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Properties

Label 471.8.a.d
Level $471$
Weight $8$
Character orbit 471.a
Self dual yes
Analytic conductor $147.133$
Analytic rank $0$
Dimension $51$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [471,8,Mod(1,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(147.133347003\)
Analytic rank: \(0\)
Dimension: \(51\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q + 8 q^{2} + 1377 q^{3} + 3790 q^{4} + 1072 q^{5} + 216 q^{6} + 4122 q^{7} + 1035 q^{8} + 37179 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 51 q + 8 q^{2} + 1377 q^{3} + 3790 q^{4} + 1072 q^{5} + 216 q^{6} + 4122 q^{7} + 1035 q^{8} + 37179 q^{9} + 13273 q^{10} + 4097 q^{11} + 102330 q^{12} + 32759 q^{13} + 24393 q^{14} + 28944 q^{15} + 295614 q^{16} + 92804 q^{17} + 5832 q^{18} + 146770 q^{19} + 142567 q^{20} + 111294 q^{21} + 159896 q^{22} + 118058 q^{23} + 27945 q^{24} + 1074951 q^{25} + 151037 q^{26} + 1003833 q^{27} + 457305 q^{28} + 401147 q^{29} + 358371 q^{30} + 513321 q^{31} + 212148 q^{32} + 110619 q^{33} + 1302157 q^{34} + 787512 q^{35} + 2762910 q^{36} + 724064 q^{37} + 510776 q^{38} + 884493 q^{39} + 1282884 q^{40} + 841934 q^{41} + 658611 q^{42} + 2716746 q^{43} - 1581895 q^{44} + 781488 q^{45} + 1833346 q^{46} + 1279499 q^{47} + 7981578 q^{48} + 12150981 q^{49} + 2871013 q^{50} + 2505708 q^{51} + 6857246 q^{52} + 3563040 q^{53} + 157464 q^{54} + 6725405 q^{55} + 15451920 q^{56} + 3962790 q^{57} + 12567310 q^{58} + 7417490 q^{59} + 3849309 q^{60} + 14536020 q^{61} + 13157716 q^{62} + 3004938 q^{63} + 27018577 q^{64} + 9947799 q^{65} + 4317192 q^{66} + 12923543 q^{67} + 26798493 q^{68} + 3187566 q^{69} + 28005079 q^{70} + 15529331 q^{71} + 754515 q^{72} + 19492631 q^{73} + 26009386 q^{74} + 29023677 q^{75} + 37592954 q^{76} + 18505647 q^{77} + 4077999 q^{78} + 23768052 q^{79} + 39279447 q^{80} + 27103491 q^{81} + 21196574 q^{82} + 17755141 q^{83} + 12347235 q^{84} + 31023224 q^{85} + 20012797 q^{86} + 10830969 q^{87} + 39366240 q^{88} + 17161207 q^{89} + 9676017 q^{90} + 38583289 q^{91} + 26647832 q^{92} + 13859667 q^{93} + 77157324 q^{94} + 25249920 q^{95} + 5727996 q^{96} + 31771818 q^{97} + 41847673 q^{98} + 2986713 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −22.4819 27.0000 377.438 −172.901 −607.012 244.747 −5607.84 729.000 3887.14
1.2 −21.9952 27.0000 355.787 285.885 −593.869 −1251.96 −5010.21 729.000 −6288.08
1.3 −21.4136 27.0000 330.543 211.487 −578.168 −1254.17 −4337.18 729.000 −4528.71
1.4 −20.6359 27.0000 297.842 156.370 −557.170 1202.73 −3504.85 729.000 −3226.84
1.5 −19.4815 27.0000 251.528 −544.105 −526.000 44.0744 −2406.50 729.000 10600.0
1.6 −18.8744 27.0000 228.244 −342.090 −509.610 1713.62 −1892.05 729.000 6456.74
1.7 −18.5008 27.0000 214.280 98.5484 −499.522 −790.827 −1596.25 729.000 −1823.23
1.8 −16.1540 27.0000 132.951 126.620 −436.157 −586.528 −79.9733 729.000 −2045.41
1.9 −16.1271 27.0000 132.083 285.949 −435.432 1677.50 −65.8495 729.000 −4611.52
1.10 −15.9543 27.0000 126.539 544.758 −430.766 −375.555 23.3016 729.000 −8691.23
1.11 −14.5480 27.0000 83.6457 −382.100 −392.797 1005.88 645.269 729.000 5558.81
1.12 −13.6442 27.0000 58.1651 −214.857 −368.394 239.211 952.843 729.000 2931.56
1.13 −13.3597 27.0000 50.4822 47.7053 −360.712 69.0849 1035.62 729.000 −637.330
1.14 −13.2243 27.0000 46.8834 −132.653 −357.057 −1765.84 1072.71 729.000 1754.25
1.15 −12.3826 27.0000 25.3291 282.607 −334.331 778.426 1271.33 729.000 −3499.41
1.16 −11.2255 27.0000 −1.98717 −521.350 −303.090 −715.023 1459.18 729.000 5852.44
1.17 −10.6779 27.0000 −13.9815 −378.028 −288.305 303.576 1516.07 729.000 4036.56
1.18 −8.61465 27.0000 −53.7878 242.481 −232.596 −805.596 1566.04 729.000 −2088.89
1.19 −8.50745 27.0000 −55.6233 492.706 −229.701 1680.33 1562.17 729.000 −4191.67
1.20 −5.53591 27.0000 −97.3537 228.314 −149.470 345.995 1247.54 729.000 −1263.92
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
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.d ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.8.a.d ✓ 51 1.a even 1 1 trivial