LMFDB - Embedded newform 9360.2.a.dc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.20147 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.20147 q^{7} -5.24943 q^{11} +1.00000 q^{13} +4.20147 q^{17} -0.952047 q^{19} -4.20147 q^{23} +1.00000 q^{25} +1.04795 q^{29} -2.00000 q^{31} +2.20147 q^{35} -1.15352 q^{37} -11.6524 q^{41} +4.40294 q^{43} -8.40294 q^{47} -2.15352 q^{49} -5.24943 q^{53} -5.24943 q^{55} +2.09591 q^{59} +5.15352 q^{61} +1.00000 q^{65} -10.4029 q^{67} +7.55646 q^{71} -5.35499 q^{73} -11.5565 q^{77} +5.65237 q^{79} +4.20147 q^{85} +7.65237 q^{89} +2.20147 q^{91} -0.952047 q^{95} +16.7003 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 3 * q^7	$ 3 q + 3 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{31} - 3 q^{35} + 3 q^{37} - 3 q^{41} - 6 q^{43} - 6 q^{47} - 3 q^{53} - 3 q^{55} + 9 q^{61} + 3 q^{65} - 12 q^{67} - 3 q^{71} - 9 q^{77} - 15 q^{79} + 3 q^{85} - 9 q^{89} - 3 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 - 3 * q^23 + 3 * q^25 - 6 * q^31 - 3 * q^35 + 3 * q^37 - 3 * q^41 - 6 * q^43 - 6 * q^47 - 3 * q^53 - 3 * q^55 + 9 * q^61 + 3 * q^65 - 12 * q^67 - 3 * q^71 - 9 * q^77 - 15 * q^79 + 3 * q^85 - 9 * q^89 - 3 * q^91 - 6 * q^95 + 15 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.20147	0.832078	0.416039	−	0.909347i	$-0.363418\pi$
$7$	2.20147	0.832078	0.416039	+	0.909347i	$0.363418\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.24943	−1.58276	−0.791381	−	0.611324i	$-0.790637\pi$
$11$	−5.24943	−1.58276	−0.791381	+	0.611324i	$0.790637\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.20147	1.01901	0.509503	−	0.860469i	$-0.329829\pi$
$17$	4.20147	1.01901	0.509503	+	0.860469i	$0.329829\pi$
$18$	0	0
$19$	−0.952047	−0.218415	−0.109207	−	0.994019i	$-0.534831\pi$
$19$	−0.952047	−0.218415	−0.109207	+	0.994019i	$0.534831\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.20147	−0.876068	−0.438034	−	0.898959i	$-0.644325\pi$
$23$	−4.20147	−0.876068	−0.438034	+	0.898959i	$0.644325\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.04795	0.194600	0.0973000	−	0.995255i	$-0.468979\pi$
$29$	1.04795	0.194600	0.0973000	+	0.995255i	$0.468979\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.20147	0.372117
$36$	0	0
$37$	−1.15352	−0.189637	−0.0948187	−	0.995495i	$-0.530227\pi$
$37$	−1.15352	−0.189637	−0.0948187	+	0.995495i	$0.530227\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.6524	−1.81979	−0.909897	−	0.414834i	$-0.863840\pi$
$41$	−11.6524	−1.81979	−0.909897	+	0.414834i	$0.863840\pi$
$42$	0	0
$43$	4.40294	0.671443	0.335721	−	0.941961i	$-0.391020\pi$
$43$	4.40294	0.671443	0.335721	+	0.941961i	$0.391020\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.40294	−1.22570	−0.612848	−	0.790201i	$-0.709976\pi$
$47$	−8.40294	−1.22570	−0.612848	+	0.790201i	$0.709976\pi$
$48$	0	0
$49$	−2.15352	−0.307646
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.24943	−0.721064	−0.360532	−	0.932747i	$-0.617405\pi$
$53$	−5.24943	−0.721064	−0.360532	+	0.932747i	$0.617405\pi$
$54$	0	0
$55$	−5.24943	−0.707832
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.09591	0.272864	0.136432	−	0.990649i	$-0.456437\pi$
$59$	2.09591	0.272864	0.136432	+	0.990649i	$0.456437\pi$
$60$	0	0
$61$	5.15352	0.659841	0.329920	−	0.944009i	$-0.392978\pi$
$61$	5.15352	0.659841	0.329920	+	0.944009i	$0.392978\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.4029	−1.27092	−0.635461	−	0.772133i	$-0.719190\pi$
$67$	−10.4029	−1.27092	−0.635461	+	0.772133i	$0.719190\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.55646	0.896787	0.448394	−	0.893836i	$-0.351996\pi$
$71$	7.55646	0.896787	0.448394	+	0.893836i	$0.351996\pi$
$72$	0	0
$73$	−5.35499	−0.626754	−0.313377	−	0.949629i	$-0.601460\pi$
$73$	−5.35499	−0.626754	−0.313377	+	0.949629i	$0.601460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.5565	−1.31698
$78$	0	0
$79$	5.65237	0.635941	0.317971	−	0.948101i	$-0.396999\pi$
$79$	5.65237	0.635941	0.317971	+	0.948101i	$0.396999\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.20147	0.455714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.65237	0.811150	0.405575	−	0.914062i	$-0.367071\pi$
$89$	7.65237	0.811150	0.405575	+	0.914062i	$0.367071\pi$
$90$	0	0
$91$	2.20147	0.230777
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.952047	−0.0976780
$96$	0	0
$97$	16.7003	1.69566	0.847830	−	0.530267i	$-0.177908\pi$
$97$	16.7003	1.69566	0.847830	+	0.530267i	$0.177908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.35499	−0.731849	−0.365925	−	0.930645i	$-0.619247\pi$
$101$	−7.35499	−0.731849	−0.365925	+	0.930645i	$0.619247\pi$
$102$	0	0
$103$	−10.3070	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$103$	−10.3070	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.5565	−1.50390	−0.751950	−	0.659220i	$-0.770887\pi$
$107$	−15.5565	−1.50390	−0.751950	+	0.659220i	$0.770887\pi$
$108$	0	0
$109$	3.25909	0.312164	0.156082	−	0.987744i	$-0.450114\pi$
$109$	3.25909	0.312164	0.156082	+	0.987744i	$0.450114\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.14386	−0.672038	−0.336019	−	0.941855i	$-0.609081\pi$
$113$	−7.14386	−0.672038	−0.336019	+	0.941855i	$0.609081\pi$
$114$	0	0
$115$	−4.20147	−0.391789
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.24943	0.847893
$120$	0	0
$121$	16.5565	1.50513
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.9018	−1.67726	−0.838632	−	0.544699i	$-0.816644\pi$
$127$	−18.9018	−1.67726	−0.838632	+	0.544699i	$0.816644\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.3550	−0.992090	−0.496045	−	0.868297i	$-0.665215\pi$
$131$	−11.3550	−0.992090	−0.496045	+	0.868297i	$0.665215\pi$
$132$	0	0
$133$	−2.09591	−0.181738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.40294	0.205297	0.102649	−	0.994718i	$-0.467268\pi$
$137$	2.40294	0.205297	0.102649	+	0.994718i	$0.467268\pi$
$138$	0	0
$139$	1.24943	0.105975	0.0529874	−	0.998595i	$-0.483126\pi$
$139$	1.24943	0.105975	0.0529874	+	0.998595i	$0.483126\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.24943	−0.438979
$144$	0	0
$145$	1.04795	0.0870277
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.55646	0.455203	0.227602	−	0.973754i	$-0.426912\pi$
$149$	5.55646	0.455203	0.227602	+	0.973754i	$0.426912\pi$
$150$	0	0
$151$	0.307039	0.0249865	0.0124932	−	0.999922i	$-0.496023\pi$
$151$	0.307039	0.0249865	0.0124932	+	0.999922i	$0.496023\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	1.59706	0.127459	0.0637294	−	0.997967i	$-0.479701\pi$
$157$	1.59706	0.127459	0.0637294	+	0.997967i	$0.479701\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.24943	−0.728957
$162$	0	0
$163$	−7.24943	−0.567819	−0.283909	−	0.958851i	$-0.591631\pi$
$163$	−7.24943	−0.567819	−0.283909	+	0.958851i	$0.591631\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.4989	−0.812425	−0.406213	−	0.913779i	$-0.633151\pi$
$167$	−10.4989	−0.812425	−0.406213	+	0.913779i	$0.633151\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.3070	−0.783630	−0.391815	−	0.920044i	$-0.628153\pi$
$173$	−10.3070	−0.783630	−0.391815	+	0.920044i	$0.628153\pi$
$174$	0	0
$175$	2.20147	0.166416
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.66203	0.722174	0.361087	−	0.932532i	$-0.382406\pi$
$179$	9.66203	0.722174	0.361087	+	0.932532i	$0.382406\pi$
$180$	0	0
$181$	−24.4583	−1.81797	−0.908984	−	0.416831i	$-0.863141\pi$
$181$	−24.4583	−1.81797	−0.908984	+	0.416831i	$0.863141\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.15352	−0.0848084
$186$	0	0
$187$	−22.0553	−1.61284
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.49885	0.470240	0.235120	−	0.971966i	$-0.424452\pi$
$191$	6.49885	0.470240	0.235120	+	0.971966i	$0.424452\pi$
$192$	0	0
$193$	24.9115	1.79317	0.896583	−	0.442876i	$-0.146042\pi$
$193$	24.9115	1.79317	0.896583	+	0.442876i	$0.146042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.49885	−0.605518	−0.302759	−	0.953067i	$-0.597908\pi$
$197$	−8.49885	−0.605518	−0.302759	+	0.953067i	$0.597908\pi$
$198$	0	0
$199$	−14.0959	−0.999232	−0.499616	−	0.866247i	$-0.666526\pi$
$199$	−14.0959	−0.999232	−0.499616	+	0.866247i	$0.666526\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.30704	0.161922
$204$	0	0
$205$	−11.6524	−0.813837
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.99770	0.345698
$210$	0	0
$211$	−27.7077	−1.90748	−0.953738	−	0.300640i	$-0.902800\pi$
$211$	−27.7077	−1.90748	−0.953738	+	0.300640i	$0.902800\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.40294	0.300278
$216$	0	0
$217$	−4.40294	−0.298891
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.20147	0.282622
$222$	0	0
$223$	7.85384	0.525932	0.262966	−	0.964805i	$-0.415299\pi$
$223$	7.85384	0.525932	0.262966	+	0.964805i	$0.415299\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−3.04795	−0.201414	−0.100707	−	0.994916i	$-0.532111\pi$
$229$	−3.04795	−0.201414	−0.100707	+	0.994916i	$0.532111\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.5062	0.753797	0.376899	−	0.926255i	$-0.376991\pi$
$233$	11.5062	0.753797	0.376899	+	0.926255i	$0.376991\pi$
$234$	0	0
$235$	−8.40294	−0.548148
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.94239	0.449066	0.224533	−	0.974467i	$-0.427914\pi$
$239$	6.94239	0.449066	0.224533	+	0.974467i	$0.427914\pi$
$240$	0	0
$241$	26.9977	1.73907	0.869537	−	0.493867i	$-0.164417\pi$
$241$	26.9977	1.73907	0.869537	+	0.493867i	$0.164417\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.15352	−0.137583
$246$	0	0
$247$	−0.952047	−0.0605773
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.64501	−0.545668	−0.272834	−	0.962061i	$-0.587961\pi$
$251$	−8.64501	−0.545668	−0.272834	+	0.962061i	$0.587961\pi$
$252$	0	0
$253$	22.0553	1.38661
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.6597	1.91250	0.956251	−	0.292548i	$-0.0945032\pi$
$257$	30.6597	1.91250	0.956251	+	0.292548i	$0.0945032\pi$
$258$	0	0
$259$	−2.53944	−0.157793
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.54680	−0.465356	−0.232678	−	0.972554i	$-0.574749\pi$
$263$	−7.54680	−0.465356	−0.232678	+	0.972554i	$0.574749\pi$
$264$	0	0
$265$	−5.24943	−0.322470
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.8538	−1.57634	−0.788168	−	0.615460i	$-0.788970\pi$
$269$	−25.8538	−1.57634	−0.788168	+	0.615460i	$0.788970\pi$
$270$	0	0
$271$	8.49885	0.516268	0.258134	−	0.966109i	$-0.416892\pi$
$271$	8.49885	0.516268	0.258134	+	0.966109i	$0.416892\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.24943	−0.316552
$276$	0	0
$277$	−17.7077	−1.06395	−0.531976	−	0.846760i	$-0.678550\pi$
$277$	−17.7077	−1.06395	−0.531976	+	0.846760i	$0.678550\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0959	−0.721581	−0.360791	−	0.932647i	$-0.617493\pi$
$281$	−12.0959	−0.721581	−0.360791	+	0.932647i	$0.617493\pi$
$282$	0	0
$283$	14.7100	0.874418	0.437209	−	0.899360i	$-0.355967\pi$
$283$	14.7100	0.874418	0.437209	+	0.899360i	$0.355967\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.6524	−1.51421
$288$	0	0
$289$	0.652370	0.0383747
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.8059	−1.56602	−0.783008	−	0.622011i	$-0.786316\pi$
$293$	−26.8059	−1.56602	−0.783008	+	0.622011i	$0.786316\pi$
$294$	0	0
$295$	2.09591	0.122028
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.20147	−0.242977
$300$	0	0
$301$	9.69296	0.558693
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.15352	0.295090
$306$	0	0
$307$	15.6524	0.893328	0.446664	−	0.894702i	$-0.352612\pi$
$307$	15.6524	0.893328	0.446664	+	0.894702i	$0.352612\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.1129	1.08379	0.541897	−	0.840445i	$-0.317706\pi$
$311$	19.1129	1.08379	0.541897	+	0.840445i	$0.317706\pi$
$312$	0	0
$313$	−7.20883	−0.407467	−0.203734	−	0.979026i	$-0.565308\pi$
$313$	−7.20883	−0.407467	−0.203734	+	0.979026i	$0.565308\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.80819	0.101558	0.0507790	−	0.998710i	$-0.483830\pi$
$317$	1.80819	0.101558	0.0507790	+	0.998710i	$0.483830\pi$
$318$	0	0
$319$	−5.50115	−0.308005
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.4989	−1.01987
$330$	0	0
$331$	16.0457	0.881949	0.440974	−	0.897520i	$-0.354633\pi$
$331$	16.0457	0.881949	0.440974	+	0.897520i	$0.354633\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.4029	−0.568374
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.4989	0.568545
$342$	0	0
$343$	−20.1512	−1.08806
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.9571	0.910305	0.455153	−	0.890413i	$-0.349585\pi$
$347$	16.9571	0.910305	0.455153	+	0.890413i	$0.349585\pi$
$348$	0	0
$349$	21.9497	1.17494	0.587472	−	0.809245i	$-0.300123\pi$
$349$	21.9497	1.17494	0.587472	+	0.809245i	$0.300123\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.09591	−0.430902	−0.215451	−	0.976515i	$-0.569122\pi$
$353$	−8.09591	−0.430902	−0.215451	+	0.976515i	$0.569122\pi$
$354$	0	0
$355$	7.55646	0.401055
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.191811	0.0101234	0.00506170	−	0.999987i	$-0.498389\pi$
$359$	0.191811	0.0101234	0.00506170	+	0.999987i	$0.498389\pi$
$360$	0	0
$361$	−18.0936	−0.952295
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.35499	−0.280293
$366$	0	0
$367$	14.2877	0.745813	0.372906	−	0.927869i	$-0.378361\pi$
$367$	14.2877	0.745813	0.372906	+	0.927869i	$0.378361\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.5565	−0.599982
$372$	0	0
$373$	21.0936	1.09219	0.546093	−	0.837725i	$-0.316115\pi$
$373$	21.0936	1.09219	0.546093	+	0.837725i	$0.316115\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.04795	0.0539723
$378$	0	0
$379$	−0.549103	−0.0282055	−0.0141028	−	0.999901i	$-0.504489\pi$
$379$	−0.549103	−0.0282055	−0.0141028	+	0.999901i	$0.504489\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−11.5565	−0.588972
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.24206	−0.215081	−0.107541	−	0.994201i	$-0.534298\pi$
$389$	−4.24206	−0.215081	−0.107541	+	0.994201i	$0.534298\pi$
$390$	0	0
$391$	−17.6524	−0.892719
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.65237	0.284402
$396$	0	0
$397$	−14.1512	−0.710229	−0.355115	−	0.934823i	$-0.615558\pi$
$397$	−14.1512	−0.710229	−0.355115	+	0.934823i	$0.615558\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.9018	−1.04379	−0.521893	−	0.853011i	$-0.674774\pi$
$401$	−20.9018	−1.04379	−0.521893	+	0.853011i	$0.674774\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.05531	0.300151
$408$	0	0
$409$	27.4006	1.35487	0.677437	−	0.735580i	$-0.263090\pi$
$409$	27.4006	1.35487	0.677437	+	0.735580i	$0.263090\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.61408	0.227044
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.8538	−0.676804	−0.338402	−	0.941002i	$-0.609886\pi$
$419$	−13.8538	−0.676804	−0.338402	+	0.941002i	$0.609886\pi$
$420$	0	0
$421$	−0.952047	−0.0463999	−0.0232000	−	0.999731i	$-0.507385\pi$
$421$	−0.952047	−0.0463999	−0.0232000	+	0.999731i	$0.507385\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.20147	0.203801
$426$	0	0
$427$	11.3453	0.549039
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.19181	0.201912	0.100956	−	0.994891i	$-0.467810\pi$
$431$	4.19181	0.201912	0.100956	+	0.994891i	$0.467810\pi$
$432$	0	0
$433$	3.50115	0.168255	0.0841273	−	0.996455i	$-0.473190\pi$
$433$	3.50115	0.168255	0.0841273	+	0.996455i	$0.473190\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−36.5542	−1.74464	−0.872318	−	0.488940i	$-0.837384\pi$
$439$	−36.5542	−1.74464	−0.872318	+	0.488940i	$0.837384\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.15122	0.197230	0.0986152	−	0.995126i	$-0.468559\pi$
$443$	4.15122	0.197230	0.0986152	+	0.995126i	$0.468559\pi$
$444$	0	0
$445$	7.65237	0.362757
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.4606	−1.10717	−0.553586	−	0.832792i	$-0.686741\pi$
$449$	−23.4606	−1.10717	−0.553586	+	0.832792i	$0.686741\pi$
$450$	0	0
$451$	61.1682	2.88030
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.20147	0.103207
$456$	0	0
$457$	3.70262	0.173201	0.0866007	−	0.996243i	$-0.472400\pi$
$457$	3.70262	0.173201	0.0866007	+	0.996243i	$0.472400\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.1512	0.659088	0.329544	−	0.944140i	$-0.393105\pi$
$461$	14.1512	0.659088	0.329544	+	0.944140i	$0.393105\pi$
$462$	0	0
$463$	29.1033	1.35254	0.676272	−	0.736652i	$-0.263595\pi$
$463$	29.1033	1.35254	0.676272	+	0.736652i	$0.263595\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.7483	−1.83933	−0.919665	−	0.392703i	$-0.871540\pi$
$467$	−39.7483	−1.83933	−0.919665	+	0.392703i	$0.871540\pi$
$468$	0	0
$469$	−22.9018	−1.05751
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−23.1129	−1.06273
$474$	0	0
$475$	−0.952047	−0.0436829
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.0576	1.14491	0.572456	−	0.819936i	$-0.305991\pi$
$479$	25.0576	1.14491	0.572456	+	0.819936i	$0.305991\pi$
$480$	0	0
$481$	−1.15352	−0.0525960
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.7003	0.758323
$486$	0	0
$487$	−17.1033	−0.775023	−0.387512	−	0.921865i	$-0.626665\pi$
$487$	−17.1033	−0.775023	−0.387512	+	0.921865i	$0.626665\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.8538	−1.52780	−0.763901	−	0.645333i	$-0.776719\pi$
$491$	−33.8538	−1.52780	−0.763901	+	0.645333i	$0.776719\pi$
$492$	0	0
$493$	4.40294	0.198299
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.6353	0.746197
$498$	0	0
$499$	−26.3527	−1.17971	−0.589854	−	0.807510i	$-0.700815\pi$
$499$	−26.3527	−1.17971	−0.589854	+	0.807510i	$0.700815\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.3550	−0.506294	−0.253147	−	0.967428i	$-0.581466\pi$
$503$	−11.3550	−0.506294	−0.253147	+	0.967428i	$0.581466\pi$
$504$	0	0
$505$	−7.35499	−0.327293
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.9594	−0.973334	−0.486667	−	0.873588i	$-0.661787\pi$
$509$	−21.9594	−0.973334	−0.486667	+	0.873588i	$0.661787\pi$
$510$	0	0
$511$	−11.7889	−0.521509
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.3070	−0.454182
$516$	0	0
$517$	44.1106	1.93998
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.79117	−0.122283	−0.0611416	−	0.998129i	$-0.519474\pi$
$521$	−2.79117	−0.122283	−0.0611416	+	0.998129i	$0.519474\pi$
$522$	0	0
$523$	11.1941	0.489484	0.244742	−	0.969588i	$-0.421297\pi$
$523$	11.1941	0.489484	0.244742	+	0.969588i	$0.421297\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.40294	−0.366038
$528$	0	0
$529$	−5.34763	−0.232506
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.6524	−0.504720
$534$	0	0
$535$	−15.5565	−0.672565
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.3047	0.486930
$540$	0	0
$541$	16.6597	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$541$	16.6597	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.25909	0.139604
$546$	0	0
$547$	19.1129	0.817210	0.408605	−	0.912711i	$-0.366015\pi$
$547$	19.1129	0.817210	0.408605	+	0.912711i	$0.366015\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.997701	−0.0425035
$552$	0	0
$553$	12.4435	0.529153
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.99770	−0.127017	−0.0635083	−	0.997981i	$-0.520229\pi$
$557$	−2.99770	−0.127017	−0.0635083	+	0.997981i	$0.520229\pi$
$558$	0	0
$559$	4.40294	0.186225
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.0723	−1.64670	−0.823351	−	0.567532i	$-0.807898\pi$
$563$	−39.0723	−1.64670	−0.823351	+	0.567532i	$0.807898\pi$
$564$	0	0
$565$	−7.14386	−0.300544
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.5971	−0.486174	−0.243087	−	0.970005i	$-0.578160\pi$
$569$	−11.5971	−0.486174	−0.243087	+	0.970005i	$0.578160\pi$
$570$	0	0
$571$	23.9401	1.00186	0.500931	−	0.865487i	$-0.332991\pi$
$571$	23.9401	1.00186	0.500931	+	0.865487i	$0.332991\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.20147	−0.175214
$576$	0	0
$577$	−26.3933	−1.09877	−0.549383	−	0.835570i	$-0.685137\pi$
$577$	−26.3933	−1.09877	−0.549383	+	0.835570i	$0.685137\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27.5565	1.14127
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	1.90409	0.0784568
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.3241	−1.20419	−0.602097	−	0.798423i	$-0.705668\pi$
$593$	−29.3241	−1.20419	−0.602097	+	0.798423i	$0.705668\pi$
$594$	0	0
$595$	9.24943	0.379189
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.1129	0.944369	0.472184	−	0.881500i	$-0.343466\pi$
$599$	23.1129	0.944369	0.472184	+	0.881500i	$0.343466\pi$
$600$	0	0
$601$	33.4560	1.36470	0.682349	−	0.731027i	$-0.260959\pi$
$601$	33.4560	1.36470	0.682349	+	0.731027i	$0.260959\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.5565	0.673116
$606$	0	0
$607$	21.9041	0.889060	0.444530	−	0.895764i	$-0.353371\pi$
$607$	21.9041	0.889060	0.444530	+	0.895764i	$0.353371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.40294	−0.339947
$612$	0	0
$613$	18.1512	0.733121	0.366560	−	0.930394i	$-0.380535\pi$
$613$	18.1512	0.733121	0.366560	+	0.930394i	$0.380535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.1918	0.893409	0.446704	−	0.894682i	$-0.352598\pi$
$617$	22.1918	0.893409	0.446704	+	0.894682i	$0.352598\pi$
$618$	0	0
$619$	17.9497	0.721461	0.360731	−	0.932670i	$-0.382527\pi$
$619$	17.9497	0.721461	0.360731	+	0.932670i	$0.382527\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.8465	0.674940
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.84648	−0.193242
$630$	0	0
$631$	−29.5159	−1.17501	−0.587504	−	0.809221i	$-0.699889\pi$
$631$	−29.5159	−1.17501	−0.587504	+	0.809221i	$0.699889\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.9018	−0.750095
$636$	0	0
$637$	−2.15352	−0.0853256
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.9977	−0.513378	−0.256689	−	0.966494i	$-0.582632\pi$
$641$	−12.9977	−0.513378	−0.256689	+	0.966494i	$0.582632\pi$
$642$	0	0
$643$	−5.15352	−0.203235	−0.101617	−	0.994824i	$-0.532402\pi$
$643$	−5.15352	−0.203235	−0.101617	+	0.994824i	$0.532402\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.6191	1.36102	0.680509	−	0.732739i	$-0.261759\pi$
$647$	34.6191	1.36102	0.680509	+	0.732739i	$0.261759\pi$
$648$	0	0
$649$	−11.0023	−0.431878
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.40294	−0.172301	−0.0861503	−	0.996282i	$-0.527457\pi$
$653$	−4.40294	−0.172301	−0.0861503	+	0.996282i	$0.527457\pi$
$654$	0	0
$655$	−11.3550	−0.443676
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.6574	−1.54483	−0.772417	−	0.635116i	$-0.780952\pi$
$659$	−39.6574	−1.54483	−0.772417	+	0.635116i	$0.780952\pi$
$660$	0	0
$661$	34.5445	1.34363	0.671813	−	0.740721i	$-0.265516\pi$
$661$	34.5445	1.34363	0.671813	+	0.740721i	$0.265516\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.09591	−0.0812757
$666$	0	0
$667$	−4.40294	−0.170483
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27.0530	−1.04437
$672$	0	0
$673$	−33.8995	−1.30673	−0.653365	−	0.757043i	$-0.726643\pi$
$673$	−33.8995	−1.30673	−0.653365	+	0.757043i	$0.726643\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.9571	−0.959180	−0.479590	−	0.877493i	$-0.659215\pi$
$677$	−24.9571	−0.959180	−0.479590	+	0.877493i	$0.659215\pi$
$678$	0	0
$679$	36.7653	1.41092
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.80589	0.336948	0.168474	−	0.985706i	$-0.446116\pi$
$683$	8.80589	0.336948	0.168474	+	0.985706i	$0.446116\pi$
$684$	0	0
$685$	2.40294	0.0918118
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.24943	−0.199987
$690$	0	0
$691$	−47.7533	−1.81662	−0.908311	−	0.418295i	$-0.862628\pi$
$691$	−47.7533	−1.81662	−0.908311	+	0.418295i	$0.862628\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.24943	0.0473934
$696$	0	0
$697$	−48.9571	−1.85438
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.4486	0.394638	0.197319	−	0.980339i	$-0.436777\pi$
$701$	10.4486	0.394638	0.197319	+	0.980339i	$0.436777\pi$
$702$	0	0
$703$	1.09821	0.0414196
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.1918	−0.608956
$708$	0	0
$709$	5.75794	0.216244	0.108122	−	0.994138i	$-0.465516\pi$
$709$	5.75794	0.216244	0.108122	+	0.994138i	$0.465516\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.40294	0.314693
$714$	0	0
$715$	−5.24943	−0.196317
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35.8995	−1.33883	−0.669413	−	0.742891i	$-0.733454\pi$
$719$	−35.8995	−1.33883	−0.669413	+	0.742891i	$0.733454\pi$
$720$	0	0
$721$	−22.6907	−0.845044
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.04795	0.0389200
$726$	0	0
$727$	8.01932	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$727$	8.01932	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.4989	0.684205
$732$	0	0
$733$	11.6524	0.430390	0.215195	−	0.976571i	$-0.430961\pi$
$733$	11.6524	0.430390	0.215195	+	0.976571i	$0.430961\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.6095	2.01157
$738$	0	0
$739$	−18.1609	−0.668059	−0.334029	−	0.942563i	$-0.608408\pi$
$739$	−18.1609	−0.668059	−0.334029	+	0.942563i	$0.608408\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.5136	−0.752570	−0.376285	−	0.926504i	$-0.622799\pi$
$743$	−20.5136	−0.752570	−0.376285	+	0.926504i	$0.622799\pi$
$744$	0	0
$745$	5.55646	0.203573
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−34.2471	−1.25136
$750$	0	0
$751$	17.6330	0.643439	0.321720	−	0.946835i	$-0.395739\pi$
$751$	17.6330	0.643439	0.321720	+	0.946835i	$0.395739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.307039	0.0111743
$756$	0	0
$757$	−50.9977	−1.85354	−0.926771	−	0.375626i	$-0.877428\pi$
$757$	−50.9977	−1.85354	−0.926771	+	0.375626i	$0.877428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.9018	0.467690	0.233845	−	0.972274i	$-0.424869\pi$
$761$	12.9018	0.467690	0.233845	+	0.972274i	$0.424869\pi$
$762$	0	0
$763$	7.17479	0.259745
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.09591	0.0756788
$768$	0	0
$769$	−39.9954	−1.44227	−0.721136	−	0.692794i	$-0.756380\pi$
$769$	−39.9954	−1.44227	−0.721136	+	0.692794i	$0.756380\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.11293	0.183899	0.0919496	−	0.995764i	$-0.470690\pi$
$773$	5.11293	0.183899	0.0919496	+	0.995764i	$0.470690\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.0936	0.397470
$780$	0	0
$781$	−39.6671	−1.41940
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.59706	0.0570013
$786$	0	0
$787$	−28.3070	−1.00904	−0.504518	−	0.863401i	$-0.668330\pi$
$787$	−28.3070	−1.00904	−0.504518	+	0.863401i	$0.668330\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.7270	−0.559188
$792$	0	0
$793$	5.15352	0.183007
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.4412	0.617800	0.308900	−	0.951094i	$-0.400039\pi$
$797$	17.4412	0.617800	0.308900	+	0.951094i	$0.400039\pi$
$798$	0	0
$799$	−35.3047	−1.24899
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.1106	0.992003
$804$	0	0
$805$	−9.24943	−0.325999
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.8013	−1.92671	−0.963355	−	0.268228i	$-0.913562\pi$
$809$	−54.8013	−1.92671	−0.963355	+	0.268228i	$0.913562\pi$
$810$	0	0
$811$	8.35729	0.293464	0.146732	−	0.989176i	$-0.453124\pi$
$811$	8.35729	0.293464	0.146732	+	0.989176i	$0.453124\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.24943	−0.253936
$816$	0	0
$817$	−4.19181	−0.146653
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.1705	−0.773757	−0.386879	−	0.922131i	$-0.626447\pi$
$821$	−22.1705	−0.773757	−0.386879	+	0.922131i	$0.626447\pi$
$822$	0	0
$823$	−41.8036	−1.45718	−0.728591	−	0.684949i	$-0.759824\pi$
$823$	−41.8036	−1.45718	−0.728591	+	0.684949i	$0.759824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.4989	−1.47783	−0.738915	−	0.673799i	$-0.764661\pi$
$827$	−42.4989	−1.47783	−0.738915	+	0.673799i	$0.764661\pi$
$828$	0	0
$829$	−30.6141	−1.06327	−0.531635	−	0.846973i	$-0.678422\pi$
$829$	−30.6141	−1.06327	−0.531635	+	0.846973i	$0.678422\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.04795	−0.313493
$834$	0	0
$835$	−10.4989	−0.363328
$836$	0	0
$837$	0	0
$838$	0	0
$839$	43.5565	1.50374	0.751868	−	0.659314i	$-0.229153\pi$
$839$	43.5565	1.50374	0.751868	+	0.659314i	$0.229153\pi$
$840$	0	0
$841$	−27.9018	−0.962131
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	36.4486	1.25239
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.84648	0.166135
$852$	0	0
$853$	−35.6524	−1.22071	−0.610357	−	0.792127i	$-0.708974\pi$
$853$	−35.6524	−1.22071	−0.610357	+	0.792127i	$0.708974\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.1969	−1.16814	−0.584071	−	0.811702i	$-0.698541\pi$
$857$	−34.1969	−1.16814	−0.584071	+	0.811702i	$0.698541\pi$
$858$	0	0
$859$	7.76760	0.265027	0.132514	−	0.991181i	$-0.457695\pi$
$859$	7.76760	0.265027	0.132514	+	0.991181i	$0.457695\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.1106	1.77387	0.886933	−	0.461898i	$-0.152831\pi$
$863$	52.1106	1.77387	0.886933	+	0.461898i	$0.152831\pi$
$864$	0	0
$865$	−10.3070	−0.350450
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29.6717	−1.00654
$870$	0	0
$871$	−10.4029	−0.352490
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.20147	0.0744233
$876$	0	0
$877$	18.3836	0.620771	0.310385	−	0.950611i	$-0.399542\pi$
$877$	18.3836	0.620771	0.310385	+	0.950611i	$0.399542\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.78887	−0.262414	−0.131207	−	0.991355i	$-0.541885\pi$
$881$	−7.78887	−0.262414	−0.131207	+	0.991355i	$0.541885\pi$
$882$	0	0
$883$	38.9211	1.30980	0.654900	−	0.755716i	$-0.272711\pi$
$883$	38.9211	1.30980	0.654900	+	0.755716i	$0.272711\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.2951	0.916479	0.458240	−	0.888829i	$-0.348480\pi$
$887$	27.2951	0.916479	0.458240	+	0.888829i	$0.348480\pi$
$888$	0	0
$889$	−41.6118	−1.39561
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	9.66203	0.322966
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.09591	−0.0699024
$900$	0	0
$901$	−22.0553	−0.734769
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.4583	−0.813020
$906$	0	0
$907$	−29.4006	−0.976232	−0.488116	−	0.872779i	$-0.662316\pi$
$907$	−29.4006	−0.976232	−0.488116	+	0.872779i	$0.662316\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.9018	1.55393	0.776963	−	0.629546i	$-0.216759\pi$
$911$	46.9018	1.55393	0.776963	+	0.629546i	$0.216759\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.9977	−0.825497
$918$	0	0
$919$	26.9424	0.888747	0.444374	−	0.895842i	$-0.353426\pi$
$919$	26.9424	0.888747	0.444374	+	0.895842i	$0.353426\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.55646	0.248724
$924$	0	0
$925$	−1.15352	−0.0379275
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.4606	0.507244	0.253622	−	0.967303i	$-0.418378\pi$
$929$	15.4606	0.507244	0.253622	+	0.967303i	$0.418378\pi$
$930$	0	0
$931$	2.05025	0.0671943
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22.0553	−0.721286
$936$	0	0
$937$	24.0147	0.784527	0.392263	−	0.919853i	$-0.371692\pi$
$937$	24.0147	0.784527	0.392263	+	0.919853i	$0.371692\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.8782	1.75638	0.878190	−	0.478312i	$-0.158751\pi$
$941$	53.8782	1.75638	0.878190	+	0.478312i	$0.158751\pi$
$942$	0	0
$943$	48.9571	1.59426
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.3070	−0.464916	−0.232458	−	0.972606i	$-0.574677\pi$
$947$	−14.3070	−0.464916	−0.232458	+	0.972606i	$0.574677\pi$
$948$	0	0
$949$	−5.35499	−0.173830
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17.8133	0.577028	0.288514	−	0.957476i	$-0.406839\pi$
$953$	17.8133	0.577028	0.288514	+	0.957476i	$0.406839\pi$
$954$	0	0
$955$	6.49885	0.210298
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.29002	0.170824
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.9115	0.801928
$966$	0	0
$967$	18.3720	0.590804	0.295402	−	0.955373i	$-0.404546\pi$
$967$	18.3720	0.590804	0.295402	+	0.955373i	$0.404546\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.6450	−0.534164	−0.267082	−	0.963674i	$-0.586059\pi$
$971$	−16.6450	−0.534164	−0.267082	+	0.963674i	$0.586059\pi$
$972$	0	0
$973$	2.75057	0.0881794
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.2900	1.00106	0.500528	−	0.865720i	$-0.333139\pi$
$977$	31.2900	1.00106	0.500528	+	0.865720i	$0.333139\pi$
$978$	0	0
$979$	−40.1705	−1.28386
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.7077	1.13890	0.569449	−	0.822027i	$-0.307157\pi$
$983$	35.7077	1.13890	0.569449	+	0.822027i	$0.307157\pi$
$984$	0	0
$985$	−8.49885	−0.270796
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.4989	−0.588229
$990$	0	0
$991$	18.9617	0.602339	0.301169	−	0.953571i	$-0.402623\pi$
$991$	18.9617	0.602339	0.301169	+	0.953571i	$0.402623\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.0959	−0.446870
$996$	0	0
$997$	13.8995	0.440201	0.220101	−	0.975477i	$-0.429361\pi$
$997$	13.8995	0.440201	0.220101	+	0.975477i	$0.429361\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.dc.1.3		3
3.2	odd	2		9360.2.a.cx.1.3		3
4.3	odd	2		4680.2.a.bk.1.1	yes	3
12.11	even	2		4680.2.a.bi.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4680.2.a.bi.1.1	✓	3	12.11	even	2
4680.2.a.bk.1.1	yes	3	4.3	odd	2
9360.2.a.cx.1.3		3	3.2	odd	2
9360.2.a.dc.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.20147 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.20147 q^{7} -5.24943 q^{11} +1.00000 q^{13} +4.20147 q^{17} -0.952047 q^{19} -4.20147 q^{23} +1.00000 q^{25} +1.04795 q^{29} -2.00000 q^{31} +2.20147 q^{35} -1.15352 q^{37} -11.6524 q^{41} +4.40294 q^{43} -8.40294 q^{47} -2.15352 q^{49} -5.24943 q^{53} -5.24943 q^{55} +2.09591 q^{59} +5.15352 q^{61} +1.00000 q^{65} -10.4029 q^{67} +7.55646 q^{71} -5.35499 q^{73} -11.5565 q^{77} +5.65237 q^{79} +4.20147 q^{85} +7.65237 q^{89} +2.20147 q^{91} -0.952047 q^{95} +16.7003 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 3 * q^7	\( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{31} - 3 q^{35} + 3 q^{37} - 3 q^{41} - 6 q^{43} - 6 q^{47} - 3 q^{53} - 3 q^{55} + 9 q^{61} + 3 q^{65} - 12 q^{67} - 3 q^{71} - 9 q^{77} - 15 q^{79} + 3 q^{85} - 9 q^{89} - 3 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 - 3 * q^23 + 3 * q^25 - 6 * q^31 - 3 * q^35 + 3 * q^37 - 3 * q^41 - 6 * q^43 - 6 * q^47 - 3 * q^53 - 3 * q^55 + 9 * q^61 + 3 * q^65 - 12 * q^67 - 3 * q^71 - 9 * q^77 - 15 * q^79 + 3 * q^85 - 9 * q^89 - 3 * q^91 - 6 * q^95 + 15 * q^97

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