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Properties

Label 220.3.x.a
Level $220$
Weight $3$
Character orbit 220.x
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(37,220)]
 
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([0, 5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.x (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: \(96\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} + 4 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} + 4 q^{5} - 2 q^{7} - 20 q^{11} - 8 q^{13} + 88 q^{15} + 42 q^{17} + 56 q^{21} - 104 q^{23} - 126 q^{25} - 14 q^{27} - 32 q^{31} + 52 q^{33} + 56 q^{35} - 134 q^{37} + 24 q^{41} + 332 q^{43} + 344 q^{45} + 258 q^{47} - 320 q^{51} + 222 q^{53} - 46 q^{55} + 504 q^{57} - 168 q^{61} - 298 q^{63} - 416 q^{65} - 632 q^{67} - 252 q^{71} + 56 q^{73} - 352 q^{75} - 820 q^{77} + 336 q^{81} + 236 q^{83} + 88 q^{85} - 176 q^{87} - 756 q^{91} - 770 q^{93} - 334 q^{95} - 310 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0 −2.43165 − 4.77239i 0 −4.22520 − 2.67351i 0 −2.84186 + 5.57747i 0 −11.5727 + 15.9284i 0
37.2 0 −1.87775 − 3.68530i 0 3.24376 − 3.80500i 0 2.52559 − 4.95675i 0 −4.76541 + 6.55902i 0
37.3 0 −1.74541 − 3.42555i 0 3.92601 + 3.09620i 0 −5.38135 + 10.5615i 0 −3.39790 + 4.67681i 0
37.4 0 −1.72462 − 3.38475i 0 −0.450384 + 4.97967i 0 5.09251 − 9.99462i 0 −3.19218 + 4.39365i 0
37.5 0 −0.558955 − 1.09701i 0 −3.68048 + 3.38438i 0 0.455994 − 0.894939i 0 4.39907 − 6.05480i 0
37.6 0 −0.182447 − 0.358073i 0 −4.08347 − 2.88535i 0 −1.29828 + 2.54802i 0 5.19514 − 7.15049i 0
37.7 0 0.0872094 + 0.171158i 0 2.30993 − 4.43444i 0 −3.04336 + 5.97294i 0 5.26838 − 7.25130i 0
37.8 0 0.509699 + 1.00034i 0 4.99414 + 0.242016i 0 3.01016 − 5.90777i 0 4.54918 − 6.26141i 0
37.9 0 1.34992 + 2.64938i 0 −3.11539 + 3.91080i 0 −3.58267 + 7.03139i 0 0.0931732 − 0.128242i 0
37.10 0 1.43092 + 2.80834i 0 −3.68211 − 3.38262i 0 5.41546 − 10.6284i 0 −0.549155 + 0.755847i 0
37.11 0 1.88028 + 3.69025i 0 3.28123 + 3.77273i 0 0.0424515 − 0.0833157i 0 −4.79243 + 6.59621i 0
37.12 0 2.62077 + 5.14354i 0 0.218614 − 4.99522i 0 −3.35961 + 6.59360i 0 −14.2976 + 19.6789i 0
53.1 0 −5.61948 − 0.890038i 0 −2.13745 − 4.52010i 0 −10.3212 + 1.63472i 0 22.2269 + 7.22194i 0
53.2 0 −4.20075 − 0.665333i 0 −0.400380 + 4.98394i 0 7.93154 − 1.25623i 0 8.64412 + 2.80864i 0
53.3 0 −3.18350 − 0.504217i 0 4.79251 + 1.42544i 0 −1.39356 + 0.220719i 0 1.32095 + 0.429202i 0
53.4 0 −2.80761 − 0.444681i 0 −4.83437 + 1.27628i 0 3.00361 − 0.475726i 0 −0.874593 − 0.284172i 0
53.5 0 −1.93758 − 0.306883i 0 −1.07120 − 4.88391i 0 4.67201 − 0.739973i 0 −4.89946 − 1.59193i 0
53.6 0 −0.450444 − 0.0713433i 0 3.44613 − 3.62274i 0 −1.83487 + 0.290614i 0 −8.36170 − 2.71688i 0
53.7 0 −0.350343 − 0.0554888i 0 2.67198 + 4.22617i 0 −12.2603 + 1.94184i 0 −8.43985 − 2.74227i 0
53.8 0 1.70499 + 0.270045i 0 −4.99007 + 0.315041i 0 −5.56146 + 0.880848i 0 −5.72543 − 1.86030i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.c even 5 1 inner
55.k 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.x.a ✓ 96
5.c odd 4 1 inner 220.3.x.a ✓ 96
11.c even 5 1 inner 220.3.x.a ✓ 96
55.k odd 20 1 inner 220.3.x.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.x.a ✓ 96 1.a even 1 1 trivial
220.3.x.a ✓ 96 5.c odd 4 1 inner
220.3.x.a ✓ 96 11.c even 5 1 inner
220.3.x.a ✓ 96 55.k odd 20 1 inner

Hecke kernels

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