# Properties

 Label 912.2.q.b Level $912$ Weight $2$ Character orbit 912.q 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(49,912)]

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([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("912.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.q (of order $$3$$, 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: $$7.28235666434$$ 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: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} - q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 - q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} - q^{7} - \zeta_{6} q^{9} + 2 q^{11} + 3 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{17} + ( - 2 \zeta_{6} - 3) q^{19} + ( - \zeta_{6} + 1) q^{21} + 4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} + q^{27} + 3 q^{31} + (2 \zeta_{6} - 2) q^{33} - 5 q^{37} - 3 q^{39} + (4 \zeta_{6} - 4) q^{41} + (9 \zeta_{6} - 9) q^{43} + 10 \zeta_{6} q^{47} - 6 q^{49} - 4 \zeta_{6} q^{51} + 4 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 5) q^{57} + (14 \zeta_{6} - 14) q^{59} - 11 \zeta_{6} q^{61} + \zeta_{6} q^{63} + 3 \zeta_{6} q^{67} - 4 q^{69} + ( - 14 \zeta_{6} + 14) q^{71} + ( - 11 \zeta_{6} + 11) q^{73} - 5 q^{75} - 2 q^{77} + ( - \zeta_{6} + 1) q^{79} + (\zeta_{6} - 1) q^{81} - 8 q^{83} + 14 \zeta_{6} q^{89} - 3 \zeta_{6} q^{91} + (3 \zeta_{6} - 3) q^{93} + (2 \zeta_{6} - 2) q^{97} - 2 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - q^7 - z * q^9 + 2 * q^11 + 3*z * q^13 + (4*z - 4) * q^17 + (-2*z - 3) * q^19 + (-z + 1) * q^21 + 4*z * q^23 + 5*z * q^25 + q^27 + 3 * q^31 + (2*z - 2) * q^33 - 5 * q^37 - 3 * q^39 + (4*z - 4) * q^41 + (9*z - 9) * q^43 + 10*z * q^47 - 6 * q^49 - 4*z * q^51 + 4*z * q^53 + (-3*z + 5) * q^57 + (14*z - 14) * q^59 - 11*z * q^61 + z * q^63 + 3*z * q^67 - 4 * q^69 + (-14*z + 14) * q^71 + (-11*z + 11) * q^73 - 5 * q^75 - 2 * q^77 + (-z + 1) * q^79 + (z - 1) * q^81 - 8 * q^83 + 14*z * q^89 - 3*z * q^91 + (3*z - 3) * q^93 + (2*z - 2) * q^97 - 2*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^7 - q^9 $$2 q - q^{3} - 2 q^{7} - q^{9} + 4 q^{11} + 3 q^{13} - 4 q^{17} - 8 q^{19} + q^{21} + 4 q^{23} + 5 q^{25} + 2 q^{27} + 6 q^{31} - 2 q^{33} - 10 q^{37} - 6 q^{39} - 4 q^{41} - 9 q^{43} + 10 q^{47} - 12 q^{49} - 4 q^{51} + 4 q^{53} + 7 q^{57} - 14 q^{59} - 11 q^{61} + q^{63} + 3 q^{67} - 8 q^{69} + 14 q^{71} + 11 q^{73} - 10 q^{75} - 4 q^{77} + q^{79} - q^{81} - 16 q^{83} + 14 q^{89} - 3 q^{91} - 3 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^7 - q^9 + 4 * q^11 + 3 * q^13 - 4 * q^17 - 8 * q^19 + q^21 + 4 * q^23 + 5 * q^25 + 2 * q^27 + 6 * q^31 - 2 * q^33 - 10 * q^37 - 6 * q^39 - 4 * q^41 - 9 * q^43 + 10 * q^47 - 12 * q^49 - 4 * q^51 + 4 * q^53 + 7 * q^57 - 14 * q^59 - 11 * q^61 + q^63 + 3 * q^67 - 8 * q^69 + 14 * q^71 + 11 * q^73 - 10 * q^75 - 4 * q^77 + q^79 - q^81 - 16 * q^83 + 14 * q^89 - 3 * q^91 - 3 * q^93 - 2 * q^97 - 2 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 0 0 −1.00000 0 −0.500000 0.866025i 0
577.1 0 −0.500000 0.866025i 0 0 0 −1.00000 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.q.b 2
3.b odd 2 1 2736.2.s.k 2
4.b odd 2 1 114.2.e.b 2
12.b even 2 1 342.2.g.c 2
19.c even 3 1 inner 912.2.q.b 2
57.h odd 6 1 2736.2.s.k 2
76.f even 6 1 2166.2.a.h 1
76.g odd 6 1 114.2.e.b 2
76.g odd 6 1 2166.2.a.b 1
228.m even 6 1 342.2.g.c 2
228.m even 6 1 6498.2.a.u 1
228.n odd 6 1 6498.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 4.b odd 2 1
114.2.e.b 2 76.g odd 6 1
342.2.g.c 2 12.b even 2 1
342.2.g.c 2 228.m even 6 1
912.2.q.b 2 1.a even 1 1 trivial
912.2.q.b 2 19.c even 3 1 inner
2166.2.a.b 1 76.g odd 6 1
2166.2.a.h 1 76.f even 6 1
2736.2.s.k 2 3.b odd 2 1
2736.2.s.k 2 57.h odd 6 1
6498.2.a.g 1 228.n odd 6 1
6498.2.a.u 1 228.m even 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 $$T_{7} + 1$$ T7 + 1 $$T_{13}^{2} - 3T_{13} + 9$$ T13^2 - 3*T13 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 3T + 9$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2}$$
$31$ $$(T - 3)^{2}$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2} + 4T + 16$$
$43$ $$T^{2} + 9T + 81$$
$47$ $$T^{2} - 10T + 100$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} + 14T + 196$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$T^{2} - 14T + 196$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$T^{2} + 2T + 4$$