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Properties

Label 666.2.bf.a
Level $666$
Weight $2$
Character orbit 666.bf
Analytic conductor $5.318$
Analytic rank $0$
Dimension $152$
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] = [666,2,Mod(245,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.245");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.bf (of order \(12\), degree \(4\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 12 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 12 q^{5} - 4 q^{9} + 4 q^{12} + 8 q^{13} - 12 q^{15} + 76 q^{16} + 4 q^{19} - 12 q^{20} + 36 q^{23} - 36 q^{27} - 12 q^{28} - 16 q^{31} - 20 q^{37} - 28 q^{39} - 48 q^{41} - 20 q^{42} + 4 q^{43} + 24 q^{47} + 160 q^{49} + 48 q^{50} - 28 q^{51} - 8 q^{52} - 36 q^{54} + 12 q^{55} + 64 q^{57} + 12 q^{59} + 56 q^{61} + 36 q^{62} - 36 q^{63} - 64 q^{66} - 36 q^{67} + 16 q^{69} - 48 q^{71} + 100 q^{75} + 8 q^{76} + 4 q^{79} + 4 q^{81} + 24 q^{82} - 120 q^{83} + 76 q^{87} - 36 q^{89} - 60 q^{90} + 60 q^{91} - 12 q^{93} - 96 q^{95} - 76 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −0.258819 + 0.965926i −1.68648 + 0.394690i −0.866025 − 0.500000i −0.743082 − 2.77322i 0.0552525 − 1.73117i 1.89577 0.707107 − 0.707107i 2.68844 − 1.33127i 2.87105
245.2 −0.258819 + 0.965926i −1.67147 + 0.454091i −0.866025 − 0.500000i 0.00333892 + 0.0124610i −0.00601099 − 1.73204i −4.65378 0.707107 − 0.707107i 2.58760 − 1.51800i −0.0129006
245.3 −0.258819 + 0.965926i −1.64361 − 0.546395i −0.866025 − 0.500000i 0.989800 + 3.69398i 0.953175 − 1.44619i −0.478437 0.707107 − 0.707107i 2.40290 + 1.79612i −3.82429
245.4 −0.258819 + 0.965926i −1.47491 + 0.908100i −0.866025 − 0.500000i 0.428798 + 1.60029i −0.495423 − 1.65969i 3.69937 0.707107 − 0.707107i 1.35071 − 2.67873i −1.65675
245.5 −0.258819 + 0.965926i −1.23676 + 1.21261i −0.866025 − 0.500000i 0.116223 + 0.433749i −0.851189 − 1.50847i −2.94935 0.707107 − 0.707107i 0.0591727 − 2.99942i −0.449050
245.6 −0.258819 + 0.965926i −1.17589 − 1.27172i −0.866025 − 0.500000i 0.105093 + 0.392212i 1.53273 − 0.806680i 0.173800 0.707107 − 0.707107i −0.234549 + 2.99082i −0.406048
245.7 −0.258819 + 0.965926i −0.938216 − 1.45594i −0.866025 − 0.500000i −0.170947 − 0.637983i 1.64915 − 0.529424i 5.00666 0.707107 − 0.707107i −1.23950 + 2.73197i 0.660488
245.8 −0.258819 + 0.965926i −0.671145 − 1.59674i −0.866025 − 0.500000i −1.11236 − 4.15139i 1.71603 − 0.235011i −2.55667 0.707107 − 0.707107i −2.09913 + 2.14328i 4.29784
245.9 −0.258819 + 0.965926i −0.362689 + 1.69365i −0.866025 − 0.500000i −0.395774 − 1.47705i −1.54207 − 0.788680i 3.39212 0.707107 − 0.707107i −2.73691 − 1.22854i 1.52915
245.10 −0.258819 + 0.965926i −0.0267639 + 1.73184i −0.866025 − 0.500000i −0.522511 − 1.95004i −1.66591 − 0.474086i −0.788271 0.707107 − 0.707107i −2.99857 − 0.0927017i 2.01883
245.11 −0.258819 + 0.965926i 0.0613418 − 1.73096i −0.866025 − 0.500000i 0.123865 + 0.462270i 1.65611 + 0.507258i −1.46006 0.707107 − 0.707107i −2.99247 − 0.212361i −0.478577
245.12 −0.258819 + 0.965926i 0.654998 + 1.60343i −0.866025 − 0.500000i 0.689228 + 2.57223i −1.71832 + 0.217682i −3.14302 0.707107 − 0.707107i −2.14196 + 2.10048i −2.66297
245.13 −0.258819 + 0.965926i 1.03191 − 1.39110i −0.866025 − 0.500000i −0.859123 − 3.20629i 1.07663 + 1.35679i 4.59798 0.707107 − 0.707107i −0.870336 − 2.87098i 3.31940
245.14 −0.258819 + 0.965926i 1.03692 − 1.38737i −0.866025 − 0.500000i 0.991075 + 3.69874i 1.07172 + 1.36067i 3.24760 0.707107 − 0.707107i −0.849596 − 2.87718i −3.82922
245.15 −0.258819 + 0.965926i 1.07814 − 1.35559i −0.866025 − 0.500000i −0.0382118 − 0.142609i 1.03035 + 1.39225i −2.32035 0.707107 − 0.707107i −0.675230 − 2.92302i 0.147639
245.16 −0.258819 + 0.965926i 1.31219 + 1.13056i −0.866025 − 0.500000i −0.882455 − 3.29337i −1.43166 + 0.974863i −2.73457 0.707107 − 0.707107i 0.443660 + 2.96701i 3.40954
245.17 −0.258819 + 0.965926i 1.39398 + 1.02802i −0.866025 − 0.500000i 0.536739 + 2.00314i −1.35378 + 1.08041i 1.37855 0.707107 − 0.707107i 0.886340 + 2.86608i −2.07380
245.18 −0.258819 + 0.965926i 1.72311 + 0.175759i −0.866025 − 0.500000i 0.413064 + 1.54158i −0.615743 + 1.61891i 2.55244 0.707107 − 0.707107i 2.93822 + 0.605703i −1.59596
245.19 −0.258819 + 0.965926i 1.72934 − 0.0969404i −0.866025 − 0.500000i −0.306731 − 1.14474i −0.353948 + 1.69550i −3.12771 0.707107 − 0.707107i 2.98121 − 0.335285i 1.18512
245.20 0.258819 − 0.965926i −1.68208 − 0.413050i −0.866025 − 0.500000i 0.563358 + 2.10248i −0.834330 + 1.51786i −3.27112 −0.707107 + 0.707107i 2.65878 + 1.38957i 2.17665
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.bf.a yes 152
9.d odd 6 1 666.2.ba.a ✓ 152
37.g odd 12 1 666.2.ba.a ✓ 152
333.bf even 12 1 inner 666.2.bf.a yes 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.ba.a ✓ 152 9.d odd 6 1
666.2.ba.a ✓ 152 37.g odd 12 1
666.2.bf.a yes 152 1.a even 1 1 trivial
666.2.bf.a yes 152 333.bf even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(666, [\chi])\).