# Properties

 Label 2450.2.c.h 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 350) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} + 2 q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - i * q^8 + 2 * q^9 $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} + 2 q^{9} + 3 q^{11} - i q^{12} + 2 i q^{13} + q^{16} - 3 i q^{17} + 2 i q^{18} - 7 q^{19} + 3 i q^{22} + q^{24} - 2 q^{26} + 5 i q^{27} + 6 q^{29} + 4 q^{31} + i q^{32} + 3 i q^{33} + 3 q^{34} - 2 q^{36} + 8 i q^{37} - 7 i q^{38} - 2 q^{39} + 9 q^{41} - 8 i q^{43} - 3 q^{44} + 6 i q^{47} + i q^{48} + 3 q^{51} - 2 i q^{52} + 12 i q^{53} - 5 q^{54} - 7 i q^{57} + 6 i q^{58} + 12 q^{59} + 10 q^{61} + 4 i q^{62} - q^{64} - 3 q^{66} - 7 i q^{67} + 3 i q^{68} + 6 q^{71} - 2 i q^{72} + 5 i q^{73} - 8 q^{74} + 7 q^{76} - 2 i q^{78} - 14 q^{79} + q^{81} + 9 i q^{82} - 9 i q^{83} + 8 q^{86} + 6 i q^{87} - 3 i q^{88} - 15 q^{89} + 4 i q^{93} - 6 q^{94} - q^{96} + 10 i q^{97} + 6 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - i * q^8 + 2 * q^9 + 3 * q^11 - i * q^12 + 2*i * q^13 + q^16 - 3*i * q^17 + 2*i * q^18 - 7 * q^19 + 3*i * q^22 + q^24 - 2 * q^26 + 5*i * q^27 + 6 * q^29 + 4 * q^31 + i * q^32 + 3*i * q^33 + 3 * q^34 - 2 * q^36 + 8*i * q^37 - 7*i * q^38 - 2 * q^39 + 9 * q^41 - 8*i * q^43 - 3 * q^44 + 6*i * q^47 + i * q^48 + 3 * q^51 - 2*i * q^52 + 12*i * q^53 - 5 * q^54 - 7*i * q^57 + 6*i * q^58 + 12 * q^59 + 10 * q^61 + 4*i * q^62 - q^64 - 3 * q^66 - 7*i * q^67 + 3*i * q^68 + 6 * q^71 - 2*i * q^72 + 5*i * q^73 - 8 * q^74 + 7 * q^76 - 2*i * q^78 - 14 * q^79 + q^81 + 9*i * q^82 - 9*i * q^83 + 8 * q^86 + 6*i * q^87 - 3*i * q^88 - 15 * q^89 + 4*i * q^93 - 6 * q^94 - q^96 + 10*i * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 6 q^{11} + 2 q^{16} - 14 q^{19} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{36} - 4 q^{39} + 18 q^{41} - 6 q^{44} + 6 q^{51} - 10 q^{54} + 24 q^{59} + 20 q^{61} - 2 q^{64} - 6 q^{66} + 12 q^{71} - 16 q^{74} + 14 q^{76} - 28 q^{79} + 2 q^{81} + 16 q^{86} - 30 q^{89} - 12 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 + 6 * q^11 + 2 * q^16 - 14 * q^19 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 8 * q^31 + 6 * q^34 - 4 * q^36 - 4 * q^39 + 18 * q^41 - 6 * q^44 + 6 * q^51 - 10 * q^54 + 24 * q^59 + 20 * q^61 - 2 * q^64 - 6 * q^66 + 12 * q^71 - 16 * q^74 + 14 * q^76 - 28 * q^79 + 2 * q^81 + 16 * q^86 - 30 * q^89 - 12 * q^94 - 2 * q^96 + 12 * 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 1.00000i −1.00000 0 −1.00000 0 1.00000i 2.00000 0
99.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i 2.00000 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.h 2
5.b even 2 1 inner 2450.2.c.h 2
5.c odd 4 1 2450.2.a.m 1
5.c odd 4 1 2450.2.a.x 1
7.b odd 2 1 350.2.c.c 2
21.c even 2 1 3150.2.g.f 2
28.d even 2 1 2800.2.g.i 2
35.c odd 2 1 350.2.c.c 2
35.f even 4 1 350.2.a.a 1
35.f even 4 1 350.2.a.e yes 1
105.g even 2 1 3150.2.g.f 2
105.k odd 4 1 3150.2.a.m 1
105.k odd 4 1 3150.2.a.x 1
140.c even 2 1 2800.2.g.i 2
140.j odd 4 1 2800.2.a.h 1
140.j odd 4 1 2800.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 35.f even 4 1
350.2.a.e yes 1 35.f even 4 1
350.2.c.c 2 7.b odd 2 1
350.2.c.c 2 35.c odd 2 1
2450.2.a.m 1 5.c odd 4 1
2450.2.a.x 1 5.c odd 4 1
2450.2.c.h 2 1.a even 1 1 trivial
2450.2.c.h 2 5.b even 2 1 inner
2800.2.a.h 1 140.j odd 4 1
2800.2.a.x 1 140.j odd 4 1
2800.2.g.i 2 28.d even 2 1
2800.2.g.i 2 140.c even 2 1
3150.2.a.m 1 105.k odd 4 1
3150.2.a.x 1 105.k odd 4 1
3150.2.g.f 2 21.c even 2 1
3150.2.g.f 2 105.g even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{19} + 7$$ T19 + 7 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 9)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 25$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 100$$