Properties

Label 882.3.s.b
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} -2 \beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} -2 \beta_{3} q^{8} -6 \beta_{2} q^{10} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{11} + 8 q^{13} + ( -4 + 4 \beta_{2} ) q^{16} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{17} + ( 16 - 16 \beta_{2} ) q^{19} + 6 \beta_{3} q^{20} -24 q^{22} + 12 \beta_{1} q^{23} -7 \beta_{2} q^{25} -8 \beta_{1} q^{26} + 3 \beta_{3} q^{29} -44 \beta_{2} q^{31} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{32} + 18 q^{34} + ( 34 - 34 \beta_{2} ) q^{37} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{38} + ( 12 - 12 \beta_{2} ) q^{40} + 33 \beta_{3} q^{41} -40 q^{43} + 24 \beta_{1} q^{44} -24 \beta_{2} q^{46} + 60 \beta_{1} q^{47} + 7 \beta_{3} q^{50} + 16 \beta_{2} q^{52} + ( 27 \beta_{1} - 27 \beta_{3} ) q^{53} + 72 q^{55} + ( 6 - 6 \beta_{2} ) q^{58} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{59} + ( -50 + 50 \beta_{2} ) q^{61} + 44 \beta_{3} q^{62} -8 q^{64} + 24 \beta_{1} q^{65} -8 \beta_{2} q^{67} -18 \beta_{1} q^{68} -36 \beta_{3} q^{71} + 16 \beta_{2} q^{73} + ( -34 \beta_{1} + 34 \beta_{3} ) q^{74} + 32 q^{76} + ( 76 - 76 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{80} + ( 66 - 66 \beta_{2} ) q^{82} + 84 \beta_{3} q^{83} -54 q^{85} + 40 \beta_{1} q^{86} -48 \beta_{2} q^{88} -9 \beta_{1} q^{89} + 24 \beta_{3} q^{92} -120 \beta_{2} q^{94} + ( 48 \beta_{1} - 48 \beta_{3} ) q^{95} + 176 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 12q^{10} + 32q^{13} - 8q^{16} + 32q^{19} - 96q^{22} - 14q^{25} - 88q^{31} + 72q^{34} + 68q^{37} + 24q^{40} - 160q^{43} - 48q^{46} + 32q^{52} + 288q^{55} + 12q^{58} - 100q^{61} - 32q^{64} - 16q^{67} + 32q^{73} + 128q^{76} + 152q^{79} + 132q^{82} - 216q^{85} - 96q^{88} - 240q^{94} + 704q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\)

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
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 3.67423 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
557.2 1.22474 0.707107i 0 1.00000 1.73205i −3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
863.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.67423 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 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.b 4
3.b odd 2 1 inner 882.3.s.b 4
7.b odd 2 1 882.3.s.d 4
7.c even 3 1 18.3.b.a 2
7.c even 3 1 inner 882.3.s.b 4
7.d odd 6 1 882.3.b.a 2
7.d odd 6 1 882.3.s.d 4
21.c even 2 1 882.3.s.d 4
21.g even 6 1 882.3.b.a 2
21.g even 6 1 882.3.s.d 4
21.h odd 6 1 18.3.b.a 2
21.h odd 6 1 inner 882.3.s.b 4
28.g odd 6 1 144.3.e.b 2
35.j even 6 1 450.3.d.f 2
35.l odd 12 2 450.3.b.b 4
56.k odd 6 1 576.3.e.f 2
56.p even 6 1 576.3.e.c 2
63.g even 3 1 162.3.d.b 4
63.h even 3 1 162.3.d.b 4
63.j odd 6 1 162.3.d.b 4
63.n odd 6 1 162.3.d.b 4
77.h odd 6 1 2178.3.c.d 2
84.n even 6 1 144.3.e.b 2
91.r even 6 1 3042.3.c.e 2
91.z odd 12 2 3042.3.d.a 4
105.o odd 6 1 450.3.d.f 2
105.x even 12 2 450.3.b.b 4
112.u odd 12 2 2304.3.h.c 4
112.w even 12 2 2304.3.h.f 4
140.p odd 6 1 3600.3.l.d 2
140.w even 12 2 3600.3.c.b 4
168.s odd 6 1 576.3.e.c 2
168.v even 6 1 576.3.e.f 2
231.l even 6 1 2178.3.c.d 2
252.o even 6 1 1296.3.q.f 4
252.u odd 6 1 1296.3.q.f 4
252.bb even 6 1 1296.3.q.f 4
252.bl odd 6 1 1296.3.q.f 4
273.w odd 6 1 3042.3.c.e 2
273.cd even 12 2 3042.3.d.a 4
336.bt odd 12 2 2304.3.h.f 4
336.bu even 12 2 2304.3.h.c 4
420.ba even 6 1 3600.3.l.d 2
420.bp odd 12 2 3600.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 7.c even 3 1
18.3.b.a 2 21.h odd 6 1
144.3.e.b 2 28.g odd 6 1
144.3.e.b 2 84.n even 6 1
162.3.d.b 4 63.g even 3 1
162.3.d.b 4 63.h even 3 1
162.3.d.b 4 63.j odd 6 1
162.3.d.b 4 63.n odd 6 1
450.3.b.b 4 35.l odd 12 2
450.3.b.b 4 105.x even 12 2
450.3.d.f 2 35.j even 6 1
450.3.d.f 2 105.o odd 6 1
576.3.e.c 2 56.p even 6 1
576.3.e.c 2 168.s odd 6 1
576.3.e.f 2 56.k odd 6 1
576.3.e.f 2 168.v even 6 1
882.3.b.a 2 7.d odd 6 1
882.3.b.a 2 21.g even 6 1
882.3.s.b 4 1.a even 1 1 trivial
882.3.s.b 4 3.b odd 2 1 inner
882.3.s.b 4 7.c even 3 1 inner
882.3.s.b 4 21.h odd 6 1 inner
882.3.s.d 4 7.b odd 2 1
882.3.s.d 4 7.d odd 6 1
882.3.s.d 4 21.c even 2 1
882.3.s.d 4 21.g even 6 1
1296.3.q.f 4 252.o even 6 1
1296.3.q.f 4 252.u odd 6 1
1296.3.q.f 4 252.bb even 6 1
1296.3.q.f 4 252.bl odd 6 1
2178.3.c.d 2 77.h odd 6 1
2178.3.c.d 2 231.l even 6 1
2304.3.h.c 4 112.u odd 12 2
2304.3.h.c 4 336.bu even 12 2
2304.3.h.f 4 112.w even 12 2
2304.3.h.f 4 336.bt odd 12 2
3042.3.c.e 2 91.r even 6 1
3042.3.c.e 2 273.w odd 6 1
3042.3.d.a 4 91.z odd 12 2
3042.3.d.a 4 273.cd even 12 2
3600.3.c.b 4 140.w even 12 2
3600.3.c.b 4 420.bp odd 12 2
3600.3.l.d 2 140.p odd 6 1
3600.3.l.d 2 420.ba even 6 1

Hecke kernels

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

\( T_{5}^{4} - 18 T_{5}^{2} + 324 \)
\( T_{11}^{4} - 288 T_{11}^{2} + 82944 \)
\( T_{13} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 - 18 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 82944 - 288 T^{2} + T^{4} \)
$13$ \( ( -8 + T )^{4} \)
$17$ \( 26244 - 162 T^{2} + T^{4} \)
$19$ \( ( 256 - 16 T + T^{2} )^{2} \)
$23$ \( 82944 - 288 T^{2} + T^{4} \)
$29$ \( ( 18 + T^{2} )^{2} \)
$31$ \( ( 1936 + 44 T + T^{2} )^{2} \)
$37$ \( ( 1156 - 34 T + T^{2} )^{2} \)
$41$ \( ( 2178 + T^{2} )^{2} \)
$43$ \( ( 40 + T )^{4} \)
$47$ \( 51840000 - 7200 T^{2} + T^{4} \)
$53$ \( 2125764 - 1458 T^{2} + T^{4} \)
$59$ \( 1327104 - 1152 T^{2} + T^{4} \)
$61$ \( ( 2500 + 50 T + T^{2} )^{2} \)
$67$ \( ( 64 + 8 T + T^{2} )^{2} \)
$71$ \( ( 2592 + T^{2} )^{2} \)
$73$ \( ( 256 - 16 T + T^{2} )^{2} \)
$79$ \( ( 5776 - 76 T + T^{2} )^{2} \)
$83$ \( ( 14112 + T^{2} )^{2} \)
$89$ \( 26244 - 162 T^{2} + T^{4} \)
$97$ \( ( -176 + T )^{4} \)
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