# Properties

 Label 3680.2.a.g Level $3680$ Weight $2$ Character orbit 3680.a Self dual yes Analytic conductor $29.385$ 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] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3680 = 2^{5} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3680.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: $$29.3849479438$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - q^{5} - 3 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - q^5 - 3 * q^7 + q^9 $$q + 2 q^{3} - q^{5} - 3 q^{7} + q^{9} - 6 q^{11} + 4 q^{13} - 2 q^{15} + 7 q^{17} - 8 q^{19} - 6 q^{21} - q^{23} + q^{25} - 4 q^{27} + 3 q^{29} + 11 q^{31} - 12 q^{33} + 3 q^{35} + 7 q^{37} + 8 q^{39} + 5 q^{41} + 12 q^{43} - q^{45} + 12 q^{47} + 2 q^{49} + 14 q^{51} + 7 q^{53} + 6 q^{55} - 16 q^{57} - 5 q^{59} + 2 q^{61} - 3 q^{63} - 4 q^{65} + q^{67} - 2 q^{69} - 3 q^{71} + 2 q^{75} + 18 q^{77} - 12 q^{79} - 11 q^{81} + 5 q^{83} - 7 q^{85} + 6 q^{87} - 6 q^{89} - 12 q^{91} + 22 q^{93} + 8 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100})$$ q + 2 * q^3 - q^5 - 3 * q^7 + q^9 - 6 * q^11 + 4 * q^13 - 2 * q^15 + 7 * q^17 - 8 * q^19 - 6 * q^21 - q^23 + q^25 - 4 * q^27 + 3 * q^29 + 11 * q^31 - 12 * q^33 + 3 * q^35 + 7 * q^37 + 8 * q^39 + 5 * q^41 + 12 * q^43 - q^45 + 12 * q^47 + 2 * q^49 + 14 * q^51 + 7 * q^53 + 6 * q^55 - 16 * q^57 - 5 * q^59 + 2 * q^61 - 3 * q^63 - 4 * q^65 + q^67 - 2 * q^69 - 3 * q^71 + 2 * q^75 + 18 * q^77 - 12 * q^79 - 11 * q^81 + 5 * q^83 - 7 * q^85 + 6 * q^87 - 6 * q^89 - 12 * q^91 + 22 * q^93 + 8 * q^95 + 10 * 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 2.00000 0 −1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 3680.2.a.g yes 1
4.b odd 2 1 3680.2.a.c 1
8.b even 2 1 7360.2.a.d 1
8.d odd 2 1 7360.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.c 1 4.b odd 2 1
3680.2.a.g yes 1 1.a even 1 1 trivial
7360.2.a.d 1 8.b even 2 1
7360.2.a.z 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3680))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 6$$ T11 + 6 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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