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Properties

Label 440.2.bi.c
Level $440$
Weight $2$
Character orbit 440.bi
Analytic conductor $3.513$
Analytic rank $0$
Dimension $176$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(141,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.141");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.bi (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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 8 q^{2} - 4 q^{4} + 2 q^{6} + 18 q^{7} - 4 q^{8} + 60 q^{9} - 16 q^{10} + 24 q^{12} + 12 q^{14} + 8 q^{15} - 32 q^{16} - 4 q^{17} + 12 q^{18} + 4 q^{20} + 56 q^{22} - 28 q^{23} - 48 q^{24} + 44 q^{25}+ \cdots + 64 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1 −1.40731 + 0.139522i −1.64066 − 2.25818i 1.96107 − 0.392703i 0.951057 − 0.309017i 2.62400 + 2.94906i −2.50625 − 1.82090i −2.70505 + 0.826268i −1.48055 + 4.55666i −1.29532 + 0.567577i
141.2 −1.40089 − 0.193691i −0.181740 − 0.250144i 1.92497 + 0.542678i 0.951057 − 0.309017i 0.206147 + 0.385625i 1.70838 + 1.24121i −2.59155 − 1.13308i 0.897508 − 2.76225i −1.39218 + 0.248687i
141.3 −1.38798 − 0.271150i 0.737421 + 1.01497i 1.85296 + 0.752699i −0.951057 + 0.309017i −0.748313 − 1.60871i −1.82369 − 1.32499i −2.36776 − 1.54716i 0.440670 − 1.35624i 1.40383 − 0.171029i
141.4 −1.38740 − 0.274076i −1.15082 − 1.58396i 1.84976 + 0.760508i −0.951057 + 0.309017i 1.16252 + 2.51301i 0.982305 + 0.713686i −2.35793 − 1.56211i −0.257512 + 0.792540i 1.40419 − 0.168069i
141.5 −1.33308 + 0.472107i −1.10200 − 1.51677i 1.55423 − 1.25872i −0.951057 + 0.309017i 2.18513 + 1.50172i −0.250287 − 0.181844i −1.47767 + 2.41174i −0.159139 + 0.489779i 1.12195 − 0.860946i
141.6 −1.30840 + 0.536747i 0.629032 + 0.865789i 1.42381 − 1.40456i 0.951057 − 0.309017i −1.28773 − 0.795164i −2.90328 − 2.10936i −1.10901 + 2.60194i 0.573143 − 1.76395i −1.07850 + 0.914794i
141.7 −1.30393 + 0.547521i 1.63487 + 2.25021i 1.40044 − 1.42785i 0.951057 − 0.309017i −3.36378 − 2.03898i 1.14662 + 0.833067i −1.04429 + 2.62858i −1.46358 + 4.50442i −1.07091 + 0.923658i
141.8 −1.16760 + 0.797936i 0.447728 + 0.616245i 0.726598 − 1.86335i −0.951057 + 0.309017i −1.01449 − 0.362272i −3.07379 − 2.23324i 0.638451 + 2.75543i 0.747754 − 2.30135i 0.863882 − 1.11969i
141.9 −1.16620 − 0.799992i 1.24182 + 1.70922i 0.720027 + 1.86589i 0.951057 − 0.309017i −0.0808452 − 2.98672i 4.19870 + 3.05053i 0.653007 − 2.75201i −0.452256 + 1.39190i −1.35633 − 0.400463i
141.10 −1.11143 − 0.874482i 0.817965 + 1.12583i 0.470561 + 1.94385i −0.951057 + 0.309017i 0.0754081 − 1.96658i 1.51723 + 1.10233i 1.17687 − 2.57196i 0.328619 − 1.01139i 1.32726 + 0.488231i
141.11 −1.07634 + 0.917325i −1.43949 − 1.98129i 0.317028 − 1.97471i 0.951057 − 0.309017i 3.36687 + 0.812064i 2.79417 + 2.03008i 1.47022 + 2.41629i −0.926319 + 2.85092i −0.740194 + 1.20504i
141.12 −0.936190 − 1.05998i −1.76822 − 2.43375i −0.247097 + 1.98468i −0.951057 + 0.309017i −0.924324 + 4.15273i −3.56481 − 2.58998i 2.33504 − 1.59612i −1.86948 + 5.75367i 1.21792 + 0.718798i
141.13 −0.928705 − 1.06654i 0.510216 + 0.702252i −0.275014 + 1.98100i 0.951057 − 0.309017i 0.275139 − 1.19635i −3.04744 − 2.21410i 2.36822 − 1.54645i 0.694213 − 2.13657i −1.21283 − 0.727354i
141.14 −0.867603 + 1.11681i 1.68053 + 2.31305i −0.494531 − 1.93790i −0.951057 + 0.309017i −4.04128 − 0.129976i 1.91912 + 1.39432i 2.59332 + 1.12903i −1.59898 + 4.92115i 0.480026 − 1.33025i
141.15 −0.753980 + 1.19646i 0.146254 + 0.201302i −0.863029 − 1.80421i −0.951057 + 0.309017i −0.351122 + 0.0232099i 0.460550 + 0.334609i 2.80937 + 0.327759i 0.907919 − 2.79429i 0.347351 − 1.37089i
141.16 −0.687331 − 1.23595i −1.58735 − 2.18480i −1.05515 + 1.69902i 0.951057 − 0.309017i −1.60927 + 3.46357i 2.62183 + 1.90487i 2.82514 + 0.136332i −1.32662 + 4.08290i −1.03562 − 0.963063i
141.17 −0.630440 + 1.26592i 1.00423 + 1.38221i −1.20509 − 1.59617i 0.951057 − 0.309017i −2.38287 + 0.399877i 1.69771 + 1.23346i 2.78035 − 0.519256i 0.0250325 − 0.0770420i −0.208394 + 1.39878i
141.18 −0.373026 − 1.36413i −0.645500 − 0.888455i −1.72170 + 1.01771i 0.951057 − 0.309017i −0.971180 + 1.21196i 0.366696 + 0.266420i 2.03053 + 1.96899i 0.554369 − 1.70617i −0.776308 − 1.18209i
141.19 −0.265232 − 1.38912i 1.08077 + 1.48755i −1.85930 + 0.736877i 0.951057 − 0.309017i 1.77973 − 1.89586i −0.397798 − 0.289017i 1.51676 + 2.38735i −0.117688 + 0.362207i −0.681512 − 1.23917i
141.20 −0.234051 + 1.39471i −1.00423 − 1.38221i −1.89044 − 0.652866i −0.951057 + 0.309017i 2.16283 − 1.07711i 1.69771 + 1.23346i 1.35302 − 2.48382i 0.0250325 − 0.0770420i −0.208394 − 1.39878i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.c even 5 1 inner
88.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.bi.c ✓ 176
8.b even 2 1 inner 440.2.bi.c ✓ 176
11.c even 5 1 inner 440.2.bi.c ✓ 176
88.o even 10 1 inner 440.2.bi.c ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.bi.c ✓ 176 1.a even 1 1 trivial
440.2.bi.c ✓ 176 8.b even 2 1 inner
440.2.bi.c ✓ 176 11.c even 5 1 inner
440.2.bi.c ✓ 176 88.o even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{176} - 96 T_{3}^{174} + 4862 T_{3}^{172} - 173153 T_{3}^{170} + 4880921 T_{3}^{168} + \cdots + 27\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display