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Properties

Label 351.2.w.c
Level $351$
Weight $2$
Character orbit 351.w
Analytic conductor $2.803$
Analytic rank $0$
Dimension $108$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(40,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.40"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 3 q^{5} + 9 q^{6} + 18 q^{8} - 3 q^{11} - 9 q^{12} - 21 q^{14} - 33 q^{15} + 24 q^{17} - 15 q^{18} + 3 q^{20} - 36 q^{21} - 9 q^{22} + 18 q^{23} + 36 q^{24} - 9 q^{25} + 12 q^{26} - 15 q^{27} - 18 q^{29}+ \cdots + 123 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −0.474271 − 2.68972i 0.0138992 − 1.73200i −5.13030 + 1.86728i 0.692001 + 0.580658i −4.66518 + 0.784050i −1.29701 − 0.472074i 4.72439 + 8.18289i −2.99961 − 0.0481468i 1.23361 − 2.13668i
40.2 −0.454147 − 2.57560i −1.66166 + 0.488766i −4.54806 + 1.65536i −3.21339 − 2.69635i 2.01350 + 4.05779i −2.77376 − 1.00957i 3.71370 + 6.43231i 2.52222 − 1.62432i −5.48536 + 9.50093i
40.3 −0.407915 − 2.31340i −1.64786 − 0.533443i −3.30605 + 1.20330i 1.66322 + 1.39561i −0.561882 + 4.02976i 3.96479 + 1.44307i 1.78322 + 3.08863i 2.43088 + 1.75808i 2.55014 − 4.41698i
40.4 −0.335767 − 1.90423i 1.68012 − 0.420951i −1.63396 + 0.594714i −0.367675 − 0.308516i −1.36571 − 3.05799i −4.32926 − 1.57572i −0.252503 − 0.437349i 2.64560 − 1.41449i −0.464031 + 0.803726i
40.5 −0.265829 − 1.50759i 1.06285 + 1.36760i −0.322775 + 0.117480i 3.03687 + 2.54823i 1.77925 − 1.96589i −1.05609 − 0.384386i −1.26793 − 2.19612i −0.740685 + 2.90713i 3.03440 − 5.25574i
40.6 −0.218353 − 1.23834i −0.839582 − 1.51496i 0.393569 − 0.143248i −1.95819 − 1.64311i −1.69271 + 1.37049i −0.151500 − 0.0551416i −1.52077 − 2.63406i −1.59020 + 2.54387i −1.60716 + 2.78369i
40.7 −0.199657 − 1.13231i 0.805016 + 1.53361i 0.637115 − 0.231891i −1.59335 − 1.33698i 1.57580 − 1.21773i 3.13337 + 1.14045i −1.53956 − 2.66659i −1.70390 + 2.46916i −1.19575 + 2.07110i
40.8 −0.0966103 − 0.547904i −0.427593 + 1.67844i 1.58852 − 0.578174i −2.41140 − 2.02340i 0.960935 + 0.0721256i −2.70519 − 0.984608i −1.02661 − 1.77814i −2.63433 − 1.43538i −0.875665 + 1.51670i
40.9 −0.0645564 − 0.366118i 1.10179 − 1.33644i 1.74951 − 0.636770i 1.06233 + 0.891400i −0.560421 − 0.317107i −0.539045 − 0.196196i −0.717840 − 1.24334i −0.572140 − 2.94494i 0.257777 − 0.446483i
40.10 −0.0531763 − 0.301578i −1.57955 + 0.710656i 1.79126 − 0.651967i 2.76513 + 2.32022i 0.298312 + 0.438566i 0.960792 + 0.349700i −0.598101 − 1.03594i 1.98994 − 2.24503i 0.552686 − 0.957281i
40.11 0.0994229 + 0.563855i 0.936471 − 1.45706i 1.57134 − 0.571920i −2.57242 − 2.15851i 0.914677 + 0.383169i 4.47311 + 1.62808i 1.05126 + 1.82084i −1.24604 − 2.72899i 0.961332 − 1.66508i
40.12 0.111970 + 0.635014i 1.51505 + 0.839421i 1.48868 − 0.541835i 0.752112 + 0.631097i −0.363404 + 1.05607i −2.92807 − 1.06573i 1.15557 + 2.00151i 1.59074 + 2.54353i −0.316541 + 0.548265i
40.13 0.160916 + 0.912600i −1.33058 + 1.10886i 1.07244 − 0.390337i −0.798571 − 0.670081i −1.22606 − 1.03585i 3.23158 + 1.17620i 1.45547 + 2.52095i 0.540865 − 2.95084i 0.483013 − 0.836603i
40.14 0.297507 + 1.68724i 0.0493675 − 1.73135i −0.878897 + 0.319893i 1.06448 + 0.893205i 2.93589 − 0.431792i 0.318876 + 0.116061i 0.912058 + 1.57973i −2.99513 − 0.170945i −1.19037 + 2.06177i
40.15 0.354444 + 2.01015i 1.71092 + 0.269707i −2.03571 + 0.740936i −0.707019 − 0.593259i 0.0642738 + 3.53482i 1.57446 + 0.573056i −0.169776 − 0.294061i 2.85452 + 0.922897i 0.941944 − 1.63149i
40.16 0.376880 + 2.13740i −1.16285 − 1.28366i −2.54703 + 0.927044i 0.504624 + 0.423430i 2.30543 − 2.96926i −0.518516 − 0.188725i −0.771020 − 1.33545i −0.295555 + 2.98541i −0.714854 + 1.23816i
40.17 0.386059 + 2.18945i −0.442686 + 1.67452i −2.76527 + 1.00648i 2.01217 + 1.68841i −3.83719 − 0.322774i 2.70241 + 0.983598i −1.04796 − 1.81513i −2.60806 − 1.48258i −2.91988 + 5.05738i
40.18 0.435786 + 2.47147i 1.15656 − 1.28933i −4.03885 + 1.47002i −2.90271 − 2.43566i 3.69054 + 2.29653i −4.06095 − 1.47807i −2.88359 − 4.99453i −0.324733 − 2.98237i 4.75470 − 8.23538i
79.1 −0.474271 + 2.68972i 0.0138992 + 1.73200i −5.13030 − 1.86728i 0.692001 − 0.580658i −4.66518 − 0.784050i −1.29701 + 0.472074i 4.72439 − 8.18289i −2.99961 + 0.0481468i 1.23361 + 2.13668i
79.2 −0.454147 + 2.57560i −1.66166 − 0.488766i −4.54806 − 1.65536i −3.21339 + 2.69635i 2.01350 − 4.05779i −2.77376 + 1.00957i 3.71370 − 6.43231i 2.52222 + 1.62432i −5.48536 − 9.50093i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 40.18
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 351.2.w.c ✓ 108
27.e even 9 1 inner 351.2.w.c ✓ 108
27.e even 9 1 9477.2.a.j 54
27.f odd 18 1 9477.2.a.k 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.w.c ✓ 108 1.a even 1 1 trivial
351.2.w.c ✓ 108 27.e even 9 1 inner
9477.2.a.j 54 27.e even 9 1
9477.2.a.k 54 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} - 14 T_{2}^{105} + 12 T_{2}^{103} + 872 T_{2}^{102} - 132 T_{2}^{101} - 168 T_{2}^{100} + \cdots + 37246609 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display