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Properties

Label 1098.2.e.e
Level $1098$
Weight $2$
Character orbit 1098.e
Analytic conductor $8.768$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 2 q^{3} - 12 q^{4} + 3 q^{5} - q^{6} + 11 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 2 q^{3} - 12 q^{4} + 3 q^{5} - q^{6} + 11 q^{7} + 24 q^{8} - 6 q^{10} - 3 q^{11} - q^{12} + 12 q^{13} + 11 q^{14} + q^{15} - 12 q^{16} - 4 q^{17} + 3 q^{18} - 14 q^{19} + 3 q^{20} - 29 q^{21} - 3 q^{22} + 9 q^{23} + 2 q^{24} + 3 q^{25} - 24 q^{26} + 11 q^{27} - 22 q^{28} + 26 q^{29} - 5 q^{30} + 2 q^{31} - 12 q^{32} + 28 q^{33} + 2 q^{34} - 16 q^{35} - 3 q^{36} - 34 q^{37} + 7 q^{38} - 40 q^{39} + 3 q^{40} + 8 q^{41} + 22 q^{42} + 20 q^{43} + 6 q^{44} + 46 q^{45} - 18 q^{46} + 2 q^{47} - q^{48} + 17 q^{49} + 3 q^{50} + 7 q^{51} + 12 q^{52} - 4 q^{53} - 10 q^{54} - 26 q^{55} + 11 q^{56} - 77 q^{57} + 26 q^{58} + 3 q^{59} + 4 q^{60} - 12 q^{61} - 4 q^{62} + 31 q^{63} + 24 q^{64} + 4 q^{65} - 32 q^{66} + 5 q^{67} + 2 q^{68} + 29 q^{69} + 8 q^{70} - 86 q^{71} - 80 q^{73} + 17 q^{74} - 42 q^{75} + 7 q^{76} + 11 q^{77} + 26 q^{78} + 6 q^{79} - 6 q^{80} + 48 q^{81} - 16 q^{82} + 20 q^{83} + 7 q^{84} + 39 q^{85} + 20 q^{86} + 32 q^{87} - 3 q^{88} - 72 q^{89} - 5 q^{90} - 20 q^{91} + 9 q^{92} - 102 q^{93} + 2 q^{94} + 31 q^{95} - q^{96} + 20 q^{97} - 34 q^{98} + 21 q^{99}+O(q^{100}) \) 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 −1.72764 + 0.123550i −0.500000 − 0.866025i 0.384881 + 0.666633i 0.756822 − 1.55795i −0.210586 + 0.364746i 1.00000 2.96947 − 0.426901i −0.769762
367.2 −0.500000 + 0.866025i −1.60634 − 0.647809i −0.500000 − 0.866025i −0.837561 − 1.45070i 1.36419 − 1.06723i 2.41425 − 4.18161i 1.00000 2.16069 + 2.08121i 1.67512
367.3 −0.500000 + 0.866025i −1.12976 + 1.31288i −0.500000 − 0.866025i 1.78247 + 3.08732i −0.572108 − 1.63484i 0.332938 − 0.576666i 1.00000 −0.447302 − 2.96647i −3.56493
367.4 −0.500000 + 0.866025i −0.901675 + 1.47884i −0.500000 − 0.866025i −0.798867 − 1.38368i −0.829879 − 1.52030i 0.116450 − 0.201697i 1.00000 −1.37396 − 2.66688i 1.59773
367.5 −0.500000 + 0.866025i −0.714333 − 1.57789i −0.500000 − 0.866025i −0.528430 − 0.915268i 1.72366 + 0.170313i 1.59017 − 2.75425i 1.00000 −1.97946 + 2.25427i 1.05686
367.6 −0.500000 + 0.866025i 0.336965 − 1.69896i −0.500000 − 0.866025i −0.632814 − 1.09607i 1.30286 + 1.14130i 1.24184 − 2.15092i 1.00000 −2.77291 − 1.14498i 1.26563
367.7 −0.500000 + 0.866025i 0.346342 + 1.69707i −0.500000 − 0.866025i 0.995031 + 1.72345i −1.64288 − 0.548594i −1.52192 + 2.63604i 1.00000 −2.76009 + 1.17553i −1.99006
367.8 −0.500000 + 0.866025i 0.390031 − 1.68757i −0.500000 − 0.866025i 1.15948 + 2.00828i 1.26646 + 1.18156i −0.795385 + 1.37765i 1.00000 −2.69575 − 1.31641i −2.31896
367.9 −0.500000 + 0.866025i 0.938048 + 1.45604i −0.500000 − 0.866025i −1.44464 − 2.50219i −1.73000 + 0.0843511i 0.0700883 − 0.121397i 1.00000 −1.24013 + 2.73168i 2.88928
367.10 −0.500000 + 0.866025i 1.62180 − 0.608076i −0.500000 − 0.866025i 1.77168 + 3.06864i −0.284292 + 1.70856i 1.51104 − 2.61719i 1.00000 2.26049 − 1.97236i −3.54336
367.11 −0.500000 + 0.866025i 1.71638 + 0.232446i −0.500000 − 0.866025i 0.625418 + 1.08326i −1.05950 + 1.37021i 1.13804 − 1.97114i 1.00000 2.89194 + 0.797933i −1.25084
367.12 −0.500000 + 0.866025i 1.73018 − 0.0805403i −0.500000 − 0.866025i −0.976641 − 1.69159i −0.795339 + 1.53865i −0.386915 + 0.670156i 1.00000 2.98703 − 0.278698i 1.95328
733.1 −0.500000 − 0.866025i −1.72764 − 0.123550i −0.500000 + 0.866025i 0.384881 − 0.666633i 0.756822 + 1.55795i −0.210586 − 0.364746i 1.00000 2.96947 + 0.426901i −0.769762
733.2 −0.500000 − 0.866025i −1.60634 + 0.647809i −0.500000 + 0.866025i −0.837561 + 1.45070i 1.36419 + 1.06723i 2.41425 + 4.18161i 1.00000 2.16069 − 2.08121i 1.67512
733.3 −0.500000 − 0.866025i −1.12976 − 1.31288i −0.500000 + 0.866025i 1.78247 − 3.08732i −0.572108 + 1.63484i 0.332938 + 0.576666i 1.00000 −0.447302 + 2.96647i −3.56493
733.4 −0.500000 − 0.866025i −0.901675 − 1.47884i −0.500000 + 0.866025i −0.798867 + 1.38368i −0.829879 + 1.52030i 0.116450 + 0.201697i 1.00000 −1.37396 + 2.66688i 1.59773
733.5 −0.500000 − 0.866025i −0.714333 + 1.57789i −0.500000 + 0.866025i −0.528430 + 0.915268i 1.72366 − 0.170313i 1.59017 + 2.75425i 1.00000 −1.97946 − 2.25427i 1.05686
733.6 −0.500000 − 0.866025i 0.336965 + 1.69896i −0.500000 + 0.866025i −0.632814 + 1.09607i 1.30286 − 1.14130i 1.24184 + 2.15092i 1.00000 −2.77291 + 1.14498i 1.26563
733.7 −0.500000 − 0.866025i 0.346342 − 1.69707i −0.500000 + 0.866025i 0.995031 − 1.72345i −1.64288 + 0.548594i −1.52192 − 2.63604i 1.00000 −2.76009 − 1.17553i −1.99006
733.8 −0.500000 − 0.866025i 0.390031 + 1.68757i −0.500000 + 0.866025i 1.15948 − 2.00828i 1.26646 − 1.18156i −0.795385 − 1.37765i 1.00000 −2.69575 + 1.31641i −2.31896
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.12
Significant digits:
Format:

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.e ✓ 24
9.c even 3 1 inner 1098.2.e.e ✓ 24
9.c even 3 1 9882.2.a.bj 12
9.d odd 6 1 9882.2.a.bg 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.e.e ✓ 24 1.a even 1 1 trivial
1098.2.e.e ✓ 24 9.c even 3 1 inner
9882.2.a.bg 12 9.d odd 6 1
9882.2.a.bj 12 9.c even 3 1

Hecke kernels

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

\( T_{5}^{24} - 3 T_{5}^{23} + 33 T_{5}^{22} - 44 T_{5}^{21} + 517 T_{5}^{20} - 378 T_{5}^{19} + \cdots + 1285956 \) Copy content Toggle raw display
\( T_{7}^{24} - 11 T_{7}^{23} + 94 T_{7}^{22} - 479 T_{7}^{21} + 2195 T_{7}^{20} - 7537 T_{7}^{19} + \cdots + 81 \) Copy content Toggle raw display