# Properties

 Label 288.4.d.d Level $288$ Weight $4$ Character orbit 288.d Analytic conductor $16.993$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 288.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.9925500817$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8248384.1 Defining polynomial: $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ x^6 + x^4 - 12*x^3 + 4*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{15}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + ( - \beta_1 - 5) q^{7}+O(q^{10})$$ q - b2 * q^5 + (-b1 - 5) * q^7 $$q - \beta_{2} q^{5} + ( - \beta_1 - 5) q^{7} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{11} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{13} + (\beta_{5} + 3 \beta_1 - 8) q^{17} + ( - 3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{19} + ( - 2 \beta_1 + 54) q^{23} + (2 \beta_{5} - 2 \beta_1 - 19) q^{25} + (4 \beta_{4} + 2 \beta_{3} + 5 \beta_{2}) q^{29} + (2 \beta_{5} - \beta_1 + 105) q^{31} + ( - 7 \beta_{4} - 2 \beta_{3} + 14 \beta_{2}) q^{35} + (6 \beta_{4} - 11 \beta_{3} - 3 \beta_{2}) q^{37} + (\beta_{5} - 13 \beta_1 - 44) q^{41} + ( - 5 \beta_{4} - 10 \beta_{3} - 18 \beta_{2}) q^{43} + (2 \beta_{5} - 4 \beta_1 - 70) q^{47} + (4 \beta_{5} + 4 \beta_1 + 109) q^{49} + ( - 36 \beta_{4} - 10 \beta_{3} + 11 \beta_{2}) q^{53} + (6 \beta_{5} + 2 \beta_1 - 172) q^{55} + ( - \beta_{4} + 8 \beta_{3} - 8 \beta_{2}) q^{59} + ( - 46 \beta_{4} - 9 \beta_{3} + 3 \beta_{2}) q^{61} + ( - 3 \beta_{5} + 7 \beta_1 + 294) q^{65} + (5 \beta_{4} + 48 \beta_{2}) q^{67} + (6 \beta_{5} + 12 \beta_1 - 282) q^{71} + ( - 4 \beta_{5} - 20 \beta_1 + 154) q^{73} + (64 \beta_{4} - 20 \beta_{3} - 20 \beta_{2}) q^{77} + ( - 2 \beta_{5} - 9 \beta_1 + 5) q^{79} + (36 \beta_{4} + 2 \beta_{3} - 14 \beta_{2}) q^{83} + (72 \beta_{4} + 28 \beta_{3} + 42 \beta_{2}) q^{85} + (2 \beta_{5} + 6 \beta_1 + 38) q^{89} + ( - 2 \beta_{4} + 38 \beta_{3} - 18 \beta_{2}) q^{91} + ( - 4 \beta_{5} + 24 \beta_1 + 860) q^{95} + ( - 6 \beta_{5} - 2 \beta_1 - 406) q^{97}+O(q^{100})$$ q - b2 * q^5 + (-b1 - 5) * q^7 + (-2*b3 - 2*b2) * q^11 + (-2*b4 + b3 + 3*b2) * q^13 + (b5 + 3*b1 - 8) * q^17 + (-3*b4 - 2*b3 + 6*b2) * q^19 + (-2*b1 + 54) * q^23 + (2*b5 - 2*b1 - 19) * q^25 + (4*b4 + 2*b3 + 5*b2) * q^29 + (2*b5 - b1 + 105) * q^31 + (-7*b4 - 2*b3 + 14*b2) * q^35 + (6*b4 - 11*b3 - 3*b2) * q^37 + (b5 - 13*b1 - 44) * q^41 + (-5*b4 - 10*b3 - 18*b2) * q^43 + (2*b5 - 4*b1 - 70) * q^47 + (4*b5 + 4*b1 + 109) * q^49 + (-36*b4 - 10*b3 + 11*b2) * q^53 + (6*b5 + 2*b1 - 172) * q^55 + (-b4 + 8*b3 - 8*b2) * q^59 + (-46*b4 - 9*b3 + 3*b2) * q^61 + (-3*b5 + 7*b1 + 294) * q^65 + (5*b4 + 48*b2) * q^67 + (6*b5 + 12*b1 - 282) * q^71 + (-4*b5 - 20*b1 + 154) * q^73 + (64*b4 - 20*b3 - 20*b2) * q^77 + (-2*b5 - 9*b1 + 5) * q^79 + (36*b4 + 2*b3 - 14*b2) * q^83 + (72*b4 + 28*b3 + 42*b2) * q^85 + (2*b5 + 6*b1 + 38) * q^89 + (-2*b4 + 38*b3 - 18*b2) * q^91 + (-4*b5 + 24*b1 + 860) * q^95 + (-6*b5 - 2*b1 - 406) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 28 q^{7}+O(q^{10})$$ 6 * q - 28 * q^7 $$6 q - 28 q^{7} - 52 q^{17} + 328 q^{23} - 106 q^{25} + 636 q^{31} - 236 q^{41} - 408 q^{47} + 654 q^{49} - 1024 q^{55} + 1744 q^{65} - 1704 q^{71} + 956 q^{73} + 44 q^{79} + 220 q^{89} + 5104 q^{95} - 2444 q^{97}+O(q^{100})$$ 6 * q - 28 * q^7 - 52 * q^17 + 328 * q^23 - 106 * q^25 + 636 * q^31 - 236 * q^41 - 408 * q^47 + 654 * q^49 - 1024 * q^55 + 1744 * q^65 - 1704 * q^71 + 956 * q^73 + 44 * q^79 + 220 * q^89 + 5104 * q^95 - 2444 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} - 3\nu^{2} - 12\nu + 1$$ v^4 - 3*v^2 - 12*v + 1 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 6\nu^{4} - 13\nu^{3} + 42\nu^{2} - 104\nu + 96 ) / 16$$ (-v^5 + 6*v^4 - 13*v^3 + 42*v^2 - 104*v + 96) / 16 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 6\nu^{4} - 51\nu^{3} - 42\nu^{2} + 40\nu + 288 ) / 16$$ (v^5 - 6*v^4 - 51*v^3 - 42*v^2 + 40*v + 288) / 16 $$\beta_{4}$$ $$=$$ $$( 3\nu^{5} + 6\nu^{4} - 9\nu^{3} - 6\nu^{2} - 24\nu + 96 ) / 8$$ (3*v^5 + 6*v^4 - 9*v^3 - 6*v^2 - 24*v + 96) / 8 $$\beta_{5}$$ $$=$$ $$-3\nu^{5} + 5\nu^{4} - 3\nu^{3} + 21\nu^{2} - 24\nu + 7$$ -3*v^5 + 5*v^4 - 3*v^3 + 21*v^2 - 24*v + 7
 $$\nu$$ $$=$$ $$( \beta_{5} + 6\beta_{4} - 12\beta_{2} - 5\beta _1 - 2 ) / 96$$ (b5 + 6*b4 - 12*b2 - 5*b1 - 2) / 96 $$\nu^{2}$$ $$=$$ $$( 2\beta_{4} - 3\beta_{3} + 9\beta_{2} - 6\beta _1 - 18 ) / 48$$ (2*b4 - 3*b3 + 9*b2 - 6*b1 - 18) / 48 $$\nu^{3}$$ $$=$$ $$( -\beta_{5} - 6\beta_{4} - 24\beta_{3} - 12\beta_{2} + 5\beta _1 + 578 ) / 96$$ (-b5 - 6*b4 - 24*b3 - 12*b2 + 5*b1 + 578) / 96 $$\nu^{4}$$ $$=$$ $$( 2\beta_{5} + 14\beta_{4} - 3\beta_{3} - 15\beta_{2} - 38 ) / 16$$ (2*b5 + 14*b4 - 3*b3 - 15*b2 - 38) / 16 $$\nu^{5}$$ $$=$$ $$( -19\beta_{5} + 126\beta_{4} - 48\beta_{3} + 84\beta_{2} - 49\beta _1 - 970 ) / 96$$ (-19*b5 + 126*b4 - 48*b3 + 84*b2 - 49*b1 - 970) / 96

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −1.24181 − 1.56777i −0.641412 − 1.89436i 1.88322 + 0.673417i 1.88322 − 0.673417i −0.641412 + 1.89436i −1.24181 + 1.56777i
0 0 0 18.5422i 0 −9.32669 0 0 0
145.2 0 0 0 9.15486i 0 −27.4175 0 0 0
145.3 0 0 0 0.612661i 0 22.7441 0 0 0
145.4 0 0 0 0.612661i 0 22.7441 0 0 0
145.5 0 0 0 9.15486i 0 −27.4175 0 0 0
145.6 0 0 0 18.5422i 0 −9.32669 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.4.d.d 6
3.b odd 2 1 96.4.d.a 6
4.b odd 2 1 72.4.d.d 6
8.b even 2 1 inner 288.4.d.d 6
8.d odd 2 1 72.4.d.d 6
12.b even 2 1 24.4.d.a 6
16.e even 4 1 2304.4.a.bu 3
16.e even 4 1 2304.4.a.bw 3
16.f odd 4 1 2304.4.a.bt 3
16.f odd 4 1 2304.4.a.bv 3
24.f even 2 1 24.4.d.a 6
24.h odd 2 1 96.4.d.a 6
48.i odd 4 1 768.4.a.q 3
48.i odd 4 1 768.4.a.t 3
48.k even 4 1 768.4.a.r 3
48.k even 4 1 768.4.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 12.b even 2 1
24.4.d.a 6 24.f even 2 1
72.4.d.d 6 4.b odd 2 1
72.4.d.d 6 8.d odd 2 1
96.4.d.a 6 3.b odd 2 1
96.4.d.a 6 24.h odd 2 1
288.4.d.d 6 1.a even 1 1 trivial
288.4.d.d 6 8.b even 2 1 inner
768.4.a.q 3 48.i odd 4 1
768.4.a.r 3 48.k even 4 1
768.4.a.s 3 48.k even 4 1
768.4.a.t 3 48.i odd 4 1
2304.4.a.bt 3 16.f odd 4 1
2304.4.a.bu 3 16.e even 4 1
2304.4.a.bv 3 16.f odd 4 1
2304.4.a.bw 3 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 428T_{5}^{4} + 28976T_{5}^{2} + 10816$$ acting on $$S_{4}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 428 T^{4} + 28976 T^{2} + \cdots + 10816$$
$7$ $$(T^{3} + 14 T^{2} - 580 T - 5816)^{2}$$
$11$ $$T^{6} + 5632 T^{4} + \cdots + 2415919104$$
$13$ $$T^{6} + 4912 T^{4} + \cdots + 3121680384$$
$17$ $$(T^{3} + 26 T^{2} - 11124 T - 477576)^{2}$$
$19$ $$T^{6} + 22960 T^{4} + \cdots + 75488661504$$
$23$ $$(T^{3} - 164 T^{2} + 6384 T - 45504)^{2}$$
$29$ $$T^{6} + 22348 T^{4} + \cdots + 3766031424$$
$31$ $$(T^{3} - 318 T^{2} + 4476 T + 3749624)^{2}$$
$37$ $$T^{6} + 179776 T^{4} + \cdots + 6879707136$$
$41$ $$(T^{3} + 118 T^{2} - 117300 T - 19985976)^{2}$$
$43$ $$T^{6} + 229552 T^{4} + \cdots + 73984219582464$$
$47$ $$(T^{3} + 204 T^{2} - 27792 T - 1964736)^{2}$$
$53$ $$T^{6} + \cdots + 427051482970176$$
$59$ $$T^{6} + 138416 T^{4} + \cdots + 72651484205056$$
$61$ $$T^{6} + 902016 T^{4} + \cdots + 10\!\cdots\!56$$
$67$ $$T^{6} + 1054512 T^{4} + \cdots + 10\!\cdots\!84$$
$71$ $$(T^{3} + 852 T^{2} - 66960 T - 85084992)^{2}$$
$73$ $$(T^{3} - 478 T^{2} - 255956 T + 120833304)^{2}$$
$79$ $$(T^{3} - 22 T^{2} - 71524 T + 7902616)^{2}$$
$83$ $$T^{6} + 520448 T^{4} + \cdots + 14\!\cdots\!96$$
$89$ $$(T^{3} - 110 T^{2} - 41364 T - 1423656)^{2}$$
$97$ $$(T^{3} + 1222 T^{2} + 251660 T - 74802424)^{2}$$