Properties

Label 1690.2.a.g
Level $1690$
Weight $2$
Character orbit 1690.a
Self dual yes
Analytic conductor $13.495$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4947179416\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 3 q^{11} - 2 q^{12} + q^{14} - 2 q^{15} + q^{16} - 6 q^{17} + q^{18} - 5 q^{19} + q^{20} - 2 q^{21} - 3 q^{22} - 2 q^{24} + q^{25} + 4 q^{27} + q^{28} - 2 q^{30} + 4 q^{31} + q^{32} + 6 q^{33} - 6 q^{34} + q^{35} + q^{36} - 11 q^{37} - 5 q^{38} + q^{40} - 6 q^{41} - 2 q^{42} + 2 q^{43} - 3 q^{44} + q^{45} + 3 q^{47} - 2 q^{48} - 6 q^{49} + q^{50} + 12 q^{51} - 9 q^{53} + 4 q^{54} - 3 q^{55} + q^{56} + 10 q^{57} - 2 q^{60} + 8 q^{61} + 4 q^{62} + q^{63} + q^{64} + 6 q^{66} + 16 q^{67} - 6 q^{68} + q^{70} - 6 q^{71} + q^{72} - 14 q^{73} - 11 q^{74} - 2 q^{75} - 5 q^{76} - 3 q^{77} - 16 q^{79} + q^{80} - 11 q^{81} - 6 q^{82} + 6 q^{83} - 2 q^{84} - 6 q^{85} + 2 q^{86} - 3 q^{88} - 9 q^{89} + q^{90} - 8 q^{93} + 3 q^{94} - 5 q^{95} - 2 q^{96} + 10 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) 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
1.00000 −2.00000 1.00000 1.00000 −2.00000 1.00000 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 1690.2.a.g 1
5.b even 2 1 8450.2.a.k 1
13.b even 2 1 1690.2.a.a 1
13.c even 3 2 1690.2.e.e 2
13.d odd 4 2 1690.2.d.a 2
13.e even 6 2 130.2.e.b 2
13.f odd 12 4 1690.2.l.i 4
39.h odd 6 2 1170.2.i.f 2
52.i odd 6 2 1040.2.q.c 2
65.d even 2 1 8450.2.a.w 1
65.l even 6 2 650.2.e.a 2
65.r odd 12 4 650.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 13.e even 6 2
650.2.e.a 2 65.l even 6 2
650.2.o.b 4 65.r odd 12 4
1040.2.q.c 2 52.i odd 6 2
1170.2.i.f 2 39.h odd 6 2
1690.2.a.a 1 13.b even 2 1
1690.2.a.g 1 1.a even 1 1 trivial
1690.2.d.a 2 13.d odd 4 2
1690.2.e.e 2 13.c even 3 2
1690.2.l.i 4 13.f odd 12 4
8450.2.a.k 1 5.b even 2 1
8450.2.a.w 1 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{31} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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