LMFDB - Embedded newform 600.3.u.d.457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(193,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("600.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.u (of order $4$, degree $2$, 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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			457.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	600.457
Dual form			600.3.u.d.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(0.775255 + 0.775255i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(0.775255 + 0.775255i) q^{7} -3.00000i q^{9} +2.89898 q^{11} +(5.87628 - 5.87628i) q^{13} +(4.44949 + 4.44949i) q^{17} +0.101021i q^{19} +1.89898 q^{21} +(25.3485 - 25.3485i) q^{23} +(-3.67423 - 3.67423i) q^{27} -32.2929i q^{29} -3.69694 q^{31} +(3.55051 - 3.55051i) q^{33} +(42.6969 + 42.6969i) q^{37} -14.3939i q^{39} -12.8990 q^{41} +(49.2702 - 49.2702i) q^{43} +(2.85357 + 2.85357i) q^{47} -47.7980i q^{49} +10.8990 q^{51} +(-13.1918 + 13.1918i) q^{53} +(0.123724 + 0.123724i) q^{57} -76.3837i q^{59} -103.788 q^{61} +(2.32577 - 2.32577i) q^{63} +(47.6288 + 47.6288i) q^{67} -62.0908i q^{69} +29.7071 q^{71} +(3.50510 - 3.50510i) q^{73} +(2.24745 + 2.24745i) q^{77} +87.7980i q^{79} -9.00000 q^{81} +(81.7321 - 81.7321i) q^{83} +(-39.5505 - 39.5505i) q^{87} -96.5857i q^{89} +9.11123 q^{91} +(-4.52781 + 4.52781i) q^{93} +(54.2804 + 54.2804i) q^{97} -8.69694i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} - 8 q^{11} + 48 q^{13} + 8 q^{17} - 12 q^{21} + 72 q^{23} + 44 q^{31} + 24 q^{33} + 112 q^{37} - 32 q^{41} + 104 q^{43} + 80 q^{47} + 24 q^{51} + 104 q^{53} - 24 q^{57} - 180 q^{61} + 24 q^{63} + 264 q^{67} + 256 q^{71} + 112 q^{73} - 40 q^{77} - 36 q^{81} - 16 q^{83} - 168 q^{87} + 252 q^{91} - 72 q^{93} + 320 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 - 8 * q^11 + 48 * q^13 + 8 * q^17 - 12 * q^21 + 72 * q^23 + 44 * q^31 + 24 * q^33 + 112 * q^37 - 32 * q^41 + 104 * q^43 + 80 * q^47 + 24 * q^51 + 104 * q^53 - 24 * q^57 - 180 * q^61 + 24 * q^63 + 264 * q^67 + 256 * q^71 + 112 * q^73 - 40 * q^77 - 36 * q^81 - 16 * q^83 - 168 * q^87 + 252 * q^91 - 72 * q^93 + 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$2$		0			0
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.775255	+	0.775255i	0.110751	+	0.110751i	0.760311	−	0.649560i	$-0.225047\pi$
$7$	0.775255	+	0.775255i	0.110751	+	0.110751i	−0.649560	+	0.760311i	$0.725047\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	2.89898			0.263544			0.131772	−	0.991280i	$-0.457933\pi$
$11$	2.89898			0.263544			0.131772	+	0.991280i	$0.457933\pi$
$12$		0			0
$13$	5.87628	−	5.87628i	0.452021	−	0.452021i	−0.444004	−	0.896025i	$-0.646442\pi$
$13$	5.87628	−	5.87628i	0.452021	−	0.452021i	0.896025	+	0.444004i	$0.146442\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.44949	+	4.44949i	0.261735	+	0.261735i	0.825759	−	0.564024i	$-0.190747\pi$
$17$	4.44949	+	4.44949i	0.261735	+	0.261735i	−0.564024	+	0.825759i	$0.690747\pi$
$18$		0			0
$19$			0.101021i			0.00531687i	0.999996	+	0.00265843i	$0.000846207\pi$
$19$			0.101021i			0.00531687i	−0.999996	+	0.00265843i	$0.999154\pi$
$20$		0			0
$21$	1.89898			0.0904276
$22$		0			0
$23$	25.3485	−	25.3485i	1.10211	−	1.10211i	0.107951	−	0.994156i	$-0.465571\pi$
$23$	25.3485	−	25.3485i	1.10211	−	1.10211i	0.994156	−	0.107951i	$-0.0344290\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$		−	32.2929i		−	1.11355i	−0.830664	−	0.556773i	$-0.812039\pi$
$29$		−	32.2929i		−	1.11355i	0.830664	−	0.556773i	$-0.187961\pi$
$30$		0			0
$31$	−3.69694			−0.119256			−0.0596280	−	0.998221i	$-0.518991\pi$
$31$	−3.69694			−0.119256			−0.0596280	+	0.998221i	$0.518991\pi$
$32$		0			0
$33$	3.55051	−	3.55051i	0.107591	−	0.107591i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	42.6969	+	42.6969i	1.15397	+	1.15397i	0.985749	+	0.168222i	$0.0538026\pi$
$37$	42.6969	+	42.6969i	1.15397	+	1.15397i	0.168222	+	0.985749i	$0.446197\pi$
$38$		0			0
$39$		−	14.3939i		−	0.369074i
$40$		0			0
$41$	−12.8990			−0.314609			−0.157305	−	0.987550i	$-0.550280\pi$
$41$	−12.8990			−0.314609			−0.157305	+	0.987550i	$0.550280\pi$
$42$		0			0
$43$	49.2702	−	49.2702i	1.14582	−	1.14582i	0.158451	−	0.987367i	$-0.449350\pi$
$43$	49.2702	−	49.2702i	1.14582	−	1.14582i	0.987367	−	0.158451i	$-0.0506499\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.85357	+	2.85357i	0.0607143	+	0.0607143i	0.736812	−	0.676098i	$-0.236330\pi$
$47$	2.85357	+	2.85357i	0.0607143	+	0.0607143i	−0.676098	+	0.736812i	$0.736330\pi$
$48$		0			0
$49$		−	47.7980i		−	0.975469i
$50$		0			0
$51$	10.8990			0.213705
$52$		0			0
$53$	−13.1918	+	13.1918i	−0.248903	+	0.248903i	−0.820520	−	0.571618i	$-0.806316\pi$
$53$	−13.1918	+	13.1918i	−0.248903	+	0.248903i	0.571618	+	0.820520i	$0.306316\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.123724	+	0.123724i	0.00217060	+	0.00217060i
$58$		0			0
$59$		−	76.3837i		−	1.29464i	−0.762219	−	0.647319i	$-0.775890\pi$
$59$		−	76.3837i		−	1.29464i	0.762219	−	0.647319i	$-0.224110\pi$
$60$		0			0
$61$	−103.788			−1.70144			−0.850719	−	0.525620i	$-0.823833\pi$
$61$	−103.788			−1.70144			−0.850719	+	0.525620i	$0.823833\pi$
$62$		0			0
$63$	2.32577	−	2.32577i	0.0369169	−	0.0369169i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	47.6288	+	47.6288i	0.710878	+	0.710878i	0.966719	−	0.255841i	$-0.0823523\pi$
$67$	47.6288	+	47.6288i	0.710878	+	0.710878i	−0.255841	+	0.966719i	$0.582352\pi$
$68$		0			0
$69$		−	62.0908i		−	0.899867i
$70$		0			0
$71$	29.7071			0.418410			0.209205	−	0.977872i	$-0.432912\pi$
$71$	29.7071			0.418410			0.209205	+	0.977872i	$0.432912\pi$
$72$		0			0
$73$	3.50510	−	3.50510i	0.0480151	−	0.0480151i	−0.682692	−	0.730707i	$-0.739191\pi$
$73$	3.50510	−	3.50510i	0.0480151	−	0.0480151i	0.730707	+	0.682692i	$0.239191\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.24745	+	2.24745i	0.0291876	+	0.0291876i
$78$		0			0
$79$			87.7980i			1.11137i	0.831394	+	0.555683i	$0.187543\pi$
$79$			87.7980i			1.11137i	−0.831394	+	0.555683i	$0.812457\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	81.7321	−	81.7321i	0.984725	−	0.984725i	−0.0151605	−	0.999885i	$-0.504826\pi$
$83$	81.7321	−	81.7321i	0.984725	−	0.984725i	0.999885	+	0.0151605i	$0.00482592\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−39.5505	−	39.5505i	−0.454604	−	0.454604i
$88$		0			0
$89$		−	96.5857i		−	1.08523i	−0.839981	−	0.542616i	$-0.817434\pi$
$89$		−	96.5857i		−	1.08523i	0.839981	−	0.542616i	$-0.182566\pi$
$90$		0			0
$91$	9.11123			0.100123
$92$		0			0
$93$	−4.52781	+	4.52781i	−0.0486861	+	0.0486861i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	54.2804	+	54.2804i	0.559591	+	0.559591i	0.929191	−	0.369600i	$-0.120505\pi$
$97$	54.2804	+	54.2804i	0.559591	+	0.559591i	−0.369600	+	0.929191i	$0.620505\pi$
$98$		0			0
$99$		−	8.69694i		−	0.0878479i
$100$		0			0
$101$	−50.5153			−0.500152			−0.250076	−	0.968226i	$-0.580456\pi$
$101$	−50.5153			−0.500152			−0.250076	+	0.968226i	$0.580456\pi$
$102$		0			0
$103$	26.2020	−	26.2020i	0.254389	−	0.254389i	−0.568378	−	0.822767i	$-0.692429\pi$
$103$	26.2020	−	26.2020i	0.254389	−	0.254389i	0.822767	+	0.568378i	$0.192429\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.48469	+	7.48469i	0.0699504	+	0.0699504i	0.741216	−	0.671266i	$-0.234249\pi$
$107$	7.48469	+	7.48469i	0.0699504	+	0.0699504i	−0.671266	+	0.741216i	$0.734249\pi$
$108$		0			0
$109$			94.5755i			0.867665i	0.900993	+	0.433833i	$0.142839\pi$
$109$			94.5755i			0.867665i	−0.900993	+	0.433833i	$0.857161\pi$
$110$		0			0
$111$	104.586			0.942214
$112$		0			0
$113$	−89.9796	+	89.9796i	−0.796280	+	0.796280i	−0.982507	−	0.186227i	$-0.940374\pi$
$113$	−89.9796	+	89.9796i	−0.796280	+	0.796280i	0.186227	+	0.982507i	$0.440374\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−17.6288	−	17.6288i	−0.150674	−	0.150674i
$118$		0			0
$119$			6.89898i			0.0579746i
$120$		0			0
$121$	−112.596			−0.930545
$122$		0			0
$123$	−15.7980	+	15.7980i	−0.128439	+	0.128439i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	37.3031	+	37.3031i	0.293725	+	0.293725i	0.838550	−	0.544825i	$-0.183404\pi$
$127$	37.3031	+	37.3031i	0.293725	+	0.293725i	−0.544825	+	0.838550i	$0.683404\pi$
$128$		0			0
$129$		−	120.687i		−	0.935556i
$130$		0			0
$131$	192.677			1.47081			0.735407	−	0.677626i	$-0.236991\pi$
$131$	192.677			1.47081			0.735407	+	0.677626i	$0.236991\pi$
$132$		0			0
$133$	−0.0783167	+	0.0783167i	−0.000588847	+	0.000588847i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−165.328	−	165.328i	−1.20677	−	1.20677i	−0.972066	−	0.234708i	$-0.924587\pi$
$137$	−165.328	−	165.328i	−1.20677	−	1.20677i	−0.234708	−	0.972066i	$-0.575413\pi$
$138$		0			0
$139$			256.747i			1.84710i	0.383478	+	0.923550i	$0.374726\pi$
$139$			256.747i			1.84710i	−0.383478	+	0.923550i	$0.625274\pi$
$140$		0			0
$141$	6.98979			0.0495730
$142$		0			0
$143$	17.0352	−	17.0352i	0.119127	−	0.119127i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−58.5403	−	58.5403i	−0.398233	−	0.398233i
$148$		0			0
$149$			192.788i			1.29388i	0.762542	+	0.646939i	$0.223951\pi$
$149$			192.788i			1.29388i	−0.762542	+	0.646939i	$0.776049\pi$
$150$		0			0
$151$	98.9092			0.655028			0.327514	−	0.944846i	$-0.393789\pi$
$151$	98.9092			0.655028			0.327514	+	0.944846i	$0.393789\pi$
$152$		0			0
$153$	13.3485	−	13.3485i	0.0872449	−	0.0872449i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	103.452	+	103.452i	0.658929	+	0.658929i	0.955127	−	0.296198i	$-0.0957188\pi$
$157$	103.452	+	103.452i	0.658929	+	0.658929i	−0.296198	+	0.955127i	$0.595719\pi$
$158$		0			0
$159$			32.3133i			0.203228i
$160$		0			0
$161$	39.3031			0.244118
$162$		0			0
$163$	−19.1339	+	19.1339i	−0.117386	+	0.117386i	−0.763360	−	0.645974i	$-0.776452\pi$
$163$	−19.1339	+	19.1339i	−0.117386	+	0.117386i	0.645974	+	0.763360i	$0.276452\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−96.6969	−	96.6969i	−0.579024	−	0.579024i	0.355611	−	0.934634i	$-0.384273\pi$
$167$	−96.6969	−	96.6969i	−0.579024	−	0.579024i	−0.934634	+	0.355611i	$0.884273\pi$
$168$		0			0
$169$			99.9388i			0.591354i
$170$		0			0
$171$	0.303062			0.00177229
$172$		0			0
$173$	−180.136	+	180.136i	−1.04125	+	1.04125i	−0.0421380	+	0.999112i	$0.513417\pi$
$173$	−180.136	+	180.136i	−1.04125	+	1.04125i	−0.999112	+	0.0421380i	$0.986583\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−93.5505	−	93.5505i	−0.528534	−	0.528534i
$178$		0			0
$179$		−	134.000i		−	0.748603i	−0.927307	−	0.374302i	$-0.877882\pi$
$179$		−	134.000i		−	0.748603i	0.927307	−	0.374302i	$-0.122118\pi$
$180$		0			0
$181$	−171.586			−0.947987			−0.473994	−	0.880528i	$-0.657188\pi$
$181$	−171.586			−0.947987			−0.473994	+	0.880528i	$0.657188\pi$
$182$		0			0
$183$	−127.114	+	127.114i	−0.694609	+	0.694609i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	12.8990	+	12.8990i	0.0689785	+	0.0689785i
$188$		0			0
$189$		−	5.69694i		−	0.0301425i
$190$		0			0
$191$	26.2724			0.137552			0.0687760	−	0.997632i	$-0.478091\pi$
$191$	26.2724			0.137552			0.0687760	+	0.997632i	$0.478091\pi$
$192$		0			0
$193$	−74.1237	+	74.1237i	−0.384061	+	0.384061i	−0.872563	−	0.488502i	$-0.837543\pi$
$193$	−74.1237	+	74.1237i	−0.384061	+	0.384061i	0.488502	+	0.872563i	$0.337543\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−47.7526	−	47.7526i	−0.242399	−	0.242399i	0.575443	−	0.817842i	$-0.304830\pi$
$197$	−47.7526	−	47.7526i	−0.242399	−	0.242399i	−0.817842	+	0.575443i	$0.804830\pi$
$198$		0			0
$199$			355.454i			1.78620i	0.449857	+	0.893101i	$0.351475\pi$
$199$			355.454i			1.78620i	−0.449857	+	0.893101i	$0.648525\pi$
$200$		0			0
$201$	116.666			0.580429
$202$		0			0
$203$	25.0352	−	25.0352i	0.123326	−	0.123326i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−76.0454	−	76.0454i	−0.367369	−	0.367369i
$208$		0			0
$209$			0.292856i			0.00140123i
$210$		0			0
$211$	−145.474			−0.689453			−0.344726	−	0.938703i	$-0.612028\pi$
$211$	−145.474			−0.689453			−0.344726	+	0.938703i	$0.612028\pi$
$212$		0			0
$213$	36.3837	−	36.3837i	0.170815	−	0.170815i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.86607	−	2.86607i	−0.0132077	−	0.0132077i
$218$		0			0
$219$		−	8.58571i		−	0.0392042i
$220$		0			0
$221$	52.2929			0.236619
$222$		0			0
$223$	−230.351	+	230.351i	−1.03296	+	1.03296i	−0.0335252	+	0.999438i	$0.510673\pi$
$223$	−230.351	+	230.351i	−1.03296	+	1.03296i	−0.999438	+	0.0335252i	$0.989327\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	40.4745	+	40.4745i	0.178302	+	0.178302i	0.790615	−	0.612313i	$-0.209761\pi$
$227$	40.4745	+	40.4745i	0.178302	+	0.178302i	−0.612313	+	0.790615i	$0.709761\pi$
$228$		0			0
$229$			210.192i			0.917868i	0.888470	+	0.458934i	$0.151769\pi$
$229$			210.192i			0.917868i	−0.888470	+	0.458934i	$0.848231\pi$
$230$		0			0
$231$	5.50510			0.0238316
$232$		0			0
$233$	−296.384	+	296.384i	−1.27203	+	1.27203i	−0.327013	+	0.945020i	$0.606042\pi$
$233$	−296.384	+	296.384i	−1.27203	+	1.27203i	−0.945020	+	0.327013i	$0.893958\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	107.530	+	107.530i	0.453714	+	0.453714i
$238$		0			0
$239$		−	89.3031i		−	0.373653i	−0.982393	−	0.186826i	$-0.940180\pi$
$239$		−	89.3031i		−	0.373653i	0.982393	−	0.186826i	$-0.0598202\pi$
$240$		0			0
$241$	120.616			0.500483			0.250241	−	0.968183i	$-0.419490\pi$
$241$	120.616			0.500483			0.250241	+	0.968183i	$0.419490\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.593624	+	0.593624i	0.00240334	+	0.00240334i
$248$		0			0
$249$		−	200.202i		−	0.804024i
$250$		0			0
$251$	−197.576			−0.787153			−0.393577	−	0.919292i	$-0.628762\pi$
$251$	−197.576			−0.787153			−0.393577	+	0.919292i	$0.628762\pi$
$252$		0			0
$253$	73.4847	−	73.4847i	0.290453	−	0.290453i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−223.868	−	223.868i	−0.871083	−	0.871083i	0.121507	−	0.992591i	$-0.461227\pi$
$257$	−223.868	−	223.868i	−0.871083	−	0.871083i	−0.992591	+	0.121507i	$0.961227\pi$
$258$		0			0
$259$			66.2020i			0.255606i
$260$		0			0
$261$	−96.8786			−0.371182
$262$		0			0
$263$	90.7673	−	90.7673i	0.345123	−	0.345123i	−0.513166	−	0.858289i	$-0.671528\pi$
$263$	90.7673	−	90.7673i	0.345123	−	0.345123i	0.858289	+	0.513166i	$0.171528\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−118.293	−	118.293i	−0.443044	−	0.443044i
$268$		0			0
$269$			361.909i			1.34539i	0.739921	+	0.672694i	$0.234863\pi$
$269$			361.909i			1.34539i	−0.739921	+	0.672694i	$0.765137\pi$
$270$		0			0
$271$	−216.788			−0.799955			−0.399977	−	0.916525i	$-0.630982\pi$
$271$	−216.788			−0.799955			−0.399977	+	0.916525i	$0.630982\pi$
$272$		0			0
$273$	11.1589	−	11.1589i	0.0408752	−	0.0408752i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	205.406	+	205.406i	0.741539	+	0.741539i	0.972874	−	0.231335i	$-0.0743093\pi$
$277$	205.406	+	205.406i	0.741539	+	0.741539i	−0.231335	+	0.972874i	$0.574309\pi$
$278$		0			0
$279$			11.0908i			0.0397520i
$280$		0			0
$281$	334.899			1.19181			0.595906	−	0.803054i	$-0.296793\pi$
$281$	334.899			1.19181			0.595906	+	0.803054i	$0.296793\pi$
$282$		0			0
$283$	−74.3508	+	74.3508i	−0.262724	+	0.262724i	−0.826160	−	0.563436i	$-0.809479\pi$
$283$	−74.3508	+	74.3508i	−0.262724	+	0.262724i	0.563436	+	0.826160i	$0.309479\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0000	−	10.0000i	−0.0348432	−	0.0348432i
$288$		0			0
$289$		−	249.404i		−	0.862990i
$290$		0			0
$291$	132.959			0.456904
$292$		0			0
$293$	116.874	−	116.874i	0.398887	−	0.398887i	−0.478953	−	0.877840i	$-0.658984\pi$
$293$	116.874	−	116.874i	0.398887	−	0.398887i	0.877840	+	0.478953i	$0.158984\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−10.6515	−	10.6515i	−0.0358637	−	0.0358637i
$298$		0			0
$299$		−	297.909i		−	0.996352i
$300$		0			0
$301$	76.3939			0.253800
$302$		0			0
$303$	−61.8684	+	61.8684i	−0.204186	+	0.204186i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−135.250	−	135.250i	−0.440553	−	0.440553i	0.451645	−	0.892198i	$-0.350837\pi$
$307$	−135.250	−	135.250i	−0.440553	−	0.440553i	−0.892198	+	0.451645i	$0.850837\pi$
$308$		0			0
$309$		−	64.1816i		−	0.207708i
$310$		0			0
$311$	−239.212			−0.769171			−0.384586	−	0.923089i	$-0.625656\pi$
$311$	−239.212			−0.769171			−0.384586	+	0.923089i	$0.625656\pi$
$312$		0			0
$313$	271.386	−	271.386i	0.867048	−	0.867048i	−0.125097	−	0.992145i	$-0.539924\pi$
$313$	271.386	−	271.386i	0.867048	−	0.867048i	0.992145	+	0.125097i	$0.0399241\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	175.060	+	175.060i	0.552240	+	0.552240i	0.927087	−	0.374847i	$-0.122305\pi$
$317$	175.060	+	175.060i	0.552240	+	0.552240i	−0.374847	+	0.927087i	$0.622305\pi$
$318$		0			0
$319$		−	93.6163i		−	0.293468i
$320$		0			0
$321$	18.3337			0.0571143
$322$		0			0
$323$	−0.449490	+	0.449490i	−0.00139161	+	0.00139161i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	115.831	+	115.831i	0.354223	+	0.354223i
$328$		0			0
$329$			4.42449i			0.0134483i
$330$		0			0
$331$	−134.445			−0.406178			−0.203089	−	0.979160i	$-0.565098\pi$
$331$	−134.445			−0.406178			−0.203089	+	0.979160i	$0.565098\pi$
$332$		0			0
$333$	128.091	−	128.091i	0.384657	−	0.384657i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−137.654	−	137.654i	−0.408468	−	0.408468i	0.472736	−	0.881204i	$-0.343266\pi$
$337$	−137.654	−	137.654i	−0.408468	−	0.408468i	−0.881204	+	0.472736i	$0.843266\pi$
$338$		0			0
$339$			220.404i			0.650160i
$340$		0			0
$341$	−10.7173			−0.0314292
$342$		0			0
$343$	75.0431	−	75.0431i	0.218785	−	0.218785i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	365.823	+	365.823i	1.05424	+	1.05424i	0.998442	+	0.0558030i	$0.0177719\pi$
$347$	365.823	+	365.823i	1.05424	+	1.05424i	0.0558030	+	0.998442i	$0.482228\pi$
$348$		0			0
$349$			140.020i			0.401205i	0.979673	+	0.200602i	$0.0642899\pi$
$349$			140.020i			0.401205i	−0.979673	+	0.200602i	$0.935710\pi$
$350$		0			0
$351$	−43.1816			−0.123025
$352$		0			0
$353$	208.672	−	208.672i	0.591139	−	0.591139i	−0.346800	−	0.937939i	$-0.612732\pi$
$353$	208.672	−	208.672i	0.591139	−	0.591139i	0.937939	+	0.346800i	$0.112732\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	8.44949	+	8.44949i	0.0236680	+	0.0236680i
$358$		0			0
$359$		−	583.019i		−	1.62401i	−0.583651	−	0.812005i	$-0.698376\pi$
$359$		−	583.019i		−	1.62401i	0.583651	−	0.812005i	$-0.301624\pi$
$360$		0			0
$361$	360.990			0.999972
$362$		0			0
$363$	−137.901	+	137.901i	−0.379893	+	0.379893i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	123.341	+	123.341i	0.336078	+	0.336078i	0.854889	−	0.518811i	$-0.173625\pi$
$367$	123.341	+	123.341i	0.336078	+	0.336078i	−0.518811	+	0.854889i	$0.673625\pi$
$368$		0			0
$369$			38.6969i			0.104870i
$370$		0			0
$371$	−20.4541			−0.0551323
$372$		0			0
$373$	345.052	−	345.052i	0.925073	−	0.925073i	−0.0723091	−	0.997382i	$-0.523037\pi$
$373$	345.052	−	345.052i	0.925073	−	0.925073i	0.997382	+	0.0723091i	$0.0230368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−189.762	−	189.762i	−0.503347	−	0.503347i
$378$		0			0
$379$		−	421.393i		−	1.11185i	−0.831231	−	0.555927i	$-0.812363\pi$
$379$		−	421.393i		−	1.11185i	0.831231	−	0.555927i	$-0.187637\pi$
$380$		0			0
$381$	91.3735			0.239825
$382$		0			0
$383$	344.586	−	344.586i	0.899702	−	0.899702i	−0.0957079	−	0.995409i	$-0.530511\pi$
$383$	344.586	−	344.586i	0.899702	−	0.899702i	0.995409	+	0.0957079i	$0.0305115\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−147.810	−	147.810i	−0.381939	−	0.381939i
$388$		0			0
$389$			486.111i			1.24964i	0.780768	+	0.624822i	$0.214828\pi$
$389$			486.111i			1.24964i	−0.780768	+	0.624822i	$0.785172\pi$
$390$		0			0
$391$	225.576			0.576919
$392$		0			0
$393$	235.980	−	235.980i	0.600457	−	0.600457i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	312.062	+	312.062i	0.786052	+	0.786052i	0.980844	−	0.194793i	$-0.0624034\pi$
$397$	312.062	+	312.062i	0.786052	+	0.786052i	−0.194793	+	0.980844i	$0.562403\pi$
$398$		0			0
$399$			0.191836i			0.000480792i
$400$		0			0
$401$	590.252			1.47195			0.735975	−	0.677009i	$-0.236724\pi$
$401$	590.252			1.47195			0.735975	+	0.677009i	$0.236724\pi$
$402$		0			0
$403$	−21.7242	+	21.7242i	−0.0539063	+	0.0539063i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	123.778	+	123.778i	0.304122	+	0.304122i
$408$		0			0
$409$		−	736.696i		−	1.80121i	−0.434636	−	0.900606i	$-0.643123\pi$
$409$		−	736.696i		−	1.80121i	0.434636	−	0.900606i	$-0.356877\pi$
$410$		0			0
$411$	−404.969			−0.985327
$412$		0			0
$413$	59.2168	−	59.2168i	0.143382	−	0.143382i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	314.449	+	314.449i	0.754076	+	0.754076i
$418$		0			0
$419$			38.0296i			0.0907627i	0.998970	+	0.0453814i	$0.0144503\pi$
$419$			38.0296i			0.0907627i	−0.998970	+	0.0453814i	$0.985550\pi$
$420$		0			0
$421$	−331.394			−0.787159			−0.393579	−	0.919291i	$-0.628763\pi$
$421$	−331.394			−0.787159			−0.393579	+	0.919291i	$0.628763\pi$
$422$		0			0
$423$	8.56072	−	8.56072i	0.0202381	−	0.0202381i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−80.4620	−	80.4620i	−0.188436	−	0.188436i
$428$		0			0
$429$		−	41.7276i		−	0.0972670i
$430$		0			0
$431$	−843.040			−1.95601			−0.978004	−	0.208584i	$-0.933114\pi$
$431$	−843.040			−1.95601			−0.978004	+	0.208584i	$0.933114\pi$
$432$		0			0
$433$	149.381	−	149.381i	0.344992	−	0.344992i	−0.513248	−	0.858240i	$-0.671558\pi$
$433$	149.381	−	149.381i	0.344992	−	0.344992i	0.858240	+	0.513248i	$0.171558\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.56072	+	2.56072i	0.00585976	+	0.00585976i
$438$		0			0
$439$		−	545.999i		−	1.24373i	−0.783123	−	0.621867i	$-0.786375\pi$
$439$		−	545.999i		−	1.24373i	0.783123	−	0.621867i	$-0.213625\pi$
$440$		0			0
$441$	−143.394			−0.325156
$442$		0			0
$443$	−45.0398	+	45.0398i	−0.101670	+	0.101670i	−0.756112	−	0.654442i	$-0.772904\pi$
$443$	−45.0398	+	45.0398i	−0.101670	+	0.101670i	0.654442	+	0.756112i	$0.272904\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	236.116	+	236.116i	0.528223	+	0.528223i
$448$		0			0
$449$		−	320.767i		−	0.714404i	−0.934027	−	0.357202i	$-0.883731\pi$
$449$		−	320.767i		−	0.714404i	0.934027	−	0.357202i	$-0.116269\pi$
$450$		0			0
$451$	−37.3939			−0.0829133
$452$		0			0
$453$	121.139	−	121.139i	0.267414	−	0.267414i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−599.918	−	599.918i	−1.31273	−	1.31273i	−0.919394	−	0.393337i	$-0.871320\pi$
$457$	−599.918	−	599.918i	−1.31273	−	1.31273i	−0.393337	−	0.919394i	$-0.628680\pi$
$458$		0			0
$459$		−	32.6969i		−	0.0712352i
$460$		0			0
$461$	−376.595			−0.816909			−0.408454	−	0.912779i	$-0.633932\pi$
$461$	−376.595			−0.816909			−0.408454	+	0.912779i	$0.633932\pi$
$462$		0			0
$463$	−214.838	+	214.838i	−0.464012	+	0.464012i	−0.899968	−	0.435956i	$-0.856410\pi$
$463$	−214.838	+	214.838i	−0.464012	+	0.464012i	0.435956	+	0.899968i	$0.356410\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	463.419	+	463.419i	0.992332	+	0.992332i	0.999971	−	0.00763918i	$-0.00243165\pi$
$467$	463.419	+	463.419i	0.992332	+	0.992332i	−0.00763918	+	0.999971i	$0.502432\pi$
$468$		0			0
$469$			73.8490i			0.157461i
$470$		0			0
$471$	253.404			0.538013
$472$		0			0
$473$	142.833	−	142.833i	0.301973	−	0.301973i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	39.5755	+	39.5755i	0.0829675	+	0.0829675i
$478$		0			0
$479$		−	415.394i		−	0.867211i	−0.901103	−	0.433605i	$-0.857241\pi$
$479$		−	415.394i		−	0.867211i	0.901103	−	0.433605i	$-0.142759\pi$
$480$		0			0
$481$	501.798			1.04324
$482$		0			0
$483$	48.1362	−	48.1362i	0.0996609	−	0.0996609i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	38.1033	+	38.1033i	0.0782409	+	0.0782409i	0.745144	−	0.666903i	$-0.232381\pi$
$487$	38.1033	+	38.1033i	0.0782409	+	0.0782409i	−0.666903	+	0.745144i	$0.732381\pi$
$488$		0			0
$489$			46.8684i			0.0958453i
$490$		0			0
$491$	383.514			0.781088			0.390544	−	0.920584i	$-0.372287\pi$
$491$	383.514			0.781088			0.390544	+	0.920584i	$0.372287\pi$
$492$		0			0
$493$	143.687	−	143.687i	0.291454	−	0.291454i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	23.0306	+	23.0306i	0.0463393	+	0.0463393i
$498$		0			0
$499$		−	81.4133i		−	0.163153i	−0.996667	−	0.0815764i	$-0.974005\pi$
$499$		−	81.4133i		−	0.163153i	0.996667	−	0.0815764i	$-0.0259955\pi$
$500$		0			0
$501$	−236.858			−0.472771
$502$		0			0
$503$	171.626	−	171.626i	0.341204	−	0.341204i	−0.515616	−	0.856820i	$-0.672437\pi$
$503$	171.626	−	171.626i	0.341204	−	0.341204i	0.856820	+	0.515616i	$0.172437\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	122.399	+	122.399i	0.241419	+	0.241419i
$508$		0			0
$509$			452.202i			0.888413i	0.895925	+	0.444206i	$0.146514\pi$
$509$			452.202i			0.888413i	−0.895925	+	0.444206i	$0.853486\pi$
$510$		0			0
$511$	5.43470			0.0106354
$512$		0			0
$513$	0.371173	−	0.371173i	0.000723534	−	0.000723534i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	8.27245	+	8.27245i	0.0160009	+	0.0160009i
$518$		0			0
$519$			441.242i			0.850177i
$520$		0			0
$521$	−773.928			−1.48547			−0.742733	−	0.669588i	$-0.766471\pi$
$521$	−773.928			−1.48547			−0.742733	+	0.669588i	$0.766471\pi$
$522$		0			0
$523$	−474.507	+	474.507i	−0.907280	+	0.907280i	−0.996052	−	0.0887721i	$-0.971706\pi$
$523$	−474.507	+	474.507i	−0.907280	+	0.907280i	0.0887721	+	0.996052i	$0.471706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.4495	−	16.4495i	−0.0312135	−	0.0312135i
$528$		0			0
$529$		−	756.090i		−	1.42928i
$530$		0			0
$531$	−229.151			−0.431546
$532$		0			0
$533$	−75.7980	+	75.7980i	−0.142210	+	0.142210i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−164.116	−	164.116i	−0.305616	−	0.305616i
$538$		0			0
$539$		−	138.565i		−	0.257078i
$540$		0			0
$541$	511.867			0.946150			0.473075	−	0.881022i	$-0.343144\pi$
$541$	511.867			0.946150			0.473075	+	0.881022i	$0.343144\pi$
$542$		0			0
$543$	−210.149	+	210.149i	−0.387014	+	0.387014i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−373.980	−	373.980i	−0.683692	−	0.683692i	0.277138	−	0.960830i	$-0.410614\pi$
$547$	−373.980	−	373.980i	−0.683692	−	0.683692i	−0.960830	+	0.277138i	$0.910614\pi$
$548$		0			0
$549$			311.363i			0.567146i
$550$		0			0
$551$	3.26224			0.00592058
$552$		0			0
$553$	−68.0658	+	68.0658i	−0.123085	+	0.123085i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	421.015	+	421.015i	0.755861	+	0.755861i	0.975566	−	0.219705i	$-0.0705095\pi$
$557$	421.015	+	421.015i	0.755861	+	0.755861i	−0.219705	+	0.975566i	$0.570509\pi$
$558$		0			0
$559$		−	579.050i		−	1.03587i
$560$		0			0
$561$	31.5959			0.0563207
$562$		0			0
$563$	7.77296	−	7.77296i	0.0138063	−	0.0138063i	−0.700170	−	0.713976i	$-0.746892\pi$
$563$	7.77296	−	7.77296i	0.0138063	−	0.0138063i	0.713976	+	0.700170i	$0.246892\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.97730	−	6.97730i	−0.0123056	−	0.0123056i
$568$		0			0
$569$			473.787i			0.832666i	0.909212	+	0.416333i	$0.136685\pi$
$569$			473.787i			0.832666i	−0.909212	+	0.416333i	$0.863315\pi$
$570$		0			0
$571$	−120.344			−0.210760			−0.105380	−	0.994432i	$-0.533606\pi$
$571$	−120.344			−0.210760			−0.105380	+	0.994432i	$0.533606\pi$
$572$		0			0
$573$	32.1770	−	32.1770i	0.0561554	−	0.0561554i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−266.033	−	266.033i	−0.461062	−	0.461062i	0.437941	−	0.899004i	$-0.355708\pi$
$577$	−266.033	−	266.033i	−0.461062	−	0.461062i	−0.899004	+	0.437941i	$0.855708\pi$
$578$		0			0
$579$			181.565i			0.313584i
$580$		0			0
$581$	126.727			0.218118
$582$		0			0
$583$	−38.2429	+	38.2429i	−0.0655967	+	0.0655967i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	114.141	+	114.141i	0.194448	+	0.194448i	0.797615	−	0.603167i	$-0.206095\pi$
$587$	114.141	+	114.141i	0.194448	+	0.194448i	−0.603167	+	0.797615i	$0.706095\pi$
$588$		0			0
$589$		−	0.373467i		−	0.000634069i
$590$		0			0
$591$	−116.969			−0.197918
$592$		0			0
$593$	−566.636	+	566.636i	−0.955541	+	0.955541i	−0.999053	−	0.0435121i	$-0.986145\pi$
$593$	−566.636	+	566.636i	−0.955541	+	0.955541i	0.0435121	+	0.999053i	$0.486145\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	435.341	+	435.341i	0.729214	+	0.729214i
$598$		0			0
$599$			1002.44i			1.67353i	0.547564	+	0.836764i	$0.315555\pi$
$599$			1002.44i			1.67353i	−0.547564	+	0.836764i	$0.684445\pi$
$600$		0			0
$601$	−20.8796			−0.0347414			−0.0173707	−	0.999849i	$-0.505530\pi$
$601$	−20.8796			−0.0347414			−0.0173707	+	0.999849i	$0.505530\pi$
$602$		0			0
$603$	142.886	−	142.886i	0.236959	−	0.236959i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−305.748	−	305.748i	−0.503703	−	0.503703i	0.408883	−	0.912587i	$-0.365918\pi$
$607$	−305.748	−	305.748i	−0.503703	−	0.503703i	−0.912587	+	0.408883i	$0.865918\pi$
$608$		0			0
$609$		−	61.3235i		−	0.100695i
$610$		0			0
$611$	33.5367			0.0548883
$612$		0			0
$613$	−153.303	+	153.303i	−0.250087	+	0.250087i	−0.821006	−	0.570919i	$-0.806587\pi$
$613$	−153.303	+	153.303i	−0.250087	+	0.250087i	0.570919	+	0.821006i	$0.306587\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−276.879	−	276.879i	−0.448750	−	0.448750i	0.446189	−	0.894939i	$-0.352781\pi$
$617$	−276.879	−	276.879i	−0.448750	−	0.448750i	−0.894939	+	0.446189i	$0.852781\pi$
$618$		0			0
$619$			389.352i			0.629002i	0.949257	+	0.314501i	$0.101837\pi$
$619$			389.352i			0.629002i	−0.949257	+	0.314501i	$0.898163\pi$
$620$		0			0
$621$	−186.272			−0.299956
$622$		0			0
$623$	74.8786	−	74.8786i	0.120190	−	0.120190i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	0.358674	+	0.358674i	0.000572048	+	0.000572048i
$628$		0			0
$629$			379.959i			0.604069i
$630$		0			0
$631$	576.201			0.913155			0.456578	−	0.889684i	$-0.349075\pi$
$631$	576.201			0.913155			0.456578	+	0.889684i	$0.349075\pi$
$632$		0			0
$633$	−178.169	+	178.169i	−0.281468	+	0.281468i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−280.874	−	280.874i	−0.440932	−	0.440932i
$638$		0			0
$639$		−	89.1214i		−	0.139470i
$640$		0			0
$641$	780.827			1.21814			0.609069	−	0.793117i	$-0.291543\pi$
$641$	780.827			1.21814			0.609069	+	0.793117i	$0.291543\pi$
$642$		0			0
$643$	403.787	−	403.787i	0.627973	−	0.627973i	−0.319585	−	0.947558i	$-0.603543\pi$
$643$	403.787	−	403.787i	0.627973	−	0.627973i	0.947558	+	0.319585i	$0.103543\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−381.898	−	381.898i	−0.590260	−	0.590260i	0.347442	−	0.937702i	$-0.387050\pi$
$647$	−381.898	−	381.898i	−0.590260	−	0.590260i	−0.937702	+	0.347442i	$0.887050\pi$
$648$		0			0
$649$		−	221.435i		−	0.341194i
$650$		0			0
$651$	−7.02041			−0.0107840
$652$		0			0
$653$	149.864	−	149.864i	0.229500	−	0.229500i	−0.582984	−	0.812484i	$-0.698115\pi$
$653$	149.864	−	149.864i	0.229500	−	0.229500i	0.812484	+	0.582984i	$0.198115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−10.5153	−	10.5153i	−0.0160050	−	0.0160050i
$658$		0			0
$659$			552.495i			0.838384i	0.907898	+	0.419192i	$0.137687\pi$
$659$			552.495i			0.838384i	−0.907898	+	0.419192i	$0.862313\pi$
$660$		0			0
$661$	−274.767			−0.415684			−0.207842	−	0.978162i	$-0.566644\pi$
$661$	−274.767			−0.415684			−0.207842	+	0.978162i	$0.566644\pi$
$662$		0			0
$663$	64.0454	−	64.0454i	0.0965994	−	0.0965994i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−818.574	−	818.574i	−1.22725	−	1.22725i
$668$		0			0
$669$			564.242i			0.843411i
$670$		0			0
$671$	−300.879			−0.448403
$672$		0			0
$673$	−624.272	+	624.272i	−0.927597	+	0.927597i	−0.997550	−	0.0699537i	$-0.977715\pi$
$673$	−624.272	+	624.272i	−0.927597	+	0.927597i	0.0699537	+	0.997550i	$0.477715\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.04541	−	4.04541i	−0.00597549	−	0.00597549i	0.704113	−	0.710088i	$-0.251345\pi$
$677$	−4.04541	−	4.04541i	−0.00597549	−	0.00597549i	−0.710088	+	0.704113i	$0.751345\pi$
$678$		0			0
$679$			84.1623i			0.123950i
$680$		0			0
$681$	99.1418			0.145583
$682$		0			0
$683$	−913.757	+	913.757i	−1.33786	+	1.33786i	−0.439726	+	0.898132i	$0.644925\pi$
$683$	−913.757	+	913.757i	−1.33786	+	1.33786i	−0.898132	+	0.439726i	$0.855075\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	257.431	+	257.431i	0.374718	+	0.374718i
$688$		0			0
$689$			155.038i			0.225018i
$690$		0			0
$691$	1286.24			1.86142			0.930710	−	0.365759i	$-0.119190\pi$
$691$	1286.24			1.86142			0.930710	+	0.365759i	$0.119190\pi$
$692$		0			0
$693$	6.74235	−	6.74235i	0.00972922	−	0.00972922i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−57.3939	−	57.3939i	−0.0823442	−	0.0823442i
$698$		0			0
$699$			725.989i			1.03861i
$700$		0			0
$701$	−527.181			−0.752041			−0.376020	−	0.926611i	$-0.622708\pi$
$701$	−527.181			−0.752041			−0.376020	+	0.926611i	$0.622708\pi$
$702$		0			0
$703$	−4.31327	+	4.31327i	−0.00613551	+	0.00613551i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−39.1623	−	39.1623i	−0.0553922	−	0.0553922i
$708$		0			0
$709$		−	204.514i		−	0.288455i	−0.989545	−	0.144227i	$-0.953930\pi$
$709$		−	204.514i		−	0.288455i	0.989545	−	0.144227i	$-0.0460696\pi$
$710$		0			0
$711$	263.394			0.370456
$712$		0			0
$713$	−93.7117	+	93.7117i	−0.131433	+	0.131433i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−109.373	−	109.373i	−0.152543	−	0.152543i
$718$		0			0
$719$			511.989i			0.712085i	0.934470	+	0.356042i	$0.115874\pi$
$719$			511.989i			0.712085i	−0.934470	+	0.356042i	$0.884126\pi$
$720$		0			0
$721$	40.6265			0.0563475
$722$		0			0
$723$	147.724	−	147.724i	0.204321	−	0.204321i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	211.992	+	211.992i	0.291598	+	0.291598i	0.837712	−	0.546113i	$-0.183893\pi$
$727$	211.992	+	211.992i	0.291598	+	0.291598i	−0.546113	+	0.837712i	$0.683893\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	438.454			0.599800
$732$		0			0
$733$	−702.979	+	702.979i	−0.959043	+	0.959043i	−0.999194	−	0.0401506i	$-0.987216\pi$
$733$	−702.979	+	702.979i	−0.959043	+	0.959043i	0.0401506	+	0.999194i	$0.487216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	138.075	+	138.075i	0.187347	+	0.187347i
$738$		0			0
$739$			838.586i			1.13476i	0.823457	+	0.567379i	$0.192042\pi$
$739$			838.586i			1.13476i	−0.823457	+	0.567379i	$0.807958\pi$
$740$		0			0
$741$	1.45408			0.00196232
$742$		0			0
$743$	962.534	−	962.534i	1.29547	−	1.29547i	0.364115	−	0.931354i	$-0.381371\pi$
$743$	962.534	−	962.534i	1.29547	−	1.29547i	0.931354	−	0.364115i	$-0.118629\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−245.196	−	245.196i	−0.328242	−	0.328242i
$748$		0			0
$749$			11.6051i			0.0154941i
$750$		0			0
$751$	594.241			0.791266			0.395633	−	0.918409i	$-0.370525\pi$
$751$	594.241			0.791266			0.395633	+	0.918409i	$0.370525\pi$
$752$		0			0
$753$	−241.980	+	241.980i	−0.321354	+	0.321354i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−399.402	−	399.402i	−0.527611	−	0.527611i	0.392248	−	0.919859i	$-0.371697\pi$
$757$	−399.402	−	399.402i	−0.527611	−	0.527611i	−0.919859	+	0.392248i	$0.871697\pi$
$758$		0			0
$759$		−	180.000i		−	0.237154i
$760$		0			0
$761$	127.292			0.167269			0.0836346	−	0.996496i	$-0.473347\pi$
$761$	127.292			0.167269			0.0836346	+	0.996496i	$0.473347\pi$
$762$		0			0
$763$	−73.3201	+	73.3201i	−0.0960946	+	0.0960946i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−448.852	−	448.852i	−0.585204	−	0.585204i
$768$		0			0
$769$			699.847i			0.910074i	0.890473	+	0.455037i	$0.150374\pi$
$769$			699.847i			0.910074i	−0.890473	+	0.455037i	$0.849626\pi$
$770$		0			0
$771$	−548.363			−0.711236
$772$		0			0
$773$	−627.485	+	627.485i	−0.811753	+	0.811753i	−0.984897	−	0.173144i	$-0.944607\pi$
$773$	−627.485	+	627.485i	−0.811753	+	0.811753i	0.173144	+	0.984897i	$0.444607\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	81.0806	+	81.0806i	0.104351	+	0.104351i
$778$		0			0
$779$		−	1.30306i		−	0.00167274i
$780$		0			0
$781$	86.1204			0.110269
$782$		0			0
$783$	−118.652	+	118.652i	−0.151535	+	0.151535i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−363.992	−	363.992i	−0.462506	−	0.462506i	0.436970	−	0.899476i	$-0.356052\pi$
$787$	−363.992	−	363.992i	−0.462506	−	0.462506i	−0.899476	+	0.436970i	$0.856052\pi$
$788$		0			0
$789$		−	222.334i		−	0.281792i
$790$		0			0
$791$	−139.514			−0.176377
$792$		0			0
$793$	−609.885	+	609.885i	−0.769086	+	0.769086i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	506.080	+	506.080i	0.634981	+	0.634981i	0.949313	−	0.314332i	$-0.101781\pi$
$797$	506.080	+	506.080i	0.634981	+	0.634981i	−0.314332	+	0.949313i	$0.601781\pi$
$798$		0			0
$799$			25.3939i			0.0317821i
$800$		0			0
$801$	−289.757			−0.361744
$802$		0			0
$803$	10.1612	−	10.1612i	0.0126541	−	0.0126541i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	443.246	+	443.246i	0.549252	+	0.549252i
$808$		0			0
$809$		−	1165.28i		−	1.44040i	−0.693768	−	0.720198i	$-0.744051\pi$
$809$		−	1165.28i		−	1.44040i	0.693768	−	0.720198i	$-0.255949\pi$
$810$		0			0
$811$	−1507.67			−1.85903			−0.929516	−	0.368783i	$-0.879775\pi$
$811$	−1507.67			−1.85903			−0.929516	+	0.368783i	$0.879775\pi$
$812$		0			0
$813$	−265.510	+	265.510i	−0.326580	+	0.326580i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.97730	+	4.97730i	0.00609216	+	0.00609216i
$818$		0			0
$819$		−	27.3337i		−	0.0333745i
$820$		0			0
$821$	−965.959			−1.17656			−0.588282	−	0.808656i	$-0.700196\pi$
$821$	−965.959			−1.17656			−0.588282	+	0.808656i	$0.700196\pi$
$822$		0			0
$823$	−88.9727	+	88.9727i	−0.108108	+	0.108108i	−0.759092	−	0.650984i	$-0.774357\pi$
$823$	−88.9727	+	88.9727i	−0.108108	+	0.108108i	0.650984	+	0.759092i	$0.274357\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	275.271	+	275.271i	0.332855	+	0.332855i	0.853670	−	0.520814i	$-0.174372\pi$
$827$	275.271	+	275.271i	0.332855	+	0.332855i	−0.520814	+	0.853670i	$0.674372\pi$
$828$		0			0
$829$		−	351.980i		−	0.424583i	−0.977206	−	0.212292i	$-0.931907\pi$
$829$		−	351.980i		−	0.424583i	0.977206	−	0.212292i	$-0.0680927\pi$
$830$		0			0
$831$	503.141			0.605464
$832$		0			0
$833$	212.677	−	212.677i	0.255314	−	0.255314i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	13.5834	+	13.5834i	0.0162287	+	0.0162287i
$838$		0			0
$839$			1408.43i			1.67871i	0.543587	+	0.839353i	$0.317066\pi$
$839$			1408.43i			1.67871i	−0.543587	+	0.839353i	$0.682934\pi$
$840$		0			0
$841$	−201.829			−0.239986
$842$		0			0
$843$	410.166	−	410.166i	0.486555	−	0.486555i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−87.2906	−	87.2906i	−0.103059	−	0.103059i
$848$		0			0
$849$			182.121i			0.214513i
$850$		0			0
$851$	2164.60			2.54360
$852$		0			0
$853$	50.5528	−	50.5528i	0.0592647	−	0.0592647i	−0.676853	−	0.736118i	$-0.736657\pi$
$853$	50.5528	−	50.5528i	0.0592647	−	0.0592647i	0.736118	+	0.676853i	$0.236657\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−210.434	−	210.434i	−0.245547	−	0.245547i	0.573593	−	0.819140i	$-0.305549\pi$
$857$	−210.434	−	210.434i	−0.245547	−	0.245547i	−0.819140	+	0.573593i	$0.805549\pi$
$858$		0			0
$859$		−	1255.45i		−	1.46153i	−0.682630	−	0.730764i	$-0.739164\pi$
$859$		−	1255.45i		−	1.46153i	0.682630	−	0.730764i	$-0.260836\pi$
$860$		0			0
$861$	−24.4949			−0.0284494
$862$		0			0
$863$	−1007.70	+	1007.70i	−1.16767	+	1.16767i	−0.184911	+	0.982755i	$0.559200\pi$
$863$	−1007.70	+	1007.70i	−1.16767	+	1.16767i	−0.982755	+	0.184911i	$0.940800\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−305.456	−	305.456i	−0.352314	−	0.352314i
$868$		0			0
$869$			254.524i			0.292894i
$870$		0			0
$871$	559.760			0.642664
$872$		0			0
$873$	162.841	−	162.841i	0.186530	−	0.186530i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	282.346	+	282.346i	0.321945	+	0.321945i	0.849513	−	0.527568i	$-0.176896\pi$
$877$	282.346	+	282.346i	0.321945	+	0.321945i	−0.527568	+	0.849513i	$0.676896\pi$
$878$		0			0
$879$		−	286.282i		−	0.325690i
$880$		0			0
$881$	−1420.34			−1.61219			−0.806096	−	0.591785i	$-0.798423\pi$
$881$	−1420.34			−1.61219			−0.806096	+	0.591785i	$0.798423\pi$
$882$		0			0
$883$	1170.15	−	1170.15i	1.32519	−	1.32519i	0.415686	−	0.909508i	$-0.363541\pi$
$883$	1170.15	−	1170.15i	1.32519	−	1.32519i	0.909508	−	0.415686i	$-0.136459\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−756.711	−	756.711i	−0.853112	−	0.853112i	0.137403	−	0.990515i	$-0.456125\pi$
$887$	−756.711	−	756.711i	−0.853112	−	0.853112i	−0.990515	+	0.137403i	$0.956125\pi$
$888$		0			0
$889$			57.8388i			0.0650605i
$890$		0			0
$891$	−26.0908			−0.0292826
$892$		0			0
$893$	−0.288269	+	0.288269i	−0.000322810	+	0.000322810i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−364.863	−	364.863i	−0.406759	−	0.406759i
$898$		0			0
$899$			119.385i			0.132797i
$900$		0			0
$901$	−117.394			−0.130293
$902$		0			0
$903$	93.5630	−	93.5630i	0.103614	−	0.103614i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	942.979	+	942.979i	1.03967	+	1.03967i	0.999180	+	0.0404876i	$0.0128911\pi$
$907$	942.979	+	942.979i	1.03967	+	1.03967i	0.0404876	+	0.999180i	$0.487109\pi$
$908$		0			0
$909$			151.546i			0.166717i
$910$		0			0
$911$	−1181.15			−1.29654			−0.648272	−	0.761409i	$-0.724508\pi$
$911$	−1181.15			−1.29654			−0.648272	+	0.761409i	$0.724508\pi$
$912$		0			0
$913$	236.940	−	236.940i	0.259518	−	0.259518i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	149.373	+	149.373i	0.162894	+	0.162894i
$918$		0			0
$919$			473.595i			0.515337i	0.966233	+	0.257669i	$0.0829543\pi$
$919$			473.595i			0.515337i	−0.966233	+	0.257669i	$0.917046\pi$
$920$		0			0
$921$	−331.293			−0.359710
$922$		0			0
$923$	174.567	−	174.567i	0.189130	−	0.189130i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−78.6061	−	78.6061i	−0.0847962	−	0.0847962i
$928$		0			0
$929$		−	1094.43i		−	1.17807i	−0.808106	−	0.589037i	$-0.799507\pi$
$929$		−	1094.43i		−	1.17807i	0.808106	−	0.589037i	$-0.200493\pi$
$930$		0			0
$931$	4.82857			0.00518644
$932$		0			0
$933$	−292.974	+	292.974i	−0.314013	+	0.314013i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	654.891	+	654.891i	0.698923	+	0.698923i	0.964178	−	0.265255i	$-0.0854562\pi$
$937$	654.891	+	654.891i	0.698923	+	0.698923i	−0.265255	+	0.964178i	$0.585456\pi$
$938$		0			0
$939$		−	664.757i		−	0.707942i
$940$		0			0
$941$	766.382			0.814433			0.407217	−	0.913332i	$-0.366499\pi$
$941$	766.382			0.814433			0.407217	+	0.913332i	$0.366499\pi$
$942$		0			0
$943$	−326.969	+	326.969i	−0.346733	+	0.346733i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	325.348	+	325.348i	0.343557	+	0.343557i	0.857703	−	0.514146i	$-0.171891\pi$
$947$	325.348	+	325.348i	0.343557	+	0.343557i	−0.514146	+	0.857703i	$0.671891\pi$
$948$		0			0
$949$		−	41.1939i		−	0.0434077i
$950$		0			0
$951$	428.808			0.450902
$952$		0			0
$953$	−31.1010	+	31.1010i	−0.0326349	+	0.0326349i	−0.723236	−	0.690601i	$-0.757346\pi$
$953$	−31.1010	+	31.1010i	−0.0326349	+	0.0326349i	0.690601	+	0.723236i	$0.257346\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−114.656	−	114.656i	−0.119808	−	0.119808i
$958$		0			0
$959$		−	256.343i		−	0.267302i
$960$		0			0
$961$	−947.333			−0.985778
$962$		0			0
$963$	22.4541	−	22.4541i	0.0233168	−	0.0233168i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	314.656	+	314.656i	0.325394	+	0.325394i	0.850832	−	0.525438i	$-0.176098\pi$
$967$	314.656	+	314.656i	0.325394	+	0.325394i	−0.525438	+	0.850832i	$0.676098\pi$
$968$		0			0
$969$			1.10102i			0.00113624i
$970$		0			0
$971$	−253.614			−0.261189			−0.130594	−	0.991436i	$-0.541689\pi$
$971$	−253.614			−0.261189			−0.130594	+	0.991436i	$0.541689\pi$
$972$		0			0
$973$	−199.044	+	199.044i	−0.204568	+	0.204568i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	580.974	+	580.974i	0.594651	+	0.594651i	0.938884	−	0.344233i	$-0.111861\pi$
$977$	580.974	+	580.974i	0.594651	+	0.594651i	−0.344233	+	0.938884i	$0.611861\pi$
$978$		0			0
$979$		−	280.000i		−	0.286006i
$980$		0			0
$981$	283.727			0.289222
$982$		0			0
$983$	734.070	−	734.070i	0.746765	−	0.746765i	−0.227105	−	0.973870i	$-0.572926\pi$
$983$	734.070	−	734.070i	0.746765	−	0.746765i	0.973870	+	0.227105i	$0.0729260\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	5.41887	+	5.41887i	0.00549025	+	0.00549025i
$988$		0			0
$989$		−	2497.85i		−	2.52563i
$990$		0			0
$991$	616.887			0.622489			0.311245	−	0.950330i	$-0.399254\pi$
$991$	616.887			0.622489			0.311245	+	0.950330i	$0.399254\pi$
$992$		0			0
$993$	−164.661	+	164.661i	−0.165821	+	0.165821i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−460.191	−	460.191i	−0.461576	−	0.461576i	0.437596	−	0.899172i	$-0.355830\pi$
$997$	−460.191	−	460.191i	−0.461576	−	0.461576i	−0.899172	+	0.437596i	$0.855830\pi$
$998$		0			0
$999$		−	313.757i		−	0.314071i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.u.d.457.2	yes	4
3.2	odd	2		1800.3.v.m.1657.1		4
4.3	odd	2		1200.3.bg.f.1057.1		4
5.2	odd	4		600.3.u.c.193.1	✓	4
5.3	odd	4	inner	600.3.u.d.193.2	yes	4
5.4	even	2		600.3.u.c.457.1	yes	4
15.2	even	4		1800.3.v.l.793.2		4
15.8	even	4		1800.3.v.m.793.1		4
15.14	odd	2		1800.3.v.l.1657.2		4
20.3	even	4		1200.3.bg.f.193.1		4
20.7	even	4		1200.3.bg.l.193.2		4
20.19	odd	2		1200.3.bg.l.1057.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.c.193.1	✓	4	5.2	odd	4
600.3.u.c.457.1	yes	4	5.4	even	2
600.3.u.d.193.2	yes	4	5.3	odd	4	inner
600.3.u.d.457.2	yes	4	1.1	even	1	trivial
1200.3.bg.f.193.1		4	20.3	even	4
1200.3.bg.f.1057.1		4	4.3	odd	2
1200.3.bg.l.193.2		4	20.7	even	4
1200.3.bg.l.1057.2		4	20.19	odd	2
1800.3.v.l.793.2		4	15.2	even	4
1800.3.v.l.1657.2		4	15.14	odd	2
1800.3.v.m.793.1		4	15.8	even	4
1800.3.v.m.1657.1		4	3.2	odd	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(0.775255 + 0.775255i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(0.775255 + 0.775255i) q^{7} -3.00000i q^{9} +2.89898 q^{11} +(5.87628 - 5.87628i) q^{13} +(4.44949 + 4.44949i) q^{17} +0.101021i q^{19} +1.89898 q^{21} +(25.3485 - 25.3485i) q^{23} +(-3.67423 - 3.67423i) q^{27} -32.2929i q^{29} -3.69694 q^{31} +(3.55051 - 3.55051i) q^{33} +(42.6969 + 42.6969i) q^{37} -14.3939i q^{39} -12.8990 q^{41} +(49.2702 - 49.2702i) q^{43} +(2.85357 + 2.85357i) q^{47} -47.7980i q^{49} +10.8990 q^{51} +(-13.1918 + 13.1918i) q^{53} +(0.123724 + 0.123724i) q^{57} -76.3837i q^{59} -103.788 q^{61} +(2.32577 - 2.32577i) q^{63} +(47.6288 + 47.6288i) q^{67} -62.0908i q^{69} +29.7071 q^{71} +(3.50510 - 3.50510i) q^{73} +(2.24745 + 2.24745i) q^{77} +87.7980i q^{79} -9.00000 q^{81} +(81.7321 - 81.7321i) q^{83} +(-39.5505 - 39.5505i) q^{87} -96.5857i q^{89} +9.11123 q^{91} +(-4.52781 + 4.52781i) q^{93} +(54.2804 + 54.2804i) q^{97} -8.69694i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} - 8 q^{11} + 48 q^{13} + 8 q^{17} - 12 q^{21} + 72 q^{23} + 44 q^{31} + 24 q^{33} + 112 q^{37} - 32 q^{41} + 104 q^{43} + 80 q^{47} + 24 q^{51} + 104 q^{53} - 24 q^{57} - 180 q^{61} + 24 q^{63} + 264 q^{67} + 256 q^{71} + 112 q^{73} - 40 q^{77} - 36 q^{81} - 16 q^{83} - 168 q^{87} + 252 q^{91} - 72 q^{93} + 320 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 - 8 * q^11 + 48 * q^13 + 8 * q^17 - 12 * q^21 + 72 * q^23 + 44 * q^31 + 24 * q^33 + 112 * q^37 - 32 * q^41 + 104 * q^43 + 80 * q^47 + 24 * q^51 + 104 * q^53 - 24 * q^57 - 180 * q^61 + 24 * q^63 + 264 * q^67 + 256 * q^71 + 112 * q^73 - 40 * q^77 - 36 * q^81 - 16 * q^83 - 168 * q^87 + 252 * q^91 - 72 * q^93 + 320 * q^97

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