LMFDB - Embedded newform 6039.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$1$
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} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 $ x^13 - 4x^12 - 11x^11 + 55x^10 + 32x^9 - 266x^8 + 13x^7 + 534x^6 - 141x^5 - 404x^4 + 98x^3 + 118x^2 - 16x - 11
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			1.2
Root			$2.60074$ 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)

$f(q)$	$=$	$q-2.60074 q^{2} +4.76385 q^{4} +0.540623 q^{5} +4.31275 q^{7} -7.18805 q^{8} +O(q^{10})$	\(q-2.60074 q^{2} +4.76385 q^{4} +0.540623 q^{5} +4.31275 q^{7} -7.18805 q^{8} -1.40602 q^{10} +1.00000 q^{11} -6.04924 q^{13} -11.2163 q^{14} +9.16656 q^{16} +1.53220 q^{17} +5.71304 q^{19} +2.57545 q^{20} -2.60074 q^{22} -2.05269 q^{23} -4.70773 q^{25} +15.7325 q^{26} +20.5453 q^{28} +0.0227019 q^{29} -7.86398 q^{31} -9.46374 q^{32} -3.98485 q^{34} +2.33157 q^{35} -6.22862 q^{37} -14.8581 q^{38} -3.88603 q^{40} -7.86011 q^{41} -2.59570 q^{43} +4.76385 q^{44} +5.33851 q^{46} +6.08525 q^{47} +11.5998 q^{49} +12.2436 q^{50} -28.8177 q^{52} +5.04706 q^{53} +0.540623 q^{55} -31.0003 q^{56} -0.0590417 q^{58} -5.62146 q^{59} -1.00000 q^{61} +20.4522 q^{62} +6.27960 q^{64} -3.27036 q^{65} +8.08014 q^{67} +7.29916 q^{68} -6.06382 q^{70} +8.56121 q^{71} +13.6253 q^{73} +16.1990 q^{74} +27.2161 q^{76} +4.31275 q^{77} -8.84218 q^{79} +4.95566 q^{80} +20.4421 q^{82} -9.24421 q^{83} +0.828342 q^{85} +6.75073 q^{86} -7.18805 q^{88} -11.7669 q^{89} -26.0889 q^{91} -9.77869 q^{92} -15.8262 q^{94} +3.08860 q^{95} -9.60720 q^{97} -30.1681 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + 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 $

$2$	−2.60074	−1.83900	−0.919500	−	0.393089i	$-0.871406\pi$
$2$	−2.60074	−1.83900	−0.919500	+	0.393089i	$0.871406\pi$
$3$	0	0
$3$	0	0
$4$	4.76385	2.38192
$5$	0.540623	0.241774	0.120887	−	0.992666i	$-0.461426\pi$
$5$	0.540623	0.241774	0.120887	+	0.992666i	$0.461426\pi$
$6$	0	0
$7$	4.31275	1.63007	0.815033	−	0.579414i	$-0.196719\pi$
$7$	4.31275	1.63007	0.815033	+	0.579414i	$0.196719\pi$
$8$	−7.18805	−2.54136
$9$	0	0
$10$	−1.40602	−0.444623
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.04924	−1.67776	−0.838879	−	0.544319i	$-0.816788\pi$
$13$	−6.04924	−1.67776	−0.838879	+	0.544319i	$0.816788\pi$
$14$	−11.2163	−2.99769
$15$	0	0
$16$	9.16656	2.29164
$17$	1.53220	0.371613	0.185806	−	0.982586i	$-0.440510\pi$
$17$	1.53220	0.371613	0.185806	+	0.982586i	$0.440510\pi$
$18$	0	0
$19$	5.71304	1.31066	0.655331	−	0.755342i	$-0.272529\pi$
$19$	5.71304	1.31066	0.655331	+	0.755342i	$0.272529\pi$
$20$	2.57545	0.575888
$21$	0	0
$22$	−2.60074	−0.554480
$23$	−2.05269	−0.428015	−0.214007	−	0.976832i	$-0.568652\pi$
$23$	−2.05269	−0.428015	−0.214007	+	0.976832i	$0.568652\pi$
$24$	0	0
$25$	−4.70773	−0.941545
$26$	15.7325	3.08540
$27$	0	0
$28$	20.5453	3.88270
$29$	0.0227019	0.00421563	0.00210782	−	0.999998i	$-0.499329\pi$
$29$	0.0227019	0.00421563	0.00210782	+	0.999998i	$0.499329\pi$
$30$	0	0
$31$	−7.86398	−1.41241	−0.706206	−	0.708006i	$-0.749595\pi$
$31$	−7.86398	−1.41241	−0.706206	+	0.708006i	$0.749595\pi$
$32$	−9.46374	−1.67297
$33$	0	0
$34$	−3.98485	−0.683396
$35$	2.33157	0.394108
$36$	0	0
$37$	−6.22862	−1.02398	−0.511990	−	0.858992i	$-0.671091\pi$
$37$	−6.22862	−1.02398	−0.511990	+	0.858992i	$0.671091\pi$
$38$	−14.8581	−2.41031
$39$	0	0
$40$	−3.88603	−0.614435
$41$	−7.86011	−1.22754	−0.613772	−	0.789483i	$-0.710349\pi$
$41$	−7.86011	−1.22754	−0.613772	+	0.789483i	$0.710349\pi$
$42$	0	0
$43$	−2.59570	−0.395840	−0.197920	−	0.980218i	$-0.563419\pi$
$43$	−2.59570	−0.395840	−0.197920	+	0.980218i	$0.563419\pi$
$44$	4.76385	0.718177
$45$	0	0
$46$	5.33851	0.787120
$47$	6.08525	0.887625	0.443813	−	0.896120i	$-0.353626\pi$
$47$	6.08525	0.887625	0.443813	+	0.896120i	$0.353626\pi$
$48$	0	0
$49$	11.5998	1.65712
$50$	12.2436	1.73150
$51$	0	0
$52$	−28.8177	−3.99629
$53$	5.04706	0.693268	0.346634	−	0.938001i	$-0.387325\pi$
$53$	5.04706	0.693268	0.346634	+	0.938001i	$0.387325\pi$
$54$	0	0
$55$	0.540623	0.0728976
$56$	−31.0003	−4.14259
$57$	0	0
$58$	−0.0590417	−0.00775255
$59$	−5.62146	−0.731852	−0.365926	−	0.930644i	$-0.619248\pi$
$59$	−5.62146	−0.731852	−0.365926	+	0.930644i	$0.619248\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	20.4522	2.59743
$63$	0	0
$64$	6.27960	0.784950
$65$	−3.27036	−0.405638
$66$	0	0
$67$	8.08014	0.987146	0.493573	−	0.869704i	$-0.335691\pi$
$67$	8.08014	0.987146	0.493573	+	0.869704i	$0.335691\pi$
$68$	7.29916	0.885153
$69$	0	0
$70$	−6.06382	−0.724765
$71$	8.56121	1.01603	0.508014	−	0.861349i	$-0.330380\pi$
$71$	8.56121	1.01603	0.508014	+	0.861349i	$0.330380\pi$
$72$	0	0
$73$	13.6253	1.59472	0.797358	−	0.603506i	$-0.206230\pi$
$73$	13.6253	1.59472	0.797358	+	0.603506i	$0.206230\pi$
$74$	16.1990	1.88310
$75$	0	0
$76$	27.2161	3.12190
$77$	4.31275	0.491483
$78$	0	0
$79$	−8.84218	−0.994823	−0.497411	−	0.867515i	$-0.665716\pi$
$79$	−8.84218	−0.994823	−0.497411	+	0.867515i	$0.665716\pi$
$80$	4.95566	0.554059
$81$	0	0
$82$	20.4421	2.25745
$83$	−9.24421	−1.01468	−0.507342	−	0.861745i	$-0.669372\pi$
$83$	−9.24421	−1.01468	−0.507342	+	0.861745i	$0.669372\pi$
$84$	0	0
$85$	0.828342	0.0898463
$86$	6.75073	0.727950
$87$	0	0
$88$	−7.18805	−0.766249
$89$	−11.7669	−1.24729	−0.623644	−	0.781708i	$-0.714349\pi$
$89$	−11.7669	−1.24729	−0.623644	+	0.781708i	$0.714349\pi$
$90$	0	0
$91$	−26.0889	−2.73486
$92$	−9.77869	−1.01950
$93$	0	0
$94$	−15.8262	−1.63234
$95$	3.08860	0.316884
$96$	0	0
$97$	−9.60720	−0.975463	−0.487732	−	0.872994i	$-0.662175\pi$
$97$	−9.60720	−0.975463	−0.487732	+	0.872994i	$0.662175\pi$
$98$	−30.1681	−3.04744
$99$	0	0
$100$	−22.4269	−2.24269
$101$	−15.3839	−1.53075	−0.765377	−	0.643582i	$-0.777448\pi$
$101$	−15.3839	−1.53075	−0.765377	+	0.643582i	$0.777448\pi$
$102$	0	0
$103$	−10.3446	−1.01928	−0.509642	−	0.860386i	$-0.670222\pi$
$103$	−10.3446	−1.01928	−0.509642	+	0.860386i	$0.670222\pi$
$104$	43.4823	4.26379
$105$	0	0
$106$	−13.1261	−1.27492
$107$	−5.66352	−0.547514	−0.273757	−	0.961799i	$-0.588266\pi$
$107$	−5.66352	−0.547514	−0.273757	+	0.961799i	$0.588266\pi$
$108$	0	0
$109$	−14.4612	−1.38513	−0.692564	−	0.721357i	$-0.743519\pi$
$109$	−14.4612	−1.38513	−0.692564	+	0.721357i	$0.743519\pi$
$110$	−1.40602	−0.134059
$111$	0	0
$112$	39.5331	3.73553
$113$	−3.56615	−0.335475	−0.167738	−	0.985832i	$-0.553646\pi$
$113$	−3.56615	−0.335475	−0.167738	+	0.985832i	$0.553646\pi$
$114$	0	0
$115$	−1.10973	−0.103483
$116$	0.108148	0.0100413
$117$	0	0
$118$	14.6200	1.34588
$119$	6.60799	0.605753
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.60074	0.235460
$123$	0	0
$124$	−37.4628	−3.36426
$125$	−5.24822	−0.469415
$126$	0	0
$127$	−15.5050	−1.37585	−0.687925	−	0.725782i	$-0.741478\pi$
$127$	−15.5050	−1.37585	−0.687925	+	0.725782i	$0.741478\pi$
$128$	2.59587	0.229445
$129$	0	0
$130$	8.50535	0.745969
$131$	−18.8326	−1.64541	−0.822707	−	0.568465i	$-0.807537\pi$
$131$	−18.8326	−1.64541	−0.822707	+	0.568465i	$0.807537\pi$
$132$	0	0
$133$	24.6389	2.13646
$134$	−21.0143	−1.81536
$135$	0	0
$136$	−11.0135	−0.944402
$137$	16.6626	1.42359	0.711793	−	0.702390i	$-0.247884\pi$
$137$	16.6626	1.42359	0.711793	+	0.702390i	$0.247884\pi$
$138$	0	0
$139$	−17.2228	−1.46082	−0.730411	−	0.683008i	$-0.760671\pi$
$139$	−17.2228	−1.46082	−0.730411	+	0.683008i	$0.760671\pi$
$140$	11.1073	0.938735
$141$	0	0
$142$	−22.2655	−1.86848
$143$	−6.04924	−0.505863
$144$	0	0
$145$	0.0122732	0.00101923
$146$	−35.4358	−2.93269
$147$	0	0
$148$	−29.6722	−2.43904
$149$	8.95498	0.733620	0.366810	−	0.930296i	$-0.380450\pi$
$149$	8.95498	0.733620	0.366810	+	0.930296i	$0.380450\pi$
$150$	0	0
$151$	−12.6384	−1.02849	−0.514247	−	0.857642i	$-0.671929\pi$
$151$	−12.6384	−1.02849	−0.514247	+	0.857642i	$0.671929\pi$
$152$	−41.0656	−3.33086
$153$	0	0
$154$	−11.2163	−0.903839
$155$	−4.25145	−0.341485
$156$	0	0
$157$	16.8355	1.34362	0.671810	−	0.740723i	$-0.265517\pi$
$157$	16.8355	1.34362	0.671810	+	0.740723i	$0.265517\pi$
$158$	22.9962	1.82948
$159$	0	0
$160$	−5.11632	−0.404480
$161$	−8.85273	−0.697693
$162$	0	0
$163$	6.53716	0.512030	0.256015	−	0.966673i	$-0.417590\pi$
$163$	6.53716	0.512030	0.256015	+	0.966673i	$0.417590\pi$
$164$	−37.4444	−2.92392
$165$	0	0
$166$	24.0418	1.86600
$167$	1.74070	0.134699	0.0673497	−	0.997729i	$-0.478546\pi$
$167$	1.74070	0.134699	0.0673497	+	0.997729i	$0.478546\pi$
$168$	0	0
$169$	23.5933	1.81487
$170$	−2.15430	−0.165227
$171$	0	0
$172$	−12.3655	−0.942861
$173$	−8.99614	−0.683964	−0.341982	−	0.939707i	$-0.611098\pi$
$173$	−8.99614	−0.683964	−0.341982	+	0.939707i	$0.611098\pi$
$174$	0	0
$175$	−20.3032	−1.53478
$176$	9.16656	0.690956
$177$	0	0
$178$	30.6026	2.29376
$179$	12.0429	0.900129	0.450064	−	0.892996i	$-0.351401\pi$
$179$	12.0429	0.900129	0.450064	+	0.892996i	$0.351401\pi$
$180$	0	0
$181$	22.4118	1.66585	0.832926	−	0.553384i	$-0.186664\pi$
$181$	22.4118	1.66585	0.832926	+	0.553384i	$0.186664\pi$
$182$	67.8503	5.02940
$183$	0	0
$184$	14.7548	1.08774
$185$	−3.36734	−0.247572
$186$	0	0
$187$	1.53220	0.112045
$188$	28.9892	2.11426
$189$	0	0
$190$	−8.03265	−0.582750
$191$	−19.2565	−1.39335	−0.696677	−	0.717385i	$-0.745339\pi$
$191$	−19.2565	−1.39335	−0.696677	+	0.717385i	$0.745339\pi$
$192$	0	0
$193$	11.2997	0.813370	0.406685	−	0.913569i	$-0.366685\pi$
$193$	11.2997	0.813370	0.406685	+	0.913569i	$0.366685\pi$
$194$	24.9858	1.79388
$195$	0	0
$196$	55.2598	3.94713
$197$	−11.8065	−0.841176	−0.420588	−	0.907252i	$-0.638176\pi$
$197$	−11.8065	−0.841176	−0.420588	+	0.907252i	$0.638176\pi$
$198$	0	0
$199$	5.62741	0.398916	0.199458	−	0.979906i	$-0.436082\pi$
$199$	5.62741	0.398916	0.199458	+	0.979906i	$0.436082\pi$
$200$	33.8394	2.39281
$201$	0	0
$202$	40.0095	2.81506
$203$	0.0979075	0.00687176
$204$	0	0
$205$	−4.24936	−0.296788
$206$	26.9036	1.87446
$207$	0	0
$208$	−55.4507	−3.84482
$209$	5.71304	0.395179
$210$	0	0
$211$	−7.91724	−0.545045	−0.272523	−	0.962149i	$-0.587858\pi$
$211$	−7.91724	−0.545045	−0.272523	+	0.962149i	$0.587858\pi$
$212$	24.0435	1.65131
$213$	0	0
$214$	14.7294	1.00688
$215$	−1.40329	−0.0957038
$216$	0	0
$217$	−33.9154	−2.30233
$218$	37.6097	2.54725
$219$	0	0
$220$	2.57545	0.173637
$221$	−9.26863	−0.623476
$222$	0	0
$223$	14.3852	0.963302	0.481651	−	0.876363i	$-0.340037\pi$
$223$	14.3852	0.963302	0.481651	+	0.876363i	$0.340037\pi$
$224$	−40.8147	−2.72705
$225$	0	0
$226$	9.27463	0.616940
$227$	−1.89780	−0.125961	−0.0629806	−	0.998015i	$-0.520061\pi$
$227$	−1.89780	−0.125961	−0.0629806	+	0.998015i	$0.520061\pi$
$228$	0	0
$229$	−15.0613	−0.995276	−0.497638	−	0.867385i	$-0.665799\pi$
$229$	−15.0613	−0.995276	−0.497638	+	0.867385i	$0.665799\pi$
$230$	2.88612	0.190305
$231$	0	0
$232$	−0.163182	−0.0107134
$233$	−11.0715	−0.725318	−0.362659	−	0.931922i	$-0.618131\pi$
$233$	−11.0715	−0.725318	−0.362659	+	0.931922i	$0.618131\pi$
$234$	0	0
$235$	3.28983	0.214605
$236$	−26.7798	−1.74322
$237$	0	0
$238$	−17.1857	−1.11398
$239$	−29.2019	−1.88891	−0.944457	−	0.328635i	$-0.893411\pi$
$239$	−29.2019	−1.88891	−0.944457	+	0.328635i	$0.893411\pi$
$240$	0	0
$241$	−13.1945	−0.849932	−0.424966	−	0.905209i	$-0.639714\pi$
$241$	−13.1945	−0.849932	−0.424966	+	0.905209i	$0.639714\pi$
$242$	−2.60074	−0.167182
$243$	0	0
$244$	−4.76385	−0.304974
$245$	6.27113	0.400648
$246$	0	0
$247$	−34.5595	−2.19897
$248$	56.5267	3.58945
$249$	0	0
$250$	13.6493	0.863255
$251$	21.4744	1.35545	0.677727	−	0.735314i	$-0.262965\pi$
$251$	21.4744	1.35545	0.677727	+	0.735314i	$0.262965\pi$
$252$	0	0
$253$	−2.05269	−0.129051
$254$	40.3246	2.53019
$255$	0	0
$256$	−19.3104	−1.20690
$257$	−17.8382	−1.11271	−0.556357	−	0.830943i	$-0.687801\pi$
$257$	−17.8382	−1.11271	−0.556357	+	0.830943i	$0.687801\pi$
$258$	0	0
$259$	−26.8625	−1.66915
$260$	−15.5795	−0.966200
$261$	0	0
$262$	48.9788	3.02592
$263$	29.8044	1.83782	0.918908	−	0.394472i	$-0.129072\pi$
$263$	29.8044	1.83782	0.918908	+	0.394472i	$0.129072\pi$
$264$	0	0
$265$	2.72856	0.167614
$266$	−64.0794	−3.92896
$267$	0	0
$268$	38.4926	2.35131
$269$	25.4442	1.55136	0.775681	−	0.631125i	$-0.217407\pi$
$269$	25.4442	1.55136	0.775681	+	0.631125i	$0.217407\pi$
$270$	0	0
$271$	−4.56237	−0.277144	−0.138572	−	0.990352i	$-0.544251\pi$
$271$	−4.56237	−0.277144	−0.138572	+	0.990352i	$0.544251\pi$
$272$	14.0450	0.851602
$273$	0	0
$274$	−43.3352	−2.61797
$275$	−4.70773	−0.283887
$276$	0	0
$277$	17.4846	1.05055	0.525273	−	0.850934i	$-0.323963\pi$
$277$	17.4846	1.05055	0.525273	+	0.850934i	$0.323963\pi$
$278$	44.7921	2.68645
$279$	0	0
$280$	−16.7595	−1.00157
$281$	23.6079	1.40833	0.704165	−	0.710036i	$-0.251321\pi$
$281$	23.6079	1.40833	0.704165	+	0.710036i	$0.251321\pi$
$282$	0	0
$283$	−2.59000	−0.153960	−0.0769799	−	0.997033i	$-0.524528\pi$
$283$	−2.59000	−0.153960	−0.0769799	+	0.997033i	$0.524528\pi$
$284$	40.7843	2.42010
$285$	0	0
$286$	15.7325	0.930282
$287$	−33.8987	−2.00098
$288$	0	0
$289$	−14.6524	−0.861904
$290$	−0.0319193	−0.00187437
$291$	0	0
$292$	64.9087	3.79849
$293$	30.1955	1.76404	0.882020	−	0.471213i	$-0.156184\pi$
$293$	30.1955	1.76404	0.882020	+	0.471213i	$0.156184\pi$
$294$	0	0
$295$	−3.03909	−0.176943
$296$	44.7717	2.60230
$297$	0	0
$298$	−23.2896	−1.34913
$299$	12.4172	0.718105
$300$	0	0
$301$	−11.1946	−0.645245
$302$	32.8691	1.89140
$303$	0	0
$304$	52.3689	3.00356
$305$	−0.540623	−0.0309560
$306$	0	0
$307$	15.3667	0.877022	0.438511	−	0.898726i	$-0.355506\pi$
$307$	15.3667	0.877022	0.438511	+	0.898726i	$0.355506\pi$
$308$	20.5453	1.17068
$309$	0	0
$310$	11.0569	0.627991
$311$	22.5782	1.28029	0.640145	−	0.768254i	$-0.278874\pi$
$311$	22.5782	1.28029	0.640145	+	0.768254i	$0.278874\pi$
$312$	0	0
$313$	−1.93752	−0.109515	−0.0547576	−	0.998500i	$-0.517439\pi$
$313$	−1.93752	−0.109515	−0.0547576	+	0.998500i	$0.517439\pi$
$314$	−43.7848	−2.47092
$315$	0	0
$316$	−42.1228	−2.36959
$317$	−17.4823	−0.981901	−0.490950	−	0.871187i	$-0.663350\pi$
$317$	−17.4823	−0.981901	−0.490950	+	0.871187i	$0.663350\pi$
$318$	0	0
$319$	0.0227019	0.00127106
$320$	3.39490	0.189781
$321$	0	0
$322$	23.0236	1.28306
$323$	8.75351	0.487058
$324$	0	0
$325$	28.4782	1.57968
$326$	−17.0014	−0.941623
$327$	0	0
$328$	56.4989	3.11963
$329$	26.2442	1.44689
$330$	0	0
$331$	−3.64398	−0.200291	−0.100146	−	0.994973i	$-0.531931\pi$
$331$	−3.64398	−0.200291	−0.100146	+	0.994973i	$0.531931\pi$
$332$	−44.0380	−2.41690
$333$	0	0
$334$	−4.52711	−0.247712
$335$	4.36831	0.238666
$336$	0	0
$337$	−25.1928	−1.37234	−0.686169	−	0.727442i	$-0.740709\pi$
$337$	−25.1928	−1.37234	−0.686169	+	0.727442i	$0.740709\pi$
$338$	−61.3600	−3.33755
$339$	0	0
$340$	3.94610	0.214007
$341$	−7.86398	−0.425858
$342$	0	0
$343$	19.8378	1.07114
$344$	18.6580	1.00597
$345$	0	0
$346$	23.3966	1.25781
$347$	−15.0795	−0.809510	−0.404755	−	0.914425i	$-0.632643\pi$
$347$	−15.0795	−0.809510	−0.404755	+	0.914425i	$0.632643\pi$
$348$	0	0
$349$	18.4363	0.986875	0.493437	−	0.869781i	$-0.335740\pi$
$349$	18.4363	0.986875	0.493437	+	0.869781i	$0.335740\pi$
$350$	52.8035	2.82246
$351$	0	0
$352$	−9.46374	−0.504419
$353$	−0.328084	−0.0174622	−0.00873109	−	0.999962i	$-0.502779\pi$
$353$	−0.328084	−0.0174622	−0.00873109	+	0.999962i	$0.502779\pi$
$354$	0	0
$355$	4.62839	0.245649
$356$	−56.0557	−2.97095
$357$	0	0
$358$	−31.3205	−1.65534
$359$	−9.73592	−0.513842	−0.256921	−	0.966432i	$-0.582708\pi$
$359$	−9.73592	−0.513842	−0.256921	+	0.966432i	$0.582708\pi$
$360$	0	0
$361$	13.6388	0.717833
$362$	−58.2872	−3.06351
$363$	0	0
$364$	−124.283	−6.51422
$365$	7.36613	0.385561
$366$	0	0
$367$	−7.31107	−0.381635	−0.190817	−	0.981626i	$-0.561114\pi$
$367$	−7.31107	−0.381635	−0.190817	+	0.981626i	$0.561114\pi$
$368$	−18.8161	−0.980856
$369$	0	0
$370$	8.75757	0.455284
$371$	21.7667	1.13007
$372$	0	0
$373$	17.6033	0.911464	0.455732	−	0.890117i	$-0.349377\pi$
$373$	17.6033	0.911464	0.455732	+	0.890117i	$0.349377\pi$
$374$	−3.98485	−0.206052
$375$	0	0
$376$	−43.7411	−2.25578
$377$	−0.137329	−0.00707280
$378$	0	0
$379$	23.1208	1.18763	0.593817	−	0.804600i	$-0.297620\pi$
$379$	23.1208	1.18763	0.593817	+	0.804600i	$0.297620\pi$
$380$	14.7136	0.754794
$381$	0	0
$382$	50.0812	2.56238
$383$	−10.0134	−0.511663	−0.255831	−	0.966721i	$-0.582349\pi$
$383$	−10.0134	−0.511663	−0.255831	+	0.966721i	$0.582349\pi$
$384$	0	0
$385$	2.33157	0.118828
$386$	−29.3876	−1.49579
$387$	0	0
$388$	−45.7672	−2.32348
$389$	−2.12966	−0.107978	−0.0539889	−	0.998542i	$-0.517194\pi$
$389$	−2.12966	−0.107978	−0.0539889	+	0.998542i	$0.517194\pi$
$390$	0	0
$391$	−3.14512	−0.159056
$392$	−83.3801	−4.21133
$393$	0	0
$394$	30.7056	1.54692
$395$	−4.78029	−0.240522
$396$	0	0
$397$	0.991316	0.0497527	0.0248764	−	0.999691i	$-0.492081\pi$
$397$	0.991316	0.0497527	0.0248764	+	0.999691i	$0.492081\pi$
$398$	−14.6354	−0.733608
$399$	0	0
$400$	−43.1537	−2.15768
$401$	−4.94978	−0.247180	−0.123590	−	0.992333i	$-0.539441\pi$
$401$	−4.94978	−0.247180	−0.123590	+	0.992333i	$0.539441\pi$
$402$	0	0
$403$	47.5711	2.36968
$404$	−73.2865	−3.64614
$405$	0	0
$406$	−0.254632	−0.0126372
$407$	−6.22862	−0.308741
$408$	0	0
$409$	−18.5646	−0.917963	−0.458981	−	0.888446i	$-0.651785\pi$
$409$	−18.5646	−0.917963	−0.458981	+	0.888446i	$0.651785\pi$
$410$	11.0515	0.545794
$411$	0	0
$412$	−49.2801	−2.42786
$413$	−24.2439	−1.19297
$414$	0	0
$415$	−4.99763	−0.245324
$416$	57.2484	2.80683
$417$	0	0
$418$	−14.8581	−0.726735
$419$	22.7594	1.11187	0.555935	−	0.831225i	$-0.312360\pi$
$419$	22.7594	1.11187	0.555935	+	0.831225i	$0.312360\pi$
$420$	0	0
$421$	−10.7741	−0.525096	−0.262548	−	0.964919i	$-0.584563\pi$
$421$	−10.7741	−0.525096	−0.262548	+	0.964919i	$0.584563\pi$
$422$	20.5907	1.00234
$423$	0	0
$424$	−36.2786	−1.76184
$425$	−7.21317	−0.349890
$426$	0	0
$427$	−4.31275	−0.208709
$428$	−26.9802	−1.30414
$429$	0	0
$430$	3.64960	0.175999
$431$	−7.60900	−0.366512	−0.183256	−	0.983065i	$-0.558664\pi$
$431$	−7.60900	−0.366512	−0.183256	+	0.983065i	$0.558664\pi$
$432$	0	0
$433$	2.06386	0.0991829	0.0495914	−	0.998770i	$-0.484208\pi$
$433$	2.06386	0.0991829	0.0495914	+	0.998770i	$0.484208\pi$
$434$	88.2051	4.23398
$435$	0	0
$436$	−68.8908	−3.29927
$437$	−11.7271	−0.560983
$438$	0	0
$439$	7.60854	0.363136	0.181568	−	0.983378i	$-0.441883\pi$
$439$	7.60854	0.363136	0.181568	+	0.983378i	$0.441883\pi$
$440$	−3.88603	−0.185259
$441$	0	0
$442$	24.1053	1.14657
$443$	−9.27130	−0.440493	−0.220246	−	0.975444i	$-0.570686\pi$
$443$	−9.27130	−0.440493	−0.220246	+	0.975444i	$0.570686\pi$
$444$	0	0
$445$	−6.36146	−0.301562
$446$	−37.4121	−1.77151
$447$	0	0
$448$	27.0823	1.27952
$449$	−7.28337	−0.343724	−0.171862	−	0.985121i	$-0.554978\pi$
$449$	−7.28337	−0.343724	−0.171862	+	0.985121i	$0.554978\pi$
$450$	0	0
$451$	−7.86011	−0.370118
$452$	−16.9886	−0.799077
$453$	0	0
$454$	4.93568	0.231643
$455$	−14.1042	−0.661217
$456$	0	0
$457$	9.93757	0.464860	0.232430	−	0.972613i	$-0.425332\pi$
$457$	9.93757	0.464860	0.232430	+	0.972613i	$0.425332\pi$
$458$	39.1704	1.83031
$459$	0	0
$460$	−5.28659	−0.246488
$461$	11.3804	0.530038	0.265019	−	0.964243i	$-0.414622\pi$
$461$	11.3804	0.530038	0.265019	+	0.964243i	$0.414622\pi$
$462$	0	0
$463$	31.7317	1.47470	0.737349	−	0.675512i	$-0.236077\pi$
$463$	31.7317	1.47470	0.737349	+	0.675512i	$0.236077\pi$
$464$	0.208098	0.00966071
$465$	0	0
$466$	28.7941	1.33386
$467$	−34.2226	−1.58363	−0.791815	−	0.610760i	$-0.790864\pi$
$467$	−34.2226	−1.58363	−0.791815	+	0.610760i	$0.790864\pi$
$468$	0	0
$469$	34.8476	1.60911
$470$	−8.55599	−0.394658
$471$	0	0
$472$	40.4073	1.85990
$473$	−2.59570	−0.119350
$474$	0	0
$475$	−26.8954	−1.23405
$476$	31.4795	1.44286
$477$	0	0
$478$	75.9465	3.47371
$479$	0.0317724	0.00145172	0.000725859	−	1.00000i	$-0.499769\pi$
$479$	0.0317724	0.00145172	0.000725859		1.00000i	$0.499769\pi$
$480$	0	0
$481$	37.6784	1.71799
$482$	34.3154	1.56302
$483$	0	0
$484$	4.76385	0.216539
$485$	−5.19387	−0.235842
$486$	0	0
$487$	−6.71443	−0.304260	−0.152130	−	0.988360i	$-0.548613\pi$
$487$	−6.71443	−0.304260	−0.152130	+	0.988360i	$0.548613\pi$
$488$	7.18805	0.325388
$489$	0	0
$490$	−16.3096	−0.736792
$491$	−15.6320	−0.705464	−0.352732	−	0.935724i	$-0.614747\pi$
$491$	−15.6320	−0.705464	−0.352732	+	0.935724i	$0.614747\pi$
$492$	0	0
$493$	0.0347838	0.00156658
$494$	89.8804	4.04391
$495$	0	0
$496$	−72.0856	−3.23674
$497$	36.9224	1.65619
$498$	0	0
$499$	−42.6023	−1.90714	−0.953570	−	0.301170i	$-0.902623\pi$
$499$	−42.6023	−1.90714	−0.953570	+	0.301170i	$0.902623\pi$
$500$	−25.0017	−1.11811
$501$	0	0
$502$	−55.8494	−2.49268
$503$	−20.7317	−0.924380	−0.462190	−	0.886781i	$-0.652936\pi$
$503$	−20.7317	−0.924380	−0.462190	+	0.886781i	$0.652936\pi$
$504$	0	0
$505$	−8.31689	−0.370097
$506$	5.33851	0.237326
$507$	0	0
$508$	−73.8637	−3.27717
$509$	10.5993	0.469807	0.234904	−	0.972019i	$-0.424523\pi$
$509$	10.5993	0.469807	0.234904	+	0.972019i	$0.424523\pi$
$510$	0	0
$511$	58.7624	2.59949
$512$	45.0295	1.99004
$513$	0	0
$514$	46.3925	2.04628
$515$	−5.59253	−0.246437
$516$	0	0
$517$	6.08525	0.267629
$518$	69.8624	3.06958
$519$	0	0
$520$	23.5075	1.03087
$521$	−17.7172	−0.776203	−0.388101	−	0.921617i	$-0.626869\pi$
$521$	−17.7172	−0.776203	−0.388101	+	0.921617i	$0.626869\pi$
$522$	0	0
$523$	33.2492	1.45389	0.726944	−	0.686697i	$-0.240940\pi$
$523$	33.2492	1.45389	0.726944	+	0.686697i	$0.240940\pi$
$524$	−89.7158	−3.91925
$525$	0	0
$526$	−77.5134	−3.37975
$527$	−12.0492	−0.524870
$528$	0	0
$529$	−18.7865	−0.816803
$530$	−7.09628	−0.308243
$531$	0	0
$532$	117.376	5.08890
$533$	47.5477	2.05952
$534$	0	0
$535$	−3.06183	−0.132375
$536$	−58.0805	−2.50869
$537$	0	0
$538$	−66.1738	−2.85296
$539$	11.5998	0.499639
$540$	0	0
$541$	0.0519933	0.00223537	0.00111768	−	0.999999i	$-0.499644\pi$
$541$	0.0519933	0.00223537	0.00111768	+	0.999999i	$0.499644\pi$
$542$	11.8655	0.509668
$543$	0	0
$544$	−14.5003	−0.621696
$545$	−7.81804	−0.334888
$546$	0	0
$547$	−41.7142	−1.78357	−0.891785	−	0.452458i	$-0.850547\pi$
$547$	−41.7142	−1.78357	−0.891785	+	0.452458i	$0.850547\pi$
$548$	79.3783	3.39087
$549$	0	0
$550$	12.2436	0.522068
$551$	0.129697	0.00552526
$552$	0	0
$553$	−38.1341	−1.62163
$554$	−45.4728	−1.93195
$555$	0	0
$556$	−82.0470	−3.47957
$557$	−7.84719	−0.332496	−0.166248	−	0.986084i	$-0.553165\pi$
$557$	−7.84719	−0.332496	−0.166248	+	0.986084i	$0.553165\pi$
$558$	0	0
$559$	15.7020	0.664123
$560$	21.3725	0.903153
$561$	0	0
$562$	−61.3981	−2.58992
$563$	−11.3050	−0.476450	−0.238225	−	0.971210i	$-0.576566\pi$
$563$	−11.3050	−0.476450	−0.238225	+	0.971210i	$0.576566\pi$
$564$	0	0
$565$	−1.92794	−0.0811092
$566$	6.73593	0.283132
$567$	0	0
$568$	−61.5385	−2.58210
$569$	−24.2745	−1.01764	−0.508820	−	0.860873i	$-0.669918\pi$
$569$	−24.2745	−1.01764	−0.508820	+	0.860873i	$0.669918\pi$
$570$	0	0
$571$	−3.48083	−0.145668	−0.0728342	−	0.997344i	$-0.523204\pi$
$571$	−3.48083	−0.145668	−0.0728342	+	0.997344i	$0.523204\pi$
$572$	−28.8177	−1.20493
$573$	0	0
$574$	88.1617	3.67980
$575$	9.66349	0.402995
$576$	0	0
$577$	4.02370	0.167509	0.0837545	−	0.996486i	$-0.473309\pi$
$577$	4.02370	0.167509	0.0837545	+	0.996486i	$0.473309\pi$
$578$	38.1070	1.58504
$579$	0	0
$580$	0.0584675	0.00242773
$581$	−39.8680	−1.65400
$582$	0	0
$583$	5.04706	0.209028
$584$	−97.9391	−4.05275
$585$	0	0
$586$	−78.5306	−3.24407
$587$	−23.1317	−0.954746	−0.477373	−	0.878701i	$-0.658411\pi$
$587$	−23.1317	−0.954746	−0.477373	+	0.878701i	$0.658411\pi$
$588$	0	0
$589$	−44.9272	−1.85119
$590$	7.90389	0.325398
$591$	0	0
$592$	−57.0951	−2.34659
$593$	26.1833	1.07522	0.537610	−	0.843194i	$-0.319327\pi$
$593$	26.1833	1.07522	0.537610	+	0.843194i	$0.319327\pi$
$594$	0	0
$595$	3.57243	0.146455
$596$	42.6602	1.74743
$597$	0	0
$598$	−32.2939	−1.32060
$599$	22.7916	0.931240	0.465620	−	0.884985i	$-0.345831\pi$
$599$	22.7916	0.931240	0.465620	+	0.884985i	$0.345831\pi$
$600$	0	0
$601$	37.8468	1.54380	0.771902	−	0.635741i	$-0.219306\pi$
$601$	37.8468	1.54380	0.771902	+	0.635741i	$0.219306\pi$
$602$	29.1142	1.18661
$603$	0	0
$604$	−60.2072	−2.44980
$605$	0.540623	0.0219795
$606$	0	0
$607$	16.0621	0.651939	0.325969	−	0.945380i	$-0.394309\pi$
$607$	16.0621	0.651939	0.325969	+	0.945380i	$0.394309\pi$
$608$	−54.0667	−2.19269
$609$	0	0
$610$	1.40602	0.0569281
$611$	−36.8111	−1.48922
$612$	0	0
$613$	36.9475	1.49229	0.746147	−	0.665781i	$-0.231901\pi$
$613$	36.9475	1.49229	0.746147	+	0.665781i	$0.231901\pi$
$614$	−39.9647	−1.61284
$615$	0	0
$616$	−31.0003	−1.24904
$617$	13.0177	0.524072	0.262036	−	0.965058i	$-0.415606\pi$
$617$	13.0177	0.524072	0.262036	+	0.965058i	$0.415606\pi$
$618$	0	0
$619$	22.0427	0.885971	0.442985	−	0.896529i	$-0.353919\pi$
$619$	22.0427	0.885971	0.442985	+	0.896529i	$0.353919\pi$
$620$	−20.2533	−0.813391
$621$	0	0
$622$	−58.7199	−2.35445
$623$	−50.7477	−2.03316
$624$	0	0
$625$	20.7013	0.828053
$626$	5.03899	0.201399
$627$	0	0
$628$	80.2019	3.20040
$629$	−9.54348	−0.380524
$630$	0	0
$631$	−21.1563	−0.842220	−0.421110	−	0.907010i	$-0.638359\pi$
$631$	−21.1563	−0.842220	−0.421110	+	0.907010i	$0.638359\pi$
$632$	63.5581	2.52820
$633$	0	0
$634$	45.4668	1.80572
$635$	−8.38239	−0.332645
$636$	0	0
$637$	−70.1700	−2.78024
$638$	−0.0590417	−0.00233748
$639$	0	0
$640$	1.40339	0.0554738
$641$	−25.2664	−0.997965	−0.498982	−	0.866612i	$-0.666293\pi$
$641$	−25.2664	−0.997965	−0.498982	+	0.866612i	$0.666293\pi$
$642$	0	0
$643$	−20.5251	−0.809430	−0.404715	−	0.914443i	$-0.632629\pi$
$643$	−20.5251	−0.809430	−0.404715	+	0.914443i	$0.632629\pi$
$644$	−42.1731	−1.66185
$645$	0	0
$646$	−22.7656	−0.895701
$647$	3.31668	0.130392	0.0651960	−	0.997872i	$-0.479233\pi$
$647$	3.31668	0.130392	0.0651960	+	0.997872i	$0.479233\pi$
$648$	0	0
$649$	−5.62146	−0.220662
$650$	−74.0643	−2.90504
$651$	0	0
$652$	31.1420	1.21962
$653$	−40.2518	−1.57517	−0.787587	−	0.616204i	$-0.788670\pi$
$653$	−40.2518	−1.57517	−0.787587	+	0.616204i	$0.788670\pi$
$654$	0	0
$655$	−10.1814	−0.397819
$656$	−72.0502	−2.81309
$657$	0	0
$658$	−68.2543	−2.66083
$659$	11.2229	0.437183	0.218591	−	0.975816i	$-0.429854\pi$
$659$	11.2229	0.437183	0.218591	+	0.975816i	$0.429854\pi$
$660$	0	0
$661$	18.4223	0.716546	0.358273	−	0.933617i	$-0.383366\pi$
$661$	18.4223	0.716546	0.358273	+	0.933617i	$0.383366\pi$
$662$	9.47704	0.368336
$663$	0	0
$664$	66.4479	2.57868
$665$	13.3204	0.516542
$666$	0	0
$667$	−0.0465998	−0.00180435
$668$	8.29243	0.320844
$669$	0	0
$670$	−11.3608	−0.438907
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−32.2878	−1.24460	−0.622301	−	0.782778i	$-0.713802\pi$
$673$	−32.2878	−1.24460	−0.622301	+	0.782778i	$0.713802\pi$
$674$	65.5199	2.52373
$675$	0	0
$676$	112.395	4.32288
$677$	45.6192	1.75329	0.876644	−	0.481139i	$-0.159777\pi$
$677$	45.6192	1.75329	0.876644	+	0.481139i	$0.159777\pi$
$678$	0	0
$679$	−41.4334	−1.59007
$680$	−5.95417	−0.228332
$681$	0	0
$682$	20.4522	0.783154
$683$	−6.70817	−0.256681	−0.128340	−	0.991730i	$-0.540965\pi$
$683$	−6.70817	−0.256681	−0.128340	+	0.991730i	$0.540965\pi$
$684$	0	0
$685$	9.00821	0.344186
$686$	−51.5931	−1.96983
$687$	0	0
$688$	−23.7936	−0.907123
$689$	−30.5309	−1.16313
$690$	0	0
$691$	6.61809	0.251764	0.125882	−	0.992045i	$-0.459824\pi$
$691$	6.61809	0.251764	0.125882	+	0.992045i	$0.459824\pi$
$692$	−42.8562	−1.62915
$693$	0	0
$694$	39.2179	1.48869
$695$	−9.31106	−0.353189
$696$	0	0
$697$	−12.0433	−0.456171
$698$	−47.9481	−1.81486
$699$	0	0
$700$	−96.7216	−3.65573
$701$	−36.1896	−1.36686	−0.683432	−	0.730014i	$-0.739513\pi$
$701$	−36.1896	−1.36686	−0.683432	+	0.730014i	$0.739513\pi$
$702$	0	0
$703$	−35.5844	−1.34209
$704$	6.27960	0.236671
$705$	0	0
$706$	0.853263	0.0321130
$707$	−66.3469	−2.49523
$708$	0	0
$709$	42.6357	1.60122	0.800609	−	0.599187i	$-0.204510\pi$
$709$	42.6357	1.60122	0.800609	+	0.599187i	$0.204510\pi$
$710$	−12.0372	−0.451750
$711$	0	0
$712$	84.5811	3.16981
$713$	16.1423	0.604533
$714$	0	0
$715$	−3.27036	−0.122305
$716$	57.3706	2.14404
$717$	0	0
$718$	25.3206	0.944957
$719$	18.9095	0.705206	0.352603	−	0.935773i	$-0.385297\pi$
$719$	18.9095	0.705206	0.352603	+	0.935773i	$0.385297\pi$
$720$	0	0
$721$	−44.6137	−1.66150
$722$	−35.4710	−1.32010
$723$	0	0
$724$	106.766	3.96794
$725$	−0.106874	−0.00396921
$726$	0	0
$727$	7.37299	0.273449	0.136725	−	0.990609i	$-0.456342\pi$
$727$	7.37299	0.273449	0.136725	+	0.990609i	$0.456342\pi$
$728$	187.528	6.95025
$729$	0	0
$730$	−19.1574	−0.709047
$731$	−3.97712	−0.147099
$732$	0	0
$733$	12.6913	0.468764	0.234382	−	0.972145i	$-0.424693\pi$
$733$	12.6913	0.468764	0.234382	+	0.972145i	$0.424693\pi$
$734$	19.0142	0.701827
$735$	0	0
$736$	19.4261	0.716055
$737$	8.08014	0.297636
$738$	0	0
$739$	−9.50571	−0.349673	−0.174837	−	0.984597i	$-0.555940\pi$
$739$	−9.50571	−0.349673	−0.174837	+	0.984597i	$0.555940\pi$
$740$	−16.0415	−0.589697
$741$	0	0
$742$	−56.6096	−2.07820
$743$	−1.76760	−0.0648470	−0.0324235	−	0.999474i	$-0.510323\pi$
$743$	−1.76760	−0.0648470	−0.0324235	+	0.999474i	$0.510323\pi$
$744$	0	0
$745$	4.84127	0.177370
$746$	−45.7816	−1.67618
$747$	0	0
$748$	7.29916	0.266884
$749$	−24.4254	−0.892483
$750$	0	0
$751$	−21.8139	−0.796001	−0.398000	−	0.917385i	$-0.630296\pi$
$751$	−21.8139	−0.796001	−0.398000	+	0.917385i	$0.630296\pi$
$752$	55.7808	2.03412
$753$	0	0
$754$	0.357157	0.0130069
$755$	−6.83259	−0.248663
$756$	0	0
$757$	20.5403	0.746552	0.373276	−	0.927720i	$-0.378235\pi$
$757$	20.5403	0.746552	0.373276	+	0.927720i	$0.378235\pi$
$758$	−60.1311	−2.18406
$759$	0	0
$760$	−22.2010	−0.805316
$761$	−42.2493	−1.53154	−0.765768	−	0.643116i	$-0.777641\pi$
$761$	−42.2493	−1.53154	−0.765768	+	0.643116i	$0.777641\pi$
$762$	0	0
$763$	−62.3674	−2.25785
$764$	−91.7352	−3.31886
$765$	0	0
$766$	26.0423	0.940948
$767$	34.0055	1.22787
$768$	0	0
$769$	12.3073	0.443814	0.221907	−	0.975068i	$-0.428772\pi$
$769$	12.3073	0.443814	0.221907	+	0.975068i	$0.428772\pi$
$770$	−6.06382	−0.218525
$771$	0	0
$772$	53.8300	1.93739
$773$	−41.7428	−1.50138	−0.750692	−	0.660652i	$-0.770280\pi$
$773$	−41.7428	−1.50138	−0.750692	+	0.660652i	$0.770280\pi$
$774$	0	0
$775$	37.0215	1.32985
$776$	69.0570	2.47900
$777$	0	0
$778$	5.53868	0.198571
$779$	−44.9051	−1.60889
$780$	0	0
$781$	8.56121	0.306344
$782$	8.17965	0.292504
$783$	0	0
$784$	106.330	3.79751
$785$	9.10167	0.324853
$786$	0	0
$787$	−20.4654	−0.729512	−0.364756	−	0.931103i	$-0.618848\pi$
$787$	−20.4654	−0.729512	−0.364756	+	0.931103i	$0.618848\pi$
$788$	−56.2442	−2.00362
$789$	0	0
$790$	12.4323	0.442321
$791$	−15.3799	−0.546847
$792$	0	0
$793$	6.04924	0.214815
$794$	−2.57816	−0.0914953
$795$	0	0
$796$	26.8081	0.950189
$797$	−17.3900	−0.615986	−0.307993	−	0.951389i	$-0.599657\pi$
$797$	−17.3900	−0.615986	−0.307993	+	0.951389i	$0.599657\pi$
$798$	0	0
$799$	9.32381	0.329853
$800$	44.5527	1.57518
$801$	0	0
$802$	12.8731	0.454565
$803$	13.6253	0.480825
$804$	0	0
$805$	−4.78599	−0.168684
$806$	−123.720	−4.35785
$807$	0	0
$808$	110.580	3.89020
$809$	−28.4398	−0.999891	−0.499946	−	0.866057i	$-0.666647\pi$
$809$	−28.4398	−0.999891	−0.499946	+	0.866057i	$0.666647\pi$
$810$	0	0
$811$	−12.1286	−0.425892	−0.212946	−	0.977064i	$-0.568306\pi$
$811$	−12.1286	−0.425892	−0.212946	+	0.977064i	$0.568306\pi$
$812$	0.466416	0.0163680
$813$	0	0
$814$	16.1990	0.567776
$815$	3.53414	0.123795
$816$	0	0
$817$	−14.8293	−0.518812
$818$	48.2818	1.68813
$819$	0	0
$820$	−20.2433	−0.706927
$821$	2.72354	0.0950521	0.0475261	−	0.998870i	$-0.484866\pi$
$821$	2.72354	0.0950521	0.0475261	+	0.998870i	$0.484866\pi$
$822$	0	0
$823$	−10.1098	−0.352404	−0.176202	−	0.984354i	$-0.556381\pi$
$823$	−10.1098	−0.352404	−0.176202	+	0.984354i	$0.556381\pi$
$824$	74.3576	2.59037
$825$	0	0
$826$	63.0522	2.19387
$827$	−27.8472	−0.968341	−0.484170	−	0.874974i	$-0.660878\pi$
$827$	−27.8472	−0.968341	−0.484170	+	0.874974i	$0.660878\pi$
$828$	0	0
$829$	32.8263	1.14010	0.570052	−	0.821608i	$-0.306923\pi$
$829$	32.8263	1.14010	0.570052	+	0.821608i	$0.306923\pi$
$830$	12.9975	0.451151
$831$	0	0
$832$	−37.9868	−1.31696
$833$	17.7732	0.615805
$834$	0	0
$835$	0.941063	0.0325668
$836$	27.2161	0.941287
$837$	0	0
$838$	−59.1914	−2.04473
$839$	38.5543	1.33104	0.665521	−	0.746379i	$-0.268209\pi$
$839$	38.5543	1.33104	0.665521	+	0.746379i	$0.268209\pi$
$840$	0	0
$841$	−28.9995	−0.999982
$842$	28.0205	0.965652
$843$	0	0
$844$	−37.7165	−1.29826
$845$	12.7551	0.438788
$846$	0	0
$847$	4.31275	0.148188
$848$	46.2642	1.58872
$849$	0	0
$850$	18.7596	0.643448
$851$	12.7854	0.438278
$852$	0	0
$853$	−14.1974	−0.486111	−0.243055	−	0.970012i	$-0.578150\pi$
$853$	−14.1974	−0.486111	−0.243055	+	0.970012i	$0.578150\pi$
$854$	11.2163	0.383815
$855$	0	0
$856$	40.7097	1.39143
$857$	46.1146	1.57524	0.787622	−	0.616159i	$-0.211312\pi$
$857$	46.1146	1.57524	0.787622	+	0.616159i	$0.211312\pi$
$858$	0	0
$859$	57.2683	1.95397	0.976985	−	0.213306i	$-0.0684231\pi$
$859$	57.2683	1.95397	0.976985	+	0.213306i	$0.0684231\pi$
$860$	−6.68508	−0.227959
$861$	0	0
$862$	19.7890	0.674017
$863$	21.7311	0.739737	0.369868	−	0.929084i	$-0.379403\pi$
$863$	21.7311	0.739737	0.369868	+	0.929084i	$0.379403\pi$
$864$	0	0
$865$	−4.86352	−0.165365
$866$	−5.36757	−0.182397
$867$	0	0
$868$	−161.568	−5.48397
$869$	−8.84218	−0.299950
$870$	0	0
$871$	−48.8787	−1.65619
$872$	103.948	3.52011
$873$	0	0
$874$	30.4991	1.03165
$875$	−22.6343	−0.765178
$876$	0	0
$877$	−44.3014	−1.49595	−0.747976	−	0.663726i	$-0.768974\pi$
$877$	−44.3014	−1.49595	−0.747976	+	0.663726i	$0.768974\pi$
$878$	−19.7878	−0.667807
$879$	0	0
$880$	4.95566	0.167055
$881$	−20.2667	−0.682802	−0.341401	−	0.939918i	$-0.610901\pi$
$881$	−20.2667	−0.682802	−0.341401	+	0.939918i	$0.610901\pi$
$882$	0	0
$883$	49.6889	1.67216	0.836081	−	0.548605i	$-0.184841\pi$
$883$	49.6889	1.67216	0.836081	+	0.548605i	$0.184841\pi$
$884$	−44.1544	−1.48507
$885$	0	0
$886$	24.1123	0.810067
$887$	21.0928	0.708228	0.354114	−	0.935202i	$-0.384782\pi$
$887$	21.0928	0.708228	0.354114	+	0.935202i	$0.384782\pi$
$888$	0	0
$889$	−66.8694	−2.24273
$890$	16.5445	0.554573
$891$	0	0
$892$	68.5287	2.29451
$893$	34.7653	1.16338
$894$	0	0
$895$	6.51067	0.217628
$896$	11.1953	0.374010
$897$	0	0
$898$	18.9422	0.632108
$899$	−0.178527	−0.00595421
$900$	0	0
$901$	7.73310	0.257627
$902$	20.4421	0.680648
$903$	0	0
$904$	25.6337	0.852564
$905$	12.1163	0.402760
$906$	0	0
$907$	24.8398	0.824794	0.412397	−	0.911004i	$-0.364692\pi$
$907$	24.8398	0.824794	0.412397	+	0.911004i	$0.364692\pi$
$908$	−9.04082	−0.300030
$909$	0	0
$910$	36.6815	1.21598
$911$	−23.9432	−0.793275	−0.396637	−	0.917975i	$-0.629823\pi$
$911$	−23.9432	−0.793275	−0.396637	+	0.917975i	$0.629823\pi$
$912$	0	0
$913$	−9.24421	−0.305939
$914$	−25.8450	−0.854878
$915$	0	0
$916$	−71.7495	−2.37067
$917$	−81.2204	−2.68214
$918$	0	0
$919$	14.3672	0.473931	0.236965	−	0.971518i	$-0.423847\pi$
$919$	14.3672	0.473931	0.236965	+	0.971518i	$0.423847\pi$
$920$	7.97680	0.262987
$921$	0	0
$922$	−29.5974	−0.974740
$923$	−51.7888	−1.70465
$924$	0	0
$925$	29.3227	0.964123
$926$	−82.5259	−2.71197
$927$	0	0
$928$	−0.214845	−0.00705262
$929$	39.5612	1.29796	0.648980	−	0.760806i	$-0.275196\pi$
$929$	39.5612	1.29796	0.648980	+	0.760806i	$0.275196\pi$
$930$	0	0
$931$	66.2702	2.17192
$932$	−52.7429	−1.72765
$933$	0	0
$934$	89.0040	2.91230
$935$	0.828342	0.0270897
$936$	0	0
$937$	−48.6379	−1.58893	−0.794465	−	0.607310i	$-0.792249\pi$
$937$	−48.6379	−1.58893	−0.794465	+	0.607310i	$0.792249\pi$
$938$	−90.6296	−2.95916
$939$	0	0
$940$	15.6722	0.511172
$941$	−28.6981	−0.935530	−0.467765	−	0.883853i	$-0.654941\pi$
$941$	−28.6981	−0.935530	−0.467765	+	0.883853i	$0.654941\pi$
$942$	0	0
$943$	16.1344	0.525407
$944$	−51.5294	−1.67714
$945$	0	0
$946$	6.75073	0.219485
$947$	9.54462	0.310158	0.155079	−	0.987902i	$-0.450437\pi$
$947$	9.54462	0.310158	0.155079	+	0.987902i	$0.450437\pi$
$948$	0	0
$949$	−82.4225	−2.67555
$950$	69.9480	2.26941
$951$	0	0
$952$	−47.4986	−1.53944
$953$	−31.2116	−1.01104	−0.505521	−	0.862814i	$-0.668700\pi$
$953$	−31.2116	−1.01104	−0.505521	+	0.862814i	$0.668700\pi$
$954$	0	0
$955$	−10.4105	−0.336877
$956$	−139.113	−4.49925
$957$	0	0
$958$	−0.0826318	−0.00266971
$959$	71.8618	2.32054
$960$	0	0
$961$	30.8422	0.994908
$962$	−97.9918	−3.15938
$963$	0	0
$964$	−62.8565	−2.02447
$965$	6.10888	0.196652
$966$	0	0
$967$	49.0672	1.57790	0.788948	−	0.614460i	$-0.210626\pi$
$967$	49.0672	1.57790	0.788948	+	0.614460i	$0.210626\pi$
$968$	−7.18805	−0.231033
$969$	0	0
$970$	13.5079	0.433713
$971$	−30.9733	−0.993980	−0.496990	−	0.867756i	$-0.665561\pi$
$971$	−30.9733	−0.993980	−0.496990	+	0.867756i	$0.665561\pi$
$972$	0	0
$973$	−74.2778	−2.38124
$974$	17.4625	0.559534
$975$	0	0
$976$	−9.16656	−0.293414
$977$	2.27911	0.0729153	0.0364577	−	0.999335i	$-0.488393\pi$
$977$	2.27911	0.0729153	0.0364577	+	0.999335i	$0.488393\pi$
$978$	0	0
$979$	−11.7669	−0.376072
$980$	29.8747	0.954313
$981$	0	0
$982$	40.6549	1.29735
$983$	−24.0060	−0.765671	−0.382835	−	0.923817i	$-0.625052\pi$
$983$	−24.0060	−0.765671	−0.382835	+	0.923817i	$0.625052\pi$
$984$	0	0
$985$	−6.38285	−0.203375
$986$	−0.0904635	−0.00288094
$987$	0	0
$988$	−164.636	−5.23778
$989$	5.32815	0.169425
$990$	0	0
$991$	−15.1701	−0.481894	−0.240947	−	0.970538i	$-0.577458\pi$
$991$	−15.1701	−0.481894	−0.240947	+	0.970538i	$0.577458\pi$
$992$	74.4226	2.36292
$993$	0	0
$994$	−96.0255	−3.04574
$995$	3.04231	0.0964476
$996$	0	0
$997$	−39.1763	−1.24072	−0.620362	−	0.784315i	$-0.713014\pi$
$997$	−39.1763	−1.24072	−0.620362	+	0.784315i	$0.713014\pi$
$998$	110.798	3.50723
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.60074 q^{2} +4.76385 q^{4} +0.540623 q^{5} +4.31275 q^{7} -7.18805 q^{8} +O(q^{10})\)	\(q-2.60074 q^{2} +4.76385 q^{4} +0.540623 q^{5} +4.31275 q^{7} -7.18805 q^{8} -1.40602 q^{10} +1.00000 q^{11} -6.04924 q^{13} -11.2163 q^{14} +9.16656 q^{16} +1.53220 q^{17} +5.71304 q^{19} +2.57545 q^{20} -2.60074 q^{22} -2.05269 q^{23} -4.70773 q^{25} +15.7325 q^{26} +20.5453 q^{28} +0.0227019 q^{29} -7.86398 q^{31} -9.46374 q^{32} -3.98485 q^{34} +2.33157 q^{35} -6.22862 q^{37} -14.8581 q^{38} -3.88603 q^{40} -7.86011 q^{41} -2.59570 q^{43} +4.76385 q^{44} +5.33851 q^{46} +6.08525 q^{47} +11.5998 q^{49} +12.2436 q^{50} -28.8177 q^{52} +5.04706 q^{53} +0.540623 q^{55} -31.0003 q^{56} -0.0590417 q^{58} -5.62146 q^{59} -1.00000 q^{61} +20.4522 q^{62} +6.27960 q^{64} -3.27036 q^{65} +8.08014 q^{67} +7.29916 q^{68} -6.06382 q^{70} +8.56121 q^{71} +13.6253 q^{73} +16.1990 q^{74} +27.2161 q^{76} +4.31275 q^{77} -8.84218 q^{79} +4.95566 q^{80} +20.4421 q^{82} -9.24421 q^{83} +0.828342 q^{85} +6.75073 q^{86} -7.18805 q^{88} -11.7669 q^{89} -26.0889 q^{91} -9.77869 q^{92} -15.8262 q^{94} +3.08860 q^{95} -9.60720 q^{97} -30.1681 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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