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Properties

Label 510.2.bf.b
Level $510$
Weight $2$
Character orbit 510.bf
Analytic conductor $4.072$
Analytic rank $0$
Dimension $96$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [510,2,Mod(11,510)]
 
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, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.bf (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: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{12} + 32 q^{18} + 48 q^{21} - 32 q^{29} + 16 q^{31} + 32 q^{33} + 32 q^{37} - 48 q^{38} - 16 q^{39} + 48 q^{42} + 16 q^{43} - 32 q^{45} - 32 q^{46} - 16 q^{47} - 32 q^{49} - 96 q^{50} - 32 q^{55} - 32 q^{58} - 128 q^{59} - 16 q^{60} - 32 q^{61} + 32 q^{62} - 16 q^{66} + 32 q^{69} + 16 q^{72} - 32 q^{77} - 16 q^{78} + 80 q^{81} + 144 q^{83} - 32 q^{84} - 48 q^{87} + 96 q^{91} + 32 q^{92} - 32 q^{97} - 32 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 0.382683 − 0.923880i −1.73133 + 0.0500797i −0.707107 − 0.707107i −0.555570 + 0.831470i −0.616282 + 1.61870i −1.26342 + 0.844193i −0.923880 + 0.382683i 2.99498 − 0.173409i 0.555570 + 0.831470i
11.2 0.382683 − 0.923880i −1.45681 − 0.936859i −0.707107 − 0.707107i 0.555570 − 0.831470i −1.42304 + 0.987397i −3.46720 + 2.31671i −0.923880 + 0.382683i 1.24459 + 2.72965i −0.555570 − 0.831470i
11.3 0.382683 − 0.923880i −1.35823 − 1.07481i −0.707107 − 0.707107i −0.555570 + 0.831470i −1.51276 + 0.843533i 4.27868 − 2.85892i −0.923880 + 0.382683i 0.689588 + 2.91967i 0.555570 + 0.831470i
11.4 0.382683 − 0.923880i −1.12440 + 1.31747i −0.707107 − 0.707107i −0.555570 + 0.831470i 0.786898 + 1.54298i −0.594677 + 0.397350i −0.923880 + 0.382683i −0.471466 − 2.96272i 0.555570 + 0.831470i
11.5 0.382683 − 0.923880i −0.915034 + 1.47062i −0.707107 − 0.707107i 0.555570 − 0.831470i 1.00850 + 1.40816i −2.82621 + 1.88841i −0.923880 + 0.382683i −1.32543 − 2.69133i −0.555570 − 0.831470i
11.6 0.382683 − 0.923880i −0.646452 − 1.60689i −0.707107 − 0.707107i −0.555570 + 0.831470i −1.73196 − 0.0176871i −2.00838 + 1.34196i −0.923880 + 0.382683i −2.16420 + 2.07756i 0.555570 + 0.831470i
11.7 0.382683 − 0.923880i 0.136957 − 1.72663i −0.707107 − 0.707107i 0.555570 − 0.831470i −1.54278 − 0.787283i −0.0401891 + 0.0268535i −0.923880 + 0.382683i −2.96249 − 0.472947i −0.555570 − 0.831470i
11.8 0.382683 − 0.923880i 0.190167 + 1.72158i −0.707107 − 0.707107i 0.555570 − 0.831470i 1.66331 + 0.483128i 3.94469 − 2.63576i −0.923880 + 0.382683i −2.92767 + 0.654776i −0.555570 − 0.831470i
11.9 0.382683 − 0.923880i 0.842373 + 1.51341i −0.707107 − 0.707107i −0.555570 + 0.831470i 1.72057 − 0.199095i −1.80413 + 1.20548i −0.923880 + 0.382683i −1.58081 + 2.54971i 0.555570 + 0.831470i
11.10 0.382683 − 0.923880i 1.49867 + 0.868322i −0.707107 − 0.707107i −0.555570 + 0.831470i 1.37574 − 1.05230i 1.39193 − 0.930061i −0.923880 + 0.382683i 1.49203 + 2.60266i 0.555570 + 0.831470i
11.11 0.382683 − 0.923880i 1.53367 − 0.804899i −0.707107 − 0.707107i 0.555570 − 0.831470i −0.156720 − 1.72495i 0.836933 − 0.559221i −0.923880 + 0.382683i 1.70428 − 2.46890i −0.555570 − 0.831470i
11.12 0.382683 − 0.923880i 1.61620 + 0.622816i −0.707107 − 0.707107i 0.555570 − 0.831470i 1.19390 − 1.25483i 1.55197 − 1.03699i −0.923880 + 0.382683i 2.22420 + 2.01319i −0.555570 − 0.831470i
41.1 −0.923880 − 0.382683i −1.70834 + 0.285591i 0.707107 + 0.707107i −0.980785 + 0.195090i 1.68759 + 0.389903i −0.440840 + 2.21625i −0.382683 − 0.923880i 2.83688 − 0.975774i 0.980785 + 0.195090i
41.2 −0.923880 − 0.382683i −1.36552 + 1.06553i 0.707107 + 0.707107i 0.980785 − 0.195090i 1.66934 − 0.461860i −0.00432258 + 0.0217311i −0.382683 − 0.923880i 0.729291 − 2.91001i −0.980785 − 0.195090i
41.3 −0.923880 − 0.382683i −1.32611 − 1.11419i 0.707107 + 0.707107i 0.980785 − 0.195090i 0.798786 + 1.53686i −0.214406 + 1.07789i −0.382683 − 0.923880i 0.517151 + 2.95509i −0.980785 − 0.195090i
41.4 −0.923880 − 0.382683i −0.902435 − 1.47838i 0.707107 + 0.707107i −0.980785 + 0.195090i 0.267990 + 1.71119i −0.138425 + 0.695908i −0.382683 − 0.923880i −1.37122 + 2.66829i 0.980785 + 0.195090i
41.5 −0.923880 − 0.382683i −0.0952166 + 1.72943i 0.707107 + 0.707107i −0.980785 + 0.195090i 0.749793 − 1.56135i 0.366395 − 1.84199i −0.382683 − 0.923880i −2.98187 − 0.329341i 0.980785 + 0.195090i
41.6 −0.923880 − 0.382683i 0.0194320 − 1.73194i 0.707107 + 0.707107i 0.980785 − 0.195090i −0.680738 + 1.59267i 0.991285 − 4.98352i −0.382683 − 0.923880i −2.99924 − 0.0673103i −0.980785 − 0.195090i
41.7 −0.923880 − 0.382683i 0.312929 − 1.70355i 0.707107 + 0.707107i 0.980785 − 0.195090i −0.941029 + 1.45412i −0.826172 + 4.15345i −0.382683 − 0.923880i −2.80415 − 1.06618i −0.980785 − 0.195090i
41.8 −0.923880 − 0.382683i 0.679506 − 1.59320i 0.707107 + 0.707107i −0.980785 + 0.195090i −1.23747 + 1.21189i 0.329765 − 1.65784i −0.382683 − 0.923880i −2.07654 − 2.16517i 0.980785 + 0.195090i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i 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.bf.b yes 96
3.b odd 2 1 510.2.bf.a ✓ 96
17.e odd 16 1 510.2.bf.a ✓ 96
51.i even 16 1 inner 510.2.bf.b yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.bf.a ✓ 96 3.b odd 2 1
510.2.bf.a ✓ 96 17.e odd 16 1
510.2.bf.b yes 96 1.a even 1 1 trivial
510.2.bf.b yes 96 51.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{96} - 24 T_{11}^{94} + 628 T_{11}^{92} + 336 T_{11}^{91} + 520 T_{11}^{90} + \cdots + 10\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display