Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,Mod(37,126)] 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("126.37"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,96,0,-3072,-1045] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 1516x^{4} + 1461x^{3} + 2295252x^{2} - 40905x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.3
Root \(0.00891079 - 0.0154339i\) of defining polynomial
Character \(\chi\) \(=\) 126.109
Dual form 126.12.g.b.37.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 + 27.7128i) q^{2} +(-512.000 + 886.810i) q^{4} +(1098.21 + 1902.16i) q^{5} +(39703.5 - 20023.9i) q^{7} -32768.0 q^{8} +(-35142.9 + 60869.3i) q^{10} +(206569. - 357788. i) q^{11} +1.89569e6 q^{13} +(1.19018e6 + 779913. i) q^{14} +(-524288. - 908093. i) q^{16} +(-4.55417e6 + 7.88805e6i) q^{17} +(-1.24340e6 - 2.15364e6i) q^{19} -2.24914e6 q^{20} +1.32204e7 q^{22} +(-2.95467e7 - 5.11765e7i) q^{23} +(2.20019e7 - 3.81084e7i) q^{25} +(3.03310e7 + 5.25349e7i) q^{26} +(-2.57079e6 + 4.54617e7i) q^{28} +7.92564e7 q^{29} +(5.46551e7 - 9.46654e7i) q^{31} +(1.67772e7 - 2.90590e7i) q^{32} -2.91467e8 q^{34} +(8.16918e7 + 5.35320e7i) q^{35} +(-3.38292e8 - 5.85938e8i) q^{37} +(3.97889e7 - 6.89163e7i) q^{38} +(-3.59863e7 - 6.23301e7i) q^{40} -2.52468e8 q^{41} -1.75201e9 q^{43} +(2.11526e8 + 3.66374e8i) q^{44} +(9.45496e8 - 1.63765e9i) q^{46} +(1.29188e9 + 2.23760e9i) q^{47} +(1.17541e9 - 1.59004e9i) q^{49} +1.40812e9 q^{50} +(-9.70593e8 + 1.68112e9i) q^{52} +(2.44697e9 - 4.23828e9i) q^{53} +9.07428e8 q^{55} +(-1.30100e9 + 6.56144e8i) q^{56} +(1.26810e9 + 2.19642e9i) q^{58} +(-1.46530e9 + 2.53797e9i) q^{59} +(-5.72843e8 - 9.92193e8i) q^{61} +3.49793e9 q^{62} +1.07374e9 q^{64} +(2.08187e9 + 3.60591e9i) q^{65} +(5.82061e9 - 1.00816e10i) q^{67} +(-4.66347e9 - 8.07736e9i) q^{68} +(-1.76455e8 + 3.12042e9i) q^{70} -5.24957e9 q^{71} +(8.75111e9 - 1.51574e10i) q^{73} +(1.08253e10 - 1.87500e10i) q^{74} +2.54649e9 q^{76} +(1.03720e9 - 1.83417e10i) q^{77} +(2.12040e10 + 3.67263e10i) q^{79} +(1.15156e9 - 1.99456e9i) q^{80} +(-4.03949e9 - 6.99661e9i) q^{82} +8.46684e9 q^{83} -2.00058e10 q^{85} +(-2.80322e10 - 4.85531e10i) q^{86} +(-6.76885e9 + 1.17240e10i) q^{88} +(-9.36269e9 - 1.62167e10i) q^{89} +(7.52655e10 - 3.79591e10i) q^{91} +6.05117e10 q^{92} +(-4.13402e10 + 7.16034e10i) q^{94} +(2.73104e9 - 4.73031e9i) q^{95} +1.03324e11 q^{97} +(6.28711e10 + 7.13333e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 96 q^{2} - 3072 q^{4} - 1045 q^{5} + 45731 q^{7} - 196608 q^{8} + 33440 q^{10} - 181565 q^{11} + 1186364 q^{13} + 703808 q^{14} - 3145728 q^{16} + 701848 q^{17} - 7893102 q^{19} + 2140160 q^{20} - 11620160 q^{22}+ \cdots - 30564771552 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 + 27.7128i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −512.000 + 886.810i −0.250000 + 0.433013i
\(5\) 1098.21 + 1902.16i 0.157164 + 0.272216i 0.933845 − 0.357678i \(-0.116432\pi\)
−0.776681 + 0.629894i \(0.783098\pi\)
\(6\) 0 0
\(7\) 39703.5 − 20023.9i 0.892873 − 0.450308i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) −35142.9 + 60869.3i −0.111132 + 0.192485i
\(11\) 206569. − 357788.i 0.386727 − 0.669831i −0.605280 − 0.796013i \(-0.706939\pi\)
0.992007 + 0.126181i \(0.0402721\pi\)
\(12\) 0 0
\(13\) 1.89569e6 1.41605 0.708025 − 0.706187i \(-0.249586\pi\)
0.708025 + 0.706187i \(0.249586\pi\)
\(14\) 1.19018e6 + 779913.i 0.591435 + 0.387563i
\(15\) 0 0
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) −4.55417e6 + 7.88805e6i −0.777929 + 1.34741i 0.155205 + 0.987882i \(0.450396\pi\)
−0.933134 + 0.359530i \(0.882937\pi\)
\(18\) 0 0
\(19\) −1.24340e6 − 2.15364e6i −0.115204 − 0.199539i 0.802657 − 0.596440i \(-0.203419\pi\)
−0.917861 + 0.396902i \(0.870085\pi\)
\(20\) −2.24914e6 −0.157164
\(21\) 0 0
\(22\) 1.32204e7 0.546915
\(23\) −2.95467e7 − 5.11765e7i −0.957208 − 1.65793i −0.729232 − 0.684267i \(-0.760122\pi\)
−0.227977 − 0.973667i \(-0.573211\pi\)
\(24\) 0 0
\(25\) 2.20019e7 − 3.81084e7i 0.450599 − 0.780461i
\(26\) 3.03310e7 + 5.25349e7i 0.500649 + 0.867150i
\(27\) 0 0
\(28\) −2.57079e6 + 4.54617e7i −0.0282291 + 0.499202i
\(29\) 7.92564e7 0.717539 0.358769 − 0.933426i \(-0.383196\pi\)
0.358769 + 0.933426i \(0.383196\pi\)
\(30\) 0 0
\(31\) 5.46551e7 − 9.46654e7i 0.342879 − 0.593884i −0.642087 − 0.766632i \(-0.721931\pi\)
0.984966 + 0.172747i \(0.0552645\pi\)
\(32\) 1.67772e7 − 2.90590e7i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.91467e8 −1.10016
\(35\) 8.16918e7 + 5.35320e7i 0.262908 + 0.172282i
\(36\) 0 0
\(37\) −3.38292e8 − 5.85938e8i −0.802014 − 1.38913i −0.918289 − 0.395912i \(-0.870429\pi\)
0.116275 − 0.993217i \(-0.462905\pi\)
\(38\) 3.97889e7 − 6.89163e7i 0.0814614 − 0.141095i
\(39\) 0 0
\(40\) −3.59863e7 − 6.23301e7i −0.0555658 − 0.0962427i
\(41\) −2.52468e8 −0.340327 −0.170163 − 0.985416i \(-0.554430\pi\)
−0.170163 + 0.985416i \(0.554430\pi\)
\(42\) 0 0
\(43\) −1.75201e9 −1.81744 −0.908720 − 0.417407i \(-0.862939\pi\)
−0.908720 + 0.417407i \(0.862939\pi\)
\(44\) 2.11526e8 + 3.66374e8i 0.193364 + 0.334916i
\(45\) 0 0
\(46\) 9.45496e8 − 1.63765e9i 0.676849 − 1.17234i
\(47\) 1.29188e9 + 2.23760e9i 0.821646 + 1.42313i 0.904456 + 0.426567i \(0.140277\pi\)
−0.0828105 + 0.996565i \(0.526390\pi\)
\(48\) 0 0
\(49\) 1.17541e9 − 1.59004e9i 0.594445 − 0.804136i
\(50\) 1.40812e9 0.637243
\(51\) 0 0
\(52\) −9.70593e8 + 1.68112e9i −0.354013 + 0.613168i
\(53\) 2.44697e9 − 4.23828e9i 0.803732 − 1.39210i −0.113412 − 0.993548i \(-0.536178\pi\)
0.917144 − 0.398557i \(-0.130489\pi\)
\(54\) 0 0
\(55\) 9.07428e8 0.243118
\(56\) −1.30100e9 + 6.56144e8i −0.315678 + 0.159208i
\(57\) 0 0
\(58\) 1.26810e9 + 2.19642e9i 0.253688 + 0.439401i
\(59\) −1.46530e9 + 2.53797e9i −0.266833 + 0.462168i −0.968042 − 0.250787i \(-0.919310\pi\)
0.701209 + 0.712955i \(0.252644\pi\)
\(60\) 0 0
\(61\) −5.72843e8 − 9.92193e8i −0.0868403 − 0.150412i 0.819334 − 0.573317i \(-0.194344\pi\)
−0.906174 + 0.422905i \(0.861010\pi\)
\(62\) 3.49793e9 0.484905
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 2.08187e9 + 3.60591e9i 0.222552 + 0.385471i
\(66\) 0 0
\(67\) 5.82061e9 − 1.00816e10i 0.526692 − 0.912258i −0.472824 − 0.881157i \(-0.656765\pi\)
0.999516 − 0.0311008i \(-0.00990130\pi\)
\(68\) −4.66347e9 − 8.07736e9i −0.388964 − 0.673706i
\(69\) 0 0
\(70\) −1.76455e8 + 3.12042e9i −0.0125486 + 0.221909i
\(71\) −5.24957e9 −0.345305 −0.172653 − 0.984983i \(-0.555234\pi\)
−0.172653 + 0.984983i \(0.555234\pi\)
\(72\) 0 0
\(73\) 8.75111e9 − 1.51574e10i 0.494069 − 0.855752i −0.505908 − 0.862587i \(-0.668842\pi\)
0.999977 + 0.00683535i \(0.00217578\pi\)
\(74\) 1.08253e10 − 1.87500e10i 0.567109 − 0.982262i
\(75\) 0 0
\(76\) 2.54649e9 0.115204
\(77\) 1.03720e9 − 1.83417e10i 0.0436679 − 0.772221i
\(78\) 0 0
\(79\) 2.12040e10 + 3.67263e10i 0.775297 + 1.34285i 0.934627 + 0.355628i \(0.115733\pi\)
−0.159331 + 0.987225i \(0.550934\pi\)
\(80\) 1.15156e9 − 1.99456e9i 0.0392909 − 0.0680539i
\(81\) 0 0
\(82\) −4.03949e9 − 6.99661e9i −0.120324 − 0.208407i
\(83\) 8.46684e9 0.235935 0.117967 − 0.993017i \(-0.462362\pi\)
0.117967 + 0.993017i \(0.462362\pi\)
\(84\) 0 0
\(85\) −2.00058e10 −0.489049
\(86\) −2.80322e10 − 4.85531e10i −0.642562 − 1.11295i
\(87\) 0 0
\(88\) −6.76885e9 + 1.17240e10i −0.136729 + 0.236821i
\(89\) −9.36269e9 − 1.62167e10i −0.177728 − 0.307834i 0.763374 − 0.645957i \(-0.223541\pi\)
−0.941102 + 0.338123i \(0.890208\pi\)
\(90\) 0 0
\(91\) 7.52655e10 − 3.79591e10i 1.26435 − 0.637659i
\(92\) 6.05117e10 0.957208
\(93\) 0 0
\(94\) −4.13402e10 + 7.16034e10i −0.580991 + 1.00631i
\(95\) 2.73104e9 − 4.73031e9i 0.0362117 − 0.0627205i
\(96\) 0 0
\(97\) 1.03324e11 1.22168 0.610839 − 0.791755i \(-0.290832\pi\)
0.610839 + 0.791755i \(0.290832\pi\)
\(98\) 6.28711e10 + 7.13333e9i 0.702599 + 0.0797167i
\(99\) 0 0
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 126.12.g.b.109.3 6
3.2 odd 2 42.12.e.a.25.1 ✓ 6
7.2 even 3 inner 126.12.g.b.37.3 6
21.2 odd 6 42.12.e.a.37.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.a.25.1 ✓ 6 3.2 odd 2
42.12.e.a.37.1 yes 6 21.2 odd 6
126.12.g.b.37.3 6 7.2 even 3 inner
126.12.g.b.109.3 6 1.1 even 1 trivial