LMFDB - Embedded newform 9801.2.a.cb.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$1$
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} - 15x^{8} - x^{7} + 75x^{6} + 15x^{5} - 148x^{4} - 54x^{3} + 96x^{2} + 55x + 3 $ x^10 - 15x^8 - x^7 + 75x^6 + 15x^5 - 148x^4 - 54x^3 + 96x^2 + 55*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 1089)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.71687$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71687 q^{2} +5.38138 q^{4} -1.67141 q^{5} -2.59211 q^{7} +9.18678 q^{8} +O(q^{10})$	$q+2.71687 q^{2} +5.38138 q^{4} -1.67141 q^{5} -2.59211 q^{7} +9.18678 q^{8} -4.54100 q^{10} -0.199351 q^{13} -7.04241 q^{14} +14.1965 q^{16} -7.79195 q^{17} -1.13361 q^{19} -8.99449 q^{20} -1.79747 q^{23} -2.20639 q^{25} -0.541612 q^{26} -13.9491 q^{28} +4.02665 q^{29} +4.43134 q^{31} +20.1965 q^{32} -21.1697 q^{34} +4.33247 q^{35} +4.55523 q^{37} -3.07987 q^{38} -15.3549 q^{40} -4.49550 q^{41} +3.36629 q^{43} -4.88349 q^{46} -8.35219 q^{47} -0.280991 q^{49} -5.99449 q^{50} -1.07279 q^{52} -0.660920 q^{53} -23.8131 q^{56} +10.9399 q^{58} -4.53865 q^{59} +0.455719 q^{61} +12.0394 q^{62} +26.4783 q^{64} +0.333198 q^{65} -11.5434 q^{67} -41.9315 q^{68} +11.7707 q^{70} -13.4057 q^{71} -10.7548 q^{73} +12.3760 q^{74} -6.10038 q^{76} +11.9014 q^{79} -23.7282 q^{80} -12.2137 q^{82} +2.89275 q^{83} +13.0235 q^{85} +9.14576 q^{86} -11.6422 q^{89} +0.516740 q^{91} -9.67286 q^{92} -22.6918 q^{94} +1.89472 q^{95} -4.72812 q^{97} -0.763417 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{4} - 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 10 * q^4 - 3 * q^5 - q^7 - 3 * q^8	$ 10 q + 10 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - q^{13} - 15 q^{14} + 10 q^{16} - 3 q^{17} + 2 q^{19} - 15 q^{20} - 18 q^{23} + 7 q^{25} - 6 q^{26} - 4 q^{28} + 12 q^{29} - 7 q^{31} + 24 q^{32} - 6 q^{34} - 18 q^{35} - q^{37} + 12 q^{38} + 12 q^{40} - 6 q^{41} + 8 q^{43} - 18 q^{46} - 27 q^{47} + 15 q^{49} + 15 q^{50} + 5 q^{52} - 6 q^{53} - 60 q^{56} + 9 q^{58} - 21 q^{59} + 20 q^{61} + 3 q^{62} + 31 q^{64} + 12 q^{65} - 10 q^{67} - 69 q^{68} + 6 q^{70} - 18 q^{71} - 7 q^{73} - 6 q^{74} + 2 q^{76} - q^{79} - 51 q^{80} - 30 q^{82} + 30 q^{83} + 24 q^{85} - 36 q^{86} - 54 q^{89} + 10 q^{91} - 6 q^{92} + 15 q^{94} + 12 q^{95} - 16 q^{97} + 60 q^{98}+O(q^{100}) $ 10 * q + 10 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - q^13 - 15 * q^14 + 10 * q^16 - 3 * q^17 + 2 * q^19 - 15 * q^20 - 18 * q^23 + 7 * q^25 - 6 * q^26 - 4 * q^28 + 12 * q^29 - 7 * q^31 + 24 * q^32 - 6 * q^34 - 18 * q^35 - q^37 + 12 * q^38 + 12 * q^40 - 6 * q^41 + 8 * q^43 - 18 * q^46 - 27 * q^47 + 15 * q^49 + 15 * q^50 + 5 * q^52 - 6 * q^53 - 60 * q^56 + 9 * q^58 - 21 * q^59 + 20 * q^61 + 3 * q^62 + 31 * q^64 + 12 * q^65 - 10 * q^67 - 69 * q^68 + 6 * q^70 - 18 * q^71 - 7 * q^73 - 6 * q^74 + 2 * q^76 - q^79 - 51 * q^80 - 30 * q^82 + 30 * q^83 + 24 * q^85 - 36 * q^86 - 54 * q^89 + 10 * q^91 - 6 * q^92 + 15 * q^94 + 12 * q^95 - 16 * q^97 + 60 * 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 $

$2$	2.71687	1.92112	0.960559	−	0.278078i	$-0.0896972\pi$
$2$	2.71687	1.92112	0.960559	+	0.278078i	$0.0896972\pi$
$3$	0	0
$3$	0	0
$4$	5.38138	2.69069
$5$	−1.67141	−0.747476	−0.373738	−	0.927534i	$-0.621924\pi$
$5$	−1.67141	−0.747476	−0.373738	+	0.927534i	$0.621924\pi$
$6$	0	0
$7$	−2.59211	−0.979724	−0.489862	−	0.871800i	$-0.662953\pi$
$7$	−2.59211	−0.979724	−0.489862	+	0.871800i	$0.662953\pi$
$8$	9.18678	3.24802
$9$	0	0
$10$	−4.54100	−1.43599
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.199351	−0.0552902	−0.0276451	−	0.999618i	$-0.508801\pi$
$13$	−0.199351	−0.0552902	−0.0276451	+	0.999618i	$0.508801\pi$
$14$	−7.04241	−1.88216
$15$	0	0
$16$	14.1965	3.54913
$17$	−7.79195	−1.88983	−0.944913	−	0.327321i	$-0.893854\pi$
$17$	−7.79195	−1.88983	−0.944913	+	0.327321i	$0.893854\pi$
$18$	0	0
$19$	−1.13361	−0.260067	−0.130034	−	0.991510i	$-0.541509\pi$
$19$	−1.13361	−0.260067	−0.130034	+	0.991510i	$0.541509\pi$
$20$	−8.99449	−2.01123
$21$	0	0
$22$	0	0
$23$	−1.79747	−0.374798	−0.187399	−	0.982284i	$-0.560006\pi$
$23$	−1.79747	−0.374798	−0.187399	+	0.982284i	$0.560006\pi$
$24$	0	0
$25$	−2.20639	−0.441279
$26$	−0.541612	−0.106219
$27$	0	0
$28$	−13.9491	−2.63613
$29$	4.02665	0.747730	0.373865	−	0.927483i	$-0.378032\pi$
$29$	4.02665	0.747730	0.373865	+	0.927483i	$0.378032\pi$
$30$	0	0
$31$	4.43134	0.795892	0.397946	−	0.917409i	$-0.369723\pi$
$31$	4.43134	0.795892	0.397946	+	0.917409i	$0.369723\pi$
$32$	20.1965	3.57028
$33$	0	0
$34$	−21.1697	−3.63058
$35$	4.33247	0.732320
$36$	0	0
$37$	4.55523	0.748876	0.374438	−	0.927252i	$-0.377836\pi$
$37$	4.55523	0.748876	0.374438	+	0.927252i	$0.377836\pi$
$38$	−3.07987	−0.499620
$39$	0	0
$40$	−15.3549	−2.42782
$41$	−4.49550	−0.702080	−0.351040	−	0.936360i	$-0.614172\pi$
$41$	−4.49550	−0.702080	−0.351040	+	0.936360i	$0.614172\pi$
$42$	0	0
$43$	3.36629	0.513354	0.256677	−	0.966497i	$-0.417372\pi$
$43$	3.36629	0.513354	0.256677	+	0.966497i	$0.417372\pi$
$44$	0	0
$45$	0	0
$46$	−4.88349	−0.720031
$47$	−8.35219	−1.21829	−0.609146	−	0.793058i	$-0.708488\pi$
$47$	−8.35219	−1.21829	−0.609146	+	0.793058i	$0.708488\pi$
$48$	0	0
$49$	−0.280991	−0.0401416
$50$	−5.99449	−0.847749
$51$	0	0
$52$	−1.07279	−0.148769
$53$	−0.660920	−0.0907844	−0.0453922	−	0.998969i	$-0.514454\pi$
$53$	−0.660920	−0.0907844	−0.0453922	+	0.998969i	$0.514454\pi$
$54$	0	0
$55$	0	0
$56$	−23.8131	−3.18216
$57$	0	0
$58$	10.9399	1.43648
$59$	−4.53865	−0.590882	−0.295441	−	0.955361i	$-0.595467\pi$
$59$	−4.53865	−0.590882	−0.295441	+	0.955361i	$0.595467\pi$
$60$	0	0
$61$	0.455719	0.0583489	0.0291745	−	0.999574i	$-0.490712\pi$
$61$	0.455719	0.0583489	0.0291745	+	0.999574i	$0.490712\pi$
$62$	12.0394	1.52900
$63$	0	0
$64$	26.4783	3.30979
$65$	0.333198	0.0413281
$66$	0	0
$67$	−11.5434	−1.41025	−0.705123	−	0.709085i	$-0.749108\pi$
$67$	−11.5434	−1.41025	−0.705123	+	0.709085i	$0.749108\pi$
$68$	−41.9315	−5.08494
$69$	0	0
$70$	11.7707	1.40687
$71$	−13.4057	−1.59096	−0.795481	−	0.605979i	$-0.792782\pi$
$71$	−13.4057	−1.59096	−0.795481	+	0.605979i	$0.792782\pi$
$72$	0	0
$73$	−10.7548	−1.25875	−0.629375	−	0.777101i	$-0.716689\pi$
$73$	−10.7548	−1.25875	−0.629375	+	0.777101i	$0.716689\pi$
$74$	12.3760	1.43868
$75$	0	0
$76$	−6.10038	−0.699761
$77$	0	0
$78$	0	0
$79$	11.9014	1.33901	0.669505	−	0.742808i	$-0.266506\pi$
$79$	11.9014	1.33901	0.669505	+	0.742808i	$0.266506\pi$
$80$	−23.7282	−2.65289
$81$	0	0
$82$	−12.2137	−1.34878
$83$	2.89275	0.317521	0.158760	−	0.987317i	$-0.449250\pi$
$83$	2.89275	0.317521	0.158760	+	0.987317i	$0.449250\pi$
$84$	0	0
$85$	13.0235	1.41260
$86$	9.14576	0.986213
$87$	0	0
$88$	0	0
$89$	−11.6422	−1.23408	−0.617038	−	0.786933i	$-0.711667\pi$
$89$	−11.6422	−1.23408	−0.617038	+	0.786933i	$0.711667\pi$
$90$	0	0
$91$	0.516740	0.0541691
$92$	−9.67286	−1.00847
$93$	0	0
$94$	−22.6918	−2.34048
$95$	1.89472	0.194394
$96$	0	0
$97$	−4.72812	−0.480068	−0.240034	−	0.970764i	$-0.577159\pi$
$97$	−4.72812	−0.480068	−0.240034	+	0.970764i	$0.577159\pi$
$98$	−0.763417	−0.0771168
$99$	0	0
$100$	−11.8735	−1.18735
$101$	0.620511	0.0617432	0.0308716	−	0.999523i	$-0.490172\pi$
$101$	0.620511	0.0617432	0.0308716	+	0.999523i	$0.490172\pi$
$102$	0	0
$103$	−13.1420	−1.29492	−0.647461	−	0.762099i	$-0.724169\pi$
$103$	−13.1420	−1.29492	−0.647461	+	0.762099i	$0.724169\pi$
$104$	−1.83140	−0.179583
$105$	0	0
$106$	−1.79563	−0.174407
$107$	−0.141493	−0.0136787	−0.00683933	−	0.999977i	$-0.502177\pi$
$107$	−0.141493	−0.0136787	−0.00683933	+	0.999977i	$0.502177\pi$
$108$	0	0
$109$	−16.0135	−1.53381	−0.766906	−	0.641759i	$-0.778205\pi$
$109$	−16.0135	−1.53381	−0.766906	+	0.641759i	$0.778205\pi$
$110$	0	0
$111$	0	0
$112$	−36.7989	−3.47717
$113$	−6.23635	−0.586666	−0.293333	−	0.956010i	$-0.594764\pi$
$113$	−6.23635	−0.586666	−0.293333	+	0.956010i	$0.594764\pi$
$114$	0	0
$115$	3.00430	0.280153
$116$	21.6690	2.01191
$117$	0	0
$118$	−12.3309	−1.13515
$119$	20.1976	1.85151
$120$	0	0
$121$	0	0
$122$	1.23813	0.112095
$123$	0	0
$124$	23.8467	2.14150
$125$	12.0448	1.07732
$126$	0	0
$127$	3.73490	0.331419	0.165709	−	0.986175i	$-0.447009\pi$
$127$	3.73490	0.331419	0.165709	+	0.986175i	$0.447009\pi$
$128$	31.5451	2.78822
$129$	0	0
$130$	0.905255	0.0793961
$131$	−13.0364	−1.13900	−0.569498	−	0.821992i	$-0.692863\pi$
$131$	−13.0364	−1.13900	−0.569498	+	0.821992i	$0.692863\pi$
$132$	0	0
$133$	2.93843	0.254794
$134$	−31.3618	−2.70925
$135$	0	0
$136$	−71.5830	−6.13819
$137$	10.9025	0.931464	0.465732	−	0.884926i	$-0.345791\pi$
$137$	10.9025	0.931464	0.465732	+	0.884926i	$0.345791\pi$
$138$	0	0
$139$	−13.7372	−1.16517	−0.582585	−	0.812770i	$-0.697959\pi$
$139$	−13.7372	−1.16517	−0.582585	+	0.812770i	$0.697959\pi$
$140$	23.3147	1.97045
$141$	0	0
$142$	−36.4215	−3.05642
$143$	0	0
$144$	0	0
$145$	−6.73018	−0.558911
$146$	−29.2193	−2.41821
$147$	0	0
$148$	24.5135	2.01499
$149$	−5.00287	−0.409851	−0.204926	−	0.978778i	$-0.565695\pi$
$149$	−5.00287	−0.409851	−0.204926	+	0.978778i	$0.565695\pi$
$150$	0	0
$151$	18.2484	1.48503	0.742516	−	0.669828i	$-0.233632\pi$
$151$	18.2484	1.48503	0.742516	+	0.669828i	$0.233632\pi$
$152$	−10.4142	−0.844703
$153$	0	0
$154$	0	0
$155$	−7.40658	−0.594911
$156$	0	0
$157$	−3.17899	−0.253711	−0.126856	−	0.991921i	$-0.540488\pi$
$157$	−3.17899	−0.253711	−0.126856	+	0.991921i	$0.540488\pi$
$158$	32.3345	2.57239
$159$	0	0
$160$	−33.7567	−2.66870
$161$	4.65922	0.367198
$162$	0	0
$163$	−15.0397	−1.17800	−0.588999	−	0.808134i	$-0.700478\pi$
$163$	−15.0397	−1.17800	−0.588999	+	0.808134i	$0.700478\pi$
$164$	−24.1920	−1.88908
$165$	0	0
$166$	7.85923	0.609995
$167$	2.30344	0.178246	0.0891229	−	0.996021i	$-0.471594\pi$
$167$	2.30344	0.178246	0.0891229	+	0.996021i	$0.471594\pi$
$168$	0	0
$169$	−12.9603	−0.996943
$170$	35.3833	2.71377
$171$	0	0
$172$	18.1153	1.38128
$173$	19.6981	1.49762	0.748808	−	0.662787i	$-0.230627\pi$
$173$	19.6981	1.49762	0.748808	+	0.662787i	$0.230627\pi$
$174$	0	0
$175$	5.71921	0.432331
$176$	0	0
$177$	0	0
$178$	−31.6305	−2.37080
$179$	14.2894	1.06804	0.534022	−	0.845471i	$-0.320680\pi$
$179$	14.2894	1.06804	0.534022	+	0.845471i	$0.320680\pi$
$180$	0	0
$181$	14.3355	1.06555	0.532776	−	0.846256i	$-0.321149\pi$
$181$	14.3355	1.06555	0.532776	+	0.846256i	$0.321149\pi$
$182$	1.40392	0.104065
$183$	0	0
$184$	−16.5129	−1.21735
$185$	−7.61366	−0.559767
$186$	0	0
$187$	0	0
$188$	−44.9464	−3.27805
$189$	0	0
$190$	5.14771	0.373454
$191$	6.70138	0.484895	0.242448	−	0.970165i	$-0.422050\pi$
$191$	6.70138	0.484895	0.242448	+	0.970165i	$0.422050\pi$
$192$	0	0
$193$	24.7162	1.77911	0.889555	−	0.456828i	$-0.151015\pi$
$193$	24.7162	1.77911	0.889555	+	0.456828i	$0.151015\pi$
$194$	−12.8457	−0.922267
$195$	0	0
$196$	−1.51212	−0.108009
$197$	−3.16677	−0.225623	−0.112811	−	0.993616i	$-0.535986\pi$
$197$	−3.16677	−0.225623	−0.112811	+	0.993616i	$0.535986\pi$
$198$	0	0
$199$	−8.30214	−0.588523	−0.294261	−	0.955725i	$-0.595074\pi$
$199$	−8.30214	−0.588523	−0.294261	+	0.955725i	$0.595074\pi$
$200$	−20.2697	−1.43328
$201$	0	0
$202$	1.68585	0.118616
$203$	−10.4375	−0.732569
$204$	0	0
$205$	7.51382	0.524788
$206$	−35.7052	−2.48770
$207$	0	0
$208$	−2.83010	−0.196232
$209$	0	0
$210$	0	0
$211$	22.4934	1.54851	0.774254	−	0.632875i	$-0.218125\pi$
$211$	22.4934	1.54851	0.774254	+	0.632875i	$0.218125\pi$
$212$	−3.55667	−0.244273
$213$	0	0
$214$	−0.384419	−0.0262783
$215$	−5.62644	−0.383720
$216$	0	0
$217$	−11.4865	−0.779755
$218$	−43.5065	−2.94663
$219$	0	0
$220$	0	0
$221$	1.55334	0.104489
$222$	0	0
$223$	12.6694	0.848405	0.424203	−	0.905567i	$-0.360554\pi$
$223$	12.6694	0.848405	0.424203	+	0.905567i	$0.360554\pi$
$224$	−52.3515	−3.49788
$225$	0	0
$226$	−16.9433	−1.12705
$227$	−5.69621	−0.378071	−0.189035	−	0.981970i	$-0.560536\pi$
$227$	−5.69621	−0.378071	−0.189035	+	0.981970i	$0.560536\pi$
$228$	0	0
$229$	−6.72848	−0.444630	−0.222315	−	0.974975i	$-0.571361\pi$
$229$	−6.72848	−0.444630	−0.222315	+	0.974975i	$0.571361\pi$
$230$	8.16230	0.538206
$231$	0	0
$232$	36.9920	2.42864
$233$	−6.62352	−0.433921	−0.216961	−	0.976180i	$-0.569614\pi$
$233$	−6.62352	−0.433921	−0.216961	+	0.976180i	$0.569614\pi$
$234$	0	0
$235$	13.9599	0.910645
$236$	−24.4242	−1.58988
$237$	0	0
$238$	54.8742	3.55696
$239$	4.68001	0.302725	0.151362	−	0.988478i	$-0.451634\pi$
$239$	4.68001	0.302725	0.151362	+	0.988478i	$0.451634\pi$
$240$	0	0
$241$	3.79666	0.244564	0.122282	−	0.992495i	$-0.460979\pi$
$241$	3.79666	0.244564	0.122282	+	0.992495i	$0.460979\pi$
$242$	0	0
$243$	0	0
$244$	2.45240	0.156999
$245$	0.469651	0.0300049
$246$	0	0
$247$	0.225986	0.0143792
$248$	40.7097	2.58507
$249$	0	0
$250$	32.7242	2.06966
$251$	−11.5694	−0.730253	−0.365126	−	0.930958i	$-0.618974\pi$
$251$	−11.5694	−0.730253	−0.365126	+	0.930958i	$0.618974\pi$
$252$	0	0
$253$	0	0
$254$	10.1472	0.636694
$255$	0	0
$256$	32.7473	2.04671
$257$	13.2795	0.828350	0.414175	−	0.910197i	$-0.364070\pi$
$257$	13.2795	0.828350	0.414175	+	0.910197i	$0.364070\pi$
$258$	0	0
$259$	−11.8076	−0.733691
$260$	1.79306	0.111201
$261$	0	0
$262$	−35.4182	−2.18815
$263$	16.4771	1.01602	0.508011	−	0.861351i	$-0.330381\pi$
$263$	16.4771	1.01602	0.508011	+	0.861351i	$0.330381\pi$
$264$	0	0
$265$	1.10467	0.0678592
$266$	7.98333	0.489490
$267$	0	0
$268$	−62.1193	−3.79454
$269$	−16.8816	−1.02929	−0.514643	−	0.857404i	$-0.672076\pi$
$269$	−16.8816	−1.02929	−0.514643	+	0.857404i	$0.672076\pi$
$270$	0	0
$271$	28.0885	1.70625	0.853126	−	0.521704i	$-0.174704\pi$
$271$	28.0885	1.70625	0.853126	+	0.521704i	$0.174704\pi$
$272$	−110.619	−6.70724
$273$	0	0
$274$	29.6207	1.78945
$275$	0	0
$276$	0	0
$277$	11.4650	0.688867	0.344433	−	0.938811i	$-0.388071\pi$
$277$	11.4650	0.688867	0.344433	+	0.938811i	$0.388071\pi$
$278$	−37.3221	−2.23843
$279$	0	0
$280$	39.8014	2.37859
$281$	−12.9124	−0.770290	−0.385145	−	0.922856i	$-0.625849\pi$
$281$	−12.9124	−0.770290	−0.385145	+	0.922856i	$0.625849\pi$
$282$	0	0
$283$	25.2669	1.50196	0.750981	−	0.660324i	$-0.229581\pi$
$283$	25.2669	1.50196	0.750981	+	0.660324i	$0.229581\pi$
$284$	−72.1411	−4.28079
$285$	0	0
$286$	0	0
$287$	11.6528	0.687844
$288$	0	0
$289$	43.7146	2.57144
$290$	−18.2850	−1.07373
$291$	0	0
$292$	−57.8755	−3.38691
$293$	−2.28505	−0.133494	−0.0667471	−	0.997770i	$-0.521262\pi$
$293$	−2.28505	−0.133494	−0.0667471	+	0.997770i	$0.521262\pi$
$294$	0	0
$295$	7.58594	0.441670
$296$	41.8479	2.43236
$297$	0	0
$298$	−13.5922	−0.787372
$299$	0.358328	0.0207226
$300$	0	0
$301$	−8.72577	−0.502945
$302$	49.5785	2.85292
$303$	0	0
$304$	−16.0933	−0.923013
$305$	−0.761693	−0.0436144
$306$	0	0
$307$	2.56109	0.146169	0.0730847	−	0.997326i	$-0.476716\pi$
$307$	2.56109	0.146169	0.0730847	+	0.997326i	$0.476716\pi$
$308$	0	0
$309$	0	0
$310$	−20.1227	−1.14289
$311$	3.41091	0.193415	0.0967075	−	0.995313i	$-0.469169\pi$
$311$	3.41091	0.193415	0.0967075	+	0.995313i	$0.469169\pi$
$312$	0	0
$313$	27.9763	1.58131	0.790657	−	0.612259i	$-0.209739\pi$
$313$	27.9763	1.58131	0.790657	+	0.612259i	$0.209739\pi$
$314$	−8.63690	−0.487409
$315$	0	0
$316$	64.0458	3.60286
$317$	30.7328	1.72613	0.863064	−	0.505095i	$-0.168543\pi$
$317$	30.7328	1.72613	0.863064	+	0.505095i	$0.168543\pi$
$318$	0	0
$319$	0	0
$320$	−44.2561	−2.47399
$321$	0	0
$322$	12.6585	0.705431
$323$	8.83302	0.491482
$324$	0	0
$325$	0.439848	0.0243984
$326$	−40.8608	−2.26307
$327$	0	0
$328$	−41.2992	−2.28037
$329$	21.6498	1.19359
$330$	0	0
$331$	−5.97755	−0.328556	−0.164278	−	0.986414i	$-0.552529\pi$
$331$	−5.97755	−0.328556	−0.164278	+	0.986414i	$0.552529\pi$
$332$	15.5670	0.854350
$333$	0	0
$334$	6.25816	0.342431
$335$	19.2937	1.05413
$336$	0	0
$337$	1.31492	0.0716280	0.0358140	−	0.999358i	$-0.488598\pi$
$337$	1.31492	0.0716280	0.0358140	+	0.999358i	$0.488598\pi$
$338$	−35.2113	−1.91524
$339$	0	0
$340$	70.0846	3.80087
$341$	0	0
$342$	0	0
$343$	18.8731	1.01905
$344$	30.9253	1.66738
$345$	0	0
$346$	53.5171	2.87709
$347$	−17.9358	−0.962842	−0.481421	−	0.876489i	$-0.659879\pi$
$347$	−17.9358	−0.962842	−0.481421	+	0.876489i	$0.659879\pi$
$348$	0	0
$349$	−25.7232	−1.37693	−0.688466	−	0.725269i	$-0.741715\pi$
$349$	−25.7232	−1.37693	−0.688466	+	0.725269i	$0.741715\pi$
$350$	15.5383	0.830559
$351$	0	0
$352$	0	0
$353$	19.9067	1.05953	0.529763	−	0.848145i	$-0.322281\pi$
$353$	19.9067	1.05953	0.529763	+	0.848145i	$0.322281\pi$
$354$	0	0
$355$	22.4064	1.18921
$356$	−62.6514	−3.32052
$357$	0	0
$358$	38.8226	2.05184
$359$	−29.2180	−1.54207	−0.771033	−	0.636796i	$-0.780260\pi$
$359$	−29.2180	−1.54207	−0.771033	+	0.636796i	$0.780260\pi$
$360$	0	0
$361$	−17.7149	−0.932365
$362$	38.9478	2.04705
$363$	0	0
$364$	2.78078	0.145752
$365$	17.9756	0.940887
$366$	0	0
$367$	17.1477	0.895101	0.447550	−	0.894259i	$-0.352297\pi$
$367$	17.1477	0.895101	0.447550	+	0.894259i	$0.352297\pi$
$368$	−25.5178	−1.33021
$369$	0	0
$370$	−20.6853	−1.07538
$371$	1.71318	0.0889436
$372$	0	0
$373$	−20.2416	−1.04807	−0.524034	−	0.851697i	$-0.675574\pi$
$373$	−20.2416	−1.04807	−0.524034	+	0.851697i	$0.675574\pi$
$374$	0	0
$375$	0	0
$376$	−76.7297	−3.95703
$377$	−0.802719	−0.0413421
$378$	0	0
$379$	−20.1090	−1.03293	−0.516465	−	0.856308i	$-0.672752\pi$
$379$	−20.1090	−1.03293	−0.516465	+	0.856308i	$0.672752\pi$
$380$	10.1962	0.523055
$381$	0	0
$382$	18.2068	0.931540
$383$	21.3083	1.08880	0.544402	−	0.838824i	$-0.316757\pi$
$383$	21.3083	1.08880	0.544402	+	0.838824i	$0.316757\pi$
$384$	0	0
$385$	0	0
$386$	67.1507	3.41788
$387$	0	0
$388$	−25.4438	−1.29171
$389$	−34.5424	−1.75137	−0.875685	−	0.482883i	$-0.839590\pi$
$389$	−34.5424	−1.75137	−0.875685	+	0.482883i	$0.839590\pi$
$390$	0	0
$391$	14.0058	0.708303
$392$	−2.58141	−0.130381
$393$	0	0
$394$	−8.60369	−0.433448
$395$	−19.8921	−1.00088
$396$	0	0
$397$	−2.61262	−0.131124	−0.0655619	−	0.997849i	$-0.520884\pi$
$397$	−2.61262	−0.131124	−0.0655619	+	0.997849i	$0.520884\pi$
$398$	−22.5558	−1.13062
$399$	0	0
$400$	−31.3231	−1.56616
$401$	−10.2520	−0.511962	−0.255981	−	0.966682i	$-0.582398\pi$
$401$	−10.2520	−0.511962	−0.255981	+	0.966682i	$0.582398\pi$
$402$	0	0
$403$	−0.883395	−0.0440050
$404$	3.33921	0.166132
$405$	0	0
$406$	−28.3573	−1.40735
$407$	0	0
$408$	0	0
$409$	3.96459	0.196036	0.0980181	−	0.995185i	$-0.468750\pi$
$409$	3.96459	0.196036	0.0980181	+	0.995185i	$0.468750\pi$
$410$	20.4141	1.00818
$411$	0	0
$412$	−70.7222	−3.48423
$413$	11.7647	0.578901
$414$	0	0
$415$	−4.83497	−0.237339
$416$	−4.02621	−0.197401
$417$	0	0
$418$	0	0
$419$	−32.7769	−1.60126	−0.800628	−	0.599162i	$-0.795501\pi$
$419$	−32.7769	−1.60126	−0.800628	+	0.599162i	$0.795501\pi$
$420$	0	0
$421$	−37.5922	−1.83213	−0.916066	−	0.401029i	$-0.868653\pi$
$421$	−37.5922	−1.83213	−0.916066	+	0.401029i	$0.868653\pi$
$422$	61.1116	2.97487
$423$	0	0
$424$	−6.07173	−0.294869
$425$	17.1921	0.833941
$426$	0	0
$427$	−1.18127	−0.0571658
$428$	−0.761429	−0.0368051
$429$	0	0
$430$	−15.2863	−0.737171
$431$	30.0260	1.44630	0.723152	−	0.690689i	$-0.242693\pi$
$431$	30.0260	1.44630	0.723152	+	0.690689i	$0.242693\pi$
$432$	0	0
$433$	17.8078	0.855788	0.427894	−	0.903829i	$-0.359256\pi$
$433$	17.8078	0.855788	0.427894	+	0.903829i	$0.359256\pi$
$434$	−31.2073	−1.49800
$435$	0	0
$436$	−86.1746	−4.12702
$437$	2.03762	0.0974727
$438$	0	0
$439$	−2.34051	−0.111707	−0.0558533	−	0.998439i	$-0.517788\pi$
$439$	−2.34051	−0.111707	−0.0558533	+	0.998439i	$0.517788\pi$
$440$	0	0
$441$	0	0
$442$	4.22022	0.200735
$443$	−6.40676	−0.304394	−0.152197	−	0.988350i	$-0.548635\pi$
$443$	−6.40676	−0.304394	−0.152197	+	0.988350i	$0.548635\pi$
$444$	0	0
$445$	19.4589	0.922443
$446$	34.4211	1.62989
$447$	0	0
$448$	−68.6346	−3.24268
$449$	−21.7677	−1.02728	−0.513640	−	0.858006i	$-0.671703\pi$
$449$	−21.7677	−1.02728	−0.513640	+	0.858006i	$0.671703\pi$
$450$	0	0
$451$	0	0
$452$	−33.5602	−1.57854
$453$	0	0
$454$	−15.4759	−0.726318
$455$	−0.863684	−0.0404901
$456$	0	0
$457$	−7.14228	−0.334102	−0.167051	−	0.985948i	$-0.553424\pi$
$457$	−7.14228	−0.334102	−0.167051	+	0.985948i	$0.553424\pi$
$458$	−18.2804	−0.854187
$459$	0	0
$460$	16.1673	0.753804
$461$	−2.88780	−0.134498	−0.0672491	−	0.997736i	$-0.521422\pi$
$461$	−2.88780	−0.134498	−0.0672491	+	0.997736i	$0.521422\pi$
$462$	0	0
$463$	1.45208	0.0674838	0.0337419	−	0.999431i	$-0.489258\pi$
$463$	1.45208	0.0674838	0.0337419	+	0.999431i	$0.489258\pi$
$464$	57.1644	2.65379
$465$	0	0
$466$	−17.9953	−0.833614
$467$	−5.50281	−0.254640	−0.127320	−	0.991862i	$-0.540637\pi$
$467$	−5.50281	−0.254640	−0.127320	+	0.991862i	$0.540637\pi$
$468$	0	0
$469$	29.9216	1.38165
$470$	37.9273	1.74946
$471$	0	0
$472$	−41.6956	−1.91919
$473$	0	0
$474$	0	0
$475$	2.50119	0.114762
$476$	108.691	4.98184
$477$	0	0
$478$	12.7150	0.581570
$479$	10.4549	0.477695	0.238848	−	0.971057i	$-0.423230\pi$
$479$	10.4549	0.477695	0.238848	+	0.971057i	$0.423230\pi$
$480$	0	0
$481$	−0.908093	−0.0414055
$482$	10.3150	0.469837
$483$	0	0
$484$	0	0
$485$	7.90262	0.358840
$486$	0	0
$487$	5.69868	0.258232	0.129116	−	0.991630i	$-0.458786\pi$
$487$	5.69868	0.258232	0.129116	+	0.991630i	$0.458786\pi$
$488$	4.18659	0.189518
$489$	0	0
$490$	1.27598	0.0576430
$491$	8.30036	0.374590	0.187295	−	0.982304i	$-0.440028\pi$
$491$	8.30036	0.374590	0.187295	+	0.982304i	$0.440028\pi$
$492$	0	0
$493$	−31.3755	−1.41308
$494$	0.613976	0.0276241
$495$	0	0
$496$	62.9096	2.82472
$497$	34.7489	1.55870
$498$	0	0
$499$	13.9052	0.622484	0.311242	−	0.950331i	$-0.399255\pi$
$499$	13.9052	0.622484	0.311242	+	0.950331i	$0.399255\pi$
$500$	64.8178	2.89874
$501$	0	0
$502$	−31.4325	−1.40290
$503$	−29.2592	−1.30460	−0.652302	−	0.757959i	$-0.726197\pi$
$503$	−29.2592	−1.30460	−0.652302	+	0.757959i	$0.726197\pi$
$504$	0	0
$505$	−1.03713	−0.0461516
$506$	0	0
$507$	0	0
$508$	20.0989	0.891746
$509$	36.6693	1.62534	0.812669	−	0.582726i	$-0.198014\pi$
$509$	36.6693	1.62534	0.812669	+	0.582726i	$0.198014\pi$
$510$	0	0
$511$	27.8775	1.23323
$512$	25.8799	1.14374
$513$	0	0
$514$	36.0785	1.59136
$515$	21.9657	0.967923
$516$	0	0
$517$	0	0
$518$	−32.0798	−1.40951
$519$	0	0
$520$	3.06101	0.134234
$521$	−42.0462	−1.84208	−0.921039	−	0.389469i	$-0.872658\pi$
$521$	−42.0462	−1.84208	−0.921039	+	0.389469i	$0.872658\pi$
$522$	0	0
$523$	−36.3284	−1.58853	−0.794264	−	0.607573i	$-0.792143\pi$
$523$	−36.3284	−1.58853	−0.794264	+	0.607573i	$0.792143\pi$
$524$	−70.1539	−3.06469
$525$	0	0
$526$	44.7661	1.95190
$527$	−34.5288	−1.50410
$528$	0	0
$529$	−19.7691	−0.859527
$530$	3.00124	0.130365
$531$	0	0
$532$	15.8128	0.685573
$533$	0.896186	0.0388181
$534$	0	0
$535$	0.236493	0.0102245
$536$	−106.046	−4.58050
$537$	0	0
$538$	−45.8650	−1.97738
$539$	0	0
$540$	0	0
$541$	16.4772	0.708409	0.354205	−	0.935168i	$-0.384752\pi$
$541$	16.4772	0.708409	0.354205	+	0.935168i	$0.384752\pi$
$542$	76.3127	3.27791
$543$	0	0
$544$	−157.370	−6.74720
$545$	26.7650	1.14649
$546$	0	0
$547$	−13.4828	−0.576484	−0.288242	−	0.957558i	$-0.593071\pi$
$547$	−13.4828	−0.576484	−0.288242	+	0.957558i	$0.593071\pi$
$548$	58.6706	2.50628
$549$	0	0
$550$	0	0
$551$	−4.56464	−0.194460
$552$	0	0
$553$	−30.8496	−1.31186
$554$	31.1490	1.32339
$555$	0	0
$556$	−73.9249	−3.13511
$557$	41.2760	1.74892	0.874460	−	0.485098i	$-0.161216\pi$
$557$	41.2760	1.74892	0.874460	+	0.485098i	$0.161216\pi$
$558$	0	0
$559$	−0.671074	−0.0283834
$560$	61.5059	2.59910
$561$	0	0
$562$	−35.0814	−1.47982
$563$	−1.43165	−0.0603369	−0.0301685	−	0.999545i	$-0.509604\pi$
$563$	−1.43165	−0.0603369	−0.0301685	+	0.999545i	$0.509604\pi$
$564$	0	0
$565$	10.4235	0.438519
$566$	68.6469	2.88545
$567$	0	0
$568$	−123.155	−5.16747
$569$	−29.3751	−1.23147	−0.615734	−	0.787954i	$-0.711141\pi$
$569$	−29.3751	−1.23147	−0.615734	+	0.787954i	$0.711141\pi$
$570$	0	0
$571$	−23.9706	−1.00314	−0.501569	−	0.865118i	$-0.667244\pi$
$571$	−23.9706	−1.00314	−0.501569	+	0.865118i	$0.667244\pi$
$572$	0	0
$573$	0	0
$574$	31.6592	1.32143
$575$	3.96592	0.165390
$576$	0	0
$577$	41.8351	1.74162	0.870810	−	0.491620i	$-0.163595\pi$
$577$	41.8351	1.74162	0.870810	+	0.491620i	$0.163595\pi$
$578$	118.767	4.94005
$579$	0	0
$580$	−36.2177	−1.50386
$581$	−7.49832	−0.311083
$582$	0	0
$583$	0	0
$584$	−98.8017	−4.08844
$585$	0	0
$586$	−6.20819	−0.256458
$587$	−3.15857	−0.130368	−0.0651841	−	0.997873i	$-0.520763\pi$
$587$	−3.15857	−0.130368	−0.0651841	+	0.997873i	$0.520763\pi$
$588$	0	0
$589$	−5.02340	−0.206986
$590$	20.6100	0.848500
$591$	0	0
$592$	64.6685	2.65786
$593$	−42.1283	−1.73000	−0.865001	−	0.501770i	$-0.832682\pi$
$593$	−42.1283	−1.73000	−0.865001	+	0.501770i	$0.832682\pi$
$594$	0	0
$595$	−33.7584	−1.38396
$596$	−26.9224	−1.10278
$597$	0	0
$598$	0.973530	0.0398106
$599$	−22.9023	−0.935762	−0.467881	−	0.883792i	$-0.654982\pi$
$599$	−22.9023	−0.935762	−0.467881	+	0.883792i	$0.654982\pi$
$600$	0	0
$601$	2.95721	0.120627	0.0603136	−	0.998179i	$-0.480790\pi$
$601$	2.95721	0.120627	0.0603136	+	0.998179i	$0.480790\pi$
$602$	−23.7068	−0.966216
$603$	0	0
$604$	98.2015	3.99576
$605$	0	0
$606$	0	0
$607$	16.7620	0.680348	0.340174	−	0.940362i	$-0.389514\pi$
$607$	16.7620	0.680348	0.340174	+	0.940362i	$0.389514\pi$
$608$	−22.8949	−0.928513
$609$	0	0
$610$	−2.06942	−0.0837884
$611$	1.66502	0.0673596
$612$	0	0
$613$	−27.4806	−1.10993	−0.554965	−	0.831874i	$-0.687268\pi$
$613$	−27.4806	−1.10993	−0.554965	+	0.831874i	$0.687268\pi$
$614$	6.95816	0.280808
$615$	0	0
$616$	0	0
$617$	35.9473	1.44718	0.723592	−	0.690228i	$-0.242490\pi$
$617$	35.9473	1.44718	0.723592	+	0.690228i	$0.242490\pi$
$618$	0	0
$619$	30.9891	1.24556	0.622778	−	0.782398i	$-0.286004\pi$
$619$	30.9891	1.24556	0.622778	+	0.782398i	$0.286004\pi$
$620$	−39.8576	−1.60072
$621$	0	0
$622$	9.26700	0.371573
$623$	30.1779	1.20905
$624$	0	0
$625$	−9.09985	−0.363994
$626$	76.0080	3.03789
$627$	0	0
$628$	−17.1074	−0.682658
$629$	−35.4942	−1.41525
$630$	0	0
$631$	17.8066	0.708868	0.354434	−	0.935081i	$-0.384674\pi$
$631$	17.8066	0.708868	0.354434	+	0.935081i	$0.384674\pi$
$632$	109.335	4.34912
$633$	0	0
$634$	83.4971	3.31609
$635$	−6.24254	−0.247728
$636$	0	0
$637$	0.0560160	0.00221944
$638$	0	0
$639$	0	0
$640$	−52.7247	−2.08413
$641$	−12.8415	−0.507208	−0.253604	−	0.967308i	$-0.581616\pi$
$641$	−12.8415	−0.507208	−0.253604	+	0.967308i	$0.581616\pi$
$642$	0	0
$643$	16.1684	0.637620	0.318810	−	0.947819i	$-0.396717\pi$
$643$	16.1684	0.637620	0.318810	+	0.947819i	$0.396717\pi$
$644$	25.0731	0.988017
$645$	0	0
$646$	23.9982	0.944195
$647$	−0.875926	−0.0344362	−0.0172181	−	0.999852i	$-0.505481\pi$
$647$	−0.875926	−0.0344362	−0.0172181	+	0.999852i	$0.505481\pi$
$648$	0	0
$649$	0	0
$650$	1.19501	0.0468722
$651$	0	0
$652$	−80.9343	−3.16963
$653$	−31.7262	−1.24154	−0.620771	−	0.783992i	$-0.713180\pi$
$653$	−31.7262	−1.24154	−0.620771	+	0.783992i	$0.713180\pi$
$654$	0	0
$655$	21.7892	0.851373
$656$	−63.8205	−2.49177
$657$	0	0
$658$	58.8196	2.29303
$659$	−11.9937	−0.467209	−0.233605	−	0.972332i	$-0.575052\pi$
$659$	−11.9937	−0.467209	−0.233605	+	0.972332i	$0.575052\pi$
$660$	0	0
$661$	−23.8469	−0.927536	−0.463768	−	0.885957i	$-0.653503\pi$
$661$	−23.8469	−0.927536	−0.463768	+	0.885957i	$0.653503\pi$
$662$	−16.2402	−0.631194
$663$	0	0
$664$	26.5751	1.03131
$665$	−4.91132	−0.190453
$666$	0	0
$667$	−7.23777	−0.280248
$668$	12.3957	0.479605
$669$	0	0
$670$	52.4184	2.02510
$671$	0	0
$672$	0	0
$673$	32.9721	1.27098	0.635491	−	0.772108i	$-0.280798\pi$
$673$	32.9721	1.27098	0.635491	+	0.772108i	$0.280798\pi$
$674$	3.57245	0.137606
$675$	0	0
$676$	−69.7441	−2.68247
$677$	−26.5286	−1.01958	−0.509789	−	0.860299i	$-0.670277\pi$
$677$	−26.5286	−1.01958	−0.509789	+	0.860299i	$0.670277\pi$
$678$	0	0
$679$	12.2558	0.470334
$680$	119.644	4.58815
$681$	0	0
$682$	0	0
$683$	−17.4229	−0.666669	−0.333334	−	0.942809i	$-0.608174\pi$
$683$	−17.4229	−0.666669	−0.333334	+	0.942809i	$0.608174\pi$
$684$	0	0
$685$	−18.2225	−0.696247
$686$	51.2757	1.95772
$687$	0	0
$688$	47.7895	1.82196
$689$	0.131755	0.00501948
$690$	0	0
$691$	43.4638	1.65344	0.826721	−	0.562613i	$-0.190204\pi$
$691$	43.4638	1.65344	0.826721	+	0.562613i	$0.190204\pi$
$692$	106.003	4.02962
$693$	0	0
$694$	−48.7292	−1.84973
$695$	22.9604	0.870938
$696$	0	0
$697$	35.0288	1.32681
$698$	−69.8866	−2.64525
$699$	0	0
$700$	30.7772	1.16327
$701$	16.2104	0.612258	0.306129	−	0.951990i	$-0.400966\pi$
$701$	16.2104	0.612258	0.306129	+	0.951990i	$0.400966\pi$
$702$	0	0
$703$	−5.16385	−0.194758
$704$	0	0
$705$	0	0
$706$	54.0839	2.03548
$707$	−1.60843	−0.0604912
$708$	0	0
$709$	9.12098	0.342546	0.171273	−	0.985224i	$-0.445212\pi$
$709$	9.12098	0.342546	0.171273	+	0.985224i	$0.445212\pi$
$710$	60.8752	2.28460
$711$	0	0
$712$	−106.955	−4.00830
$713$	−7.96519	−0.298299
$714$	0	0
$715$	0	0
$716$	76.8970	2.87378
$717$	0	0
$718$	−79.3814	−2.96249
$719$	−30.5511	−1.13936	−0.569682	−	0.821865i	$-0.692934\pi$
$719$	−30.5511	−1.13936	−0.569682	+	0.821865i	$0.692934\pi$
$720$	0	0
$721$	34.0655	1.26867
$722$	−48.1292	−1.79118
$723$	0	0
$724$	77.1451	2.86707
$725$	−8.88438	−0.329958
$726$	0	0
$727$	16.6974	0.619271	0.309635	−	0.950855i	$-0.399793\pi$
$727$	16.6974	0.619271	0.309635	+	0.950855i	$0.399793\pi$
$728$	4.74718	0.175942
$729$	0	0
$730$	48.8374	1.80755
$731$	−26.2299	−0.970150
$732$	0	0
$733$	−32.3643	−1.19540	−0.597702	−	0.801718i	$-0.703919\pi$
$733$	−32.3643	−1.19540	−0.597702	+	0.801718i	$0.703919\pi$
$734$	46.5880	1.71959
$735$	0	0
$736$	−36.3026	−1.33813
$737$	0	0
$738$	0	0
$739$	27.6489	1.01708	0.508541	−	0.861038i	$-0.330185\pi$
$739$	27.6489	1.01708	0.508541	+	0.861038i	$0.330185\pi$
$740$	−40.9720	−1.50616
$741$	0	0
$742$	4.65447	0.170871
$743$	−6.85387	−0.251444	−0.125722	−	0.992066i	$-0.540125\pi$
$743$	−6.85387	−0.251444	−0.125722	+	0.992066i	$0.540125\pi$
$744$	0	0
$745$	8.36184	0.306354
$746$	−54.9937	−2.01346
$747$	0	0
$748$	0	0
$749$	0.366765	0.0134013
$750$	0	0
$751$	22.8183	0.832653	0.416327	−	0.909215i	$-0.363317\pi$
$751$	22.8183	0.832653	0.416327	+	0.909215i	$0.363317\pi$
$752$	−118.572	−4.32388
$753$	0	0
$754$	−2.18088	−0.0794231
$755$	−30.5005	−1.11003
$756$	0	0
$757$	−2.07971	−0.0755884	−0.0377942	−	0.999286i	$-0.512033\pi$
$757$	−2.07971	−0.0755884	−0.0377942	+	0.999286i	$0.512033\pi$
$758$	−54.6335	−1.98438
$759$	0	0
$760$	17.4064	0.631396
$761$	21.8445	0.791863	0.395931	−	0.918280i	$-0.370422\pi$
$761$	21.8445	0.791863	0.395931	+	0.918280i	$0.370422\pi$
$762$	0	0
$763$	41.5086	1.50271
$764$	36.0627	1.30470
$765$	0	0
$766$	57.8920	2.09172
$767$	0.904787	0.0326699
$768$	0	0
$769$	3.56817	0.128671	0.0643357	−	0.997928i	$-0.479507\pi$
$769$	3.56817	0.128671	0.0643357	+	0.997928i	$0.479507\pi$
$770$	0	0
$771$	0	0
$772$	133.007	4.78704
$773$	9.24340	0.332462	0.166231	−	0.986087i	$-0.446840\pi$
$773$	9.24340	0.332462	0.166231	+	0.986087i	$0.446840\pi$
$774$	0	0
$775$	−9.77729	−0.351211
$776$	−43.4362	−1.55927
$777$	0	0
$778$	−93.8472	−3.36459
$779$	5.09614	0.182588
$780$	0	0
$781$	0	0
$782$	38.0519	1.36073
$783$	0	0
$784$	−3.98910	−0.142468
$785$	5.31339	0.189643
$786$	0	0
$787$	−14.7058	−0.524204	−0.262102	−	0.965040i	$-0.584416\pi$
$787$	−14.7058	−0.524204	−0.262102	+	0.965040i	$0.584416\pi$
$788$	−17.0416	−0.607081
$789$	0	0
$790$	−54.0441	−1.92280
$791$	16.1653	0.574771
$792$	0	0
$793$	−0.0908484	−0.00322612
$794$	−7.09815	−0.251904
$795$	0	0
$796$	−44.6770	−1.58353
$797$	21.1269	0.748352	0.374176	−	0.927358i	$-0.377926\pi$
$797$	21.1269	0.748352	0.374176	+	0.927358i	$0.377926\pi$
$798$	0	0
$799$	65.0799	2.30236
$800$	−44.5615	−1.57549
$801$	0	0
$802$	−27.8534	−0.983539
$803$	0	0
$804$	0	0
$805$	−7.78747	−0.274472
$806$	−2.40007	−0.0845388
$807$	0	0
$808$	5.70050	0.200543
$809$	29.6574	1.04270	0.521349	−	0.853344i	$-0.325429\pi$
$809$	29.6574	1.04270	0.521349	+	0.853344i	$0.325429\pi$
$810$	0	0
$811$	6.19031	0.217371	0.108686	−	0.994076i	$-0.465336\pi$
$811$	6.19031	0.217371	0.108686	+	0.994076i	$0.465336\pi$
$812$	−56.1682	−1.97112
$813$	0	0
$814$	0	0
$815$	25.1374	0.880526
$816$	0	0
$817$	−3.81605	−0.133507
$818$	10.7713	0.376608
$819$	0	0
$820$	40.4348	1.41204
$821$	21.2591	0.741948	0.370974	−	0.928643i	$-0.379024\pi$
$821$	21.2591	0.741948	0.370974	+	0.928643i	$0.379024\pi$
$822$	0	0
$823$	−40.0182	−1.39495	−0.697474	−	0.716611i	$-0.745693\pi$
$823$	−40.0182	−1.39495	−0.697474	+	0.716611i	$0.745693\pi$
$824$	−120.733	−4.20593
$825$	0	0
$826$	31.9630	1.11214
$827$	22.0260	0.765919	0.382960	−	0.923765i	$-0.374905\pi$
$827$	22.0260	0.765919	0.382960	+	0.923765i	$0.374905\pi$
$828$	0	0
$829$	−30.6206	−1.06350	−0.531749	−	0.846902i	$-0.678465\pi$
$829$	−30.6206	−1.06350	−0.531749	+	0.846902i	$0.678465\pi$
$830$	−13.1360	−0.455957
$831$	0	0
$832$	−5.27849	−0.182999
$833$	2.18947	0.0758607
$834$	0	0
$835$	−3.85000	−0.133235
$836$	0	0
$837$	0	0
$838$	−89.0506	−3.07620
$839$	−8.64337	−0.298402	−0.149201	−	0.988807i	$-0.547670\pi$
$839$	−8.64337	−0.298402	−0.149201	+	0.988807i	$0.547670\pi$
$840$	0	0
$841$	−12.7861	−0.440899
$842$	−102.133	−3.51974
$843$	0	0
$844$	121.045	4.16656
$845$	21.6619	0.745191
$846$	0	0
$847$	0	0
$848$	−9.38277	−0.322206
$849$	0	0
$850$	46.7088	1.60210
$851$	−8.18789	−0.280677
$852$	0	0
$853$	5.24417	0.179557	0.0897784	−	0.995962i	$-0.471384\pi$
$853$	5.24417	0.179557	0.0897784	+	0.995962i	$0.471384\pi$
$854$	−3.20936	−0.109822
$855$	0	0
$856$	−1.29987	−0.0444285
$857$	22.4040	0.765306	0.382653	−	0.923892i	$-0.375011\pi$
$857$	22.4040	0.765306	0.382653	+	0.923892i	$0.375011\pi$
$858$	0	0
$859$	0.380602	0.0129860	0.00649299	−	0.999979i	$-0.497933\pi$
$859$	0.380602	0.0129860	0.00649299	+	0.999979i	$0.497933\pi$
$860$	−30.2780	−1.03247
$861$	0	0
$862$	81.5769	2.77852
$863$	−14.6703	−0.499382	−0.249691	−	0.968326i	$-0.580329\pi$
$863$	−14.6703	−0.499382	−0.249691	+	0.968326i	$0.580329\pi$
$864$	0	0
$865$	−32.9235	−1.11943
$866$	48.3814	1.64407
$867$	0	0
$868$	−61.8133	−2.09808
$869$	0	0
$870$	0	0
$871$	2.30119	0.0779727
$872$	−147.112	−4.98185
$873$	0	0
$874$	5.53596	0.187257
$875$	−31.2215	−1.05548
$876$	0	0
$877$	27.7817	0.938123	0.469061	−	0.883166i	$-0.344592\pi$
$877$	27.7817	0.938123	0.469061	+	0.883166i	$0.344592\pi$
$878$	−6.35887	−0.214601
$879$	0	0
$880$	0	0
$881$	−12.5329	−0.422243	−0.211122	−	0.977460i	$-0.567712\pi$
$881$	−12.5329	−0.422243	−0.211122	+	0.977460i	$0.567712\pi$
$882$	0	0
$883$	−51.2943	−1.72619	−0.863095	−	0.505042i	$-0.831477\pi$
$883$	−51.2943	−1.72619	−0.863095	+	0.505042i	$0.831477\pi$
$884$	8.35911	0.281147
$885$	0	0
$886$	−17.4063	−0.584777
$887$	2.56217	0.0860291	0.0430146	−	0.999074i	$-0.486304\pi$
$887$	2.56217	0.0860291	0.0430146	+	0.999074i	$0.486304\pi$
$888$	0	0
$889$	−9.68126	−0.324699
$890$	52.8674	1.77212
$891$	0	0
$892$	68.1789	2.28280
$893$	9.46811	0.316838
$894$	0	0
$895$	−23.8835	−0.798337
$896$	−81.7682	−2.73168
$897$	0	0
$898$	−59.1400	−1.97353
$899$	17.8435	0.595113
$900$	0	0
$901$	5.14986	0.171567
$902$	0	0
$903$	0	0
$904$	−57.2919	−1.90550
$905$	−23.9606	−0.796476
$906$	0	0
$907$	47.9430	1.59192	0.795960	−	0.605349i	$-0.206966\pi$
$907$	47.9430	1.59192	0.795960	+	0.605349i	$0.206966\pi$
$908$	−30.6535	−1.01727
$909$	0	0
$910$	−2.34652	−0.0777862
$911$	−54.4206	−1.80303	−0.901517	−	0.432743i	$-0.857546\pi$
$911$	−54.4206	−1.80303	−0.901517	+	0.432743i	$0.857546\pi$
$912$	0	0
$913$	0	0
$914$	−19.4046	−0.641848
$915$	0	0
$916$	−36.2085	−1.19636
$917$	33.7918	1.11590
$918$	0	0
$919$	−3.69832	−0.121996	−0.0609982	−	0.998138i	$-0.519428\pi$
$919$	−3.69832	−0.121996	−0.0609982	+	0.998138i	$0.519428\pi$
$920$	27.5999	0.909940
$921$	0	0
$922$	−7.84577	−0.258387
$923$	2.67244	0.0879645
$924$	0	0
$925$	−10.0506	−0.330463
$926$	3.94511	0.129644
$927$	0	0
$928$	81.3244	2.66960
$929$	25.8407	0.847807	0.423903	−	0.905707i	$-0.360660\pi$
$929$	25.8407	0.847807	0.423903	+	0.905707i	$0.360660\pi$
$930$	0	0
$931$	0.318534	0.0104395
$932$	−35.6437	−1.16755
$933$	0	0
$934$	−14.9504	−0.489192
$935$	0	0
$936$	0	0
$937$	28.1689	0.920239	0.460120	−	0.887857i	$-0.347806\pi$
$937$	28.1689	0.920239	0.460120	+	0.887857i	$0.347806\pi$
$938$	81.2931	2.65431
$939$	0	0
$940$	75.1237	2.45026
$941$	−30.4871	−0.993851	−0.496925	−	0.867793i	$-0.665538\pi$
$941$	−30.4871	−0.993851	−0.496925	+	0.867793i	$0.665538\pi$
$942$	0	0
$943$	8.08052	0.263138
$944$	−64.4330	−2.09712
$945$	0	0
$946$	0	0
$947$	54.8142	1.78122	0.890611	−	0.454766i	$-0.150277\pi$
$947$	54.8142	1.78122	0.890611	+	0.454766i	$0.150277\pi$
$948$	0	0
$949$	2.14398	0.0695965
$950$	6.79540	0.220472
$951$	0	0
$952$	185.551	6.01373
$953$	38.8502	1.25848	0.629240	−	0.777211i	$-0.283366\pi$
$953$	38.8502	1.25848	0.629240	+	0.777211i	$0.283366\pi$
$954$	0	0
$955$	−11.2007	−0.362448
$956$	25.1849	0.814539
$957$	0	0
$958$	28.4045	0.917709
$959$	−28.2604	−0.912577
$960$	0	0
$961$	−11.3632	−0.366555
$962$	−2.46717	−0.0795448
$963$	0	0
$964$	20.4313	0.658047
$965$	−41.3108	−1.32984
$966$	0	0
$967$	25.8498	0.831272	0.415636	−	0.909531i	$-0.363559\pi$
$967$	25.8498	0.831272	0.415636	+	0.909531i	$0.363559\pi$
$968$	0	0
$969$	0	0
$970$	21.4704	0.689373
$971$	−14.7459	−0.473219	−0.236609	−	0.971605i	$-0.576036\pi$
$971$	−14.7459	−0.473219	−0.236609	+	0.971605i	$0.576036\pi$
$972$	0	0
$973$	35.6082	1.14155
$974$	15.4826	0.496094
$975$	0	0
$976$	6.46963	0.207088
$977$	−37.6396	−1.20420	−0.602099	−	0.798422i	$-0.705669\pi$
$977$	−37.6396	−1.20420	−0.602099	+	0.798422i	$0.705669\pi$
$978$	0	0
$979$	0	0
$980$	2.52737	0.0807340
$981$	0	0
$982$	22.5510	0.719631
$983$	−6.42972	−0.205076	−0.102538	−	0.994729i	$-0.532696\pi$
$983$	−6.42972	−0.205076	−0.102538	+	0.994729i	$0.532696\pi$
$984$	0	0
$985$	5.29296	0.168648
$986$	−85.2431	−2.71469
$987$	0	0
$988$	1.21612	0.0386899
$989$	−6.05079	−0.192404
$990$	0	0
$991$	35.1704	1.11722	0.558612	−	0.829429i	$-0.311334\pi$
$991$	35.1704	1.11722	0.558612	+	0.829429i	$0.311334\pi$
$992$	89.4977	2.84156
$993$	0	0
$994$	94.4083	2.99445
$995$	13.8763	0.439907
$996$	0	0
$997$	−0.128713	−0.00407637	−0.00203818	−	0.999998i	$-0.500649\pi$
$997$	−0.128713	−0.00407637	−0.00203818	+	0.999998i	$0.500649\pi$
$998$	37.7787	1.19587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cb.1.10		10
3.2	odd	2		9801.2.a.cd.1.1		10
9.2	odd	6		1089.2.e.l.364.10	✓	20
9.5	odd	6		1089.2.e.l.727.10	yes	20
11.10	odd	2		9801.2.a.cc.1.1		10
33.32	even	2		9801.2.a.ce.1.10		10
99.32	even	6		1089.2.e.m.727.1	yes	20
99.65	even	6		1089.2.e.m.364.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.2.e.l.364.10	✓	20	9.2	odd	6
1089.2.e.l.727.10	yes	20	9.5	odd	6
1089.2.e.m.364.1	yes	20	99.65	even	6
1089.2.e.m.727.1	yes	20	99.32	even	6
9801.2.a.cb.1.10		10	1.1	even	1	trivial
9801.2.a.cc.1.1		10	11.10	odd	2
9801.2.a.cd.1.1		10	3.2	odd	2
9801.2.a.ce.1.10		10	33.32	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71687 q^{2} +5.38138 q^{4} -1.67141 q^{5} -2.59211 q^{7} +9.18678 q^{8} +O(q^{10})\)	\(q+2.71687 q^{2} +5.38138 q^{4} -1.67141 q^{5} -2.59211 q^{7} +9.18678 q^{8} -4.54100 q^{10} -0.199351 q^{13} -7.04241 q^{14} +14.1965 q^{16} -7.79195 q^{17} -1.13361 q^{19} -8.99449 q^{20} -1.79747 q^{23} -2.20639 q^{25} -0.541612 q^{26} -13.9491 q^{28} +4.02665 q^{29} +4.43134 q^{31} +20.1965 q^{32} -21.1697 q^{34} +4.33247 q^{35} +4.55523 q^{37} -3.07987 q^{38} -15.3549 q^{40} -4.49550 q^{41} +3.36629 q^{43} -4.88349 q^{46} -8.35219 q^{47} -0.280991 q^{49} -5.99449 q^{50} -1.07279 q^{52} -0.660920 q^{53} -23.8131 q^{56} +10.9399 q^{58} -4.53865 q^{59} +0.455719 q^{61} +12.0394 q^{62} +26.4783 q^{64} +0.333198 q^{65} -11.5434 q^{67} -41.9315 q^{68} +11.7707 q^{70} -13.4057 q^{71} -10.7548 q^{73} +12.3760 q^{74} -6.10038 q^{76} +11.9014 q^{79} -23.7282 q^{80} -12.2137 q^{82} +2.89275 q^{83} +13.0235 q^{85} +9.14576 q^{86} -11.6422 q^{89} +0.516740 q^{91} -9.67286 q^{92} -22.6918 q^{94} +1.89472 q^{95} -4.72812 q^{97} -0.763417 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{4} - 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 10 * q^4 - 3 * q^5 - q^7 - 3 * q^8	\( 10 q + 10 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - q^{13} - 15 q^{14} + 10 q^{16} - 3 q^{17} + 2 q^{19} - 15 q^{20} - 18 q^{23} + 7 q^{25} - 6 q^{26} - 4 q^{28} + 12 q^{29} - 7 q^{31} + 24 q^{32} - 6 q^{34} - 18 q^{35} - q^{37} + 12 q^{38} + 12 q^{40} - 6 q^{41} + 8 q^{43} - 18 q^{46} - 27 q^{47} + 15 q^{49} + 15 q^{50} + 5 q^{52} - 6 q^{53} - 60 q^{56} + 9 q^{58} - 21 q^{59} + 20 q^{61} + 3 q^{62} + 31 q^{64} + 12 q^{65} - 10 q^{67} - 69 q^{68} + 6 q^{70} - 18 q^{71} - 7 q^{73} - 6 q^{74} + 2 q^{76} - q^{79} - 51 q^{80} - 30 q^{82} + 30 q^{83} + 24 q^{85} - 36 q^{86} - 54 q^{89} + 10 q^{91} - 6 q^{92} + 15 q^{94} + 12 q^{95} - 16 q^{97} + 60 q^{98}+O(q^{100}) \) 10 * q + 10 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - q^13 - 15 * q^14 + 10 * q^16 - 3 * q^17 + 2 * q^19 - 15 * q^20 - 18 * q^23 + 7 * q^25 - 6 * q^26 - 4 * q^28 + 12 * q^29 - 7 * q^31 + 24 * q^32 - 6 * q^34 - 18 * q^35 - q^37 + 12 * q^38 + 12 * q^40 - 6 * q^41 + 8 * q^43 - 18 * q^46 - 27 * q^47 + 15 * q^49 + 15 * q^50 + 5 * q^52 - 6 * q^53 - 60 * q^56 + 9 * q^58 - 21 * q^59 + 20 * q^61 + 3 * q^62 + 31 * q^64 + 12 * q^65 - 10 * q^67 - 69 * q^68 + 6 * q^70 - 18 * q^71 - 7 * q^73 - 6 * q^74 + 2 * q^76 - q^79 - 51 * q^80 - 30 * q^82 + 30 * q^83 + 24 * q^85 - 36 * q^86 - 54 * q^89 + 10 * q^91 - 6 * q^92 + 15 * q^94 + 12 * q^95 - 16 * q^97 + 60 * q^98

Introduction

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