# Properties

 Label 3234.2.a.o 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + 3 q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + 3 * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + 3 q^{5} - q^{6} - q^{8} + q^{9} - 3 q^{10} - q^{11} + q^{12} + 6 q^{13} + 3 q^{15} + q^{16} - 5 q^{17} - q^{18} + 6 q^{19} + 3 q^{20} + q^{22} + 5 q^{23} - q^{24} + 4 q^{25} - 6 q^{26} + q^{27} - 6 q^{29} - 3 q^{30} + 4 q^{31} - q^{32} - q^{33} + 5 q^{34} + q^{36} - 2 q^{37} - 6 q^{38} + 6 q^{39} - 3 q^{40} + 5 q^{41} - 10 q^{43} - q^{44} + 3 q^{45} - 5 q^{46} + 9 q^{47} + q^{48} - 4 q^{50} - 5 q^{51} + 6 q^{52} + 2 q^{53} - q^{54} - 3 q^{55} + 6 q^{57} + 6 q^{58} - 12 q^{59} + 3 q^{60} - 5 q^{61} - 4 q^{62} + q^{64} + 18 q^{65} + q^{66} + 5 q^{67} - 5 q^{68} + 5 q^{69} + 4 q^{71} - q^{72} + 12 q^{73} + 2 q^{74} + 4 q^{75} + 6 q^{76} - 6 q^{78} - q^{79} + 3 q^{80} + q^{81} - 5 q^{82} + q^{83} - 15 q^{85} + 10 q^{86} - 6 q^{87} + q^{88} + 6 q^{89} - 3 q^{90} + 5 q^{92} + 4 q^{93} - 9 q^{94} + 18 q^{95} - q^{96} + 9 q^{97} - q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + 3 * q^5 - q^6 - q^8 + q^9 - 3 * q^10 - q^11 + q^12 + 6 * q^13 + 3 * q^15 + q^16 - 5 * q^17 - q^18 + 6 * q^19 + 3 * q^20 + q^22 + 5 * q^23 - q^24 + 4 * q^25 - 6 * q^26 + q^27 - 6 * q^29 - 3 * q^30 + 4 * q^31 - q^32 - q^33 + 5 * q^34 + q^36 - 2 * q^37 - 6 * q^38 + 6 * q^39 - 3 * q^40 + 5 * q^41 - 10 * q^43 - q^44 + 3 * q^45 - 5 * q^46 + 9 * q^47 + q^48 - 4 * q^50 - 5 * q^51 + 6 * q^52 + 2 * q^53 - q^54 - 3 * q^55 + 6 * q^57 + 6 * q^58 - 12 * q^59 + 3 * q^60 - 5 * q^61 - 4 * q^62 + q^64 + 18 * q^65 + q^66 + 5 * q^67 - 5 * q^68 + 5 * q^69 + 4 * q^71 - q^72 + 12 * q^73 + 2 * q^74 + 4 * q^75 + 6 * q^76 - 6 * q^78 - q^79 + 3 * q^80 + q^81 - 5 * q^82 + q^83 - 15 * q^85 + 10 * q^86 - 6 * q^87 + q^88 + 6 * q^89 - 3 * q^90 + 5 * q^92 + 4 * q^93 - 9 * q^94 + 18 * q^95 - q^96 + 9 * 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 3.00000 −1.00000 0 −1.00000 1.00000 −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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$+1$$
$$11$$ $$+1$$

## 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.o 1
3.b odd 2 1 9702.2.a.be 1
7.b odd 2 1 3234.2.a.a 1
7.c even 3 2 462.2.i.a 2
21.c even 2 1 9702.2.a.ce 1
21.h odd 6 2 1386.2.k.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.a 2 7.c even 3 2
1386.2.k.j 2 21.h odd 6 2
3234.2.a.a 1 7.b odd 2 1
3234.2.a.o 1 1.a even 1 1 trivial
9702.2.a.be 1 3.b odd 2 1
9702.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{13} - 6$$ T13 - 6 $$T_{17} + 5$$ T17 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 3$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 6$$
$17$ $$T + 5$$
$19$ $$T - 6$$
$23$ $$T - 5$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T - 5$$
$43$ $$T + 10$$
$47$ $$T - 9$$
$53$ $$T - 2$$
$59$ $$T + 12$$
$61$ $$T + 5$$
$67$ $$T - 5$$
$71$ $$T - 4$$
$73$ $$T - 12$$
$79$ $$T + 1$$
$83$ $$T - 1$$
$89$ $$T - 6$$
$97$ $$T - 9$$