Properties

Label 441.2.y.a
Level $441$
Weight $2$
Character orbit 441.y
Analytic conductor $3.521$
Analytic rank $0$
Dimension $648$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(648\)
Relative dimension: \(54\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648q - 5q^{2} - 13q^{3} - 109q^{4} - 12q^{5} + 2q^{6} - 7q^{7} - 16q^{8} - 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 648q - 5q^{2} - 13q^{3} - 109q^{4} - 12q^{5} + 2q^{6} - 7q^{7} - 16q^{8} - 3q^{9} - 22q^{10} - 8q^{11} - 64q^{12} - 4q^{13} - 26q^{14} - 4q^{15} - 97q^{16} - 37q^{17} - q^{18} - 14q^{19} - 11q^{20} - 33q^{21} - q^{22} + 31q^{23} - 20q^{24} + 38q^{25} - 44q^{26} - 49q^{27} - 22q^{28} + 73q^{29} - 12q^{30} - 20q^{31} - q^{32} - 43q^{33} - 7q^{34} - 32q^{35} + 20q^{36} - 39q^{37} - 40q^{38} + q^{39} - q^{40} - 17q^{41} + 82q^{42} - q^{43} - 31q^{44} - 177q^{45} + 56q^{46} + 5q^{47} - 23q^{48} - q^{49} - 21q^{50} - 18q^{51} - 48q^{52} + 50q^{53} - 15q^{54} + 20q^{55} + 127q^{56} + 33q^{57} - 82q^{58} + 53q^{59} + 43q^{60} + 42q^{61} - 30q^{62} + 57q^{63} - 88q^{64} - 11q^{65} - 90q^{66} - 26q^{67} + 145q^{68} + 64q^{69} - 46q^{70} - 19q^{71} - 46q^{72} - 40q^{73} + 18q^{74} - 114q^{75} + 43q^{76} + 7q^{77} - 129q^{78} - 26q^{79} + 163q^{80} + 85q^{81} - 28q^{82} - 221q^{83} - 113q^{84} - 10q^{85} - 18q^{86} + 37q^{87} - 4q^{88} - 104q^{89} - 60q^{90} - 49q^{91} + 119q^{92} - 149q^{93} - 13q^{94} + 37q^{95} + 186q^{96} - 11q^{97} - 33q^{98} + 62q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.620557 + 2.71884i −1.73004 + 0.0835206i −5.20506 2.50662i −0.446610 1.13794i 0.846508 4.75552i 2.43233 + 1.04104i 6.56762 8.23553i 2.98605 0.288987i 3.37104 0.508102i
25.2 −0.603411 + 2.64372i 0.233708 + 1.71621i −4.82320 2.32273i −0.317680 0.809436i −4.67820 0.417724i −2.63556 0.231947i 5.66957 7.10941i −2.89076 + 0.802183i 2.33161 0.351434i
25.3 −0.596104 + 2.61170i 0.768590 1.55218i −4.66370 2.24592i 1.23942 + 3.15800i 3.59567 + 2.93259i 1.25551 + 2.32888i 5.30523 6.65255i −1.81854 2.38598i −8.98657 + 1.35451i
25.4 −0.533217 + 2.33617i 1.50562 + 0.856211i −3.37146 1.62361i 0.810456 + 2.06501i −2.80308 + 3.06086i 1.78727 1.95082i 2.60267 3.26364i 1.53381 + 2.57826i −5.25637 + 0.792271i
25.5 −0.524991 + 2.30014i −1.18605 1.26225i −3.21307 1.54733i 1.00605 + 2.56336i 3.52602 2.06541i −1.26982 2.32111i 2.30393 2.88904i −0.186564 + 2.99419i −6.42425 + 0.968300i
25.6 −0.524692 + 2.29882i 1.63349 + 0.575929i −3.20735 1.54458i −1.57024 4.00089i −2.18104 + 3.45293i 2.51000 + 0.836609i 2.29328 2.87568i 2.33661 + 1.88155i 10.0212 1.51046i
25.7 −0.523139 + 2.29202i 1.32112 1.12011i −3.17776 1.53033i −0.871294 2.22002i 1.87618 + 3.61401i −2.54574 + 0.720548i 2.23835 2.80680i 0.490715 2.95959i 5.54415 0.835646i
25.8 −0.502908 + 2.20339i −0.976404 1.43061i −2.80006 1.34844i −0.712094 1.81439i 3.64322 1.43193i −2.27396 + 1.35244i 1.56106 1.95751i −1.09327 + 2.79370i 4.35591 0.656548i
25.9 −0.474737 + 2.07996i −1.49636 + 0.872292i −2.29891 1.10710i −0.320939 0.817739i −1.10395 3.52648i −0.354961 2.62183i 0.733726 0.920064i 1.47821 2.61053i 1.85322 0.279328i
25.10 −0.464927 + 2.03698i 0.999043 1.41489i −2.13119 1.02633i −0.231038 0.588676i 2.41762 + 2.69285i 1.57123 2.12867i 0.476059 0.596959i −1.00382 2.82707i 1.30654 0.196929i
25.11 −0.436037 + 1.91040i 0.660299 + 1.60125i −1.65757 0.798243i 0.532974 + 1.35800i −3.34695 + 0.563231i 0.00125911 + 2.64575i −0.195769 + 0.245486i −2.12801 + 2.11461i −2.82672 + 0.426059i
25.12 −0.416291 + 1.82389i −0.611111 + 1.62066i −1.35134 0.650771i −0.0698517 0.177979i −2.70151 1.78927i 2.35192 + 1.21181i −0.583357 + 0.731506i −2.25309 1.98081i 0.353693 0.0533107i
25.13 −0.395139 + 1.73122i 1.72082 + 0.196905i −1.03904 0.500374i 1.06496 + 2.71348i −1.02085 + 2.90131i −2.62846 + 0.301952i −0.937490 + 1.17557i 2.92246 + 0.677677i −5.11842 + 0.771478i
25.14 −0.322970 + 1.41503i −1.49665 0.871802i −0.0960492 0.0462548i −1.42246 3.62437i 1.71700 1.83623i 1.71510 2.01455i −1.71341 + 2.14855i 1.47992 + 2.60956i 5.58799 0.842254i
25.15 −0.316624 + 1.38722i −1.35055 + 1.08445i −0.0221929 0.0106875i −1.32101 3.36589i −1.07675 2.21687i −1.86586 + 1.87578i −1.75247 + 2.19753i 0.647946 2.92919i 5.08750 0.766817i
25.16 −0.302994 + 1.32750i −1.73138 0.0482649i 0.131481 + 0.0633181i 1.28284 + 3.26863i 0.588668 2.28378i 2.63656 + 0.220319i −1.82183 + 2.28450i 2.99534 + 0.167130i −4.72781 + 0.712603i
25.17 −0.279438 + 1.22430i −0.359714 1.69429i 0.381117 + 0.183536i 0.462772 + 1.17912i 2.17483 + 0.0330508i 2.42137 1.06628i −1.89714 + 2.37894i −2.74121 + 1.21892i −1.57292 + 0.237079i
25.18 −0.263181 + 1.15307i 1.58469 0.699101i 0.541631 + 0.260836i −0.0198537 0.0505863i 0.389051 + 2.01125i 1.10457 + 2.40415i −1.91814 + 2.40527i 2.02252 2.21572i 0.0635547 0.00957933i
25.19 −0.234242 + 1.02628i −0.449041 + 1.67283i 0.803556 + 0.386972i 0.973235 + 2.47976i −1.61161 0.852689i −1.43872 2.22038i −1.89803 + 2.38005i −2.59672 1.50234i −2.77290 + 0.417948i
25.20 −0.222631 + 0.975409i −0.211834 1.71905i 0.900079 + 0.433455i 0.692294 + 1.76394i 1.72394 + 0.176088i −0.942297 + 2.47226i −1.87078 + 2.34588i −2.91025 + 0.728307i −1.87469 + 0.282563i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 436.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.y even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.y.a 648
9.c even 3 1 441.2.z.a yes 648
49.g even 21 1 441.2.z.a yes 648
441.y even 21 1 inner 441.2.y.a 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.y.a 648 1.a even 1 1 trivial
441.2.y.a 648 441.y even 21 1 inner
441.2.z.a yes 648 9.c even 3 1
441.2.z.a yes 648 49.g even 21 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).