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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 2 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 2 \zeta_{6} q^{5} - 8 q^{8} + (4 \zeta_{6} - 4) q^{10} + (8 \zeta_{6} - 8) q^{11} - 42 q^{13} - 16 \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + 124 \zeta_{6} q^{19} - 8 q^{20} - 16 q^{22} + 76 \zeta_{6} q^{23} + ( - 121 \zeta_{6} + 121) q^{25} - 84 \zeta_{6} q^{26} - 254 q^{29} + ( - 72 \zeta_{6} + 72) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 4 q^{34} - 398 \zeta_{6} q^{37} + (248 \zeta_{6} - 248) q^{38} - 16 \zeta_{6} q^{40} - 462 q^{41} + 212 q^{43} - 32 \zeta_{6} q^{44} + (152 \zeta_{6} - 152) q^{46} - 264 \zeta_{6} q^{47} + 242 q^{50} + ( - 168 \zeta_{6} + 168) q^{52} + (162 \zeta_{6} - 162) q^{53} - 16 q^{55} - 508 \zeta_{6} q^{58} + (772 \zeta_{6} - 772) q^{59} - 30 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} - 84 \zeta_{6} q^{65} + ( - 764 \zeta_{6} + 764) q^{67} - 8 \zeta_{6} q^{68} + 236 q^{71} + (418 \zeta_{6} - 418) q^{73} + ( - 796 \zeta_{6} + 796) q^{74} - 496 q^{76} - 552 \zeta_{6} q^{79} + ( - 32 \zeta_{6} + 32) q^{80} - 924 \zeta_{6} q^{82} - 1036 q^{83} - 4 q^{85} + 424 \zeta_{6} q^{86} + ( - 64 \zeta_{6} + 64) q^{88} + 30 \zeta_{6} q^{89} - 304 q^{92} + ( - 528 \zeta_{6} + 528) q^{94} + (248 \zeta_{6} - 248) q^{95} - 1190 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 2 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 2 q^{5} - 16 q^{8} - 4 q^{10} - 8 q^{11} - 84 q^{13} - 16 q^{16} - 2 q^{17} + 124 q^{19} - 16 q^{20} - 32 q^{22} + 76 q^{23} + 121 q^{25} - 84 q^{26} - 508 q^{29} + 72 q^{31} + 32 q^{32} - 8 q^{34} - 398 q^{37} - 248 q^{38} - 16 q^{40} - 924 q^{41} + 424 q^{43} - 32 q^{44} - 152 q^{46} - 264 q^{47} + 484 q^{50} + 168 q^{52} - 162 q^{53} - 32 q^{55} - 508 q^{58} - 772 q^{59} - 30 q^{61} + 288 q^{62} + 128 q^{64} - 84 q^{65} + 764 q^{67} - 8 q^{68} + 472 q^{71} - 418 q^{73} + 796 q^{74} - 992 q^{76} - 552 q^{79} + 32 q^{80} - 924 q^{82} - 2072 q^{83} - 8 q^{85} + 424 q^{86} + 64 q^{88} + 30 q^{89} - 608 q^{92} + 528 q^{94} - 248 q^{95} - 2380 q^{97}+O(q^{100}) \) 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 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

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