Properties

Label 138.6.a.c
Level $138$
Weight $6$
Character orbit 138.a
Self dual yes
Analytic conductor $22.133$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 10 q^{5} - 36 q^{6} - 8 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 10 q^{5} - 36 q^{6} - 8 q^{7} + 64 q^{8} + 81 q^{9} - 40 q^{10} - 300 q^{11} - 144 q^{12} + 142 q^{13} - 32 q^{14} + 90 q^{15} + 256 q^{16} - 1110 q^{17} + 324 q^{18} - 1548 q^{19} - 160 q^{20} + 72 q^{21} - 1200 q^{22} + 529 q^{23} - 576 q^{24} - 3025 q^{25} + 568 q^{26} - 729 q^{27} - 128 q^{28} - 8106 q^{29} + 360 q^{30} + 1736 q^{31} + 1024 q^{32} + 2700 q^{33} - 4440 q^{34} + 80 q^{35} + 1296 q^{36} + 3030 q^{37} - 6192 q^{38} - 1278 q^{39} - 640 q^{40} - 14086 q^{41} + 288 q^{42} - 12228 q^{43} - 4800 q^{44} - 810 q^{45} + 2116 q^{46} + 80 q^{47} - 2304 q^{48} - 16743 q^{49} - 12100 q^{50} + 9990 q^{51} + 2272 q^{52} + 23782 q^{53} - 2916 q^{54} + 3000 q^{55} - 512 q^{56} + 13932 q^{57} - 32424 q^{58} + 19092 q^{59} + 1440 q^{60} + 34030 q^{61} + 6944 q^{62} - 648 q^{63} + 4096 q^{64} - 1420 q^{65} + 10800 q^{66} + 10884 q^{67} - 17760 q^{68} - 4761 q^{69} + 320 q^{70} - 13560 q^{71} + 5184 q^{72} - 20278 q^{73} + 12120 q^{74} + 27225 q^{75} - 24768 q^{76} + 2400 q^{77} - 5112 q^{78} + 16320 q^{79} - 2560 q^{80} + 6561 q^{81} - 56344 q^{82} - 10100 q^{83} + 1152 q^{84} + 11100 q^{85} - 48912 q^{86} + 72954 q^{87} - 19200 q^{88} + 17682 q^{89} - 3240 q^{90} - 1136 q^{91} + 8464 q^{92} - 15624 q^{93} + 320 q^{94} + 15480 q^{95} - 9216 q^{96} + 48690 q^{97} - 66972 q^{98} - 24300 q^{99}+O(q^{100}) \) 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 −10.0000 −36.0000 −8.00000 64.0000 81.0000 −40.0000
\(n\): 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.c 1
3.b odd 2 1 414.6.a.a 1
4.b odd 2 1 1104.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.c 1 1.a even 1 1 trivial
414.6.a.a 1 3.b odd 2 1
1104.6.a.d 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 10 \) 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
$5$ \( T + 10 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T + 300 \) Copy content Toggle raw display
$13$ \( T - 142 \) Copy content Toggle raw display
$17$ \( T + 1110 \) Copy content Toggle raw display
$19$ \( T + 1548 \) Copy content Toggle raw display
$23$ \( T - 529 \) Copy content Toggle raw display
$29$ \( T + 8106 \) Copy content Toggle raw display
$31$ \( T - 1736 \) Copy content Toggle raw display
$37$ \( T - 3030 \) Copy content Toggle raw display
$41$ \( T + 14086 \) Copy content Toggle raw display
$43$ \( T + 12228 \) Copy content Toggle raw display
$47$ \( T - 80 \) Copy content Toggle raw display
$53$ \( T - 23782 \) Copy content Toggle raw display
$59$ \( T - 19092 \) Copy content Toggle raw display
$61$ \( T - 34030 \) Copy content Toggle raw display
$67$ \( T - 10884 \) Copy content Toggle raw display
$71$ \( T + 13560 \) Copy content Toggle raw display
$73$ \( T + 20278 \) Copy content Toggle raw display
$79$ \( T - 16320 \) Copy content Toggle raw display
$83$ \( T + 10100 \) Copy content Toggle raw display
$89$ \( T - 17682 \) Copy content Toggle raw display
$97$ \( T - 48690 \) Copy content Toggle raw display
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