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Properties

Label 273.3.bo.c
Level $273$
Weight $3$
Character orbit 273.bo
Analytic conductor $7.439$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,3,Mod(160,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.160");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 273.bo (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.43871121704\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 54 q^{3} + 44 q^{4} - 4 q^{5} + 10 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 54 q^{3} + 44 q^{4} - 4 q^{5} + 10 q^{7} + 54 q^{9} + 42 q^{11} - 36 q^{13} + 16 q^{14} + 6 q^{15} - 96 q^{16} - 12 q^{17} + 12 q^{19} - 10 q^{20} - 18 q^{22} + 24 q^{23} + 264 q^{25} + 114 q^{26} - 104 q^{28} + 76 q^{29} - 160 q^{31} - 42 q^{33} - 192 q^{34} - 100 q^{35} - 132 q^{36} + 6 q^{37} + 60 q^{39} + 200 q^{41} + 18 q^{42} + 48 q^{43} - 6 q^{45} + 396 q^{46} + 56 q^{47} + 288 q^{48} - 154 q^{49} - 102 q^{50} + 24 q^{51} - 360 q^{52} + 76 q^{53} + 192 q^{55} - 132 q^{56} - 162 q^{58} + 128 q^{59} - 120 q^{61} + 24 q^{62} - 30 q^{63} - 484 q^{64} - 284 q^{65} - 144 q^{67} + 234 q^{68} - 72 q^{69} + 300 q^{70} - 96 q^{71} + 728 q^{73} - 144 q^{74} - 396 q^{75} - 516 q^{76} - 160 q^{77} - 144 q^{78} + 68 q^{79} - 58 q^{80} - 162 q^{81} + 72 q^{82} + 368 q^{83} + 108 q^{84} - 324 q^{85} - 228 q^{87} + 186 q^{88} + 92 q^{89} + 176 q^{91} - 1044 q^{92} + 240 q^{93} - 336 q^{94} - 2 q^{95} - 72 q^{97} + 234 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1 −3.42934 − 1.97993i −1.50000 − 0.866025i 5.84025 + 10.1156i 3.37699 3.42934 + 5.93979i −6.27422 − 3.10389i − 30.4138i 1.50000 + 2.59808i −11.5809 − 6.68621i
160.2 −2.91220 − 1.68136i −1.50000 − 0.866025i 3.65393 + 6.32879i −9.11132 2.91220 + 5.04407i 6.88035 + 1.28869i − 11.1234i 1.50000 + 2.59808i 26.5340 + 15.3194i
160.3 −2.70725 − 1.56303i −1.50000 − 0.866025i 2.88613 + 4.99893i −2.70278 2.70725 + 4.68909i −3.66082 + 5.96644i − 5.54020i 1.50000 + 2.59808i 7.31711 + 4.22453i
160.4 −2.50186 − 1.44445i −1.50000 − 0.866025i 2.17288 + 3.76353i 0.513528 2.50186 + 4.33335i 1.60791 − 6.81283i − 0.998850i 1.50000 + 2.59808i −1.28478 − 0.741765i
160.5 −2.23101 − 1.28808i −1.50000 − 0.866025i 1.31828 + 2.28333i 8.28362 2.23101 + 3.86423i −1.15847 + 6.90347i 3.51244i 1.50000 + 2.59808i −18.4809 − 10.6699i
160.6 −1.57897 − 0.911620i −1.50000 − 0.866025i −0.337897 − 0.585254i 5.47364 1.57897 + 2.73486i 4.91355 − 4.98568i 8.52510i 1.50000 + 2.59808i −8.64273 − 4.98988i
160.7 −1.28706 − 0.743087i −1.50000 − 0.866025i −0.895643 − 1.55130i −6.18658 1.28706 + 2.22926i −6.65420 + 2.17293i 8.60686i 1.50000 + 2.59808i 7.96253 + 4.59717i
160.8 −0.753808 − 0.435211i −1.50000 − 0.866025i −1.62118 − 2.80797i 2.33010 0.753808 + 1.30563i 2.98985 + 6.32936i 6.30392i 1.50000 + 2.59808i −1.75644 − 1.01408i
160.9 −0.149303 − 0.0862001i −1.50000 − 0.866025i −1.98514 − 3.43836i −9.04137 0.149303 + 0.258600i 1.40516 − 6.85752i 1.37408i 1.50000 + 2.59808i 1.34990 + 0.779367i
160.10 0.181190 + 0.104610i −1.50000 − 0.866025i −1.97811 − 3.42619i 6.37482 −0.181190 − 0.313831i −3.53577 − 6.04139i − 1.66461i 1.50000 + 2.59808i 1.15505 + 0.666871i
160.11 0.853578 + 0.492814i −1.50000 − 0.866025i −1.51427 − 2.62279i −0.237276 −0.853578 − 1.47844i 6.93605 + 0.944019i − 6.92752i 1.50000 + 2.59808i −0.202534 − 0.116933i
160.12 1.15840 + 0.668803i −1.50000 − 0.866025i −1.10541 − 1.91462i −3.82472 −1.15840 − 2.00641i −0.0208330 + 6.99997i − 8.30761i 1.50000 + 2.59808i −4.43056 − 2.55798i
160.13 1.63641 + 0.944780i −1.50000 − 0.866025i −0.214780 − 0.372010i 8.59608 −1.63641 − 2.83434i 0.630577 + 6.97154i − 8.36992i 1.50000 + 2.59808i 14.0667 + 8.12141i
160.14 2.15997 + 1.24706i −1.50000 − 0.866025i 1.11031 + 1.92311i −2.93884 −2.15997 − 3.74118i 6.81641 − 1.59267i − 4.43798i 1.50000 + 2.59808i −6.34780 − 3.66491i
160.15 2.24805 + 1.29791i −1.50000 − 0.866025i 1.36914 + 2.37142i −3.07282 −2.24805 − 3.89373i −4.26630 − 5.54966i − 3.27519i 1.50000 + 2.59808i −6.90784 − 3.98824i
160.16 3.01987 + 1.74353i −1.50000 − 0.866025i 4.07976 + 7.06636i 5.35167 −3.01987 − 5.23058i −6.87138 + 1.33570i 14.5045i 1.50000 + 2.59808i 16.1614 + 9.33077i
160.17 3.02517 + 1.74658i −1.50000 − 0.866025i 4.10109 + 7.10329i −8.36326 −3.02517 − 5.23974i −1.39364 + 6.85987i 14.6789i 1.50000 + 2.59808i −25.3002 − 14.6071i
160.18 3.26818 + 1.88688i −1.50000 − 0.866025i 5.12066 + 8.86924i 3.17854 −3.26818 − 5.66065i 6.65578 − 2.16809i 23.5533i 1.50000 + 2.59808i 10.3880 + 5.99754i
244.1 −3.42934 + 1.97993i −1.50000 + 0.866025i 5.84025 − 10.1156i 3.37699 3.42934 − 5.93979i −6.27422 + 3.10389i 30.4138i 1.50000 − 2.59808i −11.5809 + 6.68621i
244.2 −2.91220 + 1.68136i −1.50000 + 0.866025i 3.65393 − 6.32879i −9.11132 2.91220 − 5.04407i 6.88035 − 1.28869i 11.1234i 1.50000 − 2.59808i 26.5340 − 15.3194i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.3.bo.c ✓ 36
7.b odd 2 1 273.3.bo.d yes 36
13.e even 6 1 273.3.bo.d yes 36
91.t odd 6 1 inner 273.3.bo.c ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.3.bo.c ✓ 36 1.a even 1 1 trivial
273.3.bo.c ✓ 36 91.t odd 6 1 inner
273.3.bo.d yes 36 7.b odd 2 1
273.3.bo.d yes 36 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{36} - 58 T_{2}^{34} + 1982 T_{2}^{32} - 45166 T_{2}^{30} - 138 T_{2}^{29} + 768186 T_{2}^{28} + \cdots + 381772521 \) Copy content Toggle raw display
\( T_{5}^{18} + 2 T_{5}^{17} - 289 T_{5}^{16} - 386 T_{5}^{15} + 33608 T_{5}^{14} + 24562 T_{5}^{13} + \cdots - 16123887168 \) Copy content Toggle raw display