LMFDB - Embedded newform 6039.2.a.i.1.3

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:	$13$
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} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
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			1.3
Root			$2.14727$ 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)

$f(q)$	$=$	$q-2.14727 q^{2} +2.61077 q^{4} +3.62125 q^{5} +2.48742 q^{7} -1.31149 q^{8} +O(q^{10})$	\(q-2.14727 q^{2} +2.61077 q^{4} +3.62125 q^{5} +2.48742 q^{7} -1.31149 q^{8} -7.77580 q^{10} -1.00000 q^{11} +0.181722 q^{13} -5.34117 q^{14} -2.40542 q^{16} +1.02658 q^{17} +8.45421 q^{19} +9.45425 q^{20} +2.14727 q^{22} +4.42711 q^{23} +8.11345 q^{25} -0.390206 q^{26} +6.49409 q^{28} -5.84092 q^{29} +9.14893 q^{31} +7.78806 q^{32} -2.20434 q^{34} +9.00758 q^{35} +1.08915 q^{37} -18.1535 q^{38} -4.74923 q^{40} -6.75705 q^{41} +3.54999 q^{43} -2.61077 q^{44} -9.50620 q^{46} -3.99461 q^{47} -0.812722 q^{49} -17.4218 q^{50} +0.474435 q^{52} -10.0865 q^{53} -3.62125 q^{55} -3.26223 q^{56} +12.5420 q^{58} +1.12062 q^{59} -1.00000 q^{61} -19.6452 q^{62} -11.9122 q^{64} +0.658061 q^{65} -2.55535 q^{67} +2.68016 q^{68} -19.3417 q^{70} +13.1336 q^{71} +8.20991 q^{73} -2.33869 q^{74} +22.0720 q^{76} -2.48742 q^{77} +5.35933 q^{79} -8.71063 q^{80} +14.5092 q^{82} -9.24024 q^{83} +3.71749 q^{85} -7.62280 q^{86} +1.31149 q^{88} +8.09587 q^{89} +0.452020 q^{91} +11.5582 q^{92} +8.57750 q^{94} +30.6148 q^{95} +14.9871 q^{97} +1.74513 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	−2.14727	−1.51835	−0.759175	−	0.650887i	$-0.774397\pi$
$2$	−2.14727	−1.51835	−0.759175	+	0.650887i	$0.774397\pi$
$3$	0	0
$3$	0	0
$4$	2.61077	1.30539
$5$	3.62125	1.61947	0.809736	−	0.586794i	$-0.199610\pi$
$5$	3.62125	1.61947	0.809736	+	0.586794i	$0.199610\pi$
$6$	0	0
$7$	2.48742	0.940158	0.470079	−	0.882624i	$-0.344225\pi$
$7$	2.48742	0.940158	0.470079	+	0.882624i	$0.344225\pi$
$8$	−1.31149	−0.463681
$9$	0	0
$10$	−7.77580	−2.45892
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.181722	0.0504006	0.0252003	−	0.999682i	$-0.491978\pi$
$13$	0.181722	0.0504006	0.0252003	+	0.999682i	$0.491978\pi$
$14$	−5.34117	−1.42749
$15$	0	0
$16$	−2.40542	−0.601355
$17$	1.02658	0.248982	0.124491	−	0.992221i	$-0.460270\pi$
$17$	1.02658	0.248982	0.124491	+	0.992221i	$0.460270\pi$
$18$	0	0
$19$	8.45421	1.93953	0.969765	−	0.244042i	$-0.0784736\pi$
$19$	8.45421	1.93953	0.969765	+	0.244042i	$0.0784736\pi$
$20$	9.45425	2.11403
$21$	0	0
$22$	2.14727	0.457800
$23$	4.42711	0.923116	0.461558	−	0.887110i	$-0.347291\pi$
$23$	4.42711	0.923116	0.461558	+	0.887110i	$0.347291\pi$
$24$	0	0
$25$	8.11345	1.62269
$26$	−0.390206	−0.0765258
$27$	0	0
$28$	6.49409	1.22727
$29$	−5.84092	−1.08463	−0.542316	−	0.840175i	$-0.682452\pi$
$29$	−5.84092	−1.08463	−0.542316	+	0.840175i	$0.682452\pi$
$30$	0	0
$31$	9.14893	1.64320	0.821598	−	0.570067i	$-0.193083\pi$
$31$	9.14893	1.64320	0.821598	+	0.570067i	$0.193083\pi$
$32$	7.78806	1.37675
$33$	0	0
$34$	−2.20434	−0.378041
$35$	9.00758	1.52256
$36$	0	0
$37$	1.08915	0.179054	0.0895272	−	0.995984i	$-0.471464\pi$
$37$	1.08915	0.179054	0.0895272	+	0.995984i	$0.471464\pi$
$38$	−18.1535	−2.94488
$39$	0	0
$40$	−4.74923	−0.750919
$41$	−6.75705	−1.05527	−0.527637	−	0.849470i	$-0.676922\pi$
$41$	−6.75705	−1.05527	−0.527637	+	0.849470i	$0.676922\pi$
$42$	0	0
$43$	3.54999	0.541369	0.270684	−	0.962668i	$-0.412750\pi$
$43$	3.54999	0.541369	0.270684	+	0.962668i	$0.412750\pi$
$44$	−2.61077	−0.393588
$45$	0	0
$46$	−9.50620	−1.40161
$47$	−3.99461	−0.582674	−0.291337	−	0.956621i	$-0.594100\pi$
$47$	−3.99461	−0.582674	−0.291337	+	0.956621i	$0.594100\pi$
$48$	0	0
$49$	−0.812722	−0.116103
$50$	−17.4218	−2.46381
$51$	0	0
$52$	0.474435	0.0657922
$53$	−10.0865	−1.38548	−0.692741	−	0.721187i	$-0.743597\pi$
$53$	−10.0865	−1.38548	−0.692741	+	0.721187i	$0.743597\pi$
$54$	0	0
$55$	−3.62125	−0.488289
$56$	−3.26223	−0.435934
$57$	0	0
$58$	12.5420	1.64685
$59$	1.12062	0.145892	0.0729459	−	0.997336i	$-0.476760\pi$
$59$	1.12062	0.145892	0.0729459	+	0.997336i	$0.476760\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−19.6452	−2.49495
$63$	0	0
$64$	−11.9122	−1.48903
$65$	0.658061	0.0816224
$66$	0	0
$67$	−2.55535	−0.312185	−0.156093	−	0.987742i	$-0.549890\pi$
$67$	−2.55535	−0.312185	−0.156093	+	0.987742i	$0.549890\pi$
$68$	2.68016	0.325017
$69$	0	0
$70$	−19.3417	−2.31178
$71$	13.1336	1.55867	0.779336	−	0.626606i	$-0.215557\pi$
$71$	13.1336	1.55867	0.779336	+	0.626606i	$0.215557\pi$
$72$	0	0
$73$	8.20991	0.960897	0.480449	−	0.877023i	$-0.340474\pi$
$73$	8.20991	0.960897	0.480449	+	0.877023i	$0.340474\pi$
$74$	−2.33869	−0.271867
$75$	0	0
$76$	22.0720	2.53183
$77$	−2.48742	−0.283468
$78$	0	0
$79$	5.35933	0.602972	0.301486	−	0.953471i	$-0.402517\pi$
$79$	5.35933	0.602972	0.301486	+	0.953471i	$0.402517\pi$
$80$	−8.71063	−0.973878
$81$	0	0
$82$	14.5092	1.60228
$83$	−9.24024	−1.01425	−0.507124	−	0.861873i	$-0.669291\pi$
$83$	−9.24024	−1.01425	−0.507124	+	0.861873i	$0.669291\pi$
$84$	0	0
$85$	3.71749	0.403219
$86$	−7.62280	−0.821987
$87$	0	0
$88$	1.31149	0.139805
$89$	8.09587	0.858161	0.429080	−	0.903266i	$-0.358838\pi$
$89$	8.09587	0.858161	0.429080	+	0.903266i	$0.358838\pi$
$90$	0	0
$91$	0.452020	0.0473846
$92$	11.5582	1.20502
$93$	0	0
$94$	8.57750	0.884702
$95$	30.6148	3.14101
$96$	0	0
$97$	14.9871	1.52171	0.760855	−	0.648922i	$-0.224780\pi$
$97$	14.9871	1.52171	0.760855	+	0.648922i	$0.224780\pi$
$98$	1.74513	0.176285
$99$	0	0
$100$	21.1824	2.11824
$101$	−9.37540	−0.932888	−0.466444	−	0.884551i	$-0.654465\pi$
$101$	−9.37540	−0.932888	−0.466444	+	0.884551i	$0.654465\pi$
$102$	0	0
$103$	−1.55392	−0.153112	−0.0765561	−	0.997065i	$-0.524392\pi$
$103$	−1.55392	−0.153112	−0.0765561	+	0.997065i	$0.524392\pi$
$104$	−0.238326	−0.0233698
$105$	0	0
$106$	21.6583	2.10364
$107$	16.1691	1.56313	0.781564	−	0.623825i	$-0.214422\pi$
$107$	16.1691	1.56313	0.781564	+	0.623825i	$0.214422\pi$
$108$	0	0
$109$	−3.55078	−0.340103	−0.170051	−	0.985435i	$-0.554393\pi$
$109$	−3.55078	−0.340103	−0.170051	+	0.985435i	$0.554393\pi$
$110$	7.77580	0.741394
$111$	0	0
$112$	−5.98330	−0.565369
$113$	6.78075	0.637879	0.318939	−	0.947775i	$-0.396673\pi$
$113$	6.78075	0.637879	0.318939	+	0.947775i	$0.396673\pi$
$114$	0	0
$115$	16.0317	1.49496
$116$	−15.2493	−1.41586
$117$	0	0
$118$	−2.40627	−0.221515
$119$	2.55353	0.234082
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.14727	0.194405
$123$	0	0
$124$	23.8857	2.14500
$125$	11.2746	1.00843
$126$	0	0
$127$	2.97038	0.263579	0.131789	−	0.991278i	$-0.457928\pi$
$127$	2.97038	0.263579	0.131789	+	0.991278i	$0.457928\pi$
$128$	10.0027	0.884120
$129$	0	0
$130$	−1.41303	−0.123931
$131$	−12.2455	−1.06989	−0.534945	−	0.844887i	$-0.679668\pi$
$131$	−12.2455	−1.06989	−0.534945	+	0.844887i	$0.679668\pi$
$132$	0	0
$133$	21.0292	1.82346
$134$	5.48702	0.474006
$135$	0	0
$136$	−1.34634	−0.115448
$137$	−4.62204	−0.394887	−0.197444	−	0.980314i	$-0.563264\pi$
$137$	−4.62204	−0.394887	−0.197444	+	0.980314i	$0.563264\pi$
$138$	0	0
$139$	−18.2752	−1.55008	−0.775040	−	0.631912i	$-0.782270\pi$
$139$	−18.2752	−1.55008	−0.775040	+	0.631912i	$0.782270\pi$
$140$	23.5167	1.98753
$141$	0	0
$142$	−28.2014	−2.36661
$143$	−0.181722	−0.0151964
$144$	0	0
$145$	−21.1514	−1.75653
$146$	−17.6289	−1.45898
$147$	0	0
$148$	2.84351	0.233735
$149$	19.4843	1.59622	0.798109	−	0.602513i	$-0.205834\pi$
$149$	19.4843	1.59622	0.798109	+	0.602513i	$0.205834\pi$
$150$	0	0
$151$	−22.0551	−1.79482	−0.897408	−	0.441202i	$-0.854552\pi$
$151$	−22.0551	−1.79482	−0.897408	+	0.441202i	$0.854552\pi$
$152$	−11.0876	−0.899323
$153$	0	0
$154$	5.34117	0.430404
$155$	33.1306	2.66111
$156$	0	0
$157$	−12.2301	−0.976071	−0.488035	−	0.872824i	$-0.662286\pi$
$157$	−12.2301	−0.976071	−0.488035	+	0.872824i	$0.662286\pi$
$158$	−11.5079	−0.915522
$159$	0	0
$160$	28.2025	2.22961
$161$	11.0121	0.867875
$162$	0	0
$163$	20.2586	1.58677	0.793387	−	0.608718i	$-0.208316\pi$
$163$	20.2586	1.58677	0.793387	+	0.608718i	$0.208316\pi$
$164$	−17.6411	−1.37754
$165$	0	0
$166$	19.8413	1.53998
$167$	−1.97347	−0.152712	−0.0763558	−	0.997081i	$-0.524328\pi$
$167$	−1.97347	−0.152712	−0.0763558	+	0.997081i	$0.524328\pi$
$168$	0	0
$169$	−12.9670	−0.997460
$170$	−7.98246	−0.612227
$171$	0	0
$172$	9.26822	0.706695
$173$	−15.7980	−1.20110	−0.600549	−	0.799588i	$-0.705051\pi$
$173$	−15.7980	−1.20110	−0.600549	+	0.799588i	$0.705051\pi$
$174$	0	0
$175$	20.1816	1.52558
$176$	2.40542	0.181315
$177$	0	0
$178$	−17.3840	−1.30299
$179$	−3.95408	−0.295542	−0.147771	−	0.989022i	$-0.547210\pi$
$179$	−3.95408	−0.295542	−0.147771	+	0.989022i	$0.547210\pi$
$180$	0	0
$181$	21.0521	1.56479	0.782397	−	0.622780i	$-0.213997\pi$
$181$	21.0521	1.56479	0.782397	+	0.622780i	$0.213997\pi$
$182$	−0.970609	−0.0719463
$183$	0	0
$184$	−5.80610	−0.428031
$185$	3.94407	0.289974
$186$	0	0
$187$	−1.02658	−0.0750708
$188$	−10.4290	−0.760613
$189$	0	0
$190$	−65.7383	−4.76916
$191$	16.1292	1.16707	0.583536	−	0.812088i	$-0.301669\pi$
$191$	16.1292	1.16707	0.583536	+	0.812088i	$0.301669\pi$
$192$	0	0
$193$	16.0649	1.15638	0.578189	−	0.815903i	$-0.303760\pi$
$193$	16.0649	1.15638	0.578189	+	0.815903i	$0.303760\pi$
$194$	−32.1814	−2.31049
$195$	0	0
$196$	−2.12183	−0.151559
$197$	−10.7311	−0.764558	−0.382279	−	0.924047i	$-0.624861\pi$
$197$	−10.7311	−0.764558	−0.382279	+	0.924047i	$0.624861\pi$
$198$	0	0
$199$	−1.75556	−0.124448	−0.0622242	−	0.998062i	$-0.519819\pi$
$199$	−1.75556	−0.124448	−0.0622242	+	0.998062i	$0.519819\pi$
$200$	−10.6407	−0.752411
$201$	0	0
$202$	20.1315	1.41645
$203$	−14.5288	−1.01973
$204$	0	0
$205$	−24.4690	−1.70899
$206$	3.33669	0.232478
$207$	0	0
$208$	−0.437118	−0.0303087
$209$	−8.45421	−0.584790
$210$	0	0
$211$	−4.40424	−0.303201	−0.151600	−	0.988442i	$-0.548443\pi$
$211$	−4.40424	−0.303201	−0.151600	+	0.988442i	$0.548443\pi$
$212$	−26.3334	−1.80859
$213$	0	0
$214$	−34.7195	−2.37337
$215$	12.8554	0.876732
$216$	0	0
$217$	22.7573	1.54486
$218$	7.62448	0.516395
$219$	0	0
$220$	−9.45425	−0.637405
$221$	0.186552	0.0125488
$222$	0	0
$223$	−9.81727	−0.657413	−0.328707	−	0.944432i	$-0.606613\pi$
$223$	−9.81727	−0.657413	−0.328707	+	0.944432i	$0.606613\pi$
$224$	19.3722	1.29436
$225$	0	0
$226$	−14.5601	−0.968523
$227$	−18.5665	−1.23230	−0.616151	−	0.787628i	$-0.711309\pi$
$227$	−18.5665	−1.23230	−0.616151	+	0.787628i	$0.711309\pi$
$228$	0	0
$229$	−18.6934	−1.23529	−0.617647	−	0.786455i	$-0.711914\pi$
$229$	−18.6934	−1.23529	−0.617647	+	0.786455i	$0.711914\pi$
$230$	−34.4243	−2.26987
$231$	0	0
$232$	7.66030	0.502924
$233$	13.1659	0.862524	0.431262	−	0.902227i	$-0.358068\pi$
$233$	13.1659	0.862524	0.431262	+	0.902227i	$0.358068\pi$
$234$	0	0
$235$	−14.4655	−0.943624
$236$	2.92567	0.190445
$237$	0	0
$238$	−5.48313	−0.355418
$239$	18.0130	1.16516	0.582581	−	0.812772i	$-0.302043\pi$
$239$	18.0130	1.16516	0.582581	+	0.812772i	$0.302043\pi$
$240$	0	0
$241$	−28.6918	−1.84820	−0.924102	−	0.382146i	$-0.875185\pi$
$241$	−28.6918	−1.84820	−0.924102	+	0.382146i	$0.875185\pi$
$242$	−2.14727	−0.138032
$243$	0	0
$244$	−2.61077	−0.167137
$245$	−2.94307	−0.188026
$246$	0	0
$247$	1.53632	0.0977535
$248$	−11.9987	−0.761919
$249$	0	0
$250$	−24.2096	−1.53115
$251$	1.25826	0.0794204	0.0397102	−	0.999211i	$-0.487357\pi$
$251$	1.25826	0.0794204	0.0397102	+	0.999211i	$0.487357\pi$
$252$	0	0
$253$	−4.42711	−0.278330
$254$	−6.37821	−0.400204
$255$	0	0
$256$	2.34604	0.146627
$257$	10.9135	0.680763	0.340382	−	0.940287i	$-0.389444\pi$
$257$	10.9135	0.680763	0.340382	+	0.940287i	$0.389444\pi$
$258$	0	0
$259$	2.70917	0.168339
$260$	1.71805	0.106549
$261$	0	0
$262$	26.2943	1.62447
$263$	−21.9507	−1.35354	−0.676768	−	0.736197i	$-0.736620\pi$
$263$	−21.9507	−1.35354	−0.676768	+	0.736197i	$0.736620\pi$
$264$	0	0
$265$	−36.5256	−2.24375
$266$	−45.1554	−2.76865
$267$	0	0
$268$	−6.67142	−0.407522
$269$	1.26259	0.0769817	0.0384909	−	0.999259i	$-0.487745\pi$
$269$	1.26259	0.0769817	0.0384909	+	0.999259i	$0.487745\pi$
$270$	0	0
$271$	1.70719	0.103704	0.0518522	−	0.998655i	$-0.483488\pi$
$271$	1.70719	0.103704	0.0518522	+	0.998655i	$0.483488\pi$
$272$	−2.46935	−0.149726
$273$	0	0
$274$	9.92477	0.599577
$275$	−8.11345	−0.489259
$276$	0	0
$277$	−0.510990	−0.0307024	−0.0153512	−	0.999882i	$-0.504887\pi$
$277$	−0.510990	−0.0307024	−0.0153512	+	0.999882i	$0.504887\pi$
$278$	39.2417	2.35356
$279$	0	0
$280$	−11.8133	−0.705982
$281$	16.3798	0.977135	0.488567	−	0.872526i	$-0.337520\pi$
$281$	16.3798	0.977135	0.488567	+	0.872526i	$0.337520\pi$
$282$	0	0
$283$	−14.7590	−0.877332	−0.438666	−	0.898650i	$-0.644549\pi$
$283$	−14.7590	−0.877332	−0.438666	+	0.898650i	$0.644549\pi$
$284$	34.2888	2.03467
$285$	0	0
$286$	0.390206	0.0230734
$287$	−16.8077	−0.992125
$288$	0	0
$289$	−15.9461	−0.938008
$290$	45.4179	2.66703
$291$	0	0
$292$	21.4342	1.25434
$293$	−11.5784	−0.676417	−0.338208	−	0.941071i	$-0.609821\pi$
$293$	−11.5784	−0.676417	−0.338208	+	0.941071i	$0.609821\pi$
$294$	0	0
$295$	4.05803	0.236268
$296$	−1.42840	−0.0830242
$297$	0	0
$298$	−41.8381	−2.42362
$299$	0.804503	0.0465256
$300$	0	0
$301$	8.83034	0.508972
$302$	47.3582	2.72516
$303$	0	0
$304$	−20.3359	−1.16635
$305$	−3.62125	−0.207352
$306$	0	0
$307$	−26.5991	−1.51809	−0.759047	−	0.651036i	$-0.774335\pi$
$307$	−26.5991	−1.51809	−0.759047	+	0.651036i	$0.774335\pi$
$308$	−6.49409	−0.370035
$309$	0	0
$310$	−71.1403	−4.04049
$311$	20.9090	1.18564	0.592819	−	0.805336i	$-0.298015\pi$
$311$	20.9090	1.18564	0.592819	+	0.805336i	$0.298015\pi$
$312$	0	0
$313$	24.9511	1.41032	0.705160	−	0.709048i	$-0.250875\pi$
$313$	24.9511	1.41032	0.705160	+	0.709048i	$0.250875\pi$
$314$	26.2614	1.48202
$315$	0	0
$316$	13.9920	0.787111
$317$	−15.8388	−0.889596	−0.444798	−	0.895631i	$-0.646724\pi$
$317$	−15.8388	−0.889596	−0.444798	+	0.895631i	$0.646724\pi$
$318$	0	0
$319$	5.84092	0.327029
$320$	−43.1372	−2.41144
$321$	0	0
$322$	−23.6459	−1.31774
$323$	8.67890	0.482907
$324$	0	0
$325$	1.47439	0.0817846
$326$	−43.5006	−2.40928
$327$	0	0
$328$	8.86180	0.489311
$329$	−9.93629	−0.547805
$330$	0	0
$331$	16.5911	0.911928	0.455964	−	0.889998i	$-0.349294\pi$
$331$	16.5911	0.911928	0.455964	+	0.889998i	$0.349294\pi$
$332$	−24.1241	−1.32398
$333$	0	0
$334$	4.23757	0.231870
$335$	−9.25355	−0.505575
$336$	0	0
$337$	15.9284	0.867673	0.433836	−	0.900992i	$-0.357160\pi$
$337$	15.9284	0.867673	0.433836	+	0.900992i	$0.357160\pi$
$338$	27.8436	1.51449
$339$	0	0
$340$	9.70552	0.526356
$341$	−9.14893	−0.495442
$342$	0	0
$343$	−19.4336	−1.04931
$344$	−4.65578	−0.251023
$345$	0	0
$346$	33.9225	1.82369
$347$	−10.0215	−0.537983	−0.268991	−	0.963143i	$-0.586690\pi$
$347$	−10.0215	−0.537983	−0.268991	+	0.963143i	$0.586690\pi$
$348$	0	0
$349$	10.7818	0.577135	0.288567	−	0.957460i	$-0.406821\pi$
$349$	10.7818	0.577135	0.288567	+	0.957460i	$0.406821\pi$
$350$	−43.3353	−2.31637
$351$	0	0
$352$	−7.78806	−0.415105
$353$	14.5658	0.775258	0.387629	−	0.921816i	$-0.373294\pi$
$353$	14.5658	0.775258	0.387629	+	0.921816i	$0.373294\pi$
$354$	0	0
$355$	47.5601	2.52423
$356$	21.1365	1.12023
$357$	0	0
$358$	8.49049	0.448736
$359$	8.13977	0.429601	0.214800	−	0.976658i	$-0.431090\pi$
$359$	8.13977	0.429601	0.214800	+	0.976658i	$0.431090\pi$
$360$	0	0
$361$	52.4737	2.76177
$362$	−45.2046	−2.37590
$363$	0	0
$364$	1.18012	0.0618551
$365$	29.7301	1.55615
$366$	0	0
$367$	10.4128	0.543545	0.271773	−	0.962361i	$-0.412390\pi$
$367$	10.4128	0.543545	0.271773	+	0.962361i	$0.412390\pi$
$368$	−10.6491	−0.555120
$369$	0	0
$370$	−8.46898	−0.440281
$371$	−25.0893	−1.30257
$372$	0	0
$373$	14.4160	0.746434	0.373217	−	0.927744i	$-0.378255\pi$
$373$	14.4160	0.746434	0.373217	+	0.927744i	$0.378255\pi$
$374$	2.20434	0.113984
$375$	0	0
$376$	5.23888	0.270175
$377$	−1.06142	−0.0546661
$378$	0	0
$379$	12.7476	0.654802	0.327401	−	0.944886i	$-0.393827\pi$
$379$	12.7476	0.654802	0.327401	+	0.944886i	$0.393827\pi$
$380$	79.9282	4.10023
$381$	0	0
$382$	−34.6339	−1.77202
$383$	−10.1427	−0.518270	−0.259135	−	0.965841i	$-0.583437\pi$
$383$	−10.1427	−0.518270	−0.259135	+	0.965841i	$0.583437\pi$
$384$	0	0
$385$	−9.00758	−0.459069
$386$	−34.4957	−1.75579
$387$	0	0
$388$	39.1279	1.98642
$389$	−17.0449	−0.864208	−0.432104	−	0.901824i	$-0.642229\pi$
$389$	−17.0449	−0.864208	−0.432104	+	0.901824i	$0.642229\pi$
$390$	0	0
$391$	4.54477	0.229839
$392$	1.06588	0.0538348
$393$	0	0
$394$	23.0425	1.16087
$395$	19.4075	0.976496
$396$	0	0
$397$	−10.6886	−0.536444	−0.268222	−	0.963357i	$-0.586436\pi$
$397$	−10.6886	−0.536444	−0.268222	+	0.963357i	$0.586436\pi$
$398$	3.76966	0.188956
$399$	0	0
$400$	−19.5163	−0.975813
$401$	3.16010	0.157808	0.0789039	−	0.996882i	$-0.474858\pi$
$401$	3.16010	0.157808	0.0789039	+	0.996882i	$0.474858\pi$
$402$	0	0
$403$	1.66256	0.0828181
$404$	−24.4770	−1.21778
$405$	0	0
$406$	31.1974	1.54830
$407$	−1.08915	−0.0539869
$408$	0	0
$409$	−13.0980	−0.647655	−0.323828	−	0.946116i	$-0.604970\pi$
$409$	−13.0980	−0.647655	−0.323828	+	0.946116i	$0.604970\pi$
$410$	52.5415	2.59484
$411$	0	0
$412$	−4.05693	−0.199870
$413$	2.78745	0.137161
$414$	0	0
$415$	−33.4612	−1.64255
$416$	1.41526	0.0693890
$417$	0	0
$418$	18.1535	0.887916
$419$	−21.4749	−1.04912	−0.524558	−	0.851375i	$-0.675769\pi$
$419$	−21.4749	−1.04912	−0.524558	+	0.851375i	$0.675769\pi$
$420$	0	0
$421$	−14.0457	−0.684546	−0.342273	−	0.939601i	$-0.611197\pi$
$421$	−14.0457	−0.684546	−0.342273	+	0.939601i	$0.611197\pi$
$422$	9.45710	0.460364
$423$	0	0
$424$	13.2283	0.642422
$425$	8.32909	0.404020
$426$	0	0
$427$	−2.48742	−0.120375
$428$	42.2139	2.04048
$429$	0	0
$430$	−27.6041	−1.33119
$431$	−23.1921	−1.11712	−0.558561	−	0.829463i	$-0.688646\pi$
$431$	−23.1921	−1.11712	−0.558561	+	0.829463i	$0.688646\pi$
$432$	0	0
$433$	14.7223	0.707507	0.353753	−	0.935339i	$-0.384905\pi$
$433$	14.7223	0.707507	0.353753	+	0.935339i	$0.384905\pi$
$434$	−48.8660	−2.34564
$435$	0	0
$436$	−9.27027	−0.443965
$437$	37.4277	1.79041
$438$	0	0
$439$	−36.6864	−1.75095	−0.875474	−	0.483266i	$-0.839450\pi$
$439$	−36.6864	−1.75095	−0.875474	+	0.483266i	$0.839450\pi$
$440$	4.74923	0.226411
$441$	0	0
$442$	−0.400577	−0.0190535
$443$	−41.4579	−1.96972	−0.984862	−	0.173338i	$-0.944545\pi$
$443$	−41.4579	−1.96972	−0.984862	+	0.173338i	$0.944545\pi$
$444$	0	0
$445$	29.3172	1.38977
$446$	21.0803	0.998183
$447$	0	0
$448$	−29.6308	−1.39992
$449$	7.78679	0.367481	0.183741	−	0.982975i	$-0.441179\pi$
$449$	7.78679	0.367481	0.183741	+	0.982975i	$0.441179\pi$
$450$	0	0
$451$	6.75705	0.318177
$452$	17.7030	0.832678
$453$	0	0
$454$	39.8673	1.87106
$455$	1.63688	0.0767380
$456$	0	0
$457$	36.5049	1.70763	0.853814	−	0.520578i	$-0.174284\pi$
$457$	36.5049	1.70763	0.853814	+	0.520578i	$0.174284\pi$
$458$	40.1398	1.87561
$459$	0	0
$460$	41.8550	1.95150
$461$	−2.24092	−0.104370	−0.0521850	−	0.998637i	$-0.516619\pi$
$461$	−2.24092	−0.104370	−0.0521850	+	0.998637i	$0.516619\pi$
$462$	0	0
$463$	−9.86522	−0.458476	−0.229238	−	0.973370i	$-0.573623\pi$
$463$	−9.86522	−0.458476	−0.229238	+	0.973370i	$0.573623\pi$
$464$	14.0499	0.652249
$465$	0	0
$466$	−28.2707	−1.30961
$467$	8.97808	0.415456	0.207728	−	0.978187i	$-0.433393\pi$
$467$	8.97808	0.415456	0.207728	+	0.978187i	$0.433393\pi$
$468$	0	0
$469$	−6.35623	−0.293503
$470$	31.0613	1.43275
$471$	0	0
$472$	−1.46968	−0.0676473
$473$	−3.54999	−0.163229
$474$	0	0
$475$	68.5928	3.14725
$476$	6.66669	0.305567
$477$	0	0
$478$	−38.6787	−1.76912
$479$	−16.0004	−0.731078	−0.365539	−	0.930796i	$-0.619115\pi$
$479$	−16.0004	−0.731078	−0.365539	+	0.930796i	$0.619115\pi$
$480$	0	0
$481$	0.197922	0.00902446
$482$	61.6092	2.80622
$483$	0	0
$484$	2.61077	0.118671
$485$	54.2720	2.46437
$486$	0	0
$487$	3.78986	0.171735	0.0858676	−	0.996307i	$-0.472634\pi$
$487$	3.78986	0.171735	0.0858676	+	0.996307i	$0.472634\pi$
$488$	1.31149	0.0593683
$489$	0	0
$490$	6.31957	0.285489
$491$	8.31228	0.375128	0.187564	−	0.982252i	$-0.439941\pi$
$491$	8.31228	0.375128	0.187564	+	0.982252i	$0.439941\pi$
$492$	0	0
$493$	−5.99616	−0.270053
$494$	−3.29889	−0.148424
$495$	0	0
$496$	−22.0070	−0.988144
$497$	32.6688	1.46540
$498$	0	0
$499$	32.3218	1.44692	0.723461	−	0.690365i	$-0.242550\pi$
$499$	32.3218	1.44692	0.723461	+	0.690365i	$0.242550\pi$
$500$	29.4353	1.31639
$501$	0	0
$502$	−2.70182	−0.120588
$503$	5.08665	0.226803	0.113401	−	0.993549i	$-0.463825\pi$
$503$	5.08665	0.226803	0.113401	+	0.993549i	$0.463825\pi$
$504$	0	0
$505$	−33.9507	−1.51079
$506$	9.50620	0.422602
$507$	0	0
$508$	7.75498	0.344071
$509$	−4.62312	−0.204916	−0.102458	−	0.994737i	$-0.532671\pi$
$509$	−4.62312	−0.204916	−0.102458	+	0.994737i	$0.532671\pi$
$510$	0	0
$511$	20.4215	0.903395
$512$	−25.0429	−1.10675
$513$	0	0
$514$	−23.4342	−1.03364
$515$	−5.62713	−0.247961
$516$	0	0
$517$	3.99461	0.175683
$518$	−5.81731	−0.255598
$519$	0	0
$520$	−0.863040	−0.0378468
$521$	−8.80274	−0.385655	−0.192828	−	0.981233i	$-0.561766\pi$
$521$	−8.80274	−0.385655	−0.192828	+	0.981233i	$0.561766\pi$
$522$	0	0
$523$	28.2658	1.23598	0.617989	−	0.786186i	$-0.287947\pi$
$523$	28.2658	1.23598	0.617989	+	0.786186i	$0.287947\pi$
$524$	−31.9701	−1.39662
$525$	0	0
$526$	47.1340	2.05514
$527$	9.39208	0.409126
$528$	0	0
$529$	−3.40072	−0.147857
$530$	78.4303	3.40679
$531$	0	0
$532$	54.9024	2.38032
$533$	−1.22791	−0.0531865
$534$	0	0
$535$	58.5524	2.53144
$536$	3.35131	0.144754
$537$	0	0
$538$	−2.71113	−0.116885
$539$	0.812722	0.0350064
$540$	0	0
$541$	−43.2186	−1.85811	−0.929056	−	0.369940i	$-0.879378\pi$
$541$	−43.2186	−1.85811	−0.929056	+	0.369940i	$0.879378\pi$
$542$	−3.66580	−0.157459
$543$	0	0
$544$	7.99505	0.342785
$545$	−12.8583	−0.550787
$546$	0	0
$547$	−21.7658	−0.930638	−0.465319	−	0.885143i	$-0.654060\pi$
$547$	−21.7658	−0.930638	−0.465319	+	0.885143i	$0.654060\pi$
$548$	−12.0671	−0.515480
$549$	0	0
$550$	17.4218	0.742867
$551$	−49.3804	−2.10368
$552$	0	0
$553$	13.3309	0.566889
$554$	1.09723	0.0466170
$555$	0	0
$556$	−47.7123	−2.02345
$557$	33.1891	1.40627	0.703134	−	0.711057i	$-0.251783\pi$
$557$	33.1891	1.40627	0.703134	+	0.711057i	$0.251783\pi$
$558$	0	0
$559$	0.645112	0.0272853
$560$	−21.6670	−0.915599
$561$	0	0
$562$	−35.1718	−1.48363
$563$	−16.7017	−0.703892	−0.351946	−	0.936020i	$-0.614480\pi$
$563$	−16.7017	−0.703892	−0.351946	+	0.936020i	$0.614480\pi$
$564$	0	0
$565$	24.5548	1.03303
$566$	31.6916	1.33210
$567$	0	0
$568$	−17.2246	−0.722727
$569$	46.4050	1.94540	0.972700	−	0.232068i	$-0.0745493\pi$
$569$	46.4050	1.94540	0.972700	+	0.232068i	$0.0745493\pi$
$570$	0	0
$571$	5.17531	0.216580	0.108290	−	0.994119i	$-0.465462\pi$
$571$	5.17531	0.216580	0.108290	+	0.994119i	$0.465462\pi$
$572$	−0.474435	−0.0198371
$573$	0	0
$574$	36.0906	1.50639
$575$	35.9191	1.49793
$576$	0	0
$577$	11.8916	0.495055	0.247528	−	0.968881i	$-0.420382\pi$
$577$	11.8916	0.495055	0.247528	+	0.968881i	$0.420382\pi$
$578$	34.2407	1.42422
$579$	0	0
$580$	−55.2215	−2.29295
$581$	−22.9844	−0.953553
$582$	0	0
$583$	10.0865	0.417738
$584$	−10.7672	−0.445550
$585$	0	0
$586$	24.8619	1.02704
$587$	−32.9881	−1.36157	−0.680783	−	0.732485i	$-0.738360\pi$
$587$	−32.9881	−1.36157	−0.680783	+	0.732485i	$0.738360\pi$
$588$	0	0
$589$	77.3470	3.18703
$590$	−8.71369	−0.358737
$591$	0	0
$592$	−2.61985	−0.107675
$593$	−2.08016	−0.0854221	−0.0427111	−	0.999087i	$-0.513599\pi$
$593$	−2.08016	−0.0854221	−0.0427111	+	0.999087i	$0.513599\pi$
$594$	0	0
$595$	9.24698	0.379089
$596$	50.8691	2.08368
$597$	0	0
$598$	−1.72749	−0.0706421
$599$	−3.43085	−0.140181	−0.0700903	−	0.997541i	$-0.522329\pi$
$599$	−3.43085	−0.140181	−0.0700903	+	0.997541i	$0.522329\pi$
$600$	0	0
$601$	−11.5256	−0.470141	−0.235070	−	0.971978i	$-0.575532\pi$
$601$	−11.5256	−0.470141	−0.235070	+	0.971978i	$0.575532\pi$
$602$	−18.9611	−0.772798
$603$	0	0
$604$	−57.5807	−2.34293
$605$	3.62125	0.147225
$606$	0	0
$607$	−29.6459	−1.20329	−0.601645	−	0.798764i	$-0.705488\pi$
$607$	−29.6459	−1.20329	−0.601645	+	0.798764i	$0.705488\pi$
$608$	65.8419	2.67024
$609$	0	0
$610$	7.77580	0.314833
$611$	−0.725909	−0.0293671
$612$	0	0
$613$	−32.1201	−1.29732	−0.648659	−	0.761079i	$-0.724670\pi$
$613$	−32.1201	−1.29732	−0.648659	+	0.761079i	$0.724670\pi$
$614$	57.1156	2.30500
$615$	0	0
$616$	3.26223	0.131439
$617$	−3.29006	−0.132453	−0.0662265	−	0.997805i	$-0.521096\pi$
$617$	−3.29006	−0.132453	−0.0662265	+	0.997805i	$0.521096\pi$
$618$	0	0
$619$	30.1376	1.21133	0.605666	−	0.795719i	$-0.292907\pi$
$619$	30.1376	1.21133	0.605666	+	0.795719i	$0.292907\pi$
$620$	86.4963	3.47377
$621$	0	0
$622$	−44.8972	−1.80021
$623$	20.1379	0.806806
$624$	0	0
$625$	0.260823	0.0104329
$626$	−53.5768	−2.14136
$627$	0	0
$628$	−31.9301	−1.27415
$629$	1.11809	0.0445813
$630$	0	0
$631$	−12.1776	−0.484782	−0.242391	−	0.970179i	$-0.577932\pi$
$631$	−12.1776	−0.484782	−0.242391	+	0.970179i	$0.577932\pi$
$632$	−7.02870	−0.279587
$633$	0	0
$634$	34.0102	1.35072
$635$	10.7565	0.426858
$636$	0	0
$637$	−0.147690	−0.00585167
$638$	−12.5420	−0.496544
$639$	0	0
$640$	36.2222	1.43181
$641$	−16.9243	−0.668469	−0.334234	−	0.942490i	$-0.608478\pi$
$641$	−16.9243	−0.668469	−0.334234	+	0.942490i	$0.608478\pi$
$642$	0	0
$643$	−19.6269	−0.774009	−0.387004	−	0.922078i	$-0.626490\pi$
$643$	−19.6269	−0.774009	−0.387004	+	0.922078i	$0.626490\pi$
$644$	28.7500	1.13291
$645$	0	0
$646$	−18.6360	−0.733222
$647$	−24.6067	−0.967389	−0.483695	−	0.875237i	$-0.660705\pi$
$647$	−24.6067	−0.967389	−0.483695	+	0.875237i	$0.660705\pi$
$648$	0	0
$649$	−1.12062	−0.0439880
$650$	−3.16592	−0.124178
$651$	0	0
$652$	52.8905	2.07135
$653$	−23.1376	−0.905446	−0.452723	−	0.891651i	$-0.649547\pi$
$653$	−23.1376	−0.905446	−0.452723	+	0.891651i	$0.649547\pi$
$654$	0	0
$655$	−44.3439	−1.73266
$656$	16.2535	0.634594
$657$	0	0
$658$	21.3359	0.831760
$659$	9.65499	0.376105	0.188053	−	0.982159i	$-0.439782\pi$
$659$	9.65499	0.376105	0.188053	+	0.982159i	$0.439782\pi$
$660$	0	0
$661$	42.9636	1.67109	0.835545	−	0.549421i	$-0.185152\pi$
$661$	42.9636	1.67109	0.835545	+	0.549421i	$0.185152\pi$
$662$	−35.6255	−1.38463
$663$	0	0
$664$	12.1185	0.470288
$665$	76.1520	2.95305
$666$	0	0
$667$	−25.8584	−1.00124
$668$	−5.15227	−0.199347
$669$	0	0
$670$	19.8699	0.767640
$671$	1.00000	0.0386046
$672$	0	0
$673$	39.0023	1.50343	0.751715	−	0.659489i	$-0.229227\pi$
$673$	39.0023	1.50343	0.751715	+	0.659489i	$0.229227\pi$
$674$	−34.2025	−1.31743
$675$	0	0
$676$	−33.8538	−1.30207
$677$	49.2672	1.89349	0.946746	−	0.321981i	$-0.104349\pi$
$677$	49.2672	1.89349	0.946746	+	0.321981i	$0.104349\pi$
$678$	0	0
$679$	37.2793	1.43065
$680$	−4.87545	−0.186965
$681$	0	0
$682$	19.6452	0.752254
$683$	18.1420	0.694183	0.347092	−	0.937831i	$-0.387169\pi$
$683$	18.1420	0.694183	0.347092	+	0.937831i	$0.387169\pi$
$684$	0	0
$685$	−16.7376	−0.639509
$686$	41.7291	1.59322
$687$	0	0
$688$	−8.53922	−0.325555
$689$	−1.83293	−0.0698291
$690$	0	0
$691$	39.8244	1.51499	0.757496	−	0.652840i	$-0.226423\pi$
$691$	39.8244	1.51499	0.757496	+	0.652840i	$0.226423\pi$
$692$	−41.2449	−1.56790
$693$	0	0
$694$	21.5189	0.816846
$695$	−66.1789	−2.51031
$696$	0	0
$697$	−6.93664	−0.262744
$698$	−23.1514	−0.876292
$699$	0	0
$700$	52.6895	1.99148
$701$	1.05971	0.0400247	0.0200123	−	0.999800i	$-0.493629\pi$
$701$	1.05971	0.0400247	0.0200123	+	0.999800i	$0.493629\pi$
$702$	0	0
$703$	9.20787	0.347281
$704$	11.9122	0.448959
$705$	0	0
$706$	−31.2766	−1.17711
$707$	−23.3206	−0.877062
$708$	0	0
$709$	−16.3607	−0.614440	−0.307220	−	0.951639i	$-0.599399\pi$
$709$	−16.3607	−0.614440	−0.307220	+	0.951639i	$0.599399\pi$
$710$	−102.124	−3.83266
$711$	0	0
$712$	−10.6176	−0.397913
$713$	40.5033	1.51686
$714$	0	0
$715$	−0.658061	−0.0246101
$716$	−10.3232	−0.385796
$717$	0	0
$718$	−17.4783	−0.652284
$719$	19.9404	0.743651	0.371826	−	0.928303i	$-0.378732\pi$
$719$	19.9404	0.743651	0.371826	+	0.928303i	$0.378732\pi$
$720$	0	0
$721$	−3.86526	−0.143950
$722$	−112.675	−4.19334
$723$	0	0
$724$	54.9623	2.04266
$725$	−47.3900	−1.76002
$726$	0	0
$727$	−42.7616	−1.58594	−0.792969	−	0.609262i	$-0.791466\pi$
$727$	−42.7616	−1.58594	−0.792969	+	0.609262i	$0.791466\pi$
$728$	−0.592819	−0.0219713
$729$	0	0
$730$	−63.8386	−2.36277
$731$	3.64434	0.134791
$732$	0	0
$733$	−47.7154	−1.76241	−0.881204	−	0.472736i	$-0.843267\pi$
$733$	−47.7154	−1.76241	−0.881204	+	0.472736i	$0.843267\pi$
$734$	−22.3592	−0.825292
$735$	0	0
$736$	34.4786	1.27090
$737$	2.55535	0.0941274
$738$	0	0
$739$	−43.5566	−1.60226	−0.801128	−	0.598494i	$-0.795766\pi$
$739$	−43.5566	−1.60226	−0.801128	+	0.598494i	$0.795766\pi$
$740$	10.2971	0.378527
$741$	0	0
$742$	53.8735	1.97776
$743$	−10.8289	−0.397275	−0.198637	−	0.980073i	$-0.563652\pi$
$743$	−10.8289	−0.397275	−0.198637	+	0.980073i	$0.563652\pi$
$744$	0	0
$745$	70.5576	2.58503
$746$	−30.9551	−1.13335
$747$	0	0
$748$	−2.68016	−0.0979963
$749$	40.2195	1.46959
$750$	0	0
$751$	40.1160	1.46385	0.731926	−	0.681384i	$-0.238621\pi$
$751$	40.1160	1.46385	0.731926	+	0.681384i	$0.238621\pi$
$752$	9.60871	0.350394
$753$	0	0
$754$	2.27917	0.0830023
$755$	−79.8669	−2.90665
$756$	0	0
$757$	−27.3129	−0.992705	−0.496353	−	0.868121i	$-0.665328\pi$
$757$	−27.3129	−0.992705	−0.496353	+	0.868121i	$0.665328\pi$
$758$	−27.3726	−0.994218
$759$	0	0
$760$	−40.1510	−1.45643
$761$	−1.27815	−0.0463331	−0.0231665	−	0.999732i	$-0.507375\pi$
$761$	−1.27815	−0.0463331	−0.0231665	+	0.999732i	$0.507375\pi$
$762$	0	0
$763$	−8.83229	−0.319750
$764$	42.1098	1.52348
$765$	0	0
$766$	21.7792	0.786914
$767$	0.203641	0.00735304
$768$	0	0
$769$	16.2719	0.586781	0.293391	−	0.955993i	$-0.405216\pi$
$769$	16.2719	0.586781	0.293391	+	0.955993i	$0.405216\pi$
$770$	19.3417	0.697027
$771$	0	0
$772$	41.9418	1.50952
$773$	43.2949	1.55721	0.778604	−	0.627515i	$-0.215928\pi$
$773$	43.2949	1.55721	0.778604	+	0.627515i	$0.215928\pi$
$774$	0	0
$775$	74.2294	2.66640
$776$	−19.6554	−0.705588
$777$	0	0
$778$	36.5999	1.31217
$779$	−57.1256	−2.04674
$780$	0	0
$781$	−13.1336	−0.469957
$782$	−9.75885	−0.348976
$783$	0	0
$784$	1.95494	0.0698192
$785$	−44.2884	−1.58072
$786$	0	0
$787$	1.65622	0.0590380	0.0295190	−	0.999564i	$-0.490602\pi$
$787$	1.65622	0.0590380	0.0295190	+	0.999564i	$0.490602\pi$
$788$	−28.0164	−0.998042
$789$	0	0
$790$	−41.6731	−1.48266
$791$	16.8666	0.599707
$792$	0	0
$793$	−0.181722	−0.00645314
$794$	22.9513	0.814510
$795$	0	0
$796$	−4.58336	−0.162453
$797$	−3.43162	−0.121554	−0.0607772	−	0.998151i	$-0.519358\pi$
$797$	−3.43162	−0.121554	−0.0607772	+	0.998151i	$0.519358\pi$
$798$	0	0
$799$	−4.10078	−0.145075
$800$	63.1881	2.23404
$801$	0	0
$802$	−6.78558	−0.239607
$803$	−8.20991	−0.289721
$804$	0	0
$805$	39.8775	1.40550
$806$	−3.56997	−0.125747
$807$	0	0
$808$	12.2957	0.432562
$809$	17.6124	0.619221	0.309610	−	0.950864i	$-0.399801\pi$
$809$	17.6124	0.619221	0.309610	+	0.950864i	$0.399801\pi$
$810$	0	0
$811$	−33.2624	−1.16800	−0.584001	−	0.811753i	$-0.698514\pi$
$811$	−33.2624	−1.16800	−0.584001	+	0.811753i	$0.698514\pi$
$812$	−37.9315	−1.33113
$813$	0	0
$814$	2.33869	0.0819710
$815$	73.3614	2.56974
$816$	0	0
$817$	30.0124	1.05000
$818$	28.1250	0.983367
$819$	0	0
$820$	−63.8829	−2.23089
$821$	38.6262	1.34806	0.674032	−	0.738702i	$-0.264561\pi$
$821$	38.6262	1.34806	0.674032	+	0.738702i	$0.264561\pi$
$822$	0	0
$823$	33.8042	1.17834	0.589170	−	0.808009i	$-0.299455\pi$
$823$	33.8042	1.17834	0.589170	+	0.808009i	$0.299455\pi$
$824$	2.03795	0.0709953
$825$	0	0
$826$	−5.98540	−0.208259
$827$	−0.557114	−0.0193727	−0.00968637	−	0.999953i	$-0.503083\pi$
$827$	−0.557114	−0.0193727	−0.00968637	+	0.999953i	$0.503083\pi$
$828$	0	0
$829$	5.19937	0.180581	0.0902907	−	0.995915i	$-0.471220\pi$
$829$	5.19937	0.180581	0.0902907	+	0.995915i	$0.471220\pi$
$830$	71.8503	2.49396
$831$	0	0
$832$	−2.16472	−0.0750481
$833$	−0.834322	−0.0289075
$834$	0	0
$835$	−7.14642	−0.247312
$836$	−22.0720	−0.763376
$837$	0	0
$838$	46.1123	1.59292
$839$	11.9950	0.414112	0.207056	−	0.978329i	$-0.433612\pi$
$839$	11.9950	0.414112	0.207056	+	0.978329i	$0.433612\pi$
$840$	0	0
$841$	5.11637	0.176427
$842$	30.1599	1.03938
$843$	0	0
$844$	−11.4985	−0.395793
$845$	−46.9567	−1.61536
$846$	0	0
$847$	2.48742	0.0854689
$848$	24.2622	0.833166
$849$	0	0
$850$	−17.8848	−0.613444
$851$	4.82176	0.165288
$852$	0	0
$853$	−22.8770	−0.783294	−0.391647	−	0.920115i	$-0.628095\pi$
$853$	−22.8770	−0.783294	−0.391647	+	0.920115i	$0.628095\pi$
$854$	5.34117	0.182771
$855$	0	0
$856$	−21.2056	−0.724793
$857$	−42.2589	−1.44354	−0.721768	−	0.692135i	$-0.756670\pi$
$857$	−42.2589	−1.44354	−0.721768	+	0.692135i	$0.756670\pi$
$858$	0	0
$859$	14.4661	0.493576	0.246788	−	0.969070i	$-0.420625\pi$
$859$	14.4661	0.493576	0.246788	+	0.969070i	$0.420625\pi$
$860$	33.5625	1.14447
$861$	0	0
$862$	49.7996	1.69618
$863$	34.2785	1.16685	0.583427	−	0.812166i	$-0.301711\pi$
$863$	34.2785	1.16685	0.583427	+	0.812166i	$0.301711\pi$
$864$	0	0
$865$	−57.2084	−1.94515
$866$	−31.6127	−1.07424
$867$	0	0
$868$	59.4140	2.01664
$869$	−5.35933	−0.181803
$870$	0	0
$871$	−0.464363	−0.0157343
$872$	4.65681	0.157699
$873$	0	0
$874$	−80.3674	−2.71847
$875$	28.0447	0.948083
$876$	0	0
$877$	50.0011	1.68842	0.844209	−	0.536014i	$-0.180070\pi$
$877$	50.0011	1.68842	0.844209	+	0.536014i	$0.180070\pi$
$878$	78.7757	2.65855
$879$	0	0
$880$	8.71063	0.293635
$881$	5.16237	0.173925	0.0869624	−	0.996212i	$-0.472284\pi$
$881$	5.16237	0.173925	0.0869624	+	0.996212i	$0.472284\pi$
$882$	0	0
$883$	54.9858	1.85042	0.925209	−	0.379457i	$-0.123889\pi$
$883$	54.9858	1.85042	0.925209	+	0.379457i	$0.123889\pi$
$884$	0.487044	0.0163811
$885$	0	0
$886$	89.0213	2.99073
$887$	19.7988	0.664779	0.332389	−	0.943142i	$-0.392145\pi$
$887$	19.7988	0.664779	0.332389	+	0.943142i	$0.392145\pi$
$888$	0	0
$889$	7.38859	0.247805
$890$	−62.9519	−2.11015
$891$	0	0
$892$	−25.6306	−0.858177
$893$	−33.7713	−1.13011
$894$	0	0
$895$	−14.3187	−0.478622
$896$	24.8809	0.831212
$897$	0	0
$898$	−16.7203	−0.557965
$899$	−53.4382	−1.78226
$900$	0	0
$901$	−10.3545	−0.344959
$902$	−14.5092	−0.483104
$903$	0	0
$904$	−8.89287	−0.295772
$905$	76.2351	2.53414
$906$	0	0
$907$	−46.3300	−1.53836	−0.769181	−	0.639031i	$-0.779336\pi$
$907$	−46.3300	−1.53836	−0.769181	+	0.639031i	$0.779336\pi$
$908$	−48.4729	−1.60863
$909$	0	0
$910$	−3.51482	−0.116515
$911$	44.9211	1.48830	0.744151	−	0.668011i	$-0.232854\pi$
$911$	44.9211	1.48830	0.744151	+	0.668011i	$0.232854\pi$
$912$	0	0
$913$	9.24024	0.305807
$914$	−78.3859	−2.59278
$915$	0	0
$916$	−48.8042	−1.61254
$917$	−30.4596	−1.00587
$918$	0	0
$919$	−30.9670	−1.02151	−0.510754	−	0.859727i	$-0.670634\pi$
$919$	−30.9670	−1.02151	−0.510754	+	0.859727i	$0.670634\pi$
$920$	−21.0253	−0.693185
$921$	0	0
$922$	4.81186	0.158470
$923$	2.38667	0.0785581
$924$	0	0
$925$	8.83673	0.290550
$926$	21.1833	0.696126
$927$	0	0
$928$	−45.4895	−1.49327
$929$	34.3813	1.12801	0.564007	−	0.825770i	$-0.309259\pi$
$929$	34.3813	1.12801	0.564007	+	0.825770i	$0.309259\pi$
$930$	0	0
$931$	−6.87092	−0.225185
$932$	34.3730	1.12593
$933$	0	0
$934$	−19.2784	−0.630807
$935$	−3.71749	−0.121575
$936$	0	0
$937$	−8.57984	−0.280291	−0.140146	−	0.990131i	$-0.544757\pi$
$937$	−8.57984	−0.280291	−0.140146	+	0.990131i	$0.544757\pi$
$938$	13.6485	0.445641
$939$	0	0
$940$	−37.7660	−1.23179
$941$	42.1060	1.37262	0.686309	−	0.727310i	$-0.259230\pi$
$941$	42.1060	1.37262	0.686309	+	0.727310i	$0.259230\pi$
$942$	0	0
$943$	−29.9142	−0.974140
$944$	−2.69555	−0.0877328
$945$	0	0
$946$	7.62280	0.247838
$947$	54.0758	1.75723	0.878614	−	0.477532i	$-0.158469\pi$
$947$	54.0758	1.75723	0.878614	+	0.477532i	$0.158469\pi$
$948$	0	0
$949$	1.49192	0.0484298
$950$	−147.287	−4.77863
$951$	0	0
$952$	−3.34893	−0.108539
$953$	23.6590	0.766392	0.383196	−	0.923667i	$-0.374823\pi$
$953$	23.6590	0.766392	0.383196	+	0.923667i	$0.374823\pi$
$954$	0	0
$955$	58.4080	1.89004
$956$	47.0277	1.52099
$957$	0	0
$958$	34.3572	1.11003
$959$	−11.4970	−0.371257
$960$	0	0
$961$	52.7029	1.70009
$962$	−0.424992	−0.0137023
$963$	0	0
$964$	−74.9078	−2.41262
$965$	58.1751	1.87272
$966$	0	0
$967$	−36.9302	−1.18759	−0.593797	−	0.804615i	$-0.702372\pi$
$967$	−36.9302	−1.18759	−0.593797	+	0.804615i	$0.702372\pi$
$968$	−1.31149	−0.0421528
$969$	0	0
$970$	−116.537	−3.74177
$971$	−9.30267	−0.298537	−0.149268	−	0.988797i	$-0.547692\pi$
$971$	−9.30267	−0.298537	−0.149268	+	0.988797i	$0.547692\pi$
$972$	0	0
$973$	−45.4581	−1.45732
$974$	−8.13786	−0.260754
$975$	0	0
$976$	2.40542	0.0769956
$977$	5.96172	0.190732	0.0953662	−	0.995442i	$-0.469598\pi$
$977$	5.96172	0.190732	0.0953662	+	0.995442i	$0.469598\pi$
$978$	0	0
$979$	−8.09587	−0.258745
$980$	−7.68368	−0.245446
$981$	0	0
$982$	−17.8487	−0.569575
$983$	−33.8665	−1.08018	−0.540088	−	0.841609i	$-0.681609\pi$
$983$	−33.8665	−1.08018	−0.540088	+	0.841609i	$0.681609\pi$
$984$	0	0
$985$	−38.8599	−1.23818
$986$	12.8754	0.410035
$987$	0	0
$988$	4.01097	0.127606
$989$	15.7162	0.499746
$990$	0	0
$991$	−7.02387	−0.223121	−0.111560	−	0.993758i	$-0.535585\pi$
$991$	−7.02387	−0.223121	−0.111560	+	0.993758i	$0.535585\pi$
$992$	71.2524	2.26227
$993$	0	0
$994$	−70.1488	−2.22499
$995$	−6.35732	−0.201541
$996$	0	0
$997$	28.9360	0.916412	0.458206	−	0.888846i	$-0.348492\pi$
$997$	28.9360	0.916412	0.458206	+	0.888846i	$0.348492\pi$
$998$	−69.4036	−2.19693
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.3		13
3.2	odd	2		2013.2.a.e.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.11	✓	13	3.2	odd	2
6039.2.a.i.1.3		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)

\(f(q)\)	\(=\)	\(q-2.14727 q^{2} +2.61077 q^{4} +3.62125 q^{5} +2.48742 q^{7} -1.31149 q^{8} +O(q^{10})\)	\(q-2.14727 q^{2} +2.61077 q^{4} +3.62125 q^{5} +2.48742 q^{7} -1.31149 q^{8} -7.77580 q^{10} -1.00000 q^{11} +0.181722 q^{13} -5.34117 q^{14} -2.40542 q^{16} +1.02658 q^{17} +8.45421 q^{19} +9.45425 q^{20} +2.14727 q^{22} +4.42711 q^{23} +8.11345 q^{25} -0.390206 q^{26} +6.49409 q^{28} -5.84092 q^{29} +9.14893 q^{31} +7.78806 q^{32} -2.20434 q^{34} +9.00758 q^{35} +1.08915 q^{37} -18.1535 q^{38} -4.74923 q^{40} -6.75705 q^{41} +3.54999 q^{43} -2.61077 q^{44} -9.50620 q^{46} -3.99461 q^{47} -0.812722 q^{49} -17.4218 q^{50} +0.474435 q^{52} -10.0865 q^{53} -3.62125 q^{55} -3.26223 q^{56} +12.5420 q^{58} +1.12062 q^{59} -1.00000 q^{61} -19.6452 q^{62} -11.9122 q^{64} +0.658061 q^{65} -2.55535 q^{67} +2.68016 q^{68} -19.3417 q^{70} +13.1336 q^{71} +8.20991 q^{73} -2.33869 q^{74} +22.0720 q^{76} -2.48742 q^{77} +5.35933 q^{79} -8.71063 q^{80} +14.5092 q^{82} -9.24024 q^{83} +3.71749 q^{85} -7.62280 q^{86} +1.31149 q^{88} +8.09587 q^{89} +0.452020 q^{91} +11.5582 q^{92} +8.57750 q^{94} +30.6148 q^{95} +14.9871 q^{97} +1.74513 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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