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Properties

Label 1098.2.e.f
Level $1098$
Weight $2$
Character orbit 1098.e
Analytic conductor $8.768$
Analytic rank $0$
Dimension $34$
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: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 17 q^{2} + 2 q^{3} - 17 q^{4} - 5 q^{5} - q^{6} - 11 q^{7} + 34 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 17 q^{2} + 2 q^{3} - 17 q^{4} - 5 q^{5} - q^{6} - 11 q^{7} + 34 q^{8} + 10 q^{10} - 3 q^{11} - q^{12} - 10 q^{13} - 11 q^{14} + q^{15} - 17 q^{16} + 16 q^{17} + 3 q^{18} + 22 q^{19} - 5 q^{20} - q^{21} - 3 q^{22} - 5 q^{23} + 2 q^{24} - 32 q^{25} + 20 q^{26} - q^{27} + 22 q^{28} - 6 q^{29} - 5 q^{30} - 20 q^{31} - 17 q^{32} - 14 q^{33} - 8 q^{34} - 16 q^{35} - 3 q^{36} + 58 q^{37} - 11 q^{38} + 44 q^{39} - 5 q^{40} - 20 q^{41} + 8 q^{42} - 30 q^{43} + 6 q^{44} - 22 q^{45} + 10 q^{46} - 10 q^{47} - q^{48} - 42 q^{49} - 32 q^{50} - 49 q^{51} - 10 q^{52} + 4 q^{53} - 22 q^{54} + 30 q^{55} - 11 q^{56} + 27 q^{57} - 6 q^{58} + 3 q^{59} + 4 q^{60} + 17 q^{61} + 40 q^{62} - 45 q^{63} + 34 q^{64} - 28 q^{65} + 28 q^{66} - 11 q^{67} - 8 q^{68} - 32 q^{69} + 8 q^{70} + 126 q^{71} + 56 q^{73} - 29 q^{74} + 100 q^{75} - 11 q^{76} - 17 q^{77} - 16 q^{78} - 17 q^{79} + 10 q^{80} - 8 q^{81} + 40 q^{82} + 20 q^{83} - 7 q^{84} - 33 q^{85} - 30 q^{86} - 86 q^{87} - 3 q^{88} + 4 q^{89} + 23 q^{90} + 60 q^{91} - 5 q^{92} + 121 q^{93} - 10 q^{94} - q^{95} - q^{96} - 32 q^{97} + 84 q^{98} - 31 q^{99}+O(q^{100}) \) 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 −1.70593 + 0.299644i −0.500000 − 0.866025i −2.02139 − 3.50116i 0.593468 − 1.62720i 1.31426 − 2.27636i 1.00000 2.82043 − 1.02235i 4.04279
367.2 −0.500000 + 0.866025i −1.68083 + 0.418122i −0.500000 − 0.866025i −1.31261 − 2.27350i 0.478309 − 1.66470i −2.37545 + 4.11441i 1.00000 2.65035 − 1.40558i 2.62522
367.3 −0.500000 + 0.866025i −1.53282 + 0.806520i −0.500000 − 0.866025i 1.59654 + 2.76529i 0.0679412 − 1.73072i −2.01441 + 3.48906i 1.00000 1.69905 − 2.47249i −3.19308
367.4 −0.500000 + 0.866025i −1.37904 − 1.04798i −0.500000 − 0.866025i 1.65700 + 2.87001i 1.59709 − 0.670294i 0.254712 − 0.441174i 1.00000 0.803491 + 2.89040i −3.31400
367.5 −0.500000 + 0.866025i −1.02303 + 1.39765i −0.500000 − 0.866025i 0.182999 + 0.316964i −0.698884 − 1.58479i 1.22896 − 2.12861i 1.00000 −0.906832 − 2.85966i −0.365998
367.6 −0.500000 + 0.866025i −0.887881 − 1.48717i −0.500000 − 0.866025i −2.04970 − 3.55018i 1.73187 − 0.0253439i −1.01857 + 1.76421i 1.00000 −1.42333 + 2.64086i 4.09940
367.7 −0.500000 + 0.866025i −0.177122 − 1.72297i −0.500000 − 0.866025i −0.951924 − 1.64878i 1.58070 + 0.708094i −0.486861 + 0.843268i 1.00000 −2.93726 + 0.610350i 1.90385
367.8 −0.500000 + 0.866025i −0.143239 + 1.72612i −0.500000 − 0.866025i −0.792589 − 1.37280i −1.42324 − 0.987108i −2.02040 + 3.49944i 1.00000 −2.95897 − 0.494496i 1.58518
367.9 −0.500000 + 0.866025i −0.0462544 + 1.73143i −0.500000 − 0.866025i 0.379433 + 0.657197i −1.47634 − 0.905774i 1.58694 − 2.74866i 1.00000 −2.99572 − 0.160173i −0.758865
367.10 −0.500000 + 0.866025i −0.0352391 − 1.73169i −0.500000 − 0.866025i 1.66276 + 2.87999i 1.51731 + 0.835328i −1.00835 + 1.74651i 1.00000 −2.99752 + 0.122047i −3.32553
367.11 −0.500000 + 0.866025i 0.935531 − 1.45766i −0.500000 − 0.866025i 0.939094 + 1.62656i 0.794608 + 1.53903i 1.66888 − 2.89058i 1.00000 −1.24956 − 2.72738i −1.87819
367.12 −0.500000 + 0.866025i 1.20191 − 1.24716i −0.500000 − 0.866025i −0.417846 − 0.723730i 0.479118 + 1.66447i −0.864413 + 1.49721i 1.00000 −0.110822 − 2.99795i 0.835692
367.13 −0.500000 + 0.866025i 1.24732 + 1.20175i −0.500000 − 0.866025i −1.50626 − 2.60891i −1.66440 + 0.479334i −0.0606085 + 0.104977i 1.00000 0.111601 + 2.99792i 3.01251
367.14 −0.500000 + 0.866025i 1.40109 + 1.01830i −0.500000 − 0.866025i 0.830957 + 1.43926i −1.58242 + 0.704234i 0.867433 − 1.50244i 1.00000 0.926133 + 2.85347i −1.66191
367.15 −0.500000 + 0.866025i 1.53940 + 0.793888i −0.500000 − 0.866025i 1.46258 + 2.53326i −1.45723 + 0.936212i −2.14457 + 3.71451i 1.00000 1.73948 + 2.44422i −2.92515
367.16 −0.500000 + 0.866025i 1.55524 − 0.762377i −0.500000 − 0.866025i −1.78057 − 3.08403i −0.117384 + 1.72807i 1.95142 − 3.37997i 1.00000 1.83756 − 2.37136i 3.56113
367.17 −0.500000 + 0.866025i 1.73088 + 0.0635888i −0.500000 − 0.866025i −0.378482 − 0.655549i −0.920511 + 1.46719i −2.37896 + 4.12049i 1.00000 2.99191 + 0.220130i 0.756963
733.1 −0.500000 − 0.866025i −1.70593 − 0.299644i −0.500000 + 0.866025i −2.02139 + 3.50116i 0.593468 + 1.62720i 1.31426 + 2.27636i 1.00000 2.82043 + 1.02235i 4.04279
733.2 −0.500000 − 0.866025i −1.68083 − 0.418122i −0.500000 + 0.866025i −1.31261 + 2.27350i 0.478309 + 1.66470i −2.37545 − 4.11441i 1.00000 2.65035 + 1.40558i 2.62522
733.3 −0.500000 − 0.866025i −1.53282 − 0.806520i −0.500000 + 0.866025i 1.59654 − 2.76529i 0.0679412 + 1.73072i −2.01441 − 3.48906i 1.00000 1.69905 + 2.47249i −3.19308
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.f ✓ 34
9.c even 3 1 inner 1098.2.e.f ✓ 34
9.c even 3 1 9882.2.a.bm 17
9.d odd 6 1 9882.2.a.bk 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.e.f ✓ 34 1.a even 1 1 trivial
1098.2.e.f ✓ 34 9.c even 3 1 inner
9882.2.a.bk 17 9.d odd 6 1
9882.2.a.bm 17 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 \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{34} + 5 T_{5}^{33} + 71 T_{5}^{32} + 250 T_{5}^{31} + 2477 T_{5}^{30} + 7296 T_{5}^{29} + \cdots + 6323430400 \) Copy content Toggle raw display
\( T_{7}^{34} + 11 T_{7}^{33} + 141 T_{7}^{32} + 920 T_{7}^{31} + 7281 T_{7}^{30} + 36593 T_{7}^{29} + \cdots + 97575641641 \) Copy content Toggle raw display