# Properties

 Label 2100.2.ce.b Level 2100 Weight 2 Character orbit 2100.ce 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.ce (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24} q^{3} + ( -3 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{24} q^{3} + ( -3 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{2} q^{9} + ( \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{13} + ( \zeta_{24} - 2 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{17} + ( 5 - 2 \zeta_{24}^{2} - 5 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{19} + ( 3 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{21} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{3} q^{27} + ( 4 - 8 \zeta_{24}^{4} ) q^{29} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{33} + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{39} + ( -3 + 6 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{41} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{43} + ( -6 \zeta_{24} + 5 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{47} + ( 5 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{49} + ( 5 - \zeta_{24}^{2} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{51} + ( -\zeta_{24} + 2 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{53} + ( -5 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{57} + ( -\zeta_{24}^{2} - 11 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{59} + ( -1 + 8 \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{61} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{63} -7 \zeta_{24} q^{67} + ( 4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{69} -8 q^{71} + ( 5 \zeta_{24} - 12 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{73} + ( 2 \zeta_{24} - 5 \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{77} + ( -1 - 4 \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{79} + \zeta_{24}^{4} q^{81} + ( -3 \zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{83} + ( -4 \zeta_{24} + 8 \zeta_{24}^{5} ) q^{87} + ( 9 + \zeta_{24}^{2} - 9 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{89} + ( \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} + 7 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{97} + ( -1 + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{11} + 20q^{19} + 20q^{51} - 44q^{59} - 12q^{61} + 32q^{69} - 64q^{71} - 12q^{79} + 4q^{81} + 36q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{24}^{6}$$ $$1 - \zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i
0 −0.965926 + 0.258819i 0 0 0 −2.38014 1.15539i 0 0.866025 0.500000i 0
157.2 0 0.965926 0.258819i 0 0 0 2.38014 + 1.15539i 0 0.866025 0.500000i 0
493.1 0 −0.258819 0.965926i 0 0 0 1.15539 2.38014i 0 −0.866025 + 0.500000i 0
493.2 0 0.258819 + 0.965926i 0 0 0 −1.15539 + 2.38014i 0 −0.866025 + 0.500000i 0
1657.1 0 −0.258819 + 0.965926i 0 0 0 1.15539 + 2.38014i 0 −0.866025 0.500000i 0
1657.2 0 0.258819 0.965926i 0 0 0 −1.15539 2.38014i 0 −0.866025 0.500000i 0
1993.1 0 −0.965926 0.258819i 0 0 0 −2.38014 + 1.15539i 0 0.866025 + 0.500000i 0
1993.2 0 0.965926 + 0.258819i 0 0 0 2.38014 1.15539i 0 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1993.2 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
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.ce.b yes 8
5.b even 2 1 inner 2100.2.ce.b yes 8
5.c odd 4 2 2100.2.ce.a 8
7.d odd 6 1 2100.2.ce.a 8
35.i odd 6 1 2100.2.ce.a 8
35.k even 12 2 inner 2100.2.ce.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.ce.a 8 5.c odd 4 2
2100.2.ce.a 8 7.d odd 6 1
2100.2.ce.a 8 35.i odd 6 1
2100.2.ce.b yes 8 1.a even 1 1 trivial
2100.2.ce.b yes 8 5.b even 2 1 inner
2100.2.ce.b yes 8 35.k even 12 2 inner

## Hecke kernels

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

 $$T_{11}^{4} + 2 T_{11}^{3} + 6 T_{11}^{2} - 4 T_{11} + 4$$ $$T_{17}^{8} - 60 T_{17}^{6} + 716 T_{17}^{4} + 29040 T_{17}^{2} + 234256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{4} + T^{8}$$
$5$ 1
$7$ $$1 + 23 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 287 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$1 - 60 T^{2} + 2042 T^{4} - 50520 T^{6} + 972243 T^{8} - 14600280 T^{10} + 170549882 T^{12} - 1448254140 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 10 T + 49 T^{2} - 130 T^{3} + 340 T^{4} - 2470 T^{5} + 17689 T^{6} - 68590 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 + 96 T^{2} + 4574 T^{4} + 144192 T^{6} + 3601251 T^{8} + 76277568 T^{10} + 1279992734 T^{12} + 14211445344 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 10 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 58 T^{2} + 2403 T^{4} + 55738 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$1 - 216 T^{2} + 22057 T^{4} - 1405080 T^{6} + 61731552 T^{8} - 1923554520 T^{10} + 41338369177 T^{12} - 554196904344 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 - 108 T^{2} + 6170 T^{4} - 181548 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$1 + 868 T^{4} - 3308250 T^{8} + 2967519268 T^{12} + 11688200277601 T^{16}$$
$47$ $$1 + 180 T^{2} + 16154 T^{4} + 963720 T^{6} + 47642835 T^{8} + 2128857480 T^{10} + 78826366874 T^{12} + 1940258759220 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 - 84 T^{2} + 5642 T^{4} - 276360 T^{6} + 9540387 T^{8} - 776295240 T^{10} + 44518093802 T^{12} - 1861806334836 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 22 T + 248 T^{2} + 2596 T^{3} + 23659 T^{4} + 153164 T^{5} + 863288 T^{6} + 4518338 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 6 T + 73 T^{2} + 366 T^{3} + 732 T^{4} + 22326 T^{5} + 271633 T^{6} + 1361886 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 1753 T^{4} - 17078112 T^{8} + 35324915113 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{8}$$
$73$ $$1 + 360 T^{2} + 64489 T^{4} + 7664040 T^{6} + 655036080 T^{8} + 40841669160 T^{10} + 1831374163849 T^{12} + 54480321464040 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 6 T + 157 T^{2} + 870 T^{3} + 15732 T^{4} + 68730 T^{5} + 979837 T^{6} + 2958234 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 20092 T^{4} + 185593446 T^{8} + 953532585532 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 18 T + 68 T^{2} - 1404 T^{3} + 28779 T^{4} - 124956 T^{5} + 538628 T^{6} - 12689442 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 2446 T^{4} + 12136179 T^{8} + 216542621326 T^{12} + 7837433594376961 T^{16}$$