# Properties

 Label 576.3.q.e Level $576$ Weight $3$ Character orbit 576.q Analytic conductor $15.695$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 576.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6948632272$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 2 \beta_1 - 1) q^{3} + (3 \beta_1 + 3) q^{5} + (3 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + (b3 + 2*b1 - 1) * q^3 + (3*b1 + 3) * q^5 + (3*b2 - b1) * q^7 + (-2*b3 + 4*b2 + 3) * q^9 $$q + (\beta_{3} + 2 \beta_1 - 1) q^{3} + (3 \beta_1 + 3) q^{5} + (3 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} + 3) q^{9} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 6) q^{11} + (6 \beta_{3} - 6 \beta_{2} - 5 \beta_1 + 5) q^{13} + (3 \beta_{3} + 3 \beta_{2} + 9 \beta_1 - 9) q^{15} + (2 \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 6) q^{17} + (6 \beta_{3} - 10) q^{19} + ( - 6 \beta_{3} + 2 \beta_{2} + 17 \beta_1 + 2) q^{21} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 3) q^{23} + 2 \beta_1 q^{25} + ( - 3 \beta_{3} + 30 \beta_1 - 15) q^{27} + (4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 6) q^{29} + ( - 9 \beta_{3} + 9 \beta_{2} + 19 \beta_1 - 19) q^{31} + (6 \beta_{3} + 3 \beta_1 + 12) q^{33} + ( - 9 \beta_{3} + 18 \beta_{2} - 6 \beta_1 + 3) q^{35} + (6 \beta_{3} - 32) q^{37} + (11 \beta_{3} + \beta_{2} - 31 \beta_1 + 41) q^{39} + ( - 6 \beta_{3} - 6 \beta_{2} - 21 \beta_1 - 21) q^{41} + (9 \beta_{2} - 23 \beta_1) q^{43} + ( - 18 \beta_{3} + 18 \beta_{2} + 9 \beta_1 + 9) q^{45} + (14 \beta_{3} - 7 \beta_{2} + 9 \beta_1 - 18) q^{47} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 6) q^{49} + (12 \beta_{2} - 24 \beta_1 - 6) q^{51} + ( - 10 \beta_{3} + 20 \beta_{2} - 60 \beta_1 + 30) q^{53} + (9 \beta_{3} + 27) q^{55} + ( - 16 \beta_{3} + 12 \beta_{2} - 20 \beta_1 + 46) q^{57} + (13 \beta_{3} + 13 \beta_{2} + 21 \beta_1 + 21) q^{59} + ( - 18 \beta_{2} - 31 \beta_1) q^{61} + (4 \beta_{3} + 7 \beta_{2} + 33 \beta_1 - 72) q^{63} + (36 \beta_{3} - 18 \beta_{2} - 15 \beta_1 + 30) q^{65} + (9 \beta_{3} - 9 \beta_{2} + 53 \beta_1 - 53) q^{67} + ( - 6 \beta_{2} - 15 \beta_1 + 3) q^{69} + ( - 8 \beta_{3} + 16 \beta_{2} + 60 \beta_1 - 30) q^{71} + ( - 18 \beta_{3} - 52) q^{73} + (2 \beta_{2} + 2 \beta_1 - 4) q^{75} + (8 \beta_{3} + 8 \beta_{2} + 15 \beta_1 + 15) q^{77} + (15 \beta_{2} - 7 \beta_1) q^{79} + ( - 12 \beta_{3} + 24 \beta_{2} - 63) q^{81} + (10 \beta_{3} - 5 \beta_{2} + 63 \beta_1 - 126) q^{83} + (18 \beta_{3} - 18 \beta_{2} + 54 \beta_1 - 54) q^{85} + ( - 6 \beta_{3} + 9 \beta_{2} - 21 \beta_1 + 24) q^{87} + ( - 22 \beta_{3} + 44 \beta_{2} - 60 \beta_1 + 30) q^{89} + (9 \beta_{3} + 103) q^{91} + ( - 28 \beta_{3} + 10 \beta_{2} + 35 \beta_1 - 73) q^{93} + (18 \beta_{3} + 18 \beta_{2} - 30 \beta_1 - 30) q^{95} + ( - 42 \beta_{2} + 7 \beta_1) q^{97} + (6 \beta_{3} + 15 \beta_{2} + 27 \beta_1 + 18) q^{99}+O(q^{100})$$ q + (b3 + 2*b1 - 1) * q^3 + (3*b1 + 3) * q^5 + (3*b2 - b1) * q^7 + (-2*b3 + 4*b2 + 3) * q^9 + (2*b3 - b2 - 3*b1 + 6) * q^11 + (6*b3 - 6*b2 - 5*b1 + 5) * q^13 + (3*b3 + 3*b2 + 9*b1 - 9) * q^15 + (2*b3 - 4*b2 + 12*b1 - 6) * q^17 + (6*b3 - 10) * q^19 + (-6*b3 + 2*b2 + 17*b1 + 2) * q^21 + (-b3 - b2 - 3*b1 - 3) * q^23 + 2*b1 * q^25 + (-3*b3 + 30*b1 - 15) * q^27 + (4*b3 - 2*b2 + 3*b1 - 6) * q^29 + (-9*b3 + 9*b2 + 19*b1 - 19) * q^31 + (6*b3 + 3*b1 + 12) * q^33 + (-9*b3 + 18*b2 - 6*b1 + 3) * q^35 + (6*b3 - 32) * q^37 + (11*b3 + b2 - 31*b1 + 41) * q^39 + (-6*b3 - 6*b2 - 21*b1 - 21) * q^41 + (9*b2 - 23*b1) * q^43 + (-18*b3 + 18*b2 + 9*b1 + 9) * q^45 + (14*b3 - 7*b2 + 9*b1 - 18) * q^47 + (6*b3 - 6*b2 + 6*b1 - 6) * q^49 + (12*b2 - 24*b1 - 6) * q^51 + (-10*b3 + 20*b2 - 60*b1 + 30) * q^53 + (9*b3 + 27) * q^55 + (-16*b3 + 12*b2 - 20*b1 + 46) * q^57 + (13*b3 + 13*b2 + 21*b1 + 21) * q^59 + (-18*b2 - 31*b1) * q^61 + (4*b3 + 7*b2 + 33*b1 - 72) * q^63 + (36*b3 - 18*b2 - 15*b1 + 30) * q^65 + (9*b3 - 9*b2 + 53*b1 - 53) * q^67 + (-6*b2 - 15*b1 + 3) * q^69 + (-8*b3 + 16*b2 + 60*b1 - 30) * q^71 + (-18*b3 - 52) * q^73 + (2*b2 + 2*b1 - 4) * q^75 + (8*b3 + 8*b2 + 15*b1 + 15) * q^77 + (15*b2 - 7*b1) * q^79 + (-12*b3 + 24*b2 - 63) * q^81 + (10*b3 - 5*b2 + 63*b1 - 126) * q^83 + (18*b3 - 18*b2 + 54*b1 - 54) * q^85 + (-6*b3 + 9*b2 - 21*b1 + 24) * q^87 + (-22*b3 + 44*b2 - 60*b1 + 30) * q^89 + (9*b3 + 103) * q^91 + (-28*b3 + 10*b2 + 35*b1 - 73) * q^93 + (18*b3 + 18*b2 - 30*b1 - 30) * q^95 + (-42*b2 + 7*b1) * q^97 + (6*b3 + 15*b2 + 27*b1 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18 q^{5} - 2 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q + 18 * q^5 - 2 * q^7 + 12 * q^9 $$4 q + 18 q^{5} - 2 q^{7} + 12 q^{9} + 18 q^{11} + 10 q^{13} - 18 q^{15} - 40 q^{19} + 42 q^{21} - 18 q^{23} + 4 q^{25} - 18 q^{29} - 38 q^{31} + 54 q^{33} - 128 q^{37} + 102 q^{39} - 126 q^{41} - 46 q^{43} + 54 q^{45} - 54 q^{47} - 12 q^{49} - 72 q^{51} + 108 q^{55} + 144 q^{57} + 126 q^{59} - 62 q^{61} - 222 q^{63} + 90 q^{65} - 106 q^{67} - 18 q^{69} - 208 q^{73} - 12 q^{75} + 90 q^{77} - 14 q^{79} - 252 q^{81} - 378 q^{83} - 108 q^{85} + 54 q^{87} + 412 q^{91} - 222 q^{93} - 180 q^{95} + 14 q^{97} + 126 q^{99}+O(q^{100})$$ 4 * q + 18 * q^5 - 2 * q^7 + 12 * q^9 + 18 * q^11 + 10 * q^13 - 18 * q^15 - 40 * q^19 + 42 * q^21 - 18 * q^23 + 4 * q^25 - 18 * q^29 - 38 * q^31 + 54 * q^33 - 128 * q^37 + 102 * q^39 - 126 * q^41 - 46 * q^43 + 54 * q^45 - 54 * q^47 - 12 * q^49 - 72 * q^51 + 108 * q^55 + 144 * q^57 + 126 * q^59 - 62 * q^61 - 222 * q^63 + 90 * q^65 - 106 * q^67 - 18 * q^69 - 208 * q^73 - 12 * q^75 + 90 * q^77 - 14 * q^79 - 252 * q^81 - 378 * q^83 - 108 * q^85 + 54 * q^87 + 412 * q^91 - 222 * q^93 - 180 * q^95 + 14 * q^97 + 126 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1 - \beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 −2.44949 1.73205i 0 4.50000 2.59808i 0 −4.17423 + 7.22999i 0 3.00000 + 8.48528i 0
65.2 0 2.44949 1.73205i 0 4.50000 2.59808i 0 3.17423 5.49794i 0 3.00000 8.48528i 0
257.1 0 −2.44949 + 1.73205i 0 4.50000 + 2.59808i 0 −4.17423 7.22999i 0 3.00000 8.48528i 0
257.2 0 2.44949 + 1.73205i 0 4.50000 + 2.59808i 0 3.17423 + 5.49794i 0 3.00000 + 8.48528i 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.q.e 4
3.b odd 2 1 1728.3.q.c 4
4.b odd 2 1 576.3.q.f 4
8.b even 2 1 144.3.q.c 4
8.d odd 2 1 18.3.d.a 4
9.c even 3 1 1728.3.q.c 4
9.d odd 6 1 inner 576.3.q.e 4
12.b even 2 1 1728.3.q.d 4
24.f even 2 1 54.3.d.a 4
24.h odd 2 1 432.3.q.d 4
36.f odd 6 1 1728.3.q.d 4
36.h even 6 1 576.3.q.f 4
40.e odd 2 1 450.3.i.b 4
40.k even 4 2 450.3.k.a 8
72.j odd 6 1 144.3.q.c 4
72.j odd 6 1 1296.3.e.g 4
72.l even 6 1 18.3.d.a 4
72.l even 6 1 162.3.b.a 4
72.n even 6 1 432.3.q.d 4
72.n even 6 1 1296.3.e.g 4
72.p odd 6 1 54.3.d.a 4
72.p odd 6 1 162.3.b.a 4
120.m even 2 1 1350.3.i.b 4
120.q odd 4 2 1350.3.k.a 8
360.z odd 6 1 1350.3.i.b 4
360.bd even 6 1 450.3.i.b 4
360.bo even 12 2 1350.3.k.a 8
360.bt odd 12 2 450.3.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 8.d odd 2 1
18.3.d.a 4 72.l even 6 1
54.3.d.a 4 24.f even 2 1
54.3.d.a 4 72.p odd 6 1
144.3.q.c 4 8.b even 2 1
144.3.q.c 4 72.j odd 6 1
162.3.b.a 4 72.l even 6 1
162.3.b.a 4 72.p odd 6 1
432.3.q.d 4 24.h odd 2 1
432.3.q.d 4 72.n even 6 1
450.3.i.b 4 40.e odd 2 1
450.3.i.b 4 360.bd even 6 1
450.3.k.a 8 40.k even 4 2
450.3.k.a 8 360.bt odd 12 2
576.3.q.e 4 1.a even 1 1 trivial
576.3.q.e 4 9.d odd 6 1 inner
576.3.q.f 4 4.b odd 2 1
576.3.q.f 4 36.h even 6 1
1296.3.e.g 4 72.j odd 6 1
1296.3.e.g 4 72.n even 6 1
1350.3.i.b 4 120.m even 2 1
1350.3.i.b 4 360.z odd 6 1
1350.3.k.a 8 120.q odd 4 2
1350.3.k.a 8 360.bo even 12 2
1728.3.q.c 4 3.b odd 2 1
1728.3.q.c 4 9.c even 3 1
1728.3.q.d 4 12.b even 2 1
1728.3.q.d 4 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{2} - 9T_{5} + 27$$ T5^2 - 9*T5 + 27 $$T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809$$ T7^4 + 2*T7^3 + 57*T7^2 - 106*T7 + 2809

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 6T^{2} + 81$$
$5$ $$(T^{2} - 9 T + 27)^{2}$$
$7$ $$T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$11$ $$T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81$$
$13$ $$T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481$$
$17$ $$T^{4} + 360T^{2} + 1296$$
$19$ $$(T^{2} + 20 T - 116)^{2}$$
$23$ $$T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81$$
$29$ $$T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025$$
$31$ $$T^{4} + 38 T^{3} + 1569 T^{2} + \cdots + 15625$$
$37$ $$(T^{2} + 64 T + 808)^{2}$$
$41$ $$T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625$$
$43$ $$T^{4} + 46 T^{3} + 2073 T^{2} + \cdots + 1849$$
$47$ $$T^{4} + 54 T^{3} + 333 T^{2} + \cdots + 408321$$
$53$ $$T^{4} + 9000 T^{2} + 810000$$
$59$ $$T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961$$
$61$ $$T^{4} + 62 T^{3} + 4827 T^{2} + \cdots + 966289$$
$67$ $$T^{4} + 106 T^{3} + 8913 T^{2} + \cdots + 5396329$$
$71$ $$T^{4} + 7704 T^{2} + \cdots + 2396304$$
$73$ $$(T^{2} + 104 T + 760)^{2}$$
$79$ $$T^{4} + 14 T^{3} + 1497 T^{2} + \cdots + 1692601$$
$83$ $$T^{4} + 378 T^{3} + \cdots + 131262849$$
$89$ $$T^{4} + 22824 T^{2} + \cdots + 36144144$$
$97$ $$T^{4} - 14 T^{3} + \cdots + 110986225$$