Properties

Label 4263.2.a.j
Level $4263$
Weight $2$
Character orbit 4263.a
Self dual yes
Analytic conductor $34.040$
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] = [4263,2,Mod(1,4263)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4263, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4263.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4263 = 3 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4263.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: \(34.0402263817\)
Analytic rank: \(0\)
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: no (minimal twist has level 87)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} + (2 \beta - 2) q^{5} - \beta q^{6} + ( - 2 \beta + 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} + (2 \beta - 2) q^{5} - \beta q^{6} + ( - 2 \beta + 1) q^{8} + q^{9} + 2 q^{10} + (2 \beta + 1) q^{11} + ( - \beta + 1) q^{12} + ( - 4 \beta + 3) q^{13} + ( - 2 \beta + 2) q^{15} - 3 \beta q^{16} - 3 q^{17} + \beta q^{18} + ( - 2 \beta + 6) q^{19} + ( - 2 \beta + 4) q^{20} + (3 \beta + 2) q^{22} + (6 \beta - 4) q^{23} + (2 \beta - 1) q^{24} + ( - 4 \beta + 3) q^{25} + ( - \beta - 4) q^{26} - q^{27} - q^{29} - 2 q^{30} + 6 \beta q^{31} + (\beta - 5) q^{32} + ( - 2 \beta - 1) q^{33} - 3 \beta q^{34} + (\beta - 1) q^{36} + ( - 2 \beta + 4) q^{37} + (4 \beta - 2) q^{38} + (4 \beta - 3) q^{39} + (2 \beta - 6) q^{40} - 2 q^{41} + 4 q^{43} + (\beta + 1) q^{44} + (2 \beta - 2) q^{45} + (2 \beta + 6) q^{46} + (6 \beta - 1) q^{47} + 3 \beta q^{48} + ( - \beta - 4) q^{50} + 3 q^{51} + (3 \beta - 7) q^{52} + ( - 2 \beta + 10) q^{53} - \beta q^{54} + (2 \beta + 2) q^{55} + (2 \beta - 6) q^{57} - \beta q^{58} + (4 \beta - 2) q^{59} + (2 \beta - 4) q^{60} + ( - 2 \beta + 4) q^{61} + (6 \beta + 6) q^{62} + (2 \beta + 1) q^{64} + (6 \beta - 14) q^{65} + ( - 3 \beta - 2) q^{66} + (10 \beta - 7) q^{67} + ( - 3 \beta + 3) q^{68} + ( - 6 \beta + 4) q^{69} + (2 \beta - 4) q^{71} + ( - 2 \beta + 1) q^{72} + ( - 2 \beta - 8) q^{73} + (2 \beta - 2) q^{74} + (4 \beta - 3) q^{75} + (6 \beta - 8) q^{76} + (\beta + 4) q^{78} + ( - 2 \beta - 14) q^{79} - 6 q^{80} + q^{81} - 2 \beta q^{82} + (8 \beta + 2) q^{83} + ( - 6 \beta + 6) q^{85} + 4 \beta q^{86} + q^{87} + ( - 4 \beta - 3) q^{88} - 5 q^{89} + 2 q^{90} + ( - 4 \beta + 10) q^{92} - 6 \beta q^{93} + (5 \beta + 6) q^{94} + (12 \beta - 16) q^{95} + ( - \beta + 5) q^{96} + (14 \beta - 10) q^{97} + (2 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9} + 4 q^{10} + 4 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} - 6 q^{17} + q^{18} + 10 q^{19} + 6 q^{20} + 7 q^{22} - 2 q^{23} + 2 q^{25} - 9 q^{26} - 2 q^{27} - 2 q^{29} - 4 q^{30} + 6 q^{31} - 9 q^{32} - 4 q^{33} - 3 q^{34} - q^{36} + 6 q^{37} - 2 q^{39} - 10 q^{40} - 4 q^{41} + 8 q^{43} + 3 q^{44} - 2 q^{45} + 14 q^{46} + 4 q^{47} + 3 q^{48} - 9 q^{50} + 6 q^{51} - 11 q^{52} + 18 q^{53} - q^{54} + 6 q^{55} - 10 q^{57} - q^{58} - 6 q^{60} + 6 q^{61} + 18 q^{62} + 4 q^{64} - 22 q^{65} - 7 q^{66} - 4 q^{67} + 3 q^{68} + 2 q^{69} - 6 q^{71} - 18 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} + 9 q^{78} - 30 q^{79} - 12 q^{80} + 2 q^{81} - 2 q^{82} + 12 q^{83} + 6 q^{85} + 4 q^{86} + 2 q^{87} - 10 q^{88} - 10 q^{89} + 4 q^{90} + 16 q^{92} - 6 q^{93} + 17 q^{94} - 20 q^{95} + 9 q^{96} - 6 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 −1.00000 −1.61803 −3.23607 0.618034 0 2.23607 1.00000 2.00000
1.2 1.61803 −1.00000 0.618034 1.23607 −1.61803 0 −2.23607 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(29\) \(1\)

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 4263.2.a.j 2
7.b odd 2 1 87.2.a.a 2
21.c even 2 1 261.2.a.b 2
28.d even 2 1 1392.2.a.q 2
35.c odd 2 1 2175.2.a.l 2
35.f even 4 2 2175.2.c.k 4
56.e even 2 1 5568.2.a.bs 2
56.h odd 2 1 5568.2.a.bl 2
84.h odd 2 1 4176.2.a.bn 2
105.g even 2 1 6525.2.a.ba 2
203.c odd 2 1 2523.2.a.c 2
609.h even 2 1 7569.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.a.a 2 7.b odd 2 1
261.2.a.b 2 21.c even 2 1
1392.2.a.q 2 28.d even 2 1
2175.2.a.l 2 35.c odd 2 1
2175.2.c.k 4 35.f even 4 2
2523.2.a.c 2 203.c odd 2 1
4176.2.a.bn 2 84.h odd 2 1
4263.2.a.j 2 1.a even 1 1 trivial
5568.2.a.bl 2 56.h odd 2 1
5568.2.a.bs 2 56.e even 2 1
6525.2.a.ba 2 105.g even 2 1
7569.2.a.k 2 609.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4263))\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} - 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$79$ \( T^{2} + 30T + 220 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$89$ \( (T + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 236 \) Copy content Toggle raw display
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