Properties

Label 1088.2.a.s
Level $1088$
Weight $2$
Character orbit 1088.a
Self dual yes
Analytic conductor $8.688$
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] = [1088,2,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, 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("1088.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(8.68772373992\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 136)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - 2 q^{5} + (\beta + 1) q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - 2 q^{5} + (\beta + 1) q^{7} + (2 \beta + 3) q^{9} + ( - \beta - 1) q^{11} + 2 \beta q^{13} + ( - 2 \beta - 2) q^{15} + q^{17} + ( - 2 \beta + 2) q^{19} + (2 \beta + 6) q^{21} + (\beta + 1) q^{23} - q^{25} + (2 \beta + 10) q^{27} - 2 q^{29} + ( - \beta - 1) q^{31} + ( - 2 \beta - 6) q^{33} + ( - 2 \beta - 2) q^{35} + ( - 4 \beta + 2) q^{37} + (2 \beta + 10) q^{39} + 2 q^{41} + (2 \beta + 6) q^{43} + ( - 4 \beta - 6) q^{45} + ( - 4 \beta + 4) q^{47} + (2 \beta - 1) q^{49} + (\beta + 1) q^{51} + 2 q^{53} + (2 \beta + 2) q^{55} - 8 q^{57} + (2 \beta - 10) q^{59} + (4 \beta + 2) q^{61} + (5 \beta + 13) q^{63} - 4 \beta q^{65} + 12 q^{67} + (2 \beta + 6) q^{69} + ( - \beta + 7) q^{71} + ( - 4 \beta + 6) q^{73} + ( - \beta - 1) q^{75} + ( - 2 \beta - 6) q^{77} + ( - 3 \beta + 5) q^{79} + (6 \beta + 11) q^{81} + ( - 2 \beta - 6) q^{83} - 2 q^{85} + ( - 2 \beta - 2) q^{87} + ( - 2 \beta - 12) q^{89} + (2 \beta + 10) q^{91} + ( - 2 \beta - 6) q^{93} + (4 \beta - 4) q^{95} + 2 q^{97} + ( - 5 \beta - 13) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{17} + 4 q^{19} + 12 q^{21} + 2 q^{23} - 2 q^{25} + 20 q^{27} - 4 q^{29} - 2 q^{31} - 12 q^{33} - 4 q^{35} + 4 q^{37} + 20 q^{39} + 4 q^{41} + 12 q^{43} - 12 q^{45} + 8 q^{47} - 2 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} - 16 q^{57} - 20 q^{59} + 4 q^{61} + 26 q^{63} + 24 q^{67} + 12 q^{69} + 14 q^{71} + 12 q^{73} - 2 q^{75} - 12 q^{77} + 10 q^{79} + 22 q^{81} - 12 q^{83} - 4 q^{85} - 4 q^{87} - 24 q^{89} + 20 q^{91} - 12 q^{93} - 8 q^{95} + 4 q^{97} - 26 q^{99}+O(q^{100}) \) 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 −1.23607 0 −2.00000 0 −1.23607 0 −1.47214 0
1.2 0 3.23607 0 −2.00000 0 3.23607 0 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)

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 1088.2.a.s 2
3.b odd 2 1 9792.2.a.db 2
4.b odd 2 1 1088.2.a.o 2
8.b even 2 1 136.2.a.c 2
8.d odd 2 1 272.2.a.f 2
12.b even 2 1 9792.2.a.da 2
24.f even 2 1 2448.2.a.u 2
24.h odd 2 1 1224.2.a.i 2
40.e odd 2 1 6800.2.a.bd 2
40.f even 2 1 3400.2.a.i 2
40.i odd 4 2 3400.2.e.f 4
56.h odd 2 1 6664.2.a.i 2
136.e odd 2 1 4624.2.a.h 2
136.h even 2 1 2312.2.a.m 2
136.i even 4 2 2312.2.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.c 2 8.b even 2 1
272.2.a.f 2 8.d odd 2 1
1088.2.a.o 2 4.b odd 2 1
1088.2.a.s 2 1.a even 1 1 trivial
1224.2.a.i 2 24.h odd 2 1
2312.2.a.m 2 136.h even 2 1
2312.2.b.g 4 136.i even 4 2
2448.2.a.u 2 24.f even 2 1
3400.2.a.i 2 40.f even 2 1
3400.2.e.f 4 40.i odd 4 2
4624.2.a.h 2 136.e odd 2 1
6664.2.a.i 2 56.h odd 2 1
6800.2.a.bd 2 40.e odd 2 1
9792.2.a.da 2 12.b even 2 1
9792.2.a.db 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1088))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 124 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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