Properties

Label 48.24.a.e
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: \( x^{2} - x - 324160122 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 5184\sqrt{1296640489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 177147 q^{3} + ( - \beta + 12624078) q^{5} + ( - 25 \beta - 2882231384) q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 177147 q^{3} + ( - \beta + 12624078) q^{5} + ( - 25 \beta - 2882231384) q^{7} + 31381059609 q^{9} + ( - 6050 \beta - 508560735012) q^{11} + ( - 15550 \beta + 1377630542438) q^{13} + (177147 \beta - 2236317545466) q^{15} + (801550 \beta + 29152391885874) q^{17} + (1361350 \beta + 209169475284580) q^{19} + (4428675 \beta + 510578642981448) q^{21} + (7057550 \beta - 75\!\cdots\!68) q^{23}+ \cdots + ( - 189855410634450 \beta - 15\!\cdots\!08) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 354294 q^{3} + 25248156 q^{5} - 5764462768 q^{7} + 62762119218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 354294 q^{3} + 25248156 q^{5} - 5764462768 q^{7} + 62762119218 q^{9} - 1017121470024 q^{11} + 2755261084876 q^{13} - 4472635090932 q^{15} + 58304783771748 q^{17} + 418338950569160 q^{19} + 10\!\cdots\!96 q^{21}+ \cdots - 31\!\cdots\!16 q^{99}+O(q^{100}) \) 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
18004.9
−18003.9
0 −177147. 0 −1.74046e8 0 −7.54898e9 0 3.13811e10 0
1.2 0 −177147. 0 1.99294e8 0 1.78452e9 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.e 2
4.b odd 2 1 6.24.a.d 2
12.b even 2 1 18.24.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.24.a.d 2 4.b odd 2 1
18.24.a.e 2 12.b even 2 1
48.24.a.e 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 25248156T_{5} - 34686362439805500 \) 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
$5$ \( T^{2} - 25248156 T - 34\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 5764462768 T - 13\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{2} + 1017121470024 T - 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{2} - 2755261084876 T - 65\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{2} - 58304783771748 T - 21\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 418338950569160 T - 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 97\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 84\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 86\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 24\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 97\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
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