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Properties

Label 950.2.n.a
Level $950$
Weight $2$
Character orbit 950.n
Analytic conductor $7.586$
Analytic rank $0$
Dimension $88$
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] = [950,2,Mod(39,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.n (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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 22 q^{4} + 10 q^{5} + 2 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 22 q^{4} + 10 q^{5} + 2 q^{6} + 24 q^{9} + 26 q^{11} + 10 q^{12} - 10 q^{14} + 12 q^{15} - 22 q^{16} - 40 q^{17} - 22 q^{19} + 10 q^{23} + 8 q^{24} + 6 q^{25} - 28 q^{26} - 30 q^{27} - 10 q^{28} - 4 q^{29} - 4 q^{30} + 2 q^{31} - 8 q^{34} - 48 q^{35} - 24 q^{36} + 50 q^{37} + 8 q^{39} + 32 q^{41} + 10 q^{42} + 4 q^{44} - 8 q^{45} + 10 q^{46} + 10 q^{48} - 56 q^{49} + 28 q^{50} - 60 q^{51} - 70 q^{53} - 8 q^{54} + 4 q^{55} + 10 q^{56} - 60 q^{58} - 28 q^{59} - 12 q^{60} - 58 q^{61} + 60 q^{63} + 22 q^{64} - 24 q^{65} + 4 q^{66} - 70 q^{67} - 8 q^{69} - 4 q^{70} + 48 q^{71} + 40 q^{73} + 52 q^{74} + 108 q^{75} - 88 q^{76} - 50 q^{78} - 20 q^{79} + 24 q^{81} - 80 q^{83} + 30 q^{85} + 20 q^{86} + 70 q^{87} + 10 q^{88} - 62 q^{89} - 104 q^{90} + 20 q^{91} - 10 q^{92} - 10 q^{94} + 2 q^{96} - 10 q^{97} + 60 q^{98} + 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −0.587785 − 0.809017i −2.56558 − 0.833609i −0.309017 + 0.951057i 0.0605232 + 2.23525i 0.833609 + 2.56558i − 3.84106i 0.951057 − 0.309017i 3.46027 + 2.51403i 1.77278 − 1.36281i
39.2 −0.587785 − 0.809017i −2.20724 − 0.717177i −0.309017 + 0.951057i 2.23210 − 0.133199i 0.717177 + 2.20724i − 2.34643i 0.951057 − 0.309017i 1.93053 + 1.40261i −1.41975 − 1.72751i
39.3 −0.587785 − 0.809017i −2.20015 − 0.714873i −0.309017 + 0.951057i −2.17200 − 0.531443i 0.714873 + 2.20015i 2.86995i 0.951057 − 0.309017i 1.90257 + 1.38230i 0.846721 + 2.06956i
39.4 −0.587785 − 0.809017i −1.06179 − 0.344997i −0.309017 + 0.951057i −0.518181 − 2.17520i 0.344997 + 1.06179i 0.133310i 0.951057 − 0.309017i −1.41867 − 1.03073i −1.45519 + 1.69777i
39.5 −0.587785 − 0.809017i −0.812784 − 0.264090i −0.309017 + 0.951057i 0.775807 + 2.09717i 0.264090 + 0.812784i 3.74573i 0.951057 − 0.309017i −1.83618 − 1.33406i 1.24064 − 1.86033i
39.6 −0.587785 − 0.809017i −0.273661 − 0.0889179i −0.309017 + 0.951057i −1.60391 + 1.55803i 0.0889179 + 0.273661i − 0.224849i 0.951057 − 0.309017i −2.36007 − 1.71469i 2.20323 + 0.381801i
39.7 −0.587785 − 0.809017i 0.236785 + 0.0769362i −0.309017 + 0.951057i −1.89595 − 1.18549i −0.0769362 − 0.236785i − 4.79883i 0.951057 − 0.309017i −2.37690 − 1.72692i 0.155332 + 2.23067i
39.8 −0.587785 − 0.809017i 1.25142 + 0.406611i −0.309017 + 0.951057i 1.47560 − 1.68006i −0.406611 − 1.25142i 2.56926i 0.951057 − 0.309017i −1.02633 − 0.745671i −2.22653 − 0.206269i
39.9 −0.587785 − 0.809017i 1.63623 + 0.531642i −0.309017 + 0.951057i 1.99525 − 1.00944i −0.531642 − 1.63623i − 3.69408i 0.951057 − 0.309017i −0.0324603 − 0.0235838i −1.98943 − 1.02086i
39.10 −0.587785 − 0.809017i 2.87975 + 0.935689i −0.309017 + 0.951057i 0.587147 + 2.15760i −0.935689 − 2.87975i 1.61619i 0.951057 − 0.309017i 4.99042 + 3.62575i 1.40042 − 1.74322i
39.11 −0.587785 − 0.809017i 2.95005 + 0.958531i −0.309017 + 0.951057i 1.23590 − 1.86348i −0.958531 − 2.95005i − 2.65378i 0.951057 − 0.309017i 5.35699 + 3.89208i −2.23403 + 0.0954627i
39.12 0.587785 + 0.809017i −3.09795 − 1.00658i −0.309017 + 0.951057i −0.367092 + 2.20573i −1.00658 − 3.09795i 4.43412i −0.951057 + 0.309017i 6.15701 + 4.47333i −2.00024 + 0.999512i
39.13 0.587785 + 0.809017i −1.88808 − 0.613474i −0.309017 + 0.951057i 2.21981 + 0.269141i −0.613474 − 1.88808i 0.715850i −0.951057 + 0.309017i 0.761435 + 0.553215i 1.08703 + 1.95406i
39.14 0.587785 + 0.809017i −1.78167 − 0.578899i −0.309017 + 0.951057i 0.452104 − 2.18989i −0.578899 − 1.78167i − 4.62971i −0.951057 + 0.309017i 0.412164 + 0.299454i 2.03740 − 0.921423i
39.15 0.587785 + 0.809017i −1.70705 − 0.554656i −0.309017 + 0.951057i −2.20234 + 0.386932i −0.554656 − 1.70705i − 2.40506i −0.951057 + 0.309017i 0.179341 + 0.130299i −1.60753 − 1.55429i
39.16 0.587785 + 0.809017i −1.33501 − 0.433772i −0.309017 + 0.951057i 0.407477 + 2.19863i −0.433772 − 1.33501i − 1.42943i −0.951057 + 0.309017i −0.832948 − 0.605172i −1.53922 + 1.62198i
39.17 0.587785 + 0.809017i −0.284260 − 0.0923618i −0.309017 + 0.951057i 0.810167 − 2.08414i −0.0923618 − 0.284260i 5.14607i −0.951057 + 0.309017i −2.35478 − 1.71085i 2.16231 − 0.569586i
39.18 0.587785 + 0.809017i 0.285827 + 0.0928709i −0.309017 + 0.951057i 2.23251 − 0.126065i 0.0928709 + 0.285827i − 0.304227i −0.951057 + 0.309017i −2.35398 − 1.71027i 1.41423 + 1.73204i
39.19 0.587785 + 0.809017i 0.718550 + 0.233471i −0.309017 + 0.951057i −1.61612 − 1.54537i 0.233471 + 0.718550i − 0.567373i −0.951057 + 0.309017i −1.96525 − 1.42783i 0.300299 − 2.21581i
39.20 0.587785 + 0.809017i 1.55730 + 0.505997i −0.309017 + 0.951057i −0.854432 + 2.06638i 0.505997 + 1.55730i 0.994490i −0.951057 + 0.309017i −0.257907 − 0.187380i −2.17396 + 0.523340i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.n.a ✓ 88
25.e even 10 1 inner 950.2.n.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.n.a ✓ 88 1.a even 1 1 trivial
950.2.n.a ✓ 88 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 45 T_{3}^{86} + 30 T_{3}^{85} + 1137 T_{3}^{84} - 1350 T_{3}^{83} - 20983 T_{3}^{82} + \cdots + 9468569026816 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display