Properties

Label 231.3.m.b
Level $231$
Weight $3$
Character orbit 231.m
Analytic conductor $6.294$
Analytic rank $0$
Dimension $28$
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] = [231,3,Mod(166,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.166");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 231.m (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: \(6.29429410672\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 42 q^{3} - 32 q^{4} + 16 q^{7} - 12 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 42 q^{3} - 32 q^{4} + 16 q^{7} - 12 q^{8} + 42 q^{9} + 30 q^{10} - 96 q^{12} - 56 q^{14} - 20 q^{16} - 24 q^{17} + 36 q^{19} + 30 q^{21} - 20 q^{23} - 18 q^{24} + 54 q^{25} + 72 q^{26} - 96 q^{28} + 68 q^{29} + 30 q^{30} + 138 q^{31} + 124 q^{32} + 26 q^{35} - 192 q^{36} - 126 q^{37} - 168 q^{38} - 24 q^{39} - 234 q^{40} - 108 q^{42} - 196 q^{43} + 44 q^{44} + 56 q^{46} + 240 q^{47} + 82 q^{49} - 360 q^{50} - 24 q^{51} + 330 q^{52} + 188 q^{53} + 64 q^{56} + 72 q^{57} - 32 q^{58} - 108 q^{59} + 132 q^{60} + 150 q^{61} + 42 q^{63} - 80 q^{64} - 164 q^{65} - 242 q^{67} - 534 q^{68} + 476 q^{70} + 112 q^{71} - 18 q^{72} - 276 q^{73} - 206 q^{74} + 162 q^{75} + 22 q^{77} + 144 q^{78} - 4 q^{79} + 480 q^{80} - 126 q^{81} - 72 q^{82} - 126 q^{84} + 460 q^{85} + 476 q^{86} + 102 q^{87} + 66 q^{88} + 180 q^{89} - 22 q^{91} + 656 q^{92} + 138 q^{93} + 378 q^{94} - 238 q^{95} + 372 q^{96} - 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −1.77117 + 3.06776i 1.50000 0.866025i −4.27410 7.40296i −7.27902 4.20254i 6.13552i 6.51694 + 2.55529i 16.1113 1.50000 2.59808i 25.7848 14.8869i
166.2 −1.58858 + 2.75151i 1.50000 0.866025i −3.04719 5.27789i −0.961730 0.555255i 5.50301i −6.99783 0.174356i 6.65421 1.50000 2.59808i 3.05558 1.76414i
166.3 −1.56641 + 2.71310i 1.50000 0.866025i −2.90728 5.03555i 1.45988 + 0.842865i 5.42620i 3.76448 5.90159i 5.68466 1.50000 2.59808i −4.57355 + 2.64054i
166.4 −1.19231 + 2.06513i 1.50000 0.866025i −0.843183 1.46044i 2.80248 + 1.61801i 4.13027i −6.38656 + 2.86564i −5.51712 1.50000 2.59808i −6.68282 + 3.85833i
166.5 −0.960419 + 1.66349i 1.50000 0.866025i 0.155192 + 0.268801i 5.90040 + 3.40660i 3.32699i 5.01823 + 4.88030i −8.27955 1.50000 2.59808i −11.3337 + 6.54352i
166.6 −0.702448 + 1.21667i 1.50000 0.866025i 1.01313 + 1.75480i −7.60842 4.39272i 2.43335i −0.216994 + 6.99664i −8.46628 1.50000 2.59808i 10.6890 6.17131i
166.7 −0.0824997 + 0.142894i 1.50000 0.866025i 1.98639 + 3.44052i 6.61692 + 3.82028i 0.285787i 2.84959 6.39373i −1.31550 1.50000 2.59808i −1.09179 + 0.630344i
166.8 0.235283 0.407522i 1.50000 0.866025i 1.88928 + 3.27234i 1.22197 + 0.705503i 0.815044i −1.82511 + 6.75788i 3.66033 1.50000 2.59808i 0.575016 0.331985i
166.9 0.482489 0.835695i 1.50000 0.866025i 1.53441 + 2.65767i −3.16765 1.82885i 1.67139i 6.98831 0.404391i 6.82125 1.50000 2.59808i −3.05672 + 1.76480i
166.10 1.06901 1.85157i 1.50000 0.866025i −0.285549 0.494586i −5.40395 3.11997i 3.70315i 0.976308 6.93158i 7.33104 1.50000 2.59808i −11.5537 + 6.67054i
166.11 1.10214 1.90897i 1.50000 0.866025i −0.429434 0.743802i 5.58603 + 3.22509i 3.81793i −6.22295 3.20545i 6.92395 1.50000 2.59808i 12.3132 7.10903i
166.12 1.37561 2.38263i 1.50000 0.866025i −1.78461 3.09104i 3.80869 + 2.19895i 4.76526i 5.12759 + 4.76527i 1.18516 1.50000 2.59808i 10.4786 6.04979i
166.13 1.68556 2.91948i 1.50000 0.866025i −3.68224 6.37782i −3.91191 2.25854i 5.83896i −6.46810 + 2.67650i −11.3421 1.50000 2.59808i −13.1875 + 7.61382i
166.14 1.91374 3.31470i 1.50000 0.866025i −5.32482 9.22286i 0.936300 + 0.540573i 6.62940i 4.87609 5.02232i −25.4514 1.50000 2.59808i 3.58367 2.06904i
199.1 −1.77117 3.06776i 1.50000 + 0.866025i −4.27410 + 7.40296i −7.27902 + 4.20254i 6.13552i 6.51694 2.55529i 16.1113 1.50000 + 2.59808i 25.7848 + 14.8869i
199.2 −1.58858 2.75151i 1.50000 + 0.866025i −3.04719 + 5.27789i −0.961730 + 0.555255i 5.50301i −6.99783 + 0.174356i 6.65421 1.50000 + 2.59808i 3.05558 + 1.76414i
199.3 −1.56641 2.71310i 1.50000 + 0.866025i −2.90728 + 5.03555i 1.45988 0.842865i 5.42620i 3.76448 + 5.90159i 5.68466 1.50000 + 2.59808i −4.57355 2.64054i
199.4 −1.19231 2.06513i 1.50000 + 0.866025i −0.843183 + 1.46044i 2.80248 1.61801i 4.13027i −6.38656 2.86564i −5.51712 1.50000 + 2.59808i −6.68282 3.85833i
199.5 −0.960419 1.66349i 1.50000 + 0.866025i 0.155192 0.268801i 5.90040 3.40660i 3.32699i 5.01823 4.88030i −8.27955 1.50000 + 2.59808i −11.3337 6.54352i
199.6 −0.702448 1.21667i 1.50000 + 0.866025i 1.01313 1.75480i −7.60842 + 4.39272i 2.43335i −0.216994 6.99664i −8.46628 1.50000 + 2.59808i 10.6890 + 6.17131i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 166.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.3.m.b 28
7.d odd 6 1 inner 231.3.m.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.3.m.b 28 1.a even 1 1 trivial
231.3.m.b 28 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 44 T_{2}^{26} + 4 T_{2}^{25} + 1181 T_{2}^{24} + 132 T_{2}^{23} + 20428 T_{2}^{22} + 2094 T_{2}^{21} + 260734 T_{2}^{20} + 19404 T_{2}^{19} + 2437672 T_{2}^{18} + 71432 T_{2}^{17} + 17439685 T_{2}^{16} + \cdots + 8088336 \) acting on \(S_{3}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display