# Properties

 Label 2100.2.k.f 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: yes 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} -i q^{7} - q^{9} + q^{11} + 4 i q^{13} -2 i q^{17} + 4 q^{19} - q^{21} + 7 i q^{23} + i q^{27} + 9 q^{29} -2 q^{31} -i q^{33} -i q^{37} + 4 q^{39} + 8 q^{41} -9 i q^{43} -4 i q^{47} - q^{49} -2 q^{51} + 6 i q^{53} -4 i q^{57} -4 q^{59} + 4 q^{61} + i q^{63} -9 i q^{67} + 7 q^{69} + 5 q^{71} + 10 i q^{73} -i q^{77} + 15 q^{79} + q^{81} -6 i q^{83} -9 i q^{87} -8 q^{89} + 4 q^{91} + 2 i q^{93} -10 i q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{11} + 8q^{19} - 2q^{21} + 18q^{29} - 4q^{31} + 8q^{39} + 16q^{41} - 2q^{49} - 4q^{51} - 8q^{59} + 8q^{61} + 14q^{69} + 10q^{71} + 30q^{79} + 2q^{81} - 16q^{89} + 8q^{91} - 2q^{99} + O(q^{100})$$

## 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.f 2
3.b odd 2 1 6300.2.k.h 2
5.b even 2 1 inner 2100.2.k.f 2
5.c odd 4 1 2100.2.a.g 1
5.c odd 4 1 2100.2.a.m yes 1
15.d odd 2 1 6300.2.k.h 2
15.e even 4 1 6300.2.a.f 1
15.e even 4 1 6300.2.a.x 1
20.e even 4 1 8400.2.a.x 1
20.e even 4 1 8400.2.a.bv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.g 1 5.c odd 4 1
2100.2.a.m yes 1 5.c odd 4 1
2100.2.k.f 2 1.a even 1 1 trivial
2100.2.k.f 2 5.b even 2 1 inner
6300.2.a.f 1 15.e even 4 1
6300.2.a.x 1 15.e even 4 1
6300.2.k.h 2 3.b odd 2 1
6300.2.k.h 2 15.d odd 2 1
8400.2.a.x 1 20.e even 4 1
8400.2.a.bv 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 $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11} - 1$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 - T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 3 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{2}$$
$37$ $$1 - 73 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 8 T + 41 T^{2} )^{2}$$
$43$ $$1 - 5 T^{2} + 1849 T^{4}$$
$47$ $$1 - 78 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 4 T + 61 T^{2} )^{2}$$
$67$ $$1 - 53 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 5 T + 71 T^{2} )^{2}$$
$73$ $$1 - 46 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 15 T + 79 T^{2} )^{2}$$
$83$ $$1 - 130 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 8 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$