# Properties

 Label 42.2.f.a Level $42$ Weight $2$ Character orbit 42.f Analytic conductor $0.335$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 42.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.335371688489$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 + \zeta_{12}^{2} ) q^{10} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{12} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} + 3 q^{15} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} + ( 2 + 2 \zeta_{12}^{2} ) q^{19} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{20} + ( \zeta_{12} + 4 \zeta_{12}^{3} ) q^{21} + 3 q^{22} -6 \zeta_{12} q^{23} + ( 2 - \zeta_{12}^{2} ) q^{24} + 2 \zeta_{12}^{2} q^{25} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 1 - 3 \zeta_{12}^{2} ) q^{28} -3 \zeta_{12}^{3} q^{29} + 3 \zeta_{12} q^{30} + ( -2 + \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -3 - 3 \zeta_{12}^{2} ) q^{33} + ( -2 + 4 \zeta_{12}^{2} ) q^{34} + ( 4 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{35} -3 q^{36} + ( 2 - 2 \zeta_{12}^{2} ) q^{37} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{38} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{39} + ( -1 - \zeta_{12}^{2} ) q^{40} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{41} + ( -4 + 5 \zeta_{12}^{2} ) q^{42} -8 q^{43} + 3 \zeta_{12} q^{44} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{45} -6 \zeta_{12}^{2} q^{46} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{48} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} + 2 \zeta_{12}^{3} q^{50} + ( 6 - 6 \zeta_{12}^{2} ) q^{51} + ( 4 - 2 \zeta_{12}^{2} ) q^{52} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{53} + ( 3 + 3 \zeta_{12}^{2} ) q^{54} + ( -3 + 6 \zeta_{12}^{2} ) q^{55} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{56} -6 \zeta_{12}^{3} q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{58} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + 3 \zeta_{12}^{2} q^{60} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{62} + ( 9 - 6 \zeta_{12}^{2} ) q^{63} - q^{64} + 6 \zeta_{12} q^{65} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{66} -2 \zeta_{12}^{2} q^{67} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{68} + ( -6 + 12 \zeta_{12}^{2} ) q^{69} + ( 5 - \zeta_{12}^{2} ) q^{70} + 12 \zeta_{12}^{3} q^{71} -3 \zeta_{12} q^{72} + ( 8 - 4 \zeta_{12}^{2} ) q^{73} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{74} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( -2 + 4 \zeta_{12}^{2} ) q^{76} + ( -9 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -6 q^{78} + ( 1 - \zeta_{12}^{2} ) q^{79} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( -4 - 4 \zeta_{12}^{2} ) q^{82} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} + ( -4 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{84} -6 q^{85} -8 \zeta_{12} q^{86} + ( -6 + 3 \zeta_{12}^{2} ) q^{87} + 3 \zeta_{12}^{2} q^{88} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{89} + ( 3 - 6 \zeta_{12}^{2} ) q^{90} + ( -8 + 10 \zeta_{12}^{2} ) q^{91} -6 \zeta_{12}^{3} q^{92} + 3 \zeta_{12} q^{93} + ( -8 + 4 \zeta_{12}^{2} ) q^{94} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{95} + ( 1 + \zeta_{12}^{2} ) q^{96} + ( 3 - 6 \zeta_{12}^{2} ) q^{97} + ( 3 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{98} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 10q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 10q^{7} - 6q^{9} - 6q^{10} + 12q^{15} - 2q^{16} + 12q^{19} + 12q^{22} + 6q^{24} + 4q^{25} - 2q^{28} - 6q^{31} - 18q^{33} - 12q^{36} + 4q^{37} - 6q^{40} - 6q^{42} - 32q^{43} - 12q^{46} + 22q^{49} + 12q^{51} + 12q^{52} + 18q^{54} + 6q^{58} + 6q^{60} + 24q^{63} - 4q^{64} - 4q^{67} + 18q^{70} + 24q^{73} - 24q^{78} + 2q^{79} - 18q^{81} - 24q^{82} - 24q^{85} - 18q^{87} + 6q^{88} - 12q^{91} - 24q^{94} + 6q^{96} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i −2.50000 0.866025i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
5.2 0.866025 + 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i −2.50000 0.866025i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
17.1 −0.866025 + 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i −2.50000 + 0.866025i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
17.2 0.866025 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i −2.50000 + 0.866025i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
 $$n$$: 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.2.f.a 4
3.b odd 2 1 inner 42.2.f.a 4
4.b odd 2 1 336.2.bc.e 4
5.b even 2 1 1050.2.s.b 4
5.c odd 4 1 1050.2.u.a 4
5.c odd 4 1 1050.2.u.d 4
7.b odd 2 1 294.2.f.a 4
7.c even 3 1 294.2.d.a 4
7.c even 3 1 294.2.f.a 4
7.d odd 6 1 inner 42.2.f.a 4
7.d odd 6 1 294.2.d.a 4
9.c even 3 1 1134.2.l.c 4
9.c even 3 1 1134.2.t.d 4
9.d odd 6 1 1134.2.l.c 4
9.d odd 6 1 1134.2.t.d 4
12.b even 2 1 336.2.bc.e 4
15.d odd 2 1 1050.2.s.b 4
15.e even 4 1 1050.2.u.a 4
15.e even 4 1 1050.2.u.d 4
21.c even 2 1 294.2.f.a 4
21.g even 6 1 inner 42.2.f.a 4
21.g even 6 1 294.2.d.a 4
21.h odd 6 1 294.2.d.a 4
21.h odd 6 1 294.2.f.a 4
28.f even 6 1 336.2.bc.e 4
28.f even 6 1 2352.2.k.e 4
28.g odd 6 1 2352.2.k.e 4
35.i odd 6 1 1050.2.s.b 4
35.k even 12 1 1050.2.u.a 4
35.k even 12 1 1050.2.u.d 4
63.i even 6 1 1134.2.t.d 4
63.k odd 6 1 1134.2.l.c 4
63.s even 6 1 1134.2.l.c 4
63.t odd 6 1 1134.2.t.d 4
84.j odd 6 1 336.2.bc.e 4
84.j odd 6 1 2352.2.k.e 4
84.n even 6 1 2352.2.k.e 4
105.p even 6 1 1050.2.s.b 4
105.w odd 12 1 1050.2.u.a 4
105.w odd 12 1 1050.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 1.a even 1 1 trivial
42.2.f.a 4 3.b odd 2 1 inner
42.2.f.a 4 7.d odd 6 1 inner
42.2.f.a 4 21.g even 6 1 inner
294.2.d.a 4 7.c even 3 1
294.2.d.a 4 7.d odd 6 1
294.2.d.a 4 21.g even 6 1
294.2.d.a 4 21.h odd 6 1
294.2.f.a 4 7.b odd 2 1
294.2.f.a 4 7.c even 3 1
294.2.f.a 4 21.c even 2 1
294.2.f.a 4 21.h odd 6 1
336.2.bc.e 4 4.b odd 2 1
336.2.bc.e 4 12.b even 2 1
336.2.bc.e 4 28.f even 6 1
336.2.bc.e 4 84.j odd 6 1
1050.2.s.b 4 5.b even 2 1
1050.2.s.b 4 15.d odd 2 1
1050.2.s.b 4 35.i odd 6 1
1050.2.s.b 4 105.p even 6 1
1050.2.u.a 4 5.c odd 4 1
1050.2.u.a 4 15.e even 4 1
1050.2.u.a 4 35.k even 12 1
1050.2.u.a 4 105.w odd 12 1
1050.2.u.d 4 5.c odd 4 1
1050.2.u.d 4 15.e even 4 1
1050.2.u.d 4 35.k even 12 1
1050.2.u.d 4 105.w odd 12 1
1134.2.l.c 4 9.c even 3 1
1134.2.l.c 4 9.d odd 6 1
1134.2.l.c 4 63.k odd 6 1
1134.2.l.c 4 63.s even 6 1
1134.2.t.d 4 9.c even 3 1
1134.2.t.d 4 9.d odd 6 1
1134.2.t.d 4 63.i even 6 1
1134.2.t.d 4 63.t odd 6 1
2352.2.k.e 4 28.f even 6 1
2352.2.k.e 4 28.g odd 6 1
2352.2.k.e 4 84.j odd 6 1
2352.2.k.e 4 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$9 + 3 T^{2} + T^{4}$$
$7$ $$( 7 + 5 T + T^{2} )^{2}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$144 + 12 T^{2} + T^{4}$$
$19$ $$( 12 - 6 T + T^{2} )^{2}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 9 + T^{2} )^{2}$$
$31$ $$( 3 + 3 T + T^{2} )^{2}$$
$37$ $$( 4 - 2 T + T^{2} )^{2}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$2304 + 48 T^{2} + T^{4}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$9 + 3 T^{2} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 4 + 2 T + T^{2} )^{2}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$( 48 - 12 T + T^{2} )^{2}$$
$79$ $$( 1 - T + T^{2} )^{2}$$
$83$ $$( -75 + T^{2} )^{2}$$
$89$ $$11664 + 108 T^{2} + T^{4}$$
$97$ $$( 27 + T^{2} )^{2}$$