Properties

Label 1617.2.a.ba
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.3676752.1
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 14x^{2} + 11x - 10 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 8x^{3} + 14x^{2} + 11x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 23 \) 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
−2.49291
−1.06884
0.614936
2.35232
2.59450
−2.49291 −1.00000 4.21460 −0.880926 2.49291 0 −5.52081 1.00000 2.19607
1.2 −1.06884 −1.00000 −0.857576 −2.69184 1.06884 0 3.05430 1.00000 2.87715
1.3 0.614936 −1.00000 −1.62185 1.49597 −0.614936 0 −2.22721 1.00000 0.919926
1.4 2.35232 −1.00000 3.53341 −4.11523 −2.35232 0 3.60708 1.00000 −9.68034
1.5 2.59450 −1.00000 4.73141 2.19202 −2.59450 0 7.08664 1.00000 5.68719
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.ba 5
3.b odd 2 1 4851.2.a.ca 5
7.b odd 2 1 1617.2.a.bb 5
7.c even 3 2 231.2.i.f 10
21.c even 2 1 4851.2.a.bz 5
21.h odd 6 2 693.2.i.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.f 10 7.c even 3 2
693.2.i.j 10 21.h odd 6 2
1617.2.a.ba 5 1.a even 1 1 trivial
1617.2.a.bb 5 7.b odd 2 1
4851.2.a.bz 5 21.c even 2 1
4851.2.a.ca 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{5} - 2T_{2}^{4} - 8T_{2}^{3} + 14T_{2}^{2} + 11T_{2} - 10 \) Copy content Toggle raw display
\( T_{5}^{5} + 4T_{5}^{4} - 8T_{5}^{3} - 28T_{5}^{2} + 20T_{5} + 32 \) Copy content Toggle raw display
\( T_{13}^{5} - 5T_{13}^{4} - 38T_{13}^{3} + 80T_{13}^{2} + 524T_{13} + 476 \) Copy content Toggle raw display
\( T_{17}^{5} - 2T_{17}^{4} - 8T_{17}^{3} + 14T_{17}^{2} + 11T_{17} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 8 T^{3} + 14 T^{2} + \cdots - 10 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 4 T^{4} - 8 T^{3} - 28 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 38 T^{3} + 80 T^{2} + \cdots + 476 \) Copy content Toggle raw display
$17$ \( T^{5} - 2 T^{4} - 8 T^{3} + 14 T^{2} + \cdots - 10 \) Copy content Toggle raw display
$19$ \( T^{5} + 3 T^{4} - 42 T^{3} - 108 T^{2} + \cdots + 603 \) Copy content Toggle raw display
$23$ \( T^{5} - 16 T^{4} + 70 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( T^{5} - 96 T^{3} + 24 T^{2} + \cdots - 1290 \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$37$ \( T^{5} - 15 T^{4} + 30 T^{3} + \cdots + 201 \) Copy content Toggle raw display
$41$ \( T^{5} + 22 T^{4} + 124 T^{3} + \cdots - 11536 \) Copy content Toggle raw display
$43$ \( T^{5} - 3 T^{4} - 108 T^{3} + \cdots - 2973 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} - 194 T^{3} + \cdots + 27994 \) Copy content Toggle raw display
$53$ \( T^{5} - 6 T^{4} - 96 T^{3} + \cdots - 6504 \) Copy content Toggle raw display
$59$ \( T^{5} - 16 T^{4} + 70 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$61$ \( T^{5} - 12 T^{4} - 180 T^{3} + \cdots - 48600 \) Copy content Toggle raw display
$67$ \( T^{5} - 7 T^{4} - 92 T^{3} + \cdots - 3044 \) Copy content Toggle raw display
$71$ \( T^{5} - 24 T^{4} + 66 T^{3} + \cdots + 5232 \) Copy content Toggle raw display
$73$ \( T^{5} - 17 T^{4} - 98 T^{3} + \cdots + 3044 \) Copy content Toggle raw display
$79$ \( T^{5} - 7 T^{4} - 254 T^{3} + \cdots - 94868 \) Copy content Toggle raw display
$83$ \( T^{5} + 12 T^{4} - 120 T^{3} + \cdots + 2688 \) Copy content Toggle raw display
$89$ \( T^{5} + 6 T^{4} - 168 T^{3} + \cdots + 17760 \) Copy content Toggle raw display
$97$ \( T^{5} + 14 T^{4} - 326 T^{3} + \cdots - 38906 \) Copy content Toggle raw display
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