# Properties

 Label 2100.2.d.a Level 2100 Weight 2 Character orbit 2100.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 6 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{13} + 3 q^{17} + ( -2 + 4 \zeta_{6} ) q^{19} + ( 5 - \zeta_{6} ) q^{21} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 3 - 6 \zeta_{6} ) q^{29} + ( -6 + 12 \zeta_{6} ) q^{31} + ( -9 + 9 \zeta_{6} ) q^{33} + 8 q^{37} + ( -3 + 3 \zeta_{6} ) q^{39} -6 q^{41} -10 q^{43} + 3 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -3 - 3 \zeta_{6} ) q^{51} + ( 6 - 12 \zeta_{6} ) q^{53} + ( 6 - 6 \zeta_{6} ) q^{57} -6 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} + ( -6 - 3 \zeta_{6} ) q^{63} -2 q^{67} + ( 6 - 12 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} + ( 3 + 12 \zeta_{6} ) q^{77} -13 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -12 q^{83} + ( -9 + 9 \zeta_{6} ) q^{87} + ( 1 + 4 \zeta_{6} ) q^{91} + ( 18 - 18 \zeta_{6} ) q^{93} + ( 1 - 2 \zeta_{6} ) q^{97} + ( 18 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 4q^{7} + 3q^{9} + 6q^{17} + 9q^{21} - 9q^{33} + 16q^{37} - 3q^{39} - 12q^{41} - 20q^{43} + 6q^{47} + 2q^{49} - 9q^{51} + 6q^{57} - 12q^{59} - 15q^{63} - 4q^{67} + 18q^{77} - 26q^{79} - 9q^{81} - 24q^{83} - 9q^{87} + 6q^{91} + 18q^{93} + 27q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1301.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 0 0 −2.00000 + 1.73205i 0 1.50000 + 2.59808i 0
1301.2 0 −1.50000 + 0.866025i 0 0 0 −2.00000 1.73205i 0 1.50000 2.59808i 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
21.c 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.d.a 2
3.b odd 2 1 2100.2.d.e 2
5.b even 2 1 420.2.d.b yes 2
5.c odd 4 2 2100.2.f.d 4
7.b odd 2 1 2100.2.d.e 2
15.d odd 2 1 420.2.d.a 2
15.e even 4 2 2100.2.f.c 4
20.d odd 2 1 1680.2.f.a 2
21.c even 2 1 inner 2100.2.d.a 2
35.c odd 2 1 420.2.d.a 2
35.f even 4 2 2100.2.f.c 4
60.h even 2 1 1680.2.f.d 2
105.g even 2 1 420.2.d.b yes 2
105.k odd 4 2 2100.2.f.d 4
140.c even 2 1 1680.2.f.d 2
420.o odd 2 1 1680.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.a 2 15.d odd 2 1
420.2.d.a 2 35.c odd 2 1
420.2.d.b yes 2 5.b even 2 1
420.2.d.b yes 2 105.g even 2 1
1680.2.f.a 2 20.d odd 2 1
1680.2.f.a 2 420.o odd 2 1
1680.2.f.d 2 60.h even 2 1
1680.2.f.d 2 140.c even 2 1
2100.2.d.a 2 1.a even 1 1 trivial
2100.2.d.a 2 21.c even 2 1 inner
2100.2.d.e 2 3.b odd 2 1
2100.2.d.e 2 7.b odd 2 1
2100.2.f.c 4 15.e even 4 2
2100.2.f.c 4 35.f even 4 2
2100.2.f.d 4 5.c odd 4 2
2100.2.f.d 4 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{2} + 27$$ $$T_{13}^{2} + 3$$ $$T_{17} - 3$$ $$T_{37} - 8$$ $$T_{41} + 6$$ $$T_{43} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ 1
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 + 5 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$( 1 - 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$1 - 31 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 - 8 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 10 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 3 T + 47 T^{2} )^{2}$$
$53$ $$1 + 2 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 6 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$( 1 + 2 T + 67 T^{2} )^{2}$$
$71$ $$1 - 34 T^{2} + 5041 T^{4}$$
$73$ $$1 - 98 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 - 191 T^{2} + 9409 T^{4}$$