Properties

Label 1694.4.a.c
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 18 q^{5} + 4 q^{6} - 7 q^{7} - 8 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 18 q^{5} + 4 q^{6} - 7 q^{7} - 8 q^{8} - 23 q^{9} - 36 q^{10} - 8 q^{12} - 56 q^{13} + 14 q^{14} - 36 q^{15} + 16 q^{16} - 36 q^{17} + 46 q^{18} + 28 q^{19} + 72 q^{20} + 14 q^{21} + 180 q^{23} + 16 q^{24} + 199 q^{25} + 112 q^{26} + 100 q^{27} - 28 q^{28} + 54 q^{29} + 72 q^{30} - 334 q^{31} - 32 q^{32} + 72 q^{34} - 126 q^{35} - 92 q^{36} + 386 q^{37} - 56 q^{38} + 112 q^{39} - 144 q^{40} + 444 q^{41} - 28 q^{42} + 316 q^{43} - 414 q^{45} - 360 q^{46} - 402 q^{47} - 32 q^{48} + 49 q^{49} - 398 q^{50} + 72 q^{51} - 224 q^{52} - 486 q^{53} - 200 q^{54} + 56 q^{56} - 56 q^{57} - 108 q^{58} - 282 q^{59} - 144 q^{60} - 380 q^{61} + 668 q^{62} + 161 q^{63} + 64 q^{64} - 1008 q^{65} + 176 q^{67} - 144 q^{68} - 360 q^{69} + 252 q^{70} - 324 q^{71} + 184 q^{72} - 800 q^{73} - 772 q^{74} - 398 q^{75} + 112 q^{76} - 224 q^{78} + 1144 q^{79} + 288 q^{80} + 421 q^{81} - 888 q^{82} - 468 q^{83} + 56 q^{84} - 648 q^{85} - 632 q^{86} - 108 q^{87} - 870 q^{89} + 828 q^{90} + 392 q^{91} + 720 q^{92} + 668 q^{93} + 804 q^{94} + 504 q^{95} + 64 q^{96} - 1330 q^{97} - 98 q^{98}+O(q^{100}) \) 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 −2.00000 4.00000 18.0000 4.00000 −7.00000 −8.00000 −23.0000 −36.0000
\(n\): 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.c 1
11.b odd 2 1 154.4.a.d 1
33.d even 2 1 1386.4.a.a 1
44.c even 2 1 1232.4.a.f 1
77.b even 2 1 1078.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 11.b odd 2 1
1078.4.a.g 1 77.b even 2 1
1232.4.a.f 1 44.c even 2 1
1386.4.a.a 1 33.d even 2 1
1694.4.a.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} - 18 \) Copy content Toggle raw display
\( T_{13} + 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 18 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 56 \) Copy content Toggle raw display
$17$ \( T + 36 \) Copy content Toggle raw display
$19$ \( T - 28 \) Copy content Toggle raw display
$23$ \( T - 180 \) Copy content Toggle raw display
$29$ \( T - 54 \) Copy content Toggle raw display
$31$ \( T + 334 \) Copy content Toggle raw display
$37$ \( T - 386 \) Copy content Toggle raw display
$41$ \( T - 444 \) Copy content Toggle raw display
$43$ \( T - 316 \) Copy content Toggle raw display
$47$ \( T + 402 \) Copy content Toggle raw display
$53$ \( T + 486 \) Copy content Toggle raw display
$59$ \( T + 282 \) Copy content Toggle raw display
$61$ \( T + 380 \) Copy content Toggle raw display
$67$ \( T - 176 \) Copy content Toggle raw display
$71$ \( T + 324 \) Copy content Toggle raw display
$73$ \( T + 800 \) Copy content Toggle raw display
$79$ \( T - 1144 \) Copy content Toggle raw display
$83$ \( T + 468 \) Copy content Toggle raw display
$89$ \( T + 870 \) Copy content Toggle raw display
$97$ \( T + 1330 \) Copy content Toggle raw display
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