Properties

Label 882.4.g.k
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
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: \(0\)
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 14)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 14 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 14 \zeta_{6} q^{5} + 8 q^{8} + ( 28 - 28 \zeta_{6} ) q^{10} + ( -28 + 28 \zeta_{6} ) q^{11} -18 q^{13} -16 \zeta_{6} q^{16} + ( -74 + 74 \zeta_{6} ) q^{17} + 80 \zeta_{6} q^{19} -56 q^{20} + 56 q^{22} -112 \zeta_{6} q^{23} + ( -71 + 71 \zeta_{6} ) q^{25} + 36 \zeta_{6} q^{26} -190 q^{29} + ( 72 - 72 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 148 q^{34} + 346 \zeta_{6} q^{37} + ( 160 - 160 \zeta_{6} ) q^{38} + 112 \zeta_{6} q^{40} + 162 q^{41} -412 q^{43} -112 \zeta_{6} q^{44} + ( -224 + 224 \zeta_{6} ) q^{46} -24 \zeta_{6} q^{47} + 142 q^{50} + ( 72 - 72 \zeta_{6} ) q^{52} + ( 318 - 318 \zeta_{6} ) q^{53} -392 q^{55} + 380 \zeta_{6} q^{58} + ( 200 - 200 \zeta_{6} ) q^{59} -198 \zeta_{6} q^{61} -144 q^{62} + 64 q^{64} -252 \zeta_{6} q^{65} + ( 716 - 716 \zeta_{6} ) q^{67} -296 \zeta_{6} q^{68} -392 q^{71} + ( 538 - 538 \zeta_{6} ) q^{73} + ( 692 - 692 \zeta_{6} ) q^{74} -320 q^{76} -240 \zeta_{6} q^{79} + ( 224 - 224 \zeta_{6} ) q^{80} -324 \zeta_{6} q^{82} -1072 q^{83} -1036 q^{85} + 824 \zeta_{6} q^{86} + ( -224 + 224 \zeta_{6} ) q^{88} -810 \zeta_{6} q^{89} + 448 q^{92} + ( -48 + 48 \zeta_{6} ) q^{94} + ( -1120 + 1120 \zeta_{6} ) q^{95} -1354 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} + 28q^{10} - 28q^{11} - 36q^{13} - 16q^{16} - 74q^{17} + 80q^{19} - 112q^{20} + 112q^{22} - 112q^{23} - 71q^{25} + 36q^{26} - 380q^{29} + 72q^{31} - 32q^{32} + 296q^{34} + 346q^{37} + 160q^{38} + 112q^{40} + 324q^{41} - 824q^{43} - 112q^{44} - 224q^{46} - 24q^{47} + 284q^{50} + 72q^{52} + 318q^{53} - 784q^{55} + 380q^{58} + 200q^{59} - 198q^{61} - 288q^{62} + 128q^{64} - 252q^{65} + 716q^{67} - 296q^{68} - 784q^{71} + 538q^{73} + 692q^{74} - 640q^{76} - 240q^{79} + 224q^{80} - 324q^{82} - 2144q^{83} - 2072q^{85} + 824q^{86} - 224q^{88} - 810q^{89} + 896q^{92} - 48q^{94} - 1120q^{95} - 2708q^{97} + O(q^{100}) \)

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 7.00000 + 12.1244i 0 0 8.00000 0 14.0000 24.2487i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.00000 12.1244i 0 0 8.00000 0 14.0000 + 24.2487i
\(n\): 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.k 2
3.b odd 2 1 98.4.c.f 2
7.b odd 2 1 882.4.g.b 2
7.c even 3 1 882.4.a.i 1
7.c even 3 1 inner 882.4.g.k 2
7.d odd 6 1 126.4.a.h 1
7.d odd 6 1 882.4.g.b 2
21.c even 2 1 98.4.c.d 2
21.g even 6 1 14.4.a.a 1
21.g even 6 1 98.4.c.d 2
21.h odd 6 1 98.4.a.a 1
21.h odd 6 1 98.4.c.f 2
28.f even 6 1 1008.4.a.s 1
84.j odd 6 1 112.4.a.a 1
84.n even 6 1 784.4.a.s 1
105.o odd 6 1 2450.4.a.bo 1
105.p even 6 1 350.4.a.l 1
105.w odd 12 2 350.4.c.b 2
168.ba even 6 1 448.4.a.b 1
168.be odd 6 1 448.4.a.o 1
231.k odd 6 1 1694.4.a.g 1
273.ba even 6 1 2366.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 21.g even 6 1
98.4.a.a 1 21.h odd 6 1
98.4.c.d 2 21.c even 2 1
98.4.c.d 2 21.g even 6 1
98.4.c.f 2 3.b odd 2 1
98.4.c.f 2 21.h odd 6 1
112.4.a.a 1 84.j odd 6 1
126.4.a.h 1 7.d odd 6 1
350.4.a.l 1 105.p even 6 1
350.4.c.b 2 105.w odd 12 2
448.4.a.b 1 168.ba even 6 1
448.4.a.o 1 168.be odd 6 1
784.4.a.s 1 84.n even 6 1
882.4.a.i 1 7.c even 3 1
882.4.g.b 2 7.b odd 2 1
882.4.g.b 2 7.d odd 6 1
882.4.g.k 2 1.a even 1 1 trivial
882.4.g.k 2 7.c even 3 1 inner
1008.4.a.s 1 28.f even 6 1
1694.4.a.g 1 231.k odd 6 1
2366.4.a.h 1 273.ba even 6 1
2450.4.a.bo 1 105.o odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 14 T_{5} + 196 \)
\( T_{11}^{2} + 28 T_{11} + 784 \)
\( T_{13} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 196 - 14 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 784 + 28 T + T^{2} \)
$13$ \( ( 18 + T )^{2} \)
$17$ \( 5476 + 74 T + T^{2} \)
$19$ \( 6400 - 80 T + T^{2} \)
$23$ \( 12544 + 112 T + T^{2} \)
$29$ \( ( 190 + T )^{2} \)
$31$ \( 5184 - 72 T + T^{2} \)
$37$ \( 119716 - 346 T + T^{2} \)
$41$ \( ( -162 + T )^{2} \)
$43$ \( ( 412 + T )^{2} \)
$47$ \( 576 + 24 T + T^{2} \)
$53$ \( 101124 - 318 T + T^{2} \)
$59$ \( 40000 - 200 T + T^{2} \)
$61$ \( 39204 + 198 T + T^{2} \)
$67$ \( 512656 - 716 T + T^{2} \)
$71$ \( ( 392 + T )^{2} \)
$73$ \( 289444 - 538 T + T^{2} \)
$79$ \( 57600 + 240 T + T^{2} \)
$83$ \( ( 1072 + T )^{2} \)
$89$ \( 656100 + 810 T + T^{2} \)
$97$ \( ( 1354 + T )^{2} \)
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