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Properties

Label 15.18.a.b.1.1
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $1$
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,-442] 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: \(1\)
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: \( x^{3} - 37234x - 350700 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-188.067\) 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)
 
\(f(q)\) \(=\) \(q-670.613 q^{2} -6561.00 q^{3} +318650. q^{4} -390625. q^{5} +4.39989e6 q^{6} -2.70804e7 q^{7} -1.25792e8 q^{8} +4.30467e7 q^{9} +2.61958e8 q^{10} +1.01644e9 q^{11} -2.09066e9 q^{12} -1.04834e9 q^{13} +1.81605e10 q^{14} +2.56289e9 q^{15} +4.25918e10 q^{16} +2.00099e10 q^{17} -2.88677e10 q^{18} +1.64417e10 q^{19} -1.24473e11 q^{20} +1.77675e11 q^{21} -6.81636e11 q^{22} +5.92721e11 q^{23} +8.25322e11 q^{24} +1.52588e11 q^{25} +7.03032e11 q^{26} -2.82430e11 q^{27} -8.62917e12 q^{28} +4.83212e11 q^{29} -1.71871e12 q^{30} -5.20802e12 q^{31} -1.20748e13 q^{32} -6.66885e12 q^{33} -1.34189e13 q^{34} +1.05783e13 q^{35} +1.37168e13 q^{36} +3.82367e13 q^{37} -1.10260e13 q^{38} +6.87817e12 q^{39} +4.91375e13 q^{40} -6.56402e13 q^{41} -1.19151e14 q^{42} -5.83804e13 q^{43} +3.23888e14 q^{44} -1.68151e13 q^{45} -3.97486e14 q^{46} +7.43658e12 q^{47} -2.79445e14 q^{48} +5.00719e14 q^{49} -1.02327e14 q^{50} -1.31285e14 q^{51} -3.34054e14 q^{52} -3.01466e14 q^{53} +1.89401e14 q^{54} -3.97046e14 q^{55} +3.40650e15 q^{56} -1.07874e14 q^{57} -3.24048e14 q^{58} -3.44389e14 q^{59} +8.16665e14 q^{60} -2.43237e15 q^{61} +3.49257e15 q^{62} -1.16572e15 q^{63} +2.51491e15 q^{64} +4.09508e14 q^{65} +4.47222e15 q^{66} +1.19632e15 q^{67} +6.37614e15 q^{68} -3.88884e15 q^{69} -7.09394e15 q^{70} +5.42254e15 q^{71} -5.41494e15 q^{72} -5.84732e15 q^{73} -2.56420e16 q^{74} -1.00113e15 q^{75} +5.23914e15 q^{76} -2.75256e16 q^{77} -4.61259e15 q^{78} +2.03455e15 q^{79} -1.66374e16 q^{80} +1.85302e15 q^{81} +4.40192e16 q^{82} -1.18343e16 q^{83} +5.66160e16 q^{84} -7.81636e15 q^{85} +3.91507e16 q^{86} -3.17035e15 q^{87} -1.27860e17 q^{88} -2.43170e16 q^{89} +1.12764e16 q^{90} +2.83895e16 q^{91} +1.88870e17 q^{92} +3.41698e16 q^{93} -4.98707e15 q^{94} -6.42253e15 q^{95} +7.92225e16 q^{96} -1.06105e17 q^{97} -3.35788e17 q^{98} +4.37543e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 442 q^{2} - 19683 q^{3} + 298148 q^{4} - 1171875 q^{5} + 2899962 q^{6} + 4962644 q^{7} - 108831912 q^{8} + 129140163 q^{9} + 172656250 q^{10} + 1049849720 q^{11} - 1956149028 q^{12} - 3091742090 q^{13}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −670.613 −1.85232 −0.926162 − 0.377126i \(-0.876912\pi\)
−0.926162 + 0.377126i \(0.876912\pi\)
\(3\) −6561.00 −0.577350
\(4\) 318650. 2.43110
\(5\) −390625. −0.447214
\(6\) 4.39989e6 1.06944
\(7\) −2.70804e7 −1.77551 −0.887753 − 0.460320i \(-0.847735\pi\)
−0.887753 + 0.460320i \(0.847735\pi\)
\(8\) −1.25792e8 −2.65087
\(9\) 4.30467e7 0.333333
\(10\) 2.61958e8 0.828385
\(11\) 1.01644e9 1.42969 0.714847 − 0.699281i \(-0.246496\pi\)
0.714847 + 0.699281i \(0.246496\pi\)
\(12\) −2.09066e9 −1.40360
\(13\) −1.04834e9 −0.356438 −0.178219 − 0.983991i \(-0.557034\pi\)
−0.178219 + 0.983991i \(0.557034\pi\)
\(14\) 1.81605e10 3.28881
\(15\) 2.56289e9 0.258199
\(16\) 4.25918e10 2.47917
\(17\) 2.00099e10 0.695711 0.347855 − 0.937548i \(-0.386910\pi\)
0.347855 + 0.937548i \(0.386910\pi\)
\(18\) −2.88677e10 −0.617441
\(19\) 1.64417e10 0.222096 0.111048 − 0.993815i \(-0.464579\pi\)
0.111048 + 0.993815i \(0.464579\pi\)
\(20\) −1.24473e11 −1.08722
\(21\) 1.77675e11 1.02509
\(22\) −6.81636e11 −2.64826
\(23\) 5.92721e11 1.57821 0.789103 − 0.614261i \(-0.210546\pi\)
0.789103 + 0.614261i \(0.210546\pi\)
\(24\) 8.25322e11 1.53048
\(25\) 1.52588e11 0.200000
\(26\) 7.03032e11 0.660239
\(27\) −2.82430e11 −0.192450
\(28\) −8.62917e12 −4.31644
\(29\) 4.83212e11 0.179372 0.0896860 − 0.995970i \(-0.471414\pi\)
0.0896860 + 0.995970i \(0.471414\pi\)
\(30\) −1.71871e12 −0.478268
\(31\) −5.20802e12 −1.09673 −0.548363 − 0.836240i \(-0.684749\pi\)
−0.548363 + 0.836240i \(0.684749\pi\)
\(32\) −1.20748e13 −1.94135
\(33\) −6.66885e12 −0.825434
\(34\) −1.34189e13 −1.28868
\(35\) 1.05783e13 0.794030
\(36\) 1.37168e13 0.810368
\(37\) 3.82367e13 1.78964 0.894820 − 0.446427i \(-0.147304\pi\)
0.894820 + 0.446427i \(0.147304\pi\)
\(38\) −1.10260e13 −0.411393
\(39\) 6.87817e12 0.205790
\(40\) 4.91375e13 1.18551
\(41\) −6.56402e13 −1.28383 −0.641915 − 0.766776i \(-0.721860\pi\)
−0.641915 + 0.766776i \(0.721860\pi\)
\(42\) −1.19151e14 −1.89880
\(43\) −5.83804e13 −0.761703 −0.380851 − 0.924636i \(-0.624369\pi\)
−0.380851 + 0.924636i \(0.624369\pi\)
\(44\) 3.23888e14 3.47573
\(45\) −1.68151e13 −0.149071
\(46\) −3.97486e14 −2.92335
\(47\) 7.43658e12 0.0455556 0.0227778 − 0.999741i \(-0.492749\pi\)
0.0227778 + 0.999741i \(0.492749\pi\)
\(48\) −2.79445e14 −1.43135
\(49\) 5.00719e14 2.15242
\(50\) −1.02327e14 −0.370465
\(51\) −1.31285e14 −0.401669
\(52\) −3.34054e14 −0.866539
\(53\) −3.01466e14 −0.665109 −0.332555 − 0.943084i \(-0.607911\pi\)
−0.332555 + 0.943084i \(0.607911\pi\)
\(54\) 1.89401e14 0.356480
\(55\) −3.97046e14 −0.639378
\(56\) 3.40650e15 4.70664
\(57\) −1.07874e14 −0.128227
\(58\) −3.24048e14 −0.332255
\(59\) −3.44389e14 −0.305357 −0.152678 − 0.988276i \(-0.548790\pi\)
−0.152678 + 0.988276i \(0.548790\pi\)
\(60\) 8.16665e14 0.627709
\(61\) −2.43237e15 −1.62452 −0.812261 − 0.583294i \(-0.801763\pi\)
−0.812261 + 0.583294i \(0.801763\pi\)
\(62\) 3.49257e15 2.03149
\(63\) −1.16572e15 −0.591835
\(64\) 2.51491e15 1.11684
\(65\) 4.09508e14 0.159404
\(66\) 4.47222e15 1.52897
\(67\) 1.19632e15 0.359924 0.179962 − 0.983674i \(-0.442403\pi\)
0.179962 + 0.983674i \(0.442403\pi\)
\(68\) 6.37614e15 1.69135
\(69\) −3.88884e15 −0.911177
\(70\) −7.09394e15 −1.47080
\(71\) 5.42254e15 0.996567 0.498283 − 0.867014i \(-0.333964\pi\)
0.498283 + 0.867014i \(0.333964\pi\)
\(72\) −5.41494e15 −0.883623
\(73\) −5.84732e15 −0.848619 −0.424309 − 0.905517i \(-0.639483\pi\)
−0.424309 + 0.905517i \(0.639483\pi\)
\(74\) −2.56420e16 −3.31499
\(75\) −1.00113e15 −0.115470
\(76\) 5.23914e15 0.539938
\(77\) −2.75256e16 −2.53843
\(78\) −4.61259e15 −0.381189
\(79\) 2.03455e15 0.150882 0.0754411 − 0.997150i \(-0.475964\pi\)
0.0754411 + 0.997150i \(0.475964\pi\)
\(80\) −1.66374e16 −1.10872
\(81\) 1.85302e15 0.111111
\(82\) 4.40192e16 2.37807
\(83\) −1.18343e16 −0.576740 −0.288370 − 0.957519i \(-0.593113\pi\)
−0.288370 + 0.957519i \(0.593113\pi\)
\(84\) 5.66160e16 2.49210
\(85\) −7.81636e15 −0.311131
\(86\) 3.91507e16 1.41092
\(87\) −3.17035e15 −0.103560
\(88\) −1.27860e17 −3.78993
\(89\) −2.43170e16 −0.654780 −0.327390 − 0.944889i \(-0.606169\pi\)
−0.327390 + 0.944889i \(0.606169\pi\)
\(90\) 1.12764e16 0.276128
\(91\) 2.83895e16 0.632858
\(92\) 1.88870e17 3.83678
\(93\) 3.41698e16 0.633195
\(94\) −4.98707e15 −0.0843837
\(95\) −6.42253e15 −0.0993243
\(96\) 7.92225e16 1.12084
\(97\) −1.06105e17 −1.37459 −0.687296 − 0.726377i \(-0.741203\pi\)
−0.687296 + 0.726377i \(0.741203\pi\)
\(98\) −3.35788e17 −3.98698
\(99\) 4.37543e16 0.476564
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.b.1.1 ✓ 3
3.2 odd 2 45.18.a.e.1.3 3
5.2 odd 4 75.18.b.d.49.1 6
5.3 odd 4 75.18.b.d.49.6 6
5.4 even 2 75.18.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.1 ✓ 3 1.1 even 1 trivial
45.18.a.e.1.3 3 3.2 odd 2
75.18.a.e.1.3 3 5.4 even 2
75.18.b.d.49.1 6 5.2 odd 4
75.18.b.d.49.6 6 5.3 odd 4