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Properties

Label 572.2.bv.a
Level $572$
Weight $2$
Character orbit 572.bv
Analytic conductor $4.567$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [572,2,Mod(41,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 18, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.bv (of order \(60\), degree \(16\), 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: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{9} + 16 q^{11} + 10 q^{13} - 28 q^{15} - 48 q^{23} + 24 q^{27} + 20 q^{29} + 4 q^{31} + 60 q^{33} + 50 q^{35} + 12 q^{37} - 40 q^{39} + 20 q^{41} + 64 q^{45} - 62 q^{47} + 100 q^{53} - 22 q^{55} + 12 q^{59} - 40 q^{61} - 80 q^{63} - 44 q^{67} - 152 q^{71} + 30 q^{73} - 120 q^{75} + 80 q^{79} + 72 q^{81} + 90 q^{83} - 40 q^{85} - 8 q^{89} - 36 q^{91} - 90 q^{93} - 42 q^{97} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −2.12932 + 2.36485i 0 1.33109 − 0.210824i 0 −1.93450 − 0.101383i 0 −0.744921 − 7.08745i 0
41.2 0 −1.60983 + 1.78790i 0 −1.34020 + 0.212267i 0 3.84208 + 0.201355i 0 −0.291440 − 2.77287i 0
41.3 0 −1.57791 + 1.75245i 0 −0.871666 + 0.138058i 0 1.64722 + 0.0863272i 0 −0.267683 − 2.54684i 0
41.4 0 −1.00688 + 1.11826i 0 3.41629 − 0.541087i 0 −1.08459 − 0.0568410i 0 0.0769002 + 0.731656i 0
41.5 0 −0.765371 + 0.850031i 0 −3.62842 + 0.574686i 0 −0.942248 − 0.0493811i 0 0.176826 + 1.68239i 0
41.6 0 −0.389674 + 0.432777i 0 3.67270 − 0.581698i 0 4.87664 + 0.255574i 0 0.278135 + 2.64628i 0
41.7 0 −0.170530 + 0.189393i 0 −0.481251 + 0.0762227i 0 −0.670222 − 0.0351249i 0 0.306796 + 2.91897i 0
41.8 0 0.183828 − 0.204162i 0 2.17308 − 0.344183i 0 −2.74555 − 0.143888i 0 0.305696 + 2.90850i 0
41.9 0 0.555859 − 0.617343i 0 −3.44415 + 0.545500i 0 1.95070 + 0.102232i 0 0.241451 + 2.29725i 0
41.10 0 0.632506 − 0.702469i 0 −1.21053 + 0.191729i 0 −4.38981 − 0.230060i 0 0.220187 + 2.09493i 0
41.11 0 1.13877 − 1.26473i 0 −0.297595 + 0.0471344i 0 3.35932 + 0.176054i 0 0.0108356 + 0.103094i 0
41.12 0 1.43269 − 1.59116i 0 −0.186319 + 0.0295100i 0 0.938903 + 0.0492058i 0 −0.165612 − 1.57570i 0
41.13 0 1.52983 − 1.69905i 0 3.46848 − 0.549353i 0 −0.174489 − 0.00914460i 0 −0.232798 − 2.21493i 0
41.14 0 2.17604 − 2.41674i 0 −2.60150 + 0.412038i 0 −4.67346 − 0.244925i 0 −0.791886 − 7.53429i 0
85.1 0 −3.18098 − 0.676138i 0 −0.412038 − 2.60150i 0 −3.92487 − 2.54884i 0 6.92083 + 3.08135i 0
85.2 0 −2.23633 − 0.475347i 0 0.549353 + 3.46848i 0 −0.146540 − 0.0951642i 0 2.03458 + 0.905855i 0
85.3 0 −2.09433 − 0.445163i 0 −0.0295100 − 0.186319i 0 0.788511 + 0.512065i 0 1.44740 + 0.644424i 0
85.4 0 −1.66467 − 0.353837i 0 −0.0471344 − 0.297595i 0 2.82123 + 1.83213i 0 −0.0946995 − 0.0421630i 0
85.5 0 −0.924609 − 0.196532i 0 −0.191729 − 1.21053i 0 −3.68665 − 2.39414i 0 −1.92436 − 0.856780i 0
85.6 0 −0.812564 − 0.172716i 0 −0.545500 − 3.44415i 0 1.63824 + 1.06389i 0 −2.11021 − 0.939524i 0
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.f odd 12 1 inner
143.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bv.a ✓ 224
11.d odd 10 1 inner 572.2.bv.a ✓ 224
13.f odd 12 1 inner 572.2.bv.a ✓ 224
143.w even 60 1 inner 572.2.bv.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bv.a ✓ 224 1.a even 1 1 trivial
572.2.bv.a ✓ 224 11.d odd 10 1 inner
572.2.bv.a ✓ 224 13.f odd 12 1 inner
572.2.bv.a ✓ 224 143.w even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).