LMFDB - Embedded newform 7440.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.48119 q^{7} +1.00000 q^{9} -2.00000 q^{11} -5.83146 q^{13} -1.00000 q^{15} +3.76845 q^{17} +2.48119 q^{21} -0.806063 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.06300 q^{29} -1.00000 q^{31} +2.00000 q^{33} -2.48119 q^{35} +12.1441 q^{37} +5.83146 q^{39} +9.92478 q^{41} +10.3127 q^{43} +1.00000 q^{45} +1.58181 q^{47} -0.843675 q^{49} -3.76845 q^{51} -7.11871 q^{53} -2.00000 q^{55} -9.41327 q^{59} +6.31265 q^{61} -2.48119 q^{63} -5.83146 q^{65} +14.4060 q^{67} +0.806063 q^{69} -12.2496 q^{71} +1.25694 q^{73} -1.00000 q^{75} +4.96239 q^{77} -12.8568 q^{79} +1.00000 q^{81} +0.0303172 q^{83} +3.76845 q^{85} -4.06300 q^{87} -13.7259 q^{89} +14.4690 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * q^99

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.48119	−0.937803	−0.468902	−	0.883250i	$-0.655350\pi$
$7$	−2.48119	−0.937803	−0.468902	+	0.883250i	$0.655350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−5.83146	−1.61735	−0.808677	−	0.588252i	$-0.799816\pi$
$13$	−5.83146	−1.61735	−0.808677	+	0.588252i	$0.799816\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.76845	0.913984	0.456992	−	0.889471i	$-0.348927\pi$
$17$	3.76845	0.913984	0.456992	+	0.889471i	$0.348927\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.48119	0.541441
$22$	0	0
$23$	−0.806063	−0.168076	−0.0840379	−	0.996463i	$-0.526782\pi$
$23$	−0.806063	−0.168076	−0.0840379	+	0.996463i	$0.526782\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.06300	0.754481	0.377240	−	0.926115i	$-0.376873\pi$
$29$	4.06300	0.754481	0.377240	+	0.926115i	$0.376873\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−2.48119	−0.419398
$36$	0	0
$37$	12.1441	1.99648	0.998239	−	0.0593136i	$-0.0188912\pi$
$37$	12.1441	1.99648	0.998239	+	0.0593136i	$0.0188912\pi$
$38$	0	0
$39$	5.83146	0.933780
$40$	0	0
$41$	9.92478	1.54999	0.774995	−	0.631967i	$-0.217752\pi$
$41$	9.92478	1.54999	0.774995	+	0.631967i	$0.217752\pi$
$42$	0	0
$43$	10.3127	1.57266	0.786332	−	0.617804i	$-0.211977\pi$
$43$	10.3127	1.57266	0.786332	+	0.617804i	$0.211977\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.58181	0.230731	0.115365	−	0.993323i	$-0.463196\pi$
$47$	1.58181	0.230731	0.115365	+	0.993323i	$0.463196\pi$
$48$	0	0
$49$	−0.843675	−0.120525
$50$	0	0
$51$	−3.76845	−0.527689
$52$	0	0
$53$	−7.11871	−0.977831	−0.488915	−	0.872331i	$-0.662607\pi$
$53$	−7.11871	−0.977831	−0.488915	+	0.872331i	$0.662607\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.41327	−1.22550	−0.612751	−	0.790276i	$-0.709937\pi$
$59$	−9.41327	−1.22550	−0.612751	+	0.790276i	$0.709937\pi$
$60$	0	0
$61$	6.31265	0.808252	0.404126	−	0.914703i	$-0.367576\pi$
$61$	6.31265	0.808252	0.404126	+	0.914703i	$0.367576\pi$
$62$	0	0
$63$	−2.48119	−0.312601
$64$	0	0
$65$	−5.83146	−0.723303
$66$	0	0
$67$	14.4060	1.75997	0.879985	−	0.475002i	$-0.157553\pi$
$67$	14.4060	1.75997	0.879985	+	0.475002i	$0.157553\pi$
$68$	0	0
$69$	0.806063	0.0970386
$70$	0	0
$71$	−12.2496	−1.45377	−0.726883	−	0.686762i	$-0.759032\pi$
$71$	−12.2496	−1.45377	−0.726883	+	0.686762i	$0.759032\pi$
$72$	0	0
$73$	1.25694	0.147114	0.0735569	−	0.997291i	$-0.476565\pi$
$73$	1.25694	0.147114	0.0735569	+	0.997291i	$0.476565\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.96239	0.565517
$78$	0	0
$79$	−12.8568	−1.44651	−0.723254	−	0.690582i	$-0.757355\pi$
$79$	−12.8568	−1.44651	−0.723254	+	0.690582i	$0.757355\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0303172	0.00332774	0.00166387	−	0.999999i	$-0.499470\pi$
$83$	0.0303172	0.00332774	0.00166387	+	0.999999i	$0.499470\pi$
$84$	0	0
$85$	3.76845	0.408746
$86$	0	0
$87$	−4.06300	−0.435600
$88$	0	0
$89$	−13.7259	−1.45494	−0.727472	−	0.686137i	$-0.759305\pi$
$89$	−13.7259	−1.45494	−0.727472	+	0.686137i	$0.759305\pi$
$90$	0	0
$91$	14.4690	1.51676
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−10.2374	−1.01866	−0.509331	−	0.860571i	$-0.670107\pi$
$101$	−10.2374	−1.01866	−0.509331	+	0.860571i	$0.670107\pi$
$102$	0	0
$103$	13.7562	1.35544	0.677721	−	0.735319i	$-0.262968\pi$
$103$	13.7562	1.35544	0.677721	+	0.735319i	$0.262968\pi$
$104$	0	0
$105$	2.48119	0.242140
$106$	0	0
$107$	4.15633	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$107$	4.15633	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$108$	0	0
$109$	−13.8945	−1.33085	−0.665424	−	0.746466i	$-0.731749\pi$
$109$	−13.8945	−1.33085	−0.665424	+	0.746466i	$0.731749\pi$
$110$	0	0
$111$	−12.1441	−1.15267
$112$	0	0
$113$	−8.57452	−0.806623	−0.403311	−	0.915063i	$-0.632141\pi$
$113$	−8.57452	−0.806623	−0.403311	+	0.915063i	$0.632141\pi$
$114$	0	0
$115$	−0.806063	−0.0751658
$116$	0	0
$117$	−5.83146	−0.539118
$118$	0	0
$119$	−9.35026	−0.857137
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−9.92478	−0.894887
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.66291	−0.679973	−0.339987	−	0.940430i	$-0.610423\pi$
$127$	−7.66291	−0.679973	−0.339987	+	0.940430i	$0.610423\pi$
$128$	0	0
$129$	−10.3127	−0.907978
$130$	0	0
$131$	6.37565	0.557044	0.278522	−	0.960430i	$-0.410156\pi$
$131$	6.37565	0.557044	0.278522	+	0.960430i	$0.410156\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.93207	0.250504	0.125252	−	0.992125i	$-0.460026\pi$
$137$	2.93207	0.250504	0.125252	+	0.992125i	$0.460026\pi$
$138$	0	0
$139$	1.73813	0.147427	0.0737133	−	0.997279i	$-0.476515\pi$
$139$	1.73813	0.147427	0.0737133	+	0.997279i	$0.476515\pi$
$140$	0	0
$141$	−1.58181	−0.133212
$142$	0	0
$143$	11.6629	0.975302
$144$	0	0
$145$	4.06300	0.337414
$146$	0	0
$147$	0.843675	0.0695851
$148$	0	0
$149$	−15.8496	−1.29845	−0.649223	−	0.760598i	$-0.724906\pi$
$149$	−15.8496	−1.29845	−0.649223	+	0.760598i	$0.724906\pi$
$150$	0	0
$151$	−15.8945	−1.29347	−0.646736	−	0.762714i	$-0.723867\pi$
$151$	−15.8945	−1.29347	−0.646736	+	0.762714i	$0.723867\pi$
$152$	0	0
$153$	3.76845	0.304661
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−0.463096	−0.0369591	−0.0184795	−	0.999829i	$-0.505883\pi$
$157$	−0.463096	−0.0369591	−0.0184795	+	0.999829i	$0.505883\pi$
$158$	0	0
$159$	7.11871	0.564551
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−14.5320	−1.13823	−0.569116	−	0.822257i	$-0.692715\pi$
$163$	−14.5320	−1.13823	−0.569116	+	0.822257i	$0.692715\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	3.87399	0.299779	0.149889	−	0.988703i	$-0.452108\pi$
$167$	3.87399	0.299779	0.149889	+	0.988703i	$0.452108\pi$
$168$	0	0
$169$	21.0059	1.61584
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.8265	0.975183	0.487592	−	0.873072i	$-0.337876\pi$
$173$	12.8265	0.975183	0.487592	+	0.873072i	$0.337876\pi$
$174$	0	0
$175$	−2.48119	−0.187561
$176$	0	0
$177$	9.41327	0.707544
$178$	0	0
$179$	2.23743	0.167233	0.0836166	−	0.996498i	$-0.473353\pi$
$179$	2.23743	0.167233	0.0836166	+	0.996498i	$0.473353\pi$
$180$	0	0
$181$	−20.2374	−1.50424	−0.752118	−	0.659028i	$-0.770968\pi$
$181$	−20.2374	−1.50424	−0.752118	+	0.659028i	$0.770968\pi$
$182$	0	0
$183$	−6.31265	−0.466645
$184$	0	0
$185$	12.1441	0.892852
$186$	0	0
$187$	−7.53690	−0.551153
$188$	0	0
$189$	2.48119	0.180480
$190$	0	0
$191$	9.78655	0.708130	0.354065	−	0.935221i	$-0.384799\pi$
$191$	9.78655	0.708130	0.354065	+	0.935221i	$0.384799\pi$
$192$	0	0
$193$	4.83638	0.348130	0.174065	−	0.984734i	$-0.444310\pi$
$193$	4.83638	0.348130	0.174065	+	0.984734i	$0.444310\pi$
$194$	0	0
$195$	5.83146	0.417599
$196$	0	0
$197$	−10.0059	−0.712889	−0.356445	−	0.934316i	$-0.616011\pi$
$197$	−10.0059	−0.712889	−0.356445	+	0.934316i	$0.616011\pi$
$198$	0	0
$199$	3.05808	0.216782	0.108391	−	0.994108i	$-0.465430\pi$
$199$	3.05808	0.216782	0.108391	+	0.994108i	$0.465430\pi$
$200$	0	0
$201$	−14.4060	−1.01612
$202$	0	0
$203$	−10.0811	−0.707555
$204$	0	0
$205$	9.92478	0.693177
$206$	0	0
$207$	−0.806063	−0.0560253
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.49929	0.585115	0.292558	−	0.956248i	$-0.405494\pi$
$211$	8.49929	0.585115	0.292558	+	0.956248i	$0.405494\pi$
$212$	0	0
$213$	12.2496	0.839332
$214$	0	0
$215$	10.3127	0.703317
$216$	0	0
$217$	2.48119	0.168434
$218$	0	0
$219$	−1.25694	−0.0849362
$220$	0	0
$221$	−21.9756	−1.47824
$222$	0	0
$223$	−21.7235	−1.45472	−0.727358	−	0.686258i	$-0.759252\pi$
$223$	−21.7235	−1.45472	−0.727358	+	0.686258i	$0.759252\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.0205	0.930571	0.465286	−	0.885161i	$-0.345952\pi$
$227$	14.0205	0.930571	0.465286	+	0.885161i	$0.345952\pi$
$228$	0	0
$229$	−7.02302	−0.464094	−0.232047	−	0.972705i	$-0.574542\pi$
$229$	−7.02302	−0.464094	−0.232047	+	0.972705i	$0.574542\pi$
$230$	0	0
$231$	−4.96239	−0.326501
$232$	0	0
$233$	−10.9321	−0.716184	−0.358092	−	0.933686i	$-0.616573\pi$
$233$	−10.9321	−0.716184	−0.358092	+	0.933686i	$0.616573\pi$
$234$	0	0
$235$	1.58181	0.103186
$236$	0	0
$237$	12.8568	0.835142
$238$	0	0
$239$	−7.42548	−0.480315	−0.240157	−	0.970734i	$-0.577199\pi$
$239$	−7.42548	−0.480315	−0.240157	+	0.970734i	$0.577199\pi$
$240$	0	0
$241$	−10.4387	−0.672413	−0.336207	−	0.941788i	$-0.609144\pi$
$241$	−10.4387	−0.672413	−0.336207	+	0.941788i	$0.609144\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.843675	−0.0539004
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.0303172	−0.00192127
$250$	0	0
$251$	−10.7612	−0.679238	−0.339619	−	0.940563i	$-0.610298\pi$
$251$	−10.7612	−0.679238	−0.339619	+	0.940563i	$0.610298\pi$
$252$	0	0
$253$	1.61213	0.101354
$254$	0	0
$255$	−3.76845	−0.235990
$256$	0	0
$257$	−21.9452	−1.36891	−0.684453	−	0.729057i	$-0.739959\pi$
$257$	−21.9452	−1.36891	−0.684453	+	0.729057i	$0.739959\pi$
$258$	0	0
$259$	−30.1319	−1.87230
$260$	0	0
$261$	4.06300	0.251494
$262$	0	0
$263$	−5.73813	−0.353829	−0.176914	−	0.984226i	$-0.556612\pi$
$263$	−5.73813	−0.353829	−0.176914	+	0.984226i	$0.556612\pi$
$264$	0	0
$265$	−7.11871	−0.437299
$266$	0	0
$267$	13.7259	0.840012
$268$	0	0
$269$	13.3625	0.814725	0.407362	−	0.913267i	$-0.366449\pi$
$269$	13.3625	0.814725	0.407362	+	0.913267i	$0.366449\pi$
$270$	0	0
$271$	−3.03761	−0.184522	−0.0922609	−	0.995735i	$-0.529409\pi$
$271$	−3.03761	−0.184522	−0.0922609	+	0.995735i	$0.529409\pi$
$272$	0	0
$273$	−14.4690	−0.875702
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−16.1441	−0.970005	−0.485003	−	0.874513i	$-0.661181\pi$
$277$	−16.1441	−0.970005	−0.485003	+	0.874513i	$0.661181\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	28.2130	1.68305	0.841523	−	0.540221i	$-0.181660\pi$
$281$	28.2130	1.68305	0.841523	+	0.540221i	$0.181660\pi$
$282$	0	0
$283$	13.0557	0.776081	0.388041	−	0.921642i	$-0.373152\pi$
$283$	13.0557	0.776081	0.388041	+	0.921642i	$0.373152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.6253	−1.45359
$288$	0	0
$289$	−2.79877	−0.164633
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	2.00588	0.117185	0.0585924	−	0.998282i	$-0.481339\pi$
$293$	2.00588	0.117185	0.0585924	+	0.998282i	$0.481339\pi$
$294$	0	0
$295$	−9.41327	−0.548062
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	4.70052	0.271838
$300$	0	0
$301$	−25.5877	−1.47485
$302$	0	0
$303$	10.2374	0.588125
$304$	0	0
$305$	6.31265	0.361461
$306$	0	0
$307$	−0.743059	−0.0424086	−0.0212043	−	0.999775i	$-0.506750\pi$
$307$	−0.743059	−0.0424086	−0.0212043	+	0.999775i	$0.506750\pi$
$308$	0	0
$309$	−13.7562	−0.782565
$310$	0	0
$311$	−19.1368	−1.08515	−0.542575	−	0.840008i	$-0.682550\pi$
$311$	−19.1368	−1.08515	−0.542575	+	0.840008i	$0.682550\pi$
$312$	0	0
$313$	−11.3176	−0.639707	−0.319854	−	0.947467i	$-0.603634\pi$
$313$	−11.3176	−0.639707	−0.319854	+	0.947467i	$0.603634\pi$
$314$	0	0
$315$	−2.48119	−0.139799
$316$	0	0
$317$	30.3185	1.70286	0.851429	−	0.524470i	$-0.175736\pi$
$317$	30.3185	1.70286	0.851429	+	0.524470i	$0.175736\pi$
$318$	0	0
$319$	−8.12601	−0.454969
$320$	0	0
$321$	−4.15633	−0.231983
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5.83146	−0.323471
$326$	0	0
$327$	13.8945	0.768365
$328$	0	0
$329$	−3.92478	−0.216380
$330$	0	0
$331$	6.68006	0.367169	0.183585	−	0.983004i	$-0.441230\pi$
$331$	6.68006	0.367169	0.183585	+	0.983004i	$0.441230\pi$
$332$	0	0
$333$	12.1441	0.665493
$334$	0	0
$335$	14.4060	0.787082
$336$	0	0
$337$	14.2193	0.774576	0.387288	−	0.921959i	$-0.373412\pi$
$337$	14.2193	0.774576	0.387288	+	0.921959i	$0.373412\pi$
$338$	0	0
$339$	8.57452	0.465704
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	19.4617	1.05083
$344$	0	0
$345$	0.806063	0.0433970
$346$	0	0
$347$	−17.5271	−0.940902	−0.470451	−	0.882426i	$-0.655909\pi$
$347$	−17.5271	−0.940902	−0.470451	+	0.882426i	$0.655909\pi$
$348$	0	0
$349$	−14.7308	−0.788524	−0.394262	−	0.918998i	$-0.629000\pi$
$349$	−14.7308	−0.788524	−0.394262	+	0.918998i	$0.629000\pi$
$350$	0	0
$351$	5.83146	0.311260
$352$	0	0
$353$	−30.9321	−1.64635	−0.823174	−	0.567789i	$-0.807799\pi$
$353$	−30.9321	−1.64635	−0.823174	+	0.567789i	$0.807799\pi$
$354$	0	0
$355$	−12.2496	−0.650144
$356$	0	0
$357$	9.35026	0.494868
$358$	0	0
$359$	3.02539	0.159674	0.0798371	−	0.996808i	$-0.474560\pi$
$359$	3.02539	0.159674	0.0798371	+	0.996808i	$0.474560\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	1.25694	0.0657913
$366$	0	0
$367$	31.3258	1.63519	0.817597	−	0.575790i	$-0.195306\pi$
$367$	31.3258	1.63519	0.817597	+	0.575790i	$0.195306\pi$
$368$	0	0
$369$	9.92478	0.516663
$370$	0	0
$371$	17.6629	0.917013
$372$	0	0
$373$	−33.4518	−1.73207	−0.866035	−	0.499983i	$-0.833340\pi$
$373$	−33.4518	−1.73207	−0.866035	+	0.499983i	$0.833340\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−23.6932	−1.22026
$378$	0	0
$379$	1.40105	0.0719669	0.0359835	−	0.999352i	$-0.488544\pi$
$379$	1.40105	0.0719669	0.0359835	+	0.999352i	$0.488544\pi$
$380$	0	0
$381$	7.66291	0.392583
$382$	0	0
$383$	32.1319	1.64186	0.820931	−	0.571027i	$-0.193455\pi$
$383$	32.1319	1.64186	0.820931	+	0.571027i	$0.193455\pi$
$384$	0	0
$385$	4.96239	0.252907
$386$	0	0
$387$	10.3127	0.524221
$388$	0	0
$389$	25.7621	1.30619	0.653095	−	0.757276i	$-0.273470\pi$
$389$	25.7621	1.30619	0.653095	+	0.757276i	$0.273470\pi$
$390$	0	0
$391$	−3.03761	−0.153619
$392$	0	0
$393$	−6.37565	−0.321609
$394$	0	0
$395$	−12.8568	−0.646898
$396$	0	0
$397$	4.38787	0.220221	0.110111	−	0.993919i	$-0.464880\pi$
$397$	4.38787	0.220221	0.110111	+	0.993919i	$0.464880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.3479	−0.866312	−0.433156	−	0.901319i	$-0.642600\pi$
$401$	−17.3479	−0.866312	−0.433156	+	0.901319i	$0.642600\pi$
$402$	0	0
$403$	5.83146	0.290486
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−24.2882	−1.20392
$408$	0	0
$409$	14.3127	0.707715	0.353858	−	0.935299i	$-0.384870\pi$
$409$	14.3127	0.707715	0.353858	+	0.935299i	$0.384870\pi$
$410$	0	0
$411$	−2.93207	−0.144628
$412$	0	0
$413$	23.3561	1.14928
$414$	0	0
$415$	0.0303172	0.00148821
$416$	0	0
$417$	−1.73813	−0.0851168
$418$	0	0
$419$	8.31028	0.405984	0.202992	−	0.979180i	$-0.434933\pi$
$419$	8.31028	0.405984	0.202992	+	0.979180i	$0.434933\pi$
$420$	0	0
$421$	0.00587961	0.000286555 0	0.000143277	−	1.00000i	$-0.499954\pi$
$421$	0.00587961	0.000286555 0	0.000143277		1.00000i	$0.499954\pi$
$422$	0	0
$423$	1.58181	0.0769102
$424$	0	0
$425$	3.76845	0.182797
$426$	0	0
$427$	−15.6629	−0.757981
$428$	0	0
$429$	−11.6629	−0.563091
$430$	0	0
$431$	1.86177	0.0896785	0.0448392	−	0.998994i	$-0.485722\pi$
$431$	1.86177	0.0896785	0.0448392	+	0.998994i	$0.485722\pi$
$432$	0	0
$433$	23.6566	1.13686	0.568431	−	0.822731i	$-0.307551\pi$
$433$	23.6566	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$434$	0	0
$435$	−4.06300	−0.194806
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.67276	−0.0798365	−0.0399183	−	0.999203i	$-0.512710\pi$
$439$	−1.67276	−0.0798365	−0.0399183	+	0.999203i	$0.512710\pi$
$440$	0	0
$441$	−0.843675	−0.0401750
$442$	0	0
$443$	10.2315	0.486116	0.243058	−	0.970012i	$-0.421850\pi$
$443$	10.2315	0.486116	0.243058	+	0.970012i	$0.421850\pi$
$444$	0	0
$445$	−13.7259	−0.650671
$446$	0	0
$447$	15.8496	0.749658
$448$	0	0
$449$	11.9224	0.562653	0.281327	−	0.959612i	$-0.409226\pi$
$449$	11.9224	0.562653	0.281327	+	0.959612i	$0.409226\pi$
$450$	0	0
$451$	−19.8496	−0.934679
$452$	0	0
$453$	15.8945	0.746787
$454$	0	0
$455$	14.4690	0.678316
$456$	0	0
$457$	9.16713	0.428820	0.214410	−	0.976744i	$-0.431217\pi$
$457$	9.16713	0.428820	0.214410	+	0.976744i	$0.431217\pi$
$458$	0	0
$459$	−3.76845	−0.175896
$460$	0	0
$461$	27.2628	1.26976	0.634878	−	0.772612i	$-0.281050\pi$
$461$	27.2628	1.26976	0.634878	+	0.772612i	$0.281050\pi$
$462$	0	0
$463$	−6.77575	−0.314896	−0.157448	−	0.987527i	$-0.550327\pi$
$463$	−6.77575	−0.314896	−0.157448	+	0.987527i	$0.550327\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−0.655618	−0.0303384	−0.0151692	−	0.999885i	$-0.504829\pi$
$467$	−0.655618	−0.0303384	−0.0151692	+	0.999885i	$0.504829\pi$
$468$	0	0
$469$	−35.7440	−1.65051
$470$	0	0
$471$	0.463096	0.0213383
$472$	0	0
$473$	−20.6253	−0.948352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.11871	−0.325944
$478$	0	0
$479$	−8.31028	−0.379706	−0.189853	−	0.981812i	$-0.560801\pi$
$479$	−8.31028	−0.379706	−0.189853	+	0.981812i	$0.560801\pi$
$480$	0	0
$481$	−70.8178	−3.22901
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−9.17347	−0.415690	−0.207845	−	0.978162i	$-0.566645\pi$
$487$	−9.17347	−0.415690	−0.207845	+	0.978162i	$0.566645\pi$
$488$	0	0
$489$	14.5320	0.657159
$490$	0	0
$491$	−30.4241	−1.37302	−0.686510	−	0.727121i	$-0.740858\pi$
$491$	−30.4241	−1.37302	−0.686510	+	0.727121i	$0.740858\pi$
$492$	0	0
$493$	15.3112	0.689583
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	30.3938	1.36335
$498$	0	0
$499$	25.1490	1.12583	0.562913	−	0.826516i	$-0.309681\pi$
$499$	25.1490	1.12583	0.562913	+	0.826516i	$0.309681\pi$
$500$	0	0
$501$	−3.87399	−0.173077
$502$	0	0
$503$	−39.2955	−1.75210	−0.876050	−	0.482220i	$-0.839831\pi$
$503$	−39.2955	−1.75210	−0.876050	+	0.482220i	$0.839831\pi$
$504$	0	0
$505$	−10.2374	−0.455560
$506$	0	0
$507$	−21.0059	−0.932904
$508$	0	0
$509$	26.9356	1.19390	0.596949	−	0.802279i	$-0.296379\pi$
$509$	26.9356	1.19390	0.596949	+	0.802279i	$0.296379\pi$
$510$	0	0
$511$	−3.11871	−0.137964
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.7562	0.606172
$516$	0	0
$517$	−3.16362	−0.139136
$518$	0	0
$519$	−12.8265	−0.563022
$520$	0	0
$521$	9.25060	0.405276	0.202638	−	0.979254i	$-0.435048\pi$
$521$	9.25060	0.405276	0.202638	+	0.979254i	$0.435048\pi$
$522$	0	0
$523$	5.79877	0.253562	0.126781	−	0.991931i	$-0.459535\pi$
$523$	5.79877	0.253562	0.126781	+	0.991931i	$0.459535\pi$
$524$	0	0
$525$	2.48119	0.108288
$526$	0	0
$527$	−3.76845	−0.164156
$528$	0	0
$529$	−22.3503	−0.971751
$530$	0	0
$531$	−9.41327	−0.408501
$532$	0	0
$533$	−57.8759	−2.50688
$534$	0	0
$535$	4.15633	0.179694
$536$	0	0
$537$	−2.23743	−0.0965521
$538$	0	0
$539$	1.68735	0.0726793
$540$	0	0
$541$	−12.3674	−0.531716	−0.265858	−	0.964012i	$-0.585655\pi$
$541$	−12.3674	−0.531716	−0.265858	+	0.964012i	$0.585655\pi$
$542$	0	0
$543$	20.2374	0.868471
$544$	0	0
$545$	−13.8945	−0.595173
$546$	0	0
$547$	−35.7826	−1.52995	−0.764976	−	0.644058i	$-0.777249\pi$
$547$	−35.7826	−1.52995	−0.764976	+	0.644058i	$0.777249\pi$
$548$	0	0
$549$	6.31265	0.269417
$550$	0	0
$551$	0	0
$552$	0	0
$553$	31.9003	1.35654
$554$	0	0
$555$	−12.1441	−0.515489
$556$	0	0
$557$	−12.2433	−0.518766	−0.259383	−	0.965775i	$-0.583519\pi$
$557$	−12.2433	−0.518766	−0.259383	+	0.965775i	$0.583519\pi$
$558$	0	0
$559$	−60.1378	−2.54356
$560$	0	0
$561$	7.53690	0.318208
$562$	0	0
$563$	−42.0322	−1.77145	−0.885724	−	0.464213i	$-0.846337\pi$
$563$	−42.0322	−1.77145	−0.885724	+	0.464213i	$0.846337\pi$
$564$	0	0
$565$	−8.57452	−0.360733
$566$	0	0
$567$	−2.48119	−0.104200
$568$	0	0
$569$	−32.8505	−1.37716	−0.688582	−	0.725158i	$-0.741767\pi$
$569$	−32.8505	−1.37716	−0.688582	+	0.725158i	$0.741767\pi$
$570$	0	0
$571$	−36.8119	−1.54053	−0.770266	−	0.637723i	$-0.779877\pi$
$571$	−36.8119	−1.54053	−0.770266	+	0.637723i	$0.779877\pi$
$572$	0	0
$573$	−9.78655	−0.408839
$574$	0	0
$575$	−0.806063	−0.0336152
$576$	0	0
$577$	−16.8119	−0.699890	−0.349945	−	0.936770i	$-0.613800\pi$
$577$	−16.8119	−0.699890	−0.349945	+	0.936770i	$0.613800\pi$
$578$	0	0
$579$	−4.83638	−0.200993
$580$	0	0
$581$	−0.0752228	−0.00312077
$582$	0	0
$583$	14.2374	0.589654
$584$	0	0
$585$	−5.83146	−0.241101
$586$	0	0
$587$	20.9624	0.865210	0.432605	−	0.901583i	$-0.357594\pi$
$587$	20.9624	0.865210	0.432605	+	0.901583i	$0.357594\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	10.0059	0.411587
$592$	0	0
$593$	22.6907	0.931794	0.465897	−	0.884839i	$-0.345732\pi$
$593$	22.6907	0.931794	0.465897	+	0.884839i	$0.345732\pi$
$594$	0	0
$595$	−9.35026	−0.383323
$596$	0	0
$597$	−3.05808	−0.125159
$598$	0	0
$599$	24.7245	1.01022	0.505108	−	0.863056i	$-0.331453\pi$
$599$	24.7245	1.01022	0.505108	+	0.863056i	$0.331453\pi$
$600$	0	0
$601$	−0.412311	−0.0168185	−0.00840925	−	0.999965i	$-0.502677\pi$
$601$	−0.412311	−0.0168185	−0.00840925	+	0.999965i	$0.502677\pi$
$602$	0	0
$603$	14.4060	0.586657
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−34.7186	−1.40919	−0.704593	−	0.709612i	$-0.748870\pi$
$607$	−34.7186	−1.40919	−0.704593	+	0.709612i	$0.748870\pi$
$608$	0	0
$609$	10.0811	0.408507
$610$	0	0
$611$	−9.22425	−0.373173
$612$	0	0
$613$	−28.5682	−1.15386	−0.576929	−	0.816794i	$-0.695749\pi$
$613$	−28.5682	−1.15386	−0.576929	+	0.816794i	$0.695749\pi$
$614$	0	0
$615$	−9.92478	−0.400206
$616$	0	0
$617$	−28.5501	−1.14938	−0.574691	−	0.818370i	$-0.694878\pi$
$617$	−28.5501	−1.14938	−0.574691	+	0.818370i	$0.694878\pi$
$618$	0	0
$619$	−4.23155	−0.170080	−0.0850401	−	0.996378i	$-0.527102\pi$
$619$	−4.23155	−0.170080	−0.0850401	+	0.996378i	$0.527102\pi$
$620$	0	0
$621$	0.806063	0.0323462
$622$	0	0
$623$	34.0567	1.36445
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	45.7645	1.82475
$630$	0	0
$631$	2.17679	0.0866568	0.0433284	−	0.999061i	$-0.486204\pi$
$631$	2.17679	0.0866568	0.0433284	+	0.999061i	$0.486204\pi$
$632$	0	0
$633$	−8.49929	−0.337817
$634$	0	0
$635$	−7.66291	−0.304093
$636$	0	0
$637$	4.91985	0.194932
$638$	0	0
$639$	−12.2496	−0.484589
$640$	0	0
$641$	−44.3268	−1.75080	−0.875401	−	0.483397i	$-0.839403\pi$
$641$	−44.3268	−1.75080	−0.875401	+	0.483397i	$0.839403\pi$
$642$	0	0
$643$	34.8265	1.37342	0.686712	−	0.726929i	$-0.259053\pi$
$643$	34.8265	1.37342	0.686712	+	0.726929i	$0.259053\pi$
$644$	0	0
$645$	−10.3127	−0.406060
$646$	0	0
$647$	28.4142	1.11708	0.558539	−	0.829478i	$-0.311362\pi$
$647$	28.4142	1.11708	0.558539	+	0.829478i	$0.311362\pi$
$648$	0	0
$649$	18.8265	0.739006
$650$	0	0
$651$	−2.48119	−0.0972457
$652$	0	0
$653$	−28.4544	−1.11351	−0.556753	−	0.830678i	$-0.687953\pi$
$653$	−28.4544	−1.11351	−0.556753	+	0.830678i	$0.687953\pi$
$654$	0	0
$655$	6.37565	0.249117
$656$	0	0
$657$	1.25694	0.0490379
$658$	0	0
$659$	−19.3479	−0.753687	−0.376843	−	0.926277i	$-0.622991\pi$
$659$	−19.3479	−0.753687	−0.376843	+	0.926277i	$0.622991\pi$
$660$	0	0
$661$	−20.1721	−0.784602	−0.392301	−	0.919837i	$-0.628321\pi$
$661$	−20.1721	−0.784602	−0.392301	+	0.919837i	$0.628321\pi$
$662$	0	0
$663$	21.9756	0.853460
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.27504	−0.126810
$668$	0	0
$669$	21.7235	0.839881
$670$	0	0
$671$	−12.6253	−0.487394
$672$	0	0
$673$	46.7040	1.80031	0.900154	−	0.435572i	$-0.143454\pi$
$673$	46.7040	1.80031	0.900154	+	0.435572i	$0.143454\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	44.4299	1.70758	0.853791	−	0.520616i	$-0.174298\pi$
$677$	44.4299	1.70758	0.853791	+	0.520616i	$0.174298\pi$
$678$	0	0
$679$	29.7743	1.14263
$680$	0	0
$681$	−14.0205	−0.537266
$682$	0	0
$683$	−3.21837	−0.123148	−0.0615738	−	0.998103i	$-0.519612\pi$
$683$	−3.21837	−0.123148	−0.0615738	+	0.998103i	$0.519612\pi$
$684$	0	0
$685$	2.93207	0.112029
$686$	0	0
$687$	7.02302	0.267945
$688$	0	0
$689$	41.5125	1.58150
$690$	0	0
$691$	32.9741	1.25440	0.627198	−	0.778860i	$-0.284202\pi$
$691$	32.9741	1.25440	0.627198	+	0.778860i	$0.284202\pi$
$692$	0	0
$693$	4.96239	0.188506
$694$	0	0
$695$	1.73813	0.0659312
$696$	0	0
$697$	37.4010	1.41667
$698$	0	0
$699$	10.9321	0.413489
$700$	0	0
$701$	33.3357	1.25907	0.629536	−	0.776972i	$-0.283245\pi$
$701$	33.3357	1.25907	0.629536	+	0.776972i	$0.283245\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.58181	−0.0595744
$706$	0	0
$707$	25.4010	0.955305
$708$	0	0
$709$	−34.8773	−1.30985	−0.654923	−	0.755696i	$-0.727299\pi$
$709$	−34.8773	−1.30985	−0.654923	+	0.755696i	$0.727299\pi$
$710$	0	0
$711$	−12.8568	−0.482169
$712$	0	0
$713$	0.806063	0.0301873
$714$	0	0
$715$	11.6629	0.436168
$716$	0	0
$717$	7.42548	0.277310
$718$	0	0
$719$	−1.37470	−0.0512676	−0.0256338	−	0.999671i	$-0.508160\pi$
$719$	−1.37470	−0.0512676	−0.0256338	+	0.999671i	$0.508160\pi$
$720$	0	0
$721$	−34.1319	−1.27114
$722$	0	0
$723$	10.4387	0.388218
$724$	0	0
$725$	4.06300	0.150896
$726$	0	0
$727$	−46.3209	−1.71795	−0.858974	−	0.512020i	$-0.828897\pi$
$727$	−46.3209	−1.71795	−0.858974	+	0.512020i	$0.828897\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.8627	1.43739
$732$	0	0
$733$	−29.0738	−1.07387	−0.536933	−	0.843625i	$-0.680417\pi$
$733$	−29.0738	−1.07387	−0.536933	+	0.843625i	$0.680417\pi$
$734$	0	0
$735$	0.843675	0.0311194
$736$	0	0
$737$	−28.8119	−1.06130
$738$	0	0
$739$	−18.7454	−0.689562	−0.344781	−	0.938683i	$-0.612047\pi$
$739$	−18.7454	−0.689562	−0.344781	+	0.938683i	$0.612047\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−51.8858	−1.90350	−0.951752	−	0.306869i	$-0.900719\pi$
$743$	−51.8858	−1.90350	−0.951752	+	0.306869i	$0.900719\pi$
$744$	0	0
$745$	−15.8496	−0.580683
$746$	0	0
$747$	0.0303172	0.00110925
$748$	0	0
$749$	−10.3127	−0.376816
$750$	0	0
$751$	−47.9149	−1.74844	−0.874220	−	0.485530i	$-0.838627\pi$
$751$	−47.9149	−1.74844	−0.874220	+	0.485530i	$0.838627\pi$
$752$	0	0
$753$	10.7612	0.392158
$754$	0	0
$755$	−15.8945	−0.578459
$756$	0	0
$757$	27.6302	1.00424	0.502119	−	0.864799i	$-0.332554\pi$
$757$	27.6302	1.00424	0.502119	+	0.864799i	$0.332554\pi$
$758$	0	0
$759$	−1.61213	−0.0585165
$760$	0	0
$761$	11.0108	0.399141	0.199571	−	0.979883i	$-0.436045\pi$
$761$	11.0108	0.399141	0.199571	+	0.979883i	$0.436045\pi$
$762$	0	0
$763$	34.4749	1.24807
$764$	0	0
$765$	3.76845	0.136249
$766$	0	0
$767$	54.8930	1.98207
$768$	0	0
$769$	−1.52118	−0.0548550	−0.0274275	−	0.999624i	$-0.508732\pi$
$769$	−1.52118	−0.0548550	−0.0274275	+	0.999624i	$0.508732\pi$
$770$	0	0
$771$	21.9452	0.790339
$772$	0	0
$773$	−47.5085	−1.70876	−0.854381	−	0.519647i	$-0.826063\pi$
$773$	−47.5085	−1.70876	−0.854381	+	0.519647i	$0.826063\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	30.1319	1.08098
$778$	0	0
$779$	0	0
$780$	0	0
$781$	24.4993	0.876654
$782$	0	0
$783$	−4.06300	−0.145200
$784$	0	0
$785$	−0.463096	−0.0165286
$786$	0	0
$787$	27.2144	0.970089	0.485044	−	0.874490i	$-0.338803\pi$
$787$	27.2144	0.970089	0.485044	+	0.874490i	$0.338803\pi$
$788$	0	0
$789$	5.73813	0.204283
$790$	0	0
$791$	21.2750	0.756453
$792$	0	0
$793$	−36.8119	−1.30723
$794$	0	0
$795$	7.11871	0.252475
$796$	0	0
$797$	−1.94525	−0.0689041	−0.0344521	−	0.999406i	$-0.510969\pi$
$797$	−1.94525	−0.0689041	−0.0344521	+	0.999406i	$0.510969\pi$
$798$	0	0
$799$	5.96097	0.210884
$800$	0	0
$801$	−13.7259	−0.484981
$802$	0	0
$803$	−2.51388	−0.0887129
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−13.3625	−0.470382
$808$	0	0
$809$	9.81573	0.345103	0.172551	−	0.985001i	$-0.444799\pi$
$809$	9.81573	0.345103	0.172551	+	0.985001i	$0.444799\pi$
$810$	0	0
$811$	−3.71179	−0.130338	−0.0651692	−	0.997874i	$-0.520759\pi$
$811$	−3.71179	−0.130338	−0.0651692	+	0.997874i	$0.520759\pi$
$812$	0	0
$813$	3.03761	0.106534
$814$	0	0
$815$	−14.5320	−0.509033
$816$	0	0
$817$	0	0
$818$	0	0
$819$	14.4690	0.505587
$820$	0	0
$821$	30.9403	1.07982	0.539912	−	0.841721i	$-0.318457\pi$
$821$	30.9403	1.07982	0.539912	+	0.841721i	$0.318457\pi$
$822$	0	0
$823$	−29.4471	−1.02646	−0.513231	−	0.858251i	$-0.671551\pi$
$823$	−29.4471	−1.02646	−0.513231	+	0.858251i	$0.671551\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−9.11871	−0.317089	−0.158544	−	0.987352i	$-0.550680\pi$
$827$	−9.11871	−0.317089	−0.158544	+	0.987352i	$0.550680\pi$
$828$	0	0
$829$	1.19982	0.0416713	0.0208357	−	0.999783i	$-0.493367\pi$
$829$	1.19982	0.0416713	0.0208357	+	0.999783i	$0.493367\pi$
$830$	0	0
$831$	16.1441	0.560033
$832$	0	0
$833$	−3.17935	−0.110158
$834$	0	0
$835$	3.87399	0.134065
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−20.7753	−0.717243	−0.358621	−	0.933483i	$-0.616753\pi$
$839$	−20.7753	−0.717243	−0.358621	+	0.933483i	$0.616753\pi$
$840$	0	0
$841$	−12.4920	−0.430759
$842$	0	0
$843$	−28.2130	−0.971707
$844$	0	0
$845$	21.0059	0.722624
$846$	0	0
$847$	17.3684	0.596784
$848$	0	0
$849$	−13.0557	−0.448071
$850$	0	0
$851$	−9.78892	−0.335560
$852$	0	0
$853$	41.0757	1.40641	0.703203	−	0.710989i	$-0.251753\pi$
$853$	41.0757	1.40641	0.703203	+	0.710989i	$0.251753\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.3766	−0.935167	−0.467584	−	0.883949i	$-0.654875\pi$
$857$	−27.3766	−0.935167	−0.467584	+	0.883949i	$0.654875\pi$
$858$	0	0
$859$	32.1016	1.09529	0.547646	−	0.836710i	$-0.315524\pi$
$859$	32.1016	1.09529	0.547646	+	0.836710i	$0.315524\pi$
$860$	0	0
$861$	24.6253	0.839228
$862$	0	0
$863$	−55.2966	−1.88232	−0.941160	−	0.337962i	$-0.890263\pi$
$863$	−55.2966	−1.88232	−0.941160	+	0.337962i	$0.890263\pi$
$864$	0	0
$865$	12.8265	0.436115
$866$	0	0
$867$	2.79877	0.0950512
$868$	0	0
$869$	25.7137	0.872277
$870$	0	0
$871$	−84.0078	−2.84650
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	−2.48119	−0.0838797
$876$	0	0
$877$	51.1392	1.72685	0.863424	−	0.504479i	$-0.168316\pi$
$877$	51.1392	1.72685	0.863424	+	0.504479i	$0.168316\pi$
$878$	0	0
$879$	−2.00588	−0.0676566
$880$	0	0
$881$	39.1632	1.31944	0.659720	−	0.751511i	$-0.270675\pi$
$881$	39.1632	1.31944	0.659720	+	0.751511i	$0.270675\pi$
$882$	0	0
$883$	−40.6155	−1.36682	−0.683409	−	0.730035i	$-0.739504\pi$
$883$	−40.6155	−1.36682	−0.683409	+	0.730035i	$0.739504\pi$
$884$	0	0
$885$	9.41327	0.316423
$886$	0	0
$887$	6.16617	0.207040	0.103520	−	0.994627i	$-0.466989\pi$
$887$	6.16617	0.207040	0.103520	+	0.994627i	$0.466989\pi$
$888$	0	0
$889$	19.0132	0.637681
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	0	0
$894$	0	0
$895$	2.23743	0.0747890
$896$	0	0
$897$	−4.70052	−0.156946
$898$	0	0
$899$	−4.06300	−0.135509
$900$	0	0
$901$	−26.8265	−0.893721
$902$	0	0
$903$	25.5877	0.851505
$904$	0	0
$905$	−20.2374	−0.672715
$906$	0	0
$907$	21.4944	0.713709	0.356854	−	0.934160i	$-0.383849\pi$
$907$	21.4944	0.713709	0.356854	+	0.934160i	$0.383849\pi$
$908$	0	0
$909$	−10.2374	−0.339554
$910$	0	0
$911$	−17.4471	−0.578048	−0.289024	−	0.957322i	$-0.593331\pi$
$911$	−17.4471	−0.578048	−0.289024	+	0.957322i	$0.593331\pi$
$912$	0	0
$913$	−0.0606343	−0.00200670
$914$	0	0
$915$	−6.31265	−0.208690
$916$	0	0
$917$	−15.8192	−0.522397
$918$	0	0
$919$	−32.5355	−1.07325	−0.536623	−	0.843822i	$-0.680300\pi$
$919$	−32.5355	−1.07325	−0.536623	+	0.843822i	$0.680300\pi$
$920$	0	0
$921$	0.743059	0.0244846
$922$	0	0
$923$	71.4333	2.35125
$924$	0	0
$925$	12.1441	0.399296
$926$	0	0
$927$	13.7562	0.451814
$928$	0	0
$929$	45.4641	1.49163	0.745814	−	0.666155i	$-0.232061\pi$
$929$	45.4641	1.49163	0.745814	+	0.666155i	$0.232061\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19.1368	0.626511
$934$	0	0
$935$	−7.53690	−0.246483
$936$	0	0
$937$	−56.2031	−1.83608	−0.918038	−	0.396492i	$-0.870228\pi$
$937$	−56.2031	−1.83608	−0.918038	+	0.396492i	$0.870228\pi$
$938$	0	0
$939$	11.3176	0.369335
$940$	0	0
$941$	13.1251	0.427864	0.213932	−	0.976849i	$-0.431373\pi$
$941$	13.1251	0.427864	0.213932	+	0.976849i	$0.431373\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	2.48119	0.0807133
$946$	0	0
$947$	55.1939	1.79356	0.896781	−	0.442475i	$-0.145899\pi$
$947$	55.1939	1.79356	0.896781	+	0.442475i	$0.145899\pi$
$948$	0	0
$949$	−7.32979	−0.237935
$950$	0	0
$951$	−30.3185	−0.983146
$952$	0	0
$953$	10.4690	0.339123	0.169562	−	0.985520i	$-0.445765\pi$
$953$	10.4690	0.339123	0.169562	+	0.985520i	$0.445765\pi$
$954$	0	0
$955$	9.78655	0.316685
$956$	0	0
$957$	8.12601	0.262677
$958$	0	0
$959$	−7.27504	−0.234923
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.15633	0.133936
$964$	0	0
$965$	4.83638	0.155689
$966$	0	0
$967$	7.37661	0.237216	0.118608	−	0.992941i	$-0.462157\pi$
$967$	7.37661	0.237216	0.118608	+	0.992941i	$0.462157\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.4020	−0.975647	−0.487823	−	0.872942i	$-0.662209\pi$
$971$	−30.4020	−0.975647	−0.487823	+	0.872942i	$0.662209\pi$
$972$	0	0
$973$	−4.31265	−0.138257
$974$	0	0
$975$	5.83146	0.186756
$976$	0	0
$977$	12.8510	0.411139	0.205569	−	0.978643i	$-0.434095\pi$
$977$	12.8510	0.411139	0.205569	+	0.978643i	$0.434095\pi$
$978$	0	0
$979$	27.4518	0.877364
$980$	0	0
$981$	−13.8945	−0.443616
$982$	0	0
$983$	−30.1162	−0.960556	−0.480278	−	0.877116i	$-0.659464\pi$
$983$	−30.1162	−0.960556	−0.480278	+	0.877116i	$0.659464\pi$
$984$	0	0
$985$	−10.0059	−0.318814
$986$	0	0
$987$	3.92478	0.124927
$988$	0	0
$989$	−8.31265	−0.264327
$990$	0	0
$991$	27.9873	0.889047	0.444523	−	0.895767i	$-0.353373\pi$
$991$	27.9873	0.889047	0.444523	+	0.895767i	$0.353373\pi$
$992$	0	0
$993$	−6.68006	−0.211985
$994$	0	0
$995$	3.05808	0.0969476
$996$	0	0
$997$	−22.0968	−0.699814	−0.349907	−	0.936785i	$-0.613787\pi$
$997$	−22.0968	−0.699814	−0.349907	+	0.936785i	$0.613787\pi$
$998$	0	0
$999$	−12.1441	−0.384223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bp.1.1		3
4.3	odd	2		465.2.a.f.1.1	✓	3
12.11	even	2		1395.2.a.i.1.3		3
20.3	even	4		2325.2.c.o.1024.5		6
20.7	even	4		2325.2.c.o.1024.2		6
20.19	odd	2		2325.2.a.q.1.3		3
60.59	even	2		6975.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.1	✓	3	4.3	odd	2
1395.2.a.i.1.3		3	12.11	even	2
2325.2.a.q.1.3		3	20.19	odd	2
2325.2.c.o.1024.2		6	20.7	even	4
2325.2.c.o.1024.5		6	20.3	even	4
6975.2.a.be.1.1		3	60.59	even	2
7440.2.a.bp.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.48119 q^{7} +1.00000 q^{9} -2.00000 q^{11} -5.83146 q^{13} -1.00000 q^{15} +3.76845 q^{17} +2.48119 q^{21} -0.806063 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.06300 q^{29} -1.00000 q^{31} +2.00000 q^{33} -2.48119 q^{35} +12.1441 q^{37} +5.83146 q^{39} +9.92478 q^{41} +10.3127 q^{43} +1.00000 q^{45} +1.58181 q^{47} -0.843675 q^{49} -3.76845 q^{51} -7.11871 q^{53} -2.00000 q^{55} -9.41327 q^{59} +6.31265 q^{61} -2.48119 q^{63} -5.83146 q^{65} +14.4060 q^{67} +0.806063 q^{69} -12.2496 q^{71} +1.25694 q^{73} -1.00000 q^{75} +4.96239 q^{77} -12.8568 q^{79} +1.00000 q^{81} +0.0303172 q^{83} +3.76845 q^{85} -4.06300 q^{87} -13.7259 q^{89} +14.4690 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * q^99

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