# Properties

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

# Related objects

## 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: $$\Q(\sqrt{-3}, \sqrt{58})$$ Defining polynomial: $$x^{4} + 58x^{2} + 3364$$ x^4 + 58*x^2 + 3364 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} + 8 q^{8}+O(q^{10})$$ q + (-2*b2 - 2) * q^2 + 4*b2 * q^4 + b1 * q^5 + 8 * q^8 $$q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} + 8 q^{8} + ( - 2 \beta_{3} - 2 \beta_1) q^{10} + 2 \beta_{2} q^{11} - 2 \beta_{3} q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (3 \beta_{3} + 3 \beta_1) q^{17} + 10 \beta_1 q^{19} + 4 \beta_{3} q^{20} + 4 q^{22} + ( - 30 \beta_{2} - 30) q^{23} + 107 \beta_{2} q^{25} - 4 \beta_1 q^{26} + 212 q^{29} + ( - 14 \beta_{3} - 14 \beta_1) q^{31} + 32 \beta_{2} q^{32} - 6 \beta_{3} q^{34} + ( - 246 \beta_{2} - 246) q^{37} + ( - 20 \beta_{3} - 20 \beta_1) q^{38} + 8 \beta_1 q^{40} - 21 \beta_{3} q^{41} - 284 q^{43} + ( - 8 \beta_{2} - 8) q^{44} + 60 \beta_{2} q^{46} + 4 \beta_1 q^{47} + 214 q^{50} + (8 \beta_{3} + 8 \beta_1) q^{52} + 548 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( - 424 \beta_{2} - 424) q^{58} + (44 \beta_{3} + 44 \beta_1) q^{59} - 34 \beta_1 q^{61} + 28 \beta_{3} q^{62} + 64 q^{64} + (464 \beta_{2} + 464) q^{65} + 652 \beta_{2} q^{67} - 12 \beta_1 q^{68} + 770 q^{71} + (64 \beta_{3} + 64 \beta_1) q^{73} + 492 \beta_{2} q^{74} + 40 \beta_{3} q^{76} + ( - 472 \beta_{2} - 472) q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} - 42 \beta_1 q^{82} + 12 \beta_{3} q^{83} - 696 q^{85} + (568 \beta_{2} + 568) q^{86} + 16 \beta_{2} q^{88} + 47 \beta_1 q^{89} + 120 q^{92} + ( - 8 \beta_{3} - 8 \beta_1) q^{94} + 2320 \beta_{2} q^{95} + 20 \beta_{3} q^{97}+O(q^{100})$$ q + (-2*b2 - 2) * q^2 + 4*b2 * q^4 + b1 * q^5 + 8 * q^8 + (-2*b3 - 2*b1) * q^10 + 2*b2 * q^11 - 2*b3 * q^13 + (-16*b2 - 16) * q^16 + (3*b3 + 3*b1) * q^17 + 10*b1 * q^19 + 4*b3 * q^20 + 4 * q^22 + (-30*b2 - 30) * q^23 + 107*b2 * q^25 - 4*b1 * q^26 + 212 * q^29 + (-14*b3 - 14*b1) * q^31 + 32*b2 * q^32 - 6*b3 * q^34 + (-246*b2 - 246) * q^37 + (-20*b3 - 20*b1) * q^38 + 8*b1 * q^40 - 21*b3 * q^41 - 284 * q^43 + (-8*b2 - 8) * q^44 + 60*b2 * q^46 + 4*b1 * q^47 + 214 * q^50 + (8*b3 + 8*b1) * q^52 + 548*b2 * q^53 + 2*b3 * q^55 + (-424*b2 - 424) * q^58 + (44*b3 + 44*b1) * q^59 - 34*b1 * q^61 + 28*b3 * q^62 + 64 * q^64 + (464*b2 + 464) * q^65 + 652*b2 * q^67 - 12*b1 * q^68 + 770 * q^71 + (64*b3 + 64*b1) * q^73 + 492*b2 * q^74 + 40*b3 * q^76 + (-472*b2 - 472) * q^79 + (-16*b3 - 16*b1) * q^80 - 42*b1 * q^82 + 12*b3 * q^83 - 696 * q^85 + (568*b2 + 568) * q^86 + 16*b2 * q^88 + 47*b1 * q^89 + 120 * q^92 + (-8*b3 - 8*b1) * q^94 + 2320*b2 * q^95 + 20*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 $$4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 4 q^{11} - 32 q^{16} + 16 q^{22} - 60 q^{23} - 214 q^{25} + 848 q^{29} - 64 q^{32} - 492 q^{37} - 1136 q^{43} - 16 q^{44} - 120 q^{46} + 856 q^{50} - 1096 q^{53} - 848 q^{58} + 256 q^{64} + 928 q^{65} - 1304 q^{67} + 3080 q^{71} - 984 q^{74} - 944 q^{79} - 2784 q^{85} + 1136 q^{86} - 32 q^{88} + 480 q^{92} - 4640 q^{95}+O(q^{100})$$ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 4 * q^11 - 32 * q^16 + 16 * q^22 - 60 * q^23 - 214 * q^25 + 848 * q^29 - 64 * q^32 - 492 * q^37 - 1136 * q^43 - 16 * q^44 - 120 * q^46 + 856 * q^50 - 1096 * q^53 - 848 * q^58 + 256 * q^64 + 928 * q^65 - 1304 * q^67 + 3080 * q^71 - 984 * q^74 - 944 * q^79 - 2784 * q^85 + 1136 * q^86 - 32 * q^88 + 480 * q^92 - 4640 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 58x^{2} + 3364$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 58$$ (v^2) / 58 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 29$$ (v^3) / 29
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$58\beta_{2}$$ 58*b2 $$\nu^{3}$$ $$=$$ $$29\beta_{3}$$ 29*b3

## 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
 −3.80789 − 6.59545i 3.80789 + 6.59545i −3.80789 + 6.59545i 3.80789 − 6.59545i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −7.61577 13.1909i 0 0 8.00000 0 −15.2315 + 26.3818i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 7.61577 + 13.1909i 0 0 8.00000 0 15.2315 26.3818i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −7.61577 + 13.1909i 0 0 8.00000 0 −15.2315 26.3818i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.61577 13.1909i 0 0 8.00000 0 15.2315 + 26.3818i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.bb 4
3.b odd 2 1 882.4.g.bh 4
7.b odd 2 1 inner 882.4.g.bb 4
7.c even 3 1 882.4.a.bf yes 2
7.c even 3 1 inner 882.4.g.bb 4
7.d odd 6 1 882.4.a.bf yes 2
7.d odd 6 1 inner 882.4.g.bb 4
21.c even 2 1 882.4.g.bh 4
21.g even 6 1 882.4.a.x 2
21.g even 6 1 882.4.g.bh 4
21.h odd 6 1 882.4.a.x 2
21.h odd 6 1 882.4.g.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x 2 21.g even 6 1
882.4.a.x 2 21.h odd 6 1
882.4.a.bf yes 2 7.c even 3 1
882.4.a.bf yes 2 7.d odd 6 1
882.4.g.bb 4 1.a even 1 1 trivial
882.4.g.bb 4 7.b odd 2 1 inner
882.4.g.bb 4 7.c even 3 1 inner
882.4.g.bb 4 7.d odd 6 1 inner
882.4.g.bh 4 3.b odd 2 1
882.4.g.bh 4 21.c even 2 1
882.4.g.bh 4 21.g even 6 1
882.4.g.bh 4 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 232T_{5}^{2} + 53824$$ T5^4 + 232*T5^2 + 53824 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13}^{2} - 928$$ T13^2 - 928

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 232 T^{2} + 53824$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$(T^{2} - 928)^{2}$$
$17$ $$T^{4} + 2088 T^{2} + \cdots + 4359744$$
$19$ $$T^{4} + 23200 T^{2} + \cdots + 538240000$$
$23$ $$(T^{2} + 30 T + 900)^{2}$$
$29$ $$(T - 212)^{4}$$
$31$ $$T^{4} + 45472 T^{2} + \cdots + 2067702784$$
$37$ $$(T^{2} + 246 T + 60516)^{2}$$
$41$ $$(T^{2} - 102312)^{2}$$
$43$ $$(T + 284)^{4}$$
$47$ $$T^{4} + 3712 T^{2} + \cdots + 13778944$$
$53$ $$(T^{2} + 548 T + 300304)^{2}$$
$59$ $$T^{4} + 449152 T^{2} + \cdots + 201737519104$$
$61$ $$T^{4} + 268192 T^{2} + \cdots + 71926948864$$
$67$ $$(T^{2} + 652 T + 425104)^{2}$$
$71$ $$(T - 770)^{4}$$
$73$ $$T^{4} + 950272 T^{2} + \cdots + 903016873984$$
$79$ $$(T^{2} + 472 T + 222784)^{2}$$
$83$ $$(T^{2} - 33408)^{2}$$
$89$ $$T^{4} + 512488 T^{2} + \cdots + 262643950144$$
$97$ $$(T^{2} - 92800)^{2}$$