LMFDB - Embedded newform 6039.2.a.g.1.8

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			1.8
Root			$-0.329361$ 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)

$f(q)$	$=$	$q+0.329361 q^{2} -1.89152 q^{4} -2.08793 q^{5} -4.32416 q^{7} -1.28171 q^{8} +O(q^{10})$	\(q+0.329361 q^{2} -1.89152 q^{4} -2.08793 q^{5} -4.32416 q^{7} -1.28171 q^{8} -0.687681 q^{10} +1.00000 q^{11} +2.30428 q^{13} -1.42421 q^{14} +3.36090 q^{16} +0.847902 q^{17} -0.842952 q^{19} +3.94936 q^{20} +0.329361 q^{22} -4.59119 q^{23} -0.640561 q^{25} +0.758939 q^{26} +8.17925 q^{28} +4.26540 q^{29} +6.92500 q^{31} +3.67038 q^{32} +0.279266 q^{34} +9.02854 q^{35} +4.24680 q^{37} -0.277635 q^{38} +2.67613 q^{40} -2.01259 q^{41} +5.36766 q^{43} -1.89152 q^{44} -1.51216 q^{46} +0.441959 q^{47} +11.6984 q^{49} -0.210976 q^{50} -4.35859 q^{52} +5.16024 q^{53} -2.08793 q^{55} +5.54234 q^{56} +1.40486 q^{58} -7.76000 q^{59} -1.00000 q^{61} +2.28082 q^{62} -5.51291 q^{64} -4.81116 q^{65} +15.0506 q^{67} -1.60383 q^{68} +2.97365 q^{70} -8.68938 q^{71} +2.20155 q^{73} +1.39873 q^{74} +1.59446 q^{76} -4.32416 q^{77} +5.20677 q^{79} -7.01731 q^{80} -0.662867 q^{82} -12.0842 q^{83} -1.77036 q^{85} +1.76790 q^{86} -1.28171 q^{88} -0.539840 q^{89} -9.96407 q^{91} +8.68433 q^{92} +0.145564 q^{94} +1.76002 q^{95} -13.1853 q^{97} +3.85299 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.329361	0.232893	0.116447	−	0.993197i	$-0.462850\pi$
$2$	0.329361	0.232893	0.116447	+	0.993197i	$0.462850\pi$
$3$	0	0
$3$	0	0
$4$	−1.89152	−0.945761
$5$	−2.08793	−0.933749	−0.466875	−	0.884323i	$-0.654620\pi$
$5$	−2.08793	−0.933749	−0.466875	+	0.884323i	$0.654620\pi$
$6$	0	0
$7$	−4.32416	−1.63438	−0.817190	−	0.576368i	$-0.804469\pi$
$7$	−4.32416	−1.63438	−0.817190	+	0.576368i	$0.804469\pi$
$8$	−1.28171	−0.453155
$9$	0	0
$10$	−0.687681	−0.217464
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.30428	0.639092	0.319546	−	0.947571i	$-0.396470\pi$
$13$	2.30428	0.639092	0.319546	+	0.947571i	$0.396470\pi$
$14$	−1.42421	−0.380636
$15$	0	0
$16$	3.36090	0.840224
$17$	0.847902	0.205647	0.102823	−	0.994700i	$-0.467212\pi$
$17$	0.847902	0.205647	0.102823	+	0.994700i	$0.467212\pi$
$18$	0	0
$19$	−0.842952	−0.193386	−0.0966932	−	0.995314i	$-0.530827\pi$
$19$	−0.842952	−0.193386	−0.0966932	+	0.995314i	$0.530827\pi$
$20$	3.94936	0.883103
$21$	0	0
$22$	0.329361	0.0702200
$23$	−4.59119	−0.957329	−0.478664	−	0.877998i	$-0.658879\pi$
$23$	−4.59119	−0.957329	−0.478664	+	0.877998i	$0.658879\pi$
$24$	0	0
$25$	−0.640561	−0.128112
$26$	0.758939	0.148840
$27$	0	0
$28$	8.17925	1.54573
$29$	4.26540	0.792065	0.396032	−	0.918237i	$-0.370387\pi$
$29$	4.26540	0.792065	0.396032	+	0.918237i	$0.370387\pi$
$30$	0	0
$31$	6.92500	1.24377	0.621883	−	0.783110i	$-0.286368\pi$
$31$	6.92500	1.24377	0.621883	+	0.783110i	$0.286368\pi$
$32$	3.67038	0.648837
$33$	0	0
$34$	0.279266	0.0478937
$35$	9.02854	1.52610
$36$	0	0
$37$	4.24680	0.698170	0.349085	−	0.937091i	$-0.386492\pi$
$37$	4.24680	0.698170	0.349085	+	0.937091i	$0.386492\pi$
$38$	−0.277635	−0.0450384
$39$	0	0
$40$	2.67613	0.423133
$41$	−2.01259	−0.314313	−0.157156	−	0.987574i	$-0.550233\pi$
$41$	−2.01259	−0.314313	−0.157156	+	0.987574i	$0.550233\pi$
$42$	0	0
$43$	5.36766	0.818560	0.409280	−	0.912409i	$-0.365780\pi$
$43$	5.36766	0.818560	0.409280	+	0.912409i	$0.365780\pi$
$44$	−1.89152	−0.285158
$45$	0	0
$46$	−1.51216	−0.222955
$47$	0.441959	0.0644664	0.0322332	−	0.999480i	$-0.489738\pi$
$47$	0.441959	0.0644664	0.0322332	+	0.999480i	$0.489738\pi$
$48$	0	0
$49$	11.6984	1.67120
$50$	−0.210976	−0.0298365
$51$	0	0
$52$	−4.35859	−0.604428
$53$	5.16024	0.708814	0.354407	−	0.935091i	$-0.384683\pi$
$53$	5.16024	0.708814	0.354407	+	0.935091i	$0.384683\pi$
$54$	0	0
$55$	−2.08793	−0.281536
$56$	5.54234	0.740627
$57$	0	0
$58$	1.40486	0.184467
$59$	−7.76000	−1.01027	−0.505133	−	0.863042i	$-0.668556\pi$
$59$	−7.76000	−1.01027	−0.505133	+	0.863042i	$0.668556\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	2.28082	0.289665
$63$	0	0
$64$	−5.51291	−0.689114
$65$	−4.81116	−0.596751
$66$	0	0
$67$	15.0506	1.83872	0.919362	−	0.393412i	$-0.128705\pi$
$67$	15.0506	1.83872	0.919362	+	0.393412i	$0.128705\pi$
$68$	−1.60383	−0.194492
$69$	0	0
$70$	2.97365	0.355419
$71$	−8.68938	−1.03124	−0.515620	−	0.856818i	$-0.672438\pi$
$71$	−8.68938	−1.03124	−0.515620	+	0.856818i	$0.672438\pi$
$72$	0	0
$73$	2.20155	0.257672	0.128836	−	0.991666i	$-0.458876\pi$
$73$	2.20155	0.257672	0.128836	+	0.991666i	$0.458876\pi$
$74$	1.39873	0.162599
$75$	0	0
$76$	1.59446	0.182897
$77$	−4.32416	−0.492784
$78$	0	0
$79$	5.20677	0.585807	0.292904	−	0.956142i	$-0.405378\pi$
$79$	5.20677	0.585807	0.292904	+	0.956142i	$0.405378\pi$
$80$	−7.01731	−0.784559
$81$	0	0
$82$	−0.662867	−0.0732014
$83$	−12.0842	−1.32642	−0.663209	−	0.748434i	$-0.730806\pi$
$83$	−12.0842	−1.32642	−0.663209	+	0.748434i	$0.730806\pi$
$84$	0	0
$85$	−1.77036	−0.192022
$86$	1.76790	0.190637
$87$	0	0
$88$	−1.28171	−0.136631
$89$	−0.539840	−0.0572229	−0.0286114	−	0.999591i	$-0.509109\pi$
$89$	−0.539840	−0.0572229	−0.0286114	+	0.999591i	$0.509109\pi$
$90$	0	0
$91$	−9.96407	−1.04452
$92$	8.68433	0.905404
$93$	0	0
$94$	0.145564	0.0150138
$95$	1.76002	0.180574
$96$	0	0
$97$	−13.1853	−1.33877	−0.669383	−	0.742917i	$-0.733442\pi$
$97$	−13.1853	−1.33877	−0.669383	+	0.742917i	$0.733442\pi$
$98$	3.85299	0.389211
$99$	0	0
$100$	1.21163	0.121163
$101$	−10.1739	−1.01234	−0.506171	−	0.862433i	$-0.668940\pi$
$101$	−10.1739	−1.01234	−0.506171	+	0.862433i	$0.668940\pi$
$102$	0	0
$103$	10.5843	1.04290	0.521451	−	0.853281i	$-0.325391\pi$
$103$	10.5843	1.04290	0.521451	+	0.853281i	$0.325391\pi$
$104$	−2.95343	−0.289607
$105$	0	0
$106$	1.69958	0.165078
$107$	2.78318	0.269060	0.134530	−	0.990909i	$-0.457047\pi$
$107$	2.78318	0.269060	0.134530	+	0.990909i	$0.457047\pi$
$108$	0	0
$109$	−2.49317	−0.238802	−0.119401	−	0.992846i	$-0.538097\pi$
$109$	−2.49317	−0.238802	−0.119401	+	0.992846i	$0.538097\pi$
$110$	−0.687681	−0.0655679
$111$	0	0
$112$	−14.5331	−1.37325
$113$	−2.15593	−0.202813	−0.101406	−	0.994845i	$-0.532334\pi$
$113$	−2.15593	−0.202813	−0.101406	+	0.994845i	$0.532334\pi$
$114$	0	0
$115$	9.58607	0.893905
$116$	−8.06809	−0.749104
$117$	0	0
$118$	−2.55584	−0.235284
$119$	−3.66647	−0.336105
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.329361	−0.0298189
$123$	0	0
$124$	−13.0988	−1.17631
$125$	11.7771	1.05337
$126$	0	0
$127$	−5.82408	−0.516803	−0.258402	−	0.966038i	$-0.583196\pi$
$127$	−5.82408	−0.516803	−0.258402	+	0.966038i	$0.583196\pi$
$128$	−9.15649	−0.809327
$129$	0	0
$130$	−1.58461	−0.138979
$131$	11.1761	0.976460	0.488230	−	0.872715i	$-0.337643\pi$
$131$	11.1761	0.976460	0.488230	+	0.872715i	$0.337643\pi$
$132$	0	0
$133$	3.64506	0.316067
$134$	4.95708	0.428227
$135$	0	0
$136$	−1.08677	−0.0931897
$137$	−11.7160	−1.00097	−0.500483	−	0.865746i	$-0.666844\pi$
$137$	−11.7160	−1.00097	−0.500483	+	0.865746i	$0.666844\pi$
$138$	0	0
$139$	22.0583	1.87096	0.935481	−	0.353378i	$-0.114967\pi$
$139$	22.0583	1.87096	0.935481	+	0.353378i	$0.114967\pi$
$140$	−17.0777	−1.44333
$141$	0	0
$142$	−2.86194	−0.240169
$143$	2.30428	0.192693
$144$	0	0
$145$	−8.90584	−0.739590
$146$	0.725104	0.0600100
$147$	0	0
$148$	−8.03292	−0.660302
$149$	−4.04616	−0.331475	−0.165737	−	0.986170i	$-0.553000\pi$
$149$	−4.04616	−0.331475	−0.165737	+	0.986170i	$0.553000\pi$
$150$	0	0
$151$	−3.95386	−0.321760	−0.160880	−	0.986974i	$-0.551433\pi$
$151$	−3.95386	−0.321760	−0.160880	+	0.986974i	$0.551433\pi$
$152$	1.08042	0.0876339
$153$	0	0
$154$	−1.42421	−0.114766
$155$	−14.4589	−1.16137
$156$	0	0
$157$	−13.3288	−1.06376	−0.531879	−	0.846821i	$-0.678514\pi$
$157$	−13.3288	−1.06376	−0.531879	+	0.846821i	$0.678514\pi$
$158$	1.71491	0.136431
$159$	0	0
$160$	−7.66348	−0.605851
$161$	19.8530	1.56464
$162$	0	0
$163$	−8.27237	−0.647942	−0.323971	−	0.946067i	$-0.605018\pi$
$163$	−8.27237	−0.647942	−0.323971	+	0.946067i	$0.605018\pi$
$164$	3.80685	0.297265
$165$	0	0
$166$	−3.98008	−0.308914
$167$	17.3978	1.34628	0.673142	−	0.739513i	$-0.264944\pi$
$167$	17.3978	1.34628	0.673142	+	0.739513i	$0.264944\pi$
$168$	0	0
$169$	−7.69030	−0.591562
$170$	−0.583087	−0.0447207
$171$	0	0
$172$	−10.1530	−0.774162
$173$	−7.80601	−0.593480	−0.296740	−	0.954958i	$-0.595899\pi$
$173$	−7.80601	−0.593480	−0.296740	+	0.954958i	$0.595899\pi$
$174$	0	0
$175$	2.76989	0.209384
$176$	3.36090	0.253337
$177$	0	0
$178$	−0.177802	−0.0133268
$179$	−11.9302	−0.891702	−0.445851	−	0.895107i	$-0.647099\pi$
$179$	−11.9302	−0.891702	−0.445851	+	0.895107i	$0.647099\pi$
$180$	0	0
$181$	1.16588	0.0866594	0.0433297	−	0.999061i	$-0.486203\pi$
$181$	1.16588	0.0866594	0.0433297	+	0.999061i	$0.486203\pi$
$182$	−3.28178	−0.243261
$183$	0	0
$184$	5.88459	0.433818
$185$	−8.86702	−0.651916
$186$	0	0
$187$	0.847902	0.0620048
$188$	−0.835975	−0.0609698
$189$	0	0
$190$	0.579682	0.0420546
$191$	−5.77475	−0.417846	−0.208923	−	0.977932i	$-0.566996\pi$
$191$	−5.77475	−0.417846	−0.208923	+	0.977932i	$0.566996\pi$
$192$	0	0
$193$	4.75543	0.342303	0.171152	−	0.985245i	$-0.445251\pi$
$193$	4.75543	0.342303	0.171152	+	0.985245i	$0.445251\pi$
$194$	−4.34273	−0.311790
$195$	0	0
$196$	−22.1277	−1.58055
$197$	−9.54151	−0.679805	−0.339902	−	0.940461i	$-0.610394\pi$
$197$	−9.54151	−0.679805	−0.339902	+	0.940461i	$0.610394\pi$
$198$	0	0
$199$	19.6958	1.39620	0.698098	−	0.716002i	$-0.254030\pi$
$199$	19.6958	1.39620	0.698098	+	0.716002i	$0.254030\pi$
$200$	0.821016	0.0580546
$201$	0	0
$202$	−3.35089	−0.235768
$203$	−18.4443	−1.29453
$204$	0	0
$205$	4.20213	0.293490
$206$	3.48606	0.242885
$207$	0	0
$208$	7.74444	0.536980
$209$	−0.842952	−0.0583082
$210$	0	0
$211$	−8.26763	−0.569167	−0.284584	−	0.958651i	$-0.591855\pi$
$211$	−8.26763	−0.569167	−0.284584	+	0.958651i	$0.591855\pi$
$212$	−9.76071	−0.670368
$213$	0	0
$214$	0.916672	0.0626624
$215$	−11.2073	−0.764330
$216$	0	0
$217$	−29.9448	−2.03279
$218$	−0.821153	−0.0556155
$219$	0	0
$220$	3.94936	0.266266
$221$	1.95380	0.131427
$222$	0	0
$223$	14.7795	0.989710	0.494855	−	0.868976i	$-0.335221\pi$
$223$	14.7795	0.989710	0.494855	+	0.868976i	$0.335221\pi$
$224$	−15.8713	−1.06045
$225$	0	0
$226$	−0.710079	−0.0472337
$227$	−13.8237	−0.917513	−0.458757	−	0.888562i	$-0.651705\pi$
$227$	−13.8237	−0.917513	−0.458757	+	0.888562i	$0.651705\pi$
$228$	0	0
$229$	−19.6064	−1.29563	−0.647813	−	0.761800i	$-0.724316\pi$
$229$	−19.6064	−1.29563	−0.647813	+	0.761800i	$0.724316\pi$
$230$	3.15727	0.208185
$231$	0	0
$232$	−5.46703	−0.358928
$233$	1.13926	0.0746351	0.0373176	−	0.999303i	$-0.488119\pi$
$233$	1.13926	0.0746351	0.0373176	+	0.999303i	$0.488119\pi$
$234$	0	0
$235$	−0.922779	−0.0601954
$236$	14.6782	0.955469
$237$	0	0
$238$	−1.20759	−0.0782765
$239$	6.10323	0.394785	0.197393	−	0.980324i	$-0.436753\pi$
$239$	6.10323	0.394785	0.197393	+	0.980324i	$0.436753\pi$
$240$	0	0
$241$	−15.1879	−0.978341	−0.489171	−	0.872188i	$-0.662701\pi$
$241$	−15.1879	−0.978341	−0.489171	+	0.872188i	$0.662701\pi$
$242$	0.329361	0.0211721
$243$	0	0
$244$	1.89152	0.121092
$245$	−24.4254	−1.56048
$246$	0	0
$247$	−1.94239	−0.123592
$248$	−8.87588	−0.563619
$249$	0	0
$250$	3.87891	0.245324
$251$	−10.8342	−0.683849	−0.341924	−	0.939728i	$-0.611079\pi$
$251$	−10.8342	−0.683849	−0.341924	+	0.939728i	$0.611079\pi$
$252$	0	0
$253$	−4.59119	−0.288646
$254$	−1.91822	−0.120360
$255$	0	0
$256$	8.01004	0.500627
$257$	11.1230	0.693832	0.346916	−	0.937896i	$-0.387229\pi$
$257$	11.1230	0.693832	0.346916	+	0.937896i	$0.387229\pi$
$258$	0	0
$259$	−18.3639	−1.14108
$260$	9.10042	0.564384
$261$	0	0
$262$	3.68097	0.227411
$263$	−23.8445	−1.47031	−0.735156	−	0.677898i	$-0.762891\pi$
$263$	−23.8445	−1.47031	−0.735156	+	0.677898i	$0.762891\pi$
$264$	0	0
$265$	−10.7742	−0.661854
$266$	1.20054	0.0736098
$267$	0	0
$268$	−28.4686	−1.73899
$269$	−16.3601	−0.997496	−0.498748	−	0.866747i	$-0.666207\pi$
$269$	−16.3601	−0.997496	−0.498748	+	0.866747i	$0.666207\pi$
$270$	0	0
$271$	−11.1059	−0.674638	−0.337319	−	0.941390i	$-0.609520\pi$
$271$	−11.1059	−0.674638	−0.337319	+	0.941390i	$0.609520\pi$
$272$	2.84971	0.172789
$273$	0	0
$274$	−3.85880	−0.233118
$275$	−0.640561	−0.0386273
$276$	0	0
$277$	−15.3696	−0.923467	−0.461734	−	0.887019i	$-0.652772\pi$
$277$	−15.3696	−0.923467	−0.461734	+	0.887019i	$0.652772\pi$
$278$	7.26514	0.435734
$279$	0	0
$280$	−11.5720	−0.691560
$281$	18.4473	1.10047	0.550237	−	0.835008i	$-0.314537\pi$
$281$	18.4473	1.10047	0.550237	+	0.835008i	$0.314537\pi$
$282$	0	0
$283$	−0.965328	−0.0573828	−0.0286914	−	0.999588i	$-0.509134\pi$
$283$	−0.965328	−0.0573828	−0.0286914	+	0.999588i	$0.509134\pi$
$284$	16.4361	0.975305
$285$	0	0
$286$	0.758939	0.0448770
$287$	8.70275	0.513707
$288$	0	0
$289$	−16.2811	−0.957710
$290$	−2.93324	−0.172246
$291$	0	0
$292$	−4.16428	−0.243696
$293$	1.12409	0.0656698	0.0328349	−	0.999461i	$-0.489546\pi$
$293$	1.12409	0.0656698	0.0328349	+	0.999461i	$0.489546\pi$
$294$	0	0
$295$	16.2023	0.943335
$296$	−5.44319	−0.316379
$297$	0	0
$298$	−1.33265	−0.0771982
$299$	−10.5794	−0.611821
$300$	0	0
$301$	−23.2106	−1.33784
$302$	−1.30225	−0.0749359
$303$	0	0
$304$	−2.83307	−0.162488
$305$	2.08793	0.119554
$306$	0	0
$307$	−9.26362	−0.528703	−0.264351	−	0.964426i	$-0.585158\pi$
$307$	−9.26362	−0.528703	−0.264351	+	0.964426i	$0.585158\pi$
$308$	8.17925	0.466056
$309$	0	0
$310$	−4.76219	−0.270474
$311$	−13.4089	−0.760349	−0.380174	−	0.924915i	$-0.624136\pi$
$311$	−13.4089	−0.760349	−0.380174	+	0.924915i	$0.624136\pi$
$312$	0	0
$313$	30.3582	1.71595	0.857973	−	0.513694i	$-0.171723\pi$
$313$	30.3582	1.71595	0.857973	+	0.513694i	$0.171723\pi$
$314$	−4.39000	−0.247742
$315$	0	0
$316$	−9.84872	−0.554034
$317$	1.67653	0.0941635	0.0470817	−	0.998891i	$-0.485008\pi$
$317$	1.67653	0.0941635	0.0470817	+	0.998891i	$0.485008\pi$
$318$	0	0
$319$	4.26540	0.238817
$320$	11.5106	0.643460
$321$	0	0
$322$	6.53882	0.364394
$323$	−0.714741	−0.0397692
$324$	0	0
$325$	−1.47603	−0.0818754
$326$	−2.72459	−0.150901
$327$	0	0
$328$	2.57956	0.142432
$329$	−1.91110	−0.105363
$330$	0	0
$331$	−1.94883	−0.107117	−0.0535586	−	0.998565i	$-0.517056\pi$
$331$	−1.94883	−0.107117	−0.0535586	+	0.998565i	$0.517056\pi$
$332$	22.8576	1.25447
$333$	0	0
$334$	5.73016	0.313540
$335$	−31.4246	−1.71691
$336$	0	0
$337$	−19.6545	−1.07065	−0.535325	−	0.844646i	$-0.679811\pi$
$337$	−19.6545	−1.07065	−0.535325	+	0.844646i	$0.679811\pi$
$338$	−2.53289	−0.137771
$339$	0	0
$340$	3.34867	0.181607
$341$	6.92500	0.375010
$342$	0	0
$343$	−20.3166	−1.09699
$344$	−6.87980	−0.370934
$345$	0	0
$346$	−2.57099	−0.138218
$347$	−18.0621	−0.969623	−0.484812	−	0.874619i	$-0.661112\pi$
$347$	−18.0621	−0.969623	−0.484812	+	0.874619i	$0.661112\pi$
$348$	0	0
$349$	1.78200	0.0953881	0.0476940	−	0.998862i	$-0.484813\pi$
$349$	1.78200	0.0953881	0.0476940	+	0.998862i	$0.484813\pi$
$350$	0.912293	0.0487641
$351$	0	0
$352$	3.67038	0.195632
$353$	31.0791	1.65418	0.827088	−	0.562073i	$-0.189996\pi$
$353$	31.0791	1.65418	0.827088	+	0.562073i	$0.189996\pi$
$354$	0	0
$355$	18.1428	0.962919
$356$	1.02112	0.0541192
$357$	0	0
$358$	−3.92933	−0.207671
$359$	20.3494	1.07400	0.537001	−	0.843582i	$-0.319557\pi$
$359$	20.3494	1.07400	0.537001	+	0.843582i	$0.319557\pi$
$360$	0	0
$361$	−18.2894	−0.962602
$362$	0.383996	0.0201824
$363$	0	0
$364$	18.8473	0.987865
$365$	−4.59667	−0.240601
$366$	0	0
$367$	25.0716	1.30873	0.654364	−	0.756179i	$-0.272936\pi$
$367$	25.0716	1.30873	0.654364	+	0.756179i	$0.272936\pi$
$368$	−15.4305	−0.804371
$369$	0	0
$370$	−2.92045	−0.151827
$371$	−22.3137	−1.15847
$372$	0	0
$373$	16.3776	0.848000	0.424000	−	0.905662i	$-0.360626\pi$
$373$	16.3776	0.848000	0.424000	+	0.905662i	$0.360626\pi$
$374$	0.279266	0.0144405
$375$	0	0
$376$	−0.566466	−0.0292132
$377$	9.82866	0.506202
$378$	0	0
$379$	−9.04693	−0.464709	−0.232355	−	0.972631i	$-0.574643\pi$
$379$	−9.04693	−0.464709	−0.232355	+	0.972631i	$0.574643\pi$
$380$	−3.32912	−0.170780
$381$	0	0
$382$	−1.90198	−0.0973136
$383$	31.7938	1.62459	0.812295	−	0.583247i	$-0.198218\pi$
$383$	31.7938	1.62459	0.812295	+	0.583247i	$0.198218\pi$
$384$	0	0
$385$	9.02854	0.460137
$386$	1.56625	0.0797201
$387$	0	0
$388$	24.9403	1.26615
$389$	23.2632	1.17949	0.589746	−	0.807589i	$-0.299228\pi$
$389$	23.2632	1.17949	0.589746	+	0.807589i	$0.299228\pi$
$390$	0	0
$391$	−3.89288	−0.196871
$392$	−14.9940	−0.757311
$393$	0	0
$394$	−3.14260	−0.158322
$395$	−10.8714	−0.546997
$396$	0	0
$397$	−19.2573	−0.966494	−0.483247	−	0.875484i	$-0.660543\pi$
$397$	−19.2573	−0.966494	−0.483247	+	0.875484i	$0.660543\pi$
$398$	6.48702	0.325165
$399$	0	0
$400$	−2.15286	−0.107643
$401$	−21.1245	−1.05491	−0.527453	−	0.849584i	$-0.676853\pi$
$401$	−21.1245	−1.05491	−0.527453	+	0.849584i	$0.676853\pi$
$402$	0	0
$403$	15.9571	0.794881
$404$	19.2442	0.957433
$405$	0	0
$406$	−6.07482	−0.301488
$407$	4.24680	0.210506
$408$	0	0
$409$	5.21456	0.257844	0.128922	−	0.991655i	$-0.458848\pi$
$409$	5.21456	0.257844	0.128922	+	0.991655i	$0.458848\pi$
$410$	1.38402	0.0683517
$411$	0	0
$412$	−20.0204	−0.986337
$413$	33.5555	1.65116
$414$	0	0
$415$	25.2310	1.23854
$416$	8.45757	0.414666
$417$	0	0
$418$	−0.277635	−0.0135796
$419$	23.6700	1.15635	0.578177	−	0.815911i	$-0.303764\pi$
$419$	23.6700	1.15635	0.578177	+	0.815911i	$0.303764\pi$
$420$	0	0
$421$	−13.7273	−0.669027	−0.334514	−	0.942391i	$-0.608572\pi$
$421$	−13.7273	−0.669027	−0.334514	+	0.942391i	$0.608572\pi$
$422$	−2.72303	−0.132555
$423$	0	0
$424$	−6.61396	−0.321202
$425$	−0.543133	−0.0263458
$426$	0	0
$427$	4.32416	0.209261
$428$	−5.26445	−0.254467
$429$	0	0
$430$	−3.69124	−0.178007
$431$	13.6115	0.655641	0.327820	−	0.944740i	$-0.393686\pi$
$431$	13.6115	0.655641	0.327820	+	0.944740i	$0.393686\pi$
$432$	0	0
$433$	21.7940	1.04735	0.523677	−	0.851917i	$-0.324560\pi$
$433$	21.7940	1.04735	0.523677	+	0.851917i	$0.324560\pi$
$434$	−9.86266	−0.473423
$435$	0	0
$436$	4.71588	0.225850
$437$	3.87015	0.185134
$438$	0	0
$439$	−11.5719	−0.552297	−0.276149	−	0.961115i	$-0.589058\pi$
$439$	−11.5719	−0.552297	−0.276149	+	0.961115i	$0.589058\pi$
$440$	2.67613	0.127579
$441$	0	0
$442$	0.643506	0.0306085
$443$	35.5444	1.68876	0.844382	−	0.535741i	$-0.179968\pi$
$443$	35.5444	1.68876	0.844382	+	0.535741i	$0.179968\pi$
$444$	0	0
$445$	1.12715	0.0534318
$446$	4.86780	0.230497
$447$	0	0
$448$	23.8387	1.12627
$449$	−14.5506	−0.686685	−0.343343	−	0.939210i	$-0.611559\pi$
$449$	−14.5506	−0.686685	−0.343343	+	0.939210i	$0.611559\pi$
$450$	0	0
$451$	−2.01259	−0.0947689
$452$	4.07799	0.191812
$453$	0	0
$454$	−4.55300	−0.213683
$455$	20.8043	0.975319
$456$	0	0
$457$	−28.4816	−1.33231	−0.666156	−	0.745812i	$-0.732062\pi$
$457$	−28.4816	−1.33231	−0.666156	+	0.745812i	$0.732062\pi$
$458$	−6.45757	−0.301742
$459$	0	0
$460$	−18.1322	−0.845420
$461$	−17.9644	−0.836686	−0.418343	−	0.908289i	$-0.637389\pi$
$461$	−17.9644	−0.836686	−0.418343	+	0.908289i	$0.637389\pi$
$462$	0	0
$463$	6.52056	0.303036	0.151518	−	0.988454i	$-0.451584\pi$
$463$	6.52056	0.303036	0.151518	+	0.988454i	$0.451584\pi$
$464$	14.3356	0.665512
$465$	0	0
$466$	0.375226	0.0173820
$467$	−35.2132	−1.62947	−0.814737	−	0.579831i	$-0.803119\pi$
$467$	−35.2132	−1.62947	−0.814737	+	0.579831i	$0.803119\pi$
$468$	0	0
$469$	−65.0813	−3.00518
$470$	−0.303927	−0.0140191
$471$	0	0
$472$	9.94610	0.457806
$473$	5.36766	0.246805
$474$	0	0
$475$	0.539962	0.0247751
$476$	6.93520	0.317874
$477$	0	0
$478$	2.01017	0.0919429
$479$	2.53836	0.115981	0.0579904	−	0.998317i	$-0.481531\pi$
$479$	2.53836	0.115981	0.0579904	+	0.998317i	$0.481531\pi$
$480$	0	0
$481$	9.78581	0.446195
$482$	−5.00231	−0.227849
$483$	0	0
$484$	−1.89152	−0.0859782
$485$	27.5300	1.25007
$486$	0	0
$487$	24.2622	1.09943	0.549714	−	0.835353i	$-0.314737\pi$
$487$	24.2622	1.09943	0.549714	+	0.835353i	$0.314737\pi$
$488$	1.28171	0.0580205
$489$	0	0
$490$	−8.04476	−0.363425
$491$	22.6338	1.02145	0.510726	−	0.859744i	$-0.329377\pi$
$491$	22.6338	1.02145	0.510726	+	0.859744i	$0.329377\pi$
$492$	0	0
$493$	3.61664	0.162885
$494$	−0.639749	−0.0287837
$495$	0	0
$496$	23.2742	1.04504
$497$	37.5743	1.68544
$498$	0	0
$499$	−11.6740	−0.522599	−0.261299	−	0.965258i	$-0.584151\pi$
$499$	−11.6740	−0.522599	−0.261299	+	0.965258i	$0.584151\pi$
$500$	−22.2766	−0.996240
$501$	0	0
$502$	−3.56836	−0.159264
$503$	12.3067	0.548727	0.274363	−	0.961626i	$-0.411533\pi$
$503$	12.3067	0.548727	0.274363	+	0.961626i	$0.411533\pi$
$504$	0	0
$505$	21.2424	0.945274
$506$	−1.51216	−0.0672236
$507$	0	0
$508$	11.0164	0.488772
$509$	−5.60103	−0.248261	−0.124131	−	0.992266i	$-0.539614\pi$
$509$	−5.60103	−0.248261	−0.124131	+	0.992266i	$0.539614\pi$
$510$	0	0
$511$	−9.51986	−0.421134
$512$	20.9512	0.925920
$513$	0	0
$514$	3.66347	0.161589
$515$	−22.0993	−0.973810
$516$	0	0
$517$	0.441959	0.0194373
$518$	−6.04834	−0.265749
$519$	0	0
$520$	6.16654	0.270421
$521$	17.7699	0.778515	0.389257	−	0.921129i	$-0.372732\pi$
$521$	17.7699	0.778515	0.389257	+	0.921129i	$0.372732\pi$
$522$	0	0
$523$	−23.2872	−1.01828	−0.509140	−	0.860684i	$-0.670036\pi$
$523$	−23.2872	−1.01828	−0.509140	+	0.860684i	$0.670036\pi$
$524$	−21.1398	−0.923497
$525$	0	0
$526$	−7.85343	−0.342426
$527$	5.87172	0.255776
$528$	0	0
$529$	−1.92099	−0.0835214
$530$	−3.54860	−0.154141
$531$	0	0
$532$	−6.89471	−0.298924
$533$	−4.63756	−0.200875
$534$	0	0
$535$	−5.81108	−0.251235
$536$	−19.2906	−0.833227
$537$	0	0
$538$	−5.38839	−0.232310
$539$	11.6984	0.503885
$540$	0	0
$541$	35.0299	1.50605	0.753027	−	0.657990i	$-0.228593\pi$
$541$	35.0299	1.50605	0.753027	+	0.657990i	$0.228593\pi$
$542$	−3.65786	−0.157119
$543$	0	0
$544$	3.11212	0.133431
$545$	5.20556	0.222982
$546$	0	0
$547$	3.50473	0.149851	0.0749256	−	0.997189i	$-0.476128\pi$
$547$	3.50473	0.149851	0.0749256	+	0.997189i	$0.476128\pi$
$548$	22.1611	0.946675
$549$	0	0
$550$	−0.210976	−0.00899603
$551$	−3.59553	−0.153175
$552$	0	0
$553$	−22.5149	−0.957432
$554$	−5.06213	−0.215069
$555$	0	0
$556$	−41.7238	−1.76948
$557$	−31.7692	−1.34610	−0.673052	−	0.739595i	$-0.735017\pi$
$557$	−31.7692	−1.34610	−0.673052	+	0.739595i	$0.735017\pi$
$558$	0	0
$559$	12.3686	0.523135
$560$	30.3440	1.28227
$561$	0	0
$562$	6.07582	0.256293
$563$	−11.8264	−0.498421	−0.249211	−	0.968449i	$-0.580171\pi$
$563$	−11.8264	−0.498421	−0.249211	+	0.968449i	$0.580171\pi$
$564$	0	0
$565$	4.50142	0.189376
$566$	−0.317941	−0.0133641
$567$	0	0
$568$	11.1373	0.467311
$569$	−37.0328	−1.55250	−0.776248	−	0.630427i	$-0.782880\pi$
$569$	−37.0328	−1.55250	−0.776248	+	0.630427i	$0.782880\pi$
$570$	0	0
$571$	−10.2834	−0.430348	−0.215174	−	0.976576i	$-0.569032\pi$
$571$	−10.2834	−0.430348	−0.215174	+	0.976576i	$0.569032\pi$
$572$	−4.35859	−0.182242
$573$	0	0
$574$	2.86634	0.119639
$575$	2.94093	0.122645
$576$	0	0
$577$	−13.7210	−0.571211	−0.285605	−	0.958347i	$-0.592195\pi$
$577$	−13.7210	−0.571211	−0.285605	+	0.958347i	$0.592195\pi$
$578$	−5.36234	−0.223044
$579$	0	0
$580$	16.8456	0.699475
$581$	52.2543	2.16787
$582$	0	0
$583$	5.16024	0.213715
$584$	−2.82176	−0.116765
$585$	0	0
$586$	0.370230	0.0152940
$587$	25.3528	1.04642	0.523210	−	0.852204i	$-0.324734\pi$
$587$	25.3528	1.04642	0.523210	+	0.852204i	$0.324734\pi$
$588$	0	0
$589$	−5.83744	−0.240528
$590$	5.33640	0.219696
$591$	0	0
$592$	14.2731	0.586619
$593$	−0.723615	−0.0297153	−0.0148577	−	0.999890i	$-0.504730\pi$
$593$	−0.723615	−0.0297153	−0.0148577	+	0.999890i	$0.504730\pi$
$594$	0	0
$595$	7.65532	0.313837
$596$	7.65341	0.313496
$597$	0	0
$598$	−3.48443	−0.142489
$599$	−19.3534	−0.790760	−0.395380	−	0.918518i	$-0.629387\pi$
$599$	−19.3534	−0.790760	−0.395380	+	0.918518i	$0.629387\pi$
$600$	0	0
$601$	−26.1008	−1.06468	−0.532338	−	0.846532i	$-0.678686\pi$
$601$	−26.1008	−1.06468	−0.532338	+	0.846532i	$0.678686\pi$
$602$	−7.64467	−0.311573
$603$	0	0
$604$	7.47881	0.304308
$605$	−2.08793	−0.0848863
$606$	0	0
$607$	−32.7723	−1.33018	−0.665092	−	0.746761i	$-0.731608\pi$
$607$	−32.7723	−1.33018	−0.665092	+	0.746761i	$0.731608\pi$
$608$	−3.09395	−0.125476
$609$	0	0
$610$	0.687681	0.0278434
$611$	1.01840	0.0411999
$612$	0	0
$613$	−13.2831	−0.536501	−0.268251	−	0.963349i	$-0.586446\pi$
$613$	−13.2831	−0.536501	−0.268251	+	0.963349i	$0.586446\pi$
$614$	−3.05107	−0.123131
$615$	0	0
$616$	5.54234	0.223307
$617$	−35.9399	−1.44689	−0.723443	−	0.690385i	$-0.757441\pi$
$617$	−35.9399	−1.44689	−0.723443	+	0.690385i	$0.757441\pi$
$618$	0	0
$619$	−41.2369	−1.65745	−0.828726	−	0.559655i	$-0.810934\pi$
$619$	−41.2369	−1.65745	−0.828726	+	0.559655i	$0.810934\pi$
$620$	27.3493	1.09837
$621$	0	0
$622$	−4.41636	−0.177080
$623$	2.33436	0.0935240
$624$	0	0
$625$	−21.3869	−0.855475
$626$	9.99880	0.399633
$627$	0	0
$628$	25.2118	1.00606
$629$	3.60087	0.143576
$630$	0	0
$631$	14.2696	0.568065	0.284033	−	0.958815i	$-0.408328\pi$
$631$	14.2696	0.568065	0.284033	+	0.958815i	$0.408328\pi$
$632$	−6.67359	−0.265461
$633$	0	0
$634$	0.552184	0.0219300
$635$	12.1603	0.482565
$636$	0	0
$637$	26.9563	1.06805
$638$	1.40486	0.0556188
$639$	0	0
$640$	19.1181	0.755709
$641$	−36.1653	−1.42844	−0.714222	−	0.699919i	$-0.753219\pi$
$641$	−36.1653	−1.42844	−0.714222	+	0.699919i	$0.753219\pi$
$642$	0	0
$643$	−12.8249	−0.505764	−0.252882	−	0.967497i	$-0.581379\pi$
$643$	−12.8249	−0.505764	−0.252882	+	0.967497i	$0.581379\pi$
$644$	−37.5525	−1.47977
$645$	0	0
$646$	−0.235408	−0.00926199
$647$	−3.31842	−0.130460	−0.0652302	−	0.997870i	$-0.520778\pi$
$647$	−3.31842	−0.130460	−0.0652302	+	0.997870i	$0.520778\pi$
$648$	0	0
$649$	−7.76000	−0.304606
$650$	−0.486146	−0.0190682
$651$	0	0
$652$	15.6474	0.612798
$653$	−10.3625	−0.405514	−0.202757	−	0.979229i	$-0.564990\pi$
$653$	−10.3625	−0.405514	−0.202757	+	0.979229i	$0.564990\pi$
$654$	0	0
$655$	−23.3349	−0.911768
$656$	−6.76409	−0.264093
$657$	0	0
$658$	−0.629443	−0.0245382
$659$	18.1902	0.708589	0.354295	−	0.935134i	$-0.384721\pi$
$659$	18.1902	0.708589	0.354295	+	0.935134i	$0.384721\pi$
$660$	0	0
$661$	16.7230	0.650449	0.325225	−	0.945637i	$-0.394560\pi$
$661$	16.7230	0.650449	0.325225	+	0.945637i	$0.394560\pi$
$662$	−0.641868	−0.0249469
$663$	0	0
$664$	15.4886	0.601073
$665$	−7.61062	−0.295127
$666$	0	0
$667$	−19.5833	−0.758267
$668$	−32.9084	−1.27326
$669$	0	0
$670$	−10.3500	−0.399856
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	47.4612	1.82950	0.914748	−	0.404025i	$-0.132389\pi$
$673$	47.4612	1.82950	0.914748	+	0.404025i	$0.132389\pi$
$674$	−6.47343	−0.249347
$675$	0	0
$676$	14.5464	0.559476
$677$	15.9217	0.611922	0.305961	−	0.952044i	$-0.401022\pi$
$677$	15.9217	0.611922	0.305961	+	0.952044i	$0.401022\pi$
$678$	0	0
$679$	57.0155	2.18805
$680$	2.26909	0.0870158
$681$	0	0
$682$	2.28082	0.0873373
$683$	16.1416	0.617640	0.308820	−	0.951121i	$-0.400066\pi$
$683$	16.1416	0.617640	0.308820	+	0.951121i	$0.400066\pi$
$684$	0	0
$685$	24.4622	0.934652
$686$	−6.69149	−0.255482
$687$	0	0
$688$	18.0401	0.687774
$689$	11.8906	0.452997
$690$	0	0
$691$	−7.65006	−0.291022	−0.145511	−	0.989357i	$-0.546483\pi$
$691$	−7.65006	−0.291022	−0.145511	+	0.989357i	$0.546483\pi$
$692$	14.7652	0.561290
$693$	0	0
$694$	−5.94894	−0.225819
$695$	−46.0561	−1.74701
$696$	0	0
$697$	−1.70648	−0.0646374
$698$	0.586920	0.0222152
$699$	0	0
$700$	−5.23930	−0.198027
$701$	47.8107	1.80578	0.902892	−	0.429867i	$-0.141440\pi$
$701$	47.8107	1.80578	0.902892	+	0.429867i	$0.141440\pi$
$702$	0	0
$703$	−3.57985	−0.135017
$704$	−5.51291	−0.207776
$705$	0	0
$706$	10.2363	0.385246
$707$	43.9936	1.65455
$708$	0	0
$709$	−23.0730	−0.866524	−0.433262	−	0.901268i	$-0.642637\pi$
$709$	−23.0730	−0.866524	−0.433262	+	0.901268i	$0.642637\pi$
$710$	5.97552	0.224257
$711$	0	0
$712$	0.691921	0.0259308
$713$	−31.7940	−1.19069
$714$	0	0
$715$	−4.81116	−0.179927
$716$	22.5662	0.843337
$717$	0	0
$718$	6.70230	0.250128
$719$	30.8571	1.15077	0.575387	−	0.817881i	$-0.304851\pi$
$719$	30.8571	1.15077	0.575387	+	0.817881i	$0.304851\pi$
$720$	0	0
$721$	−45.7683	−1.70450
$722$	−6.02382	−0.224183
$723$	0	0
$724$	−2.20529	−0.0819591
$725$	−2.73225	−0.101473
$726$	0	0
$727$	−2.82088	−0.104621	−0.0523104	−	0.998631i	$-0.516659\pi$
$727$	−2.82088	−0.104621	−0.0523104	+	0.998631i	$0.516659\pi$
$728$	12.7711	0.473328
$729$	0	0
$730$	−1.51396	−0.0560343
$731$	4.55125	0.168334
$732$	0	0
$733$	18.0122	0.665297	0.332648	−	0.943051i	$-0.392058\pi$
$733$	18.0122	0.665297	0.332648	+	0.943051i	$0.392058\pi$
$734$	8.25762	0.304794
$735$	0	0
$736$	−16.8514	−0.621151
$737$	15.0506	0.554396
$738$	0	0
$739$	49.4573	1.81931	0.909657	−	0.415359i	$-0.136344\pi$
$739$	49.4573	1.81931	0.909657	+	0.415359i	$0.136344\pi$
$740$	16.7721	0.616556
$741$	0	0
$742$	−7.34927	−0.269800
$743$	53.5302	1.96383	0.981916	−	0.189318i	$-0.0606275\pi$
$743$	53.5302	1.96383	0.981916	+	0.189318i	$0.0606275\pi$
$744$	0	0
$745$	8.44809	0.309514
$746$	5.39414	0.197493
$747$	0	0
$748$	−1.60383	−0.0586417
$749$	−12.0349	−0.439747
$750$	0	0
$751$	−24.4357	−0.891671	−0.445836	−	0.895115i	$-0.647093\pi$
$751$	−24.4357	−0.891671	−0.445836	+	0.895115i	$0.647093\pi$
$752$	1.48538	0.0541662
$753$	0	0
$754$	3.23718	0.117891
$755$	8.25537	0.300444
$756$	0	0
$757$	26.7299	0.971513	0.485757	−	0.874094i	$-0.338544\pi$
$757$	26.7299	0.971513	0.485757	+	0.874094i	$0.338544\pi$
$758$	−2.97970	−0.108228
$759$	0	0
$760$	−2.25585	−0.0818281
$761$	−22.2642	−0.807076	−0.403538	−	0.914963i	$-0.632220\pi$
$761$	−22.2642	−0.807076	−0.403538	+	0.914963i	$0.632220\pi$
$762$	0	0
$763$	10.7809	0.390294
$764$	10.9231	0.395183
$765$	0	0
$766$	10.4716	0.378356
$767$	−17.8812	−0.645652
$768$	0	0
$769$	52.3202	1.88672	0.943358	−	0.331777i	$-0.107648\pi$
$769$	52.3202	1.88672	0.943358	+	0.331777i	$0.107648\pi$
$770$	2.97365	0.107163
$771$	0	0
$772$	−8.99499	−0.323737
$773$	−32.2771	−1.16093	−0.580464	−	0.814286i	$-0.697129\pi$
$773$	−32.2771	−1.16093	−0.580464	+	0.814286i	$0.697129\pi$
$774$	0	0
$775$	−4.43588	−0.159342
$776$	16.8998	0.606668
$777$	0	0
$778$	7.66200	0.274696
$779$	1.69651	0.0607839
$780$	0	0
$781$	−8.68938	−0.310930
$782$	−1.28216	−0.0458500
$783$	0	0
$784$	39.3171	1.40418
$785$	27.8296	0.993283
$786$	0	0
$787$	−0.413158	−0.0147275	−0.00736375	−	0.999973i	$-0.502344\pi$
$787$	−0.413158	−0.0147275	−0.00736375	+	0.999973i	$0.502344\pi$
$788$	18.0480	0.642933
$789$	0	0
$790$	−3.58060	−0.127392
$791$	9.32259	0.331473
$792$	0	0
$793$	−2.30428	−0.0818273
$794$	−6.34259	−0.225090
$795$	0	0
$796$	−37.2550	−1.32047
$797$	18.7709	0.664900	0.332450	−	0.943121i	$-0.392125\pi$
$797$	18.7709	0.664900	0.332450	+	0.943121i	$0.392125\pi$
$798$	0	0
$799$	0.374738	0.0132573
$800$	−2.35110	−0.0831239
$801$	0	0
$802$	−6.95757	−0.245680
$803$	2.20155	0.0776910
$804$	0	0
$805$	−41.4517	−1.46098
$806$	5.25565	0.185122
$807$	0	0
$808$	13.0401	0.458747
$809$	−15.2590	−0.536477	−0.268239	−	0.963352i	$-0.586442\pi$
$809$	−15.2590	−0.536477	−0.268239	+	0.963352i	$0.586442\pi$
$810$	0	0
$811$	−34.2538	−1.20281	−0.601407	−	0.798943i	$-0.705393\pi$
$811$	−34.2538	−1.20281	−0.601407	+	0.798943i	$0.705393\pi$
$812$	34.8878	1.22432
$813$	0	0
$814$	1.39873	0.0490255
$815$	17.2721	0.605015
$816$	0	0
$817$	−4.52467	−0.158298
$818$	1.71747	0.0600500
$819$	0	0
$820$	−7.94842	−0.277571
$821$	−36.6114	−1.27775	−0.638874	−	0.769311i	$-0.720600\pi$
$821$	−36.6114	−1.27775	−0.638874	+	0.769311i	$0.720600\pi$
$822$	0	0
$823$	15.7864	0.550279	0.275139	−	0.961404i	$-0.411276\pi$
$823$	15.7864	0.550279	0.275139	+	0.961404i	$0.411276\pi$
$824$	−13.5661	−0.472596
$825$	0	0
$826$	11.0519	0.384544
$827$	−6.51591	−0.226581	−0.113290	−	0.993562i	$-0.536139\pi$
$827$	−6.51591	−0.226581	−0.113290	+	0.993562i	$0.536139\pi$
$828$	0	0
$829$	−25.2256	−0.876122	−0.438061	−	0.898945i	$-0.644335\pi$
$829$	−25.2256	−0.876122	−0.438061	+	0.898945i	$0.644335\pi$
$830$	8.31011	0.288448
$831$	0	0
$832$	−12.7033	−0.440407
$833$	9.91909	0.343676
$834$	0	0
$835$	−36.3254	−1.25709
$836$	1.59446	0.0551456
$837$	0	0
$838$	7.79596	0.269307
$839$	−28.2774	−0.976244	−0.488122	−	0.872775i	$-0.662318\pi$
$839$	−28.2774	−0.976244	−0.488122	+	0.872775i	$0.662318\pi$
$840$	0	0
$841$	−10.8064	−0.372633
$842$	−4.52123	−0.155812
$843$	0	0
$844$	15.6384	0.538296
$845$	16.0568	0.552371
$846$	0	0
$847$	−4.32416	−0.148580
$848$	17.3430	0.595562
$849$	0	0
$850$	−0.178887	−0.00613576
$851$	−19.4979	−0.668378
$852$	0	0
$853$	−38.3057	−1.31156	−0.655781	−	0.754951i	$-0.727661\pi$
$853$	−38.3057	−1.31156	−0.655781	+	0.754951i	$0.727661\pi$
$854$	1.42421	0.0487355
$855$	0	0
$856$	−3.56725	−0.121926
$857$	−49.4250	−1.68833	−0.844163	−	0.536086i	$-0.819902\pi$
$857$	−49.4250	−1.68833	−0.844163	+	0.536086i	$0.819902\pi$
$858$	0	0
$859$	13.0579	0.445531	0.222766	−	0.974872i	$-0.428492\pi$
$859$	13.0579	0.445531	0.222766	+	0.974872i	$0.428492\pi$
$860$	21.1988	0.722873
$861$	0	0
$862$	4.48308	0.152694
$863$	25.0308	0.852058	0.426029	−	0.904710i	$-0.359912\pi$
$863$	25.0308	0.852058	0.426029	+	0.904710i	$0.359912\pi$
$864$	0	0
$865$	16.2984	0.554162
$866$	7.17809	0.243922
$867$	0	0
$868$	56.6413	1.92253
$869$	5.20677	0.176628
$870$	0	0
$871$	34.6808	1.17511
$872$	3.19553	0.108214
$873$	0	0
$874$	1.27468	0.0431166
$875$	−50.9260	−1.72161
$876$	0	0
$877$	−37.2777	−1.25878	−0.629389	−	0.777091i	$-0.716695\pi$
$877$	−37.2777	−1.25878	−0.629389	+	0.777091i	$0.716695\pi$
$878$	−3.81134	−0.128626
$879$	0	0
$880$	−7.01731	−0.236553
$881$	−15.4388	−0.520146	−0.260073	−	0.965589i	$-0.583747\pi$
$881$	−15.4388	−0.520146	−0.260073	+	0.965589i	$0.583747\pi$
$882$	0	0
$883$	9.43107	0.317381	0.158690	−	0.987328i	$-0.449273\pi$
$883$	9.43107	0.317381	0.158690	+	0.987328i	$0.449273\pi$
$884$	−3.69566	−0.124298
$885$	0	0
$886$	11.7069	0.393302
$887$	−8.93298	−0.299940	−0.149970	−	0.988691i	$-0.547918\pi$
$887$	−8.93298	−0.299940	−0.149970	+	0.988691i	$0.547918\pi$
$888$	0	0
$889$	25.1843	0.844653
$890$	0.371238	0.0124439
$891$	0	0
$892$	−27.9558	−0.936029
$893$	−0.372550	−0.0124669
$894$	0	0
$895$	24.9093	0.832626
$896$	39.5942	1.32275
$897$	0	0
$898$	−4.79240	−0.159924
$899$	29.5379	0.985144
$900$	0	0
$901$	4.37538	0.145765
$902$	−0.662867	−0.0220710
$903$	0	0
$904$	2.76329	0.0919055
$905$	−2.43428	−0.0809182
$906$	0	0
$907$	29.2126	0.969988	0.484994	−	0.874518i	$-0.338822\pi$
$907$	29.2126	0.969988	0.484994	+	0.874518i	$0.338822\pi$
$908$	26.1479	0.867748
$909$	0	0
$910$	6.85211	0.227145
$911$	1.52462	0.0505130	0.0252565	−	0.999681i	$-0.491960\pi$
$911$	1.52462	0.0505130	0.0252565	+	0.999681i	$0.491960\pi$
$912$	0	0
$913$	−12.0842	−0.399930
$914$	−9.38072	−0.310287
$915$	0	0
$916$	37.0859	1.22535
$917$	−48.3272	−1.59591
$918$	0	0
$919$	−25.6046	−0.844618	−0.422309	−	0.906452i	$-0.638780\pi$
$919$	−25.6046	−0.844618	−0.422309	+	0.906452i	$0.638780\pi$
$920$	−12.2866	−0.405077
$921$	0	0
$922$	−5.91677	−0.194859
$923$	−20.0227	−0.659056
$924$	0	0
$925$	−2.72033	−0.0894441
$926$	2.14762	0.0705751
$927$	0	0
$928$	15.6556	0.513921
$929$	−42.6822	−1.40036	−0.700179	−	0.713967i	$-0.746896\pi$
$929$	−42.6822	−1.40036	−0.700179	+	0.713967i	$0.746896\pi$
$930$	0	0
$931$	−9.86117	−0.323187
$932$	−2.15493	−0.0705869
$933$	0	0
$934$	−11.5979	−0.379493
$935$	−1.77036	−0.0578969
$936$	0	0
$937$	19.9302	0.651092	0.325546	−	0.945526i	$-0.394452\pi$
$937$	19.9302	0.651092	0.325546	+	0.945526i	$0.394452\pi$
$938$	−21.4352	−0.699885
$939$	0	0
$940$	1.74546	0.0569305
$941$	−10.1909	−0.332214	−0.166107	−	0.986108i	$-0.553120\pi$
$941$	−10.1909	−0.332214	−0.166107	+	0.986108i	$0.553120\pi$
$942$	0	0
$943$	9.24016	0.300901
$944$	−26.0805	−0.848849
$945$	0	0
$946$	1.76790	0.0574792
$947$	−41.7252	−1.35589	−0.677943	−	0.735115i	$-0.737128\pi$
$947$	−41.7252	−1.35589	−0.677943	+	0.735115i	$0.737128\pi$
$948$	0	0
$949$	5.07298	0.164676
$950$	0.177842	0.00576996
$951$	0	0
$952$	4.69937	0.152307
$953$	20.2056	0.654524	0.327262	−	0.944934i	$-0.393874\pi$
$953$	20.2056	0.654524	0.327262	+	0.944934i	$0.393874\pi$
$954$	0	0
$955$	12.0573	0.390164
$956$	−11.5444	−0.373373
$957$	0	0
$958$	0.836037	0.0270111
$959$	50.6620	1.63596
$960$	0	0
$961$	16.9556	0.546956
$962$	3.22306	0.103916
$963$	0	0
$964$	28.7283	0.925277
$965$	−9.92899	−0.319625
$966$	0	0
$967$	14.3443	0.461283	0.230641	−	0.973039i	$-0.425918\pi$
$967$	14.3443	0.461283	0.230641	+	0.973039i	$0.425918\pi$
$968$	−1.28171	−0.0411959
$969$	0	0
$970$	9.06730	0.291133
$971$	−20.4581	−0.656530	−0.328265	−	0.944586i	$-0.606464\pi$
$971$	−20.4581	−0.656530	−0.328265	+	0.944586i	$0.606464\pi$
$972$	0	0
$973$	−95.3837	−3.05786
$974$	7.99103	0.256049
$975$	0	0
$976$	−3.36090	−0.107580
$977$	−49.2661	−1.57616	−0.788081	−	0.615572i	$-0.788925\pi$
$977$	−49.2661	−1.57616	−0.788081	+	0.615572i	$0.788925\pi$
$978$	0	0
$979$	−0.539840	−0.0172534
$980$	46.2011	1.47584
$981$	0	0
$982$	7.45470	0.237889
$983$	−36.9607	−1.17886	−0.589431	−	0.807818i	$-0.700648\pi$
$983$	−36.9607	−1.17886	−0.589431	+	0.807818i	$0.700648\pi$
$984$	0	0
$985$	19.9220	0.634767
$986$	1.19118	0.0379349
$987$	0	0
$988$	3.67408	0.116888
$989$	−24.6439	−0.783631
$990$	0	0
$991$	8.02248	0.254842	0.127421	−	0.991849i	$-0.459330\pi$
$991$	8.02248	0.254842	0.127421	+	0.991849i	$0.459330\pi$
$992$	25.4174	0.807002
$993$	0	0
$994$	12.3755	0.392527
$995$	−41.1233	−1.30370
$996$	0	0
$997$	−42.4687	−1.34500	−0.672499	−	0.740098i	$-0.734779\pi$
$997$	−42.4687	−1.34500	−0.672499	+	0.740098i	$0.734779\pi$
$998$	−3.84495	−0.121710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.g.1.8		13
3.2	odd	2		2013.2.a.f.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.f.1.6	✓	13	3.2	odd	2
6039.2.a.g.1.8		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)

\(f(q)\)	\(=\)	\(q+0.329361 q^{2} -1.89152 q^{4} -2.08793 q^{5} -4.32416 q^{7} -1.28171 q^{8} +O(q^{10})\)	\(q+0.329361 q^{2} -1.89152 q^{4} -2.08793 q^{5} -4.32416 q^{7} -1.28171 q^{8} -0.687681 q^{10} +1.00000 q^{11} +2.30428 q^{13} -1.42421 q^{14} +3.36090 q^{16} +0.847902 q^{17} -0.842952 q^{19} +3.94936 q^{20} +0.329361 q^{22} -4.59119 q^{23} -0.640561 q^{25} +0.758939 q^{26} +8.17925 q^{28} +4.26540 q^{29} +6.92500 q^{31} +3.67038 q^{32} +0.279266 q^{34} +9.02854 q^{35} +4.24680 q^{37} -0.277635 q^{38} +2.67613 q^{40} -2.01259 q^{41} +5.36766 q^{43} -1.89152 q^{44} -1.51216 q^{46} +0.441959 q^{47} +11.6984 q^{49} -0.210976 q^{50} -4.35859 q^{52} +5.16024 q^{53} -2.08793 q^{55} +5.54234 q^{56} +1.40486 q^{58} -7.76000 q^{59} -1.00000 q^{61} +2.28082 q^{62} -5.51291 q^{64} -4.81116 q^{65} +15.0506 q^{67} -1.60383 q^{68} +2.97365 q^{70} -8.68938 q^{71} +2.20155 q^{73} +1.39873 q^{74} +1.59446 q^{76} -4.32416 q^{77} +5.20677 q^{79} -7.01731 q^{80} -0.662867 q^{82} -12.0842 q^{83} -1.77036 q^{85} +1.76790 q^{86} -1.28171 q^{88} -0.539840 q^{89} -9.96407 q^{91} +8.68433 q^{92} +0.145564 q^{94} +1.76002 q^{95} -13.1853 q^{97} +3.85299 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 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

Representations

Groups

Database

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