Properties

Label 23.5.b.a
Level $23$
Weight $5$
Character orbit 23.b
Self dual yes
Analytic conductor $2.378$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.37750915093\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} + 6 \beta_1) q^{3} + (\beta_{2} - 11 \beta_1 + 16) q^{4} + (17 \beta_{2} - 11 \beta_1 + 49) q^{6} + ( - 32 \beta_{2} + 16 \beta_1 - 79) q^{8} + (47 \beta_{2} + 22 \beta_1 + 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} + 6 \beta_1) q^{3} + (\beta_{2} - 11 \beta_1 + 16) q^{4} + (17 \beta_{2} - 11 \beta_1 + 49) q^{6} + ( - 32 \beta_{2} + 16 \beta_1 - 79) q^{8} + (47 \beta_{2} + 22 \beta_1 + 81) q^{9} + ( - 98 \beta_{2} + 49 \beta_1 - 287) q^{12} + ( - 17 \beta_{2} - 122 \beta_1) q^{13} + (158 \beta_{2} - 79 \beta_1 + 256) q^{16} + ( - 49 \beta_{2} + 294 \beta_1 - 479) q^{18} + 529 q^{23} + (351 \beta_{2} - 650 \beta_1 + 784) q^{24} + 625 q^{25} + ( - 383 \beta_{2} + 37 \beta_1 - 511) q^{26} + ( - 81 \beta_{2} + 486 \beta_1 - 14) q^{27} + ( - 353 \beta_{2} - 506 \beta_1) q^{29} + (127 \beta_{2} + 694 \beta_1) q^{31} + ( - 79 \beta_{2} + 869 \beta_1 - 1264) q^{32} + (1039 \beta_{2} - 1370 \beta_1 + 1105) q^{36} + ( - 769 \beta_{2} - 746 \beta_1 - 2846) q^{39} + (991 \beta_{2} + 214 \beta_1) q^{41} + ( - 1058 \beta_{2} + 529 \beta_1) q^{46} + (943 \beta_{2} - 842 \beta_1) q^{47} + ( - 1599 \beta_{2} + 2405 \beta_1 - 3871) q^{48} + 2401 q^{49} + ( - 1250 \beta_{2} + 625 \beta_1) q^{50} + (1022 \beta_{2} - 511 \beta_1 + 5201) q^{52} + (1405 \beta_{2} - 905 \beta_1 + 3969) q^{54} + ( - 1871 \beta_{2} - 1259 \beta_1 + 1553) q^{58} - 6286 q^{59} + (2209 \beta_{2} - 59 \beta_1 + 2513) q^{62} + (2528 \beta_{2} - 1264 \beta_1 + 2145) q^{64} + ( - 529 \beta_{2} + 3174 \beta_1) q^{69} + (1327 \beta_{2} - 2858 \beta_1) q^{71} + ( - 4497 \beta_{2} + 2966 \beta_1 - 14063) q^{72} + ( - 3137 \beta_{2} - 554 \beta_1) q^{73} + ( - 625 \beta_{2} + 3750 \beta_1) q^{75} + (2685 \beta_{2} - 5945 \beta_1 + 5521) q^{78} + (14 \beta_{2} - 84 \beta_1 + 6561) q^{81} + (1633 \beta_{2} + 4741 \beta_1 - 11599) q^{82} + ( - 1777 \beta_{2} - 5354 \beta_1 - 8414) q^{87} + (529 \beta_{2} - 5819 \beta_1 + 8464) q^{92} + (4223 \beta_{2} + 4486 \beta_1 + 15826) q^{93} + ( - 1583 \beta_{2} + 5557 \beta_1 - 17311) q^{94} + (7742 \beta_{2} - 3871 \beta_1 + 22673) q^{96} + ( - 4802 \beta_{2} + 2401 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) 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
−2.14510
2.66908
−0.523976
−7.63824 −15.6172 42.3427 0 119.288 0 −201.212 162.896 0
22.2 1.75927 15.5596 −12.9050 0 27.3735 0 −50.8517 161.100 0
22.3 5.87897 0.0576140 18.5623 0 0.338711 0 15.0635 −80.9967 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 23.5.b.a 3
3.b odd 2 1 207.5.d.a 3
4.b odd 2 1 368.5.f.a 3
23.b odd 2 1 CM 23.5.b.a 3
69.c even 2 1 207.5.d.a 3
92.b even 2 1 368.5.f.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.5.b.a 3 1.a even 1 1 trivial
23.5.b.a 3 23.b odd 2 1 CM
207.5.d.a 3 3.b odd 2 1
207.5.d.a 3 69.c even 2 1
368.5.f.a 3 4.b odd 2 1
368.5.f.a 3 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 48T_{2} + 79 \) acting on \(S_{5}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 48T + 79 \) Copy content Toggle raw display
$3$ \( T^{3} - 243T + 14 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 85683 T + 8482894 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T - 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2121843 T + 244330126 \) Copy content Toggle raw display
$31$ \( T^{3} - 2770563 T - 1677025154 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 8477283 T + 7596282526 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 14639043 T - 20606906306 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T + 6286)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 76235043 T - 188893891874 \) Copy content Toggle raw display
$73$ \( T^{3} - 85194723 T - 223017449186 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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