Properties

Label 867.2.a.p
Level $867$
Weight $2$
Character orbit 867.a
Self dual yes
Analytic conductor $6.923$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.3418281.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + q^{3} + (2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} - \beta_{4} q^{6} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{7} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + q^{3} + (2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} - \beta_{4} q^{6} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{7} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{10} + (2 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{12} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + ( - 3 \beta_{5} + \beta_{3} - 1) q^{14} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{15} + (3 \beta_{5} - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{16} - \beta_{4} q^{18} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{19} + ( - 2 \beta_{5} - 5 \beta_{2} + \beta_1 + 1) q^{20} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{21} + (\beta_{5} + 3 \beta_{3} + 4 \beta_{2} - 3) q^{22} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{23} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{24} + ( - 4 \beta_{5} + \beta_{4} - 4 \beta_{2} - 2 \beta_1 + 3) q^{25} + ( - 5 \beta_{5} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{26} + q^{27} + (\beta_{5} + 3 \beta_{4} - \beta_{3} - 1) q^{28} + ( - 2 \beta_{5} - \beta_{2} - \beta_1 - 1) q^{29} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{30} + (\beta_{5} + 2 \beta_{4} + \beta_1 + 5) q^{31} + (5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 5 \beta_{2} - \beta_1 + 6) q^{32} + (2 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{33} + ( - 2 \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + \beta_1 - 1) q^{35} + (2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{37} + ( - \beta_{5} - \beta_{3} - 4 \beta_{2} - 1) q^{38} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 7 \beta_{2} + \beta_1) q^{40} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{41} + ( - 3 \beta_{5} + \beta_{3} - 1) q^{42} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{43} + (3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 1) q^{44} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{45} + (8 \beta_{5} + \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - \beta_1 + 3) q^{46} + (2 \beta_{4} - 3 \beta_{2} + \beta_1 + 5) q^{47} + (3 \beta_{5} - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{48} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} + 5) q^{49} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{50} + ( - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + \beta_1 - 1) q^{52} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 4) q^{53} - \beta_{4} q^{54} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{55} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 8) q^{56} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{57} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{58} + (\beta_{4} - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{59} + ( - 2 \beta_{5} - 5 \beta_{2} + \beta_1 + 1) q^{60} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 4 \beta_{5} - 4 \beta_{4} - 3 \beta_{2} - \beta_1 - 7) q^{62} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{63} + (6 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} + \beta_1 + 1) q^{64} + ( - 4 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{65} + (\beta_{5} + 3 \beta_{3} + 4 \beta_{2} - 3) q^{66} + (5 \beta_{5} - \beta_{2} + 3 \beta_1 - 1) q^{67} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{69} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1) q^{70} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 4) q^{71} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{72} + ( - \beta_{5} - 4 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{73} + ( - 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 5) q^{74} + ( - 4 \beta_{5} + \beta_{4} - 4 \beta_{2} - 2 \beta_1 + 3) q^{75} + ( - 3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{76} + (5 \beta_{5} + 3 \beta_{4} + \beta_{3} + 6 \beta_1 + 7) q^{77} + ( - 5 \beta_{5} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{78} + (4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 3) q^{79} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{80} + q^{81} + ( - 5 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 1) q^{82} + ( - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} - \beta_1 + 3) q^{83} + (\beta_{5} + 3 \beta_{4} - \beta_{3} - 1) q^{84} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 2) q^{86} + ( - 2 \beta_{5} - \beta_{2} - \beta_1 - 1) q^{87} + (\beta_{5} + \beta_{3} + 7 \beta_{2} - 7 \beta_1 - 4) q^{88} + ( - \beta_{5} - 3 \beta_{3} + 5 \beta_{2} - \beta_1) q^{89} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{90} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 5 \beta_1 + 7) q^{91} + (7 \beta_{5} - 6 \beta_{4} + 5 \beta_{3} + 14 \beta_{2} - 2 \beta_1 - 3) q^{92} + (\beta_{5} + 2 \beta_{4} + \beta_1 + 5) q^{93} + ( - 5 \beta_{5} - 3 \beta_{4} - \beta_{3} - 7 \beta_{2} + \beta_1 - 4) q^{94} + (6 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 9) q^{95} + (5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 5 \beta_{2} - \beta_1 + 6) q^{96} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_1 - 7) q^{97} + ( - 7 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 10) q^{98} + (2 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 6 q^{3} + 9 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 6 q^{3} + 9 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 12 q^{8} + 6 q^{9} + 12 q^{10} - 9 q^{11} + 9 q^{12} + 9 q^{13} - 6 q^{14} - 3 q^{15} + 15 q^{16} + 3 q^{18} + 9 q^{19} + 6 q^{20} + 3 q^{21} - 18 q^{22} - 9 q^{23} + 12 q^{24} + 15 q^{25} - 12 q^{26} + 6 q^{27} - 15 q^{28} - 6 q^{29} + 12 q^{30} + 24 q^{31} + 42 q^{32} - 9 q^{33} + 9 q^{36} - 3 q^{37} - 6 q^{38} + 9 q^{39} - 3 q^{40} - 18 q^{41} - 6 q^{42} - 3 q^{44} - 3 q^{45} + 15 q^{46} + 24 q^{47} + 15 q^{48} + 21 q^{49} + 12 q^{50} - 18 q^{52} + 24 q^{53} + 3 q^{54} - 24 q^{55} - 54 q^{56} + 9 q^{57} + 3 q^{58} - 9 q^{59} + 6 q^{60} + 21 q^{61} - 30 q^{62} + 3 q^{63} + 24 q^{64} + 9 q^{65} - 18 q^{66} - 6 q^{67} - 9 q^{69} - 3 q^{70} - 27 q^{71} + 12 q^{72} + 18 q^{73} - 36 q^{74} + 15 q^{75} - 3 q^{76} + 33 q^{77} - 12 q^{78} + 24 q^{79} - 3 q^{80} + 6 q^{81} - 15 q^{82} + 6 q^{83} - 15 q^{84} + 6 q^{86} - 6 q^{87} - 24 q^{88} + 12 q^{90} + 39 q^{91} + 24 q^{93} - 15 q^{94} - 42 q^{95} + 42 q^{96} - 33 q^{97} - 57 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 4\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 3\nu^{2} + 14\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 7\beta_{3} + 2\beta_{2} + 8\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 12\beta_{3} + 18\beta_{2} + 31\beta _1 + 21 \) 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
0.0750494
2.73700
−2.06104
−0.857616
1.71374
−1.60714
−2.09548 1.00000 2.39104 −1.55815 −2.09548 4.13541 −0.819422 1.00000 3.26508
1.2 −1.43915 1.00000 0.0711653 −2.31394 −1.43915 −4.44173 2.77589 1.00000 3.33012
1.3 0.435433 1.00000 −1.81040 −4.22078 0.435433 2.90981 −1.65917 1.00000 −1.83787
1.4 0.907065 1.00000 −1.17723 3.19333 0.907065 3.56234 −2.88196 1.00000 2.89656
1.5 2.44395 1.00000 3.97290 2.87349 2.44395 −1.56252 4.82168 1.00000 7.02266
1.6 2.74819 1.00000 5.55252 −0.973936 2.74819 −1.60333 9.76298 1.00000 −2.67656
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.p yes 6
3.b odd 2 1 2601.2.a.bi 6
17.b even 2 1 867.2.a.o 6
17.c even 4 2 867.2.d.g 12
17.d even 8 4 867.2.e.k 24
17.e odd 16 8 867.2.h.m 48
51.c odd 2 1 2601.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.o 6 17.b even 2 1
867.2.a.p yes 6 1.a even 1 1 trivial
867.2.d.g 12 17.c even 4 2
867.2.e.k 24 17.d even 8 4
867.2.h.m 48 17.e odd 16 8
2601.2.a.bh 6 51.c odd 2 1
2601.2.a.bi 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(867))\):

\( T_{2}^{6} - 3T_{2}^{5} - 6T_{2}^{4} + 19T_{2}^{3} + 6T_{2}^{2} - 24T_{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} - 18T_{5}^{4} - 51T_{5}^{3} + 60T_{5}^{2} + 228T_{5} + 136 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} - 27T_{7}^{4} + 75T_{7}^{3} + 171T_{7}^{2} - 297T_{7} - 477 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} - 6 T^{4} + 19 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} - 18 T^{4} - 51 T^{3} + \cdots + 136 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} - 27 T^{4} + 75 T^{3} + \cdots - 477 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{5} - 6 T^{4} - 199 T^{3} + \cdots + 296 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + 9 T^{4} + 103 T^{3} + \cdots - 109 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} - 9 T^{4} + 207 T^{3} + \cdots + 333 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} - 36 T^{4} + \cdots + 1576 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} - 3 T^{4} - 57 T^{3} + \cdots + 136 \) Copy content Toggle raw display
$31$ \( T^{6} - 24 T^{5} + 186 T^{4} + \cdots - 127 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{5} - 75 T^{4} - 281 T^{3} + \cdots - 109 \) Copy content Toggle raw display
$41$ \( T^{6} + 18 T^{5} + 63 T^{4} + \cdots - 1592 \) Copy content Toggle raw display
$43$ \( T^{6} - 117 T^{4} + 297 T^{3} + \cdots - 11736 \) Copy content Toggle raw display
$47$ \( T^{6} - 24 T^{5} + 171 T^{4} + \cdots + 20312 \) Copy content Toggle raw display
$53$ \( T^{6} - 24 T^{5} + 171 T^{4} + \cdots - 5608 \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} - 60 T^{4} + \cdots + 1432 \) Copy content Toggle raw display
$61$ \( T^{6} - 21 T^{5} + 45 T^{4} + \cdots + 31841 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} - 153 T^{4} + \cdots - 14552 \) Copy content Toggle raw display
$71$ \( T^{6} + 27 T^{5} + 207 T^{4} + \cdots - 8704 \) Copy content Toggle raw display
$73$ \( T^{6} - 18 T^{5} - 30 T^{4} + \cdots + 92429 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 9 T^{4} + \cdots + 31419 \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{5} - 243 T^{4} + \cdots - 205336 \) Copy content Toggle raw display
$89$ \( T^{6} - 327 T^{4} + 585 T^{3} + \cdots - 39896 \) Copy content Toggle raw display
$97$ \( T^{6} + 33 T^{5} + 324 T^{4} + \cdots - 17389 \) Copy content Toggle raw display
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