↑

Properties

Label 1860.2.bb.b
Level $1860$
Weight $2$
Character orbit 1860.bb
Analytic conductor $14.852$
Analytic rank $0$
Dimension $80$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(161,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 12 q^{7} + 6 q^{9} - 6 q^{13} + 24 q^{19} + 6 q^{21} + 40 q^{25} - 4 q^{31} + 4 q^{33} - 30 q^{37} + 8 q^{39} - 30 q^{43} + 4 q^{45} - 36 q^{49} - 14 q^{51} + 24 q^{55} + 60 q^{57} - 32 q^{63} - 24 q^{67}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 −1.69215 − 0.369617i 0 −0.866025 − 0.500000i 0 −2.40489 − 4.16539i 0 2.72677 + 1.25090i 0
161.2 0 −1.68463 + 0.402527i 0 −0.866025 − 0.500000i 0 −0.802383 − 1.38977i 0 2.67594 − 1.35622i 0
161.3 0 −1.66857 − 0.464639i 0 0.866025 + 0.500000i 0 2.54324 + 4.40502i 0 2.56822 + 1.55056i 0
161.4 0 −1.66347 + 0.482578i 0 0.866025 + 0.500000i 0 −1.20595 − 2.08877i 0 2.53424 − 1.60550i 0
161.5 0 −1.64408 − 0.544971i 0 0.866025 + 0.500000i 0 0.0826514 + 0.143156i 0 2.40601 + 1.79196i 0
161.6 0 −1.64388 − 0.545584i 0 −0.866025 − 0.500000i 0 1.67675 + 2.90422i 0 2.40468 + 1.79375i 0
161.7 0 −1.53171 + 0.808615i 0 0.866025 + 0.500000i 0 1.40992 + 2.44206i 0 1.69229 − 2.47713i 0
161.8 0 −1.46614 + 0.922194i 0 −0.866025 − 0.500000i 0 1.40992 + 2.44206i 0 1.29911 − 2.70413i 0
161.9 0 −1.45888 − 0.933627i 0 −0.866025 − 0.500000i 0 0.292136 + 0.505995i 0 1.25668 + 2.72410i 0
161.10 0 −1.43474 − 0.970326i 0 0.866025 + 0.500000i 0 −0.548131 − 0.949390i 0 1.11693 + 2.78432i 0
161.11 0 −1.24966 + 1.19931i 0 −0.866025 − 0.500000i 0 −1.20595 − 2.08877i 0 0.123290 − 2.99747i 0
161.12 0 −1.19091 + 1.25767i 0 0.866025 + 0.500000i 0 −0.802383 − 1.38977i 0 −0.163456 − 2.99554i 0
161.13 0 −0.981290 − 1.42726i 0 −0.866025 − 0.500000i 0 0.737701 + 1.27774i 0 −1.07414 + 2.80111i 0
161.14 0 −0.525979 + 1.65026i 0 0.866025 + 0.500000i 0 −2.40489 − 4.16539i 0 −2.44669 − 1.73600i 0
161.15 0 −0.506436 − 1.65636i 0 −0.866025 − 0.500000i 0 −1.39213 − 2.41124i 0 −2.48705 + 1.67768i 0
161.16 0 −0.446093 − 1.67362i 0 0.866025 + 0.500000i 0 2.26268 + 3.91908i 0 −2.60200 + 1.49318i 0
161.17 0 −0.439338 − 1.67540i 0 0.866025 + 0.500000i 0 −2.04474 − 3.54160i 0 −2.61396 + 1.47214i 0
161.18 0 −0.431893 + 1.67734i 0 −0.866025 − 0.500000i 0 2.54324 + 4.40502i 0 −2.62694 − 1.44886i 0
161.19 0 −0.350082 + 1.69630i 0 −0.866025 − 0.500000i 0 0.0826514 + 0.143156i 0 −2.75489 − 1.18769i 0
161.20 0 −0.349450 + 1.69643i 0 0.866025 + 0.500000i 0 1.67675 + 2.90422i 0 −2.75577 − 1.18564i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 161.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.bb.b ✓ 80
3.b odd 2 1 inner 1860.2.bb.b ✓ 80
31.e odd 6 1 inner 1860.2.bb.b ✓ 80
93.g even 6 1 inner 1860.2.bb.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.bb.b ✓ 80 1.a even 1 1 trivial
1860.2.bb.b ✓ 80 3.b odd 2 1 inner
1860.2.bb.b ✓ 80 31.e odd 6 1 inner
1860.2.bb.b ✓ 80 93.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{40} - 6 T_{7}^{39} + 97 T_{7}^{38} - 430 T_{7}^{37} + 4746 T_{7}^{36} - 18204 T_{7}^{35} + \cdots + 20259536896 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display