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 37.2
Root \(-19.2176 - 33.2859i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.12.g.b.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)
 
\(f(q)\) \(=\) \(q+(16.0000 - 27.7128i) q^{2} +(-512.000 - 886.810i) q^{4} +(405.223 - 701.867i) q^{5} +(24393.5 - 37179.1i) q^{7} -32768.0 q^{8} +(-12967.1 - 22459.7i) q^{10} +(-244705. - 423841. i) q^{11} -1.44732e6 q^{13} +(-640041. - 1.27088e6i) q^{14} +(-524288. + 908093. i) q^{16} +(1.35771e6 + 2.35163e6i) q^{17} +(-5.56973e6 + 9.64706e6i) q^{19} -829897. q^{20} -1.56611e7 q^{22} +(1.90394e7 - 3.29772e7i) q^{23} +(2.40857e7 + 4.17176e7i) q^{25} +(-2.31571e7 + 4.01093e7i) q^{26} +(-4.54603e7 - 2.59670e6i) q^{28} -5.21283e7 q^{29} +(-1.48670e8 - 2.57505e8i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} +8.68937e7 q^{34} +(-1.62100e7 - 3.21868e7i) q^{35} +(2.39642e7 - 4.15072e7i) q^{37} +(1.78231e8 + 3.08706e8i) q^{38} +(-1.32783e7 + 2.29988e7i) q^{40} +1.06860e9 q^{41} -1.74933e9 q^{43} +(-2.50578e8 + 4.34013e8i) q^{44} +(-6.09260e8 - 1.05527e9i) q^{46} +(-7.34828e8 + 1.27276e9i) q^{47} +(-7.87242e8 - 1.81386e9i) q^{49} +1.54148e9 q^{50} +(7.41027e8 + 1.28350e9i) q^{52} +(-3.29273e8 - 5.70317e8i) q^{53} -3.96640e8 q^{55} +(-7.99326e8 + 1.21828e9i) q^{56} +(-8.34053e8 + 1.44462e9i) q^{58} +(2.49057e9 + 4.31379e9i) q^{59} +(-4.24192e9 + 7.34722e9i) q^{61} -9.51490e9 q^{62} +1.07374e9 q^{64} +(-5.86487e8 + 1.01583e9i) q^{65} +(-7.76132e8 - 1.34430e9i) q^{67} +(1.39030e9 - 2.40807e9i) q^{68} +(-1.15135e9 - 6.57652e7i) q^{70} +8.97099e9 q^{71} +(9.91735e9 + 1.71774e10i) q^{73} +(-7.66854e8 - 1.32823e9i) q^{74} +1.14068e10 q^{76} +(-2.17272e10 - 1.24106e9i) q^{77} +(-9.01333e9 + 1.56115e10i) q^{79} +(4.24907e8 + 7.35961e8i) q^{80} +(1.70975e10 - 2.96138e10i) q^{82} -3.77221e10 q^{83} +2.20071e9 q^{85} +(-2.79893e10 + 4.84788e10i) q^{86} +(8.01848e9 + 1.38884e10i) q^{88} +(5.93170e9 - 1.02740e10i) q^{89} +(-3.53052e10 + 5.38100e10i) q^{91} -3.89926e10 q^{92} +(2.35145e10 + 4.07283e10i) q^{94} +(4.51397e9 + 7.81842e9i) q^{95} -2.67922e9 q^{97} +(-6.28629e10 - 7.20500e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 96 q^{2} - 3072 q^{4} - 1045 q^{5} + 45731 q^{7} - 196608 q^{8} + 33440 q^{10} - 181565 q^{11} + 1186364 q^{13} + 703808 q^{14} - 3145728 q^{16} + 701848 q^{17} - 7893102 q^{19} + 2140160 q^{20} - 11620160 q^{22}+ \cdots - 30564771552 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 405.223 − 701.867i 0.0579908 − 0.100443i −0.835573 − 0.549380i \(-0.814864\pi\)
0.893563 + 0.448937i \(0.148197\pi\)
\(6\) 0 0
\(7\) 24393.5 − 37179.1i 0.548573 − 0.836102i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) −12967.1 − 22459.7i −0.0410057 − 0.0710239i
\(11\) −244705. − 423841.i −0.458123 − 0.793493i 0.540738 − 0.841191i \(-0.318145\pi\)
−0.998862 + 0.0476979i \(0.984812\pi\)
\(12\) 0 0
\(13\) −1.44732e6 −1.08112 −0.540562 − 0.841304i \(-0.681789\pi\)
−0.540562 + 0.841304i \(0.681789\pi\)
\(14\) −640041. − 1.27088e6i −0.318056 − 0.631538i
\(15\) 0 0
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 1.35771e6 + 2.35163e6i 0.231920 + 0.401698i 0.958373 − 0.285518i \(-0.0921658\pi\)
−0.726453 + 0.687216i \(0.758832\pi\)
\(18\) 0 0
\(19\) −5.56973e6 + 9.64706e6i −0.516047 + 0.893820i 0.483779 + 0.875190i \(0.339264\pi\)
−0.999826 + 0.0186301i \(0.994069\pi\)
\(20\) −829897. −0.0579908
\(21\) 0 0
\(22\) −1.56611e7 −0.647884
\(23\) 1.90394e7 − 3.29772e7i 0.616808 − 1.06834i −0.373257 − 0.927728i \(-0.621759\pi\)
0.990064 − 0.140614i \(-0.0449077\pi\)
\(24\) 0 0
\(25\) 2.40857e7 + 4.17176e7i 0.493274 + 0.854376i
\(26\) −2.31571e7 + 4.01093e7i −0.382235 + 0.662051i
\(27\) 0 0
\(28\) −4.54603e7 − 2.59670e6i −0.499186 − 0.0285137i
\(29\) −5.21283e7 −0.471938 −0.235969 − 0.971761i \(-0.575826\pi\)
−0.235969 + 0.971761i \(0.575826\pi\)
\(30\) 0 0
\(31\) −1.48670e8 − 2.57505e8i −0.932685 − 1.61546i −0.778711 − 0.627383i \(-0.784126\pi\)
−0.153974 − 0.988075i \(-0.549207\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 8.68937e7 0.327985
\(35\) −1.62100e7 − 3.21868e7i −0.0521684 − 0.103587i
\(36\) 0 0
\(37\) 2.39642e7 − 4.15072e7i 0.0568137 − 0.0984043i −0.836220 − 0.548395i \(-0.815239\pi\)
0.893033 + 0.449990i \(0.148573\pi\)
\(38\) 1.78231e8 + 3.08706e8i 0.364901 + 0.632026i
\(39\) 0 0
\(40\) −1.32783e7 + 2.29988e7i −0.0205028 + 0.0355120i
\(41\) 1.06860e9 1.44046 0.720232 − 0.693733i \(-0.244035\pi\)
0.720232 + 0.693733i \(0.244035\pi\)
\(42\) 0 0
\(43\) −1.74933e9 −1.81466 −0.907329 − 0.420421i \(-0.861882\pi\)
−0.907329 + 0.420421i \(0.861882\pi\)
\(44\) −2.50578e8 + 4.34013e8i −0.229062 + 0.396746i
\(45\) 0 0
\(46\) −6.09260e8 − 1.05527e9i −0.436149 − 0.755432i
\(47\) −7.34828e8 + 1.27276e9i −0.467356 + 0.809484i −0.999304 − 0.0372927i \(-0.988127\pi\)
0.531949 + 0.846777i \(0.321460\pi\)
\(48\) 0 0
\(49\) −7.87242e8 − 1.81386e9i −0.398134 − 0.917327i
\(50\) 1.54148e9 0.697595
\(51\) 0 0
\(52\) 7.41027e8 + 1.28350e9i 0.270281 + 0.468141i
\(53\) −3.29273e8 − 5.70317e8i −0.108153 − 0.187326i 0.806869 − 0.590730i \(-0.201160\pi\)
−0.915022 + 0.403404i \(0.867827\pi\)
\(54\) 0 0
\(55\) −3.96640e8 −0.106268
\(56\) −7.99326e8 + 1.21828e9i −0.193950 + 0.295607i
\(57\) 0 0
\(58\) −8.34053e8 + 1.44462e9i −0.166855 + 0.289002i
\(59\) 2.49057e9 + 4.31379e9i 0.453536 + 0.785547i 0.998603 − 0.0528452i \(-0.0168290\pi\)
−0.545067 + 0.838393i \(0.683496\pi\)
\(60\) 0 0
\(61\) −4.24192e9 + 7.34722e9i −0.643055 + 1.11380i 0.341692 + 0.939812i \(0.389000\pi\)
−0.984747 + 0.173992i \(0.944333\pi\)
\(62\) −9.51490e9 −1.31902
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −5.86487e8 + 1.01583e9i −0.0626953 + 0.108591i
\(66\) 0 0
\(67\) −7.76132e8 − 1.34430e9i −0.0702302 − 0.121642i 0.828772 − 0.559587i \(-0.189040\pi\)
−0.899002 + 0.437944i \(0.855707\pi\)
\(68\) 1.39030e9 − 2.40807e9i 0.115960 − 0.200849i
\(69\) 0 0
\(70\) −1.15135e9 − 6.57652e7i −0.0818779 − 0.00467689i
\(71\) 8.97099e9 0.590091 0.295046 − 0.955483i \(-0.404665\pi\)
0.295046 + 0.955483i \(0.404665\pi\)
\(72\) 0 0
\(73\) 9.91735e9 + 1.71774e10i 0.559912 + 0.969797i 0.997503 + 0.0706223i \(0.0224985\pi\)
−0.437591 + 0.899174i \(0.644168\pi\)
\(74\) −7.66854e8 − 1.32823e9i −0.0401734 − 0.0695823i
\(75\) 0 0
\(76\) 1.14068e10 0.516047
\(77\) −2.17272e10 − 1.24106e9i −0.914756 − 0.0522511i
\(78\) 0 0
\(79\) −9.01333e9 + 1.56115e10i −0.329561 + 0.570817i −0.982425 − 0.186659i \(-0.940234\pi\)
0.652864 + 0.757475i \(0.273567\pi\)
\(80\) 4.24907e8 + 7.35961e8i 0.0144977 + 0.0251107i
\(81\) 0 0
\(82\) 1.70975e10 − 2.96138e10i 0.509281 − 0.882101i
\(83\) −3.77221e10 −1.05115 −0.525577 − 0.850746i \(-0.676151\pi\)
−0.525577 + 0.850746i \(0.676151\pi\)
\(84\) 0 0
\(85\) 2.20071e9 0.0537970
\(86\) −2.79893e10 + 4.84788e10i −0.641578 + 1.11125i
\(87\) 0 0
\(88\) 8.01848e9 + 1.38884e10i 0.161971 + 0.280542i
\(89\) 5.93170e9 − 1.02740e10i 0.112599 − 0.195027i −0.804218 − 0.594334i \(-0.797416\pi\)
0.916817 + 0.399307i \(0.130749\pi\)
\(90\) 0 0
\(91\) −3.53052e10 + 5.38100e10i −0.593076 + 0.903931i
\(92\) −3.89926e10 −0.616808
\(93\) 0 0
\(94\) 2.35145e10 + 4.07283e10i 0.330470 + 0.572392i
\(95\) 4.51397e9 + 7.81842e9i 0.0598520 + 0.103667i
\(96\) 0 0
\(97\) −2.67922e9 −0.0316784 −0.0158392 − 0.999875i \(-0.505042\pi\)
−0.0158392 + 0.999875i \(0.505042\pi\)
\(98\) −6.28629e10 − 7.20500e9i −0.702508 − 0.0805176i
\(99\) 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.37.2 6
3.2 odd 2 42.12.e.a.37.2 yes 6
7.4 even 3 inner 126.12.g.b.109.2 6
21.11 odd 6 42.12.e.a.25.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.a.25.2 ✓ 6 21.11 odd 6
42.12.e.a.37.2 yes 6 3.2 odd 2
126.12.g.b.37.2 6 1.1 even 1 trivial
126.12.g.b.109.2 6 7.4 even 3 inner