# Properties

 Label 4200.2.t.r Level $4200$ Weight $2$ Character orbit 4200.t Analytic conductor $33.537$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4200.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.5371688489$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + i q^{7} - q^{9} + 4 q^{11} -2 i q^{13} -2 i q^{17} -4 q^{19} - q^{21} -i q^{27} + 2 q^{29} + 8 q^{31} + 4 i q^{33} + 2 i q^{37} + 2 q^{39} + 2 q^{41} + 4 i q^{43} - q^{49} + 2 q^{51} -10 i q^{53} -4 i q^{57} + 12 q^{59} + 6 q^{61} -i q^{63} -12 i q^{67} -6 i q^{73} + 4 i q^{77} + 8 q^{79} + q^{81} + 4 i q^{83} + 2 i q^{87} -2 q^{89} + 2 q^{91} + 8 i q^{93} + 14 i q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 8q^{11} - 8q^{19} - 2q^{21} + 4q^{29} + 16q^{31} + 4q^{39} + 4q^{41} - 2q^{49} + 4q^{51} + 24q^{59} + 12q^{61} + 16q^{79} + 2q^{81} - 4q^{89} + 4q^{91} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4200\mathbb{Z}\right)^\times$$.

 $$n$$ $$1177$$ $$2101$$ $$2801$$ $$3151$$ $$3601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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 4200.2.t.r 2
5.b even 2 1 inner 4200.2.t.r 2
5.c odd 4 1 840.2.a.j 1
5.c odd 4 1 4200.2.a.n 1
15.e even 4 1 2520.2.a.a 1
20.e even 4 1 1680.2.a.h 1
20.e even 4 1 8400.2.a.bl 1
35.f even 4 1 5880.2.a.g 1
40.i odd 4 1 6720.2.a.b 1
40.k even 4 1 6720.2.a.bw 1
60.l odd 4 1 5040.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.j 1 5.c odd 4 1
1680.2.a.h 1 20.e even 4 1
2520.2.a.a 1 15.e even 4 1
4200.2.a.n 1 5.c odd 4 1
4200.2.t.r 2 1.a even 1 1 trivial
4200.2.t.r 2 5.b even 2 1 inner
5040.2.a.s 1 60.l odd 4 1
5880.2.a.g 1 35.f even 4 1
6720.2.a.b 1 40.i odd 4 1
6720.2.a.bw 1 40.k even 4 1
8400.2.a.bl 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4200, [\chi])$$:

 $$T_{11} - 4$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 4$$ $$T_{19} + 4$$ $$T_{29} - 2$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$196 + T^{2}$$