# Properties

 Label 2100.2.bc.d 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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} + 4 \zeta_{12}^{2} q^{11} + 7 \zeta_{12}^{3} q^{13} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{17} + ( 3 - 3 \zeta_{12}^{2} ) q^{19} + ( -1 - 2 \zeta_{12}^{2} ) q^{21} + 2 \zeta_{12} q^{23} + \zeta_{12}^{3} q^{27} + 2 q^{29} -7 \zeta_{12}^{2} q^{31} -4 \zeta_{12} q^{33} -7 \zeta_{12} q^{37} -7 \zeta_{12}^{2} q^{39} -8 q^{41} + 5 \zeta_{12}^{3} q^{43} + 10 \zeta_{12} q^{47} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + ( -6 + 6 \zeta_{12}^{2} ) q^{51} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{53} + 3 \zeta_{12}^{3} q^{57} + 10 \zeta_{12}^{2} q^{59} + ( 6 - 6 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{63} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{67} -2 q^{69} + ( 15 \zeta_{12} - 15 \zeta_{12}^{3} ) q^{73} + ( -8 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} + ( 1 - \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} + 8 \zeta_{12}^{3} q^{83} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 2 - 2 \zeta_{12}^{2} ) q^{89} + ( -21 + 7 \zeta_{12}^{2} ) q^{91} + 7 \zeta_{12} q^{93} + 10 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} + 8q^{11} + 6q^{19} - 8q^{21} + 8q^{29} - 14q^{31} - 14q^{39} - 32q^{41} - 22q^{49} - 12q^{51} + 20q^{59} + 12q^{61} - 8q^{69} + 2q^{79} - 2q^{81} + 4q^{89} - 70q^{91} + 16q^{99} + O(q^{100})$$

## Character values

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

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

## 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.d 4
5.b even 2 1 inner 2100.2.bc.d 4
5.c odd 4 1 420.2.q.b 2
5.c odd 4 1 2100.2.q.d 2
7.c even 3 1 inner 2100.2.bc.d 4
15.e even 4 1 1260.2.s.a 2
20.e even 4 1 1680.2.bg.j 2
35.f even 4 1 2940.2.q.c 2
35.j even 6 1 inner 2100.2.bc.d 4
35.k even 12 1 2940.2.a.k 1
35.k even 12 1 2940.2.q.c 2
35.l odd 12 1 420.2.q.b 2
35.l odd 12 1 2100.2.q.d 2
35.l odd 12 1 2940.2.a.b 1
105.w odd 12 1 8820.2.a.l 1
105.x even 12 1 1260.2.s.a 2
105.x even 12 1 8820.2.a.bb 1
140.w even 12 1 1680.2.bg.j 2

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

## Hecke kernels

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

 $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{2} + T^{4}$$
$5$ 1
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 23 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 8 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 61 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 6 T^{2} - 2173 T^{4} - 13254 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 42 T^{2} - 1045 T^{4} + 117978 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$1 - 79 T^{2} + 912 T^{4} - 420991 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 102 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 94 T^{2} + 9409 T^{4} )^{2}$$