# Properties

 Label 471.4.b.a Level $471$ Weight $4$ Character orbit 471.b Analytic conductor $27.790$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 471.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.7898996127$$ Analytic rank: $$0$$ Dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
313.1 5.39623i −3.00000 −21.1193 16.5163i 16.1887i 25.8596i 70.7949i 9.00000 −89.1257
313.2 5.31164i −3.00000 −20.2135 13.7454i 15.9349i 18.9295i 64.8739i 9.00000 73.0104
313.3 5.19343i −3.00000 −18.9718 15.3454i 15.5803i 23.0710i 56.9811i 9.00000 −79.6953
313.4 5.06578i −3.00000 −17.6621 10.9807i 15.1973i 24.6884i 48.9461i 9.00000 55.6257
313.5 4.67506i −3.00000 −13.8562 4.61864i 14.0252i 11.3073i 27.3780i 9.00000 −21.5924
313.6 4.30375i −3.00000 −10.5223 4.32730i 12.9113i 21.9616i 10.8553i 9.00000 −18.6236
313.7 4.28965i −3.00000 −10.4011 1.35162i 12.8690i 9.03182i 10.3000i 9.00000 −5.79797
313.8 3.78788i −3.00000 −6.34802 9.26420i 11.3636i 4.30898i 6.25748i 9.00000 −35.0916
313.9 3.68048i −3.00000 −5.54594 17.9346i 11.0414i 17.3085i 9.03212i 9.00000 66.0080
313.10 3.30421i −3.00000 −2.91779 13.7388i 9.91262i 33.0358i 16.7927i 9.00000 45.3958
313.11 3.18058i −3.00000 −2.11606 2.38612i 9.54173i 14.3272i 18.7143i 9.00000 −7.58923
313.12 2.64007i −3.00000 1.03005 13.5204i 7.92020i 34.8670i 23.8399i 9.00000 −35.6946
313.13 2.63265i −3.00000 1.06918 20.9406i 7.89794i 20.4039i 23.8759i 9.00000 −55.1291
313.14 2.09701i −3.00000 3.60255 1.98891i 6.29103i 9.73610i 24.3307i 9.00000 4.17077
313.15 1.46936i −3.00000 5.84098 4.94652i 4.40809i 12.8754i 20.3374i 9.00000 −7.26823
313.16 1.38921i −3.00000 6.07011 4.30021i 4.16762i 36.7534i 19.5463i 9.00000 5.97388
313.17 0.999086i −3.00000 7.00183 10.0353i 2.99726i 4.69571i 14.9881i 9.00000 10.0261
313.18 0.783695i −3.00000 7.38582 11.1397i 2.35108i 5.77258i 12.0578i 9.00000 8.73009
313.19 0.500423i −3.00000 7.74958 10.0481i 1.50127i 27.3201i 7.88145i 9.00000 5.02829
313.20 0.275605i −3.00000 7.92404 19.4532i 0.826816i 15.2471i 4.38875i 9.00000 −5.36141
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.4.b.a 40
157.b even 2 1 inner 471.4.b.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.4.b.a 40 1.a even 1 1 trivial
471.4.b.a 40 157.b even 2 1 inner