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Properties

Label 574.2.ba.b
Level $574$
Weight $2$
Character orbit 574.ba
Analytic conductor $4.583$
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] = [574,2,Mod(13,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.ba (of order \(40\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 4 q^{7} + 8 q^{9} + 8 q^{11} + 4 q^{14} - 24 q^{15} + 56 q^{16} + 56 q^{18} - 16 q^{21} - 8 q^{22} + 120 q^{23} - 4 q^{28} - 24 q^{29} - 24 q^{30} + 8 q^{36} - 24 q^{37} + 8 q^{39} + 72 q^{42} + 72 q^{43} - 32 q^{44} + 24 q^{46} + 40 q^{49} - 8 q^{50} - 96 q^{51} + 72 q^{53} + 4 q^{56} + 16 q^{57} - 24 q^{58} - 24 q^{60} - 48 q^{63} + 40 q^{65} - 128 q^{67} + 4 q^{70} - 32 q^{71} - 24 q^{74} - 24 q^{77} - 120 q^{78} + 48 q^{79} - 16 q^{84} - 16 q^{85} + 8 q^{88} - 148 q^{91} + 16 q^{92} - 88 q^{93} + 200 q^{95} + 72 q^{98} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.891007 − 0.453990i −3.06051 − 1.26770i 0.587785 + 0.809017i 0.480653 − 3.03472i 2.15141 + 2.51897i 2.60529 + 0.460964i −0.156434 − 0.987688i 5.63832 + 5.63832i −1.80600 + 2.48575i
13.2 −0.891007 − 0.453990i −2.61536 − 1.08332i 0.587785 + 0.809017i −0.0382490 + 0.241494i 1.83848 + 2.15259i −2.64544 + 0.0403505i −0.156434 − 0.987688i 3.54519 + 3.54519i 0.143716 − 0.197808i
13.3 −0.891007 − 0.453990i −2.14141 − 0.887000i 0.587785 + 0.809017i −0.545078 + 3.44149i 1.50532 + 1.76250i 2.06786 + 1.65044i −0.156434 − 0.987688i 1.67754 + 1.67754i 2.04807 − 2.81893i
13.4 −0.891007 − 0.453990i −1.68199 − 0.696704i 0.587785 + 0.809017i −0.367014 + 2.31723i 1.18237 + 1.38438i −1.48959 − 2.18658i −0.156434 − 0.987688i 0.222379 + 0.222379i 1.37901 − 1.89805i
13.5 −0.891007 − 0.453990i −0.896341 − 0.371277i 0.587785 + 0.809017i 0.388574 − 2.45336i 0.630090 + 0.737740i 0.803278 − 2.52086i −0.156434 − 0.987688i −1.45574 − 1.45574i −1.46002 + 2.00955i
13.6 −0.891007 − 0.453990i −0.392034 − 0.162386i 0.587785 + 0.809017i −0.168798 + 1.06575i 0.275583 + 0.322666i −0.646124 + 2.56564i −0.156434 − 0.987688i −1.99400 − 1.99400i 0.634238 − 0.872954i
13.7 −0.891007 − 0.453990i −0.309629 − 0.128253i 0.587785 + 0.809017i 0.339396 − 2.14286i 0.217656 + 0.254843i −1.19699 + 2.35950i −0.156434 − 0.987688i −2.04190 − 2.04190i −1.27524 + 1.75522i
13.8 −0.891007 − 0.453990i 0.309629 + 0.128253i 0.587785 + 0.809017i −0.339396 + 2.14286i −0.217656 − 0.254843i 2.14320 − 1.55136i −0.156434 − 0.987688i −2.04190 − 2.04190i 1.27524 − 1.75522i
13.9 −0.891007 − 0.453990i 0.392034 + 0.162386i 0.587785 + 0.809017i 0.168798 − 1.06575i −0.275583 − 0.322666i 2.43298 − 1.03952i −0.156434 − 0.987688i −1.99400 − 1.99400i −0.634238 + 0.872954i
13.10 −0.891007 − 0.453990i 0.896341 + 0.371277i 0.587785 + 0.809017i −0.388574 + 2.45336i −0.630090 − 0.737740i −2.36417 + 1.18774i −0.156434 − 0.987688i −1.45574 − 1.45574i 1.46002 − 2.00955i
13.11 −0.891007 − 0.453990i 1.68199 + 0.696704i 0.587785 + 0.809017i 0.367014 − 2.31723i −1.18237 − 1.38438i −2.39268 − 1.12920i −0.156434 − 0.987688i 0.222379 + 0.222379i −1.37901 + 1.89805i
13.12 −0.891007 − 0.453990i 2.14141 + 0.887000i 0.587785 + 0.809017i 0.545078 − 3.44149i −1.50532 − 1.76250i 1.95361 + 1.78422i −0.156434 − 0.987688i 1.67754 + 1.67754i −2.04807 + 2.81893i
13.13 −0.891007 − 0.453990i 2.61536 + 1.08332i 0.587785 + 0.809017i 0.0382490 − 0.241494i −1.83848 − 2.15259i −0.373985 − 2.61919i −0.156434 − 0.987688i 3.54519 + 3.54519i −0.143716 + 0.197808i
13.14 −0.891007 − 0.453990i 3.06051 + 1.26770i 0.587785 + 0.809017i −0.480653 + 3.03472i −2.15141 − 2.51897i 0.862845 + 2.50110i −0.156434 − 0.987688i 5.63832 + 5.63832i 1.80600 − 2.48575i
69.1 0.891007 + 0.453990i −1.29112 + 3.11704i 0.587785 + 0.809017i 0.00104644 − 0.00660694i −2.56550 + 2.19114i −0.718404 − 2.54635i 0.156434 + 0.987688i −5.92760 − 5.92760i 0.00393187 − 0.00541175i
69.2 0.891007 + 0.453990i −1.06948 + 2.58196i 0.587785 + 0.809017i −0.400556 + 2.52901i −2.12510 + 1.81501i 0.427898 + 2.61092i 0.156434 + 0.987688i −3.40139 − 3.40139i −1.50505 + 2.07152i
69.3 0.891007 + 0.453990i −0.619735 + 1.49617i 0.587785 + 0.809017i 0.200809 − 1.26786i −1.23144 + 1.05175i 2.64104 − 0.157843i 0.156434 + 0.987688i 0.266862 + 0.266862i 0.754518 − 1.03850i
69.4 0.891007 + 0.453990i −0.579941 + 1.40010i 0.587785 + 0.809017i 0.514039 − 3.24552i −1.15236 + 0.984213i −1.14545 − 2.38494i 0.156434 + 0.987688i 0.497366 + 0.497366i 1.93145 − 2.65841i
69.5 0.891007 + 0.453990i −0.554765 + 1.33932i 0.587785 + 0.809017i −0.610559 + 3.85492i −1.10234 + 0.941486i 1.18762 − 2.36423i 0.156434 + 0.987688i 0.635303 + 0.635303i −2.29411 + 3.15757i
69.6 0.891007 + 0.453990i −0.305709 + 0.738048i 0.587785 + 0.809017i −0.241300 + 1.52351i −0.607456 + 0.518816i −2.51115 + 0.833135i 0.156434 + 0.987688i 1.67006 + 1.67006i −0.906657 + 1.24791i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.h odd 40 1 inner
287.bb even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.ba.b ✓ 224
7.b odd 2 1 inner 574.2.ba.b ✓ 224
41.h odd 40 1 inner 574.2.ba.b ✓ 224
287.bb even 40 1 inner 574.2.ba.b ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.ba.b ✓ 224 1.a even 1 1 trivial
574.2.ba.b ✓ 224 7.b odd 2 1 inner
574.2.ba.b ✓ 224 41.h odd 40 1 inner
574.2.ba.b ✓ 224 287.bb even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{224} - 4 T_{3}^{222} + 8 T_{3}^{220} + 120 T_{3}^{218} + 59466 T_{3}^{216} + \cdots + 15\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display