Properties

Label 76.8.e.a
Level $76$
Weight $8$
Character orbit 76.e
Analytic conductor $23.741$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22q + 13q^{3} + q^{5} + 560q^{7} - 6002q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q + 13q^{3} + q^{5} + 560q^{7} - 6002q^{9} + 472q^{11} - 567q^{13} + 2995q^{15} + 5589q^{17} + 80912q^{19} + 44412q^{21} - 15425q^{23} - 32806q^{25} + 50290q^{27} - 18919q^{29} + 150296q^{31} + 314618q^{33} + 92808q^{35} + 350100q^{37} + 948810q^{39} + 698891q^{41} + 402545q^{43} + 1477508q^{45} - 653621q^{47} - 1938490q^{49} - 1386401q^{51} - 106763q^{53} + 414508q^{55} + 1267563q^{57} + 3136737q^{59} + 2004581q^{61} + 1465000q^{63} - 7397638q^{65} + 4344391q^{67} + 1732238q^{69} - 133823q^{71} - 8349685q^{73} - 12136824q^{75} + 9147480q^{77} - 94679q^{79} - 838595q^{81} - 2884080q^{83} - 1421409q^{85} - 31740598q^{87} - 7039347q^{89} + 1520096q^{91} - 1993628q^{93} + 1707587q^{95} + 13308115q^{97} + 6011488q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 0 −40.1712 69.5786i 0 −97.7376 169.287i 0 −587.742 0 −2133.95 + 3696.12i 0
45.2 0 −34.7112 60.1216i 0 55.3969 + 95.9502i 0 314.254 0 −1316.24 + 2279.79i 0
45.3 0 −22.8105 39.5090i 0 216.474 + 374.944i 0 −508.391 0 52.8600 91.5562i 0
45.4 0 −18.8127 32.5845i 0 −125.931 218.119i 0 965.440 0 385.666 667.994i 0
45.5 0 −3.42876 5.93879i 0 −200.054 346.504i 0 −1032.75 0 1069.99 1853.27i 0
45.6 0 1.69911 + 2.94294i 0 105.538 + 182.797i 0 1469.14 0 1087.73 1884.00i 0
45.7 0 7.71048 + 13.3549i 0 60.2613 + 104.376i 0 −1390.32 0 974.597 1688.05i 0
45.8 0 20.2338 + 35.0460i 0 −90.2780 156.366i 0 231.633 0 274.684 475.766i 0
45.9 0 20.3841 + 35.3063i 0 172.055 + 298.008i 0 763.824 0 262.475 454.621i 0
45.10 0 36.8075 + 63.7525i 0 −206.713 358.037i 0 601.253 0 −1616.09 + 2799.14i 0
45.11 0 39.5993 + 68.5881i 0 111.488 + 193.103i 0 −546.344 0 −2042.72 + 3538.09i 0
49.1 0 −40.1712 + 69.5786i 0 −97.7376 + 169.287i 0 −587.742 0 −2133.95 3696.12i 0
49.2 0 −34.7112 + 60.1216i 0 55.3969 95.9502i 0 314.254 0 −1316.24 2279.79i 0
49.3 0 −22.8105 + 39.5090i 0 216.474 374.944i 0 −508.391 0 52.8600 + 91.5562i 0
49.4 0 −18.8127 + 32.5845i 0 −125.931 + 218.119i 0 965.440 0 385.666 + 667.994i 0
49.5 0 −3.42876 + 5.93879i 0 −200.054 + 346.504i 0 −1032.75 0 1069.99 + 1853.27i 0
49.6 0 1.69911 2.94294i 0 105.538 182.797i 0 1469.14 0 1087.73 + 1884.00i 0
49.7 0 7.71048 13.3549i 0 60.2613 104.376i 0 −1390.32 0 974.597 + 1688.05i 0
49.8 0 20.2338 35.0460i 0 −90.2780 + 156.366i 0 231.633 0 274.684 + 475.766i 0
49.9 0 20.3841 35.3063i 0 172.055 298.008i 0 763.824 0 262.475 + 454.621i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.8.e.a 22
19.c even 3 1 inner 76.8.e.a 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.e.a 22 1.a even 1 1 trivial
76.8.e.a 22 19.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(76, [\chi])\).