LMFDB - Embedded newform 1300.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1,1300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, 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("1300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	1300.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32088 q^{3} +5.02827 q^{7} +8.02827 q^{9} +O(q^{10})$	$q-3.32088 q^{3} +5.02827 q^{7} +8.02827 q^{9} +1.70739 q^{11} -1.00000 q^{13} +4.64177 q^{17} -4.34916 q^{19} -16.6983 q^{21} +0.679116 q^{23} -16.6983 q^{27} -1.02827 q^{29} -2.29261 q^{31} -5.67004 q^{33} +1.61350 q^{37} +3.32088 q^{39} +4.64177 q^{41} +3.32088 q^{43} -1.02827 q^{47} +18.2835 q^{49} -15.4148 q^{51} +9.41478 q^{53} +14.4431 q^{57} -8.93438 q^{59} +9.02827 q^{61} +40.3684 q^{63} -5.61350 q^{67} -2.25526 q^{69} +1.70739 q^{71} -12.4431 q^{73} +8.58522 q^{77} -2.64177 q^{79} +31.3684 q^{81} +8.25526 q^{83} +3.41478 q^{87} +1.22699 q^{89} -5.02827 q^{91} +7.61350 q^{93} +0.0565477 q^{97} +13.7074 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 2 * q^7 + 11 * q^9	$ 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{23} - 8 q^{27} + 10 q^{29} - 12 q^{31} + 12 q^{33} + 2 q^{37} + 2 q^{39} - 2 q^{41} + 2 q^{43} + 10 q^{47} + 23 q^{49} - 36 q^{51} + 18 q^{53} + 20 q^{57} - 16 q^{59} + 14 q^{61} + 50 q^{63} - 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} + 8 q^{79} + 23 q^{81} + 6 q^{83} - 2 q^{89} - 2 q^{91} + 20 q^{93} - 26 q^{97} + 36 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^17 + 8 * q^19 - 8 * q^21 + 10 * q^23 - 8 * q^27 + 10 * q^29 - 12 * q^31 + 12 * q^33 + 2 * q^37 + 2 * q^39 - 2 * q^41 + 2 * q^43 + 10 * q^47 + 23 * q^49 - 36 * q^51 + 18 * q^53 + 20 * q^57 - 16 * q^59 + 14 * q^61 + 50 * q^63 - 14 * q^67 + 12 * q^69 - 14 * q^73 + 36 * q^77 + 8 * q^79 + 23 * q^81 + 6 * q^83 - 2 * q^89 - 2 * q^91 + 20 * q^93 - 26 * q^97 + 36 * 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
$3$	−3.32088	−1.91731	−0.958657	−	0.284565i	$-0.908151\pi$
$3$	−3.32088	−1.91731	−0.958657	+	0.284565i	$0.908151\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.02827	1.90051	0.950254	−	0.311475i	$-0.100823\pi$
$7$	5.02827	1.90051	0.950254	+	0.311475i	$0.100823\pi$
$8$	0	0
$9$	8.02827	2.67609
$10$	0	0
$11$	1.70739	0.514797	0.257399	−	0.966305i	$-0.417135\pi$
$11$	1.70739	0.514797	0.257399	+	0.966305i	$0.417135\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.64177	1.12579	0.562897	−	0.826527i	$-0.309687\pi$
$17$	4.64177	1.12579	0.562897	+	0.826527i	$0.309687\pi$
$18$	0	0
$19$	−4.34916	−0.997765	−0.498883	−	0.866670i	$-0.666256\pi$
$19$	−4.34916	−0.997765	−0.498883	+	0.866670i	$0.666256\pi$
$20$	0	0
$21$	−16.6983	−3.64387
$22$	0	0
$23$	0.679116	0.141605	0.0708027	−	0.997490i	$-0.477444\pi$
$23$	0.679116	0.141605	0.0708027	+	0.997490i	$0.477444\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−16.6983	−3.21359
$28$	0	0
$29$	−1.02827	−0.190946	−0.0954728	−	0.995432i	$-0.530436\pi$
$29$	−1.02827	−0.190946	−0.0954728	+	0.995432i	$0.530436\pi$
$30$	0	0
$31$	−2.29261	−0.411765	−0.205883	−	0.978577i	$-0.566006\pi$
$31$	−2.29261	−0.411765	−0.205883	+	0.978577i	$0.566006\pi$
$32$	0	0
$33$	−5.67004	−0.987028
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.61350	0.265257	0.132628	−	0.991166i	$-0.457658\pi$
$37$	1.61350	0.265257	0.132628	+	0.991166i	$0.457658\pi$
$38$	0	0
$39$	3.32088	0.531767
$40$	0	0
$41$	4.64177	0.724923	0.362461	−	0.931999i	$-0.381937\pi$
$41$	4.64177	0.724923	0.362461	+	0.931999i	$0.381937\pi$
$42$	0	0
$43$	3.32088	0.506430	0.253215	−	0.967410i	$-0.418512\pi$
$43$	3.32088	0.506430	0.253215	+	0.967410i	$0.418512\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.02827	−0.149989	−0.0749946	−	0.997184i	$-0.523894\pi$
$47$	−1.02827	−0.149989	−0.0749946	+	0.997184i	$0.523894\pi$
$48$	0	0
$49$	18.2835	2.61193
$50$	0	0
$51$	−15.4148	−2.15850
$52$	0	0
$53$	9.41478	1.29322	0.646610	−	0.762821i	$-0.276186\pi$
$53$	9.41478	1.29322	0.646610	+	0.762821i	$0.276186\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.4431	1.91303
$58$	0	0
$59$	−8.93438	−1.16316	−0.581579	−	0.813490i	$-0.697565\pi$
$59$	−8.93438	−1.16316	−0.581579	+	0.813490i	$0.697565\pi$
$60$	0	0
$61$	9.02827	1.15595	0.577976	−	0.816054i	$-0.303843\pi$
$61$	9.02827	1.15595	0.577976	+	0.816054i	$0.303843\pi$
$62$	0	0
$63$	40.3684	5.08594
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.61350	−0.685798	−0.342899	−	0.939372i	$-0.611409\pi$
$67$	−5.61350	−0.685798	−0.342899	+	0.939372i	$0.611409\pi$
$68$	0	0
$69$	−2.25526	−0.271502
$70$	0	0
$71$	1.70739	0.202630	0.101315	−	0.994854i	$-0.467695\pi$
$71$	1.70739	0.202630	0.101315	+	0.994854i	$0.467695\pi$
$72$	0	0
$73$	−12.4431	−1.45635	−0.728175	−	0.685392i	$-0.759631\pi$
$73$	−12.4431	−1.45635	−0.728175	+	0.685392i	$0.759631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.58522	0.978377
$78$	0	0
$79$	−2.64177	−0.297222	−0.148611	−	0.988896i	$-0.547480\pi$
$79$	−2.64177	−0.297222	−0.148611	+	0.988896i	$0.547480\pi$
$80$	0	0
$81$	31.3684	3.48537
$82$	0	0
$83$	8.25526	0.906133	0.453066	−	0.891477i	$-0.350330\pi$
$83$	8.25526	0.906133	0.453066	+	0.891477i	$0.350330\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.41478	0.366103
$88$	0	0
$89$	1.22699	0.130061	0.0650304	−	0.997883i	$-0.479286\pi$
$89$	1.22699	0.130061	0.0650304	+	0.997883i	$0.479286\pi$
$90$	0	0
$91$	−5.02827	−0.527106
$92$	0	0
$93$	7.61350	0.789483
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.0565477	0.00574155	0.00287078	−	0.999996i	$-0.499086\pi$
$97$	0.0565477	0.00574155	0.00287078	+	0.999996i	$0.499086\pi$
$98$	0	0
$99$	13.7074	1.37764
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−9.37743	−0.923986	−0.461993	−	0.886884i	$-0.652865\pi$
$103$	−9.37743	−0.923986	−0.461993	+	0.886884i	$0.652865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3774	1.29325	0.646623	−	0.762810i	$-0.276181\pi$
$107$	13.3774	1.29325	0.646623	+	0.762810i	$0.276181\pi$
$108$	0	0
$109$	11.2835	1.08077	0.540383	−	0.841419i	$-0.318279\pi$
$109$	11.2835	1.08077	0.540383	+	0.841419i	$0.318279\pi$
$110$	0	0
$111$	−5.35823	−0.508581
$112$	0	0
$113$	18.6983	1.75899	0.879495	−	0.475908i	$-0.157881\pi$
$113$	18.6983	1.75899	0.879495	+	0.475908i	$0.157881\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.02827	−0.742214
$118$	0	0
$119$	23.3401	2.13958
$120$	0	0
$121$	−8.08482	−0.734984
$122$	0	0
$123$	−15.4148	−1.38990
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.7357	−0.952636	−0.476318	−	0.879273i	$-0.658029\pi$
$127$	−10.7357	−0.952636	−0.476318	+	0.879273i	$0.658029\pi$
$128$	0	0
$129$	−11.0283	−0.970985
$130$	0	0
$131$	7.22699	0.631425	0.315713	−	0.948855i	$-0.397756\pi$
$131$	7.22699	0.631425	0.315713	+	0.948855i	$0.397756\pi$
$132$	0	0
$133$	−21.8688	−1.89626
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.3401	−1.48146	−0.740732	−	0.671801i	$-0.765521\pi$
$137$	−17.3401	−1.48146	−0.740732	+	0.671801i	$0.765521\pi$
$138$	0	0
$139$	9.35823	0.793755	0.396877	−	0.917872i	$-0.370094\pi$
$139$	9.35823	0.793755	0.396877	+	0.917872i	$0.370094\pi$
$140$	0	0
$141$	3.41478	0.287576
$142$	0	0
$143$	−1.70739	−0.142779
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−60.7175	−5.00790
$148$	0	0
$149$	17.3401	1.42056	0.710278	−	0.703922i	$-0.248569\pi$
$149$	17.3401	1.42056	0.710278	+	0.703922i	$0.248569\pi$
$150$	0	0
$151$	3.57615	0.291023	0.145511	−	0.989357i	$-0.453517\pi$
$151$	3.57615	0.291023	0.145511	+	0.989357i	$0.453517\pi$
$152$	0	0
$153$	37.2654	3.01273
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.28354	0.581290	0.290645	−	0.956831i	$-0.406130\pi$
$157$	7.28354	0.581290	0.290645	+	0.956831i	$0.406130\pi$
$158$	0	0
$159$	−31.2654	−2.47951
$160$	0	0
$161$	3.41478	0.269122
$162$	0	0
$163$	−0.840485	−0.0658319	−0.0329159	−	0.999458i	$-0.510479\pi$
$163$	−0.840485	−0.0658319	−0.0329159	+	0.999458i	$0.510479\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.0283	1.00816	0.504079	−	0.863658i	$-0.331832\pi$
$167$	13.0283	1.00816	0.504079	+	0.863658i	$0.331832\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−34.9162	−2.67011
$172$	0	0
$173$	−25.9253	−1.97106	−0.985532	−	0.169488i	$-0.945789\pi$
$173$	−25.9253	−1.97106	−0.985532	+	0.169488i	$0.945789\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	29.6700	2.23014
$178$	0	0
$179$	−4.77301	−0.356751	−0.178376	−	0.983962i	$-0.557084\pi$
$179$	−4.77301	−0.356751	−0.178376	+	0.983962i	$0.557084\pi$
$180$	0	0
$181$	−14.3118	−1.06379	−0.531894	−	0.846811i	$-0.678520\pi$
$181$	−14.3118	−1.06379	−0.531894	+	0.846811i	$0.678520\pi$
$182$	0	0
$183$	−29.9819	−2.21632
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.92531	0.579556
$188$	0	0
$189$	−83.9637	−6.10746
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−4.05655	−0.291997	−0.145998	−	0.989285i	$-0.546639\pi$
$193$	−4.05655	−0.291997	−0.145998	+	0.989285i	$0.546639\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.2835	1.08891	0.544453	−	0.838791i	$-0.316737\pi$
$197$	15.2835	1.08891	0.544453	+	0.838791i	$0.316737\pi$
$198$	0	0
$199$	−8.77301	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$199$	−8.77301	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$200$	0	0
$201$	18.6418	1.31489
$202$	0	0
$203$	−5.17044	−0.362894
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.45213	0.378949
$208$	0	0
$209$	−7.42571	−0.513647
$210$	0	0
$211$	0.773010	0.0532162	0.0266081	−	0.999646i	$-0.491529\pi$
$211$	0.773010	0.0532162	0.0266081	+	0.999646i	$0.491529\pi$
$212$	0	0
$213$	−5.67004	−0.388505
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.5279	−0.782563
$218$	0	0
$219$	41.3219	2.79228
$220$	0	0
$221$	−4.64177	−0.312239
$222$	0	0
$223$	14.3118	0.958390	0.479195	−	0.877709i	$-0.340929\pi$
$223$	14.3118	0.958390	0.479195	+	0.877709i	$0.340929\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.7266	0.911066	0.455533	−	0.890219i	$-0.349449\pi$
$227$	13.7266	0.911066	0.455533	+	0.890219i	$0.349449\pi$
$228$	0	0
$229$	21.9253	1.44887	0.724433	−	0.689346i	$-0.242102\pi$
$229$	21.9253	1.44887	0.724433	+	0.689346i	$0.242102\pi$
$230$	0	0
$231$	−28.5105	−1.87586
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.77301	0.569868
$238$	0	0
$239$	−15.7639	−1.01968	−0.509842	−	0.860268i	$-0.670296\pi$
$239$	−15.7639	−1.01968	−0.509842	+	0.860268i	$0.670296\pi$
$240$	0	0
$241$	9.92531	0.639345	0.319673	−	0.947528i	$-0.396427\pi$
$241$	9.92531	0.639345	0.319673	+	0.947528i	$0.396427\pi$
$242$	0	0
$243$	−54.0757	−3.46896
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.34916	0.276730
$248$	0	0
$249$	−27.4148	−1.73734
$250$	0	0
$251$	24.6983	1.55894	0.779472	−	0.626437i	$-0.215487\pi$
$251$	24.6983	1.55894	0.779472	+	0.626437i	$0.215487\pi$
$252$	0	0
$253$	1.15951	0.0728981
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0565	1.25109	0.625547	−	0.780187i	$-0.284876\pi$
$257$	20.0565	1.25109	0.625547	+	0.780187i	$0.284876\pi$
$258$	0	0
$259$	8.11310	0.504123
$260$	0	0
$261$	−8.25526	−0.510988
$262$	0	0
$263$	−26.0757	−1.60790	−0.803950	−	0.594697i	$-0.797272\pi$
$263$	−26.0757	−1.60790	−0.803950	+	0.594697i	$0.797272\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.07469	−0.249367
$268$	0	0
$269$	−12.8296	−0.782232	−0.391116	−	0.920341i	$-0.627911\pi$
$269$	−12.8296	−0.782232	−0.391116	+	0.920341i	$0.627911\pi$
$270$	0	0
$271$	−17.7074	−1.07565	−0.537824	−	0.843057i	$-0.680753\pi$
$271$	−17.7074	−1.07565	−0.537824	+	0.843057i	$0.680753\pi$
$272$	0	0
$273$	16.6983	1.01063
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.4713	0.929582	0.464791	−	0.885420i	$-0.346129\pi$
$277$	15.4713	0.929582	0.464791	+	0.885420i	$0.346129\pi$
$278$	0	0
$279$	−18.4057	−1.10192
$280$	0	0
$281$	7.35823	0.438955	0.219478	−	0.975618i	$-0.429565\pi$
$281$	7.35823	0.438955	0.219478	+	0.975618i	$0.429565\pi$
$282$	0	0
$283$	0.604422	0.0359292	0.0179646	−	0.999839i	$-0.494281\pi$
$283$	0.604422	0.0359292	0.0179646	+	0.999839i	$0.494281\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.3401	1.37772
$288$	0	0
$289$	4.54602	0.267413
$290$	0	0
$291$	−0.187788	−0.0110084
$292$	0	0
$293$	−23.0101	−1.34427	−0.672133	−	0.740430i	$-0.734622\pi$
$293$	−23.0101	−1.34427	−0.672133	+	0.740430i	$0.734622\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−28.5105	−1.65435
$298$	0	0
$299$	−0.679116	−0.0392743
$300$	0	0
$301$	16.6983	0.962475
$302$	0	0
$303$	−19.9253	−1.14468
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.3684	−1.16248	−0.581242	−	0.813731i	$-0.697433\pi$
$307$	−20.3684	−1.16248	−0.581242	+	0.813731i	$0.697433\pi$
$308$	0	0
$309$	31.1414	1.77157
$310$	0	0
$311$	−19.9253	−1.12986	−0.564930	−	0.825139i	$-0.691097\pi$
$311$	−19.9253	−1.12986	−0.564930	+	0.825139i	$0.691097\pi$
$312$	0	0
$313$	−31.4713	−1.77886	−0.889432	−	0.457067i	$-0.848900\pi$
$313$	−31.4713	−1.77886	−0.889432	+	0.457067i	$0.848900\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.72659	−0.0969750	−0.0484875	−	0.998824i	$-0.515440\pi$
$317$	−1.72659	−0.0969750	−0.0484875	+	0.998824i	$0.515440\pi$
$318$	0	0
$319$	−1.75566	−0.0982983
$320$	0	0
$321$	−44.4249	−2.47956
$322$	0	0
$323$	−20.1878	−1.12328
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−37.4713	−2.07217
$328$	0	0
$329$	−5.17044	−0.285056
$330$	0	0
$331$	10.4057	0.571949	0.285975	−	0.958237i	$-0.407683\pi$
$331$	10.4057	0.571949	0.285975	+	0.958237i	$0.407683\pi$
$332$	0	0
$333$	12.9536	0.709852
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.6983	0.582774	0.291387	−	0.956605i	$-0.405883\pi$
$337$	10.6983	0.582774	0.291387	+	0.956605i	$0.405883\pi$
$338$	0	0
$339$	−62.0950	−3.37253
$340$	0	0
$341$	−3.91438	−0.211976
$342$	0	0
$343$	56.7367	3.06349
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.6044	1.75030	0.875149	−	0.483854i	$-0.160764\pi$
$347$	32.6044	1.75030	0.875149	+	0.483854i	$0.160764\pi$
$348$	0	0
$349$	−13.4148	−0.718077	−0.359038	−	0.933323i	$-0.616895\pi$
$349$	−13.4148	−0.718077	−0.359038	+	0.933323i	$0.616895\pi$
$350$	0	0
$351$	16.6983	0.891290
$352$	0	0
$353$	35.0101	1.86340	0.931701	−	0.363227i	$-0.118325\pi$
$353$	35.0101	1.86340	0.931701	+	0.363227i	$0.118325\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−77.5097	−4.10225
$358$	0	0
$359$	20.9344	1.10487	0.552437	−	0.833555i	$-0.313698\pi$
$359$	20.9344	1.10487	0.552437	+	0.833555i	$0.313698\pi$
$360$	0	0
$361$	−0.0848216	−0.00446429
$362$	0	0
$363$	26.8488	1.40919
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.4340	−0.596849	−0.298424	−	0.954433i	$-0.596461\pi$
$367$	−11.4340	−0.596849	−0.298424	+	0.954433i	$0.596461\pi$
$368$	0	0
$369$	37.2654	1.93996
$370$	0	0
$371$	47.3401	2.45777
$372$	0	0
$373$	14.1131	0.730748	0.365374	−	0.930861i	$-0.380941\pi$
$373$	14.1131	0.730748	0.365374	+	0.930861i	$0.380941\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.02827	0.0529588
$378$	0	0
$379$	−1.89518	−0.0973487	−0.0486744	−	0.998815i	$-0.515500\pi$
$379$	−1.89518	−0.0973487	−0.0486744	+	0.998815i	$0.515500\pi$
$380$	0	0
$381$	35.6519	1.82650
$382$	0	0
$383$	22.3118	1.14008	0.570040	−	0.821617i	$-0.306928\pi$
$383$	22.3118	1.14008	0.570040	+	0.821617i	$0.306928\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.6610	1.35525
$388$	0	0
$389$	−28.6802	−1.45414	−0.727071	−	0.686562i	$-0.759119\pi$
$389$	−28.6802	−1.45414	−0.727071	+	0.686562i	$0.759119\pi$
$390$	0	0
$391$	3.15230	0.159419
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.5569	−0.780781	−0.390390	−	0.920649i	$-0.627660\pi$
$397$	−15.5569	−0.780781	−0.390390	+	0.920649i	$0.627660\pi$
$398$	0	0
$399$	72.6236	3.63573
$400$	0	0
$401$	−24.1696	−1.20697	−0.603487	−	0.797373i	$-0.706223\pi$
$401$	−24.1696	−1.20697	−0.603487	+	0.797373i	$0.706223\pi$
$402$	0	0
$403$	2.29261	0.114203
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.75486	0.136554
$408$	0	0
$409$	−29.9253	−1.47971	−0.739856	−	0.672766i	$-0.765106\pi$
$409$	−29.9253	−1.47971	−0.739856	+	0.672766i	$0.765106\pi$
$410$	0	0
$411$	57.5844	2.84043
$412$	0	0
$413$	−44.9245	−2.21059
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−31.0776	−1.52188
$418$	0	0
$419$	−8.58522	−0.419416	−0.209708	−	0.977764i	$-0.567251\pi$
$419$	−8.58522	−0.419416	−0.209708	+	0.977764i	$0.567251\pi$
$420$	0	0
$421$	−21.7375	−1.05942	−0.529711	−	0.848178i	$-0.677700\pi$
$421$	−21.7375	−1.05942	−0.529711	+	0.848178i	$0.677700\pi$
$422$	0	0
$423$	−8.25526	−0.401385
$424$	0	0
$425$	0	0
$426$	0	0
$427$	45.3966	2.19690
$428$	0	0
$429$	5.67004	0.273752
$430$	0	0
$431$	13.7074	0.660262	0.330131	−	0.943935i	$-0.392907\pi$
$431$	13.7074	0.660262	0.330131	+	0.943935i	$0.392907\pi$
$432$	0	0
$433$	40.5671	1.94953	0.974765	−	0.223235i	$-0.0716618\pi$
$433$	40.5671	1.94953	0.974765	+	0.223235i	$0.0716618\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.95358	−0.141289
$438$	0	0
$439$	26.5671	1.26798	0.633989	−	0.773342i	$-0.281417\pi$
$439$	26.5671	1.26798	0.633989	+	0.773342i	$0.281417\pi$
$440$	0	0
$441$	146.785	6.98977
$442$	0	0
$443$	16.7922	0.797822	0.398911	−	0.916990i	$-0.369388\pi$
$443$	16.7922	0.797822	0.398911	+	0.916990i	$0.369388\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−57.5844	−2.72365
$448$	0	0
$449$	−16.2443	−0.766618	−0.383309	−	0.923620i	$-0.625215\pi$
$449$	−16.2443	−0.766618	−0.383309	+	0.923620i	$0.625215\pi$
$450$	0	0
$451$	7.92531	0.373188
$452$	0	0
$453$	−11.8760	−0.557982
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.77301	−0.316828	−0.158414	−	0.987373i	$-0.550638\pi$
$457$	−6.77301	−0.316828	−0.158414	+	0.987373i	$0.550638\pi$
$458$	0	0
$459$	−77.5097	−3.61784
$460$	0	0
$461$	−22.8114	−1.06243	−0.531217	−	0.847236i	$-0.678265\pi$
$461$	−22.8114	−1.06243	−0.531217	+	0.847236i	$0.678265\pi$
$462$	0	0
$463$	16.3300	0.758917	0.379459	−	0.925209i	$-0.376110\pi$
$463$	16.3300	0.758917	0.379459	+	0.925209i	$0.376110\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.6610	−0.493331	−0.246665	−	0.969101i	$-0.579335\pi$
$467$	−10.6610	−0.493331	−0.246665	+	0.969101i	$0.579335\pi$
$468$	0	0
$469$	−28.2262	−1.30336
$470$	0	0
$471$	−24.1878	−1.11451
$472$	0	0
$473$	5.67004	0.260709
$474$	0	0
$475$	0	0
$476$	0	0
$477$	75.5844	3.46077
$478$	0	0
$479$	6.48040	0.296097	0.148048	−	0.988980i	$-0.452701\pi$
$479$	6.48040	0.296097	0.148048	+	0.988980i	$0.452701\pi$
$480$	0	0
$481$	−1.61350	−0.0735690
$482$	0	0
$483$	−11.3401	−0.515992
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.7831	−1.07772	−0.538858	−	0.842396i	$-0.681144\pi$
$487$	−23.7831	−1.07772	−0.538858	+	0.842396i	$0.681144\pi$
$488$	0	0
$489$	2.79116	0.126220
$490$	0	0
$491$	−6.13124	−0.276699	−0.138350	−	0.990383i	$-0.544180\pi$
$491$	−6.13124	−0.276699	−0.138350	+	0.990383i	$0.544180\pi$
$492$	0	0
$493$	−4.77301	−0.214966
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.58522	0.385100
$498$	0	0
$499$	21.0475	0.942214	0.471107	−	0.882076i	$-0.343854\pi$
$499$	21.0475	0.942214	0.471107	+	0.882076i	$0.343854\pi$
$500$	0	0
$501$	−43.2654	−1.93296
$502$	0	0
$503$	−33.5652	−1.49660	−0.748300	−	0.663361i	$-0.769129\pi$
$503$	−33.5652	−1.49660	−0.748300	+	0.663361i	$0.769129\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.32088	−0.147486
$508$	0	0
$509$	−14.5852	−0.646479	−0.323239	−	0.946317i	$-0.604772\pi$
$509$	−14.5852	−0.646479	−0.323239	+	0.946317i	$0.604772\pi$
$510$	0	0
$511$	−62.5671	−2.76780
$512$	0	0
$513$	72.6236	3.20641
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.75566	−0.0772140
$518$	0	0
$519$	86.0950	3.77915
$520$	0	0
$521$	−3.48225	−0.152560	−0.0762802	−	0.997086i	$-0.524304\pi$
$521$	−3.48225	−0.152560	−0.0762802	+	0.997086i	$0.524304\pi$
$522$	0	0
$523$	11.5087	0.503239	0.251620	−	0.967826i	$-0.419037\pi$
$523$	11.5087	0.503239	0.251620	+	0.967826i	$0.419037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.6418	−0.463563
$528$	0	0
$529$	−22.5388	−0.979948
$530$	0	0
$531$	−71.7276	−3.11271
$532$	0	0
$533$	−4.64177	−0.201057
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.8506	0.684004
$538$	0	0
$539$	31.2171	1.34462
$540$	0	0
$541$	−13.8122	−0.593833	−0.296917	−	0.954903i	$-0.595958\pi$
$541$	−13.8122	−0.593833	−0.296917	+	0.954903i	$0.595958\pi$
$542$	0	0
$543$	47.5279	2.03962
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.37743	0.229922	0.114961	−	0.993370i	$-0.463326\pi$
$547$	5.37743	0.229922	0.114961	+	0.993370i	$0.463326\pi$
$548$	0	0
$549$	72.4815	3.09343
$550$	0	0
$551$	4.47213	0.190519
$552$	0	0
$553$	−13.2835	−0.564873
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.0674757	0.00285904	0.00142952	−	0.999999i	$-0.499545\pi$
$557$	0.0674757	0.00285904	0.00142952	+	0.999999i	$0.499545\pi$
$558$	0	0
$559$	−3.32088	−0.140458
$560$	0	0
$561$	−26.3190	−1.11119
$562$	0	0
$563$	7.20779	0.303772	0.151886	−	0.988398i	$-0.451465\pi$
$563$	7.20779	0.303772	0.151886	+	0.988398i	$0.451465\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	157.729	6.62398
$568$	0	0
$569$	−7.85783	−0.329417	−0.164709	−	0.986342i	$-0.552668\pi$
$569$	−7.85783	−0.329417	−0.164709	+	0.986342i	$0.552668\pi$
$570$	0	0
$571$	−23.9253	−1.00124	−0.500621	−	0.865666i	$-0.666895\pi$
$571$	−23.9253	−1.00124	−0.500621	+	0.865666i	$0.666895\pi$
$572$	0	0
$573$	−39.8506	−1.66478
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.0101	−1.29097	−0.645484	−	0.763774i	$-0.723344\pi$
$577$	−31.0101	−1.29097	−0.645484	+	0.763774i	$0.723344\pi$
$578$	0	0
$579$	13.4713	0.559849
$580$	0	0
$581$	41.5097	1.72211
$582$	0	0
$583$	16.0747	0.665746
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−47.4076	−1.95672	−0.978360	−	0.206911i	$-0.933659\pi$
$587$	−47.4076	−1.95672	−0.978360	+	0.206911i	$0.933659\pi$
$588$	0	0
$589$	9.97093	0.410845
$590$	0	0
$591$	−50.7549	−2.08778
$592$	0	0
$593$	14.8861	0.611299	0.305650	−	0.952144i	$-0.401126\pi$
$593$	14.8861	0.611299	0.305650	+	0.952144i	$0.401126\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.1342	1.19238
$598$	0	0
$599$	42.8680	1.75154	0.875769	−	0.482731i	$-0.160355\pi$
$599$	42.8680	1.75154	0.875769	+	0.482731i	$0.160355\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−45.0667	−1.83526
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.7357	0.760457	0.380229	−	0.924893i	$-0.375845\pi$
$607$	18.7357	0.760457	0.380229	+	0.924893i	$0.375845\pi$
$608$	0	0
$609$	17.1704	0.695781
$610$	0	0
$611$	1.02827	0.0415995
$612$	0	0
$613$	−36.6802	−1.48150	−0.740749	−	0.671782i	$-0.765529\pi$
$613$	−36.6802	−1.48150	−0.740749	+	0.671782i	$0.765529\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.05655	0.324344	0.162172	−	0.986762i	$-0.448150\pi$
$617$	8.05655	0.324344	0.162172	+	0.986762i	$0.448150\pi$
$618$	0	0
$619$	−33.1606	−1.33284	−0.666418	−	0.745578i	$-0.732173\pi$
$619$	−33.1606	−1.33284	−0.666418	+	0.745578i	$0.732173\pi$
$620$	0	0
$621$	−11.3401	−0.455062
$622$	0	0
$623$	6.16964	0.247182
$624$	0	0
$625$	0	0
$626$	0	0
$627$	24.6599	0.984822
$628$	0	0
$629$	7.48947	0.298625
$630$	0	0
$631$	−33.8205	−1.34637	−0.673186	−	0.739473i	$-0.735075\pi$
$631$	−33.8205	−1.34637	−0.673186	+	0.739473i	$0.735075\pi$
$632$	0	0
$633$	−2.56708	−0.102032
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18.2835	−0.724420
$638$	0	0
$639$	13.7074	0.542256
$640$	0	0
$641$	12.8296	0.506737	0.253369	−	0.967370i	$-0.418461\pi$
$641$	12.8296	0.506737	0.253369	+	0.967370i	$0.418461\pi$
$642$	0	0
$643$	−43.7084	−1.72369	−0.861846	−	0.507169i	$-0.830692\pi$
$643$	−43.7084	−1.72369	−0.861846	+	0.507169i	$0.830692\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.4158	−0.999200	−0.499600	−	0.866256i	$-0.666520\pi$
$647$	−25.4158	−0.999200	−0.499600	+	0.866256i	$0.666520\pi$
$648$	0	0
$649$	−15.2545	−0.598790
$650$	0	0
$651$	38.2827	1.50042
$652$	0	0
$653$	3.94345	0.154319	0.0771596	−	0.997019i	$-0.475415\pi$
$653$	3.94345	0.154319	0.0771596	+	0.997019i	$0.475415\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−99.8962	−3.89732
$658$	0	0
$659$	−38.0950	−1.48397	−0.741984	−	0.670417i	$-0.766115\pi$
$659$	−38.0950	−1.48397	−0.741984	+	0.670417i	$0.766115\pi$
$660$	0	0
$661$	18.1131	0.704518	0.352259	−	0.935903i	$-0.385414\pi$
$661$	18.1131	0.704518	0.352259	+	0.935903i	$0.385414\pi$
$662$	0	0
$663$	15.4148	0.598660
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.698317	−0.0270389
$668$	0	0
$669$	−47.5279	−1.83753
$670$	0	0
$671$	15.4148	0.595081
$672$	0	0
$673$	−43.2088	−1.66558	−0.832789	−	0.553590i	$-0.813257\pi$
$673$	−43.2088	−1.66558	−0.832789	+	0.553590i	$0.813257\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.6983	1.17983	0.589916	−	0.807465i	$-0.299161\pi$
$677$	30.6983	1.17983	0.589916	+	0.807465i	$0.299161\pi$
$678$	0	0
$679$	0.284337	0.0109119
$680$	0	0
$681$	−45.5844	−1.74680
$682$	0	0
$683$	32.9536	1.26093	0.630467	−	0.776216i	$-0.282863\pi$
$683$	32.9536	1.26093	0.630467	+	0.776216i	$0.282863\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−72.8114	−2.77793
$688$	0	0
$689$	−9.41478	−0.358675
$690$	0	0
$691$	37.1222	1.41219	0.706097	−	0.708115i	$-0.250454\pi$
$691$	37.1222	1.41219	0.706097	+	0.708115i	$0.250454\pi$
$692$	0	0
$693$	68.9245	2.61823
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.5460	0.816114
$698$	0	0
$699$	−19.9253	−0.753644
$700$	0	0
$701$	−7.39663	−0.279367	−0.139683	−	0.990196i	$-0.544609\pi$
$701$	−7.39663	−0.279367	−0.139683	+	0.990196i	$0.544609\pi$
$702$	0	0
$703$	−7.01735	−0.264664
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.1696	1.13465
$708$	0	0
$709$	28.7549	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$709$	28.7549	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$710$	0	0
$711$	−21.2088	−0.795394
$712$	0	0
$713$	−1.55695	−0.0583081
$714$	0	0
$715$	0	0
$716$	0	0
$717$	52.3502	1.95505
$718$	0	0
$719$	−39.4532	−1.47136	−0.735678	−	0.677332i	$-0.763136\pi$
$719$	−39.4532	−1.47136	−0.735678	+	0.677332i	$0.763136\pi$
$720$	0	0
$721$	−47.1523	−1.75604
$722$	0	0
$723$	−32.9608	−1.22583
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.866904	0.0321517	0.0160758	−	0.999871i	$-0.494883\pi$
$727$	0.866904	0.0321517	0.0160758	+	0.999871i	$0.494883\pi$
$728$	0	0
$729$	85.4742	3.16571
$730$	0	0
$731$	15.4148	0.570136
$732$	0	0
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.58442	−0.353047
$738$	0	0
$739$	−31.5015	−1.15880	−0.579400	−	0.815043i	$-0.696713\pi$
$739$	−31.5015	−1.15880	−0.579400	+	0.815043i	$0.696713\pi$
$740$	0	0
$741$	−14.4431	−0.530579
$742$	0	0
$743$	31.8578	1.16875	0.584375	−	0.811484i	$-0.301340\pi$
$743$	31.8578	1.16875	0.584375	+	0.811484i	$0.301340\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	66.2755	2.42489
$748$	0	0
$749$	67.2654	2.45782
$750$	0	0
$751$	46.0950	1.68203	0.841014	−	0.541013i	$-0.181959\pi$
$751$	46.0950	1.68203	0.841014	+	0.541013i	$0.181959\pi$
$752$	0	0
$753$	−82.0203	−2.98898
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.01735	−0.182359	−0.0911793	−	0.995834i	$-0.529064\pi$
$757$	−5.01735	−0.182359	−0.0911793	+	0.995834i	$0.529064\pi$
$758$	0	0
$759$	−3.85061	−0.139768
$760$	0	0
$761$	24.8296	0.900071	0.450035	−	0.893011i	$-0.351411\pi$
$761$	24.8296	0.900071	0.450035	+	0.893011i	$0.351411\pi$
$762$	0	0
$763$	56.7367	2.05401
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.93438	0.322602
$768$	0	0
$769$	36.9427	1.33219	0.666093	−	0.745869i	$-0.267965\pi$
$769$	36.9427	1.33219	0.666093	+	0.745869i	$0.267965\pi$
$770$	0	0
$771$	−66.6055	−2.39874
$772$	0	0
$773$	16.4431	0.591415	0.295708	−	0.955278i	$-0.404445\pi$
$773$	16.4431	0.591415	0.295708	+	0.955278i	$0.404445\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−26.9427	−0.966562
$778$	0	0
$779$	−20.1878	−0.723303
$780$	0	0
$781$	2.91518	0.104313
$782$	0	0
$783$	17.1704	0.613622
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40.2553	−1.43495	−0.717473	−	0.696587i	$-0.754701\pi$
$787$	−40.2553	−1.43495	−0.717473	+	0.696587i	$0.754701\pi$
$788$	0	0
$789$	86.5946	3.08285
$790$	0	0
$791$	94.0203	3.34298
$792$	0	0
$793$	−9.02827	−0.320603
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.2835	−0.966432	−0.483216	−	0.875501i	$-0.660532\pi$
$797$	−27.2835	−0.966432	−0.483216	+	0.875501i	$0.660532\pi$
$798$	0	0
$799$	−4.77301	−0.168857
$800$	0	0
$801$	9.85061	0.348054
$802$	0	0
$803$	−21.2451	−0.749725
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.6055	1.49978
$808$	0	0
$809$	−28.4815	−1.00135	−0.500677	−	0.865634i	$-0.666916\pi$
$809$	−28.4815	−1.00135	−0.500677	+	0.865634i	$0.666916\pi$
$810$	0	0
$811$	31.6892	1.11276	0.556380	−	0.830928i	$-0.312190\pi$
$811$	31.6892	1.11276	0.556380	+	0.830928i	$0.312190\pi$
$812$	0	0
$813$	58.8042	2.06235
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.4431	−0.505298
$818$	0	0
$819$	−40.3684	−1.41058
$820$	0	0
$821$	−6.65991	−0.232433	−0.116216	−	0.993224i	$-0.537077\pi$
$821$	−6.65991	−0.232433	−0.116216	+	0.993224i	$0.537077\pi$
$822$	0	0
$823$	7.79301	0.271647	0.135824	−	0.990733i	$-0.456632\pi$
$823$	7.79301	0.271647	0.135824	+	0.990733i	$0.456632\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.4249	−0.918884	−0.459442	−	0.888208i	$-0.651951\pi$
$827$	−26.4249	−0.918884	−0.459442	+	0.888208i	$0.651951\pi$
$828$	0	0
$829$	11.7447	0.407912	0.203956	−	0.978980i	$-0.434620\pi$
$829$	11.7447	0.407912	0.203956	+	0.978980i	$0.434620\pi$
$830$	0	0
$831$	−51.3785	−1.78230
$832$	0	0
$833$	84.8680	2.94050
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.2827	1.32325
$838$	0	0
$839$	−44.5753	−1.53891	−0.769456	−	0.638700i	$-0.779473\pi$
$839$	−44.5753	−1.53891	−0.769456	+	0.638700i	$0.779473\pi$
$840$	0	0
$841$	−27.9427	−0.963540
$842$	0	0
$843$	−24.4358	−0.841615
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−40.6527	−1.39684
$848$	0	0
$849$	−2.00722	−0.0688875
$850$	0	0
$851$	1.09575	0.0375618
$852$	0	0
$853$	−29.6135	−1.01395	−0.506973	−	0.861962i	$-0.669236\pi$
$853$	−29.6135	−1.01395	−0.506973	+	0.861962i	$0.669236\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.5105	−0.768945	−0.384472	−	0.923136i	$-0.625617\pi$
$857$	−22.5105	−0.768945	−0.384472	+	0.923136i	$0.625617\pi$
$858$	0	0
$859$	−0.585221	−0.0199675	−0.00998375	−	0.999950i	$-0.503178\pi$
$859$	−0.585221	−0.0199675	−0.00998375	+	0.999950i	$0.503178\pi$
$860$	0	0
$861$	−77.5097	−2.64152
$862$	0	0
$863$	−45.6519	−1.55401	−0.777004	−	0.629495i	$-0.783262\pi$
$863$	−45.6519	−1.55401	−0.777004	+	0.629495i	$0.783262\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.0968	−0.512714
$868$	0	0
$869$	−4.51053	−0.153009
$870$	0	0
$871$	5.61350	0.190206
$872$	0	0
$873$	0.453981	0.0153649
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	76.4140	2.57738
$880$	0	0
$881$	42.5380	1.43314	0.716571	−	0.697514i	$-0.245711\pi$
$881$	42.5380	1.43314	0.716571	+	0.697514i	$0.245711\pi$
$882$	0	0
$883$	−6.22513	−0.209492	−0.104746	−	0.994499i	$-0.533403\pi$
$883$	−6.22513	−0.209492	−0.104746	+	0.994499i	$0.533403\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.0011	−0.738723	−0.369362	−	0.929286i	$-0.620424\pi$
$887$	−22.0011	−0.738723	−0.369362	+	0.929286i	$0.620424\pi$
$888$	0	0
$889$	−53.9819	−1.81049
$890$	0	0
$891$	53.5580	1.79426
$892$	0	0
$893$	4.47213	0.149654
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.25526	0.0753011
$898$	0	0
$899$	2.35743	0.0786247
$900$	0	0
$901$	43.7012	1.45590
$902$	0	0
$903$	−55.4532	−1.84537
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.2070	−0.936598	−0.468299	−	0.883570i	$-0.655133\pi$
$907$	−28.2070	−0.936598	−0.468299	+	0.883570i	$0.655133\pi$
$908$	0	0
$909$	48.1696	1.59769
$910$	0	0
$911$	40.5105	1.34217	0.671087	−	0.741379i	$-0.265828\pi$
$911$	40.5105	1.34217	0.671087	+	0.741379i	$0.265828\pi$
$912$	0	0
$913$	14.0950	0.466475
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.3393	1.20003
$918$	0	0
$919$	0.0746930	0.00246389	0.00123195	−	0.999999i	$-0.499608\pi$
$919$	0.0746930	0.00246389	0.00123195	+	0.999999i	$0.499608\pi$
$920$	0	0
$921$	67.6410	2.22885
$922$	0	0
$923$	−1.70739	−0.0561994
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−75.2846	−2.47267
$928$	0	0
$929$	53.6410	1.75990	0.879952	−	0.475063i	$-0.157575\pi$
$929$	53.6410	1.75990	0.879952	+	0.475063i	$0.157575\pi$
$930$	0	0
$931$	−79.5180	−2.60610
$932$	0	0
$933$	66.1696	2.16630
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	104.513	3.41064
$940$	0	0
$941$	−49.9253	−1.62752	−0.813759	−	0.581202i	$-0.802583\pi$
$941$	−49.9253	−1.62752	−0.813759	+	0.581202i	$0.802583\pi$
$942$	0	0
$943$	3.15230	0.102653
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.80128	0.188516	0.0942582	−	0.995548i	$-0.469952\pi$
$947$	5.80128	0.188516	0.0942582	+	0.995548i	$0.469952\pi$
$948$	0	0
$949$	12.4431	0.403919
$950$	0	0
$951$	5.73381	0.185931
$952$	0	0
$953$	13.2270	0.428464	0.214232	−	0.976783i	$-0.431275\pi$
$953$	13.2270	0.428464	0.214232	+	0.976783i	$0.431275\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.83036	0.188469
$958$	0	0
$959$	−87.1907	−2.81553
$960$	0	0
$961$	−25.7439	−0.830450
$962$	0	0
$963$	107.398	3.46084
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.61350	0.0518865	0.0259433	−	0.999663i	$-0.491741\pi$
$967$	1.61350	0.0518865	0.0259433	+	0.999663i	$0.491741\pi$
$968$	0	0
$969$	67.0413	2.15368
$970$	0	0
$971$	16.7730	0.538271	0.269136	−	0.963102i	$-0.413262\pi$
$971$	16.7730	0.538271	0.269136	+	0.963102i	$0.413262\pi$
$972$	0	0
$973$	47.0557	1.50854
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.6700	1.52510	0.762550	−	0.646929i	$-0.223947\pi$
$977$	47.6700	1.52510	0.762550	+	0.646929i	$0.223947\pi$
$978$	0	0
$979$	2.09495	0.0669549
$980$	0	0
$981$	90.5873	2.89223
$982$	0	0
$983$	−30.4996	−0.972786	−0.486393	−	0.873740i	$-0.661688\pi$
$983$	−30.4996	−0.972786	−0.486393	+	0.873740i	$0.661688\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.1704	0.546541
$988$	0	0
$989$	2.25526	0.0717132
$990$	0	0
$991$	−19.8506	−0.630576	−0.315288	−	0.948996i	$-0.602101\pi$
$991$	−19.8506	−0.630576	−0.315288	+	0.948996i	$0.602101\pi$
$992$	0	0
$993$	−34.5561	−1.09661
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.6410	1.06542	0.532710	−	0.846298i	$-0.321174\pi$
$997$	33.6410	1.06542	0.532710	+	0.846298i	$0.321174\pi$
$998$	0	0
$999$	−26.9427	−0.852428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.i.1.1		3
4.3	odd	2		5200.2.a.ci.1.3		3
5.2	odd	4		1300.2.c.f.1249.6		6
5.3	odd	4		1300.2.c.f.1249.1		6
5.4	even	2		260.2.a.b.1.3	✓	3
15.14	odd	2		2340.2.a.n.1.1		3
20.19	odd	2		1040.2.a.o.1.1		3
40.19	odd	2		4160.2.a.br.1.3		3
40.29	even	2		4160.2.a.bo.1.1		3
60.59	even	2		9360.2.a.da.1.3		3
65.34	odd	4		3380.2.f.h.3041.6		6
65.44	odd	4		3380.2.f.h.3041.5		6
65.64	even	2		3380.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.3	✓	3	5.4	even	2
1040.2.a.o.1.1		3	20.19	odd	2
1300.2.a.i.1.1		3	1.1	even	1	trivial
1300.2.c.f.1249.1		6	5.3	odd	4
1300.2.c.f.1249.6		6	5.2	odd	4
2340.2.a.n.1.1		3	15.14	odd	2
3380.2.a.o.1.3		3	65.64	even	2
3380.2.f.h.3041.5		6	65.44	odd	4
3380.2.f.h.3041.6		6	65.34	odd	4
4160.2.a.bo.1.1		3	40.29	even	2
4160.2.a.br.1.3		3	40.19	odd	2
5200.2.a.ci.1.3		3	4.3	odd	2
9360.2.a.da.1.3		3	60.59	even	2

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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q-3.32088 q^{3} +5.02827 q^{7} +8.02827 q^{9} +O(q^{10})\)	\(q-3.32088 q^{3} +5.02827 q^{7} +8.02827 q^{9} +1.70739 q^{11} -1.00000 q^{13} +4.64177 q^{17} -4.34916 q^{19} -16.6983 q^{21} +0.679116 q^{23} -16.6983 q^{27} -1.02827 q^{29} -2.29261 q^{31} -5.67004 q^{33} +1.61350 q^{37} +3.32088 q^{39} +4.64177 q^{41} +3.32088 q^{43} -1.02827 q^{47} +18.2835 q^{49} -15.4148 q^{51} +9.41478 q^{53} +14.4431 q^{57} -8.93438 q^{59} +9.02827 q^{61} +40.3684 q^{63} -5.61350 q^{67} -2.25526 q^{69} +1.70739 q^{71} -12.4431 q^{73} +8.58522 q^{77} -2.64177 q^{79} +31.3684 q^{81} +8.25526 q^{83} +3.41478 q^{87} +1.22699 q^{89} -5.02827 q^{91} +7.61350 q^{93} +0.0565477 q^{97} +13.7074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 2 * q^7 + 11 * q^9	\( 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{23} - 8 q^{27} + 10 q^{29} - 12 q^{31} + 12 q^{33} + 2 q^{37} + 2 q^{39} - 2 q^{41} + 2 q^{43} + 10 q^{47} + 23 q^{49} - 36 q^{51} + 18 q^{53} + 20 q^{57} - 16 q^{59} + 14 q^{61} + 50 q^{63} - 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} + 8 q^{79} + 23 q^{81} + 6 q^{83} - 2 q^{89} - 2 q^{91} + 20 q^{93} - 26 q^{97} + 36 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^17 + 8 * q^19 - 8 * q^21 + 10 * q^23 - 8 * q^27 + 10 * q^29 - 12 * q^31 + 12 * q^33 + 2 * q^37 + 2 * q^39 - 2 * q^41 + 2 * q^43 + 10 * q^47 + 23 * q^49 - 36 * q^51 + 18 * q^53 + 20 * q^57 - 16 * q^59 + 14 * q^61 + 50 * q^63 - 14 * q^67 + 12 * q^69 - 14 * q^73 + 36 * q^77 + 8 * q^79 + 23 * q^81 + 6 * q^83 - 2 * q^89 - 2 * q^91 + 20 * q^93 - 26 * q^97 + 36 * q^99

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