# Properties

 Label 2475.2.a.g Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ 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] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - 4 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8} - q^{11} + 2 q^{13} - 4 q^{14} - q^{16} - 2 q^{17} - q^{22} + 8 q^{23} + 2 q^{26} + 4 q^{28} + 6 q^{29} - 8 q^{31} + 5 q^{32} - 2 q^{34} - 6 q^{37} + 2 q^{41} + q^{44} + 8 q^{46} + 8 q^{47} + 9 q^{49} - 2 q^{52} + 6 q^{53} + 12 q^{56} + 6 q^{58} + 4 q^{59} + 6 q^{61} - 8 q^{62} + 7 q^{64} + 4 q^{67} + 2 q^{68} + 14 q^{73} - 6 q^{74} + 4 q^{77} - 4 q^{79} + 2 q^{82} + 12 q^{83} + 3 q^{88} + 6 q^{89} - 8 q^{91} - 8 q^{92} + 8 q^{94} - 2 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^4 - 4 * q^7 - 3 * q^8 - q^11 + 2 * q^13 - 4 * q^14 - q^16 - 2 * q^17 - q^22 + 8 * q^23 + 2 * q^26 + 4 * q^28 + 6 * q^29 - 8 * q^31 + 5 * q^32 - 2 * q^34 - 6 * q^37 + 2 * q^41 + q^44 + 8 * q^46 + 8 * q^47 + 9 * q^49 - 2 * q^52 + 6 * q^53 + 12 * q^56 + 6 * q^58 + 4 * q^59 + 6 * q^61 - 8 * q^62 + 7 * q^64 + 4 * q^67 + 2 * q^68 + 14 * q^73 - 6 * q^74 + 4 * q^77 - 4 * q^79 + 2 * q^82 + 12 * q^83 + 3 * q^88 + 6 * q^89 - 8 * q^91 - 8 * q^92 + 8 * q^94 - 2 * q^97 + 9 * 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 0 0 −4.00000 −3.00000 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$+1$$

## 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 2475.2.a.g 1
3.b odd 2 1 825.2.a.a 1
5.b even 2 1 99.2.a.b 1
5.c odd 4 2 2475.2.c.d 2
15.d odd 2 1 33.2.a.a 1
15.e even 4 2 825.2.c.a 2
20.d odd 2 1 1584.2.a.o 1
33.d even 2 1 9075.2.a.q 1
35.c odd 2 1 4851.2.a.b 1
40.e odd 2 1 6336.2.a.n 1
40.f even 2 1 6336.2.a.x 1
45.h odd 6 2 891.2.e.e 2
45.j even 6 2 891.2.e.g 2
55.d odd 2 1 1089.2.a.j 1
60.h even 2 1 528.2.a.g 1
105.g even 2 1 1617.2.a.j 1
120.i odd 2 1 2112.2.a.bb 1
120.m even 2 1 2112.2.a.j 1
165.d even 2 1 363.2.a.b 1
165.o odd 10 4 363.2.e.e 4
165.r even 10 4 363.2.e.g 4
195.e odd 2 1 5577.2.a.a 1
255.h odd 2 1 9537.2.a.m 1
660.g odd 2 1 5808.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 15.d odd 2 1
99.2.a.b 1 5.b even 2 1
363.2.a.b 1 165.d even 2 1
363.2.e.e 4 165.o odd 10 4
363.2.e.g 4 165.r even 10 4
528.2.a.g 1 60.h even 2 1
825.2.a.a 1 3.b odd 2 1
825.2.c.a 2 15.e even 4 2
891.2.e.e 2 45.h odd 6 2
891.2.e.g 2 45.j even 6 2
1089.2.a.j 1 55.d odd 2 1
1584.2.a.o 1 20.d odd 2 1
1617.2.a.j 1 105.g even 2 1
2112.2.a.j 1 120.m even 2 1
2112.2.a.bb 1 120.i odd 2 1
2475.2.a.g 1 1.a even 1 1 trivial
2475.2.c.d 2 5.c odd 4 2
4851.2.a.b 1 35.c odd 2 1
5577.2.a.a 1 195.e odd 2 1
5808.2.a.t 1 660.g odd 2 1
6336.2.a.n 1 40.e odd 2 1
6336.2.a.x 1 40.f even 2 1
9075.2.a.q 1 33.d even 2 1
9537.2.a.m 1 255.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} + 4$$ T7 + 4 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T - 4$$
$61$ $$T - 6$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T + 4$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 2$$