Properties

Label 882.3.s.h
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.621801639936.1
Defining polynomial: \(x^{8} - 4 x^{7} - 34 x^{6} + 116 x^{5} + 413 x^{4} - 1024 x^{3} - 1664 x^{2} + 2196 x + 4467\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -\beta_{6} - \beta_{7} ) q^{10} + 2 \beta_{5} q^{11} -\beta_{6} q^{13} + ( -4 + 4 \beta_{1} ) q^{16} + 3 \beta_{3} q^{17} + 2 \beta_{7} q^{19} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{20} + 4 q^{22} -30 \beta_{4} q^{23} + 49 \beta_{1} q^{25} -2 \beta_{2} q^{26} + ( -11 \beta_{4} - 11 \beta_{5} ) q^{29} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{31} -4 \beta_{5} q^{32} + 3 \beta_{6} q^{34} + ( 6 - 6 \beta_{1} ) q^{37} -4 \beta_{3} q^{38} -2 \beta_{7} q^{40} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{41} -68 q^{43} -4 \beta_{4} q^{44} + 60 \beta_{1} q^{46} + 8 \beta_{2} q^{47} + ( -49 \beta_{4} - 49 \beta_{5} ) q^{50} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{52} + 29 \beta_{5} q^{53} -2 \beta_{6} q^{55} + ( -22 + 22 \beta_{1} ) q^{58} -8 \beta_{3} q^{59} -8 \beta_{7} q^{61} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{62} -8 q^{64} -74 \beta_{4} q^{65} + 104 \beta_{1} q^{67} + 6 \beta_{2} q^{68} + ( -50 \beta_{4} - 50 \beta_{5} ) q^{71} + ( 5 \beta_{6} + 5 \beta_{7} ) q^{73} + 6 \beta_{5} q^{74} -4 \beta_{6} q^{76} + ( 20 - 20 \beta_{1} ) q^{79} + 4 \beta_{3} q^{80} + 3 \beta_{7} q^{82} -222 q^{85} + 68 \beta_{4} q^{86} + 8 \beta_{1} q^{88} -11 \beta_{2} q^{89} + ( -60 \beta_{4} - 60 \beta_{5} ) q^{92} + ( 8 \beta_{6} + 8 \beta_{7} ) q^{94} + 148 \beta_{5} q^{95} + 13 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} - 16q^{16} + 32q^{22} + 196q^{25} + 24q^{37} - 544q^{43} + 240q^{46} - 88q^{58} - 64q^{64} + 416q^{67} + 80q^{79} - 1776q^{85} + 32q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 34 x^{6} + 116 x^{5} + 413 x^{4} - 1024 x^{3} - 1664 x^{2} + 2196 x + 4467\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} + 6 \nu^{5} + 91 \nu^{4} - 192 \nu^{3} - 1187 \nu^{2} + 1284 \nu + 4425 \)\()/2418\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{6} - 6 \nu^{5} - 91 \nu^{4} + 192 \nu^{3} + 2396 \nu^{2} - 2493 \nu - 15306 \)\()/1209\)
\(\beta_{3}\)\(=\)\((\)\( -35 \nu^{6} + 105 \nu^{5} + 988 \nu^{4} - 2151 \nu^{3} - 8078 \nu^{2} + 9171 \nu + 4293 \)\()/2418\)
\(\beta_{4}\)\(=\)\((\)\( -724 \nu^{7} + 2534 \nu^{6} + 27020 \nu^{5} - 73885 \nu^{4} - 387688 \nu^{3} + 656684 \nu^{2} + 1520571 \nu - 872256 \)\()/1374633\)
\(\beta_{5}\)\(=\)\((\)\( -1838 \nu^{7} + 6433 \nu^{6} + 45811 \nu^{5} - 130610 \nu^{4} - 262721 \nu^{3} + 527908 \nu^{2} - 1269957 \nu + 542487 \)\()/2749266\)
\(\beta_{6}\)\(=\)\((\)\( -2896 \nu^{7} + 10136 \nu^{6} + 108080 \nu^{5} - 295540 \nu^{4} - 1550752 \nu^{3} + 2626736 \nu^{2} + 11580816 \nu - 6238290 \)\()/1374633\)
\(\beta_{7}\)\(=\)\((\)\( -8224 \nu^{7} + 28784 \nu^{6} + 291734 \nu^{5} - 801295 \nu^{4} - 3006376 \nu^{3} + 5325251 \nu^{2} + 564096 \nu - 1196985 \)\()/1374633\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - 4 \beta_{4} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 4 \beta_{4} + 4 \beta_{2} + 8 \beta_{1} + 38\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 61 \beta_{4} + 3 \beta_{2} + 6 \beta_{1} + 28\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{7} + 31 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} - 16 \beta_{3} + 96 \beta_{2} + 472 \beta_{1} + 382\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(220 \beta_{7} + 309 \beta_{6} - 776 \beta_{5} - 2750 \beta_{4} - 40 \beta_{3} + 230 \beta_{2} + 1160 \beta_{1} + 862\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(315 \beta_{7} + 425 \beta_{6} - 1144 \beta_{5} - 3826 \beta_{4} - 424 \beta_{3} + 1054 \beta_{2} + 7110 \beta_{1} + 1086\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(5978 \beta_{7} + 5783 \beta_{6} - 31052 \beta_{5} - 59216 \beta_{4} - 2828 \beta_{3} + 6580 \beta_{2} + 45724 \beta_{1} + 4650\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−3.76613 + 0.707107i
2.31664 + 0.707107i
4.76613 0.707107i
−1.31664 0.707107i
−3.76613 0.707107i
2.31664 0.707107i
4.76613 + 0.707107i
−1.31664 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 10.5357i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 7.44983 4.30116i 0 0 2.82843i 0 −6.08276 + 10.5357i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −7.44983 + 4.30116i 0 0 2.82843i 0 −6.08276 + 10.5357i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 7.44983 4.30116i 0 0 2.82843i 0 6.08276 10.5357i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −7.44983 4.30116i 0 0 2.82843i 0 6.08276 + 10.5357i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 2.82843i 0 −6.08276 10.5357i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.44983 4.30116i 0 0 2.82843i 0 −6.08276 10.5357i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 + 10.5357i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.h 8
3.b odd 2 1 inner 882.3.s.h 8
7.b odd 2 1 inner 882.3.s.h 8
7.c even 3 1 882.3.b.g 4
7.c even 3 1 inner 882.3.s.h 8
7.d odd 6 1 882.3.b.g 4
7.d odd 6 1 inner 882.3.s.h 8
21.c even 2 1 inner 882.3.s.h 8
21.g even 6 1 882.3.b.g 4
21.g even 6 1 inner 882.3.s.h 8
21.h odd 6 1 882.3.b.g 4
21.h odd 6 1 inner 882.3.s.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.g 4 7.c even 3 1
882.3.b.g 4 7.d odd 6 1
882.3.b.g 4 21.g even 6 1
882.3.b.g 4 21.h odd 6 1
882.3.s.h 8 1.a even 1 1 trivial
882.3.s.h 8 3.b odd 2 1 inner
882.3.s.h 8 7.b odd 2 1 inner
882.3.s.h 8 7.c even 3 1 inner
882.3.s.h 8 7.d odd 6 1 inner
882.3.s.h 8 21.c even 2 1 inner
882.3.s.h 8 21.g even 6 1 inner
882.3.s.h 8 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 74 T_{5}^{2} + 5476 \)
\( T_{11}^{4} - 8 T_{11}^{2} + 64 \)
\( T_{13}^{2} - 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 5476 - 74 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$13$ \( ( -148 + T^{2} )^{4} \)
$17$ \( ( 443556 - 666 T^{2} + T^{4} )^{2} \)
$19$ \( ( 350464 + 592 T^{2} + T^{4} )^{2} \)
$23$ \( ( 3240000 - 1800 T^{2} + T^{4} )^{2} \)
$29$ \( ( 242 + T^{2} )^{4} \)
$31$ \( ( 350464 + 592 T^{2} + T^{4} )^{2} \)
$37$ \( ( 36 - 6 T + T^{2} )^{4} \)
$41$ \( ( 666 + T^{2} )^{4} \)
$43$ \( ( 68 + T )^{8} \)
$47$ \( ( 22429696 - 4736 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2829124 - 1682 T^{2} + T^{4} )^{2} \)
$59$ \( ( 22429696 - 4736 T^{2} + T^{4} )^{2} \)
$61$ \( ( 89718784 + 9472 T^{2} + T^{4} )^{2} \)
$67$ \( ( 10816 - 104 T + T^{2} )^{4} \)
$71$ \( ( 5000 + T^{2} )^{4} \)
$73$ \( ( 13690000 + 3700 T^{2} + T^{4} )^{2} \)
$79$ \( ( 400 - 20 T + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( ( 80174116 - 8954 T^{2} + T^{4} )^{2} \)
$97$ \( ( -25012 + T^{2} )^{4} \)
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