Properties

 Label 2100.2.x.a Level 2100 Weight 2 Character orbit 2100.x Analytic conductor 16.769 Analytic rank 0 Dimension 8 CM no Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.x (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{9} -3 q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{13} -6 \zeta_{24}^{3} q^{17} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{19} + ( -1 - 2 \zeta_{24}^{4} ) q^{21} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{23} + \zeta_{24}^{3} q^{27} -9 \zeta_{24}^{6} q^{29} + ( 4 - 8 \zeta_{24}^{4} ) q^{31} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{33} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{37} + 2 \zeta_{24}^{6} q^{39} + ( 7 \zeta_{24} + 7 \zeta_{24}^{5} ) q^{43} -6 \zeta_{24}^{3} q^{47} + ( -8 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{49} + 6 q^{51} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{57} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59} + ( -4 + 8 \zeta_{24}^{4} ) q^{61} + ( 3 \zeta_{24} - \zeta_{24}^{5} ) q^{63} + ( 7 \zeta_{24}^{3} - 14 \zeta_{24}^{7} ) q^{67} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{69} + 3 q^{71} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{73} + ( -3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{77} -\zeta_{24}^{6} q^{79} - q^{81} + ( 12 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{83} + 9 \zeta_{24}^{3} q^{87} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{89} + ( 2 + 4 \zeta_{24}^{4} ) q^{91} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{93} -4 \zeta_{24}^{3} q^{97} + 3 \zeta_{24}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{11} - 16q^{21} + 48q^{51} + 24q^{71} - 8q^{81} + 32q^{91} + O(q^{100})$$

Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{24}^{3}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1357.1
 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 −0.965926 − 0.258819i
0 −0.707107 0.707107i 0 0 0 0.189469 2.63896i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 2.63896 0.189469i 0 1.00000i 0
1357.3 0 0.707107 + 0.707107i 0 0 0 −2.63896 + 0.189469i 0 1.00000i 0
1357.4 0 0.707107 + 0.707107i 0 0 0 −0.189469 + 2.63896i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 0.189469 + 2.63896i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 2.63896 + 0.189469i 0 1.00000i 0
1693.3 0 0.707107 0.707107i 0 0 0 −2.63896 0.189469i 0 1.00000i 0
1693.4 0 0.707107 0.707107i 0 0 0 −0.189469 2.63896i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1693.4 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
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.x.a 8
5.b even 2 1 inner 2100.2.x.a 8
5.c odd 4 2 inner 2100.2.x.a 8
7.b odd 2 1 inner 2100.2.x.a 8
35.c odd 2 1 inner 2100.2.x.a 8
35.f even 4 2 inner 2100.2.x.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.x.a 8 1.a even 1 1 trivial
2100.2.x.a 8 5.b even 2 1 inner
2100.2.x.a 8 5.c odd 4 2 inner
2100.2.x.a 8 7.b odd 2 1 inner
2100.2.x.a 8 35.c odd 2 1 inner
2100.2.x.a 8 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} + 3$$ acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{4} )^{2}$$
$5$ 1
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{8}$$
$13$ $$( 1 + 146 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 26 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 697 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 23 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 14 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 529 T^{4} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{8}$$
$43$ $$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 1054 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 10 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 8809 T^{4} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{8}$$
$73$ $$( 1 - 24 T + 288 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 157 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 13294 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 12866 T^{4} + 88529281 T^{8} )^{2}$$