Properties

Label 92.2.b.a
Level $92$
Weight $2$
Character orbit 92.b
Analytic conductor $0.735$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{14})\)
Defining polynomial: \(x^{4} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( 1 - \beta_{1} ) q^{6} + \beta_{3} q^{7} + ( 2 - 2 \beta_{1} ) q^{8} + 2 q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( 1 - \beta_{1} ) q^{6} + \beta_{3} q^{7} + ( 2 - 2 \beta_{1} ) q^{8} + 2 q^{9} + ( \beta_{2} - \beta_{3} ) q^{10} -\beta_{3} q^{11} -2 q^{12} - q^{13} + ( -\beta_{2} - \beta_{3} ) q^{14} + \beta_{3} q^{15} -4 q^{16} + \beta_{2} q^{17} + ( -2 - 2 \beta_{1} ) q^{18} + 2 \beta_{3} q^{20} + \beta_{2} q^{21} + ( \beta_{2} + \beta_{3} ) q^{22} + ( 3 \beta_{1} - \beta_{3} ) q^{23} + ( 2 + 2 \beta_{1} ) q^{24} -9 q^{25} + ( 1 + \beta_{1} ) q^{26} + 5 \beta_{1} q^{27} + 2 \beta_{2} q^{28} + 5 q^{29} + ( -\beta_{2} - \beta_{3} ) q^{30} -5 \beta_{1} q^{31} + ( 4 + 4 \beta_{1} ) q^{32} -\beta_{2} q^{33} + ( -\beta_{2} + \beta_{3} ) q^{34} -14 \beta_{1} q^{35} + 4 \beta_{1} q^{36} + \beta_{2} q^{37} -\beta_{1} q^{39} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{40} -3 q^{41} + ( -\beta_{2} + \beta_{3} ) q^{42} -2 \beta_{3} q^{43} -2 \beta_{2} q^{44} -2 \beta_{2} q^{45} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + 3 \beta_{1} q^{47} -4 \beta_{1} q^{48} + 7 q^{49} + ( 9 + 9 \beta_{1} ) q^{50} -\beta_{3} q^{51} -2 \beta_{1} q^{52} + 2 \beta_{2} q^{53} + ( 5 - 5 \beta_{1} ) q^{54} + 14 \beta_{1} q^{55} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -5 - 5 \beta_{1} ) q^{58} -6 \beta_{1} q^{59} + 2 \beta_{2} q^{60} + ( -5 + 5 \beta_{1} ) q^{62} + 2 \beta_{3} q^{63} -8 \beta_{1} q^{64} + \beta_{2} q^{65} + ( \beta_{2} - \beta_{3} ) q^{66} + \beta_{3} q^{67} -2 \beta_{3} q^{68} + ( -3 - \beta_{2} ) q^{69} + ( -14 + 14 \beta_{1} ) q^{70} + 5 \beta_{1} q^{71} + ( 4 - 4 \beta_{1} ) q^{72} - q^{73} + ( -\beta_{2} + \beta_{3} ) q^{74} -9 \beta_{1} q^{75} -14 q^{77} + ( -1 + \beta_{1} ) q^{78} + 4 \beta_{2} q^{80} + q^{81} + ( 3 + 3 \beta_{1} ) q^{82} + 3 \beta_{3} q^{83} -2 \beta_{3} q^{84} + 14 q^{85} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{86} + 5 \beta_{1} q^{87} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{88} -2 \beta_{2} q^{89} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{90} -\beta_{3} q^{91} + ( -6 - 2 \beta_{2} ) q^{92} + 5 q^{93} + ( 3 - 3 \beta_{1} ) q^{94} + ( -4 + 4 \beta_{1} ) q^{96} + \beta_{2} q^{97} + ( -7 - 7 \beta_{1} ) q^{98} -2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{6} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{6} + 8q^{8} + 8q^{9} - 8q^{12} - 4q^{13} - 16q^{16} - 8q^{18} + 8q^{24} - 36q^{25} + 4q^{26} + 20q^{29} + 16q^{32} - 12q^{41} + 12q^{46} + 28q^{49} + 36q^{50} + 20q^{54} - 20q^{58} - 20q^{62} - 12q^{69} - 56q^{70} + 16q^{72} - 4q^{73} - 56q^{77} - 4q^{78} + 4q^{81} + 12q^{82} + 56q^{85} - 24q^{92} + 20q^{93} + 12q^{94} - 16q^{96} - 28q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 7 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 7 \nu \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(7 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{3} + 7 \beta_{2}\)\()/2\)

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
1.87083 + 1.87083i
−1.87083 1.87083i
−1.87083 + 1.87083i
1.87083 1.87083i
−1.00000 1.00000i 1.00000i 2.00000i 3.74166i 1.00000 1.00000i 3.74166 2.00000 2.00000i 2.00000 −3.74166 + 3.74166i
91.2 −1.00000 1.00000i 1.00000i 2.00000i 3.74166i 1.00000 1.00000i −3.74166 2.00000 2.00000i 2.00000 3.74166 3.74166i
91.3 −1.00000 + 1.00000i 1.00000i 2.00000i 3.74166i 1.00000 + 1.00000i −3.74166 2.00000 + 2.00000i 2.00000 3.74166 + 3.74166i
91.4 −1.00000 + 1.00000i 1.00000i 2.00000i 3.74166i 1.00000 + 1.00000i 3.74166 2.00000 + 2.00000i 2.00000 −3.74166 3.74166i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(92, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 14 + T^{2} )^{2} \)
$7$ \( ( -14 + T^{2} )^{2} \)
$11$ \( ( -14 + T^{2} )^{2} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( ( 14 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( 529 - 10 T^{2} + T^{4} \)
$29$ \( ( -5 + T )^{4} \)
$31$ \( ( 25 + T^{2} )^{2} \)
$37$ \( ( 14 + T^{2} )^{2} \)
$41$ \( ( 3 + T )^{4} \)
$43$ \( ( -56 + T^{2} )^{2} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( ( 56 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -14 + T^{2} )^{2} \)
$71$ \( ( 25 + T^{2} )^{2} \)
$73$ \( ( 1 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -126 + T^{2} )^{2} \)
$89$ \( ( 56 + T^{2} )^{2} \)
$97$ \( ( 14 + T^{2} )^{2} \)
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