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Properties

Label 15.18.a.d.1.3
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $0$
Dimension $4$
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 = [4,-33] 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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-265.574\) 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)
 
\(f(q)\) \(=\) \(q+257.574 q^{2} -6561.00 q^{3} -64727.8 q^{4} +390625. q^{5} -1.68994e6 q^{6} +8.43659e6 q^{7} -5.04329e7 q^{8} +4.30467e7 q^{9} +1.00615e8 q^{10} -1.27592e9 q^{11} +4.24679e8 q^{12} +2.16831e9 q^{13} +2.17304e9 q^{14} -2.56289e9 q^{15} -4.50617e9 q^{16} +3.94044e10 q^{17} +1.10877e10 q^{18} +1.12109e11 q^{19} -2.52843e10 q^{20} -5.53525e10 q^{21} -3.28642e11 q^{22} +5.33219e11 q^{23} +3.30890e11 q^{24} +1.52588e11 q^{25} +5.58501e11 q^{26} -2.82430e11 q^{27} -5.46082e11 q^{28} -1.70330e12 q^{29} -6.60133e11 q^{30} -1.16317e11 q^{31} +5.44967e12 q^{32} +8.37128e12 q^{33} +1.01495e13 q^{34} +3.29554e12 q^{35} -2.78632e12 q^{36} +3.69532e13 q^{37} +2.88763e13 q^{38} -1.42263e13 q^{39} -1.97003e13 q^{40} +6.01774e13 q^{41} -1.42573e13 q^{42} -3.10214e13 q^{43} +8.25872e13 q^{44} +1.68151e13 q^{45} +1.37343e14 q^{46} -2.34600e14 q^{47} +2.95650e13 q^{48} -1.61454e14 q^{49} +3.93026e13 q^{50} -2.58532e14 q^{51} -1.40350e14 q^{52} +5.42186e14 q^{53} -7.27464e13 q^{54} -4.98404e14 q^{55} -4.25482e14 q^{56} -7.35546e14 q^{57} -4.38725e14 q^{58} -1.85401e14 q^{59} +1.65890e14 q^{60} -1.71339e15 q^{61} -2.99603e13 q^{62} +3.63168e14 q^{63} +1.99432e15 q^{64} +8.46998e14 q^{65} +2.15622e15 q^{66} -6.75194e12 q^{67} -2.55056e15 q^{68} -3.49845e15 q^{69} +8.48845e14 q^{70} +9.21473e13 q^{71} -2.17097e15 q^{72} +1.14713e16 q^{73} +9.51816e15 q^{74} -1.00113e15 q^{75} -7.25656e15 q^{76} -1.07644e16 q^{77} -3.66432e15 q^{78} +8.98946e15 q^{79} -1.76022e15 q^{80} +1.85302e15 q^{81} +1.55001e16 q^{82} +4.25580e15 q^{83} +3.58285e15 q^{84} +1.53923e16 q^{85} -7.99030e15 q^{86} +1.11753e16 q^{87} +6.43481e16 q^{88} -3.54476e16 q^{89} +4.33113e15 q^{90} +1.82932e16 q^{91} -3.45141e16 q^{92} +7.63158e14 q^{93} -6.04269e16 q^{94} +4.37925e16 q^{95} -3.57553e16 q^{96} +3.08428e16 q^{97} -4.15864e16 q^{98} -5.49240e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 33 q^{2} - 26244 q^{3} + 439357 q^{4} + 1562500 q^{5} + 216513 q^{6} + 17583104 q^{7} + 63651621 q^{8} + 172186884 q^{9} - 12890625 q^{10} - 575495184 q^{11} - 2882621277 q^{12} - 5049645832 q^{13}+ \cdots - 24\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 257.574 0.711453 0.355727 − 0.934590i \(-0.384233\pi\)
0.355727 + 0.934590i \(0.384233\pi\)
\(3\) −6561.00 −0.577350
\(4\) −64727.8 −0.493834
\(5\) 390625. 0.447214
\(6\) −1.68994e6 −0.410758
\(7\) 8.43659e6 0.553138 0.276569 − 0.960994i \(-0.410803\pi\)
0.276569 + 0.960994i \(0.410803\pi\)
\(8\) −5.04329e7 −1.06279
\(9\) 4.30467e7 0.333333
\(10\) 1.00615e8 0.318172
\(11\) −1.27592e9 −1.79467 −0.897334 − 0.441353i \(-0.854499\pi\)
−0.897334 + 0.441353i \(0.854499\pi\)
\(12\) 4.24679e8 0.285115
\(13\) 2.16831e9 0.737231 0.368616 − 0.929582i \(-0.379832\pi\)
0.368616 + 0.929582i \(0.379832\pi\)
\(14\) 2.17304e9 0.393532
\(15\) −2.56289e9 −0.258199
\(16\) −4.50617e9 −0.262294
\(17\) 3.94044e10 1.37003 0.685013 − 0.728531i \(-0.259797\pi\)
0.685013 + 0.728531i \(0.259797\pi\)
\(18\) 1.10877e10 0.237151
\(19\) 1.12109e11 1.51438 0.757189 − 0.653196i \(-0.226572\pi\)
0.757189 + 0.653196i \(0.226572\pi\)
\(20\) −2.52843e10 −0.220849
\(21\) −5.53525e10 −0.319354
\(22\) −3.28642e11 −1.27682
\(23\) 5.33219e11 1.41977 0.709886 − 0.704317i \(-0.248746\pi\)
0.709886 + 0.704317i \(0.248746\pi\)
\(24\) 3.30890e11 0.613604
\(25\) 1.52588e11 0.200000
\(26\) 5.58501e11 0.524506
\(27\) −2.82430e11 −0.192450
\(28\) −5.46082e11 −0.273159
\(29\) −1.70330e12 −0.632278 −0.316139 − 0.948713i \(-0.602387\pi\)
−0.316139 + 0.948713i \(0.602387\pi\)
\(30\) −6.60133e11 −0.183696
\(31\) −1.16317e11 −0.0244946 −0.0122473 − 0.999925i \(-0.503899\pi\)
−0.0122473 + 0.999925i \(0.503899\pi\)
\(32\) 5.44967e12 0.876184
\(33\) 8.37128e12 1.03615
\(34\) 1.01495e13 0.974709
\(35\) 3.29554e12 0.247371
\(36\) −2.78632e12 −0.164611
\(37\) 3.69532e13 1.72957 0.864783 − 0.502146i \(-0.167456\pi\)
0.864783 + 0.502146i \(0.167456\pi\)
\(38\) 2.88763e13 1.07741
\(39\) −1.42263e13 −0.425641
\(40\) −1.97003e13 −0.475296
\(41\) 6.01774e13 1.17698 0.588492 − 0.808503i \(-0.299722\pi\)
0.588492 + 0.808503i \(0.299722\pi\)
\(42\) −1.42573e13 −0.227206
\(43\) −3.10214e13 −0.404744 −0.202372 − 0.979309i \(-0.564865\pi\)
−0.202372 + 0.979309i \(0.564865\pi\)
\(44\) 8.25872e13 0.886268
\(45\) 1.68151e13 0.149071
\(46\) 1.37343e14 1.01010
\(47\) −2.34600e14 −1.43713 −0.718567 − 0.695458i \(-0.755201\pi\)
−0.718567 + 0.695458i \(0.755201\pi\)
\(48\) 2.95650e13 0.151435
\(49\) −1.61454e14 −0.694038
\(50\) 3.93026e13 0.142291
\(51\) −2.58532e14 −0.790985
\(52\) −1.40350e14 −0.364070
\(53\) 5.42186e14 1.19620 0.598100 − 0.801422i \(-0.295923\pi\)
0.598100 + 0.801422i \(0.295923\pi\)
\(54\) −7.27464e13 −0.136919
\(55\) −4.98404e14 −0.802600
\(56\) −4.25482e14 −0.587872
\(57\) −7.35546e14 −0.874327
\(58\) −4.38725e14 −0.449836
\(59\) −1.85401e14 −0.164388 −0.0821941 − 0.996616i \(-0.526193\pi\)
−0.0821941 + 0.996616i \(0.526193\pi\)
\(60\) 1.65890e14 0.127507
\(61\) −1.71339e15 −1.14433 −0.572167 − 0.820137i \(-0.693897\pi\)
−0.572167 + 0.820137i \(0.693897\pi\)
\(62\) −2.99603e13 −0.0174268
\(63\) 3.63168e14 0.184379
\(64\) 1.99432e15 0.885657
\(65\) 8.46998e14 0.329700
\(66\) 2.15622e15 0.737173
\(67\) −6.75194e12 −0.00203139 −0.00101569 − 0.999999i \(-0.500323\pi\)
−0.00101569 + 0.999999i \(0.500323\pi\)
\(68\) −2.55056e15 −0.676566
\(69\) −3.49845e15 −0.819706
\(70\) 8.48845e14 0.175993
\(71\) 9.21473e13 0.0169351 0.00846753 − 0.999964i \(-0.497305\pi\)
0.00846753 + 0.999964i \(0.497305\pi\)
\(72\) −2.17097e15 −0.354264
\(73\) 1.14713e16 1.66483 0.832415 − 0.554153i \(-0.186958\pi\)
0.832415 + 0.554153i \(0.186958\pi\)
\(74\) 9.51816e15 1.23050
\(75\) −1.00113e15 −0.115470
\(76\) −7.25656e15 −0.747852
\(77\) −1.07644e16 −0.992699
\(78\) −3.66432e15 −0.302824
\(79\) 8.98946e15 0.666658 0.333329 − 0.942810i \(-0.391828\pi\)
0.333329 + 0.942810i \(0.391828\pi\)
\(80\) −1.76022e15 −0.117301
\(81\) 1.85302e15 0.111111
\(82\) 1.55001e16 0.837370
\(83\) 4.25580e15 0.207404 0.103702 − 0.994608i \(-0.466931\pi\)
0.103702 + 0.994608i \(0.466931\pi\)
\(84\) 3.58285e15 0.157708
\(85\) 1.53923e16 0.612694
\(86\) −7.99030e15 −0.287956
\(87\) 1.11753e16 0.365046
\(88\) 6.43481e16 1.90736
\(89\) −3.54476e16 −0.954490 −0.477245 − 0.878770i \(-0.658365\pi\)
−0.477245 + 0.878770i \(0.658365\pi\)
\(90\) 4.33113e15 0.106057
\(91\) 1.82932e16 0.407791
\(92\) −3.45141e16 −0.701132
\(93\) 7.63158e14 0.0141420
\(94\) −6.04269e16 −1.02245
\(95\) 4.37925e16 0.677250
\(96\) −3.57553e16 −0.505865
\(97\) 3.08428e16 0.399570 0.199785 − 0.979840i \(-0.435976\pi\)
0.199785 + 0.979840i \(0.435976\pi\)
\(98\) −4.15864e16 −0.493776
\(99\) −5.49240e16 −0.598222
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.d.1.3 ✓ 4
3.2 odd 2 45.18.a.f.1.2 4
5.2 odd 4 75.18.b.f.49.5 8
5.3 odd 4 75.18.b.f.49.4 8
5.4 even 2 75.18.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.3 ✓ 4 1.1 even 1 trivial
45.18.a.f.1.2 4 3.2 odd 2
75.18.a.f.1.2 4 5.4 even 2
75.18.b.f.49.4 8 5.3 odd 4
75.18.b.f.49.5 8 5.2 odd 4