LMFDB - Embedded newform 4600.2.e.p.4049.5

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:	$6$
Coefficient field:	6.0.431642176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 19x^{4} + 97x^{2} + 64 $ x^6 + 19x^4 + 97x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.5
Root			$2.95759i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.p.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} +O(q^{10})$	$q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} -0.957587 q^{11} +2.74732i q^{13} +5.74732i q^{17} +6.74732 q^{19} -11.7049 q^{21} -1.00000i q^{23} -8.12544i q^{27} -5.21027 q^{29} -5.95759 q^{31} -2.83215i q^{33} +9.12544i q^{37} -8.12544 q^{39} +0.252679 q^{41} -8.00000i q^{43} +5.49464i q^{47} -8.66249 q^{49} -16.9982 q^{51} +7.12544i q^{53} +19.9558i q^{57} +4.78973 q^{59} +12.4522 q^{61} -22.7455i q^{63} +9.12544i q^{67} +2.95759 q^{69} +1.66249 q^{71} -12.3357i q^{73} -3.78973i q^{77} -11.8303 q^{79} +6.78973 q^{81} -0.704908i q^{83} -15.4098i q^{87} +15.8303 q^{89} -10.8728 q^{91} -17.6201i q^{93} -10.8728i q^{97} +5.50356 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{9}+O(q^{10}) $ 6 * q - 20 * q^9	$ 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) $ 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * 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
$3$	2.95759i	1.70756i	0.520631	+	0.853782i	$0.325697\pi$
$3$	2.95759i	1.70756i	−0.520631	+	0.853782i	$0.674303\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.95759i	1.49583i	0.663796	+	0.747914i	$0.268944\pi$
$7$	3.95759i	1.49583i	−0.663796	+	0.747914i	$0.731056\pi$
$8$	0	0
$9$	−5.74732	−1.91577
$10$	0	0
$11$	−0.957587	−0.288723	−0.144362	−	0.989525i	$-0.546113\pi$
$11$	−0.957587	−0.288723	−0.144362	+	0.989525i	$0.546113\pi$
$12$	0	0
$13$	2.74732i	0.761970i	0.924581	+	0.380985i	$0.124415\pi$
$13$	2.74732i	0.761970i	−0.924581	+	0.380985i	$0.875585\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.74732i	1.39393i	0.717105	+	0.696965i	$0.245467\pi$
$17$	5.74732i	1.39393i	−0.717105	+	0.696965i	$0.754533\pi$
$18$	0	0
$19$	6.74732	1.54794	0.773971	−	0.633221i	$-0.218268\pi$
$19$	6.74732	1.54794	0.773971	+	0.633221i	$0.218268\pi$
$20$	0	0
$21$	−11.7049	−2.55422
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 8.12544i	− 1.56374i
$28$	0	0
$29$	−5.21027	−0.967522	−0.483761	−	0.875200i	$-0.660730\pi$
$29$	−5.21027	−0.967522	−0.483761	+	0.875200i	$0.660730\pi$
$30$	0	0
$31$	−5.95759	−1.07001	−0.535007	−	0.844848i	$-0.679691\pi$
$31$	−5.95759	−1.07001	−0.535007	+	0.844848i	$0.679691\pi$
$32$	0	0
$33$	− 2.83215i	− 0.493013i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.12544i	1.50021i	0.661317	+	0.750107i	$0.269998\pi$
$37$	9.12544i	1.50021i	−0.661317	+	0.750107i	$0.730002\pi$
$38$	0	0
$39$	−8.12544	−1.30111
$40$	0	0
$41$	0.252679	0.0394619	0.0197309	−	0.999805i	$-0.493719\pi$
$41$	0.252679	0.0394619	0.0197309	+	0.999805i	$0.493719\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.49464i	0.801476i	0.916193	+	0.400738i	$0.131246\pi$
$47$	5.49464i	0.801476i	−0.916193	+	0.400738i	$0.868754\pi$
$48$	0	0
$49$	−8.66249	−1.23750
$50$	0	0
$51$	−16.9982	−2.38022
$52$	0	0
$53$	7.12544i	0.978754i	0.872072	+	0.489377i	$0.162776\pi$
$53$	7.12544i	0.978754i	−0.872072	+	0.489377i	$0.837224\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19.9558i	2.64321i
$58$	0	0
$59$	4.78973	0.623570	0.311785	−	0.950153i	$-0.399073\pi$
$59$	4.78973	0.623570	0.311785	+	0.950153i	$0.399073\pi$
$60$	0	0
$61$	12.4522	1.59434	0.797172	−	0.603752i	$-0.206328\pi$
$61$	12.4522	1.59434	0.797172	+	0.603752i	$0.206328\pi$
$62$	0	0
$63$	− 22.7455i	− 2.86567i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.12544i	1.11485i	0.830227	+	0.557425i	$0.188211\pi$
$67$	9.12544i	1.11485i	−0.830227	+	0.557425i	$0.811789\pi$
$68$	0	0
$69$	2.95759	0.356052
$70$	0	0
$71$	1.66249	0.197302	0.0986509	−	0.995122i	$-0.468547\pi$
$71$	1.66249	0.197302	0.0986509	+	0.995122i	$0.468547\pi$
$72$	0	0
$73$	− 12.3357i	− 1.44379i	−0.692005	−	0.721893i	$-0.743272\pi$
$73$	− 12.3357i	− 1.44379i	0.692005	−	0.721893i	$-0.256728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.78973i	− 0.431880i
$78$	0	0
$79$	−11.8303	−1.33102	−0.665509	−	0.746390i	$-0.731786\pi$
$79$	−11.8303	−1.33102	−0.665509	+	0.746390i	$0.731786\pi$
$80$	0	0
$81$	6.78973	0.754415
$82$	0	0
$83$	− 0.704908i	− 0.0773737i	−0.999251	−	0.0386868i	$-0.987683\pi$
$83$	− 0.704908i	− 0.0773737i	0.999251	−	0.0386868i	$-0.0123175\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 15.4098i	− 1.65211i
$88$	0	0
$89$	15.8303	1.67801	0.839007	−	0.544121i	$-0.183137\pi$
$89$	15.8303	1.67801	0.839007	+	0.544121i	$0.183137\pi$
$90$	0	0
$91$	−10.8728	−1.13978
$92$	0	0
$93$	− 17.6201i	− 1.82712i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.8728i	− 1.10396i	−0.833857	−	0.551981i	$-0.813872\pi$
$97$	− 10.8728i	− 1.10396i	0.833857	−	0.551981i	$-0.186128\pi$
$98$	0	0
$99$	5.50356	0.553129
$100$	0	0
$101$	9.21027	0.916456	0.458228	−	0.888835i	$-0.348484\pi$
$101$	9.21027	0.916456	0.458228	+	0.888835i	$0.348484\pi$
$102$	0	0
$103$	− 12.4522i	− 1.22695i	−0.789712	−	0.613477i	$-0.789770\pi$
$103$	− 12.4522i	− 1.22695i	0.789712	−	0.613477i	$-0.210230\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.63080i	0.351003i	0.984479	+	0.175501i	$0.0561546\pi$
$107$	3.63080i	0.351003i	−0.984479	+	0.175501i	$0.943845\pi$
$108$	0	0
$109$	−16.6625	−1.59598	−0.797989	−	0.602672i	$-0.794103\pi$
$109$	−16.6625	−1.59598	−0.797989	+	0.602672i	$0.794103\pi$
$110$	0	0
$111$	−26.9893	−2.56171
$112$	0	0
$113$	18.1147i	1.70409i	0.523469	+	0.852045i	$0.324638\pi$
$113$	18.1147i	1.70409i	−0.523469	+	0.852045i	$0.675362\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 15.7897i	− 1.45976i
$118$	0	0
$119$	−22.7455	−2.08508
$120$	0	0
$121$	−10.0830	−0.916639
$122$	0	0
$123$	0.747321i	0.0673836i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.91517i	− 0.524887i	−0.964947	−	0.262443i	$-0.915472\pi$
$127$	− 5.91517i	− 0.524887i	0.964947	−	0.262443i	$-0.0845283\pi$
$128$	0	0
$129$	23.6607	2.08321
$130$	0	0
$131$	3.40982	0.297917	0.148958	−	0.988843i	$-0.452408\pi$
$131$	3.40982	0.297917	0.148958	+	0.988843i	$0.452408\pi$
$132$	0	0
$133$	26.7031i	2.31545i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.67321i	− 0.142952i	−0.997442	−	0.0714761i	$-0.977229\pi$
$137$	− 1.67321i	− 0.142952i	0.997442	−	0.0714761i	$-0.0227710\pi$
$138$	0	0
$139$	−8.62008	−0.731146	−0.365573	−	0.930783i	$-0.619127\pi$
$139$	−8.62008	−0.731146	−0.365573	+	0.930783i	$0.619127\pi$
$140$	0	0
$141$	−16.2509	−1.36857
$142$	0	0
$143$	− 2.63080i	− 0.219998i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 25.6201i	− 2.11311i
$148$	0	0
$149$	23.0830	1.89104	0.945518	−	0.325571i	$-0.105556\pi$
$149$	23.0830	1.89104	0.945518	+	0.325571i	$0.105556\pi$
$150$	0	0
$151$	10.7879	0.877910	0.438955	−	0.898509i	$-0.355349\pi$
$151$	10.7879	0.877910	0.438955	+	0.898509i	$0.355349\pi$
$152$	0	0
$153$	− 33.0317i	− 2.67045i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.19955i	0.654395i	0.944956	+	0.327198i	$0.106104\pi$
$157$	8.19955i	0.654395i	−0.944956	+	0.327198i	$0.893896\pi$
$158$	0	0
$159$	−21.0741	−1.67129
$160$	0	0
$161$	3.95759	0.311902
$162$	0	0
$163$	− 16.6625i	− 1.30511i	−0.757743	−	0.652554i	$-0.773698\pi$
$163$	− 16.6625i	− 1.30511i	0.757743	−	0.652554i	$-0.226302\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	5.45223	0.419402
$170$	0	0
$171$	−38.7790	−2.96551
$172$	0	0
$173$	− 8.87276i	− 0.674584i	−0.941400	−	0.337292i	$-0.890489\pi$
$173$	− 8.87276i	− 0.674584i	0.941400	−	0.337292i	$-0.109511\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.1661i	1.06479i
$178$	0	0
$179$	−1.49464	−0.111715	−0.0558574	−	0.998439i	$-0.517789\pi$
$179$	−1.49464	−0.111715	−0.0558574	+	0.998439i	$0.517789\pi$
$180$	0	0
$181$	−12.5777	−0.934891	−0.467445	−	0.884022i	$-0.654826\pi$
$181$	−12.5777	−0.934891	−0.467445	+	0.884022i	$0.654826\pi$
$182$	0	0
$183$	36.8285i	2.72244i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.50356i	− 0.402460i
$188$	0	0
$189$	32.1571	2.33909
$190$	0	0
$191$	16.9045	1.22316	0.611582	−	0.791181i	$-0.290534\pi$
$191$	16.9045	1.22316	0.611582	+	0.791181i	$0.290534\pi$
$192$	0	0
$193$	− 9.91517i	− 0.713710i	−0.934160	−	0.356855i	$-0.883849\pi$
$193$	− 9.91517i	− 0.713710i	0.934160	−	0.356855i	$-0.116151\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.53705i	0.323252i	0.986852	+	0.161626i	$0.0516738\pi$
$197$	4.53705i	0.323252i	−0.986852	+	0.161626i	$0.948326\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−26.9893	−1.90368
$202$	0	0
$203$	− 20.6201i	− 1.44725i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.74732i	0.399466i
$208$	0	0
$209$	−6.46115	−0.446927
$210$	0	0
$211$	8.70491	0.599271	0.299635	−	0.954054i	$-0.403135\pi$
$211$	8.70491	0.599271	0.299635	+	0.954054i	$0.403135\pi$
$212$	0	0
$213$	4.91697i	0.336905i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 23.5777i	− 1.60056i
$218$	0	0
$219$	36.4839	2.46536
$220$	0	0
$221$	−15.7897	−1.06213
$222$	0	0
$223$	5.40982i	0.362268i	0.983458	+	0.181134i	$0.0579768\pi$
$223$	5.40982i	0.362268i	−0.983458	+	0.181134i	$0.942023\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.3250i	− 1.14990i	−0.818189	−	0.574950i	$-0.805022\pi$
$227$	− 17.3250i	− 1.14990i	0.818189	−	0.574950i	$-0.194978\pi$
$228$	0	0
$229$	11.4098	0.753982	0.376991	−	0.926217i	$-0.376959\pi$
$229$	11.4098	0.753982	0.376991	+	0.926217i	$0.376959\pi$
$230$	0	0
$231$	11.2085	0.737463
$232$	0	0
$233$	− 2.08483i	− 0.136581i	−0.997665	−	0.0682907i	$-0.978245\pi$
$233$	− 2.08483i	− 0.136581i	0.997665	−	0.0682907i	$-0.0217546\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 34.9893i	− 2.27280i
$238$	0	0
$239$	18.5353	1.19895	0.599473	−	0.800395i	$-0.295377\pi$
$239$	18.5353	1.19895	0.599473	+	0.800395i	$0.295377\pi$
$240$	0	0
$241$	10.2509	0.660317	0.330159	−	0.943925i	$-0.392898\pi$
$241$	10.2509	0.660317	0.330159	+	0.943925i	$0.392898\pi$
$242$	0	0
$243$	− 4.29509i	− 0.275530i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.5371i	1.17948i
$248$	0	0
$249$	2.08483	0.132120
$250$	0	0
$251$	−17.2527	−1.08898	−0.544490	−	0.838768i	$-0.683277\pi$
$251$	−17.2527	−1.08898	−0.544490	+	0.838768i	$0.683277\pi$
$252$	0	0
$253$	0.957587i	0.0602030i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.9045i	1.80301i	0.432768	+	0.901505i	$0.357537\pi$
$257$	28.9045i	1.80301i	−0.432768	+	0.901505i	$0.642463\pi$
$258$	0	0
$259$	−36.1147	−2.24406
$260$	0	0
$261$	29.9451	1.85355
$262$	0	0
$263$	− 5.24196i	− 0.323233i	−0.986854	−	0.161617i	$-0.948329\pi$
$263$	− 5.24196i	− 0.323233i	0.986854	−	0.161617i	$-0.0516708\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	46.8196i	2.86531i
$268$	0	0
$269$	5.37992	0.328019	0.164010	−	0.986459i	$-0.447557\pi$
$269$	5.37992	0.328019	0.164010	+	0.986459i	$0.447557\pi$
$270$	0	0
$271$	−11.5371	−0.700826	−0.350413	−	0.936595i	$-0.613959\pi$
$271$	−11.5371	−0.700826	−0.350413	+	0.936595i	$0.613959\pi$
$272$	0	0
$273$	− 32.1571i	− 1.94624i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.83035i	0.350312i	0.984541	+	0.175156i	$0.0560429\pi$
$277$	5.83035i	0.350312i	−0.984541	+	0.175156i	$0.943957\pi$
$278$	0	0
$279$	34.2402	2.04990
$280$	0	0
$281$	11.0741	0.660626	0.330313	−	0.943871i	$-0.392846\pi$
$281$	11.0741	0.660626	0.330313	+	0.943871i	$0.392846\pi$
$282$	0	0
$283$	3.86384i	0.229682i	0.993384	+	0.114841i	$0.0366358\pi$
$283$	3.86384i	0.229682i	−0.993384	+	0.114841i	$0.963364\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000i	0.0590281i
$288$	0	0
$289$	−16.0317	−0.943041
$290$	0	0
$291$	32.1571	1.88508
$292$	0	0
$293$	1.54597i	0.0903167i	0.998980	+	0.0451583i	$0.0143792\pi$
$293$	1.54597i	0.0903167i	−0.998980	+	0.0451583i	$0.985621\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.78082i	0.451489i
$298$	0	0
$299$	2.74732	0.158882
$300$	0	0
$301$	31.6607	1.82489
$302$	0	0
$303$	27.2402i	1.56491i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.45223i	− 0.368248i	−0.982903	−	0.184124i	$-0.941055\pi$
$307$	− 6.45223i	− 0.368248i	0.982903	−	0.184124i	$-0.0589448\pi$
$308$	0	0
$309$	36.8285	2.09510
$310$	0	0
$311$	22.8196	1.29398	0.646991	−	0.762497i	$-0.276027\pi$
$311$	22.8196	1.29398	0.646991	+	0.762497i	$0.276027\pi$
$312$	0	0
$313$	16.8620i	0.953099i	0.879148	+	0.476550i	$0.158113\pi$
$313$	16.8620i	0.953099i	−0.879148	+	0.476550i	$0.841887\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.33751i	− 0.0751218i	−0.999294	−	0.0375609i	$-0.988041\pi$
$317$	− 1.33751i	− 0.0751218i	0.999294	−	0.0375609i	$-0.0119588\pi$
$318$	0	0
$319$	4.98928	0.279346
$320$	0	0
$321$	−10.7384	−0.599359
$322$	0	0
$323$	38.7790i	2.15772i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 49.2808i	− 2.72523i
$328$	0	0
$329$	−21.7455	−1.19887
$330$	0	0
$331$	−32.2808	−1.77431	−0.887156	−	0.461470i	$-0.847322\pi$
$331$	−32.2808	−1.77431	−0.887156	+	0.461470i	$0.847322\pi$
$332$	0	0
$333$	− 52.4468i	− 2.87407i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.83215i	− 0.263224i	−0.991301	−	0.131612i	$-0.957985\pi$
$337$	− 4.83215i	− 0.263224i	0.991301	−	0.131612i	$-0.0420153\pi$
$338$	0	0
$339$	−53.5759	−2.90984
$340$	0	0
$341$	5.70491	0.308938
$342$	0	0
$343$	− 6.57947i	− 0.355258i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 34.6183i	− 1.85841i	−0.369569	−	0.929203i	$-0.620495\pi$
$347$	− 34.6183i	− 1.85841i	0.369569	−	0.929203i	$-0.379505\pi$
$348$	0	0
$349$	−26.9558	−1.44291	−0.721455	−	0.692461i	$-0.756526\pi$
$349$	−26.9558	−1.44291	−0.721455	+	0.692461i	$0.756526\pi$
$350$	0	0
$351$	22.3232	1.19152
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 67.2719i	− 3.56040i
$358$	0	0
$359$	−0.420532	−0.0221949	−0.0110974	−	0.999938i	$-0.503532\pi$
$359$	−0.420532	−0.0221949	−0.0110974	+	0.999938i	$0.503532\pi$
$360$	0	0
$361$	26.5263	1.39612
$362$	0	0
$363$	− 29.8214i	− 1.56522i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.7156i	0.611551i	0.952104	+	0.305775i	$0.0989156\pi$
$367$	11.7156i	0.611551i	−0.952104	+	0.305775i	$0.901084\pi$
$368$	0	0
$369$	−1.45223	−0.0756000
$370$	0	0
$371$	−28.1995	−1.46405
$372$	0	0
$373$	28.1661i	1.45838i	0.684310	+	0.729192i	$0.260104\pi$
$373$	28.1661i	1.45838i	−0.684310	+	0.729192i	$0.739896\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.3143i	− 0.737223i
$378$	0	0
$379$	−27.3781	−1.40632	−0.703160	−	0.711032i	$-0.748228\pi$
$379$	−27.3781	−1.40632	−0.703160	+	0.711032i	$0.748228\pi$
$380$	0	0
$381$	17.4946	0.896278
$382$	0	0
$383$	31.7754i	1.62365i	0.583902	+	0.811824i	$0.301525\pi$
$383$	31.7754i	1.62365i	−0.583902	+	0.811824i	$0.698475\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	45.9786i	2.33722i
$388$	0	0
$389$	18.7031	0.948285	0.474143	−	0.880448i	$-0.342758\pi$
$389$	18.7031	0.948285	0.474143	+	0.880448i	$0.342758\pi$
$390$	0	0
$391$	5.74732	0.290655
$392$	0	0
$393$	10.0848i	0.508712i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 39.6924i	− 1.99210i	−0.0887714	−	0.996052i	$-0.528294\pi$
$397$	− 39.6924i	− 1.99210i	0.0887714	−	0.996052i	$-0.471706\pi$
$398$	0	0
$399$	−78.9768	−3.95378
$400$	0	0
$401$	−11.3250	−0.565543	−0.282771	−	0.959187i	$-0.591254\pi$
$401$	−11.3250	−0.565543	−0.282771	+	0.959187i	$0.591254\pi$
$402$	0	0
$403$	− 16.3674i	− 0.815318i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.73840i	− 0.433147i
$408$	0	0
$409$	−9.45223	−0.467383	−0.233691	−	0.972311i	$-0.575081\pi$
$409$	−9.45223	−0.467383	−0.233691	+	0.972311i	$0.575081\pi$
$410$	0	0
$411$	4.94867	0.244100
$412$	0	0
$413$	18.9558i	0.932753i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 25.4946i	− 1.24848i
$418$	0	0
$419$	33.2402	1.62389	0.811944	−	0.583735i	$-0.198409\pi$
$419$	33.2402	1.62389	0.811944	+	0.583735i	$0.198409\pi$
$420$	0	0
$421$	−38.8285	−1.89239	−0.946194	−	0.323600i	$-0.895107\pi$
$421$	−38.8285	−1.89239	−0.946194	+	0.323600i	$0.895107\pi$
$422$	0	0
$423$	− 31.5795i	− 1.53545i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	49.2808i	2.38486i
$428$	0	0
$429$	7.78082	0.375661
$430$	0	0
$431$	−12.4839	−0.601329	−0.300665	−	0.953730i	$-0.597209\pi$
$431$	−12.4839	−0.601329	−0.300665	+	0.953730i	$0.597209\pi$
$432$	0	0
$433$	5.78793i	0.278150i	0.990282	+	0.139075i	$0.0444130\pi$
$433$	5.78793i	0.278150i	−0.990282	+	0.139075i	$0.955587\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.74732i	− 0.322768i
$438$	0	0
$439$	11.9241	0.569106	0.284553	−	0.958660i	$-0.408155\pi$
$439$	11.9241	0.569106	0.284553	+	0.958660i	$0.408155\pi$
$440$	0	0
$441$	49.7861	2.37077
$442$	0	0
$443$	− 39.3973i	− 1.87182i	−0.352236	−	0.935911i	$-0.614579\pi$
$443$	− 39.3973i	− 1.87182i	0.352236	−	0.935911i	$-0.385421\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	68.2701i	3.22906i
$448$	0	0
$449$	−21.3674	−1.00839	−0.504195	−	0.863590i	$-0.668211\pi$
$449$	−21.3674	−1.00839	−0.504195	+	0.863590i	$0.668211\pi$
$450$	0	0
$451$	−0.241962	−0.0113936
$452$	0	0
$453$	31.9063i	1.49909i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.3656i	− 0.952663i	−0.879266	−	0.476331i	$-0.841966\pi$
$457$	− 20.3656i	− 0.952663i	0.879266	−	0.476331i	$-0.158034\pi$
$458$	0	0
$459$	46.6995	2.17975
$460$	0	0
$461$	−27.4946	−1.28055	−0.640277	−	0.768144i	$-0.721180\pi$
$461$	−27.4946	−1.28055	−0.640277	+	0.768144i	$0.721180\pi$
$462$	0	0
$463$	16.8196i	0.781675i	0.920460	+	0.390837i	$0.127814\pi$
$463$	16.8196i	0.781675i	−0.920460	+	0.390837i	$0.872186\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.4504i	− 1.13143i	−0.824601	−	0.565715i	$-0.808600\pi$
$467$	− 24.4504i	− 1.13143i	0.824601	−	0.565715i	$-0.191400\pi$
$468$	0	0
$469$	−36.1147	−1.66762
$470$	0	0
$471$	−24.2509	−1.11742
$472$	0	0
$473$	7.66070i	0.352239i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 40.9522i	− 1.87507i
$478$	0	0
$479$	−10.5688	−0.482899	−0.241449	−	0.970413i	$-0.577623\pi$
$479$	−10.5688	−0.482899	−0.241449	+	0.970413i	$0.577623\pi$
$480$	0	0
$481$	−25.0705	−1.14312
$482$	0	0
$483$	11.7049i	0.532592i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13.0741i	− 0.592444i	−0.955119	−	0.296222i	$-0.904273\pi$
$487$	− 13.0741i	− 0.592444i	0.955119	−	0.296222i	$-0.0957269\pi$
$488$	0	0
$489$	49.2808	2.22855
$490$	0	0
$491$	2.22098	0.100232	0.0501158	−	0.998743i	$-0.484041\pi$
$491$	2.22098	0.100232	0.0501158	+	0.998743i	$0.484041\pi$
$492$	0	0
$493$	− 29.9451i	− 1.34866i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.57947i	0.295129i
$498$	0	0
$499$	−2.87456	−0.128683	−0.0643415	−	0.997928i	$-0.520495\pi$
$499$	−2.87456	−0.128683	−0.0643415	+	0.997928i	$0.520495\pi$
$500$	0	0
$501$	23.6607	1.05708
$502$	0	0
$503$	− 9.36740i	− 0.417672i	−0.977951	−	0.208836i	$-0.933033\pi$
$503$	− 9.36740i	− 0.417672i	0.977951	−	0.208836i	$-0.0669675\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.1254i	0.716156i
$508$	0	0
$509$	40.5866	1.79897	0.899484	−	0.436953i	$-0.143942\pi$
$509$	40.5866	1.79897	0.899484	+	0.436953i	$0.143942\pi$
$510$	0	0
$511$	48.8196	2.15965
$512$	0	0
$513$	− 54.8250i	− 2.42058i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.26160i	− 0.231405i
$518$	0	0
$519$	26.2420	1.15189
$520$	0	0
$521$	34.6714	1.51898	0.759491	−	0.650518i	$-0.225448\pi$
$521$	34.6714	1.51898	0.759491	+	0.650518i	$0.225448\pi$
$522$	0	0
$523$	− 35.4098i	− 1.54836i	−0.632964	−	0.774182i	$-0.718162\pi$
$523$	− 35.4098i	− 1.54836i	0.632964	−	0.774182i	$-0.281838\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 34.2402i	− 1.49152i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−27.5281	−1.19462
$532$	0	0
$533$	0.694191i	0.0300687i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.42053i	− 0.190760i
$538$	0	0
$539$	8.29509	0.357295
$540$	0	0
$541$	37.0705	1.59379	0.796893	−	0.604121i	$-0.206475\pi$
$541$	37.0705	1.59379	0.796893	+	0.604121i	$0.206475\pi$
$542$	0	0
$543$	− 37.1995i	− 1.59639i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.4187i	0.915799i	0.889004	+	0.457899i	$0.151398\pi$
$547$	21.4187i	0.915799i	−0.889004	+	0.457899i	$0.848602\pi$
$548$	0	0
$549$	−71.5670	−3.05440
$550$	0	0
$551$	−35.1553	−1.49767
$552$	0	0
$553$	− 46.8196i	− 1.99097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.9558i	0.887925i	0.896045	+	0.443963i	$0.146428\pi$
$557$	20.9558i	0.887925i	−0.896045	+	0.443963i	$0.853572\pi$
$558$	0	0
$559$	21.9786	0.929594
$560$	0	0
$561$	16.2773	0.687226
$562$	0	0
$563$	− 6.70491i	− 0.282578i	−0.989968	−	0.141289i	$-0.954875\pi$
$563$	− 6.70491i	− 0.282578i	0.989968	−	0.141289i	$-0.0451247\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	26.8710i	1.12847i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	13.2933	0.556307	0.278154	−	0.960537i	$-0.410278\pi$
$571$	13.2933	0.556307	0.278154	+	0.960537i	$0.410278\pi$
$572$	0	0
$573$	49.9964i	2.08863i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.2402i	0.967501i	0.875206	+	0.483750i	$0.160726\pi$
$577$	23.2402i	0.967501i	−0.875206	+	0.483750i	$0.839274\pi$
$578$	0	0
$579$	29.3250	1.21870
$580$	0	0
$581$	2.78973	0.115738
$582$	0	0
$583$	− 6.82323i	− 0.282589i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.6112i	0.726891i	0.931616	+	0.363445i	$0.118400\pi$
$587$	17.6112i	0.726891i	−0.931616	+	0.363445i	$0.881600\pi$
$588$	0	0
$589$	−40.1978	−1.65632
$590$	0	0
$591$	−13.4187	−0.551973
$592$	0	0
$593$	41.5759i	1.70732i	0.520834	+	0.853658i	$0.325621\pi$
$593$	41.5759i	1.70732i	−0.520834	+	0.853658i	$0.674379\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.4062i	1.69464i
$598$	0	0
$599$	−17.7138	−0.723767	−0.361884	−	0.932223i	$-0.617866\pi$
$599$	−17.7138	−0.723767	−0.361884	+	0.932223i	$0.617866\pi$
$600$	0	0
$601$	−46.7772	−1.90808	−0.954041	−	0.299675i	$-0.903122\pi$
$601$	−46.7772	−1.90808	−0.954041	+	0.299675i	$0.903122\pi$
$602$	0	0
$603$	− 52.4468i	− 2.13580i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.9893i	1.74488i	0.488720	+	0.872441i	$0.337464\pi$
$607$	42.9893i	1.74488i	−0.488720	+	0.872441i	$0.662536\pi$
$608$	0	0
$609$	60.9857	2.47126
$610$	0	0
$611$	−15.0955	−0.610700
$612$	0	0
$613$	32.6500i	1.31872i	0.751827	+	0.659360i	$0.229173\pi$
$613$	32.6500i	1.31872i	−0.751827	+	0.659360i	$0.770827\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.3232i	− 0.455854i	−0.973678	−	0.227927i	$-0.926805\pi$
$617$	− 11.3232i	− 0.455854i	0.973678	−	0.227927i	$-0.0731948\pi$
$618$	0	0
$619$	10.6625	0.428562	0.214281	−	0.976772i	$-0.431259\pi$
$619$	10.6625	0.428562	0.214281	+	0.976772i	$0.431259\pi$
$620$	0	0
$621$	−8.12544	−0.326063
$622$	0	0
$623$	62.6500i	2.51002i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 19.1094i	− 0.763156i
$628$	0	0
$629$	−52.4468	−2.09119
$630$	0	0
$631$	5.24376	0.208751	0.104375	−	0.994538i	$-0.466716\pi$
$631$	5.24376	0.208751	0.104375	+	0.994538i	$0.466716\pi$
$632$	0	0
$633$	25.7455i	1.02329i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 23.7987i	− 0.942937i
$638$	0	0
$639$	−9.55489	−0.377986
$640$	0	0
$641$	30.4205	1.20154	0.600769	−	0.799422i	$-0.294861\pi$
$641$	30.4205	1.20154	0.600769	+	0.799422i	$0.294861\pi$
$642$	0	0
$643$	9.37992i	0.369908i	0.982747	+	0.184954i	$0.0592136\pi$
$643$	9.37992i	0.369908i	−0.982747	+	0.184954i	$0.940786\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 18.6714i	− 0.734049i	−0.930211	−	0.367024i	$-0.880377\pi$
$647$	− 18.6714i	− 0.734049i	0.930211	−	0.367024i	$-0.119623\pi$
$648$	0	0
$649$	−4.58659	−0.180039
$650$	0	0
$651$	69.7330	2.73305
$652$	0	0
$653$	− 12.0723i	− 0.472426i	−0.971701	−	0.236213i	$-0.924094\pi$
$653$	− 12.0723i	− 0.472426i	0.971701	−	0.236213i	$-0.0759063\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	70.8973i	2.76597i
$658$	0	0
$659$	−21.2402	−0.827399	−0.413700	−	0.910413i	$-0.635764\pi$
$659$	−21.2402	−0.827399	−0.413700	+	0.910413i	$0.635764\pi$
$660$	0	0
$661$	−26.9362	−1.04769	−0.523847	−	0.851812i	$-0.675504\pi$
$661$	−26.9362	−1.04769	−0.523847	+	0.851812i	$0.675504\pi$
$662$	0	0
$663$	− 46.6995i	− 1.81366i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.21027i	0.201742i
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−11.9241	−0.460324
$672$	0	0
$673$	11.4098i	0.439816i	0.975521	+	0.219908i	$0.0705757\pi$
$673$	11.4098i	0.439816i	−0.975521	+	0.219908i	$0.929424\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11.8638i	− 0.455965i	−0.973665	−	0.227982i	$-0.926787\pi$
$677$	− 11.8638i	− 0.455965i	0.973665	−	0.227982i	$-0.0732128\pi$
$678$	0	0
$679$	43.0299	1.65134
$680$	0	0
$681$	51.2402	1.96353
$682$	0	0
$683$	− 10.0723i	− 0.385406i	−0.981257	−	0.192703i	$-0.938275\pi$
$683$	− 10.0723i	− 0.385406i	0.981257	−	0.192703i	$-0.0617254\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	33.7455i	1.28747i
$688$	0	0
$689$	−19.5759	−0.745781
$690$	0	0
$691$	−45.9964	−1.74979	−0.874893	−	0.484317i	$-0.839068\pi$
$691$	−45.9964	−1.74979	−0.874893	+	0.484317i	$0.839068\pi$
$692$	0	0
$693$	21.7808i	0.827385i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.45223i	0.0550071i
$698$	0	0
$699$	6.16605	0.233222
$700$	0	0
$701$	−21.2312	−0.801893	−0.400947	−	0.916101i	$-0.631319\pi$
$701$	−21.2312	−0.801893	−0.400947	+	0.916101i	$0.631319\pi$
$702$	0	0
$703$	61.5723i	2.32224i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.4504i	1.37086i
$708$	0	0
$709$	−31.3781	−1.17843	−0.589215	−	0.807976i	$-0.700563\pi$
$709$	−31.3781	−1.17843	−0.589215	+	0.807976i	$0.700563\pi$
$710$	0	0
$711$	67.9928	2.54993
$712$	0	0
$713$	5.95759i	0.223113i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	54.8196i	2.04728i
$718$	0	0
$719$	20.4629	0.763139	0.381570	−	0.924340i	$-0.375384\pi$
$719$	20.4629	0.763139	0.381570	+	0.924340i	$0.375384\pi$
$720$	0	0
$721$	49.2808	1.83531
$722$	0	0
$723$	30.3179i	1.12753i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.9875i	0.555855i	0.960602	+	0.277928i	$0.0896475\pi$
$727$	14.9875i	0.555855i	−0.960602	+	0.277928i	$0.910352\pi$
$728$	0	0
$729$	33.0723	1.22490
$730$	0	0
$731$	45.9786	1.70058
$732$	0	0
$733$	22.2844i	0.823092i	0.911389	+	0.411546i	$0.135011\pi$
$733$	22.2844i	0.823092i	−0.911389	+	0.411546i	$0.864989\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.73840i	− 0.321883i
$738$	0	0
$739$	−30.9344	−1.13794	−0.568969	−	0.822359i	$-0.692658\pi$
$739$	−30.9344	−1.13794	−0.568969	+	0.822359i	$0.692658\pi$
$740$	0	0
$741$	−54.8250	−2.01404
$742$	0	0
$743$	44.8285i	1.64460i	0.569054	+	0.822300i	$0.307309\pi$
$743$	44.8285i	1.64460i	−0.569054	+	0.822300i	$0.692691\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.05133i	0.148230i
$748$	0	0
$749$	−14.3692	−0.525039
$750$	0	0
$751$	32.1661	1.17376	0.586878	−	0.809675i	$-0.300357\pi$
$751$	32.1661	1.17376	0.586878	+	0.809675i	$0.300357\pi$
$752$	0	0
$753$	− 51.0263i	− 1.85950i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.6835i	1.47867i	0.673340	+	0.739333i	$0.264859\pi$
$757$	40.6835i	1.47867i	−0.673340	+	0.739333i	$0.735141\pi$
$758$	0	0
$759$	−2.83215	−0.102800
$760$	0	0
$761$	28.4415	1.03100	0.515502	−	0.856888i	$-0.327605\pi$
$761$	28.4415	1.03100	0.515502	+	0.856888i	$0.327605\pi$
$762$	0	0
$763$	− 65.9433i	− 2.38731i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.1589i	0.475142i
$768$	0	0
$769$	40.4205	1.45760	0.728801	−	0.684726i	$-0.240078\pi$
$769$	40.4205	1.45760	0.728801	+	0.684726i	$0.240078\pi$
$770$	0	0
$771$	−85.4874	−3.07876
$772$	0	0
$773$	− 24.5688i	− 0.883677i	−0.897095	−	0.441838i	$-0.854327\pi$
$773$	− 24.5688i	− 0.883677i	0.897095	−	0.441838i	$-0.145673\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 106.812i	− 3.83187i
$778$	0	0
$779$	1.70491	0.0610847
$780$	0	0
$781$	−1.59198	−0.0569656
$782$	0	0
$783$	42.3357i	1.51295i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 11.7790i	− 0.419877i	−0.977715	−	0.209938i	$-0.932674\pi$
$787$	− 11.7790i	− 0.419877i	0.977715	−	0.209938i	$-0.0673263\pi$
$788$	0	0
$789$	15.5036	0.551941
$790$	0	0
$791$	−71.6906	−2.54902
$792$	0	0
$793$	34.2103i	1.21484i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.8602i	− 0.490955i	−0.969402	−	0.245478i	$-0.921055\pi$
$797$	− 13.8602i	− 0.490955i	0.969402	−	0.245478i	$-0.0789448\pi$
$798$	0	0
$799$	−31.5795	−1.11720
$800$	0	0
$801$	−90.9821	−3.21469
$802$	0	0
$803$	11.8125i	0.416854i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.9116i	0.560114i
$808$	0	0
$809$	52.0830	1.83114	0.915571	−	0.402157i	$-0.131739\pi$
$809$	52.0830	1.83114	0.915571	+	0.402157i	$0.131739\pi$
$810$	0	0
$811$	−40.7013	−1.42922	−0.714608	−	0.699525i	$-0.753395\pi$
$811$	−40.7013	−1.42922	−0.714608	+	0.699525i	$0.753395\pi$
$812$	0	0
$813$	− 34.1218i	− 1.19671i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 53.9786i	− 1.88847i
$818$	0	0
$819$	62.4892	2.18355
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	42.9009i	1.49543i	0.664020	+	0.747715i	$0.268849\pi$
$823$	42.9009i	1.49543i	−0.664020	+	0.747715i	$0.731151\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.4397i	1.37145i	0.727859	+	0.685727i	$0.240515\pi$
$827$	39.4397i	1.37145i	−0.727859	+	0.685727i	$0.759485\pi$
$828$	0	0
$829$	−29.5460	−1.02617	−0.513087	−	0.858337i	$-0.671498\pi$
$829$	−29.5460	−1.02617	−0.513087	+	0.858337i	$0.671498\pi$
$830$	0	0
$831$	−17.2438	−0.598179
$832$	0	0
$833$	− 49.7861i	− 1.72499i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	48.4080i	1.67323i
$838$	0	0
$839$	1.09554	0.0378223	0.0189112	−	0.999821i	$-0.493980\pi$
$839$	1.09554	0.0378223	0.0189112	+	0.999821i	$0.493980\pi$
$840$	0	0
$841$	−1.85313	−0.0639009
$842$	0	0
$843$	32.7526i	1.12806i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 39.9045i	− 1.37113i
$848$	0	0
$849$	−11.4277	−0.392196
$850$	0	0
$851$	9.12544	0.312816
$852$	0	0
$853$	10.0937i	0.345603i	0.984957	+	0.172802i	$0.0552819\pi$
$853$	10.0937i	0.345603i	−0.984957	+	0.172802i	$0.944718\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.8303i	1.42890i	0.699688	+	0.714449i	$0.253322\pi$
$857$	41.8303i	1.42890i	−0.699688	+	0.714449i	$0.746678\pi$
$858$	0	0
$859$	3.35848	0.114590	0.0572950	−	0.998357i	$-0.481752\pi$
$859$	3.35848	0.114590	0.0572950	+	0.998357i	$0.481752\pi$
$860$	0	0
$861$	−2.95759	−0.100794
$862$	0	0
$863$	3.83395i	0.130509i	0.997869	+	0.0652545i	$0.0207859\pi$
$863$	3.83395i	0.130509i	−0.997869	+	0.0652545i	$0.979214\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 47.4151i	− 1.61030i
$868$	0	0
$869$	11.3286	0.384296
$870$	0	0
$871$	−25.0705	−0.849482
$872$	0	0
$873$	62.4892i	2.11494i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.5263i	0.524287i	0.965029	+	0.262144i	$0.0844294\pi$
$877$	15.5263i	0.524287i	−0.965029	+	0.262144i	$0.915571\pi$
$878$	0	0
$879$	−4.57235	−0.154221
$880$	0	0
$881$	26.9893	0.909292	0.454646	−	0.890672i	$-0.349766\pi$
$881$	26.9893	0.909292	0.454646	+	0.890672i	$0.349766\pi$
$882$	0	0
$883$	− 21.5884i	− 0.726507i	−0.931690	−	0.363254i	$-0.881666\pi$
$883$	− 21.5884i	− 0.726507i	0.931690	−	0.363254i	$-0.118334\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.4098i	1.65902i	0.558492	+	0.829510i	$0.311380\pi$
$887$	49.4098i	1.65902i	−0.558492	+	0.829510i	$0.688620\pi$
$888$	0	0
$889$	23.4098	0.785140
$890$	0	0
$891$	−6.50176	−0.217817
$892$	0	0
$893$	37.0741i	1.24064i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.12544i	0.271301i
$898$	0	0
$899$	31.0406	1.03526
$900$	0	0
$901$	−40.9522	−1.36432
$902$	0	0
$903$	93.6393i	3.11612i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 9.54957i	− 0.317088i	−0.987352	−	0.158544i	$-0.949320\pi$
$907$	− 9.54957i	− 0.317088i	0.987352	−	0.158544i	$-0.0506800\pi$
$908$	0	0
$909$	−52.9344	−1.75572
$910$	0	0
$911$	−47.3036	−1.56724	−0.783618	−	0.621243i	$-0.786628\pi$
$911$	−47.3036	−1.56724	−0.783618	+	0.621243i	$0.786628\pi$
$912$	0	0
$913$	0.675011i	0.0223396i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.4946i	0.445632i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	19.0830	0.628807
$922$	0	0
$923$	4.56741i	0.150338i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	71.5670i	2.35057i
$928$	0	0
$929$	19.6942	0.646145	0.323073	−	0.946374i	$-0.395284\pi$
$929$	19.6942	0.646145	0.323073	+	0.946374i	$0.395284\pi$
$930$	0	0
$931$	−58.4486	−1.91558
$932$	0	0
$933$	67.4910i	2.20956i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31.8214i	− 1.03956i	−0.854300	−	0.519780i	$-0.826014\pi$
$937$	− 31.8214i	− 1.03956i	0.854300	−	0.519780i	$-0.173986\pi$
$938$	0	0
$939$	−49.8710	−1.62748
$940$	0	0
$941$	−2.36740	−0.0771751	−0.0385876	−	0.999255i	$-0.512286\pi$
$941$	−2.36740	−0.0771751	−0.0385876	+	0.999255i	$0.512286\pi$
$942$	0	0
$943$	− 0.252679i	− 0.00822837i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.90266i	− 0.321793i	−0.986971	−	0.160897i	$-0.948561\pi$
$947$	− 9.90266i	− 0.321793i	0.986971	−	0.160897i	$-0.0514386\pi$
$948$	0	0
$949$	33.8901	1.10012
$950$	0	0
$951$	3.95579	0.128275
$952$	0	0
$953$	31.5442i	1.02182i	0.859635	+	0.510908i	$0.170691\pi$
$953$	31.5442i	1.02182i	−0.859635	+	0.510908i	$0.829309\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.7562i	0.477001i
$958$	0	0
$959$	6.62188	0.213832
$960$	0	0
$961$	4.49284	0.144930
$962$	0	0
$963$	− 20.8674i	− 0.672441i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 15.5973i	− 0.501575i	−0.968042	−	0.250788i	$-0.919310\pi$
$967$	− 15.5973i	− 0.501575i	0.968042	−	0.250788i	$-0.0806896\pi$
$968$	0	0
$969$	−114.692	−3.68445
$970$	0	0
$971$	−47.1022	−1.51158	−0.755791	−	0.654813i	$-0.772747\pi$
$971$	−47.1022	−1.51158	−0.755791	+	0.654813i	$0.772747\pi$
$972$	0	0
$973$	− 34.1147i	− 1.09367i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.9469i	0.926092i	0.886334	+	0.463046i	$0.153244\pi$
$977$	28.9469i	0.926092i	−0.886334	+	0.463046i	$0.846756\pi$
$978$	0	0
$979$	−15.1589	−0.484482
$980$	0	0
$981$	95.7647	3.05753
$982$	0	0
$983$	36.2312i	1.15560i	0.816179	+	0.577799i	$0.196088\pi$
$983$	36.2312i	1.15560i	−0.816179	+	0.577799i	$0.803912\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 64.3143i	− 2.04715i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−35.1750	−1.11737	−0.558685	−	0.829380i	$-0.688694\pi$
$991$	−35.1750	−1.11737	−0.558685	+	0.829380i	$0.688694\pi$
$992$	0	0
$993$	− 95.4732i	− 3.02975i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.2402i	0.609342i	0.952458	+	0.304671i	$0.0985465\pi$
$997$	19.2402i	0.609342i	−0.952458	+	0.304671i	$0.901453\pi$
$998$	0	0
$999$	74.1482	2.34595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.p.4049.5		6
5.2	odd	4		4600.2.a.x.1.3		3
5.3	odd	4		920.2.a.h.1.1	✓	3
5.4	even	2	inner	4600.2.e.p.4049.2		6
15.8	even	4		8280.2.a.bj.1.3		3
20.3	even	4		1840.2.a.s.1.3		3
20.7	even	4		9200.2.a.ce.1.1		3
40.3	even	4		7360.2.a.cc.1.1		3
40.13	odd	4		7360.2.a.by.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.1	✓	3	5.3	odd	4
1840.2.a.s.1.3		3	20.3	even	4
4600.2.a.x.1.3		3	5.2	odd	4
4600.2.e.p.4049.2		6	5.4	even	2	inner
4600.2.e.p.4049.5		6	1.1	even	1	trivial
7360.2.a.by.1.3		3	40.13	odd	4
7360.2.a.cc.1.1		3	40.3	even	4
8280.2.a.bj.1.3		3	15.8	even	4
9200.2.a.ce.1.1		3	20.7	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)

\(f(q)\)	\(=\)	\(q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} +O(q^{10})\)	\(q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} -0.957587 q^{11} +2.74732i q^{13} +5.74732i q^{17} +6.74732 q^{19} -11.7049 q^{21} -1.00000i q^{23} -8.12544i q^{27} -5.21027 q^{29} -5.95759 q^{31} -2.83215i q^{33} +9.12544i q^{37} -8.12544 q^{39} +0.252679 q^{41} -8.00000i q^{43} +5.49464i q^{47} -8.66249 q^{49} -16.9982 q^{51} +7.12544i q^{53} +19.9558i q^{57} +4.78973 q^{59} +12.4522 q^{61} -22.7455i q^{63} +9.12544i q^{67} +2.95759 q^{69} +1.66249 q^{71} -12.3357i q^{73} -3.78973i q^{77} -11.8303 q^{79} +6.78973 q^{81} -0.704908i q^{83} -15.4098i q^{87} +15.8303 q^{89} -10.8728 q^{91} -17.6201i q^{93} -10.8728i q^{97} +5.50356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{9}+O(q^{10}) \) 6 * q - 20 * q^9	\( 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) \) 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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