↑

Properties

Label 261.2.e.a
Level $261$
Weight $2$
Character orbit 261.e
Analytic conductor $2.084$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [261,2,Mod(88,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, 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("261.88");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(2.08409549276\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - q^{6} + 7 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - q^{6} + 7 q^{7} + 6 q^{8} - 2 q^{9} - 20 q^{10} - 3 q^{11} + 10 q^{12} + 7 q^{13} + 10 q^{14} - 8 q^{15} + 7 q^{16} + 2 q^{17} + 2 q^{18} - 56 q^{19} + 4 q^{20} + 16 q^{21} + 13 q^{22} - 4 q^{23} - 33 q^{24} + 4 q^{25} - 12 q^{26} + 13 q^{27} - 32 q^{28} + 11 q^{29} + 46 q^{30} + 27 q^{31} - 7 q^{32} - 10 q^{33} + 19 q^{34} - 22 q^{35} + 19 q^{36} - 28 q^{37} + 5 q^{38} + 17 q^{39} + 24 q^{40} + 9 q^{41} - 50 q^{42} + 25 q^{43} + 46 q^{44} - 56 q^{45} - 20 q^{46} - 6 q^{47} + 33 q^{48} + 12 q^{49} - 41 q^{50} - 15 q^{51} + 20 q^{52} - 12 q^{53} + 5 q^{54} - 70 q^{55} + 10 q^{56} + 25 q^{57} + q^{58} - 48 q^{60} + 25 q^{61} - 12 q^{62} - 11 q^{63} - 50 q^{64} - 5 q^{65} + 29 q^{66} + 29 q^{67} + 50 q^{68} + 20 q^{69} + 7 q^{70} - 2 q^{71} + 30 q^{72} - 26 q^{73} - 14 q^{74} + 46 q^{75} + 38 q^{76} - 9 q^{77} - 39 q^{78} + 27 q^{79} - 74 q^{80} - 38 q^{81} - 34 q^{82} + 12 q^{83} + 60 q^{84} + 7 q^{85} - q^{86} - q^{87} - 5 q^{88} + 46 q^{89} - 35 q^{90} - 86 q^{91} - 19 q^{92} + 78 q^{93} - 10 q^{94} - 8 q^{95} - 75 q^{96} + 36 q^{97} + 86 q^{98} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1 −1.20917 + 2.09435i −1.43915 − 0.963770i −1.92419 − 3.33280i 2.02330 + 3.50445i 3.75865 − 1.84871i 0.403314 − 0.698560i 4.47002 1.14229 + 2.77402i −9.78605
88.2 −1.10224 + 1.90914i −1.06870 + 1.36304i −1.42988 − 2.47662i −0.463291 − 0.802444i −1.42428 − 3.54270i 2.26860 − 3.92934i 1.89533 −0.715771 − 2.91336i 2.04264
88.3 −0.764092 + 1.32345i 1.06160 + 1.36858i −0.167673 − 0.290418i 0.0756080 + 0.130957i −2.62240 + 0.359245i 0.0289713 − 0.0501797i −2.54390 −0.746025 + 2.90576i −0.231086
88.4 −0.540070 + 0.935429i 1.44567 − 0.953961i 0.416648 + 0.721656i −1.91815 − 3.32233i 0.111600 + 1.86753i 1.07867 − 1.86831i −3.06036 1.17992 − 2.75822i 4.14374
88.5 −0.391918 + 0.678822i −0.203885 − 1.72001i 0.692800 + 1.19996i 0.728601 + 1.26197i 1.24749 + 0.535701i −2.16737 + 3.75400i −2.65376 −2.91686 + 0.701368i −1.14221
88.6 0.0549262 − 0.0951349i 0.0678358 + 1.73072i 0.993966 + 1.72160i 1.31653 + 2.28029i 0.168378 + 0.0886084i 1.68615 − 2.92049i 0.438084 −2.99080 + 0.234810i 0.289247
88.7 0.100158 − 0.173479i 1.19785 − 1.25106i 0.979937 + 1.69730i 0.556635 + 0.964120i −0.0970582 − 0.333105i 0.359329 − 0.622375i 0.793225 −0.130309 − 2.99717i 0.223006
88.8 0.489415 − 0.847692i −1.70010 + 0.331155i 0.520946 + 0.902304i −1.17228 − 2.03045i −0.551337 + 1.60323i 0.172733 − 0.299183i 2.97750 2.78067 − 1.12599i −2.29493
88.9 0.788768 − 1.36619i −0.834239 + 1.51791i −0.244311 − 0.423159i 0.409773 + 0.709748i 1.41572 + 2.33700i −1.13728 + 1.96983i 2.38425 −1.60809 − 2.53260i 1.29286
88.10 0.942480 − 1.63242i 1.71821 − 0.218551i −0.776536 − 1.34500i −0.325626 − 0.564001i 1.26261 − 3.01082i −0.0750596 + 0.130007i 0.842439 2.90447 − 0.751031i −1.22758
88.11 1.13175 − 1.96024i −1.24509 − 1.20406i −1.56171 − 2.70495i −0.731092 − 1.26629i −3.76937 + 1.07799i 0.881947 − 1.52758i −2.54284 0.100498 + 2.99832i −3.30965
175.1 −1.20917 − 2.09435i −1.43915 + 0.963770i −1.92419 + 3.33280i 2.02330 − 3.50445i 3.75865 + 1.84871i 0.403314 + 0.698560i 4.47002 1.14229 − 2.77402i −9.78605
175.2 −1.10224 − 1.90914i −1.06870 − 1.36304i −1.42988 + 2.47662i −0.463291 + 0.802444i −1.42428 + 3.54270i 2.26860 + 3.92934i 1.89533 −0.715771 + 2.91336i 2.04264
175.3 −0.764092 − 1.32345i 1.06160 − 1.36858i −0.167673 + 0.290418i 0.0756080 − 0.130957i −2.62240 − 0.359245i 0.0289713 + 0.0501797i −2.54390 −0.746025 − 2.90576i −0.231086
175.4 −0.540070 − 0.935429i 1.44567 + 0.953961i 0.416648 − 0.721656i −1.91815 + 3.32233i 0.111600 − 1.86753i 1.07867 + 1.86831i −3.06036 1.17992 + 2.75822i 4.14374
175.5 −0.391918 − 0.678822i −0.203885 + 1.72001i 0.692800 − 1.19996i 0.728601 − 1.26197i 1.24749 − 0.535701i −2.16737 − 3.75400i −2.65376 −2.91686 − 0.701368i −1.14221
175.6 0.0549262 + 0.0951349i 0.0678358 − 1.73072i 0.993966 − 1.72160i 1.31653 − 2.28029i 0.168378 − 0.0886084i 1.68615 + 2.92049i 0.438084 −2.99080 − 0.234810i 0.289247
175.7 0.100158 + 0.173479i 1.19785 + 1.25106i 0.979937 − 1.69730i 0.556635 − 0.964120i −0.0970582 + 0.333105i 0.359329 + 0.622375i 0.793225 −0.130309 + 2.99717i 0.223006
175.8 0.489415 + 0.847692i −1.70010 − 0.331155i 0.520946 − 0.902304i −1.17228 + 2.03045i −0.551337 − 1.60323i 0.172733 + 0.299183i 2.97750 2.78067 + 1.12599i −2.29493
175.9 0.788768 + 1.36619i −0.834239 − 1.51791i −0.244311 + 0.423159i 0.409773 − 0.709748i 1.41572 − 2.33700i −1.13728 − 1.96983i 2.38425 −1.60809 + 2.53260i 1.29286
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.11
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 261.2.e.a ✓ 22
3.b odd 2 1 783.2.e.a 22
9.c even 3 1 inner 261.2.e.a ✓ 22
9.c even 3 1 2349.2.a.f 11
9.d odd 6 1 783.2.e.a 22
9.d odd 6 1 2349.2.a.e 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.e.a ✓ 22 1.a even 1 1 trivial
261.2.e.a ✓ 22 9.c even 3 1 inner
783.2.e.a 22 3.b odd 2 1
783.2.e.a 22 9.d odd 6 1
2349.2.a.e 11 9.d odd 6 1
2349.2.a.f 11 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + T_{2}^{21} + 14 T_{2}^{20} + 9 T_{2}^{19} + 121 T_{2}^{18} + 66 T_{2}^{17} + 627 T_{2}^{16} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display