# Properties

 Label 5850.2.e.o Level $5850$ Weight $2$ Character orbit 5850.e Analytic conductor $46.712$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 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

 Self dual: no Analytic conductor: $$46.7124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} -i q^{13} -2 q^{14} + q^{16} -2 q^{19} -6 i q^{23} + q^{26} -2 i q^{28} + 8 q^{31} + i q^{32} + 2 i q^{37} -2 i q^{38} -6 q^{41} + 4 i q^{43} + 6 q^{46} + 3 q^{49} + i q^{52} -6 i q^{53} + 2 q^{56} + 14 q^{61} + 8 i q^{62} - q^{64} -4 i q^{67} + 4 i q^{73} -2 q^{74} + 2 q^{76} + 16 q^{79} -6 i q^{82} -12 i q^{83} -4 q^{86} -6 q^{89} + 2 q^{91} + 6 i q^{92} -4 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{14} + 2q^{16} - 4q^{19} + 2q^{26} + 16q^{31} - 12q^{41} + 12q^{46} + 6q^{49} + 4q^{56} + 28q^{61} - 2q^{64} - 4q^{74} + 4q^{76} + 32q^{79} - 8q^{86} - 12q^{89} + 4q^{91} + O(q^{100})$$

## 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.

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.o 2
3.b odd 2 1 1950.2.e.d 2
5.b even 2 1 inner 5850.2.e.o 2
5.c odd 4 1 1170.2.a.k 1
5.c odd 4 1 5850.2.a.g 1
15.d odd 2 1 1950.2.e.d 2
15.e even 4 1 390.2.a.d 1
15.e even 4 1 1950.2.a.o 1
20.e even 4 1 9360.2.a.g 1
60.l odd 4 1 3120.2.a.j 1
195.j odd 4 1 5070.2.b.m 2
195.s even 4 1 5070.2.a.t 1
195.u odd 4 1 5070.2.b.m 2

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

 $$T_{7}^{2} + 4$$ $$T_{11}$$ $$T_{17}$$ $$T_{19} + 2$$ $$T_{29}$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$16 + T^{2}$$
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