Properties

Label 23.17.b.a
Level $23$
Weight $17$
Character orbit 23.b
Self dual yes
Analytic conductor $37.335$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(37.3346450870\)
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: \( 7 \)
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.

\(f(q)\) \(=\) \( q + ( - 47 \beta_{2} + 48 \beta_1) q^{2} + (1359 \beta_{2} - 1075 \beta_1) q^{3} + (15934 \beta_{2} - 6435 \beta_1 + 65536) q^{4} + ( - 386882 \beta_{2} + 216733 \beta_1 - 3322751) q^{6} + ( - 3080192 \beta_{2} + 3145728 \beta_1 - 29748031) q^{8} + (9149263 \beta_{2} - 6761843 \beta_1 + 43046721) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 47 \beta_{2} + 48 \beta_1) q^{2} + (1359 \beta_{2} - 1075 \beta_1) q^{3} + (15934 \beta_{2} - 6435 \beta_1 + 65536) q^{4} + ( - 386882 \beta_{2} + 216733 \beta_1 - 3322751) q^{6} + ( - 3080192 \beta_{2} + 3145728 \beta_1 - 29748031) q^{8} + (9149263 \beta_{2} - 6761843 \beta_1 + 43046721) q^{9} + (156169297 \beta_{2} - 159492048 \beta_1 + 812601217) q^{12} + (138621583 \beta_{2} + 123075597 \beta_1) q^{13} + (1398157457 \beta_{2} - 1427905488 \beta_1 + 4294967296) q^{16} + ( - 4515618609 \beta_{2} + 3571957325 \beta_1 - 21660100031) q^{18} + 78310985281 q^{23} + ( - 65782272881 \beta_{2} + 46182947213 \beta_1 - 217759809536) q^{24} + 152587890625 q^{25} + (15460925758 \beta_{2} + 44914723677 \beta_1 + 8534185409) q^{26} + (58500493839 \beta_{2} - 46275225075 \beta_1 + 564442461634) q^{27} + ( - 124533468977 \beta_{2} - 3107988147 \beta_1) q^{29} + (106919567503 \beta_{2} + 159710427213 \beta_1) q^{31} + ( - 474005125954 \beta_{2} + \cdots - 1949566959616) q^{32}+ \cdots + ( - 15\!\cdots\!47 \beta_{2} + 15\!\cdots\!48 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 196608 q^{4} - 9968253 q^{6} - 89244093 q^{8} + 129140163 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 196608 q^{4} - 9968253 q^{6} - 89244093 q^{8} + 129140163 q^{9} + 2437803651 q^{12} + 12884901888 q^{16} - 64980300093 q^{18} + 234932955843 q^{23} - 653279428608 q^{24} + 457763671875 q^{25} + 25602556227 q^{26} + 1693327384902 q^{27} - 5848700878848 q^{32} + 24658692901635 q^{36} + 4069672411782 q^{39} + 296535899259843 q^{48} + 99698791708803 q^{49} + 142545216874947 q^{52} - 429100605748413 q^{54} + 458310235450371 q^{58} - 180632318733498 q^{59} + 309904003515651 q^{62} + 18\!\cdots\!15 q^{64}+ \cdots - 72\!\cdots\!81 q^{96}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{2} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 3\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{2} + 3\beta _1 + 28 ) / 7 \) 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
−505.435 13120.9 189928. 0 −6.63177e6 0 −6.28722e7 1.29112e8 0
22.2 181.938 −6414.97 −32434.7 0 −1.16713e6 0 −1.78246e7 −1.89486e6 0
22.3 323.497 −6705.95 39114.3 0 −2.16936e6 0 −8.54733e6 1.92308e6 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 23.17.b.a 3
23.b odd 2 1 CM 23.17.b.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.17.b.a 3 1.a even 1 1 trivial
23.17.b.a 3 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 196608T_{2} + 29748031 \) acting on \(S_{17}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 196608 T + 29748031 \) Copy content Toggle raw display
$3$ \( T^{3} - 129140163 T - 564442461634 \) 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
$13$ \( T^{3} + \cdots + 44\!\cdots\!46 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T - 78310985281)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 16\!\cdots\!74 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 28\!\cdots\!94 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 86\!\cdots\!26 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 99\!\cdots\!54 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T + 60210772911166)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 53\!\cdots\!26 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 10\!\cdots\!66 \) 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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