# Properties

 Label 4050.2.c.g 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: $$x^{2} + 1$$ 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 $$q + i q^{2} - q^{4} + 4 i q^{7} - i q^{8} - i q^{13} - 4 q^{14} + q^{16} + 3 i q^{17} + 4 q^{19} + q^{26} - 4 i q^{28} - 9 q^{29} - 4 q^{31} + i q^{32} - 3 q^{34} + i q^{37} + 4 i q^{38} + 6 q^{41} + 8 i q^{43} + 12 i q^{47} - 9 q^{49} + i q^{52} - 6 i q^{53} + 4 q^{56} - 9 i q^{58} - q^{61} - 4 i q^{62} - q^{64} + 4 i q^{67} - 3 i q^{68} - 12 q^{71} + 11 i q^{73} - q^{74} - 4 q^{76} + 16 q^{79} + 6 i q^{82} - 12 i q^{83} - 8 q^{86} + 3 q^{89} + 4 q^{91} - 12 q^{94} - 2 i q^{97} - 9 i q^{98} +O(q^{100})$$ 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 8 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{26} - 18 q^{29} - 8 q^{31} - 6 q^{34} + 12 q^{41} - 18 q^{49} + 8 q^{56} - 2 q^{61} - 2 q^{64} - 24 q^{71} - 2 q^{74} - 8 q^{76} + 32 q^{79} - 16 q^{86} + 6 q^{89} + 8 q^{91} - 24 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 8 * q^14 + 2 * q^16 + 8 * q^19 + 2 * q^26 - 18 * q^29 - 8 * q^31 - 6 * q^34 + 12 * q^41 - 18 * q^49 + 8 * q^56 - 2 * q^61 - 2 * q^64 - 24 * q^71 - 2 * q^74 - 8 * q^76 + 32 * q^79 - 16 * q^86 + 6 * q^89 + 8 * q^91 - 24 * q^94

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

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

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}$$ T11 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{17}^{2} + 9$$ T17^2 + 9 $$T_{19} - 4$$ T19 - 4 $$T_{29} + 9$$ T29 + 9 $$T_{41} - 6$$ T41 - 6 $$T_{71} + 12$$ T71 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 121$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 3)^{2}$$
$97$ $$T^{2} + 4$$