LMFDB - Embedded newform 7920.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, 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("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.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:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{11} -1.46410 q^{13} +1.46410 q^{19} -6.92820 q^{23} +1.00000 q^{25} -3.46410 q^{29} -2.92820 q^{31} -2.00000 q^{35} +8.92820 q^{37} +3.46410 q^{41} -8.92820 q^{43} +6.92820 q^{47} -3.00000 q^{49} +12.9282 q^{53} -1.00000 q^{55} +6.92820 q^{59} +2.00000 q^{61} -1.46410 q^{65} -8.00000 q^{67} -13.8564 q^{71} +12.3923 q^{73} +2.00000 q^{77} +13.4641 q^{79} +15.4641 q^{83} +12.9282 q^{89} +2.92820 q^{91} +1.46410 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7} - 2 q^{11} + 4 q^{13} - 4 q^{19} + 2 q^{25} + 8 q^{31} - 4 q^{35} + 4 q^{37} - 4 q^{43} - 6 q^{49} + 12 q^{53} - 2 q^{55} + 4 q^{61} + 4 q^{65} - 16 q^{67} + 4 q^{73} + 4 q^{77}+ \cdots - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 4 * q^13 - 4 * q^19 + 2 * q^25 + 8 * q^31 - 4 * q^35 + 4 * q^37 - 4 * q^43 - 6 * q^49 + 12 * q^53 - 2 * q^55 + 4 * q^61 + 4 * q^65 - 16 * q^67 + 4 * q^73 + 4 * q^77 + 20 * q^79 + 24 * q^83 + 12 * q^89 - 8 * q^91 - 4 * q^95 - 20 * q^97

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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.46410	−0.181599
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	12.3923	1.45041	0.725205	−	0.688533i	$-0.241745\pi$
$73$	12.3923	1.45041	0.725205	+	0.688533i	$0.241745\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	13.4641	1.51483	0.757415	−	0.652934i	$-0.226462\pi$
$79$	13.4641	1.51483	0.757415	+	0.652934i	$0.226462\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	2.92820	0.306959
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.46410	0.150214
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4641	1.49497	0.747486	−	0.664278i	$-0.231261\pi$
$107$	15.4641	1.49497	0.747486	+	0.664278i	$0.231261\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	−6.92820	−0.646058
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.92820	0.437307	0.218654	−	0.975803i	$-0.429834\pi$
$127$	4.92820	0.437307	0.218654	+	0.975803i	$0.429834\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	0	0
$133$	−2.92820	−0.253907
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.46410	0.122434
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.53590	0.699288	0.349644	−	0.936883i	$-0.386303\pi$
$149$	8.53590	0.699288	0.349644	+	0.936883i	$0.386303\pi$
$150$	0	0
$151$	−0.392305	−0.0319253	−0.0159627	−	0.999873i	$-0.505081\pi$
$151$	−0.392305	−0.0319253	−0.0159627	+	0.999873i	$0.505081\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.92820	−0.235199
$156$	0	0
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.8564	1.09204
$162$	0	0
$163$	17.8564	1.39862	0.699311	−	0.714818i	$-0.253490\pi$
$163$	17.8564	1.39862	0.699311	+	0.714818i	$0.253490\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.92820	0.656415
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	0	0
$193$	3.60770	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$193$	3.60770	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.92820	0.486265
$204$	0	0
$205$	3.46410	0.241943
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.46410	−0.101274
$210$	0	0
$211$	−12.3923	−0.853121	−0.426561	−	0.904459i	$-0.640275\pi$
$211$	−12.3923	−0.853121	−0.426561	+	0.904459i	$0.640275\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.92820	−0.608898
$216$	0	0
$217$	5.85641	0.397559
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.8564	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.53590	0.566547	0.283274	−	0.959039i	$-0.408580\pi$
$227$	8.53590	0.566547	0.283274	+	0.959039i	$0.408580\pi$
$228$	0	0
$229$	3.85641	0.254839	0.127419	−	0.991849i	$-0.459331\pi$
$229$	3.85641	0.254839	0.127419	+	0.991849i	$0.459331\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	6.92820	0.451946
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−2.14359	−0.136394
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.8564	1.63204	0.816021	−	0.578022i	$-0.196175\pi$
$251$	25.8564	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.85641	−0.490069	−0.245035	−	0.969514i	$-0.578799\pi$
$257$	−7.85641	−0.490069	−0.245035	+	0.969514i	$0.578799\pi$
$258$	0	0
$259$	−17.8564	−1.10954
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.4641	1.69351	0.846755	−	0.531984i	$-0.178553\pi$
$263$	27.4641	1.69351	0.846755	+	0.531984i	$0.178553\pi$
$264$	0	0
$265$	12.9282	0.794173
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	0	0
$271$	32.3923	1.96769	0.983846	−	0.179016i	$-0.0572913\pi$
$271$	32.3923	1.96769	0.983846	+	0.179016i	$0.0572913\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.5359	1.35405	0.677025	−	0.735960i	$-0.263269\pi$
$277$	22.5359	1.35405	0.677025	+	0.735960i	$0.263269\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.46410	0.206651	0.103325	−	0.994648i	$-0.467052\pi$
$281$	3.46410	0.206651	0.103325	+	0.994648i	$0.467052\pi$
$282$	0	0
$283$	−8.92820	−0.530727	−0.265363	−	0.964148i	$-0.585492\pi$
$283$	−8.92820	−0.530727	−0.265363	+	0.964148i	$0.585492\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	6.92820	0.403376
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.1436	0.586619
$300$	0	0
$301$	17.8564	1.02923
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.9282	1.07332	0.536660	−	0.843799i	$-0.319686\pi$
$311$	18.9282	1.07332	0.536660	+	0.843799i	$0.319686\pi$
$312$	0	0
$313$	7.07180	0.399722	0.199861	−	0.979824i	$-0.435951\pi$
$313$	7.07180	0.399722	0.199861	+	0.979824i	$0.435951\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.0718	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$317$	11.0718	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$318$	0	0
$319$	3.46410	0.193952
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.8564	−0.763928
$330$	0	0
$331$	17.8564	0.981477	0.490738	−	0.871307i	$-0.336727\pi$
$331$	17.8564	0.981477	0.490738	+	0.871307i	$0.336727\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−29.1769	−1.58937	−0.794684	−	0.607023i	$-0.792363\pi$
$337$	−29.1769	−1.58937	−0.794684	+	0.607023i	$0.792363\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.92820	0.158571
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.60770	0.0863056	0.0431528	−	0.999068i	$-0.486260\pi$
$347$	1.60770	0.0863056	0.0431528	+	0.999068i	$0.486260\pi$
$348$	0	0
$349$	−35.8564	−1.91935	−0.959675	−	0.281113i	$-0.909296\pi$
$349$	−35.8564	−1.91935	−0.959675	+	0.281113i	$0.909296\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	−13.8564	−0.735422
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.3923	0.648643
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.8564	−1.34240
$372$	0	0
$373$	0.392305	0.0203128	0.0101564	−	0.999948i	$-0.496767\pi$
$373$	0.392305	0.0203128	0.0101564	+	0.999948i	$0.496767\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.07180	0.261211
$378$	0	0
$379$	−9.85641	−0.506290	−0.253145	−	0.967428i	$-0.581465\pi$
$379$	−9.85641	−0.506290	−0.253145	+	0.967428i	$0.581465\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.9282	−1.26391	−0.631955	−	0.775005i	$-0.717747\pi$
$389$	−24.9282	−1.26391	−0.631955	+	0.775005i	$0.717747\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.4641	0.677452
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.8564	−0.991582	−0.495791	−	0.868442i	$-0.665122\pi$
$401$	−19.8564	−0.991582	−0.495791	+	0.868442i	$0.665122\pi$
$402$	0	0
$403$	4.28719	0.213560
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.92820	−0.442555
$408$	0	0
$409$	34.7846	1.71999	0.859994	−	0.510304i	$-0.170467\pi$
$409$	34.7846	1.71999	0.859994	+	0.510304i	$0.170467\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.8564	−0.681829
$414$	0	0
$415$	15.4641	0.759103
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.0718	0.834012	0.417006	−	0.908904i	$-0.363079\pi$
$419$	17.0718	0.834012	0.417006	+	0.908904i	$0.363079\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.7846	−1.57918	−0.789590	−	0.613635i	$-0.789706\pi$
$431$	−32.7846	−1.57918	−0.789590	+	0.613635i	$0.789706\pi$
$432$	0	0
$433$	27.8564	1.33869	0.669347	−	0.742950i	$-0.266574\pi$
$433$	27.8564	1.33869	0.669347	+	0.742950i	$0.266574\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.1436	−0.485234
$438$	0	0
$439$	29.1769	1.39254	0.696269	−	0.717781i	$-0.254842\pi$
$439$	29.1769	1.39254	0.696269	+	0.717781i	$0.254842\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.7846	−0.697729	−0.348864	−	0.937173i	$-0.613433\pi$
$449$	−14.7846	−0.697729	−0.348864	+	0.937173i	$0.613433\pi$
$450$	0	0
$451$	−3.46410	−0.163118
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.92820	0.137276
$456$	0	0
$457$	−8.39230	−0.392575	−0.196288	−	0.980546i	$-0.562889\pi$
$457$	−8.39230	−0.392575	−0.196288	+	0.980546i	$0.562889\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.2487	0.570479	0.285240	−	0.958456i	$-0.407927\pi$
$461$	12.2487	0.570479	0.285240	+	0.958456i	$0.407927\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9282	0.875893	0.437946	−	0.899001i	$-0.355706\pi$
$467$	18.9282	0.875893	0.437946	+	0.899001i	$0.355706\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.92820	0.410519
$474$	0	0
$475$	1.46410	0.0671776
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−13.0718	−0.596023
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−23.7128	−1.07453	−0.537265	−	0.843413i	$-0.680543\pi$
$487$	−23.7128	−1.07453	−0.537265	+	0.843413i	$0.680543\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.0718	−0.770439	−0.385220	−	0.922825i	$-0.625874\pi$
$491$	−17.0718	−0.770439	−0.385220	+	0.922825i	$0.625874\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.7128	1.24309
$498$	0	0
$499$	12.7846	0.572318	0.286159	−	0.958182i	$-0.407621\pi$
$499$	12.7846	0.572318	0.286159	+	0.958182i	$0.407621\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.1769	−1.39011	−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.85641	0.348229	0.174115	−	0.984725i	$-0.444294\pi$
$509$	7.85641	0.348229	0.174115	+	0.984725i	$0.444294\pi$
$510$	0	0
$511$	−24.7846	−1.09641
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−6.92820	−0.304702
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.07180	−0.219684
$534$	0	0
$535$	15.4641	0.668571
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.07180	−0.216066
$552$	0	0
$553$	−26.9282	−1.14510
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.7846	−1.89758	−0.948792	−	0.315900i	$-0.897694\pi$
$557$	−44.7846	−1.89758	−0.948792	+	0.315900i	$0.897694\pi$
$558$	0	0
$559$	13.0718	0.552878
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.3923	−0.437983	−0.218992	−	0.975727i	$-0.570277\pi$
$563$	−10.3923	−0.437983	−0.218992	+	0.975727i	$0.570277\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	0	0
$571$	−24.3923	−1.02079	−0.510393	−	0.859941i	$-0.670500\pi$
$571$	−24.3923	−1.02079	−0.510393	+	0.859941i	$0.670500\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	22.7846	0.948536	0.474268	−	0.880381i	$-0.342713\pi$
$577$	22.7846	0.948536	0.474268	+	0.880381i	$0.342713\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.9282	−1.28312
$582$	0	0
$583$	−12.9282	−0.535431
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.07180	−0.209335	−0.104668	−	0.994507i	$-0.533378\pi$
$587$	−5.07180	−0.209335	−0.104668	+	0.994507i	$0.533378\pi$
$588$	0	0
$589$	−4.28719	−0.176650
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.7846	1.34630	0.673151	−	0.739505i	$-0.264940\pi$
$593$	32.7846	1.34630	0.673151	+	0.739505i	$0.264940\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.1436	−0.414456	−0.207228	−	0.978293i	$-0.566444\pi$
$599$	−10.1436	−0.414456	−0.207228	+	0.978293i	$0.566444\pi$
$600$	0	0
$601$	36.6410	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$601$	36.6410	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−22.7846	−0.924799	−0.462399	−	0.886672i	$-0.653011\pi$
$607$	−22.7846	−0.924799	−0.462399	+	0.886672i	$0.653011\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.1436	−0.410366
$612$	0	0
$613$	0.392305	0.0158450	0.00792252	−	0.999969i	$-0.497478\pi$
$613$	0.392305	0.0158450	0.00792252	+	0.999969i	$0.497478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.0718	0.928836	0.464418	−	0.885616i	$-0.346264\pi$
$617$	23.0718	0.928836	0.464418	+	0.885616i	$0.346264\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.8564	−1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	34.9282	1.39047	0.695235	−	0.718783i	$-0.255300\pi$
$631$	34.9282	1.39047	0.695235	+	0.718783i	$0.255300\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.92820	0.195570
$636$	0	0
$637$	4.39230	0.174029
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0	0
$643$	45.5692	1.79707	0.898537	−	0.438897i	$-0.144631\pi$
$643$	45.5692	1.79707	0.898537	+	0.438897i	$0.144631\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.7128	−1.08950	−0.544752	−	0.838597i	$-0.683376\pi$
$647$	−27.7128	−1.08950	−0.544752	+	0.838597i	$0.683376\pi$
$648$	0	0
$649$	−6.92820	−0.271956
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	0	0
$655$	5.07180	0.198171
$656$	0	0
$657$	0	0
$658$	0	0
$659$	39.7128	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$659$	39.7128	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.92820	−0.113551
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	31.3205	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$673$	31.3205	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.7846	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$677$	32.7846	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.78461	0.336134	0.168067	−	0.985776i	$-0.446248\pi$
$683$	8.78461	0.336134	0.168067	+	0.985776i	$0.446248\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.9282	−0.721107
$690$	0	0
$691$	7.71281	0.293409	0.146705	−	0.989180i	$-0.453133\pi$
$691$	7.71281	0.293409	0.146705	+	0.989180i	$0.453133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.39230	0.318338
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.5359	−1.22886	−0.614432	−	0.788970i	$-0.710615\pi$
$701$	−32.5359	−1.22886	−0.614432	+	0.788970i	$0.710615\pi$
$702$	0	0
$703$	13.0718	0.493012
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	15.8564	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$709$	15.8564	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.2872	0.759761
$714$	0	0
$715$	1.46410	0.0547543
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.9282	−0.705903	−0.352951	−	0.935642i	$-0.614822\pi$
$719$	−18.9282	−0.705903	−0.352951	+	0.935642i	$0.614822\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−49.9615	−1.84537	−0.922686	−	0.385553i	$-0.874011\pi$
$733$	−49.9615	−1.84537	−0.922686	+	0.385553i	$0.874011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−10.5359	−0.387569	−0.193785	−	0.981044i	$-0.562076\pi$
$739$	−10.5359	−0.387569	−0.193785	+	0.981044i	$0.562076\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.3923	1.70197	0.850984	−	0.525191i	$-0.176006\pi$
$743$	46.3923	1.70197	0.850984	+	0.525191i	$0.176006\pi$
$744$	0	0
$745$	8.53590	0.312731
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−30.9282	−1.13009
$750$	0	0
$751$	−13.0718	−0.476997	−0.238498	−	0.971143i	$-0.576655\pi$
$751$	−13.0718	−0.476997	−0.238498	+	0.971143i	$0.576655\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.392305	−0.0142774
$756$	0	0
$757$	−6.78461	−0.246591	−0.123295	−	0.992370i	$-0.539346\pi$
$757$	−6.78461	−0.246591	−0.123295	+	0.992370i	$0.539346\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.4641	1.43057	0.715286	−	0.698832i	$-0.246296\pi$
$761$	39.4641	1.43057	0.715286	+	0.698832i	$0.246296\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.1436	−0.366264
$768$	0	0
$769$	−46.4974	−1.67674	−0.838370	−	0.545102i	$-0.816491\pi$
$769$	−46.4974	−1.67674	−0.838370	+	0.545102i	$0.816491\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.8564	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$773$	31.8564	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$774$	0	0
$775$	−2.92820	−0.105184
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.07180	0.181716
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.9282	−0.604193
$786$	0	0
$787$	18.7846	0.669599	0.334800	−	0.942289i	$-0.391331\pi$
$787$	18.7846	0.669599	0.334800	+	0.942289i	$0.391331\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.85641	0.0660062
$792$	0	0
$793$	−2.92820	−0.103984
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.6410	−0.589455	−0.294728	−	0.955581i	$-0.595229\pi$
$797$	−16.6410	−0.589455	−0.294728	+	0.955581i	$0.595229\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12.3923	−0.437315
$804$	0	0
$805$	13.8564	0.488374
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.53590	0.300106	0.150053	−	0.988678i	$-0.452056\pi$
$809$	8.53590	0.300106	0.150053	+	0.988678i	$0.452056\pi$
$810$	0	0
$811$	8.39230	0.294694	0.147347	−	0.989085i	$-0.452927\pi$
$811$	8.39230	0.294694	0.147347	+	0.989085i	$0.452927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.8564	0.625483
$816$	0	0
$817$	−13.0718	−0.457324
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.4641	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$821$	−27.4641	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$822$	0	0
$823$	−49.5692	−1.72787	−0.863937	−	0.503600i	$-0.832009\pi$
$823$	−49.5692	−1.72787	−0.863937	+	0.503600i	$0.832009\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.60770	0.0559050	0.0279525	−	0.999609i	$-0.491101\pi$
$827$	1.60770	0.0559050	0.0279525	+	0.999609i	$0.491101\pi$
$828$	0	0
$829$	−25.7128	−0.893043	−0.446521	−	0.894773i	$-0.647337\pi$
$829$	−25.7128	−0.893043	−0.446521	+	0.894773i	$0.647337\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	10.3923	0.359641
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.2154	0.525294	0.262647	−	0.964892i	$-0.415405\pi$
$839$	15.2154	0.525294	0.262647	+	0.964892i	$0.415405\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−61.8564	−2.12041
$852$	0	0
$853$	24.3923	0.835177	0.417588	−	0.908636i	$-0.362875\pi$
$853$	24.3923	0.835177	0.417588	+	0.908636i	$0.362875\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.1436	0.346499	0.173249	−	0.984878i	$-0.444573\pi$
$857$	10.1436	0.346499	0.173249	+	0.984878i	$0.444573\pi$
$858$	0	0
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.1436	−0.345292	−0.172646	−	0.984984i	$-0.555232\pi$
$863$	−10.1436	−0.345292	−0.172646	+	0.984984i	$0.555232\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.4641	−0.456738
$870$	0	0
$871$	11.7128	0.396874
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	14.2487	0.481145	0.240572	−	0.970631i	$-0.422665\pi$
$877$	14.2487	0.481145	0.240572	+	0.970631i	$0.422665\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.9282	−0.435562	−0.217781	−	0.975998i	$-0.569882\pi$
$881$	−12.9282	−0.435562	−0.217781	+	0.975998i	$0.569882\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.2487	−1.21711	−0.608556	−	0.793511i	$-0.708251\pi$
$887$	−36.2487	−1.21711	−0.608556	+	0.793511i	$0.708251\pi$
$888$	0	0
$889$	−9.85641	−0.330573
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.1436	0.339442
$894$	0	0
$895$	−6.92820	−0.231584
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.1436	0.338308
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.8564	−0.394120
$906$	0	0
$907$	−45.8564	−1.52264	−0.761318	−	0.648378i	$-0.775448\pi$
$907$	−45.8564	−1.52264	−0.761318	+	0.648378i	$0.775448\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.07180	0.168036	0.0840181	−	0.996464i	$-0.473225\pi$
$911$	5.07180	0.168036	0.0840181	+	0.996464i	$0.473225\pi$
$912$	0	0
$913$	−15.4641	−0.511787
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.1436	−0.334971
$918$	0	0
$919$	11.6077	0.382903	0.191451	−	0.981502i	$-0.438681\pi$
$919$	11.6077	0.382903	0.191451	+	0.981502i	$0.438681\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.2872	0.667761
$924$	0	0
$925$	8.92820	0.293558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.7846	1.27248	0.636241	−	0.771490i	$-0.280488\pi$
$929$	38.7846	1.27248	0.636241	+	0.771490i	$0.280488\pi$
$930$	0	0
$931$	−4.39230	−0.143952
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.392305	0.0128160	0.00640802	−	0.999979i	$-0.497960\pi$
$937$	0.392305	0.0128160	0.00640802	+	0.999979i	$0.497960\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.5359	−0.669451	−0.334726	−	0.942316i	$-0.608644\pi$
$941$	−20.5359	−0.669451	−0.334726	+	0.942316i	$0.608644\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.07180	−0.164811	−0.0824056	−	0.996599i	$-0.526260\pi$
$947$	−5.07180	−0.164811	−0.0824056	+	0.996599i	$0.526260\pi$
$948$	0	0
$949$	−18.1436	−0.588966
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.7846	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$953$	44.7846	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$954$	0	0
$955$	−5.07180	−0.164119
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−22.4256	−0.723407
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.60770	0.116136
$966$	0	0
$967$	18.7846	0.604072	0.302036	−	0.953296i	$-0.402334\pi$
$967$	18.7846	0.604072	0.302036	+	0.953296i	$0.402334\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.85641	−0.0595749	−0.0297875	−	0.999556i	$-0.509483\pi$
$971$	−1.85641	−0.0595749	−0.0297875	+	0.999556i	$0.509483\pi$
$972$	0	0
$973$	−16.7846	−0.538090
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.5692	1.13796	0.568980	−	0.822351i	$-0.307338\pi$
$977$	35.5692	1.13796	0.568980	+	0.822351i	$0.307338\pi$
$978$	0	0
$979$	−12.9282	−0.413187
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	61.8564	1.96692
$990$	0	0
$991$	48.7846	1.54969	0.774847	−	0.632149i	$-0.217827\pi$
$991$	48.7846	1.54969	0.774847	+	0.632149i	$0.217827\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.7846	−0.532108
$996$	0	0
$997$	48.3923	1.53260	0.766300	−	0.642483i	$-0.222096\pi$
$997$	48.3923	1.53260	0.766300	+	0.642483i	$0.222096\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.bz.1.1		2
3.2	odd	2		2640.2.a.x.1.1		2
4.3	odd	2		495.2.a.c.1.1		2
12.11	even	2		165.2.a.b.1.2	✓	2
20.3	even	4		2475.2.c.n.199.3		4
20.7	even	4		2475.2.c.n.199.2		4
20.19	odd	2		2475.2.a.r.1.2		2
44.43	even	2		5445.2.a.s.1.2		2
60.23	odd	4		825.2.c.c.199.2		4
60.47	odd	4		825.2.c.c.199.3		4
60.59	even	2		825.2.a.e.1.1		2
84.83	odd	2		8085.2.a.bd.1.2		2
132.131	odd	2		1815.2.a.i.1.1		2
660.659	odd	2		9075.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	12.11	even	2
495.2.a.c.1.1		2	4.3	odd	2
825.2.a.e.1.1		2	60.59	even	2
825.2.c.c.199.2		4	60.23	odd	4
825.2.c.c.199.3		4	60.47	odd	4
1815.2.a.i.1.1		2	132.131	odd	2
2475.2.a.r.1.2		2	20.19	odd	2
2475.2.c.n.199.2		4	20.7	even	4
2475.2.c.n.199.3		4	20.3	even	4
2640.2.a.x.1.1		2	3.2	odd	2
5445.2.a.s.1.2		2	44.43	even	2
7920.2.a.bz.1.1		2	1.1	even	1	trivial
8085.2.a.bd.1.2		2	84.83	odd	2
9075.2.a.bh.1.2		2	660.659	odd	2

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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{11} -1.46410 q^{13} +1.46410 q^{19} -6.92820 q^{23} +1.00000 q^{25} -3.46410 q^{29} -2.92820 q^{31} -2.00000 q^{35} +8.92820 q^{37} +3.46410 q^{41} -8.92820 q^{43} +6.92820 q^{47} -3.00000 q^{49} +12.9282 q^{53} -1.00000 q^{55} +6.92820 q^{59} +2.00000 q^{61} -1.46410 q^{65} -8.00000 q^{67} -13.8564 q^{71} +12.3923 q^{73} +2.00000 q^{77} +13.4641 q^{79} +15.4641 q^{83} +12.9282 q^{89} +2.92820 q^{91} +1.46410 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} - 2 q^{11} + 4 q^{13} - 4 q^{19} + 2 q^{25} + 8 q^{31} - 4 q^{35} + 4 q^{37} - 4 q^{43} - 6 q^{49} + 12 q^{53} - 2 q^{55} + 4 q^{61} + 4 q^{65} - 16 q^{67} + 4 q^{73} + 4 q^{77}+ \cdots - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 4 * q^13 - 4 * q^19 + 2 * q^25 + 8 * q^31 - 4 * q^35 + 4 * q^37 - 4 * q^43 - 6 * q^49 + 12 * q^53 - 2 * q^55 + 4 * q^61 + 4 * q^65 - 16 * q^67 + 4 * q^73 + 4 * q^77 + 20 * q^79 + 24 * q^83 + 12 * q^89 - 8 * q^91 - 4 * q^95 - 20 * q^97

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