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Properties

Label 572.2.n.b
Level $572$
Weight $2$
Character orbit 572.n
Analytic conductor $4.567$
Analytic rank $0$
Dimension $28$
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] = [572,2,Mod(53,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.n (of order \(5\), degree \(4\), 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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{3} - 7 q^{5} + 11 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{3} - 7 q^{5} + 11 q^{7} - 22 q^{9} + q^{11} + 7 q^{13} - 24 q^{15} + 7 q^{17} - 7 q^{19} - 12 q^{21} + 34 q^{23} - 26 q^{25} - 11 q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} - 15 q^{37} + 4 q^{39} + 29 q^{41} - 8 q^{43} + 62 q^{45} - 27 q^{47} - 4 q^{49} + 69 q^{51} - 32 q^{53} + 19 q^{55} + 9 q^{57} - 37 q^{59} - 19 q^{61} + 46 q^{63} - 18 q^{65} + 10 q^{67} - 32 q^{69} - 27 q^{71} - 79 q^{75} + 13 q^{77} + 21 q^{79} - 18 q^{81} + 27 q^{83} + 7 q^{85} + 56 q^{87} + 82 q^{89} - 11 q^{91} - 53 q^{93} + 51 q^{95} - 88 q^{97} + 86 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 0 −2.46647 + 1.79199i 0 0.518170 + 1.59476i 0 −2.24130 − 1.62840i 0 1.94517 − 5.98661i 0
53.2 0 −1.94312 + 1.41176i 0 0.879947 + 2.70820i 0 3.92186 + 2.84939i 0 0.855592 − 2.63324i 0
53.3 0 −0.634331 + 0.460868i 0 −0.799372 − 2.46021i 0 0.289986 + 0.210687i 0 −0.737075 + 2.26848i 0
53.4 0 −0.0637055 + 0.0462848i 0 −0.454508 − 1.39883i 0 1.50470 + 1.09323i 0 −0.925135 + 2.84727i 0
53.5 0 0.178116 − 0.129409i 0 1.35047 + 4.15632i 0 −3.13912 − 2.28071i 0 −0.912072 + 2.80707i 0
53.6 0 2.12584 − 1.54451i 0 0.674037 + 2.07447i 0 1.61142 + 1.17077i 0 1.20663 − 3.71362i 0
53.7 0 2.49464 − 1.81246i 0 −1.12366 − 3.45827i 0 1.36148 + 0.989171i 0 2.01117 − 6.18974i 0
157.1 0 −1.05349 − 3.24229i 0 −2.42026 − 1.75842i 0 1.07034 − 3.29416i 0 −6.97559 + 5.06806i 0
157.2 0 −0.527342 − 1.62299i 0 1.55733 + 1.13147i 0 −0.0663952 + 0.204343i 0 0.0710371 − 0.0516114i 0
157.3 0 −0.344293 − 1.05962i 0 −0.665019 − 0.483164i 0 0.816463 − 2.51281i 0 1.42278 − 1.03371i 0
157.4 0 0.288033 + 0.886474i 0 −2.56853 − 1.86615i 0 −0.780072 + 2.40081i 0 1.72418 − 1.25269i 0
157.5 0 0.525215 + 1.61645i 0 1.11280 + 0.808497i 0 1.47468 − 4.53859i 0 0.0900014 − 0.0653899i 0
157.6 0 0.955130 + 2.93959i 0 −3.52306 − 2.55966i 0 0.357931 − 1.10160i 0 −5.30186 + 3.85203i 0
157.7 0 0.965759 + 2.97230i 0 1.96166 + 1.42523i 0 −0.681960 + 2.09886i 0 −5.47482 + 3.97769i 0
313.1 0 −2.46647 − 1.79199i 0 0.518170 − 1.59476i 0 −2.24130 + 1.62840i 0 1.94517 + 5.98661i 0
313.2 0 −1.94312 − 1.41176i 0 0.879947 − 2.70820i 0 3.92186 − 2.84939i 0 0.855592 + 2.63324i 0
313.3 0 −0.634331 − 0.460868i 0 −0.799372 + 2.46021i 0 0.289986 − 0.210687i 0 −0.737075 − 2.26848i 0
313.4 0 −0.0637055 − 0.0462848i 0 −0.454508 + 1.39883i 0 1.50470 − 1.09323i 0 −0.925135 − 2.84727i 0
313.5 0 0.178116 + 0.129409i 0 1.35047 − 4.15632i 0 −3.13912 + 2.28071i 0 −0.912072 − 2.80707i 0
313.6 0 2.12584 + 1.54451i 0 0.674037 − 2.07447i 0 1.61142 − 1.17077i 0 1.20663 + 3.71362i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.n.b ✓ 28
11.c even 5 1 inner 572.2.n.b ✓ 28
11.c even 5 1 6292.2.a.z 14
11.d odd 10 1 6292.2.a.y 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.n.b ✓ 28 1.a even 1 1 trivial
572.2.n.b ✓ 28 11.c even 5 1 inner
6292.2.a.y 14 11.d odd 10 1
6292.2.a.z 14 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - T_{3}^{27} + 22 T_{3}^{26} - 10 T_{3}^{25} + 282 T_{3}^{24} - 53 T_{3}^{23} + 2986 T_{3}^{22} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\). Copy content Toggle raw display