# Properties

 Label 12.3.c.a Level $12$ Weight $3$ Character orbit 12.c Self dual yes Analytic conductor $0.327$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,3,Mod(5,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("12.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 12.c (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: yes Analytic conductor: $$0.326976317232$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 2 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 2 * q^7 + 9 * q^9 $$q - 3 q^{3} + 2 q^{7} + 9 q^{9} - 22 q^{13} + 26 q^{19} - 6 q^{21} + 25 q^{25} - 27 q^{27} - 46 q^{31} + 26 q^{37} + 66 q^{39} - 22 q^{43} - 45 q^{49} - 78 q^{57} + 74 q^{61} + 18 q^{63} + 122 q^{67} - 46 q^{73} - 75 q^{75} - 142 q^{79} + 81 q^{81} - 44 q^{91} + 138 q^{93} + 2 q^{97}+O(q^{100})$$ q - 3 * q^3 + 2 * q^7 + 9 * q^9 - 22 * q^13 + 26 * q^19 - 6 * q^21 + 25 * q^25 - 27 * q^27 - 46 * q^31 + 26 * q^37 + 66 * q^39 - 22 * q^43 - 45 * q^49 - 78 * q^57 + 74 * q^61 + 18 * q^63 + 122 * q^67 - 46 * q^73 - 75 * q^75 - 142 * q^79 + 81 * q^81 - 44 * q^91 + 138 * q^93 + 2 * q^97

## Expression as an eta quotient

$$f(z) = \eta(2z)^{3}\eta(6z)^{3}=q\prod_{n=1}^\infty(1 - q^{2n})^{3}(1 - q^{6n})^{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0
0 −3.00000 0 0 0 2.00000 0 9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.3.c.a 1
3.b odd 2 1 CM 12.3.c.a 1
4.b odd 2 1 48.3.e.a 1
5.b even 2 1 300.3.g.b 1
5.c odd 4 2 300.3.b.a 2
7.b odd 2 1 588.3.c.c 1
7.c even 3 2 588.3.p.c 2
7.d odd 6 2 588.3.p.b 2
8.b even 2 1 192.3.e.b 1
8.d odd 2 1 192.3.e.a 1
9.c even 3 2 324.3.g.b 2
9.d odd 6 2 324.3.g.b 2
11.b odd 2 1 1452.3.e.b 1
12.b even 2 1 48.3.e.a 1
15.d odd 2 1 300.3.g.b 1
15.e even 4 2 300.3.b.a 2
16.e even 4 2 768.3.h.a 2
16.f odd 4 2 768.3.h.b 2
20.d odd 2 1 1200.3.l.b 1
20.e even 4 2 1200.3.c.c 2
21.c even 2 1 588.3.c.c 1
21.g even 6 2 588.3.p.b 2
21.h odd 6 2 588.3.p.c 2
24.f even 2 1 192.3.e.a 1
24.h odd 2 1 192.3.e.b 1
33.d even 2 1 1452.3.e.b 1
36.f odd 6 2 1296.3.q.b 2
36.h even 6 2 1296.3.q.b 2
48.i odd 4 2 768.3.h.a 2
48.k even 4 2 768.3.h.b 2
60.h even 2 1 1200.3.l.b 1
60.l odd 4 2 1200.3.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 1.a even 1 1 trivial
12.3.c.a 1 3.b odd 2 1 CM
48.3.e.a 1 4.b odd 2 1
48.3.e.a 1 12.b even 2 1
192.3.e.a 1 8.d odd 2 1
192.3.e.a 1 24.f even 2 1
192.3.e.b 1 8.b even 2 1
192.3.e.b 1 24.h odd 2 1
300.3.b.a 2 5.c odd 4 2
300.3.b.a 2 15.e even 4 2
300.3.g.b 1 5.b even 2 1
300.3.g.b 1 15.d odd 2 1
324.3.g.b 2 9.c even 3 2
324.3.g.b 2 9.d odd 6 2
588.3.c.c 1 7.b odd 2 1
588.3.c.c 1 21.c even 2 1
588.3.p.b 2 7.d odd 6 2
588.3.p.b 2 21.g even 6 2
588.3.p.c 2 7.c even 3 2
588.3.p.c 2 21.h odd 6 2
768.3.h.a 2 16.e even 4 2
768.3.h.a 2 48.i odd 4 2
768.3.h.b 2 16.f odd 4 2
768.3.h.b 2 48.k even 4 2
1200.3.c.c 2 20.e even 4 2
1200.3.c.c 2 60.l odd 4 2
1200.3.l.b 1 20.d odd 2 1
1200.3.l.b 1 60.h even 2 1
1296.3.q.b 2 36.f odd 6 2
1296.3.q.b 2 36.h even 6 2
1452.3.e.b 1 11.b odd 2 1
1452.3.e.b 1 33.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 22$$
$17$ $$T$$
$19$ $$T - 26$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 46$$
$37$ $$T - 26$$
$41$ $$T$$
$43$ $$T + 22$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 74$$
$67$ $$T - 122$$
$71$ $$T$$
$73$ $$T + 46$$
$79$ $$T + 142$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 2$$
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