# Properties

 Label 288.2.i.f Level 288 Weight 2 Character orbit 288.i Analytic conductor 2.300 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 288.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.170772624.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{3} + ( 1 - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{7} + ( \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{3} + ( 1 - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{7} + ( \beta_{6} + \beta_{7} ) q^{9} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{11} + ( -1 + \beta_{4} + \beta_{7} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{15} + ( 1 + \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{19} + ( 3 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{21} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{23} + ( 1 - 4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{25} + ( \beta_{2} - \beta_{3} - \beta_{5} ) q^{27} + ( -1 + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{29} + ( 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( 3 - 5 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{33} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{35} + 4 q^{37} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{39} + ( -1 + \beta_{4} ) q^{41} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{43} + ( -9 + \beta_{4} - \beta_{7} ) q^{45} + ( -\beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{47} + ( -8 + 8 \beta_{4} - 3 \beta_{7} ) q^{49} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{51} -4 q^{53} + ( -\beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{55} + ( -3 + 8 \beta_{4} + \beta_{6} - \beta_{7} ) q^{57} + ( -\beta_{2} + 2 \beta_{3} + 4 \beta_{5} ) q^{59} + ( -3 - 5 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{61} + ( -6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} ) q^{63} + ( -1 - 7 \beta_{4} - \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{2} - 2 \beta_{3} - 4 \beta_{5} ) q^{67} + ( 3 - 5 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 7 - \beta_{6} ) q^{73} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( 3 - 3 \beta_{4} - 3 \beta_{7} ) q^{77} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{79} + ( -9 + 8 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{81} + ( 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{83} + ( -8 + 8 \beta_{4} ) q^{85} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{87} + ( 6 - 2 \beta_{6} ) q^{89} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{91} + ( 9 + 3 \beta_{4} - 3 \beta_{7} ) q^{93} + ( 4 \beta_{1} - 4 \beta_{3} - 8 \beta_{5} ) q^{95} -9 \beta_{4} q^{97} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{5} - 6q^{9} + O(q^{10})$$ $$8q + 2q^{5} - 6q^{9} - 6q^{13} + 4q^{17} + 30q^{21} - 14q^{25} + 10q^{29} + 12q^{33} + 32q^{37} - 4q^{41} - 66q^{45} - 26q^{49} - 32q^{53} + 6q^{57} - 26q^{61} - 30q^{65} - 6q^{69} + 60q^{73} + 18q^{77} - 30q^{81} - 32q^{85} + 56q^{89} + 90q^{93} - 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} + 2 \nu^{3} + 4 \nu^{2} + 4 \nu - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - \nu^{5} + 6 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 4 \nu + 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 3 \nu^{5} - 6 \nu^{3} + 4 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{7} + 5 \nu^{6} - 7 \nu^{5} + 6 \nu^{4} - 10 \nu^{3} + 24 \nu^{2} - 28 \nu + 32$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 3 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 16 \nu + 20$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} + 6 \nu^{5} - 5 \nu^{4} + 6 \nu^{3} - 14 \nu^{2} + 20 \nu - 24$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} - 7 \nu^{6} + 13 \nu^{5} - 10 \nu^{4} + 10 \nu^{3} - 28 \nu^{2} + 44 \nu - 56$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{7} + 4 \beta_{5} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - \beta_{6} + 3 \beta_{5} - 8 \beta_{4} - 3 \beta_{3} + 3 \beta_{1} + 3$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} - 2 \beta_{3} + 5 \beta_{2} + 4 \beta_{1} - 3$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{6} - 10 \beta_{5} + 16 \beta_{4} - 5 \beta_{3} - \beta_{2} + 7 \beta_{1} + 18$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{6} - 3 \beta_{5} + \beta_{3} - 2 \beta_{2} + 7 \beta_{1} + 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 13 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 4 \beta_{3} + \beta_{2} - 10 \beta_{1} + 3$$$$)/6$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.335728 − 1.37379i 1.41203 + 0.0786378i 0.774115 + 1.18353i −1.02187 + 0.977642i 0.335728 + 1.37379i 1.41203 − 0.0786378i 0.774115 − 1.18353i −1.02187 − 0.977642i
0 −1.35760 + 1.07561i 0 −1.18614 2.05446i 0 −1.10489 + 1.91373i 0 0.686141 2.92048i 0
97.2 0 −0.637910 1.61030i 0 1.68614 + 2.92048i 0 −2.35143 + 4.07279i 0 −2.18614 + 2.05446i 0
97.3 0 0.637910 + 1.61030i 0 1.68614 + 2.92048i 0 2.35143 4.07279i 0 −2.18614 + 2.05446i 0
97.4 0 1.35760 1.07561i 0 −1.18614 2.05446i 0 1.10489 1.91373i 0 0.686141 2.92048i 0
193.1 0 −1.35760 1.07561i 0 −1.18614 + 2.05446i 0 −1.10489 1.91373i 0 0.686141 + 2.92048i 0
193.2 0 −0.637910 + 1.61030i 0 1.68614 2.92048i 0 −2.35143 4.07279i 0 −2.18614 2.05446i 0
193.3 0 0.637910 1.61030i 0 1.68614 2.92048i 0 2.35143 + 4.07279i 0 −2.18614 2.05446i 0
193.4 0 1.35760 + 1.07561i 0 −1.18614 + 2.05446i 0 1.10489 + 1.91373i 0 0.686141 + 2.92048i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.i.f 8
3.b odd 2 1 864.2.i.f 8
4.b odd 2 1 inner 288.2.i.f 8
8.b even 2 1 576.2.i.n 8
8.d odd 2 1 576.2.i.n 8
9.c even 3 1 inner 288.2.i.f 8
9.c even 3 1 2592.2.a.u 4
9.d odd 6 1 864.2.i.f 8
9.d odd 6 1 2592.2.a.x 4
12.b even 2 1 864.2.i.f 8
24.f even 2 1 1728.2.i.n 8
24.h odd 2 1 1728.2.i.n 8
36.f odd 6 1 inner 288.2.i.f 8
36.f odd 6 1 2592.2.a.u 4
36.h even 6 1 864.2.i.f 8
36.h even 6 1 2592.2.a.x 4
72.j odd 6 1 1728.2.i.n 8
72.j odd 6 1 5184.2.a.cc 4
72.l even 6 1 1728.2.i.n 8
72.l even 6 1 5184.2.a.cc 4
72.n even 6 1 576.2.i.n 8
72.n even 6 1 5184.2.a.cf 4
72.p odd 6 1 576.2.i.n 8
72.p odd 6 1 5184.2.a.cf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 1.a even 1 1 trivial
288.2.i.f 8 4.b odd 2 1 inner
288.2.i.f 8 9.c even 3 1 inner
288.2.i.f 8 36.f odd 6 1 inner
576.2.i.n 8 8.b even 2 1
576.2.i.n 8 8.d odd 2 1
576.2.i.n 8 72.n even 6 1
576.2.i.n 8 72.p odd 6 1
864.2.i.f 8 3.b odd 2 1
864.2.i.f 8 9.d odd 6 1
864.2.i.f 8 12.b even 2 1
864.2.i.f 8 36.h even 6 1
1728.2.i.n 8 24.f even 2 1
1728.2.i.n 8 24.h odd 2 1
1728.2.i.n 8 72.j odd 6 1
1728.2.i.n 8 72.l even 6 1
2592.2.a.u 4 9.c even 3 1
2592.2.a.u 4 36.f odd 6 1
2592.2.a.x 4 9.d odd 6 1
2592.2.a.x 4 36.h even 6 1
5184.2.a.cc 4 72.j odd 6 1
5184.2.a.cc 4 72.l even 6 1
5184.2.a.cf 4 72.n even 6 1
5184.2.a.cf 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64$$ $$T_{7}^{8} + 27 T_{7}^{6} + 621 T_{7}^{4} + 2916 T_{7}^{2} + 11664$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T^{2} + 12 T^{4} + 27 T^{6} + 81 T^{8}$$
$5$ $$( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8} )^{2}$$
$7$ $$1 - T^{2} - 23 T^{4} + 74 T^{6} - 1874 T^{8} + 3626 T^{10} - 55223 T^{12} - 117649 T^{14} + 5764801 T^{16}$$
$11$ $$1 - 8 T^{2} + 103 T^{4} + 2248 T^{6} - 20864 T^{8} + 272008 T^{10} + 1508023 T^{12} - 14172488 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 3 T - 11 T^{2} - 18 T^{3} + 114 T^{4} - 234 T^{5} - 1859 T^{6} + 6591 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - T + 26 T^{2} - 17 T^{3} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 31 T^{2} + 888 T^{4} + 11191 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 - 65 T^{2} + 2185 T^{4} - 63830 T^{6} + 1646734 T^{8} - 33766070 T^{10} + 611452585 T^{12} - 9622332785 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 5 T - 31 T^{2} + 10 T^{3} + 1570 T^{4} + 290 T^{5} - 26071 T^{6} - 121945 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 + 11 T^{2} - 1163 T^{4} - 7018 T^{6} + 608854 T^{8} - 6744298 T^{10} - 1074054923 T^{12} + 9762540491 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{8}$$
$41$ $$( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$1 - 136 T^{2} + 10471 T^{4} - 588472 T^{6} + 26782720 T^{8} - 1088084728 T^{10} + 35798265271 T^{12} - 859705374664 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 - 161 T^{2} + 15097 T^{4} - 1031366 T^{6} + 55019806 T^{8} - 2278287494 T^{10} + 73668544057 T^{12} - 1735453667969 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{8}$$
$59$ $$1 - 56 T^{2} - 4313 T^{4} - 27272 T^{6} + 32453824 T^{8} - 94933832 T^{10} - 52262177993 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 + 13 T + 79 T^{2} - 416 T^{3} - 5930 T^{4} - 25376 T^{5} + 293959 T^{6} + 2950753 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 88 T^{2} - 2873 T^{4} - 144232 T^{6} + 57806752 T^{8} - 647457448 T^{10} - 57894170633 T^{12} - 7960337630872 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 + 176 T^{2} + 16638 T^{4} + 887216 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 15 T + 194 T^{2} - 1095 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$1 - 181 T^{2} + 12757 T^{4} - 1361482 T^{6} + 156748534 T^{8} - 8497009162 T^{10} + 496886183317 T^{12} - 43998829449301 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 - 125 T^{2} + 6925 T^{4} + 634750 T^{6} - 79408946 T^{8} + 4372792750 T^{10} + 328648872925 T^{12} - 40867546671125 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 14 T + 194 T^{2} - 1246 T^{3} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4} )^{4}$$