# Properties

 Label 2100.2.q.e Level 2100 Weight 2 Character orbit 2100.q Analytic conductor 16.769 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + 2 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} -8 \zeta_{6} q^{23} - q^{27} + 4 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} -9 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} + 6 q^{41} + q^{43} -6 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} -4 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + ( 2 + \zeta_{6} ) q^{63} + ( 12 - 12 \zeta_{6} ) q^{67} -8 q^{69} + 10 q^{71} + ( 1 - \zeta_{6} ) q^{73} + ( 2 - 6 \zeta_{6} ) q^{77} -7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -18 q^{83} + ( 4 - 4 \zeta_{6} ) q^{87} -10 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} + 3 \zeta_{6} q^{93} -5 q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 4q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 4q^{7} - q^{9} - 2q^{11} + 4q^{13} + 2q^{17} - 4q^{19} + q^{21} - 8q^{23} - 2q^{27} + 8q^{29} - 3q^{31} + 2q^{33} - 9q^{37} + 2q^{39} + 12q^{41} + 2q^{43} - 6q^{47} + 2q^{49} - 2q^{51} + 2q^{53} - 8q^{57} + 6q^{59} + q^{61} + 5q^{63} + 12q^{67} - 16q^{69} + 20q^{71} + q^{73} - 2q^{77} - 7q^{79} - q^{81} - 36q^{83} + 4q^{87} - 10q^{89} - 8q^{91} + 3q^{93} - 10q^{97} + 4q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1201.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0 0 −2.00000 + 1.73205i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −2.00000 1.73205i 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
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ 1
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 4 T + 29 T^{2} )^{2}$$
$31$ $$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$
$37$ $$1 + 9 T + 44 T^{2} + 333 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 10 T + 71 T^{2} )^{2}$$
$73$ $$1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4}$$
$79$ $$1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 5 T + 97 T^{2} )^{2}$$