# Properties

 Label 2100.2.a.d Level $2100$ Weight $2$ Character orbit 2100.a Self dual yes Analytic conductor $16.769$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{7} + q^{9} + 6q^{11} + 4q^{13} - 6q^{17} + 2q^{19} + q^{21} - q^{27} + 6q^{29} - 10q^{31} - 6q^{33} - 2q^{37} - 4q^{39} - 6q^{41} + 4q^{43} + q^{49} + 6q^{51} + 12q^{53} - 2q^{57} + 14q^{61} - q^{63} + 4q^{67} + 6q^{71} + 4q^{73} - 6q^{77} - 16q^{79} + q^{81} + 12q^{83} - 6q^{87} + 6q^{89} - 4q^{91} + 10q^{93} + 16q^{97} + 6q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 0 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.a.d 1
3.b odd 2 1 6300.2.a.a 1
4.b odd 2 1 8400.2.a.cj 1
5.b even 2 1 420.2.a.c 1
5.c odd 4 2 2100.2.k.j 2
15.d odd 2 1 1260.2.a.i 1
15.e even 4 2 6300.2.k.a 2
20.d odd 2 1 1680.2.a.a 1
35.c odd 2 1 2940.2.a.f 1
35.i odd 6 2 2940.2.q.i 2
35.j even 6 2 2940.2.q.e 2
40.e odd 2 1 6720.2.a.ch 1
40.f even 2 1 6720.2.a.x 1
60.h even 2 1 5040.2.a.bc 1
105.g even 2 1 8820.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 5.b even 2 1
1260.2.a.i 1 15.d odd 2 1
1680.2.a.a 1 20.d odd 2 1
2100.2.a.d 1 1.a even 1 1 trivial
2100.2.k.j 2 5.c odd 4 2
2940.2.a.f 1 35.c odd 2 1
2940.2.q.e 2 35.j even 6 2
2940.2.q.i 2 35.i odd 6 2
5040.2.a.bc 1 60.h even 2 1
6300.2.a.a 1 3.b odd 2 1
6300.2.k.a 2 15.e even 4 2
6720.2.a.x 1 40.f even 2 1
6720.2.a.ch 1 40.e odd 2 1
8400.2.a.cj 1 4.b odd 2 1
8820.2.a.b 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2100))$$:

 $$T_{11} - 6$$ $$T_{13} - 4$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-6 + T$$
$13$ $$-4 + T$$
$17$ $$6 + T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$10 + T$$
$37$ $$2 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$-12 + T$$
$59$ $$T$$
$61$ $$-14 + T$$
$67$ $$-4 + T$$
$71$ $$-6 + T$$
$73$ $$-4 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$-16 + T$$