Properties

 Label 42.2.d.a Level $42$ Weight $2$ Character orbit 42.d 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.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.335371688489$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{8} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{8} + 3 \beta_{2} q^{9} + ( \beta_{1} + \beta_{3} ) q^{10} -\beta_{1} q^{12} + ( \beta_{1} + \beta_{3} ) q^{13} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{14} + ( -3 - 3 \beta_{2} ) q^{15} + q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} + 3 q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{3} ) q^{20} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{21} -6 \beta_{2} q^{23} + \beta_{3} q^{24} + q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} + 3 \beta_{3} q^{27} + ( 1 + \beta_{1} + \beta_{3} ) q^{28} + 6 \beta_{2} q^{29} + ( -3 + 3 \beta_{2} ) q^{30} -\beta_{2} q^{32} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{34} + ( \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{35} -3 \beta_{2} q^{36} -2 q^{37} + ( -\beta_{1} + \beta_{3} ) q^{38} + ( -3 + 3 \beta_{2} ) q^{39} + ( -\beta_{1} - \beta_{3} ) q^{40} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -3 - 3 \beta_{2} + \beta_{3} ) q^{42} + 4 q^{43} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{45} -6 q^{46} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -5 + 2 \beta_{1} + 2 \beta_{3} ) q^{49} -\beta_{2} q^{50} + ( 6 + 6 \beta_{2} ) q^{51} + ( -\beta_{1} - \beta_{3} ) q^{52} -6 \beta_{2} q^{53} + 3 \beta_{1} q^{54} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{56} + ( 3 - 3 \beta_{2} ) q^{57} + 6 q^{58} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{2} ) q^{60} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{63} - q^{64} -6 \beta_{2} q^{65} + 8 q^{67} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{68} -6 \beta_{3} q^{69} + ( 6 - \beta_{1} - \beta_{3} ) q^{70} -3 q^{72} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{73} + 2 \beta_{2} q^{74} + \beta_{1} q^{75} + ( \beta_{1} + \beta_{3} ) q^{76} + ( 3 + 3 \beta_{2} ) q^{78} -10 q^{79} + ( -\beta_{1} + \beta_{3} ) q^{80} -9 q^{81} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{82} + ( \beta_{1} - \beta_{3} ) q^{83} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{84} -12 q^{85} -4 \beta_{2} q^{86} + 6 \beta_{3} q^{87} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{90} + ( 6 - \beta_{1} - \beta_{3} ) q^{91} + 6 \beta_{2} q^{92} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{94} + 6 \beta_{2} q^{95} -\beta_{3} q^{96} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{7} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{7} - 12q^{15} + 4q^{16} + 12q^{18} + 12q^{21} + 4q^{25} + 4q^{28} - 12q^{30} - 8q^{37} - 12q^{39} - 12q^{42} + 16q^{43} - 24q^{46} - 20q^{49} + 24q^{51} + 12q^{57} + 24q^{58} + 12q^{60} - 4q^{64} + 32q^{67} + 24q^{70} - 12q^{72} + 12q^{78} - 40q^{79} - 36q^{81} - 12q^{84} - 48q^{85} + 24q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

Character values

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

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000i −1.22474 1.22474i −1.00000 2.44949 −1.22474 + 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
41.2 1.00000i 1.22474 + 1.22474i −1.00000 −2.44949 1.22474 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
41.3 1.00000i −1.22474 + 1.22474i −1.00000 2.44949 −1.22474 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
41.4 1.00000i 1.22474 1.22474i −1.00000 −2.44949 1.22474 + 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
 $$n$$: 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.2.d.a 4
3.b odd 2 1 inner 42.2.d.a 4
4.b odd 2 1 336.2.k.b 4
5.b even 2 1 1050.2.b.b 4
5.c odd 4 1 1050.2.d.b 4
5.c odd 4 1 1050.2.d.e 4
7.b odd 2 1 inner 42.2.d.a 4
7.c even 3 2 294.2.f.b 8
7.d odd 6 2 294.2.f.b 8
8.b even 2 1 1344.2.k.c 4
8.d odd 2 1 1344.2.k.d 4
9.c even 3 2 1134.2.m.g 8
9.d odd 6 2 1134.2.m.g 8
12.b even 2 1 336.2.k.b 4
15.d odd 2 1 1050.2.b.b 4
15.e even 4 1 1050.2.d.b 4
15.e even 4 1 1050.2.d.e 4
21.c even 2 1 inner 42.2.d.a 4
21.g even 6 2 294.2.f.b 8
21.h odd 6 2 294.2.f.b 8
24.f even 2 1 1344.2.k.d 4
24.h odd 2 1 1344.2.k.c 4
28.d even 2 1 336.2.k.b 4
35.c odd 2 1 1050.2.b.b 4
35.f even 4 1 1050.2.d.b 4
35.f even 4 1 1050.2.d.e 4
56.e even 2 1 1344.2.k.d 4
56.h odd 2 1 1344.2.k.c 4
63.l odd 6 2 1134.2.m.g 8
63.o even 6 2 1134.2.m.g 8
84.h odd 2 1 336.2.k.b 4
105.g even 2 1 1050.2.b.b 4
105.k odd 4 1 1050.2.d.b 4
105.k odd 4 1 1050.2.d.e 4
168.e odd 2 1 1344.2.k.d 4
168.i even 2 1 1344.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 1.a even 1 1 trivial
42.2.d.a 4 3.b odd 2 1 inner
42.2.d.a 4 7.b odd 2 1 inner
42.2.d.a 4 21.c even 2 1 inner
294.2.f.b 8 7.c even 3 2
294.2.f.b 8 7.d odd 6 2
294.2.f.b 8 21.g even 6 2
294.2.f.b 8 21.h odd 6 2
336.2.k.b 4 4.b odd 2 1
336.2.k.b 4 12.b even 2 1
336.2.k.b 4 28.d even 2 1
336.2.k.b 4 84.h odd 2 1
1050.2.b.b 4 5.b even 2 1
1050.2.b.b 4 15.d odd 2 1
1050.2.b.b 4 35.c odd 2 1
1050.2.b.b 4 105.g even 2 1
1050.2.d.b 4 5.c odd 4 1
1050.2.d.b 4 15.e even 4 1
1050.2.d.b 4 35.f even 4 1
1050.2.d.b 4 105.k odd 4 1
1050.2.d.e 4 5.c odd 4 1
1050.2.d.e 4 15.e even 4 1
1050.2.d.e 4 35.f even 4 1
1050.2.d.e 4 105.k odd 4 1
1134.2.m.g 8 9.c even 3 2
1134.2.m.g 8 9.d odd 6 2
1134.2.m.g 8 63.l odd 6 2
1134.2.m.g 8 63.o even 6 2
1344.2.k.c 4 8.b even 2 1
1344.2.k.c 4 24.h odd 2 1
1344.2.k.c 4 56.h odd 2 1
1344.2.k.c 4 168.i even 2 1
1344.2.k.d 4 8.d odd 2 1
1344.2.k.d 4 24.f even 2 1
1344.2.k.d 4 56.e even 2 1
1344.2.k.d 4 168.e odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$9 + T^{4}$$
$5$ $$( -6 + T^{2} )^{2}$$
$7$ $$( 7 + 2 T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 6 + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 2 + T )^{4}$$
$41$ $$( -24 + T^{2} )^{2}$$
$43$ $$( -4 + T )^{4}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( -150 + T^{2} )^{2}$$
$61$ $$( 150 + T^{2} )^{2}$$
$67$ $$( -8 + T )^{4}$$
$71$ $$T^{4}$$
$73$ $$( 96 + T^{2} )^{2}$$
$79$ $$( 10 + T )^{4}$$
$83$ $$( -6 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 24 + T^{2} )^{2}$$