LMFDB - Embedded newform 6039.2.a.e.1.8

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 $ x^12 - x^11 - 16x^10 + 13x^9 + 93x^8 - 59x^7 - 238x^6 + 108x^5 + 257x^4 - 71x^3 - 93x^2 + 13x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.811354$ 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)

$f(q)$	$=$	$q+0.811354 q^{2} -1.34170 q^{4} -0.350684 q^{5} +3.06527 q^{7} -2.71131 q^{8} +O(q^{10})$	\(q+0.811354 q^{2} -1.34170 q^{4} -0.350684 q^{5} +3.06527 q^{7} -2.71131 q^{8} -0.284529 q^{10} +1.00000 q^{11} -5.45552 q^{13} +2.48702 q^{14} +0.483577 q^{16} +2.06199 q^{17} +1.40317 q^{19} +0.470515 q^{20} +0.811354 q^{22} -0.147981 q^{23} -4.87702 q^{25} -4.42636 q^{26} -4.11269 q^{28} -2.86506 q^{29} +4.16462 q^{31} +5.81497 q^{32} +1.67301 q^{34} -1.07494 q^{35} -0.144684 q^{37} +1.13847 q^{38} +0.950813 q^{40} -3.77643 q^{41} +2.48169 q^{43} -1.34170 q^{44} -0.120065 q^{46} +5.49942 q^{47} +2.39589 q^{49} -3.95699 q^{50} +7.31969 q^{52} +4.06954 q^{53} -0.350684 q^{55} -8.31089 q^{56} -2.32458 q^{58} -3.27787 q^{59} -1.00000 q^{61} +3.37898 q^{62} +3.75084 q^{64} +1.91317 q^{65} -12.0165 q^{67} -2.76658 q^{68} -0.872160 q^{70} -10.3117 q^{71} -5.13147 q^{73} -0.117390 q^{74} -1.88264 q^{76} +3.06527 q^{77} -7.67884 q^{79} -0.169583 q^{80} -3.06402 q^{82} -15.8628 q^{83} -0.723108 q^{85} +2.01353 q^{86} -2.71131 q^{88} -3.35464 q^{89} -16.7227 q^{91} +0.198547 q^{92} +4.46198 q^{94} -0.492070 q^{95} -5.17684 q^{97} +1.94392 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.811354	0.573714	0.286857	−	0.957973i	$-0.407390\pi$
$2$	0.811354	0.573714	0.286857	+	0.957973i	$0.407390\pi$
$3$	0	0
$3$	0	0
$4$	−1.34170	−0.670852
$5$	−0.350684	−0.156831	−0.0784154	−	0.996921i	$-0.524986\pi$
$5$	−0.350684	−0.156831	−0.0784154	+	0.996921i	$0.524986\pi$
$6$	0	0
$7$	3.06527	1.15856	0.579282	−	0.815127i	$-0.303333\pi$
$7$	3.06527	1.15856	0.579282	+	0.815127i	$0.303333\pi$
$8$	−2.71131	−0.958592
$9$	0	0
$10$	−0.284529	−0.0899761
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.45552	−1.51309	−0.756544	−	0.653942i	$-0.773114\pi$
$13$	−5.45552	−1.51309	−0.756544	+	0.653942i	$0.773114\pi$
$14$	2.48702	0.664685
$15$	0	0
$16$	0.483577	0.120894
$17$	2.06199	0.500106	0.250053	−	0.968232i	$-0.419552\pi$
$17$	2.06199	0.500106	0.250053	+	0.968232i	$0.419552\pi$
$18$	0	0
$19$	1.40317	0.321910	0.160955	−	0.986962i	$-0.448543\pi$
$19$	1.40317	0.321910	0.160955	+	0.986962i	$0.448543\pi$
$20$	0.470515	0.105210
$21$	0	0
$22$	0.811354	0.172981
$23$	−0.147981	−0.0308562	−0.0154281	−	0.999881i	$-0.504911\pi$
$23$	−0.147981	−0.0308562	−0.0154281	+	0.999881i	$0.504911\pi$
$24$	0	0
$25$	−4.87702	−0.975404
$26$	−4.42636	−0.868080
$27$	0	0
$28$	−4.11269	−0.777225
$29$	−2.86506	−0.532028	−0.266014	−	0.963969i	$-0.585707\pi$
$29$	−2.86506	−0.532028	−0.266014	+	0.963969i	$0.585707\pi$
$30$	0	0
$31$	4.16462	0.747988	0.373994	−	0.927431i	$-0.377988\pi$
$31$	4.16462	0.747988	0.373994	+	0.927431i	$0.377988\pi$
$32$	5.81497	1.02795
$33$	0	0
$34$	1.67301	0.286918
$35$	−1.07494	−0.181699
$36$	0	0
$37$	−0.144684	−0.0237860	−0.0118930	−	0.999929i	$-0.503786\pi$
$37$	−0.144684	−0.0237860	−0.0118930	+	0.999929i	$0.503786\pi$
$38$	1.13847	0.184684
$39$	0	0
$40$	0.950813	0.150337
$41$	−3.77643	−0.589779	−0.294890	−	0.955531i	$-0.595283\pi$
$41$	−3.77643	−0.589779	−0.294890	+	0.955531i	$0.595283\pi$
$42$	0	0
$43$	2.48169	0.378454	0.189227	−	0.981933i	$-0.439402\pi$
$43$	2.48169	0.378454	0.189227	+	0.981933i	$0.439402\pi$
$44$	−1.34170	−0.202269
$45$	0	0
$46$	−0.120065	−0.0177027
$47$	5.49942	0.802173	0.401087	−	0.916040i	$-0.368633\pi$
$47$	5.49942	0.802173	0.401087	+	0.916040i	$0.368633\pi$
$48$	0	0
$49$	2.39589	0.342271
$50$	−3.95699	−0.559603
$51$	0	0
$52$	7.31969	1.01506
$53$	4.06954	0.558995	0.279497	−	0.960146i	$-0.409832\pi$
$53$	4.06954	0.558995	0.279497	+	0.960146i	$0.409832\pi$
$54$	0	0
$55$	−0.350684	−0.0472863
$56$	−8.31089	−1.11059
$57$	0	0
$58$	−2.32458	−0.305232
$59$	−3.27787	−0.426742	−0.213371	−	0.976971i	$-0.568444\pi$
$59$	−3.27787	−0.426742	−0.213371	+	0.976971i	$0.568444\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	3.37898	0.429131
$63$	0	0
$64$	3.75084	0.468855
$65$	1.91317	0.237299
$66$	0	0
$67$	−12.0165	−1.46804	−0.734022	−	0.679126i	$-0.762359\pi$
$67$	−12.0165	−1.46804	−0.734022	+	0.679126i	$0.762359\pi$
$68$	−2.76658	−0.335497
$69$	0	0
$70$	−0.872160	−0.104243
$71$	−10.3117	−1.22378	−0.611888	−	0.790944i	$-0.709590\pi$
$71$	−10.3117	−1.22378	−0.611888	+	0.790944i	$0.709590\pi$
$72$	0	0
$73$	−5.13147	−0.600593	−0.300296	−	0.953846i	$-0.597086\pi$
$73$	−5.13147	−0.600593	−0.300296	+	0.953846i	$0.597086\pi$
$74$	−0.117390	−0.0136463
$75$	0	0
$76$	−1.88264	−0.215954
$77$	3.06527	0.349320
$78$	0	0
$79$	−7.67884	−0.863937	−0.431968	−	0.901889i	$-0.642181\pi$
$79$	−7.67884	−0.863937	−0.431968	+	0.901889i	$0.642181\pi$
$80$	−0.169583	−0.0189600
$81$	0	0
$82$	−3.06402	−0.338365
$83$	−15.8628	−1.74117	−0.870585	−	0.492017i	$-0.836260\pi$
$83$	−15.8628	−1.74117	−0.870585	+	0.492017i	$0.836260\pi$
$84$	0	0
$85$	−0.723108	−0.0784321
$86$	2.01353	0.217124
$87$	0	0
$88$	−2.71131	−0.289026
$89$	−3.35464	−0.355591	−0.177796	−	0.984067i	$-0.556897\pi$
$89$	−3.35464	−0.355591	−0.177796	+	0.984067i	$0.556897\pi$
$90$	0	0
$91$	−16.7227	−1.75301
$92$	0.198547	0.0207000
$93$	0	0
$94$	4.46198	0.460218
$95$	−0.492070	−0.0504854
$96$	0	0
$97$	−5.17684	−0.525628	−0.262814	−	0.964847i	$-0.584651\pi$
$97$	−5.17684	−0.525628	−0.262814	+	0.964847i	$0.584651\pi$
$98$	1.94392	0.196366
$99$	0	0
$100$	6.54352	0.654352
$101$	8.46305	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$101$	8.46305	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$102$	0	0
$103$	7.06816	0.696447	0.348223	−	0.937412i	$-0.386785\pi$
$103$	7.06816	0.696447	0.348223	+	0.937412i	$0.386785\pi$
$104$	14.7916	1.45043
$105$	0	0
$106$	3.30184	0.320703
$107$	11.9968	1.15978	0.579889	−	0.814696i	$-0.303096\pi$
$107$	11.9968	1.15978	0.579889	+	0.814696i	$0.303096\pi$
$108$	0	0
$109$	−5.28742	−0.506443	−0.253222	−	0.967408i	$-0.581490\pi$
$109$	−5.28742	−0.506443	−0.253222	+	0.967408i	$0.581490\pi$
$110$	−0.284529	−0.0271288
$111$	0	0
$112$	1.48230	0.140064
$113$	7.22992	0.680133	0.340067	−	0.940401i	$-0.389550\pi$
$113$	7.22992	0.680133	0.340067	+	0.940401i	$0.389550\pi$
$114$	0	0
$115$	0.0518947	0.00483921
$116$	3.84406	0.356912
$117$	0	0
$118$	−2.65952	−0.244828
$119$	6.32056	0.579405
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.811354	−0.0734566
$123$	0	0
$124$	−5.58769	−0.501789
$125$	3.46372	0.309804
$126$	0	0
$127$	−19.2693	−1.70988	−0.854939	−	0.518729i	$-0.826405\pi$
$127$	−19.2693	−1.70988	−0.854939	+	0.518729i	$0.826405\pi$
$128$	−8.58667	−0.758961
$129$	0	0
$130$	1.55226	0.136142
$131$	19.0697	1.66613	0.833063	−	0.553178i	$-0.186585\pi$
$131$	19.0697	1.66613	0.833063	+	0.553178i	$0.186585\pi$
$132$	0	0
$133$	4.30110	0.372953
$134$	−9.74960	−0.842238
$135$	0	0
$136$	−5.59069	−0.479398
$137$	5.26924	0.450182	0.225091	−	0.974338i	$-0.427732\pi$
$137$	5.26924	0.450182	0.225091	+	0.974338i	$0.427732\pi$
$138$	0	0
$139$	−14.8768	−1.26184	−0.630918	−	0.775849i	$-0.717322\pi$
$139$	−14.8768	−1.26184	−0.630918	+	0.775849i	$0.717322\pi$
$140$	1.44226	0.121893
$141$	0	0
$142$	−8.36646	−0.702098
$143$	−5.45552	−0.456213
$144$	0	0
$145$	1.00473	0.0834385
$146$	−4.16344	−0.344569
$147$	0	0
$148$	0.194124	0.0159569
$149$	23.4135	1.91811	0.959054	−	0.283222i	$-0.0914034\pi$
$149$	23.4135	1.91811	0.959054	+	0.283222i	$0.0914034\pi$
$150$	0	0
$151$	−11.9086	−0.969111	−0.484556	−	0.874761i	$-0.661019\pi$
$151$	−11.9086	−0.969111	−0.484556	+	0.874761i	$0.661019\pi$
$152$	−3.80443	−0.308580
$153$	0	0
$154$	2.48702	0.200410
$155$	−1.46047	−0.117308
$156$	0	0
$157$	−12.0808	−0.964150	−0.482075	−	0.876130i	$-0.660117\pi$
$157$	−12.0808	−0.964150	−0.482075	+	0.876130i	$0.660117\pi$
$158$	−6.23026	−0.495653
$159$	0	0
$160$	−2.03922	−0.161214
$161$	−0.453603	−0.0357489
$162$	0	0
$163$	−1.04500	−0.0818510	−0.0409255	−	0.999162i	$-0.513031\pi$
$163$	−1.04500	−0.0818510	−0.0409255	+	0.999162i	$0.513031\pi$
$164$	5.06685	0.395654
$165$	0	0
$166$	−12.8704	−0.998935
$167$	−3.91541	−0.302983	−0.151492	−	0.988459i	$-0.548408\pi$
$167$	−3.91541	−0.302983	−0.151492	+	0.988459i	$0.548408\pi$
$168$	0	0
$169$	16.7627	1.28944
$170$	−0.586697	−0.0449976
$171$	0	0
$172$	−3.32969	−0.253886
$173$	−6.91306	−0.525590	−0.262795	−	0.964852i	$-0.584644\pi$
$173$	−6.91306	−0.525590	−0.262795	+	0.964852i	$0.584644\pi$
$174$	0	0
$175$	−14.9494	−1.13007
$176$	0.483577	0.0364510
$177$	0	0
$178$	−2.72180	−0.204008
$179$	8.22908	0.615070	0.307535	−	0.951537i	$-0.400496\pi$
$179$	8.22908	0.615070	0.307535	+	0.951537i	$0.400496\pi$
$180$	0	0
$181$	−20.2042	−1.50177	−0.750883	−	0.660435i	$-0.770372\pi$
$181$	−20.2042	−1.50177	−0.750883	+	0.660435i	$0.770372\pi$
$182$	−13.5680	−1.00573
$183$	0	0
$184$	0.401223	0.0295785
$185$	0.0507386	0.00373037
$186$	0	0
$187$	2.06199	0.150788
$188$	−7.37860	−0.538140
$189$	0	0
$190$	−0.399243	−0.0289642
$191$	−18.4767	−1.33693	−0.668463	−	0.743745i	$-0.733048\pi$
$191$	−18.4767	−1.33693	−0.668463	+	0.743745i	$0.733048\pi$
$192$	0	0
$193$	−15.8196	−1.13872	−0.569359	−	0.822089i	$-0.692808\pi$
$193$	−15.8196	−1.13872	−0.569359	+	0.822089i	$0.692808\pi$
$194$	−4.20025	−0.301560
$195$	0	0
$196$	−3.21458	−0.229613
$197$	−14.3733	−1.02405	−0.512026	−	0.858970i	$-0.671105\pi$
$197$	−14.3733	−1.02405	−0.512026	+	0.858970i	$0.671105\pi$
$198$	0	0
$199$	−8.22066	−0.582747	−0.291374	−	0.956609i	$-0.594112\pi$
$199$	−8.22066	−0.582747	−0.291374	+	0.956609i	$0.594112\pi$
$200$	13.2231	0.935014
$201$	0	0
$202$	6.86653	0.483128
$203$	−8.78219	−0.616389
$204$	0	0
$205$	1.32433	0.0924955
$206$	5.73479	0.399562
$207$	0	0
$208$	−2.63817	−0.182924
$209$	1.40317	0.0970594
$210$	0	0
$211$	12.6667	0.872013	0.436007	−	0.899943i	$-0.356392\pi$
$211$	12.6667	0.872013	0.436007	+	0.899943i	$0.356392\pi$
$212$	−5.46012	−0.375003
$213$	0	0
$214$	9.73368	0.665381
$215$	−0.870289	−0.0593532
$216$	0	0
$217$	12.7657	0.866592
$218$	−4.28997	−0.290554
$219$	0	0
$220$	0.470515	0.0317221
$221$	−11.2492	−0.756705
$222$	0	0
$223$	0.384419	0.0257426	0.0128713	−	0.999917i	$-0.495903\pi$
$223$	0.384419	0.0257426	0.0128713	+	0.999917i	$0.495903\pi$
$224$	17.8245	1.19095
$225$	0	0
$226$	5.86603	0.390202
$227$	−10.5003	−0.696930	−0.348465	−	0.937322i	$-0.613297\pi$
$227$	−10.5003	−0.696930	−0.348465	+	0.937322i	$0.613297\pi$
$228$	0	0
$229$	15.4219	1.01911	0.509555	−	0.860438i	$-0.329810\pi$
$229$	15.4219	1.01911	0.509555	+	0.860438i	$0.329810\pi$
$230$	0.0421050	0.00277632
$231$	0	0
$232$	7.76806	0.509998
$233$	−12.1580	−0.796496	−0.398248	−	0.917278i	$-0.630382\pi$
$233$	−12.1580	−0.796496	−0.398248	+	0.917278i	$0.630382\pi$
$234$	0	0
$235$	−1.92856	−0.125806
$236$	4.39793	0.286281
$237$	0	0
$238$	5.12822	0.332413
$239$	5.12659	0.331612	0.165806	−	0.986158i	$-0.446977\pi$
$239$	5.12659	0.331612	0.165806	+	0.986158i	$0.446977\pi$
$240$	0	0
$241$	4.26966	0.275033	0.137516	−	0.990499i	$-0.456088\pi$
$241$	4.26966	0.275033	0.137516	+	0.990499i	$0.456088\pi$
$242$	0.811354	0.0521558
$243$	0	0
$244$	1.34170	0.0858938
$245$	−0.840203	−0.0536786
$246$	0	0
$247$	−7.65503	−0.487078
$248$	−11.2916	−0.717015
$249$	0	0
$250$	2.81030	0.177739
$251$	−24.4387	−1.54256	−0.771278	−	0.636498i	$-0.780382\pi$
$251$	−24.4387	−1.54256	−0.771278	+	0.636498i	$0.780382\pi$
$252$	0	0
$253$	−0.147981	−0.00930350
$254$	−15.6343	−0.980981
$255$	0	0
$256$	−14.4685	−0.904282
$257$	11.0393	0.688616	0.344308	−	0.938857i	$-0.388114\pi$
$257$	11.0393	0.688616	0.344308	+	0.938857i	$0.388114\pi$
$258$	0	0
$259$	−0.443497	−0.0275576
$260$	−2.56690	−0.159192
$261$	0	0
$262$	15.4723	0.955880
$263$	−13.5468	−0.835331	−0.417665	−	0.908601i	$-0.637152\pi$
$263$	−13.5468	−0.835331	−0.417665	+	0.908601i	$0.637152\pi$
$264$	0	0
$265$	−1.42713	−0.0876676
$266$	3.48972	0.213968
$267$	0	0
$268$	16.1225	0.984840
$269$	8.89405	0.542280	0.271140	−	0.962540i	$-0.412599\pi$
$269$	8.89405	0.542280	0.271140	+	0.962540i	$0.412599\pi$
$270$	0	0
$271$	−28.9586	−1.75911	−0.879555	−	0.475798i	$-0.842159\pi$
$271$	−28.9586	−1.75911	−0.879555	+	0.475798i	$0.842159\pi$
$272$	0.997132	0.0604600
$273$	0	0
$274$	4.27522	0.258276
$275$	−4.87702	−0.294095
$276$	0	0
$277$	16.3364	0.981558	0.490779	−	0.871284i	$-0.336712\pi$
$277$	16.3364	0.981558	0.490779	+	0.871284i	$0.336712\pi$
$278$	−12.0704	−0.723934
$279$	0	0
$280$	2.91450	0.174175
$281$	−21.1539	−1.26194	−0.630968	−	0.775809i	$-0.717342\pi$
$281$	−21.1539	−1.26194	−0.630968	+	0.775809i	$0.717342\pi$
$282$	0	0
$283$	−29.7641	−1.76929	−0.884646	−	0.466263i	$-0.845600\pi$
$283$	−29.7641	−1.76929	−0.884646	+	0.466263i	$0.845600\pi$
$284$	13.8353	0.820973
$285$	0	0
$286$	−4.42636	−0.261736
$287$	−11.5758	−0.683297
$288$	0	0
$289$	−12.7482	−0.749894
$290$	0.815194	0.0478698
$291$	0	0
$292$	6.88491	0.402909
$293$	−14.1561	−0.827006	−0.413503	−	0.910503i	$-0.635695\pi$
$293$	−14.1561	−0.827006	−0.413503	+	0.910503i	$0.635695\pi$
$294$	0	0
$295$	1.14950	0.0669264
$296$	0.392284	0.0228010
$297$	0	0
$298$	18.9966	1.10045
$299$	0.807314	0.0466882
$300$	0	0
$301$	7.60704	0.438463
$302$	−9.66213	−0.555993
$303$	0	0
$304$	0.678542	0.0389170
$305$	0.350684	0.0200801
$306$	0	0
$307$	−27.2699	−1.55638	−0.778188	−	0.628032i	$-0.783861\pi$
$307$	−27.2699	−1.55638	−0.778188	+	0.628032i	$0.783861\pi$
$308$	−4.11269	−0.234342
$309$	0	0
$310$	−1.18496	−0.0673010
$311$	−4.17556	−0.236774	−0.118387	−	0.992968i	$-0.537772\pi$
$311$	−4.17556	−0.236774	−0.118387	+	0.992968i	$0.537772\pi$
$312$	0	0
$313$	−3.37580	−0.190812	−0.0954059	−	0.995438i	$-0.530415\pi$
$313$	−3.37580	−0.190812	−0.0954059	+	0.995438i	$0.530415\pi$
$314$	−9.80179	−0.553147
$315$	0	0
$316$	10.3027	0.579574
$317$	16.3928	0.920711	0.460355	−	0.887735i	$-0.347722\pi$
$317$	16.3928	0.920711	0.460355	+	0.887735i	$0.347722\pi$
$318$	0	0
$319$	−2.86506	−0.160413
$320$	−1.31536	−0.0735310
$321$	0	0
$322$	−0.368033	−0.0205097
$323$	2.89333	0.160989
$324$	0	0
$325$	26.6067	1.47587
$326$	−0.847868	−0.0469591
$327$	0	0
$328$	10.2391	0.565357
$329$	16.8572	0.929369
$330$	0	0
$331$	−5.86712	−0.322486	−0.161243	−	0.986915i	$-0.551550\pi$
$331$	−5.86712	−0.322486	−0.161243	+	0.986915i	$0.551550\pi$
$332$	21.2832	1.16807
$333$	0	0
$334$	−3.17678	−0.173826
$335$	4.21398	0.230235
$336$	0	0
$337$	21.3314	1.16199	0.580997	−	0.813906i	$-0.302663\pi$
$337$	21.3314	1.16199	0.580997	+	0.813906i	$0.302663\pi$
$338$	13.6005	0.739768
$339$	0	0
$340$	0.970197	0.0526163
$341$	4.16462	0.225527
$342$	0	0
$343$	−14.1128	−0.762022
$344$	−6.72861	−0.362782
$345$	0	0
$346$	−5.60894	−0.301539
$347$	35.8160	1.92270	0.961351	−	0.275327i	$-0.0887862\pi$
$347$	35.8160	1.92270	0.961351	+	0.275327i	$0.0887862\pi$
$348$	0	0
$349$	−12.5671	−0.672701	−0.336351	−	0.941737i	$-0.609193\pi$
$349$	−12.5671	−0.672701	−0.336351	+	0.941737i	$0.609193\pi$
$350$	−12.1293	−0.648336
$351$	0	0
$352$	5.81497	0.309939
$353$	−15.3693	−0.818023	−0.409012	−	0.912529i	$-0.634126\pi$
$353$	−15.3693	−0.818023	−0.409012	+	0.912529i	$0.634126\pi$
$354$	0	0
$355$	3.61616	0.191926
$356$	4.50094	0.238549
$357$	0	0
$358$	6.67670	0.352874
$359$	−15.4909	−0.817580	−0.408790	−	0.912629i	$-0.634049\pi$
$359$	−15.4909	−0.817580	−0.408790	+	0.912629i	$0.634049\pi$
$360$	0	0
$361$	−17.0311	−0.896374
$362$	−16.3928	−0.861585
$363$	0	0
$364$	22.4368	1.17601
$365$	1.79953	0.0941915
$366$	0	0
$367$	−23.8478	−1.24485	−0.622423	−	0.782681i	$-0.713852\pi$
$367$	−23.8478	−1.24485	−0.622423	+	0.782681i	$0.713852\pi$
$368$	−0.0715604	−0.00373034
$369$	0	0
$370$	0.0411670	0.00214017
$371$	12.4743	0.647631
$372$	0	0
$373$	10.7371	0.555945	0.277973	−	0.960589i	$-0.410338\pi$
$373$	10.7371	0.555945	0.277973	+	0.960589i	$0.410338\pi$
$374$	1.67301	0.0865091
$375$	0	0
$376$	−14.9106	−0.768957
$377$	15.6304	0.805006
$378$	0	0
$379$	10.4369	0.536108	0.268054	−	0.963404i	$-0.413619\pi$
$379$	10.4369	0.536108	0.268054	+	0.963404i	$0.413619\pi$
$380$	0.660213	0.0338682
$381$	0	0
$382$	−14.9911	−0.767014
$383$	32.7915	1.67557	0.837784	−	0.546002i	$-0.183851\pi$
$383$	32.7915	1.67557	0.837784	+	0.546002i	$0.183851\pi$
$384$	0	0
$385$	−1.07494	−0.0547842
$386$	−12.8353	−0.653298
$387$	0	0
$388$	6.94578	0.352619
$389$	−3.79669	−0.192500	−0.0962499	−	0.995357i	$-0.530685\pi$
$389$	−3.79669	−0.192500	−0.0962499	+	0.995357i	$0.530685\pi$
$390$	0	0
$391$	−0.305136	−0.0154314
$392$	−6.49600	−0.328098
$393$	0	0
$394$	−11.6618	−0.587513
$395$	2.69285	0.135492
$396$	0	0
$397$	−5.74208	−0.288187	−0.144093	−	0.989564i	$-0.546027\pi$
$397$	−5.74208	−0.288187	−0.144093	+	0.989564i	$0.546027\pi$
$398$	−6.66987	−0.334330
$399$	0	0
$400$	−2.35842	−0.117921
$401$	−12.8703	−0.642710	−0.321355	−	0.946959i	$-0.604138\pi$
$401$	−12.8703	−0.642710	−0.321355	+	0.946959i	$0.604138\pi$
$402$	0	0
$403$	−22.7202	−1.13177
$404$	−11.3549	−0.564928
$405$	0	0
$406$	−7.12547	−0.353631
$407$	−0.144684	−0.00717174
$408$	0	0
$409$	17.6215	0.871328	0.435664	−	0.900109i	$-0.356514\pi$
$409$	17.6215	0.871328	0.435664	+	0.900109i	$0.356514\pi$
$410$	1.07450	0.0530660
$411$	0	0
$412$	−9.48338	−0.467213
$413$	−10.0476	−0.494409
$414$	0	0
$415$	5.56284	0.273069
$416$	−31.7237	−1.55538
$417$	0	0
$418$	1.13847	0.0556844
$419$	10.1071	0.493762	0.246881	−	0.969046i	$-0.420594\pi$
$419$	10.1071	0.493762	0.246881	+	0.969046i	$0.420594\pi$
$420$	0	0
$421$	−6.08544	−0.296586	−0.148293	−	0.988943i	$-0.547378\pi$
$421$	−6.08544	−0.296586	−0.148293	+	0.988943i	$0.547378\pi$
$422$	10.2772	0.500286
$423$	0	0
$424$	−11.0338	−0.535848
$425$	−10.0564	−0.487806
$426$	0	0
$427$	−3.06527	−0.148339
$428$	−16.0962	−0.778039
$429$	0	0
$430$	−0.706113	−0.0340518
$431$	−10.2081	−0.491707	−0.245853	−	0.969307i	$-0.579068\pi$
$431$	−10.2081	−0.491707	−0.245853	+	0.969307i	$0.579068\pi$
$432$	0	0
$433$	−2.53260	−0.121709	−0.0608545	−	0.998147i	$-0.519383\pi$
$433$	−2.53260	−0.121709	−0.0608545	+	0.998147i	$0.519383\pi$
$434$	10.3575	0.497176
$435$	0	0
$436$	7.09416	0.339748
$437$	−0.207643	−0.00993291
$438$	0	0
$439$	37.0940	1.77040	0.885201	−	0.465209i	$-0.154021\pi$
$439$	37.0940	1.77040	0.885201	+	0.465209i	$0.154021\pi$
$440$	0.950813	0.0453282
$441$	0	0
$442$	−9.12711	−0.434132
$443$	−6.46823	−0.307315	−0.153657	−	0.988124i	$-0.549105\pi$
$443$	−6.46823	−0.307315	−0.153657	+	0.988124i	$0.549105\pi$
$444$	0	0
$445$	1.17642	0.0557677
$446$	0.311900	0.0147689
$447$	0	0
$448$	11.4974	0.543199
$449$	−17.8163	−0.840801	−0.420400	−	0.907339i	$-0.638110\pi$
$449$	−17.8163	−0.840801	−0.420400	+	0.907339i	$0.638110\pi$
$450$	0	0
$451$	−3.77643	−0.177825
$452$	−9.70041	−0.456269
$453$	0	0
$454$	−8.51948	−0.399839
$455$	5.86437	0.274926
$456$	0	0
$457$	35.0188	1.63811	0.819055	−	0.573715i	$-0.194498\pi$
$457$	35.0188	1.63811	0.819055	+	0.573715i	$0.194498\pi$
$458$	12.5127	0.584678
$459$	0	0
$460$	−0.0696274	−0.00324639
$461$	−0.991967	−0.0462005	−0.0231002	−	0.999733i	$-0.507354\pi$
$461$	−0.991967	−0.0462005	−0.0231002	+	0.999733i	$0.507354\pi$
$462$	0	0
$463$	−15.6278	−0.726286	−0.363143	−	0.931733i	$-0.618296\pi$
$463$	−15.6278	−0.726286	−0.363143	+	0.931733i	$0.618296\pi$
$464$	−1.38548	−0.0643192
$465$	0	0
$466$	−9.86443	−0.456961
$467$	25.6358	1.18628	0.593142	−	0.805098i	$-0.297887\pi$
$467$	25.6358	1.18628	0.593142	+	0.805098i	$0.297887\pi$
$468$	0	0
$469$	−36.8337	−1.70082
$470$	−1.56475	−0.0721764
$471$	0	0
$472$	8.88731	0.409072
$473$	2.48169	0.114108
$474$	0	0
$475$	−6.84330	−0.313992
$476$	−8.48032	−0.388695
$477$	0	0
$478$	4.15948	0.190250
$479$	−23.4825	−1.07294	−0.536472	−	0.843918i	$-0.680243\pi$
$479$	−23.4825	−1.07294	−0.536472	+	0.843918i	$0.680243\pi$
$480$	0	0
$481$	0.789328	0.0359903
$482$	3.46420	0.157790
$483$	0	0
$484$	−1.34170	−0.0609865
$485$	1.81544	0.0824347
$486$	0	0
$487$	−5.68347	−0.257543	−0.128771	−	0.991674i	$-0.541103\pi$
$487$	−5.68347	−0.257543	−0.128771	+	0.991674i	$0.541103\pi$
$488$	2.71131	0.122735
$489$	0	0
$490$	−0.681702	−0.0307962
$491$	−1.93005	−0.0871018	−0.0435509	−	0.999051i	$-0.513867\pi$
$491$	−1.93005	−0.0871018	−0.0435509	+	0.999051i	$0.513867\pi$
$492$	0	0
$493$	−5.90773	−0.266071
$494$	−6.21094	−0.279443
$495$	0	0
$496$	2.01392	0.0904275
$497$	−31.6082	−1.41782
$498$	0	0
$499$	4.02644	0.180248	0.0901242	−	0.995931i	$-0.471274\pi$
$499$	4.02644	0.180248	0.0901242	+	0.995931i	$0.471274\pi$
$500$	−4.64728	−0.207833
$501$	0	0
$502$	−19.8284	−0.884987
$503$	2.06560	0.0921004	0.0460502	−	0.998939i	$-0.485337\pi$
$503$	2.06560	0.0921004	0.0460502	+	0.998939i	$0.485337\pi$
$504$	0	0
$505$	−2.96786	−0.132068
$506$	−0.120065	−0.00533755
$507$	0	0
$508$	25.8537	1.14707
$509$	31.3994	1.39175	0.695876	−	0.718162i	$-0.255016\pi$
$509$	31.3994	1.39175	0.695876	+	0.718162i	$0.255016\pi$
$510$	0	0
$511$	−15.7293	−0.695825
$512$	5.43424	0.240162
$513$	0	0
$514$	8.95683	0.395069
$515$	−2.47870	−0.109224
$516$	0	0
$517$	5.49942	0.241864
$518$	−0.359833	−0.0158102
$519$	0	0
$520$	−5.18718	−0.227473
$521$	−17.7173	−0.776210	−0.388105	−	0.921615i	$-0.626870\pi$
$521$	−17.7173	−0.776210	−0.388105	+	0.921615i	$0.626870\pi$
$522$	0	0
$523$	−7.36710	−0.322141	−0.161070	−	0.986943i	$-0.551495\pi$
$523$	−7.36710	−0.322141	−0.161070	+	0.986943i	$0.551495\pi$
$524$	−25.5859	−1.11772
$525$	0	0
$526$	−10.9912	−0.479241
$527$	8.58741	0.374073
$528$	0	0
$529$	−22.9781	−0.999048
$530$	−1.15790	−0.0502962
$531$	0	0
$532$	−5.77081	−0.250196
$533$	20.6024	0.892388
$534$	0	0
$535$	−4.20710	−0.181889
$536$	32.5803	1.40725
$537$	0	0
$538$	7.21623	0.311114
$539$	2.39589	0.103198
$540$	0	0
$541$	4.67387	0.200945	0.100473	−	0.994940i	$-0.467965\pi$
$541$	4.67387	0.200945	0.100473	+	0.994940i	$0.467965\pi$
$542$	−23.4957	−1.00923
$543$	0	0
$544$	11.9904	0.514084
$545$	1.85422	0.0794259
$546$	0	0
$547$	15.6137	0.667596	0.333798	−	0.942645i	$-0.391670\pi$
$547$	15.6137	0.667596	0.333798	+	0.942645i	$0.391670\pi$
$548$	−7.06976	−0.302005
$549$	0	0
$550$	−3.95699	−0.168727
$551$	−4.02017	−0.171265
$552$	0	0
$553$	−23.5377	−1.00093
$554$	13.2546	0.563134
$555$	0	0
$556$	19.9603	0.846506
$557$	−32.6435	−1.38315	−0.691575	−	0.722304i	$-0.743083\pi$
$557$	−32.6435	−1.38315	−0.691575	+	0.722304i	$0.743083\pi$
$558$	0	0
$559$	−13.5389	−0.572634
$560$	−0.519818	−0.0219663
$561$	0	0
$562$	−17.1633	−0.723991
$563$	44.4977	1.87535	0.937676	−	0.347509i	$-0.112973\pi$
$563$	44.4977	1.87535	0.937676	+	0.347509i	$0.112973\pi$
$564$	0	0
$565$	−2.53542	−0.106666
$566$	−24.1492	−1.01507
$567$	0	0
$568$	27.9582	1.17310
$569$	14.4354	0.605162	0.302581	−	0.953124i	$-0.402152\pi$
$569$	14.4354	0.605162	0.302581	+	0.953124i	$0.402152\pi$
$570$	0	0
$571$	27.2352	1.13976	0.569879	−	0.821729i	$-0.306990\pi$
$571$	27.2352	1.13976	0.569879	+	0.821729i	$0.306990\pi$
$572$	7.31969	0.306052
$573$	0	0
$574$	−9.39206	−0.392017
$575$	0.721708	0.0300973
$576$	0	0
$577$	18.4252	0.767050	0.383525	−	0.923531i	$-0.374710\pi$
$577$	18.4252	0.767050	0.383525	+	0.923531i	$0.374710\pi$
$578$	−10.3433	−0.430225
$579$	0	0
$580$	−1.34805	−0.0559749
$581$	−48.6239	−2.01726
$582$	0	0
$583$	4.06954	0.168543
$584$	13.9130	0.575723
$585$	0	0
$586$	−11.4856	−0.474465
$587$	17.0540	0.703893	0.351946	−	0.936020i	$-0.385520\pi$
$587$	17.0540	0.703893	0.351946	+	0.936020i	$0.385520\pi$
$588$	0	0
$589$	5.84367	0.240784
$590$	0.932651	0.0383966
$591$	0	0
$592$	−0.0699661	−0.00287559
$593$	−8.28898	−0.340388	−0.170194	−	0.985411i	$-0.554439\pi$
$593$	−8.28898	−0.340388	−0.170194	+	0.985411i	$0.554439\pi$
$594$	0	0
$595$	−2.21652	−0.0908686
$596$	−31.4140	−1.28677
$597$	0	0
$598$	0.655018	0.0267857
$599$	−16.6039	−0.678418	−0.339209	−	0.940711i	$-0.610159\pi$
$599$	−16.6039	−0.678418	−0.339209	+	0.940711i	$0.610159\pi$
$600$	0	0
$601$	28.2882	1.15390	0.576950	−	0.816780i	$-0.304243\pi$
$601$	28.2882	1.15390	0.576950	+	0.816780i	$0.304243\pi$
$602$	6.17201	0.251552
$603$	0	0
$604$	15.9779	0.650130
$605$	−0.350684	−0.0142573
$606$	0	0
$607$	15.1496	0.614901	0.307451	−	0.951564i	$-0.400524\pi$
$607$	15.1496	0.614901	0.307451	+	0.951564i	$0.400524\pi$
$608$	8.15939	0.330907
$609$	0	0
$610$	0.284529	0.0115203
$611$	−30.0022	−1.21376
$612$	0	0
$613$	−38.2662	−1.54556	−0.772779	−	0.634675i	$-0.781134\pi$
$613$	−38.2662	−1.54556	−0.772779	+	0.634675i	$0.781134\pi$
$614$	−22.1256	−0.892915
$615$	0	0
$616$	−8.31089	−0.334855
$617$	−16.9404	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$617$	−16.9404	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$618$	0	0
$619$	−42.4150	−1.70480	−0.852401	−	0.522889i	$-0.824854\pi$
$619$	−42.4150	−1.70480	−0.852401	+	0.522889i	$0.824854\pi$
$620$	1.95951	0.0786960
$621$	0	0
$622$	−3.38786	−0.135841
$623$	−10.2829	−0.411975
$624$	0	0
$625$	23.1704	0.926817
$626$	−2.73897	−0.109471
$627$	0	0
$628$	16.2088	0.646802
$629$	−0.298338	−0.0118955
$630$	0	0
$631$	−10.5987	−0.421927	−0.210963	−	0.977494i	$-0.567660\pi$
$631$	−10.5987	−0.421927	−0.210963	+	0.977494i	$0.567660\pi$
$632$	20.8197	0.828162
$633$	0	0
$634$	13.3004	0.528225
$635$	6.75746	0.268161
$636$	0	0
$637$	−13.0708	−0.517886
$638$	−2.32458	−0.0920310
$639$	0	0
$640$	3.01121	0.119029
$641$	13.4569	0.531515	0.265758	−	0.964040i	$-0.414378\pi$
$641$	13.4569	0.531515	0.265758	+	0.964040i	$0.414378\pi$
$642$	0	0
$643$	41.9992	1.65629	0.828143	−	0.560517i	$-0.189398\pi$
$643$	41.9992	1.65629	0.828143	+	0.560517i	$0.189398\pi$
$644$	0.608601	0.0239822
$645$	0	0
$646$	2.34751	0.0923617
$647$	22.4826	0.883883	0.441941	−	0.897044i	$-0.354290\pi$
$647$	22.4826	0.883883	0.441941	+	0.897044i	$0.354290\pi$
$648$	0	0
$649$	−3.27787	−0.128668
$650$	21.5874	0.846729
$651$	0	0
$652$	1.40208	0.0549099
$653$	32.6465	1.27756	0.638779	−	0.769390i	$-0.279440\pi$
$653$	32.6465	1.27756	0.638779	+	0.769390i	$0.279440\pi$
$654$	0	0
$655$	−6.68744	−0.261300
$656$	−1.82620	−0.0713009
$657$	0	0
$658$	13.6772	0.533192
$659$	44.5231	1.73437	0.867186	−	0.497984i	$-0.165926\pi$
$659$	44.5231	1.73437	0.867186	+	0.497984i	$0.165926\pi$
$660$	0	0
$661$	22.1696	0.862297	0.431149	−	0.902281i	$-0.358108\pi$
$661$	22.1696	0.862297	0.431149	+	0.902281i	$0.358108\pi$
$662$	−4.76031	−0.185015
$663$	0	0
$664$	43.0090	1.66907
$665$	−1.50833	−0.0584905
$666$	0	0
$667$	0.423975	0.0164164
$668$	5.25332	0.203257
$669$	0	0
$670$	3.41903	0.132089
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	37.7568	1.45542	0.727708	−	0.685887i	$-0.240585\pi$
$673$	37.7568	1.45542	0.727708	+	0.685887i	$0.240585\pi$
$674$	17.3073	0.666652
$675$	0	0
$676$	−22.4906	−0.865021
$677$	20.5477	0.789711	0.394855	−	0.918743i	$-0.370795\pi$
$677$	20.5477	0.789711	0.394855	+	0.918743i	$0.370795\pi$
$678$	0	0
$679$	−15.8684	−0.608974
$680$	1.96057	0.0751843
$681$	0	0
$682$	3.37898	0.129388
$683$	5.07792	0.194301	0.0971506	−	0.995270i	$-0.469027\pi$
$683$	5.07792	0.194301	0.0971506	+	0.995270i	$0.469027\pi$
$684$	0	0
$685$	−1.84784	−0.0706024
$686$	−11.4505	−0.437183
$687$	0	0
$688$	1.20009	0.0457529
$689$	−22.2015	−0.845808
$690$	0	0
$691$	34.7498	1.32194	0.660972	−	0.750410i	$-0.270144\pi$
$691$	34.7498	1.32194	0.660972	+	0.750410i	$0.270144\pi$
$692$	9.27528	0.352593
$693$	0	0
$694$	29.0594	1.10308
$695$	5.21707	0.197895
$696$	0	0
$697$	−7.78696	−0.294952
$698$	−10.1964	−0.385938
$699$	0	0
$700$	20.0577	0.758108
$701$	6.15899	0.232622	0.116311	−	0.993213i	$-0.462893\pi$
$701$	6.15899	0.232622	0.116311	+	0.993213i	$0.462893\pi$
$702$	0	0
$703$	−0.203017	−0.00765693
$704$	3.75084	0.141365
$705$	0	0
$706$	−12.4699	−0.469312
$707$	25.9416	0.975633
$708$	0	0
$709$	5.81568	0.218412	0.109206	−	0.994019i	$-0.465169\pi$
$709$	5.81568	0.218412	0.109206	+	0.994019i	$0.465169\pi$
$710$	2.93399	0.110111
$711$	0	0
$712$	9.09546	0.340867
$713$	−0.616286	−0.0230801
$714$	0	0
$715$	1.91317	0.0715483
$716$	−11.0410	−0.412621
$717$	0	0
$718$	−12.5686	−0.469057
$719$	22.1738	0.826944	0.413472	−	0.910517i	$-0.364316\pi$
$719$	22.1738	0.826944	0.413472	+	0.910517i	$0.364316\pi$
$720$	0	0
$721$	21.6659	0.806878
$722$	−13.8183	−0.514263
$723$	0	0
$724$	27.1081	1.00746
$725$	13.9730	0.518943
$726$	0	0
$727$	−25.0474	−0.928955	−0.464478	−	0.885585i	$-0.653758\pi$
$727$	−25.0474	−0.928955	−0.464478	+	0.885585i	$0.653758\pi$
$728$	45.3402	1.68042
$729$	0	0
$730$	1.46005	0.0540390
$731$	5.11721	0.189267
$732$	0	0
$733$	1.09363	0.0403940	0.0201970	−	0.999796i	$-0.493571\pi$
$733$	1.09363	0.0403940	0.0201970	+	0.999796i	$0.493571\pi$
$734$	−19.3490	−0.714186
$735$	0	0
$736$	−0.860506	−0.0317187
$737$	−12.0165	−0.442632
$738$	0	0
$739$	35.5704	1.30848	0.654239	−	0.756288i	$-0.272989\pi$
$739$	35.5704	1.30848	0.654239	+	0.756288i	$0.272989\pi$
$740$	−0.0680761	−0.00250253
$741$	0	0
$742$	10.1210	0.371555
$743$	−42.9021	−1.57392	−0.786962	−	0.617001i	$-0.788347\pi$
$743$	−42.9021	−1.57392	−0.786962	+	0.617001i	$0.788347\pi$
$744$	0	0
$745$	−8.21075	−0.300819
$746$	8.71158	0.318954
$747$	0	0
$748$	−2.76658	−0.101156
$749$	36.7736	1.34368
$750$	0	0
$751$	24.4082	0.890667	0.445333	−	0.895365i	$-0.353085\pi$
$751$	24.4082	0.890667	0.445333	+	0.895365i	$0.353085\pi$
$752$	2.65940	0.0969782
$753$	0	0
$754$	12.6818	0.461843
$755$	4.17617	0.151987
$756$	0	0
$757$	−0.348726	−0.0126747	−0.00633733	−	0.999980i	$-0.502017\pi$
$757$	−0.348726	−0.0126747	−0.00633733	+	0.999980i	$0.502017\pi$
$758$	8.46803	0.307573
$759$	0	0
$760$	1.33415	0.0483948
$761$	−18.6006	−0.674271	−0.337135	−	0.941456i	$-0.609458\pi$
$761$	−18.6006	−0.674271	−0.337135	+	0.941456i	$0.609458\pi$
$762$	0	0
$763$	−16.2074	−0.586747
$764$	24.7903	0.896880
$765$	0	0
$766$	26.6055	0.961297
$767$	17.8825	0.645699
$768$	0	0
$769$	11.4172	0.411714	0.205857	−	0.978582i	$-0.434002\pi$
$769$	11.4172	0.411714	0.205857	+	0.978582i	$0.434002\pi$
$770$	−0.872160	−0.0314305
$771$	0	0
$772$	21.2252	0.763911
$773$	19.2564	0.692604	0.346302	−	0.938123i	$-0.387437\pi$
$773$	19.2564	0.692604	0.346302	+	0.938123i	$0.387437\pi$
$774$	0	0
$775$	−20.3109	−0.729590
$776$	14.0360	0.503863
$777$	0	0
$778$	−3.08046	−0.110440
$779$	−5.29898	−0.189856
$780$	0	0
$781$	−10.3117	−0.368982
$782$	−0.247573	−0.00885321
$783$	0	0
$784$	1.15860	0.0413786
$785$	4.23654	0.151209
$786$	0	0
$787$	−28.7433	−1.02459	−0.512294	−	0.858810i	$-0.671204\pi$
$787$	−28.7433	−1.02459	−0.512294	+	0.858810i	$0.671204\pi$
$788$	19.2846	0.686987
$789$	0	0
$790$	2.18485	0.0777336
$791$	22.1617	0.787978
$792$	0	0
$793$	5.45552	0.193731
$794$	−4.65886	−0.165337
$795$	0	0
$796$	11.0297	0.390937
$797$	−15.3196	−0.542647	−0.271323	−	0.962488i	$-0.587461\pi$
$797$	−15.3196	−0.542647	−0.271323	+	0.962488i	$0.587461\pi$
$798$	0	0
$799$	11.3398	0.401172
$800$	−28.3597	−1.00267
$801$	0	0
$802$	−10.4423	−0.368732
$803$	−5.13147	−0.181086
$804$	0	0
$805$	0.159071	0.00560653
$806$	−18.4341	−0.649314
$807$	0	0
$808$	−22.9459	−0.807235
$809$	14.8613	0.522494	0.261247	−	0.965272i	$-0.415866\pi$
$809$	14.8613	0.522494	0.261247	+	0.965272i	$0.415866\pi$
$810$	0	0
$811$	−43.3181	−1.52110	−0.760552	−	0.649277i	$-0.775071\pi$
$811$	−43.3181	−1.52110	−0.760552	+	0.649277i	$0.775071\pi$
$812$	11.7831	0.413506
$813$	0	0
$814$	−0.117390	−0.00411453
$815$	0.366466	0.0128368
$816$	0	0
$817$	3.48223	0.121828
$818$	14.2973	0.499893
$819$	0	0
$820$	−1.77686	−0.0620508
$821$	−42.6323	−1.48788	−0.743939	−	0.668248i	$-0.767044\pi$
$821$	−42.6323	−1.48788	−0.743939	+	0.668248i	$0.767044\pi$
$822$	0	0
$823$	44.1113	1.53762	0.768811	−	0.639476i	$-0.220849\pi$
$823$	44.1113	1.53762	0.768811	+	0.639476i	$0.220849\pi$
$824$	−19.1640	−0.667608
$825$	0	0
$826$	−8.15214	−0.283649
$827$	0.210906	0.00733390	0.00366695	−	0.999993i	$-0.498833\pi$
$827$	0.210906	0.00733390	0.00366695	+	0.999993i	$0.498833\pi$
$828$	0	0
$829$	19.5308	0.678332	0.339166	−	0.940727i	$-0.389855\pi$
$829$	19.5308	0.678332	0.339166	+	0.940727i	$0.389855\pi$
$830$	4.51344	0.156664
$831$	0	0
$832$	−20.4628	−0.709420
$833$	4.94031	0.171172
$834$	0	0
$835$	1.37307	0.0475171
$836$	−1.88264	−0.0651125
$837$	0	0
$838$	8.20041	0.283278
$839$	51.4138	1.77500	0.887501	−	0.460806i	$-0.152440\pi$
$839$	51.4138	1.77500	0.887501	+	0.460806i	$0.152440\pi$
$840$	0	0
$841$	−20.7914	−0.716946
$842$	−4.93745	−0.170156
$843$	0	0
$844$	−16.9950	−0.584992
$845$	−5.87841	−0.202223
$846$	0	0
$847$	3.06527	0.105324
$848$	1.96794	0.0675793
$849$	0	0
$850$	−8.15928	−0.279861
$851$	0.0214106	0.000733945 0
$852$	0	0
$853$	28.7095	0.982994	0.491497	−	0.870879i	$-0.336450\pi$
$853$	28.7095	0.982994	0.491497	+	0.870879i	$0.336450\pi$
$854$	−2.48702	−0.0851042
$855$	0	0
$856$	−32.5271	−1.11175
$857$	29.3799	1.00360	0.501799	−	0.864985i	$-0.332672\pi$
$857$	29.3799	1.00360	0.501799	+	0.864985i	$0.332672\pi$
$858$	0	0
$859$	11.3543	0.387405	0.193702	−	0.981060i	$-0.437950\pi$
$859$	11.3543	0.387405	0.193702	+	0.981060i	$0.437950\pi$
$860$	1.16767	0.0398172
$861$	0	0
$862$	−8.28239	−0.282099
$863$	−13.4097	−0.456471	−0.228235	−	0.973606i	$-0.573296\pi$
$863$	−13.4097	−0.456471	−0.228235	+	0.973606i	$0.573296\pi$
$864$	0	0
$865$	2.42430	0.0824288
$866$	−2.05484	−0.0698262
$867$	0	0
$868$	−17.1278	−0.581355
$869$	−7.67884	−0.260487
$870$	0	0
$871$	65.5560	2.22128
$872$	14.3358	0.485472
$873$	0	0
$874$	−0.168472	−0.00569865
$875$	10.6172	0.358928
$876$	0	0
$877$	30.9542	1.04525	0.522624	−	0.852563i	$-0.324953\pi$
$877$	30.9542	1.04525	0.522624	+	0.852563i	$0.324953\pi$
$878$	30.0964	1.01570
$879$	0	0
$880$	−0.169583	−0.00571664
$881$	−21.7719	−0.733515	−0.366757	−	0.930317i	$-0.619532\pi$
$881$	−21.7719	−0.733515	−0.366757	+	0.930317i	$0.619532\pi$
$882$	0	0
$883$	−12.9715	−0.436527	−0.218264	−	0.975890i	$-0.570039\pi$
$883$	−12.9715	−0.436527	−0.218264	+	0.975890i	$0.570039\pi$
$884$	15.0931	0.507637
$885$	0	0
$886$	−5.24803	−0.176311
$887$	−19.9273	−0.669092	−0.334546	−	0.942379i	$-0.608583\pi$
$887$	−19.9273	−0.669092	−0.334546	+	0.942379i	$0.608583\pi$
$888$	0	0
$889$	−59.0658	−1.98100
$890$	0.954494	0.0319947
$891$	0	0
$892$	−0.515777	−0.0172695
$893$	7.71664	0.258227
$894$	0	0
$895$	−2.88581	−0.0964620
$896$	−26.3205	−0.879305
$897$	0	0
$898$	−14.4553	−0.482379
$899$	−11.9319	−0.397951
$900$	0	0
$901$	8.39136	0.279557
$902$	−3.06402	−0.102021
$903$	0	0
$904$	−19.6025	−0.651970
$905$	7.08530	0.235523
$906$	0	0
$907$	−31.0013	−1.02938	−0.514691	−	0.857376i	$-0.672093\pi$
$907$	−31.0013	−1.02938	−0.514691	+	0.857376i	$0.672093\pi$
$908$	14.0883	0.467537
$909$	0	0
$910$	4.75809	0.157729
$911$	31.3158	1.03754	0.518770	−	0.854914i	$-0.326390\pi$
$911$	31.3158	1.03754	0.518770	+	0.854914i	$0.326390\pi$
$912$	0	0
$913$	−15.8628	−0.524983
$914$	28.4126	0.939807
$915$	0	0
$916$	−20.6917	−0.683673
$917$	58.4538	1.93031
$918$	0	0
$919$	22.9889	0.758333	0.379166	−	0.925329i	$-0.376211\pi$
$919$	22.9889	0.758333	0.379166	+	0.925329i	$0.376211\pi$
$920$	−0.140702	−0.00463882
$921$	0	0
$922$	−0.804837	−0.0265059
$923$	56.2558	1.85168
$924$	0	0
$925$	0.705629	0.0232009
$926$	−12.6797	−0.416681
$927$	0	0
$928$	−16.6602	−0.546899
$929$	−26.5210	−0.870125	−0.435063	−	0.900400i	$-0.643274\pi$
$929$	−26.5210	−0.870125	−0.435063	+	0.900400i	$0.643274\pi$
$930$	0	0
$931$	3.36185	0.110180
$932$	16.3124	0.534331
$933$	0	0
$934$	20.7997	0.680588
$935$	−0.723108	−0.0236482
$936$	0	0
$937$	−25.2739	−0.825663	−0.412831	−	0.910807i	$-0.635460\pi$
$937$	−25.2739	−0.825663	−0.412831	+	0.910807i	$0.635460\pi$
$938$	−29.8852	−0.975786
$939$	0	0
$940$	2.58756	0.0843969
$941$	−10.3509	−0.337429	−0.168715	−	0.985665i	$-0.553962\pi$
$941$	−10.3509	−0.337429	−0.168715	+	0.985665i	$0.553962\pi$
$942$	0	0
$943$	0.558841	0.0181984
$944$	−1.58510	−0.0515907
$945$	0	0
$946$	2.01353	0.0654654
$947$	−7.06389	−0.229545	−0.114773	−	0.993392i	$-0.536614\pi$
$947$	−7.06389	−0.229545	−0.114773	+	0.993392i	$0.536614\pi$
$948$	0	0
$949$	27.9948	0.908750
$950$	−5.55234	−0.180142
$951$	0	0
$952$	−17.1370	−0.555413
$953$	−14.0331	−0.454578	−0.227289	−	0.973827i	$-0.572986\pi$
$953$	−14.0331	−0.454578	−0.227289	+	0.973827i	$0.572986\pi$
$954$	0	0
$955$	6.47949	0.209671
$956$	−6.87837	−0.222462
$957$	0	0
$958$	−19.0526	−0.615563
$959$	16.1517	0.521564
$960$	0	0
$961$	−13.6559	−0.440514
$962$	0.640425	0.0206481
$963$	0	0
$964$	−5.72861	−0.184506
$965$	5.54768	0.178586
$966$	0	0
$967$	28.8586	0.928029	0.464015	−	0.885827i	$-0.346408\pi$
$967$	28.8586	0.928029	0.464015	+	0.885827i	$0.346408\pi$
$968$	−2.71131	−0.0871447
$969$	0	0
$970$	1.47296	0.0472940
$971$	−11.5064	−0.369259	−0.184630	−	0.982808i	$-0.559109\pi$
$971$	−11.5064	−0.369259	−0.184630	+	0.982808i	$0.559109\pi$
$972$	0	0
$973$	−45.6016	−1.46192
$974$	−4.61131	−0.147756
$975$	0	0
$976$	−0.483577	−0.0154789
$977$	−48.8381	−1.56247	−0.781234	−	0.624238i	$-0.785410\pi$
$977$	−48.8381	−1.56247	−0.781234	+	0.624238i	$0.785410\pi$
$978$	0	0
$979$	−3.35464	−0.107215
$980$	1.12730	0.0360104
$981$	0	0
$982$	−1.56595	−0.0499716
$983$	−19.1197	−0.609823	−0.304911	−	0.952381i	$-0.598627\pi$
$983$	−19.1197	−0.609823	−0.304911	+	0.952381i	$0.598627\pi$
$984$	0	0
$985$	5.04048	0.160603
$986$	−4.79326	−0.152649
$987$	0	0
$988$	10.2708	0.326757
$989$	−0.367243	−0.0116776
$990$	0	0
$991$	30.2963	0.962393	0.481197	−	0.876613i	$-0.340202\pi$
$991$	30.2963	0.962393	0.481197	+	0.876613i	$0.340202\pi$
$992$	24.2171	0.768894
$993$	0	0
$994$	−25.6455	−0.813425
$995$	2.88286	0.0913928
$996$	0	0
$997$	44.3179	1.40356	0.701782	−	0.712392i	$-0.252388\pi$
$997$	44.3179	1.40356	0.701782	+	0.712392i	$0.252388\pi$
$998$	3.26687	0.103411
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.8		12
3.2	odd	2		2013.2.a.d.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.5	✓	12	3.2	odd	2
6039.2.a.e.1.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.811354 q^{2} -1.34170 q^{4} -0.350684 q^{5} +3.06527 q^{7} -2.71131 q^{8} +O(q^{10})\)	\(q+0.811354 q^{2} -1.34170 q^{4} -0.350684 q^{5} +3.06527 q^{7} -2.71131 q^{8} -0.284529 q^{10} +1.00000 q^{11} -5.45552 q^{13} +2.48702 q^{14} +0.483577 q^{16} +2.06199 q^{17} +1.40317 q^{19} +0.470515 q^{20} +0.811354 q^{22} -0.147981 q^{23} -4.87702 q^{25} -4.42636 q^{26} -4.11269 q^{28} -2.86506 q^{29} +4.16462 q^{31} +5.81497 q^{32} +1.67301 q^{34} -1.07494 q^{35} -0.144684 q^{37} +1.13847 q^{38} +0.950813 q^{40} -3.77643 q^{41} +2.48169 q^{43} -1.34170 q^{44} -0.120065 q^{46} +5.49942 q^{47} +2.39589 q^{49} -3.95699 q^{50} +7.31969 q^{52} +4.06954 q^{53} -0.350684 q^{55} -8.31089 q^{56} -2.32458 q^{58} -3.27787 q^{59} -1.00000 q^{61} +3.37898 q^{62} +3.75084 q^{64} +1.91317 q^{65} -12.0165 q^{67} -2.76658 q^{68} -0.872160 q^{70} -10.3117 q^{71} -5.13147 q^{73} -0.117390 q^{74} -1.88264 q^{76} +3.06527 q^{77} -7.67884 q^{79} -0.169583 q^{80} -3.06402 q^{82} -15.8628 q^{83} -0.723108 q^{85} +2.01353 q^{86} -2.71131 q^{88} -3.35464 q^{89} -16.7227 q^{91} +0.198547 q^{92} +4.46198 q^{94} -0.492070 q^{95} -5.17684 q^{97} +1.94392 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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