LMFDB - Embedded newform 2352.2.b.e.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1567,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.b (of order $2$, degree $1$, 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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.e.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +3.46410i q^{11} -1.73205i q^{13} +3.46410i q^{15} -5.00000 q^{19} +6.92820i q^{23} -7.00000 q^{25} +1.00000 q^{27} +5.00000 q^{31} +3.46410i q^{33} -11.0000 q^{37} -1.73205i q^{39} -3.46410i q^{41} +8.66025i q^{43} +3.46410i q^{45} +6.00000 q^{47} +12.0000 q^{53} -12.0000 q^{55} -5.00000 q^{57} -12.0000 q^{59} -13.8564i q^{61} +6.00000 q^{65} -8.66025i q^{67} +6.92820i q^{69} +3.46410i q^{71} +5.19615i q^{73} -7.00000 q^{75} +12.1244i q^{79} +1.00000 q^{81} -18.0000 q^{83} +6.92820i q^{89} +5.00000 q^{93} -17.3205i q^{95} +6.92820i q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 10 q^{19} - 14 q^{25} + 2 q^{27} + 10 q^{31} - 22 q^{37} + 12 q^{47} + 24 q^{53} - 24 q^{55} - 10 q^{57} - 24 q^{59} + 12 q^{65} - 14 q^{75} + 2 q^{81} - 36 q^{83} + 10 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 10 * q^19 - 14 * q^25 + 2 * q^27 + 10 * q^31 - 22 * q^37 + 12 * q^47 + 24 * q^53 - 24 * q^55 - 10 * q^57 - 24 * q^59 + 12 * q^65 - 14 * q^75 + 2 * q^81 - 36 * q^83 + 10 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.46410i	1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$	3.46410i	1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410i	1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$	3.46410i	1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	− 1.73205i	− 0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	− 1.73205i	− 0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$	0	0
$15$	3.46410i	0.894427i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.92820i	1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$	6.92820i	1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	3.46410i	0.603023i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	− 1.73205i	− 0.277350i
$40$	0	0
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	0	0
$43$	8.66025i	1.32068i	0.750968	+	0.660338i	$0.229587\pi$
$43$	8.66025i	1.32068i	−0.750968	+	0.660338i	$0.770413\pi$
$44$	0	0
$45$	3.46410i	0.516398i
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	− 13.8564i	− 1.77413i	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	− 13.8564i	− 1.77413i	0.461644	−	0.887066i	$-0.347260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	− 8.66025i	− 1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$	− 8.66025i	− 1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$	0	0
$69$	6.92820i	0.834058i
$70$	0	0
$71$	3.46410i	0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$	3.46410i	0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$	0	0
$73$	5.19615i	0.608164i	0.952646	+	0.304082i	$0.0983496\pi$
$73$	5.19615i	0.608164i	−0.952646	+	0.304082i	$0.901650\pi$
$74$	0	0
$75$	−7.00000	−0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.1244i	1.36410i	0.731307	+	0.682048i	$0.238911\pi$
$79$	12.1244i	1.36410i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.92820i	0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$	6.92820i	0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	− 17.3205i	− 1.77705i
$96$	0	0
$97$	6.92820i	0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$	6.92820i	0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	0	0
$99$	3.46410i	0.348155i
$100$	0	0
$101$	− 10.3923i	− 1.03407i	−0.855963	−	0.517036i	$-0.827035\pi$
$101$	− 10.3923i	− 1.03407i	0.855963	−	0.517036i	$-0.172965\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	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
$115$	−24.0000	−2.23801
$116$	0	0
$117$	− 1.73205i	− 0.160128i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 3.46410i	− 0.312348i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	12.1244i	1.07586i	0.842989	+	0.537931i	$0.180794\pi$
$127$	12.1244i	1.07586i	−0.842989	+	0.537931i	$0.819206\pi$
$128$	0	0
$129$	8.66025i	0.762493i
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.46410i	0.298142i
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	− 3.46410i	− 0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$	− 3.46410i	− 0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.3205i	1.39122i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.46410i	− 0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	− 3.46410i	− 0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	10.0000	0.769231
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	6.92820i	0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$	6.92820i	0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	10.3923i	0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$	10.3923i	0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$	0	0
$181$	− 5.19615i	− 0.386227i	−0.981176	−	0.193113i	$-0.938141\pi$
$181$	− 5.19615i	− 0.386227i	0.981176	−	0.193113i	$-0.0618586\pi$
$182$	0	0
$183$	− 13.8564i	− 1.02430i
$184$	0	0
$185$	− 38.1051i	− 2.80154i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.3205i	1.25327i	0.779314	+	0.626634i	$0.215568\pi$
$191$	17.3205i	1.25327i	−0.779314	+	0.626634i	$0.784432\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	− 8.66025i	− 0.610847i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	6.92820i	0.481543i
$208$	0	0
$209$	− 17.3205i	− 1.19808i
$210$	0	0
$211$	3.46410i	0.238479i	0.992866	+	0.119239i	$0.0380456\pi$
$211$	3.46410i	0.238479i	−0.992866	+	0.119239i	$0.961954\pi$
$212$	0	0
$213$	3.46410i	0.237356i
$214$	0	0
$215$	−30.0000	−2.04598
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.19615i	0.351123i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−7.00000	−0.466667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	8.66025i	0.572286i	0.958187	+	0.286143i	$0.0923732\pi$
$229$	8.66025i	0.572286i	−0.958187	+	0.286143i	$0.907627\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	20.7846i	1.35584i
$236$	0	0
$237$	12.1244i	0.787562i
$238$	0	0
$239$	10.3923i	0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$	10.3923i	0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$	0	0
$241$	13.8564i	0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$	13.8564i	0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.66025i	0.551039i
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 17.3205i	− 1.08042i	−0.841529	−	0.540212i	$-0.818344\pi$
$257$	− 17.3205i	− 1.08042i	0.841529	−	0.540212i	$-0.181656\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 13.8564i	− 0.854423i	−0.904152	−	0.427211i	$-0.859496\pi$
$263$	− 13.8564i	− 0.854423i	0.904152	−	0.427211i	$-0.140504\pi$
$264$	0	0
$265$	41.5692i	2.55358i
$266$	0	0
$267$	6.92820i	0.423999i
$268$	0	0
$269$	17.3205i	1.05605i	0.849229	+	0.528025i	$0.177067\pi$
$269$	17.3205i	1.05605i	−0.849229	+	0.528025i	$0.822933\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 24.2487i	− 1.46225i
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	7.00000	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$283$	7.00000	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$284$	0	0
$285$	− 17.3205i	− 1.02598i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	6.92820i	0.406138i
$292$	0	0
$293$	− 20.7846i	− 1.21425i	−0.794606	−	0.607125i	$-0.792323\pi$
$293$	− 20.7846i	− 1.21425i	0.794606	−	0.607125i	$-0.207677\pi$
$294$	0	0
$295$	− 41.5692i	− 2.42025i
$296$	0	0
$297$	3.46410i	0.201008i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 10.3923i	− 0.597022i
$304$	0	0
$305$	48.0000	2.74847
$306$	0	0
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	0	0
$309$	5.00000	0.284440
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	15.5885i	0.881112i	0.897725	+	0.440556i	$0.145219\pi$
$313$	15.5885i	0.881112i	−0.897725	+	0.440556i	$0.854781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12.1244i	0.672538i
$326$	0	0
$327$	−7.00000	−0.387101
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.5885i	0.856819i	0.903585	+	0.428410i	$0.140926\pi$
$331$	15.5885i	0.856819i	−0.903585	+	0.428410i	$0.859074\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	30.0000	1.63908
$336$	0	0
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	17.3205i	0.937958i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	27.7128i	1.48770i	0.668346	+	0.743851i	$0.267003\pi$
$347$	27.7128i	1.48770i	−0.668346	+	0.743851i	$0.732997\pi$
$348$	0	0
$349$	− 13.8564i	− 0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$	− 13.8564i	− 0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	0	0
$351$	− 1.73205i	− 0.0924500i
$352$	0	0
$353$	− 24.2487i	− 1.29063i	−0.763917	−	0.645314i	$-0.776726\pi$
$353$	− 24.2487i	− 1.29063i	0.763917	−	0.645314i	$-0.223274\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 20.7846i	− 1.09697i	−0.836160	−	0.548485i	$-0.815205\pi$
$359$	− 20.7846i	− 1.09697i	0.836160	−	0.548485i	$-0.184795\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−18.0000	−0.942163
$366$	0	0
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	0	0
$369$	− 3.46410i	− 0.180334i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	0	0
$375$	− 6.92820i	− 0.357771i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.66025i	0.444847i	0.974950	+	0.222424i	$0.0713968\pi$
$379$	8.66025i	0.444847i	−0.974950	+	0.222424i	$0.928603\pi$
$380$	0	0
$381$	12.1244i	0.621150i
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.66025i	0.440225i
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−42.0000	−2.11325
$396$	0	0
$397$	− 1.73205i	− 0.0869291i	−0.999055	−	0.0434646i	$-0.986160\pi$
$397$	− 1.73205i	− 0.0869291i	0.999055	−	0.0434646i	$-0.0138396\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	− 8.66025i	− 0.431398i
$404$	0	0
$405$	3.46410i	0.172133i
$406$	0	0
$407$	− 38.1051i	− 1.88880i
$408$	0	0
$409$	22.5167i	1.11338i	0.830721	+	0.556689i	$0.187928\pi$
$409$	22.5167i	1.11338i	−0.830721	+	0.556689i	$0.812072\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 62.3538i	− 3.06083i
$416$	0	0
$417$	−7.00000	−0.342791
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	− 24.2487i	− 1.16802i	−0.811747	−	0.584010i	$-0.801483\pi$
$431$	− 24.2487i	− 1.16802i	0.811747	−	0.584010i	$-0.198517\pi$
$432$	0	0
$433$	− 1.73205i	− 0.0832370i	−0.999134	−	0.0416185i	$-0.986749\pi$
$433$	− 1.73205i	− 0.0832370i	0.999134	−	0.0416185i	$-0.0132514\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 34.6410i	− 1.65710i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.8564i	0.658338i	0.944271	+	0.329169i	$0.106769\pi$
$443$	13.8564i	0.658338i	−0.944271	+	0.329169i	$0.893231\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	12.0000	0.567581
$448$	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
$451$	12.0000	0.565058
$452$	0	0
$453$	− 3.46410i	− 0.162758i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.0000	−0.888783	−0.444391	−	0.895833i	$-0.646580\pi$
$457$	−19.0000	−0.888783	−0.444391	+	0.895833i	$0.646580\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 13.8564i	− 0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$	− 13.8564i	− 0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$	0	0
$463$	− 15.5885i	− 0.724457i	−0.932089	−	0.362229i	$-0.882016\pi$
$463$	− 15.5885i	− 0.724457i	0.932089	−	0.362229i	$-0.117984\pi$
$464$	0	0
$465$	17.3205i	0.803219i
$466$	0	0
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.0000	−1.37940
$474$	0	0
$475$	35.0000	1.60591
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	19.0526i	0.868722i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	− 25.9808i	− 1.17730i	−0.808388	−	0.588650i	$-0.799659\pi$
$487$	− 25.9808i	− 1.17730i	0.808388	−	0.588650i	$-0.200341\pi$
$488$	0	0
$489$	− 3.46410i	− 0.156652i
$490$	0	0
$491$	− 34.6410i	− 1.56333i	−0.623700	−	0.781664i	$-0.714371\pi$
$491$	− 34.6410i	− 1.56333i	0.623700	−	0.781664i	$-0.285629\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−12.0000	−0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 32.9090i	− 1.47321i	−0.676325	−	0.736604i	$-0.736428\pi$
$499$	− 32.9090i	− 1.47321i	0.676325	−	0.736604i	$-0.263572\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	10.0000	0.444116
$508$	0	0
$509$	31.1769i	1.38189i	0.722906	+	0.690946i	$0.242806\pi$
$509$	31.1769i	1.38189i	−0.722906	+	0.690946i	$0.757194\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	17.3205i	0.763233i
$516$	0	0
$517$	20.7846i	0.914106i
$518$	0	0
$519$	6.92820i	0.304114i
$520$	0	0
$521$	13.8564i	0.607060i	0.952822	+	0.303530i	$0.0981653\pi$
$521$	13.8564i	0.607060i	−0.952822	+	0.303530i	$0.901835\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−25.0000	−1.08696
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.3923i	0.448461i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.00000	−0.214967	−0.107483	−	0.994207i	$-0.534279\pi$
$541$	−5.00000	−0.214967	−0.107483	+	0.994207i	$0.534279\pi$
$542$	0	0
$543$	− 5.19615i	− 0.222988i
$544$	0	0
$545$	− 24.2487i	− 1.03870i
$546$	0	0
$547$	− 3.46410i	− 0.148114i	−0.997254	−	0.0740571i	$-0.976405\pi$
$547$	− 3.46410i	− 0.148114i	0.997254	−	0.0740571i	$-0.0235947\pi$
$548$	0	0
$549$	− 13.8564i	− 0.591377i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 38.1051i	− 1.61747i
$556$	0	0
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	15.0000	0.634432
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	− 20.7846i	− 0.874415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	46.7654i	1.95707i	0.206081	+	0.978535i	$0.433929\pi$
$571$	46.7654i	1.95707i	−0.206081	+	0.978535i	$0.566071\pi$
$572$	0	0
$573$	17.3205i	0.723575i
$574$	0	0
$575$	− 48.4974i	− 2.02248i
$576$	0	0
$577$	− 12.1244i	− 0.504744i	−0.967630	−	0.252372i	$-0.918789\pi$
$577$	− 12.1244i	− 0.504744i	0.967630	−	0.252372i	$-0.0812106\pi$
$578$	0	0
$579$	−11.0000	−0.457144
$580$	0	0
$581$	0	0
$582$	0	0
$583$	41.5692i	1.72162i
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−25.0000	−1.03011
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	3.46410i	0.142254i	0.997467	+	0.0711268i	$0.0226595\pi$
$593$	3.46410i	0.142254i	−0.997467	+	0.0711268i	$0.977341\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	48.4974i	1.98155i	0.135514	+	0.990775i	$0.456732\pi$
$599$	48.4974i	1.98155i	−0.135514	+	0.990775i	$0.543268\pi$
$600$	0	0
$601$	− 39.8372i	− 1.62499i	−0.582967	−	0.812496i	$-0.698108\pi$
$601$	− 39.8372i	− 1.62499i	0.582967	−	0.812496i	$-0.301892\pi$
$602$	0	0
$603$	− 8.66025i	− 0.352673i
$604$	0	0
$605$	− 3.46410i	− 0.140836i
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 10.3923i	− 0.420428i
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	0	0
$621$	6.92820i	0.278019i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	− 17.3205i	− 0.691714i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	3.46410i	0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$	3.46410i	0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$	0	0
$633$	3.46410i	0.137686i
$634$	0	0
$635$	−42.0000	−1.66672
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.46410i	0.137038i
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	0	0
$645$	−30.0000	−1.18125
$646$	0	0
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	− 41.5692i	− 1.63173i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	− 20.7846i	− 0.812122i
$656$	0	0
$657$	5.19615i	0.202721i
$658$	0	0
$659$	13.8564i	0.539769i	0.962893	+	0.269884i	$0.0869855\pi$
$659$	13.8564i	0.539769i	−0.962893	+	0.269884i	$0.913014\pi$
$660$	0	0
$661$	− 25.9808i	− 1.01053i	−0.862963	−	0.505267i	$-0.831394\pi$
$661$	− 25.9808i	− 1.01053i	0.862963	−	0.505267i	$-0.168606\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	−7.00000	−0.269430
$676$	0	0
$677$	− 6.92820i	− 0.266272i	−0.991098	−	0.133136i	$-0.957495\pi$
$677$	− 6.92820i	− 0.266272i	0.991098	−	0.133136i	$-0.0425048\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	− 6.92820i	− 0.265100i	−0.991176	−	0.132550i	$-0.957683\pi$
$683$	− 6.92820i	− 0.265100i	0.991176	−	0.132550i	$-0.0423166\pi$
$684$	0	0
$685$	41.5692i	1.58828i
$686$	0	0
$687$	8.66025i	0.330409i
$688$	0	0
$689$	− 20.7846i	− 0.791831i
$690$	0	0
$691$	23.0000	0.874961	0.437481	−	0.899228i	$-0.355871\pi$
$691$	23.0000	0.874961	0.437481	+	0.899228i	$0.355871\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 24.2487i	− 0.919806i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	55.0000	2.07436
$704$	0	0
$705$	20.7846i	0.782794i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	12.1244i	0.454699i
$712$	0	0
$713$	34.6410i	1.29732i
$714$	0	0
$715$	20.7846i	0.777300i
$716$	0	0
$717$	10.3923i	0.388108i
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	13.8564i	0.515325i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 29.4449i	− 1.08757i	−0.839224	−	0.543785i	$-0.816991\pi$
$733$	− 29.4449i	− 1.08757i	0.839224	−	0.543785i	$-0.183009\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	53.6936i	1.97515i	0.157148	+	0.987575i	$0.449770\pi$
$739$	53.6936i	1.97515i	−0.157148	+	0.987575i	$0.550230\pi$
$740$	0	0
$741$	8.66025i	0.318142i
$742$	0	0
$743$	3.46410i	0.127086i	0.997979	+	0.0635428i	$0.0202399\pi$
$743$	3.46410i	0.127086i	−0.997979	+	0.0635428i	$0.979760\pi$
$744$	0	0
$745$	41.5692i	1.52298i
$746$	0	0
$747$	−18.0000	−0.658586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 5.19615i	− 0.189610i	−0.995496	−	0.0948051i	$-0.969777\pi$
$751$	− 5.19615i	− 0.189610i	0.995496	−	0.0948051i	$-0.0302228\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	− 41.5692i	− 1.50688i	−0.657515	−	0.753442i	$-0.728392\pi$
$761$	− 41.5692i	− 1.50688i	0.657515	−	0.753442i	$-0.271608\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.7846i	0.750489i
$768$	0	0
$769$	− 15.5885i	− 0.562134i	−0.959688	−	0.281067i	$-0.909312\pi$
$769$	− 15.5885i	− 0.562134i	0.959688	−	0.281067i	$-0.0906883\pi$
$770$	0	0
$771$	− 17.3205i	− 0.623783i
$772$	0	0
$773$	45.0333i	1.61974i	0.586612	+	0.809868i	$0.300461\pi$
$773$	45.0333i	1.61974i	−0.586612	+	0.809868i	$0.699539\pi$
$774$	0	0
$775$	−35.0000	−1.25724
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.3205i	0.620572i
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	0	0
$789$	− 13.8564i	− 0.493301i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	41.5692i	1.47431i
$796$	0	0
$797$	34.6410i	1.22705i	0.789676	+	0.613524i	$0.210249\pi$
$797$	34.6410i	1.22705i	−0.789676	+	0.613524i	$0.789751\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.92820i	0.244796i
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.3205i	0.609711i
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	− 43.3013i	− 1.51492i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	3.46410i	0.120751i	0.998176	+	0.0603755i	$0.0192298\pi$
$823$	3.46410i	0.120751i	−0.998176	+	0.0603755i	$0.980770\pi$
$824$	0	0
$825$	− 24.2487i	− 0.844232i
$826$	0	0
$827$	38.1051i	1.32504i	0.749042	+	0.662522i	$0.230514\pi$
$827$	38.1051i	1.32504i	−0.749042	+	0.662522i	$0.769486\pi$
$828$	0	0
$829$	− 25.9808i	− 0.902349i	−0.892436	−	0.451175i	$-0.851005\pi$
$829$	− 25.9808i	− 0.902349i	0.892436	−	0.451175i	$-0.148995\pi$
$830$	0	0
$831$	19.0000	0.659103
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 20.7846i	− 0.719281i
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34.6410i	1.19169i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.00000	0.240239
$850$	0	0
$851$	− 76.2102i	− 2.61245i
$852$	0	0
$853$	− 5.19615i	− 0.177913i	−0.996036	−	0.0889564i	$-0.971647\pi$
$853$	− 5.19615i	− 0.177913i	0.996036	−	0.0889564i	$-0.0283532\pi$
$854$	0	0
$855$	− 17.3205i	− 0.592349i
$856$	0	0
$857$	− 13.8564i	− 0.473326i	−0.971592	−	0.236663i	$-0.923946\pi$
$857$	− 13.8564i	− 0.473326i	0.971592	−	0.236663i	$-0.0760537\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.3923i	0.353758i	0.984233	+	0.176879i	$0.0566002\pi$
$863$	10.3923i	0.353758i	−0.984233	+	0.176879i	$0.943400\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−42.0000	−1.42475
$870$	0	0
$871$	−15.0000	−0.508256
$872$	0	0
$873$	6.92820i	0.234484i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	− 20.7846i	− 0.701047i
$880$	0	0
$881$	6.92820i	0.233417i	0.993166	+	0.116709i	$0.0372343\pi$
$881$	6.92820i	0.233417i	−0.993166	+	0.116709i	$0.962766\pi$
$882$	0	0
$883$	− 46.7654i	− 1.57378i	−0.617093	−	0.786890i	$-0.711690\pi$
$883$	− 46.7654i	− 1.57378i	0.617093	−	0.786890i	$-0.288310\pi$
$884$	0	0
$885$	− 41.5692i	− 1.39733i
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.46410i	0.116052i
$892$	0	0
$893$	−30.0000	−1.00391
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	− 32.9090i	− 1.09272i	−0.837549	−	0.546362i	$-0.816012\pi$
$907$	− 32.9090i	− 1.09272i	0.837549	−	0.546362i	$-0.183988\pi$
$908$	0	0
$909$	− 10.3923i	− 0.344691i
$910$	0	0
$911$	41.5692i	1.37725i	0.725118	+	0.688625i	$0.241785\pi$
$911$	41.5692i	1.37725i	−0.725118	+	0.688625i	$0.758215\pi$
$912$	0	0
$913$	− 62.3538i	− 2.06361i
$914$	0	0
$915$	48.0000	1.58683
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.3013i	1.42838i	0.699953	+	0.714189i	$0.253204\pi$
$919$	43.3013i	1.42838i	−0.699953	+	0.714189i	$0.746796\pi$
$920$	0	0
$921$	−11.0000	−0.362462
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	77.0000	2.53174
$926$	0	0
$927$	5.00000	0.164222
$928$	0	0
$929$	10.3923i	0.340960i	0.985361	+	0.170480i	$0.0545319\pi$
$929$	10.3923i	0.340960i	−0.985361	+	0.170480i	$0.945468\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.0000	0.589294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 53.6936i	− 1.75409i	−0.480406	−	0.877046i	$-0.659511\pi$
$937$	− 53.6936i	− 1.75409i	0.480406	−	0.877046i	$-0.340489\pi$
$938$	0	0
$939$	15.5885i	0.508710i
$940$	0	0
$941$	13.8564i	0.451706i	0.974161	+	0.225853i	$0.0725169\pi$
$941$	13.8564i	0.451706i	−0.974161	+	0.225853i	$0.927483\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45.0333i	− 1.46339i	−0.681634	−	0.731693i	$-0.738730\pi$
$947$	− 45.0333i	− 1.46339i	0.681634	−	0.731693i	$-0.261270\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 38.1051i	− 1.22665i
$966$	0	0
$967$	− 8.66025i	− 0.278495i	−0.990258	−	0.139247i	$-0.955532\pi$
$967$	− 8.66025i	− 0.278495i	0.990258	−	0.139247i	$-0.0444684\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.1244i	0.388290i
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−7.00000	−0.223493
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	83.1384i	2.64901i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	− 39.8372i	− 1.26547i	−0.774369	−	0.632735i	$-0.781932\pi$
$991$	− 39.8372i	− 1.26547i	0.774369	−	0.632735i	$-0.218068\pi$
$992$	0	0
$993$	15.5885i	0.494685i
$994$	0	0
$995$	55.4256i	1.75711i
$996$	0	0
$997$	19.0526i	0.603401i	0.953403	+	0.301700i	$0.0975542\pi$
$997$	19.0526i	0.603401i	−0.953403	+	0.301700i	$0.902446\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.b.e.1567.2		2
3.2	odd	2		7056.2.b.d.1567.1		2
4.3	odd	2		2352.2.b.d.1567.2		2
7.2	even	3		336.2.bl.d.31.1	✓	2
7.3	odd	6		336.2.bl.h.271.1	yes	2
7.4	even	3		2352.2.bl.a.607.1		2
7.5	odd	6		2352.2.bl.g.31.1		2
7.6	odd	2		2352.2.b.d.1567.1		2
12.11	even	2		7056.2.b.k.1567.1		2
21.2	odd	6		1008.2.cs.b.703.1		2
21.17	even	6		1008.2.cs.a.271.1		2
21.20	even	2		7056.2.b.k.1567.2		2
28.3	even	6		336.2.bl.d.271.1	yes	2
28.11	odd	6		2352.2.bl.g.607.1		2
28.19	even	6		2352.2.bl.a.31.1		2
28.23	odd	6		336.2.bl.h.31.1	yes	2
28.27	even	2	inner	2352.2.b.e.1567.1		2
56.3	even	6		1344.2.bl.e.1279.1		2
56.37	even	6		1344.2.bl.e.703.1		2
56.45	odd	6		1344.2.bl.a.1279.1		2
56.51	odd	6		1344.2.bl.a.703.1		2
84.23	even	6		1008.2.cs.a.703.1		2
84.59	odd	6		1008.2.cs.b.271.1		2
84.83	odd	2		7056.2.b.d.1567.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.d.31.1	✓	2	7.2	even	3
336.2.bl.d.271.1	yes	2	28.3	even	6
336.2.bl.h.31.1	yes	2	28.23	odd	6
336.2.bl.h.271.1	yes	2	7.3	odd	6
1008.2.cs.a.271.1		2	21.17	even	6
1008.2.cs.a.703.1		2	84.23	even	6
1008.2.cs.b.271.1		2	84.59	odd	6
1008.2.cs.b.703.1		2	21.2	odd	6
1344.2.bl.a.703.1		2	56.51	odd	6
1344.2.bl.a.1279.1		2	56.45	odd	6
1344.2.bl.e.703.1		2	56.37	even	6
1344.2.bl.e.1279.1		2	56.3	even	6
2352.2.b.d.1567.1		2	7.6	odd	2
2352.2.b.d.1567.2		2	4.3	odd	2
2352.2.b.e.1567.1		2	28.27	even	2	inner
2352.2.b.e.1567.2		2	1.1	even	1	trivial
2352.2.bl.a.31.1		2	28.19	even	6
2352.2.bl.a.607.1		2	7.4	even	3
2352.2.bl.g.31.1		2	7.5	odd	6
2352.2.bl.g.607.1		2	28.11	odd	6
7056.2.b.d.1567.1		2	3.2	odd	2
7056.2.b.d.1567.2		2	84.83	odd	2
7056.2.b.k.1567.1		2	12.11	even	2
7056.2.b.k.1567.2		2	21.20	even	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +3.46410i q^{11} -1.73205i q^{13} +3.46410i q^{15} -5.00000 q^{19} +6.92820i q^{23} -7.00000 q^{25} +1.00000 q^{27} +5.00000 q^{31} +3.46410i q^{33} -11.0000 q^{37} -1.73205i q^{39} -3.46410i q^{41} +8.66025i q^{43} +3.46410i q^{45} +6.00000 q^{47} +12.0000 q^{53} -12.0000 q^{55} -5.00000 q^{57} -12.0000 q^{59} -13.8564i q^{61} +6.00000 q^{65} -8.66025i q^{67} +6.92820i q^{69} +3.46410i q^{71} +5.19615i q^{73} -7.00000 q^{75} +12.1244i q^{79} +1.00000 q^{81} -18.0000 q^{83} +6.92820i q^{89} +5.00000 q^{93} -17.3205i q^{95} +6.92820i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 10 q^{19} - 14 q^{25} + 2 q^{27} + 10 q^{31} - 22 q^{37} + 12 q^{47} + 24 q^{53} - 24 q^{55} - 10 q^{57} - 24 q^{59} + 12 q^{65} - 14 q^{75} + 2 q^{81} - 36 q^{83} + 10 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 10 * q^19 - 14 * q^25 + 2 * q^27 + 10 * q^31 - 22 * q^37 + 12 * q^47 + 24 * q^53 - 24 * q^55 - 10 * q^57 - 24 * q^59 + 12 * q^65 - 14 * q^75 + 2 * q^81 - 36 * q^83 + 10 * q^93

Introduction

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