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Properties

Label 608.2.bt.a
Level $608$
Weight $2$
Character orbit 608.bt
Analytic conductor $4.855$
Analytic rank $0$
Dimension $1872$
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(3,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(72))
 
chi = DirichletCharacter(H, H._module([36, 27, 52]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.bt (of order \(72\), degree \(24\), 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: \(1872\)
Relative dimension: \(78\) over \(\Q(\zeta_{72})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1872 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 12 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1872 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 12 q^{7} - 36 q^{8} - 24 q^{9} - 12 q^{11} - 36 q^{12} - 24 q^{13} - 24 q^{14} - 48 q^{15} - 24 q^{16} - 24 q^{19} - 48 q^{20} - 24 q^{21} - 24 q^{22} - 24 q^{23} - 24 q^{24} - 24 q^{25} - 12 q^{26} - 36 q^{27} - 24 q^{28} - 24 q^{29} - 12 q^{30} - 24 q^{32} - 48 q^{33} + 96 q^{34} - 24 q^{35} - 24 q^{36} + 36 q^{38} - 48 q^{39} - 84 q^{40} - 24 q^{41} - 264 q^{42} - 24 q^{43} - 156 q^{44} - 12 q^{45} - 36 q^{46} - 48 q^{47} - 24 q^{48} + 72 q^{50} + 24 q^{51} + 72 q^{52} - 24 q^{53} + 12 q^{54} - 24 q^{55} - 24 q^{57} - 48 q^{58} - 24 q^{59} - 108 q^{60} - 24 q^{61} - 12 q^{64} - 72 q^{65} - 24 q^{66} - 24 q^{67} - 84 q^{68} - 324 q^{69} + 120 q^{70} - 24 q^{71} - 24 q^{72} - 24 q^{73} - 24 q^{74} - 24 q^{76} - 48 q^{77} - 24 q^{78} - 48 q^{79} - 144 q^{80} + 96 q^{82} - 12 q^{83} - 36 q^{84} - 24 q^{85} - 24 q^{86} - 12 q^{87} - 36 q^{88} - 24 q^{89} + 192 q^{90} - 24 q^{91} - 24 q^{92} + 12 q^{93} - 648 q^{96} - 48 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41420 − 0.00599287i −2.95435 − 0.128990i 1.99993 + 0.0169502i −1.46856 + 0.325572i 4.17728 + 0.200123i −1.29616 + 4.83734i −2.82820 − 0.0359564i 5.72298 + 0.500696i 2.07879 − 0.451624i
3.2 −1.41319 − 0.0537406i 0.144419 + 0.00630545i 1.99422 + 0.151892i 1.89683 − 0.420517i −0.203752 − 0.0166719i −0.288554 + 1.07690i −2.81006 − 0.321823i −2.96777 − 0.259646i −2.70318 + 0.492335i
3.3 −1.41209 − 0.0775588i −1.00445 − 0.0438554i 1.98797 + 0.219039i 3.46068 − 0.767213i 1.41497 + 0.139832i −0.453458 + 1.69233i −2.79019 − 0.463487i −1.98158 − 0.173366i −4.94627 + 0.814965i
3.4 −1.40207 + 0.184921i −0.835522 − 0.0364797i 1.93161 − 0.518544i −2.36007 + 0.523215i 1.17821 − 0.103358i −0.0690746 + 0.257790i −2.61236 + 1.08423i −2.29182 − 0.200508i 3.21223 − 1.17001i
3.5 −1.39547 − 0.229487i 3.13637 + 0.136937i 1.89467 + 0.640485i −4.11458 + 0.912180i −4.34529 − 0.910848i 0.839810 − 3.13421i −2.49697 − 1.32858i 6.82949 + 0.597503i 5.95110 − 0.328676i
3.6 −1.38057 + 0.306656i 0.316991 + 0.0138401i 1.81192 − 0.846716i −1.24990 + 0.277096i −0.441871 + 0.0780998i 1.30156 − 4.85749i −2.24183 + 1.72458i −2.88829 − 0.252693i 1.64059 − 0.765837i
3.7 −1.37377 − 0.335775i −3.03340 − 0.132441i 1.77451 + 0.922557i 2.48803 − 0.551582i 4.12274 + 1.20048i 0.956824 − 3.57091i −2.12801 − 1.86322i 6.19540 + 0.542027i −3.60319 − 0.0776670i
3.8 −1.36670 − 0.363514i 2.00011 + 0.0873269i 1.73572 + 0.993626i 2.07831 − 0.460750i −2.70180 − 0.846419i 0.140029 − 0.522596i −2.01100 − 1.98894i 1.00425 + 0.0878601i −3.00790 − 0.125789i
3.9 −1.36112 + 0.383874i 2.39134 + 0.104408i 1.70528 − 1.04500i 2.99113 − 0.663117i −3.29497 + 0.775861i 0.863676 − 3.22328i −1.91994 + 2.07698i 2.71900 + 0.237882i −3.81672 + 2.05080i
3.10 −1.33476 + 0.467340i 3.26125 + 0.142389i 1.56319 − 1.24758i 0.562002 − 0.124593i −4.41955 + 1.33406i −1.14540 + 4.27470i −1.50344 + 2.39576i 7.62692 + 0.667269i −0.691912 + 0.428948i
3.11 −1.32293 − 0.499847i 1.57440 + 0.0687396i 1.50031 + 1.32253i −1.09063 + 0.241787i −2.04846 − 0.877895i −0.763407 + 2.84907i −1.32374 − 2.49954i −0.514587 − 0.0450205i 1.56369 + 0.225281i
3.12 −1.31841 − 0.511660i −1.96342 − 0.0857249i 1.47641 + 1.34916i −3.09931 + 0.687100i 2.54473 + 1.11763i 0.467899 − 1.74622i −1.25620 − 2.53416i 0.859098 + 0.0751614i 4.43772 + 0.679914i
3.13 −1.27144 + 0.619227i 1.55498 + 0.0678917i 1.23312 − 1.57462i −1.17268 + 0.259977i −2.01910 + 0.876564i 0.00888125 − 0.0331453i −0.592784 + 2.76561i −0.575242 − 0.0503271i 1.33001 − 1.05670i
3.14 −1.20136 + 0.746151i −1.01023 − 0.0441078i 0.886518 − 1.79279i 0.718680 − 0.159328i 1.24656 − 0.700798i −0.365353 + 1.36352i 0.272664 + 2.81525i −1.96996 − 0.172349i −0.744509 + 0.727653i
3.15 −1.17960 + 0.780089i −2.19601 − 0.0958797i 0.782923 − 1.84039i 2.75781 − 0.611392i 2.66521 − 1.59998i 0.281459 − 1.05042i 0.512129 + 2.78168i 1.82467 + 0.159638i −2.77618 + 2.87254i
3.16 −1.12136 − 0.861713i −2.32450 − 0.101490i 0.514901 + 1.93258i 1.09080 − 0.241824i 2.51915 + 2.11686i −1.05255 + 3.92816i 1.08794 − 2.61082i 2.40442 + 0.210359i −1.43156 − 0.668782i
3.17 −1.09637 − 0.893292i 0.224569 + 0.00980491i 0.404058 + 1.95876i −2.72299 + 0.603673i −0.237453 − 0.211356i −0.564193 + 2.10560i 1.30675 − 2.50847i −2.93825 − 0.257063i 3.52467 + 1.77058i
3.18 −1.08056 + 0.912358i 1.81843 + 0.0793944i 0.335204 − 1.97171i −2.10667 + 0.467038i −2.03735 + 1.57327i −0.0976280 + 0.364353i 1.43670 + 2.43637i 0.311805 + 0.0272794i 1.85027 − 2.42670i
3.19 −1.06565 + 0.929727i −3.14594 − 0.137355i 0.271215 − 1.98153i −3.43962 + 0.762546i 3.48017 − 2.77850i 0.930294 − 3.47191i 1.55326 + 2.36377i 6.88951 + 0.602754i 2.95647 − 4.01052i
3.20 −1.00798 + 0.991961i −1.02254 − 0.0446451i 0.0320273 − 1.99974i −3.72716 + 0.826291i 1.07498 − 0.969319i −0.996639 + 3.71951i 1.95138 + 2.04746i −1.94499 − 0.170164i 2.93723 − 4.53007i
See next 80 embeddings (of 1872 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner
32.h odd 8 1 inner
608.bt even 72 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.bt.a ✓ 1872
19.f odd 18 1 inner 608.2.bt.a ✓ 1872
32.h odd 8 1 inner 608.2.bt.a ✓ 1872
608.bt even 72 1 inner 608.2.bt.a ✓ 1872
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.bt.a ✓ 1872 1.a even 1 1 trivial
608.2.bt.a ✓ 1872 19.f odd 18 1 inner
608.2.bt.a ✓ 1872 32.h odd 8 1 inner
608.2.bt.a ✓ 1872 608.bt even 72 1 inner

Hecke kernels

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