Properties

Label 165.2.r.a
Level $165$
Weight $2$
Character orbit 165.r
Analytic conductor $1.318$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80q + 4q^{4} - 10q^{6} - 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q + 4q^{4} - 10q^{6} - 10q^{9} - 2q^{15} - 60q^{16} - 20q^{19} - 30q^{24} + 6q^{25} + 10q^{30} - 20q^{31} - 56q^{34} + 2q^{36} - 50q^{39} + 60q^{40} - 92q^{45} - 40q^{46} + 72q^{49} + 30q^{51} + 4q^{55} + 54q^{60} + 96q^{64} - 42q^{66} + 30q^{69} - 86q^{70} + 2q^{75} - 66q^{81} + 140q^{84} - 30q^{85} + 120q^{90} + 48q^{91} + 60q^{94} - 70q^{96} + 60q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.55596 + 2.14159i −0.928513 1.46214i −1.54738 4.76233i 1.59907 + 1.56301i 4.57604 + 0.286538i 0.391826 + 1.20592i 7.57144 + 2.46011i −1.27573 + 2.71524i −5.83540 + 0.992578i
29.2 −1.55596 + 2.14159i 1.61061 0.637133i −1.54738 4.76233i −0.992368 2.00380i −1.14156 + 4.44062i −0.391826 1.20592i 7.57144 + 2.46011i 2.18812 2.05234i 5.83540 + 0.992578i
29.3 −1.26516 + 1.74134i −1.42080 + 0.990613i −0.813603 2.50401i 0.737221 2.11104i 0.0725479 3.72738i −1.15619 3.55838i 1.29554 + 0.420945i 1.03737 2.81493i 2.74334 + 3.95455i
29.4 −1.26516 + 1.74134i 0.567188 + 1.63655i −0.813603 2.50401i 2.23554 0.0487905i −3.56737 1.08283i 1.15619 + 3.55838i 1.29554 + 0.420945i −2.35660 + 1.85646i −2.74334 + 3.95455i
29.5 −0.875212 + 1.20463i −0.577976 1.63277i −0.0670941 0.206494i −2.16710 0.551056i 2.47273 + 0.732776i −0.259482 0.798604i −2.52478 0.820350i −2.33189 + 1.88741i 2.56049 2.12826i
29.6 −0.875212 + 1.20463i 1.42731 0.981214i −0.0670941 0.206494i −0.145586 + 2.23132i −0.0672046 + 2.57815i 0.259482 + 0.798604i −2.52478 0.820350i 1.07444 2.80100i −2.56049 2.12826i
29.7 −0.413334 + 0.568905i −1.53256 0.807004i 0.465226 + 1.43182i 1.55748 1.60445i 1.09257 0.538320i 0.615368 + 1.89391i −2.34444 0.761754i 1.69749 + 2.47357i 0.269018 + 1.54923i
29.8 −0.413334 + 0.568905i 1.71421 + 0.247937i 0.465226 + 1.43182i 2.00721 0.985452i −0.849595 + 0.872744i −0.615368 1.89391i −2.34444 0.761754i 2.87705 + 0.850033i −0.269018 + 1.54923i
29.9 −0.303223 + 0.417350i −0.725625 + 1.57273i 0.535797 + 1.64901i −1.78836 1.34229i −0.436352 0.779726i 0.781546 + 2.40535i −1.83193 0.595230i −1.94694 2.28242i 1.10248 0.339360i
29.10 −0.303223 + 0.417350i −0.337382 + 1.69887i 0.535797 + 1.64901i 0.723962 + 2.11563i −0.606723 0.655944i −0.781546 2.40535i −1.83193 0.595230i −2.77235 1.14634i −1.10248 0.339360i
29.11 0.303223 0.417350i 0.337382 1.69887i 0.535797 + 1.64901i 1.78836 + 1.34229i −0.606723 0.655944i 0.781546 + 2.40535i 1.83193 + 0.595230i −2.77235 1.14634i 1.10248 0.339360i
29.12 0.303223 0.417350i 0.725625 1.57273i 0.535797 + 1.64901i −0.723962 2.11563i −0.436352 0.779726i −0.781546 2.40535i 1.83193 + 0.595230i −1.94694 2.28242i −1.10248 0.339360i
29.13 0.413334 0.568905i −1.71421 0.247937i 0.465226 + 1.43182i −1.55748 + 1.60445i −0.849595 + 0.872744i 0.615368 + 1.89391i 2.34444 + 0.761754i 2.87705 + 0.850033i 0.269018 + 1.54923i
29.14 0.413334 0.568905i 1.53256 + 0.807004i 0.465226 + 1.43182i −2.00721 + 0.985452i 1.09257 0.538320i −0.615368 1.89391i 2.34444 + 0.761754i 1.69749 + 2.47357i −0.269018 + 1.54923i
29.15 0.875212 1.20463i −1.42731 + 0.981214i −0.0670941 0.206494i 2.16710 + 0.551056i −0.0672046 + 2.57815i −0.259482 0.798604i 2.52478 + 0.820350i 1.07444 2.80100i 2.56049 2.12826i
29.16 0.875212 1.20463i 0.577976 + 1.63277i −0.0670941 0.206494i 0.145586 2.23132i 2.47273 + 0.732776i 0.259482 + 0.798604i 2.52478 + 0.820350i −2.33189 + 1.88741i −2.56049 2.12826i
29.17 1.26516 1.74134i −0.567188 1.63655i −0.813603 2.50401i −0.737221 + 2.11104i −3.56737 1.08283i −1.15619 3.55838i −1.29554 0.420945i −2.35660 + 1.85646i 2.74334 + 3.95455i
29.18 1.26516 1.74134i 1.42080 0.990613i −0.813603 2.50401i −2.23554 + 0.0487905i 0.0725479 3.72738i 1.15619 + 3.55838i −1.29554 0.420945i 1.03737 2.81493i −2.74334 + 3.95455i
29.19 1.55596 2.14159i −1.61061 + 0.637133i −1.54738 4.76233i −1.59907 1.56301i −1.14156 + 4.44062i 0.391826 + 1.20592i −7.57144 2.46011i 2.18812 2.05234i −5.83540 + 0.992578i
29.20 1.55596 2.14159i 0.928513 + 1.46214i −1.54738 4.76233i 0.992368 + 2.00380i 4.57604 + 0.286538i −0.391826 1.20592i −7.57144 2.46011i −1.27573 + 2.71524i 5.83540 + 0.992578i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.r.a 80
3.b odd 2 1 inner 165.2.r.a 80
5.b even 2 1 inner 165.2.r.a 80
5.c odd 4 2 825.2.bi.h 80
11.d odd 10 1 inner 165.2.r.a 80
15.d odd 2 1 inner 165.2.r.a 80
15.e even 4 2 825.2.bi.h 80
33.f even 10 1 inner 165.2.r.a 80
55.h odd 10 1 inner 165.2.r.a 80
55.l even 20 2 825.2.bi.h 80
165.r even 10 1 inner 165.2.r.a 80
165.u odd 20 2 825.2.bi.h 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.r.a 80 1.a even 1 1 trivial
165.2.r.a 80 3.b odd 2 1 inner
165.2.r.a 80 5.b even 2 1 inner
165.2.r.a 80 11.d odd 10 1 inner
165.2.r.a 80 15.d odd 2 1 inner
165.2.r.a 80 33.f even 10 1 inner
165.2.r.a 80 55.h odd 10 1 inner
165.2.r.a 80 165.r even 10 1 inner
825.2.bi.h 80 5.c odd 4 2
825.2.bi.h 80 15.e even 4 2
825.2.bi.h 80 55.l even 20 2
825.2.bi.h 80 165.u odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(165, [\chi])\).