# Properties

 Label 588.4.i.f Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -6 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -6 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} + ( -36 + 36 \zeta_{6} ) q^{11} + 62 q^{13} -18 q^{15} + ( -114 + 114 \zeta_{6} ) q^{17} + 76 \zeta_{6} q^{19} + 24 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} -27 q^{27} + 54 q^{29} + ( 112 - 112 \zeta_{6} ) q^{31} + 108 \zeta_{6} q^{33} + 178 \zeta_{6} q^{37} + ( 186 - 186 \zeta_{6} ) q^{39} + 378 q^{41} -172 q^{43} + ( -54 + 54 \zeta_{6} ) q^{45} + 192 \zeta_{6} q^{47} + 342 \zeta_{6} q^{51} + ( 402 - 402 \zeta_{6} ) q^{53} + 216 q^{55} + 228 q^{57} + ( -396 + 396 \zeta_{6} ) q^{59} -254 \zeta_{6} q^{61} -372 \zeta_{6} q^{65} + ( 1012 - 1012 \zeta_{6} ) q^{67} + 72 q^{69} + 840 q^{71} + ( -890 + 890 \zeta_{6} ) q^{73} -267 \zeta_{6} q^{75} -80 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -108 q^{83} + 684 q^{85} + ( 162 - 162 \zeta_{6} ) q^{87} + 1638 \zeta_{6} q^{89} -336 \zeta_{6} q^{93} + ( 456 - 456 \zeta_{6} ) q^{95} + 1010 q^{97} + 324 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 6q^{5} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 6q^{5} - 9q^{9} - 36q^{11} + 124q^{13} - 36q^{15} - 114q^{17} + 76q^{19} + 24q^{23} + 89q^{25} - 54q^{27} + 108q^{29} + 112q^{31} + 108q^{33} + 178q^{37} + 186q^{39} + 756q^{41} - 344q^{43} - 54q^{45} + 192q^{47} + 342q^{51} + 402q^{53} + 432q^{55} + 456q^{57} - 396q^{59} - 254q^{61} - 372q^{65} + 1012q^{67} + 144q^{69} + 1680q^{71} - 890q^{73} - 267q^{75} - 80q^{79} - 81q^{81} - 216q^{83} + 1368q^{85} + 162q^{87} + 1638q^{89} - 336q^{93} + 456q^{95} + 2020q^{97} + 648q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 2.59808i 0 −3.00000 5.19615i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −3.00000 + 5.19615i 0 0 0 −4.50000 + 7.79423i 0
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.i.f 2
3.b odd 2 1 1764.4.k.l 2
7.b odd 2 1 588.4.i.c 2
7.c even 3 1 84.4.a.a 1
7.c even 3 1 inner 588.4.i.f 2
7.d odd 6 1 588.4.a.d 1
7.d odd 6 1 588.4.i.c 2
21.c even 2 1 1764.4.k.f 2
21.g even 6 1 1764.4.a.j 1
21.g even 6 1 1764.4.k.f 2
21.h odd 6 1 252.4.a.b 1
21.h odd 6 1 1764.4.k.l 2
28.f even 6 1 2352.4.a.d 1
28.g odd 6 1 336.4.a.k 1
35.j even 6 1 2100.4.a.l 1
35.l odd 12 2 2100.4.k.j 2
56.k odd 6 1 1344.4.a.d 1
56.p even 6 1 1344.4.a.q 1
84.n even 6 1 1008.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 7.c even 3 1
252.4.a.b 1 21.h odd 6 1
336.4.a.k 1 28.g odd 6 1
588.4.a.d 1 7.d odd 6 1
588.4.i.c 2 7.b odd 2 1
588.4.i.c 2 7.d odd 6 1
588.4.i.f 2 1.a even 1 1 trivial
588.4.i.f 2 7.c even 3 1 inner
1008.4.a.h 1 84.n even 6 1
1344.4.a.d 1 56.k odd 6 1
1344.4.a.q 1 56.p even 6 1
1764.4.a.j 1 21.g even 6 1
1764.4.k.f 2 21.c even 2 1
1764.4.k.f 2 21.g even 6 1
1764.4.k.l 2 3.b odd 2 1
1764.4.k.l 2 21.h odd 6 1
2100.4.a.l 1 35.j even 6 1
2100.4.k.j 2 35.l odd 12 2
2352.4.a.d 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6 T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$36 + 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1296 + 36 T + T^{2}$$
$13$ $$( -62 + T )^{2}$$
$17$ $$12996 + 114 T + T^{2}$$
$19$ $$5776 - 76 T + T^{2}$$
$23$ $$576 - 24 T + T^{2}$$
$29$ $$( -54 + T )^{2}$$
$31$ $$12544 - 112 T + T^{2}$$
$37$ $$31684 - 178 T + T^{2}$$
$41$ $$( -378 + T )^{2}$$
$43$ $$( 172 + T )^{2}$$
$47$ $$36864 - 192 T + T^{2}$$
$53$ $$161604 - 402 T + T^{2}$$
$59$ $$156816 + 396 T + T^{2}$$
$61$ $$64516 + 254 T + T^{2}$$
$67$ $$1024144 - 1012 T + T^{2}$$
$71$ $$( -840 + T )^{2}$$
$73$ $$792100 + 890 T + T^{2}$$
$79$ $$6400 + 80 T + T^{2}$$
$83$ $$( 108 + T )^{2}$$
$89$ $$2683044 - 1638 T + T^{2}$$
$97$ $$( -1010 + T )^{2}$$