Properties

Label 1176.4.a.u
Level $1176$
Weight $4$
Character orbit 1176.a
Self dual yes
Analytic conductor $69.386$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{505}) \)
Defining polynomial: \(x^{2} - x - 126\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{505})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( -4 - \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( -4 - \beta ) q^{5} + 9 q^{9} + 5 \beta q^{11} + ( 9 + \beta ) q^{13} + ( -12 - 3 \beta ) q^{15} + ( 4 - 4 \beta ) q^{17} + ( -55 - 7 \beta ) q^{19} + ( 76 - 4 \beta ) q^{23} + ( 17 + 9 \beta ) q^{25} + 27 q^{27} + ( -50 + 5 \beta ) q^{29} + ( 179 + 2 \beta ) q^{31} + 15 \beta q^{33} + ( -27 + \beta ) q^{37} + ( 27 + 3 \beta ) q^{39} + ( 94 - 18 \beta ) q^{41} + ( 215 - 27 \beta ) q^{43} + ( -36 - 9 \beta ) q^{45} + ( -166 - 36 \beta ) q^{47} + ( 12 - 12 \beta ) q^{51} + ( 346 + 5 \beta ) q^{53} + ( -630 - 25 \beta ) q^{55} + ( -165 - 21 \beta ) q^{57} + ( 306 - 27 \beta ) q^{59} + ( 566 + 28 \beta ) q^{61} + ( -162 - 14 \beta ) q^{65} + ( 153 - 73 \beta ) q^{67} + ( 228 - 12 \beta ) q^{69} + ( 270 + 76 \beta ) q^{71} + ( 377 + 63 \beta ) q^{73} + ( 51 + 27 \beta ) q^{75} + ( -425 + 48 \beta ) q^{79} + 81 q^{81} + ( 108 + 67 \beta ) q^{83} + ( 488 + 16 \beta ) q^{85} + ( -150 + 15 \beta ) q^{87} + ( 940 - 22 \beta ) q^{89} + ( 537 + 6 \beta ) q^{93} + ( 1102 + 90 \beta ) q^{95} + ( -892 + 55 \beta ) q^{97} + 45 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 9q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 9q^{5} + 18q^{9} + 5q^{11} + 19q^{13} - 27q^{15} + 4q^{17} - 117q^{19} + 148q^{23} + 43q^{25} + 54q^{27} - 95q^{29} + 360q^{31} + 15q^{33} - 53q^{37} + 57q^{39} + 170q^{41} + 403q^{43} - 81q^{45} - 368q^{47} + 12q^{51} + 697q^{53} - 1285q^{55} - 351q^{57} + 585q^{59} + 1160q^{61} - 338q^{65} + 233q^{67} + 444q^{69} + 616q^{71} + 817q^{73} + 129q^{75} - 802q^{79} + 162q^{81} + 283q^{83} + 992q^{85} - 285q^{87} + 1858q^{89} + 1080q^{93} + 2294q^{95} - 1729q^{97} + 45q^{99} + O(q^{100}) \)

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
11.7361
−10.7361
0 3.00000 0 −15.7361 0 0 0 9.00000 0
1.2 0 3.00000 0 6.73610 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.u 2
4.b odd 2 1 2352.4.a.bo 2
7.b odd 2 1 1176.4.a.r 2
7.d odd 6 2 168.4.q.d 4
21.g even 6 2 504.4.s.f 4
28.d even 2 1 2352.4.a.cc 2
28.f even 6 2 336.4.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 7.d odd 6 2
336.4.q.g 4 28.f even 6 2
504.4.s.f 4 21.g even 6 2
1176.4.a.r 2 7.b odd 2 1
1176.4.a.u 2 1.a even 1 1 trivial
2352.4.a.bo 2 4.b odd 2 1
2352.4.a.cc 2 28.d even 2 1

Hecke kernels

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

\( T_{5}^{2} + 9 T_{5} - 106 \)
\( T_{11}^{2} - 5 T_{11} - 3150 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( -106 + 9 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -3150 - 5 T + T^{2} \)
$13$ \( -36 - 19 T + T^{2} \)
$17$ \( -2016 - 4 T + T^{2} \)
$19$ \( -2764 + 117 T + T^{2} \)
$23$ \( 3456 - 148 T + T^{2} \)
$29$ \( -900 + 95 T + T^{2} \)
$31$ \( 31895 - 360 T + T^{2} \)
$37$ \( 576 + 53 T + T^{2} \)
$41$ \( -33680 - 170 T + T^{2} \)
$43$ \( -51434 - 403 T + T^{2} \)
$47$ \( -129764 + 368 T + T^{2} \)
$53$ \( 118296 - 697 T + T^{2} \)
$59$ \( -6480 - 585 T + T^{2} \)
$61$ \( 237420 - 1160 T + T^{2} \)
$67$ \( -659214 - 233 T + T^{2} \)
$71$ \( -634356 - 616 T + T^{2} \)
$73$ \( -334214 - 817 T + T^{2} \)
$79$ \( -130079 + 802 T + T^{2} \)
$83$ \( -546714 - 283 T + T^{2} \)
$89$ \( 801936 - 1858 T + T^{2} \)
$97$ \( 365454 + 1729 T + T^{2} \)
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