Properties

Label 273.2.cd.e
Level $273$
Weight $2$
Character orbit 273.cd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112q - 12q^{3} - 8q^{6} - 4q^{7} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 112q - 12q^{3} - 8q^{6} - 4q^{7} + 8q^{9} - 48q^{13} - 12q^{15} + 40q^{16} - 26q^{18} + 40q^{19} - 10q^{21} + 16q^{22} + 32q^{24} - 24q^{27} - 52q^{28} - 12q^{31} - 44q^{33} + 16q^{34} - 8q^{37} - 42q^{39} - 160q^{40} - 80q^{42} + 6q^{45} + 32q^{46} + 72q^{48} - 12q^{52} + 34q^{54} - 48q^{55} - 24q^{57} - 28q^{58} + 44q^{60} + 78q^{63} + 4q^{66} + 24q^{67} - 12q^{70} - 26q^{72} - 40q^{73} + 112q^{76} + 32q^{78} + 48q^{79} + 128q^{81} - 150q^{84} + 160q^{85} - 48q^{87} + 24q^{91} + 10q^{93} - 8q^{94} - 106q^{96} + 56q^{97} - 36q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −2.69539 + 0.722228i 1.57685 + 0.716628i 5.01147 2.89338i 1.51996 0.407273i −4.76779 0.792751i −0.780863 + 2.52790i −7.47188 + 7.47188i 1.97289 + 2.26003i −3.80275 + 2.19552i
44.2 −2.42391 + 0.649484i −1.28483 1.16155i 3.72144 2.14858i 3.58252 0.959933i 3.86873 + 1.98102i 1.55291 2.14207i −4.07612 + 4.07612i 0.301593 + 2.98480i −8.06023 + 4.65358i
44.3 −2.41247 + 0.646418i 0.872737 1.49611i 3.67009 2.11892i −1.51647 + 0.406338i −1.13834 + 4.17346i −1.79039 1.94795i −3.95215 + 3.95215i −1.47666 2.61141i 3.39578 1.96055i
44.4 −2.19229 + 0.587422i −1.51343 + 0.842335i 2.72902 1.57560i 0.959762 0.257167i 2.82307 2.73567i −2.15459 + 1.53549i −1.84753 + 1.84753i 1.58094 2.54963i −1.95301 + 1.12757i
44.5 −2.10280 + 0.563443i −0.755319 1.55868i 2.37223 1.36961i −2.37528 + 0.636456i 2.46651 + 2.85201i 1.72404 + 2.00691i −1.13792 + 1.13792i −1.85899 + 2.35461i 4.63613 2.67667i
44.6 −2.00566 + 0.537415i −0.123517 + 1.72764i 2.00181 1.15574i 0.878767 0.235465i −0.680727 3.53144i −1.67646 2.04682i −0.457340 + 0.457340i −2.96949 0.426786i −1.63596 + 0.944525i
44.7 −1.87036 + 0.501163i 1.45895 + 0.933521i 1.51505 0.874714i −1.33892 + 0.358762i −3.19662 1.01485i 1.90555 1.83545i 0.343083 0.343083i 1.25708 + 2.72392i 2.32447 1.34203i
44.8 −1.64181 + 0.439921i 1.02109 1.39906i 0.769949 0.444530i 3.49677 0.936956i −1.06096 + 2.74619i 0.612738 + 2.57382i 1.33522 1.33522i −0.914735 2.85714i −5.32883 + 3.07660i
44.9 −1.55291 + 0.416100i −1.66835 + 0.465425i 0.506326 0.292327i 0.0914679 0.0245087i 2.39712 1.41696i 2.58235 + 0.575745i 1.60897 1.60897i 2.56676 1.55298i −0.131843 + 0.0761195i
44.10 −1.27346 + 0.341222i 1.73165 0.0374536i −0.226792 + 0.130939i −3.02659 + 0.810973i −2.19240 + 0.638570i −0.631669 + 2.56924i 2.10860 2.10860i 2.99719 0.129713i 3.57751 2.06548i
44.11 −1.25001 + 0.334940i −1.60871 0.641920i −0.281700 + 0.162639i −3.70264 + 0.992121i 2.22591 + 0.263589i −1.90960 1.83124i 2.12780 2.12780i 2.17588 + 2.06532i 4.29606 2.48033i
44.12 −0.623254 + 0.167000i −1.63214 0.579750i −1.37149 + 0.791833i 2.98988 0.801137i 1.11406 + 0.0887629i −2.60516 + 0.461652i 1.63506 1.63506i 2.32778 + 1.89247i −1.72966 + 0.998622i
44.13 −0.578797 + 0.155088i 0.590016 1.62846i −1.42110 + 0.820471i −0.609213 + 0.163238i −0.0889444 + 1.03405i 2.12540 1.57565i 1.54270 1.54270i −2.30376 1.92163i 0.327294 0.188963i
44.14 −0.246180 + 0.0659636i −0.753906 + 1.55937i −1.67580 + 0.967522i −2.71744 + 0.728135i 0.0827348 0.433615i 1.41179 2.23760i 0.709158 0.709158i −1.86325 2.35123i 0.620947 0.358504i
44.15 0.246180 0.0659636i 1.72740 + 0.126782i −1.67580 + 0.967522i 2.71744 0.728135i 0.433615 0.0827348i 1.41179 2.23760i −0.709158 + 0.709158i 2.96785 + 0.438007i 0.620947 0.358504i
44.16 0.578797 0.155088i −1.70530 0.303261i −1.42110 + 0.820471i 0.609213 0.163238i −1.03405 + 0.0889444i 2.12540 1.57565i −1.54270 + 1.54270i 2.81607 + 1.03430i 0.327294 0.188963i
44.17 0.623254 0.167000i 0.313993 1.70335i −1.37149 + 0.791833i −2.98988 + 0.801137i −0.0887629 1.11406i −2.60516 + 0.461652i −1.63506 + 1.63506i −2.80282 1.06968i −1.72966 + 0.998622i
44.18 1.25001 0.334940i 0.248434 1.71414i −0.281700 + 0.162639i 3.70264 0.992121i −0.263589 2.22591i −1.90960 1.83124i −2.12780 + 2.12780i −2.87656 0.851703i 4.29606 2.48033i
44.19 1.27346 0.341222i −0.898259 + 1.48092i −0.226792 + 0.130939i 3.02659 0.810973i −0.638570 + 2.19240i −0.631669 + 2.56924i −2.10860 + 2.10860i −1.38626 2.66050i 3.57751 2.06548i
44.20 1.55291 0.416100i 1.23724 1.21212i 0.506326 0.292327i −0.0914679 + 0.0245087i 1.41696 2.39712i 2.58235 + 0.575745i −1.60897 + 1.60897i 0.0615393 2.99937i −0.131843 + 0.0761195i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 242.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
13.d odd 4 1 inner
21.h odd 6 1 inner
39.f even 4 1 inner
91.z odd 12 1 inner
273.cd 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.cd.e 112
3.b odd 2 1 inner 273.2.cd.e 112
7.c even 3 1 inner 273.2.cd.e 112
13.d odd 4 1 inner 273.2.cd.e 112
21.h odd 6 1 inner 273.2.cd.e 112
39.f even 4 1 inner 273.2.cd.e 112
91.z odd 12 1 inner 273.2.cd.e 112
273.cd even 12 1 inner 273.2.cd.e 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.e 112 1.a even 1 1 trivial
273.2.cd.e 112 3.b odd 2 1 inner
273.2.cd.e 112 7.c even 3 1 inner
273.2.cd.e 112 13.d odd 4 1 inner
273.2.cd.e 112 21.h odd 6 1 inner
273.2.cd.e 112 39.f even 4 1 inner
273.2.cd.e 112 91.z odd 12 1 inner
273.2.cd.e 112 273.cd even 12 1 inner

Hecke kernels

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

\(19\!\cdots\!38\)\( T_{2}^{84} + \)\(57\!\cdots\!64\)\( T_{2}^{80} - \)\(14\!\cdots\!36\)\( T_{2}^{76} + \)\(28\!\cdots\!48\)\( T_{2}^{72} - \)\(49\!\cdots\!22\)\( T_{2}^{68} + \)\(70\!\cdots\!12\)\( T_{2}^{64} - \)\(82\!\cdots\!18\)\( T_{2}^{60} + \)\(79\!\cdots\!87\)\( T_{2}^{56} - \)\(62\!\cdots\!66\)\( T_{2}^{52} + \)\(39\!\cdots\!39\)\( T_{2}^{48} - \)\(19\!\cdots\!32\)\( T_{2}^{44} + \)\(77\!\cdots\!37\)\( T_{2}^{40} - \)\(22\!\cdots\!38\)\( T_{2}^{36} + \)\(49\!\cdots\!38\)\( T_{2}^{32} - \)\(72\!\cdots\!52\)\( T_{2}^{28} + \)\(68\!\cdots\!30\)\( T_{2}^{24} - \)\(18\!\cdots\!78\)\( T_{2}^{20} + \)\(37\!\cdots\!41\)\( T_{2}^{16} - \)\(36\!\cdots\!68\)\( T_{2}^{12} + \)\(25\!\cdots\!34\)\( T_{2}^{8} - \)\(10\!\cdots\!24\)\( T_{2}^{4} + \)\(42\!\cdots\!41\)\( \)">\(T_{2}^{112} - \cdots\)
\(80\!\cdots\!00\)\( T_{5}^{92} + \)\(12\!\cdots\!85\)\( T_{5}^{88} - \)\(15\!\cdots\!84\)\( T_{5}^{84} + \)\(15\!\cdots\!83\)\( T_{5}^{80} - \)\(12\!\cdots\!32\)\( T_{5}^{76} + \)\(75\!\cdots\!61\)\( T_{5}^{72} - \)\(36\!\cdots\!88\)\( T_{5}^{68} + \)\(13\!\cdots\!01\)\( T_{5}^{64} - \)\(35\!\cdots\!38\)\( T_{5}^{60} + \)\(68\!\cdots\!46\)\( T_{5}^{56} - \)\(81\!\cdots\!06\)\( T_{5}^{52} + \)\(69\!\cdots\!06\)\( T_{5}^{48} - \)\(41\!\cdots\!76\)\( T_{5}^{44} + \)\(18\!\cdots\!96\)\( T_{5}^{40} - \)\(54\!\cdots\!48\)\( T_{5}^{36} + \)\(11\!\cdots\!61\)\( T_{5}^{32} - \)\(14\!\cdots\!80\)\( T_{5}^{28} + \)\(13\!\cdots\!24\)\( T_{5}^{24} - \)\(75\!\cdots\!88\)\( T_{5}^{20} + \)\(28\!\cdots\!64\)\( T_{5}^{16} - \)\(40\!\cdots\!96\)\( T_{5}^{12} + \)\(42\!\cdots\!48\)\( T_{5}^{8} - \)\(34\!\cdots\!92\)\( T_{5}^{4} + \)\(27\!\cdots\!76\)\( \)">\(T_{5}^{112} - \cdots\)
\(27\!\cdots\!78\)\( T_{19}^{41} + \)\(46\!\cdots\!45\)\( T_{19}^{40} + \)\(22\!\cdots\!12\)\( T_{19}^{39} - \)\(15\!\cdots\!26\)\( T_{19}^{38} + \)\(10\!\cdots\!32\)\( T_{19}^{37} - \)\(31\!\cdots\!98\)\( T_{19}^{36} - \)\(49\!\cdots\!12\)\( T_{19}^{35} + \)\(47\!\cdots\!50\)\( T_{19}^{34} - \)\(41\!\cdots\!80\)\( T_{19}^{33} + \)\(20\!\cdots\!29\)\( T_{19}^{32} - \)\(43\!\cdots\!24\)\( T_{19}^{31} + \)\(14\!\cdots\!56\)\( T_{19}^{30} - \)\(28\!\cdots\!84\)\( T_{19}^{29} - \)\(10\!\cdots\!53\)\( T_{19}^{28} + \)\(69\!\cdots\!12\)\( T_{19}^{27} + \)\(87\!\cdots\!58\)\( T_{19}^{26} + \)\(29\!\cdots\!90\)\( T_{19}^{25} + \)\(48\!\cdots\!77\)\( T_{19}^{24} - \)\(13\!\cdots\!76\)\( T_{19}^{23} - \)\(31\!\cdots\!34\)\( T_{19}^{22} - \)\(50\!\cdots\!96\)\( T_{19}^{21} - \)\(87\!\cdots\!74\)\( T_{19}^{20} + \)\(10\!\cdots\!24\)\( T_{19}^{19} + \)\(74\!\cdots\!18\)\( T_{19}^{18} + \)\(18\!\cdots\!88\)\( T_{19}^{17} + \)\(40\!\cdots\!09\)\( T_{19}^{16} + \)\(73\!\cdots\!68\)\( T_{19}^{15} + \)\(10\!\cdots\!84\)\( T_{19}^{14} + \)\(14\!\cdots\!08\)\( T_{19}^{13} + \)\(17\!\cdots\!39\)\( T_{19}^{12} + \)\(17\!\cdots\!48\)\( T_{19}^{11} + \)\(16\!\cdots\!46\)\( T_{19}^{10} + \)\(13\!\cdots\!94\)\( T_{19}^{9} + \)\(95\!\cdots\!72\)\( T_{19}^{8} + \)\(56\!\cdots\!12\)\( T_{19}^{7} + \)\(29\!\cdots\!62\)\( T_{19}^{6} + \)\(11\!\cdots\!04\)\( T_{19}^{5} + \)\(16\!\cdots\!49\)\( T_{19}^{4} - \)\(25\!\cdots\!60\)\( T_{19}^{3} + \)\(92\!\cdots\!50\)\( T_{19}^{2} - \)\(12\!\cdots\!50\)\( T_{19} + \)\(88\!\cdots\!25\)\( \)">\(T_{19}^{56} - \cdots\)