# Properties

 Label 2100.2.bc.b 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: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} + 2 \zeta_{12} q^{17} + 4 \zeta_{12}^{2} q^{19} + ( -1 + 3 \zeta_{12}^{2} ) q^{21} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{3} q^{27} -4 q^{29} + ( -3 + 3 \zeta_{12}^{2} ) q^{31} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{37} + ( -2 + 2 \zeta_{12}^{2} ) q^{39} + 6 q^{41} -\zeta_{12}^{3} q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -2 \zeta_{12}^{2} q^{51} -2 \zeta_{12} q^{53} -4 \zeta_{12}^{3} q^{57} + ( -6 + 6 \zeta_{12}^{2} ) q^{59} + \zeta_{12}^{2} q^{61} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + 12 \zeta_{12} q^{67} + 8 q^{69} + 10 q^{71} -\zeta_{12} q^{73} + ( 6 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + 7 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 18 \zeta_{12}^{3} q^{83} + 4 \zeta_{12} q^{87} + 10 \zeta_{12}^{2} q^{89} + ( -6 + 4 \zeta_{12}^{2} ) q^{91} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{93} -5 \zeta_{12}^{3} q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} - 4q^{11} + 8q^{19} + 2q^{21} - 16q^{29} - 6q^{31} - 4q^{39} + 24q^{41} - 4q^{49} - 4q^{51} - 12q^{59} + 2q^{61} + 32q^{69} + 40q^{71} + 14q^{79} - 2q^{81} + 20q^{89} - 16q^{91} - 8q^{99} + O(q^{100})$$

## 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 −1.73205 2.00000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 1.73205 + 2.00000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −1.73205 + 2.00000i 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 0 0 1.73205 2.00000i 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.b 4
5.b even 2 1 inner 2100.2.bc.b 4
5.c odd 4 1 2100.2.q.c 2
5.c odd 4 1 2100.2.q.e yes 2
7.c even 3 1 inner 2100.2.bc.b 4
35.j even 6 1 inner 2100.2.bc.b 4
35.l odd 12 1 2100.2.q.c 2
35.l odd 12 1 2100.2.q.e yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.c 2 5.c odd 4 1
2100.2.q.c 2 35.l odd 12 1
2100.2.q.e yes 2 5.c odd 4 1
2100.2.q.e yes 2 35.l odd 12 1
2100.2.bc.b 4 1.a even 1 1 trivial
2100.2.bc.b 4 5.b even 2 1 inner
2100.2.bc.b 4 7.c even 3 1 inner
2100.2.bc.b 4 35.j even 6 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{2} + T^{4}$$
$5$ 1
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 4 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 7 T^{2} - 1320 T^{4} - 9583 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 85 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$
$67$ $$1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 10 T + 71 T^{2} )^{4}$$
$73$ $$1 + 145 T^{2} + 15696 T^{4} + 772705 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 158 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 169 T^{2} + 9409 T^{4} )^{2}$$