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Properties

Label 273.2.cg.a
Level $273$
Weight $2$
Character orbit 273.cg
Analytic conductor $2.180$
Analytic rank $0$
Dimension $36$
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] = [273,2,Mod(19,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cg (of order \(12\), degree \(4\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} - 36 q^{9} + 4 q^{11} + 16 q^{12} + 42 q^{14} + 12 q^{16} - 4 q^{17} - 24 q^{19} - 14 q^{20} + 4 q^{22} - 12 q^{23} - 24 q^{25} - 28 q^{26} - 12 q^{28} + 8 q^{29} - 6 q^{31} + 46 q^{32} + 4 q^{33} + 24 q^{34} - 10 q^{35} - 20 q^{37} + 8 q^{38} - 2 q^{39} - 30 q^{40} - 34 q^{41} + 24 q^{42} + 30 q^{43} - 32 q^{44} - 26 q^{46} + 4 q^{47} - 24 q^{48} - 20 q^{50} + 24 q^{51} + 98 q^{52} - 8 q^{53} + 30 q^{55} - 10 q^{56} - 24 q^{57} - 96 q^{58} - 14 q^{59} - 46 q^{60} + 48 q^{62} - 4 q^{63} + 28 q^{65} + 18 q^{66} + 62 q^{67} - 54 q^{68} - 4 q^{69} - 148 q^{70} + 42 q^{71} - 52 q^{73} - 20 q^{74} - 10 q^{75} - 12 q^{76} - 24 q^{77} - 16 q^{78} + 76 q^{80} + 36 q^{81} + 48 q^{82} + 60 q^{83} + 50 q^{84} + 2 q^{85} + 12 q^{86} + 18 q^{87} + 50 q^{89} + 40 q^{91} - 100 q^{92} - 6 q^{93} + 24 q^{95} - 4 q^{96} - 36 q^{97} + 16 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.38279 − 0.638467i − 1.00000i 3.53801 + 2.04267i 0.488495 − 0.130892i −0.638467 + 2.38279i −2.15177 + 1.53945i −3.63751 − 3.63751i −1.00000 −1.24755
19.2 −1.53002 − 0.409967i − 1.00000i 0.440828 + 0.254512i −3.63307 + 0.973479i −0.409967 + 1.53002i −1.31124 − 2.29797i 1.66997 + 1.66997i −1.00000 5.95776
19.3 −1.50246 − 0.402582i − 1.00000i 0.363252 + 0.209723i 2.78312 − 0.745735i −0.402582 + 1.50246i 0.599883 + 2.57685i 1.73841 + 1.73841i −1.00000 −4.48173
19.4 −0.759625 − 0.203541i − 1.00000i −1.19645 − 0.690770i 2.02645 − 0.542987i −0.203541 + 0.759625i 1.65876 − 2.06120i 1.88042 + 1.88042i −1.00000 −1.64986
19.5 −0.511829 − 0.137144i − 1.00000i −1.48889 − 0.859611i −2.03763 + 0.545981i −0.137144 + 0.511829i 0.917679 + 2.48150i 1.39354 + 1.39354i −1.00000 1.11780
19.6 0.562494 + 0.150720i − 1.00000i −1.43837 − 0.830442i −0.672922 + 0.180309i 0.150720 − 0.562494i −2.49458 − 0.881516i −1.50746 − 1.50746i −1.00000 −0.405691
19.7 1.64664 + 0.441217i − 1.00000i 0.784712 + 0.453054i 1.35835 − 0.363968i 0.441217 − 1.64664i 0.501823 − 2.59772i −1.31861 − 1.31861i −1.00000 2.39731
19.8 2.02536 + 0.542694i − 1.00000i 2.07553 + 1.19831i 1.89270 − 0.507149i 0.542694 − 2.02536i −1.04247 + 2.43172i 0.588043 + 0.588043i −1.00000 4.10864
19.9 2.45222 + 0.657070i − 1.00000i 3.84958 + 2.22256i −2.20549 + 0.590961i 0.657070 − 2.45222i 2.58986 + 0.540936i 4.38935 + 4.38935i −1.00000 −5.79666
115.1 −2.38279 + 0.638467i 1.00000i 3.53801 − 2.04267i 0.488495 + 0.130892i −0.638467 − 2.38279i −2.15177 − 1.53945i −3.63751 + 3.63751i −1.00000 −1.24755
115.2 −1.53002 + 0.409967i 1.00000i 0.440828 − 0.254512i −3.63307 − 0.973479i −0.409967 − 1.53002i −1.31124 + 2.29797i 1.66997 − 1.66997i −1.00000 5.95776
115.3 −1.50246 + 0.402582i 1.00000i 0.363252 − 0.209723i 2.78312 + 0.745735i −0.402582 − 1.50246i 0.599883 − 2.57685i 1.73841 − 1.73841i −1.00000 −4.48173
115.4 −0.759625 + 0.203541i 1.00000i −1.19645 + 0.690770i 2.02645 + 0.542987i −0.203541 − 0.759625i 1.65876 + 2.06120i 1.88042 − 1.88042i −1.00000 −1.64986
115.5 −0.511829 + 0.137144i 1.00000i −1.48889 + 0.859611i −2.03763 − 0.545981i −0.137144 − 0.511829i 0.917679 − 2.48150i 1.39354 − 1.39354i −1.00000 1.11780
115.6 0.562494 − 0.150720i 1.00000i −1.43837 + 0.830442i −0.672922 − 0.180309i 0.150720 + 0.562494i −2.49458 + 0.881516i −1.50746 + 1.50746i −1.00000 −0.405691
115.7 1.64664 − 0.441217i 1.00000i 0.784712 − 0.453054i 1.35835 + 0.363968i 0.441217 + 1.64664i 0.501823 + 2.59772i −1.31861 + 1.31861i −1.00000 2.39731
115.8 2.02536 − 0.542694i 1.00000i 2.07553 − 1.19831i 1.89270 + 0.507149i 0.542694 + 2.02536i −1.04247 − 2.43172i 0.588043 − 0.588043i −1.00000 4.10864
115.9 2.45222 − 0.657070i 1.00000i 3.84958 − 2.22256i −2.20549 − 0.590961i 0.657070 + 2.45222i 2.58986 − 0.540936i 4.38935 − 4.38935i −1.00000 −5.79666
124.1 −0.639011 + 2.38482i 1.00000i −3.54699 − 2.04786i −0.746344 − 2.78539i −2.38482 − 0.639011i −2.52355 − 0.794791i 3.65872 − 3.65872i −1.00000 7.11959
124.2 −0.478662 + 1.78639i 1.00000i −1.23003 − 0.710156i −0.0199621 − 0.0744995i −1.78639 − 0.478662i 1.85948 + 1.88211i −0.758075 + 0.758075i −1.00000 0.142640
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cg.a yes 36
3.b odd 2 1 819.2.gh.c 36
7.d odd 6 1 273.2.bt.a ✓ 36
13.f odd 12 1 273.2.bt.a ✓ 36
21.g even 6 1 819.2.et.c 36
39.k even 12 1 819.2.et.c 36
91.w even 12 1 inner 273.2.cg.a yes 36
273.ch odd 12 1 819.2.gh.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.a ✓ 36 7.d odd 6 1
273.2.bt.a ✓ 36 13.f odd 12 1
273.2.cg.a yes 36 1.a even 1 1 trivial
273.2.cg.a yes 36 91.w even 12 1 inner
819.2.et.c 36 21.g even 6 1
819.2.et.c 36 39.k even 12 1
819.2.gh.c 36 3.b odd 2 1
819.2.gh.c 36 273.ch odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 63 T_{2}^{32} - 38 T_{2}^{31} + 224 T_{2}^{29} + 3079 T_{2}^{28} + 1750 T_{2}^{27} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display