# Properties

 Label 2475.2.c.d Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} +O(q^{10})$$ q + i * q^2 + q^4 - 4*i * q^7 + 3*i * q^8 $$q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} - q^{11} - 2 i q^{13} + 4 q^{14} - q^{16} - 2 i q^{17} - i q^{22} - 8 i q^{23} + 2 q^{26} - 4 i q^{28} - 6 q^{29} - 8 q^{31} + 5 i q^{32} + 2 q^{34} - 6 i q^{37} + 2 q^{41} - q^{44} + 8 q^{46} + 8 i q^{47} - 9 q^{49} - 2 i q^{52} - 6 i q^{53} + 12 q^{56} - 6 i q^{58} - 4 q^{59} + 6 q^{61} - 8 i q^{62} - 7 q^{64} + 4 i q^{67} - 2 i q^{68} - 14 i q^{73} + 6 q^{74} + 4 i q^{77} + 4 q^{79} + 2 i q^{82} - 12 i q^{83} - 3 i q^{88} - 6 q^{89} - 8 q^{91} - 8 i q^{92} - 8 q^{94} - 2 i q^{97} - 9 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 - 4*i * q^7 + 3*i * q^8 - q^11 - 2*i * q^13 + 4 * q^14 - q^16 - 2*i * q^17 - i * q^22 - 8*i * q^23 + 2 * q^26 - 4*i * q^28 - 6 * q^29 - 8 * q^31 + 5*i * q^32 + 2 * q^34 - 6*i * q^37 + 2 * q^41 - q^44 + 8 * q^46 + 8*i * q^47 - 9 * q^49 - 2*i * q^52 - 6*i * q^53 + 12 * q^56 - 6*i * q^58 - 4 * q^59 + 6 * q^61 - 8*i * q^62 - 7 * q^64 + 4*i * q^67 - 2*i * q^68 - 14*i * q^73 + 6 * q^74 + 4*i * q^77 + 4 * q^79 + 2*i * q^82 - 12*i * q^83 - 3*i * q^88 - 6 * q^89 - 8 * q^91 - 8*i * q^92 - 8 * q^94 - 2*i * q^97 - 9*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 2 q^{11} + 8 q^{14} - 2 q^{16} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 4 q^{41} - 2 q^{44} + 16 q^{46} - 18 q^{49} + 24 q^{56} - 8 q^{59} + 12 q^{61} - 14 q^{64} + 12 q^{74} + 8 q^{79} - 12 q^{89} - 16 q^{91} - 16 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^11 + 8 * q^14 - 2 * q^16 + 4 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 4 * q^41 - 2 * q^44 + 16 * q^46 - 18 * q^49 + 24 * q^56 - 8 * q^59 + 12 * q^61 - 14 * q^64 + 12 * q^74 + 8 * q^79 - 12 * q^89 - 16 * q^91 - 16 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 4.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 4.00000i 3.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.c.d 2
3.b odd 2 1 825.2.c.a 2
5.b even 2 1 inner 2475.2.c.d 2
5.c odd 4 1 99.2.a.b 1
5.c odd 4 1 2475.2.a.g 1
15.d odd 2 1 825.2.c.a 2
15.e even 4 1 33.2.a.a 1
15.e even 4 1 825.2.a.a 1
20.e even 4 1 1584.2.a.o 1
35.f even 4 1 4851.2.a.b 1
40.i odd 4 1 6336.2.a.x 1
40.k even 4 1 6336.2.a.n 1
45.k odd 12 2 891.2.e.g 2
45.l even 12 2 891.2.e.e 2
55.e even 4 1 1089.2.a.j 1
60.l odd 4 1 528.2.a.g 1
105.k odd 4 1 1617.2.a.j 1
120.q odd 4 1 2112.2.a.j 1
120.w even 4 1 2112.2.a.bb 1
165.l odd 4 1 363.2.a.b 1
165.l odd 4 1 9075.2.a.q 1
165.u odd 20 4 363.2.e.g 4
165.v even 20 4 363.2.e.e 4
195.s even 4 1 5577.2.a.a 1
255.o even 4 1 9537.2.a.m 1
660.q even 4 1 5808.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 15.e even 4 1
99.2.a.b 1 5.c odd 4 1
363.2.a.b 1 165.l odd 4 1
363.2.e.e 4 165.v even 20 4
363.2.e.g 4 165.u odd 20 4
528.2.a.g 1 60.l odd 4 1
825.2.a.a 1 15.e even 4 1
825.2.c.a 2 3.b odd 2 1
825.2.c.a 2 15.d odd 2 1
891.2.e.e 2 45.l even 12 2
891.2.e.g 2 45.k odd 12 2
1089.2.a.j 1 55.e even 4 1
1584.2.a.o 1 20.e even 4 1
1617.2.a.j 1 105.k odd 4 1
2112.2.a.j 1 120.q odd 4 1
2112.2.a.bb 1 120.w even 4 1
2475.2.a.g 1 5.c odd 4 1
2475.2.c.d 2 1.a even 1 1 trivial
2475.2.c.d 2 5.b even 2 1 inner
4851.2.a.b 1 35.f even 4 1
5577.2.a.a 1 195.s even 4 1
5808.2.a.t 1 660.q even 4 1
6336.2.a.n 1 40.k even 4 1
6336.2.a.x 1 40.i odd 4 1
9075.2.a.q 1 165.l odd 4 1
9537.2.a.m 1 255.o even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4$$