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Properties

Label 338.2.g.a
Level $338$
Weight $2$
Character orbit 338.g
Analytic conductor $2.699$
Analytic rank $0$
Dimension $96$
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] = [338,2,Mod(27,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.g (of order \(13\), 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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(8\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{2} - q^{3} - 8 q^{4} + 3 q^{5} + q^{6} - 13 q^{7} + 8 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{2} - q^{3} - 8 q^{4} + 3 q^{5} + q^{6} - 13 q^{7} + 8 q^{8} - 5 q^{9} - 3 q^{10} - 7 q^{11} - q^{12} + 25 q^{13} + 26 q^{15} - 8 q^{16} + 4 q^{17} + 5 q^{18} - 42 q^{19} + 3 q^{20} - 2 q^{21} + 20 q^{22} + 2 q^{23} + q^{24} - 9 q^{25} + q^{26} - 31 q^{27} - 3 q^{29} + 65 q^{30} + 26 q^{31} + 8 q^{32} - 10 q^{33} - 17 q^{34} + 2 q^{35} - 31 q^{36} + 3 q^{37} - 49 q^{38} - 40 q^{39} + 10 q^{40} - 2 q^{41} - 24 q^{42} - 49 q^{43} - 7 q^{44} - 13 q^{45} - 15 q^{46} + 98 q^{47} - q^{48} + 17 q^{49} + 9 q^{50} + 75 q^{51} - q^{52} - 36 q^{53} + 31 q^{54} - 17 q^{55} - 26 q^{56} + 31 q^{57} + 3 q^{58} - 51 q^{59} + 26 q^{60} - 3 q^{61} - 26 q^{62} - 36 q^{63} - 8 q^{64} - 75 q^{65} + 10 q^{66} + 61 q^{67} - 22 q^{68} - 18 q^{69} + 76 q^{70} + 9 q^{71} - 21 q^{72} - 8 q^{73} + 23 q^{74} + 163 q^{75} + 10 q^{76} - 179 q^{77} + q^{78} - 52 q^{79} - 10 q^{80} + 4 q^{81} - 37 q^{82} - 6 q^{83} - 67 q^{84} - 12 q^{85} + 10 q^{86} - 61 q^{87} + 7 q^{88} - 74 q^{89} + 13 q^{90} - 38 q^{91} - 37 q^{92} - 32 q^{93} - 20 q^{94} - 60 q^{95} + q^{96} + 47 q^{97} - 17 q^{98} - 246 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 0.354605 + 0.935016i −1.27313 + 1.84444i −0.748511 + 0.663123i 0.533964 + 4.39759i −2.17604 − 0.536346i −1.61764 − 0.849004i −0.885456 − 0.464723i −0.717303 − 1.89137i −3.92247 + 2.05867i
27.2 0.354605 + 0.935016i −0.943457 + 1.36683i −0.748511 + 0.663123i 0.133458 + 1.09913i −1.61257 − 0.397462i 3.81467 + 2.00209i −0.885456 − 0.464723i 0.0856918 + 0.225951i −0.980378 + 0.514542i
27.3 0.354605 + 0.935016i −0.892540 + 1.29307i −0.748511 + 0.663123i −0.532751 − 4.38760i −1.52554 − 0.376011i −1.47052 − 0.771788i −0.885456 − 0.464723i 0.188419 + 0.496819i 3.91356 − 2.05399i
27.4 0.354605 + 0.935016i −0.225984 + 0.327395i −0.748511 + 0.663123i 0.0333270 + 0.274473i −0.386254 − 0.0952032i −4.06632 − 2.13417i −0.885456 − 0.464723i 1.00770 + 2.65708i −0.244819 + 0.128491i
27.5 0.354605 + 0.935016i −0.137526 + 0.199240i −0.748511 + 0.663123i −0.204588 − 1.68493i −0.235060 − 0.0579371i 1.63883 + 0.860124i −0.885456 − 0.464723i 1.04303 + 2.75025i 1.50289 − 0.788778i
27.6 0.354605 + 0.935016i 1.09346 − 1.58416i −0.748511 + 0.663123i 0.370268 + 3.04943i 1.86896 + 0.460657i 0.755731 + 0.396638i −0.885456 − 0.464723i −0.250071 − 0.659383i −2.71997 + 1.42755i
27.7 0.354605 + 0.935016i 1.39590 − 2.02231i −0.748511 + 0.663123i −0.185649 − 1.52896i 2.38588 + 0.588067i −3.39620 − 1.78247i −0.885456 − 0.464723i −1.07738 − 2.84082i 1.36377 − 0.715760i
27.8 0.354605 + 0.935016i 1.55134 − 2.24750i −0.748511 + 0.663123i −0.183081 − 1.50781i 2.65156 + 0.653551i 3.01446 + 1.58211i −0.885456 − 0.464723i −1.58080 − 4.16822i 1.34490 − 0.705860i
53.1 0.748511 − 0.663123i −1.04848 − 2.76462i 0.120537 − 0.992709i −3.78486 + 0.932884i −2.61808 − 1.37407i 0.369186 + 0.534858i −0.568065 − 0.822984i −4.29826 + 3.80792i −2.21439 + 3.20810i
53.2 0.748511 − 0.663123i −0.857853 − 2.26197i 0.120537 − 0.992709i 2.30408 − 0.567904i −2.14208 − 1.12425i −1.05333 − 1.52601i −0.568065 − 0.822984i −2.13507 + 1.89151i 1.34804 − 1.95297i
53.3 0.748511 − 0.663123i −0.516622 − 1.36222i 0.120537 − 0.992709i −0.300734 + 0.0741243i −1.29002 − 0.677053i −0.326405 − 0.472879i −0.568065 − 0.822984i 0.656786 − 0.581861i −0.175949 + 0.254906i
53.4 0.748511 − 0.663123i −0.0735898 − 0.194040i 0.120537 − 0.992709i −0.0818378 + 0.0201712i −0.183755 − 0.0964422i 2.35891 + 3.41748i −0.568065 − 0.822984i 2.21330 − 1.96081i −0.0478805 + 0.0693669i
53.5 0.748511 − 0.663123i 0.261721 + 0.690101i 0.120537 − 0.992709i 1.60575 − 0.395783i 0.653523 + 0.342995i −1.88781 − 2.73496i −0.568065 − 0.822984i 1.83779 − 1.62814i 0.939472 − 1.36106i
53.6 0.748511 − 0.663123i 0.269164 + 0.709726i 0.120537 − 0.992709i −4.14333 + 1.02124i 0.672107 + 0.352749i −2.95846 − 4.28607i −0.568065 − 0.822984i 1.81427 − 1.60730i −2.42412 + 3.51195i
53.7 0.748511 − 0.663123i 0.743842 + 1.96135i 0.120537 − 0.992709i 0.276459 − 0.0681411i 1.85739 + 0.974833i 0.0160166 + 0.0232041i −0.568065 − 0.822984i −1.04806 + 0.928500i 0.161747 − 0.234331i
53.8 0.748511 − 0.663123i 0.867214 + 2.28666i 0.120537 − 0.992709i 3.64190 − 0.897647i 2.16545 + 1.13652i 0.894608 + 1.29606i −0.568065 − 0.822984i −2.23120 + 1.97667i 2.13075 − 3.08692i
79.1 −0.885456 − 0.464723i −2.97363 − 0.732933i 0.568065 + 0.822984i 1.20072 + 3.16603i 2.29240 + 2.03089i −0.350703 − 2.88830i −0.120537 − 0.992709i 5.64889 + 2.96477i 0.408146 − 3.36138i
79.2 −0.885456 − 0.464723i −2.44981 − 0.603825i 0.568065 + 0.822984i −1.34679 − 3.55120i 1.88859 + 1.67315i −0.430610 − 3.54640i −0.120537 − 0.992709i 2.98062 + 1.56435i −0.457800 + 3.77032i
79.3 −0.885456 − 0.464723i −1.77262 − 0.436911i 0.568065 + 0.822984i −0.441162 − 1.16325i 1.36653 + 1.21064i 0.279739 + 2.30386i −0.120537 − 0.992709i 0.294918 + 0.154785i −0.149959 + 1.23502i
79.4 −0.885456 − 0.464723i −0.263672 − 0.0649894i 0.568065 + 0.822984i 0.541063 + 1.42667i 0.203268 + 0.180080i −0.353221 − 2.90903i −0.120537 − 0.992709i −2.59107 − 1.35990i 0.183917 − 1.51469i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.g.a ✓ 96
169.g even 13 1 inner 338.2.g.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.g.a ✓ 96 1.a even 1 1 trivial
338.2.g.a ✓ 96 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + T_{3}^{95} + 15 T_{3}^{94} + 27 T_{3}^{93} + 171 T_{3}^{92} + 417 T_{3}^{91} + \cdots + 103330745401 \) acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display