# Properties

 Label 2100.2.q.d 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: no (minimal twist has level 420) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} -7 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + q^{27} -2 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + 4 \zeta_{6} q^{33} -7 \zeta_{6} q^{37} + ( 7 - 7 \zeta_{6} ) q^{39} -8 q^{41} -5 q^{43} + 10 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( -8 + 8 \zeta_{6} ) q^{53} + 3 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} + ( 3 - 3 \zeta_{6} ) q^{67} + 2 q^{69} + ( 15 - 15 \zeta_{6} ) q^{73} + ( 12 - 8 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -8 q^{83} + ( 2 - 2 \zeta_{6} ) q^{87} -2 \zeta_{6} q^{89} + ( -14 - 7 \zeta_{6} ) q^{91} -7 \zeta_{6} q^{93} + 10 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + 5q^{7} - q^{9} + 4q^{11} - 14q^{13} - 6q^{17} - 3q^{19} - 4q^{21} - 2q^{23} + 2q^{27} - 4q^{29} - 7q^{31} + 4q^{33} - 7q^{37} + 7q^{39} - 16q^{41} - 10q^{43} + 10q^{47} + 11q^{49} - 6q^{51} - 8q^{53} + 6q^{57} - 10q^{59} + 6q^{61} - q^{63} + 3q^{67} + 4q^{69} + 15q^{73} + 16q^{77} - q^{79} - q^{81} - 16q^{83} + 2q^{87} - 2q^{89} - 35q^{91} - 7q^{93} + 20q^{97} - 8q^{99} + O(q^{100})$$

## 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 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
1801.1 0 −0.500000 0.866025i 0 0 0 2.50000 0.866025i 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.d 2
5.b even 2 1 420.2.q.b 2
5.c odd 4 2 2100.2.bc.d 4
7.c even 3 1 inner 2100.2.q.d 2
15.d odd 2 1 1260.2.s.a 2
20.d odd 2 1 1680.2.bg.j 2
35.c odd 2 1 2940.2.q.c 2
35.i odd 6 1 2940.2.a.k 1
35.i odd 6 1 2940.2.q.c 2
35.j even 6 1 420.2.q.b 2
35.j even 6 1 2940.2.a.b 1
35.l odd 12 2 2100.2.bc.d 4
105.o odd 6 1 1260.2.s.a 2
105.o odd 6 1 8820.2.a.bb 1
105.p even 6 1 8820.2.a.l 1
140.p odd 6 1 1680.2.bg.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.b 2 5.b even 2 1
420.2.q.b 2 35.j even 6 1
1260.2.s.a 2 15.d odd 2 1
1260.2.s.a 2 105.o odd 6 1
1680.2.bg.j 2 20.d odd 2 1
1680.2.bg.j 2 140.p odd 6 1
2100.2.q.d 2 1.a even 1 1 trivial
2100.2.q.d 2 7.c even 3 1 inner
2100.2.bc.d 4 5.c odd 4 2
2100.2.bc.d 4 35.l odd 12 2
2940.2.a.b 1 35.j even 6 1
2940.2.a.k 1 35.i odd 6 1
2940.2.q.c 2 35.c odd 2 1
2940.2.q.c 2 35.i odd 6 1
8820.2.a.l 1 105.p even 6 1
8820.2.a.bb 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T + T^{2}$$
$5$ 1
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 7 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 8 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 5 T + 43 T^{2} )^{2}$$
$47$ $$1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4}$$
$53$ $$1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4}$$
$59$ $$1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 15 T + 152 T^{2} - 1095 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 8 T + 83 T^{2} )^{2}$$
$89$ $$1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 10 T + 97 T^{2} )^{2}$$