# Properties

 Label 4050.2.c.n 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 162) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} - q^{4} + 4 i q^{7} + i q^{8} +O(q^{10})$$ $$q -i q^{2} - q^{4} + 4 i q^{7} + i q^{8} -i q^{13} + 4 q^{14} + q^{16} -3 i q^{17} + 4 q^{19} - q^{26} -4 i q^{28} + 9 q^{29} -4 q^{31} -i q^{32} -3 q^{34} + i q^{37} -4 i q^{38} -6 q^{41} + 8 i q^{43} -12 i q^{47} -9 q^{49} + i q^{52} + 6 i q^{53} -4 q^{56} -9 i q^{58} - q^{61} + 4 i q^{62} - q^{64} + 4 i q^{67} + 3 i q^{68} + 12 q^{71} + 11 i q^{73} + q^{74} -4 q^{76} + 16 q^{79} + 6 i q^{82} + 12 i q^{83} + 8 q^{86} -3 q^{89} + 4 q^{91} -12 q^{94} -2 i q^{97} + 9 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 8q^{14} + 2q^{16} + 8q^{19} - 2q^{26} + 18q^{29} - 8q^{31} - 6q^{34} - 12q^{41} - 18q^{49} - 8q^{56} - 2q^{61} - 2q^{64} + 24q^{71} + 2q^{74} - 8q^{76} + 32q^{79} + 16q^{86} - 6q^{89} + 8q^{91} - 24q^{94} + O(q^{100})$$

## 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 4.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 4.00000i 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.n 2
3.b odd 2 1 4050.2.c.g 2
5.b even 2 1 inner 4050.2.c.n 2
5.c odd 4 1 162.2.a.d yes 1
5.c odd 4 1 4050.2.a.r 1
15.d odd 2 1 4050.2.c.g 2
15.e even 4 1 162.2.a.a 1
15.e even 4 1 4050.2.a.bh 1
20.e even 4 1 1296.2.a.l 1
35.f even 4 1 7938.2.a.s 1
40.i odd 4 1 5184.2.a.c 1
40.k even 4 1 5184.2.a.h 1
45.k odd 12 2 162.2.c.a 2
45.l even 12 2 162.2.c.d 2
60.l odd 4 1 1296.2.a.c 1
105.k odd 4 1 7938.2.a.n 1
120.q odd 4 1 5184.2.a.bd 1
120.w even 4 1 5184.2.a.y 1
180.v odd 12 2 1296.2.i.n 2
180.x even 12 2 1296.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 15.e even 4 1
162.2.a.d yes 1 5.c odd 4 1
162.2.c.a 2 45.k odd 12 2
162.2.c.d 2 45.l even 12 2
1296.2.a.c 1 60.l odd 4 1
1296.2.a.l 1 20.e even 4 1
1296.2.i.b 2 180.x even 12 2
1296.2.i.n 2 180.v odd 12 2
4050.2.a.r 1 5.c odd 4 1
4050.2.a.bh 1 15.e even 4 1
4050.2.c.g 2 3.b odd 2 1
4050.2.c.g 2 15.d odd 2 1
4050.2.c.n 2 1.a even 1 1 trivial
4050.2.c.n 2 5.b even 2 1 inner
5184.2.a.c 1 40.i odd 4 1
5184.2.a.h 1 40.k even 4 1
5184.2.a.y 1 120.w even 4 1
5184.2.a.bd 1 120.q odd 4 1
7938.2.a.n 1 105.k odd 4 1
7938.2.a.s 1 35.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11}$$ $$T_{13}^{2} + 1$$ $$T_{17}^{2} + 9$$ $$T_{19} - 4$$ $$T_{29} - 9$$ $$T_{41} + 6$$ $$T_{71} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ 1
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 73 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 + 50 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + T + 61 T^{2} )^{2}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 25 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 16 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 3 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$