LMFDB - Embedded newform 6039.2.a.h.1.12

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} + \cdots - 35 $ x^13 - 4x^12 - 11x^11 + 57x^10 + 28x^9 - 290x^8 + 51x^7 + 644x^6 - 259x^5 - 640x^4 + 274x^3 + 256x^2 - 74x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.17077$ 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)

$f(q)$	$=$	$q+2.17077 q^{2} +2.71223 q^{4} -2.18891 q^{5} +3.77754 q^{7} +1.54609 q^{8} +O(q^{10})$	\(q+2.17077 q^{2} +2.71223 q^{4} -2.18891 q^{5} +3.77754 q^{7} +1.54609 q^{8} -4.75162 q^{10} -1.00000 q^{11} -2.80832 q^{13} +8.20017 q^{14} -2.06827 q^{16} -0.740007 q^{17} -0.533054 q^{19} -5.93684 q^{20} -2.17077 q^{22} -2.10330 q^{23} -0.208662 q^{25} -6.09621 q^{26} +10.2456 q^{28} -8.30782 q^{29} -3.15993 q^{31} -7.58190 q^{32} -1.60638 q^{34} -8.26871 q^{35} -2.81222 q^{37} -1.15714 q^{38} -3.38425 q^{40} -8.19571 q^{41} +9.36633 q^{43} -2.71223 q^{44} -4.56577 q^{46} +1.30995 q^{47} +7.26984 q^{49} -0.452956 q^{50} -7.61681 q^{52} +4.44723 q^{53} +2.18891 q^{55} +5.84041 q^{56} -18.0344 q^{58} +0.127920 q^{59} +1.00000 q^{61} -6.85948 q^{62} -12.3220 q^{64} +6.14717 q^{65} -0.175386 q^{67} -2.00707 q^{68} -17.9495 q^{70} +0.0782005 q^{71} -0.0568582 q^{73} -6.10468 q^{74} -1.44577 q^{76} -3.77754 q^{77} -10.5905 q^{79} +4.52726 q^{80} -17.7910 q^{82} -4.37497 q^{83} +1.61981 q^{85} +20.3321 q^{86} -1.54609 q^{88} -10.9500 q^{89} -10.6086 q^{91} -5.70463 q^{92} +2.84361 q^{94} +1.16681 q^{95} +4.47645 q^{97} +15.7811 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * 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 $

$2$	2.17077	1.53496	0.767482	−	0.641070i	$-0.221509\pi$
$2$	2.17077	1.53496	0.767482	+	0.641070i	$0.221509\pi$
$3$	0	0
$3$	0	0
$4$	2.71223	1.35612
$5$	−2.18891	−0.978911	−0.489456	−	0.872028i	$-0.662805\pi$
$5$	−2.18891	−0.978911	−0.489456	+	0.872028i	$0.662805\pi$
$6$	0	0
$7$	3.77754	1.42778	0.713889	−	0.700259i	$-0.246932\pi$
$7$	3.77754	1.42778	0.713889	+	0.700259i	$0.246932\pi$
$8$	1.54609	0.546624
$9$	0	0
$10$	−4.75162	−1.50259
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.80832	−0.778888	−0.389444	−	0.921050i	$-0.627333\pi$
$13$	−2.80832	−0.778888	−0.389444	+	0.921050i	$0.627333\pi$
$14$	8.20017	2.19159
$15$	0	0
$16$	−2.06827	−0.517067
$17$	−0.740007	−0.179478	−0.0897390	−	0.995965i	$-0.528603\pi$
$17$	−0.740007	−0.179478	−0.0897390	+	0.995965i	$0.528603\pi$
$18$	0	0
$19$	−0.533054	−0.122291	−0.0611455	−	0.998129i	$-0.519475\pi$
$19$	−0.533054	−0.122291	−0.0611455	+	0.998129i	$0.519475\pi$
$20$	−5.93684	−1.32752
$21$	0	0
$22$	−2.17077	−0.462809
$23$	−2.10330	−0.438568	−0.219284	−	0.975661i	$-0.570372\pi$
$23$	−2.10330	−0.438568	−0.219284	+	0.975661i	$0.570372\pi$
$24$	0	0
$25$	−0.208662	−0.0417324
$26$	−6.09621	−1.19556
$27$	0	0
$28$	10.2456	1.93623
$29$	−8.30782	−1.54272	−0.771362	−	0.636396i	$-0.780424\pi$
$29$	−8.30782	−1.54272	−0.771362	+	0.636396i	$0.780424\pi$
$30$	0	0
$31$	−3.15993	−0.567541	−0.283771	−	0.958892i	$-0.591585\pi$
$31$	−3.15993	−0.567541	−0.283771	+	0.958892i	$0.591585\pi$
$32$	−7.58190	−1.34030
$33$	0	0
$34$	−1.60638	−0.275492
$35$	−8.26871	−1.39767
$36$	0	0
$37$	−2.81222	−0.462327	−0.231163	−	0.972915i	$-0.574253\pi$
$37$	−2.81222	−0.462327	−0.231163	+	0.972915i	$0.574253\pi$
$38$	−1.15714	−0.187712
$39$	0	0
$40$	−3.38425	−0.535097
$41$	−8.19571	−1.27995	−0.639977	−	0.768394i	$-0.721056\pi$
$41$	−8.19571	−1.27995	−0.639977	+	0.768394i	$0.721056\pi$
$42$	0	0
$43$	9.36633	1.42835	0.714176	−	0.699966i	$-0.246802\pi$
$43$	9.36633	1.42835	0.714176	+	0.699966i	$0.246802\pi$
$44$	−2.71223	−0.408884
$45$	0	0
$46$	−4.56577	−0.673186
$47$	1.30995	0.191077	0.0955383	−	0.995426i	$-0.469543\pi$
$47$	1.30995	0.191077	0.0955383	+	0.995426i	$0.469543\pi$
$48$	0	0
$49$	7.26984	1.03855
$50$	−0.452956	−0.0640577
$51$	0	0
$52$	−7.61681	−1.05626
$53$	4.44723	0.610874	0.305437	−	0.952212i	$-0.401197\pi$
$53$	4.44723	0.610874	0.305437	+	0.952212i	$0.401197\pi$
$54$	0	0
$55$	2.18891	0.295153
$56$	5.84041	0.780458
$57$	0	0
$58$	−18.0344	−2.36803
$59$	0.127920	0.0166538	0.00832691	−	0.999965i	$-0.497349\pi$
$59$	0.127920	0.0166538	0.00832691	+	0.999965i	$0.497349\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−6.85948	−0.871155
$63$	0	0
$64$	−12.3220	−1.54025
$65$	6.14717	0.762462
$66$	0	0
$67$	−0.175386	−0.0214268	−0.0107134	−	0.999943i	$-0.503410\pi$
$67$	−0.175386	−0.0214268	−0.0107134	+	0.999943i	$0.503410\pi$
$68$	−2.00707	−0.243393
$69$	0	0
$70$	−17.9495	−2.14537
$71$	0.0782005	0.00928069	0.00464034	−	0.999989i	$-0.498523\pi$
$71$	0.0782005	0.00928069	0.00464034	+	0.999989i	$0.498523\pi$
$72$	0	0
$73$	−0.0568582	−0.00665475	−0.00332737	−	0.999994i	$-0.501059\pi$
$73$	−0.0568582	−0.00665475	−0.00332737	+	0.999994i	$0.501059\pi$
$74$	−6.10468	−0.709655
$75$	0	0
$76$	−1.44577	−0.165841
$77$	−3.77754	−0.430491
$78$	0	0
$79$	−10.5905	−1.19153	−0.595763	−	0.803161i	$-0.703150\pi$
$79$	−10.5905	−1.19153	−0.595763	+	0.803161i	$0.703150\pi$
$80$	4.52726	0.506163
$81$	0	0
$82$	−17.7910	−1.96468
$83$	−4.37497	−0.480215	−0.240107	−	0.970746i	$-0.577183\pi$
$83$	−4.37497	−0.480215	−0.240107	+	0.970746i	$0.577183\pi$
$84$	0	0
$85$	1.61981	0.175693
$86$	20.3321	2.19247
$87$	0	0
$88$	−1.54609	−0.164813
$89$	−10.9500	−1.16070	−0.580350	−	0.814367i	$-0.697084\pi$
$89$	−10.9500	−1.16070	−0.580350	+	0.814367i	$0.697084\pi$
$90$	0	0
$91$	−10.6086	−1.11208
$92$	−5.70463	−0.594749
$93$	0	0
$94$	2.84361	0.293296
$95$	1.16681	0.119712
$96$	0	0
$97$	4.47645	0.454515	0.227257	−	0.973835i	$-0.427024\pi$
$97$	4.47645	0.454515	0.227257	+	0.973835i	$0.427024\pi$
$98$	15.7811	1.59413
$99$	0	0
$100$	−0.565939	−0.0565939
$101$	10.8564	1.08025	0.540124	−	0.841585i	$-0.318377\pi$
$101$	10.8564	1.08025	0.540124	+	0.841585i	$0.318377\pi$
$102$	0	0
$103$	−6.67205	−0.657417	−0.328708	−	0.944431i	$-0.606613\pi$
$103$	−6.67205	−0.657417	−0.328708	+	0.944431i	$0.606613\pi$
$104$	−4.34191	−0.425759
$105$	0	0
$106$	9.65390	0.937670
$107$	−9.65698	−0.933576	−0.466788	−	0.884369i	$-0.654589\pi$
$107$	−9.65698	−0.933576	−0.466788	+	0.884369i	$0.654589\pi$
$108$	0	0
$109$	2.74312	0.262743	0.131372	−	0.991333i	$-0.458062\pi$
$109$	2.74312	0.262743	0.131372	+	0.991333i	$0.458062\pi$
$110$	4.75162	0.453049
$111$	0	0
$112$	−7.81297	−0.738256
$113$	8.15205	0.766881	0.383440	−	0.923566i	$-0.374739\pi$
$113$	8.15205	0.766881	0.383440	+	0.923566i	$0.374739\pi$
$114$	0	0
$115$	4.60394	0.429319
$116$	−22.5327	−2.09211
$117$	0	0
$118$	0.277686	0.0255630
$119$	−2.79541	−0.256255
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.17077	0.196532
$123$	0	0
$124$	−8.57047	−0.769651
$125$	11.4013	1.01976
$126$	0	0
$127$	8.01895	0.711567	0.355783	−	0.934568i	$-0.384214\pi$
$127$	8.01895	0.711567	0.355783	+	0.934568i	$0.384214\pi$
$128$	−11.5844	−1.02393
$129$	0	0
$130$	13.3441	1.17035
$131$	11.2602	0.983808	0.491904	−	0.870650i	$-0.336301\pi$
$131$	11.2602	0.983808	0.491904	+	0.870650i	$0.336301\pi$
$132$	0	0
$133$	−2.01364	−0.174604
$134$	−0.380721	−0.0328893
$135$	0	0
$136$	−1.14411	−0.0981070
$137$	−4.52047	−0.386210	−0.193105	−	0.981178i	$-0.561856\pi$
$137$	−4.52047	−0.386210	−0.193105	+	0.981178i	$0.561856\pi$
$138$	0	0
$139$	−0.921389	−0.0781512	−0.0390756	−	0.999236i	$-0.512441\pi$
$139$	−0.921389	−0.0781512	−0.0390756	+	0.999236i	$0.512441\pi$
$140$	−22.4267	−1.89540
$141$	0	0
$142$	0.169755	0.0142455
$143$	2.80832	0.234844
$144$	0	0
$145$	18.1851	1.51019
$146$	−0.123426	−0.0102148
$147$	0	0
$148$	−7.62740	−0.626968
$149$	−8.25743	−0.676475	−0.338237	−	0.941061i	$-0.609831\pi$
$149$	−8.25743	−0.676475	−0.338237	+	0.941061i	$0.609831\pi$
$150$	0	0
$151$	−4.30559	−0.350384	−0.175192	−	0.984534i	$-0.556055\pi$
$151$	−4.30559	−0.350384	−0.175192	+	0.984534i	$0.556055\pi$
$152$	−0.824148	−0.0668472
$153$	0	0
$154$	−8.20017	−0.660788
$155$	6.91682	0.555572
$156$	0	0
$157$	−16.7808	−1.33925	−0.669625	−	0.742699i	$-0.733545\pi$
$157$	−16.7808	−1.33925	−0.669625	+	0.742699i	$0.733545\pi$
$158$	−22.9895	−1.82895
$159$	0	0
$160$	16.5961	1.31204
$161$	−7.94530	−0.626178
$162$	0	0
$163$	−11.7466	−0.920063	−0.460032	−	0.887903i	$-0.652162\pi$
$163$	−11.7466	−0.920063	−0.460032	+	0.887903i	$0.652162\pi$
$164$	−22.2286	−1.73577
$165$	0	0
$166$	−9.49703	−0.737113
$167$	4.87620	0.377332	0.188666	−	0.982041i	$-0.439584\pi$
$167$	4.87620	0.377332	0.188666	+	0.982041i	$0.439584\pi$
$168$	0	0
$169$	−5.11334	−0.393334
$170$	3.51623	0.269683
$171$	0	0
$172$	25.4036	1.93701
$173$	−8.84196	−0.672242	−0.336121	−	0.941819i	$-0.609115\pi$
$173$	−8.84196	−0.672242	−0.336121	+	0.941819i	$0.609115\pi$
$174$	0	0
$175$	−0.788229	−0.0595845
$176$	2.06827	0.155901
$177$	0	0
$178$	−23.7700	−1.78163
$179$	−10.9302	−0.816959	−0.408479	−	0.912768i	$-0.633941\pi$
$179$	−10.9302	−0.816959	−0.408479	+	0.912768i	$0.633941\pi$
$180$	0	0
$181$	14.2878	1.06200	0.531000	−	0.847372i	$-0.321816\pi$
$181$	14.2878	1.06200	0.531000	+	0.847372i	$0.321816\pi$
$182$	−23.0287	−1.70700
$183$	0	0
$184$	−3.25188	−0.239732
$185$	6.15571	0.452577
$186$	0	0
$187$	0.740007	0.0541147
$188$	3.55290	0.259122
$189$	0	0
$190$	2.53287	0.183754
$191$	−13.1958	−0.954817	−0.477409	−	0.878681i	$-0.658424\pi$
$191$	−13.1958	−0.954817	−0.477409	+	0.878681i	$0.658424\pi$
$192$	0	0
$193$	−18.4016	−1.32458	−0.662288	−	0.749249i	$-0.730415\pi$
$193$	−18.4016	−1.32458	−0.662288	+	0.749249i	$0.730415\pi$
$194$	9.71733	0.697664
$195$	0	0
$196$	19.7175	1.40839
$197$	−0.691903	−0.0492961	−0.0246480	−	0.999696i	$-0.507847\pi$
$197$	−0.691903	−0.0492961	−0.0246480	+	0.999696i	$0.507847\pi$
$198$	0	0
$199$	−12.4450	−0.882202	−0.441101	−	0.897458i	$-0.645412\pi$
$199$	−12.4450	−0.882202	−0.441101	+	0.897458i	$0.645412\pi$
$200$	−0.322609	−0.0228119
$201$	0	0
$202$	23.5666	1.65814
$203$	−31.3832	−2.20267
$204$	0	0
$205$	17.9397	1.25296
$206$	−14.4835	−1.00911
$207$	0	0
$208$	5.80836	0.402737
$209$	0.533054	0.0368721
$210$	0	0
$211$	4.11085	0.283003	0.141501	−	0.989938i	$-0.454807\pi$
$211$	4.11085	0.283003	0.141501	+	0.989938i	$0.454807\pi$
$212$	12.0619	0.828415
$213$	0	0
$214$	−20.9631	−1.43301
$215$	−20.5021	−1.39823
$216$	0	0
$217$	−11.9368	−0.810322
$218$	5.95468	0.403302
$219$	0	0
$220$	5.93684	0.400261
$221$	2.07818	0.139793
$222$	0	0
$223$	−4.98112	−0.333561	−0.166780	−	0.985994i	$-0.553337\pi$
$223$	−4.98112	−0.333561	−0.166780	+	0.985994i	$0.553337\pi$
$224$	−28.6410	−1.91365
$225$	0	0
$226$	17.6962	1.17713
$227$	17.8209	1.18281	0.591406	−	0.806374i	$-0.298573\pi$
$227$	17.8209	1.18281	0.591406	+	0.806374i	$0.298573\pi$
$228$	0	0
$229$	20.0521	1.32508	0.662539	−	0.749027i	$-0.269479\pi$
$229$	20.0521	1.32508	0.662539	+	0.749027i	$0.269479\pi$
$230$	9.99408	0.658990
$231$	0	0
$232$	−12.8446	−0.843290
$233$	−14.1446	−0.926643	−0.463321	−	0.886190i	$-0.653342\pi$
$233$	−14.1446	−0.926643	−0.463321	+	0.886190i	$0.653342\pi$
$234$	0	0
$235$	−2.86738	−0.187047
$236$	0.346950	0.0225845
$237$	0	0
$238$	−6.06818	−0.393342
$239$	−7.46151	−0.482645	−0.241322	−	0.970445i	$-0.577581\pi$
$239$	−7.46151	−0.482645	−0.241322	+	0.970445i	$0.577581\pi$
$240$	0	0
$241$	11.1377	0.717439	0.358720	−	0.933445i	$-0.383213\pi$
$241$	11.1377	0.717439	0.358720	+	0.933445i	$0.383213\pi$
$242$	2.17077	0.139542
$243$	0	0
$244$	2.71223	0.173633
$245$	−15.9130	−1.01665
$246$	0	0
$247$	1.49699	0.0952510
$248$	−4.88553	−0.310232
$249$	0	0
$250$	24.7496	1.56530
$251$	17.9266	1.13152	0.565758	−	0.824571i	$-0.308584\pi$
$251$	17.9266	1.13152	0.565758	+	0.824571i	$0.308584\pi$
$252$	0	0
$253$	2.10330	0.132233
$254$	17.4073	1.09223
$255$	0	0
$256$	−0.503040	−0.0314400
$257$	−16.5638	−1.03322	−0.516611	−	0.856220i	$-0.672807\pi$
$257$	−16.5638	−1.03322	−0.516611	+	0.856220i	$0.672807\pi$
$258$	0	0
$259$	−10.6233	−0.660100
$260$	16.6725	1.03399
$261$	0	0
$262$	24.4433	1.51011
$263$	−8.56762	−0.528302	−0.264151	−	0.964481i	$-0.585092\pi$
$263$	−8.56762	−0.528302	−0.264151	+	0.964481i	$0.585092\pi$
$264$	0	0
$265$	−9.73460	−0.597992
$266$	−4.37113	−0.268011
$267$	0	0
$268$	−0.475686	−0.0290571
$269$	2.88907	0.176150	0.0880748	−	0.996114i	$-0.471929\pi$
$269$	2.88907	0.176150	0.0880748	+	0.996114i	$0.471929\pi$
$270$	0	0
$271$	19.8520	1.20592	0.602962	−	0.797770i	$-0.293987\pi$
$271$	19.8520	1.20592	0.602962	+	0.797770i	$0.293987\pi$
$272$	1.53053	0.0928021
$273$	0	0
$274$	−9.81290	−0.592819
$275$	0.208662	0.0125828
$276$	0	0
$277$	13.0392	0.783452	0.391726	−	0.920082i	$-0.371878\pi$
$277$	13.0392	0.783452	0.391726	+	0.920082i	$0.371878\pi$
$278$	−2.00012	−0.119959
$279$	0	0
$280$	−12.7841	−0.763999
$281$	14.2722	0.851406	0.425703	−	0.904863i	$-0.360027\pi$
$281$	14.2722	0.851406	0.425703	+	0.904863i	$0.360027\pi$
$282$	0	0
$283$	8.01505	0.476445	0.238223	−	0.971211i	$-0.423435\pi$
$283$	8.01505	0.476445	0.238223	+	0.971211i	$0.423435\pi$
$284$	0.212098	0.0125857
$285$	0	0
$286$	6.09621	0.360476
$287$	−30.9596	−1.82749
$288$	0	0
$289$	−16.4524	−0.967788
$290$	39.4756	2.31809
$291$	0	0
$292$	−0.154212	−0.00902460
$293$	−5.86479	−0.342625	−0.171312	−	0.985217i	$-0.554801\pi$
$293$	−5.86479	−0.342625	−0.171312	+	0.985217i	$0.554801\pi$
$294$	0	0
$295$	−0.280007	−0.0163026
$296$	−4.34794	−0.252719
$297$	0	0
$298$	−17.9249	−1.03836
$299$	5.90674	0.341595
$300$	0	0
$301$	35.3817	2.03937
$302$	−9.34643	−0.537827
$303$	0	0
$304$	1.10250	0.0632326
$305$	−2.18891	−0.125337
$306$	0	0
$307$	−23.8758	−1.36267	−0.681333	−	0.731974i	$-0.738599\pi$
$307$	−23.8758	−1.36267	−0.681333	+	0.731974i	$0.738599\pi$
$308$	−10.2456	−0.583796
$309$	0	0
$310$	15.0148	0.852784
$311$	−22.3084	−1.26499	−0.632496	−	0.774564i	$-0.717969\pi$
$311$	−22.3084	−1.26499	−0.632496	+	0.774564i	$0.717969\pi$
$312$	0	0
$313$	32.7871	1.85324	0.926619	−	0.376001i	$-0.122701\pi$
$313$	32.7871	1.85324	0.926619	+	0.376001i	$0.122701\pi$
$314$	−36.4271	−2.05570
$315$	0	0
$316$	−28.7239	−1.61585
$317$	26.6621	1.49749	0.748747	−	0.662856i	$-0.230656\pi$
$317$	26.6621	1.49749	0.748747	+	0.662856i	$0.230656\pi$
$318$	0	0
$319$	8.30782	0.465149
$320$	26.9718	1.50777
$321$	0	0
$322$	−17.2474	−0.961160
$323$	0.394464	0.0219485
$324$	0	0
$325$	0.585989	0.0325048
$326$	−25.4991	−1.41226
$327$	0	0
$328$	−12.6713	−0.699654
$329$	4.94841	0.272815
$330$	0	0
$331$	−14.7884	−0.812846	−0.406423	−	0.913685i	$-0.633224\pi$
$331$	−14.7884	−0.812846	−0.406423	+	0.913685i	$0.633224\pi$
$332$	−11.8659	−0.651227
$333$	0	0
$334$	10.5851	0.579190
$335$	0.383904	0.0209749
$336$	0	0
$337$	8.60146	0.468551	0.234276	−	0.972170i	$-0.424728\pi$
$337$	8.60146	0.468551	0.234276	+	0.972170i	$0.424728\pi$
$338$	−11.0999	−0.603753
$339$	0	0
$340$	4.39330	0.238260
$341$	3.15993	0.171120
$342$	0	0
$343$	1.01933	0.0550386
$344$	14.4812	0.780772
$345$	0	0
$346$	−19.1938	−1.03187
$347$	−22.1720	−1.19025	−0.595127	−	0.803632i	$-0.702898\pi$
$347$	−22.1720	−1.19025	−0.595127	+	0.803632i	$0.702898\pi$
$348$	0	0
$349$	−5.01093	−0.268229	−0.134114	−	0.990966i	$-0.542819\pi$
$349$	−5.01093	−0.268229	−0.134114	+	0.990966i	$0.542819\pi$
$350$	−1.71106	−0.0914601
$351$	0	0
$352$	7.58190	0.404117
$353$	14.4198	0.767490	0.383745	−	0.923439i	$-0.374634\pi$
$353$	14.4198	0.767490	0.383745	+	0.923439i	$0.374634\pi$
$354$	0	0
$355$	−0.171174	−0.00908497
$356$	−29.6990	−1.57404
$357$	0	0
$358$	−23.7268	−1.25400
$359$	32.7641	1.72922	0.864612	−	0.502440i	$-0.167564\pi$
$359$	32.7641	1.72922	0.864612	+	0.502440i	$0.167564\pi$
$360$	0	0
$361$	−18.7159	−0.985045
$362$	31.0154	1.63013
$363$	0	0
$364$	−28.7728	−1.50811
$365$	0.124458	0.00651441
$366$	0	0
$367$	27.4911	1.43503	0.717513	−	0.696545i	$-0.245280\pi$
$367$	27.4911	1.43503	0.717513	+	0.696545i	$0.245280\pi$
$368$	4.35018	0.226769
$369$	0	0
$370$	13.3626	0.694690
$371$	16.7996	0.872192
$372$	0	0
$373$	−17.1315	−0.887036	−0.443518	−	0.896265i	$-0.646270\pi$
$373$	−17.1315	−0.887036	−0.443518	+	0.896265i	$0.646270\pi$
$374$	1.60638	0.0830641
$375$	0	0
$376$	2.02530	0.104447
$377$	23.3310	1.20161
$378$	0	0
$379$	−3.95374	−0.203090	−0.101545	−	0.994831i	$-0.532379\pi$
$379$	−3.95374	−0.203090	−0.101545	+	0.994831i	$0.532379\pi$
$380$	3.16465	0.162343
$381$	0	0
$382$	−28.6451	−1.46561
$383$	−8.04411	−0.411035	−0.205517	−	0.978653i	$-0.565888\pi$
$383$	−8.04411	−0.411035	−0.205517	+	0.978653i	$0.565888\pi$
$384$	0	0
$385$	8.26871	0.421413
$386$	−39.9456	−2.03318
$387$	0	0
$388$	12.1412	0.616374
$389$	36.9149	1.87166	0.935830	−	0.352450i	$-0.114652\pi$
$389$	36.9149	1.87166	0.935830	+	0.352450i	$0.114652\pi$
$390$	0	0
$391$	1.55646	0.0787134
$392$	11.2398	0.567696
$393$	0	0
$394$	−1.50196	−0.0756677
$395$	23.1817	1.16640
$396$	0	0
$397$	5.71192	0.286673	0.143336	−	0.989674i	$-0.454217\pi$
$397$	5.71192	0.286673	0.143336	+	0.989674i	$0.454217\pi$
$398$	−27.0152	−1.35415
$399$	0	0
$400$	0.431568	0.0215784
$401$	7.60506	0.379779	0.189889	−	0.981805i	$-0.439187\pi$
$401$	7.60506	0.379779	0.189889	+	0.981805i	$0.439187\pi$
$402$	0	0
$403$	8.87411	0.442051
$404$	29.4449	1.46494
$405$	0	0
$406$	−68.1256	−3.38101
$407$	2.81222	0.139397
$408$	0	0
$409$	10.4994	0.519162	0.259581	−	0.965721i	$-0.416416\pi$
$409$	10.4994	0.519162	0.259581	+	0.965721i	$0.416416\pi$
$410$	38.9429	1.92325
$411$	0	0
$412$	−18.0961	−0.891533
$413$	0.483225	0.0237780
$414$	0	0
$415$	9.57642	0.470088
$416$	21.2924	1.04395
$417$	0	0
$418$	1.15714	0.0565974
$419$	26.6073	1.29985	0.649925	−	0.759998i	$-0.274800\pi$
$419$	26.6073	1.29985	0.649925	+	0.759998i	$0.274800\pi$
$420$	0	0
$421$	−13.6666	−0.666067	−0.333034	−	0.942915i	$-0.608072\pi$
$421$	−13.6666	−0.666067	−0.333034	+	0.942915i	$0.608072\pi$
$422$	8.92371	0.434399
$423$	0	0
$424$	6.87580	0.333918
$425$	0.154411	0.00749004
$426$	0	0
$427$	3.77754	0.182808
$428$	−26.1920	−1.26604
$429$	0	0
$430$	−44.5052	−2.14623
$431$	31.1346	1.49970	0.749851	−	0.661606i	$-0.230125\pi$
$431$	31.1346	1.49970	0.749851	+	0.661606i	$0.230125\pi$
$432$	0	0
$433$	−18.2618	−0.877607	−0.438803	−	0.898583i	$-0.644598\pi$
$433$	−18.2618	−0.877607	−0.438803	+	0.898583i	$0.644598\pi$
$434$	−25.9120	−1.24382
$435$	0	0
$436$	7.43998	0.356310
$437$	1.12117	0.0536329
$438$	0	0
$439$	−29.3662	−1.40157	−0.700785	−	0.713372i	$-0.747167\pi$
$439$	−29.3662	−1.40157	−0.700785	+	0.713372i	$0.747167\pi$
$440$	3.38425	0.161338
$441$	0	0
$442$	4.51124	0.214578
$443$	−8.96167	−0.425782	−0.212891	−	0.977076i	$-0.568288\pi$
$443$	−8.96167	−0.425782	−0.212891	+	0.977076i	$0.568288\pi$
$444$	0	0
$445$	23.9686	1.13622
$446$	−10.8129	−0.512004
$447$	0	0
$448$	−46.5469	−2.19914
$449$	−15.9696	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$449$	−15.9696	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$450$	0	0
$451$	8.19571	0.385921
$452$	22.1102	1.03998
$453$	0	0
$454$	38.6849	1.81557
$455$	23.2212	1.08863
$456$	0	0
$457$	−4.01634	−0.187877	−0.0939383	−	0.995578i	$-0.529946\pi$
$457$	−4.01634	−0.187877	−0.0939383	+	0.995578i	$0.529946\pi$
$458$	43.5284	2.03395
$459$	0	0
$460$	12.4869	0.582206
$461$	14.9808	0.697725	0.348863	−	0.937174i	$-0.386568\pi$
$461$	14.9808	0.697725	0.348863	+	0.937174i	$0.386568\pi$
$462$	0	0
$463$	15.5676	0.723489	0.361744	−	0.932277i	$-0.382181\pi$
$463$	15.5676	0.723489	0.361744	+	0.932277i	$0.382181\pi$
$464$	17.1828	0.797691
$465$	0	0
$466$	−30.7046	−1.42236
$467$	−16.9130	−0.782641	−0.391320	−	0.920255i	$-0.627982\pi$
$467$	−16.9130	−0.782641	−0.391320	+	0.920255i	$0.627982\pi$
$468$	0	0
$469$	−0.662526	−0.0305926
$470$	−6.22441	−0.287110
$471$	0	0
$472$	0.197776	0.00910338
$473$	−9.36633	−0.430664
$474$	0	0
$475$	0.111228	0.00510349
$476$	−7.58179	−0.347511
$477$	0	0
$478$	−16.1972	−0.740842
$479$	9.48542	0.433400	0.216700	−	0.976238i	$-0.430471\pi$
$479$	9.48542	0.433400	0.216700	+	0.976238i	$0.430471\pi$
$480$	0	0
$481$	7.89763	0.360101
$482$	24.1773	1.10124
$483$	0	0
$484$	2.71223	0.123283
$485$	−9.79856	−0.444930
$486$	0	0
$487$	−33.8675	−1.53468	−0.767342	−	0.641238i	$-0.778421\pi$
$487$	−33.8675	−1.53468	−0.767342	+	0.641238i	$0.778421\pi$
$488$	1.54609	0.0699881
$489$	0	0
$490$	−34.5435	−1.56052
$491$	−0.760590	−0.0343250	−0.0171625	−	0.999853i	$-0.505463\pi$
$491$	−0.760590	−0.0343250	−0.0171625	+	0.999853i	$0.505463\pi$
$492$	0	0
$493$	6.14785	0.276885
$494$	3.24961	0.146207
$495$	0	0
$496$	6.53559	0.293457
$497$	0.295406	0.0132508
$498$	0	0
$499$	40.7712	1.82517	0.912584	−	0.408890i	$-0.134084\pi$
$499$	40.7712	1.82517	0.912584	+	0.408890i	$0.134084\pi$
$500$	30.9230	1.38292
$501$	0	0
$502$	38.9144	1.73684
$503$	−34.0342	−1.51751	−0.758754	−	0.651377i	$-0.774192\pi$
$503$	−34.0342	−1.51751	−0.758754	+	0.651377i	$0.774192\pi$
$504$	0	0
$505$	−23.7636	−1.05747
$506$	4.56577	0.202973
$507$	0	0
$508$	21.7492	0.964967
$509$	5.05965	0.224265	0.112132	−	0.993693i	$-0.464232\pi$
$509$	5.05965	0.224265	0.112132	+	0.993693i	$0.464232\pi$
$510$	0	0
$511$	−0.214784	−0.00950150
$512$	22.0768	0.975667
$513$	0	0
$514$	−35.9562	−1.58596
$515$	14.6045	0.643553
$516$	0	0
$517$	−1.30995	−0.0576117
$518$	−23.0607	−1.01323
$519$	0	0
$520$	9.50405	0.416780
$521$	23.4009	1.02521	0.512607	−	0.858624i	$-0.328680\pi$
$521$	23.4009	1.02521	0.512607	+	0.858624i	$0.328680\pi$
$522$	0	0
$523$	−24.2767	−1.06155	−0.530774	−	0.847514i	$-0.678099\pi$
$523$	−24.2767	−1.06155	−0.530774	+	0.847514i	$0.678099\pi$
$524$	30.5402	1.33416
$525$	0	0
$526$	−18.5983	−0.810925
$527$	2.33837	0.101861
$528$	0	0
$529$	−18.5761	−0.807658
$530$	−21.1315	−0.917896
$531$	0	0
$532$	−5.46144	−0.236784
$533$	23.0162	0.996941
$534$	0	0
$535$	21.1383	0.913888
$536$	−0.271161	−0.0117124
$537$	0	0
$538$	6.27149	0.270383
$539$	−7.26984	−0.313134
$540$	0	0
$541$	2.44476	0.105108	0.0525541	−	0.998618i	$-0.483264\pi$
$541$	2.44476	0.105108	0.0525541	+	0.998618i	$0.483264\pi$
$542$	43.0941	1.85105
$543$	0	0
$544$	5.61066	0.240555
$545$	−6.00446	−0.257203
$546$	0	0
$547$	29.8262	1.27527	0.637637	−	0.770337i	$-0.279912\pi$
$547$	29.8262	1.27527	0.637637	+	0.770337i	$0.279912\pi$
$548$	−12.2606	−0.523746
$549$	0	0
$550$	0.452956	0.0193141
$551$	4.42852	0.188661
$552$	0	0
$553$	−40.0061	−1.70123
$554$	28.3051	1.20257
$555$	0	0
$556$	−2.49902	−0.105982
$557$	−13.7893	−0.584271	−0.292135	−	0.956377i	$-0.594366\pi$
$557$	−13.7893	−0.584271	−0.292135	+	0.956377i	$0.594366\pi$
$558$	0	0
$559$	−26.3036	−1.11253
$560$	17.1019	0.722687
$561$	0	0
$562$	30.9815	1.30688
$563$	−28.5442	−1.20299	−0.601497	−	0.798875i	$-0.705429\pi$
$563$	−28.5442	−1.20299	−0.601497	+	0.798875i	$0.705429\pi$
$564$	0	0
$565$	−17.8441	−0.750708
$566$	17.3988	0.731326
$567$	0	0
$568$	0.120905	0.00507305
$569$	25.6554	1.07553	0.537765	−	0.843095i	$-0.319269\pi$
$569$	25.6554	1.07553	0.537765	+	0.843095i	$0.319269\pi$
$570$	0	0
$571$	30.9209	1.29400	0.646999	−	0.762491i	$-0.276024\pi$
$571$	30.9209	1.29400	0.646999	+	0.762491i	$0.276024\pi$
$572$	7.61681	0.318475
$573$	0	0
$574$	−67.2062	−2.80513
$575$	0.438878	0.0183025
$576$	0	0
$577$	−18.3309	−0.763127	−0.381563	−	0.924343i	$-0.624614\pi$
$577$	−18.3309	−0.763127	−0.381563	+	0.924343i	$0.624614\pi$
$578$	−35.7143	−1.48552
$579$	0	0
$580$	49.3222	2.04799
$581$	−16.5266	−0.685640
$582$	0	0
$583$	−4.44723	−0.184185
$584$	−0.0879076	−0.00363764
$585$	0	0
$586$	−12.7311	−0.525917
$587$	−14.0652	−0.580533	−0.290266	−	0.956946i	$-0.593744\pi$
$587$	−14.0652	−0.580533	−0.290266	+	0.956946i	$0.593744\pi$
$588$	0	0
$589$	1.68442	0.0694052
$590$	−0.607829	−0.0250239
$591$	0	0
$592$	5.81643	0.239054
$593$	−13.3411	−0.547855	−0.273927	−	0.961750i	$-0.588323\pi$
$593$	−13.3411	−0.547855	−0.273927	+	0.961750i	$0.588323\pi$
$594$	0	0
$595$	6.11891	0.250851
$596$	−22.3960	−0.917378
$597$	0	0
$598$	12.8221	0.524337
$599$	−3.50127	−0.143058	−0.0715290	−	0.997439i	$-0.522788\pi$
$599$	−3.50127	−0.143058	−0.0715290	+	0.997439i	$0.522788\pi$
$600$	0	0
$601$	−21.8698	−0.892087	−0.446043	−	0.895011i	$-0.647167\pi$
$601$	−21.8698	−0.892087	−0.446043	+	0.895011i	$0.647167\pi$
$602$	76.8055	3.13036
$603$	0	0
$604$	−11.6778	−0.475161
$605$	−2.18891	−0.0889920
$606$	0	0
$607$	4.28621	0.173972	0.0869859	−	0.996210i	$-0.472276\pi$
$607$	4.28621	0.173972	0.0869859	+	0.996210i	$0.472276\pi$
$608$	4.04156	0.163907
$609$	0	0
$610$	−4.75162	−0.192387
$611$	−3.67877	−0.148827
$612$	0	0
$613$	22.6195	0.913592	0.456796	−	0.889572i	$-0.348997\pi$
$613$	22.6195	0.913592	0.456796	+	0.889572i	$0.348997\pi$
$614$	−51.8289	−2.09164
$615$	0	0
$616$	−5.84041	−0.235317
$617$	30.8333	1.24130	0.620650	−	0.784088i	$-0.286869\pi$
$617$	30.8333	1.24130	0.620650	+	0.784088i	$0.286869\pi$
$618$	0	0
$619$	−36.8754	−1.48215	−0.741074	−	0.671423i	$-0.765683\pi$
$619$	−36.8754	−1.48215	−0.741074	+	0.671423i	$0.765683\pi$
$620$	18.7600	0.753420
$621$	0	0
$622$	−48.4263	−1.94172
$623$	−41.3642	−1.65722
$624$	0	0
$625$	−23.9132	−0.956526
$626$	71.1732	2.84465
$627$	0	0
$628$	−45.5133	−1.81618
$629$	2.08107	0.0829775
$630$	0	0
$631$	−41.6108	−1.65650	−0.828249	−	0.560360i	$-0.810663\pi$
$631$	−41.6108	−1.65650	−0.828249	+	0.560360i	$0.810663\pi$
$632$	−16.3738	−0.651317
$633$	0	0
$634$	57.8772	2.29860
$635$	−17.5528	−0.696561
$636$	0	0
$637$	−20.4160	−0.808913
$638$	18.0344	0.713987
$639$	0	0
$640$	25.3573	1.00233
$641$	31.8655	1.25861	0.629305	−	0.777158i	$-0.283339\pi$
$641$	31.8655	1.25861	0.629305	+	0.777158i	$0.283339\pi$
$642$	0	0
$643$	32.2883	1.27333	0.636663	−	0.771142i	$-0.280314\pi$
$643$	32.2883	1.27333	0.636663	+	0.771142i	$0.280314\pi$
$644$	−21.5495	−0.849169
$645$	0	0
$646$	0.856289	0.0336902
$647$	3.24980	0.127763	0.0638815	−	0.997957i	$-0.479652\pi$
$647$	3.24980	0.127763	0.0638815	+	0.997957i	$0.479652\pi$
$648$	0	0
$649$	−0.127920	−0.00502132
$650$	1.27205	0.0498937
$651$	0	0
$652$	−31.8594	−1.24771
$653$	−11.9774	−0.468711	−0.234355	−	0.972151i	$-0.575298\pi$
$653$	−11.9774	−0.468711	−0.234355	+	0.972151i	$0.575298\pi$
$654$	0	0
$655$	−24.6476	−0.963061
$656$	16.9509	0.661822
$657$	0	0
$658$	10.7418	0.418761
$659$	15.2638	0.594594	0.297297	−	0.954785i	$-0.403915\pi$
$659$	15.2638	0.594594	0.297297	+	0.954785i	$0.403915\pi$
$660$	0	0
$661$	−25.2153	−0.980761	−0.490380	−	0.871508i	$-0.663142\pi$
$661$	−25.2153	−0.980761	−0.490380	+	0.871508i	$0.663142\pi$
$662$	−32.1023	−1.24769
$663$	0	0
$664$	−6.76408	−0.262497
$665$	4.40767	0.170922
$666$	0	0
$667$	17.4738	0.676590
$668$	13.2254	0.511705
$669$	0	0
$670$	0.833365	0.0321957
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−13.9437	−0.537488	−0.268744	−	0.963212i	$-0.586609\pi$
$673$	−13.9437	−0.537488	−0.268744	+	0.963212i	$0.586609\pi$
$674$	18.6718	0.719210
$675$	0	0
$676$	−13.8686	−0.533406
$677$	38.0344	1.46178	0.730891	−	0.682494i	$-0.239105\pi$
$677$	38.0344	1.46178	0.730891	+	0.682494i	$0.239105\pi$
$678$	0	0
$679$	16.9100	0.648946
$680$	2.50437	0.0960381
$681$	0	0
$682$	6.85948	0.262663
$683$	−24.2537	−0.928040	−0.464020	−	0.885825i	$-0.653594\pi$
$683$	−24.2537	−0.928040	−0.464020	+	0.885825i	$0.653594\pi$
$684$	0	0
$685$	9.89492	0.378066
$686$	2.21273	0.0844823
$687$	0	0
$688$	−19.3721	−0.738553
$689$	−12.4892	−0.475802
$690$	0	0
$691$	29.6425	1.12765	0.563826	−	0.825894i	$-0.309329\pi$
$691$	29.6425	1.12765	0.563826	+	0.825894i	$0.309329\pi$
$692$	−23.9814	−0.911637
$693$	0	0
$694$	−48.1302	−1.82700
$695$	2.01684	0.0765031
$696$	0	0
$697$	6.06488	0.229724
$698$	−10.8776	−0.411722
$699$	0	0
$700$	−2.13786	−0.0808035
$701$	23.5394	0.889072	0.444536	−	0.895761i	$-0.353369\pi$
$701$	23.5394	0.889072	0.444536	+	0.895761i	$0.353369\pi$
$702$	0	0
$703$	1.49907	0.0565384
$704$	12.3220	0.464403
$705$	0	0
$706$	31.3021	1.17807
$707$	41.0104	1.54235
$708$	0	0
$709$	9.68253	0.363635	0.181817	−	0.983332i	$-0.441802\pi$
$709$	9.68253	0.363635	0.181817	+	0.983332i	$0.441802\pi$
$710$	−0.371579	−0.0139451
$711$	0	0
$712$	−16.9297	−0.634467
$713$	6.64629	0.248905
$714$	0	0
$715$	−6.14717	−0.229891
$716$	−29.6451	−1.10789
$717$	0	0
$718$	71.1233	2.65430
$719$	26.2464	0.978827	0.489414	−	0.872052i	$-0.337211\pi$
$719$	26.2464	0.978827	0.489414	+	0.872052i	$0.337211\pi$
$720$	0	0
$721$	−25.2040	−0.938645
$722$	−40.6278	−1.51201
$723$	0	0
$724$	38.7517	1.44020
$725$	1.73353	0.0643815
$726$	0	0
$727$	32.9897	1.22352	0.611760	−	0.791043i	$-0.290462\pi$
$727$	32.9897	1.22352	0.611760	+	0.791043i	$0.290462\pi$
$728$	−16.4017	−0.607889
$729$	0	0
$730$	0.270168	0.00999938
$731$	−6.93115	−0.256358
$732$	0	0
$733$	52.5254	1.94007	0.970035	−	0.242966i	$-0.0781205\pi$
$733$	52.5254	1.94007	0.970035	+	0.242966i	$0.0781205\pi$
$734$	59.6769	2.20271
$735$	0	0
$736$	15.9470	0.587814
$737$	0.175386	0.00646041
$738$	0	0
$739$	−5.52992	−0.203421	−0.101711	−	0.994814i	$-0.532432\pi$
$739$	−5.52992	−0.203421	−0.101711	+	0.994814i	$0.532432\pi$
$740$	16.6957	0.613747
$741$	0	0
$742$	36.4680	1.33878
$743$	−40.3659	−1.48088	−0.740440	−	0.672123i	$-0.765383\pi$
$743$	−40.3659	−1.48088	−0.740440	+	0.672123i	$0.765383\pi$
$744$	0	0
$745$	18.0748	0.662209
$746$	−37.1885	−1.36157
$747$	0	0
$748$	2.00707	0.0733857
$749$	−36.4797	−1.33294
$750$	0	0
$751$	5.46383	0.199378	0.0996889	−	0.995019i	$-0.468215\pi$
$751$	5.46383	0.199378	0.0996889	+	0.995019i	$0.468215\pi$
$752$	−2.70934	−0.0987993
$753$	0	0
$754$	50.6462	1.84443
$755$	9.42456	0.342995
$756$	0	0
$757$	24.1846	0.879005	0.439503	−	0.898241i	$-0.355155\pi$
$757$	24.1846	0.879005	0.439503	+	0.898241i	$0.355155\pi$
$758$	−8.58265	−0.311736
$759$	0	0
$760$	1.80399	0.0654375
$761$	−0.558992	−0.0202634	−0.0101317	−	0.999949i	$-0.503225\pi$
$761$	−0.558992	−0.0202634	−0.0101317	+	0.999949i	$0.503225\pi$
$762$	0	0
$763$	10.3623	0.375139
$764$	−35.7902	−1.29484
$765$	0	0
$766$	−17.4619	−0.630924
$767$	−0.359242	−0.0129715
$768$	0	0
$769$	8.51951	0.307221	0.153611	−	0.988131i	$-0.450910\pi$
$769$	8.51951	0.307221	0.153611	+	0.988131i	$0.450910\pi$
$770$	17.9495	0.646853
$771$	0	0
$772$	−49.9094	−1.79628
$773$	−34.6086	−1.24479	−0.622393	−	0.782705i	$-0.713839\pi$
$773$	−34.6086	−1.24479	−0.622393	+	0.782705i	$0.713839\pi$
$774$	0	0
$775$	0.659358	0.0236848
$776$	6.92098	0.248449
$777$	0	0
$778$	80.1337	2.87293
$779$	4.36875	0.156527
$780$	0	0
$781$	−0.0782005	−0.00279823
$782$	3.37870	0.120822
$783$	0	0
$784$	−15.0360	−0.536999
$785$	36.7316	1.31101
$786$	0	0
$787$	−2.55675	−0.0911382	−0.0455691	−	0.998961i	$-0.514510\pi$
$787$	−2.55675	−0.0911382	−0.0455691	+	0.998961i	$0.514510\pi$
$788$	−1.87660	−0.0668511
$789$	0	0
$790$	50.3221	1.79038
$791$	30.7947	1.09493
$792$	0	0
$793$	−2.80832	−0.0997264
$794$	12.3992	0.440033
$795$	0	0
$796$	−33.7537	−1.19637
$797$	46.9340	1.66249	0.831244	−	0.555908i	$-0.187629\pi$
$797$	46.9340	1.66249	0.831244	+	0.555908i	$0.187629\pi$
$798$	0	0
$799$	−0.969376	−0.0342940
$800$	1.58205	0.0559340
$801$	0	0
$802$	16.5088	0.582947
$803$	0.0568582	0.00200648
$804$	0	0
$805$	17.3916	0.612972
$806$	19.2636	0.678532
$807$	0	0
$808$	16.7849	0.590490
$809$	−37.5492	−1.32016	−0.660080	−	0.751195i	$-0.729478\pi$
$809$	−37.5492	−1.32016	−0.660080	+	0.751195i	$0.729478\pi$
$810$	0	0
$811$	−33.9404	−1.19181	−0.595904	−	0.803055i	$-0.703206\pi$
$811$	−33.9404	−1.19181	−0.595904	+	0.803055i	$0.703206\pi$
$812$	−85.1184	−2.98707
$813$	0	0
$814$	6.10468	0.213969
$815$	25.7122	0.900660
$816$	0	0
$817$	−4.99276	−0.174675
$818$	22.7918	0.796895
$819$	0	0
$820$	48.6566	1.69916
$821$	−16.0170	−0.558996	−0.279498	−	0.960146i	$-0.590168\pi$
$821$	−16.0170	−0.558996	−0.279498	+	0.960146i	$0.590168\pi$
$822$	0	0
$823$	22.9171	0.798839	0.399420	−	0.916768i	$-0.369212\pi$
$823$	22.9171	0.798839	0.399420	+	0.916768i	$0.369212\pi$
$824$	−10.3156	−0.359360
$825$	0	0
$826$	1.04897	0.0364983
$827$	−40.3625	−1.40354	−0.701771	−	0.712403i	$-0.747607\pi$
$827$	−40.3625	−1.40354	−0.701771	+	0.712403i	$0.747607\pi$
$828$	0	0
$829$	−20.5565	−0.713957	−0.356978	−	0.934113i	$-0.616193\pi$
$829$	−20.5565	−0.713957	−0.356978	+	0.934113i	$0.616193\pi$
$830$	20.7882	0.721568
$831$	0	0
$832$	34.6041	1.19968
$833$	−5.37973	−0.186397
$834$	0	0
$835$	−10.6736	−0.369374
$836$	1.44577	0.0500028
$837$	0	0
$838$	57.7582	1.99522
$839$	16.3332	0.563885	0.281942	−	0.959431i	$-0.409021\pi$
$839$	16.3332	0.563885	0.281942	+	0.959431i	$0.409021\pi$
$840$	0	0
$841$	40.0199	1.38000
$842$	−29.6669	−1.02239
$843$	0	0
$844$	11.1496	0.383784
$845$	11.1927	0.385039
$846$	0	0
$847$	3.77754	0.129798
$848$	−9.19806	−0.315863
$849$	0	0
$850$	0.335191	0.0114969
$851$	5.91495	0.202762
$852$	0	0
$853$	41.9773	1.43727	0.718637	−	0.695385i	$-0.244766\pi$
$853$	41.9773	1.43727	0.718637	+	0.695385i	$0.244766\pi$
$854$	8.20017	0.280604
$855$	0	0
$856$	−14.9305	−0.510315
$857$	44.1272	1.50736	0.753679	−	0.657243i	$-0.228278\pi$
$857$	44.1272	1.50736	0.753679	+	0.657243i	$0.228278\pi$
$858$	0	0
$859$	48.2412	1.64597	0.822984	−	0.568065i	$-0.192308\pi$
$859$	48.2412	1.64597	0.822984	+	0.568065i	$0.192308\pi$
$860$	−55.6063	−1.89616
$861$	0	0
$862$	67.5861	2.30199
$863$	26.5101	0.902414	0.451207	−	0.892419i	$-0.350994\pi$
$863$	26.5101	0.902414	0.451207	+	0.892419i	$0.350994\pi$
$864$	0	0
$865$	19.3543	0.658065
$866$	−39.6421	−1.34709
$867$	0	0
$868$	−32.3753	−1.09889
$869$	10.5905	0.359258
$870$	0	0
$871$	0.492539	0.0166890
$872$	4.24110	0.143622
$873$	0	0
$874$	2.43380	0.0823246
$875$	43.0689	1.45600
$876$	0	0
$877$	−39.6832	−1.34001	−0.670004	−	0.742358i	$-0.733707\pi$
$877$	−39.6832	−1.34001	−0.670004	+	0.742358i	$0.733707\pi$
$878$	−63.7471	−2.15136
$879$	0	0
$880$	−4.52726	−0.152614
$881$	−29.6233	−0.998034	−0.499017	−	0.866592i	$-0.666305\pi$
$881$	−29.6233	−0.998034	−0.499017	+	0.866592i	$0.666305\pi$
$882$	0	0
$883$	−16.2386	−0.546473	−0.273236	−	0.961947i	$-0.588094\pi$
$883$	−16.2386	−0.546473	−0.273236	+	0.961947i	$0.588094\pi$
$884$	5.63649	0.189576
$885$	0	0
$886$	−19.4537	−0.653560
$887$	7.54951	0.253488	0.126744	−	0.991935i	$-0.459547\pi$
$887$	7.54951	0.253488	0.126744	+	0.991935i	$0.459547\pi$
$888$	0	0
$889$	30.2919	1.01596
$890$	52.0303	1.74406
$891$	0	0
$892$	−13.5099	−0.452347
$893$	−0.698277	−0.0233669
$894$	0	0
$895$	23.9252	0.799730
$896$	−43.7606	−1.46194
$897$	0	0
$898$	−34.6662	−1.15683
$899$	26.2522	0.875559
$900$	0	0
$901$	−3.29098	−0.109638
$902$	17.7910	0.592374
$903$	0	0
$904$	12.6038	0.419196
$905$	−31.2746	−1.03960
$906$	0	0
$907$	−42.3589	−1.40650	−0.703252	−	0.710940i	$-0.748270\pi$
$907$	−42.3589	−1.40650	−0.703252	+	0.710940i	$0.748270\pi$
$908$	48.3343	1.60403
$909$	0	0
$910$	50.4078	1.67100
$911$	−21.3038	−0.705825	−0.352913	−	0.935656i	$-0.614809\pi$
$911$	−21.3038	−0.705825	−0.352913	+	0.935656i	$0.614809\pi$
$912$	0	0
$913$	4.37497	0.144790
$914$	−8.71854	−0.288384
$915$	0	0
$916$	54.3859	1.79696
$917$	42.5359	1.40466
$918$	0	0
$919$	18.0951	0.596902	0.298451	−	0.954425i	$-0.403530\pi$
$919$	18.0951	0.596902	0.298451	+	0.954425i	$0.403530\pi$
$920$	7.11808	0.234676
$921$	0	0
$922$	32.5198	1.07098
$923$	−0.219612	−0.00722862
$924$	0	0
$925$	0.586804	0.0192940
$926$	33.7937	1.11053
$927$	0	0
$928$	62.9891	2.06772
$929$	−7.45873	−0.244713	−0.122357	−	0.992486i	$-0.539045\pi$
$929$	−7.45873	−0.244713	−0.122357	+	0.992486i	$0.539045\pi$
$930$	0	0
$931$	−3.87522	−0.127005
$932$	−38.3634	−1.25663
$933$	0	0
$934$	−36.7142	−1.20133
$935$	−1.61981	−0.0529735
$936$	0	0
$937$	−4.65256	−0.151993	−0.0759963	−	0.997108i	$-0.524214\pi$
$937$	−4.65256	−0.151993	−0.0759963	+	0.997108i	$0.524214\pi$
$938$	−1.43819	−0.0469586
$939$	0	0
$940$	−7.77699	−0.253657
$941$	34.8368	1.13565	0.567823	−	0.823150i	$-0.307786\pi$
$941$	34.8368	1.13565	0.567823	+	0.823150i	$0.307786\pi$
$942$	0	0
$943$	17.2380	0.561347
$944$	−0.264574	−0.00861114
$945$	0	0
$946$	−20.3321	−0.661054
$947$	14.6230	0.475185	0.237593	−	0.971365i	$-0.423642\pi$
$947$	14.6230	0.475185	0.237593	+	0.971365i	$0.423642\pi$
$948$	0	0
$949$	0.159676	0.00518330
$950$	0.241450	0.00783368
$951$	0	0
$952$	−4.32194	−0.140075
$953$	38.1011	1.23422	0.617109	−	0.786878i	$-0.288304\pi$
$953$	38.1011	1.23422	0.617109	+	0.786878i	$0.288304\pi$
$954$	0	0
$955$	28.8845	0.934682
$956$	−20.2373	−0.654522
$957$	0	0
$958$	20.5906	0.665254
$959$	−17.0763	−0.551422
$960$	0	0
$961$	−21.0148	−0.677897
$962$	17.1439	0.552742
$963$	0	0
$964$	30.2079	0.972931
$965$	40.2795	1.29664
$966$	0	0
$967$	−45.6234	−1.46715	−0.733575	−	0.679609i	$-0.762150\pi$
$967$	−45.6234	−1.46715	−0.733575	+	0.679609i	$0.762150\pi$
$968$	1.54609	0.0496931
$969$	0	0
$970$	−21.2704	−0.682951
$971$	−52.7906	−1.69413	−0.847066	−	0.531488i	$-0.821633\pi$
$971$	−52.7906	−1.69413	−0.847066	+	0.531488i	$0.821633\pi$
$972$	0	0
$973$	−3.48059	−0.111583
$974$	−73.5185	−2.35568
$975$	0	0
$976$	−2.06827	−0.0662036
$977$	61.9976	1.98348	0.991739	−	0.128269i	$-0.0409421\pi$
$977$	61.9976	1.98348	0.991739	+	0.128269i	$0.0409421\pi$
$978$	0	0
$979$	10.9500	0.349964
$980$	−43.1598	−1.37869
$981$	0	0
$982$	−1.65106	−0.0526876
$983$	−48.0026	−1.53104	−0.765522	−	0.643410i	$-0.777519\pi$
$983$	−48.0026	−1.53104	−0.765522	+	0.643410i	$0.777519\pi$
$984$	0	0
$985$	1.51452	0.0482565
$986$	13.3455	0.425009
$987$	0	0
$988$	4.06017	0.129171
$989$	−19.7002	−0.626429
$990$	0	0
$991$	−18.3616	−0.583276	−0.291638	−	0.956529i	$-0.594200\pi$
$991$	−18.3616	−0.583276	−0.291638	+	0.956529i	$0.594200\pi$
$992$	23.9583	0.760677
$993$	0	0
$994$	0.641257	0.0203394
$995$	27.2410	0.863597
$996$	0	0
$997$	1.54992	0.0490864	0.0245432	−	0.999699i	$-0.492187\pi$
$997$	1.54992	0.0490864	0.0245432	+	0.999699i	$0.492187\pi$
$998$	88.5047	2.80157
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.12		13
3.2	odd	2		2013.2.a.g.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.2	✓	13	3.2	odd	2
6039.2.a.h.1.12		13	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)

\(f(q)\)	\(=\)	\(q+2.17077 q^{2} +2.71223 q^{4} -2.18891 q^{5} +3.77754 q^{7} +1.54609 q^{8} +O(q^{10})\)	\(q+2.17077 q^{2} +2.71223 q^{4} -2.18891 q^{5} +3.77754 q^{7} +1.54609 q^{8} -4.75162 q^{10} -1.00000 q^{11} -2.80832 q^{13} +8.20017 q^{14} -2.06827 q^{16} -0.740007 q^{17} -0.533054 q^{19} -5.93684 q^{20} -2.17077 q^{22} -2.10330 q^{23} -0.208662 q^{25} -6.09621 q^{26} +10.2456 q^{28} -8.30782 q^{29} -3.15993 q^{31} -7.58190 q^{32} -1.60638 q^{34} -8.26871 q^{35} -2.81222 q^{37} -1.15714 q^{38} -3.38425 q^{40} -8.19571 q^{41} +9.36633 q^{43} -2.71223 q^{44} -4.56577 q^{46} +1.30995 q^{47} +7.26984 q^{49} -0.452956 q^{50} -7.61681 q^{52} +4.44723 q^{53} +2.18891 q^{55} +5.84041 q^{56} -18.0344 q^{58} +0.127920 q^{59} +1.00000 q^{61} -6.85948 q^{62} -12.3220 q^{64} +6.14717 q^{65} -0.175386 q^{67} -2.00707 q^{68} -17.9495 q^{70} +0.0782005 q^{71} -0.0568582 q^{73} -6.10468 q^{74} -1.44577 q^{76} -3.77754 q^{77} -10.5905 q^{79} +4.52726 q^{80} -17.7910 q^{82} -4.37497 q^{83} +1.61981 q^{85} +20.3321 q^{86} -1.54609 q^{88} -10.9500 q^{89} -10.6086 q^{91} -5.70463 q^{92} +2.84361 q^{94} +1.16681 q^{95} +4.47645 q^{97} +15.7811 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more