Properties

Label 273.2.bf.b
Level $273$
Weight $2$
Character orbit 273.bf
Analytic conductor $2.180$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64q + 32q^{4} - 12q^{6} - 4q^{7} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q + 32q^{4} - 12q^{6} - 4q^{7} + 8q^{9} + 6q^{12} - 12q^{13} - 9q^{15} - 16q^{16} + 2q^{18} + 10q^{21} + 10q^{22} - 24q^{25} - 50q^{28} - 16q^{30} - 24q^{31} - 33q^{39} + 90q^{40} - 48q^{42} - 20q^{43} - 3q^{45} + 6q^{48} - 10q^{51} + 30q^{52} - 27q^{54} + 18q^{55} + 4q^{57} - 60q^{58} + 55q^{60} - 74q^{63} - 84q^{64} + 75q^{66} - 88q^{67} - 33q^{69} + 20q^{70} - 34q^{72} + 84q^{73} + 33q^{75} + 18q^{76} - 71q^{78} + 20q^{79} - 32q^{81} - 6q^{84} - 2q^{85} + 3q^{87} + 92q^{88} - 76q^{91} + 28q^{93} + 30q^{96} + 24q^{97} + 22q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
152.1 −2.36551 + 1.36573i 1.47077 + 0.914786i 2.73043 4.72924i 0.121221 0.209960i −4.72847 0.155263i −0.288550 2.62997i 9.45317i 1.32633 + 2.69088i 0.662218i
152.2 −2.25245 + 1.30045i −1.59264 + 0.680804i 2.38234 4.12633i 1.54658 2.67876i 2.70199 3.60462i −2.63374 0.251835i 7.19066i 2.07301 2.16855i 8.04502i
152.3 −2.07561 + 1.19835i −0.232998 1.71631i 1.87210 3.24258i 1.10064 1.90637i 2.54036 + 3.28317i 2.62075 0.362837i 4.18035i −2.89142 + 0.799793i 5.27584i
152.4 −2.07482 + 1.19790i 1.22164 1.22784i 1.86992 3.23879i −1.50632 + 2.60901i −1.06387 + 4.01095i −1.19447 + 2.36077i 4.16828i −0.0151701 2.99996i 7.21764i
152.5 −1.70697 + 0.985518i −1.16871 1.27832i 0.942492 1.63244i −0.272603 + 0.472162i 3.25477 + 1.03026i −2.22004 + 1.43924i 0.226699i −0.268213 + 2.98799i 1.07462i
152.6 −1.66598 + 0.961856i −0.628043 + 1.61418i 0.850333 1.47282i −1.94643 + 3.37132i −0.506295 3.29327i −1.91023 1.83058i 0.575834i −2.21112 2.02754i 7.48874i
152.7 −1.60522 + 0.926776i 1.72065 + 0.198407i 0.717829 1.24332i −0.852778 + 1.47706i −2.94591 + 1.27617i 2.63557 0.231921i 1.04604i 2.92127 + 0.682779i 3.16134i
152.8 −1.60258 + 0.925247i 0.929062 + 1.46179i 0.712166 1.23351i 1.21731 2.10844i −2.84141 1.48302i −0.567699 + 2.58413i 1.06527i −1.27369 + 2.71620i 4.50525i
152.9 −1.33345 + 0.769868i −0.674223 + 1.59544i 0.185393 0.321109i 0.956175 1.65614i −0.329235 2.64650i 1.55646 2.13949i 2.50856i −2.09085 2.15136i 2.94451i
152.10 −1.21710 + 0.702692i 1.65544 0.509428i −0.0124468 + 0.0215586i 1.48377 2.56997i −1.65686 + 1.78329i −2.30005 1.30759i 2.84575i 2.48097 1.68666i 4.17054i
152.11 −0.913280 + 0.527282i −1.53174 0.808559i −0.443947 + 0.768938i −0.000964697 0.00167090i 1.82525 0.0692194i 1.27194 2.31995i 3.04547i 1.69246 + 2.47701i 0.00203467i
152.12 −0.765058 + 0.441707i 0.162694 1.72439i −0.609790 + 1.05619i −1.72131 + 2.98140i 0.637205 + 1.39112i −0.0494112 2.64529i 2.84422i −2.94706 0.561097i 3.04126i
152.13 −0.730177 + 0.421568i −1.64300 + 0.548211i −0.644561 + 1.11641i −1.06280 + 1.84082i 0.968576 1.09293i −0.110845 + 2.64343i 2.77318i 2.39893 1.80143i 1.79216i
152.14 −0.667801 + 0.385555i 1.00894 1.40785i −0.702694 + 1.21710i 0.907647 1.57209i −0.130967 + 1.32917i 1.94952 + 1.78868i 2.62593i −0.964085 2.84087i 1.39979i
152.15 −0.473030 + 0.273104i 0.966120 + 1.43757i −0.850828 + 1.47368i −1.25657 + 2.17644i −0.849611 0.416164i 2.53535 + 0.756296i 2.02188i −1.13323 + 2.77773i 1.37270i
152.16 −0.0436821 + 0.0252199i 1.71783 + 0.221508i −0.998728 + 1.72985i −1.19579 + 2.07117i −0.0806247 + 0.0336474i −2.29456 1.31719i 0.201631i 2.90187 + 0.761026i 0.120631i
152.17 0.0436821 0.0252199i −1.71783 + 0.221508i −0.998728 + 1.72985i 1.19579 2.07117i −0.0694519 + 0.0529993i −2.29456 1.31719i 0.201631i 2.90187 0.761026i 0.120631i
152.18 0.473030 0.273104i −0.966120 + 1.43757i −0.850828 + 1.47368i 1.25657 2.17644i −0.0643970 + 0.943866i 2.53535 + 0.756296i 2.02188i −1.13323 2.77773i 1.37270i
152.19 0.667801 0.385555i −1.00894 1.40785i −0.702694 + 1.21710i −0.907647 + 1.57209i −1.21657 0.551162i 1.94952 + 1.78868i 2.62593i −0.964085 + 2.84087i 1.39979i
152.20 0.730177 0.421568i 1.64300 + 0.548211i −0.644561 + 1.11641i 1.06280 1.84082i 1.43079 0.292347i −0.110845 + 2.64343i 2.77318i 2.39893 + 1.80143i 1.79216i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 185.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.m odd 6 1 inner
273.bf even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bf.b yes 64
3.b odd 2 1 inner 273.2.bf.b yes 64
7.d odd 6 1 273.2.r.b 64
13.c even 3 1 273.2.r.b 64
21.g even 6 1 273.2.r.b 64
39.i odd 6 1 273.2.r.b 64
91.m odd 6 1 inner 273.2.bf.b yes 64
273.bf even 6 1 inner 273.2.bf.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.r.b 64 7.d odd 6 1
273.2.r.b 64 13.c even 3 1
273.2.r.b 64 21.g even 6 1
273.2.r.b 64 39.i odd 6 1
273.2.bf.b yes 64 1.a even 1 1 trivial
273.2.bf.b yes 64 3.b odd 2 1 inner
273.2.bf.b yes 64 91.m odd 6 1 inner
273.2.bf.b yes 64 273.bf even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(25\!\cdots\!19\)\( T_{2}^{38} + \)\(75\!\cdots\!31\)\( T_{2}^{36} - \)\(19\!\cdots\!66\)\( T_{2}^{34} + \)\(42\!\cdots\!80\)\( T_{2}^{32} - \)\(81\!\cdots\!88\)\( T_{2}^{30} + \)\(13\!\cdots\!51\)\( T_{2}^{28} - \)\(19\!\cdots\!18\)\( T_{2}^{26} + \)\(23\!\cdots\!90\)\( T_{2}^{24} - \)\(25\!\cdots\!34\)\( T_{2}^{22} + \)\(22\!\cdots\!89\)\( T_{2}^{20} - \)\(16\!\cdots\!73\)\( T_{2}^{18} + \)\(10\!\cdots\!31\)\( T_{2}^{16} - \)\(52\!\cdots\!76\)\( T_{2}^{14} + \)\(21\!\cdots\!42\)\( T_{2}^{12} - \)\(66\!\cdots\!25\)\( T_{2}^{10} + \)\(15\!\cdots\!35\)\( T_{2}^{8} - 238854029115 T_{2}^{6} + 21416104784 T_{2}^{4} - 54460965 T_{2}^{2} + 134689 \)">\(T_{2}^{64} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).