Properties

Label 882.4.g.b
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: \(\Z[a_1, \ldots, a_{25}]\)
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 8 q^{8} + (28 \zeta_{6} - 28) q^{10} + (28 \zeta_{6} - 28) q^{11} + 18 q^{13} - 16 \zeta_{6} q^{16} + ( - 74 \zeta_{6} + 74) q^{17} - 80 \zeta_{6} q^{19} + 56 q^{20} + 56 q^{22} - 112 \zeta_{6} q^{23} + (71 \zeta_{6} - 71) q^{25} - 36 \zeta_{6} q^{26} - 190 q^{29} + (72 \zeta_{6} - 72) q^{31} + (32 \zeta_{6} - 32) q^{32} - 148 q^{34} + 346 \zeta_{6} q^{37} + (160 \zeta_{6} - 160) q^{38} - 112 \zeta_{6} q^{40} - 162 q^{41} - 412 q^{43} - 112 \zeta_{6} q^{44} + (224 \zeta_{6} - 224) q^{46} + 24 \zeta_{6} q^{47} + 142 q^{50} + (72 \zeta_{6} - 72) q^{52} + ( - 318 \zeta_{6} + 318) q^{53} + 392 q^{55} + 380 \zeta_{6} q^{58} + (200 \zeta_{6} - 200) q^{59} + 198 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} - 252 \zeta_{6} q^{65} + ( - 716 \zeta_{6} + 716) q^{67} + 296 \zeta_{6} q^{68} - 392 q^{71} + (538 \zeta_{6} - 538) q^{73} + ( - 692 \zeta_{6} + 692) q^{74} + 320 q^{76} - 240 \zeta_{6} q^{79} + (224 \zeta_{6} - 224) q^{80} + 324 \zeta_{6} q^{82} + 1072 q^{83} - 1036 q^{85} + 824 \zeta_{6} q^{86} + (224 \zeta_{6} - 224) q^{88} + 810 \zeta_{6} q^{89} + 448 q^{92} + ( - 48 \zeta_{6} + 48) q^{94} + (1120 \zeta_{6} - 1120) q^{95} + 1354 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 14 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 14 q^{5} + 16 q^{8} - 28 q^{10} - 28 q^{11} + 36 q^{13} - 16 q^{16} + 74 q^{17} - 80 q^{19} + 112 q^{20} + 112 q^{22} - 112 q^{23} - 71 q^{25} - 36 q^{26} - 380 q^{29} - 72 q^{31} - 32 q^{32} - 296 q^{34} + 346 q^{37} - 160 q^{38} - 112 q^{40} - 324 q^{41} - 824 q^{43} - 112 q^{44} - 224 q^{46} + 24 q^{47} + 284 q^{50} - 72 q^{52} + 318 q^{53} + 784 q^{55} + 380 q^{58} - 200 q^{59} + 198 q^{61} + 288 q^{62} + 128 q^{64} - 252 q^{65} + 716 q^{67} + 296 q^{68} - 784 q^{71} - 538 q^{73} + 692 q^{74} + 640 q^{76} - 240 q^{79} - 224 q^{80} + 324 q^{82} + 2144 q^{83} - 2072 q^{85} + 824 q^{86} - 224 q^{88} + 810 q^{89} + 896 q^{92} + 48 q^{94} - 1120 q^{95} + 2708 q^{97}+O(q^{100}) \) 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 −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.b 2
3.b odd 2 1 98.4.c.d 2
7.b odd 2 1 882.4.g.k 2
7.c even 3 1 126.4.a.h 1
7.c even 3 1 inner 882.4.g.b 2
7.d odd 6 1 882.4.a.i 1
7.d odd 6 1 882.4.g.k 2
21.c even 2 1 98.4.c.f 2
21.g even 6 1 98.4.a.a 1
21.g even 6 1 98.4.c.f 2
21.h odd 6 1 14.4.a.a 1
21.h odd 6 1 98.4.c.d 2
28.g odd 6 1 1008.4.a.s 1
84.j odd 6 1 784.4.a.s 1
84.n even 6 1 112.4.a.a 1
105.o odd 6 1 350.4.a.l 1
105.p even 6 1 2450.4.a.bo 1
105.x even 12 2 350.4.c.b 2
168.s odd 6 1 448.4.a.b 1
168.v even 6 1 448.4.a.o 1
231.l even 6 1 1694.4.a.g 1
273.w odd 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.h odd 6 1
98.4.a.a 1 21.g even 6 1
98.4.c.d 2 3.b odd 2 1
98.4.c.d 2 21.h odd 6 1
98.4.c.f 2 21.c even 2 1
98.4.c.f 2 21.g even 6 1
112.4.a.a 1 84.n even 6 1
126.4.a.h 1 7.c even 3 1
350.4.a.l 1 105.o odd 6 1
350.4.c.b 2 105.x even 12 2
448.4.a.b 1 168.s odd 6 1
448.4.a.o 1 168.v even 6 1
784.4.a.s 1 84.j odd 6 1
882.4.a.i 1 7.d odd 6 1
882.4.g.b 2 1.a even 1 1 trivial
882.4.g.b 2 7.c even 3 1 inner
882.4.g.k 2 7.b odd 2 1
882.4.g.k 2 7.d odd 6 1
1008.4.a.s 1 28.g odd 6 1
1694.4.a.g 1 231.l even 6 1
2366.4.a.h 1 273.w odd 6 1
2450.4.a.bo 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

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