Properties

Label 1694.4.a.a
Level $1694$
Weight $4$
Character orbit 1694.a
Self dual yes
Analytic conductor $99.949$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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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

\(f(q)\) \(=\) \( q - 2 q^{2} - 10 q^{3} + 4 q^{4} - 14 q^{5} + 20 q^{6} - 7 q^{7} - 8 q^{8} + 73 q^{9} + O(q^{10}) \) \( q - 2 q^{2} - 10 q^{3} + 4 q^{4} - 14 q^{5} + 20 q^{6} - 7 q^{7} - 8 q^{8} + 73 q^{9} + 28 q^{10} - 40 q^{12} + 16 q^{13} + 14 q^{14} + 140 q^{15} + 16 q^{16} - 108 q^{17} - 146 q^{18} - 116 q^{19} - 56 q^{20} + 70 q^{21} + 68 q^{23} + 80 q^{24} + 71 q^{25} - 32 q^{26} - 460 q^{27} - 28 q^{28} - 122 q^{29} - 280 q^{30} - 262 q^{31} - 32 q^{32} + 216 q^{34} + 98 q^{35} + 292 q^{36} + 130 q^{37} + 232 q^{38} - 160 q^{39} + 112 q^{40} - 204 q^{41} - 140 q^{42} + 396 q^{43} - 1022 q^{45} - 136 q^{46} + 166 q^{47} - 160 q^{48} + 49 q^{49} - 142 q^{50} + 1080 q^{51} + 64 q^{52} + 442 q^{53} + 920 q^{54} + 56 q^{56} + 1160 q^{57} + 244 q^{58} + 702 q^{59} + 560 q^{60} - 196 q^{61} + 524 q^{62} - 511 q^{63} + 64 q^{64} - 224 q^{65} - 416 q^{67} - 432 q^{68} - 680 q^{69} - 196 q^{70} + 492 q^{71} - 584 q^{72} - 408 q^{73} - 260 q^{74} - 710 q^{75} - 464 q^{76} + 320 q^{78} - 600 q^{79} - 224 q^{80} + 2629 q^{81} + 408 q^{82} + 1212 q^{83} + 280 q^{84} + 1512 q^{85} - 792 q^{86} + 1220 q^{87} + 1146 q^{89} + 2044 q^{90} - 112 q^{91} + 272 q^{92} + 2620 q^{93} - 332 q^{94} + 1624 q^{95} + 320 q^{96} - 482 q^{97} - 98 q^{98} + O(q^{100}) \)

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
−2.00000 −10.0000 4.00000 −14.0000 20.0000 −7.00000 −8.00000 73.0000 28.0000
\(n\): 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.a 1
11.b odd 2 1 154.4.a.c 1
33.d even 2 1 1386.4.a.g 1
44.c even 2 1 1232.4.a.i 1
77.b even 2 1 1078.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.c 1 11.b odd 2 1
1078.4.a.h 1 77.b even 2 1
1232.4.a.i 1 44.c even 2 1
1386.4.a.g 1 33.d even 2 1
1694.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} + 10 \)
\( T_{5} + 14 \)
\( T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 10 + T \)
$5$ \( 14 + T \)
$7$ \( 7 + T \)
$11$ \( T \)
$13$ \( -16 + T \)
$17$ \( 108 + T \)
$19$ \( 116 + T \)
$23$ \( -68 + T \)
$29$ \( 122 + T \)
$31$ \( 262 + T \)
$37$ \( -130 + T \)
$41$ \( 204 + T \)
$43$ \( -396 + T \)
$47$ \( -166 + T \)
$53$ \( -442 + T \)
$59$ \( -702 + T \)
$61$ \( 196 + T \)
$67$ \( 416 + T \)
$71$ \( -492 + T \)
$73$ \( 408 + T \)
$79$ \( 600 + T \)
$83$ \( -1212 + T \)
$89$ \( -1146 + T \)
$97$ \( 482 + T \)
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