# Properties

 Label 567.2.f.i Level $567$ Weight $2$ Character orbit 567.f Analytic conductor $4.528$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + 3 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 5*b2 * q^4 + (b3 + b1) * q^5 + (b2 + 1) * q^7 + 3*b3 * q^8 $$q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + 3 \beta_{3} q^{8} - 7 q^{10} - \beta_1 q^{11} - 2 \beta_{2} q^{13} + (\beta_{3} + \beta_1) q^{14} + ( - 11 \beta_{2} - 11) q^{16} + 7 q^{19} - 5 \beta_1 q^{20} - 7 \beta_{2} q^{22} + (3 \beta_{3} + 3 \beta_1) q^{23} + ( - 2 \beta_{2} - 2) q^{25} - 2 \beta_{3} q^{26} - 5 q^{28} + 2 \beta_1 q^{29} + 3 \beta_{2} q^{31} + ( - 5 \beta_{3} - 5 \beta_1) q^{32} + \beta_{3} q^{35} - 3 q^{37} + 7 \beta_1 q^{38} - 21 \beta_{2} q^{40} + ( - \beta_{3} - \beta_1) q^{41} + ( - 8 \beta_{2} - 8) q^{43} - 5 \beta_{3} q^{44} - 21 q^{46} + \beta_{2} q^{49} + ( - 2 \beta_{3} - 2 \beta_1) q^{50} + (10 \beta_{2} + 10) q^{52} + 7 q^{55} - 3 \beta_1 q^{56} + 14 \beta_{2} q^{58} + (8 \beta_{2} + 8) q^{61} + 3 \beta_{3} q^{62} + 13 q^{64} + 2 \beta_1 q^{65} - 2 \beta_{2} q^{67} + ( - 7 \beta_{2} - 7) q^{70} + 3 \beta_{3} q^{71} - 3 \beta_1 q^{74} + 35 \beta_{2} q^{76} + ( - \beta_{3} - \beta_1) q^{77} + (4 \beta_{2} + 4) q^{79} - 11 \beta_{3} q^{80} + 7 q^{82} + 6 \beta_1 q^{83} + ( - 8 \beta_{3} - 8 \beta_1) q^{86} + (21 \beta_{2} + 21) q^{88} - 7 \beta_{3} q^{89} + 2 q^{91} - 15 \beta_1 q^{92} + (7 \beta_{3} + 7 \beta_1) q^{95} + (12 \beta_{2} + 12) q^{97} + \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 5*b2 * q^4 + (b3 + b1) * q^5 + (b2 + 1) * q^7 + 3*b3 * q^8 - 7 * q^10 - b1 * q^11 - 2*b2 * q^13 + (b3 + b1) * q^14 + (-11*b2 - 11) * q^16 + 7 * q^19 - 5*b1 * q^20 - 7*b2 * q^22 + (3*b3 + 3*b1) * q^23 + (-2*b2 - 2) * q^25 - 2*b3 * q^26 - 5 * q^28 + 2*b1 * q^29 + 3*b2 * q^31 + (-5*b3 - 5*b1) * q^32 + b3 * q^35 - 3 * q^37 + 7*b1 * q^38 - 21*b2 * q^40 + (-b3 - b1) * q^41 + (-8*b2 - 8) * q^43 - 5*b3 * q^44 - 21 * q^46 + b2 * q^49 + (-2*b3 - 2*b1) * q^50 + (10*b2 + 10) * q^52 + 7 * q^55 - 3*b1 * q^56 + 14*b2 * q^58 + (8*b2 + 8) * q^61 + 3*b3 * q^62 + 13 * q^64 + 2*b1 * q^65 - 2*b2 * q^67 + (-7*b2 - 7) * q^70 + 3*b3 * q^71 - 3*b1 * q^74 + 35*b2 * q^76 + (-b3 - b1) * q^77 + (4*b2 + 4) * q^79 - 11*b3 * q^80 + 7 * q^82 + 6*b1 * q^83 + (-8*b3 - 8*b1) * q^86 + (21*b2 + 21) * q^88 - 7*b3 * q^89 + 2 * q^91 - 15*b1 * q^92 + (7*b3 + 7*b1) * q^95 + (12*b2 + 12) * q^97 + b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4} + 2 q^{7}+O(q^{10})$$ 4 * q - 10 * q^4 + 2 * q^7 $$4 q - 10 q^{4} + 2 q^{7} - 28 q^{10} + 4 q^{13} - 22 q^{16} + 28 q^{19} + 14 q^{22} - 4 q^{25} - 20 q^{28} - 6 q^{31} - 12 q^{37} + 42 q^{40} - 16 q^{43} - 84 q^{46} - 2 q^{49} + 20 q^{52} + 28 q^{55} - 28 q^{58} + 16 q^{61} + 52 q^{64} + 4 q^{67} - 14 q^{70} - 70 q^{76} + 8 q^{79} + 28 q^{82} + 42 q^{88} + 8 q^{91} + 24 q^{97}+O(q^{100})$$ 4 * q - 10 * q^4 + 2 * q^7 - 28 * q^10 + 4 * q^13 - 22 * q^16 + 28 * q^19 + 14 * q^22 - 4 * q^25 - 20 * q^28 - 6 * q^31 - 12 * q^37 + 42 * q^40 - 16 * q^43 - 84 * q^46 - 2 * q^49 + 20 * q^52 + 28 * q^55 - 28 * q^58 + 16 * q^61 + 52 * q^64 + 4 * q^67 - 14 * q^70 - 70 * q^76 + 8 * q^79 + 28 * q^82 + 42 * q^88 + 8 * q^91 + 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
190.1
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
−1.32288 2.29129i 0 −2.50000 + 4.33013i 1.32288 2.29129i 0 0.500000 + 0.866025i 7.93725 0 −7.00000
190.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i −1.32288 + 2.29129i 0 0.500000 + 0.866025i −7.93725 0 −7.00000
379.1 −1.32288 + 2.29129i 0 −2.50000 4.33013i 1.32288 + 2.29129i 0 0.500000 0.866025i 7.93725 0 −7.00000
379.2 1.32288 2.29129i 0 −2.50000 4.33013i −1.32288 2.29129i 0 0.500000 0.866025i −7.93725 0 −7.00000
 $$n$$: 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.f.i 4
3.b odd 2 1 inner 567.2.f.i 4
9.c even 3 1 189.2.a.f 2
9.c even 3 1 inner 567.2.f.i 4
9.d odd 6 1 189.2.a.f 2
9.d odd 6 1 inner 567.2.f.i 4
36.f odd 6 1 3024.2.a.bi 2
36.h even 6 1 3024.2.a.bi 2
45.h odd 6 1 4725.2.a.bb 2
45.j even 6 1 4725.2.a.bb 2
63.l odd 6 1 1323.2.a.w 2
63.o even 6 1 1323.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 9.c even 3 1
189.2.a.f 2 9.d odd 6 1
567.2.f.i 4 1.a even 1 1 trivial
567.2.f.i 4 3.b odd 2 1 inner
567.2.f.i 4 9.c even 3 1 inner
567.2.f.i 4 9.d odd 6 1 inner
1323.2.a.w 2 63.l odd 6 1
1323.2.a.w 2 63.o even 6 1
3024.2.a.bi 2 36.f odd 6 1
3024.2.a.bi 2 36.h even 6 1
4725.2.a.bb 2 45.h odd 6 1
4725.2.a.bb 2 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2}^{4} + 7T_{2}^{2} + 49$$ T2^4 + 7*T2^2 + 49 $$T_{5}^{4} + 7T_{5}^{2} + 49$$ T5^4 + 7*T5^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 49$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 7T^{2} + 49$$
$7$ $$(T^{2} - T + 1)^{2}$$
$11$ $$T^{4} + 7T^{2} + 49$$
$13$ $$(T^{2} - 2 T + 4)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T - 7)^{4}$$
$23$ $$T^{4} + 63T^{2} + 3969$$
$29$ $$T^{4} + 28T^{2} + 784$$
$31$ $$(T^{2} + 3 T + 9)^{2}$$
$37$ $$(T + 3)^{4}$$
$41$ $$T^{4} + 7T^{2} + 49$$
$43$ $$(T^{2} + 8 T + 64)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 8 T + 64)^{2}$$
$67$ $$(T^{2} - 2 T + 4)^{2}$$
$71$ $$(T^{2} - 63)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 4 T + 16)^{2}$$
$83$ $$T^{4} + 252 T^{2} + 63504$$
$89$ $$(T^{2} - 343)^{2}$$
$97$ $$(T^{2} - 12 T + 144)^{2}$$