Properties

Label 92.2.b.b
Level $92$
Weight $2$
Character orbit 92.b
Analytic conductor $0.735$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 92.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.734623698596\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
Defining polynomial: \(x^{6} - 3 x^{3} + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -\beta_{2} - \beta_{4} ) q^{3} + \beta_{2} q^{4} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{6} + ( -1 - \beta_{3} ) q^{8} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -\beta_{2} - \beta_{4} ) q^{3} + \beta_{2} q^{4} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{6} + ( -1 - \beta_{3} ) q^{8} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{9} + ( 3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{12} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{16} + ( 3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{18} + ( 1 - 2 \beta_{3} ) q^{23} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{24} + 5 q^{25} + ( -5 - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{26} + ( -2 + 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{27} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{29} + ( 4 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{31} + ( -3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{32} + ( -5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{36} + ( 2 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{39} + ( -2 \beta_{1} + 3 \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{41} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{46} + ( -4 \beta_{1} + 3 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{47} + ( 7 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{48} -7 q^{49} -5 \beta_{1} q^{50} + ( -1 + 5 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{52} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 7 \beta_{4} - \beta_{5} ) q^{54} + ( -1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -2 + 4 \beta_{3} ) q^{59} + ( 7 - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{62} + ( -5 + 3 \beta_{3} ) q^{64} + ( -6 \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{69} + ( 4 \beta_{1} + 3 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{71} + ( 3 + 5 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{72} + ( -2 \beta_{1} - 5 \beta_{2} + \beta_{4} + 4 \beta_{5} ) q^{73} + ( -5 \beta_{2} - 5 \beta_{4} ) q^{75} + ( -9 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{78} + ( 9 + 12 \beta_{1} + 2 \beta_{2} - 6 \beta_{4} + 4 \beta_{5} ) q^{81} + ( 7 + 5 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{82} + ( 2 + 4 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{87} + ( -3 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{92} + ( -2 + 2 \beta_{1} - 5 \beta_{2} - \beta_{4} + 6 \beta_{5} ) q^{93} + ( -5 + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{94} + ( -9 - 7 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{96} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{6} - 9q^{8} - 18q^{9} + O(q^{10}) \) \( 6q - 3q^{6} - 9q^{8} - 18q^{9} + 15q^{12} + 21q^{18} + 30q^{25} - 27q^{26} - 33q^{36} + 39q^{48} - 42q^{49} + 3q^{52} + 9q^{54} - 15q^{58} + 45q^{62} - 21q^{64} + 27q^{72} - 51q^{78} + 54q^{81} + 33q^{82} - 12q^{93} - 39q^{94} - 57q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{3} + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 1 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 3 \nu^{2} - 2 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 3 \nu^{2} - 2 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + 3 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.33454 + 0.467979i
1.33454 0.467979i
−0.261988 + 1.38973i
−0.261988 1.38973i
−1.07255 + 0.921756i
−1.07255 0.921756i
−1.33454 0.467979i 3.43410i 1.56199 + 1.24907i 0 −1.60709 + 4.58295i 0 −1.50000 2.39792i −8.79306 0
91.2 −1.33454 + 0.467979i 3.43410i 1.56199 1.24907i 0 −1.60709 4.58295i 0 −1.50000 + 2.39792i −8.79306 0
91.3 0.261988 1.38973i 1.32309i −1.86272 0.728188i 0 −1.83875 0.346635i 0 −1.50000 + 2.39792i 1.24943 0
91.4 0.261988 + 1.38973i 1.32309i −1.86272 + 0.728188i 0 −1.83875 + 0.346635i 0 −1.50000 2.39792i 1.24943 0
91.5 1.07255 0.921756i 2.11101i 0.300733 1.97726i 0 1.94584 + 2.26417i 0 −1.50000 2.39792i −1.45636 0
91.6 1.07255 + 0.921756i 2.11101i 0.300733 + 1.97726i 0 1.94584 2.26417i 0 −1.50000 + 2.39792i −1.45636 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
4.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.2.b.b 6
3.b odd 2 1 828.2.e.b 6
4.b odd 2 1 inner 92.2.b.b 6
8.b even 2 1 1472.2.c.c 6
8.d odd 2 1 1472.2.c.c 6
12.b even 2 1 828.2.e.b 6
23.b odd 2 1 CM 92.2.b.b 6
69.c even 2 1 828.2.e.b 6
92.b even 2 1 inner 92.2.b.b 6
184.e odd 2 1 1472.2.c.c 6
184.h even 2 1 1472.2.c.c 6
276.h odd 2 1 828.2.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.b.b 6 1.a even 1 1 trivial
92.2.b.b 6 4.b odd 2 1 inner
92.2.b.b 6 23.b odd 2 1 CM
92.2.b.b 6 92.b even 2 1 inner
828.2.e.b 6 3.b odd 2 1
828.2.e.b 6 12.b even 2 1
828.2.e.b 6 69.c even 2 1
828.2.e.b 6 276.h odd 2 1
1472.2.c.c 6 8.b even 2 1
1472.2.c.c 6 8.d odd 2 1
1472.2.c.c 6 184.e odd 2 1
1472.2.c.c 6 184.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 18 T_{3}^{4} + 81 T_{3}^{2} + 92 \) acting on \(S_{2}^{\mathrm{new}}(92, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 3 T^{3} + T^{6} \)
$3$ \( 92 + 81 T^{2} + 18 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( T^{6} \)
$13$ \( ( 74 - 39 T + T^{3} )^{2} \)
$17$ \( T^{6} \)
$19$ \( T^{6} \)
$23$ \( ( 23 + T^{2} )^{3} \)
$29$ \( ( 282 - 87 T + T^{3} )^{2} \)
$31$ \( 828 + 8649 T^{2} + 186 T^{4} + T^{6} \)
$37$ \( T^{6} \)
$41$ \( ( -426 - 123 T + T^{3} )^{2} \)
$43$ \( T^{6} \)
$47$ \( 412988 + 19881 T^{2} + 282 T^{4} + T^{6} \)
$53$ \( T^{6} \)
$59$ \( ( 92 + T^{2} )^{3} \)
$61$ \( T^{6} \)
$67$ \( T^{6} \)
$71$ \( 48668 + 45369 T^{2} + 426 T^{4} + T^{6} \)
$73$ \( ( -1226 - 219 T + T^{3} )^{2} \)
$79$ \( T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( T^{6} \)
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