LMFDB - Embedded newform 1344.4.p.c.223.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (of order $2$, degree $1$, 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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.4
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +25.0429 q^{11} +64.1604 q^{13} -38.4675i q^{15} -77.4251i q^{17} +49.9995i q^{19} +(-52.2158 + 18.9872i) q^{21} +80.5687i q^{23} +39.4169 q^{25} -27.0000i q^{27} -159.100i q^{29} +96.5596 q^{31} +75.1287i q^{33} +(-81.1547 - 223.179i) q^{35} -274.101i q^{37} +192.481i q^{39} -299.796i q^{41} -385.824 q^{43} +115.403 q^{45} +418.362 q^{47} +(-262.886 + 220.318i) q^{49} +232.275 q^{51} -665.385i q^{53} -321.113 q^{55} -149.999 q^{57} -445.655i q^{59} +599.783 q^{61} +(-56.9617 - 156.647i) q^{63} -822.697 q^{65} +675.504 q^{67} -241.706 q^{69} +877.093i q^{71} +696.060i q^{73} +118.251i q^{75} +(158.498 + 435.878i) q^{77} -1248.56i q^{79} +81.0000 q^{81} +238.303i q^{83} +992.784i q^{85} +477.299 q^{87} +743.503i q^{89} +(406.076 + 1116.73i) q^{91} +289.679i q^{93} -641.120i q^{95} +1552.16i q^{97} -225.386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 288 q^{9}+O(q^{10}) $ 32 * q - 288 * q^9	$ 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) $ 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−12.8225			−1.14688			−0.573440	−	0.819247i	$-0.694392\pi$
$5$	−12.8225			−1.14688			−0.573440	+	0.819247i	$0.694392\pi$
$6$		0			0
$7$	6.32908	+	17.4053i	0.341738	+	0.939795i
$7$	6.32908	+	17.4053i	0.341738	+	0.939795i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	25.0429			0.686429			0.343214	−	0.939257i	$-0.388484\pi$
$11$	25.0429			0.686429			0.343214	+	0.939257i	$0.388484\pi$
$12$		0			0
$13$	64.1604			1.36884			0.684419	−	0.729089i	$-0.260056\pi$
$13$	64.1604			1.36884			0.684419	+	0.729089i	$0.260056\pi$
$14$		0			0
$15$		−	38.4675i		−	0.662152i
$16$		0			0
$17$		−	77.4251i		−	1.10461i	−0.833643	−	0.552304i	$-0.813749\pi$
$17$		−	77.4251i		−	1.10461i	0.833643	−	0.552304i	$-0.186251\pi$
$18$		0			0
$19$			49.9995i			0.603720i	0.953352	+	0.301860i	$0.0976075\pi$
$19$			49.9995i			0.603720i	−0.953352	+	0.301860i	$0.902393\pi$
$20$		0			0
$21$	−52.2158	+	18.9872i	−0.542591	+	0.197303i
$22$		0			0
$23$			80.5687i			0.730423i	0.930925	+	0.365212i	$0.119003\pi$
$23$			80.5687i			0.730423i	−0.930925	+	0.365212i	$0.880997\pi$
$24$		0			0
$25$	39.4169			0.315335
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$		−	159.100i		−	1.01876i	−0.860541	−	0.509381i	$-0.829874\pi$
$29$		−	159.100i		−	1.01876i	0.860541	−	0.509381i	$-0.170126\pi$
$30$		0			0
$31$	96.5596			0.559440			0.279720	−	0.960082i	$-0.409758\pi$
$31$	96.5596			0.559440			0.279720	+	0.960082i	$0.409758\pi$
$32$		0			0
$33$			75.1287i			0.396310i
$34$		0			0
$35$	−81.1547	−	223.179i	−0.391933	−	1.07783i
$36$		0			0
$37$		−	274.101i		−	1.21789i	−0.793212	−	0.608946i	$-0.791593\pi$
$37$		−	274.101i		−	1.21789i	0.793212	−	0.608946i	$-0.208407\pi$
$38$		0			0
$39$			192.481i			0.790299i
$40$		0			0
$41$		−	299.796i		−	1.14196i	−0.820964	−	0.570980i	$-0.806564\pi$
$41$		−	299.796i		−	1.14196i	0.820964	−	0.570980i	$-0.193436\pi$
$42$		0			0
$43$	−385.824			−1.36832			−0.684158	−	0.729334i	$-0.739830\pi$
$43$	−385.824			−1.36832			−0.684158	+	0.729334i	$0.739830\pi$
$44$		0			0
$45$	115.403			0.382294
$46$		0			0
$47$	418.362			1.29839			0.649196	−	0.760622i	$-0.275106\pi$
$47$	418.362			1.29839			0.649196	+	0.760622i	$0.275106\pi$
$48$		0			0
$49$	−262.886	+	220.318i	−0.766430	+	0.642327i
$50$		0			0
$51$	232.275			0.637746
$52$		0			0
$53$		−	665.385i		−	1.72448i	−0.506497	−	0.862242i	$-0.669060\pi$
$53$		−	665.385i		−	1.72448i	0.506497	−	0.862242i	$-0.330940\pi$
$54$		0			0
$55$	−321.113			−0.787252
$56$		0			0
$57$	−149.999			−0.348558
$58$		0			0
$59$		−	445.655i		−	0.983380i	−0.870770	−	0.491690i	$-0.836379\pi$
$59$		−	445.655i		−	0.983380i	0.870770	−	0.491690i	$-0.163621\pi$
$60$		0			0
$61$	599.783			1.25892			0.629461	−	0.777032i	$-0.283276\pi$
$61$	599.783			1.25892			0.629461	+	0.777032i	$0.283276\pi$
$62$		0			0
$63$	−56.9617	−	156.647i	−0.113913	−	0.313265i
$64$		0			0
$65$	−822.697			−1.56989
$66$		0			0
$67$	675.504			1.23173			0.615866	−	0.787851i	$-0.288806\pi$
$67$	675.504			1.23173			0.615866	+	0.787851i	$0.288806\pi$
$68$		0			0
$69$	−241.706			−0.421710
$70$		0			0
$71$			877.093i			1.46608i	0.680185	+	0.733041i	$0.261900\pi$
$71$			877.093i			1.46608i	−0.680185	+	0.733041i	$0.738100\pi$
$72$		0			0
$73$			696.060i			1.11600i	0.829842	+	0.557998i	$0.188430\pi$
$73$			696.060i			1.11600i	−0.829842	+	0.557998i	$0.811570\pi$
$74$		0			0
$75$			118.251i			0.182059i
$76$		0			0
$77$	158.498	+	435.878i	0.234579	+	0.645102i
$78$		0			0
$79$		−	1248.56i		−	1.77815i	−0.457760	−	0.889076i	$-0.651348\pi$
$79$		−	1248.56i		−	1.77815i	0.457760	−	0.889076i	$-0.348652\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			238.303i			0.315146i	0.987507	+	0.157573i	$0.0503670\pi$
$83$			238.303i			0.315146i	−0.987507	+	0.157573i	$0.949633\pi$
$84$		0			0
$85$			992.784i			1.26685i
$86$		0			0
$87$	477.299			0.588182
$88$		0			0
$89$			743.503i			0.885519i	0.896640	+	0.442759i	$0.146000\pi$
$89$			743.503i			0.885519i	−0.896640	+	0.442759i	$0.854000\pi$
$90$		0			0
$91$	406.076	+	1116.73i	0.467784	+	1.28643i
$92$		0			0
$93$			289.679i			0.322993i
$94$		0			0
$95$		−	641.120i		−	0.692395i
$96$		0			0
$97$			1552.16i			1.62472i	0.583153	+	0.812362i	$0.301819\pi$
$97$			1552.16i			1.62472i	−0.583153	+	0.812362i	$0.698181\pi$
$98$		0			0
$99$	−225.386			−0.228810
$100$		0			0
$101$	1202.46			1.18464			0.592322	−	0.805701i	$-0.298211\pi$
$101$	1202.46			1.18464			0.592322	+	0.805701i	$0.298211\pi$
$102$		0			0
$103$	404.562			0.387016			0.193508	−	0.981099i	$-0.438013\pi$
$103$	404.562			0.387016			0.193508	+	0.981099i	$0.438013\pi$
$104$		0			0
$105$	669.537	−	243.464i	0.622287	−	0.226282i
$106$		0			0
$107$	1065.72			0.962871			0.481436	−	0.876481i	$-0.340116\pi$
$107$	1065.72			0.962871			0.481436	+	0.876481i	$0.340116\pi$
$108$		0			0
$109$			359.515i			0.315920i	0.987445	+	0.157960i	$0.0504918\pi$
$109$			359.515i			0.315920i	−0.987445	+	0.157960i	$0.949508\pi$
$110$		0			0
$111$	822.304			0.703150
$112$		0			0
$113$	828.461			0.689691			0.344845	−	0.938660i	$-0.387931\pi$
$113$	828.461			0.689691			0.344845	+	0.938660i	$0.387931\pi$
$114$		0			0
$115$		−	1033.09i		−	0.837708i
$116$		0			0
$117$	−577.443			−0.456279
$118$		0			0
$119$	1347.60	−	490.029i	1.03811	−	0.377486i
$120$		0			0
$121$	−703.854			−0.528816
$122$		0			0
$123$	899.389			0.659310
$124$		0			0
$125$	1097.39			0.785229
$126$		0			0
$127$			876.572i			0.612466i	0.951957	+	0.306233i	$0.0990687\pi$
$127$			876.572i			0.612466i	−0.951957	+	0.306233i	$0.900931\pi$
$128$		0			0
$129$		−	1157.47i		−	0.789998i
$130$		0			0
$131$		−	1083.04i		−	0.722335i	−0.932501	−	0.361167i	$-0.882378\pi$
$131$		−	1083.04i		−	0.722335i	0.932501	−	0.361167i	$-0.117622\pi$
$132$		0			0
$133$	−870.254	+	316.451i	−0.567373	+	0.206314i
$134$		0			0
$135$			346.208i			0.220717i
$136$		0			0
$137$	−751.836			−0.468859			−0.234430	−	0.972133i	$-0.575322\pi$
$137$	−751.836			−0.468859			−0.234430	+	0.972133i	$0.575322\pi$
$138$		0			0
$139$		−	543.555i		−	0.331681i	−0.986153	−	0.165841i	$-0.946966\pi$
$139$		−	543.555i		−	0.331681i	0.986153	−	0.165841i	$-0.0530337\pi$
$140$		0			0
$141$			1255.09i			0.749626i
$142$		0			0
$143$	1606.76			0.939609
$144$		0			0
$145$			2040.06i			1.16840i
$146$		0			0
$147$	−660.955	−	788.657i	−0.370848	−	0.442499i
$148$		0			0
$149$		−	1056.44i		−	0.580855i	−0.956897	−	0.290427i	$-0.906203\pi$
$149$		−	1056.44i		−	0.580855i	0.956897	−	0.290427i	$-0.0937974\pi$
$150$		0			0
$151$			2092.13i			1.12752i	0.825939	+	0.563759i	$0.190645\pi$
$151$			2092.13i			1.12752i	−0.825939	+	0.563759i	$0.809355\pi$
$152$		0			0
$153$			696.826i			0.368203i
$154$		0			0
$155$	−1238.14			−0.641610
$156$		0			0
$157$	1185.70			0.602735			0.301367	−	0.953508i	$-0.402557\pi$
$157$	1185.70			0.602735			0.301367	+	0.953508i	$0.402557\pi$
$158$		0			0
$159$	1996.15			0.995631
$160$		0			0
$161$	−1402.32	+	509.925i	−0.686448	+	0.249613i
$162$		0			0
$163$	903.635			0.434222			0.217111	−	0.976147i	$-0.430337\pi$
$163$	903.635			0.434222			0.217111	+	0.976147i	$0.430337\pi$
$164$		0			0
$165$		−	963.338i		−	0.454520i
$166$		0			0
$167$	−474.632			−0.219929			−0.109964	−	0.993936i	$-0.535074\pi$
$167$	−474.632			−0.219929			−0.109964	+	0.993936i	$0.535074\pi$
$168$		0			0
$169$	1919.55			0.873715
$170$		0			0
$171$		−	449.996i		−	0.201240i
$172$		0			0
$173$	−1641.33			−0.721317			−0.360658	−	0.932698i	$-0.617448\pi$
$173$	−1641.33			−0.721317			−0.360658	+	0.932698i	$0.617448\pi$
$174$		0			0
$175$	249.473	+	686.061i	0.107762	+	0.296351i
$176$		0			0
$177$	1336.97			0.567755
$178$		0			0
$179$	1802.11			0.752491			0.376246	−	0.926520i	$-0.377215\pi$
$179$	1802.11			0.752491			0.376246	+	0.926520i	$0.377215\pi$
$180$		0			0
$181$	2068.45			0.849429			0.424715	−	0.905327i	$-0.360374\pi$
$181$	2068.45			0.849429			0.424715	+	0.905327i	$0.360374\pi$
$182$		0			0
$183$			1799.35i			0.726840i
$184$		0			0
$185$			3514.67i			1.39678i
$186$		0			0
$187$		−	1938.95i		−	0.758234i
$188$		0			0
$189$	469.942	−	170.885i	0.180864	−	0.0657675i
$190$		0			0
$191$			2179.61i			0.825711i	0.910797	+	0.412855i	$0.135469\pi$
$191$			2179.61i			0.825711i	−0.910797	+	0.412855i	$0.864531\pi$
$192$		0			0
$193$	−4489.81			−1.67453			−0.837263	−	0.546801i	$-0.815845\pi$
$193$	−4489.81			−1.67453			−0.837263	+	0.546801i	$0.815845\pi$
$194$		0			0
$195$		−	2468.09i		−	0.906378i
$196$		0			0
$197$			4519.08i			1.63437i	0.576376	+	0.817185i	$0.304467\pi$
$197$			4519.08i			1.63437i	−0.576376	+	0.817185i	$0.695533\pi$
$198$		0			0
$199$	826.598			0.294452			0.147226	−	0.989103i	$-0.452966\pi$
$199$	826.598			0.294452			0.147226	+	0.989103i	$0.452966\pi$
$200$		0			0
$201$			2026.51i			0.711140i
$202$		0			0
$203$	2769.17	−	1006.95i	0.957428	−	0.348150i
$204$		0			0
$205$			3844.14i			1.30969i
$206$		0			0
$207$		−	725.118i		−	0.243474i
$208$		0			0
$209$			1252.13i			0.414411i
$210$		0			0
$211$	−4709.17			−1.53646			−0.768229	−	0.640175i	$-0.778862\pi$
$211$	−4709.17			−1.53646			−0.768229	+	0.640175i	$0.778862\pi$
$212$		0			0
$213$	−2631.28			−0.846443
$214$		0			0
$215$	4947.23			1.56930
$216$		0			0
$217$	611.133	+	1680.65i	0.191182	+	0.525759i
$218$		0			0
$219$	−2088.18			−0.644321
$220$		0			0
$221$		−	4967.62i		−	1.51203i
$222$		0			0
$223$	5775.09			1.73421			0.867104	−	0.498127i	$-0.165979\pi$
$223$	5775.09			1.73421			0.867104	+	0.498127i	$0.165979\pi$
$224$		0			0
$225$	−354.752			−0.105112
$226$		0			0
$227$		−	2280.11i		−	0.666678i	−0.942807	−	0.333339i	$-0.891825\pi$
$227$		−	2280.11i		−	0.666678i	0.942807	−	0.333339i	$-0.108175\pi$
$228$		0			0
$229$	3692.99			1.06567			0.532837	−	0.846218i	$-0.321126\pi$
$229$	3692.99			1.06567			0.532837	+	0.846218i	$0.321126\pi$
$230$		0			0
$231$	−1307.63	+	475.495i	−0.372450	+	0.135434i
$232$		0			0
$233$	3140.07			0.882887			0.441443	−	0.897289i	$-0.354467\pi$
$233$	3140.07			0.882887			0.441443	+	0.897289i	$0.354467\pi$
$234$		0			0
$235$	−5364.45			−1.48910
$236$		0			0
$237$	3745.68			1.02662
$238$		0			0
$239$		−	310.784i		−	0.0841127i	−0.999115	−	0.0420563i	$-0.986609\pi$
$239$		−	310.784i		−	0.0841127i	0.999115	−	0.0420563i	$-0.0133909\pi$
$240$		0			0
$241$		−	457.833i		−	0.122372i	−0.998126	−	0.0611859i	$-0.980512\pi$
$241$		−	457.833i		−	0.122372i	0.998126	−	0.0611859i	$-0.0194883\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	3370.85	−	2825.04i	0.879004	−	0.736673i
$246$		0			0
$247$			3207.99i			0.826394i
$248$		0			0
$249$	−714.909			−0.181950
$250$		0			0
$251$			807.916i			0.203168i	0.994827	+	0.101584i	$0.0323911\pi$
$251$			807.916i			0.203168i	−0.994827	+	0.101584i	$0.967609\pi$
$252$		0			0
$253$			2017.67i			0.501383i
$254$		0			0
$255$	−2978.35			−0.731418
$256$		0			0
$257$		−	1749.36i		−	0.424599i	−0.977205	−	0.212299i	$-0.931905\pi$
$257$		−	1749.36i		−	0.424599i	0.977205	−	0.212299i	$-0.0680952\pi$
$258$		0			0
$259$	4770.80	−	1734.81i	1.14457	−	0.416200i
$260$		0			0
$261$			1431.90i			0.339587i
$262$		0			0
$263$		−	1360.35i		−	0.318945i	−0.987202	−	0.159473i	$-0.949021\pi$
$263$		−	1360.35i		−	0.318945i	0.987202	−	0.159473i	$-0.0509793\pi$
$264$		0			0
$265$			8531.91i			1.97778i
$266$		0			0
$267$	−2230.51			−0.511255
$268$		0			0
$269$	2224.06			0.504102			0.252051	−	0.967714i	$-0.418895\pi$
$269$	2224.06			0.504102			0.252051	+	0.967714i	$0.418895\pi$
$270$		0			0
$271$	−2918.59			−0.654212			−0.327106	−	0.944988i	$-0.606073\pi$
$271$	−2918.59			−0.654212			−0.327106	+	0.944988i	$0.606073\pi$
$272$		0			0
$273$	−3350.18	+	1218.23i	−0.742719	+	0.270075i
$274$		0			0
$275$	987.113			0.216455
$276$		0			0
$277$		−	1632.84i		−	0.354180i	−0.984195	−	0.177090i	$-0.943332\pi$
$277$		−	1632.84i		−	0.354180i	0.984195	−	0.177090i	$-0.0566684\pi$
$278$		0			0
$279$	−869.037			−0.186480
$280$		0			0
$281$	−7615.42			−1.61672			−0.808360	−	0.588689i	$-0.799644\pi$
$281$	−7615.42			−1.61672			−0.808360	+	0.588689i	$0.799644\pi$
$282$		0			0
$283$		−	2867.07i		−	0.602226i	−0.953589	−	0.301113i	$-0.902642\pi$
$283$		−	2867.07i		−	0.602226i	0.953589	−	0.301113i	$-0.0973581\pi$
$284$		0			0
$285$	1923.36			0.399754
$286$		0			0
$287$	5218.03	−	1897.43i	1.07321	−	0.390251i
$288$		0			0
$289$	−1081.64			−0.220159
$290$		0			0
$291$	−4656.49			−0.938035
$292$		0			0
$293$	−1133.07			−0.225920			−0.112960	−	0.993600i	$-0.536033\pi$
$293$	−1133.07			−0.225920			−0.112960	+	0.993600i	$0.536033\pi$
$294$		0			0
$295$			5714.42i			1.12782i
$296$		0			0
$297$		−	676.158i		−	0.132103i
$298$		0			0
$299$			5169.32i			0.999831i
$300$		0			0
$301$	−2441.91	−	6715.36i	−0.467606	−	1.28594i
$302$		0			0
$303$			3607.37i			0.683954i
$304$		0			0
$305$	−7690.72			−1.44383
$306$		0			0
$307$			206.149i			0.0383242i	0.999816	+	0.0191621i	$0.00609985\pi$
$307$			206.149i			0.0383242i	−0.999816	+	0.0191621i	$0.993900\pi$
$308$		0			0
$309$			1213.69i			0.223444i
$310$		0			0
$311$	6296.00			1.14795			0.573977	−	0.818871i	$-0.305400\pi$
$311$	6296.00			1.14795			0.573977	+	0.818871i	$0.305400\pi$
$312$		0			0
$313$		−	7997.75i		−	1.44428i	−0.691747	−	0.722140i	$-0.743158\pi$
$313$		−	7997.75i		−	1.44428i	0.691747	−	0.722140i	$-0.256842\pi$
$314$		0			0
$315$	730.392	+	2008.61i	0.130644	+	0.359278i
$316$		0			0
$317$			5830.42i			1.03303i	0.856279	+	0.516513i	$0.172770\pi$
$317$			5830.42i			1.03303i	−0.856279	+	0.516513i	$0.827230\pi$
$318$		0			0
$319$		−	3984.32i		−	0.699307i
$320$		0			0
$321$			3197.17i			0.555914i
$322$		0			0
$323$	3871.22			0.666874
$324$		0			0
$325$	2529.00			0.431643
$326$		0			0
$327$	−1078.55			−0.182397
$328$		0			0
$329$	2647.85	+	7281.70i	0.443710	+	1.22022i
$330$		0			0
$331$	−5022.44			−0.834013			−0.417007	−	0.908903i	$-0.636921\pi$
$331$	−5022.44			−0.834013			−0.417007	+	0.908903i	$0.636921\pi$
$332$		0			0
$333$			2466.91i			0.405964i
$334$		0			0
$335$	−8661.66			−1.41265
$336$		0			0
$337$	8480.49			1.37081			0.685403	−	0.728164i	$-0.259626\pi$
$337$	8480.49			1.37081			0.685403	+	0.728164i	$0.259626\pi$
$338$		0			0
$339$			2485.38i			0.398193i
$340$		0			0
$341$	2418.13			0.384015
$342$		0			0
$343$	−5498.52	−	3181.18i	−0.865575	−	0.500780i
$344$		0			0
$345$	3099.28			0.483651
$346$		0			0
$347$	9890.27			1.53008			0.765040	−	0.643983i	$-0.222719\pi$
$347$	9890.27			1.53008			0.765040	+	0.643983i	$0.222719\pi$
$348$		0			0
$349$	−6913.69			−1.06041			−0.530203	−	0.847871i	$-0.677884\pi$
$349$	−6913.69			−1.06041			−0.530203	+	0.847871i	$0.677884\pi$
$350$		0			0
$351$		−	1732.33i		−	0.263433i
$352$		0			0
$353$			5062.99i			0.763388i	0.924289	+	0.381694i	$0.124659\pi$
$353$			5062.99i			0.763388i	−0.924289	+	0.381694i	$0.875341\pi$
$354$		0			0
$355$		−	11246.5i		−	1.68142i
$356$		0			0
$357$	1470.09	+	4042.81i	0.217942	+	0.599350i
$358$		0			0
$359$		−	7551.38i		−	1.11016i	−0.831798	−	0.555079i	$-0.812688\pi$
$359$		−	7551.38i		−	1.11016i	0.831798	−	0.555079i	$-0.187312\pi$
$360$		0			0
$361$	4359.05			0.635522
$362$		0			0
$363$		−	2111.56i		−	0.305312i
$364$		0			0
$365$		−	8925.25i		−	1.27991i
$366$		0			0
$367$	12597.1			1.79173			0.895864	−	0.444329i	$-0.146558\pi$
$367$	12597.1			1.79173			0.895864	+	0.444329i	$0.146558\pi$
$368$		0			0
$369$			2698.17i			0.380653i
$370$		0			0
$371$	11581.2	−	4211.27i	1.62066	−	0.589322i
$372$		0			0
$373$			9549.28i			1.32558i	0.748803	+	0.662792i	$0.230629\pi$
$373$			9549.28i			1.32558i	−0.748803	+	0.662792i	$0.769371\pi$
$374$		0			0
$375$			3292.17i			0.453352i
$376$		0			0
$377$		−	10207.9i		−	1.39452i
$378$		0			0
$379$	1603.13			0.217276			0.108638	−	0.994081i	$-0.465351\pi$
$379$	1603.13			0.217276			0.108638	+	0.994081i	$0.465351\pi$
$380$		0			0
$381$	−2629.72			−0.353608
$382$		0			0
$383$	402.889			0.0537511			0.0268756	−	0.999639i	$-0.491444\pi$
$383$	402.889			0.0537511			0.0268756	+	0.999639i	$0.491444\pi$
$384$		0			0
$385$	−2032.35	−	5589.05i	−0.269034	−	0.739855i
$386$		0			0
$387$	3472.42			0.456105
$388$		0			0
$389$		−	8540.21i		−	1.11313i	−0.830806	−	0.556563i	$-0.812120\pi$
$389$		−	8540.21i		−	1.11313i	0.830806	−	0.556563i	$-0.187880\pi$
$390$		0			0
$391$	6238.04			0.806831
$392$		0			0
$393$	3249.13			0.417040
$394$		0			0
$395$			16009.7i			2.03933i
$396$		0			0
$397$	−11977.7			−1.51421			−0.757106	−	0.653292i	$-0.773387\pi$
$397$	−11977.7			−1.51421			−0.757106	+	0.653292i	$0.773387\pi$
$398$		0			0
$399$	−949.352	−	2610.76i	−0.119115	−	0.327573i
$400$		0			0
$401$	58.8089			0.00732364			0.00366182	−	0.999993i	$-0.498834\pi$
$401$	58.8089			0.00732364			0.00366182	+	0.999993i	$0.498834\pi$
$402$		0			0
$403$	6195.30			0.765782
$404$		0			0
$405$	−1038.62			−0.127431
$406$		0			0
$407$		−	6864.29i		−	0.835996i
$408$		0			0
$409$			805.417i			0.0973723i	0.998814	+	0.0486862i	$0.0155034\pi$
$409$			805.417i			0.0973723i	−0.998814	+	0.0486862i	$0.984497\pi$
$410$		0			0
$411$		−	2255.51i		−	0.270696i
$412$		0			0
$413$	7756.75	−	2820.59i	0.924176	−	0.336058i
$414$		0			0
$415$		−	3055.64i		−	0.361435i
$416$		0			0
$417$	1630.66			0.191496
$418$		0			0
$419$			7784.37i			0.907616i	0.891099	+	0.453808i	$0.149935\pi$
$419$			7784.37i			0.907616i	−0.891099	+	0.453808i	$0.850065\pi$
$420$		0			0
$421$			4998.41i			0.578640i	0.957232	+	0.289320i	$0.0934292\pi$
$421$			4998.41i			0.578640i	−0.957232	+	0.289320i	$0.906571\pi$
$422$		0			0
$423$	−3765.26			−0.432797
$424$		0			0
$425$		−	3051.86i		−	0.348322i
$426$		0			0
$427$	3796.07	+	10439.4i	0.430222	+	1.18313i
$428$		0			0
$429$			4820.28i			0.542484i
$430$		0			0
$431$			5983.17i			0.668675i	0.942453	+	0.334338i	$0.108513\pi$
$431$			5983.17i			0.668675i	−0.942453	+	0.334338i	$0.891487\pi$
$432$		0			0
$433$		−	3739.75i		−	0.415060i	−0.978229	−	0.207530i	$-0.933458\pi$
$433$		−	3739.75i		−	0.415060i	0.978229	−	0.207530i	$-0.0665425\pi$
$434$		0			0
$435$	−6120.18			−0.674575
$436$		0			0
$437$	−4028.40			−0.440971
$438$		0			0
$439$	8032.79			0.873313			0.436656	−	0.899628i	$-0.356163\pi$
$439$	8032.79			0.873313			0.436656	+	0.899628i	$0.356163\pi$
$440$		0			0
$441$	2365.97	−	1982.86i	0.255477	−	0.214109i
$442$		0			0
$443$	10025.1			1.07519			0.537595	−	0.843203i	$-0.319333\pi$
$443$	10025.1			1.07519			0.537595	+	0.843203i	$0.319333\pi$
$444$		0			0
$445$		−	9533.58i		−	1.01558i
$446$		0			0
$447$	3169.33			0.335357
$448$		0			0
$449$	4911.85			0.516268			0.258134	−	0.966109i	$-0.416892\pi$
$449$	4911.85			0.516268			0.258134	+	0.966109i	$0.416892\pi$
$450$		0			0
$451$		−	7507.77i		−	0.783873i
$452$		0			0
$453$	−6276.40			−0.650973
$454$		0			0
$455$	−5206.91	−	14319.3i	−0.536492	−	1.47538i
$456$		0			0
$457$	10828.7			1.10842			0.554208	−	0.832378i	$-0.313021\pi$
$457$	10828.7			1.10842			0.554208	+	0.832378i	$0.313021\pi$
$458$		0			0
$459$	−2090.48			−0.212582
$460$		0			0
$461$	7105.06			0.717821			0.358911	−	0.933372i	$-0.383148\pi$
$461$	7105.06			0.717821			0.358911	+	0.933372i	$0.383148\pi$
$462$		0			0
$463$			14241.0i			1.42945i	0.699403	+	0.714727i	$0.253449\pi$
$463$			14241.0i			1.42945i	−0.699403	+	0.714727i	$0.746551\pi$
$464$		0			0
$465$		−	3714.41i		−	0.370434i
$466$		0			0
$467$		−	5568.41i		−	0.551767i	−0.961191	−	0.275884i	$-0.911030\pi$
$467$		−	5568.41i		−	0.551767i	0.961191	−	0.275884i	$-0.0889705\pi$
$468$		0			0
$469$	4275.32	+	11757.3i	0.420929	+	1.15757i
$470$		0			0
$471$			3557.11i			0.347989i
$472$		0			0
$473$	−9662.15			−0.939252
$474$		0			0
$475$			1970.83i			0.190374i
$476$		0			0
$477$			5988.46i			0.574828i
$478$		0			0
$479$	−14755.8			−1.40754			−0.703768	−	0.710430i	$-0.748501\pi$
$479$	−14755.8			−1.40754			−0.703768	+	0.710430i	$0.748501\pi$
$480$		0			0
$481$		−	17586.4i		−	1.66710i
$482$		0			0
$483$	−1529.78	−	4206.96i	−0.144114	−	0.396321i
$484$		0			0
$485$		−	19902.6i		−	1.86336i
$486$		0			0
$487$			2870.19i			0.267065i	0.991044	+	0.133533i	$0.0426321\pi$
$487$			2870.19i			0.267065i	−0.991044	+	0.133533i	$0.957368\pi$
$488$		0			0
$489$			2710.91i			0.250698i
$490$		0			0
$491$	−13050.2			−1.19949			−0.599745	−	0.800191i	$-0.704731\pi$
$491$	−13050.2			−1.19949			−0.599745	+	0.800191i	$0.704731\pi$
$492$		0			0
$493$	−12318.3			−1.12533
$494$		0			0
$495$	2890.02			0.262417
$496$		0			0
$497$	−15266.0	+	5551.19i	−1.37782	+	0.501016i
$498$		0			0
$499$	−10021.7			−0.899063			−0.449531	−	0.893265i	$-0.648409\pi$
$499$	−10021.7			−0.899063			−0.449531	+	0.893265i	$0.648409\pi$
$500$		0			0
$501$		−	1423.90i		−	0.126976i
$502$		0			0
$503$	−13625.2			−1.20779			−0.603894	−	0.797065i	$-0.706385\pi$
$503$	−13625.2			−1.20779			−0.603894	+	0.797065i	$0.706385\pi$
$504$		0			0
$505$	−15418.5			−1.35865
$506$		0			0
$507$			5758.66i			0.504440i
$508$		0			0
$509$	6931.67			0.603617			0.301808	−	0.953369i	$-0.402410\pi$
$509$	6931.67			0.603617			0.301808	+	0.953369i	$0.402410\pi$
$510$		0			0
$511$	−12115.1	+	4405.42i	−1.04881	+	0.381378i
$512$		0			0
$513$	1349.99			0.116186
$514$		0			0
$515$	−5187.50			−0.443861
$516$		0			0
$517$	10477.0			0.891253
$518$		0			0
$519$		−	4923.98i		−	0.416452i
$520$		0			0
$521$		−	20497.7i		−	1.72365i	−0.507207	−	0.861824i	$-0.669322\pi$
$521$		−	20497.7i		−	1.72365i	0.507207	−	0.861824i	$-0.330678\pi$
$522$		0			0
$523$			17220.5i			1.43977i	0.694094	+	0.719885i	$0.255805\pi$
$523$			17220.5i			1.43977i	−0.694094	+	0.719885i	$0.744195\pi$
$524$		0			0
$525$	−2058.18	+	748.418i	−0.171098	+	0.0622165i
$526$		0			0
$527$		−	7476.14i		−	0.617961i
$528$		0			0
$529$	5675.68			0.466482
$530$		0			0
$531$			4010.90i			0.327793i
$532$		0			0
$533$		−	19235.0i		−	1.56316i
$534$		0			0
$535$	−13665.2			−1.10430
$536$		0			0
$537$			5406.33i			0.434451i
$538$		0			0
$539$	−6583.41	+	5517.41i	−0.526100	+	0.440912i
$540$		0			0
$541$			6118.24i			0.486218i	0.969999	+	0.243109i	$0.0781672\pi$
$541$			6118.24i			0.486218i	−0.969999	+	0.243109i	$0.921833\pi$
$542$		0			0
$543$			6205.35i			0.490418i
$544$		0			0
$545$		−	4609.89i		−	0.362323i
$546$		0			0
$547$	−391.181			−0.0305772			−0.0152886	−	0.999883i	$-0.504867\pi$
$547$	−391.181			−0.0305772			−0.0152886	+	0.999883i	$0.504867\pi$
$548$		0			0
$549$	−5398.04			−0.419641
$550$		0			0
$551$	7954.91			0.615047
$552$		0			0
$553$	21731.5	−	7902.23i	1.67110	−	0.607662i
$554$		0			0
$555$	−10544.0			−0.806429
$556$		0			0
$557$		−	14903.1i		−	1.13369i	−0.823824	−	0.566846i	$-0.808163\pi$
$557$		−	14903.1i		−	1.13369i	0.823824	−	0.566846i	$-0.191837\pi$
$558$		0			0
$559$	−24754.6			−1.87300
$560$		0			0
$561$	5816.84			0.437767
$562$		0			0
$563$		−	18651.1i		−	1.39618i	−0.716010	−	0.698090i	$-0.754034\pi$
$563$		−	18651.1i		−	1.39618i	0.716010	−	0.698090i	$-0.245966\pi$
$564$		0			0
$565$	−10623.0			−0.790993
$566$		0			0
$567$	512.655	+	1409.83i	0.0379709	+	0.104422i
$568$		0			0
$569$	−14380.9			−1.05954			−0.529770	−	0.848141i	$-0.677722\pi$
$569$	−14380.9			−1.05954			−0.529770	+	0.848141i	$0.677722\pi$
$570$		0			0
$571$	−26145.9			−1.91623			−0.958117	−	0.286376i	$-0.907549\pi$
$571$	−26145.9			−1.91623			−0.958117	+	0.286376i	$0.907549\pi$
$572$		0			0
$573$	−6538.82			−0.476724
$574$		0			0
$575$			3175.77i			0.230328i
$576$		0			0
$577$			18408.8i			1.32820i	0.747645	+	0.664099i	$0.231185\pi$
$577$			18408.8i			1.32820i	−0.747645	+	0.664099i	$0.768815\pi$
$578$		0			0
$579$		−	13469.4i		−	0.966788i
$580$		0			0
$581$	−4147.72	+	1508.24i	−0.296173	+	0.107697i
$582$		0			0
$583$		−	16663.2i		−	1.18374i
$584$		0			0
$585$	7404.28			0.523298
$586$		0			0
$587$			25942.5i			1.82412i	0.410053	+	0.912062i	$0.365510\pi$
$587$			25942.5i			1.82412i	−0.410053	+	0.912062i	$0.634490\pi$
$588$		0			0
$589$			4827.94i			0.337745i
$590$		0			0
$591$	−13557.2			−0.943604
$592$		0			0
$593$		−	19221.9i		−	1.33111i	−0.746347	−	0.665557i	$-0.768194\pi$
$593$		−	19221.9i		−	1.33111i	0.746347	−	0.665557i	$-0.231806\pi$
$594$		0			0
$595$	−17279.7	+	6283.41i	−1.19058	+	0.432932i
$596$		0			0
$597$			2479.79i			0.170002i
$598$		0			0
$599$		−	14280.0i		−	0.974068i	−0.873383	−	0.487034i	$-0.838079\pi$
$599$		−	14280.0i		−	0.974068i	0.873383	−	0.487034i	$-0.161921\pi$
$600$		0			0
$601$			5677.17i			0.385319i	0.981266	+	0.192659i	$0.0617113\pi$
$601$			5677.17i			0.385319i	−0.981266	+	0.192659i	$0.938289\pi$
$602$		0			0
$603$	−6079.54			−0.410577
$604$		0			0
$605$	9025.18			0.606489
$606$		0			0
$607$	−16394.3			−1.09625			−0.548124	−	0.836397i	$-0.684658\pi$
$607$	−16394.3			−1.09625			−0.548124	+	0.836397i	$0.684658\pi$
$608$		0			0
$609$	3020.86	+	8307.52i	0.201004	+	0.552771i
$610$		0			0
$611$	26842.3			1.77729
$612$		0			0
$613$		−	2171.28i		−	0.143063i	−0.997438	−	0.0715313i	$-0.977211\pi$
$613$		−	2171.28i		−	0.143063i	0.997438	−	0.0715313i	$-0.0227886\pi$
$614$		0			0
$615$	−11532.4			−0.756150
$616$		0			0
$617$	−6024.75			−0.393108			−0.196554	−	0.980493i	$-0.562975\pi$
$617$	−6024.75			−0.393108			−0.196554	+	0.980493i	$0.562975\pi$
$618$		0			0
$619$			13135.5i			0.852926i	0.904505	+	0.426463i	$0.140240\pi$
$619$			13135.5i			0.852926i	−0.904505	+	0.426463i	$0.859760\pi$
$620$		0			0
$621$	2175.36			0.140570
$622$		0			0
$623$	−12940.9	+	4705.69i	−0.832206	+	0.302615i
$624$		0			0
$625$	−18998.4			−1.21590
$626$		0			0
$627$	−3756.40			−0.239260
$628$		0			0
$629$	−21222.3			−1.34529
$630$		0			0
$631$		−	13957.4i		−	0.880562i	−0.897860	−	0.440281i	$-0.854879\pi$
$631$		−	13957.4i		−	0.880562i	0.897860	−	0.440281i	$-0.145121\pi$
$632$		0			0
$633$		−	14127.5i		−	0.887074i
$634$		0			0
$635$		−	11239.9i		−	0.702426i
$636$		0			0
$637$	−16866.8	+	14135.7i	−1.04912	+	0.879242i
$638$		0			0
$639$		−	7893.84i		−	0.488694i
$640$		0			0
$641$	24561.9			1.51348			0.756738	−	0.653718i	$-0.226792\pi$
$641$	24561.9			1.51348			0.756738	+	0.653718i	$0.226792\pi$
$642$		0			0
$643$		−	15778.3i		−	0.967708i	−0.875149	−	0.483854i	$-0.839237\pi$
$643$		−	15778.3i		−	0.967708i	0.875149	−	0.483854i	$-0.160763\pi$
$644$		0			0
$645$			14841.7i			0.906033i
$646$		0			0
$647$	11167.7			0.678588			0.339294	−	0.940680i	$-0.389812\pi$
$647$	11167.7			0.678588			0.339294	+	0.940680i	$0.389812\pi$
$648$		0			0
$649$		−	11160.5i		−	0.675020i
$650$		0			0
$651$	−5041.94	+	1833.40i	−0.303547	+	0.110379i
$652$		0			0
$653$		−	15738.6i		−	0.943185i	−0.881817	−	0.471592i	$-0.843679\pi$
$653$		−	15738.6i		−	0.943185i	0.881817	−	0.471592i	$-0.156321\pi$
$654$		0			0
$655$			13887.3i			0.828432i
$656$		0			0
$657$		−	6264.54i		−	0.371999i
$658$		0			0
$659$	21821.3			1.28989			0.644944	−	0.764230i	$-0.276881\pi$
$659$	21821.3			1.28989			0.644944	+	0.764230i	$0.276881\pi$
$660$		0			0
$661$	5703.63			0.335621			0.167810	−	0.985819i	$-0.446330\pi$
$661$	5703.63			0.335621			0.167810	+	0.985819i	$0.446330\pi$
$662$		0			0
$663$	14902.9			0.872970
$664$		0			0
$665$	11158.8	−	4057.69i	0.650709	−	0.236618i
$666$		0			0
$667$	12818.5			0.744127
$668$		0			0
$669$			17325.3i			1.00125i
$670$		0			0
$671$	15020.3			0.864161
$672$		0			0
$673$	30185.3			1.72891			0.864456	−	0.502708i	$-0.167663\pi$
$673$	30185.3			1.72891			0.864456	+	0.502708i	$0.167663\pi$
$674$		0			0
$675$		−	1064.26i		−	0.0606863i
$676$		0			0
$677$	−12990.9			−0.737488			−0.368744	−	0.929531i	$-0.620212\pi$
$677$	−12990.9			−0.737488			−0.368744	+	0.929531i	$0.620212\pi$
$678$		0			0
$679$	−27015.8	+	9823.76i	−1.52691	+	0.555230i
$680$		0			0
$681$	6840.32			0.384907
$682$		0			0
$683$	−3636.79			−0.203745			−0.101872	−	0.994797i	$-0.532483\pi$
$683$	−3636.79			−0.203745			−0.101872	+	0.994797i	$0.532483\pi$
$684$		0			0
$685$	9640.43			0.537725
$686$		0			0
$687$			11079.0i			0.615267i
$688$		0			0
$689$		−	42691.3i		−	2.36054i
$690$		0			0
$691$		−	869.636i		−	0.0478763i	−0.999713	−	0.0239381i	$-0.992380\pi$
$691$		−	869.636i		−	0.0478763i	0.999713	−	0.0239381i	$-0.00762048\pi$
$692$		0			0
$693$	−1426.49	−	3922.90i	−0.0781929	−	0.215034i
$694$		0			0
$695$			6969.74i			0.380399i
$696$		0			0
$697$	−23211.7			−1.26142
$698$		0			0
$699$			9420.20i			0.509735i
$700$		0			0
$701$		−	8128.07i		−	0.437936i	−0.975732	−	0.218968i	$-0.929731\pi$
$701$		−	8128.07i		−	0.437936i	0.975732	−	0.218968i	$-0.0702690\pi$
$702$		0			0
$703$	13704.9			0.735265
$704$		0			0
$705$		−	16093.4i		−	0.859732i
$706$		0			0
$707$	7610.45	+	20929.1i	0.404838	+	1.11332i
$708$		0			0
$709$			6488.87i			0.343716i	0.985122	+	0.171858i	$0.0549770\pi$
$709$			6488.87i			0.343716i	−0.985122	+	0.171858i	$0.945023\pi$
$710$		0			0
$711$			11237.0i			0.592717i
$712$		0			0
$713$			7779.69i			0.408628i
$714$		0			0
$715$	−20602.7			−1.07762
$716$		0			0
$717$	932.351			0.0485625
$718$		0			0
$719$	19419.9			1.00729			0.503645	−	0.863911i	$-0.331992\pi$
$719$	19419.9			1.00729			0.503645	+	0.863911i	$0.331992\pi$
$720$		0			0
$721$	2560.50	+	7041.50i	0.132258	+	0.363716i
$722$		0			0
$723$	1373.50			0.0706514
$724$		0			0
$725$		−	6271.22i		−	0.321252i
$726$		0			0
$727$	−29343.3			−1.49695			−0.748475	−	0.663163i	$-0.769214\pi$
$727$	−29343.3			−1.49695			−0.748475	+	0.663163i	$0.769214\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			29872.4i			1.51145i
$732$		0			0
$733$	−27788.9			−1.40028			−0.700140	−	0.714006i	$-0.746879\pi$
$733$	−27788.9			−1.40028			−0.700140	+	0.714006i	$0.746879\pi$
$734$		0			0
$735$	8475.11	+	10112.6i	0.425318	+	0.507493i
$736$		0			0
$737$	16916.6			0.845495
$738$		0			0
$739$	−12588.2			−0.626607			−0.313304	−	0.949653i	$-0.601436\pi$
$739$	−12588.2			−0.626607			−0.313304	+	0.949653i	$0.601436\pi$
$740$		0			0
$741$	−9623.96			−0.477119
$742$		0			0
$743$		−	8137.53i		−	0.401799i	−0.979612	−	0.200900i	$-0.935613\pi$
$743$		−	8137.53i		−	0.401799i	0.979612	−	0.200900i	$-0.0643865\pi$
$744$		0			0
$745$			13546.3i			0.666171i
$746$		0			0
$747$		−	2144.73i		−	0.105049i
$748$		0			0
$749$	6745.03	+	18549.2i	0.329050	+	0.904902i
$750$		0			0
$751$			2723.45i			0.132331i	0.997809	+	0.0661653i	$0.0210765\pi$
$751$			2723.45i			0.132331i	−0.997809	+	0.0661653i	$0.978924\pi$
$752$		0			0
$753$	−2423.75			−0.117299
$754$		0			0
$755$		−	26826.4i		−	1.29313i
$756$		0			0
$757$		−	8168.46i		−	0.392190i	−0.980585	−	0.196095i	$-0.937174\pi$
$757$		−	8168.46i		−	0.392190i	0.980585	−	0.196095i	$-0.0628261\pi$
$758$		0			0
$759$	−6053.02			−0.289474
$760$		0			0
$761$		−	24255.5i		−	1.15540i	−0.816248	−	0.577702i	$-0.803950\pi$
$761$		−	24255.5i		−	1.15540i	0.816248	−	0.577702i	$-0.196050\pi$
$762$		0			0
$763$	−6257.46	+	2275.40i	−0.296901	+	0.107962i
$764$		0			0
$765$		−	8935.06i		−	0.422285i
$766$		0			0
$767$		−	28593.4i		−	1.34609i
$768$		0			0
$769$			13226.7i			0.620241i	0.950697	+	0.310121i	$0.100369\pi$
$769$			13226.7i			0.620241i	−0.950697	+	0.310121i	$0.899631\pi$
$770$		0			0
$771$	5248.07			0.245142
$772$		0			0
$773$	−24347.2			−1.13287			−0.566434	−	0.824107i	$-0.691677\pi$
$773$	−24347.2			−1.13287			−0.566434	+	0.824107i	$0.691677\pi$
$774$		0			0
$775$	3806.08			0.176411
$776$		0			0
$777$	5204.42	+	14312.4i	0.240293	+	0.660817i
$778$		0			0
$779$	14989.7			0.689423
$780$		0			0
$781$			21964.9i			1.00636i
$782$		0			0
$783$	−4295.69			−0.196061
$784$		0			0
$785$	−15203.7			−0.691265
$786$		0			0
$787$		−	31493.4i		−	1.42645i	−0.700933	−	0.713227i	$-0.747233\pi$
$787$		−	31493.4i		−	1.42645i	0.700933	−	0.713227i	$-0.252767\pi$
$788$		0			0
$789$	4081.04			0.184143
$790$		0			0
$791$	5243.39	+	14419.6i	0.235694	+	0.648168i
$792$		0			0
$793$	38482.3			1.72326
$794$		0			0
$795$	−25595.7			−1.14187
$796$		0			0
$797$	27985.5			1.24378			0.621892	−	0.783103i	$-0.286364\pi$
$797$	27985.5			1.24378			0.621892	+	0.783103i	$0.286364\pi$
$798$		0			0
$799$		−	32391.7i		−	1.43421i
$800$		0			0
$801$		−	6691.53i		−	0.295173i
$802$		0			0
$803$			17431.4i			0.766052i
$804$		0			0
$805$	17981.3	−	6538.53i	0.787274	−	0.286277i
$806$		0			0
$807$			6672.18i			0.291043i
$808$		0			0
$809$	−31789.5			−1.38153			−0.690766	−	0.723079i	$-0.742726\pi$
$809$	−31789.5			−1.38153			−0.690766	+	0.723079i	$0.742726\pi$
$810$		0			0
$811$			24872.1i			1.07692i	0.842652	+	0.538458i	$0.180993\pi$
$811$			24872.1i			1.07692i	−0.842652	+	0.538458i	$0.819007\pi$
$812$		0			0
$813$		−	8755.76i		−	0.377709i
$814$		0			0
$815$	−11586.9			−0.498001
$816$		0			0
$817$		−	19291.0i		−	0.826080i
$818$		0			0
$819$	−3654.68	−	10050.5i	−0.155928	−	0.428809i
$820$		0			0
$821$			7964.00i			0.338545i	0.985569	+	0.169272i	$0.0541418\pi$
$821$			7964.00i			0.338545i	−0.985569	+	0.169272i	$0.945858\pi$
$822$		0			0
$823$			2998.32i			0.126993i	0.997982	+	0.0634963i	$0.0202251\pi$
$823$			2998.32i			0.126993i	−0.997982	+	0.0634963i	$0.979775\pi$
$824$		0			0
$825$			2961.34i			0.124970i
$826$		0			0
$827$	43518.7			1.82986			0.914930	−	0.403612i	$-0.132245\pi$
$827$	43518.7			1.82986			0.914930	+	0.403612i	$0.132245\pi$
$828$		0			0
$829$	35520.8			1.48816			0.744082	−	0.668089i	$-0.232887\pi$
$829$	35520.8			1.48816			0.744082	+	0.668089i	$0.232887\pi$
$830$		0			0
$831$	4898.52			0.204486
$832$		0			0
$833$	17058.2	+	20353.9i	0.709520	+	0.846605i
$834$		0			0
$835$	6085.98			0.252232
$836$		0			0
$837$		−	2607.11i		−	0.107664i
$838$		0			0
$839$	−30875.0			−1.27047			−0.635234	−	0.772319i	$-0.719097\pi$
$839$	−30875.0			−1.27047			−0.635234	+	0.772319i	$0.719097\pi$
$840$		0			0
$841$	−923.747			−0.0378756
$842$		0			0
$843$		−	22846.3i		−	0.933413i
$844$		0			0
$845$	−24613.5			−1.00205
$846$		0			0
$847$	−4454.74	−	12250.8i	−0.180716	−	0.496979i
$848$		0			0
$849$	8601.22			0.347695
$850$		0			0
$851$	22084.0			0.889576
$852$		0			0
$853$	8198.79			0.329099			0.164549	−	0.986369i	$-0.447383\pi$
$853$	8198.79			0.329099			0.164549	+	0.986369i	$0.447383\pi$
$854$		0			0
$855$			5770.08i			0.230798i
$856$		0			0
$857$		−	13833.3i		−	0.551384i	−0.961246	−	0.275692i	$-0.911093\pi$
$857$		−	13833.3i		−	0.551384i	0.961246	−	0.275692i	$-0.0889070\pi$
$858$		0			0
$859$			38843.1i			1.54285i	0.636319	+	0.771426i	$0.280456\pi$
$859$			38843.1i			1.54285i	−0.636319	+	0.771426i	$0.719544\pi$
$860$		0			0
$861$	5692.30	+	15654.1i	0.225311	+	0.619617i
$862$		0			0
$863$		−	45262.8i		−	1.78536i	−0.450693	−	0.892679i	$-0.648823\pi$
$863$		−	45262.8i		−	1.78536i	0.450693	−	0.892679i	$-0.351177\pi$
$864$		0			0
$865$	21045.9			0.827264
$866$		0			0
$867$		−	3244.92i		−	0.127109i
$868$		0			0
$869$		−	31267.5i		−	1.22057i
$870$		0			0
$871$	43340.6			1.68604
$872$		0			0
$873$		−	13969.5i		−	0.541575i
$874$		0			0
$875$	6945.47	+	19100.4i	0.268342	+	0.737954i
$876$		0			0
$877$		−	15537.9i		−	0.598262i	−0.954212	−	0.299131i	$-0.903303\pi$
$877$		−	15537.9i		−	0.598262i	0.954212	−	0.299131i	$-0.0966968\pi$
$878$		0			0
$879$		−	3399.21i		−	0.130435i
$880$		0			0
$881$			39484.6i			1.50995i	0.655751	+	0.754977i	$0.272352\pi$
$881$			39484.6i			1.50995i	−0.655751	+	0.754977i	$0.727648\pi$
$882$		0			0
$883$	30198.1			1.15090			0.575452	−	0.817835i	$-0.304826\pi$
$883$	30198.1			1.15090			0.575452	+	0.817835i	$0.304826\pi$
$884$		0			0
$885$	−17143.3			−0.651147
$886$		0			0
$887$	24805.1			0.938976			0.469488	−	0.882939i	$-0.344438\pi$
$887$	24805.1			0.938976			0.469488	+	0.882939i	$0.344438\pi$
$888$		0			0
$889$	−15257.0	+	5547.89i	−0.575593	+	0.209303i
$890$		0			0
$891$	2028.47			0.0762698
$892$		0			0
$893$			20917.9i			0.783864i
$894$		0			0
$895$	−23107.6			−0.863018
$896$		0			0
$897$	−15508.0			−0.577253
$898$		0			0
$899$		−	15362.6i		−	0.569936i
$900$		0			0
$901$	−51517.5			−1.90488
$902$		0			0
$903$	20146.1	−	7325.73i	0.742436	−	0.269972i
$904$		0			0
$905$	−26522.7			−0.974194
$906$		0			0
$907$	−1393.42			−0.0510119			−0.0255059	−	0.999675i	$-0.508120\pi$
$907$	−1393.42			−0.0510119			−0.0255059	+	0.999675i	$0.508120\pi$
$908$		0			0
$909$	−10822.1			−0.394881
$910$		0			0
$911$		−	31078.8i		−	1.13028i	−0.824995	−	0.565141i	$-0.808822\pi$
$911$		−	31078.8i		−	1.13028i	0.824995	−	0.565141i	$-0.191178\pi$
$912$		0			0
$913$			5967.79i			0.216326i
$914$		0			0
$915$		−	23072.2i		−	0.833598i
$916$		0			0
$917$	18850.6	−	6854.66i	0.678847	−	0.246849i
$918$		0			0
$919$			46707.6i			1.67654i	0.545253	+	0.838272i	$0.316434\pi$
$919$			46707.6i			1.67654i	−0.545253	+	0.838272i	$0.683566\pi$
$920$		0			0
$921$	−618.446			−0.0221265
$922$		0			0
$923$			56274.6i			2.00683i
$924$		0			0
$925$		−	10804.2i		−	0.384044i
$926$		0			0
$927$	−3641.06			−0.129005
$928$		0			0
$929$			35085.6i			1.23910i	0.784958	+	0.619549i	$0.212684\pi$
$929$			35085.6i			1.23910i	−0.784958	+	0.619549i	$0.787316\pi$
$930$		0			0
$931$	−11015.8	−	13144.2i	−0.387786	−	0.462709i
$932$		0			0
$933$			18888.0i			0.662771i
$934$		0			0
$935$			24862.2i			0.869604i
$936$		0			0
$937$			8314.62i			0.289890i	0.989440	+	0.144945i	$0.0463005\pi$
$937$			8314.62i			0.289890i	−0.989440	+	0.144945i	$0.953699\pi$
$938$		0			0
$939$	23993.3			0.833856
$940$		0			0
$941$	−40367.5			−1.39845			−0.699225	−	0.714902i	$-0.746471\pi$
$941$	−40367.5			−1.39845			−0.699225	+	0.714902i	$0.746471\pi$
$942$		0			0
$943$	24154.2			0.834114
$944$		0			0
$945$	−6025.84	+	2191.18i	−0.207429	+	0.0754275i
$946$		0			0
$947$	−9649.94			−0.331131			−0.165565	−	0.986199i	$-0.552945\pi$
$947$	−9649.94			−0.331131			−0.165565	+	0.986199i	$0.552945\pi$
$948$		0			0
$949$			44659.5i			1.52762i
$950$		0			0
$951$	−17491.3			−0.596418
$952$		0			0
$953$	−58010.8			−1.97183			−0.985915	−	0.167250i	$-0.946511\pi$
$953$	−58010.8			−1.97183			−0.985915	+	0.167250i	$0.946511\pi$
$954$		0			0
$955$		−	27948.0i		−	0.946992i
$956$		0			0
$957$	11953.0			0.403745
$958$		0			0
$959$	−4758.43	−	13085.9i	−0.160227	−	0.440631i
$960$		0			0
$961$	−20467.2			−0.687027
$962$		0			0
$963$	−9591.50			−0.320957
$964$		0			0
$965$	57570.6			1.92048
$966$		0			0
$967$			1452.77i			0.0483124i	0.999708	+	0.0241562i	$0.00768990\pi$
$967$			1452.77i			0.0483124i	−0.999708	+	0.0241562i	$0.992310\pi$
$968$		0			0
$969$			11613.6i			0.385020i
$970$		0			0
$971$		−	57626.6i		−	1.90456i	−0.305227	−	0.952280i	$-0.598732\pi$
$971$		−	57626.6i		−	1.90456i	0.305227	−	0.952280i	$-0.401268\pi$
$972$		0			0
$973$	9460.71	−	3440.20i	0.311713	−	0.113348i
$974$		0			0
$975$			7587.01i			0.249209i
$976$		0			0
$977$	45343.0			1.48480			0.742401	−	0.669956i	$-0.233687\pi$
$977$	45343.0			1.48480			0.742401	+	0.669956i	$0.233687\pi$
$978$		0			0
$979$			18619.5i			0.607846i
$980$		0			0
$981$		−	3235.64i		−	0.105307i
$982$		0			0
$983$	−46702.1			−1.51533			−0.757663	−	0.652646i	$-0.773659\pi$
$983$	−46702.1			−1.51533			−0.757663	+	0.652646i	$0.773659\pi$
$984$		0			0
$985$		−	57945.9i		−	1.87443i
$986$		0			0
$987$	−21845.1	+	7943.54i	−0.704495	+	0.256176i
$988$		0			0
$989$		−	31085.3i		−	0.999450i
$990$		0			0
$991$			6796.04i			0.217844i	0.994050	+	0.108922i	$0.0347399\pi$
$991$			6796.04i			0.217844i	−0.994050	+	0.108922i	$0.965260\pi$
$992$		0			0
$993$		−	15067.3i		−	0.481518i
$994$		0			0
$995$	−10599.1			−0.337702
$996$		0			0
$997$	24424.9			0.775872			0.387936	−	0.921686i	$-0.373188\pi$
$997$	24424.9			0.775872			0.387936	+	0.921686i	$0.373188\pi$
$998$		0			0
$999$	−7400.74			−0.234383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.c.223.4	yes	32
4.3	odd	2	inner	1344.4.p.c.223.11	yes	32
7.6	odd	2		1344.4.p.d.223.5	yes	32
8.3	odd	2		1344.4.p.d.223.6	yes	32
8.5	even	2		1344.4.p.d.223.15	yes	32
28.27	even	2		1344.4.p.d.223.16	yes	32
56.13	odd	2	inner	1344.4.p.c.223.12	yes	32
56.27	even	2	inner	1344.4.p.c.223.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.3	✓	32	56.27	even	2	inner
1344.4.p.c.223.4	yes	32	1.1	even	1	trivial
1344.4.p.c.223.11	yes	32	4.3	odd	2	inner
1344.4.p.c.223.12	yes	32	56.13	odd	2	inner
1344.4.p.d.223.5	yes	32	7.6	odd	2
1344.4.p.d.223.6	yes	32	8.3	odd	2
1344.4.p.d.223.15	yes	32	8.5	even	2
1344.4.p.d.223.16	yes	32	28.27	even	2

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +25.0429 q^{11} +64.1604 q^{13} -38.4675i q^{15} -77.4251i q^{17} +49.9995i q^{19} +(-52.2158 + 18.9872i) q^{21} +80.5687i q^{23} +39.4169 q^{25} -27.0000i q^{27} -159.100i q^{29} +96.5596 q^{31} +75.1287i q^{33} +(-81.1547 - 223.179i) q^{35} -274.101i q^{37} +192.481i q^{39} -299.796i q^{41} -385.824 q^{43} +115.403 q^{45} +418.362 q^{47} +(-262.886 + 220.318i) q^{49} +232.275 q^{51} -665.385i q^{53} -321.113 q^{55} -149.999 q^{57} -445.655i q^{59} +599.783 q^{61} +(-56.9617 - 156.647i) q^{63} -822.697 q^{65} +675.504 q^{67} -241.706 q^{69} +877.093i q^{71} +696.060i q^{73} +118.251i q^{75} +(158.498 + 435.878i) q^{77} -1248.56i q^{79} +81.0000 q^{81} +238.303i q^{83} +992.784i q^{85} +477.299 q^{87} +743.503i q^{89} +(406.076 + 1116.73i) q^{91} +289.679i q^{93} -641.120i q^{95} +1552.16i q^{97} -225.386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 32 * q - 288 * q^9	\( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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