# Properties

 Label 2450.2.c.f Level $2450$ Weight $2$ Character orbit 2450.c Analytic conductor $19.563$ 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] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.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: $$19.5633484952$$ 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 70) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 - i * q^8 - q^9 $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} - i q^{8} - q^{9} + 3 q^{11} - 2 i q^{12} - 5 i q^{13} + q^{16} + 6 i q^{17} - i q^{18} + q^{19} + 3 i q^{22} - 3 i q^{23} + 2 q^{24} + 5 q^{26} + 4 i q^{27} + 6 q^{29} - 4 q^{31} + i q^{32} + 6 i q^{33} - 6 q^{34} + q^{36} + 11 i q^{37} + i q^{38} + 10 q^{39} + 3 q^{41} + 10 i q^{43} - 3 q^{44} + 3 q^{46} + 3 i q^{47} + 2 i q^{48} - 12 q^{51} + 5 i q^{52} - 3 i q^{53} - 4 q^{54} + 2 i q^{57} + 6 i q^{58} - 4 q^{61} - 4 i q^{62} - q^{64} - 6 q^{66} - 4 i q^{67} - 6 i q^{68} + 6 q^{69} + 12 q^{71} + i q^{72} + 4 i q^{73} - 11 q^{74} - q^{76} + 10 i q^{78} + 10 q^{79} - 11 q^{81} + 3 i q^{82} + 12 i q^{83} - 10 q^{86} + 12 i q^{87} - 3 i q^{88} - 6 q^{89} + 3 i q^{92} - 8 i q^{93} - 3 q^{94} - 2 q^{96} + 14 i q^{97} - 3 q^{99} +O(q^{100})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 - i * q^8 - q^9 + 3 * q^11 - 2*i * q^12 - 5*i * q^13 + q^16 + 6*i * q^17 - i * q^18 + q^19 + 3*i * q^22 - 3*i * q^23 + 2 * q^24 + 5 * q^26 + 4*i * q^27 + 6 * q^29 - 4 * q^31 + i * q^32 + 6*i * q^33 - 6 * q^34 + q^36 + 11*i * q^37 + i * q^38 + 10 * q^39 + 3 * q^41 + 10*i * q^43 - 3 * q^44 + 3 * q^46 + 3*i * q^47 + 2*i * q^48 - 12 * q^51 + 5*i * q^52 - 3*i * q^53 - 4 * q^54 + 2*i * q^57 + 6*i * q^58 - 4 * q^61 - 4*i * q^62 - q^64 - 6 * q^66 - 4*i * q^67 - 6*i * q^68 + 6 * q^69 + 12 * q^71 + i * q^72 + 4*i * q^73 - 11 * q^74 - q^76 + 10*i * q^78 + 10 * q^79 - 11 * q^81 + 3*i * q^82 + 12*i * q^83 - 10 * q^86 + 12*i * q^87 - 3*i * q^88 - 6 * q^89 + 3*i * q^92 - 8*i * q^93 - 3 * q^94 - 2 * q^96 + 14*i * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{16} + 2 q^{19} + 4 q^{24} + 10 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} + 2 q^{36} + 20 q^{39} + 6 q^{41} - 6 q^{44} + 6 q^{46} - 24 q^{51} - 8 q^{54} - 8 q^{61} - 2 q^{64} - 12 q^{66} + 12 q^{69} + 24 q^{71} - 22 q^{74} - 2 q^{76} + 20 q^{79} - 22 q^{81} - 20 q^{86} - 12 q^{89} - 6 q^{94} - 4 q^{96} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^16 + 2 * q^19 + 4 * q^24 + 10 * q^26 + 12 * q^29 - 8 * q^31 - 12 * q^34 + 2 * q^36 + 20 * q^39 + 6 * q^41 - 6 * q^44 + 6 * q^46 - 24 * q^51 - 8 * q^54 - 8 * q^61 - 2 * q^64 - 12 * q^66 + 12 * q^69 + 24 * q^71 - 22 * q^74 - 2 * q^76 + 20 * q^79 - 22 * q^81 - 20 * q^86 - 12 * q^89 - 6 * q^94 - 4 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 0 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 −2.00000 0 1.00000i −1.00000 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 2450.2.c.f 2
5.b even 2 1 inner 2450.2.c.f 2
5.c odd 4 1 490.2.a.g 1
5.c odd 4 1 2450.2.a.p 1
7.b odd 2 1 2450.2.c.p 2
7.c even 3 2 350.2.j.a 4
15.e even 4 1 4410.2.a.m 1
20.e even 4 1 3920.2.a.be 1
35.c odd 2 1 2450.2.c.p 2
35.f even 4 1 490.2.a.j 1
35.f even 4 1 2450.2.a.f 1
35.j even 6 2 350.2.j.a 4
35.k even 12 2 490.2.e.a 2
35.l odd 12 2 70.2.e.b 2
35.l odd 12 2 350.2.e.h 2
105.k odd 4 1 4410.2.a.c 1
105.x even 12 2 630.2.k.e 2
140.j odd 4 1 3920.2.a.g 1
140.w even 12 2 560.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 35.l odd 12 2
350.2.e.h 2 35.l odd 12 2
350.2.j.a 4 7.c even 3 2
350.2.j.a 4 35.j even 6 2
490.2.a.g 1 5.c odd 4 1
490.2.a.j 1 35.f even 4 1
490.2.e.a 2 35.k even 12 2
560.2.q.d 2 140.w even 12 2
630.2.k.e 2 105.x even 12 2
2450.2.a.f 1 35.f even 4 1
2450.2.a.p 1 5.c odd 4 1
2450.2.c.f 2 1.a even 1 1 trivial
2450.2.c.f 2 5.b even 2 1 inner
2450.2.c.p 2 7.b odd 2 1
2450.2.c.p 2 35.c odd 2 1
3920.2.a.g 1 140.j odd 4 1
3920.2.a.be 1 20.e even 4 1
4410.2.a.c 1 105.k odd 4 1
4410.2.a.m 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2450, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} + 25$$ T13^2 + 25 $$T_{19} - 1$$ T19 - 1 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 121$$
$41$ $$(T - 3)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 9$$
$59$ $$T^{2}$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 196$$