# Properties

 Label 38.3.b.a Level $38$ Weight $3$ Character orbit 38.b Analytic conductor $1.035$ 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] = [38,3,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 38.b (of order $$2$$, degree $$1$$, 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: $$1.03542500457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 \beta q^{3} - 2 q^{4} - q^{5} - 4 q^{6} + 5 q^{7} - 2 \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 + 2*b * q^3 - 2 * q^4 - q^5 - 4 * q^6 + 5 * q^7 - 2*b * q^8 + q^9 $$q + \beta q^{2} + 2 \beta q^{3} - 2 q^{4} - q^{5} - 4 q^{6} + 5 q^{7} - 2 \beta q^{8} + q^{9} - \beta q^{10} + 5 q^{11} - 4 \beta q^{12} - 12 \beta q^{13} + 5 \beta q^{14} - 2 \beta q^{15} + 4 q^{16} - 25 q^{17} + \beta q^{18} + 19 q^{19} + 2 q^{20} + 10 \beta q^{21} + 5 \beta q^{22} - 10 q^{23} + 8 q^{24} - 24 q^{25} + 24 q^{26} + 20 \beta q^{27} - 10 q^{28} + 30 \beta q^{29} + 4 q^{30} - 30 \beta q^{31} + 4 \beta q^{32} + 10 \beta q^{33} - 25 \beta q^{34} - 5 q^{35} - 2 q^{36} - 18 \beta q^{37} + 19 \beta q^{38} + 48 q^{39} + 2 \beta q^{40} - 30 \beta q^{41} - 20 q^{42} + 5 q^{43} - 10 q^{44} - q^{45} - 10 \beta q^{46} + 5 q^{47} + 8 \beta q^{48} - 24 q^{49} - 24 \beta q^{50} - 50 \beta q^{51} + 24 \beta q^{52} - 18 \beta q^{53} - 40 q^{54} - 5 q^{55} - 10 \beta q^{56} + 38 \beta q^{57} - 60 q^{58} + 60 \beta q^{59} + 4 \beta q^{60} + 95 q^{61} + 60 q^{62} + 5 q^{63} - 8 q^{64} + 12 \beta q^{65} - 20 q^{66} + 78 \beta q^{67} + 50 q^{68} - 20 \beta q^{69} - 5 \beta q^{70} - 2 \beta q^{72} - 25 q^{73} + 36 q^{74} - 48 \beta q^{75} - 38 q^{76} + 25 q^{77} + 48 \beta q^{78} + 30 \beta q^{79} - 4 q^{80} - 71 q^{81} + 60 q^{82} - 130 q^{83} - 20 \beta q^{84} + 25 q^{85} + 5 \beta q^{86} - 120 q^{87} - 10 \beta q^{88} - 90 \beta q^{89} - \beta q^{90} - 60 \beta q^{91} + 20 q^{92} + 120 q^{93} + 5 \beta q^{94} - 19 q^{95} - 16 q^{96} + 12 \beta q^{97} - 24 \beta q^{98} + 5 q^{99} +O(q^{100})$$ q + b * q^2 + 2*b * q^3 - 2 * q^4 - q^5 - 4 * q^6 + 5 * q^7 - 2*b * q^8 + q^9 - b * q^10 + 5 * q^11 - 4*b * q^12 - 12*b * q^13 + 5*b * q^14 - 2*b * q^15 + 4 * q^16 - 25 * q^17 + b * q^18 + 19 * q^19 + 2 * q^20 + 10*b * q^21 + 5*b * q^22 - 10 * q^23 + 8 * q^24 - 24 * q^25 + 24 * q^26 + 20*b * q^27 - 10 * q^28 + 30*b * q^29 + 4 * q^30 - 30*b * q^31 + 4*b * q^32 + 10*b * q^33 - 25*b * q^34 - 5 * q^35 - 2 * q^36 - 18*b * q^37 + 19*b * q^38 + 48 * q^39 + 2*b * q^40 - 30*b * q^41 - 20 * q^42 + 5 * q^43 - 10 * q^44 - q^45 - 10*b * q^46 + 5 * q^47 + 8*b * q^48 - 24 * q^49 - 24*b * q^50 - 50*b * q^51 + 24*b * q^52 - 18*b * q^53 - 40 * q^54 - 5 * q^55 - 10*b * q^56 + 38*b * q^57 - 60 * q^58 + 60*b * q^59 + 4*b * q^60 + 95 * q^61 + 60 * q^62 + 5 * q^63 - 8 * q^64 + 12*b * q^65 - 20 * q^66 + 78*b * q^67 + 50 * q^68 - 20*b * q^69 - 5*b * q^70 - 2*b * q^72 - 25 * q^73 + 36 * q^74 - 48*b * q^75 - 38 * q^76 + 25 * q^77 + 48*b * q^78 + 30*b * q^79 - 4 * q^80 - 71 * q^81 + 60 * q^82 - 130 * q^83 - 20*b * q^84 + 25 * q^85 + 5*b * q^86 - 120 * q^87 - 10*b * q^88 - 90*b * q^89 - b * q^90 - 60*b * q^91 + 20 * q^92 + 120 * q^93 + 5*b * q^94 - 19 * q^95 - 16 * q^96 + 12*b * q^97 - 24*b * q^98 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9 $$2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9} + 10 q^{11} + 8 q^{16} - 50 q^{17} + 38 q^{19} + 4 q^{20} - 20 q^{23} + 16 q^{24} - 48 q^{25} + 48 q^{26} - 20 q^{28} + 8 q^{30} - 10 q^{35} - 4 q^{36} + 96 q^{39} - 40 q^{42} + 10 q^{43} - 20 q^{44} - 2 q^{45} + 10 q^{47} - 48 q^{49} - 80 q^{54} - 10 q^{55} - 120 q^{58} + 190 q^{61} + 120 q^{62} + 10 q^{63} - 16 q^{64} - 40 q^{66} + 100 q^{68} - 50 q^{73} + 72 q^{74} - 76 q^{76} + 50 q^{77} - 8 q^{80} - 142 q^{81} + 120 q^{82} - 260 q^{83} + 50 q^{85} - 240 q^{87} + 40 q^{92} + 240 q^{93} - 38 q^{95} - 32 q^{96} + 10 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9 + 10 * q^11 + 8 * q^16 - 50 * q^17 + 38 * q^19 + 4 * q^20 - 20 * q^23 + 16 * q^24 - 48 * q^25 + 48 * q^26 - 20 * q^28 + 8 * q^30 - 10 * q^35 - 4 * q^36 + 96 * q^39 - 40 * q^42 + 10 * q^43 - 20 * q^44 - 2 * q^45 + 10 * q^47 - 48 * q^49 - 80 * q^54 - 10 * q^55 - 120 * q^58 + 190 * q^61 + 120 * q^62 + 10 * q^63 - 16 * q^64 - 40 * q^66 + 100 * q^68 - 50 * q^73 + 72 * q^74 - 76 * q^76 + 50 * q^77 - 8 * q^80 - 142 * q^81 + 120 * q^82 - 260 * q^83 + 50 * q^85 - 240 * q^87 + 40 * q^92 + 240 * q^93 - 38 * q^95 - 32 * q^96 + 10 * q^99

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 1.41421i 1.41421i
1.41421i 2.82843i −2.00000 −1.00000 −4.00000 5.00000 2.82843i 1.00000 1.41421i
37.2 1.41421i 2.82843i −2.00000 −1.00000 −4.00000 5.00000 2.82843i 1.00000 1.41421i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.3.b.a 2
3.b odd 2 1 342.3.d.a 2
4.b odd 2 1 304.3.e.c 2
5.b even 2 1 950.3.c.a 2
5.c odd 4 2 950.3.d.a 4
8.b even 2 1 1216.3.e.j 2
8.d odd 2 1 1216.3.e.i 2
12.b even 2 1 2736.3.o.h 2
19.b odd 2 1 inner 38.3.b.a 2
57.d even 2 1 342.3.d.a 2
76.d even 2 1 304.3.e.c 2
95.d odd 2 1 950.3.c.a 2
95.g even 4 2 950.3.d.a 4
152.b even 2 1 1216.3.e.i 2
152.g odd 2 1 1216.3.e.j 2
228.b odd 2 1 2736.3.o.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 1.a even 1 1 trivial
38.3.b.a 2 19.b odd 2 1 inner
304.3.e.c 2 4.b odd 2 1
304.3.e.c 2 76.d even 2 1
342.3.d.a 2 3.b odd 2 1
342.3.d.a 2 57.d even 2 1
950.3.c.a 2 5.b even 2 1
950.3.c.a 2 95.d odd 2 1
950.3.d.a 4 5.c odd 4 2
950.3.d.a 4 95.g even 4 2
1216.3.e.i 2 8.d odd 2 1
1216.3.e.i 2 152.b even 2 1
1216.3.e.j 2 8.b even 2 1
1216.3.e.j 2 152.g odd 2 1
2736.3.o.h 2 12.b even 2 1
2736.3.o.h 2 228.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2} + 8$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 5)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 288$$
$17$ $$(T + 25)^{2}$$
$19$ $$(T - 19)^{2}$$
$23$ $$(T + 10)^{2}$$
$29$ $$T^{2} + 1800$$
$31$ $$T^{2} + 1800$$
$37$ $$T^{2} + 648$$
$41$ $$T^{2} + 1800$$
$43$ $$(T - 5)^{2}$$
$47$ $$(T - 5)^{2}$$
$53$ $$T^{2} + 648$$
$59$ $$T^{2} + 7200$$
$61$ $$(T - 95)^{2}$$
$67$ $$T^{2} + 12168$$
$71$ $$T^{2}$$
$73$ $$(T + 25)^{2}$$
$79$ $$T^{2} + 1800$$
$83$ $$(T + 130)^{2}$$
$89$ $$T^{2} + 16200$$
$97$ $$T^{2} + 288$$