Properties

Label 1617.2.a.k
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - q^{3} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + (\beta + 1) q^{6} + ( - 4 \beta - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} - q^{3} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + (\beta + 1) q^{6} + ( - 4 \beta - 1) q^{8} + q^{9} + (3 \beta + 1) q^{10} - q^{11} - 3 \beta q^{12} + (4 \beta - 3) q^{13} + (2 \beta - 1) q^{15} + (3 \beta + 5) q^{16} + 6 q^{17} + ( - \beta - 1) q^{18} + ( - 2 \beta - 1) q^{19} + ( - 3 \beta - 6) q^{20} + (\beta + 1) q^{22} + ( - 4 \beta + 4) q^{23} + (4 \beta + 1) q^{24} + ( - 5 \beta - 1) q^{26} - q^{27} + ( - 2 \beta - 7) q^{29} + ( - 3 \beta - 1) q^{30} + (4 \beta + 2) q^{31} + ( - 3 \beta - 6) q^{32} + q^{33} + ( - 6 \beta - 6) q^{34} + 3 \beta q^{36} + (4 \beta - 7) q^{37} + (5 \beta + 3) q^{38} + ( - 4 \beta + 3) q^{39} + (6 \beta + 7) q^{40} + 6 q^{41} + ( - 4 \beta + 6) q^{43} - 3 \beta q^{44} + ( - 2 \beta + 1) q^{45} + 4 \beta q^{46} + (4 \beta + 5) q^{47} + ( - 3 \beta - 5) q^{48} - 6 q^{51} + (3 \beta + 12) q^{52} - 8 \beta q^{53} + (\beta + 1) q^{54} + (2 \beta - 1) q^{55} + (2 \beta + 1) q^{57} + (11 \beta + 9) q^{58} + ( - 8 \beta + 1) q^{59} + (3 \beta + 6) q^{60} + ( - 8 \beta + 6) q^{61} + ( - 10 \beta - 6) q^{62} + (6 \beta - 1) q^{64} + (2 \beta - 11) q^{65} + ( - \beta - 1) q^{66} + (6 \beta - 9) q^{67} + 18 \beta q^{68} + (4 \beta - 4) q^{69} + (4 \beta - 2) q^{71} + ( - 4 \beta - 1) q^{72} - q^{73} + ( - \beta + 3) q^{74} + ( - 9 \beta - 6) q^{76} + (5 \beta + 1) q^{78} + ( - 4 \beta + 4) q^{79} + ( - 13 \beta - 1) q^{80} + q^{81} + ( - 6 \beta - 6) q^{82} + (4 \beta + 6) q^{83} + ( - 12 \beta + 6) q^{85} + (2 \beta - 2) q^{86} + (2 \beta + 7) q^{87} + (4 \beta + 1) q^{88} + ( - 12 \beta + 6) q^{89} + (3 \beta + 1) q^{90} - 12 q^{92} + ( - 4 \beta - 2) q^{93} + ( - 13 \beta - 9) q^{94} + (4 \beta + 3) q^{95} + (3 \beta + 6) q^{96} + ( - 8 \beta + 6) q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{8} + 2 q^{9} + 5 q^{10} - 2 q^{11} - 3 q^{12} - 2 q^{13} + 13 q^{16} + 12 q^{17} - 3 q^{18} - 4 q^{19} - 15 q^{20} + 3 q^{22} + 4 q^{23} + 6 q^{24} - 7 q^{26} - 2 q^{27} - 16 q^{29} - 5 q^{30} + 8 q^{31} - 15 q^{32} + 2 q^{33} - 18 q^{34} + 3 q^{36} - 10 q^{37} + 11 q^{38} + 2 q^{39} + 20 q^{40} + 12 q^{41} + 8 q^{43} - 3 q^{44} + 4 q^{46} + 14 q^{47} - 13 q^{48} - 12 q^{51} + 27 q^{52} - 8 q^{53} + 3 q^{54} + 4 q^{57} + 29 q^{58} - 6 q^{59} + 15 q^{60} + 4 q^{61} - 22 q^{62} + 4 q^{64} - 20 q^{65} - 3 q^{66} - 12 q^{67} + 18 q^{68} - 4 q^{69} - 6 q^{72} - 2 q^{73} + 5 q^{74} - 21 q^{76} + 7 q^{78} + 4 q^{79} - 15 q^{80} + 2 q^{81} - 18 q^{82} + 16 q^{83} - 2 q^{86} + 16 q^{87} + 6 q^{88} + 5 q^{90} - 24 q^{92} - 8 q^{93} - 31 q^{94} + 10 q^{95} + 15 q^{96} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 −1.00000 4.85410 −2.23607 2.61803 0 −7.47214 1.00000 5.85410
1.2 −0.381966 −1.00000 −1.85410 2.23607 0.381966 0 1.47214 1.00000 −0.854102
\(n\): 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.k 2
3.b odd 2 1 4851.2.a.bh 2
7.b odd 2 1 1617.2.a.l yes 2
21.c even 2 1 4851.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.k 2 1.a even 1 1 trivial
1617.2.a.l yes 2 7.b odd 2 1
4851.2.a.bg 2 21.c even 2 1
4851.2.a.bh 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 19 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 29 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 71 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} - 20 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$89$ \( T^{2} - 180 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
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