# Properties

 Label 912.2.bn.d 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} - 2) q^{5} + 2 q^{7} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (-z + 2) * q^3 + (-2*z - 2) * q^5 + 2 * q^7 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} - 2) q^{5} + 2 q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 2 \zeta_{6} + 1) q^{11} + (2 \zeta_{6} - 4) q^{13} - 6 q^{15} + ( - 4 \zeta_{6} - 4) q^{17} + ( - 5 \zeta_{6} + 2) q^{19} + ( - 2 \zeta_{6} + 4) q^{21} + 7 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + 6 \zeta_{6} q^{29} + (8 \zeta_{6} - 4) q^{31} - 3 \zeta_{6} q^{33} + ( - 4 \zeta_{6} - 4) q^{35} + ( - 8 \zeta_{6} + 4) q^{37} + (6 \zeta_{6} - 6) q^{39} + ( - 3 \zeta_{6} + 3) q^{41} + ( - 8 \zeta_{6} + 8) q^{43} + (6 \zeta_{6} - 12) q^{45} + (2 \zeta_{6} - 4) q^{47} - 3 q^{49} - 12 q^{51} - 6 \zeta_{6} q^{53} + (6 \zeta_{6} - 6) q^{55} + ( - 7 \zeta_{6} - 1) q^{57} + ( - 3 \zeta_{6} + 3) q^{59} - 10 \zeta_{6} q^{61} + ( - 6 \zeta_{6} + 6) q^{63} + 12 q^{65} + (3 \zeta_{6} - 6) q^{67} + ( - 6 \zeta_{6} + 6) q^{71} + (5 \zeta_{6} - 5) q^{73} + (7 \zeta_{6} + 7) q^{75} + ( - 4 \zeta_{6} + 2) q^{77} + (8 \zeta_{6} + 8) q^{79} - 9 \zeta_{6} q^{81} + ( - 6 \zeta_{6} + 3) q^{83} + 24 \zeta_{6} q^{85} + (6 \zeta_{6} + 6) q^{87} + 6 \zeta_{6} q^{89} + (4 \zeta_{6} - 8) q^{91} + 12 \zeta_{6} q^{93} + (16 \zeta_{6} - 14) q^{95} + (5 \zeta_{6} + 5) q^{97} + ( - 3 \zeta_{6} - 3) q^{99} +O(q^{100})$$ q + (-z + 2) * q^3 + (-2*z - 2) * q^5 + 2 * q^7 + (-3*z + 3) * q^9 + (-2*z + 1) * q^11 + (2*z - 4) * q^13 - 6 * q^15 + (-4*z - 4) * q^17 + (-5*z + 2) * q^19 + (-2*z + 4) * q^21 + 7*z * q^25 + (-6*z + 3) * q^27 + 6*z * q^29 + (8*z - 4) * q^31 - 3*z * q^33 + (-4*z - 4) * q^35 + (-8*z + 4) * q^37 + (6*z - 6) * q^39 + (-3*z + 3) * q^41 + (-8*z + 8) * q^43 + (6*z - 12) * q^45 + (2*z - 4) * q^47 - 3 * q^49 - 12 * q^51 - 6*z * q^53 + (6*z - 6) * q^55 + (-7*z - 1) * q^57 + (-3*z + 3) * q^59 - 10*z * q^61 + (-6*z + 6) * q^63 + 12 * q^65 + (3*z - 6) * q^67 + (-6*z + 6) * q^71 + (5*z - 5) * q^73 + (7*z + 7) * q^75 + (-4*z + 2) * q^77 + (8*z + 8) * q^79 - 9*z * q^81 + (-6*z + 3) * q^83 + 24*z * q^85 + (6*z + 6) * q^87 + 6*z * q^89 + (4*z - 8) * q^91 + 12*z * q^93 + (16*z - 14) * q^95 + (5*z + 5) * q^97 + (-3*z - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 6 * q^5 + 4 * q^7 + 3 * q^9 $$2 q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 3 q^{9} - 6 q^{13} - 12 q^{15} - 12 q^{17} - q^{19} + 6 q^{21} + 7 q^{25} + 6 q^{29} - 3 q^{33} - 12 q^{35} - 6 q^{39} + 3 q^{41} + 8 q^{43} - 18 q^{45} - 6 q^{47} - 6 q^{49} - 24 q^{51} - 6 q^{53} - 6 q^{55} - 9 q^{57} + 3 q^{59} - 10 q^{61} + 6 q^{63} + 24 q^{65} - 9 q^{67} + 6 q^{71} - 5 q^{73} + 21 q^{75} + 24 q^{79} - 9 q^{81} + 24 q^{85} + 18 q^{87} + 6 q^{89} - 12 q^{91} + 12 q^{93} - 12 q^{95} + 15 q^{97} - 9 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 6 * q^5 + 4 * q^7 + 3 * q^9 - 6 * q^13 - 12 * q^15 - 12 * q^17 - q^19 + 6 * q^21 + 7 * q^25 + 6 * q^29 - 3 * q^33 - 12 * q^35 - 6 * q^39 + 3 * q^41 + 8 * q^43 - 18 * q^45 - 6 * q^47 - 6 * q^49 - 24 * q^51 - 6 * q^53 - 6 * q^55 - 9 * q^57 + 3 * q^59 - 10 * q^61 + 6 * q^63 + 24 * q^65 - 9 * q^67 + 6 * q^71 - 5 * q^73 + 21 * q^75 + 24 * q^79 - 9 * q^81 + 24 * q^85 + 18 * q^87 + 6 * q^89 - 12 * q^91 + 12 * q^93 - 12 * q^95 + 15 * q^97 - 9 * 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 2.00000 0 1.50000 2.59808i 0
449.1 0 1.50000 + 0.866025i 0 −3.00000 + 1.73205i 0 2.00000 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.d 2
3.b odd 2 1 912.2.bn.b 2
4.b odd 2 1 114.2.h.a 2
12.b even 2 1 114.2.h.d yes 2
19.d odd 6 1 912.2.bn.b 2
57.f even 6 1 inner 912.2.bn.d 2
76.f even 6 1 114.2.h.d yes 2
228.n odd 6 1 114.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.a 2 4.b odd 2 1
114.2.h.a 2 228.n odd 6 1
114.2.h.d yes 2 12.b even 2 1
114.2.h.d yes 2 76.f even 6 1
912.2.bn.b 2 3.b odd 2 1
912.2.bn.b 2 19.d odd 6 1
912.2.bn.d 2 1.a even 1 1 trivial
912.2.bn.d 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{2} + 6T_{5} + 12$$ T5^2 + 6*T5 + 12 $$T_{7} - 2$$ T7 - 2 $$T_{17}^{2} + 12T_{17} + 48$$ T17^2 + 12*T17 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 6T + 12$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 3$$
$13$ $$T^{2} + 6T + 12$$
$17$ $$T^{2} + 12T + 48$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 48$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 6T + 12$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} + 9T + 27$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} + 5T + 25$$
$79$ $$T^{2} - 24T + 192$$
$83$ $$T^{2} + 27$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} - 15T + 75$$