Properties

Label 2695.2.a.h
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
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: \(0\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + ( - \beta_{2} - 2 \beta_1) q^{6} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) 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) q^{6} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - 1) q^{10} + q^{11} + (\beta_1 - 1) q^{12} + ( - \beta_1 + 3) q^{13} + (\beta_1 + 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - \beta_1 + 5) q^{17} - 5 \beta_1 q^{18} + ( - 4 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} - \beta_1 + 2) q^{20} + ( - \beta_{2} - 1) q^{22} + (\beta_{2} - 3 \beta_1 - 1) q^{23} + (3 \beta_{2} + 2 \beta_1 + 2) q^{24} + q^{25} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{26} + (4 \beta_{2} + 4 \beta_1 + 2) q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{2} - 2 \beta_1) q^{30} + (3 \beta_{2} + 4 \beta_1) q^{31} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{32} + (\beta_1 + 1) q^{33} + ( - 5 \beta_{2} + 2 \beta_1 - 6) q^{34} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{36} + ( - \beta_{2} + 5 \beta_1 + 1) q^{37} + 10 q^{38} + ( - \beta_{2} + \beta_1 + 1) q^{39} + (3 \beta_1 - 4) q^{40} + (\beta_{2} - 2 \beta_1 + 4) q^{41} + ( - 3 \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1 + 2) q^{44} + (\beta_{2} + 3 \beta_1) q^{45} + (\beta_{2} + 7 \beta_1 - 5) q^{46} + ( - \beta_1 + 9) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{48} + ( - \beta_{2} - 1) q^{50} + ( - \beta_{2} + 3 \beta_1 + 3) q^{51} + (4 \beta_{2} - 5 \beta_1 + 9) q^{52} + (\beta_{2} - 5 \beta_1 + 3) q^{53} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{54} + q^{55} + ( - 6 \beta_{2} - 8 \beta_1) q^{57} + ( - 6 \beta_1 + 8) q^{58} + (3 \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_1 - 1) q^{60} + (\beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 5 \beta_1 - 5) q^{62} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_1 + 3) q^{65} + ( - \beta_{2} - 2 \beta_1) q^{66} + (5 \beta_{2} + 3 \beta_1 - 3) q^{67} + (6 \beta_{2} - 7 \beta_1 + 13) q^{68} + ( - 2 \beta_{2} - 6 \beta_1 - 8) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{71} + (5 \beta_{2} + 15) q^{72} + ( - 4 \beta_{2} - 3 \beta_1 + 3) q^{73} + ( - \beta_{2} - 11 \beta_1 + 7) q^{74} + (\beta_1 + 1) q^{75} + ( - 2 \beta_{2} + 4 \beta_1 - 10) q^{76} + ( - \beta_{2} - 3 \beta_1 + 3) q^{78} + ( - 5 \beta_{2} - 5 \beta_1 - 1) q^{79} + (2 \beta_{2} - 4 \beta_1 + 3) q^{80} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + ( - 4 \beta_{2} + 5 \beta_1 - 9) q^{82} + ( - 6 \beta_{2} + 2 \beta_1 - 4) q^{83} + ( - \beta_1 + 5) q^{85} + (\beta_{2} - 5 \beta_1 + 11) q^{86} + (2 \beta_1 + 6) q^{87} + (3 \beta_1 - 4) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} - 5 \beta_1 q^{90} + (3 \beta_{2} - 7 \beta_1 + 11) q^{92} + (7 \beta_{2} + 11 \beta_1 + 5) q^{93} + ( - 9 \beta_{2} + 2 \beta_1 - 10) q^{94} + ( - 4 \beta_{2} - 2 \beta_1) q^{95} + (\beta_{2} + 6) q^{96} + 4 \beta_{2} q^{97} + (\beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{5} - 2 q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{5} - 2 q^{6} - 9 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{11} - 2 q^{12} + 8 q^{13} + 4 q^{15} + 5 q^{16} + 14 q^{17} - 5 q^{18} - 2 q^{19} + 5 q^{20} - 3 q^{22} - 6 q^{23} + 8 q^{24} + 3 q^{25} - 10 q^{26} + 10 q^{27} + 2 q^{29} - 2 q^{30} + 4 q^{31} - 11 q^{32} + 4 q^{33} - 16 q^{34} - 11 q^{36} + 8 q^{37} + 30 q^{38} + 4 q^{39} - 9 q^{40} + 10 q^{41} - 2 q^{43} + 5 q^{44} + 3 q^{45} - 8 q^{46} + 26 q^{47} - 24 q^{48} - 3 q^{50} + 12 q^{51} + 22 q^{52} + 4 q^{53} - 34 q^{54} + 3 q^{55} - 8 q^{57} + 18 q^{58} + 8 q^{59} - 2 q^{60} + 8 q^{61} - 20 q^{62} + 33 q^{64} + 8 q^{65} - 2 q^{66} - 6 q^{67} + 32 q^{68} - 30 q^{69} - 20 q^{71} + 45 q^{72} + 6 q^{73} + 10 q^{74} + 4 q^{75} - 26 q^{76} + 6 q^{78} - 8 q^{79} + 5 q^{80} + 23 q^{81} - 22 q^{82} - 10 q^{83} + 14 q^{85} + 28 q^{86} + 20 q^{87} - 9 q^{88} + 16 q^{89} - 5 q^{90} + 26 q^{92} + 26 q^{93} - 28 q^{94} - 2 q^{95} + 18 q^{96} + 3 q^{99}+O(q^{100}) \) 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 −0.481194 5.15633 1.00000 1.28726 0 −8.44358 −2.76845 −2.67513
1.2 −1.53919 3.17009 0.369102 1.00000 −4.87936 0 2.51026 7.04945 −1.53919
1.3 1.21432 1.31111 −0.525428 1.00000 1.59210 0 −3.06668 −1.28100 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.h 3
7.b odd 2 1 385.2.a.e 3
21.c even 2 1 3465.2.a.bi 3
28.d even 2 1 6160.2.a.bo 3
35.c odd 2 1 1925.2.a.w 3
35.f even 4 2 1925.2.b.m 6
77.b even 2 1 4235.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.e 3 7.b odd 2 1
1925.2.a.w 3 35.c odd 2 1
1925.2.b.m 6 35.f even 4 2
2695.2.a.h 3 1.a even 1 1 trivial
3465.2.a.bi 3 21.c even 2 1
4235.2.a.p 3 77.b even 2 1
6160.2.a.bo 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
\( T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( T^{3} - 4 T^{2} + 2 T + 2 \) 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
$13$ \( T^{3} - 8 T^{2} + 18 T - 10 \) Copy content Toggle raw display
$17$ \( T^{3} - 14 T^{2} + 62 T - 86 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 60 T - 200 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 28 T - 148 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 60 T - 50 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 76 T + 436 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 44 T - 20 \) Copy content Toggle raw display
$47$ \( T^{3} - 26 T^{2} + 222 T - 622 \) Copy content Toggle raw display
$53$ \( T^{3} - 4 T^{2} - 92 T - 68 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 16 T + 130 \) Copy content Toggle raw display
$61$ \( T^{3} - 8 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 88 T + 76 \) Copy content Toggle raw display
$71$ \( T^{3} + 20 T^{2} + 112 T + 160 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 58 T + 46 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} - 112 T - 244 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 148 T - 488 \) Copy content Toggle raw display
$89$ \( T^{3} - 16 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$97$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
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