Properties

Label 882.4.g.u
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 9 \zeta_{6} q^{5} -8 q^{8} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 9 \zeta_{6} q^{5} -8 q^{8} + ( -18 + 18 \zeta_{6} ) q^{10} + ( -57 + 57 \zeta_{6} ) q^{11} + 70 q^{13} -16 \zeta_{6} q^{16} + ( -51 + 51 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} -36 q^{20} -114 q^{22} + 69 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} + 140 \zeta_{6} q^{26} -114 q^{29} + ( 23 - 23 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -102 q^{34} + 253 \zeta_{6} q^{37} + ( -10 + 10 \zeta_{6} ) q^{38} -72 \zeta_{6} q^{40} -42 q^{41} -124 q^{43} -228 \zeta_{6} q^{44} + ( -138 + 138 \zeta_{6} ) q^{46} -201 \zeta_{6} q^{47} + 88 q^{50} + ( -280 + 280 \zeta_{6} ) q^{52} + ( -393 + 393 \zeta_{6} ) q^{53} -513 q^{55} -228 \zeta_{6} q^{58} + ( -219 + 219 \zeta_{6} ) q^{59} -709 \zeta_{6} q^{61} + 46 q^{62} + 64 q^{64} + 630 \zeta_{6} q^{65} + ( -419 + 419 \zeta_{6} ) q^{67} -204 \zeta_{6} q^{68} + 96 q^{71} + ( -313 + 313 \zeta_{6} ) q^{73} + ( -506 + 506 \zeta_{6} ) q^{74} -20 q^{76} -461 \zeta_{6} q^{79} + ( 144 - 144 \zeta_{6} ) q^{80} -84 \zeta_{6} q^{82} -588 q^{83} -459 q^{85} -248 \zeta_{6} q^{86} + ( 456 - 456 \zeta_{6} ) q^{88} + 1017 \zeta_{6} q^{89} -276 q^{92} + ( 402 - 402 \zeta_{6} ) q^{94} + ( -45 + 45 \zeta_{6} ) q^{95} + 1834 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} + 9q^{5} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} + 9q^{5} - 16q^{8} - 18q^{10} - 57q^{11} + 140q^{13} - 16q^{16} - 51q^{17} + 5q^{19} - 72q^{20} - 228q^{22} + 69q^{23} + 44q^{25} + 140q^{26} - 228q^{29} + 23q^{31} + 32q^{32} - 204q^{34} + 253q^{37} - 10q^{38} - 72q^{40} - 84q^{41} - 248q^{43} - 228q^{44} - 138q^{46} - 201q^{47} + 176q^{50} - 280q^{52} - 393q^{53} - 1026q^{55} - 228q^{58} - 219q^{59} - 709q^{61} + 92q^{62} + 128q^{64} + 630q^{65} - 419q^{67} - 204q^{68} + 192q^{71} - 313q^{73} - 506q^{74} - 40q^{76} - 461q^{79} + 144q^{80} - 84q^{82} - 1176q^{83} - 918q^{85} - 248q^{86} + 456q^{88} + 1017q^{89} - 552q^{92} + 402q^{94} - 45q^{95} + 3668q^{97} + O(q^{100}) \)

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 4.50000 + 7.79423i 0 0 −8.00000 0 −9.00000 + 15.5885i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 4.50000 7.79423i 0 0 −8.00000 0 −9.00000 15.5885i
\(n\): 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.u 2
3.b odd 2 1 98.4.c.a 2
7.b odd 2 1 126.4.g.d 2
7.c even 3 1 882.4.a.c 1
7.c even 3 1 inner 882.4.g.u 2
7.d odd 6 1 126.4.g.d 2
7.d odd 6 1 882.4.a.f 1
21.c even 2 1 14.4.c.a 2
21.g even 6 1 14.4.c.a 2
21.g even 6 1 98.4.a.d 1
21.h odd 6 1 98.4.a.f 1
21.h odd 6 1 98.4.c.a 2
84.h odd 2 1 112.4.i.a 2
84.j odd 6 1 112.4.i.a 2
84.j odd 6 1 784.4.a.p 1
84.n even 6 1 784.4.a.c 1
105.g even 2 1 350.4.e.e 2
105.k odd 4 2 350.4.j.b 4
105.o odd 6 1 2450.4.a.d 1
105.p even 6 1 350.4.e.e 2
105.p even 6 1 2450.4.a.q 1
105.w odd 12 2 350.4.j.b 4
168.e odd 2 1 448.4.i.e 2
168.i even 2 1 448.4.i.b 2
168.ba even 6 1 448.4.i.b 2
168.be odd 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.c even 2 1
14.4.c.a 2 21.g even 6 1
98.4.a.d 1 21.g even 6 1
98.4.a.f 1 21.h odd 6 1
98.4.c.a 2 3.b odd 2 1
98.4.c.a 2 21.h odd 6 1
112.4.i.a 2 84.h odd 2 1
112.4.i.a 2 84.j odd 6 1
126.4.g.d 2 7.b odd 2 1
126.4.g.d 2 7.d odd 6 1
350.4.e.e 2 105.g even 2 1
350.4.e.e 2 105.p even 6 1
350.4.j.b 4 105.k odd 4 2
350.4.j.b 4 105.w odd 12 2
448.4.i.b 2 168.i even 2 1
448.4.i.b 2 168.ba even 6 1
448.4.i.e 2 168.e odd 2 1
448.4.i.e 2 168.be odd 6 1
784.4.a.c 1 84.n even 6 1
784.4.a.p 1 84.j odd 6 1
882.4.a.c 1 7.c even 3 1
882.4.a.f 1 7.d odd 6 1
882.4.g.u 2 1.a even 1 1 trivial
882.4.g.u 2 7.c even 3 1 inner
2450.4.a.d 1 105.o odd 6 1
2450.4.a.q 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} - 9 T_{5} + 81 \)
\( T_{11}^{2} + 57 T_{11} + 3249 \)
\( T_{13} - 70 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ \( \)
$5$ \( 1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4} \)
$7$ \( \)
$11$ \( 1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 70 T + 2197 T^{2} )^{2} \)
$17$ \( 1 + 51 T - 2312 T^{2} + 250563 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 5 T - 6834 T^{2} - 34295 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 69 T - 7406 T^{2} - 839523 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 + 114 T + 24389 T^{2} )^{2} \)
$31$ \( 1 - 23 T - 29262 T^{2} - 685193 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 253 T + 13356 T^{2} - 12815209 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 42 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 124 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 201 T - 63422 T^{2} + 20868423 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 393 T + 5572 T^{2} + 58508661 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 219 T - 157418 T^{2} + 44978001 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 709 T + 275700 T^{2} + 160929529 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 419 T - 125202 T^{2} + 126019697 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 96 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 313 T - 291048 T^{2} + 121762321 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 588 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 1017 T + 329320 T^{2} - 716953473 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 1834 T + 912673 T^{2} )^{2} \)
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