Properties

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

Related objects

Newspace parameters

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

Newform invariants

 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 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} + 6q^{11} + q^{13} + q^{16} + 6q^{19} - 6q^{22} - 6q^{23} - q^{26} - 2q^{29} + 4q^{31} - q^{32} + 10q^{37} - 6q^{38} + 6q^{41} + 8q^{43} + 6q^{44} + 6q^{46} - 8q^{47} - 7q^{49} + q^{52} + 6q^{53} + 2q^{58} - 10q^{59} - 6q^{61} - 4q^{62} + q^{64} + 4q^{67} + 8q^{71} + 6q^{73} - 10q^{74} + 6q^{76} + 16q^{79} - 6q^{82} - 4q^{83} - 8q^{86} - 6q^{88} + 10q^{89} - 6q^{92} + 8q^{94} + 2q^{97} + 7q^{98} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 1.00000 0 0 0 −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.o 1
3.b odd 2 1 1950.2.a.r 1
5.b even 2 1 5850.2.a.bs 1
5.c odd 4 2 1170.2.e.d 2
15.d odd 2 1 1950.2.a.i 1
15.e even 4 2 390.2.e.a 2
60.l odd 4 2 3120.2.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.a 2 15.e even 4 2
1170.2.e.d 2 5.c odd 4 2
1950.2.a.i 1 15.d odd 2 1
1950.2.a.r 1 3.b odd 2 1
3120.2.l.a 2 60.l odd 4 2
5850.2.a.o 1 1.a even 1 1 trivial
5850.2.a.bs 1 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5850))$$:

 $$T_{7}$$ $$T_{11} - 6$$ $$T_{17}$$ $$T_{23} + 6$$ $$T_{31} - 4$$

Hecke characteristic polynomials

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