Properties

Label 268.2.m.a
Level $268$
Weight $2$
Character orbit 268.m
Analytic conductor $2.140$
Analytic rank $0$
Dimension $120$
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] = [268,2,Mod(17,268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(268, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("268.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 268 = 2^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 268.m (of order \(33\), degree \(20\), 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: \(120\)
Relative dimension: \(6\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 4 q^{3} - 6 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 4 q^{3} - 6 q^{5} + 3 q^{7} - 4 q^{9} + 5 q^{11} - q^{13} + 20 q^{15} - 14 q^{17} + 19 q^{19} - 16 q^{21} + 6 q^{23} - 26 q^{25} - 2 q^{27} + 2 q^{29} - 43 q^{31} + 9 q^{33} + q^{35} + 21 q^{37} - 49 q^{39} + 44 q^{41} + 50 q^{43} - 28 q^{45} + 37 q^{47} + q^{49} + 26 q^{51} - 10 q^{53} - 40 q^{55} - 41 q^{57} - 93 q^{59} - 111 q^{61} - 115 q^{63} - 88 q^{65} - 26 q^{67} - 31 q^{69} - 104 q^{71} - 110 q^{73} - 103 q^{75} - 56 q^{77} - 87 q^{79} - 10 q^{81} - 34 q^{83} + 2 q^{85} - 45 q^{87} - 23 q^{89} + 58 q^{91} + 17 q^{93} - 6 q^{95} - 13 q^{97} + 71 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.63746 + 1.69499i 0 −0.492907 + 3.42824i 0 −0.397629 0.558392i 0 2.83695 6.21205i 0
17.2 0 −1.67230 + 1.07472i 0 0.298559 2.07653i 0 1.54983 + 2.17643i 0 0.395311 0.865610i 0
17.3 0 −0.0148630 + 0.00955186i 0 0.395959 2.75395i 0 −2.83633 3.98307i 0 −1.24612 + 2.72861i 0
17.4 0 0.505110 0.324615i 0 −0.422041 + 2.93536i 0 −0.322193 0.452456i 0 −1.09648 + 2.40096i 0
17.5 0 1.68771 1.08462i 0 0.308909 2.14851i 0 2.06618 + 2.90154i 0 0.425703 0.932160i 0
17.6 0 2.67078 1.71641i 0 −0.167112 + 1.16229i 0 −1.95872 2.75064i 0 2.94078 6.43941i 0
21.1 0 −1.13773 + 2.49127i 0 −3.83671 1.12656i 0 1.55028 4.47923i 0 −2.94743 3.40151i 0
21.2 0 −1.02902 + 2.25325i 0 2.64873 + 0.777737i 0 −0.432784 + 1.25045i 0 −2.05366 2.37005i 0
21.3 0 −0.160887 + 0.352293i 0 −1.66715 0.489520i 0 −1.05867 + 3.05882i 0 1.86636 + 2.15389i 0
21.4 0 0.368818 0.807599i 0 −0.0737807 0.0216640i 0 0.660618 1.90873i 0 1.44839 + 1.67153i 0
21.5 0 0.796606 1.74432i 0 3.57137 + 1.04865i 0 0.0733205 0.211846i 0 −0.443504 0.511831i 0
21.6 0 1.38323 3.02886i 0 −2.47014 0.725297i 0 −0.0923903 + 0.266944i 0 −5.29607 6.11199i 0
33.1 0 −2.99106 + 0.878254i 0 −0.544051 0.627868i 0 −0.623535 0.321455i 0 5.65133 3.63189i 0
33.2 0 −0.748191 + 0.219689i 0 −1.58768 1.83229i 0 −1.29012 0.665104i 0 −2.01223 + 1.29318i 0
33.3 0 −0.680838 + 0.199912i 0 2.09126 + 2.41344i 0 −0.652698 0.336489i 0 −2.10019 + 1.34971i 0
33.4 0 1.26064 0.370157i 0 0.454892 + 0.524973i 0 3.21543 + 1.65767i 0 −1.07157 + 0.688655i 0
33.5 0 2.12465 0.623854i 0 −1.97351 2.27755i 0 0.0956759 + 0.0493244i 0 1.60119 1.02903i 0
33.6 0 3.19808 0.939041i 0 1.46527 + 1.69102i 0 −3.25284 1.67696i 0 6.82216 4.38434i 0
49.1 0 −2.26129 1.45324i 0 0.354018 + 2.46225i 0 −0.314804 0.0300601i 0 1.75526 + 3.84348i 0
49.2 0 −1.35538 0.871048i 0 −0.136981 0.952723i 0 1.40117 + 0.133795i 0 −0.167923 0.367700i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 268.2.m.a 120
67.g even 33 1 inner 268.2.m.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
268.2.m.a 120 1.a even 1 1 trivial
268.2.m.a 120 67.g even 33 1 inner

Hecke kernels

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