Properties

Label 273.2.bw.b
Level $273$
Weight $2$
Character orbit 273.bw
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.bw (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.

\(\operatorname{Tr}(f)(q) = \) \( 128q - 4q^{3} - 12q^{4} - 4q^{6} - 16q^{7} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q - 4q^{3} - 12q^{4} - 4q^{6} - 16q^{7} - 16q^{9} - 48q^{12} - 16q^{13} - 6q^{15} + 32q^{16} + 22q^{18} - 16q^{19} - 18q^{21} - 8q^{22} - 4q^{24} - 40q^{27} - 76q^{28} - 4q^{31} + 50q^{33} - 48q^{34} - 60q^{36} + 28q^{37} + 40q^{39} + 44q^{40} + 44q^{42} - 144q^{43} + 58q^{45} + 48q^{46} - 64q^{48} + 24q^{49} + 36q^{51} - 22q^{54} - 16q^{55} + 40q^{57} - 28q^{58} - 4q^{60} - 40q^{61} + 20q^{63} - 34q^{66} + 96q^{67} - 54q^{69} + 64q^{70} - 98q^{72} + 48q^{73} - 12q^{75} + 144q^{76} + 82q^{78} - 24q^{79} - 48q^{81} + 4q^{84} + 56q^{85} - 2q^{87} - 24q^{91} + 10q^{93} + 32q^{94} - 54q^{96} + 52q^{97} - 10q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.700434 2.61406i 1.08921 + 1.34671i −4.61063 + 2.66195i −2.04884 0.548984i 2.75745 3.79054i −0.908491 + 2.48488i 6.36068 + 6.36068i −0.627240 + 2.93370i 5.74030i
11.2 −0.679659 2.53652i −1.73115 0.0557742i −4.23995 + 2.44794i 2.98598 + 0.800092i 1.03512 + 4.42901i 2.50266 + 0.858313i 5.37724 + 5.37724i 2.99378 + 0.193107i 8.11779i
11.3 −0.614189 2.29218i −0.572173 1.63481i −3.14483 + 1.81567i 0.203686 + 0.0545776i −3.39587 + 2.31561i −2.22424 1.43274i 2.73738 + 2.73738i −2.34524 + 1.87079i 0.500408i
11.4 −0.590022 2.20199i 1.60162 0.659394i −2.76859 + 1.59845i −1.90854 0.511391i −2.39697 3.13771i 0.185390 2.63925i 1.92936 + 1.92936i 2.13040 2.11220i 4.50431i
11.5 −0.588963 2.19804i 1.52334 0.824281i −2.75245 + 1.58913i 3.96614 + 1.06272i −2.70899 2.86289i −1.45144 + 2.21209i 1.89590 + 1.89590i 1.64112 2.51132i 9.34364i
11.6 −0.548543 2.04719i −0.721270 + 1.57473i −2.15804 + 1.24594i −0.706275 0.189246i 3.61942 + 0.612770i 2.22477 1.43193i 0.737165 + 0.737165i −1.95954 2.27161i 1.54969i
11.7 −0.504901 1.88432i −1.22779 1.22170i −1.56368 + 0.902789i −3.20968 0.860032i −1.68215 + 2.93037i 0.850713 + 2.50525i −0.268188 0.268188i 0.0149148 + 2.99996i 6.48229i
11.8 −0.428681 1.59986i 1.25532 + 1.19339i −0.643729 + 0.371657i 0.428701 + 0.114870i 1.37112 2.51992i 2.29163 + 1.32228i −1.47180 1.47180i 0.151659 + 2.99616i 0.735103i
11.9 −0.375380 1.40094i −1.57757 + 0.715025i −0.0896663 + 0.0517688i 0.283932 + 0.0760792i 1.59390 + 1.94168i −2.00629 + 1.72476i −1.94493 1.94493i 1.97748 2.25601i 0.426329i
11.10 −0.329383 1.22927i 0.639267 1.60976i 0.329428 0.190195i 1.27845 + 0.342559i −2.18940 0.255606i 2.63584 + 0.228814i −2.14209 2.14209i −2.18268 2.05814i 1.68440i
11.11 −0.305471 1.14003i 1.47664 + 0.905280i 0.525688 0.303506i 1.64653 + 0.441185i 0.580979 1.95996i −1.12852 2.39300i −2.17571 2.17571i 1.36093 + 2.67355i 2.01186i
11.12 −0.250255 0.933963i 0.138617 + 1.72650i 0.922391 0.532543i −4.14513 1.11068i 1.57779 0.561527i −2.32521 1.26230i −2.09562 2.09562i −2.96157 + 0.478645i 4.14935i
11.13 −0.248591 0.927753i −1.67503 0.440774i 0.933123 0.538739i 2.67091 + 0.715669i 0.00746719 + 1.66358i −1.47717 2.19498i −2.09011 2.09011i 2.61144 + 1.47662i 2.65586i
11.14 −0.179441 0.669682i 0.594756 1.62673i 1.31578 0.759664i −1.48044 0.396684i −1.19612 0.106395i −2.58806 + 0.549511i −1.72532 1.72532i −2.29253 1.93502i 1.06261i
11.15 −0.0262469 0.0979549i −0.748033 + 1.56219i 1.72314 0.994858i 2.20563 + 0.590996i 0.172658 + 0.0322707i 1.71000 2.01889i −0.286094 0.286094i −1.88089 2.33714i 0.231564i
11.16 −0.00573734 0.0214121i −1.43178 0.974676i 1.73163 0.999754i 2.15869 + 0.578419i −0.0126552 + 0.0362495i 1.44048 + 2.21924i −0.0626912 0.0626912i 1.10001 + 2.79105i 0.0495405i
11.17 0.00573734 + 0.0214121i −1.43178 + 0.974676i 1.73163 0.999754i −2.15869 0.578419i −0.0290845 0.0250654i 1.44048 + 2.21924i 0.0626912 + 0.0626912i 1.10001 2.79105i 0.0495405i
11.18 0.0262469 + 0.0979549i −0.748033 1.56219i 1.72314 0.994858i −2.20563 0.590996i 0.133391 0.114276i 1.71000 2.01889i 0.286094 + 0.286094i −1.88089 + 2.33714i 0.231564i
11.19 0.179441 + 0.669682i 0.594756 + 1.62673i 1.31578 0.759664i 1.48044 + 0.396684i −0.982671 + 0.690199i −2.58806 + 0.549511i 1.72532 + 1.72532i −2.29253 + 1.93502i 1.06261i
11.20 0.248591 + 0.927753i −1.67503 + 0.440774i 0.933123 0.538739i −2.67091 0.715669i −0.825325 1.44444i −1.47717 2.19498i 2.09011 + 2.09011i 2.61144 1.47662i 2.65586i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 254.32
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!38\)\( T_{2}^{102} - \)\(78\!\cdots\!10\)\( T_{2}^{100} - \)\(51\!\cdots\!02\)\( T_{2}^{98} + \)\(20\!\cdots\!16\)\( T_{2}^{96} + \)\(13\!\cdots\!10\)\( T_{2}^{94} - \)\(44\!\cdots\!69\)\( T_{2}^{92} - \)\(28\!\cdots\!42\)\( T_{2}^{90} + \)\(81\!\cdots\!77\)\( T_{2}^{88} + \)\(51\!\cdots\!96\)\( T_{2}^{86} - \)\(12\!\cdots\!33\)\( T_{2}^{84} - \)\(74\!\cdots\!98\)\( T_{2}^{82} + \)\(14\!\cdots\!32\)\( T_{2}^{80} + \)\(87\!\cdots\!96\)\( T_{2}^{78} - \)\(14\!\cdots\!85\)\( T_{2}^{76} - \)\(84\!\cdots\!36\)\( T_{2}^{74} + \)\(11\!\cdots\!47\)\( T_{2}^{72} + \)\(65\!\cdots\!12\)\( T_{2}^{70} - \)\(72\!\cdots\!98\)\( T_{2}^{68} - \)\(39\!\cdots\!12\)\( T_{2}^{66} + \)\(34\!\cdots\!48\)\( T_{2}^{64} + \)\(18\!\cdots\!48\)\( T_{2}^{62} - \)\(12\!\cdots\!72\)\( T_{2}^{60} - \)\(67\!\cdots\!52\)\( T_{2}^{58} + \)\(37\!\cdots\!03\)\( T_{2}^{56} + \)\(18\!\cdots\!54\)\( T_{2}^{54} - \)\(87\!\cdots\!98\)\( T_{2}^{52} - \)\(39\!\cdots\!52\)\( T_{2}^{50} + \)\(17\!\cdots\!44\)\( T_{2}^{48} + \)\(62\!\cdots\!44\)\( T_{2}^{46} - \)\(28\!\cdots\!10\)\( T_{2}^{44} - \)\(75\!\cdots\!58\)\( T_{2}^{42} + \)\(38\!\cdots\!51\)\( T_{2}^{40} + \)\(65\!\cdots\!46\)\( T_{2}^{38} - \)\(38\!\cdots\!49\)\( T_{2}^{36} - \)\(39\!\cdots\!56\)\( T_{2}^{34} + \)\(29\!\cdots\!09\)\( T_{2}^{32} + \)\(14\!\cdots\!14\)\( T_{2}^{30} - \)\(15\!\cdots\!98\)\( T_{2}^{28} - \)\(23\!\cdots\!30\)\( T_{2}^{26} + \)\(53\!\cdots\!20\)\( T_{2}^{24} - \)\(89\!\cdots\!14\)\( T_{2}^{22} - \)\(69\!\cdots\!99\)\( T_{2}^{20} + \)\(25\!\cdots\!98\)\( T_{2}^{18} + \)\(56\!\cdots\!69\)\( T_{2}^{16} - \)\(53\!\cdots\!52\)\( T_{2}^{14} + \)\(13\!\cdots\!51\)\( T_{2}^{12} - \)\(16\!\cdots\!06\)\( T_{2}^{10} + \)\(70\!\cdots\!24\)\( T_{2}^{8} + \)\(17\!\cdots\!40\)\( T_{2}^{6} + \)\(12\!\cdots\!73\)\( T_{2}^{4} + 990405301344 T_{2}^{2} + 269517889 \)">\(T_{2}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).