LMFDB - Embedded newform 6039.2.a.h.1.13

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			1.13
Root			$-2.33467$ 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)

$f(q)$	$=$	$q+2.33467 q^{2} +3.45069 q^{4} -0.0293558 q^{5} -2.70286 q^{7} +3.38688 q^{8} +O(q^{10})$	\(q+2.33467 q^{2} +3.45069 q^{4} -0.0293558 q^{5} -2.70286 q^{7} +3.38688 q^{8} -0.0685361 q^{10} -1.00000 q^{11} +2.89225 q^{13} -6.31030 q^{14} +1.00587 q^{16} -7.17621 q^{17} +4.17563 q^{19} -0.101298 q^{20} -2.33467 q^{22} +2.06467 q^{23} -4.99914 q^{25} +6.75246 q^{26} -9.32674 q^{28} -4.39436 q^{29} -4.71288 q^{31} -4.42539 q^{32} -16.7541 q^{34} +0.0793448 q^{35} -2.29994 q^{37} +9.74873 q^{38} -0.0994245 q^{40} +2.67070 q^{41} -8.92207 q^{43} -3.45069 q^{44} +4.82032 q^{46} -10.7872 q^{47} +0.305477 q^{49} -11.6713 q^{50} +9.98027 q^{52} +4.86692 q^{53} +0.0293558 q^{55} -9.15427 q^{56} -10.2594 q^{58} +0.181028 q^{59} +1.00000 q^{61} -11.0030 q^{62} -12.3436 q^{64} -0.0849045 q^{65} +14.4055 q^{67} -24.7628 q^{68} +0.185244 q^{70} -4.73226 q^{71} +7.51871 q^{73} -5.36960 q^{74} +14.4088 q^{76} +2.70286 q^{77} +11.1725 q^{79} -0.0295280 q^{80} +6.23522 q^{82} -12.9734 q^{83} +0.210663 q^{85} -20.8301 q^{86} -3.38688 q^{88} +10.1884 q^{89} -7.81737 q^{91} +7.12452 q^{92} -25.1846 q^{94} -0.122579 q^{95} -16.7207 q^{97} +0.713188 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.33467	1.65086	0.825431	−	0.564503i	$-0.190932\pi$
$2$	2.33467	1.65086	0.825431	+	0.564503i	$0.190932\pi$
$3$	0	0
$3$	0	0
$4$	3.45069	1.72534
$5$	−0.0293558	−0.0131283	−0.00656416	−	0.999978i	$-0.502089\pi$
$5$	−0.0293558	−0.0131283	−0.00656416	+	0.999978i	$0.502089\pi$
$6$	0	0
$7$	−2.70286	−1.02159	−0.510793	−	0.859703i	$-0.670648\pi$
$7$	−2.70286	−1.02159	−0.510793	+	0.859703i	$0.670648\pi$
$8$	3.38688	1.19744
$9$	0	0
$10$	−0.0685361	−0.0216730
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.89225	0.802167	0.401084	−	0.916041i	$-0.368634\pi$
$13$	2.89225	0.802167	0.401084	+	0.916041i	$0.368634\pi$
$14$	−6.31030	−1.68650
$15$	0	0
$16$	1.00587	0.251467
$17$	−7.17621	−1.74049	−0.870243	−	0.492623i	$-0.836038\pi$
$17$	−7.17621	−1.74049	−0.870243	+	0.492623i	$0.836038\pi$
$18$	0	0
$19$	4.17563	0.957956	0.478978	−	0.877827i	$-0.341007\pi$
$19$	4.17563	0.957956	0.478978	+	0.877827i	$0.341007\pi$
$20$	−0.101298	−0.0226508
$21$	0	0
$22$	−2.33467	−0.497753
$23$	2.06467	0.430513	0.215256	−	0.976558i	$-0.430941\pi$
$23$	2.06467	0.430513	0.215256	+	0.976558i	$0.430941\pi$
$24$	0	0
$25$	−4.99914	−0.999828
$26$	6.75246	1.32427
$27$	0	0
$28$	−9.32674	−1.76259
$29$	−4.39436	−0.816012	−0.408006	−	0.912979i	$-0.633776\pi$
$29$	−4.39436	−0.816012	−0.408006	+	0.912979i	$0.633776\pi$
$30$	0	0
$31$	−4.71288	−0.846457	−0.423229	−	0.906023i	$-0.639103\pi$
$31$	−4.71288	−0.846457	−0.423229	+	0.906023i	$0.639103\pi$
$32$	−4.42539	−0.782305
$33$	0	0
$34$	−16.7541	−2.87330
$35$	0.0793448	0.0134117
$36$	0	0
$37$	−2.29994	−0.378108	−0.189054	−	0.981967i	$-0.560542\pi$
$37$	−2.29994	−0.378108	−0.189054	+	0.981967i	$0.560542\pi$
$38$	9.74873	1.58145
$39$	0	0
$40$	−0.0994245	−0.0157204
$41$	2.67070	0.417094	0.208547	−	0.978012i	$-0.433127\pi$
$41$	2.67070	0.417094	0.208547	+	0.978012i	$0.433127\pi$
$42$	0	0
$43$	−8.92207	−1.36060	−0.680301	−	0.732932i	$-0.738151\pi$
$43$	−8.92207	−1.36060	−0.680301	+	0.732932i	$0.738151\pi$
$44$	−3.45069	−0.520211
$45$	0	0
$46$	4.82032	0.710717
$47$	−10.7872	−1.57348	−0.786738	−	0.617287i	$-0.788232\pi$
$47$	−10.7872	−1.57348	−0.786738	+	0.617287i	$0.788232\pi$
$48$	0	0
$49$	0.305477	0.0436396
$50$	−11.6713	−1.65058
$51$	0	0
$52$	9.98027	1.38401
$53$	4.86692	0.668523	0.334261	−	0.942480i	$-0.391513\pi$
$53$	4.86692	0.668523	0.334261	+	0.942480i	$0.391513\pi$
$54$	0	0
$55$	0.0293558	0.00395834
$56$	−9.15427	−1.22329
$57$	0	0
$58$	−10.2594	−1.34712
$59$	0.181028	0.0235678	0.0117839	−	0.999931i	$-0.496249\pi$
$59$	0.181028	0.0235678	0.0117839	+	0.999931i	$0.496249\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−11.0030	−1.39738
$63$	0	0
$64$	−12.3436	−1.54294
$65$	−0.0849045	−0.0105311
$66$	0	0
$67$	14.4055	1.75991	0.879955	−	0.475057i	$-0.157573\pi$
$67$	14.4055	1.75991	0.879955	+	0.475057i	$0.157573\pi$
$68$	−24.7628	−3.00294
$69$	0	0
$70$	0.185244	0.0221409
$71$	−4.73226	−0.561616	−0.280808	−	0.959764i	$-0.590602\pi$
$71$	−4.73226	−0.561616	−0.280808	+	0.959764i	$0.590602\pi$
$72$	0	0
$73$	7.51871	0.879999	0.439999	−	0.897998i	$-0.354979\pi$
$73$	7.51871	0.879999	0.439999	+	0.897998i	$0.354979\pi$
$74$	−5.36960	−0.624204
$75$	0	0
$76$	14.4088	1.65280
$77$	2.70286	0.308020
$78$	0	0
$79$	11.1725	1.25701	0.628503	−	0.777807i	$-0.283668\pi$
$79$	11.1725	1.25701	0.628503	+	0.777807i	$0.283668\pi$
$80$	−0.0295280	−0.00330133
$81$	0	0
$82$	6.23522	0.688564
$83$	−12.9734	−1.42402	−0.712008	−	0.702171i	$-0.752214\pi$
$83$	−12.9734	−1.42402	−0.712008	+	0.702171i	$0.752214\pi$
$84$	0	0
$85$	0.210663	0.0228496
$86$	−20.8301	−2.24617
$87$	0	0
$88$	−3.38688	−0.361042
$89$	10.1884	1.07997	0.539983	−	0.841676i	$-0.318431\pi$
$89$	10.1884	1.07997	0.539983	+	0.841676i	$0.318431\pi$
$90$	0	0
$91$	−7.81737	−0.819483
$92$	7.12452	0.742783
$93$	0	0
$94$	−25.1846	−2.59759
$95$	−0.122579	−0.0125763
$96$	0	0
$97$	−16.7207	−1.69773	−0.848867	−	0.528606i	$-0.822715\pi$
$97$	−16.7207	−1.69773	−0.848867	+	0.528606i	$0.822715\pi$
$98$	0.713188	0.0720429
$99$	0	0
$100$	−17.2505	−1.72505
$101$	−1.97502	−0.196522	−0.0982610	−	0.995161i	$-0.531328\pi$
$101$	−1.97502	−0.196522	−0.0982610	+	0.995161i	$0.531328\pi$
$102$	0	0
$103$	8.68750	0.856005	0.428003	−	0.903777i	$-0.359217\pi$
$103$	8.68750	0.856005	0.428003	+	0.903777i	$0.359217\pi$
$104$	9.79571	0.960548
$105$	0	0
$106$	11.3627	1.10364
$107$	6.12215	0.591851	0.295925	−	0.955211i	$-0.404372\pi$
$107$	6.12215	0.591851	0.295925	+	0.955211i	$0.404372\pi$
$108$	0	0
$109$	−3.59659	−0.344491	−0.172245	−	0.985054i	$-0.555102\pi$
$109$	−3.59659	−0.344491	−0.172245	+	0.985054i	$0.555102\pi$
$110$	0.0685361	0.00653466
$111$	0	0
$112$	−2.71872	−0.256895
$113$	−6.81766	−0.641352	−0.320676	−	0.947189i	$-0.603910\pi$
$113$	−6.81766	−0.641352	−0.320676	+	0.947189i	$0.603910\pi$
$114$	0	0
$115$	−0.0606100	−0.00565191
$116$	−15.1636	−1.40790
$117$	0	0
$118$	0.422640	0.0389072
$119$	19.3963	1.77806
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.33467	0.211371
$123$	0	0
$124$	−16.2627	−1.46043
$125$	0.293533	0.0262544
$126$	0	0
$127$	5.40526	0.479639	0.239819	−	0.970817i	$-0.422912\pi$
$127$	5.40526	0.479639	0.239819	+	0.970817i	$0.422912\pi$
$128$	−19.9674	−1.76488
$129$	0	0
$130$	−0.198224	−0.0173854
$131$	−19.3818	−1.69340	−0.846698	−	0.532075i	$-0.821413\pi$
$131$	−19.3818	−1.69340	−0.846698	+	0.532075i	$0.821413\pi$
$132$	0	0
$133$	−11.2862	−0.978635
$134$	33.6321	2.90537
$135$	0	0
$136$	−24.3049	−2.08413
$137$	−15.7164	−1.34275	−0.671373	−	0.741119i	$-0.734295\pi$
$137$	−15.7164	−1.34275	−0.671373	+	0.741119i	$0.734295\pi$
$138$	0	0
$139$	1.57642	0.133711	0.0668553	−	0.997763i	$-0.478703\pi$
$139$	1.57642	0.133711	0.0668553	+	0.997763i	$0.478703\pi$
$140$	0.273794	0.0231398
$141$	0	0
$142$	−11.0483	−0.927150
$143$	−2.89225	−0.241862
$144$	0	0
$145$	0.129000	0.0107129
$146$	17.5537	1.45276
$147$	0	0
$148$	−7.93637	−0.652366
$149$	−1.10490	−0.0905172	−0.0452586	−	0.998975i	$-0.514411\pi$
$149$	−1.10490	−0.0905172	−0.0452586	+	0.998975i	$0.514411\pi$
$150$	0	0
$151$	−15.0032	−1.22095	−0.610473	−	0.792037i	$-0.709021\pi$
$151$	−15.0032	−1.22095	−0.610473	+	0.792037i	$0.709021\pi$
$152$	14.1424	1.14710
$153$	0	0
$154$	6.31030	0.508498
$155$	0.138350	0.0111126
$156$	0	0
$157$	−9.75971	−0.778910	−0.389455	−	0.921046i	$-0.627337\pi$
$157$	−9.75971	−0.778910	−0.389455	+	0.921046i	$0.627337\pi$
$158$	26.0842	2.07514
$159$	0	0
$160$	0.129911	0.0102703
$161$	−5.58052	−0.439806
$162$	0	0
$163$	21.3224	1.67010	0.835050	−	0.550174i	$-0.185439\pi$
$163$	21.3224	1.67010	0.835050	+	0.550174i	$0.185439\pi$
$164$	9.21577	0.719630
$165$	0	0
$166$	−30.2886	−2.35085
$167$	−19.0985	−1.47789	−0.738943	−	0.673768i	$-0.764675\pi$
$167$	−19.0985	−1.47789	−0.738943	+	0.673768i	$0.764675\pi$
$168$	0	0
$169$	−4.63486	−0.356528
$170$	0.491829	0.0377216
$171$	0	0
$172$	−30.7873	−2.34751
$173$	17.4746	1.32857	0.664285	−	0.747479i	$-0.268736\pi$
$173$	17.4746	1.32857	0.664285	+	0.747479i	$0.268736\pi$
$174$	0	0
$175$	13.5120	1.02141
$176$	−1.00587	−0.0758200
$177$	0	0
$178$	23.7865	1.78288
$179$	−15.9382	−1.19127	−0.595637	−	0.803253i	$-0.703100\pi$
$179$	−15.9382	−1.19127	−0.595637	+	0.803253i	$0.703100\pi$
$180$	0	0
$181$	7.41273	0.550984	0.275492	−	0.961303i	$-0.411159\pi$
$181$	7.41273	0.550984	0.275492	+	0.961303i	$0.411159\pi$
$182$	−18.2510	−1.35285
$183$	0	0
$184$	6.99277	0.515514
$185$	0.0675166	0.00496392
$186$	0	0
$187$	7.17621	0.524776
$188$	−37.2233	−2.71479
$189$	0	0
$190$	−0.286182	−0.0207618
$191$	−11.6075	−0.839886	−0.419943	−	0.907550i	$-0.637950\pi$
$191$	−11.6075	−0.839886	−0.419943	+	0.907550i	$0.637950\pi$
$192$	0	0
$193$	20.4885	1.47479	0.737396	−	0.675460i	$-0.236055\pi$
$193$	20.4885	1.47479	0.737396	+	0.675460i	$0.236055\pi$
$194$	−39.0374	−2.80272
$195$	0	0
$196$	1.05411	0.0752933
$197$	−7.27132	−0.518060	−0.259030	−	0.965869i	$-0.583403\pi$
$197$	−7.27132	−0.518060	−0.259030	+	0.965869i	$0.583403\pi$
$198$	0	0
$199$	−10.0401	−0.711723	−0.355862	−	0.934539i	$-0.615813\pi$
$199$	−10.0401	−0.711723	−0.355862	+	0.934539i	$0.615813\pi$
$200$	−16.9315	−1.19724
$201$	0	0
$202$	−4.61103	−0.324431
$203$	11.8774	0.833627
$204$	0	0
$205$	−0.0784007	−0.00547574
$206$	20.2825	1.41315
$207$	0	0
$208$	2.90922	0.201718
$209$	−4.17563	−0.288835
$210$	0	0
$211$	−7.36422	−0.506974	−0.253487	−	0.967339i	$-0.581577\pi$
$211$	−7.36422	−0.506974	−0.253487	+	0.967339i	$0.581577\pi$
$212$	16.7942	1.15343
$213$	0	0
$214$	14.2932	0.977063
$215$	0.261915	0.0178624
$216$	0	0
$217$	12.7383	0.864730
$218$	−8.39685	−0.568706
$219$	0	0
$220$	0.101298	0.00682949
$221$	−20.7554	−1.39616
$222$	0	0
$223$	1.02663	0.0687482	0.0343741	−	0.999409i	$-0.489056\pi$
$223$	1.02663	0.0687482	0.0343741	+	0.999409i	$0.489056\pi$
$224$	11.9612	0.799193
$225$	0	0
$226$	−15.9170	−1.05878
$227$	−21.6433	−1.43651	−0.718257	−	0.695778i	$-0.755060\pi$
$227$	−21.6433	−1.43651	−0.718257	+	0.695778i	$0.755060\pi$
$228$	0	0
$229$	9.88816	0.653428	0.326714	−	0.945123i	$-0.394059\pi$
$229$	9.88816	0.653428	0.326714	+	0.945123i	$0.394059\pi$
$230$	−0.141504	−0.00933052
$231$	0	0
$232$	−14.8832	−0.977127
$233$	24.8039	1.62496	0.812480	−	0.582989i	$-0.198117\pi$
$233$	24.8039	1.62496	0.812480	+	0.582989i	$0.198117\pi$
$234$	0	0
$235$	0.316667	0.0206571
$236$	0.624670	0.0406626
$237$	0	0
$238$	45.2840	2.93533
$239$	−15.9365	−1.03084	−0.515422	−	0.856936i	$-0.672365\pi$
$239$	−15.9365	−1.03084	−0.515422	+	0.856936i	$0.672365\pi$
$240$	0	0
$241$	−10.6415	−0.685481	−0.342740	−	0.939430i	$-0.611355\pi$
$241$	−10.6415	−0.685481	−0.342740	+	0.939430i	$0.611355\pi$
$242$	2.33467	0.150078
$243$	0	0
$244$	3.45069	0.220908
$245$	−0.00896753	−0.000572914 0
$246$	0	0
$247$	12.0770	0.768441
$248$	−15.9619	−1.01358
$249$	0	0
$250$	0.685302	0.0433423
$251$	24.6375	1.55510	0.777552	−	0.628819i	$-0.216461\pi$
$251$	24.6375	1.55510	0.777552	+	0.628819i	$0.216461\pi$
$252$	0	0
$253$	−2.06467	−0.129805
$254$	12.6195	0.791818
$255$	0	0
$256$	−21.9301	−1.37063
$257$	−8.56747	−0.534424	−0.267212	−	0.963638i	$-0.586102\pi$
$257$	−8.56747	−0.534424	−0.267212	+	0.963638i	$0.586102\pi$
$258$	0	0
$259$	6.21643	0.386270
$260$	−0.292979	−0.0181698
$261$	0	0
$262$	−45.2501	−2.79556
$263$	7.12601	0.439408	0.219704	−	0.975567i	$-0.429491\pi$
$263$	7.12601	0.439408	0.219704	+	0.975567i	$0.429491\pi$
$264$	0	0
$265$	−0.142872	−0.00877658
$266$	−26.3495	−1.61559
$267$	0	0
$268$	49.7088	3.03645
$269$	−14.1919	−0.865295	−0.432648	−	0.901563i	$-0.642421\pi$
$269$	−14.1919	−0.865295	−0.432648	+	0.901563i	$0.642421\pi$
$270$	0	0
$271$	29.2260	1.77535	0.887677	−	0.460466i	$-0.152318\pi$
$271$	29.2260	1.77535	0.887677	+	0.460466i	$0.152318\pi$
$272$	−7.21830	−0.437674
$273$	0	0
$274$	−36.6927	−2.21669
$275$	4.99914	0.301459
$276$	0	0
$277$	−13.9600	−0.838774	−0.419387	−	0.907808i	$-0.637755\pi$
$277$	−13.9600	−0.838774	−0.419387	+	0.907808i	$0.637755\pi$
$278$	3.68043	0.220738
$279$	0	0
$280$	0.268731	0.0160597
$281$	5.84690	0.348797	0.174398	−	0.984675i	$-0.444202\pi$
$281$	5.84690	0.348797	0.174398	+	0.984675i	$0.444202\pi$
$282$	0	0
$283$	25.6344	1.52380	0.761902	−	0.647692i	$-0.224266\pi$
$283$	25.6344	1.52380	0.761902	+	0.647692i	$0.224266\pi$
$284$	−16.3296	−0.968981
$285$	0	0
$286$	−6.75246	−0.399281
$287$	−7.21855	−0.426098
$288$	0	0
$289$	34.4979	2.02929
$290$	0.301172	0.0176855
$291$	0	0
$292$	25.9447	1.51830
$293$	−1.56499	−0.0914279	−0.0457140	−	0.998955i	$-0.514556\pi$
$293$	−1.56499	−0.0914279	−0.0457140	+	0.998955i	$0.514556\pi$
$294$	0	0
$295$	−0.00531422	−0.000309406 0
$296$	−7.78961	−0.452762
$297$	0	0
$298$	−2.57959	−0.149431
$299$	5.97154	0.345343
$300$	0	0
$301$	24.1151	1.38997
$302$	−35.0276	−2.01561
$303$	0	0
$304$	4.20013	0.240894
$305$	−0.0293558	−0.00168091
$306$	0	0
$307$	8.60524	0.491127	0.245563	−	0.969381i	$-0.421027\pi$
$307$	8.60524	0.491127	0.245563	+	0.969381i	$0.421027\pi$
$308$	9.32674	0.531440
$309$	0	0
$310$	0.323002	0.0183453
$311$	31.7733	1.80170	0.900850	−	0.434130i	$-0.142944\pi$
$311$	31.7733	1.80170	0.900850	+	0.434130i	$0.142944\pi$
$312$	0	0
$313$	26.8892	1.51987	0.759934	−	0.650000i	$-0.225231\pi$
$313$	26.8892	1.51987	0.759934	+	0.650000i	$0.225231\pi$
$314$	−22.7857	−1.28587
$315$	0	0
$316$	38.5529	2.16877
$317$	21.6911	1.21830	0.609148	−	0.793057i	$-0.291512\pi$
$317$	21.6911	1.21830	0.609148	+	0.793057i	$0.291512\pi$
$318$	0	0
$319$	4.39436	0.246037
$320$	0.362355	0.0202562
$321$	0	0
$322$	−13.0287	−0.726059
$323$	−29.9652	−1.66731
$324$	0	0
$325$	−14.4588	−0.802029
$326$	49.7808	2.75710
$327$	0	0
$328$	9.04535	0.499446
$329$	29.1564	1.60744
$330$	0	0
$331$	−3.16662	−0.174053	−0.0870265	−	0.996206i	$-0.527736\pi$
$331$	−3.16662	−0.174053	−0.0870265	+	0.996206i	$0.527736\pi$
$332$	−44.7672	−2.45692
$333$	0	0
$334$	−44.5887	−2.43979
$335$	−0.422884	−0.0231046
$336$	0	0
$337$	29.0756	1.58385	0.791924	−	0.610619i	$-0.209079\pi$
$337$	29.0756	1.58385	0.791924	+	0.610619i	$0.209079\pi$
$338$	−10.8209	−0.588578
$339$	0	0
$340$	0.726933	0.0394235
$341$	4.71288	0.255217
$342$	0	0
$343$	18.0944	0.977005
$344$	−30.2179	−1.62924
$345$	0	0
$346$	40.7975	2.19329
$347$	5.76588	0.309529	0.154764	−	0.987951i	$-0.450538\pi$
$347$	5.76588	0.309529	0.154764	+	0.987951i	$0.450538\pi$
$348$	0	0
$349$	−29.8833	−1.59961	−0.799807	−	0.600257i	$-0.795065\pi$
$349$	−29.8833	−1.59961	−0.799807	+	0.600257i	$0.795065\pi$
$350$	31.5461	1.68621
$351$	0	0
$352$	4.42539	0.235874
$353$	−8.68646	−0.462334	−0.231167	−	0.972914i	$-0.574254\pi$
$353$	−8.68646	−0.462334	−0.231167	+	0.972914i	$0.574254\pi$
$354$	0	0
$355$	0.138919	0.00737307
$356$	35.1569	1.86331
$357$	0	0
$358$	−37.2104	−1.96663
$359$	0.320645	0.0169230	0.00846150	−	0.999964i	$-0.497307\pi$
$359$	0.320645	0.0169230	0.00846150	+	0.999964i	$0.497307\pi$
$360$	0	0
$361$	−1.56409	−0.0823205
$362$	17.3063	0.909598
$363$	0	0
$364$	−26.9753	−1.41389
$365$	−0.220718	−0.0115529
$366$	0	0
$367$	−14.7522	−0.770060	−0.385030	−	0.922904i	$-0.625809\pi$
$367$	−14.7522	−0.770060	−0.385030	+	0.922904i	$0.625809\pi$
$368$	2.07678	0.108260
$369$	0	0
$370$	0.157629	0.00819474
$371$	−13.1546	−0.682954
$372$	0	0
$373$	−7.54533	−0.390683	−0.195341	−	0.980735i	$-0.562581\pi$
$373$	−7.54533	−0.390683	−0.195341	+	0.980735i	$0.562581\pi$
$374$	16.7541	0.866333
$375$	0	0
$376$	−36.5349	−1.88415
$377$	−12.7096	−0.654578
$378$	0	0
$379$	−5.90833	−0.303491	−0.151745	−	0.988420i	$-0.548489\pi$
$379$	−5.90833	−0.303491	−0.151745	+	0.988420i	$0.548489\pi$
$380$	−0.422982	−0.0216985
$381$	0	0
$382$	−27.0996	−1.38654
$383$	−28.3128	−1.44672	−0.723358	−	0.690473i	$-0.757402\pi$
$383$	−28.3128	−1.44672	−0.723358	+	0.690473i	$0.757402\pi$
$384$	0	0
$385$	−0.0793448	−0.00404378
$386$	47.8338	2.43468
$387$	0	0
$388$	−57.6981	−2.92918
$389$	16.1105	0.816833	0.408417	−	0.912796i	$-0.366081\pi$
$389$	16.1105	0.816833	0.408417	+	0.912796i	$0.366081\pi$
$390$	0	0
$391$	−14.8165	−0.749302
$392$	1.03461	0.0522559
$393$	0	0
$394$	−16.9761	−0.855245
$395$	−0.327978	−0.0165024
$396$	0	0
$397$	−12.9139	−0.648128	−0.324064	−	0.946035i	$-0.605049\pi$
$397$	−12.9139	−0.648128	−0.324064	+	0.946035i	$0.605049\pi$
$398$	−23.4403	−1.17496
$399$	0	0
$400$	−5.02846	−0.251423
$401$	−21.6181	−1.07956	−0.539778	−	0.841808i	$-0.681492\pi$
$401$	−21.6181	−1.07956	−0.539778	+	0.841808i	$0.681492\pi$
$402$	0	0
$403$	−13.6308	−0.679000
$404$	−6.81518	−0.339068
$405$	0	0
$406$	27.7297	1.37620
$407$	2.29994	0.114004
$408$	0	0
$409$	−10.5185	−0.520108	−0.260054	−	0.965594i	$-0.583740\pi$
$409$	−10.5185	−0.520108	−0.260054	+	0.965594i	$0.583740\pi$
$410$	−0.183040	−0.00903969
$411$	0	0
$412$	29.9779	1.47690
$413$	−0.489294	−0.0240766
$414$	0	0
$415$	0.380845	0.0186949
$416$	−12.7993	−0.627539
$417$	0	0
$418$	−9.74873	−0.476826
$419$	−5.19468	−0.253777	−0.126888	−	0.991917i	$-0.540499\pi$
$419$	−5.19468	−0.253777	−0.126888	+	0.991917i	$0.540499\pi$
$420$	0	0
$421$	−22.0265	−1.07351	−0.536753	−	0.843740i	$-0.680349\pi$
$421$	−22.0265	−1.07351	−0.536753	+	0.843740i	$0.680349\pi$
$422$	−17.1930	−0.836943
$423$	0	0
$424$	16.4837	0.800517
$425$	35.8748	1.74019
$426$	0	0
$427$	−2.70286	−0.130801
$428$	21.1256	1.02115
$429$	0	0
$430$	0.611484	0.0294884
$431$	0.844284	0.0406677	0.0203339	−	0.999793i	$-0.493527\pi$
$431$	0.844284	0.0406677	0.0203339	+	0.999793i	$0.493527\pi$
$432$	0	0
$433$	28.8544	1.38665	0.693326	−	0.720624i	$-0.256144\pi$
$433$	28.8544	1.38665	0.693326	+	0.720624i	$0.256144\pi$
$434$	29.7397	1.42755
$435$	0	0
$436$	−12.4107	−0.594365
$437$	8.62129	0.412412
$438$	0	0
$439$	−21.6203	−1.03188	−0.515941	−	0.856624i	$-0.672557\pi$
$439$	−21.6203	−1.03188	−0.515941	+	0.856624i	$0.672557\pi$
$440$	0.0994245	0.00473987
$441$	0	0
$442$	−48.4571	−2.30487
$443$	3.90144	0.185363	0.0926815	−	0.995696i	$-0.470456\pi$
$443$	3.90144	0.185363	0.0926815	+	0.995696i	$0.470456\pi$
$444$	0	0
$445$	−0.299088	−0.0141781
$446$	2.39684	0.113494
$447$	0	0
$448$	33.3629	1.57625
$449$	7.16531	0.338152	0.169076	−	0.985603i	$-0.445922\pi$
$449$	7.16531	0.338152	0.169076	+	0.985603i	$0.445922\pi$
$450$	0	0
$451$	−2.67070	−0.125759
$452$	−23.5256	−1.10655
$453$	0	0
$454$	−50.5299	−2.37149
$455$	0.229485	0.0107584
$456$	0	0
$457$	3.35821	0.157091	0.0785453	−	0.996911i	$-0.474972\pi$
$457$	3.35821	0.157091	0.0785453	+	0.996911i	$0.474972\pi$
$458$	23.0856	1.07872
$459$	0	0
$460$	−0.209146	−0.00975148
$461$	41.2484	1.92113	0.960565	−	0.278055i	$-0.0896896\pi$
$461$	41.2484	1.92113	0.960565	+	0.278055i	$0.0896896\pi$
$462$	0	0
$463$	18.7387	0.870862	0.435431	−	0.900222i	$-0.356596\pi$
$463$	18.7387	0.870862	0.435431	+	0.900222i	$0.356596\pi$
$464$	−4.42014	−0.205200
$465$	0	0
$466$	57.9091	2.68259
$467$	24.4499	1.13141	0.565704	−	0.824608i	$-0.308604\pi$
$467$	24.4499	1.13141	0.565704	+	0.824608i	$0.308604\pi$
$468$	0	0
$469$	−38.9361	−1.79790
$470$	0.739314	0.0341020
$471$	0	0
$472$	0.613119	0.0282211
$473$	8.92207	0.410237
$474$	0	0
$475$	−20.8746	−0.957791
$476$	66.9306	3.06776
$477$	0	0
$478$	−37.2064	−1.70178
$479$	34.9779	1.59818	0.799089	−	0.601212i	$-0.205315\pi$
$479$	34.9779	1.59818	0.799089	+	0.601212i	$0.205315\pi$
$480$	0	0
$481$	−6.65201	−0.303306
$482$	−24.8445	−1.13163
$483$	0	0
$484$	3.45069	0.156849
$485$	0.490851	0.0222884
$486$	0	0
$487$	11.8692	0.537847	0.268923	−	0.963162i	$-0.413332\pi$
$487$	11.8692	0.537847	0.268923	+	0.963162i	$0.413332\pi$
$488$	3.38688	0.153317
$489$	0	0
$490$	−0.0209362	−0.000945802 0
$491$	−34.2782	−1.54695	−0.773477	−	0.633824i	$-0.781484\pi$
$491$	−34.2782	−1.54695	−0.773477	+	0.633824i	$0.781484\pi$
$492$	0	0
$493$	31.5348	1.42026
$494$	28.1958	1.26859
$495$	0	0
$496$	−4.74052	−0.212856
$497$	12.7907	0.573740
$498$	0	0
$499$	22.9767	1.02858	0.514289	−	0.857617i	$-0.328056\pi$
$499$	22.9767	1.02858	0.514289	+	0.857617i	$0.328056\pi$
$500$	1.01289	0.0452978
$501$	0	0
$502$	57.5204	2.56726
$503$	31.8586	1.42051	0.710253	−	0.703946i	$-0.248580\pi$
$503$	31.8586	1.42051	0.710253	+	0.703946i	$0.248580\pi$
$504$	0	0
$505$	0.0579784	0.00258000
$506$	−4.82032	−0.214289
$507$	0	0
$508$	18.6518	0.827542
$509$	14.4168	0.639012	0.319506	−	0.947584i	$-0.396483\pi$
$509$	14.4168	0.639012	0.319506	+	0.947584i	$0.396483\pi$
$510$	0	0
$511$	−20.3221	−0.898995
$512$	−11.2648	−0.497840
$513$	0	0
$514$	−20.0022	−0.882261
$515$	−0.255029	−0.0112379
$516$	0	0
$517$	10.7872	0.474421
$518$	14.5133	0.637678
$519$	0	0
$520$	−0.287561	−0.0126104
$521$	−5.61958	−0.246198	−0.123099	−	0.992394i	$-0.539283\pi$
$521$	−5.61958	−0.246198	−0.123099	+	0.992394i	$0.539283\pi$
$522$	0	0
$523$	13.6790	0.598142	0.299071	−	0.954231i	$-0.403323\pi$
$523$	13.6790	0.598142	0.299071	+	0.954231i	$0.403323\pi$
$524$	−66.8805	−2.92169
$525$	0	0
$526$	16.6369	0.725402
$527$	33.8206	1.47325
$528$	0	0
$529$	−18.7371	−0.814659
$530$	−0.333560	−0.0144889
$531$	0	0
$532$	−38.9450	−1.68848
$533$	7.72436	0.334579
$534$	0	0
$535$	−0.179721	−0.00777000
$536$	48.7896	2.10739
$537$	0	0
$538$	−33.1334	−1.42848
$539$	−0.305477	−0.0131578
$540$	0	0
$541$	7.06159	0.303602	0.151801	−	0.988411i	$-0.451493\pi$
$541$	7.06159	0.303602	0.151801	+	0.988411i	$0.451493\pi$
$542$	68.2331	2.93086
$543$	0	0
$544$	31.7575	1.36159
$545$	0.105581	0.00452258
$546$	0	0
$547$	−21.3505	−0.912881	−0.456441	−	0.889754i	$-0.650876\pi$
$547$	−21.3505	−0.912881	−0.456441	+	0.889754i	$0.650876\pi$
$548$	−54.2325	−2.31670
$549$	0	0
$550$	11.6713	0.497668
$551$	−18.3492	−0.781704
$552$	0	0
$553$	−30.1978	−1.28414
$554$	−32.5919	−1.38470
$555$	0	0
$556$	5.43975	0.230697
$557$	−12.8162	−0.543039	−0.271520	−	0.962433i	$-0.587526\pi$
$557$	−12.8162	−0.543039	−0.271520	+	0.962433i	$0.587526\pi$
$558$	0	0
$559$	−25.8049	−1.09143
$560$	0.0798102	0.00337260
$561$	0	0
$562$	13.6506	0.575815
$563$	30.0977	1.26846	0.634232	−	0.773142i	$-0.281316\pi$
$563$	30.0977	1.26846	0.634232	+	0.773142i	$0.281316\pi$
$564$	0	0
$565$	0.200138	0.00841986
$566$	59.8478	2.51559
$567$	0	0
$568$	−16.0276	−0.672502
$569$	−33.1992	−1.39178	−0.695891	−	0.718148i	$-0.744990\pi$
$569$	−33.1992	−1.39178	−0.695891	+	0.718148i	$0.744990\pi$
$570$	0	0
$571$	−21.8687	−0.915177	−0.457589	−	0.889164i	$-0.651287\pi$
$571$	−21.8687	−0.915177	−0.457589	+	0.889164i	$0.651287\pi$
$572$	−9.98027	−0.417296
$573$	0	0
$574$	−16.8529	−0.703428
$575$	−10.3216	−0.430439
$576$	0	0
$577$	−10.2833	−0.428101	−0.214050	−	0.976823i	$-0.568666\pi$
$577$	−10.2833	−0.428101	−0.214050	+	0.976823i	$0.568666\pi$
$578$	80.5413	3.35008
$579$	0	0
$580$	0.445139	0.0184834
$581$	35.0654	1.45476
$582$	0	0
$583$	−4.86692	−0.201567
$584$	25.4650	1.05375
$585$	0	0
$586$	−3.65374	−0.150935
$587$	21.6507	0.893622	0.446811	−	0.894628i	$-0.352560\pi$
$587$	21.6507	0.893622	0.446811	+	0.894628i	$0.352560\pi$
$588$	0	0
$589$	−19.6792	−0.810869
$590$	−0.0124069	−0.000510786 0
$591$	0	0
$592$	−2.31343	−0.0950815
$593$	12.8896	0.529312	0.264656	−	0.964343i	$-0.414742\pi$
$593$	12.8896	0.529312	0.264656	+	0.964343i	$0.414742\pi$
$594$	0	0
$595$	−0.569394	−0.0233429
$596$	−3.81268	−0.156173
$597$	0	0
$598$	13.9416	0.570114
$599$	22.2972	0.911039	0.455520	−	0.890226i	$-0.349453\pi$
$599$	22.2972	0.911039	0.455520	+	0.890226i	$0.349453\pi$
$600$	0	0
$601$	−7.06663	−0.288254	−0.144127	−	0.989559i	$-0.546037\pi$
$601$	−7.06663	−0.288254	−0.144127	+	0.989559i	$0.546037\pi$
$602$	56.3009	2.29465
$603$	0	0
$604$	−51.7715	−2.10655
$605$	−0.0293558	−0.00119348
$606$	0	0
$607$	−10.6898	−0.433886	−0.216943	−	0.976184i	$-0.569609\pi$
$607$	−10.6898	−0.433886	−0.216943	+	0.976184i	$0.569609\pi$
$608$	−18.4788	−0.749414
$609$	0	0
$610$	−0.0685361	−0.00277495
$611$	−31.1994	−1.26219
$612$	0	0
$613$	24.7225	0.998533	0.499267	−	0.866448i	$-0.333603\pi$
$613$	24.7225	0.998533	0.499267	+	0.866448i	$0.333603\pi$
$614$	20.0904	0.810782
$615$	0	0
$616$	9.15427	0.368836
$617$	−47.5636	−1.91484	−0.957420	−	0.288700i	$-0.906777\pi$
$617$	−47.5636	−1.91484	−0.957420	+	0.288700i	$0.906777\pi$
$618$	0	0
$619$	19.1635	0.770246	0.385123	−	0.922865i	$-0.374159\pi$
$619$	19.1635	0.770246	0.385123	+	0.922865i	$0.374159\pi$
$620$	0.477403	0.0191730
$621$	0	0
$622$	74.1803	2.97436
$623$	−27.5378	−1.10328
$624$	0	0
$625$	24.9871	0.999483
$626$	62.7775	2.50909
$627$	0	0
$628$	−33.6777	−1.34389
$629$	16.5048	0.658091
$630$	0	0
$631$	4.32941	0.172351	0.0861755	−	0.996280i	$-0.472535\pi$
$631$	4.32941	0.172351	0.0861755	+	0.996280i	$0.472535\pi$
$632$	37.8399	1.50519
$633$	0	0
$634$	50.6417	2.01124
$635$	−0.158676	−0.00629685
$636$	0	0
$637$	0.883518	0.0350062
$638$	10.2594	0.406173
$639$	0	0
$640$	0.586158	0.0231699
$641$	−28.9219	−1.14234	−0.571172	−	0.820830i	$-0.693511\pi$
$641$	−28.9219	−1.14234	−0.571172	+	0.820830i	$0.693511\pi$
$642$	0	0
$643$	−33.7472	−1.33086	−0.665430	−	0.746460i	$-0.731752\pi$
$643$	−33.7472	−1.33086	−0.665430	+	0.746460i	$0.731752\pi$
$644$	−19.2566	−0.758817
$645$	0	0
$646$	−69.9589	−2.75250
$647$	43.6251	1.71508	0.857540	−	0.514418i	$-0.171992\pi$
$647$	43.6251	1.71508	0.857540	+	0.514418i	$0.171992\pi$
$648$	0	0
$649$	−0.181028	−0.00710597
$650$	−33.7565	−1.32404
$651$	0	0
$652$	73.5770	2.88150
$653$	−13.5296	−0.529455	−0.264727	−	0.964323i	$-0.585282\pi$
$653$	−13.5296	−0.529455	−0.264727	+	0.964323i	$0.585282\pi$
$654$	0	0
$655$	0.568968	0.0222314
$656$	2.68637	0.104885
$657$	0	0
$658$	68.0705	2.65367
$659$	3.51759	0.137026	0.0685129	−	0.997650i	$-0.478175\pi$
$659$	3.51759	0.137026	0.0685129	+	0.997650i	$0.478175\pi$
$660$	0	0
$661$	−43.5404	−1.69353	−0.846763	−	0.531971i	$-0.821452\pi$
$661$	−43.5404	−1.69353	−0.846763	+	0.531971i	$0.821452\pi$
$662$	−7.39301	−0.287337
$663$	0	0
$664$	−43.9393	−1.70518
$665$	0.331315	0.0128478
$666$	0	0
$667$	−9.07289	−0.351304
$668$	−65.9030	−2.54986
$669$	0	0
$670$	−0.987296	−0.0381426
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	31.6788	1.22113	0.610563	−	0.791968i	$-0.290943\pi$
$673$	31.6788	1.22113	0.610563	+	0.791968i	$0.290943\pi$
$674$	67.8819	2.61471
$675$	0	0
$676$	−15.9935	−0.615133
$677$	−0.472792	−0.0181709	−0.00908543	−	0.999959i	$-0.502892\pi$
$677$	−0.472792	−0.0181709	−0.00908543	+	0.999959i	$0.502892\pi$
$678$	0	0
$679$	45.1939	1.73438
$680$	0.713490	0.0273611
$681$	0	0
$682$	11.0030	0.421327
$683$	−1.89167	−0.0723828	−0.0361914	−	0.999345i	$-0.511523\pi$
$683$	−1.89167	−0.0723828	−0.0361914	+	0.999345i	$0.511523\pi$
$684$	0	0
$685$	0.461369	0.0176280
$686$	42.2444	1.61290
$687$	0	0
$688$	−8.97441	−0.342146
$689$	14.0764	0.536267
$690$	0	0
$691$	47.6079	1.81109	0.905545	−	0.424250i	$-0.139462\pi$
$691$	47.6079	1.81109	0.905545	+	0.424250i	$0.139462\pi$
$692$	60.2994	2.29224
$693$	0	0
$694$	13.4614	0.510989
$695$	−0.0462772	−0.00175539
$696$	0	0
$697$	−19.1655	−0.725946
$698$	−69.7676	−2.64074
$699$	0	0
$700$	46.6257	1.76228
$701$	−29.5010	−1.11424	−0.557119	−	0.830433i	$-0.688093\pi$
$701$	−29.5010	−1.11424	−0.557119	+	0.830433i	$0.688093\pi$
$702$	0	0
$703$	−9.60370	−0.362211
$704$	12.3436	0.465215
$705$	0	0
$706$	−20.2800	−0.763249
$707$	5.33822	0.200764
$708$	0	0
$709$	−24.1831	−0.908215	−0.454107	−	0.890947i	$-0.650042\pi$
$709$	−24.1831	−0.908215	−0.454107	+	0.890947i	$0.650042\pi$
$710$	0.324331	0.0121719
$711$	0	0
$712$	34.5068	1.29320
$713$	−9.73052	−0.364411
$714$	0	0
$715$	0.0849045	0.00317525
$716$	−54.9976	−2.05536
$717$	0	0
$718$	0.748601	0.0279375
$719$	−5.78255	−0.215653	−0.107826	−	0.994170i	$-0.534389\pi$
$719$	−5.78255	−0.215653	−0.107826	+	0.994170i	$0.534389\pi$
$720$	0	0
$721$	−23.4811	−0.874484
$722$	−3.65163	−0.135900
$723$	0	0
$724$	25.5790	0.950636
$725$	21.9680	0.815872
$726$	0	0
$727$	11.9924	0.444774	0.222387	−	0.974958i	$-0.428615\pi$
$727$	11.9924	0.444774	0.222387	+	0.974958i	$0.428615\pi$
$728$	−26.4765	−0.981283
$729$	0	0
$730$	−0.515304	−0.0190722
$731$	64.0266	2.36811
$732$	0	0
$733$	6.07851	0.224515	0.112257	−	0.993679i	$-0.464192\pi$
$733$	6.07851	0.224515	0.112257	+	0.993679i	$0.464192\pi$
$734$	−34.4416	−1.27126
$735$	0	0
$736$	−9.13695	−0.336792
$737$	−14.4055	−0.530633
$738$	0	0
$739$	27.1397	0.998350	0.499175	−	0.866501i	$-0.333636\pi$
$739$	27.1397	0.998350	0.499175	+	0.866501i	$0.333636\pi$
$740$	0.232979	0.00856446
$741$	0	0
$742$	−30.7117	−1.12746
$743$	−1.32191	−0.0484963	−0.0242482	−	0.999706i	$-0.507719\pi$
$743$	−1.32191	−0.0484963	−0.0242482	+	0.999706i	$0.507719\pi$
$744$	0	0
$745$	0.0324353	0.00118834
$746$	−17.6159	−0.644963
$747$	0	0
$748$	24.7628	0.905419
$749$	−16.5473	−0.604627
$750$	0	0
$751$	5.66385	0.206677	0.103338	−	0.994646i	$-0.467048\pi$
$751$	5.66385	0.206677	0.103338	+	0.994646i	$0.467048\pi$
$752$	−10.8505	−0.395677
$753$	0	0
$754$	−29.6728	−1.08062
$755$	0.440432	0.0160290
$756$	0	0
$757$	−43.1109	−1.56689	−0.783447	−	0.621459i	$-0.786540\pi$
$757$	−43.1109	−1.56689	−0.783447	+	0.621459i	$0.786540\pi$
$758$	−13.7940	−0.501021
$759$	0	0
$760$	−0.415160	−0.0150594
$761$	−15.5334	−0.563086	−0.281543	−	0.959549i	$-0.590846\pi$
$761$	−15.5334	−0.563086	−0.281543	+	0.959549i	$0.590846\pi$
$762$	0	0
$763$	9.72109	0.351927
$764$	−40.0537	−1.44909
$765$	0	0
$766$	−66.1011	−2.38833
$767$	0.523579	0.0189053
$768$	0	0
$769$	−44.5062	−1.60493	−0.802467	−	0.596697i	$-0.796479\pi$
$769$	−44.5062	−1.60493	−0.802467	+	0.596697i	$0.796479\pi$
$770$	−0.185244	−0.00667573
$771$	0	0
$772$	70.6993	2.54452
$773$	3.75321	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$773$	3.75321	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$774$	0	0
$775$	23.5603	0.846312
$776$	−56.6311	−2.03294
$777$	0	0
$778$	37.6126	1.34848
$779$	11.1519	0.399558
$780$	0	0
$781$	4.73226	0.169334
$782$	−34.5916	−1.23699
$783$	0	0
$784$	0.307269	0.0109739
$785$	0.286504	0.0102258
$786$	0	0
$787$	−44.9918	−1.60379	−0.801893	−	0.597468i	$-0.796174\pi$
$787$	−44.9918	−1.60379	−0.801893	+	0.597468i	$0.796174\pi$
$788$	−25.0910	−0.893832
$789$	0	0
$790$	−0.765721	−0.0272431
$791$	18.4272	0.655196
$792$	0	0
$793$	2.89225	0.102707
$794$	−30.1496	−1.06997
$795$	0	0
$796$	−34.6452	−1.22797
$797$	−43.6737	−1.54700	−0.773501	−	0.633795i	$-0.781496\pi$
$797$	−43.6737	−1.54700	−0.773501	+	0.633795i	$0.781496\pi$
$798$	0	0
$799$	77.4112	2.73861
$800$	22.1231	0.782170
$801$	0	0
$802$	−50.4711	−1.78220
$803$	−7.51871	−0.265330
$804$	0	0
$805$	0.163821	0.00577392
$806$	−31.8235	−1.12094
$807$	0	0
$808$	−6.68916	−0.235324
$809$	21.0958	0.741690	0.370845	−	0.928695i	$-0.379068\pi$
$809$	21.0958	0.741690	0.370845	+	0.928695i	$0.379068\pi$
$810$	0	0
$811$	47.7913	1.67818	0.839090	−	0.543992i	$-0.183088\pi$
$811$	47.7913	1.67818	0.839090	+	0.543992i	$0.183088\pi$
$812$	40.9851	1.43829
$813$	0	0
$814$	5.36960	0.188204
$815$	−0.625936	−0.0219256
$816$	0	0
$817$	−37.2553	−1.30340
$818$	−24.5573	−0.858626
$819$	0	0
$820$	−0.270536	−0.00944753
$821$	−8.42000	−0.293860	−0.146930	−	0.989147i	$-0.546939\pi$
$821$	−8.42000	−0.293860	−0.146930	+	0.989147i	$0.546939\pi$
$822$	0	0
$823$	21.4926	0.749186	0.374593	−	0.927189i	$-0.377782\pi$
$823$	21.4926	0.749186	0.374593	+	0.927189i	$0.377782\pi$
$824$	29.4235	1.02502
$825$	0	0
$826$	−1.14234	−0.0397471
$827$	−32.0863	−1.11575	−0.557875	−	0.829925i	$-0.688383\pi$
$827$	−32.0863	−1.11575	−0.557875	+	0.829925i	$0.688383\pi$
$828$	0	0
$829$	−43.5722	−1.51332	−0.756662	−	0.653806i	$-0.773171\pi$
$829$	−43.5722	−1.51332	−0.756662	+	0.653806i	$0.773171\pi$
$830$	0.889147	0.0308627
$831$	0	0
$832$	−35.7007	−1.23770
$833$	−2.19217	−0.0759541
$834$	0	0
$835$	0.560652	0.0194022
$836$	−14.4088	−0.498339
$837$	0	0
$838$	−12.1279	−0.418951
$839$	9.71778	0.335495	0.167748	−	0.985830i	$-0.446351\pi$
$839$	9.71778	0.335495	0.167748	+	0.985830i	$0.446351\pi$
$840$	0	0
$841$	−9.68959	−0.334124
$842$	−51.4246	−1.77221
$843$	0	0
$844$	−25.4116	−0.874704
$845$	0.136060	0.00468061
$846$	0	0
$847$	−2.70286	−0.0928715
$848$	4.89547	0.168111
$849$	0	0
$850$	83.7559	2.87281
$851$	−4.74861	−0.162780
$852$	0	0
$853$	10.1506	0.347551	0.173776	−	0.984785i	$-0.444403\pi$
$853$	10.1506	0.347551	0.173776	+	0.984785i	$0.444403\pi$
$854$	−6.31030	−0.215934
$855$	0	0
$856$	20.7350	0.708707
$857$	−1.00340	−0.0342754	−0.0171377	−	0.999853i	$-0.505455\pi$
$857$	−1.00340	−0.0342754	−0.0171377	+	0.999853i	$0.505455\pi$
$858$	0	0
$859$	−30.9914	−1.05741	−0.528706	−	0.848805i	$-0.677323\pi$
$859$	−30.9914	−1.05741	−0.528706	+	0.848805i	$0.677323\pi$
$860$	0.903785	0.0308188
$861$	0	0
$862$	1.97113	0.0671368
$863$	36.5669	1.24475	0.622377	−	0.782718i	$-0.286167\pi$
$863$	36.5669	1.24475	0.622377	+	0.782718i	$0.286167\pi$
$864$	0	0
$865$	−0.512981	−0.0174419
$866$	67.3655	2.28917
$867$	0	0
$868$	43.9558	1.49196
$869$	−11.1725	−0.379002
$870$	0	0
$871$	41.6643	1.41174
$872$	−12.1812	−0.412507
$873$	0	0
$874$	20.1279	0.680836
$875$	−0.793379	−0.0268211
$876$	0	0
$877$	4.64834	0.156963	0.0784816	−	0.996916i	$-0.474993\pi$
$877$	4.64834	0.156963	0.0784816	+	0.996916i	$0.474993\pi$
$878$	−50.4763	−1.70349
$879$	0	0
$880$	0.0295280	0.000995389 0
$881$	29.7857	1.00351	0.501753	−	0.865011i	$-0.332689\pi$
$881$	29.7857	1.00351	0.501753	+	0.865011i	$0.332689\pi$
$882$	0	0
$883$	−43.9469	−1.47893	−0.739466	−	0.673194i	$-0.764922\pi$
$883$	−43.9469	−1.47893	−0.739466	+	0.673194i	$0.764922\pi$
$884$	−71.6204	−2.40886
$885$	0	0
$886$	9.10858	0.306009
$887$	31.7216	1.06511	0.532554	−	0.846396i	$-0.321233\pi$
$887$	31.7216	1.06511	0.532554	+	0.846396i	$0.321233\pi$
$888$	0	0
$889$	−14.6097	−0.489993
$890$	−0.698272	−0.0234061
$891$	0	0
$892$	3.54258	0.118614
$893$	−45.0434	−1.50732
$894$	0	0
$895$	0.467878	0.0156394
$896$	53.9691	1.80298
$897$	0	0
$898$	16.7286	0.558242
$899$	20.7101	0.690720
$900$	0	0
$901$	−34.9260	−1.16355
$902$	−6.23522	−0.207610
$903$	0	0
$904$	−23.0906	−0.767981
$905$	−0.217607	−0.00723349
$906$	0	0
$907$	−43.3090	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$907$	−43.3090	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$908$	−74.6841	−2.47848
$909$	0	0
$910$	0.535772	0.0177607
$911$	39.7332	1.31642	0.658210	−	0.752834i	$-0.271314\pi$
$911$	39.7332	1.31642	0.658210	+	0.752834i	$0.271314\pi$
$912$	0	0
$913$	12.9734	0.429357
$914$	7.84032	0.259335
$915$	0	0
$916$	34.1209	1.12739
$917$	52.3864	1.72995
$918$	0	0
$919$	−47.8215	−1.57749	−0.788744	−	0.614722i	$-0.789268\pi$
$919$	−47.8215	−1.57749	−0.788744	+	0.614722i	$0.789268\pi$
$920$	−0.205278	−0.00676783
$921$	0	0
$922$	96.3015	3.17152
$923$	−13.6869	−0.450510
$924$	0	0
$925$	11.4977	0.378043
$926$	43.7487	1.43767
$927$	0	0
$928$	19.4467	0.638371
$929$	−33.7455	−1.10715	−0.553576	−	0.832798i	$-0.686737\pi$
$929$	−33.7455	−1.10715	−0.553576	+	0.832798i	$0.686737\pi$
$930$	0	0
$931$	1.27556	0.0418048
$932$	85.5907	2.80362
$933$	0	0
$934$	57.0826	1.86780
$935$	−0.210663	−0.00688943
$936$	0	0
$937$	28.0948	0.917817	0.458908	−	0.888484i	$-0.348241\pi$
$937$	28.0948	0.917817	0.458908	+	0.888484i	$0.348241\pi$
$938$	−90.9029	−2.96808
$939$	0	0
$940$	1.09272	0.0356406
$941$	2.34219	0.0763531	0.0381765	−	0.999271i	$-0.487845\pi$
$941$	2.34219	0.0763531	0.0381765	+	0.999271i	$0.487845\pi$
$942$	0	0
$943$	5.51412	0.179564
$944$	0.182090	0.00592652
$945$	0	0
$946$	20.8301	0.677245
$947$	−0.361908	−0.0117604	−0.00588021	−	0.999983i	$-0.501872\pi$
$947$	−0.361908	−0.0117604	−0.00588021	+	0.999983i	$0.501872\pi$
$948$	0	0
$949$	21.7460	0.705906
$950$	−48.7352	−1.58118
$951$	0	0
$952$	65.6929	2.12912
$953$	−20.2966	−0.657473	−0.328736	−	0.944422i	$-0.606623\pi$
$953$	−20.2966	−0.657473	−0.328736	+	0.944422i	$0.606623\pi$
$954$	0	0
$955$	0.340746	0.0110263
$956$	−54.9918	−1.77856
$957$	0	0
$958$	81.6618	2.63837
$959$	42.4794	1.37173
$960$	0	0
$961$	−8.78880	−0.283510
$962$	−15.5303	−0.500716
$963$	0	0
$964$	−36.7206	−1.18269
$965$	−0.601455	−0.0193615
$966$	0	0
$967$	−18.7607	−0.603303	−0.301651	−	0.953418i	$-0.597538\pi$
$967$	−18.7607	−0.603303	−0.301651	+	0.953418i	$0.597538\pi$
$968$	3.38688	0.108858
$969$	0	0
$970$	1.14598	0.0367950
$971$	−56.5296	−1.81412	−0.907061	−	0.420999i	$-0.861680\pi$
$971$	−56.5296	−1.81412	−0.907061	+	0.420999i	$0.861680\pi$
$972$	0	0
$973$	−4.26086	−0.136597
$974$	27.7108	0.887911
$975$	0	0
$976$	1.00587	0.0321970
$977$	−3.17054	−0.101435	−0.0507173	−	0.998713i	$-0.516151\pi$
$977$	−3.17054	−0.101435	−0.0507173	+	0.998713i	$0.516151\pi$
$978$	0	0
$979$	−10.1884	−0.325622
$980$	−0.0309441	−0.000988474 0
$981$	0	0
$982$	−80.0284	−2.55381
$983$	40.8147	1.30179	0.650893	−	0.759169i	$-0.274394\pi$
$983$	40.8147	1.30179	0.650893	+	0.759169i	$0.274394\pi$
$984$	0	0
$985$	0.213455	0.00680125
$986$	73.6235	2.34465
$987$	0	0
$988$	41.6739	1.32582
$989$	−18.4211	−0.585757
$990$	0	0
$991$	−56.3255	−1.78924	−0.894619	−	0.446829i	$-0.852553\pi$
$991$	−56.3255	−1.78924	−0.894619	+	0.446829i	$0.852553\pi$
$992$	20.8563	0.662188
$993$	0	0
$994$	29.8620	0.947164
$995$	0.294735	0.00934372
$996$	0	0
$997$	−44.4996	−1.40932	−0.704658	−	0.709547i	$-0.748900\pi$
$997$	−44.4996	−1.40932	−0.704658	+	0.709547i	$0.748900\pi$
$998$	53.6430	1.69804
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.13		13
3.2	odd	2		2013.2.a.g.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.1	✓	13	3.2	odd	2
6039.2.a.h.1.13		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)

\(f(q)\)	\(=\)	\(q+2.33467 q^{2} +3.45069 q^{4} -0.0293558 q^{5} -2.70286 q^{7} +3.38688 q^{8} +O(q^{10})\)	\(q+2.33467 q^{2} +3.45069 q^{4} -0.0293558 q^{5} -2.70286 q^{7} +3.38688 q^{8} -0.0685361 q^{10} -1.00000 q^{11} +2.89225 q^{13} -6.31030 q^{14} +1.00587 q^{16} -7.17621 q^{17} +4.17563 q^{19} -0.101298 q^{20} -2.33467 q^{22} +2.06467 q^{23} -4.99914 q^{25} +6.75246 q^{26} -9.32674 q^{28} -4.39436 q^{29} -4.71288 q^{31} -4.42539 q^{32} -16.7541 q^{34} +0.0793448 q^{35} -2.29994 q^{37} +9.74873 q^{38} -0.0994245 q^{40} +2.67070 q^{41} -8.92207 q^{43} -3.45069 q^{44} +4.82032 q^{46} -10.7872 q^{47} +0.305477 q^{49} -11.6713 q^{50} +9.98027 q^{52} +4.86692 q^{53} +0.0293558 q^{55} -9.15427 q^{56} -10.2594 q^{58} +0.181028 q^{59} +1.00000 q^{61} -11.0030 q^{62} -12.3436 q^{64} -0.0849045 q^{65} +14.4055 q^{67} -24.7628 q^{68} +0.185244 q^{70} -4.73226 q^{71} +7.51871 q^{73} -5.36960 q^{74} +14.4088 q^{76} +2.70286 q^{77} +11.1725 q^{79} -0.0295280 q^{80} +6.23522 q^{82} -12.9734 q^{83} +0.210663 q^{85} -20.8301 q^{86} -3.38688 q^{88} +10.1884 q^{89} -7.81737 q^{91} +7.12452 q^{92} -25.1846 q^{94} -0.122579 q^{95} -16.7207 q^{97} +0.713188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 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

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Groups

Database

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