Properties

Label 177.2.a.a
Level $177$
Weight $2$
Character orbit 177.a
Self dual yes
Analytic conductor $1.413$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} - 3 q^{5} + ( - \beta - 1) q^{6} + (\beta - 4) q^{7} + ( - 4 \beta - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} - 3 q^{5} + ( - \beta - 1) q^{6} + (\beta - 4) q^{7} + ( - 4 \beta - 1) q^{8} + q^{9} + (3 \beta + 3) q^{10} + (4 \beta - 3) q^{11} + 3 \beta q^{12} + ( - 6 \beta + 3) q^{13} + (2 \beta + 3) q^{14} - 3 q^{15} + (3 \beta + 5) q^{16} + ( - \beta - 1) q^{17} + ( - \beta - 1) q^{18} + ( - 3 \beta - 1) q^{19} - 9 \beta q^{20} + (\beta - 4) q^{21} + ( - 5 \beta - 1) q^{22} + (\beta - 3) q^{23} + ( - 4 \beta - 1) q^{24} + 4 q^{25} + (9 \beta + 3) q^{26} + q^{27} + ( - 9 \beta + 3) q^{28} + (\beta - 6) q^{29} + (3 \beta + 3) q^{30} + (\beta - 4) q^{31} + ( - 3 \beta - 6) q^{32} + (4 \beta - 3) q^{33} + (3 \beta + 2) q^{34} + ( - 3 \beta + 12) q^{35} + 3 \beta q^{36} + (5 \beta - 3) q^{37} + (7 \beta + 4) q^{38} + ( - 6 \beta + 3) q^{39} + (12 \beta + 3) q^{40} + ( - 5 \beta + 2) q^{41} + (2 \beta + 3) q^{42} + (6 \beta + 3) q^{43} + (3 \beta + 12) q^{44} - 3 q^{45} + (\beta + 2) q^{46} - 3 \beta q^{47} + (3 \beta + 5) q^{48} + ( - 7 \beta + 10) q^{49} + ( - 4 \beta - 4) q^{50} + ( - \beta - 1) q^{51} + ( - 9 \beta - 18) q^{52} + (4 \beta - 1) q^{53} + ( - \beta - 1) q^{54} + ( - 12 \beta + 9) q^{55} + 11 \beta q^{56} + ( - 3 \beta - 1) q^{57} + (4 \beta + 5) q^{58} + q^{59} - 9 \beta q^{60} + (5 \beta + 4) q^{61} + (2 \beta + 3) q^{62} + (\beta - 4) q^{63} + (6 \beta - 1) q^{64} + (18 \beta - 9) q^{65} + ( - 5 \beta - 1) q^{66} + ( - 2 \beta - 1) q^{67} + ( - 6 \beta - 3) q^{68} + (\beta - 3) q^{69} + ( - 6 \beta - 9) q^{70} + (2 \beta + 1) q^{71} + ( - 4 \beta - 1) q^{72} + 5 \beta q^{73} + ( - 7 \beta - 2) q^{74} + 4 q^{75} + ( - 12 \beta - 9) q^{76} + ( - 15 \beta + 16) q^{77} + (9 \beta + 3) q^{78} - 3 q^{79} + ( - 9 \beta - 15) q^{80} + q^{81} + (8 \beta + 3) q^{82} + (3 \beta - 6) q^{83} + ( - 9 \beta + 3) q^{84} + (3 \beta + 3) q^{85} + ( - 15 \beta - 9) q^{86} + (\beta - 6) q^{87} + ( - 8 \beta - 13) q^{88} + (5 \beta - 2) q^{89} + (3 \beta + 3) q^{90} + (21 \beta - 18) q^{91} + ( - 6 \beta + 3) q^{92} + (\beta - 4) q^{93} + (6 \beta + 3) q^{94} + (9 \beta + 3) q^{95} + ( - 3 \beta - 6) q^{96} + ( - 4 \beta - 3) q^{97} + (4 \beta - 3) q^{98} + (4 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9} + 9 q^{10} - 2 q^{11} + 3 q^{12} + 8 q^{14} - 6 q^{15} + 13 q^{16} - 3 q^{17} - 3 q^{18} - 5 q^{19} - 9 q^{20} - 7 q^{21} - 7 q^{22} - 5 q^{23} - 6 q^{24} + 8 q^{25} + 15 q^{26} + 2 q^{27} - 3 q^{28} - 11 q^{29} + 9 q^{30} - 7 q^{31} - 15 q^{32} - 2 q^{33} + 7 q^{34} + 21 q^{35} + 3 q^{36} - q^{37} + 15 q^{38} + 18 q^{40} - q^{41} + 8 q^{42} + 12 q^{43} + 27 q^{44} - 6 q^{45} + 5 q^{46} - 3 q^{47} + 13 q^{48} + 13 q^{49} - 12 q^{50} - 3 q^{51} - 45 q^{52} + 2 q^{53} - 3 q^{54} + 6 q^{55} + 11 q^{56} - 5 q^{57} + 14 q^{58} + 2 q^{59} - 9 q^{60} + 13 q^{61} + 8 q^{62} - 7 q^{63} + 4 q^{64} - 7 q^{66} - 4 q^{67} - 12 q^{68} - 5 q^{69} - 24 q^{70} + 4 q^{71} - 6 q^{72} + 5 q^{73} - 11 q^{74} + 8 q^{75} - 30 q^{76} + 17 q^{77} + 15 q^{78} - 6 q^{79} - 39 q^{80} + 2 q^{81} + 14 q^{82} - 9 q^{83} - 3 q^{84} + 9 q^{85} - 33 q^{86} - 11 q^{87} - 34 q^{88} + q^{89} + 9 q^{90} - 15 q^{91} - 7 q^{93} + 12 q^{94} + 15 q^{95} - 15 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 1.00000 4.85410 −3.00000 −2.61803 −2.38197 −7.47214 1.00000 7.85410
1.2 −0.381966 1.00000 −1.85410 −3.00000 −0.381966 −4.61803 1.47214 1.00000 1.14590
\(n\): 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.a 2
3.b odd 2 1 531.2.a.c 2
4.b odd 2 1 2832.2.a.h 2
5.b even 2 1 4425.2.a.u 2
7.b odd 2 1 8673.2.a.j 2
12.b even 2 1 8496.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.a 2 1.a even 1 1 trivial
531.2.a.c 2 3.b odd 2 1
2832.2.a.h 2 4.b odd 2 1
4425.2.a.u 2 5.b even 2 1
8496.2.a.bg 2 12.b even 2 1
8673.2.a.j 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$13$ \( T^{2} - 45 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$29$ \( T^{2} + 11T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$37$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T - 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$59$ \( (T - 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 11 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$79$ \( (T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
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