# Properties

 Label 234.2.a.d Level $234$ Weight $2$ Character orbit 234.a Self dual yes Analytic conductor $1.868$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.a (trivial)

## 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: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8} + 2 q^{10} + 4 q^{11} - q^{13} - 2 q^{14} + q^{16} - 6 q^{19} + 2 q^{20} + 4 q^{22} - 4 q^{23} - q^{25} - q^{26} - 2 q^{28} + 8 q^{29} - 2 q^{31} + q^{32} - 4 q^{35} + 6 q^{37} - 6 q^{38} + 2 q^{40} - 6 q^{41} - 8 q^{43} + 4 q^{44} - 4 q^{46} - 8 q^{47} - 3 q^{49} - q^{50} - q^{52} - 12 q^{53} + 8 q^{55} - 2 q^{56} + 8 q^{58} - 4 q^{59} + 10 q^{61} - 2 q^{62} + q^{64} - 2 q^{65} - 2 q^{67} - 4 q^{70} + 16 q^{71} + 14 q^{73} + 6 q^{74} - 6 q^{76} - 8 q^{77} - 4 q^{79} + 2 q^{80} - 6 q^{82} + 12 q^{83} - 8 q^{86} + 4 q^{88} + 6 q^{89} + 2 q^{91} - 4 q^{92} - 8 q^{94} - 12 q^{95} - 10 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 + 2 * q^10 + 4 * q^11 - q^13 - 2 * q^14 + q^16 - 6 * q^19 + 2 * q^20 + 4 * q^22 - 4 * q^23 - q^25 - q^26 - 2 * q^28 + 8 * q^29 - 2 * q^31 + q^32 - 4 * q^35 + 6 * q^37 - 6 * q^38 + 2 * q^40 - 6 * q^41 - 8 * q^43 + 4 * q^44 - 4 * q^46 - 8 * q^47 - 3 * q^49 - q^50 - q^52 - 12 * q^53 + 8 * q^55 - 2 * q^56 + 8 * q^58 - 4 * q^59 + 10 * q^61 - 2 * q^62 + q^64 - 2 * q^65 - 2 * q^67 - 4 * q^70 + 16 * q^71 + 14 * q^73 + 6 * q^74 - 6 * q^76 - 8 * q^77 - 4 * q^79 + 2 * q^80 - 6 * q^82 + 12 * q^83 - 8 * q^86 + 4 * q^88 + 6 * q^89 + 2 * q^91 - 4 * q^92 - 8 * q^94 - 12 * q^95 - 10 * q^97 - 3 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 2.00000 0 −2.00000 1.00000 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$13$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.a.d yes 1
3.b odd 2 1 234.2.a.a 1
4.b odd 2 1 1872.2.a.p 1
5.b even 2 1 5850.2.a.v 1
5.c odd 4 2 5850.2.e.bd 2
8.b even 2 1 7488.2.a.j 1
8.d odd 2 1 7488.2.a.s 1
9.c even 3 2 2106.2.e.e 2
9.d odd 6 2 2106.2.e.z 2
12.b even 2 1 1872.2.a.g 1
13.b even 2 1 3042.2.a.b 1
13.d odd 4 2 3042.2.b.b 2
15.d odd 2 1 5850.2.a.bv 1
15.e even 4 2 5850.2.e.d 2
24.f even 2 1 7488.2.a.bu 1
24.h odd 2 1 7488.2.a.bp 1
39.d odd 2 1 3042.2.a.o 1
39.f even 4 2 3042.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.a.a 1 3.b odd 2 1
234.2.a.d yes 1 1.a even 1 1 trivial
1872.2.a.g 1 12.b even 2 1
1872.2.a.p 1 4.b odd 2 1
2106.2.e.e 2 9.c even 3 2
2106.2.e.z 2 9.d odd 6 2
3042.2.a.b 1 13.b even 2 1
3042.2.a.o 1 39.d odd 2 1
3042.2.b.b 2 13.d odd 4 2
3042.2.b.c 2 39.f even 4 2
5850.2.a.v 1 5.b even 2 1
5850.2.a.bv 1 15.d odd 2 1
5850.2.e.d 2 15.e even 4 2
5850.2.e.bd 2 5.c odd 4 2
7488.2.a.j 1 8.b even 2 1
7488.2.a.s 1 8.d odd 2 1
7488.2.a.bp 1 24.h odd 2 1
7488.2.a.bu 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(234))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T + 2$$
$11$ $$T - 4$$
$13$ $$T + 1$$
$17$ $$T$$
$19$ $$T + 6$$
$23$ $$T + 4$$
$29$ $$T - 8$$
$31$ $$T + 2$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T + 8$$
$47$ $$T + 8$$
$53$ $$T + 12$$
$59$ $$T + 4$$
$61$ $$T - 10$$
$67$ $$T + 2$$
$71$ $$T - 16$$
$73$ $$T - 14$$
$79$ $$T + 4$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 10$$