LMFDB - Embedded newform 10000.2.a.w.1.1

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:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1250)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.82709$ 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)

$f(q)$	$=$	$q-1.95630 q^{3} -3.91259 q^{7} +0.827091 q^{9} +O(q^{10})$	$q-1.95630 q^{3} -3.91259 q^{7} +0.827091 q^{9} -1.67652 q^{11} +5.14866 q^{13} -6.68119 q^{17} -6.15622 q^{19} +7.65418 q^{21} +1.84029 q^{23} +4.25085 q^{27} +5.07636 q^{29} +5.56677 q^{31} +3.27977 q^{33} -6.21373 q^{37} -10.0723 q^{39} +5.56210 q^{41} -8.11251 q^{43} +10.2433 q^{47} +8.30836 q^{49} +13.0704 q^{51} -0.836228 q^{53} +12.0434 q^{57} +3.82242 q^{59} +5.29902 q^{61} -3.23607 q^{63} -5.63940 q^{67} -3.60016 q^{69} -1.60016 q^{71} +0.128228 q^{73} +6.55955 q^{77} +9.47936 q^{79} -10.7972 q^{81} +9.45702 q^{83} -9.93086 q^{87} +15.6356 q^{89} -20.1446 q^{91} -10.8902 q^{93} -10.5623 q^{97} -1.38664 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^7 - 3 * q^9	$ 4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 6 * q^13 - 7 * q^17 - 15 * q^19 + 18 * q^21 + 6 * q^23 - 5 * q^27 + 10 * q^29 - 8 * q^31 + 13 * q^33 - 12 * q^37 - 24 * q^39 + 3 * q^41 - 14 * q^43 + 2 * q^47 + 8 * q^49 + 7 * q^51 + 4 * q^53 + 10 * q^57 + 20 * q^59 + 18 * q^61 - 4 * q^63 - 23 * q^67 - 16 * q^69 - 8 * q^71 - q^73 + 26 * q^77 - 10 * q^79 - 16 * q^81 - 14 * q^83 - 20 * q^87 + 5 * q^89 - 48 * q^91 - 22 * q^93 - 2 * q^97 + 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
$3$	−1.95630	−1.12947	−0.564734	−	0.825273i	$-0.691021\pi$
$3$	−1.95630	−1.12947	−0.564734	+	0.825273i	$0.691021\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.91259	−1.47882	−0.739410	−	0.673255i	$-0.764895\pi$
$7$	−3.91259	−1.47882	−0.739410	+	0.673255i	$0.764895\pi$
$8$	0	0
$9$	0.827091	0.275697
$10$	0	0
$11$	−1.67652	−0.505491	−0.252745	−	0.967533i	$-0.581333\pi$
$11$	−1.67652	−0.505491	−0.252745	+	0.967533i	$0.581333\pi$
$12$	0	0
$13$	5.14866	1.42798	0.713990	−	0.700155i	$-0.246886\pi$
$13$	5.14866	1.42798	0.713990	+	0.700155i	$0.246886\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.68119	−1.62043	−0.810214	−	0.586135i	$-0.800649\pi$
$17$	−6.68119	−1.62043	−0.810214	+	0.586135i	$0.800649\pi$
$18$	0	0
$19$	−6.15622	−1.41233	−0.706166	−	0.708046i	$-0.749577\pi$
$19$	−6.15622	−1.41233	−0.706166	+	0.708046i	$0.749577\pi$
$20$	0	0
$21$	7.65418	1.67028
$22$	0	0
$23$	1.84029	0.383728	0.191864	−	0.981422i	$-0.438547\pi$
$23$	1.84029	0.383728	0.191864	+	0.981422i	$0.438547\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.25085	0.818077
$28$	0	0
$29$	5.07636	0.942657	0.471328	−	0.881958i	$-0.343775\pi$
$29$	5.07636	0.942657	0.471328	+	0.881958i	$0.343775\pi$
$30$	0	0
$31$	5.56677	0.999822	0.499911	−	0.866077i	$-0.333366\pi$
$31$	5.56677	0.999822	0.499911	+	0.866077i	$0.333366\pi$
$32$	0	0
$33$	3.27977	0.570935
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.21373	−1.02153	−0.510765	−	0.859720i	$-0.670638\pi$
$37$	−6.21373	−1.02153	−0.510765	+	0.859720i	$0.670638\pi$
$38$	0	0
$39$	−10.0723	−1.61286
$40$	0	0
$41$	5.56210	0.868654	0.434327	−	0.900755i	$-0.356986\pi$
$41$	5.56210	0.868654	0.434327	+	0.900755i	$0.356986\pi$
$42$	0	0
$43$	−8.11251	−1.23715	−0.618573	−	0.785727i	$-0.712289\pi$
$43$	−8.11251	−1.23715	−0.618573	+	0.785727i	$0.712289\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.2433	1.49414	0.747069	−	0.664746i	$-0.231460\pi$
$47$	10.2433	1.49414	0.747069	+	0.664746i	$0.231460\pi$
$48$	0	0
$49$	8.30836	1.18691
$50$	0	0
$51$	13.0704	1.83022
$52$	0	0
$53$	−0.836228	−0.114865	−0.0574324	−	0.998349i	$-0.518291\pi$
$53$	−0.836228	−0.114865	−0.0574324	+	0.998349i	$0.518291\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0434	1.59518
$58$	0	0
$59$	3.82242	0.497637	0.248818	−	0.968550i	$-0.419958\pi$
$59$	3.82242	0.497637	0.248818	+	0.968550i	$0.419958\pi$
$60$	0	0
$61$	5.29902	0.678470	0.339235	−	0.940702i	$-0.389832\pi$
$61$	5.29902	0.678470	0.339235	+	0.940702i	$0.389832\pi$
$62$	0	0
$63$	−3.23607	−0.407706
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.63940	−0.688962	−0.344481	−	0.938793i	$-0.611945\pi$
$67$	−5.63940	−0.688962	−0.344481	+	0.938793i	$0.611945\pi$
$68$	0	0
$69$	−3.60016	−0.433408
$70$	0	0
$71$	−1.60016	−0.189904	−0.0949520	−	0.995482i	$-0.530270\pi$
$71$	−1.60016	−0.189904	−0.0949520	+	0.995482i	$0.530270\pi$
$72$	0	0
$73$	0.128228	0.0150079	0.00750397	−	0.999972i	$-0.497611\pi$
$73$	0.128228	0.0150079	0.00750397	+	0.999972i	$0.497611\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.55955	0.747530
$78$	0	0
$79$	9.47936	1.06651	0.533256	−	0.845954i	$-0.320968\pi$
$79$	9.47936	1.06651	0.533256	+	0.845954i	$0.320968\pi$
$80$	0	0
$81$	−10.7972	−1.19969
$82$	0	0
$83$	9.45702	1.03804	0.519022	−	0.854761i	$-0.326296\pi$
$83$	9.45702	1.03804	0.519022	+	0.854761i	$0.326296\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.93086	−1.06470
$88$	0	0
$89$	15.6356	1.65737	0.828684	−	0.559717i	$-0.189090\pi$
$89$	15.6356	1.65737	0.828684	+	0.559717i	$0.189090\pi$
$90$	0	0
$91$	−20.1446	−2.11173
$92$	0	0
$93$	−10.8902	−1.12927
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.5623	−1.07244	−0.536220	−	0.844078i	$-0.680148\pi$
$97$	−10.5623	−1.07244	−0.536220	+	0.844078i	$0.680148\pi$
$98$	0	0
$99$	−1.38664	−0.139362
$100$	0	0
$101$	14.0389	1.39692	0.698462	−	0.715647i	$-0.253868\pi$
$101$	14.0389	1.39692	0.698462	+	0.715647i	$0.253868\pi$
$102$	0	0
$103$	−1.52786	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$103$	−1.52786	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.18014	−0.887477	−0.443739	−	0.896156i	$-0.646348\pi$
$107$	−9.18014	−0.887477	−0.443739	+	0.896156i	$0.646348\pi$
$108$	0	0
$109$	−4.40300	−0.421731	−0.210865	−	0.977515i	$-0.567628\pi$
$109$	−4.40300	−0.421731	−0.210865	+	0.977515i	$0.567628\pi$
$110$	0	0
$111$	12.1559	1.15379
$112$	0	0
$113$	−8.01636	−0.754116	−0.377058	−	0.926190i	$-0.623064\pi$
$113$	−8.01636	−0.754116	−0.377058	+	0.926190i	$0.623064\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.25841	0.393690
$118$	0	0
$119$	26.1408	2.39632
$120$	0	0
$121$	−8.18927	−0.744479
$122$	0	0
$123$	−10.8811	−0.981117
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.46807	0.130270	0.0651350	−	0.997876i	$-0.479252\pi$
$127$	1.46807	0.130270	0.0651350	+	0.997876i	$0.479252\pi$
$128$	0	0
$129$	15.8705	1.39732
$130$	0	0
$131$	−3.71267	−0.324378	−0.162189	−	0.986760i	$-0.551855\pi$
$131$	−3.71267	−0.324378	−0.162189	+	0.986760i	$0.551855\pi$
$132$	0	0
$133$	24.0867	2.08859
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.7061	−1.00012	−0.500059	−	0.865991i	$-0.666688\pi$
$137$	−11.7061	−1.00012	−0.500059	+	0.865991i	$0.666688\pi$
$138$	0	0
$139$	2.29923	0.195018	0.0975089	−	0.995235i	$-0.468913\pi$
$139$	2.29923	0.195018	0.0975089	+	0.995235i	$0.468913\pi$
$140$	0	0
$141$	−20.0389	−1.68758
$142$	0	0
$143$	−8.63184	−0.721831
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.2536	−1.34058
$148$	0	0
$149$	10.2616	0.840660	0.420330	−	0.907371i	$-0.361914\pi$
$149$	10.2616	0.840660	0.420330	+	0.907371i	$0.361914\pi$
$150$	0	0
$151$	12.1559	0.989232	0.494616	−	0.869112i	$-0.335309\pi$
$151$	12.1559	0.989232	0.494616	+	0.869112i	$0.335309\pi$
$152$	0	0
$153$	−5.52595	−0.446747
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.42941	−0.193888	−0.0969439	−	0.995290i	$-0.530907\pi$
$157$	−2.42941	−0.193888	−0.0969439	+	0.995290i	$0.530907\pi$
$158$	0	0
$159$	1.63591	0.129736
$160$	0	0
$161$	−7.20032	−0.567465
$162$	0	0
$163$	2.47681	0.193998	0.0969992	−	0.995284i	$-0.469076\pi$
$163$	2.47681	0.193998	0.0969992	+	0.995284i	$0.469076\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.0389	−1.08636	−0.543182	−	0.839615i	$-0.682781\pi$
$167$	−14.0389	−1.08636	−0.543182	+	0.839615i	$0.682781\pi$
$168$	0	0
$169$	13.5087	1.03913
$170$	0	0
$171$	−5.09175	−0.389376
$172$	0	0
$173$	16.2616	1.23634	0.618172	−	0.786043i	$-0.287873\pi$
$173$	16.2616	1.23634	0.618172	+	0.786043i	$0.287873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.47778	−0.562065
$178$	0	0
$179$	−12.4255	−0.928729	−0.464364	−	0.885644i	$-0.653717\pi$
$179$	−12.4255	−0.928729	−0.464364	+	0.885644i	$0.653717\pi$
$180$	0	0
$181$	22.0055	1.63566	0.817829	−	0.575462i	$-0.195178\pi$
$181$	22.0055	1.63566	0.817829	+	0.575462i	$0.195178\pi$
$182$	0	0
$183$	−10.3665	−0.766310
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.2012	0.819111
$188$	0	0
$189$	−16.6318	−1.20979
$190$	0	0
$191$	3.25434	0.235476	0.117738	−	0.993045i	$-0.462436\pi$
$191$	3.25434	0.235476	0.117738	+	0.993045i	$0.462436\pi$
$192$	0	0
$193$	9.76572	0.702952	0.351476	−	0.936197i	$-0.385680\pi$
$193$	9.76572	0.702952	0.351476	+	0.936197i	$0.385680\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2656	0.945137	0.472569	−	0.881294i	$-0.343327\pi$
$197$	13.2656	0.945137	0.472569	+	0.881294i	$0.343327\pi$
$198$	0	0
$199$	14.6412	1.03789	0.518943	−	0.854809i	$-0.326326\pi$
$199$	14.6412	1.03789	0.518943	+	0.854809i	$0.326326\pi$
$200$	0	0
$201$	11.0323	0.778160
$202$	0	0
$203$	−19.8617	−1.39402
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.52209	0.105793
$208$	0	0
$209$	10.3210	0.713921
$210$	0	0
$211$	−4.28384	−0.294912	−0.147456	−	0.989069i	$-0.547108\pi$
$211$	−4.28384	−0.294912	−0.147456	+	0.989069i	$0.547108\pi$
$212$	0	0
$213$	3.13038	0.214490
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−21.7805	−1.47856
$218$	0	0
$219$	−0.250852	−0.0169510
$220$	0	0
$221$	−34.3992	−2.31394
$222$	0	0
$223$	20.7726	1.39104	0.695519	−	0.718508i	$-0.255175\pi$
$223$	20.7726	1.39104	0.695519	+	0.718508i	$0.255175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.41535	0.0939403	0.0469702	−	0.998896i	$-0.485043\pi$
$227$	1.41535	0.0939403	0.0469702	+	0.998896i	$0.485043\pi$
$228$	0	0
$229$	−10.0836	−0.666342	−0.333171	−	0.942866i	$-0.608119\pi$
$229$	−10.0836	−0.666342	−0.333171	+	0.942866i	$0.608119\pi$
$230$	0	0
$231$	−12.8324	−0.844310
$232$	0	0
$233$	−23.6456	−1.54908	−0.774539	−	0.632526i	$-0.782018\pi$
$233$	−23.6456	−1.54908	−0.774539	+	0.632526i	$0.782018\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−18.5444	−1.20459
$238$	0	0
$239$	23.1782	1.49927	0.749637	−	0.661849i	$-0.230228\pi$
$239$	23.1782	1.49927	0.749637	+	0.661849i	$0.230228\pi$
$240$	0	0
$241$	−19.4559	−1.25326	−0.626632	−	0.779315i	$-0.715567\pi$
$241$	−19.4559	−1.25326	−0.626632	+	0.779315i	$0.715567\pi$
$242$	0	0
$243$	8.36994	0.536932
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−31.6962	−2.01678
$248$	0	0
$249$	−18.5007	−1.17244
$250$	0	0
$251$	−2.93711	−0.185389	−0.0926945	−	0.995695i	$-0.529548\pi$
$251$	−2.93711	−0.185389	−0.0926945	+	0.995695i	$0.529548\pi$
$252$	0	0
$253$	−3.08530	−0.193971
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.8630	−0.864752	−0.432376	−	0.901693i	$-0.642325\pi$
$257$	−13.8630	−0.864752	−0.432376	+	0.901693i	$0.642325\pi$
$258$	0	0
$259$	24.3118	1.51066
$260$	0	0
$261$	4.19861	0.259888
$262$	0	0
$263$	−2.39366	−0.147599	−0.0737997	−	0.997273i	$-0.523513\pi$
$263$	−2.39366	−0.147599	−0.0737997	+	0.997273i	$0.523513\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−30.5878	−1.87194
$268$	0	0
$269$	−22.9311	−1.39813	−0.699067	−	0.715056i	$-0.746401\pi$
$269$	−22.9311	−1.39813	−0.699067	+	0.715056i	$0.746401\pi$
$270$	0	0
$271$	−4.20057	−0.255166	−0.127583	−	0.991828i	$-0.540722\pi$
$271$	−4.20057	−0.255166	−0.127583	+	0.991828i	$0.540722\pi$
$272$	0	0
$273$	39.4088	2.38513
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.2677	−1.33794	−0.668970	−	0.743290i	$-0.733264\pi$
$277$	−22.2677	−1.33794	−0.668970	+	0.743290i	$0.733264\pi$
$278$	0	0
$279$	4.60423	0.275648
$280$	0	0
$281$	−29.9298	−1.78546	−0.892731	−	0.450590i	$-0.851214\pi$
$281$	−29.9298	−1.78546	−0.892731	+	0.450590i	$0.851214\pi$
$282$	0	0
$283$	6.70800	0.398749	0.199375	−	0.979923i	$-0.436109\pi$
$283$	6.70800	0.398749	0.199375	+	0.979923i	$0.436109\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.7622	−1.28458
$288$	0	0
$289$	27.6383	1.62578
$290$	0	0
$291$	20.6630	1.21129
$292$	0	0
$293$	−4.47791	−0.261602	−0.130801	−	0.991409i	$-0.541755\pi$
$293$	−4.47791	−0.261602	−0.130801	+	0.991409i	$0.541755\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.12665	−0.413530
$298$	0	0
$299$	9.47505	0.547956
$300$	0	0
$301$	31.7409	1.82952
$302$	0	0
$303$	−27.4642	−1.57778
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.9396	0.738502	0.369251	−	0.929330i	$-0.379614\pi$
$307$	12.9396	0.738502	0.369251	+	0.929330i	$0.379614\pi$
$308$	0	0
$309$	2.98895	0.170036
$310$	0	0
$311$	−6.04589	−0.342831	−0.171415	−	0.985199i	$-0.554834\pi$
$311$	−6.04589	−0.342831	−0.171415	+	0.985199i	$0.554834\pi$
$312$	0	0
$313$	15.9264	0.900213	0.450107	−	0.892975i	$-0.351386\pi$
$313$	15.9264	0.900213	0.450107	+	0.892975i	$0.351386\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6878	0.712619	0.356309	−	0.934368i	$-0.384035\pi$
$317$	12.6878	0.712619	0.356309	+	0.934368i	$0.384035\pi$
$318$	0	0
$319$	−8.51064	−0.476504
$320$	0	0
$321$	17.9591	1.00238
$322$	0	0
$323$	41.1309	2.28858
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.61357	0.476331
$328$	0	0
$329$	−40.0778	−2.20956
$330$	0	0
$331$	−35.4896	−1.95068	−0.975341	−	0.220703i	$-0.929165\pi$
$331$	−35.4896	−1.95068	−0.975341	+	0.220703i	$0.929165\pi$
$332$	0	0
$333$	−5.13932	−0.281633
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.94537	0.160445	0.0802224	−	0.996777i	$-0.474437\pi$
$337$	2.94537	0.160445	0.0802224	+	0.996777i	$0.474437\pi$
$338$	0	0
$339$	15.6824	0.851750
$340$	0	0
$341$	−9.33282	−0.505400
$342$	0	0
$343$	−5.11909	−0.276405
$344$	0	0
$345$	0	0
$346$	0	0
$347$	35.4790	1.90461	0.952307	−	0.305141i	$-0.0987037\pi$
$347$	35.4790	1.90461	0.952307	+	0.305141i	$0.0987037\pi$
$348$	0	0
$349$	−13.3045	−0.712176	−0.356088	−	0.934452i	$-0.615890\pi$
$349$	−13.3045	−0.712176	−0.356088	+	0.934452i	$0.615890\pi$
$350$	0	0
$351$	21.8862	1.16820
$352$	0	0
$353$	−26.9766	−1.43582	−0.717910	−	0.696136i	$-0.754901\pi$
$353$	−26.9766	−1.43582	−0.717910	+	0.696136i	$0.754901\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−51.1391	−2.70657
$358$	0	0
$359$	−1.83452	−0.0968224	−0.0484112	−	0.998827i	$-0.515416\pi$
$359$	−1.83452	−0.0968224	−0.0484112	+	0.998827i	$0.515416\pi$
$360$	0	0
$361$	18.8990	0.994684
$362$	0	0
$363$	16.0206	0.840865
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.7174	0.663842	0.331921	−	0.943307i	$-0.392303\pi$
$367$	12.7174	0.663842	0.331921	+	0.943307i	$0.392303\pi$
$368$	0	0
$369$	4.60036	0.239485
$370$	0	0
$371$	3.27182	0.169864
$372$	0	0
$373$	−25.7743	−1.33454	−0.667272	−	0.744814i	$-0.732538\pi$
$373$	−25.7743	−1.33454	−0.667272	+	0.744814i	$0.732538\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.1365	1.34610
$378$	0	0
$379$	11.6796	0.599941	0.299971	−	0.953948i	$-0.403023\pi$
$379$	11.6796	0.599941	0.299971	+	0.953948i	$0.403023\pi$
$380$	0	0
$381$	−2.87198	−0.147136
$382$	0	0
$383$	8.81729	0.450543	0.225271	−	0.974296i	$-0.427673\pi$
$383$	8.81729	0.450543	0.225271	+	0.974296i	$0.427673\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.70978	−0.341077
$388$	0	0
$389$	−17.3197	−0.878141	−0.439071	−	0.898452i	$-0.644692\pi$
$389$	−17.3197	−0.878141	−0.439071	+	0.898452i	$0.644692\pi$
$390$	0	0
$391$	−12.2954	−0.621803
$392$	0	0
$393$	7.26308	0.366374
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.69716	−0.336121	−0.168060	−	0.985777i	$-0.553750\pi$
$397$	−6.69716	−0.336121	−0.168060	+	0.985777i	$0.553750\pi$
$398$	0	0
$399$	−47.1208	−2.35899
$400$	0	0
$401$	−29.7795	−1.48712	−0.743559	−	0.668670i	$-0.766864\pi$
$401$	−29.7795	−1.48712	−0.743559	+	0.668670i	$0.766864\pi$
$402$	0	0
$403$	28.6614	1.42773
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4175	0.516374
$408$	0	0
$409$	−22.9030	−1.13248	−0.566240	−	0.824240i	$-0.691603\pi$
$409$	−22.9030	−1.13248	−0.566240	+	0.824240i	$0.691603\pi$
$410$	0	0
$411$	22.9006	1.12960
$412$	0	0
$413$	−14.9556	−0.735915
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.49797	−0.220266
$418$	0	0
$419$	13.6104	0.664909	0.332455	−	0.943119i	$-0.392123\pi$
$419$	13.6104	0.664909	0.332455	+	0.943119i	$0.392123\pi$
$420$	0	0
$421$	20.3665	0.992600	0.496300	−	0.868151i	$-0.334692\pi$
$421$	20.3665	0.992600	0.496300	+	0.868151i	$0.334692\pi$
$422$	0	0
$423$	8.47214	0.411929
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−20.7329	−1.00334
$428$	0	0
$429$	16.8864	0.815285
$430$	0	0
$431$	−24.5476	−1.18242	−0.591208	−	0.806519i	$-0.701349\pi$
$431$	−24.5476	−1.18242	−0.591208	+	0.806519i	$0.701349\pi$
$432$	0	0
$433$	5.83866	0.280588	0.140294	−	0.990110i	$-0.455195\pi$
$433$	5.83866	0.280588	0.140294	+	0.990110i	$0.455195\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.3293	−0.541952
$438$	0	0
$439$	−17.8403	−0.851471	−0.425735	−	0.904848i	$-0.639985\pi$
$439$	−17.8403	−0.851471	−0.425735	+	0.904848i	$0.639985\pi$
$440$	0	0
$441$	6.87177	0.327227
$442$	0	0
$443$	23.3881	1.11120	0.555601	−	0.831449i	$-0.312488\pi$
$443$	23.3881	1.11120	0.555601	+	0.831449i	$0.312488\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−20.0747	−0.949499
$448$	0	0
$449$	0.261252	0.0123293	0.00616463	−	0.999981i	$-0.498038\pi$
$449$	0.261252	0.0123293	0.00616463	+	0.999981i	$0.498038\pi$
$450$	0	0
$451$	−9.32499	−0.439097
$452$	0	0
$453$	−23.7805	−1.11731
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.6047	0.823513	0.411757	−	0.911294i	$-0.364915\pi$
$457$	17.6047	0.823513	0.411757	+	0.911294i	$0.364915\pi$
$458$	0	0
$459$	−28.4008	−1.32563
$460$	0	0
$461$	33.0348	1.53859	0.769293	−	0.638896i	$-0.220609\pi$
$461$	33.0348	1.53859	0.769293	+	0.638896i	$0.220609\pi$
$462$	0	0
$463$	−34.1928	−1.58908	−0.794538	−	0.607214i	$-0.792287\pi$
$463$	−34.1928	−1.58908	−0.794538	+	0.607214i	$0.792287\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.81519	0.0839972	0.0419986	−	0.999118i	$-0.486628\pi$
$467$	1.81519	0.0839972	0.0419986	+	0.999118i	$0.486628\pi$
$468$	0	0
$469$	22.0647	1.01885
$470$	0	0
$471$	4.75264	0.218990
$472$	0	0
$473$	13.6008	0.625366
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.691636	−0.0316679
$478$	0	0
$479$	−15.6025	−0.712897	−0.356449	−	0.934315i	$-0.616013\pi$
$479$	−15.6025	−0.712897	−0.356449	+	0.934315i	$0.616013\pi$
$480$	0	0
$481$	−31.9924	−1.45873
$482$	0	0
$483$	14.0860	0.640933
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.0885	−0.502466	−0.251233	−	0.967927i	$-0.580836\pi$
$487$	−11.0885	−0.502466	−0.251233	+	0.967927i	$0.580836\pi$
$488$	0	0
$489$	−4.84536	−0.219115
$490$	0	0
$491$	−15.7353	−0.710122	−0.355061	−	0.934843i	$-0.615540\pi$
$491$	−15.7353	−0.710122	−0.355061	+	0.934843i	$0.615540\pi$
$492$	0	0
$493$	−33.9162	−1.52751
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.26077	0.280834
$498$	0	0
$499$	16.7374	0.749268	0.374634	−	0.927173i	$-0.377768\pi$
$499$	16.7374	0.749268	0.374634	+	0.927173i	$0.377768\pi$
$500$	0	0
$501$	27.4642	1.22701
$502$	0	0
$503$	−34.2029	−1.52503	−0.762517	−	0.646969i	$-0.776036\pi$
$503$	−34.2029	−1.52503	−0.762517	+	0.646969i	$0.776036\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−26.4270	−1.17366
$508$	0	0
$509$	1.33477	0.0591627	0.0295813	−	0.999562i	$-0.490583\pi$
$509$	1.33477	0.0591627	0.0295813	+	0.999562i	$0.490583\pi$
$510$	0	0
$511$	−0.501703	−0.0221940
$512$	0	0
$513$	−26.1692	−1.15540
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−17.1731	−0.755273
$518$	0	0
$519$	−31.8124	−1.39641
$520$	0	0
$521$	−30.0775	−1.31772	−0.658859	−	0.752266i	$-0.728961\pi$
$521$	−30.0775	−1.31772	−0.658859	+	0.752266i	$0.728961\pi$
$522$	0	0
$523$	27.5162	1.20320	0.601599	−	0.798798i	$-0.294531\pi$
$523$	27.5162	1.20320	0.601599	+	0.798798i	$0.294531\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.1927	−1.62014
$528$	0	0
$529$	−19.6133	−0.852753
$530$	0	0
$531$	3.16149	0.137197
$532$	0	0
$533$	28.6374	1.24042
$534$	0	0
$535$	0	0
$536$	0	0
$537$	24.3080	1.04897
$538$	0	0
$539$	−13.9292	−0.599971
$540$	0	0
$541$	28.6607	1.23222	0.616111	−	0.787659i	$-0.288707\pi$
$541$	28.6607	1.23222	0.616111	+	0.787659i	$0.288707\pi$
$542$	0	0
$543$	−43.0493	−1.84742
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.6021	0.453314	0.226657	−	0.973975i	$-0.427220\pi$
$547$	10.6021	0.453314	0.226657	+	0.973975i	$0.427220\pi$
$548$	0	0
$549$	4.38277	0.187052
$550$	0	0
$551$	−31.2512	−1.33135
$552$	0	0
$553$	−37.0889	−1.57718
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.6646	0.706100	0.353050	−	0.935604i	$-0.385144\pi$
$557$	16.6646	0.706100	0.353050	+	0.935604i	$0.385144\pi$
$558$	0	0
$559$	−41.7685	−1.76662
$560$	0	0
$561$	−21.9128	−0.925159
$562$	0	0
$563$	−6.97303	−0.293878	−0.146939	−	0.989146i	$-0.546942\pi$
$563$	−6.97303	−0.293878	−0.146939	+	0.989146i	$0.546942\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	42.2450	1.77412
$568$	0	0
$569$	46.5034	1.94952	0.974761	−	0.223250i	$-0.0716667\pi$
$569$	46.5034	1.94952	0.974761	+	0.223250i	$0.0716667\pi$
$570$	0	0
$571$	−30.7089	−1.28513	−0.642563	−	0.766233i	$-0.722129\pi$
$571$	−30.7089	−1.28513	−0.642563	+	0.766233i	$0.722129\pi$
$572$	0	0
$573$	−6.36645	−0.265962
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−16.2637	−0.677066	−0.338533	−	0.940955i	$-0.609931\pi$
$577$	−16.2637	−0.677066	−0.338533	+	0.940955i	$0.609931\pi$
$578$	0	0
$579$	−19.1046	−0.793961
$580$	0	0
$581$	−37.0015	−1.53508
$582$	0	0
$583$	1.40195	0.0580630
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.1784	−1.03922	−0.519612	−	0.854403i	$-0.673923\pi$
$587$	−25.1784	−1.03922	−0.519612	+	0.854403i	$0.673923\pi$
$588$	0	0
$589$	−34.2702	−1.41208
$590$	0	0
$591$	−25.9515	−1.06750
$592$	0	0
$593$	−44.3295	−1.82039	−0.910197	−	0.414176i	$-0.864070\pi$
$593$	−44.3295	−1.82039	−0.910197	+	0.414176i	$0.864070\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−28.6425	−1.17226
$598$	0	0
$599$	1.34370	0.0549023	0.0274511	−	0.999623i	$-0.491261\pi$
$599$	1.34370	0.0549023	0.0274511	+	0.999623i	$0.491261\pi$
$600$	0	0
$601$	−12.6988	−0.517994	−0.258997	−	0.965878i	$-0.583392\pi$
$601$	−12.6988	−0.517994	−0.258997	+	0.965878i	$0.583392\pi$
$602$	0	0
$603$	−4.66430	−0.189945
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.6246	−0.553006	−0.276503	−	0.961013i	$-0.589176\pi$
$607$	−13.6246	−0.553006	−0.276503	+	0.961013i	$0.589176\pi$
$608$	0	0
$609$	38.8554	1.57450
$610$	0	0
$611$	52.7392	2.13360
$612$	0	0
$613$	15.7320	0.635409	0.317705	−	0.948190i	$-0.397088\pi$
$613$	15.7320	0.635409	0.317705	+	0.948190i	$0.397088\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.30154	−0.132915	−0.0664574	−	0.997789i	$-0.521170\pi$
$617$	−3.30154	−0.132915	−0.0664574	+	0.997789i	$0.521170\pi$
$618$	0	0
$619$	−43.5881	−1.75195	−0.875976	−	0.482355i	$-0.839782\pi$
$619$	−43.5881	−1.75195	−0.875976	+	0.482355i	$0.839782\pi$
$620$	0	0
$621$	7.82282	0.313919
$622$	0	0
$623$	−61.1756	−2.45095
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.1910	−0.806350
$628$	0	0
$629$	41.5151	1.65532
$630$	0	0
$631$	−19.7448	−0.786026	−0.393013	−	0.919533i	$-0.628567\pi$
$631$	−19.7448	−0.786026	−0.393013	+	0.919533i	$0.628567\pi$
$632$	0	0
$633$	8.38046	0.333093
$634$	0	0
$635$	0	0
$636$	0	0
$637$	42.7769	1.69488
$638$	0	0
$639$	−1.32348	−0.0523560
$640$	0	0
$641$	10.5134	0.415254	0.207627	−	0.978208i	$-0.433426\pi$
$641$	10.5134	0.415254	0.207627	+	0.978208i	$0.433426\pi$
$642$	0	0
$643$	−45.5202	−1.79514	−0.897571	−	0.440870i	$-0.854670\pi$
$643$	−45.5202	−1.79514	−0.897571	+	0.440870i	$0.854670\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.9373	−1.53078	−0.765391	−	0.643565i	$-0.777455\pi$
$647$	−38.9373	−1.53078	−0.765391	+	0.643565i	$0.777455\pi$
$648$	0	0
$649$	−6.40837	−0.251551
$650$	0	0
$651$	42.6091	1.66998
$652$	0	0
$653$	33.7106	1.31920	0.659598	−	0.751618i	$-0.270726\pi$
$653$	33.7106	1.31920	0.659598	+	0.751618i	$0.270726\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.106056	0.00413764
$658$	0	0
$659$	−2.21657	−0.0863454	−0.0431727	−	0.999068i	$-0.513747\pi$
$659$	−2.21657	−0.0863454	−0.0431727	+	0.999068i	$0.513747\pi$
$660$	0	0
$661$	5.01105	0.194907	0.0974536	−	0.995240i	$-0.468930\pi$
$661$	5.01105	0.194907	0.0974536	+	0.995240i	$0.468930\pi$
$662$	0	0
$663$	67.2950	2.61352
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.34200	0.361724
$668$	0	0
$669$	−40.6374	−1.57113
$670$	0	0
$671$	−8.88393	−0.342960
$672$	0	0
$673$	31.0599	1.19727	0.598636	−	0.801021i	$-0.295709\pi$
$673$	31.0599	1.19727	0.598636	+	0.801021i	$0.295709\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.5344	−1.32727	−0.663633	−	0.748058i	$-0.730986\pi$
$677$	−34.5344	−1.32727	−0.663633	+	0.748058i	$0.730986\pi$
$678$	0	0
$679$	41.3260	1.58595
$680$	0	0
$681$	−2.76885	−0.106103
$682$	0	0
$683$	17.7984	0.681037	0.340519	−	0.940238i	$-0.389397\pi$
$683$	17.7984	0.681037	0.340519	+	0.940238i	$0.389397\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19.7265	0.752612
$688$	0	0
$689$	−4.30545	−0.164025
$690$	0	0
$691$	24.2712	0.923318	0.461659	−	0.887057i	$-0.347254\pi$
$691$	24.2712	0.923318	0.461659	+	0.887057i	$0.347254\pi$
$692$	0	0
$693$	5.42534	0.206092
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−37.1615	−1.40759
$698$	0	0
$699$	46.2579	1.74963
$700$	0	0
$701$	−37.0761	−1.40035	−0.700173	−	0.713974i	$-0.746894\pi$
$701$	−37.0761	−1.40035	−0.700173	+	0.713974i	$0.746894\pi$
$702$	0	0
$703$	38.2530	1.44274
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−54.9285	−2.06580
$708$	0	0
$709$	−22.8872	−0.859548	−0.429774	−	0.902937i	$-0.641407\pi$
$709$	−22.8872	−0.859548	−0.429774	+	0.902937i	$0.641407\pi$
$710$	0	0
$711$	7.84029	0.294034
$712$	0	0
$713$	10.2445	0.383660
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−45.3435	−1.69338
$718$	0	0
$719$	2.76393	0.103077	0.0515386	−	0.998671i	$-0.483587\pi$
$719$	2.76393	0.103077	0.0515386	+	0.998671i	$0.483587\pi$
$720$	0	0
$721$	5.97791	0.222629
$722$	0	0
$723$	38.0615	1.41552
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20.5500	−0.762156	−0.381078	−	0.924543i	$-0.624447\pi$
$727$	−20.5500	−0.762156	−0.381078	+	0.924543i	$0.624447\pi$
$728$	0	0
$729$	16.0175	0.593241
$730$	0	0
$731$	54.2012	2.00471
$732$	0	0
$733$	33.8692	1.25099	0.625494	−	0.780229i	$-0.284898\pi$
$733$	33.8692	1.25099	0.625494	+	0.780229i	$0.284898\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.45458	0.348264
$738$	0	0
$739$	10.8097	0.397640	0.198820	−	0.980036i	$-0.436289\pi$
$739$	10.8097	0.397640	0.198820	+	0.980036i	$0.436289\pi$
$740$	0	0
$741$	62.0072	2.27789
$742$	0	0
$743$	5.27116	0.193380	0.0966900	−	0.995315i	$-0.469174\pi$
$743$	5.27116	0.193380	0.0966900	+	0.995315i	$0.469174\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.82182	0.286185
$748$	0	0
$749$	35.9181	1.31242
$750$	0	0
$751$	42.3821	1.54655	0.773273	−	0.634074i	$-0.218618\pi$
$751$	42.3821	1.54655	0.773273	+	0.634074i	$0.218618\pi$
$752$	0	0
$753$	5.74586	0.209391
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.9431	−0.397732	−0.198866	−	0.980027i	$-0.563726\pi$
$757$	−10.9431	−0.397732	−0.198866	+	0.980027i	$0.563726\pi$
$758$	0	0
$759$	6.03575	0.219084
$760$	0	0
$761$	37.0423	1.34278	0.671391	−	0.741104i	$-0.265697\pi$
$761$	37.0423	1.34278	0.671391	+	0.741104i	$0.265697\pi$
$762$	0	0
$763$	17.2271	0.623664
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.6803	0.710616
$768$	0	0
$769$	−15.3665	−0.554131	−0.277065	−	0.960851i	$-0.589362\pi$
$769$	−15.3665	−0.554131	−0.277065	+	0.960851i	$0.589362\pi$
$770$	0	0
$771$	27.1202	0.976710
$772$	0	0
$773$	−5.79521	−0.208439	−0.104220	−	0.994554i	$-0.533234\pi$
$773$	−5.79521	−0.208439	−0.104220	+	0.994554i	$0.533234\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−47.5610	−1.70624
$778$	0	0
$779$	−34.2415	−1.22683
$780$	0	0
$781$	2.68270	0.0959947
$782$	0	0
$783$	21.5789	0.771166
$784$	0	0
$785$	0	0
$786$	0	0
$787$	37.1969	1.32593	0.662963	−	0.748652i	$-0.269299\pi$
$787$	37.1969	1.32593	0.662963	+	0.748652i	$0.269299\pi$
$788$	0	0
$789$	4.68270	0.166709
$790$	0	0
$791$	31.3647	1.11520
$792$	0	0
$793$	27.2829	0.968843
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.17823	0.183422	0.0917111	−	0.995786i	$-0.470766\pi$
$797$	5.17823	0.183422	0.0917111	+	0.995786i	$0.470766\pi$
$798$	0	0
$799$	−68.4374	−2.42114
$800$	0	0
$801$	12.9320	0.456931
$802$	0	0
$803$	−0.214977	−0.00758637
$804$	0	0
$805$	0	0
$806$	0	0
$807$	44.8600	1.57915
$808$	0	0
$809$	20.7214	0.728524	0.364262	−	0.931297i	$-0.381321\pi$
$809$	20.7214	0.728524	0.364262	+	0.931297i	$0.381321\pi$
$810$	0	0
$811$	−5.72251	−0.200945	−0.100472	−	0.994940i	$-0.532035\pi$
$811$	−5.72251	−0.200945	−0.100472	+	0.994940i	$0.532035\pi$
$812$	0	0
$813$	8.21755	0.288202
$814$	0	0
$815$	0	0
$816$	0	0
$817$	49.9424	1.74726
$818$	0	0
$819$	−16.6614	−0.582197
$820$	0	0
$821$	−0.0597951	−0.00208686	−0.00104343	−	0.999999i	$-0.500332\pi$
$821$	−0.0597951	−0.00208686	−0.00104343	+	0.999999i	$0.500332\pi$
$822$	0	0
$823$	2.95873	0.103135	0.0515673	−	0.998670i	$-0.483578\pi$
$823$	2.95873	0.103135	0.0515673	+	0.998670i	$0.483578\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.9761	−1.14669	−0.573347	−	0.819313i	$-0.694355\pi$
$827$	−32.9761	−1.14669	−0.573347	+	0.819313i	$0.694355\pi$
$828$	0	0
$829$	0.851342	0.0295683	0.0147842	−	0.999891i	$-0.495294\pi$
$829$	0.851342	0.0295683	0.0147842	+	0.999891i	$0.495294\pi$
$830$	0	0
$831$	43.5623	1.51116
$832$	0	0
$833$	−55.5098	−1.92330
$834$	0	0
$835$	0	0
$836$	0	0
$837$	23.6635	0.817931
$838$	0	0
$839$	−10.9777	−0.378991	−0.189495	−	0.981882i	$-0.560685\pi$
$839$	−10.9777	−0.378991	−0.189495	+	0.981882i	$0.560685\pi$
$840$	0	0
$841$	−3.23054	−0.111398
$842$	0	0
$843$	58.5515	2.01662
$844$	0	0
$845$	0	0
$846$	0	0
$847$	32.0413	1.10095
$848$	0	0
$849$	−13.1228	−0.450374
$850$	0	0
$851$	−11.4351	−0.391990
$852$	0	0
$853$	−8.44386	−0.289112	−0.144556	−	0.989497i	$-0.546175\pi$
$853$	−8.44386	−0.289112	−0.144556	+	0.989497i	$0.546175\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.9735	−1.33131	−0.665655	−	0.746260i	$-0.731848\pi$
$857$	−38.9735	−1.33131	−0.665655	+	0.746260i	$0.731848\pi$
$858$	0	0
$859$	−25.0251	−0.853844	−0.426922	−	0.904288i	$-0.640402\pi$
$859$	−25.0251	−0.853844	−0.426922	+	0.904288i	$0.640402\pi$
$860$	0	0
$861$	42.5733	1.45090
$862$	0	0
$863$	−29.7646	−1.01320	−0.506599	−	0.862182i	$-0.669098\pi$
$863$	−29.7646	−1.01320	−0.506599	+	0.862182i	$0.669098\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−54.0687	−1.83627
$868$	0	0
$869$	−15.8924	−0.539112
$870$	0	0
$871$	−29.0353	−0.983825
$872$	0	0
$873$	−8.73599	−0.295668
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.5234	1.53722	0.768608	−	0.639720i	$-0.220950\pi$
$877$	45.5234	1.53722	0.768608	+	0.639720i	$0.220950\pi$
$878$	0	0
$879$	8.76011	0.295471
$880$	0	0
$881$	−3.56310	−0.120044	−0.0600220	−	0.998197i	$-0.519117\pi$
$881$	−3.56310	−0.120044	−0.0600220	+	0.998197i	$0.519117\pi$
$882$	0	0
$883$	17.8066	0.599239	0.299620	−	0.954059i	$-0.403140\pi$
$883$	17.8066	0.599239	0.299620	+	0.954059i	$0.403140\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.9302	−1.60934	−0.804669	−	0.593723i	$-0.797657\pi$
$887$	−47.9302	−1.60934	−0.804669	+	0.593723i	$0.797657\pi$
$888$	0	0
$889$	−5.74395	−0.192646
$890$	0	0
$891$	18.1017	0.606431
$892$	0	0
$893$	−63.0599	−2.11022
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−18.5360	−0.618899
$898$	0	0
$899$	28.2590	0.942489
$900$	0	0
$901$	5.58700	0.186130
$902$	0	0
$903$	−62.0946	−2.06638
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.2814	−0.474206	−0.237103	−	0.971485i	$-0.576198\pi$
$907$	−14.2814	−0.474206	−0.237103	+	0.971485i	$0.576198\pi$
$908$	0	0
$909$	11.6115	0.385128
$910$	0	0
$911$	34.6036	1.14647	0.573234	−	0.819392i	$-0.305689\pi$
$911$	34.6036	1.14647	0.573234	+	0.819392i	$0.305689\pi$
$912$	0	0
$913$	−15.8549	−0.524721
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5262	0.479696
$918$	0	0
$919$	47.4251	1.56441	0.782205	−	0.623022i	$-0.214095\pi$
$919$	47.4251	1.56441	0.782205	+	0.623022i	$0.214095\pi$
$920$	0	0
$921$	−25.3137	−0.834114
$922$	0	0
$923$	−8.23868	−0.271179
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.26368	−0.0415048
$928$	0	0
$929$	−19.6738	−0.645475	−0.322738	−	0.946488i	$-0.604603\pi$
$929$	−19.6738	−0.645475	−0.322738	+	0.946488i	$0.604603\pi$
$930$	0	0
$931$	−51.1481	−1.67631
$932$	0	0
$933$	11.8275	0.387216
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.6356	−1.42551	−0.712756	−	0.701412i	$-0.752553\pi$
$937$	−43.6356	−1.42551	−0.712756	+	0.701412i	$0.752553\pi$
$938$	0	0
$939$	−31.1567	−1.01676
$940$	0	0
$941$	−10.1668	−0.331427	−0.165714	−	0.986174i	$-0.552993\pi$
$941$	−10.1668	−0.331427	−0.165714	+	0.986174i	$0.552993\pi$
$942$	0	0
$943$	10.2359	0.333327
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.7994	−0.903360	−0.451680	−	0.892180i	$-0.649175\pi$
$947$	−27.7994	−0.903360	−0.451680	+	0.892180i	$0.649175\pi$
$948$	0	0
$949$	0.660202	0.0214311
$950$	0	0
$951$	−24.8211	−0.804880
$952$	0	0
$953$	33.3672	1.08087	0.540434	−	0.841386i	$-0.318260\pi$
$953$	33.3672	1.08087	0.540434	+	0.841386i	$0.318260\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.6493	0.538196
$958$	0	0
$959$	45.8011	1.47900
$960$	0	0
$961$	−0.0110469	−0.000356352 0
$962$	0	0
$963$	−7.59281	−0.244675
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.3755	0.365813	0.182906	−	0.983130i	$-0.441449\pi$
$967$	11.3755	0.365813	0.182906	+	0.983130i	$0.441449\pi$
$968$	0	0
$969$	−80.4641	−2.58488
$970$	0	0
$971$	1.51385	0.0485818	0.0242909	−	0.999705i	$-0.492267\pi$
$971$	1.51385	0.0485818	0.0242909	+	0.999705i	$0.492267\pi$
$972$	0	0
$973$	−8.99593	−0.288396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.3319	−0.490512	−0.245256	−	0.969458i	$-0.578872\pi$
$977$	−15.3319	−0.490512	−0.245256	+	0.969458i	$0.578872\pi$
$978$	0	0
$979$	−26.2134	−0.837784
$980$	0	0
$981$	−3.64168	−0.116270
$982$	0	0
$983$	15.7709	0.503014	0.251507	−	0.967856i	$-0.419074\pi$
$983$	15.7709	0.503014	0.251507	+	0.967856i	$0.419074\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	78.4040	2.49563
$988$	0	0
$989$	−14.9294	−0.474728
$990$	0	0
$991$	11.3080	0.359209	0.179605	−	0.983739i	$-0.442518\pi$
$991$	11.3080	0.359209	0.179605	+	0.983739i	$0.442518\pi$
$992$	0	0
$993$	69.4280	2.20323
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.65273	−0.147353	−0.0736767	−	0.997282i	$-0.523473\pi$
$997$	−4.65273	−0.147353	−0.0736767	+	0.997282i	$0.523473\pi$
$998$	0	0
$999$	−26.4136	−0.835690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.w.1.1		4
4.3	odd	2		1250.2.a.e.1.4	✓	4
5.4	even	2		10000.2.a.s.1.4		4
20.3	even	4		1250.2.b.f.1249.8		8
20.7	even	4		1250.2.b.f.1249.1		8
20.19	odd	2		1250.2.a.k.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.2.a.e.1.4	✓	4	4.3	odd	2
1250.2.a.k.1.1	yes	4	20.19	odd	2
1250.2.b.f.1249.1		8	20.7	even	4
1250.2.b.f.1249.8		8	20.3	even	4
10000.2.a.s.1.4		4	5.4	even	2
10000.2.a.w.1.1		4	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.95630 q^{3} -3.91259 q^{7} +0.827091 q^{9} +O(q^{10})\)	\(q-1.95630 q^{3} -3.91259 q^{7} +0.827091 q^{9} -1.67652 q^{11} +5.14866 q^{13} -6.68119 q^{17} -6.15622 q^{19} +7.65418 q^{21} +1.84029 q^{23} +4.25085 q^{27} +5.07636 q^{29} +5.56677 q^{31} +3.27977 q^{33} -6.21373 q^{37} -10.0723 q^{39} +5.56210 q^{41} -8.11251 q^{43} +10.2433 q^{47} +8.30836 q^{49} +13.0704 q^{51} -0.836228 q^{53} +12.0434 q^{57} +3.82242 q^{59} +5.29902 q^{61} -3.23607 q^{63} -5.63940 q^{67} -3.60016 q^{69} -1.60016 q^{71} +0.128228 q^{73} +6.55955 q^{77} +9.47936 q^{79} -10.7972 q^{81} +9.45702 q^{83} -9.93086 q^{87} +15.6356 q^{89} -20.1446 q^{91} -10.8902 q^{93} -10.5623 q^{97} -1.38664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^7 - 3 * q^9	\( 4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 6 * q^13 - 7 * q^17 - 15 * q^19 + 18 * q^21 + 6 * q^23 - 5 * q^27 + 10 * q^29 - 8 * q^31 + 13 * q^33 - 12 * q^37 - 24 * q^39 + 3 * q^41 - 14 * q^43 + 2 * q^47 + 8 * q^49 + 7 * q^51 + 4 * q^53 + 10 * q^57 + 20 * q^59 + 18 * q^61 - 4 * q^63 - 23 * q^67 - 16 * q^69 - 8 * q^71 - q^73 + 26 * q^77 - 10 * q^79 - 16 * q^81 - 14 * q^83 - 20 * q^87 + 5 * q^89 - 48 * q^91 - 22 * q^93 - 2 * q^97 + q^99

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