Properties

Label 3.71.b.a
Level 3
Weight 71
Character orbit 3.b
Analytic conductor 93.095
Analytic rank 0
Dimension 22
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 71 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(93.0951693564\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22q - 57760946614928610q^{3} - \)\(12\!\cdots\!72\)\(q^{4} - \)\(25\!\cdots\!40\)\(q^{6} + \)\(34\!\cdots\!20\)\(q^{7} - \)\(31\!\cdots\!82\)\(q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q - 57760946614928610q^{3} - \)\(12\!\cdots\!72\)\(q^{4} - \)\(25\!\cdots\!40\)\(q^{6} + \)\(34\!\cdots\!20\)\(q^{7} - \)\(31\!\cdots\!82\)\(q^{9} - \)\(20\!\cdots\!60\)\(q^{10} + \)\(16\!\cdots\!40\)\(q^{12} - \)\(14\!\cdots\!00\)\(q^{13} - \)\(22\!\cdots\!40\)\(q^{15} + \)\(77\!\cdots\!32\)\(q^{16} - \)\(15\!\cdots\!40\)\(q^{18} + \)\(20\!\cdots\!76\)\(q^{19} - \)\(53\!\cdots\!16\)\(q^{21} - \)\(17\!\cdots\!40\)\(q^{22} + \)\(22\!\cdots\!60\)\(q^{24} - \)\(47\!\cdots\!90\)\(q^{25} - \)\(32\!\cdots\!90\)\(q^{27} + \)\(19\!\cdots\!00\)\(q^{28} + \)\(19\!\cdots\!20\)\(q^{30} + \)\(12\!\cdots\!04\)\(q^{31} + \)\(91\!\cdots\!20\)\(q^{33} + \)\(60\!\cdots\!20\)\(q^{34} + \)\(90\!\cdots\!12\)\(q^{36} - \)\(29\!\cdots\!20\)\(q^{37} - \)\(83\!\cdots\!04\)\(q^{39} + \)\(27\!\cdots\!20\)\(q^{40} + \)\(15\!\cdots\!80\)\(q^{42} - \)\(45\!\cdots\!80\)\(q^{43} - \)\(13\!\cdots\!40\)\(q^{45} + \)\(34\!\cdots\!80\)\(q^{46} + \)\(18\!\cdots\!60\)\(q^{48} + \)\(53\!\cdots\!94\)\(q^{49} + \)\(93\!\cdots\!40\)\(q^{51} - \)\(25\!\cdots\!60\)\(q^{52} + \)\(56\!\cdots\!00\)\(q^{54} - \)\(35\!\cdots\!80\)\(q^{55} + \)\(62\!\cdots\!80\)\(q^{57} + \)\(11\!\cdots\!60\)\(q^{58} - \)\(28\!\cdots\!60\)\(q^{60} - \)\(10\!\cdots\!76\)\(q^{61} + \)\(22\!\cdots\!20\)\(q^{63} - \)\(69\!\cdots\!32\)\(q^{64} + \)\(98\!\cdots\!40\)\(q^{66} - \)\(95\!\cdots\!60\)\(q^{67} + \)\(68\!\cdots\!80\)\(q^{69} - \)\(18\!\cdots\!40\)\(q^{70} + \)\(80\!\cdots\!40\)\(q^{72} - \)\(49\!\cdots\!20\)\(q^{73} + \)\(21\!\cdots\!30\)\(q^{75} - \)\(52\!\cdots\!16\)\(q^{76} + \)\(12\!\cdots\!00\)\(q^{78} - \)\(17\!\cdots\!64\)\(q^{79} + \)\(13\!\cdots\!42\)\(q^{81} - \)\(39\!\cdots\!60\)\(q^{82} + \)\(13\!\cdots\!36\)\(q^{84} - \)\(61\!\cdots\!80\)\(q^{85} - \)\(14\!\cdots\!00\)\(q^{87} + \)\(42\!\cdots\!60\)\(q^{88} - \)\(95\!\cdots\!40\)\(q^{90} + \)\(13\!\cdots\!68\)\(q^{91} - \)\(14\!\cdots\!80\)\(q^{93} + \)\(10\!\cdots\!60\)\(q^{94} - \)\(12\!\cdots\!20\)\(q^{96} + \)\(59\!\cdots\!80\)\(q^{97} - \)\(21\!\cdots\!80\)\(q^{99} + O(q^{100}) \)

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 6.63817e10i 2.36972e16 4.40636e16i −3.22594e21 6.32737e23i −2.92502e27 1.57306e27i −2.54650e29 1.35774e32i −1.38004e33 2.08836e33i 4.20021e34
2.2 6.15211e10i −4.79288e16 + 1.43522e16i −2.60426e21 5.00764e24i 8.82962e26 + 2.94863e27i 5.12912e29 8.75855e31i 2.09119e33 1.37577e33i −3.08075e35
2.3 5.67567e10i 2.16075e15 + 4.99849e16i −2.04073e21 2.61507e24i 2.83697e27 1.22637e26i −2.08517e29 4.88185e31i −2.49382e33 + 2.16009e32i 1.48422e35
2.4 4.67615e10i −4.93039e16 8.50161e15i −1.00605e21 4.28418e24i −3.97548e26 + 2.30553e27i −1.75590e29 8.16202e30i 2.35860e33 + 8.38326e32i 2.00335e35
2.5 4.57279e10i 4.91081e16 + 9.56812e15i −9.10451e20 1.29561e24i 4.37530e26 2.24561e27i 1.42646e29 1.23530e31i 2.32006e33 + 9.39745e32i −5.92457e34
2.6 3.74673e10i −7.56180e15 4.94568e16i −2.23206e20 1.07403e24i −1.85301e27 + 2.83320e26i 6.73954e29 3.58706e31i −2.38879e33 + 7.47965e32i 4.02409e34
2.7 3.59410e10i −2.70537e16 4.20862e16i −1.11161e20 4.52111e24i −1.51262e27 + 9.72336e26i −7.40509e29 3.84364e31i −1.03935e33 + 2.27718e33i −1.62493e35
2.8 2.60562e10i −2.92624e15 + 4.99459e16i 5.01667e20 2.60973e24i 1.30140e27 + 7.62467e25i −1.01844e29 4.38332e31i −2.48603e33 2.92308e32i −6.79996e34
2.9 1.54812e10i 3.84365e16 3.20279e16i 9.40925e20 2.49363e24i −4.95830e26 5.95043e26i −4.12745e29 3.28436e31i 4.51579e32 2.46209e33i 3.86044e34
2.10 1.12481e10i −4.22339e16 + 2.68226e16i 1.05407e21 2.30186e23i 3.01705e26 + 4.75053e26i 8.16038e28 2.51358e31i 1.06425e33 2.26565e33i 2.58917e33
2.11 4.22234e9i 3.47254e16 + 3.60181e16i 1.16276e21 5.48695e24i 1.52081e26 1.46622e26i 4.84471e29 9.89443e30i −9.14522e31 + 2.50148e33i 2.31678e34
2.12 4.22234e9i 3.47254e16 3.60181e16i 1.16276e21 5.48695e24i 1.52081e26 + 1.46622e26i 4.84471e29 9.89443e30i −9.14522e31 2.50148e33i 2.31678e34
2.13 1.12481e10i −4.22339e16 2.68226e16i 1.05407e21 2.30186e23i 3.01705e26 4.75053e26i 8.16038e28 2.51358e31i 1.06425e33 + 2.26565e33i 2.58917e33
2.14 1.54812e10i 3.84365e16 + 3.20279e16i 9.40925e20 2.49363e24i −4.95830e26 + 5.95043e26i −4.12745e29 3.28436e31i 4.51579e32 + 2.46209e33i 3.86044e34
2.15 2.60562e10i −2.92624e15 4.99459e16i 5.01667e20 2.60973e24i 1.30140e27 7.62467e25i −1.01844e29 4.38332e31i −2.48603e33 + 2.92308e32i −6.79996e34
2.16 3.59410e10i −2.70537e16 + 4.20862e16i −1.11161e20 4.52111e24i −1.51262e27 9.72336e26i −7.40509e29 3.84364e31i −1.03935e33 2.27718e33i −1.62493e35
2.17 3.74673e10i −7.56180e15 + 4.94568e16i −2.23206e20 1.07403e24i −1.85301e27 2.83320e26i 6.73954e29 3.58706e31i −2.38879e33 7.47965e32i 4.02409e34
2.18 4.57279e10i 4.91081e16 9.56812e15i −9.10451e20 1.29561e24i 4.37530e26 + 2.24561e27i 1.42646e29 1.23530e31i 2.32006e33 9.39745e32i −5.92457e34
2.19 4.67615e10i −4.93039e16 + 8.50161e15i −1.00605e21 4.28418e24i −3.97548e26 2.30553e27i −1.75590e29 8.16202e30i 2.35860e33 8.38326e32i 2.00335e35
2.20 5.67567e10i 2.16075e15 4.99849e16i −2.04073e21 2.61507e24i 2.83697e27 + 1.22637e26i −2.08517e29 4.88185e31i −2.49382e33 2.16009e32i 1.48422e35
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

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{71}^{\mathrm{new}}(3, [\chi])\).