Properties

Label 471.4.a.c
Level $471$
Weight $4$
Character orbit 471.a
Self dual yes
Analytic conductor $27.790$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.7898996127\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99} + O(q^{100}) \)

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 −5.40228 −3.00000 21.1846 6.01793 16.2068 −1.25342 −71.2270 9.00000 −32.5106
1.2 −5.06198 −3.00000 17.6236 12.0952 15.1859 −27.2034 −48.7145 9.00000 −61.2257
1.3 −4.42661 −3.00000 11.5949 −9.85596 13.2798 −17.0380 −15.9132 9.00000 43.6285
1.4 −3.69988 −3.00000 5.68908 8.40381 11.0996 21.0620 8.55011 9.00000 −31.0931
1.5 −3.62090 −3.00000 5.11092 −9.00865 10.8627 5.02048 10.4611 9.00000 32.6194
1.6 −3.40822 −3.00000 3.61596 −8.48893 10.2247 10.4594 14.9418 9.00000 28.9321
1.7 −2.31678 −3.00000 −2.63252 −1.90474 6.95035 −14.6237 24.6332 9.00000 4.41286
1.8 −1.56491 −3.00000 −5.55104 8.64342 4.69474 19.5992 21.2062 9.00000 −13.5262
1.9 −1.38179 −3.00000 −6.09066 20.1959 4.14537 −9.60234 19.4703 9.00000 −27.9065
1.10 −1.22585 −3.00000 −6.49728 −2.10116 3.67756 35.0092 17.7716 9.00000 2.57572
1.11 0.196646 −3.00000 −7.96133 −2.23494 −0.589937 −24.0345 −3.13873 9.00000 −0.439492
1.12 1.29561 −3.00000 −6.32139 −17.0439 −3.88684 13.0789 −18.5550 9.00000 −22.0823
1.13 1.36089 −3.00000 −6.14798 17.6223 −4.08267 31.9947 −19.2538 9.00000 23.9820
1.14 1.51003 −3.00000 −5.71981 14.7843 −4.53009 −36.4705 −20.7173 9.00000 22.3248
1.15 2.29403 −3.00000 −2.73741 −14.4094 −6.88210 −20.5162 −24.6320 9.00000 −33.0556
1.16 2.43477 −3.00000 −2.07188 −12.7413 −7.30432 −29.4801 −24.5227 9.00000 −31.0221
1.17 2.99489 −3.00000 0.969360 5.29125 −8.98467 8.08775 −21.0560 9.00000 15.8467
1.18 4.33495 −3.00000 10.7918 −21.4542 −13.0048 −11.7470 12.1021 9.00000 −93.0027
1.19 4.50022 −3.00000 12.2520 17.0216 −13.5007 15.6710 19.1349 9.00000 76.6007
1.20 5.00169 −3.00000 17.0169 16.1971 −15.0051 −23.8262 45.0996 9.00000 81.0129
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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 471.4.a.c 22
3.b odd 2 1 1413.4.a.e 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.4.a.c 22 1.a even 1 1 trivial
1413.4.a.e 22 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{22} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(471))\).