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Properties

Label 220.3.s.a
Level $220$
Weight $3$
Character orbit 220.s
Analytic conductor $5.995$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(31,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.s (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: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} - 4 q^{4} + 120 q^{5} + 20 q^{6} - 10 q^{8} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} - 4 q^{4} + 120 q^{5} + 20 q^{6} - 10 q^{8} + 64 q^{9} + 10 q^{10} + 70 q^{12} - 20 q^{13} - 37 q^{14} - 4 q^{16} + 24 q^{17} + 45 q^{18} + 5 q^{20} - 15 q^{22} - 62 q^{24} - 120 q^{25} + 12 q^{26} - 200 q^{28} + 72 q^{29} + 40 q^{30} - 348 q^{32} - 60 q^{33} - 246 q^{34} + 26 q^{36} + 144 q^{37} - 105 q^{38} + 132 q^{41} - 25 q^{42} - 67 q^{44} - 40 q^{45} - 85 q^{46} + 213 q^{48} - 28 q^{49} + 10 q^{50} + 312 q^{52} - 224 q^{53} + 752 q^{54} + 660 q^{56} - 204 q^{57} + 88 q^{58} + 105 q^{60} - 48 q^{61} + 88 q^{62} + 44 q^{64} - 40 q^{65} + 75 q^{66} - 117 q^{68} + 32 q^{69} - 75 q^{70} - 198 q^{72} - 64 q^{73} - 672 q^{74} - 1280 q^{76} + 80 q^{77} - 1132 q^{78} - 125 q^{80} - 44 q^{81} - 301 q^{82} - 348 q^{84} - 60 q^{85} + 470 q^{86} + 5 q^{88} + 176 q^{89} - 162 q^{92} + 280 q^{93} + 394 q^{94} + 1143 q^{96} + 204 q^{97} + 158 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.99616 − 0.123819i 1.59683 + 2.19784i 3.96934 + 0.494327i 0.690983 + 2.12663i −2.91539 − 4.58497i 7.19852 − 9.90792i −7.86224 − 1.47824i 0.500496 − 1.54037i −1.11600 − 4.33065i
31.2 −1.94953 − 0.446488i 2.97721 + 4.09778i 3.60130 + 1.74088i 0.690983 + 2.12663i −3.97454 − 9.31801i −5.14097 + 7.07593i −6.24354 − 5.00182i −5.14685 + 15.8404i −0.397576 − 4.45443i
31.3 −1.94089 + 0.482647i −1.83428 − 2.52467i 3.53410 − 1.87353i 0.690983 + 2.12663i 4.77866 + 4.01480i −2.53678 + 3.49158i −5.95505 + 5.34204i −0.228228 + 0.702415i −2.36753 − 3.79405i
31.4 −1.82595 − 0.816028i −1.81835 − 2.50274i 2.66820 + 2.98006i 0.690983 + 2.12663i 1.27791 + 6.05370i −0.390935 + 0.538076i −2.44019 − 7.61876i −0.176165 + 0.542179i 0.473686 − 4.44698i
31.5 −1.50655 + 1.31541i −2.29101 − 3.15331i 0.539403 − 3.96346i 0.690983 + 2.12663i 7.59942 + 1.73701i 5.93140 − 8.16388i 4.40093 + 6.68070i −1.91346 + 5.88902i −3.83838 − 2.29495i
31.6 −1.49291 − 1.33087i 0.367286 + 0.505525i 0.457554 + 3.97374i 0.690983 + 2.12663i 0.124466 − 1.24351i −0.904411 + 1.24481i 4.60546 − 6.54138i 2.66050 − 8.18816i 1.79870 − 4.09447i
31.7 −1.38558 + 1.44228i 2.61105 + 3.59381i −0.160338 − 3.99679i 0.690983 + 2.12663i −8.80109 − 1.21364i −0.802115 + 1.10402i 5.98664 + 5.30661i −3.31669 + 10.2077i −4.02460 − 1.95002i
31.8 −0.874978 + 1.79845i 0.667552 + 0.918806i −2.46883 − 3.14721i 0.690983 + 2.12663i −2.23652 + 0.396621i 4.67079 − 6.42880i 7.82025 − 1.68632i 2.38257 − 7.33281i −4.42922 − 0.618056i
31.9 −0.707726 − 1.87059i −3.23278 − 4.44954i −2.99825 + 2.64774i 0.690983 + 2.12663i −6.03537 + 9.19628i 1.08319 − 1.49088i 7.07478 + 3.73464i −6.56639 + 20.2093i 3.48903 − 2.79762i
31.10 −0.594717 − 1.90953i 0.0826374 + 0.113741i −3.29262 + 2.27126i 0.690983 + 2.12663i 0.168045 − 0.225442i −1.63068 + 2.24443i 6.29523 + 4.93661i 2.77504 − 8.54071i 3.64992 − 2.58420i
31.11 −0.349228 + 1.96927i −0.667552 − 0.918806i −3.75608 − 1.37545i 0.690983 + 2.12663i 2.04251 − 0.993719i −4.67079 + 6.42880i 4.02037 − 6.91640i 2.38257 − 7.33281i −4.42922 + 0.618056i
31.12 −0.117720 − 1.99653i 2.91538 + 4.01268i −3.97228 + 0.470065i 0.690983 + 2.12663i 7.66825 − 6.29303i 7.54604 − 10.3862i 1.40612 + 7.87546i −4.82100 + 14.8375i 4.16454 − 1.62992i
31.13 0.273207 + 1.98125i −2.61105 − 3.59381i −3.85072 + 1.08258i 0.690983 + 2.12663i 6.40688 − 6.15501i 0.802115 − 1.10402i −3.19691 − 7.33347i −3.31669 + 10.2077i −4.02460 + 1.95002i
31.14 0.284167 − 1.97971i 1.08637 + 1.49525i −3.83850 − 1.12514i 0.690983 + 2.12663i 3.26888 − 1.72579i −6.46077 + 8.89248i −3.31822 + 7.27938i 1.72556 − 5.31072i 4.40646 − 0.763628i
31.15 0.445649 + 1.94972i 2.29101 + 3.15331i −3.60279 + 1.73778i 0.690983 + 2.12663i −5.12707 + 5.87209i −5.93140 + 8.16388i −4.99376 − 6.24999i −1.91346 + 5.88902i −3.83838 + 2.29495i
31.16 0.933748 − 1.76865i −1.08637 − 1.49525i −2.25623 − 3.30294i 0.690983 + 2.12663i −3.65897 + 0.525208i 6.46077 − 8.89248i −7.94849 + 0.906360i 1.72556 − 5.31072i 4.40646 + 0.763628i
31.17 1.26877 − 1.54603i −2.91538 − 4.01268i −0.780445 − 3.92312i 0.690983 + 2.12663i −9.90270 − 0.583887i −7.54604 + 10.3862i −7.05549 − 3.77095i −4.82100 + 14.8375i 4.16454 + 1.62992i
31.18 1.28652 + 1.53130i 1.83428 + 2.52467i −0.689733 + 3.94008i 0.690983 + 2.12663i −1.50618 + 6.05687i 2.53678 − 3.49158i −6.92079 + 4.01281i −0.228228 + 0.702415i −2.36753 + 3.79405i
31.19 1.60353 − 1.19528i −0.0826374 − 0.113741i 1.14262 − 3.83333i 0.690983 + 2.12663i −0.268463 − 0.0836120i 1.63068 − 2.24443i −2.74966 − 7.51261i 2.77504 − 8.54071i 3.64992 + 2.58420i
31.20 1.67207 − 1.09735i 3.23278 + 4.44954i 1.59164 − 3.66970i 0.690983 + 2.12663i 10.2881 + 3.89245i −1.08319 + 1.49088i −1.36562 − 7.88258i −6.56639 + 20.2093i 3.48903 + 2.79762i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.c even 5 1 inner
44.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.3.s.a ✓ 96
4.b odd 2 1 inner 220.3.s.a ✓ 96
11.c even 5 1 inner 220.3.s.a ✓ 96
44.h odd 10 1 inner 220.3.s.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.s.a ✓ 96 1.a even 1 1 trivial
220.3.s.a ✓ 96 4.b odd 2 1 inner
220.3.s.a ✓ 96 11.c even 5 1 inner
220.3.s.a ✓ 96 44.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} - 140 T_{3}^{94} + 11350 T_{3}^{92} - 698554 T_{3}^{90} + 36300215 T_{3}^{88} + \cdots + 34\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display