# Properties

 Label 2100.2.bc.g Level 2100 Weight 2 Character orbit 2100.bc Analytic conductor 16.769 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{3} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{3} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{11} + 5 \zeta_{24}^{6} q^{13} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} - \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{19} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{21} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{23} -\zeta_{24}^{6} q^{27} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{29} + ( 4 - 4 \zeta_{24}^{4} ) q^{31} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{33} + ( -5 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{37} + ( 5 - 5 \zeta_{24}^{4} ) q^{39} + ( -6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{43} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{47} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( 3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{51} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{53} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{57} + ( 6 - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{59} + \zeta_{24}^{4} q^{61} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} + ( -\zeta_{24}^{2} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 9 \zeta_{24}^{7} ) q^{67} -6 q^{69} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} + ( -5 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 5 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{79} + ( -1 + \zeta_{24}^{4} ) q^{81} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{83} + ( -6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{87} + ( 3 \zeta_{24} + 12 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{89} + ( -5 + 10 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{91} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{93} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{97} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} - 4q^{19} - 4q^{21} + 16q^{31} + 20q^{39} - 48q^{41} - 20q^{49} + 24q^{59} + 4q^{61} - 48q^{69} + 20q^{79} - 4q^{81} + 48q^{89} - 20q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\zeta_{24}^{2}$$

## 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
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
0 −0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.09077 1.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1549.4 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.g 8
5.b even 2 1 inner 2100.2.bc.g 8
5.c odd 4 1 2100.2.q.f 4
5.c odd 4 1 2100.2.q.j yes 4
7.c even 3 1 inner 2100.2.bc.g 8
35.j even 6 1 inner 2100.2.bc.g 8
35.l odd 12 1 2100.2.q.f 4
35.l odd 12 1 2100.2.q.j yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.f 4 5.c odd 4 1
2100.2.q.f 4 35.l odd 12 1
2100.2.q.j yes 4 5.c odd 4 1
2100.2.q.j yes 4 35.l odd 12 1
2100.2.bc.g 8 1.a even 1 1 trivial
2100.2.bc.g 8 5.b even 2 1 inner
2100.2.bc.g 8 7.c even 3 1 inner
2100.2.bc.g 8 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 $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{4} + 18 T_{11}^{2} + 324$$ $$T_{13}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ 1
$7$ $$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 16 T^{2} - 33 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 2 T - 17 T^{2} - 34 T^{3} + 4 T^{4} - 646 T^{5} - 6137 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 14 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4}$$
$37$ $$( 1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 4 T^{2} - 906 T^{4} + 7396 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 80 T^{2} + 2974 T^{4} - 79360 T^{6} - 7279805 T^{8} - 175306240 T^{10} + 14512171294 T^{12} + 862337226320 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 + 88 T^{2} + 4935 T^{4} + 247192 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 12744 T^{5} + 27848 T^{6} - 2464548 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4}$$
$67$ $$1 - 58 T^{2} - 5807 T^{4} - 11194 T^{6} + 48855124 T^{8} - 50249866 T^{10} - 117017559647 T^{12} - 5246586165802 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 + 70 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$1 + 98 T^{2} + 3745 T^{4} - 470302 T^{6} - 45250076 T^{8} - 2506239358 T^{10} + 106351412545 T^{12} + 14830754176322 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 - 10 T - 65 T^{2} - 70 T^{3} + 13084 T^{4} - 5530 T^{5} - 405665 T^{6} - 4930390 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 224 T^{2} + 23730 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 24 T + 272 T^{2} - 3024 T^{3} + 33231 T^{4} - 269136 T^{5} + 2154512 T^{6} - 16919256 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} )^{2}$$