# Properties

 Label 126.4.a.d 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 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 4q^{4} + 12q^{5} + 7q^{7} - 8q^{8} + O(q^{10})$$ $$q - 2q^{2} + 4q^{4} + 12q^{5} + 7q^{7} - 8q^{8} - 24q^{10} - 48q^{11} + 56q^{13} - 14q^{14} + 16q^{16} + 114q^{17} + 2q^{19} + 48q^{20} + 96q^{22} + 120q^{23} + 19q^{25} - 112q^{26} + 28q^{28} + 54q^{29} + 236q^{31} - 32q^{32} - 228q^{34} + 84q^{35} + 146q^{37} - 4q^{38} - 96q^{40} - 126q^{41} - 376q^{43} - 192q^{44} - 240q^{46} + 12q^{47} + 49q^{49} - 38q^{50} + 224q^{52} - 174q^{53} - 576q^{55} - 56q^{56} - 108q^{58} - 138q^{59} + 380q^{61} - 472q^{62} + 64q^{64} + 672q^{65} - 484q^{67} + 456q^{68} - 168q^{70} - 576q^{71} - 1150q^{73} - 292q^{74} + 8q^{76} - 336q^{77} + 776q^{79} + 192q^{80} + 252q^{82} - 378q^{83} + 1368q^{85} + 752q^{86} + 384q^{88} + 390q^{89} + 392q^{91} + 480q^{92} - 24q^{94} + 24q^{95} - 1330q^{97} - 98q^{98} + O(q^{100})$$

## 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 12.0000 0 7.00000 −8.00000 0 −24.0000
 $$n$$: 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.d 1
3.b odd 2 1 14.4.a.b 1
4.b odd 2 1 1008.4.a.r 1
7.b odd 2 1 882.4.a.b 1
7.c even 3 2 882.4.g.p 2
7.d odd 6 2 882.4.g.v 2
12.b even 2 1 112.4.a.e 1
15.d odd 2 1 350.4.a.f 1
15.e even 4 2 350.4.c.g 2
21.c even 2 1 98.4.a.e 1
21.g even 6 2 98.4.c.b 2
21.h odd 6 2 98.4.c.c 2
24.f even 2 1 448.4.a.g 1
24.h odd 2 1 448.4.a.k 1
33.d even 2 1 1694.4.a.b 1
39.d odd 2 1 2366.4.a.c 1
84.h odd 2 1 784.4.a.h 1
105.g even 2 1 2450.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 3.b odd 2 1
98.4.a.e 1 21.c even 2 1
98.4.c.b 2 21.g even 6 2
98.4.c.c 2 21.h odd 6 2
112.4.a.e 1 12.b even 2 1
126.4.a.d 1 1.a even 1 1 trivial
350.4.a.f 1 15.d odd 2 1
350.4.c.g 2 15.e even 4 2
448.4.a.g 1 24.f even 2 1
448.4.a.k 1 24.h odd 2 1
784.4.a.h 1 84.h odd 2 1
882.4.a.b 1 7.b odd 2 1
882.4.g.p 2 7.c even 3 2
882.4.g.v 2 7.d odd 6 2
1008.4.a.r 1 4.b odd 2 1
1694.4.a.b 1 33.d even 2 1
2366.4.a.c 1 39.d odd 2 1
2450.4.a.i 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 12$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(126))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$-12 + T$$
$7$ $$-7 + T$$
$11$ $$48 + T$$
$13$ $$-56 + T$$
$17$ $$-114 + T$$
$19$ $$-2 + T$$
$23$ $$-120 + T$$
$29$ $$-54 + T$$
$31$ $$-236 + T$$
$37$ $$-146 + T$$
$41$ $$126 + T$$
$43$ $$376 + T$$
$47$ $$-12 + T$$
$53$ $$174 + T$$
$59$ $$138 + T$$
$61$ $$-380 + T$$
$67$ $$484 + T$$
$71$ $$576 + T$$
$73$ $$1150 + T$$
$79$ $$-776 + T$$
$83$ $$378 + T$$
$89$ $$-390 + T$$
$97$ $$1330 + T$$