Properties

Label 471.2.s.a
Level $471$
Weight $2$
Character orbit 471.s
Analytic conductor $3.761$
Analytic rank $0$
Dimension $1200$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(1200\)
Relative dimension: \(50\) over \(\Q(\zeta_{52})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{52}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9} - 52 q^{10} - 80 q^{12} - 12 q^{15} + 16 q^{16} + 12 q^{18} - 52 q^{19} - 26 q^{21} - 64 q^{22} + 18 q^{24} + 104 q^{25} - 26 q^{27} - 84 q^{28} - 34 q^{30} - 52 q^{31} - 26 q^{33} - 80 q^{34} - 26 q^{36} - 308 q^{37} + 52 q^{39} - 340 q^{40} - 26 q^{42} - 76 q^{43} - 52 q^{45} - 60 q^{46} - 130 q^{48} - 52 q^{49} - 26 q^{51} + 84 q^{52} - 246 q^{54} - 36 q^{55} - 26 q^{57} - 52 q^{58} + 14 q^{60} - 4 q^{61} - 46 q^{63} - 52 q^{64} - 22 q^{66} + 28 q^{67} - 72 q^{69} - 40 q^{70} + 46 q^{72} - 48 q^{73} - 46 q^{75} - 52 q^{76} + 146 q^{78} - 100 q^{79} + 58 q^{81} - 52 q^{82} - 156 q^{84} - 20 q^{85} + 12 q^{87} - 116 q^{88} - 26 q^{90} + 360 q^{91} - 310 q^{93} + 96 q^{94} + 176 q^{96} + 116 q^{97} - 66 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.64211 + 0.484185i 0.497232 1.65914i 4.87629 1.84933i −0.627219 2.01282i −0.510409 + 4.62440i 0.231121 0.104019i −7.39084 + 4.46792i −2.50552 1.64996i 2.63176 + 5.01439i
2.2 −2.63103 + 0.482154i −1.15550 1.29028i 4.81980 1.82791i 1.16001 + 3.72259i 3.66226 + 2.83763i −2.99849 + 1.34951i −7.22155 + 4.36558i −0.329644 + 2.98183i −4.84687 9.23493i
2.3 −2.58276 + 0.473308i −0.893751 + 1.48365i 4.57659 1.73567i −0.115271 0.369916i 1.60612 4.25492i −0.994360 + 0.447525i −6.50457 + 3.93215i −1.40242 2.65202i 0.472800 + 0.900846i
2.4 −2.31579 + 0.424384i 1.59880 0.666207i 3.31274 1.25636i −0.0565505 0.181477i −3.41976 + 2.22130i −2.19753 + 0.989028i −3.10880 + 1.87934i 2.11234 2.13027i 0.207975 + 0.396263i
2.5 −2.27735 + 0.417339i 1.73033 0.0771921i 3.14210 1.19164i 0.837635 + 2.68807i −3.90835 + 0.897928i 1.67927 0.755779i −2.69561 + 1.62955i 2.98808 0.267135i −3.02942 5.77208i
2.6 −2.25488 + 0.413222i −1.68169 + 0.414612i 3.04369 1.15432i 0.325864 + 1.04573i 3.62069 1.62981i 2.00200 0.901027i −2.46252 + 1.48865i 2.65619 1.39450i −1.16690 2.22335i
2.7 −2.20517 + 0.404112i 1.03114 + 1.39167i 2.82943 1.07306i −0.562752 1.80593i −2.83623 2.65217i −2.85622 + 1.28548i −1.96860 + 1.19006i −0.873498 + 2.87002i 1.97076 + 3.75497i
2.8 −2.14879 + 0.393780i −1.33115 1.10817i 2.59219 0.983089i −0.374862 1.20298i 3.29674 + 1.85703i 3.78902 1.70530i −1.44392 + 0.872879i 0.543937 + 2.95028i 1.27921 + 2.43733i
2.9 −1.89926 + 0.348052i 0.522736 + 1.65129i 1.61602 0.612874i 0.859972 + 2.75975i −1.56755 2.95428i −0.504691 + 0.227143i 0.448912 0.271377i −2.45349 + 1.72637i −2.59385 4.94216i
2.10 −1.87532 + 0.343664i −0.909746 1.47389i 1.52867 0.579748i −0.603303 1.93607i 2.21258 + 2.45137i −3.07153 + 1.38238i 0.595672 0.360097i −1.34473 + 2.68174i 1.79674 + 3.42340i
2.11 −1.76476 + 0.323404i −0.748594 + 1.56192i 1.13976 0.432253i −1.29547 4.15732i 0.815956 2.99852i 1.29997 0.585071i 1.19919 0.724937i −1.87921 2.33849i 3.63069 + 6.91770i
2.12 −1.45245 + 0.266172i −1.71334 + 0.253888i 0.168739 0.0639944i −0.893202 2.86638i 2.42097 0.824803i −2.00969 + 0.904488i 2.29931 1.38998i 2.87108 0.869993i 2.06028 + 3.92554i
2.13 −1.36706 + 0.250523i 1.10695 1.33216i −0.0639385 + 0.0242487i −0.772673 2.47960i −1.17953 + 2.09846i 2.24254 1.00929i 2.46011 1.48719i −0.549313 2.94928i 1.67749 + 3.19619i
2.14 −1.34852 + 0.247125i 1.12953 1.31307i −0.112603 + 0.0427047i 0.441900 + 1.41811i −1.19870 + 2.04984i −4.30939 + 1.93950i 2.48780 1.50393i −0.448318 2.96631i −0.946359 1.80314i
2.15 −1.22356 + 0.224226i 1.58808 + 0.691369i −0.423212 + 0.160503i −0.208958 0.670569i −2.09814 0.489842i 2.87237 1.29275i 2.61091 1.57835i 2.04402 + 2.19590i 0.406031 + 0.773628i
2.16 −1.17172 + 0.214726i −0.496480 1.65937i −0.543213 + 0.206013i 0.486414 + 1.56096i 0.938044 + 1.83771i 0.350360 0.157684i 2.63112 1.59057i −2.50702 + 1.64769i −0.905119 1.72456i
2.17 −1.16594 + 0.213666i −0.896153 + 1.48220i −0.556274 + 0.210967i 0.649823 + 2.08535i 0.728163 1.91963i 3.50250 1.57635i 2.63231 1.59129i −1.39382 2.65655i −1.20322 2.29255i
2.18 −1.12589 + 0.206326i −1.71242 0.260028i −0.644984 + 0.244610i 0.953289 + 3.05921i 1.98164 0.0605555i −0.513805 + 0.231245i 2.63482 1.59281i 2.86477 + 0.890555i −1.70449 3.24764i
2.19 −0.932577 + 0.170901i 1.58987 + 0.687239i −1.02954 + 0.390453i −0.457736 1.46893i −1.60013 0.369192i −2.14157 + 0.963841i 2.51614 1.52106i 2.05540 + 2.18525i 0.677916 + 1.29166i
2.20 −0.467978 + 0.0857602i −0.751208 + 1.56067i −1.65838 + 0.628942i −0.232282 0.745421i 0.217706 0.794782i −3.20493 + 1.44242i 1.53646 0.928823i −1.87137 2.34477i 0.172631 + 0.328920i
See next 80 embeddings (of 1200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 464.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
157.j odd 52 1 inner
471.s even 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.s.a 1200
3.b odd 2 1 inner 471.2.s.a 1200
157.j odd 52 1 inner 471.2.s.a 1200
471.s even 52 1 inner 471.2.s.a 1200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.s.a 1200 1.a even 1 1 trivial
471.2.s.a 1200 3.b odd 2 1 inner
471.2.s.a 1200 157.j odd 52 1 inner
471.2.s.a 1200 471.s even 52 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(471, [\chi])\).