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Properties

Label 77.3.h.b
Level $77$
Weight $3$
Character orbit 77.h
Analytic conductor $2.098$
Analytic rank $0$
Dimension $24$
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] = [77,3,Mod(32,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.32");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 77.h (of order \(6\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} + 10 q^{4} - 16 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} + 10 q^{4} - 16 q^{5} - 4 q^{9} - 10 q^{11} + 12 q^{12} + 34 q^{14} + 60 q^{15} - 18 q^{16} - 104 q^{20} - 108 q^{22} - 20 q^{23} + 16 q^{25} + 64 q^{26} - 172 q^{27} - 10 q^{31} - 72 q^{33} - 136 q^{34} + 316 q^{36} + 68 q^{37} - 88 q^{38} - 146 q^{42} - 2 q^{44} + 62 q^{45} - 70 q^{47} + 684 q^{48} - 202 q^{49} + 258 q^{53} - 32 q^{55} + 196 q^{56} - 20 q^{58} + 14 q^{59} + 84 q^{60} - 200 q^{64} - 54 q^{66} + 14 q^{67} + 164 q^{69} - 302 q^{70} - 188 q^{71} + 192 q^{75} + 302 q^{77} + 268 q^{78} - 62 q^{80} - 284 q^{81} - 382 q^{82} + 18 q^{86} - 392 q^{88} + 338 q^{89} - 256 q^{91} - 860 q^{92} + 580 q^{93} - 408 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −3.13137 + 1.80790i −0.818029 + 1.41687i 4.53698 − 7.85828i −1.87825 − 3.25322i − 5.91565i −0.261106 − 6.99513i 18.3464i 3.16166 + 5.47615i 11.7630 + 6.79135i
32.2 −2.49331 + 1.43952i 0.611732 − 1.05955i 2.14441 − 3.71423i 0.569648 + 0.986660i 3.52239i 5.17011 + 4.71910i 0.831531i 3.75157 + 6.49791i −2.84062 − 1.64004i
32.3 −1.63178 + 0.942110i 2.71949 − 4.71029i −0.224857 + 0.389464i 1.42714 + 2.47188i 10.2482i 2.73375 − 6.44412i − 8.38424i −10.2912 − 17.8249i −4.65756 − 2.68905i
32.4 −1.41448 + 0.816650i 1.29392 − 2.24113i −0.666164 + 1.15383i −4.59222 − 7.95395i 4.22672i −6.20852 + 3.23331i − 8.70930i 1.15155 + 1.99455i 12.9912 + 7.50047i
32.5 −0.977807 + 0.564537i −1.96515 + 3.40374i −1.36260 + 2.36008i −1.81382 − 3.14162i − 4.43760i −0.995213 − 6.92889i − 7.59324i −3.22361 − 5.58346i 3.54712 + 2.04793i
32.6 −0.329148 + 0.190034i 0.158041 − 0.273735i −1.92777 + 3.33900i 2.28749 + 3.96205i 0.120132i −6.92403 + 1.02847i − 2.98564i 4.45005 + 7.70771i −1.50584 − 0.869399i
32.7 0.329148 − 0.190034i 0.158041 − 0.273735i −1.92777 + 3.33900i 2.28749 + 3.96205i − 0.120132i 6.92403 − 1.02847i 2.98564i 4.45005 + 7.70771i 1.50584 + 0.869399i
32.8 0.977807 − 0.564537i −1.96515 + 3.40374i −1.36260 + 2.36008i −1.81382 − 3.14162i 4.43760i 0.995213 + 6.92889i 7.59324i −3.22361 − 5.58346i −3.54712 − 2.04793i
32.9 1.41448 − 0.816650i 1.29392 − 2.24113i −0.666164 + 1.15383i −4.59222 − 7.95395i − 4.22672i 6.20852 − 3.23331i 8.70930i 1.15155 + 1.99455i −12.9912 − 7.50047i
32.10 1.63178 − 0.942110i 2.71949 − 4.71029i −0.224857 + 0.389464i 1.42714 + 2.47188i − 10.2482i −2.73375 + 6.44412i 8.38424i −10.2912 − 17.8249i 4.65756 + 2.68905i
32.11 2.49331 − 1.43952i 0.611732 − 1.05955i 2.14441 − 3.71423i 0.569648 + 0.986660i − 3.52239i −5.17011 − 4.71910i − 0.831531i 3.75157 + 6.49791i 2.84062 + 1.64004i
32.12 3.13137 − 1.80790i −0.818029 + 1.41687i 4.53698 − 7.85828i −1.87825 − 3.25322i 5.91565i 0.261106 + 6.99513i − 18.3464i 3.16166 + 5.47615i −11.7630 − 6.79135i
65.1 −3.13137 − 1.80790i −0.818029 − 1.41687i 4.53698 + 7.85828i −1.87825 + 3.25322i 5.91565i −0.261106 + 6.99513i − 18.3464i 3.16166 − 5.47615i 11.7630 − 6.79135i
65.2 −2.49331 − 1.43952i 0.611732 + 1.05955i 2.14441 + 3.71423i 0.569648 − 0.986660i − 3.52239i 5.17011 − 4.71910i − 0.831531i 3.75157 − 6.49791i −2.84062 + 1.64004i
65.3 −1.63178 − 0.942110i 2.71949 + 4.71029i −0.224857 − 0.389464i 1.42714 − 2.47188i − 10.2482i 2.73375 + 6.44412i 8.38424i −10.2912 + 17.8249i −4.65756 + 2.68905i
65.4 −1.41448 − 0.816650i 1.29392 + 2.24113i −0.666164 − 1.15383i −4.59222 + 7.95395i − 4.22672i −6.20852 − 3.23331i 8.70930i 1.15155 − 1.99455i 12.9912 − 7.50047i
65.5 −0.977807 − 0.564537i −1.96515 − 3.40374i −1.36260 − 2.36008i −1.81382 + 3.14162i 4.43760i −0.995213 + 6.92889i 7.59324i −3.22361 + 5.58346i 3.54712 − 2.04793i
65.6 −0.329148 − 0.190034i 0.158041 + 0.273735i −1.92777 − 3.33900i 2.28749 − 3.96205i − 0.120132i −6.92403 − 1.02847i 2.98564i 4.45005 − 7.70771i −1.50584 + 0.869399i
65.7 0.329148 + 0.190034i 0.158041 + 0.273735i −1.92777 − 3.33900i 2.28749 − 3.96205i 0.120132i 6.92403 + 1.02847i − 2.98564i 4.45005 − 7.70771i 1.50584 − 0.869399i
65.8 0.977807 + 0.564537i −1.96515 − 3.40374i −1.36260 − 2.36008i −1.81382 + 3.14162i − 4.43760i 0.995213 − 6.92889i − 7.59324i −3.22361 + 5.58346i −3.54712 + 2.04793i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.b odd 2 1 inner
77.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.3.h.b ✓ 24
7.c even 3 1 inner 77.3.h.b ✓ 24
7.c even 3 1 539.3.c.j 12
7.d odd 6 1 539.3.c.k 12
11.b odd 2 1 inner 77.3.h.b ✓ 24
77.h odd 6 1 inner 77.3.h.b ✓ 24
77.h odd 6 1 539.3.c.j 12
77.i even 6 1 539.3.c.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.3.h.b ✓ 24 1.a even 1 1 trivial
77.3.h.b ✓ 24 7.c even 3 1 inner
77.3.h.b ✓ 24 11.b odd 2 1 inner
77.3.h.b ✓ 24 77.h odd 6 1 inner
539.3.c.j 12 7.c even 3 1
539.3.c.j 12 77.h odd 6 1
539.3.c.k 12 7.d odd 6 1
539.3.c.k 12 77.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 29 T_{2}^{22} + 551 T_{2}^{20} - 5936 T_{2}^{18} + 45911 T_{2}^{16} - 226020 T_{2}^{14} + \cdots + 35721 \) acting on \(S_{3}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display