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Properties

Label 297.2.x.a
Level $297$
Weight $2$
Character orbit 297.x
Analytic conductor $2.372$
Analytic rank $0$
Dimension $816$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(2,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.x (of order \(90\), 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: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(816\)
Relative dimension: \(34\) over \(\Q(\zeta_{90})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 816 q - 30 q^{2} - 18 q^{3} - 18 q^{4} - 21 q^{5} - 30 q^{6} - 30 q^{7} - 45 q^{8} - 36 q^{9} - 33 q^{11} - 30 q^{12} - 30 q^{13} - 18 q^{14} - 21 q^{15} - 30 q^{16} - 45 q^{17} - 15 q^{19} - 60 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.35990 − 1.25478i 1.70844 − 0.285014i 2.87626 + 4.26423i −0.324115 + 0.518692i −4.38938 − 1.47111i −0.825769 + 2.87980i −0.878253 − 8.35602i 2.83753 − 0.973859i 1.41572 − 0.817368i
2.2 −2.26253 − 1.20301i 0.515632 + 1.65352i 2.55343 + 3.78562i 0.859531 − 1.37554i 0.822565 − 4.36145i 0.489304 − 1.70640i −0.687383 − 6.54001i −2.46825 + 1.70521i −3.59950 + 2.07817i
2.3 −2.26092 − 1.20215i −0.483507 − 1.66320i 2.54822 + 3.77789i 1.25715 − 2.01187i −0.906247 + 4.34161i −0.310667 + 1.08342i −0.684398 − 6.51161i −2.53244 + 1.60833i −5.26090 + 3.03738i
2.4 −2.19675 − 1.16803i −1.73177 − 0.0312456i 2.34303 + 3.47368i −0.743142 + 1.18928i 3.76777 + 2.09140i 1.21182 − 4.22614i −0.569546 − 5.41887i 2.99805 + 0.108220i 3.02161 − 1.74453i
2.5 −1.96350 − 1.04401i −1.03570 + 1.38828i 1.64697 + 2.44174i −1.86245 + 2.98054i 3.48297 − 1.64461i −1.27752 + 4.45523i −0.219728 − 2.09057i −0.854668 − 2.87568i 6.76862 − 3.90787i
2.6 −1.85576 − 0.986723i −1.57006 + 0.731366i 1.35182 + 2.00416i 2.23335 − 3.57411i 3.63531 + 0.191982i −0.641535 + 2.23730i −0.0917117 − 0.872579i 1.93021 − 2.29658i −7.67120 + 4.42897i
2.7 −1.85524 − 0.986448i −0.117170 − 1.72808i 1.35045 + 2.00212i −1.51818 + 2.42960i −1.48729 + 3.32159i 0.513847 − 1.79200i −0.0911464 − 0.867200i −2.97254 + 0.404959i 5.21326 − 3.00988i
2.8 −1.52582 − 0.811295i 1.55110 − 0.770772i 0.551554 + 0.817713i −1.39756 + 2.23657i −2.99203 − 0.0823374i 0.208948 − 0.728689i 0.183104 + 1.74211i 1.81182 − 2.39109i 3.94695 − 2.27877i
2.9 −1.46798 − 0.780540i 1.71058 + 0.271862i 0.427343 + 0.633562i 1.29552 − 2.07327i −2.29890 − 1.73427i 1.10448 − 3.85179i 0.214765 + 2.04335i 2.85218 + 0.930084i −3.52007 + 2.03232i
2.10 −1.31955 − 0.701616i −0.367723 + 1.69257i 0.130555 + 0.193555i −0.165773 + 0.265291i 1.67276 − 1.97542i 0.396429 − 1.38251i 0.275960 + 2.62558i −2.72956 − 1.24479i 0.404877 − 0.233756i
2.11 −1.17173 − 0.623019i −1.24456 − 1.20460i −0.133592 − 0.198058i −0.0930248 + 0.148871i 0.707795 + 2.18685i −0.585190 + 2.04080i 0.310571 + 2.95489i 0.0978601 + 2.99840i 0.201749 − 0.116480i
2.12 −1.03336 − 0.549449i 1.50566 + 0.856142i −0.352441 − 0.522516i 0.280072 − 0.448209i −1.08549 − 1.71199i −0.906185 + 3.16024i 0.321775 + 3.06149i 1.53404 + 2.57812i −0.535684 + 0.309277i
2.13 −0.707588 − 0.376231i −1.13129 − 1.31156i −0.759255 − 1.12564i 1.77592 − 2.84206i 0.307039 + 1.35367i 1.17780 − 4.10748i 0.281275 + 2.67615i −0.440362 + 2.96750i −2.32589 + 1.34285i
2.14 −0.496209 − 0.263839i 1.09688 − 1.34046i −0.941773 − 1.39624i 1.91409 − 3.06319i −0.897951 + 0.375750i −0.560137 + 1.95343i 0.216423 + 2.05913i −0.593689 − 2.94067i −1.75798 + 1.01497i
2.15 −0.479124 − 0.254755i 1.10485 + 1.33390i −0.953726 − 1.41396i −1.90249 + 3.04462i −0.189543 − 0.920571i 0.294306 − 1.02637i 0.210184 + 1.99976i −0.558602 + 2.94754i 1.68716 − 0.974083i
2.16 −0.330318 − 0.175633i −0.396513 + 1.68605i −1.04012 − 1.54205i 1.19100 − 1.90599i 0.427103 − 0.487294i −0.158877 + 0.554068i 0.150947 + 1.43616i −2.68556 − 1.33708i −0.728164 + 0.420406i
2.17 −0.269305 − 0.143192i −1.63163 + 0.581205i −1.06636 − 1.58095i −1.80542 + 2.88927i 0.522628 + 0.0771141i 0.611788 − 2.13356i 0.124561 + 1.18512i 2.32440 − 1.89662i 0.899926 − 0.519573i
2.18 −0.214028 − 0.113801i 0.440063 − 1.67521i −1.08553 − 1.60936i −0.565155 + 0.904436i −0.284826 + 0.308463i 0.830386 − 2.89590i 0.0998625 + 0.950128i −2.61269 − 1.47440i 0.223884 − 0.129259i
2.19 −0.206000 − 0.109532i −1.69914 + 0.336021i −1.08795 − 1.61295i 0.239455 − 0.383208i 0.386828 + 0.116890i −0.317762 + 1.10817i 0.0962223 + 0.915494i 2.77418 − 1.14190i −0.0913011 + 0.0527127i
2.20 0.170131 + 0.0904602i −0.411206 − 1.68253i −1.09762 − 1.62729i −1.62004 + 2.59261i 0.0822432 − 0.323448i −1.38803 + 4.84064i −0.0798166 − 0.759404i −2.66182 + 1.38373i −0.510147 + 0.294533i
See next 80 embeddings (of 816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
27.f odd 18 1 inner
297.x even 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.x.a ✓ 816
3.b odd 2 1 891.2.bb.a 816
11.d odd 10 1 inner 297.2.x.a ✓ 816
27.e even 9 1 891.2.bb.a 816
27.f odd 18 1 inner 297.2.x.a ✓ 816
33.f even 10 1 891.2.bb.a 816
297.w odd 90 1 891.2.bb.a 816
297.x even 90 1 inner 297.2.x.a ✓ 816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.x.a ✓ 816 1.a even 1 1 trivial
297.2.x.a ✓ 816 11.d odd 10 1 inner
297.2.x.a ✓ 816 27.f odd 18 1 inner
297.2.x.a ✓ 816 297.x even 90 1 inner
891.2.bb.a 816 3.b odd 2 1
891.2.bb.a 816 27.e even 9 1
891.2.bb.a 816 33.f even 10 1
891.2.bb.a 816 297.w odd 90 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(297, [\chi])\).