# Properties

 Label 325.2.a.c Level $325$ Weight $2$ Character orbit 325.a Self dual yes Analytic conductor $2.595$ 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] = [325,2,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.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: $$2.59513806569$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} - 4 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 - 4 * q^7 - 2 * q^9 $$q + q^{3} - 2 q^{4} - 4 q^{7} - 2 q^{9} - 6 q^{11} - 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} - 4 q^{19} - 4 q^{21} + 3 q^{23} - 5 q^{27} + 8 q^{28} - 3 q^{29} - 4 q^{31} - 6 q^{33} + 4 q^{36} + 2 q^{37} + q^{39} + 6 q^{41} - 7 q^{43} + 12 q^{44} + 4 q^{48} + 9 q^{49} + 6 q^{51} - 2 q^{52} - 9 q^{53} - 4 q^{57} - 6 q^{59} - q^{61} + 8 q^{63} - 8 q^{64} + 14 q^{67} - 12 q^{68} + 3 q^{69} - 6 q^{71} - 4 q^{73} + 8 q^{76} + 24 q^{77} + 11 q^{79} + q^{81} - 6 q^{83} + 8 q^{84} - 3 q^{87} - 4 q^{91} - 6 q^{92} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^4 - 4 * q^7 - 2 * q^9 - 6 * q^11 - 2 * q^12 + q^13 + 4 * q^16 + 6 * q^17 - 4 * q^19 - 4 * q^21 + 3 * q^23 - 5 * q^27 + 8 * q^28 - 3 * q^29 - 4 * q^31 - 6 * q^33 + 4 * q^36 + 2 * q^37 + q^39 + 6 * q^41 - 7 * q^43 + 12 * q^44 + 4 * q^48 + 9 * q^49 + 6 * q^51 - 2 * q^52 - 9 * q^53 - 4 * q^57 - 6 * q^59 - q^61 + 8 * q^63 - 8 * q^64 + 14 * q^67 - 12 * q^68 + 3 * q^69 - 6 * q^71 - 4 * q^73 + 8 * q^76 + 24 * q^77 + 11 * q^79 + q^81 - 6 * q^83 + 8 * q^84 - 3 * q^87 - 4 * q^91 - 6 * q^92 - 4 * q^93 - 10 * q^97 + 12 * 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 −2.00000 0 0 −4.00000 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
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 325.2.a.c yes 1
3.b odd 2 1 2925.2.a.g 1
4.b odd 2 1 5200.2.a.n 1
5.b even 2 1 325.2.a.b 1
5.c odd 4 2 325.2.b.c 2
13.b even 2 1 4225.2.a.j 1
15.d odd 2 1 2925.2.a.l 1
15.e even 4 2 2925.2.c.n 2
20.d odd 2 1 5200.2.a.v 1
65.d even 2 1 4225.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.a.b 1 5.b even 2 1
325.2.a.c yes 1 1.a even 1 1 trivial
325.2.b.c 2 5.c odd 4 2
2925.2.a.g 1 3.b odd 2 1
2925.2.a.l 1 15.d odd 2 1
2925.2.c.n 2 15.e even 4 2
4225.2.a.i 1 65.d even 2 1
4225.2.a.j 1 13.b even 2 1
5200.2.a.n 1 4.b odd 2 1
5200.2.a.v 1 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$:

 $$T_{2}$$ T2 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 6$$
$13$ $$T - 1$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T - 3$$
$29$ $$T + 3$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T + 7$$
$47$ $$T$$
$53$ $$T + 9$$
$59$ $$T + 6$$
$61$ $$T + 1$$
$67$ $$T - 14$$
$71$ $$T + 6$$
$73$ $$T + 4$$
$79$ $$T - 11$$
$83$ $$T + 6$$
$89$ $$T$$
$97$ $$T + 10$$