LMFDB - Embedded newform 6039.2.a.k.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-1.82302$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82302 q^{2} +1.32341 q^{4} +3.15385 q^{5} -0.287436 q^{7} -1.23344 q^{8} +O(q^{10})$	\(q+1.82302 q^{2} +1.32341 q^{4} +3.15385 q^{5} -0.287436 q^{7} -1.23344 q^{8} +5.74953 q^{10} -1.00000 q^{11} +0.910644 q^{13} -0.524002 q^{14} -4.89541 q^{16} -0.621790 q^{17} +6.81024 q^{19} +4.17384 q^{20} -1.82302 q^{22} +7.16003 q^{23} +4.94675 q^{25} +1.66012 q^{26} -0.380396 q^{28} -7.26934 q^{29} +5.73531 q^{31} -6.45756 q^{32} -1.13354 q^{34} -0.906529 q^{35} +9.51534 q^{37} +12.4152 q^{38} -3.89007 q^{40} -6.96842 q^{41} +1.62703 q^{43} -1.32341 q^{44} +13.0529 q^{46} +11.4314 q^{47} -6.91738 q^{49} +9.01804 q^{50} +1.20516 q^{52} -2.14938 q^{53} -3.15385 q^{55} +0.354534 q^{56} -13.2522 q^{58} -4.08607 q^{59} -1.00000 q^{61} +10.4556 q^{62} -1.98147 q^{64} +2.87203 q^{65} +11.2048 q^{67} -0.822884 q^{68} -1.65262 q^{70} -4.90057 q^{71} +9.71284 q^{73} +17.3467 q^{74} +9.01275 q^{76} +0.287436 q^{77} -4.88947 q^{79} -15.4394 q^{80} -12.7036 q^{82} +8.93050 q^{83} -1.96103 q^{85} +2.96610 q^{86} +1.23344 q^{88} +13.4860 q^{89} -0.261752 q^{91} +9.47567 q^{92} +20.8398 q^{94} +21.4785 q^{95} +3.68860 q^{97} -12.6105 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	1.82302	1.28907	0.644536	−	0.764574i	$-0.277051\pi$
$2$	1.82302	1.28907	0.644536	+	0.764574i	$0.277051\pi$
$3$	0	0
$3$	0	0
$4$	1.32341	0.661705
$5$	3.15385	1.41044	0.705222	−	0.708987i	$-0.250848\pi$
$5$	3.15385	1.41044	0.705222	+	0.708987i	$0.250848\pi$
$6$	0	0
$7$	−0.287436	−0.108641	−0.0543203	−	0.998524i	$-0.517299\pi$
$7$	−0.287436	−0.108641	−0.0543203	+	0.998524i	$0.517299\pi$
$8$	−1.23344	−0.436086
$9$	0	0
$10$	5.74953	1.81816
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.910644	0.252567	0.126284	−	0.991994i	$-0.459695\pi$
$13$	0.910644	0.252567	0.126284	+	0.991994i	$0.459695\pi$
$14$	−0.524002	−0.140045
$15$	0	0
$16$	−4.89541	−1.22385
$17$	−0.621790	−0.150806	−0.0754031	−	0.997153i	$-0.524024\pi$
$17$	−0.621790	−0.150806	−0.0754031	+	0.997153i	$0.524024\pi$
$18$	0	0
$19$	6.81024	1.56238	0.781188	−	0.624295i	$-0.214614\pi$
$19$	6.81024	1.56238	0.781188	+	0.624295i	$0.214614\pi$
$20$	4.17384	0.933298
$21$	0	0
$22$	−1.82302	−0.388670
$23$	7.16003	1.49297	0.746485	−	0.665402i	$-0.231740\pi$
$23$	7.16003	1.49297	0.746485	+	0.665402i	$0.231740\pi$
$24$	0	0
$25$	4.94675	0.989350
$26$	1.66012	0.325577
$27$	0	0
$28$	−0.380396	−0.0718881
$29$	−7.26934	−1.34988	−0.674942	−	0.737871i	$-0.735831\pi$
$29$	−7.26934	−1.34988	−0.674942	+	0.737871i	$0.735831\pi$
$30$	0	0
$31$	5.73531	1.03009	0.515046	−	0.857162i	$-0.327775\pi$
$31$	5.73531	1.03009	0.515046	+	0.857162i	$0.327775\pi$
$32$	−6.45756	−1.14155
$33$	0	0
$34$	−1.13354	−0.194400
$35$	−0.906529	−0.153231
$36$	0	0
$37$	9.51534	1.56431	0.782156	−	0.623082i	$-0.214120\pi$
$37$	9.51534	1.56431	0.782156	+	0.623082i	$0.214120\pi$
$38$	12.4152	2.01402
$39$	0	0
$40$	−3.89007	−0.615074
$41$	−6.96842	−1.08828	−0.544142	−	0.838993i	$-0.683145\pi$
$41$	−6.96842	−1.08828	−0.544142	+	0.838993i	$0.683145\pi$
$42$	0	0
$43$	1.62703	0.248119	0.124060	−	0.992275i	$-0.460409\pi$
$43$	1.62703	0.248119	0.124060	+	0.992275i	$0.460409\pi$
$44$	−1.32341	−0.199512
$45$	0	0
$46$	13.0529	1.92455
$47$	11.4314	1.66745	0.833724	−	0.552181i	$-0.186204\pi$
$47$	11.4314	1.66745	0.833724	+	0.552181i	$0.186204\pi$
$48$	0	0
$49$	−6.91738	−0.988197
$50$	9.01804	1.27534
$51$	0	0
$52$	1.20516	0.167125
$53$	−2.14938	−0.295240	−0.147620	−	0.989044i	$-0.547161\pi$
$53$	−2.14938	−0.295240	−0.147620	+	0.989044i	$0.547161\pi$
$54$	0	0
$55$	−3.15385	−0.425265
$56$	0.354534	0.0473766
$57$	0	0
$58$	−13.2522	−1.74010
$59$	−4.08607	−0.531961	−0.265981	−	0.963978i	$-0.585696\pi$
$59$	−4.08607	−0.531961	−0.265981	+	0.963978i	$0.585696\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	10.4556	1.32786
$63$	0	0
$64$	−1.98147	−0.247683
$65$	2.87203	0.356232
$66$	0	0
$67$	11.2048	1.36888	0.684442	−	0.729067i	$-0.260046\pi$
$67$	11.2048	1.36888	0.684442	+	0.729067i	$0.260046\pi$
$68$	−0.822884	−0.0997893
$69$	0	0
$70$	−1.65262	−0.197526
$71$	−4.90057	−0.581591	−0.290795	−	0.956785i	$-0.593920\pi$
$71$	−4.90057	−0.581591	−0.290795	+	0.956785i	$0.593920\pi$
$72$	0	0
$73$	9.71284	1.13680	0.568401	−	0.822752i	$-0.307562\pi$
$73$	9.71284	1.13680	0.568401	+	0.822752i	$0.307562\pi$
$74$	17.3467	2.01651
$75$	0	0
$76$	9.01275	1.03383
$77$	0.287436	0.0327564
$78$	0	0
$79$	−4.88947	−0.550109	−0.275054	−	0.961429i	$-0.588696\pi$
$79$	−4.88947	−0.550109	−0.275054	+	0.961429i	$0.588696\pi$
$80$	−15.4394	−1.72617
$81$	0	0
$82$	−12.7036	−1.40288
$83$	8.93050	0.980250	0.490125	−	0.871652i	$-0.336951\pi$
$83$	8.93050	0.980250	0.490125	+	0.871652i	$0.336951\pi$
$84$	0	0
$85$	−1.96103	−0.212704
$86$	2.96610	0.319843
$87$	0	0
$88$	1.23344	0.131485
$89$	13.4860	1.42952	0.714758	−	0.699372i	$-0.246537\pi$
$89$	13.4860	1.42952	0.714758	+	0.699372i	$0.246537\pi$
$90$	0	0
$91$	−0.261752	−0.0274390
$92$	9.47567	0.987906
$93$	0	0
$94$	20.8398	2.14946
$95$	21.4785	2.20364
$96$	0	0
$97$	3.68860	0.374520	0.187260	−	0.982310i	$-0.440039\pi$
$97$	3.68860	0.374520	0.187260	+	0.982310i	$0.440039\pi$
$98$	−12.6105	−1.27386
$99$	0	0
$100$	6.54658	0.654658
$101$	10.0944	1.00443	0.502217	−	0.864742i	$-0.332518\pi$
$101$	10.0944	1.00443	0.502217	+	0.864742i	$0.332518\pi$
$102$	0	0
$103$	−10.5365	−1.03819	−0.519097	−	0.854715i	$-0.673732\pi$
$103$	−10.5365	−1.03819	−0.519097	+	0.854715i	$0.673732\pi$
$104$	−1.12322	−0.110141
$105$	0	0
$106$	−3.91836	−0.380585
$107$	−16.3056	−1.57632	−0.788160	−	0.615470i	$-0.788966\pi$
$107$	−16.3056	−1.57632	−0.788160	+	0.615470i	$0.788966\pi$
$108$	0	0
$109$	5.49319	0.526152	0.263076	−	0.964775i	$-0.415263\pi$
$109$	5.49319	0.526152	0.263076	+	0.964775i	$0.415263\pi$
$110$	−5.74953	−0.548197
$111$	0	0
$112$	1.40712	0.132960
$113$	11.7663	1.10688	0.553442	−	0.832887i	$-0.313314\pi$
$113$	11.7663	1.10688	0.553442	+	0.832887i	$0.313314\pi$
$114$	0	0
$115$	22.5816	2.10575
$116$	−9.62033	−0.893225
$117$	0	0
$118$	−7.44900	−0.685736
$119$	0.178725	0.0163837
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.82302	−0.165049
$123$	0	0
$124$	7.59017	0.681618
$125$	−0.167943	−0.0150213
$126$	0	0
$127$	−17.6684	−1.56781	−0.783907	−	0.620878i	$-0.786776\pi$
$127$	−17.6684	−1.56781	−0.783907	+	0.620878i	$0.786776\pi$
$128$	9.30286	0.822265
$129$	0	0
$130$	5.23578	0.459208
$131$	6.52874	0.570419	0.285209	−	0.958465i	$-0.407937\pi$
$131$	6.52874	0.570419	0.285209	+	0.958465i	$0.407937\pi$
$132$	0	0
$133$	−1.95751	−0.169738
$134$	20.4266	1.76459
$135$	0	0
$136$	0.766939	0.0657645
$137$	14.2878	1.22069	0.610345	−	0.792136i	$-0.291031\pi$
$137$	14.2878	1.22069	0.610345	+	0.792136i	$0.291031\pi$
$138$	0	0
$139$	−0.596965	−0.0506339	−0.0253169	−	0.999679i	$-0.508059\pi$
$139$	−0.596965	−0.0506339	−0.0253169	+	0.999679i	$0.508059\pi$
$140$	−1.19971	−0.101394
$141$	0	0
$142$	−8.93385	−0.749712
$143$	−0.910644	−0.0761519
$144$	0	0
$145$	−22.9264	−1.90393
$146$	17.7067	1.46542
$147$	0	0
$148$	12.5927	1.03511
$149$	−15.4628	−1.26676	−0.633382	−	0.773839i	$-0.718334\pi$
$149$	−15.4628	−1.26676	−0.633382	+	0.773839i	$0.718334\pi$
$150$	0	0
$151$	7.37707	0.600338	0.300169	−	0.953886i	$-0.402957\pi$
$151$	7.37707	0.600338	0.300169	+	0.953886i	$0.402957\pi$
$152$	−8.40001	−0.681330
$153$	0	0
$154$	0.524002	0.0422253
$155$	18.0883	1.45289
$156$	0	0
$157$	7.79705	0.622272	0.311136	−	0.950365i	$-0.399291\pi$
$157$	7.79705	0.622272	0.311136	+	0.950365i	$0.399291\pi$
$158$	−8.91362	−0.709129
$159$	0	0
$160$	−20.3662	−1.61009
$161$	−2.05805	−0.162197
$162$	0	0
$163$	−0.591572	−0.0463355	−0.0231677	−	0.999732i	$-0.507375\pi$
$163$	−0.591572	−0.0463355	−0.0231677	+	0.999732i	$0.507375\pi$
$164$	−9.22208	−0.720123
$165$	0	0
$166$	16.2805	1.26361
$167$	−3.67811	−0.284621	−0.142310	−	0.989822i	$-0.545453\pi$
$167$	−3.67811	−0.284621	−0.142310	+	0.989822i	$0.545453\pi$
$168$	0	0
$169$	−12.1707	−0.936210
$170$	−3.57500	−0.274190
$171$	0	0
$172$	2.15322	0.164182
$173$	12.9607	0.985383	0.492692	−	0.870204i	$-0.336013\pi$
$173$	12.9607	0.985383	0.492692	+	0.870204i	$0.336013\pi$
$174$	0	0
$175$	−1.42187	−0.107484
$176$	4.89541	0.369005
$177$	0	0
$178$	24.5853	1.84275
$179$	10.0940	0.754458	0.377229	−	0.926120i	$-0.376877\pi$
$179$	10.0940	0.754458	0.377229	+	0.926120i	$0.376877\pi$
$180$	0	0
$181$	−16.4988	−1.22635	−0.613174	−	0.789948i	$-0.710108\pi$
$181$	−16.4988	−1.22635	−0.613174	+	0.789948i	$0.710108\pi$
$182$	−0.477179	−0.0353709
$183$	0	0
$184$	−8.83145	−0.651063
$185$	30.0099	2.20637
$186$	0	0
$187$	0.621790	0.0454698
$188$	15.1285	1.10336
$189$	0	0
$190$	39.1557	2.84065
$191$	−7.51056	−0.543445	−0.271722	−	0.962376i	$-0.587593\pi$
$191$	−7.51056	−0.543445	−0.271722	+	0.962376i	$0.587593\pi$
$192$	0	0
$193$	−20.3457	−1.46451	−0.732257	−	0.681029i	$-0.761533\pi$
$193$	−20.3457	−1.46451	−0.732257	+	0.681029i	$0.761533\pi$
$194$	6.72439	0.482783
$195$	0	0
$196$	−9.15454	−0.653896
$197$	10.2107	0.727481	0.363741	−	0.931500i	$-0.381499\pi$
$197$	10.2107	0.727481	0.363741	+	0.931500i	$0.381499\pi$
$198$	0	0
$199$	9.42808	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$199$	9.42808	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$200$	−6.10150	−0.431441
$201$	0	0
$202$	18.4024	1.29479
$203$	2.08947	0.146652
$204$	0	0
$205$	−21.9773	−1.53496
$206$	−19.2083	−1.33831
$207$	0	0
$208$	−4.45797	−0.309105
$209$	−6.81024	−0.471074
$210$	0	0
$211$	−12.0384	−0.828761	−0.414380	−	0.910104i	$-0.636002\pi$
$211$	−12.0384	−0.828761	−0.414380	+	0.910104i	$0.636002\pi$
$212$	−2.84451	−0.195362
$213$	0	0
$214$	−29.7254	−2.03199
$215$	5.13139	0.349958
$216$	0	0
$217$	−1.64853	−0.111910
$218$	10.0142	0.678248
$219$	0	0
$220$	−4.17384	−0.281400
$221$	−0.566229	−0.0380887
$222$	0	0
$223$	16.0854	1.07715	0.538577	−	0.842576i	$-0.318962\pi$
$223$	16.0854	1.07715	0.538577	+	0.842576i	$0.318962\pi$
$224$	1.85613	0.124018
$225$	0	0
$226$	21.4503	1.42685
$227$	−8.88217	−0.589530	−0.294765	−	0.955570i	$-0.595241\pi$
$227$	−8.88217	−0.589530	−0.294765	+	0.955570i	$0.595241\pi$
$228$	0	0
$229$	−11.2325	−0.742264	−0.371132	−	0.928580i	$-0.621030\pi$
$229$	−11.2325	−0.742264	−0.371132	+	0.928580i	$0.621030\pi$
$230$	41.1668	2.71446
$231$	0	0
$232$	8.96628	0.588665
$233$	−15.7110	−1.02926	−0.514631	−	0.857412i	$-0.672071\pi$
$233$	−15.7110	−1.02926	−0.514631	+	0.857412i	$0.672071\pi$
$234$	0	0
$235$	36.0530	2.35184
$236$	−5.40755	−0.352002
$237$	0	0
$238$	0.325819	0.0211197
$239$	−6.28509	−0.406549	−0.203274	−	0.979122i	$-0.565158\pi$
$239$	−6.28509	−0.406549	−0.203274	+	0.979122i	$0.565158\pi$
$240$	0	0
$241$	3.70185	0.238457	0.119229	−	0.992867i	$-0.461958\pi$
$241$	3.70185	0.238457	0.119229	+	0.992867i	$0.461958\pi$
$242$	1.82302	0.117188
$243$	0	0
$244$	−1.32341	−0.0847227
$245$	−21.8164	−1.39380
$246$	0	0
$247$	6.20171	0.394605
$248$	−7.07414	−0.449209
$249$	0	0
$250$	−0.306165	−0.0193636
$251$	−7.05635	−0.445393	−0.222696	−	0.974888i	$-0.571486\pi$
$251$	−7.05635	−0.445393	−0.222696	+	0.974888i	$0.571486\pi$
$252$	0	0
$253$	−7.16003	−0.450147
$254$	−32.2098	−2.02102
$255$	0	0
$256$	20.9223	1.30764
$257$	4.59045	0.286345	0.143172	−	0.989698i	$-0.454270\pi$
$257$	4.59045	0.286345	0.143172	+	0.989698i	$0.454270\pi$
$258$	0	0
$259$	−2.73505	−0.169948
$260$	3.80088	0.235720
$261$	0	0
$262$	11.9020	0.735310
$263$	−0.961078	−0.0592626	−0.0296313	−	0.999561i	$-0.509433\pi$
$263$	−0.961078	−0.0592626	−0.0296313	+	0.999561i	$0.509433\pi$
$264$	0	0
$265$	−6.77881	−0.416419
$266$	−3.56858	−0.218804
$267$	0	0
$268$	14.8286	0.905799
$269$	−13.6257	−0.830773	−0.415386	−	0.909645i	$-0.636354\pi$
$269$	−13.6257	−0.830773	−0.415386	+	0.909645i	$0.636354\pi$
$270$	0	0
$271$	−29.0169	−1.76265	−0.881327	−	0.472507i	$-0.843349\pi$
$271$	−29.0169	−1.76265	−0.881327	+	0.472507i	$0.843349\pi$
$272$	3.04391	0.184564
$273$	0	0
$274$	26.0470	1.57356
$275$	−4.94675	−0.298300
$276$	0	0
$277$	−16.5957	−0.997137	−0.498569	−	0.866850i	$-0.666141\pi$
$277$	−16.5957	−0.997137	−0.498569	+	0.866850i	$0.666141\pi$
$278$	−1.08828	−0.0652707
$279$	0	0
$280$	1.11815	0.0668220
$281$	−7.06643	−0.421548	−0.210774	−	0.977535i	$-0.567598\pi$
$281$	−7.06643	−0.421548	−0.210774	+	0.977535i	$0.567598\pi$
$282$	0	0
$283$	−14.4028	−0.856160	−0.428080	−	0.903741i	$-0.640810\pi$
$283$	−14.4028	−0.856160	−0.428080	+	0.903741i	$0.640810\pi$
$284$	−6.48547	−0.384842
$285$	0	0
$286$	−1.66012	−0.0981652
$287$	2.00297	0.118232
$288$	0	0
$289$	−16.6134	−0.977257
$290$	−41.7953	−2.45431
$291$	0	0
$292$	12.8541	0.752228
$293$	28.4883	1.66431	0.832153	−	0.554547i	$-0.187108\pi$
$293$	28.4883	1.66431	0.832153	+	0.554547i	$0.187108\pi$
$294$	0	0
$295$	−12.8868	−0.750301
$296$	−11.7366	−0.682175
$297$	0	0
$298$	−28.1891	−1.63295
$299$	6.52024	0.377075
$300$	0	0
$301$	−0.467666	−0.0269558
$302$	13.4486	0.773878
$303$	0	0
$304$	−33.3389	−1.91212
$305$	−3.15385	−0.180589
$306$	0	0
$307$	10.5356	0.601297	0.300649	−	0.953735i	$-0.402797\pi$
$307$	10.5356	0.601297	0.300649	+	0.953735i	$0.402797\pi$
$308$	0.380396	0.0216751
$309$	0	0
$310$	32.9754	1.87287
$311$	−16.5963	−0.941087	−0.470544	−	0.882377i	$-0.655942\pi$
$311$	−16.5963	−0.941087	−0.470544	+	0.882377i	$0.655942\pi$
$312$	0	0
$313$	9.85264	0.556904	0.278452	−	0.960450i	$-0.410179\pi$
$313$	9.85264	0.556904	0.278452	+	0.960450i	$0.410179\pi$
$314$	14.2142	0.802153
$315$	0	0
$316$	−6.47078	−0.364010
$317$	29.8035	1.67393	0.836964	−	0.547257i	$-0.184328\pi$
$317$	29.8035	1.67393	0.836964	+	0.547257i	$0.184328\pi$
$318$	0	0
$319$	7.26934	0.407005
$320$	−6.24924	−0.349343
$321$	0	0
$322$	−3.75187	−0.209084
$323$	−4.23454	−0.235616
$324$	0	0
$325$	4.50473	0.249877
$326$	−1.07845	−0.0597297
$327$	0	0
$328$	8.59510	0.474585
$329$	−3.28581	−0.181153
$330$	0	0
$331$	−33.2914	−1.82986	−0.914931	−	0.403611i	$-0.867755\pi$
$331$	−33.2914	−1.82986	−0.914931	+	0.403611i	$0.867755\pi$
$332$	11.8187	0.648637
$333$	0	0
$334$	−6.70528	−0.366897
$335$	35.3382	1.93073
$336$	0	0
$337$	17.5453	0.955753	0.477876	−	0.878427i	$-0.341407\pi$
$337$	17.5453	0.955753	0.477876	+	0.878427i	$0.341407\pi$
$338$	−22.1875	−1.20684
$339$	0	0
$340$	−2.59525	−0.140747
$341$	−5.73531	−0.310585
$342$	0	0
$343$	4.00035	0.215999
$344$	−2.00683	−0.108201
$345$	0	0
$346$	23.6276	1.27023
$347$	−19.7792	−1.06180	−0.530902	−	0.847433i	$-0.678147\pi$
$347$	−19.7792	−1.06180	−0.530902	+	0.847433i	$0.678147\pi$
$348$	0	0
$349$	−19.0217	−1.01821	−0.509103	−	0.860705i	$-0.670023\pi$
$349$	−19.0217	−1.01821	−0.509103	+	0.860705i	$0.670023\pi$
$350$	−2.59211	−0.138554
$351$	0	0
$352$	6.45756	0.344189
$353$	28.2882	1.50563	0.752814	−	0.658233i	$-0.228696\pi$
$353$	28.2882	1.50563	0.752814	+	0.658233i	$0.228696\pi$
$354$	0	0
$355$	−15.4557	−0.820301
$356$	17.8476	0.945918
$357$	0	0
$358$	18.4015	0.972551
$359$	19.0153	1.00359	0.501796	−	0.864986i	$-0.332673\pi$
$359$	19.0153	1.00359	0.501796	+	0.864986i	$0.332673\pi$
$360$	0	0
$361$	27.3794	1.44102
$362$	−30.0777	−1.58085
$363$	0	0
$364$	−0.346405	−0.0181566
$365$	30.6328	1.60339
$366$	0	0
$367$	1.78686	0.0932731	0.0466365	−	0.998912i	$-0.485150\pi$
$367$	1.78686	0.0932731	0.0466365	+	0.998912i	$0.485150\pi$
$368$	−35.0513	−1.82717
$369$	0	0
$370$	54.7088	2.84417
$371$	0.617808	0.0320750
$372$	0	0
$373$	19.1615	0.992146	0.496073	−	0.868281i	$-0.334775\pi$
$373$	19.1615	0.992146	0.496073	+	0.868281i	$0.334775\pi$
$374$	1.13354	0.0586138
$375$	0	0
$376$	−14.1000	−0.727151
$377$	−6.61979	−0.340936
$378$	0	0
$379$	−22.8534	−1.17390	−0.586949	−	0.809624i	$-0.699671\pi$
$379$	−22.8534	−1.17390	−0.586949	+	0.809624i	$0.699671\pi$
$380$	28.4248	1.45816
$381$	0	0
$382$	−13.6919	−0.700539
$383$	−7.28326	−0.372157	−0.186079	−	0.982535i	$-0.559578\pi$
$383$	−7.28326	−0.372157	−0.186079	+	0.982535i	$0.559578\pi$
$384$	0	0
$385$	0.906529	0.0462010
$386$	−37.0906	−1.88786
$387$	0	0
$388$	4.88153	0.247822
$389$	−20.4440	−1.03655	−0.518276	−	0.855214i	$-0.673426\pi$
$389$	−20.4440	−1.03655	−0.518276	+	0.855214i	$0.673426\pi$
$390$	0	0
$391$	−4.45204	−0.225149
$392$	8.53215	0.430939
$393$	0	0
$394$	18.6143	0.937775
$395$	−15.4206	−0.775897
$396$	0	0
$397$	15.9369	0.799852	0.399926	−	0.916547i	$-0.369036\pi$
$397$	15.9369	0.799852	0.399926	+	0.916547i	$0.369036\pi$
$398$	17.1876	0.861537
$399$	0	0
$400$	−24.2163	−1.21082
$401$	−37.1941	−1.85738	−0.928692	−	0.370852i	$-0.879066\pi$
$401$	−37.1941	−1.85738	−0.928692	+	0.370852i	$0.879066\pi$
$402$	0	0
$403$	5.22283	0.260168
$404$	13.3591	0.664640
$405$	0	0
$406$	3.80915	0.189045
$407$	−9.51534	−0.471658
$408$	0	0
$409$	33.4964	1.65629	0.828145	−	0.560513i	$-0.189396\pi$
$409$	33.4964	1.65629	0.828145	+	0.560513i	$0.189396\pi$
$410$	−40.0651	−1.97868
$411$	0	0
$412$	−13.9441	−0.686978
$413$	1.17448	0.0577926
$414$	0	0
$415$	28.1654	1.38259
$416$	−5.88054	−0.288317
$417$	0	0
$418$	−12.4152	−0.607249
$419$	10.6606	0.520803	0.260402	−	0.965500i	$-0.416145\pi$
$419$	10.6606	0.520803	0.260402	+	0.965500i	$0.416145\pi$
$420$	0	0
$421$	34.4306	1.67804	0.839021	−	0.544098i	$-0.183128\pi$
$421$	34.4306	1.67804	0.839021	+	0.544098i	$0.183128\pi$
$422$	−21.9464	−1.06833
$423$	0	0
$424$	2.65112	0.128750
$425$	−3.07584	−0.149200
$426$	0	0
$427$	0.287436	0.0139100
$428$	−21.5790	−1.04306
$429$	0	0
$430$	9.35464	0.451121
$431$	10.8437	0.522324	0.261162	−	0.965295i	$-0.415894\pi$
$431$	10.8437	0.522324	0.261162	+	0.965295i	$0.415894\pi$
$432$	0	0
$433$	−1.14549	−0.0550485	−0.0275243	−	0.999621i	$-0.508762\pi$
$433$	−1.14549	−0.0550485	−0.0275243	+	0.999621i	$0.508762\pi$
$434$	−3.00532	−0.144260
$435$	0	0
$436$	7.26975	0.348158
$437$	48.7616	2.33258
$438$	0	0
$439$	30.5933	1.46014	0.730069	−	0.683373i	$-0.239488\pi$
$439$	30.5933	1.46014	0.730069	+	0.683373i	$0.239488\pi$
$440$	3.89007	0.185452
$441$	0	0
$442$	−1.03225	−0.0490991
$443$	9.53948	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$443$	9.53948	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$444$	0	0
$445$	42.5329	2.01625
$446$	29.3240	1.38853
$447$	0	0
$448$	0.569545	0.0269085
$449$	−16.6546	−0.785979	−0.392990	−	0.919543i	$-0.628559\pi$
$449$	−16.6546	−0.785979	−0.392990	+	0.919543i	$0.628559\pi$
$450$	0	0
$451$	6.96842	0.328130
$452$	15.5717	0.732432
$453$	0	0
$454$	−16.1924	−0.759947
$455$	−0.825525	−0.0387012
$456$	0	0
$457$	6.60614	0.309022	0.154511	−	0.987991i	$-0.450620\pi$
$457$	6.60614	0.309022	0.154511	+	0.987991i	$0.450620\pi$
$458$	−20.4771	−0.956832
$459$	0	0
$460$	29.8848	1.39339
$461$	−9.91877	−0.461963	−0.230982	−	0.972958i	$-0.574194\pi$
$461$	−9.91877	−0.461963	−0.230982	+	0.972958i	$0.574194\pi$
$462$	0	0
$463$	−19.7305	−0.916953	−0.458477	−	0.888707i	$-0.651605\pi$
$463$	−19.7305	−0.916953	−0.458477	+	0.888707i	$0.651605\pi$
$464$	35.5864	1.65206
$465$	0	0
$466$	−28.6415	−1.32679
$467$	−24.2804	−1.12356	−0.561782	−	0.827285i	$-0.689884\pi$
$467$	−24.2804	−1.12356	−0.561782	+	0.827285i	$0.689884\pi$
$468$	0	0
$469$	−3.22066	−0.148716
$470$	65.7255	3.03169
$471$	0	0
$472$	5.03991	0.231981
$473$	−1.62703	−0.0748107
$474$	0	0
$475$	33.6886	1.54574
$476$	0.236526	0.0108412
$477$	0	0
$478$	−11.4579	−0.524070
$479$	−3.15535	−0.144172	−0.0720859	−	0.997398i	$-0.522966\pi$
$479$	−3.15535	−0.144172	−0.0720859	+	0.997398i	$0.522966\pi$
$480$	0	0
$481$	8.66509	0.395094
$482$	6.74856	0.307388
$483$	0	0
$484$	1.32341	0.0601550
$485$	11.6333	0.528239
$486$	0	0
$487$	−37.8261	−1.71406	−0.857031	−	0.515265i	$-0.827694\pi$
$487$	−37.8261	−1.71406	−0.857031	+	0.515265i	$0.827694\pi$
$488$	1.23344	0.0558351
$489$	0	0
$490$	−39.7717	−1.79670
$491$	−32.9576	−1.48736	−0.743678	−	0.668538i	$-0.766920\pi$
$491$	−32.9576	−1.48736	−0.743678	+	0.668538i	$0.766920\pi$
$492$	0	0
$493$	4.52001	0.203571
$494$	11.3059	0.508674
$495$	0	0
$496$	−28.0767	−1.26068
$497$	1.40860	0.0631844
$498$	0	0
$499$	−16.4934	−0.738345	−0.369173	−	0.929361i	$-0.620359\pi$
$499$	−16.4934	−0.738345	−0.369173	+	0.929361i	$0.620359\pi$
$500$	−0.222258	−0.00993969
$501$	0	0
$502$	−12.8639	−0.574143
$503$	−27.8051	−1.23977	−0.619883	−	0.784694i	$-0.712820\pi$
$503$	−27.8051	−1.23977	−0.619883	+	0.784694i	$0.712820\pi$
$504$	0	0
$505$	31.8363	1.41670
$506$	−13.0529	−0.580272
$507$	0	0
$508$	−23.3825	−1.03743
$509$	−22.1434	−0.981487	−0.490744	−	0.871304i	$-0.663275\pi$
$509$	−22.1434	−0.981487	−0.490744	+	0.871304i	$0.663275\pi$
$510$	0	0
$511$	−2.79182	−0.123503
$512$	19.5360	0.863378
$513$	0	0
$514$	8.36850	0.369119
$515$	−33.2306	−1.46431
$516$	0	0
$517$	−11.4314	−0.502755
$518$	−4.98606	−0.219075
$519$	0	0
$520$	−3.54247	−0.155348
$521$	−5.33123	−0.233565	−0.116783	−	0.993157i	$-0.537258\pi$
$521$	−5.33123	−0.233565	−0.116783	+	0.993157i	$0.537258\pi$
$522$	0	0
$523$	14.4681	0.632645	0.316322	−	0.948652i	$-0.397552\pi$
$523$	14.4681	0.632645	0.316322	+	0.948652i	$0.397552\pi$
$524$	8.64020	0.377449
$525$	0	0
$526$	−1.75207	−0.0763938
$527$	−3.56616	−0.155344
$528$	0	0
$529$	28.2661	1.22896
$530$	−12.3579	−0.536794
$531$	0	0
$532$	−2.59059	−0.112316
$533$	−6.34575	−0.274865
$534$	0	0
$535$	−51.4253	−2.22331
$536$	−13.8204	−0.596951
$537$	0	0
$538$	−24.8399	−1.07093
$539$	6.91738	0.297953
$540$	0	0
$541$	18.4464	0.793073	0.396536	−	0.918019i	$-0.370212\pi$
$541$	18.4464	0.793073	0.396536	+	0.918019i	$0.370212\pi$
$542$	−52.8985	−2.27219
$543$	0	0
$544$	4.01525	0.172152
$545$	17.3247	0.742108
$546$	0	0
$547$	−36.5079	−1.56097	−0.780483	−	0.625177i	$-0.785027\pi$
$547$	−36.5079	−1.56097	−0.780483	+	0.625177i	$0.785027\pi$
$548$	18.9086	0.807737
$549$	0	0
$550$	−9.01804	−0.384530
$551$	−49.5060	−2.10903
$552$	0	0
$553$	1.40541	0.0597641
$554$	−30.2543	−1.28538
$555$	0	0
$556$	−0.790030	−0.0335047
$557$	−15.5872	−0.660449	−0.330225	−	0.943902i	$-0.607124\pi$
$557$	−15.5872	−0.660449	−0.330225	+	0.943902i	$0.607124\pi$
$558$	0	0
$559$	1.48164	0.0626667
$560$	4.43783	0.187532
$561$	0	0
$562$	−12.8823	−0.543406
$563$	30.8511	1.30022	0.650110	−	0.759840i	$-0.274723\pi$
$563$	30.8511	1.30022	0.650110	+	0.759840i	$0.274723\pi$
$564$	0	0
$565$	37.1093	1.56120
$566$	−26.2567	−1.10365
$567$	0	0
$568$	6.04455	0.253624
$569$	−40.2875	−1.68894	−0.844469	−	0.535604i	$-0.820084\pi$
$569$	−40.2875	−1.68894	−0.844469	+	0.535604i	$0.820084\pi$
$570$	0	0
$571$	−34.7792	−1.45546	−0.727732	−	0.685861i	$-0.759426\pi$
$571$	−34.7792	−1.45546	−0.727732	+	0.685861i	$0.759426\pi$
$572$	−1.20516	−0.0503901
$573$	0	0
$574$	3.65146	0.152409
$575$	35.4189	1.47707
$576$	0	0
$577$	−3.33773	−0.138951	−0.0694757	−	0.997584i	$-0.522133\pi$
$577$	−3.33773	−0.138951	−0.0694757	+	0.997584i	$0.522133\pi$
$578$	−30.2866	−1.25975
$579$	0	0
$580$	−30.3410	−1.25984
$581$	−2.56695	−0.106495
$582$	0	0
$583$	2.14938	0.0890181
$584$	−11.9802	−0.495743
$585$	0	0
$586$	51.9348	2.14541
$587$	27.5509	1.13715	0.568575	−	0.822632i	$-0.307495\pi$
$587$	27.5509	1.13715	0.568575	+	0.822632i	$0.307495\pi$
$588$	0	0
$589$	39.0589	1.60939
$590$	−23.4930	−0.967192
$591$	0	0
$592$	−46.5815	−1.91449
$593$	−0.856831	−0.0351858	−0.0175929	−	0.999845i	$-0.505600\pi$
$593$	−0.856831	−0.0351858	−0.0175929	+	0.999845i	$0.505600\pi$
$594$	0	0
$595$	0.563671	0.0231082
$596$	−20.4637	−0.838225
$597$	0	0
$598$	11.8865	0.486077
$599$	18.1092	0.739920	0.369960	−	0.929048i	$-0.379371\pi$
$599$	18.1092	0.739920	0.369960	+	0.929048i	$0.379371\pi$
$600$	0	0
$601$	18.2819	0.745736	0.372868	−	0.927884i	$-0.378375\pi$
$601$	18.2819	0.745736	0.372868	+	0.927884i	$0.378375\pi$
$602$	−0.852565	−0.0347479
$603$	0	0
$604$	9.76290	0.397247
$605$	3.15385	0.128222
$606$	0	0
$607$	6.78910	0.275561	0.137781	−	0.990463i	$-0.456003\pi$
$607$	6.78910	0.275561	0.137781	+	0.990463i	$0.456003\pi$
$608$	−43.9776	−1.78353
$609$	0	0
$610$	−5.74953	−0.232792
$611$	10.4100	0.421143
$612$	0	0
$613$	−25.5339	−1.03131	−0.515653	−	0.856797i	$-0.672451\pi$
$613$	−25.5339	−1.03131	−0.515653	+	0.856797i	$0.672451\pi$
$614$	19.2066	0.775115
$615$	0	0
$616$	−0.354534	−0.0142846
$617$	−0.104760	−0.00421748	−0.00210874	−	0.999998i	$-0.500671\pi$
$617$	−0.104760	−0.00421748	−0.00210874	+	0.999998i	$0.500671\pi$
$618$	0	0
$619$	26.2158	1.05370	0.526852	−	0.849957i	$-0.323372\pi$
$619$	26.2158	1.05370	0.526852	+	0.849957i	$0.323372\pi$
$620$	23.9382	0.961383
$621$	0	0
$622$	−30.2553	−1.21313
$623$	−3.87637	−0.155303
$624$	0	0
$625$	−25.2634	−1.01054
$626$	17.9616	0.717889
$627$	0	0
$628$	10.3187	0.411761
$629$	−5.91655	−0.235908
$630$	0	0
$631$	20.1814	0.803408	0.401704	−	0.915769i	$-0.368418\pi$
$631$	20.1814	0.803408	0.401704	+	0.915769i	$0.368418\pi$
$632$	6.03086	0.239895
$633$	0	0
$634$	54.3324	2.15781
$635$	−55.7233	−2.21131
$636$	0	0
$637$	−6.29927	−0.249586
$638$	13.2522	0.524659
$639$	0	0
$640$	29.3398	1.15976
$641$	−3.60690	−0.142464	−0.0712319	−	0.997460i	$-0.522693\pi$
$641$	−3.60690	−0.142464	−0.0712319	+	0.997460i	$0.522693\pi$
$642$	0	0
$643$	−33.7876	−1.33245	−0.666226	−	0.745750i	$-0.732091\pi$
$643$	−33.7876	−1.33245	−0.666226	+	0.745750i	$0.732091\pi$
$644$	−2.72365	−0.107327
$645$	0	0
$646$	−7.71966	−0.303726
$647$	0.768454	0.0302110	0.0151055	−	0.999886i	$-0.495192\pi$
$647$	0.768454	0.0302110	0.0151055	+	0.999886i	$0.495192\pi$
$648$	0	0
$649$	4.08607	0.160392
$650$	8.21222	0.322110
$651$	0	0
$652$	−0.782893	−0.0306604
$653$	36.4030	1.42456	0.712280	−	0.701896i	$-0.247663\pi$
$653$	36.4030	1.42456	0.712280	+	0.701896i	$0.247663\pi$
$654$	0	0
$655$	20.5906	0.804543
$656$	34.1132	1.33190
$657$	0	0
$658$	−5.99010	−0.233519
$659$	13.4552	0.524142	0.262071	−	0.965049i	$-0.415595\pi$
$659$	13.4552	0.524142	0.262071	+	0.965049i	$0.415595\pi$
$660$	0	0
$661$	−28.1101	−1.09335	−0.546677	−	0.837343i	$-0.684108\pi$
$661$	−28.1101	−1.09335	−0.546677	+	0.837343i	$0.684108\pi$
$662$	−60.6910	−2.35882
$663$	0	0
$664$	−11.0152	−0.427473
$665$	−6.17368	−0.239405
$666$	0	0
$667$	−52.0487	−2.01534
$668$	−4.86766	−0.188335
$669$	0	0
$670$	64.4224	2.48885
$671$	1.00000	0.0386046
$672$	0	0
$673$	−33.4248	−1.28843	−0.644216	−	0.764844i	$-0.722816\pi$
$673$	−33.4248	−1.28843	−0.644216	+	0.764844i	$0.722816\pi$
$674$	31.9855	1.23203
$675$	0	0
$676$	−16.1069	−0.619495
$677$	33.9964	1.30659	0.653294	−	0.757104i	$-0.273387\pi$
$677$	33.9964	1.30659	0.653294	+	0.757104i	$0.273387\pi$
$678$	0	0
$679$	−1.06023	−0.0406881
$680$	2.41881	0.0927570
$681$	0	0
$682$	−10.4556	−0.400366
$683$	−14.0398	−0.537218	−0.268609	−	0.963249i	$-0.586564\pi$
$683$	−14.0398	−0.537218	−0.268609	+	0.963249i	$0.586564\pi$
$684$	0	0
$685$	45.0615	1.72171
$686$	7.29274	0.278438
$687$	0	0
$688$	−7.96495	−0.303661
$689$	−1.95732	−0.0745679
$690$	0	0
$691$	−8.49435	−0.323140	−0.161570	−	0.986861i	$-0.551656\pi$
$691$	−8.49435	−0.323140	−0.161570	+	0.986861i	$0.551656\pi$
$692$	17.1523	0.652033
$693$	0	0
$694$	−36.0580	−1.36874
$695$	−1.88274	−0.0714162
$696$	0	0
$697$	4.33289	0.164120
$698$	−34.6769	−1.31254
$699$	0	0
$700$	−1.88172	−0.0711224
$701$	−42.8308	−1.61770	−0.808849	−	0.588016i	$-0.799909\pi$
$701$	−42.8308	−1.61770	−0.808849	+	0.588016i	$0.799909\pi$
$702$	0	0
$703$	64.8018	2.44405
$704$	1.98147	0.0746793
$705$	0	0
$706$	51.5700	1.94086
$707$	−2.90150	−0.109122
$708$	0	0
$709$	43.5929	1.63716	0.818582	−	0.574389i	$-0.194760\pi$
$709$	43.5929	1.63716	0.818582	+	0.574389i	$0.194760\pi$
$710$	−28.1760	−1.05743
$711$	0	0
$712$	−16.6342	−0.623392
$713$	41.0650	1.53790
$714$	0	0
$715$	−2.87203	−0.107408
$716$	13.3585	0.499229
$717$	0	0
$718$	34.6654	1.29370
$719$	−29.8724	−1.11405	−0.557026	−	0.830495i	$-0.688058\pi$
$719$	−29.8724	−1.11405	−0.557026	+	0.830495i	$0.688058\pi$
$720$	0	0
$721$	3.02857	0.112790
$722$	49.9133	1.85758
$723$	0	0
$724$	−21.8347	−0.811482
$725$	−35.9596	−1.33551
$726$	0	0
$727$	9.05098	0.335682	0.167841	−	0.985814i	$-0.446320\pi$
$727$	9.05098	0.335682	0.167841	+	0.985814i	$0.446320\pi$
$728$	0.322854	0.0119658
$729$	0	0
$730$	55.8443	2.06689
$731$	−1.01167	−0.0374179
$732$	0	0
$733$	34.7099	1.28204	0.641019	−	0.767525i	$-0.278512\pi$
$733$	34.7099	1.28204	0.641019	+	0.767525i	$0.278512\pi$
$734$	3.25748	0.120236
$735$	0	0
$736$	−46.2363	−1.70429
$737$	−11.2048	−0.412734
$738$	0	0
$739$	−5.55188	−0.204229	−0.102115	−	0.994773i	$-0.532561\pi$
$739$	−5.55188	−0.204229	−0.102115	+	0.994773i	$0.532561\pi$
$740$	39.7155	1.45997
$741$	0	0
$742$	1.12628	0.0413470
$743$	2.08882	0.0766315	0.0383158	−	0.999266i	$-0.487801\pi$
$743$	2.08882	0.0766315	0.0383158	+	0.999266i	$0.487801\pi$
$744$	0	0
$745$	−48.7674	−1.78670
$746$	34.9319	1.27895
$747$	0	0
$748$	0.822884	0.0300876
$749$	4.68681	0.171252
$750$	0	0
$751$	10.8303	0.395203	0.197601	−	0.980282i	$-0.436685\pi$
$751$	10.8303	0.395203	0.197601	+	0.980282i	$0.436685\pi$
$752$	−55.9616	−2.04071
$753$	0	0
$754$	−12.0680	−0.439491
$755$	23.2662	0.846742
$756$	0	0
$757$	−23.2966	−0.846730	−0.423365	−	0.905959i	$-0.639151\pi$
$757$	−23.2966	−0.846730	−0.423365	+	0.905959i	$0.639151\pi$
$758$	−41.6622	−1.51324
$759$	0	0
$760$	−26.4923	−0.960978
$761$	53.5346	1.94063	0.970314	−	0.241849i	$-0.0777538\pi$
$761$	53.5346	1.94063	0.970314	+	0.241849i	$0.0777538\pi$
$762$	0	0
$763$	−1.57894	−0.0571615
$764$	−9.93956	−0.359601
$765$	0	0
$766$	−13.2775	−0.479737
$767$	−3.72096	−0.134356
$768$	0	0
$769$	26.4098	0.952363	0.476182	−	0.879347i	$-0.342020\pi$
$769$	26.4098	0.952363	0.476182	+	0.879347i	$0.342020\pi$
$770$	1.65262	0.0595564
$771$	0	0
$772$	−26.9257	−0.969076
$773$	−23.6677	−0.851269	−0.425635	−	0.904895i	$-0.639949\pi$
$773$	−23.6677	−0.851269	−0.425635	+	0.904895i	$0.639949\pi$
$774$	0	0
$775$	28.3711	1.01912
$776$	−4.54965	−0.163323
$777$	0	0
$778$	−37.2698	−1.33619
$779$	−47.4566	−1.70031
$780$	0	0
$781$	4.90057	0.175356
$782$	−8.11616	−0.290233
$783$	0	0
$784$	33.8634	1.20941
$785$	24.5907	0.877679
$786$	0	0
$787$	−45.9007	−1.63618	−0.818092	−	0.575088i	$-0.804968\pi$
$787$	−45.9007	−1.63618	−0.818092	+	0.575088i	$0.804968\pi$
$788$	13.5129	0.481378
$789$	0	0
$790$	−28.1122	−1.00019
$791$	−3.38207	−0.120253
$792$	0	0
$793$	−0.910644	−0.0323379
$794$	29.0534	1.03107
$795$	0	0
$796$	12.4772	0.442243
$797$	32.7460	1.15992	0.579962	−	0.814644i	$-0.303067\pi$
$797$	32.7460	1.15992	0.579962	+	0.814644i	$0.303067\pi$
$798$	0	0
$799$	−7.10796	−0.251462
$800$	−31.9439	−1.12939
$801$	0	0
$802$	−67.8057	−2.39430
$803$	−9.71284	−0.342759
$804$	0	0
$805$	−6.49078	−0.228770
$806$	9.52133	0.335375
$807$	0	0
$808$	−12.4509	−0.438020
$809$	48.2215	1.69538	0.847688	−	0.530495i	$-0.177994\pi$
$809$	48.2215	1.69538	0.847688	+	0.530495i	$0.177994\pi$
$810$	0	0
$811$	−28.8635	−1.01353	−0.506767	−	0.862083i	$-0.669160\pi$
$811$	−28.8635	−1.01353	−0.506767	+	0.862083i	$0.669160\pi$
$812$	2.76523	0.0970405
$813$	0	0
$814$	−17.3467	−0.608001
$815$	−1.86573	−0.0653536
$816$	0	0
$817$	11.0804	0.387655
$818$	61.0647	2.13508
$819$	0	0
$820$	−29.0850	−1.01569
$821$	−50.5520	−1.76428	−0.882139	−	0.470989i	$-0.843897\pi$
$821$	−50.5520	−1.76428	−0.882139	+	0.470989i	$0.843897\pi$
$822$	0	0
$823$	−29.2399	−1.01924	−0.509619	−	0.860400i	$-0.670214\pi$
$823$	−29.2399	−1.01924	−0.509619	+	0.860400i	$0.670214\pi$
$824$	12.9961	0.452742
$825$	0	0
$826$	2.14111	0.0744988
$827$	−17.1084	−0.594918	−0.297459	−	0.954735i	$-0.596139\pi$
$827$	−17.1084	−0.594918	−0.297459	+	0.954735i	$0.596139\pi$
$828$	0	0
$829$	−35.4183	−1.23013	−0.615064	−	0.788477i	$-0.710870\pi$
$829$	−35.4183	−1.23013	−0.615064	+	0.788477i	$0.710870\pi$
$830$	51.3462	1.78225
$831$	0	0
$832$	−1.80441	−0.0625567
$833$	4.30116	0.149026
$834$	0	0
$835$	−11.6002	−0.401442
$836$	−9.01275	−0.311712
$837$	0	0
$838$	19.4345	0.671353
$839$	32.8599	1.13445	0.567224	−	0.823564i	$-0.308017\pi$
$839$	32.8599	1.13445	0.567224	+	0.823564i	$0.308017\pi$
$840$	0	0
$841$	23.8434	0.822185
$842$	62.7677	2.16312
$843$	0	0
$844$	−15.9318	−0.548396
$845$	−38.3846	−1.32047
$846$	0	0
$847$	−0.287436	−0.00987641
$848$	10.5221	0.361329
$849$	0	0
$850$	−5.60733	−0.192330
$851$	68.1302	2.33547
$852$	0	0
$853$	1.85750	0.0635996	0.0317998	−	0.999494i	$-0.489876\pi$
$853$	1.85750	0.0635996	0.0317998	+	0.999494i	$0.489876\pi$
$854$	0.524002	0.0179310
$855$	0	0
$856$	20.1119	0.687411
$857$	−43.9149	−1.50010	−0.750052	−	0.661379i	$-0.769971\pi$
$857$	−43.9149	−1.50010	−0.750052	+	0.661379i	$0.769971\pi$
$858$	0	0
$859$	14.8800	0.507698	0.253849	−	0.967244i	$-0.418303\pi$
$859$	14.8800	0.507698	0.253849	+	0.967244i	$0.418303\pi$
$860$	6.79094	0.231569
$861$	0	0
$862$	19.7684	0.673313
$863$	44.5369	1.51605	0.758027	−	0.652224i	$-0.226164\pi$
$863$	44.5369	1.51605	0.758027	+	0.652224i	$0.226164\pi$
$864$	0	0
$865$	40.8760	1.38983
$866$	−2.08825	−0.0709615
$867$	0	0
$868$	−2.18169	−0.0740513
$869$	4.88947	0.165864
$870$	0	0
$871$	10.2036	0.345735
$872$	−6.77550	−0.229447
$873$	0	0
$874$	88.8934	3.00686
$875$	0.0482730	0.00163192
$876$	0	0
$877$	−20.2794	−0.684787	−0.342393	−	0.939557i	$-0.611238\pi$
$877$	−20.2794	−0.684787	−0.342393	+	0.939557i	$0.611238\pi$
$878$	55.7723	1.88222
$879$	0	0
$880$	15.4394	0.520461
$881$	40.5359	1.36569	0.682844	−	0.730564i	$-0.260743\pi$
$881$	40.5359	1.36569	0.682844	+	0.730564i	$0.260743\pi$
$882$	0	0
$883$	8.55778	0.287992	0.143996	−	0.989578i	$-0.454005\pi$
$883$	8.55778	0.287992	0.143996	+	0.989578i	$0.454005\pi$
$884$	−0.749354	−0.0252035
$885$	0	0
$886$	17.3907	0.584251
$887$	−10.5486	−0.354188	−0.177094	−	0.984194i	$-0.556670\pi$
$887$	−10.5486	−0.354188	−0.177094	+	0.984194i	$0.556670\pi$
$888$	0	0
$889$	5.07852	0.170328
$890$	77.5384	2.59909
$891$	0	0
$892$	21.2875	0.712759
$893$	77.8509	2.60518
$894$	0	0
$895$	31.8348	1.06412
$896$	−2.67398	−0.0893313
$897$	0	0
$898$	−30.3617	−1.01318
$899$	−41.6920	−1.39050
$900$	0	0
$901$	1.33646	0.0445240
$902$	12.7036	0.422983
$903$	0	0
$904$	−14.5130	−0.482697
$905$	−52.0348	−1.72970
$906$	0	0
$907$	3.88830	0.129109	0.0645544	−	0.997914i	$-0.479437\pi$
$907$	3.88830	0.129109	0.0645544	+	0.997914i	$0.479437\pi$
$908$	−11.7548	−0.390095
$909$	0	0
$910$	−1.50495	−0.0498886
$911$	−36.2379	−1.20061	−0.600307	−	0.799769i	$-0.704955\pi$
$911$	−36.2379	−1.20061	−0.600307	+	0.799769i	$0.704955\pi$
$912$	0	0
$913$	−8.93050	−0.295556
$914$	12.0431	0.398352
$915$	0	0
$916$	−14.8652	−0.491160
$917$	−1.87659	−0.0619706
$918$	0	0
$919$	−43.8784	−1.44741	−0.723707	−	0.690107i	$-0.757563\pi$
$919$	−43.8784	−1.44741	−0.723707	+	0.690107i	$0.757563\pi$
$920$	−27.8530	−0.918287
$921$	0	0
$922$	−18.0821	−0.595503
$923$	−4.46268	−0.146891
$924$	0	0
$925$	47.0700	1.54765
$926$	−35.9691	−1.18202
$927$	0	0
$928$	46.9422	1.54095
$929$	−29.9320	−0.982036	−0.491018	−	0.871149i	$-0.663375\pi$
$929$	−29.9320	−0.982036	−0.491018	+	0.871149i	$0.663375\pi$
$930$	0	0
$931$	−47.1090	−1.54394
$932$	−20.7921	−0.681068
$933$	0	0
$934$	−44.2638	−1.44836
$935$	1.96103	0.0641326
$936$	0	0
$937$	−0.358168	−0.0117008	−0.00585041	−	0.999983i	$-0.501862\pi$
$937$	−0.358168	−0.0117008	−0.00585041	+	0.999983i	$0.501862\pi$
$938$	−5.87134	−0.191706
$939$	0	0
$940$	47.7130	1.55623
$941$	46.6762	1.52160	0.760801	−	0.648985i	$-0.224806\pi$
$941$	46.6762	1.52160	0.760801	+	0.648985i	$0.224806\pi$
$942$	0	0
$943$	−49.8941	−1.62478
$944$	20.0030	0.651041
$945$	0	0
$946$	−2.96610	−0.0964364
$947$	−26.4645	−0.859981	−0.429990	−	0.902833i	$-0.641483\pi$
$947$	−26.4645	−0.859981	−0.429990	+	0.902833i	$0.641483\pi$
$948$	0	0
$949$	8.84494	0.287119
$950$	61.4150	1.99257
$951$	0	0
$952$	−0.220446	−0.00714469
$953$	41.6677	1.34975	0.674874	−	0.737933i	$-0.264198\pi$
$953$	41.6677	1.34975	0.674874	+	0.737933i	$0.264198\pi$
$954$	0	0
$955$	−23.6872	−0.766498
$956$	−8.31776	−0.269016
$957$	0	0
$958$	−5.75228	−0.185848
$959$	−4.10683	−0.132616
$960$	0	0
$961$	1.89379	0.0610901
$962$	15.7967	0.509305
$963$	0	0
$964$	4.89907	0.157788
$965$	−64.1671	−2.06561
$966$	0	0
$967$	−26.2577	−0.844390	−0.422195	−	0.906505i	$-0.638740\pi$
$967$	−26.2577	−0.844390	−0.422195	+	0.906505i	$0.638740\pi$
$968$	−1.23344	−0.0396442
$969$	0	0
$970$	21.2077	0.680938
$971$	35.2185	1.13021	0.565107	−	0.825017i	$-0.308835\pi$
$971$	35.2185	1.13021	0.565107	+	0.825017i	$0.308835\pi$
$972$	0	0
$973$	0.171589	0.00550090
$974$	−68.9577	−2.20955
$975$	0	0
$976$	4.89541	0.156698
$977$	−61.8806	−1.97974	−0.989868	−	0.141987i	$-0.954651\pi$
$977$	−61.8806	−1.97974	−0.989868	+	0.141987i	$0.954651\pi$
$978$	0	0
$979$	−13.4860	−0.431015
$980$	−28.8720	−0.922282
$981$	0	0
$982$	−60.0825	−1.91731
$983$	−28.0245	−0.893842	−0.446921	−	0.894573i	$-0.647479\pi$
$983$	−28.0245	−0.893842	−0.446921	+	0.894573i	$0.647479\pi$
$984$	0	0
$985$	32.2029	1.02607
$986$	8.24007	0.262417
$987$	0	0
$988$	8.20741	0.261112
$989$	11.6496	0.370434
$990$	0	0
$991$	2.64549	0.0840367	0.0420184	−	0.999117i	$-0.486621\pi$
$991$	2.64549	0.0840367	0.0420184	+	0.999117i	$0.486621\pi$
$992$	−37.0361	−1.17590
$993$	0	0
$994$	2.56791	0.0814492
$995$	29.7347	0.942654
$996$	0	0
$997$	57.5224	1.82175	0.910876	−	0.412680i	$-0.135407\pi$
$997$	57.5224	1.82175	0.910876	+	0.412680i	$0.135407\pi$
$998$	−30.0678	−0.951780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.16		19
3.2	odd	2		671.2.a.c.1.4	✓	19
33.32	even	2		7381.2.a.i.1.16		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.4	✓	19	3.2	odd	2
6039.2.a.k.1.16		19	1.1	even	1	trivial
7381.2.a.i.1.16		19	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82302 q^{2} +1.32341 q^{4} +3.15385 q^{5} -0.287436 q^{7} -1.23344 q^{8} +O(q^{10})\)	\(q+1.82302 q^{2} +1.32341 q^{4} +3.15385 q^{5} -0.287436 q^{7} -1.23344 q^{8} +5.74953 q^{10} -1.00000 q^{11} +0.910644 q^{13} -0.524002 q^{14} -4.89541 q^{16} -0.621790 q^{17} +6.81024 q^{19} +4.17384 q^{20} -1.82302 q^{22} +7.16003 q^{23} +4.94675 q^{25} +1.66012 q^{26} -0.380396 q^{28} -7.26934 q^{29} +5.73531 q^{31} -6.45756 q^{32} -1.13354 q^{34} -0.906529 q^{35} +9.51534 q^{37} +12.4152 q^{38} -3.89007 q^{40} -6.96842 q^{41} +1.62703 q^{43} -1.32341 q^{44} +13.0529 q^{46} +11.4314 q^{47} -6.91738 q^{49} +9.01804 q^{50} +1.20516 q^{52} -2.14938 q^{53} -3.15385 q^{55} +0.354534 q^{56} -13.2522 q^{58} -4.08607 q^{59} -1.00000 q^{61} +10.4556 q^{62} -1.98147 q^{64} +2.87203 q^{65} +11.2048 q^{67} -0.822884 q^{68} -1.65262 q^{70} -4.90057 q^{71} +9.71284 q^{73} +17.3467 q^{74} +9.01275 q^{76} +0.287436 q^{77} -4.88947 q^{79} -15.4394 q^{80} -12.7036 q^{82} +8.93050 q^{83} -1.96103 q^{85} +2.96610 q^{86} +1.23344 q^{88} +13.4860 q^{89} -0.261752 q^{91} +9.47567 q^{92} +20.8398 q^{94} +21.4785 q^{95} +3.68860 q^{97} -12.6105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

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Database

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