Properties

Label 25.26.b.a
Level $25$
Weight $26$
Character orbit 25.b
Analytic conductor $98.999$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(98.9991949881\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 24 \beta q^{2} - 97902 \beta q^{3} + 33552128 q^{4} + 9398592 q^{6} - 19540298596 \beta q^{7} + 1610557440 \beta q^{8} + 808949403027 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 24 \beta q^{2} - 97902 \beta q^{3} + 33552128 q^{4} + 9398592 q^{6} - 19540298596 \beta q^{7} + 1610557440 \beta q^{8} + 808949403027 q^{9} + 8419515299052 q^{11} - 3284820435456 \beta q^{12} - 40825522667657 \beta q^{13} + 1875868665216 q^{14} + 11\!\cdots\!56 q^{16} + \cdots + 68\!\cdots\!04 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 67104256 q^{4} + 18797184 q^{6} + 1617898806054 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 67104256 q^{4} + 18797184 q^{6} + 1617898806054 q^{9} + 16839030598104 q^{11} + 3751737330432 q^{14} + 22\!\cdots\!12 q^{16}+ \cdots + 13\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.00000i
1.00000i
48.0000i 195804.i 3.35521e7 0 9.39859e6 3.90806e10i 3.22111e9i 8.08949e11 0
24.2 48.0000i 195804.i 3.35521e7 0 9.39859e6 3.90806e10i 3.22111e9i 8.08949e11 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.26.b.a 2
5.b even 2 1 inner 25.26.b.a 2
5.c odd 4 1 1.26.a.a 1
5.c odd 4 1 25.26.a.a 1
15.e even 4 1 9.26.a.a 1
20.e even 4 1 16.26.a.b 1
35.f even 4 1 49.26.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.26.a.a 1 5.c odd 4 1
9.26.a.a 1 15.e even 4 1
16.26.a.b 1 20.e even 4 1
25.26.a.a 1 5.c odd 4 1
25.26.b.a 2 1.a even 1 1 trivial
25.26.b.a 2 5.b even 2 1 inner
49.26.a.a 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2304 \) acting on \(S_{26}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2304 \) Copy content Toggle raw display
$3$ \( T^{2} + 38339206416 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 15\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( (T - 8419515299052)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 66\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{2} + 63\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T - 60\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 90\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T - 27\!\cdots\!10)^{2} \) Copy content Toggle raw display
$31$ \( (T - 42\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 41\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T + 18\!\cdots\!98)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 90\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + 85\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + 98\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T + 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T - 90\!\cdots\!02)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 71\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T + 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T - 27\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 86\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T - 17\!\cdots\!30)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 80\!\cdots\!44 \) Copy content Toggle raw display
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