Properties

Label 931.2.a.e
Level $931$
Weight $2$
Character orbit 931.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.43407242818\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + \beta q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 2 - \beta ) q^{6} + ( -3 + \beta ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + \beta q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 2 - \beta ) q^{6} + ( -3 + \beta ) q^{8} - q^{9} + ( 1 - \beta ) q^{10} + ( 1 + \beta ) q^{11} + ( -4 + \beta ) q^{12} + ( -2 - 3 \beta ) q^{13} -\beta q^{15} + 3 q^{16} + 4 q^{17} + ( 1 - \beta ) q^{18} + q^{19} + ( -1 + 2 \beta ) q^{20} + q^{22} + ( -7 - \beta ) q^{23} + ( 2 - 3 \beta ) q^{24} -4 q^{25} + ( -4 + \beta ) q^{26} -4 \beta q^{27} + ( -8 - \beta ) q^{29} + ( -2 + \beta ) q^{30} + ( 2 - 3 \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( 2 + \beta ) q^{33} + ( -4 + 4 \beta ) q^{34} + ( -1 + 2 \beta ) q^{36} + ( 2 - 5 \beta ) q^{37} + ( -1 + \beta ) q^{38} + ( -6 - 2 \beta ) q^{39} + ( 3 - \beta ) q^{40} + ( 6 + 2 \beta ) q^{41} + ( 1 - 5 \beta ) q^{43} + ( -3 - \beta ) q^{44} + q^{45} + ( 5 - 6 \beta ) q^{46} + ( 3 - \beta ) q^{47} + 3 \beta q^{48} + ( 4 - 4 \beta ) q^{50} + 4 \beta q^{51} + ( 10 + \beta ) q^{52} + 3 \beta q^{53} + ( -8 + 4 \beta ) q^{54} + ( -1 - \beta ) q^{55} + \beta q^{57} + ( 6 - 7 \beta ) q^{58} + ( -4 + 2 \beta ) q^{59} + ( 4 - \beta ) q^{60} + ( -9 + 2 \beta ) q^{61} + ( -8 + 5 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 2 + 3 \beta ) q^{65} + \beta q^{66} + ( 4 + 2 \beta ) q^{67} + ( 4 - 8 \beta ) q^{68} + ( -2 - 7 \beta ) q^{69} + ( -10 + 3 \beta ) q^{71} + ( 3 - \beta ) q^{72} + ( -1 + 10 \beta ) q^{73} + ( -12 + 7 \beta ) q^{74} -4 \beta q^{75} + ( 1 - 2 \beta ) q^{76} + ( 2 - 4 \beta ) q^{78} -\beta q^{79} -3 q^{80} -5 q^{81} + ( -2 + 4 \beta ) q^{82} + ( -5 + 3 \beta ) q^{83} -4 q^{85} + ( -11 + 6 \beta ) q^{86} + ( -2 - 8 \beta ) q^{87} + ( -1 - 2 \beta ) q^{88} + ( 10 - \beta ) q^{89} + ( -1 + \beta ) q^{90} + ( -3 + 13 \beta ) q^{92} + ( -6 + 2 \beta ) q^{93} + ( -5 + 4 \beta ) q^{94} - q^{95} + ( 2 + 3 \beta ) q^{96} + ( 6 + 2 \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{6} - 6q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{6} - 6q^{8} - 2q^{9} + 2q^{10} + 2q^{11} - 8q^{12} - 4q^{13} + 6q^{16} + 8q^{17} + 2q^{18} + 2q^{19} - 2q^{20} + 2q^{22} - 14q^{23} + 4q^{24} - 8q^{25} - 8q^{26} - 16q^{29} - 4q^{30} + 4q^{31} + 6q^{32} + 4q^{33} - 8q^{34} - 2q^{36} + 4q^{37} - 2q^{38} - 12q^{39} + 6q^{40} + 12q^{41} + 2q^{43} - 6q^{44} + 2q^{45} + 10q^{46} + 6q^{47} + 8q^{50} + 20q^{52} - 16q^{54} - 2q^{55} + 12q^{58} - 8q^{59} + 8q^{60} - 18q^{61} - 16q^{62} - 14q^{64} + 4q^{65} + 8q^{67} + 8q^{68} - 4q^{69} - 20q^{71} + 6q^{72} - 2q^{73} - 24q^{74} + 2q^{76} + 4q^{78} - 6q^{80} - 10q^{81} - 4q^{82} - 10q^{83} - 8q^{85} - 22q^{86} - 4q^{87} - 2q^{88} + 20q^{89} - 2q^{90} - 6q^{92} - 12q^{93} - 10q^{94} - 2q^{95} + 4q^{96} + 12q^{97} - 2q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 −1.41421 3.82843 −1.00000 3.41421 0 −4.41421 −1.00000 2.41421
1.2 0.414214 1.41421 −1.82843 −1.00000 0.585786 0 −1.58579 −1.00000 −0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.a.e 2
3.b odd 2 1 8379.2.a.bl 2
7.b odd 2 1 931.2.a.f 2
7.c even 3 2 931.2.f.i 4
7.d odd 6 2 133.2.f.c 4
21.c even 2 1 8379.2.a.bi 2
21.g even 6 2 1197.2.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.c 4 7.d odd 6 2
931.2.a.e 2 1.a even 1 1 trivial
931.2.a.f 2 7.b odd 2 1
931.2.f.i 4 7.c even 3 2
1197.2.j.e 4 21.g even 6 2
8379.2.a.bi 2 21.c even 2 1
8379.2.a.bl 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(931))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{3}^{2} - 2 \)
\( T_{5} + 1 \)
\( T_{13}^{2} + 4 T_{13} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -1 - 2 T + T^{2} \)
$13$ \( -14 + 4 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 47 + 14 T + T^{2} \)
$29$ \( 62 + 16 T + T^{2} \)
$31$ \( -14 - 4 T + T^{2} \)
$37$ \( -46 - 4 T + T^{2} \)
$41$ \( 28 - 12 T + T^{2} \)
$43$ \( -49 - 2 T + T^{2} \)
$47$ \( 7 - 6 T + T^{2} \)
$53$ \( -18 + T^{2} \)
$59$ \( 8 + 8 T + T^{2} \)
$61$ \( 73 + 18 T + T^{2} \)
$67$ \( 8 - 8 T + T^{2} \)
$71$ \( 82 + 20 T + T^{2} \)
$73$ \( -199 + 2 T + T^{2} \)
$79$ \( -2 + T^{2} \)
$83$ \( 7 + 10 T + T^{2} \)
$89$ \( 98 - 20 T + T^{2} \)
$97$ \( 28 - 12 T + T^{2} \)
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