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Properties

Label 12.51.c.b
Level $12$
Weight $51$
Character orbit 12.c
Analytic conductor $190.003$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,51,Mod(5,12)] 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("12.5"); S:= CuspForms(chi, 51); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 51, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 51 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(190.002544227\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{247}\cdot 3^{179}\cdot 5^{20}\cdot 7^{9}\cdot 17^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 47428184265) q^{3} + (\beta_{2} + 8013 \beta_1) q^{5} + (\beta_{3} + 5 \beta_{2} + \cdots + 15\!\cdots\!30) q^{7} + ( - \beta_{5} - 6 \beta_{4} + \cdots - 11\!\cdots\!11) q^{9} + ( - \beta_{6} + \cdots + 9411398236060 \beta_1) q^{11}+ \cdots + (33\!\cdots\!33 \beta_{15} + \cdots + 39\!\cdots\!20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 758850948240 q^{3} + 24\!\cdots\!80 q^{7} - 18\!\cdots\!76 q^{9} + 87\!\cdots\!00 q^{13} - 91\!\cdots\!20 q^{15} - 16\!\cdots\!44 q^{19} - 98\!\cdots\!56 q^{21} - 50\!\cdots\!20 q^{25} + 18\!\cdots\!40 q^{27}+ \cdots + 63\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + \cdots + 77\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!44 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!24 \nu^{15} + \cdots - 30\!\cdots\!00 ) / 53\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39\!\cdots\!84 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 53\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!76 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\!\cdots\!68 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!84 \nu^{15} + \cdots + 88\!\cdots\!00 ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!08 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77\!\cdots\!24 \nu^{15} + \cdots - 69\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!12 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!52 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 94\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 67\!\cdots\!08 \nu^{15} + \cdots + 52\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!12 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38\!\cdots\!12 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 32\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 29\!\cdots\!84 \nu^{15} + \cdots + 55\!\cdots\!00 ) / 55\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28\!\cdots\!92 \nu^{15} + \cdots - 83\!\cdots\!00 ) / 16\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 8013\beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2913 \beta_{12} + 40631 \beta_{11} + 5706 \beta_{10} - 13043 \beta_{9} + 5366 \beta_{8} + \cdots - 12\!\cdots\!20 ) / 46656 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\!\cdots\!96 \beta_{15} + \cdots + 25\!\cdots\!79 \beta_1 ) / 10077696 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 98\!\cdots\!80 \beta_{12} + \cdots + 49\!\cdots\!00 ) / 362797056 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18\!\cdots\!80 \beta_{15} + \cdots - 25\!\cdots\!95 \beta_1 ) / 39182082048 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\!\cdots\!00 \beta_{12} + \cdots - 12\!\cdots\!00 ) / 156728328192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23\!\cdots\!00 \beta_{15} + \cdots + 12\!\cdots\!75 \beta_1 ) / 1880739938304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 32\!\cdots\!00 \beta_{12} + \cdots + 21\!\cdots\!00 ) / 4231664861184 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 30\!\cdots\!75 \beta_{15} + \cdots - 60\!\cdots\!50 \beta_1 ) / 101559956668416 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12\!\cdots\!00 \beta_{12} + \cdots - 10\!\cdots\!00 ) / 304679870005248 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 69\!\cdots\!75 \beta_{15} + \cdots + 39\!\cdots\!00 \beta_1 ) / 812479653347328 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 25\!\cdots\!00 \beta_{12} + \cdots + 26\!\cdots\!00 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30\!\cdots\!25 \beta_{15} + \cdots - 12\!\cdots\!00 \beta_1 ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 22\!\cdots\!00 \beta_{12} + \cdots - 28\!\cdots\!00 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 22\!\cdots\!75 \beta_{15} + \cdots + 73\!\cdots\!75 \beta_1 ) / 25\!\cdots\!96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
3.18714e14i
− 3.18714e14i
− 2.27434e15i
2.27434e15i
2.67719e15i
− 2.67719e15i
− 1.20638e15i
1.20638e15i
7.68529e13i
− 7.68529e13i
7.12722e14i
− 7.12722e14i
9.42496e14i
− 9.42496e14i
− 2.30426e15i
2.30426e15i
0 −6.75644e11 − 5.11276e11i 0 6.88423e16i 0 1.94686e21 0 1.95092e23 + 6.90881e23i 0
5.2 0 −6.75644e11 + 5.11276e11i 0 − 6.88423e16i 0 1.94686e21 0 1.95092e23 − 6.90881e23i 0
5.3 0 −5.70987e11 − 6.25997e11i 0 − 4.91259e17i 0 4.46517e20 0 −6.58467e22 + 7.14872e23i 0
5.4 0 −5.70987e11 + 6.25997e11i 0 4.91259e17i 0 4.46517e20 0 −6.58467e22 − 7.14872e23i 0
5.5 0 −4.58850e11 − 7.12288e11i 0 5.78273e17i 0 −8.74653e20 0 −2.96811e23 + 6.53667e23i 0
5.6 0 −4.58850e11 + 7.12288e11i 0 − 5.78273e17i 0 −8.74653e20 0 −2.96811e23 − 6.53667e23i 0
5.7 0 3.78720e10 − 8.46442e11i 0 − 2.60577e17i 0 −2.13340e21 0 −7.15029e23 − 6.41130e22i 0
5.8 0 3.78720e10 + 8.46442e11i 0 2.60577e17i 0 −2.13340e21 0 −7.15029e23 + 6.41130e22i 0
5.9 0 4.48355e10 − 8.46102e11i 0 1.66002e16i 0 2.86432e20 0 −7.13878e23 − 7.58708e22i 0
5.10 0 4.48355e10 + 8.46102e11i 0 − 1.66002e16i 0 2.86432e20 0 −7.13878e23 + 7.58708e22i 0
5.11 0 4.47184e11 − 7.19670e11i 0 1.53948e17i 0 1.49627e21 0 −3.17951e23 − 6.43649e23i 0
5.12 0 4.47184e11 + 7.19670e11i 0 − 1.53948e17i 0 1.49627e21 0 −3.17951e23 + 6.43649e23i 0
5.13 0 7.71810e11 − 3.49581e11i 0 2.03579e17i 0 −1.22926e21 0 4.73485e23 − 5.39620e23i 0
5.14 0 7.71810e11 + 3.49581e11i 0 − 2.03579e17i 0 −1.22926e21 0 4.73485e23 + 5.39620e23i 0
5.15 0 7.83205e11 − 3.23247e11i 0 − 4.97721e17i 0 1.30841e21 0 5.08921e23 − 5.06337e23i 0
5.16 0 7.83205e11 + 3.23247e11i 0 4.97721e17i 0 1.30841e21 0 5.08921e23 + 5.06337e23i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.51.c.b ✓ 16
3.b odd 2 1 inner 12.51.c.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.51.c.b ✓ 16 1.a even 1 1 trivial
12.51.c.b ✓ 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + \cdots + 17\!\cdots\!00 \) acting on \(S_{51}^{\mathrm{new}}(12, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 70\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots - 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 68\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 35\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 33\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 54\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 55\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 76\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 68\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 99\!\cdots\!56)^{2} \) Copy content Toggle raw display
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