# Properties

 Label 2100.2.bc.a 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 84) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{3} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} -3 \zeta_{12}^{3} q^{13} + 8 \zeta_{12} q^{17} -\zeta_{12}^{2} q^{19} + ( -1 - 2 \zeta_{12}^{2} ) q^{21} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{3} q^{27} -4 q^{29} + ( -3 + 3 \zeta_{12}^{2} ) q^{31} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{33} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{37} + ( 3 - 3 \zeta_{12}^{2} ) q^{39} + 6 q^{41} + 11 \zeta_{12}^{3} q^{43} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{47} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + 8 \zeta_{12}^{2} q^{51} + 12 \zeta_{12} q^{53} -\zeta_{12}^{3} q^{57} + ( 4 - 4 \zeta_{12}^{2} ) q^{59} + 6 \zeta_{12}^{2} q^{61} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} + 13 \zeta_{12} q^{67} + 8 q^{69} -10 q^{71} + 11 \zeta_{12} q^{73} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} -3 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 2 \zeta_{12}^{3} q^{83} -4 \zeta_{12} q^{87} + ( -6 + 9 \zeta_{12}^{2} ) q^{91} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{93} -10 \zeta_{12}^{3} q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} - 4q^{11} - 2q^{19} - 8q^{21} - 16q^{29} - 6q^{31} + 6q^{39} + 24q^{41} + 26q^{49} + 16q^{51} + 8q^{59} + 12q^{61} + 32q^{69} - 40q^{71} - 6q^{79} - 2q^{81} - 6q^{91} - 8q^{99} + O(q^{100})$$

## 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 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 2.59808 0.500000i 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 0 0 −2.59808 + 0.500000i 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.a 4
5.b even 2 1 inner 2100.2.bc.a 4
5.c odd 4 1 84.2.i.a 2
5.c odd 4 1 2100.2.q.b 2
7.c even 3 1 inner 2100.2.bc.a 4
15.e even 4 1 252.2.k.a 2
20.e even 4 1 336.2.q.c 2
35.f even 4 1 588.2.i.b 2
35.j even 6 1 inner 2100.2.bc.a 4
35.k even 12 1 588.2.a.f 1
35.k even 12 1 588.2.i.b 2
35.l odd 12 1 84.2.i.a 2
35.l odd 12 1 588.2.a.a 1
35.l odd 12 1 2100.2.q.b 2
40.i odd 4 1 1344.2.q.b 2
40.k even 4 1 1344.2.q.n 2
45.k odd 12 1 2268.2.i.g 2
45.k odd 12 1 2268.2.l.b 2
45.l even 12 1 2268.2.i.b 2
45.l even 12 1 2268.2.l.g 2
60.l odd 4 1 1008.2.s.c 2
105.k odd 4 1 1764.2.k.j 2
105.w odd 12 1 1764.2.a.c 1
105.w odd 12 1 1764.2.k.j 2
105.x even 12 1 252.2.k.a 2
105.x even 12 1 1764.2.a.h 1
140.j odd 4 1 2352.2.q.q 2
140.w even 12 1 336.2.q.c 2
140.w even 12 1 2352.2.a.o 1
140.x odd 12 1 2352.2.a.k 1
140.x odd 12 1 2352.2.q.q 2
280.bp odd 12 1 9408.2.a.bx 1
280.br even 12 1 1344.2.q.n 2
280.br even 12 1 9408.2.a.bi 1
280.bt odd 12 1 1344.2.q.b 2
280.bt odd 12 1 9408.2.a.cx 1
280.bv even 12 1 9408.2.a.i 1
315.bt odd 12 1 2268.2.l.b 2
315.bv even 12 1 2268.2.l.g 2
315.bx even 12 1 2268.2.i.b 2
315.ch odd 12 1 2268.2.i.g 2
420.bp odd 12 1 1008.2.s.c 2
420.bp odd 12 1 7056.2.a.bs 1
420.br even 12 1 7056.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 5.c odd 4 1
84.2.i.a 2 35.l odd 12 1
252.2.k.a 2 15.e even 4 1
252.2.k.a 2 105.x even 12 1
336.2.q.c 2 20.e even 4 1
336.2.q.c 2 140.w even 12 1
588.2.a.a 1 35.l odd 12 1
588.2.a.f 1 35.k even 12 1
588.2.i.b 2 35.f even 4 1
588.2.i.b 2 35.k even 12 1
1008.2.s.c 2 60.l odd 4 1
1008.2.s.c 2 420.bp odd 12 1
1344.2.q.b 2 40.i odd 4 1
1344.2.q.b 2 280.bt odd 12 1
1344.2.q.n 2 40.k even 4 1
1344.2.q.n 2 280.br even 12 1
1764.2.a.c 1 105.w odd 12 1
1764.2.a.h 1 105.x even 12 1
1764.2.k.j 2 105.k odd 4 1
1764.2.k.j 2 105.w odd 12 1
2100.2.q.b 2 5.c odd 4 1
2100.2.q.b 2 35.l odd 12 1
2100.2.bc.a 4 1.a even 1 1 trivial
2100.2.bc.a 4 5.b even 2 1 inner
2100.2.bc.a 4 7.c even 3 1 inner
2100.2.bc.a 4 35.j even 6 1 inner
2268.2.i.b 2 45.l even 12 1
2268.2.i.b 2 315.bx even 12 1
2268.2.i.g 2 45.k odd 12 1
2268.2.i.g 2 315.ch odd 12 1
2268.2.l.b 2 45.k odd 12 1
2268.2.l.b 2 315.bt odd 12 1
2268.2.l.g 2 45.l even 12 1
2268.2.l.g 2 315.bv even 12 1
2352.2.a.k 1 140.x odd 12 1
2352.2.a.o 1 140.w even 12 1
2352.2.q.q 2 140.j odd 4 1
2352.2.q.q 2 140.x odd 12 1
7056.2.a.o 1 420.br even 12 1
7056.2.a.bs 1 420.bp odd 12 1
9408.2.a.i 1 280.bv even 12 1
9408.2.a.bi 1 280.br even 12 1
9408.2.a.bx 1 280.bp odd 12 1
9408.2.a.cx 1 280.bt odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13}^{2} + 9$$