# Properties

 Label 912.2.k.g Level $912$ Weight $2$ Character orbit 912.k 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(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: $$4$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - b3 * q^5 + b1 * q^7 + q^9 $$q - q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} + q^{9} + \beta_1 q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} + \beta_{3} q^{15} + (\beta_{3} + 2) q^{17} + (\beta_{2} - 4) q^{19} - \beta_1 q^{21} + (4 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{3} + 3) q^{25} - q^{27} + (4 \beta_{2} - 4 \beta_1) q^{29} + ( - 2 \beta_{3} - 2) q^{31} - \beta_1 q^{33} + (2 \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{2} - 2 \beta_1) q^{39} + 4 \beta_{2} q^{41} + (4 \beta_{2} - 3 \beta_1) q^{43} - \beta_{3} q^{45} + (2 \beta_{2} - \beta_1) q^{47} + (\beta_{3} + 3) q^{49} + ( - \beta_{3} - 2) q^{51} + 4 \beta_1 q^{53} + (2 \beta_{2} + \beta_1) q^{55} + ( - \beta_{2} + 4) q^{57} + 4 q^{59} + (\beta_{3} + 2) q^{61} + \beta_1 q^{63} + (8 \beta_{2} - 4 \beta_1) q^{65} + (4 \beta_{3} - 4) q^{67} + ( - 4 \beta_{2} + 2 \beta_1) q^{69} + 4 q^{71} + ( - 3 \beta_{3} + 6) q^{73} + ( - \beta_{3} - 3) q^{75} + (\beta_{3} - 4) q^{77} + ( - 2 \beta_{3} + 2) q^{79} + q^{81} + 2 \beta_{2} q^{83} + ( - 3 \beta_{3} - 8) q^{85} + ( - 4 \beta_{2} + 4 \beta_1) q^{87} + ( - 4 \beta_{2} + 8 \beta_1) q^{89} - 4 q^{91} + (2 \beta_{3} + 2) q^{93} + (4 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{95} + (4 \beta_{2} - 8 \beta_1) q^{97} + \beta_1 q^{99}+O(q^{100})$$ q - q^3 - b3 * q^5 + b1 * q^7 + q^9 + b1 * q^11 + (-2*b2 + 2*b1) * q^13 + b3 * q^15 + (b3 + 2) * q^17 + (b2 - 4) * q^19 - b1 * q^21 + (4*b2 - 2*b1) * q^23 + (b3 + 3) * q^25 - q^27 + (4*b2 - 4*b1) * q^29 + (-2*b3 - 2) * q^31 - b1 * q^33 + (2*b2 + b1) * q^35 + (-2*b2 + 2*b1) * q^37 + (2*b2 - 2*b1) * q^39 + 4*b2 * q^41 + (4*b2 - 3*b1) * q^43 - b3 * q^45 + (2*b2 - b1) * q^47 + (b3 + 3) * q^49 + (-b3 - 2) * q^51 + 4*b1 * q^53 + (2*b2 + b1) * q^55 + (-b2 + 4) * q^57 + 4 * q^59 + (b3 + 2) * q^61 + b1 * q^63 + (8*b2 - 4*b1) * q^65 + (4*b3 - 4) * q^67 + (-4*b2 + 2*b1) * q^69 + 4 * q^71 + (-3*b3 + 6) * q^73 + (-b3 - 3) * q^75 + (b3 - 4) * q^77 + (-2*b3 + 2) * q^79 + q^81 + 2*b2 * q^83 + (-3*b3 - 8) * q^85 + (-4*b2 + 4*b1) * q^87 + (-4*b2 + 8*b1) * q^89 - 4 * q^91 + (2*b3 + 2) * q^93 + (4*b3 - 2*b2 + 3*b1) * q^95 + (4*b2 - 8*b1) * q^97 + b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 $$4 q - 4 q^{3} - 2 q^{5} + 4 q^{9} + 2 q^{15} + 10 q^{17} - 16 q^{19} + 14 q^{25} - 4 q^{27} - 12 q^{31} - 2 q^{45} + 14 q^{49} - 10 q^{51} + 16 q^{57} + 16 q^{59} + 10 q^{61} - 8 q^{67} + 16 q^{71} + 18 q^{73} - 14 q^{75} - 14 q^{77} + 4 q^{79} + 4 q^{81} - 38 q^{85} - 16 q^{91} + 12 q^{93} + 8 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 + 2 * q^15 + 10 * q^17 - 16 * q^19 + 14 * q^25 - 4 * q^27 - 12 * q^31 - 2 * q^45 + 14 * q^49 - 10 * q^51 + 16 * q^57 + 16 * q^59 + 10 * q^61 - 8 * q^67 + 16 * q^71 + 18 * q^73 - 14 * q^75 - 14 * q^77 + 4 * q^79 + 4 * q^81 - 38 * q^85 - 16 * q^91 + 12 * q^93 + 8 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + \nu^{2} - \nu + 3 ) / 3$$ (-v^3 + v^2 - v + 3) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3$$ (v^3 + 2*v^2 - 2*v - 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu + 3 ) / 3$$ (-v^3 + v^2 + 5*v + 3) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 2$$ (b3 + 2*b2 + b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} - 2\beta _1 + 4$$ b2 - 2*b1 + 4

## 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
 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 + 1.26217i −1.18614 − 1.26217i
0 −1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.2 0 −1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.3 0 −1.00000 0 2.37228 0 2.52434i 0 1.00000 0
607.4 0 −1.00000 0 2.37228 0 2.52434i 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.g 4
3.b odd 2 1 2736.2.k.n 4
4.b odd 2 1 912.2.k.h yes 4
8.b even 2 1 3648.2.k.h 4
8.d odd 2 1 3648.2.k.g 4
12.b even 2 1 2736.2.k.o 4
19.b odd 2 1 912.2.k.h yes 4
57.d even 2 1 2736.2.k.o 4
76.d even 2 1 inner 912.2.k.g 4
152.b even 2 1 3648.2.k.h 4
152.g odd 2 1 3648.2.k.g 4
228.b odd 2 1 2736.2.k.n 4

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

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} - 8$$ T5^2 + T5 - 8 $$T_{31}^{2} + 6T_{31} - 24$$ T31^2 + 6*T31 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + T - 8)^{2}$$
$7$ $$T^{4} + 7T^{2} + 4$$
$11$ $$T^{4} + 7T^{2} + 4$$
$13$ $$T^{4} + 28T^{2} + 64$$
$17$ $$(T^{2} - 5 T - 2)^{2}$$
$19$ $$(T^{2} + 8 T + 19)^{2}$$
$23$ $$T^{4} + 76T^{2} + 256$$
$29$ $$T^{4} + 112T^{2} + 1024$$
$31$ $$(T^{2} + 6 T - 24)^{2}$$
$37$ $$T^{4} + 28T^{2} + 64$$
$41$ $$(T^{2} + 48)^{2}$$
$43$ $$T^{4} + 87T^{2} + 36$$
$47$ $$T^{4} + 19T^{2} + 16$$
$53$ $$T^{4} + 112T^{2} + 1024$$
$59$ $$(T - 4)^{4}$$
$61$ $$(T^{2} - 5 T - 2)^{2}$$
$67$ $$(T^{2} + 4 T - 128)^{2}$$
$71$ $$(T - 4)^{4}$$
$73$ $$(T^{2} - 9 T - 54)^{2}$$
$79$ $$(T^{2} - 2 T - 32)^{2}$$
$83$ $$(T^{2} + 12)^{2}$$
$89$ $$(T^{2} + 176)^{2}$$
$97$ $$(T^{2} + 176)^{2}$$