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Properties

Label 77.3.k.a
Level $77$
Weight $3$
Character orbit 77.k
Analytic conductor $2.098$
Analytic rank $0$
Dimension $48$
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] = [77,3,Mod(8,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 77.k (of order \(10\), degree \(4\), 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: \(2.09809803557\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} + 30 q^{4} + 6 q^{5} - 40 q^{6} - 20 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} + 30 q^{4} + 6 q^{5} - 40 q^{6} - 20 q^{8} - 8 q^{9} + 6 q^{11} + 64 q^{12} + 30 q^{13} - 21 q^{14} + 72 q^{15} - 142 q^{16} + 5 q^{18} - 60 q^{19} - 78 q^{20} - 24 q^{22} - 36 q^{23} - 240 q^{24} - 90 q^{25} + 262 q^{26} + 90 q^{27} + 175 q^{28} - 90 q^{29} + 240 q^{30} - 10 q^{31} - 338 q^{33} + 100 q^{34} - 118 q^{36} - 118 q^{37} - 42 q^{38} + 50 q^{39} - 210 q^{40} + 281 q^{44} + 240 q^{45} - 215 q^{46} + 228 q^{47} - 336 q^{48} + 84 q^{49} + 330 q^{50} + 180 q^{51} - 230 q^{52} + 34 q^{53} - 142 q^{55} - 154 q^{56} + 540 q^{57} - 141 q^{58} - 260 q^{59} + 376 q^{60} - 160 q^{61} - 130 q^{62} + 406 q^{64} + 442 q^{66} + 404 q^{67} + 390 q^{68} + 26 q^{69} + 28 q^{70} - 102 q^{71} + 675 q^{72} + 320 q^{73} - 650 q^{74} - 76 q^{75} - 28 q^{77} - 1028 q^{78} + 160 q^{79} + 1076 q^{80} - 310 q^{81} - 152 q^{82} - 670 q^{83} - 560 q^{84} + 60 q^{85} - 645 q^{86} + 36 q^{88} - 476 q^{89} - 320 q^{90} - 126 q^{91} - 125 q^{92} - 540 q^{93} - 1050 q^{94} + 120 q^{95} + 580 q^{96} + 544 q^{97} - 574 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.29889 − 3.16415i −1.22583 − 3.77270i −3.49088 + 10.7438i −1.93908 − 1.40883i −9.11936 + 12.5517i −2.51626 − 0.817582i 27.1415 − 8.81882i −5.44949 + 3.95929i 9.37429i
8.2 −1.79202 − 2.46650i 0.356265 + 1.09647i −1.63623 + 5.03580i 7.42706 + 5.39607i 2.06602 − 2.84363i 2.51626 + 0.817582i 3.75476 − 1.22000i 6.20583 − 4.50880i − 27.9887i
8.3 −1.08064 − 1.48738i −0.525678 − 1.61787i 0.191566 − 0.589580i −1.44449 − 1.04949i −1.83831 + 2.53022i −2.51626 − 0.817582i −8.07801 + 2.62470i 4.93998 − 3.58911i 3.28262i
8.4 −0.827130 − 1.13845i −1.67117 − 5.14334i 0.624150 − 1.92094i 2.27093 + 1.64993i −4.47314 + 6.15676i 2.51626 + 0.817582i −8.05644 + 2.61770i −16.3800 + 11.9008i − 3.95004i
8.5 −0.266709 − 0.367093i 0.208599 + 0.642001i 1.17244 − 3.60841i −7.51387 − 5.45915i 0.180039 − 0.247802i 2.51626 + 0.817582i −3.36350 + 1.09287i 6.91250 − 5.02223i 4.21430i
8.6 −0.215317 − 0.296358i 0.326605 + 1.00519i 1.19460 − 3.67660i 4.41565 + 3.20816i 0.227572 − 0.313226i −2.51626 − 0.817582i −2.74037 + 0.890400i 6.37742 − 4.63347i − 1.99939i
8.7 0.0256207 + 0.0352638i 1.64464 + 5.06169i 1.23548 − 3.80242i 1.04076 + 0.756155i −0.136358 + 0.187680i 2.51626 + 0.817582i 0.331562 − 0.107731i −15.6347 + 11.3593i 0.0560743i
8.8 0.909197 + 1.25140i −0.639838 − 1.96922i 0.496699 − 1.52868i 2.81297 + 2.04374i 1.88255 − 2.59111i 2.51626 + 0.817582i 8.24904 − 2.68028i 3.81272 − 2.77010i 5.37832i
8.9 1.06954 + 1.47209i −1.33967 − 4.12308i 0.212921 − 0.655302i −4.43999 − 3.22585i 4.63673 − 6.38191i −2.51626 − 0.817582i 8.11459 − 2.63659i −7.92390 + 5.75705i − 9.98626i
8.10 1.40269 + 1.93064i 1.07115 + 3.29665i −0.523764 + 1.61198i −0.684591 − 0.497384i −4.86217 + 6.69220i −2.51626 − 0.817582i 5.23159 − 1.69985i −2.43941 + 1.77234i − 2.01938i
8.11 2.02949 + 2.79336i 0.347622 + 1.06987i −2.44794 + 7.53397i −3.61079 − 2.62339i −2.28303 + 3.14232i 2.51626 + 0.817582i −12.8780 + 4.18431i 6.25737 − 4.54625i − 15.4104i
8.12 2.16220 + 2.97601i −1.17073 − 3.60314i −2.94546 + 9.06519i 6.51956 + 4.73674i 8.19161 − 11.2748i −2.51626 − 0.817582i −19.3527 + 6.28807i −4.33084 + 3.14654i 29.6440i
29.1 −2.29889 + 3.16415i −1.22583 + 3.77270i −3.49088 − 10.7438i −1.93908 + 1.40883i −9.11936 − 12.5517i −2.51626 + 0.817582i 27.1415 + 8.81882i −5.44949 − 3.95929i − 9.37429i
29.2 −1.79202 + 2.46650i 0.356265 − 1.09647i −1.63623 − 5.03580i 7.42706 − 5.39607i 2.06602 + 2.84363i 2.51626 − 0.817582i 3.75476 + 1.22000i 6.20583 + 4.50880i 27.9887i
29.3 −1.08064 + 1.48738i −0.525678 + 1.61787i 0.191566 + 0.589580i −1.44449 + 1.04949i −1.83831 − 2.53022i −2.51626 + 0.817582i −8.07801 − 2.62470i 4.93998 + 3.58911i − 3.28262i
29.4 −0.827130 + 1.13845i −1.67117 + 5.14334i 0.624150 + 1.92094i 2.27093 − 1.64993i −4.47314 − 6.15676i 2.51626 − 0.817582i −8.05644 − 2.61770i −16.3800 − 11.9008i 3.95004i
29.5 −0.266709 + 0.367093i 0.208599 − 0.642001i 1.17244 + 3.60841i −7.51387 + 5.45915i 0.180039 + 0.247802i 2.51626 − 0.817582i −3.36350 − 1.09287i 6.91250 + 5.02223i − 4.21430i
29.6 −0.215317 + 0.296358i 0.326605 − 1.00519i 1.19460 + 3.67660i 4.41565 − 3.20816i 0.227572 + 0.313226i −2.51626 + 0.817582i −2.74037 − 0.890400i 6.37742 + 4.63347i 1.99939i
29.7 0.0256207 − 0.0352638i 1.64464 − 5.06169i 1.23548 + 3.80242i 1.04076 − 0.756155i −0.136358 − 0.187680i 2.51626 − 0.817582i 0.331562 + 0.107731i −15.6347 − 11.3593i − 0.0560743i
29.8 0.909197 − 1.25140i −0.639838 + 1.96922i 0.496699 + 1.52868i 2.81297 − 2.04374i 1.88255 + 2.59111i 2.51626 − 0.817582i 8.24904 + 2.68028i 3.81272 + 2.77010i − 5.37832i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.3.k.a ✓ 48
11.d odd 10 1 inner 77.3.k.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.3.k.a ✓ 48 1.a even 1 1 trivial
77.3.k.a ✓ 48 11.d odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(77, [\chi])\).