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Properties

Label 273.2.by.c
Level $273$
Weight $2$
Character orbit 273.by
Analytic conductor $2.180$
Analytic rank $0$
Dimension $32$
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(76,273)]
 
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, 6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.by (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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 6 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 6 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 16 q^{9} - 2 q^{10} - 4 q^{11} - 32 q^{12} - 6 q^{13} + 34 q^{14} + 4 q^{15} + 14 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 44 q^{20} + 2 q^{21} - 4 q^{22} - 18 q^{23} - 4 q^{24} + 28 q^{26} - 18 q^{28} - 18 q^{29} + 14 q^{31} - 8 q^{32} - 4 q^{33} + 66 q^{34} - 20 q^{35} + 6 q^{36} - 24 q^{37} - 24 q^{38} + 8 q^{39} + 16 q^{42} - 6 q^{43} - 20 q^{44} - 4 q^{45} - 58 q^{46} + 28 q^{47} + 60 q^{48} + 10 q^{49} + 70 q^{50} - 28 q^{52} - 80 q^{53} + 4 q^{54} - 60 q^{55} - 120 q^{56} + 16 q^{57} - 4 q^{58} + 42 q^{59} - 58 q^{60} - 36 q^{61} - 52 q^{62} + 2 q^{63} + 14 q^{65} + 26 q^{67} + 72 q^{68} - 2 q^{69} + 68 q^{70} - 4 q^{71} + 4 q^{72} - 12 q^{73} - 18 q^{74} - 16 q^{75} + 48 q^{76} - 28 q^{77} - 14 q^{78} - 4 q^{79} + 98 q^{80} - 16 q^{81} - 20 q^{82} + 36 q^{83} + 32 q^{84} - 10 q^{85} - 40 q^{86} + 96 q^{88} + 54 q^{89} - 4 q^{90} - 54 q^{91} - 4 q^{92} + 2 q^{93} + 60 q^{95} - 22 q^{96} + 40 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −0.706652 + 2.63726i 0.866025 − 0.500000i −4.72373 − 2.72725i 2.18431 − 2.18431i 0.706652 + 2.63726i −1.85732 − 1.88424i 6.66928 − 6.66928i 0.500000 − 0.866025i 4.21705 + 7.30414i
76.2 −0.500642 + 1.86842i 0.866025 − 0.500000i −1.50830 − 0.870817i −2.44068 + 2.44068i 0.500642 + 1.86842i −2.60515 − 0.461729i −0.353388 + 0.353388i 0.500000 − 0.866025i −3.33831 − 5.78213i
76.3 −0.281068 + 1.04896i 0.866025 − 0.500000i 0.710736 + 0.410344i 1.02734 − 1.02734i 0.281068 + 1.04896i −1.22381 + 2.34570i −2.16598 + 2.16598i 0.500000 − 0.866025i 0.788887 + 1.36639i
76.4 −0.189683 + 0.707908i 0.866025 − 0.500000i 1.26690 + 0.731443i 1.23329 − 1.23329i 0.189683 + 0.707908i 2.32720 − 1.25862i −1.79455 + 1.79455i 0.500000 − 0.866025i 0.639122 + 1.10699i
76.5 0.0473445 − 0.176692i 0.866025 − 0.500000i 1.70307 + 0.983269i −2.80040 + 2.80040i −0.0473445 − 0.176692i 1.70697 + 2.02145i 0.513062 − 0.513062i 0.500000 − 0.866025i 0.362225 + 0.627392i
76.6 0.0578232 − 0.215799i 0.866025 − 0.500000i 1.68883 + 0.975044i −1.08760 + 1.08760i −0.0578232 − 0.215799i −0.643420 − 2.56632i 0.624019 − 0.624019i 0.500000 − 0.866025i 0.171815 + 0.297592i
76.7 0.385482 − 1.43864i 0.866025 − 0.500000i −0.189037 − 0.109141i 1.07205 − 1.07205i −0.385482 − 1.43864i −0.612570 + 2.57386i 1.87643 − 1.87643i 0.500000 − 0.866025i −1.12904 − 1.95556i
76.8 0.687394 − 2.56539i 0.866025 − 0.500000i −4.37666 − 2.52687i 1.17771 − 1.17771i −0.687394 − 2.56539i 2.40809 + 1.09594i −5.73490 + 5.73490i 0.500000 − 0.866025i −2.21174 − 3.83085i
97.1 −0.706652 − 2.63726i 0.866025 + 0.500000i −4.72373 + 2.72725i 2.18431 + 2.18431i 0.706652 − 2.63726i −1.85732 + 1.88424i 6.66928 + 6.66928i 0.500000 + 0.866025i 4.21705 − 7.30414i
97.2 −0.500642 − 1.86842i 0.866025 + 0.500000i −1.50830 + 0.870817i −2.44068 − 2.44068i 0.500642 − 1.86842i −2.60515 + 0.461729i −0.353388 − 0.353388i 0.500000 + 0.866025i −3.33831 + 5.78213i
97.3 −0.281068 − 1.04896i 0.866025 + 0.500000i 0.710736 − 0.410344i 1.02734 + 1.02734i 0.281068 − 1.04896i −1.22381 − 2.34570i −2.16598 − 2.16598i 0.500000 + 0.866025i 0.788887 − 1.36639i
97.4 −0.189683 − 0.707908i 0.866025 + 0.500000i 1.26690 − 0.731443i 1.23329 + 1.23329i 0.189683 − 0.707908i 2.32720 + 1.25862i −1.79455 − 1.79455i 0.500000 + 0.866025i 0.639122 − 1.10699i
97.5 0.0473445 + 0.176692i 0.866025 + 0.500000i 1.70307 − 0.983269i −2.80040 − 2.80040i −0.0473445 + 0.176692i 1.70697 − 2.02145i 0.513062 + 0.513062i 0.500000 + 0.866025i 0.362225 − 0.627392i
97.6 0.0578232 + 0.215799i 0.866025 + 0.500000i 1.68883 − 0.975044i −1.08760 − 1.08760i −0.0578232 + 0.215799i −0.643420 + 2.56632i 0.624019 + 0.624019i 0.500000 + 0.866025i 0.171815 − 0.297592i
97.7 0.385482 + 1.43864i 0.866025 + 0.500000i −0.189037 + 0.109141i 1.07205 + 1.07205i −0.385482 + 1.43864i −0.612570 − 2.57386i 1.87643 + 1.87643i 0.500000 + 0.866025i −1.12904 + 1.95556i
97.8 0.687394 + 2.56539i 0.866025 + 0.500000i −4.37666 + 2.52687i 1.17771 + 1.17771i −0.687394 + 2.56539i 2.40809 − 1.09594i −5.73490 − 5.73490i 0.500000 + 0.866025i −2.21174 + 3.83085i
202.1 −2.16866 + 0.581092i −0.866025 − 0.500000i 2.63339 − 1.52039i −1.87790 + 1.87790i 2.16866 + 0.581092i 2.25300 − 1.38707i −1.65230 + 1.65230i 0.500000 + 0.866025i 2.98130 − 5.16377i
202.2 −2.11902 + 0.567791i −0.866025 − 0.500000i 2.43582 − 1.40632i 3.00219 − 3.00219i 2.11902 + 0.567791i −2.49394 − 0.883331i −1.26060 + 1.26060i 0.500000 + 0.866025i −4.65710 + 8.06633i
202.3 −1.38083 + 0.369991i −0.866025 − 0.500000i 0.0377371 − 0.0217876i −0.512287 + 0.512287i 1.38083 + 0.369991i −1.54136 − 2.15040i 1.97762 − 1.97762i 0.500000 + 0.866025i 0.517838 − 0.896921i
202.4 −1.11595 + 0.299019i −0.866025 − 0.500000i −0.576113 + 0.332619i −0.549341 + 0.549341i 1.11595 + 0.299019i 0.347225 + 2.62287i 2.17732 − 2.17732i 0.500000 + 0.866025i 0.448776 − 0.777302i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bc 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.by.c ✓ 32
3.b odd 2 1 819.2.fm.f 32
7.b odd 2 1 273.2.by.d yes 32
13.f odd 12 1 273.2.by.d yes 32
21.c even 2 1 819.2.fm.e 32
39.k even 12 1 819.2.fm.e 32
91.bc even 12 1 inner 273.2.by.c ✓ 32
273.ca odd 12 1 819.2.fm.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.c ✓ 32 1.a even 1 1 trivial
273.2.by.c ✓ 32 91.bc even 12 1 inner
273.2.by.d yes 32 7.b odd 2 1
273.2.by.d yes 32 13.f odd 12 1
819.2.fm.e 32 21.c even 2 1
819.2.fm.e 32 39.k even 12 1
819.2.fm.f 32 3.b odd 2 1
819.2.fm.f 32 273.ca odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{32} + 2 T_{2}^{31} - T_{2}^{30} - 8 T_{2}^{29} - 67 T_{2}^{28} - 96 T_{2}^{27} + 112 T_{2}^{26} + \cdots + 576 \) Copy content Toggle raw display
\( T_{5}^{32} + 2 T_{5}^{31} + 2 T_{5}^{30} - 18 T_{5}^{29} + 501 T_{5}^{28} + 916 T_{5}^{27} + \cdots + 404653456 \) Copy content Toggle raw display