Properties

Label 268.2.h.a
Level $268$
Weight $2$
Character orbit 268.h
Analytic conductor $2.140$
Analytic rank $0$
Dimension $64$
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] = [268,2,Mod(231,268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("268.231");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 268 = 2^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 268.h (of order \(6\), degree \(2\), 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: \(2.13999077417\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} + q^{4} - 3 q^{6} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} + q^{4} - 3 q^{6} + 44 q^{9} - 5 q^{10} - 12 q^{12} - 18 q^{13} - 10 q^{14} + 9 q^{16} - 4 q^{17} - 18 q^{18} + 33 q^{20} + 4 q^{21} + 20 q^{22} - 44 q^{24} - 56 q^{25} + 13 q^{26} + 3 q^{28} + 10 q^{29} + 27 q^{30} + 12 q^{32} - 16 q^{33} - 3 q^{34} - 18 q^{37} + 6 q^{38} - 10 q^{40} - 6 q^{41} - 9 q^{44} - 39 q^{46} - 51 q^{48} + 2 q^{49} - 33 q^{50} - 33 q^{54} + 6 q^{56} - 54 q^{57} + 18 q^{60} - 6 q^{61} + 4 q^{62} - 2 q^{64} + 8 q^{65} - 56 q^{68} - 24 q^{69} - 20 q^{73} + 18 q^{74} + 22 q^{76} + 30 q^{77} - 12 q^{78} + 123 q^{80} - 32 q^{81} - 6 q^{82} - 11 q^{84} + 24 q^{85} + 52 q^{86} + 3 q^{88} + 76 q^{89} - 75 q^{90} - 22 q^{92} + 8 q^{93} - 40 q^{96} - 24 q^{97} + 126 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
231.1 −1.40647 + 0.147767i 0.455030 1.95633 0.415660i 2.16145i −0.639988 + 0.0672385i 0.987783 1.71089i −2.69010 + 0.873696i −2.79295 0.319391 + 3.04002i
231.2 −1.38407 + 0.290422i 0.541462 1.83131 0.803930i 1.10815i −0.749423 + 0.157253i −1.59264 + 2.75853i −2.30119 + 1.64455i −2.70682 −0.321830 1.53375i
231.3 −1.37412 0.334372i −2.28807 1.77639 + 0.918932i 2.12424i 3.14407 + 0.765066i 2.00882 3.47937i −2.13370 1.85670i 2.23525 0.710288 2.91896i
231.4 −1.31834 0.511846i −2.39012 1.47603 + 1.34957i 2.15536i 3.15098 + 1.22337i −2.12266 + 3.67656i −1.25513 2.53469i 2.71266 −1.10321 + 2.84149i
231.5 −1.30208 + 0.551888i 2.91499 1.39084 1.43721i 3.25556i −3.79555 + 1.60875i 1.81038 3.13566i −1.01781 + 2.63895i 5.49714 −1.79671 4.23901i
231.6 −1.27284 0.616344i 3.05470 1.24024 + 1.56901i 0.378054i −3.88814 1.88275i −1.13094 + 1.95884i −0.611571 2.76152i 6.33119 −0.233012 + 0.481202i
231.7 −1.22124 0.713141i 0.817393 0.982860 + 1.74183i 3.21972i −0.998234 0.582916i 1.30581 2.26173i 0.0418643 2.82812i −2.33187 −2.29611 + 3.93205i
231.8 −1.12899 + 0.851692i −2.91499 0.549240 1.92311i 3.25556i 3.29099 2.48267i −1.81038 + 3.13566i 1.01781 + 2.63895i 5.49714 −2.77274 3.67550i
231.9 −0.955026 1.04304i 1.04429 −0.175852 + 1.99225i 3.14744i −0.997321 1.08923i −0.0992949 + 0.171984i 2.24594 1.71923i −1.90946 3.28289 3.00588i
231.10 −0.943549 + 1.05343i −0.541462 −0.219431 1.98793i 1.10815i 0.510896 0.570393i 1.59264 2.75853i 2.30119 + 1.64455i −2.70682 −1.16735 1.04559i
231.11 −0.831206 + 1.14416i −0.455030 −0.618192 1.90206i 2.16145i 0.378224 0.520626i −0.987783 + 1.71089i 2.69010 + 0.873696i −2.79295 2.47304 + 1.79661i
231.12 −0.730066 1.21120i −1.87421 −0.934007 + 1.76851i 1.10069i 1.36830 + 2.27004i 0.173063 0.299754i 2.82391 0.159861i 0.512664 1.33315 0.803575i
231.13 −0.397483 + 1.35721i 2.28807 −1.68401 1.07893i 2.12424i −0.909469 + 3.10538i −2.00882 + 3.47937i 2.13370 1.85670i 2.23525 −2.88304 0.844351i
231.14 −0.327458 1.37578i 1.92148 −1.78554 + 0.901020i 0.452542i −0.629203 2.64353i 1.25960 2.18169i 1.82430 + 2.16147i 0.692079 −0.622599 + 0.148189i
231.15 −0.215897 + 1.39764i 2.39012 −1.90678 0.603491i 2.15536i −0.516019 + 3.34052i 2.12266 3.67656i 1.25513 2.53469i 2.71266 3.01241 + 0.465335i
231.16 −0.193483 1.40092i −2.17140 −1.92513 + 0.542107i 4.02184i 0.420130 + 3.04195i 0.863606 1.49581i 1.13193 + 2.59205i 1.71500 −5.63426 + 0.778158i
231.17 −0.102650 + 1.41048i −3.05470 −1.97893 0.289571i 0.378054i 0.313564 4.30860i 1.13094 1.95884i 0.611571 2.76152i 6.33119 0.533239 + 0.0388071i
231.18 0.00697761 + 1.41420i −0.817393 −1.99990 + 0.0197354i 3.21972i −0.00570345 1.15595i −1.30581 + 2.26173i −0.0418643 2.82812i −2.33187 4.55331 0.0224659i
231.19 0.165823 1.40446i −0.657653 −1.94501 0.465783i 1.64739i −0.109054 + 0.923646i −1.75445 + 3.03879i −0.976699 + 2.65444i −2.56749 2.31369 + 0.273175i
231.20 0.425784 + 1.34859i −1.04429 −1.63742 + 1.14842i 3.14744i −0.444640 1.40832i 0.0992949 0.171984i −2.24594 1.71923i −1.90946 −4.24462 + 1.34013i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 231.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
67.d odd 6 1 inner
268.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 268.2.h.a 64
4.b odd 2 1 inner 268.2.h.a 64
67.d odd 6 1 inner 268.2.h.a 64
268.h even 6 1 inner 268.2.h.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
268.2.h.a 64 1.a even 1 1 trivial
268.2.h.a 64 4.b odd 2 1 inner
268.2.h.a 64 67.d odd 6 1 inner
268.2.h.a 64 268.h even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(268, [\chi])\).