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Properties

Label 334.2.c.b
Level $334$
Weight $2$
Character orbit 334.c
Analytic conductor $2.667$
Analytic rank $0$
Dimension $574$
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] = [334,2,Mod(3,334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(334, base_ring=CyclotomicField(166))
 
chi = DirichletCharacter(H, H._module([94]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("334.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 334.c (of order \(83\), degree \(82\), 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: \(2.66700342751\)
Analytic rank: \(0\)
Dimension: \(574\)
Relative dimension: \(7\) over \(\Q(\zeta_{83})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{83}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 574 q + 7 q^{2} + 2 q^{3} - 7 q^{4} - 2 q^{6} + 6 q^{7} + 7 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 574 q + 7 q^{2} + 2 q^{3} - 7 q^{4} - 2 q^{6} + 6 q^{7} + 7 q^{8} - 9 q^{9} - 6 q^{11} + 2 q^{12} + 2 q^{13} - 6 q^{14} - 4 q^{15} - 7 q^{16} - 8 q^{17} + 9 q^{18} - 6 q^{21} + 6 q^{22} + 2 q^{23} - 2 q^{24} - 165 q^{25} - 2 q^{26} - 4 q^{27} + 6 q^{28} - 14 q^{29} + 4 q^{30} + 10 q^{31} + 7 q^{32} - 12 q^{33} + 8 q^{34} - 20 q^{35} - 9 q^{36} + 2 q^{37} - 83 q^{38} - 4 q^{39} - 20 q^{41} + 6 q^{42} - 2 q^{43} - 6 q^{44} + 2 q^{45} - 2 q^{46} + 14 q^{47} + 2 q^{48} - 171 q^{49} - q^{50} - 417 q^{51} + 2 q^{52} - 4 q^{53} + 4 q^{54} + 18 q^{55} - 6 q^{56} - 6 q^{57} + 14 q^{58} - 2 q^{59} - 4 q^{60} + 16 q^{61} - 10 q^{62} + 40 q^{63} - 7 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} - 8 q^{68} + 16 q^{69} + 20 q^{70} - 4 q^{71} + 9 q^{72} + 22 q^{73} - 2 q^{74} - 533 q^{75} - 239 q^{77} + 4 q^{78} - 10 q^{79} + 17 q^{81} + 20 q^{82} - 14 q^{83} - 6 q^{84} + 6 q^{85} + 2 q^{86} + 10 q^{87} - 160 q^{88} - 16 q^{89} - 417 q^{90} + 10 q^{91} + 2 q^{92} + 10 q^{93} - 14 q^{94} - 38 q^{95} - 2 q^{96} - 8 q^{97} + 5 q^{98} - 32 q^{99}+O(q^{100}) \) 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 0.455676 − 0.890146i −2.00413 − 1.75478i −0.584719 − 0.811236i 0.650892 − 3.08132i −2.47524 + 0.984355i 4.34081 − 2.76861i −0.988561 + 0.150824i 0.541009 + 4.05988i −2.44622 − 1.98347i
3.2 0.455676 − 0.890146i −1.70416 − 1.49213i −0.584719 − 0.811236i −0.460795 + 2.18140i −2.10476 + 0.837022i −1.20467 + 0.768349i −0.988561 + 0.150824i 0.281433 + 2.11195i 1.73179 + 1.40419i
3.3 0.455676 − 0.890146i −0.571244 − 0.500172i −0.584719 − 0.811236i −0.682585 + 3.23135i −0.705528 + 0.280575i 2.94947 − 1.88120i −0.988561 + 0.150824i −0.320120 − 2.40227i 2.56534 + 2.08005i
3.4 0.455676 − 0.890146i −0.503049 − 0.440461i −0.584719 − 0.811236i 0.272296 − 1.28905i −0.621302 + 0.247080i −1.43923 + 0.917957i −0.988561 + 0.150824i −0.337216 − 2.53057i −1.02336 − 0.829771i
3.5 0.455676 − 0.890146i 0.667080 + 0.584083i −0.584719 − 0.811236i 0.522875 − 2.47528i 0.823892 − 0.327646i 0.958742 − 0.611495i −0.988561 + 0.150824i −0.292427 − 2.19445i −1.96510 − 1.59336i
3.6 0.455676 − 0.890146i 1.37953 + 1.20789i −0.584719 − 0.811236i −0.714286 + 3.38143i 1.70382 − 0.677575i −3.44480 + 2.19713i −0.988561 + 0.150824i 0.0478301 + 0.358930i 2.68448 + 2.17665i
3.7 0.455676 − 0.890146i 2.26017 + 1.97896i −0.584719 − 0.811236i 0.411604 − 1.94853i 2.79147 − 1.11011i −0.330294 + 0.210665i −0.988561 + 0.150824i 0.795793 + 5.97186i −1.54692 − 1.25429i
7.1 −0.206677 − 0.978409i −2.35442 + 1.50167i −0.914569 + 0.404430i 1.46195 − 1.87446i 1.95586 + 1.99323i −1.31600 + 0.523348i 0.584719 + 0.811236i 2.02329 − 4.35090i −2.13614 − 1.04298i
7.2 −0.206677 − 0.978409i −2.32229 + 1.48118i −0.914569 + 0.404430i −1.86822 + 2.39537i 1.92916 + 1.96602i 2.32197 − 0.923403i 0.584719 + 0.811236i 1.93413 − 4.15917i 2.72977 + 1.33282i
7.3 −0.206677 − 0.978409i −0.142703 + 0.0910175i −0.914569 + 0.404430i −0.840453 + 1.07760i 0.118546 + 0.120811i 1.50763 − 0.599556i 0.584719 + 0.811236i −1.25292 + 2.69428i 1.22803 + 0.599592i
7.4 −0.206677 − 0.978409i −0.0752411 + 0.0479895i −0.914569 + 0.404430i −0.665259 + 0.852970i 0.0625040 + 0.0636982i −1.26359 + 0.502506i 0.584719 + 0.811236i −1.26164 + 2.71303i 0.972048 + 0.474606i
7.5 −0.206677 − 0.978409i 1.19061 − 0.759384i −0.914569 + 0.404430i 2.26755 − 2.90737i −0.989061 − 1.00796i 1.19104 − 0.473653i 0.584719 + 0.811236i −0.424102 + 0.911991i −3.31325 − 1.61771i
7.6 −0.206677 − 0.978409i 2.39352 − 1.52661i −0.914569 + 0.404430i −0.771229 + 0.988842i −1.98834 − 2.02633i 4.46447 − 1.77543i 0.584719 + 0.811236i 2.13341 − 4.58769i 1.12689 + 0.550207i
7.7 −0.206677 − 0.978409i 2.44513 − 1.55953i −0.914569 + 0.404430i 0.415662 − 0.532946i −2.03121 − 2.07002i −2.72105 + 1.08211i 0.584719 + 0.811236i 2.28154 − 4.90624i −0.607347 − 0.296539i
9.1 0.584719 + 0.811236i −0.407417 − 3.05737i −0.316208 + 0.948690i 0.496265 + 0.219452i 2.24202 − 2.11821i 0.843128 − 1.81307i −0.954504 + 0.298198i −6.28620 + 1.70565i 0.112148 + 0.530906i
9.2 0.584719 + 0.811236i −0.215681 − 1.61853i −0.316208 + 0.948690i 2.93446 + 1.29764i 1.18690 − 1.12135i −1.68280 + 3.61870i −0.954504 + 0.298198i 0.322191 − 0.0874209i 0.663138 + 3.13929i
9.3 0.584719 + 0.811236i −0.0372028 − 0.279180i −0.316208 + 0.948690i −3.70856 − 1.63996i 0.204728 − 0.193422i 1.46085 − 3.14141i −0.954504 + 0.298198i 2.81876 − 0.764821i −0.838073 − 3.96743i
9.4 0.584719 + 0.811236i 0.0494350 + 0.370974i −0.316208 + 0.948690i 2.04202 + 0.902997i −0.272042 + 0.257019i 1.35508 − 2.91397i −0.954504 + 0.298198i 2.76014 − 0.748915i 0.461462 + 2.18456i
9.5 0.584719 + 0.811236i 0.0825641 + 0.619584i −0.316208 + 0.948690i −2.10165 − 0.929370i −0.454352 + 0.429262i −2.03565 + 4.37747i −0.954504 + 0.298198i 2.51825 − 0.683283i −0.474939 − 2.24836i
9.6 0.584719 + 0.811236i 0.366452 + 2.74996i −0.316208 + 0.948690i 2.38380 + 1.05414i −2.01659 + 1.90523i −0.379501 + 0.816081i −0.954504 + 0.298198i −4.53265 + 1.22986i 0.538700 + 2.55020i
See next 80 embeddings (of 574 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.c even 83 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 334.2.c.b ✓ 574
167.c even 83 1 inner 334.2.c.b ✓ 574
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
334.2.c.b ✓ 574 1.a even 1 1 trivial
334.2.c.b ✓ 574 167.c even 83 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{574} - 2 T_{3}^{573} + 17 T_{3}^{572} - 34 T_{3}^{571} + 196 T_{3}^{570} - 394 T_{3}^{569} + \cdots + 55\!\cdots\!69 \) acting on \(S_{2}^{\mathrm{new}}(334, [\chi])\). Copy content Toggle raw display