# Properties

 Label 576.2.k.b Level $576$ Weight $2$ Character orbit 576.k Analytic conductor $4.599$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(145,576)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.k (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q + (b6 + b4) * q^5 + (b7 - b6 - b5 - b4 + b3 + b1) * q^7 $$q + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{7} + ( - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{11} + ( - 2 \beta_{7} - \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{17} + ( - 2 \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + ( - \beta_{7} + \beta_{6}) q^{23} + (\beta_{5} + 2 \beta_1) q^{25} + ( - \beta_{7} - 2 \beta_{5} + \beta_{3} + 2) q^{29} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{31} + ( - \beta_{7} - 3 \beta_{5} - \beta_{2} - \beta_1 + 3) q^{35} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{37} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{41} + (\beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{43} + ( - \beta_{7} - \beta_{6}) q^{47} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{49} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2) q^{53} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5}) q^{55} + (4 \beta_{5} + 4) q^{59} + (4 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{61} + (4 \beta_{7} + 4 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{65} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{67} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{71} + (4 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{73} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 3 \beta_{2} + 3) q^{79} + ( - 3 \beta_{7} + 5 \beta_{5} - 4 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{83} + (6 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 2) q^{85} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{89} + ( - 4 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{91} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 6) q^{95} + ( - 4 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{97}+O(q^{100})$$ q + (b6 + b4) * q^5 + (b7 - b6 - b5 - b4 + b3 + b1) * q^7 + (-b6 - b5 - b2 + b1 - 1) * q^11 + (-2*b7 - b2 - b1) * q^13 + (-2*b7 - 2*b6 - b4 - b3 - 2*b2) * q^17 + (-2*b7 - b5 - 2*b3 - b2 - b1 + 1) * q^19 + (-b7 + b6) * q^23 + (b5 + 2*b1) * q^25 + (-b7 - 2*b5 + b3 + 2) * q^29 + (b7 + b6 + b4 + b3 + b2 - 3) * q^31 + (-b7 - 3*b5 - b2 - b1 + 3) * q^35 + (-2*b5 + 2*b4 + b2 - b1 - 2) * q^37 + (-b4 + b3 + 2*b1) * q^41 + (b5 + 2*b4 - b2 + b1 + 1) * q^43 + (-b7 - b6) * q^47 + (-2*b4 - 2*b3 - 2*b2 - 1) * q^49 + (-3*b6 - 2*b5 - b4 - 2) * q^53 + (-2*b7 + 2*b6 + 4*b5) * q^55 + (4*b5 + 4) * q^59 + (4*b7 - 2*b5 + 2*b3 + b2 + b1 + 2) * q^61 + (4*b7 + 4*b6 + b4 + b3 + 2*b2 + 2) * q^65 + (-2*b5 - 2*b2 - 2*b1 + 2) * q^67 + (2*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 + 2*b1) * q^71 + (4*b7 - 4*b6 + 2*b5 - 2*b4 + 2*b3 + 2*b1) * q^73 + (2*b5 - 2*b3 - 2*b2 - 2*b1 - 2) * q^77 + (-b7 - b6 - b4 - b3 + 3*b2 + 3) * q^79 + (-3*b7 + 5*b5 - 4*b3 - b2 - b1 - 5) * q^83 + (6*b6 - 2*b5 + 2*b4 + 2*b2 - 2*b1 - 2) * q^85 + (2*b7 - 2*b6 + 2*b5 + 2*b4 - 2*b3) * q^89 + (-4*b6 + b5 - 2*b4 - b2 + b1 + 1) * q^91 + (3*b7 + 3*b6 + 2*b4 + 2*b3 + 2*b2 - 6) * q^95 + (-4*b7 - 4*b6 - 2*b4 - 2*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 24 q^{31} + 24 q^{35} - 16 q^{37} + 8 q^{43} - 8 q^{49} - 16 q^{53} + 32 q^{59} + 16 q^{61} + 16 q^{65} + 16 q^{67} - 16 q^{77} + 24 q^{79} - 40 q^{83} - 16 q^{85} + 8 q^{91} - 48 q^{95}+O(q^{100})$$ 8 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 24 * q^31 + 24 * q^35 - 16 * q^37 + 8 * q^43 - 8 * q^49 - 16 * q^53 + 32 * q^59 + 16 * q^61 + 16 * q^65 + 16 * q^67 - 16 * q^77 + 24 * q^79 - 40 * q^83 - 16 * q^85 + 8 * q^91 - 48 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} - 5\nu^{2} + 4\nu - 1$$ -v^4 + 2*v^3 - 5*v^2 + 4*v - 1 $$\beta_{3}$$ $$=$$ $$4\nu^{7} - 13\nu^{6} + 45\nu^{5} - 75\nu^{4} + 105\nu^{3} - 84\nu^{2} + 42\nu - 10$$ 4*v^7 - 13*v^6 + 45*v^5 - 75*v^4 + 105*v^3 - 84*v^2 + 42*v - 10 $$\beta_{4}$$ $$=$$ $$-4\nu^{7} + 15\nu^{6} - 51\nu^{5} + 95\nu^{4} - 135\nu^{3} + 120\nu^{2} - 64\nu + 14$$ -4*v^7 + 15*v^6 - 51*v^5 + 95*v^4 - 135*v^3 + 120*v^2 - 64*v + 14 $$\beta_{5}$$ $$=$$ $$8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31$$ 8*v^7 - 28*v^6 + 98*v^5 - 175*v^4 + 256*v^3 - 223*v^2 + 126*v - 31 $$\beta_{6}$$ $$=$$ $$-10\nu^{7} + 34\nu^{6} - 120\nu^{5} + 210\nu^{4} - 310\nu^{3} + 266\nu^{2} - 154\nu + 38$$ -10*v^7 + 34*v^6 - 120*v^5 + 210*v^4 - 310*v^3 + 266*v^2 - 154*v + 38 $$\beta_{7}$$ $$=$$ $$10\nu^{7} - 36\nu^{6} + 126\nu^{5} - 230\nu^{4} + 340\nu^{3} - 304\nu^{2} + 178\nu - 46$$ 10*v^7 - 36*v^6 + 126*v^5 - 230*v^4 + 340*v^3 - 304*v^2 + 178*v - 46
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta _1 - 3 ) / 2$$ (-b7 - b6 - b4 - b3 + b1 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} - 5\beta_{6} - 6\beta_{5} - 4\beta_{4} - 2\beta_{3} - 4\beta _1 - 10 ) / 4$$ (-b7 - 5*b6 - 6*b5 - 4*b4 - 2*b3 - 4*b1 - 10) / 4 $$\nu^{4}$$ $$=$$ $$( 4\beta_{7} - 6\beta_{5} + \beta_{4} + 3\beta_{3} - 2\beta_{2} - 5\beta _1 + 7 ) / 2$$ (4*b7 - 6*b5 + b4 + 3*b3 - 2*b2 - 5*b1 + 7) / 2 $$\nu^{5}$$ $$=$$ $$( 9\beta_{7} + 21\beta_{6} + 20\beta_{5} + 20\beta_{4} + 10\beta_{3} - 10\beta_{2} + 6\beta _1 + 52 ) / 4$$ (9*b7 + 21*b6 + 20*b5 + 20*b4 + 10*b3 - 10*b2 + 6*b1 + 52) / 4 $$\nu^{6}$$ $$=$$ $$( -16\beta_{7} + 12\beta_{6} + 45\beta_{5} + 9\beta_{4} - 11\beta_{3} + 5\beta_{2} + 22\beta _1 - 6 ) / 2$$ (-16*b7 + 12*b6 + 45*b5 + 9*b4 - 11*b3 + 5*b2 + 22*b1 - 6) / 2 $$\nu^{7}$$ $$=$$ $$( -71\beta_{7} - 69\beta_{6} - 66\beta_{4} - 60\beta_{3} + 70\beta_{2} + 14\beta _1 - 236 ) / 4$$ (-71*b7 - 69*b6 - 66*b4 - 60*b3 + 70*b2 + 14*b1 - 236) / 4

## Character values

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

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{5}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.0297061i 0.5 + 0.691860i 0.5 + 1.44392i 0.5 − 2.10607i 0.5 + 0.0297061i 0.5 − 0.691860i 0.5 − 1.44392i 0.5 + 2.10607i
0 0 0 −1.74912 + 1.74912i 0 2.55765i 0 0 0
145.2 0 0 0 −1.27133 + 1.27133i 0 0.158942i 0 0 0
145.3 0 0 0 0.334904 0.334904i 0 4.55765i 0 0 0
145.4 0 0 0 2.68554 2.68554i 0 2.15894i 0 0 0
433.1 0 0 0 −1.74912 1.74912i 0 2.55765i 0 0 0
433.2 0 0 0 −1.27133 1.27133i 0 0.158942i 0 0 0
433.3 0 0 0 0.334904 + 0.334904i 0 4.55765i 0 0 0
433.4 0 0 0 2.68554 + 2.68554i 0 2.15894i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.k.b 8
3.b odd 2 1 192.2.j.a 8
4.b odd 2 1 144.2.k.b 8
8.b even 2 1 1152.2.k.f 8
8.d odd 2 1 1152.2.k.c 8
12.b even 2 1 48.2.j.a 8
16.e even 4 1 inner 576.2.k.b 8
16.e even 4 1 1152.2.k.f 8
16.f odd 4 1 144.2.k.b 8
16.f odd 4 1 1152.2.k.c 8
24.f even 2 1 384.2.j.b 8
24.h odd 2 1 384.2.j.a 8
32.g even 8 1 9216.2.a.x 4
32.g even 8 1 9216.2.a.bn 4
32.h odd 8 1 9216.2.a.y 4
32.h odd 8 1 9216.2.a.bo 4
48.i odd 4 1 192.2.j.a 8
48.i odd 4 1 384.2.j.a 8
48.k even 4 1 48.2.j.a 8
48.k even 4 1 384.2.j.b 8
96.o even 8 1 3072.2.a.i 4
96.o even 8 1 3072.2.a.t 4
96.o even 8 2 3072.2.d.f 8
96.p odd 8 1 3072.2.a.n 4
96.p odd 8 1 3072.2.a.o 4
96.p odd 8 2 3072.2.d.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 12.b even 2 1
48.2.j.a 8 48.k even 4 1
144.2.k.b 8 4.b odd 2 1
144.2.k.b 8 16.f odd 4 1
192.2.j.a 8 3.b odd 2 1
192.2.j.a 8 48.i odd 4 1
384.2.j.a 8 24.h odd 2 1
384.2.j.a 8 48.i odd 4 1
384.2.j.b 8 24.f even 2 1
384.2.j.b 8 48.k even 4 1
576.2.k.b 8 1.a even 1 1 trivial
576.2.k.b 8 16.e even 4 1 inner
1152.2.k.c 8 8.d odd 2 1
1152.2.k.c 8 16.f odd 4 1
1152.2.k.f 8 8.b even 2 1
1152.2.k.f 8 16.e even 4 1
3072.2.a.i 4 96.o even 8 1
3072.2.a.n 4 96.p odd 8 1
3072.2.a.o 4 96.p odd 8 1
3072.2.a.t 4 96.o even 8 1
3072.2.d.f 8 96.o even 8 2
3072.2.d.i 8 96.p odd 8 2
9216.2.a.x 4 32.g even 8 1
9216.2.a.y 4 32.h odd 8 1
9216.2.a.bn 4 32.g even 8 1
9216.2.a.bo 4 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 16T_{5}^{5} + 128T_{5}^{4} + 192T_{5}^{3} + 128T_{5}^{2} - 128T_{5} + 64$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 16 T^{5} + 128 T^{4} + \cdots + 64$$
$7$ $$T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16$$
$11$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 1024$$
$13$ $$T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16$$
$17$ $$(T^{4} - 32 T^{2} - 64 T + 16)^{2}$$
$19$ $$T^{8} - 8 T^{7} + 32 T^{6} + 32 T^{5} + \cdots + 256$$
$23$ $$(T^{2} + 8)^{4}$$
$29$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 61504$$
$31$ $$(T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28)^{2}$$
$37$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$41$ $$T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544$$
$43$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 12544$$
$47$ $$(T^{2} - 8)^{4}$$
$53$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 18496$$
$59$ $$(T^{2} - 8 T + 32)^{4}$$
$61$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704$$
$67$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 65536$$
$71$ $$T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096$$
$73$ $$T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096$$
$79$ $$(T^{4} - 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2}$$
$83$ $$T^{8} + 40 T^{7} + 800 T^{6} + \cdots + 1024$$
$89$ $$T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216$$
$97$ $$(T^{4} - 224 T^{2} + 768 T + 512)^{2}$$