# Properties

 Label 3380.2.a.f Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ 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] = [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: $$0$$ 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)

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

## 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.f 1
13.b even 2 1 3380.2.a.i 1
13.c even 3 2 260.2.i.a 2
13.d odd 4 2 3380.2.f.d 2
39.i odd 6 2 2340.2.q.f 2
52.j odd 6 2 1040.2.q.i 2
65.n even 6 2 1300.2.i.d 2
65.q odd 12 4 1300.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 13.c even 3 2
1040.2.q.i 2 52.j odd 6 2
1300.2.i.d 2 65.n even 6 2
1300.2.bb.b 4 65.q odd 12 4
2340.2.q.f 2 39.i odd 6 2
3380.2.a.f 1 1.a even 1 1 trivial
3380.2.a.i 1 13.b even 2 1
3380.2.f.d 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3380))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 1$$ T7 + 1 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

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