# Properties

 Label 572.2.j.a Level $572$ Weight $2$ Character orbit 572.j Analytic conductor $4.567$ Analytic rank $0$ Dimension $140$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$70$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$140q + 4q^{5} - 12q^{6} + 12q^{8} - 140q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q + 4q^{5} - 12q^{6} + 12q^{8} - 140q^{9} + 16q^{14} - 12q^{18} - 8q^{20} - 16q^{21} - 32q^{26} - 20q^{28} + 20q^{32} - 8q^{34} + 36q^{37} - 80q^{40} - 20q^{41} - 20q^{42} - 8q^{44} - 20q^{45} + 60q^{46} + 20q^{48} + 88q^{50} - 8q^{53} + 88q^{54} - 80q^{57} - 60q^{58} + 12q^{60} - 40q^{61} - 20q^{65} - 20q^{66} - 80q^{68} - 28q^{70} + 20q^{72} + 100q^{73} - 136q^{74} - 32q^{76} - 88q^{78} - 72q^{80} + 140q^{81} + 92q^{84} + 24q^{85} - 32q^{86} - 60q^{89} - 68q^{92} - 80q^{93} + 64q^{94} + 100q^{96} - 20q^{97} + 92q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
463.1 −1.40964 0.113659i 0.946143i 1.97416 + 0.320437i −1.09559 1.09559i −0.107538 + 1.33372i 0.184005 + 0.184005i −2.74644 0.676083i 2.10481 1.41986 + 1.66891i
463.2 −1.40874 0.124340i 2.77989i 1.96908 + 0.350325i −0.0767607 0.0767607i −0.345652 + 3.91613i 2.31083 + 2.31083i −2.73035 0.738352i −4.72779 0.0985912 + 0.117680i
463.3 −1.39906 + 0.206499i 2.87009i 1.91472 0.577808i 0.956348 + 0.956348i −0.592670 4.01541i 3.01968 + 3.01968i −2.55948 + 1.20377i −5.23740 −1.53547 1.14050i
463.4 −1.38865 + 0.267664i 2.20360i 1.85671 0.743385i −2.49116 2.49116i 0.589826 + 3.06004i −2.07384 2.07384i −2.37935 + 1.52928i −1.85587 4.12615 + 2.79256i
463.5 −1.37854 + 0.315622i 0.826677i 1.80077 0.870198i 1.70772 + 1.70772i 0.260917 + 1.13961i 0.257567 + 0.257567i −2.20778 + 1.76797i 2.31661 −2.89317 1.81518i
463.6 −1.35126 0.417264i 0.449813i 1.65178 + 1.12766i −1.24390 1.24390i 0.187691 0.607812i −2.23254 2.23254i −1.76145 2.21299i 2.79767 1.16179 + 2.19986i
463.7 −1.34707 0.430589i 1.83416i 1.62919 + 1.16007i 3.05786 + 3.05786i −0.789770 + 2.47074i 0.449413 + 0.449413i −1.69511 2.26420i −0.364149 −2.80247 5.43583i
463.8 −1.34567 + 0.434926i 0.845393i 1.62168 1.17054i 1.03191 + 1.03191i −0.367683 1.13762i −2.79241 2.79241i −1.67316 + 2.28047i 2.28531 −1.83742 0.939812i
463.9 −1.31145 + 0.529251i 2.04664i 1.43979 1.38817i −1.50803 1.50803i −1.08318 2.68406i 0.828927 + 0.828927i −1.15351 + 2.58252i −1.18872 2.77583 + 1.17957i
463.10 −1.28898 0.581825i 3.13637i 1.32296 + 1.49993i 2.34094 + 2.34094i 1.82482 4.04273i −2.63806 2.63806i −0.832580 2.70311i −6.83681 −1.65542 4.37946i
463.11 −1.22033 0.714703i 0.940195i 0.978400 + 1.74434i −2.27069 2.27069i −0.671960 + 1.14735i 2.86451 + 2.86451i 0.0527177 2.82794i 2.11603 1.14812 + 4.39385i
463.12 −1.19924 0.749544i 2.85994i 0.876368 + 1.79777i −2.69418 2.69418i 2.14365 3.42977i 1.37814 + 1.37814i 0.296529 2.81284i −5.17927 1.21157 + 5.25038i
463.13 −1.16815 + 0.797142i 3.05966i 0.729128 1.86236i 2.32382 + 2.32382i 2.43899 + 3.57413i −2.18069 2.18069i 0.632835 + 2.75672i −6.36154 −4.56698 0.862145i
463.14 −1.11823 0.865774i 0.913063i 0.500870 + 1.93627i 1.64493 + 1.64493i −0.790506 + 1.02101i −3.07399 3.07399i 1.11628 2.59883i 2.16632 −0.415270 3.26355i
463.15 −1.11565 + 0.869100i 0.202427i 0.489330 1.93922i −1.00276 1.00276i 0.175929 + 0.225836i 0.926254 + 0.926254i 1.13945 + 2.58875i 2.95902 1.99022 + 0.247225i
463.16 −1.09876 + 0.890354i 3.11188i 0.414539 1.95657i −0.115988 0.115988i −2.77067 3.41920i −2.35663 2.35663i 1.28656 + 2.51888i −6.68380 0.230714 + 0.0241724i
463.17 −1.03582 + 0.962851i 0.112664i 0.145835 1.99468i 2.00381 + 2.00381i −0.108479 0.116699i 3.01400 + 3.01400i 1.76952 + 2.20654i 2.98731 −4.00495 0.146211i
463.18 −1.01828 0.981374i 1.60363i 0.0738084 + 1.99864i 0.228079 + 0.228079i 1.57376 1.63295i 0.445735 + 0.445735i 1.88625 2.10762i 0.428375 −0.00841849 0.456080i
463.19 −0.981374 1.01828i 1.60363i −0.0738084 + 1.99864i 0.228079 + 0.228079i −1.63295 + 1.57376i −0.445735 0.445735i 2.10762 1.88625i 0.428375 0.00841849 0.456080i
463.20 −0.865774 1.11823i 0.913063i −0.500870 + 1.93627i 1.64493 + 1.64493i 1.02101 0.790506i 3.07399 + 3.07399i 2.59883 1.11628i 2.16632 0.415270 3.26355i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 551.70 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.j.a 140
4.b odd 2 1 inner 572.2.j.a 140
13.d odd 4 1 inner 572.2.j.a 140
52.f even 4 1 inner 572.2.j.a 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.j.a 140 1.a even 1 1 trivial
572.2.j.a 140 4.b odd 2 1 inner
572.2.j.a 140 13.d odd 4 1 inner
572.2.j.a 140 52.f even 4 1 inner

## Hecke kernels

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