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Properties

Label 663.2.n.b
Level $663$
Weight $2$
Character orbit 663.n
Analytic conductor $5.294$
Analytic rank $0$
Dimension $76$
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] = [663,2,Mod(86,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.86");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.n (of order \(4\), degree \(2\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 2 q^{5} + 6 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 2 q^{5} + 6 q^{6} - 8 q^{7} - 14 q^{11} - 2 q^{13} + 4 q^{15} - 80 q^{16} + 76 q^{17} - 34 q^{18} - 10 q^{19} - 8 q^{20} - 16 q^{21} + 8 q^{22} + 30 q^{24} - 28 q^{26} - 12 q^{27} + 28 q^{28} - 112 q^{30} + 24 q^{31} + 52 q^{33} + 16 q^{37} + 2 q^{39} + 48 q^{40} - 18 q^{41} - 20 q^{42} - 28 q^{44} + 22 q^{45} - 20 q^{46} + 16 q^{47} + 4 q^{48} - 52 q^{50} + 50 q^{54} - 32 q^{55} + 256 q^{56} + 32 q^{57} + 12 q^{58} - 24 q^{59} - 72 q^{60} + 42 q^{63} - 50 q^{65} + 12 q^{66} - 16 q^{67} - 88 q^{69} + 44 q^{70} + 8 q^{71} + 20 q^{72} - 32 q^{73} - 16 q^{76} - 48 q^{77} + 96 q^{78} + 8 q^{79} - 88 q^{80} + 16 q^{83} - 18 q^{84} + 2 q^{85} + 24 q^{86} - 8 q^{87} - 12 q^{89} + 56 q^{90} - 32 q^{91} + 42 q^{93} - 16 q^{94} + 84 q^{95} - 28 q^{96} + 28 q^{97} - 56 q^{98} - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −1.93515 + 1.93515i 1.69598 + 0.351660i − 5.48961i 2.21406 − 2.21406i −3.96248 + 2.60145i 1.40349 − 1.40349i 6.75292 + 6.75292i 2.75267 + 1.19281i 8.56906i
86.2 −1.93161 + 1.93161i −1.53593 − 0.800577i − 5.46227i −1.03356 + 1.03356i 4.51323 − 1.42042i 1.77979 − 1.77979i 6.68778 + 6.68778i 1.71815 + 2.45926i − 3.99289i
86.3 −1.76405 + 1.76405i 0.163106 + 1.72435i − 4.22377i 0.696234 − 0.696234i −3.32958 − 2.75413i −0.361248 + 0.361248i 3.92285 + 3.92285i −2.94679 + 0.562504i 2.45639i
86.4 −1.71530 + 1.71530i 1.43204 − 0.974302i − 3.88451i −1.72487 + 1.72487i −0.785155 + 4.12760i −0.338235 + 0.338235i 3.23250 + 3.23250i 1.10147 − 2.79048i − 5.91734i
86.5 −1.59955 + 1.59955i −1.67277 − 0.449283i − 3.11714i 1.05262 − 1.05262i 3.39433 − 1.95702i −2.21835 + 2.21835i 1.78692 + 1.78692i 2.59629 + 1.50309i 3.36744i
86.6 −1.50055 + 1.50055i 0.573786 − 1.63425i − 2.50333i 1.58842 − 1.58842i 1.59128 + 3.31328i 2.86640 − 2.86640i 0.755273 + 0.755273i −2.34154 − 1.87542i 4.76702i
86.7 −1.44395 + 1.44395i −0.662748 + 1.60024i − 2.17000i 2.61598 − 2.61598i −1.35369 − 3.26765i −1.47453 + 1.47453i 0.245470 + 0.245470i −2.12153 − 2.12111i 7.55471i
86.8 −1.30908 + 1.30908i −1.56116 + 0.750182i − 1.42740i −1.97011 + 1.97011i 1.06164 − 3.02574i 1.77260 − 1.77260i −0.749582 − 0.749582i 1.87446 − 2.34231i − 5.15808i
86.9 −1.30079 + 1.30079i 0.112117 − 1.72842i − 1.38411i −2.84219 + 2.84219i 2.10247 + 2.39415i −2.66674 + 2.66674i −0.801142 − 0.801142i −2.97486 − 0.387570i − 7.39418i
86.10 −1.15887 + 1.15887i 0.718055 + 1.57620i − 0.685938i −1.19478 + 1.19478i −2.65873 − 0.994472i 0.144908 − 0.144908i −1.52282 − 1.52282i −1.96880 + 2.26359i − 2.76918i
86.11 −0.977947 + 0.977947i −1.47613 − 0.906116i 0.0872391i −0.790072 + 0.790072i 2.32971 − 0.557442i −3.21923 + 3.21923i −2.04121 − 2.04121i 1.35791 + 2.67509i − 1.54530i
86.12 −0.969033 + 0.969033i 1.03474 − 1.38900i 0.121949i 1.26591 − 1.26591i 0.343290 + 2.34868i −2.15472 + 2.15472i −2.05624 − 2.05624i −0.858634 − 2.87450i 2.45341i
86.13 −0.957019 + 0.957019i 1.64579 + 0.539780i 0.168231i 0.111175 − 0.111175i −2.09164 + 1.05848i 2.26063 − 2.26063i −2.07504 − 2.07504i 2.41728 + 1.77673i 0.212794i
86.14 −0.685372 + 0.685372i 1.55108 + 0.770812i 1.06053i 0.837656 − 0.837656i −1.59136 + 0.534775i −3.05964 + 3.05964i −2.09760 − 2.09760i 1.81170 + 2.39118i 1.14821i
86.15 −0.618992 + 0.618992i −1.26773 − 1.18020i 1.23370i −1.18345 + 1.18345i 1.51525 − 0.0541813i 1.94186 − 1.94186i −2.00163 − 2.00163i 0.214271 + 2.99234i − 1.46509i
86.16 −0.380678 + 0.380678i −1.39381 + 1.02825i 1.71017i −1.81925 + 1.81925i 0.139159 − 0.922025i −1.56441 + 1.56441i −1.41238 − 1.41238i 0.885398 − 2.86637i − 1.38510i
86.17 −0.308340 + 0.308340i −1.69665 − 0.348414i 1.80985i 1.76166 − 1.76166i 0.630573 − 0.415713i 2.99194 − 2.99194i −1.17473 − 1.17473i 2.75721 + 1.18227i 1.08638i
86.18 −0.106470 + 0.106470i −1.65687 + 0.504765i 1.97733i 2.31621 − 2.31621i 0.122665 − 0.230150i −1.69486 + 1.69486i −0.423468 − 0.423468i 2.49042 − 1.67266i 0.493215i
86.19 0.0125554 − 0.0125554i 1.47780 − 0.903393i 1.99968i 0.618345 − 0.618345i 0.00721185 − 0.0298967i 1.69294 − 1.69294i 0.0502175 + 0.0502175i 1.36776 − 2.67006i − 0.0155271i
86.20 0.110794 − 0.110794i 0.370292 − 1.69201i 1.97545i −0.814102 + 0.814102i −0.146438 − 0.228490i −1.63462 + 1.63462i 0.440455 + 0.440455i −2.72577 − 1.25307i 0.180395i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.n.b yes 76
3.b odd 2 1 663.2.n.a ✓ 76
13.d odd 4 1 663.2.n.a ✓ 76
39.f even 4 1 inner 663.2.n.b yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.n.a ✓ 76 3.b odd 2 1
663.2.n.a ✓ 76 13.d odd 4 1
663.2.n.b yes 76 1.a even 1 1 trivial
663.2.n.b yes 76 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{76} + 292 T_{2}^{72} + 20 T_{2}^{69} + 37390 T_{2}^{68} + 268 T_{2}^{67} + 3252 T_{2}^{65} + \cdots + 24336 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display