# Properties

 Label 2100.4.a.l Level $2100$ Weight $4$ Character orbit 2100.a Self dual yes Analytic conductor $123.904$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2100.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$123.904011012$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} - 7q^{7} + 9q^{9} + 36q^{11} - 62q^{13} - 114q^{17} - 76q^{19} - 21q^{21} + 24q^{23} + 27q^{27} + 54q^{29} - 112q^{31} + 108q^{33} + 178q^{37} - 186q^{39} + 378q^{41} + 172q^{43} + 192q^{47} + 49q^{49} - 342q^{51} + 402q^{53} - 228q^{57} + 396q^{59} + 254q^{61} - 63q^{63} + 1012q^{67} + 72q^{69} + 840q^{71} - 890q^{73} - 252q^{77} + 80q^{79} + 81q^{81} + 108q^{83} + 162q^{87} - 1638q^{89} + 434q^{91} - 336q^{93} - 1010q^{97} + 324q^{99} + O(q^{100})$$

## 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 3.00000 0 0 0 −7.00000 0 9.00000 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 2100.4.a.l 1
5.b even 2 1 84.4.a.a 1
5.c odd 4 2 2100.4.k.j 2
15.d odd 2 1 252.4.a.b 1
20.d odd 2 1 336.4.a.k 1
35.c odd 2 1 588.4.a.d 1
35.i odd 6 2 588.4.i.c 2
35.j even 6 2 588.4.i.f 2
40.e odd 2 1 1344.4.a.d 1
40.f even 2 1 1344.4.a.q 1
60.h even 2 1 1008.4.a.h 1
105.g even 2 1 1764.4.a.j 1
105.o odd 6 2 1764.4.k.l 2
105.p even 6 2 1764.4.k.f 2
140.c even 2 1 2352.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 5.b even 2 1
252.4.a.b 1 15.d odd 2 1
336.4.a.k 1 20.d odd 2 1
588.4.a.d 1 35.c odd 2 1
588.4.i.c 2 35.i odd 6 2
588.4.i.f 2 35.j even 6 2
1008.4.a.h 1 60.h even 2 1
1344.4.a.d 1 40.e odd 2 1
1344.4.a.q 1 40.f even 2 1
1764.4.a.j 1 105.g even 2 1
1764.4.k.f 2 105.p even 6 2
1764.4.k.l 2 105.o odd 6 2
2100.4.a.l 1 1.a even 1 1 trivial
2100.4.k.j 2 5.c odd 4 2
2352.4.a.d 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2100))$$:

 $$T_{11} - 36$$ $$T_{13} + 62$$ $$T_{17} + 114$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$7 + T$$
$11$ $$-36 + T$$
$13$ $$62 + T$$
$17$ $$114 + T$$
$19$ $$76 + T$$
$23$ $$-24 + T$$
$29$ $$-54 + T$$
$31$ $$112 + T$$
$37$ $$-178 + T$$
$41$ $$-378 + T$$
$43$ $$-172 + T$$
$47$ $$-192 + T$$
$53$ $$-402 + T$$
$59$ $$-396 + T$$
$61$ $$-254 + T$$
$67$ $$-1012 + T$$
$71$ $$-840 + T$$
$73$ $$890 + T$$
$79$ $$-80 + T$$
$83$ $$-108 + T$$
$89$ $$1638 + T$$
$97$ $$1010 + T$$