Properties

Label 76.6.a.a
Level $76$
Weight $6$
Character orbit 76.a
Self dual yes
Analytic conductor $12.189$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1891703058\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.272193.1
Defining polynomial: \(x^{3} - x^{2} - 74 x + 168\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{1} - \beta_{2} ) q^{3} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{5} + ( -4 + \beta_{1} + 12 \beta_{2} ) q^{7} + ( 70 + 21 \beta_{1} + 9 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{1} - \beta_{2} ) q^{3} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{5} + ( -4 + \beta_{1} + 12 \beta_{2} ) q^{7} + ( 70 + 21 \beta_{1} + 9 \beta_{2} ) q^{9} + ( -416 - 19 \beta_{1} - 5 \beta_{2} ) q^{11} + ( -8 + 32 \beta_{1} - 7 \beta_{2} ) q^{13} + ( -578 - 36 \beta_{1} - 26 \beta_{2} ) q^{15} + ( -1061 - 34 \beta_{1} - 50 \beta_{2} ) q^{17} + 361 q^{19} + ( -2096 - 102 \beta_{1} + 53 \beta_{2} ) q^{21} + ( -1088 - 52 \beta_{1} - 101 \beta_{2} ) q^{23} + ( -1469 + 15 \beta_{1} + 75 \beta_{2} ) q^{25} + ( -3897 - 109 \beta_{1} - 13 \beta_{2} ) q^{27} + ( 976 + 414 \beta_{1} - 5 \beta_{2} ) q^{29} + ( -3618 - 70 \beta_{1} - 152 \beta_{2} ) q^{31} + ( 4728 + 646 \beta_{1} + 600 \beta_{2} ) q^{33} + ( 1142 + 345 \beta_{1} - 25 \beta_{2} ) q^{35} + ( -284 - 326 \beta_{1} + 344 \beta_{2} ) q^{37} + ( -3308 - 256 \beta_{1} - 379 \beta_{2} ) q^{39} + ( -5008 - 778 \beta_{1} - 226 \beta_{2} ) q^{41} + ( -270 - 733 \beta_{1} + 325 \beta_{2} ) q^{43} + ( 11524 + 417 \beta_{1} + 601 \beta_{2} ) q^{45} + ( 876 - 265 \beta_{1} - 1041 \beta_{2} ) q^{47} + ( 13862 - 180 \beta_{1} - 1600 \beta_{2} ) q^{49} + ( 16143 + 1801 \beta_{1} + 1185 \beta_{2} ) q^{51} + ( 9924 + 172 \beta_{1} - 927 \beta_{2} ) q^{53} + ( -9564 - 1341 \beta_{1} - 909 \beta_{2} ) q^{55} + ( -1083 - 361 \beta_{1} - 361 \beta_{2} ) q^{57} + ( -943 - 217 \beta_{1} + 2075 \beta_{2} ) q^{59} + ( 19288 - 2031 \beta_{1} - 61 \beta_{2} ) q^{61} + ( 12848 + 2449 \beta_{1} + 567 \beta_{2} ) q^{63} + ( 16754 - 330 \beta_{1} + 884 \beta_{2} ) q^{65} + ( -595 + 1265 \beta_{1} - 339 \beta_{2} ) q^{67} + ( 27108 + 2416 \beta_{1} + 1155 \beta_{2} ) q^{69} + ( -11612 - 1274 \beta_{1} + 2284 \beta_{2} ) q^{71} + ( 15337 + 1984 \beta_{1} + 320 \beta_{2} ) q^{73} + ( -9993 + 719 \beta_{1} + 1679 \beta_{2} ) q^{75} + ( -2058 - 2647 \beta_{1} - 5163 \beta_{2} ) q^{77} + ( 8640 + 976 \beta_{1} - 3314 \beta_{2} ) q^{79} + ( 12073 - 12 \beta_{1} + 2844 \beta_{2} ) q^{81} + ( -73264 + 364 \beta_{1} + 2030 \beta_{2} ) q^{83} + ( -18534 - 4581 \beta_{1} - 1779 \beta_{2} ) q^{85} + ( -60068 - 5076 \beta_{1} - 5555 \beta_{2} ) q^{87} + ( -38718 - 34 \beta_{1} + 240 \beta_{2} ) q^{89} + ( -33717 + 4135 \beta_{1} + 2243 \beta_{2} ) q^{91} + ( 45582 + 5534 \beta_{1} + 3628 \beta_{2} ) q^{93} + ( -722 + 1083 \beta_{1} + 361 \beta_{2} ) q^{95} + ( 59102 + 6102 \beta_{1} + 3750 \beta_{2} ) q^{97} + ( -101936 - 11371 \beta_{1} - 7619 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 8q^{3} - 9q^{5} - 13q^{7} + 189q^{9} + O(q^{10}) \) \( 3q - 8q^{3} - 9q^{5} - 13q^{7} + 189q^{9} - 1229q^{11} - 56q^{13} - 1698q^{15} - 3149q^{17} + 1083q^{19} - 6186q^{21} - 3212q^{23} - 4422q^{25} - 11582q^{27} + 2514q^{29} - 10784q^{31} + 13538q^{33} + 3081q^{35} - 526q^{37} - 9668q^{39} - 14246q^{41} - 77q^{43} + 34155q^{45} + 2893q^{47} + 41766q^{49} + 46628q^{51} + 29600q^{53} - 27351q^{55} - 2888q^{57} - 2612q^{59} + 59895q^{61} + 36095q^{63} + 50592q^{65} - 3050q^{67} + 78908q^{69} - 33562q^{71} + 44027q^{73} - 30698q^{75} - 3527q^{77} + 24944q^{79} + 36231q^{81} - 220156q^{83} - 51021q^{85} - 175128q^{87} - 116120q^{89} - 105286q^{91} + 131212q^{93} - 3249q^{95} + 171204q^{97} - 294437q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 74 x + 168\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + \nu - 50 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{2} - \beta_{1} + 99\)\()/2\)

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
7.75086
−9.12596
2.37509
0 −26.4151 0 50.4185 0 117.462 0 454.756 0
1.2 0 4.17335 0 −47.6772 0 121.691 0 −225.583 0
1.3 0 14.2417 0 −11.7413 0 −252.153 0 −40.1733 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 76.6.a.a 3
3.b odd 2 1 684.6.a.b 3
4.b odd 2 1 304.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.a.a 3 1.a even 1 1 trivial
304.6.a.j 3 4.b odd 2 1
684.6.a.b 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 8 T_{3}^{2} - 427 T_{3} + 1570 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(76))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 1570 - 427 T + 8 T^{2} + T^{3} \)
$5$ \( -28224 - 2436 T + 9 T^{2} + T^{3} \)
$7$ \( 3604283 - 46009 T + 13 T^{2} + T^{3} \)
$11$ \( 31125372 + 405108 T + 1229 T^{2} + T^{3} \)
$13$ \( 72234732 - 360509 T + 56 T^{2} + T^{3} \)
$17$ \( 280662027 + 2438583 T + 3149 T^{2} + T^{3} \)
$19$ \( ( -361 + T )^{3} \)
$23$ \( -3000249264 + 193923 T + 3212 T^{2} + T^{3} \)
$29$ \( 128846557278 - 49240077 T - 2514 T^{2} + T^{3} \)
$31$ \( 16959632256 + 31550800 T + 10784 T^{2} + T^{3} \)
$37$ \( -172285751608 - 91323268 T + 526 T^{2} + T^{3} \)
$41$ \( -421643137536 - 97329216 T + 14246 T^{2} + T^{3} \)
$43$ \( -1399322853024 - 238282028 T + 77 T^{2} + T^{3} \)
$47$ \( -1755590562048 - 328748760 T - 2893 T^{2} + T^{3} \)
$53$ \( 572042936316 - 31918341 T - 29600 T^{2} + T^{3} \)
$59$ \( 18472105095894 - 1527450123 T + 2612 T^{2} + T^{3} \)
$61$ \( 7996188319084 - 9375660 T - 59895 T^{2} + T^{3} \)
$67$ \( 4818723981540 - 589016063 T + 3050 T^{2} + T^{3} \)
$71$ \( -25351387808040 - 2373858276 T + 33562 T^{2} + T^{3} \)
$73$ \( 14324767129415 - 442805581 T - 44027 T^{2} + T^{3} \)
$79$ \( -27708927088096 - 4321708468 T - 24944 T^{2} + T^{3} \)
$83$ \( 318522133126128 + 14879049708 T + 220156 T^{2} + T^{3} \)
$89$ \( 57207326543232 + 4473608496 T + 116120 T^{2} + T^{3} \)
$97$ \( 7731995443552 - 1818957084 T - 171204 T^{2} + T^{3} \)
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