LMFDB - Embedded newform 7800.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.461844.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} - 12x + 6 $ x^4 - x^3 - 14x^2 - 12x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.352593$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.647407 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.647407 q^{7} +1.00000 q^{9} +0.502467 q^{11} +1.00000 q^{13} -4.73074 q^{17} -6.58086 q^{19} +0.647407 q^{21} +2.85013 q^{23} +1.00000 q^{27} +0.294814 q^{29} +2.20272 q^{31} +0.502467 q^{33} +7.28605 q^{37} +1.00000 q^{39} +7.58580 q^{41} +0.444688 q^{43} +8.07840 q^{47} -6.58086 q^{49} -4.73074 q^{51} -12.7220 q^{53} -6.58086 q^{57} +10.8141 q^{59} +11.4359 q^{61} +0.647407 q^{63} -8.63864 q^{67} +2.85013 q^{69} -2.58086 q^{71} +10.6114 q^{73} +0.325301 q^{77} +5.28605 q^{79} +1.00000 q^{81} +5.35259 q^{83} +0.294814 q^{87} +4.99507 q^{89} +0.647407 q^{91} +2.20272 q^{93} -1.29481 q^{97} +0.502467 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} + 7 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} + 4 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - q^{37} + 4 q^{39} + 7 q^{41} + 6 q^{43} + 3 q^{49} + 7 q^{51} + 6 q^{53} + 3 q^{57} - 9 q^{59} + 19 q^{61} + 3 q^{63} - 4 q^{67} + 8 q^{69} + 19 q^{71} - 6 q^{73} - 17 q^{77} - 9 q^{79} + 4 q^{81} + 21 q^{83} + 2 q^{87} + 14 q^{89} + 3 q^{91} + 5 q^{93} - 6 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 + 5 * q^11 + 4 * q^13 + 7 * q^17 + 3 * q^19 + 3 * q^21 + 8 * q^23 + 4 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - q^37 + 4 * q^39 + 7 * q^41 + 6 * q^43 + 3 * q^49 + 7 * q^51 + 6 * q^53 + 3 * q^57 - 9 * q^59 + 19 * q^61 + 3 * q^63 - 4 * q^67 + 8 * q^69 + 19 * q^71 - 6 * q^73 - 17 * q^77 - 9 * q^79 + 4 * q^81 + 21 * q^83 + 2 * q^87 + 14 * q^89 + 3 * q^91 + 5 * q^93 - 6 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.647407	0.244697	0.122348	−	0.992487i	$-0.460957\pi$
$7$	0.647407	0.244697	0.122348	+	0.992487i	$0.460957\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.502467	0.151500	0.0757498	−	0.997127i	$-0.475865\pi$
$11$	0.502467	0.151500	0.0757498	+	0.997127i	$0.475865\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.73074	−1.14737	−0.573686	−	0.819075i	$-0.694487\pi$
$17$	−4.73074	−1.14737	−0.573686	+	0.819075i	$0.694487\pi$
$18$	0	0
$19$	−6.58086	−1.50975	−0.754877	−	0.655867i	$-0.772303\pi$
$19$	−6.58086	−1.50975	−0.754877	+	0.655867i	$0.772303\pi$
$20$	0	0
$21$	0.647407	0.141276
$22$	0	0
$23$	2.85013	0.594292	0.297146	−	0.954832i	$-0.403965\pi$
$23$	2.85013	0.594292	0.297146	+	0.954832i	$0.403965\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.294814	0.0547456	0.0273728	−	0.999625i	$-0.491286\pi$
$29$	0.294814	0.0547456	0.0273728	+	0.999625i	$0.491286\pi$
$30$	0	0
$31$	2.20272	0.395620	0.197810	−	0.980240i	$-0.436617\pi$
$31$	2.20272	0.395620	0.197810	+	0.980240i	$0.436617\pi$
$32$	0	0
$33$	0.502467	0.0874683
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.28605	1.19782	0.598910	−	0.800817i	$-0.295601\pi$
$37$	7.28605	1.19782	0.598910	+	0.800817i	$0.295601\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.58580	1.18470	0.592351	−	0.805680i	$-0.298200\pi$
$41$	7.58580	1.18470	0.592351	+	0.805680i	$0.298200\pi$
$42$	0	0
$43$	0.444688	0.0678143	0.0339072	−	0.999425i	$-0.489205\pi$
$43$	0.444688	0.0678143	0.0339072	+	0.999425i	$0.489205\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.07840	1.17836	0.589178	−	0.808004i	$-0.299452\pi$
$47$	8.07840	1.17836	0.589178	+	0.808004i	$0.299452\pi$
$48$	0	0
$49$	−6.58086	−0.940123
$50$	0	0
$51$	−4.73074	−0.662436
$52$	0	0
$53$	−12.7220	−1.74750	−0.873749	−	0.486377i	$-0.838318\pi$
$53$	−12.7220	−1.74750	−0.873749	+	0.486377i	$0.838318\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.58086	−0.871657
$58$	0	0
$59$	10.8141	1.40787	0.703936	−	0.710263i	$-0.251424\pi$
$59$	10.8141	1.40787	0.703936	+	0.710263i	$0.251424\pi$
$60$	0	0
$61$	11.4359	1.46422	0.732110	−	0.681186i	$-0.238536\pi$
$61$	11.4359	1.46422	0.732110	+	0.681186i	$0.238536\pi$
$62$	0	0
$63$	0.647407	0.0815656
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.63864	−1.05538	−0.527689	−	0.849438i	$-0.676941\pi$
$67$	−8.63864	−1.05538	−0.527689	+	0.849438i	$0.676941\pi$
$68$	0	0
$69$	2.85013	0.343115
$70$	0	0
$71$	−2.58086	−0.306292	−0.153146	−	0.988204i	$-0.548941\pi$
$71$	−2.58086	−0.306292	−0.153146	+	0.988204i	$0.548941\pi$
$72$	0	0
$73$	10.6114	1.24196	0.620982	−	0.783825i	$-0.286734\pi$
$73$	10.6114	1.24196	0.620982	+	0.783825i	$0.286734\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.325301	0.0370715
$78$	0	0
$79$	5.28605	0.594727	0.297364	−	0.954764i	$-0.403893\pi$
$79$	5.28605	0.594727	0.297364	+	0.954764i	$0.403893\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.35259	0.587523	0.293762	−	0.955879i	$-0.405093\pi$
$83$	5.35259	0.587523	0.293762	+	0.955879i	$0.405093\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.294814	0.0316074
$88$	0	0
$89$	4.99507	0.529476	0.264738	−	0.964320i	$-0.414715\pi$
$89$	4.99507	0.529476	0.264738	+	0.964320i	$0.414715\pi$
$90$	0	0
$91$	0.647407	0.0678667
$92$	0	0
$93$	2.20272	0.228411
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.29481	−0.131468	−0.0657342	−	0.997837i	$-0.520939\pi$
$97$	−1.29481	−0.131468	−0.0657342	+	0.997837i	$0.520939\pi$
$98$	0	0
$99$	0.502467	0.0504999
$100$	0	0
$101$	−4.28605	−0.426478	−0.213239	−	0.977000i	$-0.568401\pi$
$101$	−4.28605	−0.426478	−0.213239	+	0.977000i	$0.568401\pi$
$102$	0	0
$103$	7.25556	0.714912	0.357456	−	0.933930i	$-0.383644\pi$
$103$	7.25556	0.714912	0.357456	+	0.933930i	$0.383644\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1362	0.979901	0.489951	−	0.871750i	$-0.337015\pi$
$107$	10.1362	0.979901	0.489951	+	0.871750i	$0.337015\pi$
$108$	0	0
$109$	17.4527	1.67167	0.835833	−	0.548983i	$-0.184985\pi$
$109$	17.4527	1.67167	0.835833	+	0.548983i	$0.184985\pi$
$110$	0	0
$111$	7.28605	0.691561
$112$	0	0
$113$	15.8669	1.49263	0.746317	−	0.665591i	$-0.231820\pi$
$113$	15.8669	1.49263	0.746317	+	0.665591i	$0.231820\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−3.06271	−0.280758
$120$	0	0
$121$	−10.7475	−0.977048
$122$	0	0
$123$	7.58580	0.683988
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.99124	0.886579	0.443289	−	0.896379i	$-0.353811\pi$
$127$	9.99124	0.886579	0.443289	+	0.896379i	$0.353811\pi$
$128$	0	0
$129$	0.444688	0.0391526
$130$	0	0
$131$	16.2811	1.42249	0.711244	−	0.702945i	$-0.248132\pi$
$131$	16.2811	1.42249	0.711244	+	0.702945i	$0.248132\pi$
$132$	0	0
$133$	−4.26050	−0.369432
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.73567	0.404596	0.202298	−	0.979324i	$-0.435159\pi$
$137$	4.73567	0.404596	0.202298	+	0.979324i	$0.435159\pi$
$138$	0	0
$139$	−17.4615	−1.48106	−0.740532	−	0.672022i	$-0.765426\pi$
$139$	−17.4615	−1.48106	−0.740532	+	0.672022i	$0.765426\pi$
$140$	0	0
$141$	8.07840	0.680324
$142$	0	0
$143$	0.502467	0.0420184
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.58086	−0.542781
$148$	0	0
$149$	5.29481	0.433768	0.216884	−	0.976197i	$-0.430411\pi$
$149$	5.29481	0.433768	0.216884	+	0.976197i	$0.430411\pi$
$150$	0	0
$151$	18.5143	1.50667	0.753337	−	0.657635i	$-0.228443\pi$
$151$	18.5143	1.50667	0.753337	+	0.657635i	$0.228443\pi$
$152$	0	0
$153$	−4.73074	−0.382458
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.5809	−1.40311	−0.701553	−	0.712617i	$-0.747510\pi$
$157$	−17.5809	−1.40311	−0.701553	+	0.712617i	$0.747510\pi$
$158$	0	0
$159$	−12.7220	−1.00892
$160$	0	0
$161$	1.84519	0.145421
$162$	0	0
$163$	−10.0119	−0.784189	−0.392094	−	0.919925i	$-0.628249\pi$
$163$	−10.0119	−0.784189	−0.392094	+	0.919925i	$0.628249\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.69642	−0.672949	−0.336475	−	0.941693i	$-0.609235\pi$
$167$	−8.69642	−0.672949	−0.336475	+	0.941693i	$0.609235\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.58086	−0.503251
$172$	0	0
$173$	18.0069	1.36904	0.684520	−	0.728994i	$-0.260012\pi$
$173$	18.0069	1.36904	0.684520	+	0.728994i	$0.260012\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.8141	0.812835
$178$	0	0
$179$	−2.73457	−0.204391	−0.102196	−	0.994764i	$-0.532587\pi$
$179$	−2.73457	−0.204391	−0.102196	+	0.994764i	$0.532587\pi$
$180$	0	0
$181$	16.8757	1.25436	0.627180	−	0.778875i	$-0.284209\pi$
$181$	16.8757	1.25436	0.627180	+	0.778875i	$0.284209\pi$
$182$	0	0
$183$	11.4359	0.845368
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.37704	−0.173826
$188$	0	0
$189$	0.647407	0.0470919
$190$	0	0
$191$	12.5896	0.910954	0.455477	−	0.890248i	$-0.349469\pi$
$191$	12.5896	0.910954	0.455477	+	0.890248i	$0.349469\pi$
$192$	0	0
$193$	−20.0728	−1.44487	−0.722437	−	0.691437i	$-0.756978\pi$
$193$	−20.0728	−1.44487	−0.722437	+	0.691437i	$0.756978\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.6163	−1.11261	−0.556307	−	0.830977i	$-0.687782\pi$
$197$	−15.6163	−1.11261	−0.556307	+	0.830977i	$0.687782\pi$
$198$	0	0
$199$	−15.7307	−1.11512	−0.557561	−	0.830136i	$-0.688263\pi$
$199$	−15.7307	−1.11512	−0.557561	+	0.830136i	$0.688263\pi$
$200$	0	0
$201$	−8.63864	−0.609323
$202$	0	0
$203$	0.190865	0.0133961
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.85013	0.198097
$208$	0	0
$209$	−3.30667	−0.228727
$210$	0	0
$211$	6.85013	0.471582	0.235791	−	0.971804i	$-0.424232\pi$
$211$	6.85013	0.471582	0.235791	+	0.971804i	$0.424232\pi$
$212$	0	0
$213$	−2.58086	−0.176838
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.42606	0.0968070
$218$	0	0
$219$	10.6114	0.717049
$220$	0	0
$221$	−4.73074	−0.318224
$222$	0	0
$223$	−1.15974	−0.0776621	−0.0388311	−	0.999246i	$-0.512363\pi$
$223$	−1.15974	−0.0776621	−0.0388311	+	0.999246i	$0.512363\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.1207	−1.73369	−0.866847	−	0.498574i	$-0.833857\pi$
$227$	−26.1207	−1.73369	−0.866847	+	0.498574i	$0.833857\pi$
$228$	0	0
$229$	−4.02062	−0.265690	−0.132845	−	0.991137i	$-0.542411\pi$
$229$	−4.02062	−0.265690	−0.132845	+	0.991137i	$0.542411\pi$
$230$	0	0
$231$	0.325301	0.0214032
$232$	0	0
$233$	6.99507	0.458262	0.229131	−	0.973396i	$-0.426412\pi$
$233$	6.99507	0.458262	0.229131	+	0.973396i	$0.426412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.28605	0.343366
$238$	0	0
$239$	−15.4037	−0.996382	−0.498191	−	0.867067i	$-0.666002\pi$
$239$	−15.4037	−0.996382	−0.498191	+	0.867067i	$0.666002\pi$
$240$	0	0
$241$	5.14987	0.331733	0.165866	−	0.986148i	$-0.446958\pi$
$241$	5.14987	0.331733	0.165866	+	0.986148i	$0.446958\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.58086	−0.418730
$248$	0	0
$249$	5.35259	0.339207
$250$	0	0
$251$	17.5858	1.11001	0.555003	−	0.831848i	$-0.312717\pi$
$251$	17.5858	1.11001	0.555003	+	0.831848i	$0.312717\pi$
$252$	0	0
$253$	1.43209	0.0900350
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.3421	−0.957013	−0.478507	−	0.878084i	$-0.658822\pi$
$257$	−15.3421	−0.957013	−0.478507	+	0.878084i	$0.658822\pi$
$258$	0	0
$259$	4.71704	0.293103
$260$	0	0
$261$	0.294814	0.0182485
$262$	0	0
$263$	21.0168	1.29595	0.647975	−	0.761661i	$-0.275616\pi$
$263$	21.0168	1.29595	0.647975	+	0.761661i	$0.275616\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.99507	0.305693
$268$	0	0
$269$	−5.57210	−0.339737	−0.169868	−	0.985467i	$-0.554334\pi$
$269$	−5.57210	−0.339737	−0.169868	+	0.985467i	$0.554334\pi$
$270$	0	0
$271$	−5.50939	−0.334671	−0.167336	−	0.985900i	$-0.553516\pi$
$271$	−5.50939	−0.334671	−0.167336	+	0.985900i	$0.553516\pi$
$272$	0	0
$273$	0.647407	0.0391829
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.5672	−1.77652	−0.888259	−	0.459342i	$-0.848085\pi$
$277$	−29.5672	−1.77652	−0.888259	+	0.459342i	$0.848085\pi$
$278$	0	0
$279$	2.20272	0.131873
$280$	0	0
$281$	−5.14111	−0.306693	−0.153346	−	0.988172i	$-0.549005\pi$
$281$	−5.14111	−0.306693	−0.153346	+	0.988172i	$0.549005\pi$
$282$	0	0
$283$	14.3509	0.853070	0.426535	−	0.904471i	$-0.359734\pi$
$283$	14.3509	0.853070	0.426535	+	0.904471i	$0.359734\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.91110	0.289893
$288$	0	0
$289$	5.37989	0.316464
$290$	0	0
$291$	−1.29481	−0.0759033
$292$	0	0
$293$	−15.4516	−0.902693	−0.451346	−	0.892349i	$-0.649056\pi$
$293$	−15.4516	−0.902693	−0.451346	+	0.892349i	$0.649056\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.502467	0.0291561
$298$	0	0
$299$	2.85013	0.164827
$300$	0	0
$301$	0.287894	0.0165940
$302$	0	0
$303$	−4.28605	−0.246227
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.12432	−0.121241	−0.0606207	−	0.998161i	$-0.519308\pi$
$307$	−2.12432	−0.121241	−0.0606207	+	0.998161i	$0.519308\pi$
$308$	0	0
$309$	7.25556	0.412755
$310$	0	0
$311$	32.6331	1.85045	0.925226	−	0.379417i	$-0.123875\pi$
$311$	32.6331	1.85045	0.925226	+	0.379417i	$0.123875\pi$
$312$	0	0
$313$	18.7318	1.05879	0.529393	−	0.848377i	$-0.322420\pi$
$313$	18.7318	1.05879	0.529393	+	0.848377i	$0.322420\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.3452	1.47969	0.739846	−	0.672776i	$-0.234898\pi$
$317$	26.3452	1.47969	0.739846	+	0.672776i	$0.234898\pi$
$318$	0	0
$319$	0.148134	0.00829393
$320$	0	0
$321$	10.1362	0.565746
$322$	0	0
$323$	31.1323	1.73225
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.4527	0.965137
$328$	0	0
$329$	5.23001	0.288340
$330$	0	0
$331$	24.3116	1.33629	0.668143	−	0.744033i	$-0.267089\pi$
$331$	24.3116	1.33629	0.668143	+	0.744033i	$0.267089\pi$
$332$	0	0
$333$	7.28605	0.399273
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.72382	−0.0939024	−0.0469512	−	0.998897i	$-0.514951\pi$
$337$	−1.72382	−0.0939024	−0.0469512	+	0.998897i	$0.514951\pi$
$338$	0	0
$339$	15.8669	0.861772
$340$	0	0
$341$	1.10679	0.0599362
$342$	0	0
$343$	−8.79235	−0.474742
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.44086	0.184715	0.0923575	−	0.995726i	$-0.470560\pi$
$347$	3.44086	0.184715	0.0923575	+	0.995726i	$0.470560\pi$
$348$	0	0
$349$	12.2059	0.653368	0.326684	−	0.945134i	$-0.394069\pi$
$349$	12.2059	0.653368	0.326684	+	0.945134i	$0.394069\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−16.3967	−0.872707	−0.436353	−	0.899775i	$-0.643730\pi$
$353$	−16.3967	−0.872707	−0.436353	+	0.899775i	$0.643730\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.06271	−0.162096
$358$	0	0
$359$	0.511231	0.0269817	0.0134909	−	0.999909i	$-0.495706\pi$
$359$	0.511231	0.0269817	0.0134909	+	0.999909i	$0.495706\pi$
$360$	0	0
$361$	24.3078	1.27936
$362$	0	0
$363$	−10.7475	−0.564099
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.3028	−0.851001	−0.425501	−	0.904958i	$-0.639902\pi$
$367$	−16.3028	−0.851001	−0.425501	+	0.904958i	$0.639902\pi$
$368$	0	0
$369$	7.58580	0.394901
$370$	0	0
$371$	−8.23630	−0.427607
$372$	0	0
$373$	−32.3372	−1.67435	−0.837177	−	0.546932i	$-0.815796\pi$
$373$	−32.3372	−1.67435	−0.837177	+	0.546932i	$0.815796\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.294814	0.0151837
$378$	0	0
$379$	9.36445	0.481019	0.240510	−	0.970647i	$-0.422685\pi$
$379$	9.36445	0.481019	0.240510	+	0.970647i	$0.422685\pi$
$380$	0	0
$381$	9.99124	0.511867
$382$	0	0
$383$	−10.0423	−0.513140	−0.256570	−	0.966526i	$-0.582592\pi$
$383$	−10.0423	−0.513140	−0.256570	+	0.966526i	$0.582592\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.444688	0.0226048
$388$	0	0
$389$	35.8974	1.82007	0.910035	−	0.414531i	$-0.136054\pi$
$389$	35.8974	1.82007	0.910035	+	0.414531i	$0.136054\pi$
$390$	0	0
$391$	−13.4832	−0.681875
$392$	0	0
$393$	16.2811	0.821274
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−39.2979	−1.97231	−0.986153	−	0.165840i	$-0.946966\pi$
$397$	−39.2979	−1.97231	−0.986153	+	0.165840i	$0.946966\pi$
$398$	0	0
$399$	−4.26050	−0.213292
$400$	0	0
$401$	−2.69333	−0.134499	−0.0672493	−	0.997736i	$-0.521422\pi$
$401$	−2.69333	−0.134499	−0.0672493	+	0.997736i	$0.521422\pi$
$402$	0	0
$403$	2.20272	0.109725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.66100	0.181469
$408$	0	0
$409$	25.8963	1.28049	0.640245	−	0.768171i	$-0.278833\pi$
$409$	25.8963	1.28049	0.640245	+	0.768171i	$0.278833\pi$
$410$	0	0
$411$	4.73567	0.233594
$412$	0	0
$413$	7.00110	0.344502
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.4615	−0.855092
$418$	0	0
$419$	−19.0374	−0.930038	−0.465019	−	0.885301i	$-0.653953\pi$
$419$	−19.0374	−0.930038	−0.465019	+	0.885301i	$0.653953\pi$
$420$	0	0
$421$	22.5721	1.10010	0.550048	−	0.835133i	$-0.314609\pi$
$421$	22.5721	1.10010	0.550048	+	0.835133i	$0.314609\pi$
$422$	0	0
$423$	8.07840	0.392785
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.40370	0.358290
$428$	0	0
$429$	0.502467	0.0242593
$430$	0	0
$431$	−9.75246	−0.469760	−0.234880	−	0.972024i	$-0.575470\pi$
$431$	−9.75246	−0.469760	−0.234880	+	0.972024i	$0.575470\pi$
$432$	0	0
$433$	−8.14604	−0.391474	−0.195737	−	0.980656i	$-0.562710\pi$
$433$	−8.14604	−0.391474	−0.195737	+	0.980656i	$0.562710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−18.7563	−0.897235
$438$	0	0
$439$	0.870744	0.0415583	0.0207792	−	0.999784i	$-0.493385\pi$
$439$	0.870744	0.0415583	0.0207792	+	0.999784i	$0.493385\pi$
$440$	0	0
$441$	−6.58086	−0.313374
$442$	0	0
$443$	33.3470	1.58436	0.792182	−	0.610284i	$-0.208945\pi$
$443$	33.3470	1.58436	0.792182	+	0.610284i	$0.208945\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.29481	0.250436
$448$	0	0
$449$	38.0641	1.79635	0.898177	−	0.439634i	$-0.144892\pi$
$449$	38.0641	1.79635	0.898177	+	0.439634i	$0.144892\pi$
$450$	0	0
$451$	3.81161	0.179482
$452$	0	0
$453$	18.5143	0.869879
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.0938351	−0.00438942	−0.00219471	−	0.999998i	$-0.500699\pi$
$457$	−0.0938351	−0.00438942	−0.00219471	+	0.999998i	$0.500699\pi$
$458$	0	0
$459$	−4.73074	−0.220812
$460$	0	0
$461$	−31.1617	−1.45135	−0.725673	−	0.688040i	$-0.758472\pi$
$461$	−31.1617	−1.45135	−0.725673	+	0.688040i	$0.758472\pi$
$462$	0	0
$463$	−16.7748	−0.779592	−0.389796	−	0.920901i	$-0.627454\pi$
$463$	−16.7748	−0.779592	−0.389796	+	0.920901i	$0.627454\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.7594	1.14573	0.572864	−	0.819651i	$-0.305832\pi$
$467$	24.7594	1.14573	0.572864	+	0.819651i	$0.305832\pi$
$468$	0	0
$469$	−5.59272	−0.258248
$470$	0	0
$471$	−17.5809	−0.810083
$472$	0	0
$473$	0.223441	0.0102738
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−12.7220	−0.582499
$478$	0	0
$479$	16.8130	0.768204	0.384102	−	0.923291i	$-0.374511\pi$
$479$	16.8130	0.768204	0.384102	+	0.923291i	$0.374511\pi$
$480$	0	0
$481$	7.28605	0.332215
$482$	0	0
$483$	1.84519	0.0839591
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.78741	0.262253	0.131126	−	0.991366i	$-0.458141\pi$
$487$	5.78741	0.262253	0.131126	+	0.991366i	$0.458141\pi$
$488$	0	0
$489$	−10.0119	−0.452752
$490$	0	0
$491$	−0.650602	−0.0293612	−0.0146806	−	0.999892i	$-0.504673\pi$
$491$	−0.650602	−0.0293612	−0.0146806	+	0.999892i	$0.504673\pi$
$492$	0	0
$493$	−1.39469	−0.0628136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.67087	−0.0749487
$498$	0	0
$499$	−31.5300	−1.41148	−0.705738	−	0.708472i	$-0.749385\pi$
$499$	−31.5300	−1.41148	−0.705738	+	0.708472i	$0.749385\pi$
$500$	0	0
$501$	−8.69642	−0.388527
$502$	0	0
$503$	2.61135	0.116434	0.0582172	−	0.998304i	$-0.481458\pi$
$503$	2.61135	0.116434	0.0582172	+	0.998304i	$0.481458\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−2.27037	−0.100632	−0.0503161	−	0.998733i	$-0.516023\pi$
$509$	−2.27037	−0.100632	−0.0503161	+	0.998733i	$0.516023\pi$
$510$	0	0
$511$	6.86986	0.303905
$512$	0	0
$513$	−6.58086	−0.290552
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.05913	0.178520
$518$	0	0
$519$	18.0069	0.790416
$520$	0	0
$521$	2.87185	0.125818	0.0629090	−	0.998019i	$-0.479962\pi$
$521$	2.87185	0.125818	0.0629090	+	0.998019i	$0.479962\pi$
$522$	0	0
$523$	14.1469	0.618602	0.309301	−	0.950964i	$-0.399905\pi$
$523$	14.1469	0.618602	0.309301	+	0.950964i	$0.399905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.4205	−0.453924
$528$	0	0
$529$	−14.8768	−0.646817
$530$	0	0
$531$	10.8141	0.469291
$532$	0	0
$533$	7.58580	0.328577
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.73457	−0.118005
$538$	0	0
$539$	−3.30667	−0.142428
$540$	0	0
$541$	14.5504	0.625570	0.312785	−	0.949824i	$-0.398738\pi$
$541$	14.5504	0.625570	0.312785	+	0.949824i	$0.398738\pi$
$542$	0	0
$543$	16.8757	0.724205
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.5378	0.578834	0.289417	−	0.957203i	$-0.406539\pi$
$547$	13.5378	0.578834	0.289417	+	0.957203i	$0.406539\pi$
$548$	0	0
$549$	11.4359	0.488073
$550$	0	0
$551$	−1.94013	−0.0826524
$552$	0	0
$553$	3.42223	0.145528
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.20098	−0.135630	−0.0678149	−	0.997698i	$-0.521603\pi$
$557$	−3.20098	−0.135630	−0.0678149	+	0.997698i	$0.521603\pi$
$558$	0	0
$559$	0.444688	0.0188083
$560$	0	0
$561$	−2.37704	−0.100359
$562$	0	0
$563$	14.0088	0.590399	0.295200	−	0.955436i	$-0.404614\pi$
$563$	14.0088	0.590399	0.295200	+	0.955436i	$0.404614\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.647407	0.0271885
$568$	0	0
$569$	28.0571	1.17622	0.588108	−	0.808782i	$-0.299873\pi$
$569$	28.0571	1.17622	0.588108	+	0.808782i	$0.299873\pi$
$570$	0	0
$571$	43.4733	1.81930	0.909651	−	0.415373i	$-0.136349\pi$
$571$	43.4733	1.81930	0.909651	+	0.415373i	$0.136349\pi$
$572$	0	0
$573$	12.5896	0.525939
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.6281	−1.73300	−0.866501	−	0.499175i	$-0.833636\pi$
$577$	−41.6281	−1.73300	−0.866501	+	0.499175i	$0.833636\pi$
$578$	0	0
$579$	−20.0728	−0.834198
$580$	0	0
$581$	3.46531	0.143765
$582$	0	0
$583$	−6.39237	−0.264745
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.2489	−1.70252	−0.851262	−	0.524741i	$-0.824162\pi$
$587$	−41.2489	−1.70252	−0.851262	+	0.524741i	$0.824162\pi$
$588$	0	0
$589$	−14.4958	−0.597289
$590$	0	0
$591$	−15.6163	−0.642368
$592$	0	0
$593$	20.5416	0.843543	0.421771	−	0.906702i	$-0.361408\pi$
$593$	20.5416	0.843543	0.421771	+	0.906702i	$0.361408\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.7307	−0.643816
$598$	0	0
$599$	14.0316	0.573315	0.286658	−	0.958033i	$-0.407456\pi$
$599$	14.0316	0.573315	0.286658	+	0.958033i	$0.407456\pi$
$600$	0	0
$601$	−14.6610	−0.598035	−0.299017	−	0.954248i	$-0.596659\pi$
$601$	−14.6610	−0.598035	−0.299017	+	0.954248i	$0.596659\pi$
$602$	0	0
$603$	−8.63864	−0.351793
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.8875	0.888388	0.444194	−	0.895931i	$-0.353490\pi$
$607$	21.8875	0.888388	0.444194	+	0.895931i	$0.353490\pi$
$608$	0	0
$609$	0.190865	0.00773423
$610$	0	0
$611$	8.07840	0.326817
$612$	0	0
$613$	−24.0099	−0.969749	−0.484875	−	0.874584i	$-0.661135\pi$
$613$	−24.0099	−0.969749	−0.484875	+	0.874584i	$0.661135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.21468	0.0891595	0.0445798	−	0.999006i	$-0.485805\pi$
$617$	2.21468	0.0891595	0.0445798	+	0.999006i	$0.485805\pi$
$618$	0	0
$619$	12.4348	0.499798	0.249899	−	0.968272i	$-0.419603\pi$
$619$	12.4348	0.499798	0.249899	+	0.968272i	$0.419603\pi$
$620$	0	0
$621$	2.85013	0.114372
$622$	0	0
$623$	3.23384	0.129561
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.30667	−0.132056
$628$	0	0
$629$	−34.4684	−1.37434
$630$	0	0
$631$	26.4684	1.05369	0.526845	−	0.849961i	$-0.323375\pi$
$631$	26.4684	1.05369	0.526845	+	0.849961i	$0.323375\pi$
$632$	0	0
$633$	6.85013	0.272268
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.58086	−0.260743
$638$	0	0
$639$	−2.58086	−0.102097
$640$	0	0
$641$	−6.44852	−0.254701	−0.127351	−	0.991858i	$-0.540647\pi$
$641$	−6.44852	−0.254701	−0.127351	+	0.991858i	$0.540647\pi$
$642$	0	0
$643$	−2.01863	−0.0796071	−0.0398035	−	0.999208i	$-0.512673\pi$
$643$	−2.01863	−0.0796071	−0.0398035	+	0.999208i	$0.512673\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.142954	0.00562012	0.00281006	−	0.999996i	$-0.499106\pi$
$647$	0.142954	0.00562012	0.00281006	+	0.999996i	$0.499106\pi$
$648$	0	0
$649$	5.43371	0.213292
$650$	0	0
$651$	1.42606	0.0558915
$652$	0	0
$653$	−22.1823	−0.868062	−0.434031	−	0.900898i	$-0.642909\pi$
$653$	−22.1823	−0.868062	−0.434031	+	0.900898i	$0.642909\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.6114	0.413988
$658$	0	0
$659$	28.4989	1.11016	0.555079	−	0.831797i	$-0.312688\pi$
$659$	28.4989	1.11016	0.555079	+	0.831797i	$0.312688\pi$
$660$	0	0
$661$	−21.8680	−0.850567	−0.425284	−	0.905060i	$-0.639826\pi$
$661$	−21.8680	−0.850567	−0.425284	+	0.905060i	$0.639826\pi$
$662$	0	0
$663$	−4.73074	−0.183727
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.840257	0.0325349
$668$	0	0
$669$	−1.15974	−0.0448382
$670$	0	0
$671$	5.74618	0.221829
$672$	0	0
$673$	−10.0679	−0.388089	−0.194044	−	0.980993i	$-0.562161\pi$
$673$	−10.0679	−0.388089	−0.194044	+	0.980993i	$0.562161\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.7003	−1.06461	−0.532304	−	0.846554i	$-0.678674\pi$
$677$	−27.7003	−1.06461	−0.532304	+	0.846554i	$0.678674\pi$
$678$	0	0
$679$	−0.838272	−0.0321699
$680$	0	0
$681$	−26.1207	−1.00095
$682$	0	0
$683$	7.73631	0.296022	0.148011	−	0.988986i	$-0.452713\pi$
$683$	7.73631	0.296022	0.148011	+	0.988986i	$0.452713\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.02062	−0.153396
$688$	0	0
$689$	−12.7220	−0.484669
$690$	0	0
$691$	46.7115	1.77699	0.888494	−	0.458888i	$-0.151752\pi$
$691$	46.7115	1.77699	0.888494	+	0.458888i	$0.151752\pi$
$692$	0	0
$693$	0.325301	0.0123572
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−35.8864	−1.35930
$698$	0	0
$699$	6.99507	0.264578
$700$	0	0
$701$	−39.7670	−1.50198	−0.750990	−	0.660313i	$-0.770423\pi$
$701$	−39.7670	−1.50198	−0.750990	+	0.660313i	$0.770423\pi$
$702$	0	0
$703$	−47.9485	−1.80841
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.77482	−0.104358
$708$	0	0
$709$	−35.5988	−1.33694	−0.668470	−	0.743739i	$-0.733050\pi$
$709$	−35.5988	−1.33694	−0.668470	+	0.743739i	$0.733050\pi$
$710$	0	0
$711$	5.28605	0.198242
$712$	0	0
$713$	6.27803	0.235114
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.4037	−0.575262
$718$	0	0
$719$	38.6995	1.44325	0.721624	−	0.692285i	$-0.243396\pi$
$719$	38.6995	1.44325	0.721624	+	0.692285i	$0.243396\pi$
$720$	0	0
$721$	4.69730	0.174937
$722$	0	0
$723$	5.14987	0.191526
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.8788	−0.737263	−0.368631	−	0.929576i	$-0.620174\pi$
$727$	−19.8788	−0.737263	−0.368631	+	0.929576i	$0.620174\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.10370	−0.0778083
$732$	0	0
$733$	−14.2693	−0.527047	−0.263524	−	0.964653i	$-0.584885\pi$
$733$	−14.2693	−0.527047	−0.263524	+	0.964653i	$0.584885\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.34063	−0.159889
$738$	0	0
$739$	−12.3801	−0.455410	−0.227705	−	0.973730i	$-0.573122\pi$
$739$	−12.3801	−0.455410	−0.227705	+	0.973730i	$0.573122\pi$
$740$	0	0
$741$	−6.58086	−0.241754
$742$	0	0
$743$	−4.23321	−0.155301	−0.0776506	−	0.996981i	$-0.524742\pi$
$743$	−4.23321	−0.155301	−0.0776506	+	0.996981i	$0.524742\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.35259	0.195841
$748$	0	0
$749$	6.56223	0.239779
$750$	0	0
$751$	−20.6418	−0.753231	−0.376616	−	0.926370i	$-0.622912\pi$
$751$	−20.6418	−0.753231	−0.376616	+	0.926370i	$0.622912\pi$
$752$	0	0
$753$	17.5858	0.640862
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.7033	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$757$	28.7033	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$758$	0	0
$759$	1.43209	0.0519817
$760$	0	0
$761$	−6.53963	−0.237061	−0.118531	−	0.992950i	$-0.537818\pi$
$761$	−6.53963	−0.237061	−0.118531	+	0.992950i	$0.537818\pi$
$762$	0	0
$763$	11.2990	0.409052
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.8141	0.390473
$768$	0	0
$769$	7.58690	0.273591	0.136795	−	0.990599i	$-0.456320\pi$
$769$	7.58690	0.273591	0.136795	+	0.990599i	$0.456320\pi$
$770$	0	0
$771$	−15.3421	−0.552532
$772$	0	0
$773$	−20.7151	−0.745069	−0.372534	−	0.928018i	$-0.621511\pi$
$773$	−20.7151	−0.745069	−0.372534	+	0.928018i	$0.621511\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.71704	0.169223
$778$	0	0
$779$	−49.9211	−1.78861
$780$	0	0
$781$	−1.29680	−0.0464031
$782$	0	0
$783$	0.294814	0.0105358
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.0486	1.85533	0.927667	−	0.373410i	$-0.121811\pi$
$787$	52.0486	1.85533	0.927667	+	0.373410i	$0.121811\pi$
$788$	0	0
$789$	21.0168	0.748217
$790$	0	0
$791$	10.2724	0.365243
$792$	0	0
$793$	11.4359	0.406102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.1796	0.679377	0.339689	−	0.940538i	$-0.389678\pi$
$797$	19.1796	0.679377	0.339689	+	0.940538i	$0.389678\pi$
$798$	0	0
$799$	−38.2168	−1.35201
$800$	0	0
$801$	4.99507	0.176492
$802$	0	0
$803$	5.33186	0.188157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.57210	−0.196147
$808$	0	0
$809$	42.4803	1.49353	0.746763	−	0.665090i	$-0.231607\pi$
$809$	42.4803	1.49353	0.746763	+	0.665090i	$0.231607\pi$
$810$	0	0
$811$	−21.5094	−0.755297	−0.377648	−	0.925949i	$-0.623267\pi$
$811$	−21.5094	−0.755297	−0.377648	+	0.925949i	$0.623267\pi$
$812$	0	0
$813$	−5.50939	−0.193223
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.92643	−0.102383
$818$	0	0
$819$	0.647407	0.0226222
$820$	0	0
$821$	21.3067	0.743608	0.371804	−	0.928311i	$-0.378739\pi$
$821$	21.3067	0.743608	0.371804	+	0.928311i	$0.378739\pi$
$822$	0	0
$823$	7.24680	0.252608	0.126304	−	0.991992i	$-0.459689\pi$
$823$	7.24680	0.252608	0.126304	+	0.991992i	$0.459689\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43.7076	−1.51986	−0.759932	−	0.650003i	$-0.774768\pi$
$827$	−43.7076	−1.51986	−0.759932	+	0.650003i	$0.774768\pi$
$828$	0	0
$829$	−24.3665	−0.846285	−0.423142	−	0.906063i	$-0.639073\pi$
$829$	−24.3665	−0.846285	−0.423142	+	0.906063i	$0.639073\pi$
$830$	0	0
$831$	−29.5672	−1.02567
$832$	0	0
$833$	31.1323	1.07867
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.20272	0.0761371
$838$	0	0
$839$	−18.0553	−0.623338	−0.311669	−	0.950191i	$-0.600888\pi$
$839$	−18.0553	−0.623338	−0.311669	+	0.950191i	$0.600888\pi$
$840$	0	0
$841$	−28.9131	−0.997003
$842$	0	0
$843$	−5.14111	−0.177069
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.95802	−0.239081
$848$	0	0
$849$	14.3509	0.492520
$850$	0	0
$851$	20.7662	0.711855
$852$	0	0
$853$	−6.28679	−0.215256	−0.107628	−	0.994191i	$-0.534325\pi$
$853$	−6.28679	−0.215256	−0.107628	+	0.994191i	$0.534325\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.82074	0.164673	0.0823367	−	0.996605i	$-0.473762\pi$
$857$	4.82074	0.164673	0.0823367	+	0.996605i	$0.473762\pi$
$858$	0	0
$859$	7.32420	0.249898	0.124949	−	0.992163i	$-0.460123\pi$
$859$	7.32420	0.249898	0.124949	+	0.992163i	$0.460123\pi$
$860$	0	0
$861$	4.91110	0.167370
$862$	0	0
$863$	−14.0567	−0.478495	−0.239247	−	0.970959i	$-0.576901\pi$
$863$	−14.0567	−0.478495	−0.239247	+	0.970959i	$0.576901\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.37989	0.182710
$868$	0	0
$869$	2.65607	0.0901009
$870$	0	0
$871$	−8.63864	−0.292709
$872$	0	0
$873$	−1.29481	−0.0438228
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.6747	−0.427994	−0.213997	−	0.976834i	$-0.568648\pi$
$877$	−12.6747	−0.427994	−0.213997	+	0.976834i	$0.568648\pi$
$878$	0	0
$879$	−15.4516	−0.521170
$880$	0	0
$881$	−27.0325	−0.910747	−0.455374	−	0.890300i	$-0.650494\pi$
$881$	−27.0325	−0.910747	−0.455374	+	0.890300i	$0.650494\pi$
$882$	0	0
$883$	−8.14693	−0.274166	−0.137083	−	0.990560i	$-0.543773\pi$
$883$	−8.14693	−0.274166	−0.137083	+	0.990560i	$0.543773\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.4173	0.887006	0.443503	−	0.896273i	$-0.353736\pi$
$887$	26.4173	0.887006	0.443503	+	0.896273i	$0.353736\pi$
$888$	0	0
$889$	6.46840	0.216943
$890$	0	0
$891$	0.502467	0.0168333
$892$	0	0
$893$	−53.1628	−1.77903
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.85013	0.0951629
$898$	0	0
$899$	0.649392	0.0216585
$900$	0	0
$901$	60.1843	2.00503
$902$	0	0
$903$	0.287894	0.00958052
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.12815	0.236686	0.118343	−	0.992973i	$-0.462242\pi$
$907$	7.12815	0.236686	0.118343	+	0.992973i	$0.462242\pi$
$908$	0	0
$909$	−4.28605	−0.142159
$910$	0	0
$911$	17.1911	0.569567	0.284783	−	0.958592i	$-0.408078\pi$
$911$	17.1911	0.569567	0.284783	+	0.958592i	$0.408078\pi$
$912$	0	0
$913$	2.68950	0.0890095
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.5405	0.348078
$918$	0	0
$919$	3.35888	0.110799	0.0553996	−	0.998464i	$-0.482357\pi$
$919$	3.35888	0.110799	0.0553996	+	0.998464i	$0.482357\pi$
$920$	0	0
$921$	−2.12432	−0.0699988
$922$	0	0
$923$	−2.58086	−0.0849502
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.25556	0.238304
$928$	0	0
$929$	7.75048	0.254285	0.127142	−	0.991884i	$-0.459419\pi$
$929$	7.75048	0.254285	0.127142	+	0.991884i	$0.459419\pi$
$930$	0	0
$931$	43.3078	1.41935
$932$	0	0
$933$	32.6331	1.06836
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.75320	−0.187949	−0.0939744	−	0.995575i	$-0.529957\pi$
$937$	−5.75320	−0.187949	−0.0939744	+	0.995575i	$0.529957\pi$
$938$	0	0
$939$	18.7318	0.611291
$940$	0	0
$941$	1.43209	0.0466850	0.0233425	−	0.999728i	$-0.492569\pi$
$941$	1.43209	0.0466850	0.0233425	+	0.999728i	$0.492569\pi$
$942$	0	0
$943$	21.6205	0.704060
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6830	1.51699	0.758496	−	0.651677i	$-0.225934\pi$
$947$	46.6830	1.51699	0.758496	+	0.651677i	$0.225934\pi$
$948$	0	0
$949$	10.6114	0.344459
$950$	0	0
$951$	26.3452	0.854301
$952$	0	0
$953$	13.1145	0.424819	0.212409	−	0.977181i	$-0.431869\pi$
$953$	13.1145	0.424819	0.212409	+	0.977181i	$0.431869\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.148134	0.00478850
$958$	0	0
$959$	3.06591	0.0990033
$960$	0	0
$961$	−26.1480	−0.843485
$962$	0	0
$963$	10.1362	0.326634
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−23.4765	−0.754954	−0.377477	−	0.926019i	$-0.623208\pi$
$967$	−23.4765	−0.754954	−0.377477	+	0.926019i	$0.623208\pi$
$968$	0	0
$969$	31.1323	1.00012
$970$	0	0
$971$	57.3687	1.84105	0.920525	−	0.390683i	$-0.127761\pi$
$971$	57.3687	1.84105	0.920525	+	0.390683i	$0.127761\pi$
$972$	0	0
$973$	−11.3047	−0.362411
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.2961	1.67310	0.836550	−	0.547891i	$-0.184569\pi$
$977$	52.2961	1.67310	0.836550	+	0.547891i	$0.184569\pi$
$978$	0	0
$979$	2.50986	0.0802154
$980$	0	0
$981$	17.4527	0.557222
$982$	0	0
$983$	17.5074	0.558399	0.279200	−	0.960233i	$-0.409931\pi$
$983$	17.5074	0.558399	0.279200	+	0.960233i	$0.409931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.23001	0.166473
$988$	0	0
$989$	1.26742	0.0403015
$990$	0	0
$991$	−49.4527	−1.57092	−0.785459	−	0.618914i	$-0.787573\pi$
$991$	−49.4527	−1.57092	−0.785459	+	0.618914i	$0.787573\pi$
$992$	0	0
$993$	24.3116	0.771505
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−40.0493	−1.26837	−0.634186	−	0.773180i	$-0.718665\pi$
$997$	−40.0493	−1.26837	−0.634186	+	0.773180i	$0.718665\pi$
$998$	0	0
$999$	7.28605	0.230520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bx.1.2	yes	4
5.4	even	2		7800.2.a.bu.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bu.1.3	✓	4	5.4	even	2
7800.2.a.bx.1.2	yes	4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.647407 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.647407 q^{7} +1.00000 q^{9} +0.502467 q^{11} +1.00000 q^{13} -4.73074 q^{17} -6.58086 q^{19} +0.647407 q^{21} +2.85013 q^{23} +1.00000 q^{27} +0.294814 q^{29} +2.20272 q^{31} +0.502467 q^{33} +7.28605 q^{37} +1.00000 q^{39} +7.58580 q^{41} +0.444688 q^{43} +8.07840 q^{47} -6.58086 q^{49} -4.73074 q^{51} -12.7220 q^{53} -6.58086 q^{57} +10.8141 q^{59} +11.4359 q^{61} +0.647407 q^{63} -8.63864 q^{67} +2.85013 q^{69} -2.58086 q^{71} +10.6114 q^{73} +0.325301 q^{77} +5.28605 q^{79} +1.00000 q^{81} +5.35259 q^{83} +0.294814 q^{87} +4.99507 q^{89} +0.647407 q^{91} +2.20272 q^{93} -1.29481 q^{97} +0.502467 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} + 7 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} + 4 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - q^{37} + 4 q^{39} + 7 q^{41} + 6 q^{43} + 3 q^{49} + 7 q^{51} + 6 q^{53} + 3 q^{57} - 9 q^{59} + 19 q^{61} + 3 q^{63} - 4 q^{67} + 8 q^{69} + 19 q^{71} - 6 q^{73} - 17 q^{77} - 9 q^{79} + 4 q^{81} + 21 q^{83} + 2 q^{87} + 14 q^{89} + 3 q^{91} + 5 q^{93} - 6 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 + 5 * q^11 + 4 * q^13 + 7 * q^17 + 3 * q^19 + 3 * q^21 + 8 * q^23 + 4 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - q^37 + 4 * q^39 + 7 * q^41 + 6 * q^43 + 3 * q^49 + 7 * q^51 + 6 * q^53 + 3 * q^57 - 9 * q^59 + 19 * q^61 + 3 * q^63 - 4 * q^67 + 8 * q^69 + 19 * q^71 - 6 * q^73 - 17 * q^77 - 9 * q^79 + 4 * q^81 + 21 * q^83 + 2 * q^87 + 14 * q^89 + 3 * q^91 + 5 * q^93 - 6 * q^97 + 5 * q^99

Introduction

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