# Properties

 Label 588.4.i.l Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ 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: $$k$$ $$=$$ $$4$$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 27 x^{6} + 10 x^{5} + 446 x^{4} + 62 x^{3} + 3061 x^{2} + 2142 x + 14161$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} -\beta_{6} q^{5} + ( -9 + 9 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} -\beta_{6} q^{5} + ( -9 + 9 \beta_{1} ) q^{9} + ( 3 \beta_{2} + \beta_{4} ) q^{11} + ( 3 \beta_{4} + 3 \beta_{7} ) q^{13} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{15} + ( 12 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{17} + ( 48 - 48 \beta_{1} + 5 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{19} + ( -48 + 48 \beta_{1} + 6 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{23} + ( -81 \beta_{1} + 6 \beta_{3} - 6 \beta_{4} ) q^{25} -27 q^{27} + ( 24 - 9 \beta_{3} + \beta_{4} + 9 \beta_{5} + \beta_{7} ) q^{29} + ( 12 \beta_{1} + 6 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} ) q^{31} + ( -9 \beta_{6} - 3 \beta_{7} ) q^{33} + ( -64 + 64 \beta_{1} + 24 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{37} + 9 \beta_{4} q^{39} + ( -252 + \beta_{2} + 15 \beta_{3} - 3 \beta_{4} - 15 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{41} + ( -28 + 6 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{43} -9 \beta_{2} q^{45} + ( 216 - 216 \beta_{1} + 3 \beta_{5} - 10 \beta_{6} - 3 \beta_{7} ) q^{47} + ( -36 + 36 \beta_{1} + 9 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} ) q^{51} + ( 162 \beta_{1} - 24 \beta_{2} + 18 \beta_{3} + 10 \beta_{4} ) q^{53} + ( -588 - 6 \beta_{2} - 5 \beta_{3} - 21 \beta_{4} + 5 \beta_{5} - 6 \beta_{6} - 21 \beta_{7} ) q^{55} + ( 144 + 18 \beta_{2} - 15 \beta_{3} + 9 \beta_{4} + 15 \beta_{5} + 18 \beta_{6} + 9 \beta_{7} ) q^{57} + ( 84 \beta_{1} - 10 \beta_{2} + 15 \beta_{3} - 21 \beta_{4} ) q^{59} + ( 240 - 240 \beta_{1} - 14 \beta_{5} + 30 \beta_{6} - 15 \beta_{7} ) q^{61} + ( 90 - 90 \beta_{1} + 69 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} ) q^{65} + ( -180 \beta_{1} - 6 \beta_{2} + 48 \beta_{3} - 30 \beta_{4} ) q^{67} + ( -144 + 9 \beta_{2} - 18 \beta_{3} - 3 \beta_{4} + 18 \beta_{5} + 9 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -336 - 3 \beta_{2} - 30 \beta_{3} + 9 \beta_{4} + 30 \beta_{5} - 3 \beta_{6} + 9 \beta_{7} ) q^{71} + ( 168 \beta_{1} + 30 \beta_{2} - 49 \beta_{3} + 18 \beta_{4} ) q^{73} + ( 243 - 243 \beta_{1} + 18 \beta_{5} + 18 \beta_{7} ) q^{75} + ( 496 - 496 \beta_{1} + 96 \beta_{5} + 36 \beta_{6} + 24 \beta_{7} ) q^{79} -81 \beta_{1} q^{81} + ( -780 + 8 \beta_{2} + 18 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} + 8 \beta_{6} + 18 \beta_{7} ) q^{83} + ( 170 - 30 \beta_{2} - 66 \beta_{3} - 24 \beta_{4} + 66 \beta_{5} - 30 \beta_{6} - 24 \beta_{7} ) q^{85} + ( 72 \beta_{1} - 27 \beta_{3} + 3 \beta_{4} ) q^{87} + ( 540 - 540 \beta_{1} - 69 \beta_{5} - 11 \beta_{6} + 15 \beta_{7} ) q^{89} + ( -36 + 36 \beta_{1} + 27 \beta_{5} - 18 \beta_{6} + 9 \beta_{7} ) q^{93} + ( 936 \beta_{1} + 66 \beta_{2} + 48 \beta_{3} + 10 \beta_{4} ) q^{95} + ( -504 - 18 \beta_{2} - 49 \beta_{3} + 48 \beta_{4} + 49 \beta_{5} - 18 \beta_{6} + 48 \beta_{7} ) q^{97} + ( -27 \beta_{2} - 9 \beta_{4} - 27 \beta_{6} - 9 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{3} - 36q^{9} + O(q^{10})$$ $$8q + 12q^{3} - 36q^{9} + 48q^{17} + 192q^{19} - 192q^{23} - 324q^{25} - 216q^{27} + 192q^{29} + 48q^{31} - 256q^{37} - 2016q^{41} - 224q^{43} + 864q^{47} - 144q^{51} + 648q^{53} - 4704q^{55} + 1152q^{57} + 336q^{59} + 960q^{61} + 360q^{65} - 720q^{67} - 1152q^{69} - 2688q^{71} + 672q^{73} + 972q^{75} + 1984q^{79} - 324q^{81} - 6240q^{83} + 1360q^{85} + 288q^{87} + 2160q^{89} - 144q^{93} + 3744q^{95} - 4032q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 27 x^{6} + 10 x^{5} + 446 x^{4} + 62 x^{3} + 3061 x^{2} + 2142 x + 14161$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-21796 \nu^{7} + 134695 \nu^{6} - 629156 \nu^{5} + 1627866 \nu^{4} - 5942222 \nu^{3} + 45151502 \nu^{2} - 33983852 \nu + 283035907$$$$)/ 256662294$$ $$\beta_{2}$$ $$=$$ $$($$$$-1205557 \nu^{7} - 11454035 \nu^{6} + 53501428 \nu^{5} - 460363998 \nu^{4} + 505307686 \nu^{3} - 3839540326 \nu^{2} + 17697242605 \nu - 24068474591$$$$)/ 2309960646$$ $$\beta_{3}$$ $$=$$ $$($$$$-319577 \nu^{7} - 1805905 \nu^{6} + 8435324 \nu^{5} - 86212482 \nu^{4} + 79669538 \nu^{3} - 605362658 \nu^{2} + 1569138137 \nu - 3794765653$$$$)/ 329994378$$ $$\beta_{4}$$ $$=$$ $$($$$$-1120543 \nu^{7} + 13834525 \nu^{6} - 64620620 \nu^{5} + 391076178 \nu^{4} - 610325690 \nu^{3} + 4637511290 \nu^{2} - 2020738937 \nu + 29070621265$$$$)/ 769986882$$ $$\beta_{5}$$ $$=$$ $$($$$$553591 \nu^{7} - 2047537 \nu^{6} + 18671351 \nu^{5} - 13516500 \nu^{4} + 207897296 \nu^{3} - 46360973 \nu^{2} + 1356668411 \nu + 431668097$$$$)/ 164997189$$ $$\beta_{6}$$ $$=$$ $$($$$$-4603973 \nu^{7} + 12181892 \nu^{6} - 132244027 \nu^{5} + 14215002 \nu^{4} - 1930017850 \nu^{3} + 48961711 \nu^{2} - 13024209625 \nu - 7476734776$$$$)/ 1154980323$$ $$\beta_{7}$$ $$=$$ $$($$$$1576285 \nu^{7} - 11501593 \nu^{6} + 52895627 \nu^{5} - 153395616 \nu^{4} + 356720156 \nu^{3} - 1953697577 \nu^{2} + 1825168121 \nu - 3319150975$$$$)/ 384993441$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-5 \beta_{3} + 7 \beta_{2} + 14 \beta_{1}$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{7} - 7 \beta_{6} - 8 \beta_{5} + 350 \beta_{1} - 350$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$21 \beta_{7} - 98 \beta_{6} - 115 \beta_{5} + 21 \beta_{4} + 115 \beta_{3} - 98 \beta_{2} - 644$$$$)/28$$ $$\nu^{4}$$ $$=$$ $$($$$$203 \beta_{4} + 324 \beta_{3} - 231 \beta_{2} - 5754 \beta_{1}$$$$)/28$$ $$\nu^{5}$$ $$=$$ $$($$$$-763 \beta_{7} + 1757 \beta_{6} + 2554 \beta_{5} - 18354 \beta_{1} + 18354$$$$)/28$$ $$\nu^{6}$$ $$=$$ $$($$$$-2492 \beta_{7} + 3115 \beta_{6} + 4769 \beta_{5} - 2492 \beta_{4} - 4769 \beta_{3} + 3115 \beta_{2} + 57904$$$$)/14$$ $$\nu^{7}$$ $$=$$ $$($$$$-21364 \beta_{4} - 58301 \beta_{3} + 37051 \beta_{2} + 473550 \beta_{1}$$$$)/28$$

## 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
 1.44795 − 2.50793i 2.46576 − 4.27083i −1.65506 + 2.86665i −1.25866 + 2.18006i 1.44795 + 2.50793i 2.46576 + 4.27083i −1.65506 − 2.86665i −1.25866 − 2.18006i
0 1.50000 2.59808i 0 −8.32734 14.4234i 0 0 0 −4.50000 7.79423i 0
361.2 0 1.50000 2.59808i 0 −5.32752 9.22754i 0 0 0 −4.50000 7.79423i 0
361.3 0 1.50000 2.59808i 0 4.08470 + 7.07491i 0 0 0 −4.50000 7.79423i 0
361.4 0 1.50000 2.59808i 0 9.57016 + 16.5760i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −8.32734 + 14.4234i 0 0 0 −4.50000 + 7.79423i 0
373.2 0 1.50000 + 2.59808i 0 −5.32752 + 9.22754i 0 0 0 −4.50000 + 7.79423i 0
373.3 0 1.50000 + 2.59808i 0 4.08470 7.07491i 0 0 0 −4.50000 + 7.79423i 0
373.4 0 1.50000 + 2.59808i 0 9.57016 16.5760i 0 0 0 −4.50000 + 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.4 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.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.l 8
3.b odd 2 1 1764.4.k.bb 8
7.b odd 2 1 588.4.i.k 8
7.c even 3 1 588.4.a.j 4
7.c even 3 1 inner 588.4.i.l 8
7.d odd 6 1 588.4.a.k yes 4
7.d odd 6 1 588.4.i.k 8
21.c even 2 1 1764.4.k.bd 8
21.g even 6 1 1764.4.a.ba 4
21.g even 6 1 1764.4.k.bd 8
21.h odd 6 1 1764.4.a.bc 4
21.h odd 6 1 1764.4.k.bb 8
28.f even 6 1 2352.4.a.cl 4
28.g odd 6 1 2352.4.a.cq 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.j 4 7.c even 3 1
588.4.a.k yes 4 7.d odd 6 1
588.4.i.k 8 7.b odd 2 1
588.4.i.k 8 7.d odd 6 1
588.4.i.l 8 1.a even 1 1 trivial
588.4.i.l 8 7.c even 3 1 inner
1764.4.a.ba 4 21.g even 6 1
1764.4.a.bc 4 21.h odd 6 1
1764.4.k.bb 8 3.b odd 2 1
1764.4.k.bb 8 21.h odd 6 1
1764.4.k.bd 8 21.c even 2 1
1764.4.k.bd 8 21.g even 6 1
2352.4.a.cl 4 28.f even 6 1
2352.4.a.cq 4 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 412 T_{5}^{6} + 1152 T_{5}^{5} + 141996 T_{5}^{4} + 237312 T_{5}^{3} + 11763952 T_{5}^{2} - 15982848 T_{5} + 769951504$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 - 3 T + T^{2} )^{4}$$
$5$ $$769951504 - 15982848 T + 11763952 T^{2} + 237312 T^{3} + 141996 T^{4} + 1152 T^{5} + 412 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$537394557184 - 60803923968 T + 9911692928 T^{2} + 343056384 T^{3} + 16373424 T^{4} + 165888 T^{5} + 4136 T^{6} + T^{8}$$
$13$ $$( 10008036 + 31104 T - 7092 T^{2} + T^{4} )^{2}$$
$17$ $$23669042327056 - 2872812641664 T + 301143925168 T^{2} - 6237374976 T^{3} + 128700876 T^{4} - 711936 T^{5} + 12076 T^{6} - 48 T^{7} + T^{8}$$
$19$ $$1761576257130496 - 65047873880064 T + 2264289104896 T^{2} - 21200338944 T^{3} + 350295744 T^{4} - 2469888 T^{5} + 40144 T^{6} - 192 T^{7} + T^{8}$$
$23$ $$1644676230033664 + 36845645770752 T + 984750700160 T^{2} + 12004208640 T^{3} + 230424240 T^{4} + 2571264 T^{5} + 32936 T^{6} + 192 T^{7} + T^{8}$$
$29$ $$( 38719552 + 648960 T - 11696 T^{2} - 96 T^{3} + T^{4} )^{2}$$
$31$ $$36060150523170816 - 1083514118529024 T + 24271289515008 T^{2} - 267187838976 T^{3} + 2367527616 T^{4} - 9317376 T^{5} + 45936 T^{6} - 48 T^{7} + T^{8}$$
$37$ $$2188246218024484864 + 60887884830343168 T + 1493259219214336 T^{2} + 6348657393664 T^{3} + 30468918016 T^{4} + 47546368 T^{5} + 201376 T^{6} + 256 T^{7} + T^{8}$$
$41$ $$( 1611829828 + 41616864 T + 334100 T^{2} + 1008 T^{3} + T^{4} )^{2}$$
$43$ $$( -789373952 - 14991872 T - 71040 T^{2} + 112 T^{3} + T^{4} )^{2}$$
$47$ $$1365341173005684736 + 15606530612969472 T + 443550837400576 T^{2} - 5050045956096 T^{3} + 41124962496 T^{4} - 169353216 T^{5} + 519568 T^{6} - 864 T^{7} + T^{8}$$
$53$ $$42\!\cdots\!56$$$$- 3583422767117984256 T + 26263579147817600 T^{2} - 63032005550592 T^{3} + 177270158640 T^{4} - 214337664 T^{5} + 628520 T^{6} - 648 T^{7} + T^{8}$$
$59$ $$36\!\cdots\!64$$$$+ 904986722371264512 T + 7723924544711680 T^{2} - 928537141248 T^{3} + 79485705408 T^{4} + 1152000 T^{5} + 400048 T^{6} - 336 T^{7} + T^{8}$$
$61$ $$11\!\cdots\!16$$$$- 53297896411990510080 T + 214499086259755984 T^{2} - 369784423526400 T^{3} + 695413490412 T^{4} - 682444800 T^{5} + 1251796 T^{6} - 960 T^{7} + T^{8}$$
$67$ $$20\!\cdots\!24$$$$- 4740889990276841472 T + 118947517219602432 T^{2} + 200685970096128 T^{3} + 671734342656 T^{4} + 179463168 T^{5} + 1187712 T^{6} + 720 T^{7} + T^{8}$$
$71$ $$( -17989567344 - 15945984 T + 460440 T^{2} + 1344 T^{3} + T^{4} )^{2}$$
$73$ $$82\!\cdots\!36$$$$- 41770460619361616640 T + 168227011259367568 T^{2} - 336874324156416 T^{3} + 617979231948 T^{4} - 604053504 T^{5} + 919012 T^{6} - 672 T^{7} + T^{8}$$
$79$ $$10\!\cdots\!36$$$$-$$$$22\!\cdots\!60$$$$T + 4895492744600289280 T^{2} - 4579430342852608 T^{3} + 5574336080896 T^{4} - 4047216640 T^{5} + 4182976 T^{6} - 1984 T^{7} + T^{8}$$
$83$ $$( 74256064768 + 1203548928 T + 3243872 T^{2} + 3120 T^{3} + T^{4} )^{2}$$
$89$ $$97\!\cdots\!84$$$$-$$$$30\!\cdots\!60$$$$T + 1152332964022618672 T^{2} - 825195889889280 T^{3} + 2733644241804 T^{4} - 3119316480 T^{5} + 4142476 T^{6} - 2160 T^{7} + T^{8}$$
$97$ $$( -580580611196 - 2679572160 T - 752260 T^{2} + 2016 T^{3} + T^{4} )^{2}$$