Properties

Label 48.24.a.h
Level $48$
Weight $24$
Character orbit 48.a
Self dual yes
Analytic conductor $160.898$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(160.897937926\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1792561}) \)
Defining polynomial: \( x^{2} - x - 448140 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3456\sqrt{1792561}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 177147 q^{3} + ( - 19 \beta - 17111034) q^{5} + ( - 525 \beta - 10817408) q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 177147 q^{3} + ( - 19 \beta - 17111034) q^{5} + ( - 525 \beta - 10817408) q^{7} + 31381059609 q^{9} + ( - 316250 \beta + 28827120564) q^{11} + ( - 1866450 \beta + 3574033893590) q^{13} + ( - 3365793 \beta - 3031168339998) q^{15} + ( - 20541350 \beta + 90842157810018) q^{17} + ( - 111494850 \beta - 156872914889300) q^{19} + ( - 93002175 \beta - 1916271374976) q^{21} + (462222950 \beta + 21\!\cdots\!28) q^{23}+ \cdots + ( - 99\!\cdots\!50 \beta + 90\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 354294 q^{3} - 34222068 q^{5} - 21634816 q^{7} + 62762119218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 354294 q^{3} - 34222068 q^{5} - 21634816 q^{7} + 62762119218 q^{9} + 57654241128 q^{11} + 7148067787180 q^{13} - 6062336679996 q^{15} + 181684315620036 q^{17} - 313745829778600 q^{19} - 3832542749952 q^{21} + 43\!\cdots\!56 q^{23}+ \cdots + 18\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
669.933
−668.933
0 177147. 0 −1.05026e8 0 −2.44006e9 0 3.13811e10 0
1.2 0 177147. 0 7.08042e7 0 2.41842e9 0 3.13811e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.24.a.h 2
4.b odd 2 1 12.24.a.a 2
12.b even 2 1 36.24.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.24.a.a 2 4.b odd 2 1
36.24.a.d 2 12.b even 2 1
48.24.a.h 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 34222068T_{5} - 7436306938945500 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 177147)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 34222068 T - 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 21634816 T - 59\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{2} - 57654241128 T - 21\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{2} - 7148067787180 T - 61\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{2} - 181684315620036 T - 78\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + 313745829778600 T - 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 99\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 16\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 16\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 49\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 38\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 88\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
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