Properties

Label 76.3.g.b
Level $76$
Weight $3$
Character orbit 76.g
Analytic conductor $2.071$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-10})\)
Defining polynomial: \(x^{4} - 10 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + 4 q^{4} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 4 - 8 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 2 q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + 4 q^{4} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 4 - 8 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{10} + ( -1 + 2 \beta_{2} - 5 \beta_{3} ) q^{11} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{12} -6 \beta_{2} q^{13} + ( 8 - 16 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -16 + \beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{15} + 16 q^{16} + ( -16 - 2 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{18} + ( 6 + 3 \beta_{1} + 9 \beta_{2} - 5 \beta_{3} ) q^{19} + ( -8 - 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{20} + ( -22 + 5 \beta_{1} + 22 \beta_{2} - 10 \beta_{3} ) q^{21} + ( -2 + 4 \beta_{2} - 10 \beta_{3} ) q^{22} + ( 4 - 9 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} ) q^{23} + ( -8 + 8 \beta_{1} - 8 \beta_{2} ) q^{24} + ( -4 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} ) q^{25} -12 \beta_{2} q^{26} + ( 15 - 30 \beta_{2} + \beta_{3} ) q^{27} + ( 16 - 32 \beta_{2} + 4 \beta_{3} ) q^{28} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{29} + ( -32 + 2 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{30} + ( -10 + 20 \beta_{2} + 7 \beta_{3} ) q^{31} + 32 q^{32} + ( 53 - 6 \beta_{1} - 53 \beta_{2} + 12 \beta_{3} ) q^{33} + ( -32 - 4 \beta_{1} + 32 \beta_{2} + 8 \beta_{3} ) q^{34} + ( -2 + 10 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{36} + ( -24 + 10 \beta_{1} - 5 \beta_{3} ) q^{37} + ( 12 + 6 \beta_{1} + 18 \beta_{2} - 10 \beta_{3} ) q^{38} + ( -6 + 12 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -16 - 8 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} ) q^{40} + ( 13 - 4 \beta_{1} - 13 \beta_{2} + 8 \beta_{3} ) q^{41} + ( -44 + 10 \beta_{1} + 44 \beta_{2} - 20 \beta_{3} ) q^{42} + ( -12 - 10 \beta_{1} - 12 \beta_{2} ) q^{43} + ( -4 + 8 \beta_{2} - 20 \beta_{3} ) q^{44} + 52 q^{45} + ( 8 - 18 \beta_{1} - 4 \beta_{2} + 18 \beta_{3} ) q^{46} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{47} + ( -16 + 16 \beta_{1} - 16 \beta_{2} ) q^{48} + ( -9 + 16 \beta_{1} - 8 \beta_{3} ) q^{49} + ( -8 \beta_{1} - 18 \beta_{2} - 8 \beta_{3} ) q^{50} + ( -8 - 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{51} -24 \beta_{2} q^{52} + ( 4 \beta_{1} + 62 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 30 - 60 \beta_{2} + 2 \beta_{3} ) q^{54} + ( 48 + 7 \beta_{1} + 48 \beta_{2} ) q^{55} + ( 32 - 64 \beta_{2} + 8 \beta_{3} ) q^{56} + ( 53 - 2 \beta_{1} - 44 \beta_{2} + 16 \beta_{3} ) q^{57} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -13 - \beta_{1} - 13 \beta_{2} ) q^{59} + ( -64 + 4 \beta_{1} + 32 \beta_{2} - 4 \beta_{3} ) q^{60} + ( -\beta_{1} - 22 \beta_{2} - \beta_{3} ) q^{61} + ( -20 + 40 \beta_{2} + 14 \beta_{3} ) q^{62} + ( 72 - 28 \beta_{1} - 36 \beta_{2} + 28 \beta_{3} ) q^{63} + 64 q^{64} + ( 12 + 12 \beta_{1} - 6 \beta_{3} ) q^{65} + ( 106 - 12 \beta_{1} - 106 \beta_{2} + 24 \beta_{3} ) q^{66} + ( -34 + 21 \beta_{1} + 17 \beta_{2} - 21 \beta_{3} ) q^{67} + ( -64 - 8 \beta_{1} + 64 \beta_{2} + 16 \beta_{3} ) q^{68} + ( -96 + 22 \beta_{1} - 11 \beta_{3} ) q^{69} + ( -4 + 20 \beta_{1} - 4 \beta_{2} ) q^{70} + ( 34 + 14 \beta_{1} + 34 \beta_{2} ) q^{71} + ( -16 \beta_{1} + 32 \beta_{2} - 16 \beta_{3} ) q^{72} + ( -13 + 4 \beta_{1} + 13 \beta_{2} - 8 \beta_{3} ) q^{73} + ( -48 + 20 \beta_{1} - 10 \beta_{3} ) q^{74} + ( 31 - 62 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 24 + 12 \beta_{1} + 36 \beta_{2} - 20 \beta_{3} ) q^{76} + ( 62 - 42 \beta_{1} + 21 \beta_{3} ) q^{77} + ( -12 + 24 \beta_{2} - 12 \beta_{3} ) q^{78} + ( -60 - 8 \beta_{1} - 60 \beta_{2} ) q^{79} + ( -32 - 16 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{80} + ( -19 - 2 \beta_{1} + 19 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 26 - 8 \beta_{1} - 26 \beta_{2} + 16 \beta_{3} ) q^{82} + ( -19 + 38 \beta_{2} - 19 \beta_{3} ) q^{83} + ( -88 + 20 \beta_{1} + 88 \beta_{2} - 40 \beta_{3} ) q^{84} + ( -20 \beta_{1} - 92 \beta_{2} - 20 \beta_{3} ) q^{85} + ( -24 - 20 \beta_{1} - 24 \beta_{2} ) q^{86} + ( -12 + 24 \beta_{2} - 5 \beta_{3} ) q^{87} + ( -8 + 16 \beta_{2} - 40 \beta_{3} ) q^{88} + ( -22 \beta_{1} - 8 \beta_{2} - 22 \beta_{3} ) q^{89} + 104 q^{90} + ( -48 + 6 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} ) q^{91} + ( 16 - 36 \beta_{1} - 8 \beta_{2} + 36 \beta_{3} ) q^{92} + ( -40 - 3 \beta_{1} + 40 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 14 \beta_{1} - 14 \beta_{3} ) q^{94} + ( -40 - 20 \beta_{1} + 92 \beta_{2} + 27 \beta_{3} ) q^{95} + ( -32 + 32 \beta_{1} - 32 \beta_{2} ) q^{96} + ( -117 + 2 \beta_{1} + 117 \beta_{2} - 4 \beta_{3} ) q^{97} + ( -18 + 32 \beta_{1} - 16 \beta_{3} ) q^{98} + ( -208 + 26 \beta_{1} + 104 \beta_{2} - 26 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{2} - 6q^{3} + 16q^{4} - 4q^{5} - 12q^{6} + 32q^{8} + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{2} - 6q^{3} + 16q^{4} - 4q^{5} - 12q^{6} + 32q^{8} + 8q^{9} - 8q^{10} - 24q^{12} - 12q^{13} - 48q^{15} + 64q^{16} - 32q^{17} + 16q^{18} + 42q^{19} - 16q^{20} - 44q^{21} + 12q^{23} - 48q^{24} - 18q^{25} - 24q^{26} - 4q^{29} - 96q^{30} + 128q^{32} + 106q^{33} - 64q^{34} - 12q^{35} + 32q^{36} - 96q^{37} + 84q^{38} - 32q^{40} + 26q^{41} - 88q^{42} - 72q^{43} + 208q^{45} + 24q^{46} - 96q^{48} - 36q^{49} - 36q^{50} - 24q^{51} - 48q^{52} + 124q^{53} + 288q^{55} + 124q^{57} - 8q^{58} - 78q^{59} - 192q^{60} - 44q^{61} + 216q^{63} + 256q^{64} + 48q^{65} + 212q^{66} - 102q^{67} - 128q^{68} - 384q^{69} - 24q^{70} + 204q^{71} + 64q^{72} - 26q^{73} - 192q^{74} + 168q^{76} + 248q^{77} - 360q^{79} - 64q^{80} - 38q^{81} + 52q^{82} - 176q^{84} - 184q^{85} - 144q^{86} - 16q^{89} + 416q^{90} - 144q^{91} + 48q^{92} - 80q^{93} + 24q^{95} - 192q^{96} - 234q^{97} - 72q^{98} - 624q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/10\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(10 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−2.73861 + 1.58114i
2.73861 1.58114i
−2.73861 1.58114i
2.73861 + 1.58114i
2.00000 −4.23861 + 2.44716i 4.00000 1.73861 + 3.01137i −8.47723 + 4.89433i 10.0905i 8.00000 7.47723 12.9509i 3.47723 + 6.02273i
7.2 2.00000 1.23861 0.715113i 4.00000 −3.73861 6.47547i 2.47723 1.43023i 3.76593i 8.00000 −3.47723 + 6.02273i −7.47723 12.9509i
11.1 2.00000 −4.23861 2.44716i 4.00000 1.73861 3.01137i −8.47723 4.89433i 10.0905i 8.00000 7.47723 + 12.9509i 3.47723 6.02273i
11.2 2.00000 1.23861 + 0.715113i 4.00000 −3.73861 + 6.47547i 2.47723 + 1.43023i 3.76593i 8.00000 −3.47723 6.02273i −7.47723 + 12.9509i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.g.b yes 4
4.b odd 2 1 76.3.g.a 4
19.c even 3 1 76.3.g.a 4
76.g odd 6 1 inner 76.3.g.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.g.a 4 4.b odd 2 1
76.3.g.a 4 19.c even 3 1
76.3.g.b yes 4 1.a even 1 1 trivial
76.3.g.b yes 4 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6 T_{3}^{3} + 5 T_{3}^{2} - 42 T_{3} + 49 \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{4} \)
$3$ \( 49 - 42 T + 5 T^{2} + 6 T^{3} + T^{4} \)
$5$ \( 676 - 104 T + 42 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 1444 + 116 T^{2} + T^{4} \)
$11$ \( 61009 + 506 T^{2} + T^{4} \)
$13$ \( ( 36 + 6 T + T^{2} )^{2} \)
$17$ \( 18496 + 4352 T + 888 T^{2} + 32 T^{3} + T^{4} \)
$19$ \( 130321 - 15162 T + 893 T^{2} - 42 T^{3} + T^{4} \)
$23$ \( 636804 + 9576 T - 750 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 676 - 104 T + 42 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( 36100 + 1580 T^{2} + T^{4} \)
$37$ \( ( -174 + 48 T + T^{2} )^{2} \)
$41$ \( 96721 + 8086 T + 987 T^{2} - 26 T^{3} + T^{4} \)
$43$ \( 322624 - 40896 T + 1160 T^{2} + 72 T^{3} + T^{4} \)
$47$ \( 240100 - 490 T^{2} + T^{4} \)
$53$ \( 11316496 - 417136 T + 12012 T^{2} - 124 T^{3} + T^{4} \)
$59$ \( 247009 + 38766 T + 2525 T^{2} + 78 T^{3} + T^{4} \)
$61$ \( 206116 + 19976 T + 1482 T^{2} + 44 T^{3} + T^{4} \)
$67$ \( 12552849 - 361386 T - 75 T^{2} + 102 T^{3} + T^{4} \)
$71$ \( 2274064 - 307632 T + 15380 T^{2} - 204 T^{3} + T^{4} \)
$73$ \( 96721 - 8086 T + 987 T^{2} + 26 T^{3} + T^{4} \)
$79$ \( 103225600 + 3657600 T + 53360 T^{2} + 360 T^{3} + T^{4} \)
$83$ \( 6385729 + 9386 T^{2} + T^{4} \)
$89$ \( 208975936 - 231296 T + 14712 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( 184117761 + 3175146 T + 41187 T^{2} + 234 T^{3} + T^{4} \)
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