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Properties

Label 287.2.j.a
Level $287$
Weight $2$
Character orbit 287.j
Analytic conductor $2.292$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(81,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.j (of order \(6\), 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: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9} + 16 q^{10} - 26 q^{16} + 12 q^{18} + 16 q^{20} - 48 q^{21} - 32 q^{25} + 6 q^{31} + 20 q^{32} - 20 q^{33} - 4 q^{36} - 24 q^{37} + 14 q^{39} + 44 q^{40} + 24 q^{41} - 2 q^{42} - 32 q^{43} + 26 q^{45} + 12 q^{46} + 16 q^{49} - 12 q^{51} + 8 q^{57} + 2 q^{59} + 14 q^{61} - 20 q^{62} - 20 q^{64} - 4 q^{66} - 42 q^{72} - 16 q^{73} - 22 q^{74} - 32 q^{77} + 88 q^{78} - 52 q^{80} - 46 q^{81} - 42 q^{82} - 112 q^{83} + 72 q^{84} + 16 q^{86} - 18 q^{87} + 224 q^{90} + 86 q^{91} - 32 q^{92} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −1.30798 − 2.26549i −0.306987 − 0.177239i −2.42163 + 4.19439i 0.326729 + 0.565911i 0.927301i 0.262924 + 2.63265i 7.43789 −1.43717 − 2.48926i 0.854711 − 1.48040i
81.2 −1.30798 − 2.26549i 0.306987 + 0.177239i −2.42163 + 4.19439i 0.326729 + 0.565911i − 0.927301i −0.262924 − 2.63265i 7.43789 −1.43717 − 2.48926i 0.854711 − 1.48040i
81.3 −1.10471 − 1.91341i −2.73490 − 1.57900i −1.44075 + 2.49545i 1.43765 + 2.49009i 6.97730i 2.25170 − 1.38919i 1.94760 3.48645 + 6.03872i 3.17637 − 5.50163i
81.4 −1.10471 − 1.91341i 2.73490 + 1.57900i −1.44075 + 2.49545i 1.43765 + 2.49009i − 6.97730i −2.25170 + 1.38919i 1.94760 3.48645 + 6.03872i 3.17637 − 5.50163i
81.5 −1.02851 − 1.78143i −1.38029 − 0.796913i −1.11566 + 1.93238i −2.03101 − 3.51781i 3.27853i 2.61591 − 0.396236i 0.475827 −0.229860 − 0.398129i −4.17782 + 7.23619i
81.6 −1.02851 − 1.78143i 1.38029 + 0.796913i −1.11566 + 1.93238i −2.03101 − 3.51781i − 3.27853i −2.61591 + 0.396236i 0.475827 −0.229860 − 0.398129i −4.17782 + 7.23619i
81.7 −0.749549 − 1.29826i −1.70608 − 0.985006i −0.123649 + 0.214166i −0.142717 − 0.247192i 2.95324i −1.33336 + 2.28520i −2.62747 0.440474 + 0.762924i −0.213946 + 0.370566i
81.8 −0.749549 − 1.29826i 1.70608 + 0.985006i −0.123649 + 0.214166i −0.142717 − 0.247192i − 2.95324i 1.33336 − 2.28520i −2.62747 0.440474 + 0.762924i −0.213946 + 0.370566i
81.9 −0.655113 − 1.13469i −0.913309 − 0.527299i 0.141655 − 0.245353i 1.55927 + 2.70074i 1.38176i −2.26438 − 1.36842i −2.99165 −0.943911 − 1.63490i 2.04300 − 3.53858i
81.10 −0.655113 − 1.13469i 0.913309 + 0.527299i 0.141655 − 0.245353i 1.55927 + 2.70074i − 1.38176i 2.26438 + 1.36842i −2.99165 −0.943911 − 1.63490i 2.04300 − 3.53858i
81.11 −0.158705 − 0.274885i −0.453525 − 0.261843i 0.949625 − 1.64480i −0.708472 − 1.22711i 0.166223i 1.98670 + 1.74729i −1.23766 −1.36288 − 2.36057i −0.224876 + 0.389497i
81.12 −0.158705 − 0.274885i 0.453525 + 0.261843i 0.949625 − 1.64480i −0.708472 − 1.22711i − 0.166223i −1.98670 − 1.74729i −1.23766 −1.36288 − 2.36057i −0.224876 + 0.389497i
81.13 −0.143305 − 0.248211i −2.80985 − 1.62226i 0.958928 − 1.66091i −1.41048 − 2.44302i 0.929911i −1.71093 − 2.01810i −1.12289 3.76349 + 6.51855i −0.404256 + 0.700192i
81.14 −0.143305 − 0.248211i 2.80985 + 1.62226i 0.958928 − 1.66091i −1.41048 − 2.44302i − 0.929911i 1.71093 + 2.01810i −1.12289 3.76349 + 6.51855i −0.404256 + 0.700192i
81.15 0.0647109 + 0.112083i −1.92881 − 1.11360i 0.991625 − 1.71754i 0.451146 + 0.781409i − 0.288247i 2.43969 − 1.02368i 0.515519 0.980198 + 1.69775i −0.0583882 + 0.101131i
81.16 0.0647109 + 0.112083i 1.92881 + 1.11360i 0.991625 − 1.71754i 0.451146 + 0.781409i 0.288247i −2.43969 + 1.02368i 0.515519 0.980198 + 1.69775i −0.0583882 + 0.101131i
81.17 0.467463 + 0.809670i −0.691950 − 0.399498i 0.562956 − 0.975068i 1.41567 + 2.45201i − 0.747002i −0.561364 − 2.58551i 2.92250 −1.18080 − 2.04521i −1.32355 + 2.29245i
81.18 0.467463 + 0.809670i 0.691950 + 0.399498i 0.562956 − 0.975068i 1.41567 + 2.45201i 0.747002i 0.561364 + 2.58551i 2.92250 −1.18080 − 2.04521i −1.32355 + 2.29245i
81.19 0.649153 + 1.12437i −0.800607 − 0.462230i 0.157202 − 0.272281i −1.62887 − 2.82129i − 1.20023i −2.18878 + 1.48635i 3.00480 −1.07269 − 1.85795i 2.11478 − 3.66290i
81.20 0.649153 + 1.12437i 0.800607 + 0.462230i 0.157202 − 0.272281i −1.62887 − 2.82129i 1.20023i 2.18878 − 1.48635i 3.00480 −1.07269 − 1.85795i 2.11478 − 3.66290i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.b even 2 1 inner
287.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.j.a ✓ 52
7.c even 3 1 inner 287.2.j.a ✓ 52
41.b even 2 1 inner 287.2.j.a ✓ 52
287.j even 6 1 inner 287.2.j.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.j.a ✓ 52 1.a even 1 1 trivial
287.2.j.a ✓ 52 7.c even 3 1 inner
287.2.j.a ✓ 52 41.b even 2 1 inner
287.2.j.a ✓ 52 287.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(287, [\chi])\).