# Properties

 Label 126.4.a.f Level $126$ Weight $4$ Character orbit 126.a Self dual yes Analytic conductor $7.434$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.43424066072$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + 8q^{8} + O(q^{10})$$ $$q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + 8q^{8} - 44q^{10} - 26q^{11} - 54q^{13} - 14q^{14} + 16q^{16} - 74q^{17} + 116q^{19} - 88q^{20} - 52q^{22} + 58q^{23} + 359q^{25} - 108q^{26} - 28q^{28} - 208q^{29} - 252q^{31} + 32q^{32} - 148q^{34} + 154q^{35} + 50q^{37} + 232q^{38} - 176q^{40} - 126q^{41} + 164q^{43} - 104q^{44} + 116q^{46} + 444q^{47} + 49q^{49} + 718q^{50} - 216q^{52} - 12q^{53} + 572q^{55} - 56q^{56} - 416q^{58} - 124q^{59} - 162q^{61} - 504q^{62} + 64q^{64} + 1188q^{65} - 860q^{67} - 296q^{68} + 308q^{70} + 238q^{71} - 146q^{73} + 100q^{74} + 464q^{76} + 182q^{77} - 984q^{79} - 352q^{80} - 252q^{82} - 656q^{83} + 1628q^{85} + 328q^{86} - 208q^{88} + 954q^{89} + 378q^{91} + 232q^{92} + 888q^{94} - 2552q^{95} + 526q^{97} + 98q^{98} + O(q^{100})$$

## 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
2.00000 0 4.00000 −22.0000 0 −7.00000 8.00000 0 −44.0000
 $$n$$: 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 126.4.a.f yes 1
3.b odd 2 1 126.4.a.e 1
4.b odd 2 1 1008.4.a.a 1
7.b odd 2 1 882.4.a.s 1
7.c even 3 2 882.4.g.m 2
7.d odd 6 2 882.4.g.a 2
12.b even 2 1 1008.4.a.w 1
21.c even 2 1 882.4.a.a 1
21.g even 6 2 882.4.g.x 2
21.h odd 6 2 882.4.g.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 3.b odd 2 1
126.4.a.f yes 1 1.a even 1 1 trivial
882.4.a.a 1 21.c even 2 1
882.4.a.s 1 7.b odd 2 1
882.4.g.a 2 7.d odd 6 2
882.4.g.m 2 7.c even 3 2
882.4.g.n 2 21.h odd 6 2
882.4.g.x 2 21.g even 6 2
1008.4.a.a 1 4.b odd 2 1
1008.4.a.w 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 22$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(126))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$22 + T$$
$7$ $$7 + T$$
$11$ $$26 + T$$
$13$ $$54 + T$$
$17$ $$74 + T$$
$19$ $$-116 + T$$
$23$ $$-58 + T$$
$29$ $$208 + T$$
$31$ $$252 + T$$
$37$ $$-50 + T$$
$41$ $$126 + T$$
$43$ $$-164 + T$$
$47$ $$-444 + T$$
$53$ $$12 + T$$
$59$ $$124 + T$$
$61$ $$162 + T$$
$67$ $$860 + T$$
$71$ $$-238 + T$$
$73$ $$146 + T$$
$79$ $$984 + T$$
$83$ $$656 + T$$
$89$ $$-954 + T$$
$97$ $$-526 + T$$