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Properties

Label 1005.2.i.e
Level $1005$
Weight $2$
Character orbit 1005.i
Analytic conductor $8.025$
Analytic rank $0$
Dimension $22$
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] = [1005,2,Mod(766,1005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1005, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1005.766");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1005.i (of order \(3\), 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: \(8.02496540314\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} + 22 q^{3} - 9 q^{4} + 22 q^{5} + 3 q^{6} + 9 q^{7} - 12 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} + 22 q^{3} - 9 q^{4} + 22 q^{5} + 3 q^{6} + 9 q^{7} - 12 q^{8} + 22 q^{9} + 3 q^{10} - 9 q^{12} + 9 q^{13} - 10 q^{14} + 22 q^{15} - 9 q^{16} - 6 q^{17} + 3 q^{18} + 4 q^{19} - 9 q^{20} + 9 q^{21} - 28 q^{22} + 2 q^{23} - 12 q^{24} + 22 q^{25} - 7 q^{26} + 22 q^{27} + 7 q^{28} + 5 q^{29} + 3 q^{30} + 5 q^{31} + 30 q^{32} + 15 q^{34} + 9 q^{35} - 9 q^{36} - 8 q^{37} - 19 q^{38} + 9 q^{39} - 12 q^{40} - 2 q^{41} - 10 q^{42} - 26 q^{43} - 30 q^{44} + 22 q^{45} + 11 q^{46} + 3 q^{47} - 9 q^{48} + 4 q^{49} + 3 q^{50} - 6 q^{51} - 70 q^{52} + 8 q^{53} + 3 q^{54} - 3 q^{56} + 4 q^{57} + 4 q^{58} - 18 q^{59} - 9 q^{60} + 8 q^{61} - 28 q^{62} + 9 q^{63} + 52 q^{64} + 9 q^{65} - 28 q^{66} + 3 q^{67} + 52 q^{68} + 2 q^{69} - 10 q^{70} + 8 q^{71} - 12 q^{72} - 7 q^{73} + 16 q^{74} + 22 q^{75} - 16 q^{76} - 7 q^{77} - 7 q^{78} - 9 q^{79} - 9 q^{80} + 22 q^{81} - 28 q^{82} + 3 q^{83} + 7 q^{84} - 6 q^{85} - 46 q^{86} + 5 q^{87} + 11 q^{88} - 32 q^{89} + 3 q^{90} + 74 q^{91} + 10 q^{92} + 5 q^{93} - 28 q^{94} + 4 q^{95} + 30 q^{96} - 10 q^{97} - 15 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
766.1 −1.18431 + 2.05128i 1.00000 −1.80517 − 3.12664i 1.00000 −1.18431 + 2.05128i 1.47524 + 2.55519i 3.81426 1.00000 −1.18431 + 2.05128i
766.2 −1.00228 + 1.73600i 1.00000 −1.00913 − 1.74787i 1.00000 −1.00228 + 1.73600i −0.657819 − 1.13938i 0.0366212 1.00000 −1.00228 + 1.73600i
766.3 −0.601197 + 1.04130i 1.00000 0.277123 + 0.479992i 1.00000 −0.601197 + 1.04130i 2.31261 + 4.00556i −3.07121 1.00000 −0.601197 + 1.04130i
766.4 −0.507594 + 0.879179i 1.00000 0.484696 + 0.839519i 1.00000 −0.507594 + 0.879179i −0.349260 − 0.604937i −3.01449 1.00000 −0.507594 + 0.879179i
766.5 0.0466708 − 0.0808363i 1.00000 0.995644 + 1.72451i 1.00000 0.0466708 − 0.0808363i −0.121207 − 0.209937i 0.372553 1.00000 0.0466708 − 0.0808363i
766.6 0.201167 − 0.348431i 1.00000 0.919064 + 1.59187i 1.00000 0.201167 − 0.348431i 0.289722 + 0.501813i 1.54421 1.00000 0.201167 − 0.348431i
766.7 0.461181 − 0.798788i 1.00000 0.574625 + 0.995279i 1.00000 0.461181 − 0.798788i 2.18321 + 3.78142i 2.90475 1.00000 0.461181 − 0.798788i
766.8 0.632085 − 1.09480i 1.00000 0.200937 + 0.348033i 1.00000 0.632085 − 1.09480i −1.30686 − 2.26355i 3.03638 1.00000 0.632085 − 1.09480i
766.9 1.02471 − 1.77484i 1.00000 −1.10004 − 1.90533i 1.00000 1.02471 − 1.77484i 1.46581 + 2.53886i −0.410061 1.00000 1.02471 − 1.77484i
766.10 1.03112 − 1.78595i 1.00000 −1.12642 − 1.95101i 1.00000 1.03112 − 1.78595i −1.14940 − 1.99082i −0.521405 1.00000 1.03112 − 1.78595i
766.11 1.39845 − 2.42219i 1.00000 −2.91133 − 5.04257i 1.00000 1.39845 − 2.42219i 0.357960 + 0.620006i −10.6916 1.00000 1.39845 − 2.42219i
841.1 −1.18431 − 2.05128i 1.00000 −1.80517 + 3.12664i 1.00000 −1.18431 − 2.05128i 1.47524 − 2.55519i 3.81426 1.00000 −1.18431 − 2.05128i
841.2 −1.00228 − 1.73600i 1.00000 −1.00913 + 1.74787i 1.00000 −1.00228 − 1.73600i −0.657819 + 1.13938i 0.0366212 1.00000 −1.00228 − 1.73600i
841.3 −0.601197 − 1.04130i 1.00000 0.277123 − 0.479992i 1.00000 −0.601197 − 1.04130i 2.31261 − 4.00556i −3.07121 1.00000 −0.601197 − 1.04130i
841.4 −0.507594 − 0.879179i 1.00000 0.484696 − 0.839519i 1.00000 −0.507594 − 0.879179i −0.349260 + 0.604937i −3.01449 1.00000 −0.507594 − 0.879179i
841.5 0.0466708 + 0.0808363i 1.00000 0.995644 − 1.72451i 1.00000 0.0466708 + 0.0808363i −0.121207 + 0.209937i 0.372553 1.00000 0.0466708 + 0.0808363i
841.6 0.201167 + 0.348431i 1.00000 0.919064 − 1.59187i 1.00000 0.201167 + 0.348431i 0.289722 − 0.501813i 1.54421 1.00000 0.201167 + 0.348431i
841.7 0.461181 + 0.798788i 1.00000 0.574625 − 0.995279i 1.00000 0.461181 + 0.798788i 2.18321 − 3.78142i 2.90475 1.00000 0.461181 + 0.798788i
841.8 0.632085 + 1.09480i 1.00000 0.200937 − 0.348033i 1.00000 0.632085 + 1.09480i −1.30686 + 2.26355i 3.03638 1.00000 0.632085 + 1.09480i
841.9 1.02471 + 1.77484i 1.00000 −1.10004 + 1.90533i 1.00000 1.02471 + 1.77484i 1.46581 − 2.53886i −0.410061 1.00000 1.02471 + 1.77484i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 766.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1005.2.i.e ✓ 22
67.c even 3 1 inner 1005.2.i.e ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1005.2.i.e ✓ 22 1.a even 1 1 trivial
1005.2.i.e ✓ 22 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 3 T_{2}^{21} + 20 T_{2}^{20} - 39 T_{2}^{19} + 195 T_{2}^{18} - 333 T_{2}^{17} + 1219 T_{2}^{16} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(1005, [\chi])\). Copy content Toggle raw display