# Properties

 Label 38.2.c.b Level $38$ Weight $2$ Character orbit 38.c Analytic conductor $0.303$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} + ( - 4 \beta_{2} - 4) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b3 + b1) * q^3 + (-b2 - 1) * q^4 + (-b3 + b2 - b1) * q^5 - b1 * q^6 + (-b3 + 1) * q^7 + q^8 + (-4*b2 - 4) * q^9 $$q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} + ( - 4 \beta_{2} - 4) q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{10} + (\beta_{3} - 2) q^{11} - \beta_{3} q^{12} + ( - 2 \beta_{2} - 2) q^{13} + (\beta_{3} + \beta_{2} + \beta_1) q^{14} + (7 \beta_{2} - \beta_1 + 7) q^{15} + \beta_{2} q^{16} + 4 q^{18} + ( - 2 \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{3} + 1) q^{20} + (\beta_{3} + 7 \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{3} + \beta_1) q^{24} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{25} + 2 q^{26} - \beta_{3} q^{27} + ( - \beta_{2} - \beta_1 - 1) q^{28} + (\beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{3} - 7) q^{30} + (\beta_{3} - 3) q^{31} + ( - \beta_{2} - 1) q^{32} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{33} - 6 \beta_{2} q^{35} + 4 \beta_{2} q^{36} + (\beta_{3} + 3) q^{37} + ( - \beta_{3} + 4 \beta_{2} + 2) q^{38} - 2 \beta_{3} q^{39} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{40} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1) q^{41} + ( - 7 \beta_{2} - \beta_1 - 7) q^{42} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{2} + \beta_1 + 2) q^{44} + (4 \beta_{3} + 4) q^{45} + (\beta_{3} + 1) q^{46} + ( - 7 \beta_{2} + \beta_1 - 7) q^{47} - \beta_1 q^{48} + ( - 2 \beta_{3} + 1) q^{49} + (2 \beta_{3} + 3) q^{50} + 2 \beta_{2} q^{52} + (2 \beta_{2} + 4 \beta_1 + 2) q^{53} + (\beta_{3} + \beta_1) q^{54} + (\beta_{3} + 5 \beta_{2} + \beta_1) q^{55} + ( - \beta_{3} + 1) q^{56} + (2 \beta_{3} + 4 \beta_1 + 7) q^{57} + ( - \beta_{3} - 1) q^{58} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} + (\beta_{3} - 7 \beta_{2} + \beta_1) q^{60} + (7 \beta_{2} - 3 \beta_1 + 7) q^{61} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{62} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{63} + q^{64} + (2 \beta_{3} + 2) q^{65} + (7 \beta_{2} + 2 \beta_1 + 7) q^{66} + (2 \beta_{2} - \beta_1 + 2) q^{67} + ( - \beta_{3} - 7) q^{69} + (6 \beta_{2} + 6) q^{70} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{71} + ( - 4 \beta_{2} - 4) q^{72} + ( - 2 \beta_{3} + 7 \beta_{2} - 2 \beta_1) q^{73} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{74} + ( - 3 \beta_{3} - 14) q^{75} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{76} + (3 \beta_{3} - 9) q^{77} + (2 \beta_{3} + 2 \beta_1) q^{78} - 4 \beta_{2} q^{79} + ( - \beta_{2} + \beta_1 - 1) q^{80} - 5 \beta_{2} q^{81} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{82} - 3 \beta_{3} q^{83} + ( - \beta_{3} + 7) q^{84} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{86} + (\beta_{3} + 7) q^{87} + (\beta_{3} - 2) q^{88} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{90} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{91} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{92} + ( - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{93} + (\beta_{3} + 7) q^{94} + ( - 3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 5) q^{95} - \beta_{3} q^{96} + (2 \beta_{3} - 9 \beta_{2} + 2 \beta_1) q^{97} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{98} + (8 \beta_{2} + 4 \beta_1 + 8) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b3 + b1) * q^3 + (-b2 - 1) * q^4 + (-b3 + b2 - b1) * q^5 - b1 * q^6 + (-b3 + 1) * q^7 + q^8 + (-4*b2 - 4) * q^9 + (-b2 + b1 - 1) * q^10 + (b3 - 2) * q^11 - b3 * q^12 + (-2*b2 - 2) * q^13 + (b3 + b2 + b1) * q^14 + (7*b2 - b1 + 7) * q^15 + b2 * q^16 + 4 * q^18 + (-2*b2 - b1 + 2) * q^19 + (b3 + 1) * q^20 + (b3 + 7*b2 + b1) * q^21 + (-b3 - 2*b2 - b1) * q^22 + (-b2 + b1 - 1) * q^23 + (b3 + b1) * q^24 + (-3*b2 + 2*b1 - 3) * q^25 + 2 * q^26 - b3 * q^27 + (-b2 - b1 - 1) * q^28 + (b2 - b1 + 1) * q^29 + (-b3 - 7) * q^30 + (b3 - 3) * q^31 + (-b2 - 1) * q^32 + (-2*b3 - 7*b2 - 2*b1) * q^33 - 6*b2 * q^35 + 4*b2 * q^36 + (b3 + 3) * q^37 + (-b3 + 4*b2 + 2) * q^38 - 2*b3 * q^39 + (-b3 + b2 - b1) * q^40 + (-2*b3 + 5*b2 - 2*b1) * q^41 + (-7*b2 - b1 - 7) * q^42 + (2*b3 + 6*b2 + 2*b1) * q^43 + (2*b2 + b1 + 2) * q^44 + (4*b3 + 4) * q^45 + (b3 + 1) * q^46 + (-7*b2 + b1 - 7) * q^47 - b1 * q^48 + (-2*b3 + 1) * q^49 + (2*b3 + 3) * q^50 + 2*b2 * q^52 + (2*b2 + 4*b1 + 2) * q^53 + (b3 + b1) * q^54 + (b3 + 5*b2 + b1) * q^55 + (-b3 + 1) * q^56 + (2*b3 + 4*b1 + 7) * q^57 + (-b3 - 1) * q^58 + (-3*b3 - 3*b1) * q^59 + (b3 - 7*b2 + b1) * q^60 + (7*b2 - 3*b1 + 7) * q^61 + (-b3 - 3*b2 - b1) * q^62 + (-4*b2 - 4*b1 - 4) * q^63 + q^64 + (2*b3 + 2) * q^65 + (7*b2 + 2*b1 + 7) * q^66 + (2*b2 - b1 + 2) * q^67 + (-b3 - 7) * q^69 + (6*b2 + 6) * q^70 + (2*b3 - 8*b2 + 2*b1) * q^71 + (-4*b2 - 4) * q^72 + (-2*b3 + 7*b2 - 2*b1) * q^73 + (-b3 + 3*b2 - b1) * q^74 + (-3*b3 - 14) * q^75 + (b3 - 2*b2 + b1 - 4) * q^76 + (3*b3 - 9) * q^77 + (2*b3 + 2*b1) * q^78 - 4*b2 * q^79 + (-b2 + b1 - 1) * q^80 - 5*b2 * q^81 + (-5*b2 + 2*b1 - 5) * q^82 - 3*b3 * q^83 + (-b3 + 7) * q^84 + (-6*b2 - 2*b1 - 6) * q^86 + (b3 + 7) * q^87 + (b3 - 2) * q^88 + (-4*b3 + 4*b2 - 4*b1) * q^90 + (-2*b2 - 2*b1 - 2) * q^91 + (-b3 + b2 - b1) * q^92 + (-3*b3 - 7*b2 - 3*b1) * q^93 + (b3 + 7) * q^94 + (-3*b3 + 4*b2 - 4*b1 - 5) * q^95 - b3 * q^96 + (2*b3 - 9*b2 + 2*b1) * q^97 + (2*b3 + b2 + 2*b1) * q^98 + (8*b2 + 4*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{8} - 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^7 + 4 * q^8 - 8 * q^9 $$4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{8} - 8 q^{9} - 2 q^{10} - 8 q^{11} - 4 q^{13} - 2 q^{14} + 14 q^{15} - 2 q^{16} + 16 q^{18} + 12 q^{19} + 4 q^{20} - 14 q^{21} + 4 q^{22} - 2 q^{23} - 6 q^{25} + 8 q^{26} - 2 q^{28} + 2 q^{29} - 28 q^{30} - 12 q^{31} - 2 q^{32} + 14 q^{33} + 12 q^{35} - 8 q^{36} + 12 q^{37} - 2 q^{40} - 10 q^{41} - 14 q^{42} - 12 q^{43} + 4 q^{44} + 16 q^{45} + 4 q^{46} - 14 q^{47} + 4 q^{49} + 12 q^{50} - 4 q^{52} + 4 q^{53} - 10 q^{55} + 4 q^{56} + 28 q^{57} - 4 q^{58} + 14 q^{60} + 14 q^{61} + 6 q^{62} - 8 q^{63} + 4 q^{64} + 8 q^{65} + 14 q^{66} + 4 q^{67} - 28 q^{69} + 12 q^{70} + 16 q^{71} - 8 q^{72} - 14 q^{73} - 6 q^{74} - 56 q^{75} - 12 q^{76} - 36 q^{77} + 8 q^{79} - 2 q^{80} + 10 q^{81} - 10 q^{82} + 28 q^{84} - 12 q^{86} + 28 q^{87} - 8 q^{88} - 8 q^{90} - 4 q^{91} - 2 q^{92} + 14 q^{93} + 28 q^{94} - 28 q^{95} + 18 q^{97} - 2 q^{98} + 16 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^7 + 4 * q^8 - 8 * q^9 - 2 * q^10 - 8 * q^11 - 4 * q^13 - 2 * q^14 + 14 * q^15 - 2 * q^16 + 16 * q^18 + 12 * q^19 + 4 * q^20 - 14 * q^21 + 4 * q^22 - 2 * q^23 - 6 * q^25 + 8 * q^26 - 2 * q^28 + 2 * q^29 - 28 * q^30 - 12 * q^31 - 2 * q^32 + 14 * q^33 + 12 * q^35 - 8 * q^36 + 12 * q^37 - 2 * q^40 - 10 * q^41 - 14 * q^42 - 12 * q^43 + 4 * q^44 + 16 * q^45 + 4 * q^46 - 14 * q^47 + 4 * q^49 + 12 * q^50 - 4 * q^52 + 4 * q^53 - 10 * q^55 + 4 * q^56 + 28 * q^57 - 4 * q^58 + 14 * q^60 + 14 * q^61 + 6 * q^62 - 8 * q^63 + 4 * q^64 + 8 * q^65 + 14 * q^66 + 4 * q^67 - 28 * q^69 + 12 * q^70 + 16 * q^71 - 8 * q^72 - 14 * q^73 - 6 * q^74 - 56 * q^75 - 12 * q^76 - 36 * q^77 + 8 * q^79 - 2 * q^80 + 10 * q^81 - 10 * q^82 + 28 * q^84 - 12 * q^86 + 28 * q^87 - 8 * q^88 - 8 * q^90 - 4 * q^91 - 2 * q^92 + 14 * q^93 + 28 * q^94 - 28 * q^95 + 18 * q^97 - 2 * q^98 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

## 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.

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
 1.32288 − 2.29129i −1.32288 + 2.29129i 1.32288 + 2.29129i −1.32288 − 2.29129i
−0.500000 0.866025i −1.32288 2.29129i −0.500000 + 0.866025i 0.822876 + 1.42526i −1.32288 + 2.29129i 3.64575 1.00000 −2.00000 + 3.46410i 0.822876 1.42526i
7.2 −0.500000 0.866025i 1.32288 + 2.29129i −0.500000 + 0.866025i −1.82288 3.15731i 1.32288 2.29129i −1.64575 1.00000 −2.00000 + 3.46410i −1.82288 + 3.15731i
11.1 −0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 0.866025i 0.822876 1.42526i −1.32288 2.29129i 3.64575 1.00000 −2.00000 3.46410i 0.822876 + 1.42526i
11.2 −0.500000 + 0.866025i 1.32288 2.29129i −0.500000 0.866025i −1.82288 + 3.15731i 1.32288 + 2.29129i −1.64575 1.00000 −2.00000 3.46410i −1.82288 3.15731i
 $$n$$: 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.b 4
3.b odd 2 1 342.2.g.f 4
4.b odd 2 1 304.2.i.e 4
5.b even 2 1 950.2.e.k 4
5.c odd 4 2 950.2.j.g 8
8.b even 2 1 1216.2.i.l 4
8.d odd 2 1 1216.2.i.k 4
12.b even 2 1 2736.2.s.v 4
19.b odd 2 1 722.2.c.j 4
19.c even 3 1 inner 38.2.c.b 4
19.c even 3 1 722.2.a.j 2
19.d odd 6 1 722.2.a.g 2
19.d odd 6 1 722.2.c.j 4
19.e even 9 6 722.2.e.n 12
19.f odd 18 6 722.2.e.o 12
57.f even 6 1 6498.2.a.bg 2
57.h odd 6 1 342.2.g.f 4
57.h odd 6 1 6498.2.a.ba 2
76.f even 6 1 5776.2.a.z 2
76.g odd 6 1 304.2.i.e 4
76.g odd 6 1 5776.2.a.ba 2
95.i even 6 1 950.2.e.k 4
95.m odd 12 2 950.2.j.g 8
152.k odd 6 1 1216.2.i.k 4
152.p even 6 1 1216.2.i.l 4
228.m even 6 1 2736.2.s.v 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$7$ $$(T^{2} - 2 T - 6)^{2}$$
$11$ $$(T^{2} + 4 T - 3)^{2}$$
$13$ $$(T^{2} + 2 T + 4)^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 12 T^{3} + 67 T^{2} + \cdots + 361$$
$23$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$29$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$31$ $$(T^{2} + 6 T + 2)^{2}$$
$37$ $$(T^{2} - 6 T + 2)^{2}$$
$41$ $$T^{4} + 10 T^{3} + 103 T^{2} - 30 T + 9$$
$43$ $$T^{4} + 12 T^{3} + 136 T^{2} + \cdots + 64$$
$47$ $$T^{4} + 14 T^{3} + 154 T^{2} + \cdots + 1764$$
$53$ $$T^{4} - 4 T^{3} + 124 T^{2} + \cdots + 11664$$
$59$ $$T^{4} + 63T^{2} + 3969$$
$61$ $$T^{4} - 14 T^{3} + 210 T^{2} + \cdots + 196$$
$67$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$71$ $$T^{4} - 16 T^{3} + 220 T^{2} + \cdots + 1296$$
$73$ $$T^{4} + 14 T^{3} + 175 T^{2} + \cdots + 441$$
$79$ $$(T^{2} - 4 T + 16)^{2}$$
$83$ $$(T^{2} - 63)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 18 T^{3} + 271 T^{2} + \cdots + 2809$$