# Properties

 Label 3648.2.k.e Level $3648$ Weight $2$ Character orbit 3648.k Analytic conductor $29.129$ Analytic rank $1$ 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] = [3648,2,Mod(2431,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, 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("3648.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.k (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: $$29.1294266574$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 912) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 - 2*b * q^7 + q^9 $$q + q^{3} - 2 \beta q^{7} + q^{9} + 2 \beta q^{11} - 2 \beta q^{13} - 6 q^{17} + ( - \beta - 4) q^{19} - 2 \beta q^{21} - 5 q^{25} + q^{27} + 4 \beta q^{29} - 10 q^{31} + 2 \beta q^{33} - 2 \beta q^{37} - 2 \beta q^{39} + 4 \beta q^{41} + 6 \beta q^{43} - 4 \beta q^{47} - 5 q^{49} - 6 q^{51} + 8 \beta q^{53} + ( - \beta - 4) q^{57} + 12 q^{59} - 10 q^{61} - 2 \beta q^{63} + 4 q^{67} - 12 q^{71} - 2 q^{73} - 5 q^{75} + 12 q^{77} + 10 q^{79} + q^{81} - 2 \beta q^{83} + 4 \beta q^{87} - 4 \beta q^{89} - 12 q^{91} - 10 q^{93} + 4 \beta q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + q^3 - 2*b * q^7 + q^9 + 2*b * q^11 - 2*b * q^13 - 6 * q^17 + (-b - 4) * q^19 - 2*b * q^21 - 5 * q^25 + q^27 + 4*b * q^29 - 10 * q^31 + 2*b * q^33 - 2*b * q^37 - 2*b * q^39 + 4*b * q^41 + 6*b * q^43 - 4*b * q^47 - 5 * q^49 - 6 * q^51 + 8*b * q^53 + (-b - 4) * q^57 + 12 * q^59 - 10 * q^61 - 2*b * q^63 + 4 * q^67 - 12 * q^71 - 2 * q^73 - 5 * q^75 + 12 * q^77 + 10 * q^79 + q^81 - 2*b * q^83 + 4*b * q^87 - 4*b * q^89 - 12 * q^91 - 10 * q^93 + 4*b * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} - 12 q^{17} - 8 q^{19} - 10 q^{25} + 2 q^{27} - 20 q^{31} - 10 q^{49} - 12 q^{51} - 8 q^{57} + 24 q^{59} - 20 q^{61} + 8 q^{67} - 24 q^{71} - 4 q^{73} - 10 q^{75} + 24 q^{77} + 20 q^{79} + 2 q^{81} - 24 q^{91} - 20 q^{93}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^17 - 8 * q^19 - 10 * q^25 + 2 * q^27 - 20 * q^31 - 10 * q^49 - 12 * q^51 - 8 * q^57 + 24 * q^59 - 20 * q^61 + 8 * q^67 - 24 * q^71 - 4 * q^73 - 10 * q^75 + 24 * q^77 + 20 * q^79 + 2 * q^81 - 24 * q^91 - 20 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$$.

 $$n$$ $$1217$$ $$1921$$ $$2053$$ $$2623$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 0 0 0 3.46410i 0 1.00000 0
2431.2 0 1.00000 0 0 0 3.46410i 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 3648.2.k.e 2
4.b odd 2 1 3648.2.k.b 2
8.b even 2 1 912.2.k.b 2
8.d odd 2 1 912.2.k.e yes 2
19.b odd 2 1 3648.2.k.b 2
24.f even 2 1 2736.2.k.e 2
24.h odd 2 1 2736.2.k.f 2
76.d even 2 1 inner 3648.2.k.e 2
152.b even 2 1 912.2.k.b 2
152.g odd 2 1 912.2.k.e yes 2
456.l odd 2 1 2736.2.k.f 2
456.p even 2 1 2736.2.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.b 2 8.b even 2 1
912.2.k.b 2 152.b even 2 1
912.2.k.e yes 2 8.d odd 2 1
912.2.k.e yes 2 152.g odd 2 1
2736.2.k.e 2 24.f even 2 1
2736.2.k.e 2 456.p even 2 1
2736.2.k.f 2 24.h odd 2 1
2736.2.k.f 2 456.l odd 2 1
3648.2.k.b 2 4.b odd 2 1
3648.2.k.b 2 19.b odd 2 1
3648.2.k.e 2 1.a even 1 1 trivial
3648.2.k.e 2 76.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3648, [\chi])$$:

 $$T_{5}$$ T5 $$T_{31} + 10$$ T31 + 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 12$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 48$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 12$$
$41$ $$T^{2} + 48$$
$43$ $$T^{2} + 108$$
$47$ $$T^{2} + 48$$
$53$ $$T^{2} + 192$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 48$$