LMFDB - Embedded newform 224.3.v.a.69.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(13,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("224.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.v (of order $8$, degree $4$, 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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label			69.2
Root			$0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	224.69
Dual form			224.3.v.a.13.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.125246 - 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(1.49509 + 7.85905i) q^{8} +(-6.36396 + 6.36396i) q^{9} +O(q^{10})$	\(q+(-0.125246 - 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(1.49509 + 7.85905i) q^{8} +(-6.36396 + 6.36396i) q^{9} +(-5.22101 + 12.6046i) q^{11} +(9.26013 - 10.5000i) q^{14} +(15.5000 - 3.96863i) q^{16} +(13.5000 + 11.9059i) q^{18} +(25.8137 + 8.84285i) q^{22} +(12.1660 - 12.1660i) q^{23} +(17.6777 + 17.6777i) q^{25} +(-22.1186 - 17.1688i) q^{28} +(22.1916 + 53.5752i) q^{29} +(-9.86299 - 30.4421i) q^{32} +(22.0742 - 28.4382i) q^{36} +(-67.6340 - 28.0149i) q^{37} +(13.9560 - 33.6929i) q^{43} +(14.4179 - 52.6336i) q^{44} +(-25.8080 - 22.7605i) q^{46} +49.0000i q^{49} +(33.0719 - 37.5000i) q^{50} +(-36.5379 + 88.2103i) q^{53} +(-31.5000 + 46.3006i) q^{56} +(104.161 - 51.0061i) q^{58} -63.0000 q^{63} +(-59.5294 + 23.5000i) q^{64} +(-39.7306 - 95.9183i) q^{67} +(59.8665 + 59.8665i) q^{71} +(-59.5294 - 40.5000i) q^{72} +(-47.4490 + 138.511i) q^{74} +(-88.2324 + 36.5471i) q^{77} +23.3317i q^{79} -81.0000i q^{81} +(-69.0014 - 23.6374i) q^{86} +(-106.866 - 22.1871i) q^{88} +(-42.1994 + 54.3654i) q^{92} +(97.8077 - 6.13705i) q^{98} +(-46.9891 - 113.442i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 124 q^{16} + 108 q^{18} + 148 q^{22} - 72 q^{23} - 232 q^{43} + 324 q^{44} + 24 q^{53} - 252 q^{56} - 504 q^{63} - 472 q^{67} - 108 q^{74} - 168 q^{77} - 708 q^{92}+O(q^{100}) $ 8 * q + 124 * q^16 + 108 * q^18 + 148 * q^22 - 72 * q^23 - 232 * q^43 + 324 * q^44 + 24 * q^53 - 252 * q^56 - 504 * q^63 - 472 * q^67 - 108 * q^74 - 168 * q^77 - 708 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{8}\right)$

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 $

$2$	−0.125246	−	1.99607i	−0.0626229	−	0.998037i
$2$	−0.125246	−	1.99607i	−0.0626229	−	0.998037i
$3$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$3$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$4$	−3.96863	+	0.500000i	−0.992157	+	0.125000i
$5$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$5$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$6$		0			0
$7$	4.94975	+	4.94975i	0.707107	+	0.707107i
$7$	4.94975	+	4.94975i	0.707107	+	0.707107i
$8$	1.49509	+	7.85905i	0.186886	+	0.982382i
$9$	−6.36396	+	6.36396i	−0.707107	+	0.707107i
$10$		0			0
$11$	−5.22101	+	12.6046i	−0.474637	+	1.14588i	0.487454	+	0.873149i	$0.337926\pi$
$11$	−5.22101	+	12.6046i	−0.474637	+	1.14588i	−0.962091	+	0.272727i	$0.912074\pi$
$12$		0			0
$13$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$13$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$14$	9.26013	−	10.5000i	0.661438	−	0.750000i
$15$		0			0
$16$	15.5000	−	3.96863i	0.968750	−	0.248039i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	13.5000	+	11.9059i	0.750000	+	0.661438i
$19$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$	25.8137	+	8.84285i	1.17335	+	0.401948i
$23$	12.1660	−	12.1660i	0.528957	−	0.528957i	−0.391304	−	0.920261i	$-0.627976\pi$
$23$	12.1660	−	12.1660i	0.528957	−	0.528957i	0.920261	+	0.391304i	$0.127976\pi$
$24$		0			0
$25$	17.6777	+	17.6777i	0.707107	+	0.707107i
$26$		0			0
$27$		0			0
$28$	−22.1186	−	17.1688i	−0.789949	−	0.613172i
$29$	22.1916	+	53.5752i	0.765227	+	1.84742i	0.400295	+	0.916386i	$0.368907\pi$
$29$	22.1916	+	53.5752i	0.765227	+	1.84742i	0.364931	+	0.931034i	$0.381093\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−9.86299	−	30.4421i	−0.308218	−	0.951316i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	22.0742	−	28.4382i	0.613172	−	0.789949i
$37$	−67.6340	−	28.0149i	−1.82795	−	0.757160i	−0.969864	−	0.243646i	$-0.921656\pi$
$37$	−67.6340	−	28.0149i	−1.82795	−	0.757160i	−0.858082	−	0.513514i	$-0.828344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	13.9560	−	33.6929i	0.324559	−	0.783555i	−0.674419	−	0.738349i	$-0.735606\pi$
$43$	13.9560	−	33.6929i	0.324559	−	0.783555i	0.998978	−	0.0452058i	$-0.0143943\pi$
$44$	14.4179	−	52.6336i	0.327680	−	1.19622i
$45$		0			0
$46$	−25.8080	−	22.7605i	−0.561044	−	0.494794i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$			49.0000i			1.00000i
$50$	33.0719	−	37.5000i	0.661438	−	0.750000i
$51$		0			0
$52$		0			0
$53$	−36.5379	+	88.2103i	−0.689394	+	1.66434i	0.0566038	+	0.998397i	$0.481973\pi$
$53$	−36.5379	+	88.2103i	−0.689394	+	1.66434i	−0.745998	+	0.665948i	$0.768027\pi$
$54$		0			0
$55$		0			0
$56$	−31.5000	+	46.3006i	−0.562500	+	0.826797i
$57$		0			0
$58$	104.161	−	51.0061i	1.79587	−	0.879416i
$59$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$59$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$60$		0			0
$61$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$61$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$62$		0			0
$63$	−63.0000			−1.00000
$64$	−59.5294	+	23.5000i	−0.930147	+	0.367188i
$65$		0			0
$66$		0			0
$67$	−39.7306	−	95.9183i	−0.592995	−	1.43162i	−0.880597	−	0.473866i	$-0.842858\pi$
$67$	−39.7306	−	95.9183i	−0.592995	−	1.43162i	0.287602	−	0.957750i	$-0.407142\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	59.8665	+	59.8665i	0.843190	+	0.843190i	0.989272	−	0.146082i	$-0.0466664\pi$
$71$	59.8665	+	59.8665i	0.843190	+	0.843190i	−0.146082	+	0.989272i	$0.546666\pi$
$72$	−59.5294	−	40.5000i	−0.826797	−	0.562500i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	−47.4490	+	138.511i	−0.641202	+	1.87177i
$75$		0			0
$76$		0			0
$77$	−88.2324	+	36.5471i	−1.14588	+	0.474637i
$78$		0			0
$79$			23.3317i			0.295338i	0.989037	+	0.147669i	$0.0471771\pi$
$79$			23.3317i			0.295338i	−0.989037	+	0.147669i	$0.952823\pi$
$80$		0			0
$81$		−	81.0000i		−	1.00000i
$82$		0			0
$83$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$83$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$		0			0
$86$	−69.0014	−	23.6374i	−0.802342	−	0.274854i
$87$		0			0
$88$	−106.866	−	22.1871i	−1.21439	−	0.252126i
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$	−42.1994	+	54.3654i	−0.458689	+	0.590928i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	97.8077	−	6.13705i	0.998037	−	0.0626229i
$99$	−46.9891	−	113.442i	−0.474637	−	1.14588i
$100$	−78.9949	−	61.3172i	−0.789949	−	0.613172i
$101$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$101$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	180.651	+	61.8844i	1.70425	+	0.583815i
$107$	64.6556	−	156.093i	0.604258	−	1.45881i	−0.264901	−	0.964276i	$-0.585339\pi$
$107$	64.6556	−	156.093i	0.604258	−	1.45881i	0.869159	−	0.494533i	$-0.164661\pi$
$108$		0			0
$109$	23.1268	−	9.57945i	0.212173	−	0.0878848i	−0.274066	−	0.961711i	$-0.588369\pi$
$109$	23.1268	−	9.57945i	0.212173	−	0.0878848i	0.486239	+	0.873826i	$0.338369\pi$
$110$		0			0
$111$		0			0
$112$	96.3648	+	57.0774i	0.860400	+	0.509620i
$113$			186.911i			1.65408i	0.562144	+	0.827040i	$0.309977\pi$
$113$			186.911i			1.65408i	−0.562144	+	0.827040i	$0.690023\pi$
$114$		0			0
$115$		0			0
$116$	−114.858	−	201.524i	−0.990152	−	1.73728i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−46.0580	−	46.0580i	−0.380644	−	0.380644i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	7.89049	+	125.753i	0.0626229	+	0.998037i
$127$	253.992			1.99994			0.999969	−	0.00787402i	$-0.00250640\pi$
$127$	253.992			1.99994			0.999969	+	0.00787402i	$0.00250640\pi$
$128$	54.3636	+	115.882i	0.424715	+	0.905327i
$129$		0			0
$130$		0			0
$131$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$		0			0
$134$	−186.484	+	91.3187i	−1.39167	+	0.681483i
$135$		0			0
$136$		0			0
$137$	192.830	−	192.830i	1.40752	−	1.40752i	0.635036	−	0.772482i	$-0.280985\pi$
$137$	192.830	−	192.830i	1.40752	−	1.40752i	0.772482	−	0.635036i	$-0.219015\pi$
$138$		0			0
$139$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$		0			0
$142$	112.000	−	126.996i	0.788732	−	0.894338i
$143$		0			0
$144$	−73.3852	+	123.898i	−0.509620	+	0.860400i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	282.422	+	77.3638i	1.90825	+	0.522728i
$149$	−55.0774	+	132.969i	−0.369647	+	0.892407i	0.624161	+	0.781296i	$0.285441\pi$
$149$	−55.0774	+	132.969i	−0.369647	+	0.892407i	−0.993808	+	0.111111i	$0.964559\pi$
$150$		0			0
$151$	200.498	−	200.498i	1.32780	−	1.32780i	0.420517	−	0.907285i	$-0.361849\pi$
$151$	200.498	−	200.498i	1.32780	−	1.32780i	0.907285	−	0.420517i	$-0.138151\pi$
$152$		0			0
$153$		0			0
$154$	84.0014	+	171.541i	0.545464	+	1.11390i
$155$		0			0
$156$		0			0
$157$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$157$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$158$	46.5719	−	2.92220i	0.294759	−	0.0184950i
$159$		0			0
$160$		0			0
$161$	120.437			0.748058
$162$	−161.682	+	10.1449i	−0.998037	+	0.0626229i
$163$	−72.6583	−	175.413i	−0.445756	−	1.07615i	−0.973896	−	0.226994i	$-0.927110\pi$
$163$	−72.6583	−	175.413i	−0.445756	−	1.07615i	0.528140	−	0.849158i	$-0.322890\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$	119.501	−	119.501i	0.707107	−	0.707107i
$170$		0			0
$171$		0			0
$172$	−38.5399	+	140.692i	−0.224069	+	0.817979i
$173$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$173$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$174$		0			0
$175$			175.000i			1.00000i
$176$	−30.9026	+	216.092i	−0.175583	+	1.22780i
$177$		0			0
$178$		0			0
$179$	30.5837	−	12.6682i	0.170858	−	0.0707719i	−0.295615	−	0.955307i	$-0.595524\pi$
$179$	30.5837	−	12.6682i	0.170858	−	0.0707719i	0.466473	+	0.884535i	$0.345524\pi$
$180$		0			0
$181$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$181$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$182$		0			0
$183$		0			0
$184$	113.803	+	77.4240i	0.618492	+	0.420783i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−374.526			−1.96087			−0.980435	−	0.196842i	$-0.936931\pi$
$191$	−374.526			−1.96087			−0.980435	+	0.196842i	$0.936931\pi$
$192$		0			0
$193$	313.240			1.62300			0.811502	−	0.584349i	$-0.198650\pi$
$193$	313.240			1.62300			0.811502	+	0.584349i	$0.198650\pi$
$194$		0			0
$195$		0			0
$196$	−24.5000	−	194.463i	−0.125000	−	0.992157i
$197$	−334.564	−	138.581i	−1.69830	−	0.703457i	−0.698371	−	0.715736i	$-0.746091\pi$
$197$	−334.564	−	138.581i	−1.69830	−	0.703457i	−0.999925	+	0.0122790i	$0.996091\pi$
$198$	−220.553	+	108.002i	−1.11390	+	0.545464i
$199$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$199$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$200$	−112.500	+	165.359i	−0.562500	+	0.826797i
$201$		0			0
$202$		0			0
$203$	−155.341	+	375.026i	−0.765227	+	1.84742i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			154.848i			0.748058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	125.038	−	51.7925i	0.592598	−	0.245462i	−0.0661700	−	0.997808i	$-0.521078\pi$
$211$	125.038	−	51.7925i	0.592598	−	0.245462i	0.658768	+	0.752346i	$0.271078\pi$
$212$	100.900	−	368.343i	0.475944	−	1.73747i
$213$		0			0
$214$	−319.670	−	109.508i	−1.49379	−	0.511717i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−22.0178	−	44.9631i	−0.100999	−	0.206253i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	101.861	−	199.500i	0.454739	−	0.890625i
$225$	−225.000			−1.00000
$226$	373.088	−	23.4098i	1.65083	−	0.103583i
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$229$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$	−387.872	+	254.505i	−1.67186	+	1.09700i
$233$	241.826	−	241.826i	1.03788	−	1.03788i	0.0386266	−	0.999254i	$-0.487702\pi$
$233$	241.826	−	241.826i	1.03788	−	1.03788i	0.999254	−	0.0386266i	$-0.0122983\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			423.320i			1.77121i	0.464435	+	0.885607i	$0.346257\pi$
$239$			423.320i			1.77121i	−0.464435	+	0.885607i	$0.653743\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−86.1665	+	97.7037i	−0.356060	+	0.403734i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$251$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$252$	250.023	−	31.5000i	0.992157	−	0.125000i
$253$	89.8293	+	216.867i	0.355056	+	0.857182i
$254$	−31.8115	−	506.987i	−0.125242	−	1.99601i
$255$		0			0
$256$	224.500	−	123.027i	0.876953	−	0.480576i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−196.104	−	473.438i	−0.757160	−	1.82795i
$260$		0			0
$261$	−482.177	−	199.724i	−1.84742	−	0.765227i
$262$		0			0
$263$	352.139	+	352.139i	1.33893	+	1.33893i	0.897095	+	0.441837i	$0.145673\pi$
$263$	352.139	+	352.139i	1.33893	+	1.33893i	0.441837	+	0.897095i	$0.354327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	205.635	+	360.798i	0.767296	+	1.34626i
$269$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$269$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−409.054	−	360.752i	−1.49290	−	1.31661i
$275$	−315.116	+	130.525i	−1.14588	+	0.474637i
$276$		0			0
$277$	114.018	−	275.264i	0.411617	−	0.993731i	−0.573087	−	0.819495i	$-0.694254\pi$
$277$	114.018	−	275.264i	0.411617	−	0.993731i	0.984704	−	0.174237i	$-0.0557457\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−218.158	−	218.158i	−0.776363	−	0.776363i	0.202847	−	0.979210i	$-0.434981\pi$
$281$	−218.158	−	218.158i	−0.776363	−	0.776363i	−0.979210	+	0.202847i	$0.934981\pi$
$282$		0			0
$283$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$	−267.521	−	207.655i	−0.941976	−	0.731178i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	256.500	+	130.965i	0.890625	+	0.454739i
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$293$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$294$		0			0
$295$		0			0
$296$	119.052	−	573.424i	0.402202	−	1.93724i
$297$		0			0
$298$	272.313	+	93.2848i	0.913803	+	0.313036i
$299$		0			0
$300$		0			0
$301$	235.850	−	97.6923i	0.783555	−	0.324559i
$302$	−425.321	−	375.097i	−1.40835	−	1.24204i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$307$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$308$	331.888	−	189.158i	1.07756	−	0.614149i
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	−11.6659	−	92.5950i	−0.0369173	−	0.293022i
$317$	131.860	+	318.339i	0.415963	+	1.00422i	0.983505	+	0.180879i	$0.0578941\pi$
$317$	131.860	+	318.339i	0.415963	+	1.00422i	−0.567543	+	0.823344i	$0.692106\pi$
$318$		0			0
$319$	−791.158			−2.48012
$320$		0			0
$321$		0			0
$322$	−15.0843	−	240.402i	−0.0468456	−	0.746590i
$323$		0			0
$324$	40.5000	+	321.459i	0.125000	+	0.992157i
$325$		0			0
$326$	−341.037	+	167.001i	−1.04612	+	0.512273i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−196.256	+	473.803i	−0.592917	+	1.43143i	0.287755	+	0.957704i	$0.407091\pi$
$331$	−196.256	+	473.803i	−0.592917	+	1.43143i	−0.880673	+	0.473725i	$0.842909\pi$
$332$		0			0
$333$	608.706	−	252.134i	1.82795	−	0.757160i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			608.805i			1.80654i	0.429069	+	0.903272i	$0.358842\pi$
$337$			608.805i			1.80654i	−0.429069	+	0.903272i	$0.641158\pi$
$338$	−253.500	−	223.566i	−0.750000	−	0.661438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−242.538	+	242.538i	−0.707107	+	0.707107i
$344$	285.660	+	59.3074i	0.830406	+	0.172405i
$345$		0			0
$346$		0			0
$347$	526.326	+	218.011i	1.51679	+	0.628275i	0.976945	−	0.213490i	$-0.0684831\pi$
$347$	526.326	+	218.011i	1.51679	+	0.628275i	0.539844	+	0.841765i	$0.318483\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$	349.313	−	21.9180i	0.998037	−	0.0626229i
$351$		0			0
$352$	435.206	+	34.6192i	1.23638	+	0.0983500i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−29.1171	−	59.4606i	−0.0813326	−	0.166091i
$359$	178.838	+	178.838i	0.498156	+	0.498156i	0.910864	−	0.412708i	$-0.135417\pi$
$359$	178.838	+	178.838i	0.498156	+	0.498156i	−0.412708	+	0.910864i	$0.635417\pi$
$360$		0			0
$361$	255.266	−	255.266i	0.707107	−	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	140.291	−	236.856i	0.381225	−	0.643629i
$369$		0			0
$370$		0			0
$371$	−617.472	+	255.765i	−1.66434	+	0.689394i
$372$		0			0
$373$	−194.921	+	470.580i	−0.522576	+	1.26161i	0.413722	+	0.910403i	$0.364228\pi$
$373$	−194.921	+	470.580i	−0.522576	+	1.26161i	−0.936298	+	0.351206i	$0.885772\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	162.311	+	67.2314i	0.428261	+	0.177392i	0.586393	−	0.810026i	$-0.300547\pi$
$379$	162.311	+	67.2314i	0.428261	+	0.177392i	−0.158132	+	0.987418i	$0.550547\pi$
$380$		0			0
$381$		0			0
$382$	46.9078	+	747.582i	0.122795	+	1.95702i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−39.2320	−	625.250i	−0.101637	−	1.61982i
$387$	125.604	+	303.236i	0.324559	+	0.783555i
$388$		0			0
$389$	107.794	+	44.6495i	0.277104	+	0.114780i	0.516908	−	0.856041i	$-0.327083\pi$
$389$	107.794	+	44.6495i	0.277104	+	0.114780i	−0.239804	+	0.970821i	$0.577083\pi$
$390$		0			0
$391$		0			0
$392$	−385.094	+	73.2595i	−0.982382	+	0.186886i
$393$		0			0
$394$	−234.715	+	685.172i	−0.595724	+	1.73901i
$395$		0			0
$396$	243.203	+	426.713i	0.614149	+	1.07756i
$397$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$397$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$398$		0			0
$399$		0			0
$400$	344.160	+	203.848i	0.860400	+	0.509620i
$401$		−	258.550i		−	0.644762i	−0.946610	−	0.322381i	$-0.895517\pi$
$401$		−	258.550i		−	0.644762i	0.946610	−	0.322381i	$-0.104483\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	768.036	+	263.102i	1.89172	+	0.648034i
$407$	706.236	−	706.236i	1.73522	−	1.73522i
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	309.088	−	19.3941i	0.746590	−	0.0468456i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$	−150.159	−	62.1981i	−0.356673	−	0.147739i	0.197150	−	0.980373i	$-0.436832\pi$
$421$	−150.159	−	62.1981i	−0.356673	−	0.147739i	−0.553823	+	0.832635i	$0.686832\pi$
$422$	−119.042	−	243.099i	−0.282090	−	0.576063i
$423$		0			0
$424$	−747.877	−	155.271i	−1.76386	−	0.366205i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−178.548	+	651.801i	−0.417168	+	1.52290i
$429$		0			0
$430$		0			0
$431$			162.000i			0.375870i	0.982181	+	0.187935i	$0.0601794\pi$
$431$			162.000i			0.375870i	−0.982181	+	0.187935i	$0.939821\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−86.9920	+	49.5807i	−0.199523	+	0.113717i
$437$		0			0
$438$		0			0
$439$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$439$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	−311.834	−	311.834i	−0.707107	−	0.707107i
$442$		0			0
$443$	804.148	+	333.089i	1.81523	+	0.751894i	0.979104	+	0.203361i	$0.0651866\pi$
$443$	804.148	+	333.089i	1.81523	+	0.751894i	0.836129	+	0.548533i	$0.184813\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−410.975	−	178.336i	−0.917354	−	0.398072i
$449$	−84.6640			−0.188561			−0.0942807	−	0.995546i	$-0.530055\pi$
$449$	−84.6640			−0.188561			−0.0942807	+	0.995546i	$0.530055\pi$
$450$	28.1803	+	449.117i	0.0626229	+	0.998037i
$451$		0			0
$452$	−93.4555	−	741.780i	−0.206760	−	1.64111i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−179.600	+	179.600i	−0.392997	+	0.392997i	−0.875754	−	0.482757i	$-0.839635\pi$
$457$	−179.600	+	179.600i	−0.392997	+	0.392997i	0.482757	+	0.875754i	$0.339635\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$461$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$462$		0			0
$463$		−	925.589i		−	1.99911i	−0.0297960	−	0.999556i	$-0.509486\pi$
$463$		−	925.589i		−	1.99911i	0.0297960	−	0.999556i	$-0.490514\pi$
$464$	556.589	+	742.345i	1.19955	+	1.59988i
$465$		0			0
$466$	−512.991	−	452.415i	−1.10084	−	0.970848i
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$		0			0
$469$	278.115	−	671.428i	0.592995	−	1.43162i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	351.822	+	351.822i	0.743809	+	0.743809i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−328.841	−	793.893i	−0.689394	−	1.66434i
$478$	844.979	−	53.0191i	1.76774	−	0.110919i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	205.816	+	159.758i	0.425239	+	0.330078i
$485$		0			0
$486$		0			0
$487$	−643.486	−	643.486i	−1.32133	−	1.32133i	−0.912703	−	0.408624i	$-0.866009\pi$
$487$	−643.486	−	643.486i	−1.32133	−	1.32133i	−0.408624	−	0.912703i	$-0.633991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	57.3936	−	138.560i	0.116891	−	0.282200i	−0.854596	−	0.519294i	$-0.826195\pi$
$491$	57.3936	−	138.560i	0.116891	−	0.282200i	0.971487	+	0.237094i	$0.0761949\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			592.648i			1.19245i
$498$		0			0
$499$	918.343	−	380.390i	1.84037	−	0.762305i	0.885988	−	0.463708i	$-0.153481\pi$
$499$	918.343	−	380.390i	1.84037	−	0.762305i	0.954379	−	0.298597i	$-0.0965187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	−94.1907	−	495.120i	−0.186886	−	0.982382i
$505$		0			0
$506$	421.632	−	206.468i	0.833265	−	0.408039i
$507$		0			0
$508$	−1008.00	+	126.996i	−1.98425	+	0.249992i
$509$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$509$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$510$		0			0
$511$		0			0
$512$	−273.690	−	432.710i	−0.534550	−	0.845137i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−920.456	+	450.735i	−1.77694	+	0.870145i
$519$		0			0
$520$		0			0
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$	−338.274	+	987.475i	−0.648034	+	1.89172i
$523$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$523$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$524$		0			0
$525$		0			0
$526$	658.792	−	747.000i	1.25246	−	1.42015i
$527$		0			0
$528$		0			0
$529$			232.976i			0.440409i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	694.426	−	455.652i	1.29557	−	0.850097i
$537$		0			0
$538$		0			0
$539$	−617.627	−	255.830i	−1.14588	−	0.474637i
$540$		0			0
$541$	370.812	+	895.219i	0.685420	+	1.65475i	0.753811	+	0.657091i	$0.228213\pi$
$541$	370.812	+	895.219i	0.685420	+	1.65475i	−0.0683919	+	0.997659i	$0.521787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−195.402	−	471.743i	−0.357225	−	0.862418i	−0.995688	−	0.0927652i	$-0.970429\pi$
$547$	−195.402	−	471.743i	−0.357225	−	0.862418i	0.638463	−	0.769653i	$-0.279571\pi$
$548$	−668.856	+	861.686i	−1.22054	+	1.57242i
$549$		0			0
$550$	300.005	+	612.647i	0.545464	+	1.11390i
$551$		0			0
$552$		0			0
$553$	−115.486	+	115.486i	−0.208836	+	0.208836i
$554$	−563.727	−	193.113i	−1.01756	−	0.348579i
$555$		0			0
$556$		0			0
$557$	1027.38	−	425.553i	1.84448	−	0.764010i	0.899461	−	0.437000i	$-0.143959\pi$
$557$	1027.38	−	425.553i	1.84448	−	0.764010i	0.945021	−	0.327009i	$-0.106041\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−408.137	+	462.783i	−0.726222	+	0.823458i
$563$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$563$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	400.930	−	400.930i	0.707107	−	0.707107i
$568$	−380.988	+	560.000i	−0.670754	+	0.985915i
$569$	−658.532	−	658.532i	−1.15735	−	1.15735i	−0.985044	−	0.172306i	$-0.944878\pi$
$569$	−658.532	−	658.532i	−1.15735	−	1.15735i	−0.172306	−	0.985044i	$-0.555122\pi$
$570$		0			0
$571$	−1028.45	−	425.998i	−1.80114	−	0.746057i	−0.985989	−	0.166807i	$-0.946654\pi$
$571$	−1028.45	−	425.998i	−1.80114	−	0.746057i	−0.815151	−	0.579249i	$-0.803346\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	430.133			0.748058
$576$	229.290	−	528.396i	0.398072	−	0.917354i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	36.1960	+	576.866i	0.0626229	+	0.998037i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−921.094	−	921.094i	−1.57992	−	1.57992i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1159.51	−	165.817i	−1.95863	−	0.280096i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	152.097	−	555.241i	0.255197	−	0.931613i
$597$		0			0
$598$		0			0
$599$	123.037	−	123.037i	0.205403	−	0.205403i	−0.596907	−	0.802310i	$-0.703604\pi$
$599$	123.037	−	123.037i	0.205403	−	0.205403i	0.802310	+	0.596907i	$0.203604\pi$
$600$		0			0
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$	−224.540	−	458.539i	−0.372991	−	0.761692i
$603$	863.264	+	357.576i	1.43162	+	0.592995i
$604$	−695.453	+	895.951i	−1.15141	+	1.48336i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	240.474	+	99.6077i	0.392291	+	0.162492i	0.570105	−	0.821572i	$-0.306903\pi$
$613$	240.474	+	99.6077i	0.392291	+	0.162492i	−0.177814	+	0.984064i	$0.556903\pi$
$614$		0			0
$615$		0			0
$616$	−419.141	−	638.782i	−0.680424	−	1.03698i
$617$	−778.265	+	778.265i	−1.26137	+	1.26137i	−0.310939	+	0.950430i	$0.600644\pi$
$617$	−778.265	+	778.265i	−1.26137	+	1.26137i	−0.950430	+	0.310939i	$0.899356\pi$
$618$		0			0
$619$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$619$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$			625.000i			1.00000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	538.799	−	538.799i	0.853881	−	0.853881i	−0.136728	−	0.990609i	$-0.543659\pi$
$631$	538.799	−	538.799i	0.853881	−	0.853881i	0.990609	+	0.136728i	$0.0436586\pi$
$632$	−183.365	+	34.8831i	−0.290135	+	0.0551947i
$633$		0			0
$634$	618.912	−	303.073i	0.976203	−	0.478033i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	99.0893	+	1579.21i	0.155312	+	2.47525i
$639$	−761.976			−1.19245
$640$		0			0
$641$	−1278.19			−1.99406			−0.997030	−	0.0770186i	$-0.975460\pi$
$641$	−1278.19			−1.99406			−0.997030	+	0.0770186i	$0.975460\pi$
$642$		0			0
$643$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$643$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$644$	−477.971	+	60.2187i	−0.742191	+	0.0935073i
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	636.583	−	121.102i	0.982382	−	0.186886i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	376.060	+	659.818i	0.576779	+	1.01199i
$653$	−832.060	+	344.650i	−1.27421	+	0.527796i	−0.914242	−	0.405169i	$-0.867213\pi$
$653$	−832.060	+	344.650i	−1.27421	+	0.527796i	−0.359969	+	0.932964i	$0.617213\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	939.078	−	388.979i	1.42501	−	0.590256i	0.468892	−	0.883255i	$-0.344653\pi$
$659$	939.078	−	388.979i	1.42501	−	0.590256i	0.956113	+	0.292999i	$0.0946533\pi$
$660$		0			0
$661$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$661$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$662$	970.326	+	332.399i	1.46575	+	0.502113i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−579.517	−	1183.44i	−0.870145	−	1.77694i
$667$	921.779	+	381.814i	1.38198	+	0.572434i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1269.96			−1.88701			−0.943507	−	0.331352i	$-0.892495\pi$
$673$	−1269.96			−1.88701			−0.943507	+	0.331352i	$0.892495\pi$
$674$	1215.22	−	76.2503i	1.80300	−	0.113131i
$675$		0			0
$676$	−414.505	+	534.006i	−0.613172	+	0.789949i
$677$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$677$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−293.229	+	707.917i	−0.429325	+	1.03648i	0.550178	+	0.835048i	$0.314560\pi$
$683$	−293.229	+	707.917i	−0.429325	+	1.03648i	−0.979502	+	0.201433i	$0.935440\pi$
$684$		0			0
$685$		0			0
$686$	514.500	+	453.746i	0.750000	+	0.661438i
$687$		0			0
$688$	82.6042	−	577.626i	0.120064	−	0.839572i
$689$		0			0
$690$		0			0
$691$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$691$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$692$		0			0
$693$	328.924	−	794.092i	0.474637	−	1.14588i
$694$	369.247	−	1077.89i	0.532056	−	1.55316i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−87.5000	−	694.510i	−0.125000	−	0.992157i
$701$	167.316	+	403.936i	0.238682	+	0.576228i	0.997147	−	0.0754851i	$-0.0240505\pi$
$701$	167.316	+	403.936i	0.238682	+	0.576228i	−0.758465	+	0.651713i	$0.774051\pi$
$702$		0			0
$703$		0			0
$704$	14.5947	−	873.040i	0.0207312	−	1.24011i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1067.36	+	442.115i	1.50545	+	0.623576i	0.974612	−	0.223900i	$-0.0718789\pi$
$709$	1067.36	+	442.115i	1.50545	+	0.623576i	0.530834	+	0.847476i	$0.321879\pi$
$710$		0			0
$711$	−148.482	−	148.482i	−0.208836	−	0.208836i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−115.041	+	65.5671i	−0.160672	+	0.0915741i
$717$		0			0
$718$	334.575	−	379.373i	0.465982	−	0.528374i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−541.500	−	477.558i	−0.750000	−	0.661438i
$723$		0			0
$724$		0			0
$725$	−554.789	+	1339.38i	−0.765227	+	1.84742i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	515.481	+	515.481i	0.707107	+	0.707107i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$	−490.352	−	250.366i	−0.666239	−	0.340171i
$737$	1416.45			1.92191
$738$		0			0
$739$	554.344	+	1338.31i	0.750128	+	1.81097i	0.558508	+	0.829499i	$0.311374\pi$
$739$	554.344	+	1338.31i	0.750128	+	1.81097i	0.191620	+	0.981469i	$0.438626\pi$
$740$		0			0
$741$		0			0
$742$	587.862	+	1200.49i	0.792267	+	1.61791i
$743$	797.810	+	797.810i	1.07377	+	1.07377i	0.997053	+	0.0767160i	$0.0244435\pi$
$743$	797.810	+	797.810i	1.07377	+	1.07377i	0.0767160	+	0.997053i	$0.475557\pi$
$744$		0			0
$745$		0			0
$746$	963.727	+	330.138i	1.29186	+	0.442544i
$747$		0			0
$748$		0			0
$749$	1092.65	−	452.590i	1.45881	−	0.604258i
$750$		0			0
$751$		−	1269.96i		−	1.69103i	−0.533955	−	0.845513i	$-0.679295\pi$
$751$		−	1269.96i		−	1.69103i	0.533955	−	0.845513i	$-0.320705\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.26865	+	7.89121i	−0.00431790	+	0.0104243i	−0.926024	−	0.377465i	$-0.876796\pi$
$757$	−3.26865	+	7.89121i	−0.00431790	+	0.0104243i	0.921706	+	0.387890i	$0.126796\pi$
$758$	113.870	−	332.405i	0.150224	−	0.438529i
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$	161.888	+	67.0561i	0.212173	+	0.0878848i
$764$	1486.35	−	187.263i	1.94549	−	0.245109i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−1243.13	+	156.620i	−1.61028	+	0.202876i
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$	589.550	−	288.695i	0.761692	−	0.372991i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	75.6231	−	220.756i	0.0972020	−	0.283748i
$779$		0			0
$780$		0			0
$781$	−1067.16	+	442.032i	−1.36640	+	0.565982i
$782$		0			0
$783$		0			0
$784$	194.463	+	759.500i	0.248039	+	0.968750i
$785$		0			0
$786$		0			0
$787$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$787$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$788$	1397.05	+	382.694i	1.77291	+	0.485653i
$789$		0			0
$790$		0			0
$791$	−925.162	+	925.162i	−1.16961	+	1.16961i
$792$	821.291	−	538.895i	1.03698	−	0.680424i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$797$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$798$		0			0
$799$		0			0
$800$	363.791	−	712.500i	0.454739	−	0.890625i
$801$		0			0
$802$	−516.084	+	32.3823i	−0.643497	+	0.0403769i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1137.46	−	1137.46i	1.40601	−	1.40601i	0.626964	−	0.779048i	$-0.284297\pi$
$809$	1137.46	−	1137.46i	1.40601	−	1.40601i	0.779048	−	0.626964i	$-0.215703\pi$
$810$		0			0
$811$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$811$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$812$	428.977	−	1566.01i	0.528297	−	1.92858i
$813$		0			0
$814$	−1498.15	−	1321.25i	−1.84048	−	1.62315i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	238.932	−	576.833i	0.291025	−	0.702597i	−0.708971	−	0.705238i	$-0.750840\pi$
$821$	238.932	−	576.833i	0.291025	−	0.702597i	0.999997	+	0.00264006i	$0.000840357\pi$
$822$		0			0
$823$	−439.820	+	439.820i	−0.534411	+	0.534411i	−0.921882	−	0.387471i	$-0.873349\pi$
$823$	−439.820	+	439.820i	−0.534411	+	0.534411i	0.387471	+	0.921882i	$0.373349\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1490.81	−	617.513i	−1.80267	−	0.746691i	−0.985358	−	0.170496i	$-0.945463\pi$
$827$	−1490.81	−	617.513i	−1.80267	−	0.746691i	−0.817312	−	0.576195i	$-0.804537\pi$
$828$	−77.4240	−	614.534i	−0.0935073	−	0.742191i
$829$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$829$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	−1783.16	+	1783.16i	−2.12028	+	2.12028i
$842$	−105.345	+	307.519i	−0.125113	+	0.365225i
$843$		0			0
$844$	−470.333	+	268.064i	−0.557267	+	0.317612i
$845$		0			0
$846$		0			0
$847$		−	455.950i		−	0.538312i
$848$	−216.264	+	1512.26i	−0.255028	+	1.78333i
$849$		0			0
$850$		0			0
$851$	−1163.67	+	482.006i	−1.36741	+	0.566400i
$852$		0			0
$853$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$853$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$854$		0			0
$855$		0			0
$856$	1323.41	+	274.760i	1.54603	+	0.320981i
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$		0			0
$861$		0			0
$862$	323.364	−	20.2898i	0.375132	−	0.0235381i
$863$	845.946			0.980238			0.490119	−	0.871655i	$-0.336953\pi$
$863$	845.946			0.980238			0.490119	+	0.871655i	$0.336953\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−294.088	−	121.815i	−0.338421	−	0.140179i
$870$		0			0
$871$		0			0
$872$	109.862	+	167.433i	0.125989	+	0.192010i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1091.22	+	451.999i	−1.24427	+	0.515393i	−0.905046	−	0.425314i	$-0.860164\pi$
$877$	−1091.22	+	451.999i	−1.24427	+	0.515393i	−0.339222	+	0.940706i	$0.610164\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−583.388	+	661.500i	−0.661438	+	0.750000i
$883$	−1631.41	+	675.753i	−1.84758	+	0.765293i	−0.918460	+	0.395514i	$0.870566\pi$
$883$	−1631.41	+	675.753i	−1.84758	+	0.765293i	−0.929120	+	0.369778i	$0.879434\pi$
$884$		0			0
$885$		0			0
$886$	564.154	−	1646.86i	0.636743	−	1.85876i
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$	1257.20	+	1257.20i	1.41417	+	1.41417i
$890$		0			0
$891$	1020.98	+	422.902i	1.14588	+	0.474637i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−304.500	+	842.672i	−0.339844	+	0.940482i
$897$		0			0
$898$	10.6038	+	168.996i	0.0118083	+	0.188191i
$899$		0			0
$900$	892.941	−	112.500i	0.992157	−	0.125000i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1468.94	+	279.449i	−1.62494	+	0.309125i
$905$		0			0
$906$		0			0
$907$	677.942	−	1636.70i	0.747455	−	1.80452i	0.175022	−	0.984564i	$-0.444000\pi$
$907$	677.942	−	1636.70i	0.747455	−	1.80452i	0.572433	−	0.819952i	$-0.306000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			931.304i			1.02229i	0.859495	+	0.511144i	$0.170778\pi$
$911$			931.304i			1.02229i	−0.859495	+	0.511144i	$0.829222\pi$
$912$		0			0
$913$		0			0
$914$	380.988	+	336.000i	0.416836	+	0.367615i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1257.20	+	1257.20i	−1.36801	+	1.36801i	−0.504725	+	0.863280i	$0.668406\pi$
$919$	−1257.20	+	1257.20i	−1.36801	+	1.36801i	−0.863280	+	0.504725i	$0.831594\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−700.373	−	1690.85i	−0.757160	−	1.82795i
$926$	−1847.54	+	115.926i	−1.99519	+	0.125190i
$927$		0			0
$928$	1412.07	−	1203.97i	1.52162	−	1.29738i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−838.805	+	1080.63i	−0.900005	+	1.15948i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	−1375.05	−	471.044i	−1.46594	−	0.502179i
$939$		0			0
$940$		0			0
$941$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$941$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	658.198	−	746.326i	0.695770	−	0.788928i
$947$	863.626	−	357.726i	0.911960	−	0.377746i	0.123154	−	0.992388i	$-0.460699\pi$
$947$	863.626	−	357.726i	0.911960	−	0.377746i	0.788807	+	0.614641i	$0.210699\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−115.186	−	115.186i	−0.120866	−	0.120866i	0.644086	−	0.764953i	$-0.277238\pi$
$953$	−115.186	−	115.186i	−0.120866	−	0.120866i	−0.764953	+	0.644086i	$0.777238\pi$
$954$	−1543.48	+	755.823i	−1.61791	+	0.792267i
$955$		0			0
$956$	−211.660	−	1680.00i	−0.221402	−	1.75732i
$957$		0			0
$958$		0			0
$959$	1908.92			1.99053
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	581.901	+	1404.83i	0.604258	+	1.45881i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1119.47	+	1119.47i	1.15767	+	1.15767i	0.984975	+	0.172699i	$0.0552488\pi$
$967$	1119.47	+	1119.47i	1.15767	+	1.15767i	0.172699	+	0.984975i	$0.444751\pi$
$968$	293.111	−	430.833i	0.302801	−	0.445075i
$969$		0			0
$970$		0			0
$971$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$971$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$972$		0			0
$973$		0			0
$974$	−1203.85	+	1365.04i	−1.23599	+	1.40148i
$975$		0			0
$976$		0			0
$977$		−	1947.27i		−	1.99311i	−0.0829069	−	0.996557i	$-0.526420\pi$
$977$		−	1947.27i		−	1.99311i	0.0829069	−	0.996557i	$-0.473580\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−86.2150	+	208.141i	−0.0878848	+	0.212173i
$982$	−283.765	−	97.2078i	−0.288967	−	0.0989896i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−240.118	−	579.697i	−0.242789	−	0.586145i
$990$		0			0
$991$	−1981.99			−1.99999			−0.999995	−	0.00322633i	$-0.998973\pi$
$991$	−1981.99			−1.99999			−0.999995	+	0.00322633i	$0.998973\pi$
$992$		0			0
$993$		0			0
$994$	1182.97	−	74.2267i	1.19011	−	0.0746748i
$995$		0			0
$996$		0			0
$997$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$997$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$998$	−874.306	−	1785.44i	−0.876058	−	1.78902i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.a.69.2	yes	8
7.6	odd	2	CM	224.3.v.a.69.2	yes	8
32.13	even	8	inner	224.3.v.a.13.2	✓	8
224.13	odd	8	inner	224.3.v.a.13.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.a.13.2	✓	8	32.13	even	8	inner
224.3.v.a.13.2	✓	8	224.13	odd	8	inner
224.3.v.a.69.2	yes	8	1.1	even	1	trivial
224.3.v.a.69.2	yes	8	7.6	odd	2	CM

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

\(f(q)\)	\(=\)	\(q+(-0.125246 - 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(1.49509 + 7.85905i) q^{8} +(-6.36396 + 6.36396i) q^{9} +O(q^{10})\)	\(q+(-0.125246 - 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(1.49509 + 7.85905i) q^{8} +(-6.36396 + 6.36396i) q^{9} +(-5.22101 + 12.6046i) q^{11} +(9.26013 - 10.5000i) q^{14} +(15.5000 - 3.96863i) q^{16} +(13.5000 + 11.9059i) q^{18} +(25.8137 + 8.84285i) q^{22} +(12.1660 - 12.1660i) q^{23} +(17.6777 + 17.6777i) q^{25} +(-22.1186 - 17.1688i) q^{28} +(22.1916 + 53.5752i) q^{29} +(-9.86299 - 30.4421i) q^{32} +(22.0742 - 28.4382i) q^{36} +(-67.6340 - 28.0149i) q^{37} +(13.9560 - 33.6929i) q^{43} +(14.4179 - 52.6336i) q^{44} +(-25.8080 - 22.7605i) q^{46} +49.0000i q^{49} +(33.0719 - 37.5000i) q^{50} +(-36.5379 + 88.2103i) q^{53} +(-31.5000 + 46.3006i) q^{56} +(104.161 - 51.0061i) q^{58} -63.0000 q^{63} +(-59.5294 + 23.5000i) q^{64} +(-39.7306 - 95.9183i) q^{67} +(59.8665 + 59.8665i) q^{71} +(-59.5294 - 40.5000i) q^{72} +(-47.4490 + 138.511i) q^{74} +(-88.2324 + 36.5471i) q^{77} +23.3317i q^{79} -81.0000i q^{81} +(-69.0014 - 23.6374i) q^{86} +(-106.866 - 22.1871i) q^{88} +(-42.1994 + 54.3654i) q^{92} +(97.8077 - 6.13705i) q^{98} +(-46.9891 - 113.442i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 124 q^{16} + 108 q^{18} + 148 q^{22} - 72 q^{23} - 232 q^{43} + 324 q^{44} + 24 q^{53} - 252 q^{56} - 504 q^{63} - 472 q^{67} - 108 q^{74} - 168 q^{77} - 708 q^{92}+O(q^{100}) \) 8 * q + 124 * q^16 + 108 * q^18 + 148 * q^22 - 72 * q^23 - 232 * q^43 + 324 * q^44 + 24 * q^53 - 252 * q^56 - 504 * q^63 - 472 * q^67 - 108 * q^74 - 168 * q^77 - 708 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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