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Properties

Label 503.3.b.b
Level $503$
Weight $3$
Character orbit 503.b
Analytic conductor $13.706$
Analytic rank $0$
Dimension $62$
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] = [503,3,Mod(502,503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("503.502");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 503.b (of order \(2\), degree \(1\), 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: \(13.7057572973\)
Analytic rank: \(0\)
Dimension: \(62\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 62 q - 2 q^{2} - 2 q^{3} + 82 q^{4} + 10 q^{6} - 8 q^{7} - 16 q^{8} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q - 2 q^{2} - 2 q^{3} + 82 q^{4} + 10 q^{6} - 8 q^{7} - 16 q^{8} + 64 q^{9} - 32 q^{11} - 28 q^{12} - 6 q^{13} - 40 q^{14} - 6 q^{16} - 30 q^{18} + 32 q^{21} + 2 q^{22} - 34 q^{23} + 34 q^{24} - 876 q^{25} - 44 q^{26} - 26 q^{27} - 218 q^{28} - 166 q^{32} - 150 q^{33} - 136 q^{36} - 66 q^{39} - 146 q^{42} - 138 q^{43} - 252 q^{44} + 352 q^{46} + 136 q^{47} + 82 q^{48} - 434 q^{49} + 158 q^{50} + 108 q^{52} - 40 q^{54} - 114 q^{56} + 58 q^{59} + 282 q^{61} - 240 q^{63} - 592 q^{64} - 54 q^{66} - 266 q^{67} + 422 q^{69} - 226 q^{72} - 106 q^{73} + 86 q^{75} + 140 q^{77} + 564 q^{78} - 124 q^{79} - 726 q^{81} - 110 q^{83} + 182 q^{84} - 64 q^{85} + 370 q^{86} - 252 q^{88} - 370 q^{91} + 172 q^{92} + 28 q^{94} + 118 q^{95} - 470 q^{96} - 38 q^{97} - 140 q^{98} - 158 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
502.1 −3.52302 2.86378 8.41166 − 6.98742i −10.0891 −3.23995 −15.5424 −0.798771 24.6168i
502.2 −3.52302 2.86378 8.41166 6.98742i −10.0891 −3.23995 −15.5424 −0.798771 − 24.6168i
502.3 −3.51330 −3.02548 8.34328 − 7.61908i 10.6294 6.87783 −15.2592 0.153555 26.7681i
502.4 −3.51330 −3.02548 8.34328 7.61908i 10.6294 6.87783 −15.2592 0.153555 − 26.7681i
502.5 −3.46818 0.818945 8.02830 − 4.21907i −2.84025 4.33090 −13.9709 −8.32933 14.6325i
502.6 −3.46818 0.818945 8.02830 4.21907i −2.84025 4.33090 −13.9709 −8.32933 − 14.6325i
502.7 −3.11267 −2.83555 5.68873 − 1.36139i 8.82615 −3.28446 −5.25647 −0.959645 4.23756i
502.8 −3.11267 −2.83555 5.68873 1.36139i 8.82615 −3.28446 −5.25647 −0.959645 − 4.23756i
502.9 −3.06582 4.43579 5.39926 − 7.11224i −13.5993 −4.47162 −4.28989 10.6762 21.8049i
502.10 −3.06582 4.43579 5.39926 7.11224i −13.5993 −4.47162 −4.28989 10.6762 − 21.8049i
502.11 −2.99324 −4.77159 4.95948 − 9.66648i 14.2825 −12.1286 −2.87196 13.7680 28.9341i
502.12 −2.99324 −4.77159 4.95948 9.66648i 14.2825 −12.1286 −2.87196 13.7680 − 28.9341i
502.13 −2.22739 1.66393 0.961260 − 4.36601i −3.70622 6.43491 6.76845 −6.23134 9.72481i
502.14 −2.22739 1.66393 0.961260 4.36601i −3.70622 6.43491 6.76845 −6.23134 − 9.72481i
502.15 −2.09587 −1.77748 0.392658 6.94208i 3.72536 10.1288 7.56051 −5.84057 − 14.5497i
502.16 −2.09587 −1.77748 0.392658 − 6.94208i 3.72536 10.1288 7.56051 −5.84057 14.5497i
502.17 −1.78017 0.451876 −0.831011 − 6.90793i −0.804413 −4.80244 8.60000 −8.79581 12.2973i
502.18 −1.78017 0.451876 −0.831011 6.90793i −0.804413 −4.80244 8.60000 −8.79581 − 12.2973i
502.19 −1.37976 −2.01004 −2.09627 − 5.99690i 2.77337 −2.01960 8.41137 −4.95974 8.27427i
502.20 −1.37976 −2.01004 −2.09627 5.99690i 2.77337 −2.01960 8.41137 −4.95974 − 8.27427i
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 502.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
503.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 503.3.b.b ✓ 62
503.b odd 2 1 inner 503.3.b.b ✓ 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.3.b.b ✓ 62 1.a even 1 1 trivial
503.3.b.b ✓ 62 503.b odd 2 1 inner