Properties

Label 882.4.g.s
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(52.0396846251\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -8 q^{8} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -8 q^{8} + ( -4 + 4 \zeta_{6} ) q^{10} + ( -8 + 8 \zeta_{6} ) q^{11} -42 q^{13} -16 \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + 124 \zeta_{6} q^{19} -8 q^{20} -16 q^{22} + 76 \zeta_{6} q^{23} + ( 121 - 121 \zeta_{6} ) q^{25} -84 \zeta_{6} q^{26} -254 q^{29} + ( 72 - 72 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -4 q^{34} -398 \zeta_{6} q^{37} + ( -248 + 248 \zeta_{6} ) q^{38} -16 \zeta_{6} q^{40} -462 q^{41} + 212 q^{43} -32 \zeta_{6} q^{44} + ( -152 + 152 \zeta_{6} ) q^{46} -264 \zeta_{6} q^{47} + 242 q^{50} + ( 168 - 168 \zeta_{6} ) q^{52} + ( -162 + 162 \zeta_{6} ) q^{53} -16 q^{55} -508 \zeta_{6} q^{58} + ( -772 + 772 \zeta_{6} ) q^{59} -30 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} -84 \zeta_{6} q^{65} + ( 764 - 764 \zeta_{6} ) q^{67} -8 \zeta_{6} q^{68} + 236 q^{71} + ( -418 + 418 \zeta_{6} ) q^{73} + ( 796 - 796 \zeta_{6} ) q^{74} -496 q^{76} -552 \zeta_{6} q^{79} + ( 32 - 32 \zeta_{6} ) q^{80} -924 \zeta_{6} q^{82} -1036 q^{83} -4 q^{85} + 424 \zeta_{6} q^{86} + ( 64 - 64 \zeta_{6} ) q^{88} + 30 \zeta_{6} q^{89} -304 q^{92} + ( 528 - 528 \zeta_{6} ) q^{94} + ( -248 + 248 \zeta_{6} ) q^{95} -1190 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} + 2q^{5} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} + 2q^{5} - 16q^{8} - 4q^{10} - 8q^{11} - 84q^{13} - 16q^{16} - 2q^{17} + 124q^{19} - 16q^{20} - 32q^{22} + 76q^{23} + 121q^{25} - 84q^{26} - 508q^{29} + 72q^{31} + 32q^{32} - 8q^{34} - 398q^{37} - 248q^{38} - 16q^{40} - 924q^{41} + 424q^{43} - 32q^{44} - 152q^{46} - 264q^{47} + 484q^{50} + 168q^{52} - 162q^{53} - 32q^{55} - 508q^{58} - 772q^{59} - 30q^{61} + 288q^{62} + 128q^{64} - 84q^{65} + 764q^{67} - 8q^{68} + 472q^{71} - 418q^{73} + 796q^{74} - 992q^{76} - 552q^{79} + 32q^{80} - 924q^{82} - 2072q^{83} - 8q^{85} + 424q^{86} + 64q^{88} + 30q^{89} - 608q^{92} + 528q^{94} - 248q^{95} - 2380q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 1.00000 + 1.73205i 0 0 −8.00000 0 −2.00000 + 3.46410i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 1.00000 1.73205i 0 0 −8.00000 0 −2.00000 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.s 2
3.b odd 2 1 294.4.e.a 2
7.b odd 2 1 882.4.g.r 2
7.c even 3 1 126.4.a.c 1
7.c even 3 1 inner 882.4.g.s 2
7.d odd 6 1 882.4.a.d 1
7.d odd 6 1 882.4.g.r 2
21.c even 2 1 294.4.e.d 2
21.g even 6 1 294.4.a.h 1
21.g even 6 1 294.4.e.d 2
21.h odd 6 1 42.4.a.b 1
21.h odd 6 1 294.4.e.a 2
28.g odd 6 1 1008.4.a.j 1
84.j odd 6 1 2352.4.a.ba 1
84.n even 6 1 336.4.a.d 1
105.o odd 6 1 1050.4.a.d 1
105.x even 12 2 1050.4.g.n 2
168.s odd 6 1 1344.4.a.f 1
168.v even 6 1 1344.4.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 21.h odd 6 1
126.4.a.c 1 7.c even 3 1
294.4.a.h 1 21.g even 6 1
294.4.e.a 2 3.b odd 2 1
294.4.e.a 2 21.h odd 6 1
294.4.e.d 2 21.c even 2 1
294.4.e.d 2 21.g even 6 1
336.4.a.d 1 84.n even 6 1
882.4.a.d 1 7.d odd 6 1
882.4.g.r 2 7.b odd 2 1
882.4.g.r 2 7.d odd 6 1
882.4.g.s 2 1.a even 1 1 trivial
882.4.g.s 2 7.c even 3 1 inner
1008.4.a.j 1 28.g odd 6 1
1050.4.a.d 1 105.o odd 6 1
1050.4.g.n 2 105.x even 12 2
1344.4.a.f 1 168.s odd 6 1
1344.4.a.t 1 168.v even 6 1
2352.4.a.ba 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} + 8 T_{11} + 64 \)
\( T_{13} + 42 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 64 + 8 T + T^{2} \)
$13$ \( ( 42 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 15376 - 124 T + T^{2} \)
$23$ \( 5776 - 76 T + T^{2} \)
$29$ \( ( 254 + T )^{2} \)
$31$ \( 5184 - 72 T + T^{2} \)
$37$ \( 158404 + 398 T + T^{2} \)
$41$ \( ( 462 + T )^{2} \)
$43$ \( ( -212 + T )^{2} \)
$47$ \( 69696 + 264 T + T^{2} \)
$53$ \( 26244 + 162 T + T^{2} \)
$59$ \( 595984 + 772 T + T^{2} \)
$61$ \( 900 + 30 T + T^{2} \)
$67$ \( 583696 - 764 T + T^{2} \)
$71$ \( ( -236 + T )^{2} \)
$73$ \( 174724 + 418 T + T^{2} \)
$79$ \( 304704 + 552 T + T^{2} \)
$83$ \( ( 1036 + T )^{2} \)
$89$ \( 900 - 30 T + T^{2} \)
$97$ \( ( 1190 + T )^{2} \)
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