# Properties

 Label 5850.2.e.h Level $5850$ Weight $2$ Character orbit 5850.e Analytic conductor $46.712$ 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] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.e (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: $$46.7124851824$$ 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 390) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + 2 i q^{7} + i q^{8} +O(q^{10})$$ q - i * q^2 - q^4 + 2*i * q^7 + i * q^8 $$q - i q^{2} - q^{4} + 2 i q^{7} + i q^{8} - 4 q^{11} - i q^{13} + 2 q^{14} + q^{16} + 4 i q^{17} + 2 q^{19} + 4 i q^{22} - 2 i q^{23} - q^{26} - 2 i q^{28} + 8 q^{29} + 4 q^{31} - i q^{32} + 4 q^{34} - 6 i q^{37} - 2 i q^{38} - 10 q^{41} + 4 i q^{43} + 4 q^{44} - 2 q^{46} + 3 q^{49} + i q^{52} - 6 i q^{53} - 2 q^{56} - 8 i q^{58} - 12 q^{59} - 2 q^{61} - 4 i q^{62} - q^{64} + 8 i q^{67} - 4 i q^{68} - 6 q^{74} - 2 q^{76} - 8 i q^{77} + 8 q^{79} + 10 i q^{82} + 12 i q^{83} + 4 q^{86} - 4 i q^{88} - 10 q^{89} + 2 q^{91} + 2 i q^{92} + 8 i q^{97} - 3 i q^{98} +O(q^{100})$$ q - i * q^2 - q^4 + 2*i * q^7 + i * q^8 - 4 * q^11 - i * q^13 + 2 * q^14 + q^16 + 4*i * q^17 + 2 * q^19 + 4*i * q^22 - 2*i * q^23 - q^26 - 2*i * q^28 + 8 * q^29 + 4 * q^31 - i * q^32 + 4 * q^34 - 6*i * q^37 - 2*i * q^38 - 10 * q^41 + 4*i * q^43 + 4 * q^44 - 2 * q^46 + 3 * q^49 + i * q^52 - 6*i * q^53 - 2 * q^56 - 8*i * q^58 - 12 * q^59 - 2 * q^61 - 4*i * q^62 - q^64 + 8*i * q^67 - 4*i * q^68 - 6 * q^74 - 2 * q^76 - 8*i * q^77 + 8 * q^79 + 10*i * q^82 + 12*i * q^83 + 4 * q^86 - 4*i * q^88 - 10 * q^89 + 2 * q^91 + 2*i * q^92 + 8*i * q^97 - 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{26} + 16 q^{29} + 8 q^{31} + 8 q^{34} - 20 q^{41} + 8 q^{44} - 4 q^{46} + 6 q^{49} - 4 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 12 q^{74} - 4 q^{76} + 16 q^{79} + 8 q^{86} - 20 q^{89} + 4 q^{91}+O(q^{100})$$ 2 * q - 2 * q^4 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^26 + 16 * q^29 + 8 * q^31 + 8 * q^34 - 20 * q^41 + 8 * q^44 - 4 * q^46 + 6 * q^49 - 4 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 12 * q^74 - 4 * q^76 + 16 * q^79 + 8 * q^86 - 20 * q^89 + 4 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$$.

 $$n$$ $$2251$$ $$3251$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5149.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
5149.2 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 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 5850.2.e.h 2
3.b odd 2 1 1950.2.e.m 2
5.b even 2 1 inner 5850.2.e.h 2
5.c odd 4 1 1170.2.a.j 1
5.c odd 4 1 5850.2.a.s 1
15.d odd 2 1 1950.2.e.m 2
15.e even 4 1 390.2.a.b 1
15.e even 4 1 1950.2.a.ba 1
20.e even 4 1 9360.2.a.v 1
60.l odd 4 1 3120.2.a.y 1
195.j odd 4 1 5070.2.b.f 2
195.s even 4 1 5070.2.a.n 1
195.u odd 4 1 5070.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.b 1 15.e even 4 1
1170.2.a.j 1 5.c odd 4 1
1950.2.a.ba 1 15.e even 4 1
1950.2.e.m 2 3.b odd 2 1
1950.2.e.m 2 15.d odd 2 1
3120.2.a.y 1 60.l odd 4 1
5070.2.a.n 1 195.s even 4 1
5070.2.b.f 2 195.j odd 4 1
5070.2.b.f 2 195.u odd 4 1
5850.2.a.s 1 5.c odd 4 1
5850.2.e.h 2 1.a even 1 1 trivial
5850.2.e.h 2 5.b even 2 1 inner
9360.2.a.v 1 20.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 $$S_{2}^{\mathrm{new}}(5850, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 4$$ T11 + 4 $$T_{17}^{2} + 16$$ T17^2 + 16 $$T_{19} - 2$$ T19 - 2 $$T_{29} - 8$$ T29 - 8 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 8)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 64$$