# Properties

 Label 288.7.b.b Level 288 Weight 7 Character orbit 288.b Analytic conductor 66.256 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 288.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$66.2555760825$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.3803625.2 Defining polynomial: $$x^{4} - x^{3} + 6 x^{2} - 16 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} + ( 244 + 57 \beta_{1} ) q^{11} + ( \beta_{2} - 8 \beta_{3} ) q^{13} + ( -1042 + 200 \beta_{1} ) q^{17} + ( 364 - 65 \beta_{1} ) q^{19} + ( -91 \beta_{2} - 5 \beta_{3} ) q^{23} + ( -5975 + 880 \beta_{1} ) q^{25} + ( -13 \beta_{2} - 64 \beta_{3} ) q^{29} + ( -28 \beta_{2} + 60 \beta_{3} ) q^{31} + ( -12480 - 800 \beta_{1} ) q^{35} + ( 207 \beta_{2} - 56 \beta_{3} ) q^{37} + ( 29486 - 1104 \beta_{1} ) q^{41} + ( -49364 - 2121 \beta_{1} ) q^{43} + ( -638 \beta_{2} + 286 \beta_{3} ) q^{47} + ( 529 - 5696 \beta_{1} ) q^{49} + ( 1053 \beta_{2} - 56 \beta_{3} ) q^{53} + ( -611 \beta_{2} - 285 \beta_{3} ) q^{55} + ( 135508 - 327 \beta_{1} ) q^{59} + ( 845 \beta_{2} - 328 \beta_{3} ) q^{61} + ( 51360 - 12560 \beta_{1} ) q^{65} + ( 197548 + 5967 \beta_{1} ) q^{67} + ( -1977 \beta_{2} - 743 \beta_{3} ) q^{71} + ( 110978 - 8536 \beta_{1} ) q^{73} + ( 1612 \beta_{2} - 1384 \beta_{3} ) q^{77} + ( -1118 \beta_{2} - 546 \beta_{3} ) q^{79} + ( -866252 + 3089 \beta_{1} ) q^{83} + ( -4042 \beta_{2} - 1000 \beta_{3} ) q^{85} + ( -190306 + 11928 \beta_{1} ) q^{89} + ( -849600 - 39968 \beta_{1} ) q^{91} + ( 1339 \beta_{2} + 325 \beta_{3} ) q^{95} + ( -231694 - 42696 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 976q^{11} - 4168q^{17} + 1456q^{19} - 23900q^{25} - 49920q^{35} + 117944q^{41} - 197456q^{43} + 2116q^{49} + 542032q^{59} + 205440q^{65} + 790192q^{67} + 443912q^{73} - 3465008q^{83} - 761224q^{89} - 3398400q^{91} - 926776q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 6 x^{2} - 16 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 10 \nu + 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 17 \nu^{2} + 38 \nu - 64$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{3} + 9 \nu^{2} + 138 \nu - 64$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 16 \beta_{1} + 32$$$$)/128$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{3} - 13 \beta_{2} + 16 \beta_{1} - 352$$$$)/128$$ $$\nu^{3}$$ $$=$$ $$($$$$13 \beta_{3} - 3 \beta_{2} - 336 \beta_{1} + 992$$$$)/128$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 −2.31174 − 3.26433i 2.81174 + 2.84502i 2.81174 − 2.84502i −2.31174 + 3.26433i
0 0 0 199.084i 0 19.6656i 0 0 0
271.2 0 0 0 59.7107i 0 483.584i 0 0 0
271.3 0 0 0 59.7107i 0 483.584i 0 0 0
271.4 0 0 0 199.084i 0 19.6656i 0 0 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
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.7.b.b 4
3.b odd 2 1 32.7.d.b 4
4.b odd 2 1 72.7.b.b 4
8.b even 2 1 72.7.b.b 4
8.d odd 2 1 inner 288.7.b.b 4
12.b even 2 1 8.7.d.b 4
24.f even 2 1 32.7.d.b 4
24.h odd 2 1 8.7.d.b 4
48.i odd 4 2 256.7.c.l 8
48.k even 4 2 256.7.c.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 12.b even 2 1
8.7.d.b 4 24.h odd 2 1
32.7.d.b 4 3.b odd 2 1
32.7.d.b 4 24.f even 2 1
72.7.b.b 4 4.b odd 2 1
72.7.b.b 4 8.b even 2 1
256.7.c.l 8 48.i odd 4 2
256.7.c.l 8 48.k even 4 2
288.7.b.b 4 1.a even 1 1 trivial
288.7.b.b 4 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 43200 T_{5}^{2} + 141312000$$ acting on $$S_{7}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 19300 T^{2} + 256155750 T^{4} - 4711914062500 T^{6} + 59604644775390625 T^{8}$$
$7$ $$1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 488 T + 2238078 T^{2} - 864521768 T^{3} + 3138428376721 T^{4} )^{2}$$
$13$ $$1 - 4994596 T^{2} + 30260873415846 T^{4} -$$$$11\!\cdots\!76$$$$T^{6} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$( 1 + 2084 T + 32560902 T^{2} + 50302693796 T^{3} + 582622237229761 T^{4} )^{2}$$
$19$ $$( 1 - 728 T + 92449758 T^{2} - 34249401368 T^{3} + 2213314919066161 T^{4} )^{2}$$
$23$ $$1 - 212117956 T^{2} + 23026671552237126 T^{4} -$$$$46\!\cdots\!76$$$$T^{6} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$1 - 1409719204 T^{2} + 1160515330165289766 T^{4} -$$$$49\!\cdots\!64$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$31$ $$1 - 2758360324 T^{2} + 3362667870952277766 T^{4} -$$$$21\!\cdots\!64$$$$T^{6} +$$$$62\!\cdots\!21$$$$T^{8}$$
$37$ $$1 - 8121202276 T^{2} + 29600495645847907686 T^{4} -$$$$53\!\cdots\!56$$$$T^{6} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 - 58972 T + 9857729958 T^{2} - 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{2}$$
$43$ $$( 1 + 98728 T + 13190101374 T^{2} + 624095531101672 T^{3} + 39959630797262576401 T^{4} )^{2}$$
$47$ $$1 - 13578767236 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} -$$$$15\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 - 42194545636 T^{2} +$$$$11\!\cdots\!86$$$$T^{4} -$$$$20\!\cdots\!76$$$$T^{6} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$( 1 - 271016 T + 102678575166 T^{2} - 11431599505249256 T^{3} +$$$$17\!\cdots\!81$$$$T^{4} )^{2}$$
$61$ $$1 - 160868902564 T^{2} +$$$$11\!\cdots\!46$$$$T^{4} -$$$$42\!\cdots\!44$$$$T^{6} +$$$$70\!\cdots\!41$$$$T^{8}$$
$67$ $$( 1 - 395096 T + 204987839262 T^{2} - 35739744961443224 T^{3} +$$$$81\!\cdots\!61$$$$T^{4} )^{2}$$
$71$ $$1 - 164364621124 T^{2} +$$$$21\!\cdots\!06$$$$T^{4} -$$$$26\!\cdots\!84$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8}$$
$73$ $$( 1 - 221956 T + 284381984742 T^{2} - 33589539530201284 T^{3} +$$$$22\!\cdots\!21$$$$T^{4} )^{2}$$
$79$ $$1 - 828257339524 T^{2} +$$$$28\!\cdots\!06$$$$T^{4} -$$$$48\!\cdots\!84$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$( 1 + 1732504 T + 1400265667422 T^{2} + 566425504623285976 T^{3} +$$$$10\!\cdots\!61$$$$T^{4} )^{2}$$
$89$ $$( 1 + 380612 T + 970422538278 T^{2} + 189157043115248132 T^{3} +$$$$24\!\cdots\!21$$$$T^{4} )^{2}$$
$97$ $$( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$