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 = [14,224,0,-7168,-7527] 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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} + 234028822 x^{12} + 368819651895 x^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{17}\cdot 7^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 32 \beta_{2} q^{2} + ( - 1024 \beta_{2} - 1024) q^{4} + (\beta_{3} + 1075 \beta_{2} - \beta_1) q^{5} + (\beta_{5} - 6807 \beta_{2} + \cdots - 509) q^{7} - 32768 q^{8} + (34400 \beta_{2} - 32 \beta_1 + 34400) q^{10}+ \cdots + ( - 24640 \beta_{13} + \cdots + 2703488032) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 224 q^{2} - 7168 q^{4} - 7527 q^{5} + 40523 q^{7} - 458752 q^{8} + 240864 q^{10} - 216279 q^{11} - 1981052 q^{13} - 1638784 q^{14} - 7340032 q^{16} - 4758882 q^{17} + 13850542 q^{19} + 15415296 q^{20}+ \cdots + 100554558944 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} + 234028822 x^{12} + 368819651895 x^{11} + \cdots + 11\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\!\cdots\!73 \nu^{13} + \cdots - 49\!\cdots\!80 ) / 59\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!73 \nu^{13} + \cdots - 54\!\cdots\!88 ) / 59\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!31 \nu^{13} + \cdots - 12\!\cdots\!20 ) / 59\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{13} + \cdots + 46\!\cdots\!72 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59\!\cdots\!51 \nu^{13} + \cdots + 88\!\cdots\!64 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!03 \nu^{13} + \cdots + 25\!\cdots\!20 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!90 \nu^{13} + \cdots + 57\!\cdots\!96 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!99 \nu^{13} + \cdots + 10\!\cdots\!96 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58\!\cdots\!33 \nu^{13} + \cdots + 29\!\cdots\!48 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!21 \nu^{13} + \cdots - 73\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59\!\cdots\!27 \nu^{13} + \cdots + 65\!\cdots\!32 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\!\cdots\!65 \nu^{13} + \cdots - 33\!\cdots\!08 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31\!\cdots\!93 \nu^{13} + \cdots + 75\!\cdots\!56 ) / 10\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 16 \beta_{13} + 50 \beta_{12} + 50 \beta_{11} + 33 \beta_{9} - 16 \beta_{8} - 78 \beta_{7} + 86 \beta_{6} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 71265 \beta_{11} - 65043 \beta_{10} + 41114 \beta_{8} - 3128147 \beta_{7} + 2681353 \beta_{6} + \cdots - 79316058913 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3519448918 \beta_{13} - 8922113480 \beta_{12} - 6139491559 \beta_{10} - 6139491559 \beta_{9} + \cdots - 87\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7586427941442 \beta_{13} - 19736850222625 \beta_{12} - 19736850222625 \beta_{11} + \cdots - 622399860909271 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15\!\cdots\!20 \beta_{11} + \cdots + 13\!\cdots\!95 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 88\!\cdots\!78 \beta_{13} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10\!\cdots\!50 \beta_{13} + \cdots + 46\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10\!\cdots\!05 \beta_{11} + \cdots - 95\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16\!\cdots\!86 \beta_{13} + \cdots - 39\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 48\!\cdots\!26 \beta_{13} + \cdots - 32\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 76\!\cdots\!60 \beta_{11} + \cdots + 68\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 34\!\cdots\!58 \beta_{13} + \cdots + 39\!\cdots\!88 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−6238.75 − 10805.8i
−3128.01 − 5417.87i
−2129.12 − 3687.74i
−1117.59 − 1935.73i
2715.68 + 4703.70i
3197.57 + 5538.35i
6702.72 + 11609.5i
−6238.75 + 10805.8i
−3128.01 + 5417.87i
−2129.12 + 3687.74i
−1117.59 + 1935.73i
2715.68 − 4703.70i
3197.57 − 5538.35i
6702.72 − 11609.5i
16.0000 − 27.7128i 0 −512.000 − 886.810i −6776.75 + 11737.7i 0 −31433.8 − 31452.2i −32768.0 0 216856. + 375606.i
37.2 16.0000 − 27.7128i 0 −512.000 − 886.810i −3666.01 + 6349.72i 0 −1751.68 + 44432.6i −32768.0 0 117312. + 203191.i
37.3 16.0000 − 27.7128i 0 −512.000 − 886.810i −2667.12 + 4619.58i 0 38608.4 − 22061.7i −32768.0 0 85347.7 + 147827.i
37.4 16.0000 − 27.7128i 0 −512.000 − 886.810i −1655.59 + 2867.57i 0 38832.9 − 21664.0i −32768.0 0 52978.9 + 91762.2i
37.5 16.0000 − 27.7128i 0 −512.000 − 886.810i 2177.68 − 3771.85i 0 −22436.9 − 38391.6i −32768.0 0 −69685.7 − 120699.i
37.6 16.0000 − 27.7128i 0 −512.000 − 886.810i 2659.57 − 4606.50i 0 −42932.8 + 11580.4i −32768.0 0 −85106.1 − 147408.i
37.7 16.0000 − 27.7128i 0 −512.000 − 886.810i 6164.72 − 10677.6i 0 41375.4 + 16291.3i −32768.0 0 −197271. − 341684.i
109.1 16.0000 + 27.7128i 0 −512.000 + 886.810i −6776.75 − 11737.7i 0 −31433.8 + 31452.2i −32768.0 0 216856. − 375606.i
109.2 16.0000 + 27.7128i 0 −512.000 + 886.810i −3666.01 − 6349.72i 0 −1751.68 − 44432.6i −32768.0 0 117312. − 203191.i
109.3 16.0000 + 27.7128i 0 −512.000 + 886.810i −2667.12 − 4619.58i 0 38608.4 + 22061.7i −32768.0 0 85347.7 − 147827.i
109.4 16.0000 + 27.7128i 0 −512.000 + 886.810i −1655.59 − 2867.57i 0 38832.9 + 21664.0i −32768.0 0 52978.9 − 91762.2i
109.5 16.0000 + 27.7128i 0 −512.000 + 886.810i 2177.68 + 3771.85i 0 −22436.9 + 38391.6i −32768.0 0 −69685.7 + 120699.i
109.6 16.0000 + 27.7128i 0 −512.000 + 886.810i 2659.57 + 4606.50i 0 −42932.8 − 11580.4i −32768.0 0 −85106.1 + 147408.i
109.7 16.0000 + 27.7128i 0 −512.000 + 886.810i 6164.72 + 10677.6i 0 41375.4 − 16291.3i −32768.0 0 −197271. + 341684.i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.12.g.h yes 14
3.b odd 2 1 126.12.g.g ✓ 14
7.c even 3 1 inner 126.12.g.h yes 14
21.h odd 6 1 126.12.g.g ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.12.g.g ✓ 14 3.b odd 2 1
126.12.g.g ✓ 14 21.h odd 6 1
126.12.g.h yes 14 1.a even 1 1 trivial
126.12.g.h yes 14 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 7527 T_{5}^{13} + 266403510 T_{5}^{12} + 1219997794123 T_{5}^{11} + \cdots + 25\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32 T + 1024)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 90\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 75\!\cdots\!52)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 99\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 11\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 17\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 25\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 37\!\cdots\!04)^{2} \) Copy content Toggle raw display
show more
show less