# Properties

 Label 42.4.e.a Level $42$ Weight $4$ Character orbit 42.e Analytic conductor $2.478$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 42.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.47808022024$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} -3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( 6 - 6 \zeta_{6} ) q^{5} -6 q^{6} + ( 7 - 21 \zeta_{6} ) q^{7} -8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} -3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( 6 - 6 \zeta_{6} ) q^{5} -6 q^{6} + ( 7 - 21 \zeta_{6} ) q^{7} -8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} -12 \zeta_{6} q^{10} + 30 \zeta_{6} q^{11} + ( -12 + 12 \zeta_{6} ) q^{12} + 53 q^{13} + ( -28 - 14 \zeta_{6} ) q^{14} -18 q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} + 84 \zeta_{6} q^{17} + 18 \zeta_{6} q^{18} + ( 97 - 97 \zeta_{6} ) q^{19} -24 q^{20} + ( -63 + 42 \zeta_{6} ) q^{21} + 60 q^{22} + ( -84 + 84 \zeta_{6} ) q^{23} + 24 \zeta_{6} q^{24} + 89 \zeta_{6} q^{25} + ( 106 - 106 \zeta_{6} ) q^{26} + 27 q^{27} + ( -84 + 56 \zeta_{6} ) q^{28} -180 q^{29} + ( -36 + 36 \zeta_{6} ) q^{30} -179 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( 90 - 90 \zeta_{6} ) q^{33} + 168 q^{34} + ( -84 - 42 \zeta_{6} ) q^{35} + 36 q^{36} + ( 145 - 145 \zeta_{6} ) q^{37} -194 \zeta_{6} q^{38} -159 \zeta_{6} q^{39} + ( -48 + 48 \zeta_{6} ) q^{40} + 126 q^{41} + ( -42 + 126 \zeta_{6} ) q^{42} -325 q^{43} + ( 120 - 120 \zeta_{6} ) q^{44} + 54 \zeta_{6} q^{45} + 168 \zeta_{6} q^{46} + ( 366 - 366 \zeta_{6} ) q^{47} + 48 q^{48} + ( -392 + 147 \zeta_{6} ) q^{49} + 178 q^{50} + ( 252 - 252 \zeta_{6} ) q^{51} -212 \zeta_{6} q^{52} + 768 \zeta_{6} q^{53} + ( 54 - 54 \zeta_{6} ) q^{54} + 180 q^{55} + ( -56 + 168 \zeta_{6} ) q^{56} -291 q^{57} + ( -360 + 360 \zeta_{6} ) q^{58} + 264 \zeta_{6} q^{59} + 72 \zeta_{6} q^{60} + ( -818 + 818 \zeta_{6} ) q^{61} -358 q^{62} + ( 126 + 63 \zeta_{6} ) q^{63} + 64 q^{64} + ( 318 - 318 \zeta_{6} ) q^{65} -180 \zeta_{6} q^{66} + 523 \zeta_{6} q^{67} + ( 336 - 336 \zeta_{6} ) q^{68} + 252 q^{69} + ( -252 + 168 \zeta_{6} ) q^{70} -342 q^{71} + ( 72 - 72 \zeta_{6} ) q^{72} + 43 \zeta_{6} q^{73} -290 \zeta_{6} q^{74} + ( 267 - 267 \zeta_{6} ) q^{75} -388 q^{76} + ( 630 - 420 \zeta_{6} ) q^{77} -318 q^{78} + ( 1171 - 1171 \zeta_{6} ) q^{79} + 96 \zeta_{6} q^{80} -81 \zeta_{6} q^{81} + ( 252 - 252 \zeta_{6} ) q^{82} -810 q^{83} + ( 168 + 84 \zeta_{6} ) q^{84} + 504 q^{85} + ( -650 + 650 \zeta_{6} ) q^{86} + 540 \zeta_{6} q^{87} -240 \zeta_{6} q^{88} + ( 600 - 600 \zeta_{6} ) q^{89} + 108 q^{90} + ( 371 - 1113 \zeta_{6} ) q^{91} + 336 q^{92} + ( -537 + 537 \zeta_{6} ) q^{93} -732 \zeta_{6} q^{94} -582 \zeta_{6} q^{95} + ( 96 - 96 \zeta_{6} ) q^{96} + 386 q^{97} + ( -490 + 784 \zeta_{6} ) q^{98} -270 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} + 6q^{5} - 12q^{6} - 7q^{7} - 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} + 6q^{5} - 12q^{6} - 7q^{7} - 16q^{8} - 9q^{9} - 12q^{10} + 30q^{11} - 12q^{12} + 106q^{13} - 70q^{14} - 36q^{15} - 16q^{16} + 84q^{17} + 18q^{18} + 97q^{19} - 48q^{20} - 84q^{21} + 120q^{22} - 84q^{23} + 24q^{24} + 89q^{25} + 106q^{26} + 54q^{27} - 112q^{28} - 360q^{29} - 36q^{30} - 179q^{31} + 32q^{32} + 90q^{33} + 336q^{34} - 210q^{35} + 72q^{36} + 145q^{37} - 194q^{38} - 159q^{39} - 48q^{40} + 252q^{41} + 42q^{42} - 650q^{43} + 120q^{44} + 54q^{45} + 168q^{46} + 366q^{47} + 96q^{48} - 637q^{49} + 356q^{50} + 252q^{51} - 212q^{52} + 768q^{53} + 54q^{54} + 360q^{55} + 56q^{56} - 582q^{57} - 360q^{58} + 264q^{59} + 72q^{60} - 818q^{61} - 716q^{62} + 315q^{63} + 128q^{64} + 318q^{65} - 180q^{66} + 523q^{67} + 336q^{68} + 504q^{69} - 336q^{70} - 684q^{71} + 72q^{72} + 43q^{73} - 290q^{74} + 267q^{75} - 776q^{76} + 840q^{77} - 636q^{78} + 1171q^{79} + 96q^{80} - 81q^{81} + 252q^{82} - 1620q^{83} + 420q^{84} + 1008q^{85} - 650q^{86} + 540q^{87} - 240q^{88} + 600q^{89} + 216q^{90} - 371q^{91} + 672q^{92} - 537q^{93} - 732q^{94} - 582q^{95} + 96q^{96} + 772q^{97} - 196q^{98} - 540q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 3.00000 + 5.19615i −6.00000 −3.50000 + 18.1865i −8.00000 −4.50000 7.79423i −6.00000 + 10.3923i
37.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i 3.00000 5.19615i −6.00000 −3.50000 18.1865i −8.00000 −4.50000 + 7.79423i −6.00000 10.3923i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.4.e.a 2
3.b odd 2 1 126.4.g.b 2
4.b odd 2 1 336.4.q.f 2
7.b odd 2 1 294.4.e.i 2
7.c even 3 1 inner 42.4.e.a 2
7.c even 3 1 294.4.a.d 1
7.d odd 6 1 294.4.a.c 1
7.d odd 6 1 294.4.e.i 2
21.c even 2 1 882.4.g.g 2
21.g even 6 1 882.4.a.l 1
21.g even 6 1 882.4.g.g 2
21.h odd 6 1 126.4.g.b 2
21.h odd 6 1 882.4.a.o 1
28.f even 6 1 2352.4.a.bf 1
28.g odd 6 1 336.4.q.f 2
28.g odd 6 1 2352.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 1.a even 1 1 trivial
42.4.e.a 2 7.c even 3 1 inner
126.4.g.b 2 3.b odd 2 1
126.4.g.b 2 21.h odd 6 1
294.4.a.c 1 7.d odd 6 1
294.4.a.d 1 7.c even 3 1
294.4.e.i 2 7.b odd 2 1
294.4.e.i 2 7.d odd 6 1
336.4.q.f 2 4.b odd 2 1
336.4.q.f 2 28.g odd 6 1
882.4.a.l 1 21.g even 6 1
882.4.a.o 1 21.h odd 6 1
882.4.g.g 2 21.c even 2 1
882.4.g.g 2 21.g even 6 1
2352.4.a.f 1 28.g odd 6 1
2352.4.a.bf 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6 T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$36 - 6 T + T^{2}$$
$7$ $$343 + 7 T + T^{2}$$
$11$ $$900 - 30 T + T^{2}$$
$13$ $$( -53 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$9409 - 97 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( 180 + T )^{2}$$
$31$ $$32041 + 179 T + T^{2}$$
$37$ $$21025 - 145 T + T^{2}$$
$41$ $$( -126 + T )^{2}$$
$43$ $$( 325 + T )^{2}$$
$47$ $$133956 - 366 T + T^{2}$$
$53$ $$589824 - 768 T + T^{2}$$
$59$ $$69696 - 264 T + T^{2}$$
$61$ $$669124 + 818 T + T^{2}$$
$67$ $$273529 - 523 T + T^{2}$$
$71$ $$( 342 + T )^{2}$$
$73$ $$1849 - 43 T + T^{2}$$
$79$ $$1371241 - 1171 T + T^{2}$$
$83$ $$( 810 + T )^{2}$$
$89$ $$360000 - 600 T + T^{2}$$
$97$ $$( -386 + T )^{2}$$