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Properties

Label 369.2.e.a
Level $369$
Weight $2$
Character orbit 369.e
Analytic conductor $2.946$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,2,Mod(124,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.e (of order \(3\), degree \(2\), 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: \(2.94647983459\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} - 8 q^{4} + q^{5} + 13 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} - 8 q^{4} + q^{5} + 13 q^{7} + 3 q^{9} - 16 q^{10} - 2 q^{11} - 23 q^{12} + 5 q^{13} + q^{14} + 9 q^{15} + 8 q^{16} + 2 q^{17} + 21 q^{18} - 38 q^{19} + 11 q^{20} - 6 q^{21} + 14 q^{22} + 2 q^{23} + 15 q^{24} + 3 q^{25} + 30 q^{26} + 11 q^{27} - 48 q^{28} - 10 q^{29} - 59 q^{30} + 39 q^{31} - 10 q^{32} + 6 q^{33} + 18 q^{34} + 4 q^{35} + 14 q^{36} - 52 q^{37} - 14 q^{38} - 20 q^{39} + 18 q^{40} + 16 q^{41} + 11 q^{42} + 20 q^{43} - 46 q^{44} + 31 q^{45} - 52 q^{46} + 5 q^{47} - 72 q^{48} + 7 q^{49} + 9 q^{50} - 31 q^{51} + 14 q^{52} + 14 q^{53} + 36 q^{54} - 74 q^{55} - 2 q^{56} - 6 q^{57} + 24 q^{58} + 6 q^{59} - 17 q^{60} + 21 q^{61} - 10 q^{62} + 56 q^{63} - 68 q^{64} + 2 q^{65} - 23 q^{66} + 32 q^{67} - 8 q^{68} - 29 q^{69} + 22 q^{70} - 40 q^{71} + 87 q^{72} - 34 q^{73} + 28 q^{74} + 24 q^{75} + 40 q^{76} - 22 q^{77} + 30 q^{78} + 69 q^{79} + 78 q^{80} + 63 q^{81} + 10 q^{83} - 70 q^{84} + 42 q^{85} - 8 q^{86} - 58 q^{87} + 42 q^{88} + 22 q^{89} + 49 q^{90} - 102 q^{91} - 10 q^{92} - 5 q^{93} + 13 q^{94} - 22 q^{95} + 21 q^{96} + 25 q^{97} - 96 q^{98} + 23 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1 −1.26536 + 2.19167i 1.69326 − 0.364528i −2.20228 − 3.81446i 0.592169 + 1.02567i −1.34366 + 4.17232i 1.45439 − 2.51909i 6.08526 2.73424 − 1.23448i −2.99723
124.2 −0.987647 + 1.71066i −1.61969 + 0.613673i −0.950894 − 1.64700i −0.800059 − 1.38574i 0.549903 − 3.37683i 0.367319 − 0.636215i −0.193998 2.24681 − 1.98792i 3.16071
124.3 −0.966816 + 1.67457i 1.03213 − 1.39094i −0.869467 − 1.50596i 1.86893 + 3.23707i 1.33135 + 3.07316i −0.233357 + 0.404186i −0.504807 −0.869426 − 2.87125i −7.22763
124.4 −0.954563 + 1.65335i −0.947667 − 1.44980i −0.822382 − 1.42441i 0.112697 + 0.195198i 3.30164 − 0.182900i 0.630622 − 1.09227i −0.678190 −1.20385 + 2.74786i −0.430307
124.5 −0.484849 + 0.839783i −0.272051 + 1.71055i 0.529843 + 0.917715i 0.894340 + 1.54904i −1.30459 − 1.05782i −0.911059 + 1.57800i −2.96697 −2.85198 − 0.930716i −1.73448
124.6 −0.395185 + 0.684480i −1.70940 − 0.279205i 0.687658 + 1.19106i −0.0693987 − 0.120202i 0.866638 − 1.05971i 2.32472 − 4.02654i −2.66775 2.84409 + 0.954544i 0.109701
124.7 −0.373128 + 0.646277i −0.160252 + 1.72462i 0.721551 + 1.24976i −1.18163 − 2.04664i −1.05479 − 0.747072i 0.193114 − 0.334484i −2.56944 −2.94864 − 0.552748i 1.76359
124.8 −0.111854 + 0.193736i 1.67652 − 0.435046i 0.974977 + 1.68871i 0.0946835 + 0.163997i −0.103241 + 0.373465i −1.37723 + 2.38544i −0.883634 2.62147 − 1.45873i −0.0423628
124.9 0.0766551 − 0.132771i 1.63859 + 0.561255i 0.988248 + 1.71170i −1.89658 − 3.28497i 0.200125 − 0.174534i 1.48279 − 2.56827i 0.609637 2.36999 + 1.83934i −0.581529
124.10 0.178971 − 0.309987i −1.39349 + 1.02868i 0.935939 + 1.62109i 1.28139 + 2.21943i 0.0694843 + 0.616068i 0.644358 − 1.11606i 1.38591 0.883621 − 2.86692i 0.917326
124.11 0.474444 − 0.821762i −0.261726 − 1.71216i 0.549805 + 0.952290i −1.41059 − 2.44321i −1.53116 − 0.597249i −0.217294 + 0.376365i 2.94119 −2.86300 + 0.896235i −2.67698
124.12 0.779106 − 1.34945i −1.72925 − 0.0985029i −0.214012 − 0.370680i 0.276060 + 0.478149i −1.48019 + 2.25679i −1.91437 + 3.31579i 2.44947 2.98059 + 0.340672i 0.860319
124.13 0.820512 − 1.42117i 1.55494 + 0.763004i −0.346479 − 0.600120i 0.171853 + 0.297657i 2.36020 − 1.58377i 0.00181771 − 0.00314836i 2.14489 1.83565 + 2.37285i 0.564028
124.14 0.881207 − 1.52630i −0.938111 − 1.45600i −0.553053 − 0.957916i 2.03169 + 3.51900i −3.04896 + 0.148793i 2.50362 − 4.33640i 1.57541 −1.23990 + 2.73179i 7.16138
124.15 1.12092 − 1.94148i 0.323299 − 1.70161i −1.51291 − 2.62043i −0.671092 − 1.16236i −2.94126 − 2.53504i −0.114415 + 0.198173i −2.29971 −2.79096 − 1.10026i −3.00895
124.16 1.20759 − 2.09161i 0.612897 + 1.61999i −1.91655 − 3.31956i −0.794470 − 1.37606i 4.12851 + 0.674340i 1.66497 − 2.88381i −4.42726 −2.24871 + 1.98577i −3.83758
247.1 −1.26536 − 2.19167i 1.69326 + 0.364528i −2.20228 + 3.81446i 0.592169 − 1.02567i −1.34366 − 4.17232i 1.45439 + 2.51909i 6.08526 2.73424 + 1.23448i −2.99723
247.2 −0.987647 − 1.71066i −1.61969 − 0.613673i −0.950894 + 1.64700i −0.800059 + 1.38574i 0.549903 + 3.37683i 0.367319 + 0.636215i −0.193998 2.24681 + 1.98792i 3.16071
247.3 −0.966816 − 1.67457i 1.03213 + 1.39094i −0.869467 + 1.50596i 1.86893 − 3.23707i 1.33135 − 3.07316i −0.233357 − 0.404186i −0.504807 −0.869426 + 2.87125i −7.22763
247.4 −0.954563 − 1.65335i −0.947667 + 1.44980i −0.822382 + 1.42441i 0.112697 − 0.195198i 3.30164 + 0.182900i 0.630622 + 1.09227i −0.678190 −1.20385 − 2.74786i −0.430307
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.2.e.a ✓ 32
3.b odd 2 1 1107.2.e.a 32
9.c even 3 1 inner 369.2.e.a ✓ 32
9.c even 3 1 3321.2.a.i 16
9.d odd 6 1 1107.2.e.a 32
9.d odd 6 1 3321.2.a.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.2.e.a ✓ 32 1.a even 1 1 trivial
369.2.e.a ✓ 32 9.c even 3 1 inner
1107.2.e.a 32 3.b odd 2 1
1107.2.e.a 32 9.d odd 6 1
3321.2.a.i 16 9.c even 3 1
3321.2.a.j 16 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 20 T_{2}^{30} + 242 T_{2}^{28} + 2 T_{2}^{27} + 1902 T_{2}^{26} + 55 T_{2}^{25} + 11055 T_{2}^{24} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\). Copy content Toggle raw display