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Properties

Label 840.2.u.e
Level $840$
Weight $2$
Character orbit 840.u
Analytic conductor $6.707$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(629,840)]
 
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, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.629");
 
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.u (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: \(160\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 24 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 24 q^{4} - 32 q^{9} - 48 q^{15} - 104 q^{16} - 16 q^{25} - 32 q^{30} + 48 q^{36} - 64 q^{39} - 64 q^{46} + 144 q^{49} + 16 q^{60} - 72 q^{64} + 8 q^{70} - 96 q^{79} + 16 q^{81} - 72 q^{84}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
629.1 −1.37073 − 0.347977i −0.973146 − 1.43282i 1.75782 + 0.953969i 1.31834 + 1.80610i 0.835334 + 2.30265i 2.61441 + 0.406058i −2.07755 − 1.91932i −1.10597 + 2.78870i −1.17861 − 2.93443i
629.2 −1.37073 − 0.347977i −0.973146 + 1.43282i 1.75782 + 0.953969i 1.31834 − 1.80610i 1.83252 − 1.62539i −2.61441 − 0.406058i −2.07755 − 1.91932i −1.10597 − 2.78870i −2.43557 + 2.01693i
629.3 −1.37073 − 0.347977i 0.973146 − 1.43282i 1.75782 + 0.953969i −1.31834 + 1.80610i −1.83252 + 1.62539i 2.61441 − 0.406058i −2.07755 − 1.91932i −1.10597 − 2.78870i 2.43557 − 2.01693i
629.4 −1.37073 − 0.347977i 0.973146 + 1.43282i 1.75782 + 0.953969i −1.31834 − 1.80610i −0.835334 − 2.30265i −2.61441 + 0.406058i −2.07755 − 1.91932i −1.10597 + 2.78870i 1.17861 + 2.93443i
629.5 −1.37073 + 0.347977i −0.973146 − 1.43282i 1.75782 − 0.953969i 1.31834 + 1.80610i 1.83252 + 1.62539i −2.61441 + 0.406058i −2.07755 + 1.91932i −1.10597 + 2.78870i −2.43557 − 2.01693i
629.6 −1.37073 + 0.347977i −0.973146 + 1.43282i 1.75782 − 0.953969i 1.31834 − 1.80610i 0.835334 − 2.30265i 2.61441 − 0.406058i −2.07755 + 1.91932i −1.10597 − 2.78870i −1.17861 + 2.93443i
629.7 −1.37073 + 0.347977i 0.973146 − 1.43282i 1.75782 − 0.953969i −1.31834 + 1.80610i −0.835334 + 2.30265i −2.61441 − 0.406058i −2.07755 + 1.91932i −1.10597 − 2.78870i 1.17861 − 2.93443i
629.8 −1.37073 + 0.347977i 0.973146 + 1.43282i 1.75782 − 0.953969i −1.31834 − 1.80610i −1.83252 − 1.62539i 2.61441 + 0.406058i −2.07755 + 1.91932i −1.10597 + 2.78870i 2.43557 + 2.01693i
629.9 −1.29842 − 0.560458i −0.135991 − 1.72670i 1.37177 + 1.45542i −1.93600 − 1.11889i −0.791172 + 2.31820i −2.16054 + 1.52711i −0.965432 − 2.65856i −2.96301 + 0.469633i 1.88664 + 2.53783i
629.10 −1.29842 − 0.560458i −0.135991 + 1.72670i 1.37177 + 1.45542i −1.93600 + 1.11889i 1.14432 − 2.16576i 2.16054 − 1.52711i −0.965432 − 2.65856i −2.96301 − 0.469633i 3.14082 − 0.367741i
629.11 −1.29842 − 0.560458i 0.135991 − 1.72670i 1.37177 + 1.45542i 1.93600 − 1.11889i −1.14432 + 2.16576i −2.16054 − 1.52711i −0.965432 − 2.65856i −2.96301 − 0.469633i −3.14082 + 0.367741i
629.12 −1.29842 − 0.560458i 0.135991 + 1.72670i 1.37177 + 1.45542i 1.93600 + 1.11889i 0.791172 − 2.31820i 2.16054 + 1.52711i −0.965432 − 2.65856i −2.96301 + 0.469633i −1.88664 − 2.53783i
629.13 −1.29842 + 0.560458i −0.135991 − 1.72670i 1.37177 − 1.45542i −1.93600 − 1.11889i 1.14432 + 2.16576i 2.16054 + 1.52711i −0.965432 + 2.65856i −2.96301 + 0.469633i 3.14082 + 0.367741i
629.14 −1.29842 + 0.560458i −0.135991 + 1.72670i 1.37177 − 1.45542i −1.93600 + 1.11889i −0.791172 − 2.31820i −2.16054 − 1.52711i −0.965432 + 2.65856i −2.96301 − 0.469633i 1.88664 − 2.53783i
629.15 −1.29842 + 0.560458i 0.135991 − 1.72670i 1.37177 − 1.45542i 1.93600 − 1.11889i 0.791172 + 2.31820i 2.16054 − 1.52711i −0.965432 + 2.65856i −2.96301 − 0.469633i −1.88664 + 2.53783i
629.16 −1.29842 + 0.560458i 0.135991 + 1.72670i 1.37177 − 1.45542i 1.93600 + 1.11889i −1.14432 − 2.16576i −2.16054 + 1.52711i −0.965432 + 2.65856i −2.96301 + 0.469633i −3.14082 − 0.367741i
629.17 −1.25943 − 0.643305i −1.72207 − 0.185637i 1.17232 + 1.62039i −0.799234 + 2.08835i 2.04941 + 1.34162i −2.00372 + 1.72775i −0.434041 − 2.79493i 2.93108 + 0.639360i 2.35003 − 2.11598i
629.18 −1.25943 − 0.643305i −1.72207 + 0.185637i 1.17232 + 1.62039i −0.799234 − 2.08835i 2.28825 + 0.874023i 2.00372 − 1.72775i −0.434041 − 2.79493i 2.93108 − 0.639360i −0.336872 + 3.14428i
629.19 −1.25943 − 0.643305i 1.72207 − 0.185637i 1.17232 + 1.62039i 0.799234 + 2.08835i −2.28825 − 0.874023i −2.00372 − 1.72775i −0.434041 − 2.79493i 2.93108 − 0.639360i 0.336872 − 3.14428i
629.20 −1.25943 − 0.643305i 1.72207 + 0.185637i 1.17232 + 1.62039i 0.799234 − 2.08835i −2.04941 − 1.34162i 2.00372 + 1.72775i −0.434041 − 2.79493i 2.93108 + 0.639360i −2.35003 + 2.11598i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 629.160
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
56.h odd 2 1 inner
105.g even 2 1 inner
120.i odd 2 1 inner
168.i even 2 1 inner
280.c odd 2 1 inner
840.u 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.u.e ✓ 160
3.b odd 2 1 inner 840.2.u.e ✓ 160
5.b even 2 1 inner 840.2.u.e ✓ 160
7.b odd 2 1 inner 840.2.u.e ✓ 160
8.b even 2 1 inner 840.2.u.e ✓ 160
15.d odd 2 1 inner 840.2.u.e ✓ 160
21.c even 2 1 inner 840.2.u.e ✓ 160
24.h odd 2 1 inner 840.2.u.e ✓ 160
35.c odd 2 1 inner 840.2.u.e ✓ 160
40.f even 2 1 inner 840.2.u.e ✓ 160
56.h odd 2 1 inner 840.2.u.e ✓ 160
105.g even 2 1 inner 840.2.u.e ✓ 160
120.i odd 2 1 inner 840.2.u.e ✓ 160
168.i even 2 1 inner 840.2.u.e ✓ 160
280.c odd 2 1 inner 840.2.u.e ✓ 160
840.u even 2 1 inner 840.2.u.e ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.u.e ✓ 160 1.a even 1 1 trivial
840.2.u.e ✓ 160 3.b odd 2 1 inner
840.2.u.e ✓ 160 5.b even 2 1 inner
840.2.u.e ✓ 160 7.b odd 2 1 inner
840.2.u.e ✓ 160 8.b even 2 1 inner
840.2.u.e ✓ 160 15.d odd 2 1 inner
840.2.u.e ✓ 160 21.c even 2 1 inner
840.2.u.e ✓ 160 24.h odd 2 1 inner
840.2.u.e ✓ 160 35.c odd 2 1 inner
840.2.u.e ✓ 160 40.f even 2 1 inner
840.2.u.e ✓ 160 56.h odd 2 1 inner
840.2.u.e ✓ 160 105.g even 2 1 inner
840.2.u.e ✓ 160 120.i odd 2 1 inner
840.2.u.e ✓ 160 168.i even 2 1 inner
840.2.u.e ✓ 160 280.c odd 2 1 inner
840.2.u.e ✓ 160 840.u even 2 1 inner

Hecke kernels

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

\( T_{11}^{20} - 128 T_{11}^{18} + 6741 T_{11}^{16} - 189322 T_{11}^{14} + 3078064 T_{11}^{12} + \cdots + 7585792 \) Copy content Toggle raw display
\( T_{23}^{20} - 228 T_{23}^{18} + 20468 T_{23}^{16} - 939472 T_{23}^{14} + 24073184 T_{23}^{12} + \cdots + 79691776 \) Copy content Toggle raw display
\( T_{73}^{20} - 740 T_{73}^{18} + 210448 T_{73}^{16} - 29178848 T_{73}^{14} + 2085671680 T_{73}^{12} + \cdots + 15535702016 \) Copy content Toggle raw display