# Properties

 Label 912.2.bn.h 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_1 q^{5} + (\beta_{3} - 2 \beta_1 - 2) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b3 - b2 + b1) * q^3 + b1 * q^5 + (b3 - 2*b1 - 2) * q^7 + (-b2 - 2*b1 + 1) * q^9 $$q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_1 q^{5} + (\beta_{3} - 2 \beta_1 - 2) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9} + (\beta_{3} - 2 \beta_{2} + 1) q^{11} + (2 \beta_{2} - 4) q^{13} + ( - \beta_{3} + 2) q^{15} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{17} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (3 \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{21} + (5 \beta_{3} - 5 \beta_1) q^{23} - 3 \beta_{2} q^{25} + (\beta_{3} - 5) q^{27} + ( - \beta_{3} - \beta_1) q^{29} - 3 \beta_{3} q^{31} + ( - 2 \beta_{3} + 3 \beta_{2} - 2) q^{33} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{35} + ( - 3 \beta_{3} + 4 \beta_{2} - 2) q^{37} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{39} + (3 \beta_{2} - 3) q^{41} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{43} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{45} + ( - 7 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 4) q^{47} + ( - 4 \beta_{3} + 8 \beta_1 + 3) q^{49} + (4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 6) q^{51} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{53} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{55} + ( - 7 \beta_{2} - 2 \beta_1 + 7) q^{57} + ( - 2 \beta_{3} + 9 \beta_{2} + \beta_1 - 9) q^{59} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{61} + (2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 2) q^{63} + (2 \beta_{3} - 4 \beta_1) q^{65} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{67} + (10 \beta_{2} + 5 \beta_1 - 10) q^{69} + (6 \beta_{2} - 6) q^{71} + ( - 8 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{73} + (3 \beta_{2} - 3 \beta_1 - 3) q^{75} + \beta_{3} q^{77} - 6 \beta_1 q^{79} + (4 \beta_{3} + 7 \beta_{2} - 4 \beta_1) q^{81} + (\beta_{3} - 18 \beta_{2} + 9) q^{83} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{85} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{87} + (2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{89} + ( - 6 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 8) q^{91} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{93} + (\beta_{3} + 6) q^{95} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{97} + (4 \beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{99}+O(q^{100})$$ q + (-b3 - b2 + b1) * q^3 + b1 * q^5 + (b3 - 2*b1 - 2) * q^7 + (-b2 - 2*b1 + 1) * q^9 + (b3 - 2*b2 + 1) * q^11 + (2*b2 - 4) * q^13 + (-b3 + 2) * q^15 + (-2*b2 - 2*b1 - 2) * q^17 + (-3*b3 + b2 + 3*b1) * q^19 + (3*b3 + 4*b2 - b1 - 4) * q^21 + (5*b3 - 5*b1) * q^23 - 3*b2 * q^25 + (b3 - 5) * q^27 + (-b3 - b1) * q^29 - 3*b3 * q^31 + (-2*b3 + 3*b2 - 2) * q^33 + (-2*b2 - 2*b1 - 2) * q^35 + (-3*b3 + 4*b2 - 2) * q^37 + (4*b3 + 2*b2 - 2*b1 + 2) * q^39 + (3*b2 - 3) * q^41 + (-4*b3 + 4*b2 + 2*b1 - 4) * q^43 + (-b3 - 4*b2 + b1) * q^45 + (-7*b3 + 2*b2 + 7*b1 - 4) * q^47 + (-4*b3 + 8*b1 + 3) * q^49 + (4*b3 + 4*b2 - 4*b1 - 6) * q^51 + (2*b3 - 6*b2 + 2*b1) * q^53 + (-2*b3 + 2*b2 + b1 - 2) * q^55 + (-7*b2 - 2*b1 + 7) * q^57 + (-2*b3 + 9*b2 + b1 - 9) * q^59 + (b3 + 4*b2 + b1) * q^61 + (2*b3 + 6*b2 + 3*b1 + 2) * q^63 + (2*b3 - 4*b1) * q^65 + (-3*b3 - b2 + 3*b1 + 2) * q^67 + (10*b2 + 5*b1 - 10) * q^69 + (6*b2 - 6) * q^71 + (-8*b3 - b2 + 4*b1 + 1) * q^73 + (3*b2 - 3*b1 - 3) * q^75 + b3 * q^77 - 6*b1 * q^79 + (4*b3 + 7*b2 - 4*b1) * q^81 + (b3 - 18*b2 + 9) * q^83 + (-2*b3 - 4*b2 - 2*b1) * q^85 + (2*b3 - 2*b2 - b1 - 2) * q^87 + (2*b3 + 12*b2 + 2*b1) * q^89 + (-6*b3 - 4*b2 + 6*b1 + 8) * q^91 + (3*b3 - 6*b2 - 3*b1) * q^93 + (b3 + 6) * q^95 + (-3*b2 - 6*b1 - 3) * q^97 + (4*b3 - 5*b2 - b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 8 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 8 * q^7 + 2 * q^9 $$4 q - 2 q^{3} - 8 q^{7} + 2 q^{9} - 12 q^{13} + 8 q^{15} - 12 q^{17} + 2 q^{19} - 8 q^{21} - 6 q^{25} - 20 q^{27} - 2 q^{33} - 12 q^{35} + 12 q^{39} - 6 q^{41} - 8 q^{43} - 8 q^{45} - 12 q^{47} + 12 q^{49} - 16 q^{51} - 12 q^{53} - 4 q^{55} + 14 q^{57} - 18 q^{59} + 8 q^{61} + 20 q^{63} + 6 q^{67} - 20 q^{69} - 12 q^{71} + 2 q^{73} - 6 q^{75} + 14 q^{81} - 8 q^{85} - 12 q^{87} + 24 q^{89} + 24 q^{91} - 12 q^{93} + 24 q^{95} - 18 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 8 * q^7 + 2 * q^9 - 12 * q^13 + 8 * q^15 - 12 * q^17 + 2 * q^19 - 8 * q^21 - 6 * q^25 - 20 * q^27 - 2 * q^33 - 12 * q^35 + 12 * q^39 - 6 * q^41 - 8 * q^43 - 8 * q^45 - 12 * q^47 + 12 * q^49 - 16 * q^51 - 12 * q^53 - 4 * q^55 + 14 * q^57 - 18 * q^59 + 8 * q^61 + 20 * q^63 + 6 * q^67 - 20 * q^69 - 12 * q^71 + 2 * q^73 - 6 * q^75 + 14 * q^81 - 8 * q^85 - 12 * q^87 + 24 * q^89 + 24 * q^91 - 12 * q^93 + 24 * q^95 - 18 * q^97 + 2 * q^99

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

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

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 −1.72474 0.158919i 0 −1.22474 0.707107i 0 0.449490 0 2.94949 + 0.548188i 0
65.2 0 0.724745 1.57313i 0 1.22474 + 0.707107i 0 −4.44949 0 −1.94949 2.28024i 0
449.1 0 −1.72474 + 0.158919i 0 −1.22474 + 0.707107i 0 0.449490 0 2.94949 0.548188i 0
449.2 0 0.724745 + 1.57313i 0 1.22474 0.707107i 0 −4.44949 0 −1.94949 + 2.28024i 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.h 4
3.b odd 2 1 912.2.bn.g 4
4.b odd 2 1 114.2.h.f yes 4
12.b even 2 1 114.2.h.e 4
19.d odd 6 1 912.2.bn.g 4
57.f even 6 1 inner 912.2.bn.h 4
76.f even 6 1 114.2.h.e 4
228.n odd 6 1 114.2.h.f yes 4

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

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{7}^{2} + 4T_{7} - 2$$ T7^2 + 4*T7 - 2 $$T_{17}^{4} + 12T_{17}^{3} + 52T_{17}^{2} + 48T_{17} + 16$$ T17^4 + 12*T17^3 + 52*T17^2 + 48*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + T^{2} + 6 T + 9$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$(T^{2} + 4 T - 2)^{2}$$
$11$ $$T^{4} + 10T^{2} + 1$$
$13$ $$(T^{2} + 6 T + 12)^{2}$$
$17$ $$T^{4} + 12 T^{3} + 52 T^{2} + 48 T + 16$$
$19$ $$T^{4} - 2 T^{3} - 15 T^{2} - 38 T + 361$$
$23$ $$T^{4} - 50T^{2} + 2500$$
$29$ $$T^{4} + 6T^{2} + 36$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$T^{4} + 60T^{2} + 36$$
$41$ $$(T^{2} + 3 T + 9)^{2}$$
$43$ $$T^{4} + 8 T^{3} + 72 T^{2} - 64 T + 64$$
$47$ $$T^{4} + 12 T^{3} - 38 T^{2} + \cdots + 7396$$
$53$ $$T^{4} + 12 T^{3} + 132 T^{2} + \cdots + 144$$
$59$ $$T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625$$
$61$ $$T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100$$
$67$ $$T^{4} - 6 T^{3} - 3 T^{2} + 90 T + 225$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$T^{4} - 2 T^{3} + 99 T^{2} + \cdots + 9025$$
$79$ $$T^{4} - 72T^{2} + 5184$$
$83$ $$T^{4} + 490 T^{2} + 58081$$
$89$ $$T^{4} - 24 T^{3} + 456 T^{2} + \cdots + 14400$$
$97$ $$T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025$$