# Properties

 Label 3380.2.a.j Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 + q^5 - 2 * q^7 + q^9 $$q + 2 q^{3} + q^{5} - 2 q^{7} + q^{9} - 4 q^{11} + 2 q^{15} + 2 q^{17} - 4 q^{21} - 6 q^{23} + q^{25} - 4 q^{27} - 10 q^{29} - 8 q^{33} - 2 q^{35} - 10 q^{37} + 2 q^{41} + 2 q^{43} + q^{45} + 6 q^{47} - 3 q^{49} + 4 q^{51} + 2 q^{53} - 4 q^{55} + 8 q^{59} + 2 q^{61} - 2 q^{63} + 6 q^{67} - 12 q^{69} + 8 q^{71} - 10 q^{73} + 2 q^{75} + 8 q^{77} - 16 q^{79} - 11 q^{81} - 6 q^{83} + 2 q^{85} - 20 q^{87} - 10 q^{89} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q + 2 * q^3 + q^5 - 2 * q^7 + q^9 - 4 * q^11 + 2 * q^15 + 2 * q^17 - 4 * q^21 - 6 * q^23 + q^25 - 4 * q^27 - 10 * q^29 - 8 * q^33 - 2 * q^35 - 10 * q^37 + 2 * q^41 + 2 * q^43 + q^45 + 6 * q^47 - 3 * q^49 + 4 * q^51 + 2 * q^53 - 4 * q^55 + 8 * q^59 + 2 * q^61 - 2 * q^63 + 6 * q^67 - 12 * q^69 + 8 * q^71 - 10 * q^73 + 2 * q^75 + 8 * q^77 - 16 * q^79 - 11 * q^81 - 6 * q^83 + 2 * q^85 - 20 * q^87 - 10 * q^89 - 2 * q^97 - 4 * 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 2.00000 0 1.00000 0 −2.00000 0 1.00000 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.j 1
13.b even 2 1 260.2.a.a 1
13.d odd 4 2 3380.2.f.f 2
39.d odd 2 1 2340.2.a.i 1
52.b odd 2 1 1040.2.a.a 1
65.d even 2 1 1300.2.a.a 1
65.h odd 4 2 1300.2.c.a 2
104.e even 2 1 4160.2.a.d 1
104.h odd 2 1 4160.2.a.s 1
156.h even 2 1 9360.2.a.bi 1
260.g odd 2 1 5200.2.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.a 1 13.b even 2 1
1040.2.a.a 1 52.b odd 2 1
1300.2.a.a 1 65.d even 2 1
1300.2.c.a 2 65.h odd 4 2
2340.2.a.i 1 39.d odd 2 1
3380.2.a.j 1 1.a even 1 1 trivial
3380.2.f.f 2 13.d odd 4 2
4160.2.a.d 1 104.e even 2 1
4160.2.a.s 1 104.h odd 2 1
5200.2.a.bh 1 260.g odd 2 1
9360.2.a.bi 1 156.h even 2 1

## Hecke kernels

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

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T + 10$$
$31$ $$T$$
$37$ $$T + 10$$
$41$ $$T - 2$$
$43$ $$T - 2$$
$47$ $$T - 6$$
$53$ $$T - 2$$
$59$ $$T - 8$$
$61$ $$T - 2$$
$67$ $$T - 6$$
$71$ $$T - 8$$
$73$ $$T + 10$$
$79$ $$T + 16$$
$83$ $$T + 6$$
$89$ $$T + 10$$
$97$ $$T + 2$$