Properties

 Label 912.2.bn.e Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(65,912)]

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, 3, 1]))

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

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

chi := DirichletCharacter("912.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.bn (of order $$6$$, 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 228) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + 5 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + 5 * q^7 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + 5 q^{7} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 6) q^{13} + (2 \zeta_{6} + 3) q^{19} + (5 \zeta_{6} + 5) q^{21} - 5 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} + ( - 10 \zeta_{6} + 5) q^{31} + ( - 6 \zeta_{6} + 3) q^{37} - 9 q^{39} + (13 \zeta_{6} - 13) q^{43} + 18 q^{49} + (7 \zeta_{6} + 1) q^{57} - \zeta_{6} q^{61} + 15 \zeta_{6} q^{63} + ( - 7 \zeta_{6} + 14) q^{67} + (17 \zeta_{6} - 17) q^{73} + ( - 10 \zeta_{6} + 5) q^{75} + (3 \zeta_{6} + 3) q^{79} + (9 \zeta_{6} - 9) q^{81} + (15 \zeta_{6} - 30) q^{91} + ( - 15 \zeta_{6} + 15) q^{93} + ( - 8 \zeta_{6} - 8) q^{97} +O(q^{100})$$ q + (z + 1) * q^3 + 5 * q^7 + 3*z * q^9 + (3*z - 6) * q^13 + (2*z + 3) * q^19 + (5*z + 5) * q^21 - 5*z * q^25 + (6*z - 3) * q^27 + (-10*z + 5) * q^31 + (-6*z + 3) * q^37 - 9 * q^39 + (13*z - 13) * q^43 + 18 * q^49 + (7*z + 1) * q^57 - z * q^61 + 15*z * q^63 + (-7*z + 14) * q^67 + (17*z - 17) * q^73 + (-10*z + 5) * q^75 + (3*z + 3) * q^79 + (9*z - 9) * q^81 + (15*z - 30) * q^91 + (-15*z + 15) * q^93 + (-8*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 10 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9 $$2 q + 3 q^{3} + 10 q^{7} + 3 q^{9} - 9 q^{13} + 8 q^{19} + 15 q^{21} - 5 q^{25} - 18 q^{39} - 13 q^{43} + 36 q^{49} + 9 q^{57} - q^{61} + 15 q^{63} + 21 q^{67} - 17 q^{73} + 9 q^{79} - 9 q^{81} - 45 q^{91} + 15 q^{93} - 24 q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9 - 9 * q^13 + 8 * q^19 + 15 * q^21 - 5 * q^25 - 18 * q^39 - 13 * q^43 + 36 * q^49 + 9 * q^57 - q^61 + 15 * q^63 + 21 * q^67 - 17 * q^73 + 9 * q^79 - 9 * q^81 - 45 * q^91 + 15 * q^93 - 24 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 0 0 0 5.00000 0 1.50000 + 2.59808i 0
449.1 0 1.50000 0.866025i 0 0 0 5.00000 0 1.50000 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.d odd 6 1 inner
57.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.bn.e 2
3.b odd 2 1 CM 912.2.bn.e 2
4.b odd 2 1 228.2.p.a 2
12.b even 2 1 228.2.p.a 2
19.d odd 6 1 inner 912.2.bn.e 2
57.f even 6 1 inner 912.2.bn.e 2
76.f even 6 1 228.2.p.a 2
228.n odd 6 1 228.2.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.p.a 2 4.b odd 2 1
228.2.p.a 2 12.b even 2 1
228.2.p.a 2 76.f even 6 1
228.2.p.a 2 228.n odd 6 1
912.2.bn.e 2 1.a even 1 1 trivial
912.2.bn.e 2 3.b odd 2 1 CM
912.2.bn.e 2 19.d odd 6 1 inner
912.2.bn.e 2 57.f even 6 1 inner

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 $$T_{7} - 5$$ T7 - 5 $$T_{17}$$ T17

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2}$$
$7$ $$(T - 5)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 9T + 27$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 75$$
$37$ $$T^{2} + 27$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 13T + 169$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 21T + 147$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 17T + 289$$
$79$ $$T^{2} - 9T + 27$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 24T + 192$$