Properties

Label 1617.2.a.s
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_1 q^{6} + (2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_1 q^{6} + (2 \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{2} + 2 \beta_1 - 3) q^{10} + q^{11} + (\beta_{2} + 2) q^{12} + (\beta_{2} - \beta_1) q^{13} + (\beta_{2} - \beta_1) q^{15} + (\beta_1 + 4) q^{16} + 2 \beta_1 q^{17} + \beta_1 q^{18} + (\beta_{2} + \beta_1 - 4) q^{19} + ( - 3 \beta_1 + 7) q^{20} + \beta_1 q^{22} + ( - 2 \beta_1 - 2) q^{23} + (2 \beta_1 + 1) q^{24} + ( - \beta_{2} - 3 \beta_1 + 5) q^{25} + ( - \beta_{2} + 2 \beta_1 - 3) q^{26} + q^{27} + (\beta_{2} - \beta_1 + 4) q^{29} + ( - \beta_{2} + 2 \beta_1 - 3) q^{30} + ( - 2 \beta_{2} + 2) q^{31} + (\beta_{2} + 2) q^{32} + q^{33} + (2 \beta_{2} + 8) q^{34} + (\beta_{2} + 2) q^{36} + (\beta_{2} + 3 \beta_1) q^{37} + (\beta_{2} - 2 \beta_1 + 5) q^{38} + (\beta_{2} - \beta_1) q^{39} + ( - \beta_{2} + 3 \beta_1 - 6) q^{40} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 2 \beta_1 + 2) q^{43} + (\beta_{2} + 2) q^{44} + (\beta_{2} - \beta_1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{46} + ( - \beta_{2} - \beta_1 + 8) q^{47} + (\beta_1 + 4) q^{48} + ( - 3 \beta_{2} + 3 \beta_1 - 13) q^{50} + 2 \beta_1 q^{51} + ( - 3 \beta_1 + 7) q^{52} - 2 \beta_{2} q^{53} + \beta_1 q^{54} + (\beta_{2} - \beta_1) q^{55} + (\beta_{2} + \beta_1 - 4) q^{57} + ( - \beta_{2} + 6 \beta_1 - 3) q^{58} + (3 \beta_{2} - \beta_1 + 8) q^{59} + ( - 3 \beta_1 + 7) q^{60} - 6 q^{61} + ( - 2 \beta_1 - 2) q^{62} + (2 \beta_1 - 7) q^{64} + ( - \beta_{2} - 3 \beta_1 + 10) q^{65} + \beta_1 q^{66} + (\beta_{2} + \beta_1 + 4) q^{67} + (8 \beta_1 + 2) q^{68} + ( - 2 \beta_1 - 2) q^{69} + ( - 4 \beta_1 + 4) q^{71} + (2 \beta_1 + 1) q^{72} + ( - \beta_{2} + \beta_1 - 8) q^{73} + (3 \beta_{2} + 2 \beta_1 + 13) q^{74} + ( - \beta_{2} - 3 \beta_1 + 5) q^{75} + ( - 4 \beta_{2} + 5 \beta_1 + 1) q^{76} + ( - \beta_{2} + 2 \beta_1 - 3) q^{78} + (4 \beta_1 + 4) q^{79} + (3 \beta_{2} - 2 \beta_1 - 3) q^{80} + q^{81} + ( - 2 \beta_{2} - 6 \beta_1 - 10) q^{82} + ( - 2 \beta_{2} + 6) q^{83} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{86} + (\beta_{2} - \beta_1 + 4) q^{87} + (2 \beta_1 + 1) q^{88} + ( - 4 \beta_{2} - 6) q^{89} + ( - \beta_{2} + 2 \beta_1 - 3) q^{90} + ( - 2 \beta_{2} - 8 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} + 2) q^{93} + ( - \beta_{2} + 6 \beta_1 - 5) q^{94} + ( - 7 \beta_{2} + 5 \beta_1 + 4) q^{95} + (\beta_{2} + 2) q^{96} + ( - 2 \beta_{2} - 8) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9} - 9 q^{10} + 3 q^{11} + 6 q^{12} + 12 q^{16} - 12 q^{19} + 21 q^{20} - 6 q^{23} + 3 q^{24} + 15 q^{25} - 9 q^{26} + 3 q^{27} + 12 q^{29} - 9 q^{30} + 6 q^{31} + 6 q^{32} + 3 q^{33} + 24 q^{34} + 6 q^{36} + 15 q^{38} - 18 q^{40} - 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} - 39 q^{50} + 21 q^{52} - 12 q^{57} - 9 q^{58} + 24 q^{59} + 21 q^{60} - 18 q^{61} - 6 q^{62} - 21 q^{64} + 30 q^{65} + 12 q^{67} + 6 q^{68} - 6 q^{69} + 12 q^{71} + 3 q^{72} - 24 q^{73} + 39 q^{74} + 15 q^{75} + 3 q^{76} - 9 q^{78} + 12 q^{79} - 9 q^{80} + 3 q^{81} - 30 q^{82} + 18 q^{83} - 18 q^{85} - 24 q^{86} + 12 q^{87} + 3 q^{88} - 18 q^{89} - 9 q^{90} - 18 q^{92} + 6 q^{93} - 15 q^{94} + 12 q^{95} + 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.36147
−0.167449
2.52892
−2.36147 1.00000 3.57653 3.93800 −2.36147 0 −3.72294 1.00000 −9.29947
1.2 −0.167449 1.00000 −1.97196 −3.80451 −0.167449 0 0.665102 1.00000 0.637062
1.3 2.52892 1.00000 4.39543 −0.133492 2.52892 0 6.05784 1.00000 −0.337590
\(n\): 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.s 3
3.b odd 2 1 4851.2.a.bp 3
7.b odd 2 1 231.2.a.d 3
21.c even 2 1 693.2.a.m 3
28.d even 2 1 3696.2.a.bp 3
35.c odd 2 1 5775.2.a.bw 3
77.b even 2 1 2541.2.a.bi 3
231.h odd 2 1 7623.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 7.b odd 2 1
693.2.a.m 3 21.c even 2 1
1617.2.a.s 3 1.a even 1 1 trivial
2541.2.a.bi 3 77.b even 2 1
3696.2.a.bp 3 28.d even 2 1
4851.2.a.bp 3 3.b odd 2 1
5775.2.a.bw 3 35.c odd 2 1
7623.2.a.cb 3 231.h odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 6T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 15T_{5} - 2 \) Copy content Toggle raw display
\( T_{13}^{3} - 15T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{3} - 24T_{17} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 6T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 15T - 2 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 15T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 24T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} + 12 T^{2} + 27 T - 36 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 12 T - 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + 33 T - 6 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 36 T - 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 75T - 246 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 72 T - 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 12 T + 48 \) Copy content Toggle raw display
$47$ \( T^{3} - 24 T^{2} + 171 T - 328 \) Copy content Toggle raw display
$53$ \( T^{3} - 48T - 120 \) Copy content Toggle raw display
$59$ \( T^{3} - 24 T^{2} + 87 T + 716 \) Copy content Toggle raw display
$61$ \( (T + 6)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + 27 T + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 48 T + 384 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 177 T + 394 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 48 T + 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 18 T^{2} + 60 T - 48 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} - 84 T - 1896 \) Copy content Toggle raw display
$97$ \( T^{3} + 24 T^{2} + 144 T + 8 \) Copy content Toggle raw display
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