Properties

Label 572.2.s.a
Level $572$
Weight $2$
Character orbit 572.s
Analytic conductor $4.567$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160q - 2q^{4} + 68q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q - 2q^{4} + 68q^{9} - 36q^{12} - 8q^{14} - 6q^{16} + 30q^{20} - 20q^{22} - 144q^{25} - 40q^{26} - 6q^{33} + 46q^{36} - 12q^{37} - 56q^{38} - 2q^{42} + 48q^{45} - 40q^{48} + 44q^{49} - 32q^{53} + 30q^{56} + 78q^{58} + 64q^{64} - 4q^{66} - 44q^{69} - 44q^{77} + 16q^{78} + 24q^{80} - 40q^{81} - 6q^{82} - 10q^{88} + 24q^{89} - 24q^{92} - 96q^{93} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41333 + 0.0501117i 1.83041 + 1.05679i 1.99498 0.141648i 1.43229i −2.63992 1.40186i 1.20221 0.694098i −2.81245 + 0.300167i 0.733590 + 1.27062i 0.0717746 + 2.02430i
43.2 −1.41298 + 0.0591092i −2.54803 1.47111i 1.99301 0.167040i 3.95496i 3.68727 + 1.92803i −2.41626 + 1.39503i −2.80621 + 0.353829i 2.82830 + 4.89877i 0.233775 + 5.58827i
43.3 −1.40920 0.118969i −2.07307 1.19689i 1.97169 + 0.335302i 0.562219i 2.77898 + 1.93329i −0.112468 + 0.0649336i −2.73862 0.707079i 1.36509 + 2.36441i 0.0668866 0.792279i
43.4 −1.40794 0.133100i 1.15080 + 0.664413i 1.96457 + 0.374794i 4.34420i −1.53182 1.08862i −0.561519 + 0.324193i −2.71610 0.789171i −0.617110 1.06887i 0.578214 6.11635i
43.5 −1.40175 + 0.187369i 0.461021 + 0.266170i 1.92979 0.525287i 0.718470i −0.696106 0.286722i −3.26467 + 1.88486i −2.60665 + 1.09790i −1.35831 2.35266i 0.134619 + 1.00711i
43.6 −1.37518 + 0.329974i −0.622539 0.359423i 1.78223 0.907548i 2.91525i 0.974703 + 0.288849i 4.00752 2.31374i −2.15142 + 1.83613i −1.24163 2.15057i 0.961958 + 4.00899i
43.7 −1.36236 + 0.379439i −2.49055 1.43792i 1.71205 1.03387i 3.07981i 3.93863 + 1.01395i 1.00977 0.582994i −1.94014 + 2.05812i 2.63522 + 4.56434i −1.16860 4.19581i
43.8 −1.35805 0.394577i −1.01294 0.584824i 1.68862 + 1.07171i 1.81934i 1.14487 + 1.19391i −2.82002 + 1.62814i −1.87036 2.12173i −0.815963 1.41329i 0.717869 2.47076i
43.9 −1.32499 + 0.494359i −0.498527 0.287825i 1.51122 1.31004i 1.29030i 0.802835 + 0.134915i 0.822321 0.474767i −1.35473 + 2.48289i −1.33431 2.31110i −0.637872 1.70964i
43.10 −1.31800 0.512708i 2.39320 + 1.38172i 1.47426 + 1.35150i 1.60862i −2.44583 3.04812i 0.824322 0.475923i −1.25015 2.53715i 2.31828 + 4.01537i −0.824753 + 2.12017i
43.11 −1.30032 0.556038i 0.593637 + 0.342737i 1.38164 + 1.44605i 3.22106i −0.581342 0.775751i −1.12143 + 0.647461i −0.992516 2.64857i −1.26506 2.19115i −1.79103 + 4.18839i
43.12 −1.27678 0.608129i −1.14865 0.663171i 1.26036 + 1.55290i 1.28693i 1.06328 + 1.54525i 3.60145 2.07930i −0.664840 2.74918i −0.620407 1.07458i 0.782619 1.64313i
43.13 −1.21210 + 0.728568i 2.61335 + 1.50882i 0.938379 1.76620i 0.755146i −4.26692 + 0.0751615i 2.13122 1.23046i 0.149383 + 2.82448i 3.05306 + 5.28806i −0.550175 0.915314i
43.14 −1.20341 + 0.742836i −0.371524 0.214500i 0.896390 1.78787i 0.666498i 0.606434 0.0178505i −1.08520 + 0.626541i 0.249370 + 2.81741i −1.40798 2.43869i 0.495098 + 0.802070i
43.15 −1.20263 + 0.744103i 1.90164 + 1.09791i 0.892622 1.78976i 2.75390i −3.10392 + 0.0946368i −1.87485 + 1.08245i 0.258271 + 2.81661i 0.910822 + 1.57759i −2.04919 3.31192i
43.16 −1.16505 0.801663i 1.14865 + 0.663171i 0.714672 + 1.86795i 1.28693i −0.806588 1.69345i 3.60145 2.07930i 0.664840 2.74918i −0.620407 1.07458i 1.03168 1.49933i
43.17 −1.13170 0.848088i −0.593637 0.342737i 0.561494 + 1.91956i 3.22106i 0.381149 + 0.891332i −1.12143 + 0.647461i 0.992516 2.64857i −1.26506 2.19115i −2.73174 + 3.64528i
43.18 −1.10302 0.885069i −2.39320 1.38172i 0.433305 + 1.95250i 1.60862i 1.41683 + 3.64221i 0.824322 0.475923i 1.25015 2.53715i 2.31828 + 4.01537i −1.42374 + 1.77434i
43.19 −1.02074 0.978820i 1.01294 + 0.584824i 0.0838224 + 1.99824i 1.81934i −0.461516 1.58844i −2.82002 + 1.62814i 1.87036 2.12173i −0.815963 1.41329i 1.78080 1.85707i
43.20 −1.00832 + 0.991613i 1.38394 + 0.799017i 0.0334074 1.99972i 3.90803i −2.18776 + 0.566668i 1.87027 1.07980i 1.94926 + 2.04948i −0.223145 0.386499i 3.87525 + 3.94054i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
13.e even 6 1 inner
44.c even 2 1 inner
52.i odd 6 1 inner
143.i odd 6 1 inner
572.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.s.a 160
4.b odd 2 1 inner 572.2.s.a 160
11.b odd 2 1 inner 572.2.s.a 160
13.e even 6 1 inner 572.2.s.a 160
44.c even 2 1 inner 572.2.s.a 160
52.i odd 6 1 inner 572.2.s.a 160
143.i odd 6 1 inner 572.2.s.a 160
572.s even 6 1 inner 572.2.s.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.s.a 160 1.a even 1 1 trivial
572.2.s.a 160 4.b odd 2 1 inner
572.2.s.a 160 11.b odd 2 1 inner
572.2.s.a 160 13.e even 6 1 inner
572.2.s.a 160 44.c even 2 1 inner
572.2.s.a 160 52.i odd 6 1 inner
572.2.s.a 160 143.i odd 6 1 inner
572.2.s.a 160 572.s even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).