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Properties

Label 1110.2.bb.d
Level $1110$
Weight $2$
Character orbit 1110.bb
Analytic conductor $8.863$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(1009,1110)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.bb (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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{4} + 2 q^{5} + 28 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{4} + 2 q^{5} + 28 q^{6} + 14 q^{9} + 4 q^{10} - 12 q^{11} + 8 q^{14} + 2 q^{15} - 14 q^{16} - 20 q^{19} - 2 q^{20} + 4 q^{21} + 14 q^{24} - 8 q^{25} - 20 q^{26} - 36 q^{29} + 2 q^{30} + 24 q^{31} + 38 q^{34} - 2 q^{35} + 28 q^{36} - 10 q^{39} + 2 q^{40} - 6 q^{44} + 4 q^{45} + 8 q^{46} + 50 q^{49} - 4 q^{50} + 76 q^{51} + 14 q^{54} - 28 q^{55} + 4 q^{56} - 26 q^{59} + 4 q^{60} - 28 q^{61} - 28 q^{64} + 60 q^{65} - 12 q^{66} - 8 q^{69} - 10 q^{70} - 64 q^{71} + 24 q^{74} - 8 q^{75} + 20 q^{76} + 32 q^{79} - 4 q^{80} - 14 q^{81} + 8 q^{84} + 16 q^{85} - 8 q^{86} + 76 q^{89} + 2 q^{90} - 8 q^{91} - 38 q^{94} - 70 q^{95} - 14 q^{96} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.00835632 + 2.23605i 1.00000 3.70065 − 2.13657i − 1.00000i 0.500000 − 0.866025i 1.12526 − 1.93230i
1009.2 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.589266 − 2.15703i 1.00000 −3.59896 + 2.07786i − 1.00000i 0.500000 − 0.866025i −0.568195 + 2.16267i
1009.3 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.16486 + 0.559786i 1.00000 −2.76595 + 1.59692i − 1.00000i 0.500000 − 0.866025i 2.15472 + 0.597644i
1009.4 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.14608 − 0.627973i 1.00000 2.93485 − 1.69443i − 1.00000i 0.500000 − 0.866025i 1.54457 + 1.61688i
1009.5 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.09232 + 0.788796i 1.00000 0.268117 − 0.154798i − 1.00000i 0.500000 − 0.866025i −1.41760 − 1.72928i
1009.6 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.64155 − 1.51833i 1.00000 1.09384 − 0.631530i − 1.00000i 0.500000 − 0.866025i −2.18079 + 0.494135i
1009.7 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.808671 + 2.08472i 1.00000 −3.36459 + 1.94255i − 1.00000i 0.500000 − 0.866025i 0.342030 − 2.20975i
1009.8 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.40108 + 1.74269i 1.00000 3.36459 − 1.94255i 1.00000i 0.500000 − 0.866025i 0.342030 + 2.20975i
1009.9 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −2.13568 + 0.662460i 1.00000 −1.09384 + 0.631530i 1.00000i 0.500000 − 0.866025i −2.18079 − 0.494135i
1009.10 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −0.363043 + 2.20640i 1.00000 −0.268117 + 0.154798i 1.00000i 0.500000 − 0.866025i −1.41760 + 1.72928i
1009.11 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 0.529198 − 2.17254i 1.00000 −2.93485 + 1.69443i 1.00000i 0.500000 − 0.866025i 1.54457 − 1.61688i
1009.12 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.56722 − 1.59494i 1.00000 2.76595 − 1.59692i 1.00000i 0.500000 − 0.866025i 2.15472 − 0.597644i
1009.13 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.57341 − 1.58883i 1.00000 3.59896 − 2.07786i 1.00000i 0.500000 − 0.866025i −0.568195 − 2.16267i
1009.14 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.94066 + 1.11079i 1.00000 −3.70065 + 2.13657i 1.00000i 0.500000 − 0.866025i 1.12526 + 1.93230i
1099.1 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −0.00835632 − 2.23605i 1.00000 3.70065 + 2.13657i 1.00000i 0.500000 + 0.866025i 1.12526 + 1.93230i
1099.2 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −0.589266 + 2.15703i 1.00000 −3.59896 − 2.07786i 1.00000i 0.500000 + 0.866025i −0.568195 − 2.16267i
1099.3 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −2.16486 − 0.559786i 1.00000 −2.76595 − 1.59692i 1.00000i 0.500000 + 0.866025i 2.15472 − 0.597644i
1099.4 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −2.14608 + 0.627973i 1.00000 2.93485 + 1.69443i 1.00000i 0.500000 + 0.866025i 1.54457 − 1.61688i
1099.5 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 2.09232 − 0.788796i 1.00000 0.268117 + 0.154798i 1.00000i 0.500000 + 0.866025i −1.41760 + 1.72928i
1099.6 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 1.64155 + 1.51833i 1.00000 1.09384 + 0.631530i 1.00000i 0.500000 + 0.866025i −2.18079 − 0.494135i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n 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.bb.d ✓ 28
5.b even 2 1 inner 1110.2.bb.d ✓ 28
37.c even 3 1 inner 1110.2.bb.d ✓ 28
185.n even 6 1 inner 1110.2.bb.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.bb.d ✓ 28 1.a even 1 1 trivial
1110.2.bb.d ✓ 28 5.b even 2 1 inner
1110.2.bb.d ✓ 28 37.c even 3 1 inner
1110.2.bb.d ✓ 28 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{28} - 74 T_{7}^{26} + 3287 T_{7}^{24} - 96658 T_{7}^{22} + 2121973 T_{7}^{20} - 35020268 T_{7}^{18} + \cdots + 7269949696 \) Copy content Toggle raw display
\( T_{11}^{7} + 3T_{11}^{6} - 61T_{11}^{5} - 129T_{11}^{4} + 1256T_{11}^{3} + 1502T_{11}^{2} - 8680T_{11} - 2680 \) Copy content Toggle raw display