Properties

Label 2695.2.a.l
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 6 ) / 2 \) 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
0.723742
−1.77571
2.64119
−0.589216
−2.03967 2.19994 2.16027 −1.00000 −4.48716 0 −0.326891 1.83973 2.03967
1.2 −0.649405 −2.92887 −1.57827 −1.00000 1.90202 0 2.32375 5.57827 0.649405
1.3 1.88395 −2.33468 1.54927 −1.00000 −4.39842 0 −0.849150 2.45073 −1.88395
1.4 2.80513 1.06361 5.86874 −1.00000 2.98356 0 10.8523 −1.86874 −2.80513
\(n\): 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.l 4
7.b odd 2 1 385.2.a.h 4
21.c even 2 1 3465.2.a.bk 4
28.d even 2 1 6160.2.a.br 4
35.c odd 2 1 1925.2.a.x 4
35.f even 4 2 1925.2.b.p 8
77.b even 2 1 4235.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.h 4 7.b odd 2 1
1925.2.a.x 4 35.c odd 2 1
1925.2.b.p 8 35.f even 4 2
2695.2.a.l 4 1.a even 1 1 trivial
3465.2.a.bk 4 21.c even 2 1
4235.2.a.r 4 77.b even 2 1
6160.2.a.br 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 7 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 10T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 7 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 8 T^{2} - 10 T + 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} - 8 T^{2} + 162 T - 236 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 4 T^{2} + 14 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 28 T^{2} - 120 T + 32 \) Copy content Toggle raw display
$23$ \( T^{4} - 14 T^{3} + 28 T^{2} + \cdots - 976 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 80 T^{2} - 272 T + 304 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 30 T^{2} + \cdots + 304 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 28 T^{2} - 20 T + 8 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} - 14 T^{2} + \cdots - 428 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 36 T^{2} + \cdots - 272 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} - 72 T^{2} + \cdots - 8408 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} - 236 T^{2} + \cdots + 12728 \) Copy content Toggle raw display
$59$ \( T^{4} + 26 T^{3} + 110 T^{2} + \cdots - 11848 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 30 T^{2} + 38 T - 4 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} - 192 T^{2} + \cdots + 11168 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} - 20 T^{2} + \cdots + 124 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + 52 T^{2} + \cdots - 544 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} - 28 T^{2} + \cdots - 992 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + 16 T^{2} + 32 T - 32 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} - 192 T^{2} + \cdots + 2272 \) Copy content Toggle raw display
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