# Properties

 Label 588.4.i.i Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{3}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{9} + ( -1 - 26 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -28 + 5 \beta_{2} ) q^{13} + ( 3 - 3 \beta_{2} ) q^{15} + ( -8 - 16 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{17} + ( 85 + 85 \beta_{1} - \beta_{3} ) q^{19} + ( 96 + 96 \beta_{1} ) q^{23} + ( 3 + 7 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{25} + 27 q^{27} + ( -11 + 17 \beta_{2} ) q^{29} + ( 10 + 51 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{31} + ( 78 + 78 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 77 + 77 \beta_{1} + 19 \beta_{3} ) q^{37} + ( -15 - 99 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{39} + ( -104 - 34 \beta_{2} ) q^{41} + ( -258 - 19 \beta_{2} ) q^{43} + ( 9 + 18 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{45} + ( 98 + 98 \beta_{1} - 16 \beta_{3} ) q^{47} + ( 48 + 48 \beta_{1} - 24 \beta_{3} ) q^{51} + ( 27 + 156 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{53} + ( -153 + 27 \beta_{2} ) q^{55} + ( -252 + 3 \beta_{2} ) q^{57} + ( 3 + 636 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{59} + ( 146 + 146 \beta_{1} + 36 \beta_{3} ) q^{61} + ( 706 + 706 \beta_{1} - 38 \beta_{3} ) q^{65} + ( 9 - 433 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{67} -288 q^{69} + ( -718 - 32 \beta_{2} ) q^{71} + ( 21 + 691 \beta_{1} + 21 \beta_{2} - 21 \beta_{3} ) q^{73} + ( -21 - 21 \beta_{1} + 9 \beta_{3} ) q^{75} + ( 93 + 93 \beta_{1} - 4 \beta_{3} ) q^{79} + 81 \beta_{1} q^{81} + ( 221 + 43 \beta_{2} ) q^{83} + ( -1032 + 24 \beta_{2} ) q^{85} + ( -51 - 84 \beta_{1} - 51 \beta_{2} + 51 \beta_{3} ) q^{87} + ( 400 + 400 \beta_{1} + 22 \beta_{3} ) q^{89} + ( -153 - 153 \beta_{1} + 30 \beta_{3} ) q^{93} + ( -86 - 298 \beta_{1} - 86 \beta_{2} + 86 \beta_{3} ) q^{95} + ( -431 - 21 \beta_{2} ) q^{97} + ( -225 + 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 3q^{5} - 18q^{9} + O(q^{10})$$ $$4q - 6q^{3} - 3q^{5} - 18q^{9} + 51q^{11} - 122q^{13} + 18q^{15} + 24q^{17} + 169q^{19} + 192q^{23} - 11q^{25} + 108q^{27} - 78q^{29} - 92q^{31} + 153q^{33} + 173q^{37} + 183q^{39} - 348q^{41} - 994q^{43} - 27q^{45} + 180q^{47} + 72q^{51} - 285q^{53} - 666q^{55} - 1014q^{57} - 1269q^{59} + 328q^{61} + 1374q^{65} + 875q^{67} - 1152q^{69} - 2808q^{71} - 1361q^{73} - 33q^{75} + 182q^{79} - 162q^{81} + 798q^{83} - 4176q^{85} + 117q^{87} + 822q^{89} - 276q^{93} + 510q^{95} - 1682q^{97} - 918q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{3} + 3 \nu^{2} + 27 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$17 \nu^{3} + 8 \nu^{2} + 52 \nu - 145$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 5 \beta_{1}$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 41 \beta_{1} + 42$$$$)/9$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 57$$$$)/9$$

## Character values

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

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

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 −1.50000 + 2.59808i 0 −6.41238 11.1066i 0 0 0 −4.50000 7.79423i 0
361.2 0 −1.50000 + 2.59808i 0 4.91238 + 8.50848i 0 0 0 −4.50000 7.79423i 0
373.1 0 −1.50000 2.59808i 0 −6.41238 + 11.1066i 0 0 0 −4.50000 + 7.79423i 0
373.2 0 −1.50000 2.59808i 0 4.91238 8.50848i 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.i 4
3.b odd 2 1 1764.4.k.z 4
7.b odd 2 1 84.4.i.b 4
7.c even 3 1 588.4.a.h 2
7.c even 3 1 inner 588.4.i.i 4
7.d odd 6 1 84.4.i.b 4
7.d odd 6 1 588.4.a.g 2
21.c even 2 1 252.4.k.d 4
21.g even 6 1 252.4.k.d 4
21.g even 6 1 1764.4.a.x 2
21.h odd 6 1 1764.4.a.p 2
21.h odd 6 1 1764.4.k.z 4
28.d even 2 1 336.4.q.h 4
28.f even 6 1 336.4.q.h 4
28.f even 6 1 2352.4.a.cb 2
28.g odd 6 1 2352.4.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 7.b odd 2 1
84.4.i.b 4 7.d odd 6 1
252.4.k.d 4 21.c even 2 1
252.4.k.d 4 21.g even 6 1
336.4.q.h 4 28.d even 2 1
336.4.q.h 4 28.f even 6 1
588.4.a.g 2 7.d odd 6 1
588.4.a.h 2 7.c even 3 1
588.4.i.i 4 1.a even 1 1 trivial
588.4.i.i 4 7.c even 3 1 inner
1764.4.a.p 2 21.h odd 6 1
1764.4.a.x 2 21.g even 6 1
1764.4.k.z 4 3.b odd 2 1
1764.4.k.z 4 21.h odd 6 1
2352.4.a.bp 2 28.g odd 6 1
2352.4.a.cb 2 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3 T_{5}^{3} + 135 T_{5}^{2} - 378 T_{5} + 15876$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + 3 T + T^{2} )^{2}$$
$5$ $$15876 - 378 T + 135 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$272484 - 26622 T + 2079 T^{2} - 51 T^{3} + T^{4}$$
$13$ $$( -2276 + 61 T + T^{2} )^{2}$$
$17$ $$65028096 + 193536 T + 8640 T^{2} - 24 T^{3} + T^{4}$$
$19$ $$49168144 - 1185028 T + 21549 T^{2} - 169 T^{3} + T^{4}$$
$23$ $$( 9216 - 96 T + T^{2} )^{2}$$
$29$ $$( -36684 + 39 T + T^{2} )^{2}$$
$31$ $$114682681 - 985228 T + 19173 T^{2} + 92 T^{3} + T^{4}$$
$37$ $$1506681856 + 6715168 T + 68745 T^{2} - 173 T^{3} + T^{4}$$
$41$ $$( -140688 + 174 T + T^{2} )^{2}$$
$43$ $$( 15454 + 497 T + T^{2} )^{2}$$
$47$ $$611671824 + 4451760 T + 57132 T^{2} - 180 T^{3} + T^{4}$$
$53$ $$5356483344 - 20858580 T + 154413 T^{2} + 285 T^{3} + T^{4}$$
$59$ $$161150862096 + 509422284 T + 1208925 T^{2} + 1269 T^{3} + T^{4}$$
$61$ $$19408947856 + 45695648 T + 246900 T^{2} - 328 T^{3} + T^{4}$$
$67$ $$32767516324 - 158390750 T + 584607 T^{2} - 875 T^{3} + T^{4}$$
$71$ $$( 361476 + 1404 T + T^{2} )^{2}$$
$73$ $$165260136484 + 553276442 T + 1445799 T^{2} + 1361 T^{3} + T^{4}$$
$79$ $$38800441 - 1133678 T + 26895 T^{2} - 182 T^{3} + T^{4}$$
$83$ $$( -197334 - 399 T + T^{2} )^{2}$$
$89$ $$11416495104 - 87829056 T + 568836 T^{2} - 822 T^{3} + T^{4}$$
$97$ $$( 120262 + 841 T + T^{2} )^{2}$$