# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,2,Mod(7,38)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("38.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.303431527681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} - T + 1$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} + 17T + 289$$