Properties

Label 471.2.e.c
Level $471$
Weight $2$
Character orbit 471.e
Analytic conductor $3.761$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 14 q^{3} + 26 q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - 14 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 14 q^{3} + 26 q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - 14 q^{9} - q^{10} + 2 q^{11} - 13 q^{12} - 14 q^{14} - 2 q^{15} + 14 q^{16} + 3 q^{17} - q^{18} - 11 q^{19} + q^{20} - 2 q^{21} - 2 q^{22} - 6 q^{23} - 20 q^{25} + 12 q^{26} + 28 q^{27} + 2 q^{28} - 10 q^{29} - q^{30} - 14 q^{31} + 12 q^{32} + 2 q^{33} - 15 q^{34} - 13 q^{35} - 13 q^{36} - q^{37} + 18 q^{38} - 8 q^{40} + 16 q^{41} + 7 q^{42} - 16 q^{43} + 12 q^{44} + 4 q^{45} + 16 q^{46} - 29 q^{47} - 7 q^{48} + 44 q^{49} - 29 q^{50} + 3 q^{51} + 27 q^{52} + 6 q^{53} + 2 q^{54} - 29 q^{55} - 26 q^{56} - 11 q^{57} + 62 q^{58} - 8 q^{59} + q^{60} + 14 q^{61} + 52 q^{62} - 2 q^{63} + 16 q^{64} - 52 q^{65} - 2 q^{66} + 36 q^{67} + 6 q^{68} + 3 q^{69} - 37 q^{70} - 27 q^{71} - 10 q^{73} - 5 q^{74} + 40 q^{75} - 43 q^{76} + 11 q^{77} - 24 q^{78} - 16 q^{79} - 26 q^{80} - 14 q^{81} - 24 q^{82} + 12 q^{83} - q^{84} + 12 q^{85} - 56 q^{86} + 5 q^{87} + 47 q^{88} - 13 q^{89} + 2 q^{90} + 13 q^{91} - 50 q^{92} + 28 q^{93} - 40 q^{94} - 7 q^{95} - 6 q^{96} + 6 q^{97} - 46 q^{98} - 4 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 −2.59656 −0.500000 0.866025i 4.74215 −0.824270 1.42768i 1.29828 + 2.24869i 0.523535 −7.12017 −0.500000 + 0.866025i 2.14027 + 3.70706i
169.2 −2.26155 −0.500000 0.866025i 3.11460 1.51133 + 2.61769i 1.13077 + 1.95856i 4.51481 −2.52072 −0.500000 + 0.866025i −3.41793 5.92004i
169.3 −2.03783 −0.500000 0.866025i 2.15276 −0.0788428 0.136560i 1.01892 + 1.76481i −4.73558 −0.311299 −0.500000 + 0.866025i 0.160668 + 0.278286i
169.4 −1.21741 −0.500000 0.866025i −0.517920 −1.98968 3.44622i 0.608704 + 1.05431i 2.72799 3.06533 −0.500000 + 0.866025i 2.42225 + 4.19545i
169.5 −1.15467 −0.500000 0.866025i −0.666729 1.08032 + 1.87116i 0.577337 + 0.999977i −1.15933 3.07920 −0.500000 + 0.866025i −1.24741 2.16058i
169.6 −0.907528 −0.500000 0.866025i −1.17639 −0.0276317 0.0478596i 0.453764 + 0.785942i 1.18963 2.88267 −0.500000 + 0.866025i 0.0250766 + 0.0434339i
169.7 0.202404 −0.500000 0.866025i −1.95903 −1.48847 2.57810i −0.101202 0.175287i −1.77135 −0.801323 −0.500000 + 0.866025i −0.301271 0.521817i
169.8 0.253400 −0.500000 0.866025i −1.93579 0.879434 + 1.52322i −0.126700 0.219451i 1.78667 −0.997330 −0.500000 + 0.866025i 0.222849 + 0.385986i
169.9 1.07617 −0.500000 0.866025i −0.841867 −1.37194 2.37626i −0.538083 0.931987i 3.41476 −3.05832 −0.500000 + 0.866025i −1.47643 2.55725i
169.10 1.10791 −0.500000 0.866025i −0.772541 1.74720 + 3.02624i −0.553954 0.959476i −4.63545 −3.07172 −0.500000 + 0.866025i 1.93574 + 3.35280i
169.11 1.66438 −0.500000 0.866025i 0.770154 −0.823214 1.42585i −0.832189 1.44139i −2.35464 −2.04693 −0.500000 + 0.866025i −1.37014 2.37315i
169.12 1.80741 −0.500000 0.866025i 1.26673 0.630568 + 1.09218i −0.903705 1.56526i 4.30661 −1.32531 −0.500000 + 0.866025i 1.13970 + 1.97401i
169.13 2.49732 −0.500000 0.866025i 4.23662 1.51688 + 2.62731i −1.24866 2.16274i −1.77348 5.58556 −0.500000 + 0.866025i 3.78813 + 6.56123i
169.14 2.56656 −0.500000 0.866025i 4.58726 −1.76168 3.05133i −1.28328 2.22271i −0.0341584 6.64036 −0.500000 + 0.866025i −4.52148 7.83143i
301.1 −2.59656 −0.500000 + 0.866025i 4.74215 −0.824270 + 1.42768i 1.29828 2.24869i 0.523535 −7.12017 −0.500000 0.866025i 2.14027 3.70706i
301.2 −2.26155 −0.500000 + 0.866025i 3.11460 1.51133 2.61769i 1.13077 1.95856i 4.51481 −2.52072 −0.500000 0.866025i −3.41793 + 5.92004i
301.3 −2.03783 −0.500000 + 0.866025i 2.15276 −0.0788428 + 0.136560i 1.01892 1.76481i −4.73558 −0.311299 −0.500000 0.866025i 0.160668 0.278286i
301.4 −1.21741 −0.500000 + 0.866025i −0.517920 −1.98968 + 3.44622i 0.608704 1.05431i 2.72799 3.06533 −0.500000 0.866025i 2.42225 4.19545i
301.5 −1.15467 −0.500000 + 0.866025i −0.666729 1.08032 1.87116i 0.577337 0.999977i −1.15933 3.07920 −0.500000 0.866025i −1.24741 + 2.16058i
301.6 −0.907528 −0.500000 + 0.866025i −1.17639 −0.0276317 + 0.0478596i 0.453764 0.785942i 1.18963 2.88267 −0.500000 0.866025i 0.0250766 0.0434339i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.e.c 28
157.c even 3 1 inner 471.2.e.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.e.c 28 1.a even 1 1 trivial
471.2.e.c 28 157.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{14} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\).