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Properties

Label 15.18.a.c.1.2
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [15,18,Mod(1,15)] 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("15.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-253] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(27.4833131017\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(21.5502\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-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)
 
\(f(q)\) \(=\) \(q-105.550 q^{2} +6561.00 q^{3} -119931. q^{4} -390625. q^{5} -692515. q^{6} -2.39385e7 q^{7} +2.64934e7 q^{8} +4.30467e7 q^{9} +4.12305e7 q^{10} -5.09412e8 q^{11} -7.86868e8 q^{12} +4.71682e9 q^{13} +2.52671e9 q^{14} -2.56289e9 q^{15} +1.29232e10 q^{16} -4.44993e10 q^{17} -4.54359e9 q^{18} -4.25101e10 q^{19} +4.68481e10 q^{20} -1.57060e11 q^{21} +5.37686e10 q^{22} +4.70013e11 q^{23} +1.73823e11 q^{24} +1.52588e11 q^{25} -4.97862e11 q^{26} +2.82430e11 q^{27} +2.87097e12 q^{28} +3.15724e12 q^{29} +2.70514e11 q^{30} +3.46790e12 q^{31} -4.83660e12 q^{32} -3.34226e12 q^{33} +4.69691e12 q^{34} +9.35098e12 q^{35} -5.16264e12 q^{36} +3.18740e13 q^{37} +4.48695e12 q^{38} +3.09471e13 q^{39} -1.03490e13 q^{40} +8.13537e13 q^{41} +1.65778e13 q^{42} -1.24496e14 q^{43} +6.10944e13 q^{44} -1.68151e13 q^{45} -4.96099e13 q^{46} -7.05130e13 q^{47} +8.47893e13 q^{48} +3.40421e14 q^{49} -1.61057e13 q^{50} -2.91960e14 q^{51} -5.65694e14 q^{52} +2.51778e14 q^{53} -2.98105e13 q^{54} +1.98989e14 q^{55} -6.34213e14 q^{56} -2.78909e14 q^{57} -3.33247e14 q^{58} -6.39615e14 q^{59} +3.07370e14 q^{60} -4.96148e13 q^{61} -3.66037e14 q^{62} -1.03047e15 q^{63} -1.18337e15 q^{64} -1.84251e15 q^{65} +3.52776e14 q^{66} -2.75976e15 q^{67} +5.33686e15 q^{68} +3.08375e15 q^{69} -9.86997e14 q^{70} +2.50612e15 q^{71} +1.14046e15 q^{72} +8.85474e14 q^{73} -3.36430e15 q^{74} +1.00113e15 q^{75} +5.09829e15 q^{76} +1.21946e16 q^{77} -3.26647e15 q^{78} -6.85544e13 q^{79} -5.04814e15 q^{80} +1.85302e15 q^{81} -8.58690e15 q^{82} +3.38359e16 q^{83} +1.88364e16 q^{84} +1.73826e16 q^{85} +1.31405e16 q^{86} +2.07146e16 q^{87} -1.34961e16 q^{88} -3.37871e16 q^{89} +1.77484e15 q^{90} -1.12914e17 q^{91} -5.63692e16 q^{92} +2.27529e16 q^{93} +7.44266e15 q^{94} +1.66055e16 q^{95} -3.17329e16 q^{96} +6.48330e16 q^{97} -3.59315e16 q^{98} -2.19285e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 253 q^{2} + 19683 q^{3} - 7087 q^{4} - 1171875 q^{5} - 1659933 q^{6} - 4332484 q^{7} - 16188513 q^{8} + 129140163 q^{9} + 98828125 q^{10} + 943563680 q^{11} - 46497807 q^{12} + 4257013150 q^{13} + 1847483988 q^{14}+ \cdots + 40\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −105.550 −0.291544 −0.145772 − 0.989318i \(-0.546567\pi\)
−0.145772 + 0.989318i \(0.546567\pi\)
\(3\) 6561.00 0.577350
\(4\) −119931. −0.915002
\(5\) −390625. −0.447214
\(6\) −692515. −0.168323
\(7\) −2.39385e7 −1.56951 −0.784754 − 0.619807i \(-0.787211\pi\)
−0.784754 + 0.619807i \(0.787211\pi\)
\(8\) 2.64934e7 0.558307
\(9\) 4.30467e7 0.333333
\(10\) 4.12305e7 0.130382
\(11\) −5.09412e8 −0.716526 −0.358263 − 0.933621i \(-0.616631\pi\)
−0.358263 + 0.933621i \(0.616631\pi\)
\(12\) −7.86868e8 −0.528277
\(13\) 4.71682e9 1.60373 0.801865 − 0.597506i \(-0.203841\pi\)
0.801865 + 0.597506i \(0.203841\pi\)
\(14\) 2.52671e9 0.457580
\(15\) −2.56289e9 −0.258199
\(16\) 1.29232e10 0.752231
\(17\) −4.44993e10 −1.54717 −0.773584 − 0.633693i \(-0.781538\pi\)
−0.773584 + 0.633693i \(0.781538\pi\)
\(18\) −4.54359e9 −0.0971813
\(19\) −4.25101e10 −0.574231 −0.287116 − 0.957896i \(-0.592696\pi\)
−0.287116 + 0.957896i \(0.592696\pi\)
\(20\) 4.68481e10 0.409201
\(21\) −1.57060e11 −0.906156
\(22\) 5.37686e10 0.208899
\(23\) 4.70013e11 1.25148 0.625739 − 0.780033i \(-0.284798\pi\)
0.625739 + 0.780033i \(0.284798\pi\)
\(24\) 1.73823e11 0.322339
\(25\) 1.52588e11 0.200000
\(26\) −4.97862e11 −0.467558
\(27\) 2.82430e11 0.192450
\(28\) 2.87097e12 1.43610
\(29\) 3.15724e12 1.17199 0.585996 − 0.810314i \(-0.300704\pi\)
0.585996 + 0.810314i \(0.300704\pi\)
\(30\) 2.70514e11 0.0752763
\(31\) 3.46790e12 0.730285 0.365142 − 0.930952i \(-0.381020\pi\)
0.365142 + 0.930952i \(0.381020\pi\)
\(32\) −4.83660e12 −0.777616
\(33\) −3.34226e12 −0.413686
\(34\) 4.69691e12 0.451068
\(35\) 9.35098e12 0.701905
\(36\) −5.16264e12 −0.305001
\(37\) 3.18740e13 1.49184 0.745919 − 0.666037i \(-0.232011\pi\)
0.745919 + 0.666037i \(0.232011\pi\)
\(38\) 4.48695e12 0.167414
\(39\) 3.09471e13 0.925914
\(40\) −1.03490e13 −0.249683
\(41\) 8.13537e13 1.59116 0.795581 − 0.605847i \(-0.207166\pi\)
0.795581 + 0.605847i \(0.207166\pi\)
\(42\) 1.65778e13 0.264184
\(43\) −1.24496e14 −1.62432 −0.812161 − 0.583433i \(-0.801709\pi\)
−0.812161 + 0.583433i \(0.801709\pi\)
\(44\) 6.10944e13 0.655623
\(45\) −1.68151e13 −0.149071
\(46\) −4.96099e13 −0.364861
\(47\) −7.05130e13 −0.431954 −0.215977 − 0.976398i \(-0.569294\pi\)
−0.215977 + 0.976398i \(0.569294\pi\)
\(48\) 8.47893e13 0.434301
\(49\) 3.40421e14 1.46336
\(50\) −1.61057e13 −0.0583088
\(51\) −2.91960e14 −0.893258
\(52\) −5.65694e14 −1.46742
\(53\) 2.51778e14 0.555485 0.277743 − 0.960656i \(-0.410414\pi\)
0.277743 + 0.960656i \(0.410414\pi\)
\(54\) −2.98105e13 −0.0561076
\(55\) 1.98989e14 0.320440
\(56\) −6.34213e14 −0.876268
\(57\) −2.78909e14 −0.331533
\(58\) −3.33247e14 −0.341687
\(59\) −6.39615e14 −0.567122 −0.283561 − 0.958954i \(-0.591516\pi\)
−0.283561 + 0.958954i \(0.591516\pi\)
\(60\) 3.07370e14 0.236253
\(61\) −4.96148e13 −0.0331366 −0.0165683 − 0.999863i \(-0.505274\pi\)
−0.0165683 + 0.999863i \(0.505274\pi\)
\(62\) −3.66037e14 −0.212910
\(63\) −1.03047e15 −0.523169
\(64\) −1.18337e15 −0.525522
\(65\) −1.84251e15 −0.717210
\(66\) 3.52776e14 0.120608
\(67\) −2.75976e15 −0.830300 −0.415150 − 0.909753i \(-0.636271\pi\)
−0.415150 + 0.909753i \(0.636271\pi\)
\(68\) 5.33686e15 1.41566
\(69\) 3.08375e15 0.722541
\(70\) −9.86997e14 −0.204636
\(71\) 2.50612e15 0.460582 0.230291 − 0.973122i \(-0.426032\pi\)
0.230291 + 0.973122i \(0.426032\pi\)
\(72\) 1.14046e15 0.186102
\(73\) 8.85474e14 0.128508 0.0642542 − 0.997934i \(-0.479533\pi\)
0.0642542 + 0.997934i \(0.479533\pi\)
\(74\) −3.36430e15 −0.434936
\(75\) 1.00113e15 0.115470
\(76\) 5.09829e15 0.525423
\(77\) 1.21946e16 1.12459
\(78\) −3.26647e15 −0.269944
\(79\) −6.85544e13 −0.00508400 −0.00254200 − 0.999997i \(-0.500809\pi\)
−0.00254200 + 0.999997i \(0.500809\pi\)
\(80\) −5.04814e15 −0.336408
\(81\) 1.85302e15 0.111111
\(82\) −8.58690e15 −0.463894
\(83\) 3.38359e16 1.64897 0.824487 − 0.565881i \(-0.191464\pi\)
0.824487 + 0.565881i \(0.191464\pi\)
\(84\) 1.88364e16 0.829135
\(85\) 1.73826e16 0.691915
\(86\) 1.31405e16 0.473561
\(87\) 2.07146e16 0.676650
\(88\) −1.34961e16 −0.400041
\(89\) −3.37871e16 −0.909778 −0.454889 − 0.890548i \(-0.650321\pi\)
−0.454889 + 0.890548i \(0.650321\pi\)
\(90\) 1.77484e15 0.0434608
\(91\) −1.12914e17 −2.51707
\(92\) −5.63692e16 −1.14510
\(93\) 2.27529e16 0.421630
\(94\) 7.44266e15 0.125934
\(95\) 1.66055e16 0.256804
\(96\) −3.17329e16 −0.448957
\(97\) 6.48330e16 0.839917 0.419958 − 0.907543i \(-0.362045\pi\)
0.419958 + 0.907543i \(0.362045\pi\)
\(98\) −3.59315e16 −0.426632
\(99\) −2.19285e16 −0.238842
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 15.18.a.c.1.2 ✓ 3
3.2 odd 2 45.18.a.d.1.2 3
5.2 odd 4 75.18.b.e.49.3 6
5.3 odd 4 75.18.b.e.49.4 6
5.4 even 2 75.18.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.2 ✓ 3 1.1 even 1 trivial
45.18.a.d.1.2 3 3.2 odd 2
75.18.a.d.1.2 3 5.4 even 2
75.18.b.e.49.3 6 5.2 odd 4
75.18.b.e.49.4 6 5.3 odd 4