Properties

 Label 912.2.bn.c Level $912$ Weight $2$ Character orbit 912.bn 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(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 114) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} - 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + (-2*z - 2) * q^5 - q^7 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} - 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9} + (4 \zeta_{6} - 2) q^{11} + ( - 3 \zeta_{6} + 6) q^{13} - 6 \zeta_{6} q^{15} + (2 \zeta_{6} + 2) q^{17} + (2 \zeta_{6} + 3) q^{19} + ( - \zeta_{6} - 1) q^{21} + 7 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} + 6 \zeta_{6} q^{29} + (2 \zeta_{6} - 1) q^{31} + (6 \zeta_{6} - 6) q^{33} + (2 \zeta_{6} + 2) q^{35} + (6 \zeta_{6} - 3) q^{37} + 9 q^{39} + ( - 12 \zeta_{6} + 12) q^{41} + (\zeta_{6} - 1) q^{43} + ( - 12 \zeta_{6} + 6) q^{45} + ( - 4 \zeta_{6} + 8) q^{47} - 6 q^{49} + 6 \zeta_{6} q^{51} + 12 \zeta_{6} q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + (7 \zeta_{6} + 1) q^{57} - 7 \zeta_{6} q^{61} - 3 \zeta_{6} q^{63} - 18 q^{65} + (5 \zeta_{6} - 10) q^{67} + (6 \zeta_{6} - 6) q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + (14 \zeta_{6} - 7) q^{75} + ( - 4 \zeta_{6} + 2) q^{77} + ( - 3 \zeta_{6} - 3) q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 6) q^{83} - 12 \zeta_{6} q^{85} + (12 \zeta_{6} - 6) q^{87} + (3 \zeta_{6} - 6) q^{91} + (3 \zeta_{6} - 3) q^{93} + ( - 14 \zeta_{6} - 2) q^{95} + (4 \zeta_{6} + 4) q^{97} + (6 \zeta_{6} - 12) q^{99} +O(q^{100})$$ q + (z + 1) * q^3 + (-2*z - 2) * q^5 - q^7 + 3*z * q^9 + (4*z - 2) * q^11 + (-3*z + 6) * q^13 - 6*z * q^15 + (2*z + 2) * q^17 + (2*z + 3) * q^19 + (-z - 1) * q^21 + 7*z * q^25 + (6*z - 3) * q^27 + 6*z * q^29 + (2*z - 1) * q^31 + (6*z - 6) * q^33 + (2*z + 2) * q^35 + (6*z - 3) * q^37 + 9 * q^39 + (-12*z + 12) * q^41 + (z - 1) * q^43 + (-12*z + 6) * q^45 + (-4*z + 8) * q^47 - 6 * q^49 + 6*z * q^51 + 12*z * q^53 + (-12*z + 12) * q^55 + (7*z + 1) * q^57 - 7*z * q^61 - 3*z * q^63 - 18 * q^65 + (5*z - 10) * q^67 + (6*z - 6) * q^71 + (-7*z + 7) * q^73 + (14*z - 7) * q^75 + (-4*z + 2) * q^77 + (-3*z - 3) * q^79 + (9*z - 9) * q^81 + (-12*z + 6) * q^83 - 12*z * q^85 + (12*z - 6) * q^87 + (3*z - 6) * q^91 + (3*z - 3) * q^93 + (-14*z - 2) * q^95 + (4*z + 4) * q^97 + (6*z - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 3 * q^9 $$2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 3 q^{9} + 9 q^{13} - 6 q^{15} + 6 q^{17} + 8 q^{19} - 3 q^{21} + 7 q^{25} + 6 q^{29} - 6 q^{33} + 6 q^{35} + 18 q^{39} + 12 q^{41} - q^{43} + 12 q^{47} - 12 q^{49} + 6 q^{51} + 12 q^{53} + 12 q^{55} + 9 q^{57} - 7 q^{61} - 3 q^{63} - 36 q^{65} - 15 q^{67} - 6 q^{71} + 7 q^{73} - 9 q^{79} - 9 q^{81} - 12 q^{85} - 9 q^{91} - 3 q^{93} - 18 q^{95} + 12 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 3 * q^9 + 9 * q^13 - 6 * q^15 + 6 * q^17 + 8 * q^19 - 3 * q^21 + 7 * q^25 + 6 * q^29 - 6 * q^33 + 6 * q^35 + 18 * q^39 + 12 * q^41 - q^43 + 12 * q^47 - 12 * q^49 + 6 * q^51 + 12 * q^53 + 12 * q^55 + 9 * q^57 - 7 * q^61 - 3 * q^63 - 36 * q^65 - 15 * q^67 - 6 * q^71 + 7 * q^73 - 9 * q^79 - 9 * q^81 - 12 * q^85 - 9 * q^91 - 3 * q^93 - 18 * q^95 + 12 * q^97 - 18 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 0 −3.00000 1.73205i 0 −1.00000 0 1.50000 + 2.59808i 0
449.1 0 1.50000 0.866025i 0 −3.00000 + 1.73205i 0 −1.00000 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
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.c 2
3.b odd 2 1 912.2.bn.f 2
4.b odd 2 1 114.2.h.c yes 2
12.b even 2 1 114.2.h.b 2
19.d odd 6 1 912.2.bn.f 2
57.f even 6 1 inner 912.2.bn.c 2
76.f even 6 1 114.2.h.b 2
228.n odd 6 1 114.2.h.c yes 2

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

Hecke kernels

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

 $$T_{5}^{2} + 6T_{5} + 12$$ T5^2 + 6*T5 + 12 $$T_{7} + 1$$ T7 + 1 $$T_{17}^{2} - 6T_{17} + 12$$ T17^2 - 6*T17 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 6T + 12$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} - 9T + 27$$
$17$ $$T^{2} - 6T + 12$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2} + 27$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} - 12T + 48$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} + 15T + 75$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} + 9T + 27$$
$83$ $$T^{2} + 108$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 12T + 48$$