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Properties

Label 297.2.j.c
Level $297$
Weight $2$
Character orbit 297.j
Analytic conductor $2.372$
Analytic rank $0$
Dimension $102$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(34,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.j (of order \(9\), degree \(6\), 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: \(102\)
Relative dimension: \(17\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 6 q^{5} - 21 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 6 q^{5} - 21 q^{6} - 18 q^{10} - 9 q^{12} - 21 q^{14} - 27 q^{15} - 33 q^{18} - 30 q^{19} + 27 q^{20} + 18 q^{21} - 12 q^{23} + 105 q^{24} - 6 q^{25} + 18 q^{26} - 39 q^{27} + 72 q^{28} - 18 q^{29} + 36 q^{30} - 12 q^{31} + 96 q^{32} - 3 q^{33} - 24 q^{34} + 9 q^{35} + 39 q^{36} - 48 q^{37} - 15 q^{38} + 90 q^{40} - 54 q^{42} + 30 q^{43} - 60 q^{44} - 75 q^{45} - 42 q^{46} - 30 q^{47} - 39 q^{48} - 36 q^{49} + 66 q^{50} + 30 q^{51} - 84 q^{52} + 36 q^{53} - 36 q^{54} + 12 q^{55} - 48 q^{56} - 15 q^{57} - 75 q^{58} - 42 q^{59} - 84 q^{60} - 42 q^{61} + 105 q^{63} - 114 q^{64} - 57 q^{65} + 54 q^{67} - 6 q^{68} + 204 q^{70} - 30 q^{71} - 90 q^{72} - 69 q^{73} - 117 q^{74} - 72 q^{75} - 60 q^{76} + 189 q^{78} - 18 q^{79} + 84 q^{80} - 60 q^{81} + 84 q^{82} - 6 q^{83} - 72 q^{84} - 39 q^{85} - 132 q^{86} - 69 q^{87} - 9 q^{88} - 15 q^{89} + 138 q^{90} - 69 q^{91} + 102 q^{92} + 108 q^{93} + 123 q^{94} + 66 q^{95} - 117 q^{96} + 66 q^{97} + 57 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.60173 + 0.946954i −1.53214 − 0.807810i 4.34021 − 3.64187i 0.139232 + 0.789623i 4.75117 + 0.650844i −0.481874 − 0.404340i −5.07469 + 8.78962i 1.69489 + 2.47535i −1.10998 − 1.92254i
34.2 −2.30441 + 0.838737i 1.63750 − 0.564443i 3.07474 − 2.58002i −0.647325 − 3.67116i −3.30005 + 2.67414i 1.74400 + 1.46339i −2.46922 + 4.27681i 2.36281 − 1.84855i 4.57085 + 7.91694i
34.3 −2.27218 + 0.827005i 1.41521 + 0.998590i 2.94677 − 2.47263i 0.646856 + 3.66851i −4.04145 − 1.09859i 1.11607 + 0.936496i −2.23270 + 3.86715i 1.00564 + 2.82643i −4.50365 − 7.80054i
34.4 −1.60221 + 0.583158i −1.58045 + 0.708638i 0.694922 − 0.583109i 0.151385 + 0.858546i 2.11897 − 2.05704i 3.87388 + 3.25057i 0.931670 − 1.61370i 1.99566 − 2.23994i −0.743218 − 1.28729i
34.5 −1.20959 + 0.440255i 1.37813 − 1.04916i −0.262804 + 0.220519i 0.0689078 + 0.390796i −1.20508 + 1.87579i −0.0154059 − 0.0129271i 1.50802 − 2.61197i 0.798511 − 2.89178i −0.255400 − 0.442366i
34.6 −1.18215 + 0.430266i −0.296602 − 1.70647i −0.319750 + 0.268302i 0.736637 + 4.17768i 1.08486 + 1.88967i −1.09698 − 0.920474i 1.52056 − 2.63369i −2.82405 + 1.01228i −2.66832 − 4.62167i
34.7 −0.949421 + 0.345561i −0.253401 + 1.71341i −0.750101 + 0.629409i −0.529508 − 3.00299i −0.351505 − 1.71432i 0.254047 + 0.213171i 1.50502 − 2.60676i −2.87158 − 0.868361i 1.54044 + 2.66812i
34.8 −0.583438 + 0.212354i −1.59958 − 0.664333i −1.23678 + 1.03778i 0.0810529 + 0.459674i 1.07433 + 0.0479194i −2.02973 − 1.70315i 1.12209 − 1.94352i 2.11732 + 2.12531i −0.144903 − 0.250979i
34.9 0.172647 − 0.0628383i 1.20053 + 1.24849i −1.50623 + 1.26388i 0.119032 + 0.675063i 0.285721 + 0.140108i 0.656550 + 0.550911i −0.364353 + 0.631077i −0.117445 + 2.99770i 0.0629702 + 0.109068i
34.10 0.355283 − 0.129312i −1.11263 + 1.32743i −1.42258 + 1.19369i 0.186000 + 1.05486i −0.223644 + 0.615489i −3.05909 − 2.56688i −0.729146 + 1.26292i −0.524128 − 2.95386i 0.202489 + 0.350721i
34.11 0.877023 − 0.319210i 1.26486 − 1.18327i −0.864815 + 0.725666i 0.342989 + 1.94519i 0.731596 − 1.44151i 3.86422 + 3.24246i −1.46013 + 2.52902i 0.199728 − 2.99334i 0.921732 + 1.59649i
34.12 1.01578 − 0.369713i −0.597427 − 1.62576i −0.636972 + 0.534483i −0.488157 − 2.76848i −1.20792 − 1.43053i −1.61165 − 1.35233i −1.53038 + 2.65070i −2.28616 + 1.94254i −1.51940 − 2.63168i
34.13 1.54616 − 0.562758i 1.61499 − 0.625941i 0.541838 − 0.454656i −0.309482 − 1.75516i 2.14479 − 1.87666i −1.57643 − 1.32278i −1.06348 + 1.84201i 2.21640 − 2.02178i −1.46624 − 2.53960i
34.14 1.72658 − 0.628423i −1.71569 − 0.237488i 1.05407 − 0.884470i 0.647122 + 3.67001i −3.11152 + 0.668139i 0.682898 + 0.573019i −0.573273 + 0.992939i 2.88720 + 0.814912i 3.42363 + 5.92989i
34.15 2.16807 − 0.789111i 0.0654399 + 1.73081i 2.54572 − 2.13611i −0.640629 − 3.63319i 1.50768 + 3.70088i 2.09099 + 1.75455i 1.52645 − 2.64389i −2.99144 + 0.226529i −4.25591 − 7.37146i
34.16 2.34369 − 0.853033i −1.13344 − 1.30970i 3.23312 − 2.71291i −0.146799 − 0.832538i −3.77364 − 2.10268i 2.11009 + 1.77058i 2.76913 − 4.79628i −0.430649 + 2.96893i −1.05423 − 1.82599i
34.17 2.49990 − 0.909890i 0.304997 + 1.70499i 3.88953 − 3.26370i 0.522073 + 2.96082i 2.31381 + 3.98479i −3.45741 − 2.90111i 4.09349 − 7.09013i −2.81395 + 1.04003i 3.99915 + 6.92674i
67.1 −0.460280 − 2.61038i 1.34099 − 1.09624i −4.72282 + 1.71897i −0.426669 − 0.358018i −3.47884 − 2.99591i −3.56276 − 1.29674i 4.01032 + 6.94608i 0.596503 − 2.94010i −0.738175 + 1.27856i
67.2 −0.419820 − 2.38092i −1.28058 − 1.16625i −3.61313 + 1.31507i −3.26678 − 2.74115i −2.23912 + 3.53856i 1.39599 + 0.508098i 2.23030 + 3.86299i 0.279745 + 2.98693i −5.15500 + 8.92872i
67.3 −0.408393 − 2.31611i 1.57923 + 0.711357i −3.31821 + 1.20773i 1.30351 + 1.09377i 1.00264 − 3.94819i 2.41940 + 0.880589i 1.80053 + 3.11860i 1.98794 + 2.24679i 2.00096 − 3.46576i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.j.c ✓ 102
3.b odd 2 1 891.2.j.c 102
27.e even 9 1 inner 297.2.j.c ✓ 102
27.e even 9 1 8019.2.a.l 51
27.f odd 18 1 891.2.j.c 102
27.f odd 18 1 8019.2.a.k 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.c ✓ 102 1.a even 1 1 trivial
297.2.j.c ✓ 102 27.e even 9 1 inner
891.2.j.c 102 3.b odd 2 1
891.2.j.c 102 27.f odd 18 1
8019.2.a.k 51 27.f odd 18 1
8019.2.a.l 51 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{102} - 48 T_{2}^{97} + 849 T_{2}^{96} + 78 T_{2}^{95} - 42 T_{2}^{94} + 506 T_{2}^{93} + \cdots + 331776 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display