# Properties

 Label 3330.2.a.v Level $3330$ Weight $2$ Character orbit 3330.a Self dual yes Analytic conductor $26.590$ 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] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.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: $$26.5901838731$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - 3 q^{11} - 4 q^{13} - q^{14} + q^{16} - 3 q^{17} + 2 q^{19} + q^{20} - 3 q^{22} - 6 q^{23} + q^{25} - 4 q^{26} - q^{28} - 3 q^{29} + 5 q^{31} + q^{32} - 3 q^{34} - q^{35} + q^{37} + 2 q^{38} + q^{40} - 3 q^{41} - q^{43} - 3 q^{44} - 6 q^{46} - 12 q^{47} - 6 q^{49} + q^{50} - 4 q^{52} - 3 q^{53} - 3 q^{55} - q^{56} - 3 q^{58} - q^{61} + 5 q^{62} + q^{64} - 4 q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 6 q^{71} - 16 q^{73} + q^{74} + 2 q^{76} + 3 q^{77} + 8 q^{79} + q^{80} - 3 q^{82} + 12 q^{83} - 3 q^{85} - q^{86} - 3 q^{88} + 6 q^{89} + 4 q^{91} - 6 q^{92} - 12 q^{94} + 2 q^{95} + 17 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 + q^10 - 3 * q^11 - 4 * q^13 - q^14 + q^16 - 3 * q^17 + 2 * q^19 + q^20 - 3 * q^22 - 6 * q^23 + q^25 - 4 * q^26 - q^28 - 3 * q^29 + 5 * q^31 + q^32 - 3 * q^34 - q^35 + q^37 + 2 * q^38 + q^40 - 3 * q^41 - q^43 - 3 * q^44 - 6 * q^46 - 12 * q^47 - 6 * q^49 + q^50 - 4 * q^52 - 3 * q^53 - 3 * q^55 - q^56 - 3 * q^58 - q^61 + 5 * q^62 + q^64 - 4 * q^65 - 4 * q^67 - 3 * q^68 - q^70 - 6 * q^71 - 16 * q^73 + q^74 + 2 * q^76 + 3 * q^77 + 8 * q^79 + q^80 - 3 * q^82 + 12 * q^83 - 3 * q^85 - q^86 - 3 * q^88 + 6 * q^89 + 4 * q^91 - 6 * q^92 - 12 * q^94 + 2 * q^95 + 17 * q^97 - 6 * q^98

## 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 0 1.00000 1.00000 0 −1.00000 1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 3330.2.a.v 1
3.b odd 2 1 370.2.a.a 1
12.b even 2 1 2960.2.a.j 1
15.d odd 2 1 1850.2.a.o 1
15.e even 4 2 1850.2.b.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.a 1 3.b odd 2 1
1850.2.a.o 1 15.d odd 2 1
1850.2.b.g 2 15.e even 4 2
2960.2.a.j 1 12.b even 2 1
3330.2.a.v 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3330))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 4$$ T13 + 4 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T + 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T + 3$$
$31$ $$T - 5$$
$37$ $$T - 1$$
$41$ $$T + 3$$
$43$ $$T + 1$$
$47$ $$T + 12$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T + 1$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T + 16$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T - 17$$