Properties

Label 1710.2.a.x
Level $1710$
Weight $2$
Character orbit 1710.a
Self dual yes
Analytic conductor $13.654$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8} - q^{10} + 2 q^{11} + 4 q^{13} + (\beta + 1) q^{14} + q^{16} + ( - \beta - 1) q^{17} - q^{19} - q^{20} + 2 q^{22} + q^{25} + 4 q^{26} + (\beta + 1) q^{28} - 2 q^{29} + ( - \beta + 5) q^{31} + q^{32} + ( - \beta - 1) q^{34} + ( - \beta - 1) q^{35} - q^{38} - q^{40} + (\beta - 1) q^{41} + ( - 2 \beta + 2) q^{43} + 2 q^{44} + ( - 2 \beta + 2) q^{47} + (2 \beta + 11) q^{49} + q^{50} + 4 q^{52} + (2 \beta + 4) q^{53} - 2 q^{55} + (\beta + 1) q^{56} - 2 q^{58} + (\beta + 1) q^{59} + ( - 2 \beta + 4) q^{61} + ( - \beta + 5) q^{62} + q^{64} - 4 q^{65} + ( - 2 \beta - 2) q^{67} + ( - \beta - 1) q^{68} + ( - \beta - 1) q^{70} + ( - 2 \beta - 2) q^{71} + 6 q^{73} - q^{76} + (2 \beta + 2) q^{77} + ( - \beta + 5) q^{79} - q^{80} + (\beta - 1) q^{82} + (\beta + 11) q^{83} + (\beta + 1) q^{85} + ( - 2 \beta + 2) q^{86} + 2 q^{88} + (3 \beta + 1) q^{89} + (4 \beta + 4) q^{91} + ( - 2 \beta + 2) q^{94} + q^{95} - 6 q^{97} + (2 \beta + 11) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} + 8 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{25} + 8 q^{26} + 2 q^{28} - 4 q^{29} + 10 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{35} - 2 q^{38} - 2 q^{40} - 2 q^{41} + 4 q^{43} + 4 q^{44} + 4 q^{47} + 22 q^{49} + 2 q^{50} + 8 q^{52} + 8 q^{53} - 4 q^{55} + 2 q^{56} - 4 q^{58} + 2 q^{59} + 8 q^{61} + 10 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} - 2 q^{68} - 2 q^{70} - 4 q^{71} + 12 q^{73} - 2 q^{76} + 4 q^{77} + 10 q^{79} - 2 q^{80} - 2 q^{82} + 22 q^{83} + 2 q^{85} + 4 q^{86} + 4 q^{88} + 2 q^{89} + 8 q^{91} + 4 q^{94} + 2 q^{95} - 12 q^{97} + 22 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
1.00000 0 1.00000 −1.00000 0 −3.12311 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 5.12311 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(19\) \(1\)

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 1710.2.a.x yes 2
3.b odd 2 1 1710.2.a.v 2
5.b even 2 1 8550.2.a.bp 2
15.d odd 2 1 8550.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.v 2 3.b odd 2 1
1710.2.a.x yes 2 1.a even 1 1 trivial
8550.2.a.bp 2 5.b even 2 1
8550.2.a.bx 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7}^{2} - 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{53}^{2} - 8T_{53} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$83$ \( T^{2} - 22T + 104 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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