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Properties

Label 77.4.f.a
Level $77$
Weight $4$
Character orbit 77.f
Analytic conductor $4.543$
Analytic rank $0$
Dimension $32$
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] = [77,4,Mod(15,77)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.15");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.f (of order \(5\), 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: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9} - 72 q^{10} - 94 q^{11} - 544 q^{12} + 72 q^{13} + 56 q^{14} + 140 q^{15} + 296 q^{16} + 8 q^{17} + 422 q^{18} + 51 q^{19} - 149 q^{20} - 294 q^{21} - 66 q^{22} - 830 q^{23} + 868 q^{24} - 256 q^{25} + 775 q^{26} + 27 q^{27} + 14 q^{28} + 236 q^{29} + 1008 q^{30} + 554 q^{31} - 1836 q^{32} + 895 q^{33} - 234 q^{34} + 112 q^{35} - 2322 q^{36} + 1439 q^{37} - 267 q^{38} - 18 q^{39} - 1232 q^{40} - 42 q^{41} + 210 q^{42} - 404 q^{43} + 591 q^{44} - 3020 q^{45} + 2169 q^{46} - 714 q^{47} + 4500 q^{48} - 392 q^{49} - 1035 q^{50} + 745 q^{51} + 725 q^{52} + 1351 q^{53} + 648 q^{54} + 1708 q^{55} - 966 q^{56} + 1561 q^{57} - 2529 q^{58} + 543 q^{59} - 316 q^{60} - 1542 q^{61} - 4231 q^{62} - 567 q^{63} + 1172 q^{64} - 4084 q^{65} + 5058 q^{66} - 1744 q^{67} + 2522 q^{68} - 1584 q^{69} + 126 q^{70} - 561 q^{71} - 4810 q^{72} - 144 q^{73} + 575 q^{74} + 1623 q^{75} - 3278 q^{76} + 567 q^{77} - 6582 q^{78} + 5785 q^{79} + 3199 q^{80} + 2403 q^{81} + 1998 q^{82} - 4177 q^{83} + 1652 q^{84} - 4090 q^{85} - 184 q^{86} - 940 q^{87} + 5446 q^{88} - 11554 q^{89} + 11896 q^{90} - 826 q^{91} + 12958 q^{92} - 578 q^{93} - 2042 q^{94} - 1390 q^{95} - 10074 q^{96} - q^{97} - 588 q^{98} + 10027 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −3.31524 − 2.40867i −0.224082 + 0.689654i 2.71704 + 8.36218i −1.39631 + 1.01448i 2.40403 − 1.74663i 2.16312 + 6.65740i 1.00357 − 3.08868i 21.4180 + 15.5611i 7.07264
15.2 −3.08826 − 2.24375i −2.65557 + 8.17302i 2.03079 + 6.25011i 9.24278 − 6.71527i 26.5393 − 19.2819i 2.16312 + 6.65740i −1.68477 + 5.18518i −37.9027 − 27.5379i −43.6115
15.3 −1.29413 − 0.940240i 0.446209 − 1.37329i −1.68142 − 5.17487i −12.6489 + 9.18995i −1.86867 + 1.35767i 2.16312 + 6.65740i −6.64415 + 20.4486i 20.1566 + 14.6447i 25.0100
15.4 −0.756117 − 0.549351i 2.59084 − 7.97378i −2.20221 − 6.77770i 6.79916 − 4.93988i −6.33938 + 4.60583i 2.16312 + 6.65740i −4.36870 + 13.4455i −35.0253 − 25.4474i −7.85469
15.5 0.366429 + 0.266226i −1.58740 + 4.88552i −2.40874 − 7.41335i 13.9594 − 10.1421i −1.88232 + 1.36759i 2.16312 + 6.65740i 2.21070 − 6.80383i 0.494964 + 0.359613i 7.81520
15.6 2.35901 + 1.71392i −2.48705 + 7.65436i 0.155275 + 0.477888i −6.84043 + 4.96986i −18.9860 + 13.7941i 2.16312 + 6.65740i 6.75574 − 20.7920i −30.5603 − 22.2034i −24.6546
15.7 2.46158 + 1.78845i 1.79340 − 5.51951i 0.388722 + 1.19636i 2.07002 − 1.50396i 14.2859 − 10.3793i 2.16312 + 6.65740i 6.33917 − 19.5100i −5.40527 − 3.92716i 7.78529
15.8 3.88476 + 2.82244i −0.0845386 + 0.260183i 4.65302 + 14.3205i 1.75855 − 1.27766i −1.06276 + 0.772142i 2.16312 + 6.65740i −10.4722 + 32.2302i 21.7829 + 15.8262i 10.4376
36.1 −3.31524 + 2.40867i −0.224082 − 0.689654i 2.71704 − 8.36218i −1.39631 − 1.01448i 2.40403 + 1.74663i 2.16312 − 6.65740i 1.00357 + 3.08868i 21.4180 − 15.5611i 7.07264
36.2 −3.08826 + 2.24375i −2.65557 − 8.17302i 2.03079 − 6.25011i 9.24278 + 6.71527i 26.5393 + 19.2819i 2.16312 − 6.65740i −1.68477 − 5.18518i −37.9027 + 27.5379i −43.6115
36.3 −1.29413 + 0.940240i 0.446209 + 1.37329i −1.68142 + 5.17487i −12.6489 − 9.18995i −1.86867 − 1.35767i 2.16312 − 6.65740i −6.64415 − 20.4486i 20.1566 − 14.6447i 25.0100
36.4 −0.756117 + 0.549351i 2.59084 + 7.97378i −2.20221 + 6.77770i 6.79916 + 4.93988i −6.33938 − 4.60583i 2.16312 − 6.65740i −4.36870 − 13.4455i −35.0253 + 25.4474i −7.85469
36.5 0.366429 − 0.266226i −1.58740 − 4.88552i −2.40874 + 7.41335i 13.9594 + 10.1421i −1.88232 − 1.36759i 2.16312 − 6.65740i 2.21070 + 6.80383i 0.494964 − 0.359613i 7.81520
36.6 2.35901 − 1.71392i −2.48705 − 7.65436i 0.155275 − 0.477888i −6.84043 − 4.96986i −18.9860 − 13.7941i 2.16312 − 6.65740i 6.75574 + 20.7920i −30.5603 + 22.2034i −24.6546
36.7 2.46158 − 1.78845i 1.79340 + 5.51951i 0.388722 − 1.19636i 2.07002 + 1.50396i 14.2859 + 10.3793i 2.16312 − 6.65740i 6.33917 + 19.5100i −5.40527 + 3.92716i 7.78529
36.8 3.88476 − 2.82244i −0.0845386 − 0.260183i 4.65302 − 14.3205i 1.75855 + 1.27766i −1.06276 − 0.772142i 2.16312 − 6.65740i −10.4722 − 32.2302i 21.7829 − 15.8262i 10.4376
64.1 −1.68700 + 5.19206i 7.39468 − 5.37255i −17.6394 − 12.8158i −3.09523 − 9.52615i 15.4198 + 47.4572i −5.66312 − 4.11450i 60.9649 − 44.2936i 17.4736 − 53.7781i 54.6820
64.2 −1.30960 + 4.03054i 1.40035 − 1.01742i −8.05807 − 5.85453i 4.20134 + 12.9304i 2.26683 + 6.97660i −5.66312 − 4.11450i 6.72114 − 4.88320i −7.41760 + 22.8290i −57.6186
64.3 −1.09390 + 3.36668i −5.91742 + 4.29926i −3.66576 − 2.66333i −3.30618 − 10.1754i −8.00114 − 24.6250i −5.66312 − 4.11450i −9.93439 + 7.21775i 8.18877 − 25.2024i 37.8739
64.4 −0.241312 + 0.742683i −3.99646 + 2.90360i 5.97879 + 4.34385i 0.956848 + 2.94488i −1.19206 − 3.66877i −5.66312 − 4.11450i −9.72296 + 7.06415i −0.802655 + 2.47032i −2.41801
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.f.a ✓ 32
11.c even 5 1 inner 77.4.f.a ✓ 32
11.c even 5 1 847.4.a.o 16
11.d odd 10 1 847.4.a.p 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.f.a ✓ 32 1.a even 1 1 trivial
77.4.f.a ✓ 32 11.c even 5 1 inner
847.4.a.o 16 11.c even 5 1
847.4.a.p 16 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 2 T_{2}^{31} + 58 T_{2}^{30} + 50 T_{2}^{29} + 1724 T_{2}^{28} + 1896 T_{2}^{27} + \cdots + 113714630656 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display