Properties

Label 91.2.a.d
Level $91$
Weight $2$
Character orbit 91.a
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 + 1) q^{5} + ( - 2 \beta_1 + 2) q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + ( - 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 + 1) q^{5} + ( - 2 \beta_1 + 2) q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + ( - 2 \beta_1 + 3) q^{9} + ( - \beta_{2} + \beta_1 - 3) q^{10} + (\beta_{2} - \beta_1 + 1) q^{11} - 4 q^{12} + q^{13} - \beta_1 q^{14} + ( - \beta_{2} + 3 \beta_1 - 3) q^{15} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + (\beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{18} + ( - \beta_1 - 1) q^{19} - 2 \beta_1 q^{20} + (\beta_{2} - \beta_1 + 1) q^{21} + (2 \beta_1 - 2) q^{22} + ( - \beta_{2} - 2 \beta_1 + 4) q^{23} - 4 q^{24} + (\beta_{2} - 2 \beta_1 - 1) q^{25} + \beta_1 q^{26} + (4 \beta_1 - 4) q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_{2} + 8) q^{29} + (2 \beta_{2} - 4 \beta_1 + 8) q^{30} + (2 \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{2} - 2 \beta_1 + 3) q^{32} + (2 \beta_1 - 6) q^{33} + (2 \beta_{2} + 2 \beta_1 + 4) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} - 4 \beta_1 + 1) q^{36} + (\beta_{2} + 3 \beta_1 - 1) q^{37} + ( - \beta_{2} - \beta_1 - 3) q^{38} + ( - \beta_{2} + \beta_1 - 1) q^{39} - 2 \beta_1 q^{40} + ( - 2 \beta_{2} + 2 \beta_1) q^{41} + (2 \beta_1 - 2) q^{42} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{43} + 4 q^{44} + (2 \beta_{2} - 5 \beta_1 + 9) q^{45} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{46} + ( - 4 \beta_{2} + \beta_1 - 3) q^{47} + ( - 4 \beta_1 + 8) q^{48} + q^{49} + ( - \beta_{2} - 5) q^{50} + ( - 2 \beta_1 - 2) q^{51} + (\beta_{2} + 1) q^{52} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{53} + (4 \beta_{2} - 4 \beta_1 + 12) q^{54} + (\beta_{2} - 3 \beta_1 + 3) q^{55} + ( - \beta_{2} - 1) q^{56} + (\beta_{2} + \beta_1 - 1) q^{57} + (\beta_{2} + 9 \beta_1 + 1) q^{58} + (4 \beta_{2} + 2 \beta_1 - 2) q^{59} + (4 \beta_1 - 4) q^{60} - 2 q^{61} + (\beta_{2} + \beta_1 - 1) q^{62} + (2 \beta_1 - 3) q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + ( - \beta_1 + 1) q^{65} + (2 \beta_{2} - 6 \beta_1 + 6) q^{66} + (4 \beta_{2} - 6 \beta_1 - 2) q^{67} + (2 \beta_{2} + 4 \beta_1 + 6) q^{68} + ( - 5 \beta_{2} + 9 \beta_1 - 5) q^{69} + (\beta_{2} - \beta_1 + 3) q^{70} + ( - \beta_{2} + 3 \beta_1 - 3) q^{71} + (\beta_{2} - 4 \beta_1 + 1) q^{72} + ( - 4 \beta_{2} - \beta_1 - 3) q^{73} + (4 \beta_{2} + 10) q^{74} + (2 \beta_{2} + 2 \beta_1 - 6) q^{75} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{76} + ( - \beta_{2} + \beta_1 - 1) q^{77} + ( - 2 \beta_1 + 2) q^{78} + ( - \beta_{2} + 4 \beta_1 - 6) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{80} + (4 \beta_{2} - 6 \beta_1 + 3) q^{81} + ( - 2 \beta_1 + 4) q^{82} + (4 \beta_{2} - 9 \beta_1 - 1) q^{83} + 4 q^{84} + ( - \beta_{2} - \beta_1 - 3) q^{85} + ( - 5 \beta_{2} + \beta_1 - 9) q^{86} + ( - 7 \beta_{2} + 7 \beta_1 - 11) q^{87} + 4 q^{88} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{89} + ( - 3 \beta_{2} + 11 \beta_1 - 13) q^{90} - q^{91} + (2 \beta_{2} - 6 \beta_1 - 2) q^{92} + (3 \beta_{2} - \beta_1 - 7) q^{93} + ( - 3 \beta_{2} - 7 \beta_1 - 1) q^{94} + (\beta_{2} + 2) q^{95} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{96} + ( - \beta_1 - 3) q^{97} + \beta_1 q^{98} + (3 \beta_{2} - 7 \beta_1 + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9} - 8 q^{10} + 2 q^{11} - 12 q^{12} + 3 q^{13} - q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 15 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + q^{26} - 8 q^{27} - 3 q^{28} + 24 q^{29} + 20 q^{30} - 4 q^{31} + 7 q^{32} - 16 q^{33} + 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{39} - 2 q^{40} + 2 q^{41} - 4 q^{42} + 10 q^{43} + 12 q^{44} + 22 q^{45} - 18 q^{46} - 8 q^{47} + 20 q^{48} + 3 q^{49} - 15 q^{50} - 8 q^{51} + 3 q^{52} + 8 q^{53} + 32 q^{54} + 6 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 4 q^{59} - 8 q^{60} - 6 q^{61} - 2 q^{62} - 7 q^{63} - 17 q^{64} + 2 q^{65} + 12 q^{66} - 12 q^{67} + 22 q^{68} - 6 q^{69} + 8 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} + 30 q^{74} - 16 q^{75} - 8 q^{76} - 2 q^{77} + 4 q^{78} - 14 q^{79} - 14 q^{80} + 3 q^{81} + 10 q^{82} - 12 q^{83} + 12 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 12 q^{88} + 2 q^{89} - 28 q^{90} - 3 q^{91} - 12 q^{92} - 22 q^{93} - 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} + q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display

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
−1.81361
0.470683
2.34292
−1.81361 −3.10278 1.28917 2.81361 5.62721 −1.00000 1.28917 6.62721 −5.10278
1.2 0.470683 2.24914 −1.77846 0.529317 1.05863 −1.00000 −1.77846 2.05863 0.249141
1.3 2.34292 −1.14637 3.48929 −1.34292 −2.68585 −1.00000 3.48929 −1.68585 −3.14637
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

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 91.2.a.d 3
3.b odd 2 1 819.2.a.i 3
4.b odd 2 1 1456.2.a.t 3
5.b even 2 1 2275.2.a.m 3
7.b odd 2 1 637.2.a.j 3
7.c even 3 2 637.2.e.j 6
7.d odd 6 2 637.2.e.i 6
8.b even 2 1 5824.2.a.by 3
8.d odd 2 1 5824.2.a.bs 3
13.b even 2 1 1183.2.a.i 3
13.d odd 4 2 1183.2.c.f 6
21.c even 2 1 5733.2.a.x 3
91.b odd 2 1 8281.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 1.a even 1 1 trivial
637.2.a.j 3 7.b odd 2 1
637.2.e.i 6 7.d odd 6 2
637.2.e.j 6 7.c even 3 2
819.2.a.i 3 3.b odd 2 1
1183.2.a.i 3 13.b even 2 1
1183.2.c.f 6 13.d odd 4 2
1456.2.a.t 3 4.b odd 2 1
2275.2.a.m 3 5.b even 2 1
5733.2.a.x 3 21.c even 2 1
5824.2.a.bs 3 8.d odd 2 1
5824.2.a.by 3 8.b even 2 1
8281.2.a.bg 3 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 4T + 2 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 6 T - 8 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 3 T + 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 10 T - 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 4T^{2} + T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + T + 136 \) Copy content Toggle raw display
$29$ \( T^{3} - 24 T^{2} + 185 T - 454 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 19 T + 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 58T - 124 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} - 71 T + 628 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 79 T - 544 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 35 T - 22 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} - 156 T - 688 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} - 124 T - 976 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} - 22 T + 16 \) Copy content Toggle raw display
$73$ \( T^{3} + 10 T^{2} - 99 T - 274 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} + 5 T - 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 12 T^{2} - 271 T - 3268 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 95 T + 422 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} + 29 T + 22 \) Copy content Toggle raw display
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