Properties

Label 882.4.g.ba
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_1 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_1 q^{5} + 8 q^{8} + ( - 28 \beta_{3} - 28 \beta_1) q^{10} + 14 \beta_{2} q^{11} + 36 \beta_{3} q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (\beta_{3} + \beta_1) q^{17} - \beta_1 q^{19} + 56 \beta_{3} q^{20} + 28 q^{22} + (140 \beta_{2} + 140) q^{23} + 267 \beta_{2} q^{25} + 72 \beta_1 q^{26} + 286 q^{29} + (66 \beta_{3} + 66 \beta_1) q^{31} + 32 \beta_{2} q^{32} - 2 \beta_{3} q^{34} + (38 \beta_{2} + 38) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + 112 \beta_1 q^{40} + 89 \beta_{3} q^{41} - 34 q^{43} + ( - 56 \beta_{2} - 56) q^{44} - 280 \beta_{2} q^{46} - 370 \beta_1 q^{47} + 534 q^{50} + ( - 144 \beta_{3} - 144 \beta_1) q^{52} + 74 \beta_{2} q^{53} + 196 \beta_{3} q^{55} + ( - 572 \beta_{2} - 572) q^{58} + (307 \beta_{3} + 307 \beta_1) q^{59} + 10 \beta_1 q^{61} - 132 \beta_{3} q^{62} + 64 q^{64} + ( - 1008 \beta_{2} - 1008) q^{65} + 684 \beta_{2} q^{67} - 4 \beta_1 q^{68} - 588 q^{71} + (191 \beta_{3} + 191 \beta_1) q^{73} - 76 \beta_{2} q^{74} - 4 \beta_{3} q^{76} + ( - 1220 \beta_{2} - 1220) q^{79} + ( - 224 \beta_{3} - 224 \beta_1) q^{80} + 178 \beta_1 q^{82} - 299 \beta_{3} q^{83} - 28 q^{85} + (68 \beta_{2} + 68) q^{86} + 112 \beta_{2} q^{88} - 437 \beta_1 q^{89} - 560 q^{92} + (740 \beta_{3} + 740 \beta_1) q^{94} - 28 \beta_{2} q^{95} + 1049 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 28 q^{11} - 32 q^{16} + 112 q^{22} + 280 q^{23} - 534 q^{25} + 1144 q^{29} - 64 q^{32} + 76 q^{37} - 136 q^{43} - 112 q^{44} + 560 q^{46} + 2136 q^{50} - 148 q^{53} - 1144 q^{58} + 256 q^{64} - 2016 q^{65} - 1368 q^{67} - 2352 q^{71} + 152 q^{74} - 2440 q^{79} - 112 q^{85} + 136 q^{86} - 224 q^{88} - 2240 q^{92} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −9.89949 17.1464i 0 0 8.00000 0 −19.7990 + 34.2929i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 9.89949 + 17.1464i 0 0 8.00000 0 19.7990 34.2929i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −9.89949 + 17.1464i 0 0 8.00000 0 −19.7990 34.2929i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 9.89949 17.1464i 0 0 8.00000 0 19.7990 + 34.2929i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.ba 4
3.b odd 2 1 98.4.c.h 4
7.b odd 2 1 inner 882.4.g.ba 4
7.c even 3 1 882.4.a.bg 2
7.c even 3 1 inner 882.4.g.ba 4
7.d odd 6 1 882.4.a.bg 2
7.d odd 6 1 inner 882.4.g.ba 4
21.c even 2 1 98.4.c.h 4
21.g even 6 1 98.4.a.g 2
21.g even 6 1 98.4.c.h 4
21.h odd 6 1 98.4.a.g 2
21.h odd 6 1 98.4.c.h 4
84.j odd 6 1 784.4.a.y 2
84.n even 6 1 784.4.a.y 2
105.o odd 6 1 2450.4.a.bx 2
105.p even 6 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 21.g even 6 1
98.4.a.g 2 21.h odd 6 1
98.4.c.h 4 3.b odd 2 1
98.4.c.h 4 21.c even 2 1
98.4.c.h 4 21.g even 6 1
98.4.c.h 4 21.h odd 6 1
784.4.a.y 2 84.j odd 6 1
784.4.a.y 2 84.n even 6 1
882.4.a.bg 2 7.c even 3 1
882.4.a.bg 2 7.d odd 6 1
882.4.g.ba 4 1.a even 1 1 trivial
882.4.g.ba 4 7.b odd 2 1 inner
882.4.g.ba 4 7.c even 3 1 inner
882.4.g.ba 4 7.d odd 6 1 inner
2450.4.a.bx 2 105.o odd 6 1
2450.4.a.bx 2 105.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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 392T_{5}^{2} + 153664 \) Copy content Toggle raw display
\( T_{11}^{2} + 14T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{2} - 2592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 392 T^{2} + 153664 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2592)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T^{2} - 140 T + 19600)^{2} \) Copy content Toggle raw display
$29$ \( (T - 286)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 8712 T^{2} + \cdots + 75898944 \) Copy content Toggle raw display
$37$ \( (T^{2} - 38 T + 1444)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15842)^{2} \) Copy content Toggle raw display
$43$ \( (T + 34)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 273800 T^{2} + \cdots + 74966440000 \) Copy content Toggle raw display
$53$ \( (T^{2} + 74 T + 5476)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 188498 T^{2} + \cdots + 35531496004 \) Copy content Toggle raw display
$61$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$67$ \( (T^{2} + 684 T + 467856)^{2} \) Copy content Toggle raw display
$71$ \( (T + 588)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 72962 T^{2} + \cdots + 5323453444 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1220 T + 1488400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 178802)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 381938 T^{2} + \cdots + 145876635844 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2200802)^{2} \) Copy content Toggle raw display
show more
show less