LMFDB - Embedded newform 6039.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 $ x^12 - 5x^11 - 5x^10 + 48x^9 - 173x^7 + 29x^6 + 281x^5 - 41x^4 - 201x^3 + 8x^2 + 49x + 8
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			1.12
Root			$-1.73351$ 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)

$f(q)$	$=$	$q+2.73351 q^{2} +5.47208 q^{4} +3.64694 q^{5} -3.96128 q^{7} +9.49098 q^{8} +O(q^{10})$	\(q+2.73351 q^{2} +5.47208 q^{4} +3.64694 q^{5} -3.96128 q^{7} +9.49098 q^{8} +9.96895 q^{10} -1.00000 q^{11} +2.49181 q^{13} -10.8282 q^{14} +14.9995 q^{16} +4.84898 q^{17} -0.0465231 q^{19} +19.9564 q^{20} -2.73351 q^{22} -7.87558 q^{23} +8.30017 q^{25} +6.81139 q^{26} -21.6765 q^{28} -4.43593 q^{29} +10.7837 q^{31} +22.0194 q^{32} +13.2547 q^{34} -14.4466 q^{35} -5.41262 q^{37} -0.127171 q^{38} +34.6130 q^{40} +1.28084 q^{41} +6.56966 q^{43} -5.47208 q^{44} -21.5280 q^{46} -0.000570449 q^{47} +8.69175 q^{49} +22.6886 q^{50} +13.6354 q^{52} +9.54719 q^{53} -3.64694 q^{55} -37.5964 q^{56} -12.1257 q^{58} +7.63076 q^{59} -1.00000 q^{61} +29.4773 q^{62} +30.1913 q^{64} +9.08748 q^{65} -8.92768 q^{67} +26.5340 q^{68} -39.4898 q^{70} -10.4879 q^{71} -13.5644 q^{73} -14.7955 q^{74} -0.254578 q^{76} +3.96128 q^{77} -8.55077 q^{79} +54.7024 q^{80} +3.50118 q^{82} -3.71971 q^{83} +17.6839 q^{85} +17.9582 q^{86} -9.49098 q^{88} +13.8620 q^{89} -9.87077 q^{91} -43.0958 q^{92} -0.00155933 q^{94} -0.169667 q^{95} -6.11297 q^{97} +23.7590 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * 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 $

$2$	2.73351	1.93288	0.966442	−	0.256884i	$-0.0826960\pi$
$2$	2.73351	1.93288	0.966442	+	0.256884i	$0.0826960\pi$
$3$	0	0
$3$	0	0
$4$	5.47208	2.73604
$5$	3.64694	1.63096	0.815480	−	0.578785i	$-0.196473\pi$
$5$	3.64694	1.63096	0.815480	+	0.578785i	$0.196473\pi$
$6$	0	0
$7$	−3.96128	−1.49722	−0.748612	−	0.663009i	$-0.769279\pi$
$7$	−3.96128	−1.49722	−0.748612	+	0.663009i	$0.769279\pi$
$8$	9.49098	3.35557
$9$	0	0
$10$	9.96895	3.15246
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.49181	0.691104	0.345552	−	0.938400i	$-0.387692\pi$
$13$	2.49181	0.691104	0.345552	+	0.938400i	$0.387692\pi$
$14$	−10.8282	−2.89396
$15$	0	0
$16$	14.9995	3.74988
$17$	4.84898	1.17605	0.588026	−	0.808842i	$-0.299905\pi$
$17$	4.84898	1.17605	0.588026	+	0.808842i	$0.299905\pi$
$18$	0	0
$19$	−0.0465231	−0.0106731	−0.00533656	−	0.999986i	$-0.501699\pi$
$19$	−0.0465231	−0.0106731	−0.00533656	+	0.999986i	$0.501699\pi$
$20$	19.9564	4.46238
$21$	0	0
$22$	−2.73351	−0.582787
$23$	−7.87558	−1.64217	−0.821086	−	0.570804i	$-0.806631\pi$
$23$	−7.87558	−1.64217	−0.821086	+	0.570804i	$0.806631\pi$
$24$	0	0
$25$	8.30017	1.66003
$26$	6.81139	1.33582
$27$	0	0
$28$	−21.6765	−4.09647
$29$	−4.43593	−0.823732	−0.411866	−	0.911245i	$-0.635123\pi$
$29$	−4.43593	−0.823732	−0.411866	+	0.911245i	$0.635123\pi$
$30$	0	0
$31$	10.7837	1.93680	0.968402	−	0.249394i	$-0.0802314\pi$
$31$	10.7837	1.93680	0.968402	+	0.249394i	$0.0802314\pi$
$32$	22.0194	3.89252
$33$	0	0
$34$	13.2547	2.27317
$35$	−14.4466	−2.44191
$36$	0	0
$37$	−5.41262	−0.889830	−0.444915	−	0.895573i	$-0.646766\pi$
$37$	−5.41262	−0.889830	−0.444915	+	0.895573i	$0.646766\pi$
$38$	−0.127171	−0.0206299
$39$	0	0
$40$	34.6130	5.47280
$41$	1.28084	0.200033	0.100017	−	0.994986i	$-0.468110\pi$
$41$	1.28084	0.200033	0.100017	+	0.994986i	$0.468110\pi$
$42$	0	0
$43$	6.56966	1.00186	0.500932	−	0.865487i	$-0.332991\pi$
$43$	6.56966	1.00186	0.500932	+	0.865487i	$0.332991\pi$
$44$	−5.47208	−0.824948
$45$	0	0
$46$	−21.5280	−3.17413
$47$	−0.000570449 0	−8.32086e−5 0	−4.16043e−5	−	1.00000i	$-0.500013\pi$
$47$	−0.000570449 0	−8.32086e−5 0	−4.16043e−5		1.00000i	$0.500013\pi$
$48$	0	0
$49$	8.69175	1.24168
$50$	22.6886	3.20865
$51$	0	0
$52$	13.6354	1.89089
$53$	9.54719	1.31141	0.655704	−	0.755018i	$-0.272372\pi$
$53$	9.54719	1.31141	0.655704	+	0.755018i	$0.272372\pi$
$54$	0	0
$55$	−3.64694	−0.491753
$56$	−37.5964	−5.02403
$57$	0	0
$58$	−12.1257	−1.59218
$59$	7.63076	0.993440	0.496720	−	0.867911i	$-0.334538\pi$
$59$	7.63076	0.993440	0.496720	+	0.867911i	$0.334538\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	29.4773	3.74362
$63$	0	0
$64$	30.1913	3.77391
$65$	9.08748	1.12716
$66$	0	0
$67$	−8.92768	−1.09069	−0.545345	−	0.838212i	$-0.683601\pi$
$67$	−8.92768	−1.09069	−0.545345	+	0.838212i	$0.683601\pi$
$68$	26.5340	3.21772
$69$	0	0
$70$	−39.4898	−4.71994
$71$	−10.4879	−1.24469	−0.622343	−	0.782745i	$-0.713819\pi$
$71$	−10.4879	−1.24469	−0.622343	+	0.782745i	$0.713819\pi$
$72$	0	0
$73$	−13.5644	−1.58759	−0.793797	−	0.608183i	$-0.791899\pi$
$73$	−13.5644	−1.58759	−0.793797	+	0.608183i	$0.791899\pi$
$74$	−14.7955	−1.71994
$75$	0	0
$76$	−0.254578	−0.0292021
$77$	3.96128	0.451430
$78$	0	0
$79$	−8.55077	−0.962036	−0.481018	−	0.876711i	$-0.659733\pi$
$79$	−8.55077	−0.962036	−0.481018	+	0.876711i	$0.659733\pi$
$80$	54.7024	6.11591
$81$	0	0
$82$	3.50118	0.386641
$83$	−3.71971	−0.408291	−0.204146	−	0.978941i	$-0.565442\pi$
$83$	−3.71971	−0.408291	−0.204146	+	0.978941i	$0.565442\pi$
$84$	0	0
$85$	17.6839	1.91809
$86$	17.9582	1.93649
$87$	0	0
$88$	−9.49098	−1.01174
$89$	13.8620	1.46937	0.734684	−	0.678409i	$-0.237330\pi$
$89$	13.8620	1.46937	0.734684	+	0.678409i	$0.237330\pi$
$90$	0	0
$91$	−9.87077	−1.03474
$92$	−43.0958	−4.49305
$93$	0	0
$94$	−0.00155933	−0.000160833 0
$95$	−0.169667	−0.0174074
$96$	0	0
$97$	−6.11297	−0.620678	−0.310339	−	0.950626i	$-0.600443\pi$
$97$	−6.11297	−0.620678	−0.310339	+	0.950626i	$0.600443\pi$
$98$	23.7590	2.40002
$99$	0	0
$100$	45.4192	4.54192
$101$	15.6307	1.55531	0.777655	−	0.628691i	$-0.216409\pi$
$101$	15.6307	1.55531	0.777655	+	0.628691i	$0.216409\pi$
$102$	0	0
$103$	−9.08974	−0.895639	−0.447819	−	0.894124i	$-0.647799\pi$
$103$	−9.08974	−0.895639	−0.447819	+	0.894124i	$0.647799\pi$
$104$	23.6497	2.31905
$105$	0	0
$106$	26.0973	2.53480
$107$	−12.4016	−1.19891	−0.599453	−	0.800410i	$-0.704615\pi$
$107$	−12.4016	−1.19891	−0.599453	+	0.800410i	$0.704615\pi$
$108$	0	0
$109$	6.19519	0.593392	0.296696	−	0.954972i	$-0.404115\pi$
$109$	6.19519	0.593392	0.296696	+	0.954972i	$0.404115\pi$
$110$	−9.96895	−0.950502
$111$	0	0
$112$	−59.4173	−5.61441
$113$	−11.2098	−1.05453	−0.527263	−	0.849702i	$-0.676782\pi$
$113$	−11.2098	−1.05453	−0.527263	+	0.849702i	$0.676782\pi$
$114$	0	0
$115$	−28.7218	−2.67832
$116$	−24.2738	−2.25376
$117$	0	0
$118$	20.8588	1.92020
$119$	−19.2082	−1.76081
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.73351	−0.247480
$123$	0	0
$124$	59.0091	5.29918
$125$	12.0355	1.07649
$126$	0	0
$127$	1.57944	0.140153	0.0700763	−	0.997542i	$-0.477676\pi$
$127$	1.57944	0.140153	0.0700763	+	0.997542i	$0.477676\pi$
$128$	38.4893	3.40201
$129$	0	0
$130$	24.8407	2.17868
$131$	11.9942	1.04794	0.523970	−	0.851736i	$-0.324450\pi$
$131$	11.9942	1.04794	0.523970	+	0.851736i	$0.324450\pi$
$132$	0	0
$133$	0.184291	0.0159800
$134$	−24.4039	−2.10818
$135$	0	0
$136$	46.0216	3.94632
$137$	−9.02980	−0.771468	−0.385734	−	0.922610i	$-0.626052\pi$
$137$	−9.02980	−0.771468	−0.385734	+	0.922610i	$0.626052\pi$
$138$	0	0
$139$	−7.43255	−0.630421	−0.315210	−	0.949022i	$-0.602075\pi$
$139$	−7.43255	−0.630421	−0.315210	+	0.949022i	$0.602075\pi$
$140$	−79.0527	−6.68118
$141$	0	0
$142$	−28.6688	−2.40583
$143$	−2.49181	−0.208376
$144$	0	0
$145$	−16.1776	−1.34347
$146$	−37.0784	−3.06863
$147$	0	0
$148$	−29.6183	−2.43461
$149$	9.09929	0.745443	0.372722	−	0.927943i	$-0.378425\pi$
$149$	9.09929	0.745443	0.372722	+	0.927943i	$0.378425\pi$
$150$	0	0
$151$	10.5174	0.855891	0.427946	−	0.903804i	$-0.359237\pi$
$151$	10.5174	0.855891	0.427946	+	0.903804i	$0.359237\pi$
$152$	−0.441549	−0.0358144
$153$	0	0
$154$	10.8282	0.872562
$155$	39.3274	3.15885
$156$	0	0
$157$	−19.3242	−1.54224	−0.771118	−	0.636692i	$-0.780302\pi$
$157$	−19.3242	−1.54224	−0.771118	+	0.636692i	$0.780302\pi$
$158$	−23.3736	−1.85951
$159$	0	0
$160$	80.3035	6.34855
$161$	31.1974	2.45870
$162$	0	0
$163$	2.98353	0.233688	0.116844	−	0.993150i	$-0.462722\pi$
$163$	2.98353	0.233688	0.116844	+	0.993150i	$0.462722\pi$
$164$	7.00885	0.547299
$165$	0	0
$166$	−10.1679	−0.789180
$167$	6.61012	0.511507	0.255753	−	0.966742i	$-0.417677\pi$
$167$	6.61012	0.511507	0.255753	+	0.966742i	$0.417677\pi$
$168$	0	0
$169$	−6.79088	−0.522375
$170$	48.3393	3.70745
$171$	0	0
$172$	35.9497	2.74114
$173$	−7.02661	−0.534223	−0.267111	−	0.963666i	$-0.586069\pi$
$173$	−7.02661	−0.534223	−0.267111	+	0.963666i	$0.586069\pi$
$174$	0	0
$175$	−32.8793	−2.48544
$176$	−14.9995	−1.13063
$177$	0	0
$178$	37.8919	2.84012
$179$	−22.3293	−1.66897	−0.834486	−	0.551029i	$-0.814235\pi$
$179$	−22.3293	−1.66897	−0.834486	+	0.551029i	$0.814235\pi$
$180$	0	0
$181$	15.6618	1.16413	0.582067	−	0.813141i	$-0.302244\pi$
$181$	15.6618	1.16413	0.582067	+	0.813141i	$0.302244\pi$
$182$	−26.9818	−2.00003
$183$	0	0
$184$	−74.7470	−5.51042
$185$	−19.7395	−1.45128
$186$	0	0
$187$	−4.84898	−0.354593
$188$	−0.00312155	−0.000227662 0
$189$	0	0
$190$	−0.463786	−0.0336466
$191$	21.8177	1.57868	0.789338	−	0.613959i	$-0.210424\pi$
$191$	21.8177	1.57868	0.789338	+	0.613959i	$0.210424\pi$
$192$	0	0
$193$	0.455455	0.0327843	0.0163922	−	0.999866i	$-0.494782\pi$
$193$	0.455455	0.0327843	0.0163922	+	0.999866i	$0.494782\pi$
$194$	−16.7099	−1.19970
$195$	0	0
$196$	47.5620	3.39728
$197$	−23.1632	−1.65031	−0.825155	−	0.564906i	$-0.808912\pi$
$197$	−23.1632	−1.65031	−0.825155	+	0.564906i	$0.808912\pi$
$198$	0	0
$199$	4.75326	0.336949	0.168475	−	0.985706i	$-0.446116\pi$
$199$	4.75326	0.336949	0.168475	+	0.985706i	$0.446116\pi$
$200$	78.7767	5.57035
$201$	0	0
$202$	42.7266	3.00624
$203$	17.5720	1.23331
$204$	0	0
$205$	4.67113	0.326246
$206$	−24.8469	−1.73117
$207$	0	0
$208$	37.3760	2.59156
$209$	0.0465231	0.00321807
$210$	0	0
$211$	−25.2215	−1.73632	−0.868162	−	0.496282i	$-0.834698\pi$
$211$	−25.2215	−1.73632	−0.868162	+	0.496282i	$0.834698\pi$
$212$	52.2430	3.58806
$213$	0	0
$214$	−33.8999	−2.31735
$215$	23.9592	1.63400
$216$	0	0
$217$	−42.7172	−2.89983
$218$	16.9346	1.14696
$219$	0	0
$220$	−19.9564	−1.34546
$221$	12.0827	0.812774
$222$	0	0
$223$	−6.05952	−0.405776	−0.202888	−	0.979202i	$-0.565033\pi$
$223$	−6.05952	−0.405776	−0.202888	+	0.979202i	$0.565033\pi$
$224$	−87.2251	−5.82797
$225$	0	0
$226$	−30.6420	−2.03828
$227$	11.2156	0.744402	0.372201	−	0.928152i	$-0.378603\pi$
$227$	11.2156	0.744402	0.372201	+	0.928152i	$0.378603\pi$
$228$	0	0
$229$	22.6663	1.49783	0.748916	−	0.662665i	$-0.230575\pi$
$229$	22.6663	1.49783	0.748916	+	0.662665i	$0.230575\pi$
$230$	−78.5113	−5.17688
$231$	0	0
$232$	−42.1013	−2.76409
$233$	7.95089	0.520880	0.260440	−	0.965490i	$-0.416132\pi$
$233$	7.95089	0.520880	0.260440	+	0.965490i	$0.416132\pi$
$234$	0	0
$235$	−0.00208039	−0.000135710 0
$236$	41.7561	2.71809
$237$	0	0
$238$	−52.5058	−3.40345
$239$	−21.7846	−1.40913	−0.704563	−	0.709642i	$-0.748857\pi$
$239$	−21.7846	−1.40913	−0.704563	+	0.709642i	$0.748857\pi$
$240$	0	0
$241$	−4.08238	−0.262970	−0.131485	−	0.991318i	$-0.541974\pi$
$241$	−4.08238	−0.262970	−0.131485	+	0.991318i	$0.541974\pi$
$242$	2.73351	0.175717
$243$	0	0
$244$	−5.47208	−0.350314
$245$	31.6983	2.02513
$246$	0	0
$247$	−0.115927	−0.00737624
$248$	102.348	6.49908
$249$	0	0
$250$	32.8992	2.08073
$251$	17.4417	1.10091	0.550454	−	0.834865i	$-0.314454\pi$
$251$	17.4417	1.10091	0.550454	+	0.834865i	$0.314454\pi$
$252$	0	0
$253$	7.87558	0.495134
$254$	4.31742	0.270899
$255$	0	0
$256$	44.8285	2.80178
$257$	5.20391	0.324611	0.162306	−	0.986741i	$-0.448107\pi$
$257$	5.20391	0.324611	0.162306	+	0.986741i	$0.448107\pi$
$258$	0	0
$259$	21.4409	1.33227
$260$	49.7275	3.08397
$261$	0	0
$262$	32.7864	2.02555
$263$	−8.82731	−0.544315	−0.272158	−	0.962253i	$-0.587737\pi$
$263$	−8.82731	−0.544315	−0.272158	+	0.962253i	$0.587737\pi$
$264$	0	0
$265$	34.8180	2.13885
$266$	0.503761	0.0308876
$267$	0	0
$268$	−48.8530	−2.98417
$269$	−21.3260	−1.30027	−0.650135	−	0.759819i	$-0.725288\pi$
$269$	−21.3260	−1.30027	−0.650135	+	0.759819i	$0.725288\pi$
$270$	0	0
$271$	−17.7756	−1.07979	−0.539895	−	0.841732i	$-0.681536\pi$
$271$	−17.7756	−1.07979	−0.539895	+	0.841732i	$0.681536\pi$
$272$	72.7325	4.41005
$273$	0	0
$274$	−24.6831	−1.49116
$275$	−8.30017	−0.500519
$276$	0	0
$277$	−22.8321	−1.37185	−0.685924	−	0.727673i	$-0.740602\pi$
$277$	−22.8321	−1.37185	−0.685924	+	0.727673i	$0.740602\pi$
$278$	−20.3170	−1.21853
$279$	0	0
$280$	−137.112	−8.19400
$281$	−4.80146	−0.286431	−0.143215	−	0.989692i	$-0.545744\pi$
$281$	−4.80146	−0.286431	−0.143215	+	0.989692i	$0.545744\pi$
$282$	0	0
$283$	−21.8710	−1.30009	−0.650047	−	0.759894i	$-0.725251\pi$
$283$	−21.8710	−1.30009	−0.650047	+	0.759894i	$0.725251\pi$
$284$	−57.3907	−3.40551
$285$	0	0
$286$	−6.81139	−0.402766
$287$	−5.07376	−0.299494
$288$	0	0
$289$	6.51264	0.383096
$290$	−44.2216	−2.59678
$291$	0	0
$292$	−74.2255	−4.34372
$293$	−4.29126	−0.250698	−0.125349	−	0.992113i	$-0.540005\pi$
$293$	−4.29126	−0.250698	−0.125349	+	0.992113i	$0.540005\pi$
$294$	0	0
$295$	27.8289	1.62026
$296$	−51.3711	−2.98588
$297$	0	0
$298$	24.8730	1.44086
$299$	−19.6245	−1.13491
$300$	0	0
$301$	−26.0243	−1.50001
$302$	28.7493	1.65434
$303$	0	0
$304$	−0.697824	−0.0400229
$305$	−3.64694	−0.208823
$306$	0	0
$307$	−16.6202	−0.948566	−0.474283	−	0.880373i	$-0.657293\pi$
$307$	−16.6202	−0.948566	−0.474283	+	0.880373i	$0.657293\pi$
$308$	21.6765	1.23513
$309$	0	0
$310$	107.502	6.10569
$311$	1.40402	0.0796148	0.0398074	−	0.999207i	$-0.487326\pi$
$311$	1.40402	0.0796148	0.0398074	+	0.999207i	$0.487326\pi$
$312$	0	0
$313$	7.92475	0.447933	0.223967	−	0.974597i	$-0.428099\pi$
$313$	7.92475	0.447933	0.223967	+	0.974597i	$0.428099\pi$
$314$	−52.8228	−2.98096
$315$	0	0
$316$	−46.7905	−2.63217
$317$	−0.811864	−0.0455988	−0.0227994	−	0.999740i	$-0.507258\pi$
$317$	−0.811864	−0.0455988	−0.0227994	+	0.999740i	$0.507258\pi$
$318$	0	0
$319$	4.43593	0.248364
$320$	110.106	6.15510
$321$	0	0
$322$	85.2784	4.75238
$323$	−0.225590	−0.0125521
$324$	0	0
$325$	20.6824	1.14726
$326$	8.15551	0.451692
$327$	0	0
$328$	12.1564	0.671225
$329$	0.00225971	0.000124582 0
$330$	0	0
$331$	−0.889505	−0.0488916	−0.0244458	−	0.999701i	$-0.507782\pi$
$331$	−0.889505	−0.0488916	−0.0244458	+	0.999701i	$0.507782\pi$
$332$	−20.3546	−1.11710
$333$	0	0
$334$	18.0688	0.988683
$335$	−32.5587	−1.77887
$336$	0	0
$337$	1.39890	0.0762027	0.0381013	−	0.999274i	$-0.487869\pi$
$337$	1.39890	0.0762027	0.0381013	+	0.999274i	$0.487869\pi$
$338$	−18.5629	−1.00969
$339$	0	0
$340$	96.7680	5.24798
$341$	−10.7837	−0.583968
$342$	0	0
$343$	−6.70150	−0.361847
$344$	62.3525	3.36182
$345$	0	0
$346$	−19.2073	−1.03259
$347$	25.4056	1.36384	0.681922	−	0.731425i	$-0.261145\pi$
$347$	25.4056	1.36384	0.681922	+	0.731425i	$0.261145\pi$
$348$	0	0
$349$	−11.0467	−0.591315	−0.295658	−	0.955294i	$-0.595539\pi$
$349$	−11.0467	−0.591315	−0.295658	+	0.955294i	$0.595539\pi$
$350$	−89.8759	−4.80407
$351$	0	0
$352$	−22.0194	−1.17364
$353$	0.252241	0.0134254	0.00671272	−	0.999977i	$-0.497863\pi$
$353$	0.252241	0.0134254	0.00671272	+	0.999977i	$0.497863\pi$
$354$	0	0
$355$	−38.2488	−2.03003
$356$	75.8540	4.02025
$357$	0	0
$358$	−61.0374	−3.22593
$359$	−32.5118	−1.71591	−0.857953	−	0.513728i	$-0.828264\pi$
$359$	−32.5118	−1.71591	−0.857953	+	0.513728i	$0.828264\pi$
$360$	0	0
$361$	−18.9978	−0.999886
$362$	42.8118	2.25014
$363$	0	0
$364$	−54.0136	−2.83108
$365$	−49.4685	−2.58930
$366$	0	0
$367$	20.9788	1.09509	0.547543	−	0.836778i	$-0.315563\pi$
$367$	20.9788	1.09509	0.547543	+	0.836778i	$0.315563\pi$
$368$	−118.130	−6.15795
$369$	0	0
$370$	−53.9582	−2.80515
$371$	−37.8191	−1.96347
$372$	0	0
$373$	−26.5886	−1.37671	−0.688353	−	0.725375i	$-0.741666\pi$
$373$	−26.5886	−1.37671	−0.688353	+	0.725375i	$0.741666\pi$
$374$	−13.2547	−0.685387
$375$	0	0
$376$	−0.00541412	−0.000279212 0
$377$	−11.0535	−0.569284
$378$	0	0
$379$	−12.8737	−0.661276	−0.330638	−	0.943758i	$-0.607264\pi$
$379$	−12.8737	−0.661276	−0.330638	+	0.943758i	$0.607264\pi$
$380$	−0.928431	−0.0476275
$381$	0	0
$382$	59.6390	3.05140
$383$	19.8514	1.01436	0.507179	−	0.861841i	$-0.330688\pi$
$383$	19.8514	1.01436	0.507179	+	0.861841i	$0.330688\pi$
$384$	0	0
$385$	14.4466	0.736265
$386$	1.24499	0.0633683
$387$	0	0
$388$	−33.4507	−1.69820
$389$	35.7712	1.81367	0.906836	−	0.421484i	$-0.138490\pi$
$389$	35.7712	1.81367	0.906836	+	0.421484i	$0.138490\pi$
$390$	0	0
$391$	−38.1886	−1.93128
$392$	82.4932	4.16654
$393$	0	0
$394$	−63.3169	−3.18986
$395$	−31.1841	−1.56904
$396$	0	0
$397$	−10.7271	−0.538375	−0.269188	−	0.963088i	$-0.586755\pi$
$397$	−10.7271	−0.538375	−0.269188	+	0.963088i	$0.586755\pi$
$398$	12.9931	0.651284
$399$	0	0
$400$	124.499	6.22493
$401$	35.1051	1.75306	0.876532	−	0.481343i	$-0.159851\pi$
$401$	35.1051	1.75306	0.876532	+	0.481343i	$0.159851\pi$
$402$	0	0
$403$	26.8709	1.33853
$404$	85.5324	4.25539
$405$	0	0
$406$	48.0332	2.38385
$407$	5.41262	0.268294
$408$	0	0
$409$	−13.4048	−0.662824	−0.331412	−	0.943486i	$-0.607525\pi$
$409$	−13.4048	−0.662824	−0.331412	+	0.943486i	$0.607525\pi$
$410$	12.7686	0.630596
$411$	0	0
$412$	−49.7398	−2.45051
$413$	−30.2276	−1.48740
$414$	0	0
$415$	−13.5656	−0.665907
$416$	54.8682	2.69014
$417$	0	0
$418$	0.127171	0.00622015
$419$	−20.8940	−1.02074	−0.510369	−	0.859956i	$-0.670491\pi$
$419$	−20.8940	−1.02074	−0.510369	+	0.859956i	$0.670491\pi$
$420$	0	0
$421$	17.4927	0.852541	0.426270	−	0.904596i	$-0.359827\pi$
$421$	17.4927	0.852541	0.426270	+	0.904596i	$0.359827\pi$
$422$	−68.9434	−3.35611
$423$	0	0
$424$	90.6121	4.40051
$425$	40.2474	1.95228
$426$	0	0
$427$	3.96128	0.191700
$428$	−67.8625	−3.28026
$429$	0	0
$430$	65.4926	3.15833
$431$	−2.88306	−0.138872	−0.0694361	−	0.997586i	$-0.522120\pi$
$431$	−2.88306	−0.138872	−0.0694361	+	0.997586i	$0.522120\pi$
$432$	0	0
$433$	3.43061	0.164865	0.0824323	−	0.996597i	$-0.473731\pi$
$433$	3.43061	0.164865	0.0824323	+	0.996597i	$0.473731\pi$
$434$	−116.768	−5.60503
$435$	0	0
$436$	33.9006	1.62354
$437$	0.366396	0.0175271
$438$	0	0
$439$	16.7134	0.797687	0.398843	−	0.917019i	$-0.369412\pi$
$439$	16.7134	0.797687	0.398843	+	0.917019i	$0.369412\pi$
$440$	−34.6130	−1.65011
$441$	0	0
$442$	33.0283	1.57100
$443$	23.9890	1.13975	0.569875	−	0.821731i	$-0.306991\pi$
$443$	23.9890	1.13975	0.569875	+	0.821731i	$0.306991\pi$
$444$	0	0
$445$	50.5539	2.39648
$446$	−16.5638	−0.784317
$447$	0	0
$448$	−119.596	−5.65039
$449$	33.8675	1.59831	0.799154	−	0.601127i	$-0.205281\pi$
$449$	33.8675	1.59831	0.799154	+	0.601127i	$0.205281\pi$
$450$	0	0
$451$	−1.28084	−0.0603123
$452$	−61.3408	−2.88523
$453$	0	0
$454$	30.6578	1.43884
$455$	−35.9981	−1.68762
$456$	0	0
$457$	35.1111	1.64243	0.821215	−	0.570619i	$-0.193297\pi$
$457$	35.1111	1.64243	0.821215	+	0.570619i	$0.193297\pi$
$458$	61.9586	2.89513
$459$	0	0
$460$	−157.168	−7.32799
$461$	18.4997	0.861619	0.430809	−	0.902443i	$-0.358228\pi$
$461$	18.4997	0.861619	0.430809	+	0.902443i	$0.358228\pi$
$462$	0	0
$463$	6.23727	0.289871	0.144935	−	0.989441i	$-0.453703\pi$
$463$	6.23727	0.289871	0.144935	+	0.989441i	$0.453703\pi$
$464$	−66.5368	−3.08890
$465$	0	0
$466$	21.7338	1.00680
$467$	33.7421	1.56140	0.780700	−	0.624906i	$-0.214863\pi$
$467$	33.7421	1.56140	0.780700	+	0.624906i	$0.214863\pi$
$468$	0	0
$469$	35.3651	1.63301
$470$	−0.00568678	−0.000262312 0
$471$	0	0
$472$	72.4233	3.33355
$473$	−6.56966	−0.302073
$474$	0	0
$475$	−0.386149	−0.0177177
$476$	−105.109	−4.81765
$477$	0	0
$478$	−59.5483	−2.72368
$479$	−3.25254	−0.148612	−0.0743062	−	0.997235i	$-0.523674\pi$
$479$	−3.25254	−0.148612	−0.0743062	+	0.997235i	$0.523674\pi$
$480$	0	0
$481$	−13.4872	−0.614965
$482$	−11.1592	−0.508290
$483$	0	0
$484$	5.47208	0.248731
$485$	−22.2936	−1.01230
$486$	0	0
$487$	−29.2275	−1.32442	−0.662211	−	0.749317i	$-0.730382\pi$
$487$	−29.2275	−1.32442	−0.662211	+	0.749317i	$0.730382\pi$
$488$	−9.49098	−0.429636
$489$	0	0
$490$	86.6476	3.91434
$491$	−8.51904	−0.384459	−0.192229	−	0.981350i	$-0.561572\pi$
$491$	−8.51904	−0.384459	−0.192229	+	0.981350i	$0.561572\pi$
$492$	0	0
$493$	−21.5097	−0.968750
$494$	−0.316887	−0.0142574
$495$	0	0
$496$	161.750	7.26279
$497$	41.5456	1.86357
$498$	0	0
$499$	−11.6950	−0.523540	−0.261770	−	0.965130i	$-0.584306\pi$
$499$	−11.6950	−0.523540	−0.261770	+	0.965130i	$0.584306\pi$
$500$	65.8593	2.94532
$501$	0	0
$502$	47.6770	2.12793
$503$	−15.7000	−0.700029	−0.350014	−	0.936744i	$-0.613823\pi$
$503$	−15.7000	−0.700029	−0.350014	+	0.936744i	$0.613823\pi$
$504$	0	0
$505$	57.0041	2.53665
$506$	21.5280	0.957036
$507$	0	0
$508$	8.64283	0.383464
$509$	4.33869	0.192309	0.0961545	−	0.995366i	$-0.469346\pi$
$509$	4.33869	0.192309	0.0961545	+	0.995366i	$0.469346\pi$
$510$	0	0
$511$	53.7324	2.37698
$512$	45.5605	2.01351
$513$	0	0
$514$	14.2250	0.627436
$515$	−33.1497	−1.46075
$516$	0	0
$517$	0.000570449 0	2.50883e−5 0
$518$	58.6090	2.57513
$519$	0	0
$520$	86.2491	3.78227
$521$	24.1662	1.05874	0.529370	−	0.848391i	$-0.322428\pi$
$521$	24.1662	1.05874	0.529370	+	0.848391i	$0.322428\pi$
$522$	0	0
$523$	−1.28085	−0.0560078	−0.0280039	−	0.999608i	$-0.508915\pi$
$523$	−1.28085	−0.0560078	−0.0280039	+	0.999608i	$0.508915\pi$
$524$	65.6334	2.86721
$525$	0	0
$526$	−24.1296	−1.05210
$527$	52.2898	2.27778
$528$	0	0
$529$	39.0248	1.69673
$530$	95.1754	4.13416
$531$	0	0
$532$	1.00846	0.0437221
$533$	3.19160	0.138244
$534$	0	0
$535$	−45.2278	−1.95537
$536$	−84.7324	−3.65988
$537$	0	0
$538$	−58.2949	−2.51327
$539$	−8.69175	−0.374380
$540$	0	0
$541$	−28.2362	−1.21397	−0.606986	−	0.794713i	$-0.707621\pi$
$541$	−28.2362	−1.21397	−0.606986	+	0.794713i	$0.707621\pi$
$542$	−48.5898	−2.08711
$543$	0	0
$544$	106.772	4.57780
$545$	22.5935	0.967799
$546$	0	0
$547$	18.7087	0.799925	0.399963	−	0.916531i	$-0.369023\pi$
$547$	18.7087	0.799925	0.399963	+	0.916531i	$0.369023\pi$
$548$	−49.4118	−2.11077
$549$	0	0
$550$	−22.6886	−0.967445
$551$	0.206373	0.00879179
$552$	0	0
$553$	33.8720	1.44038
$554$	−62.4118	−2.65162
$555$	0	0
$556$	−40.6715	−1.72486
$557$	24.7519	1.04877	0.524386	−	0.851480i	$-0.324295\pi$
$557$	24.7519	1.04877	0.524386	+	0.851480i	$0.324295\pi$
$558$	0	0
$559$	16.3704	0.692392
$560$	−216.691	−9.15689
$561$	0	0
$562$	−13.1248	−0.553638
$563$	5.89926	0.248624	0.124312	−	0.992243i	$-0.460328\pi$
$563$	5.89926	0.248624	0.124312	+	0.992243i	$0.460328\pi$
$564$	0	0
$565$	−40.8813	−1.71989
$566$	−59.7845	−2.51293
$567$	0	0
$568$	−99.5406	−4.17663
$569$	21.8777	0.917161	0.458581	−	0.888653i	$-0.348358\pi$
$569$	21.8777	0.917161	0.458581	+	0.888653i	$0.348358\pi$
$570$	0	0
$571$	0.305750	0.0127953	0.00639763	−	0.999980i	$-0.497964\pi$
$571$	0.305750	0.0127953	0.00639763	+	0.999980i	$0.497964\pi$
$572$	−13.6354	−0.570125
$573$	0	0
$574$	−13.8692	−0.578888
$575$	−65.3686	−2.72606
$576$	0	0
$577$	−3.24969	−0.135286	−0.0676432	−	0.997710i	$-0.521548\pi$
$577$	−3.24969	−0.135286	−0.0676432	+	0.997710i	$0.521548\pi$
$578$	17.8024	0.740481
$579$	0	0
$580$	−88.5250	−3.67580
$581$	14.7348	0.611304
$582$	0	0
$583$	−9.54719	−0.395404
$584$	−128.739	−5.32728
$585$	0	0
$586$	−11.7302	−0.484571
$587$	−43.1427	−1.78069	−0.890345	−	0.455286i	$-0.849537\pi$
$587$	−43.1427	−1.78069	−0.890345	+	0.455286i	$0.849537\pi$
$588$	0	0
$589$	−0.501689	−0.0206717
$590$	76.0706	3.13178
$591$	0	0
$592$	−81.1868	−3.33676
$593$	−5.14139	−0.211132	−0.105566	−	0.994412i	$-0.533665\pi$
$593$	−5.14139	−0.211132	−0.105566	+	0.994412i	$0.533665\pi$
$594$	0	0
$595$	−70.0511	−2.87181
$596$	49.7921	2.03956
$597$	0	0
$598$	−53.6437	−2.19365
$599$	−2.04207	−0.0834367	−0.0417183	−	0.999129i	$-0.513283\pi$
$599$	−2.04207	−0.0834367	−0.0417183	+	0.999129i	$0.513283\pi$
$600$	0	0
$601$	7.94702	0.324166	0.162083	−	0.986777i	$-0.448179\pi$
$601$	7.94702	0.324166	0.162083	+	0.986777i	$0.448179\pi$
$602$	−71.1376	−2.89935
$603$	0	0
$604$	57.5519	2.34175
$605$	3.64694	0.148269
$606$	0	0
$607$	−18.0641	−0.733198	−0.366599	−	0.930379i	$-0.619478\pi$
$607$	−18.0641	−0.733198	−0.366599	+	0.930379i	$0.619478\pi$
$608$	−1.02441	−0.0415453
$609$	0	0
$610$	−9.96895	−0.403631
$611$	−0.00142145	−5.75058e−5 0
$612$	0	0
$613$	0.125725	0.00507800	0.00253900	−	0.999997i	$-0.499192\pi$
$613$	0.125725	0.00507800	0.00253900	+	0.999997i	$0.499192\pi$
$614$	−45.4315	−1.83347
$615$	0	0
$616$	37.5964	1.51480
$617$	−5.30198	−0.213450	−0.106725	−	0.994289i	$-0.534036\pi$
$617$	−5.30198	−0.213450	−0.106725	+	0.994289i	$0.534036\pi$
$618$	0	0
$619$	19.0810	0.766931	0.383465	−	0.923555i	$-0.374731\pi$
$619$	19.0810	0.766931	0.383465	+	0.923555i	$0.374731\pi$
$620$	215.203	8.64275
$621$	0	0
$622$	3.83791	0.153886
$623$	−54.9113	−2.19997
$624$	0	0
$625$	2.39192	0.0956769
$626$	21.6624	0.865803
$627$	0	0
$628$	−105.743	−4.21962
$629$	−26.2457	−1.04649
$630$	0	0
$631$	−28.8352	−1.14791	−0.573955	−	0.818887i	$-0.694592\pi$
$631$	−28.8352	−1.14791	−0.573955	+	0.818887i	$0.694592\pi$
$632$	−81.1551	−3.22818
$633$	0	0
$634$	−2.21924	−0.0881373
$635$	5.76012	0.228584
$636$	0	0
$637$	21.6582	0.858129
$638$	12.1257	0.480060
$639$	0	0
$640$	140.368	5.54854
$641$	36.6121	1.44609	0.723045	−	0.690801i	$-0.242742\pi$
$641$	36.6121	1.44609	0.723045	+	0.690801i	$0.242742\pi$
$642$	0	0
$643$	10.2211	0.403081	0.201540	−	0.979480i	$-0.435405\pi$
$643$	10.2211	0.403081	0.201540	+	0.979480i	$0.435405\pi$
$644$	170.715	6.72710
$645$	0	0
$646$	−0.616651	−0.0242618
$647$	−14.8266	−0.582895	−0.291448	−	0.956587i	$-0.594137\pi$
$647$	−14.8266	−0.582895	−0.291448	+	0.956587i	$0.594137\pi$
$648$	0	0
$649$	−7.63076	−0.299533
$650$	56.5357	2.21751
$651$	0	0
$652$	16.3261	0.639380
$653$	−32.8234	−1.28448	−0.642240	−	0.766503i	$-0.721995\pi$
$653$	−32.8234	−1.28448	−0.642240	+	0.766503i	$0.721995\pi$
$654$	0	0
$655$	43.7422	1.70915
$656$	19.2119	0.750100
$657$	0	0
$658$	0.00617694	0.000240802 0
$659$	−27.2740	−1.06245	−0.531223	−	0.847232i	$-0.678267\pi$
$659$	−27.2740	−1.06245	−0.531223	+	0.847232i	$0.678267\pi$
$660$	0	0
$661$	37.8805	1.47338	0.736690	−	0.676230i	$-0.236388\pi$
$661$	37.8805	1.47338	0.736690	+	0.676230i	$0.236388\pi$
$662$	−2.43147	−0.0945019
$663$	0	0
$664$	−35.3037	−1.37005
$665$	0.672098	0.0260628
$666$	0	0
$667$	34.9355	1.35271
$668$	36.1711	1.39950
$669$	0	0
$670$	−88.9996	−3.43835
$671$	1.00000	0.0386046
$672$	0	0
$673$	−15.5129	−0.597980	−0.298990	−	0.954256i	$-0.596650\pi$
$673$	−15.5129	−0.597980	−0.298990	+	0.954256i	$0.596650\pi$
$674$	3.82390	0.147291
$675$	0	0
$676$	−37.1602	−1.42924
$677$	−18.4113	−0.707603	−0.353802	−	0.935320i	$-0.615111\pi$
$677$	−18.4113	−0.707603	−0.353802	+	0.935320i	$0.615111\pi$
$678$	0	0
$679$	24.2152	0.929294
$680$	167.838	6.43629
$681$	0	0
$682$	−29.4773	−1.12874
$683$	17.0633	0.652907	0.326454	−	0.945213i	$-0.394146\pi$
$683$	17.0633	0.652907	0.326454	+	0.945213i	$0.394146\pi$
$684$	0	0
$685$	−32.9311	−1.25823
$686$	−18.3186	−0.699409
$687$	0	0
$688$	98.5418	3.75687
$689$	23.7898	0.906319
$690$	0	0
$691$	−5.56534	−0.211715	−0.105858	−	0.994381i	$-0.533759\pi$
$691$	−5.56534	−0.211715	−0.105858	+	0.994381i	$0.533759\pi$
$692$	−38.4502	−1.46166
$693$	0	0
$694$	69.4464	2.63615
$695$	−27.1061	−1.02819
$696$	0	0
$697$	6.21076	0.235249
$698$	−30.1962	−1.14294
$699$	0	0
$700$	−179.918	−6.80027
$701$	−20.5225	−0.775123	−0.387562	−	0.921844i	$-0.626683\pi$
$701$	−20.5225	−0.775123	−0.387562	+	0.921844i	$0.626683\pi$
$702$	0	0
$703$	0.251812	0.00949726
$704$	−30.1913	−1.13788
$705$	0	0
$706$	0.689504	0.0259498
$707$	−61.9175	−2.32865
$708$	0	0
$709$	10.8572	0.407751	0.203875	−	0.978997i	$-0.434646\pi$
$709$	10.8572	0.407751	0.203875	+	0.978997i	$0.434646\pi$
$710$	−104.553	−3.92382
$711$	0	0
$712$	131.564	4.93057
$713$	−84.9277	−3.18057
$714$	0	0
$715$	−9.08748	−0.339853
$716$	−122.188	−4.56638
$717$	0	0
$718$	−88.8713	−3.31665
$719$	4.63514	0.172861	0.0864307	−	0.996258i	$-0.472454\pi$
$719$	4.63514	0.172861	0.0864307	+	0.996258i	$0.472454\pi$
$720$	0	0
$721$	36.0070	1.34097
$722$	−51.9308	−1.93266
$723$	0	0
$724$	85.7028	3.18512
$725$	−36.8190	−1.36742
$726$	0	0
$727$	11.4366	0.424160	0.212080	−	0.977252i	$-0.431976\pi$
$727$	11.4366	0.424160	0.212080	+	0.977252i	$0.431976\pi$
$728$	−93.6832	−3.47213
$729$	0	0
$730$	−135.223	−5.00482
$731$	31.8562	1.17824
$732$	0	0
$733$	7.97524	0.294572	0.147286	−	0.989094i	$-0.452946\pi$
$733$	7.97524	0.294572	0.147286	+	0.989094i	$0.452946\pi$
$734$	57.3458	2.11667
$735$	0	0
$736$	−173.416	−6.39219
$737$	8.92768	0.328855
$738$	0	0
$739$	5.04646	0.185637	0.0928186	−	0.995683i	$-0.470412\pi$
$739$	5.04646	0.185637	0.0928186	+	0.995683i	$0.470412\pi$
$740$	−108.016	−3.97076
$741$	0	0
$742$	−103.379	−3.79516
$743$	7.43485	0.272758	0.136379	−	0.990657i	$-0.456453\pi$
$743$	7.43485	0.272758	0.136379	+	0.990657i	$0.456453\pi$
$744$	0	0
$745$	33.1846	1.21579
$746$	−72.6803	−2.66101
$747$	0	0
$748$	−26.5340	−0.970181
$749$	49.1262	1.79503
$750$	0	0
$751$	−33.0659	−1.20659	−0.603296	−	0.797518i	$-0.706146\pi$
$751$	−33.0659	−1.20659	−0.603296	+	0.797518i	$0.706146\pi$
$752$	−0.00855647	−0.000312022 0
$753$	0	0
$754$	−30.2149	−1.10036
$755$	38.3562	1.39593
$756$	0	0
$757$	29.1687	1.06015	0.530077	−	0.847949i	$-0.322163\pi$
$757$	29.1687	1.06015	0.530077	+	0.847949i	$0.322163\pi$
$758$	−35.1903	−1.27817
$759$	0	0
$760$	−1.61030	−0.0584118
$761$	−27.4535	−0.995190	−0.497595	−	0.867410i	$-0.665783\pi$
$761$	−27.4535	−0.995190	−0.497595	+	0.867410i	$0.665783\pi$
$762$	0	0
$763$	−24.5409	−0.888440
$764$	119.388	4.31932
$765$	0	0
$766$	54.2640	1.96064
$767$	19.0144	0.686570
$768$	0	0
$769$	2.71137	0.0977746	0.0488873	−	0.998804i	$-0.484432\pi$
$769$	2.71137	0.0977746	0.0488873	+	0.998804i	$0.484432\pi$
$770$	39.4898	1.42311
$771$	0	0
$772$	2.49228	0.0896993
$773$	31.1512	1.12043	0.560215	−	0.828347i	$-0.310719\pi$
$773$	31.1512	1.12043	0.560215	+	0.828347i	$0.310719\pi$
$774$	0	0
$775$	89.5062	3.21516
$776$	−58.0181	−2.08273
$777$	0	0
$778$	97.7810	3.50562
$779$	−0.0595884	−0.00213498
$780$	0	0
$781$	10.4879	0.375287
$782$	−104.389	−3.73294
$783$	0	0
$784$	130.372	4.65615
$785$	−70.4740	−2.51533
$786$	0	0
$787$	7.90376	0.281739	0.140869	−	0.990028i	$-0.455010\pi$
$787$	7.90376	0.281739	0.140869	+	0.990028i	$0.455010\pi$
$788$	−126.751	−4.51532
$789$	0	0
$790$	−85.2422	−3.03278
$791$	44.4050	1.57886
$792$	0	0
$793$	−2.49181	−0.0884868
$794$	−29.3225	−1.04062
$795$	0	0
$796$	26.0102	0.921908
$797$	52.9448	1.87540	0.937701	−	0.347444i	$-0.112950\pi$
$797$	52.9448	1.87540	0.937701	+	0.347444i	$0.112950\pi$
$798$	0	0
$799$	−0.00276610	−9.78575e−5 0
$800$	182.765	6.46171
$801$	0	0
$802$	95.9601	3.38847
$803$	13.5644	0.478677
$804$	0	0
$805$	113.775	4.01004
$806$	73.4518	2.58723
$807$	0	0
$808$	148.350	5.21895
$809$	33.9262	1.19278	0.596391	−	0.802694i	$-0.296601\pi$
$809$	33.9262	1.19278	0.596391	+	0.802694i	$0.296601\pi$
$810$	0	0
$811$	0.744633	0.0261476	0.0130738	−	0.999915i	$-0.495838\pi$
$811$	0.744633	0.0261476	0.0130738	+	0.999915i	$0.495838\pi$
$812$	96.1553	3.37439
$813$	0	0
$814$	14.7955	0.518581
$815$	10.8807	0.381136
$816$	0	0
$817$	−0.305641	−0.0106930
$818$	−36.6421	−1.28116
$819$	0	0
$820$	25.5608	0.892623
$821$	−31.7241	−1.10718	−0.553590	−	0.832790i	$-0.686742\pi$
$821$	−31.7241	−1.10718	−0.553590	+	0.832790i	$0.686742\pi$
$822$	0	0
$823$	11.6796	0.407124	0.203562	−	0.979062i	$-0.434748\pi$
$823$	11.6796	0.407124	0.203562	+	0.979062i	$0.434748\pi$
$824$	−86.2705	−3.00538
$825$	0	0
$826$	−82.6274	−2.87497
$827$	20.3695	0.708318	0.354159	−	0.935185i	$-0.384767\pi$
$827$	20.3695	0.708318	0.354159	+	0.935185i	$0.384767\pi$
$828$	0	0
$829$	−20.8722	−0.724922	−0.362461	−	0.931999i	$-0.618063\pi$
$829$	−20.8722	−0.724922	−0.362461	+	0.931999i	$0.618063\pi$
$830$	−37.0816	−1.28712
$831$	0	0
$832$	75.2309	2.60816
$833$	42.1462	1.46028
$834$	0	0
$835$	24.1067	0.834247
$836$	0.254578	0.00880477
$837$	0	0
$838$	−57.1139	−1.97297
$839$	20.4940	0.707530	0.353765	−	0.935334i	$-0.384901\pi$
$839$	20.4940	0.707530	0.353765	+	0.935334i	$0.384901\pi$
$840$	0	0
$841$	−9.32253	−0.321466
$842$	47.8164	1.64786
$843$	0	0
$844$	−138.014	−4.75065
$845$	−24.7659	−0.851973
$846$	0	0
$847$	−3.96128	−0.136111
$848$	143.203	4.91762
$849$	0	0
$850$	110.017	3.77354
$851$	42.6275	1.46125
$852$	0	0
$853$	20.7580	0.710739	0.355370	−	0.934726i	$-0.384355\pi$
$853$	20.7580	0.710739	0.355370	+	0.934726i	$0.384355\pi$
$854$	10.8282	0.370534
$855$	0	0
$856$	−117.703	−4.02301
$857$	−34.8179	−1.18936	−0.594679	−	0.803963i	$-0.702721\pi$
$857$	−34.8179	−1.18936	−0.594679	+	0.803963i	$0.702721\pi$
$858$	0	0
$859$	−24.7818	−0.845544	−0.422772	−	0.906236i	$-0.638943\pi$
$859$	−24.7818	−0.845544	−0.422772	+	0.906236i	$0.638943\pi$
$860$	131.106	4.47069
$861$	0	0
$862$	−7.88088	−0.268424
$863$	24.0402	0.818337	0.409168	−	0.912459i	$-0.365819\pi$
$863$	24.0402	0.818337	0.409168	+	0.912459i	$0.365819\pi$
$864$	0	0
$865$	−25.6256	−0.871297
$866$	9.37761	0.318664
$867$	0	0
$868$	−233.752	−7.93405
$869$	8.55077	0.290065
$870$	0	0
$871$	−22.2461	−0.753780
$872$	58.7984	1.99117
$873$	0	0
$874$	1.00155	0.0338779
$875$	−47.6760	−1.61174
$876$	0	0
$877$	4.16282	0.140568	0.0702842	−	0.997527i	$-0.477609\pi$
$877$	4.16282	0.140568	0.0702842	+	0.997527i	$0.477609\pi$
$878$	45.6862	1.54184
$879$	0	0
$880$	−54.7024	−1.84402
$881$	−26.8981	−0.906221	−0.453110	−	0.891454i	$-0.649686\pi$
$881$	−26.8981	−0.906221	−0.453110	+	0.891454i	$0.649686\pi$
$882$	0	0
$883$	13.4617	0.453023	0.226512	−	0.974008i	$-0.427268\pi$
$883$	13.4617	0.453023	0.226512	+	0.974008i	$0.427268\pi$
$884$	66.1178	2.22378
$885$	0	0
$886$	65.5741	2.20301
$887$	6.91141	0.232063	0.116031	−	0.993246i	$-0.462983\pi$
$887$	6.91141	0.232063	0.116031	+	0.993246i	$0.462983\pi$
$888$	0	0
$889$	−6.25661	−0.209840
$890$	138.190	4.63212
$891$	0	0
$892$	−33.1582	−1.11022
$893$	2.65390e−5 0	8.88095e−7 0
$894$	0	0
$895$	−81.4337	−2.72203
$896$	−152.467	−5.09357
$897$	0	0
$898$	92.5772	3.08934
$899$	−47.8356	−1.59541
$900$	0	0
$901$	46.2942	1.54228
$902$	−3.50118	−0.116577
$903$	0	0
$904$	−106.392	−3.53853
$905$	57.1177	1.89866
$906$	0	0
$907$	37.3712	1.24089	0.620444	−	0.784250i	$-0.286952\pi$
$907$	37.3712	1.24089	0.620444	+	0.784250i	$0.286952\pi$
$908$	61.3724	2.03672
$909$	0	0
$910$	−98.4012	−3.26197
$911$	54.0645	1.79124	0.895618	−	0.444824i	$-0.146734\pi$
$911$	54.0645	1.79124	0.895618	+	0.444824i	$0.146734\pi$
$912$	0	0
$913$	3.71971	0.123105
$914$	95.9767	3.17463
$915$	0	0
$916$	124.032	4.09813
$917$	−47.5125	−1.56900
$918$	0	0
$919$	−19.5847	−0.646041	−0.323021	−	0.946392i	$-0.604698\pi$
$919$	−19.5847	−0.646041	−0.323021	+	0.946392i	$0.604698\pi$
$920$	−272.598	−8.98728
$921$	0	0
$922$	50.5692	1.66541
$923$	−26.1339	−0.860208
$924$	0	0
$925$	−44.9257	−1.47715
$926$	17.0497	0.560287
$927$	0	0
$928$	−97.6766	−3.20639
$929$	0.822167	0.0269744	0.0134872	−	0.999909i	$-0.495707\pi$
$929$	0.822167	0.0269744	0.0134872	+	0.999909i	$0.495707\pi$
$930$	0	0
$931$	−0.404367	−0.0132526
$932$	43.5079	1.42515
$933$	0	0
$934$	92.2345	3.01801
$935$	−17.6839	−0.578327
$936$	0	0
$937$	12.5109	0.408714	0.204357	−	0.978896i	$-0.434490\pi$
$937$	12.5109	0.408714	0.204357	+	0.978896i	$0.434490\pi$
$938$	96.6708	3.15641
$939$	0	0
$940$	−0.0113841	−0.000371308 0
$941$	−31.3431	−1.02176	−0.510878	−	0.859653i	$-0.670680\pi$
$941$	−31.3431	−1.02176	−0.510878	+	0.859653i	$0.670680\pi$
$942$	0	0
$943$	−10.0873	−0.328489
$944$	114.458	3.72528
$945$	0	0
$946$	−17.9582	−0.583873
$947$	41.9864	1.36438	0.682188	−	0.731177i	$-0.261029\pi$
$947$	41.9864	1.36438	0.682188	+	0.731177i	$0.261029\pi$
$948$	0	0
$949$	−33.7999	−1.09719
$950$	−1.05554	−0.0342463
$951$	0	0
$952$	−182.304	−5.90852
$953$	−28.8688	−0.935152	−0.467576	−	0.883953i	$-0.654873\pi$
$953$	−28.8688	−0.935152	−0.467576	+	0.883953i	$0.654873\pi$
$954$	0	0
$955$	79.5679	2.57476
$956$	−119.207	−3.85543
$957$	0	0
$958$	−8.89085	−0.287250
$959$	35.7696	1.15506
$960$	0	0
$961$	85.2875	2.75121
$962$	−36.8675	−1.18866
$963$	0	0
$964$	−22.3391	−0.719495
$965$	1.66101	0.0534700
$966$	0	0
$967$	−4.71947	−0.151768	−0.0758839	−	0.997117i	$-0.524178\pi$
$967$	−4.71947	−0.151768	−0.0758839	+	0.997117i	$0.524178\pi$
$968$	9.49098	0.305052
$969$	0	0
$970$	−60.9399	−1.95666
$971$	17.6533	0.566522	0.283261	−	0.959043i	$-0.408584\pi$
$971$	17.6533	0.566522	0.283261	+	0.959043i	$0.408584\pi$
$972$	0	0
$973$	29.4424	0.943881
$974$	−79.8936	−2.55996
$975$	0	0
$976$	−14.9995	−0.480123
$977$	35.4007	1.13257	0.566285	−	0.824210i	$-0.308380\pi$
$977$	35.4007	1.13257	0.566285	+	0.824210i	$0.308380\pi$
$978$	0	0
$979$	−13.8620	−0.443031
$980$	173.456	5.54084
$981$	0	0
$982$	−23.2869	−0.743115
$983$	53.5369	1.70756	0.853781	−	0.520633i	$-0.174304\pi$
$983$	53.5369	1.70756	0.853781	+	0.520633i	$0.174304\pi$
$984$	0	0
$985$	−84.4748	−2.69159
$986$	−58.7971	−1.87248
$987$	0	0
$988$	−0.634360	−0.0201817
$989$	−51.7399	−1.64523
$990$	0	0
$991$	−53.7939	−1.70882	−0.854410	−	0.519600i	$-0.826081\pi$
$991$	−53.7939	−1.70882	−0.854410	+	0.519600i	$0.826081\pi$
$992$	237.450	7.53905
$993$	0	0
$994$	113.565	3.60207
$995$	17.3348	0.549551
$996$	0	0
$997$	14.8273	0.469585	0.234792	−	0.972046i	$-0.424559\pi$
$997$	14.8273	0.469585	0.234792	+	0.972046i	$0.424559\pi$
$998$	−31.9684	−1.01194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.f.1.12		12
3.2	odd	2		2013.2.a.c.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.c.1.1	✓	12	3.2	odd	2
6039.2.a.f.1.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.73351 q^{2} +5.47208 q^{4} +3.64694 q^{5} -3.96128 q^{7} +9.49098 q^{8} +O(q^{10})\)	\(q+2.73351 q^{2} +5.47208 q^{4} +3.64694 q^{5} -3.96128 q^{7} +9.49098 q^{8} +9.96895 q^{10} -1.00000 q^{11} +2.49181 q^{13} -10.8282 q^{14} +14.9995 q^{16} +4.84898 q^{17} -0.0465231 q^{19} +19.9564 q^{20} -2.73351 q^{22} -7.87558 q^{23} +8.30017 q^{25} +6.81139 q^{26} -21.6765 q^{28} -4.43593 q^{29} +10.7837 q^{31} +22.0194 q^{32} +13.2547 q^{34} -14.4466 q^{35} -5.41262 q^{37} -0.127171 q^{38} +34.6130 q^{40} +1.28084 q^{41} +6.56966 q^{43} -5.47208 q^{44} -21.5280 q^{46} -0.000570449 q^{47} +8.69175 q^{49} +22.6886 q^{50} +13.6354 q^{52} +9.54719 q^{53} -3.64694 q^{55} -37.5964 q^{56} -12.1257 q^{58} +7.63076 q^{59} -1.00000 q^{61} +29.4773 q^{62} +30.1913 q^{64} +9.08748 q^{65} -8.92768 q^{67} +26.5340 q^{68} -39.4898 q^{70} -10.4879 q^{71} -13.5644 q^{73} -14.7955 q^{74} -0.254578 q^{76} +3.96128 q^{77} -8.55077 q^{79} +54.7024 q^{80} +3.50118 q^{82} -3.71971 q^{83} +17.6839 q^{85} +17.9582 q^{86} -9.49098 q^{88} +13.8620 q^{89} -9.87077 q^{91} -43.0958 q^{92} -0.00155933 q^{94} -0.169667 q^{95} -6.11297 q^{97} +23.7590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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