Properties

Label 1694.4.a.b
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 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} - 2q^{3} + 4q^{4} - 12q^{5} + 4q^{6} - 7q^{7} - 8q^{8} - 23q^{9} + O(q^{10}) \) \( q - 2q^{2} - 2q^{3} + 4q^{4} - 12q^{5} + 4q^{6} - 7q^{7} - 8q^{8} - 23q^{9} + 24q^{10} - 8q^{12} - 56q^{13} + 14q^{14} + 24q^{15} + 16q^{16} + 114q^{17} + 46q^{18} - 2q^{19} - 48q^{20} + 14q^{21} - 120q^{23} + 16q^{24} + 19q^{25} + 112q^{26} + 100q^{27} - 28q^{28} + 54q^{29} - 48q^{30} + 236q^{31} - 32q^{32} - 228q^{34} + 84q^{35} - 92q^{36} + 146q^{37} + 4q^{38} + 112q^{39} + 96q^{40} - 126q^{41} - 28q^{42} + 376q^{43} + 276q^{45} + 240q^{46} - 12q^{47} - 32q^{48} + 49q^{49} - 38q^{50} - 228q^{51} - 224q^{52} + 174q^{53} - 200q^{54} + 56q^{56} + 4q^{57} - 108q^{58} + 138q^{59} + 96q^{60} - 380q^{61} - 472q^{62} + 161q^{63} + 64q^{64} + 672q^{65} - 484q^{67} + 456q^{68} + 240q^{69} - 168q^{70} + 576q^{71} + 184q^{72} + 1150q^{73} - 292q^{74} - 38q^{75} - 8q^{76} - 224q^{78} - 776q^{79} - 192q^{80} + 421q^{81} + 252q^{82} - 378q^{83} + 56q^{84} - 1368q^{85} - 752q^{86} - 108q^{87} - 390q^{89} - 552q^{90} + 392q^{91} - 480q^{92} - 472q^{93} + 24q^{94} + 24q^{95} + 64q^{96} - 1330q^{97} - 98q^{98} + O(q^{100}) \)

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 −2.00000 4.00000 −12.0000 4.00000 −7.00000 −8.00000 −23.0000 24.0000
\(n\): 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.b 1
11.b odd 2 1 14.4.a.b 1
33.d even 2 1 126.4.a.d 1
44.c even 2 1 112.4.a.e 1
55.d odd 2 1 350.4.a.f 1
55.e even 4 2 350.4.c.g 2
77.b even 2 1 98.4.a.e 1
77.h odd 6 2 98.4.c.c 2
77.i even 6 2 98.4.c.b 2
88.b odd 2 1 448.4.a.k 1
88.g even 2 1 448.4.a.g 1
132.d odd 2 1 1008.4.a.r 1
143.d odd 2 1 2366.4.a.c 1
231.h odd 2 1 882.4.a.b 1
231.k odd 6 2 882.4.g.v 2
231.l even 6 2 882.4.g.p 2
308.g odd 2 1 784.4.a.h 1
385.h even 2 1 2450.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 11.b odd 2 1
98.4.a.e 1 77.b even 2 1
98.4.c.b 2 77.i even 6 2
98.4.c.c 2 77.h odd 6 2
112.4.a.e 1 44.c even 2 1
126.4.a.d 1 33.d even 2 1
350.4.a.f 1 55.d odd 2 1
350.4.c.g 2 55.e even 4 2
448.4.a.g 1 88.g even 2 1
448.4.a.k 1 88.b odd 2 1
784.4.a.h 1 308.g odd 2 1
882.4.a.b 1 231.h odd 2 1
882.4.g.p 2 231.l even 6 2
882.4.g.v 2 231.k odd 6 2
1008.4.a.r 1 132.d odd 2 1
1694.4.a.b 1 1.a even 1 1 trivial
2366.4.a.c 1 143.d odd 2 1
2450.4.a.i 1 385.h even 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{5} + 12 \)
\( T_{13} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 2 + T \)
$5$ \( 12 + T \)
$7$ \( 7 + T \)
$11$ \( T \)
$13$ \( 56 + T \)
$17$ \( -114 + T \)
$19$ \( 2 + T \)
$23$ \( 120 + T \)
$29$ \( -54 + T \)
$31$ \( -236 + T \)
$37$ \( -146 + T \)
$41$ \( 126 + T \)
$43$ \( -376 + T \)
$47$ \( 12 + T \)
$53$ \( -174 + T \)
$59$ \( -138 + T \)
$61$ \( 380 + T \)
$67$ \( 484 + T \)
$71$ \( -576 + T \)
$73$ \( -1150 + T \)
$79$ \( 776 + T \)
$83$ \( 378 + T \)
$89$ \( 390 + T \)
$97$ \( 1330 + T \)
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