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Properties

Label 637.2.bz.a
Level $637$
Weight $2$
Character orbit 637.bz
Analytic conductor $5.086$
Analytic rank $0$
Dimension $756$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(4,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bz (of order \(42\), degree \(12\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(756\)
Relative dimension: \(63\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 756 q - 21 q^{2} - 5 q^{3} + 115 q^{4} - 15 q^{6} - 14 q^{7} + 50 q^{9} - 16 q^{10} - 24 q^{11} - 26 q^{12} - 10 q^{13} - 38 q^{14} - 6 q^{15} - 125 q^{16} + 15 q^{17} - 9 q^{18} - 27 q^{19} - 15 q^{20}+ \cdots + 105 q^{98}+O(q^{100}) \) 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 −1.21061 − 2.51386i 2.20537 − 0.680267i −3.60692 + 4.52294i 0.0440739 + 0.142884i −4.37994 − 4.72045i −1.16487 + 2.37551i 10.2962 + 2.35003i 1.92218 − 1.31052i 0.305834 − 0.283773i
4.2 −1.17308 − 2.43592i −0.945504 + 0.291649i −3.31061 + 4.15137i −0.595404 − 1.93025i 1.81958 + 1.96104i 2.50774 − 0.843362i 8.72425 + 1.99125i −1.66980 + 1.13845i −4.00349 + 3.71469i
4.3 −1.12064 − 2.32702i 0.960400 − 0.296244i −2.91224 + 3.65183i 0.762546 + 2.47211i −1.76562 − 1.90289i −0.582914 − 2.58074i 6.72535 + 1.53502i −1.64411 + 1.12093i 4.89813 − 4.54480i
4.4 −1.09947 − 2.28306i −1.42464 + 0.439442i −2.75658 + 3.45664i 1.11652 + 3.61968i 2.56961 + 2.76938i −1.77744 + 1.95977i 5.98153 + 1.36525i −0.642240 + 0.437872i 7.03638 − 6.52881i
4.5 −1.07047 − 2.22286i −3.09970 + 0.956132i −2.54820 + 3.19535i −0.389940 − 1.26415i 5.44349 + 5.86668i 1.08627 + 2.41247i 5.01993 + 1.14577i 6.21525 − 4.23749i −2.39262 + 2.22002i
4.6 −1.04741 − 2.17497i 2.86346 − 0.883262i −2.38644 + 2.99250i −1.10513 − 3.58273i −4.92028 − 5.30280i 0.696071 − 2.55254i 4.30114 + 0.981707i 4.94056 − 3.36842i −6.63480 + 6.15619i
4.7 −1.04079 − 2.16123i 0.381790 − 0.117767i −2.34067 + 2.93511i 0.210839 + 0.683523i −0.651883 − 0.702563i 2.64574 − 0.00872039i 4.10230 + 0.936324i −2.34682 + 1.60004i 1.25781 − 1.16708i
4.8 −1.02936 − 2.13750i −2.28640 + 0.705259i −2.26233 + 2.83687i −0.329479 − 1.06814i 3.86103 + 4.16120i −1.23633 − 2.33912i 3.76664 + 0.859710i 2.25150 − 1.53504i −1.94400 + 1.80377i
4.9 −0.941441 − 1.95492i −1.22564 + 0.378060i −1.68843 + 2.11723i 0.135805 + 0.440268i 1.89295 + 2.04011i −2.35928 + 1.19741i 1.49777 + 0.341857i −1.11945 + 0.763228i 0.732838 − 0.679975i
4.10 −0.938275 − 1.94835i 1.73522 − 0.535243i −1.66872 + 2.09251i −0.694611 − 2.25187i −2.67095 − 2.87860i −0.169018 + 2.64035i 1.42609 + 0.325495i 0.245774 − 0.167566i −3.73570 + 3.46622i
4.11 −0.871546 − 1.80978i −0.0408124 + 0.0125890i −1.26875 + 1.59096i −1.20544 − 3.90795i 0.0583532 + 0.0628898i 1.10936 + 2.40194i 0.0683642 + 0.0156037i −2.47721 + 1.68893i −6.02195 + 5.58755i
4.12 −0.864819 − 1.79582i 2.15696 − 0.665335i −1.23006 + 1.54245i 0.973630 + 3.15643i −3.06020 − 3.29811i 1.71343 + 2.01597i −0.0527318 − 0.0120357i 1.73110 − 1.18024i 4.82635 − 4.47820i
4.13 −0.836520 − 1.73705i 3.16602 − 0.976589i −1.07060 + 1.34249i 0.868353 + 2.81513i −4.34483 − 4.68261i −2.56241 − 0.658818i −0.531731 − 0.121364i 6.59126 − 4.49385i 4.16363 − 3.86329i
4.14 −0.782958 − 1.62583i −2.82293 + 0.870760i −0.783320 + 0.982252i 1.12803 + 3.65698i 3.62595 + 3.90784i 2.10109 − 1.60792i −1.30830 − 0.298612i 4.73202 − 3.22623i 5.06243 − 4.69725i
4.15 −0.764618 − 1.58775i −1.09944 + 0.339134i −0.689318 + 0.864377i 0.542823 + 1.75979i 1.37911 + 1.48633i 1.71305 − 2.01630i −1.53669 − 0.350739i −1.38495 + 0.944244i 2.37905 − 2.20743i
4.16 −0.754795 − 1.56735i −0.426042 + 0.131416i −0.639884 + 0.802389i −0.992234 − 3.21674i 0.527549 + 0.568563i −0.839736 − 2.50895i −1.65141 − 0.376925i −2.31448 + 1.57798i −4.29282 + 3.98316i
4.17 −0.709328 − 1.47293i 0.808193 − 0.249294i −0.419411 + 0.525924i −0.0463428 − 0.150240i −0.940468 − 1.01358i −2.52127 + 0.802000i −2.11554 − 0.482858i −1.88769 + 1.28700i −0.188421 + 0.174829i
4.18 −0.689843 − 1.43247i 2.37403 − 0.732290i −0.329116 + 0.412698i 0.0745143 + 0.241569i −2.68669 − 2.89556i 1.80877 − 1.93089i −2.28191 − 0.520830i 2.62104 − 1.78699i 0.294638 − 0.273385i
4.19 −0.553189 − 1.14871i −1.61228 + 0.497321i 0.233465 − 0.292756i −0.149862 − 0.485842i 1.46317 + 1.57692i 1.56147 + 2.13584i −2.95145 − 0.673650i −0.126610 + 0.0863216i −0.475189 + 0.440911i
4.20 −0.505856 − 1.05042i −2.35374 + 0.726034i 0.399487 − 0.500941i −1.13742 − 3.68744i 1.95329 + 2.10515i −2.48525 + 0.907488i −3.00158 − 0.685090i 2.53427 − 1.72783i −3.29799 + 3.06008i
See next 80 embeddings (of 756 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.63
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
637.bz even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bz.a yes 756
13.e even 6 1 637.2.bp.a ✓ 756
49.g even 21 1 637.2.bp.a ✓ 756
637.bz even 42 1 inner 637.2.bz.a yes 756
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bp.a ✓ 756 13.e even 6 1
637.2.bp.a ✓ 756 49.g even 21 1
637.2.bz.a yes 756 1.a even 1 1 trivial
637.2.bz.a yes 756 637.bz even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(637, [\chi])\).