Properties

Label 48.28.a.e
Level $48$
Weight $28$
Character orbit 48.a
Self dual yes
Analytic conductor $221.691$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(221.690675922\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{30001}) \)
Defining polynomial: \( x^{2} - x - 7500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 7 \)
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 \(\beta = 4032\sqrt{30001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 1594323 q^{3} + ( - 275 \beta - 885973050) q^{5} + (280971 \beta - 184832599952) q^{7} + 2541865828329 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 1594323 q^{3} + ( - 275 \beta - 885973050) q^{5} + (280971 \beta - 184832599952) q^{7} + 2541865828329 q^{9} + (126659974 \beta - 37881167834124) q^{11} + (1810703934 \beta - 51510539588794) q^{13} + ( - 438438825 \beta - 14\!\cdots\!50) q^{15}+ \cdots + (32\!\cdots\!46 \beta - 96\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9} - 75762335668248 q^{11} - 103021079177588 q^{13} - 28\!\cdots\!00 q^{15}+ \cdots - 19\!\cdots\!92 q^{99}+O(q^{100}) \) 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
87.1040
−86.1040
0 1.59432e6 0 −1.07803e9 0 1.13904e10 0 2.54187e12 0
1.2 0 1.59432e6 0 −6.93920e8 0 −3.81056e11 0 2.54187e12 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 48.28.a.e 2
4.b odd 2 1 3.28.a.b 2
12.b even 2 1 9.28.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.28.a.b 2 4.b odd 2 1
9.28.a.b 2 12.b even 2 1
48.28.a.e 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1771946100T_{5} + 748063892688862500 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1594323)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1771946100 T + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 369665199904 T - 43\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{2} + 75762335668248 T - 63\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{2} + 103021079177588 T - 15\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 34\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 41\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 36\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 45\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
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