Properties

Label 23.3.b.a
Level $23$
Weight $3$
Character orbit 23.b
Self dual yes
Analytic conductor $0.627$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.626704608029\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1 + 4) q^{4} + (2 \beta_{2} - 3 \beta_1 - 11) q^{6} + (4 \beta_{2} + 4 \beta_1 - 7) q^{8} + ( - \beta_{2} + 6 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1 + 4) q^{4} + (2 \beta_{2} - 3 \beta_1 - 11) q^{6} + (4 \beta_{2} + 4 \beta_1 - 7) q^{8} + ( - \beta_{2} + 6 \beta_1 + 9) q^{9} + ( - 11 \beta_{2} - 11 \beta_1 + 1) q^{12} + (7 \beta_{2} - 2 \beta_1) q^{13} + ( - 7 \beta_{2} - 7 \beta_1 + 16) q^{16} + (11 \beta_{2} + 22 \beta_1 + 13) q^{18} - 23 q^{23} + (15 \beta_{2} + 2 \beta_1 - 44) q^{24} + 25 q^{25} + ( - 14 \beta_{2} - 11 \beta_1 + 29) q^{26} + ( - 9 \beta_{2} - 18 \beta_1 - 38) q^{27} + ( - 17 \beta_{2} - 2 \beta_1) q^{29} + (7 \beta_{2} + 22 \beta_1) q^{31} + (14 \beta_{2} - 7 \beta_1 - 28) q^{32} + ( - 5 \beta_{2} + 22 \beta_1 + 85) q^{36} + (23 \beta_{2} + 6 \beta_1 - 14) q^{39} + (7 \beta_{2} - 26 \beta_1) q^{41} + ( - 23 \beta_{2} - 23 \beta_1) q^{46} + ( - 17 \beta_{2} + 22 \beta_1) q^{47} + ( - 30 \beta_{2} - 11 \beta_1 + 77) q^{48} + 49 q^{49} + (25 \beta_{2} + 25 \beta_1) q^{50} + (29 \beta_{2} + 29 \beta_1 - 103) q^{52} + ( - 20 \beta_{2} - 65 \beta_1 - 99) q^{54} + (34 \beta_{2} + 13 \beta_1 - 91) q^{58} + 26 q^{59} + ( - 14 \beta_{2} + 37 \beta_1 + 101) q^{62} + ( - 28 \beta_{2} - 28 \beta_1 - 15) q^{64} + (23 \beta_{2} + 46 \beta_1) q^{69} + (31 \beta_{2} - 26 \beta_1) q^{71} + (51 \beta_{2} + 46 \beta_1 - 11) q^{72} + ( - 41 \beta_{2} - 26 \beta_1) q^{73} + ( - 25 \beta_{2} - 50 \beta_1) q^{75} + ( - 60 \beta_{2} - 25 \beta_1 + 133) q^{78} + (38 \beta_{2} + 76 \beta_1 + 81) q^{81} + ( - 14 \beta_{2} - 59 \beta_1 - 43) q^{82} + ( - 49 \beta_{2} + 6 \beta_1 + 82) q^{87} + (46 \beta_{2} - 23 \beta_1 - 92) q^{92} + ( - \beta_{2} - 66 \beta_1 - 182) q^{93} + (34 \beta_{2} + 61 \beta_1 - 19) q^{94} + (77 \beta_{2} + 77 \beta_1 - 7) q^{96} + (49 \beta_{2} + 49 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9} + 3 q^{12} + 48 q^{16} + 39 q^{18} - 69 q^{23} - 132 q^{24} + 75 q^{25} + 87 q^{26} - 114 q^{27} - 84 q^{32} + 255 q^{36} - 42 q^{39} + 231 q^{48} + 147 q^{49} - 309 q^{52} - 297 q^{54} - 273 q^{58} + 78 q^{59} + 303 q^{62} - 45 q^{64} - 33 q^{72} + 399 q^{78} + 243 q^{81} - 129 q^{82} + 246 q^{87} - 276 q^{92} - 546 q^{93} - 57 q^{94} - 21 q^{96}+O(q^{100}) \) 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
−0.523976
−2.14510
2.66908
−3.72545 4.24943 9.87897 0 −15.8310 0 −21.9018 9.05761 0
22.2 0.601466 1.54364 −3.63824 0 0.928445 0 −4.59414 −6.61718 0
22.3 3.12398 −5.79306 5.75927 0 −18.0974 0 5.49593 24.5596 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 23.3.b.a 3
3.b odd 2 1 207.3.d.a 3
4.b odd 2 1 368.3.f.a 3
5.b even 2 1 575.3.d.b 3
5.c odd 4 2 575.3.c.a 6
8.b even 2 1 1472.3.f.a 3
8.d odd 2 1 1472.3.f.b 3
23.b odd 2 1 CM 23.3.b.a 3
69.c even 2 1 207.3.d.a 3
92.b even 2 1 368.3.f.a 3
115.c odd 2 1 575.3.d.b 3
115.e even 4 2 575.3.c.a 6
184.e odd 2 1 1472.3.f.a 3
184.h even 2 1 1472.3.f.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.3.b.a 3 1.a even 1 1 trivial
23.3.b.a 3 23.b odd 2 1 CM
207.3.d.a 3 3.b odd 2 1
207.3.d.a 3 69.c even 2 1
368.3.f.a 3 4.b odd 2 1
368.3.f.a 3 92.b even 2 1
575.3.c.a 6 5.c odd 4 2
575.3.c.a 6 115.e even 4 2
575.3.d.b 3 5.b even 2 1
575.3.d.b 3 115.c odd 2 1
1472.3.f.a 3 8.b even 2 1
1472.3.f.a 3 184.e odd 2 1
1472.3.f.b 3 8.d odd 2 1
1472.3.f.b 3 184.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(23, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 12T + 7 \) Copy content Toggle raw display
$3$ \( T^{3} - 27T + 38 \) 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
$13$ \( T^{3} - 507T - 1082 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2523T - 30746 \) Copy content Toggle raw display
$31$ \( T^{3} - 2883T - 58754 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 5043T - 43634 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 6627 T + 205342 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T - 26)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 15123 T - 667154 \) Copy content Toggle raw display
$73$ \( T^{3} - 15987 T - 725042 \) 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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