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Properties

Label 671.2.bn.b
Level $671$
Weight $2$
Character orbit 671.bn
Analytic conductor $5.358$
Analytic rank $0$
Dimension $208$
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] = [671,2,Mod(12,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bn (of order \(15\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q + 2 q^{2} - 2 q^{3} + 30 q^{4} - 10 q^{5} - 4 q^{6} - 8 q^{7} - 30 q^{8} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q + 2 q^{2} - 2 q^{3} + 30 q^{4} - 10 q^{5} - 4 q^{6} - 8 q^{7} - 30 q^{8} - 60 q^{9} + 27 q^{10} + 208 q^{11} + 119 q^{12} + 8 q^{13} - 17 q^{14} - 8 q^{15} - 6 q^{16} + 8 q^{17} + 27 q^{18} - 25 q^{19} + 22 q^{20} - 58 q^{21} + 2 q^{22} + 10 q^{23} - 33 q^{24} - 14 q^{25} + 12 q^{26} - 8 q^{27} - 64 q^{28} - 60 q^{29} - 40 q^{30} + 15 q^{31} - 78 q^{32} - 2 q^{33} - 78 q^{34} - 48 q^{35} - 30 q^{36} - 34 q^{37} + 15 q^{38} + 62 q^{39} - 127 q^{40} - 4 q^{41} - 34 q^{42} - 33 q^{43} + 30 q^{44} - 25 q^{45} + 50 q^{46} - 50 q^{47} + 6 q^{48} + 136 q^{50} - 77 q^{51} + 110 q^{52} - 62 q^{53} + 76 q^{54} - 10 q^{55} + 44 q^{56} + 4 q^{57} + q^{58} + 13 q^{59} - 60 q^{60} + 31 q^{61} - 86 q^{62} + 24 q^{63} - 154 q^{64} - 55 q^{65} - 4 q^{66} + 13 q^{67} - 6 q^{68} - 20 q^{69} - 97 q^{70} + 38 q^{71} + 404 q^{72} + 58 q^{73} + 108 q^{74} - 11 q^{75} - 55 q^{76} - 8 q^{77} + 80 q^{78} + 50 q^{79} - 37 q^{80} - 142 q^{81} - 126 q^{82} + 52 q^{83} - 42 q^{84} - 34 q^{85} - 113 q^{86} - 5 q^{87} - 30 q^{88} - 26 q^{89} - 131 q^{90} - 48 q^{91} - 43 q^{92} - 94 q^{93} + 79 q^{94} - 40 q^{95} - 32 q^{96} + 7 q^{97} - 83 q^{98} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.53453 − 1.12844i 2.31349 − 1.68085i 3.81218 + 4.23386i 0.401967 − 0.0854407i −7.76035 + 1.64951i −0.0462533 + 0.440071i −3.16974 − 9.75547i 1.59994 − 4.92409i −1.11521 − 0.237046i
12.2 −2.45861 − 1.09464i −0.400618 + 0.291066i 3.50824 + 3.89630i −3.94724 + 0.839012i 1.30357 − 0.277083i −0.271690 + 2.58495i −2.69703 − 8.30062i −0.851276 + 2.61996i 10.6231 + 2.25802i
12.3 −2.18489 − 0.972777i −0.337959 + 0.245541i 2.48920 + 2.76454i 0.529335 − 0.112514i 0.977261 − 0.207723i 0.170215 − 1.61949i −1.27123 − 3.91245i −0.873125 + 2.68720i −1.26599 − 0.269095i
12.4 −1.81681 − 0.808896i −2.06921 + 1.50337i 1.30823 + 1.45293i −0.440056 + 0.0935368i 4.97544 − 1.05756i −0.0466979 + 0.444301i 0.0275836 + 0.0848936i 1.09446 − 3.36841i 0.875161 + 0.186021i
12.5 −1.80349 − 0.802964i 1.17020 − 0.850197i 1.26955 + 1.40998i 3.50463 − 0.744933i −2.79311 + 0.593694i 0.361539 − 3.43982i 0.0626453 + 0.192803i −0.280527 + 0.863374i −6.91871 − 1.47062i
12.6 −1.56510 − 0.696828i −0.0263301 + 0.0191299i 0.625711 + 0.694922i −1.13259 + 0.240738i 0.0545395 − 0.0115927i −0.255972 + 2.43541i 0.563767 + 1.73510i −0.926724 + 2.85216i 1.94036 + 0.412437i
12.7 −1.19903 − 0.533842i 1.57393 − 1.14353i −0.185580 − 0.206108i 0.473477 − 0.100641i −2.49765 + 0.530892i −0.393684 + 3.74565i 0.923656 + 2.84272i 0.242551 − 0.746496i −0.621438 − 0.132091i
12.8 −0.842658 − 0.375175i 0.996445 − 0.723960i −0.768946 − 0.854001i −3.50426 + 0.744854i −1.11127 + 0.236209i 0.188886 − 1.79713i 0.897635 + 2.76264i −0.458266 + 1.41040i 3.23234 + 0.687056i
12.9 −0.789883 − 0.351679i −2.24559 + 1.63152i −0.838024 − 0.930720i −0.509922 + 0.108387i 2.34752 − 0.498982i 0.0356348 − 0.339043i 0.869000 + 2.67451i 1.45378 − 4.47428i 0.440896 + 0.0937153i
12.10 −0.734032 − 0.326812i 2.18763 − 1.58940i −0.906264 − 1.00651i 0.358155 − 0.0761282i −2.12522 + 0.451730i 0.311118 − 2.96009i 0.832877 + 2.56333i 1.33245 − 4.10087i −0.287777 − 0.0611689i
12.11 −0.553133 − 0.246271i 0.0669308 − 0.0486281i −1.09295 − 1.21385i 3.09836 − 0.658577i −0.0489973 + 0.0104147i −0.0299015 + 0.284494i 0.679820 + 2.09227i −0.924936 + 2.84666i −1.87599 − 0.398755i
12.12 −0.399268 − 0.177766i −1.11552 + 0.810470i −1.21045 − 1.34434i 0.0911235 − 0.0193689i 0.589464 − 0.125294i 0.478298 − 4.55070i 0.514430 + 1.58325i −0.339536 + 1.04498i −0.0398258 − 0.00846524i
12.13 −0.0321047 − 0.0142939i −1.22894 + 0.892880i −1.33743 − 1.48537i −3.42651 + 0.728328i 0.0522176 − 0.0110992i −0.411897 + 3.91894i 0.0434256 + 0.133650i −0.213983 + 0.658571i 0.120418 + 0.0255956i
12.14 0.452831 + 0.201613i −2.53589 + 1.84243i −1.17385 − 1.30370i 2.77297 − 0.589413i −1.51979 + 0.323041i −0.415855 + 3.95659i −0.575065 − 1.76987i 2.10913 − 6.49123i 1.37452 + 0.292163i
12.15 0.726805 + 0.323594i −0.916079 + 0.665570i −0.914730 − 1.01591i 3.21677 − 0.683745i −0.881185 + 0.187302i 0.234493 − 2.23105i −0.827787 − 2.54767i −0.530834 + 1.63374i 2.55922 + 0.543978i
12.16 0.762464 + 0.339471i −1.49909 + 1.08916i −0.872150 − 0.968621i −3.87387 + 0.823416i −1.51274 + 0.321543i 0.303392 − 2.88658i −0.851989 − 2.62215i 0.133972 − 0.412323i −3.23321 − 0.687240i
12.17 0.997012 + 0.443898i −0.648036 + 0.470826i −0.541274 − 0.601146i −0.201784 + 0.0428906i −0.855099 + 0.181757i −0.200081 + 1.90365i −0.947310 − 2.91552i −0.728777 + 2.24295i −0.220220 − 0.0468093i
12.18 1.01366 + 0.451310i 1.11708 − 0.811606i −0.514439 − 0.571343i −2.62112 + 0.557136i 1.49862 − 0.318542i 0.136753 − 1.30112i −0.949375 − 2.92188i −0.337887 + 1.03991i −2.90836 − 0.618191i
12.19 1.08826 + 0.484525i 2.51478 − 1.82709i −0.388713 − 0.431710i 1.45914 − 0.310149i 3.62201 − 0.769881i −0.475379 + 4.52293i −0.950080 − 2.92405i 2.05878 − 6.33629i 1.73820 + 0.369465i
12.20 1.16819 + 0.520111i 1.26471 − 0.918868i −0.244114 − 0.271116i 1.92692 − 0.409580i 1.95534 − 0.415619i 0.114793 − 1.09219i −0.934466 − 2.87599i −0.171870 + 0.528961i 2.46404 + 0.523747i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bn.b ✓ 208
61.i even 15 1 inner 671.2.bn.b ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bn.b ✓ 208 1.a even 1 1 trivial
671.2.bn.b ✓ 208 61.i even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{208} - 2 T_{2}^{207} - 39 T_{2}^{206} + 96 T_{2}^{205} + 648 T_{2}^{204} - 1968 T_{2}^{203} + \cdots + 4210841881 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display