# Properties

 Label 2475.2.c.a 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: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 2 q^{4} - 2 i q^{7} +O(q^{10})$$ q + 2*i * q^2 - 2 * q^4 - 2*i * q^7 $$q + 2 i q^{2} - 2 q^{4} - 2 i q^{7} - q^{11} - 4 i q^{13} + 4 q^{14} - 4 q^{16} + 2 i q^{17} - 2 i q^{22} - i q^{23} + 8 q^{26} + 4 i q^{28} + 7 q^{31} - 8 i q^{32} - 4 q^{34} + 3 i q^{37} + 8 q^{41} + 6 i q^{43} + 2 q^{44} + 2 q^{46} - 8 i q^{47} + 3 q^{49} + 8 i q^{52} - 6 i q^{53} + 5 q^{59} + 12 q^{61} + 14 i q^{62} + 8 q^{64} - 7 i q^{67} - 4 i q^{68} + 3 q^{71} - 4 i q^{73} - 6 q^{74} + 2 i q^{77} + 10 q^{79} + 16 i q^{82} - 6 i q^{83} - 12 q^{86} + 15 q^{89} - 8 q^{91} + 2 i q^{92} + 16 q^{94} - 7 i q^{97} + 6 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 2 * q^4 - 2*i * q^7 - q^11 - 4*i * q^13 + 4 * q^14 - 4 * q^16 + 2*i * q^17 - 2*i * q^22 - i * q^23 + 8 * q^26 + 4*i * q^28 + 7 * q^31 - 8*i * q^32 - 4 * q^34 + 3*i * q^37 + 8 * q^41 + 6*i * q^43 + 2 * q^44 + 2 * q^46 - 8*i * q^47 + 3 * q^49 + 8*i * q^52 - 6*i * q^53 + 5 * q^59 + 12 * q^61 + 14*i * q^62 + 8 * q^64 - 7*i * q^67 - 4*i * q^68 + 3 * q^71 - 4*i * q^73 - 6 * q^74 + 2*i * q^77 + 10 * q^79 + 16*i * q^82 - 6*i * q^83 - 12 * q^86 + 15 * q^89 - 8 * q^91 + 2*i * q^92 + 16 * q^94 - 7*i * q^97 + 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} - 2 q^{11} + 8 q^{14} - 8 q^{16} + 16 q^{26} + 14 q^{31} - 8 q^{34} + 16 q^{41} + 4 q^{44} + 4 q^{46} + 6 q^{49} + 10 q^{59} + 24 q^{61} + 16 q^{64} + 6 q^{71} - 12 q^{74} + 20 q^{79} - 24 q^{86} + 30 q^{89} - 16 q^{91} + 32 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 - 2 * q^11 + 8 * q^14 - 8 * q^16 + 16 * q^26 + 14 * q^31 - 8 * q^34 + 16 * q^41 + 4 * q^44 + 4 * q^46 + 6 * q^49 + 10 * q^59 + 24 * q^61 + 16 * q^64 + 6 * q^71 - 12 * q^74 + 20 * q^79 - 24 * q^86 + 30 * q^89 - 16 * q^91 + 32 * 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
2.00000i 0 −2.00000 0 0 2.00000i 0 0 0
199.2 2.00000i 0 −2.00000 0 0 2.00000i 0 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.a 2
3.b odd 2 1 275.2.b.a 2
5.b even 2 1 inner 2475.2.c.a 2
5.c odd 4 1 99.2.a.d 1
5.c odd 4 1 2475.2.a.a 1
12.b even 2 1 4400.2.b.h 2
15.d odd 2 1 275.2.b.a 2
15.e even 4 1 11.2.a.a 1
15.e even 4 1 275.2.a.b 1
20.e even 4 1 1584.2.a.g 1
35.f even 4 1 4851.2.a.t 1
40.i odd 4 1 6336.2.a.br 1
40.k even 4 1 6336.2.a.bu 1
45.k odd 12 2 891.2.e.b 2
45.l even 12 2 891.2.e.k 2
55.e even 4 1 1089.2.a.b 1
60.h even 2 1 4400.2.b.h 2
60.l odd 4 1 176.2.a.b 1
60.l odd 4 1 4400.2.a.i 1
105.k odd 4 1 539.2.a.a 1
105.w odd 12 2 539.2.e.g 2
105.x even 12 2 539.2.e.h 2
120.q odd 4 1 704.2.a.c 1
120.w even 4 1 704.2.a.h 1
165.l odd 4 1 121.2.a.d 1
165.l odd 4 1 3025.2.a.a 1
165.u odd 20 4 121.2.c.a 4
165.v even 20 4 121.2.c.e 4
195.s even 4 1 1859.2.a.b 1
240.z odd 4 1 2816.2.c.f 2
240.bb even 4 1 2816.2.c.j 2
240.bd odd 4 1 2816.2.c.f 2
240.bf even 4 1 2816.2.c.j 2
255.o even 4 1 3179.2.a.a 1
285.j odd 4 1 3971.2.a.b 1
345.l odd 4 1 5819.2.a.a 1
420.w even 4 1 8624.2.a.j 1
435.p even 4 1 9251.2.a.d 1
660.q even 4 1 1936.2.a.i 1
1155.t even 4 1 5929.2.a.h 1
1320.bn odd 4 1 7744.2.a.x 1
1320.bt even 4 1 7744.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 15.e even 4 1
99.2.a.d 1 5.c odd 4 1
121.2.a.d 1 165.l odd 4 1
121.2.c.a 4 165.u odd 20 4
121.2.c.e 4 165.v even 20 4
176.2.a.b 1 60.l odd 4 1
275.2.a.b 1 15.e even 4 1
275.2.b.a 2 3.b odd 2 1
275.2.b.a 2 15.d odd 2 1
539.2.a.a 1 105.k odd 4 1
539.2.e.g 2 105.w odd 12 2
539.2.e.h 2 105.x even 12 2
704.2.a.c 1 120.q odd 4 1
704.2.a.h 1 120.w even 4 1
891.2.e.b 2 45.k odd 12 2
891.2.e.k 2 45.l even 12 2
1089.2.a.b 1 55.e even 4 1
1584.2.a.g 1 20.e even 4 1
1859.2.a.b 1 195.s even 4 1
1936.2.a.i 1 660.q even 4 1
2475.2.a.a 1 5.c odd 4 1
2475.2.c.a 2 1.a even 1 1 trivial
2475.2.c.a 2 5.b even 2 1 inner
2816.2.c.f 2 240.z odd 4 1
2816.2.c.f 2 240.bd odd 4 1
2816.2.c.j 2 240.bb even 4 1
2816.2.c.j 2 240.bf even 4 1
3025.2.a.a 1 165.l odd 4 1
3179.2.a.a 1 255.o even 4 1
3971.2.a.b 1 285.j odd 4 1
4400.2.a.i 1 60.l odd 4 1
4400.2.b.h 2 12.b even 2 1
4400.2.b.h 2 60.h even 2 1
4851.2.a.t 1 35.f even 4 1
5819.2.a.a 1 345.l odd 4 1
5929.2.a.h 1 1155.t even 4 1
6336.2.a.br 1 40.i odd 4 1
6336.2.a.bu 1 40.k even 4 1
7744.2.a.k 1 1320.bt even 4 1
7744.2.a.x 1 1320.bn odd 4 1
8624.2.a.j 1 420.w even 4 1
9251.2.a.d 1 435.p even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$T^{2}$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 5)^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 49$$