# Properties

 Label 84.4.a.a Level $84$ Weight $4$ Character orbit 84.a Self dual yes Analytic conductor $4.956$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 84.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.95616044048$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 6q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 6q^{5} + 7q^{7} + 9q^{9} + 36q^{11} + 62q^{13} - 18q^{15} + 114q^{17} - 76q^{19} - 21q^{21} - 24q^{23} - 89q^{25} - 27q^{27} + 54q^{29} - 112q^{31} - 108q^{33} + 42q^{35} - 178q^{37} - 186q^{39} + 378q^{41} - 172q^{43} + 54q^{45} - 192q^{47} + 49q^{49} - 342q^{51} - 402q^{53} + 216q^{55} + 228q^{57} + 396q^{59} + 254q^{61} + 63q^{63} + 372q^{65} - 1012q^{67} + 72q^{69} + 840q^{71} + 890q^{73} + 267q^{75} + 252q^{77} + 80q^{79} + 81q^{81} - 108q^{83} + 684q^{85} - 162q^{87} - 1638q^{89} + 434q^{91} + 336q^{93} - 456q^{95} + 1010q^{97} + 324q^{99} + O(q^{100})$$

## 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 6.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 84.4.a.a 1
3.b odd 2 1 252.4.a.b 1
4.b odd 2 1 336.4.a.k 1
5.b even 2 1 2100.4.a.l 1
5.c odd 4 2 2100.4.k.j 2
7.b odd 2 1 588.4.a.d 1
7.c even 3 2 588.4.i.f 2
7.d odd 6 2 588.4.i.c 2
8.b even 2 1 1344.4.a.q 1
8.d odd 2 1 1344.4.a.d 1
12.b even 2 1 1008.4.a.h 1
21.c even 2 1 1764.4.a.j 1
21.g even 6 2 1764.4.k.f 2
21.h odd 6 2 1764.4.k.l 2
28.d 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 1.a even 1 1 trivial
252.4.a.b 1 3.b odd 2 1
336.4.a.k 1 4.b odd 2 1
588.4.a.d 1 7.b odd 2 1
588.4.i.c 2 7.d odd 6 2
588.4.i.f 2 7.c even 3 2
1008.4.a.h 1 12.b even 2 1
1344.4.a.d 1 8.d odd 2 1
1344.4.a.q 1 8.b even 2 1
1764.4.a.j 1 21.c even 2 1
1764.4.k.f 2 21.g even 6 2
1764.4.k.l 2 21.h odd 6 2
2100.4.a.l 1 5.b even 2 1
2100.4.k.j 2 5.c odd 4 2
2352.4.a.d 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 6$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(84))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-6 + T$$
$7$ $$-7 + T$$
$11$ $$-36 + T$$
$13$ $$-62 + T$$
$17$ $$-114 + T$$
$19$ $$76 + T$$
$23$ $$24 + T$$
$29$ $$-54 + T$$
$31$ $$112 + T$$
$37$ $$178 + T$$
$41$ $$-378 + T$$
$43$ $$172 + T$$
$47$ $$192 + T$$
$53$ $$402 + T$$
$59$ $$-396 + T$$
$61$ $$-254 + T$$
$67$ $$1012 + T$$
$71$ $$-840 + T$$
$73$ $$-890 + T$$
$79$ $$-80 + T$$
$83$ $$108 + T$$
$89$ $$1638 + T$$
$97$ $$-1010 + T$$