# Properties

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

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## 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 450) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 2q^{7} + q^{8} - 4q^{13} + 2q^{14} + q^{16} - 6q^{17} - 7q^{19} - 4q^{26} + 2q^{28} - 6q^{29} - 10q^{31} + q^{32} - 6q^{34} + 2q^{37} - 7q^{38} + 9q^{41} - q^{43} + 6q^{47} - 3q^{49} - 4q^{52} - 12q^{53} + 2q^{56} - 6q^{58} - 9q^{59} - 4q^{61} - 10q^{62} + q^{64} - 13q^{67} - 6q^{68} + 6q^{71} - q^{73} + 2q^{74} - 7q^{76} + 2q^{79} + 9q^{82} + 9q^{83} - q^{86} + 15q^{89} - 8q^{91} + 6q^{94} + 17q^{97} - 3q^{98} + O(q^{100})$$

## 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 2.00000 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.bg 1
3.b odd 2 1 4050.2.a.o 1
5.b even 2 1 4050.2.a.d 1
5.c odd 4 2 4050.2.c.h 2
9.c even 3 2 450.2.e.c 2
9.d odd 6 2 1350.2.e.h 2
15.d odd 2 1 4050.2.a.u 1
15.e even 4 2 4050.2.c.m 2
45.h odd 6 2 1350.2.e.d 2
45.j even 6 2 450.2.e.f yes 2
45.k odd 12 4 450.2.j.b 4
45.l even 12 4 1350.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 9.c even 3 2
450.2.e.f yes 2 45.j even 6 2
450.2.j.b 4 45.k odd 12 4
1350.2.e.d 2 45.h odd 6 2
1350.2.e.h 2 9.d odd 6 2
1350.2.j.b 4 45.l even 12 4
4050.2.a.d 1 5.b even 2 1
4050.2.a.o 1 3.b odd 2 1
4050.2.a.u 1 15.d odd 2 1
4050.2.a.bg 1 1.a even 1 1 trivial
4050.2.c.h 2 5.c odd 4 2
4050.2.c.m 2 15.e even 4 2

## Hecke kernels

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

 $$T_{7} - 2$$ $$T_{11}$$ $$T_{13} + 4$$ $$T_{17} + 6$$ $$T_{23}$$ $$T_{41} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$6 + T$$
$19$ $$7 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$10 + T$$
$37$ $$-2 + T$$
$41$ $$-9 + T$$
$43$ $$1 + T$$
$47$ $$-6 + T$$
$53$ $$12 + T$$
$59$ $$9 + T$$
$61$ $$4 + T$$
$67$ $$13 + T$$
$71$ $$-6 + T$$
$73$ $$1 + T$$
$79$ $$-2 + T$$
$83$ $$-9 + T$$
$89$ $$-15 + T$$
$97$ $$-17 + T$$
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