# Properties

 Label 2100.2.bc.c Level $2100$ Weight $2$ Character orbit 2100.bc Analytic conductor $16.769$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( 2 - 2 \zeta_{12}^{2} ) q^{11} -\zeta_{12}^{3} q^{13} + 4 \zeta_{12} q^{17} -\zeta_{12}^{2} q^{19} + ( 3 - 2 \zeta_{12}^{2} ) q^{21} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{3} q^{27} + ( 5 - 5 \zeta_{12}^{2} ) q^{31} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{33} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{37} + ( -1 + \zeta_{12}^{2} ) q^{39} + 2 q^{41} + 9 \zeta_{12}^{3} q^{43} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} -4 \zeta_{12}^{2} q^{51} + 12 \zeta_{12} q^{53} + \zeta_{12}^{3} q^{57} + ( -8 + 8 \zeta_{12}^{2} ) q^{59} + 14 \zeta_{12}^{2} q^{61} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} -9 \zeta_{12} q^{67} + 4 q^{69} + 2 q^{71} + \zeta_{12} q^{73} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{77} -3 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 18 \zeta_{12}^{3} q^{83} -4 \zeta_{12}^{2} q^{89} + ( 2 + \zeta_{12}^{2} ) q^{91} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{93} + 10 \zeta_{12}^{3} q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9} + O(q^{10})$$ $$4 q + 2 q^{9} + 4 q^{11} - 2 q^{19} + 8 q^{21} + 10 q^{31} - 2 q^{39} + 8 q^{41} - 22 q^{49} - 8 q^{51} - 16 q^{59} + 28 q^{61} + 16 q^{69} + 8 q^{71} - 6 q^{79} - 2 q^{81} - 8 q^{89} + 10 q^{91} + 8 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 0.500000i 0 0 0 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 0.866025 2.50000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −0.866025 2.50000i 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 0 0 0.866025 + 2.50000i 0 0.500000 0.866025i 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.c 4
5.b even 2 1 inner 2100.2.bc.c 4
5.c odd 4 1 420.2.q.a 2
5.c odd 4 1 2100.2.q.a 2
7.c even 3 1 inner 2100.2.bc.c 4
15.e even 4 1 1260.2.s.d 2
20.e even 4 1 1680.2.bg.a 2
35.f even 4 1 2940.2.q.h 2
35.j even 6 1 inner 2100.2.bc.c 4
35.k even 12 1 2940.2.a.h 1
35.k even 12 1 2940.2.q.h 2
35.l odd 12 1 420.2.q.a 2
35.l odd 12 1 2100.2.q.a 2
35.l odd 12 1 2940.2.a.d 1
105.w odd 12 1 8820.2.a.y 1
105.x even 12 1 1260.2.s.d 2
105.x even 12 1 8820.2.a.j 1
140.w even 12 1 1680.2.bg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 5.c odd 4 1
420.2.q.a 2 35.l odd 12 1
1260.2.s.d 2 15.e even 4 1
1260.2.s.d 2 105.x even 12 1
1680.2.bg.a 2 20.e even 4 1
1680.2.bg.a 2 140.w even 12 1
2100.2.q.a 2 5.c odd 4 1
2100.2.q.a 2 35.l odd 12 1
2100.2.bc.c 4 1.a even 1 1 trivial
2100.2.bc.c 4 5.b even 2 1 inner
2100.2.bc.c 4 7.c even 3 1 inner
2100.2.bc.c 4 35.j even 6 1 inner
2940.2.a.d 1 35.l odd 12 1
2940.2.a.h 1 35.k even 12 1
2940.2.q.h 2 35.f even 4 1
2940.2.q.h 2 35.k even 12 1
8820.2.a.j 1 105.x even 12 1
8820.2.a.y 1 105.w odd 12 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$( 4 - 2 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$256 - 16 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 25 - 5 T + T^{2} )^{2}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$( 81 + T^{2} )^{2}$$
$47$ $$16 - 4 T^{2} + T^{4}$$
$53$ $$20736 - 144 T^{2} + T^{4}$$
$59$ $$( 64 + 8 T + T^{2} )^{2}$$
$61$ $$( 196 - 14 T + T^{2} )^{2}$$
$67$ $$6561 - 81 T^{2} + T^{4}$$
$71$ $$( -2 + T )^{4}$$
$73$ $$1 - T^{2} + T^{4}$$
$79$ $$( 9 + 3 T + T^{2} )^{2}$$
$83$ $$( 324 + T^{2} )^{2}$$
$89$ $$( 16 + 4 T + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$