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Properties

Label 77.2.m.b
Level $77$
Weight $2$
Character orbit 77.m
Analytic conductor $0.615$
Analytic rank $0$
Dimension $40$
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(4,77)]
 
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([20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.m (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: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 4 q^{3} - 3 q^{4} + 4 q^{5} - 16 q^{6} - 2 q^{7} - 38 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 4 q^{3} - 3 q^{4} + 4 q^{5} - 16 q^{6} - 2 q^{7} - 38 q^{8} + 7 q^{9} + 14 q^{10} - 9 q^{11} - 18 q^{12} + 6 q^{13} - 3 q^{14} - 14 q^{15} - 5 q^{16} - 7 q^{17} + 24 q^{18} - 4 q^{19} - 30 q^{20} - 2 q^{21} + 44 q^{22} - 14 q^{23} - 12 q^{24} + 21 q^{25} - 16 q^{27} + 16 q^{28} + 16 q^{30} - 17 q^{31} - 30 q^{32} - 15 q^{33} + 48 q^{34} - 14 q^{35} + 14 q^{36} + 24 q^{37} + 12 q^{38} + 28 q^{39} + 10 q^{40} + 60 q^{41} - 70 q^{42} - 72 q^{43} + 18 q^{44} - 16 q^{45} + 8 q^{46} + 13 q^{47} + 128 q^{48} - 10 q^{49} + 6 q^{50} - 7 q^{51} + 2 q^{52} + 33 q^{53} + 34 q^{54} - 6 q^{55} + 24 q^{56} + 44 q^{57} - 17 q^{58} + 21 q^{59} - 48 q^{60} - 52 q^{62} + 24 q^{63} + 94 q^{64} - 40 q^{65} - 49 q^{66} - 38 q^{67} - 23 q^{68} - 124 q^{69} - 3 q^{70} + 20 q^{71} - 38 q^{72} + 11 q^{73} - 41 q^{74} - 11 q^{75} - 96 q^{76} + 36 q^{77} - 100 q^{78} + 21 q^{79} + 12 q^{80} - 58 q^{81} + 6 q^{82} - 46 q^{83} - 29 q^{84} - 78 q^{85} + 7 q^{86} + 48 q^{87} + 32 q^{88} - 10 q^{89} - 18 q^{90} - 14 q^{91} - 110 q^{92} + 12 q^{93} + 37 q^{94} + 7 q^{95} - 53 q^{96} - 54 q^{97} + 116 q^{98} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.233841 + 2.22485i 0.893246 + 0.992050i −2.93896 − 0.624696i −0.791435 − 0.352369i −2.41604 + 1.75535i 2.03186 − 1.69457i 0.694498 − 2.13745i 0.127310 − 1.21128i 0.969038 − 1.67842i
4.2 −0.133281 + 1.26809i −0.0932166 − 0.103528i 0.366016 + 0.0777992i 1.08262 + 0.482012i 0.143706 − 0.104408i −2.63367 + 0.252511i −0.935477 + 2.87910i 0.311557 − 2.96426i −0.755525 + 1.30861i
4.3 0.0457018 − 0.434823i −1.72872 − 1.91994i 1.76931 + 0.376079i −1.11068 − 0.494505i −0.913838 + 0.663942i −0.0819410 − 2.64448i 0.514604 − 1.58379i −0.384102 + 3.65449i −0.265782 + 0.460348i
4.4 0.234707 − 2.23309i −1.27743 − 1.41873i −2.97532 − 0.632424i 2.25193 + 1.00262i −3.46797 + 2.51963i −1.07525 + 2.41740i −0.722861 + 2.22474i −0.0673794 + 0.641072i 2.76749 − 4.79344i
4.5 0.255843 − 2.43419i 1.95053 + 2.16628i −3.90352 − 0.829718i −0.303226 − 0.135005i 5.77217 − 4.19373i −1.95978 − 1.77744i −1.50568 + 4.63401i −0.574630 + 5.46724i −0.406206 + 0.703570i
9.1 −1.23711 − 1.37395i 1.07981 + 0.480764i −0.148239 + 1.41040i 2.31107 + 0.491232i −0.675302 − 2.07837i −0.122985 − 2.64289i −0.870261 + 0.632281i −1.07253 − 1.19116i −2.18411 − 3.78299i
9.2 −0.450970 − 0.500853i 1.96516 + 0.874946i 0.161577 − 1.53730i −1.65218 − 0.351182i −0.448009 − 1.37883i 1.55484 + 2.14067i −1.93333 + 1.40464i 1.08893 + 1.20938i 0.569194 + 0.985873i
9.3 −0.0508685 − 0.0564952i −2.15146 − 0.957893i 0.208453 − 1.98330i −2.52173 − 0.536011i 0.0553254 + 0.170274i 2.37978 − 1.15613i −0.245656 + 0.178480i 1.70384 + 1.89231i 0.0979948 + 0.169732i
9.4 0.391628 + 0.434946i −1.26312 − 0.562375i 0.173251 − 1.64837i 3.87755 + 0.824199i −0.250068 − 0.769629i −1.49228 + 2.18474i 1.73180 − 1.25823i −0.728198 − 0.808746i 1.16007 + 2.00931i
9.5 1.76087 + 1.95564i −2.02209 − 0.900292i −0.514820 + 4.89819i 1.15065 + 0.244578i −1.79998 − 5.53977i 0.510555 − 2.59602i −6.22764 + 4.52465i 1.27093 + 1.41151i 1.54783 + 2.68093i
16.1 −2.57407 − 0.547134i 0.231369 − 2.20133i 4.49937 + 2.00325i −0.787136 − 0.874203i −1.79998 + 5.53977i −2.31119 − 1.28778i −6.22764 − 4.52465i −1.85787 − 0.394902i 1.54783 + 2.68093i
16.2 −0.572488 − 0.121686i 0.144526 − 1.37508i −1.51416 − 0.674145i −2.65255 − 2.94596i −0.250068 + 0.769629i 1.61668 + 2.09436i 1.73180 + 1.25823i 1.06449 + 0.226265i 1.16007 + 2.00931i
16.3 0.0743606 + 0.0158058i 0.246172 − 2.34217i −1.82181 − 0.811123i 1.72507 + 1.91588i 0.0553254 − 0.170274i −0.364157 − 2.62057i −0.245656 − 0.178480i −2.49071 − 0.529416i 0.0979948 + 0.169732i
16.4 0.659236 + 0.140125i −0.224855 + 2.13935i −1.41213 − 0.628722i 1.13022 + 1.25524i −0.448009 + 1.37883i 2.51637 − 0.817238i −1.93333 − 1.40464i −1.59182 − 0.338352i 0.569194 + 0.985873i
16.5 1.80843 + 0.384393i −0.123553 + 1.17553i 1.29556 + 0.576822i −1.58095 − 1.75583i −0.675302 + 2.07837i −2.55154 − 0.699733i −0.870261 − 0.632281i 1.56784 + 0.333255i −2.18411 − 3.78299i
25.1 −2.23599 − 0.995527i −2.85132 − 0.606067i 2.67031 + 2.96568i 0.0346954 − 0.330104i 5.77217 + 4.19373i 0.540741 + 2.58990i −1.50568 − 4.63401i 5.02208 + 2.23598i −0.406206 + 0.703570i
25.2 −2.05127 − 0.913284i 1.86737 + 0.396921i 2.03535 + 2.26049i −0.257667 + 2.45154i −3.46797 − 2.51963i 2.29081 − 1.32370i −0.722861 − 2.22474i 0.588874 + 0.262184i 2.76749 − 4.79344i
25.3 −0.399419 − 0.177833i 2.52707 + 0.537146i −1.21035 − 1.34423i 0.127084 − 1.20913i −0.913838 − 0.663942i −1.48810 + 2.18759i 0.514604 + 1.58379i 3.35693 + 1.49460i −0.265782 + 0.460348i
25.4 1.16484 + 0.518618i 0.136266 + 0.0289642i −0.250384 − 0.278080i −0.123874 + 1.17858i 0.143706 + 0.104408i 2.27911 + 1.34375i −0.935477 − 2.87910i −2.72291 − 1.21232i −0.755525 + 1.30861i
25.5 2.04369 + 0.909911i −1.30576 − 0.277549i 2.01049 + 2.23287i 0.0905565 − 0.861587i −2.41604 − 1.75535i −2.63985 + 0.176636i 0.694498 + 2.13745i −1.11265 − 0.495385i 0.969038 − 1.67842i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.m.b ✓ 40
3.b odd 2 1 693.2.by.b 40
7.b odd 2 1 539.2.q.h 40
7.c even 3 1 inner 77.2.m.b ✓ 40
7.c even 3 1 539.2.f.h 20
7.d odd 6 1 539.2.f.g 20
7.d odd 6 1 539.2.q.h 40
11.b odd 2 1 847.2.n.j 40
11.c even 5 1 inner 77.2.m.b ✓ 40
11.c even 5 1 847.2.e.i 20
11.c even 5 2 847.2.n.i 40
11.d odd 10 1 847.2.e.h 20
11.d odd 10 2 847.2.n.h 40
11.d odd 10 1 847.2.n.j 40
21.h odd 6 1 693.2.by.b 40
33.h odd 10 1 693.2.by.b 40
77.h odd 6 1 847.2.n.j 40
77.j odd 10 1 539.2.q.h 40
77.m even 15 1 inner 77.2.m.b ✓ 40
77.m even 15 1 539.2.f.h 20
77.m even 15 1 847.2.e.i 20
77.m even 15 2 847.2.n.i 40
77.m even 15 1 5929.2.a.bw 10
77.n even 30 1 5929.2.a.bz 10
77.o odd 30 1 847.2.e.h 20
77.o odd 30 2 847.2.n.h 40
77.o odd 30 1 847.2.n.j 40
77.o odd 30 1 5929.2.a.by 10
77.p odd 30 1 539.2.f.g 20
77.p odd 30 1 539.2.q.h 40
77.p odd 30 1 5929.2.a.bx 10
231.z odd 30 1 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 1.a even 1 1 trivial
77.2.m.b ✓ 40 7.c even 3 1 inner
77.2.m.b ✓ 40 11.c even 5 1 inner
77.2.m.b ✓ 40 77.m even 15 1 inner
539.2.f.g 20 7.d odd 6 1
539.2.f.g 20 77.p odd 30 1
539.2.f.h 20 7.c even 3 1
539.2.f.h 20 77.m even 15 1
539.2.q.h 40 7.b odd 2 1
539.2.q.h 40 7.d odd 6 1
539.2.q.h 40 77.j odd 10 1
539.2.q.h 40 77.p odd 30 1
693.2.by.b 40 3.b odd 2 1
693.2.by.b 40 21.h odd 6 1
693.2.by.b 40 33.h odd 10 1
693.2.by.b 40 231.z odd 30 1
847.2.e.h 20 11.d odd 10 1
847.2.e.h 20 77.o odd 30 1
847.2.e.i 20 11.c even 5 1
847.2.e.i 20 77.m even 15 1
847.2.n.h 40 11.d odd 10 2
847.2.n.h 40 77.o odd 30 2
847.2.n.i 40 11.c even 5 2
847.2.n.i 40 77.m even 15 2
847.2.n.j 40 11.b odd 2 1
847.2.n.j 40 11.d odd 10 1
847.2.n.j 40 77.h odd 6 1
847.2.n.j 40 77.o odd 30 1
5929.2.a.bw 10 77.m even 15 1
5929.2.a.bx 10 77.p odd 30 1
5929.2.a.by 10 77.o odd 30 1
5929.2.a.bz 10 77.n even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 3 T_{2}^{39} + T_{2}^{38} + 12 T_{2}^{37} + 21 T_{2}^{36} - 50 T_{2}^{35} + 238 T_{2}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display