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Properties

Label 310.2.w.a
Level $310$
Weight $2$
Character orbit 310.w
Analytic conductor $2.475$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,2,Mod(3,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([45, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.w (of order \(60\), degree \(16\), 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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(16\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 24 q^{7} - 8 q^{10} + 64 q^{16} + 4 q^{20} + 32 q^{21} - 52 q^{22} + 20 q^{23} + 20 q^{25} - 180 q^{27} + 4 q^{28} + 8 q^{31} - 44 q^{33} + 48 q^{35} + 128 q^{36} - 108 q^{37} - 52 q^{38} - 32 q^{41} + 8 q^{42} - 48 q^{43} - 28 q^{45} + 40 q^{46} + 20 q^{47} + 20 q^{48} - 48 q^{50} + 16 q^{51} + 28 q^{53} - 80 q^{55} - 24 q^{57} - 20 q^{60} + 20 q^{62} - 256 q^{63} - 8 q^{65} - 56 q^{66} + 4 q^{67} - 40 q^{70} + 88 q^{71} + 32 q^{73} - 32 q^{75} - 96 q^{76} + 20 q^{77} - 112 q^{78} + 48 q^{81} + 8 q^{82} - 108 q^{83} - 120 q^{85} - 24 q^{86} + 8 q^{87} - 108 q^{88} - 32 q^{90} - 80 q^{91} - 260 q^{93} + 64 q^{95} - 88 q^{97} - 112 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.156434 + 0.987688i −1.49400 − 1.84493i −0.951057 − 0.309017i 1.81407 + 1.30734i 2.05593 − 1.18699i −1.16615 + 1.79571i 0.453990 − 0.891007i −0.548017 + 2.57822i −1.57503 + 1.58722i
3.2 −0.156434 + 0.987688i −1.37915 − 1.70310i −0.951057 − 0.309017i 0.0427252 − 2.23566i 1.89788 − 1.09574i −1.90957 + 2.94049i 0.453990 − 0.891007i −0.374783 + 1.76322i 2.20145 + 0.391933i
3.3 −0.156434 + 0.987688i −0.781000 − 0.964454i −0.951057 − 0.309017i −1.84142 + 1.26854i 1.07476 − 0.620511i 0.183700 − 0.282874i 0.453990 − 0.891007i 0.303523 − 1.42797i −0.964859 − 2.01719i
3.4 −0.156434 + 0.987688i 0.342594 + 0.423068i −0.951057 − 0.309017i −1.13587 − 1.92608i −0.471453 + 0.272193i 1.41361 − 2.17678i 0.453990 − 0.891007i 0.562119 − 2.64456i 2.08006 − 0.820579i
3.5 −0.156434 + 0.987688i 0.448958 + 0.554416i −0.951057 − 0.309017i 2.18954 + 0.453761i −0.617823 + 0.356700i 2.29106 − 3.52793i 0.453990 − 0.891007i 0.517920 − 2.43662i −0.790695 + 2.09160i
3.6 −0.156434 + 0.987688i 0.862044 + 1.06454i −0.951057 − 0.309017i 1.15290 + 1.91594i −1.18628 + 0.684901i −2.00677 + 3.09015i 0.453990 − 0.891007i 0.233619 − 1.09909i −2.07270 + 0.838984i
3.7 −0.156434 + 0.987688i 1.69397 + 2.09188i −0.951057 − 0.309017i −1.31460 + 1.80882i −2.33112 + 1.34587i −0.0410660 + 0.0632361i 0.453990 − 0.891007i −0.882693 + 4.15274i −1.58091 − 1.58137i
3.8 −0.156434 + 0.987688i 1.99944 + 2.46911i −0.951057 − 0.309017i 1.27193 − 1.83908i −2.75149 + 1.58857i 0.204403 − 0.314753i 0.453990 − 0.891007i −1.47498 + 6.93924i 1.61746 + 1.54396i
3.9 0.156434 − 0.987688i −1.60409 − 1.98089i −0.951057 − 0.309017i −1.85572 − 1.24752i −2.20744 + 1.27447i 0.501168 − 0.771731i −0.453990 + 0.891007i −0.727078 + 3.42063i −1.52246 + 1.63772i
3.10 0.156434 − 0.987688i −1.02578 − 1.26673i −0.951057 − 0.309017i 0.0640532 + 2.23515i −1.41160 + 0.814989i −2.14566 + 3.30402i −0.453990 + 0.891007i 0.0713497 − 0.335674i 2.21765 + 0.286390i
3.11 0.156434 − 0.987688i −0.977787 − 1.20747i −0.951057 − 0.309017i 1.56826 − 1.59391i −1.34556 + 0.776859i 1.13452 − 1.74701i −0.453990 + 0.891007i 0.121828 − 0.573153i −1.32895 − 1.79830i
3.12 0.156434 − 0.987688i −0.0117328 − 0.0144888i −0.951057 − 0.309017i −0.637392 + 2.14330i −0.0161458 + 0.00932178i 1.25384 − 1.93075i −0.453990 + 0.891007i 0.623663 − 2.93410i 2.01720 + 0.964831i
3.13 0.156434 − 0.987688i 0.589039 + 0.727403i −0.951057 − 0.309017i −0.187613 − 2.22818i 0.810593 − 0.467996i −0.100019 + 0.154016i −0.453990 + 0.891007i 0.441587 − 2.07751i −2.23010 − 0.163262i
3.14 0.156434 − 0.987688i 1.14667 + 1.41602i −0.951057 − 0.309017i 2.19659 + 0.418314i 1.57796 − 0.911036i −1.88981 + 2.91005i −0.453990 + 0.891007i −0.0665202 + 0.312953i 0.756786 − 2.10411i
3.15 0.156434 − 0.987688i 1.65754 + 2.04690i −0.951057 − 0.309017i 2.03393 + 0.929048i 2.28099 − 1.31693i 2.23316 − 3.43876i −0.453990 + 0.891007i −0.818596 + 3.85119i 1.23579 − 1.86355i
3.16 0.156434 − 0.987688i 1.91901 + 2.36978i −0.951057 − 0.309017i −2.19678 + 0.417306i 2.64081 − 1.52467i −2.01799 + 3.10743i −0.453990 + 0.891007i −1.30953 + 6.16084i 0.0685157 + 2.23502i
13.1 −0.156434 + 0.987688i −1.04018 + 2.70975i −0.951057 − 0.309017i 1.06493 + 1.96619i −2.51367 − 1.45127i −3.60477 − 0.188918i 0.453990 − 0.891007i −4.03136 − 3.62985i −2.10858 + 0.744240i
13.2 −0.156434 + 0.987688i −0.570759 + 1.48688i −0.951057 − 0.309017i 2.01425 − 0.970971i −1.37929 − 0.796331i 1.41028 + 0.0739096i 0.453990 − 0.891007i 0.344395 + 0.310095i 0.643918 + 2.14135i
13.3 −0.156434 + 0.987688i −0.290852 + 0.757697i −0.951057 − 0.309017i −2.23598 + 0.0193710i −0.702869 − 0.405801i −2.60313 − 0.136424i 0.453990 − 0.891007i 1.73993 + 1.56664i 0.330652 − 2.21149i
13.4 −0.156434 + 0.987688i −0.153775 + 0.400598i −0.951057 − 0.309017i 0.0984612 − 2.23390i −0.371611 − 0.214549i 1.68380 + 0.0882441i 0.453990 − 0.891007i 2.09260 + 1.88419i 2.19099 + 0.446708i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.h odd 30 1 inner
155.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.w.a ✓ 256
5.c odd 4 1 inner 310.2.w.a ✓ 256
31.h odd 30 1 inner 310.2.w.a ✓ 256
155.x even 60 1 inner 310.2.w.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.w.a ✓ 256 1.a even 1 1 trivial
310.2.w.a ✓ 256 5.c odd 4 1 inner
310.2.w.a ✓ 256 31.h odd 30 1 inner
310.2.w.a ✓ 256 155.x even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(310, [\chi])\).