Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,12,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-266] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(86.0544362227\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 65.3
Root \(64.1153 + 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 112.65
Dual form 112.12.i.a.81.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(94.7305 + 164.078i) q^{3} +(-3211.13 + 5561.84i) q^{5} +(9575.58 + 43423.9i) q^{7} +(70625.8 - 122327. i) q^{9} +(-459958. - 796671. i) q^{11} +1.22771e6 q^{13} -1.21677e6 q^{15} +(3.31206e6 + 5.73665e6i) q^{17} +(-9.33141e6 + 1.61625e7i) q^{19} +(-6.21781e6 + 5.68471e6i) q^{21} +(-1.43390e7 + 2.48358e7i) q^{23} +(3.79132e6 + 6.56676e6i) q^{25} +6.03241e7 q^{27} -7.73286e6 q^{29} +(5.68702e7 + 9.85021e7i) q^{31} +(8.71441e7 - 1.50938e8i) q^{33} +(-2.72265e8 - 8.61820e7i) q^{35} +(9.50304e7 - 1.64597e8i) q^{37} +(1.16302e8 + 2.01441e8i) q^{39} -1.74512e8 q^{41} -1.83161e9 q^{43} +(4.53577e8 + 7.85619e8i) q^{45} +(5.75867e8 - 9.97431e8i) q^{47} +(-1.79394e9 + 8.31618e8i) q^{49} +(-6.27506e8 + 1.08687e9i) q^{51} +(-3.18521e8 - 5.51694e8i) q^{53} +5.90795e9 q^{55} -3.53588e9 q^{57} +(-1.39516e9 - 2.41649e9i) q^{59} +(2.60037e9 - 4.50397e9i) q^{61} +(5.98821e9 + 1.89549e9i) q^{63} +(-3.94234e9 + 6.82834e9i) q^{65} +(-3.19152e9 - 5.52788e9i) q^{67} -5.43335e9 q^{69} -2.41383e10 q^{71} +(8.18348e9 + 1.41742e10i) q^{73} +(-7.18307e8 + 1.24414e9i) q^{75} +(3.01902e10 - 2.76018e10i) q^{77} +(-1.91435e9 + 3.31575e9i) q^{79} +(-6.79660e9 - 1.17721e10i) q^{81} +2.84759e9 q^{83} -4.25418e10 q^{85} +(-7.32538e8 - 1.26879e9i) q^{87} +(1.35210e10 - 2.34190e10i) q^{89} +(1.17561e10 + 5.33120e10i) q^{91} +(-1.07747e10 + 1.86623e10i) q^{93} +(-5.99288e10 - 1.03800e11i) q^{95} -1.49558e11 q^{97} -1.29940e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 266 q^{3} + 7504 q^{5} + 42224 q^{7} - 123520 q^{9} - 213026 q^{11} - 2609712 q^{13} - 2275500 q^{15} + 8854244 q^{17} - 7232806 q^{19} + 49275128 q^{21} + 10649134 q^{23} - 119407256 q^{25} - 19012196 q^{27}+ \cdots - 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 94.7305 + 164.078i 0.225073 + 0.389838i 0.956341 − 0.292252i \(-0.0944047\pi\)
−0.731268 + 0.682090i \(0.761071\pi\)
\(4\) 0 0
\(5\) −3211.13 + 5561.84i −0.459540 + 0.795946i −0.998937 − 0.0461054i \(-0.985319\pi\)
0.539397 + 0.842052i \(0.318652\pi\)
\(6\) 0 0
\(7\) 9575.58 + 43423.9i 0.215341 + 0.976539i
\(8\) 0 0
\(9\) 70625.8 − 122327.i 0.398684 − 0.690542i
\(10\) 0 0
\(11\) −459958. − 796671.i −0.861110 − 1.49149i −0.870859 − 0.491533i \(-0.836437\pi\)
0.00974918 − 0.999952i \(-0.496897\pi\)
\(12\) 0 0
\(13\) 1.22771e6 0.917081 0.458541 − 0.888673i \(-0.348372\pi\)
0.458541 + 0.888673i \(0.348372\pi\)
\(14\) 0 0
\(15\) −1.21677e6 −0.413720
\(16\) 0 0
\(17\) 3.31206e6 + 5.73665e6i 0.565755 + 0.979917i 0.996979 + 0.0776717i \(0.0247486\pi\)
−0.431224 + 0.902245i \(0.641918\pi\)
\(18\) 0 0
\(19\) −9.33141e6 + 1.61625e7i −0.864574 + 1.49749i 0.00289490 + 0.999996i \(0.499079\pi\)
−0.867469 + 0.497491i \(0.834255\pi\)
\(20\) 0 0
\(21\) −6.21781e6 + 5.68471e6i −0.332224 + 0.303740i
\(22\) 0 0
\(23\) −1.43390e7 + 2.48358e7i −0.464531 + 0.804592i −0.999180 − 0.0404825i \(-0.987111\pi\)
0.534649 + 0.845074i \(0.320444\pi\)
\(24\) 0 0
\(25\) 3.79132e6 + 6.56676e6i 0.0776462 + 0.134487i
\(26\) 0 0
\(27\) 6.03241e7 0.809078
\(28\) 0 0
\(29\) −7.73286e6 −0.0700086 −0.0350043 − 0.999387i \(-0.511144\pi\)
−0.0350043 + 0.999387i \(0.511144\pi\)
\(30\) 0 0
\(31\) 5.68702e7 + 9.85021e7i 0.356776 + 0.617954i 0.987420 − 0.158118i \(-0.0505427\pi\)
−0.630644 + 0.776072i \(0.717209\pi\)
\(32\) 0 0
\(33\) 8.71441e7 − 1.50938e8i 0.387625 − 0.671386i
\(34\) 0 0
\(35\) −2.72265e8 − 8.61820e7i −0.876230 − 0.277359i
\(36\) 0 0
\(37\) 9.50304e7 − 1.64597e8i 0.225296 − 0.390224i −0.731112 − 0.682257i \(-0.760999\pi\)
0.956408 + 0.292033i \(0.0943318\pi\)
\(38\) 0 0
\(39\) 1.16302e8 + 2.01441e8i 0.206410 + 0.357513i
\(40\) 0 0
\(41\) −1.74512e8 −0.235241 −0.117621 − 0.993059i \(-0.537527\pi\)
−0.117621 + 0.993059i \(0.537527\pi\)
\(42\) 0 0
\(43\) −1.83161e9 −1.90001 −0.950005 − 0.312235i \(-0.898922\pi\)
−0.950005 + 0.312235i \(0.898922\pi\)
\(44\) 0 0
\(45\) 4.53577e8 + 7.85619e8i 0.366423 + 0.634663i
\(46\) 0 0
\(47\) 5.75867e8 − 9.97431e8i 0.366255 − 0.634373i −0.622721 − 0.782444i \(-0.713973\pi\)
0.988977 + 0.148071i \(0.0473063\pi\)
\(48\) 0 0
\(49\) −1.79394e9 + 8.31618e8i −0.907257 + 0.420577i
\(50\) 0 0
\(51\) −6.27506e8 + 1.08687e9i −0.254672 + 0.441105i
\(52\) 0 0
\(53\) −3.18521e8 − 5.51694e8i −0.104621 − 0.181209i 0.808962 − 0.587861i \(-0.200030\pi\)
−0.913583 + 0.406651i \(0.866696\pi\)
\(54\) 0 0
\(55\) 5.90795e9 1.58286
\(56\) 0 0
\(57\) −3.53588e9 −0.778369
\(58\) 0 0
\(59\) −1.39516e9 − 2.41649e9i −0.254061 − 0.440047i 0.710579 − 0.703618i \(-0.248433\pi\)
−0.964640 + 0.263571i \(0.915100\pi\)
\(60\) 0 0
\(61\) 2.60037e9 − 4.50397e9i 0.394204 − 0.682781i −0.598795 − 0.800902i \(-0.704354\pi\)
0.992999 + 0.118121i \(0.0376870\pi\)
\(62\) 0 0
\(63\) 5.98821e9 + 1.89549e9i 0.760194 + 0.240629i
\(64\) 0 0
\(65\) −3.94234e9 + 6.82834e9i −0.421435 + 0.729948i
\(66\) 0 0
\(67\) −3.19152e9 − 5.52788e9i −0.288793 − 0.500204i 0.684729 − 0.728798i \(-0.259920\pi\)
−0.973522 + 0.228594i \(0.926587\pi\)
\(68\) 0 0
\(69\) −5.43335e9 −0.418214
\(70\) 0 0
\(71\) −2.41383e10 −1.58776 −0.793880 − 0.608074i \(-0.791942\pi\)
−0.793880 + 0.608074i \(0.791942\pi\)
\(72\) 0 0
\(73\) 8.18348e9 + 1.41742e10i 0.462021 + 0.800245i 0.999062 − 0.0433121i \(-0.0137910\pi\)
−0.537040 + 0.843557i \(0.680458\pi\)
\(74\) 0 0
\(75\) −7.18307e8 + 1.24414e9i −0.0349521 + 0.0605388i
\(76\) 0 0
\(77\) 3.01902e10 − 2.76018e10i 1.27106 − 1.16208i
\(78\) 0 0
\(79\) −1.91435e9 + 3.31575e9i −0.0699958 + 0.121236i −0.898899 − 0.438155i \(-0.855632\pi\)
0.828903 + 0.559392i \(0.188965\pi\)
\(80\) 0 0
\(81\) −6.79660e9 − 1.17721e10i −0.216583 − 0.375133i
\(82\) 0 0
\(83\) 2.84759e9 0.0793502 0.0396751 − 0.999213i \(-0.487368\pi\)
0.0396751 + 0.999213i \(0.487368\pi\)
\(84\) 0 0
\(85\) −4.25418e10 −1.03995
\(86\) 0 0
\(87\) −7.32538e8 − 1.26879e9i −0.0157570 − 0.0272920i
\(88\) 0 0
\(89\) 1.35210e10 − 2.34190e10i 0.256663 − 0.444553i −0.708683 − 0.705527i \(-0.750710\pi\)
0.965346 + 0.260974i \(0.0840438\pi\)
\(90\) 0 0
\(91\) 1.17561e10 + 5.33120e10i 0.197485 + 0.895566i
\(92\) 0 0
\(93\) −1.07747e10 + 1.86623e10i −0.160601 + 0.278169i
\(94\) 0 0
\(95\) −5.99288e10 − 1.03800e11i −0.794613 − 1.37631i
\(96\) 0 0
\(97\) −1.49558e11 −1.76834 −0.884168 − 0.467169i \(-0.845274\pi\)
−0.884168 + 0.467169i \(0.845274\pi\)
\(98\) 0 0
\(99\) −1.29940e11 −1.37324
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 112.12.i.a.65.3 8
4.3 odd 2 14.12.c.a.9.2 ✓ 8
7.4 even 3 inner 112.12.i.a.81.3 8
12.11 even 2 126.12.g.e.37.3 8
28.3 even 6 98.12.c.l.67.3 8
28.11 odd 6 14.12.c.a.11.2 yes 8
28.19 even 6 98.12.a.l.1.2 4
28.23 odd 6 98.12.a.j.1.3 4
28.27 even 2 98.12.c.l.79.3 8
84.11 even 6 126.12.g.e.109.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.2 ✓ 8 4.3 odd 2
14.12.c.a.11.2 yes 8 28.11 odd 6
98.12.a.j.1.3 4 28.23 odd 6
98.12.a.l.1.2 4 28.19 even 6
98.12.c.l.67.3 8 28.3 even 6
98.12.c.l.79.3 8 28.27 even 2
112.12.i.a.65.3 8 1.1 even 1 trivial
112.12.i.a.81.3 8 7.4 even 3 inner
126.12.g.e.37.3 8 12.11 even 2
126.12.g.e.109.3 8 84.11 even 6