Properties

Label 2601.2.a.bg
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2601,2,Mod(1,2601)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, 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("2601.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.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: \(20.7690895657\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 153)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + ( - 2 \beta_{3} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + ( - 2 \beta_{3} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 3 q^{8} + ( - 2 \beta_{3} - \beta_1) q^{10} + ( - 2 \beta_{3} - 2 \beta_1) q^{11} - 3 \beta_{2} q^{13} - 2 \beta_1 q^{14} - q^{16} + 2 \beta_{2} q^{19} + (2 \beta_{3} + \beta_1) q^{20} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} + ( - 4 \beta_{3} + 2 \beta_1) q^{23} + (\beta_{2} + 5) q^{25} - 3 \beta_{2} q^{26} + 2 \beta_1 q^{28} + ( - \beta_{3} - 2 \beta_1) q^{29} + 2 \beta_{3} q^{31} + 5 q^{32} + (6 \beta_{2} + 4) q^{35} + (4 \beta_{3} - 3 \beta_1) q^{37} + 2 \beta_{2} q^{38} + (6 \beta_{3} + 3 \beta_1) q^{40} + ( - \beta_{3} - 2 \beta_1) q^{41} - 4 \beta_{2} q^{43} + (2 \beta_{3} + 2 \beta_1) q^{44} + ( - 4 \beta_{3} + 2 \beta_1) q^{46} - 2 \beta_{2} q^{47} + (4 \beta_{2} + 1) q^{49} + (\beta_{2} + 5) q^{50} + 3 \beta_{2} q^{52} + ( - 3 \beta_{2} + 4) q^{53} + (4 \beta_{2} + 12) q^{55} + 6 \beta_1 q^{56} + ( - \beta_{3} - 2 \beta_1) q^{58} + ( - 6 \beta_{2} + 4) q^{59} + ( - 3 \beta_{3} + 8 \beta_1) q^{61} + 2 \beta_{3} q^{62} + 7 q^{64} + ( - 3 \beta_{3} + 9 \beta_1) q^{65} + ( - 6 \beta_{2} + 4) q^{67} + (6 \beta_{2} + 4) q^{70} + (4 \beta_{3} - 2 \beta_1) q^{71} - 5 \beta_1 q^{73} + (4 \beta_{3} - 3 \beta_1) q^{74} - 2 \beta_{2} q^{76} + (8 \beta_{2} + 8) q^{77} + ( - 4 \beta_{3} + 6 \beta_1) q^{79} + (2 \beta_{3} + \beta_1) q^{80} + ( - \beta_{3} - 2 \beta_1) q^{82} + ( - 6 \beta_{2} - 4) q^{83} - 4 \beta_{2} q^{86} + (6 \beta_{3} + 6 \beta_1) q^{88} + ( - \beta_{2} + 12) q^{89} + (6 \beta_{3} + 6 \beta_1) q^{91} + (4 \beta_{3} - 2 \beta_1) q^{92} - 2 \beta_{2} q^{94} + (2 \beta_{3} - 6 \beta_1) q^{95} + ( - 3 \beta_{3} - 4 \beta_1) q^{97} + (4 \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{16} + 20 q^{25} + 20 q^{32} + 16 q^{35} + 4 q^{49} + 20 q^{50} + 16 q^{53} + 48 q^{55} + 16 q^{59} + 28 q^{64} + 16 q^{67} + 16 q^{70} + 32 q^{77} - 16 q^{83} + 48 q^{89} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.84776
−0.765367
0.765367
−1.84776
1.00000 0 −1.00000 −3.37849 0 −3.69552 −3.00000 0 −3.37849
1.2 1.00000 0 −1.00000 −2.93015 0 1.53073 −3.00000 0 −2.93015
1.3 1.00000 0 −1.00000 2.93015 0 −1.53073 −3.00000 0 2.93015
1.4 1.00000 0 −1.00000 3.37849 0 3.69552 −3.00000 0 3.37849
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.bg 4
3.b odd 2 1 2601.2.a.ba 4
17.b even 2 1 inner 2601.2.a.bg 4
17.e odd 16 2 153.2.l.a 4
51.c odd 2 1 2601.2.a.ba 4
51.i even 16 2 153.2.l.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.l.a 4 17.e odd 16 2
153.2.l.b yes 4 51.i even 16 2
2601.2.a.ba 4 3.b odd 2 1
2601.2.a.ba 4 51.c odd 2 1
2601.2.a.bg 4 1.a even 1 1 trivial
2601.2.a.bg 4 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 20T_{5}^{2} + 98 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 20T^{2} + 98 \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 32T^{2} + 128 \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 80T^{2} + 32 \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 2 \) Copy content Toggle raw display
$31$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$37$ \( T^{4} - 100T^{2} + 578 \) Copy content Toggle raw display
$41$ \( T^{4} - 20T^{2} + 2 \) Copy content Toggle raw display
$43$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 292 T^{2} + 21218 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 80T^{2} + 32 \) Copy content Toggle raw display
$73$ \( T^{4} - 100T^{2} + 1250 \) Copy content Toggle raw display
$79$ \( T^{4} - 208T^{2} + 9248 \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 142)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 100T^{2} + 578 \) Copy content Toggle raw display
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