# Properties

 Label 2304.2.f.b Level $2304$ Weight $2$ Character orbit 2304.f Analytic conductor $18.398$ 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] = [2304,2,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1152) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 2) q^{5} + (2 \beta_{2} - \beta_1) q^{7}+O(q^{10})$$ q + (b3 - 2) * q^5 + (2*b2 - b1) * q^7 $$q + (\beta_{3} - 2) q^{5} + (2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{11} + 2 \beta_{2} q^{13} + (\beta_{2} - 2 \beta_1) q^{17} - 4 \beta_{3} q^{19} + (2 \beta_{3} + 4) q^{23} + ( - 4 \beta_{3} + 1) q^{25} + (\beta_{3} + 2) q^{29} + ( - 2 \beta_{2} - 3 \beta_1) q^{31} + ( - 6 \beta_{2} + 4 \beta_1) q^{35} + ( - 4 \beta_{2} - \beta_1) q^{37} + (\beta_{2} + 2 \beta_1) q^{41} + (4 \beta_{3} - 4) q^{43} + (6 \beta_{3} - 4) q^{47} + (8 \beta_{3} - 5) q^{49} + ( - 5 \beta_{3} - 2) q^{53} + (4 \beta_{2} - 2 \beta_1) q^{55} + ( - 4 \beta_{2} - 4 \beta_1) q^{59} + (4 \beta_{2} - \beta_1) q^{61} + ( - 4 \beta_{2} + 2 \beta_1) q^{65} - 12 q^{67} + (6 \beta_{3} + 4) q^{71} - 4 q^{73} + ( - 4 \beta_{3} + 8) q^{77} + ( - 6 \beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + ( - 6 \beta_{2} + 5 \beta_1) q^{85} + ( - 3 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{3} - 8) q^{91} + (8 \beta_{3} - 8) q^{95} + ( - 4 \beta_{3} + 8) q^{97}+O(q^{100})$$ q + (b3 - 2) * q^5 + (2*b2 - b1) * q^7 - 2*b2 * q^11 + 2*b2 * q^13 + (b2 - 2*b1) * q^17 - 4*b3 * q^19 + (2*b3 + 4) * q^23 + (-4*b3 + 1) * q^25 + (b3 + 2) * q^29 + (-2*b2 - 3*b1) * q^31 + (-6*b2 + 4*b1) * q^35 + (-4*b2 - b1) * q^37 + (b2 + 2*b1) * q^41 + (4*b3 - 4) * q^43 + (6*b3 - 4) * q^47 + (8*b3 - 5) * q^49 + (-5*b3 - 2) * q^53 + (4*b2 - 2*b1) * q^55 + (-4*b2 - 4*b1) * q^59 + (4*b2 - b1) * q^61 + (-4*b2 + 2*b1) * q^65 - 12 * q^67 + (6*b3 + 4) * q^71 - 4 * q^73 + (-4*b3 + 8) * q^77 + (-6*b2 - b1) * q^79 + (-2*b2 - 4*b1) * q^83 + (-6*b2 + 5*b1) * q^85 + (-3*b2 + 4*b1) * q^89 + (4*b3 - 8) * q^91 + (8*b3 - 8) * q^95 + (-4*b3 + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5}+O(q^{10})$$ 4 * q - 8 * q^5 $$4 q - 8 q^{5} + 16 q^{23} + 4 q^{25} + 8 q^{29} - 16 q^{43} - 16 q^{47} - 20 q^{49} - 8 q^{53} - 48 q^{67} + 16 q^{71} - 16 q^{73} + 32 q^{77} - 32 q^{91} - 32 q^{95} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^5 + 16 * q^23 + 4 * q^25 + 8 * q^29 - 16 * q^43 - 16 * q^47 - 20 * q^49 - 8 * q^53 - 48 * q^67 + 16 * q^71 - 16 * q^73 + 32 * q^77 - 32 * q^91 - 32 * q^95 + 32 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −3.41421 0 4.82843i 0 0 0
1151.2 0 0 0 −3.41421 0 4.82843i 0 0 0
1151.3 0 0 0 −0.585786 0 0.828427i 0 0 0
1151.4 0 0 0 −0.585786 0 0.828427i 0 0 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
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.f.b 4
3.b odd 2 1 2304.2.f.g 4
4.b odd 2 1 2304.2.f.a 4
8.b even 2 1 2304.2.f.h 4
8.d odd 2 1 2304.2.f.g 4
12.b even 2 1 2304.2.f.h 4
16.e even 4 1 1152.2.c.a 4
16.e even 4 1 1152.2.c.b yes 4
16.f odd 4 1 1152.2.c.c yes 4
16.f odd 4 1 1152.2.c.d yes 4
24.f even 2 1 inner 2304.2.f.b 4
24.h odd 2 1 2304.2.f.a 4
48.i odd 4 1 1152.2.c.c yes 4
48.i odd 4 1 1152.2.c.d yes 4
48.k even 4 1 1152.2.c.a 4
48.k even 4 1 1152.2.c.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.c.a 4 16.e even 4 1
1152.2.c.a 4 48.k even 4 1
1152.2.c.b yes 4 16.e even 4 1
1152.2.c.b yes 4 48.k even 4 1
1152.2.c.c yes 4 16.f odd 4 1
1152.2.c.c yes 4 48.i odd 4 1
1152.2.c.d yes 4 16.f odd 4 1
1152.2.c.d yes 4 48.i odd 4 1
2304.2.f.a 4 4.b odd 2 1
2304.2.f.a 4 24.h odd 2 1
2304.2.f.b 4 1.a even 1 1 trivial
2304.2.f.b 4 24.f even 2 1 inner
2304.2.f.g 4 3.b odd 2 1
2304.2.f.g 4 8.d odd 2 1
2304.2.f.h 4 8.b even 2 1
2304.2.f.h 4 12.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 $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} + 4T_{5} + 2$$ T5^2 + 4*T5 + 2 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{23}^{2} - 8T_{23} + 8$$ T23^2 - 8*T23 + 8 $$T_{43}^{2} + 8T_{43} - 16$$ T43^2 + 8*T43 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 4 T + 2)^{2}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} + 8)^{2}$$
$13$ $$(T^{2} + 8)^{2}$$
$17$ $$T^{4} + 36T^{2} + 196$$
$19$ $$(T^{2} - 32)^{2}$$
$23$ $$(T^{2} - 8 T + 8)^{2}$$
$29$ $$(T^{2} - 4 T + 2)^{2}$$
$31$ $$T^{4} + 88T^{2} + 784$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$T^{4} + 36T^{2} + 196$$
$43$ $$(T^{2} + 8 T - 16)^{2}$$
$47$ $$(T^{2} + 8 T - 56)^{2}$$
$53$ $$(T^{2} + 4 T - 46)^{2}$$
$59$ $$T^{4} + 192T^{2} + 1024$$
$61$ $$T^{4} + 72T^{2} + 784$$
$67$ $$(T + 12)^{4}$$
$71$ $$(T^{2} - 8 T - 56)^{2}$$
$73$ $$(T + 4)^{4}$$
$79$ $$T^{4} + 152T^{2} + 4624$$
$83$ $$T^{4} + 144T^{2} + 3136$$
$89$ $$T^{4} + 164T^{2} + 2116$$
$97$ $$(T^{2} - 16 T + 32)^{2}$$