# Properties

 Label 42.4.a.b 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

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} + 6q^{6} - 7q^{7} + 8q^{8} + 9q^{9} + O(q^{10})$$ $$q + 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} + 6q^{6} - 7q^{7} + 8q^{8} + 9q^{9} + 4q^{10} - 8q^{11} + 12q^{12} - 42q^{13} - 14q^{14} + 6q^{15} + 16q^{16} - 2q^{17} + 18q^{18} - 124q^{19} + 8q^{20} - 21q^{21} - 16q^{22} + 76q^{23} + 24q^{24} - 121q^{25} - 84q^{26} + 27q^{27} - 28q^{28} + 254q^{29} + 12q^{30} - 72q^{31} + 32q^{32} - 24q^{33} - 4q^{34} - 14q^{35} + 36q^{36} + 398q^{37} - 248q^{38} - 126q^{39} + 16q^{40} + 462q^{41} - 42q^{42} + 212q^{43} - 32q^{44} + 18q^{45} + 152q^{46} - 264q^{47} + 48q^{48} + 49q^{49} - 242q^{50} - 6q^{51} - 168q^{52} - 162q^{53} + 54q^{54} - 16q^{55} - 56q^{56} - 372q^{57} + 508q^{58} - 772q^{59} + 24q^{60} + 30q^{61} - 144q^{62} - 63q^{63} + 64q^{64} - 84q^{65} - 48q^{66} - 764q^{67} - 8q^{68} + 228q^{69} - 28q^{70} - 236q^{71} + 72q^{72} + 418q^{73} + 796q^{74} - 363q^{75} - 496q^{76} + 56q^{77} - 252q^{78} + 552q^{79} + 32q^{80} + 81q^{81} + 924q^{82} + 1036q^{83} - 84q^{84} - 4q^{85} + 424q^{86} + 762q^{87} - 64q^{88} + 30q^{89} + 36q^{90} + 294q^{91} + 304q^{92} - 216q^{93} - 528q^{94} - 248q^{95} + 96q^{96} - 1190q^{97} + 98q^{98} - 72q^{99} + O(q^{100})$$

## 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 2.00000 6.00000 −7.00000 8.00000 9.00000 4.00000
 $$n$$: 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.b 1
3.b odd 2 1 126.4.a.c 1
4.b odd 2 1 336.4.a.d 1
5.b even 2 1 1050.4.a.d 1
5.c odd 4 2 1050.4.g.n 2
7.b odd 2 1 294.4.a.h 1
7.c even 3 2 294.4.e.a 2
7.d odd 6 2 294.4.e.d 2
8.b even 2 1 1344.4.a.f 1
8.d odd 2 1 1344.4.a.t 1
12.b even 2 1 1008.4.a.j 1
21.c even 2 1 882.4.a.d 1
21.g even 6 2 882.4.g.r 2
21.h odd 6 2 882.4.g.s 2
28.d even 2 1 2352.4.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 1.a even 1 1 trivial
126.4.a.c 1 3.b odd 2 1
294.4.a.h 1 7.b odd 2 1
294.4.e.a 2 7.c even 3 2
294.4.e.d 2 7.d odd 6 2
336.4.a.d 1 4.b odd 2 1
882.4.a.d 1 21.c even 2 1
882.4.g.r 2 21.g even 6 2
882.4.g.s 2 21.h odd 6 2
1008.4.a.j 1 12.b even 2 1
1050.4.a.d 1 5.b even 2 1
1050.4.g.n 2 5.c odd 4 2
1344.4.a.f 1 8.b even 2 1
1344.4.a.t 1 8.d odd 2 1
2352.4.a.ba 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(42))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$-3 + T$$
$5$ $$-2 + T$$
$7$ $$7 + T$$
$11$ $$8 + T$$
$13$ $$42 + T$$
$17$ $$2 + T$$
$19$ $$124 + T$$
$23$ $$-76 + T$$
$29$ $$-254 + T$$
$31$ $$72 + T$$
$37$ $$-398 + T$$
$41$ $$-462 + T$$
$43$ $$-212 + T$$
$47$ $$264 + T$$
$53$ $$162 + T$$
$59$ $$772 + T$$
$61$ $$-30 + T$$
$67$ $$764 + T$$
$71$ $$236 + T$$
$73$ $$-418 + T$$
$79$ $$-552 + T$$
$83$ $$-1036 + T$$
$89$ $$-30 + T$$
$97$ $$1190 + T$$