Properties

Label 177.2.a.c
Level $177$
Weight $2$
Character orbit 177.a
Self dual yes
Analytic conductor $1.413$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,2,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

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: yes
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + q^{5} + \beta q^{6} + ( - \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + q^{5} + \beta q^{6} + ( - \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + \beta q^{10} + ( - 2 \beta + 3) q^{11} + (\beta - 1) q^{12} - q^{13} - q^{14} + q^{15} - 3 \beta q^{16} + ( - 3 \beta + 2) q^{17} + \beta q^{18} + (3 \beta - 4) q^{19} + (\beta - 1) q^{20} + ( - \beta + 1) q^{21} + (\beta - 2) q^{22} + 3 \beta q^{23} + ( - 2 \beta + 1) q^{24} - 4 q^{25} - \beta q^{26} + q^{27} + (\beta - 2) q^{28} + ( - \beta + 3) q^{29} + \beta q^{30} + ( - 3 \beta + 1) q^{31} + (\beta - 5) q^{32} + ( - 2 \beta + 3) q^{33} + ( - \beta - 3) q^{34} + ( - \beta + 1) q^{35} + (\beta - 1) q^{36} + ( - \beta - 4) q^{37} + ( - \beta + 3) q^{38} - q^{39} + ( - 2 \beta + 1) q^{40} + (5 \beta - 3) q^{41} - q^{42} + (8 \beta - 5) q^{43} + (3 \beta - 5) q^{44} + q^{45} + (3 \beta + 3) q^{46} + (\beta + 5) q^{47} - 3 \beta q^{48} + ( - \beta - 5) q^{49} - 4 \beta q^{50} + ( - 3 \beta + 2) q^{51} + ( - \beta + 1) q^{52} + (8 \beta - 5) q^{53} + \beta q^{54} + ( - 2 \beta + 3) q^{55} + ( - \beta + 3) q^{56} + (3 \beta - 4) q^{57} + (2 \beta - 1) q^{58} - q^{59} + (\beta - 1) q^{60} + ( - 3 \beta + 1) q^{61} + ( - 2 \beta - 3) q^{62} + ( - \beta + 1) q^{63} + (2 \beta + 1) q^{64} - q^{65} + (\beta - 2) q^{66} + ( - 8 \beta + 7) q^{67} + (2 \beta - 5) q^{68} + 3 \beta q^{69} - q^{70} + (6 \beta - 1) q^{71} + ( - 2 \beta + 1) q^{72} + (\beta + 1) q^{73} + ( - 5 \beta - 1) q^{74} - 4 q^{75} + ( - 4 \beta + 7) q^{76} + ( - 3 \beta + 5) q^{77} - \beta q^{78} + ( - 8 \beta + 9) q^{79} - 3 \beta q^{80} + q^{81} + (2 \beta + 5) q^{82} + (3 \beta + 5) q^{83} + (\beta - 2) q^{84} + ( - 3 \beta + 2) q^{85} + (3 \beta + 8) q^{86} + ( - \beta + 3) q^{87} + ( - 4 \beta + 7) q^{88} + (5 \beta - 5) q^{89} + \beta q^{90} + (\beta - 1) q^{91} + 3 q^{92} + ( - 3 \beta + 1) q^{93} + (6 \beta + 1) q^{94} + (3 \beta - 4) q^{95} + (\beta - 5) q^{96} + (6 \beta - 5) q^{97} + ( - 6 \beta - 1) q^{98} + ( - 2 \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + q^{18} - 5 q^{19} - q^{20} + q^{21} - 3 q^{22} + 3 q^{23} - 8 q^{25} - q^{26} + 2 q^{27} - 3 q^{28} + 5 q^{29} + q^{30} - q^{31} - 9 q^{32} + 4 q^{33} - 7 q^{34} + q^{35} - q^{36} - 9 q^{37} + 5 q^{38} - 2 q^{39} - q^{41} - 2 q^{42} - 2 q^{43} - 7 q^{44} + 2 q^{45} + 9 q^{46} + 11 q^{47} - 3 q^{48} - 11 q^{49} - 4 q^{50} + q^{51} + q^{52} - 2 q^{53} + q^{54} + 4 q^{55} + 5 q^{56} - 5 q^{57} - 2 q^{59} - q^{60} - q^{61} - 8 q^{62} + q^{63} + 4 q^{64} - 2 q^{65} - 3 q^{66} + 6 q^{67} - 8 q^{68} + 3 q^{69} - 2 q^{70} + 4 q^{71} + 3 q^{73} - 7 q^{74} - 8 q^{75} + 10 q^{76} + 7 q^{77} - q^{78} + 10 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} + 13 q^{83} - 3 q^{84} + q^{85} + 19 q^{86} + 5 q^{87} + 10 q^{88} - 5 q^{89} + q^{90} - q^{91} + 6 q^{92} - q^{93} + 8 q^{94} - 5 q^{95} - 9 q^{96} - 4 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
−0.618034 1.00000 −1.61803 1.00000 −0.618034 1.61803 2.23607 1.00000 −0.618034
1.2 1.61803 1.00000 0.618034 1.00000 1.61803 −0.618034 −2.23607 1.00000 1.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.a.c 2
3.b odd 2 1 531.2.a.a 2
4.b odd 2 1 2832.2.a.m 2
5.b even 2 1 4425.2.a.o 2
7.b odd 2 1 8673.2.a.n 2
12.b even 2 1 8496.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.c 2 1.a even 1 1 trivial
531.2.a.a 2 3.b odd 2 1
2832.2.a.m 2 4.b odd 2 1
4425.2.a.o 2 5.b even 2 1
8496.2.a.ba 2 12.b even 2 1
8673.2.a.n 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 79 \) Copy content Toggle raw display
$47$ \( T^{2} - 11T + 29 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 79 \) Copy content Toggle raw display
$59$ \( (T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 71 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 55 \) Copy content Toggle raw display
$83$ \( T^{2} - 13T + 31 \) Copy content Toggle raw display
$89$ \( T^{2} + 5T - 25 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
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