# Properties

 Label 2100.2.k.g Level 2100 Weight 2 Character orbit 2100.k Analytic conductor 16.769 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ 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 420) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} -i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} -i q^{7} - q^{9} + 2 q^{11} -4 i q^{13} + 6 i q^{17} -6 q^{19} + q^{21} + 8 i q^{23} -i q^{27} + 2 q^{29} + 10 q^{31} + 2 i q^{33} + 2 i q^{37} + 4 q^{39} + 10 q^{41} + 4 i q^{43} -8 i q^{47} - q^{49} -6 q^{51} -4 i q^{53} -6 i q^{57} + 8 q^{59} + 6 q^{61} + i q^{63} + 12 i q^{67} -8 q^{69} -6 q^{71} + 12 i q^{73} -2 i q^{77} + 8 q^{79} + q^{81} + 4 i q^{83} + 2 i q^{87} + 10 q^{89} -4 q^{91} + 10 i q^{93} + 8 i q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{11} - 12q^{19} + 2q^{21} + 4q^{29} + 20q^{31} + 8q^{39} + 20q^{41} - 2q^{49} - 12q^{51} + 16q^{59} + 12q^{61} - 16q^{69} - 12q^{71} + 16q^{79} + 2q^{81} + 20q^{89} - 8q^{91} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 2100.2.k.g 2
3.b odd 2 1 6300.2.k.e 2
5.b even 2 1 inner 2100.2.k.g 2
5.c odd 4 1 420.2.a.a 1
5.c odd 4 1 2100.2.a.q 1
15.d odd 2 1 6300.2.k.e 2
15.e even 4 1 1260.2.a.g 1
15.e even 4 1 6300.2.a.v 1
20.e even 4 1 1680.2.a.n 1
20.e even 4 1 8400.2.a.c 1
35.f even 4 1 2940.2.a.l 1
35.k even 12 2 2940.2.q.a 2
35.l odd 12 2 2940.2.q.m 2
40.i odd 4 1 6720.2.a.cb 1
40.k even 4 1 6720.2.a.bd 1
60.l odd 4 1 5040.2.a.bl 1
105.k odd 4 1 8820.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.a 1 5.c odd 4 1
1260.2.a.g 1 15.e even 4 1
1680.2.a.n 1 20.e even 4 1
2100.2.a.q 1 5.c odd 4 1
2100.2.k.g 2 1.a even 1 1 trivial
2100.2.k.g 2 5.b even 2 1 inner
2940.2.a.l 1 35.f even 4 1
2940.2.q.a 2 35.k even 12 2
2940.2.q.m 2 35.l odd 12 2
5040.2.a.bl 1 60.l odd 4 1
6300.2.a.v 1 15.e even 4 1
6300.2.k.e 2 3.b odd 2 1
6300.2.k.e 2 15.d odd 2 1
6720.2.a.bd 1 40.k even 4 1
6720.2.a.cb 1 40.i odd 4 1
8400.2.a.c 1 20.e even 4 1
8820.2.a.f 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 $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11} - 2$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 2 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 6 T + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 10 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$
$59$ $$( 1 - 8 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 6 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$