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Properties

Label 15.18.a.b.1.3
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 1.3
Root \(-9.44141\) 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)
 
\(f(q)\) \(=\) \(q+442.567 q^{2} -6561.00 q^{3} +64793.8 q^{4} -390625. q^{5} -2.90368e6 q^{6} +2.47993e7 q^{7} -2.93326e7 q^{8} +4.30467e7 q^{9} -1.72878e8 q^{10} -7.66684e6 q^{11} -4.25112e8 q^{12} -3.40086e9 q^{13} +1.09754e10 q^{14} +2.56289e9 q^{15} -2.14743e10 q^{16} -5.35306e10 q^{17} +1.90511e10 q^{18} -1.29760e11 q^{19} -2.53101e10 q^{20} -1.62708e11 q^{21} -3.39309e9 q^{22} +2.11422e11 q^{23} +1.92451e11 q^{24} +1.52588e11 q^{25} -1.50511e12 q^{26} -2.82430e11 q^{27} +1.60684e12 q^{28} +4.13975e12 q^{29} +1.13425e12 q^{30} -5.31377e12 q^{31} -5.65914e12 q^{32} +5.03022e10 q^{33} -2.36909e13 q^{34} -9.68724e12 q^{35} +2.78916e12 q^{36} +1.85634e13 q^{37} -5.74274e13 q^{38} +2.23130e13 q^{39} +1.14580e13 q^{40} -5.57404e13 q^{41} -7.20094e13 q^{42} -7.31648e13 q^{43} -4.96764e11 q^{44} -1.68151e13 q^{45} +9.35684e13 q^{46} -1.22507e14 q^{47} +1.40893e14 q^{48} +3.82376e14 q^{49} +6.75304e13 q^{50} +3.51215e14 q^{51} -2.20355e14 q^{52} +2.74251e14 q^{53} -1.24994e14 q^{54} +2.99486e12 q^{55} -7.27428e14 q^{56} +8.51354e14 q^{57} +1.83212e15 q^{58} +6.31977e14 q^{59} +1.66059e14 q^{60} -1.14361e15 q^{61} -2.35170e15 q^{62} +1.06753e15 q^{63} +3.10129e14 q^{64} +1.32846e15 q^{65} +2.22621e13 q^{66} +6.08292e14 q^{67} -3.46845e15 q^{68} -1.38714e15 q^{69} -4.28726e15 q^{70} +1.25518e15 q^{71} -1.26267e15 q^{72} +7.71376e15 q^{73} +8.21553e15 q^{74} -1.00113e15 q^{75} -8.40763e15 q^{76} -1.90133e14 q^{77} +9.87502e15 q^{78} -2.63076e16 q^{79} +8.38839e15 q^{80} +1.85302e15 q^{81} -2.46689e16 q^{82} +1.45466e16 q^{83} -1.05425e16 q^{84} +2.09104e16 q^{85} -3.23803e16 q^{86} -2.71609e16 q^{87} +2.24888e14 q^{88} +2.61450e16 q^{89} -7.44182e15 q^{90} -8.43391e16 q^{91} +1.36988e16 q^{92} +3.48637e16 q^{93} -5.42175e16 q^{94} +5.06874e16 q^{95} +3.71296e16 q^{96} -9.64972e15 q^{97} +1.69227e17 q^{98} -3.30032e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 442 q^{2} - 19683 q^{3} + 298148 q^{4} - 1171875 q^{5} + 2899962 q^{6} + 4962644 q^{7} - 108831912 q^{8} + 129140163 q^{9} + 172656250 q^{10} + 1049849720 q^{11} - 1956149028 q^{12} - 3091742090 q^{13}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 442.567 1.22243 0.611215 − 0.791464i \(-0.290681\pi\)
0.611215 + 0.791464i \(0.290681\pi\)
\(3\) −6561.00 −0.577350
\(4\) 64793.8 0.494337
\(5\) −390625. −0.447214
\(6\) −2.90368e6 −0.705771
\(7\) 2.47993e7 1.62595 0.812974 − 0.582300i \(-0.197847\pi\)
0.812974 + 0.582300i \(0.197847\pi\)
\(8\) −2.93326e7 −0.618138
\(9\) 4.30467e7 0.333333
\(10\) −1.72878e8 −0.546688
\(11\) −7.66684e6 −0.0107840 −0.00539199 − 0.999985i \(-0.501716\pi\)
−0.00539199 + 0.999985i \(0.501716\pi\)
\(12\) −4.25112e8 −0.285406
\(13\) −3.40086e9 −1.15630 −0.578150 − 0.815931i \(-0.696225\pi\)
−0.578150 + 0.815931i \(0.696225\pi\)
\(14\) 1.09754e10 1.98761
\(15\) 2.56289e9 0.258199
\(16\) −2.14743e10 −1.24997
\(17\) −5.35306e10 −1.86117 −0.930586 − 0.366073i \(-0.880702\pi\)
−0.930586 + 0.366073i \(0.880702\pi\)
\(18\) 1.90511e10 0.407477
\(19\) −1.29760e11 −1.75281 −0.876404 − 0.481576i \(-0.840065\pi\)
−0.876404 + 0.481576i \(0.840065\pi\)
\(20\) −2.53101e10 −0.221074
\(21\) −1.62708e11 −0.938742
\(22\) −3.39309e9 −0.0131827
\(23\) 2.11422e11 0.562941 0.281471 − 0.959570i \(-0.409178\pi\)
0.281471 + 0.959570i \(0.409178\pi\)
\(24\) 1.92451e11 0.356882
\(25\) 1.52588e11 0.200000
\(26\) −1.50511e12 −1.41350
\(27\) −2.82430e11 −0.192450
\(28\) 1.60684e12 0.803767
\(29\) 4.13975e12 1.53671 0.768354 − 0.640026i \(-0.221076\pi\)
0.768354 + 0.640026i \(0.221076\pi\)
\(30\) 1.13425e12 0.315630
\(31\) −5.31377e12 −1.11900 −0.559498 − 0.828832i \(-0.689006\pi\)
−0.559498 + 0.828832i \(0.689006\pi\)
\(32\) −5.65914e12 −0.909862
\(33\) 5.03022e10 0.00622613
\(34\) −2.36909e13 −2.27515
\(35\) −9.68724e12 −0.727146
\(36\) 2.78916e12 0.164779
\(37\) 1.85634e13 0.868844 0.434422 − 0.900709i \(-0.356953\pi\)
0.434422 + 0.900709i \(0.356953\pi\)
\(38\) −5.74274e13 −2.14269
\(39\) 2.23130e13 0.667590
\(40\) 1.14580e13 0.276440
\(41\) −5.57404e13 −1.09020 −0.545101 − 0.838370i \(-0.683509\pi\)
−0.545101 + 0.838370i \(0.683509\pi\)
\(42\) −7.20094e13 −1.14755
\(43\) −7.31648e13 −0.954597 −0.477299 − 0.878741i \(-0.658384\pi\)
−0.477299 + 0.878741i \(0.658384\pi\)
\(44\) −4.96764e11 −0.00533092
\(45\) −1.68151e13 −0.149071
\(46\) 9.35684e13 0.688157
\(47\) −1.22507e14 −0.750461 −0.375231 − 0.926931i \(-0.622436\pi\)
−0.375231 + 0.926931i \(0.622436\pi\)
\(48\) 1.40893e14 0.721669
\(49\) 3.82376e14 1.64371
\(50\) 6.75304e13 0.244486
\(51\) 3.51215e14 1.07455
\(52\) −2.20355e14 −0.571602
\(53\) 2.74251e14 0.605068 0.302534 − 0.953139i \(-0.402168\pi\)
0.302534 + 0.953139i \(0.402168\pi\)
\(54\) −1.24994e14 −0.235257
\(55\) 2.99486e12 0.00482274
\(56\) −7.27428e14 −1.00506
\(57\) 8.51354e14 1.01198
\(58\) 1.83212e15 1.87852
\(59\) 6.31977e14 0.560350 0.280175 − 0.959949i \(-0.409608\pi\)
0.280175 + 0.959949i \(0.409608\pi\)
\(60\) 1.66059e14 0.127637
\(61\) −1.14361e15 −0.763791 −0.381895 − 0.924206i \(-0.624729\pi\)
−0.381895 + 0.924206i \(0.624729\pi\)
\(62\) −2.35170e15 −1.36790
\(63\) 1.06753e15 0.541983
\(64\) 3.10129e14 0.137725
\(65\) 1.32846e15 0.517113
\(66\) 2.22621e13 0.00761101
\(67\) 6.08292e14 0.183011 0.0915053 − 0.995805i \(-0.470832\pi\)
0.0915053 + 0.995805i \(0.470832\pi\)
\(68\) −3.46845e15 −0.920047
\(69\) −1.38714e15 −0.325014
\(70\) −4.28726e15 −0.888886
\(71\) 1.25518e15 0.230681 0.115340 − 0.993326i \(-0.463204\pi\)
0.115340 + 0.993326i \(0.463204\pi\)
\(72\) −1.26267e15 −0.206046
\(73\) 7.71376e15 1.11949 0.559747 − 0.828664i \(-0.310898\pi\)
0.559747 + 0.828664i \(0.310898\pi\)
\(74\) 8.21553e15 1.06210
\(75\) −1.00113e15 −0.115470
\(76\) −8.40763e15 −0.866479
\(77\) −1.90133e14 −0.0175342
\(78\) 9.87502e15 0.816082
\(79\) −2.63076e16 −1.95097 −0.975486 − 0.220061i \(-0.929374\pi\)
−0.975486 + 0.220061i \(0.929374\pi\)
\(80\) 8.38839e15 0.559003
\(81\) 1.85302e15 0.111111
\(82\) −2.46689e16 −1.33270
\(83\) 1.45466e16 0.708922 0.354461 − 0.935071i \(-0.384664\pi\)
0.354461 + 0.935071i \(0.384664\pi\)
\(84\) −1.05425e16 −0.464055
\(85\) 2.09104e16 0.832342
\(86\) −3.23803e16 −1.16693
\(87\) −2.71609e16 −0.887218
\(88\) 2.24888e14 0.00666598
\(89\) 2.61450e16 0.704001 0.352000 − 0.936000i \(-0.385502\pi\)
0.352000 + 0.936000i \(0.385502\pi\)
\(90\) −7.44182e15 −0.182229
\(91\) −8.43391e16 −1.88008
\(92\) 1.36988e16 0.278283
\(93\) 3.48637e16 0.646053
\(94\) −5.42175e16 −0.917387
\(95\) 5.06874e16 0.783880
\(96\) 3.71296e16 0.525309
\(97\) −9.64972e15 −0.125013 −0.0625064 − 0.998045i \(-0.519909\pi\)
−0.0625064 + 0.998045i \(0.519909\pi\)
\(98\) 1.69227e17 2.00932
\(99\) −3.30032e14 −0.00359466
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.1.3 ✓ 3
3.2 odd 2 45.18.a.e.1.1 3
5.2 odd 4 75.18.b.d.49.5 6
5.3 odd 4 75.18.b.d.49.2 6
5.4 even 2 75.18.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.3 ✓ 3 1.1 even 1 trivial
45.18.a.e.1.1 3 3.2 odd 2
75.18.a.e.1.1 3 5.4 even 2
75.18.b.d.49.2 6 5.3 odd 4
75.18.b.d.49.5 6 5.2 odd 4