# Properties

 Label 234.2.a.e 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: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 3 q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 3 * q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + 3 q^{5} - q^{7} + q^{8} + 3 q^{10} - 6 q^{11} + q^{13} - q^{14} + q^{16} + 3 q^{17} + 2 q^{19} + 3 q^{20} - 6 q^{22} + 4 q^{25} + q^{26} - q^{28} - 6 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} - 3 q^{35} - 7 q^{37} + 2 q^{38} + 3 q^{40} - q^{43} - 6 q^{44} - 3 q^{47} - 6 q^{49} + 4 q^{50} + q^{52} - 18 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 8 q^{61} - 4 q^{62} + q^{64} + 3 q^{65} + 14 q^{67} + 3 q^{68} - 3 q^{70} + 3 q^{71} + 2 q^{73} - 7 q^{74} + 2 q^{76} + 6 q^{77} + 8 q^{79} + 3 q^{80} - 12 q^{83} + 9 q^{85} - q^{86} - 6 q^{88} + 6 q^{89} - q^{91} - 3 q^{94} + 6 q^{95} - 10 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 3 * q^5 - q^7 + q^8 + 3 * q^10 - 6 * q^11 + q^13 - q^14 + q^16 + 3 * q^17 + 2 * q^19 + 3 * q^20 - 6 * q^22 + 4 * q^25 + q^26 - q^28 - 6 * q^29 - 4 * q^31 + q^32 + 3 * q^34 - 3 * q^35 - 7 * q^37 + 2 * q^38 + 3 * q^40 - q^43 - 6 * q^44 - 3 * q^47 - 6 * q^49 + 4 * q^50 + q^52 - 18 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 8 * q^61 - 4 * q^62 + q^64 + 3 * q^65 + 14 * q^67 + 3 * q^68 - 3 * q^70 + 3 * q^71 + 2 * q^73 - 7 * q^74 + 2 * q^76 + 6 * q^77 + 8 * q^79 + 3 * q^80 - 12 * q^83 + 9 * q^85 - q^86 - 6 * q^88 + 6 * q^89 - q^91 - 3 * q^94 + 6 * q^95 - 10 * q^97 - 6 * 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 3.00000 0 −1.00000 1.00000 0 3.00000
 $$n$$: 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.e 1
3.b odd 2 1 26.2.a.a 1
4.b odd 2 1 1872.2.a.q 1
5.b even 2 1 5850.2.a.p 1
5.c odd 4 2 5850.2.e.a 2
8.b even 2 1 7488.2.a.g 1
8.d odd 2 1 7488.2.a.h 1
9.c even 3 2 2106.2.e.b 2
9.d odd 6 2 2106.2.e.ba 2
12.b even 2 1 208.2.a.a 1
13.b even 2 1 3042.2.a.a 1
13.d odd 4 2 3042.2.b.a 2
15.d odd 2 1 650.2.a.j 1
15.e even 4 2 650.2.b.d 2
21.c even 2 1 1274.2.a.d 1
21.g even 6 2 1274.2.f.r 2
21.h odd 6 2 1274.2.f.p 2
24.f even 2 1 832.2.a.i 1
24.h odd 2 1 832.2.a.d 1
33.d even 2 1 3146.2.a.n 1
39.d odd 2 1 338.2.a.f 1
39.f even 4 2 338.2.b.c 2
39.h odd 6 2 338.2.c.a 2
39.i odd 6 2 338.2.c.d 2
39.k even 12 4 338.2.e.a 4
48.i odd 4 2 3328.2.b.m 2
48.k even 4 2 3328.2.b.j 2
51.c odd 2 1 7514.2.a.c 1
57.d even 2 1 9386.2.a.j 1
60.h even 2 1 5200.2.a.x 1
156.h even 2 1 2704.2.a.f 1
156.l odd 4 2 2704.2.f.d 2
195.e odd 2 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 3.b odd 2 1
208.2.a.a 1 12.b even 2 1
234.2.a.e 1 1.a even 1 1 trivial
338.2.a.f 1 39.d odd 2 1
338.2.b.c 2 39.f even 4 2
338.2.c.a 2 39.h odd 6 2
338.2.c.d 2 39.i odd 6 2
338.2.e.a 4 39.k even 12 4
650.2.a.j 1 15.d odd 2 1
650.2.b.d 2 15.e even 4 2
832.2.a.d 1 24.h odd 2 1
832.2.a.i 1 24.f even 2 1
1274.2.a.d 1 21.c even 2 1
1274.2.f.p 2 21.h odd 6 2
1274.2.f.r 2 21.g even 6 2
1872.2.a.q 1 4.b odd 2 1
2106.2.e.b 2 9.c even 3 2
2106.2.e.ba 2 9.d odd 6 2
2704.2.a.f 1 156.h even 2 1
2704.2.f.d 2 156.l odd 4 2
3042.2.a.a 1 13.b even 2 1
3042.2.b.a 2 13.d odd 4 2
3146.2.a.n 1 33.d even 2 1
3328.2.b.j 2 48.k even 4 2
3328.2.b.m 2 48.i odd 4 2
5200.2.a.x 1 60.h even 2 1
5850.2.a.p 1 5.b even 2 1
5850.2.e.a 2 5.c odd 4 2
7488.2.a.g 1 8.b even 2 1
7488.2.a.h 1 8.d odd 2 1
7514.2.a.c 1 51.c odd 2 1
8450.2.a.c 1 195.e odd 2 1
9386.2.a.j 1 57.d even 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ T5 - 3 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

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