# Properties

 Label 3234.2.a.i 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 $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 8 q^{19} - 2 q^{20} - q^{22} + 4 q^{23} - q^{24} - q^{25} + 2 q^{26} + q^{27} + 2 q^{29} + 2 q^{30} - 8 q^{31} - q^{32} + q^{33} + 6 q^{34} + q^{36} + 6 q^{37} - 8 q^{38} - 2 q^{39} + 2 q^{40} - 6 q^{41} + 8 q^{43} + q^{44} - 2 q^{45} - 4 q^{46} - 4 q^{47} + q^{48} + q^{50} - 6 q^{51} - 2 q^{52} + 10 q^{53} - q^{54} - 2 q^{55} + 8 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} + 14 q^{61} + 8 q^{62} + q^{64} + 4 q^{65} - q^{66} - 4 q^{67} - 6 q^{68} + 4 q^{69} - 4 q^{71} - q^{72} + 14 q^{73} - 6 q^{74} - q^{75} + 8 q^{76} + 2 q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} - 4 q^{83} + 12 q^{85} - 8 q^{86} + 2 q^{87} - q^{88} + 14 q^{89} + 2 q^{90} + 4 q^{92} - 8 q^{93} + 4 q^{94} - 16 q^{95} - q^{96} - 18 q^{97} + q^{99}+O(q^{100})$$ 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 - 6 * q^17 - q^18 + 8 * q^19 - 2 * q^20 - q^22 + 4 * q^23 - q^24 - q^25 + 2 * q^26 + q^27 + 2 * q^29 + 2 * q^30 - 8 * q^31 - q^32 + q^33 + 6 * q^34 + q^36 + 6 * q^37 - 8 * q^38 - 2 * q^39 + 2 * q^40 - 6 * q^41 + 8 * q^43 + q^44 - 2 * q^45 - 4 * q^46 - 4 * q^47 + q^48 + q^50 - 6 * q^51 - 2 * q^52 + 10 * q^53 - q^54 - 2 * q^55 + 8 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 + 14 * q^61 + 8 * q^62 + q^64 + 4 * q^65 - q^66 - 4 * q^67 - 6 * q^68 + 4 * q^69 - 4 * q^71 - q^72 + 14 * q^73 - 6 * q^74 - q^75 + 8 * q^76 + 2 * q^78 - 8 * q^79 - 2 * q^80 + q^81 + 6 * q^82 - 4 * q^83 + 12 * q^85 - 8 * q^86 + 2 * q^87 - q^88 + 14 * q^89 + 2 * q^90 + 4 * q^92 - 8 * q^93 + 4 * q^94 - 16 * q^95 - q^96 - 18 * 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.i 1
3.b odd 2 1 9702.2.a.by 1
7.b odd 2 1 462.2.a.c 1
21.c even 2 1 1386.2.a.g 1
28.d even 2 1 3696.2.a.bb 1
77.b even 2 1 5082.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.c 1 7.b odd 2 1
1386.2.a.g 1 21.c even 2 1
3234.2.a.i 1 1.a even 1 1 trivial
3696.2.a.bb 1 28.d even 2 1
5082.2.a.v 1 77.b even 2 1
9702.2.a.by 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 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T - 8$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 4$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T - 14$$
$67$ $$T + 4$$
$71$ $$T + 4$$
$73$ $$T - 14$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T - 14$$
$97$ $$T + 18$$