# Properties

 Label 2304.3.h.f Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.h (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: $$62.7794529086$$ 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: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + 4 q^{7}+O(q^{10})$$ q + b3 * q^5 + 4 * q^7 $$q + \beta_{3} q^{5} + 4 q^{7} - 4 \beta_{3} q^{11} + 4 \beta_1 q^{13} + 3 \beta_{2} q^{17} - 8 \beta_1 q^{19} - 4 \beta_{2} q^{23} - 7 q^{25} + \beta_{3} q^{29} + 44 q^{31} + 4 \beta_{3} q^{35} + 17 \beta_1 q^{37} + 11 \beta_{2} q^{41} + 20 \beta_1 q^{43} + 20 \beta_{2} q^{47} - 33 q^{49} - 9 \beta_{3} q^{53} - 72 q^{55} - 8 \beta_{3} q^{59} + 25 \beta_1 q^{61} + 8 \beta_{2} q^{65} + 4 \beta_1 q^{67} - 12 \beta_{2} q^{71} + 16 q^{73} - 16 \beta_{3} q^{77} - 76 q^{79} + 28 \beta_{3} q^{83} + 27 \beta_1 q^{85} + 3 \beta_{2} q^{89} + 16 \beta_1 q^{91} - 16 \beta_{2} q^{95} + 176 q^{97}+O(q^{100})$$ q + b3 * q^5 + 4 * q^7 - 4*b3 * q^11 + 4*b1 * q^13 + 3*b2 * q^17 - 8*b1 * q^19 - 4*b2 * q^23 - 7 * q^25 + b3 * q^29 + 44 * q^31 + 4*b3 * q^35 + 17*b1 * q^37 + 11*b2 * q^41 + 20*b1 * q^43 + 20*b2 * q^47 - 33 * q^49 - 9*b3 * q^53 - 72 * q^55 - 8*b3 * q^59 + 25*b1 * q^61 + 8*b2 * q^65 + 4*b1 * q^67 - 12*b2 * q^71 + 16 * q^73 - 16*b3 * q^77 - 76 * q^79 + 28*b3 * q^83 + 27*b1 * q^85 + 3*b2 * q^89 + 16*b1 * q^91 - 16*b2 * q^95 + 176 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{7}+O(q^{10})$$ 4 * q + 16 * q^7 $$4 q + 16 q^{7} - 28 q^{25} + 176 q^{31} - 132 q^{49} - 288 q^{55} + 64 q^{73} - 304 q^{79} + 704 q^{97}+O(q^{100})$$ 4 * q + 16 * q^7 - 28 * q^25 + 176 * q^31 - 132 * q^49 - 288 * q^55 + 64 * q^73 - 304 * q^79 + 704 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$3\zeta_{8}^{3} + 3\zeta_{8}$$ 3*v^3 + 3*v $$\beta_{3}$$ $$=$$ $$-3\zeta_{8}^{3} + 3\zeta_{8}$$ -3*v^3 + 3*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 6$$ (b3 + b2) / 6 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 6$$ (-b3 + b2) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −4.24264 0 4.00000 0 0 0
2177.2 0 0 0 −4.24264 0 4.00000 0 0 0
2177.3 0 0 0 4.24264 0 4.00000 0 0 0
2177.4 0 0 0 4.24264 0 4.00000 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.h.f 4
3.b odd 2 1 inner 2304.3.h.f 4
4.b odd 2 1 2304.3.h.c 4
8.b even 2 1 inner 2304.3.h.f 4
8.d odd 2 1 2304.3.h.c 4
12.b even 2 1 2304.3.h.c 4
16.e even 4 1 18.3.b.a 2
16.e even 4 1 576.3.e.c 2
16.f odd 4 1 144.3.e.b 2
16.f odd 4 1 576.3.e.f 2
24.f even 2 1 2304.3.h.c 4
24.h odd 2 1 inner 2304.3.h.f 4
48.i odd 4 1 18.3.b.a 2
48.i odd 4 1 576.3.e.c 2
48.k even 4 1 144.3.e.b 2
48.k even 4 1 576.3.e.f 2
80.i odd 4 1 450.3.b.b 4
80.j even 4 1 3600.3.c.b 4
80.k odd 4 1 3600.3.l.d 2
80.q even 4 1 450.3.d.f 2
80.s even 4 1 3600.3.c.b 4
80.t odd 4 1 450.3.b.b 4
112.l odd 4 1 882.3.b.a 2
112.w even 12 2 882.3.s.b 4
112.x odd 12 2 882.3.s.d 4
144.u even 12 2 1296.3.q.f 4
144.v odd 12 2 1296.3.q.f 4
144.w odd 12 2 162.3.d.b 4
144.x even 12 2 162.3.d.b 4
176.l odd 4 1 2178.3.c.d 2
208.m odd 4 1 3042.3.d.a 4
208.p even 4 1 3042.3.c.e 2
208.r odd 4 1 3042.3.d.a 4
240.t even 4 1 3600.3.l.d 2
240.z odd 4 1 3600.3.c.b 4
240.bb even 4 1 450.3.b.b 4
240.bd odd 4 1 3600.3.c.b 4
240.bf even 4 1 450.3.b.b 4
240.bm odd 4 1 450.3.d.f 2
336.y even 4 1 882.3.b.a 2
336.bo even 12 2 882.3.s.d 4
336.bt odd 12 2 882.3.s.b 4
528.x even 4 1 2178.3.c.d 2
624.u even 4 1 3042.3.d.a 4
624.bi odd 4 1 3042.3.c.e 2
624.bm even 4 1 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 16.e even 4 1
18.3.b.a 2 48.i odd 4 1
144.3.e.b 2 16.f odd 4 1
144.3.e.b 2 48.k even 4 1
162.3.d.b 4 144.w odd 12 2
162.3.d.b 4 144.x even 12 2
450.3.b.b 4 80.i odd 4 1
450.3.b.b 4 80.t odd 4 1
450.3.b.b 4 240.bb even 4 1
450.3.b.b 4 240.bf even 4 1
450.3.d.f 2 80.q even 4 1
450.3.d.f 2 240.bm odd 4 1
576.3.e.c 2 16.e even 4 1
576.3.e.c 2 48.i odd 4 1
576.3.e.f 2 16.f odd 4 1
576.3.e.f 2 48.k even 4 1
882.3.b.a 2 112.l odd 4 1
882.3.b.a 2 336.y even 4 1
882.3.s.b 4 112.w even 12 2
882.3.s.b 4 336.bt odd 12 2
882.3.s.d 4 112.x odd 12 2
882.3.s.d 4 336.bo even 12 2
1296.3.q.f 4 144.u even 12 2
1296.3.q.f 4 144.v odd 12 2
2178.3.c.d 2 176.l odd 4 1
2178.3.c.d 2 528.x even 4 1
2304.3.h.c 4 4.b odd 2 1
2304.3.h.c 4 8.d odd 2 1
2304.3.h.c 4 12.b even 2 1
2304.3.h.c 4 24.f even 2 1
2304.3.h.f 4 1.a even 1 1 trivial
2304.3.h.f 4 3.b odd 2 1 inner
2304.3.h.f 4 8.b even 2 1 inner
2304.3.h.f 4 24.h odd 2 1 inner
3042.3.c.e 2 208.p even 4 1
3042.3.c.e 2 624.bi odd 4 1
3042.3.d.a 4 208.m odd 4 1
3042.3.d.a 4 208.r odd 4 1
3042.3.d.a 4 624.u even 4 1
3042.3.d.a 4 624.bm even 4 1
3600.3.c.b 4 80.j even 4 1
3600.3.c.b 4 80.s even 4 1
3600.3.c.b 4 240.z odd 4 1
3600.3.c.b 4 240.bd odd 4 1
3600.3.l.d 2 80.k odd 4 1
3600.3.l.d 2 240.t even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} - 18$$ T5^2 - 18 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 18)^{2}$$
$7$ $$(T - 4)^{4}$$
$11$ $$(T^{2} - 288)^{2}$$
$13$ $$(T^{2} + 64)^{2}$$
$17$ $$(T^{2} + 162)^{2}$$
$19$ $$(T^{2} + 256)^{2}$$
$23$ $$(T^{2} + 288)^{2}$$
$29$ $$(T^{2} - 18)^{2}$$
$31$ $$(T - 44)^{4}$$
$37$ $$(T^{2} + 1156)^{2}$$
$41$ $$(T^{2} + 2178)^{2}$$
$43$ $$(T^{2} + 1600)^{2}$$
$47$ $$(T^{2} + 7200)^{2}$$
$53$ $$(T^{2} - 1458)^{2}$$
$59$ $$(T^{2} - 1152)^{2}$$
$61$ $$(T^{2} + 2500)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T - 16)^{4}$$
$79$ $$(T + 76)^{4}$$
$83$ $$(T^{2} - 14112)^{2}$$
$89$ $$(T^{2} + 162)^{2}$$
$97$ $$(T - 176)^{4}$$