# Properties

 Label 3344.2.a.h Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ 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] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3344.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: $$26.7019744359$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^5 + 3 * q^7 - 2 * q^9 $$q + q^{3} - 2 q^{5} + 3 q^{7} - 2 q^{9} + q^{11} + q^{13} - 2 q^{15} - 7 q^{17} - q^{19} + 3 q^{21} + 5 q^{23} - q^{25} - 5 q^{27} + q^{29} - 10 q^{31} + q^{33} - 6 q^{35} - 6 q^{37} + q^{39} + 6 q^{41} + 4 q^{43} + 4 q^{45} + 2 q^{49} - 7 q^{51} - q^{53} - 2 q^{55} - q^{57} - 3 q^{59} - 12 q^{61} - 6 q^{63} - 2 q^{65} - 3 q^{67} + 5 q^{69} + 10 q^{71} + 3 q^{73} - q^{75} + 3 q^{77} - 8 q^{79} + q^{81} - 8 q^{83} + 14 q^{85} + q^{87} - 8 q^{89} + 3 q^{91} - 10 q^{93} + 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^5 + 3 * q^7 - 2 * q^9 + q^11 + q^13 - 2 * q^15 - 7 * q^17 - q^19 + 3 * q^21 + 5 * q^23 - q^25 - 5 * q^27 + q^29 - 10 * q^31 + q^33 - 6 * q^35 - 6 * q^37 + q^39 + 6 * q^41 + 4 * q^43 + 4 * q^45 + 2 * q^49 - 7 * q^51 - q^53 - 2 * q^55 - q^57 - 3 * q^59 - 12 * q^61 - 6 * q^63 - 2 * q^65 - 3 * q^67 + 5 * q^69 + 10 * q^71 + 3 * q^73 - q^75 + 3 * q^77 - 8 * q^79 + q^81 - 8 * q^83 + 14 * q^85 + q^87 - 8 * q^89 + 3 * q^91 - 10 * q^93 + 2 * q^95 + 8 * q^97 - 2 * 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
0 1.00000 0 −2.00000 0 3.00000 0 −2.00000 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
$$2$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$1$$

## 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 3344.2.a.h 1
4.b odd 2 1 418.2.a.a 1
12.b even 2 1 3762.2.a.g 1
44.c even 2 1 4598.2.a.b 1
76.d even 2 1 7942.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.a 1 4.b odd 2 1
3344.2.a.h 1 1.a even 1 1 trivial
3762.2.a.g 1 12.b even 2 1
4598.2.a.b 1 44.c even 2 1
7942.2.a.i 1 76.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3344))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

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