# Properties

 Label 528.2.o.b Level $528$ Weight $2$ Character orbit 528.o Analytic conductor $4.216$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,2,Mod(175,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("528.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 528.o (of order $$2$$, degree $$1$$, 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: $$4.21610122672$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.454201344.7 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169$$ x^8 - 2*x^7 - 4*x^6 + 24*x^5 - 25*x^4 - 12*x^3 + 128*x^2 - 182*x + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{7} q^{7} - q^{9}+O(q^{10})$$ q + b2 * q^3 + (-b1 + 1) * q^5 - b7 * q^7 - q^9 $$q + \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{7} q^{7} - q^{9} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{11} + \beta_{5} q^{13} + (\beta_{4} + \beta_{2}) q^{15} - 2 \beta_{6} q^{17} + 2 \beta_{3} q^{19} - \beta_{5} q^{21} + ( - \beta_{4} + 3 \beta_{2}) q^{23} + ( - 2 \beta_1 - 1) q^{25} - \beta_{2} q^{27} + 2 \beta_{2} q^{31} + ( - \beta_{6} + \beta_1 - 1) q^{33} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{35} + (2 \beta_1 + 4) q^{37} - \beta_{7} q^{39} - 2 \beta_{5} q^{41} + 2 \beta_{3} q^{43} + (\beta_1 - 1) q^{45} + (3 \beta_{4} - \beta_{2}) q^{47} + ( - 6 \beta_1 + 9) q^{49} - 2 \beta_{3} q^{51} + (\beta_1 - 9) q^{53} + (\beta_{7} + 2 \beta_{4} + 4 \beta_{2}) q^{55} - 2 \beta_{6} q^{57} + (2 \beta_{4} + 6 \beta_{2}) q^{59} + (4 \beta_{6} + \beta_{5}) q^{61} + \beta_{7} q^{63} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{65} - 6 \beta_{4} q^{67} + ( - \beta_1 - 3) q^{69} + ( - \beta_{4} - 9 \beta_{2}) q^{71} + 2 \beta_{5} q^{73} + (2 \beta_{4} - \beta_{2}) q^{75} + (2 \beta_{6} - 2 \beta_{5} + 5 \beta_1 - 1) q^{77} + \beta_{7} q^{79} + q^{81} + 2 \beta_{7} q^{83} + 2 \beta_{5} q^{85} + 6 \beta_1 q^{89} + ( - 6 \beta_{4} - 16 \beta_{2}) q^{91} - 2 q^{93} + 2 \beta_{7} q^{95} + ( - 2 \beta_1 + 2) q^{97} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{99}+O(q^{100})$$ q + b2 * q^3 + (-b1 + 1) * q^5 - b7 * q^7 - q^9 + (b4 + b3 + b2) * q^11 + b5 * q^13 + (b4 + b2) * q^15 - 2*b6 * q^17 + 2*b3 * q^19 - b5 * q^21 + (-b4 + 3*b2) * q^23 + (-2*b1 - 1) * q^25 - b2 * q^27 + 2*b2 * q^31 + (-b6 + b1 - 1) * q^33 + (-2*b7 - 2*b3) * q^35 + (2*b1 + 4) * q^37 - b7 * q^39 - 2*b5 * q^41 + 2*b3 * q^43 + (b1 - 1) * q^45 + (3*b4 - b2) * q^47 + (-6*b1 + 9) * q^49 - 2*b3 * q^51 + (b1 - 9) * q^53 + (b7 + 2*b4 + 4*b2) * q^55 - 2*b6 * q^57 + (2*b4 + 6*b2) * q^59 + (4*b6 + b5) * q^61 + b7 * q^63 + (-2*b6 + 2*b5) * q^65 - 6*b4 * q^67 + (-b1 - 3) * q^69 + (-b4 - 9*b2) * q^71 + 2*b5 * q^73 + (2*b4 - b2) * q^75 + (2*b6 - 2*b5 + 5*b1 - 1) * q^77 + b7 * q^79 + q^81 + 2*b7 * q^83 + 2*b5 * q^85 + 6*b1 * q^89 + (-6*b4 - 16*b2) * q^91 - 2 * q^93 + 2*b7 * q^95 + (-2*b1 + 2) * q^97 + (-b4 - b3 - b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5} - 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^5 - 8 * q^9 $$8 q + 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} + 32 q^{37} - 8 q^{45} + 72 q^{49} - 72 q^{53} - 24 q^{69} - 8 q^{77} + 8 q^{81} - 16 q^{93} + 16 q^{97}+O(q^{100})$$ 8 * q + 8 * q^5 - 8 * q^9 - 8 * q^25 - 8 * q^33 + 32 * q^37 - 8 * q^45 + 72 * q^49 - 72 * q^53 - 24 * q^69 - 8 * q^77 + 8 * q^81 - 16 * q^93 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( -20\nu^{7} - 402\nu^{6} - 43\nu^{5} + 760\nu^{4} - 4607\nu^{3} + 5750\nu^{2} - 6903\nu - 37211 ) / 21903$$ (-20*v^7 - 402*v^6 - 43*v^5 + 760*v^4 - 4607*v^3 + 5750*v^2 - 6903*v - 37211) / 21903 $$\beta_{2}$$ $$=$$ $$( -30\nu^{7} + 34\nu^{6} + 107\nu^{5} - 330\nu^{4} - 173\nu^{3} - 342\nu^{2} - 187\nu - 1157 ) / 1911$$ (-30*v^7 + 34*v^6 + 107*v^5 - 330*v^4 - 173*v^3 - 342*v^2 - 187*v - 1157) / 1911 $$\beta_{3}$$ $$=$$ $$( - 4786 \nu^{7} + 17697 \nu^{6} + 40087 \nu^{5} - 168580 \nu^{4} + 95639 \nu^{3} + 434146 \nu^{2} + \cdots + 1041560 ) / 284739$$ (-4786*v^7 + 17697*v^6 + 40087*v^5 - 168580*v^4 + 95639*v^3 + 434146*v^2 - 813003*v + 1041560) / 284739 $$\beta_{4}$$ $$=$$ $$( -124\nu^{7} - 168\nu^{6} + 1432\nu^{5} - 1546\nu^{4} - 4336\nu^{3} + 7936\nu^{2} - 12492\nu + 3341 ) / 5811$$ (-124*v^7 - 168*v^6 + 1432*v^5 - 1546*v^4 - 4336*v^3 + 7936*v^2 - 12492*v + 3341) / 5811 $$\beta_{5}$$ $$=$$ $$( -720\nu^{7} + 130\nu^{6} + 5753\nu^{5} - 16446\nu^{4} + 2071\nu^{3} + 31776\nu^{2} - 153595\nu + 120604 ) / 21903$$ (-720*v^7 + 130*v^6 + 5753*v^5 - 16446*v^4 + 2071*v^3 + 31776*v^2 - 153595*v + 120604) / 21903 $$\beta_{6}$$ $$=$$ $$( 856\nu^{7} - 1777\nu^{6} - 1080\nu^{5} + 11278\nu^{4} - 18930\nu^{3} + 16736\nu^{2} + 29692\nu - 34032 ) / 21903$$ (856*v^7 - 1777*v^6 - 1080*v^5 + 11278*v^4 - 18930*v^3 + 16736*v^2 + 29692*v - 34032) / 21903 $$\beta_{7}$$ $$=$$ $$( - 5862 \nu^{7} + 15052 \nu^{6} + 33393 \nu^{5} - 171498 \nu^{4} + 114999 \nu^{3} + 305436 \nu^{2} + \cdots + 848796 ) / 94913$$ (-5862*v^7 + 15052*v^6 + 33393*v^5 - 171498*v^4 + 114999*v^3 + 305436*v^2 - 663093*v + 848796) / 94913
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4$$ (b7 - b5 + b4 - 2*b3 - b2 - b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 3\beta _1 + 3 ) / 2$$ (-b6 - b5 + b4 + b3 - 3*b2 + 3*b1 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - 3\beta_{6} - \beta_{5} + 3\beta_{4} - 3\beta_{3} - 10\beta_{2} - 3\beta _1 - 10 ) / 2$$ (b7 - 3*b6 - b5 + 3*b4 - 3*b3 - 10*b2 - 3*b1 - 10) / 2 $$\nu^{4}$$ $$=$$ $$( -6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 5\beta_{4} + 12\beta_{3} - 12\beta_{2} + 4\beta _1 + 5 ) / 2$$ (-6*b7 - 8*b6 - 2*b5 + 5*b4 + 12*b3 - 12*b2 + 4*b1 + 5) / 2 $$\nu^{5}$$ $$=$$ $$( -3\beta_{7} - 6\beta_{6} - \beta_{5} + 29\beta_{4} - 4\beta_{3} - 31\beta_{2} - 89\beta _1 - 149 ) / 4$$ (-3*b7 - 6*b6 - b5 + 29*b4 - 4*b3 - 31*b2 - 89*b1 - 149) / 4 $$\nu^{6}$$ $$=$$ $$-16\beta_{7} - 9\beta_{4} + 45\beta_{3} + 27\beta_{2}$$ -16*b7 - 9*b4 + 45*b3 + 27*b2 $$\nu^{7}$$ $$=$$ $$( 31\beta_{7} + 212\beta_{6} + 81\beta_{5} - 111\beta_{4} - 50\beta_{3} + 211\beta_{2} - 433\beta _1 - 755 ) / 4$$ (31*b7 + 212*b6 + 81*b5 - 111*b4 - 50*b3 + 211*b2 - 433*b1 - 755) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
175.1
 2.02641 + 1.27503i −2.39244 + 0.0909984i 1.02715 + 1.10132i 0.338876 − 1.46735i 2.02641 − 1.27503i −2.39244 − 0.0909984i 1.02715 − 1.10132i 0.338876 + 1.46735i
0 1.00000i 0 −0.732051 0 −2.36806 0 −1.00000 0
175.2 0 1.00000i 0 −0.732051 0 2.36806 0 −1.00000 0
175.3 0 1.00000i 0 2.73205 0 −5.13734 0 −1.00000 0
175.4 0 1.00000i 0 2.73205 0 5.13734 0 −1.00000 0
175.5 0 1.00000i 0 −0.732051 0 −2.36806 0 −1.00000 0
175.6 0 1.00000i 0 −0.732051 0 2.36806 0 −1.00000 0
175.7 0 1.00000i 0 2.73205 0 −5.13734 0 −1.00000 0
175.8 0 1.00000i 0 2.73205 0 5.13734 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 175.8 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
11.b odd 2 1 inner
44.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.o.b 8
3.b odd 2 1 1584.2.o.g 8
4.b odd 2 1 inner 528.2.o.b 8
8.b even 2 1 2112.2.o.e 8
8.d odd 2 1 2112.2.o.e 8
11.b odd 2 1 inner 528.2.o.b 8
12.b even 2 1 1584.2.o.g 8
33.d even 2 1 1584.2.o.g 8
44.c even 2 1 inner 528.2.o.b 8
88.b odd 2 1 2112.2.o.e 8
88.g even 2 1 2112.2.o.e 8
132.d odd 2 1 1584.2.o.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.o.b 8 1.a even 1 1 trivial
528.2.o.b 8 4.b odd 2 1 inner
528.2.o.b 8 11.b odd 2 1 inner
528.2.o.b 8 44.c even 2 1 inner
1584.2.o.g 8 3.b odd 2 1
1584.2.o.g 8 12.b even 2 1
1584.2.o.g 8 33.d even 2 1
1584.2.o.g 8 132.d odd 2 1
2112.2.o.e 8 8.b even 2 1
2112.2.o.e 8 8.d odd 2 1
2112.2.o.e 8 88.b odd 2 1
2112.2.o.e 8 88.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2T_{5} - 2$$ acting on $$S_{2}^{\mathrm{new}}(528, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$(T^{2} - 2 T - 2)^{4}$$
$7$ $$(T^{4} - 32 T^{2} + 148)^{2}$$
$11$ $$T^{8} - 12 T^{6} + \cdots + 14641$$
$13$ $$(T^{4} + 32 T^{2} + 148)^{2}$$
$17$ $$(T^{4} + 56 T^{2} + 592)^{2}$$
$19$ $$(T^{4} - 56 T^{2} + 592)^{2}$$
$23$ $$(T^{4} + 24 T^{2} + 36)^{2}$$
$29$ $$T^{8}$$
$31$ $$(T^{2} + 4)^{4}$$
$37$ $$(T^{2} - 8 T + 4)^{4}$$
$41$ $$(T^{4} + 128 T^{2} + 2368)^{2}$$
$43$ $$(T^{4} - 56 T^{2} + 592)^{2}$$
$47$ $$(T^{4} + 56 T^{2} + 676)^{2}$$
$53$ $$(T^{2} + 18 T + 78)^{4}$$
$59$ $$(T^{4} + 96 T^{2} + 576)^{2}$$
$61$ $$(T^{4} + 240 T^{2} + 1332)^{2}$$
$67$ $$(T^{2} + 108)^{4}$$
$71$ $$(T^{4} + 168 T^{2} + 6084)^{2}$$
$73$ $$(T^{4} + 128 T^{2} + 2368)^{2}$$
$79$ $$(T^{4} - 32 T^{2} + 148)^{2}$$
$83$ $$(T^{4} - 128 T^{2} + 2368)^{2}$$
$89$ $$(T^{2} - 108)^{4}$$
$97$ $$(T^{2} - 4 T - 8)^{4}$$