Properties

Label 2695.2.a.g
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5196833447\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
−2.67513 2.48119 5.15633 1.00000 −6.63752 0 −8.44358 3.15633 −2.67513
1.2 −1.53919 −1.17009 0.369102 1.00000 1.80098 0 2.51026 −1.63090 −1.53919
1.3 1.21432 0.688892 −0.525428 1.00000 0.836535 0 −3.06668 −2.52543 1.21432
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 2695.2.a.g 3
7.b odd 2 1 385.2.a.f 3
21.c even 2 1 3465.2.a.bh 3
28.d even 2 1 6160.2.a.bn 3
35.c odd 2 1 1925.2.a.v 3
35.f even 4 2 1925.2.b.n 6
77.b even 2 1 4235.2.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.f 3 7.b odd 2 1
1925.2.a.v 3 35.c odd 2 1
1925.2.b.n 6 35.f even 4 2
2695.2.a.g 3 1.a even 1 1 trivial
3465.2.a.bh 3 21.c even 2 1
4235.2.a.q 3 77.b even 2 1
6160.2.a.bn 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5 \) Copy content Toggle raw display
\( T_{3}^{3} - 2T_{3}^{2} - 2T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} - 2 T + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 22 T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 30T - 2 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$23$ \( T^{3} + 10 T^{2} + 28 T + 20 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 20 T + 26 \) Copy content Toggle raw display
$37$ \( T^{3} + 16 T^{2} + 52 T - 100 \) Copy content Toggle raw display
$41$ \( T^{3} - 36T + 54 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 92 T + 268 \) Copy content Toggle raw display
$47$ \( T^{3} - 20 T^{2} + 110 T - 158 \) Copy content Toggle raw display
$53$ \( T^{3} + 12 T^{2} + 20 T + 4 \) Copy content Toggle raw display
$59$ \( T^{3} + 14T^{2} - 74 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} - 60 T + 62 \) Copy content Toggle raw display
$67$ \( T^{3} + 2 T^{2} - 112 T - 172 \) Copy content Toggle raw display
$71$ \( T^{3} + 24 T^{2} + 80 T - 800 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 158 T - 190 \) Copy content Toggle raw display
$79$ \( T^{3} - 8 T^{2} - 112 T + 244 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} - 116 T + 1096 \) Copy content Toggle raw display
$89$ \( T^{3} + 20 T^{2} + 48 T - 320 \) Copy content Toggle raw display
$97$ \( T^{3} - 192T + 160 \) Copy content Toggle raw display
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