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Properties

Label 867.2.a.n
Level $867$
Weight $2$
Character orbit 867.a
Self dual yes
Analytic conductor $6.923$
Analytic rank $1$
Dimension $4$
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] = [867,2,Mod(1,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, 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("867.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.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: \(6.92302985525\)
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: \(\Z[a_1, a_2]\)
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)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{5} + \beta_1 q^{6} + ( - \beta_1 - 2) q^{7} + (\beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{5} + \beta_1 q^{6} + ( - \beta_1 - 2) q^{7} + (\beta_{3} - \beta_1) q^{8} + q^{9} + ( - 2 \beta_{2} - \beta_1 - 2) q^{10} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + \beta_{2} q^{12} + (2 \beta_{3} - \beta_{2} - 1) q^{13} + ( - \beta_{2} - 2 \beta_1 - 2) q^{14} + ( - \beta_{3} - \beta_1 - 1) q^{15} + ( - 2 \beta_{2} - 2) q^{16} + \beta_1 q^{18} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{19} + ( - \beta_{2} - 2 \beta_1) q^{20} + ( - \beta_1 - 2) q^{21} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{22} + (3 \beta_{3} - \beta_1 - 3) q^{23} + (\beta_{3} - \beta_1) q^{24} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{26} + q^{27} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{28} + (3 \beta_{2} - 2) q^{29} + ( - 2 \beta_{2} - \beta_1 - 2) q^{30} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 2) q^{31} + ( - 4 \beta_{3} - 2 \beta_1) q^{32} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{33} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{35} + \beta_{2} q^{36} + ( - 2 \beta_{2} + \beta_1 - 6) q^{37} + ( - \beta_{3} - 4 \beta_1 + 4) q^{38} + (2 \beta_{3} - \beta_{2} - 1) q^{39} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{40} + ( - 3 \beta_{3} + 2 \beta_{2} + \cdots - 3) q^{41}+ \cdots + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9} - 8 q^{10} - 4 q^{11} - 4 q^{13} - 8 q^{14} - 4 q^{15} - 8 q^{16} - 12 q^{19} - 8 q^{21} + 8 q^{22} - 12 q^{23} + 4 q^{27} - 8 q^{29} - 8 q^{30} - 8 q^{31} - 4 q^{33} + 16 q^{35} - 24 q^{37} + 16 q^{38} - 4 q^{39} - 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 4 q^{45} - 8 q^{46} + 8 q^{47} - 8 q^{48} - 4 q^{49} + 16 q^{50} - 8 q^{52} + 8 q^{53} - 12 q^{55} + 8 q^{56} - 12 q^{57} + 24 q^{59} - 24 q^{61} + 8 q^{62} - 8 q^{63} - 12 q^{65} + 8 q^{66} + 8 q^{67} - 12 q^{69} + 24 q^{70} + 24 q^{71} - 8 q^{73} + 8 q^{74} - 8 q^{76} + 8 q^{80} + 4 q^{81} + 8 q^{82} + 8 q^{83} - 16 q^{86} - 8 q^{87} + 16 q^{89} - 8 q^{90} + 8 q^{91} - 8 q^{93} - 16 q^{94} + 12 q^{95} + 16 q^{97} + 32 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−0.765367
0.765367
1.84776
−1.84776 1.00000 1.41421 1.61313 −1.84776 −0.152241 1.08239 1.00000 −2.98067
1.2 −0.765367 1.00000 −1.41421 −2.08239 −0.765367 −1.23463 2.61313 1.00000 1.59379
1.3 0.765367 1.00000 −1.41421 0.0823922 0.765367 −2.76537 −2.61313 1.00000 0.0630603
1.4 1.84776 1.00000 1.41421 −3.61313 1.84776 −3.84776 −1.08239 1.00000 −6.67619
\(n\): 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 867.2.a.n 4
3.b odd 2 1 2601.2.a.bd 4
17.b even 2 1 867.2.a.m 4
17.c even 4 2 867.2.d.e 8
17.d even 8 2 867.2.e.h 8
17.d even 8 2 867.2.e.i 8
17.e odd 16 2 51.2.h.a ✓ 8
17.e odd 16 2 867.2.h.b 8
17.e odd 16 2 867.2.h.f 8
17.e odd 16 2 867.2.h.g 8
51.c odd 2 1 2601.2.a.bc 4
51.i even 16 2 153.2.l.e 8
68.i even 16 2 816.2.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.h.a ✓ 8 17.e odd 16 2
153.2.l.e 8 51.i even 16 2
816.2.bq.a 8 68.i even 16 2
867.2.a.m 4 17.b even 2 1
867.2.a.n 4 1.a even 1 1 trivial
867.2.d.e 8 17.c even 4 2
867.2.e.h 8 17.d even 8 2
867.2.e.i 8 17.d even 8 2
867.2.h.b 8 17.e odd 16 2
867.2.h.f 8 17.e odd 16 2
867.2.h.g 8 17.e odd 16 2
2601.2.a.bc 4 51.c odd 2 1
2601.2.a.bd 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(867))\):

\( T_{2}^{4} - 4T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} + 20T_{7}^{2} + 16T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots - 47 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots - 367 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots - 271 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 632 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + \cdots + 706 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + \cdots + 1297 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots - 752 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} + \cdots + 496 \) Copy content Toggle raw display
$59$ \( T^{4} - 24 T^{3} + \cdots + 514 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + \cdots - 1054 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + \cdots + 68 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + \cdots + 752 \) Copy content Toggle raw display
$79$ \( T^{4} - 52 T^{2} + \cdots - 62 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} + \cdots - 272 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots - 2558 \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} + \cdots - 632 \) Copy content Toggle raw display
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