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Properties

Label 1098.2.e.d
Level $1098$
Weight $2$
Character orbit 1098.e
Analytic conductor $8.768$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(367,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.367");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.e (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: \(8.76757414194\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 11 q^{2} + 7 q^{3} - 11 q^{4} - 9 q^{5} + 5 q^{6} + 8 q^{7} - 22 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 11 q^{2} + 7 q^{3} - 11 q^{4} - 9 q^{5} + 5 q^{6} + 8 q^{7} - 22 q^{8} - 11 q^{9} - 18 q^{10} + 7 q^{11} - 2 q^{12} + 8 q^{13} - 8 q^{14} + 13 q^{15} - 11 q^{16} + 12 q^{17} - 4 q^{18} - 30 q^{19} - 9 q^{20} + 19 q^{21} - 7 q^{22} - 9 q^{23} - 7 q^{24} - 6 q^{25} + 16 q^{26} - 5 q^{27} - 16 q^{28} - q^{29} + 11 q^{30} + 3 q^{31} + 11 q^{32} - 24 q^{33} + 6 q^{34} - 16 q^{35} + 7 q^{36} - 38 q^{37} - 15 q^{38} + 36 q^{39} + 9 q^{40} - 13 q^{41} + 5 q^{42} + 22 q^{43} - 14 q^{44} - 14 q^{45} - 18 q^{46} - 4 q^{47} - 5 q^{48} - 21 q^{49} + 6 q^{50} + 3 q^{51} + 8 q^{52} + 50 q^{53} + 11 q^{54} - 2 q^{55} - 8 q^{56} + 27 q^{57} + q^{58} - q^{59} - 2 q^{60} + 11 q^{61} + 6 q^{62} - 32 q^{63} + 22 q^{64} - 24 q^{65} - 24 q^{66} + 37 q^{67} - 6 q^{68} - 13 q^{69} - 8 q^{70} - 52 q^{71} + 11 q^{72} - 50 q^{73} - 19 q^{74} - 16 q^{75} + 15 q^{76} - 21 q^{77} + 24 q^{78} + 37 q^{79} + 18 q^{80} + 25 q^{81} - 26 q^{82} - q^{83} - 14 q^{84} + 23 q^{85} - 22 q^{86} - 30 q^{87} - 7 q^{88} + 4 q^{89} + 47 q^{90} - 40 q^{91} - 9 q^{92} + 27 q^{93} + 4 q^{94} - 27 q^{95} + 2 q^{96} + 9 q^{97} - 42 q^{98} + 21 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1 0.500000 − 0.866025i −1.67533 + 0.439610i −0.500000 − 0.866025i −1.20328 − 2.08414i −0.456953 + 1.67069i −1.00342 + 1.73798i −1.00000 2.61349 − 1.47299i −2.40656
367.2 0.500000 − 0.866025i −1.12036 − 1.32091i −0.500000 − 0.866025i 0.270017 + 0.467684i −1.70412 + 0.309807i −1.26589 + 2.19259i −1.00000 −0.489589 + 2.95978i 0.540034
367.3 0.500000 − 0.866025i −0.534235 + 1.64760i −0.500000 − 0.866025i −1.91060 − 3.30926i 1.15975 + 1.28646i 2.61104 − 4.52246i −1.00000 −2.42919 − 1.76041i −3.82121
367.4 0.500000 − 0.866025i −0.282953 + 1.70878i −0.500000 − 0.866025i −0.118374 − 0.205030i 1.33837 + 1.09944i −0.432740 + 0.749528i −1.00000 −2.83988 − 0.967010i −0.236748
367.5 0.500000 − 0.866025i 0.0808991 − 1.73016i −0.500000 − 0.866025i 0.649865 + 1.12560i −1.45791 − 0.935141i 2.42179 − 4.19466i −1.00000 −2.98691 − 0.279937i 1.29973
367.6 0.500000 − 0.866025i 0.643453 − 1.60809i −0.500000 − 0.866025i −1.86459 − 3.22957i −1.07092 − 1.36129i 0.939086 − 1.62654i −1.00000 −2.17194 − 2.06947i −3.72919
367.7 0.500000 − 0.866025i 0.804672 + 1.53379i −0.500000 − 0.866025i −0.673117 − 1.16587i 1.73063 + 0.0700274i 0.519086 − 0.899084i −1.00000 −1.70501 + 2.46839i −1.34623
367.8 0.500000 − 0.866025i 0.901787 − 1.47878i −0.500000 − 0.866025i −0.939924 − 1.62800i −0.829764 − 1.52036i −1.00671 + 1.74368i −1.00000 −1.37356 − 2.66708i −1.87985
367.9 0.500000 − 0.866025i 1.26253 + 1.18576i −0.500000 − 0.866025i −0.784253 − 1.35837i 1.65816 − 0.500506i 1.48208 − 2.56703i −1.00000 0.187968 + 2.99411i −1.56851
367.10 0.500000 − 0.866025i 1.69131 + 0.373472i −0.500000 − 0.866025i 2.05273 + 3.55543i 1.16909 − 1.27798i 1.38846 − 2.40488i −1.00000 2.72104 + 1.26331i 4.10546
367.11 0.500000 − 0.866025i 1.72823 + 0.114954i −0.500000 − 0.866025i 0.0215360 + 0.0373015i 0.963669 − 1.43922i −1.65277 + 2.86268i −1.00000 2.97357 + 0.397335i 0.0430721
733.1 0.500000 + 0.866025i −1.67533 − 0.439610i −0.500000 + 0.866025i −1.20328 + 2.08414i −0.456953 − 1.67069i −1.00342 − 1.73798i −1.00000 2.61349 + 1.47299i −2.40656
733.2 0.500000 + 0.866025i −1.12036 + 1.32091i −0.500000 + 0.866025i 0.270017 − 0.467684i −1.70412 − 0.309807i −1.26589 − 2.19259i −1.00000 −0.489589 − 2.95978i 0.540034
733.3 0.500000 + 0.866025i −0.534235 − 1.64760i −0.500000 + 0.866025i −1.91060 + 3.30926i 1.15975 − 1.28646i 2.61104 + 4.52246i −1.00000 −2.42919 + 1.76041i −3.82121
733.4 0.500000 + 0.866025i −0.282953 − 1.70878i −0.500000 + 0.866025i −0.118374 + 0.205030i 1.33837 − 1.09944i −0.432740 − 0.749528i −1.00000 −2.83988 + 0.967010i −0.236748
733.5 0.500000 + 0.866025i 0.0808991 + 1.73016i −0.500000 + 0.866025i 0.649865 − 1.12560i −1.45791 + 0.935141i 2.42179 + 4.19466i −1.00000 −2.98691 + 0.279937i 1.29973
733.6 0.500000 + 0.866025i 0.643453 + 1.60809i −0.500000 + 0.866025i −1.86459 + 3.22957i −1.07092 + 1.36129i 0.939086 + 1.62654i −1.00000 −2.17194 + 2.06947i −3.72919
733.7 0.500000 + 0.866025i 0.804672 − 1.53379i −0.500000 + 0.866025i −0.673117 + 1.16587i 1.73063 − 0.0700274i 0.519086 + 0.899084i −1.00000 −1.70501 − 2.46839i −1.34623
733.8 0.500000 + 0.866025i 0.901787 + 1.47878i −0.500000 + 0.866025i −0.939924 + 1.62800i −0.829764 + 1.52036i −1.00671 − 1.74368i −1.00000 −1.37356 + 2.66708i −1.87985
733.9 0.500000 + 0.866025i 1.26253 − 1.18576i −0.500000 + 0.866025i −0.784253 + 1.35837i 1.65816 + 0.500506i 1.48208 + 2.56703i −1.00000 0.187968 − 2.99411i −1.56851
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.11
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.e.d ✓ 22
9.c even 3 1 inner 1098.2.e.d ✓ 22
9.c even 3 1 9882.2.a.be 11
9.d odd 6 1 9882.2.a.bf 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.e.d ✓ 22 1.a even 1 1 trivial
1098.2.e.d ✓ 22 9.c even 3 1 inner
9882.2.a.be 11 9.c even 3 1
9882.2.a.bf 11 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{22} + 9 T_{5}^{21} + 71 T_{5}^{20} + 354 T_{5}^{19} + 1783 T_{5}^{18} + 7184 T_{5}^{17} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{22} - 8 T_{7}^{21} + 81 T_{7}^{20} - 316 T_{7}^{19} + 2058 T_{7}^{18} - 5547 T_{7}^{17} + \cdots + 141158161 \) Copy content Toggle raw display