Properties

Label 768.4.a.t
Level $768$
Weight $4$
Character orbit 768.a
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta_{2} + 3) q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta_{2} + 3) q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9} + (2 \beta_{2} - 2 \beta_1) q^{11} + ( - 3 \beta_{2} + \beta_1 + 18) q^{13} + (3 \beta_{2} + 9) q^{15} + (6 \beta_{2} + 2 \beta_1 + 6) q^{17} + ( - 6 \beta_{2} - 2 \beta_1 + 12) q^{19} + ( - 3 \beta_1 + 15) q^{21} + (2 \beta_1 + 54) q^{23} + (12 \beta_{2} - 4 \beta_1 + 15) q^{25} + 27 q^{27} + (5 \beta_{2} - 2 \beta_1 + 57) q^{29} + ( - 12 \beta_{2} + 3 \beta_1 + 109) q^{31} + (6 \beta_{2} - 6 \beta_1) q^{33} + (14 \beta_{2} + 2 \beta_1 - 36) q^{35} + ( - 3 \beta_{2} + 11 \beta_1 + 96) q^{37} + ( - 9 \beta_{2} + 3 \beta_1 + 54) q^{39} + ( - 6 \beta_{2} + 14 \beta_1 - 42) q^{41} + ( - 18 \beta_{2} + 10 \beta_1 - 84) q^{43} + (9 \beta_{2} + 27) q^{45} + (12 \beta_{2} - 6 \beta_1 + 66) q^{47} + ( - 24 \beta_{2} + 117) q^{49} + (18 \beta_{2} + 6 \beta_1 + 18) q^{51} + ( - 11 \beta_{2} - 10 \beta_1 + 369) q^{53} + (36 \beta_{2} - 4 \beta_1 + 160) q^{55} + ( - 18 \beta_{2} - 6 \beta_1 + 36) q^{57} + (8 \beta_{2} + 8 \beta_1 + 60) q^{59} + ( - 3 \beta_{2} - 9 \beta_1 + 516) q^{61} + ( - 9 \beta_1 + 45) q^{63} + ( - 18 \beta_{2} + 10 \beta_1 - 288) q^{65} + ( - 48 \beta_{2} - 204) q^{67} + (6 \beta_1 + 162) q^{69} + ( - 36 \beta_{2} - 6 \beta_1 - 270) q^{71} + ( - 24 \beta_{2} - 16 \beta_1 - 146) q^{73} + (36 \beta_{2} - 12 \beta_1 + 45) q^{75} + ( - 20 \beta_{2} + 20 \beta_1 + 768) q^{77} + (12 \beta_{2} + 7 \beta_1 + 1) q^{79} + 81 q^{81} + ( - 14 \beta_{2} - 2 \beta_1 + 384) q^{83} + (42 \beta_{2} - 28 \beta_1 + 906) q^{85} + (15 \beta_{2} - 6 \beta_1 + 171) q^{87} + ( - 12 \beta_{2} - 4 \beta_1 + 42) q^{89} + ( - 18 \beta_{2} - 38 \beta_1 - 192) q^{91} + ( - 36 \beta_{2} + 9 \beta_1 + 327) q^{93} + ( - 24 \beta_{2} + 28 \beta_1 - 852) q^{95} + (36 \beta_{2} - 4 \beta_1 - 418) q^{97} + (18 \beta_{2} - 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 9 q^{3} + 10 q^{5} + 14 q^{7} + 27 q^{9} + 52 q^{13} + 30 q^{15} + 26 q^{17} + 28 q^{19} + 42 q^{21} + 164 q^{23} + 53 q^{25} + 81 q^{27} + 174 q^{29} + 318 q^{31} - 92 q^{35} + 296 q^{37} + 156 q^{39} - 118 q^{41} - 260 q^{43} + 90 q^{45} + 204 q^{47} + 327 q^{49} + 78 q^{51} + 1086 q^{53} + 512 q^{55} + 84 q^{57} + 196 q^{59} + 1536 q^{61} + 126 q^{63} - 872 q^{65} - 660 q^{67} + 492 q^{69} - 852 q^{71} - 478 q^{73} + 159 q^{75} + 2304 q^{77} + 22 q^{79} + 243 q^{81} + 1136 q^{83} + 2732 q^{85} + 522 q^{87} + 110 q^{89} - 632 q^{91} + 954 q^{93} - 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 11x - 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu^{2} - 29 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 8\nu - 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 + 29 ) / 4 \) 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
−1.28282
3.76644
−2.48361
0 3.00000 0 −9.15486 0 27.4175 0 9.00000 0
1.2 0 3.00000 0 0.612661 0 −22.7441 0 9.00000 0
1.3 0 3.00000 0 18.5422 0 9.32669 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 768.4.a.t 3
3.b odd 2 1 2304.4.a.bu 3
4.b odd 2 1 768.4.a.r 3
8.b even 2 1 768.4.a.q 3
8.d odd 2 1 768.4.a.s 3
12.b even 2 1 2304.4.a.bt 3
16.e even 4 2 96.4.d.a 6
16.f odd 4 2 24.4.d.a 6
24.f even 2 1 2304.4.a.bv 3
24.h odd 2 1 2304.4.a.bw 3
48.i odd 4 2 288.4.d.d 6
48.k even 4 2 72.4.d.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 16.f odd 4 2
72.4.d.d 6 48.k even 4 2
96.4.d.a 6 16.e even 4 2
288.4.d.d 6 48.i odd 4 2
768.4.a.q 3 8.b even 2 1
768.4.a.r 3 4.b odd 2 1
768.4.a.s 3 8.d odd 2 1
768.4.a.t 3 1.a even 1 1 trivial
2304.4.a.bt 3 12.b even 2 1
2304.4.a.bu 3 3.b odd 2 1
2304.4.a.bv 3 24.f even 2 1
2304.4.a.bw 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{3} - 10T_{5}^{2} - 164T_{5} + 104 \) Copy content Toggle raw display
\( T_{7}^{3} - 14T_{7}^{2} - 580T_{7} + 5816 \) Copy content Toggle raw display
\( T_{11}^{3} - 2816T_{11} + 49152 \) Copy content Toggle raw display
\( T_{19}^{3} - 28T_{19}^{2} - 11088T_{19} - 274752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 10 T^{2} - 164 T + 104 \) Copy content Toggle raw display
$7$ \( T^{3} - 14 T^{2} - 580 T + 5816 \) Copy content Toggle raw display
$11$ \( T^{3} - 2816T + 49152 \) Copy content Toggle raw display
$13$ \( T^{3} - 52 T^{2} - 1104 T + 55872 \) Copy content Toggle raw display
$17$ \( T^{3} - 26 T^{2} - 11124 T + 477576 \) Copy content Toggle raw display
$19$ \( T^{3} - 28 T^{2} - 11088 T - 274752 \) Copy content Toggle raw display
$23$ \( T^{3} - 164 T^{2} + 6384 T - 45504 \) Copy content Toggle raw display
$29$ \( T^{3} - 174 T^{2} + 3964 T + 61368 \) Copy content Toggle raw display
$31$ \( T^{3} - 318 T^{2} + 4476 T + 3749624 \) Copy content Toggle raw display
$37$ \( T^{3} - 296 T^{2} - 46080 T + 82944 \) Copy content Toggle raw display
$41$ \( T^{3} + 118 T^{2} + \cdots - 19985976 \) Copy content Toggle raw display
$43$ \( T^{3} + 260 T^{2} - 80976 T - 8601408 \) Copy content Toggle raw display
$47$ \( T^{3} - 204 T^{2} - 27792 T + 1964736 \) Copy content Toggle raw display
$53$ \( T^{3} - 1086 T^{2} + \cdots - 20665224 \) Copy content Toggle raw display
$59$ \( T^{3} - 196 T^{2} - 50000 T + 8523584 \) Copy content Toggle raw display
$61$ \( T^{3} - 1536 T^{2} + \cdots - 104841216 \) Copy content Toggle raw display
$67$ \( T^{3} + 660 T^{2} + \cdots - 32228928 \) Copy content Toggle raw display
$71$ \( T^{3} + 852 T^{2} + \cdots - 85084992 \) Copy content Toggle raw display
$73$ \( T^{3} + 478 T^{2} + \cdots - 120833304 \) Copy content Toggle raw display
$79$ \( T^{3} - 22 T^{2} - 71524 T + 7902616 \) Copy content Toggle raw display
$83$ \( T^{3} - 1136 T^{2} + \cdots - 37953536 \) Copy content Toggle raw display
$89$ \( T^{3} - 110 T^{2} - 41364 T - 1423656 \) Copy content Toggle raw display
$97$ \( T^{3} + 1222 T^{2} + \cdots - 74802424 \) Copy content Toggle raw display
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