# Properties

 Label 2450.2.a.p Level $2450$ Weight $2$ Character orbit 2450.a Self dual yes Analytic conductor $19.563$ Analytic rank $1$ 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] = [2450,2,Mod(1,2450)]

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([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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.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: $$19.5633484952$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

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

## 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 2.00000 1.00000 0 −2.00000 0 −1.00000 1.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## 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 2450.2.a.p 1
5.b even 2 1 490.2.a.g 1
5.c odd 4 2 2450.2.c.f 2
7.b odd 2 1 2450.2.a.f 1
7.c even 3 2 350.2.e.h 2
15.d odd 2 1 4410.2.a.m 1
20.d odd 2 1 3920.2.a.be 1
35.c odd 2 1 490.2.a.j 1
35.f even 4 2 2450.2.c.p 2
35.i odd 6 2 490.2.e.a 2
35.j even 6 2 70.2.e.b 2
35.l odd 12 4 350.2.j.a 4
105.g even 2 1 4410.2.a.c 1
105.o odd 6 2 630.2.k.e 2
140.c even 2 1 3920.2.a.g 1
140.p odd 6 2 560.2.q.d 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{11} - 3$$ T11 - 3 $$T_{13} + 5$$ T13 + 5 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 1$$ T19 + 1 $$T_{23} + 3$$ T23 + 3 $$T_{37} + 11$$ T37 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T + 5$$
$17$ $$T + 6$$
$19$ $$T + 1$$
$23$ $$T + 3$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 11$$
$41$ $$T - 3$$
$43$ $$T - 10$$
$47$ $$T + 3$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T + 4$$
$67$ $$T - 4$$
$71$ $$T - 12$$
$73$ $$T - 4$$
$79$ $$T + 10$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 14$$