# Properties

 Label 38.2.c.a Level $38$ Weight $2$ Character orbit 38.c Analytic conductor $0.303$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.303431527681$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -4 q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -4 q^{7} - q^{8} + 2 \zeta_{6} q^{9} + 3 q^{11} + q^{12} -2 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 2 q^{18} + ( -2 - 3 \zeta_{6} ) q^{19} + ( 4 - 4 \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{22} + 6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{25} -2 q^{26} -5 q^{27} + 4 \zeta_{6} q^{28} + 2 q^{31} + \zeta_{6} q^{32} + ( -3 + 3 \zeta_{6} ) q^{33} -6 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} -10 q^{37} + ( -5 + 2 \zeta_{6} ) q^{38} + 2 q^{39} + ( -9 + 9 \zeta_{6} ) q^{41} -4 \zeta_{6} q^{42} + ( 4 - 4 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{44} + 6 q^{46} -\zeta_{6} q^{48} + 9 q^{49} + 5 q^{50} + 6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} -6 \zeta_{6} q^{53} + ( -5 + 5 \zeta_{6} ) q^{54} + 4 q^{56} + ( 5 - 2 \zeta_{6} ) q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{62} -8 \zeta_{6} q^{63} + q^{64} + 3 \zeta_{6} q^{66} + 7 \zeta_{6} q^{67} -6 q^{68} -6 q^{69} + ( 6 - 6 \zeta_{6} ) q^{71} -2 \zeta_{6} q^{72} + ( 1 - \zeta_{6} ) q^{73} + ( -10 + 10 \zeta_{6} ) q^{74} -5 q^{75} + ( -3 + 5 \zeta_{6} ) q^{76} -12 q^{77} + ( 2 - 2 \zeta_{6} ) q^{78} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 9 \zeta_{6} q^{82} + 3 q^{83} -4 q^{84} -4 \zeta_{6} q^{86} -3 q^{88} -6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + ( -2 + 2 \zeta_{6} ) q^{93} - q^{96} + ( -17 + 17 \zeta_{6} ) q^{97} + ( 9 - 9 \zeta_{6} ) q^{98} + 6 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} - q^{4} + q^{6} - 8q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} - q^{4} + q^{6} - 8q^{7} - 2q^{8} + 2q^{9} + 6q^{11} + 2q^{12} - 2q^{13} - 4q^{14} - q^{16} + 6q^{17} + 4q^{18} - 7q^{19} + 4q^{21} + 3q^{22} + 6q^{23} + q^{24} + 5q^{25} - 4q^{26} - 10q^{27} + 4q^{28} + 4q^{31} + q^{32} - 3q^{33} - 6q^{34} + 2q^{36} - 20q^{37} - 8q^{38} + 4q^{39} - 9q^{41} - 4q^{42} + 4q^{43} - 3q^{44} + 12q^{46} - q^{48} + 18q^{49} + 10q^{50} + 6q^{51} - 2q^{52} - 6q^{53} - 5q^{54} + 8q^{56} + 8q^{57} + 9q^{59} + 4q^{61} + 2q^{62} - 8q^{63} + 2q^{64} + 3q^{66} + 7q^{67} - 12q^{68} - 12q^{69} + 6q^{71} - 2q^{72} + q^{73} - 10q^{74} - 10q^{75} - q^{76} - 24q^{77} + 2q^{78} + 4q^{79} - q^{81} + 9q^{82} + 6q^{83} - 8q^{84} - 4q^{86} - 6q^{88} - 6q^{89} + 8q^{91} + 6q^{92} - 2q^{93} - 2q^{96} - 17q^{97} + 9q^{98} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −4.00000 −1.00000 1.00000 1.73205i 0
11.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −4.00000 −1.00000 1.00000 + 1.73205i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.2.c.a 2
3.b odd 2 1 342.2.g.b 2
4.b odd 2 1 304.2.i.c 2
5.b even 2 1 950.2.e.d 2
5.c odd 4 2 950.2.j.e 4
8.b even 2 1 1216.2.i.h 2
8.d odd 2 1 1216.2.i.d 2
12.b even 2 1 2736.2.s.m 2
19.b odd 2 1 722.2.c.b 2
19.c even 3 1 inner 38.2.c.a 2
19.c even 3 1 722.2.a.c 1
19.d odd 6 1 722.2.a.d 1
19.d odd 6 1 722.2.c.b 2
19.e even 9 6 722.2.e.j 6
19.f odd 18 6 722.2.e.i 6
57.f even 6 1 6498.2.a.e 1
57.h odd 6 1 342.2.g.b 2
57.h odd 6 1 6498.2.a.s 1
76.f even 6 1 5776.2.a.n 1
76.g odd 6 1 304.2.i.c 2
76.g odd 6 1 5776.2.a.g 1
95.i even 6 1 950.2.e.d 2
95.m odd 12 2 950.2.j.e 4
152.k odd 6 1 1216.2.i.d 2
152.p even 6 1 1216.2.i.h 2
228.m even 6 1 2736.2.s.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 1.a even 1 1 trivial
38.2.c.a 2 19.c even 3 1 inner
304.2.i.c 2 4.b odd 2 1
304.2.i.c 2 76.g odd 6 1
342.2.g.b 2 3.b odd 2 1
342.2.g.b 2 57.h odd 6 1
722.2.a.c 1 19.c even 3 1
722.2.a.d 1 19.d odd 6 1
722.2.c.b 2 19.b odd 2 1
722.2.c.b 2 19.d odd 6 1
722.2.e.i 6 19.f odd 18 6
722.2.e.j 6 19.e even 9 6
950.2.e.d 2 5.b even 2 1
950.2.e.d 2 95.i even 6 1
950.2.j.e 4 5.c odd 4 2
950.2.j.e 4 95.m odd 12 2
1216.2.i.d 2 8.d odd 2 1
1216.2.i.d 2 152.k odd 6 1
1216.2.i.h 2 8.b even 2 1
1216.2.i.h 2 152.p even 6 1
2736.2.s.m 2 12.b even 2 1
2736.2.s.m 2 228.m even 6 1
5776.2.a.g 1 76.g odd 6 1
5776.2.a.n 1 76.f even 6 1
6498.2.a.e 1 57.f even 6 1
6498.2.a.s 1 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$( 1 + 4 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 7 T + 19 T^{2}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 29 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 10 T + 37 T^{2} )^{2}$$
$41$ $$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$
$43$ $$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$1 - 6 T - 35 T^{2} - 426 T^{3} + 5041 T^{4}$$
$73$ $$1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 - 3 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4}$$