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Properties

Label 510.2.bh.b
Level $510$
Weight $2$
Character orbit 510.bh
Analytic conductor $4.072$
Analytic rank $0$
Dimension $144$
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] = [510,2,Mod(29,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.bh (of order \(16\), 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: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 16 q^{5} - 24 q^{15} + 48 q^{21} - 16 q^{23} + 16 q^{25} - 16 q^{31} - 16 q^{39} + 16 q^{42} - 40 q^{45} - 16 q^{47} - 32 q^{49} - 48 q^{54} + 24 q^{60} - 16 q^{62} + 112 q^{63} - 16 q^{65} - 16 q^{70} - 16 q^{72} + 80 q^{75} - 32 q^{78} - 64 q^{79} - 32 q^{85} - 48 q^{87} - 40 q^{90} - 160 q^{91} + 32 q^{92} - 80 q^{93} - 64 q^{94} - 112 q^{95} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.382683 − 0.923880i −1.73177 − 0.0313662i −0.707107 + 0.707107i 1.32963 − 1.79780i 0.633740 + 1.61195i −0.904238 − 4.54591i 0.923880 + 0.382683i 2.99803 + 0.108638i −2.16978 − 0.540433i
29.2 −0.382683 − 0.923880i −1.71217 + 0.261704i −0.707107 + 0.707107i 1.97212 − 1.05392i 0.897000 + 1.48168i 0.646319 + 3.24927i 0.923880 + 0.382683i 2.86302 − 0.896161i −1.72839 − 1.41869i
29.3 −0.382683 − 0.923880i −1.70845 − 0.284945i −0.707107 + 0.707107i −1.35933 + 1.77545i 0.390541 + 1.68745i 0.0562617 + 0.282847i 0.923880 + 0.382683i 2.83761 + 0.973630i 2.16049 + 0.576427i
29.4 −0.382683 − 0.923880i −1.24782 − 1.20123i −0.707107 + 0.707107i −1.15292 − 1.91593i −0.632270 + 1.61252i 0.748282 + 3.76187i 0.923880 + 0.382683i 0.114103 + 2.99783i −1.32888 + 1.79835i
29.5 −0.382683 − 0.923880i −0.897000 + 1.48168i −0.707107 + 0.707107i 1.41869 + 1.72839i 1.71217 + 0.261704i −0.646319 − 3.24927i 0.923880 + 0.382683i −1.39078 − 2.65814i 1.05392 − 1.97212i
29.6 −0.382683 − 0.923880i −0.887097 − 1.48764i −0.707107 + 0.707107i 0.211027 + 2.22609i −1.03492 + 1.38886i −0.347929 − 1.74916i 0.923880 + 0.382683i −1.42612 + 2.63935i 1.97588 − 1.04685i
29.7 −0.382683 − 0.923880i −0.816426 − 1.52756i −0.707107 + 0.707107i 2.21704 + 0.291054i −1.09885 + 1.33885i 0.0600910 + 0.302098i 0.923880 + 0.382683i −1.66690 + 2.49428i −0.579528 − 2.15966i
29.8 −0.382683 − 0.923880i −0.633740 + 1.61195i −0.707107 + 0.707107i 0.540433 + 2.16978i 1.73177 − 0.0313662i 0.904238 + 4.54591i 0.923880 + 0.382683i −2.19675 − 2.04311i 1.79780 − 1.32963i
29.9 −0.382683 − 0.923880i −0.390541 + 1.68745i −0.707107 + 0.707107i −0.576427 − 2.16049i 1.70845 − 0.284945i −0.0562617 − 0.282847i 0.923880 + 0.382683i −2.69496 − 1.31803i −1.77545 + 1.35933i
29.10 −0.382683 − 0.923880i 0.187448 − 1.72188i −0.707107 + 0.707107i −2.05601 − 0.879103i −1.66254 + 0.485754i −0.551612 − 2.77314i 0.923880 + 0.382683i −2.92973 − 0.645526i −0.0253840 + 2.23592i
29.11 −0.382683 − 0.923880i 0.632270 + 1.61252i −0.707107 + 0.707107i −1.79835 + 1.32888i 1.24782 − 1.20123i −0.748282 − 3.76187i 0.923880 + 0.382683i −2.20047 + 2.03910i 1.91593 + 1.15292i
29.12 −0.382683 − 0.923880i 0.850012 − 1.50913i −0.707107 + 0.707107i 2.18577 + 0.471607i −1.71954 − 0.207789i 0.858803 + 4.31750i 0.923880 + 0.382683i −1.55496 − 2.56556i −0.400750 − 2.19986i
29.13 −0.382683 − 0.923880i 1.03492 + 1.38886i −0.707107 + 0.707107i 1.04685 − 1.97588i 0.887097 − 1.48764i 0.347929 + 1.74916i 0.923880 + 0.382683i −0.857888 + 2.87472i −2.22609 − 0.211027i
29.14 −0.382683 − 0.923880i 1.09885 + 1.33885i −0.707107 + 0.707107i 2.15966 + 0.579528i 0.816426 − 1.52756i −0.0600910 − 0.302098i 0.923880 + 0.382683i −0.585049 + 2.94240i −0.291054 − 2.21704i
29.15 −0.382683 − 0.923880i 1.25818 − 1.19037i −0.707107 + 0.707107i −0.752977 − 2.10548i −1.58125 − 0.706868i 0.386106 + 1.94108i 0.923880 + 0.382683i 0.166019 − 2.99540i −1.65705 + 1.50139i
29.16 −0.382683 − 0.923880i 1.58125 − 0.706868i −0.707107 + 0.707107i −1.50139 + 1.65705i −1.25818 − 1.19037i −0.386106 − 1.94108i 0.923880 + 0.382683i 2.00068 − 2.23546i 2.10548 + 0.752977i
29.17 −0.382683 − 0.923880i 1.66254 + 0.485754i −0.707107 + 0.707107i −2.23592 + 0.0253840i −0.187448 − 1.72188i 0.551612 + 2.77314i 0.923880 + 0.382683i 2.52809 + 1.61517i 0.879103 + 2.05601i
29.18 −0.382683 − 0.923880i 1.71954 − 0.207789i −0.707107 + 0.707107i 2.19986 + 0.400750i −0.850012 − 1.50913i −0.858803 − 4.31750i 0.923880 + 0.382683i 2.91365 − 0.714604i −0.471607 − 2.18577i
209.1 0.382683 + 0.923880i −1.73040 − 0.0755666i −0.707107 + 0.707107i −1.20263 − 1.88512i −0.592382 − 1.62760i 2.55642 − 0.508504i −0.923880 − 0.382683i 2.98858 + 0.261521i 1.28139 − 1.83249i
209.2 0.382683 + 0.923880i −1.72103 + 0.195096i −0.707107 + 0.707107i 1.86199 + 1.23814i −0.838854 − 1.51536i 2.44981 − 0.487298i −0.923880 − 0.382683i 2.92388 − 0.671531i −0.431344 + 2.19407i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
17.e odd 16 1 inner
255.be even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.bh.b yes 144
3.b odd 2 1 510.2.bh.a ✓ 144
5.b even 2 1 510.2.bh.a ✓ 144
15.d odd 2 1 inner 510.2.bh.b yes 144
17.e odd 16 1 inner 510.2.bh.b yes 144
51.i even 16 1 510.2.bh.a ✓ 144
85.p odd 16 1 510.2.bh.a ✓ 144
255.be even 16 1 inner 510.2.bh.b yes 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.bh.a ✓ 144 3.b odd 2 1
510.2.bh.a ✓ 144 5.b even 2 1
510.2.bh.a ✓ 144 51.i even 16 1
510.2.bh.a ✓ 144 85.p odd 16 1
510.2.bh.b yes 144 1.a even 1 1 trivial
510.2.bh.b yes 144 15.d odd 2 1 inner
510.2.bh.b yes 144 17.e odd 16 1 inner
510.2.bh.b yes 144 255.be even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{72} + 8 T_{23}^{71} + 12 T_{23}^{70} - 600 T_{23}^{69} - 2150 T_{23}^{68} + \cdots + 40\!\cdots\!72 \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display