Properties

Label 76.3.b.a
Level $76$
Weight $3$
Character orbit 76.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (2 \beta_1 - 2) q^{4} + (2 \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{6} + (\beta_{3} + \beta_1) q^{7} + 8 q^{8} + (\beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (2 \beta_1 - 2) q^{4} + (2 \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{6} + (\beta_{3} + \beta_1) q^{7} + 8 q^{8} + (\beta_{2} + 3) q^{9} + (6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{10} + ( - 2 \beta_{3} - 6 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{12} + (3 \beta_{2} - 12) q^{13} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{14} + (4 \beta_{3} + 12 \beta_1) q^{15} + ( - 8 \beta_1 - 8) q^{16} + (\beta_{2} - 4) q^{17} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{18} + ( - 2 \beta_{3} - \beta_1) q^{19} + ( - 12 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{20} + ( - \beta_{2} + 6) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 16) q^{22} + ( - 15 \beta_{3} - 7 \beta_1) q^{23} + ( - 8 \beta_{3} - 8 \beta_1) q^{24} + ( - 4 \beta_{2} + 35) q^{25} + (9 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 12) q^{26} + ( - 11 \beta_{3} - 7 \beta_1) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{28} + ( - 5 \beta_{2} - 12) q^{29} + ( - 4 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 32) q^{30} + 16 \beta_{3} q^{31} + (16 \beta_1 - 16) q^{32} + (6 \beta_{2} - 20) q^{33} + (3 \beta_{3} - \beta_{2} + 5 \beta_1 + 4) q^{34} + ( - 4 \beta_{3} - 12 \beta_1) q^{35} + ( - 6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{36} + ( - 8 \beta_{2} - 2) q^{37} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{38} + (15 \beta_{3} + 27 \beta_1) q^{39} + (16 \beta_{2} - 16) q^{40} + ( - 10 \beta_{2} - 26) q^{41} + ( - 3 \beta_{3} + \beta_{2} - 7 \beta_1 - 6) q^{42} + (16 \beta_{3} - 4 \beta_1) q^{43} + (4 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 32) q^{44} + (6 \beta_{2} + 22) q^{45} + (15 \beta_{3} + 15 \beta_{2} + 7 \beta_1 - 6) q^{46} + ( - 8 \beta_{3} - 28 \beta_1) q^{47} + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 16) q^{48} + (\beta_{2} + 43) q^{49} + ( - 12 \beta_{3} + 4 \beta_{2} - 39 \beta_1 - 35) q^{50} + (5 \beta_{3} + 9 \beta_1) q^{51} + ( - 18 \beta_{3} - 6 \beta_{2} - 30 \beta_1 + 24) q^{52} + (7 \beta_{2} - 4) q^{53} + (11 \beta_{3} + 11 \beta_{2} + 7 \beta_1 - 10) q^{54} + (32 \beta_{3} + 40 \beta_1) q^{55} + (8 \beta_{3} + 8 \beta_1) q^{56} + (\beta_{2} - 10) q^{57} + ( - 15 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 12) q^{58} + (7 \beta_{3} + 39 \beta_1) q^{59} + ( - 8 \beta_{3} + 8 \beta_{2} - 24 \beta_1 - 64) q^{60} + ( - 4 \beta_{2} - 30) q^{61} + ( - 16 \beta_{3} - 16 \beta_{2} - 16) q^{62} + (2 \beta_{3} - 2 \beta_1) q^{63} + 64 q^{64} + ( - 24 \beta_{2} + 108) q^{65} + (18 \beta_{3} - 6 \beta_{2} + 26 \beta_1 + 20) q^{66} + ( - 33 \beta_{3} + 7 \beta_1) q^{67} + ( - 6 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 8) q^{68} + (7 \beta_{2} - 74) q^{69} + (4 \beta_{3} + 4 \beta_{2} + 12 \beta_1 - 32) q^{70} + (22 \beta_{3} - 2 \beta_1) q^{71} + (8 \beta_{2} + 24) q^{72} + (7 \beta_{2} - 88) q^{73} + ( - 24 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 2) q^{74} + ( - 39 \beta_{3} - 55 \beta_1) q^{75} + (4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{76} + ( - 6 \beta_{2} + 20) q^{77} + ( - 15 \beta_{3} - 15 \beta_{2} - 27 \beta_1 + 66) q^{78} + (2 \beta_{3} - 62 \beta_1) q^{79} + (48 \beta_{3} - 16 \beta_{2} + 32 \beta_1 + 16) q^{80} + (16 \beta_{2} - 31) q^{81} + ( - 30 \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 26) q^{82} + ( - 34 \beta_{3} + 26 \beta_1) q^{83} + (6 \beta_{3} + 2 \beta_{2} + 14 \beta_1 - 12) q^{84} + ( - 8 \beta_{2} + 36) q^{85} + ( - 16 \beta_{3} - 16 \beta_{2} + 4 \beta_1 - 28) q^{86} + (7 \beta_{3} - 13 \beta_1) q^{87} + ( - 16 \beta_{3} - 48 \beta_1) q^{88} + (14 \beta_{2} - 2) q^{89} + (18 \beta_{3} - 6 \beta_{2} - 16 \beta_1 - 22) q^{90} + ( - 15 \beta_{3} - 27 \beta_1) q^{91} + (30 \beta_{3} - 30 \beta_{2} + 14 \beta_1 + 12) q^{92} + 64 q^{93} + (8 \beta_{3} + 8 \beta_{2} + 28 \beta_1 - 76) q^{94} + (2 \beta_{3} + 20 \beta_1) q^{95} + (16 \beta_{3} - 16 \beta_{2} + 16 \beta_1 + 32) q^{96} + ( - 28 \beta_{2} + 2) q^{97} + (3 \beta_{3} - \beta_{2} - 42 \beta_1 - 43) q^{98} + (8 \beta_{3} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} - 4 q^{5} - 6 q^{6} + 32 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} - 4 q^{5} - 6 q^{6} + 32 q^{8} + 14 q^{9} + 4 q^{10} + 12 q^{12} - 42 q^{13} + 6 q^{14} - 32 q^{16} - 14 q^{17} - 14 q^{18} + 8 q^{20} + 22 q^{21} - 60 q^{22} + 132 q^{25} + 42 q^{26} - 12 q^{28} - 58 q^{29} + 120 q^{30} - 64 q^{32} - 68 q^{33} + 14 q^{34} - 28 q^{36} - 24 q^{37} - 32 q^{40} - 124 q^{41} - 22 q^{42} + 120 q^{44} + 100 q^{45} + 6 q^{46} - 48 q^{48} + 174 q^{49} - 132 q^{50} + 84 q^{52} - 2 q^{53} - 18 q^{54} - 38 q^{57} + 58 q^{58} - 240 q^{60} - 128 q^{61} - 96 q^{62} + 256 q^{64} + 384 q^{65} + 68 q^{66} + 28 q^{68} - 282 q^{69} - 120 q^{70} + 112 q^{72} - 338 q^{73} + 24 q^{74} + 68 q^{77} + 234 q^{78} + 32 q^{80} - 92 q^{81} + 124 q^{82} - 44 q^{84} + 128 q^{85} - 144 q^{86} + 20 q^{89} - 100 q^{90} - 12 q^{92} + 256 q^{93} - 288 q^{94} + 96 q^{96} - 48 q^{97} - 174 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} - 2\beta _1 + 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
−1.63746 1.52274i
2.13746 + 0.656712i
2.13746 0.656712i
−1.63746 + 1.52274i
−1.00000 1.73205i 3.04547i −2.00000 + 3.46410i −8.54983 −5.27492 + 3.04547i 3.04547i 8.00000 −0.274917 8.54983 + 14.8087i
39.2 −1.00000 1.73205i 1.31342i −2.00000 + 3.46410i 6.54983 2.27492 1.31342i 1.31342i 8.00000 7.27492 −6.54983 11.3446i
39.3 −1.00000 + 1.73205i 1.31342i −2.00000 3.46410i 6.54983 2.27492 + 1.31342i 1.31342i 8.00000 7.27492 −6.54983 + 11.3446i
39.4 −1.00000 + 1.73205i 3.04547i −2.00000 3.46410i −8.54983 −5.27492 3.04547i 3.04547i 8.00000 −0.274917 8.54983 14.8087i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.b.a 4
3.b odd 2 1 684.3.g.a 4
4.b odd 2 1 inner 76.3.b.a 4
8.b even 2 1 1216.3.d.a 4
8.d odd 2 1 1216.3.d.a 4
12.b even 2 1 684.3.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.a 4 1.a even 1 1 trivial
76.3.b.a 4 4.b odd 2 1 inner
684.3.g.a 4 3.b odd 2 1
684.3.g.a 4 12.b even 2 1
1216.3.d.a 4 8.b even 2 1
1216.3.d.a 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 11T_{3}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 11T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T - 56)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 188T^{2} + 3136 \) Copy content Toggle raw display
$13$ \( (T^{2} + 21 T - 18)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2139 T^{2} + \cdots + 1140624 \) Copy content Toggle raw display
$29$ \( (T^{2} + 29 T - 146)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2816 T^{2} + \cdots + 1048576 \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T - 876)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 62 T - 464)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 3296 T^{2} + 614656 \) Copy content Toggle raw display
$47$ \( T^{4} + 4064 T^{2} + \cdots + 2027776 \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 698)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8027 T^{2} + \cdots + 12588304 \) Copy content Toggle raw display
$61$ \( (T^{2} + 64 T + 796)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 13659 T^{2} + \cdots + 12362256 \) Copy content Toggle raw display
$71$ \( T^{4} + 5612 T^{2} + \cdots + 3211264 \) Copy content Toggle raw display
$73$ \( (T^{2} + 169 T + 6442)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 23852 T^{2} + \cdots + 141324544 \) Copy content Toggle raw display
$83$ \( T^{4} + 22076 T^{2} + 3136 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T - 2768)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24 T - 11028)^{2} \) Copy content Toggle raw display
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