Properties

Label 69.2.g.a
Level $69$
Weight $2$
Character orbit 69.g
Analytic conductor $0.551$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60q - 11q^{3} - 10q^{4} - 14q^{6} - 22q^{7} - 11q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 11q^{3} - 10q^{4} - 14q^{6} - 22q^{7} - 11q^{9} - 22q^{10} + 4q^{12} - 22q^{13} - 46q^{16} + 12q^{18} - 22q^{19} + 22q^{21} + 50q^{24} + 8q^{25} + 10q^{27} - 22q^{28} + 33q^{30} - 22q^{31} + 22q^{36} + 22q^{37} + 13q^{39} + 132q^{40} - 11q^{42} + 22q^{43} + 66q^{46} - 58q^{48} + 68q^{49} - 11q^{51} + 94q^{52} - 33q^{54} - 44q^{57} - 8q^{58} - 121q^{60} - 66q^{61} - 66q^{63} - 20q^{64} - 66q^{66} - 44q^{67} - 66q^{69} - 132q^{70} - 101q^{72} - 44q^{73} - 44q^{75} - 110q^{76} + 84q^{78} - 66q^{79} + 77q^{81} - 132q^{82} + 77q^{84} - 44q^{85} + 73q^{87} + 66q^{88} + 176q^{90} + 116q^{93} + 100q^{94} + 85q^{96} + 44q^{97} + 121q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.694600 + 2.36559i 0.236423 + 1.71584i −3.43104 2.20500i 0.0722440 0.502468i −4.22319 0.632543i 3.47479 1.58688i 3.87278 3.35579i −2.88821 + 0.811326i 1.13845 + 0.519914i
5.2 −0.417491 + 1.42184i 1.61954 0.614080i −0.164830 0.105930i 0.394240 2.74200i 0.196983 + 2.55910i −3.84519 + 1.75604i −2.02041 + 1.75070i 2.24581 1.98905i 3.73410 + 1.70530i
5.3 −0.298617 + 1.01700i −1.66399 + 0.480761i 0.737395 + 0.473895i −0.228939 + 1.59231i 0.00796426 1.83584i −1.28446 + 0.586593i −2.30424 + 1.99663i 2.53774 1.59996i −1.55101 0.708322i
5.4 0.298617 1.01700i −0.239057 + 1.71547i 0.737395 + 0.473895i 0.228939 1.59231i 1.67325 + 0.755391i −1.28446 + 0.586593i 2.30424 1.99663i −2.88570 0.820191i −1.55101 0.708322i
5.5 0.417491 1.42184i 0.377345 1.69045i −0.164830 0.105930i −0.394240 + 2.74200i −2.24601 1.24227i −3.84519 + 1.75604i 2.02041 1.75070i −2.71522 1.27576i 3.73410 + 1.70530i
5.6 0.694600 2.36559i −1.73202 + 0.0101732i −3.43104 2.20500i −0.0722440 + 0.502468i −1.17900 + 4.10432i 3.47479 1.58688i −3.87278 + 3.35579i 2.99979 0.0352405i 1.13845 + 0.519914i
11.1 −1.13923 1.77267i 0.311887 1.70374i −1.01370 + 2.21969i 1.10513 0.324494i −3.37548 + 1.38807i −1.21996 + 1.05710i 0.918158 0.132011i −2.80545 1.06275i −1.83421 1.58935i
11.2 −0.848533 1.32034i 0.567934 + 1.63629i −0.192468 + 0.421446i 1.50619 0.442257i 1.67856 2.13832i 1.83527 1.59027i −2.38727 + 0.343238i −2.35490 + 1.85861i −1.86198 1.61342i
11.3 −0.0493131 0.0767326i 1.73042 0.0750729i 0.827374 1.81170i −2.86859 + 0.842294i −0.0910930 0.129078i −1.75762 + 1.52299i −0.360384 + 0.0518154i 2.98873 0.259816i 0.206090 + 0.178578i
11.4 0.0493131 + 0.0767326i −1.63918 0.559548i 0.827374 1.81170i 2.86859 0.842294i −0.0378973 0.153371i −1.75762 + 1.52299i 0.360384 0.0518154i 2.37381 + 1.83440i 0.206090 + 0.178578i
11.5 0.848533 + 1.32034i −1.00593 + 1.41000i −0.192468 + 0.421446i −1.50619 + 0.442257i −2.71525 0.131731i 1.83527 1.59027i 2.38727 0.343238i −0.976228 2.83672i −1.86198 1.61342i
11.6 1.13923 + 1.77267i 0.180745 1.72259i −1.01370 + 2.21969i −1.10513 + 0.324494i 3.25951 1.64203i −1.21996 + 1.05710i −0.918158 + 0.132011i −2.93466 0.622700i −1.83421 1.58935i
14.1 −0.694600 2.36559i 0.236423 1.71584i −3.43104 + 2.20500i 0.0722440 + 0.502468i −4.22319 + 0.632543i 3.47479 + 1.58688i 3.87278 + 3.35579i −2.88821 0.811326i 1.13845 0.519914i
14.2 −0.417491 1.42184i 1.61954 + 0.614080i −0.164830 + 0.105930i 0.394240 + 2.74200i 0.196983 2.55910i −3.84519 1.75604i −2.02041 1.75070i 2.24581 + 1.98905i 3.73410 1.70530i
14.3 −0.298617 1.01700i −1.66399 0.480761i 0.737395 0.473895i −0.228939 1.59231i 0.00796426 + 1.83584i −1.28446 0.586593i −2.30424 1.99663i 2.53774 + 1.59996i −1.55101 + 0.708322i
14.4 0.298617 + 1.01700i −0.239057 1.71547i 0.737395 0.473895i 0.228939 + 1.59231i 1.67325 0.755391i −1.28446 0.586593i 2.30424 + 1.99663i −2.88570 + 0.820191i −1.55101 + 0.708322i
14.5 0.417491 + 1.42184i 0.377345 + 1.69045i −0.164830 + 0.105930i −0.394240 2.74200i −2.24601 + 1.24227i −3.84519 1.75604i 2.02041 + 1.75070i −2.71522 + 1.27576i 3.73410 1.70530i
14.6 0.694600 + 2.36559i −1.73202 0.0101732i −3.43104 + 2.20500i −0.0722440 0.502468i −1.17900 4.10432i 3.47479 + 1.58688i −3.87278 3.35579i 2.99979 + 0.0352405i 1.13845 0.519914i
17.1 −1.87897 0.858095i −1.18942 1.25908i 1.48446 + 1.71316i −1.27941 + 0.822226i 1.15448 + 3.38640i −3.90579 0.561567i −0.155288 0.528861i −0.170547 + 2.99515i 3.10951 0.447081i
17.2 −1.69911 0.775958i −0.953890 + 1.44572i 0.975144 + 1.12538i 3.49515 2.24620i 2.74258 1.71625i 2.15133 + 0.309315i 0.268869 + 0.915684i −1.18019 2.75811i −7.68160 + 1.10445i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.g.a 60
3.b odd 2 1 inner 69.2.g.a 60
23.d odd 22 1 inner 69.2.g.a 60
69.g even 22 1 inner 69.2.g.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.g.a 60 1.a even 1 1 trivial
69.2.g.a 60 3.b odd 2 1 inner
69.2.g.a 60 23.d odd 22 1 inner
69.2.g.a 60 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(69, [\chi])\).