Properties

Label 138.6.a.a
Level $138$
Weight $6$
Character orbit 138.a
Self dual yes
Analytic conductor $22.133$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1329671342\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 44 q^{5} + 36 q^{6} - 70 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 44 q^{5} + 36 q^{6} - 70 q^{7} - 64 q^{8} + 81 q^{9} + 176 q^{10} - 136 q^{11} - 144 q^{12} - 1022 q^{13} + 280 q^{14} + 396 q^{15} + 256 q^{16} + 484 q^{17} - 324 q^{18} - 1046 q^{19} - 704 q^{20} + 630 q^{21} + 544 q^{22} + 529 q^{23} + 576 q^{24} - 1189 q^{25} + 4088 q^{26} - 729 q^{27} - 1120 q^{28} + 2618 q^{29} - 1584 q^{30} - 4860 q^{31} - 1024 q^{32} + 1224 q^{33} - 1936 q^{34} + 3080 q^{35} + 1296 q^{36} + 14918 q^{37} + 4184 q^{38} + 9198 q^{39} + 2816 q^{40} + 7530 q^{41} - 2520 q^{42} + 16186 q^{43} - 2176 q^{44} - 3564 q^{45} - 2116 q^{46} + 29160 q^{47} - 2304 q^{48} - 11907 q^{49} + 4756 q^{50} - 4356 q^{51} - 16352 q^{52} + 9896 q^{53} + 2916 q^{54} + 5984 q^{55} + 4480 q^{56} + 9414 q^{57} - 10472 q^{58} - 2004 q^{59} + 6336 q^{60} - 2570 q^{61} + 19440 q^{62} - 5670 q^{63} + 4096 q^{64} + 44968 q^{65} - 4896 q^{66} + 46118 q^{67} + 7744 q^{68} - 4761 q^{69} - 12320 q^{70} - 32688 q^{71} - 5184 q^{72} - 46830 q^{73} - 59672 q^{74} + 10701 q^{75} - 16736 q^{76} + 9520 q^{77} - 36792 q^{78} - 34338 q^{79} - 11264 q^{80} + 6561 q^{81} - 30120 q^{82} + 31736 q^{83} + 10080 q^{84} - 21296 q^{85} - 64744 q^{86} - 23562 q^{87} + 8704 q^{88} - 60792 q^{89} + 14256 q^{90} + 71540 q^{91} + 8464 q^{92} + 43740 q^{93} - 116640 q^{94} + 46024 q^{95} + 9216 q^{96} - 19218 q^{97} + 47628 q^{98} - 11016 q^{99}+O(q^{100}) \) 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
−4.00000 −9.00000 16.0000 −44.0000 36.0000 −70.0000 −64.0000 81.0000 176.000
\(n\): 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 138.6.a.a 1
3.b odd 2 1 414.6.a.d 1
4.b odd 2 1 1104.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.a 1 1.a even 1 1 trivial
414.6.a.d 1 3.b odd 2 1
1104.6.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 44 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T + 44 \) Copy content Toggle raw display
$7$ \( T + 70 \) Copy content Toggle raw display
$11$ \( T + 136 \) Copy content Toggle raw display
$13$ \( T + 1022 \) Copy content Toggle raw display
$17$ \( T - 484 \) Copy content Toggle raw display
$19$ \( T + 1046 \) Copy content Toggle raw display
$23$ \( T - 529 \) Copy content Toggle raw display
$29$ \( T - 2618 \) Copy content Toggle raw display
$31$ \( T + 4860 \) Copy content Toggle raw display
$37$ \( T - 14918 \) Copy content Toggle raw display
$41$ \( T - 7530 \) Copy content Toggle raw display
$43$ \( T - 16186 \) Copy content Toggle raw display
$47$ \( T - 29160 \) Copy content Toggle raw display
$53$ \( T - 9896 \) Copy content Toggle raw display
$59$ \( T + 2004 \) Copy content Toggle raw display
$61$ \( T + 2570 \) Copy content Toggle raw display
$67$ \( T - 46118 \) Copy content Toggle raw display
$71$ \( T + 32688 \) Copy content Toggle raw display
$73$ \( T + 46830 \) Copy content Toggle raw display
$79$ \( T + 34338 \) Copy content Toggle raw display
$83$ \( T - 31736 \) Copy content Toggle raw display
$89$ \( T + 60792 \) Copy content Toggle raw display
$97$ \( T + 19218 \) Copy content Toggle raw display
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