LMFDB - Embedded newform 96.2.d.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,2,Mod(49,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("96.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	96.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$0.766563859404$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	96.49
Dual form			96.2.d.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{13} -2.00000 q^{15} -2.00000 q^{17} -4.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} -2.00000 q^{31} +4.00000i q^{35} +8.00000i q^{37} +4.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +12.0000 q^{47} -3.00000 q^{49} -2.00000i q^{51} +6.00000i q^{53} +4.00000 q^{57} -4.00000i q^{59} -2.00000 q^{63} +8.00000 q^{65} +12.0000i q^{67} -4.00000i q^{69} -12.0000 q^{71} -6.00000 q^{73} +1.00000i q^{75} -10.0000 q^{79} +1.00000 q^{81} -16.0000i q^{83} -4.00000i q^{85} +6.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} -2.00000i q^{93} +8.00000 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^7 - 2 * q^9	$ 2 q + 4 q^{7} - 2 q^{9} - 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{31} + 8 q^{39} + 4 q^{41} + 24 q^{47} - 6 q^{49} + 8 q^{57} - 4 q^{63} + 16 q^{65} - 24 q^{71} - 12 q^{73} - 20 q^{79} + 2 q^{81} + 12 q^{87} - 20 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^9 - 4 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^31 + 8 * q^39 + 4 * q^41 + 24 * q^47 - 6 * q^49 + 8 * q^57 - 4 * q^63 + 16 * q^65 - 24 * q^71 - 12 * q^73 - 20 * q^79 + 2 * q^81 + 12 * q^87 - 20 * q^89 + 16 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	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
$35$	4.00000i	0.676123i
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	20.0000i	1.74741i	0.486458	+	0.873704i	$0.338289\pi$
$131$	20.0000i	1.74741i	−0.486458	+	0.873704i	$0.661711\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	2.00000	0.172133
$136$	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
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	8.00000i	0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$	8.00000i	0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	− 4.00000i	− 0.298974i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	− 4.00000i	− 0.298974i	0.988764	−	0.149487i	$-0.0477622\pi$
$180$	0	0
$181$	− 20.0000i	− 1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	− 20.0000i	− 1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	8.00000i	0.572892i
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	8.00000i	0.538138i
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	24.0000i	1.56559i
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	16.0000i	0.994192i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	8.00000i	0.473879i
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.0000i	0.925304i
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	6.00000i	0.341328i
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	0	0
$333$	− 8.00000i	− 0.438397i
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	− 8.00000i	− 0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$	− 8.00000i	− 0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$	0	0
$349$	− 16.0000i	− 0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$	− 16.0000i	− 0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	− 24.0000i	− 1.27379i
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	2.00000i	0.102463i
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	34.0000i	1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$	34.0000i	1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	− 20.0000i	− 1.00631i
$396$	0	0
$397$	− 32.0000i	− 1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$	− 32.0000i	− 1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	18.0000i	0.887875i
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	32.0000	1.57082
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	− 24.0000i	− 1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$	− 24.0000i	− 1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$	0	0
$421$	− 20.0000i	− 0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$	− 20.0000i	− 0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	16.0000i	0.765384i
$438$	0	0
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	− 20.0000i	− 0.948091i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	18.0000i	0.845714i
$454$	0	0
$455$	16.0000	0.750092
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	30.0000i	1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$	30.0000i	1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	24.0000i	1.10822i
$470$	0	0
$471$	−8.00000	−0.368621
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	20.0000i	0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$	20.0000i	0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	36.0000i	1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$	36.0000i	1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	− 8.00000i	− 0.357414i
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	12.0000i	0.528783i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	− 36.0000i	− 1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$	− 36.0000i	− 1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	0	0
$525$	2.00000i	0.0872872i
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20.0000i	0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$	20.0000i	0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	− 16.0000i	− 0.679162i
$556$	0	0
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	0	0
$579$	− 6.00000i	− 0.249351i
$580$	0	0
$581$	− 32.0000i	− 1.32758i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	− 8.00000i	− 0.327968i
$596$	0	0
$597$	− 10.0000i	− 0.409273i
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	22.0000i	0.894427i
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	− 48.0000i	− 1.94187i
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	36.0000i	1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$	36.0000i	1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.0000i	− 0.637962i
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	4.00000i	0.158735i
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	− 8.00000i	− 0.315000i
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 4.00000i	− 0.156772i
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	−40.0000	−1.56293
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	− 40.0000i	− 1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$	− 40.0000i	− 1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	− 14.0000i	− 0.541271i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	14.0000i	0.529529i
$700$	0	0
$701$	50.0000i	1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$	50.0000i	1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	4.00000i	0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$	4.00000i	0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 44.0000i	− 1.61857i	−0.587419	−	0.809283i	$-0.699856\pi$
$739$	− 44.0000i	− 1.61857i	0.587419	−	0.809283i	$-0.300144\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	24.0000i	0.876941i
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	36.0000i	1.31017i
$756$	0	0
$757$	− 12.0000i	− 0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$	− 12.0000i	− 0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	4.00000i	0.144620i
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	18.0000i	0.648254i
$772$	0	0
$773$	− 34.0000i	− 1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	− 34.0000i	− 1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	− 8.00000i	− 0.286630i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−16.0000	−0.571064
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	16.0000i	0.569615i
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 16.0000i	− 0.563926i
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	18.0000i	0.631288i
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	8.00000i	0.279543i
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	4.00000i	0.138926i	0.997585	+	0.0694629i	$0.0221285\pi$
$829$	4.00000i	0.138926i	−0.997585	+	0.0694629i	$0.977871\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	− 6.00000i	− 0.206406i
$846$	0	0
$847$	22.0000	0.755929
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 32.0000i	− 1.09695i
$852$	0	0
$853$	− 24.0000i	− 0.821744i	−0.911693	−	0.410872i	$-0.865224\pi$
$853$	− 24.0000i	− 0.821744i	0.911693	−	0.410872i	$-0.134776\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	− 44.0000i	− 1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$	− 44.0000i	− 1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$	0	0
$861$	4.00000i	0.136320i
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	8.00000i	0.268917i
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	40.0000	1.32964
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.0000i	1.32092i
$918$	0	0
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	48.0000i	1.57994i
$924$	0	0
$925$	8.00000i	0.263038i
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	12.0000i	0.393284i
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	14.0000i	0.456873i
$940$	0	0
$941$	30.0000i	0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$	30.0000i	0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	24.0000i	0.779073i
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	16.0000i	0.517748i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	− 12.0000i	− 0.386294i
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	0	0
$987$	24.0000i	0.763928i
$988$	0	0
$989$	− 16.0000i	− 0.508770i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	− 20.0000i	− 0.634043i
$996$	0	0
$997$	48.0000i	1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$	48.0000i	1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.d.a.49.2		2
3.2	odd	2		288.2.d.b.145.1		2
4.3	odd	2		24.2.d.a.13.1	✓	2
5.2	odd	4		2400.2.d.c.49.2		2
5.3	odd	4		2400.2.d.b.49.1		2
5.4	even	2		2400.2.k.a.1201.1		2
7.6	odd	2		4704.2.c.a.2353.1		2
8.3	odd	2		24.2.d.a.13.2	yes	2
8.5	even	2	inner	96.2.d.a.49.1		2
9.2	odd	6		2592.2.r.g.433.2		4
9.4	even	3		2592.2.r.f.2161.2		4
9.5	odd	6		2592.2.r.g.2161.1		4
9.7	even	3		2592.2.r.f.433.1		4
12.11	even	2		72.2.d.b.37.2		2
15.2	even	4		7200.2.d.d.2449.2		2
15.8	even	4		7200.2.d.g.2449.1		2
15.14	odd	2		7200.2.k.d.3601.2		2
16.3	odd	4		768.2.a.h.1.1		1
16.5	even	4		768.2.a.e.1.1		1
16.11	odd	4		768.2.a.a.1.1		1
16.13	even	4		768.2.a.d.1.1		1
20.3	even	4		600.2.d.b.349.2		2
20.7	even	4		600.2.d.c.349.1		2
20.19	odd	2		600.2.k.b.301.2		2
24.5	odd	2		288.2.d.b.145.2		2
24.11	even	2		72.2.d.b.37.1		2
28.27	even	2		1176.2.c.a.589.1		2
36.7	odd	6		648.2.n.k.109.1		4
36.11	even	6		648.2.n.c.109.2		4
36.23	even	6		648.2.n.c.541.1		4
36.31	odd	6		648.2.n.k.541.2		4
40.3	even	4		600.2.d.c.349.2		2
40.13	odd	4		2400.2.d.c.49.1		2
40.19	odd	2		600.2.k.b.301.1		2
40.27	even	4		600.2.d.b.349.1		2
40.29	even	2		2400.2.k.a.1201.2		2
40.37	odd	4		2400.2.d.b.49.2		2
48.5	odd	4		2304.2.a.l.1.1		1
48.11	even	4		2304.2.a.o.1.1		1
48.29	odd	4		2304.2.a.b.1.1		1
48.35	even	4		2304.2.a.e.1.1		1
56.13	odd	2		4704.2.c.a.2353.2		2
56.27	even	2		1176.2.c.a.589.2		2
60.23	odd	4		1800.2.d.i.1549.1		2
60.47	odd	4		1800.2.d.b.1549.2		2
60.59	even	2		1800.2.k.a.901.1		2
72.5	odd	6		2592.2.r.g.2161.2		4
72.11	even	6		648.2.n.c.109.1		4
72.13	even	6		2592.2.r.f.2161.1		4
72.29	odd	6		2592.2.r.g.433.1		4
72.43	odd	6		648.2.n.k.109.2		4
72.59	even	6		648.2.n.c.541.2		4
72.61	even	6		2592.2.r.f.433.2		4
72.67	odd	6		648.2.n.k.541.1		4
120.29	odd	2		7200.2.k.d.3601.1		2
120.53	even	4		7200.2.d.d.2449.1		2
120.59	even	2		1800.2.k.a.901.2		2
120.77	even	4		7200.2.d.g.2449.2		2
120.83	odd	4		1800.2.d.b.1549.1		2
120.107	odd	4		1800.2.d.i.1549.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.d.a.13.1	✓	2	4.3	odd	2
24.2.d.a.13.2	yes	2	8.3	odd	2
72.2.d.b.37.1		2	24.11	even	2
72.2.d.b.37.2		2	12.11	even	2
96.2.d.a.49.1		2	8.5	even	2	inner
96.2.d.a.49.2		2	1.1	even	1	trivial
288.2.d.b.145.1		2	3.2	odd	2
288.2.d.b.145.2		2	24.5	odd	2
600.2.d.b.349.1		2	40.27	even	4
600.2.d.b.349.2		2	20.3	even	4
600.2.d.c.349.1		2	20.7	even	4
600.2.d.c.349.2		2	40.3	even	4
600.2.k.b.301.1		2	40.19	odd	2
600.2.k.b.301.2		2	20.19	odd	2
648.2.n.c.109.1		4	72.11	even	6
648.2.n.c.109.2		4	36.11	even	6
648.2.n.c.541.1		4	36.23	even	6
648.2.n.c.541.2		4	72.59	even	6
648.2.n.k.109.1		4	36.7	odd	6
648.2.n.k.109.2		4	72.43	odd	6
648.2.n.k.541.1		4	72.67	odd	6
648.2.n.k.541.2		4	36.31	odd	6
768.2.a.a.1.1		1	16.11	odd	4
768.2.a.d.1.1		1	16.13	even	4
768.2.a.e.1.1		1	16.5	even	4
768.2.a.h.1.1		1	16.3	odd	4
1176.2.c.a.589.1		2	28.27	even	2
1176.2.c.a.589.2		2	56.27	even	2
1800.2.d.b.1549.1		2	120.83	odd	4
1800.2.d.b.1549.2		2	60.47	odd	4
1800.2.d.i.1549.1		2	60.23	odd	4
1800.2.d.i.1549.2		2	120.107	odd	4
1800.2.k.a.901.1		2	60.59	even	2
1800.2.k.a.901.2		2	120.59	even	2
2304.2.a.b.1.1		1	48.29	odd	4
2304.2.a.e.1.1		1	48.35	even	4
2304.2.a.l.1.1		1	48.5	odd	4
2304.2.a.o.1.1		1	48.11	even	4
2400.2.d.b.49.1		2	5.3	odd	4
2400.2.d.b.49.2		2	40.37	odd	4
2400.2.d.c.49.1		2	40.13	odd	4
2400.2.d.c.49.2		2	5.2	odd	4
2400.2.k.a.1201.1		2	5.4	even	2
2400.2.k.a.1201.2		2	40.29	even	2
2592.2.r.f.433.1		4	9.7	even	3
2592.2.r.f.433.2		4	72.61	even	6
2592.2.r.f.2161.1		4	72.13	even	6
2592.2.r.f.2161.2		4	9.4	even	3
2592.2.r.g.433.1		4	72.29	odd	6
2592.2.r.g.433.2		4	9.2	odd	6
2592.2.r.g.2161.1		4	9.5	odd	6
2592.2.r.g.2161.2		4	72.5	odd	6
4704.2.c.a.2353.1		2	7.6	odd	2
4704.2.c.a.2353.2		2	56.13	odd	2
7200.2.d.d.2449.1		2	120.53	even	4
7200.2.d.d.2449.2		2	15.2	even	4
7200.2.d.g.2449.1		2	15.8	even	4
7200.2.d.g.2449.2		2	120.77	even	4
7200.2.k.d.3601.1		2	120.29	odd	2
7200.2.k.d.3601.2		2	15.14	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:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	96.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{13} -2.00000 q^{15} -2.00000 q^{17} -4.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} -2.00000 q^{31} +4.00000i q^{35} +8.00000i q^{37} +4.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +12.0000 q^{47} -3.00000 q^{49} -2.00000i q^{51} +6.00000i q^{53} +4.00000 q^{57} -4.00000i q^{59} -2.00000 q^{63} +8.00000 q^{65} +12.0000i q^{67} -4.00000i q^{69} -12.0000 q^{71} -6.00000 q^{73} +1.00000i q^{75} -10.0000 q^{79} +1.00000 q^{81} -16.0000i q^{83} -4.00000i q^{85} +6.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} -2.00000i q^{93} +8.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^7 - 2 * q^9	\( 2 q + 4 q^{7} - 2 q^{9} - 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{31} + 8 q^{39} + 4 q^{41} + 24 q^{47} - 6 q^{49} + 8 q^{57} - 4 q^{63} + 16 q^{65} - 24 q^{71} - 12 q^{73} - 20 q^{79} + 2 q^{81} + 12 q^{87} - 20 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^9 - 4 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^31 + 8 * q^39 + 4 * q^41 + 24 * q^47 - 6 * q^49 + 8 * q^57 - 4 * q^63 + 16 * q^65 - 24 * q^71 - 12 * q^73 - 20 * q^79 + 2 * q^81 + 12 * q^87 - 20 * q^89 + 16 * q^95 - 4 * q^97

Introduction

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