LMFDB - Embedded newform 7440.2.a.bs.1.3

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:	3.3.564.1
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:	$\Z[a_1, \ldots, a_{7}]$
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			1.3
Root			$2.51414$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.485863 q^{7} +1.00000 q^{9} -5.02827 q^{11} +3.51414 q^{13} -1.00000 q^{15} -1.32088 q^{17} +6.64177 q^{19} -0.485863 q^{21} -0.292611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.86330 q^{29} -1.00000 q^{31} -5.02827 q^{33} +0.485863 q^{35} +5.51414 q^{37} +3.51414 q^{39} -7.02827 q^{41} +1.02827 q^{43} -1.00000 q^{45} +6.93438 q^{47} -6.76394 q^{49} -1.32088 q^{51} -1.70739 q^{53} +5.02827 q^{55} +6.64177 q^{57} +2.19325 q^{59} -2.00000 q^{61} -0.485863 q^{63} -3.51414 q^{65} -9.12763 q^{67} -0.292611 q^{69} -13.4768 q^{71} +12.5424 q^{73} +1.00000 q^{75} +2.44305 q^{77} +0.349158 q^{79} +1.00000 q^{81} -10.9344 q^{83} +1.32088 q^{85} -9.86330 q^{87} +5.03374 q^{89} -1.70739 q^{91} -1.00000 q^{93} -6.64177 q^{95} +10.4431 q^{97} -5.02827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * 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
$7$	−0.485863	−0.183639	−0.0918195	−	0.995776i	$-0.529268\pi$
$7$	−0.485863	−0.183639	−0.0918195	+	0.995776i	$0.529268\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.02827	−1.51608	−0.758041	−	0.652207i	$-0.773843\pi$
$11$	−5.02827	−1.51608	−0.758041	+	0.652207i	$0.773843\pi$
$12$	0	0
$13$	3.51414	0.974646	0.487323	−	0.873222i	$-0.337973\pi$
$13$	3.51414	0.974646	0.487323	+	0.873222i	$0.337973\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.32088	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$17$	−1.32088	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$18$	0	0
$19$	6.64177	1.52373	0.761863	−	0.647738i	$-0.224285\pi$
$19$	6.64177	1.52373	0.761863	+	0.647738i	$0.224285\pi$
$20$	0	0
$21$	−0.485863	−0.106024
$22$	0	0
$23$	−0.292611	−0.0610135	−0.0305068	−	0.999535i	$-0.509712\pi$
$23$	−0.292611	−0.0610135	−0.0305068	+	0.999535i	$0.509712\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.86330	−1.83157	−0.915784	−	0.401671i	$-0.868429\pi$
$29$	−9.86330	−1.83157	−0.915784	+	0.401671i	$0.868429\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−5.02827	−0.875310
$34$	0	0
$35$	0.485863	0.0821258
$36$	0	0
$37$	5.51414	0.906519	0.453259	−	0.891379i	$-0.350261\pi$
$37$	5.51414	0.906519	0.453259	+	0.891379i	$0.350261\pi$
$38$	0	0
$39$	3.51414	0.562712
$40$	0	0
$41$	−7.02827	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$41$	−7.02827	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$42$	0	0
$43$	1.02827	0.156810	0.0784051	−	0.996922i	$-0.475017\pi$
$43$	1.02827	0.156810	0.0784051	+	0.996922i	$0.475017\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.93438	1.01148	0.505742	−	0.862685i	$-0.331219\pi$
$47$	6.93438	1.01148	0.505742	+	0.862685i	$0.331219\pi$
$48$	0	0
$49$	−6.76394	−0.966277
$50$	0	0
$51$	−1.32088	−0.184961
$52$	0	0
$53$	−1.70739	−0.234528	−0.117264	−	0.993101i	$-0.537412\pi$
$53$	−1.70739	−0.234528	−0.117264	+	0.993101i	$0.537412\pi$
$54$	0	0
$55$	5.02827	0.678012
$56$	0	0
$57$	6.64177	0.879724
$58$	0	0
$59$	2.19325	0.285537	0.142769	−	0.989756i	$-0.454400\pi$
$59$	2.19325	0.285537	0.142769	+	0.989756i	$0.454400\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−0.485863	−0.0612130
$64$	0	0
$65$	−3.51414	−0.435875
$66$	0	0
$67$	−9.12763	−1.11512	−0.557559	−	0.830137i	$-0.688262\pi$
$67$	−9.12763	−1.11512	−0.557559	+	0.830137i	$0.688262\pi$
$68$	0	0
$69$	−0.292611	−0.0352262
$70$	0	0
$71$	−13.4768	−1.59940	−0.799700	−	0.600399i	$-0.795008\pi$
$71$	−13.4768	−1.59940	−0.799700	+	0.600399i	$0.795008\pi$
$72$	0	0
$73$	12.5424	1.46798	0.733989	−	0.679161i	$-0.237656\pi$
$73$	12.5424	1.46798	0.733989	+	0.679161i	$0.237656\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.44305	0.278412
$78$	0	0
$79$	0.349158	0.0392834	0.0196417	−	0.999807i	$-0.493747\pi$
$79$	0.349158	0.0392834	0.0196417	+	0.999807i	$0.493747\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.9344	−1.20020	−0.600102	−	0.799923i	$-0.704873\pi$
$83$	−10.9344	−1.20020	−0.600102	+	0.799923i	$0.704873\pi$
$84$	0	0
$85$	1.32088	0.143270
$86$	0	0
$87$	−9.86330	−1.05746
$88$	0	0
$89$	5.03374	0.533575	0.266788	−	0.963755i	$-0.414038\pi$
$89$	5.03374	0.533575	0.266788	+	0.963755i	$0.414038\pi$
$90$	0	0
$91$	−1.70739	−0.178983
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−6.64177	−0.681431
$96$	0	0
$97$	10.4431	1.06033	0.530166	−	0.847894i	$-0.322130\pi$
$97$	10.4431	1.06033	0.530166	+	0.847894i	$0.322130\pi$
$98$	0	0
$99$	−5.02827	−0.505361
$100$	0	0
$101$	−11.6135	−1.15559	−0.577793	−	0.816183i	$-0.696086\pi$
$101$	−11.6135	−1.15559	−0.577793	+	0.816183i	$0.696086\pi$
$102$	0	0
$103$	3.76940	0.371410	0.185705	−	0.982606i	$-0.440543\pi$
$103$	3.76940	0.371410	0.185705	+	0.982606i	$0.440543\pi$
$104$	0	0
$105$	0.485863	0.0474154
$106$	0	0
$107$	6.73566	0.651161	0.325581	−	0.945514i	$-0.394440\pi$
$107$	6.73566	0.651161	0.325581	+	0.945514i	$0.394440\pi$
$108$	0	0
$109$	−3.90611	−0.374137	−0.187069	−	0.982347i	$-0.559899\pi$
$109$	−3.90611	−0.374137	−0.187069	+	0.982347i	$0.559899\pi$
$110$	0	0
$111$	5.51414	0.523379
$112$	0	0
$113$	−15.0848	−1.41906	−0.709530	−	0.704675i	$-0.751093\pi$
$113$	−15.0848	−1.41906	−0.709530	+	0.704675i	$0.751093\pi$
$114$	0	0
$115$	0.292611	0.0272861
$116$	0	0
$117$	3.51414	0.324882
$118$	0	0
$119$	0.641769	0.0588309
$120$	0	0
$121$	14.2835	1.29850
$122$	0	0
$123$	−7.02827	−0.633718
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.25526	0.555065	0.277532	−	0.960716i	$-0.410483\pi$
$127$	6.25526	0.555065	0.277532	+	0.960716i	$0.410483\pi$
$128$	0	0
$129$	1.02827	0.0905345
$130$	0	0
$131$	22.1186	1.93251	0.966254	−	0.257592i	$-0.0829291\pi$
$131$	22.1186	1.93251	0.966254	+	0.257592i	$0.0829291\pi$
$132$	0	0
$133$	−3.22699	−0.279816
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.6044	−1.24774	−0.623870	−	0.781528i	$-0.714441\pi$
$137$	−14.6044	−1.24774	−0.623870	+	0.781528i	$0.714441\pi$
$138$	0	0
$139$	−12.8970	−1.09391	−0.546956	−	0.837161i	$-0.684214\pi$
$139$	−12.8970	−1.09391	−0.546956	+	0.837161i	$0.684214\pi$
$140$	0	0
$141$	6.93438	0.583980
$142$	0	0
$143$	−17.6700	−1.47764
$144$	0	0
$145$	9.86330	0.819102
$146$	0	0
$147$	−6.76394	−0.557880
$148$	0	0
$149$	16.3118	1.33632	0.668158	−	0.744020i	$-0.267083\pi$
$149$	16.3118	1.33632	0.668158	+	0.744020i	$0.267083\pi$
$150$	0	0
$151$	−19.3774	−1.57691	−0.788457	−	0.615090i	$-0.789119\pi$
$151$	−19.3774	−1.57691	−0.788457	+	0.615090i	$0.789119\pi$
$152$	0	0
$153$	−1.32088	−0.106787
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−3.80128	−0.303375	−0.151688	−	0.988428i	$-0.548471\pi$
$157$	−3.80128	−0.303375	−0.151688	+	0.988428i	$0.548471\pi$
$158$	0	0
$159$	−1.70739	−0.135405
$160$	0	0
$161$	0.142169	0.0112045
$162$	0	0
$163$	−12.5424	−0.982397	−0.491199	−	0.871048i	$-0.663441\pi$
$163$	−12.5424	−0.982397	−0.491199	+	0.871048i	$0.663441\pi$
$164$	0	0
$165$	5.02827	0.391451
$166$	0	0
$167$	−22.2553	−1.72216	−0.861082	−	0.508466i	$-0.830213\pi$
$167$	−22.2553	−1.72216	−0.861082	+	0.508466i	$0.830213\pi$
$168$	0	0
$169$	−0.650842	−0.0500647
$170$	0	0
$171$	6.64177	0.507909
$172$	0	0
$173$	−10.9717	−0.834165	−0.417082	−	0.908869i	$-0.636947\pi$
$173$	−10.9717	−0.834165	−0.417082	+	0.908869i	$0.636947\pi$
$174$	0	0
$175$	−0.485863	−0.0367278
$176$	0	0
$177$	2.19325	0.164855
$178$	0	0
$179$	15.9253	1.19031	0.595157	−	0.803610i	$-0.297090\pi$
$179$	15.9253	1.19031	0.595157	+	0.803610i	$0.297090\pi$
$180$	0	0
$181$	−8.05655	−0.598838	−0.299419	−	0.954122i	$-0.596793\pi$
$181$	−8.05655	−0.598838	−0.299419	+	0.954122i	$0.596793\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−5.51414	−0.405407
$186$	0	0
$187$	6.64177	0.485694
$188$	0	0
$189$	−0.485863	−0.0353413
$190$	0	0
$191$	−6.19325	−0.448128	−0.224064	−	0.974574i	$-0.571932\pi$
$191$	−6.19325	−0.448128	−0.224064	+	0.974574i	$0.571932\pi$
$192$	0	0
$193$	−2.25526	−0.162337	−0.0811687	−	0.996700i	$-0.525865\pi$
$193$	−2.25526	−0.162337	−0.0811687	+	0.996700i	$0.525865\pi$
$194$	0	0
$195$	−3.51414	−0.251653
$196$	0	0
$197$	9.32088	0.664086	0.332043	−	0.943264i	$-0.392262\pi$
$197$	9.32088	0.664086	0.332043	+	0.943264i	$0.392262\pi$
$198$	0	0
$199$	−16.5479	−1.17305	−0.586524	−	0.809932i	$-0.699504\pi$
$199$	−16.5479	−1.17305	−0.586524	+	0.809932i	$0.699504\pi$
$200$	0	0
$201$	−9.12763	−0.643814
$202$	0	0
$203$	4.79221	0.336347
$204$	0	0
$205$	7.02827	0.490876
$206$	0	0
$207$	−0.292611	−0.0203378
$208$	0	0
$209$	−33.3966	−2.31009
$210$	0	0
$211$	−25.0101	−1.72177	−0.860884	−	0.508801i	$-0.830089\pi$
$211$	−25.0101	−1.72177	−0.860884	+	0.508801i	$0.830089\pi$
$212$	0	0
$213$	−13.4768	−0.923414
$214$	0	0
$215$	−1.02827	−0.0701277
$216$	0	0
$217$	0.485863	0.0329825
$218$	0	0
$219$	12.5424	0.847538
$220$	0	0
$221$	−4.64177	−0.312239
$222$	0	0
$223$	−19.2835	−1.29132	−0.645661	−	0.763625i	$-0.723418\pi$
$223$	−19.2835	−1.29132	−0.645661	+	0.763625i	$0.723418\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−23.4340	−1.55537	−0.777684	−	0.628655i	$-0.783606\pi$
$227$	−23.4340	−1.55537	−0.777684	+	0.628655i	$0.783606\pi$
$228$	0	0
$229$	−25.0283	−1.65391	−0.826957	−	0.562265i	$-0.809930\pi$
$229$	−25.0283	−1.65391	−0.826957	+	0.562265i	$0.809930\pi$
$230$	0	0
$231$	2.44305	0.160741
$232$	0	0
$233$	18.7175	1.22623	0.613113	−	0.789995i	$-0.289917\pi$
$233$	18.7175	1.22623	0.613113	+	0.789995i	$0.289917\pi$
$234$	0	0
$235$	−6.93438	−0.452349
$236$	0	0
$237$	0.349158	0.0226803
$238$	0	0
$239$	3.86876	0.250249	0.125125	−	0.992141i	$-0.460067\pi$
$239$	3.86876	0.250249	0.125125	+	0.992141i	$0.460067\pi$
$240$	0	0
$241$	−1.61350	−0.103934	−0.0519672	−	0.998649i	$-0.516549\pi$
$241$	−1.61350	−0.103934	−0.0519672	+	0.998649i	$0.516549\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.76394	0.432132
$246$	0	0
$247$	23.3401	1.48509
$248$	0	0
$249$	−10.9344	−0.692938
$250$	0	0
$251$	25.4713	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$251$	25.4713	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$252$	0	0
$253$	1.47133	0.0925015
$254$	0	0
$255$	1.32088	0.0827170
$256$	0	0
$257$	18.0192	1.12401	0.562003	−	0.827135i	$-0.310031\pi$
$257$	18.0192	1.12401	0.562003	+	0.827135i	$0.310031\pi$
$258$	0	0
$259$	−2.67912	−0.166472
$260$	0	0
$261$	−9.86330	−0.610523
$262$	0	0
$263$	−1.15951	−0.0714987	−0.0357494	−	0.999361i	$-0.511382\pi$
$263$	−1.15951	−0.0714987	−0.0357494	+	0.999361i	$0.511382\pi$
$264$	0	0
$265$	1.70739	0.104884
$266$	0	0
$267$	5.03374	0.308060
$268$	0	0
$269$	−7.80675	−0.475986	−0.237993	−	0.971267i	$-0.576489\pi$
$269$	−7.80675	−0.475986	−0.237993	+	0.971267i	$0.576489\pi$
$270$	0	0
$271$	−17.7831	−1.08025	−0.540124	−	0.841585i	$-0.681623\pi$
$271$	−17.7831	−1.08025	−0.540124	+	0.841585i	$0.681623\pi$
$272$	0	0
$273$	−1.70739	−0.103336
$274$	0	0
$275$	−5.02827	−0.303216
$276$	0	0
$277$	25.9390	1.55853	0.779263	−	0.626697i	$-0.215594\pi$
$277$	25.9390	1.55853	0.779263	+	0.626697i	$0.215594\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−13.0283	−0.777202	−0.388601	−	0.921406i	$-0.627041\pi$
$281$	−13.0283	−0.777202	−0.388601	+	0.921406i	$0.627041\pi$
$282$	0	0
$283$	−15.9006	−0.945195	−0.472598	−	0.881278i	$-0.656684\pi$
$283$	−15.9006	−0.945195	−0.472598	+	0.881278i	$0.656684\pi$
$284$	0	0
$285$	−6.64177	−0.393424
$286$	0	0
$287$	3.41478	0.201568
$288$	0	0
$289$	−15.2553	−0.897368
$290$	0	0
$291$	10.4431	0.612183
$292$	0	0
$293$	−29.4340	−1.71955	−0.859776	−	0.510672i	$-0.829397\pi$
$293$	−29.4340	−1.71955	−0.859776	+	0.510672i	$0.829397\pi$
$294$	0	0
$295$	−2.19325	−0.127696
$296$	0	0
$297$	−5.02827	−0.291770
$298$	0	0
$299$	−1.02827	−0.0594666
$300$	0	0
$301$	−0.499600	−0.0287965
$302$	0	0
$303$	−11.6135	−0.667178
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−9.96812	−0.568911	−0.284455	−	0.958689i	$-0.591813\pi$
$307$	−9.96812	−0.568911	−0.284455	+	0.958689i	$0.591813\pi$
$308$	0	0
$309$	3.76940	0.214434
$310$	0	0
$311$	−15.9945	−0.906967	−0.453483	−	0.891265i	$-0.649819\pi$
$311$	−15.9945	−0.906967	−0.453483	+	0.891265i	$0.649819\pi$
$312$	0	0
$313$	−22.8542	−1.29180	−0.645899	−	0.763423i	$-0.723517\pi$
$313$	−22.8542	−1.29180	−0.645899	+	0.763423i	$0.723517\pi$
$314$	0	0
$315$	0.485863	0.0273753
$316$	0	0
$317$	14.6044	0.820266	0.410133	−	0.912026i	$-0.365482\pi$
$317$	14.6044	0.820266	0.410133	+	0.912026i	$0.365482\pi$
$318$	0	0
$319$	49.5953	2.77681
$320$	0	0
$321$	6.73566	0.375948
$322$	0	0
$323$	−8.77301	−0.488143
$324$	0	0
$325$	3.51414	0.194929
$326$	0	0
$327$	−3.90611	−0.216008
$328$	0	0
$329$	−3.36916	−0.185748
$330$	0	0
$331$	8.16137	0.448589	0.224295	−	0.974521i	$-0.427992\pi$
$331$	8.16137	0.448589	0.224295	+	0.974521i	$0.427992\pi$
$332$	0	0
$333$	5.51414	0.302173
$334$	0	0
$335$	9.12763	0.498696
$336$	0	0
$337$	−11.8825	−0.647281	−0.323640	−	0.946180i	$-0.604907\pi$
$337$	−11.8825	−0.647281	−0.323640	+	0.946180i	$0.604907\pi$
$338$	0	0
$339$	−15.0848	−0.819295
$340$	0	0
$341$	5.02827	0.272296
$342$	0	0
$343$	6.68739	0.361085
$344$	0	0
$345$	0.292611	0.0157536
$346$	0	0
$347$	18.3684	0.986065	0.493033	−	0.870011i	$-0.335888\pi$
$347$	18.3684	0.986065	0.493033	+	0.870011i	$0.335888\pi$
$348$	0	0
$349$	11.2462	0.601995	0.300997	−	0.953625i	$-0.402680\pi$
$349$	11.2462	0.601995	0.300997	+	0.953625i	$0.402680\pi$
$350$	0	0
$351$	3.51414	0.187571
$352$	0	0
$353$	25.3593	1.34974	0.674869	−	0.737937i	$-0.264200\pi$
$353$	25.3593	1.34974	0.674869	+	0.737937i	$0.264200\pi$
$354$	0	0
$355$	13.4768	0.715274
$356$	0	0
$357$	0.641769	0.0339660
$358$	0	0
$359$	−15.2890	−0.806923	−0.403461	−	0.914997i	$-0.632193\pi$
$359$	−15.2890	−0.806923	−0.403461	+	0.914997i	$0.632193\pi$
$360$	0	0
$361$	25.1131	1.32174
$362$	0	0
$363$	14.2835	0.749691
$364$	0	0
$365$	−12.5424	−0.656500
$366$	0	0
$367$	−6.05655	−0.316149	−0.158075	−	0.987427i	$-0.550529\pi$
$367$	−6.05655	−0.316149	−0.158075	+	0.987427i	$0.550529\pi$
$368$	0	0
$369$	−7.02827	−0.365877
$370$	0	0
$371$	0.829557	0.0430685
$372$	0	0
$373$	8.06748	0.417718	0.208859	−	0.977946i	$-0.433025\pi$
$373$	8.06748	0.417718	0.208859	+	0.977946i	$0.433025\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−34.6610	−1.78513
$378$	0	0
$379$	36.8114	1.89088	0.945438	−	0.325803i	$-0.105635\pi$
$379$	36.8114	1.89088	0.945438	+	0.325803i	$0.105635\pi$
$380$	0	0
$381$	6.25526	0.320467
$382$	0	0
$383$	3.63270	0.185622	0.0928111	−	0.995684i	$-0.470415\pi$
$383$	3.63270	0.185622	0.0928111	+	0.995684i	$0.470415\pi$
$384$	0	0
$385$	−2.44305	−0.124509
$386$	0	0
$387$	1.02827	0.0522701
$388$	0	0
$389$	15.2781	0.774629	0.387315	−	0.921948i	$-0.373403\pi$
$389$	15.2781	0.774629	0.387315	+	0.921948i	$0.373403\pi$
$390$	0	0
$391$	0.386505	0.0195464
$392$	0	0
$393$	22.1186	1.11573
$394$	0	0
$395$	−0.349158	−0.0175681
$396$	0	0
$397$	−18.5852	−0.932766	−0.466383	−	0.884583i	$-0.654443\pi$
$397$	−18.5852	−0.932766	−0.466383	+	0.884583i	$0.654443\pi$
$398$	0	0
$399$	−3.22699	−0.161552
$400$	0	0
$401$	6.19325	0.309276	0.154638	−	0.987971i	$-0.450579\pi$
$401$	6.19325	0.309276	0.154638	+	0.987971i	$0.450579\pi$
$402$	0	0
$403$	−3.51414	−0.175052
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−27.7266	−1.37436
$408$	0	0
$409$	−32.0565	−1.58509	−0.792547	−	0.609811i	$-0.791245\pi$
$409$	−32.0565	−1.58509	−0.792547	+	0.609811i	$0.791245\pi$
$410$	0	0
$411$	−14.6044	−0.720383
$412$	0	0
$413$	−1.06562	−0.0524357
$414$	0	0
$415$	10.9344	0.536748
$416$	0	0
$417$	−12.8970	−0.631570
$418$	0	0
$419$	−9.78860	−0.478205	−0.239102	−	0.970994i	$-0.576853\pi$
$419$	−9.78860	−0.478205	−0.239102	+	0.970994i	$0.576853\pi$
$420$	0	0
$421$	−14.2745	−0.695695	−0.347847	−	0.937551i	$-0.613087\pi$
$421$	−14.2745	−0.695695	−0.347847	+	0.937551i	$0.613087\pi$
$422$	0	0
$423$	6.93438	0.337161
$424$	0	0
$425$	−1.32088	−0.0640723
$426$	0	0
$427$	0.971726	0.0470251
$428$	0	0
$429$	−17.6700	−0.853118
$430$	0	0
$431$	22.8350	1.09992	0.549962	−	0.835190i	$-0.314642\pi$
$431$	22.8350	1.09992	0.549962	+	0.835190i	$0.314642\pi$
$432$	0	0
$433$	13.8259	0.664433	0.332216	−	0.943203i	$-0.392204\pi$
$433$	13.8259	0.664433	0.332216	+	0.943203i	$0.392204\pi$
$434$	0	0
$435$	9.86330	0.472909
$436$	0	0
$437$	−1.94345	−0.0929679
$438$	0	0
$439$	−39.8506	−1.90197	−0.950983	−	0.309243i	$-0.899924\pi$
$439$	−39.8506	−1.90197	−0.950983	+	0.309243i	$0.899924\pi$
$440$	0	0
$441$	−6.76394	−0.322092
$442$	0	0
$443$	10.1504	0.482262	0.241131	−	0.970493i	$-0.422482\pi$
$443$	10.1504	0.482262	0.241131	+	0.970493i	$0.422482\pi$
$444$	0	0
$445$	−5.03374	−0.238622
$446$	0	0
$447$	16.3118	0.771522
$448$	0	0
$449$	−3.75020	−0.176983	−0.0884914	−	0.996077i	$-0.528205\pi$
$449$	−3.75020	−0.176983	−0.0884914	+	0.996077i	$0.528205\pi$
$450$	0	0
$451$	35.3401	1.66410
$452$	0	0
$453$	−19.3774	−0.910431
$454$	0	0
$455$	1.70739	0.0800436
$456$	0	0
$457$	−1.70193	−0.0796127	−0.0398064	−	0.999207i	$-0.512674\pi$
$457$	−1.70193	−0.0796127	−0.0398064	+	0.999207i	$0.512674\pi$
$458$	0	0
$459$	−1.32088	−0.0616536
$460$	0	0
$461$	7.40931	0.345086	0.172543	−	0.985002i	$-0.444802\pi$
$461$	7.40931	0.345086	0.172543	+	0.985002i	$0.444802\pi$
$462$	0	0
$463$	30.3865	1.41218	0.706090	−	0.708122i	$-0.250457\pi$
$463$	30.3865	1.41218	0.706090	+	0.708122i	$0.250457\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−17.1040	−0.791480	−0.395740	−	0.918363i	$-0.629512\pi$
$467$	−17.1040	−0.791480	−0.395740	+	0.918363i	$0.629512\pi$
$468$	0	0
$469$	4.43478	0.204779
$470$	0	0
$471$	−3.80128	−0.175154
$472$	0	0
$473$	−5.17044	−0.237737
$474$	0	0
$475$	6.64177	0.304745
$476$	0	0
$477$	−1.70739	−0.0781760
$478$	0	0
$479$	−21.2890	−0.972719	−0.486360	−	0.873759i	$-0.661676\pi$
$479$	−21.2890	−0.972719	−0.486360	+	0.873759i	$0.661676\pi$
$480$	0	0
$481$	19.3774	0.883535
$482$	0	0
$483$	0.142169	0.00646890
$484$	0	0
$485$	−10.4431	−0.474195
$486$	0	0
$487$	−20.4996	−0.928926	−0.464463	−	0.885593i	$-0.653753\pi$
$487$	−20.4996	−0.928926	−0.464463	+	0.885593i	$0.653753\pi$
$488$	0	0
$489$	−12.5424	−0.567187
$490$	0	0
$491$	31.0667	1.40202	0.701010	−	0.713152i	$-0.252733\pi$
$491$	31.0667	1.40202	0.701010	+	0.713152i	$0.252733\pi$
$492$	0	0
$493$	13.0283	0.586764
$494$	0	0
$495$	5.02827	0.226004
$496$	0	0
$497$	6.54787	0.293712
$498$	0	0
$499$	−15.3401	−0.686717	−0.343358	−	0.939205i	$-0.611565\pi$
$499$	−15.3401	−0.686717	−0.343358	+	0.939205i	$0.611565\pi$
$500$	0	0
$501$	−22.2553	−0.994292
$502$	0	0
$503$	14.9344	0.665891	0.332946	−	0.942946i	$-0.391957\pi$
$503$	14.9344	0.665891	0.332946	+	0.942946i	$0.391957\pi$
$504$	0	0
$505$	11.6135	0.516794
$506$	0	0
$507$	−0.650842	−0.0289049
$508$	0	0
$509$	−1.22153	−0.0541432	−0.0270716	−	0.999633i	$-0.508618\pi$
$509$	−1.22153	−0.0541432	−0.0270716	+	0.999633i	$0.508618\pi$
$510$	0	0
$511$	−6.09389	−0.269578
$512$	0	0
$513$	6.64177	0.293241
$514$	0	0
$515$	−3.76940	−0.166100
$516$	0	0
$517$	−34.8680	−1.53349
$518$	0	0
$519$	−10.9717	−0.481605
$520$	0	0
$521$	4.57429	0.200403	0.100202	−	0.994967i	$-0.468051\pi$
$521$	4.57429	0.200403	0.100202	+	0.994967i	$0.468051\pi$
$522$	0	0
$523$	29.7831	1.30233	0.651163	−	0.758938i	$-0.274281\pi$
$523$	29.7831	1.30233	0.651163	+	0.758938i	$0.274281\pi$
$524$	0	0
$525$	−0.485863	−0.0212048
$526$	0	0
$527$	1.32088	0.0575386
$528$	0	0
$529$	−22.9144	−0.996277
$530$	0	0
$531$	2.19325	0.0951790
$532$	0	0
$533$	−24.6983	−1.06980
$534$	0	0
$535$	−6.73566	−0.291208
$536$	0	0
$537$	15.9253	0.687228
$538$	0	0
$539$	34.0109	1.46495
$540$	0	0
$541$	−3.04748	−0.131021	−0.0655106	−	0.997852i	$-0.520868\pi$
$541$	−3.04748	−0.131021	−0.0655106	+	0.997852i	$0.520868\pi$
$542$	0	0
$543$	−8.05655	−0.345740
$544$	0	0
$545$	3.90611	0.167319
$546$	0	0
$547$	−9.82595	−0.420127	−0.210064	−	0.977688i	$-0.567367\pi$
$547$	−9.82595	−0.420127	−0.210064	+	0.977688i	$0.567367\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−65.5097	−2.79081
$552$	0	0
$553$	−0.169643	−0.00721396
$554$	0	0
$555$	−5.51414	−0.234062
$556$	0	0
$557$	−38.7922	−1.64368	−0.821839	−	0.569719i	$-0.807052\pi$
$557$	−38.7922	−1.64368	−0.821839	+	0.569719i	$0.807052\pi$
$558$	0	0
$559$	3.61350	0.152835
$560$	0	0
$561$	6.64177	0.280416
$562$	0	0
$563$	−2.73566	−0.115294	−0.0576472	−	0.998337i	$-0.518360\pi$
$563$	−2.73566	−0.115294	−0.0576472	+	0.998337i	$0.518360\pi$
$564$	0	0
$565$	15.0848	0.634623
$566$	0	0
$567$	−0.485863	−0.0204043
$568$	0	0
$569$	4.39197	0.184121	0.0920605	−	0.995753i	$-0.470655\pi$
$569$	4.39197	0.184121	0.0920605	+	0.995753i	$0.470655\pi$
$570$	0	0
$571$	−9.67004	−0.404679	−0.202339	−	0.979315i	$-0.564854\pi$
$571$	−9.67004	−0.404679	−0.202339	+	0.979315i	$0.564854\pi$
$572$	0	0
$573$	−6.19325	−0.258727
$574$	0	0
$575$	−0.292611	−0.0122027
$576$	0	0
$577$	31.4148	1.30781	0.653907	−	0.756575i	$-0.273129\pi$
$577$	31.4148	1.30781	0.653907	+	0.756575i	$0.273129\pi$
$578$	0	0
$579$	−2.25526	−0.0937256
$580$	0	0
$581$	5.31261	0.220404
$582$	0	0
$583$	8.58522	0.355564
$584$	0	0
$585$	−3.51414	−0.145292
$586$	0	0
$587$	27.2161	1.12333	0.561664	−	0.827366i	$-0.310162\pi$
$587$	27.2161	1.12333	0.561664	+	0.827366i	$0.310162\pi$
$588$	0	0
$589$	−6.64177	−0.273669
$590$	0	0
$591$	9.32088	0.383410
$592$	0	0
$593$	32.5561	1.33692	0.668460	−	0.743748i	$-0.266954\pi$
$593$	32.5561	1.33692	0.668460	+	0.743748i	$0.266954\pi$
$594$	0	0
$595$	−0.641769	−0.0263100
$596$	0	0
$597$	−16.5479	−0.677259
$598$	0	0
$599$	−27.8067	−1.13615	−0.568076	−	0.822976i	$-0.692312\pi$
$599$	−27.8067	−1.13615	−0.568076	+	0.822976i	$0.692312\pi$
$600$	0	0
$601$	17.6519	0.720036	0.360018	−	0.932945i	$-0.382771\pi$
$601$	17.6519	0.720036	0.360018	+	0.932945i	$0.382771\pi$
$602$	0	0
$603$	−9.12763	−0.371706
$604$	0	0
$605$	−14.2835	−0.580708
$606$	0	0
$607$	−44.9673	−1.82517	−0.912584	−	0.408890i	$-0.865916\pi$
$607$	−44.9673	−1.82517	−0.912584	+	0.408890i	$0.865916\pi$
$608$	0	0
$609$	4.79221	0.194190
$610$	0	0
$611$	24.3684	0.985838
$612$	0	0
$613$	33.7694	1.36393	0.681967	−	0.731383i	$-0.261125\pi$
$613$	33.7694	1.36393	0.681967	+	0.731383i	$0.261125\pi$
$614$	0	0
$615$	7.02827	0.283407
$616$	0	0
$617$	−26.1131	−1.05127	−0.525637	−	0.850709i	$-0.676173\pi$
$617$	−26.1131	−1.05127	−0.525637	+	0.850709i	$0.676173\pi$
$618$	0	0
$619$	20.8597	0.838422	0.419211	−	0.907889i	$-0.362307\pi$
$619$	20.8597	0.838422	0.419211	+	0.907889i	$0.362307\pi$
$620$	0	0
$621$	−0.292611	−0.0117421
$622$	0	0
$623$	−2.44571	−0.0979852
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−33.3966	−1.33373
$628$	0	0
$629$	−7.28354	−0.290414
$630$	0	0
$631$	20.5105	0.816511	0.408256	−	0.912868i	$-0.366137\pi$
$631$	20.5105	0.816511	0.408256	+	0.912868i	$0.366137\pi$
$632$	0	0
$633$	−25.0101	−0.994063
$634$	0	0
$635$	−6.25526	−0.248233
$636$	0	0
$637$	−23.7694	−0.941778
$638$	0	0
$639$	−13.4768	−0.533134
$640$	0	0
$641$	23.9945	0.947727	0.473864	−	0.880598i	$-0.342859\pi$
$641$	23.9945	0.947727	0.473864	+	0.880598i	$0.342859\pi$
$642$	0	0
$643$	13.6700	0.539094	0.269547	−	0.962987i	$-0.413126\pi$
$643$	13.6700	0.539094	0.269547	+	0.962987i	$0.413126\pi$
$644$	0	0
$645$	−1.02827	−0.0404882
$646$	0	0
$647$	11.1523	0.438442	0.219221	−	0.975675i	$-0.429648\pi$
$647$	11.1523	0.438442	0.219221	+	0.975675i	$0.429648\pi$
$648$	0	0
$649$	−11.0283	−0.432898
$650$	0	0
$651$	0.485863	0.0190425
$652$	0	0
$653$	−39.1896	−1.53361	−0.766805	−	0.641881i	$-0.778154\pi$
$653$	−39.1896	−1.53361	−0.766805	+	0.641881i	$0.778154\pi$
$654$	0	0
$655$	−22.1186	−0.864244
$656$	0	0
$657$	12.5424	0.489326
$658$	0	0
$659$	31.6464	1.23277	0.616385	−	0.787445i	$-0.288597\pi$
$659$	31.6464	1.23277	0.616385	+	0.787445i	$0.288597\pi$
$660$	0	0
$661$	−3.59535	−0.139843	−0.0699215	−	0.997553i	$-0.522275\pi$
$661$	−3.59535	−0.139843	−0.0699215	+	0.997553i	$0.522275\pi$
$662$	0	0
$663$	−4.64177	−0.180271
$664$	0	0
$665$	3.22699	0.125137
$666$	0	0
$667$	2.88611	0.111750
$668$	0	0
$669$	−19.2835	−0.745545
$670$	0	0
$671$	10.0565	0.388229
$672$	0	0
$673$	−7.24073	−0.279110	−0.139555	−	0.990214i	$-0.544567\pi$
$673$	−7.24073	−0.279110	−0.139555	+	0.990214i	$0.544567\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	38.5188	1.48040	0.740199	−	0.672388i	$-0.234731\pi$
$677$	38.5188	1.48040	0.740199	+	0.672388i	$0.234731\pi$
$678$	0	0
$679$	−5.07389	−0.194718
$680$	0	0
$681$	−23.4340	−0.897992
$682$	0	0
$683$	−40.0192	−1.53129	−0.765646	−	0.643262i	$-0.777581\pi$
$683$	−40.0192	−1.53129	−0.765646	+	0.643262i	$0.777581\pi$
$684$	0	0
$685$	14.6044	0.558006
$686$	0	0
$687$	−25.0283	−0.954888
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	11.8013	0.448942	0.224471	−	0.974481i	$-0.427935\pi$
$691$	11.8013	0.448942	0.224471	+	0.974481i	$0.427935\pi$
$692$	0	0
$693$	2.44305	0.0928039
$694$	0	0
$695$	12.8970	0.489212
$696$	0	0
$697$	9.28354	0.351639
$698$	0	0
$699$	18.7175	0.707962
$700$	0	0
$701$	5.17044	0.195285	0.0976425	−	0.995222i	$-0.468870\pi$
$701$	5.17044	0.195285	0.0976425	+	0.995222i	$0.468870\pi$
$702$	0	0
$703$	36.6236	1.38129
$704$	0	0
$705$	−6.93438	−0.261164
$706$	0	0
$707$	5.64257	0.212211
$708$	0	0
$709$	47.7722	1.79412	0.897062	−	0.441906i	$-0.145697\pi$
$709$	47.7722	1.79412	0.897062	+	0.441906i	$0.145697\pi$
$710$	0	0
$711$	0.349158	0.0130945
$712$	0	0
$713$	0.292611	0.0109584
$714$	0	0
$715$	17.6700	0.660822
$716$	0	0
$717$	3.86876	0.144481
$718$	0	0
$719$	−23.0848	−0.860919	−0.430459	−	0.902610i	$-0.641648\pi$
$719$	−23.0848	−0.860919	−0.430459	+	0.902610i	$0.641648\pi$
$720$	0	0
$721$	−1.83141	−0.0682054
$722$	0	0
$723$	−1.61350	−0.0600065
$724$	0	0
$725$	−9.86330	−0.366314
$726$	0	0
$727$	22.2125	0.823814	0.411907	−	0.911226i	$-0.364863\pi$
$727$	22.2125	0.823814	0.411907	+	0.911226i	$0.364863\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.35823	−0.0502360
$732$	0	0
$733$	−43.9072	−1.62175	−0.810874	−	0.585221i	$-0.801008\pi$
$733$	−43.9072	−1.62175	−0.810874	+	0.585221i	$0.801008\pi$
$734$	0	0
$735$	6.76394	0.249492
$736$	0	0
$737$	45.8962	1.69061
$738$	0	0
$739$	−6.86690	−0.252603	−0.126302	−	0.991992i	$-0.540311\pi$
$739$	−6.86690	−0.252603	−0.126302	+	0.991992i	$0.540311\pi$
$740$	0	0
$741$	23.3401	0.857419
$742$	0	0
$743$	−29.2726	−1.07391	−0.536954	−	0.843612i	$-0.680425\pi$
$743$	−29.2726	−1.07391	−0.536954	+	0.843612i	$0.680425\pi$
$744$	0	0
$745$	−16.3118	−0.597619
$746$	0	0
$747$	−10.9344	−0.400068
$748$	0	0
$749$	−3.27261	−0.119579
$750$	0	0
$751$	−9.67004	−0.352865	−0.176432	−	0.984313i	$-0.556456\pi$
$751$	−9.67004	−0.352865	−0.176432	+	0.984313i	$0.556456\pi$
$752$	0	0
$753$	25.4713	0.928227
$754$	0	0
$755$	19.3774	0.705217
$756$	0	0
$757$	−49.4104	−1.79585	−0.897925	−	0.440148i	$-0.854926\pi$
$757$	−49.4104	−1.79585	−0.897925	+	0.440148i	$0.854926\pi$
$758$	0	0
$759$	1.47133	0.0534058
$760$	0	0
$761$	47.2599	1.71317	0.856586	−	0.516005i	$-0.172581\pi$
$761$	47.2599	1.71317	0.856586	+	0.516005i	$0.172581\pi$
$762$	0	0
$763$	1.89783	0.0687062
$764$	0	0
$765$	1.32088	0.0477567
$766$	0	0
$767$	7.70739	0.278298
$768$	0	0
$769$	9.49053	0.342237	0.171119	−	0.985250i	$-0.445262\pi$
$769$	9.49053	0.342237	0.171119	+	0.985250i	$0.445262\pi$
$770$	0	0
$771$	18.0192	0.648946
$772$	0	0
$773$	41.7567	1.50188	0.750942	−	0.660368i	$-0.229600\pi$
$773$	41.7567	1.50188	0.750942	+	0.660368i	$0.229600\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−2.67912	−0.0961127
$778$	0	0
$779$	−46.6802	−1.67249
$780$	0	0
$781$	67.7650	2.42482
$782$	0	0
$783$	−9.86330	−0.352485
$784$	0	0
$785$	3.80128	0.135674
$786$	0	0
$787$	40.0950	1.42923	0.714615	−	0.699518i	$-0.246602\pi$
$787$	40.0950	1.42923	0.714615	+	0.699518i	$0.246602\pi$
$788$	0	0
$789$	−1.15951	−0.0412798
$790$	0	0
$791$	7.32916	0.260595
$792$	0	0
$793$	−7.02827	−0.249581
$794$	0	0
$795$	1.70739	0.0605549
$796$	0	0
$797$	−31.1150	−1.10215	−0.551074	−	0.834456i	$-0.685782\pi$
$797$	−31.1150	−1.10215	−0.551074	+	0.834456i	$0.685782\pi$
$798$	0	0
$799$	−9.15951	−0.324040
$800$	0	0
$801$	5.03374	0.177858
$802$	0	0
$803$	−63.0667	−2.22557
$804$	0	0
$805$	−0.142169	−0.00501079
$806$	0	0
$807$	−7.80675	−0.274811
$808$	0	0
$809$	46.6291	1.63939	0.819696	−	0.572799i	$-0.194143\pi$
$809$	46.6291	1.63939	0.819696	+	0.572799i	$0.194143\pi$
$810$	0	0
$811$	47.0283	1.65139	0.825693	−	0.564120i	$-0.190784\pi$
$811$	47.0283	1.65139	0.825693	+	0.564120i	$0.190784\pi$
$812$	0	0
$813$	−17.7831	−0.623682
$814$	0	0
$815$	12.5424	0.439341
$816$	0	0
$817$	6.82956	0.238936
$818$	0	0
$819$	−1.70739	−0.0596610
$820$	0	0
$821$	35.3912	1.23516	0.617580	−	0.786508i	$-0.288113\pi$
$821$	35.3912	1.23516	0.617580	+	0.786508i	$0.288113\pi$
$822$	0	0
$823$	17.2161	0.600114	0.300057	−	0.953921i	$-0.402994\pi$
$823$	17.2161	0.600114	0.300057	+	0.953921i	$0.402994\pi$
$824$	0	0
$825$	−5.02827	−0.175062
$826$	0	0
$827$	−16.3310	−0.567885	−0.283942	−	0.958841i	$-0.591642\pi$
$827$	−16.3310	−0.567885	−0.283942	+	0.958841i	$0.591642\pi$
$828$	0	0
$829$	−29.0667	−1.00953	−0.504764	−	0.863258i	$-0.668420\pi$
$829$	−29.0667	−1.00953	−0.504764	+	0.863258i	$0.668420\pi$
$830$	0	0
$831$	25.9390	0.899815
$832$	0	0
$833$	8.93438	0.309558
$834$	0	0
$835$	22.2553	0.770175
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−27.6646	−0.955087	−0.477544	−	0.878608i	$-0.658473\pi$
$839$	−27.6646	−0.955087	−0.477544	+	0.878608i	$0.658473\pi$
$840$	0	0
$841$	68.2846	2.35464
$842$	0	0
$843$	−13.0283	−0.448718
$844$	0	0
$845$	0.650842	0.0223896
$846$	0	0
$847$	−6.93984	−0.238456
$848$	0	0
$849$	−15.9006	−0.545709
$850$	0	0
$851$	−1.61350	−0.0553099
$852$	0	0
$853$	32.5105	1.11314	0.556570	−	0.830801i	$-0.312117\pi$
$853$	32.5105	1.11314	0.556570	+	0.830801i	$0.312117\pi$
$854$	0	0
$855$	−6.64177	−0.227144
$856$	0	0
$857$	−45.9144	−1.56841	−0.784203	−	0.620505i	$-0.786928\pi$
$857$	−45.9144	−1.56841	−0.784203	+	0.620505i	$0.786928\pi$
$858$	0	0
$859$	19.6026	0.668831	0.334415	−	0.942426i	$-0.391461\pi$
$859$	19.6026	0.668831	0.334415	+	0.942426i	$0.391461\pi$
$860$	0	0
$861$	3.41478	0.116375
$862$	0	0
$863$	−46.1696	−1.57163	−0.785816	−	0.618460i	$-0.787757\pi$
$863$	−46.1696	−1.57163	−0.785816	+	0.618460i	$0.787757\pi$
$864$	0	0
$865$	10.9717	0.373050
$866$	0	0
$867$	−15.2553	−0.518096
$868$	0	0
$869$	−1.75566	−0.0595568
$870$	0	0
$871$	−32.0757	−1.08685
$872$	0	0
$873$	10.4431	0.353444
$874$	0	0
$875$	0.485863	0.0164252
$876$	0	0
$877$	11.3017	0.381631	0.190815	−	0.981626i	$-0.438887\pi$
$877$	11.3017	0.381631	0.190815	+	0.981626i	$0.438887\pi$
$878$	0	0
$879$	−29.4340	−0.992784
$880$	0	0
$881$	−9.60803	−0.323703	−0.161851	−	0.986815i	$-0.551747\pi$
$881$	−9.60803	−0.323703	−0.161851	+	0.986815i	$0.551747\pi$
$882$	0	0
$883$	33.5279	1.12830	0.564151	−	0.825671i	$-0.309203\pi$
$883$	33.5279	1.12830	0.564151	+	0.825671i	$0.309203\pi$
$884$	0	0
$885$	−2.19325	−0.0737254
$886$	0	0
$887$	9.76394	0.327841	0.163920	−	0.986474i	$-0.447586\pi$
$887$	9.76394	0.327841	0.163920	+	0.986474i	$0.447586\pi$
$888$	0	0
$889$	−3.03920	−0.101932
$890$	0	0
$891$	−5.02827	−0.168454
$892$	0	0
$893$	46.0565	1.54122
$894$	0	0
$895$	−15.9253	−0.532324
$896$	0	0
$897$	−1.02827	−0.0343331
$898$	0	0
$899$	9.86330	0.328959
$900$	0	0
$901$	2.25526	0.0751337
$902$	0	0
$903$	−0.499600	−0.0166257
$904$	0	0
$905$	8.05655	0.267809
$906$	0	0
$907$	9.18418	0.304956	0.152478	−	0.988307i	$-0.451275\pi$
$907$	9.18418	0.304956	0.152478	+	0.988307i	$0.451275\pi$
$908$	0	0
$909$	−11.6135	−0.385195
$910$	0	0
$911$	12.2553	0.406035	0.203018	−	0.979175i	$-0.434925\pi$
$911$	12.2553	0.406035	0.203018	+	0.979175i	$0.434925\pi$
$912$	0	0
$913$	54.9811	1.81961
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−10.7466	−0.354884
$918$	0	0
$919$	47.5663	1.56907	0.784533	−	0.620087i	$-0.212903\pi$
$919$	47.5663	1.56907	0.784533	+	0.620087i	$0.212903\pi$
$920$	0	0
$921$	−9.96812	−0.328461
$922$	0	0
$923$	−47.3593	−1.55885
$924$	0	0
$925$	5.51414	0.181304
$926$	0	0
$927$	3.76940	0.123803
$928$	0	0
$929$	23.5333	0.772104	0.386052	−	0.922477i	$-0.373839\pi$
$929$	23.5333	0.772104	0.386052	+	0.922477i	$0.373839\pi$
$930$	0	0
$931$	−44.9245	−1.47234
$932$	0	0
$933$	−15.9945	−0.523638
$934$	0	0
$935$	−6.64177	−0.217209
$936$	0	0
$937$	−56.7258	−1.85315	−0.926575	−	0.376109i	$-0.877262\pi$
$937$	−56.7258	−1.85315	−0.926575	+	0.376109i	$0.877262\pi$
$938$	0	0
$939$	−22.8542	−0.745819
$940$	0	0
$941$	−33.5333	−1.09316	−0.546578	−	0.837408i	$-0.684070\pi$
$941$	−33.5333	−1.09316	−0.546578	+	0.837408i	$0.684070\pi$
$942$	0	0
$943$	2.05655	0.0669704
$944$	0	0
$945$	0.485863	0.0158051
$946$	0	0
$947$	52.0685	1.69200	0.846000	−	0.533183i	$-0.179004\pi$
$947$	52.0685	1.69200	0.846000	+	0.533183i	$0.179004\pi$
$948$	0	0
$949$	44.0757	1.43076
$950$	0	0
$951$	14.6044	0.473581
$952$	0	0
$953$	−11.1896	−0.362468	−0.181234	−	0.983440i	$-0.558009\pi$
$953$	−11.1896	−0.362468	−0.181234	+	0.983440i	$0.558009\pi$
$954$	0	0
$955$	6.19325	0.200409
$956$	0	0
$957$	49.5953	1.60319
$958$	0	0
$959$	7.09575	0.229134
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	6.73566	0.217054
$964$	0	0
$965$	2.25526	0.0725995
$966$	0	0
$967$	−36.8296	−1.18436	−0.592179	−	0.805806i	$-0.701732\pi$
$967$	−36.8296	−1.18436	−0.592179	+	0.805806i	$0.701732\pi$
$968$	0	0
$969$	−8.77301	−0.281830
$970$	0	0
$971$	−18.7494	−0.601697	−0.300848	−	0.953672i	$-0.597270\pi$
$971$	−18.7494	−0.601697	−0.300848	+	0.953672i	$0.597270\pi$
$972$	0	0
$973$	6.26619	0.200885
$974$	0	0
$975$	3.51414	0.112542
$976$	0	0
$977$	−50.2262	−1.60688	−0.803439	−	0.595387i	$-0.796999\pi$
$977$	−50.2262	−1.60688	−0.803439	+	0.595387i	$0.796999\pi$
$978$	0	0
$979$	−25.3110	−0.808943
$980$	0	0
$981$	−3.90611	−0.124712
$982$	0	0
$983$	−15.5279	−0.495262	−0.247631	−	0.968854i	$-0.579652\pi$
$983$	−15.5279	−0.495262	−0.247631	+	0.968854i	$0.579652\pi$
$984$	0	0
$985$	−9.32088	−0.296988
$986$	0	0
$987$	−3.36916	−0.107242
$988$	0	0
$989$	−0.300884	−0.00956755
$990$	0	0
$991$	20.1022	0.638566	0.319283	−	0.947659i	$-0.396558\pi$
$991$	20.1022	0.638566	0.319283	+	0.947659i	$0.396558\pi$
$992$	0	0
$993$	8.16137	0.258993
$994$	0	0
$995$	16.5479	0.524603
$996$	0	0
$997$	−20.8186	−0.659333	−0.329666	−	0.944098i	$-0.606936\pi$
$997$	−20.8186	−0.659333	−0.329666	+	0.944098i	$0.606936\pi$
$998$	0	0
$999$	5.51414	0.174460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bs.1.3		3
4.3	odd	2		465.2.a.e.1.3	✓	3
12.11	even	2		1395.2.a.j.1.1		3
20.3	even	4		2325.2.c.k.1024.1		6
20.7	even	4		2325.2.c.k.1024.6		6
20.19	odd	2		2325.2.a.r.1.1		3
60.59	even	2		6975.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.3	✓	3	4.3	odd	2
1395.2.a.j.1.1		3	12.11	even	2
2325.2.a.r.1.1		3	20.19	odd	2
2325.2.c.k.1024.1		6	20.3	even	4
2325.2.c.k.1024.6		6	20.7	even	4
6975.2.a.bf.1.3		3	60.59	even	2
7440.2.a.bs.1.3		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.485863 q^{7} +1.00000 q^{9} -5.02827 q^{11} +3.51414 q^{13} -1.00000 q^{15} -1.32088 q^{17} +6.64177 q^{19} -0.485863 q^{21} -0.292611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.86330 q^{29} -1.00000 q^{31} -5.02827 q^{33} +0.485863 q^{35} +5.51414 q^{37} +3.51414 q^{39} -7.02827 q^{41} +1.02827 q^{43} -1.00000 q^{45} +6.93438 q^{47} -6.76394 q^{49} -1.32088 q^{51} -1.70739 q^{53} +5.02827 q^{55} +6.64177 q^{57} +2.19325 q^{59} -2.00000 q^{61} -0.485863 q^{63} -3.51414 q^{65} -9.12763 q^{67} -0.292611 q^{69} -13.4768 q^{71} +12.5424 q^{73} +1.00000 q^{75} +2.44305 q^{77} +0.349158 q^{79} +1.00000 q^{81} -10.9344 q^{83} +1.32088 q^{85} -9.86330 q^{87} +5.03374 q^{89} -1.70739 q^{91} -1.00000 q^{93} -6.64177 q^{95} +10.4431 q^{97} -5.02827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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