Properties

Label 2672.2.a.c
Level $2672$
Weight $2$
Character orbit 2672.a
Self dual yes
Analytic conductor $21.336$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 668)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 3 q^{5} + ( - \beta - 1) q^{7} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 3 q^{5} + ( - \beta - 1) q^{7} + \beta q^{9} + (\beta + 4) q^{13} + 3 \beta q^{15} + ( - \beta + 1) q^{17} - 2 q^{19} + (2 \beta + 3) q^{21} + ( - \beta + 1) q^{23} + 4 q^{25} + (2 \beta - 3) q^{27} + ( - 2 \beta + 5) q^{29} + (2 \beta - 4) q^{31} + (3 \beta + 3) q^{35} + (2 \beta + 3) q^{37} + ( - 5 \beta - 3) q^{39} + (4 \beta - 1) q^{41} + (2 \beta - 7) q^{43} - 3 \beta q^{45} + (2 \beta + 1) q^{47} + (3 \beta - 3) q^{49} + 3 q^{51} + (2 \beta - 2) q^{53} + 2 \beta q^{57} + 6 \beta q^{59} + ( - 6 \beta - 1) q^{61} + ( - 2 \beta - 3) q^{63} + ( - 3 \beta - 12) q^{65} + (4 \beta - 3) q^{67} + 3 q^{69} + 3 \beta q^{71} + ( - 3 \beta - 7) q^{73} - 4 \beta q^{75} + ( - 2 \beta - 9) q^{79} + ( - 2 \beta - 6) q^{81} + ( - 2 \beta - 1) q^{83} + (3 \beta - 3) q^{85} + ( - 3 \beta + 6) q^{87} + (6 \beta - 6) q^{89} + ( - 6 \beta - 7) q^{91} + (2 \beta - 6) q^{93} + 6 q^{95} + ( - 5 \beta - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9} + 9 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 8 q^{21} + q^{23} + 8 q^{25} - 4 q^{27} + 8 q^{29} - 6 q^{31} + 9 q^{35} + 8 q^{37} - 11 q^{39} + 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} - 3 q^{49} + 6 q^{51} - 2 q^{53} + 2 q^{57} + 6 q^{59} - 8 q^{61} - 8 q^{63} - 27 q^{65} - 2 q^{67} + 6 q^{69} + 3 q^{71} - 17 q^{73} - 4 q^{75} - 20 q^{79} - 14 q^{81} - 4 q^{83} - 3 q^{85} + 9 q^{87} - 6 q^{89} - 20 q^{91} - 10 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) \) 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
2.30278
−1.30278
0 −2.30278 0 −3.00000 0 −3.30278 0 2.30278 0
1.2 0 1.30278 0 −3.00000 0 0.302776 0 −1.30278 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.c 2
4.b odd 2 1 668.2.a.a 2
12.b even 2 1 6012.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.a 2 4.b odd 2 1
2672.2.a.c 2 1.a even 1 1 trivial
6012.2.a.a 2 12.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 51 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 108 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 101 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$73$ \( T^{2} + 17T + 43 \) Copy content Toggle raw display
$79$ \( T^{2} + 20T + 87 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 108 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T - 61 \) Copy content Toggle raw display
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