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Properties

Label 627.2.bo.b
Level $627$
Weight $2$
Character orbit 627.bo
Analytic conductor $5.007$
Analytic rank $0$
Dimension $480$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [627,2,Mod(4,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([0, 18, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("627.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.bo (of order \(45\), degree \(24\), 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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(20\) over \(\Q(\zeta_{45})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 480 q + 18 q^{5} + 3 q^{7} + 12 q^{10} + 3 q^{11} - 240 q^{12} + 42 q^{14} - 12 q^{15} + 72 q^{16} - 18 q^{17} - 102 q^{20} + 24 q^{22} + 54 q^{23} - 18 q^{25} - 12 q^{26} + 60 q^{27} + 36 q^{29} + 12 q^{30}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.40738 − 2.08653i 0.990268 + 0.139173i −1.62368 + 4.01874i −0.647398 + 2.59657i −1.10330 − 2.26210i −4.32507 − 0.919323i 5.74673 − 1.22151i 0.961262 + 0.275637i 6.32897 − 2.30356i
4.2 −1.35691 − 2.01170i 0.990268 + 0.139173i −1.45652 + 3.60500i 0.951720 − 3.81714i −1.06373 − 2.18096i −2.30488 − 0.489917i 4.48149 − 0.952570i 0.961262 + 0.275637i −8.97033 + 3.26493i
4.3 −1.35368 − 2.00692i 0.990268 + 0.139173i −1.44604 + 3.57908i −0.344111 + 1.38016i −1.06120 − 2.17578i 0.142915 + 0.0303776i 4.40463 − 0.936234i 0.961262 + 0.275637i 3.23567 − 1.17769i
4.4 −1.25799 − 1.86505i 0.990268 + 0.139173i −1.14665 + 2.83806i 0.172386 − 0.691403i −0.986183 − 2.02198i 2.80680 + 0.596604i 2.33459 − 0.496233i 0.961262 + 0.275637i −1.50636 + 0.548270i
4.5 −0.925154 − 1.37160i 0.990268 + 0.139173i −0.276156 + 0.683509i 0.103072 − 0.413400i −0.725261 − 1.48701i 4.70474 + 1.00002i −2.04360 + 0.434380i 0.961262 + 0.275637i −0.662375 + 0.241085i
4.6 −0.778475 − 1.15414i 0.990268 + 0.139173i 0.0232044 − 0.0574330i 0.302411 − 1.21291i −0.610274 − 1.25125i −0.175978 − 0.0374053i −2.80779 + 0.596814i 0.961262 + 0.275637i −1.63528 + 0.595193i
4.7 −0.699137 − 1.03651i 0.990268 + 0.139173i 0.163646 − 0.405038i −0.459541 + 1.84312i −0.548078 − 1.12373i −2.58516 − 0.549494i −2.98012 + 0.633444i 0.961262 + 0.275637i 2.23170 − 0.812271i
4.8 −0.424357 − 0.629135i 0.990268 + 0.139173i 0.533481 − 1.32041i 0.939506 − 3.76815i −0.332668 − 0.682071i 0.175076 + 0.0372136i −2.54168 + 0.540252i 0.961262 + 0.275637i −2.76936 + 1.00796i
4.9 −0.0551011 − 0.0816908i 0.990268 + 0.139173i 0.745576 − 1.84537i 0.0976589 − 0.391688i −0.0431957 − 0.0885643i −4.95143 − 1.05246i −0.384599 + 0.0817489i 0.961262 + 0.275637i −0.0373784 + 0.0136046i
4.10 −0.0487923 − 0.0723376i 0.990268 + 0.139173i 0.746361 − 1.84731i 0.661518 − 2.65320i −0.0382500 − 0.0784241i 2.64052 + 0.561260i −0.340743 + 0.0724271i 0.961262 + 0.275637i −0.224203 + 0.0816033i
4.11 −0.00735657 − 0.0109066i 0.990268 + 0.139173i 0.749148 − 1.85421i −0.675947 + 2.71108i −0.00576708 − 0.0118243i −0.242170 − 0.0514749i −0.0514706 + 0.0109404i 0.961262 + 0.275637i 0.0345412 − 0.0125720i
4.12 0.111836 + 0.165804i 0.990268 + 0.139173i 0.734230 − 1.81728i −0.842433 + 3.37882i 0.0876721 + 0.179754i 4.21334 + 0.895572i 0.774675 − 0.164662i 0.961262 + 0.275637i −0.654434 + 0.238194i
4.13 0.371585 + 0.550897i 0.990268 + 0.139173i 0.583801 − 1.44496i −0.234594 + 0.940907i 0.291299 + 0.597251i −1.37444 − 0.292146i 2.31292 − 0.491626i 0.961262 + 0.275637i −0.605515 + 0.220389i
4.14 0.666350 + 0.987905i 0.990268 + 0.139173i 0.217280 − 0.537786i 0.527314 − 2.11494i 0.522376 + 1.07103i −0.520032 − 0.110536i 3.00724 − 0.639209i 0.961262 + 0.275637i 2.44073 − 0.888355i
4.15 0.974615 + 1.44493i 0.990268 + 0.139173i −0.388724 + 0.962125i −0.432435 + 1.73440i 0.764035 + 1.56650i 0.443945 + 0.0943635i 1.64056 − 0.348712i 0.961262 + 0.275637i −2.92754 + 1.06554i
4.16 0.996377 + 1.47719i 0.990268 + 0.139173i −0.440110 + 1.08931i −0.343015 + 1.37576i 0.781096 + 1.60148i 3.02936 + 0.643909i 1.43812 − 0.305682i 0.961262 + 0.275637i −2.37403 + 0.864077i
4.17 1.08216 + 1.60437i 0.990268 + 0.139173i −0.653717 + 1.61801i 0.908553 − 3.64401i 0.848344 + 1.73936i −3.14193 − 0.667838i 0.482555 − 0.102570i 0.961262 + 0.275637i 6.82954 − 2.48575i
4.18 1.21010 + 1.79405i 0.990268 + 0.139173i −1.00506 + 2.48761i −0.733396 + 2.94149i 0.948644 + 1.94501i −4.34119 − 0.922749i −1.44568 + 0.307289i 0.961262 + 0.275637i −6.16468 + 2.24376i
4.19 1.39088 + 2.06207i 0.990268 + 0.139173i −1.56835 + 3.88181i 0.610603 − 2.44900i 1.09036 + 2.23557i 2.18444 + 0.464317i −5.32005 + 1.13081i 0.961262 + 0.275637i 5.89927 − 2.14716i
4.20 1.51043 + 2.23930i 0.990268 + 0.139173i −1.98386 + 4.91022i −0.111670 + 0.447883i 1.18408 + 2.42772i −0.895960 − 0.190442i −8.70779 + 1.85090i 0.961262 + 0.275637i −1.17161 + 0.426432i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.e even 9 1 inner
209.u even 45 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 627.2.bo.b ✓ 480
11.c even 5 1 inner 627.2.bo.b ✓ 480
19.e even 9 1 inner 627.2.bo.b ✓ 480
209.u even 45 1 inner 627.2.bo.b ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.bo.b ✓ 480 1.a even 1 1 trivial
627.2.bo.b ✓ 480 11.c even 5 1 inner
627.2.bo.b ✓ 480 19.e even 9 1 inner
627.2.bo.b ✓ 480 209.u even 45 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{480} - 18 T_{2}^{476} + 6 T_{2}^{475} - 870 T_{2}^{474} + 192 T_{2}^{473} + \cdots + 204374500913521 \) acting on \(S_{2}^{\mathrm{new}}(627, [\chi])\). Copy content Toggle raw display