# Properties

 Label 912.2.f.b Level $912$ Weight $2$ Character orbit 912.f 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(113,912)]

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, 1, 1]))

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

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

chi := DirichletCharacter("912.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.f (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: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + \beta q^{5} + 4 q^{7} + ( - 2 \beta - 1) q^{9}+O(q^{10})$$ q + (b - 1) * q^3 + b * q^5 + 4 * q^7 + (-2*b - 1) * q^9 $$q + (\beta - 1) q^{3} + \beta q^{5} + 4 q^{7} + ( - 2 \beta - 1) q^{9} - 4 \beta q^{11} - 3 \beta q^{13} + ( - \beta - 2) q^{15} - 2 \beta q^{17} + ( - 3 \beta - 1) q^{19} + (4 \beta - 4) q^{21} - \beta q^{23} + 3 q^{25} + (\beta + 5) q^{27} - 6 q^{29} + 3 \beta q^{31} + (4 \beta + 8) q^{33} + 4 \beta q^{35} - 3 \beta q^{37} + (3 \beta + 6) q^{39} - 2 q^{43} + ( - \beta + 4) q^{45} - \beta q^{47} + 9 q^{49} + (2 \beta + 4) q^{51} + 6 q^{53} + 8 q^{55} + (2 \beta + 7) q^{57} - 12 q^{59} + 2 q^{61} + ( - 8 \beta - 4) q^{63} + 6 q^{65} + (\beta + 2) q^{69} + 12 q^{71} - 4 q^{73} + (3 \beta - 3) q^{75} - 16 \beta q^{77} + 3 \beta q^{79} + (4 \beta - 7) q^{81} + 2 \beta q^{83} + 4 q^{85} + ( - 6 \beta + 6) q^{87} + 6 q^{89} - 12 \beta q^{91} + ( - 3 \beta - 6) q^{93} + ( - \beta + 6) q^{95} + 6 \beta q^{97} + (4 \beta - 16) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + b * q^5 + 4 * q^7 + (-2*b - 1) * q^9 - 4*b * q^11 - 3*b * q^13 + (-b - 2) * q^15 - 2*b * q^17 + (-3*b - 1) * q^19 + (4*b - 4) * q^21 - b * q^23 + 3 * q^25 + (b + 5) * q^27 - 6 * q^29 + 3*b * q^31 + (4*b + 8) * q^33 + 4*b * q^35 - 3*b * q^37 + (3*b + 6) * q^39 - 2 * q^43 + (-b + 4) * q^45 - b * q^47 + 9 * q^49 + (2*b + 4) * q^51 + 6 * q^53 + 8 * q^55 + (2*b + 7) * q^57 - 12 * q^59 + 2 * q^61 + (-8*b - 4) * q^63 + 6 * q^65 + (b + 2) * q^69 + 12 * q^71 - 4 * q^73 + (3*b - 3) * q^75 - 16*b * q^77 + 3*b * q^79 + (4*b - 7) * q^81 + 2*b * q^83 + 4 * q^85 + (-6*b + 6) * q^87 + 6 * q^89 - 12*b * q^91 + (-3*b - 6) * q^93 + (-b + 6) * q^95 + 6*b * q^97 + (4*b - 16) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 8 * q^7 - 2 * q^9 $$2 q - 2 q^{3} + 8 q^{7} - 2 q^{9} - 4 q^{15} - 2 q^{19} - 8 q^{21} + 6 q^{25} + 10 q^{27} - 12 q^{29} + 16 q^{33} + 12 q^{39} - 4 q^{43} + 8 q^{45} + 18 q^{49} + 8 q^{51} + 12 q^{53} + 16 q^{55} + 14 q^{57} - 24 q^{59} + 4 q^{61} - 8 q^{63} + 12 q^{65} + 4 q^{69} + 24 q^{71} - 8 q^{73} - 6 q^{75} - 14 q^{81} + 8 q^{85} + 12 q^{87} + 12 q^{89} - 12 q^{93} + 12 q^{95} - 32 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 8 * q^7 - 2 * q^9 - 4 * q^15 - 2 * q^19 - 8 * q^21 + 6 * q^25 + 10 * q^27 - 12 * q^29 + 16 * q^33 + 12 * q^39 - 4 * q^43 + 8 * q^45 + 18 * q^49 + 8 * q^51 + 12 * q^53 + 16 * q^55 + 14 * q^57 - 24 * q^59 + 4 * q^61 - 8 * q^63 + 12 * q^65 + 4 * q^69 + 24 * q^71 - 8 * q^73 - 6 * q^75 - 14 * q^81 + 8 * q^85 + 12 * q^87 + 12 * q^89 - 12 * q^93 + 12 * q^95 - 32 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 1.41421i 1.41421i
0 −1.00000 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 + 2.82843i 0
113.2 0 −1.00000 + 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 2.82843i 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.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.f.b 2
3.b odd 2 1 912.2.f.d 2
4.b odd 2 1 114.2.b.d yes 2
12.b even 2 1 114.2.b.a 2
19.b odd 2 1 912.2.f.d 2
57.d even 2 1 inner 912.2.f.b 2
76.d even 2 1 114.2.b.a 2
228.b odd 2 1 114.2.b.d yes 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{7} - 4$$ T7 - 4 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 3$$
$5$ $$T^{2} + 2$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 32$$
$13$ $$T^{2} + 18$$
$17$ $$T^{2} + 8$$
$19$ $$T^{2} + 2T + 19$$
$23$ $$T^{2} + 2$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 18$$
$37$ $$T^{2} + 18$$
$41$ $$T^{2}$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 2$$
$53$ $$(T - 6)^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 18$$
$83$ $$T^{2} + 8$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 72$$