Properties

Label 287.2.a.a
Level $287$
Weight $2$
Character orbit 287.a
Self dual yes
Analytic conductor $2.292$
Analytic rank $1$
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] = [287,2,Mod(1,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, 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("287.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.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: \(2.29170653801\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{5} + (2 \beta + 1) q^{6} + q^{7} + (2 \beta - 1) q^{8} + (3 \beta - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{5} + (2 \beta + 1) q^{6} + q^{7} + (2 \beta - 1) q^{8} + (3 \beta - 1) q^{9} - q^{10} + (2 \beta - 1) q^{11} - \beta q^{12} - 3 q^{13} - \beta q^{14} - \beta q^{15} - 3 \beta q^{16} + ( - 4 \beta - 1) q^{17} + ( - 2 \beta - 3) q^{18} + (3 \beta - 3) q^{19} + ( - \beta + 2) q^{20} + ( - \beta - 1) q^{21} + ( - \beta - 2) q^{22} + (\beta + 1) q^{23} + ( - 3 \beta - 1) q^{24} + ( - \beta - 3) q^{25} + 3 \beta q^{26} + ( - 2 \beta + 1) q^{27} + (\beta - 1) q^{28} + ( - 3 \beta + 6) q^{29} + (\beta + 1) q^{30} + (\beta - 9) q^{31} + ( - \beta + 5) q^{32} + ( - 3 \beta - 1) q^{33} + (5 \beta + 4) q^{34} + (\beta - 1) q^{35} + ( - \beta + 4) q^{36} + ( - 6 \beta + 4) q^{37} - 3 q^{38} + (3 \beta + 3) q^{39} + ( - \beta + 3) q^{40} + q^{41} + (2 \beta + 1) q^{42} + ( - 8 \beta + 3) q^{43} + ( - \beta + 3) q^{44} + ( - \beta + 4) q^{45} + ( - 2 \beta - 1) q^{46} + (2 \beta - 2) q^{47} + (6 \beta + 3) q^{48} + q^{49} + (4 \beta + 1) q^{50} + (9 \beta + 5) q^{51} + ( - 3 \beta + 3) q^{52} + (5 \beta - 11) q^{53} + (\beta + 2) q^{54} + ( - \beta + 3) q^{55} + (2 \beta - 1) q^{56} - 3 \beta q^{57} + ( - 3 \beta + 3) q^{58} + ( - \beta + 7) q^{59} - q^{60} - 11 q^{61} + (8 \beta - 1) q^{62} + (3 \beta - 1) q^{63} + (2 \beta + 1) q^{64} + ( - 3 \beta + 3) q^{65} + (4 \beta + 3) q^{66} + (\beta + 5) q^{67} + ( - \beta - 3) q^{68} + ( - 3 \beta - 2) q^{69} - q^{70} + ( - 2 \beta - 1) q^{71} + (\beta + 7) q^{72} - 15 q^{73} + (2 \beta + 6) q^{74} + (5 \beta + 4) q^{75} + ( - 3 \beta + 6) q^{76} + (2 \beta - 1) q^{77} + ( - 6 \beta - 3) q^{78} + (6 \beta - 4) q^{79} - 3 q^{80} + ( - 6 \beta + 4) q^{81} - \beta q^{82} + ( - 8 \beta + 7) q^{83} - \beta q^{84} + ( - \beta - 3) q^{85} + (5 \beta + 8) q^{86} - 3 q^{87} + 5 q^{88} + (9 \beta - 9) q^{89} + ( - 3 \beta + 1) q^{90} - 3 q^{91} + \beta q^{92} + (7 \beta + 8) q^{93} - 2 q^{94} + ( - 3 \beta + 6) q^{95} + ( - 3 \beta - 4) q^{96} + ( - 3 \beta - 7) q^{97} - \beta q^{98} + (\beta + 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + q^{9} - 2 q^{10} - q^{12} - 6 q^{13} - q^{14} - q^{15} - 3 q^{16} - 6 q^{17} - 8 q^{18} - 3 q^{19} + 3 q^{20} - 3 q^{21} - 5 q^{22} + 3 q^{23} - 5 q^{24} - 7 q^{25} + 3 q^{26} - q^{28} + 9 q^{29} + 3 q^{30} - 17 q^{31} + 9 q^{32} - 5 q^{33} + 13 q^{34} - q^{35} + 7 q^{36} + 2 q^{37} - 6 q^{38} + 9 q^{39} + 5 q^{40} + 2 q^{41} + 4 q^{42} - 2 q^{43} + 5 q^{44} + 7 q^{45} - 4 q^{46} - 2 q^{47} + 12 q^{48} + 2 q^{49} + 6 q^{50} + 19 q^{51} + 3 q^{52} - 17 q^{53} + 5 q^{54} + 5 q^{55} - 3 q^{57} + 3 q^{58} + 13 q^{59} - 2 q^{60} - 22 q^{61} + 6 q^{62} + q^{63} + 4 q^{64} + 3 q^{65} + 10 q^{66} + 11 q^{67} - 7 q^{68} - 7 q^{69} - 2 q^{70} - 4 q^{71} + 15 q^{72} - 30 q^{73} + 14 q^{74} + 13 q^{75} + 9 q^{76} - 12 q^{78} - 2 q^{79} - 6 q^{80} + 2 q^{81} - q^{82} + 6 q^{83} - q^{84} - 7 q^{85} + 21 q^{86} - 6 q^{87} + 10 q^{88} - 9 q^{89} - q^{90} - 6 q^{91} + q^{92} + 23 q^{93} - 4 q^{94} + 9 q^{95} - 11 q^{96} - 17 q^{97} - q^{98} + 15 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−1.61803 −2.61803 0.618034 0.618034 4.23607 1.00000 2.23607 3.85410 −1.00000
1.2 0.618034 −0.381966 −1.61803 −1.61803 −0.236068 1.00000 −2.23607 −2.85410 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(41\) \(-1\)

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 287.2.a.a 2
3.b odd 2 1 2583.2.a.h 2
4.b odd 2 1 4592.2.a.p 2
5.b even 2 1 7175.2.a.h 2
7.b odd 2 1 2009.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.a 2 1.a even 1 1 trivial
2009.2.a.b 2 7.b odd 2 1
2583.2.a.h 2 3.b odd 2 1
4592.2.a.p 2 4.b odd 2 1
7175.2.a.h 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(287))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$41$ \( (T - 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 79 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 17T + 41 \) Copy content Toggle raw display
$59$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$61$ \( (T + 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 11T + 29 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$73$ \( (T + 15)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 71 \) Copy content Toggle raw display
$89$ \( T^{2} + 9T - 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 61 \) Copy content Toggle raw display
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