Properties

Label 1694.2.a.c
Level $1694$
Weight $2$
Character orbit 1694.a
Self dual yes
Analytic conductor $13.527$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 3 q^{9} - 2 q^{10} - 2 q^{13} - q^{14} + q^{16} - 2 q^{17} + 3 q^{18} + 2 q^{20} - 8 q^{23} - q^{25} + 2 q^{26} + q^{28} + 2 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 2 q^{35} - 3 q^{36} - 2 q^{37} - 2 q^{40} - 10 q^{41} - 4 q^{43} - 6 q^{45} + 8 q^{46} + 8 q^{47} + q^{49} + q^{50} - 2 q^{52} + 6 q^{53} - q^{56} - 2 q^{58} - 10 q^{61} + 8 q^{62} - 3 q^{63} + q^{64} - 4 q^{65} - 12 q^{67} - 2 q^{68} - 2 q^{70} + 16 q^{71} + 3 q^{72} + 14 q^{73} + 2 q^{74} + 2 q^{80} + 9 q^{81} + 10 q^{82} - 4 q^{85} + 4 q^{86} - 6 q^{89} + 6 q^{90} - 2 q^{91} - 8 q^{92} - 8 q^{94} + 10 q^{97} - q^{98}+O(q^{100}) \) 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
0
−1.00000 0 1.00000 2.00000 0 1.00000 −1.00000 −3.00000 −2.00000
\(n\): 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.c 1
11.b odd 2 1 154.2.a.c 1
33.d even 2 1 1386.2.a.b 1
44.c even 2 1 1232.2.a.h 1
55.d odd 2 1 3850.2.a.f 1
55.e even 4 2 3850.2.c.l 2
77.b even 2 1 1078.2.a.j 1
77.h odd 6 2 1078.2.e.b 2
77.i even 6 2 1078.2.e.c 2
88.b odd 2 1 4928.2.a.n 1
88.g even 2 1 4928.2.a.o 1
231.h odd 2 1 9702.2.a.v 1
308.g odd 2 1 8624.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 11.b odd 2 1
1078.2.a.j 1 77.b even 2 1
1078.2.e.b 2 77.h odd 6 2
1078.2.e.c 2 77.i even 6 2
1232.2.a.h 1 44.c even 2 1
1386.2.a.b 1 33.d even 2 1
1694.2.a.c 1 1.a even 1 1 trivial
3850.2.a.f 1 55.d odd 2 1
3850.2.c.l 2 55.e even 4 2
4928.2.a.n 1 88.b odd 2 1
4928.2.a.o 1 88.g even 2 1
8624.2.a.o 1 308.g odd 2 1
9702.2.a.v 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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