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Properties

Label 495.2.i.f
Level $495$
Weight $2$
Character orbit 495.i
Analytic conductor $3.953$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(166,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.166");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.i (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: \(3.95259490005\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} - 13 q^{4} + 11 q^{5} + 13 q^{6} - 3 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} - 13 q^{4} + 11 q^{5} + 13 q^{6} - 3 q^{7} - 6 q^{8} - 4 q^{9} + 6 q^{10} + 11 q^{11} - 9 q^{12} - 9 q^{13} + 5 q^{14} - 17 q^{16} - 28 q^{17} - 7 q^{18} - 6 q^{19} + 13 q^{20} - 8 q^{21} - 3 q^{22} + 16 q^{23} - 18 q^{24} - 11 q^{25} - 28 q^{26} - 18 q^{27} - 12 q^{28} + 16 q^{29} + 11 q^{30} - 4 q^{31} + 34 q^{32} + 39 q^{34} - 6 q^{35} - 55 q^{36} + 40 q^{37} + 4 q^{38} - 4 q^{39} - 3 q^{40} + 26 q^{41} + 59 q^{42} + 3 q^{43} - 26 q^{44} - 2 q^{45} - 16 q^{46} + 8 q^{47} + 28 q^{48} - 10 q^{49} + 3 q^{50} + 7 q^{51} - 31 q^{52} - 52 q^{53} - 20 q^{54} + 22 q^{55} + 30 q^{56} + 3 q^{57} - 6 q^{58} + 10 q^{59} + 25 q^{61} + 40 q^{62} - 9 q^{63} + 26 q^{64} + 9 q^{65} + 2 q^{66} - 13 q^{67} + 20 q^{68} + 28 q^{69} - 5 q^{70} - 48 q^{71} - 114 q^{72} + 18 q^{73} + 10 q^{74} + 25 q^{76} + 3 q^{77} + 13 q^{78} + q^{79} - 34 q^{80} - 4 q^{81} + 34 q^{82} - 2 q^{83} + 89 q^{84} - 14 q^{85} + 15 q^{86} - 31 q^{87} - 3 q^{88} - 44 q^{89} + 13 q^{90} - 2 q^{91} + 44 q^{92} + 29 q^{93} + 36 q^{94} - 3 q^{95} + 42 q^{96} - 33 q^{97} - 38 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −1.26605 − 2.19286i 0.415004 + 1.68160i −2.20577 + 3.82050i 0.500000 − 0.866025i 3.16210 − 3.03904i 1.22474 + 2.12131i 6.10626 −2.65554 + 1.39574i −2.53210
166.2 −1.00630 − 1.74296i −0.598177 − 1.62548i −1.02527 + 1.77582i 0.500000 − 0.866025i −2.23120 + 2.67831i −1.47315 − 2.55156i 0.101717 −2.28437 + 1.94465i −2.01260
166.3 −0.949086 − 1.64387i −1.54093 + 0.790898i −0.801529 + 1.38829i 0.500000 − 0.866025i 2.76261 + 1.78246i 0.751191 + 1.30110i −0.753463 1.74896 − 2.43745i −1.89817
166.4 −0.350077 − 0.606351i 1.65559 − 0.508929i 0.754892 − 1.30751i 0.500000 − 0.866025i −0.888175 − 0.825707i −0.588842 − 1.01990i −2.45739 2.48198 − 1.68516i −0.700154
166.5 0.151059 + 0.261642i −1.73181 − 0.0286900i 0.954362 − 1.65300i 0.500000 − 0.866025i −0.254099 − 0.457448i −2.35692 − 4.08230i 1.18090 2.99835 + 0.0993714i 0.302118
166.6 0.167960 + 0.290915i −0.721147 + 1.57478i 0.943579 − 1.63433i 0.500000 − 0.866025i −0.579253 + 0.0547084i −0.114478 − 0.198282i 1.30577 −1.95990 − 2.27130i 0.335920
166.7 0.461676 + 0.799647i 0.817694 − 1.52688i 0.573710 − 0.993694i 0.500000 − 0.866025i 1.59848 − 0.0510597i 2.41373 + 4.18070i 2.90618 −1.66275 − 2.49705i 0.923353
166.8 0.726237 + 1.25788i 0.843757 + 1.51264i −0.0548416 + 0.0949884i 0.500000 − 0.866025i −1.28995 + 2.15988i 0.128378 + 0.222358i 2.74564 −1.57615 + 2.55260i 1.45247
166.9 0.979421 + 1.69641i 0.695938 − 1.58609i −0.918533 + 1.59095i 0.500000 − 0.866025i 3.37227 − 0.372853i −1.92591 − 3.33577i 0.319163 −2.03134 − 2.20764i 1.95884
166.10 1.19567 + 2.07097i −1.41076 − 1.00487i −1.85927 + 3.22035i 0.500000 − 0.866025i 0.394240 − 4.12313i 1.20845 + 2.09310i −4.10964 0.980483 + 2.83525i 2.39135
166.11 1.38948 + 2.40666i 1.57484 + 0.721017i −2.86133 + 4.95596i 0.500000 − 0.866025i 0.452979 + 4.79195i −0.767199 − 1.32883i −10.3451 1.96027 + 2.27098i 2.77897
331.1 −1.26605 + 2.19286i 0.415004 − 1.68160i −2.20577 − 3.82050i 0.500000 + 0.866025i 3.16210 + 3.03904i 1.22474 − 2.12131i 6.10626 −2.65554 − 1.39574i −2.53210
331.2 −1.00630 + 1.74296i −0.598177 + 1.62548i −1.02527 − 1.77582i 0.500000 + 0.866025i −2.23120 − 2.67831i −1.47315 + 2.55156i 0.101717 −2.28437 − 1.94465i −2.01260
331.3 −0.949086 + 1.64387i −1.54093 − 0.790898i −0.801529 − 1.38829i 0.500000 + 0.866025i 2.76261 − 1.78246i 0.751191 − 1.30110i −0.753463 1.74896 + 2.43745i −1.89817
331.4 −0.350077 + 0.606351i 1.65559 + 0.508929i 0.754892 + 1.30751i 0.500000 + 0.866025i −0.888175 + 0.825707i −0.588842 + 1.01990i −2.45739 2.48198 + 1.68516i −0.700154
331.5 0.151059 − 0.261642i −1.73181 + 0.0286900i 0.954362 + 1.65300i 0.500000 + 0.866025i −0.254099 + 0.457448i −2.35692 + 4.08230i 1.18090 2.99835 − 0.0993714i 0.302118
331.6 0.167960 − 0.290915i −0.721147 − 1.57478i 0.943579 + 1.63433i 0.500000 + 0.866025i −0.579253 − 0.0547084i −0.114478 + 0.198282i 1.30577 −1.95990 + 2.27130i 0.335920
331.7 0.461676 − 0.799647i 0.817694 + 1.52688i 0.573710 + 0.993694i 0.500000 + 0.866025i 1.59848 + 0.0510597i 2.41373 − 4.18070i 2.90618 −1.66275 + 2.49705i 0.923353
331.8 0.726237 − 1.25788i 0.843757 − 1.51264i −0.0548416 − 0.0949884i 0.500000 + 0.866025i −1.28995 − 2.15988i 0.128378 − 0.222358i 2.74564 −1.57615 − 2.55260i 1.45247
331.9 0.979421 − 1.69641i 0.695938 + 1.58609i −0.918533 − 1.59095i 0.500000 + 0.866025i 3.37227 + 0.372853i −1.92591 + 3.33577i 0.319163 −2.03134 + 2.20764i 1.95884
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 166.11
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 495.2.i.f ✓ 22
3.b odd 2 1 1485.2.i.f 22
9.c even 3 1 inner 495.2.i.f ✓ 22
9.c even 3 1 4455.2.a.u 11
9.d odd 6 1 1485.2.i.f 22
9.d odd 6 1 4455.2.a.v 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.i.f ✓ 22 1.a even 1 1 trivial
495.2.i.f ✓ 22 9.c even 3 1 inner
1485.2.i.f 22 3.b odd 2 1
1485.2.i.f 22 9.d odd 6 1
4455.2.a.u 11 9.c even 3 1
4455.2.a.v 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 \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{22} - 3 T_{2}^{21} + 22 T_{2}^{20} - 47 T_{2}^{19} + 251 T_{2}^{18} - 481 T_{2}^{17} + 1836 T_{2}^{16} + \cdots + 144 \) Copy content Toggle raw display
\( T_{13}^{22} + 9 T_{13}^{21} + 92 T_{13}^{20} + 505 T_{13}^{19} + 3466 T_{13}^{18} + 16561 T_{13}^{17} + \cdots + 4096 \) Copy content Toggle raw display