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: \(x^{3} - x^{2} - 3 x + 1\)
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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{3} + 3 T_{2}^{2} - T_{2} - 5 \)
\( T_{3}^{3} - 2 T_{3}^{2} - 2 T_{3} + 2 \)

Hecke characteristic polynomials

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