# Properties

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

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## 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: $$\Q(\sqrt{-3}, \sqrt{193})$$ Defining polynomial: $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} + 6 \beta_{2} ) q^{5} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} + 6 \beta_{2} ) q^{5} -9 \beta_{2} q^{9} + ( -1 + 7 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} ) q^{11} -5 \beta_{3} q^{13} + ( 15 + 3 \beta_{3} ) q^{15} + ( 52 - 4 \beta_{1} - 48 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 35 \beta_{2} ) q^{19} + ( 20 \beta_{1} - 48 \beta_{2} ) q^{23} + ( 52 - 11 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} ) q^{25} -27 q^{27} + ( 143 - 11 \beta_{3} ) q^{29} + ( 171 + 20 \beta_{1} - 191 \beta_{2} + 20 \beta_{3} ) q^{31} + ( 21 \beta_{1} - 18 \beta_{2} ) q^{33} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{37} + ( -15 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{39} + ( 72 + 18 \beta_{3} ) q^{41} + ( 362 - 3 \beta_{3} ) q^{43} + ( 45 + 9 \beta_{1} - 54 \beta_{2} + 9 \beta_{3} ) q^{45} + ( -36 \beta_{1} - 90 \beta_{2} ) q^{47} + ( -12 \beta_{1} - 144 \beta_{2} ) q^{51} + ( -243 - 9 \beta_{1} + 252 \beta_{2} - 9 \beta_{3} ) q^{53} + ( 331 + 41 \beta_{3} ) q^{55} + ( 96 + 9 \beta_{3} ) q^{57} + ( 113 - 53 \beta_{1} - 60 \beta_{2} - 53 \beta_{3} ) q^{59} + ( 40 \beta_{1} - 286 \beta_{2} ) q^{61} + ( 30 \beta_{1} - 270 \beta_{2} ) q^{65} + ( -94 + 77 \beta_{1} + 17 \beta_{2} + 77 \beta_{3} ) q^{67} + ( -84 - 60 \beta_{3} ) q^{69} + ( 778 + 44 \beta_{3} ) q^{71} + ( 634 - 53 \beta_{1} - 581 \beta_{2} - 53 \beta_{3} ) q^{73} + ( -33 \beta_{1} - 123 \beta_{2} ) q^{75} + ( 62 \beta_{1} - 761 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 755 - 101 \beta_{3} ) q^{83} + ( 68 + 28 \beta_{3} ) q^{85} + ( 429 - 33 \beta_{1} - 396 \beta_{2} - 33 \beta_{3} ) q^{87} + ( 42 \beta_{1} - 1008 \beta_{2} ) q^{89} + ( 60 \beta_{1} - 573 \beta_{2} ) q^{93} + ( -304 - 50 \beta_{1} + 354 \beta_{2} - 50 \beta_{3} ) q^{95} + ( -295 + 29 \beta_{3} ) q^{97} + ( 9 - 63 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} + 11q^{5} - 18q^{9} + O(q^{10})$$ $$4q + 6q^{3} + 11q^{5} - 18q^{9} + 5q^{11} - 10q^{13} + 66q^{15} + 100q^{17} + 67q^{19} - 76q^{23} + 93q^{25} - 108q^{27} + 550q^{29} + 362q^{31} - 15q^{33} + 5q^{37} - 15q^{39} + 324q^{41} + 1442q^{43} + 99q^{45} - 216q^{47} - 300q^{51} - 495q^{53} + 1406q^{55} + 402q^{57} + 173q^{59} - 532q^{61} - 510q^{65} - 111q^{67} - 456q^{69} + 3200q^{71} + 1215q^{73} - 279q^{75} - 1460q^{79} - 162q^{81} + 2818q^{83} + 328q^{85} + 825q^{87} - 1974q^{89} - 1086q^{93} - 658q^{95} - 1122q^{97} - 90q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 49 \nu^{2} - 49 \nu + 2304$$$$)/2352$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 97$$$$)/49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 48 \beta_{2} + \beta_{1} - 49$$ $$\nu^{3}$$ $$=$$ $$49 \beta_{3} - 97$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 3.72311 + 6.44862i −3.22311 − 5.58259i 3.72311 − 6.44862i −3.22311 + 5.58259i
0 1.50000 2.59808i 0 −0.723111 1.25246i 0 0 0 −4.50000 7.79423i 0
361.2 0 1.50000 2.59808i 0 6.22311 + 10.7787i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −0.723111 + 1.25246i 0 0 0 −4.50000 + 7.79423i 0
373.2 0 1.50000 + 2.59808i 0 6.22311 10.7787i 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.j 4
3.b odd 2 1 1764.4.k.q 4
7.b odd 2 1 84.4.i.a 4
7.c even 3 1 588.4.a.f 2
7.c even 3 1 inner 588.4.i.j 4
7.d odd 6 1 84.4.i.a 4
7.d odd 6 1 588.4.a.i 2
21.c even 2 1 252.4.k.f 4
21.g even 6 1 252.4.k.f 4
21.g even 6 1 1764.4.a.o 2
21.h odd 6 1 1764.4.a.y 2
21.h odd 6 1 1764.4.k.q 4
28.d even 2 1 336.4.q.i 4
28.f even 6 1 336.4.q.i 4
28.f even 6 1 2352.4.a.bt 2
28.g odd 6 1 2352.4.a.bx 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 11 T_{5}^{3} + 139 T_{5}^{2} + 198 T_{5} + 324$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$324 + 198 T + 139 T^{2} - 11 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$5560164 + 11790 T + 2383 T^{2} - 5 T^{3} + T^{4}$$
$13$ $$( -1200 + 5 T + T^{2} )^{2}$$
$17$ $$2985984 - 172800 T + 8272 T^{2} - 100 T^{3} + T^{4}$$
$19$ $$473344 - 46096 T + 3801 T^{2} - 67 T^{3} + T^{4}$$
$23$ $$318836736 - 1357056 T + 23632 T^{2} + 76 T^{3} + T^{4}$$
$29$ $$( 13068 - 275 T + T^{2} )^{2}$$
$31$ $$181198521 - 4872882 T + 117583 T^{2} - 362 T^{3} + T^{4}$$
$37$ $$9545290000 + 488500 T + 97725 T^{2} - 5 T^{3} + T^{4}$$
$41$ $$( -9072 - 162 T + T^{2} )^{2}$$
$43$ $$( 129526 - 721 T + T^{2} )^{2}$$
$47$ $$2587553424 - 10987488 T + 97524 T^{2} + 216 T^{3} + T^{4}$$
$53$ $$3288793104 + 28387260 T + 187677 T^{2} + 495 T^{3} + T^{4}$$
$59$ $$16397314704 + 22152996 T + 157981 T^{2} - 173 T^{3} + T^{4}$$
$61$ $$41525136 - 3428208 T + 289468 T^{2} + 532 T^{3} + T^{4}$$
$67$ $$80085604036 - 31412334 T + 295315 T^{2} + 111 T^{3} + T^{4}$$
$71$ $$( 546588 - 1600 T + T^{2} )^{2}$$
$73$ $$54532524484 - 283729230 T + 1242703 T^{2} - 1215 T^{3} + T^{4}$$
$79$ $$120705520329 + 507243420 T + 1784173 T^{2} + 1460 T^{3} + T^{4}$$
$83$ $$( 4122 - 1409 T + T^{2} )^{2}$$
$89$ $$790420571136 + 1754996544 T + 3007620 T^{2} + 1974 T^{3} + T^{4}$$
$97$ $$( 38102 + 561 T + T^{2} )^{2}$$
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