# Properties

 Label 588.3.m.f Level $588$ Weight $3$ Character orbit 588.m Analytic conductor $16.022$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.m (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0218395444$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4$$ x^8 - 4*x^6 + 14*x^4 - 8*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} - 3 \beta_{4} q^{9}+O(q^{10})$$ q + (-b4 + 1) * q^3 + (b6 + b5 + 2*b2) * q^5 - 3*b4 * q^9 $$q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} - 3 \beta_{4} q^{9} + ( - 2 \beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{7} + \beta_{6} + 8 \beta_{5} + \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{6} - \beta_{3} + 3 \beta_{2}) q^{15} + ( - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 4) q^{17} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + 4 \beta_{2} - 16) q^{19} + ( - \beta_{7} + 2 \beta_{6} + 9 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{23} + ( - 4 \beta_{7} - \beta_{5} - 9 \beta_{4} - \beta_{2} - 2 \beta_1 - 9) q^{25} + ( - 6 \beta_{4} - 3) q^{27} + (\beta_{7} + 6 \beta_{6} - 6 \beta_{3} + 5 \beta_{2} - \beta_1 + 10) q^{29} + ( - 14 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 14 \beta_{2} - 3 \beta_1 + 4) q^{31} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{2}) q^{33} + ( - 6 \beta_{7} + 4 \beta_{6} - 9 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + \cdots - 12 \beta_1) q^{37}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b4 + 1) * q^3 + (b6 + b5 + 2*b2) * q^5 - 3*b4 * q^9 + (-2*b7 + b5 + b2 - b1) * q^11 + (-2*b7 + b6 + 8*b5 + b3 + 4*b2 - 2*b1) * q^13 + (b6 - b3 + 3*b2) * q^15 + (-3*b5 - 4*b4 + 3*b3 + 3*b2 + 4) * q^17 + (b7 + 2*b6 + 2*b5 - 8*b4 + 4*b2 - 16) * q^19 + (-b7 + 2*b6 + 9*b5 - 2*b4 + 4*b3 - 2*b1) * q^23 + (-4*b7 - b5 - 9*b4 - b2 - 2*b1 - 9) * q^25 + (-6*b4 - 3) * q^27 + (b7 + 6*b6 - 6*b3 + 5*b2 - b1 + 10) * q^29 + (-14*b5 - 4*b4 - 4*b3 + 14*b2 - 3*b1 + 4) * q^31 + (-3*b7 + b5 + 2*b2) * q^33 + (-6*b7 + 4*b6 - 9*b5 + 16*b4 + 8*b3 - 12*b1) * q^37 + (-4*b7 + 2*b6 + 12*b5 + b3 + 12*b2 - 2*b1) * q^39 + (5*b7 - 7*b6 + 26*b5 - 28*b4 - 7*b3 + 13*b2 + 5*b1 - 14) * q^41 + (2*b7 + 8*b6 - 8*b3 - 8*b2 - 2*b1 - 14) * q^43 + (-3*b5 - 3*b3 + 3*b2) * q^45 + (7*b7 - 8*b6 + 2*b5 - 22*b4 + 4*b2 - 44) * q^47 + (3*b6 - 9*b5 - 12*b4 + 6*b3) * q^51 + (-12*b7 + 8*b6 - 26*b5 + 18*b4 + 4*b3 - 26*b2 - 6*b1 + 18) * q^53 + (-b7 + 6*b6 + 20*b5 + 6*b3 + 10*b2 - b1) * q^55 + (b7 + 2*b6 - 2*b3 + 6*b2 - b1 - 24) * q^57 + (-22*b5 - 14*b4 + 12*b3 + 22*b2 + 5*b1 + 14) * q^59 + (-8*b7 - 5*b6 + 14*b5 - 12*b4 + 28*b2 - 24) * q^61 + (-5*b7 + 4*b6 + 19*b5 + 30*b4 + 8*b3 - 10*b1) * q^65 + (-12*b7 - 8*b6 + 12*b5 + 8*b4 - 4*b3 + 12*b2 - 6*b1 + 8) * q^67 + (-3*b7 + 6*b6 + 18*b5 - 4*b4 + 6*b3 + 9*b2 - 3*b1 - 2) * q^69 + (-b7 + 8*b6 - 8*b3 - 43*b2 + b1 + 28) * q^71 + (-6*b5 - 28*b4 - 15*b3 + 6*b2 + 8*b1 + 28) * q^73 + (-6*b7 - b5 - 9*b4 - 2*b2 - 18) * q^75 + (4*b7 - 28*b5 - 54*b4 + 8*b1) * q^79 + (-9*b4 - 9) * q^81 + (14*b7 - 6*b6 + 12*b5 + 32*b4 - 6*b3 + 6*b2 + 14*b1 + 16) * q^83 + (4*b6 - 4*b3 + 15*b2 - 12) * q^85 + (-5*b5 - 10*b4 - 18*b3 + 5*b2 - 3*b1 + 10) * q^87 + (16*b7 + 11*b6 + 23*b5 + 8*b4 + 46*b2 + 16) * q^89 + (-3*b7 - 4*b6 - 42*b5 - 12*b4 - 8*b3 - 6*b1) * q^93 + (-8*b7 - 20*b6 - 36*b5 + 34*b4 - 10*b3 - 36*b2 - 4*b1 + 34) * q^95 + (-4*b7 - 3*b6 + 20*b5 - 56*b4 - 3*b3 + 10*b2 - 4*b1 - 28) * q^97 + (-3*b7 + 3*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{3} + 12 q^{9}+O(q^{10})$$ 8 * q + 12 * q^3 + 12 * q^9 $$8 q + 12 q^{3} + 12 q^{9} + 48 q^{17} - 96 q^{19} + 8 q^{23} - 36 q^{25} + 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} + 48 q^{51} + 72 q^{53} - 192 q^{57} + 168 q^{59} - 144 q^{61} - 120 q^{65} + 32 q^{67} + 224 q^{71} + 336 q^{73} - 108 q^{75} + 216 q^{79} - 36 q^{81} - 96 q^{85} + 120 q^{87} + 96 q^{89} + 48 q^{93} + 136 q^{95}+O(q^{100})$$ 8 * q + 12 * q^3 + 12 * q^9 + 48 * q^17 - 96 * q^19 + 8 * q^23 - 36 * q^25 + 80 * q^29 + 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 + 48 * q^51 + 72 * q^53 - 192 * q^57 + 168 * q^59 - 144 * q^61 - 120 * q^65 + 32 * q^67 + 224 * q^71 + 336 * q^73 - 108 * q^75 + 216 * q^79 - 36 * q^81 - 96 * q^85 + 120 * q^87 + 96 * q^89 + 48 * q^93 + 136 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 76\nu ) / 14$$ (v^7 + 76*v) / 14 $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 20 ) / 14$$ (v^6 + 20) / 14 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 27\nu ) / 7$$ (-v^7 - 27*v) / 7 $$\beta_{4}$$ $$=$$ $$( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14$$ (-2*v^6 + 7*v^4 - 28*v^2 + 2) / 14 $$\beta_{5}$$ $$=$$ $$( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7$$ (-2*v^6 + 7*v^4 - 21*v^2 + 2) / 7 $$\beta_{6}$$ $$=$$ $$( -8\nu^{7} + 35\nu^{5} - 112\nu^{3} + 64\nu ) / 14$$ (-8*v^7 + 35*v^5 - 112*v^3 + 64*v) / 14 $$\beta_{7}$$ $$=$$ $$( 11\nu^{7} - 42\nu^{5} + 154\nu^{3} - 88\nu ) / 14$$ (11*v^7 - 42*v^5 + 154*v^3 - 88*v) / 14
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 7$$ (b3 + 2*b1) / 7 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2\beta_{4}$$ b5 - 2*b4 $$\nu^{3}$$ $$=$$ $$( 5\beta_{7} + 6\beta_{6} + 6\beta_{3} + 5\beta_1 ) / 7$$ (5*b7 + 6*b6 + 6*b3 + 5*b1) / 7 $$\nu^{4}$$ $$=$$ $$4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6$$ 4*b5 - 6*b4 + 4*b2 - 6 $$\nu^{5}$$ $$=$$ $$( 16\beta_{7} + 22\beta_{6} ) / 7$$ (16*b7 + 22*b6) / 7 $$\nu^{6}$$ $$=$$ $$14\beta_{2} - 20$$ 14*b2 - 20 $$\nu^{7}$$ $$=$$ $$( -76\beta_{3} - 54\beta_1 ) / 7$$ (-76*b3 - 54*b1) / 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
313.1
 1.60021 − 0.923880i −0.662827 + 0.382683i −1.60021 + 0.923880i 0.662827 − 0.382683i 1.60021 + 0.923880i −0.662827 − 0.382683i −1.60021 − 0.923880i 0.662827 + 0.382683i
0 1.50000 0.866025i 0 −5.04718 2.91399i 0 0 0 1.50000 2.59808i 0
313.2 0 1.50000 0.866025i 0 −0.416265 0.240331i 0 0 0 1.50000 2.59808i 0
313.3 0 1.50000 0.866025i 0 0.804540 + 0.464502i 0 0 0 1.50000 2.59808i 0
313.4 0 1.50000 0.866025i 0 4.65891 + 2.68982i 0 0 0 1.50000 2.59808i 0
325.1 0 1.50000 + 0.866025i 0 −5.04718 + 2.91399i 0 0 0 1.50000 + 2.59808i 0
325.2 0 1.50000 + 0.866025i 0 −0.416265 + 0.240331i 0 0 0 1.50000 + 2.59808i 0
325.3 0 1.50000 + 0.866025i 0 0.804540 0.464502i 0 0 0 1.50000 + 2.59808i 0
325.4 0 1.50000 + 0.866025i 0 4.65891 2.68982i 0 0 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 325.4 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.m.f 8
3.b odd 2 1 1764.3.z.l 8
7.b odd 2 1 588.3.m.e 8
7.c even 3 1 588.3.d.c 8
7.c even 3 1 588.3.m.e 8
7.d odd 6 1 588.3.d.c 8
7.d odd 6 1 inner 588.3.m.f 8
21.c even 2 1 1764.3.z.m 8
21.g even 6 1 1764.3.d.h 8
21.g even 6 1 1764.3.z.l 8
21.h odd 6 1 1764.3.d.h 8
21.h odd 6 1 1764.3.z.m 8
28.f even 6 1 2352.3.f.j 8
28.g odd 6 1 2352.3.f.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 7.c even 3 1
588.3.d.c 8 7.d odd 6 1
588.3.m.e 8 7.b odd 2 1
588.3.m.e 8 7.c even 3 1
588.3.m.f 8 1.a even 1 1 trivial
588.3.m.f 8 7.d odd 6 1 inner
1764.3.d.h 8 21.g even 6 1
1764.3.d.h 8 21.h odd 6 1
1764.3.z.l 8 3.b odd 2 1
1764.3.z.l 8 21.g even 6 1
1764.3.z.m 8 21.c even 2 1
1764.3.z.m 8 21.h odd 6 1
2352.3.f.j 8 28.f even 6 1
2352.3.f.j 8 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 32T_{5}^{6} + 1010T_{5}^{4} - 768T_{5}^{3} - 256T_{5}^{2} + 336T_{5} + 196$$ acting on $$S_{3}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 3 T + 3)^{4}$$
$5$ $$T^{8} - 32 T^{6} + 1010 T^{4} + \cdots + 196$$
$7$ $$T^{8}$$
$11$ $$T^{8} + 124 T^{6} + \cdots + 10837264$$
$13$ $$T^{8} + 712 T^{6} + 123284 T^{4} + \cdots + 2979076$$
$17$ $$T^{8} - 48 T^{7} + \cdots + 168428484$$
$19$ $$T^{8} + 96 T^{7} + \cdots + 285745216$$
$23$ $$T^{8} - 8 T^{7} + \cdots + 1443088144$$
$29$ $$(T^{4} - 40 T^{3} - 1924 T^{2} + \cdots + 468892)^{2}$$
$31$ $$T^{8} - 48 T^{7} + \cdots + 2052452416$$
$37$ $$T^{8} + 64 T^{7} + \cdots + 298373767696$$
$41$ $$T^{8} + 9808 T^{6} + \cdots + 26697826996036$$
$43$ $$(T^{4} + 56 T^{3} - 3784 T^{2} + \cdots + 192784)^{2}$$
$47$ $$T^{8} + 264 T^{7} + \cdots + 8881401308224$$
$53$ $$T^{8} - 72 T^{7} + \cdots + 71641191948544$$
$59$ $$T^{8} + \cdots + 372481662446656$$
$61$ $$T^{8} + 144 T^{7} + 3772 T^{6} + \cdots + 454276$$
$67$ $$T^{8} - 32 T^{7} + \cdots + 306756468736$$
$71$ $$(T^{4} - 112 T^{3} - 6460 T^{2} + \cdots - 7722596)^{2}$$
$73$ $$T^{8} - 336 T^{7} + \cdots + 3712242251524$$
$79$ $$T^{8} - 216 T^{7} + \cdots + 49705658450176$$
$83$ $$T^{8} + 19712 T^{6} + \cdots + 61585579131904$$
$89$ $$T^{8} - 96 T^{7} + \cdots + 17\!\cdots\!76$$
$97$ $$T^{8} + 13640 T^{6} + \cdots + 5315948141956$$