# Properties

 Label 390.2.e.d Level $390$ Weight $2$ Character orbit 390.e Analytic conductor $3.114$ 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] = [390,2,Mod(79,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("390.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.e (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: $$3.11416567883$$ 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: $\mathrm{SU}(2)[C_{2}]$

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

 $$f(q)$$ $$=$$ $$q + i q^{2} - i q^{3} - q^{4} + ( - i + 2) q^{5} + q^{6} - 4 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - i * q^3 - q^4 + (-i + 2) * q^5 + q^6 - 4*i * q^7 - i * q^8 - q^9 $$q + i q^{2} - i q^{3} - q^{4} + ( - i + 2) q^{5} + q^{6} - 4 i q^{7} - i q^{8} - q^{9} + (2 i + 1) q^{10} - 6 q^{11} + i q^{12} - i q^{13} + 4 q^{14} + ( - 2 i - 1) q^{15} + q^{16} - 4 i q^{17} - i q^{18} - 2 q^{19} + (i - 2) q^{20} - 4 q^{21} - 6 i q^{22} + 6 i q^{23} - q^{24} + ( - 4 i + 3) q^{25} + q^{26} + i q^{27} + 4 i q^{28} + 10 q^{29} + ( - i + 2) q^{30} + 4 q^{31} + i q^{32} + 6 i q^{33} + 4 q^{34} + ( - 8 i - 4) q^{35} + q^{36} - 6 i q^{37} - 2 i q^{38} - q^{39} + ( - 2 i - 1) q^{40} + 10 q^{41} - 4 i q^{42} + 6 q^{44} + (i - 2) q^{45} - 6 q^{46} + 8 i q^{47} - i q^{48} - 9 q^{49} + (3 i + 4) q^{50} - 4 q^{51} + i q^{52} + 6 i q^{53} - q^{54} + (6 i - 12) q^{55} - 4 q^{56} + 2 i q^{57} + 10 i q^{58} + 6 q^{59} + (2 i + 1) q^{60} - 6 q^{61} + 4 i q^{62} + 4 i q^{63} - q^{64} + ( - 2 i - 1) q^{65} - 6 q^{66} - 12 i q^{67} + 4 i q^{68} + 6 q^{69} + ( - 4 i + 8) q^{70} + i q^{72} - 2 i q^{73} + 6 q^{74} + ( - 3 i - 4) q^{75} + 2 q^{76} + 24 i q^{77} - i q^{78} + 8 q^{79} + ( - i + 2) q^{80} + q^{81} + 10 i q^{82} + 4 i q^{83} + 4 q^{84} + ( - 8 i - 4) q^{85} - 10 i q^{87} + 6 i q^{88} - 14 q^{89} + ( - 2 i - 1) q^{90} - 4 q^{91} - 6 i q^{92} - 4 i q^{93} - 8 q^{94} + (2 i - 4) q^{95} + q^{96} + 14 i q^{97} - 9 i q^{98} + 6 q^{99} +O(q^{100})$$ q + i * q^2 - i * q^3 - q^4 + (-i + 2) * q^5 + q^6 - 4*i * q^7 - i * q^8 - q^9 + (2*i + 1) * q^10 - 6 * q^11 + i * q^12 - i * q^13 + 4 * q^14 + (-2*i - 1) * q^15 + q^16 - 4*i * q^17 - i * q^18 - 2 * q^19 + (i - 2) * q^20 - 4 * q^21 - 6*i * q^22 + 6*i * q^23 - q^24 + (-4*i + 3) * q^25 + q^26 + i * q^27 + 4*i * q^28 + 10 * q^29 + (-i + 2) * q^30 + 4 * q^31 + i * q^32 + 6*i * q^33 + 4 * q^34 + (-8*i - 4) * q^35 + q^36 - 6*i * q^37 - 2*i * q^38 - q^39 + (-2*i - 1) * q^40 + 10 * q^41 - 4*i * q^42 + 6 * q^44 + (i - 2) * q^45 - 6 * q^46 + 8*i * q^47 - i * q^48 - 9 * q^49 + (3*i + 4) * q^50 - 4 * q^51 + i * q^52 + 6*i * q^53 - q^54 + (6*i - 12) * q^55 - 4 * q^56 + 2*i * q^57 + 10*i * q^58 + 6 * q^59 + (2*i + 1) * q^60 - 6 * q^61 + 4*i * q^62 + 4*i * q^63 - q^64 + (-2*i - 1) * q^65 - 6 * q^66 - 12*i * q^67 + 4*i * q^68 + 6 * q^69 + (-4*i + 8) * q^70 + i * q^72 - 2*i * q^73 + 6 * q^74 + (-3*i - 4) * q^75 + 2 * q^76 + 24*i * q^77 - i * q^78 + 8 * q^79 + (-i + 2) * q^80 + q^81 + 10*i * q^82 + 4*i * q^83 + 4 * q^84 + (-8*i - 4) * q^85 - 10*i * q^87 + 6*i * q^88 - 14 * q^89 + (-2*i - 1) * q^90 - 4 * q^91 - 6*i * q^92 - 4*i * q^93 - 8 * q^94 + (2*i - 4) * q^95 + q^96 + 14*i * q^97 - 9*i * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9} + 2 q^{10} - 12 q^{11} + 8 q^{14} - 2 q^{15} + 2 q^{16} - 4 q^{19} - 4 q^{20} - 8 q^{21} - 2 q^{24} + 6 q^{25} + 2 q^{26} + 20 q^{29} + 4 q^{30} + 8 q^{31} + 8 q^{34} - 8 q^{35} + 2 q^{36} - 2 q^{39} - 2 q^{40} + 20 q^{41} + 12 q^{44} - 4 q^{45} - 12 q^{46} - 18 q^{49} + 8 q^{50} - 8 q^{51} - 2 q^{54} - 24 q^{55} - 8 q^{56} + 12 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 2 q^{65} - 12 q^{66} + 12 q^{69} + 16 q^{70} + 12 q^{74} - 8 q^{75} + 4 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} + 8 q^{84} - 8 q^{85} - 28 q^{89} - 2 q^{90} - 8 q^{91} - 16 q^{94} - 8 q^{95} + 2 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9 + 2 * q^10 - 12 * q^11 + 8 * q^14 - 2 * q^15 + 2 * q^16 - 4 * q^19 - 4 * q^20 - 8 * q^21 - 2 * q^24 + 6 * q^25 + 2 * q^26 + 20 * q^29 + 4 * q^30 + 8 * q^31 + 8 * q^34 - 8 * q^35 + 2 * q^36 - 2 * q^39 - 2 * q^40 + 20 * q^41 + 12 * q^44 - 4 * q^45 - 12 * q^46 - 18 * q^49 + 8 * q^50 - 8 * q^51 - 2 * q^54 - 24 * q^55 - 8 * q^56 + 12 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^64 - 2 * q^65 - 12 * q^66 + 12 * q^69 + 16 * q^70 + 12 * q^74 - 8 * q^75 + 4 * q^76 + 16 * q^79 + 4 * q^80 + 2 * q^81 + 8 * q^84 - 8 * q^85 - 28 * q^89 - 2 * q^90 - 8 * q^91 - 16 * q^94 - 8 * q^95 + 2 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 4.00000i 1.00000i −1.00000 1.00000 2.00000i
79.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i 1.00000 4.00000i 1.00000i −1.00000 1.00000 + 2.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.e.d 2
3.b odd 2 1 1170.2.e.c 2
4.b odd 2 1 3120.2.l.i 2
5.b even 2 1 inner 390.2.e.d 2
5.c odd 4 1 1950.2.a.d 1
5.c odd 4 1 1950.2.a.v 1
15.d odd 2 1 1170.2.e.c 2
15.e even 4 1 5850.2.a.e 1
15.e even 4 1 5850.2.a.cc 1
20.d odd 2 1 3120.2.l.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.d 2 1.a even 1 1 trivial
390.2.e.d 2 5.b even 2 1 inner
1170.2.e.c 2 3.b odd 2 1
1170.2.e.c 2 15.d odd 2 1
1950.2.a.d 1 5.c odd 4 1
1950.2.a.v 1 5.c odd 4 1
3120.2.l.i 2 4.b odd 2 1
3120.2.l.i 2 20.d odd 2 1
5850.2.a.e 1 15.e even 4 1
5850.2.a.cc 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 6$$ T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 196$$