Properties

Label 1694.2.a.g
Level $1694$
Weight $2$
Character orbit 1694.a
Self dual yes
Analytic conductor $13.527$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5266581024\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 4q^{5} + q^{7} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} - 4q^{5} + q^{7} + q^{8} - 3q^{9} - 4q^{10} - 2q^{13} + q^{14} + q^{16} + 4q^{17} - 3q^{18} + 6q^{19} - 4q^{20} + 4q^{23} + 11q^{25} - 2q^{26} + q^{28} + 2q^{29} - 2q^{31} + q^{32} + 4q^{34} - 4q^{35} - 3q^{36} + 10q^{37} + 6q^{38} - 4q^{40} - 4q^{41} + 8q^{43} + 12q^{45} + 4q^{46} + 2q^{47} + q^{49} + 11q^{50} - 2q^{52} + 6q^{53} + q^{56} + 2q^{58} - 12q^{59} + 14q^{61} - 2q^{62} - 3q^{63} + q^{64} + 8q^{65} - 12q^{67} + 4q^{68} - 4q^{70} - 8q^{71} - 3q^{72} - 4q^{73} + 10q^{74} + 6q^{76} - 4q^{80} + 9q^{81} - 4q^{82} + 6q^{83} - 16q^{85} + 8q^{86} - 6q^{89} + 12q^{90} - 2q^{91} + 4q^{92} + 2q^{94} - 24q^{95} - 14q^{97} + q^{98} + O(q^{100}) \)

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
0
1.00000 0 1.00000 −4.00000 0 1.00000 1.00000 −3.00000 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 1694.2.a.g 1
11.b odd 2 1 154.2.a.a 1
33.d even 2 1 1386.2.a.l 1
44.c even 2 1 1232.2.a.e 1
55.d odd 2 1 3850.2.a.u 1
55.e even 4 2 3850.2.c.j 2
77.b even 2 1 1078.2.a.d 1
77.h odd 6 2 1078.2.e.j 2
77.i even 6 2 1078.2.e.i 2
88.b odd 2 1 4928.2.a.v 1
88.g even 2 1 4928.2.a.w 1
231.h odd 2 1 9702.2.a.ba 1
308.g odd 2 1 8624.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 11.b odd 2 1
1078.2.a.d 1 77.b even 2 1
1078.2.e.i 2 77.i even 6 2
1078.2.e.j 2 77.h odd 6 2
1232.2.a.e 1 44.c even 2 1
1386.2.a.l 1 33.d even 2 1
1694.2.a.g 1 1.a even 1 1 trivial
3850.2.a.u 1 55.d odd 2 1
3850.2.c.j 2 55.e even 4 2
4928.2.a.v 1 88.b odd 2 1
4928.2.a.w 1 88.g even 2 1
8624.2.a.r 1 308.g odd 2 1
9702.2.a.ba 1 231.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} \)
\( T_{5} + 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( 2 + T \)
$17$ \( -4 + T \)
$19$ \( -6 + T \)
$23$ \( -4 + T \)
$29$ \( -2 + T \)
$31$ \( 2 + T \)
$37$ \( -10 + T \)
$41$ \( 4 + T \)
$43$ \( -8 + T \)
$47$ \( -2 + T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( -14 + T \)
$67$ \( 12 + T \)
$71$ \( 8 + T \)
$73$ \( 4 + T \)
$79$ \( T \)
$83$ \( -6 + T \)
$89$ \( 6 + T \)
$97$ \( 14 + T \)
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