Properties

Label 867.2.a.g
Level $867$
Weight $2$
Character orbit 867.a
Self dual yes
Analytic conductor $6.923$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} + 2 \beta q^{5} + ( - \beta - 1) q^{6} - 3 q^{7} + (\beta + 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} + 2 \beta q^{5} + ( - \beta - 1) q^{6} - 3 q^{7} + (\beta + 3) q^{8} + q^{9} + (2 \beta + 4) q^{10} + (2 \beta - 2) q^{11} + ( - 2 \beta - 1) q^{12} + (4 \beta + 1) q^{13} + ( - 3 \beta - 3) q^{14} - 2 \beta q^{15} + 3 q^{16} + (\beta + 1) q^{18} - 3 q^{19} + (2 \beta + 8) q^{20} + 3 q^{21} + 2 q^{22} + 2 \beta q^{23} + ( - \beta - 3) q^{24} + 3 q^{25} + (5 \beta + 9) q^{26} - q^{27} + ( - 6 \beta - 3) q^{28} + ( - 2 \beta + 6) q^{29} + ( - 2 \beta - 4) q^{30} + (4 \beta - 1) q^{31} + (\beta - 3) q^{32} + ( - 2 \beta + 2) q^{33} - 6 \beta q^{35} + (2 \beta + 1) q^{36} + ( - 4 \beta - 1) q^{37} + ( - 3 \beta - 3) q^{38} + ( - 4 \beta - 1) q^{39} + (6 \beta + 4) q^{40} + ( - 2 \beta - 6) q^{41} + (3 \beta + 3) q^{42} + 3 q^{43} + ( - 2 \beta + 6) q^{44} + 2 \beta q^{45} + (2 \beta + 4) q^{46} + ( - 4 \beta - 4) q^{47} - 3 q^{48} + 2 q^{49} + (3 \beta + 3) q^{50} + (6 \beta + 17) q^{52} + ( - 4 \beta + 4) q^{53} + ( - \beta - 1) q^{54} + ( - 4 \beta + 8) q^{55} + ( - 3 \beta - 9) q^{56} + 3 q^{57} + (4 \beta + 2) q^{58} + ( - 2 \beta - 6) q^{59} + ( - 2 \beta - 8) q^{60} + (4 \beta - 1) q^{61} + (3 \beta + 7) q^{62} - 3 q^{63} + ( - 2 \beta - 7) q^{64} + (2 \beta + 16) q^{65} - 2 q^{66} + q^{67} - 2 \beta q^{69} + ( - 6 \beta - 12) q^{70} + ( - 2 \beta + 8) q^{71} + (\beta + 3) q^{72} + (4 \beta + 2) q^{73} + ( - 5 \beta - 9) q^{74} - 3 q^{75} + ( - 6 \beta - 3) q^{76} + ( - 6 \beta + 6) q^{77} + ( - 5 \beta - 9) q^{78} - 12 q^{79} + 6 \beta q^{80} + q^{81} + ( - 8 \beta - 10) q^{82} + ( - 2 \beta + 4) q^{83} + (6 \beta + 3) q^{84} + (3 \beta + 3) q^{86} + (2 \beta - 6) q^{87} + (4 \beta - 2) q^{88} + (8 \beta - 6) q^{89} + (2 \beta + 4) q^{90} + ( - 12 \beta - 3) q^{91} + (2 \beta + 8) q^{92} + ( - 4 \beta + 1) q^{93} + ( - 8 \beta - 12) q^{94} - 6 \beta q^{95} + ( - \beta + 3) q^{96} + (4 \beta + 5) q^{97} + (2 \beta + 2) q^{98} + (2 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 2 q^{13} - 6 q^{14} + 6 q^{16} + 2 q^{18} - 6 q^{19} + 16 q^{20} + 6 q^{21} + 4 q^{22} - 6 q^{24} + 6 q^{25} + 18 q^{26} - 2 q^{27} - 6 q^{28} + 12 q^{29} - 8 q^{30} - 2 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{36} - 2 q^{37} - 6 q^{38} - 2 q^{39} + 8 q^{40} - 12 q^{41} + 6 q^{42} + 6 q^{43} + 12 q^{44} + 8 q^{46} - 8 q^{47} - 6 q^{48} + 4 q^{49} + 6 q^{50} + 34 q^{52} + 8 q^{53} - 2 q^{54} + 16 q^{55} - 18 q^{56} + 6 q^{57} + 4 q^{58} - 12 q^{59} - 16 q^{60} - 2 q^{61} + 14 q^{62} - 6 q^{63} - 14 q^{64} + 32 q^{65} - 4 q^{66} + 2 q^{67} - 24 q^{70} + 16 q^{71} + 6 q^{72} + 4 q^{73} - 18 q^{74} - 6 q^{75} - 6 q^{76} + 12 q^{77} - 18 q^{78} - 24 q^{79} + 2 q^{81} - 20 q^{82} + 8 q^{83} + 6 q^{84} + 6 q^{86} - 12 q^{87} - 4 q^{88} - 12 q^{89} + 8 q^{90} - 6 q^{91} + 16 q^{92} + 2 q^{93} - 24 q^{94} + 6 q^{96} + 10 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 −1.00000 −1.82843 −2.82843 0.414214 −3.00000 1.58579 1.00000 1.17157
1.2 2.41421 −1.00000 3.82843 2.82843 −2.41421 −3.00000 4.41421 1.00000 6.82843
\(n\): 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.g 2
3.b odd 2 1 2601.2.a.m 2
17.b even 2 1 867.2.a.h yes 2
17.c even 4 2 867.2.d.b 4
17.d even 8 2 867.2.e.b 4
17.d even 8 2 867.2.e.c 4
17.e odd 16 4 867.2.h.a 8
17.e odd 16 4 867.2.h.h 8
51.c odd 2 1 2601.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.g 2 1.a even 1 1 trivial
867.2.a.h yes 2 17.b even 2 1
867.2.d.b 4 17.c even 4 2
867.2.e.b 4 17.d even 8 2
867.2.e.c 4 17.d even 8 2
867.2.h.a 8 17.e odd 16 4
867.2.h.h 8 17.e odd 16 4
2601.2.a.m 2 3.b odd 2 1
2601.2.a.n 2 51.c odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 31 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$43$ \( (T - 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$67$ \( (T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 92 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
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