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Properties

Label 297.2.j.b
Level $297$
Weight $2$
Character orbit 297.j
Analytic conductor $2.372$
Analytic rank $0$
Dimension $72$
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: \(72\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{2} + 9 q^{3} + 3 q^{4} + 12 q^{5} + 3 q^{7} - 3 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{2} + 9 q^{3} + 3 q^{4} + 12 q^{5} + 3 q^{7} - 3 q^{8} - 9 q^{9} + 18 q^{10} - 9 q^{12} + 15 q^{13} - 21 q^{14} - 9 q^{15} + 15 q^{16} + 12 q^{17} + 9 q^{18} + 42 q^{19} - 48 q^{20} + 9 q^{21} - 3 q^{22} - 6 q^{23} - 54 q^{24} + 6 q^{25} - 6 q^{26} + 27 q^{27} - 84 q^{28} - 12 q^{29} + 36 q^{30} + 21 q^{31} - 87 q^{32} - 3 q^{34} + 3 q^{35} - 90 q^{36} + 48 q^{37} - 18 q^{38} - 18 q^{39} - 69 q^{40} + 6 q^{41} + 54 q^{42} - 24 q^{43} + 30 q^{44} + 9 q^{45} + 36 q^{46} + 45 q^{47} - 36 q^{48} + 15 q^{49} + 54 q^{50} - 54 q^{51} + 18 q^{52} + 78 q^{53} - 27 q^{54} - 18 q^{55} - 51 q^{56} + 9 q^{57} + 33 q^{58} + 9 q^{59} + 108 q^{60} - 18 q^{62} - 117 q^{63} + 21 q^{64} - 99 q^{65} + 9 q^{66} - 63 q^{67} + 3 q^{68} + 72 q^{69} - 141 q^{70} - 9 q^{71} + 72 q^{72} + 54 q^{73} - 27 q^{74} - 36 q^{75} + 39 q^{76} - 3 q^{77} - 207 q^{78} + 18 q^{80} + 63 q^{81} - 54 q^{82} + 33 q^{83} + 144 q^{84} + 24 q^{85} + 33 q^{86} - 63 q^{87} + 3 q^{88} - 24 q^{89} - 54 q^{90} + 63 q^{91} + 108 q^{92} - 90 q^{93} - 87 q^{94} + 27 q^{96} - 18 q^{97} + 60 q^{98}+O(q^{100}) \) 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 −2.42343 + 0.882056i −0.123860 + 1.72762i 3.56289 − 2.98962i −0.315091 − 1.78697i −1.22369 − 4.29601i 0.457807 + 0.384146i −3.41844 + 5.92092i −2.96932 − 0.427966i 2.33980 + 4.05266i
34.2 −1.97896 + 0.720283i −0.160848 − 1.72457i 1.86539 − 1.56525i 0.00594649 + 0.0337242i 1.56049 + 3.29699i 2.22172 + 1.86425i −0.458146 + 0.793532i −2.94826 + 0.554784i −0.0360589 − 0.0624558i
34.3 −1.88903 + 0.687549i −1.42329 + 0.987045i 1.56360 − 1.31202i 0.346253 + 1.96370i 2.00998 − 2.84313i −2.86771 − 2.40630i −0.0413521 + 0.0716240i 1.05148 − 2.80969i −2.00422 − 3.47141i
34.4 −0.713540 + 0.259707i 0.162484 + 1.72441i −1.09040 + 0.914953i 0.455762 + 2.58475i −0.563781 − 1.18824i 1.45929 + 1.22449i 1.29976 − 2.25124i −2.94720 + 0.560379i −0.996483 − 1.72596i
34.5 −0.255534 + 0.0930069i 0.911660 − 1.47271i −1.47544 + 1.23804i −0.261871 − 1.48515i −0.0959883 + 0.461119i −1.88600 − 1.58254i 0.533813 − 0.924592i −1.33775 − 2.68522i 0.205046 + 0.355150i
34.6 0.0185602 − 0.00675535i 1.68036 + 0.419991i −1.53179 + 1.28532i −0.588924 − 3.33995i 0.0340249 − 0.00355630i 2.70571 + 2.27036i −0.0394988 + 0.0684139i 2.64721 + 1.41147i −0.0334931 − 0.0580117i
34.7 0.405168 − 0.147469i −0.815964 − 1.52781i −1.38967 + 1.16608i 0.230534 + 1.30742i −0.555908 − 0.498691i 1.78252 + 1.49572i −0.822263 + 1.42420i −1.66841 + 2.49328i 0.286209 + 0.495729i
34.8 0.541883 − 0.197229i 1.73066 − 0.0693428i −1.27735 + 1.07182i 0.771386 + 4.37475i 0.924140 − 0.378913i −3.12771 − 2.62446i −1.05744 + 1.83154i 2.99038 − 0.240018i 1.28083 + 2.21846i
34.9 1.45898 − 0.531026i −0.722042 + 1.57437i 0.314548 − 0.263937i 0.207329 + 1.17582i −0.217412 + 2.68041i 1.08501 + 0.910433i −1.23385 + 2.13710i −1.95731 − 2.27353i 0.926879 + 1.60540i
34.10 1.88361 − 0.685576i 1.57781 + 0.714510i 1.54587 − 1.29714i −0.0785322 − 0.445378i 3.46182 + 0.264149i −0.441942 − 0.370833i 0.0180242 − 0.0312189i 1.97895 + 2.25472i −0.453265 − 0.785077i
34.11 2.14884 − 0.782114i 0.403859 − 1.68431i 2.47373 − 2.07570i 0.294249 + 1.66877i −0.449493 − 3.93518i −0.778828 − 0.653514i 1.40546 − 2.43433i −2.67380 − 1.36045i 1.93746 + 3.35578i
34.12 2.24314 − 0.816437i −1.72083 + 0.196813i 2.83303 − 2.37719i −0.425481 − 2.41302i −3.69939 + 1.84643i −3.17405 − 2.66335i 2.02695 − 3.51079i 2.92253 − 0.677364i −2.92449 − 5.06537i
67.1 −0.476843 − 2.70431i −1.65316 − 0.516794i −5.20652 + 1.89502i 2.80605 + 2.35456i −0.609274 + 4.71707i 0.111242 + 0.0404889i 4.86138 + 8.42016i 2.46585 + 1.70868i 5.02940 − 8.71118i
67.2 −0.294223 − 1.66862i 1.68594 − 0.396984i −0.818353 + 0.297856i −2.00628 − 1.68347i −1.15846 − 2.69640i 2.02584 + 0.737346i −0.956576 − 1.65684i 2.68481 − 1.33859i −2.21878 + 3.84305i
67.3 −0.195342 − 1.10784i 1.46924 − 0.917241i 0.690237 − 0.251226i 2.56160 + 2.14944i −1.30316 − 1.44851i −3.43048 − 1.24859i −1.53808 − 2.66403i 1.31734 − 2.69530i 1.88084 − 3.25771i
67.4 −0.180001 − 1.02084i 0.861915 + 1.50237i 0.869673 − 0.316535i 2.18332 + 1.83202i 1.37853 − 1.15030i 0.167861 + 0.0610962i −1.51626 − 2.62624i −1.51421 + 2.58982i 1.47720 − 2.55858i
67.5 −0.135351 − 0.767614i −1.59303 − 0.679902i 1.30847 − 0.476246i 0.155126 + 0.130166i −0.306284 + 1.31485i 4.78670 + 1.74222i −1.32213 − 2.29000i 2.07547 + 2.16620i 0.0789209 − 0.136695i
67.6 −0.0700216 − 0.397112i −0.994039 − 1.41841i 1.72659 − 0.628427i −0.847641 − 0.711255i −0.493664 + 0.494064i −3.34426 − 1.21721i −0.773693 − 1.34008i −1.02377 + 2.81991i −0.223095 + 0.386412i
67.7 0.0473674 + 0.268634i 1.12670 + 1.31550i 1.80946 − 0.658591i −1.61300 − 1.35347i −0.300018 + 0.364983i 2.12159 + 0.772196i 0.535407 + 0.927353i −0.461072 + 2.96436i 0.287183 − 0.497416i
67.8 0.190571 + 1.08078i 0.794717 − 1.53897i 0.747610 − 0.272108i 0.646840 + 0.542763i 1.81474 + 0.565633i 0.441665 + 0.160753i 1.53402 + 2.65700i −1.73685 − 2.44609i −0.463340 + 0.802529i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.12
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.b ✓ 72
3.b odd 2 1 891.2.j.b 72
27.e even 9 1 inner 297.2.j.b ✓ 72
27.e even 9 1 8019.2.a.h 36
27.f odd 18 1 891.2.j.b 72
27.f odd 18 1 8019.2.a.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.b ✓ 72 1.a even 1 1 trivial
297.2.j.b ✓ 72 27.e even 9 1 inner
891.2.j.b 72 3.b odd 2 1
891.2.j.b 72 27.f odd 18 1
8019.2.a.g 36 27.f odd 18 1
8019.2.a.h 36 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 3 T_{2}^{71} + 3 T_{2}^{70} + T_{2}^{69} - 30 T_{2}^{67} + 519 T_{2}^{66} - 1470 T_{2}^{65} + \cdots + 12321 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display