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Properties

Label 77.2.n.a
Level $77$
Weight $2$
Character orbit 77.n
Analytic conductor $0.615$
Analytic rank $0$
Dimension $48$
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] = [77,2,Mod(17,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.n (of order \(30\), 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: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 9 q^{3} - 9 q^{4} - 15 q^{5} - 5 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 9 q^{3} - 9 q^{4} - 15 q^{5} - 5 q^{7} - 11 q^{9} - q^{11} - 12 q^{12} - 8 q^{14} - 27 q^{16} + 15 q^{17} + 20 q^{18} - 15 q^{19} - 76 q^{22} + 10 q^{23} + 75 q^{24} + q^{25} + 27 q^{26} - 40 q^{28} - 40 q^{29} + 25 q^{30} + 9 q^{31} + 42 q^{33} + 5 q^{35} - 38 q^{36} - q^{37} + 33 q^{38} - 45 q^{39} + 75 q^{40} + 64 q^{42} + 30 q^{44} - 84 q^{45} - 20 q^{46} + 3 q^{47} + 59 q^{49} + 30 q^{50} + 55 q^{51} - 15 q^{52} - 3 q^{53} - 8 q^{56} + 60 q^{57} + 46 q^{58} - 3 q^{59} - 15 q^{60} - 30 q^{61} - 40 q^{63} + 12 q^{64} - 93 q^{66} + 44 q^{67} - 75 q^{68} - 27 q^{70} + 20 q^{71} - 60 q^{72} - 60 q^{73} + 45 q^{74} - 57 q^{75} + 92 q^{78} - 70 q^{79} - 75 q^{80} - 29 q^{81} - 129 q^{82} - 125 q^{84} + 10 q^{85} - 62 q^{86} + 19 q^{88} + 6 q^{89} - 12 q^{91} + 30 q^{92} - 92 q^{93} + 105 q^{94} + 30 q^{95} + 75 q^{96} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.95877 − 1.76368i 0.650787 + 1.46169i 0.517138 + 4.92024i 0.446428 + 2.10028i 1.30322 − 4.01090i −1.30623 + 2.30082i 4.56624 − 6.28489i 0.294372 − 0.326934i 2.82977 − 4.90131i
17.2 −1.33360 − 1.20078i −1.17499 − 2.63908i 0.127565 + 1.21370i 0.418270 + 1.96780i −1.60198 + 4.93040i −1.17364 − 2.37120i −0.822345 + 1.13186i −3.57672 + 3.97235i 1.80510 − 3.12652i
17.3 −1.11268 − 1.00186i 0.245339 + 0.551041i 0.0252713 + 0.240440i −0.491349 − 2.31161i 0.279082 − 0.858925i 2.00635 − 1.72470i −1.54736 + 2.12976i 1.76394 − 1.95905i −1.76920 + 3.06434i
17.4 −0.0202070 − 0.0181945i −0.500742 − 1.12469i −0.208980 − 1.98831i −0.240558 − 1.13174i −0.0103446 + 0.0318373i −0.296416 + 2.62909i −0.0639186 + 0.0879764i 0.993218 − 1.10308i −0.0157304 + 0.0272459i
17.5 0.386517 + 0.348022i 0.460952 + 1.03532i −0.180780 − 1.72001i 0.678628 + 3.19269i −0.182146 + 0.560588i −1.97073 − 1.76528i 1.14015 − 1.56929i 1.14799 − 1.27497i −0.848825 + 1.47021i
17.6 1.75157 + 1.57712i −0.821231 − 1.84451i 0.371633 + 3.53585i −0.00617973 − 0.0290734i 1.47058 − 4.52598i −2.21185 + 1.45180i −2.15474 + 2.96574i −0.720422 + 0.800109i 0.0350280 − 0.0606703i
19.1 −2.06490 + 0.217030i −0.138863 − 0.125032i 2.26043 − 0.480470i −1.04356 − 2.34387i 0.313874 + 0.228043i 2.62996 − 0.288618i −0.613987 + 0.199497i −0.309936 − 2.94884i 2.66354 + 4.61339i
19.2 −1.30277 + 0.136927i 2.02980 + 1.82764i −0.277833 + 0.0590552i 0.166444 + 0.373838i −2.89461 − 2.10306i −2.50634 + 0.847492i 2.84553 − 0.924570i 0.466233 + 4.43591i −0.268026 − 0.464235i
19.3 −0.476751 + 0.0501086i −1.66384 − 1.49813i −1.73151 + 0.368045i −0.547175 − 1.22898i 0.868306 + 0.630861i −1.63770 − 2.07796i 1.71889 − 0.558501i 0.210388 + 2.00171i 0.322449 + 0.558497i
19.4 1.11135 − 0.116808i 1.15989 + 1.04437i −0.734836 + 0.156194i −1.25246 − 2.81307i 1.41104 + 1.02518i 1.16071 + 2.37755i −2.92398 + 0.950058i −0.0589482 − 0.560855i −1.72051 − 2.98001i
19.5 1.39661 − 0.146790i 0.151995 + 0.136856i −0.0273118 + 0.00580530i 0.618266 + 1.38865i 0.232367 + 0.168824i −1.57500 − 2.12588i −2.70844 + 0.880026i −0.309213 − 2.94196i 1.06732 + 1.84865i
19.6 2.05902 − 0.216412i −2.33449 − 2.10199i 2.23645 − 0.475371i 0.483200 + 1.08528i −5.26167 − 3.82283i 2.02173 + 1.70664i 0.563952 − 0.183239i 0.717924 + 6.83059i 1.22979 + 2.13006i
24.1 −0.548010 − 2.57818i 1.59126 − 0.167248i −4.51962 + 2.01227i −1.59568 − 1.43676i −1.30322 − 4.01090i 2.59186 + 0.531305i 4.56624 + 6.28489i −0.430319 + 0.0914672i −2.82977 + 4.90131i
24.2 −0.373106 − 1.75533i −2.87300 + 0.301965i −1.11488 + 0.496374i −1.49503 − 1.34613i 1.60198 + 4.93040i −1.89247 + 1.84893i −0.822345 − 1.13186i 5.22852 − 1.11136i −1.80510 + 3.12652i
24.3 −0.311296 − 1.46453i 0.599886 − 0.0630505i −0.220863 + 0.0983344i 1.75624 + 1.58133i −0.279082 − 0.858925i −2.26028 − 1.37519i −1.54736 − 2.12976i −2.57856 + 0.548089i 1.76920 − 3.06434i
24.4 −0.00565338 − 0.0265970i −1.22438 + 0.128687i 1.82642 − 0.813173i 0.859835 + 0.774198i 0.0103446 + 0.0318373i 2.59202 − 0.530526i −0.0639186 − 0.0879764i −1.45190 + 0.308612i 0.0157304 − 0.0272459i
24.5 0.108137 + 0.508745i 1.12709 − 0.118461i 1.57996 − 0.703445i −2.42564 − 2.18406i 0.182146 + 0.560588i −1.06990 + 2.41978i 1.14015 + 1.56929i −1.67816 + 0.356703i 0.848825 − 1.47021i
24.6 0.490042 + 2.30547i −2.00801 + 0.211051i −3.24795 + 1.44608i 0.0220884 + 0.0198885i −1.47058 − 4.52598i 2.06424 + 1.65497i −2.15474 − 2.96574i 1.05313 − 0.223849i −0.0350280 + 0.0606703i
40.1 −0.842093 − 1.89137i 0.653128 + 3.07273i −1.52991 + 1.69913i 1.18148 − 0.124179i 5.26167 − 3.82283i 2.63875 + 0.192354i 0.563952 + 0.183239i −6.27443 + 2.79356i −1.22979 − 2.13006i
40.2 −0.571183 − 1.28290i −0.0425239 − 0.200059i 0.0186834 − 0.0207501i 1.51174 − 0.158890i −0.232367 + 0.168824i −2.52376 − 0.794113i −2.70844 − 0.880026i 2.70242 − 1.20320i −1.06732 − 1.84865i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.n.a ✓ 48
3.b odd 2 1 693.2.cg.a 48
7.b odd 2 1 539.2.s.d 48
7.c even 3 1 539.2.m.a 48
7.c even 3 1 539.2.s.d 48
7.d odd 6 1 inner 77.2.n.a ✓ 48
7.d odd 6 1 539.2.m.a 48
11.b odd 2 1 847.2.r.c 48
11.c even 5 1 847.2.i.b 48
11.c even 5 1 847.2.r.a 48
11.c even 5 1 847.2.r.c 48
11.c even 5 1 847.2.r.d 48
11.d odd 10 1 inner 77.2.n.a ✓ 48
11.d odd 10 1 847.2.i.b 48
11.d odd 10 1 847.2.r.a 48
11.d odd 10 1 847.2.r.d 48
21.g even 6 1 693.2.cg.a 48
33.f even 10 1 693.2.cg.a 48
77.i even 6 1 847.2.r.c 48
77.l even 10 1 539.2.s.d 48
77.n even 30 1 inner 77.2.n.a ✓ 48
77.n even 30 1 539.2.m.a 48
77.n even 30 1 847.2.i.b 48
77.n even 30 1 847.2.r.a 48
77.n even 30 1 847.2.r.d 48
77.o odd 30 1 539.2.m.a 48
77.o odd 30 1 539.2.s.d 48
77.p odd 30 1 847.2.i.b 48
77.p odd 30 1 847.2.r.a 48
77.p odd 30 1 847.2.r.c 48
77.p odd 30 1 847.2.r.d 48
231.bf odd 30 1 693.2.cg.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.n.a ✓ 48 1.a even 1 1 trivial
77.2.n.a ✓ 48 7.d odd 6 1 inner
77.2.n.a ✓ 48 11.d odd 10 1 inner
77.2.n.a ✓ 48 77.n even 30 1 inner
539.2.m.a 48 7.c even 3 1
539.2.m.a 48 7.d odd 6 1
539.2.m.a 48 77.n even 30 1
539.2.m.a 48 77.o odd 30 1
539.2.s.d 48 7.b odd 2 1
539.2.s.d 48 7.c even 3 1
539.2.s.d 48 77.l even 10 1
539.2.s.d 48 77.o odd 30 1
693.2.cg.a 48 3.b odd 2 1
693.2.cg.a 48 21.g even 6 1
693.2.cg.a 48 33.f even 10 1
693.2.cg.a 48 231.bf odd 30 1
847.2.i.b 48 11.c even 5 1
847.2.i.b 48 11.d odd 10 1
847.2.i.b 48 77.n even 30 1
847.2.i.b 48 77.p odd 30 1
847.2.r.a 48 11.c even 5 1
847.2.r.a 48 11.d odd 10 1
847.2.r.a 48 77.n even 30 1
847.2.r.a 48 77.p odd 30 1
847.2.r.c 48 11.b odd 2 1
847.2.r.c 48 11.c even 5 1
847.2.r.c 48 77.i even 6 1
847.2.r.c 48 77.p odd 30 1
847.2.r.d 48 11.c even 5 1
847.2.r.d 48 11.d odd 10 1
847.2.r.d 48 77.n even 30 1
847.2.r.d 48 77.p odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(77, [\chi])\).