# Properties

 Label 126.4.a.a Level $126$ Weight $4$ Character orbit 126.a Self dual yes Analytic conductor $7.434$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 4q^{4} - 18q^{5} + 7q^{7} - 8q^{8} + O(q^{10})$$ $$q - 2q^{2} + 4q^{4} - 18q^{5} + 7q^{7} - 8q^{8} + 36q^{10} + 72q^{11} - 34q^{13} - 14q^{14} + 16q^{16} - 6q^{17} + 92q^{19} - 72q^{20} - 144q^{22} + 180q^{23} + 199q^{25} + 68q^{26} + 28q^{28} + 114q^{29} + 56q^{31} - 32q^{32} + 12q^{34} - 126q^{35} - 34q^{37} - 184q^{38} + 144q^{40} - 6q^{41} + 164q^{43} + 288q^{44} - 360q^{46} - 168q^{47} + 49q^{49} - 398q^{50} - 136q^{52} - 654q^{53} - 1296q^{55} - 56q^{56} - 228q^{58} + 492q^{59} - 250q^{61} - 112q^{62} + 64q^{64} + 612q^{65} - 124q^{67} - 24q^{68} + 252q^{70} - 36q^{71} + 1010q^{73} + 68q^{74} + 368q^{76} + 504q^{77} + 56q^{79} - 288q^{80} + 12q^{82} - 228q^{83} + 108q^{85} - 328q^{86} - 576q^{88} - 390q^{89} - 238q^{91} + 720q^{92} + 336q^{94} - 1656q^{95} - 70q^{97} - 98q^{98} + O(q^{100})$$

## 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 −18.0000 0 7.00000 −8.00000 0 36.0000
 $$n$$: 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.a 1
3.b odd 2 1 42.4.a.a 1
4.b odd 2 1 1008.4.a.b 1
7.b odd 2 1 882.4.a.g 1
7.c even 3 2 882.4.g.w 2
7.d odd 6 2 882.4.g.o 2
12.b even 2 1 336.4.a.l 1
15.d odd 2 1 1050.4.a.g 1
15.e even 4 2 1050.4.g.a 2
21.c even 2 1 294.4.a.i 1
21.g even 6 2 294.4.e.b 2
21.h odd 6 2 294.4.e.c 2
24.f even 2 1 1344.4.a.a 1
24.h odd 2 1 1344.4.a.o 1
84.h odd 2 1 2352.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 3.b odd 2 1
126.4.a.a 1 1.a even 1 1 trivial
294.4.a.i 1 21.c even 2 1
294.4.e.b 2 21.g even 6 2
294.4.e.c 2 21.h odd 6 2
336.4.a.l 1 12.b even 2 1
882.4.a.g 1 7.b odd 2 1
882.4.g.o 2 7.d odd 6 2
882.4.g.w 2 7.c even 3 2
1008.4.a.b 1 4.b odd 2 1
1050.4.a.g 1 15.d odd 2 1
1050.4.g.a 2 15.e even 4 2
1344.4.a.a 1 24.f even 2 1
1344.4.a.o 1 24.h odd 2 1
2352.4.a.a 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(126))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$18 + T$$
$7$ $$-7 + T$$
$11$ $$-72 + T$$
$13$ $$34 + T$$
$17$ $$6 + T$$
$19$ $$-92 + T$$
$23$ $$-180 + T$$
$29$ $$-114 + T$$
$31$ $$-56 + T$$
$37$ $$34 + T$$
$41$ $$6 + T$$
$43$ $$-164 + T$$
$47$ $$168 + T$$
$53$ $$654 + T$$
$59$ $$-492 + T$$
$61$ $$250 + T$$
$67$ $$124 + T$$
$71$ $$36 + T$$
$73$ $$-1010 + T$$
$79$ $$-56 + T$$
$83$ $$228 + T$$
$89$ $$390 + T$$
$97$ $$70 + T$$