# Properties

 Label 380.2.k.a Level $380$ Weight $2$ Character orbit 380.k Analytic conductor $3.034$ 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] = [380,2,Mod(267,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("380.267");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.k (of order $$4$$, 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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{2} + ( - i - 1) q^{3} - 2 i q^{4} + ( - i - 2) q^{5} + 2 q^{6} + (2 i - 2) q^{7} + (2 i + 2) q^{8} - i q^{9} +O(q^{10})$$ q + (i - 1) * q^2 + (-i - 1) * q^3 - 2*i * q^4 + (-i - 2) * q^5 + 2 * q^6 + (2*i - 2) * q^7 + (2*i + 2) * q^8 - i * q^9 $$q + (i - 1) q^{2} + ( - i - 1) q^{3} - 2 i q^{4} + ( - i - 2) q^{5} + 2 q^{6} + (2 i - 2) q^{7} + (2 i + 2) q^{8} - i q^{9} + ( - i + 3) q^{10} + (2 i - 2) q^{12} - 4 i q^{14} + (3 i + 1) q^{15} - 4 q^{16} + (5 i + 5) q^{17} + (i + 1) q^{18} - q^{19} + (4 i - 2) q^{20} + 4 q^{21} + (4 i + 4) q^{23} - 4 i q^{24} + (4 i + 3) q^{25} + (4 i - 4) q^{27} + (4 i + 4) q^{28} + 6 i q^{29} + ( - 2 i - 4) q^{30} + ( - 4 i + 4) q^{32} - 10 q^{34} + ( - 2 i + 6) q^{35} - 2 q^{36} + ( - i + 1) q^{38} + ( - 6 i - 2) q^{40} + 2 q^{41} + (4 i - 4) q^{42} + ( - 6 i - 6) q^{43} + (2 i - 1) q^{45} - 8 q^{46} + (2 i - 2) q^{47} + (4 i + 4) q^{48} - i q^{49} + ( - i - 7) q^{50} - 10 i q^{51} + (10 i - 10) q^{53} - 8 i q^{54} - 8 q^{56} + (i + 1) q^{57} + ( - 6 i - 6) q^{58} + 10 q^{59} + ( - 2 i + 6) q^{60} + 2 q^{61} + (2 i + 2) q^{63} + 8 i q^{64} + ( - 3 i + 3) q^{67} + ( - 10 i + 10) q^{68} - 8 i q^{69} + (8 i - 4) q^{70} + ( - 2 i + 2) q^{72} + ( - 5 i + 5) q^{73} + ( - 7 i + 1) q^{75} + 2 i q^{76} - 10 q^{79} + (4 i + 8) q^{80} + 5 q^{81} + (2 i - 2) q^{82} + (4 i + 4) q^{83} - 8 i q^{84} + ( - 15 i - 5) q^{85} + 12 q^{86} + ( - 6 i + 6) q^{87} + 6 i q^{89} + ( - 3 i - 1) q^{90} + ( - 8 i + 8) q^{92} - 4 i q^{94} + (i + 2) q^{95} - 8 q^{96} + ( - 10 i - 10) q^{97} + (i + 1) q^{98} +O(q^{100})$$ q + (i - 1) * q^2 + (-i - 1) * q^3 - 2*i * q^4 + (-i - 2) * q^5 + 2 * q^6 + (2*i - 2) * q^7 + (2*i + 2) * q^8 - i * q^9 + (-i + 3) * q^10 + (2*i - 2) * q^12 - 4*i * q^14 + (3*i + 1) * q^15 - 4 * q^16 + (5*i + 5) * q^17 + (i + 1) * q^18 - q^19 + (4*i - 2) * q^20 + 4 * q^21 + (4*i + 4) * q^23 - 4*i * q^24 + (4*i + 3) * q^25 + (4*i - 4) * q^27 + (4*i + 4) * q^28 + 6*i * q^29 + (-2*i - 4) * q^30 + (-4*i + 4) * q^32 - 10 * q^34 + (-2*i + 6) * q^35 - 2 * q^36 + (-i + 1) * q^38 + (-6*i - 2) * q^40 + 2 * q^41 + (4*i - 4) * q^42 + (-6*i - 6) * q^43 + (2*i - 1) * q^45 - 8 * q^46 + (2*i - 2) * q^47 + (4*i + 4) * q^48 - i * q^49 + (-i - 7) * q^50 - 10*i * q^51 + (10*i - 10) * q^53 - 8*i * q^54 - 8 * q^56 + (i + 1) * q^57 + (-6*i - 6) * q^58 + 10 * q^59 + (-2*i + 6) * q^60 + 2 * q^61 + (2*i + 2) * q^63 + 8*i * q^64 + (-3*i + 3) * q^67 + (-10*i + 10) * q^68 - 8*i * q^69 + (8*i - 4) * q^70 + (-2*i + 2) * q^72 + (-5*i + 5) * q^73 + (-7*i + 1) * q^75 + 2*i * q^76 - 10 * q^79 + (4*i + 8) * q^80 + 5 * q^81 + (2*i - 2) * q^82 + (4*i + 4) * q^83 - 8*i * q^84 + (-15*i - 5) * q^85 + 12 * q^86 + (-6*i + 6) * q^87 + 6*i * q^89 + (-3*i - 1) * q^90 + (-8*i + 8) * q^92 - 4*i * q^94 + (i + 2) * q^95 - 8 * q^96 + (-10*i - 10) * q^97 + (i + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 $$2 q - 2 q^{2} - 2 q^{3} - 4 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{8} + 6 q^{10} - 4 q^{12} + 2 q^{15} - 8 q^{16} + 10 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{20} + 8 q^{21} + 8 q^{23} + 6 q^{25} - 8 q^{27} + 8 q^{28} - 8 q^{30} + 8 q^{32} - 20 q^{34} + 12 q^{35} - 4 q^{36} + 2 q^{38} - 4 q^{40} + 4 q^{41} - 8 q^{42} - 12 q^{43} - 2 q^{45} - 16 q^{46} - 4 q^{47} + 8 q^{48} - 14 q^{50} - 20 q^{53} - 16 q^{56} + 2 q^{57} - 12 q^{58} + 20 q^{59} + 12 q^{60} + 4 q^{61} + 4 q^{63} + 6 q^{67} + 20 q^{68} - 8 q^{70} + 4 q^{72} + 10 q^{73} + 2 q^{75} - 20 q^{79} + 16 q^{80} + 10 q^{81} - 4 q^{82} + 8 q^{83} - 10 q^{85} + 24 q^{86} + 12 q^{87} - 2 q^{90} + 16 q^{92} + 4 q^{95} - 16 q^{96} - 20 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 + 4 * q^6 - 4 * q^7 + 4 * q^8 + 6 * q^10 - 4 * q^12 + 2 * q^15 - 8 * q^16 + 10 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^20 + 8 * q^21 + 8 * q^23 + 6 * q^25 - 8 * q^27 + 8 * q^28 - 8 * q^30 + 8 * q^32 - 20 * q^34 + 12 * q^35 - 4 * q^36 + 2 * q^38 - 4 * q^40 + 4 * q^41 - 8 * q^42 - 12 * q^43 - 2 * q^45 - 16 * q^46 - 4 * q^47 + 8 * q^48 - 14 * q^50 - 20 * q^53 - 16 * q^56 + 2 * q^57 - 12 * q^58 + 20 * q^59 + 12 * q^60 + 4 * q^61 + 4 * q^63 + 6 * q^67 + 20 * q^68 - 8 * q^70 + 4 * q^72 + 10 * q^73 + 2 * q^75 - 20 * q^79 + 16 * q^80 + 10 * q^81 - 4 * q^82 + 8 * q^83 - 10 * q^85 + 24 * q^86 + 12 * q^87 - 2 * q^90 + 16 * q^92 + 4 * q^95 - 16 * q^96 - 20 * q^97 + 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$1$$ $$i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
267.1
 1.00000i − 1.00000i
−1.00000 + 1.00000i −1.00000 1.00000i 2.00000i −2.00000 1.00000i 2.00000 −2.00000 + 2.00000i 2.00000 + 2.00000i 1.00000i 3.00000 1.00000i
343.1 −1.00000 1.00000i −1.00000 + 1.00000i 2.00000i −2.00000 + 1.00000i 2.00000 −2.00000 2.00000i 2.00000 2.00000i 1.00000i 3.00000 + 1.00000i
 $$n$$: 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
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.k.a 2
4.b odd 2 1 380.2.k.b yes 2
5.c odd 4 1 380.2.k.b yes 2
20.e even 4 1 inner 380.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.k.a 2 1.a even 1 1 trivial
380.2.k.a 2 20.e even 4 1 inner
380.2.k.b yes 2 4.b odd 2 1
380.2.k.b yes 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4T + 8$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 10T + 50$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 8T + 32$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 12T + 72$$
$47$ $$T^{2} + 4T + 8$$
$53$ $$T^{2} + 20T + 200$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} - 6T + 18$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 10T + 50$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} - 8T + 32$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 20T + 200$$