# Properties

 Label 912.2.bn.l Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_1 - 3) q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^3 + (b2 - b1 + 2) * q^5 + (-b3 - b2 - b1) * q^7 + (-b3 + b1 - 3) * q^9 $$q + ( - \beta_{3} + \beta_1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_1 - 3) q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{11} + ( - \beta_{2} + 2) q^{13} + ( - 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{15} + ( - 3 \beta_{2} - \beta_1 - 2) q^{17} + ( - 2 \beta_{2} - 3) q^{19} + ( - 6 \beta_{2} - \beta_1 + 3) q^{21} + ( - \beta_{3} - 2 \beta_{2} + 3) q^{23} + (3 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 3) q^{25} + (2 \beta_{3} - 2 \beta_1 - 3) q^{27} + ( - \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 1) q^{29} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{31} + ( - 2 \beta_{3} - 6) q^{33} + ( - 4 \beta_1 + 4) q^{35} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{37} + ( - \beta_{3} + 2 \beta_1) q^{39} + ( - 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{41} + (4 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 3) q^{43} + ( - 3 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 3) q^{45} + ( - \beta_{3} + 2 \beta_{2} - 5) q^{47} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{49} + (5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{51} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{53} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{55} + (5 \beta_{3} - 3 \beta_1) q^{57} + ( - 2 \beta_{3} - 5 \beta_{2} + \beta_1 + 4) q^{59} + (2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{61} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 + 5) q^{65} + ( - \beta_{2} + 2) q^{67} + ( - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{69} + (6 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{71} + ( - 4 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 9) q^{73} + ( - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_1 + 18) q^{75} + (4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 8) q^{77} + (3 \beta_{2} + 6 \beta_1 - 3) q^{79} + (5 \beta_{3} - 5 \beta_1 + 6) q^{81} + ( - 2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 2) q^{83} + ( - \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 1) q^{85} + ( - 8 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{87} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{89} + ( - 3 \beta_{3} - \beta_{2} - 1) q^{91} + ( - 3 \beta_1 + 9) q^{93} + ( - 2 \beta_{3} - 9 \beta_{2} + 5 \beta_1 - 4) q^{95} + (3 \beta_{2} + 9 \beta_1 - 6) q^{97} + (4 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^3 + (b2 - b1 + 2) * q^5 + (-b3 - b2 - b1) * q^7 + (-b3 + b1 - 3) * q^9 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^11 + (-b2 + 2) * q^13 + (-3*b3 - 3*b2 + b1 + 3) * q^15 + (-3*b2 - b1 - 2) * q^17 + (-2*b2 - 3) * q^19 + (-6*b2 - b1 + 3) * q^21 + (-b3 - 2*b2 + 3) * q^23 + (3*b3 + 3*b2 - 6*b1 + 3) * q^25 + (2*b3 - 2*b1 - 3) * q^27 + (-b3 + 8*b2 + 2*b1 - 1) * q^29 + (3*b3 + 3*b2 - 3*b1) * q^31 + (-2*b3 - 6) * q^33 + (-4*b1 + 4) * q^35 + (-3*b3 + b2 + 3*b1 - 2) * q^37 + (-b3 + 2*b1) * q^39 + (-6*b3 - 3*b2 + 3*b1) * q^41 + (4*b3 + 5*b2 - 2*b1 - 3) * q^43 + (-3*b3 - 6*b2 + 4*b1 - 3) * q^45 + (-b3 + 2*b2 - 5) * q^47 + (b3 + b2 + b1 + 1) * q^49 + (5*b3 - 3*b2 - 3*b1 + 3) * q^51 + (-b3 - 4*b2 + 2*b1 - 1) * q^53 + (-4*b3 - 2*b2 + 2*b1) * q^55 + (5*b3 - 3*b1) * q^57 + (-2*b3 - 5*b2 + b1 + 4) * q^59 + (2*b3 + 3*b2 - 4*b1 + 2) * q^61 + (3*b3 - 3*b2 + 2*b1 + 3) * q^63 + (-b3 - b2 - b1 + 5) * q^65 + (-b2 + 2) * q^67 + (-2*b3 - 3*b2 + 3*b1) * q^69 + (6*b3 + 3*b2 - 3*b1) * q^71 + (-4*b3 + 7*b2 + 2*b1 - 9) * q^73 + (-3*b3 - 9*b2 - 3*b1 + 18) * q^75 + (4*b3 - 12*b2 - 4*b1 + 8) * q^77 + (3*b2 + 6*b1 - 3) * q^79 + (5*b3 - 5*b1 + 6) * q^81 + (-2*b3 - 6*b2 + 2*b1 + 2) * q^83 + (-b3 - 8*b2 + 2*b1 - 1) * q^85 + (-8*b3 + 3*b2 + b1 - 6) * q^87 + (-b3 - 4*b2 + 2*b1 - 1) * q^89 + (-3*b3 - b2 - 1) * q^91 + (-3*b1 + 9) * q^93 + (-2*b3 - 9*b2 + 5*b1 - 4) * q^95 + (3*b2 + 9*b1 - 6) * q^97 + (4*b3 - 6*b2 - 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 9 q^{5} - 2 q^{7} - 10 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 9 * q^5 - 2 * q^7 - 10 * q^9 $$4 q + 2 q^{3} + 9 q^{5} - 2 q^{7} - 10 q^{9} + 6 q^{13} + 10 q^{15} - 15 q^{17} - 16 q^{19} - q^{21} + 9 q^{23} + 9 q^{25} - 16 q^{27} + 15 q^{29} - 22 q^{33} + 12 q^{35} + 3 q^{39} + 3 q^{41} - 8 q^{43} - 17 q^{45} - 15 q^{47} + 6 q^{49} - 2 q^{51} - 9 q^{53} + 2 q^{55} - 8 q^{57} + 9 q^{59} + 8 q^{61} + 5 q^{63} + 18 q^{65} + 6 q^{67} - q^{69} - 3 q^{71} - 16 q^{73} + 54 q^{75} + 14 q^{81} - 17 q^{85} - 9 q^{87} - 9 q^{89} - 3 q^{91} + 33 q^{93} - 27 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 9 * q^5 - 2 * q^7 - 10 * q^9 + 6 * q^13 + 10 * q^15 - 15 * q^17 - 16 * q^19 - q^21 + 9 * q^23 + 9 * q^25 - 16 * q^27 + 15 * q^29 - 22 * q^33 + 12 * q^35 + 3 * q^39 + 3 * q^41 - 8 * q^43 - 17 * q^45 - 15 * q^47 + 6 * q^49 - 2 * q^51 - 9 * q^53 + 2 * q^55 - 8 * q^57 + 9 * q^59 + 8 * q^61 + 5 * q^63 + 18 * q^65 + 6 * q^67 - q^69 - 3 * q^71 - 16 * q^73 + 54 * q^75 + 14 * q^81 - 17 * q^85 - 9 * q^87 - 9 * q^89 - 3 * q^91 + 33 * q^93 - 27 * q^95 - 9 * q^97 - 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2}$$ b3 + 3*b2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 2\beta _1 + 3$$ -2*b3 + 2*b1 + 3

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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
 −1.18614 − 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 + 1.26217i
0 0.500000 1.65831i 0 3.68614 + 2.12819i 0 2.37228 0 −2.50000 1.65831i 0
65.2 0 0.500000 + 1.65831i 0 0.813859 + 0.469882i 0 −3.37228 0 −2.50000 + 1.65831i 0
449.1 0 0.500000 1.65831i 0 0.813859 0.469882i 0 −3.37228 0 −2.50000 1.65831i 0
449.2 0 0.500000 + 1.65831i 0 3.68614 2.12819i 0 2.37228 0 −2.50000 + 1.65831i 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.l 4
3.b odd 2 1 912.2.bn.k 4
4.b odd 2 1 228.2.p.c 4
12.b even 2 1 228.2.p.d yes 4
19.d odd 6 1 912.2.bn.k 4
57.f even 6 1 inner 912.2.bn.l 4
76.f even 6 1 228.2.p.d yes 4
228.n odd 6 1 228.2.p.c 4

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

 $$T_{5}^{4} - 9T_{5}^{3} + 31T_{5}^{2} - 36T_{5} + 16$$ T5^4 - 9*T5^3 + 31*T5^2 - 36*T5 + 16 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{17}^{4} + 15T_{17}^{3} + 91T_{17}^{2} + 240T_{17} + 256$$ T17^4 + 15*T17^3 + 91*T17^2 + 240*T17 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 3)^{2}$$
$5$ $$T^{4} - 9 T^{3} + 31 T^{2} - 36 T + 16$$
$7$ $$(T^{2} + T - 8)^{2}$$
$11$ $$T^{4} + 28T^{2} + 64$$
$13$ $$(T^{2} - 3 T + 3)^{2}$$
$17$ $$T^{4} + 15 T^{3} + 91 T^{2} + \cdots + 256$$
$19$ $$(T^{2} + 8 T + 19)^{2}$$
$23$ $$T^{4} - 9 T^{3} + 31 T^{2} - 36 T + 16$$
$29$ $$T^{4} - 15 T^{3} + 177 T^{2} + \cdots + 2304$$
$31$ $$T^{4} + 63T^{2} + 324$$
$37$ $$T^{4} + 51T^{2} + 576$$
$41$ $$T^{4} - 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$43$ $$T^{4} + 8 T^{3} + 81 T^{2} - 136 T + 289$$
$47$ $$T^{4} + 15 T^{3} + 91 T^{2} + \cdots + 256$$
$53$ $$T^{4} + 9 T^{3} + 69 T^{2} + 108 T + 144$$
$59$ $$T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144$$
$61$ $$T^{4} - 8 T^{3} + 81 T^{2} + 136 T + 289$$
$67$ $$(T^{2} - 3 T + 3)^{2}$$
$71$ $$T^{4} + 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$73$ $$T^{4} + 16 T^{3} + 225 T^{2} + \cdots + 961$$
$79$ $$T^{4} - 99T^{2} + 9801$$
$83$ $$T^{4} + 76T^{2} + 256$$
$89$ $$T^{4} + 9 T^{3} + 69 T^{2} + 108 T + 144$$
$97$ $$T^{4} + 9 T^{3} - 189 T^{2} + \cdots + 46656$$