# Properties

 Label 3234.2.a.p Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ 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] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.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: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - 2 * q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} + 2 q^{13} + 2 q^{15} + q^{16} + 2 q^{17} + q^{18} - 2 q^{20} + q^{22} - q^{24} - q^{25} + 2 q^{26} - q^{27} - 2 q^{29} + 2 q^{30} - 4 q^{31} + q^{32} - q^{33} + 2 q^{34} + q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{40} + 10 q^{41} + 4 q^{43} + q^{44} - 2 q^{45} - 4 q^{47} - q^{48} - q^{50} - 2 q^{51} + 2 q^{52} - 2 q^{53} - q^{54} - 2 q^{55} - 2 q^{58} + 12 q^{59} + 2 q^{60} + 2 q^{61} - 4 q^{62} + q^{64} - 4 q^{65} - q^{66} + 12 q^{67} + 2 q^{68} + 8 q^{71} + q^{72} - 6 q^{73} - 2 q^{74} + q^{75} - 2 q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 10 q^{82} + 8 q^{83} - 4 q^{85} + 4 q^{86} + 2 q^{87} + q^{88} + 14 q^{89} - 2 q^{90} + 4 q^{93} - 4 q^{94} - q^{96} + 14 q^{97} + q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - 2 * q^5 - q^6 + q^8 + q^9 - 2 * q^10 + q^11 - q^12 + 2 * q^13 + 2 * q^15 + q^16 + 2 * q^17 + q^18 - 2 * q^20 + q^22 - q^24 - q^25 + 2 * q^26 - q^27 - 2 * q^29 + 2 * q^30 - 4 * q^31 + q^32 - q^33 + 2 * q^34 + q^36 - 2 * q^37 - 2 * q^39 - 2 * q^40 + 10 * q^41 + 4 * q^43 + q^44 - 2 * q^45 - 4 * q^47 - q^48 - q^50 - 2 * q^51 + 2 * q^52 - 2 * q^53 - q^54 - 2 * q^55 - 2 * q^58 + 12 * q^59 + 2 * q^60 + 2 * q^61 - 4 * q^62 + q^64 - 4 * q^65 - q^66 + 12 * q^67 + 2 * q^68 + 8 * q^71 + q^72 - 6 * q^73 - 2 * q^74 + q^75 - 2 * q^78 - 8 * q^79 - 2 * q^80 + q^81 + 10 * q^82 + 8 * q^83 - 4 * q^85 + 4 * q^86 + 2 * q^87 + q^88 + 14 * q^89 - 2 * q^90 + 4 * q^93 - 4 * q^94 - q^96 + 14 * q^97 + q^99

## 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 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 3234.2.a.p 1
3.b odd 2 1 9702.2.a.r 1
7.b odd 2 1 462.2.a.g 1
21.c even 2 1 1386.2.a.a 1
28.d even 2 1 3696.2.a.m 1
77.b even 2 1 5082.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.g 1 7.b odd 2 1
1386.2.a.a 1 21.c even 2 1
3234.2.a.p 1 1.a even 1 1 trivial
3696.2.a.m 1 28.d even 2 1
5082.2.a.n 1 77.b even 2 1
9702.2.a.r 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

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