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Properties

Label 360.2.bm.b
Level $360$
Weight $2$
Character orbit 360.bm
Analytic conductor $2.875$
Analytic rank $0$
Dimension $48$
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] = [360,2,Mod(11,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bm (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{5} - q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{5} - q^{6} + 6 q^{8} + 13 q^{12} + 15 q^{14} - 12 q^{16} + 7 q^{18} + 4 q^{21} - 21 q^{22} - 4 q^{24} - 24 q^{25} + 12 q^{27} - 2 q^{30} - 8 q^{33} - 27 q^{34} - 31 q^{36} - 27 q^{38} - 16 q^{39} + 12 q^{40} + 12 q^{41} - 9 q^{42} + 24 q^{44} - 6 q^{46} - 12 q^{47} + 7 q^{48} + 24 q^{49} - 20 q^{51} + 54 q^{52} - 32 q^{54} + 21 q^{56} + 4 q^{57} + 33 q^{58} - 36 q^{59} - q^{60} - 12 q^{61} - 42 q^{62} - 56 q^{63} - 12 q^{64} - 32 q^{66} + 51 q^{68} + 40 q^{69} + 15 q^{70} + 6 q^{72} + 54 q^{74} - 51 q^{76} - 24 q^{78} - 8 q^{81} - 18 q^{82} - 60 q^{83} + 41 q^{84} + 27 q^{86} - 36 q^{87} - 57 q^{88} - 22 q^{90} - 9 q^{92} - 75 q^{94} + 13 q^{96} - 42 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.40162 + 0.188277i 1.67034 − 0.458221i 1.92910 − 0.527787i 0.500000 + 0.866025i −2.25492 + 0.956740i −3.22730 − 1.86328i −2.60451 + 1.10297i 2.58007 − 1.53077i −0.863865 − 1.11970i
11.2 −1.33590 − 0.464076i −1.29784 + 1.14700i 1.56927 + 1.23992i 0.500000 + 0.866025i 2.26609 − 0.929977i −3.14000 − 1.81288i −1.52097 − 2.38467i 0.368796 − 2.97725i −0.266049 − 1.38896i
11.3 −1.31577 − 0.518408i 0.424591 + 1.67920i 1.46251 + 1.36421i 0.500000 + 0.866025i 0.311848 − 2.42956i 3.88456 + 2.24275i −1.21710 − 2.55317i −2.63944 + 1.42595i −0.208931 − 1.39870i
11.4 −1.21368 + 0.725941i −0.452164 + 1.67199i 0.946019 − 1.76211i 0.500000 + 0.866025i −0.664985 − 2.35750i −0.550736 − 0.317967i 0.131031 + 2.82539i −2.59109 − 1.51203i −1.23552 − 0.688104i
11.5 −1.17035 + 0.793896i 1.72054 − 0.199393i 0.739459 − 1.85828i 0.500000 + 0.866025i −1.85534 + 1.59929i 3.97204 + 2.29326i 0.609850 + 2.76190i 2.92049 − 0.686124i −1.27271 − 0.616609i
11.6 −1.14743 − 0.826684i −0.952268 − 1.44678i 0.633188 + 1.89712i 0.500000 + 0.866025i −0.103374 + 2.44731i −2.56287 − 1.47968i 0.841782 − 2.70026i −1.18637 + 2.75545i 0.142215 − 1.40704i
11.7 −0.877098 − 1.10937i 0.875353 − 1.49458i −0.461398 + 1.94605i 0.500000 + 0.866025i −2.42581 + 0.339801i 1.05351 + 0.608247i 2.56358 − 1.19502i −1.46751 − 2.61656i 0.522192 − 1.31427i
11.8 −0.760571 + 1.19228i −1.71670 + 0.230091i −0.843062 − 1.81363i 0.500000 + 0.866025i 1.03134 − 2.22179i 1.06767 + 0.616420i 2.80356 + 0.374228i 2.89412 − 0.789993i −1.41283 − 0.0625343i
11.9 −0.645247 + 1.25843i 0.801968 − 1.53520i −1.16731 − 1.62400i 0.500000 + 0.866025i 1.41448 + 1.99981i −4.07138 − 2.35061i 2.79690 − 0.421104i −1.71369 − 2.46237i −1.41246 + 0.0704168i
11.10 −0.582600 − 1.28863i 1.36390 + 1.06761i −1.32115 + 1.50152i 0.500000 + 0.866025i 0.581150 − 2.37955i −1.02179 − 0.589933i 2.70461 + 0.827699i 0.720424 + 2.91221i 0.824689 − 1.14886i
11.11 −0.148442 − 1.40640i −0.857820 + 1.50471i −1.95593 + 0.417537i 0.500000 + 0.866025i 2.24356 + 0.983078i −0.661806 − 0.382094i 0.877566 + 2.68884i −1.52829 − 2.58154i 1.14376 − 0.831755i
11.12 −0.116736 − 1.40939i −1.41981 − 0.992033i −1.97275 + 0.329051i 0.500000 + 0.866025i −1.23242 + 2.11687i 4.17077 + 2.40800i 0.694050 + 2.74195i 1.03174 + 2.81700i 1.16220 − 0.805790i
11.13 −0.0418375 + 1.41359i 1.46420 + 0.925262i −1.99650 − 0.118282i 0.500000 + 0.866025i −1.36920 + 2.03108i 0.947055 + 0.546782i 0.250732 − 2.81729i 1.28778 + 2.70954i −1.24513 + 0.670565i
11.14 0.0747682 + 1.41224i −0.478797 − 1.66456i −1.98882 + 0.211181i 0.500000 + 0.866025i 2.31495 − 0.800631i 1.88846 + 1.09030i −0.446938 − 2.79289i −2.54151 + 1.59397i −1.18565 + 0.770869i
11.15 0.475189 + 1.33199i −1.73205 + 0.00397907i −1.54839 + 1.26589i 0.500000 + 0.866025i −0.828349 − 2.30518i −2.40441 − 1.38819i −2.42193 − 1.46090i 2.99997 − 0.0137839i −0.915942 + 1.07752i
11.16 0.884203 − 1.10371i −1.18636 − 1.26196i −0.436372 − 1.95181i 0.500000 + 0.866025i −2.44183 + 0.193576i −2.20775 − 1.27465i −2.54009 − 1.24417i −0.185091 + 2.99428i 1.39795 + 0.213885i
11.17 0.913976 − 1.07919i −1.45808 + 0.934888i −0.329296 − 1.97270i 0.500000 + 0.866025i −0.323726 + 2.42800i 1.98473 + 1.14588i −2.42989 − 1.44763i 1.25197 − 2.72627i 1.39159 + 0.251932i
11.18 0.951037 + 1.04667i 1.28580 − 1.16048i −0.191057 + 1.99085i 0.500000 + 0.866025i 2.43749 + 0.242161i 1.13105 + 0.653010i −2.26548 + 1.69340i 0.306582 − 2.98429i −0.430929 + 1.34696i
11.19 0.959041 + 1.03935i 0.0478668 + 1.73139i −0.160481 + 1.99355i 0.500000 + 0.866025i −1.75361 + 1.71022i −1.21691 − 0.702581i −2.22590 + 1.74510i −2.99542 + 0.165752i −0.420580 + 1.35023i
11.20 1.02877 − 0.970379i 1.62132 − 0.609351i 0.116731 − 1.99659i 0.500000 + 0.866025i 1.07667 − 2.20018i −0.518944 − 0.299612i −1.81736 − 2.16730i 2.25738 − 1.97591i 1.35476 + 0.405751i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bm.b yes 48
3.b odd 2 1 1080.2.bm.a 48
4.b odd 2 1 1440.2.cc.b 48
8.b even 2 1 1440.2.cc.a 48
8.d odd 2 1 360.2.bm.a ✓ 48
9.c even 3 1 1080.2.bm.b 48
9.d odd 6 1 360.2.bm.a ✓ 48
12.b even 2 1 4320.2.cc.a 48
24.f even 2 1 1080.2.bm.b 48
24.h odd 2 1 4320.2.cc.b 48
36.f odd 6 1 4320.2.cc.b 48
36.h even 6 1 1440.2.cc.a 48
72.j odd 6 1 1440.2.cc.b 48
72.l even 6 1 inner 360.2.bm.b yes 48
72.n even 6 1 4320.2.cc.a 48
72.p odd 6 1 1080.2.bm.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bm.a ✓ 48 8.d odd 2 1
360.2.bm.a ✓ 48 9.d odd 6 1
360.2.bm.b yes 48 1.a even 1 1 trivial
360.2.bm.b yes 48 72.l even 6 1 inner
1080.2.bm.a 48 3.b odd 2 1
1080.2.bm.a 48 72.p odd 6 1
1080.2.bm.b 48 9.c even 3 1
1080.2.bm.b 48 24.f even 2 1
1440.2.cc.a 48 8.b even 2 1
1440.2.cc.a 48 36.h even 6 1
1440.2.cc.b 48 4.b odd 2 1
1440.2.cc.b 48 72.j odd 6 1
4320.2.cc.a 48 12.b even 2 1
4320.2.cc.a 48 72.n even 6 1
4320.2.cc.b 48 24.h odd 2 1
4320.2.cc.b 48 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} - 96 T_{7}^{46} + 5325 T_{7}^{44} + 1140 T_{7}^{43} - 198840 T_{7}^{42} + \cdots + 15\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display