LMFDB - Embedded newform 2400.2.k.f.1201.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1201,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2400.1201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1201.12
Root			$-1.37729 + 0.321037i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.f.1201.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.05705 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +4.05705 q^{7} -1.00000 q^{9} +0.985939i q^{11} +4.94567i q^{13} +4.52323 q^{17} -2.60492i q^{19} +4.05705i q^{21} -3.53729 q^{23} -1.00000i q^{27} +7.59434i q^{29} +3.28415 q^{31} -0.985939 q^{33} +0.945668i q^{37} -4.94567 q^{39} +0.568295 q^{41} -8.45963i q^{43} -2.60492 q^{47} +9.45963 q^{49} +4.52323i q^{51} +0.229815i q^{53} +2.60492 q^{57} +9.10003i q^{59} +11.0183i q^{61} -4.05705 q^{63} -8.45963i q^{67} -3.53729i q^{69} -1.43171 q^{71} +11.9507 q^{73} +4.00000i q^{77} +3.28415 q^{79} +1.00000 q^{81} -9.89134i q^{83} -7.59434 q^{87} -12.3510 q^{89} +20.0648i q^{91} +3.28415i q^{93} -3.23797 q^{97} -0.985939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) $ 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.05705	1.53342	0.766710	−	0.641994i	$-0.221893\pi$
$7$	4.05705	1.53342	0.766710	+	0.641994i	$0.221893\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.985939i	0.297272i	0.988892	+	0.148636i	$0.0474882\pi$
$11$	0.985939i	0.297272i	−0.988892	+	0.148636i	$0.952512\pi$
$12$	0	0
$13$	4.94567i	1.37168i	0.727752	+	0.685841i	$0.240565\pi$
$13$	4.94567i	1.37168i	−0.727752	+	0.685841i	$0.759435\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.52323	1.09704	0.548522	−	0.836136i	$-0.315191\pi$
$17$	4.52323	1.09704	0.548522	+	0.836136i	$0.315191\pi$
$18$	0	0
$19$	− 2.60492i	− 0.597610i	−0.954314	−	0.298805i	$-0.903412\pi$
$19$	− 2.60492i	− 0.597610i	0.954314	−	0.298805i	$-0.0965881\pi$
$20$	0	0
$21$	4.05705i	0.885320i
$22$	0	0
$23$	−3.53729	−0.737577	−0.368788	−	0.929513i	$-0.620227\pi$
$23$	−3.53729	−0.737577	−0.368788	+	0.929513i	$0.620227\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	7.59434i	1.41023i	0.709091	+	0.705117i	$0.249105\pi$
$29$	7.59434i	1.41023i	−0.709091	+	0.705117i	$0.750895\pi$
$30$	0	0
$31$	3.28415	0.589850	0.294925	−	0.955520i	$-0.404705\pi$
$31$	3.28415	0.589850	0.294925	+	0.955520i	$0.404705\pi$
$32$	0	0
$33$	−0.985939	−0.171630
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.945668i	0.155467i	0.996974	+	0.0777334i	$0.0247683\pi$
$37$	0.945668i	0.155467i	−0.996974	+	0.0777334i	$0.975232\pi$
$38$	0	0
$39$	−4.94567	−0.791941
$40$	0	0
$41$	0.568295	0.0887527	0.0443763	−	0.999015i	$-0.485870\pi$
$41$	0.568295	0.0887527	0.0443763	+	0.999015i	$0.485870\pi$
$42$	0	0
$43$	− 8.45963i	− 1.29008i	−0.764148	−	0.645041i	$-0.776840\pi$
$43$	− 8.45963i	− 1.29008i	0.764148	−	0.645041i	$-0.223160\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.60492	−0.379967	−0.189984	−	0.981787i	$-0.560843\pi$
$47$	−2.60492	−0.379967	−0.189984	+	0.981787i	$0.560843\pi$
$48$	0	0
$49$	9.45963	1.35138
$50$	0	0
$51$	4.52323i	0.633379i
$52$	0	0
$53$	0.229815i	0.0315675i	0.999875	+	0.0157838i	$0.00502434\pi$
$53$	0.229815i	0.0315675i	−0.999875	+	0.0157838i	$0.994976\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.60492	0.345030
$58$	0	0
$59$	9.10003i	1.18472i	0.805672	+	0.592362i	$0.201804\pi$
$59$	9.10003i	1.18472i	−0.805672	+	0.592362i	$0.798196\pi$
$60$	0	0
$61$	11.0183i	1.41075i	0.708832	+	0.705377i	$0.249222\pi$
$61$	11.0183i	1.41075i	−0.708832	+	0.705377i	$0.750778\pi$
$62$	0	0
$63$	−4.05705	−0.511140
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.45963i	− 1.03351i	−0.856134	−	0.516754i	$-0.827140\pi$
$67$	− 8.45963i	− 1.03351i	0.856134	−	0.516754i	$-0.172860\pi$
$68$	0	0
$69$	− 3.53729i	− 0.425840i
$70$	0	0
$71$	−1.43171	−0.169912	−0.0849561	−	0.996385i	$-0.527075\pi$
$71$	−1.43171	−0.169912	−0.0849561	+	0.996385i	$0.527075\pi$
$72$	0	0
$73$	11.9507	1.39873	0.699363	−	0.714767i	$-0.253467\pi$
$73$	11.9507	1.39873	0.699363	+	0.714767i	$0.253467\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	3.28415	0.369495	0.184748	−	0.982786i	$-0.440853\pi$
$79$	3.28415	0.369495	0.184748	+	0.982786i	$0.440853\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.89134i	− 1.08572i	−0.839825	−	0.542858i	$-0.817342\pi$
$83$	− 9.89134i	− 1.08572i	0.839825	−	0.542858i	$-0.182658\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.59434	−0.814199
$88$	0	0
$89$	−12.3510	−1.30920	−0.654600	−	0.755976i	$-0.727163\pi$
$89$	−12.3510	−1.30920	−0.654600	+	0.755976i	$0.727163\pi$
$90$	0	0
$91$	20.0648i	2.10336i
$92$	0	0
$93$	3.28415i	0.340550i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.23797	−0.328766	−0.164383	−	0.986397i	$-0.552563\pi$
$97$	−3.23797	−0.328766	−0.164383	+	0.986397i	$0.552563\pi$
$98$	0	0
$99$	− 0.985939i	− 0.0990906i
$100$	0	0
$101$	4.35637i	0.433475i	0.976230	+	0.216738i	$0.0695416\pi$
$101$	4.35637i	0.433475i	−0.976230	+	0.216738i	$0.930458\pi$
$102$	0	0
$103$	15.0754	1.48542	0.742711	−	0.669612i	$-0.233540\pi$
$103$	15.0754	1.48542	0.742711	+	0.669612i	$0.233540\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 4.17034i	− 0.399446i	−0.979852	−	0.199723i	$-0.935996\pi$
$109$	− 4.17034i	− 0.399446i	0.979852	−	0.199723i	$-0.0640042\pi$
$110$	0	0
$111$	−0.945668	−0.0897588
$112$	0	0
$113$	−1.28526	−0.120907	−0.0604537	−	0.998171i	$-0.519255\pi$
$113$	−1.28526	−0.120907	−0.0604537	+	0.998171i	$0.519255\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.94567i	− 0.457227i
$118$	0	0
$119$	18.3510	1.68223
$120$	0	0
$121$	10.0279	0.911630
$122$	0	0
$123$	0.568295i	0.0512414i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.15280	0.102294	0.0511472	−	0.998691i	$-0.483712\pi$
$127$	1.15280	0.102294	0.0511472	+	0.998691i	$0.483712\pi$
$128$	0	0
$129$	8.45963	0.744829
$130$	0	0
$131$	3.89019i	0.339887i	0.985454	+	0.169944i	$0.0543586\pi$
$131$	3.89019i	0.339887i	−0.985454	+	0.169944i	$0.945641\pi$
$132$	0	0
$133$	− 10.5683i	− 0.916387i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.5135	−1.49628	−0.748138	−	0.663544i	$-0.769052\pi$
$137$	−17.5135	−1.49628	−0.748138	+	0.663544i	$0.769052\pi$
$138$	0	0
$139$	16.8612i	1.43015i	0.699047	+	0.715076i	$0.253608\pi$
$139$	16.8612i	1.43015i	−0.699047	+	0.715076i	$0.746392\pi$
$140$	0	0
$141$	− 2.60492i	− 0.219374i
$142$	0	0
$143$	−4.87613	−0.407762
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.45963i	0.780217i
$148$	0	0
$149$	10.4986i	0.860078i	0.902810	+	0.430039i	$0.141500\pi$
$149$	10.4986i	0.860078i	−0.902810	+	0.430039i	$0.858500\pi$
$150$	0	0
$151$	−4.71585	−0.383771	−0.191885	−	0.981417i	$-0.561460\pi$
$151$	−4.71585	−0.383771	−0.191885	+	0.981417i	$0.561460\pi$
$152$	0	0
$153$	−4.52323	−0.365682
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.94567i	0.713942i	0.934115	+	0.356971i	$0.116191\pi$
$157$	8.94567i	0.713942i	−0.934115	+	0.356971i	$0.883809\pi$
$158$	0	0
$159$	−0.229815	−0.0182255
$160$	0	0
$161$	−14.3510	−1.13101
$162$	0	0
$163$	15.7827i	1.23619i	0.786102	+	0.618097i	$0.212096\pi$
$163$	15.7827i	1.23619i	−0.786102	+	0.618097i	$0.787904\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.50917	−0.426312	−0.213156	−	0.977018i	$-0.568374\pi$
$167$	−5.50917	−0.426312	−0.213156	+	0.977018i	$0.568374\pi$
$168$	0	0
$169$	−11.4596	−0.881510
$170$	0	0
$171$	2.60492i	0.199203i
$172$	0	0
$173$	10.3385i	0.786020i	0.919534	+	0.393010i	$0.128566\pi$
$173$	10.3385i	0.786020i	−0.919534	+	0.393010i	$0.871434\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.10003	−0.684000
$178$	0	0
$179$	− 16.1746i	− 1.20895i	−0.796625	−	0.604474i	$-0.793383\pi$
$179$	− 16.1746i	− 1.20895i	0.796625	−	0.604474i	$-0.206617\pi$
$180$	0	0
$181$	8.11409i	0.603116i	0.953448	+	0.301558i	$0.0975067\pi$
$181$	8.11409i	0.603116i	−0.953448	+	0.301558i	$0.902493\pi$
$182$	0	0
$183$	−11.0183	−0.814499
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.45963i	0.326120i
$188$	0	0
$189$	− 4.05705i	− 0.295107i
$190$	0	0
$191$	24.9193	1.80309	0.901547	−	0.432681i	$-0.142432\pi$
$191$	24.9193	1.80309	0.901547	+	0.432681i	$0.142432\pi$
$192$	0	0
$193$	1.03951	0.0748254	0.0374127	−	0.999300i	$-0.488088\pi$
$193$	1.03951	0.0748254	0.0374127	+	0.999300i	$0.488088\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.66152i	− 0.688355i	−0.938905	−	0.344177i	$-0.888158\pi$
$197$	− 9.66152i	− 0.688355i	0.938905	−	0.344177i	$-0.111842\pi$
$198$	0	0
$199$	−23.0668	−1.63516	−0.817582	−	0.575813i	$-0.804686\pi$
$199$	−23.0668	−1.63516	−0.817582	+	0.575813i	$0.804686\pi$
$200$	0	0
$201$	8.45963	0.596696
$202$	0	0
$203$	30.8106i	2.16248i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.53729	0.245859
$208$	0	0
$209$	2.56829	0.177653
$210$	0	0
$211$	− 6.44154i	− 0.443454i	−0.975109	−	0.221727i	$-0.928831\pi$
$211$	− 6.44154i	− 0.443454i	0.975109	−	0.221727i	$-0.0711694\pi$
$212$	0	0
$213$	− 1.43171i	− 0.0980988i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	13.3239	0.904488
$218$	0	0
$219$	11.9507i	0.807554i
$220$	0	0
$221$	22.3704i	1.50480i
$222$	0	0
$223$	17.9796	1.20401	0.602003	−	0.798494i	$-0.294370\pi$
$223$	17.9796	1.20401	0.602003	+	0.798494i	$0.294370\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.02792i	0.466460i	0.972422	+	0.233230i	$0.0749295\pi$
$227$	7.02792i	0.466460i	−0.972422	+	0.233230i	$0.925071\pi$
$228$	0	0
$229$	4.17034i	0.275584i	0.990461	+	0.137792i	$0.0440005\pi$
$229$	4.17034i	0.275584i	−0.990461	+	0.137792i	$0.955999\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−23.9894	−1.57160	−0.785799	−	0.618483i	$-0.787748\pi$
$233$	−23.9894	−1.57160	−0.785799	+	0.618483i	$0.787748\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.28415i	0.213328i
$238$	0	0
$239$	8.91926	0.576939	0.288469	−	0.957489i	$-0.406854\pi$
$239$	8.91926	0.576939	0.288469	+	0.957489i	$0.406854\pi$
$240$	0	0
$241$	−16.3510	−1.05326	−0.526629	−	0.850095i	$-0.676544\pi$
$241$	−16.3510	−1.05326	−0.526629	+	0.850095i	$0.676544\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.8831	0.819731
$248$	0	0
$249$	9.89134	0.626838
$250$	0	0
$251$	4.22391i	0.266611i	0.991075	+	0.133305i	$0.0425591\pi$
$251$	4.22391i	0.266611i	−0.991075	+	0.133305i	$0.957441\pi$
$252$	0	0
$253$	− 3.48755i	− 0.219261i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.6952	1.54044	0.770221	−	0.637777i	$-0.220146\pi$
$257$	24.6952	1.54044	0.770221	+	0.637777i	$0.220146\pi$
$258$	0	0
$259$	3.83662i	0.238396i
$260$	0	0
$261$	− 7.59434i	− 0.470078i
$262$	0	0
$263$	−14.6628	−0.904145	−0.452073	−	0.891981i	$-0.649315\pi$
$263$	−14.6628	−0.904145	−0.452073	+	0.891981i	$0.649315\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.3510i	− 0.755867i
$268$	0	0
$269$	− 11.5381i	− 0.703490i	−0.936096	−	0.351745i	$-0.885588\pi$
$269$	− 11.5381i	− 0.703490i	0.936096	−	0.351745i	$-0.114412\pi$
$270$	0	0
$271$	−5.63511	−0.342309	−0.171154	−	0.985244i	$-0.554750\pi$
$271$	−5.63511	−0.342309	−0.171154	+	0.985244i	$0.554750\pi$
$272$	0	0
$273$	−20.0648	−1.21438
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 17.4053i	− 1.04578i	−0.852399	−	0.522892i	$-0.824853\pi$
$277$	− 17.4053i	− 1.04578i	0.852399	−	0.522892i	$-0.175147\pi$
$278$	0	0
$279$	−3.28415	−0.196617
$280$	0	0
$281$	21.7827	1.29945	0.649723	−	0.760171i	$-0.274885\pi$
$281$	21.7827	1.29945	0.649723	+	0.760171i	$0.274885\pi$
$282$	0	0
$283$	− 21.5962i	− 1.28376i	−0.766804	−	0.641881i	$-0.778154\pi$
$283$	− 21.5962i	− 1.28376i	0.766804	−	0.641881i	$-0.221846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.30560	0.136095
$288$	0	0
$289$	3.45963	0.203508
$290$	0	0
$291$	− 3.23797i	− 0.189813i
$292$	0	0
$293$	− 32.0125i	− 1.87019i	−0.354398	−	0.935095i	$-0.615314\pi$
$293$	− 32.0125i	− 1.87019i	0.354398	−	0.935095i	$-0.384686\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.985939	0.0572100
$298$	0	0
$299$	− 17.4943i	− 1.01172i
$300$	0	0
$301$	− 34.3211i	− 1.97824i
$302$	0	0
$303$	−4.35637	−0.250267
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.13659i	− 0.0648686i	−0.999474	−	0.0324343i	$-0.989674\pi$
$307$	− 1.13659i	− 0.0648686i	0.999474	−	0.0324343i	$-0.0103260\pi$
$308$	0	0
$309$	15.0754i	0.857609i
$310$	0	0
$311$	−5.13659	−0.291269	−0.145635	−	0.989338i	$-0.546522\pi$
$311$	−5.13659	−0.291269	−0.145635	+	0.989338i	$0.546522\pi$
$312$	0	0
$313$	23.0762	1.30434	0.652172	−	0.758071i	$-0.273858\pi$
$313$	23.0762	1.30434	0.652172	+	0.758071i	$0.273858\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.66152i	− 0.0933203i	−0.998911	−	0.0466601i	$-0.985142\pi$
$317$	− 1.66152i	− 0.0933203i	0.998911	−	0.0466601i	$-0.0148578\pi$
$318$	0	0
$319$	−7.48755	−0.419223
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 11.7827i	− 0.655605i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.17034	0.230620
$328$	0	0
$329$	−10.5683	−0.582649
$330$	0	0
$331$	− 25.9077i	− 1.42402i	−0.702171	−	0.712008i	$-0.747786\pi$
$331$	− 25.9077i	− 1.42402i	0.702171	−	0.712008i	$-0.252214\pi$
$332$	0	0
$333$	− 0.945668i	− 0.0518223i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00696	0.436167	0.218083	−	0.975930i	$-0.430020\pi$
$337$	8.00696	0.436167	0.218083	+	0.975930i	$0.430020\pi$
$338$	0	0
$339$	− 1.28526i	− 0.0698060i
$340$	0	0
$341$	3.23797i	0.175346i
$342$	0	0
$343$	9.97884	0.538806
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.0279i	− 1.23620i	−0.786098	−	0.618102i	$-0.787902\pi$
$347$	− 23.0279i	− 1.23620i	0.786098	−	0.618102i	$-0.212098\pi$
$348$	0	0
$349$	21.4380i	1.14755i	0.819012	+	0.573776i	$0.194522\pi$
$349$	21.4380i	1.14755i	−0.819012	+	0.573776i	$0.805478\pi$
$350$	0	0
$351$	4.94567	0.263980
$352$	0	0
$353$	−4.52323	−0.240747	−0.120374	−	0.992729i	$-0.538409\pi$
$353$	−4.52323	−0.240747	−0.120374	+	0.992729i	$0.538409\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.3510i	0.971236i
$358$	0	0
$359$	−10.3510	−0.546303	−0.273152	−	0.961971i	$-0.588066\pi$
$359$	−10.3510	−0.546303	−0.273152	+	0.961971i	$0.588066\pi$
$360$	0	0
$361$	12.2144	0.642862
$362$	0	0
$363$	10.0279i	0.526330i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.485359	0.0253355	0.0126678	−	0.999920i	$-0.495968\pi$
$367$	0.485359	0.0253355	0.0126678	+	0.999920i	$0.495968\pi$
$368$	0	0
$369$	−0.568295	−0.0295842
$370$	0	0
$371$	0.932371i	0.0484063i
$372$	0	0
$373$	− 30.0823i	− 1.55760i	−0.627272	−	0.778800i	$-0.715829\pi$
$373$	− 30.0823i	− 1.55760i	0.627272	−	0.778800i	$-0.284171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.5591	−1.93439
$378$	0	0
$379$	− 33.6881i	− 1.73044i	−0.501392	−	0.865220i	$-0.667179\pi$
$379$	− 33.6881i	− 1.73044i	0.501392	−	0.865220i	$-0.332821\pi$
$380$	0	0
$381$	1.15280i	0.0590597i
$382$	0	0
$383$	5.17545	0.264453	0.132227	−	0.991220i	$-0.457787\pi$
$383$	5.17545	0.264453	0.132227	+	0.991220i	$0.457787\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.45963i	0.430027i
$388$	0	0
$389$	− 16.6408i	− 0.843722i	−0.906660	−	0.421861i	$-0.861377\pi$
$389$	− 16.6408i	− 0.843722i	0.906660	−	0.421861i	$-0.138623\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−3.89019	−0.196234
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.4332i	− 1.02551i	−0.858534	−	0.512757i	$-0.828624\pi$
$397$	− 20.4332i	− 1.02551i	0.858534	−	0.512757i	$-0.171376\pi$
$398$	0	0
$399$	10.5683	0.529076
$400$	0	0
$401$	−4.56829	−0.228130	−0.114065	−	0.993473i	$-0.536387\pi$
$401$	−4.56829	−0.228130	−0.114065	+	0.993473i	$0.536387\pi$
$402$	0	0
$403$	16.2423i	0.809087i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.932371	−0.0462159
$408$	0	0
$409$	19.8913	0.983563	0.491782	−	0.870719i	$-0.336346\pi$
$409$	19.8913	0.983563	0.491782	+	0.870719i	$0.336346\pi$
$410$	0	0
$411$	− 17.5135i	− 0.863875i
$412$	0	0
$413$	36.9193i	1.81668i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.8612	−0.825698
$418$	0	0
$419$	0.387288i	0.0189203i	0.999955	+	0.00946013i	$0.00301130\pi$
$419$	0.387288i	0.0189203i	−0.999955	+	0.00946013i	$0.996989\pi$
$420$	0	0
$421$	− 12.0578i	− 0.587664i	−0.955857	−	0.293832i	$-0.905069\pi$
$421$	− 12.0578i	− 0.587664i	0.955857	−	0.293832i	$-0.0949306\pi$
$422$	0	0
$423$	2.60492	0.126656
$424$	0	0
$425$	0	0
$426$	0	0
$427$	44.7019i	2.16328i
$428$	0	0
$429$	− 4.87613i	− 0.235422i
$430$	0	0
$431$	−40.4068	−1.94633	−0.973164	−	0.230113i	$-0.926090\pi$
$431$	−40.4068	−1.94633	−0.973164	+	0.230113i	$0.926090\pi$
$432$	0	0
$433$	36.1859	1.73898	0.869491	−	0.493949i	$-0.164447\pi$
$433$	36.1859	1.73898	0.869491	+	0.493949i	$0.164447\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.21438i	0.440783i
$438$	0	0
$439$	25.4178	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$439$	25.4178	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$440$	0	0
$441$	−9.45963	−0.450459
$442$	0	0
$443$	7.02792i	0.333907i	0.985965	+	0.166953i	$0.0533929\pi$
$443$	7.02792i	0.333907i	−0.985965	+	0.166953i	$0.946607\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.4986	−0.496566
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	0.560304i	0.0263837i
$452$	0	0
$453$	− 4.71585i	− 0.221570i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.2747	1.18230	0.591149	−	0.806562i	$-0.298674\pi$
$457$	25.2747	1.18230	0.591149	+	0.806562i	$0.298674\pi$
$458$	0	0
$459$	− 4.52323i	− 0.211126i
$460$	0	0
$461$	− 41.0902i	− 1.91376i	−0.290479	−	0.956881i	$-0.593815\pi$
$461$	− 41.0902i	− 1.91376i	0.290479	−	0.956881i	$-0.406185\pi$
$462$	0	0
$463$	13.2106	0.613951	0.306975	−	0.951717i	$-0.400683\pi$
$463$	13.2106	0.613951	0.306975	+	0.951717i	$0.400683\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.89134i	− 0.0875206i	−0.999042	−	0.0437603i	$-0.986066\pi$
$467$	− 1.89134i	− 0.0875206i	0.999042	−	0.0437603i	$-0.0139338\pi$
$468$	0	0
$469$	− 34.3211i	− 1.58480i
$470$	0	0
$471$	−8.94567	−0.412195
$472$	0	0
$473$	8.34068	0.383505
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.229815i	− 0.0105225i
$478$	0	0
$479$	−31.4876	−1.43870	−0.719352	−	0.694646i	$-0.755561\pi$
$479$	−31.4876	−1.43870	−0.719352	+	0.694646i	$0.755561\pi$
$480$	0	0
$481$	−4.67696	−0.213251
$482$	0	0
$483$	− 14.3510i	− 0.652992i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.9964	−0.588922	−0.294461	−	0.955664i	$-0.595140\pi$
$487$	−12.9964	−0.588922	−0.294461	+	0.955664i	$0.595140\pi$
$488$	0	0
$489$	−15.7827	−0.713717
$490$	0	0
$491$	− 14.9085i	− 0.672812i	−0.941717	−	0.336406i	$-0.890788\pi$
$491$	− 14.9085i	− 0.672812i	0.941717	−	0.336406i	$-0.109212\pi$
$492$	0	0
$493$	34.3510i	1.54709i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.80850	−0.260547
$498$	0	0
$499$	35.6599i	1.59636i	0.602420	+	0.798179i	$0.294203\pi$
$499$	35.6599i	1.59636i	−0.602420	+	0.798179i	$0.705797\pi$
$500$	0	0
$501$	− 5.50917i	− 0.246132i
$502$	0	0
$503$	25.3090	1.12847	0.564237	−	0.825613i	$-0.309170\pi$
$503$	25.3090	1.12847	0.564237	+	0.825613i	$0.309170\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 11.4596i	− 0.508940i
$508$	0	0
$509$	− 13.7366i	− 0.608862i	−0.952534	−	0.304431i	$-0.901534\pi$
$509$	− 13.7366i	− 0.608862i	0.952534	−	0.304431i	$-0.0984663\pi$
$510$	0	0
$511$	48.4846	2.14483
$512$	0	0
$513$	−2.60492	−0.115010
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.56829i	− 0.112953i
$518$	0	0
$519$	−10.3385	−0.453809
$520$	0	0
$521$	−30.9193	−1.35460	−0.677299	−	0.735708i	$-0.736850\pi$
$521$	−30.9193	−1.35460	−0.677299	+	0.735708i	$0.736850\pi$
$522$	0	0
$523$	− 21.8385i	− 0.954932i	−0.878650	−	0.477466i	$-0.841555\pi$
$523$	− 21.8385i	− 0.954932i	0.878650	−	0.477466i	$-0.158445\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.8550	0.647092
$528$	0	0
$529$	−10.4876	−0.455981
$530$	0	0
$531$	− 9.10003i	− 0.394908i
$532$	0	0
$533$	2.81060i	0.121740i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.1746	0.697986
$538$	0	0
$539$	9.32662i	0.401726i
$540$	0	0
$541$	24.3423i	1.04656i	0.852162	+	0.523278i	$0.175291\pi$
$541$	24.3423i	1.04656i	−0.852162	+	0.523278i	$0.824709\pi$
$542$	0	0
$543$	−8.11409	−0.348209
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 33.3789i	− 1.42718i	−0.700564	−	0.713589i	$-0.747068\pi$
$547$	− 33.3789i	− 1.42718i	0.700564	−	0.713589i	$-0.252932\pi$
$548$	0	0
$549$	− 11.0183i	− 0.470251i
$550$	0	0
$551$	19.7827	0.842770
$552$	0	0
$553$	13.3239	0.566592
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.5808i	1.29575i	0.761747	+	0.647875i	$0.224342\pi$
$557$	30.5808i	1.29575i	−0.761747	+	0.647875i	$0.775658\pi$
$558$	0	0
$559$	41.8385	1.76958
$560$	0	0
$561$	−4.45963	−0.188286
$562$	0	0
$563$	− 7.02792i	− 0.296192i	−0.988973	−	0.148096i	$-0.952686\pi$
$563$	− 7.02792i	− 0.296192i	0.988973	−	0.148096i	$-0.0473144\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.05705	0.170380
$568$	0	0
$569$	24.3510	1.02085	0.510423	−	0.859924i	$-0.329489\pi$
$569$	24.3510	1.02085	0.510423	+	0.859924i	$0.329489\pi$
$570$	0	0
$571$	− 20.0992i	− 0.841125i	−0.907263	−	0.420563i	$-0.861833\pi$
$571$	− 20.0992i	− 0.841125i	0.907263	−	0.420563i	$-0.138167\pi$
$572$	0	0
$573$	24.9193i	1.04102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.76899	−0.198536	−0.0992678	−	0.995061i	$-0.531650\pi$
$577$	−4.76899	−0.198536	−0.0992678	+	0.995061i	$0.531650\pi$
$578$	0	0
$579$	1.03951i	0.0432004i
$580$	0	0
$581$	− 40.1296i	− 1.66486i
$582$	0	0
$583$	−0.226584	−0.00938413
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.21733i	0.339165i	0.985516	+	0.169583i	$0.0542420\pi$
$587$	8.21733i	0.339165i	−0.985516	+	0.169583i	$0.945758\pi$
$588$	0	0
$589$	− 8.55495i	− 0.352501i
$590$	0	0
$591$	9.66152	0.397422
$592$	0	0
$593$	−7.76120	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$593$	−7.76120	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 23.0668i	− 0.944062i
$598$	0	0
$599$	5.64903	0.230813	0.115407	−	0.993318i	$-0.463183\pi$
$599$	5.64903	0.230813	0.115407	+	0.993318i	$0.463183\pi$
$600$	0	0
$601$	−37.7299	−1.53903	−0.769516	−	0.638627i	$-0.779503\pi$
$601$	−37.7299	−1.53903	−0.769516	+	0.638627i	$0.779503\pi$
$602$	0	0
$603$	8.45963i	0.344503i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.113292	0.00459837	0.00229919	−	0.999997i	$-0.499268\pi$
$607$	0.113292	0.00459837	0.00229919	+	0.999997i	$0.499268\pi$
$608$	0	0
$609$	−30.8106	−1.24851
$610$	0	0
$611$	− 12.8831i	− 0.521194i
$612$	0	0
$613$	− 0.703366i	− 0.0284087i	−0.999899	−	0.0142044i	$-0.995478\pi$
$613$	− 0.703366i	− 0.0284087i	0.999899	−	0.0142044i	$-0.00452154\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.4809	0.985564	0.492782	−	0.870153i	$-0.335980\pi$
$617$	24.4809	0.985564	0.492782	+	0.870153i	$0.335980\pi$
$618$	0	0
$619$	− 39.4966i	− 1.58750i	−0.608243	−	0.793751i	$-0.708126\pi$
$619$	− 39.4966i	− 1.58750i	0.608243	−	0.793751i	$-0.291874\pi$
$620$	0	0
$621$	3.53729i	0.141947i
$622$	0	0
$623$	−50.1084	−2.00755
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.56829i	0.102568i
$628$	0	0
$629$	4.27748i	0.170554i
$630$	0	0
$631$	−17.3400	−0.690294	−0.345147	−	0.938549i	$-0.612171\pi$
$631$	−17.3400	−0.690294	−0.345147	+	0.938549i	$0.612171\pi$
$632$	0	0
$633$	6.44154	0.256028
$634$	0	0
$635$	0	0
$636$	0	0
$637$	46.7842i	1.85366i
$638$	0	0
$639$	1.43171	0.0566374
$640$	0	0
$641$	38.7019	1.52863	0.764317	−	0.644840i	$-0.223076\pi$
$641$	38.7019	1.52863	0.764317	+	0.644840i	$0.223076\pi$
$642$	0	0
$643$	1.13659i	0.0448227i	0.999749	+	0.0224113i	$0.00713435\pi$
$643$	1.13659i	0.0448227i	−0.999749	+	0.0224113i	$0.992866\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.54868	−0.257455	−0.128728	−	0.991680i	$-0.541089\pi$
$647$	−6.54868	−0.257455	−0.128728	+	0.991680i	$0.541089\pi$
$648$	0	0
$649$	−8.97208	−0.352185
$650$	0	0
$651$	13.3239i	0.522206i
$652$	0	0
$653$	8.74226i	0.342111i	0.985261	+	0.171056i	$0.0547178\pi$
$653$	8.74226i	0.342111i	−0.985261	+	0.171056i	$0.945282\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11.9507	−0.466242
$658$	0	0
$659$	35.5336i	1.38419i	0.721804	+	0.692097i	$0.243313\pi$
$659$	35.5336i	1.38419i	−0.721804	+	0.692097i	$0.756687\pi$
$660$	0	0
$661$	− 23.3028i	− 0.906373i	−0.891416	−	0.453186i	$-0.850287\pi$
$661$	− 23.3028i	− 0.906373i	0.891416	−	0.453186i	$-0.149713\pi$
$662$	0	0
$663$	−22.3704	−0.868795
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 26.8634i	− 1.04016i
$668$	0	0
$669$	17.9796i	0.695133i
$670$	0	0
$671$	−10.8634	−0.419377
$672$	0	0
$673$	−14.3634	−0.553670	−0.276835	−	0.960917i	$-0.589286\pi$
$673$	−14.3634	−0.553670	−0.276835	+	0.960917i	$0.589286\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 24.3076i	− 0.934217i	−0.884200	−	0.467109i	$-0.845296\pi$
$677$	− 24.3076i	− 0.934217i	0.884200	−	0.467109i	$-0.154704\pi$
$678$	0	0
$679$	−13.1366	−0.504136
$680$	0	0
$681$	−7.02792	−0.269311
$682$	0	0
$683$	− 38.5933i	− 1.47673i	−0.674401	−	0.738365i	$-0.735598\pi$
$683$	− 38.5933i	− 1.47673i	0.674401	−	0.738365i	$-0.264402\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.17034	−0.159108
$688$	0	0
$689$	−1.13659	−0.0433006
$690$	0	0
$691$	− 13.4090i	− 0.510102i	−0.966928	−	0.255051i	$-0.917908\pi$
$691$	− 13.4090i	− 0.510102i	0.966928	−	0.255051i	$-0.0820923\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.57053	0.0973657
$698$	0	0
$699$	− 23.9894i	− 0.907362i
$700$	0	0
$701$	− 15.6013i	− 0.589253i	−0.955613	−	0.294626i	$-0.904805\pi$
$701$	− 15.6013i	− 0.589253i	0.955613	−	0.294626i	$-0.0951952\pi$
$702$	0	0
$703$	2.46339	0.0929086
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.6740i	0.664699i
$708$	0	0
$709$	15.7873i	0.592906i	0.955047	+	0.296453i	$0.0958038\pi$
$709$	15.7873i	0.592906i	−0.955047	+	0.296453i	$0.904196\pi$
$710$	0	0
$711$	−3.28415	−0.123165
$712$	0	0
$713$	−11.6170	−0.435060
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.91926i	0.333096i
$718$	0	0
$719$	−22.5683	−0.841655	−0.420828	−	0.907141i	$-0.638260\pi$
$719$	−22.5683	−0.841655	−0.420828	+	0.907141i	$0.638260\pi$
$720$	0	0
$721$	61.1616	2.27778
$722$	0	0
$723$	− 16.3510i	− 0.608099i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.79096	−0.103511	−0.0517554	−	0.998660i	$-0.516482\pi$
$727$	−2.79096	−0.103511	−0.0517554	+	0.998660i	$0.516482\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 38.2649i	− 1.41528i
$732$	0	0
$733$	16.4860i	0.608926i	0.952524	+	0.304463i	$0.0984769\pi$
$733$	16.4860i	0.608926i	−0.952524	+	0.304463i	$0.901523\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.34068	0.307233
$738$	0	0
$739$	− 14.8894i	− 0.547714i	−0.961770	−	0.273857i	$-0.911701\pi$
$739$	− 14.8894i	− 0.547714i	0.961770	−	0.273857i	$-0.0882995\pi$
$740$	0	0
$741$	12.8831i	0.473272i
$742$	0	0
$743$	−41.4301	−1.51992	−0.759961	−	0.649968i	$-0.774782\pi$
$743$	−41.4301	−1.51992	−0.759961	+	0.649968i	$0.774782\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.89134i	0.361905i
$748$	0	0
$749$	16.2282i	0.592965i
$750$	0	0
$751$	−30.5544	−1.11494	−0.557472	−	0.830195i	$-0.688229\pi$
$751$	−30.5544	−1.11494	−0.557472	+	0.830195i	$0.688229\pi$
$752$	0	0
$753$	−4.22391	−0.153928
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.433223i	0.0157457i	0.999969	+	0.00787287i	$0.00250604\pi$
$757$	0.433223i	0.0157457i	−0.999969	+	0.00787287i	$0.997494\pi$
$758$	0	0
$759$	3.48755	0.126590
$760$	0	0
$761$	−14.9193	−0.540823	−0.270411	−	0.962745i	$-0.587160\pi$
$761$	−14.9193	−0.540823	−0.270411	+	0.962745i	$0.587160\pi$
$762$	0	0
$763$	− 16.9193i	− 0.612518i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.0057	−1.62506
$768$	0	0
$769$	−31.3789	−1.13155	−0.565776	−	0.824559i	$-0.691423\pi$
$769$	−31.3789	−1.13155	−0.565776	+	0.824559i	$0.691423\pi$
$770$	0	0
$771$	24.6952i	0.889375i
$772$	0	0
$773$	− 13.4192i	− 0.482656i	−0.970444	−	0.241328i	$-0.922417\pi$
$773$	− 13.4192i	− 0.482656i	0.970444	−	0.241328i	$-0.0775829\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.83662	−0.137638
$778$	0	0
$779$	− 1.48036i	− 0.0530395i
$780$	0	0
$781$	− 1.41157i	− 0.0505101i
$782$	0	0
$783$	7.59434	0.271400
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 36.4846i	− 1.30054i	−0.759705	−	0.650268i	$-0.774657\pi$
$787$	− 36.4846i	− 1.30054i	0.759705	−	0.650268i	$-0.225343\pi$
$788$	0	0
$789$	− 14.6628i	− 0.522009i
$790$	0	0
$791$	−5.21438	−0.185402
$792$	0	0
$793$	−54.4931	−1.93511
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.0683i	− 0.923388i	−0.887039	−	0.461694i	$-0.847242\pi$
$797$	− 26.0683i	− 0.923388i	0.887039	−	0.461694i	$-0.152758\pi$
$798$	0	0
$799$	−11.7827	−0.416841
$800$	0	0
$801$	12.3510	0.436400
$802$	0	0
$803$	11.7827i	0.415801i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.5381	0.406160
$808$	0	0
$809$	35.6212	1.25237	0.626187	−	0.779673i	$-0.284615\pi$
$809$	35.6212	1.25237	0.626187	+	0.779673i	$0.284615\pi$
$810$	0	0
$811$	43.8935i	1.54131i	0.637253	+	0.770654i	$0.280071\pi$
$811$	43.8935i	1.54131i	−0.637253	+	0.770654i	$0.719929\pi$
$812$	0	0
$813$	− 5.63511i	− 0.197632i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−22.0367	−0.770966
$818$	0	0
$819$	− 20.0648i	− 0.701121i
$820$	0	0
$821$	28.8058i	1.00533i	0.864482	+	0.502665i	$0.167647\pi$
$821$	28.8058i	1.00533i	−0.864482	+	0.502665i	$0.832353\pi$
$822$	0	0
$823$	−27.9585	−0.974571	−0.487286	−	0.873243i	$-0.662013\pi$
$823$	−27.9585	−0.974571	−0.487286	+	0.873243i	$0.662013\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14.8634i	− 0.516851i	−0.966031	−	0.258426i	$-0.916796\pi$
$827$	− 14.8634i	− 0.516851i	0.966031	−	0.258426i	$-0.0832037\pi$
$828$	0	0
$829$	− 41.7678i	− 1.45065i	−0.688404	−	0.725327i	$-0.741688\pi$
$829$	− 41.7678i	− 1.45065i	0.688404	−	0.725327i	$-0.258312\pi$
$830$	0	0
$831$	17.4053	0.603783
$832$	0	0
$833$	42.7881	1.48252
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.28415i	− 0.113517i
$838$	0	0
$839$	21.6490	0.747408	0.373704	−	0.927548i	$-0.378088\pi$
$839$	21.6490	0.747408	0.373704	+	0.927548i	$0.378088\pi$
$840$	0	0
$841$	−28.6740	−0.988759
$842$	0	0
$843$	21.7827i	0.750235i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	40.6838	1.39791
$848$	0	0
$849$	21.5962	0.741180
$850$	0	0
$851$	− 3.34510i	− 0.114669i
$852$	0	0
$853$	− 49.1880i	− 1.68416i	−0.539350	−	0.842082i	$-0.681330\pi$
$853$	− 49.1880i	− 1.68416i	0.539350	−	0.842082i	$-0.318670\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.65849	−0.0908123	−0.0454062	−	0.998969i	$-0.514458\pi$
$857$	−2.65849	−0.0908123	−0.0454062	+	0.998969i	$0.514458\pi$
$858$	0	0
$859$	4.57680i	0.156158i	0.996947	+	0.0780792i	$0.0248787\pi$
$859$	4.57680i	0.156158i	−0.996947	+	0.0780792i	$0.975121\pi$
$860$	0	0
$861$	2.30560i	0.0785746i
$862$	0	0
$863$	34.0218	1.15812	0.579058	−	0.815287i	$-0.303421\pi$
$863$	34.0218	1.15812	0.579058	+	0.815287i	$0.303421\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.45963i	0.117495i
$868$	0	0
$869$	3.23797i	0.109841i
$870$	0	0
$871$	41.8385	1.41764
$872$	0	0
$873$	3.23797	0.109589
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 11.7563i	− 0.396981i	−0.980103	−	0.198490i	$-0.936396\pi$
$877$	− 11.7563i	− 0.396981i	0.980103	−	0.198490i	$-0.0636039\pi$
$878$	0	0
$879$	32.0125	1.07975
$880$	0	0
$881$	13.2702	0.447085	0.223543	−	0.974694i	$-0.428238\pi$
$881$	13.2702	0.447085	0.223543	+	0.974694i	$0.428238\pi$
$882$	0	0
$883$	17.1366i	0.576692i	0.957526	+	0.288346i	$0.0931054\pi$
$883$	17.1366i	0.576692i	−0.957526	+	0.288346i	$0.906895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.0883	1.07742	0.538709	−	0.842492i	$-0.318912\pi$
$887$	32.0883	1.07742	0.538709	+	0.842492i	$0.318912\pi$
$888$	0	0
$889$	4.67696	0.156860
$890$	0	0
$891$	0.985939i	0.0330302i
$892$	0	0
$893$	6.78562i	0.227072i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.4943	0.584117
$898$	0	0
$899$	24.9409i	0.831827i
$900$	0	0
$901$	1.03951i	0.0346310i
$902$	0	0
$903$	34.3211	1.14214
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.4596i	1.34344i	0.740805	+	0.671720i	$0.234444\pi$
$907$	40.4596i	1.34344i	−0.740805	+	0.671720i	$0.765556\pi$
$908$	0	0
$909$	− 4.35637i	− 0.144492i
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	9.75225	0.322752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.7827i	0.521190i
$918$	0	0
$919$	−50.8495	−1.67737	−0.838685	−	0.544617i	$-0.816675\pi$
$919$	−50.8495	−1.67737	−0.838685	+	0.544617i	$0.816675\pi$
$920$	0	0
$921$	1.13659	0.0374519
$922$	0	0
$923$	− 7.08074i	− 0.233065i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−15.0754	−0.495141
$928$	0	0
$929$	12.6461	0.414904	0.207452	−	0.978245i	$-0.433483\pi$
$929$	12.6461	0.414904	0.207452	+	0.978245i	$0.433483\pi$
$930$	0	0
$931$	− 24.6416i	− 0.807596i
$932$	0	0
$933$	− 5.13659i	− 0.168164i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.7971	−1.33278	−0.666391	−	0.745603i	$-0.732162\pi$
$937$	−40.7971	−1.33278	−0.666391	+	0.745603i	$0.732162\pi$
$938$	0	0
$939$	23.0762i	0.753063i
$940$	0	0
$941$	14.1086i	0.459928i	0.973199	+	0.229964i	$0.0738608\pi$
$941$	14.1086i	0.459928i	−0.973199	+	0.229964i	$0.926139\pi$
$942$	0	0
$943$	−2.01022	−0.0654619
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.8385i	1.48955i	0.667315	+	0.744776i	$0.267444\pi$
$947$	45.8385i	1.48955i	−0.667315	+	0.744776i	$0.732556\pi$
$948$	0	0
$949$	59.1043i	1.91861i
$950$	0	0
$951$	1.66152	0.0538785
$952$	0	0
$953$	−21.9104	−0.709747	−0.354873	−	0.934914i	$-0.615476\pi$
$953$	−21.9104	−0.709747	−0.354873	+	0.934914i	$0.615476\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.48755i	− 0.242038i
$958$	0	0
$959$	−71.0529	−2.29442
$960$	0	0
$961$	−20.2144	−0.652077
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.0359	−0.451364	−0.225682	−	0.974201i	$-0.572461\pi$
$967$	−14.0359	−0.451364	−0.225682	+	0.974201i	$0.572461\pi$
$968$	0	0
$969$	11.7827	0.378514
$970$	0	0
$971$	21.6494i	0.694762i	0.937724	+	0.347381i	$0.112929\pi$
$971$	21.6494i	0.694762i	−0.937724	+	0.347381i	$0.887071\pi$
$972$	0	0
$973$	68.4068i	2.19302i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.60225	0.211225	0.105612	−	0.994407i	$-0.466320\pi$
$977$	6.60225	0.211225	0.105612	+	0.994407i	$0.466320\pi$
$978$	0	0
$979$	− 12.1773i	− 0.389188i
$980$	0	0
$981$	4.17034i	0.133149i
$982$	0	0
$983$	53.8600	1.71787	0.858934	−	0.512087i	$-0.171127\pi$
$983$	53.8600	1.71787	0.858934	+	0.512087i	$0.171127\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 10.5683i	− 0.336393i
$988$	0	0
$989$	29.9242i	0.951534i
$990$	0	0
$991$	−29.7129	−0.943861	−0.471931	−	0.881636i	$-0.656443\pi$
$991$	−29.7129	−0.943861	−0.471931	+	0.881636i	$0.656443\pi$
$992$	0	0
$993$	25.9077	0.822156
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 16.6506i	− 0.527328i	−0.964615	−	0.263664i	$-0.915069\pi$
$997$	− 16.6506i	− 0.527328i	0.964615	−	0.263664i	$-0.0849311\pi$
$998$	0	0
$999$	0.945668	0.0299196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.k.f.1201.12		12
3.2	odd	2		7200.2.k.u.3601.11		12
4.3	odd	2		600.2.k.f.301.1		12
5.2	odd	4		480.2.d.b.49.1		6
5.3	odd	4		480.2.d.a.49.5		6
5.4	even	2	inner	2400.2.k.f.1201.1		12
8.3	odd	2		600.2.k.f.301.2		12
8.5	even	2	inner	2400.2.k.f.1201.6		12
12.11	even	2		1800.2.k.u.901.12		12
15.2	even	4		1440.2.d.f.1009.6		6
15.8	even	4		1440.2.d.e.1009.2		6
15.14	odd	2		7200.2.k.u.3601.1		12
20.3	even	4		120.2.d.a.109.4	yes	6
20.7	even	4		120.2.d.b.109.3	yes	6
20.19	odd	2		600.2.k.f.301.12		12
24.5	odd	2		7200.2.k.u.3601.12		12
24.11	even	2		1800.2.k.u.901.11		12
40.3	even	4		120.2.d.b.109.4	yes	6
40.13	odd	4		480.2.d.b.49.2		6
40.19	odd	2		600.2.k.f.301.11		12
40.27	even	4		120.2.d.a.109.3	✓	6
40.29	even	2	inner	2400.2.k.f.1201.7		12
40.37	odd	4		480.2.d.a.49.6		6
60.23	odd	4		360.2.d.f.109.3		6
60.47	odd	4		360.2.d.e.109.4		6
60.59	even	2		1800.2.k.u.901.1		12
80.3	even	4		3840.2.f.l.769.9		12
80.13	odd	4		3840.2.f.m.769.3		12
80.27	even	4		3840.2.f.l.769.10		12
80.37	odd	4		3840.2.f.m.769.4		12
80.43	even	4		3840.2.f.l.769.4		12
80.53	odd	4		3840.2.f.m.769.10		12
80.67	even	4		3840.2.f.l.769.3		12
80.77	odd	4		3840.2.f.m.769.9		12
120.29	odd	2		7200.2.k.u.3601.2		12
120.53	even	4		1440.2.d.f.1009.5		6
120.59	even	2		1800.2.k.u.901.2		12
120.77	even	4		1440.2.d.e.1009.1		6
120.83	odd	4		360.2.d.e.109.3		6
120.107	odd	4		360.2.d.f.109.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.3	✓	6	40.27	even	4
120.2.d.a.109.4	yes	6	20.3	even	4
120.2.d.b.109.3	yes	6	20.7	even	4
120.2.d.b.109.4	yes	6	40.3	even	4
360.2.d.e.109.3		6	120.83	odd	4
360.2.d.e.109.4		6	60.47	odd	4
360.2.d.f.109.3		6	60.23	odd	4
360.2.d.f.109.4		6	120.107	odd	4
480.2.d.a.49.5		6	5.3	odd	4
480.2.d.a.49.6		6	40.37	odd	4
480.2.d.b.49.1		6	5.2	odd	4
480.2.d.b.49.2		6	40.13	odd	4
600.2.k.f.301.1		12	4.3	odd	2
600.2.k.f.301.2		12	8.3	odd	2
600.2.k.f.301.11		12	40.19	odd	2
600.2.k.f.301.12		12	20.19	odd	2
1440.2.d.e.1009.1		6	120.77	even	4
1440.2.d.e.1009.2		6	15.8	even	4
1440.2.d.f.1009.5		6	120.53	even	4
1440.2.d.f.1009.6		6	15.2	even	4
1800.2.k.u.901.1		12	60.59	even	2
1800.2.k.u.901.2		12	120.59	even	2
1800.2.k.u.901.11		12	24.11	even	2
1800.2.k.u.901.12		12	12.11	even	2
2400.2.k.f.1201.1		12	5.4	even	2	inner
2400.2.k.f.1201.6		12	8.5	even	2	inner
2400.2.k.f.1201.7		12	40.29	even	2	inner
2400.2.k.f.1201.12		12	1.1	even	1	trivial
3840.2.f.l.769.3		12	80.67	even	4
3840.2.f.l.769.4		12	80.43	even	4
3840.2.f.l.769.9		12	80.3	even	4
3840.2.f.l.769.10		12	80.27	even	4
3840.2.f.m.769.3		12	80.13	odd	4
3840.2.f.m.769.4		12	80.37	odd	4
3840.2.f.m.769.9		12	80.77	odd	4
3840.2.f.m.769.10		12	80.53	odd	4
7200.2.k.u.3601.1		12	15.14	odd	2
7200.2.k.u.3601.2		12	120.29	odd	2
7200.2.k.u.3601.11		12	3.2	odd	2
7200.2.k.u.3601.12		12	24.5	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.05705 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +4.05705 q^{7} -1.00000 q^{9} +0.985939i q^{11} +4.94567i q^{13} +4.52323 q^{17} -2.60492i q^{19} +4.05705i q^{21} -3.53729 q^{23} -1.00000i q^{27} +7.59434i q^{29} +3.28415 q^{31} -0.985939 q^{33} +0.945668i q^{37} -4.94567 q^{39} +0.568295 q^{41} -8.45963i q^{43} -2.60492 q^{47} +9.45963 q^{49} +4.52323i q^{51} +0.229815i q^{53} +2.60492 q^{57} +9.10003i q^{59} +11.0183i q^{61} -4.05705 q^{63} -8.45963i q^{67} -3.53729i q^{69} -1.43171 q^{71} +11.9507 q^{73} +4.00000i q^{77} +3.28415 q^{79} +1.00000 q^{81} -9.89134i q^{83} -7.59434 q^{87} -12.3510 q^{89} +20.0648i q^{91} +3.28415i q^{93} -3.23797 q^{97} -0.985939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) \) 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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