# Properties

 Label 912.2.bb.b Level $912$ Weight $2$ Character orbit 912.bb Analytic conductor $7.282$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(31,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))

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

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

chi := DirichletCharacter("912.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.bb (of order $$6$$, degree $$2$$, 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: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + (2*z - 1) * q^7 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{7} - \zeta_{6} q^{9} + (4 \zeta_{6} - 2) q^{11} + (3 \zeta_{6} - 6) q^{13} + (6 \zeta_{6} - 6) q^{17} + (2 \zeta_{6} + 3) q^{19} + (\zeta_{6} + 1) q^{21} + ( - 2 \zeta_{6} + 4) q^{23} + 5 \zeta_{6} q^{25} - q^{27} + (2 \zeta_{6} - 4) q^{29} - q^{31} + (2 \zeta_{6} + 2) q^{33} + ( - 10 \zeta_{6} + 5) q^{37} + (6 \zeta_{6} - 3) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} + (3 \zeta_{6} + 3) q^{43} + ( - 6 \zeta_{6} + 12) q^{47} + 4 q^{49} + 6 \zeta_{6} q^{51} + (6 \zeta_{6} - 12) q^{53} + ( - 3 \zeta_{6} + 5) q^{57} + (12 \zeta_{6} - 12) q^{59} + 5 \zeta_{6} q^{61} + ( - \zeta_{6} + 2) q^{63} + 13 \zeta_{6} q^{67} + ( - 4 \zeta_{6} + 2) q^{69} + ( - 5 \zeta_{6} + 5) q^{73} + 5 q^{75} - 6 q^{77} + (\zeta_{6} - 1) q^{79} + (\zeta_{6} - 1) q^{81} + ( - 20 \zeta_{6} + 10) q^{83} + (4 \zeta_{6} - 2) q^{87} - 9 \zeta_{6} q^{91} + (\zeta_{6} - 1) q^{93} + (8 \zeta_{6} + 8) q^{97} + ( - 2 \zeta_{6} + 4) q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (2*z - 1) * q^7 - z * q^9 + (4*z - 2) * q^11 + (3*z - 6) * q^13 + (6*z - 6) * q^17 + (2*z + 3) * q^19 + (z + 1) * q^21 + (-2*z + 4) * q^23 + 5*z * q^25 - q^27 + (2*z - 4) * q^29 - q^31 + (2*z + 2) * q^33 + (-10*z + 5) * q^37 + (6*z - 3) * q^39 + (-4*z - 4) * q^41 + (3*z + 3) * q^43 + (-6*z + 12) * q^47 + 4 * q^49 + 6*z * q^51 + (6*z - 12) * q^53 + (-3*z + 5) * q^57 + (12*z - 12) * q^59 + 5*z * q^61 + (-z + 2) * q^63 + 13*z * q^67 + (-4*z + 2) * q^69 + (-5*z + 5) * q^73 + 5 * q^75 - 6 * q^77 + (z - 1) * q^79 + (z - 1) * q^81 + (-20*z + 10) * q^83 + (4*z - 2) * q^87 - 9*z * q^91 + (z - 1) * q^93 + (8*z + 8) * q^97 + (-2*z + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^9 $$2 q + q^{3} - q^{9} - 9 q^{13} - 6 q^{17} + 8 q^{19} + 3 q^{21} + 6 q^{23} + 5 q^{25} - 2 q^{27} - 6 q^{29} - 2 q^{31} + 6 q^{33} - 12 q^{41} + 9 q^{43} + 18 q^{47} + 8 q^{49} + 6 q^{51} - 18 q^{53} + 7 q^{57} - 12 q^{59} + 5 q^{61} + 3 q^{63} + 13 q^{67} + 5 q^{73} + 10 q^{75} - 12 q^{77} - q^{79} - q^{81} - 9 q^{91} - q^{93} + 24 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^9 - 9 * q^13 - 6 * q^17 + 8 * q^19 + 3 * q^21 + 6 * q^23 + 5 * q^25 - 2 * q^27 - 6 * q^29 - 2 * q^31 + 6 * q^33 - 12 * q^41 + 9 * q^43 + 18 * q^47 + 8 * q^49 + 6 * q^51 - 18 * q^53 + 7 * q^57 - 12 * q^59 + 5 * q^61 + 3 * q^63 + 13 * q^67 + 5 * q^73 + 10 * q^75 - 12 * q^77 - q^79 - q^81 - 9 * q^91 - q^93 + 24 * q^97 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 1.73205i 0 −0.500000 + 0.866025i 0
559.1 0 0.500000 0.866025i 0 0 0 1.73205i 0 −0.500000 0.866025i 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
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bb.b yes 2
3.b odd 2 1 2736.2.bm.b 2
4.b odd 2 1 912.2.bb.a 2
12.b even 2 1 2736.2.bm.a 2
19.d odd 6 1 912.2.bb.a 2
57.f even 6 1 2736.2.bm.a 2
76.f even 6 1 inner 912.2.bb.b yes 2
228.n odd 6 1 2736.2.bm.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.a 2 4.b odd 2 1
912.2.bb.a 2 19.d odd 6 1
912.2.bb.b yes 2 1.a even 1 1 trivial
912.2.bb.b yes 2 76.f even 6 1 inner
2736.2.bm.a 2 12.b even 2 1
2736.2.bm.a 2 57.f even 6 1
2736.2.bm.b 2 3.b odd 2 1
2736.2.bm.b 2 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}$$ T5 $$T_{23}^{2} - 6T_{23} + 12$$ T23^2 - 6*T23 + 12 $$T_{31} + 1$$ T31 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 9T + 27$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2} - 6T + 12$$
$29$ $$T^{2} + 6T + 12$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 75$$
$41$ $$T^{2} + 12T + 48$$
$43$ $$T^{2} - 9T + 27$$
$47$ $$T^{2} - 18T + 108$$
$53$ $$T^{2} + 18T + 108$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 5T + 25$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2} + 300$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 24T + 192$$