Properties

Label 1682.4.a.b
Level $1682$
Weight $4$
Character orbit 1682.a
Self dual yes
Analytic conductor $99.241$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.2412126297\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} - 7 q^{3} + 4 q^{4} + 5 q^{5} - 14 q^{6} - 2 q^{7} + 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 7 q^{3} + 4 q^{4} + 5 q^{5} - 14 q^{6} - 2 q^{7} + 8 q^{8} + 22 q^{9} + 10 q^{10} - 37 q^{11} - 28 q^{12} + 27 q^{13} - 4 q^{14} - 35 q^{15} + 16 q^{16} - 24 q^{17} + 44 q^{18} + 88 q^{19} + 20 q^{20} + 14 q^{21} - 74 q^{22} - 28 q^{23} - 56 q^{24} - 100 q^{25} + 54 q^{26} + 35 q^{27} - 8 q^{28} - 70 q^{30} + 143 q^{31} + 32 q^{32} + 259 q^{33} - 48 q^{34} - 10 q^{35} + 88 q^{36} + 360 q^{37} + 176 q^{38} - 189 q^{39} + 40 q^{40} - 386 q^{41} + 28 q^{42} - 381 q^{43} - 148 q^{44} + 110 q^{45} - 56 q^{46} + 103 q^{47} - 112 q^{48} - 339 q^{49} - 200 q^{50} + 168 q^{51} + 108 q^{52} - 431 q^{53} + 70 q^{54} - 185 q^{55} - 16 q^{56} - 616 q^{57} + 288 q^{59} - 140 q^{60} + 840 q^{61} + 286 q^{62} - 44 q^{63} + 64 q^{64} + 135 q^{65} + 518 q^{66} - 180 q^{67} - 96 q^{68} + 196 q^{69} - 20 q^{70} + 706 q^{71} + 176 q^{72} - 716 q^{73} + 720 q^{74} + 700 q^{75} + 352 q^{76} + 74 q^{77} - 378 q^{78} - 931 q^{79} + 80 q^{80} - 839 q^{81} - 772 q^{82} + 1188 q^{83} + 56 q^{84} - 120 q^{85} - 762 q^{86} - 296 q^{88} + 642 q^{89} + 220 q^{90} - 54 q^{91} - 112 q^{92} - 1001 q^{93} + 206 q^{94} + 440 q^{95} - 224 q^{96} - 486 q^{97} - 678 q^{98} - 814 q^{99}+O(q^{100}) \) 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 −7.00000 4.00000 5.00000 −14.0000 −2.00000 8.00000 22.0000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 1682.4.a.b 1
29.b even 2 1 58.4.a.a 1
87.d odd 2 1 522.4.a.e 1
116.d odd 2 1 464.4.a.a 1
145.d even 2 1 1450.4.a.e 1
232.b odd 2 1 1856.4.a.d 1
232.g even 2 1 1856.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.a 1 29.b even 2 1
464.4.a.a 1 116.d odd 2 1
522.4.a.e 1 87.d odd 2 1
1450.4.a.e 1 145.d even 2 1
1682.4.a.b 1 1.a even 1 1 trivial
1856.4.a.a 1 232.g even 2 1
1856.4.a.d 1 232.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 37 \) Copy content Toggle raw display
$13$ \( T - 27 \) Copy content Toggle raw display
$17$ \( T + 24 \) Copy content Toggle raw display
$19$ \( T - 88 \) Copy content Toggle raw display
$23$ \( T + 28 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 143 \) Copy content Toggle raw display
$37$ \( T - 360 \) Copy content Toggle raw display
$41$ \( T + 386 \) Copy content Toggle raw display
$43$ \( T + 381 \) Copy content Toggle raw display
$47$ \( T - 103 \) Copy content Toggle raw display
$53$ \( T + 431 \) Copy content Toggle raw display
$59$ \( T - 288 \) Copy content Toggle raw display
$61$ \( T - 840 \) Copy content Toggle raw display
$67$ \( T + 180 \) Copy content Toggle raw display
$71$ \( T - 706 \) Copy content Toggle raw display
$73$ \( T + 716 \) Copy content Toggle raw display
$79$ \( T + 931 \) Copy content Toggle raw display
$83$ \( T - 1188 \) Copy content Toggle raw display
$89$ \( T - 642 \) Copy content Toggle raw display
$97$ \( T + 486 \) Copy content Toggle raw display
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