# Properties

 Label 4050.2.c.h Level $4050$ Weight $2$ Character orbit 4050.c Analytic conductor $32.339$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4050 = 2 \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4050.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.3394128186$$ 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 450) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} + 4 i q^{13} -2 q^{14} + q^{16} -6 i q^{17} + 7 q^{19} -4 q^{26} -2 i q^{28} + 6 q^{29} -10 q^{31} + i q^{32} + 6 q^{34} + 2 i q^{37} + 7 i q^{38} + 9 q^{41} + i q^{43} + 6 i q^{47} + 3 q^{49} -4 i q^{52} + 12 i q^{53} + 2 q^{56} + 6 i q^{58} + 9 q^{59} -4 q^{61} -10 i q^{62} - q^{64} -13 i q^{67} + 6 i q^{68} + 6 q^{71} + i q^{73} -2 q^{74} -7 q^{76} -2 q^{79} + 9 i q^{82} -9 i q^{83} - q^{86} -15 q^{89} -8 q^{91} -6 q^{94} + 17 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{14} + 2q^{16} + 14q^{19} - 8q^{26} + 12q^{29} - 20q^{31} + 12q^{34} + 18q^{41} + 6q^{49} + 4q^{56} + 18q^{59} - 8q^{61} - 2q^{64} + 12q^{71} - 4q^{74} - 14q^{76} - 4q^{79} - 2q^{86} - 30q^{89} - 16q^{91} - 12q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$$.

 $$n$$ $$2351$$ $$3727$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
649.2 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 4050.2.c.h 2
3.b odd 2 1 4050.2.c.m 2
5.b even 2 1 inner 4050.2.c.h 2
5.c odd 4 1 4050.2.a.d 1
5.c odd 4 1 4050.2.a.bg 1
9.c even 3 2 450.2.j.b 4
9.d odd 6 2 1350.2.j.b 4
15.d odd 2 1 4050.2.c.m 2
15.e even 4 1 4050.2.a.o 1
15.e even 4 1 4050.2.a.u 1
45.h odd 6 2 1350.2.j.b 4
45.j even 6 2 450.2.j.b 4
45.k odd 12 2 450.2.e.c 2
45.k odd 12 2 450.2.e.f yes 2
45.l even 12 2 1350.2.e.d 2
45.l even 12 2 1350.2.e.h 2

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

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 36$$ $$T_{19} - 7$$ $$T_{29} - 6$$ $$T_{41} - 9$$ $$T_{71} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$( -9 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$169 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$1 + T^{2}$$
$79$ $$( 2 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( 15 + T )^{2}$$
$97$ $$289 + T^{2}$$