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Properties

Label 608.2.bn.a
Level $608$
Weight $2$
Character orbit 608.bn
Analytic conductor $4.855$
Analytic rank $0$
Dimension $624$
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] = [608,2,Mod(27,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.bn (of order \(24\), 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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(78\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 12 q^{2} - 12 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 624 q - 12 q^{2} - 12 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 16 q^{7} - 4 q^{9} - 36 q^{10} - 16 q^{11} - 12 q^{13} - 12 q^{14} - 24 q^{15} - 4 q^{16} - 8 q^{19} - 16 q^{20} - 12 q^{21} - 12 q^{22} - 4 q^{23} + 12 q^{24} - 4 q^{25} + 24 q^{26} - 44 q^{28} - 12 q^{29} - 32 q^{30} - 12 q^{32} - 24 q^{33} + 48 q^{34} + 44 q^{35} - 4 q^{36} - 20 q^{38} - 16 q^{39} + 48 q^{40} - 12 q^{41} - 44 q^{42} - 4 q^{43} + 40 q^{44} - 40 q^{45} - 8 q^{47} - 12 q^{48} - 60 q^{51} - 108 q^{52} - 12 q^{53} - 16 q^{54} - 4 q^{55} - 8 q^{57} - 16 q^{58} - 12 q^{59} + 72 q^{60} + 60 q^{61} + 12 q^{62} + 56 q^{64} + 108 q^{66} - 12 q^{67} + 56 q^{68} - 156 q^{70} - 12 q^{71} - 12 q^{72} - 4 q^{73} - 4 q^{74} - 72 q^{76} - 16 q^{77} - 12 q^{78} - 24 q^{79} - 20 q^{80} - 44 q^{82} - 16 q^{83} - 44 q^{85} - 12 q^{86} - 128 q^{87} - 12 q^{89} - 228 q^{90} - 12 q^{91} - 4 q^{92} - 16 q^{93} + 192 q^{96} - 24 q^{97} - 72 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.41119 + 0.0924919i −1.67225 − 0.220156i 1.98289 − 0.261047i 1.30202 − 0.999075i 2.38022 + 0.156011i −0.0416700 + 0.0416700i −2.77408 + 0.551787i −0.149825 − 0.0401455i −1.74498 + 1.53031i
27.2 −1.40866 − 0.125196i 2.90374 + 0.382285i 1.96865 + 0.352717i −0.863399 + 0.662509i −4.04253 − 0.902046i −3.31523 + 3.31523i −2.72900 − 0.743325i 5.38779 + 1.44365i 1.29918 − 0.825157i
27.3 −1.40598 + 0.152396i −2.18711 − 0.287938i 1.95355 − 0.428531i 0.965113 − 0.740557i 3.11890 + 0.0715284i 2.45556 − 2.45556i −2.68134 + 0.900219i 1.80274 + 0.483044i −1.24407 + 1.18829i
27.4 −1.40056 + 0.196053i 1.77020 + 0.233052i 1.92313 − 0.549166i 2.80839 − 2.15495i −2.52496 + 0.0206504i −0.0979978 + 0.0979978i −2.58579 + 1.14617i 0.181525 + 0.0486394i −3.51083 + 3.56873i
27.5 −1.39634 − 0.224105i 2.24746 + 0.295884i 1.89955 + 0.625856i −0.465804 + 0.357424i −3.07192 − 0.916823i 2.27210 − 2.27210i −2.51217 − 1.29961i 2.06575 + 0.553517i 0.730523 − 0.394698i
27.6 −1.39109 + 0.254677i −0.526308 − 0.0692898i 1.87028 − 0.708559i −2.38375 + 1.82912i 0.749790 − 0.0376502i −1.68437 + 1.68437i −2.42128 + 1.46199i −2.62558 − 0.703522i 2.85019 − 3.15156i
27.7 −1.35428 − 0.407356i −1.29962 − 0.171098i 1.66812 + 1.10334i −3.39791 + 2.60731i 1.69035 + 0.761123i 2.55406 − 2.55406i −1.80964 − 2.17375i −1.23804 − 0.331732i 5.66381 − 2.14685i
27.8 −1.34192 + 0.446374i 1.27647 + 0.168051i 1.60150 − 1.19800i −1.10541 + 0.848207i −1.78794 + 0.344274i −1.86593 + 1.86593i −1.61433 + 2.32249i −1.29664 − 0.347433i 1.10475 − 1.63165i
27.9 −1.30246 − 0.550988i 0.572134 + 0.0753228i 1.39282 + 1.43528i −0.995841 + 0.764136i −0.703682 − 0.413344i −0.463978 + 0.463978i −1.02328 − 2.63683i −2.57611 − 0.690268i 1.71808 − 0.446563i
27.10 −1.30127 − 0.553795i −0.982570 − 0.129358i 1.38662 + 1.44128i 2.04598 − 1.56994i 1.20695 + 0.712472i −2.85361 + 2.85361i −1.00620 − 2.64340i −1.94907 − 0.522251i −3.53181 + 0.909861i
27.11 −1.29697 + 0.563803i −3.20342 − 0.421738i 1.36425 − 1.46247i −2.31326 + 1.77503i 4.39251 − 1.25912i −0.864931 + 0.864931i −0.944846 + 2.66595i 7.18625 + 1.92555i 1.99946 − 3.60638i
27.12 −1.28669 − 0.586886i −2.95494 − 0.389026i 1.31113 + 1.51028i −0.506327 + 0.388519i 3.57378 + 2.23477i 0.0743867 − 0.0743867i −0.800653 − 2.71274i 5.68257 + 1.52264i 0.879501 − 0.202746i
27.13 −1.26364 + 0.634991i 0.307802 + 0.0405229i 1.19357 − 1.60480i −0.0610655 + 0.0468572i −0.414683 + 0.144245i 2.48392 − 2.48392i −0.489214 + 2.78580i −2.80468 − 0.751511i 0.0474109 − 0.0979866i
27.14 −1.17173 + 0.791859i 3.32458 + 0.437689i 0.745920 − 1.85570i 0.978967 − 0.751188i −4.24211 + 2.11974i 1.84411 − 1.84411i 0.595429 + 2.76504i 7.96346 + 2.13380i −0.552254 + 1.65540i
27.15 −1.14853 + 0.825160i −1.70881 − 0.224969i 0.638222 − 1.89543i 1.95179 − 1.49766i 2.14825 − 1.15166i −2.35890 + 2.35890i 0.831023 + 2.70359i −0.0283626 − 0.00759973i −1.00587 + 3.33065i
27.16 −1.14601 − 0.828645i 0.453552 + 0.0597112i 0.626695 + 1.89928i 1.77133 − 1.35919i −0.470297 − 0.444263i −1.09942 + 1.09942i 0.855626 − 2.69591i −2.69563 − 0.722293i −3.15626 + 0.0898460i
27.17 −1.02247 − 0.977015i 2.70315 + 0.355877i 0.0908843 + 1.99793i 2.75161 − 2.11138i −2.41619 − 3.00489i −0.111665 + 0.111665i 1.85908 − 2.13162i 4.28261 + 1.14752i −4.87628 − 0.529539i
27.18 −0.989376 − 1.01051i 0.452291 + 0.0595452i −0.0422691 + 1.99955i 0.756169 − 0.580229i −0.387315 − 0.515958i 3.69348 − 3.69348i 2.06239 − 1.93560i −2.69676 − 0.722594i −1.33446 − 0.190053i
27.19 −0.964268 + 1.03450i 2.05901 + 0.271074i −0.140375 − 1.99507i 0.0430934 − 0.0330667i −2.26587 + 1.86866i −1.23053 + 1.23053i 2.19925 + 1.77856i 1.26828 + 0.339835i −0.00734608 + 0.0764652i
27.20 −0.960880 − 1.03765i 3.00591 + 0.395736i −0.153417 + 1.99411i −2.49343 + 1.91328i −2.47769 − 3.49933i 0.729043 − 0.729043i 2.21659 − 1.75691i 5.98113 + 1.60264i 4.38119 + 0.748867i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner
32.h odd 8 1 inner
608.bn even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.bn.a ✓ 624
19.d odd 6 1 inner 608.2.bn.a ✓ 624
32.h odd 8 1 inner 608.2.bn.a ✓ 624
608.bn even 24 1 inner 608.2.bn.a ✓ 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.bn.a ✓ 624 1.a even 1 1 trivial
608.2.bn.a ✓ 624 19.d odd 6 1 inner
608.2.bn.a ✓ 624 32.h odd 8 1 inner
608.2.bn.a ✓ 624 608.bn even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(608, [\chi])\).