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Properties

Label 510.2.bf.a
Level $510$
Weight $2$
Character orbit 510.bf
Analytic conductor $4.072$
Analytic rank $0$
Dimension $96$
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] = [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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 32 q^{18} - 48 q^{21} + 16 q^{24} + 32 q^{29} + 16 q^{31} - 32 q^{33} + 32 q^{37} + 48 q^{38} - 16 q^{39} - 48 q^{42} + 16 q^{43} - 32 q^{45} - 32 q^{46} + 16 q^{47} - 32 q^{49} + 96 q^{50} - 128 q^{51} - 16 q^{54} - 32 q^{55} + 16 q^{57} - 32 q^{58} + 128 q^{59} - 16 q^{60} - 32 q^{61} - 32 q^{62} - 96 q^{63} + 32 q^{69} - 16 q^{72} + 32 q^{77} + 16 q^{78} - 64 q^{81} - 144 q^{83} + 32 q^{84} - 48 q^{87} + 96 q^{91} - 32 q^{92} - 128 q^{93} - 32 q^{97} + 32 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −1.72495 − 0.156720i −0.707107 − 0.707107i −0.555570 + 0.831470i 0.804899 − 1.53367i 0.836933 − 0.559221i 0.923880 − 0.382683i 2.95088 + 0.540668i −0.555570 − 0.831470i
11.2 −0.382683 + 0.923880i −1.25483 + 1.19390i −0.707107 − 0.707107i −0.555570 + 0.831470i −0.622816 − 1.61620i 1.55197 − 1.03699i 0.923880 − 0.382683i 0.149209 − 2.99629i −0.555570 − 0.831470i
11.3 −0.382683 + 0.923880i −1.05230 + 1.37574i −0.707107 − 0.707107i 0.555570 − 0.831470i −0.868322 − 1.49867i 1.39193 − 0.930061i 0.923880 − 0.382683i −0.785331 − 2.89539i 0.555570 + 0.831470i
11.4 −0.382683 + 0.923880i −0.787283 − 1.54278i −0.707107 − 0.707107i −0.555570 + 0.831470i 1.72663 − 0.136957i −0.0401891 + 0.0268535i 0.923880 − 0.382683i −1.76037 + 2.42922i −0.555570 − 0.831470i
11.5 −0.382683 + 0.923880i −0.199095 + 1.72057i −0.707107 − 0.707107i 0.555570 − 0.831470i −1.51341 − 0.842373i −1.80413 + 1.20548i 0.923880 − 0.382683i −2.92072 − 0.685113i 0.555570 + 0.831470i
11.6 −0.382683 + 0.923880i −0.0176871 − 1.73196i −0.707107 − 0.707107i 0.555570 − 0.831470i 1.60689 + 0.646452i −2.00838 + 1.34196i 0.923880 − 0.382683i −2.99937 + 0.0612667i 0.555570 + 0.831470i
11.7 −0.382683 + 0.923880i 0.483128 + 1.66331i −0.707107 − 0.707107i −0.555570 + 0.831470i −1.72158 − 0.190167i 3.94469 − 2.63576i 0.923880 − 0.382683i −2.53317 + 1.60718i −0.555570 − 0.831470i
11.8 −0.382683 + 0.923880i 0.843533 − 1.51276i −0.707107 − 0.707107i 0.555570 − 0.831470i 1.07481 + 1.35823i 4.27868 − 2.85892i 0.923880 − 0.382683i −1.57691 − 2.55213i 0.555570 + 0.831470i
11.9 −0.382683 + 0.923880i 0.987397 − 1.42304i −0.707107 − 0.707107i −0.555570 + 0.831470i 0.936859 + 1.45681i −3.46720 + 2.31671i 0.923880 − 0.382683i −1.05010 − 2.81021i −0.555570 − 0.831470i
11.10 −0.382683 + 0.923880i 1.40816 + 1.00850i −0.707107 − 0.707107i −0.555570 + 0.831470i −1.47062 + 0.915034i −2.82621 + 1.88841i 0.923880 − 0.382683i 0.965837 + 2.84027i −0.555570 − 0.831470i
11.11 −0.382683 + 0.923880i 1.54298 + 0.786898i −0.707107 − 0.707107i 0.555570 − 0.831470i −1.31747 + 1.12440i −0.594677 + 0.397350i 0.923880 − 0.382683i 1.76158 + 2.42834i 0.555570 + 0.831470i
11.12 −0.382683 + 0.923880i 1.61870 − 0.616282i −0.707107 − 0.707107i 0.555570 − 0.831470i −0.0500797 + 1.73133i −1.26342 + 0.844193i 0.923880 − 0.382683i 2.24039 − 1.99516i 0.555570 + 0.831470i
41.1 0.923880 + 0.382683i −1.70071 − 0.327989i 0.707107 + 0.707107i −0.980785 + 0.195090i −1.44574 − 0.953857i −0.294645 + 1.48128i 0.382683 + 0.923880i 2.78485 + 1.11563i −0.980785 − 0.195090i
41.2 0.923880 + 0.382683i −1.56135 + 0.749793i 0.707107 + 0.707107i 0.980785 − 0.195090i −1.72943 + 0.0952166i 0.366395 − 1.84199i 0.382683 + 0.923880i 1.87562 − 2.34138i 0.980785 + 0.195090i
41.3 0.923880 + 0.382683i −1.40202 − 1.01703i 0.707107 + 0.707107i −0.980785 + 0.195090i −0.906092 − 1.47614i 0.348260 − 1.75082i 0.382683 + 0.923880i 0.931294 + 2.85179i −0.980785 − 0.195090i
41.4 0.923880 + 0.382683i −1.08729 − 1.34826i 0.707107 + 0.707107i 0.980785 − 0.195090i −0.488566 − 1.66172i −0.876117 + 4.40454i 0.382683 + 0.923880i −0.635611 + 2.93189i 0.980785 + 0.195090i
41.5 0.923880 + 0.382683i −0.918041 − 1.46874i 0.707107 + 0.707107i 0.980785 − 0.195090i −0.286096 − 1.70826i 0.759222 − 3.81687i 0.382683 + 0.923880i −1.31440 + 2.69673i 0.980785 + 0.195090i
41.6 0.923880 + 0.382683i −0.461860 + 1.66934i 0.707107 + 0.707107i −0.980785 + 0.195090i −1.06553 + 1.36552i −0.00432258 + 0.0217311i 0.382683 + 0.923880i −2.57337 − 1.54200i −0.980785 − 0.195090i
41.7 0.923880 + 0.382683i 0.389903 + 1.68759i 0.707107 + 0.707107i 0.980785 − 0.195090i −0.285591 + 1.70834i −0.440840 + 2.21625i 0.382683 + 0.923880i −2.69595 + 1.31600i 0.980785 + 0.195090i
41.8 0.923880 + 0.382683i 1.21189 − 1.23747i 0.707107 + 0.707107i 0.980785 − 0.195090i 1.59320 − 0.679506i 0.329765 − 1.65784i 0.382683 + 0.923880i −0.0626688 − 2.99935i 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.a ✓ 96
3.b odd 2 1 510.2.bf.b yes 96
17.e odd 16 1 510.2.bf.b yes 96
51.i even 16 1 inner 510.2.bf.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.bf.a ✓ 96 1.a even 1 1 trivial
510.2.bf.a ✓ 96 51.i even 16 1 inner
510.2.bf.b yes 96 3.b odd 2 1
510.2.bf.b yes 96 17.e odd 16 1

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