Properties

Label 882.4.g.d
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 7 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 7 \zeta_{6} q^{5} + 8 q^{8} + (14 \zeta_{6} - 14) q^{10} + ( - 35 \zeta_{6} + 35) q^{11} - 66 q^{13} - 16 \zeta_{6} q^{16} + (59 \zeta_{6} - 59) q^{17} + 137 \zeta_{6} q^{19} + 28 q^{20} - 70 q^{22} - 7 \zeta_{6} q^{23} + ( - 76 \zeta_{6} + 76) q^{25} + 132 \zeta_{6} q^{26} - 106 q^{29} + ( - 75 \zeta_{6} + 75) q^{31} + (32 \zeta_{6} - 32) q^{32} + 118 q^{34} - 11 \zeta_{6} q^{37} + ( - 274 \zeta_{6} + 274) q^{38} - 56 \zeta_{6} q^{40} - 498 q^{41} + 260 q^{43} + 140 \zeta_{6} q^{44} + (14 \zeta_{6} - 14) q^{46} + 171 \zeta_{6} q^{47} - 152 q^{50} + ( - 264 \zeta_{6} + 264) q^{52} + (417 \zeta_{6} - 417) q^{53} - 245 q^{55} + 212 \zeta_{6} q^{58} + ( - 17 \zeta_{6} + 17) q^{59} + 51 \zeta_{6} q^{61} - 150 q^{62} + 64 q^{64} + 462 \zeta_{6} q^{65} + (439 \zeta_{6} - 439) q^{67} - 236 \zeta_{6} q^{68} + 784 q^{71} + ( - 295 \zeta_{6} + 295) q^{73} + (22 \zeta_{6} - 22) q^{74} - 548 q^{76} + 495 \zeta_{6} q^{79} + (112 \zeta_{6} - 112) q^{80} + 996 \zeta_{6} q^{82} + 932 q^{83} + 413 q^{85} - 520 \zeta_{6} q^{86} + ( - 280 \zeta_{6} + 280) q^{88} + 873 \zeta_{6} q^{89} + 28 q^{92} + ( - 342 \zeta_{6} + 342) q^{94} + ( - 959 \zeta_{6} + 959) q^{95} + 290 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8} - 14 q^{10} + 35 q^{11} - 132 q^{13} - 16 q^{16} - 59 q^{17} + 137 q^{19} + 56 q^{20} - 140 q^{22} - 7 q^{23} + 76 q^{25} + 132 q^{26} - 212 q^{29} + 75 q^{31} - 32 q^{32} + 236 q^{34} - 11 q^{37} + 274 q^{38} - 56 q^{40} - 996 q^{41} + 520 q^{43} + 140 q^{44} - 14 q^{46} + 171 q^{47} - 304 q^{50} + 264 q^{52} - 417 q^{53} - 490 q^{55} + 212 q^{58} + 17 q^{59} + 51 q^{61} - 300 q^{62} + 128 q^{64} + 462 q^{65} - 439 q^{67} - 236 q^{68} + 1568 q^{71} + 295 q^{73} - 22 q^{74} - 1096 q^{76} + 495 q^{79} - 112 q^{80} + 996 q^{82} + 1864 q^{83} + 826 q^{85} - 520 q^{86} + 280 q^{88} + 873 q^{89} + 56 q^{92} + 342 q^{94} + 959 q^{95} + 580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −3.50000 6.06218i 0 0 8.00000 0 −7.00000 + 12.1244i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −3.50000 + 6.06218i 0 0 8.00000 0 −7.00000 12.1244i
\(n\): 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.d 2
3.b odd 2 1 98.4.c.e 2
7.b odd 2 1 126.4.g.c 2
7.c even 3 1 882.4.a.p 1
7.c even 3 1 inner 882.4.g.d 2
7.d odd 6 1 126.4.g.c 2
7.d odd 6 1 882.4.a.k 1
21.c even 2 1 14.4.c.b 2
21.g even 6 1 14.4.c.b 2
21.g even 6 1 98.4.a.b 1
21.h odd 6 1 98.4.a.c 1
21.h odd 6 1 98.4.c.e 2
84.h odd 2 1 112.4.i.b 2
84.j odd 6 1 112.4.i.b 2
84.j odd 6 1 784.4.a.l 1
84.n even 6 1 784.4.a.j 1
105.g even 2 1 350.4.e.b 2
105.k odd 4 2 350.4.j.d 4
105.o odd 6 1 2450.4.a.bf 1
105.p even 6 1 350.4.e.b 2
105.p even 6 1 2450.4.a.bh 1
105.w odd 12 2 350.4.j.d 4
168.e odd 2 1 448.4.i.d 2
168.i even 2 1 448.4.i.c 2
168.ba even 6 1 448.4.i.c 2
168.be odd 6 1 448.4.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 21.c even 2 1
14.4.c.b 2 21.g even 6 1
98.4.a.b 1 21.g even 6 1
98.4.a.c 1 21.h odd 6 1
98.4.c.e 2 3.b odd 2 1
98.4.c.e 2 21.h odd 6 1
112.4.i.b 2 84.h odd 2 1
112.4.i.b 2 84.j odd 6 1
126.4.g.c 2 7.b odd 2 1
126.4.g.c 2 7.d odd 6 1
350.4.e.b 2 105.g even 2 1
350.4.e.b 2 105.p even 6 1
350.4.j.d 4 105.k odd 4 2
350.4.j.d 4 105.w odd 12 2
448.4.i.c 2 168.i even 2 1
448.4.i.c 2 168.ba even 6 1
448.4.i.d 2 168.e odd 2 1
448.4.i.d 2 168.be odd 6 1
784.4.a.j 1 84.n even 6 1
784.4.a.l 1 84.j odd 6 1
882.4.a.k 1 7.d odd 6 1
882.4.a.p 1 7.c even 3 1
882.4.g.d 2 1.a even 1 1 trivial
882.4.g.d 2 7.c even 3 1 inner
2450.4.a.bf 1 105.o odd 6 1
2450.4.a.bh 1 105.p even 6 1

Hecke kernels

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

\( T_{5}^{2} + 7T_{5} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} - 35T_{11} + 1225 \) Copy content Toggle raw display
\( T_{13} + 66 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T + 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} + 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} + 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T - 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T - 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T - 290)^{2} \) Copy content Toggle raw display
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