# Properties

 Label 4050.2.a.bh Level 4050 Weight 2 Character orbit 4050.a Self dual yes Analytic conductor 32.339 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4050 = 2 \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.3394128186$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 162) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 4q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 4q^{7} + q^{8} + q^{13} + 4q^{14} + q^{16} + 3q^{17} - 4q^{19} + q^{26} + 4q^{28} + 9q^{29} - 4q^{31} + q^{32} + 3q^{34} + q^{37} - 4q^{38} + 6q^{41} - 8q^{43} + 12q^{47} + 9q^{49} + q^{52} + 6q^{53} + 4q^{56} + 9q^{58} - q^{61} - 4q^{62} + q^{64} + 4q^{67} + 3q^{68} - 12q^{71} - 11q^{73} + q^{74} - 4q^{76} - 16q^{79} + 6q^{82} + 12q^{83} - 8q^{86} - 3q^{89} + 4q^{91} + 12q^{94} - 2q^{97} + 9q^{98} + O(q^{100})$$

## 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 4.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 4050.2.a.bh 1
3.b odd 2 1 4050.2.a.r 1
5.b even 2 1 162.2.a.a 1
5.c odd 4 2 4050.2.c.g 2
15.d odd 2 1 162.2.a.d yes 1
15.e even 4 2 4050.2.c.n 2
20.d odd 2 1 1296.2.a.c 1
35.c odd 2 1 7938.2.a.n 1
40.e odd 2 1 5184.2.a.bd 1
40.f even 2 1 5184.2.a.y 1
45.h odd 6 2 162.2.c.a 2
45.j even 6 2 162.2.c.d 2
60.h even 2 1 1296.2.a.l 1
105.g even 2 1 7938.2.a.s 1
120.i odd 2 1 5184.2.a.c 1
120.m even 2 1 5184.2.a.h 1
180.n even 6 2 1296.2.i.b 2
180.p odd 6 2 1296.2.i.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 5.b even 2 1
162.2.a.d yes 1 15.d odd 2 1
162.2.c.a 2 45.h odd 6 2
162.2.c.d 2 45.j even 6 2
1296.2.a.c 1 20.d odd 2 1
1296.2.a.l 1 60.h even 2 1
1296.2.i.b 2 180.n even 6 2
1296.2.i.n 2 180.p odd 6 2
4050.2.a.r 1 3.b odd 2 1
4050.2.a.bh 1 1.a even 1 1 trivial
4050.2.c.g 2 5.c odd 4 2
4050.2.c.n 2 15.e even 4 2
5184.2.a.c 1 120.i odd 2 1
5184.2.a.h 1 120.m even 2 1
5184.2.a.y 1 40.f even 2 1
5184.2.a.bd 1 40.e odd 2 1
7938.2.a.n 1 35.c odd 2 1
7938.2.a.s 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4050))$$:

 $$T_{7} - 4$$ $$T_{11}$$ $$T_{13} - 1$$ $$T_{17} - 3$$ $$T_{23}$$ $$T_{41} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ 1
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - T + 13 T^{2}$$
$17$ $$1 - 3 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 - 9 T + 29 T^{2}$$
$31$ $$1 + 4 T + 31 T^{2}$$
$37$ $$1 - T + 37 T^{2}$$
$41$ $$1 - 6 T + 41 T^{2}$$
$43$ $$1 + 8 T + 43 T^{2}$$
$47$ $$1 - 12 T + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 + T + 61 T^{2}$$
$67$ $$1 - 4 T + 67 T^{2}$$
$71$ $$1 + 12 T + 71 T^{2}$$
$73$ $$1 + 11 T + 73 T^{2}$$
$79$ $$1 + 16 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 + 3 T + 89 T^{2}$$
$97$ $$1 + 2 T + 97 T^{2}$$