Properties

Label 931.2.a.m
Level $931$
Weight $2$
Character orbit 931.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
Defining polynomial: \(x^{4} - 5 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{10} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 2 - \beta_{2} + \beta_{3} ) q^{12} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{16} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{18} - q^{19} + ( -1 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{20} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{22} + ( -3 + 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( 3 - \beta_{3} ) q^{24} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{25} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{26} + ( -2 - \beta_{2} - \beta_{3} ) q^{27} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{30} + ( -2 \beta_{1} - \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{32} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -4 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{34} + ( -5 - 2 \beta_{1} ) q^{36} + ( 3 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{37} -\beta_{1} q^{38} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{39} + ( 4 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{40} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -1 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{44} + ( 3 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -2 + \beta_{1} - 2 \beta_{3} ) q^{46} + ( 4 + \beta_{1} + 5 \beta_{3} ) q^{47} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{48} + ( 2 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{50} + ( -5 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 6 + 6 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{52} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 1 - 4 \beta_{1} + \beta_{3} ) q^{54} + ( -3 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{55} -\beta_{2} q^{57} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 4 - \beta_{2} - 3 \beta_{3} ) q^{59} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{60} + ( 6 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{61} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( 1 + 7 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{65} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{66} + ( 3 - 4 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -7 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{68} + ( 6 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -2 - 3 \beta_{1} - 2 \beta_{3} ) q^{72} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{73} + ( -5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{74} + ( 8 + 2 \beta_{2} - 6 \beta_{3} ) q^{75} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{76} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{78} + ( -2 + 2 \beta_{1} - 4 \beta_{3} ) q^{79} + ( 7 + \beta_{3} ) q^{80} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -2 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{82} + ( 3 + 2 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{83} + ( -5 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{86} + ( -6 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{87} + ( 1 - 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{88} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -6 - 6 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{90} + ( 8 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{92} + ( 3 - 2 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 2 + 10 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{94} + ( -2 - \beta_{1} + \beta_{3} ) q^{95} + ( -4 - \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{96} + ( -2 - 2 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -1 + 4 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 2q^{4} + 8q^{5} - 4q^{6} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{4} + 8q^{5} - 4q^{6} + 6q^{8} + 2q^{9} + 10q^{10} - 6q^{11} + 6q^{12} + 2q^{13} + 4q^{15} + 2q^{16} + 8q^{17} - 6q^{18} - 4q^{19} - 14q^{22} - 8q^{23} + 12q^{24} + 20q^{25} + 16q^{26} - 10q^{27} + 2q^{29} + 2q^{30} + 2q^{32} + 18q^{33} - 22q^{34} - 20q^{36} + 10q^{37} + 2q^{39} + 22q^{40} + 12q^{41} + 4q^{43} - 14q^{44} + 8q^{45} - 8q^{46} + 16q^{47} - 12q^{48} + 12q^{50} - 10q^{51} + 28q^{52} + 12q^{53} + 4q^{54} - 8q^{55} - 2q^{57} + 4q^{58} + 14q^{59} - 2q^{60} + 20q^{61} - 20q^{62} - 20q^{64} + 10q^{65} - 8q^{66} + 2q^{67} - 24q^{68} + 14q^{69} - 2q^{71} - 8q^{72} - 16q^{73} - 26q^{74} + 36q^{75} - 2q^{76} + 14q^{78} - 8q^{79} + 28q^{80} - 20q^{81} - 12q^{82} + 20q^{83} - 14q^{85} + 8q^{86} - 20q^{87} - 2q^{88} + 16q^{89} - 32q^{90} + 26q^{92} + 12q^{93} + 10q^{94} - 8q^{95} - 12q^{96} + 2q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} - 2 x + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.92022
−0.751024
0.291367
2.37988
−1.92022 1.52077 1.68725 −2.00692 −2.92022 0 0.600553 −0.687248 3.85372
1.2 −0.751024 2.33152 −1.43596 4.26543 −1.75102 0 2.58049 2.43596 −3.20344
1.3 0.291367 −2.43210 −1.91511 2.06574 −0.708633 0 −1.14073 2.91511 0.601888
1.4 2.37988 0.579810 3.66382 3.67575 1.37988 0 3.95969 −2.66382 8.74783
\(n\): 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.m yes 4
3.b odd 2 1 8379.2.a.bu 4
7.b odd 2 1 931.2.a.l 4
7.c even 3 2 931.2.f.n 8
7.d odd 6 2 931.2.f.o 8
21.c even 2 1 8379.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.a.l 4 7.b odd 2 1
931.2.a.m yes 4 1.a even 1 1 trivial
931.2.f.n 8 7.c even 3 2
931.2.f.o 8 7.d odd 6 2
8379.2.a.bu 4 3.b odd 2 1
8379.2.a.bv 4 21.c even 2 1

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{3}^{4} - 2 T_{3}^{3} - 5 T_{3}^{2} + 12 T_{3} - 5 \)
\( T_{5}^{4} - 8 T_{5}^{3} + 12 T_{5}^{2} + 32 T_{5} - 65 \)
\( T_{13}^{4} - 2 T_{13}^{3} - 20 T_{13}^{2} - 26 T_{13} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - 5 T^{2} + T^{4} \)
$3$ \( -5 + 12 T - 5 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( -65 + 32 T + 12 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -59 - 102 T - 13 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( -5 - 26 T - 20 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( -125 + 228 T - 23 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( -59 - 144 T - 14 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( -191 + 180 T - 41 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( 35 + 22 T - 27 T^{2} + T^{4} \)
$37$ \( -691 + 358 T - 22 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( 155 + 158 T + 9 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( -448 + 352 T - 64 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( -7945 + 2232 T - 84 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( 65 + 102 T + 7 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( -305 + 234 T + 16 T^{2} - 14 T^{3} + T^{4} \)
$61$ \( -685 + 84 T + 96 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( 2293 - 260 T - 161 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( 3569 - 62 T - 130 T^{2} + 2 T^{3} + T^{4} \)
$73$ \( -305 - 48 T + 55 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( -16 - 288 T - 108 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 4555 + 1352 T - 37 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( -2240 + 736 T + 8 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( 815 + 86 T - 224 T^{2} - 2 T^{3} + T^{4} \)
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