LMFDB - Embedded newform 4600.2.e.v.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 $ x^10 + 18x^8 + 103x^6 + 239x^4 + 197x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$1.45894i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.v.4049.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21042i q^{3} -2.22487i q^{7} -1.88594 q^{9} +O(q^{10})$	$q-2.21042i q^{3} -2.22487i q^{7} -1.88594 q^{9} -1.57300 q^{11} -3.96189i q^{13} +0.294907i q^{17} +7.76572 q^{19} -4.91788 q^{21} +1.00000i q^{23} -2.46253i q^{27} +9.29233 q^{29} +9.18913 q^{31} +3.47698i q^{33} -10.5425i q^{37} -8.75744 q^{39} -2.34251 q^{41} -6.67460i q^{43} -1.38007i q^{47} +2.04998 q^{49} +0.651868 q^{51} +11.0395i q^{53} -17.1655i q^{57} +5.09378 q^{59} -8.91788 q^{61} +4.19597i q^{63} -1.12002i q^{67} +2.21042 q^{69} +7.60168 q^{71} -12.8549i q^{73} +3.49971i q^{77} -11.0211 q^{79} -11.1010 q^{81} +13.5257i q^{83} -20.5399i q^{87} -14.3475 q^{89} -8.81468 q^{91} -20.3118i q^{93} -0.199218i q^{97} +2.96658 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{9}+O(q^{10}) $ 10 * q - 8 * q^9	$ 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) $ 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 2.21042i	− 1.27618i	−0.769960	−	0.638092i	$-0.779724\pi$
$3$	− 2.21042i	− 1.27618i	0.769960	−	0.638092i	$-0.220276\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.22487i	− 0.840920i	−0.907311	−	0.420460i	$-0.861869\pi$
$7$	− 2.22487i	− 0.840920i	0.907311	−	0.420460i	$-0.138131\pi$
$8$	0	0
$9$	−1.88594	−0.628648
$10$	0	0
$11$	−1.57300	−0.474276	−0.237138	−	0.971476i	$-0.576209\pi$
$11$	−1.57300	−0.474276	−0.237138	+	0.971476i	$0.576209\pi$
$12$	0	0
$13$	− 3.96189i	− 1.09883i	−0.835549	−	0.549416i	$-0.814850\pi$
$13$	− 3.96189i	− 1.09883i	0.835549	−	0.549416i	$-0.185150\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.294907i	0.0715256i	0.999360	+	0.0357628i	$0.0113861\pi$
$17$	0.294907i	0.0715256i	−0.999360	+	0.0357628i	$0.988614\pi$
$18$	0	0
$19$	7.76572	1.78158	0.890789	−	0.454418i	$-0.150153\pi$
$19$	7.76572	1.78158	0.890789	+	0.454418i	$0.150153\pi$
$20$	0	0
$21$	−4.91788	−1.07317
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.46253i	− 0.473914i
$28$	0	0
$29$	9.29233	1.72554	0.862771	−	0.505595i	$-0.168727\pi$
$29$	9.29233	1.72554	0.862771	+	0.505595i	$0.168727\pi$
$30$	0	0
$31$	9.18913	1.65042	0.825208	−	0.564829i	$-0.191058\pi$
$31$	9.18913	1.65042	0.825208	+	0.564829i	$0.191058\pi$
$32$	0	0
$33$	3.47698i	0.605264i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.5425i	− 1.73318i	−0.499024	−	0.866588i	$-0.666308\pi$
$37$	− 10.5425i	− 1.73318i	0.499024	−	0.866588i	$-0.333692\pi$
$38$	0	0
$39$	−8.75744	−1.40231
$40$	0	0
$41$	−2.34251	−0.365839	−0.182920	−	0.983128i	$-0.558555\pi$
$41$	−2.34251	−0.365839	−0.182920	+	0.983128i	$0.558555\pi$
$42$	0	0
$43$	− 6.67460i	− 1.01787i	−0.860806	−	0.508933i	$-0.830040\pi$
$43$	− 6.67460i	− 1.01787i	0.860806	−	0.508933i	$-0.169960\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.38007i	− 0.201304i	−0.994922	−	0.100652i	$-0.967907\pi$
$47$	− 1.38007i	− 0.201304i	0.994922	−	0.100652i	$-0.0320928\pi$
$48$	0	0
$49$	2.04998	0.292854
$50$	0	0
$51$	0.651868	0.0912798
$52$	0	0
$53$	11.0395i	1.51640i	0.652023	+	0.758199i	$0.273920\pi$
$53$	11.0395i	1.51640i	−0.652023	+	0.758199i	$0.726080\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 17.1655i	− 2.27362i
$58$	0	0
$59$	5.09378	0.663154	0.331577	−	0.943428i	$-0.392419\pi$
$59$	5.09378	0.663154	0.331577	+	0.943428i	$0.392419\pi$
$60$	0	0
$61$	−8.91788	−1.14182	−0.570909	−	0.821014i	$-0.693409\pi$
$61$	−8.91788	−1.14182	−0.570909	+	0.821014i	$0.693409\pi$
$62$	0	0
$63$	4.19597i	0.528642i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.12002i	− 0.136832i	−0.997657	−	0.0684160i	$-0.978205\pi$
$67$	− 1.12002i	− 0.136832i	0.997657	−	0.0684160i	$-0.0217945\pi$
$68$	0	0
$69$	2.21042	0.266103
$70$	0	0
$71$	7.60168	0.902154	0.451077	−	0.892485i	$-0.351040\pi$
$71$	7.60168	0.902154	0.451077	+	0.892485i	$0.351040\pi$
$72$	0	0
$73$	− 12.8549i	− 1.50455i	−0.658848	−	0.752276i	$-0.728956\pi$
$73$	− 12.8549i	− 1.50455i	0.658848	−	0.752276i	$-0.271044\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.49971i	0.398828i
$78$	0	0
$79$	−11.0211	−1.23997	−0.619984	−	0.784614i	$-0.712861\pi$
$79$	−11.0211	−1.23997	−0.619984	+	0.784614i	$0.712861\pi$
$80$	0	0
$81$	−11.1010	−1.23345
$82$	0	0
$83$	13.5257i	1.48464i	0.670048	+	0.742318i	$0.266274\pi$
$83$	13.5257i	1.48464i	−0.670048	+	0.742318i	$0.733726\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 20.5399i	− 2.20211i
$88$	0	0
$89$	−14.3475	−1.52084	−0.760418	−	0.649434i	$-0.775006\pi$
$89$	−14.3475	−1.52084	−0.760418	+	0.649434i	$0.775006\pi$
$90$	0	0
$91$	−8.81468	−0.924030
$92$	0	0
$93$	− 20.3118i	− 2.10624i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.199218i	− 0.0202275i	−0.999949	−	0.0101137i	$-0.996781\pi$
$97$	− 0.199218i	− 0.0202275i	0.999949	−	0.0101137i	$-0.00321936\pi$
$98$	0	0
$99$	2.96658	0.298153
$100$	0	0
$101$	−6.07595	−0.604580	−0.302290	−	0.953216i	$-0.597751\pi$
$101$	−6.07595	−0.604580	−0.302290	+	0.953216i	$0.597751\pi$
$102$	0	0
$103$	11.9423i	1.17671i	0.808604	+	0.588353i	$0.200223\pi$
$103$	11.9423i	1.17671i	−0.808604	+	0.588353i	$0.799777\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.99058i	0.772479i	0.922399	+	0.386239i	$0.126226\pi$
$107$	7.99058i	0.772479i	−0.922399	+	0.386239i	$0.873774\pi$
$108$	0	0
$109$	4.36170	0.417775	0.208888	−	0.977940i	$-0.433016\pi$
$109$	4.36170	0.417775	0.208888	+	0.977940i	$0.433016\pi$
$110$	0	0
$111$	−23.3033	−2.21185
$112$	0	0
$113$	− 12.9156i	− 1.21500i	−0.794319	−	0.607501i	$-0.792172\pi$
$113$	− 12.9156i	− 1.21500i	0.794319	−	0.607501i	$-0.207828\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.47191i	0.690778i
$118$	0	0
$119$	0.656129	0.0601473
$120$	0	0
$121$	−8.52568	−0.775062
$122$	0	0
$123$	5.17793i	0.466878i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.3759i	1.45313i	0.687098	+	0.726565i	$0.258884\pi$
$127$	16.3759i	1.45313i	−0.687098	+	0.726565i	$0.741116\pi$
$128$	0	0
$129$	−14.7536	−1.29899
$130$	0	0
$131$	9.62242	0.840715	0.420357	−	0.907359i	$-0.361905\pi$
$131$	9.62242	0.840715	0.420357	+	0.907359i	$0.361905\pi$
$132$	0	0
$133$	− 17.2777i	− 1.49816i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.78721i	− 0.579871i	−0.957046	−	0.289935i	$-0.906366\pi$
$137$	− 6.78721i	− 0.579871i	0.957046	−	0.289935i	$-0.0936338\pi$
$138$	0	0
$139$	−6.55788	−0.556232	−0.278116	−	0.960547i	$-0.589710\pi$
$139$	−6.55788	−0.556232	−0.278116	+	0.960547i	$0.589710\pi$
$140$	0	0
$141$	−3.05053	−0.256901
$142$	0	0
$143$	6.23205i	0.521150i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.53130i	− 0.373735i
$148$	0	0
$149$	13.8647	1.13584	0.567918	−	0.823085i	$-0.307749\pi$
$149$	13.8647	1.13584	0.567918	+	0.823085i	$0.307749\pi$
$150$	0	0
$151$	2.42205	0.197104	0.0985520	−	0.995132i	$-0.468579\pi$
$151$	2.42205	0.197104	0.0985520	+	0.995132i	$0.468579\pi$
$152$	0	0
$153$	− 0.556179i	− 0.0449644i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 20.2638i	− 1.61723i	−0.588341	−	0.808613i	$-0.700219\pi$
$157$	− 20.2638i	− 1.61723i	0.588341	−	0.808613i	$-0.299781\pi$
$158$	0	0
$159$	24.4020	1.93520
$160$	0	0
$161$	2.22487	0.175344
$162$	0	0
$163$	6.91982i	0.542002i	0.962579	+	0.271001i	$0.0873547\pi$
$163$	6.91982i	0.542002i	−0.962579	+	0.271001i	$0.912645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.8296i	− 1.45708i	−0.685005	−	0.728539i	$-0.740200\pi$
$167$	− 18.8296i	− 1.45708i	0.685005	−	0.728539i	$-0.259800\pi$
$168$	0	0
$169$	−2.69661	−0.207431
$170$	0	0
$171$	−14.6457	−1.11998
$172$	0	0
$173$	23.2017i	1.76399i	0.471255	+	0.881997i	$0.343801\pi$
$173$	23.2017i	1.76399i	−0.471255	+	0.881997i	$0.656199\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.2594i	− 0.846307i
$178$	0	0
$179$	10.7011	0.799837	0.399918	−	0.916551i	$-0.369038\pi$
$179$	10.7011	0.799837	0.399918	+	0.916551i	$0.369038\pi$
$180$	0	0
$181$	−7.27484	−0.540735	−0.270367	−	0.962757i	$-0.587145\pi$
$181$	−7.27484	−0.540735	−0.270367	+	0.962757i	$0.587145\pi$
$182$	0	0
$183$	19.7122i	1.45717i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.463888i	− 0.0339229i
$188$	0	0
$189$	−5.47880	−0.398524
$190$	0	0
$191$	−3.74624	−0.271068	−0.135534	−	0.990773i	$-0.543275\pi$
$191$	−3.74624	−0.271068	−0.135534	+	0.990773i	$0.543275\pi$
$192$	0	0
$193$	11.8470i	0.852763i	0.904543	+	0.426381i	$0.140212\pi$
$193$	11.8470i	0.852763i	−0.904543	+	0.426381i	$0.859788\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0064i	0.712924i	0.934310	+	0.356462i	$0.116017\pi$
$197$	10.0064i	0.712924i	−0.934310	+	0.356462i	$0.883983\pi$
$198$	0	0
$199$	−6.19597	−0.439221	−0.219610	−	0.975588i	$-0.570479\pi$
$199$	−6.19597	−0.439221	−0.219610	+	0.975588i	$0.570479\pi$
$200$	0	0
$201$	−2.47571	−0.174623
$202$	0	0
$203$	− 20.6742i	− 1.45104i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.88594i	− 0.131082i
$208$	0	0
$209$	−12.2154	−0.844960
$210$	0	0
$211$	−0.825746	−0.0568467	−0.0284234	−	0.999596i	$-0.509049\pi$
$211$	−0.825746	−0.0568467	−0.0284234	+	0.999596i	$0.509049\pi$
$212$	0	0
$213$	− 16.8029i	− 1.15132i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.4446i	− 1.38787i
$218$	0	0
$219$	−28.4147	−1.92009
$220$	0	0
$221$	1.16839	0.0785946
$222$	0	0
$223$	18.2911i	1.22486i	0.790525	+	0.612430i	$0.209808\pi$
$223$	18.2911i	1.22486i	−0.790525	+	0.612430i	$0.790192\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.5556i	− 1.43069i	−0.698769	−	0.715347i	$-0.746268\pi$
$227$	− 21.5556i	− 1.43069i	0.698769	−	0.715347i	$-0.253732\pi$
$228$	0	0
$229$	−9.65960	−0.638324	−0.319162	−	0.947700i	$-0.603401\pi$
$229$	−9.65960	−0.638324	−0.319162	+	0.947700i	$0.603401\pi$
$230$	0	0
$231$	7.73581	0.508979
$232$	0	0
$233$	0.879774i	0.0576359i	0.999585	+	0.0288179i	$0.00917431\pi$
$233$	0.879774i	0.0576359i	−0.999585	+	0.0288179i	$0.990826\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	24.3612i	1.58243i
$238$	0	0
$239$	−22.3240	−1.44402	−0.722010	−	0.691883i	$-0.756782\pi$
$239$	−22.3240	−1.44402	−0.722010	+	0.691883i	$0.756782\pi$
$240$	0	0
$241$	−7.43761	−0.479098	−0.239549	−	0.970884i	$-0.577000\pi$
$241$	−7.43761	−0.479098	−0.239549	+	0.970884i	$0.577000\pi$
$242$	0	0
$243$	17.1504i	1.10020i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 30.7669i	− 1.95765i
$248$	0	0
$249$	29.8974	1.89467
$250$	0	0
$251$	21.4221	1.35215	0.676074	−	0.736833i	$-0.263680\pi$
$251$	21.4221	1.35215	0.676074	+	0.736833i	$0.263680\pi$
$252$	0	0
$253$	− 1.57300i	− 0.0988935i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.9788i	0.684838i	0.939547	+	0.342419i	$0.111246\pi$
$257$	10.9788i	0.684838i	−0.939547	+	0.342419i	$0.888754\pi$
$258$	0	0
$259$	−23.4556	−1.45746
$260$	0	0
$261$	−17.5248	−1.08476
$262$	0	0
$263$	11.3268i	0.698440i	0.937041	+	0.349220i	$0.113553\pi$
$263$	11.3268i	0.698440i	−0.937041	+	0.349220i	$0.886447\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	31.7141i	1.94087i
$268$	0	0
$269$	−31.1010	−1.89626	−0.948130	−	0.317883i	$-0.897028\pi$
$269$	−31.1010	−1.89626	−0.948130	+	0.317883i	$0.897028\pi$
$270$	0	0
$271$	29.1151	1.76862	0.884308	−	0.466905i	$-0.154631\pi$
$271$	29.1151	1.76862	0.884308	+	0.466905i	$0.154631\pi$
$272$	0	0
$273$	19.4841i	1.17923i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.6829i	− 1.36288i	−0.731873	−	0.681441i	$-0.761354\pi$
$277$	− 22.6829i	− 1.36288i	0.731873	−	0.681441i	$-0.238646\pi$
$278$	0	0
$279$	−17.3302	−1.03753
$280$	0	0
$281$	11.2531	0.671305	0.335653	−	0.941986i	$-0.391043\pi$
$281$	11.2531	0.671305	0.335653	+	0.941986i	$0.391043\pi$
$282$	0	0
$283$	− 0.419014i	− 0.0249078i	−0.999922	−	0.0124539i	$-0.996036\pi$
$283$	− 0.419014i	− 0.0249078i	0.999922	−	0.0124539i	$-0.00396431\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.21177i	0.307641i
$288$	0	0
$289$	16.9130	0.994884
$290$	0	0
$291$	−0.440354	−0.0258140
$292$	0	0
$293$	− 1.06844i	− 0.0624190i	−0.999513	−	0.0312095i	$-0.990064\pi$
$293$	− 1.06844i	− 0.0624190i	0.999513	−	0.0312095i	$-0.00993591\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.87355i	0.224766i
$298$	0	0
$299$	3.96189	0.229122
$300$	0	0
$301$	−14.8501	−0.855944
$302$	0	0
$303$	13.4304i	0.771555i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.3991i	1.27838i	0.769048	+	0.639191i	$0.220731\pi$
$307$	22.3991i	1.27838i	−0.769048	+	0.639191i	$0.779269\pi$
$308$	0	0
$309$	26.3974	1.50169
$310$	0	0
$311$	−19.9717	−1.13249	−0.566245	−	0.824237i	$-0.691604\pi$
$311$	−19.9717	−1.13249	−0.566245	+	0.824237i	$0.691604\pi$
$312$	0	0
$313$	− 14.1975i	− 0.802488i	−0.915971	−	0.401244i	$-0.868578\pi$
$313$	− 14.1975i	− 0.802488i	0.915971	−	0.401244i	$-0.131422\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.48205i	− 0.420234i	−0.977676	−	0.210117i	$-0.932616\pi$
$317$	− 7.48205i	− 0.420234i	0.977676	−	0.210117i	$-0.0673844\pi$
$318$	0	0
$319$	−14.6168	−0.818384
$320$	0	0
$321$	17.6625	0.985826
$322$	0	0
$323$	2.29017i	0.127428i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.64118i	− 0.533158i
$328$	0	0
$329$	−3.07047	−0.169280
$330$	0	0
$331$	6.56494	0.360842	0.180421	−	0.983590i	$-0.442254\pi$
$331$	6.56494	0.360842	0.180421	+	0.983590i	$0.442254\pi$
$332$	0	0
$333$	19.8826i	1.08956i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.2484i	0.939583i	0.882777	+	0.469791i	$0.155671\pi$
$337$	17.2484i	0.939583i	−0.882777	+	0.469791i	$0.844329\pi$
$338$	0	0
$339$	−28.5490	−1.55057
$340$	0	0
$341$	−14.4545	−0.782753
$342$	0	0
$343$	− 20.1350i	− 1.08719i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.18016i	0.224403i	0.993685	+	0.112201i	$0.0357902\pi$
$347$	4.18016i	0.224403i	−0.993685	+	0.112201i	$0.964210\pi$
$348$	0	0
$349$	−11.6150	−0.621736	−0.310868	−	0.950453i	$-0.600620\pi$
$349$	−11.6150	−0.621736	−0.310868	+	0.950453i	$0.600620\pi$
$350$	0	0
$351$	−9.75628	−0.520752
$352$	0	0
$353$	− 15.8241i	− 0.842233i	−0.907006	−	0.421117i	$-0.861638\pi$
$353$	− 15.8241i	− 0.842233i	0.907006	−	0.421117i	$-0.138362\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.45032i	− 0.0767590i
$358$	0	0
$359$	6.25850	0.330311	0.165156	−	0.986268i	$-0.447187\pi$
$359$	6.25850	0.330311	0.165156	+	0.986268i	$0.447187\pi$
$360$	0	0
$361$	41.3064	2.17402
$362$	0	0
$363$	18.8453i	0.989122i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.01500i	− 0.261781i	−0.991397	−	0.130890i	$-0.958216\pi$
$367$	− 5.01500i	− 0.261781i	0.991397	−	0.130890i	$-0.0417836\pi$
$368$	0	0
$369$	4.41785	0.229984
$370$	0	0
$371$	24.5615	1.27517
$372$	0	0
$373$	20.1804i	1.04490i	0.852670	+	0.522450i	$0.174982\pi$
$373$	20.1804i	1.04490i	−0.852670	+	0.522450i	$0.825018\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 36.8152i	− 1.89608i
$378$	0	0
$379$	−10.9860	−0.564314	−0.282157	−	0.959368i	$-0.591050\pi$
$379$	−10.9860	−0.564314	−0.282157	+	0.959368i	$0.591050\pi$
$380$	0	0
$381$	36.1977	1.85446
$382$	0	0
$383$	23.9968i	1.22618i	0.790014	+	0.613088i	$0.210073\pi$
$383$	23.9968i	1.22618i	−0.790014	+	0.613088i	$0.789927\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.5879i	0.639879i
$388$	0	0
$389$	−19.3913	−0.983180	−0.491590	−	0.870827i	$-0.663584\pi$
$389$	−19.3913	−0.983180	−0.491590	+	0.870827i	$0.663584\pi$
$390$	0	0
$391$	−0.294907	−0.0149141
$392$	0	0
$393$	− 21.2696i	− 1.07291i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.0550i	− 0.504644i	−0.967643	−	0.252322i	$-0.918806\pi$
$397$	− 10.0550i	− 0.504644i	0.967643	−	0.252322i	$-0.0811942\pi$
$398$	0	0
$399$	−38.1909	−1.91193
$400$	0	0
$401$	−16.6178	−0.829854	−0.414927	−	0.909855i	$-0.636193\pi$
$401$	−16.6178	−0.829854	−0.414927	+	0.909855i	$0.636193\pi$
$402$	0	0
$403$	− 36.4064i	− 1.81353i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.5833i	0.822005i
$408$	0	0
$409$	−7.92642	−0.391936	−0.195968	−	0.980610i	$-0.562785\pi$
$409$	−7.92642	−0.391936	−0.195968	+	0.980610i	$0.562785\pi$
$410$	0	0
$411$	−15.0026	−0.740022
$412$	0	0
$413$	− 11.3330i	− 0.557659i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.4956i	0.709855i
$418$	0	0
$419$	−40.4944	−1.97828	−0.989141	−	0.146967i	$-0.953049\pi$
$419$	−40.4944	−1.97828	−0.989141	+	0.146967i	$0.953049\pi$
$420$	0	0
$421$	7.67870	0.374237	0.187118	−	0.982337i	$-0.440085\pi$
$421$	7.67870	0.374237	0.187118	+	0.982337i	$0.440085\pi$
$422$	0	0
$423$	2.60273i	0.126549i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.8411i	0.960177i
$428$	0	0
$429$	13.7754	0.665084
$430$	0	0
$431$	−36.1674	−1.74212	−0.871061	−	0.491175i	$-0.836567\pi$
$431$	−36.1674	−1.74212	−0.871061	+	0.491175i	$0.836567\pi$
$432$	0	0
$433$	9.42856i	0.453108i	0.973999	+	0.226554i	$0.0727459\pi$
$433$	9.42856i	0.453108i	−0.973999	+	0.226554i	$0.927254\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.76572i	0.371485i
$438$	0	0
$439$	−24.4443	−1.16666	−0.583332	−	0.812234i	$-0.698251\pi$
$439$	−24.4443	−1.16666	−0.583332	+	0.812234i	$0.698251\pi$
$440$	0	0
$441$	−3.86614	−0.184102
$442$	0	0
$443$	35.5786i	1.69039i	0.534458	+	0.845195i	$0.320516\pi$
$443$	35.5786i	1.69039i	−0.534458	+	0.845195i	$0.679484\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 30.6467i	− 1.44954i
$448$	0	0
$449$	0.660651	0.0311780	0.0155890	−	0.999878i	$-0.495038\pi$
$449$	0.660651	0.0311780	0.0155890	+	0.999878i	$0.495038\pi$
$450$	0	0
$451$	3.68476	0.173509
$452$	0	0
$453$	− 5.35375i	− 0.251541i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.62187i	− 0.403314i	−0.979456	−	0.201657i	$-0.935367\pi$
$457$	− 8.62187i	− 0.403314i	0.979456	−	0.201657i	$-0.0646326\pi$
$458$	0	0
$459$	0.726219	0.0338970
$460$	0	0
$461$	−5.54580	−0.258294	−0.129147	−	0.991625i	$-0.541224\pi$
$461$	−5.54580	−0.258294	−0.129147	+	0.991625i	$0.541224\pi$
$462$	0	0
$463$	− 6.15748i	− 0.286162i	−0.989711	−	0.143081i	$-0.954299\pi$
$463$	− 6.15748i	− 0.286162i	0.989711	−	0.143081i	$-0.0457010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.93879i	0.135991i	0.997686	+	0.0679954i	$0.0216603\pi$
$467$	2.93879i	0.135991i	−0.997686	+	0.0679954i	$0.978340\pi$
$468$	0	0
$469$	−2.49189	−0.115065
$470$	0	0
$471$	−44.7914	−2.06388
$472$	0	0
$473$	10.4991i	0.482750i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 20.8200i	− 0.953280i
$478$	0	0
$479$	−13.4939	−0.616550	−0.308275	−	0.951297i	$-0.599752\pi$
$479$	−13.4939	−0.616550	−0.308275	+	0.951297i	$0.599752\pi$
$480$	0	0
$481$	−41.7683	−1.90447
$482$	0	0
$483$	− 4.91788i	− 0.223771i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.28554i	− 0.148882i	−0.997225	−	0.0744410i	$-0.976283\pi$
$487$	− 3.28554i	− 0.148882i	0.997225	−	0.0744410i	$-0.0237173\pi$
$488$	0	0
$489$	15.2957	0.691695
$490$	0	0
$491$	24.2119	1.09267	0.546335	−	0.837567i	$-0.316023\pi$
$491$	24.2119	1.09267	0.546335	+	0.837567i	$0.316023\pi$
$492$	0	0
$493$	2.74038i	0.123420i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.9127i	− 0.758639i
$498$	0	0
$499$	37.5956	1.68301	0.841506	−	0.540248i	$-0.181669\pi$
$499$	37.5956	1.68301	0.841506	+	0.540248i	$0.181669\pi$
$500$	0	0
$501$	−41.6212	−1.85950
$502$	0	0
$503$	− 11.0448i	− 0.492464i	−0.969211	−	0.246232i	$-0.920807\pi$
$503$	− 11.0448i	− 0.492464i	0.969211	−	0.246232i	$-0.0791925\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.96062i	0.264720i
$508$	0	0
$509$	31.4409	1.39359	0.696796	−	0.717269i	$-0.254608\pi$
$509$	31.4409	1.39359	0.696796	+	0.717269i	$0.254608\pi$
$510$	0	0
$511$	−28.6004	−1.26521
$512$	0	0
$513$	− 19.1233i	− 0.844315i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.17084i	0.0954735i
$518$	0	0
$519$	51.2855	2.25118
$520$	0	0
$521$	35.3591	1.54911	0.774555	−	0.632506i	$-0.217974\pi$
$521$	35.3591	1.54911	0.774555	+	0.632506i	$0.217974\pi$
$522$	0	0
$523$	− 41.8119i	− 1.82830i	−0.405371	−	0.914152i	$-0.632858\pi$
$523$	− 41.8119i	− 1.82830i	0.405371	−	0.914152i	$-0.367142\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.70994i	0.118047i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−9.60658	−0.416890
$532$	0	0
$533$	9.28079i	0.401996i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 23.6539i	− 1.02074i
$538$	0	0
$539$	−3.22460	−0.138894
$540$	0	0
$541$	1.43188	0.0615612	0.0307806	−	0.999526i	$-0.490201\pi$
$541$	1.43188	0.0615612	0.0307806	+	0.999526i	$0.490201\pi$
$542$	0	0
$543$	16.0804i	0.690077i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.8700i	1.91850i	0.282555	+	0.959251i	$0.408818\pi$
$547$	44.8700i	1.91850i	−0.282555	+	0.959251i	$0.591182\pi$
$548$	0	0
$549$	16.8186	0.717801
$550$	0	0
$551$	72.1616	3.07419
$552$	0	0
$553$	24.5204i	1.04271i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.3106i	− 0.945329i	−0.881242	−	0.472665i	$-0.843292\pi$
$557$	− 22.3106i	− 0.945329i	0.881242	−	0.472665i	$-0.156708\pi$
$558$	0	0
$559$	−26.4440	−1.11846
$560$	0	0
$561$	−1.02539	−0.0432919
$562$	0	0
$563$	1.41928i	0.0598153i	0.999553	+	0.0299077i	$0.00952132\pi$
$563$	1.41928i	0.0598153i	−0.999553	+	0.0299077i	$0.990479\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	24.6983i	1.03723i
$568$	0	0
$569$	21.4950	0.901116	0.450558	−	0.892747i	$-0.351225\pi$
$569$	21.4950	0.901116	0.450558	+	0.892747i	$0.351225\pi$
$570$	0	0
$571$	41.0272	1.71694	0.858468	−	0.512868i	$-0.171417\pi$
$571$	41.0272	1.71694	0.858468	+	0.512868i	$0.171417\pi$
$572$	0	0
$573$	8.28075i	0.345933i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.1477i	1.54648i	0.634114	+	0.773239i	$0.281365\pi$
$577$	37.1477i	1.54648i	−0.634114	+	0.773239i	$0.718635\pi$
$578$	0	0
$579$	26.1867	1.08828
$580$	0	0
$581$	30.0928	1.24846
$582$	0	0
$583$	− 17.3652i	− 0.719192i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.1838i	0.502881i	0.967873	+	0.251440i	$0.0809043\pi$
$587$	12.1838i	0.502881i	−0.967873	+	0.251440i	$0.919096\pi$
$588$	0	0
$589$	71.3602	2.94034
$590$	0	0
$591$	22.1183	0.909823
$592$	0	0
$593$	− 40.1421i	− 1.64844i	−0.566270	−	0.824220i	$-0.691614\pi$
$593$	− 40.1421i	− 1.64844i	0.566270	−	0.824220i	$-0.308386\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.6957i	0.560527i
$598$	0	0
$599$	16.4377	0.671624	0.335812	−	0.941929i	$-0.390989\pi$
$599$	16.4377	0.671624	0.335812	+	0.941929i	$0.390989\pi$
$600$	0	0
$601$	−4.98809	−0.203469	−0.101734	−	0.994812i	$-0.532439\pi$
$601$	−4.98809	−0.203469	−0.101734	+	0.994812i	$0.532439\pi$
$602$	0	0
$603$	2.11229i	0.0860191i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.09352i	− 0.0443846i	−0.999754	−	0.0221923i	$-0.992935\pi$
$607$	− 1.09352i	− 0.0443846i	0.999754	−	0.0221923i	$-0.00706461\pi$
$608$	0	0
$609$	−45.6986	−1.85180
$610$	0	0
$611$	−5.46768	−0.221199
$612$	0	0
$613$	16.4617i	0.664880i	0.943124	+	0.332440i	$0.107872\pi$
$613$	16.4617i	0.664880i	−0.943124	+	0.332440i	$0.892128\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23.9090i	− 0.962540i	−0.876572	−	0.481270i	$-0.840176\pi$
$617$	− 23.9090i	− 0.962540i	0.876572	−	0.481270i	$-0.159824\pi$
$618$	0	0
$619$	26.7498	1.07517	0.537583	−	0.843211i	$-0.319337\pi$
$619$	26.7498	1.07517	0.537583	+	0.843211i	$0.319337\pi$
$620$	0	0
$621$	2.46253	0.0988179
$622$	0	0
$623$	31.9213i	1.27890i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	27.0012i	1.07833i
$628$	0	0
$629$	3.10906	0.123966
$630$	0	0
$631$	0.317634	0.0126448	0.00632241	−	0.999980i	$-0.497988\pi$
$631$	0.317634	0.0126448	0.00632241	+	0.999980i	$0.497988\pi$
$632$	0	0
$633$	1.82524i	0.0725469i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.12179i	− 0.321797i
$638$	0	0
$639$	−14.3363	−0.567137
$640$	0	0
$641$	19.9895	0.789539	0.394769	−	0.918780i	$-0.370825\pi$
$641$	19.9895	0.789539	0.394769	+	0.918780i	$0.370825\pi$
$642$	0	0
$643$	− 20.0837i	− 0.792023i	−0.918246	−	0.396011i	$-0.870394\pi$
$643$	− 20.0837i	− 0.792023i	0.918246	−	0.396011i	$-0.129606\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22.0801i	− 0.868060i	−0.900898	−	0.434030i	$-0.857091\pi$
$647$	− 22.0801i	− 0.868060i	0.900898	−	0.434030i	$-0.142909\pi$
$648$	0	0
$649$	−8.01250	−0.314518
$650$	0	0
$651$	−45.1910	−1.77118
$652$	0	0
$653$	33.2385i	1.30072i	0.759625	+	0.650362i	$0.225383\pi$
$653$	33.2385i	1.30072i	−0.759625	+	0.650362i	$0.774617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.2436i	0.945833i
$658$	0	0
$659$	−17.3023	−0.674002	−0.337001	−	0.941504i	$-0.609413\pi$
$659$	−17.3023	−0.674002	−0.337001	+	0.941504i	$0.609413\pi$
$660$	0	0
$661$	−28.9521	−1.12611	−0.563053	−	0.826421i	$-0.690373\pi$
$661$	−28.9521	−1.12611	−0.563053	+	0.826421i	$0.690373\pi$
$662$	0	0
$663$	− 2.58263i	− 0.100301i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.29233i	0.359800i
$668$	0	0
$669$	40.4309	1.56315
$670$	0	0
$671$	14.0278	0.541537
$672$	0	0
$673$	− 10.7860i	− 0.415771i	−0.978153	−	0.207885i	$-0.933342\pi$
$673$	− 10.7860i	− 0.415771i	0.978153	−	0.207885i	$-0.0666581\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.1146i	− 0.619336i	−0.950845	−	0.309668i	$-0.899782\pi$
$677$	− 16.1146i	− 0.619336i	0.950845	−	0.309668i	$-0.100218\pi$
$678$	0	0
$679$	−0.443232	−0.0170097
$680$	0	0
$681$	−47.6468	−1.82583
$682$	0	0
$683$	− 12.1106i	− 0.463399i	−0.972787	−	0.231699i	$-0.925571\pi$
$683$	− 12.1106i	− 0.463399i	0.972787	−	0.231699i	$-0.0744286\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.3517i	0.814620i
$688$	0	0
$689$	43.7375	1.66627
$690$	0	0
$691$	−30.1463	−1.14682	−0.573410	−	0.819269i	$-0.694380\pi$
$691$	−30.1463	−1.14682	−0.573410	+	0.819269i	$0.694380\pi$
$692$	0	0
$693$	− 6.60025i	− 0.250723i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.690824i	− 0.0261668i
$698$	0	0
$699$	1.94467	0.0735540
$700$	0	0
$701$	14.8464	0.560741	0.280371	−	0.959892i	$-0.409543\pi$
$701$	14.8464	0.560741	0.280371	+	0.959892i	$0.409543\pi$
$702$	0	0
$703$	− 81.8701i	− 3.08779i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.5182i	0.508403i
$708$	0	0
$709$	−37.0163	−1.39018	−0.695088	−	0.718924i	$-0.744635\pi$
$709$	−37.0163	−1.39018	−0.695088	+	0.718924i	$0.744635\pi$
$710$	0	0
$711$	20.7851	0.779503
$712$	0	0
$713$	9.18913i	0.344136i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.3454i	1.84284i
$718$	0	0
$719$	−7.63459	−0.284722	−0.142361	−	0.989815i	$-0.545469\pi$
$719$	−7.63459	−0.284722	−0.142361	+	0.989815i	$0.545469\pi$
$720$	0	0
$721$	26.5699	0.989515
$722$	0	0
$723$	16.4402i	0.611418i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29.1278i	− 1.08029i	−0.841571	−	0.540146i	$-0.818369\pi$
$727$	− 29.1278i	− 1.08029i	0.841571	−	0.540146i	$-0.181631\pi$
$728$	0	0
$729$	4.60629	0.170603
$730$	0	0
$731$	1.96839	0.0728034
$732$	0	0
$733$	7.30466i	0.269804i	0.990859	+	0.134902i	$0.0430719\pi$
$733$	7.30466i	0.269804i	−0.990859	+	0.134902i	$0.956928\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.76178i	0.0648962i
$738$	0	0
$739$	10.5109	0.386649	0.193324	−	0.981135i	$-0.438073\pi$
$739$	10.5109	0.386649	0.193324	+	0.981135i	$0.438073\pi$
$740$	0	0
$741$	−68.0078	−2.49833
$742$	0	0
$743$	− 11.4448i	− 0.419869i	−0.977715	−	0.209934i	$-0.932675\pi$
$743$	− 11.4448i	− 0.419869i	0.977715	−	0.209934i	$-0.0673250\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 25.5087i	− 0.933313i
$748$	0	0
$749$	17.7780	0.649593
$750$	0	0
$751$	−7.62069	−0.278083	−0.139041	−	0.990287i	$-0.544402\pi$
$751$	−7.62069	−0.278083	−0.139041	+	0.990287i	$0.544402\pi$
$752$	0	0
$753$	− 47.3517i	− 1.72559i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.3725i	1.50371i	0.659329	+	0.751855i	$0.270841\pi$
$757$	41.3725i	1.50371i	−0.659329	+	0.751855i	$0.729159\pi$
$758$	0	0
$759$	−3.47698	−0.126206
$760$	0	0
$761$	12.5358	0.454421	0.227211	−	0.973846i	$-0.427039\pi$
$761$	12.5358	0.454421	0.227211	+	0.973846i	$0.427039\pi$
$762$	0	0
$763$	− 9.70420i	− 0.351315i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20.1810i	− 0.728694i
$768$	0	0
$769$	25.0182	0.902180	0.451090	−	0.892479i	$-0.351035\pi$
$769$	25.0182	0.902180	0.451090	+	0.892479i	$0.351035\pi$
$770$	0	0
$771$	24.2677	0.873980
$772$	0	0
$773$	37.1948i	1.33781i	0.743350	+	0.668903i	$0.233236\pi$
$773$	37.1948i	1.33781i	−0.743350	+	0.668903i	$0.766764\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	51.8467i	1.85999i
$778$	0	0
$779$	−18.1913	−0.651771
$780$	0	0
$781$	−11.9574	−0.427870
$782$	0	0
$783$	− 22.8826i	− 0.817759i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.64645i	− 0.272567i	−0.990670	−	0.136283i	$-0.956484\pi$
$787$	− 7.64645i	− 0.272567i	0.990670	−	0.136283i	$-0.0435157\pi$
$788$	0	0
$789$	25.0369	0.891338
$790$	0	0
$791$	−28.7356	−1.02172
$792$	0	0
$793$	35.3317i	1.25467i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.38544i	− 0.0844966i	−0.999107	−	0.0422483i	$-0.986548\pi$
$797$	− 2.38544i	− 0.0844966i	0.999107	−	0.0422483i	$-0.0134521\pi$
$798$	0	0
$799$	0.406992	0.0143984
$800$	0	0
$801$	27.0586	0.956070
$802$	0	0
$803$	20.2207i	0.713573i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	68.7461i	2.41998i
$808$	0	0
$809$	−15.4937	−0.544730	−0.272365	−	0.962194i	$-0.587806\pi$
$809$	−15.4937	−0.544730	−0.272365	+	0.962194i	$0.587806\pi$
$810$	0	0
$811$	55.3850	1.94483	0.972415	−	0.233259i	$-0.0749389\pi$
$811$	55.3850	1.94483	0.972415	+	0.233259i	$0.0749389\pi$
$812$	0	0
$813$	− 64.3565i	− 2.25708i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 51.8330i	− 1.81341i
$818$	0	0
$819$	16.6240	0.580889
$820$	0	0
$821$	−38.7241	−1.35148	−0.675740	−	0.737140i	$-0.736176\pi$
$821$	−38.7241	−1.35148	−0.675740	+	0.737140i	$0.736176\pi$
$822$	0	0
$823$	0.237837i	0.00829047i	0.999991	+	0.00414524i	$0.00131947\pi$
$823$	0.237837i	0.00829047i	−0.999991	+	0.00414524i	$0.998681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.5821i	− 0.611391i	−0.952129	−	0.305695i	$-0.901111\pi$
$827$	− 17.5821i	− 0.611391i	0.952129	−	0.305695i	$-0.0988889\pi$
$828$	0	0
$829$	−20.5607	−0.714103	−0.357051	−	0.934085i	$-0.616218\pi$
$829$	−20.5607	−0.714103	−0.357051	+	0.934085i	$0.616218\pi$
$830$	0	0
$831$	−50.1386	−1.73929
$832$	0	0
$833$	0.604553i	0.0209465i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 22.6285i	− 0.782156i
$838$	0	0
$839$	15.1584	0.523325	0.261663	−	0.965159i	$-0.415729\pi$
$839$	15.1584	0.523325	0.261663	+	0.965159i	$0.415729\pi$
$840$	0	0
$841$	57.3474	1.97750
$842$	0	0
$843$	− 24.8741i	− 0.856709i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18.9685i	0.651765i
$848$	0	0
$849$	−0.926197	−0.0317870
$850$	0	0
$851$	10.5425	0.361392
$852$	0	0
$853$	− 18.7788i	− 0.642973i	−0.946914	−	0.321486i	$-0.895818\pi$
$853$	− 18.7788i	− 0.642973i	0.946914	−	0.321486i	$-0.104182\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29.4670i	− 1.00657i	−0.864120	−	0.503286i	$-0.832124\pi$
$857$	− 29.4670i	− 1.00657i	0.864120	−	0.503286i	$-0.167876\pi$
$858$	0	0
$859$	50.7562	1.73178	0.865889	−	0.500236i	$-0.166753\pi$
$859$	50.7562	1.73178	0.865889	+	0.500236i	$0.166753\pi$
$860$	0	0
$861$	11.5202	0.392607
$862$	0	0
$863$	− 19.5166i	− 0.664351i	−0.943218	−	0.332176i	$-0.892217\pi$
$863$	− 19.5166i	− 0.664351i	0.943218	−	0.332176i	$-0.107783\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 37.3848i	− 1.26966i
$868$	0	0
$869$	17.3361	0.588088
$870$	0	0
$871$	−4.43739	−0.150355
$872$	0	0
$873$	0.375713i	0.0127160i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 6.68909i	− 0.225874i	−0.993602	−	0.112937i	$-0.963974\pi$
$877$	− 6.68909i	− 0.225874i	0.993602	−	0.112937i	$-0.0360259\pi$
$878$	0	0
$879$	−2.36170	−0.0796582
$880$	0	0
$881$	47.1211	1.58755	0.793775	−	0.608212i	$-0.208113\pi$
$881$	47.1211	1.58755	0.793775	+	0.608212i	$0.208113\pi$
$882$	0	0
$883$	40.5934i	1.36608i	0.730383	+	0.683038i	$0.239342\pi$
$883$	40.5934i	1.36608i	−0.730383	+	0.683038i	$0.760658\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.89101i	− 0.332108i	−0.986117	−	0.166054i	$-0.946897\pi$
$887$	− 9.89101i	− 0.332108i	0.986117	−	0.166054i	$-0.0531025\pi$
$888$	0	0
$889$	36.4343	1.22197
$890$	0	0
$891$	17.4619	0.584996
$892$	0	0
$893$	− 10.7172i	− 0.358638i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.75744i	− 0.292402i
$898$	0	0
$899$	85.3884	2.84786
$900$	0	0
$901$	−3.25564	−0.108461
$902$	0	0
$903$	32.8249i	1.09234i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42.3250i	1.40538i	0.711497	+	0.702689i	$0.248017\pi$
$907$	42.3250i	1.40538i	−0.711497	+	0.702689i	$0.751983\pi$
$908$	0	0
$909$	11.4589	0.380068
$910$	0	0
$911$	5.48129	0.181603	0.0908016	−	0.995869i	$-0.471057\pi$
$911$	5.48129	0.181603	0.0908016	+	0.995869i	$0.471057\pi$
$912$	0	0
$913$	− 21.2759i	− 0.704128i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21.4086i	− 0.706974i
$918$	0	0
$919$	−16.6386	−0.548856	−0.274428	−	0.961608i	$-0.588488\pi$
$919$	−16.6386	−0.548856	−0.274428	+	0.961608i	$0.588488\pi$
$920$	0	0
$921$	49.5113	1.63145
$922$	0	0
$923$	− 30.1171i	− 0.991315i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 22.5224i	− 0.739733i
$928$	0	0
$929$	30.2326	0.991900	0.495950	−	0.868351i	$-0.334820\pi$
$929$	30.2326	0.991900	0.495950	+	0.868351i	$0.334820\pi$
$930$	0	0
$931$	15.9195	0.521741
$932$	0	0
$933$	44.1457i	1.44527i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 27.8718i	− 0.910532i	−0.890356	−	0.455266i	$-0.849544\pi$
$937$	− 27.8718i	− 0.910532i	0.890356	−	0.455266i	$-0.150456\pi$
$938$	0	0
$939$	−31.3823	−1.02412
$940$	0	0
$941$	26.8768	0.876159	0.438079	−	0.898936i	$-0.355659\pi$
$941$	26.8768	0.876159	0.438079	+	0.898936i	$0.355659\pi$
$942$	0	0
$943$	− 2.34251i	− 0.0762827i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.1773i	0.753161i	0.926384	+	0.376581i	$0.122900\pi$
$947$	23.1773i	0.753161i	−0.926384	+	0.376581i	$0.877100\pi$
$948$	0	0
$949$	−50.9297	−1.65325
$950$	0	0
$951$	−16.5384	−0.536296
$952$	0	0
$953$	− 4.21870i	− 0.136657i	−0.997663	−	0.0683285i	$-0.978233\pi$
$953$	− 4.21870i	− 0.136657i	0.997663	−	0.0683285i	$-0.0217666\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.3092i	1.04441i
$958$	0	0
$959$	−15.1006	−0.487625
$960$	0	0
$961$	53.4401	1.72387
$962$	0	0
$963$	− 15.0698i	− 0.485617i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17.8055i	− 0.572588i	−0.958142	−	0.286294i	$-0.907577\pi$
$967$	− 17.8055i	− 0.572588i	0.958142	−	0.286294i	$-0.0924233\pi$
$968$	0	0
$969$	5.06223	0.162622
$970$	0	0
$971$	17.9741	0.576815	0.288408	−	0.957508i	$-0.406874\pi$
$971$	17.9741	0.576815	0.288408	+	0.957508i	$0.406874\pi$
$972$	0	0
$973$	14.5904i	0.467747i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 58.6275i	− 1.87566i	−0.347097	−	0.937829i	$-0.612833\pi$
$977$	− 58.6275i	− 1.87566i	0.347097	−	0.937829i	$-0.387167\pi$
$978$	0	0
$979$	22.5686	0.721297
$980$	0	0
$981$	−8.22592	−0.262633
$982$	0	0
$983$	7.84103i	0.250090i	0.992151	+	0.125045i	$0.0399075\pi$
$983$	7.84103i	0.250090i	−0.992151	+	0.125045i	$0.960092\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.78701i	0.216033i
$988$	0	0
$989$	6.67460	0.212240
$990$	0	0
$991$	−33.1872	−1.05423	−0.527114	−	0.849795i	$-0.676726\pi$
$991$	−33.1872	−1.05423	−0.527114	+	0.849795i	$0.676726\pi$
$992$	0	0
$993$	− 14.5112i	− 0.460500i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	60.8220i	1.92625i	0.269049	+	0.963127i	$0.413291\pi$
$997$	60.8220i	1.92625i	−0.269049	+	0.963127i	$0.586709\pi$
$998$	0	0
$999$	−25.9612	−0.821377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.v.4049.2		10
5.2	odd	4		4600.2.a.bg.1.1	yes	5
5.3	odd	4		4600.2.a.bc.1.5	✓	5
5.4	even	2	inner	4600.2.e.v.4049.9		10
20.3	even	4		9200.2.a.cw.1.1		5
20.7	even	4		9200.2.a.cs.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.5	✓	5	5.3	odd	4
4600.2.a.bg.1.1	yes	5	5.2	odd	4
4600.2.e.v.4049.2		10	1.1	even	1	trivial
4600.2.e.v.4049.9		10	5.4	even	2	inner
9200.2.a.cs.1.5		5	20.7	even	4
9200.2.a.cw.1.1		5	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.21042i q^{3} -2.22487i q^{7} -1.88594 q^{9} +O(q^{10})\)	\(q-2.21042i q^{3} -2.22487i q^{7} -1.88594 q^{9} -1.57300 q^{11} -3.96189i q^{13} +0.294907i q^{17} +7.76572 q^{19} -4.91788 q^{21} +1.00000i q^{23} -2.46253i q^{27} +9.29233 q^{29} +9.18913 q^{31} +3.47698i q^{33} -10.5425i q^{37} -8.75744 q^{39} -2.34251 q^{41} -6.67460i q^{43} -1.38007i q^{47} +2.04998 q^{49} +0.651868 q^{51} +11.0395i q^{53} -17.1655i q^{57} +5.09378 q^{59} -8.91788 q^{61} +4.19597i q^{63} -1.12002i q^{67} +2.21042 q^{69} +7.60168 q^{71} -12.8549i q^{73} +3.49971i q^{77} -11.0211 q^{79} -11.1010 q^{81} +13.5257i q^{83} -20.5399i q^{87} -14.3475 q^{89} -8.81468 q^{91} -20.3118i q^{93} -0.199218i q^{97} +2.96658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{9}+O(q^{10}) \) 10 * q - 8 * q^9	\( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) \) 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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