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Properties

Label 660.2.f.b
Level $660$
Weight $2$
Character orbit 660.f
Analytic conductor $5.270$
Analytic rank $0$
Dimension $40$
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] = [660,2,Mod(551,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.551");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.f (of order \(2\), degree \(1\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{4} + 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{4} + 16 q^{6} + 2 q^{10} + 40 q^{11} - 20 q^{12} + 8 q^{13} - 16 q^{14} - 2 q^{16} - 16 q^{18} - 4 q^{21} + 8 q^{23} - 4 q^{24} - 40 q^{25} - 16 q^{26} + 60 q^{32} - 16 q^{34} - 48 q^{36} - 8 q^{37} - 16 q^{39} - 14 q^{40} + 36 q^{42} + 2 q^{44} + 4 q^{45} + 12 q^{46} + 4 q^{48} - 64 q^{49} + 12 q^{51} + 36 q^{52} + 30 q^{54} + 32 q^{56} + 36 q^{58} - 40 q^{59} + 8 q^{61} - 28 q^{62} - 80 q^{63} - 22 q^{64} + 16 q^{66} + 4 q^{69} + 8 q^{71} - 56 q^{72} + 24 q^{73} - 20 q^{74} + 12 q^{76} - 40 q^{78} - 16 q^{80} + 20 q^{81} - 36 q^{82} - 32 q^{83} + 10 q^{84} - 16 q^{85} - 96 q^{86} - 32 q^{90} + 24 q^{92} - 48 q^{93} - 36 q^{94} - 52 q^{96} + 24 q^{97} + 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1 −1.39992 − 0.200562i −0.966677 − 1.43720i 1.91955 + 0.561540i 1.00000i 1.06502 + 2.20584i − 0.753696i −2.57459 − 1.17110i −1.13107 + 2.77861i 0.200562 − 1.39992i
551.2 −1.39992 + 0.200562i −0.966677 + 1.43720i 1.91955 − 0.561540i − 1.00000i 1.06502 − 2.20584i 0.753696i −2.57459 + 1.17110i −1.13107 − 2.77861i 0.200562 + 1.39992i
551.3 −1.33147 − 0.476654i 0.316551 + 1.70288i 1.54560 + 1.26930i 1.00000i 0.390209 − 2.41821i 4.54081i −1.45290 − 2.42674i −2.79959 + 1.07809i 0.476654 − 1.33147i
551.4 −1.33147 + 0.476654i 0.316551 − 1.70288i 1.54560 − 1.26930i − 1.00000i 0.390209 + 2.41821i − 4.54081i −1.45290 + 2.42674i −2.79959 − 1.07809i 0.476654 + 1.33147i
551.5 −1.25613 − 0.649729i −1.73200 − 0.0138729i 1.15570 + 1.63228i − 1.00000i 2.16659 + 1.14275i − 4.91023i −0.391170 − 2.80125i 2.99962 + 0.0480557i −0.649729 + 1.25613i
551.6 −1.25613 + 0.649729i −1.73200 + 0.0138729i 1.15570 − 1.63228i 1.00000i 2.16659 − 1.14275i 4.91023i −0.391170 + 2.80125i 2.99962 − 0.0480557i −0.649729 − 1.25613i
551.7 −1.22154 − 0.712627i −1.37464 − 1.05373i 0.984327 + 1.74101i − 1.00000i 0.928264 + 2.26679i 3.93333i 0.0382908 − 2.82817i 0.779286 + 2.89702i −0.712627 + 1.22154i
551.8 −1.22154 + 0.712627i −1.37464 + 1.05373i 0.984327 − 1.74101i 1.00000i 0.928264 − 2.26679i − 3.93333i 0.0382908 + 2.82817i 0.779286 − 2.89702i −0.712627 − 1.22154i
551.9 −1.06430 − 0.931272i 1.02152 − 1.39875i 0.265463 + 1.98230i 1.00000i −2.38982 + 0.537365i − 2.86889i 1.56353 − 2.35698i −0.912977 − 2.85770i 0.931272 − 1.06430i
551.10 −1.06430 + 0.931272i 1.02152 + 1.39875i 0.265463 − 1.98230i − 1.00000i −2.38982 − 0.537365i 2.86889i 1.56353 + 2.35698i −0.912977 + 2.85770i 0.931272 + 1.06430i
551.11 −1.01854 − 0.981109i 1.72538 + 0.151861i 0.0748510 + 1.99860i − 1.00000i −1.60838 − 1.84746i 2.86984i 1.88460 − 2.10909i 2.95388 + 0.524035i −0.981109 + 1.01854i
551.12 −1.01854 + 0.981109i 1.72538 − 0.151861i 0.0748510 − 1.99860i 1.00000i −1.60838 + 1.84746i − 2.86984i 1.88460 + 2.10909i 2.95388 − 0.524035i −0.981109 − 1.01854i
551.13 −0.945773 − 1.05143i −1.20164 + 1.24742i −0.211027 + 1.98884i 1.00000i 2.44806 + 0.0836622i − 0.166500i 2.29071 − 1.65911i −0.112135 − 2.99790i 1.05143 − 0.945773i
551.14 −0.945773 + 1.05143i −1.20164 − 1.24742i −0.211027 − 1.98884i − 1.00000i 2.44806 − 0.0836622i 0.166500i 2.29071 + 1.65911i −0.112135 + 2.99790i 1.05143 + 0.945773i
551.15 −0.688496 − 1.23530i 0.162284 + 1.72443i −1.05195 + 1.70100i − 1.00000i 2.01846 − 1.38773i − 2.18197i 2.82551 + 0.128337i −2.94733 + 0.559694i −1.23530 + 0.688496i
551.16 −0.688496 + 1.23530i 0.162284 − 1.72443i −1.05195 − 1.70100i 1.00000i 2.01846 + 1.38773i 2.18197i 2.82551 − 0.128337i −2.94733 − 0.559694i −1.23530 − 0.688496i
551.17 −0.227384 − 1.39581i 0.808429 + 1.53181i −1.89659 + 0.634771i 1.00000i 1.95430 − 1.47672i − 1.69537i 1.31728 + 2.50295i −1.69289 + 2.47672i 1.39581 − 0.227384i
551.18 −0.227384 + 1.39581i 0.808429 − 1.53181i −1.89659 − 0.634771i − 1.00000i 1.95430 + 1.47672i 1.69537i 1.31728 − 2.50295i −1.69289 − 2.47672i 1.39581 + 0.227384i
551.19 0.129521 − 1.40827i 1.68540 + 0.399293i −1.96645 − 0.364802i − 1.00000i 0.780607 − 2.32178i − 2.41856i −0.768437 + 2.72204i 2.68113 + 1.34593i −1.40827 − 0.129521i
551.20 0.129521 + 1.40827i 1.68540 − 0.399293i −1.96645 + 0.364802i 1.00000i 0.780607 + 2.32178i 2.41856i −0.768437 − 2.72204i 2.68113 − 1.34593i −1.40827 + 0.129521i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.f.b yes 40
3.b odd 2 1 660.2.f.a ✓ 40
4.b odd 2 1 660.2.f.a ✓ 40
12.b even 2 1 inner 660.2.f.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.f.a ✓ 40 3.b odd 2 1
660.2.f.a ✓ 40 4.b odd 2 1
660.2.f.b yes 40 1.a even 1 1 trivial
660.2.f.b yes 40 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{20} - 4 T_{23}^{19} - 230 T_{23}^{18} + 960 T_{23}^{17} + 20752 T_{23}^{16} - 91488 T_{23}^{15} + \cdots + 620232704 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\). Copy content Toggle raw display