# Properties

 Label 588.4.i.e Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ Analytic rank $1$ 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: $$1$$ 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 12) 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -18 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -18 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} + ( -36 + 36 \zeta_{6} ) q^{11} + 10 q^{13} -54 q^{15} + ( 18 - 18 \zeta_{6} ) q^{17} -100 \zeta_{6} q^{19} -72 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} -27 q^{27} -234 q^{29} + ( -16 + 16 \zeta_{6} ) q^{31} + 108 \zeta_{6} q^{33} + 226 \zeta_{6} q^{37} + ( 30 - 30 \zeta_{6} ) q^{39} -90 q^{41} + 452 q^{43} + ( -162 + 162 \zeta_{6} ) q^{45} + 432 \zeta_{6} q^{47} -54 \zeta_{6} q^{51} + ( -414 + 414 \zeta_{6} ) q^{53} + 648 q^{55} -300 q^{57} + ( -684 + 684 \zeta_{6} ) q^{59} + 422 \zeta_{6} q^{61} -180 \zeta_{6} q^{65} + ( -332 + 332 \zeta_{6} ) q^{67} -216 q^{69} -360 q^{71} + ( 26 - 26 \zeta_{6} ) q^{73} + 597 \zeta_{6} q^{75} -512 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 1188 q^{83} -324 q^{85} + ( -702 + 702 \zeta_{6} ) q^{87} -630 \zeta_{6} q^{89} + 48 \zeta_{6} q^{93} + ( -1800 + 1800 \zeta_{6} ) q^{95} + 1054 q^{97} + 324 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 18q^{5} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 18q^{5} - 9q^{9} - 36q^{11} + 20q^{13} - 108q^{15} + 18q^{17} - 100q^{19} - 72q^{23} - 199q^{25} - 54q^{27} - 468q^{29} - 16q^{31} + 108q^{33} + 226q^{37} + 30q^{39} - 180q^{41} + 904q^{43} - 162q^{45} + 432q^{47} - 54q^{51} - 414q^{53} + 1296q^{55} - 600q^{57} - 684q^{59} + 422q^{61} - 180q^{65} - 332q^{67} - 432q^{69} - 720q^{71} + 26q^{73} + 597q^{75} - 512q^{79} - 81q^{81} + 2376q^{83} - 648q^{85} - 702q^{87} - 630q^{89} + 48q^{93} - 1800q^{95} + 2108q^{97} + 648q^{99} + O(q^{100})$$

## 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 −9.00000 15.5885i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −9.00000 + 15.5885i 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.e 2
3.b odd 2 1 1764.4.k.o 2
7.b odd 2 1 588.4.i.d 2
7.c even 3 1 588.4.a.c 1
7.c even 3 1 inner 588.4.i.e 2
7.d odd 6 1 12.4.a.a 1
7.d odd 6 1 588.4.i.d 2
21.c even 2 1 1764.4.k.b 2
21.g even 6 1 36.4.a.a 1
21.g even 6 1 1764.4.k.b 2
21.h odd 6 1 1764.4.a.b 1
21.h odd 6 1 1764.4.k.o 2
28.f even 6 1 48.4.a.a 1
28.g odd 6 1 2352.4.a.bk 1
35.i odd 6 1 300.4.a.b 1
35.k even 12 2 300.4.d.e 2
56.j odd 6 1 192.4.a.f 1
56.m even 6 1 192.4.a.l 1
63.i even 6 1 324.4.e.a 2
63.k odd 6 1 324.4.e.h 2
63.s even 6 1 324.4.e.a 2
63.t odd 6 1 324.4.e.h 2
77.i even 6 1 1452.4.a.d 1
84.j odd 6 1 144.4.a.g 1
91.s odd 6 1 2028.4.a.c 1
91.bb even 12 2 2028.4.b.c 2
105.p even 6 1 900.4.a.g 1
105.w odd 12 2 900.4.d.c 2
112.v even 12 2 768.4.d.j 2
112.x odd 12 2 768.4.d.g 2
140.s even 6 1 1200.4.a.be 1
140.x odd 12 2 1200.4.f.d 2
168.ba even 6 1 576.4.a.b 1
168.be odd 6 1 576.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 7.d odd 6 1
36.4.a.a 1 21.g even 6 1
48.4.a.a 1 28.f even 6 1
144.4.a.g 1 84.j odd 6 1
192.4.a.f 1 56.j odd 6 1
192.4.a.l 1 56.m even 6 1
300.4.a.b 1 35.i odd 6 1
300.4.d.e 2 35.k even 12 2
324.4.e.a 2 63.i even 6 1
324.4.e.a 2 63.s even 6 1
324.4.e.h 2 63.k odd 6 1
324.4.e.h 2 63.t odd 6 1
576.4.a.a 1 168.be odd 6 1
576.4.a.b 1 168.ba even 6 1
588.4.a.c 1 7.c even 3 1
588.4.i.d 2 7.b odd 2 1
588.4.i.d 2 7.d odd 6 1
588.4.i.e 2 1.a even 1 1 trivial
588.4.i.e 2 7.c even 3 1 inner
768.4.d.g 2 112.x odd 12 2
768.4.d.j 2 112.v even 12 2
900.4.a.g 1 105.p even 6 1
900.4.d.c 2 105.w odd 12 2
1200.4.a.be 1 140.s even 6 1
1200.4.f.d 2 140.x odd 12 2
1452.4.a.d 1 77.i even 6 1
1764.4.a.b 1 21.h odd 6 1
1764.4.k.b 2 21.c even 2 1
1764.4.k.b 2 21.g even 6 1
1764.4.k.o 2 3.b odd 2 1
1764.4.k.o 2 21.h odd 6 1
2028.4.a.c 1 91.s odd 6 1
2028.4.b.c 2 91.bb even 12 2
2352.4.a.bk 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 18 T_{5} + 324$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$324 + 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1296 + 36 T + T^{2}$$
$13$ $$( -10 + T )^{2}$$
$17$ $$324 - 18 T + T^{2}$$
$19$ $$10000 + 100 T + T^{2}$$
$23$ $$5184 + 72 T + T^{2}$$
$29$ $$( 234 + T )^{2}$$
$31$ $$256 + 16 T + T^{2}$$
$37$ $$51076 - 226 T + T^{2}$$
$41$ $$( 90 + T )^{2}$$
$43$ $$( -452 + T )^{2}$$
$47$ $$186624 - 432 T + T^{2}$$
$53$ $$171396 + 414 T + T^{2}$$
$59$ $$467856 + 684 T + T^{2}$$
$61$ $$178084 - 422 T + T^{2}$$
$67$ $$110224 + 332 T + T^{2}$$
$71$ $$( 360 + T )^{2}$$
$73$ $$676 - 26 T + T^{2}$$
$79$ $$262144 + 512 T + T^{2}$$
$83$ $$( -1188 + T )^{2}$$
$89$ $$396900 + 630 T + T^{2}$$
$97$ $$( -1054 + T )^{2}$$