Properties

Label 224.3.w.a
Level 224
Weight 3
Character orbit 224.w
Analytic conductor 6.104
Analytic rank 0
Dimension 192
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 192q + 80q^{10} + 96q^{12} - 20q^{16} - 60q^{18} - 260q^{22} + 64q^{23} - 144q^{24} - 200q^{26} + 192q^{27} - 40q^{30} + 40q^{32} + 120q^{34} + 464q^{36} + 504q^{38} - 384q^{39} + 360q^{40} - 96q^{43} + 52q^{44} + 64q^{46} - 104q^{48} - 312q^{50} - 384q^{51} - 320q^{52} + 160q^{53} - 576q^{54} - 512q^{55} - 196q^{56} - 360q^{58} - 872q^{60} + 128q^{61} - 408q^{62} + 832q^{66} + 160q^{67} + 856q^{68} - 384q^{69} + 336q^{70} + 1488q^{72} + 308q^{74} + 768q^{75} + 1024q^{76} - 224q^{77} - 408q^{78} + 1024q^{79} - 1040q^{80} - 240q^{82} - 1384q^{86} + 896q^{87} - 560q^{88} - 1320q^{90} - 380q^{92} - 936q^{94} - 1088q^{96} - 512q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.99992 0.0176912i −0.954366 2.30404i 3.99937 + 0.0707620i −0.872291 + 2.10590i 1.86790 + 4.62479i −1.87083 + 1.87083i −7.99718 0.212272i 1.96616 1.96616i 1.78177 4.19620i
43.2 −1.99845 0.0786486i 1.73419 + 4.18671i 3.98763 + 0.314351i −2.33059 + 5.62654i −3.13642 8.50332i −1.87083 + 1.87083i −7.94437 0.941838i −8.15712 + 8.15712i 5.10010 11.0611i
43.3 −1.99185 0.180328i 0.825363 + 1.99260i 3.93496 + 0.718374i 0.360772 0.870982i −1.28468 4.11781i 1.87083 1.87083i −7.70833 2.14048i 3.07472 3.07472i −0.875668 + 1.66981i
43.4 −1.92658 + 0.536908i −0.340996 0.823236i 3.42346 2.06880i −1.83063 + 4.41953i 1.09896 + 1.40295i 1.87083 1.87083i −5.48483 + 5.82380i 5.80252 5.80252i 1.15398 9.49749i
43.5 −1.86971 0.710051i 0.216404 + 0.522445i 2.99165 + 2.65518i 2.53768 6.12651i −0.0336503 1.13048i 1.87083 1.87083i −3.70822 7.08866i 6.13784 6.13784i −9.09487 + 9.65293i
43.6 −1.86483 0.722783i −1.45943 3.52338i 2.95517 + 2.69573i −3.21097 + 7.75196i 0.174951 + 7.62534i 1.87083 1.87083i −3.56245 7.16302i −3.92028 + 3.92028i 11.5909 12.1352i
43.7 −1.80532 + 0.860701i 1.98167 + 4.78418i 2.51839 3.10769i 1.28939 3.11286i −7.69530 6.93136i 1.87083 1.87083i −1.87171 + 7.77796i −12.5974 + 12.5974i 0.351479 + 6.72950i
43.8 −1.75066 0.967050i −1.86844 4.51082i 2.12963 + 3.38595i 0.599529 1.44739i −1.09118 + 9.70379i −1.87083 + 1.87083i −0.453865 7.98712i −10.4924 + 10.4924i −2.44927 + 1.95412i
43.9 −1.72313 + 1.01530i 0.819473 + 1.97838i 1.93834 3.49898i 1.60646 3.87835i −3.42071 2.57700i −1.87083 + 1.87083i 0.212503 + 7.99718i 3.12150 3.12150i 1.16954 + 8.31393i
43.10 −1.70535 1.04488i 0.502742 + 1.21373i 1.81645 + 3.56377i −0.0438594 + 0.105886i 0.410845 2.59514i −1.87083 + 1.87083i 0.626019 7.97547i 5.14358 5.14358i 0.185434 0.134745i
43.11 −1.55094 + 1.26277i −1.89928 4.58527i 0.810816 3.91696i −2.09096 + 5.04803i 8.73581 + 4.71311i −1.87083 + 1.87083i 3.68870 + 7.09884i −11.0535 + 11.0535i −3.13156 10.4696i
43.12 −1.49125 1.33273i −1.51777 3.66421i 0.447645 + 3.97487i 2.18754 5.28118i −2.62005 + 7.48703i 1.87083 1.87083i 4.62989 6.52412i −4.75887 + 4.75887i −10.3006 + 4.96015i
43.13 −1.40435 + 1.42401i −1.23085 2.97155i −0.0555981 3.99961i 0.908665 2.19371i 5.96006 + 2.42035i 1.87083 1.87083i 5.77356 + 5.53769i −0.951124 + 0.951124i 1.84778 + 4.37468i
43.14 −1.34412 1.48100i 1.15434 + 2.78683i −0.386709 + 3.98126i −3.15510 + 7.61708i 2.57571 5.45539i 1.87083 1.87083i 6.41602 4.77856i −0.0699303 + 0.0699303i 15.5217 5.56554i
43.15 −1.03112 + 1.71371i 0.406888 + 0.982314i −1.87359 3.53407i 0.691237 1.66879i −2.10295 0.315595i −1.87083 + 1.87083i 7.98826 + 0.433260i 5.56458 5.56458i 2.14708 + 2.90530i
43.16 −1.00544 1.72890i −0.343872 0.830180i −1.97818 + 3.47661i −1.68929 + 4.07830i −1.08955 + 1.42922i −1.87083 + 1.87083i 7.99964 0.0754426i 5.79301 5.79301i 8.74944 1.17988i
43.17 −0.703873 1.87205i 0.0651792 + 0.157357i −3.00913 + 2.63537i 3.56366 8.60344i 0.248701 0.232778i −1.87083 + 1.87083i 7.05157 + 3.77827i 6.34345 6.34345i −18.6144 0.615620i
43.18 −0.521176 + 1.93090i 1.24651 + 3.00935i −3.45675 2.01268i 3.27130 7.89762i −6.46041 + 0.838495i 1.87083 1.87083i 5.68785 5.62569i −1.13845 + 1.13845i 13.5446 + 10.4326i
43.19 −0.518278 + 1.93168i −0.266183 0.642623i −3.46277 2.00230i −0.747592 + 1.80485i 1.37930 0.181123i 1.87083 1.87083i 5.66248 5.65123i 6.02185 6.02185i −3.09892 2.37952i
43.20 −0.491313 + 1.93871i −1.80355 4.35417i −3.51722 1.90503i 3.60551 8.70447i 9.32759 1.35732i −1.87083 + 1.87083i 5.42137 5.88292i −9.34199 + 9.34199i 15.1040 + 11.2667i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.w.a 192
32.h odd 8 1 inner 224.3.w.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.w.a 192 1.a even 1 1 trivial
224.3.w.a 192 32.h odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database