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Properties

Label 840.2.j.f
Level $840$
Weight $2$
Character orbit 840.j
Analytic conductor $6.707$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(589,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("840.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.j (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 32 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 32 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} + 32 q^{9} - 8 q^{10} - 2 q^{12} + 32 q^{13} + 6 q^{16} + 2 q^{18} + 24 q^{20} + 2 q^{24} - 16 q^{25} + 12 q^{26} + 32 q^{27} + 8 q^{28} - 8 q^{30} + 32 q^{31} + 22 q^{32} + 4 q^{35} - 2 q^{36} - 8 q^{37} - 32 q^{38} + 32 q^{39} - 24 q^{40} + 24 q^{41} - 8 q^{43} - 24 q^{44} + 12 q^{46} + 6 q^{48} - 32 q^{49} - 26 q^{50} - 16 q^{52} - 24 q^{53} + 2 q^{54} + 24 q^{55} - 12 q^{56} - 16 q^{58} + 24 q^{60} - 48 q^{62} + 22 q^{64} + 8 q^{65} + 24 q^{67} + 4 q^{68} + 6 q^{70} + 40 q^{71} + 2 q^{72} - 20 q^{74} - 16 q^{75} - 52 q^{76} - 16 q^{77} + 12 q^{78} - 24 q^{79} - 44 q^{80} + 32 q^{81} + 24 q^{82} + 8 q^{84} - 8 q^{85} - 76 q^{86} + 56 q^{88} + 24 q^{89} - 8 q^{90} - 96 q^{92} + 32 q^{93} + 32 q^{94} - 48 q^{95} + 22 q^{96} - 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1 −1.40102 − 0.192719i 1.00000 1.92572 + 0.540007i 2.08276 + 0.813703i −1.40102 − 0.192719i 1.00000i −2.59390 − 1.12768i 1.00000 −2.76117 − 1.54140i
589.2 −1.40102 + 0.192719i 1.00000 1.92572 − 0.540007i 2.08276 − 0.813703i −1.40102 + 0.192719i − 1.00000i −2.59390 + 1.12768i 1.00000 −2.76117 + 1.54140i
589.3 −1.35171 − 0.415804i 1.00000 1.65421 + 1.12409i −2.15737 − 0.588016i −1.35171 − 0.415804i − 1.00000i −1.76861 − 2.20727i 1.00000 2.67163 + 1.69187i
589.4 −1.35171 + 0.415804i 1.00000 1.65421 − 1.12409i −2.15737 + 0.588016i −1.35171 + 0.415804i 1.00000i −1.76861 + 2.20727i 1.00000 2.67163 − 1.69187i
589.5 −1.14916 − 0.824272i 1.00000 0.641151 + 1.89445i 0.961089 − 2.01899i −1.14916 − 0.824272i − 1.00000i 0.824751 − 2.70551i 1.00000 −2.76864 + 1.52795i
589.6 −1.14916 + 0.824272i 1.00000 0.641151 − 1.89445i 0.961089 + 2.01899i −1.14916 + 0.824272i 1.00000i 0.824751 + 2.70551i 1.00000 −2.76864 − 1.52795i
589.7 −0.941311 − 1.05543i 1.00000 −0.227866 + 1.98698i 1.54072 − 1.62055i −0.941311 − 1.05543i 1.00000i 2.31161 − 1.62987i 1.00000 −3.16067 − 0.100680i
589.8 −0.941311 + 1.05543i 1.00000 −0.227866 − 1.98698i 1.54072 + 1.62055i −0.941311 + 1.05543i − 1.00000i 2.31161 + 1.62987i 1.00000 −3.16067 + 0.100680i
589.9 −0.832752 − 1.14303i 1.00000 −0.613047 + 1.90373i −0.241469 + 2.22299i −0.832752 − 1.14303i − 1.00000i 2.68654 − 0.884600i 1.00000 2.74204 − 1.57520i
589.10 −0.832752 + 1.14303i 1.00000 −0.613047 − 1.90373i −0.241469 − 2.22299i −0.832752 + 1.14303i 1.00000i 2.68654 + 0.884600i 1.00000 2.74204 + 1.57520i
589.11 −0.653094 − 1.25438i 1.00000 −1.14694 + 1.63846i −0.980656 − 2.00956i −0.653094 − 1.25438i − 1.00000i 2.80430 + 0.368626i 1.00000 −1.88028 + 2.54254i
589.12 −0.653094 + 1.25438i 1.00000 −1.14694 − 1.63846i −0.980656 + 2.00956i −0.653094 + 1.25438i 1.00000i 2.80430 − 0.368626i 1.00000 −1.88028 − 2.54254i
589.13 −0.150941 − 1.40614i 1.00000 −1.95443 + 0.424487i −2.20890 − 0.347530i −0.150941 − 1.40614i 1.00000i 0.891891 + 2.68413i 1.00000 −0.155261 + 3.15846i
589.14 −0.150941 + 1.40614i 1.00000 −1.95443 − 0.424487i −2.20890 + 0.347530i −0.150941 + 1.40614i − 1.00000i 0.891891 − 2.68413i 1.00000 −0.155261 − 3.15846i
589.15 −0.0454850 − 1.41348i 1.00000 −1.99586 + 0.128585i 1.37352 − 1.76449i −0.0454850 − 1.41348i 1.00000i 0.272534 + 2.81527i 1.00000 −2.55655 − 1.86119i
589.16 −0.0454850 + 1.41348i 1.00000 −1.99586 − 0.128585i 1.37352 + 1.76449i −0.0454850 + 1.41348i − 1.00000i 0.272534 − 2.81527i 1.00000 −2.55655 + 1.86119i
589.17 0.406657 − 1.35449i 1.00000 −1.66926 − 1.10162i −0.836338 + 2.07377i 0.406657 − 1.35449i − 1.00000i −2.17095 + 1.81300i 1.00000 2.46879 + 1.97612i
589.18 0.406657 + 1.35449i 1.00000 −1.66926 + 1.10162i −0.836338 − 2.07377i 0.406657 + 1.35449i 1.00000i −2.17095 − 1.81300i 1.00000 2.46879 − 1.97612i
589.19 0.453365 − 1.33957i 1.00000 −1.58892 − 1.21463i 2.12585 + 0.693382i 0.453365 − 1.33957i − 1.00000i −2.34745 + 1.57780i 1.00000 1.89262 − 2.53337i
589.20 0.453365 + 1.33957i 1.00000 −1.58892 + 1.21463i 2.12585 − 0.693382i 0.453365 + 1.33957i 1.00000i −2.34745 − 1.57780i 1.00000 1.89262 + 2.53337i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.j.f yes 32
4.b odd 2 1 3360.2.j.e 32
5.b even 2 1 840.2.j.e ✓ 32
8.b even 2 1 840.2.j.e ✓ 32
8.d odd 2 1 3360.2.j.f 32
20.d odd 2 1 3360.2.j.f 32
40.e odd 2 1 3360.2.j.e 32
40.f even 2 1 inner 840.2.j.f yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.j.e ✓ 32 5.b even 2 1
840.2.j.e ✓ 32 8.b even 2 1
840.2.j.f yes 32 1.a even 1 1 trivial
840.2.j.f yes 32 40.f even 2 1 inner
3360.2.j.e 32 4.b odd 2 1
3360.2.j.e 32 40.e odd 2 1
3360.2.j.f 32 8.d odd 2 1
3360.2.j.f 32 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{32} + 200 T_{11}^{30} + 17624 T_{11}^{28} + 901760 T_{11}^{26} + 29739664 T_{11}^{24} + \cdots + 1677721600 \) Copy content Toggle raw display
\( T_{13}^{16} - 16 T_{13}^{15} + 24 T_{13}^{14} + 888 T_{13}^{13} - 5072 T_{13}^{12} - 6112 T_{13}^{11} + \cdots - 262144 \) Copy content Toggle raw display