Properties

Label 267.2.a.f
Level $267$
Weight $2$
Character orbit 267.a
Self dual yes
Analytic conductor $2.132$
Analytic rank $0$
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] = [267,2,Mod(1,267)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(267, 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("267.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 267 = 3 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 267.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: \(2.13200573397\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.23377.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 6x + 7 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} + 1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} + 1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{8} + q^{9} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{10} + ( - \beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{3} - \beta_1 + 2) q^{13} + ( - 2 \beta_{3} + \beta_1 - 1) q^{14} + ( - \beta_{2} - 1) q^{15} + (\beta_{3} - \beta_1 + 1) q^{16} + (\beta_{3} + 1) q^{17} + \beta_1 q^{18} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{19} + (\beta_{3} + \beta_{2} - \beta_1 + 7) q^{20} + (\beta_{3} - 1) q^{21} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{22} + (\beta_{2} - 2 \beta_1 + 1) q^{23} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{24} + (\beta_{3} - \beta_1 + 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{26} - q^{27} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{28} + ( - \beta_{3} - 1) q^{29} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{30} + ( - \beta_{2} - 3) q^{31} + (\beta_{2} - \beta_1 - 3) q^{32} + (\beta_{2} - \beta_1 + 2) q^{33} + (2 \beta_{3} + \beta_1 + 1) q^{34} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{2} + 2) q^{36} + (2 \beta_{3} - \beta_1 + 7) q^{37} + (\beta_{3} + 2 \beta_1 - 3) q^{38} + (\beta_{3} + \beta_1 - 2) q^{39} + (\beta_{3} + 4 \beta_1 - 3) q^{40} + ( - 3 \beta_1 + 3) q^{41} + (2 \beta_{3} - \beta_1 + 1) q^{42} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 - 9) q^{44} + (\beta_{2} + 1) q^{45} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 8) q^{46} + (3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{3} + \beta_1 - 1) q^{48} + ( - 4 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{49} + (2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{50} + ( - \beta_{3} - 1) q^{51} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{52} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{53} - \beta_1 q^{54} + ( - 2 \beta_{2} + 3 \beta_1 - 7) q^{55} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{56} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{57} + ( - 2 \beta_{3} - \beta_1 - 1) q^{58} + ( - 4 \beta_{3} + 5 \beta_{2} - \beta_1 + 2) q^{59} + ( - \beta_{3} - \beta_{2} + \beta_1 - 7) q^{60} + (4 \beta_{3} - 3 \beta_{2} + \beta_1 + 6) q^{61} + ( - \beta_{3} + \beta_{2} - 4 \beta_1) q^{62} + ( - \beta_{3} + 1) q^{63} + ( - \beta_{3} - 2 \beta_{2} - 6) q^{64} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{65} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{66} + (\beta_{3} + \beta_1) q^{67} + (2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{68} + ( - \beta_{2} + 2 \beta_1 - 1) q^{69} + ( - \beta_{3} - 2 \beta_{2} - 5) q^{70} + ( - 2 \beta_{2} - 3 \beta_1 - 5) q^{71} + (\beta_{3} - \beta_{2} + \beta_1) q^{72} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 5) q^{73} + (4 \beta_{3} - \beta_{2} + 7 \beta_1 - 2) q^{74} + ( - \beta_{3} + \beta_1 - 1) q^{75} + (4 \beta_{2} - \beta_1 + 3) q^{76} + ( - \beta_{2} + 2 \beta_1 - 1) q^{77} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{78} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{79} + (2 \beta_{2} - \beta_1 + 3) q^{80} + q^{81} + ( - 3 \beta_{2} + 3 \beta_1 - 12) q^{82} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{83} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{84} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{85} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 9) q^{86} + (\beta_{3} + 1) q^{87} + ( - \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 8) q^{88} + q^{89} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{90} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{91} + ( - \beta_{3} + 3 \beta_{2} - 7 \beta_1 + 7) q^{92} + (\beta_{2} + 3) q^{93} + (5 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 7) q^{94} + ( - \beta_{3} + 5 \beta_{2}) q^{95} + ( - \beta_{2} + \beta_1 + 3) q^{96} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{97} + ( - 7 \beta_{3} + 2 \beta_{2} + 8) q^{98} + ( - \beta_{2} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 6 q^{7} + 4 q^{9} + q^{10} - 6 q^{11} - 7 q^{12} + 9 q^{13} + q^{14} - 3 q^{15} + q^{16} + 2 q^{17} + q^{18} + 10 q^{19} + 24 q^{20} - 6 q^{21} + 13 q^{22} + q^{23} + q^{25} - 13 q^{26} - 4 q^{27} + 2 q^{28} - 2 q^{29} - q^{30} - 11 q^{31} - 14 q^{32} + 6 q^{33} + q^{34} - 4 q^{35} + 7 q^{36} + 23 q^{37} - 12 q^{38} - 9 q^{39} - 10 q^{40} + 9 q^{41} - q^{42} + q^{43} - 29 q^{44} + 3 q^{45} - 29 q^{46} - 8 q^{47} - q^{48} + 6 q^{49} - 14 q^{50} - 2 q^{51} + 7 q^{52} + 7 q^{53} - q^{54} - 23 q^{55} - 17 q^{56} - 10 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 20 q^{61} - 3 q^{62} + 6 q^{63} - 20 q^{64} - 2 q^{65} - 13 q^{66} - q^{67} + 12 q^{68} - q^{69} - 16 q^{70} - 21 q^{71} + 24 q^{73} - 8 q^{74} - q^{75} + 7 q^{76} - q^{77} + 13 q^{78} - q^{79} + 9 q^{80} + 4 q^{81} - 42 q^{82} + 10 q^{83} - 2 q^{84} + 10 q^{85} - 30 q^{86} + 2 q^{87} + 25 q^{88} + 4 q^{89} + q^{90} + 39 q^{91} + 20 q^{92} + 11 q^{93} + 14 q^{94} - 3 q^{95} + 14 q^{96} + 2 q^{97} + 44 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 7x^{2} + 6x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 5\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 5\beta_1 \) 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
−2.42130
−0.696379
1.76330
2.35438
−2.42130 −1.00000 3.86271 2.86271 2.42130 1.22618 −4.51019 1.00000 −6.93150
1.2 −0.696379 −1.00000 −1.51506 −2.51506 0.696379 1.37087 2.44781 1.00000 1.75143
1.3 1.76330 −1.00000 1.10923 0.109229 −1.76330 5.22477 −1.57070 1.00000 0.192604
1.4 2.35438 −1.00000 3.54311 2.54311 −2.35438 −1.82181 3.63308 1.00000 5.98746
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(89\) \(-1\)

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 267.2.a.f 4
3.b odd 2 1 801.2.a.h 4
4.b odd 2 1 4272.2.a.u 4
5.b even 2 1 6675.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
267.2.a.f 4 1.a even 1 1 trivial
801.2.a.h 4 3.b odd 2 1
4272.2.a.u 4 4.b odd 2 1
6675.2.a.q 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(267))\):

\( T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{4} - 3T_{5}^{3} - 6T_{5}^{2} + 19T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 7 T^{2} + 6 T + 7 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} - 6 T^{2} + 19 T - 2 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + T^{2} + 19 T - 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} - 3 T^{2} - 7 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} + 10 T^{2} + 91 T - 202 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 11 T^{2} + 17 T - 6 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 17 T^{2} + 45 T - 92 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{3} - 38 T^{2} - 97 T - 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 11 T^{2} - 17 T - 6 \) Copy content Toggle raw display
$31$ \( T^{4} + 11 T^{3} + 36 T^{2} + 25 T - 24 \) Copy content Toggle raw display
$37$ \( T^{4} - 23 T^{3} + 142 T^{2} + \cdots - 2062 \) Copy content Toggle raw display
$41$ \( T^{4} - 9 T^{3} - 36 T^{2} + 189 T + 486 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} - 61 T^{2} + 249 T - 244 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 81 T^{2} - 491 T + 384 \) Copy content Toggle raw display
$53$ \( T^{4} - 7 T^{3} - 22 T^{2} - T + 6 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} - 235 T^{2} + \cdots + 9764 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} - 41 T^{2} + \cdots + 4062 \) Copy content Toggle raw display
$67$ \( T^{4} + T^{3} - 20 T^{2} - 55 T - 36 \) Copy content Toggle raw display
$71$ \( T^{4} + 21 T^{3} + 60 T^{2} + \cdots - 776 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 65 T^{2} + \cdots + 694 \) Copy content Toggle raw display
$79$ \( T^{4} + T^{3} - 130 T^{2} - 443 T + 1248 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} - T^{2} + 35 T + 12 \) Copy content Toggle raw display
$89$ \( (T - 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} - 93 T^{2} + 233 T - 126 \) Copy content Toggle raw display
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