LMFDB - Embedded newform 7800.2.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.504568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52064$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.45939 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.45939 q^{7} +1.00000 q^{9} +3.77463 q^{11} +1.00000 q^{13} +0.793451 q^{17} +3.15795 q^{19} +2.45939 q^{21} +1.20655 q^{23} +1.00000 q^{27} +5.04127 q^{29} -4.94639 q^{31} +3.77463 q^{33} +9.91376 q^{37} +1.00000 q^{39} -10.6043 q^{41} -6.58472 q^{43} +9.60935 q^{47} -0.951404 q^{49} +0.793451 q^{51} -0.168784 q^{53} +3.15795 q^{57} +12.0419 q^{59} +15.2491 q^{61} +2.45939 q^{63} -3.33188 q^{67} +1.20655 q^{69} -16.0564 q^{71} -10.9064 q^{73} +9.28328 q^{77} +6.46520 q^{79} +1.00000 q^{81} +3.10367 q^{83} +5.04127 q^{87} +1.89131 q^{89} +2.45939 q^{91} -4.94639 q^{93} +9.62315 q^{97} +3.77463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^7 + 5 * q^9	$ 5 q + 5 q^{3} + q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + 7 q^{17} + 6 q^{19} + q^{21} + 3 q^{23} + 5 q^{27} + 2 q^{29} + 5 q^{33} + 9 q^{37} + 5 q^{39} - q^{41} + 4 q^{43} + 14 q^{47} + 2 q^{49} + 7 q^{51} + 5 q^{53} + 6 q^{57} + 20 q^{59} - 19 q^{61} + q^{63} + 12 q^{67} + 3 q^{69} + 13 q^{71} + 16 q^{73} + 11 q^{77} + 7 q^{79} + 5 q^{81} - 2 q^{83} + 2 q^{87} + 9 q^{89} + q^{91} + 13 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + 7 * q^17 + 6 * q^19 + q^21 + 3 * q^23 + 5 * q^27 + 2 * q^29 + 5 * q^33 + 9 * q^37 + 5 * q^39 - q^41 + 4 * q^43 + 14 * q^47 + 2 * q^49 + 7 * q^51 + 5 * q^53 + 6 * q^57 + 20 * q^59 - 19 * q^61 + q^63 + 12 * q^67 + 3 * q^69 + 13 * q^71 + 16 * q^73 + 11 * q^77 + 7 * q^79 + 5 * q^81 - 2 * q^83 + 2 * q^87 + 9 * q^89 + q^91 + 13 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.45939	0.929562	0.464781	−	0.885426i	$-0.346133\pi$
$7$	2.45939	0.929562	0.464781	+	0.885426i	$0.346133\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.77463	1.13809	0.569047	−	0.822305i	$-0.307312\pi$
$11$	3.77463	1.13809	0.569047	+	0.822305i	$0.307312\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.793451	0.192440	0.0962200	−	0.995360i	$-0.469325\pi$
$17$	0.793451	0.192440	0.0962200	+	0.995360i	$0.469325\pi$
$18$	0	0
$19$	3.15795	0.724484	0.362242	−	0.932084i	$-0.382011\pi$
$19$	3.15795	0.724484	0.362242	+	0.932084i	$0.382011\pi$
$20$	0	0
$21$	2.45939	0.536683
$22$	0	0
$23$	1.20655	0.251583	0.125791	−	0.992057i	$-0.459853\pi$
$23$	1.20655	0.251583	0.125791	+	0.992057i	$0.459853\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.04127	0.936141	0.468070	−	0.883691i	$-0.344949\pi$
$29$	5.04127	0.936141	0.468070	+	0.883691i	$0.344949\pi$
$30$	0	0
$31$	−4.94639	−0.888397	−0.444199	−	0.895928i	$-0.646512\pi$
$31$	−4.94639	−0.888397	−0.444199	+	0.895928i	$0.646512\pi$
$32$	0	0
$33$	3.77463	0.657079
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.91376	1.62981	0.814906	−	0.579593i	$-0.196788\pi$
$37$	9.91376	1.62981	0.814906	+	0.579593i	$0.196788\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.6043	−1.65612	−0.828059	−	0.560641i	$-0.810555\pi$
$41$	−10.6043	−1.65612	−0.828059	+	0.560641i	$0.810555\pi$
$42$	0	0
$43$	−6.58472	−1.00416	−0.502080	−	0.864821i	$-0.667432\pi$
$43$	−6.58472	−1.00416	−0.502080	+	0.864821i	$0.667432\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.60935	1.40167	0.700834	−	0.713324i	$-0.252811\pi$
$47$	9.60935	1.40167	0.700834	+	0.713324i	$0.252811\pi$
$48$	0	0
$49$	−0.951404	−0.135915
$50$	0	0
$51$	0.793451	0.111105
$52$	0	0
$53$	−0.168784	−0.0231842	−0.0115921	−	0.999933i	$-0.503690\pi$
$53$	−0.168784	−0.0231842	−0.0115921	+	0.999933i	$0.503690\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.15795	0.418281
$58$	0	0
$59$	12.0419	1.56773	0.783863	−	0.620933i	$-0.213246\pi$
$59$	12.0419	1.56773	0.783863	+	0.620933i	$0.213246\pi$
$60$	0	0
$61$	15.2491	1.95245	0.976226	−	0.216753i	$-0.0695468\pi$
$61$	15.2491	1.95245	0.976226	+	0.216753i	$0.0695468\pi$
$62$	0	0
$63$	2.45939	0.309854
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.33188	−0.407054	−0.203527	−	0.979069i	$-0.565240\pi$
$67$	−3.33188	−0.407054	−0.203527	+	0.979069i	$0.565240\pi$
$68$	0	0
$69$	1.20655	0.145251
$70$	0	0
$71$	−16.0564	−1.90554	−0.952772	−	0.303687i	$-0.901782\pi$
$71$	−16.0564	−1.90554	−0.952772	+	0.303687i	$0.901782\pi$
$72$	0	0
$73$	−10.9064	−1.27650	−0.638251	−	0.769828i	$-0.720342\pi$
$73$	−10.9064	−1.27650	−0.638251	+	0.769828i	$0.720342\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.28328	1.05793
$78$	0	0
$79$	6.46520	0.727392	0.363696	−	0.931518i	$-0.381515\pi$
$79$	6.46520	0.727392	0.363696	+	0.931518i	$0.381515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.10367	0.340672	0.170336	−	0.985386i	$-0.445515\pi$
$83$	3.10367	0.340672	0.170336	+	0.985386i	$0.445515\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.04127	0.540481
$88$	0	0
$89$	1.89131	0.200478	0.100239	−	0.994963i	$-0.468039\pi$
$89$	1.89131	0.200478	0.100239	+	0.994963i	$0.468039\pi$
$90$	0	0
$91$	2.45939	0.257814
$92$	0	0
$93$	−4.94639	−0.512916
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.62315	0.977083	0.488542	−	0.872541i	$-0.337529\pi$
$97$	9.62315	0.977083	0.488542	+	0.872541i	$0.337529\pi$
$98$	0	0
$99$	3.77463	0.379364
$100$	0	0
$101$	2.62817	0.261513	0.130756	−	0.991415i	$-0.458259\pi$
$101$	2.62817	0.261513	0.130756	+	0.991415i	$0.458259\pi$
$102$	0	0
$103$	−19.5346	−1.92480	−0.962401	−	0.271632i	$-0.912437\pi$
$103$	−19.5346	−1.92480	−0.962401	+	0.271632i	$0.912437\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.9963	−1.35307	−0.676537	−	0.736409i	$-0.736520\pi$
$107$	−13.9963	−1.35307	−0.676537	+	0.736409i	$0.736520\pi$
$108$	0	0
$109$	−12.8274	−1.22864	−0.614321	−	0.789056i	$-0.710570\pi$
$109$	−12.8274	−1.22864	−0.614321	+	0.789056i	$0.710570\pi$
$110$	0	0
$111$	9.91376	0.940972
$112$	0	0
$113$	10.2941	0.968389	0.484194	−	0.874961i	$-0.339113\pi$
$113$	10.2941	0.968389	0.484194	+	0.874961i	$0.339113\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.95140	0.178885
$120$	0	0
$121$	3.24782	0.295256
$122$	0	0
$123$	−10.6043	−0.956160
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.89929	−0.434742	−0.217371	−	0.976089i	$-0.569748\pi$
$127$	−4.89929	−0.434742	−0.217371	+	0.976089i	$0.569748\pi$
$128$	0	0
$129$	−6.58472	−0.579752
$130$	0	0
$131$	6.94869	0.607110	0.303555	−	0.952814i	$-0.401826\pi$
$131$	6.94869	0.607110	0.303555	+	0.952814i	$0.401826\pi$
$132$	0	0
$133$	7.76664	0.673453
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.50080	−0.213658	−0.106829	−	0.994277i	$-0.534070\pi$
$137$	−2.50080	−0.213658	−0.106829	+	0.994277i	$0.534070\pi$
$138$	0	0
$139$	2.04860	0.173760	0.0868798	−	0.996219i	$-0.472310\pi$
$139$	2.04860	0.173760	0.0868798	+	0.996219i	$0.472310\pi$
$140$	0	0
$141$	9.60935	0.809253
$142$	0	0
$143$	3.77463	0.315650
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.951404	−0.0784705
$148$	0	0
$149$	−0.828243	−0.0678523	−0.0339262	−	0.999424i	$-0.510801\pi$
$149$	−0.828243	−0.0678523	−0.0339262	+	0.999424i	$0.510801\pi$
$150$	0	0
$151$	−7.35948	−0.598906	−0.299453	−	0.954111i	$-0.596804\pi$
$151$	−7.35948	−0.598906	−0.299453	+	0.954111i	$0.596804\pi$
$152$	0	0
$153$	0.793451	0.0641467
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.2804	1.05990	0.529948	−	0.848030i	$-0.322212\pi$
$157$	13.2804	1.05990	0.529948	+	0.848030i	$0.322212\pi$
$158$	0	0
$159$	−0.168784	−0.0133854
$160$	0	0
$161$	2.96737	0.233862
$162$	0	0
$163$	17.8065	1.39472	0.697358	−	0.716723i	$-0.254359\pi$
$163$	17.8065	1.39472	0.697358	+	0.716723i	$0.254359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.593380	−0.0459171	−0.0229586	−	0.999736i	$-0.507309\pi$
$167$	−0.593380	−0.0459171	−0.0229586	+	0.999736i	$0.507309\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.15795	0.241495
$172$	0	0
$173$	−12.4304	−0.945065	−0.472533	−	0.881313i	$-0.656660\pi$
$173$	−12.4304	−0.945065	−0.472533	+	0.881313i	$0.656660\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0419	0.905128
$178$	0	0
$179$	2.76301	0.206517	0.103258	−	0.994655i	$-0.467073\pi$
$179$	2.76301	0.206517	0.103258	+	0.994655i	$0.467073\pi$
$180$	0	0
$181$	−24.2137	−1.79979	−0.899894	−	0.436108i	$-0.856357\pi$
$181$	−24.2137	−1.79979	−0.899894	+	0.436108i	$0.856357\pi$
$182$	0	0
$183$	15.2491	1.12725
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.99498	0.219015
$188$	0	0
$189$	2.45939	0.178894
$190$	0	0
$191$	−14.7563	−1.06773	−0.533866	−	0.845569i	$-0.679261\pi$
$191$	−14.7563	−1.06773	−0.533866	+	0.845569i	$0.679261\pi$
$192$	0	0
$193$	1.82891	0.131648	0.0658239	−	0.997831i	$-0.479032\pi$
$193$	1.82891	0.131648	0.0658239	+	0.997831i	$0.479032\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.4585	1.45761	0.728805	−	0.684721i	$-0.240076\pi$
$197$	20.4585	1.45761	0.728805	+	0.684721i	$0.240076\pi$
$198$	0	0
$199$	−1.52977	−0.108442	−0.0542212	−	0.998529i	$-0.517268\pi$
$199$	−1.52977	−0.108442	−0.0542212	+	0.998529i	$0.517268\pi$
$200$	0	0
$201$	−3.33188	−0.235013
$202$	0	0
$203$	12.3985	0.870201
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.20655	0.0838610
$208$	0	0
$209$	11.9201	0.824531
$210$	0	0
$211$	−9.33538	−0.642674	−0.321337	−	0.946965i	$-0.604132\pi$
$211$	−9.33538	−0.642674	−0.321337	+	0.946965i	$0.604132\pi$
$212$	0	0
$213$	−16.0564	−1.10017
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.1651	−0.825820
$218$	0	0
$219$	−10.9064	−0.736989
$220$	0	0
$221$	0.793451	0.0533733
$222$	0	0
$223$	7.96005	0.533044	0.266522	−	0.963829i	$-0.414125\pi$
$223$	7.96005	0.533044	0.266522	+	0.963829i	$0.414125\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.4456	0.826043	0.413021	−	0.910721i	$-0.364474\pi$
$227$	12.4456	0.826043	0.413021	+	0.910721i	$0.364474\pi$
$228$	0	0
$229$	−18.9955	−1.25526	−0.627629	−	0.778512i	$-0.715975\pi$
$229$	−18.9955	−1.25526	−0.627629	+	0.778512i	$0.715975\pi$
$230$	0	0
$231$	9.28328	0.610795
$232$	0	0
$233$	25.5674	1.67497	0.837487	−	0.546458i	$-0.184024\pi$
$233$	25.5674	1.67497	0.837487	+	0.546458i	$0.184024\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.46520	0.419960
$238$	0	0
$239$	18.1912	1.17669	0.588347	−	0.808609i	$-0.299779\pi$
$239$	18.1912	1.17669	0.588347	+	0.808609i	$0.299779\pi$
$240$	0	0
$241$	−11.8100	−0.760746	−0.380373	−	0.924833i	$-0.624204\pi$
$241$	−11.8100	−0.760746	−0.380373	+	0.924833i	$0.624204\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.15795	0.200936
$248$	0	0
$249$	3.10367	0.196687
$250$	0	0
$251$	1.20371	0.0759778	0.0379889	−	0.999278i	$-0.487905\pi$
$251$	1.20371	0.0759778	0.0379889	+	0.999278i	$0.487905\pi$
$252$	0	0
$253$	4.55428	0.286325
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.94190	−0.495402	−0.247701	−	0.968837i	$-0.579675\pi$
$257$	−7.94190	−0.495402	−0.247701	+	0.968837i	$0.579675\pi$
$258$	0	0
$259$	24.3818	1.51501
$260$	0	0
$261$	5.04127	0.312047
$262$	0	0
$263$	4.45789	0.274885	0.137443	−	0.990510i	$-0.456112\pi$
$263$	4.45789	0.274885	0.137443	+	0.990510i	$0.456112\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.89131	0.115746
$268$	0	0
$269$	−30.1687	−1.83942	−0.919710	−	0.392599i	$-0.871576\pi$
$269$	−30.1687	−1.83942	−0.919710	+	0.392599i	$0.871576\pi$
$270$	0	0
$271$	9.98535	0.606567	0.303283	−	0.952900i	$-0.401917\pi$
$271$	9.98535	0.606567	0.303283	+	0.952900i	$0.401917\pi$
$272$	0	0
$273$	2.45939	0.148849
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.78745	−0.407818	−0.203909	−	0.978990i	$-0.565365\pi$
$277$	−6.78745	−0.407818	−0.203909	+	0.978990i	$0.565365\pi$
$278$	0	0
$279$	−4.94639	−0.296132
$280$	0	0
$281$	17.6689	1.05404	0.527019	−	0.849853i	$-0.323310\pi$
$281$	17.6689	1.05404	0.527019	+	0.849853i	$0.323310\pi$
$282$	0	0
$283$	−18.1434	−1.07851	−0.539257	−	0.842141i	$-0.681295\pi$
$283$	−18.1434	−1.07851	−0.539257	+	0.842141i	$0.681295\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.0802	−1.53946
$288$	0	0
$289$	−16.3704	−0.962967
$290$	0	0
$291$	9.62315	0.564119
$292$	0	0
$293$	−0.277473	−0.0162102	−0.00810508	−	0.999967i	$-0.502580\pi$
$293$	−0.277473	−0.0162102	−0.00810508	+	0.999967i	$0.502580\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.77463	0.219026
$298$	0	0
$299$	1.20655	0.0697766
$300$	0	0
$301$	−16.1944	−0.933429
$302$	0	0
$303$	2.62817	0.150985
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.6515	0.722057	0.361029	−	0.932555i	$-0.382426\pi$
$307$	12.6515	0.722057	0.361029	+	0.932555i	$0.382426\pi$
$308$	0	0
$309$	−19.5346	−1.11129
$310$	0	0
$311$	−10.3056	−0.584377	−0.292188	−	0.956361i	$-0.594383\pi$
$311$	−10.3056	−0.584377	−0.292188	+	0.956361i	$0.594383\pi$
$312$	0	0
$313$	22.7601	1.28648	0.643239	−	0.765665i	$-0.277590\pi$
$313$	22.7601	1.28648	0.643239	+	0.765665i	$0.277590\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.0324873	−0.00182467	−0.000912335	−	1.00000i	$-0.500290\pi$
$317$	−0.0324873	−0.00182467	−0.000912335		1.00000i	$0.500290\pi$
$318$	0	0
$319$	19.0289	1.06542
$320$	0	0
$321$	−13.9963	−0.781197
$322$	0	0
$323$	2.50568	0.139420
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.8274	−0.709357
$328$	0	0
$329$	23.6331	1.30294
$330$	0	0
$331$	31.3679	1.72414	0.862069	−	0.506791i	$-0.169168\pi$
$331$	31.3679	1.72414	0.862069	+	0.506791i	$0.169168\pi$
$332$	0	0
$333$	9.91376	0.543271
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.5674	0.630117	0.315059	−	0.949072i	$-0.397976\pi$
$337$	11.5674	0.630117	0.315059	+	0.949072i	$0.397976\pi$
$338$	0	0
$339$	10.2941	0.559099
$340$	0	0
$341$	−18.6708	−1.01108
$342$	0	0
$343$	−19.5556	−1.05590
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.1471	1.34997	0.674984	−	0.737833i	$-0.264151\pi$
$347$	25.1471	1.34997	0.674984	+	0.737833i	$0.264151\pi$
$348$	0	0
$349$	−19.1709	−1.02620	−0.513098	−	0.858330i	$-0.671502\pi$
$349$	−19.1709	−1.02620	−0.513098	+	0.858330i	$0.671502\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−5.00648	−0.266468	−0.133234	−	0.991085i	$-0.542536\pi$
$353$	−5.00648	−0.266468	−0.133234	+	0.991085i	$0.542536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.95140	0.103279
$358$	0	0
$359$	7.96724	0.420495	0.210247	−	0.977648i	$-0.432573\pi$
$359$	7.96724	0.420495	0.210247	+	0.977648i	$0.432573\pi$
$360$	0	0
$361$	−9.02733	−0.475123
$362$	0	0
$363$	3.24782	0.170466
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.0403	0.732897	0.366449	−	0.930438i	$-0.380574\pi$
$367$	14.0403	0.732897	0.366449	+	0.930438i	$0.380574\pi$
$368$	0	0
$369$	−10.6043	−0.552039
$370$	0	0
$371$	−0.415104	−0.0215512
$372$	0	0
$373$	16.8489	0.872404	0.436202	−	0.899849i	$-0.356323\pi$
$373$	16.8489	0.872404	0.436202	+	0.899849i	$0.356323\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.04127	0.259639
$378$	0	0
$379$	−28.8174	−1.48025	−0.740124	−	0.672470i	$-0.765233\pi$
$379$	−28.8174	−1.48025	−0.740124	+	0.672470i	$0.765233\pi$
$380$	0	0
$381$	−4.89929	−0.250998
$382$	0	0
$383$	31.9149	1.63078	0.815389	−	0.578914i	$-0.196523\pi$
$383$	31.9149	1.63078	0.815389	+	0.578914i	$0.196523\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.58472	−0.334720
$388$	0	0
$389$	−23.1934	−1.17595	−0.587976	−	0.808878i	$-0.700075\pi$
$389$	−23.1934	−1.17595	−0.587976	+	0.808878i	$0.700075\pi$
$390$	0	0
$391$	0.957337	0.0484146
$392$	0	0
$393$	6.94869	0.350515
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.13247	−0.408157	−0.204079	−	0.978955i	$-0.565420\pi$
$397$	−8.13247	−0.408157	−0.204079	+	0.978955i	$0.565420\pi$
$398$	0	0
$399$	7.76664	0.388818
$400$	0	0
$401$	29.0398	1.45018	0.725088	−	0.688656i	$-0.241799\pi$
$401$	29.0398	1.45018	0.725088	+	0.688656i	$0.241799\pi$
$402$	0	0
$403$	−4.94639	−0.246397
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.4208	1.85488
$408$	0	0
$409$	−12.9564	−0.640654	−0.320327	−	0.947307i	$-0.603793\pi$
$409$	−12.9564	−0.640654	−0.320327	+	0.947307i	$0.603793\pi$
$410$	0	0
$411$	−2.50080	−0.123355
$412$	0	0
$413$	29.6158	1.45730
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.04860	0.100320
$418$	0	0
$419$	23.7267	1.15912	0.579562	−	0.814928i	$-0.303224\pi$
$419$	23.7267	1.15912	0.579562	+	0.814928i	$0.303224\pi$
$420$	0	0
$421$	27.5590	1.34315	0.671573	−	0.740938i	$-0.265619\pi$
$421$	27.5590	1.34315	0.671573	+	0.740938i	$0.265619\pi$
$422$	0	0
$423$	9.60935	0.467223
$424$	0	0
$425$	0	0
$426$	0	0
$427$	37.5036	1.81493
$428$	0	0
$429$	3.77463	0.182241
$430$	0	0
$431$	9.79042	0.471588	0.235794	−	0.971803i	$-0.424231\pi$
$431$	9.79042	0.471588	0.235794	+	0.971803i	$0.424231\pi$
$432$	0	0
$433$	27.2558	1.30983	0.654914	−	0.755703i	$-0.272705\pi$
$433$	27.2558	1.30983	0.654914	+	0.755703i	$0.272705\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.81023	0.182268
$438$	0	0
$439$	−5.71343	−0.272687	−0.136344	−	0.990662i	$-0.543535\pi$
$439$	−5.71343	−0.272687	−0.136344	+	0.990662i	$0.543535\pi$
$440$	0	0
$441$	−0.951404	−0.0453049
$442$	0	0
$443$	−12.2616	−0.582567	−0.291283	−	0.956637i	$-0.594082\pi$
$443$	−12.2616	−0.582567	−0.291283	+	0.956637i	$0.594082\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.828243	−0.0391746
$448$	0	0
$449$	15.5185	0.732363	0.366182	−	0.930543i	$-0.380665\pi$
$449$	15.5185	0.732363	0.366182	+	0.930543i	$0.380665\pi$
$450$	0	0
$451$	−40.0274	−1.88482
$452$	0	0
$453$	−7.35948	−0.345779
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.61614	0.262712	0.131356	−	0.991335i	$-0.458067\pi$
$457$	5.61614	0.262712	0.131356	+	0.991335i	$0.458067\pi$
$458$	0	0
$459$	0.793451	0.0370351
$460$	0	0
$461$	7.18905	0.334827	0.167414	−	0.985887i	$-0.446458\pi$
$461$	7.18905	0.334827	0.167414	+	0.985887i	$0.446458\pi$
$462$	0	0
$463$	−18.1477	−0.843397	−0.421698	−	0.906736i	$-0.638566\pi$
$463$	−18.1477	−0.843397	−0.421698	+	0.906736i	$0.638566\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.564589	0.0261261	0.0130630	−	0.999915i	$-0.495842\pi$
$467$	0.564589	0.0261261	0.0130630	+	0.999915i	$0.495842\pi$
$468$	0	0
$469$	−8.19438	−0.378382
$470$	0	0
$471$	13.2804	0.611931
$472$	0	0
$473$	−24.8549	−1.14283
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.168784	−0.00772807
$478$	0	0
$479$	−35.8793	−1.63937	−0.819684	−	0.572816i	$-0.805851\pi$
$479$	−35.8793	−1.63937	−0.819684	+	0.572816i	$0.805851\pi$
$480$	0	0
$481$	9.91376	0.452029
$482$	0	0
$483$	2.96737	0.135020
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.31888	−0.376965	−0.188482	−	0.982077i	$-0.560357\pi$
$487$	−8.31888	−0.376965	−0.188482	+	0.982077i	$0.560357\pi$
$488$	0	0
$489$	17.8065	0.805239
$490$	0	0
$491$	30.3685	1.37051	0.685256	−	0.728302i	$-0.259690\pi$
$491$	30.3685	1.37051	0.685256	+	0.728302i	$0.259690\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.4889	−1.77132
$498$	0	0
$499$	−13.2129	−0.591490	−0.295745	−	0.955267i	$-0.595568\pi$
$499$	−13.2129	−0.591490	−0.295745	+	0.955267i	$0.595568\pi$
$500$	0	0
$501$	−0.593380	−0.0265103
$502$	0	0
$503$	36.7258	1.63752	0.818762	−	0.574133i	$-0.194661\pi$
$503$	36.7258	1.63752	0.818762	+	0.574133i	$0.194661\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−16.1555	−0.716080	−0.358040	−	0.933706i	$-0.616555\pi$
$509$	−16.1555	−0.716080	−0.358040	+	0.933706i	$0.616555\pi$
$510$	0	0
$511$	−26.8232	−1.18659
$512$	0	0
$513$	3.15795	0.139427
$514$	0	0
$515$	0	0
$516$	0	0
$517$	36.2717	1.59523
$518$	0	0
$519$	−12.4304	−0.545634
$520$	0	0
$521$	−1.07252	−0.0469879	−0.0234939	−	0.999724i	$-0.507479\pi$
$521$	−1.07252	−0.0469879	−0.0234939	+	0.999724i	$0.507479\pi$
$522$	0	0
$523$	2.49214	0.108974	0.0544868	−	0.998514i	$-0.482648\pi$
$523$	2.49214	0.108974	0.0544868	+	0.998514i	$0.482648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.92471	−0.170963
$528$	0	0
$529$	−21.5442	−0.936706
$530$	0	0
$531$	12.0419	0.522576
$532$	0	0
$533$	−10.6043	−0.459325
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.76301	0.119233
$538$	0	0
$539$	−3.59120	−0.154684
$540$	0	0
$541$	−24.8158	−1.06691	−0.533457	−	0.845827i	$-0.679107\pi$
$541$	−24.8158	−1.06691	−0.533457	+	0.845827i	$0.679107\pi$
$542$	0	0
$543$	−24.2137	−1.03911
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.7614	−0.802181	−0.401090	−	0.916038i	$-0.631369\pi$
$547$	−18.7614	−0.802181	−0.401090	+	0.916038i	$0.631369\pi$
$548$	0	0
$549$	15.2491	0.650818
$550$	0	0
$551$	15.9201	0.678219
$552$	0	0
$553$	15.9004	0.676156
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.7948	−1.43193	−0.715965	−	0.698136i	$-0.754013\pi$
$557$	−33.7948	−1.43193	−0.715965	+	0.698136i	$0.754013\pi$
$558$	0	0
$559$	−6.58472	−0.278504
$560$	0	0
$561$	2.99498	0.126448
$562$	0	0
$563$	−8.02934	−0.338396	−0.169198	−	0.985582i	$-0.554118\pi$
$563$	−8.02934	−0.338396	−0.169198	+	0.985582i	$0.554118\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.45939	0.103285
$568$	0	0
$569$	−11.6754	−0.489458	−0.244729	−	0.969592i	$-0.578699\pi$
$569$	−11.6754	−0.489458	−0.244729	+	0.969592i	$0.578699\pi$
$570$	0	0
$571$	10.4606	0.437762	0.218881	−	0.975752i	$-0.429759\pi$
$571$	10.4606	0.437762	0.218881	+	0.975752i	$0.429759\pi$
$572$	0	0
$573$	−14.7563	−0.616455
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.921755	0.0383732	0.0191866	−	0.999816i	$-0.493892\pi$
$577$	0.921755	0.0383732	0.0191866	+	0.999816i	$0.493892\pi$
$578$	0	0
$579$	1.82891	0.0760069
$580$	0	0
$581$	7.63314	0.316676
$582$	0	0
$583$	−0.637095	−0.0263858
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1210	0.582835	0.291418	−	0.956596i	$-0.405873\pi$
$587$	14.1210	0.582835	0.291418	+	0.956596i	$0.405873\pi$
$588$	0	0
$589$	−15.6205	−0.643630
$590$	0	0
$591$	20.4585	0.841552
$592$	0	0
$593$	−15.7238	−0.645701	−0.322850	−	0.946450i	$-0.604641\pi$
$593$	−15.7238	−0.645701	−0.322850	+	0.946450i	$0.604641\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.52977	−0.0626092
$598$	0	0
$599$	−4.03923	−0.165038	−0.0825192	−	0.996589i	$-0.526297\pi$
$599$	−4.03923	−0.165038	−0.0825192	+	0.996589i	$0.526297\pi$
$600$	0	0
$601$	0.631979	0.0257790	0.0128895	−	0.999917i	$-0.495897\pi$
$601$	0.631979	0.0257790	0.0128895	+	0.999917i	$0.495897\pi$
$602$	0	0
$603$	−3.33188	−0.135685
$604$	0	0
$605$	0	0
$606$	0	0
$607$	29.5083	1.19771	0.598853	−	0.800859i	$-0.295623\pi$
$607$	29.5083	1.19771	0.598853	+	0.800859i	$0.295623\pi$
$608$	0	0
$609$	12.3985	0.502411
$610$	0	0
$611$	9.60935	0.388753
$612$	0	0
$613$	18.2443	0.736881	0.368440	−	0.929651i	$-0.379892\pi$
$613$	18.2443	0.736881	0.368440	+	0.929651i	$0.379892\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.3510	0.497234	0.248617	−	0.968602i	$-0.420024\pi$
$617$	12.3510	0.497234	0.248617	+	0.968602i	$0.420024\pi$
$618$	0	0
$619$	4.44652	0.178721	0.0893603	−	0.995999i	$-0.471518\pi$
$619$	4.44652	0.178721	0.0893603	+	0.995999i	$0.471518\pi$
$620$	0	0
$621$	1.20655	0.0484172
$622$	0	0
$623$	4.65147	0.186357
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.9201	0.476043
$628$	0	0
$629$	7.86608	0.313641
$630$	0	0
$631$	−47.8258	−1.90392	−0.951958	−	0.306228i	$-0.900933\pi$
$631$	−47.8258	−1.90392	−0.951958	+	0.306228i	$0.900933\pi$
$632$	0	0
$633$	−9.33538	−0.371048
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.951404	−0.0376960
$638$	0	0
$639$	−16.0564	−0.635181
$640$	0	0
$641$	−36.0752	−1.42488	−0.712441	−	0.701732i	$-0.752411\pi$
$641$	−36.0752	−1.42488	−0.712441	+	0.701732i	$0.752411\pi$
$642$	0	0
$643$	42.9381	1.69331	0.846656	−	0.532140i	$-0.178612\pi$
$643$	42.9381	1.69331	0.846656	+	0.532140i	$0.178612\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.9558	1.64945	0.824726	−	0.565533i	$-0.191329\pi$
$647$	41.9558	1.64945	0.824726	+	0.565533i	$0.191329\pi$
$648$	0	0
$649$	45.4538	1.78422
$650$	0	0
$651$	−12.1651	−0.476787
$652$	0	0
$653$	−16.8898	−0.660950	−0.330475	−	0.943815i	$-0.607209\pi$
$653$	−16.8898	−0.660950	−0.330475	+	0.943815i	$0.607209\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.9064	−0.425501
$658$	0	0
$659$	28.8701	1.12462	0.562310	−	0.826926i	$-0.309913\pi$
$659$	28.8701	1.12462	0.562310	+	0.826926i	$0.309913\pi$
$660$	0	0
$661$	−14.6466	−0.569686	−0.284843	−	0.958574i	$-0.591941\pi$
$661$	−14.6466	−0.569686	−0.284843	+	0.958574i	$0.591941\pi$
$662$	0	0
$663$	0.793451	0.0308151
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.08254	0.235517
$668$	0	0
$669$	7.96005	0.307753
$670$	0	0
$671$	57.5599	2.22207
$672$	0	0
$673$	−32.0289	−1.23462	−0.617312	−	0.786718i	$-0.711778\pi$
$673$	−32.0289	−1.23462	−0.617312	+	0.786718i	$0.711778\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.8452	1.18548	0.592739	−	0.805395i	$-0.298047\pi$
$677$	30.8452	1.18548	0.592739	+	0.805395i	$0.298047\pi$
$678$	0	0
$679$	23.6671	0.908259
$680$	0	0
$681$	12.4456	0.476916
$682$	0	0
$683$	−19.8454	−0.759362	−0.379681	−	0.925118i	$-0.623966\pi$
$683$	−19.8454	−0.759362	−0.379681	+	0.925118i	$0.623966\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.9955	−0.724724
$688$	0	0
$689$	−0.168784	−0.00643014
$690$	0	0
$691$	−24.3913	−0.927889	−0.463944	−	0.885864i	$-0.653566\pi$
$691$	−24.3913	−0.927889	−0.463944	+	0.885864i	$0.653566\pi$
$692$	0	0
$693$	9.28328	0.352643
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.41401	−0.318703
$698$	0	0
$699$	25.5674	0.967046
$700$	0	0
$701$	35.9397	1.35742	0.678712	−	0.734405i	$-0.262539\pi$
$701$	35.9397	1.35742	0.678712	+	0.734405i	$0.262539\pi$
$702$	0	0
$703$	31.3072	1.18077
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.46370	0.243092
$708$	0	0
$709$	3.68093	0.138240	0.0691200	−	0.997608i	$-0.477981\pi$
$709$	3.68093	0.138240	0.0691200	+	0.997608i	$0.477981\pi$
$710$	0	0
$711$	6.46520	0.242464
$712$	0	0
$713$	−5.96806	−0.223506
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.1912	0.679364
$718$	0	0
$719$	20.1973	0.753232	0.376616	−	0.926369i	$-0.377088\pi$
$719$	20.1973	0.753232	0.376616	+	0.926369i	$0.377088\pi$
$720$	0	0
$721$	−48.0432	−1.78922
$722$	0	0
$723$	−11.8100	−0.439217
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.6513	0.543386	0.271693	−	0.962384i	$-0.412417\pi$
$727$	14.6513	0.543386	0.271693	+	0.962384i	$0.412417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.22465	−0.193241
$732$	0	0
$733$	−44.8092	−1.65507	−0.827533	−	0.561418i	$-0.810256\pi$
$733$	−44.8092	−1.65507	−0.827533	+	0.561418i	$0.810256\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.5766	−0.463265
$738$	0	0
$739$	17.5551	0.645774	0.322887	−	0.946438i	$-0.395347\pi$
$739$	17.5551	0.645774	0.322887	+	0.946438i	$0.395347\pi$
$740$	0	0
$741$	3.15795	0.116010
$742$	0	0
$743$	−17.7398	−0.650811	−0.325405	−	0.945575i	$-0.605501\pi$
$743$	−17.7398	−0.650811	−0.325405	+	0.945575i	$0.605501\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.10367	0.113557
$748$	0	0
$749$	−34.4224	−1.25777
$750$	0	0
$751$	41.2161	1.50400	0.751998	−	0.659166i	$-0.229090\pi$
$751$	41.2161	1.50400	0.751998	+	0.659166i	$0.229090\pi$
$752$	0	0
$753$	1.20371	0.0438658
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.7855	−0.937191	−0.468596	−	0.883413i	$-0.655240\pi$
$757$	−25.7855	−0.937191	−0.468596	+	0.883413i	$0.655240\pi$
$758$	0	0
$759$	4.55428	0.165310
$760$	0	0
$761$	−29.5584	−1.07149	−0.535745	−	0.844380i	$-0.679969\pi$
$761$	−29.5584	−1.07149	−0.535745	+	0.844380i	$0.679969\pi$
$762$	0	0
$763$	−31.5476	−1.14210
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0419	0.434809
$768$	0	0
$769$	−26.5998	−0.959215	−0.479607	−	0.877483i	$-0.659221\pi$
$769$	−26.5998	−0.959215	−0.479607	+	0.877483i	$0.659221\pi$
$770$	0	0
$771$	−7.94190	−0.286020
$772$	0	0
$773$	−31.5960	−1.13643	−0.568215	−	0.822880i	$-0.692366\pi$
$773$	−31.5960	−1.13643	−0.568215	+	0.822880i	$0.692366\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.3818	0.874692
$778$	0	0
$779$	−33.4880	−1.19983
$780$	0	0
$781$	−60.6069	−2.16869
$782$	0	0
$783$	5.04127	0.180160
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.379546	0.0135294	0.00676468	−	0.999977i	$-0.497847\pi$
$787$	0.379546	0.0135294	0.00676468	+	0.999977i	$0.497847\pi$
$788$	0	0
$789$	4.45789	0.158705
$790$	0	0
$791$	25.3172	0.900177
$792$	0	0
$793$	15.2491	0.541513
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.1817	−1.06909	−0.534546	−	0.845140i	$-0.679517\pi$
$797$	−30.1817	−1.06909	−0.534546	+	0.845140i	$0.679517\pi$
$798$	0	0
$799$	7.62455	0.269737
$800$	0	0
$801$	1.89131	0.0668262
$802$	0	0
$803$	−41.1677	−1.45278
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.1687	−1.06199
$808$	0	0
$809$	31.2430	1.09845	0.549223	−	0.835676i	$-0.314924\pi$
$809$	31.2430	1.09845	0.549223	+	0.835676i	$0.314924\pi$
$810$	0	0
$811$	−31.2562	−1.09755	−0.548777	−	0.835969i	$-0.684906\pi$
$811$	−31.2562	−1.09755	−0.548777	+	0.835969i	$0.684906\pi$
$812$	0	0
$813$	9.98535	0.350202
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.7942	−0.727498
$818$	0	0
$819$	2.45939	0.0859380
$820$	0	0
$821$	−33.9936	−1.18638	−0.593192	−	0.805061i	$-0.702133\pi$
$821$	−33.9936	−1.18638	−0.593192	+	0.805061i	$0.702133\pi$
$822$	0	0
$823$	0.989950	0.0345075	0.0172537	−	0.999851i	$-0.494508\pi$
$823$	0.989950	0.0345075	0.0172537	+	0.999851i	$0.494508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.0641	−0.802016	−0.401008	−	0.916075i	$-0.631340\pi$
$827$	−23.0641	−0.802016	−0.401008	+	0.916075i	$0.631340\pi$
$828$	0	0
$829$	−16.0615	−0.557838	−0.278919	−	0.960315i	$-0.589976\pi$
$829$	−16.0615	−0.557838	−0.278919	+	0.960315i	$0.589976\pi$
$830$	0	0
$831$	−6.78745	−0.235454
$832$	0	0
$833$	−0.754892	−0.0261555
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.94639	−0.170972
$838$	0	0
$839$	35.6563	1.23099	0.615495	−	0.788141i	$-0.288956\pi$
$839$	35.6563	1.23099	0.615495	+	0.788141i	$0.288956\pi$
$840$	0	0
$841$	−3.58558	−0.123641
$842$	0	0
$843$	17.6689	0.608549
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.98766	0.274459
$848$	0	0
$849$	−18.1434	−0.622681
$850$	0	0
$851$	11.9614	0.410033
$852$	0	0
$853$	39.5905	1.35555	0.677776	−	0.735269i	$-0.262944\pi$
$853$	39.5905	1.35555	0.677776	+	0.735269i	$0.262944\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.6906	0.433501	0.216751	−	0.976227i	$-0.430454\pi$
$857$	12.6906	0.433501	0.216751	+	0.976227i	$0.430454\pi$
$858$	0	0
$859$	−41.0997	−1.40230	−0.701151	−	0.713013i	$-0.747330\pi$
$859$	−41.0997	−1.40230	−0.701151	+	0.713013i	$0.747330\pi$
$860$	0	0
$861$	−26.0802	−0.888810
$862$	0	0
$863$	−27.3992	−0.932681	−0.466340	−	0.884605i	$-0.654428\pi$
$863$	−27.3992	−0.932681	−0.466340	+	0.884605i	$0.654428\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.3704	−0.555969
$868$	0	0
$869$	24.4037	0.827840
$870$	0	0
$871$	−3.33188	−0.112896
$872$	0	0
$873$	9.62315	0.325694
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0013	−1.28321	−0.641607	−	0.767034i	$-0.721732\pi$
$877$	−38.0013	−1.28321	−0.641607	+	0.767034i	$0.721732\pi$
$878$	0	0
$879$	−0.277473	−0.00935894
$880$	0	0
$881$	22.9970	0.774788	0.387394	−	0.921914i	$-0.373375\pi$
$881$	22.9970	0.774788	0.387394	+	0.921914i	$0.373375\pi$
$882$	0	0
$883$	12.3013	0.413970	0.206985	−	0.978344i	$-0.433635\pi$
$883$	12.3013	0.413970	0.206985	+	0.978344i	$0.433635\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.3541	−0.784153	−0.392076	−	0.919933i	$-0.628243\pi$
$887$	−23.3541	−0.784153	−0.392076	+	0.919933i	$0.628243\pi$
$888$	0	0
$889$	−12.0493	−0.404119
$890$	0	0
$891$	3.77463	0.126455
$892$	0	0
$893$	30.3459	1.01549
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.20655	0.0402855
$898$	0	0
$899$	−24.9361	−0.831665
$900$	0	0
$901$	−0.133921	−0.00446157
$902$	0	0
$903$	−16.1944	−0.538915
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.81155	0.0933560	0.0466780	−	0.998910i	$-0.485137\pi$
$907$	2.81155	0.0933560	0.0466780	+	0.998910i	$0.485137\pi$
$908$	0	0
$909$	2.62817	0.0871710
$910$	0	0
$911$	7.32621	0.242728	0.121364	−	0.992608i	$-0.461273\pi$
$911$	7.32621	0.242728	0.121364	+	0.992608i	$0.461273\pi$
$912$	0	0
$913$	11.7152	0.387717
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.0895	0.564346
$918$	0	0
$919$	13.6579	0.450533	0.225266	−	0.974297i	$-0.427675\pi$
$919$	13.6579	0.450533	0.225266	+	0.974297i	$0.427675\pi$
$920$	0	0
$921$	12.6515	0.416880
$922$	0	0
$923$	−16.0564	−0.528503
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−19.5346	−0.641601
$928$	0	0
$929$	31.5593	1.03543	0.517714	−	0.855554i	$-0.326783\pi$
$929$	31.5593	1.03543	0.517714	+	0.855554i	$0.326783\pi$
$930$	0	0
$931$	−3.00449	−0.0984682
$932$	0	0
$933$	−10.3056	−0.337390
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.8408	1.23620	0.618102	−	0.786098i	$-0.287902\pi$
$937$	37.8408	1.23620	0.618102	+	0.786098i	$0.287902\pi$
$938$	0	0
$939$	22.7601	0.742748
$940$	0	0
$941$	0.107829	0.00351511	0.00175755	−	0.999998i	$-0.499441\pi$
$941$	0.107829	0.00351511	0.00175755	+	0.999998i	$0.499441\pi$
$942$	0	0
$943$	−12.7947	−0.416651
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.3483	1.99355	0.996775	−	0.0802416i	$-0.0255692\pi$
$947$	61.3483	1.99355	0.996775	+	0.0802416i	$0.0255692\pi$
$948$	0	0
$949$	−10.9064	−0.354038
$950$	0	0
$951$	−0.0324873	−0.00105347
$952$	0	0
$953$	−38.5741	−1.24954	−0.624769	−	0.780810i	$-0.714807\pi$
$953$	−38.5741	−1.24954	−0.624769	+	0.780810i	$0.714807\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.0289	0.615118
$958$	0	0
$959$	−6.15044	−0.198608
$960$	0	0
$961$	−6.53327	−0.210751
$962$	0	0
$963$	−13.9963	−0.451025
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−57.1847	−1.83894	−0.919468	−	0.393165i	$-0.871380\pi$
$967$	−57.1847	−1.83894	−0.919468	+	0.393165i	$0.871380\pi$
$968$	0	0
$969$	2.50568	0.0804940
$970$	0	0
$971$	17.1571	0.550597	0.275299	−	0.961359i	$-0.411223\pi$
$971$	17.1571	0.550597	0.275299	+	0.961359i	$0.411223\pi$
$972$	0	0
$973$	5.03830	0.161520
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.4062	0.684846	0.342423	−	0.939546i	$-0.388752\pi$
$977$	21.4062	0.684846	0.342423	+	0.939546i	$0.388752\pi$
$978$	0	0
$979$	7.13899	0.228163
$980$	0	0
$981$	−12.8274	−0.409547
$982$	0	0
$983$	29.8048	0.950625	0.475313	−	0.879817i	$-0.342335\pi$
$983$	29.8048	0.950625	0.475313	+	0.879817i	$0.342335\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	23.6331	0.752251
$988$	0	0
$989$	−7.94479	−0.252630
$990$	0	0
$991$	−34.6684	−1.10128	−0.550639	−	0.834743i	$-0.685616\pi$
$991$	−34.6684	−1.10128	−0.550639	+	0.834743i	$0.685616\pi$
$992$	0	0
$993$	31.3679	0.995431
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.8453	1.16690	0.583451	−	0.812148i	$-0.301702\pi$
$997$	36.8453	1.16690	0.583451	+	0.812148i	$0.301702\pi$
$998$	0	0
$999$	9.91376	0.313657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.ca.1.4		5
5.2	odd	4		1560.2.l.f.1249.1	✓	10
5.3	odd	4		1560.2.l.f.1249.6	yes	10
5.4	even	2		7800.2.a.bz.1.2		5
15.2	even	4		4680.2.l.h.2809.9		10
15.8	even	4		4680.2.l.h.2809.10		10
20.3	even	4		3120.2.l.q.1249.1		10
20.7	even	4		3120.2.l.q.1249.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.f.1249.1	✓	10	5.2	odd	4
1560.2.l.f.1249.6	yes	10	5.3	odd	4
3120.2.l.q.1249.1		10	20.3	even	4
3120.2.l.q.1249.6		10	20.7	even	4
4680.2.l.h.2809.9		10	15.2	even	4
4680.2.l.h.2809.10		10	15.8	even	4
7800.2.a.bz.1.2		5	5.4	even	2
7800.2.a.ca.1.4		5	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.45939 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.45939 q^{7} +1.00000 q^{9} +3.77463 q^{11} +1.00000 q^{13} +0.793451 q^{17} +3.15795 q^{19} +2.45939 q^{21} +1.20655 q^{23} +1.00000 q^{27} +5.04127 q^{29} -4.94639 q^{31} +3.77463 q^{33} +9.91376 q^{37} +1.00000 q^{39} -10.6043 q^{41} -6.58472 q^{43} +9.60935 q^{47} -0.951404 q^{49} +0.793451 q^{51} -0.168784 q^{53} +3.15795 q^{57} +12.0419 q^{59} +15.2491 q^{61} +2.45939 q^{63} -3.33188 q^{67} +1.20655 q^{69} -16.0564 q^{71} -10.9064 q^{73} +9.28328 q^{77} +6.46520 q^{79} +1.00000 q^{81} +3.10367 q^{83} +5.04127 q^{87} +1.89131 q^{89} +2.45939 q^{91} -4.94639 q^{93} +9.62315 q^{97} +3.77463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^7 + 5 * q^9	\( 5 q + 5 q^{3} + q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + 7 q^{17} + 6 q^{19} + q^{21} + 3 q^{23} + 5 q^{27} + 2 q^{29} + 5 q^{33} + 9 q^{37} + 5 q^{39} - q^{41} + 4 q^{43} + 14 q^{47} + 2 q^{49} + 7 q^{51} + 5 q^{53} + 6 q^{57} + 20 q^{59} - 19 q^{61} + q^{63} + 12 q^{67} + 3 q^{69} + 13 q^{71} + 16 q^{73} + 11 q^{77} + 7 q^{79} + 5 q^{81} - 2 q^{83} + 2 q^{87} + 9 q^{89} + q^{91} + 13 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + 7 * q^17 + 6 * q^19 + q^21 + 3 * q^23 + 5 * q^27 + 2 * q^29 + 5 * q^33 + 9 * q^37 + 5 * q^39 - q^41 + 4 * q^43 + 14 * q^47 + 2 * q^49 + 7 * q^51 + 5 * q^53 + 6 * q^57 + 20 * q^59 - 19 * q^61 + q^63 + 12 * q^67 + 3 * q^69 + 13 * q^71 + 16 * q^73 + 11 * q^77 + 7 * q^79 + 5 * q^81 - 2 * q^83 + 2 * q^87 + 9 * q^89 + q^91 + 13 * q^97 + 5 * q^99

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