Properties

Label 138.8.a.a
Level $138$
Weight $8$
Character orbit 138.a
Self dual yes
Analytic conductor $43.109$
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: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1091335168\)
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

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 230 q^{5} + 216 q^{6} + 106 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 230 q^{5} + 216 q^{6} + 106 q^{7} + 512 q^{8} + 729 q^{9} - 1840 q^{10} - 4326 q^{11} + 1728 q^{12} - 5426 q^{13} + 848 q^{14} - 6210 q^{15} + 4096 q^{16} - 3300 q^{17} + 5832 q^{18} - 4140 q^{19} - 14720 q^{20} + 2862 q^{21} - 34608 q^{22} + 12167 q^{23} + 13824 q^{24} - 25225 q^{25} - 43408 q^{26} + 19683 q^{27} + 6784 q^{28} - 150186 q^{29} - 49680 q^{30} - 307192 q^{31} + 32768 q^{32} - 116802 q^{33} - 26400 q^{34} - 24380 q^{35} + 46656 q^{36} - 55200 q^{37} - 33120 q^{38} - 146502 q^{39} - 117760 q^{40} + 270130 q^{41} + 22896 q^{42} + 36264 q^{43} - 276864 q^{44} - 167670 q^{45} + 97336 q^{46} + 494224 q^{47} + 110592 q^{48} - 812307 q^{49} - 201800 q^{50} - 89100 q^{51} - 347264 q^{52} + 646646 q^{53} + 157464 q^{54} + 994980 q^{55} + 54272 q^{56} - 111780 q^{57} - 1201488 q^{58} - 387948 q^{59} - 397440 q^{60} - 2060876 q^{61} - 2457536 q^{62} + 77274 q^{63} + 262144 q^{64} + 1247980 q^{65} - 934416 q^{66} - 17664 q^{67} - 211200 q^{68} + 328509 q^{69} - 195040 q^{70} - 3580320 q^{71} + 373248 q^{72} + 484550 q^{73} - 441600 q^{74} - 681075 q^{75} - 264960 q^{76} - 458556 q^{77} - 1172016 q^{78} - 2167314 q^{79} - 942080 q^{80} + 531441 q^{81} + 2161040 q^{82} + 381182 q^{83} + 183168 q^{84} + 759000 q^{85} + 290112 q^{86} - 4055022 q^{87} - 2214912 q^{88} - 628620 q^{89} - 1341360 q^{90} - 575156 q^{91} + 778688 q^{92} - 8294184 q^{93} + 3953792 q^{94} + 952200 q^{95} + 884736 q^{96} + 13964418 q^{97} - 6498456 q^{98} - 3153654 q^{99}+O(q^{100}) \) 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
8.00000 27.0000 64.0000 −230.000 216.000 106.000 512.000 729.000 −1840.00
\(n\): 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 138.8.a.a 1
3.b odd 2 1 414.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.a 1 1.a even 1 1 trivial
414.8.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 230 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T + 230 \) Copy content Toggle raw display
$7$ \( T - 106 \) Copy content Toggle raw display
$11$ \( T + 4326 \) Copy content Toggle raw display
$13$ \( T + 5426 \) Copy content Toggle raw display
$17$ \( T + 3300 \) Copy content Toggle raw display
$19$ \( T + 4140 \) Copy content Toggle raw display
$23$ \( T - 12167 \) Copy content Toggle raw display
$29$ \( T + 150186 \) Copy content Toggle raw display
$31$ \( T + 307192 \) Copy content Toggle raw display
$37$ \( T + 55200 \) Copy content Toggle raw display
$41$ \( T - 270130 \) Copy content Toggle raw display
$43$ \( T - 36264 \) Copy content Toggle raw display
$47$ \( T - 494224 \) Copy content Toggle raw display
$53$ \( T - 646646 \) Copy content Toggle raw display
$59$ \( T + 387948 \) Copy content Toggle raw display
$61$ \( T + 2060876 \) Copy content Toggle raw display
$67$ \( T + 17664 \) Copy content Toggle raw display
$71$ \( T + 3580320 \) Copy content Toggle raw display
$73$ \( T - 484550 \) Copy content Toggle raw display
$79$ \( T + 2167314 \) Copy content Toggle raw display
$83$ \( T - 381182 \) Copy content Toggle raw display
$89$ \( T + 628620 \) Copy content Toggle raw display
$97$ \( T - 13964418 \) Copy content Toggle raw display
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