# Properties

 Label 5850.2.a.bn Level $5850$ Weight $2$ Character orbit 5850.a Self dual yes Analytic conductor $46.712$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(1,5850)]

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, 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("5850.1");

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.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: $$46.7124851824$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

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

## 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 0 1.00000 0 0 −1.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$13$$ $$-1$$

## 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 5850.2.a.bn 1
3.b odd 2 1 650.2.a.g 1
5.b even 2 1 234.2.a.b 1
5.c odd 4 2 5850.2.e.v 2
12.b even 2 1 5200.2.a.c 1
15.d odd 2 1 26.2.a.b 1
15.e even 4 2 650.2.b.a 2
20.d odd 2 1 1872.2.a.m 1
39.d odd 2 1 8450.2.a.y 1
40.e odd 2 1 7488.2.a.v 1
40.f even 2 1 7488.2.a.w 1
45.h odd 6 2 2106.2.e.h 2
45.j even 6 2 2106.2.e.t 2
60.h even 2 1 208.2.a.d 1
65.d even 2 1 3042.2.a.l 1
65.g odd 4 2 3042.2.b.f 2
105.g even 2 1 1274.2.a.o 1
105.o odd 6 2 1274.2.f.l 2
105.p even 6 2 1274.2.f.a 2
120.i odd 2 1 832.2.a.j 1
120.m even 2 1 832.2.a.a 1
165.d even 2 1 3146.2.a.a 1
195.e odd 2 1 338.2.a.a 1
195.n even 4 2 338.2.b.a 2
195.x odd 6 2 338.2.c.c 2
195.y odd 6 2 338.2.c.g 2
195.bh even 12 4 338.2.e.d 4
240.t even 4 2 3328.2.b.k 2
240.bm odd 4 2 3328.2.b.g 2
255.h odd 2 1 7514.2.a.i 1
285.b even 2 1 9386.2.a.f 1
780.d even 2 1 2704.2.a.n 1
780.bb odd 4 2 2704.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 15.d odd 2 1
208.2.a.d 1 60.h even 2 1
234.2.a.b 1 5.b even 2 1
338.2.a.a 1 195.e odd 2 1
338.2.b.a 2 195.n even 4 2
338.2.c.c 2 195.x odd 6 2
338.2.c.g 2 195.y odd 6 2
338.2.e.d 4 195.bh even 12 4
650.2.a.g 1 3.b odd 2 1
650.2.b.a 2 15.e even 4 2
832.2.a.a 1 120.m even 2 1
832.2.a.j 1 120.i odd 2 1
1274.2.a.o 1 105.g even 2 1
1274.2.f.a 2 105.p even 6 2
1274.2.f.l 2 105.o odd 6 2
1872.2.a.m 1 20.d odd 2 1
2106.2.e.h 2 45.h odd 6 2
2106.2.e.t 2 45.j even 6 2
2704.2.a.n 1 780.d even 2 1
2704.2.f.j 2 780.bb odd 4 2
3042.2.a.l 1 65.d even 2 1
3042.2.b.f 2 65.g odd 4 2
3146.2.a.a 1 165.d even 2 1
3328.2.b.g 2 240.bm odd 4 2
3328.2.b.k 2 240.t even 4 2
5200.2.a.c 1 12.b even 2 1
5850.2.a.bn 1 1.a even 1 1 trivial
5850.2.e.v 2 5.c odd 4 2
7488.2.a.v 1 40.e odd 2 1
7488.2.a.w 1 40.f even 2 1
7514.2.a.i 1 255.h odd 2 1
8450.2.a.y 1 39.d odd 2 1
9386.2.a.f 1 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5850))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 2$$ T11 - 2 $$T_{17} + 3$$ T17 + 3 $$T_{23} + 4$$ T23 + 4 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 2$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 6$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T + 3$$
$41$ $$T$$
$43$ $$T - 5$$
$47$ $$T - 13$$
$53$ $$T - 12$$
$59$ $$T - 10$$
$61$ $$T + 8$$
$67$ $$T - 2$$
$71$ $$T - 5$$
$73$ $$T - 10$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 14$$
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