Properties

Label 1617.2.a.p
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
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: \(1\)
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: 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - q^{5} - \beta q^{6} + ( - 2 \beta + 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - q^{5} - \beta q^{6} + ( - 2 \beta + 1) q^{8} + q^{9} - \beta q^{10} + q^{11} + ( - \beta + 1) q^{12} + (4 \beta - 1) q^{13} + q^{15} - 3 \beta q^{16} + (2 \beta - 4) q^{17} + \beta q^{18} + ( - 6 \beta + 3) q^{19} + ( - \beta + 1) q^{20} + \beta q^{22} + ( - 6 \beta + 2) q^{23} + (2 \beta - 1) q^{24} - 4 q^{25} + (3 \beta + 4) q^{26} - q^{27} + 5 q^{29} + \beta q^{30} + ( - 2 \beta + 4) q^{31} + (\beta - 5) q^{32} - q^{33} + ( - 2 \beta + 2) q^{34} + (\beta - 1) q^{36} - 7 q^{37} + ( - 3 \beta - 6) q^{38} + ( - 4 \beta + 1) q^{39} + (2 \beta - 1) q^{40} - 4 \beta q^{41} + ( - 6 \beta + 2) q^{43} + (\beta - 1) q^{44} - q^{45} + ( - 4 \beta - 6) q^{46} + (2 \beta + 1) q^{47} + 3 \beta q^{48} - 4 \beta q^{50} + ( - 2 \beta + 4) q^{51} + ( - \beta + 5) q^{52} + (10 \beta - 6) q^{53} - \beta q^{54} - q^{55} + (6 \beta - 3) q^{57} + 5 \beta q^{58} + ( - 10 \beta + 5) q^{59} + (\beta - 1) q^{60} - 2 q^{61} + (2 \beta - 2) q^{62} + (2 \beta + 1) q^{64} + ( - 4 \beta + 1) q^{65} - \beta q^{66} + ( - 2 \beta - 11) q^{67} + ( - 4 \beta + 6) q^{68} + (6 \beta - 2) q^{69} + 4 \beta q^{71} + ( - 2 \beta + 1) q^{72} + ( - 4 \beta - 7) q^{73} - 7 \beta q^{74} + 4 q^{75} + (3 \beta - 9) q^{76} + ( - 3 \beta - 4) q^{78} + (4 \beta - 12) q^{79} + 3 \beta q^{80} + q^{81} + ( - 4 \beta - 4) q^{82} + ( - 2 \beta - 8) q^{83} + ( - 2 \beta + 4) q^{85} + ( - 4 \beta - 6) q^{86} - 5 q^{87} + ( - 2 \beta + 1) q^{88} + ( - 4 \beta + 2) q^{89} - \beta q^{90} + (2 \beta - 8) q^{92} + (2 \beta - 4) q^{93} + (3 \beta + 2) q^{94} + (6 \beta - 3) q^{95} + ( - \beta + 5) q^{96} + (6 \beta - 6) q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9} - q^{10} + 2 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} - 6 q^{17} + q^{18} + q^{20} + q^{22} - 2 q^{23} - 8 q^{25} + 11 q^{26} - 2 q^{27} + 10 q^{29} + q^{30} + 6 q^{31} - 9 q^{32} - 2 q^{33} + 2 q^{34} - q^{36} - 14 q^{37} - 15 q^{38} - 2 q^{39} - 4 q^{41} - 2 q^{43} - q^{44} - 2 q^{45} - 16 q^{46} + 4 q^{47} + 3 q^{48} - 4 q^{50} + 6 q^{51} + 9 q^{52} - 2 q^{53} - q^{54} - 2 q^{55} + 5 q^{58} - q^{60} - 4 q^{61} - 2 q^{62} + 4 q^{64} - 2 q^{65} - q^{66} - 24 q^{67} + 8 q^{68} + 2 q^{69} + 4 q^{71} - 18 q^{73} - 7 q^{74} + 8 q^{75} - 15 q^{76} - 11 q^{78} - 20 q^{79} + 3 q^{80} + 2 q^{81} - 12 q^{82} - 18 q^{83} + 6 q^{85} - 16 q^{86} - 10 q^{87} - q^{90} - 14 q^{92} - 6 q^{93} + 7 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 −1.00000 −1.61803 −1.00000 0.618034 0 2.23607 1.00000 0.618034
1.2 1.61803 −1.00000 0.618034 −1.00000 −1.61803 0 −2.23607 1.00000 −1.61803
\(n\): 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.p 2
3.b odd 2 1 4851.2.a.w 2
7.b odd 2 1 231.2.a.c 2
21.c even 2 1 693.2.a.f 2
28.d even 2 1 3696.2.a.be 2
35.c odd 2 1 5775.2.a.be 2
77.b even 2 1 2541.2.a.t 2
231.h odd 2 1 7623.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 7.b odd 2 1
693.2.a.f 2 21.c even 2 1
1617.2.a.p 2 1.a even 1 1 trivial
2541.2.a.t 2 77.b even 2 1
3696.2.a.be 2 28.d even 2 1
4851.2.a.w 2 3.b odd 2 1
5775.2.a.be 2 35.c odd 2 1
7623.2.a.bm 2 231.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 19 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) 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
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 45 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} - 125 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 24T + 139 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 61 \) Copy content Toggle raw display
$79$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$89$ \( T^{2} - 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
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