# Properties

 Label 2100.2.x.b 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: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 1$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/24$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{2}$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 13 \nu^{3}$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 5 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} + 9 \beta_{3}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} - 13 \beta_{5}$$$$)/2$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{3}$$ $$-1$$

## 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
 0.581861 − 1.28897i −1.28897 + 0.581861i 1.28897 − 0.581861i −0.581861 + 1.28897i 0.581861 + 1.28897i −1.28897 − 0.581861i 1.28897 + 0.581861i −0.581861 − 1.28897i
0 −0.707107 0.707107i 0 0 0 −1.87083 1.87083i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 1.87083 + 1.87083i 0 1.00000i 0
1357.3 0 0.707107 + 0.707107i 0 0 0 −1.87083 1.87083i 0 1.00000i 0
1357.4 0 0.707107 + 0.707107i 0 0 0 1.87083 + 1.87083i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 −1.87083 + 1.87083i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 1.87083 1.87083i 0 1.00000i 0
1693.3 0 0.707107 0.707107i 0 0 0 −1.87083 + 1.87083i 0 1.00000i 0
1693.4 0 0.707107 0.707107i 0 0 0 1.87083 1.87083i 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.b 8
5.b even 2 1 inner 2100.2.x.b 8
5.c odd 4 2 inner 2100.2.x.b 8
7.b odd 2 1 inner 2100.2.x.b 8
35.c odd 2 1 inner 2100.2.x.b 8
35.f even 4 2 inner 2100.2.x.b 8

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{4} )^{2}$$
$5$ 1
$7$ $$( 1 + 49 T^{4} )^{2}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{8}$$
$13$ $$( 1 + 146 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 254 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 734 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 22 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 34 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 30 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 334 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 3518 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 466 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 6 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 10 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 2258 T^{4} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{8}$$
$73$ $$( 1 - 8158 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 + 17282 T^{4} + 88529281 T^{8} )^{2}$$