Properties

Label 3.69.b.a
Level 3
Weight 69
Character orbit 3.b
Analytic conductor 87.852
Analytic rank 0
Dimension 22
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 69 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(87.8517980619\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22q + 18789723640115478q^{3} - \)\(29\!\cdots\!16\)\(q^{4} + \)\(60\!\cdots\!44\)\(q^{6} - \)\(36\!\cdots\!44\)\(q^{7} + \)\(84\!\cdots\!78\)\(q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q + 18789723640115478q^{3} - \)\(29\!\cdots\!16\)\(q^{4} + \)\(60\!\cdots\!44\)\(q^{6} - \)\(36\!\cdots\!44\)\(q^{7} + \)\(84\!\cdots\!78\)\(q^{9} + \)\(20\!\cdots\!80\)\(q^{10} - \)\(64\!\cdots\!32\)\(q^{12} - \)\(10\!\cdots\!04\)\(q^{13} + \)\(14\!\cdots\!60\)\(q^{15} + \)\(37\!\cdots\!20\)\(q^{16} + \)\(17\!\cdots\!60\)\(q^{18} - \)\(79\!\cdots\!24\)\(q^{19} - \)\(12\!\cdots\!80\)\(q^{21} + \)\(67\!\cdots\!60\)\(q^{22} - \)\(75\!\cdots\!48\)\(q^{24} - \)\(12\!\cdots\!50\)\(q^{25} + \)\(26\!\cdots\!18\)\(q^{27} + \)\(42\!\cdots\!76\)\(q^{28} + \)\(30\!\cdots\!00\)\(q^{30} - \)\(46\!\cdots\!16\)\(q^{31} + \)\(20\!\cdots\!00\)\(q^{33} + \)\(13\!\cdots\!72\)\(q^{34} - \)\(65\!\cdots\!04\)\(q^{36} - \)\(16\!\cdots\!44\)\(q^{37} + \)\(16\!\cdots\!20\)\(q^{39} + \)\(60\!\cdots\!40\)\(q^{40} - \)\(12\!\cdots\!60\)\(q^{42} - \)\(18\!\cdots\!64\)\(q^{43} + \)\(36\!\cdots\!00\)\(q^{45} + \)\(14\!\cdots\!88\)\(q^{46} + \)\(20\!\cdots\!88\)\(q^{48} + \)\(46\!\cdots\!18\)\(q^{49} + \)\(21\!\cdots\!84\)\(q^{51} + \)\(24\!\cdots\!36\)\(q^{52} - \)\(11\!\cdots\!84\)\(q^{54} + \)\(27\!\cdots\!00\)\(q^{55} + \)\(10\!\cdots\!16\)\(q^{57} - \)\(20\!\cdots\!40\)\(q^{58} - \)\(13\!\cdots\!80\)\(q^{60} + \)\(63\!\cdots\!44\)\(q^{61} - \)\(11\!\cdots\!04\)\(q^{63} + \)\(74\!\cdots\!76\)\(q^{64} - \)\(38\!\cdots\!00\)\(q^{66} + \)\(69\!\cdots\!56\)\(q^{67} - \)\(10\!\cdots\!64\)\(q^{69} + \)\(43\!\cdots\!00\)\(q^{70} - \)\(88\!\cdots\!60\)\(q^{72} + \)\(88\!\cdots\!56\)\(q^{73} - \)\(35\!\cdots\!50\)\(q^{75} + \)\(50\!\cdots\!32\)\(q^{76} - \)\(79\!\cdots\!20\)\(q^{78} - \)\(30\!\cdots\!64\)\(q^{79} - \)\(88\!\cdots\!38\)\(q^{81} + \)\(41\!\cdots\!40\)\(q^{82} + \)\(99\!\cdots\!20\)\(q^{84} - \)\(86\!\cdots\!20\)\(q^{85} + \)\(36\!\cdots\!80\)\(q^{87} - \)\(46\!\cdots\!40\)\(q^{88} + \)\(14\!\cdots\!20\)\(q^{90} - \)\(69\!\cdots\!60\)\(q^{91} + \)\(22\!\cdots\!56\)\(q^{93} - \)\(49\!\cdots\!68\)\(q^{94} - \)\(55\!\cdots\!68\)\(q^{96} + \)\(90\!\cdots\!96\)\(q^{97} - \)\(11\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 3.12148e10i −1.31335e16 1.02781e16i −6.79217e20 1.89531e23i −3.20828e26 + 4.09961e26i 1.94003e27 1.19887e31i 6.68516e31 + 2.69975e32i 5.91618e33
2.2 3.11028e10i 9.90505e15 + 1.34171e16i −6.72234e20 1.00008e24i 4.17309e26 3.08075e26i −7.71928e28 1.17284e31i −8.19083e31 + 2.65794e32i −3.11052e34
2.3 2.79362e10i 1.66655e16 + 6.25230e14i −4.85282e20 1.09211e24i 1.74665e25 4.65569e26i 3.32586e27 5.31162e30i 2.77347e32 + 2.08395e31i 3.05093e34
2.4 2.53895e10i −5.69070e15 + 1.56762e16i −3.49477e20 2.74876e23i 3.98011e26 + 1.44484e26i 9.96672e28 1.37939e30i −2.13360e32 1.78417e32i 6.97895e33
2.5 2.34876e10i 6.92552e15 1.51712e16i −2.56517e20 4.36463e23i −3.56335e26 1.62663e26i 1.55709e28 9.07342e29i −1.82203e32 2.10137e32i −1.02514e34
2.6 1.98633e10i −1.52299e16 + 6.79538e15i −9.94018e19 1.15188e23i 1.34979e26 + 3.02517e26i −6.02651e28 3.88816e30i 1.85774e32 2.06987e32i −2.28800e33
2.7 1.12696e10i 1.62840e16 3.59996e15i 1.68144e20 1.29685e23i −4.05701e25 1.83514e26i −3.11566e28 5.22111e30i 2.52209e32 1.17244e32i −1.46150e33
2.8 9.15147e9i 1.20188e16 + 1.15619e16i 2.11398e20 4.85869e23i 1.05808e26 1.09989e26i 8.15893e28 4.63565e30i 1.07731e31 + 2.77920e32i −4.44641e33
2.9 9.08296e9i −6.37287e15 1.54115e16i 2.12648e20 8.19633e23i −1.39982e26 + 5.78846e25i −1.46000e28 4.61229e30i −1.96901e32 + 1.96431e32i 7.44469e33
2.10 8.19205e9i 3.18264e15 + 1.63707e16i 2.28038e20 5.93220e23i 1.34109e26 2.60723e25i −6.83050e28 4.28597e30i −2.57870e32 + 1.04204e32i 4.85968e33
2.11 6.69746e9i −1.51595e16 6.95107e15i 2.50292e20 7.66457e23i −4.65545e25 + 1.01530e26i 4.76233e28 3.65306e30i 1.81494e32 + 2.10750e32i −5.13331e33
2.12 6.69746e9i −1.51595e16 + 6.95107e15i 2.50292e20 7.66457e23i −4.65545e25 1.01530e26i 4.76233e28 3.65306e30i 1.81494e32 2.10750e32i −5.13331e33
2.13 8.19205e9i 3.18264e15 1.63707e16i 2.28038e20 5.93220e23i 1.34109e26 + 2.60723e25i −6.83050e28 4.28597e30i −2.57870e32 1.04204e32i 4.85968e33
2.14 9.08296e9i −6.37287e15 + 1.54115e16i 2.12648e20 8.19633e23i −1.39982e26 5.78846e25i −1.46000e28 4.61229e30i −1.96901e32 1.96431e32i 7.44469e33
2.15 9.15147e9i 1.20188e16 1.15619e16i 2.11398e20 4.85869e23i 1.05808e26 + 1.09989e26i 8.15893e28 4.63565e30i 1.07731e31 2.77920e32i −4.44641e33
2.16 1.12696e10i 1.62840e16 + 3.59996e15i 1.68144e20 1.29685e23i −4.05701e25 + 1.83514e26i −3.11566e28 5.22111e30i 2.52209e32 + 1.17244e32i −1.46150e33
2.17 1.98633e10i −1.52299e16 6.79538e15i −9.94018e19 1.15188e23i 1.34979e26 3.02517e26i −6.02651e28 3.88816e30i 1.85774e32 + 2.06987e32i −2.28800e33
2.18 2.34876e10i 6.92552e15 + 1.51712e16i −2.56517e20 4.36463e23i −3.56335e26 + 1.62663e26i 1.55709e28 9.07342e29i −1.82203e32 + 2.10137e32i −1.02514e34
2.19 2.53895e10i −5.69070e15 1.56762e16i −3.49477e20 2.74876e23i 3.98011e26 1.44484e26i 9.96672e28 1.37939e30i −2.13360e32 + 1.78417e32i 6.97895e33
2.20 2.79362e10i 1.66655e16 6.25230e14i −4.85282e20 1.09211e24i 1.74665e25 + 4.65569e26i 3.32586e27 5.31162e30i 2.77347e32 2.08395e31i 3.05093e34
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.69.b.a 22
3.b odd 2 1 inner 3.69.b.a 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.69.b.a 22 1.a even 1 1 trivial
3.69.b.a 22 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{69}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database