LMFDB - Embedded newform 9405.2.a.bq.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 $ x^14 - x^13 - 19x^12 + 15x^11 + 137x^10 - 80x^9 - 467x^8 + 193x^7 + 766x^6 - 240x^5 - 567x^4 + 142x^3 + 142x^2 - 25x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.814716$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.814716 q^{2} -1.33624 q^{4} +1.00000 q^{5} +2.87567 q^{7} +2.71809 q^{8} +O(q^{10})$	\(q-0.814716 q^{2} -1.33624 q^{4} +1.00000 q^{5} +2.87567 q^{7} +2.71809 q^{8} -0.814716 q^{10} +1.00000 q^{11} +0.395222 q^{13} -2.34285 q^{14} +0.458008 q^{16} -2.38254 q^{17} -1.00000 q^{19} -1.33624 q^{20} -0.814716 q^{22} -5.25814 q^{23} +1.00000 q^{25} -0.321993 q^{26} -3.84257 q^{28} +10.4441 q^{29} +2.07018 q^{31} -5.80932 q^{32} +1.94110 q^{34} +2.87567 q^{35} -10.0797 q^{37} +0.814716 q^{38} +2.71809 q^{40} +9.62740 q^{41} -8.14758 q^{43} -1.33624 q^{44} +4.28389 q^{46} -4.17176 q^{47} +1.26945 q^{49} -0.814716 q^{50} -0.528110 q^{52} -10.8648 q^{53} +1.00000 q^{55} +7.81631 q^{56} -8.50901 q^{58} -9.96412 q^{59} -3.18327 q^{61} -1.68661 q^{62} +3.81693 q^{64} +0.395222 q^{65} -11.8473 q^{67} +3.18364 q^{68} -2.34285 q^{70} -14.6297 q^{71} +9.50123 q^{73} +8.21213 q^{74} +1.33624 q^{76} +2.87567 q^{77} +8.73705 q^{79} +0.458008 q^{80} -7.84359 q^{82} -2.37039 q^{83} -2.38254 q^{85} +6.63796 q^{86} +2.71809 q^{88} +12.1601 q^{89} +1.13653 q^{91} +7.02613 q^{92} +3.39880 q^{94} -1.00000 q^{95} -7.04099 q^{97} -1.03424 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * 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 $

$2$	−0.814716	−0.576091	−0.288046	−	0.957617i	$-0.593005\pi$
$2$	−0.814716	−0.576091	−0.288046	+	0.957617i	$0.593005\pi$
$3$	0	0
$3$	0	0
$4$	−1.33624	−0.668119
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.87567	1.08690	0.543450	−	0.839442i	$-0.317118\pi$
$7$	2.87567	1.08690	0.543450	+	0.839442i	$0.317118\pi$
$8$	2.71809	0.960989
$9$	0	0
$10$	−0.814716	−0.257636
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.395222	0.109615	0.0548074	−	0.998497i	$-0.482546\pi$
$13$	0.395222	0.109615	0.0548074	+	0.998497i	$0.482546\pi$
$14$	−2.34285	−0.626153
$15$	0	0
$16$	0.458008	0.114502
$17$	−2.38254	−0.577851	−0.288926	−	0.957352i	$-0.593298\pi$
$17$	−2.38254	−0.577851	−0.288926	+	0.957352i	$0.593298\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.33624	−0.298792
$21$	0	0
$22$	−0.814716	−0.173698
$23$	−5.25814	−1.09640	−0.548199	−	0.836348i	$-0.684686\pi$
$23$	−5.25814	−1.09640	−0.548199	+	0.836348i	$0.684686\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.321993	−0.0631481
$27$	0	0
$28$	−3.84257	−0.726178
$29$	10.4441	1.93943	0.969715	−	0.244241i	$-0.0785387\pi$
$29$	10.4441	1.93943	0.969715	+	0.244241i	$0.0785387\pi$
$30$	0	0
$31$	2.07018	0.371815	0.185907	−	0.982567i	$-0.440478\pi$
$31$	2.07018	0.371815	0.185907	+	0.982567i	$0.440478\pi$
$32$	−5.80932	−1.02695
$33$	0	0
$34$	1.94110	0.332895
$35$	2.87567	0.486076
$36$	0	0
$37$	−10.0797	−1.65710	−0.828550	−	0.559915i	$-0.810834\pi$
$37$	−10.0797	−1.65710	−0.828550	+	0.559915i	$0.810834\pi$
$38$	0.814716	0.132164
$39$	0	0
$40$	2.71809	0.429767
$41$	9.62740	1.50355	0.751774	−	0.659421i	$-0.229199\pi$
$41$	9.62740	1.50355	0.751774	+	0.659421i	$0.229199\pi$
$42$	0	0
$43$	−8.14758	−1.24249	−0.621247	−	0.783615i	$-0.713374\pi$
$43$	−8.14758	−1.24249	−0.621247	+	0.783615i	$0.713374\pi$
$44$	−1.33624	−0.201445
$45$	0	0
$46$	4.28389	0.631625
$47$	−4.17176	−0.608514	−0.304257	−	0.952590i	$-0.598408\pi$
$47$	−4.17176	−0.608514	−0.304257	+	0.952590i	$0.598408\pi$
$48$	0	0
$49$	1.26945	0.181350
$50$	−0.814716	−0.115218
$51$	0	0
$52$	−0.528110	−0.0732357
$53$	−10.8648	−1.49239	−0.746194	−	0.665728i	$-0.768121\pi$
$53$	−10.8648	−1.49239	−0.746194	+	0.665728i	$0.768121\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	7.81631	1.04450
$57$	0	0
$58$	−8.50901	−1.11729
$59$	−9.96412	−1.29722	−0.648609	−	0.761122i	$-0.724649\pi$
$59$	−9.96412	−1.29722	−0.648609	+	0.761122i	$0.724649\pi$
$60$	0	0
$61$	−3.18327	−0.407576	−0.203788	−	0.979015i	$-0.565325\pi$
$61$	−3.18327	−0.407576	−0.203788	+	0.979015i	$0.565325\pi$
$62$	−1.68661	−0.214199
$63$	0	0
$64$	3.81693	0.477116
$65$	0.395222	0.0490212
$66$	0	0
$67$	−11.8473	−1.44738	−0.723688	−	0.690127i	$-0.757555\pi$
$67$	−11.8473	−1.44738	−0.723688	+	0.690127i	$0.757555\pi$
$68$	3.18364	0.386074
$69$	0	0
$70$	−2.34285	−0.280024
$71$	−14.6297	−1.73622	−0.868112	−	0.496369i	$-0.834666\pi$
$71$	−14.6297	−1.73622	−0.868112	+	0.496369i	$0.834666\pi$
$72$	0	0
$73$	9.50123	1.11203	0.556017	−	0.831171i	$-0.312329\pi$
$73$	9.50123	1.11203	0.556017	+	0.831171i	$0.312329\pi$
$74$	8.21213	0.954641
$75$	0	0
$76$	1.33624	0.153277
$77$	2.87567	0.327713
$78$	0	0
$79$	8.73705	0.982995	0.491497	−	0.870879i	$-0.336450\pi$
$79$	8.73705	0.982995	0.491497	+	0.870879i	$0.336450\pi$
$80$	0.458008	0.0512069
$81$	0	0
$82$	−7.84359	−0.866180
$83$	−2.37039	−0.260185	−0.130092	−	0.991502i	$-0.541527\pi$
$83$	−2.37039	−0.260185	−0.130092	+	0.991502i	$0.541527\pi$
$84$	0	0
$85$	−2.38254	−0.258423
$86$	6.63796	0.715790
$87$	0	0
$88$	2.71809	0.289749
$89$	12.1601	1.28897	0.644486	−	0.764616i	$-0.277072\pi$
$89$	12.1601	1.28897	0.644486	+	0.764616i	$0.277072\pi$
$90$	0	0
$91$	1.13653	0.119140
$92$	7.02613	0.732525
$93$	0	0
$94$	3.39880	0.350560
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−7.04099	−0.714904	−0.357452	−	0.933931i	$-0.616354\pi$
$97$	−7.04099	−0.714904	−0.357452	+	0.933931i	$0.616354\pi$
$98$	−1.03424	−0.104474
$99$	0	0
$100$	−1.33624	−0.133624
$101$	−16.2240	−1.61435	−0.807173	−	0.590314i	$-0.799004\pi$
$101$	−16.2240	−1.61435	−0.807173	+	0.590314i	$0.799004\pi$
$102$	0	0
$103$	−0.448112	−0.0441538	−0.0220769	−	0.999756i	$-0.507028\pi$
$103$	−0.448112	−0.0441538	−0.0220769	+	0.999756i	$0.507028\pi$
$104$	1.07425	0.105339
$105$	0	0
$106$	8.85169	0.859752
$107$	−3.44941	−0.333467	−0.166733	−	0.986002i	$-0.553322\pi$
$107$	−3.44941	−0.333467	−0.166733	+	0.986002i	$0.553322\pi$
$108$	0	0
$109$	−14.5373	−1.39242	−0.696210	−	0.717838i	$-0.745132\pi$
$109$	−14.5373	−1.39242	−0.696210	+	0.717838i	$0.745132\pi$
$110$	−0.814716	−0.0776801
$111$	0	0
$112$	1.31708	0.124452
$113$	6.21721	0.584866	0.292433	−	0.956286i	$-0.405535\pi$
$113$	6.21721	0.584866	0.292433	+	0.956286i	$0.405535\pi$
$114$	0	0
$115$	−5.25814	−0.490324
$116$	−13.9559	−1.29577
$117$	0	0
$118$	8.11793	0.747316
$119$	−6.85140	−0.628066
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.59346	0.234801
$123$	0	0
$124$	−2.76625	−0.248416
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.35796	−0.830385	−0.415192	−	0.909734i	$-0.636286\pi$
$127$	−9.35796	−0.830385	−0.415192	+	0.909734i	$0.636286\pi$
$128$	8.50893	0.752090
$129$	0	0
$130$	−0.321993	−0.0282407
$131$	−14.3775	−1.25617	−0.628084	−	0.778145i	$-0.716161\pi$
$131$	−14.3775	−1.25617	−0.628084	+	0.778145i	$0.716161\pi$
$132$	0	0
$133$	−2.87567	−0.249352
$134$	9.65217	0.833821
$135$	0	0
$136$	−6.47596	−0.555309
$137$	−2.40548	−0.205514	−0.102757	−	0.994707i	$-0.532766\pi$
$137$	−2.40548	−0.205514	−0.102757	+	0.994707i	$0.532766\pi$
$138$	0	0
$139$	−2.81138	−0.238458	−0.119229	−	0.992867i	$-0.538042\pi$
$139$	−2.81138	−0.238458	−0.119229	+	0.992867i	$0.538042\pi$
$140$	−3.84257	−0.324757
$141$	0	0
$142$	11.9190	1.00022
$143$	0.395222	0.0330501
$144$	0	0
$145$	10.4441	0.867339
$146$	−7.74080	−0.640633
$147$	0	0
$148$	13.4689	1.10714
$149$	12.4677	1.02139	0.510696	−	0.859761i	$-0.329388\pi$
$149$	12.4677	1.02139	0.510696	+	0.859761i	$0.329388\pi$
$150$	0	0
$151$	18.1381	1.47606	0.738030	−	0.674768i	$-0.235756\pi$
$151$	18.1381	1.47606	0.738030	+	0.674768i	$0.235756\pi$
$152$	−2.71809	−0.220466
$153$	0	0
$154$	−2.34285	−0.188792
$155$	2.07018	0.166281
$156$	0	0
$157$	−15.2358	−1.21595	−0.607976	−	0.793955i	$-0.708019\pi$
$157$	−15.2358	−1.21595	−0.607976	+	0.793955i	$0.708019\pi$
$158$	−7.11821	−0.566294
$159$	0	0
$160$	−5.80932	−0.459267
$161$	−15.1207	−1.19167
$162$	0	0
$163$	0.101489	0.00794920	0.00397460	−	0.999992i	$-0.498735\pi$
$163$	0.101489	0.00794920	0.00397460	+	0.999992i	$0.498735\pi$
$164$	−12.8645	−1.00455
$165$	0	0
$166$	1.93120	0.149890
$167$	−2.16495	−0.167529	−0.0837646	−	0.996486i	$-0.526694\pi$
$167$	−2.16495	−0.167529	−0.0837646	+	0.996486i	$0.526694\pi$
$168$	0	0
$169$	−12.8438	−0.987985
$170$	1.94110	0.148875
$171$	0	0
$172$	10.8871	0.830134
$173$	19.0144	1.44564	0.722820	−	0.691036i	$-0.242845\pi$
$173$	19.0144	1.44564	0.722820	+	0.691036i	$0.242845\pi$
$174$	0	0
$175$	2.87567	0.217380
$176$	0.458008	0.0345237
$177$	0	0
$178$	−9.90705	−0.742565
$179$	−12.7841	−0.955525	−0.477762	−	0.878489i	$-0.658552\pi$
$179$	−12.7841	−0.955525	−0.477762	+	0.878489i	$0.658552\pi$
$180$	0	0
$181$	−20.1825	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$181$	−20.1825	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$182$	−0.925945	−0.0686356
$183$	0	0
$184$	−14.2921	−1.05363
$185$	−10.0797	−0.741078
$186$	0	0
$187$	−2.38254	−0.174229
$188$	5.57447	0.406560
$189$	0	0
$190$	0.814716	0.0591057
$191$	21.1731	1.53203	0.766016	−	0.642822i	$-0.222237\pi$
$191$	21.1731	1.53203	0.766016	+	0.642822i	$0.222237\pi$
$192$	0	0
$193$	−0.812169	−0.0584612	−0.0292306	−	0.999573i	$-0.509306\pi$
$193$	−0.812169	−0.0584612	−0.0292306	+	0.999573i	$0.509306\pi$
$194$	5.73641	0.411850
$195$	0	0
$196$	−1.69629	−0.121164
$197$	15.6580	1.11558	0.557791	−	0.829981i	$-0.311649\pi$
$197$	15.6580	1.11558	0.557791	+	0.829981i	$0.311649\pi$
$198$	0	0
$199$	−15.7938	−1.11959	−0.559797	−	0.828630i	$-0.689121\pi$
$199$	−15.7938	−1.11959	−0.559797	+	0.828630i	$0.689121\pi$
$200$	2.71809	0.192198
$201$	0	0
$202$	13.2179	0.930011
$203$	30.0339	2.10796
$204$	0	0
$205$	9.62740	0.672407
$206$	0.365084	0.0254366
$207$	0	0
$208$	0.181015	0.0125511
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−11.7543	−0.809199	−0.404600	−	0.914494i	$-0.632589\pi$
$211$	−11.7543	−0.809199	−0.404600	+	0.914494i	$0.632589\pi$
$212$	14.5179	0.997093
$213$	0	0
$214$	2.81029	0.192107
$215$	−8.14758	−0.555661
$216$	0	0
$217$	5.95313	0.404125
$218$	11.8438	0.802161
$219$	0	0
$220$	−1.33624	−0.0900892
$221$	−0.941633	−0.0633411
$222$	0	0
$223$	−17.2712	−1.15656	−0.578281	−	0.815837i	$-0.696276\pi$
$223$	−17.2712	−1.15656	−0.578281	+	0.815837i	$0.696276\pi$
$224$	−16.7057	−1.11619
$225$	0	0
$226$	−5.06526	−0.336936
$227$	−3.69997	−0.245576	−0.122788	−	0.992433i	$-0.539184\pi$
$227$	−3.69997	−0.245576	−0.122788	+	0.992433i	$0.539184\pi$
$228$	0	0
$229$	−1.18819	−0.0785180	−0.0392590	−	0.999229i	$-0.512500\pi$
$229$	−1.18819	−0.0785180	−0.0392590	+	0.999229i	$0.512500\pi$
$230$	4.28389	0.282471
$231$	0	0
$232$	28.3881	1.86377
$233$	13.0205	0.852999	0.426499	−	0.904488i	$-0.359747\pi$
$233$	13.0205	0.852999	0.426499	+	0.904488i	$0.359747\pi$
$234$	0	0
$235$	−4.17176	−0.272136
$236$	13.3144	0.866696
$237$	0	0
$238$	5.58194	0.361823
$239$	13.7427	0.888941	0.444471	−	0.895793i	$-0.353392\pi$
$239$	13.7427	0.888941	0.444471	+	0.895793i	$0.353392\pi$
$240$	0	0
$241$	30.7766	1.98249	0.991247	−	0.132024i	$-0.0421475\pi$
$241$	30.7766	1.98249	0.991247	+	0.132024i	$0.0421475\pi$
$242$	−0.814716	−0.0523719
$243$	0	0
$244$	4.25361	0.272309
$245$	1.26945	0.0811023
$246$	0	0
$247$	−0.395222	−0.0251474
$248$	5.62692	0.357310
$249$	0	0
$250$	−0.814716	−0.0515272
$251$	5.65765	0.357107	0.178554	−	0.983930i	$-0.442858\pi$
$251$	5.65765	0.357107	0.178554	+	0.983930i	$0.442858\pi$
$252$	0	0
$253$	−5.25814	−0.330577
$254$	7.62408	0.478377
$255$	0	0
$256$	−14.5662	−0.910388
$257$	13.3320	0.831628	0.415814	−	0.909450i	$-0.363497\pi$
$257$	13.3320	0.831628	0.415814	+	0.909450i	$0.363497\pi$
$258$	0	0
$259$	−28.9860	−1.80110
$260$	−0.528110	−0.0327520
$261$	0	0
$262$	11.7136	0.723667
$263$	−1.17431	−0.0724111	−0.0362055	−	0.999344i	$-0.511527\pi$
$263$	−1.17431	−0.0724111	−0.0362055	+	0.999344i	$0.511527\pi$
$264$	0	0
$265$	−10.8648	−0.667416
$266$	2.34285	0.143649
$267$	0	0
$268$	15.8308	0.967020
$269$	30.7006	1.87185	0.935923	−	0.352204i	$-0.114568\pi$
$269$	30.7006	1.87185	0.935923	+	0.352204i	$0.114568\pi$
$270$	0	0
$271$	24.0322	1.45985	0.729926	−	0.683527i	$-0.239555\pi$
$271$	24.0322	1.45985	0.729926	+	0.683527i	$0.239555\pi$
$272$	−1.09122	−0.0661652
$273$	0	0
$274$	1.95978	0.118395
$275$	1.00000	0.0603023
$276$	0	0
$277$	−15.5231	−0.932690	−0.466345	−	0.884603i	$-0.654429\pi$
$277$	−15.5231	−0.932690	−0.466345	+	0.884603i	$0.654429\pi$
$278$	2.29048	0.137374
$279$	0	0
$280$	7.81631	0.467114
$281$	14.0552	0.838461	0.419231	−	0.907880i	$-0.362300\pi$
$281$	14.0552	0.838461	0.419231	+	0.907880i	$0.362300\pi$
$282$	0	0
$283$	2.06370	0.122674	0.0613371	−	0.998117i	$-0.480464\pi$
$283$	2.06370	0.122674	0.0613371	+	0.998117i	$0.480464\pi$
$284$	19.5487	1.16000
$285$	0	0
$286$	−0.321993	−0.0190399
$287$	27.6852	1.63420
$288$	0	0
$289$	−11.3235	−0.666088
$290$	−8.50901	−0.499666
$291$	0	0
$292$	−12.6959	−0.742971
$293$	20.6719	1.20767	0.603833	−	0.797111i	$-0.293640\pi$
$293$	20.6719	1.20767	0.603833	+	0.797111i	$0.293640\pi$
$294$	0	0
$295$	−9.96412	−0.580134
$296$	−27.3976	−1.59245
$297$	0	0
$298$	−10.1576	−0.588415
$299$	−2.07813	−0.120181
$300$	0	0
$301$	−23.4297	−1.35047
$302$	−14.7774	−0.850345
$303$	0	0
$304$	−0.458008	−0.0262686
$305$	−3.18327	−0.182274
$306$	0	0
$307$	−21.6699	−1.23677	−0.618383	−	0.785877i	$-0.712212\pi$
$307$	−21.6699	−1.23677	−0.618383	+	0.785877i	$0.712212\pi$
$308$	−3.84257	−0.218951
$309$	0	0
$310$	−1.68661	−0.0957927
$311$	0.345204	0.0195747	0.00978737	−	0.999952i	$-0.496885\pi$
$311$	0.345204	0.0195747	0.00978737	+	0.999952i	$0.496885\pi$
$312$	0	0
$313$	8.72493	0.493162	0.246581	−	0.969122i	$-0.420693\pi$
$313$	8.72493	0.493162	0.246581	+	0.969122i	$0.420693\pi$
$314$	12.4129	0.700500
$315$	0	0
$316$	−11.6748	−0.656757
$317$	−6.38738	−0.358751	−0.179376	−	0.983781i	$-0.557408\pi$
$317$	−6.38738	−0.358751	−0.179376	+	0.983781i	$0.557408\pi$
$318$	0	0
$319$	10.4441	0.584760
$320$	3.81693	0.213373
$321$	0	0
$322$	12.3190	0.686513
$323$	2.38254	0.132568
$324$	0	0
$325$	0.395222	0.0219230
$326$	−0.0826843	−0.00457946
$327$	0	0
$328$	26.1681	1.44489
$329$	−11.9966	−0.661394
$330$	0	0
$331$	−12.8438	−0.705960	−0.352980	−	0.935631i	$-0.614832\pi$
$331$	−12.8438	−0.705960	−0.352980	+	0.935631i	$0.614832\pi$
$332$	3.16741	0.173834
$333$	0	0
$334$	1.76382	0.0965120
$335$	−11.8473	−0.647286
$336$	0	0
$337$	−29.2500	−1.59335	−0.796674	−	0.604409i	$-0.793409\pi$
$337$	−29.2500	−1.59335	−0.796674	+	0.604409i	$0.793409\pi$
$338$	10.4640	0.569169
$339$	0	0
$340$	3.18364	0.172657
$341$	2.07018	0.112106
$342$	0	0
$343$	−16.4791	−0.889790
$344$	−22.1458	−1.19402
$345$	0	0
$346$	−15.4914	−0.832821
$347$	2.44789	0.131410	0.0657048	−	0.997839i	$-0.479070\pi$
$347$	2.44789	0.131410	0.0657048	+	0.997839i	$0.479070\pi$
$348$	0	0
$349$	−2.51293	−0.134514	−0.0672570	−	0.997736i	$-0.521425\pi$
$349$	−2.51293	−0.134514	−0.0672570	+	0.997736i	$0.521425\pi$
$350$	−2.34285	−0.125231
$351$	0	0
$352$	−5.80932	−0.309638
$353$	−22.1884	−1.18097	−0.590486	−	0.807048i	$-0.701064\pi$
$353$	−22.1884	−1.18097	−0.590486	+	0.807048i	$0.701064\pi$
$354$	0	0
$355$	−14.6297	−0.776463
$356$	−16.2488	−0.861186
$357$	0	0
$358$	10.4154	0.550469
$359$	−13.0662	−0.689606	−0.344803	−	0.938675i	$-0.612054\pi$
$359$	−13.0662	−0.689606	−0.344803	+	0.938675i	$0.612054\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.4430	0.864223
$363$	0	0
$364$	−1.51867	−0.0795999
$365$	9.50123	0.497317
$366$	0	0
$367$	24.8628	1.29783	0.648915	−	0.760861i	$-0.275223\pi$
$367$	24.8628	1.29783	0.648915	+	0.760861i	$0.275223\pi$
$368$	−2.40827	−0.125540
$369$	0	0
$370$	8.21213	0.426928
$371$	−31.2434	−1.62208
$372$	0	0
$373$	36.1300	1.87074	0.935371	−	0.353667i	$-0.115065\pi$
$373$	36.1300	1.87074	0.935371	+	0.353667i	$0.115065\pi$
$374$	1.94110	0.100372
$375$	0	0
$376$	−11.3392	−0.584775
$377$	4.12775	0.212590
$378$	0	0
$379$	−9.79413	−0.503091	−0.251545	−	0.967846i	$-0.580939\pi$
$379$	−9.79413	−0.503091	−0.251545	+	0.967846i	$0.580939\pi$
$380$	1.33624	0.0685476
$381$	0	0
$382$	−17.2501	−0.882590
$383$	7.96428	0.406955	0.203478	−	0.979080i	$-0.434776\pi$
$383$	7.96428	0.406955	0.203478	+	0.979080i	$0.434776\pi$
$384$	0	0
$385$	2.87567	0.146557
$386$	0.661687	0.0336790
$387$	0	0
$388$	9.40844	0.477641
$389$	−12.5291	−0.635250	−0.317625	−	0.948216i	$-0.602885\pi$
$389$	−12.5291	−0.635250	−0.317625	+	0.948216i	$0.602885\pi$
$390$	0	0
$391$	12.5277	0.633555
$392$	3.45048	0.174276
$393$	0	0
$394$	−12.7568	−0.642677
$395$	8.73705	0.439609
$396$	0	0
$397$	−7.75003	−0.388963	−0.194481	−	0.980906i	$-0.562302\pi$
$397$	−7.75003	−0.388963	−0.194481	+	0.980906i	$0.562302\pi$
$398$	12.8675	0.644988
$399$	0	0
$400$	0.458008	0.0229004
$401$	11.0304	0.550830	0.275415	−	0.961325i	$-0.411185\pi$
$401$	11.0304	0.550830	0.275415	+	0.961325i	$0.411185\pi$
$402$	0	0
$403$	0.818178	0.0407564
$404$	21.6791	1.07858
$405$	0	0
$406$	−24.4691	−1.21438
$407$	−10.0797	−0.499634
$408$	0	0
$409$	10.7655	0.532320	0.266160	−	0.963929i	$-0.414245\pi$
$409$	10.7655	0.532320	0.266160	+	0.963929i	$0.414245\pi$
$410$	−7.84359	−0.387368
$411$	0	0
$412$	0.598785	0.0295000
$413$	−28.6535	−1.40995
$414$	0	0
$415$	−2.37039	−0.116358
$416$	−2.29597	−0.112569
$417$	0	0
$418$	0.814716	0.0398491
$419$	15.4378	0.754186	0.377093	−	0.926175i	$-0.376924\pi$
$419$	15.4378	0.754186	0.377093	+	0.926175i	$0.376924\pi$
$420$	0	0
$421$	−12.4720	−0.607847	−0.303924	−	0.952696i	$-0.598297\pi$
$421$	−12.4720	−0.607847	−0.303924	+	0.952696i	$0.598297\pi$
$422$	9.57641	0.466172
$423$	0	0
$424$	−29.5313	−1.43417
$425$	−2.38254	−0.115570
$426$	0	0
$427$	−9.15402	−0.442994
$428$	4.60923	0.222795
$429$	0	0
$430$	6.63796	0.320111
$431$	23.9854	1.15534	0.577668	−	0.816272i	$-0.303963\pi$
$431$	23.9854	1.15534	0.577668	+	0.816272i	$0.303963\pi$
$432$	0	0
$433$	−1.56561	−0.0752382	−0.0376191	−	0.999292i	$-0.511977\pi$
$433$	−1.56561	−0.0752382	−0.0376191	+	0.999292i	$0.511977\pi$
$434$	−4.85011	−0.232813
$435$	0	0
$436$	19.4253	0.930302
$437$	5.25814	0.251531
$438$	0	0
$439$	16.7062	0.797342	0.398671	−	0.917094i	$-0.369472\pi$
$439$	16.7062	0.797342	0.398671	+	0.917094i	$0.369472\pi$
$440$	2.71809	0.129580
$441$	0	0
$442$	0.767163	0.0364902
$443$	13.5115	0.641953	0.320976	−	0.947087i	$-0.395989\pi$
$443$	13.5115	0.641953	0.320976	+	0.947087i	$0.395989\pi$
$444$	0	0
$445$	12.1601	0.576445
$446$	14.0711	0.666286
$447$	0	0
$448$	10.9762	0.518577
$449$	4.65264	0.219572	0.109786	−	0.993955i	$-0.464983\pi$
$449$	4.65264	0.219572	0.109786	+	0.993955i	$0.464983\pi$
$450$	0	0
$451$	9.62740	0.453336
$452$	−8.30768	−0.390760
$453$	0	0
$454$	3.01443	0.141474
$455$	1.13653	0.0532811
$456$	0	0
$457$	−29.7955	−1.39378	−0.696888	−	0.717180i	$-0.745432\pi$
$457$	−29.7955	−1.39378	−0.696888	+	0.717180i	$0.745432\pi$
$458$	0.968040	0.0452335
$459$	0	0
$460$	7.02613	0.327595
$461$	−31.6235	−1.47285	−0.736426	−	0.676518i	$-0.763488\pi$
$461$	−31.6235	−1.47285	−0.736426	+	0.676518i	$0.763488\pi$
$462$	0	0
$463$	−16.7432	−0.778121	−0.389060	−	0.921212i	$-0.627200\pi$
$463$	−16.7432	−0.778121	−0.389060	+	0.921212i	$0.627200\pi$
$464$	4.78351	0.222069
$465$	0	0
$466$	−10.6080	−0.491405
$467$	−13.6652	−0.632351	−0.316176	−	0.948701i	$-0.602399\pi$
$467$	−13.6652	−0.632351	−0.316176	+	0.948701i	$0.602399\pi$
$468$	0	0
$469$	−34.0688	−1.57315
$470$	3.39880	0.156775
$471$	0	0
$472$	−27.0833	−1.24661
$473$	−8.14758	−0.374626
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	9.15510	0.419623
$477$	0	0
$478$	−11.1964	−0.512111
$479$	15.5171	0.708994	0.354497	−	0.935057i	$-0.384652\pi$
$479$	15.5171	0.708994	0.354497	+	0.935057i	$0.384652\pi$
$480$	0	0
$481$	−3.98373	−0.181643
$482$	−25.0742	−1.14210
$483$	0	0
$484$	−1.33624	−0.0607381
$485$	−7.04099	−0.319715
$486$	0	0
$487$	−1.45076	−0.0657402	−0.0328701	−	0.999460i	$-0.510465\pi$
$487$	−1.45076	−0.0657402	−0.0328701	+	0.999460i	$0.510465\pi$
$488$	−8.65241	−0.391676
$489$	0	0
$490$	−1.03424	−0.0467223
$491$	−25.1520	−1.13509	−0.567547	−	0.823341i	$-0.692108\pi$
$491$	−25.1520	−1.13509	−0.567547	+	0.823341i	$0.692108\pi$
$492$	0	0
$493$	−24.8836	−1.12070
$494$	0.321993	0.0144872
$495$	0	0
$496$	0.948158	0.0425735
$497$	−42.0701	−1.88710
$498$	0	0
$499$	14.2463	0.637750	0.318875	−	0.947797i	$-0.396695\pi$
$499$	14.2463	0.637750	0.318875	+	0.947797i	$0.396695\pi$
$500$	−1.33624	−0.0597584
$501$	0	0
$502$	−4.60937	−0.205726
$503$	−12.8738	−0.574014	−0.287007	−	0.957929i	$-0.592660\pi$
$503$	−12.8738	−0.574014	−0.287007	+	0.957929i	$0.592660\pi$
$504$	0	0
$505$	−16.2240	−0.721958
$506$	4.28389	0.190442
$507$	0	0
$508$	12.5045	0.554796
$509$	−18.5857	−0.823797	−0.411898	−	0.911230i	$-0.635134\pi$
$509$	−18.5857	−0.823797	−0.411898	+	0.911230i	$0.635134\pi$
$510$	0	0
$511$	27.3223	1.20867
$512$	−5.15053	−0.227623
$513$	0	0
$514$	−10.8618	−0.479094
$515$	−0.448112	−0.0197462
$516$	0	0
$517$	−4.17176	−0.183474
$518$	23.6153	1.03760
$519$	0	0
$520$	1.07425	0.0471088
$521$	18.4405	0.807894	0.403947	−	0.914782i	$-0.367638\pi$
$521$	18.4405	0.807894	0.403947	+	0.914782i	$0.367638\pi$
$522$	0	0
$523$	−11.4504	−0.500691	−0.250345	−	0.968157i	$-0.580544\pi$
$523$	−11.4504	−0.500691	−0.250345	+	0.968157i	$0.580544\pi$
$524$	19.2118	0.839270
$525$	0	0
$526$	0.956729	0.0417154
$527$	−4.93228	−0.214854
$528$	0	0
$529$	4.64806	0.202090
$530$	8.85169	0.384493
$531$	0	0
$532$	3.84257	0.166597
$533$	3.80496	0.164811
$534$	0	0
$535$	−3.44941	−0.149131
$536$	−32.2019	−1.39091
$537$	0	0
$538$	−25.0122	−1.07835
$539$	1.26945	0.0546792
$540$	0	0
$541$	−3.24207	−0.139388	−0.0696938	−	0.997568i	$-0.522202\pi$
$541$	−3.24207	−0.139388	−0.0696938	+	0.997568i	$0.522202\pi$
$542$	−19.5794	−0.841007
$543$	0	0
$544$	13.8409	0.593426
$545$	−14.5373	−0.622709
$546$	0	0
$547$	−29.3831	−1.25633	−0.628166	−	0.778080i	$-0.716194\pi$
$547$	−29.3831	−1.25633	−0.628166	+	0.778080i	$0.716194\pi$
$548$	3.21429	0.137308
$549$	0	0
$550$	−0.814716	−0.0347396
$551$	−10.4441	−0.444936
$552$	0	0
$553$	25.1248	1.06842
$554$	12.6469	0.537314
$555$	0	0
$556$	3.75668	0.159319
$557$	5.02323	0.212841	0.106421	−	0.994321i	$-0.466061\pi$
$557$	5.02323	0.212841	0.106421	+	0.994321i	$0.466061\pi$
$558$	0	0
$559$	−3.22010	−0.136196
$560$	1.31708	0.0556567
$561$	0	0
$562$	−11.4510	−0.483030
$563$	9.71580	0.409472	0.204736	−	0.978817i	$-0.434366\pi$
$563$	9.71580	0.409472	0.204736	+	0.978817i	$0.434366\pi$
$564$	0	0
$565$	6.21721	0.261560
$566$	−1.68133	−0.0706715
$567$	0	0
$568$	−39.7647	−1.66849
$569$	5.33449	0.223634	0.111817	−	0.993729i	$-0.464333\pi$
$569$	5.33449	0.223634	0.111817	+	0.993729i	$0.464333\pi$
$570$	0	0
$571$	23.6961	0.991651	0.495825	−	0.868422i	$-0.334866\pi$
$571$	23.6961	0.991651	0.495825	+	0.868422i	$0.334866\pi$
$572$	−0.528110	−0.0220814
$573$	0	0
$574$	−22.5556	−0.941451
$575$	−5.25814	−0.219280
$576$	0	0
$577$	8.32652	0.346637	0.173319	−	0.984866i	$-0.444551\pi$
$577$	8.32652	0.346637	0.173319	+	0.984866i	$0.444551\pi$
$578$	9.22543	0.383727
$579$	0	0
$580$	−13.9559	−0.579486
$581$	−6.81646	−0.282795
$582$	0	0
$583$	−10.8648	−0.449972
$584$	25.8252	1.06865
$585$	0	0
$586$	−16.8417	−0.695725
$587$	−12.2642	−0.506197	−0.253098	−	0.967441i	$-0.581450\pi$
$587$	−12.2642	−0.506197	−0.253098	+	0.967441i	$0.581450\pi$
$588$	0	0
$589$	−2.07018	−0.0853001
$590$	8.11793	0.334210
$591$	0	0
$592$	−4.61661	−0.189741
$593$	−18.1727	−0.746262	−0.373131	−	0.927779i	$-0.621716\pi$
$593$	−18.1727	−0.746262	−0.373131	+	0.927779i	$0.621716\pi$
$594$	0	0
$595$	−6.85140	−0.280880
$596$	−16.6598	−0.682412
$597$	0	0
$598$	1.69309	0.0692355
$599$	−20.9145	−0.854543	−0.427271	−	0.904123i	$-0.640525\pi$
$599$	−20.9145	−0.854543	−0.427271	+	0.904123i	$0.640525\pi$
$600$	0	0
$601$	−17.1967	−0.701469	−0.350735	−	0.936475i	$-0.614068\pi$
$601$	−17.1967	−0.701469	−0.350735	+	0.936475i	$0.614068\pi$
$602$	19.0886	0.777992
$603$	0	0
$604$	−24.2369	−0.986184
$605$	1.00000	0.0406558
$606$	0	0
$607$	9.39960	0.381518	0.190759	−	0.981637i	$-0.438905\pi$
$607$	9.39960	0.381518	0.190759	+	0.981637i	$0.438905\pi$
$608$	5.80932	0.235599
$609$	0	0
$610$	2.59346	0.105006
$611$	−1.64877	−0.0667022
$612$	0	0
$613$	−19.3353	−0.780945	−0.390472	−	0.920615i	$-0.627688\pi$
$613$	−19.3353	−0.780945	−0.390472	+	0.920615i	$0.627688\pi$
$614$	17.6548	0.712490
$615$	0	0
$616$	7.81631	0.314928
$617$	−33.3973	−1.34452	−0.672262	−	0.740313i	$-0.734677\pi$
$617$	−33.3973	−1.34452	−0.672262	+	0.740313i	$0.734677\pi$
$618$	0	0
$619$	−0.135908	−0.00546262	−0.00273131	−	0.999996i	$-0.500869\pi$
$619$	−0.135908	−0.00546262	−0.00273131	+	0.999996i	$0.500869\pi$
$620$	−2.76625	−0.111095
$621$	0	0
$622$	−0.281243	−0.0112768
$623$	34.9685	1.40098
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.10834	−0.284106
$627$	0	0
$628$	20.3587	0.812401
$629$	24.0154	0.957558
$630$	0	0
$631$	−35.1891	−1.40086	−0.700428	−	0.713723i	$-0.747008\pi$
$631$	−35.1891	−1.40086	−0.700428	+	0.713723i	$0.747008\pi$
$632$	23.7480	0.944647
$633$	0	0
$634$	5.20390	0.206673
$635$	−9.35796	−0.371359
$636$	0	0
$637$	0.501715	0.0198787
$638$	−8.50901	−0.336875
$639$	0	0
$640$	8.50893	0.336345
$641$	−39.9163	−1.57660	−0.788299	−	0.615293i	$-0.789038\pi$
$641$	−39.9163	−1.57660	−0.788299	+	0.615293i	$0.789038\pi$
$642$	0	0
$643$	29.4558	1.16162	0.580812	−	0.814038i	$-0.302735\pi$
$643$	29.4558	1.16162	0.580812	+	0.814038i	$0.302735\pi$
$644$	20.2048	0.796181
$645$	0	0
$646$	−1.94110	−0.0763714
$647$	−10.9142	−0.429080	−0.214540	−	0.976715i	$-0.568825\pi$
$647$	−10.9142	−0.429080	−0.214540	+	0.976715i	$0.568825\pi$
$648$	0	0
$649$	−9.96412	−0.391126
$650$	−0.321993	−0.0126296
$651$	0	0
$652$	−0.135613	−0.00531101
$653$	−10.9828	−0.429791	−0.214896	−	0.976637i	$-0.568941\pi$
$653$	−10.9828	−0.429791	−0.214896	+	0.976637i	$0.568941\pi$
$654$	0	0
$655$	−14.3775	−0.561776
$656$	4.40943	0.172159
$657$	0	0
$658$	9.77382	0.381023
$659$	−34.1879	−1.33177	−0.665885	−	0.746054i	$-0.731946\pi$
$659$	−34.1879	−1.33177	−0.665885	+	0.746054i	$0.731946\pi$
$660$	0	0
$661$	−28.4076	−1.10493	−0.552464	−	0.833537i	$-0.686312\pi$
$661$	−28.4076	−1.10493	−0.552464	+	0.833537i	$0.686312\pi$
$662$	10.4641	0.406697
$663$	0	0
$664$	−6.44294	−0.250034
$665$	−2.87567	−0.111514
$666$	0	0
$667$	−54.9168	−2.12639
$668$	2.89289	0.111929
$669$	0	0
$670$	9.65217	0.372896
$671$	−3.18327	−0.122889
$672$	0	0
$673$	−34.5379	−1.33134	−0.665669	−	0.746247i	$-0.731854\pi$
$673$	−34.5379	−1.33134	−0.665669	+	0.746247i	$0.731854\pi$
$674$	23.8304	0.917913
$675$	0	0
$676$	17.1624	0.660091
$677$	8.70123	0.334415	0.167208	−	0.985922i	$-0.446525\pi$
$677$	8.70123	0.334415	0.167208	+	0.985922i	$0.446525\pi$
$678$	0	0
$679$	−20.2475	−0.777029
$680$	−6.47596	−0.248342
$681$	0	0
$682$	−1.68661	−0.0645834
$683$	−44.7446	−1.71210	−0.856052	−	0.516889i	$-0.827090\pi$
$683$	−44.7446	−1.71210	−0.856052	+	0.516889i	$0.827090\pi$
$684$	0	0
$685$	−2.40548	−0.0919085
$686$	13.4258	0.512600
$687$	0	0
$688$	−3.73166	−0.142268
$689$	−4.29399	−0.163588
$690$	0	0
$691$	51.3502	1.95345	0.976727	−	0.214486i	$-0.0688078\pi$
$691$	51.3502	1.95345	0.976727	+	0.214486i	$0.0688078\pi$
$692$	−25.4078	−0.965860
$693$	0	0
$694$	−1.99434	−0.0757039
$695$	−2.81138	−0.106642
$696$	0	0
$697$	−22.9377	−0.868827
$698$	2.04732	0.0774923
$699$	0	0
$700$	−3.84257	−0.145236
$701$	33.4313	1.26268	0.631341	−	0.775506i	$-0.282505\pi$
$701$	33.4313	1.26268	0.631341	+	0.775506i	$0.282505\pi$
$702$	0	0
$703$	10.0797	0.380165
$704$	3.81693	0.143856
$705$	0	0
$706$	18.0773	0.680347
$707$	−46.6548	−1.75463
$708$	0	0
$709$	−36.7993	−1.38203	−0.691014	−	0.722841i	$-0.742836\pi$
$709$	−36.7993	−1.38203	−0.691014	+	0.722841i	$0.742836\pi$
$710$	11.9190	0.447313
$711$	0	0
$712$	33.0523	1.23869
$713$	−10.8853	−0.407657
$714$	0	0
$715$	0.395222	0.0147805
$716$	17.0825	0.638404
$717$	0	0
$718$	10.6452	0.397276
$719$	17.4738	0.651664	0.325832	−	0.945428i	$-0.394356\pi$
$719$	17.4738	0.651664	0.325832	+	0.945428i	$0.394356\pi$
$720$	0	0
$721$	−1.28862	−0.0479908
$722$	−0.814716	−0.0303206
$723$	0	0
$724$	26.9686	1.00228
$725$	10.4441	0.387886
$726$	0	0
$727$	−28.7311	−1.06558	−0.532789	−	0.846248i	$-0.678856\pi$
$727$	−28.7311	−1.06558	−0.532789	+	0.846248i	$0.678856\pi$
$728$	3.08917	0.114492
$729$	0	0
$730$	−7.74080	−0.286500
$731$	19.4120	0.717977
$732$	0	0
$733$	−28.3653	−1.04770	−0.523848	−	0.851812i	$-0.675504\pi$
$733$	−28.3653	−1.04770	−0.523848	+	0.851812i	$0.675504\pi$
$734$	−20.2561	−0.747668
$735$	0	0
$736$	30.5462	1.12595
$737$	−11.8473	−0.436400
$738$	0	0
$739$	31.1741	1.14676	0.573379	−	0.819290i	$-0.305632\pi$
$739$	31.1741	1.14676	0.573379	+	0.819290i	$0.305632\pi$
$740$	13.4689	0.495128
$741$	0	0
$742$	25.4545	0.934464
$743$	8.48554	0.311304	0.155652	−	0.987812i	$-0.450252\pi$
$743$	8.48554	0.311304	0.155652	+	0.987812i	$0.450252\pi$
$744$	0	0
$745$	12.4677	0.456781
$746$	−29.4357	−1.07772
$747$	0	0
$748$	3.18364	0.116406
$749$	−9.91934	−0.362445
$750$	0	0
$751$	18.0977	0.660396	0.330198	−	0.943912i	$-0.392884\pi$
$751$	18.0977	0.660396	0.330198	+	0.943912i	$0.392884\pi$
$752$	−1.91070	−0.0696762
$753$	0	0
$754$	−3.36295	−0.122471
$755$	18.1381	0.660114
$756$	0	0
$757$	12.4912	0.454000	0.227000	−	0.973895i	$-0.427108\pi$
$757$	12.4912	0.454000	0.227000	+	0.973895i	$0.427108\pi$
$758$	7.97943	0.289826
$759$	0	0
$760$	−2.71809	−0.0985953
$761$	2.35448	0.0853500	0.0426750	−	0.999089i	$-0.486412\pi$
$761$	2.35448	0.0853500	0.0426750	+	0.999089i	$0.486412\pi$
$762$	0	0
$763$	−41.8044	−1.51342
$764$	−28.2923	−1.02358
$765$	0	0
$766$	−6.48862	−0.234443
$767$	−3.93804	−0.142194
$768$	0	0
$769$	35.8937	1.29436	0.647180	−	0.762337i	$-0.275948\pi$
$769$	35.8937	1.29436	0.647180	+	0.762337i	$0.275948\pi$
$770$	−2.34285	−0.0844305
$771$	0	0
$772$	1.08525	0.0390590
$773$	43.7007	1.57180	0.785902	−	0.618351i	$-0.212199\pi$
$773$	43.7007	1.57180	0.785902	+	0.618351i	$0.212199\pi$
$774$	0	0
$775$	2.07018	0.0743629
$776$	−19.1380	−0.687015
$777$	0	0
$778$	10.2076	0.365962
$779$	−9.62740	−0.344937
$780$	0	0
$781$	−14.6297	−0.523491
$782$	−10.2066	−0.364986
$783$	0	0
$784$	0.581420	0.0207650
$785$	−15.2358	−0.543791
$786$	0	0
$787$	37.9709	1.35352	0.676758	−	0.736205i	$-0.263384\pi$
$787$	37.9709	1.35352	0.676758	+	0.736205i	$0.263384\pi$
$788$	−20.9228	−0.745342
$789$	0	0
$790$	−7.11821	−0.253255
$791$	17.8786	0.635691
$792$	0	0
$793$	−1.25810	−0.0446764
$794$	6.31407	0.224078
$795$	0	0
$796$	21.1043	0.748022
$797$	−5.56282	−0.197045	−0.0985225	−	0.995135i	$-0.531412\pi$
$797$	−5.56282	−0.197045	−0.0985225	+	0.995135i	$0.531412\pi$
$798$	0	0
$799$	9.93941	0.351631
$800$	−5.80932	−0.205390
$801$	0	0
$802$	−8.98661	−0.317328
$803$	9.50123	0.335291
$804$	0	0
$805$	−15.1207	−0.532933
$806$	−0.666583	−0.0234794
$807$	0	0
$808$	−44.0982	−1.55137
$809$	24.7310	0.869497	0.434748	−	0.900552i	$-0.356837\pi$
$809$	24.7310	0.869497	0.434748	+	0.900552i	$0.356837\pi$
$810$	0	0
$811$	−48.7292	−1.71111	−0.855557	−	0.517709i	$-0.826785\pi$
$811$	−48.7292	−1.71111	−0.855557	+	0.517709i	$0.826785\pi$
$812$	−40.1324	−1.40837
$813$	0	0
$814$	8.21213	0.287835
$815$	0.101489	0.00355499
$816$	0	0
$817$	8.14758	0.285048
$818$	−8.77083	−0.306665
$819$	0	0
$820$	−12.8645	−0.449248
$821$	−19.7198	−0.688227	−0.344113	−	0.938928i	$-0.611820\pi$
$821$	−19.7198	−0.688227	−0.344113	+	0.938928i	$0.611820\pi$
$822$	0	0
$823$	−51.7677	−1.80451	−0.902254	−	0.431205i	$-0.858089\pi$
$823$	−51.7677	−1.80451	−0.902254	+	0.431205i	$0.858089\pi$
$824$	−1.21801	−0.0424313
$825$	0	0
$826$	23.3444	0.812257
$827$	−10.1425	−0.352690	−0.176345	−	0.984328i	$-0.556427\pi$
$827$	−10.1425	−0.352690	−0.176345	+	0.984328i	$0.556427\pi$
$828$	0	0
$829$	−47.0391	−1.63374	−0.816868	−	0.576825i	$-0.804292\pi$
$829$	−47.0391	−1.63374	−0.816868	+	0.576825i	$0.804292\pi$
$830$	1.93120	0.0670329
$831$	0	0
$832$	1.50853	0.0522990
$833$	−3.02452	−0.104794
$834$	0	0
$835$	−2.16495	−0.0749213
$836$	1.33624	0.0462148
$837$	0	0
$838$	−12.5774	−0.434480
$839$	−36.2804	−1.25254	−0.626270	−	0.779606i	$-0.715419\pi$
$839$	−36.2804	−1.25254	−0.626270	+	0.779606i	$0.715419\pi$
$840$	0	0
$841$	80.0802	2.76139
$842$	10.1611	0.350175
$843$	0	0
$844$	15.7065	0.540641
$845$	−12.8438	−0.441840
$846$	0	0
$847$	2.87567	0.0988090
$848$	−4.97615	−0.170882
$849$	0	0
$850$	1.94110	0.0665790
$851$	53.0007	1.81684
$852$	0	0
$853$	−1.04609	−0.0358174	−0.0179087	−	0.999840i	$-0.505701\pi$
$853$	−1.04609	−0.0358174	−0.0179087	+	0.999840i	$0.505701\pi$
$854$	7.45793	0.255205
$855$	0	0
$856$	−9.37578	−0.320458
$857$	14.1239	0.482462	0.241231	−	0.970468i	$-0.422449\pi$
$857$	14.1239	0.482462	0.241231	+	0.970468i	$0.422449\pi$
$858$	0	0
$859$	−31.1123	−1.06154	−0.530769	−	0.847516i	$-0.678097\pi$
$859$	−31.1123	−1.06154	−0.530769	+	0.847516i	$0.678097\pi$
$860$	10.8871	0.371247
$861$	0	0
$862$	−19.5413	−0.665579
$863$	18.4276	0.627284	0.313642	−	0.949541i	$-0.398451\pi$
$863$	18.4276	0.627284	0.313642	+	0.949541i	$0.398451\pi$
$864$	0	0
$865$	19.0144	0.646510
$866$	1.27552	0.0433441
$867$	0	0
$868$	−7.95480	−0.270004
$869$	8.73705	0.296384
$870$	0	0
$871$	−4.68230	−0.158654
$872$	−39.5136	−1.33810
$873$	0	0
$874$	−4.28389	−0.144905
$875$	2.87567	0.0972152
$876$	0	0
$877$	2.52851	0.0853818	0.0426909	−	0.999088i	$-0.486407\pi$
$877$	2.52851	0.0853818	0.0426909	+	0.999088i	$0.486407\pi$
$878$	−13.6108	−0.459342
$879$	0	0
$880$	0.458008	0.0154395
$881$	10.6008	0.357150	0.178575	−	0.983926i	$-0.442851\pi$
$881$	10.6008	0.357150	0.178575	+	0.983926i	$0.442851\pi$
$882$	0	0
$883$	−32.3642	−1.08914	−0.544571	−	0.838715i	$-0.683308\pi$
$883$	−32.3642	−1.08914	−0.544571	+	0.838715i	$0.683308\pi$
$884$	1.25825	0.0423194
$885$	0	0
$886$	−11.0081	−0.369823
$887$	−2.43260	−0.0816787	−0.0408393	−	0.999166i	$-0.513003\pi$
$887$	−2.43260	−0.0816787	−0.0408393	+	0.999166i	$0.513003\pi$
$888$	0	0
$889$	−26.9104	−0.902545
$890$	−9.90705	−0.332085
$891$	0	0
$892$	23.0784	0.772722
$893$	4.17176	0.139603
$894$	0	0
$895$	−12.7841	−0.427324
$896$	24.4688	0.817446
$897$	0	0
$898$	−3.79058	−0.126493
$899$	21.6212	0.721108
$900$	0	0
$901$	25.8857	0.862379
$902$	−7.84359	−0.261163
$903$	0	0
$904$	16.8989	0.562050
$905$	−20.1825	−0.670887
$906$	0	0
$907$	−36.4973	−1.21187	−0.605936	−	0.795513i	$-0.707201\pi$
$907$	−36.4973	−1.21187	−0.605936	+	0.795513i	$0.707201\pi$
$908$	4.94405	0.164074
$909$	0	0
$910$	−0.925945	−0.0306948
$911$	9.89311	0.327773	0.163887	−	0.986479i	$-0.447597\pi$
$911$	9.89311	0.327773	0.163887	+	0.986479i	$0.447597\pi$
$912$	0	0
$913$	−2.37039	−0.0784486
$914$	24.2749	0.802941
$915$	0	0
$916$	1.58771	0.0524594
$917$	−41.3449	−1.36533
$918$	0	0
$919$	45.3873	1.49719	0.748595	−	0.663028i	$-0.230729\pi$
$919$	45.3873	1.49719	0.748595	+	0.663028i	$0.230729\pi$
$920$	−14.2921	−0.471196
$921$	0	0
$922$	25.7641	0.848497
$923$	−5.78197	−0.190316
$924$	0	0
$925$	−10.0797	−0.331420
$926$	13.6409	0.448269
$927$	0	0
$928$	−60.6734	−1.99170
$929$	−2.53064	−0.0830275	−0.0415137	−	0.999138i	$-0.513218\pi$
$929$	−2.53064	−0.0830275	−0.0415137	+	0.999138i	$0.513218\pi$
$930$	0	0
$931$	−1.26945	−0.0416046
$932$	−17.3984	−0.569905
$933$	0	0
$934$	11.1333	0.364292
$935$	−2.38254	−0.0779175
$936$	0	0
$937$	20.3367	0.664370	0.332185	−	0.943214i	$-0.392214\pi$
$937$	20.3367	0.664370	0.332185	+	0.943214i	$0.392214\pi$
$938$	27.7564	0.906279
$939$	0	0
$940$	5.57447	0.181819
$941$	−30.9251	−1.00813	−0.504065	−	0.863665i	$-0.668163\pi$
$941$	−30.9251	−1.00813	−0.504065	+	0.863665i	$0.668163\pi$
$942$	0	0
$943$	−50.6222	−1.64849
$944$	−4.56365	−0.148534
$945$	0	0
$946$	6.63796	0.215819
$947$	31.9467	1.03813	0.519065	−	0.854735i	$-0.326280\pi$
$947$	31.9467	1.03813	0.519065	+	0.854735i	$0.326280\pi$
$948$	0	0
$949$	3.75509	0.121895
$950$	0.814716	0.0264329
$951$	0	0
$952$	−18.6227	−0.603565
$953$	39.5142	1.27999	0.639995	−	0.768379i	$-0.278936\pi$
$953$	39.5142	1.27999	0.639995	+	0.768379i	$0.278936\pi$
$954$	0	0
$955$	21.1731	0.685145
$956$	−18.3635	−0.593919
$957$	0	0
$958$	−12.6420	−0.408445
$959$	−6.91735	−0.223373
$960$	0	0
$961$	−26.7144	−0.861754
$962$	3.24561	0.104643
$963$	0	0
$964$	−41.1248	−1.32454
$965$	−0.812169	−0.0261446
$966$	0	0
$967$	−15.1134	−0.486015	−0.243007	−	0.970024i	$-0.578134\pi$
$967$	−15.1134	−0.486015	−0.243007	+	0.970024i	$0.578134\pi$
$968$	2.71809	0.0873626
$969$	0	0
$970$	5.73641	0.184185
$971$	23.7985	0.763729	0.381865	−	0.924218i	$-0.375282\pi$
$971$	23.7985	0.763729	0.381865	+	0.924218i	$0.375282\pi$
$972$	0	0
$973$	−8.08460	−0.259180
$974$	1.18196	0.0378723
$975$	0	0
$976$	−1.45796	−0.0466683
$977$	−20.2759	−0.648682	−0.324341	−	0.945940i	$-0.605143\pi$
$977$	−20.2759	−0.648682	−0.324341	+	0.945940i	$0.605143\pi$
$978$	0	0
$979$	12.1601	0.388639
$980$	−1.69629	−0.0541860
$981$	0	0
$982$	20.4917	0.653918
$983$	−11.9122	−0.379939	−0.189970	−	0.981790i	$-0.560839\pi$
$983$	−11.9122	−0.379939	−0.189970	+	0.981790i	$0.560839\pi$
$984$	0	0
$985$	15.6580	0.498904
$986$	20.2731	0.645626
$987$	0	0
$988$	0.528110	0.0168014
$989$	42.8411	1.36227
$990$	0	0
$991$	0.673919	0.0214077	0.0107039	−	0.999943i	$-0.496593\pi$
$991$	0.673919	0.0214077	0.0107039	+	0.999943i	$0.496593\pi$
$992$	−12.0263	−0.381836
$993$	0	0
$994$	34.2751	1.08714
$995$	−15.7938	−0.500698
$996$	0	0
$997$	24.2577	0.768248	0.384124	−	0.923282i	$-0.374504\pi$
$997$	24.2577	0.768248	0.384124	+	0.923282i	$0.374504\pi$
$998$	−11.6067	−0.367402
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bq.1.6	✓	14
3.2	odd	2		9405.2.a.br.1.9	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9405.2.a.bq.1.6	✓	14	1.1	even	1	trivial
9405.2.a.br.1.9	yes	14	3.2	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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-0.814716 q^{2} -1.33624 q^{4} +1.00000 q^{5} +2.87567 q^{7} +2.71809 q^{8} +O(q^{10})\)	\(q-0.814716 q^{2} -1.33624 q^{4} +1.00000 q^{5} +2.87567 q^{7} +2.71809 q^{8} -0.814716 q^{10} +1.00000 q^{11} +0.395222 q^{13} -2.34285 q^{14} +0.458008 q^{16} -2.38254 q^{17} -1.00000 q^{19} -1.33624 q^{20} -0.814716 q^{22} -5.25814 q^{23} +1.00000 q^{25} -0.321993 q^{26} -3.84257 q^{28} +10.4441 q^{29} +2.07018 q^{31} -5.80932 q^{32} +1.94110 q^{34} +2.87567 q^{35} -10.0797 q^{37} +0.814716 q^{38} +2.71809 q^{40} +9.62740 q^{41} -8.14758 q^{43} -1.33624 q^{44} +4.28389 q^{46} -4.17176 q^{47} +1.26945 q^{49} -0.814716 q^{50} -0.528110 q^{52} -10.8648 q^{53} +1.00000 q^{55} +7.81631 q^{56} -8.50901 q^{58} -9.96412 q^{59} -3.18327 q^{61} -1.68661 q^{62} +3.81693 q^{64} +0.395222 q^{65} -11.8473 q^{67} +3.18364 q^{68} -2.34285 q^{70} -14.6297 q^{71} +9.50123 q^{73} +8.21213 q^{74} +1.33624 q^{76} +2.87567 q^{77} +8.73705 q^{79} +0.458008 q^{80} -7.84359 q^{82} -2.37039 q^{83} -2.38254 q^{85} +6.63796 q^{86} +2.71809 q^{88} +12.1601 q^{89} +1.13653 q^{91} +7.02613 q^{92} +3.39880 q^{94} -1.00000 q^{95} -7.04099 q^{97} -1.03424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

Introduction

L-functions

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