# Properties

 Label 224.2.be.a Level 224 Weight 2 Character orbit 224.be Analytic conductor 1.789 Analytic rank 0 Dimension 240 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 224.be (of order $$24$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{24})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 4q^{2} - 12q^{3} - 4q^{4} - 12q^{5} - 8q^{7} - 16q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 4q^{2} - 12q^{3} - 4q^{4} - 12q^{5} - 8q^{7} - 16q^{8} - 4q^{9} - 12q^{10} - 4q^{11} - 12q^{12} - 40q^{14} - 32q^{15} - 24q^{16} + 16q^{18} - 12q^{19} - 8q^{21} - 8q^{22} + 4q^{23} - 12q^{24} - 4q^{25} - 12q^{26} + 12q^{28} - 16q^{29} - 52q^{30} - 4q^{32} - 24q^{33} - 32q^{35} + 64q^{36} - 4q^{37} - 12q^{38} - 4q^{39} - 12q^{40} - 28q^{42} - 32q^{43} - 52q^{44} - 48q^{45} - 4q^{46} - 24q^{47} - 40q^{50} + 20q^{51} + 60q^{52} - 20q^{53} - 12q^{54} - 48q^{56} - 16q^{57} - 36q^{58} + 84q^{59} + 28q^{60} - 12q^{61} - 136q^{64} - 8q^{65} + 132q^{66} + 36q^{67} - 12q^{68} + 28q^{70} - 80q^{71} - 4q^{72} - 12q^{73} - 20q^{74} - 72q^{75} - 8q^{77} - 216q^{78} - 8q^{79} + 24q^{80} + 108q^{82} + 12q^{84} + 24q^{85} - 4q^{86} - 12q^{87} - 48q^{88} - 12q^{89} + 40q^{91} - 80q^{92} + 20q^{93} + 60q^{94} + 312q^{96} - 16q^{98} - 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.40664 0.146150i −0.173475 + 0.133112i 1.95728 + 0.411162i 1.47800 + 1.13411i 0.263472 0.161888i 1.10840 2.40238i −2.69310 0.864414i −0.764082 + 2.85159i −1.91327 1.81130i
3.2 −1.37290 + 0.339345i −1.61210 + 1.23700i 1.76969 0.931772i 0.108918 + 0.0835753i 1.79347 2.24534i −2.62908 + 0.296550i −2.11341 + 1.87976i 0.292215 1.09056i −0.177893 0.0777796i
3.3 −1.32601 + 0.491631i 1.63883 1.25752i 1.51660 1.30381i 2.28738 + 1.75517i −1.55487 + 2.47319i 1.14998 + 2.38276i −1.37003 + 2.47448i 0.327959 1.22396i −3.89598 1.20282i
3.4 −1.28037 0.600540i −2.17370 + 1.66793i 1.27870 + 1.53783i 0.725858 + 0.556970i 3.78480 0.830186i 1.28317 + 2.31376i −0.713688 2.73691i 1.16649 4.35339i −0.594885 1.14904i
3.5 −1.26996 + 0.622259i −0.630556 + 0.483843i 1.22559 1.58049i −3.20409 2.45858i 0.499704 1.00683i 2.00959 + 1.72092i −0.572972 + 2.76978i −0.612960 + 2.28760i 5.59893 + 1.12852i
3.6 −1.12720 + 0.854063i 2.03554 1.56193i 0.541153 1.92540i −1.56026 1.19723i −0.960477 + 3.49909i −1.31328 2.29680i 1.03442 + 2.63248i 0.927364 3.46097i 2.78123 + 0.0169550i
3.7 −1.08124 0.911550i 0.158962 0.121976i 0.338152 + 1.97121i −2.84975 2.18669i −0.283063 0.0130170i 1.92735 1.81254i 1.43123 2.43959i −0.766066 + 2.85900i 1.08798 + 4.96203i
3.8 −1.06453 0.931006i 2.25308 1.72885i 0.266455 + 1.98217i 1.40553 + 1.07850i −4.00805 0.257217i −2.00939 1.72115i 1.56176 2.35816i 1.31100 4.89272i −0.492141 2.45666i
3.9 −0.886934 1.10152i 0.450846 0.345946i −0.426696 + 1.95395i 0.813478 + 0.624204i −0.780937 0.189784i 0.0551167 + 2.64518i 2.53077 1.26301i −0.692874 + 2.58584i −0.0339283 1.44969i
3.10 −0.810400 + 1.15899i −2.13924 + 1.64150i −0.686504 1.87849i −0.625729 0.480139i −0.168834 3.80962i 0.349257 2.62260i 2.73349 + 0.726676i 1.10538 4.12534i 1.06357 0.336108i
3.11 −0.670369 + 1.24523i 0.102906 0.0789622i −1.10121 1.66953i 2.38833 + 1.83263i 0.0293417 + 0.181075i −2.62469 0.333180i 2.81717 0.252064i −0.772103 + 2.88153i −3.88312 + 1.74549i
3.12 −0.229686 + 1.39544i 1.43074 1.09785i −1.89449 0.641024i −0.628945 0.482607i 1.20336 + 2.24867i 2.64136 0.152394i 1.32965 2.49641i 0.0652997 0.243702i 0.817907 0.766806i
3.13 −0.210188 1.39851i −2.33910 + 1.79485i −1.91164 + 0.587899i −1.19071 0.913663i 3.00177 + 2.89399i 2.51525 0.820673i 1.22399 + 2.54987i 1.47343 5.49891i −1.02749 + 1.85726i
3.14 −0.201712 1.39975i −0.700660 + 0.537636i −1.91862 + 0.564694i −0.634957 0.487220i 0.893889 + 0.872305i −2.64410 0.0934084i 1.17744 + 2.57170i −0.574584 + 2.14438i −0.553910 + 0.987063i
3.15 −0.124966 1.40868i 2.48500 1.90681i −1.96877 + 0.352074i −2.71407 2.08258i −2.99663 3.26229i 1.12555 + 2.39440i 0.741988 + 2.72937i 1.76286 6.57908i −2.59452 + 4.08351i
3.16 0.157697 1.40539i 1.49249 1.14523i −1.95026 0.443252i 2.44964 + 1.87968i −1.37414 2.27814i 1.65828 2.06158i −0.930493 + 2.67099i 0.139527 0.520721i 3.02799 3.14630i
3.17 0.316867 + 1.37826i −1.12518 + 0.863381i −1.79919 + 0.873450i 2.90549 + 2.22946i −1.54649 1.27721i 2.61828 0.380261i −1.77394 2.20298i −0.255854 + 0.954861i −2.15212 + 4.71095i
3.18 0.337816 + 1.37327i −0.983428 + 0.754611i −1.77176 + 0.927827i −2.45933 1.88711i −1.36851 1.09560i −0.990618 2.45330i −1.87269 2.11968i −0.378764 + 1.41357i 1.76072 4.01483i
3.19 0.510534 + 1.31885i 2.43405 1.86771i −1.47871 + 1.34663i 1.80920 + 1.38825i 3.70589 + 2.25660i −1.87347 + 1.86819i −2.53093 1.26269i 1.65979 6.19441i −0.907226 + 3.09480i
3.20 0.612401 1.27474i −1.50340 + 1.15360i −1.24993 1.56131i 3.01130 + 2.31065i 0.549854 + 2.62290i −0.677804 + 2.55746i −2.75572 + 0.637190i 0.152957 0.570843i 4.78960 2.42358i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 187.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
32.h odd 8 1 inner
224.be even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.be.a 240
4.b odd 2 1 896.2.bi.a 240
7.d odd 6 1 inner 224.2.be.a 240
28.f even 6 1 896.2.bi.a 240
32.g even 8 1 896.2.bi.a 240
32.h odd 8 1 inner 224.2.be.a 240
224.bc odd 24 1 896.2.bi.a 240
224.be even 24 1 inner 224.2.be.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.be.a 240 1.a even 1 1 trivial
224.2.be.a 240 7.d odd 6 1 inner
224.2.be.a 240 32.h odd 8 1 inner
224.2.be.a 240 224.be even 24 1 inner
896.2.bi.a 240 4.b odd 2 1
896.2.bi.a 240 28.f even 6 1
896.2.bi.a 240 32.g even 8 1
896.2.bi.a 240 224.bc odd 24 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database