Properties

Label 471.8.a.c
Level $471$
Weight $8$
Character orbit 471.a
Self dual yes
Analytic conductor $147.133$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(147.133347003\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + 6185 q^{10} + 11989 q^{11} - 86778 q^{12} - 2393 q^{13} + 18201 q^{14} - 11556 q^{15} + 208150 q^{16} + 62538 q^{17} + 1458 q^{18} - 39882 q^{19} + 113423 q^{20} + 18360 q^{21} - 93716 q^{22} + 195618 q^{23} - 63585 q^{24} + 886490 q^{25} + 294399 q^{26} - 944784 q^{27} - 60819 q^{28} + 421501 q^{29} - 166995 q^{30} + 392689 q^{31} - 341578 q^{32} - 323703 q^{33} + 50837 q^{34} + 697874 q^{35} + 2343006 q^{36} - 410396 q^{37} + 677216 q^{38} + 64611 q^{39} + 3232376 q^{40} + 3832958 q^{41} - 491427 q^{42} - 1751932 q^{43} + 4888297 q^{44} + 312012 q^{45} + 1163150 q^{46} + 106461 q^{47} - 5620050 q^{48} + 8202048 q^{49} - 2159111 q^{50} - 1688526 q^{51} - 3605030 q^{52} + 1755534 q^{53} - 39366 q^{54} - 1220729 q^{55} - 4430622 q^{56} + 1076814 q^{57} - 10000202 q^{58} - 2037752 q^{59} - 3062421 q^{60} + 1274098 q^{61} + 97748 q^{62} - 495720 q^{63} + 15135201 q^{64} + 6139645 q^{65} + 2530332 q^{66} - 7751257 q^{67} + 1700631 q^{68} - 5281686 q^{69} - 20935703 q^{70} - 12592217 q^{71} + 1716795 q^{72} + 12508355 q^{73} - 14999956 q^{74} - 23935230 q^{75} - 23946874 q^{76} + 1874177 q^{77} - 7948773 q^{78} - 5103480 q^{79} + 3128449 q^{80} + 25509168 q^{81} + 11622426 q^{82} + 3040643 q^{83} + 1642113 q^{84} - 13756076 q^{85} + 964635 q^{86} - 11380527 q^{87} - 29653500 q^{88} + 28462995 q^{89} + 4508865 q^{90} + 3016621 q^{91} + 22938254 q^{92} - 10602603 q^{93} - 10070348 q^{94} - 2579984 q^{95} + 9222606 q^{96} + 16208760 q^{97} + 6323227 q^{98} + 8739981 q^{99} + O(q^{100}) \)

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 −22.3612 −27.0000 372.021 −432.818 603.751 −1328.71 −5456.60 729.000 9678.32
1.2 −22.2958 −27.0000 369.104 234.355 601.987 1270.63 −5375.60 729.000 −5225.14
1.3 −20.6187 −27.0000 297.131 165.459 556.705 −1385.76 −3487.25 729.000 −3411.55
1.4 −19.9953 −27.0000 271.812 −231.869 539.873 342.382 −2875.57 729.000 4636.29
1.5 −18.2609 −27.0000 205.462 63.1999 493.045 499.385 −1414.52 729.000 −1154.09
1.6 −17.8012 −27.0000 188.881 −168.224 480.631 −317.259 −1083.76 729.000 2994.59
1.7 −16.8482 −27.0000 155.860 189.080 454.900 904.411 −469.396 729.000 −3185.64
1.8 −16.2410 −27.0000 135.769 −491.991 438.506 940.580 −126.175 729.000 7990.41
1.9 −16.0952 −27.0000 131.055 414.780 434.570 1067.44 −49.1712 729.000 −6675.96
1.10 −15.7094 −27.0000 118.786 126.488 424.154 1592.00 144.749 729.000 −1987.05
1.11 −14.4638 −27.0000 81.2016 346.642 390.523 −1595.50 676.883 729.000 −5013.77
1.12 −13.7948 −27.0000 62.2971 −136.436 372.460 −1182.74 906.360 729.000 1882.11
1.13 −12.5583 −27.0000 29.7115 553.645 339.075 −406.725 1234.34 729.000 −6952.86
1.14 −10.6994 −27.0000 −13.5238 −456.397 288.883 −1238.76 1514.21 729.000 4883.15
1.15 −9.95430 −27.0000 −28.9118 −469.613 268.766 −412.443 1561.95 729.000 4674.67
1.16 −9.40303 −27.0000 −39.5830 163.955 253.882 68.7987 1575.79 729.000 −1541.67
1.17 −8.95758 −27.0000 −47.7618 476.788 241.855 984.094 1574.40 729.000 −4270.87
1.18 −8.93486 −27.0000 −48.1683 7.15974 241.241 −664.570 1574.04 729.000 −63.9712
1.19 −6.65005 −27.0000 −83.7769 355.102 179.551 −1545.65 1408.33 729.000 −2361.45
1.20 −3.69501 −27.0000 −114.347 −349.671 99.7651 479.777 895.473 729.000 1292.04
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.48
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 471.8.a.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.8.a.c 48 1.a even 1 1 trivial