# Properties

 Label 42.4.a.a Level $42$ Weight $4$ Character orbit 42.a Self dual yes Analytic conductor $2.478$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 42.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.47808022024$$ 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

 $$f(q)$$ $$=$$ $$q + 2q^{2} - 3q^{3} + 4q^{4} + 18q^{5} - 6q^{6} + 7q^{7} + 8q^{8} + 9q^{9} + O(q^{10})$$ $$q + 2q^{2} - 3q^{3} + 4q^{4} + 18q^{5} - 6q^{6} + 7q^{7} + 8q^{8} + 9q^{9} + 36q^{10} - 72q^{11} - 12q^{12} - 34q^{13} + 14q^{14} - 54q^{15} + 16q^{16} + 6q^{17} + 18q^{18} + 92q^{19} + 72q^{20} - 21q^{21} - 144q^{22} - 180q^{23} - 24q^{24} + 199q^{25} - 68q^{26} - 27q^{27} + 28q^{28} - 114q^{29} - 108q^{30} + 56q^{31} + 32q^{32} + 216q^{33} + 12q^{34} + 126q^{35} + 36q^{36} - 34q^{37} + 184q^{38} + 102q^{39} + 144q^{40} + 6q^{41} - 42q^{42} + 164q^{43} - 288q^{44} + 162q^{45} - 360q^{46} + 168q^{47} - 48q^{48} + 49q^{49} + 398q^{50} - 18q^{51} - 136q^{52} + 654q^{53} - 54q^{54} - 1296q^{55} + 56q^{56} - 276q^{57} - 228q^{58} - 492q^{59} - 216q^{60} - 250q^{61} + 112q^{62} + 63q^{63} + 64q^{64} - 612q^{65} + 432q^{66} - 124q^{67} + 24q^{68} + 540q^{69} + 252q^{70} + 36q^{71} + 72q^{72} + 1010q^{73} - 68q^{74} - 597q^{75} + 368q^{76} - 504q^{77} + 204q^{78} + 56q^{79} + 288q^{80} + 81q^{81} + 12q^{82} + 228q^{83} - 84q^{84} + 108q^{85} + 328q^{86} + 342q^{87} - 576q^{88} + 390q^{89} + 324q^{90} - 238q^{91} - 720q^{92} - 168q^{93} + 336q^{94} + 1656q^{95} - 96q^{96} - 70q^{97} + 98q^{98} - 648q^{99} + O(q^{100})$$

## 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
2.00000 −3.00000 4.00000 18.0000 −6.00000 7.00000 8.00000 9.00000 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 42.4.a.a 1
3.b odd 2 1 126.4.a.a 1
4.b odd 2 1 336.4.a.l 1
5.b even 2 1 1050.4.a.g 1
5.c odd 4 2 1050.4.g.a 2
7.b odd 2 1 294.4.a.i 1
7.c even 3 2 294.4.e.c 2
7.d odd 6 2 294.4.e.b 2
8.b even 2 1 1344.4.a.o 1
8.d odd 2 1 1344.4.a.a 1
12.b even 2 1 1008.4.a.b 1
21.c even 2 1 882.4.a.g 1
21.g even 6 2 882.4.g.o 2
21.h odd 6 2 882.4.g.w 2
28.d even 2 1 2352.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 1.a even 1 1 trivial
126.4.a.a 1 3.b odd 2 1
294.4.a.i 1 7.b odd 2 1
294.4.e.b 2 7.d odd 6 2
294.4.e.c 2 7.c even 3 2
336.4.a.l 1 4.b odd 2 1
882.4.a.g 1 21.c even 2 1
882.4.g.o 2 21.g even 6 2
882.4.g.w 2 21.h odd 6 2
1008.4.a.b 1 12.b even 2 1
1050.4.a.g 1 5.b even 2 1
1050.4.g.a 2 5.c odd 4 2
1344.4.a.a 1 8.d odd 2 1
1344.4.a.o 1 8.b even 2 1
2352.4.a.a 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(42))$$.

## Hecke characteristic polynomials

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