# Properties

 Label 364.2.a.a Level $364$ Weight $2$ Character orbit 364.a Self dual yes Analytic conductor $2.907$ 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] = [364,2,Mod(1,364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(364, 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("364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$364 = 2^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 364.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: $$2.90655463357$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^5 - q^7 + q^9 $$q - 2 q^{3} + q^{5} - q^{7} + q^{9} - 4 q^{11} + q^{13} - 2 q^{15} - 2 q^{17} - q^{19} + 2 q^{21} - 7 q^{23} - 4 q^{25} + 4 q^{27} - 5 q^{29} - 9 q^{31} + 8 q^{33} - q^{35} - 2 q^{37} - 2 q^{39} + 2 q^{41} + q^{43} + q^{45} + 9 q^{47} + q^{49} + 4 q^{51} + 3 q^{53} - 4 q^{55} + 2 q^{57} + 14 q^{61} - q^{63} + q^{65} + 10 q^{67} + 14 q^{69} - 14 q^{71} + 3 q^{73} + 8 q^{75} + 4 q^{77} + 5 q^{79} - 11 q^{81} + 5 q^{83} - 2 q^{85} + 10 q^{87} - 9 q^{89} - q^{91} + 18 q^{93} - q^{95} - q^{97} - 4 q^{99}+O(q^{100})$$ q - 2 * q^3 + q^5 - q^7 + q^9 - 4 * q^11 + q^13 - 2 * q^15 - 2 * q^17 - q^19 + 2 * q^21 - 7 * q^23 - 4 * q^25 + 4 * q^27 - 5 * q^29 - 9 * q^31 + 8 * q^33 - q^35 - 2 * q^37 - 2 * q^39 + 2 * q^41 + q^43 + q^45 + 9 * q^47 + q^49 + 4 * q^51 + 3 * q^53 - 4 * q^55 + 2 * q^57 + 14 * q^61 - q^63 + q^65 + 10 * q^67 + 14 * q^69 - 14 * q^71 + 3 * q^73 + 8 * q^75 + 4 * q^77 + 5 * q^79 - 11 * q^81 + 5 * q^83 - 2 * q^85 + 10 * q^87 - 9 * q^89 - q^91 + 18 * q^93 - q^95 - 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 −1.00000 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
$$2$$ $$-1$$
$$7$$ $$+1$$
$$13$$ $$-1$$

## 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 364.2.a.a 1
3.b odd 2 1 3276.2.a.b 1
4.b odd 2 1 1456.2.a.m 1
5.b even 2 1 9100.2.a.l 1
7.b odd 2 1 2548.2.a.i 1
7.c even 3 2 2548.2.j.j 2
7.d odd 6 2 2548.2.j.c 2
8.b even 2 1 5824.2.a.bb 1
8.d odd 2 1 5824.2.a.d 1
13.b even 2 1 4732.2.a.a 1
13.d odd 4 2 4732.2.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.a.a 1 1.a even 1 1 trivial
1456.2.a.m 1 4.b odd 2 1
2548.2.a.i 1 7.b odd 2 1
2548.2.j.c 2 7.d odd 6 2
2548.2.j.j 2 7.c even 3 2
3276.2.a.b 1 3.b odd 2 1
4732.2.a.a 1 13.b even 2 1
4732.2.g.a 2 13.d odd 4 2
5824.2.a.d 1 8.d odd 2 1
5824.2.a.bb 1 8.b even 2 1
9100.2.a.l 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(364))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T + 2$$
$19$ $$T + 1$$
$23$ $$T + 7$$
$29$ $$T + 5$$
$31$ $$T + 9$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T - 1$$
$47$ $$T - 9$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T - 14$$
$67$ $$T - 10$$
$71$ $$T + 14$$
$73$ $$T - 3$$
$79$ $$T - 5$$
$83$ $$T - 5$$
$89$ $$T + 9$$
$97$ $$T + 1$$