# Properties

 Label 3380.2.a.a Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ Analytic rank $2$ 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] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.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: $$26.9894358832$$ Analytic rank: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 - q^5 - 3 * q^7 + 6 * q^9 $$q - 3 q^{3} - q^{5} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{15} - 7 q^{17} - q^{19} + 9 q^{21} - 7 q^{23} + q^{25} - 9 q^{27} - 5 q^{29} + 4 q^{31} + 9 q^{33} + 3 q^{35} + 3 q^{37} - 7 q^{41} - 9 q^{43} - 6 q^{45} - 8 q^{47} + 2 q^{49} + 21 q^{51} - 6 q^{53} + 3 q^{55} + 3 q^{57} - 5 q^{59} - 5 q^{61} - 18 q^{63} - 13 q^{67} + 21 q^{69} + 3 q^{71} + 14 q^{73} - 3 q^{75} + 9 q^{77} - 8 q^{79} + 9 q^{81} - 12 q^{83} + 7 q^{85} + 15 q^{87} - 7 q^{89} - 12 q^{93} + q^{95} + 11 q^{97} - 18 q^{99}+O(q^{100})$$ q - 3 * q^3 - q^5 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^15 - 7 * q^17 - q^19 + 9 * q^21 - 7 * q^23 + q^25 - 9 * q^27 - 5 * q^29 + 4 * q^31 + 9 * q^33 + 3 * q^35 + 3 * q^37 - 7 * q^41 - 9 * q^43 - 6 * q^45 - 8 * q^47 + 2 * q^49 + 21 * q^51 - 6 * q^53 + 3 * q^55 + 3 * q^57 - 5 * q^59 - 5 * q^61 - 18 * q^63 - 13 * q^67 + 21 * q^69 + 3 * q^71 + 14 * q^73 - 3 * q^75 + 9 * q^77 - 8 * q^79 + 9 * q^81 - 12 * q^83 + 7 * q^85 + 15 * q^87 - 7 * q^89 - 12 * q^93 + q^95 + 11 * q^97 - 18 * 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 −3.00000 0 −1.00000 0 −3.00000 0 6.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 3380.2.a.a 1
13.b even 2 1 3380.2.a.b 1
13.d odd 4 2 3380.2.f.a 2
13.e even 6 2 260.2.i.d 2
39.h odd 6 2 2340.2.q.a 2
52.i odd 6 2 1040.2.q.b 2
65.l even 6 2 1300.2.i.a 2
65.r odd 12 4 1300.2.bb.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 13.e even 6 2
1040.2.q.b 2 52.i odd 6 2
1300.2.i.a 2 65.l even 6 2
1300.2.bb.e 4 65.r odd 12 4
2340.2.q.a 2 39.h odd 6 2
3380.2.a.a 1 1.a even 1 1 trivial
3380.2.a.b 1 13.b even 2 1
3380.2.f.a 2 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3380))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{7} + 3$$ T7 + 3 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 1$$
$7$ $$T + 3$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T + 7$$
$19$ $$T + 1$$
$23$ $$T + 7$$
$29$ $$T + 5$$
$31$ $$T - 4$$
$37$ $$T - 3$$
$41$ $$T + 7$$
$43$ $$T + 9$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T + 5$$
$61$ $$T + 5$$
$67$ $$T + 13$$
$71$ $$T - 3$$
$73$ $$T - 14$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T + 7$$
$97$ $$T - 11$$