# Properties

 Label 912.3.o.a Level $912$ Weight $3$ Character orbit 912.o Analytic conductor $24.850$ 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] = [912,3,Mod(721,912)]

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([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 912.o (of order $$2$$, degree $$1$$, 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: $$24.8502001097$$ 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: $$2$$ Twist minimal: no (minimal twist has level 57) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 4 q^{5} + 10 q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 + 4 * q^5 + 10 * q^7 - 3 * q^9 $$q - \beta q^{3} + 4 q^{5} + 10 q^{7} - 3 q^{9} - 10 q^{11} - 14 \beta q^{13} - 4 \beta q^{15} + 10 q^{17} - 19 q^{19} - 10 \beta q^{21} + 20 q^{23} - 9 q^{25} + 3 \beta q^{27} - 20 \beta q^{29} - 10 \beta q^{31} + 10 \beta q^{33} + 40 q^{35} - 6 \beta q^{37} - 42 q^{39} + 20 \beta q^{41} + 10 q^{43} - 12 q^{45} + 80 q^{47} + 51 q^{49} - 10 \beta q^{51} + 24 \beta q^{53} - 40 q^{55} + 19 \beta q^{57} - 20 \beta q^{59} - 10 q^{61} - 30 q^{63} - 56 \beta q^{65} + 44 \beta q^{67} - 20 \beta q^{69} - 60 \beta q^{71} - 10 q^{73} + 9 \beta q^{75} - 100 q^{77} + 10 \beta q^{79} + 9 q^{81} - 70 q^{83} + 40 q^{85} - 60 q^{87} - 60 \beta q^{89} - 140 \beta q^{91} - 30 q^{93} - 76 q^{95} + 44 \beta q^{97} + 30 q^{99} +O(q^{100})$$ q - b * q^3 + 4 * q^5 + 10 * q^7 - 3 * q^9 - 10 * q^11 - 14*b * q^13 - 4*b * q^15 + 10 * q^17 - 19 * q^19 - 10*b * q^21 + 20 * q^23 - 9 * q^25 + 3*b * q^27 - 20*b * q^29 - 10*b * q^31 + 10*b * q^33 + 40 * q^35 - 6*b * q^37 - 42 * q^39 + 20*b * q^41 + 10 * q^43 - 12 * q^45 + 80 * q^47 + 51 * q^49 - 10*b * q^51 + 24*b * q^53 - 40 * q^55 + 19*b * q^57 - 20*b * q^59 - 10 * q^61 - 30 * q^63 - 56*b * q^65 + 44*b * q^67 - 20*b * q^69 - 60*b * q^71 - 10 * q^73 + 9*b * q^75 - 100 * q^77 + 10*b * q^79 + 9 * q^81 - 70 * q^83 + 40 * q^85 - 60 * q^87 - 60*b * q^89 - 140*b * q^91 - 30 * q^93 - 76 * q^95 + 44*b * q^97 + 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{5} + 20 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 8 * q^5 + 20 * q^7 - 6 * q^9 $$2 q + 8 q^{5} + 20 q^{7} - 6 q^{9} - 20 q^{11} + 20 q^{17} - 38 q^{19} + 40 q^{23} - 18 q^{25} + 80 q^{35} - 84 q^{39} + 20 q^{43} - 24 q^{45} + 160 q^{47} + 102 q^{49} - 80 q^{55} - 20 q^{61} - 60 q^{63} - 20 q^{73} - 200 q^{77} + 18 q^{81} - 140 q^{83} + 80 q^{85} - 120 q^{87} - 60 q^{93} - 152 q^{95} + 60 q^{99}+O(q^{100})$$ 2 * q + 8 * q^5 + 20 * q^7 - 6 * q^9 - 20 * q^11 + 20 * q^17 - 38 * q^19 + 40 * q^23 - 18 * q^25 + 80 * q^35 - 84 * q^39 + 20 * q^43 - 24 * q^45 + 160 * q^47 + 102 * q^49 - 80 * q^55 - 20 * q^61 - 60 * q^63 - 20 * q^73 - 200 * q^77 + 18 * q^81 - 140 * q^83 + 80 * q^85 - 120 * q^87 - 60 * q^93 - 152 * q^95 + 60 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 4.00000 0 10.0000 0 −3.00000 0
721.2 0 1.73205i 0 4.00000 0 10.0000 0 −3.00000 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.o.a 2
3.b odd 2 1 2736.3.o.e 2
4.b odd 2 1 57.3.c.a 2
12.b even 2 1 171.3.c.c 2
19.b odd 2 1 inner 912.3.o.a 2
57.d even 2 1 2736.3.o.e 2
76.d even 2 1 57.3.c.a 2
228.b odd 2 1 171.3.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.a 2 4.b odd 2 1
57.3.c.a 2 76.d even 2 1
171.3.c.c 2 12.b even 2 1
171.3.c.c 2 228.b odd 2 1
912.3.o.a 2 1.a even 1 1 trivial
912.3.o.a 2 19.b odd 2 1 inner
2736.3.o.e 2 3.b odd 2 1
2736.3.o.e 2 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 4$$ acting on $$S_{3}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$(T - 4)^{2}$$
$7$ $$(T - 10)^{2}$$
$11$ $$(T + 10)^{2}$$
$13$ $$T^{2} + 588$$
$17$ $$(T - 10)^{2}$$
$19$ $$(T + 19)^{2}$$
$23$ $$(T - 20)^{2}$$
$29$ $$T^{2} + 1200$$
$31$ $$T^{2} + 300$$
$37$ $$T^{2} + 108$$
$41$ $$T^{2} + 1200$$
$43$ $$(T - 10)^{2}$$
$47$ $$(T - 80)^{2}$$
$53$ $$T^{2} + 1728$$
$59$ $$T^{2} + 1200$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 5808$$
$71$ $$T^{2} + 10800$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 300$$
$83$ $$(T + 70)^{2}$$
$89$ $$T^{2} + 10800$$
$97$ $$T^{2} + 5808$$