# Properties

 Label 5850.2.e.v Level $5850$ Weight $2$ Character orbit 5850.e Analytic conductor $46.712$ 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] = [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 26) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + i q^{7} + i q^{8} +O(q^{10})$$ q - i * q^2 - q^4 + i * q^7 + i * q^8 $$q - i q^{2} - q^{4} + i q^{7} + i q^{8} + 2 q^{11} + i q^{13} + q^{14} + q^{16} + 3 i q^{17} - 6 q^{19} - 2 i q^{22} - 4 i q^{23} + q^{26} - i q^{28} + 2 q^{29} + 4 q^{31} - i q^{32} + 3 q^{34} + 3 i q^{37} + 6 i q^{38} + 5 i q^{43} - 2 q^{44} - 4 q^{46} - 13 i q^{47} + 6 q^{49} - i q^{52} + 12 i q^{53} - q^{56} - 2 i q^{58} - 10 q^{59} - 8 q^{61} - 4 i q^{62} - q^{64} - 2 i q^{67} - 3 i q^{68} + 5 q^{71} + 10 i q^{73} + 3 q^{74} + 6 q^{76} + 2 i q^{77} + 4 q^{79} + 5 q^{86} + 2 i q^{88} + 6 q^{89} - q^{91} + 4 i q^{92} - 13 q^{94} + 14 i q^{97} - 6 i q^{98} +O(q^{100})$$ q - i * q^2 - q^4 + i * q^7 + i * q^8 + 2 * q^11 + i * q^13 + q^14 + q^16 + 3*i * q^17 - 6 * q^19 - 2*i * q^22 - 4*i * q^23 + q^26 - i * q^28 + 2 * q^29 + 4 * q^31 - i * q^32 + 3 * q^34 + 3*i * q^37 + 6*i * q^38 + 5*i * q^43 - 2 * q^44 - 4 * q^46 - 13*i * q^47 + 6 * q^49 - i * q^52 + 12*i * q^53 - q^56 - 2*i * q^58 - 10 * q^59 - 8 * q^61 - 4*i * q^62 - q^64 - 2*i * q^67 - 3*i * q^68 + 5 * q^71 + 10*i * q^73 + 3 * q^74 + 6 * q^76 + 2*i * q^77 + 4 * q^79 + 5 * q^86 + 2*i * q^88 + 6 * q^89 - q^91 + 4*i * q^92 - 13 * q^94 + 14*i * q^97 - 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{26} + 4 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{44} - 8 q^{46} + 12 q^{49} - 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} + 10 q^{71} + 6 q^{74} + 12 q^{76} + 8 q^{79} + 10 q^{86} + 12 q^{89} - 2 q^{91} - 26 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 2 * q^26 + 4 * q^29 + 8 * q^31 + 6 * q^34 - 4 * q^44 - 8 * q^46 + 12 * q^49 - 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 + 10 * q^71 + 6 * q^74 + 12 * q^76 + 8 * q^79 + 10 * q^86 + 12 * q^89 - 2 * q^91 - 26 * q^94

## 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 1.00000i 1.00000i 0 0
5149.2 1.00000i 0 −1.00000 0 0 1.00000i 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.v 2
3.b odd 2 1 650.2.b.a 2
5.b even 2 1 inner 5850.2.e.v 2
5.c odd 4 1 234.2.a.b 1
5.c odd 4 1 5850.2.a.bn 1
15.d odd 2 1 650.2.b.a 2
15.e even 4 1 26.2.a.b 1
15.e even 4 1 650.2.a.g 1
20.e even 4 1 1872.2.a.m 1
40.i odd 4 1 7488.2.a.w 1
40.k even 4 1 7488.2.a.v 1
45.k odd 12 2 2106.2.e.t 2
45.l even 12 2 2106.2.e.h 2
60.l odd 4 1 208.2.a.d 1
60.l odd 4 1 5200.2.a.c 1
65.f even 4 1 3042.2.b.f 2
65.h odd 4 1 3042.2.a.l 1
65.k even 4 1 3042.2.b.f 2
105.k odd 4 1 1274.2.a.o 1
105.w odd 12 2 1274.2.f.a 2
105.x even 12 2 1274.2.f.l 2
120.q odd 4 1 832.2.a.a 1
120.w even 4 1 832.2.a.j 1
165.l odd 4 1 3146.2.a.a 1
195.j odd 4 1 338.2.b.a 2
195.s even 4 1 338.2.a.a 1
195.s even 4 1 8450.2.a.y 1
195.u odd 4 1 338.2.b.a 2
195.bc odd 12 2 338.2.e.d 4
195.bf even 12 2 338.2.c.g 2
195.bl even 12 2 338.2.c.c 2
195.bn odd 12 2 338.2.e.d 4
240.z odd 4 1 3328.2.b.k 2
240.bb even 4 1 3328.2.b.g 2
240.bd odd 4 1 3328.2.b.k 2
240.bf even 4 1 3328.2.b.g 2
255.o even 4 1 7514.2.a.i 1
285.j odd 4 1 9386.2.a.f 1
780.u even 4 1 2704.2.f.j 2
780.w odd 4 1 2704.2.a.n 1
780.bn even 4 1 2704.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 15.e even 4 1
208.2.a.d 1 60.l odd 4 1
234.2.a.b 1 5.c odd 4 1
338.2.a.a 1 195.s even 4 1
338.2.b.a 2 195.j odd 4 1
338.2.b.a 2 195.u odd 4 1
338.2.c.c 2 195.bl even 12 2
338.2.c.g 2 195.bf even 12 2
338.2.e.d 4 195.bc odd 12 2
338.2.e.d 4 195.bn odd 12 2
650.2.a.g 1 15.e even 4 1
650.2.b.a 2 3.b odd 2 1
650.2.b.a 2 15.d odd 2 1
832.2.a.a 1 120.q odd 4 1
832.2.a.j 1 120.w even 4 1
1274.2.a.o 1 105.k odd 4 1
1274.2.f.a 2 105.w odd 12 2
1274.2.f.l 2 105.x even 12 2
1872.2.a.m 1 20.e even 4 1
2106.2.e.h 2 45.l even 12 2
2106.2.e.t 2 45.k odd 12 2
2704.2.a.n 1 780.w odd 4 1
2704.2.f.j 2 780.u even 4 1
2704.2.f.j 2 780.bn even 4 1
3042.2.a.l 1 65.h odd 4 1
3042.2.b.f 2 65.f even 4 1
3042.2.b.f 2 65.k even 4 1
3146.2.a.a 1 165.l odd 4 1
3328.2.b.g 2 240.bb even 4 1
3328.2.b.g 2 240.bf even 4 1
3328.2.b.k 2 240.z odd 4 1
3328.2.b.k 2 240.bd odd 4 1
5200.2.a.c 1 60.l odd 4 1
5850.2.a.bn 1 5.c odd 4 1
5850.2.e.v 2 1.a even 1 1 trivial
5850.2.e.v 2 5.b even 2 1 inner
7488.2.a.v 1 40.k even 4 1
7488.2.a.w 1 40.i odd 4 1
7514.2.a.i 1 255.o even 4 1
8450.2.a.y 1 195.s even 4 1
9386.2.a.f 1 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5850, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} - 2$$ T11 - 2 $$T_{17}^{2} + 9$$ T17^2 + 9 $$T_{19} + 6$$ T19 + 6 $$T_{29} - 2$$ T29 - 2 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2} + 169$$
$53$ $$T^{2} + 144$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 5)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 196$$