Properties

Label 8954.2.a.j
Level $8954$
Weight $2$
Character orbit 8954.a
Self dual yes
Analytic conductor $71.498$
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] = [8954,2,Mod(1,8954)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8954, 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("8954.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8954 = 2 \cdot 11^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8954.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: \(71.4980499699\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} + \beta q^{6} + 2 \beta q^{7} - q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} + \beta q^{6} + 2 \beta q^{7} - q^{8} + (\beta - 2) q^{9} + ( - 3 \beta + 1) q^{10} - \beta q^{12} + (3 \beta - 2) q^{13} - 2 \beta q^{14} + ( - 2 \beta - 3) q^{15} + q^{16} + (4 \beta - 2) q^{17} + ( - \beta + 2) q^{18} + ( - 4 \beta + 2) q^{19} + (3 \beta - 1) q^{20} + ( - 2 \beta - 2) q^{21} + (3 \beta - 2) q^{23} + \beta q^{24} + (3 \beta + 5) q^{25} + ( - 3 \beta + 2) q^{26} + (4 \beta - 1) q^{27} + 2 \beta q^{28} + (7 \beta - 2) q^{29} + (2 \beta + 3) q^{30} + ( - \beta + 9) q^{31} - q^{32} + ( - 4 \beta + 2) q^{34} + (4 \beta + 6) q^{35} + (\beta - 2) q^{36} - q^{37} + (4 \beta - 2) q^{38} + ( - \beta - 3) q^{39} + ( - 3 \beta + 1) q^{40} + ( - \beta - 8) q^{41} + (2 \beta + 2) q^{42} + (2 \beta + 2) q^{43} + ( - 4 \beta + 5) q^{45} + ( - 3 \beta + 2) q^{46} + ( - 2 \beta + 2) q^{47} - \beta q^{48} + (4 \beta - 3) q^{49} + ( - 3 \beta - 5) q^{50} + ( - 2 \beta - 4) q^{51} + (3 \beta - 2) q^{52} + (4 \beta - 6) q^{53} + ( - 4 \beta + 1) q^{54} - 2 \beta q^{56} + (2 \beta + 4) q^{57} + ( - 7 \beta + 2) q^{58} + (2 \beta - 8) q^{59} + ( - 2 \beta - 3) q^{60} + ( - \beta - 9) q^{61} + (\beta - 9) q^{62} + ( - 2 \beta + 2) q^{63} + q^{64} + 11 q^{65} + (5 \beta - 7) q^{67} + (4 \beta - 2) q^{68} + ( - \beta - 3) q^{69} + ( - 4 \beta - 6) q^{70} + (8 \beta - 10) q^{71} + ( - \beta + 2) q^{72} + ( - 5 \beta + 1) q^{73} + q^{74} + ( - 8 \beta - 3) q^{75} + ( - 4 \beta + 2) q^{76} + (\beta + 3) q^{78} + (9 \beta - 6) q^{79} + (3 \beta - 1) q^{80} + ( - 6 \beta + 2) q^{81} + (\beta + 8) q^{82} + (4 \beta + 8) q^{83} + ( - 2 \beta - 2) q^{84} + (2 \beta + 14) q^{85} + ( - 2 \beta - 2) q^{86} + ( - 5 \beta - 7) q^{87} + (4 \beta - 8) q^{89} + (4 \beta - 5) q^{90} + (2 \beta + 6) q^{91} + (3 \beta - 2) q^{92} + ( - 8 \beta + 1) q^{93} + (2 \beta - 2) q^{94} + ( - 2 \beta - 14) q^{95} + \beta q^{96} + ( - 4 \beta + 6) q^{97} + ( - 4 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - q^{10} - q^{12} - q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} + 3 q^{18} + q^{20} - 6 q^{21} - q^{23} + q^{24} + 13 q^{25} + q^{26} + 2 q^{27} + 2 q^{28} + 3 q^{29} + 8 q^{30} + 17 q^{31} - 2 q^{32} + 16 q^{35} - 3 q^{36} - 2 q^{37} - 7 q^{39} - q^{40} - 17 q^{41} + 6 q^{42} + 6 q^{43} + 6 q^{45} + q^{46} + 2 q^{47} - q^{48} - 2 q^{49} - 13 q^{50} - 10 q^{51} - q^{52} - 8 q^{53} - 2 q^{54} - 2 q^{56} + 10 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} - 19 q^{61} - 17 q^{62} + 2 q^{63} + 2 q^{64} + 22 q^{65} - 9 q^{67} - 7 q^{69} - 16 q^{70} - 12 q^{71} + 3 q^{72} - 3 q^{73} + 2 q^{74} - 14 q^{75} + 7 q^{78} - 3 q^{79} + q^{80} - 2 q^{81} + 17 q^{82} + 20 q^{83} - 6 q^{84} + 30 q^{85} - 6 q^{86} - 19 q^{87} - 12 q^{89} - 6 q^{90} + 14 q^{91} - q^{92} - 6 q^{93} - 2 q^{94} - 30 q^{95} + q^{96} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 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
−1.00000 −1.61803 1.00000 3.85410 1.61803 3.23607 −1.00000 −0.381966 −3.85410
1.2 −1.00000 0.618034 1.00000 −2.85410 −0.618034 −1.23607 −1.00000 −2.61803 2.85410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( -1 \)
\(37\) \( +1 \)

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 8954.2.a.j 2
11.b odd 2 1 74.2.a.b 2
33.d even 2 1 666.2.a.i 2
44.c even 2 1 592.2.a.g 2
55.d odd 2 1 1850.2.a.t 2
55.e even 4 2 1850.2.b.j 4
77.b even 2 1 3626.2.a.s 2
88.b odd 2 1 2368.2.a.y 2
88.g even 2 1 2368.2.a.u 2
132.d odd 2 1 5328.2.a.bc 2
407.d odd 2 1 2738.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 11.b odd 2 1
592.2.a.g 2 44.c even 2 1
666.2.a.i 2 33.d even 2 1
1850.2.a.t 2 55.d odd 2 1
1850.2.b.j 4 55.e even 4 2
2368.2.a.u 2 88.g even 2 1
2368.2.a.y 2 88.b odd 2 1
2738.2.a.g 2 407.d odd 2 1
3626.2.a.s 2 77.b even 2 1
5328.2.a.bc 2 132.d odd 2 1
8954.2.a.j 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 20 \) Copy content Toggle raw display
$23$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$31$ \( T^{2} - 17T + 71 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} + 19T + 89 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 99 \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
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