LMFDB - Embedded newform 6039.2.a.e.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:	$12$
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} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 $ x^12 - x^11 - 16x^10 + 13x^9 + 93x^8 - 59x^7 - 238x^6 + 108x^5 + 257x^4 - 71x^3 - 93x^2 + 13x + 1
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			$2.07880$ 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)

$f(q)$	$=$	$q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} +O(q^{10})$	\(q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} -5.22935 q^{10} +1.00000 q^{11} -4.69636 q^{13} -6.89089 q^{14} -3.25391 q^{16} -2.84744 q^{17} -7.33856 q^{19} +5.83963 q^{20} -2.07880 q^{22} +2.08156 q^{23} +1.32807 q^{25} +9.76278 q^{26} +7.69508 q^{28} -0.764221 q^{29} +2.67183 q^{31} +8.10046 q^{32} +5.91926 q^{34} +8.33871 q^{35} +2.93137 q^{37} +15.2554 q^{38} -1.68071 q^{40} -3.67795 q^{41} -9.30987 q^{43} +2.32140 q^{44} -4.32715 q^{46} +0.466652 q^{47} +3.98820 q^{49} -2.76079 q^{50} -10.9021 q^{52} +7.82503 q^{53} +2.51557 q^{55} -2.21472 q^{56} +1.58866 q^{58} +7.75368 q^{59} -1.00000 q^{61} -5.55420 q^{62} -10.3314 q^{64} -11.8140 q^{65} -4.65473 q^{67} -6.61005 q^{68} -17.3345 q^{70} -7.30894 q^{71} -6.71256 q^{73} -6.09372 q^{74} -17.0357 q^{76} +3.31485 q^{77} -6.44921 q^{79} -8.18541 q^{80} +7.64571 q^{82} -2.42650 q^{83} -7.16293 q^{85} +19.3533 q^{86} -0.668123 q^{88} +2.54789 q^{89} -15.5677 q^{91} +4.83214 q^{92} -0.970075 q^{94} -18.4606 q^{95} -11.8241 q^{97} -8.29066 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * 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 $

$2$	−2.07880	−1.46993	−0.734966	−	0.678104i	$-0.762802\pi$
$2$	−2.07880	−1.46993	−0.734966	+	0.678104i	$0.762802\pi$
$3$	0	0
$3$	0	0
$4$	2.32140	1.16070
$5$	2.51557	1.12500	0.562498	−	0.826799i	$-0.309841\pi$
$5$	2.51557	1.12500	0.562498	+	0.826799i	$0.309841\pi$
$6$	0	0
$7$	3.31485	1.25289	0.626447	−	0.779464i	$-0.284509\pi$
$7$	3.31485	1.25289	0.626447	+	0.779464i	$0.284509\pi$
$8$	−0.668123	−0.236217
$9$	0	0
$10$	−5.22935	−1.65367
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.69636	−1.30254	−0.651268	−	0.758848i	$-0.725762\pi$
$13$	−4.69636	−1.30254	−0.651268	+	0.758848i	$0.725762\pi$
$14$	−6.89089	−1.84167
$15$	0	0
$16$	−3.25391	−0.813476
$17$	−2.84744	−0.690606	−0.345303	−	0.938491i	$-0.612224\pi$
$17$	−2.84744	−0.690606	−0.345303	+	0.938491i	$0.612224\pi$
$18$	0	0
$19$	−7.33856	−1.68358	−0.841790	−	0.539805i	$-0.818498\pi$
$19$	−7.33856	−1.68358	−0.841790	+	0.539805i	$0.818498\pi$
$20$	5.83963	1.30578
$21$	0	0
$22$	−2.07880	−0.443201
$23$	2.08156	0.434036	0.217018	−	0.976168i	$-0.430367\pi$
$23$	2.08156	0.434036	0.217018	+	0.976168i	$0.430367\pi$
$24$	0	0
$25$	1.32807	0.265614
$26$	9.76278	1.91464
$27$	0	0
$28$	7.69508	1.45423
$29$	−0.764221	−0.141912	−0.0709561	−	0.997479i	$-0.522605\pi$
$29$	−0.764221	−0.141912	−0.0709561	+	0.997479i	$0.522605\pi$
$30$	0	0
$31$	2.67183	0.479875	0.239938	−	0.970788i	$-0.422873\pi$
$31$	2.67183	0.479875	0.239938	+	0.970788i	$0.422873\pi$
$32$	8.10046	1.43197
$33$	0	0
$34$	5.91926	1.01514
$35$	8.33871	1.40950
$36$	0	0
$37$	2.93137	0.481914	0.240957	−	0.970536i	$-0.422539\pi$
$37$	2.93137	0.481914	0.240957	+	0.970536i	$0.422539\pi$
$38$	15.2554	2.47475
$39$	0	0
$40$	−1.68071	−0.265743
$41$	−3.67795	−0.574399	−0.287199	−	0.957871i	$-0.592724\pi$
$41$	−3.67795	−0.574399	−0.287199	+	0.957871i	$0.592724\pi$
$42$	0	0
$43$	−9.30987	−1.41974	−0.709871	−	0.704332i	$-0.751247\pi$
$43$	−9.30987	−1.41974	−0.709871	+	0.704332i	$0.751247\pi$
$44$	2.32140	0.349964
$45$	0	0
$46$	−4.32715	−0.638003
$47$	0.466652	0.0680682	0.0340341	−	0.999421i	$-0.489165\pi$
$47$	0.466652	0.0680682	0.0340341	+	0.999421i	$0.489165\pi$
$48$	0	0
$49$	3.98820	0.569743
$50$	−2.76079	−0.390435
$51$	0	0
$52$	−10.9021	−1.51185
$53$	7.82503	1.07485	0.537425	−	0.843311i	$-0.319397\pi$
$53$	7.82503	1.07485	0.537425	+	0.843311i	$0.319397\pi$
$54$	0	0
$55$	2.51557	0.339199
$56$	−2.21472	−0.295955
$57$	0	0
$58$	1.58866	0.208601
$59$	7.75368	1.00944	0.504722	−	0.863282i	$-0.331595\pi$
$59$	7.75368	1.00944	0.504722	+	0.863282i	$0.331595\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−5.55420	−0.705384
$63$	0	0
$64$	−10.3314	−1.29142
$65$	−11.8140	−1.46535
$66$	0	0
$67$	−4.65473	−0.568666	−0.284333	−	0.958726i	$-0.591772\pi$
$67$	−4.65473	−0.568666	−0.284333	+	0.958726i	$0.591772\pi$
$68$	−6.61005	−0.801586
$69$	0	0
$70$	−17.3345	−2.07187
$71$	−7.30894	−0.867412	−0.433706	−	0.901054i	$-0.642794\pi$
$71$	−7.30894	−0.867412	−0.433706	+	0.901054i	$0.642794\pi$
$72$	0	0
$73$	−6.71256	−0.785645	−0.392823	−	0.919614i	$-0.628501\pi$
$73$	−6.71256	−0.785645	−0.392823	+	0.919614i	$0.628501\pi$
$74$	−6.09372	−0.708381
$75$	0	0
$76$	−17.0357	−1.95413
$77$	3.31485	0.377762
$78$	0	0
$79$	−6.44921	−0.725593	−0.362797	−	0.931868i	$-0.618178\pi$
$79$	−6.44921	−0.725593	−0.362797	+	0.931868i	$0.618178\pi$
$80$	−8.18541	−0.915157
$81$	0	0
$82$	7.64571	0.844327
$83$	−2.42650	−0.266343	−0.133172	−	0.991093i	$-0.542516\pi$
$83$	−2.42650	−0.266343	−0.133172	+	0.991093i	$0.542516\pi$
$84$	0	0
$85$	−7.16293	−0.776929
$86$	19.3533	2.08692
$87$	0	0
$88$	−0.668123	−0.0712222
$89$	2.54789	0.270076	0.135038	−	0.990840i	$-0.456884\pi$
$89$	2.54789	0.270076	0.135038	+	0.990840i	$0.456884\pi$
$90$	0	0
$91$	−15.5677	−1.63194
$92$	4.83214	0.503785
$93$	0	0
$94$	−0.970075	−0.100056
$95$	−18.4606	−1.89402
$96$	0	0
$97$	−11.8241	−1.20055	−0.600277	−	0.799793i	$-0.704943\pi$
$97$	−11.8241	−1.20055	−0.600277	+	0.799793i	$0.704943\pi$
$98$	−8.29066	−0.837483
$99$	0	0
$100$	3.08298	0.308298
$101$	12.8545	1.27907	0.639537	−	0.768760i	$-0.279126\pi$
$101$	12.8545	1.27907	0.639537	+	0.768760i	$0.279126\pi$
$102$	0	0
$103$	−7.74643	−0.763279	−0.381639	−	0.924311i	$-0.624640\pi$
$103$	−7.74643	−0.763279	−0.381639	+	0.924311i	$0.624640\pi$
$104$	3.13774	0.307681
$105$	0	0
$106$	−16.2667	−1.57996
$107$	−15.8933	−1.53646	−0.768230	−	0.640174i	$-0.778862\pi$
$107$	−15.8933	−1.53646	−0.768230	+	0.640174i	$0.778862\pi$
$108$	0	0
$109$	−3.51888	−0.337048	−0.168524	−	0.985698i	$-0.553900\pi$
$109$	−3.51888	−0.337048	−0.168524	+	0.985698i	$0.553900\pi$
$110$	−5.22935	−0.498599
$111$	0	0
$112$	−10.7862	−1.01920
$113$	−8.00502	−0.753049	−0.376524	−	0.926407i	$-0.622881\pi$
$113$	−8.00502	−0.753049	−0.376524	+	0.926407i	$0.622881\pi$
$114$	0	0
$115$	5.23631	0.488288
$116$	−1.77406	−0.164717
$117$	0	0
$118$	−16.1183	−1.48381
$119$	−9.43883	−0.865256
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.07880	0.188205
$123$	0	0
$124$	6.20239	0.556991
$125$	−9.23698	−0.826180
$126$	0	0
$127$	5.63027	0.499605	0.249803	−	0.968297i	$-0.419634\pi$
$127$	5.63027	0.499605	0.249803	+	0.968297i	$0.419634\pi$
$128$	5.27597	0.466334
$129$	0	0
$130$	24.5589	2.15396
$131$	−4.25978	−0.372179	−0.186089	−	0.982533i	$-0.559581\pi$
$131$	−4.25978	−0.372179	−0.186089	+	0.982533i	$0.559581\pi$
$132$	0	0
$133$	−24.3262	−2.10935
$134$	9.67624	0.835900
$135$	0	0
$136$	1.90244	0.163133
$137$	−0.938042	−0.0801423	−0.0400712	−	0.999197i	$-0.512758\pi$
$137$	−0.938042	−0.0801423	−0.0400712	+	0.999197i	$0.512758\pi$
$138$	0	0
$139$	20.0152	1.69767	0.848834	−	0.528660i	$-0.177305\pi$
$139$	20.0152	1.69767	0.848834	+	0.528660i	$0.177305\pi$
$140$	19.3575	1.63601
$141$	0	0
$142$	15.1938	1.27504
$143$	−4.69636	−0.392729
$144$	0	0
$145$	−1.92245	−0.159651
$146$	13.9540	1.15484
$147$	0	0
$148$	6.80487	0.559357
$149$	−13.2774	−1.08773	−0.543864	−	0.839173i	$-0.683039\pi$
$149$	−13.2774	−1.08773	−0.543864	+	0.839173i	$0.683039\pi$
$150$	0	0
$151$	11.8307	0.962771	0.481385	−	0.876509i	$-0.340134\pi$
$151$	11.8307	0.962771	0.481385	+	0.876509i	$0.340134\pi$
$152$	4.90306	0.397691
$153$	0	0
$154$	−6.89089	−0.555284
$155$	6.72117	0.539858
$156$	0	0
$157$	−12.3071	−0.982216	−0.491108	−	0.871099i	$-0.663408\pi$
$157$	−12.3071	−0.982216	−0.491108	+	0.871099i	$0.663408\pi$
$158$	13.4066	1.06657
$159$	0	0
$160$	20.3772	1.61096
$161$	6.90006	0.543801
$162$	0	0
$163$	20.8550	1.63349	0.816744	−	0.577001i	$-0.195777\pi$
$163$	20.8550	1.63349	0.816744	+	0.577001i	$0.195777\pi$
$164$	−8.53798	−0.666704
$165$	0	0
$166$	5.04421	0.391506
$167$	3.99278	0.308970	0.154485	−	0.987995i	$-0.450628\pi$
$167$	3.99278	0.308970	0.154485	+	0.987995i	$0.450628\pi$
$168$	0	0
$169$	9.05577	0.696598
$170$	14.8903	1.14203
$171$	0	0
$172$	−21.6119	−1.64789
$173$	25.5150	1.93987	0.969937	−	0.243358i	$-0.0782490\pi$
$173$	25.5150	1.93987	0.969937	+	0.243358i	$0.0782490\pi$
$174$	0	0
$175$	4.40235	0.332787
$176$	−3.25391	−0.245272
$177$	0	0
$178$	−5.29655	−0.396993
$179$	−20.3940	−1.52432	−0.762158	−	0.647391i	$-0.775860\pi$
$179$	−20.3940	−1.52432	−0.762158	+	0.647391i	$0.775860\pi$
$180$	0	0
$181$	−10.3544	−0.769633	−0.384816	−	0.922993i	$-0.625735\pi$
$181$	−10.3544	−0.769633	−0.384816	+	0.922993i	$0.625735\pi$
$182$	32.3621	2.39884
$183$	0	0
$184$	−1.39074	−0.102527
$185$	7.37405	0.542151
$186$	0	0
$187$	−2.84744	−0.208226
$188$	1.08329	0.0790067
$189$	0	0
$190$	38.3759	2.78408
$191$	−2.57556	−0.186361	−0.0931805	−	0.995649i	$-0.529703\pi$
$191$	−2.57556	−0.186361	−0.0931805	+	0.995649i	$0.529703\pi$
$192$	0	0
$193$	21.0757	1.51706	0.758530	−	0.651638i	$-0.225918\pi$
$193$	21.0757	1.51706	0.758530	+	0.651638i	$0.225918\pi$
$194$	24.5799	1.76473
$195$	0	0
$196$	9.25820	0.661300
$197$	−21.1509	−1.50694	−0.753469	−	0.657484i	$-0.771621\pi$
$197$	−21.1509	−1.50694	−0.753469	+	0.657484i	$0.771621\pi$
$198$	0	0
$199$	−7.63056	−0.540916	−0.270458	−	0.962732i	$-0.587175\pi$
$199$	−7.63056	−0.540916	−0.270458	+	0.962732i	$0.587175\pi$
$200$	−0.887315	−0.0627427
$201$	0	0
$202$	−26.7220	−1.88015
$203$	−2.53327	−0.177801
$204$	0	0
$205$	−9.25212	−0.646196
$206$	16.1033	1.12197
$207$	0	0
$208$	15.2815	1.05958
$209$	−7.33856	−0.507619
$210$	0	0
$211$	−25.6471	−1.76562	−0.882811	−	0.469728i	$-0.844352\pi$
$211$	−25.6471	−1.76562	−0.882811	+	0.469728i	$0.844352\pi$
$212$	18.1650	1.24758
$213$	0	0
$214$	33.0389	2.25849
$215$	−23.4196	−1.59720
$216$	0	0
$217$	8.85671	0.601233
$218$	7.31505	0.495437
$219$	0	0
$220$	5.83963	0.393708
$221$	13.3726	0.899539
$222$	0	0
$223$	−21.6223	−1.44794	−0.723968	−	0.689834i	$-0.757684\pi$
$223$	−21.6223	−1.44794	−0.723968	+	0.689834i	$0.757684\pi$
$224$	26.8518	1.79411
$225$	0	0
$226$	16.6408	1.10693
$227$	−7.41119	−0.491898	−0.245949	−	0.969283i	$-0.579100\pi$
$227$	−7.41119	−0.491898	−0.245949	+	0.969283i	$0.579100\pi$
$228$	0	0
$229$	12.2381	0.808718	0.404359	−	0.914600i	$-0.367495\pi$
$229$	12.2381	0.808718	0.404359	+	0.914600i	$0.367495\pi$
$230$	−10.8852	−0.717750
$231$	0	0
$232$	0.510594	0.0335221
$233$	−9.92929	−0.650489	−0.325245	−	0.945630i	$-0.605447\pi$
$233$	−9.92929	−0.650489	−0.325245	+	0.945630i	$0.605447\pi$
$234$	0	0
$235$	1.17389	0.0765764
$236$	17.9994	1.17166
$237$	0	0
$238$	19.6214	1.27187
$239$	−11.4528	−0.740823	−0.370412	−	0.928868i	$-0.620783\pi$
$239$	−11.4528	−0.740823	−0.370412	+	0.928868i	$0.620783\pi$
$240$	0	0
$241$	1.38803	0.0894111	0.0447055	−	0.999000i	$-0.485765\pi$
$241$	1.38803	0.0894111	0.0447055	+	0.999000i	$0.485765\pi$
$242$	−2.07880	−0.133630
$243$	0	0
$244$	−2.32140	−0.148612
$245$	10.0326	0.640958
$246$	0	0
$247$	34.4645	2.19292
$248$	−1.78511	−0.113355
$249$	0	0
$250$	19.2018	1.21443
$251$	10.9919	0.693801	0.346900	−	0.937902i	$-0.387234\pi$
$251$	10.9919	0.693801	0.346900	+	0.937902i	$0.387234\pi$
$252$	0	0
$253$	2.08156	0.130867
$254$	−11.7042	−0.734386
$255$	0	0
$256$	9.69512	0.605945
$257$	−24.1795	−1.50827	−0.754137	−	0.656718i	$-0.771944\pi$
$257$	−24.1795	−1.50827	−0.754137	+	0.656718i	$0.771944\pi$
$258$	0	0
$259$	9.71703	0.603787
$260$	−27.4250	−1.70083
$261$	0	0
$262$	8.85522	0.547077
$263$	−0.0653591	−0.00403021	−0.00201511	−	0.999998i	$-0.500641\pi$
$263$	−0.0653591	−0.00403021	−0.00201511	+	0.999998i	$0.500641\pi$
$264$	0	0
$265$	19.6844	1.20920
$266$	50.5692	3.10060
$267$	0	0
$268$	−10.8055	−0.660050
$269$	−5.24648	−0.319884	−0.159942	−	0.987126i	$-0.551131\pi$
$269$	−5.24648	−0.319884	−0.159942	+	0.987126i	$0.551131\pi$
$270$	0	0
$271$	−7.61564	−0.462617	−0.231308	−	0.972880i	$-0.574301\pi$
$271$	−7.61564	−0.462617	−0.231308	+	0.972880i	$0.574301\pi$
$272$	9.26531	0.561792
$273$	0	0
$274$	1.95000	0.117804
$275$	1.32807	0.0800857
$276$	0	0
$277$	16.4198	0.986570	0.493285	−	0.869868i	$-0.335796\pi$
$277$	16.4198	0.986570	0.493285	+	0.869868i	$0.335796\pi$
$278$	−41.6075	−2.49545
$279$	0	0
$280$	−5.57129	−0.332948
$281$	6.84038	0.408063	0.204032	−	0.978964i	$-0.434595\pi$
$281$	6.84038	0.408063	0.204032	+	0.978964i	$0.434595\pi$
$282$	0	0
$283$	−6.59236	−0.391875	−0.195937	−	0.980616i	$-0.562775\pi$
$283$	−6.59236	−0.391875	−0.195937	+	0.980616i	$0.562775\pi$
$284$	−16.9670	−1.00680
$285$	0	0
$286$	9.76278	0.577285
$287$	−12.1918	−0.719661
$288$	0	0
$289$	−8.89207	−0.523063
$290$	3.99638	0.234676
$291$	0	0
$292$	−15.5825	−0.911898
$293$	0.825657	0.0482354	0.0241177	−	0.999709i	$-0.492322\pi$
$293$	0.825657	0.0482354	0.0241177	+	0.999709i	$0.492322\pi$
$294$	0	0
$295$	19.5049	1.13562
$296$	−1.95851	−0.113836
$297$	0	0
$298$	27.6010	1.59889
$299$	−9.77576	−0.565347
$300$	0	0
$301$	−30.8608	−1.77879
$302$	−24.5937	−1.41521
$303$	0	0
$304$	23.8790	1.36955
$305$	−2.51557	−0.144041
$306$	0	0
$307$	−10.6787	−0.609463	−0.304732	−	0.952438i	$-0.598567\pi$
$307$	−10.6787	−0.609463	−0.304732	+	0.952438i	$0.598567\pi$
$308$	7.69508	0.438468
$309$	0	0
$310$	−13.9720	−0.793554
$311$	3.76872	0.213704	0.106852	−	0.994275i	$-0.465923\pi$
$311$	3.76872	0.213704	0.106852	+	0.994275i	$0.465923\pi$
$312$	0	0
$313$	18.2136	1.02950	0.514748	−	0.857342i	$-0.327886\pi$
$313$	18.2136	1.02950	0.514748	+	0.857342i	$0.327886\pi$
$314$	25.5840	1.44379
$315$	0	0
$316$	−14.9712	−0.842196
$317$	−18.1257	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$317$	−18.1257	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$318$	0	0
$319$	−0.764221	−0.0427882
$320$	−25.9893	−1.45285
$321$	0	0
$322$	−14.3438	−0.799350
$323$	20.8961	1.16269
$324$	0	0
$325$	−6.23710	−0.345972
$326$	−43.3533	−2.40111
$327$	0	0
$328$	2.45732	0.135683
$329$	1.54688	0.0852822
$330$	0	0
$331$	−23.3048	−1.28095	−0.640475	−	0.767979i	$-0.721262\pi$
$331$	−23.3048	−1.28095	−0.640475	+	0.767979i	$0.721262\pi$
$332$	−5.63288	−0.309144
$333$	0	0
$334$	−8.30017	−0.454165
$335$	−11.7093	−0.639746
$336$	0	0
$337$	−16.0798	−0.875924	−0.437962	−	0.898993i	$-0.644300\pi$
$337$	−16.0798	−0.875924	−0.437962	+	0.898993i	$0.644300\pi$
$338$	−18.8251	−1.02395
$339$	0	0
$340$	−16.6280	−0.901781
$341$	2.67183	0.144688
$342$	0	0
$343$	−9.98366	−0.539067
$344$	6.22014	0.335368
$345$	0	0
$346$	−53.0406	−2.85148
$347$	−2.04396	−0.109725	−0.0548627	−	0.998494i	$-0.517472\pi$
$347$	−2.04396	−0.109725	−0.0548627	+	0.998494i	$0.517472\pi$
$348$	0	0
$349$	−27.4212	−1.46782	−0.733912	−	0.679244i	$-0.762308\pi$
$349$	−27.4212	−1.46782	−0.733912	+	0.679244i	$0.762308\pi$
$350$	−9.15160	−0.489173
$351$	0	0
$352$	8.10046	0.431756
$353$	26.7877	1.42576	0.712882	−	0.701284i	$-0.247390\pi$
$353$	26.7877	1.42576	0.712882	+	0.701284i	$0.247390\pi$
$354$	0	0
$355$	−18.3861	−0.975835
$356$	5.91467	0.313477
$357$	0	0
$358$	42.3949	2.24064
$359$	20.1894	1.06556	0.532778	−	0.846255i	$-0.321148\pi$
$359$	20.1894	1.06556	0.532778	+	0.846255i	$0.321148\pi$
$360$	0	0
$361$	34.8544	1.83444
$362$	21.5246	1.13131
$363$	0	0
$364$	−36.1388	−1.89419
$365$	−16.8859	−0.883847
$366$	0	0
$367$	−8.71927	−0.455142	−0.227571	−	0.973761i	$-0.573078\pi$
$367$	−8.71927	−0.455142	−0.227571	+	0.973761i	$0.573078\pi$
$368$	−6.77321	−0.353078
$369$	0	0
$370$	−15.3292	−0.796925
$371$	25.9388	1.34667
$372$	0	0
$373$	8.10112	0.419460	0.209730	−	0.977759i	$-0.432742\pi$
$373$	8.10112	0.419460	0.209730	+	0.977759i	$0.432742\pi$
$374$	5.91926	0.306077
$375$	0	0
$376$	−0.311781	−0.0160789
$377$	3.58905	0.184846
$378$	0	0
$379$	16.1829	0.831259	0.415629	−	0.909534i	$-0.363561\pi$
$379$	16.1829	0.831259	0.415629	+	0.909534i	$0.363561\pi$
$380$	−42.8545	−2.19839
$381$	0	0
$382$	5.35407	0.273938
$383$	11.5062	0.587940	0.293970	−	0.955815i	$-0.405023\pi$
$383$	11.5062	0.587940	0.293970	+	0.955815i	$0.405023\pi$
$384$	0	0
$385$	8.33871	0.424980
$386$	−43.8120	−2.22997
$387$	0	0
$388$	−27.4484	−1.39348
$389$	17.6156	0.893147	0.446573	−	0.894747i	$-0.352644\pi$
$389$	17.6156	0.893147	0.446573	+	0.894747i	$0.352644\pi$
$390$	0	0
$391$	−5.92713	−0.299748
$392$	−2.66461	−0.134583
$393$	0	0
$394$	43.9684	2.21509
$395$	−16.2234	−0.816289
$396$	0	0
$397$	18.6556	0.936298	0.468149	−	0.883649i	$-0.344921\pi$
$397$	18.6556	0.936298	0.468149	+	0.883649i	$0.344921\pi$
$398$	15.8624	0.795110
$399$	0	0
$400$	−4.32142	−0.216071
$401$	26.6759	1.33213	0.666066	−	0.745893i	$-0.267977\pi$
$401$	26.6759	1.33213	0.666066	+	0.745893i	$0.267977\pi$
$402$	0	0
$403$	−12.5479	−0.625055
$404$	29.8405	1.48462
$405$	0	0
$406$	5.26616	0.261355
$407$	2.93137	0.145303
$408$	0	0
$409$	−38.2979	−1.89371	−0.946856	−	0.321658i	$-0.895760\pi$
$409$	−38.2979	−1.89371	−0.946856	+	0.321658i	$0.895760\pi$
$410$	19.2333	0.949864
$411$	0	0
$412$	−17.9826	−0.885937
$413$	25.7023	1.26473
$414$	0	0
$415$	−6.10402	−0.299635
$416$	−38.0426	−1.86519
$417$	0	0
$418$	15.2554	0.746165
$419$	4.67442	0.228360	0.114180	−	0.993460i	$-0.463576\pi$
$419$	4.67442	0.228360	0.114180	+	0.993460i	$0.463576\pi$
$420$	0	0
$421$	29.7448	1.44967	0.724836	−	0.688922i	$-0.241916\pi$
$421$	29.7448	1.44967	0.724836	+	0.688922i	$0.241916\pi$
$422$	53.3152	2.59534
$423$	0	0
$424$	−5.22808	−0.253898
$425$	−3.78161	−0.183435
$426$	0	0
$427$	−3.31485	−0.160417
$428$	−36.8946	−1.78337
$429$	0	0
$430$	48.6846	2.34778
$431$	−14.2615	−0.686950	−0.343475	−	0.939162i	$-0.611604\pi$
$431$	−14.2615	−0.686950	−0.343475	+	0.939162i	$0.611604\pi$
$432$	0	0
$433$	40.0835	1.92629	0.963145	−	0.268983i	$-0.0866875\pi$
$433$	40.0835	1.92629	0.963145	+	0.268983i	$0.0866875\pi$
$434$	−18.4113	−0.883771
$435$	0	0
$436$	−8.16873	−0.391211
$437$	−15.2757	−0.730734
$438$	0	0
$439$	−41.7801	−1.99405	−0.997027	−	0.0770534i	$-0.975449\pi$
$439$	−41.7801	−1.99405	−0.997027	+	0.0770534i	$0.975449\pi$
$440$	−1.68071	−0.0801246
$441$	0	0
$442$	−27.7989	−1.32226
$443$	22.6161	1.07452	0.537262	−	0.843415i	$-0.319459\pi$
$443$	22.6161	1.07452	0.537262	+	0.843415i	$0.319459\pi$
$444$	0	0
$445$	6.40939	0.303834
$446$	44.9484	2.12837
$447$	0	0
$448$	−34.2470	−1.61802
$449$	−0.296481	−0.0139918	−0.00699590	−	0.999976i	$-0.502227\pi$
$449$	−0.296481	−0.0139918	−0.00699590	+	0.999976i	$0.502227\pi$
$450$	0	0
$451$	−3.67795	−0.173188
$452$	−18.5828	−0.874063
$453$	0	0
$454$	15.4064	0.723056
$455$	−39.1616	−1.83592
$456$	0	0
$457$	−35.4014	−1.65601	−0.828004	−	0.560722i	$-0.810524\pi$
$457$	−35.4014	−1.65601	−0.828004	+	0.560722i	$0.810524\pi$
$458$	−25.4406	−1.18876
$459$	0	0
$460$	12.1556	0.566756
$461$	−24.8643	−1.15804	−0.579022	−	0.815312i	$-0.696566\pi$
$461$	−24.8643	−1.15804	−0.579022	+	0.815312i	$0.696566\pi$
$462$	0	0
$463$	−23.6296	−1.09816	−0.549080	−	0.835770i	$-0.685022\pi$
$463$	−23.6296	−1.09816	−0.549080	+	0.835770i	$0.685022\pi$
$464$	2.48670	0.115442
$465$	0	0
$466$	20.6410	0.956175
$467$	24.7168	1.14376	0.571879	−	0.820338i	$-0.306215\pi$
$467$	24.7168	1.14376	0.571879	+	0.820338i	$0.306215\pi$
$468$	0	0
$469$	−15.4297	−0.712478
$470$	−2.44029	−0.112562
$471$	0	0
$472$	−5.18042	−0.238448
$473$	−9.30987	−0.428068
$474$	0	0
$475$	−9.74613	−0.447183
$476$	−21.9113	−1.00430
$477$	0	0
$478$	23.8081	1.08896
$479$	3.86168	0.176445	0.0882223	−	0.996101i	$-0.471881\pi$
$479$	3.86168	0.176445	0.0882223	+	0.996101i	$0.471881\pi$
$480$	0	0
$481$	−13.7668	−0.627710
$482$	−2.88544	−0.131428
$483$	0	0
$484$	2.32140	0.105518
$485$	−29.7442	−1.35062
$486$	0	0
$487$	4.58072	0.207572	0.103786	−	0.994600i	$-0.466904\pi$
$487$	4.58072	0.207572	0.103786	+	0.994600i	$0.466904\pi$
$488$	0.668123	0.0302445
$489$	0	0
$490$	−20.8557	−0.942164
$491$	33.5912	1.51595	0.757974	−	0.652284i	$-0.226189\pi$
$491$	33.5912	1.51595	0.757974	+	0.652284i	$0.226189\pi$
$492$	0	0
$493$	2.17608	0.0980055
$494$	−71.6447	−3.22345
$495$	0	0
$496$	−8.69389	−0.390367
$497$	−24.2280	−1.08678
$498$	0	0
$499$	32.2036	1.44163	0.720816	−	0.693127i	$-0.243767\pi$
$499$	32.2036	1.44163	0.720816	+	0.693127i	$0.243767\pi$
$500$	−21.4427	−0.958947
$501$	0	0
$502$	−22.8499	−1.01984
$503$	39.4574	1.75932	0.879660	−	0.475603i	$-0.157770\pi$
$503$	39.4574	1.75932	0.879660	+	0.475603i	$0.157770\pi$
$504$	0	0
$505$	32.3364	1.43895
$506$	−4.32715	−0.192365
$507$	0	0
$508$	13.0701	0.579892
$509$	−21.0465	−0.932871	−0.466436	−	0.884555i	$-0.654462\pi$
$509$	−21.0465	−0.932871	−0.466436	+	0.884555i	$0.654462\pi$
$510$	0	0
$511$	−22.2511	−0.984330
$512$	−30.7061	−1.35703
$513$	0	0
$514$	50.2642	2.21706
$515$	−19.4867	−0.858685
$516$	0	0
$517$	0.466652	0.0205233
$518$	−20.1997	−0.887526
$519$	0	0
$520$	7.89320	0.346140
$521$	−25.8419	−1.13216	−0.566078	−	0.824352i	$-0.691540\pi$
$521$	−25.8419	−1.13216	−0.566078	+	0.824352i	$0.691540\pi$
$522$	0	0
$523$	11.0550	0.483400	0.241700	−	0.970351i	$-0.422295\pi$
$523$	11.0550	0.483400	0.241700	+	0.970351i	$0.422295\pi$
$524$	−9.88865	−0.431988
$525$	0	0
$526$	0.135868	0.00592414
$527$	−7.60789	−0.331405
$528$	0	0
$529$	−18.6671	−0.811613
$530$	−40.9198	−1.77744
$531$	0	0
$532$	−56.4708	−2.44832
$533$	17.2730	0.748175
$534$	0	0
$535$	−39.9805	−1.72851
$536$	3.10993	0.134329
$537$	0	0
$538$	10.9064	0.470207
$539$	3.98820	0.171784
$540$	0	0
$541$	−32.1200	−1.38095	−0.690473	−	0.723358i	$-0.742598\pi$
$541$	−32.1200	−1.38095	−0.690473	+	0.723358i	$0.742598\pi$
$542$	15.8314	0.680015
$543$	0	0
$544$	−23.0656	−0.988929
$545$	−8.85198	−0.379177
$546$	0	0
$547$	−21.4050	−0.915211	−0.457605	−	0.889155i	$-0.651293\pi$
$547$	−21.4050	−0.915211	−0.457605	+	0.889155i	$0.651293\pi$
$548$	−2.17757	−0.0930211
$549$	0	0
$550$	−2.76079	−0.117721
$551$	5.60828	0.238921
$552$	0	0
$553$	−21.3781	−0.909091
$554$	−34.1334	−1.45019
$555$	0	0
$556$	46.4633	1.97048
$557$	14.8084	0.627453	0.313726	−	0.949513i	$-0.398423\pi$
$557$	14.8084	0.627453	0.313726	+	0.949513i	$0.398423\pi$
$558$	0	0
$559$	43.7225	1.84926
$560$	−27.1334	−1.14659
$561$	0	0
$562$	−14.2198	−0.599825
$563$	−41.4376	−1.74639	−0.873193	−	0.487374i	$-0.837955\pi$
$563$	−41.4376	−1.74639	−0.873193	+	0.487374i	$0.837955\pi$
$564$	0	0
$565$	−20.1371	−0.847176
$566$	13.7042	0.576029
$567$	0	0
$568$	4.88327	0.204898
$569$	24.3931	1.02261	0.511306	−	0.859399i	$-0.329162\pi$
$569$	24.3931	1.02261	0.511306	+	0.859399i	$0.329162\pi$
$570$	0	0
$571$	−28.7749	−1.20419	−0.602097	−	0.798423i	$-0.705668\pi$
$571$	−28.7749	−1.20419	−0.602097	+	0.798423i	$0.705668\pi$
$572$	−10.9021	−0.455840
$573$	0	0
$574$	25.3443	1.05785
$575$	2.76446	0.115286
$576$	0	0
$577$	5.15803	0.214732	0.107366	−	0.994220i	$-0.465758\pi$
$577$	5.15803	0.214732	0.107366	+	0.994220i	$0.465758\pi$
$578$	18.4848	0.768867
$579$	0	0
$580$	−4.46277	−0.185306
$581$	−8.04348	−0.333700
$582$	0	0
$583$	7.82503	0.324080
$584$	4.48481	0.185583
$585$	0	0
$586$	−1.71637	−0.0709028
$587$	36.0284	1.48705	0.743525	−	0.668708i	$-0.233152\pi$
$587$	36.0284	1.48705	0.743525	+	0.668708i	$0.233152\pi$
$588$	0	0
$589$	−19.6074	−0.807909
$590$	−40.5467	−1.66928
$591$	0	0
$592$	−9.53839	−0.392026
$593$	18.1163	0.743946	0.371973	−	0.928243i	$-0.378681\pi$
$593$	18.1163	0.743946	0.371973	+	0.928243i	$0.378681\pi$
$594$	0	0
$595$	−23.7440	−0.973409
$596$	−30.8222	−1.26252
$597$	0	0
$598$	20.3218	0.831021
$599$	8.70761	0.355783	0.177892	−	0.984050i	$-0.443072\pi$
$599$	8.70761	0.355783	0.177892	+	0.984050i	$0.443072\pi$
$600$	0	0
$601$	23.7422	0.968464	0.484232	−	0.874940i	$-0.339099\pi$
$601$	23.7422	0.968464	0.484232	+	0.874940i	$0.339099\pi$
$602$	64.1533	2.61469
$603$	0	0
$604$	27.4638	1.11749
$605$	2.51557	0.102272
$606$	0	0
$607$	0.604695	0.0245438	0.0122719	−	0.999925i	$-0.496094\pi$
$607$	0.604695	0.0245438	0.0122719	+	0.999925i	$0.496094\pi$
$608$	−59.4457	−2.41084
$609$	0	0
$610$	5.22935	0.211730
$611$	−2.19156	−0.0886612
$612$	0	0
$613$	19.6301	0.792853	0.396426	−	0.918067i	$-0.370250\pi$
$613$	19.6301	0.792853	0.396426	+	0.918067i	$0.370250\pi$
$614$	22.1988	0.895870
$615$	0	0
$616$	−2.21472	−0.0892338
$617$	10.4332	0.420025	0.210012	−	0.977699i	$-0.432650\pi$
$617$	10.4332	0.420025	0.210012	+	0.977699i	$0.432650\pi$
$618$	0	0
$619$	21.6981	0.872121	0.436060	−	0.899917i	$-0.356373\pi$
$619$	21.6981	0.872121	0.436060	+	0.899917i	$0.356373\pi$
$620$	15.6025	0.626612
$621$	0	0
$622$	−7.83440	−0.314131
$623$	8.44586	0.338376
$624$	0	0
$625$	−29.8766	−1.19506
$626$	−37.8625	−1.51329
$627$	0	0
$628$	−28.5697	−1.14006
$629$	−8.34690	−0.332813
$630$	0	0
$631$	2.07442	0.0825814	0.0412907	−	0.999147i	$-0.486853\pi$
$631$	2.07442	0.0825814	0.0412907	+	0.999147i	$0.486853\pi$
$632$	4.30887	0.171398
$633$	0	0
$634$	37.6796	1.49645
$635$	14.1633	0.562054
$636$	0	0
$637$	−18.7300	−0.742110
$638$	1.58866	0.0628957
$639$	0	0
$640$	13.2720	0.524624
$641$	13.9661	0.551628	0.275814	−	0.961211i	$-0.411053\pi$
$641$	13.9661	0.551628	0.275814	+	0.961211i	$0.411053\pi$
$642$	0	0
$643$	−20.4874	−0.807943	−0.403971	−	0.914772i	$-0.632370\pi$
$643$	−20.4874	−0.807943	−0.403971	+	0.914772i	$0.632370\pi$
$644$	16.0178	0.631189
$645$	0	0
$646$	−43.4388	−1.70908
$647$	−11.4213	−0.449018	−0.224509	−	0.974472i	$-0.572078\pi$
$647$	−11.4213	−0.449018	−0.224509	+	0.974472i	$0.572078\pi$
$648$	0	0
$649$	7.75368	0.304359
$650$	12.9657	0.508555
$651$	0	0
$652$	48.4127	1.89599
$653$	−17.7057	−0.692876	−0.346438	−	0.938073i	$-0.612609\pi$
$653$	−17.7057	−0.692876	−0.346438	+	0.938073i	$0.612609\pi$
$654$	0	0
$655$	−10.7158	−0.418699
$656$	11.9677	0.467260
$657$	0	0
$658$	−3.21565	−0.125359
$659$	−17.5526	−0.683753	−0.341877	−	0.939745i	$-0.611063\pi$
$659$	−17.5526	−0.683753	−0.341877	+	0.939745i	$0.611063\pi$
$660$	0	0
$661$	−37.1442	−1.44474	−0.722370	−	0.691506i	$-0.756947\pi$
$661$	−37.1442	−1.44474	−0.722370	+	0.691506i	$0.756947\pi$
$662$	48.4460	1.88291
$663$	0	0
$664$	1.62120	0.0629148
$665$	−61.1941	−2.37301
$666$	0	0
$667$	−1.59077	−0.0615950
$668$	9.26883	0.358622
$669$	0	0
$670$	24.3412	0.940384
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−24.8198	−0.956733	−0.478366	−	0.878160i	$-0.658771\pi$
$673$	−24.8198	−0.956733	−0.478366	+	0.878160i	$0.658771\pi$
$674$	33.4267	1.28755
$675$	0	0
$676$	21.0221	0.808541
$677$	−20.8171	−0.800064	−0.400032	−	0.916501i	$-0.631001\pi$
$677$	−20.8171	−0.800064	−0.400032	+	0.916501i	$0.631001\pi$
$678$	0	0
$679$	−39.1950	−1.50417
$680$	4.78572	0.183524
$681$	0	0
$682$	−5.55420	−0.212681
$683$	21.9543	0.840058	0.420029	−	0.907511i	$-0.362020\pi$
$683$	21.9543	0.840058	0.420029	+	0.907511i	$0.362020\pi$
$684$	0	0
$685$	−2.35971	−0.0901597
$686$	20.7540	0.792391
$687$	0	0
$688$	30.2934	1.15493
$689$	−36.7491	−1.40003
$690$	0	0
$691$	−0.279975	−0.0106508	−0.00532538	−	0.999986i	$-0.501695\pi$
$691$	−0.279975	−0.0106508	−0.00532538	+	0.999986i	$0.501695\pi$
$692$	59.2306	2.25161
$693$	0	0
$694$	4.24897	0.161289
$695$	50.3496	1.90987
$696$	0	0
$697$	10.4727	0.396683
$698$	57.0032	2.15760
$699$	0	0
$700$	10.2196	0.386265
$701$	−26.9968	−1.01965	−0.509827	−	0.860277i	$-0.670290\pi$
$701$	−26.9968	−1.01965	−0.509827	+	0.860277i	$0.670290\pi$
$702$	0	0
$703$	−21.5120	−0.811341
$704$	−10.3314	−0.389379
$705$	0	0
$706$	−55.6861	−2.09578
$707$	42.6108	1.60254
$708$	0	0
$709$	17.4492	0.655319	0.327659	−	0.944796i	$-0.393740\pi$
$709$	17.4492	0.655319	0.327659	+	0.944796i	$0.393740\pi$
$710$	38.2210	1.43441
$711$	0	0
$712$	−1.70230	−0.0637966
$713$	5.56159	0.208283
$714$	0	0
$715$	−11.8140	−0.441818
$716$	−47.3425	−1.76927
$717$	0	0
$718$	−41.9697	−1.56629
$719$	−24.9061	−0.928839	−0.464420	−	0.885615i	$-0.653737\pi$
$719$	−24.9061	−0.928839	−0.464420	+	0.885615i	$0.653737\pi$
$720$	0	0
$721$	−25.6782	−0.956307
$722$	−72.4553	−2.69651
$723$	0	0
$724$	−24.0366	−0.893312
$725$	−1.01494	−0.0376939
$726$	0	0
$727$	−21.4670	−0.796166	−0.398083	−	0.917349i	$-0.630324\pi$
$727$	−21.4670	−0.796166	−0.398083	+	0.917349i	$0.630324\pi$
$728$	10.4011	0.385492
$729$	0	0
$730$	35.1023	1.29919
$731$	26.5093	0.980483
$732$	0	0
$733$	−2.10577	−0.0777784	−0.0388892	−	0.999244i	$-0.512382\pi$
$733$	−2.10577	−0.0777784	−0.0388892	+	0.999244i	$0.512382\pi$
$734$	18.1256	0.669028
$735$	0	0
$736$	16.8616	0.621527
$737$	−4.65473	−0.171459
$738$	0	0
$739$	12.0592	0.443604	0.221802	−	0.975092i	$-0.428806\pi$
$739$	12.0592	0.443604	0.221802	+	0.975092i	$0.428806\pi$
$740$	17.1181	0.629274
$741$	0	0
$742$	−53.9214	−1.97952
$743$	−5.58320	−0.204828	−0.102414	−	0.994742i	$-0.532657\pi$
$743$	−5.58320	−0.204828	−0.102414	+	0.994742i	$0.532657\pi$
$744$	0	0
$745$	−33.4002	−1.22369
$746$	−16.8406	−0.616577
$747$	0	0
$748$	−6.61005	−0.241687
$749$	−52.6837	−1.92502
$750$	0	0
$751$	40.1470	1.46499	0.732493	−	0.680775i	$-0.238357\pi$
$751$	40.1470	1.46499	0.732493	+	0.680775i	$0.238357\pi$
$752$	−1.51844	−0.0553719
$753$	0	0
$754$	−7.46092	−0.271711
$755$	29.7610	1.08311
$756$	0	0
$757$	10.2101	0.371091	0.185546	−	0.982636i	$-0.440595\pi$
$757$	10.2101	0.371091	0.185546	+	0.982636i	$0.440595\pi$
$758$	−33.6409	−1.22189
$759$	0	0
$760$	12.3340	0.447400
$761$	−41.3818	−1.50009	−0.750045	−	0.661387i	$-0.769968\pi$
$761$	−41.3818	−1.50009	−0.750045	+	0.661387i	$0.769968\pi$
$762$	0	0
$763$	−11.6646	−0.422285
$764$	−5.97890	−0.216309
$765$	0	0
$766$	−23.9191	−0.864232
$767$	−36.4141	−1.31484
$768$	0	0
$769$	−1.86294	−0.0671793	−0.0335896	−	0.999436i	$-0.510694\pi$
$769$	−1.86294	−0.0671793	−0.0335896	+	0.999436i	$0.510694\pi$
$770$	−17.3345	−0.624692
$771$	0	0
$772$	48.9250	1.76085
$773$	8.28658	0.298048	0.149024	−	0.988834i	$-0.452387\pi$
$773$	8.28658	0.298048	0.149024	+	0.988834i	$0.452387\pi$
$774$	0	0
$775$	3.54839	0.127462
$776$	7.89994	0.283591
$777$	0	0
$778$	−36.6193	−1.31286
$779$	26.9908	0.967046
$780$	0	0
$781$	−7.30894	−0.261535
$782$	12.3213	0.440609
$783$	0	0
$784$	−12.9772	−0.463472
$785$	−30.9594	−1.10499
$786$	0	0
$787$	1.03420	0.0368654	0.0184327	−	0.999830i	$-0.494132\pi$
$787$	1.03420	0.0368654	0.0184327	+	0.999830i	$0.494132\pi$
$788$	−49.0996	−1.74910
$789$	0	0
$790$	33.7252	1.19989
$791$	−26.5354	−0.943490
$792$	0	0
$793$	4.69636	0.166773
$794$	−38.7812	−1.37629
$795$	0	0
$796$	−17.7136	−0.627841
$797$	22.6893	0.803696	0.401848	−	0.915706i	$-0.368368\pi$
$797$	22.6893	0.803696	0.401848	+	0.915706i	$0.368368\pi$
$798$	0	0
$799$	−1.32877	−0.0470083
$800$	10.7580	0.380352
$801$	0	0
$802$	−55.4538	−1.95814
$803$	−6.71256	−0.236881
$804$	0	0
$805$	17.3575	0.611773
$806$	26.0845	0.918788
$807$	0	0
$808$	−8.58841	−0.302139
$809$	19.9728	0.702206	0.351103	−	0.936337i	$-0.385807\pi$
$809$	19.9728	0.702206	0.351103	+	0.936337i	$0.385807\pi$
$810$	0	0
$811$	51.5452	1.81000	0.904999	−	0.425414i	$-0.139871\pi$
$811$	51.5452	1.81000	0.904999	+	0.425414i	$0.139871\pi$
$812$	−5.88074	−0.206374
$813$	0	0
$814$	−6.09372	−0.213585
$815$	52.4621	1.83767
$816$	0	0
$817$	68.3210	2.39025
$818$	79.6137	2.78363
$819$	0	0
$820$	−21.4779	−0.750039
$821$	−14.9141	−0.520506	−0.260253	−	0.965541i	$-0.583806\pi$
$821$	−14.9141	−0.520506	−0.260253	+	0.965541i	$0.583806\pi$
$822$	0	0
$823$	−55.5932	−1.93786	−0.968929	−	0.247340i	$-0.920444\pi$
$823$	−55.5932	−1.93786	−0.968929	+	0.247340i	$0.920444\pi$
$824$	5.17557	0.180300
$825$	0	0
$826$	−53.4298	−1.85906
$827$	9.76170	0.339448	0.169724	−	0.985492i	$-0.445712\pi$
$827$	9.76170	0.339448	0.169724	+	0.985492i	$0.445712\pi$
$828$	0	0
$829$	−10.8861	−0.378090	−0.189045	−	0.981968i	$-0.560539\pi$
$829$	−10.8861	−0.378090	−0.189045	+	0.981968i	$0.560539\pi$
$830$	12.6890	0.440443
$831$	0	0
$832$	48.5199	1.68213
$833$	−11.3562	−0.393468
$834$	0	0
$835$	10.0441	0.347590
$836$	−17.0357	−0.589193
$837$	0	0
$838$	−9.71717	−0.335674
$839$	−30.5135	−1.05344	−0.526722	−	0.850038i	$-0.676579\pi$
$839$	−30.5135	−1.05344	−0.526722	+	0.850038i	$0.676579\pi$
$840$	0	0
$841$	−28.4160	−0.979861
$842$	−61.8334	−2.13092
$843$	0	0
$844$	−59.5373	−2.04936
$845$	22.7804	0.783669
$846$	0	0
$847$	3.31485	0.113899
$848$	−25.4619	−0.874365
$849$	0	0
$850$	7.86120	0.269637
$851$	6.10183	0.209168
$852$	0	0
$853$	46.5861	1.59508	0.797539	−	0.603267i	$-0.206135\pi$
$853$	46.5861	1.59508	0.797539	+	0.603267i	$0.206135\pi$
$854$	6.89089	0.235801
$855$	0	0
$856$	10.6187	0.362938
$857$	23.3679	0.798233	0.399116	−	0.916900i	$-0.369317\pi$
$857$	23.3679	0.798233	0.399116	+	0.916900i	$0.369317\pi$
$858$	0	0
$859$	−29.1184	−0.993506	−0.496753	−	0.867892i	$-0.665475\pi$
$859$	−29.1184	−0.993506	−0.496753	+	0.867892i	$0.665475\pi$
$860$	−54.3662	−1.85387
$861$	0	0
$862$	29.6467	1.00977
$863$	−20.8386	−0.709353	−0.354676	−	0.934989i	$-0.615409\pi$
$863$	−20.8386	−0.709353	−0.354676	+	0.934989i	$0.615409\pi$
$864$	0	0
$865$	64.1848	2.18235
$866$	−83.3255	−2.83151
$867$	0	0
$868$	20.5600	0.697851
$869$	−6.44921	−0.218775
$870$	0	0
$871$	21.8603	0.740707
$872$	2.35105	0.0796165
$873$	0	0
$874$	31.7550	1.07413
$875$	−30.6192	−1.03512
$876$	0	0
$877$	40.3955	1.36406	0.682030	−	0.731324i	$-0.261097\pi$
$877$	40.3955	1.36406	0.682030	+	0.731324i	$0.261097\pi$
$878$	86.8523	2.93112
$879$	0	0
$880$	−8.18541	−0.275930
$881$	−28.4606	−0.958863	−0.479432	−	0.877579i	$-0.659157\pi$
$881$	−28.4606	−0.958863	−0.479432	+	0.877579i	$0.659157\pi$
$882$	0	0
$883$	37.3212	1.25596	0.627980	−	0.778230i	$-0.283882\pi$
$883$	37.3212	1.25596	0.627980	+	0.778230i	$0.283882\pi$
$884$	31.0432	1.04409
$885$	0	0
$886$	−47.0143	−1.57948
$887$	5.69722	0.191294	0.0956470	−	0.995415i	$-0.469508\pi$
$887$	5.69722	0.191294	0.0956470	+	0.995415i	$0.469508\pi$
$888$	0	0
$889$	18.6635	0.625953
$890$	−13.3238	−0.446615
$891$	0	0
$892$	−50.1940	−1.68062
$893$	−3.42455	−0.114598
$894$	0	0
$895$	−51.3024	−1.71485
$896$	17.4890	0.584267
$897$	0	0
$898$	0.616324	0.0205670
$899$	−2.04187	−0.0681002
$900$	0	0
$901$	−22.2813	−0.742299
$902$	7.64571	0.254574
$903$	0	0
$904$	5.34834	0.177883
$905$	−26.0470	−0.865833
$906$	0	0
$907$	−28.0847	−0.932536	−0.466268	−	0.884644i	$-0.654402\pi$
$907$	−28.0847	−0.932536	−0.466268	+	0.884644i	$0.654402\pi$
$908$	−17.2043	−0.570946
$909$	0	0
$910$	81.4090	2.69868
$911$	−5.56419	−0.184350	−0.0921749	−	0.995743i	$-0.529382\pi$
$911$	−5.56419	−0.184350	−0.0921749	+	0.995743i	$0.529382\pi$
$912$	0	0
$913$	−2.42650	−0.0803055
$914$	73.5924	2.43422
$915$	0	0
$916$	28.4096	0.938679
$917$	−14.1205	−0.466301
$918$	0	0
$919$	9.22080	0.304166	0.152083	−	0.988368i	$-0.451402\pi$
$919$	9.22080	0.304166	0.152083	+	0.988368i	$0.451402\pi$
$920$	−3.49850	−0.115342
$921$	0	0
$922$	51.6878	1.70225
$923$	34.3254	1.12983
$924$	0	0
$925$	3.89307	0.128003
$926$	49.1211	1.61422
$927$	0	0
$928$	−6.19054	−0.203214
$929$	18.9686	0.622339	0.311170	−	0.950354i	$-0.399279\pi$
$929$	18.9686	0.622339	0.311170	+	0.950354i	$0.399279\pi$
$930$	0	0
$931$	−29.2676	−0.959208
$932$	−23.0498	−0.755022
$933$	0	0
$934$	−51.3812	−1.68124
$935$	−7.16293	−0.234253
$936$	0	0
$937$	30.0732	0.982449	0.491224	−	0.871033i	$-0.336550\pi$
$937$	30.0732	0.982449	0.491224	+	0.871033i	$0.336550\pi$
$938$	32.0752	1.04729
$939$	0	0
$940$	2.72508	0.0888822
$941$	23.3429	0.760956	0.380478	−	0.924790i	$-0.375760\pi$
$941$	23.3429	0.760956	0.380478	+	0.924790i	$0.375760\pi$
$942$	0	0
$943$	−7.65588	−0.249310
$944$	−25.2298	−0.821159
$945$	0	0
$946$	19.3533	0.629231
$947$	16.1738	0.525579	0.262790	−	0.964853i	$-0.415358\pi$
$947$	16.1738	0.525579	0.262790	+	0.964853i	$0.415358\pi$
$948$	0	0
$949$	31.5246	1.02333
$950$	20.2602	0.657329
$951$	0	0
$952$	6.30630	0.204388
$953$	53.5723	1.73538	0.867689	−	0.497107i	$-0.165604\pi$
$953$	53.5723	1.73538	0.867689	+	0.497107i	$0.165604\pi$
$954$	0	0
$955$	−6.47899	−0.209655
$956$	−26.5866	−0.859873
$957$	0	0
$958$	−8.02765	−0.259361
$959$	−3.10946	−0.100410
$960$	0	0
$961$	−23.8613	−0.769720
$962$	28.6183	0.922691
$963$	0	0
$964$	3.22218	0.103779
$965$	53.0172	1.70668
$966$	0	0
$967$	−26.8918	−0.864781	−0.432390	−	0.901686i	$-0.642330\pi$
$967$	−26.8918	−0.864781	−0.432390	+	0.901686i	$0.642330\pi$
$968$	−0.668123	−0.0214743
$969$	0	0
$970$	61.8323	1.98531
$971$	−4.19888	−0.134749	−0.0673743	−	0.997728i	$-0.521462\pi$
$971$	−4.19888	−0.134749	−0.0673743	+	0.997728i	$0.521462\pi$
$972$	0	0
$973$	66.3473	2.12700
$974$	−9.52238	−0.305117
$975$	0	0
$976$	3.25391	0.104155
$977$	−4.88596	−0.156316	−0.0781578	−	0.996941i	$-0.524904\pi$
$977$	−4.88596	−0.156316	−0.0781578	+	0.996941i	$0.524904\pi$
$978$	0	0
$979$	2.54789	0.0814309
$980$	23.2896	0.743959
$981$	0	0
$982$	−69.8293	−2.22834
$983$	41.8812	1.33580	0.667902	−	0.744250i	$-0.267193\pi$
$983$	41.8812	1.33580	0.667902	+	0.744250i	$0.267193\pi$
$984$	0	0
$985$	−53.2064	−1.69530
$986$	−4.52362	−0.144061
$987$	0	0
$988$	80.0058	2.54532
$989$	−19.3791	−0.616219
$990$	0	0
$991$	20.1386	0.639724	0.319862	−	0.947464i	$-0.396363\pi$
$991$	20.1386	0.639724	0.319862	+	0.947464i	$0.396363\pi$
$992$	21.6431	0.687168
$993$	0	0
$994$	50.3651	1.59749
$995$	−19.1952	−0.608528
$996$	0	0
$997$	−2.25885	−0.0715385	−0.0357692	−	0.999360i	$-0.511388\pi$
$997$	−2.25885	−0.0715385	−0.0357692	+	0.999360i	$0.511388\pi$
$998$	−66.9448	−2.11910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.2		12
3.2	odd	2		2013.2.a.d.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.11	✓	12	3.2	odd	2
6039.2.a.e.1.2		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)

\(f(q)\)	\(=\)	\(q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} +O(q^{10})\)	\(q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} -5.22935 q^{10} +1.00000 q^{11} -4.69636 q^{13} -6.89089 q^{14} -3.25391 q^{16} -2.84744 q^{17} -7.33856 q^{19} +5.83963 q^{20} -2.07880 q^{22} +2.08156 q^{23} +1.32807 q^{25} +9.76278 q^{26} +7.69508 q^{28} -0.764221 q^{29} +2.67183 q^{31} +8.10046 q^{32} +5.91926 q^{34} +8.33871 q^{35} +2.93137 q^{37} +15.2554 q^{38} -1.68071 q^{40} -3.67795 q^{41} -9.30987 q^{43} +2.32140 q^{44} -4.32715 q^{46} +0.466652 q^{47} +3.98820 q^{49} -2.76079 q^{50} -10.9021 q^{52} +7.82503 q^{53} +2.51557 q^{55} -2.21472 q^{56} +1.58866 q^{58} +7.75368 q^{59} -1.00000 q^{61} -5.55420 q^{62} -10.3314 q^{64} -11.8140 q^{65} -4.65473 q^{67} -6.61005 q^{68} -17.3345 q^{70} -7.30894 q^{71} -6.71256 q^{73} -6.09372 q^{74} -17.0357 q^{76} +3.31485 q^{77} -6.44921 q^{79} -8.18541 q^{80} +7.64571 q^{82} -2.42650 q^{83} -7.16293 q^{85} +19.3533 q^{86} -0.668123 q^{88} +2.54789 q^{89} -15.5677 q^{91} +4.83214 q^{92} -0.970075 q^{94} -18.4606 q^{95} -11.8241 q^{97} -8.29066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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