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Properties

Label 273.2.bv.b
Level $273$
Weight $2$
Character orbit 273.bv
Analytic conductor $2.180$
Analytic rank $0$
Dimension $128$
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.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128q + 2q^{3} - 4q^{6} - 16q^{7} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 2q^{3} - 4q^{6} - 16q^{7} + 8q^{9} - 12q^{10} + 48q^{12} - 16q^{13} - 6q^{15} - 64q^{16} - 2q^{18} - 4q^{19} - 6q^{21} - 8q^{22} + 2q^{24} - 40q^{27} + 68q^{28} + 18q^{30} + 20q^{31} - 16q^{33} - 48q^{34} - 60q^{36} - 8q^{37} + 4q^{39} + 44q^{40} + 2q^{42} - 144q^{43} - 2q^{45} - 24q^{46} - 64q^{48} - 60q^{49} - 36q^{51} + 48q^{52} + 14q^{54} - 16q^{55} + 40q^{57} + 44q^{58} - 58q^{60} + 20q^{61} + 14q^{63} - 34q^{66} - 84q^{67} - 54q^{69} - 104q^{70} + 46q^{72} - 48q^{73} + 144q^{76} + 82q^{78} - 24q^{79} + 24q^{81} + 36q^{82} + 184q^{84} + 56q^{85} + 4q^{87} + 132q^{88} + 24q^{91} + 16q^{93} - 16q^{94} - 90q^{96} + 52q^{97} - 10q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.83790 − 1.83790i 1.29065 + 1.15509i 4.75574i 2.20098 − 0.589752i −0.249144 − 4.49501i −2.64509 − 0.0592679i 5.06476 − 5.06476i 0.331543 + 2.98162i −5.12909 − 2.96128i
2.2 −1.81876 − 1.81876i −0.267828 − 1.71122i 4.61580i 2.15482 − 0.577382i −2.62519 + 3.59942i 1.30976 − 2.29881i 4.75753 − 4.75753i −2.85654 + 0.916623i −4.96923 − 2.86898i
2.3 −1.73689 − 1.73689i 1.00077 − 1.41367i 4.03355i −2.06624 + 0.553646i −4.19360 + 0.717172i −1.93609 + 1.80321i 3.53204 − 3.53204i −0.996937 − 2.82951i 4.55044 + 2.62720i
2.4 −1.52398 − 1.52398i −1.72942 − 0.0953716i 2.64502i −2.04381 + 0.547638i 2.49026 + 2.78095i −2.26018 − 1.37535i 0.982996 − 0.982996i 2.98181 + 0.329876i 3.94931 + 2.28014i
2.5 −1.48334 − 1.48334i 1.73149 − 0.0439971i 2.40062i −0.600039 + 0.160780i −2.63366 − 2.50313i 2.54988 + 0.705761i 0.594253 − 0.594253i 2.99613 − 0.152361i 1.12856 + 0.651572i
2.6 −1.44872 − 1.44872i −0.783380 + 1.54477i 2.19760i 0.629445 − 0.168659i 3.37285 − 1.10304i −0.719902 + 2.54593i 0.286268 − 0.286268i −1.77263 − 2.42029i −1.15623 − 0.667551i
2.7 −1.32954 − 1.32954i −1.62246 − 0.606325i 1.53537i 3.58803 − 0.961410i 1.35099 + 2.96326i 0.951973 + 2.46855i −0.617751 + 0.617751i 2.26474 + 1.96747i −6.04868 − 3.49220i
2.8 −1.21487 − 1.21487i 0.917597 + 1.46902i 0.951801i −2.79534 + 0.749010i 0.669904 − 2.89942i −0.107896 − 2.64355i −1.27342 + 1.27342i −1.31603 + 2.69593i 4.30591 + 2.48602i
2.9 −0.879582 − 0.879582i 1.39839 − 1.02202i − 0.452670i 2.08749 − 0.559342i −2.12894 − 0.331049i 0.755530 − 2.53558i −2.15733 + 2.15733i 0.910969 − 2.85835i −2.32811 − 1.34414i
2.10 −0.824043 − 0.824043i −0.772097 − 1.55044i − 0.641906i −0.114643 + 0.0307184i −0.641388 + 1.91387i −2.54084 − 0.737668i −2.17704 + 2.17704i −1.80773 + 2.39418i 0.119784 + 0.0691572i
2.11 −0.748547 − 0.748547i −1.33611 + 1.10219i − 0.879356i −2.02207 + 0.541811i 1.82518 + 0.175098i 2.61787 + 0.383108i −2.15533 + 2.15533i 0.570358 − 2.94528i 1.91918 + 1.10804i
2.12 −0.583259 − 0.583259i −0.386481 + 1.68838i − 1.31962i 3.27322 − 0.877056i 1.21018 − 0.759346i −0.829143 − 2.51247i −1.93620 + 1.93620i −2.70127 − 1.30505i −2.42069 − 1.39758i
2.13 −0.554014 − 0.554014i 0.671048 + 1.59678i − 1.38614i −1.68690 + 0.452004i 0.512866 − 1.25641i −1.01706 + 2.44246i −1.87597 + 1.87597i −2.09939 + 2.14303i 1.18499 + 0.684151i
2.14 −0.397102 − 0.397102i 1.57374 − 0.723421i − 1.68462i −3.29155 + 0.881968i −0.912208 − 0.337664i −2.64283 + 0.124223i −1.46317 + 1.46317i 1.95332 − 2.27696i 1.65731 + 0.956850i
2.15 −0.363959 − 0.363959i −1.62227 − 0.606835i − 1.73507i 0.585566 − 0.156902i 0.369576 + 0.811302i 2.22038 − 1.43871i −1.35941 + 1.35941i 2.26350 + 1.96890i −0.270228 − 0.156016i
2.16 −0.340464 − 0.340464i 1.62034 + 0.611966i − 1.76817i 3.37589 − 0.904567i −0.343315 − 0.760019i −0.304456 + 2.62818i −1.28292 + 1.28292i 2.25100 + 1.98318i −1.45734 − 0.841396i
2.17 0.340464 + 0.340464i −1.34015 − 1.09727i − 1.76817i −3.37589 + 0.904567i −0.0826902 − 0.829853i −0.304456 + 2.62818i 1.28292 − 1.28292i 0.591989 + 2.94101i −1.45734 − 0.841396i
2.18 0.363959 + 0.363959i 1.33667 + 1.10151i − 1.73507i −0.585566 + 0.156902i 0.0855890 + 0.887396i 2.22038 − 1.43871i 1.35941 − 1.35941i 0.573364 + 2.94470i −0.270228 − 0.156016i
2.19 0.397102 + 0.397102i −0.160369 − 1.72461i − 1.68462i 3.29155 − 0.881968i 0.621163 − 0.748529i −2.64283 + 0.124223i 1.46317 − 1.46317i −2.94856 + 0.553150i 1.65731 + 0.956850i
2.20 0.554014 + 0.554014i −1.71837 + 0.217243i − 1.38614i 1.68690 − 0.452004i −1.07236 − 0.831647i −1.01706 + 2.44246i 1.87597 − 1.87597i 2.90561 − 0.746610i 1.18499 + 0.684151i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.x odd 12 1 inner
273.bv 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.bv.b ✓ 128
3.b odd 2 1 inner 273.2.bv.b ✓ 128
7.c even 3 1 273.2.bw.b yes 128
13.f odd 12 1 273.2.bw.b yes 128
21.h odd 6 1 273.2.bw.b yes 128
39.k even 12 1 273.2.bw.b yes 128
91.x odd 12 1 inner 273.2.bv.b ✓ 128
273.bv even 12 1 inner 273.2.bv.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bv.b ✓ 128 1.a even 1 1 trivial
273.2.bv.b ✓ 128 3.b odd 2 1 inner
273.2.bv.b ✓ 128 91.x odd 12 1 inner
273.2.bv.b ✓ 128 273.bv even 12 1 inner
273.2.bw.b yes 128 7.c even 3 1
273.2.bw.b yes 128 13.f odd 12 1
273.2.bw.b yes 128 21.h odd 6 1
273.2.bw.b yes 128 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(34\!\cdots\!83\)\( T_{2}^{104} + \)\(13\!\cdots\!20\)\( T_{2}^{100} + \)\(40\!\cdots\!55\)\( T_{2}^{96} + \)\(97\!\cdots\!22\)\( T_{2}^{92} + \)\(18\!\cdots\!89\)\( T_{2}^{88} + \)\(29\!\cdots\!76\)\( T_{2}^{84} + \)\(36\!\cdots\!59\)\( T_{2}^{80} + \)\(35\!\cdots\!22\)\( T_{2}^{76} + \)\(28\!\cdots\!37\)\( T_{2}^{72} + \)\(17\!\cdots\!78\)\( T_{2}^{68} + \)\(85\!\cdots\!12\)\( T_{2}^{64} + \)\(32\!\cdots\!62\)\( T_{2}^{60} + \)\(96\!\cdots\!37\)\( T_{2}^{56} + \)\(21\!\cdots\!34\)\( T_{2}^{52} + \)\(37\!\cdots\!08\)\( T_{2}^{48} + \)\(48\!\cdots\!76\)\( T_{2}^{44} + \)\(46\!\cdots\!06\)\( T_{2}^{40} + \)\(31\!\cdots\!48\)\( T_{2}^{36} + \)\(15\!\cdots\!83\)\( T_{2}^{32} + \)\(53\!\cdots\!28\)\( T_{2}^{28} + \)\(12\!\cdots\!31\)\( T_{2}^{24} + \)\(17\!\cdots\!12\)\( T_{2}^{20} + \)\(14\!\cdots\!77\)\( T_{2}^{16} + \)\(61\!\cdots\!72\)\( T_{2}^{12} + \)\(10\!\cdots\!27\)\( T_{2}^{8} + \)\(11\!\cdots\!78\)\( T_{2}^{4} + 269517889 \)">\(T_{2}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).