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Properties

Label 2601.2.a.bb
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
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: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + (2 \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + ( - \beta_{2} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + (2 \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + ( - \beta_{2} - 3) q^{8} - \beta_1 q^{10} + ( - \beta_{3} - \beta_1) q^{11} + \beta_{2} q^{13} + (\beta_{3} + 3 \beta_1) q^{14} + 3 q^{16} + ( - 2 \beta_{2} + 2) q^{19} + ( - \beta_{3} + 2 \beta_1) q^{20} + (\beta_{3} + 3 \beta_1) q^{22} + ( - \beta_{3} + 3 \beta_1) q^{23} + ( - \beta_{2} - 3) q^{25} + ( - \beta_{2} - 2) q^{26} + ( - \beta_{3} - 5 \beta_1) q^{28} + (2 \beta_{3} - \beta_1) q^{29} + (3 \beta_{3} + 3 \beta_1) q^{31} + ( - \beta_{2} + 3) q^{32} - 2 q^{35} + 5 \beta_1 q^{37} + 2 q^{38} + ( - 2 \beta_{3} - \beta_1) q^{40} + ( - 4 \beta_{3} + \beta_1) q^{41} + (2 \beta_{2} - 2) q^{43} + ( - \beta_{3} - 5 \beta_1) q^{44} + ( - 3 \beta_{3} - 5 \beta_1) q^{46} + (2 \beta_{2} - 8) q^{47} + (2 \beta_{2} - 3) q^{49} + (4 \beta_{2} + 5) q^{50} + (\beta_{2} + 4) q^{52} - \beta_{2} q^{53} - 2 q^{55} + (3 \beta_{3} + 5 \beta_1) q^{56} + \beta_{3} q^{58} - 6 q^{59} + 5 \beta_{3} q^{61} + ( - 3 \beta_{3} - 9 \beta_1) q^{62} + ( - 2 \beta_{2} - 7) q^{64} + ( - \beta_{3} + \beta_1) q^{65} + (2 \beta_{2} - 4) q^{67} + (2 \beta_{2} + 2) q^{70} + ( - 5 \beta_{3} + 5 \beta_1) q^{71} - 7 \beta_1 q^{73} + ( - 5 \beta_{3} - 10 \beta_1) q^{74} + (2 \beta_{2} - 6) q^{76} + (2 \beta_{2} + 4) q^{77} + (\beta_{3} - 3 \beta_1) q^{79} + 3 \beta_{3} q^{80} + ( - \beta_{3} + 2 \beta_1) q^{82} + ( - 4 \beta_{2} - 6) q^{83} - 2 q^{86} + (3 \beta_{3} + 5 \beta_1) q^{88} + (\beta_{2} - 8) q^{89} - 2 \beta_1 q^{91} + (7 \beta_{3} + 7 \beta_1) q^{92} + (6 \beta_{2} + 4) q^{94} + (4 \beta_{3} - 2 \beta_1) q^{95} + (\beta_{3} - 6 \beta_1) q^{97} + (\beta_{2} - 1) q^{98}+O(q^{100}) \) 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 \( 4 q - 4 q^{2} + 4 q^{4} - 12 q^{8} + 12 q^{16} + 8 q^{19} - 12 q^{25} - 8 q^{26} + 12 q^{32} - 8 q^{35} + 8 q^{38} - 8 q^{43} - 32 q^{47} - 12 q^{49} + 20 q^{50} + 16 q^{52} - 8 q^{55} - 24 q^{59} - 28 q^{64} - 16 q^{67} + 8 q^{70} - 24 q^{76} + 16 q^{77} - 24 q^{83} - 8 q^{86} - 32 q^{89} + 16 q^{94} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.84776
0.765367
−0.765367
−2.41421 0 3.82843 −0.765367 0 2.61313 −4.41421 0 1.84776
1.2 −2.41421 0 3.82843 0.765367 0 −2.61313 −4.41421 0 −1.84776
1.3 0.414214 0 −1.82843 −1.84776 0 1.08239 −1.58579 0 −0.765367
1.4 0.414214 0 −1.82843 1.84776 0 −1.08239 −1.58579 0 0.765367
\(n\): 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.bb 4
3.b odd 2 1 289.2.a.f 4
12.b even 2 1 4624.2.a.bp 4
15.d odd 2 1 7225.2.a.u 4
17.b even 2 1 inner 2601.2.a.bb 4
17.e odd 16 2 153.2.l.c 4
51.c odd 2 1 289.2.a.f 4
51.f odd 4 2 289.2.b.b 4
51.g odd 8 4 289.2.c.c 8
51.i even 16 2 17.2.d.a ✓ 4
51.i even 16 2 289.2.d.a 4
51.i even 16 2 289.2.d.b 4
51.i even 16 2 289.2.d.c 4
204.h even 2 1 4624.2.a.bp 4
204.t odd 16 2 272.2.v.d 4
255.h odd 2 1 7225.2.a.u 4
255.bc odd 16 2 425.2.n.b 4
255.be even 16 2 425.2.m.a 4
255.bj odd 16 2 425.2.n.a 4
357.be odd 16 2 833.2.l.a 4
357.bm even 48 4 833.2.v.b 8
357.bn odd 48 4 833.2.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a ✓ 4 51.i even 16 2
153.2.l.c 4 17.e odd 16 2
272.2.v.d 4 204.t odd 16 2
289.2.a.f 4 3.b odd 2 1
289.2.a.f 4 51.c odd 2 1
289.2.b.b 4 51.f odd 4 2
289.2.c.c 8 51.g odd 8 4
289.2.d.a 4 51.i even 16 2
289.2.d.b 4 51.i even 16 2
289.2.d.c 4 51.i even 16 2
425.2.m.a 4 255.be even 16 2
425.2.n.a 4 255.bj odd 16 2
425.2.n.b 4 255.bc odd 16 2
833.2.l.a 4 357.be odd 16 2
833.2.v.a 8 357.bn odd 48 4
833.2.v.b 8 357.bm even 48 4
2601.2.a.bb 4 1.a even 1 1 trivial
2601.2.a.bb 4 17.b even 2 1 inner
4624.2.a.bp 4 12.b even 2 1
4624.2.a.bp 4 204.h even 2 1
7225.2.a.u 4 15.d odd 2 1
7225.2.a.u 4 255.h odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 40T^{2} + 392 \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 2 \) Copy content Toggle raw display
$31$ \( T^{4} - 72T^{2} + 648 \) Copy content Toggle raw display
$37$ \( T^{4} - 100T^{2} + 1250 \) Copy content Toggle raw display
$41$ \( T^{4} - 68T^{2} + 98 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 100T^{2} + 1250 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 200T^{2} + 5000 \) Copy content Toggle raw display
$73$ \( T^{4} - 196T^{2} + 4802 \) Copy content Toggle raw display
$79$ \( T^{4} - 40T^{2} + 392 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 16 T + 62)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 148T^{2} + 4418 \) Copy content Toggle raw display
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