	$-0.560829\pi$
$31$	−2.11507	−0.379878	−0.189939	+	0.981796i	$0.560829\pi$
$32$	0	0
$33$	5.99885	1.04427
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.69572	−1.42957	−0.714784	−	0.699346i	$-0.753475\pi$
$37$	−8.69572	−1.42957	−0.714784	+	0.699346i	$0.753475\pi$
$38$	0	0
$39$	−16.8137	−2.69235
$40$	0	0
$41$	−6.57408	−1.02670	−0.513349	−	0.858180i	$-0.671596\pi$
$41$	−6.57408	−1.02670	−0.513349	+	0.858180i	$0.671596\pi$
$42$	0	0
$43$	7.47684	1.14021	0.570103	−	0.821573i	$-0.306903\pi$
$43$	7.47684	1.14021	0.570103	+	0.821573i	$0.306903\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.65176	1.26199	0.630995	−	0.775787i	$-0.282647\pi$
$47$	8.65176	1.26199	0.630995	+	0.775787i	$0.282647\pi$
$48$	0	0
$49$	−5.83860	−0.834085
$50$	0	0
$51$	−2.39259	−0.335029
$52$	0	0
$53$	−2.56816	−0.352764	−0.176382	−	0.984322i	$-0.556439\pi$
$53$	−2.56816	−0.352764	−0.176382	+	0.984322i	$0.556439\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.90096	−1.31141
$58$	0	0
$59$	−4.70228	−0.612185	−0.306092	−	0.952002i	$-0.599022\pi$
$59$	−4.70228	−0.612185	−0.306092	+	0.952002i	$0.599022\pi$
$60$	0	0
$61$	−9.28293	−1.18856	−0.594278	−	0.804259i	$-0.702562\pi$
$61$	−9.28293	−1.18856	−0.594278	+	0.804259i	$0.702562\pi$
$62$	0	0
$63$	−5.84348	−0.736209
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.36176	−0.532874	−0.266437	−	0.963852i	$-0.585846\pi$
$67$	−4.36176	−0.532874	−0.266437	+	0.963852i	$0.585846\pi$
$68$	0	0
$69$	−5.69572	−0.685684
$70$	0	0
$71$	−4.90398	−0.581995	−0.290998	−	0.956724i	$-0.593987\pi$
$71$	−4.90398	−0.581995	−0.290998	+	0.956724i	$0.593987\pi$
$72$	0	0
$73$	−1.29772	−0.151886	−0.0759432	−	0.997112i	$-0.524197\pi$
$73$	−1.29772	−0.151886	−0.0759432	+	0.997112i	$0.524197\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.22764	−0.253863
$78$	0	0
$79$	7.19566	0.809575	0.404788	−	0.914411i	$-0.367345\pi$
$79$	7.19566	0.809575	0.404788	+	0.914411i	$0.367345\pi$
$80$	0	0
$81$	4.13412	0.459347
$82$	0	0
$83$	14.9680	1.64295	0.821477	−	0.570242i	$-0.193150\pi$
$83$	14.9680	1.64295	0.821477	+	0.570242i	$0.193150\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.04145	−0.326077
$88$	0	0
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	6.24367	0.654515
$92$	0	0
$93$	−6.13818	−0.636500
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.510678	−0.0518515	−0.0259257	−	0.999664i	$-0.508253\pi$
$97$	−0.510678	−0.0518515	−0.0259257	+	0.999664i	$0.508253\pi$
$98$	0	0
$99$	11.2081	1.12646
$100$	0	0
$101$	−3.50766	−0.349025	−0.174513	−	0.984655i	$-0.555835\pi$
$101$	−3.50766	−0.349025	−0.174513	+	0.984655i	$0.555835\pi$
$102$	0	0
$103$	4.64178	0.457369	0.228684	−	0.973501i	$-0.426558\pi$
$103$	4.64178	0.457369	0.228684	+	0.973501i	$0.426558\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.60189	−0.734902	−0.367451	−	0.930043i	$-0.619769\pi$
$107$	−7.60189	−0.734902	−0.367451	+	0.930043i	$0.619769\pi$
$108$	0	0
$109$	−2.98636	−0.286042	−0.143021	−	0.989720i	$-0.545682\pi$
$109$	−2.98636	−0.286042	−0.143021	+	0.989720i	$0.545682\pi$
$110$	0	0
$111$	−25.2360	−2.39529
$112$	0	0
$113$	20.1986	1.90012	0.950061	−	0.312065i	$-0.101021\pi$
$113$	20.1986	1.90012	0.950061	+	0.312065i	$0.101021\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−31.4144	−2.90427
$118$	0	0
$119$	0.888474	0.0814463
$120$	0	0
$121$	−6.72725	−0.611569
$122$	0	0
$123$	−19.0787	−1.72027
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−21.3510	−1.89460	−0.947299	−	0.320352i	$-0.896199\pi$
$127$	−21.3510	−1.89460	−0.947299	+	0.320352i	$0.896199\pi$
$128$	0	0
$129$	21.6986	1.91046
$130$	0	0
$131$	9.00760	0.786998	0.393499	−	0.919325i	$-0.371264\pi$
$131$	9.00760	0.786998	0.393499	+	0.919325i	$0.371264\pi$
$132$	0	0
$133$	3.67667	0.318807
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.49151	0.383736	0.191868	−	0.981421i	$-0.438545\pi$
$137$	4.49151	0.383736	0.191868	+	0.981421i	$0.438545\pi$
$138$	0	0
$139$	−13.4039	−1.13691	−0.568453	−	0.822716i	$-0.692458\pi$
$139$	−13.4039	−1.13691	−0.568453	+	0.822716i	$0.692458\pi$
$140$	0	0
$141$	25.1084	2.11451
$142$	0	0
$143$	−11.9757	−1.00146
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.9443	−1.39754
$148$	0	0
$149$	13.6139	1.11529	0.557646	−	0.830079i	$-0.311705\pi$
$149$	13.6139	1.11529	0.557646	+	0.830079i	$0.311705\pi$
$150$	0	0
$151$	−8.79830	−0.715996	−0.357998	−	0.933722i	$-0.616541\pi$
$151$	−8.79830	−0.715996	−0.357998	+	0.933722i	$0.616541\pi$
$152$	0	0
$153$	−4.47027	−0.361400
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.8558	−0.946195	−0.473097	−	0.881010i	$-0.656864\pi$
$157$	−11.8558	−0.946195	−0.473097	+	0.881010i	$0.656864\pi$
$158$	0	0
$159$	−7.45309	−0.591068
$160$	0	0
$161$	2.11507	0.166691
$162$	0	0
$163$	14.9590	1.17168	0.585838	−	0.810428i	$-0.300766\pi$
$163$	14.9590	1.17168	0.585838	+	0.810428i	$0.300766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.5699	−1.66913	−0.834565	−	0.550910i	$-0.814281\pi$
$167$	−21.5699	−1.66913	−0.834565	+	0.550910i	$0.814281\pi$
$168$	0	0
$169$	20.5659	1.58199
$170$	0	0
$171$	−18.4988	−1.41464
$172$	0	0
$173$	4.48526	0.341008	0.170504	−	0.985357i	$-0.445460\pi$
$173$	4.48526	0.341008	0.170504	+	0.985357i	$0.445460\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.6466	−1.02574
$178$	0	0
$179$	15.1323	1.13104	0.565522	−	0.824733i	$-0.308675\pi$
$179$	15.1323	1.13104	0.565522	+	0.824733i	$0.308675\pi$
$180$	0	0
$181$	−9.82108	−0.729995	−0.364998	−	0.931008i	$-0.618930\pi$
$181$	−9.82108	−0.729995	−0.364998	+	0.931008i	$0.618930\pi$
$182$	0	0
$183$	−26.9401	−1.99147
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.70415	−0.124620
$188$	0	0
$189$	−7.57576	−0.551056
$190$	0	0
$191$	−19.4530	−1.40757	−0.703784	−	0.710414i	$-0.748508\pi$
$191$	−19.4530	−1.40757	−0.703784	+	0.710414i	$0.748508\pi$
$192$	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
$197$	1.40687	0.100235	0.0501176	−	0.998743i	$-0.484040\pi$
$197$	1.40687	0.100235	0.0501176	+	0.998743i	$0.484040\pi$
$198$	0	0
$199$	−13.7199	−0.972575	−0.486288	−	0.873799i	$-0.661649\pi$
$199$	−13.7199	−0.972575	−0.486288	+	0.873799i	$0.661649\pi$
$200$	0	0
$201$	−12.6583	−0.892850
$202$	0	0
$203$	1.12942	0.0792700
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.6418	−0.739655
$208$	0	0
$209$	−7.05207	−0.487802
$210$	0	0
$211$	16.7785	1.15508	0.577540	−	0.816363i	$-0.304013\pi$
$211$	16.7785	1.15508	0.577540	+	0.816363i	$0.304013\pi$
$212$	0	0
$213$	−14.2319	−0.975154
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.27938	0.154734
$218$	0	0
$219$	−3.76612	−0.254491
$220$	0	0
$221$	4.77642	0.321297
$222$	0	0
$223$	11.6839	0.782415	0.391207	−	0.920303i	$-0.372058\pi$
$223$	11.6839	0.782415	0.391207	+	0.920303i	$0.372058\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.712904	0.0473171	0.0236585	−	0.999720i	$-0.492469\pi$
$227$	0.712904	0.0473171	0.0236585	+	0.999720i	$0.492469\pi$
$228$	0	0
$229$	20.4169	1.34919	0.674595	−	0.738188i	$-0.264318\pi$
$229$	20.4169	1.34919	0.674595	+	0.738188i	$0.264318\pi$
$230$	0	0
$231$	−6.46486	−0.425357
$232$	0	0
$233$	15.0403	0.985322	0.492661	−	0.870221i	$-0.336024\pi$
$233$	15.0403	0.985322	0.492661	+	0.870221i	$0.336024\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.8826	1.35647
$238$	0	0
$239$	−4.18630	−0.270790	−0.135395	−	0.990792i	$-0.543230\pi$
$239$	−4.18630	−0.270790	−0.135395	+	0.990792i	$0.543230\pi$
$240$	0	0
$241$	−29.4162	−1.89487	−0.947433	−	0.319955i	$-0.896332\pi$
$241$	−29.4162	−1.89487	−0.947433	+	0.319955i	$0.896332\pi$
$242$	0	0
$243$	−9.09132	−0.583209
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.7657	1.25766
$248$	0	0
$249$	43.4389	2.75283
$250$	0	0
$251$	23.6695	1.49400	0.747002	−	0.664822i	$-0.231492\pi$
$251$	23.6695	1.49400	0.747002	+	0.664822i	$0.231492\pi$
$252$	0	0
$253$	−4.05684	−0.255051
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.34928	0.271300	0.135650	−	0.990757i	$-0.456688\pi$
$257$	4.34928	0.271300	0.135650	+	0.990757i	$0.456688\pi$
$258$	0	0
$259$	9.37123	0.582300
$260$	0	0
$261$	−5.68259	−0.351743
$262$	0	0
$263$	2.25698	0.139172	0.0695858	−	0.997576i	$-0.477832\pi$
$263$	2.25698	0.139172	0.0695858	+	0.997576i	$0.477832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−33.9785	−2.07945
$268$	0	0
$269$	−8.55940	−0.521876	−0.260938	−	0.965356i	$-0.584032\pi$
$269$	−8.55940	−0.521876	−0.260938	+	0.965356i	$0.584032\pi$
$270$	0	0
$271$	23.9559	1.45522	0.727610	−	0.685991i	$-0.240631\pi$
$271$	23.9559	1.45522	0.727610	+	0.685991i	$0.240631\pi$
$272$	0	0
$273$	18.1198	1.09666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.12692	0.428215	0.214107	−	0.976810i	$-0.431316\pi$
$277$	7.12692	0.428215	0.214107	+	0.976810i	$0.431316\pi$
$278$	0	0
$279$	−11.4685	−0.686600
$280$	0	0
$281$	2.25731	0.134660	0.0673299	−	0.997731i	$-0.478552\pi$
$281$	2.25731	0.134660	0.0673299	+	0.997731i	$0.478552\pi$
$282$	0	0
$283$	−9.71760	−0.577652	−0.288826	−	0.957382i	$-0.593265\pi$
$283$	−9.71760	−0.577652	−0.288826	+	0.957382i	$0.593265\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.08478	0.418201
$288$	0	0
$289$	−16.3203	−0.960019
$290$	0	0
$291$	−1.48205	−0.0868790
$292$	0	0
$293$	−6.52369	−0.381118	−0.190559	−	0.981676i	$-0.561030\pi$
$293$	−6.52369	−0.381118	−0.190559	+	0.981676i	$0.561030\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.5308	0.843161
$298$	0	0
$299$	11.3706	0.657578
$300$	0	0
$301$	−8.05766	−0.464436
$302$	0	0
$303$	−10.1796	−0.584804
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.7081	−1.18187	−0.590937	−	0.806718i	$-0.701242\pi$
$307$	−20.7081	−1.18187	−0.590937	+	0.806718i	$0.701242\pi$
$308$	0	0
$309$	13.4710	0.766337
$310$	0	0
$311$	−7.96917	−0.451890	−0.225945	−	0.974140i	$-0.572547\pi$
$311$	−7.96917	−0.451890	−0.225945	+	0.974140i	$0.572547\pi$
$312$	0	0
$313$	4.69031	0.265112	0.132556	−	0.991176i	$-0.457682\pi$
$313$	4.69031	0.265112	0.132556	+	0.991176i	$0.457682\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.2996	0.803145	0.401572	−	0.915827i	$-0.368464\pi$
$317$	14.2996	0.803145	0.401572	+	0.915827i	$0.368464\pi$
$318$	0	0
$319$	−2.16630	−0.121290
$320$	0	0
$321$	−22.0615	−1.23135
$322$	0	0
$323$	2.81266	0.156500
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.66676	−0.479273
$328$	0	0
$329$	−9.32386	−0.514041
$330$	0	0
$331$	−35.5705	−1.95513	−0.977566	−	0.210629i	$-0.932449\pi$
$331$	−35.5705	−1.95513	−0.977566	+	0.210629i	$0.932449\pi$
$332$	0	0
$333$	−47.1504	−2.58383
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.56224	0.411941	0.205971	−	0.978558i	$-0.433965\pi$
$337$	7.56224	0.411941	0.205971	+	0.978558i	$0.433965\pi$
$338$	0	0
$339$	58.6185	3.18372
$340$	0	0
$341$	−4.37199	−0.236756
$342$	0	0
$343$	13.8359	0.747071
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.7457	−0.791589	−0.395794	−	0.918339i	$-0.629531\pi$
$347$	−14.7457	−0.791589	−0.395794	+	0.918339i	$0.629531\pi$
$348$	0	0
$349$	−7.61178	−0.407449	−0.203725	−	0.979028i	$-0.565305\pi$
$349$	−7.61178	−0.407449	−0.203725	+	0.979028i	$0.565305\pi$
$350$	0	0
$351$	−40.7271	−2.17385
$352$	0	0
$353$	32.1662	1.71203	0.856017	−	0.516948i	$-0.172932\pi$
$353$	32.1662	1.71203	0.856017	+	0.516948i	$0.172932\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.57845	0.136466
$358$	0	0
$359$	34.9455	1.84435	0.922177	−	0.386768i	$-0.126409\pi$
$359$	34.9455	1.84435	0.922177	+	0.386768i	$0.126409\pi$
$360$	0	0
$361$	−7.36072	−0.387406
$362$	0	0
$363$	−19.5233	−1.02471
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0433	0.576456	0.288228	−	0.957562i	$-0.406934\pi$
$367$	11.0433	0.576456	0.288228	+	0.957562i	$0.406934\pi$
$368$	0	0
$369$	−35.6464	−1.85568
$370$	0	0
$371$	2.76766	0.143690
$372$	0	0
$373$	−20.4400	−1.05835	−0.529173	−	0.848514i	$-0.677498\pi$
$373$	−20.4400	−1.05835	−0.529173	+	0.848514i	$0.677498\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.07176	0.312712
$378$	0	0
$379$	3.49389	0.179469	0.0897345	−	0.995966i	$-0.471398\pi$
$379$	3.49389	0.179469	0.0897345	+	0.995966i	$0.471398\pi$
$380$	0	0
$381$	−61.9631	−3.17447
$382$	0	0
$383$	19.5881	1.00091	0.500453	−	0.865763i	$-0.333167\pi$
$383$	19.5881	1.00091	0.500453	+	0.865763i	$0.333167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	40.5413	2.06083
$388$	0	0
$389$	10.2195	0.518148	0.259074	−	0.965857i	$-0.416583\pi$
$389$	10.2195	0.518148	0.259074	+	0.965857i	$0.416583\pi$
$390$	0	0
$391$	1.61803	0.0818275
$392$	0	0
$393$	26.1411	1.31864
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.5054	1.12951	0.564757	−	0.825257i	$-0.308970\pi$
$397$	22.5054	1.12951	0.564757	+	0.825257i	$0.308970\pi$
$398$	0	0
$399$	10.6701	0.534173
$400$	0	0
$401$	30.3084	1.51353	0.756765	−	0.653687i	$-0.226779\pi$
$401$	30.3084	1.51353	0.756765	+	0.653687i	$0.226779\pi$
$402$	0	0
$403$	12.2539	0.610410
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.9746	−0.890967
$408$	0	0
$409$	−15.4128	−0.762113	−0.381057	−	0.924552i	$-0.624440\pi$
$409$	−15.4128	−0.762113	−0.381057	+	0.924552i	$0.624440\pi$
$410$	0	0
$411$	13.0349	0.642963
$412$	0	0
$413$	5.06757	0.249359
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−38.8997	−1.90493
$418$	0	0
$419$	−21.2300	−1.03715	−0.518577	−	0.855031i	$-0.673538\pi$
$419$	−21.2300	−1.03715	−0.518577	+	0.855031i	$0.673538\pi$
$420$	0	0
$421$	32.8220	1.59965	0.799823	−	0.600235i	$-0.204926\pi$
$421$	32.8220	1.59965	0.799823	+	0.600235i	$0.204926\pi$
$422$	0	0
$423$	46.9121	2.28094
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.0041	0.484130
$428$	0	0
$429$	−34.7549	−1.67798
$430$	0	0
$431$	0.840790	0.0404995	0.0202497	−	0.999795i	$-0.493554\pi$
$431$	0.840790	0.0404995	0.0202497	+	0.999795i	$0.493554\pi$
$432$	0	0
$433$	−34.0351	−1.63562	−0.817811	−	0.575487i	$-0.804813\pi$
$433$	−34.0351	−1.63562	−0.817811	+	0.575487i	$0.804813\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.69572	0.320300
$438$	0	0
$439$	1.59313	0.0760360	0.0380180	−	0.999277i	$-0.487896\pi$
$439$	1.59313	0.0760360	0.0380180	+	0.999277i	$0.487896\pi$
$440$	0	0
$441$	−31.6584	−1.50754
$442$	0	0
$443$	24.6238	1.16991	0.584955	−	0.811065i	$-0.301112\pi$
$443$	24.6238	1.16991	0.584955	+	0.811065i	$0.301112\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.5090	1.86871
$448$	0	0
$449$	−34.1186	−1.61016	−0.805078	−	0.593169i	$-0.797876\pi$
$449$	−34.1186	−1.61016	−0.805078	+	0.593169i	$0.797876\pi$
$450$	0	0
$451$	−13.5890	−0.639882
$452$	0	0
$453$	−25.5337	−1.19968
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.48932	−0.163224	−0.0816118	−	0.996664i	$-0.526007\pi$
$457$	−3.48932	−0.163224	−0.0816118	+	0.996664i	$0.526007\pi$
$458$	0	0
$459$	−5.79547	−0.270509
$460$	0	0
$461$	13.4852	0.628066	0.314033	−	0.949412i	$-0.398320\pi$
$461$	13.4852	0.628066	0.314033	+	0.949412i	$0.398320\pi$
$462$	0	0
$463$	1.86401	0.0866279	0.0433140	−	0.999062i	$-0.486208\pi$
$463$	1.86401	0.0866279	0.0433140	+	0.999062i	$0.486208\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.0415	0.603489	0.301745	−	0.953389i	$-0.402431\pi$
$467$	13.0415	0.603489	0.301745	+	0.953389i	$0.402431\pi$
$468$	0	0
$469$	4.70060	0.217054
$470$	0	0
$471$	−34.4068	−1.58538
$472$	0	0
$473$	15.4551	0.710625
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.9252	−0.637592
$478$	0	0
$479$	−12.3819	−0.565741	−0.282871	−	0.959158i	$-0.591287\pi$
$479$	−12.3819	−0.565741	−0.282871	+	0.959158i	$0.591287\pi$
$480$	0	0
$481$	50.3795	2.29711
$482$	0	0
$483$	6.13818	0.279297
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.51906	−0.204778	−0.102389	−	0.994744i	$-0.532649\pi$
$487$	−4.51906	−0.204778	−0.102389	+	0.994744i	$0.532649\pi$
$488$	0	0
$489$	43.4126	1.96318
$490$	0	0
$491$	−27.4676	−1.23960	−0.619799	−	0.784761i	$-0.712786\pi$
$491$	−27.4676	−1.23960	−0.619799	+	0.784761i	$0.712786\pi$
$492$	0	0
$493$	0.864011	0.0389131
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.28494	0.237062
$498$	0	0
$499$	−13.4814	−0.603510	−0.301755	−	0.953385i	$-0.597573\pi$
$499$	−13.4814	−0.603510	−0.301755	+	0.953385i	$0.597573\pi$
$500$	0	0
$501$	−62.5983	−2.79669
$502$	0	0
$503$	−27.8294	−1.24085	−0.620426	−	0.784265i	$-0.713040\pi$
$503$	−27.8294	−1.24085	−0.620426	+	0.784265i	$0.713040\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	59.6844	2.65068
$508$	0	0
$509$	−20.0695	−0.889566	−0.444783	−	0.895638i	$-0.646719\pi$
$509$	−20.0695	−0.889566	−0.444783	+	0.895638i	$0.646719\pi$
$510$	0	0
$511$	1.39853	0.0618673
$512$	0	0
$513$	−23.9827	−1.05886
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.8837	0.786526
$518$	0	0
$519$	13.0167	0.571372
$520$	0	0
$521$	−29.1589	−1.27748	−0.638738	−	0.769425i	$-0.720543\pi$
$521$	−29.1589	−1.27748	−0.638738	+	0.769425i	$0.720543\pi$
$522$	0	0
$523$	−11.7059	−0.511863	−0.255932	−	0.966695i	$-0.582382\pi$
$523$	−11.7059	−0.511863	−0.255932	+	0.966695i	$0.582382\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.74373	0.0759580
$528$	0	0
$529$	−19.1482	−0.832529
$530$	0	0
$531$	−25.4970	−1.10648
$532$	0	0
$533$	38.0876	1.64976
$534$	0	0
$535$	0	0
$536$	0	0
$537$	43.9157	1.89510
$538$	0	0
$539$	−12.0687	−0.519838
$540$	0	0
$541$	−29.9196	−1.28634	−0.643172	−	0.765722i	$-0.722382\pi$
$541$	−29.9196	−1.28634	−0.643172	+	0.765722i	$0.722382\pi$
$542$	0	0
$543$	−28.5019	−1.22313
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.89181	0.0808878	0.0404439	−	0.999182i	$-0.487123\pi$
$547$	1.89181	0.0808878	0.0404439	+	0.999182i	$0.487123\pi$
$548$	0	0
$549$	−50.3344	−2.14822
$550$	0	0
$551$	3.57543	0.152319
$552$	0	0
$553$	−7.75465	−0.329761
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.95618	0.125257	0.0626287	−	0.998037i	$-0.480052\pi$
$557$	2.95618	0.125257	0.0626287	+	0.998037i	$0.480052\pi$
$558$	0	0
$559$	−43.3178	−1.83215
$560$	0	0
$561$	−4.94563	−0.208805
$562$	0	0
$563$	−9.28304	−0.391233	−0.195617	−	0.980680i	$-0.562671\pi$
$563$	−9.28304	−0.391233	−0.195617	+	0.980680i	$0.562671\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.45528	−0.187104
$568$	0	0
$569$	−14.4699	−0.606612	−0.303306	−	0.952893i	$-0.598090\pi$
$569$	−14.4699	−0.606612	−0.303306	+	0.952893i	$0.598090\pi$
$570$	0	0
$571$	5.08844	0.212944	0.106472	−	0.994316i	$-0.466044\pi$
$571$	5.08844	0.212944	0.106472	+	0.994316i	$0.466044\pi$
$572$	0	0
$573$	−56.4547	−2.35843
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.8746	0.702498	0.351249	−	0.936282i	$-0.385757\pi$
$577$	16.8746	0.702498	0.351249	+	0.936282i	$0.385757\pi$
$578$	0	0
$579$	−17.4127	−0.723646
$580$	0	0
$581$	−16.1308	−0.669218
$582$	0	0
$583$	−5.30854	−0.219857
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.0945	1.81998	0.909988	−	0.414635i	$-0.136091\pi$
$587$	44.0945	1.81998	0.909988	+	0.414635i	$0.136091\pi$
$588$	0	0
$589$	7.21586	0.297325
$590$	0	0
$591$	4.08289	0.167948
$592$	0	0
$593$	31.1398	1.27876	0.639380	−	0.768891i	$-0.279191\pi$
$593$	31.1398	1.27876	0.639380	+	0.768891i	$0.279191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39.8166	−1.62958
$598$	0	0
$599$	−28.6194	−1.16936	−0.584678	−	0.811265i	$-0.698779\pi$
$599$	−28.6194	−1.16936	−0.584678	+	0.811265i	$0.698779\pi$
$600$	0	0
$601$	38.8122	1.58318	0.791592	−	0.611050i	$-0.209253\pi$
$601$	38.8122	1.58318	0.791592	+	0.611050i	$0.209253\pi$
$602$	0	0
$603$	−23.6506	−0.963127
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.78331	0.275326	0.137663	−	0.990479i	$-0.456041\pi$
$607$	6.78331	0.275326	0.137663	+	0.990479i	$0.456041\pi$
$608$	0	0
$609$	3.27772	0.132820
$610$	0	0
$611$	−50.1249	−2.02784
$612$	0	0
$613$	−33.8251	−1.36618	−0.683092	−	0.730332i	$-0.739365\pi$
$613$	−33.8251	−1.36618	−0.683092	+	0.730332i	$0.739365\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.1377	−0.528906	−0.264453	−	0.964399i	$-0.585191\pi$
$617$	−13.1377	−0.528906	−0.264453	+	0.964399i	$0.585191\pi$
$618$	0	0
$619$	28.0385	1.12696	0.563481	−	0.826129i	$-0.309462\pi$
$619$	28.0385	1.12696	0.563481	+	0.826129i	$0.309462\pi$
$620$	0	0
$621$	−13.7965	−0.553635
$622$	0	0
$623$	12.6177	0.505519
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.4659	−0.817329
$628$	0	0
$629$	7.16901	0.285847
$630$	0	0
$631$	28.4473	1.13247	0.566235	−	0.824244i	$-0.308400\pi$
$631$	28.4473	1.13247	0.566235	+	0.824244i	$0.308400\pi$
$632$	0	0
$633$	48.6931	1.93538
$634$	0	0
$635$	0	0
$636$	0	0
$637$	33.8265	1.34026
$638$	0	0
$639$	−26.5906	−1.05191
$640$	0	0
$641$	25.4933	1.00692	0.503462	−	0.864018i	$-0.332059\pi$
$641$	25.4933	1.00692	0.503462	+	0.864018i	$0.332059\pi$
$642$	0	0
$643$	13.7340	0.541617	0.270809	−	0.962633i	$-0.412709\pi$
$643$	13.7340	0.541617	0.270809	+	0.962633i	$0.412709\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.5285	−1.51471	−0.757355	−	0.653003i	$-0.773509\pi$
$647$	−38.5285	−1.51471	−0.757355	+	0.653003i	$0.773509\pi$
$648$	0	0
$649$	−9.71991	−0.381540
$650$	0	0
$651$	6.61502	0.259263
$652$	0	0
$653$	−13.8623	−0.542476	−0.271238	−	0.962512i	$-0.587433\pi$
$653$	−13.8623	−0.542476	−0.271238	+	0.962512i	$0.587433\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.03656	−0.274523
$658$	0	0
$659$	−27.0822	−1.05497	−0.527486	−	0.849564i	$-0.676865\pi$
$659$	−27.0822	−1.05497	−0.527486	+	0.849564i	$0.676865\pi$
$660$	0	0
$661$	−23.4629	−0.912600	−0.456300	−	0.889826i	$-0.650826\pi$
$661$	−23.4629	−0.912600	−0.456300	+	0.889826i	$0.650826\pi$
$662$	0	0
$663$	13.8617	0.538344
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.05684	0.0796411
$668$	0	0
$669$	33.9081	1.31096
$670$	0	0
$671$	−19.1884	−0.740759
$672$	0	0
$673$	10.9949	0.423821	0.211911	−	0.977289i	$-0.432031\pi$
$673$	10.9949	0.423821	0.211911	+	0.977289i	$0.432031\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−36.6580	−1.40888	−0.704440	−	0.709764i	$-0.748802\pi$
$677$	−36.6580	−1.40888	−0.704440	+	0.709764i	$0.748802\pi$
$678$	0	0
$679$	0.550349	0.0211205
$680$	0	0
$681$	2.06893	0.0792814
$682$	0	0
$683$	10.2158	0.390897	0.195448	−	0.980714i	$-0.437384\pi$
$683$	10.2158	0.390897	0.195448	+	0.980714i	$0.437384\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	59.2523	2.26061
$688$	0	0
$689$	14.8789	0.566841
$690$	0	0
$691$	−5.44433	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$691$	−5.44433	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$692$	0	0
$693$	−12.0788	−0.458837
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.41987	0.205292
$698$	0	0
$699$	43.6486	1.65094
$700$	0	0
$701$	13.9633	0.527387	0.263694	−	0.964606i	$-0.415059\pi$
$701$	13.9633	0.527387	0.263694	+	0.964606i	$0.415059\pi$
$702$	0	0
$703$	29.6666	1.11890
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.78015	0.142167
$708$	0	0
$709$	19.3787	0.727782	0.363891	−	0.931441i	$-0.381448\pi$
$709$	19.3787	0.727782	0.363891	+	0.931441i	$0.381448\pi$
$710$	0	0
$711$	39.0167	1.46324
$712$	0	0
$713$	4.15106	0.155459
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.1491	−0.453718
$718$	0	0
$719$	−9.85421	−0.367500	−0.183750	−	0.982973i	$-0.558824\pi$
$719$	−9.85421	−0.367500	−0.183750	+	0.982973i	$0.558824\pi$
$720$	0	0
$721$	−5.00237	−0.186298
$722$	0	0
$723$	−85.3692	−3.17492
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.4862	1.31611	0.658055	−	0.752970i	$-0.271380\pi$
$727$	35.4862	1.31611	0.658055	+	0.752970i	$0.271380\pi$
$728$	0	0
$729$	−38.7864	−1.43653
$730$	0	0
$731$	−6.16412	−0.227988
$732$	0	0
$733$	−27.5799	−1.01869	−0.509343	−	0.860563i	$-0.670112\pi$
$733$	−27.5799	−1.01869	−0.509343	+	0.860563i	$0.670112\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.01603	−0.332110
$738$	0	0
$739$	38.7168	1.42422	0.712111	−	0.702067i	$-0.247739\pi$
$739$	38.7168	1.42422	0.712111	+	0.702067i	$0.247739\pi$
$740$	0	0
$741$	57.3622	2.10725
$742$	0	0
$743$	−13.3434	−0.489520	−0.244760	−	0.969584i	$-0.578709\pi$
$743$	−13.3434	−0.489520	−0.244760	+	0.969584i	$0.578709\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	81.1605	2.96951
$748$	0	0
$749$	8.19243	0.299345
$750$	0	0
$751$	−13.3487	−0.487099	−0.243550	−	0.969888i	$-0.578312\pi$
$751$	−13.3487	−0.487099	−0.243550	+	0.969888i	$0.578312\pi$
$752$	0	0
$753$	68.6915	2.50326
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.3735	−1.21298	−0.606490	−	0.795091i	$-0.707423\pi$
$757$	−33.3735	−1.21298	−0.606490	+	0.795091i	$0.707423\pi$
$758$	0	0
$759$	−11.7734	−0.427347
$760$	0	0
$761$	−2.19559	−0.0795901	−0.0397951	−	0.999208i	$-0.512671\pi$
$761$	−2.19559	−0.0795901	−0.0397951	+	0.999208i	$0.512671\pi$
$762$	0	0
$763$	3.21835	0.116512
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.2432	0.983694
$768$	0	0
$769$	17.3240	0.624719	0.312359	−	0.949964i	$-0.398881\pi$
$769$	17.3240	0.624719	0.312359	+	0.949964i	$0.398881\pi$
$770$	0	0
$771$	12.6221	0.454574
$772$	0	0
$773$	16.3460	0.587925	0.293962	−	0.955817i	$-0.405026\pi$
$773$	16.3460	0.587925	0.293962	+	0.955817i	$0.405026\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.1964	0.975664
$778$	0	0
$779$	22.4284	0.803580
$780$	0	0
$781$	−10.1368	−0.362724
$782$	0	0
$783$	−7.36717	−0.263281
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−49.0498	−1.74844	−0.874219	−	0.485532i	$-0.838626\pi$
$787$	−49.0498	−1.74844	−0.874219	+	0.485532i	$0.838626\pi$
$788$	0	0
$789$	6.55002	0.233187
$790$	0	0
$791$	−21.7677	−0.773969
$792$	0	0
$793$	53.7816	1.90984
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.7523	1.37268	0.686339	−	0.727282i	$-0.259217\pi$
$797$	38.7523	1.37268	0.686339	+	0.727282i	$0.259217\pi$
$798$	0	0
$799$	−7.13277	−0.252339
$800$	0	0
$801$	−63.4849	−2.24313
$802$	0	0
$803$	−2.68246	−0.0946621
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.8404	−0.874422
$808$	0	0
$809$	11.5682	0.406715	0.203357	−	0.979105i	$-0.434815\pi$
$809$	11.5682	0.406715	0.203357	+	0.979105i	$0.434815\pi$
$810$	0	0
$811$	−44.6023	−1.56620	−0.783099	−	0.621897i	$-0.786362\pi$
$811$	−44.6023	−1.56620	−0.783099	+	0.621897i	$0.786362\pi$
$812$	0	0
$813$	69.5228	2.43827
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25.5083	−0.892421
$818$	0	0
$819$	33.8548	1.18298
$820$	0	0
$821$	38.1518	1.33151	0.665753	−	0.746172i	$-0.268110\pi$
$821$	38.1518	1.33151	0.665753	+	0.746172i	$0.268110\pi$
$822$	0	0
$823$	50.9265	1.77519	0.887594	−	0.460627i	$-0.152376\pi$
$823$	50.9265	1.77519	0.887594	+	0.460627i	$0.152376\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.9354	1.11050	0.555251	−	0.831683i	$-0.312622\pi$
$827$	31.9354	1.11050	0.555251	+	0.831683i	$0.312622\pi$
$828$	0	0
$829$	−32.1729	−1.11741	−0.558705	−	0.829366i	$-0.688702\pi$
$829$	−32.1729	−1.11741	−0.558705	+	0.829366i	$0.688702\pi$
$830$	0	0
$831$	20.6831	0.717489
$832$	0	0
$833$	4.81351	0.166778
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−14.8683	−0.513923
$838$	0	0
$839$	30.1317	1.04026	0.520131	−	0.854086i	$-0.325883\pi$
$839$	30.1317	1.04026	0.520131	+	0.854086i	$0.325883\pi$
$840$	0	0
$841$	−27.9017	−0.962127
$842$	0	0
$843$	6.55097	0.225627
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.24985	0.249108
$848$	0	0
$849$	−28.2016	−0.967876
$850$	0	0
$851$	17.0663	0.585025
$852$	0	0
$853$	21.5034	0.736263	0.368132	−	0.929774i	$-0.379998\pi$
$853$	21.5034	0.736263	0.368132	+	0.929774i	$0.379998\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.1894	1.37284	0.686422	−	0.727204i	$-0.259181\pi$
$857$	40.1894	1.37284	0.686422	+	0.727204i	$0.259181\pi$
$858$	0	0
$859$	53.2136	1.81562	0.907812	−	0.419378i	$-0.137752\pi$
$859$	53.2136	1.81562	0.907812	+	0.419378i	$0.137752\pi$
$860$	0	0
$861$	20.5608	0.700711
$862$	0	0
$863$	10.9118	0.371443	0.185722	−	0.982602i	$-0.440538\pi$
$863$	10.9118	0.371443	0.185722	+	0.982602i	$0.440538\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−47.3634	−1.60855
$868$	0	0
$869$	14.8739	0.504562
$870$	0	0
$871$	25.2703	0.856252
$872$	0	0
$873$	−2.76903	−0.0937174
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.65915	0.191096	0.0955480	−	0.995425i	$-0.469540\pi$
$877$	5.65915	0.191096	0.0955480	+	0.995425i	$0.469540\pi$
$878$	0	0
$879$	−18.9325	−0.638577
$880$	0	0
$881$	15.2786	0.514748	0.257374	−	0.966312i	$-0.417143\pi$
$881$	15.2786	0.514748	0.257374	+	0.966312i	$0.417143\pi$
$882$	0	0
$883$	25.5448	0.859652	0.429826	−	0.902912i	$-0.358575\pi$
$883$	25.5448	0.859652	0.429826	+	0.902912i	$0.358575\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.68855	−0.0566961	−0.0283480	−	0.999598i	$-0.509025\pi$
$887$	−1.68855	−0.0566961	−0.0283480	+	0.999598i	$0.509025\pi$
$888$	0	0
$889$	23.0097	0.771719
$890$	0	0
$891$	8.54549	0.286285
$892$	0	0
$893$	−29.5167	−0.987738
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.9987	1.10180
$898$	0	0
$899$	2.21662	0.0739284
$900$	0	0
$901$	2.11727	0.0705363
$902$	0	0
$903$	−23.3842	−0.778179
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.58315	−0.0525677	−0.0262838	−	0.999655i	$-0.508367\pi$
$907$	−1.58315	−0.0525677	−0.0262838	+	0.999655i	$0.508367\pi$
$908$	0	0
$909$	−19.0194	−0.630835
$910$	0	0
$911$	4.16710	0.138062	0.0690311	−	0.997615i	$-0.478009\pi$
$911$	4.16710	0.138062	0.0690311	+	0.997615i	$0.478009\pi$
$912$	0	0
$913$	30.9398	1.02396
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.70735	−0.320565
$918$	0	0
$919$	36.9454	1.21872	0.609359	−	0.792895i	$-0.291427\pi$
$919$	36.9454	1.21872	0.609359	+	0.792895i	$0.291427\pi$
$920$	0	0
$921$	−60.0972	−1.98027
$922$	0	0
$923$	28.4117	0.935183
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.1690	0.826657
$928$	0	0
$929$	−13.3263	−0.437222	−0.218611	−	0.975812i	$-0.570153\pi$
$929$	−13.3263	−0.437222	−0.218611	+	0.975812i	$0.570153\pi$
$930$	0	0
$931$	19.9192	0.652825
$932$	0	0
$933$	−23.1274	−0.757159
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−47.3006	−1.54524	−0.772622	−	0.634866i	$-0.781055\pi$
$937$	−47.3006	−1.54524	−0.772622	+	0.634866i	$0.781055\pi$
$938$	0	0
$939$	13.6118	0.444204
$940$	0	0
$941$	20.0960	0.655112	0.327556	−	0.944832i	$-0.393775\pi$
$941$	20.0960	0.655112	0.327556	+	0.944832i	$0.393775\pi$
$942$	0	0
$943$	12.9024	0.420159
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.6114	1.02723	0.513617	−	0.858020i	$-0.328305\pi$
$947$	31.6114	1.02723	0.513617	+	0.858020i	$0.328305\pi$
$948$	0	0
$949$	7.51846	0.244060
$950$	0	0
$951$	41.4990	1.34570
$952$	0	0
$953$	−30.6423	−0.992601	−0.496300	−	0.868151i	$-0.665308\pi$
$953$	−30.6423	−0.992601	−0.496300	+	0.868151i	$0.665308\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.28686	−0.203225
$958$	0	0
$959$	−4.84043	−0.156306
$960$	0	0
$961$	−26.5265	−0.855692
$962$	0	0
$963$	−41.2194	−1.32828
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.9179	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$967$	−15.9179	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$968$	0	0
$969$	8.16264	0.262222
$970$	0	0
$971$	−50.8591	−1.63215	−0.816073	−	0.577949i	$-0.803853\pi$
$971$	−50.8591	−1.63215	−0.816073	+	0.577949i	$0.803853\pi$
$972$	0	0
$973$	14.4452	0.463091
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.5827	−0.530526	−0.265263	−	0.964176i	$-0.585459\pi$
$977$	−16.5827	−0.530526	−0.265263	+	0.964176i	$0.585459\pi$
$978$	0	0
$979$	−24.2016	−0.773486
$980$	0	0
$981$	−16.1928	−0.516997
$982$	0	0
$983$	2.68779	0.0857271	0.0428635	−	0.999081i	$-0.486352\pi$
$983$	2.68779	0.0857271	0.0428635	+	0.999081i	$0.486352\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−27.0589	−0.861294
$988$	0	0
$989$	−14.6741	−0.466610
$990$	0	0
$991$	19.2493	0.611473	0.305737	−	0.952116i	$-0.401097\pi$
$991$	19.2493	0.611473	0.305737	+	0.952116i	$0.401097\pi$
$992$	0	0
$993$	−103.230	−3.27589
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.9478	1.36017	0.680086	−	0.733132i	$-0.261942\pi$
$997$	42.9478	1.36017	0.680086	+	0.733132i	$0.261942\pi$
$998$	0	0
$999$	−61.1280	−1.93401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bb.1.4		4
4.3	odd	2		1250.2.a.i.1.1		4
5.4	even	2		10000.2.a.o.1.1		4
20.3	even	4		1250.2.b.c.1249.1		8
20.7	even	4		1250.2.b.c.1249.8		8
20.19	odd	2		1250.2.a.h.1.4		4
25.12	odd	20		400.2.y.a.369.2		8
25.23	odd	20		400.2.y.a.129.2		8
100.11	odd	10		250.2.d.b.101.2		8
100.23	even	20		50.2.e.a.29.1	yes	8
100.27	even	20		250.2.e.a.149.2		8
100.39	odd	10		250.2.d.c.101.1		8
100.59	odd	10		250.2.d.c.151.1		8
100.63	even	20		250.2.e.a.99.2		8
100.87	even	20		50.2.e.a.19.1	✓	8
100.91	odd	10		250.2.d.b.151.2		8
300.23	odd	20		450.2.l.b.379.2		8
300.287	odd	20		450.2.l.b.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.e.a.19.1	✓	8	100.87	even	20
50.2.e.a.29.1	yes	8	100.23	even	20
250.2.d.b.101.2		8	100.11	odd	10
250.2.d.b.151.2		8	100.91	odd	10
250.2.d.c.101.1		8	100.39	odd	10
250.2.d.c.151.1		8	100.59	odd	10
250.2.e.a.99.2		8	100.63	even	20
250.2.e.a.149.2		8	100.27	even	20
400.2.y.a.129.2		8	25.23	odd	20
400.2.y.a.369.2		8	25.12	odd	20
450.2.l.b.19.2		8	300.287	odd	20
450.2.l.b.379.2		8	300.23	odd	20
1250.2.a.h.1.4		4	20.19	odd	2
1250.2.a.i.1.1		4	4.3	odd	2
1250.2.b.c.1249.1		8	20.3	even	4
1250.2.b.c.1249.8		8	20.7	even	4
10000.2.a.o.1.1		4	5.4	even	2
10000.2.a.bb.1.4		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)

\(f(q)\)	\(=\)	\(q+2.90211 q^{3} -1.07768 q^{7} +5.42226 q^{9} +O(q^{10})\)	\(q+2.90211 q^{3} -1.07768 q^{7} +5.42226 q^{9} +2.06706 q^{11} -5.79360 q^{13} -0.824429 q^{17} -3.41164 q^{19} -3.12756 q^{21} -1.96261 q^{23} +7.02967 q^{27} -1.04801 q^{29} -2.11507 q^{31} +5.99885 q^{33} -8.69572 q^{37} -16.8137 q^{39} -6.57408 q^{41} +7.47684 q^{43} +8.65176 q^{47} -5.83860 q^{49} -2.39259 q^{51} -2.56816 q^{53} -9.90096 q^{57} -4.70228 q^{59} -9.28293 q^{61} -5.84348 q^{63} -4.36176 q^{67} -5.69572 q^{69} -4.90398 q^{71} -1.29772 q^{73} -2.22764 q^{77} +7.19566 q^{79} +4.13412 q^{81} +14.9680 q^{83} -3.04145 q^{87} -11.7082 q^{89} +6.24367 q^{91} -6.13818 q^{93} -0.510678 q^{97} +11.2081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) 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

L-functions

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