Properties

Label 23.13.b.a
Level $23$
Weight $13$
Character orbit 23.b
Self dual yes
Analytic conductor $21.022$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0218577974\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 79 q^{2} - 14 q^{3} + 2145 q^{4} + 1106 q^{6} + 154129 q^{8} - 531245 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 79 q^{2} - 14 q^{3} + 2145 q^{4} + 1106 q^{6} + 154129 q^{8} - 531245 q^{9} - 30030 q^{12} - 8482894 q^{13} - 20962111 q^{16} + 41968355 q^{18} + 148035889 q^{23} - 2157806 q^{24} + 244140625 q^{25} + 670148626 q^{26} + 14877604 q^{27} - 244330126 q^{29} + 1677025154 q^{31} + 1024694385 q^{32} - 1139520525 q^{36} + 118760516 q^{39} - 7596282526 q^{41} - 11694835231 q^{46} + 20606906306 q^{47} + 293469554 q^{48} + 13841287201 q^{49} - 19287109375 q^{50} - 18195807630 q^{52} - 1175330716 q^{54} + 19302079954 q^{58} - 19874527918 q^{59} - 132484987166 q^{62} + 4909950241 q^{64} - 2072502446 q^{69} + 188893891874 q^{71} - 81880260605 q^{72} + 223017449186 q^{73} - 3417968750 q^{75} - 9382080764 q^{78} + 282117087589 q^{81} + 600106319554 q^{82} + 3420621764 q^{87} + 317536981905 q^{92} - 23478352156 q^{93} - 1627945598174 q^{94} - 14345721390 q^{96} - 1093461688879 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0
−79.0000 −14.0000 2145.00 0 1106.00 0 154129. −531245. 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.13.b.a 1
23.b odd 2 1 CM 23.13.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.13.b.a 1 1.a even 1 1 trivial
23.13.b.a 1 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 79 \) acting on \(S_{13}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 79 \) Copy content Toggle raw display
$3$ \( T + 14 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 8482894 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 148035889 \) Copy content Toggle raw display
$29$ \( T + 244330126 \) Copy content Toggle raw display
$31$ \( T - 1677025154 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 7596282526 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 20606906306 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 19874527918 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 188893891874 \) Copy content Toggle raw display
$73$ \( T - 223017449186 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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