# 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 Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$93.0951693564$$ 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 3.71.b.a 22
3.b odd 2 1 inner 3.71.b.a 22

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{71}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database