Properties

Label 1617.2.a.q
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} - \beta_1 q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} - \beta_1 q^{6} + q^{8} + q^{9} + ( - \beta_{2} - 2) q^{10} + q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{2} + \beta_1 + 1) q^{15} + ( - 2 \beta_{2} + \beta_1 - 2) q^{16} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + \beta_1 q^{18} + (\beta_{2} - \beta_1 - 1) q^{19} + ( - 2 \beta_{2} - \beta_1 + 1) q^{20} + \beta_1 q^{22} - q^{24} + ( - 3 \beta_{2} + \beta_1) q^{25} + ( - \beta_{2} - 2 \beta_1 - 4) q^{26} - q^{27} + ( - 3 \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{2} + 2) q^{30} + (2 \beta_{2} + 2 \beta_1 - 2) q^{31} + (\beta_{2} - 4 \beta_1 - 1) q^{32} - q^{33} + (2 \beta_{2} - 4 \beta_1 + 4) q^{34} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - \beta_1 - 3) q^{37} + ( - \beta_{2} - 2) q^{38} + (\beta_{2} + \beta_1 + 1) q^{39} + (\beta_{2} - \beta_1 - 1) q^{40} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{2} - \beta_1 - 1) q^{45} + (\beta_{2} - 3 \beta_1 - 3) q^{47} + (2 \beta_{2} - \beta_1 + 2) q^{48} + (\beta_{2} - 3 \beta_1) q^{50} + (2 \beta_{2} - 2 \beta_1 + 2) q^{51} + ( - 3 \beta_1 - 5) q^{52} + (2 \beta_{2} - 2 \beta_1 + 4) q^{53} - \beta_1 q^{54} + (\beta_{2} - \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - 2 \beta_1 - 6) q^{58} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{59} + (2 \beta_{2} + \beta_1 - 1) q^{60} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{61} + (2 \beta_{2} + 8) q^{62} + ( - 2 \beta_1 - 7) q^{64} + (3 \beta_{2} + \beta_1 + 1) q^{65} - \beta_1 q^{66} + ( - 5 \beta_{2} + 5 \beta_1 + 1) q^{67} + (2 \beta_1 - 6) q^{68} + (6 \beta_{2} - 6 \beta_1 + 2) q^{71} + q^{72} + (5 \beta_{2} + \beta_1 + 1) q^{73} + ( - \beta_{2} - 4 \beta_1 - 4) q^{74} + (3 \beta_{2} - \beta_1) q^{75} + ( - 2 \beta_{2} - \beta_1 + 1) q^{76} + (\beta_{2} + 2 \beta_1 + 4) q^{78} + ( - 4 \beta_1 - 4) q^{79} + (3 \beta_{2} + 2 \beta_1 - 4) q^{80} + q^{81} + ( - 2 \beta_{2} - 8) q^{82} + (2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{2} + 8) q^{86} + (3 \beta_{2} + \beta_1 - 1) q^{87} + q^{88} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{89} + ( - \beta_{2} - 2) q^{90} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - 3 \beta_{2} - 2 \beta_1 - 8) q^{94} + ( - 3 \beta_{2} + \beta_1 + 5) q^{95} + ( - \beta_{2} + 4 \beta_1 + 1) q^{96} + (2 \beta_{2} + 6 \beta_1 - 2) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{15} - 4 q^{16} - 4 q^{17} - 4 q^{19} + 5 q^{20} - 3 q^{24} + 3 q^{25} - 11 q^{26} - 3 q^{27} + 6 q^{29} + 5 q^{30} - 8 q^{31} - 4 q^{32} - 3 q^{33} + 10 q^{34} + 2 q^{36} - 8 q^{37} - 5 q^{38} + 2 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 2 q^{44} - 4 q^{45} - 10 q^{47} + 4 q^{48} - q^{50} + 4 q^{51} - 15 q^{52} + 10 q^{53} - 4 q^{55} + 4 q^{57} - 17 q^{58} - 6 q^{59} - 5 q^{60} - 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 3 q^{72} - 2 q^{73} - 11 q^{74} - 3 q^{75} + 5 q^{76} + 11 q^{78} - 12 q^{79} - 15 q^{80} + 3 q^{81} - 22 q^{82} + 4 q^{83} - 20 q^{85} + 22 q^{86} - 6 q^{87} + 3 q^{88} - 16 q^{89} - 5 q^{90} + 8 q^{93} - 21 q^{94} + 18 q^{95} + 4 q^{96} - 8 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
−1.86081
−0.254102
2.11491
−1.86081 −1.00000 1.46260 1.32340 1.86081 0 1.00000 1.00000 −2.46260
1.2 −0.254102 −1.00000 −1.93543 −3.68133 0.254102 0 1.00000 1.00000 0.935432
1.3 2.11491 −1.00000 2.47283 −1.64207 −2.11491 0 1.00000 1.00000 −3.47283
\(n\): 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.q 3
3.b odd 2 1 4851.2.a.bm 3
7.b odd 2 1 1617.2.a.r yes 3
21.c even 2 1 4851.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.q 3 1.a even 1 1 trivial
1617.2.a.r yes 3 7.b odd 2 1
4851.2.a.bl 3 21.c even 2 1
4851.2.a.bm 3 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - T_{5} - 8 \) Copy content Toggle raw display
\( T_{13}^{3} + 2T_{13}^{2} - 11T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 20T_{17} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4T^{2} - T - 8 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 11 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 20 T - 16 \) Copy content Toggle raw display
$19$ \( T^{3} + 4T^{2} - T - 8 \) Copy content Toggle raw display
$23$ \( T^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 49 T + 82 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 28 T - 208 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} + 9 T - 2 \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} - 28 T + 208 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 28 T - 208 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} + T - 124 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + 8 T + 32 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 91 T + 196 \) Copy content Toggle raw display
$61$ \( T^{3} + 16 T^{2} + 36 T - 16 \) Copy content Toggle raw display
$67$ \( T^{3} - 8 T^{2} - 137 T + 644 \) Copy content Toggle raw display
$71$ \( T^{3} - 228T - 416 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 151 T + 212 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} - 16 T - 128 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 44 T - 32 \) Copy content Toggle raw display
$89$ \( T^{3} + 16 T^{2} + 60 T + 32 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 180 T - 1568 \) Copy content Toggle raw display
show more
show less