Properties

Label 1694.4.a.f
Level $1694$
Weight $4$
Character orbit 1694.a
Self dual yes
Analytic conductor $99.949$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.9492355497\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 7 q^{7} + 8 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 7 q^{7} + 8 q^{8} - 27 q^{9} + 4 q^{10} - 26 q^{13} + 14 q^{14} + 16 q^{16} + 46 q^{17} - 54 q^{18} + 48 q^{19} + 8 q^{20} - 128 q^{23} - 121 q^{25} - 52 q^{26} + 28 q^{28} + 146 q^{29} - 128 q^{31} + 32 q^{32} + 92 q^{34} + 14 q^{35} - 108 q^{36} - 26 q^{37} + 96 q^{38} + 16 q^{40} - 10 q^{41} - 52 q^{43} - 54 q^{45} - 256 q^{46} - 544 q^{47} + 49 q^{49} - 242 q^{50} - 104 q^{52} + 318 q^{53} + 56 q^{56} + 292 q^{58} - 48 q^{59} - 466 q^{61} - 256 q^{62} - 189 q^{63} + 64 q^{64} - 52 q^{65} + 516 q^{67} + 184 q^{68} + 28 q^{70} - 392 q^{71} - 216 q^{72} - 754 q^{73} - 52 q^{74} + 192 q^{76} + 32 q^{80} + 729 q^{81} - 20 q^{82} - 624 q^{83} + 92 q^{85} - 104 q^{86} - 1590 q^{89} - 108 q^{90} - 182 q^{91} - 512 q^{92} - 1088 q^{94} + 96 q^{95} + 1018 q^{97} + 98 q^{98}+O(q^{100}) \) 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
2.00000 0 4.00000 2.00000 0 7.00000 8.00000 −27.0000 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 1694.4.a.f 1
11.b odd 2 1 154.4.a.b 1
33.d even 2 1 1386.4.a.j 1
44.c even 2 1 1232.4.a.e 1
77.b even 2 1 1078.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.b 1 11.b odd 2 1
1078.4.a.b 1 77.b even 2 1
1232.4.a.e 1 44.c even 2 1
1386.4.a.j 1 33.d even 2 1
1694.4.a.f 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} + 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 26 \) Copy content Toggle raw display
$17$ \( T - 46 \) Copy content Toggle raw display
$19$ \( T - 48 \) Copy content Toggle raw display
$23$ \( T + 128 \) Copy content Toggle raw display
$29$ \( T - 146 \) Copy content Toggle raw display
$31$ \( T + 128 \) Copy content Toggle raw display
$37$ \( T + 26 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 52 \) Copy content Toggle raw display
$47$ \( T + 544 \) Copy content Toggle raw display
$53$ \( T - 318 \) Copy content Toggle raw display
$59$ \( T + 48 \) Copy content Toggle raw display
$61$ \( T + 466 \) Copy content Toggle raw display
$67$ \( T - 516 \) Copy content Toggle raw display
$71$ \( T + 392 \) Copy content Toggle raw display
$73$ \( T + 754 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 624 \) Copy content Toggle raw display
$89$ \( T + 1590 \) Copy content Toggle raw display
$97$ \( T - 1018 \) Copy content Toggle raw display
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