Properties

Label 882.4.g.f
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: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 8 q^{8} + ( -12 + 12 \zeta_{6} ) q^{10} + ( 12 - 12 \zeta_{6} ) q^{11} -38 q^{13} -16 \zeta_{6} q^{16} + ( 126 - 126 \zeta_{6} ) q^{17} + 20 \zeta_{6} q^{19} + 24 q^{20} -24 q^{22} + 168 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} + 76 \zeta_{6} q^{26} -30 q^{29} + ( -88 + 88 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -252 q^{34} -254 \zeta_{6} q^{37} + ( 40 - 40 \zeta_{6} ) q^{38} -48 \zeta_{6} q^{40} + 42 q^{41} -52 q^{43} + 48 \zeta_{6} q^{44} + ( 336 - 336 \zeta_{6} ) q^{46} + 96 \zeta_{6} q^{47} -178 q^{50} + ( 152 - 152 \zeta_{6} ) q^{52} + ( 198 - 198 \zeta_{6} ) q^{53} -72 q^{55} + 60 \zeta_{6} q^{58} + ( 660 - 660 \zeta_{6} ) q^{59} -538 \zeta_{6} q^{61} + 176 q^{62} + 64 q^{64} + 228 \zeta_{6} q^{65} + ( -884 + 884 \zeta_{6} ) q^{67} + 504 \zeta_{6} q^{68} -792 q^{71} + ( 218 - 218 \zeta_{6} ) q^{73} + ( -508 + 508 \zeta_{6} ) q^{74} -80 q^{76} + 520 \zeta_{6} q^{79} + ( -96 + 96 \zeta_{6} ) q^{80} -84 \zeta_{6} q^{82} -492 q^{83} -756 q^{85} + 104 \zeta_{6} q^{86} + ( 96 - 96 \zeta_{6} ) q^{88} -810 \zeta_{6} q^{89} -672 q^{92} + ( 192 - 192 \zeta_{6} ) q^{94} + ( 120 - 120 \zeta_{6} ) q^{95} -1154 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} - 6q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} - 6q^{5} + 16q^{8} - 12q^{10} + 12q^{11} - 76q^{13} - 16q^{16} + 126q^{17} + 20q^{19} + 48q^{20} - 48q^{22} + 168q^{23} + 89q^{25} + 76q^{26} - 60q^{29} - 88q^{31} - 32q^{32} - 504q^{34} - 254q^{37} + 40q^{38} - 48q^{40} + 84q^{41} - 104q^{43} + 48q^{44} + 336q^{46} + 96q^{47} - 356q^{50} + 152q^{52} + 198q^{53} - 144q^{55} + 60q^{58} + 660q^{59} - 538q^{61} + 352q^{62} + 128q^{64} + 228q^{65} - 884q^{67} + 504q^{68} - 1584q^{71} + 218q^{73} - 508q^{74} - 160q^{76} + 520q^{79} - 96q^{80} - 84q^{82} - 984q^{83} - 1512q^{85} + 104q^{86} + 96q^{88} - 810q^{89} - 1344q^{92} + 192q^{94} + 120q^{95} - 2308q^{97} + O(q^{100}) \)

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 −3.00000 5.19615i 0 0 8.00000 0 −6.00000 + 10.3923i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −3.00000 + 5.19615i 0 0 8.00000 0 −6.00000 10.3923i
\(n\): 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.f 2
3.b odd 2 1 294.4.e.g 2
7.b odd 2 1 882.4.g.i 2
7.c even 3 1 882.4.a.n 1
7.c even 3 1 inner 882.4.g.f 2
7.d odd 6 1 18.4.a.a 1
7.d odd 6 1 882.4.g.i 2
21.c even 2 1 294.4.e.h 2
21.g even 6 1 6.4.a.a 1
21.g even 6 1 294.4.e.h 2
21.h odd 6 1 294.4.a.e 1
21.h odd 6 1 294.4.e.g 2
28.f even 6 1 144.4.a.c 1
35.i odd 6 1 450.4.a.h 1
35.k even 12 2 450.4.c.e 2
56.j odd 6 1 576.4.a.q 1
56.m even 6 1 576.4.a.r 1
63.i even 6 1 162.4.c.f 2
63.k odd 6 1 162.4.c.c 2
63.s even 6 1 162.4.c.f 2
63.t odd 6 1 162.4.c.c 2
77.i even 6 1 2178.4.a.e 1
84.j odd 6 1 48.4.a.c 1
84.n even 6 1 2352.4.a.e 1
105.p even 6 1 150.4.a.i 1
105.w odd 12 2 150.4.c.d 2
168.ba even 6 1 192.4.a.i 1
168.be odd 6 1 192.4.a.c 1
231.k odd 6 1 726.4.a.f 1
273.ba even 6 1 1014.4.a.g 1
273.cb odd 12 2 1014.4.b.d 2
336.bo even 12 2 768.4.d.n 2
336.br odd 12 2 768.4.d.c 2
357.s even 6 1 1734.4.a.d 1
399.s odd 6 1 2166.4.a.i 1
420.be odd 6 1 1200.4.a.b 1
420.br even 12 2 1200.4.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 21.g even 6 1
18.4.a.a 1 7.d odd 6 1
48.4.a.c 1 84.j odd 6 1
144.4.a.c 1 28.f even 6 1
150.4.a.i 1 105.p even 6 1
150.4.c.d 2 105.w odd 12 2
162.4.c.c 2 63.k odd 6 1
162.4.c.c 2 63.t odd 6 1
162.4.c.f 2 63.i even 6 1
162.4.c.f 2 63.s even 6 1
192.4.a.c 1 168.be odd 6 1
192.4.a.i 1 168.ba even 6 1
294.4.a.e 1 21.h odd 6 1
294.4.e.g 2 3.b odd 2 1
294.4.e.g 2 21.h odd 6 1
294.4.e.h 2 21.c even 2 1
294.4.e.h 2 21.g even 6 1
450.4.a.h 1 35.i odd 6 1
450.4.c.e 2 35.k even 12 2
576.4.a.q 1 56.j odd 6 1
576.4.a.r 1 56.m even 6 1
726.4.a.f 1 231.k odd 6 1
768.4.d.c 2 336.br odd 12 2
768.4.d.n 2 336.bo even 12 2
882.4.a.n 1 7.c even 3 1
882.4.g.f 2 1.a even 1 1 trivial
882.4.g.f 2 7.c even 3 1 inner
882.4.g.i 2 7.b odd 2 1
882.4.g.i 2 7.d odd 6 1
1014.4.a.g 1 273.ba even 6 1
1014.4.b.d 2 273.cb odd 12 2
1200.4.a.b 1 420.be odd 6 1
1200.4.f.j 2 420.br even 12 2
1734.4.a.d 1 357.s even 6 1
2166.4.a.i 1 399.s odd 6 1
2178.4.a.e 1 77.i even 6 1
2352.4.a.e 1 84.n even 6 1

Hecke kernels

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

\( T_{5}^{2} + 6 T_{5} + 36 \)
\( T_{11}^{2} - 12 T_{11} + 144 \)
\( T_{13} + 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} \)
$3$ 1
$5$ \( 1 + 6 T - 89 T^{2} + 750 T^{3} + 15625 T^{4} \)
$7$ 1
$11$ \( 1 - 12 T - 1187 T^{2} - 15972 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 + 38 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 126 T + 10963 T^{2} - 619038 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 20 T - 6459 T^{2} - 137180 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 168 T + 16057 T^{2} - 2044056 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 + 30 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 88 T - 22047 T^{2} + 2621608 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 254 T + 13863 T^{2} + 12865862 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 42 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 52 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 96 T - 94607 T^{2} - 9967008 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 198 T - 109673 T^{2} - 29477646 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 538 T + 62463 T^{2} + 122115778 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 792 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 218 T - 341493 T^{2} - 84805706 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 492 T + 571787 T^{2} )^{2} \)
$89$ \( 1 + 810 T - 48869 T^{2} + 571024890 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 1154 T + 912673 T^{2} )^{2} \)
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