Properties

Label 471.6.a.b
Level $471$
Weight $6$
Character orbit 471.a
Self dual yes
Analytic conductor $75.541$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.5407791319\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 8 q^{2} - 270 q^{3} + 470 q^{4} - 136 q^{5} + 72 q^{6} + 68 q^{7} - 261 q^{8} + 2430 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 8 q^{2} - 270 q^{3} + 470 q^{4} - 136 q^{5} + 72 q^{6} + 68 q^{7} - 261 q^{8} + 2430 q^{9} - 383 q^{10} - 875 q^{11} - 4230 q^{12} + 101 q^{13} - 2279 q^{14} + 1224 q^{15} + 7454 q^{16} - 4042 q^{17} - 648 q^{18} + 846 q^{19} - 5089 q^{20} - 612 q^{21} - 700 q^{22} - 5902 q^{23} + 2349 q^{24} + 12880 q^{25} - 7567 q^{26} - 21870 q^{27} - 375 q^{28} - 10301 q^{29} + 3447 q^{30} - 4099 q^{31} - 1560 q^{32} + 7875 q^{33} - 3683 q^{34} - 20686 q^{35} + 38070 q^{36} + 8468 q^{37} - 11848 q^{38} - 909 q^{39} - 5132 q^{40} - 47958 q^{41} + 20511 q^{42} + 63916 q^{43} + 3101 q^{44} - 11016 q^{45} + 19654 q^{46} + 8589 q^{47} - 67086 q^{48} + 27834 q^{49} + 121727 q^{50} + 36378 q^{51} + 56510 q^{52} + 10134 q^{53} + 5832 q^{54} - 11473 q^{55} - 68192 q^{56} - 7614 q^{57} + 32006 q^{58} - 64236 q^{59} + 45801 q^{60} - 98194 q^{61} - 67276 q^{62} + 5508 q^{63} + 138849 q^{64} - 155917 q^{65} + 6300 q^{66} + 62323 q^{67} - 117531 q^{68} + 53118 q^{69} - 220939 q^{70} - 179713 q^{71} - 21141 q^{72} - 148343 q^{73} - 214732 q^{74} - 115920 q^{75} - 189758 q^{76} - 142357 q^{77} + 68103 q^{78} + 26916 q^{79} - 463727 q^{80} + 196830 q^{81} - 206514 q^{82} - 89285 q^{83} + 3375 q^{84} - 23932 q^{85} - 477235 q^{86} + 92709 q^{87} - 114708 q^{88} - 474411 q^{89} - 31023 q^{90} + 51305 q^{91} - 1030074 q^{92} + 36891 q^{93} - 485800 q^{94} - 169960 q^{95} + 14040 q^{96} - 169188 q^{97} - 629739 q^{98} - 70875 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.3107 −9.00000 95.9310 17.2447 101.796 −21.0168 −723.101 81.0000 −195.049
1.2 −10.5389 −9.00000 79.0692 −79.1689 94.8504 161.660 −496.059 81.0000 834.356
1.3 −9.78552 −9.00000 63.7563 53.0529 88.0696 −2.88714 −310.752 81.0000 −519.150
1.4 −9.06695 −9.00000 50.2095 −0.166632 81.6025 −235.849 −165.105 81.0000 1.51084
1.5 −8.01065 −9.00000 32.1705 −50.4239 72.0959 130.088 −1.36618 81.0000 403.929
1.6 −7.62164 −9.00000 26.0894 −4.86745 68.5948 215.306 45.0481 81.0000 37.0980
1.7 −7.13980 −9.00000 18.9767 54.3556 64.2582 −91.4021 92.9836 81.0000 −388.088
1.8 −6.95119 −9.00000 16.3190 −101.673 62.5607 −73.4089 109.002 81.0000 706.746
1.9 −6.49004 −9.00000 10.1206 53.7142 58.4103 −12.0983 141.998 81.0000 −348.607
1.10 −5.25286 −9.00000 −4.40749 14.5253 47.2757 156.395 191.243 81.0000 −76.2993
1.11 −4.96468 −9.00000 −7.35194 −22.5622 44.6821 −186.074 195.370 81.0000 112.014
1.12 −4.55243 −9.00000 −11.2754 65.2180 40.9718 201.026 197.008 81.0000 −296.900
1.13 −1.89378 −9.00000 −28.4136 13.1753 17.0440 −20.4742 114.410 81.0000 −24.9510
1.14 −1.04956 −9.00000 −30.8984 80.7800 9.44602 79.1448 66.0156 81.0000 −84.7833
1.15 0.00358760 −9.00000 −32.0000 −98.0149 −0.0322884 −183.236 −0.229607 81.0000 −0.351639
1.16 0.614444 −9.00000 −31.6225 −56.4863 −5.52999 −140.573 −39.0924 81.0000 −34.7077
1.17 0.892048 −9.00000 −31.2043 −60.6723 −8.02843 242.261 −56.3812 81.0000 −54.1226
1.18 1.56244 −9.00000 −29.5588 49.5691 −14.0620 −144.113 −96.1822 81.0000 77.4490
1.19 2.78050 −9.00000 −24.2688 24.4329 −25.0245 −110.716 −156.455 81.0000 67.9358
1.20 3.44137 −9.00000 −20.1570 −70.8289 −30.9724 −20.9998 −179.492 81.0000 −243.749
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(157\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.6.a.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.6.a.b 30 1.a even 1 1 trivial