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Properties

Label 671.2.x.a
Level $671$
Weight $2$
Character orbit 671.x
Analytic conductor $5.358$
Analytic rank $0$
Dimension $240$
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] = [671,2,Mod(60,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.60");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.x (of order \(10\), degree \(4\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{3} + 52 q^{4} - 10 q^{5} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{3} + 52 q^{4} - 10 q^{5} - 62 q^{9} - 28 q^{12} - 10 q^{13} - 22 q^{14} + 6 q^{15} - 92 q^{16} + 2 q^{19} + 38 q^{20} - 24 q^{22} - 54 q^{25} + 36 q^{27} - 44 q^{34} - 14 q^{36} + 24 q^{39} + 28 q^{41} + 28 q^{42} - 16 q^{45} - 2 q^{46} - 22 q^{47} - 48 q^{48} + 16 q^{49} - 8 q^{52} - 12 q^{56} - 86 q^{57} - 42 q^{58} - 16 q^{60} - 14 q^{61} - 120 q^{62} + 92 q^{64} - 56 q^{65} + 60 q^{66} + 82 q^{70} + 16 q^{73} - 52 q^{74} + 14 q^{75} + 8 q^{76} - 38 q^{77} + 284 q^{80} - 54 q^{81} - 86 q^{83} + 72 q^{86} - 200 q^{88} + 44 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
60.1 −2.60729 + 0.847161i −2.30056 − 1.67146i 4.46226 − 3.24202i −0.382234 + 1.17640i 7.41423 + 2.40903i 0.568534 + 0.782520i −5.66510 + 7.79735i 1.57177 + 4.83740i − 3.39102i
60.2 −2.51592 + 0.817471i 1.45597 + 1.05783i 4.04354 − 2.93781i −1.17946 + 3.62999i −4.52784 − 1.47119i 2.08017 + 2.86311i −4.66181 + 6.41643i 0.0738076 + 0.227156i − 10.0969i
60.3 −2.47793 + 0.805127i 0.0715911 + 0.0520140i 3.87386 − 2.81452i 0.387254 − 1.19185i −0.219275 − 0.0712469i 0.427052 + 0.587786i −4.27020 + 5.87743i −0.924631 − 2.84572i 3.26509i
60.4 −2.43284 + 0.790479i 0.249958 + 0.181605i 3.67583 − 2.67065i −1.02358 + 3.15026i −0.751665 − 0.244231i −2.70316 − 3.72058i −3.82448 + 5.26394i −0.897552 − 2.76238i − 8.47320i
60.5 −2.35051 + 0.763728i −1.01104 − 0.734560i 3.32360 − 2.41473i 0.490368 − 1.50920i 2.93746 + 0.954437i −2.32261 − 3.19680i −3.06256 + 4.21525i −0.444437 − 1.36784i 3.92190i
60.6 −2.20430 + 0.716220i 2.37701 + 1.72700i 2.72792 − 1.98195i −0.110772 + 0.340921i −6.47654 − 2.10436i 0.369406 + 0.508444i −1.86898 + 2.57243i 1.74060 + 5.35701i − 0.830828i
60.7 −2.16134 + 0.702262i −0.584104 − 0.424376i 2.56019 − 1.86009i −0.265324 + 0.816583i 1.56047 + 0.507028i 2.17499 + 2.99361i −1.55561 + 2.14111i −0.765969 − 2.35741i − 1.95124i
60.8 −2.16037 + 0.701947i −1.78495 − 1.29684i 2.55643 − 1.85736i 1.16147 − 3.57464i 4.76646 + 1.54872i −0.0472233 − 0.0649973i −1.54871 + 2.13162i 0.577192 + 1.77641i 8.53783i
60.9 −1.94928 + 0.633359i 1.44148 + 1.04730i 1.78051 − 1.29362i 0.577706 − 1.77800i −3.47317 − 1.12850i −1.27825 − 1.75936i −0.241943 + 0.333006i 0.0539872 + 0.166156i 3.83170i
60.10 −1.91777 + 0.623120i 1.85593 + 1.34841i 1.67152 − 1.21443i 1.07729 − 3.31556i −4.39946 − 1.42947i 2.47070 + 3.40062i −0.0783571 + 0.107849i 0.699210 + 2.15195i 7.02976i
60.11 −1.83449 + 0.596063i −2.07036 − 1.50421i 1.39204 − 1.01138i 0.450958 − 1.38790i 4.69467 + 1.52539i 2.52359 + 3.47343i 0.316714 − 0.435920i 1.09672 + 3.37534i 2.81490i
60.12 −1.72954 + 0.561963i −2.75928 − 2.00473i 1.05748 − 0.768306i −0.0884230 + 0.272138i 5.89888 + 1.91666i −1.66296 − 2.28887i 0.740630 − 1.01939i 2.66762 + 8.21008i − 0.520365i
60.13 −1.70819 + 0.555024i −0.956094 − 0.694643i 0.991824 − 0.720602i −1.12853 + 3.47326i 2.01873 + 0.655926i 1.14864 + 1.58096i 0.817167 − 1.12473i −0.495464 − 1.52488i − 6.55935i
60.14 −1.68193 + 0.546491i 0.850090 + 0.617626i 0.912191 − 0.662746i −0.245801 + 0.756498i −1.76732 − 0.574236i −1.75353 − 2.41353i 0.906922 − 1.24827i −0.585861 − 1.80309i − 1.40670i
60.15 −1.68044 + 0.546008i −1.61800 − 1.17554i 0.907717 − 0.659495i −0.710083 + 2.18541i 3.36080 + 1.09199i −0.822391 − 1.13192i 0.911863 − 1.25507i 0.308958 + 0.950876i − 4.06016i
60.16 −1.57752 + 0.512567i 1.27513 + 0.926435i 0.607804 − 0.441595i −0.707460 + 2.17734i −2.48640 − 0.807880i −0.227660 − 0.313348i 1.21744 − 1.67567i −0.159380 − 0.490521i − 3.79741i
60.17 −1.57259 + 0.510967i −0.426906 − 0.310166i 0.593930 − 0.431515i 1.15839 − 3.56517i 0.829834 + 0.269630i −0.767863 − 1.05687i 1.23031 − 1.69338i −0.841005 − 2.58835i 6.19846i
60.18 −1.15420 + 0.375023i 2.23204 + 1.62167i −0.426493 + 0.309865i −1.09743 + 3.37753i −3.18439 − 1.03467i 0.676882 + 0.931648i 1.80273 − 2.48124i 1.42514 + 4.38612i − 4.30991i
60.19 −1.09385 + 0.355415i 0.613208 + 0.445522i −0.547835 + 0.398026i 0.256819 − 0.790408i −0.829106 − 0.269393i 2.87860 + 3.96205i 1.80987 − 2.49107i −0.749516 − 2.30677i 0.955869i
60.20 −1.02449 + 0.332878i −1.12142 − 0.814758i −0.679255 + 0.493508i 0.391788 − 1.20580i 1.42010 + 0.461419i −2.20150 − 3.03010i 1.79796 − 2.47468i −0.333303 − 1.02580i 1.36575i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 60.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
61.b even 2 1 inner
671.x even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.x.a ✓ 240
11.c even 5 1 inner 671.2.x.a ✓ 240
61.b even 2 1 inner 671.2.x.a ✓ 240
671.x even 10 1 inner 671.2.x.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.x.a ✓ 240 1.a even 1 1 trivial
671.2.x.a ✓ 240 11.c even 5 1 inner
671.2.x.a ✓ 240 61.b even 2 1 inner
671.2.x.a ✓ 240 671.x even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).