Properties

Label 48.22.a.g
Level $48$
Weight $22$
Character orbit 48.a
Self dual yes
Analytic conductor $134.149$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Defining polynomial: \( x^{2} - x - 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 16128\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 59049 q^{3} + (53 \beta + 498438) q^{5} + (711 \beta - 339948056) q^{7} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 59049 q^{3} + (53 \beta + 498438) q^{5} + (711 \beta - 339948056) q^{7} + 3486784401 q^{9} + ( - 122122 \beta - 109934561484) q^{11} + ( - 1856394 \beta - 24234454978) q^{13} + ( - 3129597 \beta - 29432265462) q^{15} + (6551970 \beta - 5666764520718) q^{17} + (87438798 \beta - 5980292505812) q^{19} + ( - 41983839 \beta + 20073592758744) q^{21} + (287553838 \beta + 73254195031752) q^{23} + (52834428 \beta - 2393177123537) q^{25} - 205891132094649 q^{27} + (4545078919 \beta - 899260021837026) q^{29} + (6766071651 \beta - 55\!\cdots\!96) q^{31}+ \cdots + ( - 425813084618922 \beta - 38\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100}) \) 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
−12.2377
13.2377
0 −59049.0 0 −2.12776e7 0 −6.32076e8 0 3.48678e9 0
1.2 0 −59049.0 0 2.22745e7 0 −4.78205e7 0 3.48678e9 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 48.22.a.g 2
4.b odd 2 1 3.22.a.c 2
12.b even 2 1 9.22.a.e 2
20.d odd 2 1 75.22.a.d 2
20.e even 4 2 75.22.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 4.b odd 2 1
9.22.a.e 2 12.b even 2 1
48.22.a.g 2 1.a even 1 1 trivial
75.22.a.d 2 20.d odd 2 1
75.22.b.d 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 996876T_{5} - 473947100199900 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 59049)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 473947100199900 \) Copy content Toggle raw display
$7$ \( T^{2} + 679896112 T + 30\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{2} + 219869122968 T + 95\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{2} + 48468909956 T - 58\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{2} + 11333529041436 T + 24\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} + 11960585011624 T - 12\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 146508390063504 T - 85\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 54\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 83\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 33\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
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