Properties

Label 273.2.bt.a
Level $273$
Weight $2$
Character orbit 273.bt
Analytic conductor $2.180$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bt (of order \(12\), degree \(4\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 36q + 6q^{7} + 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 6q^{7} + 18q^{9} - 8q^{11} - 16q^{12} + 42q^{14} - 24q^{16} - 8q^{17} - 18q^{19} + 14q^{20} - 4q^{21} + 4q^{22} + 18q^{24} + 24q^{25} - 50q^{26} + 34q^{28} + 8q^{29} + 6q^{31} - 50q^{32} + 8q^{33} - 24q^{34} + 14q^{35} - 14q^{37} - 8q^{38} - 2q^{39} - 30q^{40} + 34q^{41} - 18q^{42} + 30q^{43} + 28q^{44} - 32q^{46} - 10q^{47} + 24q^{48} + 6q^{49} - 20q^{50} - 24q^{51} + 4q^{52} - 8q^{53} - 30q^{55} - 92q^{56} - 24q^{57} + 72q^{58} - 70q^{59} + 14q^{60} - 60q^{61} - 48q^{62} + 6q^{63} - 44q^{65} + 18q^{66} - 46q^{67} + 4q^{69} + 80q^{70} + 42q^{71} + 18q^{72} - 56q^{73} + 40q^{74} - 20q^{75} + 12q^{76} + 24q^{77} - 16q^{78} + 170q^{80} - 18q^{81} + 24q^{82} - 60q^{83} + 2q^{85} + 12q^{86} + 84q^{88} + 64q^{89} - 86q^{91} - 100q^{92} + 12q^{93} - 66q^{94} + 46q^{96} + 36q^{97} - 22q^{98} - 4q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.79515 + 1.79515i 0.866025 0.500000i 4.44512i −0.590961 + 2.20549i −0.657070 + 2.45222i −2.51335 0.826467i 4.38935 + 4.38935i 0.500000 0.866025i −2.89833 5.02005i
136.2 −1.48267 + 1.48267i 0.866025 0.500000i 2.39661i 0.507149 1.89270i −0.542694 + 2.02536i −0.313052 + 2.62717i 0.588043 + 0.588043i 0.500000 0.866025i 2.05432 + 3.55819i
136.3 −1.20543 + 1.20543i 0.866025 0.500000i 0.906108i 0.363968 1.35835i −0.441217 + 1.64664i 0.864271 2.50061i −1.31861 1.31861i 0.500000 0.866025i 1.19865 + 2.07613i
136.4 −0.411775 + 0.411775i 0.866025 0.500000i 1.66088i −0.180309 + 0.672922i −0.150720 + 0.562494i 2.60113 + 0.483875i −1.50746 1.50746i 0.500000 0.866025i −0.202846 0.351339i
136.5 0.374685 0.374685i 0.866025 0.500000i 1.71922i −0.545981 + 2.03763i 0.137144 0.511829i −2.03549 + 1.69021i 1.39354 + 1.39354i 0.500000 0.866025i 0.558898 + 0.968040i
136.6 0.556084 0.556084i 0.866025 0.500000i 1.38154i 0.542987 2.02645i 0.203541 0.759625i −0.405927 2.61443i 1.88042 + 1.88042i 0.500000 0.866025i −0.824932 1.42882i
136.7 1.09987 1.09987i 0.866025 0.500000i 0.419447i 0.745735 2.78312i 0.402582 1.50246i −1.80794 + 1.93167i 1.73841 + 1.73841i 0.500000 0.866025i −2.24087 3.88130i
136.8 1.12005 1.12005i 0.866025 0.500000i 0.509024i −0.973479 + 3.63307i 0.409967 1.53002i 2.28455 1.33448i 1.66997 + 1.66997i 0.500000 0.866025i 2.97888 + 5.15957i
136.9 1.74432 1.74432i 0.866025 0.500000i 4.08534i 0.130892 0.488495i 0.638467 2.38279i 1.09376 + 2.40909i −3.63751 3.63751i 0.500000 0.866025i −0.623777 1.08041i
145.1 −1.74581 + 1.74581i −0.866025 0.500000i 4.09571i −2.78539 + 0.746344i 2.38482 0.639011i −2.58285 + 0.573466i 3.65872 + 3.65872i 0.500000 + 0.866025i 3.55979 6.16574i
145.2 −1.30773 + 1.30773i −0.866025 0.500000i 1.42031i −0.0744995 + 0.0199621i 1.78639 0.478662i 2.55141 + 0.700217i −0.758075 0.758075i 0.500000 + 0.866025i 0.0713202 0.123530i
145.3 −0.926196 + 0.926196i −0.866025 0.500000i 0.284323i 0.409991 0.109857i 1.26521 0.339011i −2.25606 + 1.38209i −2.11573 2.11573i 0.500000 + 0.866025i −0.277983 + 0.481481i
145.4 −0.745928 + 0.745928i −0.866025 0.500000i 0.887184i 3.80456 1.01943i 1.01896 0.273028i 0.148943 2.64156i −2.15363 2.15363i 0.500000 + 0.866025i −2.07751 + 3.59835i
145.5 0.111217 0.111217i −0.866025 0.500000i 1.97526i −2.13060 + 0.570893i −0.151925 + 0.0407083i −0.399954 2.61535i 0.442117 + 0.442117i 0.500000 + 0.866025i −0.173466 + 0.300452i
145.6 0.430820 0.430820i −0.866025 0.500000i 1.62879i 1.97745 0.529856i −0.588511 + 0.157691i −1.23433 + 2.34018i 1.56335 + 1.56335i 0.500000 + 0.866025i 0.623652 1.08020i
145.7 0.698661 0.698661i −0.866025 0.500000i 1.02375i −0.912136 + 0.244406i −0.954388 + 0.255728i 2.61412 0.407922i 2.11257 + 2.11257i 0.500000 + 0.866025i −0.466517 + 0.808031i
145.8 1.55654 1.55654i −0.866025 0.500000i 2.84566i 3.12520 0.837395i −2.12628 + 0.569735i 1.93981 + 1.79920i −1.31631 1.31631i 0.500000 + 0.866025i 3.56107 6.16796i
145.9 1.92842 1.92842i −0.866025 0.500000i 5.43762i −3.41458 + 0.914933i −2.63427 + 0.705851i 2.45097 0.996354i −6.62917 6.62917i 0.500000 + 0.866025i −4.82036 + 8.34912i
241.1 −1.74581 1.74581i −0.866025 + 0.500000i 4.09571i −2.78539 0.746344i 2.38482 + 0.639011i −2.58285 0.573466i 3.65872 3.65872i 0.500000 0.866025i 3.55979 + 6.16574i
241.2 −1.30773 1.30773i −0.866025 + 0.500000i 1.42031i −0.0744995 0.0199621i 1.78639 + 0.478662i 2.55141 0.700217i −0.758075 + 0.758075i 0.500000 0.866025i 0.0713202 + 0.123530i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.9
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{36} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).