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Properties

Label 221.2.z.a
Level $221$
Weight $2$
Character orbit 221.z
Analytic conductor $1.765$
Analytic rank $0$
Dimension $152$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [221,2,Mod(44,221)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(221, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([12, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("221.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.z (of order \(16\), 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: \(1.76469388467\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(19\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9} - 8 q^{11} - 24 q^{15} - 16 q^{17} + 16 q^{18} - 8 q^{19} - 80 q^{20} - 32 q^{22} - 40 q^{24} - 8 q^{26} - 16 q^{27} - 24 q^{28} + 24 q^{29} - 40 q^{31} + 120 q^{32} + 48 q^{33} - 40 q^{34} - 32 q^{35} - 8 q^{37} + 80 q^{38} - 8 q^{39} - 16 q^{40} + 32 q^{42} + 64 q^{43} + 24 q^{44} - 16 q^{45} + 24 q^{46} - 96 q^{47} + 32 q^{48} - 16 q^{49} - 16 q^{52} - 40 q^{53} + 16 q^{54} - 48 q^{55} + 32 q^{57} + 88 q^{58} + 56 q^{59} + 16 q^{60} + 32 q^{61} - 96 q^{62} + 64 q^{63} + 48 q^{64} + 32 q^{65} - 224 q^{66} - 64 q^{67} - 16 q^{68} - 88 q^{70} - 72 q^{71} + 72 q^{73} + 104 q^{74} + 112 q^{75} - 120 q^{76} + 56 q^{78} - 80 q^{79} - 16 q^{81} - 8 q^{83} + 160 q^{84} + 24 q^{85} - 16 q^{86} + 80 q^{87} - 8 q^{90} - 128 q^{91} - 16 q^{92} - 16 q^{94} - 64 q^{95} + 64 q^{96} - 56 q^{97} - 88 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −0.984825 − 2.37758i −2.11739 − 1.41479i −3.26879 + 3.26879i −0.563241 + 0.842950i −1.27853 + 6.42758i −0.960771 − 1.43790i 6.23582 + 2.58296i 1.33364 + 3.21970i 2.55887 + 0.508992i
44.2 −0.841331 − 2.03115i 0.600920 + 0.401522i −2.00353 + 2.00353i 0.250002 − 0.374154i 0.309980 − 1.55837i −0.555096 − 0.830760i 1.69279 + 0.701176i −0.948165 − 2.28907i −0.970297 − 0.193004i
44.3 −0.779496 − 1.88187i −1.16465 − 0.778196i −1.51961 + 1.51961i 2.30627 − 3.45157i −0.556621 + 2.79832i 0.508755 + 0.761405i 0.280487 + 0.116182i −0.397224 − 0.958985i −8.29313 − 1.64961i
44.4 −0.728016 − 1.75759i −0.779718 − 0.520991i −1.14489 + 1.14489i −1.76865 + 2.64697i −0.348039 + 1.74971i 2.64641 + 3.96063i −0.669433 − 0.277288i −0.811522 − 1.95919i 5.93988 + 1.18152i
44.5 −0.697165 − 1.68310i 2.59086 + 1.73116i −0.932588 + 0.932588i −2.00990 + 3.00802i 1.10746 − 5.56759i 0.727170 + 1.08829i −1.14640 − 0.474853i 2.56760 + 6.19874i 6.46405 + 1.28578i
44.6 −0.392573 − 0.947754i 1.23769 + 0.827000i 0.670089 − 0.670089i 0.936343 − 1.40134i 0.297908 − 1.49769i −0.783053 − 1.17192i −2.79365 − 1.15717i −0.300096 − 0.724496i −1.69571 − 0.337297i
44.7 −0.306289 − 0.739446i −0.471645 − 0.315143i 0.961245 − 0.961245i −1.97622 + 2.95762i −0.0885718 + 0.445281i −2.57309 − 3.85090i −2.48410 − 1.02895i −1.02492 − 2.47437i 2.79230 + 0.555422i
44.8 −0.192390 − 0.464472i 1.59700 + 1.06708i 1.23549 − 1.23549i 0.191542 − 0.286663i 0.188381 − 0.947055i 1.43415 + 2.14636i −1.74049 − 0.720935i 0.263689 + 0.636602i −0.169998 − 0.0338147i
44.9 −0.139805 − 0.337519i −2.15216 − 1.43803i 1.31984 − 1.31984i 1.49666 − 2.23991i −0.184479 + 0.927438i 0.0376968 + 0.0564173i −1.30503 − 0.540561i 1.41582 + 3.41808i −0.965254 − 0.192001i
44.10 0.0161892 + 0.0390843i −1.17844 − 0.787411i 1.41295 − 1.41295i −0.213750 + 0.319900i 0.0116973 − 0.0588062i 2.71459 + 4.06267i 0.156267 + 0.0647279i −0.379336 − 0.915798i −0.0159635 − 0.00317533i
44.11 0.271988 + 0.656637i −0.671616 − 0.448759i 1.05702 − 1.05702i 0.388698 − 0.581728i 0.112001 − 0.563065i −1.33635 − 1.99999i 2.29485 + 0.950557i −0.898367 − 2.16885i 0.487705 + 0.0970106i
44.12 0.341299 + 0.823968i 2.47943 + 1.65670i 0.851775 − 0.851775i −0.678056 + 1.01478i −0.518843 + 2.60840i −2.60342 − 3.89629i 2.64048 + 1.09372i 2.25486 + 5.44371i −1.06757 − 0.212352i
44.13 0.387074 + 0.934480i −2.72959 − 1.82386i 0.690787 − 0.690787i −2.09932 + 3.14186i 0.647801 − 3.25672i −0.0152419 − 0.0228111i 2.78187 + 1.15229i 2.97618 + 7.18512i −3.74860 − 0.745643i
44.14 0.509634 + 1.23037i 0.464497 + 0.310367i 0.160142 − 0.160142i −1.48137 + 2.21702i −0.145141 + 0.729674i 0.874722 + 1.30911i 2.73938 + 1.13469i −1.02862 − 2.48331i −3.48271 − 0.692753i
44.15 0.691922 + 1.67045i 0.178554 + 0.119306i −0.897427 + 0.897427i 1.86415 − 2.78990i −0.0757490 + 0.380816i 1.17598 + 1.75998i 1.22084 + 0.505689i −1.13040 − 2.72903i 5.95022 + 1.18357i
44.16 0.775433 + 1.87206i −2.10259 − 1.40490i −1.48910 + 1.48910i 1.29487 − 1.93791i 0.999651 − 5.02558i −2.23914 − 3.35111i −0.198263 − 0.0821234i 1.29907 + 3.13624i 4.63196 + 0.921354i
44.17 0.813483 + 1.96392i 1.03290 + 0.690159i −1.78102 + 1.78102i −1.32730 + 1.98644i −0.515175 + 2.58996i 0.0574971 + 0.0860505i −1.01878 − 0.421991i −0.557496 − 1.34591i −4.98095 − 0.990772i
44.18 0.996319 + 2.40533i 1.79121 + 1.19685i −3.37874 + 3.37874i 1.00493 − 1.50398i −1.09419 + 5.50088i −1.79926 − 2.69278i −6.68261 − 2.76803i 0.627934 + 1.51597i 4.61880 + 0.918736i
44.19 1.05544 + 2.54807i −1.37062 − 0.915818i −3.96447 + 3.96447i −0.680190 + 1.01798i 0.886955 − 4.45902i 1.14724 + 1.71697i −9.18989 − 3.80658i −0.108178 − 0.261164i −3.31177 − 0.658753i
57.1 −2.44282 + 1.01185i −0.336323 + 0.0668989i 3.52932 − 3.52932i 0.0589569 + 0.296397i 0.753886 − 0.503730i 0.562652 − 2.82864i −3.02666 + 7.30700i −2.66300 + 1.10305i −0.443930 − 0.664388i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
221.z even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 221.2.z.a ✓ 152
13.d odd 4 1 221.2.ba.a yes 152
17.e odd 16 1 221.2.ba.a yes 152
221.z even 16 1 inner 221.2.z.a ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.z.a ✓ 152 1.a even 1 1 trivial
221.2.z.a ✓ 152 221.z even 16 1 inner
221.2.ba.a yes 152 13.d odd 4 1
221.2.ba.a yes 152 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(221, [\chi])\).