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Properties

Label 289.10.a.f.1.13
Level $289$
Weight $10$
Character 289.1
Self dual yes
Analytic conductor $148.845$
Analytic rank $1$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [289,10,Mod(1,289)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("289.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(289, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 289.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.42963 q^{2} -213.375 q^{3} -456.801 q^{4} +298.376 q^{5} -1585.29 q^{6} -7723.69 q^{7} -7197.83 q^{8} +25845.7 q^{9} +2216.82 q^{10} -9286.87 q^{11} +97469.6 q^{12} -122491. q^{13} -57384.1 q^{14} -63665.7 q^{15} +180405. q^{16} +192024. q^{18} -464145. q^{19} -136298. q^{20} +1.64804e6 q^{21} -68998.0 q^{22} -1.17349e6 q^{23} +1.53583e6 q^{24} -1.86410e6 q^{25} -910061. q^{26} -1.31496e6 q^{27} +3.52819e6 q^{28} +1.31507e6 q^{29} -473013. q^{30} +7.45472e6 q^{31} +5.02563e6 q^{32} +1.98158e6 q^{33} -2.30456e6 q^{35} -1.18063e7 q^{36} -1.57683e6 q^{37} -3.44843e6 q^{38} +2.61364e7 q^{39} -2.14766e6 q^{40} +2.43826e7 q^{41} +1.22443e7 q^{42} +7.29437e6 q^{43} +4.24225e6 q^{44} +7.71172e6 q^{45} -8.71858e6 q^{46} +5.46941e7 q^{47} -3.84938e7 q^{48} +1.93017e7 q^{49} -1.38496e7 q^{50} +5.59538e7 q^{52} +6.10696e7 q^{53} -9.76967e6 q^{54} -2.77097e6 q^{55} +5.55938e7 q^{56} +9.90368e7 q^{57} +9.77051e6 q^{58} -8.90678e7 q^{59} +2.90825e7 q^{60} +2.63450e7 q^{61} +5.53858e7 q^{62} -1.99624e8 q^{63} -5.50286e7 q^{64} -3.65482e7 q^{65} +1.47224e7 q^{66} -2.68240e8 q^{67} +2.50392e8 q^{69} -1.71220e7 q^{70} -3.93211e8 q^{71} -1.86033e8 q^{72} +2.80165e8 q^{73} -1.17152e7 q^{74} +3.97751e8 q^{75} +2.12022e8 q^{76} +7.17289e7 q^{77} +1.94184e8 q^{78} +6.07464e8 q^{79} +5.38283e7 q^{80} -2.28142e8 q^{81} +1.81154e8 q^{82} +5.22381e8 q^{83} -7.52825e8 q^{84} +5.41945e7 q^{86} -2.80603e8 q^{87} +6.68453e7 q^{88} +2.71477e8 q^{89} +5.72952e7 q^{90} +9.46080e8 q^{91} +5.36050e8 q^{92} -1.59065e9 q^{93} +4.06357e8 q^{94} -1.38490e8 q^{95} -1.07234e9 q^{96} +8.85422e8 q^{97} +1.43405e8 q^{98} -2.40026e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 5124 q^{4} + 108368 q^{9} - 244832 q^{13} - 127332 q^{15} - 279932 q^{16} + 888764 q^{18} - 2280212 q^{19} + 775748 q^{21} - 2762628 q^{25} - 3334452 q^{26} - 39084792 q^{30} + 2010240 q^{32} - 30349992 q^{33}+ \cdots + 4413444720 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.42963 0.328346 0.164173 − 0.986432i \(-0.447504\pi\)
0.164173 + 0.986432i \(0.447504\pi\)
\(3\) −213.375 −1.52089 −0.760443 − 0.649404i \(-0.775018\pi\)
−0.760443 + 0.649404i \(0.775018\pi\)
\(4\) −456.801 −0.892189
\(5\) 298.376 0.213500 0.106750 − 0.994286i \(-0.465956\pi\)
0.106750 + 0.994286i \(0.465956\pi\)
\(6\) −1585.29 −0.499378
\(7\) −7723.69 −1.21586 −0.607930 − 0.793991i \(-0.708000\pi\)
−0.607930 + 0.793991i \(0.708000\pi\)
\(8\) −7197.83 −0.621293
\(9\) 25845.7 1.31310
\(10\) 2216.82 0.0701020
\(11\) −9286.87 −0.191250 −0.0956252 − 0.995417i \(-0.530485\pi\)
−0.0956252 + 0.995417i \(0.530485\pi\)
\(12\) 97469.6 1.35692
\(13\) −122491. −1.18948 −0.594741 − 0.803917i \(-0.702746\pi\)
−0.594741 + 0.803917i \(0.702746\pi\)
\(14\) −57384.1 −0.399223
\(15\) −63665.7 −0.324710
\(16\) 180405. 0.688189
\(17\) 0 0
\(18\) 192024. 0.431151
\(19\) −464145. −0.817077 −0.408538 − 0.912741i \(-0.633961\pi\)
−0.408538 + 0.912741i \(0.633961\pi\)
\(20\) −136298. −0.190482
\(21\) 1.64804e6 1.84919
\(22\) −68998.0 −0.0627964
\(23\) −1.17349e6 −0.874386 −0.437193 − 0.899368i \(-0.644027\pi\)
−0.437193 + 0.899368i \(0.644027\pi\)
\(24\) 1.53583e6 0.944917
\(25\) −1.86410e6 −0.954418
\(26\) −910061. −0.390562
\(27\) −1.31496e6 −0.476185
\(28\) 3.52819e6 1.08478
\(29\) 1.31507e6 0.345270 0.172635 − 0.984986i \(-0.444772\pi\)
0.172635 + 0.984986i \(0.444772\pi\)
\(30\) −473013. −0.106617
\(31\) 7.45472e6 1.44979 0.724893 − 0.688861i \(-0.241889\pi\)
0.724893 + 0.688861i \(0.241889\pi\)
\(32\) 5.02563e6 0.847258
\(33\) 1.98158e6 0.290870
\(34\) 0 0
\(35\) −2.30456e6 −0.259586
\(36\) −1.18063e7 −1.17153
\(37\) −1.57683e6 −0.138317 −0.0691585 − 0.997606i \(-0.522031\pi\)
−0.0691585 + 0.997606i \(0.522031\pi\)
\(38\) −3.44843e6 −0.268284
\(39\) 2.61364e7 1.80907
\(40\) −2.14766e6 −0.132646
\(41\) 2.43826e7 1.34758 0.673788 − 0.738925i \(-0.264666\pi\)
0.673788 + 0.738925i \(0.264666\pi\)
\(42\) 1.22443e7 0.607173
\(43\) 7.29437e6 0.325372 0.162686 − 0.986678i \(-0.447984\pi\)
0.162686 + 0.986678i \(0.447984\pi\)
\(44\) 4.24225e6 0.170631
\(45\) 7.71172e6 0.280346
\(46\) −8.71858e6 −0.287102
\(47\) 5.46941e7 1.63493 0.817466 − 0.575977i \(-0.195378\pi\)
0.817466 + 0.575977i \(0.195378\pi\)
\(48\) −3.84938e7 −1.04666
\(49\) 1.93017e7 0.478315
\(50\) −1.38496e7 −0.313380
\(51\) 0 0
\(52\) 5.59538e7 1.06124
\(53\) 6.10696e7 1.06312 0.531561 − 0.847020i \(-0.321606\pi\)
0.531561 + 0.847020i \(0.321606\pi\)
\(54\) −9.76967e6 −0.156354
\(55\) −2.77097e6 −0.0408320
\(56\) 5.55938e7 0.755406
\(57\) 9.90368e7 1.24268
\(58\) 9.77051e6 0.113368
\(59\) −8.90678e7 −0.956944 −0.478472 − 0.878103i \(-0.658809\pi\)
−0.478472 + 0.878103i \(0.658809\pi\)
\(60\) 2.90825e7 0.289702
\(61\) 2.63450e7 0.243621 0.121810 − 0.992553i \(-0.461130\pi\)
0.121810 + 0.992553i \(0.461130\pi\)
\(62\) 5.53858e7 0.476032
\(63\) −1.99624e8 −1.59654
\(64\) −5.50286e7 −0.409995
\(65\) −3.65482e7 −0.253955
\(66\) 1.47224e7 0.0955062
\(67\) −2.68240e8 −1.62625 −0.813124 − 0.582091i \(-0.802235\pi\)
−0.813124 + 0.582091i \(0.802235\pi\)
\(68\) 0 0
\(69\) 2.50392e8 1.32984
\(70\) −1.71220e7 −0.0852342
\(71\) −3.93211e8 −1.83638 −0.918191 − 0.396139i \(-0.870350\pi\)
−0.918191 + 0.396139i \(0.870350\pi\)
\(72\) −1.86033e8 −0.815818
\(73\) 2.80165e8 1.15468 0.577339 − 0.816505i \(-0.304091\pi\)
0.577339 + 0.816505i \(0.304091\pi\)
\(74\) −1.17152e7 −0.0454159
\(75\) 3.97751e8 1.45156
\(76\) 2.12022e8 0.728986
\(77\) 7.17289e7 0.232534
\(78\) 1.94184e8 0.594001
\(79\) 6.07464e8 1.75468 0.877342 − 0.479866i \(-0.159315\pi\)
0.877342 + 0.479866i \(0.159315\pi\)
\(80\) 5.38283e7 0.146929
\(81\) −2.28142e8 −0.588874
\(82\) 1.81154e8 0.442472
\(83\) 5.22381e8 1.20819 0.604096 − 0.796912i \(-0.293534\pi\)
0.604096 + 0.796912i \(0.293534\pi\)
\(84\) −7.52825e8 −1.64982
\(85\) 0 0
\(86\) 5.41945e7 0.106835
\(87\) −2.80603e8 −0.525117
\(88\) 6.68453e7 0.118823
\(89\) 2.71477e8 0.458647 0.229324 − 0.973350i \(-0.426349\pi\)
0.229324 + 0.973350i \(0.426349\pi\)
\(90\) 5.72952e7 0.0920507
\(91\) 9.46080e8 1.44624
\(92\) 5.36050e8 0.780118
\(93\) −1.59065e9 −2.20496
\(94\) 4.06357e8 0.536824
\(95\) −1.38490e8 −0.174446
\(96\) −1.07234e9 −1.28858
\(97\) 8.85422e8 1.01549 0.507747 − 0.861506i \(-0.330478\pi\)
0.507747 + 0.861506i \(0.330478\pi\)
\(98\) 1.43405e8 0.157053
\(99\) −2.40026e8 −0.251130
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 289.10.a.f.1.13 24
17.8 even 8 17.10.c.a.13.6 yes 24
17.15 even 8 17.10.c.a.4.7 ✓ 24
17.16 even 2 inner 289.10.a.f.1.14 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.c.a.4.7 ✓ 24 17.15 even 8
17.10.c.a.13.6 yes 24 17.8 even 8
289.10.a.f.1.13 24 1.1 even 1 trivial
289.10.a.f.1.14 24 17.16 even 2 inner