# Properties

 Label 3234.2.a.e Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} - 4 q^{13} + q^{16} + 4 q^{17} - q^{18} - q^{22} + q^{24} - 5 q^{25} + 4 q^{26} - q^{27} + 2 q^{29} - 4 q^{31} - q^{32} - q^{33} - 4 q^{34} + q^{36} + 10 q^{37} + 4 q^{39} - 12 q^{41} + 4 q^{43} + q^{44} + 4 q^{47} - q^{48} + 5 q^{50} - 4 q^{51} - 4 q^{52} - 10 q^{53} + q^{54} - 2 q^{58} - 4 q^{59} + 4 q^{61} + 4 q^{62} + q^{64} + q^{66} + 4 q^{67} + 4 q^{68} - 8 q^{71} - q^{72} + 4 q^{73} - 10 q^{74} + 5 q^{75} - 4 q^{78} + 8 q^{79} + q^{81} + 12 q^{82} - 16 q^{83} - 4 q^{86} - 2 q^{87} - q^{88} + 8 q^{89} + 4 q^{93} - 4 q^{94} + q^{96} - 8 q^{97} + q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 + q^11 - q^12 - 4 * q^13 + q^16 + 4 * q^17 - q^18 - q^22 + q^24 - 5 * q^25 + 4 * q^26 - q^27 + 2 * q^29 - 4 * q^31 - q^32 - q^33 - 4 * q^34 + q^36 + 10 * q^37 + 4 * q^39 - 12 * q^41 + 4 * q^43 + q^44 + 4 * q^47 - q^48 + 5 * q^50 - 4 * q^51 - 4 * q^52 - 10 * q^53 + q^54 - 2 * q^58 - 4 * q^59 + 4 * q^61 + 4 * q^62 + q^64 + q^66 + 4 * q^67 + 4 * q^68 - 8 * q^71 - q^72 + 4 * q^73 - 10 * q^74 + 5 * q^75 - 4 * q^78 + 8 * q^79 + q^81 + 12 * q^82 - 16 * q^83 - 4 * q^86 - 2 * q^87 - q^88 + 8 * q^89 + 4 * q^93 - 4 * q^94 + q^96 - 8 * 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 0 1.00000 0 −1.00000 1.00000 0
 $$n$$: 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.e 1
3.b odd 2 1 9702.2.a.bk 1
7.b odd 2 1 3234.2.a.m yes 1
21.c even 2 1 9702.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.e 1 1.a even 1 1 trivial
3234.2.a.m yes 1 7.b odd 2 1
9702.2.a.bk 1 3.b odd 2 1
9702.2.a.bo 1 21.c even 2 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

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