Properties

Label 2601.2.a.j
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $1$
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] = [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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + 4 q^{7} - 3 q^{8} + 4 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 4 q^{19} + 4 q^{22} - 4 q^{23} - 5 q^{25} + 2 q^{26} - 4 q^{28} - 4 q^{31} + 5 q^{32} + 8 q^{37} + 4 q^{38} - 8 q^{41} + 4 q^{43} - 4 q^{44} - 4 q^{46} + 8 q^{47} + 9 q^{49} - 5 q^{50} - 2 q^{52} + 6 q^{53} - 12 q^{56} - 12 q^{59} + 8 q^{61} - 4 q^{62} + 7 q^{64} + 12 q^{67} - 12 q^{71} + 8 q^{74} - 4 q^{76} + 16 q^{77} + 4 q^{79} - 8 q^{82} + 12 q^{83} + 4 q^{86} - 12 q^{88} + 10 q^{89} + 8 q^{91} + 4 q^{92} + 8 q^{94} + 16 q^{97} + 9 q^{98}+O(q^{100}) \) 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
1.00000 0 −1.00000 0 0 4.00000 −3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2601.2.a.j 1
3.b odd 2 1 867.2.a.b 1
17.b even 2 1 2601.2.a.i 1
17.c even 4 2 153.2.d.a 2
51.c odd 2 1 867.2.a.a 1
51.f odd 4 2 51.2.d.b 2
51.g odd 8 4 867.2.e.d 4
51.i even 16 8 867.2.h.d 8
68.f odd 4 2 2448.2.c.j 2
204.l even 4 2 816.2.c.c 2
255.i odd 4 2 1275.2.g.a 2
255.k even 4 2 1275.2.d.b 2
255.r even 4 2 1275.2.d.d 2
408.q even 4 2 3264.2.c.d 2
408.t odd 4 2 3264.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 51.f odd 4 2
153.2.d.a 2 17.c even 4 2
816.2.c.c 2 204.l even 4 2
867.2.a.a 1 51.c odd 2 1
867.2.a.b 1 3.b odd 2 1
867.2.e.d 4 51.g odd 8 4
867.2.h.d 8 51.i even 16 8
1275.2.d.b 2 255.k even 4 2
1275.2.d.d 2 255.r even 4 2
1275.2.g.a 2 255.i odd 4 2
2448.2.c.j 2 68.f odd 4 2
2601.2.a.i 1 17.b even 2 1
2601.2.a.j 1 1.a even 1 1 trivial
3264.2.c.d 2 408.q even 4 2
3264.2.c.e 2 408.t odd 4 2

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
\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T + 8 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T - 16 \) Copy content Toggle raw display
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