Properties

Label 690.2.a.l
Level $690$
Weight $2$
Character orbit 690.a
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( -1 - \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( -1 - \beta ) q^{7} + q^{8} + q^{9} + q^{10} + ( 1 + \beta ) q^{11} - q^{12} + 2 q^{13} + ( -1 - \beta ) q^{14} - q^{15} + q^{16} + ( 3 + \beta ) q^{17} + q^{18} + 4 q^{19} + q^{20} + ( 1 + \beta ) q^{21} + ( 1 + \beta ) q^{22} + q^{23} - q^{24} + q^{25} + 2 q^{26} - q^{27} + ( -1 - \beta ) q^{28} + 2 q^{29} - q^{30} + q^{32} + ( -1 - \beta ) q^{33} + ( 3 + \beta ) q^{34} + ( -1 - \beta ) q^{35} + q^{36} + ( -3 - \beta ) q^{37} + 4 q^{38} -2 q^{39} + q^{40} + 2 q^{41} + ( 1 + \beta ) q^{42} + ( 1 + \beta ) q^{44} + q^{45} + q^{46} -8 q^{47} - q^{48} + ( 11 + 2 \beta ) q^{49} + q^{50} + ( -3 - \beta ) q^{51} + 2 q^{52} + ( 4 - 2 \beta ) q^{53} - q^{54} + ( 1 + \beta ) q^{55} + ( -1 - \beta ) q^{56} -4 q^{57} + 2 q^{58} + ( -6 - 2 \beta ) q^{59} - q^{60} + ( 5 - \beta ) q^{61} + ( -1 - \beta ) q^{63} + q^{64} + 2 q^{65} + ( -1 - \beta ) q^{66} -8 q^{67} + ( 3 + \beta ) q^{68} - q^{69} + ( -1 - \beta ) q^{70} + ( -2 + 2 \beta ) q^{71} + q^{72} + ( 4 + 2 \beta ) q^{73} + ( -3 - \beta ) q^{74} - q^{75} + 4 q^{76} + ( -18 - 2 \beta ) q^{77} -2 q^{78} + ( -1 - \beta ) q^{79} + q^{80} + q^{81} + 2 q^{82} + ( -1 + 3 \beta ) q^{83} + ( 1 + \beta ) q^{84} + ( 3 + \beta ) q^{85} -2 q^{87} + ( 1 + \beta ) q^{88} + ( -1 + \beta ) q^{89} + q^{90} + ( -2 - 2 \beta ) q^{91} + q^{92} -8 q^{94} + 4 q^{95} - q^{96} + ( -8 + 2 \beta ) q^{97} + ( 11 + 2 \beta ) q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{11} - 2q^{12} + 4q^{13} - 2q^{14} - 2q^{15} + 2q^{16} + 6q^{17} + 2q^{18} + 8q^{19} + 2q^{20} + 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} + 4q^{26} - 2q^{27} - 2q^{28} + 4q^{29} - 2q^{30} + 2q^{32} - 2q^{33} + 6q^{34} - 2q^{35} + 2q^{36} - 6q^{37} + 8q^{38} - 4q^{39} + 2q^{40} + 4q^{41} + 2q^{42} + 2q^{44} + 2q^{45} + 2q^{46} - 16q^{47} - 2q^{48} + 22q^{49} + 2q^{50} - 6q^{51} + 4q^{52} + 8q^{53} - 2q^{54} + 2q^{55} - 2q^{56} - 8q^{57} + 4q^{58} - 12q^{59} - 2q^{60} + 10q^{61} - 2q^{63} + 2q^{64} + 4q^{65} - 2q^{66} - 16q^{67} + 6q^{68} - 2q^{69} - 2q^{70} - 4q^{71} + 2q^{72} + 8q^{73} - 6q^{74} - 2q^{75} + 8q^{76} - 36q^{77} - 4q^{78} - 2q^{79} + 2q^{80} + 2q^{81} + 4q^{82} - 2q^{83} + 2q^{84} + 6q^{85} - 4q^{87} + 2q^{88} - 2q^{89} + 2q^{90} - 4q^{91} + 2q^{92} - 16q^{94} + 8q^{95} - 2q^{96} - 16q^{97} + 22q^{98} + 2q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
1.00000 −1.00000 1.00000 1.00000 −1.00000 −5.12311 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.12311 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 690.2.a.l 2
3.b odd 2 1 2070.2.a.t 2
4.b odd 2 1 5520.2.a.bs 2
5.b even 2 1 3450.2.a.bi 2
5.c odd 4 2 3450.2.d.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.l 2 1.a even 1 1 trivial
2070.2.a.t 2 3.b odd 2 1
3450.2.a.bi 2 5.b even 2 1
3450.2.d.v 4 5.c odd 4 2
5520.2.a.bs 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(690))\):

\( T_{7}^{2} + 2 T_{7} - 16 \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{17}^{2} - 6 T_{17} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -16 + 2 T + T^{2} \)
$11$ \( -16 - 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -8 - 6 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( -8 + 6 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -32 + 12 T + T^{2} \)
$61$ \( 8 - 10 T + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -64 + 4 T + T^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( -16 + 2 T + T^{2} \)
$83$ \( -152 + 2 T + T^{2} \)
$89$ \( -16 + 2 T + T^{2} \)
$97$ \( -4 + 16 T + T^{2} \)
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