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Properties

Label 574.2.ba.a
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
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 4 q^{7} + 8 q^{9} - 8 q^{11} + 4 q^{14} - 24 q^{15} + 56 q^{16} - 56 q^{18} - 16 q^{21} - 8 q^{22} - 120 q^{23} + 4 q^{28} + 8 q^{29} - 24 q^{30} + 56 q^{35} + 8 q^{36} - 120 q^{37} - 8 q^{39} + 88 q^{42} - 104 q^{43} - 32 q^{44} - 40 q^{46} + 40 q^{49} + 40 q^{50} - 48 q^{51} - 72 q^{53} + 4 q^{56} + 16 q^{57} - 8 q^{58} + 24 q^{60} - 104 q^{63} + 72 q^{65} + 144 q^{67} - 28 q^{70} - 96 q^{71} - 8 q^{74} - 72 q^{77} + 120 q^{78} - 32 q^{79} - 16 q^{84} - 16 q^{85} - 8 q^{88} + 36 q^{91} - 16 q^{92} + 24 q^{93} - 152 q^{95} - 56 q^{98} + 160 q^{99}+O(q^{100}) \) 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 −2.91944 − 1.20927i 0.587785 + 0.809017i 0.193497 − 1.22169i −2.05224 − 2.40287i −0.552136 − 2.58750i 0.156434 + 0.987688i 4.93947 + 4.93947i 0.727044 − 1.00069i
13.2 0.891007 + 0.453990i −2.16508 − 0.896807i 0.587785 + 0.809017i −0.640181 + 4.04194i −1.52196 − 1.78199i −2.61685 − 0.389967i 0.156434 + 0.987688i 1.76201 + 1.76201i −2.40541 + 3.31076i
13.3 0.891007 + 0.453990i −2.16220 − 0.895612i 0.587785 + 0.809017i 0.183115 − 1.15615i −1.51993 − 1.77961i −1.70153 + 2.02603i 0.156434 + 0.987688i 1.75167 + 1.75167i 0.688036 − 0.947001i
13.4 0.891007 + 0.453990i −1.90685 − 0.789844i 0.587785 + 0.809017i −0.156092 + 0.985523i −1.34044 − 1.56945i 2.34652 − 1.22223i 0.156434 + 0.987688i 0.890908 + 0.890908i −0.586497 + 0.807244i
13.5 0.891007 + 0.453990i −1.32744 − 0.549843i 0.587785 + 0.809017i 0.603040 − 3.80745i −0.933133 − 1.09256i 1.92982 + 1.80991i 0.156434 + 0.987688i −0.661554 − 0.661554i 2.26586 − 3.11869i
13.6 0.891007 + 0.453990i −1.13172 − 0.468773i 0.587785 + 0.809017i −0.287239 + 1.81355i −0.795550 − 0.931469i 0.669988 + 2.55951i 0.156434 + 0.987688i −1.06028 − 1.06028i −1.07927 + 1.48549i
13.7 0.891007 + 0.453990i −0.331235 − 0.137202i 0.587785 + 0.809017i 0.415677 − 2.62448i −0.232844 − 0.272626i −1.90538 − 1.83563i 0.156434 + 0.987688i −2.03043 − 2.03043i 1.56186 − 2.14972i
13.8 0.891007 + 0.453990i 0.331235 + 0.137202i 0.587785 + 0.809017i −0.415677 + 2.62448i 0.232844 + 0.272626i −2.11110 − 1.59477i 0.156434 + 0.987688i −2.03043 − 2.03043i −1.56186 + 2.14972i
13.9 0.891007 + 0.453990i 1.13172 + 0.468773i 0.587785 + 0.809017i 0.287239 − 1.81355i 0.795550 + 0.931469i 2.63281 + 0.261343i 0.156434 + 0.987688i −1.06028 − 1.06028i 1.07927 − 1.48549i
13.10 0.891007 + 0.453990i 1.32744 + 0.549843i 0.587785 + 0.809017i −0.603040 + 3.80745i 0.933133 + 1.09256i 2.08952 + 1.62293i 0.156434 + 0.987688i −0.661554 − 0.661554i −2.26586 + 3.11869i
13.11 0.891007 + 0.453990i 1.90685 + 0.789844i 0.587785 + 0.809017i 0.156092 − 0.985523i 1.34044 + 1.56945i −0.840104 + 2.50883i 0.156434 + 0.987688i 0.890908 + 0.890908i 0.586497 − 0.807244i
13.12 0.891007 + 0.453990i 2.16220 + 0.895612i 0.587785 + 0.809017i −0.183115 + 1.15615i 1.51993 + 1.77961i 1.73491 − 1.99752i 0.156434 + 0.987688i 1.75167 + 1.75167i −0.688036 + 0.947001i
13.13 0.891007 + 0.453990i 2.16508 + 0.896807i 0.587785 + 0.809017i 0.640181 − 4.04194i 1.52196 + 1.78199i −0.794532 − 2.52363i 0.156434 + 0.987688i 1.76201 + 1.76201i 2.40541 − 3.31076i
13.14 0.891007 + 0.453990i 2.91944 + 1.20927i 0.587785 + 0.809017i −0.193497 + 1.22169i 2.05224 + 2.40287i −2.64201 − 0.140564i 0.156434 + 0.987688i 4.93947 + 4.93947i −0.727044 + 1.00069i
69.1 −0.891007 − 0.453990i −1.10828 + 2.67562i 0.587785 + 0.809017i −0.561912 + 3.54777i 2.20219 − 1.88085i −2.63280 − 0.261516i −0.156434 − 0.987688i −3.80935 − 3.80935i 2.11132 − 2.90599i
69.2 −0.891007 − 0.453990i −0.904903 + 2.18463i 0.587785 + 0.809017i −0.205706 + 1.29878i 1.79808 − 1.53570i 1.79176 − 1.94669i −0.156434 − 0.987688i −1.83244 − 1.83244i 0.772919 − 1.06383i
69.3 −0.891007 − 0.453990i −0.893060 + 2.15604i 0.587785 + 0.809017i 0.446045 − 2.81622i 1.77454 − 1.51560i 1.11733 + 2.39824i −0.156434 − 0.987688i −1.72962 − 1.72962i −1.67597 + 2.30677i
69.4 −0.891007 − 0.453990i −0.873203 + 2.10810i 0.587785 + 0.809017i 0.399976 − 2.52535i 1.73509 − 1.48190i 0.333070 − 2.62470i −0.156434 − 0.987688i −1.56028 − 1.56028i −1.50286 + 2.06851i
69.5 −0.891007 − 0.453990i −0.490637 + 1.18450i 0.587785 + 0.809017i −0.00701053 + 0.0442628i 0.974914 − 0.832656i −1.78601 + 1.95196i −0.156434 − 0.987688i 0.958997 + 0.958997i 0.0263413 − 0.0362557i
69.6 −0.891007 − 0.453990i −0.342791 + 0.827570i 0.587785 + 0.809017i −0.637438 + 4.02462i 0.681137 − 0.581746i −0.00873645 + 2.64574i −0.156434 − 0.987688i 1.55395 + 1.55395i 2.39510 − 3.29657i
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.a ✓ 224
7.b odd 2 1 inner 574.2.ba.a ✓ 224
41.h odd 40 1 inner 574.2.ba.a ✓ 224
287.bb even 40 1 inner 574.2.ba.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.ba.a ✓ 224 1.a even 1 1 trivial
574.2.ba.a ✓ 224 7.b odd 2 1 inner
574.2.ba.a ✓ 224 41.h odd 40 1 inner
574.2.ba.a ✓ 224 287.bb even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{224} - 4 T_{3}^{222} + 8 T_{3}^{220} + 424 T_{3}^{218} + 53482 T_{3}^{216} + \cdots + 15\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display