Properties

Label 1176.4.a.ba
Level $1176$
Weight $4$
Character orbit 1176.a
Self dual yes
Analytic conductor $69.386$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
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 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.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -1 - \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -1 - \beta_{1} ) q^{5} + 9 q^{9} + ( 3 + 2 \beta_{1} + \beta_{3} ) q^{11} + ( -6 - 4 \beta_{1} + \beta_{3} ) q^{13} + ( 3 + 3 \beta_{1} ) q^{15} + ( -25 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{19} + ( 23 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + ( 28 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} -27 q^{27} + ( -18 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{29} + ( -57 + 7 \beta_{1} - 5 \beta_{3} ) q^{31} + ( -9 - 6 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 140 - 14 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{37} + ( 18 + 12 \beta_{1} - 3 \beta_{3} ) q^{39} + ( -106 - 14 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -65 - 15 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{43} + ( -9 - 9 \beta_{1} ) q^{45} + ( 17 + 25 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{47} + ( 75 + 9 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 1 - 11 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -370 - 18 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{55} + ( -21 + 9 \beta_{1} - 3 \beta_{2} ) q^{57} + ( -68 - 3 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} ) q^{59} + ( 68 - 36 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{61} + ( 551 + 9 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{65} + ( 33 + 5 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -69 + 9 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{69} + ( -211 - 31 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{71} + ( 54 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} + ( -84 - 12 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 441 + 25 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{79} + 81 q^{81} + ( -862 + 27 \beta_{1} + \beta_{2} ) q^{83} + ( 364 + 94 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 54 + 18 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{87} + ( 86 + 34 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{89} + ( 171 - 21 \beta_{1} + 15 \beta_{3} ) q^{93} + ( 503 - 67 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 151 + 90 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{97} + ( 27 + 18 \beta_{1} + 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} - 4q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} - 4q^{5} + 36q^{9} + 14q^{11} - 22q^{13} + 12q^{15} - 96q^{17} + 26q^{19} + 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} - 42q^{33} + 562q^{37} + 66q^{39} - 428q^{41} - 258q^{43} - 36q^{45} + 80q^{47} + 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} + 276q^{61} + 2196q^{65} + 150q^{67} - 288q^{69} - 848q^{71} + 218q^{73} - 330q^{75} + 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} + 344q^{89} + 714q^{93} + 2004q^{95} + 622q^{97} + 126q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 7 \nu^{2} - 213 \nu - 1158 \)\()/105\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} - 98 \nu^{2} + 942 \nu + 6297 \)\()/105\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 77 \nu^{2} + 677 \nu - 3348 \)\()/35\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 15\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(11 \beta_{3} + 5 \beta_{2} + 127 \beta_{1} + 2153\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(34 \beta_{3} + 151 \beta_{2} + 992 \beta_{1} + 5137\)\()/14\)

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
13.2349
−6.46638
−8.69515
3.92665
0 −3.00000 0 −18.9580 0 0 0 9.00000 0
1.2 0 −3.00000 0 −0.726342 0 0 0 9.00000 0
1.3 0 −3.00000 0 −0.128591 0 0 0 9.00000 0
1.4 0 −3.00000 0 15.8130 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.ba 4
4.b odd 2 1 2352.4.a.cp 4
7.b odd 2 1 1176.4.a.bd 4
7.d odd 6 2 168.4.q.f 8
21.g even 6 2 504.4.s.j 8
28.d even 2 1 2352.4.a.cm 4
28.f even 6 2 336.4.q.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.f 8 7.d odd 6 2
336.4.q.m 8 28.f even 6 2
504.4.s.j 8 21.g even 6 2
1176.4.a.ba 4 1.a even 1 1 trivial
1176.4.a.bd 4 7.b odd 2 1
2352.4.a.cm 4 28.d even 2 1
2352.4.a.cp 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\):

\( T_{5}^{4} + 4 T_{5}^{3} - 297 T_{5}^{2} - 256 T_{5} - 28 \)
\( T_{11}^{4} - 14 T_{11}^{3} - 5395 T_{11}^{2} + 67916 T_{11} + 5765844 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T )^{4} \)
$5$ \( -28 - 256 T - 297 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 5765844 + 67916 T - 5395 T^{2} - 14 T^{3} + T^{4} \)
$13$ \( 13795008 - 84004 T - 7423 T^{2} + 22 T^{3} + T^{4} \)
$17$ \( -44203008 - 1716032 T - 14304 T^{2} + 96 T^{3} + T^{4} \)
$19$ \( 5487296 - 258652 T - 14235 T^{2} - 26 T^{3} + T^{4} \)
$23$ \( -98304 - 121664 T - 14304 T^{2} - 96 T^{3} + T^{4} \)
$29$ \( -802800 - 3696384 T - 52293 T^{2} + 76 T^{3} + T^{4} \)
$31$ \( -165027985 - 16758574 T - 78768 T^{2} + 238 T^{3} + T^{4} \)
$37$ \( -4624450848 + 34102020 T + 16305 T^{2} - 562 T^{3} + T^{4} \)
$41$ \( -3866949120 - 36803808 T - 41132 T^{2} + 428 T^{3} + T^{4} \)
$43$ \( -2654719484 - 46672932 T - 179543 T^{2} + 258 T^{3} + T^{4} \)
$47$ \( 23866588368 + 18481344 T - 336248 T^{2} - 80 T^{3} + T^{4} \)
$53$ \( -5074800 + 3013660 T - 135309 T^{2} + T^{4} \)
$59$ \( -1642011120 + 44672196 T - 276551 T^{2} + 262 T^{3} + T^{4} \)
$61$ \( 9863636400 - 42157232 T - 435696 T^{2} - 276 T^{3} + T^{4} \)
$67$ \( 26529903468 - 24707720 T - 590751 T^{2} - 150 T^{3} + T^{4} \)
$71$ \( 1940742864 - 46164032 T - 78520 T^{2} + 848 T^{3} + T^{4} \)
$73$ \( 280393876 + 5640388 T - 31515 T^{2} - 218 T^{3} + T^{4} \)
$79$ \( -93048620201 + 126613586 T + 743736 T^{2} - 1762 T^{3} + T^{4} \)
$83$ \( 352673538780 + 2140010896 T + 4237149 T^{2} + 3450 T^{3} + T^{4} \)
$89$ \( -45796564224 + 593469600 T - 1149636 T^{2} - 344 T^{3} + T^{4} \)
$97$ \( 79407506004 + 1902403156 T - 2800699 T^{2} - 622 T^{3} + T^{4} \)
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