LMFDB - Embedded newform 9306.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9306,2,Mod(1,9306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9306, 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("9306.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9306.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:	$74.3087841210$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2949696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 16x^{2} + 10x - 16 $ x^5 - 2x^4 - 10x^3 + 16x^2 + 10x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3102)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32427$ of defining polynomial
Character	$\chi$	$=$	9306.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.32427 q^{5} -0.0838763 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.32427 q^{5} -0.0838763 q^{7} -1.00000 q^{8} +1.32427 q^{10} -1.00000 q^{11} -1.48671 q^{13} +0.0838763 q^{14} +1.00000 q^{16} -6.36003 q^{17} -4.66870 q^{19} -1.32427 q^{20} +1.00000 q^{22} -7.20352 q^{23} -3.24632 q^{25} +1.48671 q^{26} -0.0838763 q^{28} -5.19820 q^{29} +2.00000 q^{31} -1.00000 q^{32} +6.36003 q^{34} +0.111075 q^{35} -6.59013 q^{37} +4.66870 q^{38} +1.32427 q^{40} -0.194951 q^{41} +5.77916 q^{43} -1.00000 q^{44} +7.20352 q^{46} -1.00000 q^{47} -6.99296 q^{49} +3.24632 q^{50} -1.48671 q^{52} +2.19228 q^{53} +1.32427 q^{55} +0.0838763 q^{56} +5.19820 q^{58} -10.4114 q^{59} +6.47111 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.96880 q^{65} +7.28208 q^{67} -6.36003 q^{68} -0.111075 q^{70} -11.8832 q^{71} -2.03515 q^{73} +6.59013 q^{74} -4.66870 q^{76} +0.0838763 q^{77} -3.93629 q^{79} -1.32427 q^{80} +0.194951 q^{82} -2.95456 q^{83} +8.42238 q^{85} -5.77916 q^{86} +1.00000 q^{88} +4.33019 q^{89} +0.124699 q^{91} -7.20352 q^{92} +1.00000 q^{94} +6.18260 q^{95} +10.6389 q^{97} +6.99296 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * 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
$5$	−1.32427	−0.592230	−0.296115	−	0.955152i	$-0.595691\pi$
$5$	−1.32427	−0.592230	−0.296115	+	0.955152i	$0.595691\pi$
$6$	0	0
$7$	−0.0838763	−0.0317023	−0.0158511	−	0.999874i	$-0.505046\pi$
$7$	−0.0838763	−0.0317023	−0.0158511	+	0.999874i	$0.505046\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.32427	0.418770
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.48671	−0.412338	−0.206169	−	0.978516i	$-0.566100\pi$
$13$	−1.48671	−0.412338	−0.206169	+	0.978516i	$0.566100\pi$
$14$	0.0838763	0.0224169
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.36003	−1.54253	−0.771267	−	0.636512i	$-0.780377\pi$
$17$	−6.36003	−1.54253	−0.771267	+	0.636512i	$0.780377\pi$
$18$	0	0
$19$	−4.66870	−1.07107	−0.535536	−	0.844512i	$-0.679891\pi$
$19$	−4.66870	−1.07107	−0.535536	+	0.844512i	$0.679891\pi$
$20$	−1.32427	−0.296115
$21$	0	0
$22$	1.00000	0.213201
$23$	−7.20352	−1.50204	−0.751019	−	0.660281i	$-0.770437\pi$
$23$	−7.20352	−1.50204	−0.751019	+	0.660281i	$0.770437\pi$
$24$	0	0
$25$	−3.24632	−0.649263
$26$	1.48671	0.291567
$27$	0	0
$28$	−0.0838763	−0.0158511
$29$	−5.19820	−0.965282	−0.482641	−	0.875818i	$-0.660322\pi$
$29$	−5.19820	−0.965282	−0.482641	+	0.875818i	$0.660322\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.36003	1.09074
$35$	0.111075	0.0187750
$36$	0	0
$37$	−6.59013	−1.08341	−0.541706	−	0.840568i	$-0.682221\pi$
$37$	−6.59013	−1.08341	−0.541706	+	0.840568i	$0.682221\pi$
$38$	4.66870	0.757363
$39$	0	0
$40$	1.32427	0.209385
$41$	−0.194951	−0.0304462	−0.0152231	−	0.999884i	$-0.504846\pi$
$41$	−0.194951	−0.0304462	−0.0152231	+	0.999884i	$0.504846\pi$
$42$	0	0
$43$	5.77916	0.881314	0.440657	−	0.897676i	$-0.354746\pi$
$43$	5.77916	0.881314	0.440657	+	0.897676i	$0.354746\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	7.20352	1.06210
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.99296	−0.998995
$50$	3.24632	0.459098
$51$	0	0
$52$	−1.48671	−0.206169
$53$	2.19228	0.301133	0.150566	−	0.988600i	$-0.451890\pi$
$53$	2.19228	0.301133	0.150566	+	0.988600i	$0.451890\pi$
$54$	0	0
$55$	1.32427	0.178564
$56$	0.0838763	0.0112084
$57$	0	0
$58$	5.19820	0.682558
$59$	−10.4114	−1.35545	−0.677724	−	0.735316i	$-0.737034\pi$
$59$	−10.4114	−1.35545	−0.677724	+	0.735316i	$0.737034\pi$
$60$	0	0
$61$	6.47111	0.828540	0.414270	−	0.910154i	$-0.364037\pi$
$61$	6.47111	0.828540	0.414270	+	0.910154i	$0.364037\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	1.96880	0.244199
$66$	0	0
$67$	7.28208	0.889648	0.444824	−	0.895618i	$-0.353266\pi$
$67$	7.28208	0.889648	0.444824	+	0.895618i	$0.353266\pi$
$68$	−6.36003	−0.771267
$69$	0	0
$70$	−0.111075	−0.0132760
$71$	−11.8832	−1.41028	−0.705138	−	0.709070i	$-0.749115\pi$
$71$	−11.8832	−1.41028	−0.705138	+	0.709070i	$0.749115\pi$
$72$	0	0
$73$	−2.03515	−0.238196	−0.119098	−	0.992882i	$-0.538000\pi$
$73$	−2.03515	−0.238196	−0.119098	+	0.992882i	$0.538000\pi$
$74$	6.59013	0.766087
$75$	0	0
$76$	−4.66870	−0.535536
$77$	0.0838763	0.00955859
$78$	0	0
$79$	−3.93629	−0.442867	−0.221433	−	0.975175i	$-0.571074\pi$
$79$	−3.93629	−0.442867	−0.221433	+	0.975175i	$0.571074\pi$
$80$	−1.32427	−0.148058
$81$	0	0
$82$	0.194951	0.0215287
$83$	−2.95456	−0.324305	−0.162153	−	0.986766i	$-0.551844\pi$
$83$	−2.95456	−0.324305	−0.162153	+	0.986766i	$0.551844\pi$
$84$	0	0
$85$	8.42238	0.913536
$86$	−5.77916	−0.623183
$87$	0	0
$88$	1.00000	0.106600
$89$	4.33019	0.458999	0.229500	−	0.973309i	$-0.426291\pi$
$89$	4.33019	0.458999	0.229500	+	0.973309i	$0.426291\pi$
$90$	0	0
$91$	0.124699	0.0130721
$92$	−7.20352	−0.751019
$93$	0	0
$94$	1.00000	0.103142
$95$	6.18260	0.634322
$96$	0	0
$97$	10.6389	1.08021	0.540106	−	0.841597i	$-0.318384\pi$
$97$	10.6389	1.08021	0.540106	+	0.841597i	$0.318384\pi$
$98$	6.99296	0.706396
$99$	0	0
$100$	−3.24632	−0.324632
$101$	11.8766	1.18176	0.590882	−	0.806758i	$-0.298780\pi$
$101$	11.8766	1.18176	0.590882	+	0.806758i	$0.298780\pi$
$102$	0	0
$103$	15.8669	1.56341	0.781706	−	0.623647i	$-0.214350\pi$
$103$	15.8669	1.56341	0.781706	+	0.623647i	$0.214350\pi$
$104$	1.48671	0.145784
$105$	0	0
$106$	−2.19228	−0.212933
$107$	6.75121	0.652664	0.326332	−	0.945255i	$-0.394187\pi$
$107$	6.75121	0.652664	0.326332	+	0.945255i	$0.394187\pi$
$108$	0	0
$109$	7.64915	0.732655	0.366328	−	0.930486i	$-0.380615\pi$
$109$	7.64915	0.732655	0.366328	+	0.930486i	$0.380615\pi$
$110$	−1.32427	−0.126264
$111$	0	0
$112$	−0.0838763	−0.00792556
$113$	13.6149	1.28078	0.640392	−	0.768049i	$-0.278772\pi$
$113$	13.6149	1.28078	0.640392	+	0.768049i	$0.278772\pi$
$114$	0	0
$115$	9.53938	0.889552
$116$	−5.19820	−0.482641
$117$	0	0
$118$	10.4114	0.958447
$119$	0.533456	0.0489018
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.47111	−0.585866
$123$	0	0
$124$	2.00000	0.179605
$125$	10.9203	0.976744
$126$	0	0
$127$	−14.4786	−1.28476	−0.642382	−	0.766384i	$-0.722054\pi$
$127$	−14.4786	−1.28476	−0.642382	+	0.766384i	$0.722054\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.96880	−0.172675
$131$	−15.2860	−1.33555	−0.667773	−	0.744365i	$-0.732752\pi$
$131$	−15.2860	−1.33555	−0.667773	+	0.744365i	$0.732752\pi$
$132$	0	0
$133$	0.391593	0.0339554
$134$	−7.28208	−0.629076
$135$	0	0
$136$	6.36003	0.545368
$137$	9.31587	0.795908	0.397954	−	0.917405i	$-0.369720\pi$
$137$	9.31587	0.795908	0.397954	+	0.917405i	$0.369720\pi$
$138$	0	0
$139$	22.7604	1.93051	0.965256	−	0.261308i	$-0.0841538\pi$
$139$	22.7604	1.93051	0.965256	+	0.261308i	$0.0841538\pi$
$140$	0.111075	0.00938752
$141$	0	0
$142$	11.8832	0.997216
$143$	1.48671	0.124325
$144$	0	0
$145$	6.88381	0.571669
$146$	2.03515	0.168430
$147$	0	0
$148$	−6.59013	−0.541706
$149$	−8.81356	−0.722035	−0.361017	−	0.932559i	$-0.617571\pi$
$149$	−8.81356	−0.722035	−0.361017	+	0.932559i	$0.617571\pi$
$150$	0	0
$151$	−9.25088	−0.752826	−0.376413	−	0.926452i	$-0.622843\pi$
$151$	−9.25088	−0.752826	−0.376413	+	0.926452i	$0.622843\pi$
$152$	4.66870	0.378681
$153$	0	0
$154$	−0.0838763	−0.00675894
$155$	−2.64853	−0.212735
$156$	0	0
$157$	−14.9216	−1.19087	−0.595437	−	0.803402i	$-0.703021\pi$
$157$	−14.9216	−1.19087	−0.595437	+	0.803402i	$0.703021\pi$
$158$	3.93629	0.313154
$159$	0	0
$160$	1.32427	0.104693
$161$	0.604204	0.0476180
$162$	0	0
$163$	1.21492	0.0951595	0.0475798	−	0.998867i	$-0.484849\pi$
$163$	1.21492	0.0951595	0.0475798	+	0.998867i	$0.484849\pi$
$164$	−0.194951	−0.0152231
$165$	0	0
$166$	2.95456	0.229318
$167$	17.9346	1.38782	0.693909	−	0.720063i	$-0.255887\pi$
$167$	17.9346	1.38782	0.693909	+	0.720063i	$0.255887\pi$
$168$	0	0
$169$	−10.7897	−0.829977
$170$	−8.42238	−0.645967
$171$	0	0
$172$	5.77916	0.440657
$173$	−25.0582	−1.90514	−0.952568	−	0.304326i	$-0.901569\pi$
$173$	−25.0582	−1.90514	−0.952568	+	0.304326i	$0.901569\pi$
$174$	0	0
$175$	0.272289	0.0205831
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−4.33019	−0.324562
$179$	−23.4497	−1.75272	−0.876358	−	0.481660i	$-0.840034\pi$
$179$	−23.4497	−1.75272	−0.876358	+	0.481660i	$0.840034\pi$
$180$	0	0
$181$	17.5609	1.30529	0.652645	−	0.757664i	$-0.273659\pi$
$181$	17.5609	1.30529	0.652645	+	0.757664i	$0.273659\pi$
$182$	−0.124699	−0.00924334
$183$	0	0
$184$	7.20352	0.531050
$185$	8.72710	0.641629
$186$	0	0
$187$	6.36003	0.465092
$188$	−1.00000	−0.0729325
$189$	0	0
$190$	−6.18260	−0.448533
$191$	8.71075	0.630287	0.315144	−	0.949044i	$-0.397947\pi$
$191$	8.71075	0.630287	0.315144	+	0.949044i	$0.397947\pi$
$192$	0	0
$193$	−4.44922	−0.320262	−0.160131	−	0.987096i	$-0.551192\pi$
$193$	−4.44922	−0.320262	−0.160131	+	0.987096i	$0.551192\pi$
$194$	−10.6389	−0.763826
$195$	0	0
$196$	−6.99296	−0.499497
$197$	−11.1087	−0.791459	−0.395729	−	0.918367i	$-0.629508\pi$
$197$	−11.1087	−0.791459	−0.395729	+	0.918367i	$0.629508\pi$
$198$	0	0
$199$	−4.89613	−0.347077	−0.173539	−	0.984827i	$-0.555520\pi$
$199$	−4.89613	−0.347077	−0.173539	+	0.984827i	$0.555520\pi$
$200$	3.24632	0.229549
$201$	0	0
$202$	−11.8766	−0.835633
$203$	0.436006	0.0306016
$204$	0	0
$205$	0.258167	0.0180312
$206$	−15.8669	−1.10550
$207$	0	0
$208$	−1.48671	−0.103085
$209$	4.66870	0.322941
$210$	0	0
$211$	−4.84649	−0.333646	−0.166823	−	0.985987i	$-0.553351\pi$
$211$	−4.84649	−0.333646	−0.166823	+	0.985987i	$0.553351\pi$
$212$	2.19228	0.150566
$213$	0	0
$214$	−6.75121	−0.461503
$215$	−7.65315	−0.521941
$216$	0	0
$217$	−0.167753	−0.0113878
$218$	−7.64915	−0.518066
$219$	0	0
$220$	1.32427	0.0892821
$221$	9.45550	0.636046
$222$	0	0
$223$	13.2452	0.886965	0.443482	−	0.896283i	$-0.353743\pi$
$223$	13.2452	0.886965	0.443482	+	0.896283i	$0.353743\pi$
$224$	0.0838763	0.00560422
$225$	0	0
$226$	−13.6149	−0.905651
$227$	−27.0217	−1.79349	−0.896746	−	0.442545i	$-0.854076\pi$
$227$	−27.0217	−1.79349	−0.896746	+	0.442545i	$0.854076\pi$
$228$	0	0
$229$	5.16483	0.341301	0.170651	−	0.985332i	$-0.445413\pi$
$229$	5.16483	0.341301	0.170651	+	0.985332i	$0.445413\pi$
$230$	−9.53938	−0.629008
$231$	0	0
$232$	5.19820	0.341279
$233$	7.18903	0.470969	0.235484	−	0.971878i	$-0.424332\pi$
$233$	7.18903	0.470969	0.235484	+	0.971878i	$0.424332\pi$
$234$	0	0
$235$	1.32427	0.0863857
$236$	−10.4114	−0.677724
$237$	0	0
$238$	−0.533456	−0.0345788
$239$	−3.26450	−0.211163	−0.105582	−	0.994411i	$-0.533670\pi$
$239$	−3.26450	−0.211163	−0.105582	+	0.994411i	$0.533670\pi$
$240$	0	0
$241$	27.2558	1.75570	0.877850	−	0.478935i	$-0.158977\pi$
$241$	27.2558	1.75570	0.877850	+	0.478935i	$0.158977\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	6.47111	0.414270
$245$	9.26055	0.591635
$246$	0	0
$247$	6.94099	0.441644
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−10.9203	−0.690662
$251$	19.7027	1.24363	0.621813	−	0.783166i	$-0.286396\pi$
$251$	19.7027	1.24363	0.621813	+	0.783166i	$0.286396\pi$
$252$	0	0
$253$	7.20352	0.452881
$254$	14.4786	0.908466
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.1457	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$257$	−18.1457	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$258$	0	0
$259$	0.552756	0.0343466
$260$	1.96880	0.122100
$261$	0	0
$262$	15.2860	0.944374
$263$	−4.79217	−0.295498	−0.147749	−	0.989025i	$-0.547203\pi$
$263$	−4.79217	−0.295498	−0.147749	+	0.989025i	$0.547203\pi$
$264$	0	0
$265$	−2.90316	−0.178340
$266$	−0.391593	−0.0240101
$267$	0	0
$268$	7.28208	0.444824
$269$	21.4345	1.30688	0.653442	−	0.756977i	$-0.273324\pi$
$269$	21.4345	1.30688	0.653442	+	0.756977i	$0.273324\pi$
$270$	0	0
$271$	−6.58482	−0.399999	−0.200000	−	0.979796i	$-0.564094\pi$
$271$	−6.58482	−0.399999	−0.200000	+	0.979796i	$0.564094\pi$
$272$	−6.36003	−0.385634
$273$	0	0
$274$	−9.31587	−0.562792
$275$	3.24632	0.195760
$276$	0	0
$277$	−14.5291	−0.872969	−0.436484	−	0.899712i	$-0.643777\pi$
$277$	−14.5291	−0.872969	−0.436484	+	0.899712i	$0.643777\pi$
$278$	−22.7604	−1.36508
$279$	0	0
$280$	−0.111075	−0.00663798
$281$	26.5979	1.58670	0.793350	−	0.608765i	$-0.208335\pi$
$281$	26.5979	1.58670	0.793350	+	0.608765i	$0.208335\pi$
$282$	0	0
$283$	−17.1670	−1.02047	−0.510235	−	0.860035i	$-0.670442\pi$
$283$	−17.1670	−1.02047	−0.510235	+	0.860035i	$0.670442\pi$
$284$	−11.8832	−0.705138
$285$	0	0
$286$	−1.48671	−0.0879108
$287$	0.0163518	0.000965214 0
$288$	0	0
$289$	23.4500	1.37941
$290$	−6.88381	−0.404231
$291$	0	0
$292$	−2.03515	−0.119098
$293$	5.73172	0.334850	0.167425	−	0.985885i	$-0.446455\pi$
$293$	5.73172	0.334850	0.167425	+	0.985885i	$0.446455\pi$
$294$	0	0
$295$	13.7875	0.802738
$296$	6.59013	0.383044
$297$	0	0
$298$	8.81356	0.510556
$299$	10.7095	0.619348
$300$	0	0
$301$	−0.484734	−0.0279396
$302$	9.25088	0.532328
$303$	0	0
$304$	−4.66870	−0.267768
$305$	−8.56947	−0.490687
$306$	0	0
$307$	−5.10729	−0.291489	−0.145744	−	0.989322i	$-0.546558\pi$
$307$	−5.10729	−0.291489	−0.145744	+	0.989322i	$0.546558\pi$
$308$	0.0838763	0.00477930
$309$	0	0
$310$	2.64853	0.150427
$311$	21.3480	1.21054	0.605268	−	0.796022i	$-0.293066\pi$
$311$	21.3480	1.21054	0.605268	+	0.796022i	$0.293066\pi$
$312$	0	0
$313$	6.29899	0.356040	0.178020	−	0.984027i	$-0.443031\pi$
$313$	6.29899	0.356040	0.178020	+	0.984027i	$0.443031\pi$
$314$	14.9216	0.842075
$315$	0	0
$316$	−3.93629	−0.221433
$317$	−4.33278	−0.243353	−0.121677	−	0.992570i	$-0.538827\pi$
$317$	−4.33278	−0.243353	−0.121677	+	0.992570i	$0.538827\pi$
$318$	0	0
$319$	5.19820	0.291044
$320$	−1.32427	−0.0740288
$321$	0	0
$322$	−0.604204	−0.0336710
$323$	29.6931	1.65217
$324$	0	0
$325$	4.82632	0.267716
$326$	−1.21492	−0.0672879
$327$	0	0
$328$	0.194951	0.0107644
$329$	0.0838763	0.00462425
$330$	0	0
$331$	−33.7852	−1.85700	−0.928500	−	0.371331i	$-0.878901\pi$
$331$	−33.7852	−1.85700	−0.928500	+	0.371331i	$0.878901\pi$
$332$	−2.95456	−0.162153
$333$	0	0
$334$	−17.9346	−0.981336
$335$	−9.64342	−0.526876
$336$	0	0
$337$	−24.0350	−1.30927	−0.654635	−	0.755945i	$-0.727178\pi$
$337$	−24.0350	−1.30927	−0.654635	+	0.755945i	$0.727178\pi$
$338$	10.7897	0.586882
$339$	0	0
$340$	8.42238	0.456768
$341$	−2.00000	−0.108306
$342$	0	0
$343$	1.17368	0.0633727
$344$	−5.77916	−0.311591
$345$	0	0
$346$	25.0582	1.34713
$347$	−22.1957	−1.19153	−0.595765	−	0.803159i	$-0.703151\pi$
$347$	−22.1957	−1.19153	−0.595765	+	0.803159i	$0.703151\pi$
$348$	0	0
$349$	−21.6517	−1.15899	−0.579495	−	0.814976i	$-0.696750\pi$
$349$	−21.6517	−1.15899	−0.579495	+	0.814976i	$0.696750\pi$
$350$	−0.272289	−0.0145545
$351$	0	0
$352$	1.00000	0.0533002
$353$	−8.65512	−0.460666	−0.230333	−	0.973112i	$-0.573982\pi$
$353$	−8.65512	−0.460666	−0.230333	+	0.973112i	$0.573982\pi$
$354$	0	0
$355$	15.7365	0.835208
$356$	4.33019	0.229500
$357$	0	0
$358$	23.4497	1.23936
$359$	17.3072	0.913439	0.456719	−	0.889611i	$-0.349024\pi$
$359$	17.3072	0.913439	0.456719	+	0.889611i	$0.349024\pi$
$360$	0	0
$361$	2.79674	0.147197
$362$	−17.5609	−0.922979
$363$	0	0
$364$	0.124699	0.00653603
$365$	2.69508	0.141067
$366$	0	0
$367$	−30.3498	−1.58425	−0.792124	−	0.610360i	$-0.791025\pi$
$367$	−30.3498	−1.58425	−0.792124	+	0.610360i	$0.791025\pi$
$368$	−7.20352	−0.375509
$369$	0	0
$370$	−8.72710	−0.453700
$371$	−0.183880	−0.00954659
$372$	0	0
$373$	9.27115	0.480042	0.240021	−	0.970768i	$-0.422846\pi$
$373$	9.27115	0.480042	0.240021	+	0.970768i	$0.422846\pi$
$374$	−6.36003	−0.328869
$375$	0	0
$376$	1.00000	0.0515711
$377$	7.72821	0.398023
$378$	0	0
$379$	32.6463	1.67693	0.838464	−	0.544957i	$-0.183454\pi$
$379$	32.6463	1.67693	0.838464	+	0.544957i	$0.183454\pi$
$380$	6.18260	0.317161
$381$	0	0
$382$	−8.71075	−0.445681
$383$	24.3310	1.24325	0.621627	−	0.783313i	$-0.286472\pi$
$383$	24.3310	1.24325	0.621627	+	0.783313i	$0.286472\pi$
$384$	0	0
$385$	−0.111075	−0.00566089
$386$	4.44922	0.226459
$387$	0	0
$388$	10.6389	0.540106
$389$	−21.1596	−1.07283	−0.536417	−	0.843953i	$-0.680223\pi$
$389$	−21.1596	−1.07283	−0.536417	+	0.843953i	$0.680223\pi$
$390$	0	0
$391$	45.8146	2.31694
$392$	6.99296	0.353198
$393$	0	0
$394$	11.1087	0.559646
$395$	5.21270	0.262279
$396$	0	0
$397$	−18.3965	−0.923293	−0.461647	−	0.887064i	$-0.652741\pi$
$397$	−18.3965	−0.923293	−0.461647	+	0.887064i	$0.652741\pi$
$398$	4.89613	0.245421
$399$	0	0
$400$	−3.24632	−0.162316
$401$	2.79785	0.139718	0.0698589	−	0.997557i	$-0.477745\pi$
$401$	2.79785	0.139718	0.0698589	+	0.997557i	$0.477745\pi$
$402$	0	0
$403$	−2.97341	−0.148116
$404$	11.8766	0.590882
$405$	0	0
$406$	−0.436006	−0.0216386
$407$	6.59013	0.326661
$408$	0	0
$409$	−24.9364	−1.23303	−0.616513	−	0.787345i	$-0.711455\pi$
$409$	−24.9364	−1.23303	−0.616513	+	0.787345i	$0.711455\pi$
$410$	−0.258167	−0.0127500
$411$	0	0
$412$	15.8669	0.781706
$413$	0.873269	0.0429708
$414$	0	0
$415$	3.91263	0.192063
$416$	1.48671	0.0728918
$417$	0	0
$418$	−4.66870	−0.228353
$419$	4.07434	0.199044	0.0995222	−	0.995035i	$-0.468269\pi$
$419$	4.07434	0.199044	0.0995222	+	0.995035i	$0.468269\pi$
$420$	0	0
$421$	−23.3146	−1.13628	−0.568141	−	0.822931i	$-0.692337\pi$
$421$	−23.3146	−1.13628	−0.568141	+	0.822931i	$0.692337\pi$
$422$	4.84649	0.235923
$423$	0	0
$424$	−2.19228	−0.106466
$425$	20.6467	1.00151
$426$	0	0
$427$	−0.542772	−0.0262666
$428$	6.75121	0.326332
$429$	0	0
$430$	7.65315	0.369068
$431$	9.14202	0.440356	0.220178	−	0.975460i	$-0.429336\pi$
$431$	9.14202	0.440356	0.220178	+	0.975460i	$0.429336\pi$
$432$	0	0
$433$	−17.9143	−0.860907	−0.430454	−	0.902613i	$-0.641646\pi$
$433$	−17.9143	−0.860907	−0.430454	+	0.902613i	$0.641646\pi$
$434$	0.167753	0.00805238
$435$	0	0
$436$	7.64915	0.366328
$437$	33.6310	1.60879
$438$	0	0
$439$	19.0124	0.907414	0.453707	−	0.891151i	$-0.350101\pi$
$439$	19.0124	0.907414	0.453707	+	0.891151i	$0.350101\pi$
$440$	−1.32427	−0.0631320
$441$	0	0
$442$	−9.45550	−0.449753
$443$	28.2217	1.34085	0.670426	−	0.741977i	$-0.266112\pi$
$443$	28.2217	1.34085	0.670426	+	0.741977i	$0.266112\pi$
$444$	0	0
$445$	−5.73433	−0.271833
$446$	−13.2452	−0.627179
$447$	0	0
$448$	−0.0838763	−0.00396278
$449$	−2.60935	−0.123143	−0.0615714	−	0.998103i	$-0.519611\pi$
$449$	−2.60935	−0.123143	−0.0615714	+	0.998103i	$0.519611\pi$
$450$	0	0
$451$	0.194951	0.00917988
$452$	13.6149	0.640392
$453$	0	0
$454$	27.0217	1.26819
$455$	−0.165135	−0.00774167
$456$	0	0
$457$	−13.2724	−0.620857	−0.310428	−	0.950597i	$-0.600473\pi$
$457$	−13.2724	−0.620857	−0.310428	+	0.950597i	$0.600473\pi$
$458$	−5.16483	−0.241337
$459$	0	0
$460$	9.53938	0.444776
$461$	−13.8762	−0.646281	−0.323141	−	0.946351i	$-0.604739\pi$
$461$	−13.8762	−0.646281	−0.323141	+	0.946351i	$0.604739\pi$
$462$	0	0
$463$	3.28350	0.152597	0.0762985	−	0.997085i	$-0.475690\pi$
$463$	3.28350	0.152597	0.0762985	+	0.997085i	$0.475690\pi$
$464$	−5.19820	−0.241321
$465$	0	0
$466$	−7.18903	−0.333025
$467$	29.2299	1.35260	0.676299	−	0.736627i	$-0.263583\pi$
$467$	29.2299	1.35260	0.676299	+	0.736627i	$0.263583\pi$
$468$	0	0
$469$	−0.610794	−0.0282038
$470$	−1.32427	−0.0610839
$471$	0	0
$472$	10.4114	0.479223
$473$	−5.77916	−0.265726
$474$	0	0
$475$	15.1561	0.695408
$476$	0.533456	0.0244509
$477$	0	0
$478$	3.26450	0.149315
$479$	11.4541	0.523349	0.261675	−	0.965156i	$-0.415725\pi$
$479$	11.4541	0.523349	0.261675	+	0.965156i	$0.415725\pi$
$480$	0	0
$481$	9.79760	0.446732
$482$	−27.2558	−1.24147
$483$	0	0
$484$	1.00000	0.0454545
$485$	−14.0887	−0.639735
$486$	0	0
$487$	−4.62039	−0.209370	−0.104685	−	0.994505i	$-0.533383\pi$
$487$	−4.62039	−0.209370	−0.104685	+	0.994505i	$0.533383\pi$
$488$	−6.47111	−0.292933
$489$	0	0
$490$	−9.26055	−0.418349
$491$	18.0171	0.813102	0.406551	−	0.913628i	$-0.366731\pi$
$491$	18.0171	0.813102	0.406551	+	0.913628i	$0.366731\pi$
$492$	0	0
$493$	33.0607	1.48898
$494$	−6.94099	−0.312290
$495$	0	0
$496$	2.00000	0.0898027
$497$	0.996719	0.0447089
$498$	0	0
$499$	10.9869	0.491841	0.245921	−	0.969290i	$-0.420910\pi$
$499$	10.9869	0.491841	0.245921	+	0.969290i	$0.420910\pi$
$500$	10.9203	0.488372
$501$	0	0
$502$	−19.7027	−0.879376
$503$	−43.2495	−1.92840	−0.964200	−	0.265177i	$-0.914570\pi$
$503$	−43.2495	−1.92840	−0.964200	+	0.265177i	$0.914570\pi$
$504$	0	0
$505$	−15.7278	−0.699876
$506$	−7.20352	−0.320235
$507$	0	0
$508$	−14.4786	−0.642382
$509$	36.2437	1.60647	0.803236	−	0.595661i	$-0.203110\pi$
$509$	36.2437	1.60647	0.803236	+	0.595661i	$0.203110\pi$
$510$	0	0
$511$	0.170701	0.00755137
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.1457	0.800372
$515$	−21.0120	−0.925900
$516$	0	0
$517$	1.00000	0.0439799
$518$	−0.552756	−0.0242867
$519$	0	0
$520$	−1.96880	−0.0863375
$521$	22.8813	1.00245	0.501223	−	0.865318i	$-0.332883\pi$
$521$	22.8813	1.00245	0.501223	+	0.865318i	$0.332883\pi$
$522$	0	0
$523$	6.72228	0.293945	0.146972	−	0.989141i	$-0.453047\pi$
$523$	6.72228	0.293945	0.146972	+	0.989141i	$0.453047\pi$
$524$	−15.2860	−0.667773
$525$	0	0
$526$	4.79217	0.208949
$527$	−12.7201	−0.554095
$528$	0	0
$529$	28.8907	1.25612
$530$	2.90316	0.126105
$531$	0	0
$532$	0.391593	0.0169777
$533$	0.289835	0.0125541
$534$	0	0
$535$	−8.94041	−0.386527
$536$	−7.28208	−0.314538
$537$	0	0
$538$	−21.4345	−0.924106
$539$	6.99296	0.301208
$540$	0	0
$541$	16.9717	0.729669	0.364834	−	0.931072i	$-0.381126\pi$
$541$	16.9717	0.729669	0.364834	+	0.931072i	$0.381126\pi$
$542$	6.58482	0.282842
$543$	0	0
$544$	6.36003	0.272684
$545$	−10.1295	−0.433901
$546$	0	0
$547$	23.9183	1.02267	0.511336	−	0.859381i	$-0.329151\pi$
$547$	23.9183	1.02267	0.511336	+	0.859381i	$0.329151\pi$
$548$	9.31587	0.397954
$549$	0	0
$550$	−3.24632	−0.138423
$551$	24.2688	1.03389
$552$	0	0
$553$	0.330161	0.0140399
$554$	14.5291	0.617282
$555$	0	0
$556$	22.7604	0.965256
$557$	−20.5690	−0.871536	−0.435768	−	0.900059i	$-0.643523\pi$
$557$	−20.5690	−0.871536	−0.435768	+	0.900059i	$0.643523\pi$
$558$	0	0
$559$	−8.59192	−0.363399
$560$	0.111075	0.00469376
$561$	0	0
$562$	−26.5979	−1.12197
$563$	−13.3510	−0.562676	−0.281338	−	0.959609i	$-0.590778\pi$
$563$	−13.3510	−0.562676	−0.281338	+	0.959609i	$0.590778\pi$
$564$	0	0
$565$	−18.0298	−0.758519
$566$	17.1670	0.721582
$567$	0	0
$568$	11.8832	0.498608
$569$	38.9580	1.63321	0.816603	−	0.577200i	$-0.195855\pi$
$569$	38.9580	1.63321	0.816603	+	0.577200i	$0.195855\pi$
$570$	0	0
$571$	−26.6477	−1.11517	−0.557586	−	0.830119i	$-0.688272\pi$
$571$	−26.6477	−1.11517	−0.557586	+	0.830119i	$0.688272\pi$
$572$	1.48671	0.0621624
$573$	0	0
$574$	−0.0163518	−0.000682509 0
$575$	23.3849	0.975218
$576$	0	0
$577$	−38.9259	−1.62051	−0.810253	−	0.586080i	$-0.800670\pi$
$577$	−38.9259	−1.62051	−0.810253	+	0.586080i	$0.800670\pi$
$578$	−23.4500	−0.975391
$579$	0	0
$580$	6.88381	0.285835
$581$	0.247818	0.0102812
$582$	0	0
$583$	−2.19228	−0.0907949
$584$	2.03515	0.0842152
$585$	0	0
$586$	−5.73172	−0.236775
$587$	−25.3233	−1.04521	−0.522603	−	0.852576i	$-0.675039\pi$
$587$	−25.3233	−1.04521	−0.522603	+	0.852576i	$0.675039\pi$
$588$	0	0
$589$	−9.33740	−0.384741
$590$	−13.7875	−0.567621
$591$	0	0
$592$	−6.59013	−0.270853
$593$	−31.2412	−1.28292	−0.641461	−	0.767156i	$-0.721671\pi$
$593$	−31.2412	−1.28292	−0.641461	+	0.767156i	$0.721671\pi$
$594$	0	0
$595$	−0.706438	−0.0289611
$596$	−8.81356	−0.361017
$597$	0	0
$598$	−10.7095	−0.437945
$599$	9.96885	0.407316	0.203658	−	0.979042i	$-0.434717\pi$
$599$	9.96885	0.407316	0.203658	+	0.979042i	$0.434717\pi$
$600$	0	0
$601$	−31.6351	−1.29042	−0.645211	−	0.764005i	$-0.723230\pi$
$601$	−31.6351	−1.29042	−0.645211	+	0.764005i	$0.723230\pi$
$602$	0.484734	0.0197563
$603$	0	0
$604$	−9.25088	−0.376413
$605$	−1.32427	−0.0538391
$606$	0	0
$607$	−31.6308	−1.28385	−0.641927	−	0.766766i	$-0.721865\pi$
$607$	−31.6308	−1.28385	−0.641927	+	0.766766i	$0.721865\pi$
$608$	4.66870	0.189341
$609$	0	0
$610$	8.56947	0.346968
$611$	1.48671	0.0601457
$612$	0	0
$613$	24.7222	0.998518	0.499259	−	0.866453i	$-0.333606\pi$
$613$	24.7222	0.998518	0.499259	+	0.866453i	$0.333606\pi$
$614$	5.10729	0.206114
$615$	0	0
$616$	−0.0838763	−0.00337947
$617$	20.4343	0.822653	0.411327	−	0.911488i	$-0.365066\pi$
$617$	20.4343	0.822653	0.411327	+	0.911488i	$0.365066\pi$
$618$	0	0
$619$	−20.3315	−0.817191	−0.408595	−	0.912716i	$-0.633981\pi$
$619$	−20.3315	−0.817191	−0.408595	+	0.912716i	$0.633981\pi$
$620$	−2.64853	−0.106368
$621$	0	0
$622$	−21.3480	−0.855978
$623$	−0.363200	−0.0145513
$624$	0	0
$625$	1.77015	0.0708061
$626$	−6.29899	−0.251758
$627$	0	0
$628$	−14.9216	−0.595437
$629$	41.9135	1.67120
$630$	0	0
$631$	−6.36712	−0.253471	−0.126736	−	0.991937i	$-0.540450\pi$
$631$	−6.36712	−0.253471	−0.126736	+	0.991937i	$0.540450\pi$
$632$	3.93629	0.156577
$633$	0	0
$634$	4.33278	0.172077
$635$	19.1735	0.760877
$636$	0	0
$637$	10.3965	0.411924
$638$	−5.19820	−0.205799
$639$	0	0
$640$	1.32427	0.0523463
$641$	7.44065	0.293888	0.146944	−	0.989145i	$-0.453056\pi$
$641$	7.44065	0.293888	0.146944	+	0.989145i	$0.453056\pi$
$642$	0	0
$643$	−35.6518	−1.40597	−0.702985	−	0.711204i	$-0.748150\pi$
$643$	−35.6518	−1.40597	−0.702985	+	0.711204i	$0.748150\pi$
$644$	0.604204	0.0238090
$645$	0	0
$646$	−29.6931	−1.16826
$647$	45.5248	1.78977	0.894883	−	0.446302i	$-0.147259\pi$
$647$	45.5248	1.78977	0.894883	+	0.446302i	$0.147259\pi$
$648$	0	0
$649$	10.4114	0.408683
$650$	−4.82632	−0.189304
$651$	0	0
$652$	1.21492	0.0475798
$653$	38.4110	1.50314	0.751569	−	0.659655i	$-0.229297\pi$
$653$	38.4110	1.50314	0.751569	+	0.659655i	$0.229297\pi$
$654$	0	0
$655$	20.2428	0.790951
$656$	−0.194951	−0.00761155
$657$	0	0
$658$	−0.0838763	−0.00326984
$659$	15.0031	0.584438	0.292219	−	0.956351i	$-0.405606\pi$
$659$	15.0031	0.584438	0.292219	+	0.956351i	$0.405606\pi$
$660$	0	0
$661$	28.2446	1.09859	0.549293	−	0.835630i	$-0.314897\pi$
$661$	28.2446	1.09859	0.549293	+	0.835630i	$0.314897\pi$
$662$	33.7852	1.31310
$663$	0	0
$664$	2.95456	0.114659
$665$	−0.518574	−0.0201094
$666$	0	0
$667$	37.4454	1.44989
$668$	17.9346	0.693909
$669$	0	0
$670$	9.64342	0.372558
$671$	−6.47111	−0.249814
$672$	0	0
$673$	−11.9511	−0.460682	−0.230341	−	0.973110i	$-0.573984\pi$
$673$	−11.9511	−0.460682	−0.230341	+	0.973110i	$0.573984\pi$
$674$	24.0350	0.925794
$675$	0	0
$676$	−10.7897	−0.414989
$677$	−25.4740	−0.979047	−0.489523	−	0.871990i	$-0.662829\pi$
$677$	−25.4740	−0.979047	−0.489523	+	0.871990i	$0.662829\pi$
$678$	0	0
$679$	−0.892348	−0.0342452
$680$	−8.42238	−0.322984
$681$	0	0
$682$	2.00000	0.0765840
$683$	17.1196	0.655064	0.327532	−	0.944840i	$-0.393783\pi$
$683$	17.1196	0.655064	0.327532	+	0.944840i	$0.393783\pi$
$684$	0	0
$685$	−12.3367	−0.471361
$686$	−1.17368	−0.0448112
$687$	0	0
$688$	5.77916	0.220328
$689$	−3.25928	−0.124169
$690$	0	0
$691$	29.4036	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$691$	29.4036	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$692$	−25.0582	−0.952568
$693$	0	0
$694$	22.1957	0.842538
$695$	−30.1408	−1.14331
$696$	0	0
$697$	1.23989	0.0469643
$698$	21.6517	0.819530
$699$	0	0
$700$	0.272289	0.0102916
$701$	41.3396	1.56137	0.780687	−	0.624922i	$-0.214869\pi$
$701$	41.3396	1.56137	0.780687	+	0.624922i	$0.214869\pi$
$702$	0	0
$703$	30.7673	1.16041
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	8.65512	0.325740
$707$	−0.996163	−0.0374646
$708$	0	0
$709$	−35.1158	−1.31880	−0.659401	−	0.751791i	$-0.729190\pi$
$709$	−35.1158	−1.31880	−0.659401	+	0.751791i	$0.729190\pi$
$710$	−15.7365	−0.590581
$711$	0	0
$712$	−4.33019	−0.162281
$713$	−14.4070	−0.539548
$714$	0	0
$715$	−1.96880	−0.0736289
$716$	−23.4497	−0.876358
$717$	0	0
$718$	−17.3072	−0.645899
$719$	−42.3981	−1.58118	−0.790592	−	0.612344i	$-0.790227\pi$
$719$	−42.3981	−1.58118	−0.790592	+	0.612344i	$0.790227\pi$
$720$	0	0
$721$	−1.33086	−0.0495637
$722$	−2.79674	−0.104084
$723$	0	0
$724$	17.5609	0.652645
$725$	16.8750	0.626722
$726$	0	0
$727$	42.7811	1.58666	0.793332	−	0.608789i	$-0.208344\pi$
$727$	42.7811	1.58666	0.793332	+	0.608789i	$0.208344\pi$
$728$	−0.124699	−0.00462167
$729$	0	0
$730$	−2.69508	−0.0997495
$731$	−36.7556	−1.35946
$732$	0	0
$733$	−53.0866	−1.96080	−0.980400	−	0.197018i	$-0.936874\pi$
$733$	−53.0866	−1.96080	−0.980400	+	0.197018i	$0.936874\pi$
$734$	30.3498	1.12023
$735$	0	0
$736$	7.20352	0.265525
$737$	−7.28208	−0.268239
$738$	0	0
$739$	16.1103	0.592627	0.296314	−	0.955091i	$-0.404243\pi$
$739$	16.1103	0.592627	0.296314	+	0.955091i	$0.404243\pi$
$740$	8.72710	0.320815
$741$	0	0
$742$	0.183880	0.00675046
$743$	−20.9638	−0.769087	−0.384543	−	0.923107i	$-0.625641\pi$
$743$	−20.9638	−0.769087	−0.384543	+	0.923107i	$0.625641\pi$
$744$	0	0
$745$	11.6715	0.427611
$746$	−9.27115	−0.339441
$747$	0	0
$748$	6.36003	0.232546
$749$	−0.566266	−0.0206909
$750$	0	0
$751$	−36.2424	−1.32250	−0.661252	−	0.750163i	$-0.729975\pi$
$751$	−36.2424	−1.32250	−0.661252	+	0.750163i	$0.729975\pi$
$752$	−1.00000	−0.0364662
$753$	0	0
$754$	−7.72821	−0.281445
$755$	12.2506	0.445846
$756$	0	0
$757$	−0.676651	−0.0245933	−0.0122966	−	0.999924i	$-0.503914\pi$
$757$	−0.676651	−0.0245933	−0.0122966	+	0.999924i	$0.503914\pi$
$758$	−32.6463	−1.18577
$759$	0	0
$760$	−6.18260	−0.224267
$761$	40.6643	1.47408	0.737040	−	0.675849i	$-0.236223\pi$
$761$	40.6643	1.47408	0.737040	+	0.675849i	$0.236223\pi$
$762$	0	0
$763$	−0.641582	−0.0232268
$764$	8.71075	0.315144
$765$	0	0
$766$	−24.3310	−0.879114
$767$	15.4787	0.558903
$768$	0	0
$769$	12.0643	0.435049	0.217524	−	0.976055i	$-0.430202\pi$
$769$	12.0643	0.435049	0.217524	+	0.976055i	$0.430202\pi$
$770$	0.111075	0.00400285
$771$	0	0
$772$	−4.44922	−0.160131
$773$	−26.1562	−0.940775	−0.470387	−	0.882460i	$-0.655886\pi$
$773$	−26.1562	−0.940775	−0.470387	+	0.882460i	$0.655886\pi$
$774$	0	0
$775$	−6.49263	−0.233222
$776$	−10.6389	−0.381913
$777$	0	0
$778$	21.1596	0.758609
$779$	0.910167	0.0326101
$780$	0	0
$781$	11.8832	0.425214
$782$	−45.8146	−1.63833
$783$	0	0
$784$	−6.99296	−0.249749
$785$	19.7602	0.705271
$786$	0	0
$787$	0.165937	0.00591501	0.00295750	−	0.999996i	$-0.499059\pi$
$787$	0.165937	0.00591501	0.00295750	+	0.999996i	$0.499059\pi$
$788$	−11.1087	−0.395729
$789$	0	0
$790$	−5.21270	−0.185459
$791$	−1.14197	−0.0406037
$792$	0	0
$793$	−9.62064	−0.341639
$794$	18.3965	0.652867
$795$	0	0
$796$	−4.89613	−0.173539
$797$	18.6995	0.662370	0.331185	−	0.943566i	$-0.392552\pi$
$797$	18.6995	0.662370	0.331185	+	0.943566i	$0.392552\pi$
$798$	0	0
$799$	6.36003	0.225002
$800$	3.24632	0.114775
$801$	0	0
$802$	−2.79785	−0.0987954
$803$	2.03515	0.0718189
$804$	0	0
$805$	−0.800128	−0.0282008
$806$	2.97341	0.104734
$807$	0	0
$808$	−11.8766	−0.417817
$809$	−9.26191	−0.325632	−0.162816	−	0.986656i	$-0.552058\pi$
$809$	−9.26191	−0.325632	−0.162816	+	0.986656i	$0.552058\pi$
$810$	0	0
$811$	−26.0894	−0.916124	−0.458062	−	0.888920i	$-0.651456\pi$
$811$	−26.0894	−0.916124	−0.458062	+	0.888920i	$0.651456\pi$
$812$	0.436006	0.0153008
$813$	0	0
$814$	−6.59013	−0.230984
$815$	−1.60887	−0.0563563
$816$	0	0
$817$	−26.9811	−0.943951
$818$	24.9364	0.871881
$819$	0	0
$820$	0.258167	0.00901559
$821$	17.3848	0.606732	0.303366	−	0.952874i	$-0.401889\pi$
$821$	17.3848	0.606732	0.303366	+	0.952874i	$0.401889\pi$
$822$	0	0
$823$	41.3725	1.44215	0.721077	−	0.692855i	$-0.243647\pi$
$823$	41.3725	1.44215	0.721077	+	0.692855i	$0.243647\pi$
$824$	−15.8669	−0.552750
$825$	0	0
$826$	−0.873269	−0.0303849
$827$	12.6943	0.441424	0.220712	−	0.975339i	$-0.429162\pi$
$827$	12.6943	0.441424	0.220712	+	0.975339i	$0.429162\pi$
$828$	0	0
$829$	−18.6243	−0.646850	−0.323425	−	0.946254i	$-0.604834\pi$
$829$	−18.6243	−0.646850	−0.323425	+	0.946254i	$0.604834\pi$
$830$	−3.91263	−0.135809
$831$	0	0
$832$	−1.48671	−0.0515423
$833$	44.4755	1.54098
$834$	0	0
$835$	−23.7502	−0.821908
$836$	4.66870	0.161470
$837$	0	0
$838$	−4.07434	−0.140746
$839$	−25.0094	−0.863420	−0.431710	−	0.902012i	$-0.642090\pi$
$839$	−25.0094	−0.863420	−0.431710	+	0.902012i	$0.642090\pi$
$840$	0	0
$841$	−1.97867	−0.0682301
$842$	23.3146	0.803473
$843$	0	0
$844$	−4.84649	−0.166823
$845$	14.2884	0.491538
$846$	0	0
$847$	−0.0838763	−0.00288202
$848$	2.19228	0.0752832
$849$	0	0
$850$	−20.6467	−0.708175
$851$	47.4721	1.62732
$852$	0	0
$853$	47.0656	1.61150	0.805748	−	0.592258i	$-0.201763\pi$
$853$	47.0656	1.61150	0.805748	+	0.592258i	$0.201763\pi$
$854$	0.542772	0.0185733
$855$	0	0
$856$	−6.75121	−0.230752
$857$	−26.5788	−0.907913	−0.453957	−	0.891024i	$-0.649988\pi$
$857$	−26.5788	−0.907913	−0.453957	+	0.891024i	$0.649988\pi$
$858$	0	0
$859$	13.2730	0.452867	0.226434	−	0.974027i	$-0.427293\pi$
$859$	13.2730	0.452867	0.226434	+	0.974027i	$0.427293\pi$
$860$	−7.65315	−0.260970
$861$	0	0
$862$	−9.14202	−0.311379
$863$	15.9881	0.544240	0.272120	−	0.962263i	$-0.412275\pi$
$863$	15.9881	0.544240	0.272120	+	0.962263i	$0.412275\pi$
$864$	0	0
$865$	33.1837	1.12828
$866$	17.9143	0.608753
$867$	0	0
$868$	−0.167753	−0.00569389
$869$	3.93629	0.133529
$870$	0	0
$871$	−10.8263	−0.366836
$872$	−7.64915	−0.259033
$873$	0	0
$874$	−33.6310	−1.13759
$875$	−0.915956	−0.0309650
$876$	0	0
$877$	17.4561	0.589449	0.294725	−	0.955582i	$-0.404772\pi$
$877$	17.4561	0.589449	0.294725	+	0.955582i	$0.404772\pi$
$878$	−19.0124	−0.641639
$879$	0	0
$880$	1.32427	0.0446410
$881$	11.7967	0.397439	0.198720	−	0.980056i	$-0.436322\pi$
$881$	11.7967	0.397439	0.198720	+	0.980056i	$0.436322\pi$
$882$	0	0
$883$	7.36801	0.247953	0.123977	−	0.992285i	$-0.460435\pi$
$883$	7.36801	0.247953	0.123977	+	0.992285i	$0.460435\pi$
$884$	9.45550	0.318023
$885$	0	0
$886$	−28.2217	−0.948125
$887$	−38.5703	−1.29506	−0.647531	−	0.762039i	$-0.724198\pi$
$887$	−38.5703	−1.29506	−0.647531	+	0.762039i	$0.724198\pi$
$888$	0	0
$889$	1.21441	0.0407299
$890$	5.73433	0.192215
$891$	0	0
$892$	13.2452	0.443482
$893$	4.66870	0.156232
$894$	0	0
$895$	31.0537	1.03801
$896$	0.0838763	0.00280211
$897$	0	0
$898$	2.60935	0.0870751
$899$	−10.3964	−0.346740
$900$	0	0
$901$	−13.9430	−0.464507
$902$	−0.194951	−0.00649116
$903$	0	0
$904$	−13.6149	−0.452825
$905$	−23.2553	−0.773032
$906$	0	0
$907$	29.0892	0.965892	0.482946	−	0.875650i	$-0.339567\pi$
$907$	29.0892	0.965892	0.482946	+	0.875650i	$0.339567\pi$
$908$	−27.0217	−0.896746
$909$	0	0
$910$	0.165135	0.00547419
$911$	−36.9157	−1.22307	−0.611536	−	0.791217i	$-0.709448\pi$
$911$	−36.9157	−1.22307	−0.611536	+	0.791217i	$0.709448\pi$
$912$	0	0
$913$	2.95456	0.0977817
$914$	13.2724	0.439012
$915$	0	0
$916$	5.16483	0.170651
$917$	1.28214	0.0423398
$918$	0	0
$919$	23.4868	0.774758	0.387379	−	0.921921i	$-0.373381\pi$
$919$	23.4868	0.774758	0.387379	+	0.921921i	$0.373381\pi$
$920$	−9.53938	−0.314504
$921$	0	0
$922$	13.8762	0.456990
$923$	17.6668	0.581511
$924$	0	0
$925$	21.3937	0.703419
$926$	−3.28350	−0.107902
$927$	0	0
$928$	5.19820	0.170639
$929$	−13.5470	−0.444462	−0.222231	−	0.974994i	$-0.571334\pi$
$929$	−13.5470	−0.444462	−0.222231	+	0.974994i	$0.571334\pi$
$930$	0	0
$931$	32.6480	1.07000
$932$	7.18903	0.235484
$933$	0	0
$934$	−29.2299	−0.956432
$935$	−8.42238	−0.275441
$936$	0	0
$937$	50.8185	1.66017	0.830084	−	0.557638i	$-0.188292\pi$
$937$	50.8185	1.66017	0.830084	+	0.557638i	$0.188292\pi$
$938$	0.610794	0.0199431
$939$	0	0
$940$	1.32427	0.0431928
$941$	26.0700	0.849859	0.424929	−	0.905227i	$-0.360299\pi$
$941$	26.0700	0.849859	0.424929	+	0.905227i	$0.360299\pi$
$942$	0	0
$943$	1.40433	0.0457313
$944$	−10.4114	−0.338862
$945$	0	0
$946$	5.77916	0.187897
$947$	32.4693	1.05511	0.527555	−	0.849521i	$-0.323109\pi$
$947$	32.4693	1.05511	0.527555	+	0.849521i	$0.323109\pi$
$948$	0	0
$949$	3.02567	0.0982175
$950$	−15.1561	−0.491728
$951$	0	0
$952$	−0.533456	−0.0172894
$953$	−40.5549	−1.31370	−0.656850	−	0.754021i	$-0.728112\pi$
$953$	−40.5549	−1.31370	−0.656850	+	0.754021i	$0.728112\pi$
$954$	0	0
$955$	−11.5354	−0.373275
$956$	−3.26450	−0.105582
$957$	0	0
$958$	−11.4541	−0.370064
$959$	−0.781380	−0.0252321
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−9.79760	−0.315887
$963$	0	0
$964$	27.2558	0.877850
$965$	5.89196	0.189669
$966$	0	0
$967$	49.5805	1.59440	0.797200	−	0.603715i	$-0.206314\pi$
$967$	49.5805	1.59440	0.797200	+	0.603715i	$0.206314\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	14.0887	0.452361
$971$	−16.7323	−0.536966	−0.268483	−	0.963284i	$-0.586522\pi$
$971$	−16.7323	−0.536966	−0.268483	+	0.963284i	$0.586522\pi$
$972$	0	0
$973$	−1.90906	−0.0612016
$974$	4.62039	0.148047
$975$	0	0
$976$	6.47111	0.207135
$977$	8.84730	0.283050	0.141525	−	0.989935i	$-0.454799\pi$
$977$	8.84730	0.283050	0.141525	+	0.989935i	$0.454799\pi$
$978$	0	0
$979$	−4.33019	−0.138394
$980$	9.26055	0.295818
$981$	0	0
$982$	−18.0171	−0.574950
$983$	−8.35708	−0.266550	−0.133275	−	0.991079i	$-0.542549\pi$
$983$	−8.35708	−0.266550	−0.133275	+	0.991079i	$0.542549\pi$
$984$	0	0
$985$	14.7108	0.468726
$986$	−33.0607	−1.05287
$987$	0	0
$988$	6.94099	0.220822
$989$	−41.6303	−1.32377
$990$	0	0
$991$	−18.7836	−0.596682	−0.298341	−	0.954459i	$-0.596433\pi$
$991$	−18.7836	−0.596682	−0.298341	+	0.954459i	$0.596433\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−0.996719	−0.0316140
$995$	6.48378	0.205550
$996$	0	0
$997$	15.9951	0.506568	0.253284	−	0.967392i	$-0.418489\pi$
$997$	15.9951	0.506568	0.253284	+	0.967392i	$0.418489\pi$
$998$	−10.9869	−0.347784
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9306.2.a.be.1.2		5
3.2	odd	2		3102.2.a.v.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3102.2.a.v.1.4	✓	5	3.2	odd	2
9306.2.a.be.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9306.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.32427 q^{5} -0.0838763 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.32427 q^{5} -0.0838763 q^{7} -1.00000 q^{8} +1.32427 q^{10} -1.00000 q^{11} -1.48671 q^{13} +0.0838763 q^{14} +1.00000 q^{16} -6.36003 q^{17} -4.66870 q^{19} -1.32427 q^{20} +1.00000 q^{22} -7.20352 q^{23} -3.24632 q^{25} +1.48671 q^{26} -0.0838763 q^{28} -5.19820 q^{29} +2.00000 q^{31} -1.00000 q^{32} +6.36003 q^{34} +0.111075 q^{35} -6.59013 q^{37} +4.66870 q^{38} +1.32427 q^{40} -0.194951 q^{41} +5.77916 q^{43} -1.00000 q^{44} +7.20352 q^{46} -1.00000 q^{47} -6.99296 q^{49} +3.24632 q^{50} -1.48671 q^{52} +2.19228 q^{53} +1.32427 q^{55} +0.0838763 q^{56} +5.19820 q^{58} -10.4114 q^{59} +6.47111 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.96880 q^{65} +7.28208 q^{67} -6.36003 q^{68} -0.111075 q^{70} -11.8832 q^{71} -2.03515 q^{73} +6.59013 q^{74} -4.66870 q^{76} +0.0838763 q^{77} -3.93629 q^{79} -1.32427 q^{80} +0.194951 q^{82} -2.95456 q^{83} +8.42238 q^{85} -5.77916 q^{86} +1.00000 q^{88} +4.33019 q^{89} +0.124699 q^{91} -7.20352 q^{92} +1.00000 q^{94} +6.18260 q^{95} +10.6389 q^{97} +6.99296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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