LMFDB - Embedded newform 6039.2.a.k.1.1

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72847$ 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)

$f(q)$	$=$	$q-2.72847 q^{2} +5.44452 q^{4} +3.87909 q^{5} +0.707409 q^{7} -9.39826 q^{8} +O(q^{10})$	\(q-2.72847 q^{2} +5.44452 q^{4} +3.87909 q^{5} +0.707409 q^{7} -9.39826 q^{8} -10.5840 q^{10} -1.00000 q^{11} -5.09083 q^{13} -1.93014 q^{14} +14.7538 q^{16} -5.13736 q^{17} -1.83047 q^{19} +21.1198 q^{20} +2.72847 q^{22} +5.52571 q^{23} +10.0473 q^{25} +13.8902 q^{26} +3.85150 q^{28} +4.81121 q^{29} -3.88155 q^{31} -21.4587 q^{32} +14.0171 q^{34} +2.74410 q^{35} +6.12073 q^{37} +4.99436 q^{38} -36.4567 q^{40} +6.78620 q^{41} +5.04075 q^{43} -5.44452 q^{44} -15.0767 q^{46} -11.7350 q^{47} -6.49957 q^{49} -27.4138 q^{50} -27.7172 q^{52} +4.77383 q^{53} -3.87909 q^{55} -6.64841 q^{56} -13.1272 q^{58} -5.48329 q^{59} -1.00000 q^{61} +10.5907 q^{62} +29.0416 q^{64} -19.7478 q^{65} +11.1597 q^{67} -27.9705 q^{68} -7.48718 q^{70} +8.09785 q^{71} -0.965908 q^{73} -16.7002 q^{74} -9.96602 q^{76} -0.707409 q^{77} +7.96622 q^{79} +57.2312 q^{80} -18.5159 q^{82} +2.76074 q^{83} -19.9283 q^{85} -13.7535 q^{86} +9.39826 q^{88} +5.89379 q^{89} -3.60130 q^{91} +30.0848 q^{92} +32.0186 q^{94} -7.10054 q^{95} -7.70416 q^{97} +17.7339 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	−2.72847	−1.92932	−0.964658	−	0.263505i	$-0.915122\pi$
$2$	−2.72847	−1.92932	−0.964658	+	0.263505i	$0.915122\pi$
$3$	0	0
$3$	0	0
$4$	5.44452	2.72226
$5$	3.87909	1.73478	0.867391	−	0.497628i	$-0.165795\pi$
$5$	3.87909	1.73478	0.867391	+	0.497628i	$0.165795\pi$
$6$	0	0
$7$	0.707409	0.267375	0.133688	−	0.991024i	$-0.457318\pi$
$7$	0.707409	0.267375	0.133688	+	0.991024i	$0.457318\pi$
$8$	−9.39826	−3.32279
$9$	0	0
$10$	−10.5840	−3.34694
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.09083	−1.41194	−0.705971	−	0.708240i	$-0.749489\pi$
$13$	−5.09083	−1.41194	−0.705971	+	0.708240i	$0.749489\pi$
$14$	−1.93014	−0.515852
$15$	0	0
$16$	14.7538	3.68844
$17$	−5.13736	−1.24599	−0.622997	−	0.782224i	$-0.714085\pi$
$17$	−5.13736	−1.24599	−0.622997	+	0.782224i	$0.714085\pi$
$18$	0	0
$19$	−1.83047	−0.419938	−0.209969	−	0.977708i	$-0.567336\pi$
$19$	−1.83047	−0.419938	−0.209969	+	0.977708i	$0.567336\pi$
$20$	21.1198	4.72253
$21$	0	0
$22$	2.72847	0.581711
$23$	5.52571	1.15219	0.576095	−	0.817383i	$-0.304576\pi$
$23$	5.52571	1.15219	0.576095	+	0.817383i	$0.304576\pi$
$24$	0	0
$25$	10.0473	2.00947
$26$	13.8902	2.72408
$27$	0	0
$28$	3.85150	0.727866
$29$	4.81121	0.893419	0.446709	−	0.894679i	$-0.352596\pi$
$29$	4.81121	0.893419	0.446709	+	0.894679i	$0.352596\pi$
$30$	0	0
$31$	−3.88155	−0.697147	−0.348573	−	0.937281i	$-0.613334\pi$
$31$	−3.88155	−0.697147	−0.348573	+	0.937281i	$0.613334\pi$
$32$	−21.4587	−3.79339
$33$	0	0
$34$	14.0171	2.40392
$35$	2.74410	0.463838
$36$	0	0
$37$	6.12073	1.00624	0.503121	−	0.864216i	$-0.332185\pi$
$37$	6.12073	1.00624	0.503121	+	0.864216i	$0.332185\pi$
$38$	4.99436	0.810193
$39$	0	0
$40$	−36.4567	−5.76431
$41$	6.78620	1.05983	0.529913	−	0.848052i	$-0.322224\pi$
$41$	6.78620	1.05983	0.529913	+	0.848052i	$0.322224\pi$
$42$	0	0
$43$	5.04075	0.768707	0.384353	−	0.923186i	$-0.374424\pi$
$43$	5.04075	0.768707	0.384353	+	0.923186i	$0.374424\pi$
$44$	−5.44452	−0.820793
$45$	0	0
$46$	−15.0767	−2.22294
$47$	−11.7350	−1.71173	−0.855865	−	0.517199i	$-0.826975\pi$
$47$	−11.7350	−1.71173	−0.855865	+	0.517199i	$0.826975\pi$
$48$	0	0
$49$	−6.49957	−0.928510
$50$	−27.4138	−3.87690
$51$	0	0
$52$	−27.7172	−3.84368
$53$	4.77383	0.655736	0.327868	−	0.944724i	$-0.393670\pi$
$53$	4.77383	0.655736	0.327868	+	0.944724i	$0.393670\pi$
$54$	0	0
$55$	−3.87909	−0.523056
$56$	−6.64841	−0.888431
$57$	0	0
$58$	−13.1272	−1.72369
$59$	−5.48329	−0.713864	−0.356932	−	0.934130i	$-0.616177\pi$
$59$	−5.48329	−0.713864	−0.356932	+	0.934130i	$0.616177\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	10.5907	1.34502
$63$	0	0
$64$	29.0416	3.63020
$65$	−19.7478	−2.44941
$66$	0	0
$67$	11.1597	1.36337	0.681685	−	0.731645i	$-0.261247\pi$
$67$	11.1597	1.36337	0.681685	+	0.731645i	$0.261247\pi$
$68$	−27.9705	−3.39192
$69$	0	0
$70$	−7.48718	−0.894890
$71$	8.09785	0.961038	0.480519	−	0.876984i	$-0.340448\pi$
$71$	8.09785	0.961038	0.480519	+	0.876984i	$0.340448\pi$
$72$	0	0
$73$	−0.965908	−0.113051	−0.0565255	−	0.998401i	$-0.518002\pi$
$73$	−0.965908	−0.113051	−0.0565255	+	0.998401i	$0.518002\pi$
$74$	−16.7002	−1.94136
$75$	0	0
$76$	−9.96602	−1.14318
$77$	−0.707409	−0.0806167
$78$	0	0
$79$	7.96622	0.896270	0.448135	−	0.893966i	$-0.352088\pi$
$79$	7.96622	0.896270	0.448135	+	0.893966i	$0.352088\pi$
$80$	57.2312	6.39864
$81$	0	0
$82$	−18.5159	−2.04474
$83$	2.76074	0.303031	0.151516	−	0.988455i	$-0.451585\pi$
$83$	2.76074	0.303031	0.151516	+	0.988455i	$0.451585\pi$
$84$	0	0
$85$	−19.9283	−2.16153
$86$	−13.7535	−1.48308
$87$	0	0
$88$	9.39826	1.00186
$89$	5.89379	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$89$	5.89379	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$90$	0	0
$91$	−3.60130	−0.377519
$92$	30.0848	3.13656
$93$	0	0
$94$	32.0186	3.30247
$95$	−7.10054	−0.728500
$96$	0	0
$97$	−7.70416	−0.782239	−0.391120	−	0.920340i	$-0.627912\pi$
$97$	−7.70416	−0.782239	−0.391120	+	0.920340i	$0.627912\pi$
$98$	17.7339	1.79139
$99$	0	0
$100$	54.7029	5.47029
$101$	5.18303	0.515731	0.257866	−	0.966181i	$-0.416981\pi$
$101$	5.18303	0.515731	0.257866	+	0.966181i	$0.416981\pi$
$102$	0	0
$103$	14.0988	1.38920	0.694598	−	0.719399i	$-0.255582\pi$
$103$	14.0988	1.38920	0.694598	+	0.719399i	$0.255582\pi$
$104$	47.8450	4.69158
$105$	0	0
$106$	−13.0252	−1.26512
$107$	−1.64594	−0.159119	−0.0795594	−	0.996830i	$-0.525351\pi$
$107$	−1.64594	−0.159119	−0.0795594	+	0.996830i	$0.525351\pi$
$108$	0	0
$109$	11.4690	1.09853	0.549267	−	0.835647i	$-0.314907\pi$
$109$	11.4690	1.09853	0.549267	+	0.835647i	$0.314907\pi$
$110$	10.5840	1.00914
$111$	0	0
$112$	10.4370	0.986199
$113$	−13.1820	−1.24006	−0.620031	−	0.784577i	$-0.712880\pi$
$113$	−13.1820	−1.24006	−0.620031	+	0.784577i	$0.712880\pi$
$114$	0	0
$115$	21.4347	1.99880
$116$	26.1947	2.43212
$117$	0	0
$118$	14.9610	1.37727
$119$	−3.63422	−0.333148
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.72847	0.247024
$123$	0	0
$124$	−21.1332	−1.89782
$125$	19.5790	1.75120
$126$	0	0
$127$	0.0386451	0.00342920	0.00171460	−	0.999999i	$-0.499454\pi$
$127$	0.0386451	0.00342920	0.00171460	+	0.999999i	$0.499454\pi$
$128$	−36.3218	−3.21042
$129$	0	0
$130$	53.8812	4.72569
$131$	−16.6885	−1.45808	−0.729041	−	0.684470i	$-0.760034\pi$
$131$	−16.6885	−1.45808	−0.729041	+	0.684470i	$0.760034\pi$
$132$	0	0
$133$	−1.29489	−0.112281
$134$	−30.4488	−2.63037
$135$	0	0
$136$	48.2823	4.14017
$137$	−20.1632	−1.72266	−0.861329	−	0.508048i	$-0.830367\pi$
$137$	−20.1632	−1.72266	−0.861329	+	0.508048i	$0.830367\pi$
$138$	0	0
$139$	4.08579	0.346552	0.173276	−	0.984873i	$-0.444565\pi$
$139$	4.08579	0.346552	0.173276	+	0.984873i	$0.444565\pi$
$140$	14.9403	1.26269
$141$	0	0
$142$	−22.0947	−1.85415
$143$	5.09083	0.425717
$144$	0	0
$145$	18.6631	1.54989
$146$	2.63545	0.218111
$147$	0	0
$148$	33.3244	2.73925
$149$	20.4786	1.67768	0.838838	−	0.544381i	$-0.183236\pi$
$149$	20.4786	1.67768	0.838838	+	0.544381i	$0.183236\pi$
$150$	0	0
$151$	21.2572	1.72988	0.864942	−	0.501871i	$-0.167355\pi$
$151$	21.2572	1.72988	0.864942	+	0.501871i	$0.167355\pi$
$152$	17.2032	1.39536
$153$	0	0
$154$	1.93014	0.155535
$155$	−15.0569	−1.20940
$156$	0	0
$157$	−8.69696	−0.694093	−0.347046	−	0.937848i	$-0.612815\pi$
$157$	−8.69696	−0.694093	−0.347046	+	0.937848i	$0.612815\pi$
$158$	−21.7356	−1.72919
$159$	0	0
$160$	−83.2400	−6.58070
$161$	3.90893	0.308067
$162$	0	0
$163$	3.87110	0.303208	0.151604	−	0.988441i	$-0.451556\pi$
$163$	3.87110	0.303208	0.151604	+	0.988441i	$0.451556\pi$
$164$	36.9476	2.88513
$165$	0	0
$166$	−7.53260	−0.584643
$167$	−5.92532	−0.458515	−0.229258	−	0.973366i	$-0.573630\pi$
$167$	−5.92532	−0.458515	−0.229258	+	0.973366i	$0.573630\pi$
$168$	0	0
$169$	12.9166	0.993583
$170$	54.3736	4.17027
$171$	0	0
$172$	27.4445	2.09262
$173$	12.8767	0.978997	0.489498	−	0.872004i	$-0.337180\pi$
$173$	12.8767	0.978997	0.489498	+	0.872004i	$0.337180\pi$
$174$	0	0
$175$	7.10757	0.537282
$176$	−14.7538	−1.11211
$177$	0	0
$178$	−16.0810	−1.20532
$179$	22.7915	1.70351	0.851757	−	0.523937i	$-0.175537\pi$
$179$	22.7915	1.70351	0.851757	+	0.523937i	$0.175537\pi$
$180$	0	0
$181$	15.1780	1.12817	0.564085	−	0.825717i	$-0.309229\pi$
$181$	15.1780	1.12817	0.564085	+	0.825717i	$0.309229\pi$
$182$	9.82602	0.728353
$183$	0	0
$184$	−51.9320	−3.82848
$185$	23.7428	1.74561
$186$	0	0
$187$	5.13736	0.375681
$188$	−63.8917	−4.65978
$189$	0	0
$190$	19.3736	1.40551
$191$	16.0220	1.15931	0.579657	−	0.814861i	$-0.303187\pi$
$191$	16.0220	1.15931	0.579657	+	0.814861i	$0.303187\pi$
$192$	0	0
$193$	18.3717	1.32242	0.661211	−	0.750200i	$-0.270043\pi$
$193$	18.3717	1.32242	0.661211	+	0.750200i	$0.270043\pi$
$194$	21.0205	1.50919
$195$	0	0
$196$	−35.3871	−2.52765
$197$	16.0456	1.14320	0.571602	−	0.820531i	$-0.306322\pi$
$197$	16.0456	1.14320	0.571602	+	0.820531i	$0.306322\pi$
$198$	0	0
$199$	−9.47729	−0.671827	−0.335913	−	0.941893i	$-0.609045\pi$
$199$	−9.47729	−0.671827	−0.335913	+	0.941893i	$0.609045\pi$
$200$	−94.4274	−6.67703
$201$	0	0
$202$	−14.1417	−0.995008
$203$	3.40349	0.238878
$204$	0	0
$205$	26.3243	1.83857
$206$	−38.4681	−2.68020
$207$	0	0
$208$	−75.1090	−5.20787
$209$	1.83047	0.126616
$210$	0	0
$211$	−16.5435	−1.13890	−0.569451	−	0.822026i	$-0.692844\pi$
$211$	−16.5435	−1.13890	−0.569451	+	0.822026i	$0.692844\pi$
$212$	25.9912	1.78508
$213$	0	0
$214$	4.49089	0.306991
$215$	19.5535	1.33354
$216$	0	0
$217$	−2.74584	−0.186400
$218$	−31.2928	−2.11942
$219$	0	0
$220$	−21.1198	−1.42390
$221$	26.1535	1.75927
$222$	0	0
$223$	−25.0408	−1.67686	−0.838429	−	0.545010i	$-0.816526\pi$
$223$	−25.0408	−1.67686	−0.838429	+	0.545010i	$0.816526\pi$
$224$	−15.1800	−1.01426
$225$	0	0
$226$	35.9667	2.39247
$227$	17.3304	1.15026	0.575130	−	0.818062i	$-0.304951\pi$
$227$	17.3304	1.15026	0.575130	+	0.818062i	$0.304951\pi$
$228$	0	0
$229$	17.6940	1.16925	0.584625	−	0.811304i	$-0.301242\pi$
$229$	17.6940	1.16925	0.584625	+	0.811304i	$0.301242\pi$
$230$	−58.4839	−3.85631
$231$	0	0
$232$	−45.2170	−2.96864
$233$	5.55895	0.364179	0.182089	−	0.983282i	$-0.441714\pi$
$233$	5.55895	0.364179	0.182089	+	0.983282i	$0.441714\pi$
$234$	0	0
$235$	−45.5212	−2.96948
$236$	−29.8539	−1.94332
$237$	0	0
$238$	9.91583	0.642748
$239$	−22.0447	−1.42595	−0.712976	−	0.701188i	$-0.752653\pi$
$239$	−22.0447	−1.42595	−0.712976	+	0.701188i	$0.752653\pi$
$240$	0	0
$241$	5.05785	0.325805	0.162902	−	0.986642i	$-0.447914\pi$
$241$	5.05785	0.325805	0.162902	+	0.986642i	$0.447914\pi$
$242$	−2.72847	−0.175392
$243$	0	0
$244$	−5.44452	−0.348550
$245$	−25.2124	−1.61076
$246$	0	0
$247$	9.31860	0.592928
$248$	36.4798	2.31647
$249$	0	0
$250$	−53.4207	−3.37862
$251$	1.09266	0.0689679	0.0344840	−	0.999405i	$-0.489021\pi$
$251$	1.09266	0.0689679	0.0344840	+	0.999405i	$0.489021\pi$
$252$	0	0
$253$	−5.52571	−0.347398
$254$	−0.105442	−0.00661602
$255$	0	0
$256$	41.0194	2.56372
$257$	−8.09625	−0.505030	−0.252515	−	0.967593i	$-0.581258\pi$
$257$	−8.09625	−0.505030	−0.252515	+	0.967593i	$0.581258\pi$
$258$	0	0
$259$	4.32986	0.269044
$260$	−107.517	−6.66794
$261$	0	0
$262$	45.5340	2.81310
$263$	−21.9406	−1.35292	−0.676459	−	0.736481i	$-0.736486\pi$
$263$	−21.9406	−1.35292	−0.676459	+	0.736481i	$0.736486\pi$
$264$	0	0
$265$	18.5181	1.13756
$266$	3.53306	0.216626
$267$	0	0
$268$	60.7591	3.71145
$269$	28.0472	1.71007	0.855035	−	0.518571i	$-0.173536\pi$
$269$	28.0472	1.71007	0.855035	+	0.518571i	$0.173536\pi$
$270$	0	0
$271$	5.41190	0.328749	0.164375	−	0.986398i	$-0.447439\pi$
$271$	5.41190	0.328749	0.164375	+	0.986398i	$0.447439\pi$
$272$	−75.7955	−4.59578
$273$	0	0
$274$	55.0145	3.32355
$275$	−10.0473	−0.605877
$276$	0	0
$277$	−4.41915	−0.265521	−0.132761	−	0.991148i	$-0.542384\pi$
$277$	−4.41915	−0.265521	−0.132761	+	0.991148i	$0.542384\pi$
$278$	−11.1479	−0.668608
$279$	0	0
$280$	−25.7898	−1.54123
$281$	13.7354	0.819388	0.409694	−	0.912223i	$-0.365635\pi$
$281$	13.7354	0.819388	0.409694	+	0.912223i	$0.365635\pi$
$282$	0	0
$283$	11.9429	0.709932	0.354966	−	0.934879i	$-0.384492\pi$
$283$	11.9429	0.709932	0.354966	+	0.934879i	$0.384492\pi$
$284$	44.0889	2.61620
$285$	0	0
$286$	−13.8902	−0.821342
$287$	4.80062	0.283372
$288$	0	0
$289$	9.39251	0.552500
$290$	−50.9216	−2.99022
$291$	0	0
$292$	−5.25891	−0.307754
$293$	11.7425	0.686006	0.343003	−	0.939334i	$-0.388556\pi$
$293$	11.7425	0.686006	0.343003	+	0.939334i	$0.388556\pi$
$294$	0	0
$295$	−21.2702	−1.23840
$296$	−57.5242	−3.34353
$297$	0	0
$298$	−55.8753	−3.23677
$299$	−28.1305	−1.62683
$300$	0	0
$301$	3.56587	0.205533
$302$	−57.9995	−3.33749
$303$	0	0
$304$	−27.0063	−1.54892
$305$	−3.87909	−0.222116
$306$	0	0
$307$	5.53566	0.315937	0.157968	−	0.987444i	$-0.449506\pi$
$307$	5.53566	0.315937	0.157968	+	0.987444i	$0.449506\pi$
$308$	−3.85150	−0.219460
$309$	0	0
$310$	41.0822	2.33331
$311$	7.11901	0.403682	0.201841	−	0.979418i	$-0.435308\pi$
$311$	7.11901	0.403682	0.201841	+	0.979418i	$0.435308\pi$
$312$	0	0
$313$	−16.9760	−0.959541	−0.479771	−	0.877394i	$-0.659280\pi$
$313$	−16.9760	−0.959541	−0.479771	+	0.877394i	$0.659280\pi$
$314$	23.7293	1.33912
$315$	0	0
$316$	43.3723	2.43988
$317$	12.3675	0.694626	0.347313	−	0.937749i	$-0.387094\pi$
$317$	12.3675	0.694626	0.347313	+	0.937749i	$0.387094\pi$
$318$	0	0
$319$	−4.81121	−0.269376
$320$	112.655	6.29761
$321$	0	0
$322$	−10.6654	−0.594359
$323$	9.40377	0.523240
$324$	0	0
$325$	−51.1493	−2.83725
$326$	−10.5622	−0.584984
$327$	0	0
$328$	−63.7785	−3.52158
$329$	−8.30146	−0.457675
$330$	0	0
$331$	2.63913	0.145060	0.0725299	−	0.997366i	$-0.476893\pi$
$331$	2.63913	0.145060	0.0725299	+	0.997366i	$0.476893\pi$
$332$	15.0309	0.824930
$333$	0	0
$334$	16.1670	0.884621
$335$	43.2894	2.36515
$336$	0	0
$337$	3.27007	0.178132	0.0890659	−	0.996026i	$-0.471612\pi$
$337$	3.27007	0.178132	0.0890659	+	0.996026i	$0.471612\pi$
$338$	−35.2424	−1.91694
$339$	0	0
$340$	−108.500	−5.88424
$341$	3.88155	0.210198
$342$	0	0
$343$	−9.54972	−0.515636
$344$	−47.3743	−2.55425
$345$	0	0
$346$	−35.1336	−1.88879
$347$	1.04149	0.0559101	0.0279551	−	0.999609i	$-0.491100\pi$
$347$	1.04149	0.0559101	0.0279551	+	0.999609i	$0.491100\pi$
$348$	0	0
$349$	21.5297	1.15246	0.576230	−	0.817287i	$-0.304523\pi$
$349$	21.5297	1.15246	0.576230	+	0.817287i	$0.304523\pi$
$350$	−19.3928	−1.03659
$351$	0	0
$352$	21.4587	1.14375
$353$	3.05042	0.162358	0.0811788	−	0.996700i	$-0.474132\pi$
$353$	3.05042	0.162358	0.0811788	+	0.996700i	$0.474132\pi$
$354$	0	0
$355$	31.4123	1.66719
$356$	32.0889	1.70071
$357$	0	0
$358$	−62.1857	−3.28662
$359$	−0.766889	−0.0404749	−0.0202374	−	0.999795i	$-0.506442\pi$
$359$	−0.766889	−0.0404749	−0.0202374	+	0.999795i	$0.506442\pi$
$360$	0	0
$361$	−15.6494	−0.823652
$362$	−41.4126	−2.17660
$363$	0	0
$364$	−19.6074	−1.02770
$365$	−3.74684	−0.196119
$366$	0	0
$367$	−10.5237	−0.549333	−0.274667	−	0.961540i	$-0.588567\pi$
$367$	−10.5237	−0.549333	−0.274667	+	0.961540i	$0.588567\pi$
$368$	81.5251	4.24979
$369$	0	0
$370$	−64.7815	−3.36783
$371$	3.37705	0.175328
$372$	0	0
$373$	−31.7219	−1.64250	−0.821250	−	0.570569i	$-0.806723\pi$
$373$	−31.7219	−1.64250	−0.821250	+	0.570569i	$0.806723\pi$
$374$	−14.0171	−0.724808
$375$	0	0
$376$	110.289	5.68772
$377$	−24.4930	−1.26146
$378$	0	0
$379$	−14.1231	−0.725454	−0.362727	−	0.931895i	$-0.618154\pi$
$379$	−14.1231	−0.725454	−0.362727	+	0.931895i	$0.618154\pi$
$380$	−38.6591	−1.98317
$381$	0	0
$382$	−43.7156	−2.23668
$383$	−17.8325	−0.911201	−0.455600	−	0.890184i	$-0.650575\pi$
$383$	−17.8325	−0.911201	−0.455600	+	0.890184i	$0.650575\pi$
$384$	0	0
$385$	−2.74410	−0.139852
$386$	−50.1265	−2.55137
$387$	0	0
$388$	−41.9455	−2.12946
$389$	−20.5027	−1.03953	−0.519764	−	0.854310i	$-0.673980\pi$
$389$	−20.5027	−1.03953	−0.519764	+	0.854310i	$0.673980\pi$
$390$	0	0
$391$	−28.3876	−1.43562
$392$	61.0847	3.08524
$393$	0	0
$394$	−43.7800	−2.20560
$395$	30.9017	1.55483
$396$	0	0
$397$	−27.6903	−1.38974	−0.694869	−	0.719136i	$-0.744538\pi$
$397$	−27.6903	−1.38974	−0.694869	+	0.719136i	$0.744538\pi$
$398$	25.8584	1.29617
$399$	0	0
$400$	148.236	7.41180
$401$	36.7065	1.83304	0.916519	−	0.399992i	$-0.130987\pi$
$401$	36.7065	1.83304	0.916519	+	0.399992i	$0.130987\pi$
$402$	0	0
$403$	19.7603	0.984332
$404$	28.2191	1.40395
$405$	0	0
$406$	−9.28630	−0.460871
$407$	−6.12073	−0.303393
$408$	0	0
$409$	25.7342	1.27247	0.636237	−	0.771494i	$-0.280490\pi$
$409$	25.7342	1.27247	0.636237	+	0.771494i	$0.280490\pi$
$410$	−71.8249	−3.54718
$411$	0	0
$412$	76.7612	3.78175
$413$	−3.87893	−0.190870
$414$	0	0
$415$	10.7092	0.525693
$416$	109.242	5.35605
$417$	0	0
$418$	−4.99436	−0.244282
$419$	−12.0656	−0.589443	−0.294721	−	0.955583i	$-0.595227\pi$
$419$	−12.0656	−0.589443	−0.294721	+	0.955583i	$0.595227\pi$
$420$	0	0
$421$	21.8925	1.06697	0.533487	−	0.845809i	$-0.320881\pi$
$421$	21.8925	1.06697	0.533487	+	0.845809i	$0.320881\pi$
$422$	45.1384	2.19730
$423$	0	0
$424$	−44.8657	−2.17887
$425$	−51.6168	−2.50378
$426$	0	0
$427$	−0.707409	−0.0342339
$428$	−8.96135	−0.433163
$429$	0	0
$430$	−53.3511	−2.57282
$431$	−17.9918	−0.866637	−0.433318	−	0.901241i	$-0.642657\pi$
$431$	−17.9918	−0.866637	−0.433318	+	0.901241i	$0.642657\pi$
$432$	0	0
$433$	−4.52318	−0.217370	−0.108685	−	0.994076i	$-0.534664\pi$
$433$	−4.52318	−0.217370	−0.108685	+	0.994076i	$0.534664\pi$
$434$	7.49193	0.359624
$435$	0	0
$436$	62.4434	2.99050
$437$	−10.1146	−0.483848
$438$	0	0
$439$	35.3967	1.68939	0.844695	−	0.535248i	$-0.179782\pi$
$439$	35.3967	1.68939	0.844695	+	0.535248i	$0.179782\pi$
$440$	36.4567	1.73800
$441$	0	0
$442$	−71.3588	−3.39419
$443$	31.6228	1.50244	0.751222	−	0.660050i	$-0.229465\pi$
$443$	31.6228	1.50244	0.751222	+	0.660050i	$0.229465\pi$
$444$	0	0
$445$	22.8625	1.08379
$446$	68.3231	3.23519
$447$	0	0
$448$	20.5443	0.970627
$449$	−20.1206	−0.949548	−0.474774	−	0.880108i	$-0.657470\pi$
$449$	−20.1206	−0.949548	−0.474774	+	0.880108i	$0.657470\pi$
$450$	0	0
$451$	−6.78620	−0.319550
$452$	−71.7699	−3.37577
$453$	0	0
$454$	−47.2855	−2.21922
$455$	−13.9698	−0.654912
$456$	0	0
$457$	18.7957	0.879224	0.439612	−	0.898188i	$-0.355116\pi$
$457$	18.7957	0.879224	0.439612	+	0.898188i	$0.355116\pi$
$458$	−48.2774	−2.25585
$459$	0	0
$460$	116.702	5.44125
$461$	31.0987	1.44841	0.724205	−	0.689584i	$-0.242207\pi$
$461$	31.0987	1.44841	0.724205	+	0.689584i	$0.242207\pi$
$462$	0	0
$463$	40.3985	1.87748	0.938740	−	0.344627i	$-0.111994\pi$
$463$	40.3985	1.87748	0.938740	+	0.344627i	$0.111994\pi$
$464$	70.9835	3.29533
$465$	0	0
$466$	−15.1674	−0.702616
$467$	19.8975	0.920744	0.460372	−	0.887726i	$-0.347716\pi$
$467$	19.8975	0.920744	0.460372	+	0.887726i	$0.347716\pi$
$468$	0	0
$469$	7.89445	0.364532
$470$	124.203	5.72906
$471$	0	0
$472$	51.5334	2.37202
$473$	−5.04075	−0.231774
$474$	0	0
$475$	−18.3913	−0.843851
$476$	−19.7866	−0.906916
$477$	0	0
$478$	60.1482	2.75111
$479$	4.39409	0.200771	0.100386	−	0.994949i	$-0.467992\pi$
$479$	4.39409	0.200771	0.100386	+	0.994949i	$0.467992\pi$
$480$	0	0
$481$	−31.1596	−1.42076
$482$	−13.8002	−0.628580
$483$	0	0
$484$	5.44452	0.247478
$485$	−29.8851	−1.35701
$486$	0	0
$487$	−2.92326	−0.132465	−0.0662327	−	0.997804i	$-0.521098\pi$
$487$	−2.92326	−0.132465	−0.0662327	+	0.997804i	$0.521098\pi$
$488$	9.39826	0.425439
$489$	0	0
$490$	68.7912	3.10767
$491$	3.68546	0.166323	0.0831613	−	0.996536i	$-0.473498\pi$
$491$	3.68546	0.166323	0.0831613	+	0.996536i	$0.473498\pi$
$492$	0	0
$493$	−24.7169	−1.11319
$494$	−25.4255	−1.14395
$495$	0	0
$496$	−57.2675	−2.57139
$497$	5.72849	0.256958
$498$	0	0
$499$	−32.5163	−1.45563	−0.727815	−	0.685773i	$-0.759464\pi$
$499$	−32.5163	−1.45563	−0.727815	+	0.685773i	$0.759464\pi$
$500$	106.599	4.76723
$501$	0	0
$502$	−2.98128	−0.133061
$503$	−23.4874	−1.04725	−0.523626	−	0.851948i	$-0.675421\pi$
$503$	−23.4874	−1.04725	−0.523626	+	0.851948i	$0.675421\pi$
$504$	0	0
$505$	20.1054	0.894681
$506$	15.0767	0.670241
$507$	0	0
$508$	0.210404	0.00933518
$509$	0.720443	0.0319331	0.0159665	−	0.999873i	$-0.494917\pi$
$509$	0.720443	0.0319331	0.0159665	+	0.999873i	$0.494917\pi$
$510$	0	0
$511$	−0.683292	−0.0302270
$512$	−39.2766	−1.73580
$513$	0	0
$514$	22.0903	0.974363
$515$	54.6905	2.40995
$516$	0	0
$517$	11.7350	0.516106
$518$	−11.8139	−0.519071
$519$	0	0
$520$	185.595	8.13887
$521$	7.41764	0.324973	0.162486	−	0.986711i	$-0.448049\pi$
$521$	7.41764	0.324973	0.162486	+	0.986711i	$0.448049\pi$
$522$	0	0
$523$	8.08243	0.353420	0.176710	−	0.984263i	$-0.443455\pi$
$523$	8.08243	0.353420	0.176710	+	0.984263i	$0.443455\pi$
$524$	−90.8610	−3.96928
$525$	0	0
$526$	59.8642	2.61020
$527$	19.9409	0.868641
$528$	0	0
$529$	7.53346	0.327542
$530$	−50.5260	−2.19471
$531$	0	0
$532$	−7.05005	−0.305658
$533$	−34.5474	−1.49641
$534$	0	0
$535$	−6.38474	−0.276036
$536$	−104.881	−4.53019
$537$	0	0
$538$	−76.5259	−3.29926
$539$	6.49957	0.279956
$540$	0	0
$541$	16.6004	0.713707	0.356853	−	0.934160i	$-0.383850\pi$
$541$	16.6004	0.713707	0.356853	+	0.934160i	$0.383850\pi$
$542$	−14.7662	−0.634262
$543$	0	0
$544$	110.241	4.72654
$545$	44.4894	1.90572
$546$	0	0
$547$	−5.15103	−0.220242	−0.110121	−	0.993918i	$-0.535124\pi$
$547$	−5.15103	−0.220242	−0.110121	+	0.993918i	$0.535124\pi$
$548$	−109.779	−4.68952
$549$	0	0
$550$	27.4138	1.16893
$551$	−8.80675	−0.375180
$552$	0	0
$553$	5.63537	0.239640
$554$	12.0575	0.512274
$555$	0	0
$556$	22.2452	0.943405
$557$	11.7383	0.497366	0.248683	−	0.968585i	$-0.420002\pi$
$557$	11.7383	0.497366	0.248683	+	0.968585i	$0.420002\pi$
$558$	0	0
$559$	−25.6616	−1.08537
$560$	40.4859	1.71084
$561$	0	0
$562$	−37.4767	−1.58086
$563$	1.30438	0.0549732	0.0274866	−	0.999622i	$-0.491250\pi$
$563$	1.30438	0.0549732	0.0274866	+	0.999622i	$0.491250\pi$
$564$	0	0
$565$	−51.1343	−2.15124
$566$	−32.5858	−1.36968
$567$	0	0
$568$	−76.1057	−3.19332
$569$	5.14844	0.215834	0.107917	−	0.994160i	$-0.465582\pi$
$569$	5.14844	0.215834	0.107917	+	0.994160i	$0.465582\pi$
$570$	0	0
$571$	−25.4392	−1.06460	−0.532298	−	0.846557i	$-0.678671\pi$
$571$	−25.4392	−1.06460	−0.532298	+	0.846557i	$0.678671\pi$
$572$	27.7172	1.15891
$573$	0	0
$574$	−13.0983	−0.546713
$575$	55.5186	2.31529
$576$	0	0
$577$	−9.02892	−0.375879	−0.187940	−	0.982181i	$-0.560181\pi$
$577$	−9.02892	−0.375879	−0.187940	+	0.982181i	$0.560181\pi$
$578$	−25.6271	−1.06595
$579$	0	0
$580$	101.612	4.21919
$581$	1.95298	0.0810231
$582$	0	0
$583$	−4.77383	−0.197712
$584$	9.07785	0.375644
$585$	0	0
$586$	−32.0391	−1.32352
$587$	−7.74587	−0.319706	−0.159853	−	0.987141i	$-0.551102\pi$
$587$	−7.74587	−0.319706	−0.159853	+	0.987141i	$0.551102\pi$
$588$	0	0
$589$	7.10505	0.292758
$590$	58.0349	2.38926
$591$	0	0
$592$	90.3039	3.71147
$593$	−14.8346	−0.609185	−0.304592	−	0.952483i	$-0.598520\pi$
$593$	−14.8346	−0.609185	−0.304592	+	0.952483i	$0.598520\pi$
$594$	0	0
$595$	−14.0974	−0.577939
$596$	111.496	4.56707
$597$	0	0
$598$	76.7530	3.13866
$599$	5.34341	0.218326	0.109163	−	0.994024i	$-0.465183\pi$
$599$	5.34341	0.218326	0.109163	+	0.994024i	$0.465183\pi$
$600$	0	0
$601$	−14.5806	−0.594757	−0.297378	−	0.954760i	$-0.596112\pi$
$601$	−14.5806	−0.594757	−0.297378	+	0.954760i	$0.596112\pi$
$602$	−9.72935	−0.396539
$603$	0	0
$604$	115.735	4.70920
$605$	3.87909	0.157707
$606$	0	0
$607$	39.4166	1.59987	0.799935	−	0.600087i	$-0.204867\pi$
$607$	39.4166	1.59987	0.799935	+	0.600087i	$0.204867\pi$
$608$	39.2794	1.59299
$609$	0	0
$610$	10.5840	0.428532
$611$	59.7411	2.41687
$612$	0	0
$613$	23.2387	0.938604	0.469302	−	0.883038i	$-0.344506\pi$
$613$	23.2387	0.938604	0.469302	+	0.883038i	$0.344506\pi$
$614$	−15.1039	−0.609542
$615$	0	0
$616$	6.64841	0.267872
$617$	31.5429	1.26987	0.634934	−	0.772566i	$-0.281027\pi$
$617$	31.5429	1.26987	0.634934	+	0.772566i	$0.281027\pi$
$618$	0	0
$619$	−1.16661	−0.0468899	−0.0234449	−	0.999725i	$-0.507463\pi$
$619$	−1.16661	−0.0468899	−0.0234449	+	0.999725i	$0.507463\pi$
$620$	−81.9775	−3.29230
$621$	0	0
$622$	−19.4240	−0.778830
$623$	4.16932	0.167040
$624$	0	0
$625$	25.7122	1.02849
$626$	46.3185	1.85126
$627$	0	0
$628$	−47.3508	−1.88950
$629$	−31.4444	−1.25377
$630$	0	0
$631$	0.0580526	0.00231104	0.00115552	−	0.999999i	$-0.499632\pi$
$631$	0.0580526	0.00231104	0.00115552	+	0.999999i	$0.499632\pi$
$632$	−74.8686	−2.97811
$633$	0	0
$634$	−33.7442	−1.34015
$635$	0.149908	0.00594892
$636$	0	0
$637$	33.0882	1.31100
$638$	13.1272	0.519711
$639$	0	0
$640$	−140.895	−5.56938
$641$	32.9927	1.30313	0.651566	−	0.758592i	$-0.274112\pi$
$641$	32.9927	1.30313	0.651566	+	0.758592i	$0.274112\pi$
$642$	0	0
$643$	−2.89633	−0.114220	−0.0571101	−	0.998368i	$-0.518189\pi$
$643$	−2.89633	−0.114220	−0.0571101	+	0.998368i	$0.518189\pi$
$644$	21.2823	0.838639
$645$	0	0
$646$	−25.6579	−1.00950
$647$	−11.3565	−0.446471	−0.223235	−	0.974765i	$-0.571662\pi$
$647$	−11.3565	−0.446471	−0.223235	+	0.974765i	$0.571662\pi$
$648$	0	0
$649$	5.48329	0.215238
$650$	139.559	5.47395
$651$	0	0
$652$	21.0763	0.825411
$653$	7.17070	0.280611	0.140305	−	0.990108i	$-0.455192\pi$
$653$	7.17070	0.280611	0.140305	+	0.990108i	$0.455192\pi$
$654$	0	0
$655$	−64.7362	−2.52945
$656$	100.122	3.90911
$657$	0	0
$658$	22.6503	0.882999
$659$	−21.2711	−0.828606	−0.414303	−	0.910139i	$-0.635975\pi$
$659$	−21.2711	−0.828606	−0.414303	+	0.910139i	$0.635975\pi$
$660$	0	0
$661$	7.92457	0.308230	0.154115	−	0.988053i	$-0.450747\pi$
$661$	7.92457	0.308230	0.154115	+	0.988053i	$0.450747\pi$
$662$	−7.20078	−0.279866
$663$	0	0
$664$	−25.9462	−1.00691
$665$	−5.02299	−0.194783
$666$	0	0
$667$	26.5853	1.02939
$668$	−32.2606	−1.24820
$669$	0	0
$670$	−118.113	−4.56312
$671$	1.00000	0.0386046
$672$	0	0
$673$	−10.8709	−0.419044	−0.209522	−	0.977804i	$-0.567191\pi$
$673$	−10.8709	−0.419044	−0.209522	+	0.977804i	$0.567191\pi$
$674$	−8.92226	−0.343673
$675$	0	0
$676$	70.3246	2.70479
$677$	−37.5670	−1.44382	−0.721908	−	0.691989i	$-0.756734\pi$
$677$	−37.5670	−1.44382	−0.721908	+	0.691989i	$0.756734\pi$
$678$	0	0
$679$	−5.44999	−0.209152
$680$	187.291	7.18229
$681$	0	0
$682$	−10.5907	−0.405538
$683$	7.92068	0.303076	0.151538	−	0.988451i	$-0.451577\pi$
$683$	7.92068	0.303076	0.151538	+	0.988451i	$0.451577\pi$
$684$	0	0
$685$	−78.2148	−2.98843
$686$	26.0561	0.994825
$687$	0	0
$688$	74.3701	2.83533
$689$	−24.3028	−0.925862
$690$	0	0
$691$	−17.5723	−0.668482	−0.334241	−	0.942488i	$-0.608480\pi$
$691$	−17.5723	−0.668482	−0.334241	+	0.942488i	$0.608480\pi$
$692$	70.1075	2.66509
$693$	0	0
$694$	−2.84167	−0.107868
$695$	15.8491	0.601192
$696$	0	0
$697$	−34.8632	−1.32054
$698$	−58.7432	−2.22346
$699$	0	0
$700$	38.6973	1.46262
$701$	43.4781	1.64215	0.821073	−	0.570824i	$-0.193376\pi$
$701$	43.4781	1.64215	0.821073	+	0.570824i	$0.193376\pi$
$702$	0	0
$703$	−11.2038	−0.422559
$704$	−29.0416	−1.09455
$705$	0	0
$706$	−8.32297	−0.313239
$707$	3.66652	0.137894
$708$	0	0
$709$	21.3072	0.800210	0.400105	−	0.916469i	$-0.368974\pi$
$709$	21.3072	0.800210	0.400105	+	0.916469i	$0.368974\pi$
$710$	−85.7073	−3.21654
$711$	0	0
$712$	−55.3913	−2.07588
$713$	−21.4483	−0.803246
$714$	0	0
$715$	19.7478	0.738525
$716$	124.089	4.63741
$717$	0	0
$718$	2.09243	0.0780888
$719$	10.2606	0.382657	0.191328	−	0.981526i	$-0.438720\pi$
$719$	10.2606	0.382657	0.191328	+	0.981526i	$0.438720\pi$
$720$	0	0
$721$	9.97361	0.371437
$722$	42.6988	1.58909
$723$	0	0
$724$	82.6368	3.07117
$725$	48.3398	1.79529
$726$	0	0
$727$	3.29112	0.122061	0.0610305	−	0.998136i	$-0.480561\pi$
$727$	3.29112	0.122061	0.0610305	+	0.998136i	$0.480561\pi$
$728$	33.8459	1.25441
$729$	0	0
$730$	10.2231	0.378375
$731$	−25.8962	−0.957804
$732$	0	0
$733$	−0.246863	−0.00911810	−0.00455905	−	0.999990i	$-0.501451\pi$
$733$	−0.246863	−0.00911810	−0.00455905	+	0.999990i	$0.501451\pi$
$734$	28.7136	1.05984
$735$	0	0
$736$	−118.574	−4.37071
$737$	−11.1597	−0.411072
$738$	0	0
$739$	−26.4887	−0.974402	−0.487201	−	0.873290i	$-0.661982\pi$
$739$	−26.4887	−0.974402	−0.487201	+	0.873290i	$0.661982\pi$
$740$	129.268	4.75200
$741$	0	0
$742$	−9.21416	−0.338262
$743$	12.4409	0.456412	0.228206	−	0.973613i	$-0.426714\pi$
$743$	12.4409	0.456412	0.228206	+	0.973613i	$0.426714\pi$
$744$	0	0
$745$	79.4385	2.91040
$746$	86.5522	3.16890
$747$	0	0
$748$	27.9705	1.02270
$749$	−1.16435	−0.0425445
$750$	0	0
$751$	−19.2062	−0.700843	−0.350421	−	0.936592i	$-0.613962\pi$
$751$	−19.2062	−0.700843	−0.350421	+	0.936592i	$0.613962\pi$
$752$	−173.136	−6.31362
$753$	0	0
$754$	66.8284	2.43375
$755$	82.4585	3.00097
$756$	0	0
$757$	20.2783	0.737026	0.368513	−	0.929623i	$-0.379867\pi$
$757$	20.2783	0.737026	0.368513	+	0.929623i	$0.379867\pi$
$758$	38.5343	1.39963
$759$	0	0
$760$	66.7327	2.42065
$761$	−32.6802	−1.18466	−0.592328	−	0.805697i	$-0.701791\pi$
$761$	−32.6802	−1.18466	−0.592328	+	0.805697i	$0.701791\pi$
$762$	0	0
$763$	8.11329	0.293721
$764$	87.2323	3.15595
$765$	0	0
$766$	48.6555	1.75799
$767$	27.9145	1.00794
$768$	0	0
$769$	30.0167	1.08243	0.541216	−	0.840884i	$-0.317964\pi$
$769$	30.0167	1.08243	0.541216	+	0.840884i	$0.317964\pi$
$770$	7.48718	0.269819
$771$	0	0
$772$	100.025	3.59998
$773$	11.1099	0.399596	0.199798	−	0.979837i	$-0.435971\pi$
$773$	11.1099	0.399596	0.199798	+	0.979837i	$0.435971\pi$
$774$	0	0
$775$	−38.9992	−1.40089
$776$	72.4057	2.59921
$777$	0	0
$778$	55.9409	2.00558
$779$	−12.4219	−0.445061
$780$	0	0
$781$	−8.09785	−0.289764
$782$	77.4545	2.76977
$783$	0	0
$784$	−95.8933	−3.42476
$785$	−33.7363	−1.20410
$786$	0	0
$787$	5.81615	0.207323	0.103662	−	0.994613i	$-0.466944\pi$
$787$	5.81615	0.207323	0.103662	+	0.994613i	$0.466944\pi$
$788$	87.3608	3.11210
$789$	0	0
$790$	−84.3142	−2.99976
$791$	−9.32509	−0.331562
$792$	0	0
$793$	5.09083	0.180781
$794$	75.5521	2.68125
$795$	0	0
$796$	−51.5993	−1.82889
$797$	−15.5187	−0.549702	−0.274851	−	0.961487i	$-0.588628\pi$
$797$	−15.5187	−0.549702	−0.274851	+	0.961487i	$0.588628\pi$
$798$	0	0
$799$	60.2871	2.13281
$800$	−215.602	−7.62269
$801$	0	0
$802$	−100.153	−3.53651
$803$	0.965908	0.0340861
$804$	0	0
$805$	15.1631	0.534429
$806$	−53.9153	−1.89909
$807$	0	0
$808$	−48.7115	−1.71366
$809$	−25.9551	−0.912533	−0.456266	−	0.889843i	$-0.650814\pi$
$809$	−25.9551	−0.912533	−0.456266	+	0.889843i	$0.650814\pi$
$810$	0	0
$811$	−20.7430	−0.728385	−0.364192	−	0.931324i	$-0.618655\pi$
$811$	−20.7430	−0.728385	−0.364192	+	0.931324i	$0.618655\pi$
$812$	18.5304	0.650289
$813$	0	0
$814$	16.7002	0.585341
$815$	15.0163	0.525999
$816$	0	0
$817$	−9.22692	−0.322809
$818$	−70.2149	−2.45501
$819$	0	0
$820$	143.323	5.00506
$821$	−25.4058	−0.886668	−0.443334	−	0.896357i	$-0.646204\pi$
$821$	−25.4058	−0.886668	−0.443334	+	0.896357i	$0.646204\pi$
$822$	0	0
$823$	−13.8440	−0.482572	−0.241286	−	0.970454i	$-0.577569\pi$
$823$	−13.8440	−0.482572	−0.241286	+	0.970454i	$0.577569\pi$
$824$	−132.504	−4.61600
$825$	0	0
$826$	10.5835	0.368248
$827$	−6.88147	−0.239292	−0.119646	−	0.992817i	$-0.538176\pi$
$827$	−6.88147	−0.239292	−0.119646	+	0.992817i	$0.538176\pi$
$828$	0	0
$829$	9.56690	0.332272	0.166136	−	0.986103i	$-0.446871\pi$
$829$	9.56690	0.332272	0.166136	+	0.986103i	$0.446871\pi$
$830$	−29.2196	−1.01423
$831$	0	0
$832$	−147.846	−5.12564
$833$	33.3907	1.15692
$834$	0	0
$835$	−22.9849	−0.795424
$836$	9.96602	0.344682
$837$	0	0
$838$	32.9205	1.13722
$839$	33.1619	1.14488	0.572439	−	0.819947i	$-0.305997\pi$
$839$	33.1619	1.14488	0.572439	+	0.819947i	$0.305997\pi$
$840$	0	0
$841$	−5.85229	−0.201803
$842$	−59.7328	−2.05853
$843$	0	0
$844$	−90.0714	−3.10039
$845$	50.1045	1.72365
$846$	0	0
$847$	0.707409	0.0243069
$848$	70.4320	2.41865
$849$	0	0
$850$	140.835	4.83059
$851$	33.8214	1.15938
$852$	0	0
$853$	−3.24963	−0.111265	−0.0556327	−	0.998451i	$-0.517718\pi$
$853$	−3.24963	−0.111265	−0.0556327	+	0.998451i	$0.517718\pi$
$854$	1.93014	0.0660480
$855$	0	0
$856$	15.4690	0.528718
$857$	−56.4139	−1.92706	−0.963531	−	0.267598i	$-0.913770\pi$
$857$	−56.4139	−1.92706	−0.963531	+	0.267598i	$0.913770\pi$
$858$	0	0
$859$	−22.9734	−0.783842	−0.391921	−	0.919999i	$-0.628189\pi$
$859$	−22.9734	−0.783842	−0.391921	+	0.919999i	$0.628189\pi$
$860$	106.460	3.63024
$861$	0	0
$862$	49.0901	1.67202
$863$	8.44250	0.287386	0.143693	−	0.989622i	$-0.454102\pi$
$863$	8.44250	0.287386	0.143693	+	0.989622i	$0.454102\pi$
$864$	0	0
$865$	49.9498	1.69835
$866$	12.3413	0.419376
$867$	0	0
$868$	−14.9498	−0.507429
$869$	−7.96622	−0.270236
$870$	0	0
$871$	−56.8120	−1.92500
$872$	−107.789	−3.65019
$873$	0	0
$874$	27.5974	0.933496
$875$	13.8504	0.468228
$876$	0	0
$877$	−31.4678	−1.06259	−0.531296	−	0.847186i	$-0.678295\pi$
$877$	−31.4678	−1.06259	−0.531296	+	0.847186i	$0.678295\pi$
$878$	−96.5785	−3.25937
$879$	0	0
$880$	−57.2312	−1.92926
$881$	28.0896	0.946362	0.473181	−	0.880965i	$-0.343106\pi$
$881$	28.0896	0.946362	0.473181	+	0.880965i	$0.343106\pi$
$882$	0	0
$883$	−14.9728	−0.503875	−0.251937	−	0.967744i	$-0.581068\pi$
$883$	−14.9728	−0.503875	−0.251937	+	0.967744i	$0.581068\pi$
$884$	142.393	4.78920
$885$	0	0
$886$	−86.2817	−2.89869
$887$	0.675570	0.0226834	0.0113417	−	0.999936i	$-0.496390\pi$
$887$	0.675570	0.0226834	0.0113417	+	0.999936i	$0.496390\pi$
$888$	0	0
$889$	0.0273379	0.000916884 0
$890$	−62.3796	−2.09097
$891$	0	0
$892$	−136.335	−4.56485
$893$	21.4806	0.718820
$894$	0	0
$895$	88.4101	2.95522
$896$	−25.6943	−0.858388
$897$	0	0
$898$	54.8982	1.83198
$899$	−18.6749	−0.622844
$900$	0	0
$901$	−24.5249	−0.817043
$902$	18.5159	0.616513
$903$	0	0
$904$	123.888	4.12046
$905$	58.8767	1.95713
$906$	0	0
$907$	−38.2733	−1.27084	−0.635421	−	0.772166i	$-0.719174\pi$
$907$	−38.2733	−1.27084	−0.635421	+	0.772166i	$0.719174\pi$
$908$	94.3559	3.13131
$909$	0	0
$910$	38.1160	1.26353
$911$	−2.79310	−0.0925394	−0.0462697	−	0.998929i	$-0.514733\pi$
$911$	−2.79310	−0.0925394	−0.0462697	+	0.998929i	$0.514733\pi$
$912$	0	0
$913$	−2.76074	−0.0913673
$914$	−51.2833	−1.69630
$915$	0	0
$916$	96.3352	3.18300
$917$	−11.8056	−0.389855
$918$	0	0
$919$	−14.6288	−0.482559	−0.241279	−	0.970456i	$-0.577567\pi$
$919$	−14.6288	−0.482559	−0.241279	+	0.970456i	$0.577567\pi$
$920$	−201.449	−6.64158
$921$	0	0
$922$	−84.8517	−2.79444
$923$	−41.2248	−1.35693
$924$	0	0
$925$	61.4970	2.02201
$926$	−110.226	−3.62225
$927$	0	0
$928$	−103.242	−3.38909
$929$	−38.8486	−1.27458	−0.637291	−	0.770623i	$-0.719945\pi$
$929$	−38.8486	−1.27458	−0.637291	+	0.770623i	$0.719945\pi$
$930$	0	0
$931$	11.8973	0.389917
$932$	30.2658	0.991390
$933$	0	0
$934$	−54.2895	−1.77641
$935$	19.9283	0.651725
$936$	0	0
$937$	14.7171	0.480787	0.240394	−	0.970676i	$-0.422724\pi$
$937$	14.7171	0.480787	0.240394	+	0.970676i	$0.422724\pi$
$938$	−21.5397	−0.703297
$939$	0	0
$940$	−247.841	−8.08369
$941$	−15.2801	−0.498119	−0.249059	−	0.968488i	$-0.580121\pi$
$941$	−15.2801	−0.498119	−0.249059	+	0.968488i	$0.580121\pi$
$942$	0	0
$943$	37.4986	1.22112
$944$	−80.8993	−2.63305
$945$	0	0
$946$	13.7535	0.447165
$947$	−24.6226	−0.800128	−0.400064	−	0.916487i	$-0.631012\pi$
$947$	−24.6226	−0.800128	−0.400064	+	0.916487i	$0.631012\pi$
$948$	0	0
$949$	4.91727	0.159621
$950$	50.1800	1.62806
$951$	0	0
$952$	34.1553	1.10698
$953$	13.0438	0.422529	0.211264	−	0.977429i	$-0.432242\pi$
$953$	13.0438	0.422529	0.211264	+	0.977429i	$0.432242\pi$
$954$	0	0
$955$	62.1509	2.01116
$956$	−120.023	−3.88182
$957$	0	0
$958$	−11.9891	−0.387351
$959$	−14.2636	−0.460596
$960$	0	0
$961$	−15.9336	−0.513986
$962$	85.0179	2.74109
$963$	0	0
$964$	27.5376	0.886925
$965$	71.2654	2.29411
$966$	0	0
$967$	−54.9810	−1.76807	−0.884034	−	0.467422i	$-0.845183\pi$
$967$	−54.9810	−1.76807	−0.884034	+	0.467422i	$0.845183\pi$
$968$	−9.39826	−0.302072
$969$	0	0
$970$	81.5406	2.61811
$971$	26.5402	0.851714	0.425857	−	0.904790i	$-0.359973\pi$
$971$	26.5402	0.851714	0.425857	+	0.904790i	$0.359973\pi$
$972$	0	0
$973$	2.89032	0.0926595
$974$	7.97600	0.255568
$975$	0	0
$976$	−14.7538	−0.472257
$977$	−32.6259	−1.04379	−0.521897	−	0.853008i	$-0.674776\pi$
$977$	−32.6259	−1.04379	−0.521897	+	0.853008i	$0.674776\pi$
$978$	0	0
$979$	−5.89379	−0.188366
$980$	−137.270	−4.38492
$981$	0	0
$982$	−10.0557	−0.320889
$983$	−37.2996	−1.18967	−0.594836	−	0.803847i	$-0.702783\pi$
$983$	−37.2996	−1.18967	−0.594836	+	0.803847i	$0.702783\pi$
$984$	0	0
$985$	62.2425	1.98321
$986$	67.4393	2.14770
$987$	0	0
$988$	50.7353	1.61411
$989$	27.8537	0.885696
$990$	0	0
$991$	44.0468	1.39919	0.699596	−	0.714538i	$-0.253363\pi$
$991$	44.0468	1.39919	0.699596	+	0.714538i	$0.253363\pi$
$992$	83.2928	2.64455
$993$	0	0
$994$	−15.6300	−0.495753
$995$	−36.7632	−1.16547
$996$	0	0
$997$	12.8623	0.407354	0.203677	−	0.979038i	$-0.434711\pi$
$997$	12.8623	0.407354	0.203677	+	0.979038i	$0.434711\pi$
$998$	88.7197	2.80837
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.1		19
3.2	odd	2		671.2.a.c.1.19	✓	19
33.32	even	2		7381.2.a.i.1.1		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.19	✓	19	3.2	odd	2
6039.2.a.k.1.1		19	1.1	even	1	trivial
7381.2.a.i.1.1		19	33.32	even	2

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)

\(f(q)\)	\(=\)	\(q-2.72847 q^{2} +5.44452 q^{4} +3.87909 q^{5} +0.707409 q^{7} -9.39826 q^{8} +O(q^{10})\)	\(q-2.72847 q^{2} +5.44452 q^{4} +3.87909 q^{5} +0.707409 q^{7} -9.39826 q^{8} -10.5840 q^{10} -1.00000 q^{11} -5.09083 q^{13} -1.93014 q^{14} +14.7538 q^{16} -5.13736 q^{17} -1.83047 q^{19} +21.1198 q^{20} +2.72847 q^{22} +5.52571 q^{23} +10.0473 q^{25} +13.8902 q^{26} +3.85150 q^{28} +4.81121 q^{29} -3.88155 q^{31} -21.4587 q^{32} +14.0171 q^{34} +2.74410 q^{35} +6.12073 q^{37} +4.99436 q^{38} -36.4567 q^{40} +6.78620 q^{41} +5.04075 q^{43} -5.44452 q^{44} -15.0767 q^{46} -11.7350 q^{47} -6.49957 q^{49} -27.4138 q^{50} -27.7172 q^{52} +4.77383 q^{53} -3.87909 q^{55} -6.64841 q^{56} -13.1272 q^{58} -5.48329 q^{59} -1.00000 q^{61} +10.5907 q^{62} +29.0416 q^{64} -19.7478 q^{65} +11.1597 q^{67} -27.9705 q^{68} -7.48718 q^{70} +8.09785 q^{71} -0.965908 q^{73} -16.7002 q^{74} -9.96602 q^{76} -0.707409 q^{77} +7.96622 q^{79} +57.2312 q^{80} -18.5159 q^{82} +2.76074 q^{83} -19.9283 q^{85} -13.7535 q^{86} +9.39826 q^{88} +5.89379 q^{89} -3.60130 q^{91} +30.0848 q^{92} +32.0186 q^{94} -7.10054 q^{95} -7.70416 q^{97} +17.7339 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more