LMFDB - Embedded newform 9450.2.a.en.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +1.16228 q^{11} -2.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -0.837722 q^{19} +1.16228 q^{22} +0.162278 q^{23} -2.16228 q^{26} -1.00000 q^{28} +2.16228 q^{29} -8.48683 q^{31} +1.00000 q^{32} -5.00000 q^{34} +9.16228 q^{37} -0.837722 q^{38} -2.32456 q^{41} +7.00000 q^{43} +1.16228 q^{44} +0.162278 q^{46} +2.00000 q^{47} +1.00000 q^{49} -2.16228 q^{52} +12.4868 q^{53} -1.00000 q^{56} +2.16228 q^{58} +7.00000 q^{59} -8.32456 q^{61} -8.48683 q^{62} +1.00000 q^{64} +15.3246 q^{67} -5.00000 q^{68} -6.48683 q^{71} +10.0000 q^{73} +9.16228 q^{74} -0.837722 q^{76} -1.16228 q^{77} +6.83772 q^{79} -2.32456 q^{82} +8.32456 q^{83} +7.00000 q^{86} +1.16228 q^{88} +15.3246 q^{89} +2.16228 q^{91} +0.162278 q^{92} +2.00000 q^{94} +1.16228 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 10 q^{17} - 8 q^{19} - 4 q^{22} - 6 q^{23} + 2 q^{26} - 2 q^{28} - 2 q^{29} + 2 q^{31} + 2 q^{32} - 10 q^{34} + 12 q^{37} - 8 q^{38} + 8 q^{41} + 14 q^{43} - 4 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} - 2 q^{56} - 2 q^{58} + 14 q^{59} - 4 q^{61} + 2 q^{62} + 2 q^{64} + 18 q^{67} - 10 q^{68} + 6 q^{71} + 20 q^{73} + 12 q^{74} - 8 q^{76} + 4 q^{77} + 20 q^{79} + 8 q^{82} + 4 q^{83} + 14 q^{86} - 4 q^{88} + 18 q^{89} - 2 q^{91} - 6 q^{92} + 4 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 10 * q^17 - 8 * q^19 - 4 * q^22 - 6 * q^23 + 2 * q^26 - 2 * q^28 - 2 * q^29 + 2 * q^31 + 2 * q^32 - 10 * q^34 + 12 * q^37 - 8 * q^38 + 8 * q^41 + 14 * q^43 - 4 * q^44 - 6 * q^46 + 4 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 - 2 * q^56 - 2 * q^58 + 14 * q^59 - 4 * q^61 + 2 * q^62 + 2 * q^64 + 18 * q^67 - 10 * q^68 + 6 * q^71 + 20 * q^73 + 12 * q^74 - 8 * q^76 + 4 * q^77 + 20 * q^79 + 8 * q^82 + 4 * q^83 + 14 * q^86 - 4 * q^88 + 18 * q^89 - 2 * q^91 - 6 * q^92 + 4 * q^94 - 4 * q^97 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	1.16228	0.350440	0.175220	−	0.984529i	$-0.443936\pi$
$11$	1.16228	0.350440	0.175220	+	0.984529i	$0.443936\pi$
$12$	0	0
$13$	−2.16228	−0.599708	−0.299854	−	0.953985i	$-0.596938\pi$
$13$	−2.16228	−0.599708	−0.299854	+	0.953985i	$0.596938\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−0.837722	−0.192187	−0.0960933	−	0.995372i	$-0.530635\pi$
$19$	−0.837722	−0.192187	−0.0960933	+	0.995372i	$0.530635\pi$
$20$	0	0
$21$	0	0
$22$	1.16228	0.247798
$23$	0.162278	0.0338372	0.0169186	−	0.999857i	$-0.494614\pi$
$23$	0.162278	0.0338372	0.0169186	+	0.999857i	$0.494614\pi$
$24$	0	0
$25$	0	0
$26$	−2.16228	−0.424058
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.16228	0.401525	0.200762	−	0.979640i	$-0.435658\pi$
$29$	2.16228	0.401525	0.200762	+	0.979640i	$0.435658\pi$
$30$	0	0
$31$	−8.48683	−1.52428	−0.762140	−	0.647412i	$-0.775851\pi$
$31$	−8.48683	−1.52428	−0.762140	+	0.647412i	$0.775851\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	0	0
$37$	9.16228	1.50627	0.753135	−	0.657866i	$-0.228541\pi$
$37$	9.16228	1.50627	0.753135	+	0.657866i	$0.228541\pi$
$38$	−0.837722	−0.135897
$39$	0	0
$40$	0	0
$41$	−2.32456	−0.363035	−0.181517	−	0.983388i	$-0.558101\pi$
$41$	−2.32456	−0.363035	−0.181517	+	0.983388i	$0.558101\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	1.16228	0.175220
$45$	0	0
$46$	0.162278	0.0239265
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−2.16228	−0.299854
$53$	12.4868	1.71520	0.857599	−	0.514319i	$-0.171955\pi$
$53$	12.4868	1.71520	0.857599	+	0.514319i	$0.171955\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.16228	0.283921
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−8.32456	−1.06585	−0.532925	−	0.846162i	$-0.678907\pi$
$61$	−8.32456	−1.06585	−0.532925	+	0.846162i	$0.678907\pi$
$62$	−8.48683	−1.07783
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	15.3246	1.87219	0.936096	−	0.351744i	$-0.114411\pi$
$67$	15.3246	1.87219	0.936096	+	0.351744i	$0.114411\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	−6.48683	−0.769845	−0.384923	−	0.922949i	$-0.625772\pi$
$71$	−6.48683	−0.769845	−0.384923	+	0.922949i	$0.625772\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	9.16228	1.06509
$75$	0	0
$76$	−0.837722	−0.0960933
$77$	−1.16228	−0.132454
$78$	0	0
$79$	6.83772	0.769304	0.384652	−	0.923062i	$-0.374321\pi$
$79$	6.83772	0.769304	0.384652	+	0.923062i	$0.374321\pi$
$80$	0	0
$81$	0	0
$82$	−2.32456	−0.256704
$83$	8.32456	0.913739	0.456869	−	0.889534i	$-0.348971\pi$
$83$	8.32456	0.913739	0.456869	+	0.889534i	$0.348971\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	0	0
$88$	1.16228	0.123899
$89$	15.3246	1.62440	0.812200	−	0.583379i	$-0.198270\pi$
$89$	15.3246	1.62440	0.812200	+	0.583379i	$0.198270\pi$
$90$	0	0
$91$	2.16228	0.226668
$92$	0.162278	0.0169186
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	1.16228	0.118011	0.0590057	−	0.998258i	$-0.481207\pi$
$97$	1.16228	0.118011	0.0590057	+	0.998258i	$0.481207\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	−1.16228	−0.115651	−0.0578255	−	0.998327i	$-0.518417\pi$
$101$	−1.16228	−0.115651	−0.0578255	+	0.998327i	$0.518417\pi$
$102$	0	0
$103$	−4.48683	−0.442101	−0.221050	−	0.975262i	$-0.570949\pi$
$103$	−4.48683	−0.442101	−0.221050	+	0.975262i	$0.570949\pi$
$104$	−2.16228	−0.212029
$105$	0	0
$106$	12.4868	1.21283
$107$	−5.16228	−0.499056	−0.249528	−	0.968368i	$-0.580275\pi$
$107$	−5.16228	−0.499056	−0.249528	+	0.968368i	$0.580275\pi$
$108$	0	0
$109$	3.16228	0.302891	0.151446	−	0.988466i	$-0.451607\pi$
$109$	3.16228	0.302891	0.151446	+	0.988466i	$0.451607\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.83772	0.643239	0.321619	−	0.946869i	$-0.395773\pi$
$113$	6.83772	0.643239	0.321619	+	0.946869i	$0.395773\pi$
$114$	0	0
$115$	0	0
$116$	2.16228	0.200762
$117$	0	0
$118$	7.00000	0.644402
$119$	5.00000	0.458349
$120$	0	0
$121$	−9.64911	−0.877192
$122$	−8.32456	−0.753670
$123$	0	0
$124$	−8.48683	−0.762140
$125$	0	0
$126$	0	0
$127$	5.67544	0.503614	0.251807	−	0.967777i	$-0.418975\pi$
$127$	5.67544	0.503614	0.251807	+	0.967777i	$0.418975\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−7.32456	−0.639949	−0.319975	−	0.947426i	$-0.603674\pi$
$131$	−7.32456	−0.639949	−0.319975	+	0.947426i	$0.603674\pi$
$132$	0	0
$133$	0.837722	0.0726397
$134$	15.3246	1.32384
$135$	0	0
$136$	−5.00000	−0.428746
$137$	−4.32456	−0.369472	−0.184736	−	0.982788i	$-0.559143\pi$
$137$	−4.32456	−0.369472	−0.184736	+	0.982788i	$0.559143\pi$
$138$	0	0
$139$	1.48683	0.126112	0.0630558	−	0.998010i	$-0.479915\pi$
$139$	1.48683	0.126112	0.0630558	+	0.998010i	$0.479915\pi$
$140$	0	0
$141$	0	0
$142$	−6.48683	−0.544363
$143$	−2.51317	−0.210162
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	9.16228	0.753135
$149$	20.8114	1.70494	0.852468	−	0.522780i	$-0.175105\pi$
$149$	20.8114	1.70494	0.852468	+	0.522780i	$0.175105\pi$
$150$	0	0
$151$	17.8114	1.44947	0.724735	−	0.689028i	$-0.241962\pi$
$151$	17.8114	1.44947	0.724735	+	0.689028i	$0.241962\pi$
$152$	−0.837722	−0.0679483
$153$	0	0
$154$	−1.16228	−0.0936590
$155$	0	0
$156$	0	0
$157$	−1.83772	−0.146666	−0.0733331	−	0.997308i	$-0.523364\pi$
$157$	−1.83772	−0.146666	−0.0733331	+	0.997308i	$0.523364\pi$
$158$	6.83772	0.543980
$159$	0	0
$160$	0	0
$161$	−0.162278	−0.0127893
$162$	0	0
$163$	13.6491	1.06908	0.534540	−	0.845143i	$-0.320485\pi$
$163$	13.6491	1.06908	0.534540	+	0.845143i	$0.320485\pi$
$164$	−2.32456	−0.181517
$165$	0	0
$166$	8.32456	0.646111
$167$	2.51317	0.194475	0.0972374	−	0.995261i	$-0.468999\pi$
$167$	2.51317	0.194475	0.0972374	+	0.995261i	$0.468999\pi$
$168$	0	0
$169$	−8.32456	−0.640350
$170$	0	0
$171$	0	0
$172$	7.00000	0.533745
$173$	−11.8114	−0.898003	−0.449002	−	0.893531i	$-0.648220\pi$
$173$	−11.8114	−0.898003	−0.449002	+	0.893531i	$0.648220\pi$
$174$	0	0
$175$	0	0
$176$	1.16228	0.0876100
$177$	0	0
$178$	15.3246	1.14862
$179$	−18.3246	−1.36964	−0.684821	−	0.728712i	$-0.740120\pi$
$179$	−18.3246	−1.36964	−0.684821	+	0.728712i	$0.740120\pi$
$180$	0	0
$181$	14.4868	1.07680	0.538399	−	0.842690i	$-0.319029\pi$
$181$	14.4868	1.07680	0.538399	+	0.842690i	$0.319029\pi$
$182$	2.16228	0.160279
$183$	0	0
$184$	0.162278	0.0119633
$185$	0	0
$186$	0	0
$187$	−5.81139	−0.424971
$188$	2.00000	0.145865
$189$	0	0
$190$	0	0
$191$	−13.6754	−0.989520	−0.494760	−	0.869030i	$-0.664744\pi$
$191$	−13.6754	−0.989520	−0.494760	+	0.869030i	$0.664744\pi$
$192$	0	0
$193$	−17.9737	−1.29377	−0.646886	−	0.762586i	$-0.723929\pi$
$193$	−17.9737	−1.29377	−0.646886	+	0.762586i	$0.723929\pi$
$194$	1.16228	0.0834467
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.3246	1.30557	0.652785	−	0.757543i	$-0.273600\pi$
$197$	18.3246	1.30557	0.652785	+	0.757543i	$0.273600\pi$
$198$	0	0
$199$	−5.51317	−0.390818	−0.195409	−	0.980722i	$-0.562603\pi$
$199$	−5.51317	−0.390818	−0.195409	+	0.980722i	$0.562603\pi$
$200$	0	0
$201$	0	0
$202$	−1.16228	−0.0817776
$203$	−2.16228	−0.151762
$204$	0	0
$205$	0	0
$206$	−4.48683	−0.312612
$207$	0	0
$208$	−2.16228	−0.149927
$209$	−0.973666	−0.0673499
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	12.4868	0.857599
$213$	0	0
$214$	−5.16228	−0.352886
$215$	0	0
$216$	0	0
$217$	8.48683	0.576124
$218$	3.16228	0.214176
$219$	0	0
$220$	0	0
$221$	10.8114	0.727253
$222$	0	0
$223$	24.6491	1.65063	0.825313	−	0.564675i	$-0.190999\pi$
$223$	24.6491	1.65063	0.825313	+	0.564675i	$0.190999\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.83772	0.454839
$227$	−0.675445	−0.0448308	−0.0224154	−	0.999749i	$-0.507136\pi$
$227$	−0.675445	−0.0448308	−0.0224154	+	0.999749i	$0.507136\pi$
$228$	0	0
$229$	18.3246	1.21092	0.605460	−	0.795875i	$-0.292989\pi$
$229$	18.3246	1.21092	0.605460	+	0.795875i	$0.292989\pi$
$230$	0	0
$231$	0	0
$232$	2.16228	0.141960
$233$	−28.1359	−1.84325	−0.921623	−	0.388085i	$-0.873137\pi$
$233$	−28.1359	−1.84325	−0.921623	+	0.388085i	$0.873137\pi$
$234$	0	0
$235$	0	0
$236$	7.00000	0.455661
$237$	0	0
$238$	5.00000	0.324102
$239$	6.32456	0.409101	0.204551	−	0.978856i	$-0.434427\pi$
$239$	6.32456	0.409101	0.204551	+	0.978856i	$0.434427\pi$
$240$	0	0
$241$	−7.16228	−0.461363	−0.230681	−	0.973029i	$-0.574096\pi$
$241$	−7.16228	−0.461363	−0.230681	+	0.973029i	$0.574096\pi$
$242$	−9.64911	−0.620268
$243$	0	0
$244$	−8.32456	−0.532925
$245$	0	0
$246$	0	0
$247$	1.81139	0.115256
$248$	−8.48683	−0.538914
$249$	0	0
$250$	0	0
$251$	−3.35089	−0.211506	−0.105753	−	0.994392i	$-0.533725\pi$
$251$	−3.35089	−0.211506	−0.105753	+	0.994392i	$0.533725\pi$
$252$	0	0
$253$	0.188612	0.0118579
$254$	5.67544	0.356109
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−9.16228	−0.569316
$260$	0	0
$261$	0	0
$262$	−7.32456	−0.452513
$263$	−4.16228	−0.256657	−0.128329	−	0.991732i	$-0.540961\pi$
$263$	−4.16228	−0.256657	−0.128329	+	0.991732i	$0.540961\pi$
$264$	0	0
$265$	0	0
$266$	0.837722	0.0513641
$267$	0	0
$268$	15.3246	0.936096
$269$	−22.3246	−1.36115	−0.680576	−	0.732677i	$-0.738270\pi$
$269$	−22.3246	−1.36115	−0.680576	+	0.732677i	$0.738270\pi$
$270$	0	0
$271$	−16.1623	−0.981788	−0.490894	−	0.871219i	$-0.663330\pi$
$271$	−16.1623	−0.981788	−0.490894	+	0.871219i	$0.663330\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	−4.32456	−0.261256
$275$	0	0
$276$	0	0
$277$	12.6491	0.760011	0.380006	−	0.924984i	$-0.375922\pi$
$277$	12.6491	0.760011	0.380006	+	0.924984i	$0.375922\pi$
$278$	1.48683	0.0891743
$279$	0	0
$280$	0	0
$281$	2.51317	0.149923	0.0749615	−	0.997186i	$-0.476117\pi$
$281$	2.51317	0.149923	0.0749615	+	0.997186i	$0.476117\pi$
$282$	0	0
$283$	21.8114	1.29655	0.648276	−	0.761405i	$-0.275490\pi$
$283$	21.8114	1.29655	0.648276	+	0.761405i	$0.275490\pi$
$284$	−6.48683	−0.384923
$285$	0	0
$286$	−2.51317	−0.148607
$287$	2.32456	0.137214
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	10.0000	0.585206
$293$	24.9737	1.45898	0.729489	−	0.683993i	$-0.239758\pi$
$293$	24.9737	1.45898	0.729489	+	0.683993i	$0.239758\pi$
$294$	0	0
$295$	0	0
$296$	9.16228	0.532547
$297$	0	0
$298$	20.8114	1.20557
$299$	−0.350889	−0.0202925
$300$	0	0
$301$	−7.00000	−0.403473
$302$	17.8114	1.02493
$303$	0	0
$304$	−0.837722	−0.0480467
$305$	0	0
$306$	0	0
$307$	11.1623	0.637065	0.318532	−	0.947912i	$-0.396810\pi$
$307$	11.1623	0.637065	0.318532	+	0.947912i	$0.396810\pi$
$308$	−1.16228	−0.0662269
$309$	0	0
$310$	0	0
$311$	21.4868	1.21841	0.609203	−	0.793014i	$-0.291489\pi$
$311$	21.4868	1.21841	0.609203	+	0.793014i	$0.291489\pi$
$312$	0	0
$313$	32.4605	1.83478	0.917388	−	0.397994i	$-0.130294\pi$
$313$	32.4605	1.83478	0.917388	+	0.397994i	$0.130294\pi$
$314$	−1.83772	−0.103709
$315$	0	0
$316$	6.83772	0.384652
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	2.51317	0.140710
$320$	0	0
$321$	0	0
$322$	−0.162278	−0.00904338
$323$	4.18861	0.233061
$324$	0	0
$325$	0	0
$326$	13.6491	0.755954
$327$	0	0
$328$	−2.32456	−0.128352
$329$	−2.00000	−0.110264
$330$	0	0
$331$	27.6491	1.51973	0.759866	−	0.650079i	$-0.225264\pi$
$331$	27.6491	1.51973	0.759866	+	0.650079i	$0.225264\pi$
$332$	8.32456	0.456869
$333$	0	0
$334$	2.51317	0.137514
$335$	0	0
$336$	0	0
$337$	−9.32456	−0.507941	−0.253970	−	0.967212i	$-0.581737\pi$
$337$	−9.32456	−0.507941	−0.253970	+	0.967212i	$0.581737\pi$
$338$	−8.32456	−0.452796
$339$	0	0
$340$	0	0
$341$	−9.86406	−0.534169
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.00000	0.377415
$345$	0	0
$346$	−11.8114	−0.634984
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	1.83772	0.0983710	0.0491855	−	0.998790i	$-0.484337\pi$
$349$	1.83772	0.0983710	0.0491855	+	0.998790i	$0.484337\pi$
$350$	0	0
$351$	0	0
$352$	1.16228	0.0619496
$353$	−5.32456	−0.283397	−0.141699	−	0.989910i	$-0.545256\pi$
$353$	−5.32456	−0.283397	−0.141699	+	0.989910i	$0.545256\pi$
$354$	0	0
$355$	0	0
$356$	15.3246	0.812200
$357$	0	0
$358$	−18.3246	−0.968483
$359$	−14.1623	−0.747456	−0.373728	−	0.927538i	$-0.621921\pi$
$359$	−14.1623	−0.747456	−0.373728	+	0.927538i	$0.621921\pi$
$360$	0	0
$361$	−18.2982	−0.963064
$362$	14.4868	0.761411
$363$	0	0
$364$	2.16228	0.113334
$365$	0	0
$366$	0	0
$367$	−0.811388	−0.0423541	−0.0211771	−	0.999776i	$-0.506741\pi$
$367$	−0.811388	−0.0423541	−0.0211771	+	0.999776i	$0.506741\pi$
$368$	0.162278	0.00845931
$369$	0	0
$370$	0	0
$371$	−12.4868	−0.648284
$372$	0	0
$373$	−1.81139	−0.0937901	−0.0468951	−	0.998900i	$-0.514933\pi$
$373$	−1.81139	−0.0937901	−0.0468951	+	0.998900i	$0.514933\pi$
$374$	−5.81139	−0.300500
$375$	0	0
$376$	2.00000	0.103142
$377$	−4.67544	−0.240798
$378$	0	0
$379$	5.35089	0.274857	0.137428	−	0.990512i	$-0.456116\pi$
$379$	5.35089	0.274857	0.137428	+	0.990512i	$0.456116\pi$
$380$	0	0
$381$	0	0
$382$	−13.6754	−0.699697
$383$	−26.3246	−1.34512	−0.672561	−	0.740042i	$-0.734806\pi$
$383$	−26.3246	−1.34512	−0.672561	+	0.740042i	$0.734806\pi$
$384$	0	0
$385$	0	0
$386$	−17.9737	−0.914836
$387$	0	0
$388$	1.16228	0.0590057
$389$	21.6754	1.09899	0.549494	−	0.835497i	$-0.314820\pi$
$389$	21.6754	1.09899	0.549494	+	0.835497i	$0.314820\pi$
$390$	0	0
$391$	−0.811388	−0.0410337
$392$	1.00000	0.0505076
$393$	0	0
$394$	18.3246	0.923178
$395$	0	0
$396$	0	0
$397$	5.67544	0.284842	0.142421	−	0.989806i	$-0.454511\pi$
$397$	5.67544	0.284842	0.142421	+	0.989806i	$0.454511\pi$
$398$	−5.51317	−0.276350
$399$	0	0
$400$	0	0
$401$	−29.6228	−1.47929	−0.739645	−	0.672997i	$-0.765007\pi$
$401$	−29.6228	−1.47929	−0.739645	+	0.672997i	$0.765007\pi$
$402$	0	0
$403$	18.3509	0.914123
$404$	−1.16228	−0.0578255
$405$	0	0
$406$	−2.16228	−0.107312
$407$	10.6491	0.527857
$408$	0	0
$409$	31.2982	1.54760	0.773799	−	0.633432i	$-0.218354\pi$
$409$	31.2982	1.54760	0.773799	+	0.633432i	$0.218354\pi$
$410$	0	0
$411$	0	0
$412$	−4.48683	−0.221050
$413$	−7.00000	−0.344447
$414$	0	0
$415$	0	0
$416$	−2.16228	−0.106014
$417$	0	0
$418$	−0.973666	−0.0476236
$419$	−28.6228	−1.39831	−0.699157	−	0.714968i	$-0.746441\pi$
$419$	−28.6228	−1.39831	−0.699157	+	0.714968i	$0.746441\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−3.00000	−0.146038
$423$	0	0
$424$	12.4868	0.606414
$425$	0	0
$426$	0	0
$427$	8.32456	0.402853
$428$	−5.16228	−0.249528
$429$	0	0
$430$	0	0
$431$	19.2982	0.929563	0.464781	−	0.885426i	$-0.346133\pi$
$431$	19.2982	0.929563	0.464781	+	0.885426i	$0.346133\pi$
$432$	0	0
$433$	−13.2982	−0.639072	−0.319536	−	0.947574i	$-0.603527\pi$
$433$	−13.2982	−0.639072	−0.319536	+	0.947574i	$0.603527\pi$
$434$	8.48683	0.407381
$435$	0	0
$436$	3.16228	0.151446
$437$	−0.135944	−0.00650307
$438$	0	0
$439$	10.8114	0.515999	0.258000	−	0.966145i	$-0.416937\pi$
$439$	10.8114	0.515999	0.258000	+	0.966145i	$0.416937\pi$
$440$	0	0
$441$	0	0
$442$	10.8114	0.514245
$443$	9.48683	0.450733	0.225367	−	0.974274i	$-0.427642\pi$
$443$	9.48683	0.450733	0.225367	+	0.974274i	$0.427642\pi$
$444$	0	0
$445$	0	0
$446$	24.6491	1.16717
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−29.2982	−1.38267	−0.691334	−	0.722535i	$-0.742977\pi$
$449$	−29.2982	−1.38267	−0.691334	+	0.722535i	$0.742977\pi$
$450$	0	0
$451$	−2.70178	−0.127222
$452$	6.83772	0.321619
$453$	0	0
$454$	−0.675445	−0.0317002
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	18.3246	0.856250
$459$	0	0
$460$	0	0
$461$	35.2982	1.64400	0.822001	−	0.569486i	$-0.192858\pi$
$461$	35.2982	1.64400	0.822001	+	0.569486i	$0.192858\pi$
$462$	0	0
$463$	−17.1623	−0.797599	−0.398799	−	0.917038i	$-0.630573\pi$
$463$	−17.1623	−0.797599	−0.398799	+	0.917038i	$0.630573\pi$
$464$	2.16228	0.100381
$465$	0	0
$466$	−28.1359	−1.30337
$467$	14.6491	0.677880	0.338940	−	0.940808i	$-0.389932\pi$
$467$	14.6491	0.677880	0.338940	+	0.940808i	$0.389932\pi$
$468$	0	0
$469$	−15.3246	−0.707622
$470$	0	0
$471$	0	0
$472$	7.00000	0.322201
$473$	8.13594	0.374091
$474$	0	0
$475$	0	0
$476$	5.00000	0.229175
$477$	0	0
$478$	6.32456	0.289278
$479$	28.6491	1.30901	0.654506	−	0.756057i	$-0.272877\pi$
$479$	28.6491	1.30901	0.654506	+	0.756057i	$0.272877\pi$
$480$	0	0
$481$	−19.8114	−0.903322
$482$	−7.16228	−0.326233
$483$	0	0
$484$	−9.64911	−0.438596
$485$	0	0
$486$	0	0
$487$	−14.6491	−0.663815	−0.331907	−	0.943312i	$-0.607692\pi$
$487$	−14.6491	−0.663815	−0.331907	+	0.943312i	$0.607692\pi$
$488$	−8.32456	−0.376835
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−10.8114	−0.486920
$494$	1.81139	0.0814982
$495$	0	0
$496$	−8.48683	−0.381070
$497$	6.48683	0.290974
$498$	0	0
$499$	−28.3246	−1.26798	−0.633991	−	0.773341i	$-0.718584\pi$
$499$	−28.3246	−1.26798	−0.633991	+	0.773341i	$0.718584\pi$
$500$	0	0
$501$	0	0
$502$	−3.35089	−0.149558
$503$	−12.8377	−0.572406	−0.286203	−	0.958169i	$-0.592393\pi$
$503$	−12.8377	−0.572406	−0.286203	+	0.958169i	$0.592393\pi$
$504$	0	0
$505$	0	0
$506$	0.188612	0.00838481
$507$	0	0
$508$	5.67544	0.251807
$509$	17.1623	0.760705	0.380352	−	0.924842i	$-0.375803\pi$
$509$	17.1623	0.760705	0.380352	+	0.924842i	$0.375803\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	2.32456	0.102234
$518$	−9.16228	−0.402567
$519$	0	0
$520$	0	0
$521$	−0.675445	−0.0295918	−0.0147959	−	0.999891i	$-0.504710\pi$
$521$	−0.675445	−0.0295918	−0.0147959	+	0.999891i	$0.504710\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	−7.32456	−0.319975
$525$	0	0
$526$	−4.16228	−0.181484
$527$	42.4342	1.84846
$528$	0	0
$529$	−22.9737	−0.998855
$530$	0	0
$531$	0	0
$532$	0.837722	0.0363199
$533$	5.02633	0.217715
$534$	0	0
$535$	0	0
$536$	15.3246	0.661920
$537$	0	0
$538$	−22.3246	−0.962480
$539$	1.16228	0.0500628
$540$	0	0
$541$	15.1623	0.651877	0.325939	−	0.945391i	$-0.394320\pi$
$541$	15.1623	0.651877	0.325939	+	0.945391i	$0.394320\pi$
$542$	−16.1623	−0.694229
$543$	0	0
$544$	−5.00000	−0.214373
$545$	0	0
$546$	0	0
$547$	−0.324555	−0.0138770	−0.00693849	−	0.999976i	$-0.502209\pi$
$547$	−0.324555	−0.0138770	−0.00693849	+	0.999976i	$0.502209\pi$
$548$	−4.32456	−0.184736
$549$	0	0
$550$	0	0
$551$	−1.81139	−0.0771677
$552$	0	0
$553$	−6.83772	−0.290770
$554$	12.6491	0.537409
$555$	0	0
$556$	1.48683	0.0630558
$557$	4.81139	0.203865	0.101933	−	0.994791i	$-0.467497\pi$
$557$	4.81139	0.203865	0.101933	+	0.994791i	$0.467497\pi$
$558$	0	0
$559$	−15.1359	−0.640182
$560$	0	0
$561$	0	0
$562$	2.51317	0.106012
$563$	41.9737	1.76898	0.884490	−	0.466560i	$-0.154507\pi$
$563$	41.9737	1.76898	0.884490	+	0.466560i	$0.154507\pi$
$564$	0	0
$565$	0	0
$566$	21.8114	0.916801
$567$	0	0
$568$	−6.48683	−0.272181
$569$	25.1623	1.05486	0.527429	−	0.849599i	$-0.323156\pi$
$569$	25.1623	1.05486	0.527429	+	0.849599i	$0.323156\pi$
$570$	0	0
$571$	28.6228	1.19783	0.598913	−	0.800814i	$-0.295600\pi$
$571$	28.6228	1.19783	0.598913	+	0.800814i	$0.295600\pi$
$572$	−2.51317	−0.105081
$573$	0	0
$574$	2.32456	0.0970251
$575$	0	0
$576$	0	0
$577$	37.4868	1.56060	0.780299	−	0.625407i	$-0.215067\pi$
$577$	37.4868	1.56060	0.780299	+	0.625407i	$0.215067\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	0	0
$581$	−8.32456	−0.345361
$582$	0	0
$583$	14.5132	0.601074
$584$	10.0000	0.413803
$585$	0	0
$586$	24.9737	1.03165
$587$	−45.2719	−1.86857	−0.934285	−	0.356526i	$-0.883961\pi$
$587$	−45.2719	−1.86857	−0.934285	+	0.356526i	$0.883961\pi$
$588$	0	0
$589$	7.10961	0.292946
$590$	0	0
$591$	0	0
$592$	9.16228	0.376567
$593$	−18.9737	−0.779155	−0.389578	−	0.920994i	$-0.627379\pi$
$593$	−18.9737	−0.779155	−0.389578	+	0.920994i	$0.627379\pi$
$594$	0	0
$595$	0	0
$596$	20.8114	0.852468
$597$	0	0
$598$	−0.350889	−0.0143489
$599$	−30.4868	−1.24566	−0.622829	−	0.782358i	$-0.714017\pi$
$599$	−30.4868	−1.24566	−0.622829	+	0.782358i	$0.714017\pi$
$600$	0	0
$601$	1.02633	0.0418650	0.0209325	−	0.999781i	$-0.493336\pi$
$601$	1.02633	0.0418650	0.0209325	+	0.999781i	$0.493336\pi$
$602$	−7.00000	−0.285299
$603$	0	0
$604$	17.8114	0.724735
$605$	0	0
$606$	0	0
$607$	−36.1623	−1.46778	−0.733891	−	0.679267i	$-0.762298\pi$
$607$	−36.1623	−1.46778	−0.733891	+	0.679267i	$0.762298\pi$
$608$	−0.837722	−0.0339741
$609$	0	0
$610$	0	0
$611$	−4.32456	−0.174953
$612$	0	0
$613$	48.7851	1.97041	0.985205	−	0.171381i	$-0.0548228\pi$
$613$	48.7851	1.97041	0.985205	+	0.171381i	$0.0548228\pi$
$614$	11.1623	0.450473
$615$	0	0
$616$	−1.16228	−0.0468295
$617$	11.6754	0.470036	0.235018	−	0.971991i	$-0.424485\pi$
$617$	11.6754	0.470036	0.235018	+	0.971991i	$0.424485\pi$
$618$	0	0
$619$	−30.6491	−1.23189	−0.615946	−	0.787788i	$-0.711226\pi$
$619$	−30.6491	−1.23189	−0.615946	+	0.787788i	$0.711226\pi$
$620$	0	0
$621$	0	0
$622$	21.4868	0.861544
$623$	−15.3246	−0.613965
$624$	0	0
$625$	0	0
$626$	32.4605	1.29738
$627$	0	0
$628$	−1.83772	−0.0733331
$629$	−45.8114	−1.82662
$630$	0	0
$631$	9.67544	0.385173	0.192587	−	0.981280i	$-0.438312\pi$
$631$	9.67544	0.385173	0.192587	+	0.981280i	$0.438312\pi$
$632$	6.83772	0.271990
$633$	0	0
$634$	10.0000	0.397151
$635$	0	0
$636$	0	0
$637$	−2.16228	−0.0856726
$638$	2.51317	0.0994972
$639$	0	0
$640$	0	0
$641$	−31.4868	−1.24366	−0.621828	−	0.783154i	$-0.713610\pi$
$641$	−31.4868	−1.24366	−0.621828	+	0.783154i	$0.713610\pi$
$642$	0	0
$643$	32.7851	1.29292	0.646458	−	0.762949i	$-0.276249\pi$
$643$	32.7851	1.29292	0.646458	+	0.762949i	$0.276249\pi$
$644$	−0.162278	−0.00639464
$645$	0	0
$646$	4.18861	0.164799
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	8.13594	0.319364
$650$	0	0
$651$	0	0
$652$	13.6491	0.534540
$653$	−41.1359	−1.60977	−0.804887	−	0.593428i	$-0.797774\pi$
$653$	−41.1359	−1.60977	−0.804887	+	0.593428i	$0.797774\pi$
$654$	0	0
$655$	0	0
$656$	−2.32456	−0.0907586
$657$	0	0
$658$	−2.00000	−0.0779681
$659$	29.8114	1.16129	0.580643	−	0.814158i	$-0.302801\pi$
$659$	29.8114	1.16129	0.580643	+	0.814158i	$0.302801\pi$
$660$	0	0
$661$	0.324555	0.0126237	0.00631187	−	0.999980i	$-0.497991\pi$
$661$	0.324555	0.0126237	0.00631187	+	0.999980i	$0.497991\pi$
$662$	27.6491	1.07461
$663$	0	0
$664$	8.32456	0.323055
$665$	0	0
$666$	0	0
$667$	0.350889	0.0135865
$668$	2.51317	0.0972374
$669$	0	0
$670$	0	0
$671$	−9.67544	−0.373516
$672$	0	0
$673$	23.6491	0.911606	0.455803	−	0.890081i	$-0.349352\pi$
$673$	23.6491	0.911606	0.455803	+	0.890081i	$0.349352\pi$
$674$	−9.32456	−0.359168
$675$	0	0
$676$	−8.32456	−0.320175
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	0	0
$679$	−1.16228	−0.0446041
$680$	0	0
$681$	0	0
$682$	−9.86406	−0.377714
$683$	0.513167	0.0196358	0.00981790	−	0.999952i	$-0.496875\pi$
$683$	0.513167	0.0196358	0.00981790	+	0.999952i	$0.496875\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	7.00000	0.266872
$689$	−27.0000	−1.02862
$690$	0	0
$691$	10.9737	0.417458	0.208729	−	0.977974i	$-0.433067\pi$
$691$	10.9737	0.417458	0.208729	+	0.977974i	$0.433067\pi$
$692$	−11.8114	−0.449002
$693$	0	0
$694$	−18.0000	−0.683271
$695$	0	0
$696$	0	0
$697$	11.6228	0.440244
$698$	1.83772	0.0695588
$699$	0	0
$700$	0	0
$701$	−30.6491	−1.15760	−0.578800	−	0.815469i	$-0.696479\pi$
$701$	−30.6491	−1.15760	−0.578800	+	0.815469i	$0.696479\pi$
$702$	0	0
$703$	−7.67544	−0.289485
$704$	1.16228	0.0438050
$705$	0	0
$706$	−5.32456	−0.200392
$707$	1.16228	0.0437120
$708$	0	0
$709$	45.4868	1.70829	0.854147	−	0.520032i	$-0.174080\pi$
$709$	45.4868	1.70829	0.854147	+	0.520032i	$0.174080\pi$
$710$	0	0
$711$	0	0
$712$	15.3246	0.574312
$713$	−1.37722	−0.0515774
$714$	0	0
$715$	0	0
$716$	−18.3246	−0.684821
$717$	0	0
$718$	−14.1623	−0.528532
$719$	−39.1623	−1.46051	−0.730253	−	0.683177i	$-0.760598\pi$
$719$	−39.1623	−1.46051	−0.730253	+	0.683177i	$0.760598\pi$
$720$	0	0
$721$	4.48683	0.167098
$722$	−18.2982	−0.680989
$723$	0	0
$724$	14.4868	0.538399
$725$	0	0
$726$	0	0
$727$	−2.16228	−0.0801944	−0.0400972	−	0.999196i	$-0.512767\pi$
$727$	−2.16228	−0.0801944	−0.0400972	+	0.999196i	$0.512767\pi$
$728$	2.16228	0.0801393
$729$	0	0
$730$	0	0
$731$	−35.0000	−1.29452
$732$	0	0
$733$	−5.18861	−0.191646	−0.0958229	−	0.995398i	$-0.530548\pi$
$733$	−5.18861	−0.191646	−0.0958229	+	0.995398i	$0.530548\pi$
$734$	−0.811388	−0.0299489
$735$	0	0
$736$	0.162278	0.00598163
$737$	17.8114	0.656091
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	0	0
$742$	−12.4868	−0.458406
$743$	−12.4868	−0.458097	−0.229049	−	0.973415i	$-0.573562\pi$
$743$	−12.4868	−0.458097	−0.229049	+	0.973415i	$0.573562\pi$
$744$	0	0
$745$	0	0
$746$	−1.81139	−0.0663196
$747$	0	0
$748$	−5.81139	−0.212485
$749$	5.16228	0.188626
$750$	0	0
$751$	40.7851	1.48827	0.744134	−	0.668031i	$-0.232862\pi$
$751$	40.7851	1.48827	0.744134	+	0.668031i	$0.232862\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	−4.67544	−0.170270
$755$	0	0
$756$	0	0
$757$	7.35089	0.267173	0.133586	−	0.991037i	$-0.457351\pi$
$757$	7.35089	0.267173	0.133586	+	0.991037i	$0.457351\pi$
$758$	5.35089	0.194353
$759$	0	0
$760$	0	0
$761$	−23.6491	−0.857280	−0.428640	−	0.903475i	$-0.641007\pi$
$761$	−23.6491	−0.857280	−0.428640	+	0.903475i	$0.641007\pi$
$762$	0	0
$763$	−3.16228	−0.114482
$764$	−13.6754	−0.494760
$765$	0	0
$766$	−26.3246	−0.951145
$767$	−15.1359	−0.546527
$768$	0	0
$769$	−1.81139	−0.0653203	−0.0326602	−	0.999467i	$-0.510398\pi$
$769$	−1.81139	−0.0653203	−0.0326602	+	0.999467i	$0.510398\pi$
$770$	0	0
$771$	0	0
$772$	−17.9737	−0.646886
$773$	5.02633	0.180785	0.0903923	−	0.995906i	$-0.471188\pi$
$773$	5.02633	0.180785	0.0903923	+	0.995906i	$0.471188\pi$
$774$	0	0
$775$	0	0
$776$	1.16228	0.0417233
$777$	0	0
$778$	21.6754	0.777102
$779$	1.94733	0.0697704
$780$	0	0
$781$	−7.53950	−0.269785
$782$	−0.811388	−0.0290152
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−53.6228	−1.91145	−0.955723	−	0.294269i	$-0.904924\pi$
$787$	−53.6228	−1.91145	−0.955723	+	0.294269i	$0.904924\pi$
$788$	18.3246	0.652785
$789$	0	0
$790$	0	0
$791$	−6.83772	−0.243121
$792$	0	0
$793$	18.0000	0.639199
$794$	5.67544	0.201414
$795$	0	0
$796$	−5.51317	−0.195409
$797$	−20.9737	−0.742925	−0.371463	−	0.928448i	$-0.621144\pi$
$797$	−20.9737	−0.742925	−0.371463	+	0.928448i	$0.621144\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	0	0
$802$	−29.6228	−1.04602
$803$	11.6228	0.410159
$804$	0	0
$805$	0	0
$806$	18.3509	0.646383
$807$	0	0
$808$	−1.16228	−0.0408888
$809$	−20.6491	−0.725984	−0.362992	−	0.931792i	$-0.618245\pi$
$809$	−20.6491	−0.725984	−0.362992	+	0.931792i	$0.618245\pi$
$810$	0	0
$811$	39.2982	1.37995	0.689974	−	0.723835i	$-0.257622\pi$
$811$	39.2982	1.37995	0.689974	+	0.723835i	$0.257622\pi$
$812$	−2.16228	−0.0758811
$813$	0	0
$814$	10.6491	0.373251
$815$	0	0
$816$	0	0
$817$	−5.86406	−0.205157
$818$	31.2982	1.09432
$819$	0	0
$820$	0	0
$821$	38.8114	1.35453	0.677263	−	0.735741i	$-0.263166\pi$
$821$	38.8114	1.35453	0.677263	+	0.735741i	$0.263166\pi$
$822$	0	0
$823$	−15.4868	−0.539837	−0.269919	−	0.962883i	$-0.586997\pi$
$823$	−15.4868	−0.539837	−0.269919	+	0.962883i	$0.586997\pi$
$824$	−4.48683	−0.156306
$825$	0	0
$826$	−7.00000	−0.243561
$827$	11.6754	0.405995	0.202997	−	0.979179i	$-0.434932\pi$
$827$	11.6754	0.405995	0.202997	+	0.979179i	$0.434932\pi$
$828$	0	0
$829$	−54.2719	−1.88494	−0.942470	−	0.334290i	$-0.891504\pi$
$829$	−54.2719	−1.88494	−0.942470	+	0.334290i	$0.891504\pi$
$830$	0	0
$831$	0	0
$832$	−2.16228	−0.0749635
$833$	−5.00000	−0.173240
$834$	0	0
$835$	0	0
$836$	−0.973666	−0.0336749
$837$	0	0
$838$	−28.6228	−0.988757
$839$	−23.4868	−0.810856	−0.405428	−	0.914127i	$-0.632877\pi$
$839$	−23.4868	−0.810856	−0.405428	+	0.914127i	$0.632877\pi$
$840$	0	0
$841$	−24.3246	−0.838778
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	−3.00000	−0.103264
$845$	0	0
$846$	0	0
$847$	9.64911	0.331547
$848$	12.4868	0.428800
$849$	0	0
$850$	0	0
$851$	1.48683	0.0509680
$852$	0	0
$853$	−0.162278	−0.00555628	−0.00277814	−	0.999996i	$-0.500884\pi$
$853$	−0.162278	−0.00555628	−0.00277814	+	0.999996i	$0.500884\pi$
$854$	8.32456	0.284860
$855$	0	0
$856$	−5.16228	−0.176443
$857$	25.3246	0.865070	0.432535	−	0.901617i	$-0.357619\pi$
$857$	25.3246	0.865070	0.432535	+	0.901617i	$0.357619\pi$
$858$	0	0
$859$	−30.7851	−1.05037	−0.525186	−	0.850987i	$-0.676004\pi$
$859$	−30.7851	−1.05037	−0.525186	+	0.850987i	$0.676004\pi$
$860$	0	0
$861$	0	0
$862$	19.2982	0.657300
$863$	8.16228	0.277847	0.138924	−	0.990303i	$-0.455636\pi$
$863$	8.16228	0.277847	0.138924	+	0.990303i	$0.455636\pi$
$864$	0	0
$865$	0	0
$866$	−13.2982	−0.451892
$867$	0	0
$868$	8.48683	0.288062
$869$	7.94733	0.269595
$870$	0	0
$871$	−33.1359	−1.12277
$872$	3.16228	0.107088
$873$	0	0
$874$	−0.135944	−0.00459836
$875$	0	0
$876$	0	0
$877$	−10.9737	−0.370554	−0.185277	−	0.982686i	$-0.559318\pi$
$877$	−10.9737	−0.370554	−0.185277	+	0.982686i	$0.559318\pi$
$878$	10.8114	0.364867
$879$	0	0
$880$	0	0
$881$	34.9473	1.17741	0.588703	−	0.808350i	$-0.299639\pi$
$881$	34.9473	1.17741	0.588703	+	0.808350i	$0.299639\pi$
$882$	0	0
$883$	56.2982	1.89459	0.947293	−	0.320369i	$-0.103807\pi$
$883$	56.2982	1.89459	0.947293	+	0.320369i	$0.103807\pi$
$884$	10.8114	0.363626
$885$	0	0
$886$	9.48683	0.318716
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	−5.67544	−0.190348
$890$	0	0
$891$	0	0
$892$	24.6491	0.825313
$893$	−1.67544	−0.0560666
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−29.2982	−0.977694
$899$	−18.3509	−0.612036
$900$	0	0
$901$	−62.4342	−2.07998
$902$	−2.70178	−0.0899594
$903$	0	0
$904$	6.83772	0.227419
$905$	0	0
$906$	0	0
$907$	−29.2982	−0.972831	−0.486416	−	0.873727i	$-0.661696\pi$
$907$	−29.2982	−0.972831	−0.486416	+	0.873727i	$0.661696\pi$
$908$	−0.675445	−0.0224154
$909$	0	0
$910$	0	0
$911$	−41.2982	−1.36827	−0.684136	−	0.729355i	$-0.739820\pi$
$911$	−41.2982	−1.36827	−0.684136	+	0.729355i	$0.739820\pi$
$912$	0	0
$913$	9.67544	0.320210
$914$	1.00000	0.0330771
$915$	0	0
$916$	18.3246	0.605460
$917$	7.32456	0.241878
$918$	0	0
$919$	−14.6491	−0.483230	−0.241615	−	0.970372i	$-0.577677\pi$
$919$	−14.6491	−0.483230	−0.241615	+	0.970372i	$0.577677\pi$
$920$	0	0
$921$	0	0
$922$	35.2982	1.16249
$923$	14.0263	0.461682
$924$	0	0
$925$	0	0
$926$	−17.1623	−0.563987
$927$	0	0
$928$	2.16228	0.0709802
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−0.837722	−0.0274552
$932$	−28.1359	−0.921623
$933$	0	0
$934$	14.6491	0.479334
$935$	0	0
$936$	0	0
$937$	35.4868	1.15930	0.579652	−	0.814864i	$-0.303188\pi$
$937$	35.4868	1.15930	0.579652	+	0.814864i	$0.303188\pi$
$938$	−15.3246	−0.500364
$939$	0	0
$940$	0	0
$941$	−42.6491	−1.39032	−0.695161	−	0.718854i	$-0.744667\pi$
$941$	−42.6491	−1.39032	−0.695161	+	0.718854i	$0.744667\pi$
$942$	0	0
$943$	−0.377223	−0.0122841
$944$	7.00000	0.227831
$945$	0	0
$946$	8.13594	0.264522
$947$	−33.4868	−1.08818	−0.544088	−	0.839028i	$-0.683124\pi$
$947$	−33.4868	−1.08818	−0.544088	+	0.839028i	$0.683124\pi$
$948$	0	0
$949$	−21.6228	−0.701905
$950$	0	0
$951$	0	0
$952$	5.00000	0.162051
$953$	35.6228	1.15393	0.576967	−	0.816767i	$-0.304236\pi$
$953$	35.6228	1.15393	0.576967	+	0.816767i	$0.304236\pi$
$954$	0	0
$955$	0	0
$956$	6.32456	0.204551
$957$	0	0
$958$	28.6491	0.925611
$959$	4.32456	0.139647
$960$	0	0
$961$	41.0263	1.32343
$962$	−19.8114	−0.638745
$963$	0	0
$964$	−7.16228	−0.230681
$965$	0	0
$966$	0	0
$967$	−19.6228	−0.631026	−0.315513	−	0.948921i	$-0.602177\pi$
$967$	−19.6228	−0.631026	−0.315513	+	0.948921i	$0.602177\pi$
$968$	−9.64911	−0.310134
$969$	0	0
$970$	0	0
$971$	−34.9473	−1.12151	−0.560757	−	0.827981i	$-0.689490\pi$
$971$	−34.9473	−1.12151	−0.560757	+	0.827981i	$0.689490\pi$
$972$	0	0
$973$	−1.48683	−0.0476657
$974$	−14.6491	−0.469388
$975$	0	0
$976$	−8.32456	−0.266463
$977$	−43.9473	−1.40600	−0.703000	−	0.711190i	$-0.748157\pi$
$977$	−43.9473	−1.40600	−0.703000	+	0.711190i	$0.748157\pi$
$978$	0	0
$979$	17.8114	0.569254
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	42.3246	1.34994	0.674972	−	0.737843i	$-0.264156\pi$
$983$	42.3246	1.34994	0.674972	+	0.737843i	$0.264156\pi$
$984$	0	0
$985$	0	0
$986$	−10.8114	−0.344305
$987$	0	0
$988$	1.81139	0.0576279
$989$	1.13594	0.0361209
$990$	0	0
$991$	−15.9473	−0.506584	−0.253292	−	0.967390i	$-0.581513\pi$
$991$	−15.9473	−0.506584	−0.253292	+	0.967390i	$0.581513\pi$
$992$	−8.48683	−0.269457
$993$	0	0
$994$	6.48683	0.205750
$995$	0	0
$996$	0	0
$997$	9.83772	0.311564	0.155782	−	0.987791i	$-0.450210\pi$
$997$	9.83772	0.311564	0.155782	+	0.987791i	$0.450210\pi$
$998$	−28.3246	−0.896598
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.en.1.2		2
3.2	odd	2		9450.2.a.ef.1.1		2
5.2	odd	4		1890.2.g.o.379.3	yes	4
5.3	odd	4		1890.2.g.o.379.1	✓	4
5.4	even	2		9450.2.a.eg.1.2		2
15.2	even	4		1890.2.g.p.379.2	yes	4
15.8	even	4		1890.2.g.p.379.4	yes	4
15.14	odd	2		9450.2.a.ew.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.g.o.379.1	✓	4	5.3	odd	4
1890.2.g.o.379.3	yes	4	5.2	odd	4
1890.2.g.p.379.2	yes	4	15.2	even	4
1890.2.g.p.379.4	yes	4	15.8	even	4
9450.2.a.ef.1.1		2	3.2	odd	2
9450.2.a.eg.1.2		2	5.4	even	2
9450.2.a.en.1.2		2	1.1	even	1	trivial
9450.2.a.ew.1.1		2	15.14	odd	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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +1.16228 q^{11} -2.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -0.837722 q^{19} +1.16228 q^{22} +0.162278 q^{23} -2.16228 q^{26} -1.00000 q^{28} +2.16228 q^{29} -8.48683 q^{31} +1.00000 q^{32} -5.00000 q^{34} +9.16228 q^{37} -0.837722 q^{38} -2.32456 q^{41} +7.00000 q^{43} +1.16228 q^{44} +0.162278 q^{46} +2.00000 q^{47} +1.00000 q^{49} -2.16228 q^{52} +12.4868 q^{53} -1.00000 q^{56} +2.16228 q^{58} +7.00000 q^{59} -8.32456 q^{61} -8.48683 q^{62} +1.00000 q^{64} +15.3246 q^{67} -5.00000 q^{68} -6.48683 q^{71} +10.0000 q^{73} +9.16228 q^{74} -0.837722 q^{76} -1.16228 q^{77} +6.83772 q^{79} -2.32456 q^{82} +8.32456 q^{83} +7.00000 q^{86} +1.16228 q^{88} +15.3246 q^{89} +2.16228 q^{91} +0.162278 q^{92} +2.00000 q^{94} +1.16228 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 10 q^{17} - 8 q^{19} - 4 q^{22} - 6 q^{23} + 2 q^{26} - 2 q^{28} - 2 q^{29} + 2 q^{31} + 2 q^{32} - 10 q^{34} + 12 q^{37} - 8 q^{38} + 8 q^{41} + 14 q^{43} - 4 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} - 2 q^{56} - 2 q^{58} + 14 q^{59} - 4 q^{61} + 2 q^{62} + 2 q^{64} + 18 q^{67} - 10 q^{68} + 6 q^{71} + 20 q^{73} + 12 q^{74} - 8 q^{76} + 4 q^{77} + 20 q^{79} + 8 q^{82} + 4 q^{83} + 14 q^{86} - 4 q^{88} + 18 q^{89} - 2 q^{91} - 6 q^{92} + 4 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 10 * q^17 - 8 * q^19 - 4 * q^22 - 6 * q^23 + 2 * q^26 - 2 * q^28 - 2 * q^29 + 2 * q^31 + 2 * q^32 - 10 * q^34 + 12 * q^37 - 8 * q^38 + 8 * q^41 + 14 * q^43 - 4 * q^44 - 6 * q^46 + 4 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 - 2 * q^56 - 2 * q^58 + 14 * q^59 - 4 * q^61 + 2 * q^62 + 2 * q^64 + 18 * q^67 - 10 * q^68 + 6 * q^71 + 20 * q^73 + 12 * q^74 - 8 * q^76 + 4 * q^77 + 20 * q^79 + 8 * q^82 + 4 * q^83 + 14 * q^86 - 4 * q^88 + 18 * q^89 - 2 * q^91 - 6 * q^92 + 4 * q^94 - 4 * q^97 + 2 * q^98

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