# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.335371688489$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{6} + ( 2 - 3 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{6} + ( 2 - 3 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{9} -\zeta_{6} q^{10} -5 \zeta_{6} q^{11} + ( 1 - \zeta_{6} ) q^{12} + ( 1 + 2 \zeta_{6} ) q^{14} - q^{15} + ( -1 + \zeta_{6} ) q^{16} + 4 \zeta_{6} q^{17} -\zeta_{6} q^{18} + ( -8 + 8 \zeta_{6} ) q^{19} + q^{20} + ( 3 - \zeta_{6} ) q^{21} + 5 q^{22} + ( 4 - 4 \zeta_{6} ) q^{23} + \zeta_{6} q^{24} + 4 \zeta_{6} q^{25} - q^{27} + ( -3 + \zeta_{6} ) q^{28} -5 q^{29} + ( 1 - \zeta_{6} ) q^{30} -3 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 5 - 5 \zeta_{6} ) q^{33} -4 q^{34} + ( 1 + 2 \zeta_{6} ) q^{35} + q^{36} + ( 4 - 4 \zeta_{6} ) q^{37} -8 \zeta_{6} q^{38} + ( -1 + \zeta_{6} ) q^{40} + ( -2 + 3 \zeta_{6} ) q^{42} + 2 q^{43} + ( -5 + 5 \zeta_{6} ) q^{44} -\zeta_{6} q^{45} + 4 \zeta_{6} q^{46} + ( 6 - 6 \zeta_{6} ) q^{47} - q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} -4 q^{50} + ( -4 + 4 \zeta_{6} ) q^{51} + 9 \zeta_{6} q^{53} + ( 1 - \zeta_{6} ) q^{54} + 5 q^{55} + ( 2 - 3 \zeta_{6} ) q^{56} -8 q^{57} + ( 5 - 5 \zeta_{6} ) q^{58} + 11 \zeta_{6} q^{59} + \zeta_{6} q^{60} + ( 6 - 6 \zeta_{6} ) q^{61} + 3 q^{62} + ( 1 + 2 \zeta_{6} ) q^{63} + q^{64} + 5 \zeta_{6} q^{66} + 2 \zeta_{6} q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} + 4 q^{69} + ( -3 + \zeta_{6} ) q^{70} + 2 q^{71} + ( -1 + \zeta_{6} ) q^{72} -10 \zeta_{6} q^{73} + 4 \zeta_{6} q^{74} + ( -4 + 4 \zeta_{6} ) q^{75} + 8 q^{76} + ( -15 + 5 \zeta_{6} ) q^{77} + ( -3 + 3 \zeta_{6} ) q^{79} -\zeta_{6} q^{80} -\zeta_{6} q^{81} -7 q^{83} + ( -1 - 2 \zeta_{6} ) q^{84} -4 q^{85} + ( -2 + 2 \zeta_{6} ) q^{86} -5 \zeta_{6} q^{87} -5 \zeta_{6} q^{88} + ( 6 - 6 \zeta_{6} ) q^{89} + q^{90} -4 q^{92} + ( 3 - 3 \zeta_{6} ) q^{93} + 6 \zeta_{6} q^{94} -8 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{96} + 7 q^{97} + ( 8 - 5 \zeta_{6} ) q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} + q^{7} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} + q^{7} + 2q^{8} - q^{9} - q^{10} - 5q^{11} + q^{12} + 4q^{14} - 2q^{15} - q^{16} + 4q^{17} - q^{18} - 8q^{19} + 2q^{20} + 5q^{21} + 10q^{22} + 4q^{23} + q^{24} + 4q^{25} - 2q^{27} - 5q^{28} - 10q^{29} + q^{30} - 3q^{31} - q^{32} + 5q^{33} - 8q^{34} + 4q^{35} + 2q^{36} + 4q^{37} - 8q^{38} - q^{40} - q^{42} + 4q^{43} - 5q^{44} - q^{45} + 4q^{46} + 6q^{47} - 2q^{48} - 13q^{49} - 8q^{50} - 4q^{51} + 9q^{53} + q^{54} + 10q^{55} + q^{56} - 16q^{57} + 5q^{58} + 11q^{59} + q^{60} + 6q^{61} + 6q^{62} + 4q^{63} + 2q^{64} + 5q^{66} + 2q^{67} + 4q^{68} + 8q^{69} - 5q^{70} + 4q^{71} - q^{72} - 10q^{73} + 4q^{74} - 4q^{75} + 16q^{76} - 25q^{77} - 3q^{79} - q^{80} - q^{81} - 14q^{83} - 4q^{84} - 8q^{85} - 2q^{86} - 5q^{87} - 5q^{88} + 6q^{89} + 2q^{90} - 8q^{92} + 3q^{93} + 6q^{94} - 8q^{95} + q^{96} + 14q^{97} + 11q^{98} + 10q^{99} + O(q^{100})$$

## 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
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0.500000 + 2.59808i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
37.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0.500000 2.59808i 1.00000 −0.500000 + 0.866025i −0.500000 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
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 42.2.e.a 2
3.b odd 2 1 126.2.g.c 2
4.b odd 2 1 336.2.q.b 2
5.b even 2 1 1050.2.i.l 2
5.c odd 4 2 1050.2.o.a 4
7.b odd 2 1 294.2.e.b 2
7.c even 3 1 inner 42.2.e.a 2
7.c even 3 1 294.2.a.e 1
7.d odd 6 1 294.2.a.f 1
7.d odd 6 1 294.2.e.b 2
8.b even 2 1 1344.2.q.g 2
8.d odd 2 1 1344.2.q.s 2
9.c even 3 1 1134.2.e.l 2
9.c even 3 1 1134.2.h.e 2
9.d odd 6 1 1134.2.e.e 2
9.d odd 6 1 1134.2.h.l 2
12.b even 2 1 1008.2.s.k 2
21.c even 2 1 882.2.g.i 2
21.g even 6 1 882.2.a.d 1
21.g even 6 1 882.2.g.i 2
21.h odd 6 1 126.2.g.c 2
21.h odd 6 1 882.2.a.c 1
28.d even 2 1 2352.2.q.u 2
28.f even 6 1 2352.2.a.f 1
28.f even 6 1 2352.2.q.u 2
28.g odd 6 1 336.2.q.b 2
28.g odd 6 1 2352.2.a.t 1
35.i odd 6 1 7350.2.a.q 1
35.j even 6 1 1050.2.i.l 2
35.j even 6 1 7350.2.a.bl 1
35.l odd 12 2 1050.2.o.a 4
56.j odd 6 1 9408.2.a.z 1
56.k odd 6 1 1344.2.q.s 2
56.k odd 6 1 9408.2.a.q 1
56.m even 6 1 9408.2.a.cr 1
56.p even 6 1 1344.2.q.g 2
56.p even 6 1 9408.2.a.ce 1
63.g even 3 1 1134.2.e.l 2
63.h even 3 1 1134.2.h.e 2
63.j odd 6 1 1134.2.h.l 2
63.n odd 6 1 1134.2.e.e 2
84.j odd 6 1 7056.2.a.bl 1
84.n even 6 1 1008.2.s.k 2
84.n even 6 1 7056.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 1.a even 1 1 trivial
42.2.e.a 2 7.c even 3 1 inner
126.2.g.c 2 3.b odd 2 1
126.2.g.c 2 21.h odd 6 1
294.2.a.e 1 7.c even 3 1
294.2.a.f 1 7.d odd 6 1
294.2.e.b 2 7.b odd 2 1
294.2.e.b 2 7.d odd 6 1
336.2.q.b 2 4.b odd 2 1
336.2.q.b 2 28.g odd 6 1
882.2.a.c 1 21.h odd 6 1
882.2.a.d 1 21.g even 6 1
882.2.g.i 2 21.c even 2 1
882.2.g.i 2 21.g even 6 1
1008.2.s.k 2 12.b even 2 1
1008.2.s.k 2 84.n even 6 1
1050.2.i.l 2 5.b even 2 1
1050.2.i.l 2 35.j even 6 1
1050.2.o.a 4 5.c odd 4 2
1050.2.o.a 4 35.l odd 12 2
1134.2.e.e 2 9.d odd 6 1
1134.2.e.e 2 63.n odd 6 1
1134.2.e.l 2 9.c even 3 1
1134.2.e.l 2 63.g even 3 1
1134.2.h.e 2 9.c even 3 1
1134.2.h.e 2 63.h even 3 1
1134.2.h.l 2 9.d odd 6 1
1134.2.h.l 2 63.j odd 6 1
1344.2.q.g 2 8.b even 2 1
1344.2.q.g 2 56.p even 6 1
1344.2.q.s 2 8.d odd 2 1
1344.2.q.s 2 56.k odd 6 1
2352.2.a.f 1 28.f even 6 1
2352.2.a.t 1 28.g odd 6 1
2352.2.q.u 2 28.d even 2 1
2352.2.q.u 2 28.f even 6 1
7056.2.a.w 1 84.n even 6 1
7056.2.a.bl 1 84.j odd 6 1
7350.2.a.q 1 35.i odd 6 1
7350.2.a.bl 1 35.j even 6 1
9408.2.a.q 1 56.k odd 6 1
9408.2.a.z 1 56.j odd 6 1
9408.2.a.ce 1 56.p even 6 1
9408.2.a.cr 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$121 - 11 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$9 + 3 T + T^{2}$$
$83$ $$( 7 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$