# Properties

 Label 8325.2.a.e Level $8325$ Weight $2$ Character orbit 8325.a Self dual yes Analytic conductor $66.475$ 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] = [8325,2,Mod(1,8325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8325, 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("8325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8325 = 3^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8325.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: $$66.4754596827$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + q^7 $$q - 2 q^{2} + 2 q^{4} + q^{7} + 5 q^{11} + 2 q^{13} - 2 q^{14} - 4 q^{16} - 10 q^{22} + 2 q^{23} - 4 q^{26} + 2 q^{28} - 6 q^{29} - 4 q^{31} + 8 q^{32} + q^{37} + 9 q^{41} - 2 q^{43} + 10 q^{44} - 4 q^{46} - 9 q^{47} - 6 q^{49} + 4 q^{52} + q^{53} + 12 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} - 8 q^{64} - 8 q^{67} - 9 q^{71} + q^{73} - 2 q^{74} + 5 q^{77} + 4 q^{79} - 18 q^{82} - 15 q^{83} + 4 q^{86} - 4 q^{89} + 2 q^{91} + 4 q^{92} + 18 q^{94} - 4 q^{97} + 12 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + q^7 + 5 * q^11 + 2 * q^13 - 2 * q^14 - 4 * q^16 - 10 * q^22 + 2 * q^23 - 4 * q^26 + 2 * q^28 - 6 * q^29 - 4 * q^31 + 8 * q^32 + q^37 + 9 * q^41 - 2 * q^43 + 10 * q^44 - 4 * q^46 - 9 * q^47 - 6 * q^49 + 4 * q^52 + q^53 + 12 * q^58 - 8 * q^59 - 8 * q^61 + 8 * q^62 - 8 * q^64 - 8 * q^67 - 9 * q^71 + q^73 - 2 * q^74 + 5 * q^77 + 4 * q^79 - 18 * q^82 - 15 * q^83 + 4 * q^86 - 4 * q^89 + 2 * q^91 + 4 * q^92 + 18 * q^94 - 4 * q^97 + 12 * 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
−2.00000 0 2.00000 0 0 1.00000 0 0 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$37$$ $$-1$$

## 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 8325.2.a.e 1
3.b odd 2 1 925.2.a.e 1
5.b even 2 1 333.2.a.d 1
15.d odd 2 1 37.2.a.a 1
15.e even 4 2 925.2.b.b 2
20.d odd 2 1 5328.2.a.r 1
60.h even 2 1 592.2.a.e 1
105.g even 2 1 1813.2.a.a 1
120.i odd 2 1 2368.2.a.q 1
120.m even 2 1 2368.2.a.b 1
165.d even 2 1 4477.2.a.b 1
195.e odd 2 1 6253.2.a.c 1
555.b odd 2 1 1369.2.a.e 1
555.m even 4 2 1369.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 15.d odd 2 1
333.2.a.d 1 5.b even 2 1
592.2.a.e 1 60.h even 2 1
925.2.a.e 1 3.b odd 2 1
925.2.b.b 2 15.e even 4 2
1369.2.a.e 1 555.b odd 2 1
1369.2.b.c 2 555.m even 4 2
1813.2.a.a 1 105.g even 2 1
2368.2.a.b 1 120.m even 2 1
2368.2.a.q 1 120.i odd 2 1
4477.2.a.b 1 165.d even 2 1
5328.2.a.r 1 20.d odd 2 1
6253.2.a.c 1 195.e odd 2 1
8325.2.a.e 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8325))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} - 1$$ T7 - 1 $$T_{11} - 5$$ T11 - 5 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 5$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 2$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 1$$
$41$ $$T - 9$$
$43$ $$T + 2$$
$47$ $$T + 9$$
$53$ $$T - 1$$
$59$ $$T + 8$$
$61$ $$T + 8$$
$67$ $$T + 8$$
$71$ $$T + 9$$
$73$ $$T - 1$$
$79$ $$T - 4$$
$83$ $$T + 15$$
$89$ $$T + 4$$
$97$ $$T + 4$$