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Properties

Label 1110.2.ba.a
Level $1110$
Weight $2$
Character orbit 1110.ba
Analytic conductor $8.863$
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] = [1110,2,Mod(529,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.ba (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: \(8.86339462436\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{2} - 18 q^{4} + 2 q^{5} + 36 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{2} - 18 q^{4} + 2 q^{5} + 36 q^{8} + 18 q^{9} + 2 q^{10} + 4 q^{11} - 14 q^{13} - 2 q^{15} - 18 q^{16} + 18 q^{18} + 6 q^{19} - 4 q^{20} - 2 q^{22} - 20 q^{23} + 4 q^{25} + 28 q^{26} - 2 q^{30} - 18 q^{32} - 6 q^{33} - 40 q^{35} - 36 q^{36} + 20 q^{37} + 6 q^{39} + 2 q^{40} + 10 q^{41} - 2 q^{44} - 2 q^{45} + 10 q^{46} + 10 q^{49} - 2 q^{50} - 14 q^{52} - 12 q^{53} + 56 q^{55} + 8 q^{57} + 30 q^{58} + 18 q^{59} + 4 q^{60} - 6 q^{61} - 12 q^{62} + 36 q^{64} + 40 q^{65} + 36 q^{67} + 12 q^{69} + 20 q^{70} - 24 q^{71} + 18 q^{72} - 34 q^{74} + 8 q^{75} - 6 q^{76} - 24 q^{77} - 6 q^{78} + 2 q^{80} - 18 q^{81} - 20 q^{82} + 36 q^{83} + 26 q^{85} - 10 q^{87} + 4 q^{88} + 4 q^{90} - 36 q^{91} + 10 q^{92} + 12 q^{93} + 12 q^{94} - 30 q^{95} + 52 q^{97} + 10 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.83366 + 1.27972i 1.00000i 2.57051 + 1.48408i 1.00000 0.500000 + 0.866025i 2.02510 + 0.948138i
529.2 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.648845 − 2.13986i 1.00000i 1.49882 + 0.865342i 1.00000 0.500000 + 0.866025i −1.52875 + 1.63185i
529.3 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.491597 + 2.18136i 1.00000i −0.916644 − 0.529225i 1.00000 0.500000 + 0.866025i 2.13491 − 0.664945i
529.4 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i 2.18544 + 0.473136i 1.00000i 0.827955 + 0.478020i 1.00000 0.500000 + 0.866025i −0.682971 − 2.12921i
529.5 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i 1.51023 − 1.64901i 1.00000i 0.0701099 + 0.0404780i 1.00000 0.500000 + 0.866025i −2.18319 − 0.483391i
529.6 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −2.21345 − 0.317246i 1.00000i −1.17143 − 0.676327i 1.00000 0.500000 + 0.866025i 0.831981 + 2.07553i
529.7 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.00696233 + 2.23606i 1.00000i 3.60632 + 2.08211i 1.00000 0.500000 + 0.866025i 1.93996 − 1.11200i
529.8 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i 1.03441 − 1.98242i 1.00000i −3.17295 − 1.83190i 1.00000 0.500000 + 0.866025i −2.23403 + 0.0953865i
529.9 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i 1.83047 + 1.28429i 1.00000i −3.31268 − 1.91258i 1.00000 0.500000 + 0.866025i 0.196990 − 2.22737i
529.10 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.05954 + 1.96911i − 1.00000i −3.29356 − 1.90154i 1.00000 0.500000 + 0.866025i 1.17553 − 1.90214i
529.11 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.17865 − 0.503482i − 1.00000i −3.20005 − 1.84755i 1.00000 0.500000 + 0.866025i −1.52535 − 1.63502i
529.12 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.12688 + 0.690199i − 1.00000i 3.89483 + 2.24868i 1.00000 0.500000 + 0.866025i 1.66117 + 1.49683i
529.13 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.668714 + 2.13373i − 1.00000i 2.13280 + 1.23137i 1.00000 0.500000 + 0.866025i 1.51351 − 1.64599i
529.14 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.16244 + 0.569075i − 1.00000i 1.99912 + 1.15419i 1.00000 0.500000 + 0.866025i −0.588388 − 2.15727i
529.15 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.18834 + 0.459547i − 1.00000i −1.07219 − 0.619032i 1.00000 0.500000 + 0.866025i 1.49215 + 1.66538i
529.16 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.603880 − 2.15298i − 1.00000i 0.998396 + 0.576424i 1.00000 0.500000 + 0.866025i −2.16648 + 0.553515i
529.17 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −0.581519 − 2.15913i − 1.00000i 1.29297 + 0.746494i 1.00000 0.500000 + 0.866025i −1.57910 + 1.58317i
529.18 −0.500000 − 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.14251 − 0.640045i − 1.00000i −2.75229 − 1.58904i 1.00000 0.500000 + 0.866025i 0.516959 + 2.17549i
619.1 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.83366 − 1.27972i − 1.00000i 2.57051 − 1.48408i 1.00000 0.500000 − 0.866025i 2.02510 − 0.948138i
619.2 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −0.648845 + 2.13986i − 1.00000i 1.49882 − 0.865342i 1.00000 0.500000 − 0.866025i −1.52875 − 1.63185i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.ba.a ✓ 36
5.b even 2 1 1110.2.ba.b yes 36
37.e even 6 1 1110.2.ba.b yes 36
185.l even 6 1 inner 1110.2.ba.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.ba.a ✓ 36 1.a even 1 1 trivial
1110.2.ba.a ✓ 36 185.l even 6 1 inner
1110.2.ba.b yes 36 5.b even 2 1
1110.2.ba.b yes 36 37.e even 6 1