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Properties

Label 338.2.i.a
Level $338$
Weight $2$
Character orbit 338.i
Analytic conductor $2.699$
Analytic rank $0$
Dimension $168$
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(3,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([62]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.i (of order \(39\), 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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(7\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 7 q^{2} + 7 q^{4} + 14 q^{7} + 14 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 7 q^{2} + 7 q^{4} + 14 q^{7} + 14 q^{8} + 7 q^{9} + q^{11} + 26 q^{13} + 2 q^{14} - 23 q^{15} + 7 q^{16} - q^{17} + 14 q^{18} + 40 q^{19} - 6 q^{21} - 14 q^{22} - 2 q^{23} - 10 q^{25} + 45 q^{27} + q^{28} - 3 q^{29} + 49 q^{30} - 34 q^{31} - 7 q^{32} + 4 q^{33} + 11 q^{34} - 5 q^{35} + 33 q^{36} + 4 q^{37} + 41 q^{38} - 156 q^{39} - 13 q^{40} + 2 q^{41} + 23 q^{42} + 50 q^{43} - 2 q^{44} - 188 q^{45} - 24 q^{46} + 7 q^{47} - 11 q^{49} - 5 q^{50} - 51 q^{51} + 45 q^{53} + 81 q^{54} - 238 q^{55} + 12 q^{56} - 51 q^{57} + 3 q^{58} + 57 q^{59} - 32 q^{60} - 5 q^{61} - 17 q^{62} - 239 q^{63} - 14 q^{64} + 8 q^{66} - 75 q^{67} + 25 q^{68} + 18 q^{69} - 166 q^{70} + 111 q^{71} - 59 q^{72} - 12 q^{73} - 30 q^{74} - 167 q^{75} - 12 q^{76} - 19 q^{77} + 68 q^{79} + 13 q^{80} - 5 q^{81} - 80 q^{82} + 168 q^{83} - 10 q^{84} + 3 q^{85} - 17 q^{86} - 23 q^{87} - q^{88} - 154 q^{89} + 14 q^{90} + 39 q^{91} + 4 q^{92} + 49 q^{93} - 94 q^{94} + 222 q^{95} - 37 q^{97} + 11 q^{98} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.799443 + 0.600742i −0.596623 + 2.92245i 0.278217 − 0.960518i 1.66214 + 0.872359i −1.27867 − 2.69475i −2.64958 − 0.430599i 0.354605 + 0.935016i −5.42483 − 2.31131i −1.85285 + 0.301117i
3.2 −0.799443 + 0.600742i −0.375407 + 1.83886i 0.278217 − 0.960518i −0.179285 − 0.0940960i −0.804567 − 1.69559i 4.59523 + 0.746798i 0.354605 + 0.935016i −0.480551 − 0.204744i 0.199856 − 0.0324797i
3.3 −0.799443 + 0.600742i −0.0809421 + 0.396481i 0.278217 − 0.960518i −1.81029 − 0.950115i −0.173474 − 0.365589i −3.01013 − 0.489193i 0.354605 + 0.935016i 2.60929 + 1.11172i 2.01800 − 0.327957i
3.4 −0.799443 + 0.600742i 0.0138336 − 0.0677613i 0.278217 − 0.960518i −3.09037 − 1.62195i 0.0296479 + 0.0624816i 0.564053 + 0.0916675i 0.354605 + 0.935016i 2.75554 + 1.17403i 3.44495 − 0.559859i
3.5 −0.799443 + 0.600742i 0.248746 − 1.21844i 0.278217 − 0.960518i 3.75085 + 1.96860i 0.533110 + 1.12350i −0.341015 − 0.0554203i 0.354605 + 0.935016i 1.33722 + 0.569736i −4.18121 + 0.679513i
3.6 −0.799443 + 0.600742i 0.272721 − 1.33588i 0.278217 − 0.960518i 0.595050 + 0.312307i 0.584493 + 1.23179i 1.91617 + 0.311407i 0.354605 + 0.935016i 1.04975 + 0.447255i −0.663324 + 0.107801i
3.7 −0.799443 + 0.600742i 0.517671 − 2.53572i 0.278217 − 0.960518i −0.184347 − 0.0967527i 1.10947 + 2.33815i −3.27796 − 0.532720i 0.354605 + 0.935016i −3.40195 − 1.44943i 0.205498 − 0.0333967i
9.1 −0.278217 + 0.960518i −2.82245 − 1.20253i −0.845190 − 0.534466i −0.746211 − 1.08107i 1.94031 − 2.37644i −3.27788 − 1.09432i 0.748511 − 0.663123i 4.44194 + 4.62455i 1.24600 − 0.415976i
9.2 −0.278217 + 0.960518i −1.36387 − 0.581093i −0.845190 − 0.534466i 0.335488 + 0.486038i 0.937604 − 1.14836i 0.682355 + 0.227804i 0.748511 − 0.663123i −0.555688 − 0.578533i −0.560187 + 0.187018i
9.3 −0.278217 + 0.960518i −0.529159 − 0.225454i −0.845190 − 0.534466i 2.18322 + 3.16294i 0.363774 − 0.445542i 1.50394 + 0.502089i 0.748511 − 0.663123i −1.84899 − 1.92501i −3.64547 + 1.21704i
9.4 −0.278217 + 0.960518i −0.0524887 − 0.0223634i −0.845190 − 0.534466i −0.665703 − 0.964437i 0.0360837 − 0.0441945i −1.13989 − 0.380552i 0.748511 − 0.663123i −2.07592 − 2.16126i 1.11157 − 0.371096i
9.5 −0.278217 + 0.960518i 0.969364 + 0.413008i −0.845190 − 0.534466i −1.50582 − 2.18156i −0.666395 + 0.816186i 4.03650 + 1.34758i 0.748511 − 0.663123i −1.30908 − 1.36290i 2.51437 − 0.839421i
9.6 −0.278217 + 0.960518i 1.53134 + 0.652444i −0.845190 − 0.534466i −2.03895 − 2.95392i −1.05273 + 1.28936i −4.38811 − 1.46497i 0.748511 − 0.663123i −0.158850 − 0.165380i 3.40457 − 1.13661i
9.7 −0.278217 + 0.960518i 2.26726 + 0.965990i −0.845190 − 0.534466i 1.08288 + 1.56883i −1.55864 + 1.90899i 1.29414 + 0.432047i 0.748511 − 0.663123i 2.12917 + 2.21670i −1.80816 + 0.603653i
29.1 −0.692724 − 0.721202i −1.89791 + 1.42619i −0.0402659 + 0.999189i −0.700093 − 1.01426i 2.34330 + 0.380823i −0.447211 − 2.19059i 0.748511 − 0.663123i 0.733401 − 2.53199i −0.246515 + 1.20751i
29.2 −0.692724 − 0.721202i −1.50068 + 1.12769i −0.0402659 + 0.999189i 2.18674 + 3.16804i 1.85285 + 0.301118i −0.0957224 − 0.468879i 0.748511 − 0.663123i 0.145713 − 0.503058i 0.769990 − 3.77166i
29.3 −0.692724 − 0.721202i −1.49312 + 1.12201i −0.0402659 + 0.999189i −1.17571 − 1.70331i 1.84352 + 0.299601i 0.727452 + 3.56330i 0.748511 − 0.663123i 0.135861 − 0.469046i −0.413989 + 2.02785i
29.4 −0.692724 − 0.721202i 0.614954 − 0.462108i −0.0402659 + 0.999189i 0.214440 + 0.310670i −0.759268 − 0.123393i 0.692871 + 3.39390i 0.748511 − 0.663123i −0.670027 + 2.31320i 0.0755080 − 0.369863i
29.5 −0.692724 − 0.721202i 0.724915 − 0.544738i −0.0402659 + 0.999189i 1.24346 + 1.80147i −0.895033 − 0.145457i −0.0936970 − 0.458958i 0.748511 − 0.663123i −0.605890 + 2.09178i 0.437845 − 2.14471i
29.6 −0.692724 − 0.721202i 1.11058 − 0.834546i −0.0402659 + 0.999189i −1.00038 − 1.44930i −1.37120 − 0.222842i −0.420286 − 2.05870i 0.748511 − 0.663123i −0.297734 + 1.02790i −0.352250 + 1.72544i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.i.a ✓ 168
169.i even 39 1 inner 338.2.i.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.i.a ✓ 168 1.a even 1 1 trivial
338.2.i.a ✓ 168 169.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{168} - 14 T_{3}^{166} - 15 T_{3}^{165} + 52 T_{3}^{164} + 132 T_{3}^{163} + 667 T_{3}^{162} + \cdots + 15\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display