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 = [8,128,0,-4096,-7504] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.2
Root \(152.619 + 264.344i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.12.g.e.109.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 - 27.7128i) q^{2} +(-512.000 - 886.810i) q^{4} +(-3667.11 + 6351.63i) q^{5} +(-28136.3 + 34433.6i) q^{7} -32768.0 q^{8} +(117348. + 203252. i) q^{10} +(24115.9 + 41769.9i) q^{11} -1.82289e6 q^{13} +(504070. + 1.33067e6i) q^{14} +(-524288. + 908093. i) q^{16} +(-3.35793e6 - 5.81611e6i) q^{17} +(-8.95659e6 + 1.55133e7i) q^{19} +7.51025e6 q^{20} +1.54342e6 q^{22} +(-1.04379e7 + 1.80789e7i) q^{23} +(-2.48139e6 - 4.29789e6i) q^{25} +(-2.91663e7 + 5.05174e7i) q^{26} +(4.49419e7 + 7.32159e6i) q^{28} -2.38808e7 q^{29} +(5.14984e7 + 8.91978e7i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} -2.14908e8 q^{34} +(-1.15530e8 - 3.04984e8i) q^{35} +(2.60667e8 - 4.51488e8i) q^{37} +(2.86611e8 + 4.96425e8i) q^{38} +(1.20164e8 - 2.08130e8i) q^{40} +1.25858e9 q^{41} -1.03048e9 q^{43} +(2.46947e7 - 4.27724e7i) q^{44} +(3.34012e8 + 5.78526e8i) q^{46} +(7.99740e8 - 1.38519e9i) q^{47} +(-3.94019e8 - 1.93767e9i) q^{49} -1.58809e8 q^{50} +(9.33320e8 + 1.61656e9i) q^{52} +(2.29236e9 + 3.97049e9i) q^{53} -3.53743e8 q^{55} +(9.21972e8 - 1.12832e9i) q^{56} +(-3.82093e8 + 6.61804e8i) q^{58} +(-2.88236e9 - 4.99240e9i) q^{59} +(2.91537e9 - 5.04957e9i) q^{61} +3.29590e9 q^{62} +1.07374e9 q^{64} +(6.68475e9 - 1.15783e10i) q^{65} +(-7.79178e9 - 1.34958e10i) q^{67} +(-3.43852e9 + 5.95570e9i) q^{68} +(-1.03004e10 - 1.67807e9i) q^{70} +7.49669e8 q^{71} +(5.51048e9 + 9.54443e9i) q^{73} +(-8.34133e9 - 1.44476e10i) q^{74} +1.83431e10 q^{76} +(-2.11682e9 - 3.44857e8i) q^{77} +(-1.78570e10 + 3.09292e10i) q^{79} +(-3.84525e9 - 6.66016e9i) q^{80} +(2.01373e10 - 3.48788e10i) q^{82} +3.77864e10 q^{83} +4.92557e10 q^{85} +(-1.64876e10 + 2.85574e10i) q^{86} +(-7.90229e8 - 1.36872e9i) q^{88} +(-8.88627e9 + 1.53915e10i) q^{89} +(5.12895e10 - 6.27687e10i) q^{91} +2.13768e10 q^{92} +(-2.55917e10 - 4.43261e10i) q^{94} +(-6.56897e10 - 1.13778e11i) q^{95} -8.79572e10 q^{97} +(-6.00026e10 - 2.00834e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 4096 q^{4} - 7504 q^{5} - 42224 q^{7} - 262144 q^{8} + 240128 q^{10} - 213026 q^{11} - 2609712 q^{13} - 1257536 q^{14} - 4194304 q^{16} - 8854244 q^{17} + 7232806 q^{19} + 15368192 q^{20}+ \cdots - 210290620288 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −3667.11 + 6351.63i −0.524795 + 0.908971i 0.474789 + 0.880100i \(0.342525\pi\)
−0.999583 + 0.0288710i \(0.990809\pi\)
\(6\) 0 0
\(7\) −28136.3 + 34433.6i −0.632745 + 0.774361i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) 117348. + 203252.i 0.371086 + 0.642740i
\(11\) 24115.9 + 41769.9i 0.0451485 + 0.0781995i 0.887717 − 0.460390i \(-0.152291\pi\)
−0.842568 + 0.538590i \(0.818957\pi\)
\(12\) 0 0
\(13\) −1.82289e6 −1.36167 −0.680836 − 0.732436i \(-0.738383\pi\)
−0.680836 + 0.732436i \(0.738383\pi\)
\(14\) 504070. + 1.33067e6i 0.250488 + 0.661253i
\(15\) 0 0
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −3.35793e6 − 5.81611e6i −0.573592 − 0.993490i −0.996193 − 0.0871746i \(-0.972216\pi\)
0.422601 − 0.906316i \(-0.361117\pi\)
\(18\) 0 0
\(19\) −8.95659e6 + 1.55133e7i −0.829847 + 1.43734i 0.0683108 + 0.997664i \(0.478239\pi\)
−0.898158 + 0.439673i \(0.855094\pi\)
\(20\) 7.51025e6 0.524795
\(21\) 0 0
\(22\) 1.54342e6 0.0638496
\(23\) −1.04379e7 + 1.80789e7i −0.338150 + 0.585693i −0.984085 − 0.177700i \(-0.943134\pi\)
0.645935 + 0.763393i \(0.276468\pi\)
\(24\) 0 0
\(25\) −2.48139e6 − 4.29789e6i −0.0508188 − 0.0880207i
\(26\) −2.91663e7 + 5.05174e7i −0.481423 + 0.833850i
\(27\) 0 0
\(28\) 4.49419e7 + 7.32159e6i 0.493494 + 0.0803963i
\(29\) −2.38808e7 −0.216202 −0.108101 − 0.994140i \(-0.534477\pi\)
−0.108101 + 0.994140i \(0.534477\pi\)
\(30\) 0 0
\(31\) 5.14984e7 + 8.91978e7i 0.323076 + 0.559583i 0.981121 − 0.193395i \(-0.0619499\pi\)
−0.658045 + 0.752978i \(0.728617\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −2.14908e8 −0.811182
\(35\) −1.15530e8 − 3.04984e8i −0.371810 − 0.981527i
\(36\) 0 0
\(37\) 2.60667e8 − 4.51488e8i 0.617982 − 1.07038i −0.371871 − 0.928284i \(-0.621284\pi\)
0.989853 − 0.142092i \(-0.0453830\pi\)
\(38\) 2.86611e8 + 4.96425e8i 0.586790 + 1.01635i
\(39\) 0 0
\(40\) 1.20164e8 − 2.08130e8i 0.185543 − 0.321370i
\(41\) 1.25858e9 1.69656 0.848281 − 0.529546i \(-0.177638\pi\)
0.848281 + 0.529546i \(0.177638\pi\)
\(42\) 0 0
\(43\) −1.03048e9 −1.06896 −0.534480 − 0.845181i \(-0.679492\pi\)
−0.534480 + 0.845181i \(0.679492\pi\)
\(44\) 2.46947e7 − 4.27724e7i 0.0225743 − 0.0390998i
\(45\) 0 0
\(46\) 3.34012e8 + 5.78526e8i 0.239108 + 0.414147i
\(47\) 7.99740e8 − 1.38519e9i 0.508640 − 0.880991i −0.491310 − 0.870985i \(-0.663482\pi\)
0.999950 − 0.0100058i \(-0.00318499\pi\)
\(48\) 0 0
\(49\) −3.94019e8 − 1.93767e9i −0.199268 − 0.979945i
\(50\) −1.58809e8 −0.0718686
\(51\) 0 0
\(52\) 9.33320e8 + 1.61656e9i 0.340418 + 0.589621i
\(53\) 2.29236e9 + 3.97049e9i 0.752949 + 1.30415i 0.946387 + 0.323034i \(0.104703\pi\)
−0.193438 + 0.981113i \(0.561964\pi\)
\(54\) 0 0
\(55\) −3.53743e8 −0.0947748
\(56\) 9.21972e8 − 1.12832e9i 0.223709 − 0.273778i
\(57\) 0 0
\(58\) −3.82093e8 + 6.61804e8i −0.0764390 + 0.132396i
\(59\) −2.88236e9 − 4.99240e9i −0.524883 − 0.909124i −0.999580 − 0.0289746i \(-0.990776\pi\)
0.474697 − 0.880149i \(-0.342558\pi\)
\(60\) 0 0
\(61\) 2.91537e9 − 5.04957e9i 0.441957 − 0.765491i −0.555878 − 0.831264i \(-0.687618\pi\)
0.997835 + 0.0657725i \(0.0209512\pi\)
\(62\) 3.29590e9 0.456898
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 6.68475e9 − 1.15783e10i 0.714598 − 1.23772i
\(66\) 0 0
\(67\) −7.79178e9 − 1.34958e10i −0.705059 − 1.22120i −0.966670 − 0.256025i \(-0.917587\pi\)
0.261611 − 0.965173i \(-0.415746\pi\)
\(68\) −3.43852e9 + 5.95570e9i −0.286796 + 0.496745i
\(69\) 0 0
\(70\) −1.03004e10 − 1.67807e9i −0.732515 − 0.119336i
\(71\) 7.49669e8 0.0493116 0.0246558 − 0.999696i \(-0.492151\pi\)
0.0246558 + 0.999696i \(0.492151\pi\)
\(72\) 0 0
\(73\) 5.51048e9 + 9.54443e9i 0.311110 + 0.538858i 0.978603 − 0.205758i \(-0.0659660\pi\)
−0.667493 + 0.744616i \(0.732633\pi\)
\(74\) −8.34133e9 − 1.44476e10i −0.436979 − 0.756870i
\(75\) 0 0
\(76\) 1.83431e10 0.829847
\(77\) −2.11682e9 − 3.44857e8i −0.0891221 − 0.0145191i
\(78\) 0 0
\(79\) −1.78570e10 + 3.09292e10i −0.652918 + 1.13089i 0.329493 + 0.944158i \(0.393122\pi\)
−0.982411 + 0.186730i \(0.940211\pi\)
\(80\) −3.84525e9 − 6.66016e9i −0.131199 − 0.227243i
\(81\) 0 0
\(82\) 2.01373e10 − 3.48788e10i 0.599825 − 1.03893i
\(83\) 3.77864e10 1.05295 0.526473 − 0.850192i \(-0.323514\pi\)
0.526473 + 0.850192i \(0.323514\pi\)
\(84\) 0 0
\(85\) 4.92557e10 1.20407
\(86\) −1.64876e10 + 2.85574e10i −0.377934 + 0.654601i
\(87\) 0 0
\(88\) −7.90229e8 − 1.36872e9i −0.0159624 − 0.0276477i
\(89\) −8.88627e9 + 1.53915e10i −0.168684 + 0.292170i −0.937957 − 0.346750i \(-0.887285\pi\)
0.769273 + 0.638920i \(0.220618\pi\)
\(90\) 0 0
\(91\) 5.12895e10 − 6.27687e10i 0.861590 − 1.05442i
\(92\) 2.13768e10 0.338150
\(93\) 0 0
\(94\) −2.55917e10 − 4.43261e10i −0.359663 − 0.622955i
\(95\) −6.56897e10 − 1.13778e11i −0.870999 − 1.50861i
\(96\) 0 0
\(97\) −8.79572e10 −1.03998 −0.519992 − 0.854171i \(-0.674065\pi\)
−0.519992 + 0.854171i \(0.674065\pi\)
\(98\) −6.00026e10 − 2.00834e10i −0.670543 − 0.224436i
\(99\) 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.e.37.2 8
3.2 odd 2 14.12.c.a.9.1 ✓ 8
7.4 even 3 inner 126.12.g.e.109.2 8
12.11 even 2 112.12.i.a.65.4 8
21.2 odd 6 98.12.a.j.1.4 4
21.5 even 6 98.12.a.l.1.1 4
21.11 odd 6 14.12.c.a.11.1 yes 8
21.17 even 6 98.12.c.l.67.4 8
21.20 even 2 98.12.c.l.79.4 8
84.11 even 6 112.12.i.a.81.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.1 ✓ 8 3.2 odd 2
14.12.c.a.11.1 yes 8 21.11 odd 6
98.12.a.j.1.4 4 21.2 odd 6
98.12.a.l.1.1 4 21.5 even 6
98.12.c.l.67.4 8 21.17 even 6
98.12.c.l.79.4 8 21.20 even 2
112.12.i.a.65.4 8 12.11 even 2
112.12.i.a.81.4 8 84.11 even 6
126.12.g.e.37.2 8 1.1 even 1 trivial
126.12.g.e.109.2 8 7.4 even 3 inner