Properties

Label 1617.2.a.t
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: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{10} - q^{11} + ( -2 - \beta_{1} ) q^{12} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{20} + ( -1 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( 1 + 5 \beta_{1} + 2 \beta_{2} ) q^{26} - q^{27} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{29} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{30} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{32} + q^{33} + ( -6 \beta_{1} - 4 \beta_{2} ) q^{34} + ( 2 + \beta_{1} ) q^{36} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{37} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{38} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{39} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{40} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 - \beta_{1} ) q^{44} + ( -1 + \beta_{1} + \beta_{2} ) q^{45} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{48} + ( -4 - \beta_{1} + 7 \beta_{2} ) q^{50} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{52} -2 \beta_{1} q^{53} + ( -1 - \beta_{2} ) q^{54} + ( 1 - \beta_{1} - \beta_{2} ) q^{55} + ( -3 + \beta_{1} - \beta_{2} ) q^{57} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{60} + 2 q^{61} + ( 12 - 2 \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{65} + ( 1 + \beta_{2} ) q^{66} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( -14 - 8 \beta_{1} ) q^{68} + ( -4 - 2 \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{74} + ( -4 - \beta_{1} + 3 \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{78} + ( 4 - 4 \beta_{1} ) q^{79} + ( 13 + \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{82} + ( -2 + 6 \beta_{1} ) q^{83} + ( -6 + 2 \beta_{1} - 12 \beta_{2} ) q^{85} + ( 6 + 10 \beta_{1} - 6 \beta_{2} ) q^{86} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{87} + ( -1 - 2 \beta_{1} ) q^{88} + ( -10 - 4 \beta_{2} ) q^{89} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{90} + ( 10 + 6 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{93} + ( -11 - 5 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -3 + 5 \beta_{1} - \beta_{2} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{96} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + 11 q^{10} - 3 q^{11} - 6 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 3 q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{24} + 15 q^{25} + q^{26} - 3 q^{27} - 4 q^{29} - 11 q^{30} + 2 q^{31} + 8 q^{32} + 3 q^{33} + 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 18 q^{40} - 14 q^{41} - 14 q^{43} - 6 q^{44} - 4 q^{45} + 28 q^{46} + 4 q^{48} - 19 q^{50} + 8 q^{51} + 29 q^{52} - 2 q^{54} + 4 q^{55} - 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} - 5 q^{64} + 14 q^{65} + 2 q^{66} - 4 q^{67} - 42 q^{68} - 10 q^{69} - 12 q^{71} + 3 q^{72} + 20 q^{73} + 29 q^{74} - 15 q^{75} + 11 q^{76} - q^{78} + 12 q^{79} + 41 q^{80} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{85} + 24 q^{86} + 4 q^{87} - 3 q^{88} - 26 q^{89} + 11 q^{90} + 26 q^{92} - 2 q^{93} - 35 q^{94} - 8 q^{95} - 8 q^{96} + 4 q^{97} - 3 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−0.254102
−1.86081
2.11491
−1.93543 −1.00000 1.74590 −4.18953 1.93543 0 0.491797 1.00000 8.10856
1.2 1.46260 −1.00000 0.139194 −2.39821 −1.46260 0 −2.72161 1.00000 −3.50761
1.3 2.47283 −1.00000 4.11491 2.58774 −2.47283 0 5.22982 1.00000 6.39905
\(n\): 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.t 3
3.b odd 2 1 4851.2.a.bi 3
7.b odd 2 1 231.2.a.e 3
21.c even 2 1 693.2.a.l 3
28.d even 2 1 3696.2.a.bo 3
35.c odd 2 1 5775.2.a.bp 3
77.b even 2 1 2541.2.a.bg 3
231.h odd 2 1 7623.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 7.b odd 2 1
693.2.a.l 3 21.c even 2 1
1617.2.a.t 3 1.a even 1 1 trivial
2541.2.a.bg 3 77.b even 2 1
3696.2.a.bo 3 28.d even 2 1
4851.2.a.bi 3 3.b odd 2 1
5775.2.a.bp 3 35.c odd 2 1
7623.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7 \)
\( T_{5}^{3} + 4 T_{5}^{2} - 7 T_{5} - 26 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 27 T_{13} + 94 \)
\( T_{17}^{3} + 8 T_{17}^{2} - 40 T_{17} - 328 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 7 - 4 T - 2 T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -26 - 7 T + 4 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 94 - 27 T - 4 T^{2} + T^{3} \)
$17$ \( -328 - 40 T + 8 T^{2} + T^{3} \)
$19$ \( -4 + 15 T - 8 T^{2} + T^{3} \)
$23$ \( 64 + 12 T - 10 T^{2} + T^{3} \)
$29$ \( -94 - 27 T + 4 T^{2} + T^{3} \)
$31$ \( 256 - 76 T - 2 T^{2} + T^{3} \)
$37$ \( 106 - 43 T + T^{3} \)
$41$ \( -32 + 40 T + 14 T^{2} + T^{3} \)
$43$ \( -848 - 44 T + 14 T^{2} + T^{3} \)
$47$ \( -32 - 61 T + T^{3} \)
$53$ \( 8 - 16 T + T^{3} \)
$59$ \( 52 - 57 T + T^{3} \)
$61$ \( ( -2 + T )^{3} \)
$67$ \( -236 - 85 T + 4 T^{2} + T^{3} \)
$71$ \( -256 - 16 T + 12 T^{2} + T^{3} \)
$73$ \( -134 + 101 T - 20 T^{2} + T^{3} \)
$79$ \( 256 - 16 T - 12 T^{2} + T^{3} \)
$83$ \( -496 - 132 T + 6 T^{2} + T^{3} \)
$89$ \( -328 + 140 T + 26 T^{2} + T^{3} \)
$97$ \( 232 - 120 T - 4 T^{2} + T^{3} \)
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