LMFDB - Embedded newform 7605.2.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

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:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.05896$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32691 q^{2} -0.239314 q^{4} -1.00000 q^{5} -1.05896 q^{7} +2.97136 q^{8} +O(q^{10})$	$q-1.32691 q^{2} -0.239314 q^{4} -1.00000 q^{5} -1.05896 q^{7} +2.97136 q^{8} +1.32691 q^{10} -2.49274 q^{11} +1.40514 q^{14} -3.46410 q^{16} -4.95209 q^{17} +6.35723 q^{19} +0.239314 q^{20} +3.30763 q^{22} +3.62518 q^{23} +1.00000 q^{25} +0.253423 q^{28} +0.234149 q^{29} +1.31755 q^{31} -1.34618 q^{32} +6.57097 q^{34} +1.05896 q^{35} -6.11792 q^{37} -8.43547 q^{38} -2.97136 q^{40} -8.85513 q^{41} +9.66962 q^{43} +0.596546 q^{44} -4.81028 q^{46} -5.70173 q^{47} -5.87861 q^{49} -1.32691 q^{50} +4.98547 q^{53} +2.49274 q^{55} -3.14655 q^{56} -0.310694 q^{58} +2.10212 q^{59} +3.05379 q^{61} -1.74826 q^{62} +8.71446 q^{64} +7.63454 q^{67} +1.18510 q^{68} -1.40514 q^{70} -8.17688 q^{71} -3.98716 q^{73} +8.11792 q^{74} -1.52137 q^{76} +2.63971 q^{77} +15.8359 q^{79} +3.46410 q^{80} +11.7500 q^{82} +15.8037 q^{83} +4.95209 q^{85} -12.8307 q^{86} -7.40683 q^{88} -15.8452 q^{89} -0.867556 q^{92} +7.56567 q^{94} -6.35723 q^{95} +10.0559 q^{97} +7.80037 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 + 6 * q^7	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 6 q^{7} + 2 q^{10} - 8 q^{11} + 2 q^{14} + 2 q^{17} - 4 q^{20} - 4 q^{23} + 4 q^{25} + 4 q^{28} - 6 q^{29} - 12 q^{32} + 24 q^{34} - 6 q^{35} - 4 q^{37} - 8 q^{38} - 10 q^{41} + 6 q^{43} - 28 q^{44} - 12 q^{46} - 38 q^{47} - 8 q^{49} - 2 q^{50} + 16 q^{53} + 8 q^{55} - 4 q^{56} - 28 q^{58} + 14 q^{59} + 30 q^{62} - 16 q^{64} + 14 q^{67} + 16 q^{68} - 2 q^{70} - 2 q^{71} + 22 q^{73} + 12 q^{74} - 16 q^{76} - 4 q^{77} + 28 q^{79} - 24 q^{82} - 20 q^{83} - 2 q^{85} - 14 q^{86} + 8 q^{88} - 30 q^{89} - 20 q^{92} + 16 q^{94} - 10 q^{97} + 16 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 + 6 * q^7 + 2 * q^10 - 8 * q^11 + 2 * q^14 + 2 * q^17 - 4 * q^20 - 4 * q^23 + 4 * q^25 + 4 * q^28 - 6 * q^29 - 12 * q^32 + 24 * q^34 - 6 * q^35 - 4 * q^37 - 8 * q^38 - 10 * q^41 + 6 * q^43 - 28 * q^44 - 12 * q^46 - 38 * q^47 - 8 * q^49 - 2 * q^50 + 16 * q^53 + 8 * q^55 - 4 * q^56 - 28 * q^58 + 14 * q^59 + 30 * q^62 - 16 * q^64 + 14 * q^67 + 16 * q^68 - 2 * q^70 - 2 * q^71 + 22 * q^73 + 12 * q^74 - 16 * q^76 - 4 * q^77 + 28 * q^79 - 24 * q^82 - 20 * q^83 - 2 * q^85 - 14 * q^86 + 8 * q^88 - 30 * q^89 - 20 * q^92 + 16 * q^94 - 10 * q^97 + 16 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.32691	−0.938266	−0.469133	−	0.883128i	$-0.655434\pi$
$2$	−1.32691	−0.938266	−0.469133	+	0.883128i	$0.655434\pi$
$3$	0	0
$3$	0	0
$4$	−0.239314	−0.119657
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.05896	−0.400249	−0.200125	−	0.979770i	$-0.564135\pi$
$7$	−1.05896	−0.400249	−0.200125	+	0.979770i	$0.564135\pi$
$8$	2.97136	1.05054
$9$	0	0
$10$	1.32691	0.419605
$11$	−2.49274	−0.751589	−0.375794	−	0.926703i	$-0.622630\pi$
$11$	−2.49274	−0.751589	−0.375794	+	0.926703i	$0.622630\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.40514	0.375540
$15$	0	0
$16$	−3.46410	−0.866025
$17$	−4.95209	−1.20106	−0.600529	−	0.799603i	$-0.705043\pi$
$17$	−4.95209	−1.20106	−0.600529	+	0.799603i	$0.705043\pi$
$18$	0	0
$19$	6.35723	1.45845	0.729225	−	0.684274i	$-0.239881\pi$
$19$	6.35723	1.45845	0.729225	+	0.684274i	$0.239881\pi$
$20$	0.239314	0.0535122
$21$	0	0
$22$	3.30763	0.705190
$23$	3.62518	0.755903	0.377951	−	0.925825i	$-0.376629\pi$
$23$	3.62518	0.755903	0.377951	+	0.925825i	$0.376629\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0.253423	0.0478925
$29$	0.234149	0.0434804	0.0217402	−	0.999764i	$-0.493079\pi$
$29$	0.234149	0.0434804	0.0217402	+	0.999764i	$0.493079\pi$
$30$	0	0
$31$	1.31755	0.236638	0.118319	−	0.992976i	$-0.462249\pi$
$31$	1.31755	0.236638	0.118319	+	0.992976i	$0.462249\pi$
$32$	−1.34618	−0.237974
$33$	0	0
$34$	6.57097	1.12691
$35$	1.05896	0.178997
$36$	0	0
$37$	−6.11792	−1.00578	−0.502890	−	0.864351i	$-0.667730\pi$
$37$	−6.11792	−1.00578	−0.502890	+	0.864351i	$0.667730\pi$
$38$	−8.43547	−1.36841
$39$	0	0
$40$	−2.97136	−0.469814
$41$	−8.85513	−1.38294	−0.691470	−	0.722405i	$-0.743036\pi$
$41$	−8.85513	−1.38294	−0.691470	+	0.722405i	$0.743036\pi$
$42$	0	0
$43$	9.66962	1.47460	0.737301	−	0.675564i	$-0.236100\pi$
$43$	9.66962	1.47460	0.737301	+	0.675564i	$0.236100\pi$
$44$	0.596546	0.0899327
$45$	0	0
$46$	−4.81028	−0.709238
$47$	−5.70173	−0.831682	−0.415841	−	0.909437i	$-0.636513\pi$
$47$	−5.70173	−0.831682	−0.415841	+	0.909437i	$0.636513\pi$
$48$	0	0
$49$	−5.87861	−0.839801
$50$	−1.32691	−0.187653
$51$	0	0
$52$	0	0
$53$	4.98547	0.684808	0.342404	−	0.939553i	$-0.388759\pi$
$53$	4.98547	0.684808	0.342404	+	0.939553i	$0.388759\pi$
$54$	0	0
$55$	2.49274	0.336121
$56$	−3.14655	−0.420476
$57$	0	0
$58$	−0.310694	−0.0407962
$59$	2.10212	0.273673	0.136836	−	0.990594i	$-0.456307\pi$
$59$	2.10212	0.273673	0.136836	+	0.990594i	$0.456307\pi$
$60$	0	0
$61$	3.05379	0.390998	0.195499	−	0.980704i	$-0.437367\pi$
$61$	3.05379	0.390998	0.195499	+	0.980704i	$0.437367\pi$
$62$	−1.74826	−0.222030
$63$	0	0
$64$	8.71446	1.08931
$65$	0	0
$66$	0	0
$67$	7.63454	0.932708	0.466354	−	0.884598i	$-0.345567\pi$
$67$	7.63454	0.932708	0.466354	+	0.884598i	$0.345567\pi$
$68$	1.18510	0.143715
$69$	0	0
$70$	−1.40514	−0.167947
$71$	−8.17688	−0.970417	−0.485208	−	0.874399i	$-0.661256\pi$
$71$	−8.17688	−0.970417	−0.485208	+	0.874399i	$0.661256\pi$
$72$	0	0
$73$	−3.98716	−0.466662	−0.233331	−	0.972397i	$-0.574963\pi$
$73$	−3.98716	−0.466662	−0.233331	+	0.972397i	$0.574963\pi$
$74$	8.11792	0.943689
$75$	0	0
$76$	−1.52137	−0.174513
$77$	2.63971	0.300823
$78$	0	0
$79$	15.8359	1.78167	0.890837	−	0.454324i	$-0.150119\pi$
$79$	15.8359	1.78167	0.890837	+	0.454324i	$0.150119\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	11.7500	1.29757
$83$	15.8037	1.73469	0.867343	−	0.497710i	$-0.165826\pi$
$83$	15.8037	1.73469	0.867343	+	0.497710i	$0.165826\pi$
$84$	0	0
$85$	4.95209	0.537130
$86$	−12.8307	−1.38357
$87$	0	0
$88$	−7.40683	−0.789571
$89$	−15.8452	−1.67959	−0.839795	−	0.542904i	$-0.817325\pi$
$89$	−15.8452	−1.67959	−0.839795	+	0.542904i	$0.817325\pi$
$90$	0	0
$91$	0	0
$92$	−0.867556	−0.0904489
$93$	0	0
$94$	7.56567	0.780339
$95$	−6.35723	−0.652238
$96$	0	0
$97$	10.0559	1.02102	0.510511	−	0.859871i	$-0.329456\pi$
$97$	10.0559	1.02102	0.510511	+	0.859871i	$0.329456\pi$
$98$	7.80037	0.787956
$99$	0	0
$100$	−0.239314	−0.0239314
$101$	11.4114	1.13548	0.567741	−	0.823208i	$-0.307818\pi$
$101$	11.4114	1.13548	0.567741	+	0.823208i	$0.307818\pi$
$102$	0	0
$103$	2.89788	0.285537	0.142768	−	0.989756i	$-0.454400\pi$
$103$	2.89788	0.285537	0.142768	+	0.989756i	$0.454400\pi$
$104$	0	0
$105$	0	0
$106$	−6.61527	−0.642532
$107$	15.1132	1.46105	0.730523	−	0.682888i	$-0.239277\pi$
$107$	15.1132	1.46105	0.730523	+	0.682888i	$0.239277\pi$
$108$	0	0
$109$	0.385868	0.0369594	0.0184797	−	0.999829i	$-0.494117\pi$
$109$	0.385868	0.0369594	0.0184797	+	0.999829i	$0.494117\pi$
$110$	−3.30763	−0.315371
$111$	0	0
$112$	3.66834	0.346626
$113$	−8.42136	−0.792215	−0.396107	−	0.918204i	$-0.629639\pi$
$113$	−8.42136	−0.792215	−0.396107	+	0.918204i	$0.629639\pi$
$114$	0	0
$115$	−3.62518	−0.338050
$116$	−0.0560351	−0.00520273
$117$	0	0
$118$	−2.78932	−0.256778
$119$	5.24406	0.480722
$120$	0	0
$121$	−4.78626	−0.435115
$122$	−4.05211	−0.366860
$123$	0	0
$124$	−0.315307	−0.0283154
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.3871	−1.01045	−0.505223	−	0.862989i	$-0.668590\pi$
$127$	−11.3871	−1.01045	−0.505223	+	0.862989i	$0.668590\pi$
$128$	−8.87093	−0.784087
$129$	0	0
$130$	0	0
$131$	11.4948	1.00431	0.502154	−	0.864778i	$-0.332541\pi$
$131$	11.4948	1.00431	0.502154	+	0.864778i	$0.332541\pi$
$132$	0	0
$133$	−6.73205	−0.583743
$134$	−10.1303	−0.875128
$135$	0	0
$136$	−14.7145	−1.26175
$137$	−17.2170	−1.47095	−0.735473	−	0.677554i	$-0.763040\pi$
$137$	−17.2170	−1.47095	−0.735473	+	0.677554i	$0.763040\pi$
$138$	0	0
$139$	2.40683	0.204145	0.102072	−	0.994777i	$-0.467453\pi$
$139$	2.40683	0.204145	0.102072	+	0.994777i	$0.467453\pi$
$140$	−0.253423	−0.0214182
$141$	0	0
$142$	10.8500	0.910509
$143$	0	0
$144$	0	0
$145$	−0.234149	−0.0194450
$146$	5.29060	0.437853
$147$	0	0
$148$	1.46410	0.120348
$149$	3.38239	0.277096	0.138548	−	0.990356i	$-0.455756\pi$
$149$	3.38239	0.277096	0.138548	+	0.990356i	$0.455756\pi$
$150$	0	0
$151$	1.46758	0.119430	0.0597149	−	0.998215i	$-0.480981\pi$
$151$	1.46758	0.119430	0.0597149	+	0.998215i	$0.480981\pi$
$152$	18.8897	1.53215
$153$	0	0
$154$	−3.50265	−0.282252
$155$	−1.31755	−0.105828
$156$	0	0
$157$	−20.7833	−1.65869	−0.829345	−	0.558736i	$-0.811286\pi$
$157$	−20.7833	−1.65869	−0.829345	+	0.558736i	$0.811286\pi$
$158$	−21.0127	−1.67168
$159$	0	0
$160$	1.34618	0.106425
$161$	−3.83892	−0.302549
$162$	0	0
$163$	−16.7273	−1.31018	−0.655092	−	0.755549i	$-0.727370\pi$
$163$	−16.7273	−1.31018	−0.655092	+	0.755549i	$0.727370\pi$
$164$	2.11915	0.165478
$165$	0	0
$166$	−20.9701	−1.62760
$167$	6.41103	0.496100	0.248050	−	0.968747i	$-0.420210\pi$
$167$	6.41103	0.496100	0.248050	+	0.968747i	$0.420210\pi$
$168$	0	0
$169$	0	0
$170$	−6.57097	−0.503970
$171$	0	0
$172$	−2.31407	−0.176446
$173$	22.5051	1.71103	0.855514	−	0.517780i	$-0.173241\pi$
$173$	22.5051	1.71103	0.855514	+	0.517780i	$0.173241\pi$
$174$	0	0
$175$	−1.05896	−0.0800498
$176$	8.63509	0.650895
$177$	0	0
$178$	21.0252	1.57590
$179$	−4.75594	−0.355475	−0.177738	−	0.984078i	$-0.556878\pi$
$179$	−4.75594	−0.355475	−0.177738	+	0.984078i	$0.556878\pi$
$180$	0	0
$181$	−11.4606	−0.851862	−0.425931	−	0.904756i	$-0.640053\pi$
$181$	−11.4606	−0.851862	−0.425931	+	0.904756i	$0.640053\pi$
$182$	0	0
$183$	0	0
$184$	10.7717	0.794103
$185$	6.11792	0.449798
$186$	0	0
$187$	12.3443	0.902702
$188$	1.36450	0.0995165
$189$	0	0
$190$	8.43547	0.611973
$191$	5.80206	0.419822	0.209911	−	0.977720i	$-0.432683\pi$
$191$	5.80206	0.419822	0.209911	+	0.977720i	$0.432683\pi$
$192$	0	0
$193$	2.43336	0.175157	0.0875786	−	0.996158i	$-0.472087\pi$
$193$	2.43336	0.175157	0.0875786	+	0.996158i	$0.472087\pi$
$194$	−13.3433	−0.957990
$195$	0	0
$196$	1.40683	0.100488
$197$	−13.3824	−0.953456	−0.476728	−	0.879051i	$-0.658177\pi$
$197$	−13.3824	−0.953456	−0.476728	+	0.879051i	$0.658177\pi$
$198$	0	0
$199$	−17.8511	−1.26543	−0.632716	−	0.774384i	$-0.718060\pi$
$199$	−17.8511	−1.26543	−0.632716	+	0.774384i	$0.718060\pi$
$200$	2.97136	0.210107
$201$	0	0
$202$	−15.1419	−1.06538
$203$	−0.247954	−0.0174030
$204$	0	0
$205$	8.85513	0.618469
$206$	−3.84522	−0.267909
$207$	0	0
$208$	0	0
$209$	−15.8469	−1.09615
$210$	0	0
$211$	9.98885	0.687661	0.343830	−	0.939032i	$-0.388275\pi$
$211$	9.98885	0.687661	0.343830	+	0.939032i	$0.388275\pi$
$212$	−1.19309	−0.0819419
$213$	0	0
$214$	−20.0538	−1.37085
$215$	−9.66962	−0.659462
$216$	0	0
$217$	−1.39523	−0.0947143
$218$	−0.512011	−0.0346778
$219$	0	0
$220$	−0.596546	−0.0402191
$221$	0	0
$222$	0	0
$223$	−20.2358	−1.35509	−0.677546	−	0.735480i	$-0.736957\pi$
$223$	−20.2358	−1.35509	−0.677546	+	0.735480i	$0.736957\pi$
$224$	1.42555	0.0952488
$225$	0	0
$226$	11.1744	0.743308
$227$	3.18793	0.211590	0.105795	−	0.994388i	$-0.466261\pi$
$227$	3.18793	0.211590	0.105795	+	0.994388i	$0.466261\pi$
$228$	0	0
$229$	27.3461	1.80708	0.903540	−	0.428504i	$-0.140959\pi$
$229$	27.3461	1.80708	0.903540	+	0.428504i	$0.140959\pi$
$230$	4.81028	0.317181
$231$	0	0
$232$	0.695742	0.0456777
$233$	12.8534	0.842057	0.421029	−	0.907047i	$-0.361669\pi$
$233$	12.8534	0.842057	0.421029	+	0.907047i	$0.361669\pi$
$234$	0	0
$235$	5.70173	0.371940
$236$	−0.503066	−0.0327468
$237$	0	0
$238$	−6.95839	−0.451046
$239$	10.3969	0.672521	0.336260	−	0.941769i	$-0.390838\pi$
$239$	10.3969	0.672521	0.336260	+	0.941769i	$0.390838\pi$
$240$	0	0
$241$	−6.12477	−0.394531	−0.197266	−	0.980350i	$-0.563206\pi$
$241$	−6.12477	−0.394531	−0.197266	+	0.980350i	$0.563206\pi$
$242$	6.35093	0.408253
$243$	0	0
$244$	−0.730815	−0.0467856
$245$	5.87861	0.375570
$246$	0	0
$247$	0	0
$248$	3.91491	0.248597
$249$	0	0
$250$	1.32691	0.0839211
$251$	7.75343	0.489392	0.244696	−	0.969600i	$-0.421312\pi$
$251$	7.75343	0.489392	0.244696	+	0.969600i	$0.421312\pi$
$252$	0	0
$253$	−9.03662	−0.568128
$254$	15.1097	0.948067
$255$	0	0
$256$	−5.65801	−0.353626
$257$	−8.04137	−0.501607	−0.250804	−	0.968038i	$-0.580695\pi$
$257$	−8.04137	−0.501607	−0.250804	+	0.968038i	$0.580695\pi$
$258$	0	0
$259$	6.47863	0.402562
$260$	0	0
$261$	0	0
$262$	−15.2526	−0.942309
$263$	17.0315	1.05020	0.525102	−	0.851039i	$-0.324027\pi$
$263$	17.0315	1.05020	0.525102	+	0.851039i	$0.324027\pi$
$264$	0	0
$265$	−4.98547	−0.306255
$266$	8.93282	0.547706
$267$	0	0
$268$	−1.82705	−0.111605
$269$	9.75174	0.594574	0.297287	−	0.954788i	$-0.403918\pi$
$269$	9.75174	0.594574	0.297287	+	0.954788i	$0.403918\pi$
$270$	0	0
$271$	−21.2859	−1.29302	−0.646512	−	0.762904i	$-0.723773\pi$
$271$	−21.2859	−1.29302	−0.646512	+	0.762904i	$0.723773\pi$
$272$	17.1545	1.04015
$273$	0	0
$274$	22.8454	1.38014
$275$	−2.49274	−0.150318
$276$	0	0
$277$	25.6648	1.54205	0.771023	−	0.636807i	$-0.219745\pi$
$277$	25.6648	1.54205	0.771023	+	0.636807i	$0.219745\pi$
$278$	−3.19364	−0.191542
$279$	0	0
$280$	3.14655	0.188043
$281$	16.8364	1.00438	0.502188	−	0.864758i	$-0.332529\pi$
$281$	16.8364	1.00438	0.502188	+	0.864758i	$0.332529\pi$
$282$	0	0
$283$	−18.0795	−1.07472	−0.537358	−	0.843355i	$-0.680577\pi$
$283$	−18.0795	−1.07472	−0.537358	+	0.843355i	$0.680577\pi$
$284$	1.95684	0.116117
$285$	0	0
$286$	0	0
$287$	9.37723	0.553520
$288$	0	0
$289$	7.52320	0.442541
$290$	0.310694	0.0182446
$291$	0	0
$292$	0.954183	0.0558393
$293$	−14.5444	−0.849695	−0.424848	−	0.905265i	$-0.639672\pi$
$293$	−14.5444	−0.849695	−0.424848	+	0.905265i	$0.639672\pi$
$294$	0	0
$295$	−2.10212	−0.122390
$296$	−18.1786	−1.05661
$297$	0	0
$298$	−4.48812	−0.259990
$299$	0	0
$300$	0	0
$301$	−10.2397	−0.590208
$302$	−1.94734	−0.112057
$303$	0	0
$304$	−22.0221	−1.26305
$305$	−3.05379	−0.174860
$306$	0	0
$307$	34.1062	1.94654	0.973272	−	0.229654i	$-0.0737595\pi$
$307$	34.1062	1.94654	0.973272	+	0.229654i	$0.0737595\pi$
$308$	−0.631718	−0.0359955
$309$	0	0
$310$	1.74826	0.0992948
$311$	−4.66962	−0.264790	−0.132395	−	0.991197i	$-0.542267\pi$
$311$	−4.66962	−0.264790	−0.132395	+	0.991197i	$0.542267\pi$
$312$	0	0
$313$	−5.04485	−0.285152	−0.142576	−	0.989784i	$-0.545538\pi$
$313$	−5.04485	−0.285152	−0.142576	+	0.989784i	$0.545538\pi$
$314$	27.5776	1.55629
$315$	0	0
$316$	−3.78974	−0.213189
$317$	−14.6107	−0.820616	−0.410308	−	0.911947i	$-0.634579\pi$
$317$	−14.6107	−0.820616	−0.410308	+	0.911947i	$0.634579\pi$
$318$	0	0
$319$	−0.583672	−0.0326794
$320$	−8.71446	−0.487153
$321$	0	0
$322$	5.09390	0.283872
$323$	−31.4816	−1.75168
$324$	0	0
$325$	0	0
$326$	22.1956	1.22930
$327$	0	0
$328$	−26.3118	−1.45283
$329$	6.03790	0.332880
$330$	0	0
$331$	15.9388	0.876078	0.438039	−	0.898956i	$-0.355673\pi$
$331$	15.9388	0.876078	0.438039	+	0.898956i	$0.355673\pi$
$332$	−3.78205	−0.207567
$333$	0	0
$334$	−8.50685	−0.465474
$335$	−7.63454	−0.417120
$336$	0	0
$337$	−19.8261	−1.08000	−0.539998	−	0.841666i	$-0.681575\pi$
$337$	−19.8261	−1.08000	−0.539998	+	0.841666i	$0.681575\pi$
$338$	0	0
$339$	0	0
$340$	−1.18510	−0.0642712
$341$	−3.28430	−0.177855
$342$	0	0
$343$	13.6379	0.736378
$344$	28.7320	1.54912
$345$	0	0
$346$	−29.8622	−1.60540
$347$	−10.3633	−0.556329	−0.278164	−	0.960534i	$-0.589726\pi$
$347$	−10.3633	−0.556329	−0.278164	+	0.960534i	$0.589726\pi$
$348$	0	0
$349$	−31.8499	−1.70488	−0.852442	−	0.522821i	$-0.824879\pi$
$349$	−31.8499	−1.70488	−0.852442	+	0.522821i	$0.824879\pi$
$350$	1.40514	0.0751080
$351$	0	0
$352$	3.35568	0.178858
$353$	−1.15605	−0.0615304	−0.0307652	−	0.999527i	$-0.509794\pi$
$353$	−1.15605	−0.0615304	−0.0307652	+	0.999527i	$0.509794\pi$
$354$	0	0
$355$	8.17688	0.433984
$356$	3.79198	0.200974
$357$	0	0
$358$	6.31069	0.333531
$359$	−21.6260	−1.14138	−0.570689	−	0.821166i	$-0.693324\pi$
$359$	−21.6260	−1.14138	−0.570689	+	0.821166i	$0.693324\pi$
$360$	0	0
$361$	21.4144	1.12707
$362$	15.2072	0.799273
$363$	0	0
$364$	0	0
$365$	3.98716	0.208698
$366$	0	0
$367$	16.8406	0.879073	0.439536	−	0.898225i	$-0.355143\pi$
$367$	16.8406	0.879073	0.439536	+	0.898225i	$0.355143\pi$
$368$	−12.5580	−0.654631
$369$	0	0
$370$	−8.11792	−0.422030
$371$	−5.27941	−0.274094
$372$	0	0
$373$	24.5009	1.26861	0.634303	−	0.773084i	$-0.281287\pi$
$373$	24.5009	1.26861	0.634303	+	0.773084i	$0.281287\pi$
$374$	−16.3797	−0.846974
$375$	0	0
$376$	−16.9419	−0.873712
$377$	0	0
$378$	0	0
$379$	−33.0678	−1.69858	−0.849290	−	0.527927i	$-0.822969\pi$
$379$	−33.0678	−1.69858	−0.849290	+	0.527927i	$0.822969\pi$
$380$	1.52137	0.0780448
$381$	0	0
$382$	−7.69880	−0.393905
$383$	1.00014	0.0511046	0.0255523	−	0.999673i	$-0.491866\pi$
$383$	1.00014	0.0511046	0.0255523	+	0.999673i	$0.491866\pi$
$384$	0	0
$385$	−2.63971	−0.134532
$386$	−3.22885	−0.164344
$387$	0	0
$388$	−2.40651	−0.122172
$389$	−1.26193	−0.0639823	−0.0319912	−	0.999488i	$-0.510185\pi$
$389$	−1.26193	−0.0639823	−0.0319912	+	0.999488i	$0.510185\pi$
$390$	0	0
$391$	−17.9522	−0.907883
$392$	−17.4675	−0.882241
$393$	0	0
$394$	17.7572	0.894595
$395$	−15.8359	−0.796789
$396$	0	0
$397$	11.1559	0.559899	0.279950	−	0.960015i	$-0.409682\pi$
$397$	11.1559	0.559899	0.279950	+	0.960015i	$0.409682\pi$
$398$	23.6868	1.18731
$399$	0	0
$400$	−3.46410	−0.173205
$401$	−33.2725	−1.66155	−0.830774	−	0.556610i	$-0.812102\pi$
$401$	−33.2725	−1.66155	−0.830774	+	0.556610i	$0.812102\pi$
$402$	0	0
$403$	0	0
$404$	−2.73091	−0.135868
$405$	0	0
$406$	0.329013	0.0163286
$407$	15.2504	0.755932
$408$	0	0
$409$	2.41756	0.119541	0.0597704	−	0.998212i	$-0.480963\pi$
$409$	2.41756	0.119541	0.0597704	+	0.998212i	$0.480963\pi$
$410$	−11.7500	−0.580289
$411$	0	0
$412$	−0.693502	−0.0341664
$413$	−2.22606	−0.109537
$414$	0	0
$415$	−15.8037	−0.775775
$416$	0	0
$417$	0	0
$418$	21.0274	1.02848
$419$	−26.6917	−1.30398	−0.651988	−	0.758229i	$-0.726065\pi$
$419$	−26.6917	−1.30398	−0.651988	+	0.758229i	$0.726065\pi$
$420$	0	0
$421$	−40.7370	−1.98540	−0.992699	−	0.120614i	$-0.961514\pi$
$421$	−40.7370	−1.98540	−0.992699	+	0.120614i	$0.961514\pi$
$422$	−13.2543	−0.645209
$423$	0	0
$424$	14.8137	0.719415
$425$	−4.95209	−0.240212
$426$	0	0
$427$	−3.23384	−0.156497
$428$	−3.61679	−0.174824
$429$	0	0
$430$	12.8307	0.618751
$431$	−7.17477	−0.345597	−0.172798	−	0.984957i	$-0.555281\pi$
$431$	−7.17477	−0.345597	−0.172798	+	0.984957i	$0.555281\pi$
$432$	0	0
$433$	−28.7342	−1.38088	−0.690438	−	0.723392i	$-0.742582\pi$
$433$	−28.7342	−1.38088	−0.690438	+	0.723392i	$0.742582\pi$
$434$	1.85134	0.0888672
$435$	0	0
$436$	−0.0923435	−0.00442245
$437$	23.0461	1.10245
$438$	0	0
$439$	−39.1690	−1.86943	−0.934716	−	0.355394i	$-0.884347\pi$
$439$	−39.1690	−1.86943	−0.934716	+	0.355394i	$0.884347\pi$
$440$	7.40683	0.353107
$441$	0	0
$442$	0	0
$443$	12.5223	0.594954	0.297477	−	0.954729i	$-0.403855\pi$
$443$	12.5223	0.594954	0.297477	+	0.954729i	$0.403855\pi$
$444$	0	0
$445$	15.8452	0.751136
$446$	26.8511	1.27144
$447$	0	0
$448$	−9.22826	−0.435994
$449$	−10.9573	−0.517105	−0.258552	−	0.965997i	$-0.583245\pi$
$449$	−10.9573	−0.517105	−0.258552	+	0.965997i	$0.583245\pi$
$450$	0	0
$451$	22.0735	1.03940
$452$	2.01535	0.0947939
$453$	0	0
$454$	−4.23009	−0.198528
$455$	0	0
$456$	0	0
$457$	16.5848	0.775806	0.387903	−	0.921700i	$-0.373199\pi$
$457$	16.5848	0.775806	0.387903	+	0.921700i	$0.373199\pi$
$458$	−36.2858	−1.69552
$459$	0	0
$460$	0.867556	0.0404500
$461$	−22.9402	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$461$	−22.9402	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$462$	0	0
$463$	−41.2096	−1.91517	−0.957587	−	0.288146i	$-0.906961\pi$
$463$	−41.2096	−1.91517	−0.957587	+	0.288146i	$0.906961\pi$
$464$	−0.811116	−0.0376551
$465$	0	0
$466$	−17.0553	−0.790074
$467$	31.1867	1.44315	0.721574	−	0.692337i	$-0.243419\pi$
$467$	31.1867	1.44315	0.721574	+	0.692337i	$0.243419\pi$
$468$	0	0
$469$	−8.08467	−0.373315
$470$	−7.56567	−0.348978
$471$	0	0
$472$	6.24617	0.287503
$473$	−24.1038	−1.10829
$474$	0	0
$475$	6.35723	0.291690
$476$	−1.25498	−0.0575217
$477$	0	0
$478$	−13.7958	−0.631003
$479$	5.15981	0.235758	0.117879	−	0.993028i	$-0.462391\pi$
$479$	5.15981	0.235758	0.117879	+	0.993028i	$0.462391\pi$
$480$	0	0
$481$	0	0
$482$	8.12701	0.370175
$483$	0	0
$484$	1.14542	0.0520644
$485$	−10.0559	−0.456615
$486$	0	0
$487$	−26.3560	−1.19430	−0.597152	−	0.802128i	$-0.703701\pi$
$487$	−26.3560	−1.19430	−0.597152	+	0.802128i	$0.703701\pi$
$488$	9.07394	0.410758
$489$	0	0
$490$	−7.80037	−0.352385
$491$	18.7223	0.844924	0.422462	−	0.906381i	$-0.361166\pi$
$491$	18.7223	0.844924	0.422462	+	0.906381i	$0.361166\pi$
$492$	0	0
$493$	−1.15953	−0.0522225
$494$	0	0
$495$	0	0
$496$	−4.56412	−0.204935
$497$	8.65898	0.388408
$498$	0	0
$499$	−9.31449	−0.416974	−0.208487	−	0.978025i	$-0.566854\pi$
$499$	−9.31449	−0.416974	−0.208487	+	0.978025i	$0.566854\pi$
$500$	0.239314	0.0107024
$501$	0	0
$502$	−10.2881	−0.459180
$503$	−14.0512	−0.626510	−0.313255	−	0.949669i	$-0.601419\pi$
$503$	−14.0512	−0.626510	−0.313255	+	0.949669i	$0.601419\pi$
$504$	0	0
$505$	−11.4114	−0.507803
$506$	11.9908	0.533055
$507$	0	0
$508$	2.72510	0.120907
$509$	22.0354	0.976701	0.488350	−	0.872648i	$-0.337599\pi$
$509$	22.0354	0.976701	0.488350	+	0.872648i	$0.337599\pi$
$510$	0	0
$511$	4.22224	0.186781
$512$	25.2495	1.11588
$513$	0	0
$514$	10.6702	0.470641
$515$	−2.89788	−0.127696
$516$	0	0
$517$	14.2129	0.625083
$518$	−8.59655	−0.377711
$519$	0	0
$520$	0	0
$521$	−29.6165	−1.29752	−0.648762	−	0.760991i	$-0.724713\pi$
$521$	−29.6165	−1.29752	−0.648762	+	0.760991i	$0.724713\pi$
$522$	0	0
$523$	11.6675	0.510185	0.255092	−	0.966917i	$-0.417894\pi$
$523$	11.6675	0.510185	0.255092	+	0.966917i	$0.417894\pi$
$524$	−2.75087	−0.120172
$525$	0	0
$526$	−22.5992	−0.985372
$527$	−6.52461	−0.284217
$528$	0	0
$529$	−9.85806	−0.428611
$530$	6.61527	0.287349
$531$	0	0
$532$	1.61107	0.0698488
$533$	0	0
$534$	0	0
$535$	−15.1132	−0.653399
$536$	22.6850	0.979843
$537$	0	0
$538$	−12.9397	−0.557869
$539$	14.6538	0.631185
$540$	0	0
$541$	−32.7583	−1.40839	−0.704196	−	0.710006i	$-0.748692\pi$
$541$	−32.7583	−1.40839	−0.704196	+	0.710006i	$0.748692\pi$
$542$	28.2444	1.21320
$543$	0	0
$544$	6.66642	0.285820
$545$	−0.385868	−0.0165288
$546$	0	0
$547$	−10.8406	−0.463511	−0.231755	−	0.972774i	$-0.574447\pi$
$547$	−10.8406	−0.463511	−0.231755	+	0.972774i	$0.574447\pi$
$548$	4.12026	0.176009
$549$	0	0
$550$	3.30763	0.141038
$551$	1.48854	0.0634140
$552$	0	0
$553$	−16.7695	−0.713113
$554$	−34.0548	−1.44685
$555$	0	0
$556$	−0.575987	−0.0244273
$557$	20.7132	0.877648	0.438824	−	0.898573i	$-0.355395\pi$
$557$	20.7132	0.877648	0.438824	+	0.898573i	$0.355395\pi$
$558$	0	0
$559$	0	0
$560$	−3.66834	−0.155016
$561$	0	0
$562$	−22.3404	−0.942372
$563$	−37.1777	−1.56686	−0.783428	−	0.621483i	$-0.786530\pi$
$563$	−37.1777	−1.56686	−0.783428	+	0.621483i	$0.786530\pi$
$564$	0	0
$565$	8.42136	0.354289
$566$	23.9899	1.00837
$567$	0	0
$568$	−24.2965	−1.01946
$569$	−46.0097	−1.92883	−0.964413	−	0.264400i	$-0.914826\pi$
$569$	−46.0097	−1.92883	−0.964413	+	0.264400i	$0.914826\pi$
$570$	0	0
$571$	−41.4861	−1.73614	−0.868070	−	0.496442i	$-0.834639\pi$
$571$	−41.4861	−1.73614	−0.868070	+	0.496442i	$0.834639\pi$
$572$	0	0
$573$	0	0
$574$	−12.4427	−0.519349
$575$	3.62518	0.151181
$576$	0	0
$577$	13.0556	0.543513	0.271756	−	0.962366i	$-0.412396\pi$
$577$	13.0556	0.543513	0.271756	+	0.962366i	$0.412396\pi$
$578$	−9.98259	−0.415221
$579$	0	0
$580$	0.0560351	0.00232673
$581$	−16.7355	−0.694307
$582$	0	0
$583$	−12.4275	−0.514694
$584$	−11.8473	−0.490245
$585$	0	0
$586$	19.2991	0.797240
$587$	16.9418	0.699261	0.349631	−	0.936888i	$-0.386307\pi$
$587$	16.9418	0.699261	0.349631	+	0.936888i	$0.386307\pi$
$588$	0	0
$589$	8.37595	0.345125
$590$	2.78932	0.114835
$591$	0	0
$592$	21.1931	0.871031
$593$	45.0621	1.85048	0.925239	−	0.379384i	$-0.123864\pi$
$593$	45.0621	1.85048	0.925239	+	0.379384i	$0.123864\pi$
$594$	0	0
$595$	−5.24406	−0.214986
$596$	−0.809453	−0.0331565
$597$	0	0
$598$	0	0
$599$	−24.1055	−0.984924	−0.492462	−	0.870334i	$-0.663903\pi$
$599$	−24.1055	−0.984924	−0.492462	+	0.870334i	$0.663903\pi$
$600$	0	0
$601$	28.1690	1.14904	0.574518	−	0.818492i	$-0.305189\pi$
$601$	28.1690	1.14904	0.574518	+	0.818492i	$0.305189\pi$
$602$	13.5872	0.553772
$603$	0	0
$604$	−0.351211	−0.0142906
$605$	4.78626	0.194589
$606$	0	0
$607$	5.74574	0.233213	0.116606	−	0.993178i	$-0.462798\pi$
$607$	5.74574	0.233213	0.116606	+	0.993178i	$0.462798\pi$
$608$	−8.55800	−0.347073
$609$	0	0
$610$	4.05211	0.164065
$611$	0	0
$612$	0	0
$613$	13.1115	0.529569	0.264785	−	0.964308i	$-0.414699\pi$
$613$	13.1115	0.529569	0.264785	+	0.964308i	$0.414699\pi$
$614$	−45.2558	−1.82638
$615$	0	0
$616$	7.84353	0.316025
$617$	−45.6403	−1.83741	−0.918705	−	0.394944i	$-0.870764\pi$
$617$	−45.6403	−1.83741	−0.918705	+	0.394944i	$0.870764\pi$
$618$	0	0
$619$	2.88239	0.115853	0.0579264	−	0.998321i	$-0.481551\pi$
$619$	2.88239	0.115853	0.0579264	+	0.998321i	$0.481551\pi$
$620$	0.315307	0.0126630
$621$	0	0
$622$	6.19615	0.248443
$623$	16.7794	0.672254
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.69405	0.267548
$627$	0	0
$628$	4.97374	0.198474
$629$	30.2965	1.20800
$630$	0	0
$631$	11.9046	0.473914	0.236957	−	0.971520i	$-0.423850\pi$
$631$	11.9046	0.473914	0.236957	+	0.971520i	$0.423850\pi$
$632$	47.0541	1.87171
$633$	0	0
$634$	19.3870	0.769956
$635$	11.3871	0.451885
$636$	0	0
$637$	0	0
$638$	0.774480	0.0306619
$639$	0	0
$640$	8.87093	0.350654
$641$	−22.2977	−0.880707	−0.440354	−	0.897824i	$-0.645147\pi$
$641$	−22.2977	−0.880707	−0.440354	+	0.897824i	$0.645147\pi$
$642$	0	0
$643$	2.54289	0.100282	0.0501408	−	0.998742i	$-0.484033\pi$
$643$	2.54289	0.100282	0.0501408	+	0.998742i	$0.484033\pi$
$644$	0.918706	0.0362021
$645$	0	0
$646$	41.7732	1.64354
$647$	−18.4251	−0.724367	−0.362183	−	0.932107i	$-0.617969\pi$
$647$	−18.4251	−0.724367	−0.362183	+	0.932107i	$0.617969\pi$
$648$	0	0
$649$	−5.24003	−0.205689
$650$	0	0
$651$	0	0
$652$	4.00307	0.156772
$653$	8.43602	0.330127	0.165063	−	0.986283i	$-0.447217\pi$
$653$	8.43602	0.330127	0.165063	+	0.986283i	$0.447217\pi$
$654$	0	0
$655$	−11.4948	−0.449141
$656$	30.6751	1.19766
$657$	0	0
$658$	−8.01174	−0.312330
$659$	−49.4605	−1.92671	−0.963354	−	0.268232i	$-0.913561\pi$
$659$	−49.4605	−1.92671	−0.963354	+	0.268232i	$0.913561\pi$
$660$	0	0
$661$	−3.27215	−0.127272	−0.0636359	−	0.997973i	$-0.520270\pi$
$661$	−3.27215	−0.127272	−0.0636359	+	0.997973i	$0.520270\pi$
$662$	−21.1494	−0.821994
$663$	0	0
$664$	46.9587	1.82235
$665$	6.73205	0.261058
$666$	0	0
$667$	0.848833	0.0328670
$668$	−1.53425	−0.0593618
$669$	0	0
$670$	10.1303	0.391369
$671$	−7.61231	−0.293870
$672$	0	0
$673$	−8.86742	−0.341814	−0.170907	−	0.985287i	$-0.554670\pi$
$673$	−8.86742	−0.341814	−0.170907	+	0.985287i	$0.554670\pi$
$674$	26.3074	1.01332
$675$	0	0
$676$	0	0
$677$	−45.0533	−1.73154	−0.865769	−	0.500444i	$-0.833170\pi$
$677$	−45.0533	−1.73154	−0.865769	+	0.500444i	$0.833170\pi$
$678$	0	0
$679$	−10.6488	−0.408663
$680$	14.7145	0.564274
$681$	0	0
$682$	4.35796	0.166875
$683$	21.3490	0.816897	0.408449	−	0.912781i	$-0.366070\pi$
$683$	21.3490	0.816897	0.408449	+	0.912781i	$0.366070\pi$
$684$	0	0
$685$	17.2170	0.657827
$686$	−18.0963	−0.690919
$687$	0	0
$688$	−33.4965	−1.27704
$689$	0	0
$690$	0	0
$691$	23.4224	0.891030	0.445515	−	0.895275i	$-0.353021\pi$
$691$	23.4224	0.891030	0.445515	+	0.895275i	$0.353021\pi$
$692$	−5.38577	−0.204736
$693$	0	0
$694$	13.7511	0.521984
$695$	−2.40683	−0.0912963
$696$	0	0
$697$	43.8514	1.66099
$698$	42.2619	1.59964
$699$	0	0
$700$	0.253423	0.00957851
$701$	−43.7290	−1.65162	−0.825811	−	0.563948i	$-0.809282\pi$
$701$	−43.7290	−1.65162	−0.825811	+	0.563948i	$0.809282\pi$
$702$	0	0
$703$	−38.8930	−1.46688
$704$	−21.7229	−0.818711
$705$	0	0
$706$	1.53397	0.0577319
$707$	−12.0843	−0.454475
$708$	0	0
$709$	9.08663	0.341255	0.170628	−	0.985336i	$-0.445420\pi$
$709$	9.08663	0.341255	0.170628	+	0.985336i	$0.445420\pi$
$710$	−10.8500	−0.407192
$711$	0	0
$712$	−47.0819	−1.76447
$713$	4.77635	0.178876
$714$	0	0
$715$	0	0
$716$	1.13816	0.0425351
$717$	0	0
$718$	28.6958	1.07092
$719$	10.8471	0.404530	0.202265	−	0.979331i	$-0.435170\pi$
$719$	10.8471	0.404530	0.202265	+	0.979331i	$0.435170\pi$
$720$	0	0
$721$	−3.06874	−0.114286
$722$	−28.4150	−1.05750
$723$	0	0
$724$	2.74268	0.101931
$725$	0.234149	0.00869608
$726$	0	0
$727$	41.2173	1.52866	0.764332	−	0.644822i	$-0.223069\pi$
$727$	41.2173	1.52866	0.764332	+	0.644822i	$0.223069\pi$
$728$	0	0
$729$	0	0
$730$	−5.29060	−0.195814
$731$	−47.8848	−1.77108
$732$	0	0
$733$	−23.4406	−0.865799	−0.432900	−	0.901442i	$-0.642510\pi$
$733$	−23.4406	−0.865799	−0.432900	+	0.901442i	$0.642510\pi$
$734$	−22.3459	−0.824804
$735$	0	0
$736$	−4.88016	−0.179885
$737$	−19.0309	−0.701013
$738$	0	0
$739$	−16.8128	−0.618470	−0.309235	−	0.950986i	$-0.600073\pi$
$739$	−16.8128	−0.618470	−0.309235	+	0.950986i	$0.600073\pi$
$740$	−1.46410	−0.0538214
$741$	0	0
$742$	7.00530	0.257173
$743$	21.2311	0.778893	0.389447	−	0.921049i	$-0.372666\pi$
$743$	21.2311	0.778893	0.389447	+	0.921049i	$0.372666\pi$
$744$	0	0
$745$	−3.38239	−0.123921
$746$	−32.5104	−1.19029
$747$	0	0
$748$	−2.95415	−0.108014
$749$	−16.0042	−0.584782
$750$	0	0
$751$	−22.3179	−0.814390	−0.407195	−	0.913341i	$-0.633493\pi$
$751$	−22.3179	−0.814390	−0.407195	+	0.913341i	$0.633493\pi$
$752$	19.7514	0.720258
$753$	0	0
$754$	0	0
$755$	−1.46758	−0.0534106
$756$	0	0
$757$	−7.92177	−0.287921	−0.143961	−	0.989583i	$-0.545984\pi$
$757$	−7.92177	−0.287921	−0.143961	+	0.989583i	$0.545984\pi$
$758$	43.8780	1.59372
$759$	0	0
$760$	−18.8897	−0.685200
$761$	−12.1706	−0.441183	−0.220592	−	0.975366i	$-0.570799\pi$
$761$	−12.1706	−0.441183	−0.220592	+	0.975366i	$0.570799\pi$
$762$	0	0
$763$	−0.408618	−0.0147930
$764$	−1.38851	−0.0502346
$765$	0	0
$766$	−1.32709	−0.0479497
$767$	0	0
$768$	0	0
$769$	−21.1591	−0.763018	−0.381509	−	0.924365i	$-0.624595\pi$
$769$	−21.1591	−0.763018	−0.381509	+	0.924365i	$0.624595\pi$
$770$	3.50265	0.126227
$771$	0	0
$772$	−0.582337	−0.0209588
$773$	4.39775	0.158176	0.0790880	−	0.996868i	$-0.474799\pi$
$773$	4.39775	0.158176	0.0790880	+	0.996868i	$0.474799\pi$
$774$	0	0
$775$	1.31755	0.0473277
$776$	29.8797	1.07262
$777$	0	0
$778$	1.67446	0.0600324
$779$	−56.2941	−2.01695
$780$	0	0
$781$	20.3828	0.729354
$782$	23.8210	0.851836
$783$	0	0
$784$	20.3641	0.727289
$785$	20.7833	0.741789
$786$	0	0
$787$	−43.1135	−1.53683	−0.768415	−	0.639952i	$-0.778954\pi$
$787$	−43.1135	−1.53683	−0.768415	+	0.639952i	$0.778954\pi$
$788$	3.20259	0.114088
$789$	0	0
$790$	21.0127	0.747600
$791$	8.91787	0.317083
$792$	0	0
$793$	0	0
$794$	−14.8029	−0.525335
$795$	0	0
$796$	4.27201	0.151418
$797$	34.0646	1.20663	0.603315	−	0.797503i	$-0.293846\pi$
$797$	34.0646	1.20663	0.603315	+	0.797503i	$0.293846\pi$
$798$	0	0
$799$	28.2355	0.998899
$800$	−1.34618	−0.0475948
$801$	0	0
$802$	44.1495	1.55897
$803$	9.93895	0.350738
$804$	0	0
$805$	3.83892	0.135304
$806$	0	0
$807$	0	0
$808$	33.9076	1.19286
$809$	−15.7359	−0.553246	−0.276623	−	0.960978i	$-0.589215\pi$
$809$	−15.7359	−0.553246	−0.276623	+	0.960978i	$0.589215\pi$
$810$	0	0
$811$	−47.5686	−1.67036	−0.835180	−	0.549976i	$-0.814637\pi$
$811$	−47.5686	−1.67036	−0.835180	+	0.549976i	$0.814637\pi$
$812$	0.0593389	0.00208239
$813$	0	0
$814$	−20.2358	−0.709266
$815$	16.7273	0.585932
$816$	0	0
$817$	61.4720	2.15063
$818$	−3.20789	−0.112161
$819$	0	0
$820$	−2.11915	−0.0740041
$821$	12.1481	0.423971	0.211985	−	0.977273i	$-0.432007\pi$
$821$	12.1481	0.423971	0.211985	+	0.977273i	$0.432007\pi$
$822$	0	0
$823$	−3.65574	−0.127431	−0.0637156	−	0.997968i	$-0.520295\pi$
$823$	−3.65574	−0.127431	−0.0637156	+	0.997968i	$0.520295\pi$
$824$	8.61066	0.299966
$825$	0	0
$826$	2.95378	0.102775
$827$	30.5084	1.06088	0.530441	−	0.847722i	$-0.322026\pi$
$827$	30.5084	1.06088	0.530441	+	0.847722i	$0.322026\pi$
$828$	0	0
$829$	52.0658	1.80832	0.904161	−	0.427192i	$-0.140497\pi$
$829$	52.0658	1.80832	0.904161	+	0.427192i	$0.140497\pi$
$830$	20.9701	0.727884
$831$	0	0
$832$	0	0
$833$	29.1114	1.00865
$834$	0	0
$835$	−6.41103	−0.221863
$836$	3.79238	0.131162
$837$	0	0
$838$	35.4175	1.22348
$839$	−43.2899	−1.49454	−0.747268	−	0.664523i	$-0.768635\pi$
$839$	−43.2899	−1.49454	−0.747268	+	0.664523i	$0.768635\pi$
$840$	0	0
$841$	−28.9452	−0.998109
$842$	54.0542	1.86283
$843$	0	0
$844$	−2.39047	−0.0822833
$845$	0	0
$846$	0	0
$847$	5.06846	0.174154
$848$	−17.2702	−0.593061
$849$	0	0
$850$	6.57097	0.225382
$851$	−22.1786	−0.760271
$852$	0	0
$853$	7.62729	0.261153	0.130577	−	0.991438i	$-0.458317\pi$
$853$	7.62729	0.261153	0.130577	+	0.991438i	$0.458317\pi$
$854$	4.29102	0.146836
$855$	0	0
$856$	44.9067	1.53488
$857$	34.6287	1.18289	0.591447	−	0.806344i	$-0.298557\pi$
$857$	34.6287	1.18289	0.591447	+	0.806344i	$0.298557\pi$
$858$	0	0
$859$	−15.1648	−0.517415	−0.258707	−	0.965956i	$-0.583297\pi$
$859$	−15.1648	−0.517415	−0.258707	+	0.965956i	$0.583297\pi$
$860$	2.31407	0.0789092
$861$	0	0
$862$	9.52027	0.324262
$863$	−17.7187	−0.603150	−0.301575	−	0.953442i	$-0.597512\pi$
$863$	−17.7187	−0.603150	−0.301575	+	0.953442i	$0.597512\pi$
$864$	0	0
$865$	−22.5051	−0.765195
$866$	38.1276	1.29563
$867$	0	0
$868$	0.333897	0.0113332
$869$	−39.4746	−1.33909
$870$	0	0
$871$	0	0
$872$	1.14655	0.0388272
$873$	0	0
$874$	−30.5801	−1.03439
$875$	1.05896	0.0357994
$876$	0	0
$877$	−2.40518	−0.0812171	−0.0406086	−	0.999175i	$-0.512930\pi$
$877$	−2.40518	−0.0812171	−0.0406086	+	0.999175i	$0.512930\pi$
$878$	51.9736	1.75403
$879$	0	0
$880$	−8.63509	−0.291089
$881$	26.4915	0.892521	0.446261	−	0.894903i	$-0.352755\pi$
$881$	26.4915	0.892521	0.446261	+	0.894903i	$0.352755\pi$
$882$	0	0
$883$	−14.5464	−0.489525	−0.244763	−	0.969583i	$-0.578710\pi$
$883$	−14.5464	−0.489525	−0.244763	+	0.969583i	$0.578710\pi$
$884$	0	0
$885$	0	0
$886$	−16.6160	−0.558225
$887$	35.2368	1.18314	0.591568	−	0.806255i	$-0.298509\pi$
$887$	35.2368	1.18314	0.591568	+	0.806255i	$0.298509\pi$
$888$	0	0
$889$	12.0585	0.404430
$890$	−21.0252	−0.704765
$891$	0	0
$892$	4.84271	0.162146
$893$	−36.2472	−1.21297
$894$	0	0
$895$	4.75594	0.158973
$896$	9.39396	0.313830
$897$	0	0
$898$	14.5393	0.485182
$899$	0.308503	0.0102891
$900$	0	0
$901$	−24.6885	−0.822494
$902$	−29.2895	−0.975235
$903$	0	0
$904$	−25.0229	−0.832250
$905$	11.4606	0.380964
$906$	0	0
$907$	−16.6388	−0.552481	−0.276241	−	0.961088i	$-0.589089\pi$
$907$	−16.6388	−0.552481	−0.276241	+	0.961088i	$0.589089\pi$
$908$	−0.762915	−0.0253182
$909$	0	0
$910$	0	0
$911$	11.3931	0.377471	0.188736	−	0.982028i	$-0.439561\pi$
$911$	11.3931	0.377471	0.188736	+	0.982028i	$0.439561\pi$
$912$	0	0
$913$	−39.3946	−1.30377
$914$	−22.0066	−0.727913
$915$	0	0
$916$	−6.54429	−0.216229
$917$	−12.1726	−0.401974
$918$	0	0
$919$	−26.7786	−0.883344	−0.441672	−	0.897177i	$-0.645615\pi$
$919$	−26.7786	−0.883344	−0.441672	+	0.897177i	$0.645615\pi$
$920$	−10.7717	−0.355134
$921$	0	0
$922$	30.4396	1.00247
$923$	0	0
$924$	0	0
$925$	−6.11792	−0.201156
$926$	54.6814	1.79694
$927$	0	0
$928$	−0.315208	−0.0103472
$929$	19.4862	0.639322	0.319661	−	0.947532i	$-0.396431\pi$
$929$	19.4862	0.639322	0.319661	+	0.947532i	$0.396431\pi$
$930$	0	0
$931$	−37.3717	−1.22481
$932$	−3.07601	−0.100758
$933$	0	0
$934$	−41.3819	−1.35406
$935$	−12.3443	−0.403700
$936$	0	0
$937$	−9.81448	−0.320625	−0.160313	−	0.987066i	$-0.551250\pi$
$937$	−9.81448	−0.320625	−0.160313	+	0.987066i	$0.551250\pi$
$938$	10.7276	0.350269
$939$	0	0
$940$	−1.36450	−0.0445051
$941$	−47.0242	−1.53295	−0.766473	−	0.642277i	$-0.777990\pi$
$941$	−47.0242	−1.53295	−0.766473	+	0.642277i	$0.777990\pi$
$942$	0	0
$943$	−32.1015	−1.04537
$944$	−7.28196	−0.237008
$945$	0	0
$946$	31.9836	1.03987
$947$	23.9226	0.777381	0.388690	−	0.921368i	$-0.372928\pi$
$947$	23.9226	0.777381	0.388690	+	0.921368i	$0.372928\pi$
$948$	0	0
$949$	0	0
$950$	−8.43547	−0.273683
$951$	0	0
$952$	15.5820	0.505016
$953$	25.6484	0.830833	0.415416	−	0.909631i	$-0.363636\pi$
$953$	25.6484	0.830833	0.415416	+	0.909631i	$0.363636\pi$
$954$	0	0
$955$	−5.80206	−0.187750
$956$	−2.48812	−0.0804717
$957$	0	0
$958$	−6.84659	−0.221203
$959$	18.2321	0.588745
$960$	0	0
$961$	−29.2641	−0.944002
$962$	0	0
$963$	0	0
$964$	1.46574	0.0472084
$965$	−2.43336	−0.0783327
$966$	0	0
$967$	−25.1221	−0.807873	−0.403936	−	0.914787i	$-0.632358\pi$
$967$	−25.1221	−0.807873	−0.403936	+	0.914787i	$0.632358\pi$
$968$	−14.2217	−0.457104
$969$	0	0
$970$	13.3433	0.428426
$971$	−39.7644	−1.27610	−0.638050	−	0.769995i	$-0.720259\pi$
$971$	−39.7644	−1.27610	−0.638050	+	0.769995i	$0.720259\pi$
$972$	0	0
$973$	−2.54874	−0.0817087
$974$	34.9720	1.12058
$975$	0	0
$976$	−10.5787	−0.338615
$977$	10.1716	0.325419	0.162709	−	0.986674i	$-0.447977\pi$
$977$	10.1716	0.325419	0.162709	+	0.986674i	$0.447977\pi$
$978$	0	0
$979$	39.4980	1.26236
$980$	−1.40683	−0.0449396
$981$	0	0
$982$	−24.8427	−0.792764
$983$	−37.0857	−1.18285	−0.591425	−	0.806360i	$-0.701434\pi$
$983$	−37.0857	−1.18285	−0.591425	+	0.806360i	$0.701434\pi$
$984$	0	0
$985$	13.3824	0.426398
$986$	1.53859	0.0489986
$987$	0	0
$988$	0	0
$989$	35.0541	1.11466
$990$	0	0
$991$	17.1580	0.545042	0.272521	−	0.962150i	$-0.412143\pi$
$991$	17.1580	0.545042	0.272521	+	0.962150i	$0.412143\pi$
$992$	−1.77366	−0.0563138
$993$	0	0
$994$	−11.4897	−0.364430
$995$	17.8511	0.565918
$996$	0	0
$997$	−34.7965	−1.10202	−0.551008	−	0.834500i	$-0.685757\pi$
$997$	−34.7965	−1.10202	−0.551008	+	0.834500i	$0.685757\pi$
$998$	12.3595	0.391232
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cg.1.2		4
3.2	odd	2		2535.2.a.bl.1.3		4
13.6	odd	12		585.2.bu.b.361.4		8
13.11	odd	12		585.2.bu.b.316.4		8
13.12	even	2		7605.2.a.ck.1.3		4
39.11	even	12		195.2.bb.c.121.1	✓	8
39.32	even	12		195.2.bb.c.166.1	yes	8
39.38	odd	2		2535.2.a.bi.1.2		4
195.32	odd	12		975.2.w.g.49.2		8
195.89	even	12		975.2.bc.i.901.4		8
195.128	odd	12		975.2.w.g.199.2		8
195.149	even	12		975.2.bc.i.751.4		8
195.167	odd	12		975.2.w.j.199.3		8
195.188	odd	12		975.2.w.j.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.c.121.1	✓	8	39.11	even	12
195.2.bb.c.166.1	yes	8	39.32	even	12
585.2.bu.b.316.4		8	13.11	odd	12
585.2.bu.b.361.4		8	13.6	odd	12
975.2.w.g.49.2		8	195.32	odd	12
975.2.w.g.199.2		8	195.128	odd	12
975.2.w.j.49.3		8	195.188	odd	12
975.2.w.j.199.3		8	195.167	odd	12
975.2.bc.i.751.4		8	195.149	even	12
975.2.bc.i.901.4		8	195.89	even	12
2535.2.a.bi.1.2		4	39.38	odd	2
2535.2.a.bl.1.3		4	3.2	odd	2
7605.2.a.cg.1.2		4	1.1	even	1	trivial
7605.2.a.ck.1.3		4	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32691 q^{2} -0.239314 q^{4} -1.00000 q^{5} -1.05896 q^{7} +2.97136 q^{8} +O(q^{10})\)	\(q-1.32691 q^{2} -0.239314 q^{4} -1.00000 q^{5} -1.05896 q^{7} +2.97136 q^{8} +1.32691 q^{10} -2.49274 q^{11} +1.40514 q^{14} -3.46410 q^{16} -4.95209 q^{17} +6.35723 q^{19} +0.239314 q^{20} +3.30763 q^{22} +3.62518 q^{23} +1.00000 q^{25} +0.253423 q^{28} +0.234149 q^{29} +1.31755 q^{31} -1.34618 q^{32} +6.57097 q^{34} +1.05896 q^{35} -6.11792 q^{37} -8.43547 q^{38} -2.97136 q^{40} -8.85513 q^{41} +9.66962 q^{43} +0.596546 q^{44} -4.81028 q^{46} -5.70173 q^{47} -5.87861 q^{49} -1.32691 q^{50} +4.98547 q^{53} +2.49274 q^{55} -3.14655 q^{56} -0.310694 q^{58} +2.10212 q^{59} +3.05379 q^{61} -1.74826 q^{62} +8.71446 q^{64} +7.63454 q^{67} +1.18510 q^{68} -1.40514 q^{70} -8.17688 q^{71} -3.98716 q^{73} +8.11792 q^{74} -1.52137 q^{76} +2.63971 q^{77} +15.8359 q^{79} +3.46410 q^{80} +11.7500 q^{82} +15.8037 q^{83} +4.95209 q^{85} -12.8307 q^{86} -7.40683 q^{88} -15.8452 q^{89} -0.867556 q^{92} +7.56567 q^{94} -6.35723 q^{95} +10.0559 q^{97} +7.80037 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 + 6 * q^7	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 6 q^{7} + 2 q^{10} - 8 q^{11} + 2 q^{14} + 2 q^{17} - 4 q^{20} - 4 q^{23} + 4 q^{25} + 4 q^{28} - 6 q^{29} - 12 q^{32} + 24 q^{34} - 6 q^{35} - 4 q^{37} - 8 q^{38} - 10 q^{41} + 6 q^{43} - 28 q^{44} - 12 q^{46} - 38 q^{47} - 8 q^{49} - 2 q^{50} + 16 q^{53} + 8 q^{55} - 4 q^{56} - 28 q^{58} + 14 q^{59} + 30 q^{62} - 16 q^{64} + 14 q^{67} + 16 q^{68} - 2 q^{70} - 2 q^{71} + 22 q^{73} + 12 q^{74} - 16 q^{76} - 4 q^{77} + 28 q^{79} - 24 q^{82} - 20 q^{83} - 2 q^{85} - 14 q^{86} + 8 q^{88} - 30 q^{89} - 20 q^{92} + 16 q^{94} - 10 q^{97} + 16 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 + 6 * q^7 + 2 * q^10 - 8 * q^11 + 2 * q^14 + 2 * q^17 - 4 * q^20 - 4 * q^23 + 4 * q^25 + 4 * q^28 - 6 * q^29 - 12 * q^32 + 24 * q^34 - 6 * q^35 - 4 * q^37 - 8 * q^38 - 10 * q^41 + 6 * q^43 - 28 * q^44 - 12 * q^46 - 38 * q^47 - 8 * q^49 - 2 * q^50 + 16 * q^53 + 8 * q^55 - 4 * q^56 - 28 * q^58 + 14 * q^59 + 30 * q^62 - 16 * q^64 + 14 * q^67 + 16 * q^68 - 2 * q^70 - 2 * q^71 + 22 * q^73 + 12 * q^74 - 16 * q^76 - 4 * q^77 + 28 * q^79 - 24 * q^82 - 20 * q^83 - 2 * q^85 - 14 * q^86 + 8 * q^88 - 30 * q^89 - 20 * q^92 + 16 * q^94 - 10 * q^97 + 16 * q^98

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