Properties

Label 25.20.b.a
Level $25$
Weight $20$
Character orbit 25.b
Analytic conductor $57.204$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.2041741391\)
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.

\(f(q)\) \(=\) \( q + 228 \beta q^{2} - 25326 \beta q^{3} + 316352 q^{4} + 23097312 q^{6} - 8458772 \beta q^{7} + 191665920 \beta q^{8} - 1403363637 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 228 \beta q^{2} - 25326 \beta q^{3} + 316352 q^{4} + 23097312 q^{6} - 8458772 \beta q^{7} + 191665920 \beta q^{8} - 1403363637 q^{9} - 16212108 q^{11} - 8011930752 \beta q^{12} - 25210807531 \beta q^{13} + 7714400064 q^{14} - 8939761664 q^{16} + 112535049753 \beta q^{17} - 319966909236 \beta q^{18} + 1710278572660 q^{19} - 856907438688 q^{21} - 3696360624 \beta q^{22} - 7018267394436 \beta q^{23} + 19416524359680 q^{24} + 22992256468272 q^{26} + 6106153557420 \beta q^{27} - 2675949439744 \beta q^{28} - 1137835269510 q^{29} - 104626880141728 q^{31} + 98449876205568 \beta q^{32} + 410587847208 \beta q^{33} - 102631965374736 q^{34} - 443956893292224 q^{36} - 84696163685297 \beta q^{37} + 389943514566480 \beta q^{38} - 25\!\cdots\!24 q^{39} + \cdots + 22\!\cdots\!96 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 632704 q^{4} + 46194624 q^{6} - 2806727274 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 632704 q^{4} + 46194624 q^{6} - 2806727274 q^{9} - 32424216 q^{11} + 15428800128 q^{14} - 17879523328 q^{16} + 3420557145320 q^{19} - 1713814877376 q^{21} + 38833048719360 q^{24} + 45984512936544 q^{26} - 2275670539020 q^{29} - 209253760283456 q^{31} - 205263930749472 q^{34} - 887913786584448 q^{36} - 51\!\cdots\!48 q^{39}+ \cdots + 45\!\cdots\!92 q^{99}+O(q^{100}) \) 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
456.000i 50652.0i 316352. 0 2.30973e7 1.69175e7i 3.83332e8i −1.40336e9 0
24.2 456.000i 50652.0i 316352. 0 2.30973e7 1.69175e7i 3.83332e8i −1.40336e9 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 25.20.b.a 2
5.b even 2 1 inner 25.20.b.a 2
5.c odd 4 1 1.20.a.a 1
5.c odd 4 1 25.20.a.a 1
15.e even 4 1 9.20.a.a 1
20.e even 4 1 16.20.a.a 1
35.f even 4 1 49.20.a.b 1
40.i odd 4 1 64.20.a.b 1
40.k even 4 1 64.20.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 5.c odd 4 1
9.20.a.a 1 15.e even 4 1
16.20.a.a 1 20.e even 4 1
25.20.a.a 1 5.c odd 4 1
25.20.b.a 2 1.a even 1 1 trivial
25.20.b.a 2 5.b even 2 1 inner
49.20.a.b 1 35.f even 4 1
64.20.a.b 1 40.i odd 4 1
64.20.a.h 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 207936 \) acting on \(S_{20}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 207936 \) Copy content Toggle raw display
$3$ \( T^{2} + 2565625104 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 286203294991936 \) Copy content Toggle raw display
$11$ \( (T + 16212108)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + 50\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T - 1710278572660)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 19\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T + 1137835269510)^{2} \) Copy content Toggle raw display
$31$ \( (T + 104626880141728)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 28\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T + 33\!\cdots\!38)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 12\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + 12\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + 89\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T + 58\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T - 23\!\cdots\!42)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 42\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T + 17\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 89\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T - 92\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 43\!\cdots\!30)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 40\!\cdots\!36 \) Copy content Toggle raw display
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