# Properties

 Label 2475.2.c.f Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no 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 55) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + 3 i q^{8}+O(q^{10})$$ q + i * q^2 + q^4 + 3*i * q^8 $$q + i q^{2} + q^{4} + 3 i q^{8} + q^{11} + 2 i q^{13} - q^{16} + 6 i q^{17} + 4 q^{19} + i q^{22} - 4 i q^{23} - 2 q^{26} + 6 q^{29} - 8 q^{31} + 5 i q^{32} - 6 q^{34} + 2 i q^{37} + 4 i q^{38} - 2 q^{41} + 4 i q^{43} + q^{44} + 4 q^{46} - 12 i q^{47} + 7 q^{49} + 2 i q^{52} + 2 i q^{53} + 6 i q^{58} + 4 q^{59} - 10 q^{61} - 8 i q^{62} - 7 q^{64} + 16 i q^{67} + 6 i q^{68} - 8 q^{71} + 14 i q^{73} - 2 q^{74} + 4 q^{76} - 8 q^{79} - 2 i q^{82} + 4 i q^{83} - 4 q^{86} + 3 i q^{88} + 10 q^{89} - 4 i q^{92} + 12 q^{94} - 10 i q^{97} + 7 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 + 3*i * q^8 + q^11 + 2*i * q^13 - q^16 + 6*i * q^17 + 4 * q^19 + i * q^22 - 4*i * q^23 - 2 * q^26 + 6 * q^29 - 8 * q^31 + 5*i * q^32 - 6 * q^34 + 2*i * q^37 + 4*i * q^38 - 2 * q^41 + 4*i * q^43 + q^44 + 4 * q^46 - 12*i * q^47 + 7 * q^49 + 2*i * q^52 + 2*i * q^53 + 6*i * q^58 + 4 * q^59 - 10 * q^61 - 8*i * q^62 - 7 * q^64 + 16*i * q^67 + 6*i * q^68 - 8 * q^71 + 14*i * q^73 - 2 * q^74 + 4 * q^76 - 8 * q^79 - 2*i * q^82 + 4*i * q^83 - 4 * q^86 + 3*i * q^88 + 10 * q^89 - 4*i * q^92 + 12 * q^94 - 10*i * q^97 + 7*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 2 q^{11} - 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 16 q^{31} - 12 q^{34} - 4 q^{41} + 2 q^{44} + 8 q^{46} + 14 q^{49} + 8 q^{59} - 20 q^{61} - 14 q^{64} - 16 q^{71} - 4 q^{74} + 8 q^{76} - 16 q^{79} - 8 q^{86} + 20 q^{89} + 24 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^11 - 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 16 * q^31 - 12 * q^34 - 4 * q^41 + 2 * q^44 + 8 * q^46 + 14 * q^49 + 8 * q^59 - 20 * q^61 - 14 * q^64 - 16 * q^71 - 4 * q^74 + 8 * q^76 - 16 * q^79 - 8 * q^86 + 20 * q^89 + 24 * 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 0 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 0 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.f 2
3.b odd 2 1 275.2.b.b 2
5.b even 2 1 inner 2475.2.c.f 2
5.c odd 4 1 495.2.a.a 1
5.c odd 4 1 2475.2.a.i 1
12.b even 2 1 4400.2.b.n 2
15.d odd 2 1 275.2.b.b 2
15.e even 4 1 55.2.a.a 1
15.e even 4 1 275.2.a.a 1
20.e even 4 1 7920.2.a.i 1
55.e even 4 1 5445.2.a.i 1
60.h even 2 1 4400.2.b.n 2
60.l odd 4 1 880.2.a.h 1
60.l odd 4 1 4400.2.a.p 1
105.k odd 4 1 2695.2.a.c 1
120.q odd 4 1 3520.2.a.n 1
120.w even 4 1 3520.2.a.p 1
165.l odd 4 1 605.2.a.b 1
165.l odd 4 1 3025.2.a.f 1
165.u odd 20 4 605.2.g.c 4
165.v even 20 4 605.2.g.a 4
195.s even 4 1 9295.2.a.b 1
660.q even 4 1 9680.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 15.e even 4 1
275.2.a.a 1 15.e even 4 1
275.2.b.b 2 3.b odd 2 1
275.2.b.b 2 15.d odd 2 1
495.2.a.a 1 5.c odd 4 1
605.2.a.b 1 165.l odd 4 1
605.2.g.a 4 165.v even 20 4
605.2.g.c 4 165.u odd 20 4
880.2.a.h 1 60.l odd 4 1
2475.2.a.i 1 5.c odd 4 1
2475.2.c.f 2 1.a even 1 1 trivial
2475.2.c.f 2 5.b even 2 1 inner
2695.2.a.c 1 105.k odd 4 1
3025.2.a.f 1 165.l odd 4 1
3520.2.a.n 1 120.q odd 4 1
3520.2.a.p 1 120.w even 4 1
4400.2.a.p 1 60.l odd 4 1
4400.2.b.n 2 12.b even 2 1
4400.2.b.n 2 60.h even 2 1
5445.2.a.i 1 55.e even 4 1
7920.2.a.i 1 20.e even 4 1
9295.2.a.b 1 195.s even 4 1
9680.2.a.r 1 660.q 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 $$T_{7}$$ T7 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 100$$