LMFDB - Embedded newform 1008.4.bt.d.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.2
Character	$\chi$	$=$	1008.593
Dual form			1008.4.bt.d.17.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.74197 + 16.8736i) q^{5} +(18.4446 + 1.67197i) q^{7} +O(q^{10})$	$q+(-9.74197 + 16.8736i) q^{5} +(18.4446 + 1.67197i) q^{7} +(15.1087 - 8.72302i) q^{11} +16.5278i q^{13} +(68.6466 + 118.899i) q^{17} +(20.6975 + 11.9497i) q^{19} +(108.910 + 62.8794i) q^{23} +(-127.312 - 220.511i) q^{25} -105.188i q^{29} +(95.0849 - 54.8973i) q^{31} +(-207.899 + 294.939i) q^{35} +(58.5919 - 101.484i) q^{37} +348.056 q^{41} +141.953 q^{43} +(-172.830 + 299.351i) q^{47} +(337.409 + 61.6778i) q^{49} +(149.292 - 86.1937i) q^{53} +339.918i q^{55} +(297.569 + 515.404i) q^{59} +(-561.746 - 324.324i) q^{61} +(-278.884 - 161.014i) q^{65} +(64.9996 + 112.583i) q^{67} -908.671i q^{71} +(-44.8988 + 25.9223i) q^{73} +(293.259 - 135.632i) q^{77} +(-474.917 + 822.580i) q^{79} -1302.27 q^{83} -2675.01 q^{85} +(-210.570 + 364.719i) q^{89} +(-27.6341 + 304.850i) q^{91} +(-403.269 + 232.827i) q^{95} +553.292i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{7}+O(q^{10}) $ 48 * q + 24 * q^7	$ 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) $ 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−9.74197	+	16.8736i	−0.871348	+	1.50922i	−0.0107451	+	0.999942i	$0.503420\pi$
$5$	−9.74197	+	16.8736i	−0.871348	+	1.50922i	−0.860603	+	0.509277i	$0.829913\pi$
$6$		0			0
$7$	18.4446	+	1.67197i	0.995917	+	0.0902780i
$7$	18.4446	+	1.67197i	0.995917	+	0.0902780i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.1087	−	8.72302i	0.414132	−	0.239099i	−0.278432	−	0.960456i	$-0.589815\pi$
$11$	15.1087	−	8.72302i	0.414132	−	0.239099i	0.692563	+	0.721357i	$0.256481\pi$
$12$		0			0
$13$			16.5278i			0.352615i	0.984335	+	0.176308i	$0.0564154\pi$
$13$			16.5278i			0.352615i	−0.984335	+	0.176308i	$0.943585\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	68.6466	+	118.899i	0.979368	+	1.69632i	0.664694	+	0.747115i	$0.268562\pi$
$17$	68.6466	+	118.899i	0.979368	+	1.69632i	0.314674	+	0.949200i	$0.398105\pi$
$18$		0			0
$19$	20.6975	+	11.9497i	0.249912	+	0.144287i	0.619724	−	0.784820i	$-0.287244\pi$
$19$	20.6975	+	11.9497i	0.249912	+	0.144287i	−0.369812	+	0.929107i	$0.620578\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	108.910	+	62.8794i	0.987365	+	0.570055i	0.904486	−	0.426504i	$-0.140255\pi$
$23$	108.910	+	62.8794i	0.987365	+	0.570055i	0.0828792	+	0.996560i	$0.473588\pi$
$24$		0			0
$25$	−127.312	−	220.511i	−1.01849	−	1.76408i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	105.188i		−	0.673548i	−0.941586	−	0.336774i	$-0.890664\pi$
$29$		−	105.188i		−	0.673548i	0.941586	−	0.336774i	$-0.109336\pi$
$30$		0			0
$31$	95.0849	−	54.8973i	0.550895	−	0.318060i	−0.198588	−	0.980083i	$-0.563635\pi$
$31$	95.0849	−	54.8973i	0.550895	−	0.318060i	0.749483	+	0.662024i	$0.230302\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−207.899	+	294.939i	−1.00404	+	1.42439i
$36$		0			0
$37$	58.5919	−	101.484i	0.260336	−	0.450916i	−0.705995	−	0.708217i	$-0.749500\pi$
$37$	58.5919	−	101.484i	0.260336	−	0.450916i	0.966331	+	0.257301i	$0.0828332\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	348.056			1.32579			0.662893	−	0.748714i	$-0.269328\pi$
$41$	348.056			1.32579			0.662893	+	0.748714i	$0.269328\pi$
$42$		0			0
$43$	141.953			0.503432			0.251716	−	0.967801i	$-0.419005\pi$
$43$	141.953			0.503432			0.251716	+	0.967801i	$0.419005\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−172.830	+	299.351i	−0.536381	+	0.929040i	0.462714	+	0.886508i	$0.346876\pi$
$47$	−172.830	+	299.351i	−0.536381	+	0.929040i	−0.999095	+	0.0425319i	$0.986458\pi$
$48$		0			0
$49$	337.409	+	61.6778i	0.983700	+	0.179819i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	149.292	−	86.1937i	0.386921	−	0.223389i	−0.293904	−	0.955835i	$-0.594955\pi$
$53$	149.292	−	86.1937i	0.386921	−	0.223389i	0.680825	+	0.732446i	$0.261621\pi$
$54$		0			0
$55$			339.918i			0.833354i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	297.569	+	515.404i	0.656613	+	1.13729i	0.981487	+	0.191530i	$0.0613448\pi$
$59$	297.569	+	515.404i	0.656613	+	1.13729i	−0.324874	+	0.945757i	$0.605322\pi$
$60$		0			0
$61$	−561.746	−	324.324i	−1.17909	−	0.680746i	−0.223283	−	0.974754i	$-0.571677\pi$
$61$	−561.746	−	324.324i	−1.17909	−	0.680746i	−0.955803	+	0.294008i	$0.905011\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−278.884	−	161.014i	−0.532174	−	0.307251i
$66$		0			0
$67$	64.9996	+	112.583i	0.118522	+	0.205286i	0.919182	−	0.393833i	$-0.128851\pi$
$67$	64.9996	+	112.583i	0.118522	+	0.205286i	−0.800660	+	0.599119i	$0.795518\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	908.671i		−	1.51886i	−0.650586	−	0.759432i	$-0.725477\pi$
$71$		−	908.671i		−	1.51886i	0.650586	−	0.759432i	$-0.274523\pi$
$72$		0			0
$73$	−44.8988	+	25.9223i	−0.0719864	+	0.0415614i	−0.535561	−	0.844496i	$-0.679900\pi$
$73$	−44.8988	+	25.9223i	−0.0719864	+	0.0415614i	0.463575	+	0.886058i	$0.346567\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	293.259	−	135.632i	0.434026	−	0.200736i
$78$		0			0
$79$	−474.917	+	822.580i	−0.676358	+	1.17149i	0.299712	+	0.954030i	$0.403109\pi$
$79$	−474.917	+	822.580i	−0.676358	+	1.17149i	−0.976070	+	0.217457i	$0.930224\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1302.27			−1.72220			−0.861100	−	0.508436i	$-0.830224\pi$
$83$	−1302.27			−1.72220			−0.861100	+	0.508436i	$0.830224\pi$
$84$		0			0
$85$	−2675.01			−3.41348
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−210.570	+	364.719i	−0.250791	+	0.434383i	−0.963744	−	0.266829i	$-0.914024\pi$
$89$	−210.570	+	364.719i	−0.250791	+	0.434383i	0.712953	+	0.701212i	$0.247357\pi$
$90$		0			0
$91$	−27.6341	+	304.850i	−0.0318334	+	0.351175i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−403.269	+	232.827i	−0.435521	+	0.251448i
$96$		0			0
$97$			553.292i			0.579158i	0.957154	+	0.289579i	$0.0935153\pi$
$97$			553.292i			0.579158i	−0.957154	+	0.289579i	$0.906485\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−115.951	−	200.833i	−0.114233	−	0.197858i	0.803240	−	0.595656i	$-0.203108\pi$
$101$	−115.951	−	200.833i	−0.114233	−	0.197858i	−0.917473	+	0.397798i	$0.869774\pi$
$102$		0			0
$103$	1674.49	+	966.766i	1.60187	+	0.924837i	0.991114	+	0.133015i	$0.0424657\pi$
$103$	1674.49	+	966.766i	1.60187	+	0.924837i	0.610751	+	0.791823i	$0.290868\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	769.974	+	444.544i	0.695665	+	0.401642i	0.805731	−	0.592282i	$-0.201773\pi$
$107$	769.974	+	444.544i	0.695665	+	0.401642i	−0.110066	+	0.993924i	$0.535106\pi$
$108$		0			0
$109$	−961.155	−	1664.77i	−0.844605	−	1.46290i	−0.885964	−	0.463755i	$-0.846502\pi$
$109$	−961.155	−	1664.77i	−0.844605	−	1.46290i	0.0413585	−	0.999144i	$-0.486831\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2070.45i			1.72365i	0.507210	+	0.861823i	$0.330677\pi$
$113$			2070.45i			1.72365i	−0.507210	+	0.861823i	$0.669323\pi$
$114$		0			0
$115$	−2122.00	+	1225.14i	−1.72068	+	0.993433i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1067.37	+	2307.83i	0.822229	+	1.77780i
$120$		0			0
$121$	−513.318	+	889.092i	−0.385663	+	0.667988i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2525.58			1.80716
$126$		0			0
$127$	−1508.37			−1.05391			−0.526955	−	0.849893i	$-0.676666\pi$
$127$	−1508.37			−1.05391			−0.526955	+	0.849893i	$0.676666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1067.52	+	1848.99i	−0.711979	+	1.23318i	0.252134	+	0.967692i	$0.418868\pi$
$131$	−1067.52	+	1848.99i	−0.711979	+	1.23318i	−0.964113	+	0.265491i	$0.914466\pi$
$132$		0			0
$133$	361.778	+	255.014i	0.235866	+	0.166259i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−494.811	+	285.679i	−0.308574	+	0.178155i	−0.646288	−	0.763094i	$-0.723680\pi$
$137$	−494.811	+	285.679i	−0.308574	+	0.178155i	0.337714	+	0.941249i	$0.390346\pi$
$138$		0			0
$139$		−	3003.94i		−	1.83302i	−0.400006	−	0.916512i	$-0.630992\pi$
$139$		−	3003.94i		−	1.83302i	0.400006	−	0.916512i	$-0.369008\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	144.173	+	249.714i	0.0843100	+	0.146029i
$144$		0			0
$145$	1774.89	+	1024.74i	1.01653	+	0.586894i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1433.52	−	827.646i	−0.788180	−	0.455056i	0.0511412	−	0.998691i	$-0.483714\pi$
$149$	−1433.52	−	827.646i	−0.788180	−	0.455056i	−0.839322	+	0.543635i	$0.817047\pi$
$150$		0			0
$151$	−396.703	−	687.109i	−0.213796	−	0.370306i	0.739103	−	0.673592i	$-0.235250\pi$
$151$	−396.703	−	687.109i	−0.213796	−	0.370306i	−0.952899	+	0.303286i	$0.901916\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2139.23i			1.10856i
$156$		0			0
$157$	−621.514	+	358.831i	−0.315938	+	0.182407i	−0.649580	−	0.760293i	$-0.725055\pi$
$157$	−621.514	+	358.831i	−0.315938	+	0.182407i	0.333643	+	0.942700i	$0.391722\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1903.68	+	1341.88i	0.931869	+	0.656865i
$162$		0			0
$163$	−1038.88	+	1799.39i	−0.499211	+	0.864659i	−1.00000		0.000910600i	$-0.999710\pi$
$163$	−1038.88	+	1799.39i	−0.499211	+	0.864659i	0.500788	+	0.865570i	$0.333043\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1017.38			0.471420			0.235710	−	0.971823i	$-0.424258\pi$
$167$	1017.38			0.471420			0.235710	+	0.971823i	$0.424258\pi$
$168$		0			0
$169$	1923.83			0.875662
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1498.46	+	2595.42i	−0.658533	+	1.14061i	0.322463	+	0.946582i	$0.395489\pi$
$173$	−1498.46	+	2595.42i	−0.658533	+	1.14061i	−0.980996	+	0.194030i	$0.937844\pi$
$174$		0			0
$175$	−1979.53	−	4280.10i	−0.855078	−	1.84883i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3473.51	−	2005.43i	1.45040	−	0.837391i	0.451900	−	0.892068i	$-0.350746\pi$
$179$	3473.51	−	2005.43i	1.45040	−	0.837391i	0.998504	+	0.0546769i	$0.0174129\pi$
$180$		0			0
$181$		−	2885.89i		−	1.18512i	−0.805527	−	0.592559i	$-0.798118\pi$
$181$		−	2885.89i		−	1.18512i	0.805527	−	0.592559i	$-0.201882\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1141.60	+	1977.31i	0.453687	+	0.785809i
$186$		0			0
$187$	2074.33	+	1197.61i	0.811175	+	0.468332i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1169.89	+	675.436i	0.443195	+	0.255879i	0.704952	−	0.709255i	$-0.250969\pi$
$191$	1169.89	+	675.436i	0.443195	+	0.255879i	−0.261757	+	0.965134i	$0.584302\pi$
$192$		0			0
$193$	−315.843	−	547.055i	−0.117797	−	0.204031i	0.801097	−	0.598534i	$-0.204250\pi$
$193$	−315.843	−	547.055i	−0.117797	−	0.204031i	−0.918894	+	0.394504i	$0.870917\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2350.91i			0.850232i	0.905139	+	0.425116i	$0.139767\pi$
$197$			2350.91i			0.850232i	−0.905139	+	0.425116i	$0.860233\pi$
$198$		0			0
$199$	970.169	−	560.127i	0.345595	−	0.199530i	−0.317148	−	0.948376i	$-0.602725\pi$
$199$	970.169	−	560.127i	0.345595	−	0.199530i	0.662744	+	0.748846i	$0.269392\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	175.871	−	1940.15i	0.0608066	−	0.670797i
$204$		0			0
$205$	−3390.75	+	5872.95i	−1.15522	+	2.00090i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	416.950			0.137996
$210$		0			0
$211$	396.657			0.129417			0.0647085	−	0.997904i	$-0.479388\pi$
$211$	396.657			0.129417			0.0647085	+	0.997904i	$0.479388\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1382.90	+	2395.25i	−0.438664	+	0.759789i
$216$		0			0
$217$	1845.59	−	853.581i	0.577360	−	0.267027i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1965.15	+	1134.58i	−0.598147	+	0.345340i
$222$		0			0
$223$			2574.52i			0.773106i	0.922267	+	0.386553i	$0.126334\pi$
$223$			2574.52i			0.773106i	−0.922267	+	0.386553i	$0.873666\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1165.21	−	2018.20i	−0.340694	−	0.590100i	0.643868	−	0.765137i	$-0.277329\pi$
$227$	−1165.21	−	2018.20i	−0.340694	−	0.590100i	−0.984562	+	0.175037i	$0.943995\pi$
$228$		0			0
$229$	1211.86	+	699.668i	0.349703	+	0.201901i	0.664554	−	0.747240i	$-0.268621\pi$
$229$	1211.86	+	699.668i	0.349703	+	0.201901i	−0.314852	+	0.949141i	$0.601955\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1950.66	−	1126.21i	−0.548463	−	0.316655i	0.200039	−	0.979788i	$-0.435893\pi$
$233$	−1950.66	−	1126.21i	−0.548463	−	0.316655i	−0.748502	+	0.663133i	$0.769227\pi$
$234$		0			0
$235$	−3367.42	−	5832.54i	−0.934749	−	1.61903i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3423.46i		−	0.926549i	−0.886215	−	0.463275i	$-0.846674\pi$
$239$		−	3423.46i		−	0.926549i	0.886215	−	0.463275i	$-0.153326\pi$
$240$		0			0
$241$	1997.78	−	1153.42i	0.533977	−	0.308292i	−0.208657	−	0.977989i	$-0.566909\pi$
$241$	1997.78	−	1153.42i	0.533977	−	0.308292i	0.742635	+	0.669697i	$0.233576\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4327.75	+	5092.43i	−1.12853	+	1.32793i
$246$		0			0
$247$	−197.503	+	342.085i	−0.0508778	+	0.0881229i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7897.26			1.98594			0.992970	−	0.118368i	$-0.0377662\pi$
$251$	7897.26			1.98594			0.992970	+	0.118368i	$0.0377662\pi$
$252$		0			0
$253$	2194.00			0.545199
$254$		0			0
$255$		0			0
$256$		0			0
$257$	217.517	−	376.750i	0.0527950	−	0.0914437i	−0.838420	−	0.545025i	$-0.816520\pi$
$257$	217.517	−	376.750i	0.0527950	−	0.0914437i	0.891215	+	0.453581i	$0.149854\pi$
$258$		0			0
$259$	1250.38	−	1773.87i	0.299981	−	0.425572i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5144.46	+	2970.16i	−1.20616	+	0.696379i	−0.961919	−	0.273335i	$-0.911873\pi$
$263$	−5144.46	+	2970.16i	−1.20616	+	0.696379i	−0.244245	+	0.969714i	$0.578540\pi$
$264$		0			0
$265$			3358.78i			0.778598i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	915.170	+	1585.12i	0.207431	+	0.359281i	0.950905	−	0.309484i	$-0.100156\pi$
$269$	915.170	+	1585.12i	0.207431	+	0.359281i	−0.743474	+	0.668765i	$0.766823\pi$
$270$		0			0
$271$	−2605.49	−	1504.28i	−0.584030	−	0.337190i	0.178703	−	0.983903i	$-0.442810\pi$
$271$	−2605.49	−	1504.28i	−0.584030	−	0.337190i	−0.762733	+	0.646713i	$0.776143\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3847.04	−	2221.09i	−0.843582	−	0.487042i
$276$		0			0
$277$	147.124	+	254.826i	0.0319127	+	0.0552744i	0.881541	−	0.472108i	$-0.156507\pi$
$277$	147.124	+	254.826i	0.0319127	+	0.0552744i	−0.849628	+	0.527383i	$0.823173\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8000.78i		−	1.69853i	−0.527967	−	0.849265i	$-0.677046\pi$
$281$		−	8000.78i		−	1.69853i	0.527967	−	0.849265i	$-0.322954\pi$
$282$		0			0
$283$	−7523.20	+	4343.52i	−1.58024	+	0.912352i	−0.585416	+	0.810733i	$0.699069\pi$
$283$	−7523.20	+	4343.52i	−1.58024	+	0.912352i	−0.994823	+	0.101619i	$0.967598\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6419.77	+	581.940i	1.32037	+	0.119689i
$288$		0			0
$289$	−6968.22	+	12069.3i	−1.41832	+	2.45661i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4055.18			−0.808554			−0.404277	−	0.914637i	$-0.632477\pi$
$293$	−4055.18			−0.808554			−0.404277	+	0.914637i	$0.632477\pi$
$294$		0			0
$295$	−11595.6			−2.28855
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1039.26	+	1800.05i	−0.201010	+	0.348160i
$300$		0			0
$301$	2618.26	+	237.341i	0.501376	+	0.0454488i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10945.0	−	6319.11i	2.05479	−	1.18633i
$306$		0			0
$307$			4591.37i			0.853562i	0.904355	+	0.426781i	$0.140352\pi$
$307$			4591.37i			0.853562i	−0.904355	+	0.426781i	$0.859648\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2849.94	−	4936.24i	−0.519631	−	0.900028i	−0.999740	−	0.0228187i	$-0.992736\pi$
$311$	−2849.94	−	4936.24i	−0.519631	−	0.900028i	0.480108	−	0.877209i	$-0.340597\pi$
$312$		0			0
$313$	−3909.34	−	2257.06i	−0.705971	−	0.407593i	0.103596	−	0.994619i	$-0.466965\pi$
$313$	−3909.34	−	2257.06i	−0.705971	−	0.407593i	−0.809568	+	0.587027i	$0.800298\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1819.78	−	1050.65i	−0.322427	−	0.186153i	0.330047	−	0.943965i	$-0.392935\pi$
$317$	−1819.78	−	1050.65i	−0.322427	−	0.186153i	−0.652474	+	0.757811i	$0.726269\pi$
$318$		0			0
$319$	−917.555	−	1589.25i	−0.161045	−	0.278937i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3281.23i			0.565240i
$324$		0			0
$325$	3644.56	−	2104.19i	0.622043	−	0.359137i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3688.30	+	5232.46i	−0.618063	+	0.876822i
$330$		0			0
$331$	3111.81	−	5389.81i	0.516739	−	0.895018i	−0.483072	−	0.875581i	$-0.660479\pi$
$331$	3111.81	−	5389.81i	0.516739	−	0.895018i	0.999811	−	0.0194375i	$-0.00618755\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2532.89			−0.413095
$336$		0			0
$337$	1599.77			0.258591			0.129295	−	0.991606i	$-0.458728\pi$
$337$	1599.77			0.258591			0.129295	+	0.991606i	$0.458728\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	957.741	−	1658.86i	0.152096	−	0.263437i
$342$		0			0
$343$	6120.26	+	1701.76i	0.963449	+	0.267891i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2237.47	+	1291.80i	−0.346149	+	0.199849i	−0.662988	−	0.748630i	$-0.730712\pi$
$347$	−2237.47	+	1291.80i	−0.346149	+	0.199849i	0.316839	+	0.948479i	$0.397379\pi$
$348$		0			0
$349$		−	6406.75i		−	0.982652i	−0.870976	−	0.491326i	$-0.836512\pi$
$349$		−	6406.75i		−	0.982652i	0.870976	−	0.491326i	$-0.163488\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3536.88	−	6126.05i	−0.533283	−	0.923673i	−0.999244	−	0.0388682i	$-0.987625\pi$
$353$	−3536.88	−	6126.05i	−0.533283	−	0.923673i	0.465961	−	0.884805i	$-0.345709\pi$
$354$		0			0
$355$	15332.5	+	8852.24i	2.29230	+	1.32346i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1355.38	−	782.532i	−0.199260	−	0.115043i	0.397050	−	0.917797i	$-0.370034\pi$
$359$	−1355.38	−	782.532i	−0.199260	−	0.115043i	−0.596310	+	0.802754i	$0.703367\pi$
$360$		0			0
$361$	−3143.91	−	5445.41i	−0.458363	−	0.793907i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1010.14i		−	0.144858i
$366$		0			0
$367$	8459.40	−	4884.03i	1.20321	−	0.694672i	0.241940	−	0.970291i	$-0.422216\pi$
$367$	8459.40	−	4884.03i	1.20321	−	0.694672i	0.961267	+	0.275620i	$0.0888830\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2897.75	−	1340.20i	0.405508	−	0.187546i
$372$		0			0
$373$	3840.66	−	6652.21i	0.533141	−	0.923427i	−0.466110	−	0.884727i	$-0.654345\pi$
$373$	3840.66	−	6652.21i	0.533141	−	0.923427i	0.999251	−	0.0387005i	$-0.0123218\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1738.53			0.237503
$378$		0			0
$379$	6320.25			0.856595			0.428297	−	0.903638i	$-0.359114\pi$
$379$	6320.25			0.856595			0.428297	+	0.903638i	$0.359114\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	91.9525	−	159.266i	0.0122678	−	0.0212484i	−0.859826	−	0.510587i	$-0.829428\pi$
$383$	91.9525	−	159.266i	0.0122678	−	0.0212484i	0.872094	+	0.489338i	$0.162762\pi$
$384$		0			0
$385$	−568.333	+	6269.65i	−0.0752336	+	0.829951i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9127.81	+	5269.94i	−1.18971	+	0.686881i	−0.958241	−	0.285961i	$-0.907687\pi$
$389$	−9127.81	+	5269.94i	−1.18971	+	0.686881i	−0.231471	+	0.972842i	$0.574354\pi$
$390$		0			0
$391$			17265.9i			2.23318i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9253.24	−	16027.1i	−1.17869	−	2.04154i
$396$		0			0
$397$	−4797.29	−	2769.72i	−0.606471	−	0.350146i	0.165112	−	0.986275i	$-0.447201\pi$
$397$	−4797.29	−	2769.72i	−0.606471	−	0.350146i	−0.771583	+	0.636129i	$0.780535\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3816.52	+	2203.47i	0.475282	+	0.274404i	0.718448	−	0.695580i	$-0.244853\pi$
$401$	3816.52	+	2203.47i	0.475282	+	0.274404i	−0.243166	+	0.969985i	$0.578186\pi$
$402$		0			0
$403$	907.334	+	1571.55i	0.112153	+	0.194254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2044.39i		−	0.248985i
$408$		0			0
$409$	8336.94	−	4813.33i	1.00791	−	0.581917i	0.0973306	−	0.995252i	$-0.468970\pi$
$409$	8336.94	−	4813.33i	1.00791	−	0.581917i	0.910579	+	0.413335i	$0.135636\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4626.81	+	10004.0i	0.551260	+	1.19192i
$414$		0			0
$415$	12686.7	−	21973.9i	1.50063	−	2.59918i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3095.82			−0.360957			−0.180478	−	0.983579i	$-0.557765\pi$
$419$	−3095.82			−0.360957			−0.180478	+	0.983579i	$0.557765\pi$
$420$		0			0
$421$	4652.51			0.538598			0.269299	−	0.963057i	$-0.413208\pi$
$421$	4652.51			0.538598			0.269299	+	0.963057i	$0.413208\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	17479.1	−	30274.6i	1.99496	−	3.45538i
$426$		0			0
$427$	−9818.94	−	6921.27i	−1.11281	−	0.784411i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6562.34	−	3788.77i	0.733404	−	0.423431i	−0.0862625	−	0.996272i	$-0.527492\pi$
$431$	6562.34	−	3788.77i	0.733404	−	0.423431i	0.819666	+	0.572842i	$0.194159\pi$
$432$		0			0
$433$		−	10028.8i		−	1.11305i	−0.830829	−	0.556527i	$-0.812133\pi$
$433$		−	10028.8i		−	1.11305i	0.830829	−	0.556527i	$-0.187867\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1502.78	+	2602.90i	0.164503	+	0.284928i
$438$		0			0
$439$	12522.8	+	7230.07i	1.36146	+	0.786042i	0.989819	−	0.142333i	$-0.0454603\pi$
$439$	12522.8	+	7230.07i	1.36146	+	0.786042i	0.371646	+	0.928375i	$0.378794\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12101.9	+	6987.05i	1.29792	+	0.749356i	0.980045	−	0.198775i	$-0.0636963\pi$
$443$	12101.9	+	6987.05i	1.29792	+	0.749356i	0.317878	+	0.948132i	$0.397030\pi$
$444$		0			0
$445$	−4102.74	−	7106.15i	−0.437053	−	0.756998i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12850.1i		−	1.35064i	−0.737527	−	0.675318i	$-0.764007\pi$
$449$		−	12850.1i		−	1.35064i	0.737527	−	0.675318i	$-0.235993\pi$
$450$		0			0
$451$	5258.68	−	3036.10i	0.549050	−	0.316994i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4874.70	−	3436.12i	−0.502263	−	0.354040i
$456$		0			0
$457$	4530.38	−	7846.86i	0.463725	−	0.803196i	−0.535418	−	0.844587i	$-0.679846\pi$
$457$	4530.38	−	7846.86i	0.463725	−	0.803196i	0.999143	+	0.0413917i	$0.0131792\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3390.73			−0.342564			−0.171282	−	0.985222i	$-0.554791\pi$
$461$	−3390.73			−0.342564			−0.171282	+	0.985222i	$0.554791\pi$
$462$		0			0
$463$	−707.176			−0.0709833			−0.0354916	−	0.999370i	$-0.511300\pi$
$463$	−707.176			−0.0709833			−0.0354916	+	0.999370i	$0.511300\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2360.37	+	4088.28i	−0.233886	+	0.405103i	−0.958948	−	0.283581i	$-0.908478\pi$
$467$	−2360.37	+	4088.28i	−0.233886	+	0.405103i	0.725062	+	0.688683i	$0.241811\pi$
$468$		0			0
$469$	1010.66	+	2185.22i	0.0995050	+	0.215147i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2144.72	−	1238.26i	0.208487	−	0.120370i
$474$		0			0
$475$		−	6085.36i		−	0.587822i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2300.35	+	3984.32i	0.219427	+	0.380059i	0.954633	−	0.297785i	$-0.0962479\pi$
$479$	2300.35	+	3984.32i	0.219427	+	0.380059i	−0.735206	+	0.677844i	$0.762915\pi$
$480$		0			0
$481$	1677.31	+	968.398i	0.159000	+	0.0917986i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9336.02	−	5390.15i	−0.874076	−	0.504648i
$486$		0			0
$487$	8189.27	+	14184.2i	0.761994	+	1.31981i	0.941821	+	0.336115i	$0.109113\pi$
$487$	8189.27	+	14184.2i	0.761994	+	1.31981i	−0.179827	+	0.983698i	$0.557554\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4293.59i		−	0.394637i	−0.980339	−	0.197319i	$-0.936777\pi$
$491$		−	4293.59i		−	0.394637i	0.980339	−	0.197319i	$-0.0632234\pi$
$492$		0			0
$493$	12506.8	−	7220.79i	1.14255	−	0.659651i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1519.27	−	16760.1i	0.137120	−	1.51266i
$498$		0			0
$499$	−2603.39	+	4509.21i	−0.233555	+	0.404529i	−0.958852	−	0.283907i	$-0.908369\pi$
$499$	−2603.39	+	4509.21i	−0.233555	+	0.404529i	0.725297	+	0.688436i	$0.241702\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16409.4			−1.45459			−0.727295	−	0.686325i	$-0.759223\pi$
$503$	−16409.4			−1.45459			−0.727295	+	0.686325i	$0.759223\pi$
$504$		0			0
$505$	4518.37			0.398148
$506$		0			0
$507$		0			0
$508$		0			0
$509$	992.966	−	1719.87i	0.0864685	−	0.149768i	−0.819548	−	0.573011i	$-0.805775\pi$
$509$	992.966	−	1719.87i	0.0864685	−	0.149768i	0.906016	+	0.423243i	$0.139108\pi$
$510$		0			0
$511$	−871.483	+	403.058i	−0.0754445	+	0.0348929i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−32625.6	+	18836.4i	−2.79156	+	1.61171i
$516$		0			0
$517$			6030.42i			0.512993i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1336.56	−	2315.00i	−0.112391	−	0.194668i	0.804343	−	0.594166i	$-0.202518\pi$
$521$	−1336.56	−	2315.00i	−0.112391	−	0.194668i	−0.916734	+	0.399498i	$0.869184\pi$
$522$		0			0
$523$	1236.34	+	713.801i	0.103368	+	0.0596794i	0.550793	−	0.834642i	$-0.314325\pi$
$523$	1236.34	+	713.801i	0.103368	+	0.0596794i	−0.447425	+	0.894322i	$0.647659\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13054.5	+	7537.03i	1.07906	+	0.622995i
$528$		0			0
$529$	1824.15	+	3159.52i	0.149926	+	0.259679i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5752.62i			0.467493i
$534$		0			0
$535$	−15002.1	+	8661.47i	−1.21233	+	0.699941i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5635.83	−	2011.35i	0.450376	−	0.160733i
$540$		0			0
$541$	−10091.7	+	17479.3i	−0.801985	+	1.38908i	0.116322	+	0.993211i	$0.462889\pi$
$541$	−10091.7	+	17479.3i	−0.801985	+	1.38908i	−0.918308	+	0.395868i	$0.870444\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	37454.2			2.94378
$546$		0			0
$547$	21275.8			1.66305			0.831524	−	0.555488i	$-0.187469\pi$
$547$	21275.8			1.66305			0.831524	+	0.555488i	$0.187469\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1256.96	−	2177.13i	0.0971842	−	0.168328i
$552$		0			0
$553$	−10135.0	+	14378.1i	−0.779356	+	1.10564i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15295.5	+	8830.84i	−1.16354	+	0.671768i	−0.952149	−	0.305634i	$-0.901132\pi$
$557$	−15295.5	+	8830.84i	−1.16354	+	0.671768i	−0.211388	+	0.977402i	$0.567798\pi$
$558$		0			0
$559$			2346.17i			0.177518i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7447.80	−	12900.0i	−0.557527	−	0.965665i	−0.997702	−	0.0677527i	$-0.978417\pi$
$563$	−7447.80	−	12900.0i	−0.557527	−	0.965665i	0.440175	−	0.897912i	$-0.354916\pi$
$564$		0			0
$565$	−34936.0	−	20170.3i	−2.60136	−	1.50189i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11813.4	−	6820.47i	−0.870375	−	0.502511i	−0.00290206	−	0.999996i	$-0.500924\pi$
$569$	−11813.4	−	6820.47i	−0.870375	−	0.502511i	−0.867473	+	0.497485i	$0.834257\pi$
$570$		0			0
$571$	7222.20	+	12509.2i	0.529317	+	0.916803i	0.999415	+	0.0341895i	$0.0108850\pi$
$571$	7222.20	+	12509.2i	0.529317	+	0.916803i	−0.470099	+	0.882614i	$0.655782\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	32021.2i		−	2.32239i
$576$		0			0
$577$	19780.9	−	11420.5i	1.42719	−	0.823989i	0.430292	−	0.902690i	$-0.358411\pi$
$577$	19780.9	−	11420.5i	1.42719	−	0.823989i	0.996898	+	0.0787009i	$0.0250772\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24019.9	−	2177.36i	−1.71517	−	0.155477i
$582$		0			0
$583$	1503.74	−	2604.55i	0.106824	−	0.185025i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21564.8			1.51631			0.758156	−	0.652073i	$-0.226100\pi$
$587$	21564.8			1.51631			0.758156	+	0.652073i	$0.226100\pi$
$588$		0			0
$589$	2624.03			0.183567
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4857.29	+	8413.08i	−0.336366	+	0.582603i	−0.983746	−	0.179564i	$-0.942531\pi$
$593$	−4857.29	+	8413.08i	−0.336366	+	0.582603i	0.647380	+	0.762167i	$0.275865\pi$
$594$		0			0
$595$	−49339.6	−	4472.55i	−3.39954	−	0.308162i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5905.86	−	3409.75i	0.402849	−	0.232585i	−0.284863	−	0.958568i	$-0.591948\pi$
$599$	5905.86	−	3409.75i	0.402849	−	0.232585i	0.687713	+	0.725983i	$0.258615\pi$
$600$		0			0
$601$			23101.9i			1.56796i	0.620783	+	0.783982i	$0.286815\pi$
$601$			23101.9i			1.56796i	−0.620783	+	0.783982i	$0.713185\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10001.4	−	17323.0i	−0.672094	−	1.16410i
$606$		0			0
$607$	−734.613	−	424.129i	−0.0491219	−	0.0283606i	0.475238	−	0.879857i	$-0.342362\pi$
$607$	−734.613	−	424.129i	−0.0491219	−	0.0283606i	−0.524360	+	0.851497i	$0.675695\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4947.63	−	2856.52i	−0.327594	−	0.189136i
$612$		0			0
$613$	−7859.10	−	13612.4i	−0.517824	−	0.896897i	−0.999786	−	0.0207049i	$-0.993409\pi$
$613$	−7859.10	−	13612.4i	−0.517824	−	0.896897i	0.481962	−	0.876192i	$-0.339924\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24534.6i			1.60086i	0.599429	+	0.800428i	$0.295394\pi$
$617$			24534.6i			1.60086i	−0.599429	+	0.800428i	$0.704606\pi$
$618$		0			0
$619$	−1330.23	+	768.007i	−0.0863754	+	0.0498689i	−0.542566	−	0.840013i	$-0.682547\pi$
$619$	−1330.23	+	768.007i	−0.0863754	+	0.0498689i	0.456190	+	0.889882i	$0.349214\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4493.69	+	6375.03i	−0.288982	+	0.409968i
$624$		0			0
$625$	−8690.12	+	15051.7i	−0.556168	+	0.963311i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16088.5			1.01986
$630$		0			0
$631$	14860.1			0.937515			0.468758	−	0.883327i	$-0.344702\pi$
$631$	14860.1			0.937515			0.468758	+	0.883327i	$0.344702\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14694.5	−	25451.7i	0.918322	−	1.59058i
$636$		0			0
$637$	−1019.40	+	5576.64i	−0.0634069	+	0.346868i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17466.3	+	10084.2i	−1.07625	+	0.621373i	−0.929882	−	0.367857i	$-0.880092\pi$
$641$	−17466.3	+	10084.2i	−1.07625	+	0.621373i	−0.146368	+	0.989230i	$0.546758\pi$
$642$		0			0
$643$		−	3124.44i		−	0.191627i	−0.995399	−	0.0958133i	$-0.969455\pi$
$643$		−	3124.44i		−	0.191627i	0.995399	−	0.0958133i	$-0.0305452\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10143.8	−	17569.5i	−0.616372	−	1.06759i	−0.990142	−	0.140066i	$-0.955268\pi$
$647$	−10143.8	−	17569.5i	−0.616372	−	1.06759i	0.373770	−	0.927521i	$-0.378065\pi$
$648$		0			0
$649$	8991.77	+	5191.40i	0.543848	+	0.313991i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2651.27	−	1530.71i	−0.158885	−	0.0917325i	0.418449	−	0.908240i	$-0.362574\pi$
$653$	−2651.27	−	1530.71i	−0.158885	−	0.0917325i	−0.577335	+	0.816508i	$0.695907\pi$
$654$		0			0
$655$	−20799.4	−	36025.6i	−1.24076	−	2.14906i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3017.77i			0.178385i	0.996014	+	0.0891924i	$0.0284286\pi$
$659$			3017.77i			0.178385i	−0.996014	+	0.0891924i	$0.971571\pi$
$660$		0			0
$661$	−4671.67	+	2697.19i	−0.274897	+	0.158712i	−0.631111	−	0.775693i	$-0.717401\pi$
$661$	−4671.67	+	2697.19i	−0.274897	+	0.158712i	0.356214	+	0.934404i	$0.384067\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7827.43	+	3620.16i	−0.456443	+	0.211104i
$666$		0			0
$667$	6614.15	−	11456.0i	0.383959	−	0.665037i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11316.4			−0.651063
$672$		0			0
$673$	21062.8			1.20641			0.603204	−	0.797587i	$-0.293890\pi$
$673$	21062.8			1.20641			0.603204	+	0.797587i	$0.293890\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13505.8	−	23392.7i	0.766718	−	1.32800i	−0.172615	−	0.984989i	$-0.555222\pi$
$677$	13505.8	−	23392.7i	0.766718	−	1.32800i	0.939333	−	0.343006i	$-0.111445\pi$
$678$		0			0
$679$	−925.089	+	10205.3i	−0.0522852	+	0.576793i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12369.6	+	7141.59i	−0.692986	+	0.400095i	−0.804730	−	0.593642i	$-0.797690\pi$
$683$	−12369.6	+	7141.59i	−0.692986	+	0.400095i	0.111744	+	0.993737i	$0.464356\pi$
$684$		0			0
$685$		−	11132.3i		−	0.620940i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1424.60	+	2467.47i	0.0787704	+	0.136434i
$690$		0			0
$691$	−2246.31	−	1296.91i	−0.123666	−	0.0713989i	0.436891	−	0.899515i	$-0.356080\pi$
$691$	−2246.31	−	1296.91i	−0.123666	−	0.0713989i	−0.560557	+	0.828116i	$0.689413\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	50687.1	+	29264.2i	2.76644	+	1.59720i
$696$		0			0
$697$	23892.9	+	41383.7i	1.29843	+	2.24895i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			30185.3i			1.62637i	0.582009	+	0.813183i	$0.302267\pi$
$701$			30185.3i			1.62637i	−0.582009	+	0.813183i	$0.697733\pi$
$702$		0			0
$703$	2425.41	−	1400.31i	0.130123	−	0.0751263i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1802.89	−	3898.16i	−0.0959047	−	0.207363i
$708$		0			0
$709$	15514.3	−	26871.5i	0.821793	−	1.42339i	−0.0825520	−	0.996587i	$-0.526307\pi$
$709$	15514.3	−	26871.5i	0.821793	−	1.42339i	0.904345	−	0.426801i	$-0.140360\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13807.7			0.725246
$714$		0			0
$715$	−5618.10			−0.293853
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5323.93	+	9221.32i	−0.276146	+	0.478299i	−0.970424	−	0.241408i	$-0.922391\pi$
$719$	−5323.93	+	9221.32i	−0.276146	+	0.478299i	0.694277	+	0.719707i	$0.255724\pi$
$720$		0			0
$721$	29268.9	+	20631.3i	1.51183	+	1.06567i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−23195.0	+	13391.6i	−1.18819	+	0.686005i
$726$		0			0
$727$			11374.6i			0.580278i	0.956985	+	0.290139i	$0.0937015\pi$
$727$			11374.6i			0.580278i	−0.956985	+	0.290139i	$0.906299\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9744.57	+	16878.1i	0.493045	+	0.853979i
$732$		0			0
$733$	−19102.2	−	11028.7i	−0.962559	−	0.555734i	−0.0655991	−	0.997846i	$-0.520896\pi$
$733$	−19102.2	−	11028.7i	−0.962559	−	0.555734i	−0.896960	+	0.442113i	$0.854229\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1964.12	+	1133.99i	0.0981673	+	0.0566769i
$738$		0			0
$739$	13160.9	+	22795.4i	0.655119	+	1.13470i	0.981864	+	0.189587i	$0.0607149\pi$
$739$	13160.9	+	22795.4i	0.655119	+	1.13470i	−0.326745	+	0.945113i	$0.605952\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4269.63i			0.210818i	0.994429	+	0.105409i	$0.0336151\pi$
$743$			4269.63i			0.210818i	−0.994429	+	0.105409i	$0.966385\pi$
$744$		0			0
$745$	27930.7	−	16125.8i	1.37356	−	0.793025i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13458.6	+	9486.83i	0.656565	+	0.462806i
$750$		0			0
$751$	14942.7	−	25881.5i	0.726054	−	1.25756i	−0.232484	−	0.972600i	$-0.574685\pi$
$751$	14942.7	−	25881.5i	0.726054	−	1.25756i	0.958539	−	0.284963i	$-0.0919813\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15458.7			0.745163
$756$		0			0
$757$	−12658.2			−0.607756			−0.303878	−	0.952711i	$-0.598281\pi$
$757$	−12658.2			−0.607756			−0.303878	+	0.952711i	$0.598281\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13590.3	−	23539.0i	0.647367	−	1.12127i	−0.336382	−	0.941726i	$-0.609203\pi$
$761$	13590.3	−	23539.0i	0.647367	−	1.12127i	0.983749	−	0.179548i	$-0.0574634\pi$
$762$		0			0
$763$	−14944.7	−	32313.1i	−0.709089	−	1.53317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8518.52	+	4918.17i	−0.401025	+	0.231532i
$768$		0			0
$769$			21028.3i			0.986087i	0.870005	+	0.493044i	$0.164116\pi$
$769$			21028.3i			0.986087i	−0.870005	+	0.493044i	$0.835884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7220.65	+	12506.5i	0.335975	+	0.581926i	0.983672	−	0.179972i	$-0.0576009\pi$
$773$	7220.65	+	12506.5i	0.335975	+	0.581926i	−0.647697	+	0.761898i	$0.724268\pi$
$774$		0			0
$775$	−24210.9	−	13978.2i	−1.12217	−	0.647884i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7203.90	+	4159.17i	0.331330	+	0.191294i
$780$		0			0
$781$	−7926.36	−	13728.9i	−0.363159	−	0.629010i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13982.9i		−	0.635759i
$786$		0			0
$787$	17907.5	−	10338.9i	0.811098	−	0.468288i	−0.0362388	−	0.999343i	$-0.511538\pi$
$787$	17907.5	−	10338.9i	0.811098	−	0.468288i	0.847337	+	0.531055i	$0.178204\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3461.74	+	38188.8i	−0.155607	+	1.71661i
$792$		0			0
$793$	5360.38	−	9284.45i	0.240041	−	0.415764i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21175.0			−0.941099			−0.470550	−	0.882374i	$-0.655944\pi$
$797$	−21175.0			−0.941099			−0.470550	+	0.882374i	$0.655944\pi$
$798$		0			0
$799$	−47456.9			−2.10126
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−452.242	+	783.306i	−0.0198746	+	0.0344238i
$804$		0			0
$805$	−41188.0	+	19049.3i	−1.80334	+	0.834037i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10628.5	+	6136.39i	−0.461903	+	0.266680i	−0.712844	−	0.701323i	$-0.752593\pi$
$809$	−10628.5	+	6136.39i	−0.461903	+	0.266680i	0.250941	+	0.968002i	$0.419260\pi$
$810$		0			0
$811$		−	12975.7i		−	0.561821i	−0.959734	−	0.280910i	$-0.909364\pi$
$811$		−	12975.7i		−	0.561821i	0.959734	−	0.280910i	$-0.0906364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20241.5	−	35059.3i	−0.869973	−	1.50684i
$816$		0			0
$817$	2938.07	+	1696.29i	0.125814	+	0.0726387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12870.4	+	7430.74i	0.547114	+	0.315876i	0.747957	−	0.663747i	$-0.231035\pi$
$821$	12870.4	+	7430.74i	0.547114	+	0.315876i	−0.200843	+	0.979623i	$0.564368\pi$
$822$		0			0
$823$	−11450.4	−	19832.6i	−0.484975	−	0.840002i	0.514876	−	0.857265i	$-0.327838\pi$
$823$	−11450.4	−	19832.6i	−0.484975	−	0.840002i	−0.999851	+	0.0172628i	$0.994505\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22216.5i			0.934151i	0.884217	+	0.467076i	$0.154692\pi$
$827$			22216.5i			0.934151i	−0.884217	+	0.467076i	$0.845308\pi$
$828$		0			0
$829$	14146.5	−	8167.47i	0.592674	−	0.342181i	−0.173480	−	0.984837i	$-0.555501\pi$
$829$	14146.5	−	8167.47i	0.592674	−	0.342181i	0.766154	+	0.642657i	$0.222168\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15828.5	+	44351.7i	0.658375	+	1.84477i
$834$		0			0
$835$	−9911.28	+	17166.8i	−0.410771	+	0.711476i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3186.11			0.131104			0.0655522	−	0.997849i	$-0.479119\pi$
$839$	3186.11			0.131104			0.0655522	+	0.997849i	$0.479119\pi$
$840$		0			0
$841$	13324.5			0.546334
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18741.9	+	32461.9i	−0.763007	+	1.32157i
$846$		0			0
$847$	−10954.5	+	15540.7i	−0.444393	+	0.630444i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12762.5	−	7368.45i	0.514094	−	0.296812i
$852$		0			0
$853$			15850.7i			0.636245i	0.948050	+	0.318123i	$0.103052\pi$
$853$			15850.7i			0.636245i	−0.948050	+	0.318123i	$0.896948\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10826.0	+	18751.2i	0.431517	+	0.747410i	0.997004	−	0.0773473i	$-0.0246450\pi$
$857$	10826.0	+	18751.2i	0.431517	+	0.747410i	−0.565487	+	0.824757i	$0.691312\pi$
$858$		0			0
$859$	11070.4	+	6391.50i	0.439717	+	0.253871i	0.703478	−	0.710717i	$-0.251630\pi$
$859$	11070.4	+	6391.50i	0.439717	+	0.253871i	−0.263760	+	0.964588i	$0.584963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21740.9	+	12552.1i	0.857555	+	0.495110i	0.863193	−	0.504874i	$-0.168461\pi$
$863$	21740.9	+	12552.1i	0.857555	+	0.495110i	−0.00563770	+	0.999984i	$0.501795\pi$
$864$		0			0
$865$	−29196.0	−	50568.9i	−1.14762	−	1.98774i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			16570.8i			0.646866i
$870$		0			0
$871$	−1860.75	+	1074.30i	−0.0723869	+	0.0417926i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	46583.4	+	4222.70i	1.79978	+	0.163147i
$876$		0			0
$877$	−25579.4	+	44304.9i	−0.984898	+	1.70589i	−0.342510	+	0.939514i	$0.611277\pi$
$877$	−25579.4	+	44304.9i	−0.984898	+	1.70589i	−0.642388	+	0.766379i	$0.722056\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21063.7			−0.805510			−0.402755	−	0.915308i	$-0.631947\pi$
$881$	−21063.7			−0.805510			−0.402755	+	0.915308i	$0.631947\pi$
$882$		0			0
$883$	−51.5775			−0.00196571			−0.000982855	−	1.00000i	$-0.500313\pi$
$883$	−51.5775			−0.00196571			−0.000982855		1.00000i	$0.500313\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4460.79	−	7726.31i	0.168860	−	0.292474i	−0.769159	−	0.639057i	$-0.779325\pi$
$887$	4460.79	−	7726.31i	0.168860	−	0.292474i	0.938019	+	0.346583i	$0.112658\pi$
$888$		0			0
$889$	−27821.4	−	2521.96i	−1.04961	−	0.0951449i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7154.32	+	4130.55i	−0.268097	+	0.154786i
$894$		0			0
$895$			78147.4i			2.91864i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5774.53	−	10001.8i	−0.214228	−	0.371054i
$900$		0			0
$901$	20496.8	+	11833.8i	0.757876	+	0.437560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	48695.2	+	28114.2i	1.78860	+	1.03265i
$906$		0			0
$907$	2808.30	+	4864.13i	0.102810	+	0.178071i	0.912841	−	0.408315i	$-0.133883\pi$
$907$	2808.30	+	4864.13i	0.102810	+	0.178071i	−0.810032	+	0.586386i	$0.800550\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			19604.1i			0.712967i	0.934302	+	0.356484i	$0.116024\pi$
$911$			19604.1i			0.712967i	−0.934302	+	0.356484i	$0.883976\pi$
$912$		0			0
$913$	−19675.6	+	11359.7i	−0.713217	+	0.411776i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22781.4	+	32319.1i	−0.820401	+	1.16387i
$918$		0			0
$919$	23979.7	−	41534.1i	0.860737	−	1.49084i	−0.0104809	−	0.999945i	$-0.503336\pi$
$919$	23979.7	−	41534.1i	0.860737	−	1.49084i	0.871218	−	0.490896i	$-0.163330\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	15018.4			0.535575
$924$		0			0
$925$	−29837.8			−1.06060
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6868.87	−	11897.2i	0.242584	−	0.420168i	−0.718866	−	0.695149i	$-0.755338\pi$
$929$	6868.87	−	11897.2i	0.242584	−	0.420168i	0.961450	+	0.274981i	$0.0886717\pi$
$930$		0			0
$931$	6246.49	+	5308.52i	0.219893	+	0.186874i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40416.0	+	23334.2i	−1.41363	+	0.816160i
$936$		0			0
$937$			13590.7i			0.473842i	0.971529	+	0.236921i	$0.0761383\pi$
$937$			13590.7i			0.473842i	−0.971529	+	0.236921i	$0.923862\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12655.2	−	21919.4i	−0.438414	−	0.759355i	0.559153	−	0.829064i	$-0.311126\pi$
$941$	−12655.2	−	21919.4i	−0.438414	−	0.759355i	−0.997567	+	0.0697090i	$0.977793\pi$
$942$		0			0
$943$	37906.9	+	21885.6i	1.30903	+	0.755771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−43071.4	−	24867.3i	−1.47796	−	0.853303i	−0.478274	−	0.878211i	$-0.658737\pi$
$947$	−43071.4	−	24867.3i	−1.47796	−	0.853303i	−0.999690	+	0.0249082i	$0.992071\pi$
$948$		0			0
$949$	−428.440	−	742.080i	−0.0146552	−	0.0253835i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26069.9i		−	0.886135i	−0.896488	−	0.443068i	$-0.853890\pi$
$953$		−	26069.9i		−	0.886135i	0.896488	−	0.443068i	$-0.146110\pi$
$954$		0			0
$955$	−22794.1	+	13160.2i	−0.772354	+	0.445919i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9604.26	+	4441.94i	−0.323397	+	0.149570i
$960$		0			0
$961$	−8868.07	+	15359.9i	−0.297676	+	0.515590i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12307.7			0.410569
$966$		0			0
$967$	−40037.9			−1.33147			−0.665736	−	0.746187i	$-0.731882\pi$
$967$	−40037.9			−1.33147			−0.665736	+	0.746187i	$0.731882\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17155.3	+	29713.8i	−0.566981	+	0.982041i	0.429881	+	0.902886i	$0.358556\pi$
$971$	−17155.3	+	29713.8i	−0.566981	+	0.982041i	−0.996862	+	0.0791551i	$0.974778\pi$
$972$		0			0
$973$	5022.50	−	55406.5i	0.165482	−	1.82554i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21799.6	−	12586.0i	0.713849	−	0.412141i	−0.0986353	−	0.995124i	$-0.531448\pi$
$977$	21799.6	−	12586.0i	0.713849	−	0.412141i	0.812485	+	0.582982i	$0.198114\pi$
$978$		0			0
$979$			7347.24i			0.239856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7087.40	+	12275.7i	0.229962	+	0.398306i	0.957797	−	0.287447i	$-0.0928064\pi$
$983$	7087.40	+	12275.7i	0.229962	+	0.398306i	−0.727834	+	0.685753i	$0.759473\pi$
$984$		0			0
$985$	−39668.3	−	22902.5i	−1.28319	−	0.740848i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	15460.1	+	8925.90i	0.497071	+	0.286984i
$990$		0			0
$991$	−27259.1	−	47214.1i	−0.873777	−	1.51343i	−0.858060	−	0.513550i	$-0.828330\pi$
$991$	−27259.1	−	47214.1i	−0.873777	−	1.51343i	−0.0157175	−	0.999876i	$-0.505003\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			21827.0i			0.695439i
$996$		0			0
$997$	−381.200	+	220.086i	−0.0121090	+	0.00699116i	−0.506042	−	0.862509i	$-0.668892\pi$
$997$	−381.200	+	220.086i	−0.0121090	+	0.00699116i	0.493933	+	0.869500i	$0.335559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.593.2		48
3.2	odd	2	inner	1008.4.bt.d.593.23		48
4.3	odd	2		504.4.bl.a.89.2	yes	48
7.3	odd	6	inner	1008.4.bt.d.17.23		48
12.11	even	2		504.4.bl.a.89.23	yes	48
21.17	even	6	inner	1008.4.bt.d.17.2		48
28.3	even	6		504.4.bl.a.17.23	yes	48
84.59	odd	6		504.4.bl.a.17.2	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.2	✓	48	84.59	odd	6
504.4.bl.a.17.23	yes	48	28.3	even	6
504.4.bl.a.89.2	yes	48	4.3	odd	2
504.4.bl.a.89.23	yes	48	12.11	even	2
1008.4.bt.d.17.2		48	21.17	even	6	inner
1008.4.bt.d.17.23		48	7.3	odd	6	inner
1008.4.bt.d.593.2		48	1.1	even	1	trivial
1008.4.bt.d.593.23		48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.74197 + 16.8736i) q^{5} +(18.4446 + 1.67197i) q^{7} +O(q^{10})\)	\(q+(-9.74197 + 16.8736i) q^{5} +(18.4446 + 1.67197i) q^{7} +(15.1087 - 8.72302i) q^{11} +16.5278i q^{13} +(68.6466 + 118.899i) q^{17} +(20.6975 + 11.9497i) q^{19} +(108.910 + 62.8794i) q^{23} +(-127.312 - 220.511i) q^{25} -105.188i q^{29} +(95.0849 - 54.8973i) q^{31} +(-207.899 + 294.939i) q^{35} +(58.5919 - 101.484i) q^{37} +348.056 q^{41} +141.953 q^{43} +(-172.830 + 299.351i) q^{47} +(337.409 + 61.6778i) q^{49} +(149.292 - 86.1937i) q^{53} +339.918i q^{55} +(297.569 + 515.404i) q^{59} +(-561.746 - 324.324i) q^{61} +(-278.884 - 161.014i) q^{65} +(64.9996 + 112.583i) q^{67} -908.671i q^{71} +(-44.8988 + 25.9223i) q^{73} +(293.259 - 135.632i) q^{77} +(-474.917 + 822.580i) q^{79} -1302.27 q^{83} -2675.01 q^{85} +(-210.570 + 364.719i) q^{89} +(-27.6341 + 304.850i) q^{91} +(-403.269 + 232.827i) q^{95} +553.292i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 48 * q + 24 * q^7	\( 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) \) 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more