# Properties

 Label 912.2.k.f Level $912$ Weight $2$ Character orbit 912.k 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(607,912)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("912.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.k (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
607.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 0 3.00000 0 1.73205i 0 1.00000 0
607.2 0 1.00000 0 3.00000 0 1.73205i 0 1.00000 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
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.k.f yes 2
3.b odd 2 1 2736.2.k.a 2
4.b odd 2 1 912.2.k.c 2
8.b even 2 1 3648.2.k.a 2
8.d odd 2 1 3648.2.k.d 2
12.b even 2 1 2736.2.k.b 2
19.b odd 2 1 912.2.k.c 2
57.d even 2 1 2736.2.k.b 2
76.d even 2 1 inner 912.2.k.f yes 2
152.b even 2 1 3648.2.k.a 2
152.g odd 2 1 3648.2.k.d 2
228.b odd 2 1 2736.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.c 2 4.b odd 2 1
912.2.k.c 2 19.b odd 2 1
912.2.k.f yes 2 1.a even 1 1 trivial
912.2.k.f yes 2 76.d even 2 1 inner
2736.2.k.a 2 3.b odd 2 1
2736.2.k.a 2 228.b odd 2 1
2736.2.k.b 2 12.b even 2 1
2736.2.k.b 2 57.d even 2 1
3648.2.k.a 2 8.b even 2 1
3648.2.k.a 2 152.b even 2 1
3648.2.k.d 2 8.d odd 2 1
3648.2.k.d 2 152.g odd 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ T5 - 3 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2} + 27$$
$13$ $$T^{2} + 48$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 12$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} + 48$$
$43$ $$T^{2} + 75$$
$47$ $$T^{2} + 75$$
$53$ $$T^{2} + 48$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$(T + 8)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 5)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 48$$