Properties

Label 8954.2.a.p
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: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta + 1) q^{3} + q^{4} - \beta q^{5} + (\beta + 1) q^{6} + (2 \beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta + 1) q^{3} + q^{4} - \beta q^{5} + (\beta + 1) q^{6} + (2 \beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9} - \beta q^{10} + (\beta + 1) q^{12} + ( - \beta + 1) q^{13} + (2 \beta - 2) q^{14} + ( - 2 \beta - 3) q^{15} + q^{16} + 6 q^{17} + (3 \beta + 1) q^{18} - 2 q^{19} - \beta q^{20} + (2 \beta + 4) q^{21} + (3 \beta - 3) q^{23} + (\beta + 1) q^{24} + (\beta - 2) q^{25} + ( - \beta + 1) q^{26} + (4 \beta + 7) q^{27} + (2 \beta - 2) q^{28} + (3 \beta - 3) q^{29} + ( - 2 \beta - 3) q^{30} + ( - \beta + 2) q^{31} + q^{32} + 6 q^{34} - 6 q^{35} + (3 \beta + 1) q^{36} + q^{37} - 2 q^{38} + ( - \beta - 2) q^{39} - \beta q^{40} + ( - 3 \beta - 3) q^{41} + (2 \beta + 4) q^{42} + ( - 2 \beta + 4) q^{43} + ( - 4 \beta - 9) q^{45} + (3 \beta - 3) q^{46} + 2 \beta q^{47} + (\beta + 1) q^{48} + ( - 4 \beta + 9) q^{49} + (\beta - 2) q^{50} + (6 \beta + 6) q^{51} + ( - \beta + 1) q^{52} - 6 q^{53} + (4 \beta + 7) q^{54} + (2 \beta - 2) q^{56} + ( - 2 \beta - 2) q^{57} + (3 \beta - 3) q^{58} + (2 \beta + 6) q^{59} + ( - 2 \beta - 3) q^{60} + ( - 5 \beta + 4) q^{61} + ( - \beta + 2) q^{62} + (2 \beta + 16) q^{63} + q^{64} + 3 q^{65} + ( - 5 \beta + 8) q^{67} + 6 q^{68} + (3 \beta + 6) q^{69} - 6 q^{70} + 6 q^{71} + (3 \beta + 1) q^{72} + (\beta + 10) q^{73} + q^{74} + q^{75} - 2 q^{76} + ( - \beta - 2) q^{78} + ( - 7 \beta + 7) q^{79} - \beta q^{80} + (6 \beta + 16) q^{81} + ( - 3 \beta - 3) q^{82} + (4 \beta - 12) q^{83} + (2 \beta + 4) q^{84} - 6 \beta q^{85} + ( - 2 \beta + 4) q^{86} + (3 \beta + 6) q^{87} - 4 \beta q^{89} + ( - 4 \beta - 9) q^{90} + (2 \beta - 8) q^{91} + (3 \beta - 3) q^{92} - q^{93} + 2 \beta q^{94} + 2 \beta q^{95} + (\beta + 1) q^{96} + ( - 8 \beta + 2) q^{97} + ( - 4 \beta + 9) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{10} + 3 q^{12} + q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} - q^{20} + 10 q^{21} - 3 q^{23} + 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} - 2 q^{28} - 3 q^{29} - 8 q^{30} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 5 q^{36} + 2 q^{37} - 4 q^{38} - 5 q^{39} - q^{40} - 9 q^{41} + 10 q^{42} + 6 q^{43} - 22 q^{45} - 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} - 3 q^{50} + 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 2 q^{56} - 6 q^{57} - 3 q^{58} + 14 q^{59} - 8 q^{60} + 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} + 15 q^{69} - 12 q^{70} + 12 q^{71} + 5 q^{72} + 21 q^{73} + 2 q^{74} + 2 q^{75} - 4 q^{76} - 5 q^{78} + 7 q^{79} - q^{80} + 38 q^{81} - 9 q^{82} - 20 q^{83} + 10 q^{84} - 6 q^{85} + 6 q^{86} + 15 q^{87} - 4 q^{89} - 22 q^{90} - 14 q^{91} - 3 q^{92} - 2 q^{93} + 2 q^{94} + 2 q^{95} + 3 q^{96} - 4 q^{97} + 14 q^{98}+O(q^{100}) \) 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
−1.30278
2.30278
1.00000 −0.302776 1.00000 1.30278 −0.302776 −4.60555 1.00000 −2.90833 1.30278
1.2 1.00000 3.30278 1.00000 −2.30278 3.30278 2.60555 1.00000 7.90833 −2.30278
\(n\): 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.p 2
11.b odd 2 1 74.2.a.a 2
33.d even 2 1 666.2.a.j 2
44.c even 2 1 592.2.a.f 2
55.d odd 2 1 1850.2.a.u 2
55.e even 4 2 1850.2.b.i 4
77.b even 2 1 3626.2.a.a 2
88.b odd 2 1 2368.2.a.s 2
88.g even 2 1 2368.2.a.ba 2
132.d odd 2 1 5328.2.a.bf 2
407.d odd 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 11.b odd 2 1
592.2.a.f 2 44.c even 2 1
666.2.a.j 2 33.d even 2 1
1850.2.a.u 2 55.d odd 2 1
1850.2.b.i 4 55.e even 4 2
2368.2.a.s 2 88.b odd 2 1
2368.2.a.ba 2 88.g even 2 1
2738.2.a.l 2 407.d odd 2 1
3626.2.a.a 2 77.b even 2 1
5328.2.a.bf 2 132.d odd 2 1
8954.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$31$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T - 79 \) Copy content Toggle raw display
$67$ \( T^{2} - 11T - 51 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 21T + 107 \) Copy content Toggle raw display
$79$ \( T^{2} - 7T - 147 \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 48 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 204 \) Copy content Toggle raw display
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