Properties

Label 138.4.a.b
Level $138$
Weight $4$
Character orbit 138.a
Self dual yes
Analytic conductor $8.142$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.14226358079\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 2 q^{5} - 6 q^{6} - 32 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 2 q^{5} - 6 q^{6} - 32 q^{7} - 8 q^{8} + 9 q^{9} - 4 q^{10} - 48 q^{11} + 12 q^{12} + 22 q^{13} + 64 q^{14} + 6 q^{15} + 16 q^{16} + 42 q^{17} - 18 q^{18} - 144 q^{19} + 8 q^{20} - 96 q^{21} + 96 q^{22} - 23 q^{23} - 24 q^{24} - 121 q^{25} - 44 q^{26} + 27 q^{27} - 128 q^{28} + 174 q^{29} - 12 q^{30} - 304 q^{31} - 32 q^{32} - 144 q^{33} - 84 q^{34} - 64 q^{35} + 36 q^{36} - 318 q^{37} + 288 q^{38} + 66 q^{39} - 16 q^{40} + 74 q^{41} + 192 q^{42} + 192 q^{43} - 192 q^{44} + 18 q^{45} + 46 q^{46} + 392 q^{47} + 48 q^{48} + 681 q^{49} + 242 q^{50} + 126 q^{51} + 88 q^{52} - 734 q^{53} - 54 q^{54} - 96 q^{55} + 256 q^{56} - 432 q^{57} - 348 q^{58} + 156 q^{59} + 24 q^{60} + 706 q^{61} + 608 q^{62} - 288 q^{63} + 64 q^{64} + 44 q^{65} + 288 q^{66} + 192 q^{67} + 168 q^{68} - 69 q^{69} + 128 q^{70} + 624 q^{71} - 72 q^{72} - 406 q^{73} + 636 q^{74} - 363 q^{75} - 576 q^{76} + 1536 q^{77} - 132 q^{78} + 696 q^{79} + 32 q^{80} + 81 q^{81} - 148 q^{82} - 800 q^{83} - 384 q^{84} + 84 q^{85} - 384 q^{86} + 522 q^{87} + 384 q^{88} - 102 q^{89} - 36 q^{90} - 704 q^{91} - 92 q^{92} - 912 q^{93} - 784 q^{94} - 288 q^{95} - 96 q^{96} - 918 q^{97} - 1362 q^{98} - 432 q^{99}+O(q^{100}) \) 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 3.00000 4.00000 2.00000 −6.00000 −32.0000 −8.00000 9.00000 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)

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 138.4.a.b 1
3.b odd 2 1 414.4.a.c 1
4.b odd 2 1 1104.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.a.b 1 1.a even 1 1 trivial
414.4.a.c 1 3.b odd 2 1
1104.4.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T + 144 \) Copy content Toggle raw display
$23$ \( T + 23 \) Copy content Toggle raw display
$29$ \( T - 174 \) Copy content Toggle raw display
$31$ \( T + 304 \) Copy content Toggle raw display
$37$ \( T + 318 \) Copy content Toggle raw display
$41$ \( T - 74 \) Copy content Toggle raw display
$43$ \( T - 192 \) Copy content Toggle raw display
$47$ \( T - 392 \) Copy content Toggle raw display
$53$ \( T + 734 \) Copy content Toggle raw display
$59$ \( T - 156 \) Copy content Toggle raw display
$61$ \( T - 706 \) Copy content Toggle raw display
$67$ \( T - 192 \) Copy content Toggle raw display
$71$ \( T - 624 \) Copy content Toggle raw display
$73$ \( T + 406 \) Copy content Toggle raw display
$79$ \( T - 696 \) Copy content Toggle raw display
$83$ \( T + 800 \) Copy content Toggle raw display
$89$ \( T + 102 \) Copy content Toggle raw display
$97$ \( T + 918 \) Copy content Toggle raw display
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