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,-4096,1420] 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: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(-73.4634 + 127.242i\) of defining polynomial
Character \(\chi\) \(=\) 126.109
Dual form 126.12.g.d.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-16.0000 - 27.7128i) q^{2} +(-512.000 + 886.810i) q^{4} +(-6385.96 - 11060.8i) q^{5} +(-43747.2 - 7969.08i) q^{7} +32768.0 q^{8} +(-204351. + 353946. i) q^{10} +(237628. - 411583. i) q^{11} -1.02512e6 q^{13} +(479110. + 1.33986e6i) q^{14} +(-524288. - 908093. i) q^{16} +(-3.50148e6 + 6.06475e6i) q^{17} +(1.99041e6 + 3.44749e6i) q^{19} +1.30784e7 q^{20} -1.52082e7 q^{22} +(7.54845e6 + 1.30743e7i) q^{23} +(-5.71469e7 + 9.89814e7i) q^{25} +(1.64019e7 + 2.84090e7i) q^{26} +(2.94656e7 - 3.47153e7i) q^{28} +5.56637e7 q^{29} +(-7.95734e7 + 1.37825e8i) q^{31} +(-1.67772e7 + 2.90590e7i) q^{32} +2.24095e8 q^{34} +(1.91224e8 + 5.34770e8i) q^{35} +(-3.59905e8 - 6.23374e8i) q^{37} +(6.36930e7 - 1.10320e8i) q^{38} +(-2.09255e8 - 3.62441e8i) q^{40} +1.45784e8 q^{41} -1.76669e9 q^{43} +(2.43331e8 + 4.21461e8i) q^{44} +(2.41550e8 - 4.18377e8i) q^{46} +(-7.02041e8 - 1.21597e9i) q^{47} +(1.85031e9 + 6.97250e8i) q^{49} +3.65740e9 q^{50} +(5.24862e8 - 9.09088e8i) q^{52} +(-6.25567e8 + 1.08351e9i) q^{53} -6.06992e9 q^{55} +(-1.43351e9 - 2.61131e8i) q^{56} +(-8.90618e8 - 1.54260e9i) q^{58} +(6.66848e8 - 1.15501e9i) q^{59} +(4.39590e9 + 7.61393e9i) q^{61} +5.09270e9 q^{62} +1.07374e9 q^{64} +(6.54639e9 + 1.13387e10i) q^{65} +(-8.49061e9 + 1.47062e10i) q^{67} +(-3.58552e9 - 6.21030e9i) q^{68} +(1.17604e10 - 1.38557e10i) q^{70} +1.62499e10 q^{71} +(1.06939e10 - 1.85223e10i) q^{73} +(-1.15170e10 + 1.99480e10i) q^{74} -4.07635e9 q^{76} +(-1.36755e10 + 1.61119e10i) q^{77} +(2.05351e10 + 3.55678e10i) q^{79} +(-6.69616e9 + 1.15981e10i) q^{80} +(-2.33255e9 - 4.04010e9i) q^{82} -1.96280e10 q^{83} +8.94413e10 q^{85} +(2.82670e10 + 4.89598e10i) q^{86} +(7.78658e9 - 1.34868e10i) q^{88} +(-3.88169e10 - 6.72329e10i) q^{89} +(4.48462e10 + 8.16928e9i) q^{91} -1.54592e10 q^{92} +(-2.24653e10 + 3.89111e10i) q^{94} +(2.54213e10 - 4.40310e10i) q^{95} -6.65701e10 q^{97} +(-1.02823e10 - 6.24334e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 4096 q^{4} + 1420 q^{5} - 66362 q^{7} + 262144 q^{8} + 45440 q^{10} + 861962 q^{11} + 836748 q^{13} + 1361216 q^{14} - 4194304 q^{16} - 5647852 q^{17} + 16668730 q^{19} - 2908160 q^{20}+ \cdots + 80526496576 q^{98}+O(q^{100}) \) 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
\(5\) −6385.96 − 11060.8i −0.913884 − 1.58289i −0.808527 − 0.588460i \(-0.799735\pi\)
−0.105358 − 0.994434i \(-0.533599\pi\)
\(6\) 0 0
\(7\) −43747.2 − 7969.08i −0.983810 − 0.179213i
\(8\) 32768.0 0.353553
\(9\) 0 0
\(10\) −204351. + 353946.i −0.646214 + 1.11928i
\(11\) 237628. − 411583.i 0.444874 − 0.770544i −0.553169 − 0.833069i \(-0.686582\pi\)
0.998043 + 0.0625244i \(0.0199151\pi\)
\(12\) 0 0
\(13\) −1.02512e6 −0.765750 −0.382875 − 0.923800i \(-0.625066\pi\)
−0.382875 + 0.923800i \(0.625066\pi\)
\(14\) 479110. + 1.33986e6i 0.238085 + 0.665820i
\(15\) 0 0
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) −3.50148e6 + 6.06475e6i −0.598113 + 1.03596i 0.394987 + 0.918687i \(0.370749\pi\)
−0.993100 + 0.117275i \(0.962584\pi\)
\(18\) 0 0
\(19\) 1.99041e6 + 3.44749e6i 0.184415 + 0.319417i 0.943379 − 0.331716i \(-0.107628\pi\)
−0.758964 + 0.651133i \(0.774294\pi\)
\(20\) 1.30784e7 0.913884
\(21\) 0 0
\(22\) −1.52082e7 −0.629147
\(23\) 7.54845e6 + 1.30743e7i 0.244543 + 0.423560i 0.962003 − 0.273039i \(-0.0880289\pi\)
−0.717460 + 0.696599i \(0.754696\pi\)
\(24\) 0 0
\(25\) −5.71469e7 + 9.89814e7i −1.17037 + 2.02714i
\(26\) 1.64019e7 + 2.84090e7i 0.270734 + 0.468924i
\(27\) 0 0
\(28\) 2.94656e7 − 3.47153e7i 0.323554 − 0.381199i
\(29\) 5.56637e7 0.503945 0.251972 − 0.967734i \(-0.418921\pi\)
0.251972 + 0.967734i \(0.418921\pi\)
\(30\) 0 0
\(31\) −7.95734e7 + 1.37825e8i −0.499204 + 0.864647i −1.00000 0.000918421i \(-0.999708\pi\)
0.500795 + 0.865566i \(0.333041\pi\)
\(32\) −1.67772e7 + 2.90590e7i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 2.24095e8 0.845859
\(35\) 1.91224e8 + 5.34770e8i 0.615414 + 1.72105i
\(36\) 0 0
\(37\) −3.59905e8 − 6.23374e8i −0.853254 − 1.47788i −0.878256 − 0.478192i \(-0.841292\pi\)
0.0250016 − 0.999687i \(-0.492041\pi\)
\(38\) 6.36930e7 − 1.10320e8i 0.130401 − 0.225862i
\(39\) 0 0
\(40\) −2.09255e8 − 3.62441e8i −0.323107 − 0.559638i
\(41\) 1.45784e8 0.196517 0.0982584 − 0.995161i \(-0.468673\pi\)
0.0982584 + 0.995161i \(0.468673\pi\)
\(42\) 0 0
\(43\) −1.76669e9 −1.83266 −0.916332 − 0.400419i \(-0.868865\pi\)
−0.916332 + 0.400419i \(0.868865\pi\)
\(44\) 2.43331e8 + 4.21461e8i 0.222437 + 0.385272i
\(45\) 0 0
\(46\) 2.41550e8 − 4.18377e8i 0.172918 − 0.299502i
\(47\) −7.02041e8 − 1.21597e9i −0.446503 − 0.773366i 0.551653 − 0.834074i \(-0.313997\pi\)
−0.998156 + 0.0607083i \(0.980664\pi\)
\(48\) 0 0
\(49\) 1.85031e9 + 6.97250e8i 0.935766 + 0.352623i
\(50\) 3.65740e9 1.65515
\(51\) 0 0
\(52\) 5.24862e8 − 9.09088e8i 0.191438 − 0.331580i
\(53\) −6.25567e8 + 1.08351e9i −0.205474 + 0.355891i −0.950284 − 0.311386i \(-0.899207\pi\)
0.744810 + 0.667277i \(0.232540\pi\)
\(54\) 0 0
\(55\) −6.06992e9 −1.62625
\(56\) −1.43351e9 − 2.61131e8i −0.347829 − 0.0633613i
\(57\) 0 0
\(58\) −8.90618e8 − 1.54260e9i −0.178171 − 0.308602i
\(59\) 6.66848e8 − 1.15501e9i 0.121434 − 0.210330i −0.798899 − 0.601465i \(-0.794584\pi\)
0.920333 + 0.391135i \(0.127917\pi\)
\(60\) 0 0
\(61\) 4.39590e9 + 7.61393e9i 0.666398 + 1.15424i 0.978904 + 0.204320i \(0.0654983\pi\)
−0.312506 + 0.949916i \(0.601168\pi\)
\(62\) 5.09270e9 0.705982
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 6.54639e9 + 1.13387e10i 0.699807 + 1.21210i
\(66\) 0 0
\(67\) −8.49061e9 + 1.47062e10i −0.768294 + 1.33072i 0.170194 + 0.985411i \(0.445561\pi\)
−0.938488 + 0.345313i \(0.887773\pi\)
\(68\) −3.58552e9 − 6.21030e9i −0.299056 − 0.517981i
\(69\) 0 0
\(70\) 1.17604e10 − 1.38557e10i 0.836340 − 0.985345i
\(71\) 1.62499e10 1.06888 0.534441 − 0.845206i \(-0.320522\pi\)
0.534441 + 0.845206i \(0.320522\pi\)
\(72\) 0 0
\(73\) 1.06939e10 − 1.85223e10i 0.603752 − 1.04573i −0.388495 − 0.921451i \(-0.627005\pi\)
0.992247 − 0.124279i \(-0.0396618\pi\)
\(74\) −1.15170e10 + 1.99480e10i −0.603342 + 1.04502i
\(75\) 0 0
\(76\) −4.07635e9 −0.184415
\(77\) −1.36755e10 + 1.61119e10i −0.575763 + 0.678342i
\(78\) 0 0
\(79\) 2.05351e10 + 3.55678e10i 0.750840 + 1.30049i 0.947416 + 0.320004i \(0.103684\pi\)
−0.196577 + 0.980488i \(0.562982\pi\)
\(80\) −6.69616e9 + 1.15981e10i −0.228471 + 0.395723i
\(81\) 0 0
\(82\) −2.33255e9 − 4.04010e9i −0.0694792 − 0.120342i
\(83\) −1.96280e10 −0.546947 −0.273474 − 0.961879i \(-0.588173\pi\)
−0.273474 + 0.961879i \(0.588173\pi\)
\(84\) 0 0
\(85\) 8.94413e10 2.18642
\(86\) 2.82670e10 + 4.89598e10i 0.647945 + 1.12227i
\(87\) 0 0
\(88\) 7.78658e9 − 1.34868e10i 0.157287 − 0.272429i
\(89\) −3.88169e10 − 6.72329e10i −0.736845 − 1.27625i −0.953909 − 0.300095i \(-0.902981\pi\)
0.217064 − 0.976157i \(-0.430352\pi\)
\(90\) 0 0
\(91\) 4.48462e10 + 8.16928e9i 0.753353 + 0.137232i
\(92\) −1.54592e10 −0.244543
\(93\) 0 0
\(94\) −2.24653e10 + 3.89111e10i −0.315725 + 0.546852i
\(95\) 2.54213e10 − 4.40310e10i 0.337069 − 0.583820i
\(96\) 0 0
\(97\) −6.65701e10 −0.787109 −0.393554 − 0.919301i \(-0.628755\pi\)
−0.393554 + 0.919301i \(0.628755\pi\)
\(98\) −1.02823e10 − 6.24334e10i −0.114907 − 0.697708i
\(99\) 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.d.109.1 8
3.2 odd 2 42.12.e.d.25.4 ✓ 8
7.2 even 3 inner 126.12.g.d.37.1 8
21.2 odd 6 42.12.e.d.37.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.4 ✓ 8 3.2 odd 2
42.12.e.d.37.4 yes 8 21.2 odd 6
126.12.g.d.37.1 8 7.2 even 3 inner
126.12.g.d.109.1 8 1.1 even 1 trivial