Properties

Label 1176.4.a.z
Level $1176$
Weight $4$
Character orbit 1176.a
Self dual yes
Analytic conductor $69.386$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.391168.1
Defining polynomial: \(x^{4} - 40 x^{2} + 382\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -2 - \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -2 - \beta_{1} ) q^{5} + 9 q^{9} + ( 10 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( -12 - \beta_{2} - \beta_{3} ) q^{13} + ( 6 + 3 \beta_{1} ) q^{15} + ( -18 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{17} + ( -8 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{19} + ( 2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{23} + ( 41 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{25} -27 q^{27} + ( 36 + 12 \beta_{1} + 5 \beta_{2} ) q^{29} + ( -12 - 6 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -30 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{33} + ( 12 + 14 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} ) q^{37} + ( 36 + 3 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -18 + 3 \beta_{1} - 11 \beta_{2} - 20 \beta_{3} ) q^{41} + ( 128 + 2 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{43} + ( -18 - 9 \beta_{1} ) q^{45} + ( -40 - 2 \beta_{1} + 11 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 54 - 9 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{51} + ( 134 + 12 \beta_{1} - 6 \beta_{2} + 24 \beta_{3} ) q^{53} + ( 84 - 6 \beta_{1} - \beta_{2} - 18 \beta_{3} ) q^{55} + ( 24 - 6 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -60 + 30 \beta_{1} - 7 \beta_{2} + 26 \beta_{3} ) q^{59} + ( -224 - 10 \beta_{1} - \beta_{2} + 29 \beta_{3} ) q^{61} + ( -34 + 14 \beta_{1} - 3 \beta_{2} - 22 \beta_{3} ) q^{65} + ( 272 - 22 \beta_{1} - 2 \beta_{2} + 38 \beta_{3} ) q^{67} + ( -6 - 9 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} ) q^{69} + ( 322 - 13 \beta_{1} + 33 \beta_{2} - 25 \beta_{3} ) q^{71} + ( -372 - 18 \beta_{1} + 26 \beta_{2} - \beta_{3} ) q^{73} + ( -123 - 12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{75} + ( 104 - 36 \beta_{1} + 20 \beta_{2} - 24 \beta_{3} ) q^{79} + 81 q^{81} + ( 28 + 32 \beta_{1} + 26 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -378 + 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{85} + ( -108 - 36 \beta_{1} - 15 \beta_{2} ) q^{87} + ( -790 + 25 \beta_{1} - 39 \beta_{2} + 28 \beta_{3} ) q^{89} + ( 36 + 18 \beta_{1} + 27 \beta_{2} - 6 \beta_{3} ) q^{93} + ( -60 - 6 \beta_{1} - 10 \beta_{2} + 102 \beta_{3} ) q^{95} + ( -596 - 22 \beta_{1} + 24 \beta_{2} - 25 \beta_{3} ) q^{97} + ( 90 - 9 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + 40q^{11} - 48q^{13} + 24q^{15} - 72q^{17} - 32q^{19} + 8q^{23} + 164q^{25} - 108q^{27} + 144q^{29} - 48q^{31} - 120q^{33} + 48q^{37} + 144q^{39} - 72q^{41} + 512q^{43} - 72q^{45} - 160q^{47} + 216q^{51} + 536q^{53} + 336q^{55} + 96q^{57} - 240q^{59} - 896q^{61} - 136q^{65} + 1088q^{67} - 24q^{69} + 1288q^{71} - 1488q^{73} - 492q^{75} + 416q^{79} + 324q^{81} + 112q^{83} - 1512q^{85} - 432q^{87} - 3160q^{89} + 144q^{93} - 240q^{95} - 2384q^{97} + 360q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 40 x^{2} + 382\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} - \nu^{2} + 40 \nu + 20 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{2} + 12 \nu + 40 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{2} - 140 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{3} + 7 \beta_{2}\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{3} + 140\)\()/7\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{3} + 70 \beta_{2} - 21 \beta_{1}\)\()/14\)

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
3.96955
−4.92368
−3.96955
4.92368
0 −3.00000 0 −14.6418 0 0 0 9.00000 0
1.2 0 −3.00000 0 −14.5121 0 0 0 9.00000 0
1.3 0 −3.00000 0 7.81338 0 0 0 9.00000 0
1.4 0 −3.00000 0 13.3405 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 1176.4.a.z 4
4.b odd 2 1 2352.4.a.co 4
7.b odd 2 1 1176.4.a.be yes 4
28.d even 2 1 2352.4.a.cn 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.z 4 1.a even 1 1 trivial
1176.4.a.be yes 4 7.b odd 2 1
2352.4.a.cn 4 28.d even 2 1
2352.4.a.co 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\):

\( T_{5}^{4} + 8 T_{5}^{3} - 300 T_{5}^{2} - 1456 T_{5} + 22148 \)
\( T_{11}^{4} - 40 T_{11}^{3} - 232 T_{11}^{2} + 4704 T_{11} + 7056 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T )^{4} \)
$5$ \( 22148 - 1456 T - 300 T^{2} + 8 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 7056 + 4704 T - 232 T^{2} - 40 T^{3} + T^{4} \)
$13$ \( 5508 - 8928 T + 124 T^{2} + 48 T^{3} + T^{4} \)
$17$ \( 695428 - 44208 T - 1324 T^{2} + 72 T^{3} + T^{4} \)
$19$ \( 50872896 - 213504 T - 14992 T^{2} + 32 T^{3} + T^{4} \)
$23$ \( -241264 + 104928 T - 3944 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( 193813056 + 1033344 T - 44720 T^{2} - 144 T^{3} + T^{4} \)
$31$ \( 280696384 + 2404992 T - 57904 T^{2} + 48 T^{3} + T^{4} \)
$37$ \( 3279104 + 11171072 T - 95392 T^{2} - 48 T^{3} + T^{4} \)
$41$ \( -110754684 + 10848720 T - 143276 T^{2} + 72 T^{3} + T^{4} \)
$43$ \( -1048085504 + 16774144 T + 18368 T^{2} - 512 T^{3} + T^{4} \)
$47$ \( 908321344 - 1189376 T - 88720 T^{2} + 160 T^{3} + T^{4} \)
$53$ \( 520036624 + 4927648 T - 88104 T^{2} - 536 T^{3} + T^{4} \)
$59$ \( -10001985984 - 176574336 T - 448304 T^{2} + 240 T^{3} + T^{4} \)
$61$ \( -2069850492 - 50346624 T + 85436 T^{2} + 896 T^{3} + T^{4} \)
$67$ \( -23013868544 + 191398912 T - 46144 T^{2} - 1088 T^{3} + T^{4} \)
$71$ \( -188096048496 + 890221536 T - 419240 T^{2} - 1288 T^{3} + T^{4} \)
$73$ \( -21509659836 - 202647456 T + 197404 T^{2} + 1488 T^{3} + T^{4} \)
$79$ \( -88431606784 + 574626816 T - 862400 T^{2} - 416 T^{3} + T^{4} \)
$83$ \( 1047073024 - 59481856 T - 608416 T^{2} - 112 T^{3} + T^{4} \)
$89$ \( -1254461582844 - 1366317456 T + 2135956 T^{2} + 3160 T^{3} + T^{4} \)
$97$ \( 13301642052 + 244151136 T + 1344796 T^{2} + 2384 T^{3} + T^{4} \)
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