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Properties

Label 660.2.q.a
Level $660$
Weight $2$
Character orbit 660.q
Analytic conductor $5.270$
Analytic rank $0$
Dimension $272$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [660,2,Mod(263,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.q (of order \(4\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(272\)
Relative dimension: \(136\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 16 q^{12} - 16 q^{16} + 12 q^{22} - 16 q^{25} - 16 q^{33} - 8 q^{36} - 16 q^{37} - 24 q^{42} - 48 q^{45} + 8 q^{48} - 16 q^{58} - 80 q^{60} + 64 q^{66} - 48 q^{70} - 16 q^{81} - 40 q^{82} - 28 q^{88} - 8 q^{93} - 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 −1.41316 + 0.0545388i −1.01245 − 1.40533i 1.99405 − 0.154144i 0.885678 − 2.05319i 1.50740 + 1.93074i −2.56545 + 2.56545i −2.80951 + 0.326584i −0.949886 + 2.84565i −1.13963 + 2.94979i
263.2 −1.41316 + 0.0545388i 1.40533 + 1.01245i 1.99405 − 0.154144i −0.885678 + 2.05319i −2.04117 − 1.35411i 2.56545 − 2.56545i −2.80951 + 0.326584i 0.949886 + 2.84565i 1.13963 − 2.94979i
263.3 −1.40679 + 0.144721i −1.45917 + 0.933178i 1.95811 − 0.407185i 0.447844 + 2.19076i 1.91769 − 1.52396i 0.996328 − 0.996328i −2.69572 + 0.856204i 1.25836 − 2.72333i −0.947072 − 3.01713i
263.4 −1.40679 + 0.144721i −0.933178 + 1.45917i 1.95811 − 0.407185i −0.447844 − 2.19076i 1.10161 − 2.18780i −0.996328 + 0.996328i −2.69572 + 0.856204i −1.25836 − 2.72333i 0.947072 + 3.01713i
263.5 −1.40556 − 0.156193i 0.294457 − 1.70684i 1.95121 + 0.439078i −0.570762 + 2.16200i −0.680474 + 2.35307i −1.16434 + 1.16434i −2.67396 − 0.921917i −2.82659 − 1.00518i 1.13993 − 2.94967i
263.6 −1.40556 − 0.156193i 1.70684 − 0.294457i 1.95121 + 0.439078i 0.570762 − 2.16200i −2.44506 + 0.147281i 1.16434 − 1.16434i −2.67396 − 0.921917i 2.82659 − 1.00518i −1.13993 + 2.94967i
263.7 −1.38038 + 0.307489i −1.69839 − 0.339787i 1.81090 − 0.848903i −2.06589 + 0.855630i 2.44891 − 0.0532017i −2.79985 + 2.79985i −2.23871 + 1.72864i 2.76909 + 1.15418i 2.58862 − 1.81633i
263.8 −1.38038 + 0.307489i 0.339787 + 1.69839i 1.81090 − 0.848903i 2.06589 − 0.855630i −0.991272 − 2.23995i 2.79985 − 2.79985i −2.23871 + 1.72864i −2.76909 + 1.15418i −2.58862 + 1.81633i
263.9 −1.37611 + 0.326057i 0.966493 − 1.43732i 1.78737 − 0.897382i 2.10401 + 0.757062i −0.861355 + 2.29305i −1.04504 + 1.04504i −2.16703 + 1.81768i −1.13178 − 2.77832i −3.14220 − 0.355776i
263.10 −1.37611 + 0.326057i 1.43732 − 0.966493i 1.78737 − 0.897382i −2.10401 − 0.757062i −1.66278 + 1.79865i 1.04504 − 1.04504i −2.16703 + 1.81768i 1.13178 − 2.77832i 3.14220 + 0.355776i
263.11 −1.36529 − 0.368772i −1.65797 − 0.501122i 1.72802 + 1.00696i −2.02092 − 0.957021i 2.07881 + 1.29559i 1.11368 − 1.11368i −1.98790 − 2.01203i 2.49775 + 1.66169i 2.40621 + 2.05186i
263.12 −1.36529 − 0.368772i 0.501122 + 1.65797i 1.72802 + 1.00696i 2.02092 + 0.957021i −0.0727618 − 2.44841i −1.11368 + 1.11368i −1.98790 − 2.01203i −2.49775 + 1.66169i −2.40621 − 2.05186i
263.13 −1.36263 − 0.378470i −1.41774 − 0.994990i 1.71352 + 1.03143i 2.23603 + 0.0137570i 1.55528 + 1.89238i 2.87910 − 2.87910i −1.94453 − 2.05397i 1.01999 + 2.82128i −3.04167 − 0.865015i
263.14 −1.36263 − 0.378470i 0.994990 + 1.41774i 1.71352 + 1.03143i −2.23603 − 0.0137570i −0.819230 − 2.30843i −2.87910 + 2.87910i −1.94453 − 2.05397i −1.01999 + 2.82128i 3.04167 + 0.865015i
263.15 −1.35866 − 0.392474i −1.54420 + 0.784496i 1.69193 + 1.06648i 2.03340 − 0.930205i 2.40595 − 0.459805i −1.16315 + 1.16315i −1.88019 − 2.11302i 1.76913 − 2.42284i −3.12779 + 0.465778i
263.16 −1.35866 − 0.392474i −0.784496 + 1.54420i 1.69193 + 1.06648i −2.03340 + 0.930205i 1.67192 − 1.79016i 1.16315 − 1.16315i −1.88019 − 2.11302i −1.76913 − 2.42284i 3.12779 − 0.465778i
263.17 −1.30005 + 0.556656i −0.894923 − 1.48294i 1.38027 − 1.44736i 1.67125 + 1.48557i 1.98893 + 1.42973i 0.953873 − 0.953873i −0.988735 + 2.64998i −1.39822 + 2.65424i −2.99967 − 1.00100i
263.18 −1.30005 + 0.556656i 1.48294 + 0.894923i 1.38027 − 1.44736i −1.67125 − 1.48557i −2.42606 − 0.337959i −0.953873 + 0.953873i −0.988735 + 2.64998i 1.39822 + 2.65424i 2.99967 + 1.00100i
263.19 −1.26919 + 0.623817i −1.71948 + 0.208336i 1.22171 − 1.58349i −0.365963 − 2.20592i 2.05238 − 1.33706i 3.60927 − 3.60927i −0.562774 + 2.77187i 2.91319 − 0.716457i 1.84057 + 2.57144i
263.20 −1.26919 + 0.623817i −0.208336 + 1.71948i 1.22171 − 1.58349i 0.365963 + 2.20592i −0.808219 − 2.31231i −3.60927 + 3.60927i −0.562774 + 2.77187i −2.91319 − 0.716457i −1.84057 − 2.57144i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.136
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
11.b odd 2 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
33.d even 2 1 inner
44.c even 2 1 inner
55.e even 4 1 inner
60.l odd 4 1 inner
132.d odd 2 1 inner
165.l odd 4 1 inner
220.i odd 4 1 inner
660.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.q.a ✓ 272
3.b odd 2 1 inner 660.2.q.a ✓ 272
4.b odd 2 1 inner 660.2.q.a ✓ 272
5.c odd 4 1 inner 660.2.q.a ✓ 272
11.b odd 2 1 inner 660.2.q.a ✓ 272
12.b even 2 1 inner 660.2.q.a ✓ 272
15.e even 4 1 inner 660.2.q.a ✓ 272
20.e even 4 1 inner 660.2.q.a ✓ 272
33.d even 2 1 inner 660.2.q.a ✓ 272
44.c even 2 1 inner 660.2.q.a ✓ 272
55.e even 4 1 inner 660.2.q.a ✓ 272
60.l odd 4 1 inner 660.2.q.a ✓ 272
132.d odd 2 1 inner 660.2.q.a ✓ 272
165.l odd 4 1 inner 660.2.q.a ✓ 272
220.i odd 4 1 inner 660.2.q.a ✓ 272
660.q even 4 1 inner 660.2.q.a ✓ 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.q.a ✓ 272 1.a even 1 1 trivial
660.2.q.a ✓ 272 3.b odd 2 1 inner
660.2.q.a ✓ 272 4.b odd 2 1 inner
660.2.q.a ✓ 272 5.c odd 4 1 inner
660.2.q.a ✓ 272 11.b odd 2 1 inner
660.2.q.a ✓ 272 12.b even 2 1 inner
660.2.q.a ✓ 272 15.e even 4 1 inner
660.2.q.a ✓ 272 20.e even 4 1 inner
660.2.q.a ✓ 272 33.d even 2 1 inner
660.2.q.a ✓ 272 44.c even 2 1 inner
660.2.q.a ✓ 272 55.e even 4 1 inner
660.2.q.a ✓ 272 60.l odd 4 1 inner
660.2.q.a ✓ 272 132.d odd 2 1 inner
660.2.q.a ✓ 272 165.l odd 4 1 inner
660.2.q.a ✓ 272 220.i odd 4 1 inner
660.2.q.a ✓ 272 660.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(660, [\chi])\).