# Properties

 Label 64.8.b.b Level $64$ Weight $8$ Character orbit 64.b Analytic conductor $19.993$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.b (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: $$19.9926416310$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{435})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 217x^{2} + 11881$$ x^4 - 217*x^2 + 11881 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 15 \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + 1287 q^{9}+O(q^{10})$$ q + 15*b1 * q^3 + b3 * q^5 + b2 * q^7 + 1287 * q^9 $$q + 15 \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + 1287 q^{9} + 2379 \beta_1 q^{11} - 13 \beta_{3} q^{13} - 15 \beta_{2} q^{15} - 13830 q^{17} + 9503 \beta_1 q^{19} + 60 \beta_{3} q^{21} + 91 \beta_{2} q^{23} - 172435 q^{25} + 52110 \beta_1 q^{27} - 65 \beta_{3} q^{29} - 260 \beta_{2} q^{31} - 142740 q^{33} + 250560 \beta_1 q^{35} - 383 \beta_{3} q^{37} + 195 \beta_{2} q^{39} - 454038 q^{41} + 226435 \beta_1 q^{43} + 1287 \beta_{3} q^{45} + 862 \beta_{2} q^{47} + 178697 q^{49} - 207450 \beta_1 q^{51} - 91 \beta_{3} q^{53} - 2379 \beta_{2} q^{55} - 570180 q^{57} - 1160133 \beta_1 q^{59} - 5265 \beta_{3} q^{61} + 1287 \beta_{2} q^{63} + 3257280 q^{65} - 154505 \beta_1 q^{67} + 5460 \beta_{3} q^{69} + 2465 \beta_{2} q^{71} + 5883410 q^{73} - 2586525 \beta_1 q^{75} + 9516 \beta_{3} q^{77} - 2470 \beta_{2} q^{79} - 311931 q^{81} + 1646775 \beta_1 q^{83} - 13830 \beta_{3} q^{85} + 975 \beta_{2} q^{87} + 3675906 q^{89} - 3257280 \beta_1 q^{91} - 15600 \beta_{3} q^{93} - 9503 \beta_{2} q^{95} - 11233430 q^{97} + 3061773 \beta_1 q^{99}+O(q^{100})$$ q + 15*b1 * q^3 + b3 * q^5 + b2 * q^7 + 1287 * q^9 + 2379*b1 * q^11 - 13*b3 * q^13 - 15*b2 * q^15 - 13830 * q^17 + 9503*b1 * q^19 + 60*b3 * q^21 + 91*b2 * q^23 - 172435 * q^25 + 52110*b1 * q^27 - 65*b3 * q^29 - 260*b2 * q^31 - 142740 * q^33 + 250560*b1 * q^35 - 383*b3 * q^37 + 195*b2 * q^39 - 454038 * q^41 + 226435*b1 * q^43 + 1287*b3 * q^45 + 862*b2 * q^47 + 178697 * q^49 - 207450*b1 * q^51 - 91*b3 * q^53 - 2379*b2 * q^55 - 570180 * q^57 - 1160133*b1 * q^59 - 5265*b3 * q^61 + 1287*b2 * q^63 + 3257280 * q^65 - 154505*b1 * q^67 + 5460*b3 * q^69 + 2465*b2 * q^71 + 5883410 * q^73 - 2586525*b1 * q^75 + 9516*b3 * q^77 - 2470*b2 * q^79 - 311931 * q^81 + 1646775*b1 * q^83 - 13830*b3 * q^85 + 975*b2 * q^87 + 3675906 * q^89 - 3257280*b1 * q^91 - 15600*b3 * q^93 - 9503*b2 * q^95 - 11233430 * q^97 + 3061773*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5148 q^{9}+O(q^{10})$$ 4 * q + 5148 * q^9 $$4 q + 5148 q^{9} - 55320 q^{17} - 689740 q^{25} - 570960 q^{33} - 1816152 q^{41} + 714788 q^{49} - 2280720 q^{57} + 13029120 q^{65} + 23533640 q^{73} - 1247724 q^{81} + 14703624 q^{89} - 44933720 q^{97}+O(q^{100})$$ 4 * q + 5148 * q^9 - 55320 * q^17 - 689740 * q^25 - 570960 * q^33 - 1816152 * q^41 + 714788 * q^49 - 2280720 * q^57 + 13029120 * q^65 + 23533640 * q^73 - 1247724 * q^81 + 14703624 * q^89 - 44933720 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 217x^{2} + 11881$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 216\nu ) / 109$$ (2*v^3 - 216*v) / 109 $$\beta_{2}$$ $$=$$ $$( -48\nu^{3} + 15648\nu ) / 109$$ (-48*v^3 + 15648*v) / 109 $$\beta_{3}$$ $$=$$ $$48\nu^{2} - 5208$$ 48*v^2 - 5208
 $$\nu$$ $$=$$ $$( \beta_{2} + 24\beta_1 ) / 96$$ (b2 + 24*b1) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 5208 ) / 48$$ (b3 + 5208) / 48 $$\nu^{3}$$ $$=$$ $$( 9\beta_{2} + 652\beta_1 ) / 8$$ (9*b2 + 652*b1) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 10.4283 − 0.500000i −10.4283 − 0.500000i −10.4283 + 0.500000i 10.4283 + 0.500000i
0 30.0000i 0 500.560i 0 1001.12 0 1287.00 0
33.2 0 30.0000i 0 500.560i 0 −1001.12 0 1287.00 0
33.3 0 30.0000i 0 500.560i 0 −1001.12 0 1287.00 0
33.4 0 30.0000i 0 500.560i 0 1001.12 0 1287.00 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.b.b 4
3.b odd 2 1 576.8.d.e 4
4.b odd 2 1 inner 64.8.b.b 4
8.b even 2 1 inner 64.8.b.b 4
8.d odd 2 1 inner 64.8.b.b 4
12.b even 2 1 576.8.d.e 4
16.e even 4 1 256.8.a.e 2
16.e even 4 1 256.8.a.i 2
16.f odd 4 1 256.8.a.e 2
16.f odd 4 1 256.8.a.i 2
24.f even 2 1 576.8.d.e 4
24.h odd 2 1 576.8.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.8.b.b 4 1.a even 1 1 trivial
64.8.b.b 4 4.b odd 2 1 inner
64.8.b.b 4 8.b even 2 1 inner
64.8.b.b 4 8.d odd 2 1 inner
256.8.a.e 2 16.e even 4 1
256.8.a.e 2 16.f odd 4 1
256.8.a.i 2 16.e even 4 1
256.8.a.i 2 16.f odd 4 1
576.8.d.e 4 3.b odd 2 1
576.8.d.e 4 12.b even 2 1
576.8.d.e 4 24.f even 2 1
576.8.d.e 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 900$$ acting on $$S_{8}^{\mathrm{new}}(64, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 900)^{2}$$
$5$ $$(T^{2} + 250560)^{2}$$
$7$ $$(T^{2} - 1002240)^{2}$$
$11$ $$(T^{2} + 22638564)^{2}$$
$13$ $$(T^{2} + 42344640)^{2}$$
$17$ $$(T + 13830)^{4}$$
$19$ $$(T^{2} + 361228036)^{2}$$
$23$ $$(T^{2} - 8299549440)^{2}$$
$29$ $$(T^{2} + 1058616000)^{2}$$
$31$ $$(T^{2} - 67751424000)^{2}$$
$37$ $$(T^{2} + 36754395840)^{2}$$
$41$ $$(T + 454038)^{4}$$
$43$ $$(T^{2} + 205091236900)^{2}$$
$47$ $$(T^{2} - 744708418560)^{2}$$
$53$ $$(T^{2} + 2074887360)^{2}$$
$59$ $$(T^{2} + 5383634310756)^{2}$$
$61$ $$(T^{2} + 6945579576000)^{2}$$
$67$ $$(T^{2} + 95487180100)^{2}$$
$71$ $$(T^{2} - 6089835744000)^{2}$$
$73$ $$(T - 5883410)^{4}$$
$79$ $$(T^{2} - 6114566016000)^{2}$$
$83$ $$(T^{2} + 10847471602500)^{2}$$
$89$ $$(T - 3675906)^{4}$$
$97$ $$(T + 11233430)^{4}$$