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Properties

Label 220.3.w.a
Level $220$
Weight $3$
Character orbit 220.w
Analytic conductor $5.995$
Analytic rank $0$
Dimension $544$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(7,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 5, 14]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.w (of order \(20\), degree \(8\), 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: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 10 q^{2} - 12 q^{5} - 20 q^{6} - 10 q^{8} - 28 q^{12} - 20 q^{13} - 36 q^{16} - 20 q^{17} - 10 q^{18} - 40 q^{20} + 86 q^{22} - 12 q^{25} + 140 q^{26} - 10 q^{28} - 370 q^{30} - 100 q^{33} - 476 q^{36}+ \cdots + 148 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.99975 − 0.0316704i 4.68739 − 2.38834i 3.99799 + 0.126666i −2.84565 − 4.11124i −9.44924 + 4.62763i −0.358764 + 0.704115i −7.99097 − 0.379918i 10.9773 − 15.1090i 5.56039 + 8.31156i
7.2 −1.99886 − 0.0675795i −1.94413 + 0.990585i 3.99087 + 0.270163i −3.70748 + 3.35479i 3.95299 − 1.84865i −1.57918 + 3.09932i −7.95892 − 0.809719i −2.49168 + 3.42950i 7.63743 − 6.45520i
7.3 −1.98295 + 0.260620i −1.14015 + 0.580936i 3.86415 − 1.03359i 1.56915 − 4.74740i 2.10946 − 1.44911i 5.70983 − 11.2062i −7.39304 + 3.05663i −4.32761 + 5.95644i −1.87427 + 9.82278i
7.4 −1.96643 + 0.364900i 1.14015 − 0.580936i 3.73370 − 1.43510i 1.56915 − 4.74740i −2.03004 + 1.55841i −5.70983 + 11.2062i −6.81838 + 4.18445i −4.32761 + 5.95644i −1.35329 + 9.90801i
7.5 −1.96237 − 0.386136i 0.676076 − 0.344478i 3.70180 + 1.51549i 3.50937 + 3.56150i −1.45973 + 0.414936i −3.62504 + 7.11454i −6.67911 − 4.40334i −4.95165 + 6.81537i −5.51146 − 8.34409i
7.6 −1.94429 − 0.468777i 4.04592 − 2.06150i 3.56050 + 1.82287i 2.33189 + 4.42293i −8.83280 + 2.11151i 3.03067 − 5.94803i −6.06810 − 5.21327i 6.82960 − 9.40014i −2.46049 − 9.69257i
7.7 −1.89633 − 0.635566i −4.39098 + 2.23732i 3.19211 + 2.41048i 4.88411 − 1.07027i 9.74869 − 1.45192i −1.29034 + 2.53243i −4.52126 − 6.59986i 8.98505 − 12.3669i −9.94210 − 1.07459i
7.8 −1.89209 + 0.648077i −4.68739 + 2.38834i 3.15999 − 2.45244i −2.84565 − 4.11124i 7.32112 − 7.55674i 0.358764 − 0.704115i −4.38962 + 6.68814i 10.9773 − 15.1090i 8.04862 + 5.93462i
7.9 −1.88014 + 0.681953i 1.94413 − 0.990585i 3.06988 − 2.56434i −3.70748 + 3.35479i −2.97971 + 3.18825i 1.57918 − 3.09932i −4.02306 + 6.91484i −2.49168 + 3.42950i 4.68278 − 8.83581i
7.10 −1.82522 − 0.817668i −2.40882 + 1.22736i 2.66284 + 2.98484i −4.64336 − 1.85452i 5.40020 − 0.270575i 0.118689 − 0.232940i −2.41965 − 7.62531i −0.994045 + 1.36819i 6.95876 + 7.18162i
7.11 −1.75740 − 0.954750i 2.01897 − 1.02872i 2.17690 + 3.35575i 4.19108 − 2.72669i −4.53030 − 0.119746i 2.63142 − 5.16446i −0.621782 − 7.97580i −2.27209 + 3.12726i −9.96871 + 0.790451i
7.12 −1.74700 + 0.973643i −0.676076 + 0.344478i 2.10404 − 3.40192i 3.50937 + 3.56150i 0.845708 − 1.26006i 3.62504 − 7.11454i −0.363505 + 7.99174i −4.95165 + 6.81537i −9.59852 − 2.80508i
7.13 −1.70427 + 1.04665i −4.04592 + 2.06150i 1.80904 − 3.56754i 2.33189 + 4.42293i 4.73765 − 7.74800i −3.03067 + 5.94803i 0.650882 + 7.97348i 6.82960 − 9.40014i −8.60342 − 5.09717i
7.14 −1.60711 + 1.19046i 4.39098 − 2.23732i 1.16563 − 3.82640i 4.88411 − 1.07027i −4.39337 + 8.82289i 1.29034 − 2.53243i 2.68187 + 7.53708i 8.98505 − 12.3669i −6.57520 + 7.53437i
7.15 −1.57651 − 1.23070i −3.31218 + 1.68764i 0.970777 + 3.88041i 0.464993 + 4.97833i 7.29866 + 1.41570i 5.48368 − 10.7623i 3.24516 − 7.31224i 2.83235 − 3.89839i 5.39374 − 8.42066i
7.16 −1.56459 − 1.24582i 3.70393 − 1.88724i 0.895857 + 3.89839i −4.57559 + 2.01593i −8.14628 − 1.66168i −4.38944 + 8.61475i 3.45505 − 7.21544i 4.86731 − 6.69928i 9.67039 + 2.54628i
7.17 −1.49892 − 1.32409i 0.172588 − 0.0879380i 0.493549 + 3.96943i −2.34222 − 4.41746i −0.375135 − 0.0967104i −1.03219 + 2.02578i 4.51611 − 6.60339i −5.26801 + 7.25080i −2.33832 + 9.72277i
7.18 −1.48321 + 1.34167i 2.40882 − 1.22736i 0.399834 − 3.97997i −4.64336 − 1.85452i −1.92608 + 5.05228i −0.118689 + 0.232940i 4.74677 + 6.43958i −0.994045 + 1.36819i 9.37524 − 3.47922i
7.19 −1.37635 + 1.45109i −2.01897 + 1.02872i −0.211311 − 3.99441i 4.19108 − 2.72669i 1.28605 − 4.34558i −2.63142 + 5.16446i 6.08708 + 5.19109i −2.27209 + 3.12726i −1.81173 + 9.83451i
7.20 −1.14597 − 1.63913i 1.96587 − 1.00166i −1.37351 + 3.75679i −3.42603 + 3.64175i −3.89468 − 2.07445i 3.39399 − 6.66107i 7.73187 − 2.05381i −2.42874 + 3.34288i 9.89543 + 1.44238i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
11.d odd 10 1 inner
20.e even 4 1 inner
44.g even 10 1 inner
55.l even 20 1 inner
220.w odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.3.w.a ✓ 544
4.b odd 2 1 inner 220.3.w.a ✓ 544
5.c odd 4 1 inner 220.3.w.a ✓ 544
11.d odd 10 1 inner 220.3.w.a ✓ 544
20.e even 4 1 inner 220.3.w.a ✓ 544
44.g even 10 1 inner 220.3.w.a ✓ 544
55.l even 20 1 inner 220.3.w.a ✓ 544
220.w odd 20 1 inner 220.3.w.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.w.a ✓ 544 1.a even 1 1 trivial
220.3.w.a ✓ 544 4.b odd 2 1 inner
220.3.w.a ✓ 544 5.c odd 4 1 inner
220.3.w.a ✓ 544 11.d odd 10 1 inner
220.3.w.a ✓ 544 20.e even 4 1 inner
220.3.w.a ✓ 544 44.g even 10 1 inner
220.3.w.a ✓ 544 55.l even 20 1 inner
220.3.w.a ✓ 544 220.w odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(220, [\chi])\).