Properties

Label 1617.2.a.y
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \) 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
2.77462
0.814115
−0.360409
−1.22833
−1.77462 1.00000 1.14929 1.50970 −1.77462 0 1.50970 1.00000 −2.67914
1.2 0.185885 1.00000 −1.96545 −0.737118 0.185885 0 −0.737118 1.00000 −0.137020
1.3 1.36041 1.00000 −0.149286 −2.92391 1.36041 0 −2.92391 1.00000 −3.97771
1.4 2.22833 1.00000 2.96545 2.15133 2.22833 0 2.15133 1.00000 4.79387
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 1617.2.a.y yes 4
3.b odd 2 1 4851.2.a.br 4
7.b odd 2 1 1617.2.a.w 4
21.c even 2 1 4851.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.w 4 7.b odd 2 1
1617.2.a.y yes 4 1.a even 1 1 trivial
4851.2.a.br 4 3.b odd 2 1
4851.2.a.bs 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 6T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 8T_{5}^{2} + 4T_{5} + 7 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} + 6T_{13}^{2} + 24T_{13} - 7 \) Copy content Toggle raw display
\( T_{17}^{4} - 8T_{17}^{3} - 6T_{17}^{2} + 48T_{17} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 3 T^{2} + 6 T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8 T^{2} + 4 T + 7 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 6 T^{2} + 24 T - 7 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} - 6 T^{2} + 48 T - 28 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 6 T^{2} + 24 T - 7 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 10 T^{2} + 40 T - 68 \) Copy content Toggle raw display
$29$ \( T^{4} - 16 T^{3} + 80 T^{2} - 140 T + 47 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} - 14 T^{2} + 40 T + 28 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} - 50 T^{2} - 236 T - 119 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 24 T^{2} + 8 T - 4 \) Copy content Toggle raw display
$43$ \( T^{4} - 30 T^{2} - 40 T - 4 \) Copy content Toggle raw display
$47$ \( T^{4} - 20 T^{3} + 112 T^{2} + \cdots + 175 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} - 70 T^{2} + 488 T + 476 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} - 48 T^{2} + 148 T + 511 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} - 56 T^{2} + \cdots - 23792 \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + 166 T^{2} + \cdots - 3871 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} - 56 T^{2} + \cdots + 412 \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + 118 T^{2} + \cdots - 223 \) Copy content Toggle raw display
$79$ \( T^{4} - 152 T^{2} - 640 T + 64 \) Copy content Toggle raw display
$83$ \( T^{4} - 32 T^{3} + 258 T^{2} + \cdots - 7100 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} - 124 T^{2} + \cdots + 3088 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} - 134 T^{2} + \cdots + 4348 \) Copy content Toggle raw display
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