# Properties

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

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 2q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - 2q^{7} + q^{8} + 3q^{11} - 2q^{13} - 2q^{14} + q^{16} - 3q^{17} - q^{19} + 3q^{22} - 6q^{23} - 2q^{26} - 2q^{28} - 6q^{29} - 4q^{31} + q^{32} - 3q^{34} + 4q^{37} - q^{38} - 9q^{41} + q^{43} + 3q^{44} - 6q^{46} - 6q^{47} - 3q^{49} - 2q^{52} + 12q^{53} - 2q^{56} - 6q^{58} - 3q^{59} + 8q^{61} - 4q^{62} + q^{64} - 5q^{67} - 3q^{68} + 12q^{71} - 11q^{73} + 4q^{74} - q^{76} - 6q^{77} - 4q^{79} - 9q^{82} + 12q^{83} + q^{86} + 3q^{88} - 6q^{89} + 4q^{91} - 6q^{92} - 6q^{94} - 5q^{97} - 3q^{98} + O(q^{100})$$

## 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.v 1
3.b odd 2 1 4050.2.a.c 1
5.b even 2 1 162.2.a.b 1
5.c odd 4 2 4050.2.c.r 2
9.c even 3 2 1350.2.e.c 2
9.d odd 6 2 450.2.e.i 2
15.d odd 2 1 162.2.a.c 1
15.e even 4 2 4050.2.c.c 2
20.d odd 2 1 1296.2.a.f 1
35.c odd 2 1 7938.2.a.i 1
40.e odd 2 1 5184.2.a.p 1
40.f even 2 1 5184.2.a.q 1
45.h odd 6 2 18.2.c.a 2
45.j even 6 2 54.2.c.a 2
45.k odd 12 4 1350.2.j.a 4
45.l even 12 4 450.2.j.e 4
60.h even 2 1 1296.2.a.g 1
105.g even 2 1 7938.2.a.x 1
120.i odd 2 1 5184.2.a.r 1
120.m even 2 1 5184.2.a.o 1
180.n even 6 2 144.2.i.c 2
180.p odd 6 2 432.2.i.b 2
315.q odd 6 2 2646.2.e.c 2
315.r even 6 2 2646.2.e.b 2
315.u even 6 2 882.2.h.b 2
315.v odd 6 2 882.2.h.c 2
315.z even 6 2 882.2.f.d 2
315.bg odd 6 2 2646.2.f.g 2
315.bn odd 6 2 2646.2.h.i 2
315.bo even 6 2 2646.2.h.h 2
315.bq even 6 2 882.2.e.g 2
315.br odd 6 2 882.2.e.i 2
360.z odd 6 2 1728.2.i.f 2
360.bd even 6 2 576.2.i.a 2
360.bh odd 6 2 576.2.i.g 2
360.bk even 6 2 1728.2.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 45.h odd 6 2
54.2.c.a 2 45.j even 6 2
144.2.i.c 2 180.n even 6 2
162.2.a.b 1 5.b even 2 1
162.2.a.c 1 15.d odd 2 1
432.2.i.b 2 180.p odd 6 2
450.2.e.i 2 9.d odd 6 2
450.2.j.e 4 45.l even 12 4
576.2.i.a 2 360.bd even 6 2
576.2.i.g 2 360.bh odd 6 2
882.2.e.g 2 315.bq even 6 2
882.2.e.i 2 315.br odd 6 2
882.2.f.d 2 315.z even 6 2
882.2.h.b 2 315.u even 6 2
882.2.h.c 2 315.v odd 6 2
1296.2.a.f 1 20.d odd 2 1
1296.2.a.g 1 60.h even 2 1
1350.2.e.c 2 9.c even 3 2
1350.2.j.a 4 45.k odd 12 4
1728.2.i.e 2 360.bk even 6 2
1728.2.i.f 2 360.z odd 6 2
2646.2.e.b 2 315.r even 6 2
2646.2.e.c 2 315.q odd 6 2
2646.2.f.g 2 315.bg odd 6 2
2646.2.h.h 2 315.bo even 6 2
2646.2.h.i 2 315.bn odd 6 2
4050.2.a.c 1 3.b odd 2 1
4050.2.a.v 1 1.a even 1 1 trivial
4050.2.c.c 2 15.e even 4 2
4050.2.c.r 2 5.c odd 4 2
5184.2.a.o 1 120.m even 2 1
5184.2.a.p 1 40.e odd 2 1
5184.2.a.q 1 40.f even 2 1
5184.2.a.r 1 120.i odd 2 1
7938.2.a.i 1 35.c odd 2 1
7938.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4050))$$:

 $$T_{7} + 2$$ $$T_{11} - 3$$ $$T_{13} + 2$$ $$T_{17} + 3$$ $$T_{23} + 6$$ $$T_{41} + 9$$

## Hecke characteristic polynomials

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