Properties

Label 48.24.a.i
Level $48$
Weight $24$
Character orbit 48.a
Self dual yes
Analytic conductor $160.898$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \( x^{2} - x - 481925607 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 576\sqrt{1927702429}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 177147 q^{3} + ( - 5 \beta + 1037990) q^{5} + (133 \beta - 2225877024) q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 177147 q^{3} + ( - 5 \beta + 1037990) q^{5} + (133 \beta - 2225877024) q^{7} + 31381059609 q^{9} + ( - 14454 \beta + 541113615988) q^{11} + (152674 \beta - 3226982360426) q^{13} + ( - 885735 \beta + 183876814530) q^{15} + (3497270 \beta + 8781663568546) q^{17} + ( - 10659598 \beta - 158862239199380) q^{19} + (23560551 \beta - 394307437170528) q^{21} + ( - 51420278 \beta - 15\!\cdots\!00) q^{23}+ \cdots + ( - 453581835588486 \beta + 16\!\cdots\!92) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 354294 q^{3} + 2075980 q^{5} - 4451754048 q^{7} + 62762119218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 354294 q^{3} + 2075980 q^{5} - 4451754048 q^{7} + 62762119218 q^{9} + 1082227231976 q^{11} - 6453964720852 q^{13} + 367753629060 q^{15} + 17563327137092 q^{17} - 317724478398760 q^{19} - 788614874341056 q^{21} - 30\!\cdots\!00 q^{23}+ \cdots + 33\!\cdots\!84 q^{99}+O(q^{100}) \) 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
21953.3
−21952.3
0 177147. 0 −1.25410e8 0 1.13764e9 0 3.13811e10 0
1.2 0 177147. 0 1.27486e8 0 −5.58940e9 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.i 2
4.b odd 2 1 24.24.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.24.a.a 2 4.b odd 2 1
48.24.a.i 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2075980T_{5} - 15988057603857500 \) 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
$5$ \( T^{2} - 2075980 T - 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 4451754048 T - 63\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{2} - 1082227231976 T + 15\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{2} + 6453964720852 T - 44\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{2} - 17563327137092 T - 77\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{2} + 317724478398760 T - 47\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 93\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 15\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 58\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 20\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 65\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 26\!\cdots\!16 \) Copy content Toggle raw display
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