LMFDB - Embedded newform 4005.2.a.r.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 $ x^10 - 4x^9 - 8x^8 + 35x^7 + 29x^6 - 103x^5 - 57x^4 + 106x^3 + 29x^2 - 39*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.43743$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.43743 q^{2} +0.0662179 q^{4} -1.00000 q^{5} +3.21347 q^{7} -2.77969 q^{8} +O(q^{10})$	\(q+1.43743 q^{2} +0.0662179 q^{4} -1.00000 q^{5} +3.21347 q^{7} -2.77969 q^{8} -1.43743 q^{10} +0.448677 q^{11} +6.82228 q^{13} +4.61915 q^{14} -4.12805 q^{16} -0.575549 q^{17} -1.07472 q^{19} -0.0662179 q^{20} +0.644944 q^{22} -5.03019 q^{23} +1.00000 q^{25} +9.80659 q^{26} +0.212789 q^{28} +5.82188 q^{29} +0.531933 q^{31} -0.374433 q^{32} -0.827314 q^{34} -3.21347 q^{35} +1.20230 q^{37} -1.54483 q^{38} +2.77969 q^{40} +0.685190 q^{41} -3.93855 q^{43} +0.0297105 q^{44} -7.23057 q^{46} +9.50881 q^{47} +3.32638 q^{49} +1.43743 q^{50} +0.451757 q^{52} +0.628816 q^{53} -0.448677 q^{55} -8.93243 q^{56} +8.36858 q^{58} +1.24088 q^{59} +1.79073 q^{61} +0.764619 q^{62} +7.71788 q^{64} -6.82228 q^{65} +6.61441 q^{67} -0.0381116 q^{68} -4.61915 q^{70} +7.73672 q^{71} +2.46799 q^{73} +1.72823 q^{74} -0.0711655 q^{76} +1.44181 q^{77} -0.0524346 q^{79} +4.12805 q^{80} +0.984915 q^{82} -0.0387270 q^{83} +0.575549 q^{85} -5.66141 q^{86} -1.24718 q^{88} -1.00000 q^{89} +21.9232 q^{91} -0.333089 q^{92} +13.6683 q^{94} +1.07472 q^{95} -7.93523 q^{97} +4.78145 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * 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 $

$2$	1.43743	1.01642	0.508210	−	0.861233i	$-0.330307\pi$
$2$	1.43743	1.01642	0.508210	+	0.861233i	$0.330307\pi$
$3$	0	0
$3$	0	0
$4$	0.0662179	0.0331090
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.21347	1.21458	0.607288	−	0.794481i	$-0.292257\pi$
$7$	3.21347	1.21458	0.607288	+	0.794481i	$0.292257\pi$
$8$	−2.77969	−0.982767
$9$	0	0
$10$	−1.43743	−0.454557
$11$	0.448677	0.135281	0.0676406	−	0.997710i	$-0.478453\pi$
$11$	0.448677	0.135281	0.0676406	+	0.997710i	$0.478453\pi$
$12$	0	0
$13$	6.82228	1.89216	0.946081	−	0.323932i	$-0.105005\pi$
$13$	6.82228	1.89216	0.946081	+	0.323932i	$0.105005\pi$
$14$	4.61915	1.23452
$15$	0	0
$16$	−4.12805	−1.03201
$17$	−0.575549	−0.139591	−0.0697955	−	0.997561i	$-0.522235\pi$
$17$	−0.575549	−0.139591	−0.0697955	+	0.997561i	$0.522235\pi$
$18$	0	0
$19$	−1.07472	−0.246557	−0.123278	−	0.992372i	$-0.539341\pi$
$19$	−1.07472	−0.246557	−0.123278	+	0.992372i	$0.539341\pi$
$20$	−0.0662179	−0.0148068
$21$	0	0
$22$	0.644944	0.137503
$23$	−5.03019	−1.04887	−0.524433	−	0.851452i	$-0.675723\pi$
$23$	−5.03019	−1.04887	−0.524433	+	0.851452i	$0.675723\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.80659	1.92323
$27$	0	0
$28$	0.212789	0.0402134
$29$	5.82188	1.08110	0.540548	−	0.841313i	$-0.318217\pi$
$29$	5.82188	1.08110	0.540548	+	0.841313i	$0.318217\pi$
$30$	0	0
$31$	0.531933	0.0955381	0.0477690	−	0.998858i	$-0.484789\pi$
$31$	0.531933	0.0955381	0.0477690	+	0.998858i	$0.484789\pi$
$32$	−0.374433	−0.0661910
$33$	0	0
$34$	−0.827314	−0.141883
$35$	−3.21347	−0.543175
$36$	0	0
$37$	1.20230	0.197657	0.0988286	−	0.995104i	$-0.468490\pi$
$37$	1.20230	0.197657	0.0988286	+	0.995104i	$0.468490\pi$
$38$	−1.54483	−0.250605
$39$	0	0
$40$	2.77969	0.439507
$41$	0.685190	0.107009	0.0535043	−	0.998568i	$-0.482961\pi$
$41$	0.685190	0.107009	0.0535043	+	0.998568i	$0.482961\pi$
$42$	0	0
$43$	−3.93855	−0.600623	−0.300312	−	0.953841i	$-0.597091\pi$
$43$	−3.93855	−0.600623	−0.300312	+	0.953841i	$0.597091\pi$
$44$	0.0297105	0.00447902
$45$	0	0
$46$	−7.23057	−1.06609
$47$	9.50881	1.38700	0.693502	−	0.720455i	$-0.256067\pi$
$47$	9.50881	1.38700	0.693502	+	0.720455i	$0.256067\pi$
$48$	0	0
$49$	3.32638	0.475197
$50$	1.43743	0.203284
$51$	0	0
$52$	0.451757	0.0626475
$53$	0.628816	0.0863745	0.0431872	−	0.999067i	$-0.486249\pi$
$53$	0.628816	0.0863745	0.0431872	+	0.999067i	$0.486249\pi$
$54$	0	0
$55$	−0.448677	−0.0604996
$56$	−8.93243	−1.19365
$57$	0	0
$58$	8.36858	1.09885
$59$	1.24088	0.161549	0.0807747	−	0.996732i	$-0.474261\pi$
$59$	1.24088	0.161549	0.0807747	+	0.996732i	$0.474261\pi$
$60$	0	0
$61$	1.79073	0.229279	0.114640	−	0.993407i	$-0.463429\pi$
$61$	1.79073	0.229279	0.114640	+	0.993407i	$0.463429\pi$
$62$	0.764619	0.0971068
$63$	0	0
$64$	7.71788	0.964735
$65$	−6.82228	−0.846200
$66$	0	0
$67$	6.61441	0.808079	0.404039	−	0.914742i	$-0.367606\pi$
$67$	6.61441	0.808079	0.404039	+	0.914742i	$0.367606\pi$
$68$	−0.0381116	−0.00462172
$69$	0	0
$70$	−4.61915	−0.552094
$71$	7.73672	0.918180	0.459090	−	0.888390i	$-0.348176\pi$
$71$	7.73672	0.918180	0.459090	+	0.888390i	$0.348176\pi$
$72$	0	0
$73$	2.46799	0.288856	0.144428	−	0.989515i	$-0.453866\pi$
$73$	2.46799	0.288856	0.144428	+	0.989515i	$0.453866\pi$
$74$	1.72823	0.200903
$75$	0	0
$76$	−0.0711655	−0.00816324
$77$	1.44181	0.164310
$78$	0	0
$79$	−0.0524346	−0.00589935	−0.00294968	−	0.999996i	$-0.500939\pi$
$79$	−0.0524346	−0.00589935	−0.00294968	+	0.999996i	$0.500939\pi$
$80$	4.12805	0.461530
$81$	0	0
$82$	0.984915	0.108766
$83$	−0.0387270	−0.00425085	−0.00212542	−	0.999998i	$-0.500677\pi$
$83$	−0.0387270	−0.00425085	−0.00212542	+	0.999998i	$0.500677\pi$
$84$	0	0
$85$	0.575549	0.0624270
$86$	−5.66141	−0.610485
$87$	0	0
$88$	−1.24718	−0.132950
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	21.9232	2.29817
$92$	−0.333089	−0.0347269
$93$	0	0
$94$	13.6683	1.40978
$95$	1.07472	0.110264
$96$	0	0
$97$	−7.93523	−0.805700	−0.402850	−	0.915266i	$-0.631980\pi$
$97$	−7.93523	−0.805700	−0.402850	+	0.915266i	$0.631980\pi$
$98$	4.78145	0.482999
$99$	0	0
$100$	0.0662179	0.00662179
$101$	−11.1668	−1.11114	−0.555570	−	0.831470i	$-0.687500\pi$
$101$	−11.1668	−1.11114	−0.555570	+	0.831470i	$0.687500\pi$
$102$	0	0
$103$	13.2714	1.30767	0.653834	−	0.756638i	$-0.273159\pi$
$103$	13.2714	1.30767	0.653834	+	0.756638i	$0.273159\pi$
$104$	−18.9638	−1.85955
$105$	0	0
$106$	0.903881	0.0877927
$107$	9.15828	0.885364	0.442682	−	0.896679i	$-0.354027\pi$
$107$	9.15828	0.885364	0.442682	+	0.896679i	$0.354027\pi$
$108$	0	0
$109$	14.9943	1.43619	0.718094	−	0.695946i	$-0.245015\pi$
$109$	14.9943	1.43619	0.718094	+	0.695946i	$0.245015\pi$
$110$	−0.644944	−0.0614930
$111$	0	0
$112$	−13.2654	−1.25346
$113$	4.90006	0.460959	0.230480	−	0.973077i	$-0.425971\pi$
$113$	4.90006	0.460959	0.230480	+	0.973077i	$0.425971\pi$
$114$	0	0
$115$	5.03019	0.469067
$116$	0.385513	0.0357940
$117$	0	0
$118$	1.78369	0.164202
$119$	−1.84951	−0.169544
$120$	0	0
$121$	−10.7987	−0.981699
$122$	2.57405	0.233044
$123$	0	0
$124$	0.0352235	0.00316317
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.4368	1.10359	0.551794	−	0.833980i	$-0.313943\pi$
$127$	12.4368	1.10359	0.551794	+	0.833980i	$0.313943\pi$
$128$	11.8428	1.04677
$129$	0	0
$130$	−9.80659	−0.860094
$131$	13.3478	1.16621	0.583103	−	0.812398i	$-0.301838\pi$
$131$	13.3478	1.16621	0.583103	+	0.812398i	$0.301838\pi$
$132$	0	0
$133$	−3.45357	−0.299462
$134$	9.50778	0.821347
$135$	0	0
$136$	1.59984	0.137186
$137$	−3.40815	−0.291178	−0.145589	−	0.989345i	$-0.546508\pi$
$137$	−3.40815	−0.291178	−0.145589	+	0.989345i	$0.546508\pi$
$138$	0	0
$139$	−7.69707	−0.652857	−0.326428	−	0.945222i	$-0.605845\pi$
$139$	−7.69707	−0.652857	−0.326428	+	0.945222i	$0.605845\pi$
$140$	−0.212789	−0.0179840
$141$	0	0
$142$	11.1210	0.933256
$143$	3.06100	0.255974
$144$	0	0
$145$	−5.82188	−0.483481
$146$	3.54757	0.293599
$147$	0	0
$148$	0.0796139	0.00654422
$149$	−6.42394	−0.526270	−0.263135	−	0.964759i	$-0.584756\pi$
$149$	−6.42394	−0.526270	−0.263135	+	0.964759i	$0.584756\pi$
$150$	0	0
$151$	10.5964	0.862323	0.431162	−	0.902275i	$-0.358104\pi$
$151$	10.5964	0.862323	0.431162	+	0.902275i	$0.358104\pi$
$152$	2.98737	0.242308
$153$	0	0
$154$	2.07251	0.167007
$155$	−0.531933	−0.0427259
$156$	0	0
$157$	12.2748	0.979636	0.489818	−	0.871825i	$-0.337063\pi$
$157$	12.2748	0.979636	0.489818	+	0.871825i	$0.337063\pi$
$158$	−0.0753713	−0.00599622
$159$	0	0
$160$	0.374433	0.0296015
$161$	−16.1643	−1.27393
$162$	0	0
$163$	−4.11048	−0.321958	−0.160979	−	0.986958i	$-0.551465\pi$
$163$	−4.11048	−0.321958	−0.160979	+	0.986958i	$0.551465\pi$
$164$	0.0453718	0.00354294
$165$	0	0
$166$	−0.0556676	−0.00432064
$167$	21.0037	1.62531	0.812657	−	0.582742i	$-0.198020\pi$
$167$	21.0037	1.62531	0.812657	+	0.582742i	$0.198020\pi$
$168$	0	0
$169$	33.5436	2.58027
$170$	0.827314	0.0634521
$171$	0	0
$172$	−0.260803	−0.0198860
$173$	−18.0263	−1.37052	−0.685258	−	0.728300i	$-0.740311\pi$
$173$	−18.0263	−1.37052	−0.685258	+	0.728300i	$0.740311\pi$
$174$	0	0
$175$	3.21347	0.242915
$176$	−1.85216	−0.139612
$177$	0	0
$178$	−1.43743	−0.107740
$179$	−0.313514	−0.0234331	−0.0117166	−	0.999931i	$-0.503730\pi$
$179$	−0.313514	−0.0234331	−0.0117166	+	0.999931i	$0.503730\pi$
$180$	0	0
$181$	−23.8192	−1.77047	−0.885233	−	0.465148i	$-0.846001\pi$
$181$	−23.8192	−1.77047	−0.885233	+	0.465148i	$0.846001\pi$
$182$	31.5132	2.33591
$183$	0	0
$184$	13.9823	1.03079
$185$	−1.20230	−0.0883950
$186$	0	0
$187$	−0.258236	−0.0188841
$188$	0.629654	0.0459222
$189$	0	0
$190$	1.54483	0.112074
$191$	−17.8485	−1.29147	−0.645737	−	0.763560i	$-0.723450\pi$
$191$	−17.8485	−1.29147	−0.645737	+	0.763560i	$0.723450\pi$
$192$	0	0
$193$	23.9240	1.72208	0.861042	−	0.508533i	$-0.169812\pi$
$193$	23.9240	1.72208	0.861042	+	0.508533i	$0.169812\pi$
$194$	−11.4064	−0.818930
$195$	0	0
$196$	0.220266	0.0157333
$197$	24.2128	1.72509	0.862546	−	0.505978i	$-0.168868\pi$
$197$	24.2128	1.72509	0.862546	+	0.505978i	$0.168868\pi$
$198$	0	0
$199$	13.2415	0.938668	0.469334	−	0.883021i	$-0.344494\pi$
$199$	13.2415	0.938668	0.469334	+	0.883021i	$0.344494\pi$
$200$	−2.77969	−0.196553
$201$	0	0
$202$	−16.0516	−1.12938
$203$	18.7084	1.31307
$204$	0	0
$205$	−0.685190	−0.0478557
$206$	19.0768	1.32914
$207$	0	0
$208$	−28.1627	−1.95273
$209$	−0.482201	−0.0333545
$210$	0	0
$211$	−17.6082	−1.21220	−0.606100	−	0.795388i	$-0.707267\pi$
$211$	−17.6082	−1.21220	−0.606100	+	0.795388i	$0.707267\pi$
$212$	0.0416389	0.00285977
$213$	0	0
$214$	13.1644	0.899901
$215$	3.93855	0.268607
$216$	0	0
$217$	1.70935	0.116038
$218$	21.5533	1.45977
$219$	0	0
$220$	−0.0297105	−0.00200308
$221$	−3.92656	−0.264129
$222$	0	0
$223$	7.84374	0.525256	0.262628	−	0.964897i	$-0.415411\pi$
$223$	7.84374	0.525256	0.262628	+	0.964897i	$0.415411\pi$
$224$	−1.20323	−0.0803940
$225$	0	0
$226$	7.04352	0.468528
$227$	−0.937950	−0.0622539	−0.0311270	−	0.999515i	$-0.509910\pi$
$227$	−0.937950	−0.0622539	−0.0311270	+	0.999515i	$0.509910\pi$
$228$	0	0
$229$	21.0598	1.39167	0.695835	−	0.718202i	$-0.255034\pi$
$229$	21.0598	1.39167	0.695835	+	0.718202i	$0.255034\pi$
$230$	7.23057	0.476769
$231$	0	0
$232$	−16.1830	−1.06247
$233$	24.0187	1.57352	0.786759	−	0.617260i	$-0.211757\pi$
$233$	24.0187	1.57352	0.786759	+	0.617260i	$0.211757\pi$
$234$	0	0
$235$	−9.50881	−0.620287
$236$	0.0821688	0.00534873
$237$	0	0
$238$	−2.65855	−0.172328
$239$	−4.27404	−0.276464	−0.138232	−	0.990400i	$-0.544142\pi$
$239$	−4.27404	−0.276464	−0.138232	+	0.990400i	$0.544142\pi$
$240$	0	0
$241$	−26.5842	−1.71244	−0.856219	−	0.516613i	$-0.827193\pi$
$241$	−26.5842	−1.71244	−0.856219	+	0.516613i	$0.827193\pi$
$242$	−15.5224	−0.997818
$243$	0	0
$244$	0.118578	0.00759120
$245$	−3.32638	−0.212514
$246$	0	0
$247$	−7.33202	−0.466525
$248$	−1.47861	−0.0938917
$249$	0	0
$250$	−1.43743	−0.0909113
$251$	2.33975	0.147684	0.0738420	−	0.997270i	$-0.476474\pi$
$251$	2.33975	0.147684	0.0738420	+	0.997270i	$0.476474\pi$
$252$	0	0
$253$	−2.25693	−0.141892
$254$	17.8771	1.12171
$255$	0	0
$256$	1.58751	0.0992192
$257$	−8.97721	−0.559983	−0.279992	−	0.960002i	$-0.590332\pi$
$257$	−8.97721	−0.559983	−0.279992	+	0.960002i	$0.590332\pi$
$258$	0	0
$259$	3.86356	0.240070
$260$	−0.451757	−0.0280168
$261$	0	0
$262$	19.1866	1.18536
$263$	−13.2157	−0.814914	−0.407457	−	0.913224i	$-0.633584\pi$
$263$	−13.2157	−0.814914	−0.407457	+	0.913224i	$0.633584\pi$
$264$	0	0
$265$	−0.628816	−0.0386278
$266$	−4.96428	−0.304379
$267$	0	0
$268$	0.437993	0.0267547
$269$	−12.6209	−0.769512	−0.384756	−	0.923018i	$-0.625714\pi$
$269$	−12.6209	−0.769512	−0.384756	+	0.923018i	$0.625714\pi$
$270$	0	0
$271$	6.52693	0.396483	0.198241	−	0.980153i	$-0.436477\pi$
$271$	6.52693	0.396483	0.198241	+	0.980153i	$0.436477\pi$
$272$	2.37589	0.144060
$273$	0	0
$274$	−4.89899	−0.295959
$275$	0.448677	0.0270563
$276$	0	0
$277$	−28.1514	−1.69145	−0.845726	−	0.533618i	$-0.820832\pi$
$277$	−28.1514	−1.69145	−0.845726	+	0.533618i	$0.820832\pi$
$278$	−11.0640	−0.663576
$279$	0	0
$280$	8.93243	0.533815
$281$	11.3122	0.674830	0.337415	−	0.941356i	$-0.390447\pi$
$281$	11.3122	0.674830	0.337415	+	0.941356i	$0.390447\pi$
$282$	0	0
$283$	−12.3568	−0.734534	−0.367267	−	0.930115i	$-0.619707\pi$
$283$	−12.3568	−0.734534	−0.367267	+	0.930115i	$0.619707\pi$
$284$	0.512309	0.0304000
$285$	0	0
$286$	4.39999	0.260177
$287$	2.20183	0.129970
$288$	0	0
$289$	−16.6687	−0.980514
$290$	−8.36858	−0.491420
$291$	0	0
$292$	0.163425	0.00956372
$293$	6.51731	0.380745	0.190373	−	0.981712i	$-0.439030\pi$
$293$	6.51731	0.380745	0.190373	+	0.981712i	$0.439030\pi$
$294$	0	0
$295$	−1.24088	−0.0722471
$296$	−3.34202	−0.194251
$297$	0	0
$298$	−9.23399	−0.534911
$299$	−34.3174	−1.98462
$300$	0	0
$301$	−12.6564	−0.729503
$302$	15.2316	0.876482
$303$	0	0
$304$	4.43649	0.254450
$305$	−1.79073	−0.102537
$306$	0	0
$307$	−20.4849	−1.16913	−0.584567	−	0.811345i	$-0.698736\pi$
$307$	−20.4849	−1.16913	−0.584567	+	0.811345i	$0.698736\pi$
$308$	0.0954737	0.00544012
$309$	0	0
$310$	−0.764619	−0.0434275
$311$	−7.11857	−0.403657	−0.201829	−	0.979421i	$-0.564688\pi$
$311$	−7.11857	−0.403657	−0.201829	+	0.979421i	$0.564688\pi$
$312$	0	0
$313$	−15.1846	−0.858286	−0.429143	−	0.903236i	$-0.641184\pi$
$313$	−15.1846	−0.858286	−0.429143	+	0.903236i	$0.641184\pi$
$314$	17.6442	0.995721
$315$	0	0
$316$	−0.00347211	−0.000195321 0
$317$	15.3561	0.862484	0.431242	−	0.902236i	$-0.358075\pi$
$317$	15.3561	0.862484	0.431242	+	0.902236i	$0.358075\pi$
$318$	0	0
$319$	2.61215	0.146252
$320$	−7.71788	−0.431443
$321$	0	0
$322$	−23.2352	−1.29485
$323$	0.618552	0.0344171
$324$	0	0
$325$	6.82228	0.378432
$326$	−5.90854	−0.327244
$327$	0	0
$328$	−1.90461	−0.105165
$329$	30.5563	1.68462
$330$	0	0
$331$	−29.0804	−1.59840	−0.799201	−	0.601065i	$-0.794743\pi$
$331$	−29.0804	−1.59840	−0.799201	+	0.601065i	$0.794743\pi$
$332$	−0.00256442	−0.000140741 0
$333$	0	0
$334$	30.1914	1.65200
$335$	−6.61441	−0.361384
$336$	0	0
$337$	−19.5508	−1.06500	−0.532500	−	0.846430i	$-0.678747\pi$
$337$	−19.5508	−1.06500	−0.532500	+	0.846430i	$0.678747\pi$
$338$	48.2167	2.62264
$339$	0	0
$340$	0.0381116	0.00206689
$341$	0.238666	0.0129245
$342$	0	0
$343$	−11.8051	−0.637414
$344$	10.9479	0.590273
$345$	0	0
$346$	−25.9117	−1.39302
$347$	−29.7414	−1.59660	−0.798301	−	0.602259i	$-0.794267\pi$
$347$	−29.7414	−1.59660	−0.798301	+	0.602259i	$0.794267\pi$
$348$	0	0
$349$	24.9884	1.33760	0.668800	−	0.743442i	$-0.266808\pi$
$349$	24.9884	1.33760	0.668800	+	0.743442i	$0.266808\pi$
$350$	4.61915	0.246904
$351$	0	0
$352$	−0.168000	−0.00895440
$353$	−29.7923	−1.58568	−0.792842	−	0.609427i	$-0.791400\pi$
$353$	−29.7923	−1.58568	−0.792842	+	0.609427i	$0.791400\pi$
$354$	0	0
$355$	−7.73672	−0.410622
$356$	−0.0662179	−0.00350954
$357$	0	0
$358$	−0.450656	−0.0238179
$359$	25.3568	1.33828	0.669141	−	0.743136i	$-0.266662\pi$
$359$	25.3568	1.33828	0.669141	+	0.743136i	$0.266662\pi$
$360$	0	0
$361$	−17.8450	−0.939210
$362$	−34.2385	−1.79954
$363$	0	0
$364$	1.45171	0.0760902
$365$	−2.46799	−0.129180
$366$	0	0
$367$	−18.1553	−0.947701	−0.473850	−	0.880605i	$-0.657136\pi$
$367$	−18.1553	−0.947701	−0.473850	+	0.880605i	$0.657136\pi$
$368$	20.7649	1.08244
$369$	0	0
$370$	−1.72823	−0.0898464
$371$	2.02068	0.104908
$372$	0	0
$373$	−14.5458	−0.753153	−0.376576	−	0.926386i	$-0.622899\pi$
$373$	−14.5458	−0.753153	−0.376576	+	0.926386i	$0.622899\pi$
$374$	−0.371197	−0.0191941
$375$	0	0
$376$	−26.4315	−1.36310
$377$	39.7185	2.04561
$378$	0	0
$379$	−8.80697	−0.452383	−0.226192	−	0.974083i	$-0.572628\pi$
$379$	−8.80697	−0.452383	−0.226192	+	0.974083i	$0.572628\pi$
$380$	0.0711655	0.00365071
$381$	0	0
$382$	−25.6561	−1.31268
$383$	−9.17296	−0.468716	−0.234358	−	0.972150i	$-0.575299\pi$
$383$	−9.17296	−0.468716	−0.234358	+	0.972150i	$0.575299\pi$
$384$	0	0
$385$	−1.44181	−0.0734814
$386$	34.3891	1.75036
$387$	0	0
$388$	−0.525454	−0.0266759
$389$	29.3969	1.49048	0.745241	−	0.666795i	$-0.232334\pi$
$389$	29.3969	1.49048	0.745241	+	0.666795i	$0.232334\pi$
$390$	0	0
$391$	2.89512	0.146412
$392$	−9.24628	−0.467008
$393$	0	0
$394$	34.8044	1.75342
$395$	0.0524346	0.00263827
$396$	0	0
$397$	6.17624	0.309977	0.154988	−	0.987916i	$-0.450466\pi$
$397$	6.17624	0.309977	0.154988	+	0.987916i	$0.450466\pi$
$398$	19.0339	0.954081
$399$	0	0
$400$	−4.12805	−0.206403
$401$	−3.16002	−0.157804	−0.0789020	−	0.996882i	$-0.525141\pi$
$401$	−3.16002	−0.157804	−0.0789020	+	0.996882i	$0.525141\pi$
$402$	0	0
$403$	3.62900	0.180773
$404$	−0.739443	−0.0367887
$405$	0	0
$406$	26.8922	1.33464
$407$	0.539446	0.0267393
$408$	0	0
$409$	−22.9479	−1.13470	−0.567350	−	0.823477i	$-0.692031\pi$
$409$	−22.9479	−1.13470	−0.567350	+	0.823477i	$0.692031\pi$
$410$	−0.984915	−0.0486415
$411$	0	0
$412$	0.878804	0.0432956
$413$	3.98754	0.196214
$414$	0	0
$415$	0.0387270	0.00190104
$416$	−2.55449	−0.125244
$417$	0	0
$418$	−0.693132	−0.0339022
$419$	14.6913	0.717717	0.358858	−	0.933392i	$-0.383166\pi$
$419$	14.6913	0.717717	0.358858	+	0.933392i	$0.383166\pi$
$420$	0	0
$421$	36.4815	1.77800	0.889000	−	0.457907i	$-0.151401\pi$
$421$	36.4815	1.77800	0.889000	+	0.457907i	$0.151401\pi$
$422$	−25.3107	−1.23210
$423$	0	0
$424$	−1.74791	−0.0848860
$425$	−0.575549	−0.0279182
$426$	0	0
$427$	5.75445	0.278477
$428$	0.606442	0.0293135
$429$	0	0
$430$	5.66141	0.273017
$431$	4.63762	0.223386	0.111693	−	0.993743i	$-0.464373\pi$
$431$	4.63762	0.223386	0.111693	+	0.993743i	$0.464373\pi$
$432$	0	0
$433$	−20.1546	−0.968569	−0.484285	−	0.874911i	$-0.660920\pi$
$433$	−20.1546	−0.968569	−0.484285	+	0.874911i	$0.660920\pi$
$434$	2.45708	0.117944
$435$	0	0
$436$	0.992888	0.0475507
$437$	5.40603	0.258605
$438$	0	0
$439$	−6.01870	−0.287257	−0.143628	−	0.989632i	$-0.545877\pi$
$439$	−6.01870	−0.287257	−0.143628	+	0.989632i	$0.545877\pi$
$440$	1.24718	0.0594571
$441$	0	0
$442$	−5.64417	−0.268466
$443$	13.5394	0.643275	0.321637	−	0.946863i	$-0.395767\pi$
$443$	13.5394	0.643275	0.321637	+	0.946863i	$0.395767\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	11.2749	0.533880
$447$	0	0
$448$	24.8012	1.17174
$449$	−4.24910	−0.200527	−0.100264	−	0.994961i	$-0.531969\pi$
$449$	−4.24910	−0.200527	−0.100264	+	0.994961i	$0.531969\pi$
$450$	0	0
$451$	0.307429	0.0144763
$452$	0.324472	0.0152619
$453$	0	0
$454$	−1.34824	−0.0632761
$455$	−21.9232	−1.02778
$456$	0	0
$457$	16.6596	0.779302	0.389651	−	0.920963i	$-0.372596\pi$
$457$	16.6596	0.779302	0.389651	+	0.920963i	$0.372596\pi$
$458$	30.2721	1.41452
$459$	0	0
$460$	0.333089	0.0155303
$461$	−26.2150	−1.22095	−0.610477	−	0.792034i	$-0.709022\pi$
$461$	−26.2150	−1.22095	−0.610477	+	0.792034i	$0.709022\pi$
$462$	0	0
$463$	14.0651	0.653660	0.326830	−	0.945083i	$-0.394019\pi$
$463$	14.0651	0.653660	0.326830	+	0.945083i	$0.394019\pi$
$464$	−24.0330	−1.11571
$465$	0	0
$466$	34.5253	1.59936
$467$	7.60919	0.352111	0.176056	−	0.984380i	$-0.443666\pi$
$467$	7.60919	0.352111	0.176056	+	0.984380i	$0.443666\pi$
$468$	0	0
$469$	21.2552	0.981474
$470$	−13.6683	−0.630472
$471$	0	0
$472$	−3.44927	−0.158765
$473$	−1.76714	−0.0812531
$474$	0	0
$475$	−1.07472	−0.0493114
$476$	−0.122471	−0.00561343
$477$	0	0
$478$	−6.14365	−0.281004
$479$	37.4499	1.71113	0.855565	−	0.517696i	$-0.173210\pi$
$479$	37.4499	1.71113	0.855565	+	0.517696i	$0.173210\pi$
$480$	0	0
$481$	8.20244	0.373999
$482$	−38.2130	−1.74056
$483$	0	0
$484$	−0.715067	−0.0325030
$485$	7.93523	0.360320
$486$	0	0
$487$	−10.8852	−0.493254	−0.246627	−	0.969110i	$-0.579322\pi$
$487$	−10.8852	−0.493254	−0.246627	+	0.969110i	$0.579322\pi$
$488$	−4.97766	−0.225328
$489$	0	0
$490$	−4.78145	−0.216004
$491$	−32.4487	−1.46439	−0.732194	−	0.681096i	$-0.761504\pi$
$491$	−32.4487	−1.46439	−0.732194	+	0.681096i	$0.761504\pi$
$492$	0	0
$493$	−3.35078	−0.150911
$494$	−10.5393	−0.474186
$495$	0	0
$496$	−2.19585	−0.0985965
$497$	24.8617	1.11520
$498$	0	0
$499$	−26.6287	−1.19206	−0.596032	−	0.802961i	$-0.703257\pi$
$499$	−26.6287	−1.19206	−0.596032	+	0.802961i	$0.703257\pi$
$500$	−0.0662179	−0.00296136
$501$	0	0
$502$	3.36324	0.150109
$503$	6.26288	0.279248	0.139624	−	0.990205i	$-0.455411\pi$
$503$	6.26288	0.279248	0.139624	+	0.990205i	$0.455411\pi$
$504$	0	0
$505$	11.1668	0.496916
$506$	−3.24419	−0.144222
$507$	0	0
$508$	0.823539	0.0365386
$509$	−20.3068	−0.900083	−0.450041	−	0.893008i	$-0.648591\pi$
$509$	−20.3068	−0.900083	−0.450041	+	0.893008i	$0.648591\pi$
$510$	0	0
$511$	7.93080	0.350838
$512$	−21.4037	−0.945918
$513$	0	0
$514$	−12.9042	−0.569178
$515$	−13.2714	−0.584807
$516$	0	0
$517$	4.26639	0.187636
$518$	5.55361	0.244012
$519$	0	0
$520$	18.9638	0.831618
$521$	−28.0560	−1.22916	−0.614578	−	0.788856i	$-0.710674\pi$
$521$	−28.0560	−1.22916	−0.614578	+	0.788856i	$0.710674\pi$
$522$	0	0
$523$	−1.71789	−0.0751182	−0.0375591	−	0.999294i	$-0.511958\pi$
$523$	−1.71789	−0.0751182	−0.0375591	+	0.999294i	$0.511958\pi$
$524$	0.883866	0.0386119
$525$	0	0
$526$	−18.9967	−0.828294
$527$	−0.306154	−0.0133363
$528$	0	0
$529$	2.30279	0.100121
$530$	−0.903881	−0.0392621
$531$	0	0
$532$	−0.228688	−0.00991489
$533$	4.67456	0.202478
$534$	0	0
$535$	−9.15828	−0.395947
$536$	−18.3860	−0.794153
$537$	0	0
$538$	−18.1418	−0.782148
$539$	1.49247	0.0642852
$540$	0	0
$541$	−9.45078	−0.406321	−0.203160	−	0.979145i	$-0.565121\pi$
$541$	−9.45078	−0.406321	−0.203160	+	0.979145i	$0.565121\pi$
$542$	9.38204	0.402993
$543$	0	0
$544$	0.215504	0.00923967
$545$	−14.9943	−0.642283
$546$	0	0
$547$	22.1619	0.947574	0.473787	−	0.880639i	$-0.342887\pi$
$547$	22.1619	0.947574	0.473787	+	0.880639i	$0.342887\pi$
$548$	−0.225680	−0.00964059
$549$	0	0
$550$	0.644944	0.0275005
$551$	−6.25688	−0.266552
$552$	0	0
$553$	−0.168497	−0.00716522
$554$	−40.4657	−1.71922
$555$	0	0
$556$	−0.509684	−0.0216154
$557$	−29.8180	−1.26343	−0.631715	−	0.775200i	$-0.717649\pi$
$557$	−29.8180	−1.26343	−0.631715	+	0.775200i	$0.717649\pi$
$558$	0	0
$559$	−26.8699	−1.13648
$560$	13.2654	0.560564
$561$	0	0
$562$	16.2606	0.685911
$563$	2.53292	0.106750	0.0533749	−	0.998575i	$-0.483002\pi$
$563$	2.53292	0.106750	0.0533749	+	0.998575i	$0.483002\pi$
$564$	0	0
$565$	−4.90006	−0.206147
$566$	−17.7621	−0.746595
$567$	0	0
$568$	−21.5056	−0.902357
$569$	−11.7303	−0.491758	−0.245879	−	0.969300i	$-0.579077\pi$
$569$	−11.7303	−0.491758	−0.245879	+	0.969300i	$0.579077\pi$
$570$	0	0
$571$	21.0374	0.880386	0.440193	−	0.897903i	$-0.354910\pi$
$571$	21.0374	0.880386	0.440193	+	0.897903i	$0.354910\pi$
$572$	0.202693	0.00847503
$573$	0	0
$574$	3.16499	0.132104
$575$	−5.03019	−0.209773
$576$	0	0
$577$	29.1630	1.21407	0.607037	−	0.794674i	$-0.292358\pi$
$577$	29.1630	1.21407	0.607037	+	0.794674i	$0.292358\pi$
$578$	−23.9602	−0.996614
$579$	0	0
$580$	−0.385513	−0.0160076
$581$	−0.124448	−0.00516298
$582$	0	0
$583$	0.282135	0.0116848
$584$	−6.86023	−0.283878
$585$	0	0
$586$	9.36821	0.386997
$587$	−8.04061	−0.331872	−0.165936	−	0.986137i	$-0.553064\pi$
$587$	−8.04061	−0.331872	−0.165936	+	0.986137i	$0.553064\pi$
$588$	0	0
$589$	−0.571678	−0.0235556
$590$	−1.78369	−0.0734333
$591$	0	0
$592$	−4.96316	−0.203985
$593$	−22.5875	−0.927557	−0.463779	−	0.885951i	$-0.653507\pi$
$593$	−22.5875	−0.927557	−0.463779	+	0.885951i	$0.653507\pi$
$594$	0	0
$595$	1.84951	0.0758224
$596$	−0.425380	−0.0174242
$597$	0	0
$598$	−49.3290	−2.01721
$599$	−26.9109	−1.09955	−0.549776	−	0.835312i	$-0.685287\pi$
$599$	−26.9109	−1.09955	−0.549776	+	0.835312i	$0.685287\pi$
$600$	0	0
$601$	29.5335	1.20470	0.602348	−	0.798234i	$-0.294232\pi$
$601$	29.5335	1.20470	0.602348	+	0.798234i	$0.294232\pi$
$602$	−18.1928	−0.741481
$603$	0	0
$604$	0.701672	0.0285506
$605$	10.7987	0.439029
$606$	0	0
$607$	5.21802	0.211793	0.105896	−	0.994377i	$-0.466229\pi$
$607$	5.21802	0.211793	0.105896	+	0.994377i	$0.466229\pi$
$608$	0.402409	0.0163198
$609$	0	0
$610$	−2.57405	−0.104220
$611$	64.8718	2.62443
$612$	0	0
$613$	8.78596	0.354862	0.177431	−	0.984133i	$-0.443221\pi$
$613$	8.78596	0.354862	0.177431	+	0.984133i	$0.443221\pi$
$614$	−29.4457	−1.18833
$615$	0	0
$616$	−4.00778	−0.161478
$617$	−24.3055	−0.978504	−0.489252	−	0.872142i	$-0.662730\pi$
$617$	−24.3055	−0.978504	−0.489252	+	0.872142i	$0.662730\pi$
$618$	0	0
$619$	−35.9893	−1.44653	−0.723266	−	0.690570i	$-0.757360\pi$
$619$	−35.9893	−1.44653	−0.723266	+	0.690570i	$0.757360\pi$
$620$	−0.0352235	−0.00141461
$621$	0	0
$622$	−10.2325	−0.410285
$623$	−3.21347	−0.128745
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.8269	−0.872379
$627$	0	0
$628$	0.812812	0.0324347
$629$	−0.691983	−0.0275912
$630$	0	0
$631$	−41.4651	−1.65070	−0.825351	−	0.564620i	$-0.809023\pi$
$631$	−41.4651	−1.65070	−0.825351	+	0.564620i	$0.809023\pi$
$632$	0.145752	0.00579769
$633$	0	0
$634$	22.0734	0.876646
$635$	−12.4368	−0.493540
$636$	0	0
$637$	22.6935	0.899149
$638$	3.75479	0.148654
$639$	0	0
$640$	−11.8428	−0.468128
$641$	−40.5680	−1.60234	−0.801169	−	0.598438i	$-0.795788\pi$
$641$	−40.5680	−1.60234	−0.801169	+	0.598438i	$0.795788\pi$
$642$	0	0
$643$	−1.97199	−0.0777677	−0.0388838	−	0.999244i	$-0.512380\pi$
$643$	−1.97199	−0.0777677	−0.0388838	+	0.999244i	$0.512380\pi$
$644$	−1.07037	−0.0421785
$645$	0	0
$646$	0.889128	0.0349823
$647$	−3.69628	−0.145316	−0.0726580	−	0.997357i	$-0.523148\pi$
$647$	−3.69628	−0.145316	−0.0726580	+	0.997357i	$0.523148\pi$
$648$	0	0
$649$	0.556756	0.0218546
$650$	9.80659	0.384646
$651$	0	0
$652$	−0.272187	−0.0106597
$653$	−29.9599	−1.17242	−0.586210	−	0.810159i	$-0.699381\pi$
$653$	−29.9599	−1.17242	−0.586210	+	0.810159i	$0.699381\pi$
$654$	0	0
$655$	−13.3478	−0.521543
$656$	−2.82850	−0.110434
$657$	0	0
$658$	43.9226	1.71228
$659$	20.0700	0.781815	0.390908	−	0.920430i	$-0.372161\pi$
$659$	20.0700	0.781815	0.390908	+	0.920430i	$0.372161\pi$
$660$	0	0
$661$	7.97219	0.310082	0.155041	−	0.987908i	$-0.450449\pi$
$661$	7.97219	0.310082	0.155041	+	0.987908i	$0.450449\pi$
$662$	−41.8011	−1.62465
$663$	0	0
$664$	0.107649	0.00417759
$665$	3.45357	0.133924
$666$	0	0
$667$	−29.2852	−1.13393
$668$	1.39082	0.0538125
$669$	0	0
$670$	−9.50778	−0.367318
$671$	0.803459	0.0310172
$672$	0	0
$673$	21.4141	0.825451	0.412726	−	0.910855i	$-0.364577\pi$
$673$	21.4141	0.825451	0.412726	+	0.910855i	$0.364577\pi$
$674$	−28.1030	−1.08249
$675$	0	0
$676$	2.22118	0.0854302
$677$	37.6605	1.44741	0.723706	−	0.690109i	$-0.242437\pi$
$677$	37.6605	1.44741	0.723706	+	0.690109i	$0.242437\pi$
$678$	0	0
$679$	−25.4996	−0.978585
$680$	−1.59984	−0.0613512
$681$	0	0
$682$	0.343067	0.0131367
$683$	4.76824	0.182452	0.0912258	−	0.995830i	$-0.470922\pi$
$683$	4.76824	0.182452	0.0912258	+	0.995830i	$0.470922\pi$
$684$	0	0
$685$	3.40815	0.130219
$686$	−16.9690	−0.647880
$687$	0	0
$688$	16.2585	0.619851
$689$	4.28996	0.163434
$690$	0	0
$691$	11.2607	0.428379	0.214189	−	0.976792i	$-0.431289\pi$
$691$	11.2607	0.428379	0.214189	+	0.976792i	$0.431289\pi$
$692$	−1.19367	−0.0453764
$693$	0	0
$694$	−42.7513	−1.62282
$695$	7.69707	0.291966
$696$	0	0
$697$	−0.394360	−0.0149375
$698$	35.9192	1.35956
$699$	0	0
$700$	0.212789	0.00804267
$701$	16.0895	0.607691	0.303845	−	0.952721i	$-0.401729\pi$
$701$	16.0895	0.607691	0.303845	+	0.952721i	$0.401729\pi$
$702$	0	0
$703$	−1.29213	−0.0487338
$704$	3.46284	0.130511
$705$	0	0
$706$	−42.8245	−1.61172
$707$	−35.8842	−1.34956
$708$	0	0
$709$	12.0241	0.451576	0.225788	−	0.974176i	$-0.427504\pi$
$709$	12.0241	0.451576	0.225788	+	0.974176i	$0.427504\pi$
$710$	−11.1210	−0.417365
$711$	0	0
$712$	2.77969	0.104173
$713$	−2.67573	−0.100207
$714$	0	0
$715$	−3.06100	−0.114475
$716$	−0.0207602	−0.000775846 0
$717$	0	0
$718$	36.4488	1.36026
$719$	38.9476	1.45250	0.726251	−	0.687430i	$-0.241261\pi$
$719$	38.9476	1.45250	0.726251	+	0.687430i	$0.241261\pi$
$720$	0	0
$721$	42.6472	1.58826
$722$	−25.6510	−0.954631
$723$	0	0
$724$	−1.57726	−0.0586183
$725$	5.82188	0.216219
$726$	0	0
$727$	9.02295	0.334643	0.167321	−	0.985902i	$-0.446488\pi$
$727$	9.02295	0.334643	0.167321	+	0.985902i	$0.446488\pi$
$728$	−60.9396	−2.25857
$729$	0	0
$730$	−3.54757	−0.131301
$731$	2.26683	0.0838416
$732$	0	0
$733$	−30.6128	−1.13071	−0.565355	−	0.824848i	$-0.691261\pi$
$733$	−30.6128	−1.13071	−0.565355	+	0.824848i	$0.691261\pi$
$734$	−26.0971	−0.963261
$735$	0	0
$736$	1.88347	0.0694255
$737$	2.96774	0.109318
$738$	0	0
$739$	−13.7122	−0.504413	−0.252207	−	0.967673i	$-0.581156\pi$
$739$	−13.7122	−0.504413	−0.252207	+	0.967673i	$0.581156\pi$
$740$	−0.0796139	−0.00292667
$741$	0	0
$742$	2.90459	0.106631
$743$	2.01286	0.0738448	0.0369224	−	0.999318i	$-0.488245\pi$
$743$	2.01286	0.0738448	0.0369224	+	0.999318i	$0.488245\pi$
$744$	0	0
$745$	6.42394	0.235355
$746$	−20.9086	−0.765519
$747$	0	0
$748$	−0.0170998	−0.000625232 0
$749$	29.4298	1.07534
$750$	0	0
$751$	1.72617	0.0629887	0.0314943	−	0.999504i	$-0.489973\pi$
$751$	1.72617	0.0629887	0.0314943	+	0.999504i	$0.489973\pi$
$752$	−39.2529	−1.43140
$753$	0	0
$754$	57.0928	2.07920
$755$	−10.5964	−0.385643
$756$	0	0
$757$	−19.1940	−0.697619	−0.348810	−	0.937194i	$-0.613414\pi$
$757$	−19.1940	−0.697619	−0.348810	+	0.937194i	$0.613414\pi$
$758$	−12.6594	−0.459811
$759$	0	0
$760$	−2.98737	−0.108363
$761$	−37.9543	−1.37584	−0.687921	−	0.725785i	$-0.741477\pi$
$761$	−37.9543	−1.37584	−0.687921	+	0.725785i	$0.741477\pi$
$762$	0	0
$763$	48.1835	1.74436
$764$	−1.18189	−0.0427594
$765$	0	0
$766$	−13.1855	−0.476413
$767$	8.46566	0.305677
$768$	0	0
$769$	14.2676	0.514504	0.257252	−	0.966344i	$-0.417183\pi$
$769$	14.2676	0.514504	0.257252	+	0.966344i	$0.417183\pi$
$770$	−2.07251	−0.0746880
$771$	0	0
$772$	1.58420	0.0570164
$773$	−1.07616	−0.0387067	−0.0193533	−	0.999813i	$-0.506161\pi$
$773$	−1.07616	−0.0387067	−0.0193533	+	0.999813i	$0.506161\pi$
$774$	0	0
$775$	0.531933	0.0191076
$776$	22.0574	0.791816
$777$	0	0
$778$	42.2561	1.51496
$779$	−0.736385	−0.0263837
$780$	0	0
$781$	3.47129	0.124213
$782$	4.16154	0.148816
$783$	0	0
$784$	−13.7315	−0.490409
$785$	−12.2748	−0.438106
$786$	0	0
$787$	26.5197	0.945325	0.472662	−	0.881244i	$-0.343293\pi$
$787$	26.5197	0.945325	0.472662	+	0.881244i	$0.343293\pi$
$788$	1.60332	0.0571160
$789$	0	0
$790$	0.0753713	0.00268159
$791$	15.7462	0.559870
$792$	0	0
$793$	12.2169	0.433833
$794$	8.87795	0.315066
$795$	0	0
$796$	0.876827	0.0310783
$797$	25.8912	0.917114	0.458557	−	0.888665i	$-0.348366\pi$
$797$	25.8912	0.917114	0.458557	+	0.888665i	$0.348366\pi$
$798$	0	0
$799$	−5.47279	−0.193613
$800$	−0.374433	−0.0132382
$801$	0	0
$802$	−4.54233	−0.160395
$803$	1.10733	0.0390768
$804$	0	0
$805$	16.1643	0.569718
$806$	5.21645	0.183742
$807$	0	0
$808$	31.0402	1.09199
$809$	−28.4915	−1.00171	−0.500854	−	0.865532i	$-0.666981\pi$
$809$	−28.4915	−1.00171	−0.500854	+	0.865532i	$0.666981\pi$
$810$	0	0
$811$	36.1828	1.27055	0.635276	−	0.772285i	$-0.280886\pi$
$811$	36.1828	1.27055	0.635276	+	0.772285i	$0.280886\pi$
$812$	1.23883	0.0434745
$813$	0	0
$814$	0.775418	0.0271784
$815$	4.11048	0.143984
$816$	0	0
$817$	4.23282	0.148088
$818$	−32.9861	−1.15333
$819$	0	0
$820$	−0.0453718	−0.00158445
$821$	51.4115	1.79427	0.897136	−	0.441754i	$-0.145644\pi$
$821$	51.4115	1.79427	0.897136	+	0.441754i	$0.145644\pi$
$822$	0	0
$823$	−39.3045	−1.37007	−0.685035	−	0.728510i	$-0.740213\pi$
$823$	−39.3045	−1.37007	−0.685035	+	0.728510i	$0.740213\pi$
$824$	−36.8903	−1.28513
$825$	0	0
$826$	5.73183	0.199436
$827$	27.2841	0.948760	0.474380	−	0.880320i	$-0.342672\pi$
$827$	27.2841	0.948760	0.474380	+	0.880320i	$0.342672\pi$
$828$	0	0
$829$	−7.89433	−0.274181	−0.137091	−	0.990558i	$-0.543775\pi$
$829$	−7.89433	−0.274181	−0.137091	+	0.990558i	$0.543775\pi$
$830$	0.0556676	0.00193225
$831$	0	0
$832$	52.6536	1.82543
$833$	−1.91449	−0.0663332
$834$	0	0
$835$	−21.0037	−0.726863
$836$	−0.0319303	−0.00110433
$837$	0	0
$838$	21.1178	0.729501
$839$	21.5443	0.743793	0.371897	−	0.928274i	$-0.378708\pi$
$839$	21.5443	0.743793	0.371897	+	0.928274i	$0.378708\pi$
$840$	0	0
$841$	4.89433	0.168770
$842$	52.4398	1.80719
$843$	0	0
$844$	−1.16598	−0.0401347
$845$	−33.5436	−1.15393
$846$	0	0
$847$	−34.7012	−1.19235
$848$	−2.59578	−0.0891395
$849$	0	0
$850$	−0.827314	−0.0283766
$851$	−6.04780	−0.207316
$852$	0	0
$853$	−12.0303	−0.411910	−0.205955	−	0.978561i	$-0.566030\pi$
$853$	−12.0303	−0.411910	−0.205955	+	0.978561i	$0.566030\pi$
$854$	8.27164	0.283050
$855$	0	0
$856$	−25.4571	−0.870107
$857$	3.00039	0.102491	0.0512457	−	0.998686i	$-0.483681\pi$
$857$	3.00039	0.102491	0.0512457	+	0.998686i	$0.483681\pi$
$858$	0	0
$859$	−45.1290	−1.53978	−0.769890	−	0.638176i	$-0.779689\pi$
$859$	−45.1290	−1.53978	−0.769890	+	0.638176i	$0.779689\pi$
$860$	0.260803	0.00889329
$861$	0	0
$862$	6.66627	0.227054
$863$	4.35626	0.148289	0.0741445	−	0.997248i	$-0.476377\pi$
$863$	4.35626	0.148289	0.0741445	+	0.997248i	$0.476377\pi$
$864$	0	0
$865$	18.0263	0.612913
$866$	−28.9709	−0.984473
$867$	0	0
$868$	0.113190	0.00384191
$869$	−0.0235262	−0.000798072 0
$870$	0	0
$871$	45.1254	1.52902
$872$	−41.6793	−1.41144
$873$	0	0
$874$	7.77081	0.262852
$875$	−3.21347	−0.108635
$876$	0	0
$877$	−30.0036	−1.01315	−0.506575	−	0.862196i	$-0.669089\pi$
$877$	−30.0036	−1.01315	−0.506575	+	0.862196i	$0.669089\pi$
$878$	−8.65149	−0.291974
$879$	0	0
$880$	1.85216	0.0624364
$881$	13.7226	0.462327	0.231164	−	0.972915i	$-0.425747\pi$
$881$	13.7226	0.462327	0.231164	+	0.972915i	$0.425747\pi$
$882$	0	0
$883$	−7.43087	−0.250069	−0.125034	−	0.992152i	$-0.539904\pi$
$883$	−7.43087	−0.250069	−0.125034	+	0.992152i	$0.539904\pi$
$884$	−0.260008	−0.00874503
$885$	0	0
$886$	19.4619	0.653837
$887$	−5.63384	−0.189166	−0.0945829	−	0.995517i	$-0.530152\pi$
$887$	−5.63384	−0.189166	−0.0945829	+	0.995517i	$0.530152\pi$
$888$	0	0
$889$	39.9653	1.34039
$890$	1.43743	0.0481829
$891$	0	0
$892$	0.519396	0.0173907
$893$	−10.2193	−0.341975
$894$	0	0
$895$	0.313514	0.0104796
$896$	38.0565	1.27138
$897$	0	0
$898$	−6.10780	−0.203820
$899$	3.09685	0.103286
$900$	0	0
$901$	−0.361914	−0.0120571
$902$	0.441909	0.0147140
$903$	0	0
$904$	−13.6206	−0.453016
$905$	23.8192	0.791776
$906$	0	0
$907$	−21.0902	−0.700288	−0.350144	−	0.936696i	$-0.613867\pi$
$907$	−21.0902	−0.700288	−0.350144	+	0.936696i	$0.613867\pi$
$908$	−0.0621091	−0.00206116
$909$	0	0
$910$	−31.5132	−1.04465
$911$	23.1959	0.768513	0.384256	−	0.923226i	$-0.374458\pi$
$911$	23.1959	0.768513	0.384256	+	0.923226i	$0.374458\pi$
$912$	0	0
$913$	−0.0173759	−0.000575060 0
$914$	23.9470	0.792098
$915$	0	0
$916$	1.39454	0.0460768
$917$	42.8929	1.41645
$918$	0	0
$919$	16.9178	0.558066	0.279033	−	0.960281i	$-0.409986\pi$
$919$	16.9178	0.558066	0.279033	+	0.960281i	$0.409986\pi$
$920$	−13.9823	−0.460984
$921$	0	0
$922$	−37.6823	−1.24100
$923$	52.7821	1.73734
$924$	0	0
$925$	1.20230	0.0395314
$926$	20.2177	0.664393
$927$	0	0
$928$	−2.17990	−0.0715589
$929$	22.2674	0.730571	0.365286	−	0.930896i	$-0.380971\pi$
$929$	22.2674	0.730571	0.365286	+	0.930896i	$0.380971\pi$
$930$	0	0
$931$	−3.57491	−0.117163
$932$	1.59047	0.0520976
$933$	0	0
$934$	10.9377	0.357893
$935$	0.258236	0.00844521
$936$	0	0
$937$	−27.9021	−0.911521	−0.455760	−	0.890103i	$-0.650633\pi$
$937$	−27.9021	−0.911521	−0.455760	+	0.890103i	$0.650633\pi$
$938$	30.5530	0.997589
$939$	0	0
$940$	−0.629654	−0.0205370
$941$	−43.7123	−1.42498	−0.712489	−	0.701683i	$-0.752432\pi$
$941$	−43.7123	−1.42498	−0.712489	+	0.701683i	$0.752432\pi$
$942$	0	0
$943$	−3.44663	−0.112238
$944$	−5.12243	−0.166721
$945$	0	0
$946$	−2.54014	−0.0825872
$947$	−0.514224	−0.0167101	−0.00835503	−	0.999965i	$-0.502660\pi$
$947$	−0.514224	−0.0167101	−0.00835503	+	0.999965i	$0.502660\pi$
$948$	0	0
$949$	16.8373	0.546562
$950$	−1.54483	−0.0501211
$951$	0	0
$952$	5.14105	0.166622
$953$	21.0525	0.681959	0.340979	−	0.940071i	$-0.389241\pi$
$953$	21.0525	0.681959	0.340979	+	0.940071i	$0.389241\pi$
$954$	0	0
$955$	17.8485	0.577565
$956$	−0.283018	−0.00915345
$957$	0	0
$958$	53.8318	1.73923
$959$	−10.9520	−0.353657
$960$	0	0
$961$	−30.7170	−0.990872
$962$	11.7905	0.380140
$963$	0	0
$964$	−1.76035	−0.0566970
$965$	−23.9240	−0.770140
$966$	0	0
$967$	27.3243	0.878691	0.439346	−	0.898318i	$-0.355210\pi$
$967$	27.3243	0.878691	0.439346	+	0.898318i	$0.355210\pi$
$968$	30.0170	0.964781
$969$	0	0
$970$	11.4064	0.366237
$971$	58.6775	1.88305	0.941526	−	0.336940i	$-0.109392\pi$
$971$	58.6775	1.88305	0.941526	+	0.336940i	$0.109392\pi$
$972$	0	0
$973$	−24.7343	−0.792945
$974$	−15.6467	−0.501353
$975$	0	0
$976$	−7.39222	−0.236619
$977$	−48.3872	−1.54804	−0.774022	−	0.633159i	$-0.781758\pi$
$977$	−48.3872	−1.54804	−0.774022	+	0.633159i	$0.781758\pi$
$978$	0	0
$979$	−0.448677	−0.0143398
$980$	−0.220266	−0.00703613
$981$	0	0
$982$	−46.6429	−1.48843
$983$	56.1182	1.78989	0.894947	−	0.446173i	$-0.147214\pi$
$983$	56.1182	1.78989	0.894947	+	0.446173i	$0.147214\pi$
$984$	0	0
$985$	−24.2128	−0.771485
$986$	−4.81652	−0.153389
$987$	0	0
$988$	−0.485511	−0.0154462
$989$	19.8116	0.629974
$990$	0	0
$991$	24.6286	0.782354	0.391177	−	0.920316i	$-0.372068\pi$
$991$	24.6286	0.782354	0.391177	+	0.920316i	$0.372068\pi$
$992$	−0.199173	−0.00632376
$993$	0	0
$994$	35.7371	1.13351
$995$	−13.2415	−0.419785
$996$	0	0
$997$	−22.2610	−0.705013	−0.352507	−	0.935809i	$-0.614671\pi$
$997$	−22.2610	−0.705013	−0.352507	+	0.935809i	$0.614671\pi$
$998$	−38.2770	−1.21164
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.r.1.9		10
3.2	odd	2		1335.2.a.k.1.2	✓	10
15.14	odd	2		6675.2.a.y.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.k.1.2	✓	10	3.2	odd	2
4005.2.a.r.1.9		10	1.1	even	1	trivial
6675.2.a.y.1.9		10	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43743 q^{2} +0.0662179 q^{4} -1.00000 q^{5} +3.21347 q^{7} -2.77969 q^{8} +O(q^{10})\)	\(q+1.43743 q^{2} +0.0662179 q^{4} -1.00000 q^{5} +3.21347 q^{7} -2.77969 q^{8} -1.43743 q^{10} +0.448677 q^{11} +6.82228 q^{13} +4.61915 q^{14} -4.12805 q^{16} -0.575549 q^{17} -1.07472 q^{19} -0.0662179 q^{20} +0.644944 q^{22} -5.03019 q^{23} +1.00000 q^{25} +9.80659 q^{26} +0.212789 q^{28} +5.82188 q^{29} +0.531933 q^{31} -0.374433 q^{32} -0.827314 q^{34} -3.21347 q^{35} +1.20230 q^{37} -1.54483 q^{38} +2.77969 q^{40} +0.685190 q^{41} -3.93855 q^{43} +0.0297105 q^{44} -7.23057 q^{46} +9.50881 q^{47} +3.32638 q^{49} +1.43743 q^{50} +0.451757 q^{52} +0.628816 q^{53} -0.448677 q^{55} -8.93243 q^{56} +8.36858 q^{58} +1.24088 q^{59} +1.79073 q^{61} +0.764619 q^{62} +7.71788 q^{64} -6.82228 q^{65} +6.61441 q^{67} -0.0381116 q^{68} -4.61915 q^{70} +7.73672 q^{71} +2.46799 q^{73} +1.72823 q^{74} -0.0711655 q^{76} +1.44181 q^{77} -0.0524346 q^{79} +4.12805 q^{80} +0.984915 q^{82} -0.0387270 q^{83} +0.575549 q^{85} -5.66141 q^{86} -1.24718 q^{88} -1.00000 q^{89} +21.9232 q^{91} -0.333089 q^{92} +13.6683 q^{94} +1.07472 q^{95} -7.93523 q^{97} +4.78145 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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