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Properties

Label 220.3.s.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} + 16 q^{4} - 120 q^{5} - 20 q^{6} - 10 q^{8} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} + 16 q^{4} - 120 q^{5} - 20 q^{6} - 10 q^{8} + 64 q^{9} - 10 q^{10} - 70 q^{12} + 20 q^{13} + 83 q^{14} - 4 q^{16} + 24 q^{17} + 45 q^{18} - 25 q^{20} + 35 q^{22} - 62 q^{24} - 120 q^{25} - 88 q^{26} - 130 q^{28} - 8 q^{29} + 40 q^{30} + 152 q^{32} + 180 q^{33} - 126 q^{34} - 394 q^{36} + 144 q^{37} - 105 q^{38} + 132 q^{41} + 275 q^{42} + 113 q^{44} + 40 q^{45} - 85 q^{46} + 353 q^{48} + 292 q^{49} + 10 q^{50} + 62 q^{52} - 224 q^{53} - 328 q^{54} + 380 q^{56} - 364 q^{57} + 598 q^{58} + 105 q^{60} - 48 q^{61} + 88 q^{62} - 356 q^{64} - 40 q^{65} - 65 q^{66} - 117 q^{68} + 32 q^{69} - 165 q^{70} - 28 q^{72} + 16 q^{73} - 72 q^{74} - 720 q^{77} - 752 q^{78} - 235 q^{80} - 44 q^{81} + 49 q^{82} - 348 q^{84} + 60 q^{85} + 70 q^{86} - 165 q^{88} - 144 q^{89} + 778 q^{92} + 40 q^{93} + 254 q^{94} + 243 q^{96} + 204 q^{97} - 1042 q^{98}+O(q^{100}) \) 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 −1.99781 + 0.0935586i −3.38992 − 4.66582i 3.98249 − 0.373825i −0.690983 − 2.12663i 7.20895 + 9.00428i −3.89563 + 5.36188i −7.92129 + 1.11943i −7.49721 + 23.0740i 1.57942 + 4.18395i
31.2 −1.96516 − 0.371672i 0.447988 + 0.616603i 3.72372 + 1.46079i −0.690983 − 2.12663i −0.651195 − 1.37823i −5.62557 + 7.74293i −6.77478 − 4.25469i 2.60165 − 8.00705i 0.567486 + 4.43598i
31.3 −1.94215 + 0.477555i 2.93429 + 4.03870i 3.54388 − 1.85497i −0.690983 − 2.12663i −7.62752 − 6.44247i 0.424350 − 0.584067i −5.99690 + 5.29502i −4.91989 + 15.1419i 2.35757 + 3.80024i
31.4 −1.66947 + 1.10131i 1.47345 + 2.02803i 1.57424 − 3.67719i −0.690983 − 2.12663i −4.69337 − 1.76301i 2.86785 − 3.94726i 1.42157 + 7.87268i 0.839295 − 2.58308i 3.49564 + 2.78935i
31.5 −1.63161 + 1.15665i −1.85697 − 2.55591i 1.32432 − 3.77441i −0.690983 − 2.12663i 5.98615 + 2.02238i 3.39664 − 4.67507i 2.20489 + 7.69015i −0.303148 + 0.932993i 3.58718 + 2.67061i
31.6 −1.57173 − 1.23680i 1.77551 + 2.44378i 0.940640 + 3.88783i −0.690983 − 2.12663i 0.231859 − 6.03691i −0.489029 + 0.673091i 3.33004 − 7.27398i −0.0384734 + 0.118409i −1.54418 + 4.19708i
31.7 −1.56762 − 1.24200i −2.15722 − 2.96915i 0.914856 + 3.89397i −0.690983 − 2.12663i −0.306006 + 7.33377i 6.01155 − 8.27419i 3.40218 − 7.24052i −1.38114 + 4.25070i −1.55808 + 4.19194i
31.8 −1.07903 − 1.68395i −2.03349 − 2.79886i −1.67137 + 3.63408i −0.690983 − 2.12663i −2.51894 + 6.44437i −6.89923 + 9.49598i 7.92307 − 1.10678i −0.917387 + 2.82343i −2.83554 + 3.45828i
31.9 −1.06001 + 1.69599i −0.573224 − 0.788975i −1.75275 − 3.59553i −0.690983 − 2.12663i 1.94572 − 0.135859i −4.04910 + 5.57311i 7.95592 + 0.838656i 2.48726 − 7.65499i 4.33918 + 1.08235i
31.10 −0.759891 − 1.85002i 0.449591 + 0.618809i −2.84513 + 2.81162i −0.690983 − 2.12663i 0.803167 − 1.30198i 3.24183 − 4.46200i 7.36354 + 3.12701i 2.60036 − 8.00309i −3.40923 + 2.89434i
31.11 −0.367625 − 1.96592i 3.13168 + 4.31039i −3.72970 + 1.44544i −0.690983 − 2.12663i 7.32262 − 7.74126i −5.92221 + 8.15122i 4.21276 + 6.80093i −5.99089 + 18.4381i −3.92676 + 2.14022i
31.12 −0.139310 + 1.99514i 0.573224 + 0.788975i −3.96119 − 0.555886i −0.690983 − 2.12663i −1.65397 + 1.03375i 4.04910 − 5.57311i 1.66091 − 7.82569i 2.48726 − 7.65499i 4.33918 − 1.08235i
31.13 0.486802 − 1.93985i −1.63335 − 2.24812i −3.52605 − 1.88865i −0.690983 − 2.12663i −5.15613 + 2.07407i −2.89334 + 3.98235i −5.38018 + 5.92061i 0.394966 − 1.21558i −4.46171 + 0.305158i
31.14 0.640141 + 1.89479i 1.85697 + 2.55591i −3.18044 + 2.42586i −0.690983 − 2.12663i −3.65417 + 5.15471i −3.39664 + 4.67507i −6.63242 − 4.47336i −0.303148 + 0.932993i 3.58718 − 2.67061i
31.15 0.703296 + 1.87226i −1.47345 − 2.02803i −3.01075 + 2.63351i −0.690983 − 2.12663i 2.76074 − 4.18500i −2.86785 + 3.94726i −7.04808 − 3.78479i 0.839295 − 2.58308i 3.49564 − 2.78935i
31.16 0.746385 − 1.85551i 1.63335 + 2.24812i −2.88582 − 2.76985i −0.690983 − 2.12663i 5.39051 − 1.35274i 2.89334 − 3.98235i −7.29340 + 3.28729i 0.394966 − 1.21558i −4.46171 − 0.305158i
31.17 1.29053 + 1.52792i −2.93429 − 4.03870i −0.669057 + 3.94365i −0.690983 − 2.12663i 2.38400 − 9.69541i −0.424350 + 0.584067i −6.88900 + 4.06714i −4.91989 + 15.1419i 2.35757 − 3.80024i
31.18 1.45296 − 1.37438i −3.13168 − 4.31039i 0.222157 − 3.99383i −0.690983 − 2.12663i −10.4743 − 1.95868i 5.92221 − 8.15122i −5.16625 − 6.10818i −5.99089 + 18.4381i −3.92676 − 2.14022i
31.19 1.56127 + 1.24997i 3.38992 + 4.66582i 0.875130 + 3.90309i −0.690983 − 2.12663i −0.539578 + 11.5219i 3.89563 − 5.36188i −3.51245 + 7.18768i −7.49721 + 23.0740i 1.57942 − 4.18395i
31.20 1.70218 − 1.05004i −0.449591 − 0.618809i 1.79482 − 3.57472i −0.690983 − 2.12663i −1.41506 − 0.581233i −3.24183 + 4.46200i −0.698506 − 7.96945i 2.60036 − 8.00309i −3.40923 − 2.89434i
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.b ✓ 96
4.b odd 2 1 inner 220.3.s.b ✓ 96
11.c even 5 1 inner 220.3.s.b ✓ 96
44.h odd 10 1 inner 220.3.s.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.s.b ✓ 96 1.a even 1 1 trivial
220.3.s.b ✓ 96 4.b odd 2 1 inner
220.3.s.b ✓ 96 11.c even 5 1 inner
220.3.s.b ✓ 96 44.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} - 140 T_{3}^{94} + 11350 T_{3}^{92} - 700874 T_{3}^{90} + 36753095 T_{3}^{88} + \cdots + 95\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display