LMFDB - Embedded newform 7200.2.k.u.3601.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(3601,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, 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("7200.3601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.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:	$57.4922894553$
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:	$\Z[a_1, \ldots, a_{53}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.4
Root			$-0.450129 - 1.34067i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.u.3601.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64265 q^{7} +O(q^{10})$	$q-2.64265 q^{7} +1.51363i q^{11} -3.87086i q^{13} -3.31415 q^{17} +7.08582i q^{19} +4.82778 q^{23} -2.18513i q^{29} +7.36266 q^{31} -7.87086i q^{37} -8.72532 q^{41} +1.01641i q^{43} -7.08582 q^{47} -0.0164068 q^{49} +4.50820i q^{53} +6.79893i q^{59} +3.60104i q^{61} +1.01641i q^{67} -6.72532 q^{71} +15.5146 q^{73} -4.00000i q^{77} +7.36266 q^{79} -7.74173i q^{83} -14.7581 q^{89} +10.2293i q^{91} +11.1444 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 32 q^{31} + 8 q^{41} + 12 q^{49} + 32 q^{71} + 32 q^{79} - 40 q^{89}+O(q^{100}) $ 12 * q + 32 * q^31 + 8 * q^41 + 12 * q^49 + 32 * q^71 + 32 * q^79 - 40 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.64265	−0.998827	−0.499414	−	0.866364i	$-0.666451\pi$
$7$	−2.64265	−0.998827	−0.499414	+	0.866364i	$0.666451\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.51363i	0.456377i	0.973617	+	0.228189i	$0.0732803\pi$
$11$	1.51363i	0.456377i	−0.973617	+	0.228189i	$0.926720\pi$
$12$	0	0
$13$	− 3.87086i	− 1.07358i	−0.843714	−	0.536792i	$-0.819636\pi$
$13$	− 3.87086i	− 1.07358i	0.843714	−	0.536792i	$-0.180364\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.31415	−0.803800	−0.401900	−	0.915684i	$-0.631650\pi$
$17$	−3.31415	−0.803800	−0.401900	+	0.915684i	$0.631650\pi$
$18$	0	0
$19$	7.08582i	1.62560i	0.582545	+	0.812799i	$0.302057\pi$
$19$	7.08582i	1.62560i	−0.582545	+	0.812799i	$0.697943\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.82778	1.00666	0.503331	−	0.864094i	$-0.332108\pi$
$23$	4.82778	1.00666	0.503331	+	0.864094i	$0.332108\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.18513i	− 0.405769i	−0.979203	−	0.202885i	$-0.934968\pi$
$29$	− 2.18513i	− 0.405769i	0.979203	−	0.202885i	$-0.0650316\pi$
$30$	0	0
$31$	7.36266	1.32237	0.661187	−	0.750222i	$-0.270053\pi$
$31$	7.36266	1.32237	0.661187	+	0.750222i	$0.270053\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.87086i	− 1.29396i	−0.762506	−	0.646981i	$-0.776031\pi$
$37$	− 7.87086i	− 1.29396i	0.762506	−	0.646981i	$-0.223969\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.72532	−1.36267	−0.681333	−	0.731973i	$-0.738600\pi$
$41$	−8.72532	−1.36267	−0.681333	+	0.731973i	$0.738600\pi$
$42$	0	0
$43$	1.01641i	0.155001i	0.996992	+	0.0775003i	$0.0246939\pi$
$43$	1.01641i	0.155001i	−0.996992	+	0.0775003i	$0.975306\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.08582	−1.03357	−0.516786	−	0.856114i	$-0.672872\pi$
$47$	−7.08582	−1.03357	−0.516786	+	0.856114i	$0.672872\pi$
$48$	0	0
$49$	−0.0164068	−0.00234382
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.50820i	0.619249i	0.950859	+	0.309625i	$0.100203\pi$
$53$	4.50820i	0.619249i	−0.950859	+	0.309625i	$0.899797\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.79893i	0.885145i	0.896733	+	0.442573i	$0.145934\pi$
$59$	6.79893i	0.885145i	−0.896733	+	0.442573i	$0.854066\pi$
$60$	0	0
$61$	3.60104i	0.461065i	0.973065	+	0.230533i	$0.0740469\pi$
$61$	3.60104i	0.461065i	−0.973065	+	0.230533i	$0.925953\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.01641i	0.124174i	0.998071	+	0.0620869i	$0.0197756\pi$
$67$	1.01641i	0.124174i	−0.998071	+	0.0620869i	$0.980224\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.72532	−0.798149	−0.399074	−	0.916919i	$-0.630669\pi$
$71$	−6.72532	−0.798149	−0.399074	+	0.916919i	$0.630669\pi$
$72$	0	0
$73$	15.5146	1.81585	0.907925	−	0.419132i	$-0.137666\pi$
$73$	15.5146	1.81585	0.907925	+	0.419132i	$0.137666\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	7.36266	0.828364	0.414182	−	0.910194i	$-0.364068\pi$
$79$	7.36266	0.828364	0.414182	+	0.910194i	$0.364068\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.74173i	− 0.849765i	−0.905248	−	0.424883i	$-0.860315\pi$
$83$	− 7.74173i	− 0.849765i	0.905248	−	0.424883i	$-0.139685\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.7581	−1.56436	−0.782180	−	0.623053i	$-0.785892\pi$
$89$	−14.7581	−1.56436	−0.782180	+	0.623053i	$0.785892\pi$
$90$	0	0
$91$	10.2293i	1.07233i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.1444	1.13154	0.565769	−	0.824563i	$-0.308579\pi$
$97$	11.1444	1.13154	0.565769	+	0.824563i	$0.308579\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 13.3295i	− 1.32633i	−0.748471	−	0.663167i	$-0.769212\pi$
$101$	− 13.3295i	− 1.32633i	0.748471	−	0.663167i	$-0.230788\pi$
$102$	0	0
$103$	0.958386	0.0944326	0.0472163	−	0.998885i	$-0.484965\pi$
$103$	0.958386	0.0944326	0.0472163	+	0.998885i	$0.484965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	− 0.769233i	− 0.0736792i	−0.999321	−	0.0368396i	$-0.988271\pi$
$109$	− 0.769233i	− 0.0736792i	0.999321	−	0.0368396i	$-0.0117291\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.4585	1.36014	0.680071	−	0.733146i	$-0.261949\pi$
$113$	14.4585	1.36014	0.680071	+	0.733146i	$0.261949\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.75814	0.802857
$120$	0	0
$121$	8.70892	0.791720
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.5290	−1.02303	−0.511516	−	0.859274i	$-0.670916\pi$
$127$	−11.5290	−1.02303	−0.511516	+	0.859274i	$0.670916\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 7.37270i	− 0.644156i	−0.946713	−	0.322078i	$-0.895619\pi$
$131$	− 7.37270i	− 0.644156i	0.946713	−	0.322078i	$-0.104381\pi$
$132$	0	0
$133$	− 18.7253i	− 1.62369i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.88792	0.332167	0.166084	−	0.986112i	$-0.446888\pi$
$137$	3.88792	0.332167	0.166084	+	0.986112i	$0.446888\pi$
$138$	0	0
$139$	− 14.6291i	− 1.24083i	−0.784275	−	0.620414i	$-0.786965\pi$
$139$	− 14.6291i	− 1.24083i	0.784275	−	0.620414i	$-0.213035\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.85907	0.489960
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.0715i	− 0.907010i	−0.891254	−	0.453505i	$-0.850173\pi$
$149$	− 11.0715i	− 0.907010i	0.891254	−	0.453505i	$-0.149827\pi$
$150$	0	0
$151$	−0.637339	−0.0518659	−0.0259329	−	0.999664i	$-0.508256\pi$
$151$	−0.637339	−0.0518659	−0.0259329	+	0.999664i	$0.508256\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.129135i	0.0103061i	0.999987	+	0.00515306i	$0.00164028\pi$
$157$	0.129135i	0.0103061i	−0.999987	+	0.00515306i	$0.998360\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.7581	−1.00548
$162$	0	0
$163$	− 19.4835i	− 1.52606i	−0.646362	−	0.763031i	$-0.723710\pi$
$163$	− 19.4835i	− 1.52606i	0.646362	−	0.763031i	$-0.276290\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.80052	0.139328	0.0696641	−	0.997571i	$-0.477807\pi$
$167$	1.80052	0.139328	0.0696641	+	0.997571i	$0.477807\pi$
$168$	0	0
$169$	−1.98359	−0.152584
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 23.2335i	− 1.76641i	−0.468985	−	0.883206i	$-0.655380\pi$
$173$	− 23.2335i	− 1.76641i	0.468985	−	0.883206i	$-0.344620\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.85664i	0.213515i	0.994285	+	0.106757i	$0.0340468\pi$
$179$	2.85664i	0.213515i	−0.994285	+	0.106757i	$0.965953\pi$
$180$	0	0
$181$	− 5.28530i	− 0.392853i	−0.980519	−	0.196427i	$-0.937066\pi$
$181$	− 5.28530i	− 0.392853i	0.980519	−	0.196427i	$-0.0629337\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.01641i	− 0.366836i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.96719	−0.431770	−0.215885	−	0.976419i	$-0.569264\pi$
$191$	−5.96719	−0.431770	−0.215885	+	0.976419i	$0.569264\pi$
$192$	0	0
$193$	−14.9409	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$193$	−14.9409	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.23353i	− 0.230379i	−0.993344	−	0.115190i	$-0.963252\pi$
$197$	− 3.23353i	− 0.230379i	0.993344	−	0.115190i	$-0.0367476\pi$
$198$	0	0
$199$	8.12080	0.575668	0.287834	−	0.957680i	$-0.407065\pi$
$199$	8.12080	0.575668	0.287834	+	0.957680i	$0.407065\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.77454i	0.405293i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.7253	−0.741886
$210$	0	0
$211$	− 13.7141i	− 0.944119i	−0.881567	−	0.472059i	$-0.843511\pi$
$211$	− 13.7141i	− 0.944119i	0.881567	−	0.472059i	$-0.156489\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.4569	−1.32082
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.8286i	0.862947i
$222$	0	0
$223$	9.84472	0.659251	0.329626	−	0.944112i	$-0.393077\pi$
$223$	9.84472	0.659251	0.329626	+	0.944112i	$0.393077\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.70892i	− 0.378914i	−0.981889	−	0.189457i	$-0.939327\pi$
$227$	− 5.70892i	− 0.378914i	0.981889	−	0.189457i	$-0.0606728\pi$
$228$	0	0
$229$	0.769233i	0.0508324i	0.999677	+	0.0254162i	$0.00809109\pi$
$229$	0.769233i	0.0508324i	−0.999677	+	0.0254162i	$0.991909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.4008	−1.20548	−0.602739	−	0.797939i	$-0.705924\pi$
$233$	−18.4008	−1.20548	−0.602739	+	0.797939i	$0.705924\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.0328	0.648969	0.324484	−	0.945891i	$-0.394809\pi$
$239$	10.0328	0.648969	0.324484	+	0.945891i	$0.394809\pi$
$240$	0	0
$241$	10.7581	0.692992	0.346496	−	0.938051i	$-0.387371\pi$
$241$	10.7581	0.692992	0.346496	+	0.938051i	$0.387371\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.4282	1.74522
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.6580i	0.798966i	0.916741	+	0.399483i	$0.130810\pi$
$251$	12.6580i	0.798966i	−0.916741	+	0.399483i	$0.869190\pi$
$252$	0	0
$253$	7.30749i	0.459418i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.3110	0.830316	0.415158	−	0.909749i	$-0.363726\pi$
$257$	13.3110	0.830316	0.415158	+	0.909749i	$0.363726\pi$
$258$	0	0
$259$	20.7999i	1.29244i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.4256	−1.13617	−0.568087	−	0.822969i	$-0.692316\pi$
$263$	−18.4256	−1.13617	−0.568087	+	0.822969i	$0.692316\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.86940i	− 0.235921i	−0.993018	−	0.117961i	$-0.962364\pi$
$269$	− 3.86940i	− 0.235921i	0.993018	−	0.117961i	$-0.0376357\pi$
$270$	0	0
$271$	17.3955	1.05670	0.528350	−	0.849027i	$-0.322811\pi$
$271$	17.3955	1.05670	0.528350	+	0.849027i	$0.322811\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.887271i	0.0533110i	0.999645	+	0.0266555i	$0.00848571\pi$
$277$	0.887271i	0.0533110i	−0.999645	+	0.0266555i	$0.991514\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.4835	0.804356	0.402178	−	0.915562i	$-0.368253\pi$
$281$	13.4835	0.804356	0.402178	+	0.915562i	$0.368253\pi$
$282$	0	0
$283$	− 28.4342i	− 1.69024i	−0.534577	−	0.845120i	$-0.679529\pi$
$283$	− 28.4342i	− 1.69024i	0.534577	−	0.845120i	$-0.320471\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.0580	1.36107
$288$	0	0
$289$	−6.01641	−0.353906
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.99166i	− 0.466878i	−0.972371	−	0.233439i	$-0.925002\pi$
$293$	− 7.99166i	− 0.466878i	0.972371	−	0.233439i	$-0.0749979\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 18.6877i	− 1.08074i
$300$	0	0
$301$	− 2.68601i	− 0.154819i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.4506i	− 0.995961i	−0.867188	−	0.497980i	$-0.834075\pi$
$307$	− 17.4506i	− 0.995961i	0.867188	−	0.497980i	$-0.165925\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.4506	1.21635	0.608177	−	0.793801i	$-0.291901\pi$
$311$	21.4506	1.21635	0.608177	+	0.793801i	$0.291901\pi$
$312$	0	0
$313$	−7.73879	−0.437422	−0.218711	−	0.975790i	$-0.570185\pi$
$313$	−7.73879	−0.437422	−0.218711	+	0.975790i	$0.570185\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.2335i	− 0.630938i	−0.948936	−	0.315469i	$-0.897838\pi$
$317$	− 11.2335i	− 0.630938i	0.948936	−	0.315469i	$-0.102162\pi$
$318$	0	0
$319$	3.30749	0.185184
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 23.4835i	− 1.30665i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18.7253	1.03236
$330$	0	0
$331$	8.00084i	0.439766i	0.975526	+	0.219883i	$0.0705676\pi$
$331$	8.00084i	0.439766i	−0.975526	+	0.219883i	$0.929432\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.5692	1.17495	0.587474	−	0.809243i	$-0.300123\pi$
$337$	21.5692	1.17495	0.587474	+	0.809243i	$0.300123\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.1444i	0.603501i
$342$	0	0
$343$	18.5419	1.00117
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.7089i	1.16540i	0.812689	+	0.582698i	$0.198003\pi$
$347$	21.7089i	1.16540i	−0.812689	+	0.582698i	$0.801997\pi$
$348$	0	0
$349$	− 24.7422i	− 1.32442i	−0.749318	−	0.662211i	$-0.769618\pi$
$349$	− 24.7422i	− 1.32442i	0.749318	−	0.662211i	$-0.230382\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.31415	0.176394	0.0881972	−	0.996103i	$-0.471889\pi$
$353$	3.31415	0.176394	0.0881972	+	0.996103i	$0.471889\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.7581	−0.884461	−0.442230	−	0.896902i	$-0.645813\pi$
$359$	−16.7581	−0.884461	−0.442230	+	0.896902i	$0.645813\pi$
$360$	0	0
$361$	−31.2088	−1.64257
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.5324	1.48938	0.744690	−	0.667411i	$-0.232597\pi$
$367$	28.5324	1.48938	0.744690	+	0.667411i	$0.232597\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11.9136i	− 0.618523i
$372$	0	0
$373$	− 37.5798i	− 1.94581i	−0.231211	−	0.972904i	$-0.574269\pi$
$373$	− 37.5798i	− 1.94581i	0.231211	−	0.972904i	$-0.425731\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.45836	−0.435628
$378$	0	0
$379$	− 6.74456i	− 0.346445i	−0.984883	−	0.173222i	$-0.944582\pi$
$379$	− 6.74456i	− 0.346445i	0.984883	−	0.173222i	$-0.0554179\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.8312	−1.11552	−0.557762	−	0.830001i	$-0.688340\pi$
$383$	−21.8312	−1.11552	−0.557762	+	0.830001i	$0.688340\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.81344i	0.446859i	0.974720	+	0.223429i	$0.0717252\pi$
$389$	8.81344i	0.446859i	−0.974720	+	0.223429i	$0.928275\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.821644i	− 0.0412372i	−0.999787	−	0.0206186i	$-0.993436\pi$
$397$	− 0.821644i	− 0.0412372i	0.999787	−	0.0206186i	$-0.00656356\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.7253	0.635472	0.317736	−	0.948179i	$-0.397077\pi$
$401$	12.7253	0.635472	0.317736	+	0.948179i	$0.397077\pi$
$402$	0	0
$403$	− 28.4999i	− 1.41968i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.9136	0.590535
$408$	0	0
$409$	2.25827	0.111664	0.0558321	−	0.998440i	$-0.482219\pi$
$409$	2.25827	0.111664	0.0558321	+	0.998440i	$0.482219\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 17.9672i	− 0.884107i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.4579i	1.63453i	0.576264	+	0.817263i	$0.304510\pi$
$419$	33.4579i	1.63453i	−0.576264	+	0.817263i	$0.695490\pi$
$420$	0	0
$421$	11.3398i	0.552669i	0.961061	+	0.276335i	$0.0891198\pi$
$421$	11.3398i	0.552669i	−0.961061	+	0.276335i	$0.910880\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 9.51627i	− 0.460525i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.6597	0.513459	0.256730	−	0.966483i	$-0.417355\pi$
$431$	10.6597	0.513459	0.256730	+	0.966483i	$0.417355\pi$
$432$	0	0
$433$	26.5132	1.27414	0.637072	−	0.770805i	$-0.280146\pi$
$433$	26.5132	1.27414	0.637072	+	0.770805i	$0.280146\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.2088i	1.63643i
$438$	0	0
$439$	−32.8789	−1.56923	−0.784613	−	0.619986i	$-0.787138\pi$
$439$	−32.8789	−1.56923	−0.784613	+	0.619986i	$0.787138\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.70892i	− 0.271239i	−0.990761	−	0.135619i	$-0.956698\pi$
$443$	− 5.70892i	− 0.271239i	0.990761	−	0.135619i	$-0.0433024\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	− 13.2069i	− 0.621890i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.94229	−0.184413	−0.0922064	−	0.995740i	$-0.529392\pi$
$457$	−3.94229	−0.184413	−0.0922064	+	0.995740i	$0.529392\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 33.8969i	− 1.57874i	−0.613920	−	0.789369i	$-0.710408\pi$
$461$	− 33.8969i	− 1.57874i	0.613920	−	0.789369i	$-0.289592\pi$
$462$	0	0
$463$	−22.8688	−1.06280	−0.531402	−	0.847120i	$-0.678335\pi$
$463$	−22.8688	−1.06280	−0.531402	+	0.847120i	$0.678335\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 15.7417i	− 0.728440i	−0.931313	−	0.364220i	$-0.881336\pi$
$467$	− 15.7417i	− 0.728440i	0.931313	−	0.364220i	$-0.118664\pi$
$468$	0	0
$469$	− 2.68601i	− 0.124028i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.53847	−0.0707388
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.6925	0.945465	0.472732	−	0.881206i	$-0.343268\pi$
$479$	20.6925	0.945465	0.472732	+	0.881206i	$0.343268\pi$
$480$	0	0
$481$	−30.4671	−1.38918
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−30.8401	−1.39750	−0.698750	−	0.715366i	$-0.746260\pi$
$487$	−30.8401	−1.39750	−0.698750	+	0.715366i	$0.746260\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.9737i	0.495238i	0.968858	+	0.247619i	$0.0796481\pi$
$491$	10.9737i	0.495238i	−0.968858	+	0.247619i	$0.920352\pi$
$492$	0	0
$493$	7.24186i	0.326157i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.7727	0.797213
$498$	0	0
$499$	3.71729i	0.166409i	0.996533	+	0.0832044i	$0.0265154\pi$
$499$	3.71729i	0.166409i	−0.996533	+	0.0832044i	$0.973485\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	39.9451	1.78107	0.890533	−	0.454919i	$-0.150332\pi$
$503$	39.9451	1.78107	0.890533	+	0.454919i	$0.150332\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 0.0728979i	− 0.00323114i	−0.999999	−	0.00161557i	$-0.999486\pi$
$509$	− 0.0728979i	− 0.00323114i	0.999999	−	0.00161557i	$-0.000514253\pi$
$510$	0	0
$511$	−40.9997	−1.81372
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 10.7253i	− 0.471699i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11.9672	0.524292	0.262146	−	0.965028i	$-0.415570\pi$
$521$	11.9672	0.524292	0.262146	+	0.965028i	$0.415570\pi$
$522$	0	0
$523$	16.0656i	0.702501i	0.936282	+	0.351250i	$0.114243\pi$
$523$	16.0656i	0.702501i	−0.936282	+	0.351250i	$0.885757\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.4010	−1.06292
$528$	0	0
$529$	0.307491	0.0133692
$530$	0	0
$531$	0	0
$532$	0	0
$533$	33.7745i	1.46294i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 0.0248338i	− 0.00106967i
$540$	0	0
$541$	− 15.8559i	− 0.681698i	−0.940118	−	0.340849i	$-0.889285\pi$
$541$	− 15.8559i	− 0.681698i	0.940118	−	0.340849i	$-0.110715\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.95078i	− 0.211680i	−0.994383	−	0.105840i	$-0.966247\pi$
$547$	− 4.95078i	− 0.211680i	0.994383	−	0.105840i	$-0.0337531\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.4835	0.659618
$552$	0	0
$553$	−19.4569	−0.827393
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.26634i	0.0536565i	0.999640	+	0.0268283i	$0.00854073\pi$
$557$	1.26634i	0.0536565i	−0.999640	+	0.0268283i	$0.991459\pi$
$558$	0	0
$559$	3.93437	0.166406
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.70892i	0.240602i	0.992737	+	0.120301i	$0.0383860\pi$
$563$	5.70892i	0.240602i	−0.992737	+	0.120301i	$0.961614\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.75814	0.115627	0.0578135	−	0.998327i	$-0.481587\pi$
$569$	2.75814	0.115627	0.0578135	+	0.998327i	$0.481587\pi$
$570$	0	0
$571$	25.7735i	1.07859i	0.842118	+	0.539294i	$0.181309\pi$
$571$	25.7735i	1.07859i	−0.842118	+	0.539294i	$0.818691\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−32.7135	−1.36188	−0.680941	−	0.732338i	$-0.738429\pi$
$577$	−32.7135	−1.36188	−0.680941	+	0.732338i	$0.738429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.4587i	0.848769i
$582$	0	0
$583$	−6.82376	−0.282611
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 43.4835i	− 1.79475i	−0.441264	−	0.897377i	$-0.645470\pi$
$587$	− 43.4835i	− 1.79475i	0.441264	−	0.897377i	$-0.354530\pi$
$588$	0	0
$589$	52.1705i	2.14965i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.83021	−0.321548	−0.160774	−	0.986991i	$-0.551399\pi$
$593$	−7.83021	−0.321548	−0.160774	+	0.986991i	$0.551399\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−32.7581	−1.33846	−0.669231	−	0.743055i	$-0.733376\pi$
$599$	−32.7581	−1.33846	−0.669231	+	0.743055i	$0.733376\pi$
$600$	0	0
$601$	17.8074	0.726377	0.363189	−	0.931716i	$-0.381688\pi$
$601$	17.8074	0.726377	0.363189	+	0.931716i	$0.381688\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.41188	0.138484	0.0692420	−	0.997600i	$-0.477942\pi$
$607$	3.41188	0.138484	0.0692420	+	0.997600i	$0.477942\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.4282i	1.10963i
$612$	0	0
$613$	− 36.6290i	− 1.47943i	−0.672920	−	0.739716i	$-0.734960\pi$
$613$	− 36.6290i	− 1.47943i	0.672920	−	0.739716i	$-0.265040\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.3979	−1.62636	−0.813180	−	0.582012i	$-0.802266\pi$
$617$	−40.3979	−1.62636	−0.813180	+	0.582012i	$0.802266\pi$
$618$	0	0
$619$	− 24.5172i	− 0.985430i	−0.870191	−	0.492715i	$-0.836004\pi$
$619$	− 24.5172i	− 0.985430i	0.870191	−	0.492715i	$-0.163996\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	39.0006	1.56252
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.0852i	1.04009i
$630$	0	0
$631$	−18.7805	−0.747640	−0.373820	−	0.927501i	$-0.621952\pi$
$631$	−18.7805	−0.747640	−0.373820	+	0.927501i	$0.621952\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.0635083i	0.00251629i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.5163	0.612856	0.306428	−	0.951894i	$-0.400866\pi$
$641$	15.5163	0.612856	0.306428	+	0.951894i	$0.400866\pi$
$642$	0	0
$643$	17.4506i	0.688186i	0.938936	+	0.344093i	$0.111814\pi$
$643$	17.4506i	0.688186i	−0.938936	+	0.344093i	$0.888186\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.1403	−0.516600	−0.258300	−	0.966065i	$-0.583162\pi$
$647$	−13.1403	−0.516600	−0.258300	+	0.966065i	$0.583162\pi$
$648$	0	0
$649$	−10.2911	−0.403960
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.7993i	− 0.579141i	−0.957157	−	0.289570i	$-0.906488\pi$
$653$	− 14.7993i	− 0.579141i	0.957157	−	0.289570i	$-0.0935124\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.99614i	− 0.311485i	−0.987798	−	0.155743i	$-0.950223\pi$
$659$	− 7.99614i	− 0.311485i	0.987798	−	0.155743i	$-0.0497771\pi$
$660$	0	0
$661$	0.915029i	0.0355905i	0.999842	+	0.0177953i	$0.00566470\pi$
$661$	0.915029i	0.0355905i	−0.999842	+	0.0177953i	$0.994335\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 10.5494i	− 0.408473i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.45065	−0.210420
$672$	0	0
$673$	34.3978	1.32594	0.662969	−	0.748647i	$-0.269296\pi$
$673$	34.3978	1.32594	0.662969	+	0.748647i	$0.269296\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 40.1676i	− 1.54377i	−0.635764	−	0.771884i	$-0.719315\pi$
$677$	− 40.1676i	− 1.54377i	0.635764	−	0.771884i	$-0.280685\pi$
$678$	0	0
$679$	−29.4506	−1.13021
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33.2580i	− 1.27258i	−0.771449	−	0.636291i	$-0.780468\pi$
$683$	− 33.2580i	− 1.27258i	0.771449	−	0.636291i	$-0.219532\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.4506	0.664817
$690$	0	0
$691$	− 50.2241i	− 1.91062i	−0.295611	−	0.955308i	$-0.595523\pi$
$691$	− 50.2241i	− 1.91062i	0.295611	−	0.955308i	$-0.404477\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	28.9170	1.09531
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.7543i	0.897188i	0.893736	+	0.448594i	$0.148075\pi$
$701$	23.7543i	0.897188i	−0.893736	+	0.448594i	$0.851925\pi$
$702$	0	0
$703$	55.7715	2.10346
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.2252i	1.32478i
$708$	0	0
$709$	36.3146i	1.36382i	0.731435	+	0.681911i	$0.238851\pi$
$709$	36.3146i	1.36382i	−0.731435	+	0.681911i	$0.761149\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.5453	1.33118
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.7253	1.14586	0.572931	−	0.819604i	$-0.305806\pi$
$719$	30.7253	1.14586	0.572931	+	0.819604i	$0.305806\pi$
$720$	0	0
$721$	−2.53268	−0.0943219
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.47445	−0.203036	−0.101518	−	0.994834i	$-0.532370\pi$
$727$	−5.47445	−0.203036	−0.101518	+	0.994834i	$0.532370\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.36852i	− 0.124589i
$732$	0	0
$733$	17.1455i	0.633285i	0.948545	+	0.316643i	$0.102556\pi$
$733$	17.1455i	0.633285i	−0.948545	+	0.316643i	$0.897444\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.53847	−0.0566701
$738$	0	0
$739$	11.6019i	0.426782i	0.976967	+	0.213391i	$0.0684508\pi$
$739$	11.6019i	0.426782i	−0.976967	+	0.213391i	$0.931549\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.6613	−0.868048	−0.434024	−	0.900901i	$-0.642907\pi$
$743$	−23.6613	−0.868048	−0.434024	+	0.900901i	$0.642907\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.5706i	0.386241i
$750$	0	0
$751$	11.4283	0.417024	0.208512	−	0.978020i	$-0.433138\pi$
$751$	11.4283	0.417024	0.208512	+	0.978020i	$0.433138\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 19.1784i	− 0.697049i	−0.937300	−	0.348525i	$-0.886683\pi$
$757$	− 19.1784i	− 0.697049i	0.937300	−	0.348525i	$-0.113317\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.03281	−0.146189	−0.0730947	−	0.997325i	$-0.523288\pi$
$761$	−4.03281	−0.146189	−0.0730947	+	0.997325i	$0.523288\pi$
$762$	0	0
$763$	2.03281i	0.0735928i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	26.3177	0.950279
$768$	0	0
$769$	−2.95078	−0.106408	−0.0532039	−	0.998584i	$-0.516943\pi$
$769$	−2.95078	−0.106408	−0.0532039	+	0.998584i	$0.516943\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	45.2663i	1.62812i	0.580783	+	0.814059i	$0.302747\pi$
$773$	45.2663i	1.62812i	−0.580783	+	0.814059i	$0.697253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 61.8260i	− 2.21515i
$780$	0	0
$781$	− 10.1797i	− 0.364257i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.9997i	1.88924i	0.328171	+	0.944618i	$0.393568\pi$
$787$	52.9997i	1.88924i	−0.328171	+	0.944618i	$0.606432\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38.2088	−1.35855
$792$	0	0
$793$	13.9391	0.494993
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.5738i	− 0.587075i	−0.955948	−	0.293538i	$-0.905167\pi$
$797$	− 16.5738i	− 0.587075i	0.955948	−	0.293538i	$-0.0948326\pi$
$798$	0	0
$799$	23.4835	0.830785
$800$	0	0
$801$	0	0
$802$	0	0
$803$	23.4835i	0.828713i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.5491	1.32016	0.660078	−	0.751197i	$-0.270523\pi$
$809$	37.5491	1.32016	0.660078	+	0.751197i	$0.270523\pi$
$810$	0	0
$811$	32.1102i	1.12754i	0.825931	+	0.563771i	$0.190650\pi$
$811$	32.1102i	1.12754i	−0.825931	+	0.563771i	$0.809350\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.20207	−0.251969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.3809i	1.02540i	0.858568	+	0.512699i	$0.171354\pi$
$821$	29.3809i	1.02540i	−0.858568	+	0.512699i	$0.828646\pi$
$822$	0	0
$823$	−28.3866	−0.989495	−0.494748	−	0.869037i	$-0.664740\pi$
$823$	−28.3866	−0.989495	−0.494748	+	0.869037i	$0.664740\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.45065i	− 0.0504439i	−0.999682	−	0.0252219i	$-0.991971\pi$
$827$	− 1.45065i	− 0.0504439i	0.999682	−	0.0252219i	$-0.00802924\pi$
$828$	0	0
$829$	− 37.4621i	− 1.30111i	−0.759458	−	0.650556i	$-0.774536\pi$
$829$	− 37.4621i	− 1.30111i	0.759458	−	0.650556i	$-0.225464\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.0543744	0.00188396
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.7581	−1.68332	−0.841659	−	0.540010i	$-0.818421\pi$
$839$	−48.7581	−1.68332	−0.841659	+	0.540010i	$0.818421\pi$
$840$	0	0
$841$	24.2252	0.835351
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−23.0146	−0.790791
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 37.9988i	− 1.30258i
$852$	0	0
$853$	4.37073i	0.149651i	0.997197	+	0.0748255i	$0.0238400\pi$
$853$	4.37073i	0.149651i	−0.997197	+	0.0748255i	$0.976160\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.5130	−0.700712	−0.350356	−	0.936617i	$-0.613939\pi$
$857$	−20.5130	−0.700712	−0.350356	+	0.936617i	$0.613939\pi$
$858$	0	0
$859$	− 10.1131i	− 0.345054i	−0.985005	−	0.172527i	$-0.944807\pi$
$859$	− 10.1131i	− 0.345054i	0.985005	−	0.172527i	$-0.0551932\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.2861	0.452266	0.226133	−	0.974096i	$-0.427392\pi$
$863$	13.2861	0.452266	0.226133	+	0.974096i	$0.427392\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.1444i	0.378047i
$870$	0	0
$871$	3.93437	0.133311
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.6454i	1.13612i	0.822986	+	0.568062i	$0.192307\pi$
$877$	33.6454i	1.13612i	−0.822986	+	0.568062i	$0.807693\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.7909	1.10476	0.552378	−	0.833594i	$-0.313721\pi$
$881$	32.7909	1.10476	0.552378	+	0.833594i	$0.313721\pi$
$882$	0	0
$883$	33.4506i	1.12570i	0.826558	+	0.562852i	$0.190296\pi$
$883$	33.4506i	1.12570i	−0.826558	+	0.562852i	$0.809704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.8924	−1.17157	−0.585785	−	0.810466i	$-0.699214\pi$
$887$	−34.8924	−1.17157	−0.585785	+	0.810466i	$0.699214\pi$
$888$	0	0
$889$	30.4671	1.02183
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 50.2088i	− 1.68017i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0884i	− 0.536578i
$900$	0	0
$901$	− 14.9409i	− 0.497752i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.9836i	1.02879i	0.857552	+	0.514397i	$0.171984\pi$
$907$	30.9836i	1.02879i	−0.857552	+	0.514397i	$0.828016\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	11.7181	0.387814
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.4835i	0.643400i
$918$	0	0
$919$	15.6043	0.514737	0.257368	−	0.966313i	$-0.417145\pi$
$919$	15.6043	0.514737	0.257368	+	0.966313i	$0.417145\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	26.0328i	0.856880i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.9341	1.27739	0.638693	−	0.769461i	$-0.279475\pi$
$929$	38.9341	1.27739	0.638693	+	0.769461i	$0.279475\pi$
$930$	0	0
$931$	− 0.116255i	− 0.00381011i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.6027	0.640393	0.320197	−	0.947351i	$-0.396251\pi$
$937$	19.6027	0.640393	0.320197	+	0.947351i	$0.396251\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 25.0476i	− 0.816530i	−0.912864	−	0.408265i	$-0.866134\pi$
$941$	− 25.0476i	− 0.816530i	0.912864	−	0.408265i	$-0.133866\pi$
$942$	0	0
$943$	−42.1240	−1.37175
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.93437i	− 0.257832i	−0.991655	−	0.128916i	$-0.958850\pi$
$947$	− 7.93437i	− 0.257832i	0.991655	−	0.128916i	$-0.0411498\pi$
$948$	0	0
$949$	− 60.0550i	− 1.94947i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.4809	0.371903	0.185952	−	0.982559i	$-0.440463\pi$
$953$	11.4809	0.371903	0.185952	+	0.982559i	$0.440463\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.2744	−0.331778
$960$	0	0
$961$	23.2088	0.748670
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.8993	−0.511285	−0.255643	−	0.966771i	$-0.582287\pi$
$967$	−15.8993	−0.511285	−0.255643	+	0.966771i	$0.582287\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 40.6600i	− 1.30484i	−0.757857	−	0.652421i	$-0.773754\pi$
$971$	− 40.6600i	− 1.30484i	0.757857	−	0.652421i	$-0.226246\pi$
$972$	0	0
$973$	38.6597i	1.23937i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.5676	0.849972	0.424986	−	0.905200i	$-0.360279\pi$
$977$	26.5676	0.849972	0.424986	+	0.905200i	$0.360279\pi$
$978$	0	0
$979$	− 22.3384i	− 0.713938i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.88057	0.315141	0.157571	−	0.987508i	$-0.449634\pi$
$983$	9.88057	0.315141	0.157571	+	0.987508i	$0.449634\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.90699i	0.156033i
$990$	0	0
$991$	53.0549	1.68534	0.842672	−	0.538427i	$-0.180981\pi$
$991$	53.0549	1.68534	0.842672	+	0.538427i	$0.180981\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 32.3051i	− 1.02311i	−0.859250	−	0.511556i	$-0.829069\pi$
$997$	− 32.3051i	− 1.02311i	0.859250	−	0.511556i	$-0.170931\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.u.3601.4		12
3.2	odd	2		2400.2.k.f.1201.8		12
4.3	odd	2		1800.2.k.u.901.8		12
5.2	odd	4		1440.2.d.f.1009.4		6
5.3	odd	4		1440.2.d.e.1009.4		6
5.4	even	2	inner	7200.2.k.u.3601.10		12
8.3	odd	2		1800.2.k.u.901.7		12
8.5	even	2	inner	7200.2.k.u.3601.3		12
12.11	even	2		600.2.k.f.301.5		12
15.2	even	4		480.2.d.b.49.3		6
15.8	even	4		480.2.d.a.49.3		6
15.14	odd	2		2400.2.k.f.1201.5		12
20.3	even	4		360.2.d.f.109.5		6
20.7	even	4		360.2.d.e.109.2		6
20.19	odd	2		1800.2.k.u.901.5		12
24.5	odd	2		2400.2.k.f.1201.2		12
24.11	even	2		600.2.k.f.301.6		12
40.3	even	4		360.2.d.e.109.1		6
40.13	odd	4		1440.2.d.f.1009.3		6
40.19	odd	2		1800.2.k.u.901.6		12
40.27	even	4		360.2.d.f.109.6		6
40.29	even	2	inner	7200.2.k.u.3601.9		12
40.37	odd	4		1440.2.d.e.1009.3		6
60.23	odd	4		120.2.d.a.109.2	yes	6
60.47	odd	4		120.2.d.b.109.5	yes	6
60.59	even	2		600.2.k.f.301.8		12
120.29	odd	2		2400.2.k.f.1201.11		12
120.53	even	4		480.2.d.b.49.4		6
120.59	even	2		600.2.k.f.301.7		12
120.77	even	4		480.2.d.a.49.4		6
120.83	odd	4		120.2.d.b.109.6	yes	6
120.107	odd	4		120.2.d.a.109.1	✓	6
240.53	even	4		3840.2.f.m.769.12		12
240.77	even	4		3840.2.f.m.769.7		12
240.83	odd	4		3840.2.f.l.769.7		12
240.107	odd	4		3840.2.f.l.769.12		12
240.173	even	4		3840.2.f.m.769.1		12
240.197	even	4		3840.2.f.m.769.6		12
240.203	odd	4		3840.2.f.l.769.6		12
240.227	odd	4		3840.2.f.l.769.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.1	✓	6	120.107	odd	4
120.2.d.a.109.2	yes	6	60.23	odd	4
120.2.d.b.109.5	yes	6	60.47	odd	4
120.2.d.b.109.6	yes	6	120.83	odd	4
360.2.d.e.109.1		6	40.3	even	4
360.2.d.e.109.2		6	20.7	even	4
360.2.d.f.109.5		6	20.3	even	4
360.2.d.f.109.6		6	40.27	even	4
480.2.d.a.49.3		6	15.8	even	4
480.2.d.a.49.4		6	120.77	even	4
480.2.d.b.49.3		6	15.2	even	4
480.2.d.b.49.4		6	120.53	even	4
600.2.k.f.301.5		12	12.11	even	2
600.2.k.f.301.6		12	24.11	even	2
600.2.k.f.301.7		12	120.59	even	2
600.2.k.f.301.8		12	60.59	even	2
1440.2.d.e.1009.3		6	40.37	odd	4
1440.2.d.e.1009.4		6	5.3	odd	4
1440.2.d.f.1009.3		6	40.13	odd	4
1440.2.d.f.1009.4		6	5.2	odd	4
1800.2.k.u.901.5		12	20.19	odd	2
1800.2.k.u.901.6		12	40.19	odd	2
1800.2.k.u.901.7		12	8.3	odd	2
1800.2.k.u.901.8		12	4.3	odd	2
2400.2.k.f.1201.2		12	24.5	odd	2
2400.2.k.f.1201.5		12	15.14	odd	2
2400.2.k.f.1201.8		12	3.2	odd	2
2400.2.k.f.1201.11		12	120.29	odd	2
3840.2.f.l.769.1		12	240.227	odd	4
3840.2.f.l.769.6		12	240.203	odd	4
3840.2.f.l.769.7		12	240.83	odd	4
3840.2.f.l.769.12		12	240.107	odd	4
3840.2.f.m.769.1		12	240.173	even	4
3840.2.f.m.769.6		12	240.197	even	4
3840.2.f.m.769.7		12	240.77	even	4
3840.2.f.m.769.12		12	240.53	even	4
7200.2.k.u.3601.3		12	8.5	even	2	inner
7200.2.k.u.3601.4		12	1.1	even	1	trivial
7200.2.k.u.3601.9		12	40.29	even	2	inner
7200.2.k.u.3601.10		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.64265 q^{7} +O(q^{10})\)	\(q-2.64265 q^{7} +1.51363i q^{11} -3.87086i q^{13} -3.31415 q^{17} +7.08582i q^{19} +4.82778 q^{23} -2.18513i q^{29} +7.36266 q^{31} -7.87086i q^{37} -8.72532 q^{41} +1.01641i q^{43} -7.08582 q^{47} -0.0164068 q^{49} +4.50820i q^{53} +6.79893i q^{59} +3.60104i q^{61} +1.01641i q^{67} -6.72532 q^{71} +15.5146 q^{73} -4.00000i q^{77} +7.36266 q^{79} -7.74173i q^{83} -14.7581 q^{89} +10.2293i q^{91} +11.1444 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 32 q^{31} + 8 q^{41} + 12 q^{49} + 32 q^{71} + 32 q^{79} - 40 q^{89}+O(q^{100}) \) 12 * q + 32 * q^31 + 8 * q^41 + 12 * q^49 + 32 * q^71 + 32 * q^79 - 40 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more