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Properties

Label 471.6.a.a
Level $471$
Weight $6$
Character orbit 471.a
Self dual yes
Analytic conductor $75.541$
Analytic rank $1$
Dimension $27$
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] = [471,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(75.5407791319\)
Analytic rank: \(1\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 2 q^{2} + 243 q^{3} + 326 q^{4} - 164 q^{5} - 18 q^{6} - 618 q^{7} - 429 q^{8} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 2 q^{2} + 243 q^{3} + 326 q^{4} - 164 q^{5} - 18 q^{6} - 618 q^{7} - 429 q^{8} + 2187 q^{9} - 919 q^{10} - 151 q^{11} + 2934 q^{12} - 2603 q^{13} - 2783 q^{14} - 1476 q^{15} + 1110 q^{16} - 4484 q^{17} - 162 q^{18} - 6886 q^{19} - 6521 q^{20} - 5562 q^{21} - 12272 q^{22} - 4358 q^{23} - 3861 q^{24} + 1851 q^{25} - 2285 q^{26} + 19683 q^{27} - 19539 q^{28} - 21211 q^{29} - 8271 q^{30} - 16059 q^{31} - 11390 q^{32} - 1359 q^{33} - 22387 q^{34} - 20832 q^{35} + 26406 q^{36} - 14312 q^{37} - 10240 q^{38} - 23427 q^{39} - 64248 q^{40} - 71094 q^{41} - 25047 q^{42} - 97878 q^{43} - 101003 q^{44} - 13284 q^{45} - 79606 q^{46} - 31701 q^{47} + 9990 q^{48} - 39867 q^{49} - 97069 q^{50} - 40356 q^{51} - 112598 q^{52} - 54172 q^{53} - 1458 q^{54} - 116987 q^{55} - 142462 q^{56} - 61974 q^{57} - 102538 q^{58} - 23106 q^{59} - 58689 q^{60} - 165996 q^{61} - 138740 q^{62} - 50058 q^{63} - 100687 q^{64} - 65639 q^{65} - 110448 q^{66} - 199901 q^{67} - 99089 q^{68} - 39222 q^{69} + 13931 q^{70} + 7083 q^{71} - 34749 q^{72} - 236027 q^{73} - 36902 q^{74} + 16659 q^{75} - 29330 q^{76} - 33147 q^{77} - 20565 q^{78} - 229104 q^{79} + 74071 q^{80} + 177147 q^{81} - 77462 q^{82} - 118611 q^{83} - 175851 q^{84} - 189968 q^{85} + 276867 q^{86} - 190899 q^{87} - 215496 q^{88} - 111327 q^{89} - 74439 q^{90} - 223771 q^{91} + 206144 q^{92} - 144531 q^{93} - 193272 q^{94} - 142056 q^{95} - 102510 q^{96} - 217878 q^{97} + 354571 q^{98} - 12231 q^{99}+O(q^{100}) \) 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 −10.9519 9.00000 87.9443 19.9896 −98.5672 16.2092 −612.697 81.0000 −218.924
1.2 −9.94794 9.00000 66.9616 77.1766 −89.5315 −149.708 −347.796 81.0000 −767.749
1.3 −8.69990 9.00000 43.6883 −37.8040 −78.2991 −58.6678 −101.687 81.0000 328.891
1.4 −8.49259 9.00000 40.1241 16.5911 −76.4333 180.869 −68.9945 81.0000 −140.901
1.5 −8.30759 9.00000 37.0161 −69.8032 −74.7684 108.675 −41.6720 81.0000 579.897
1.6 −7.99822 9.00000 31.9715 −103.345 −71.9840 −82.7918 0.227656 81.0000 826.577
1.7 −6.44054 9.00000 9.48054 3.52728 −57.9649 69.8863 145.037 81.0000 −22.7176
1.8 −4.86861 9.00000 −8.29667 70.0251 −43.8175 −29.0084 196.189 81.0000 −340.925
1.9 −4.72892 9.00000 −9.63728 91.9359 −42.5603 −136.740 196.900 81.0000 −434.758
1.10 −3.93695 9.00000 −16.5004 −8.11615 −35.4326 129.679 190.944 81.0000 31.9529
1.11 −3.46452 9.00000 −19.9971 −86.2578 −31.1807 −195.157 180.145 81.0000 298.842
1.12 −1.52459 9.00000 −29.6756 −19.7284 −13.7213 −187.198 94.0300 81.0000 30.0777
1.13 −0.217757 9.00000 −31.9526 −63.4098 −1.95981 63.3063 13.9261 81.0000 13.8079
1.14 0.189866 9.00000 −31.9640 37.0724 1.70880 122.962 −12.1446 81.0000 7.03881
1.15 0.832955 9.00000 −31.3062 24.7681 7.49660 −125.118 −52.7312 81.0000 20.6307
1.16 2.37743 9.00000 −26.3478 74.5980 21.3969 −112.358 −138.718 81.0000 177.352
1.17 2.85439 9.00000 −23.8525 −97.6990 25.6895 −10.2384 −159.425 81.0000 −278.871
1.18 3.16748 9.00000 −21.9671 −37.3170 28.5073 177.291 −170.940 81.0000 −118.201
1.19 4.36472 9.00000 −12.9493 7.56362 39.2824 −34.6663 −196.191 81.0000 33.0131
1.20 5.38317 9.00000 −3.02149 53.0580 48.4485 107.261 −188.527 81.0000 285.620
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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.a ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.6.a.a ✓ 27 1.a even 1 1 trivial