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Properties

Label 1098.2.e.g
Level $1098$
Weight $2$
Character orbit 1098.e
Analytic conductor $8.768$
Analytic rank $0$
Dimension $34$
Inner twists $2$

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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: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 17 q^{2} + 4 q^{3} - 17 q^{4} + 5 q^{5} + 5 q^{6} - 13 q^{7} - 34 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 17 q^{2} + 4 q^{3} - 17 q^{4} + 5 q^{5} + 5 q^{6} - 13 q^{7} - 34 q^{8} - 8 q^{9} + 10 q^{10} - 5 q^{11} + q^{12} - 8 q^{13} + 13 q^{14} + 7 q^{15} - 17 q^{16} - 7 q^{18} + 14 q^{19} + 5 q^{20} - 9 q^{21} + 5 q^{22} + 9 q^{23} - 4 q^{24} - 40 q^{25} - 16 q^{26} + 7 q^{27} + 26 q^{28} + 18 q^{29} - 7 q^{30} - 10 q^{31} + 17 q^{32} + 26 q^{33} + 4 q^{35} + q^{36} + 42 q^{37} + 7 q^{38} - 14 q^{39} - 5 q^{40} + 8 q^{41} - 6 q^{42} - 18 q^{43} + 10 q^{44} - 10 q^{45} + 18 q^{46} + 12 q^{47} - 5 q^{48} - 26 q^{49} + 40 q^{50} - 9 q^{51} - 8 q^{52} - 40 q^{53} + 8 q^{54} + 46 q^{55} + 13 q^{56} + 23 q^{57} - 18 q^{58} + 9 q^{59} - 14 q^{60} - 17 q^{61} - 20 q^{62} - 23 q^{63} + 34 q^{64} + 12 q^{65} + 40 q^{66} - 65 q^{67} - 8 q^{69} + 2 q^{70} + 58 q^{71} + 8 q^{72} + 88 q^{73} + 21 q^{74} - 6 q^{75} - 7 q^{76} + 27 q^{77} - 16 q^{78} - 17 q^{79} - 10 q^{80} - 8 q^{81} + 16 q^{82} + 6 q^{83} + 3 q^{84} - 61 q^{85} + 18 q^{86} + 8 q^{87} + 5 q^{88} - 80 q^{89} - 35 q^{90} + 60 q^{91} + 9 q^{92} - 19 q^{93} - 12 q^{94} + 25 q^{95} - q^{96} - 40 q^{97} - 52 q^{98} - 35 q^{99}+O(q^{100}) \) 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 −1.70491 − 0.305431i −0.500000 − 0.866025i 0.850131 + 1.47247i −1.11697 + 1.32378i −1.35146 + 2.34080i −1.00000 2.81342 + 1.04146i 1.70026
367.2 0.500000 − 0.866025i −1.66936 + 0.461789i −0.500000 − 0.866025i −0.288462 − 0.499631i −0.434757 + 1.67660i 2.06014 − 3.56826i −1.00000 2.57350 − 1.54178i −0.576924
367.3 0.500000 − 0.866025i −1.25422 + 1.19454i −0.500000 − 0.866025i −0.520829 − 0.902103i 0.407387 + 1.68346i −1.94531 + 3.36938i −1.00000 0.146161 − 2.99644i −1.04166
367.4 0.500000 − 0.866025i −1.11600 + 1.32459i −0.500000 − 0.866025i 1.51854 + 2.63019i 0.589133 + 1.62878i 1.28224 − 2.22091i −1.00000 −0.509097 − 2.95649i 3.03708
367.5 0.500000 − 0.866025i −0.770604 − 1.55118i −0.500000 − 0.866025i −1.71227 − 2.96573i −1.72867 − 0.108229i 0.785539 − 1.36059i −1.00000 −1.81234 + 2.39070i −3.42453
367.6 0.500000 − 0.866025i −0.725578 − 1.57275i −0.500000 − 0.866025i 2.07647 + 3.59655i −1.72483 − 0.158005i −1.19938 + 2.07739i −1.00000 −1.94707 + 2.28230i 4.15294
367.7 0.500000 − 0.866025i −0.663426 + 1.59996i −0.500000 − 0.866025i −0.973203 − 1.68564i 1.05389 + 1.37452i −0.858829 + 1.48753i −1.00000 −2.11973 − 2.12291i −1.94641
367.8 0.500000 − 0.866025i −0.186463 − 1.72198i −0.500000 − 0.866025i −0.491842 − 0.851895i −1.58451 − 0.699510i −2.44679 + 4.23797i −1.00000 −2.93046 + 0.642174i −0.983684
367.9 0.500000 − 0.866025i 0.366749 + 1.69278i −0.500000 − 0.866025i 1.12555 + 1.94952i 1.64936 + 0.528775i 0.171350 − 0.296787i −1.00000 −2.73099 + 1.24165i 2.25111
367.10 0.500000 − 0.866025i 0.511483 − 1.65481i −0.500000 − 0.866025i 2.04784 + 3.54696i −1.17736 − 1.27036i 0.0387937 − 0.0671926i −1.00000 −2.47677 − 1.69281i 4.09568
367.11 0.500000 − 0.866025i 0.725103 + 1.57297i −0.500000 − 0.866025i −2.17883 − 3.77385i 1.72478 + 0.158526i −1.82333 + 3.15810i −1.00000 −1.94845 + 2.28113i −4.35766
367.12 0.500000 − 0.866025i 1.01651 + 1.40239i −0.500000 − 0.866025i 0.646149 + 1.11916i 1.72276 − 0.179125i 0.000551701 0 0.000955574i −1.00000 −0.933420 + 2.85109i 1.29230
367.13 0.500000 − 0.866025i 1.09783 + 1.33969i −0.500000 − 0.866025i 1.81018 + 3.13532i 1.70912 − 0.280906i −2.17911 + 3.77434i −1.00000 −0.589530 + 2.94151i 3.62036
367.14 0.500000 − 0.866025i 1.36662 − 1.06413i −0.500000 − 0.866025i 0.980540 + 1.69834i −0.238252 − 1.71559i −1.01873 + 1.76449i −1.00000 0.735275 − 2.90850i 1.96108
367.15 0.500000 − 0.866025i 1.58854 − 0.690321i −0.500000 − 0.866025i 0.453187 + 0.784944i 0.196434 − 1.72088i 1.91255 − 3.31263i −1.00000 2.04691 − 2.19321i 0.906375
367.16 0.500000 − 0.866025i 1.68810 − 0.387694i −0.500000 − 0.866025i −1.01937 − 1.76560i 0.508299 − 1.65579i 1.29464 − 2.24238i −1.00000 2.69939 − 1.30894i −2.03874
367.17 0.500000 − 0.866025i 1.72962 + 0.0916402i −0.500000 − 0.866025i −1.82379 − 3.15889i 0.944175 − 1.45208i −1.22285 + 2.11804i −1.00000 2.98320 + 0.317007i −3.64757
733.1 0.500000 + 0.866025i −1.70491 + 0.305431i −0.500000 + 0.866025i 0.850131 − 1.47247i −1.11697 − 1.32378i −1.35146 − 2.34080i −1.00000 2.81342 − 1.04146i 1.70026
733.2 0.500000 + 0.866025i −1.66936 − 0.461789i −0.500000 + 0.866025i −0.288462 + 0.499631i −0.434757 − 1.67660i 2.06014 + 3.56826i −1.00000 2.57350 + 1.54178i −0.576924
733.3 0.500000 + 0.866025i −1.25422 − 1.19454i −0.500000 + 0.866025i −0.520829 + 0.902103i 0.407387 − 1.68346i −1.94531 − 3.36938i −1.00000 0.146161 + 2.99644i −1.04166
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.17
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.g ✓ 34
9.c even 3 1 inner 1098.2.e.g ✓ 34
9.c even 3 1 9882.2.a.bl 17
9.d odd 6 1 9882.2.a.bn 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.e.g ✓ 34 1.a even 1 1 trivial
1098.2.e.g ✓ 34 9.c even 3 1 inner
9882.2.a.bl 17 9.c even 3 1
9882.2.a.bn 17 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 \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{34} - 5 T_{5}^{33} + 75 T_{5}^{32} - 298 T_{5}^{31} + 2983 T_{5}^{30} - 10564 T_{5}^{29} + \cdots + 44079842304 \) Copy content Toggle raw display
\( T_{7}^{34} + 13 T_{7}^{33} + 157 T_{7}^{32} + 1172 T_{7}^{31} + 8689 T_{7}^{30} + 49933 T_{7}^{29} + \cdots + 6561 \) Copy content Toggle raw display