# Properties

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

# Learn more

## 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 18) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} -3 q^{11} -2 i q^{13} -2 q^{14} + q^{16} -3 i q^{17} + q^{19} -3 i q^{22} + 6 i q^{23} + 2 q^{26} -2 i q^{28} -6 q^{29} -4 q^{31} + i q^{32} + 3 q^{34} -4 i q^{37} + i q^{38} + 9 q^{41} + i q^{43} + 3 q^{44} -6 q^{46} -6 i q^{47} + 3 q^{49} + 2 i q^{52} -12 i q^{53} + 2 q^{56} -6 i q^{58} -3 q^{59} + 8 q^{61} -4 i q^{62} - q^{64} + 5 i q^{67} + 3 i q^{68} -12 q^{71} -11 i q^{73} + 4 q^{74} - q^{76} -6 i q^{77} + 4 q^{79} + 9 i q^{82} -12 i q^{83} - q^{86} + 3 i q^{88} -6 q^{89} + 4 q^{91} -6 i q^{92} + 6 q^{94} + 5 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 6q^{11} - 4q^{14} + 2q^{16} + 2q^{19} + 4q^{26} - 12q^{29} - 8q^{31} + 6q^{34} + 18q^{41} + 6q^{44} - 12q^{46} + 6q^{49} + 4q^{56} - 6q^{59} + 16q^{61} - 2q^{64} - 24q^{71} + 8q^{74} - 2q^{76} + 8q^{79} - 2q^{86} - 12q^{89} + 8q^{91} + 12q^{94} + O(q^{100})$$

## 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.c 2
3.b odd 2 1 4050.2.c.r 2
5.b even 2 1 inner 4050.2.c.c 2
5.c odd 4 1 162.2.a.c 1
5.c odd 4 1 4050.2.a.c 1
9.c even 3 2 450.2.j.e 4
9.d odd 6 2 1350.2.j.a 4
15.d odd 2 1 4050.2.c.r 2
15.e even 4 1 162.2.a.b 1
15.e even 4 1 4050.2.a.v 1
20.e even 4 1 1296.2.a.g 1
35.f even 4 1 7938.2.a.x 1
40.i odd 4 1 5184.2.a.r 1
40.k even 4 1 5184.2.a.o 1
45.h odd 6 2 1350.2.j.a 4
45.j even 6 2 450.2.j.e 4
45.k odd 12 2 18.2.c.a 2
45.k odd 12 2 450.2.e.i 2
45.l even 12 2 54.2.c.a 2
45.l even 12 2 1350.2.e.c 2
60.l odd 4 1 1296.2.a.f 1
105.k odd 4 1 7938.2.a.i 1
120.q odd 4 1 5184.2.a.p 1
120.w even 4 1 5184.2.a.q 1
180.v odd 12 2 432.2.i.b 2
180.x even 12 2 144.2.i.c 2
315.bs even 12 2 882.2.e.g 2
315.bt odd 12 2 882.2.e.i 2
315.bu odd 12 2 2646.2.e.c 2
315.bv even 12 2 2646.2.e.b 2
315.bw odd 12 2 2646.2.h.i 2
315.bx even 12 2 2646.2.h.h 2
315.cb even 12 2 882.2.f.d 2
315.cf odd 12 2 2646.2.f.g 2
315.cg even 12 2 882.2.h.b 2
315.ch odd 12 2 882.2.h.c 2
360.bo even 12 2 576.2.i.a 2
360.br even 12 2 1728.2.i.e 2
360.bt odd 12 2 1728.2.i.f 2
360.bu odd 12 2 576.2.i.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 45.k odd 12 2
54.2.c.a 2 45.l even 12 2
144.2.i.c 2 180.x even 12 2
162.2.a.b 1 15.e even 4 1
162.2.a.c 1 5.c odd 4 1
432.2.i.b 2 180.v odd 12 2
450.2.e.i 2 45.k odd 12 2
450.2.j.e 4 9.c even 3 2
450.2.j.e 4 45.j even 6 2
576.2.i.a 2 360.bo even 12 2
576.2.i.g 2 360.bu odd 12 2
882.2.e.g 2 315.bs even 12 2
882.2.e.i 2 315.bt odd 12 2
882.2.f.d 2 315.cb even 12 2
882.2.h.b 2 315.cg even 12 2
882.2.h.c 2 315.ch odd 12 2
1296.2.a.f 1 60.l odd 4 1
1296.2.a.g 1 20.e even 4 1
1350.2.e.c 2 45.l even 12 2
1350.2.j.a 4 9.d odd 6 2
1350.2.j.a 4 45.h odd 6 2
1728.2.i.e 2 360.br even 12 2
1728.2.i.f 2 360.bt odd 12 2
2646.2.e.b 2 315.bv even 12 2
2646.2.e.c 2 315.bu odd 12 2
2646.2.f.g 2 315.cf odd 12 2
2646.2.h.h 2 315.bx even 12 2
2646.2.h.i 2 315.bw odd 12 2
4050.2.a.c 1 5.c odd 4 1
4050.2.a.v 1 15.e even 4 1
4050.2.c.c 2 1.a even 1 1 trivial
4050.2.c.c 2 5.b even 2 1 inner
4050.2.c.r 2 3.b odd 2 1
4050.2.c.r 2 15.d odd 2 1
5184.2.a.o 1 40.k even 4 1
5184.2.a.p 1 120.q odd 4 1
5184.2.a.q 1 120.w even 4 1
5184.2.a.r 1 40.i odd 4 1
7938.2.a.i 1 105.k odd 4 1
7938.2.a.x 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 $$S_{2}^{\mathrm{new}}(4050, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} + 3$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 9$$ $$T_{19} - 1$$ $$T_{29} + 6$$ $$T_{41} - 9$$ $$T_{71} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$( 3 + T )^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$25 + T^{2}$$
show more
show less