Properties

Label 2672.2.a.b
Level $2672$
Weight $2$
Character orbit 2672.a
Self dual yes
Analytic conductor $21.336$
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] = [2672,2,Mod(1,2672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2672 = 2^{4} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2672.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: \(21.3360274201\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1336)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} + ( - 2 \beta + 1) q^{5} + ( - \beta + 4) q^{7} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{3} + ( - 2 \beta + 1) q^{5} + ( - \beta + 4) q^{7} + (3 \beta - 1) q^{9} + 4 q^{11} + ( - \beta + 5) q^{13} + (3 \beta + 1) q^{15} - 5 \beta q^{17} + (4 \beta + 2) q^{19} + ( - 2 \beta - 3) q^{21} + ( - 3 \beta + 4) q^{23} + ( - 2 \beta + 1) q^{27} + (2 \beta - 5) q^{29} + ( - 2 \beta - 2) q^{31} + ( - 4 \beta - 4) q^{33} + ( - 7 \beta + 6) q^{35} + (8 \beta - 3) q^{37} + ( - 3 \beta - 4) q^{39} + (2 \beta - 1) q^{41} + ( - 4 \beta + 5) q^{43} + ( - \beta - 7) q^{45} + (2 \beta + 1) q^{47} + ( - 7 \beta + 10) q^{49} + (10 \beta + 5) q^{51} + (2 \beta - 8) q^{53} + ( - 8 \beta + 4) q^{55} + ( - 10 \beta - 6) q^{57} + ( - 6 \beta - 2) q^{59} + (2 \beta - 11) q^{61} + (10 \beta - 7) q^{63} + ( - 9 \beta + 7) q^{65} + (2 \beta - 5) q^{67} + (2 \beta - 1) q^{69} + (3 \beta - 7) q^{71} + (5 \beta - 2) q^{73} + ( - 4 \beta + 16) q^{77} + 3 q^{79} + ( - 6 \beta + 4) q^{81} + (8 \beta + 3) q^{83} + (5 \beta + 10) q^{85} + (\beta + 3) q^{87} - 2 \beta q^{89} + ( - 8 \beta + 21) q^{91} + (6 \beta + 4) q^{93} + ( - 8 \beta - 6) q^{95} + (13 \beta - 5) q^{97} + (12 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 7 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 7 q^{7} + q^{9} + 8 q^{11} + 9 q^{13} + 5 q^{15} - 5 q^{17} + 8 q^{19} - 8 q^{21} + 5 q^{23} - 8 q^{29} - 6 q^{31} - 12 q^{33} + 5 q^{35} + 2 q^{37} - 11 q^{39} + 6 q^{43} - 15 q^{45} + 4 q^{47} + 13 q^{49} + 20 q^{51} - 14 q^{53} - 22 q^{57} - 10 q^{59} - 20 q^{61} - 4 q^{63} + 5 q^{65} - 8 q^{67} - 11 q^{71} + q^{73} + 28 q^{77} + 6 q^{79} + 2 q^{81} + 14 q^{83} + 25 q^{85} + 7 q^{87} - 2 q^{89} + 34 q^{91} + 14 q^{93} - 20 q^{95} + 3 q^{97} + 4 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −2.61803 0 −2.23607 0 2.38197 0 3.85410 0
1.2 0 −0.381966 0 2.23607 0 4.61803 0 −2.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(167\) \(-1\)

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 2672.2.a.b 2
4.b odd 2 1 1336.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.a 2 4.b odd 2 1
2672.2.a.b 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2672))\):

\( T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 7T_{7} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 11 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 25 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 79 \) Copy content Toggle raw display
$41$ \( T^{2} - 5 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$61$ \( T^{2} + 20T + 95 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$71$ \( T^{2} + 11T + 19 \) Copy content Toggle raw display
$73$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$79$ \( (T - 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 14T - 31 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 3T - 209 \) Copy content Toggle raw display
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