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Properties

Label 1088.2.s.a
Level $1088$
Weight $2$
Character orbit 1088.s
Analytic conductor $8.688$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1088,2,Mod(625,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1088.625");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.s (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 272)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{3} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{3} + 60 q^{9} + 4 q^{11} - 4 q^{13} - 4 q^{17} - 4 q^{21} - 52 q^{25} + 16 q^{27} - 4 q^{29} + 4 q^{31} - 8 q^{33} + 4 q^{35} + 12 q^{39} + 48 q^{47} - 4 q^{51} - 12 q^{57} + 32 q^{59} + 32 q^{63} + 4 q^{65} + 4 q^{67} + 28 q^{69} + 8 q^{73} - 4 q^{75} - 28 q^{77} - 12 q^{79} + 28 q^{81} + 24 q^{85} - 24 q^{87} - 40 q^{91} - 12 q^{93} + 4 q^{95} - 4 q^{97} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
625.1 0 −3.18433 0 − 3.46237i 0 0.548963 + 0.548963i 0 7.13997 0
625.2 0 −3.03209 0 1.47873i 0 3.44575 + 3.44575i 0 6.19356 0
625.3 0 −2.98090 0 0.726002i 0 −0.344548 − 0.344548i 0 5.88576 0
625.4 0 −2.55478 0 3.78051i 0 −1.48991 − 1.48991i 0 3.52692 0
625.5 0 −2.16143 0 1.75971i 0 0.943570 + 0.943570i 0 1.67179 0
625.6 0 −2.05711 0 − 2.40124i 0 −2.53680 − 2.53680i 0 1.23170 0
625.7 0 −2.01369 0 − 2.90977i 0 3.02165 + 3.02165i 0 1.05494 0
625.8 0 −1.97996 0 − 3.05104i 0 −2.13135 − 2.13135i 0 0.920251 0
625.9 0 −1.71343 0 − 1.98068i 0 −2.55435 − 2.55435i 0 −0.0641544 0
625.10 0 −1.70026 0 2.90401i 0 −0.453147 − 0.453147i 0 −0.109126 0
625.11 0 −1.68708 0 0.660632i 0 0.710367 + 0.710367i 0 −0.153763 0
625.12 0 −1.53523 0 − 1.14091i 0 0.381270 + 0.381270i 0 −0.643071 0
625.13 0 −0.572337 0 − 1.54731i 0 2.11387 + 2.11387i 0 −2.67243 0
625.14 0 −0.378959 0 3.16725i 0 −0.346423 − 0.346423i 0 −2.85639 0
625.15 0 −0.342255 0 1.55667i 0 −3.34124 − 3.34124i 0 −2.88286 0
625.16 0 −0.304704 0 0.222425i 0 −0.942993 − 0.942993i 0 −2.90716 0
625.17 0 −0.00542376 0 2.38432i 0 2.50734 + 2.50734i 0 −2.99997 0
625.18 0 0.410909 0 − 1.10087i 0 1.57400 + 1.57400i 0 −2.83115 0
625.19 0 0.583850 0 − 2.97401i 0 1.20406 + 1.20406i 0 −2.65912 0
625.20 0 0.696614 0 2.41672i 0 −3.18472 − 3.18472i 0 −2.51473 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 625.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.s.a 68
4.b odd 2 1 272.2.s.a yes 68
16.e even 4 1 1088.2.j.a 68
16.f odd 4 1 272.2.j.a ✓ 68
17.c even 4 1 1088.2.j.a 68
68.f odd 4 1 272.2.j.a ✓ 68
272.i odd 4 1 272.2.s.a yes 68
272.s even 4 1 inner 1088.2.s.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.2.j.a ✓ 68 16.f odd 4 1
272.2.j.a ✓ 68 68.f odd 4 1
272.2.s.a yes 68 4.b odd 2 1
272.2.s.a yes 68 272.i odd 4 1
1088.2.j.a 68 16.e even 4 1
1088.2.j.a 68 17.c even 4 1
1088.2.s.a 68 1.a even 1 1 trivial
1088.2.s.a 68 272.s even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1088, [\chi])\).