# Properties

 Label 390.2.s.b Level $390$ Weight $2$ Character orbit 390.s Analytic conductor $3.114$ Analytic rank $0$ Dimension $24$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(77,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("390.77");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.s (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: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{3} - 12 q^{9}+O(q^{10})$$ 24 * q - 8 * q^3 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{3} - 12 q^{9} + 4 q^{10} + 4 q^{12} - 8 q^{13} - 16 q^{14} - 24 q^{16} + 52 q^{17} + 16 q^{22} - 16 q^{23} + 16 q^{25} + 40 q^{27} - 80 q^{29} + 20 q^{35} - 16 q^{38} - 32 q^{39} - 4 q^{42} + 28 q^{43} + 8 q^{48} - 20 q^{51} - 8 q^{52} + 72 q^{53} + 24 q^{55} - 64 q^{61} - 32 q^{62} - 12 q^{65} + 8 q^{66} + 52 q^{68} - 16 q^{69} - 120 q^{74} - 104 q^{75} + 32 q^{77} + 44 q^{78} + 12 q^{81} + 32 q^{82} - 32 q^{87} - 16 q^{88} + 64 q^{90} + 64 q^{91} + 16 q^{92} + 96 q^{95}+O(q^{100})$$ 24 * q - 8 * q^3 - 12 * q^9 + 4 * q^10 + 4 * q^12 - 8 * q^13 - 16 * q^14 - 24 * q^16 + 52 * q^17 + 16 * q^22 - 16 * q^23 + 16 * q^25 + 40 * q^27 - 80 * q^29 + 20 * q^35 - 16 * q^38 - 32 * q^39 - 4 * q^42 + 28 * q^43 + 8 * q^48 - 20 * q^51 - 8 * q^52 + 72 * q^53 + 24 * q^55 - 64 * q^61 - 32 * q^62 - 12 * q^65 + 8 * q^66 + 52 * q^68 - 16 * q^69 - 120 * q^74 - 104 * q^75 + 32 * q^77 + 44 * q^78 + 12 * q^81 + 32 * q^82 - 32 * q^87 - 16 * q^88 + 64 * q^90 + 64 * q^91 + 16 * q^92 + 96 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1 −0.707107 0.707107i −1.53318 0.805823i 1.00000i −1.53706 + 1.62402i 0.514321 + 1.65393i 0.677204 0.677204i 0.707107 0.707107i 1.70130 + 2.47095i 2.23522 0.0614950i
77.2 −0.707107 0.707107i −1.29768 1.14719i 1.00000i 2.23605 0.00995019i 0.106413 + 1.72878i −1.03162 + 1.03162i 0.707107 0.707107i 0.367928 + 2.97735i −1.58816 1.57409i
77.3 −0.707107 0.707107i −0.854151 + 1.50679i 1.00000i 1.11448 + 1.93854i 1.66944 0.461487i 3.10713 3.10713i 0.707107 0.707107i −1.54085 2.57406i 0.582700 2.15881i
77.4 −0.707107 0.707107i −0.128848 + 1.72725i 1.00000i −1.56114 1.60089i 1.31246 1.13024i 0.236462 0.236462i 0.707107 0.707107i −2.96680 0.445105i −0.0281051 + 2.23589i
77.5 −0.707107 0.707107i 0.174068 1.72328i 1.00000i 1.17831 1.90042i −1.34163 + 1.09546i −2.28877 + 2.28877i 0.707107 0.707107i −2.93940 0.599938i −2.17699 + 0.510611i
77.6 −0.707107 0.707107i 1.63979 0.557754i 1.00000i −2.13774 + 0.655799i −1.55390 0.765115i 2.12802 2.12802i 0.707107 0.707107i 2.37782 1.82920i 1.97533 + 1.04789i
77.7 0.707107 + 0.707107i −1.53318 0.805823i 1.00000i 1.53706 1.62402i −0.514321 1.65393i −0.677204 + 0.677204i −0.707107 + 0.707107i 1.70130 + 2.47095i 2.23522 0.0614950i
77.8 0.707107 + 0.707107i −1.29768 1.14719i 1.00000i −2.23605 + 0.00995019i −0.106413 1.72878i 1.03162 1.03162i −0.707107 + 0.707107i 0.367928 + 2.97735i −1.58816 1.57409i
77.9 0.707107 + 0.707107i −0.854151 + 1.50679i 1.00000i −1.11448 1.93854i −1.66944 + 0.461487i −3.10713 + 3.10713i −0.707107 + 0.707107i −1.54085 2.57406i 0.582700 2.15881i
77.10 0.707107 + 0.707107i −0.128848 + 1.72725i 1.00000i 1.56114 + 1.60089i −1.31246 + 1.13024i −0.236462 + 0.236462i −0.707107 + 0.707107i −2.96680 0.445105i −0.0281051 + 2.23589i
77.11 0.707107 + 0.707107i 0.174068 1.72328i 1.00000i −1.17831 + 1.90042i 1.34163 1.09546i 2.28877 2.28877i −0.707107 + 0.707107i −2.93940 0.599938i −2.17699 + 0.510611i
77.12 0.707107 + 0.707107i 1.63979 0.557754i 1.00000i 2.13774 0.655799i 1.55390 + 0.765115i −2.12802 + 2.12802i −0.707107 + 0.707107i 2.37782 1.82920i 1.97533 + 1.04789i
233.1 −0.707107 + 0.707107i −1.53318 + 0.805823i 1.00000i −1.53706 1.62402i 0.514321 1.65393i 0.677204 + 0.677204i 0.707107 + 0.707107i 1.70130 2.47095i 2.23522 + 0.0614950i
233.2 −0.707107 + 0.707107i −1.29768 + 1.14719i 1.00000i 2.23605 + 0.00995019i 0.106413 1.72878i −1.03162 1.03162i 0.707107 + 0.707107i 0.367928 2.97735i −1.58816 + 1.57409i
233.3 −0.707107 + 0.707107i −0.854151 1.50679i 1.00000i 1.11448 1.93854i 1.66944 + 0.461487i 3.10713 + 3.10713i 0.707107 + 0.707107i −1.54085 + 2.57406i 0.582700 + 2.15881i
233.4 −0.707107 + 0.707107i −0.128848 1.72725i 1.00000i −1.56114 + 1.60089i 1.31246 + 1.13024i 0.236462 + 0.236462i 0.707107 + 0.707107i −2.96680 + 0.445105i −0.0281051 2.23589i
233.5 −0.707107 + 0.707107i 0.174068 + 1.72328i 1.00000i 1.17831 + 1.90042i −1.34163 1.09546i −2.28877 2.28877i 0.707107 + 0.707107i −2.93940 + 0.599938i −2.17699 0.510611i
233.6 −0.707107 + 0.707107i 1.63979 + 0.557754i 1.00000i −2.13774 0.655799i −1.55390 + 0.765115i 2.12802 + 2.12802i 0.707107 + 0.707107i 2.37782 + 1.82920i 1.97533 1.04789i
233.7 0.707107 0.707107i −1.53318 + 0.805823i 1.00000i 1.53706 + 1.62402i −0.514321 + 1.65393i −0.677204 0.677204i −0.707107 0.707107i 1.70130 2.47095i 2.23522 + 0.0614950i
233.8 0.707107 0.707107i −1.29768 + 1.14719i 1.00000i −2.23605 0.00995019i −0.106413 + 1.72878i 1.03162 + 1.03162i −0.707107 0.707107i 0.367928 2.97735i −1.58816 + 1.57409i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
15.e even 4 1 inner
195.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.s.b 24
3.b odd 2 1 390.2.s.c yes 24
5.c odd 4 1 390.2.s.c yes 24
13.b even 2 1 inner 390.2.s.b 24
15.e even 4 1 inner 390.2.s.b 24
39.d odd 2 1 390.2.s.c yes 24
65.h odd 4 1 390.2.s.c yes 24
195.s even 4 1 inner 390.2.s.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.s.b 24 1.a even 1 1 trivial
390.2.s.b 24 13.b even 2 1 inner
390.2.s.b 24 15.e even 4 1 inner
390.2.s.b 24 195.s even 4 1 inner
390.2.s.c yes 24 3.b odd 2 1
390.2.s.c yes 24 5.c odd 4 1
390.2.s.c yes 24 39.d odd 2 1
390.2.s.c yes 24 65.h odd 4 1

## Hecke kernels

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

 $$T_{7}^{24} + 570T_{7}^{20} + 83553T_{7}^{16} + 3792536T_{7}^{12} + 18386480T_{7}^{8} + 13023616T_{7}^{4} + 160000$$ T7^24 + 570*T7^20 + 83553*T7^16 + 3792536*T7^12 + 18386480*T7^8 + 13023616*T7^4 + 160000 $$T_{17}^{12} - 26 T_{17}^{11} + 338 T_{17}^{10} - 2514 T_{17}^{9} + 11301 T_{17}^{8} - 26460 T_{17}^{7} + \cdots + 26419600$$ T17^12 - 26*T17^11 + 338*T17^10 - 2514*T17^9 + 11301*T17^8 - 26460*T17^7 + 28320*T17^6 - 78472*T17^5 + 802236*T17^4 - 1702752*T17^3 + 415872*T17^2 + 4687680*T17 + 26419600