Properties

Label 2695.2.a.k
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 3\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 4\beta _1 + 1 \) 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
2.06150
−0.693822
−1.76401
0.396339
−2.57641 0.514916 4.63791 −1.00000 −1.32664 0 −6.79636 −2.73486 2.57641
1.2 −1.74747 2.44129 1.05365 −1.00000 −4.26608 0 1.65372 2.95990 1.74747
1.3 0.197126 1.56689 −1.96114 −1.00000 0.308875 0 −0.780845 −0.544860 −0.197126
1.4 1.12676 −1.52310 −0.730419 −1.00000 −1.71616 0 −3.07652 −0.680180 −1.12676
\(n\): 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.k 4
7.b odd 2 1 2695.2.a.j 4
7.d odd 6 2 385.2.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.a 8 7.d odd 6 2
2695.2.a.j 4 7.b odd 2 1
2695.2.a.k 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} - T_{3}^{2} + 7T_{3} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} - T^{2} - 5 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} - T^{2} + 7 T - 3 \) 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
$13$ \( T^{4} - 6 T^{3} - 4 T^{2} + 22 T + 3 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} - 7 T^{2} + 29 T - 21 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} - 40 T^{2} + 56 T + 373 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} - 14 T^{2} - 109 T - 127 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} - 9 T^{2} - 42 T + 43 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 14 T^{2} - 113 T - 43 \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} - 22 T^{2} - 334 T - 621 \) Copy content Toggle raw display
$41$ \( T^{4} + 15 T^{3} + 21 T^{2} + \cdots - 557 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} - 127 T^{2} + \cdots + 2851 \) Copy content Toggle raw display
$47$ \( T^{4} - 15 T^{3} + 61 T^{2} + \cdots - 301 \) Copy content Toggle raw display
$53$ \( T^{4} + 30 T^{3} + 303 T^{2} + \cdots + 1293 \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} - 21 T^{2} + \cdots - 1647 \) Copy content Toggle raw display
$61$ \( T^{4} - 46 T^{2} - 19 T + 57 \) Copy content Toggle raw display
$67$ \( T^{4} + 25 T^{3} - 58 T^{2} + \cdots - 33217 \) Copy content Toggle raw display
$71$ \( T^{4} + 13 T^{3} - 29 T^{2} + \cdots + 717 \) Copy content Toggle raw display
$73$ \( T^{4} + 3 T^{3} - 76 T^{2} + 22 T + 3 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} - 132 T^{2} + \cdots + 157 \) Copy content Toggle raw display
$83$ \( T^{4} + 18 T^{3} - 18 T^{2} + \cdots - 7629 \) Copy content Toggle raw display
$89$ \( T^{4} + 25 T^{3} - 39 T^{2} + \cdots - 29273 \) Copy content Toggle raw display
$97$ \( T^{4} - 23 T^{3} + 150 T^{2} + \cdots - 881 \) Copy content Toggle raw display
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