LMFDB - Embedded newform 6039.2.a.k.1.2

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			1.2
Root			$2.69542$ 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)

$f(q)$	$=$	$q-2.69542 q^{2} +5.26531 q^{4} -1.92020 q^{5} +2.04756 q^{7} -8.80138 q^{8} +O(q^{10})$	\(q-2.69542 q^{2} +5.26531 q^{4} -1.92020 q^{5} +2.04756 q^{7} -8.80138 q^{8} +5.17575 q^{10} -1.00000 q^{11} -5.05790 q^{13} -5.51904 q^{14} +13.1928 q^{16} +1.14759 q^{17} -8.12286 q^{19} -10.1104 q^{20} +2.69542 q^{22} -3.53584 q^{23} -1.31284 q^{25} +13.6332 q^{26} +10.7810 q^{28} -6.96857 q^{29} +3.42923 q^{31} -17.9575 q^{32} -3.09324 q^{34} -3.93172 q^{35} -7.94224 q^{37} +21.8945 q^{38} +16.9004 q^{40} -8.02845 q^{41} -0.771854 q^{43} -5.26531 q^{44} +9.53060 q^{46} +5.08985 q^{47} -2.80750 q^{49} +3.53866 q^{50} -26.6314 q^{52} -11.4851 q^{53} +1.92020 q^{55} -18.0214 q^{56} +18.7832 q^{58} +13.6413 q^{59} -1.00000 q^{61} -9.24323 q^{62} +22.0174 q^{64} +9.71217 q^{65} -11.4195 q^{67} +6.04241 q^{68} +10.5976 q^{70} +10.0671 q^{71} -5.70524 q^{73} +21.4077 q^{74} -42.7693 q^{76} -2.04756 q^{77} +3.76482 q^{79} -25.3329 q^{80} +21.6401 q^{82} +8.31723 q^{83} -2.20360 q^{85} +2.08047 q^{86} +8.80138 q^{88} -1.94450 q^{89} -10.3564 q^{91} -18.6173 q^{92} -13.7193 q^{94} +15.5975 q^{95} +15.3762 q^{97} +7.56740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.69542	−1.90595	−0.952976	−	0.303046i	$-0.901997\pi$
$2$	−2.69542	−1.90595	−0.952976	+	0.303046i	$0.901997\pi$
$3$	0	0
$3$	0	0
$4$	5.26531	2.63265
$5$	−1.92020	−0.858739	−0.429369	−	0.903129i	$-0.641264\pi$
$5$	−1.92020	−0.858739	−0.429369	+	0.903129i	$0.641264\pi$
$6$	0	0
$7$	2.04756	0.773905	0.386952	−	0.922100i	$-0.373528\pi$
$7$	2.04756	0.773905	0.386952	+	0.922100i	$0.373528\pi$
$8$	−8.80138	−3.11176
$9$	0	0
$10$	5.17575	1.63671
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.05790	−1.40281	−0.701405	−	0.712763i	$-0.747444\pi$
$13$	−5.05790	−1.40281	−0.701405	+	0.712763i	$0.747444\pi$
$14$	−5.51904	−1.47503
$15$	0	0
$16$	13.1928	3.29821
$17$	1.14759	0.278331	0.139166	−	0.990269i	$-0.455558\pi$
$17$	1.14759	0.278331	0.139166	+	0.990269i	$0.455558\pi$
$18$	0	0
$19$	−8.12286	−1.86351	−0.931756	−	0.363086i	$-0.881723\pi$
$19$	−8.12286	−1.86351	−0.931756	+	0.363086i	$0.881723\pi$
$20$	−10.1104	−2.26076
$21$	0	0
$22$	2.69542	0.574666
$23$	−3.53584	−0.737275	−0.368637	−	0.929573i	$-0.620176\pi$
$23$	−3.53584	−0.737275	−0.368637	+	0.929573i	$0.620176\pi$
$24$	0	0
$25$	−1.31284	−0.262568
$26$	13.6332	2.67369
$27$	0	0
$28$	10.7810	2.03742
$29$	−6.96857	−1.29403	−0.647015	−	0.762477i	$-0.723983\pi$
$29$	−6.96857	−1.29403	−0.647015	+	0.762477i	$0.723983\pi$
$30$	0	0
$31$	3.42923	0.615908	0.307954	−	0.951401i	$-0.400356\pi$
$31$	3.42923	0.615908	0.307954	+	0.951401i	$0.400356\pi$
$32$	−17.9575	−3.17447
$33$	0	0
$34$	−3.09324	−0.530486
$35$	−3.93172	−0.664582
$36$	0	0
$37$	−7.94224	−1.30570	−0.652848	−	0.757489i	$-0.726426\pi$
$37$	−7.94224	−1.30570	−0.652848	+	0.757489i	$0.726426\pi$
$38$	21.8945	3.55176
$39$	0	0
$40$	16.9004	2.67219
$41$	−8.02845	−1.25383	−0.626917	−	0.779086i	$-0.715683\pi$
$41$	−8.02845	−1.25383	−0.626917	+	0.779086i	$0.715683\pi$
$42$	0	0
$43$	−0.771854	−0.117707	−0.0588533	−	0.998267i	$-0.518744\pi$
$43$	−0.771854	−0.117707	−0.0588533	+	0.998267i	$0.518744\pi$
$44$	−5.26531	−0.793775
$45$	0	0
$46$	9.53060	1.40521
$47$	5.08985	0.742430	0.371215	−	0.928547i	$-0.378941\pi$
$47$	5.08985	0.742430	0.371215	+	0.928547i	$0.378941\pi$
$48$	0	0
$49$	−2.80750	−0.401071
$50$	3.53866	0.500442
$51$	0	0
$52$	−26.6314	−3.69311
$53$	−11.4851	−1.57760	−0.788801	−	0.614648i	$-0.789298\pi$
$53$	−11.4851	−1.57760	−0.788801	+	0.614648i	$0.789298\pi$
$54$	0	0
$55$	1.92020	0.258919
$56$	−18.0214	−2.40820
$57$	0	0
$58$	18.7832	2.46636
$59$	13.6413	1.77594	0.887972	−	0.459898i	$-0.152114\pi$
$59$	13.6413	1.77594	0.887972	+	0.459898i	$0.152114\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−9.24323	−1.17389
$63$	0	0
$64$	22.0174	2.75218
$65$	9.71217	1.20465
$66$	0	0
$67$	−11.4195	−1.39512	−0.697559	−	0.716528i	$-0.745730\pi$
$67$	−11.4195	−1.39512	−0.697559	+	0.716528i	$0.745730\pi$
$68$	6.04241	0.732749
$69$	0	0
$70$	10.5976	1.26666
$71$	10.0671	1.19474	0.597370	−	0.801966i	$-0.296212\pi$
$71$	10.0671	1.19474	0.597370	+	0.801966i	$0.296212\pi$
$72$	0	0
$73$	−5.70524	−0.667748	−0.333874	−	0.942618i	$-0.608356\pi$
$73$	−5.70524	−0.667748	−0.333874	+	0.942618i	$0.608356\pi$
$74$	21.4077	2.48859
$75$	0	0
$76$	−42.7693	−4.90598
$77$	−2.04756	−0.233341
$78$	0	0
$79$	3.76482	0.423575	0.211788	−	0.977316i	$-0.432072\pi$
$79$	3.76482	0.423575	0.211788	+	0.977316i	$0.432072\pi$
$80$	−25.3329	−2.83230
$81$	0	0
$82$	21.6401	2.38975
$83$	8.31723	0.912935	0.456468	−	0.889740i	$-0.349114\pi$
$83$	8.31723	0.912935	0.456468	+	0.889740i	$0.349114\pi$
$84$	0	0
$85$	−2.20360	−0.239014
$86$	2.08047	0.224343
$87$	0	0
$88$	8.80138	0.938231
$89$	−1.94450	−0.206117	−0.103058	−	0.994675i	$-0.532863\pi$
$89$	−1.94450	−0.206117	−0.103058	+	0.994675i	$0.532863\pi$
$90$	0	0
$91$	−10.3564	−1.08564
$92$	−18.6173	−1.94099
$93$	0	0
$94$	−13.7193	−1.41504
$95$	15.5975	1.60027
$96$	0	0
$97$	15.3762	1.56122	0.780611	−	0.625018i	$-0.214908\pi$
$97$	15.3762	1.56122	0.780611	+	0.625018i	$0.214908\pi$
$98$	7.56740	0.764423
$99$	0	0
$100$	−6.91251	−0.691251
$101$	−17.3436	−1.72576	−0.862878	−	0.505412i	$-0.831341\pi$
$101$	−17.3436	−1.72576	−0.862878	+	0.505412i	$0.831341\pi$
$102$	0	0
$103$	3.27640	0.322833	0.161417	−	0.986886i	$-0.448394\pi$
$103$	3.27640	0.322833	0.161417	+	0.986886i	$0.448394\pi$
$104$	44.5165	4.36521
$105$	0	0
$106$	30.9573	3.00683
$107$	−4.54584	−0.439463	−0.219732	−	0.975560i	$-0.570518\pi$
$107$	−4.54584	−0.439463	−0.219732	+	0.975560i	$0.570518\pi$
$108$	0	0
$109$	7.99283	0.765574	0.382787	−	0.923837i	$-0.374964\pi$
$109$	7.99283	0.765574	0.382787	+	0.923837i	$0.374964\pi$
$110$	−5.17575	−0.493488
$111$	0	0
$112$	27.0131	2.55250
$113$	5.30740	0.499278	0.249639	−	0.968339i	$-0.419688\pi$
$113$	5.30740	0.499278	0.249639	+	0.968339i	$0.419688\pi$
$114$	0	0
$115$	6.78952	0.633126
$116$	−36.6916	−3.40673
$117$	0	0
$118$	−36.7690	−3.38486
$119$	2.34976	0.215402
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.69542	0.244032
$123$	0	0
$124$	18.0559	1.62147
$125$	12.1219	1.08422
$126$	0	0
$127$	11.9457	1.06001	0.530006	−	0.847994i	$-0.322190\pi$
$127$	11.9457	1.06001	0.530006	+	0.847994i	$0.322190\pi$
$128$	−23.4313	−2.07105
$129$	0	0
$130$	−26.1784	−2.29600
$131$	1.68158	0.146920	0.0734600	−	0.997298i	$-0.476596\pi$
$131$	1.68158	0.146920	0.0734600	+	0.997298i	$0.476596\pi$
$132$	0	0
$133$	−16.6320	−1.44218
$134$	30.7805	2.65903
$135$	0	0
$136$	−10.1004	−0.866099
$137$	−18.2640	−1.56040	−0.780199	−	0.625532i	$-0.784882\pi$
$137$	−18.2640	−1.56040	−0.780199	+	0.625532i	$0.784882\pi$
$138$	0	0
$139$	−0.862740	−0.0731767	−0.0365883	−	0.999330i	$-0.511649\pi$
$139$	−0.862740	−0.0731767	−0.0365883	+	0.999330i	$0.511649\pi$
$140$	−20.7017	−1.74961
$141$	0	0
$142$	−27.1350	−2.27712
$143$	5.05790	0.422963
$144$	0	0
$145$	13.3810	1.11123
$146$	15.3780	1.27270
$147$	0	0
$148$	−41.8183	−3.43744
$149$	6.60379	0.541004	0.270502	−	0.962719i	$-0.412810\pi$
$149$	6.60379	0.541004	0.270502	+	0.962719i	$0.412810\pi$
$150$	0	0
$151$	11.0977	0.903119	0.451559	−	0.892241i	$-0.350868\pi$
$151$	11.0977	0.903119	0.451559	+	0.892241i	$0.350868\pi$
$152$	71.4924	5.79880
$153$	0	0
$154$	5.51904	0.444737
$155$	−6.58480	−0.528904
$156$	0	0
$157$	5.27512	0.421000	0.210500	−	0.977594i	$-0.432491\pi$
$157$	5.27512	0.421000	0.210500	+	0.977594i	$0.432491\pi$
$158$	−10.1478	−0.807314
$159$	0	0
$160$	34.4820	2.72604
$161$	−7.23985	−0.570580
$162$	0	0
$163$	−3.79414	−0.297180	−0.148590	−	0.988899i	$-0.547473\pi$
$163$	−3.79414	−0.297180	−0.148590	+	0.988899i	$0.547473\pi$
$164$	−42.2723	−3.30091
$165$	0	0
$166$	−22.4185	−1.74001
$167$	4.98070	0.385418	0.192709	−	0.981256i	$-0.438273\pi$
$167$	4.98070	0.385418	0.192709	+	0.981256i	$0.438273\pi$
$168$	0	0
$169$	12.5824	0.967876
$170$	5.93963	0.455549
$171$	0	0
$172$	−4.06405	−0.309881
$173$	13.7955	1.04885	0.524427	−	0.851455i	$-0.324279\pi$
$173$	13.7955	1.04885	0.524427	+	0.851455i	$0.324279\pi$
$174$	0	0
$175$	−2.68812	−0.203203
$176$	−13.1928	−0.994448
$177$	0	0
$178$	5.24126	0.392849
$179$	21.8696	1.63461	0.817306	−	0.576204i	$-0.195467\pi$
$179$	21.8696	1.63461	0.817306	+	0.576204i	$0.195467\pi$
$180$	0	0
$181$	11.4327	0.849787	0.424893	−	0.905243i	$-0.360312\pi$
$181$	11.4327	0.849787	0.424893	+	0.905243i	$0.360312\pi$
$182$	27.9148	2.06918
$183$	0	0
$184$	31.1203	2.29422
$185$	15.2507	1.12125
$186$	0	0
$187$	−1.14759	−0.0839200
$188$	26.7996	1.95456
$189$	0	0
$190$	−42.0418	−3.05004
$191$	−8.87583	−0.642233	−0.321116	−	0.947040i	$-0.604058\pi$
$191$	−8.87583	−0.642233	−0.321116	+	0.947040i	$0.604058\pi$
$192$	0	0
$193$	−15.3781	−1.10694	−0.553471	−	0.832868i	$-0.686697\pi$
$193$	−15.3781	−1.10694	−0.553471	+	0.832868i	$0.686697\pi$
$194$	−41.4455	−2.97561
$195$	0	0
$196$	−14.7823	−1.05588
$197$	−10.4220	−0.742540	−0.371270	−	0.928525i	$-0.621078\pi$
$197$	−10.4220	−0.742540	−0.371270	+	0.928525i	$0.621078\pi$
$198$	0	0
$199$	0.694713	0.0492469	0.0246235	−	0.999697i	$-0.492161\pi$
$199$	0.694713	0.0492469	0.0246235	+	0.999697i	$0.492161\pi$
$200$	11.5548	0.817049
$201$	0	0
$202$	46.7484	3.28921
$203$	−14.2686	−1.00146
$204$	0	0
$205$	15.4162	1.07672
$206$	−8.83128	−0.615304
$207$	0	0
$208$	−66.7281	−4.62676
$209$	8.12286	0.561870
$210$	0	0
$211$	−1.04837	−0.0721727	−0.0360863	−	0.999349i	$-0.511489\pi$
$211$	−1.04837	−0.0721727	−0.0360863	+	0.999349i	$0.511489\pi$
$212$	−60.4727	−4.15328
$213$	0	0
$214$	12.2530	0.837596
$215$	1.48211	0.101079
$216$	0	0
$217$	7.02155	0.476654
$218$	−21.5441	−1.45915
$219$	0	0
$220$	10.1104	0.681645
$221$	−5.80439	−0.390446
$222$	0	0
$223$	1.06818	0.0715306	0.0357653	−	0.999360i	$-0.488613\pi$
$223$	1.06818	0.0715306	0.0357653	+	0.999360i	$0.488613\pi$
$224$	−36.7691	−2.45674
$225$	0	0
$226$	−14.3057	−0.951601
$227$	−7.21212	−0.478685	−0.239343	−	0.970935i	$-0.576932\pi$
$227$	−7.21212	−0.478685	−0.239343	+	0.970935i	$0.576932\pi$
$228$	0	0
$229$	17.7437	1.17253	0.586267	−	0.810118i	$-0.300597\pi$
$229$	17.7437	1.17253	0.586267	+	0.810118i	$0.300597\pi$
$230$	−18.3006	−1.20671
$231$	0	0
$232$	61.3330	4.02671
$233$	−21.4789	−1.40713	−0.703566	−	0.710630i	$-0.748410\pi$
$233$	−21.4789	−1.40713	−0.703566	+	0.710630i	$0.748410\pi$
$234$	0	0
$235$	−9.77351	−0.637554
$236$	71.8255	4.67544
$237$	0	0
$238$	−6.33359	−0.410546
$239$	2.44108	0.157900	0.0789502	−	0.996879i	$-0.474843\pi$
$239$	2.44108	0.157900	0.0789502	+	0.996879i	$0.474843\pi$
$240$	0	0
$241$	−8.61322	−0.554826	−0.277413	−	0.960751i	$-0.589477\pi$
$241$	−8.61322	−0.554826	−0.277413	+	0.960751i	$0.589477\pi$
$242$	−2.69542	−0.173268
$243$	0	0
$244$	−5.26531	−0.337077
$245$	5.39095	0.344415
$246$	0	0
$247$	41.0846	2.61415
$248$	−30.1820	−1.91656
$249$	0	0
$250$	−32.6737	−2.06646
$251$	8.32074	0.525200	0.262600	−	0.964905i	$-0.415420\pi$
$251$	8.32074	0.525200	0.262600	+	0.964905i	$0.415420\pi$
$252$	0	0
$253$	3.53584	0.222297
$254$	−32.1988	−2.02033
$255$	0	0
$256$	19.1223	1.19514
$257$	20.6208	1.28629	0.643144	−	0.765745i	$-0.277630\pi$
$257$	20.6208	1.28629	0.643144	+	0.765745i	$0.277630\pi$
$258$	0	0
$259$	−16.2622	−1.01048
$260$	51.1376	3.17142
$261$	0	0
$262$	−4.53256	−0.280023
$263$	−0.976262	−0.0601989	−0.0300995	−	0.999547i	$-0.509582\pi$
$263$	−0.976262	−0.0601989	−0.0300995	+	0.999547i	$0.509582\pi$
$264$	0	0
$265$	22.0537	1.35475
$266$	44.8304	2.74873
$267$	0	0
$268$	−60.1273	−3.67286
$269$	−30.0972	−1.83506	−0.917530	−	0.397666i	$-0.869820\pi$
$269$	−30.0972	−1.83506	−0.917530	+	0.397666i	$0.869820\pi$
$270$	0	0
$271$	22.2148	1.34945	0.674726	−	0.738069i	$-0.264262\pi$
$271$	22.2148	1.34945	0.674726	+	0.738069i	$0.264262\pi$
$272$	15.1400	0.917994
$273$	0	0
$274$	49.2292	2.97404
$275$	1.31284	0.0791673
$276$	0	0
$277$	17.6386	1.05980	0.529900	−	0.848060i	$-0.322230\pi$
$277$	17.6386	1.05980	0.529900	+	0.848060i	$0.322230\pi$
$278$	2.32545	0.139471
$279$	0	0
$280$	34.6046	2.06802
$281$	−2.53996	−0.151521	−0.0757607	−	0.997126i	$-0.524139\pi$
$281$	−2.53996	−0.151521	−0.0757607	+	0.997126i	$0.524139\pi$
$282$	0	0
$283$	−10.2899	−0.611669	−0.305834	−	0.952085i	$-0.598935\pi$
$283$	−10.2899	−0.611669	−0.305834	+	0.952085i	$0.598935\pi$
$284$	53.0062	3.14534
$285$	0	0
$286$	−13.6332	−0.806147
$287$	−16.4387	−0.970348
$288$	0	0
$289$	−15.6830	−0.922532
$290$	−36.0675	−2.11796
$291$	0	0
$292$	−30.0399	−1.75795
$293$	−4.28066	−0.250079	−0.125039	−	0.992152i	$-0.539906\pi$
$293$	−4.28066	−0.250079	−0.125039	+	0.992152i	$0.539906\pi$
$294$	0	0
$295$	−26.1940	−1.52507
$296$	69.9027	4.06301
$297$	0	0
$298$	−17.8000	−1.03113
$299$	17.8840	1.03426
$300$	0	0
$301$	−1.58042	−0.0910937
$302$	−29.9130	−1.72130
$303$	0	0
$304$	−107.164	−6.14625
$305$	1.92020	0.109950
$306$	0	0
$307$	6.19669	0.353664	0.176832	−	0.984241i	$-0.443415\pi$
$307$	6.19669	0.353664	0.176832	+	0.984241i	$0.443415\pi$
$308$	−10.7810	−0.614306
$309$	0	0
$310$	17.7488	1.00807
$311$	−2.33194	−0.132232	−0.0661160	−	0.997812i	$-0.521061\pi$
$311$	−2.33194	−0.132232	−0.0661160	+	0.997812i	$0.521061\pi$
$312$	0	0
$313$	2.29322	0.129620	0.0648102	−	0.997898i	$-0.479356\pi$
$313$	2.29322	0.129620	0.0648102	+	0.997898i	$0.479356\pi$
$314$	−14.2187	−0.802407
$315$	0	0
$316$	19.8229	1.11513
$317$	−25.8628	−1.45260	−0.726299	−	0.687379i	$-0.758761\pi$
$317$	−25.8628	−1.45260	−0.726299	+	0.687379i	$0.758761\pi$
$318$	0	0
$319$	6.96857	0.390165
$320$	−42.2778	−2.36340
$321$	0	0
$322$	19.5145	1.08750
$323$	−9.32170	−0.518673
$324$	0	0
$325$	6.64022	0.368333
$326$	10.2268	0.566411
$327$	0	0
$328$	70.6615	3.90163
$329$	10.4218	0.574571
$330$	0	0
$331$	−2.52575	−0.138828	−0.0694139	−	0.997588i	$-0.522113\pi$
$331$	−2.52575	−0.138828	−0.0694139	+	0.997588i	$0.522113\pi$
$332$	43.7928	2.40344
$333$	0	0
$334$	−13.4251	−0.734588
$335$	21.9277	1.19804
$336$	0	0
$337$	11.2141	0.610871	0.305436	−	0.952213i	$-0.401198\pi$
$337$	11.2141	0.610871	0.305436	+	0.952213i	$0.401198\pi$
$338$	−33.9148	−1.84472
$339$	0	0
$340$	−11.6026	−0.629240
$341$	−3.42923	−0.185703
$342$	0	0
$343$	−20.0814	−1.08430
$344$	6.79338	0.366275
$345$	0	0
$346$	−37.1848	−1.99907
$347$	−13.7325	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$347$	−13.7325	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$348$	0	0
$349$	−24.4459	−1.30856	−0.654280	−	0.756252i	$-0.727028\pi$
$349$	−24.4459	−1.30856	−0.654280	+	0.756252i	$0.727028\pi$
$350$	7.24562	0.387295
$351$	0	0
$352$	17.9575	0.957139
$353$	−21.3254	−1.13504	−0.567519	−	0.823361i	$-0.692097\pi$
$353$	−21.3254	−1.13504	−0.567519	+	0.823361i	$0.692097\pi$
$354$	0	0
$355$	−19.3307	−1.02597
$356$	−10.2384	−0.542634
$357$	0	0
$358$	−58.9479	−3.11549
$359$	0.928492	0.0490039	0.0245020	−	0.999700i	$-0.492200\pi$
$359$	0.928492	0.0490039	0.0245020	+	0.999700i	$0.492200\pi$
$360$	0	0
$361$	46.9808	2.47267
$362$	−30.8160	−1.61965
$363$	0	0
$364$	−54.5294	−2.85812
$365$	10.9552	0.573421
$366$	0	0
$367$	−28.6651	−1.49630	−0.748152	−	0.663527i	$-0.769059\pi$
$367$	−28.6651	−1.49630	−0.748152	+	0.663527i	$0.769059\pi$
$368$	−46.6478	−2.43169
$369$	0	0
$370$	−41.1070	−2.13705
$371$	−23.5165	−1.22091
$372$	0	0
$373$	−17.8961	−0.926627	−0.463314	−	0.886194i	$-0.653340\pi$
$373$	−17.8961	−0.926627	−0.463314	+	0.886194i	$0.653340\pi$
$374$	3.09324	0.159947
$375$	0	0
$376$	−44.7977	−2.31026
$377$	35.2463	1.81528
$378$	0	0
$379$	12.1289	0.623021	0.311511	−	0.950243i	$-0.399165\pi$
$379$	12.1289	0.623021	0.311511	+	0.950243i	$0.399165\pi$
$380$	82.1256	4.21295
$381$	0	0
$382$	23.9241	1.22406
$383$	8.54411	0.436584	0.218292	−	0.975884i	$-0.429952\pi$
$383$	8.54411	0.436584	0.218292	+	0.975884i	$0.429952\pi$
$384$	0	0
$385$	3.93172	0.200379
$386$	41.4506	2.10978
$387$	0	0
$388$	80.9607	4.11015
$389$	−13.9503	−0.707310	−0.353655	−	0.935376i	$-0.615061\pi$
$389$	−13.9503	−0.707310	−0.353655	+	0.935376i	$0.615061\pi$
$390$	0	0
$391$	−4.05770	−0.205207
$392$	24.7099	1.24804
$393$	0	0
$394$	28.0918	1.41525
$395$	−7.22919	−0.363740
$396$	0	0
$397$	−38.5097	−1.93275	−0.966373	−	0.257143i	$-0.917219\pi$
$397$	−38.5097	−1.93275	−0.966373	+	0.257143i	$0.917219\pi$
$398$	−1.87255	−0.0938623
$399$	0	0
$400$	−17.3201	−0.866005
$401$	24.0504	1.20102	0.600510	−	0.799617i	$-0.294964\pi$
$401$	24.0504	1.20102	0.600510	+	0.799617i	$0.294964\pi$
$402$	0	0
$403$	−17.3447	−0.864002
$404$	−91.3196	−4.54332
$405$	0	0
$406$	38.4598	1.90873
$407$	7.94224	0.393682
$408$	0	0
$409$	19.8989	0.983938	0.491969	−	0.870613i	$-0.336277\pi$
$409$	19.8989	0.983938	0.491969	+	0.870613i	$0.336277\pi$
$410$	−41.5532	−2.05217
$411$	0	0
$412$	17.2512	0.849907
$413$	27.9313	1.37441
$414$	0	0
$415$	−15.9707	−0.783973
$416$	90.8274	4.45318
$417$	0	0
$418$	−21.8945	−1.07090
$419$	12.9815	0.634189	0.317094	−	0.948394i	$-0.397293\pi$
$419$	12.9815	0.634189	0.317094	+	0.948394i	$0.397293\pi$
$420$	0	0
$421$	−3.38577	−0.165012	−0.0825062	−	0.996591i	$-0.526292\pi$
$421$	−3.38577	−0.165012	−0.0825062	+	0.996591i	$0.526292\pi$
$422$	2.82580	0.137558
$423$	0	0
$424$	101.085	4.90912
$425$	−1.50660	−0.0730809
$426$	0	0
$427$	−2.04756	−0.0990884
$428$	−23.9353	−1.15695
$429$	0	0
$430$	−3.99492	−0.192652
$431$	−39.1250	−1.88459	−0.942293	−	0.334790i	$-0.891335\pi$
$431$	−39.1250	−1.88459	−0.942293	+	0.334790i	$0.891335\pi$
$432$	0	0
$433$	−33.7007	−1.61955	−0.809776	−	0.586740i	$-0.800411\pi$
$433$	−33.7007	−1.61955	−0.809776	+	0.586740i	$0.800411\pi$
$434$	−18.9261	−0.908480
$435$	0	0
$436$	42.0847	2.01549
$437$	28.7212	1.37392
$438$	0	0
$439$	36.3857	1.73660	0.868298	−	0.496043i	$-0.165214\pi$
$439$	36.3857	1.73660	0.868298	+	0.496043i	$0.165214\pi$
$440$	−16.9004	−0.805695
$441$	0	0
$442$	15.6453	0.744171
$443$	−27.5870	−1.31070	−0.655349	−	0.755327i	$-0.727478\pi$
$443$	−27.5870	−1.31070	−0.655349	+	0.755327i	$0.727478\pi$
$444$	0	0
$445$	3.73383	0.177000
$446$	−2.87920	−0.136334
$447$	0	0
$448$	45.0820	2.12992
$449$	33.3765	1.57513	0.787567	−	0.616229i	$-0.211340\pi$
$449$	33.3765	1.57513	0.787567	+	0.616229i	$0.211340\pi$
$450$	0	0
$451$	8.02845	0.378045
$452$	27.9451	1.31443
$453$	0	0
$454$	19.4397	0.912351
$455$	19.8863	0.932282
$456$	0	0
$457$	−10.6901	−0.500064	−0.250032	−	0.968238i	$-0.580441\pi$
$457$	−10.6901	−0.500064	−0.250032	+	0.968238i	$0.580441\pi$
$458$	−47.8267	−2.23479
$459$	0	0
$460$	35.7489	1.66680
$461$	−32.5513	−1.51607	−0.758034	−	0.652215i	$-0.773840\pi$
$461$	−32.5513	−1.51607	−0.758034	+	0.652215i	$0.773840\pi$
$462$	0	0
$463$	26.4666	1.23001	0.615004	−	0.788524i	$-0.289154\pi$
$463$	26.4666	1.23001	0.615004	+	0.788524i	$0.289154\pi$
$464$	−91.9352	−4.26798
$465$	0	0
$466$	57.8948	2.68193
$467$	31.2536	1.44624	0.723122	−	0.690720i	$-0.242706\pi$
$467$	31.2536	1.44624	0.723122	+	0.690720i	$0.242706\pi$
$468$	0	0
$469$	−23.3822	−1.07969
$470$	26.3438	1.21515
$471$	0	0
$472$	−120.062	−5.52631
$473$	0.771854	0.0354899
$474$	0	0
$475$	10.6640	0.489299
$476$	12.3722	0.567078
$477$	0	0
$478$	−6.57975	−0.300951
$479$	−38.2909	−1.74956	−0.874778	−	0.484524i	$-0.838993\pi$
$479$	−38.2909	−1.74956	−0.874778	+	0.484524i	$0.838993\pi$
$480$	0	0
$481$	40.1711	1.83164
$482$	23.2163	1.05747
$483$	0	0
$484$	5.26531	0.239332
$485$	−29.5254	−1.34068
$486$	0	0
$487$	20.3803	0.923517	0.461759	−	0.887006i	$-0.347219\pi$
$487$	20.3803	0.923517	0.461759	+	0.887006i	$0.347219\pi$
$488$	8.80138	0.398420
$489$	0	0
$490$	−14.5309	−0.656439
$491$	40.6301	1.83361	0.916805	−	0.399335i	$-0.130759\pi$
$491$	40.6301	1.83361	0.916805	+	0.399335i	$0.130759\pi$
$492$	0	0
$493$	−7.99705	−0.360169
$494$	−110.740	−4.98245
$495$	0	0
$496$	45.2413	2.03139
$497$	20.6129	0.924615
$498$	0	0
$499$	29.5333	1.32209	0.661046	−	0.750345i	$-0.270113\pi$
$499$	29.5333	1.32209	0.661046	+	0.750345i	$0.270113\pi$
$500$	63.8255	2.85436
$501$	0	0
$502$	−22.4279	−1.00101
$503$	−8.57603	−0.382386	−0.191193	−	0.981552i	$-0.561236\pi$
$503$	−8.57603	−0.382386	−0.191193	+	0.981552i	$0.561236\pi$
$504$	0	0
$505$	33.3032	1.48197
$506$	−9.53060	−0.423687
$507$	0	0
$508$	62.8980	2.79065
$509$	−25.9327	−1.14945	−0.574724	−	0.818348i	$-0.694890\pi$
$509$	−25.9327	−1.14945	−0.574724	+	0.818348i	$0.694890\pi$
$510$	0	0
$511$	−11.6818	−0.516774
$512$	−4.68019	−0.206837
$513$	0	0
$514$	−55.5817	−2.45160
$515$	−6.29133	−0.277229
$516$	0	0
$517$	−5.08985	−0.223851
$518$	43.8335	1.92593
$519$	0	0
$520$	−85.4806	−3.74857
$521$	−24.3706	−1.06770	−0.533848	−	0.845580i	$-0.679255\pi$
$521$	−24.3706	−1.06770	−0.533848	+	0.845580i	$0.679255\pi$
$522$	0	0
$523$	−33.0409	−1.44478	−0.722390	−	0.691486i	$-0.756956\pi$
$523$	−33.0409	−1.44478	−0.722390	+	0.691486i	$0.756956\pi$
$524$	8.85402	0.386790
$525$	0	0
$526$	2.63144	0.114736
$527$	3.93535	0.171426
$528$	0	0
$529$	−10.4978	−0.456426
$530$	−59.4441	−2.58208
$531$	0	0
$532$	−87.5727	−3.79676
$533$	40.6071	1.75889
$534$	0	0
$535$	8.72892	0.377384
$536$	100.508	4.34127
$537$	0	0
$538$	81.1248	3.49754
$539$	2.80750	0.120928
$540$	0	0
$541$	−3.17301	−0.136418	−0.0682091	−	0.997671i	$-0.521729\pi$
$541$	−3.17301	−0.136418	−0.0682091	+	0.997671i	$0.521729\pi$
$542$	−59.8782	−2.57199
$543$	0	0
$544$	−20.6078	−0.883554
$545$	−15.3478	−0.657428
$546$	0	0
$547$	1.72864	0.0739115	0.0369558	−	0.999317i	$-0.488234\pi$
$547$	1.72864	0.0739115	0.0369558	+	0.999317i	$0.488234\pi$
$548$	−96.1655	−4.10799
$549$	0	0
$550$	−3.53866	−0.150889
$551$	56.6047	2.41144
$552$	0	0
$553$	7.70869	0.327807
$554$	−47.5434	−2.01993
$555$	0	0
$556$	−4.54259	−0.192649
$557$	−0.868836	−0.0368138	−0.0184069	−	0.999831i	$-0.505859\pi$
$557$	−0.868836	−0.0368138	−0.0184069	+	0.999831i	$0.505859\pi$
$558$	0	0
$559$	3.90396	0.165120
$560$	−51.8705	−2.19193
$561$	0	0
$562$	6.84627	0.288792
$563$	−5.84546	−0.246357	−0.123178	−	0.992385i	$-0.539309\pi$
$563$	−5.84546	−0.246357	−0.123178	+	0.992385i	$0.539309\pi$
$564$	0	0
$565$	−10.1913	−0.428750
$566$	27.7355	1.16581
$567$	0	0
$568$	−88.6041	−3.71774
$569$	16.6614	0.698482	0.349241	−	0.937033i	$-0.386439\pi$
$569$	16.6614	0.698482	0.349241	+	0.937033i	$0.386439\pi$
$570$	0	0
$571$	−17.3260	−0.725072	−0.362536	−	0.931970i	$-0.618089\pi$
$571$	−17.3260	−0.725072	−0.362536	+	0.931970i	$0.618089\pi$
$572$	26.6314	1.11352
$573$	0	0
$574$	44.3094	1.84944
$575$	4.64200	0.193585
$576$	0	0
$577$	4.55759	0.189735	0.0948674	−	0.995490i	$-0.469757\pi$
$577$	4.55759	0.189735	0.0948674	+	0.995490i	$0.469757\pi$
$578$	42.2724	1.75830
$579$	0	0
$580$	70.4552	2.92549
$581$	17.0300	0.706525
$582$	0	0
$583$	11.4851	0.475665
$584$	50.2140	2.07787
$585$	0	0
$586$	11.5382	0.476638
$587$	26.8384	1.10774	0.553869	−	0.832604i	$-0.313151\pi$
$587$	26.8384	1.10774	0.553869	+	0.832604i	$0.313151\pi$
$588$	0	0
$589$	−27.8551	−1.14775
$590$	70.6038	2.90671
$591$	0	0
$592$	−104.781	−4.30646
$593$	7.92520	0.325449	0.162725	−	0.986672i	$-0.447972\pi$
$593$	7.92520	0.325449	0.162725	+	0.986672i	$0.447972\pi$
$594$	0	0
$595$	−4.51200	−0.184974
$596$	34.7710	1.42427
$597$	0	0
$598$	−48.2048	−1.97124
$599$	42.0884	1.71968	0.859842	−	0.510560i	$-0.170562\pi$
$599$	42.0884	1.71968	0.859842	+	0.510560i	$0.170562\pi$
$600$	0	0
$601$	22.6153	0.922497	0.461249	−	0.887271i	$-0.347402\pi$
$601$	22.6153	0.922497	0.461249	+	0.887271i	$0.347402\pi$
$602$	4.25989	0.173620
$603$	0	0
$604$	58.4328	2.37760
$605$	−1.92020	−0.0780671
$606$	0	0
$607$	−15.4013	−0.625121	−0.312560	−	0.949898i	$-0.601187\pi$
$607$	−15.4013	−0.625121	−0.312560	+	0.949898i	$0.601187\pi$
$608$	145.866	5.91566
$609$	0	0
$610$	−5.17575	−0.209560
$611$	−25.7440	−1.04149
$612$	0	0
$613$	−8.83937	−0.357019	−0.178509	−	0.983938i	$-0.557127\pi$
$613$	−8.83937	−0.357019	−0.178509	+	0.983938i	$0.557127\pi$
$614$	−16.7027	−0.674067
$615$	0	0
$616$	18.0214	0.726101
$617$	44.2618	1.78191	0.890957	−	0.454087i	$-0.150034\pi$
$617$	44.2618	1.78191	0.890957	+	0.454087i	$0.150034\pi$
$618$	0	0
$619$	−10.4503	−0.420033	−0.210017	−	0.977698i	$-0.567352\pi$
$619$	−10.4503	−0.420033	−0.210017	+	0.977698i	$0.567352\pi$
$620$	−34.6710	−1.39242
$621$	0	0
$622$	6.28556	0.252028
$623$	−3.98148	−0.159515
$624$	0	0
$625$	−16.7122	−0.668490
$626$	−6.18119	−0.247050
$627$	0	0
$628$	27.7751	1.10835
$629$	−9.11442	−0.363416
$630$	0	0
$631$	17.1012	0.680787	0.340394	−	0.940283i	$-0.389440\pi$
$631$	17.1012	0.680787	0.340394	+	0.940283i	$0.389440\pi$
$632$	−33.1356	−1.31806
$633$	0	0
$634$	69.7111	2.76858
$635$	−22.9382	−0.910274
$636$	0	0
$637$	14.2001	0.562627
$638$	−18.7832	−0.743636
$639$	0	0
$640$	44.9927	1.77849
$641$	1.95931	0.0773880	0.0386940	−	0.999251i	$-0.487680\pi$
$641$	1.95931	0.0773880	0.0386940	+	0.999251i	$0.487680\pi$
$642$	0	0
$643$	21.9759	0.866644	0.433322	−	0.901239i	$-0.357341\pi$
$643$	21.9759	0.866644	0.433322	+	0.901239i	$0.357341\pi$
$644$	−38.1200	−1.50214
$645$	0	0
$646$	25.1259	0.988566
$647$	4.74294	0.186464	0.0932321	−	0.995644i	$-0.470280\pi$
$647$	4.74294	0.186464	0.0932321	+	0.995644i	$0.470280\pi$
$648$	0	0
$649$	−13.6413	−0.535467
$650$	−17.8982	−0.702026
$651$	0	0
$652$	−19.9773	−0.782372
$653$	22.2797	0.871871	0.435935	−	0.899978i	$-0.356418\pi$
$653$	22.2797	0.871871	0.435935	+	0.899978i	$0.356418\pi$
$654$	0	0
$655$	−3.22896	−0.126166
$656$	−105.918	−4.13541
$657$	0	0
$658$	−28.0911	−1.09510
$659$	28.0012	1.09077	0.545386	−	0.838185i	$-0.316383\pi$
$659$	28.0012	1.09077	0.545386	+	0.838185i	$0.316383\pi$
$660$	0	0
$661$	41.7104	1.62235	0.811173	−	0.584806i	$-0.198830\pi$
$661$	41.7104	1.62235	0.811173	+	0.584806i	$0.198830\pi$
$662$	6.80797	0.264599
$663$	0	0
$664$	−73.2032	−2.84083
$665$	31.9368	1.23846
$666$	0	0
$667$	24.6398	0.954056
$668$	26.2249	1.01467
$669$	0	0
$670$	−59.1046	−2.28341
$671$	1.00000	0.0386046
$672$	0	0
$673$	−10.0629	−0.387898	−0.193949	−	0.981012i	$-0.562130\pi$
$673$	−10.0629	−0.387898	−0.193949	+	0.981012i	$0.562130\pi$
$674$	−30.2268	−1.16429
$675$	0	0
$676$	66.2501	2.54808
$677$	−23.1706	−0.890517	−0.445258	−	0.895402i	$-0.646888\pi$
$677$	−23.1706	−0.890517	−0.445258	+	0.895402i	$0.646888\pi$
$678$	0	0
$679$	31.4838	1.20824
$680$	19.3947	0.743753
$681$	0	0
$682$	9.24323	0.353941
$683$	44.9098	1.71842	0.859212	−	0.511619i	$-0.170954\pi$
$683$	44.9098	1.71842	0.859212	+	0.511619i	$0.170954\pi$
$684$	0	0
$685$	35.0705	1.33997
$686$	54.1280	2.06662
$687$	0	0
$688$	−10.1829	−0.388221
$689$	58.0906	2.21308
$690$	0	0
$691$	22.4821	0.855258	0.427629	−	0.903954i	$-0.359349\pi$
$691$	22.4821	0.855258	0.427629	+	0.903954i	$0.359349\pi$
$692$	72.6377	2.76127
$693$	0	0
$694$	37.0148	1.40506
$695$	1.65663	0.0628396
$696$	0	0
$697$	−9.21336	−0.348981
$698$	65.8921	2.49405
$699$	0	0
$700$	−14.1538	−0.534962
$701$	10.4901	0.396206	0.198103	−	0.980181i	$-0.436522\pi$
$701$	10.4901	0.396206	0.198103	+	0.980181i	$0.436522\pi$
$702$	0	0
$703$	64.5136	2.43318
$704$	−22.0174	−0.829813
$705$	0	0
$706$	57.4810	2.16333
$707$	−35.5121	−1.33557
$708$	0	0
$709$	49.1227	1.84484	0.922421	−	0.386186i	$-0.126208\pi$
$709$	49.1227	1.84484	0.922421	+	0.386186i	$0.126208\pi$
$710$	52.1045	1.95545
$711$	0	0
$712$	17.1143	0.641386
$713$	−12.1252	−0.454093
$714$	0	0
$715$	−9.71217	−0.363215
$716$	115.150	4.30337
$717$	0	0
$718$	−2.50268	−0.0933991
$719$	11.6568	0.434726	0.217363	−	0.976091i	$-0.430254\pi$
$719$	11.6568	0.434726	0.217363	+	0.976091i	$0.430254\pi$
$720$	0	0
$721$	6.70862	0.249842
$722$	−126.633	−4.71280
$723$	0	0
$724$	60.1967	2.23719
$725$	9.14862	0.339771
$726$	0	0
$727$	21.6498	0.802948	0.401474	−	0.915870i	$-0.368498\pi$
$727$	21.6498	0.802948	0.401474	+	0.915870i	$0.368498\pi$
$728$	91.1503	3.37825
$729$	0	0
$730$	−29.5289	−1.09291
$731$	−0.885771	−0.0327614
$732$	0	0
$733$	6.09963	0.225295	0.112647	−	0.993635i	$-0.464067\pi$
$733$	6.09963	0.225295	0.112647	+	0.993635i	$0.464067\pi$
$734$	77.2645	2.85188
$735$	0	0
$736$	63.4950	2.34046
$737$	11.4195	0.420644
$738$	0	0
$739$	−0.976933	−0.0359370	−0.0179685	−	0.999839i	$-0.505720\pi$
$739$	−0.976933	−0.0359370	−0.0179685	+	0.999839i	$0.505720\pi$
$740$	80.2994	2.95186
$741$	0	0
$742$	63.3868	2.32700
$743$	−0.821800	−0.0301489	−0.0150745	−	0.999886i	$-0.504799\pi$
$743$	−0.821800	−0.0301489	−0.0150745	+	0.999886i	$0.504799\pi$
$744$	0	0
$745$	−12.6806	−0.464581
$746$	48.2377	1.76611
$747$	0	0
$748$	−6.04241	−0.220932
$749$	−9.30789	−0.340103
$750$	0	0
$751$	−49.0864	−1.79119	−0.895594	−	0.444873i	$-0.853249\pi$
$751$	−49.0864	−1.79119	−0.895594	+	0.444873i	$0.853249\pi$
$752$	67.1495	2.44869
$753$	0	0
$754$	−95.0038	−3.45983
$755$	−21.3098	−0.775543
$756$	0	0
$757$	−32.3794	−1.17685	−0.588425	−	0.808552i	$-0.700252\pi$
$757$	−32.3794	−1.17685	−0.588425	+	0.808552i	$0.700252\pi$
$758$	−32.6926	−1.18745
$759$	0	0
$760$	−137.279	−4.97965
$761$	13.8551	0.502247	0.251124	−	0.967955i	$-0.419200\pi$
$761$	13.8551	0.502247	0.251124	+	0.967955i	$0.419200\pi$
$762$	0	0
$763$	16.3658	0.592481
$764$	−46.7340	−1.69078
$765$	0	0
$766$	−23.0300	−0.832107
$767$	−68.9963	−2.49131
$768$	0	0
$769$	18.2227	0.657128	0.328564	−	0.944482i	$-0.393435\pi$
$769$	18.2227	0.657128	0.328564	+	0.944482i	$0.393435\pi$
$770$	−10.5976	−0.381913
$771$	0	0
$772$	−80.9706	−2.91420
$773$	48.6444	1.74962	0.874809	−	0.484467i	$-0.160987\pi$
$773$	48.6444	1.74962	0.874809	+	0.484467i	$0.160987\pi$
$774$	0	0
$775$	−4.50203	−0.161718
$776$	−135.332	−4.85814
$777$	0	0
$778$	37.6020	1.34810
$779$	65.2140	2.33653
$780$	0	0
$781$	−10.0671	−0.360228
$782$	10.9372	0.391114
$783$	0	0
$784$	−37.0389	−1.32282
$785$	−10.1293	−0.361529
$786$	0	0
$787$	26.7698	0.954240	0.477120	−	0.878838i	$-0.341681\pi$
$787$	26.7698	0.954240	0.477120	+	0.878838i	$0.341681\pi$
$788$	−54.8752	−1.95485
$789$	0	0
$790$	19.4857	0.693271
$791$	10.8672	0.386394
$792$	0	0
$793$	5.05790	0.179611
$794$	103.800	3.68372
$795$	0	0
$796$	3.65788	0.129650
$797$	17.9706	0.636551	0.318276	−	0.947998i	$-0.396896\pi$
$797$	17.9706	0.636551	0.318276	+	0.947998i	$0.396896\pi$
$798$	0	0
$799$	5.84105	0.206642
$800$	23.5754	0.833515
$801$	0	0
$802$	−64.8261	−2.28909
$803$	5.70524	0.201334
$804$	0	0
$805$	13.9019	0.489979
$806$	46.7513	1.64675
$807$	0	0
$808$	152.648	5.37014
$809$	−39.0710	−1.37366	−0.686832	−	0.726816i	$-0.740999\pi$
$809$	−39.0710	−1.37366	−0.686832	+	0.726816i	$0.740999\pi$
$810$	0	0
$811$	−13.3866	−0.470068	−0.235034	−	0.971987i	$-0.575520\pi$
$811$	−13.3866	−0.470068	−0.235034	+	0.971987i	$0.575520\pi$
$812$	−75.1283	−2.63649
$813$	0	0
$814$	−21.4077	−0.750339
$815$	7.28550	0.255200
$816$	0	0
$817$	6.26966	0.219348
$818$	−53.6360	−1.87534
$819$	0	0
$820$	81.1711	2.83462
$821$	0.907897	0.0316858	0.0158429	−	0.999874i	$-0.494957\pi$
$821$	0.907897	0.0316858	0.0158429	+	0.999874i	$0.494957\pi$
$822$	0	0
$823$	−49.9622	−1.74157	−0.870787	−	0.491661i	$-0.836390\pi$
$823$	−49.9622	−1.74157	−0.870787	+	0.491661i	$0.836390\pi$
$824$	−28.8368	−1.00458
$825$	0	0
$826$	−75.2868	−2.61956
$827$	29.4460	1.02394	0.511968	−	0.859004i	$-0.328917\pi$
$827$	29.4460	1.02394	0.511968	+	0.859004i	$0.328917\pi$
$828$	0	0
$829$	7.03763	0.244427	0.122214	−	0.992504i	$-0.461001\pi$
$829$	7.03763	0.244427	0.122214	+	0.992504i	$0.461001\pi$
$830$	43.0479	1.49421
$831$	0	0
$832$	−111.362	−3.86078
$833$	−3.22186	−0.111631
$834$	0	0
$835$	−9.56392	−0.330973
$836$	42.7693	1.47921
$837$	0	0
$838$	−34.9907	−1.20873
$839$	7.08555	0.244620	0.122310	−	0.992492i	$-0.460970\pi$
$839$	7.08555	0.244620	0.122310	+	0.992492i	$0.460970\pi$
$840$	0	0
$841$	19.5609	0.674515
$842$	9.12609	0.314506
$843$	0	0
$844$	−5.51998	−0.190006
$845$	−24.1607	−0.831152
$846$	0	0
$847$	2.04756	0.0703550
$848$	−151.521	−5.20326
$849$	0	0
$850$	4.06093	0.139289
$851$	28.0825	0.962656
$852$	0	0
$853$	−14.9177	−0.510771	−0.255386	−	0.966839i	$-0.582202\pi$
$853$	−14.9177	−0.510771	−0.255386	+	0.966839i	$0.582202\pi$
$854$	5.51904	0.188858
$855$	0	0
$856$	40.0097	1.36750
$857$	−2.69868	−0.0921850	−0.0460925	−	0.998937i	$-0.514677\pi$
$857$	−2.69868	−0.0921850	−0.0460925	+	0.998937i	$0.514677\pi$
$858$	0	0
$859$	−53.7843	−1.83510	−0.917549	−	0.397623i	$-0.869835\pi$
$859$	−53.7843	−1.83510	−0.917549	+	0.397623i	$0.869835\pi$
$860$	7.80377	0.266106
$861$	0	0
$862$	105.458	3.59193
$863$	−26.3132	−0.895711	−0.447855	−	0.894106i	$-0.647812\pi$
$863$	−26.3132	−0.895711	−0.447855	+	0.894106i	$0.647812\pi$
$864$	0	0
$865$	−26.4901	−0.900692
$866$	90.8376	3.08679
$867$	0	0
$868$	36.9706	1.25487
$869$	−3.76482	−0.127713
$870$	0	0
$871$	57.7589	1.95708
$872$	−70.3479	−2.38228
$873$	0	0
$874$	−77.4157	−2.61862
$875$	24.8203	0.839080
$876$	0	0
$877$	−17.5620	−0.593027	−0.296514	−	0.955029i	$-0.595824\pi$
$877$	−17.5620	−0.593027	−0.296514	+	0.955029i	$0.595824\pi$
$878$	−98.0749	−3.30987
$879$	0	0
$880$	25.3329	0.853970
$881$	−42.3553	−1.42699	−0.713493	−	0.700662i	$-0.752888\pi$
$881$	−42.3553	−1.42699	−0.713493	+	0.700662i	$0.752888\pi$
$882$	0	0
$883$	−47.8079	−1.60886	−0.804432	−	0.594045i	$-0.797530\pi$
$883$	−47.8079	−1.60886	−0.804432	+	0.594045i	$0.797530\pi$
$884$	−30.5619	−1.02791
$885$	0	0
$886$	74.3586	2.49813
$887$	19.8353	0.666004	0.333002	−	0.942926i	$-0.391938\pi$
$887$	19.8353	0.666004	0.333002	+	0.942926i	$0.391938\pi$
$888$	0	0
$889$	24.4596	0.820349
$890$	−10.0642	−0.337354
$891$	0	0
$892$	5.62429	0.188315
$893$	−41.3441	−1.38353
$894$	0	0
$895$	−41.9940	−1.40370
$896$	−47.9769	−1.60280
$897$	0	0
$898$	−89.9638	−3.00213
$899$	−23.8968	−0.797004
$900$	0	0
$901$	−13.1802	−0.439096
$902$	−21.6401	−0.720536
$903$	0	0
$904$	−46.7125	−1.55363
$905$	−21.9531	−0.729745
$906$	0	0
$907$	−3.15204	−0.104662	−0.0523308	−	0.998630i	$-0.516665\pi$
$907$	−3.15204	−0.104662	−0.0523308	+	0.998630i	$0.516665\pi$
$908$	−37.9740	−1.26021
$909$	0	0
$910$	−53.6019	−1.77688
$911$	13.2153	0.437843	0.218921	−	0.975742i	$-0.429746\pi$
$911$	13.2153	0.437843	0.218921	+	0.975742i	$0.429746\pi$
$912$	0	0
$913$	−8.31723	−0.275260
$914$	28.8145	0.953098
$915$	0	0
$916$	93.4258	3.08688
$917$	3.44313	0.113702
$918$	0	0
$919$	52.6183	1.73572	0.867859	−	0.496810i	$-0.165495\pi$
$919$	52.6183	1.73572	0.867859	+	0.496810i	$0.165495\pi$
$920$	−59.7572	−1.97014
$921$	0	0
$922$	87.7397	2.88955
$923$	−50.9182	−1.67599
$924$	0	0
$925$	10.4269	0.342834
$926$	−71.3388	−2.34434
$927$	0	0
$928$	125.138	4.10786
$929$	39.1701	1.28513	0.642565	−	0.766231i	$-0.277870\pi$
$929$	39.1701	1.28513	0.642565	+	0.766231i	$0.277870\pi$
$930$	0	0
$931$	22.8049	0.747401
$932$	−113.093	−3.70449
$933$	0	0
$934$	−84.2417	−2.75647
$935$	2.20360	0.0720653
$936$	0	0
$937$	−50.8259	−1.66041	−0.830205	−	0.557459i	$-0.811776\pi$
$937$	−50.8259	−1.66041	−0.830205	+	0.557459i	$0.811776\pi$
$938$	63.0248	2.05783
$939$	0	0
$940$	−51.4605	−1.67846
$941$	0.673372	0.0219513	0.0109757	−	0.999940i	$-0.496506\pi$
$941$	0.673372	0.0219513	0.0109757	+	0.999940i	$0.496506\pi$
$942$	0	0
$943$	28.3874	0.924420
$944$	179.967	5.85743
$945$	0	0
$946$	−2.08047	−0.0676420
$947$	39.9715	1.29890	0.649450	−	0.760405i	$-0.274999\pi$
$947$	39.9715	1.29890	0.649450	+	0.760405i	$0.274999\pi$
$948$	0	0
$949$	28.8566	0.936724
$950$	−28.7440	−0.932580
$951$	0	0
$952$	−20.6811	−0.670278
$953$	−5.14686	−0.166723	−0.0833616	−	0.996519i	$-0.526566\pi$
$953$	−5.14686	−0.166723	−0.0833616	+	0.996519i	$0.526566\pi$
$954$	0	0
$955$	17.0433	0.551510
$956$	12.8530	0.415697
$957$	0	0
$958$	103.210	3.33457
$959$	−37.3966	−1.20760
$960$	0	0
$961$	−19.2404	−0.620657
$962$	−108.278	−3.49102
$963$	0	0
$964$	−45.3512	−1.46067
$965$	29.5291	0.950574
$966$	0	0
$967$	50.0761	1.61034	0.805170	−	0.593044i	$-0.202074\pi$
$967$	50.0761	1.61034	0.805170	+	0.593044i	$0.202074\pi$
$968$	−8.80138	−0.282887
$969$	0	0
$970$	79.5835	2.55527
$971$	19.8968	0.638519	0.319259	−	0.947667i	$-0.396566\pi$
$971$	19.8968	0.638519	0.319259	+	0.947667i	$0.396566\pi$
$972$	0	0
$973$	−1.76651	−0.0566318
$974$	−54.9334	−1.76018
$975$	0	0
$976$	−13.1928	−0.422292
$977$	−38.3757	−1.22775	−0.613874	−	0.789404i	$-0.710390\pi$
$977$	−38.3757	−1.22775	−0.613874	+	0.789404i	$0.710390\pi$
$978$	0	0
$979$	1.94450	0.0621465
$980$	28.3850	0.906726
$981$	0	0
$982$	−109.515	−3.49477
$983$	5.85548	0.186761	0.0933804	−	0.995631i	$-0.470233\pi$
$983$	5.85548	0.186761	0.0933804	+	0.995631i	$0.470233\pi$
$984$	0	0
$985$	20.0124	0.637647
$986$	21.5554	0.686465
$987$	0	0
$988$	216.323	6.88215
$989$	2.72916	0.0867821
$990$	0	0
$991$	−35.9901	−1.14326	−0.571631	−	0.820510i	$-0.693689\pi$
$991$	−35.9901	−1.14326	−0.571631	+	0.820510i	$0.693689\pi$
$992$	−61.5805	−1.95518
$993$	0	0
$994$	−55.5605	−1.76227
$995$	−1.33399	−0.0422902
$996$	0	0
$997$	−41.2696	−1.30702	−0.653510	−	0.756918i	$-0.726704\pi$
$997$	−41.2696	−1.30702	−0.653510	+	0.756918i	$0.726704\pi$
$998$	−79.6048	−2.51984
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.2		19
3.2	odd	2		671.2.a.c.1.18	✓	19
33.32	even	2		7381.2.a.i.1.2		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.18	✓	19	3.2	odd	2
6039.2.a.k.1.2		19	1.1	even	1	trivial
7381.2.a.i.1.2		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)

\(f(q)\)	\(=\)	\(q-2.69542 q^{2} +5.26531 q^{4} -1.92020 q^{5} +2.04756 q^{7} -8.80138 q^{8} +O(q^{10})\)	\(q-2.69542 q^{2} +5.26531 q^{4} -1.92020 q^{5} +2.04756 q^{7} -8.80138 q^{8} +5.17575 q^{10} -1.00000 q^{11} -5.05790 q^{13} -5.51904 q^{14} +13.1928 q^{16} +1.14759 q^{17} -8.12286 q^{19} -10.1104 q^{20} +2.69542 q^{22} -3.53584 q^{23} -1.31284 q^{25} +13.6332 q^{26} +10.7810 q^{28} -6.96857 q^{29} +3.42923 q^{31} -17.9575 q^{32} -3.09324 q^{34} -3.93172 q^{35} -7.94224 q^{37} +21.8945 q^{38} +16.9004 q^{40} -8.02845 q^{41} -0.771854 q^{43} -5.26531 q^{44} +9.53060 q^{46} +5.08985 q^{47} -2.80750 q^{49} +3.53866 q^{50} -26.6314 q^{52} -11.4851 q^{53} +1.92020 q^{55} -18.0214 q^{56} +18.7832 q^{58} +13.6413 q^{59} -1.00000 q^{61} -9.24323 q^{62} +22.0174 q^{64} +9.71217 q^{65} -11.4195 q^{67} +6.04241 q^{68} +10.5976 q^{70} +10.0671 q^{71} -5.70524 q^{73} +21.4077 q^{74} -42.7693 q^{76} -2.04756 q^{77} +3.76482 q^{79} -25.3329 q^{80} +21.6401 q^{82} +8.31723 q^{83} -2.20360 q^{85} +2.08047 q^{86} +8.80138 q^{88} -1.94450 q^{89} -10.3564 q^{91} -18.6173 q^{92} -13.7193 q^{94} +15.5975 q^{95} +15.3762 q^{97} +7.56740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more