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Properties

Label 48.22.c.b
Level $48$
Weight $22$
Character orbit 48.c
Analytic conductor $134.149$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,22,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{68}\cdot 3^{56}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{5} q^{5} + ( - \beta_{6} + 2 \beta_{3} + 3727 \beta_1) q^{7} + ( - \beta_{9} - 65 \beta_{5} + 518598189) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{5} q^{5} + ( - \beta_{6} + 2 \beta_{3} + 3727 \beta_1) q^{7} + ( - \beta_{9} - 65 \beta_{5} + 518598189) q^{9} + (10 \beta_{3} + \beta_{2} - 22928 \beta_1) q^{11} + ( - 17 \beta_{10} - 12 \beta_{9} + 22 \beta_{8} + \cdots - 129841635890) q^{13}+ \cdots + (62120770047 \beta_{7} + \cdots - 239371041556791 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6223178268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6223178268 q^{9} - 1558099630680 q^{13} - 467668605656952 q^{21} - 23\!\cdots\!04 q^{25}+ \cdots - 47\!\cdots\!00 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 61\!\cdots\!80 \nu^{11} + \cdots - 36\!\cdots\!05 ) / 48\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!60 \nu^{11} + \cdots - 12\!\cdots\!70 ) / 48\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!00 \nu^{11} + \cdots + 76\!\cdots\!45 ) / 48\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90\!\cdots\!64 \nu^{11} + \cdots - 18\!\cdots\!78 ) / 19\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!04 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 88\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76\!\cdots\!00 \nu^{11} + \cdots + 15\!\cdots\!05 ) / 48\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!44 \nu^{11} + \cdots + 64\!\cdots\!10 ) / 96\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47\!\cdots\!32 \nu^{11} + \cdots - 25\!\cdots\!80 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!32 \nu^{11} + \cdots - 45\!\cdots\!20 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 94\!\cdots\!44 \nu^{11} + \cdots - 80\!\cdots\!60 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!28 \nu^{11} + \cdots + 39\!\cdots\!80 ) / 79\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + 2\beta_{3} + 3727\beta _1 + 18 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 11262058 \beta_{10} - 32937612 \beta_{9} - 10413496 \beta_{8} + 1962 \beta_{7} - 5904 \beta_{6} - 11262040 \beta_{5} - 11772 \beta_{4} - 9681247 \beta_{3} + \cdots - 34\!\cdots\!64 ) / 648 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1488287001125 \beta_{11} + 232913784090159 \beta_{10} - 458407819047049 \beta_{9} + 94947260822108 \beta_{8} + \cdots - 92\!\cdots\!44 ) / 11664 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26789166020250 \beta_{11} + \cdots + 73\!\cdots\!84 ) / 104976 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22\!\cdots\!65 \beta_{11} + \cdots + 12\!\cdots\!12 ) / 69984 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 67\!\cdots\!45 \beta_{11} + \cdots - 71\!\cdots\!64 ) / 69984 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 91\!\cdots\!80 \beta_{11} + \cdots - 49\!\cdots\!12 ) / 139968 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 54\!\cdots\!10 \beta_{11} + \cdots + 31\!\cdots\!04 ) / 209952 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 11\!\cdots\!35 \beta_{11} + \cdots + 62\!\cdots\!64 ) / 93312 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 50\!\cdots\!55 \beta_{11} + \cdots - 18\!\cdots\!24 ) / 839808 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!35 \beta_{11} + \cdots - 66\!\cdots\!88 ) / 559872 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 − 4.47349e6i
0.500000 + 4.47349e6i
0.500000 + 1.05216e7i
0.500000 − 1.05216e7i
0.500000 − 3.90105e7i
0.500000 + 3.90105e7i
0.500000 − 3.70354e7i
0.500000 + 3.70354e7i
0.500000 + 1.11105e7i
0.500000 − 1.11105e7i
0.500000 − 5.87639e6i
0.500000 + 5.87639e6i
0 −94418.3 − 39313.4i 0 − 2.52523e7i 0 − 1.86298e8i 0 7.36926e9 + 7.42381e9i 0
47.2 0 −94418.3 + 39313.4i 0 2.52523e7i 0 1.86298e8i 0 7.36926e9 − 7.42381e9i 0
47.3 0 −81435.3 − 61876.0i 0 1.06005e7i 0 3.89377e8i 0 2.80307e9 + 1.00778e10i 0
47.4 0 −81435.3 + 61876.0i 0 − 1.06005e7i 0 − 3.89377e8i 0 2.80307e9 − 1.00778e10i 0
47.5 0 −30363.0 − 97665.0i 0 3.55520e7i 0 − 1.36883e9i 0 −8.61653e9 + 5.93080e9i 0
47.6 0 −30363.0 + 97665.0i 0 − 3.55520e7i 0 1.36883e9i 0 −8.61653e9 − 5.93080e9i 0
47.7 0 30363.0 − 97665.0i 0 − 3.55520e7i 0 − 1.36883e9i 0 −8.61653e9 − 5.93080e9i 0
47.8 0 30363.0 + 97665.0i 0 3.55520e7i 0 1.36883e9i 0 −8.61653e9 + 5.93080e9i 0
47.9 0 81435.3 − 61876.0i 0 − 1.06005e7i 0 3.89377e8i 0 2.80307e9 − 1.00778e10i 0
47.10 0 81435.3 + 61876.0i 0 1.06005e7i 0 − 3.89377e8i 0 2.80307e9 + 1.00778e10i 0
47.11 0 94418.3 − 39313.4i 0 2.52523e7i 0 − 1.86298e8i 0 7.36926e9 − 7.42381e9i 0
47.12 0 94418.3 + 39313.4i 0 − 2.52523e7i 0 1.86298e8i 0 7.36926e9 + 7.42381e9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.22.c.b ✓ 12
3.b odd 2 1 inner 48.22.c.b ✓ 12
4.b odd 2 1 inner 48.22.c.b ✓ 12
12.b even 2 1 inner 48.22.c.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.22.c.b ✓ 12 1.a even 1 1 trivial
48.22.c.b ✓ 12 3.b odd 2 1 inner
48.22.c.b ✓ 12 4.b odd 2 1 inner
48.22.c.b ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + \cdots + 90\!\cdots\!00 \) acting on \(S_{22}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3111589134 T^{10} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 98\!\cdots\!68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 389524907670 T^{2} + \cdots - 32\!\cdots\!40)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 34\!\cdots\!80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 68\!\cdots\!20)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 36\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 67\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 53\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 44\!\cdots\!80)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 96\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 34\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 19\!\cdots\!00)^{4} \) Copy content Toggle raw display
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