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Properties

Label 165.3.t.b
Level $165$
Weight $3$
Character orbit 165.t
Analytic conductor $4.496$
Analytic rank $0$
Dimension $32$
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] = [165,3,Mod(46,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.46");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.t (of order \(10\), degree \(4\), 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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 40 q^{5} - 20 q^{7} + 40 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 40 q^{5} - 20 q^{7} + 40 q^{8} - 24 q^{9} - 6 q^{11} - 90 q^{13} + 120 q^{14} - 32 q^{16} + 80 q^{17} + 30 q^{18} + 90 q^{20} - 50 q^{22} - 48 q^{23} - 270 q^{24} - 40 q^{25} + 104 q^{26} + 100 q^{28} - 220 q^{29} - 102 q^{31} + 6 q^{33} + 280 q^{34} + 72 q^{36} + 198 q^{37} + 182 q^{38} + 60 q^{40} - 150 q^{41} + 282 q^{42} - 510 q^{44} - 30 q^{46} + 20 q^{47} - 48 q^{48} + 388 q^{49} - 90 q^{51} - 1260 q^{52} - 350 q^{53} - 224 q^{56} - 120 q^{57} + 228 q^{58} + 248 q^{59} + 290 q^{61} + 460 q^{62} + 60 q^{63} + 102 q^{64} + 90 q^{66} + 152 q^{67} - 60 q^{68} - 24 q^{69} - 160 q^{70} + 422 q^{71} + 150 q^{72} + 110 q^{73} + 1310 q^{74} + 824 q^{77} - 120 q^{78} - 190 q^{79} + 310 q^{80} - 72 q^{81} - 216 q^{82} - 370 q^{83} - 300 q^{84} - 210 q^{85} + 264 q^{86} - 732 q^{88} + 8 q^{89} - 60 q^{90} - 468 q^{91} - 930 q^{92} + 72 q^{93} - 1240 q^{94} - 120 q^{95} - 90 q^{96} - 98 q^{97} - 138 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −3.32577 − 1.08061i 1.40126 − 1.01807i 6.65699 + 4.83658i −0.690983 − 2.12663i −5.76041 + 1.87167i −5.04373 + 6.94210i −8.69141 − 11.9627i 0.927051 − 2.85317i 7.81936i
46.2 −2.06413 − 0.670676i −1.40126 + 1.01807i 0.574752 + 0.417582i −0.690983 − 2.12663i 3.57518 − 1.16164i 5.15391 − 7.09375i 4.19651 + 5.77600i 0.927051 − 2.85317i 4.85306i
46.3 −1.93061 − 0.627293i −1.40126 + 1.01807i 0.0976840 + 0.0709716i −0.690983 − 2.12663i 3.34391 − 1.08650i −4.27043 + 5.87774i 4.62866 + 6.37080i 0.927051 − 2.85317i 4.53913i
46.4 −0.384792 − 0.125027i 1.40126 − 1.01807i −3.10363 − 2.25492i −0.690983 − 2.12663i −0.666480 + 0.216552i −0.126072 + 0.173523i 1.86359 + 2.56501i 0.927051 − 2.85317i 0.904701i
46.5 1.32794 + 0.431474i −1.40126 + 1.01807i −1.65881 − 1.20520i −0.690983 − 2.12663i −2.30006 + 0.747335i 3.80088 − 5.23146i −4.96564 − 6.83462i 0.927051 − 2.85317i − 3.12218i
46.6 1.63658 + 0.531755i 1.40126 − 1.01807i −0.840454 − 0.610626i −0.690983 − 2.12663i 2.83463 − 0.921027i 5.96961 − 8.21646i −5.09660 − 7.01487i 0.927051 − 2.85317i − 3.84782i
46.7 3.19203 + 1.03715i 1.40126 − 1.01807i 5.87727 + 4.27009i −0.690983 − 2.12663i 5.52875 − 1.79640i −2.03976 + 2.80749i 6.44055 + 8.86466i 0.927051 − 2.85317i − 7.50490i
46.8 3.78483 + 1.22977i −1.40126 + 1.01807i 9.57654 + 6.95777i −0.690983 − 2.12663i −6.55552 + 2.13002i 2.73594 − 3.76570i 18.3326 + 25.2326i 0.927051 − 2.85317i − 8.89867i
61.1 −3.32577 + 1.08061i 1.40126 + 1.01807i 6.65699 − 4.83658i −0.690983 + 2.12663i −5.76041 − 1.87167i −5.04373 − 6.94210i −8.69141 + 11.9627i 0.927051 + 2.85317i − 7.81936i
61.2 −2.06413 + 0.670676i −1.40126 − 1.01807i 0.574752 − 0.417582i −0.690983 + 2.12663i 3.57518 + 1.16164i 5.15391 + 7.09375i 4.19651 − 5.77600i 0.927051 + 2.85317i − 4.85306i
61.3 −1.93061 + 0.627293i −1.40126 − 1.01807i 0.0976840 − 0.0709716i −0.690983 + 2.12663i 3.34391 + 1.08650i −4.27043 − 5.87774i 4.62866 − 6.37080i 0.927051 + 2.85317i − 4.53913i
61.4 −0.384792 + 0.125027i 1.40126 + 1.01807i −3.10363 + 2.25492i −0.690983 + 2.12663i −0.666480 − 0.216552i −0.126072 − 0.173523i 1.86359 − 2.56501i 0.927051 + 2.85317i − 0.904701i
61.5 1.32794 − 0.431474i −1.40126 − 1.01807i −1.65881 + 1.20520i −0.690983 + 2.12663i −2.30006 − 0.747335i 3.80088 + 5.23146i −4.96564 + 6.83462i 0.927051 + 2.85317i 3.12218i
61.6 1.63658 − 0.531755i 1.40126 + 1.01807i −0.840454 + 0.610626i −0.690983 + 2.12663i 2.83463 + 0.921027i 5.96961 + 8.21646i −5.09660 + 7.01487i 0.927051 + 2.85317i 3.84782i
61.7 3.19203 − 1.03715i 1.40126 + 1.01807i 5.87727 − 4.27009i −0.690983 + 2.12663i 5.52875 + 1.79640i −2.03976 − 2.80749i 6.44055 − 8.86466i 0.927051 + 2.85317i 7.50490i
61.8 3.78483 − 1.22977i −1.40126 − 1.01807i 9.57654 − 6.95777i −0.690983 + 2.12663i −6.55552 − 2.13002i 2.73594 + 3.76570i 18.3326 − 25.2326i 0.927051 + 2.85317i 8.89867i
106.1 −2.26348 + 3.11541i −0.535233 + 1.64728i −3.34639 − 10.2991i −1.80902 + 1.31433i −3.92046 − 5.39605i −11.7683 + 3.82374i 25.0110 + 8.12656i −2.42705 − 1.76336i − 8.61079i
106.2 −1.78604 + 2.45827i 0.535233 − 1.64728i −1.61710 − 4.97693i −1.80902 + 1.31433i 3.09351 + 4.25786i −0.325665 + 0.105815i 3.56339 + 1.15781i −2.42705 − 1.76336i − 6.79450i
106.3 −0.919285 + 1.26529i −0.535233 + 1.64728i 0.480201 + 1.47791i −1.80902 + 1.31433i −1.59225 − 2.19154i −2.55067 + 0.828762i −8.26116 − 2.68421i −2.42705 − 1.76336i − 3.49717i
106.4 −0.819127 + 1.12743i 0.535233 − 1.64728i 0.635935 + 1.95721i −1.80902 + 1.31433i 1.41877 + 1.95277i −8.08042 + 2.62549i −8.02903 − 2.60879i −2.42705 − 1.76336i − 3.11614i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.t.b ✓ 32
11.d odd 10 1 inner 165.3.t.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.t.b ✓ 32 1.a even 1 1 trivial
165.3.t.b ✓ 32 11.d odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 28 T_{2}^{30} - 40 T_{2}^{29} + 506 T_{2}^{28} + 1120 T_{2}^{27} - 6937 T_{2}^{26} + \cdots + 2062885561 \) acting on \(S_{3}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display