LMFDB - Embedded newform 450.4.f.d.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(107,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.f (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:	$26.5508595026$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.4.f.d.143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.41421i) q^{2} -4.00000i q^{4} +(0.123724 + 0.123724i) q^{7} +(5.65685 + 5.65685i) q^{8} +O(q^{10})$	\(q+(-1.41421 + 1.41421i) q^{2} -4.00000i q^{4} +(0.123724 + 0.123724i) q^{7} +(5.65685 + 5.65685i) q^{8} +4.24264i q^{11} +(-5.62883 + 5.62883i) q^{13} -0.349945 q^{14} -16.0000 q^{16} +(-34.4660 + 34.4660i) q^{17} -110.485i q^{19} +(-6.00000 - 6.00000i) q^{22} +(111.883 + 111.883i) q^{23} -15.9207i q^{26} +(0.494897 - 0.494897i) q^{28} +53.0547 q^{29} -201.454 q^{31} +(22.6274 - 22.6274i) q^{32} -97.4847i q^{34} +(169.732 + 169.732i) q^{37} +(156.249 + 156.249i) q^{38} +371.166i q^{41} +(-202.639 + 202.639i) q^{43} +16.9706 q^{44} -316.454 q^{46} +(-106.548 + 106.548i) q^{47} -342.969i q^{49} +(22.5153 + 22.5153i) q^{52} +(243.973 + 243.973i) q^{53} +1.39978i q^{56} +(-75.0306 + 75.0306i) q^{58} -57.2540 q^{59} -609.847 q^{61} +(284.899 - 284.899i) q^{62} +64.0000i q^{64} +(340.825 + 340.825i) q^{67} +(137.864 + 137.864i) q^{68} +990.505i q^{71} +(-847.176 + 847.176i) q^{73} -480.075 q^{74} -441.939 q^{76} +(-0.524918 + 0.524918i) q^{77} -436.847i q^{79} +(-524.908 - 524.908i) q^{82} +(-127.718 - 127.718i) q^{83} -573.150i q^{86} +(-24.0000 + 24.0000i) q^{88} -1230.19 q^{89} -1.39285 q^{91} +(447.534 - 447.534i) q^{92} -301.362i q^{94} +(-202.578 - 202.578i) q^{97} +(485.032 + 485.032i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 48 q^{7}+O(q^{10}) $ 8 * q - 48 * q^7	$ 8 q - 48 q^{7} - 192 q^{13} - 128 q^{16} - 48 q^{22} - 192 q^{28} + 152 q^{31} + 672 q^{37} - 2160 q^{43} - 768 q^{46} + 768 q^{52} - 1776 q^{58} + 1000 q^{61} + 816 q^{67} - 3936 q^{73} - 1184 q^{76} - 672 q^{82} - 192 q^{88} + 4104 q^{91} + 192 q^{97}+O(q^{100}) $ 8 * q - 48 * q^7 - 192 * q^13 - 128 * q^16 - 48 * q^22 - 192 * q^28 + 152 * q^31 + 672 * q^37 - 2160 * q^43 - 768 * q^46 + 768 * q^52 - 1776 * q^58 + 1000 * q^61 + 816 * q^67 - 3936 * q^73 - 1184 * q^76 - 672 * q^82 - 192 * q^88 + 4104 * q^91 + 192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.41421	+	1.41421i	−0.500000	+	0.500000i
$2$	−1.41421	+	1.41421i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	4.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.123724	+	0.123724i	0.00668049	+	0.00668049i	0.710439	−	0.703759i	$-0.248496\pi$
$7$	0.123724	+	0.123724i	0.00668049	+	0.00668049i	−0.703759	+	0.710439i	$0.748496\pi$
$8$	5.65685	+	5.65685i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$			4.24264i			0.116291i	0.998308	+	0.0581456i	$0.0185188\pi$
$11$			4.24264i			0.116291i	−0.998308	+	0.0581456i	$0.981481\pi$
$12$		0			0
$13$	−5.62883	+	5.62883i	−0.120089	+	0.120089i	−0.764597	−	0.644508i	$-0.777062\pi$
$13$	−5.62883	+	5.62883i	−0.120089	+	0.120089i	0.644508	+	0.764597i	$0.277062\pi$
$14$	−0.349945			−0.00668049
$15$		0			0
$16$	−16.0000			−0.250000
$17$	−34.4660	+	34.4660i	−0.491720	+	0.491720i	−0.908848	−	0.417128i	$-0.863037\pi$
$17$	−34.4660	+	34.4660i	−0.491720	+	0.491720i	0.417128	+	0.908848i	$0.363037\pi$
$18$		0			0
$19$		−	110.485i		−	1.33405i	−0.745036	−	0.667024i	$-0.767568\pi$
$19$		−	110.485i		−	1.33405i	0.745036	−	0.667024i	$-0.232432\pi$
$20$		0			0
$21$		0			0
$22$	−6.00000	−	6.00000i	−0.0581456	−	0.0581456i
$23$	111.883	+	111.883i	1.01432	+	1.01432i	0.999896	+	0.0144216i	$0.00459069\pi$
$23$	111.883	+	111.883i	1.01432	+	1.01432i	0.0144216	+	0.999896i	$0.495409\pi$
$24$		0			0
$25$		0			0
$26$		−	15.9207i		−	0.120089i
$27$		0			0
$28$	0.494897	−	0.494897i	0.00334024	−	0.00334024i
$29$	53.0547			0.339724			0.169862	−	0.985468i	$-0.445668\pi$
$29$	53.0547			0.339724			0.169862	+	0.985468i	$0.445668\pi$
$30$		0			0
$31$	−201.454			−1.16717			−0.583584	−	0.812053i	$-0.698350\pi$
$31$	−201.454			−1.16717			−0.583584	+	0.812053i	$0.698350\pi$
$32$	22.6274	−	22.6274i	0.125000	−	0.125000i
$33$		0			0
$34$		−	97.4847i		−	0.491720i
$35$		0			0
$36$		0			0
$37$	169.732	+	169.732i	0.754157	+	0.754157i	0.975252	−	0.221096i	$-0.0709633\pi$
$37$	169.732	+	169.732i	0.754157	+	0.754157i	−0.221096	+	0.975252i	$0.570963\pi$
$38$	156.249	+	156.249i	0.667024	+	0.667024i
$39$		0			0
$40$		0			0
$41$			371.166i			1.41381i	0.707306	+	0.706907i	$0.249910\pi$
$41$			371.166i			1.41381i	−0.707306	+	0.706907i	$0.750090\pi$
$42$		0			0
$43$	−202.639	+	202.639i	−0.718655	+	0.718655i	−0.968330	−	0.249675i	$-0.919676\pi$
$43$	−202.639	+	202.639i	−0.718655	+	0.718655i	0.249675	+	0.968330i	$0.419676\pi$
$44$	16.9706			0.0581456
$45$		0			0
$46$	−316.454			−1.01432
$47$	−106.548	+	106.548i	−0.330672	+	0.330672i	−0.852842	−	0.522170i	$-0.825123\pi$
$47$	−106.548	+	106.548i	−0.330672	+	0.330672i	0.522170	+	0.852842i	$0.325123\pi$
$48$		0			0
$49$		−	342.969i		−	0.999911i
$50$		0			0
$51$		0			0
$52$	22.5153	+	22.5153i	0.0600445	+	0.0600445i
$53$	243.973	+	243.973i	0.632308	+	0.632308i	0.948646	−	0.316338i	$-0.102453\pi$
$53$	243.973	+	243.973i	0.632308	+	0.632308i	−0.316338	+	0.948646i	$0.602453\pi$
$54$		0			0
$55$		0			0
$56$			1.39978i			0.00334024i
$57$		0			0
$58$	−75.0306	+	75.0306i	−0.169862	+	0.169862i
$59$	−57.2540			−0.126336			−0.0631681	−	0.998003i	$-0.520120\pi$
$59$	−57.2540			−0.126336			−0.0631681	+	0.998003i	$0.520120\pi$
$60$		0			0
$61$	−609.847			−1.28005			−0.640024	−	0.768355i	$-0.721076\pi$
$61$	−609.847			−1.28005			−0.640024	+	0.768355i	$0.721076\pi$
$62$	284.899	−	284.899i	0.583584	−	0.583584i
$63$		0			0
$64$			64.0000i			0.125000i
$65$		0			0
$66$		0			0
$67$	340.825	+	340.825i	0.621469	+	0.621469i	0.945907	−	0.324438i	$-0.105175\pi$
$67$	340.825	+	340.825i	0.621469	+	0.621469i	−0.324438	+	0.945907i	$0.605175\pi$
$68$	137.864	+	137.864i	0.245860	+	0.245860i
$69$		0			0
$70$		0			0
$71$			990.505i			1.65565i	0.560985	+	0.827826i	$0.310423\pi$
$71$			990.505i			1.65565i	−0.560985	+	0.827826i	$0.689577\pi$
$72$		0			0
$73$	−847.176	+	847.176i	−1.35828	+	1.35828i	−0.482242	+	0.876038i	$0.660178\pi$
$73$	−847.176	+	847.176i	−1.35828	+	1.35828i	−0.876038	+	0.482242i	$0.839822\pi$
$74$	−480.075			−0.754157
$75$		0			0
$76$	−441.939			−0.667024
$77$	−0.524918	+	0.524918i	−0.000776883	+	0.000776883i
$78$		0			0
$79$		−	436.847i		−	0.622141i	−0.950387	−	0.311070i	$-0.899313\pi$
$79$		−	436.847i		−	0.622141i	0.950387	−	0.311070i	$-0.100687\pi$
$80$		0			0
$81$		0			0
$82$	−524.908	−	524.908i	−0.706907	−	0.706907i
$83$	−127.718	−	127.718i	−0.168901	−	0.168901i	0.617595	−	0.786496i	$-0.288107\pi$
$83$	−127.718	−	127.718i	−0.168901	−	0.168901i	−0.786496	+	0.617595i	$0.788107\pi$
$84$		0			0
$85$		0			0
$86$		−	573.150i		−	0.718655i
$87$		0			0
$88$	−24.0000	+	24.0000i	−0.0290728	+	0.0290728i
$89$	−1230.19			−1.46517			−0.732585	−	0.680675i	$-0.761686\pi$
$89$	−1230.19			−1.46517			−0.732585	+	0.680675i	$0.761686\pi$
$90$		0			0
$91$	−1.39285			−0.00160450
$92$	447.534	−	447.534i	0.507159	−	0.507159i
$93$		0			0
$94$		−	301.362i		−	0.330672i
$95$		0			0
$96$		0			0
$97$	−202.578	−	202.578i	−0.212048	−	0.212048i	0.593089	−	0.805137i	$-0.297908\pi$
$97$	−202.578	−	202.578i	−0.212048	−	0.212048i	−0.805137	+	0.593089i	$0.797908\pi$
$98$	485.032	+	485.032i	0.499955	+	0.499955i
$99$		0			0
$100$		0			0
$101$			952.538i			0.938426i	0.883085	+	0.469213i	$0.155462\pi$
$101$			952.538i			0.938426i	−0.883085	+	0.469213i	$0.844538\pi$
$102$		0			0
$103$	−862.640	+	862.640i	−0.825228	+	0.825228i	−0.986852	−	0.161624i	$-0.948327\pi$
$103$	−862.640	+	862.640i	−0.825228	+	0.825228i	0.161624	+	0.986852i	$0.448327\pi$
$104$	−63.6829			−0.0600445
$105$		0			0
$106$	−690.061			−0.632308
$107$	−594.933	+	594.933i	−0.537517	+	0.537517i	−0.922799	−	0.385282i	$-0.874104\pi$
$107$	−594.933	+	594.933i	−0.537517	+	0.537517i	0.385282	+	0.922799i	$0.374104\pi$
$108$		0			0
$109$			46.0918i			0.0405027i	0.999795	+	0.0202514i	$0.00644665\pi$
$109$			46.0918i			0.0405027i	−0.999795	+	0.0202514i	$0.993553\pi$
$110$		0			0
$111$		0			0
$112$	−1.97959	−	1.97959i	−0.00167012	−	0.00167012i
$113$	−1140.00	−	1140.00i	−0.949049	−	0.949049i	0.0497143	−	0.998763i	$-0.484169\pi$
$113$	−1140.00	−	1140.00i	−0.949049	−	0.949049i	−0.998763	+	0.0497143i	$0.984169\pi$
$114$		0			0
$115$		0			0
$116$		−	212.219i		−	0.169862i
$117$		0			0
$118$	80.9694	−	80.9694i	0.0631681	−	0.0631681i
$119$	−8.52858			−0.00656986
$120$		0			0
$121$	1313.00			0.986476
$122$	862.454	−	862.454i	0.640024	−	0.640024i
$123$		0			0
$124$			805.816i			0.583584i
$125$		0			0
$126$		0			0
$127$	629.816	+	629.816i	0.440056	+	0.440056i	0.892031	−	0.451974i	$-0.149280\pi$
$127$	629.816	+	629.816i	0.440056	+	0.440056i	−0.451974	+	0.892031i	$0.649280\pi$
$128$	−90.5097	−	90.5097i	−0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$			744.605i			0.496614i	0.968681	+	0.248307i	$0.0798742\pi$
$131$			744.605i			0.496614i	−0.968681	+	0.248307i	$0.920126\pi$
$132$		0			0
$133$	13.6696	−	13.6696i	0.00891210	−	0.00891210i
$134$	−963.999			−0.621469
$135$		0			0
$136$	−389.939			−0.245860
$137$	−629.399	+	629.399i	−0.392505	+	0.392505i	−0.875579	−	0.483074i	$-0.839520\pi$
$137$	−629.399	+	629.399i	−0.392505	+	0.392505i	0.483074	+	0.875579i	$0.339520\pi$
$138$		0			0
$139$			229.755i			0.140198i	0.997540	+	0.0700992i	$0.0223316\pi$
$139$			229.755i			0.140198i	−0.997540	+	0.0700992i	$0.977668\pi$
$140$		0			0
$141$		0			0
$142$	−1400.79	−	1400.79i	−0.827826	−	0.827826i
$143$	−23.8811	−	23.8811i	−0.0139653	−	0.0139653i
$144$		0			0
$145$		0			0
$146$		−	2396.18i		−	1.35828i
$147$		0			0
$148$	678.929	−	678.929i	0.377078	−	0.377078i
$149$	1887.54			1.03781			0.518904	−	0.854832i	$-0.326340\pi$
$149$	1887.54			1.03781			0.518904	+	0.854832i	$0.326340\pi$
$150$		0			0
$151$	803.209			0.432876			0.216438	−	0.976296i	$-0.430556\pi$
$151$	803.209			0.432876			0.216438	+	0.976296i	$0.430556\pi$
$152$	624.996	−	624.996i	0.333512	−	0.333512i
$153$		0			0
$154$		−	1.48469i		−	0.000776883i
$155$		0			0
$156$		0			0
$157$	1066.46	+	1066.46i	0.542117	+	0.542117i	0.924149	−	0.382032i	$-0.124776\pi$
$157$	1066.46	+	1066.46i	0.542117	+	0.542117i	−0.382032	+	0.924149i	$0.624776\pi$
$158$	617.795	+	617.795i	0.311070	+	0.311070i
$159$		0			0
$160$		0			0
$161$			27.6854i			0.0135523i
$162$		0			0
$163$	2084.54	−	2084.54i	1.00168	−	1.00168i	0.00167955	−	0.999999i	$-0.499465\pi$
$163$	2084.54	−	2084.54i	1.00168	−	1.00168i	0.999999	−	0.00167955i	$-0.000534619\pi$
$164$	1484.66			0.706907
$165$		0			0
$166$	361.240			0.168901
$167$	−734.724	+	734.724i	−0.340447	+	0.340447i	−0.856535	−	0.516088i	$-0.827388\pi$
$167$	−734.724	+	734.724i	−0.340447	+	0.340447i	0.516088	+	0.856535i	$0.327388\pi$
$168$		0			0
$169$			2133.63i			0.971157i
$170$		0			0
$171$		0			0
$172$	810.556	+	810.556i	0.359327	+	0.359327i
$173$	−1563.92	−	1563.92i	−0.687297	−	0.687297i	0.274337	−	0.961634i	$-0.411542\pi$
$173$	−1563.92	−	1563.92i	−0.687297	−	0.687297i	−0.961634	+	0.274337i	$0.911542\pi$
$174$		0			0
$175$		0			0
$176$		−	67.8823i		−	0.0290728i
$177$		0			0
$178$	1739.76	−	1739.76i	0.732585	−	0.732585i
$179$	2662.41			1.11172			0.555860	−	0.831276i	$-0.312389\pi$
$179$	2662.41			1.11172			0.555860	+	0.831276i	$0.312389\pi$
$180$		0			0
$181$	3228.45			1.32579			0.662897	−	0.748711i	$-0.269327\pi$
$181$	3228.45			1.32579			0.662897	+	0.748711i	$0.269327\pi$
$182$	1.96978	−	1.96978i	0.000802252	−	0.000802252i
$183$		0			0
$184$			1265.82i			0.507159i
$185$		0			0
$186$		0			0
$187$	−146.227	−	146.227i	−0.0571828	−	0.0571828i
$188$	426.191	+	426.191i	0.165336	+	0.165336i
$189$		0			0
$190$		0			0
$191$		−	2901.40i		−	1.09915i	−0.835443	−	0.549576i	$-0.814789\pi$
$191$		−	2901.40i		−	1.09915i	0.835443	−	0.549576i	$-0.185211\pi$
$192$		0			0
$193$	1073.38	−	1073.38i	0.400329	−	0.400329i	−0.478020	−	0.878349i	$-0.658645\pi$
$193$	1073.38	−	1073.38i	0.400329	−	0.400329i	0.878349	+	0.478020i	$0.158645\pi$
$194$	572.977			0.212048
$195$		0			0
$196$	−1371.88			−0.499955
$197$	−1689.97	+	1689.97i	−0.611196	+	0.611196i	−0.943258	−	0.332062i	$-0.892256\pi$
$197$	−1689.97	+	1689.97i	−0.611196	+	0.611196i	0.332062	+	0.943258i	$0.392256\pi$
$198$		0			0
$199$		−	1221.82i		−	0.435239i	−0.976034	−	0.217620i	$-0.930171\pi$
$199$		−	1221.82i		−	0.435239i	0.976034	−	0.217620i	$-0.0698292\pi$
$200$		0			0
$201$		0			0
$202$	−1347.09	−	1347.09i	−0.469213	−	0.469213i
$203$	6.56415	+	6.56415i	0.00226952	+	0.00226952i
$204$		0			0
$205$		0			0
$206$		−	2439.92i		−	0.825228i
$207$		0			0
$208$	90.0612	−	90.0612i	0.0300222	−	0.0300222i
$209$	468.747			0.155138
$210$		0			0
$211$	4685.99			1.52890			0.764448	−	0.644685i	$-0.223011\pi$
$211$	4685.99			1.52890			0.764448	+	0.644685i	$0.223011\pi$
$212$	975.894	−	975.894i	0.316154	−	0.316154i
$213$		0			0
$214$		−	1682.72i		−	0.537517i
$215$		0			0
$216$		0			0
$217$	−24.9248	−	24.9248i	−0.00779725	−	0.00779725i
$218$	−65.1837	−	65.1837i	−0.0202514	−	0.0202514i
$219$		0			0
$220$		0			0
$221$		−	388.007i		−	0.118100i
$222$		0			0
$223$	3745.86	−	3745.86i	1.12485	−	1.12485i	0.133847	−	0.991002i	$-0.457267\pi$
$223$	3745.86	−	3745.86i	1.12485	−	1.12485i	0.991002	−	0.133847i	$-0.0427332\pi$
$224$	5.59913			0.00167012
$225$		0			0
$226$	3224.42			0.949049
$227$	−3266.66	+	3266.66i	−0.955136	+	0.955136i	−0.999036	−	0.0439000i	$-0.986022\pi$
$227$	−3266.66	+	3266.66i	−0.955136	+	0.955136i	0.0439000	+	0.999036i	$0.486022\pi$
$228$		0			0
$229$			1051.91i			0.303546i	0.988415	+	0.151773i	$0.0484983\pi$
$229$			1051.91i			0.303546i	−0.988415	+	0.151773i	$0.951502\pi$
$230$		0			0
$231$		0			0
$232$	300.122	+	300.122i	0.0849311	+	0.0849311i
$233$	−972.311	−	972.311i	−0.273383	−	0.273383i	0.557077	−	0.830461i	$-0.311923\pi$
$233$	−972.311	−	972.311i	−0.273383	−	0.273383i	−0.830461	+	0.557077i	$0.811923\pi$
$234$		0			0
$235$		0			0
$236$			229.016i			0.0631681i
$237$		0			0
$238$	12.0612	−	12.0612i	0.00328493	−	0.00328493i
$239$	5521.04			1.49425			0.747126	−	0.664682i	$-0.231433\pi$
$239$	5521.04			1.49425			0.747126	+	0.664682i	$0.231433\pi$
$240$		0			0
$241$	−115.735			−0.0309342			−0.0154671	−	0.999880i	$-0.504924\pi$
$241$	−115.735			−0.0309342			−0.0154671	+	0.999880i	$0.504924\pi$
$242$	−1856.86	+	1856.86i	−0.493238	+	0.493238i
$243$		0			0
$244$			2439.39i			0.640024i
$245$		0			0
$246$		0			0
$247$	621.899	+	621.899i	0.160204	+	0.160204i
$248$	−1139.60	−	1139.60i	−0.291792	−	0.291792i
$249$		0			0
$250$		0			0
$251$		−	2230.94i		−	0.561018i	−0.959851	−	0.280509i	$-0.909497\pi$
$251$		−	2230.94i		−	0.561018i	0.959851	−	0.280509i	$-0.0905032\pi$
$252$		0			0
$253$	−474.681	+	474.681i	−0.117956	+	0.117956i
$254$	−1781.39			−0.440056
$255$		0			0
$256$	256.000			0.0625000
$257$	4373.86	−	4373.86i	1.06161	−	1.06161i	0.0636372	−	0.997973i	$-0.479730\pi$
$257$	4373.86	−	4373.86i	1.06161	−	1.06161i	0.997973	−	0.0636372i	$-0.0202701\pi$
$258$		0			0
$259$			42.0000i			0.0100763i
$260$		0			0
$261$		0			0
$262$	−1053.03	−	1053.03i	−0.248307	−	0.248307i
$263$	1040.02	+	1040.02i	0.243842	+	0.243842i	0.818437	−	0.574596i	$-0.194841\pi$
$263$	1040.02	+	1040.02i	0.243842	+	0.243842i	−0.574596	+	0.818437i	$0.694841\pi$
$264$		0			0
$265$		0			0
$266$			38.6636i			0.00891210i
$267$		0			0
$268$	1363.30	−	1363.30i	0.310735	−	0.310735i
$269$	−8334.34			−1.88905			−0.944524	−	0.328443i	$-0.893476\pi$
$269$	−8334.34			−1.88905			−0.944524	+	0.328443i	$0.893476\pi$
$270$		0			0
$271$	−417.878			−0.0936688			−0.0468344	−	0.998903i	$-0.514913\pi$
$271$	−417.878			−0.0936688			−0.0468344	+	0.998903i	$0.514913\pi$
$272$	551.457	−	551.457i	0.122930	−	0.122930i
$273$		0			0
$274$		−	1780.21i		−	0.392505i
$275$		0			0
$276$		0			0
$277$	−5795.08	−	5795.08i	−1.25701	−	1.25701i	−0.952512	−	0.304502i	$-0.901510\pi$
$277$	−5795.08	−	5795.08i	−1.25701	−	1.25701i	−0.304502	−	0.952512i	$-0.598490\pi$
$278$	−324.923	−	324.923i	−0.0700992	−	0.0700992i
$279$		0			0
$280$		0			0
$281$		−	1678.12i		−	0.356256i	−0.984007	−	0.178128i	$-0.942996\pi$
$281$		−	1678.12i		−	0.356256i	0.984007	−	0.178128i	$-0.0570042\pi$
$282$		0			0
$283$	−1934.72	+	1934.72i	−0.406386	+	0.406386i	−0.880476	−	0.474090i	$-0.842777\pi$
$283$	−1934.72	+	1934.72i	−0.406386	+	0.406386i	0.474090	+	0.880476i	$0.342777\pi$
$284$	3962.02			0.827826
$285$		0			0
$286$	67.5459			0.0139653
$287$	−45.9223	+	45.9223i	−0.00944497	+	0.00944497i
$288$		0			0
$289$			2537.18i			0.516422i
$290$		0			0
$291$		0			0
$292$	3388.70	+	3388.70i	0.679140	+	0.679140i
$293$	−3782.25	−	3782.25i	−0.754135	−	0.754135i	0.221113	−	0.975248i	$-0.429031\pi$
$293$	−3782.25	−	3782.25i	−0.754135	−	0.754135i	−0.975248	+	0.221113i	$0.929031\pi$
$294$		0			0
$295$		0			0
$296$			1920.30i			0.377078i
$297$		0			0
$298$	−2669.39	+	2669.39i	−0.518904	+	0.518904i
$299$	−1259.54			−0.243617
$300$		0			0
$301$	−50.1428			−0.00960193
$302$	−1135.91	+	1135.91i	−0.216438	+	0.216438i
$303$		0			0
$304$			1767.76i			0.333512i
$305$		0			0
$306$		0			0
$307$	3354.68	+	3354.68i	0.623654	+	0.623654i	0.946464	−	0.322810i	$-0.104627\pi$
$307$	3354.68	+	3354.68i	0.623654	+	0.623654i	−0.322810	+	0.946464i	$0.604627\pi$
$308$	2.09967	+	2.09967i	0.000388441	+	0.000388441i
$309$		0			0
$310$		0			0
$311$			4277.43i			0.779905i	0.920835	+	0.389953i	$0.127509\pi$
$311$			4277.43i			0.779905i	−0.920835	+	0.389953i	$0.872491\pi$
$312$		0			0
$313$	−2311.37	+	2311.37i	−0.417400	+	0.417400i	−0.884307	−	0.466907i	$-0.845368\pi$
$313$	−2311.37	+	2311.37i	−0.417400	+	0.417400i	0.466907	+	0.884307i	$0.345368\pi$
$314$	−3016.39			−0.542117
$315$		0			0
$316$	−1747.39			−0.311070
$317$	4202.41	−	4202.41i	0.744577	−	0.744577i	−0.228878	−	0.973455i	$-0.573506\pi$
$317$	4202.41	−	4202.41i	0.744577	−	0.744577i	0.973455	+	0.228878i	$0.0735057\pi$
$318$		0			0
$319$			225.092i			0.0395070i
$320$		0			0
$321$		0			0
$322$	−39.1531	−	39.1531i	−0.00677614	−	0.00677614i
$323$	3807.97	+	3807.97i	0.655979	+	0.655979i
$324$		0			0
$325$		0			0
$326$			5895.96i			1.00168i
$327$		0			0
$328$	−2099.63	+	2099.63i	−0.353454	+	0.353454i
$329$	−26.3651			−0.00441810
$330$		0			0
$331$	−11631.0			−1.93141			−0.965703	−	0.259650i	$-0.916393\pi$
$331$	−11631.0			−1.93141			−0.965703	+	0.259650i	$0.916393\pi$
$332$	−510.870	+	510.870i	−0.0844507	+	0.0844507i
$333$		0			0
$334$		−	2078.11i		−	0.340447i
$335$		0			0
$336$		0			0
$337$	1086.00	+	1086.00i	0.175543	+	0.175543i	0.789410	−	0.613867i	$-0.210387\pi$
$337$	1086.00	+	1086.00i	0.175543	+	0.175543i	−0.613867	+	0.789410i	$0.710387\pi$
$338$	−3017.41	−	3017.41i	−0.485579	−	0.485579i
$339$		0			0
$340$		0			0
$341$		−	854.697i		−	0.135732i
$342$		0			0
$343$	84.8711	−	84.8711i	0.0133604	−	0.0133604i
$344$	−2292.60			−0.359327
$345$		0			0
$346$	4423.42			0.687297
$347$	6152.44	−	6152.44i	0.951816	−	0.951816i	−0.0470749	−	0.998891i	$-0.514990\pi$
$347$	6152.44	−	6152.44i	0.951816	−	0.951816i	0.998891	+	0.0470749i	$0.0149899\pi$
$348$		0			0
$349$			4514.31i			0.692393i	0.938162	+	0.346197i	$0.112527\pi$
$349$			4514.31i			0.692393i	−0.938162	+	0.346197i	$0.887473\pi$
$350$		0			0
$351$		0			0
$352$	96.0000	+	96.0000i	0.0145364	+	0.0145364i
$353$	216.759	+	216.759i	0.0326825	+	0.0326825i	0.723259	−	0.690577i	$-0.242643\pi$
$353$	216.759	+	216.759i	0.0326825	+	0.0326825i	−0.690577	+	0.723259i	$0.742643\pi$
$354$		0			0
$355$		0			0
$356$			4920.77i			0.732585i
$357$		0			0
$358$	−3765.21	+	3765.21i	−0.555860	+	0.555860i
$359$	−3746.84			−0.550837			−0.275418	−	0.961324i	$-0.588816\pi$
$359$	−3746.84			−0.550837			−0.275418	+	0.961324i	$0.588816\pi$
$360$		0			0
$361$	−5347.87			−0.779686
$362$	−4565.72	+	4565.72i	−0.662897	+	0.662897i
$363$		0			0
$364$			5.57138i			0.000802252i
$365$		0			0
$366$		0			0
$367$	−4491.23	−	4491.23i	−0.638801	−	0.638801i	0.311458	−	0.950260i	$-0.399183\pi$
$367$	−4491.23	−	4491.23i	−0.638801	−	0.638801i	−0.950260	+	0.311458i	$0.899183\pi$
$368$	−1790.13	−	1790.13i	−0.253579	−	0.253579i
$369$		0			0
$370$		0			0
$371$			60.3709i			0.00844825i
$372$		0			0
$373$	−8308.64	+	8308.64i	−1.15337	+	1.15337i	−0.167492	+	0.985873i	$0.553567\pi$
$373$	−8308.64	+	8308.64i	−1.15337	+	1.15337i	−0.985873	+	0.167492i	$0.946433\pi$
$374$	413.593			0.0571828
$375$		0			0
$376$	−1205.45			−0.165336
$377$	−298.635	+	298.635i	−0.0407971	+	0.0407971i
$378$		0			0
$379$		−	11962.7i		−	1.62133i	−0.585511	−	0.810664i	$-0.699106\pi$
$379$		−	11962.7i		−	1.62133i	0.585511	−	0.810664i	$-0.300894\pi$
$380$		0			0
$381$		0			0
$382$	4103.20	+	4103.20i	0.549576	+	0.549576i
$383$	9845.04	+	9845.04i	1.31347	+	1.31347i	0.918842	+	0.394625i	$0.129125\pi$
$383$	9845.04	+	9845.04i	1.31347	+	1.31347i	0.394625	+	0.918842i	$0.370875\pi$
$384$		0			0
$385$		0			0
$386$			3035.97i			0.400329i
$387$		0			0
$388$	−810.311	+	810.311i	−0.106024	+	0.106024i
$389$	−13835.9			−1.80337			−0.901683	−	0.432398i	$-0.857667\pi$
$389$	−13835.9			−1.80337			−0.901683	+	0.432398i	$0.857667\pi$
$390$		0			0
$391$	−7712.36			−0.997521
$392$	1940.13	−	1940.13i	0.249978	−	0.249978i
$393$		0			0
$394$		−	4779.96i		−	0.611196i
$395$		0			0
$396$		0			0
$397$	−1250.91	−	1250.91i	−0.158139	−	0.158139i	0.623602	−	0.781742i	$-0.285668\pi$
$397$	−1250.91	−	1250.91i	−0.158139	−	0.158139i	−0.781742	+	0.623602i	$0.785668\pi$
$398$	1727.92	+	1727.92i	0.217620	+	0.217620i
$399$		0			0
$400$		0			0
$401$			9240.23i			1.15071i	0.817903	+	0.575356i	$0.195136\pi$
$401$			9240.23i			1.15071i	−0.817903	+	0.575356i	$0.804864\pi$
$402$		0			0
$403$	1133.95	−	1133.95i	0.140164	−	0.140164i
$404$	3810.15			0.469213
$405$		0			0
$406$	−18.5662			−0.00226952
$407$	−720.112	+	720.112i	−0.0877018	+	0.0877018i
$408$		0			0
$409$		−	3920.81i		−	0.474013i	−0.971508	−	0.237007i	$-0.923834\pi$
$409$		−	3920.81i		−	0.474013i	0.971508	−	0.237007i	$-0.0761663\pi$
$410$		0			0
$411$		0			0
$412$	3450.56	+	3450.56i	0.412614	+	0.412614i
$413$	−7.08371	−	7.08371i	−0.000843987	−	0.000843987i
$414$		0			0
$415$		0			0
$416$			254.732i			0.0300222i
$417$		0			0
$418$	−662.908	+	662.908i	−0.0775691	+	0.0775691i
$419$	11909.4			1.38858			0.694288	−	0.719698i	$-0.255720\pi$
$419$	11909.4			1.38858			0.694288	+	0.719698i	$0.255720\pi$
$420$		0			0
$421$	−10605.7			−1.22777			−0.613886	−	0.789394i	$-0.710395\pi$
$421$	−10605.7			−1.22777			−0.613886	+	0.789394i	$0.710395\pi$
$422$	−6627.00	+	6627.00i	−0.764448	+	0.764448i
$423$		0			0
$424$			2760.24i			0.316154i
$425$		0			0
$426$		0			0
$427$	−75.4529	−	75.4529i	−0.00855134	−	0.00855134i
$428$	2379.73	+	2379.73i	0.268759	+	0.268759i
$429$		0			0
$430$		0			0
$431$		−	17058.9i		−	1.90649i	−0.302198	−	0.953245i	$-0.597720\pi$
$431$		−	17058.9i		−	1.90649i	0.302198	−	0.953245i	$-0.402280\pi$
$432$		0			0
$433$	−4011.77	+	4011.77i	−0.445250	+	0.445250i	−0.893772	−	0.448522i	$-0.851951\pi$
$433$	−4011.77	+	4011.77i	−0.445250	+	0.445250i	0.448522	+	0.893772i	$0.351951\pi$
$434$	70.4979			0.00779725
$435$		0			0
$436$	184.367			0.0202514
$437$	12361.4	−	12361.4i	1.35315	−	1.35315i
$438$		0			0
$439$		−	6626.17i		−	0.720387i	−0.932878	−	0.360193i	$-0.882711\pi$
$439$		−	6626.17i		−	0.720387i	0.932878	−	0.360193i	$-0.117289\pi$
$440$		0			0
$441$		0			0
$442$	548.724	+	548.724i	0.0590501	+	0.0590501i
$443$	12510.8	+	12510.8i	1.34178	+	1.34178i	0.894291	+	0.447486i	$0.147680\pi$
$443$	12510.8	+	12510.8i	1.34178	+	1.34178i	0.447486	+	0.894291i	$0.352320\pi$
$444$		0			0
$445$		0			0
$446$			10594.9i			1.12485i
$447$		0			0
$448$	−7.91836	+	7.91836i	−0.000835061	+	0.000835061i
$449$	−9955.27			−1.04637			−0.523183	−	0.852220i	$-0.675256\pi$
$449$	−9955.27			−1.04637			−0.523183	+	0.852220i	$0.675256\pi$
$450$		0			0
$451$	−1574.72			−0.164414
$452$	−4560.02	+	4560.02i	−0.474525	+	0.474525i
$453$		0			0
$454$		−	9239.51i		−	0.955136i
$455$		0			0
$456$		0			0
$457$	−4696.35	−	4696.35i	−0.480713	−	0.480713i	0.424646	−	0.905359i	$-0.360398\pi$
$457$	−4696.35	−	4696.35i	−0.480713	−	0.480713i	−0.905359	+	0.424646i	$0.860398\pi$
$458$	−1487.62	−	1487.62i	−0.151773	−	0.151773i
$459$		0			0
$460$		0			0
$461$			1831.59i			0.185044i	0.995711	+	0.0925222i	$0.0294929\pi$
$461$			1831.59i			0.185044i	−0.995711	+	0.0925222i	$0.970507\pi$
$462$		0			0
$463$	8401.13	−	8401.13i	0.843269	−	0.843269i	−0.146014	−	0.989283i	$-0.546644\pi$
$463$	8401.13	−	8401.13i	0.843269	−	0.843269i	0.989283	+	0.146014i	$0.0466444\pi$
$464$	−848.875			−0.0849311
$465$		0			0
$466$	2750.11			0.273383
$467$	−12419.2	+	12419.2i	−1.23061	+	1.23061i	−0.266876	+	0.963731i	$0.585992\pi$
$467$	−12419.2	+	12419.2i	−1.23061	+	1.23061i	−0.963731	+	0.266876i	$0.914008\pi$
$468$		0			0
$469$			84.3368i			0.00830343i
$470$		0			0
$471$		0			0
$472$	−323.878	−	323.878i	−0.0315841	−	0.0315841i
$473$	−859.725	−	859.725i	−0.0835733	−	0.0835733i
$474$		0			0
$475$		0			0
$476$			34.1143i			0.00328493i
$477$		0			0
$478$	−7807.93	+	7807.93i	−0.747126	+	0.747126i
$479$	6740.76			0.642992			0.321496	−	0.946911i	$-0.395814\pi$
$479$	6740.76			0.642992			0.321496	+	0.946911i	$0.395814\pi$
$480$		0			0
$481$	−1910.79			−0.181132
$482$	163.674	−	163.674i	0.0154671	−	0.0154671i
$483$		0			0
$484$		−	5252.00i		−	0.493238i
$485$		0			0
$486$		0			0
$487$	7297.37	+	7297.37i	0.679005	+	0.679005i	0.959775	−	0.280770i	$-0.0905899\pi$
$487$	7297.37	+	7297.37i	0.679005	+	0.679005i	−0.280770	+	0.959775i	$0.590590\pi$
$488$	−3449.82	−	3449.82i	−0.320012	−	0.320012i
$489$		0			0
$490$		0			0
$491$			8669.79i			0.796868i	0.917197	+	0.398434i	$0.130446\pi$
$491$			8669.79i			0.796868i	−0.917197	+	0.398434i	$0.869554\pi$
$492$		0			0
$493$	−1828.58	+	1828.58i	−0.167049	+	0.167049i
$494$	−1759.00			−0.160204
$495$		0			0
$496$	3223.27			0.291792
$497$	−122.550	+	122.550i	−0.0110606	+	0.0110606i
$498$		0			0
$499$		−	6159.27i		−	0.552559i	−0.961077	−	0.276279i	$-0.910898\pi$
$499$		−	6159.27i		−	0.552559i	0.961077	−	0.276279i	$-0.0891015\pi$
$500$		0			0
$501$		0			0
$502$	3155.02	+	3155.02i	0.280509	+	0.280509i
$503$	6743.55	+	6743.55i	0.597773	+	0.597773i	0.939719	−	0.341946i	$-0.111086\pi$
$503$	6743.55	+	6743.55i	0.597773	+	0.597773i	−0.341946	+	0.939719i	$0.611086\pi$
$504$		0			0
$505$		0			0
$506$		−	1342.60i		−	0.117956i
$507$		0			0
$508$	2519.27	−	2519.27i	0.220028	−	0.220028i
$509$	14485.3			1.26140			0.630698	−	0.776028i	$-0.282769\pi$
$509$	14485.3			1.26140			0.630698	+	0.776028i	$0.282769\pi$
$510$		0			0
$511$	−209.633			−0.0181479
$512$	−362.039	+	362.039i	−0.0312500	+	0.0312500i
$513$		0			0
$514$			12371.1i			1.06161i
$515$		0			0
$516$		0			0
$517$	−452.043	−	452.043i	−0.0384542	−	0.0384542i
$518$	−59.3970	−	59.3970i	−0.00503813	−	0.00503813i
$519$		0			0
$520$		0			0
$521$		−	13211.2i		−	1.11093i	−0.831540	−	0.555464i	$-0.812541\pi$
$521$		−	13211.2i		−	1.11093i	0.831540	−	0.555464i	$-0.187459\pi$
$522$		0			0
$523$	8899.56	−	8899.56i	0.744074	−	0.744074i	−0.229285	−	0.973359i	$-0.573639\pi$
$523$	8899.56	−	8899.56i	0.744074	−	0.744074i	0.973359	+	0.229285i	$0.0736389\pi$
$524$	2978.42			0.248307
$525$		0			0
$526$	−2941.62			−0.243842
$527$	6943.32	−	6943.32i	0.573920	−	0.573920i
$528$		0			0
$529$			12868.8i			1.05768i
$530$		0			0
$531$		0			0
$532$	−54.6786	−	54.6786i	−0.00445605	−	0.00445605i
$533$	−2089.23	−	2089.23i	−0.169783	−	0.169783i
$534$		0			0
$535$		0			0
$536$			3856.00i			0.310735i
$537$		0			0
$538$	11786.5	−	11786.5i	0.944524	−	0.944524i
$539$	1455.10			0.116281
$540$		0			0
$541$	2620.33			0.208238			0.104119	−	0.994565i	$-0.466798\pi$
$541$	2620.33			0.208238			0.104119	+	0.994565i	$0.466798\pi$
$542$	590.968	−	590.968i	0.0468344	−	0.0468344i
$543$		0			0
$544$			1559.76i			0.122930i
$545$		0			0
$546$		0			0
$547$	3267.35	+	3267.35i	0.255396	+	0.255396i	0.823179	−	0.567782i	$-0.192198\pi$
$547$	3267.35	+	3267.35i	0.255396	+	0.255396i	−0.567782	+	0.823179i	$0.692198\pi$
$548$	2517.60	+	2517.60i	0.196252	+	0.196252i
$549$		0			0
$550$		0			0
$551$		−	5861.73i		−	0.453209i
$552$		0			0
$553$	54.0486	−	54.0486i	0.00415620	−	0.00415620i
$554$	16391.0			1.25701
$555$		0			0
$556$	919.020			0.0700992
$557$	981.246	−	981.246i	0.0746440	−	0.0746440i	−0.668799	−	0.743443i	$-0.733191\pi$
$557$	981.246	−	981.246i	0.0746440	−	0.0746440i	0.743443	+	0.668799i	$0.233191\pi$
$558$		0			0
$559$		−	2281.24i		−	0.172605i
$560$		0			0
$561$		0			0
$562$	2373.21	+	2373.21i	0.178128	+	0.178128i
$563$	−9237.32	−	9237.32i	−0.691486	−	0.691486i	0.271073	−	0.962559i	$-0.412622\pi$
$563$	−9237.32	−	9237.32i	−0.691486	−	0.691486i	−0.962559	+	0.271073i	$0.912622\pi$
$564$		0			0
$565$		0			0
$566$		−	5472.22i		−	0.406386i
$567$		0			0
$568$	−5603.14	+	5603.14i	−0.413913	+	0.413913i
$569$	−20917.7			−1.54115			−0.770575	−	0.637349i	$-0.780031\pi$
$569$	−20917.7			−1.54115			−0.770575	+	0.637349i	$0.780031\pi$
$570$		0			0
$571$	7734.94			0.566895			0.283448	−	0.958988i	$-0.408522\pi$
$571$	7734.94			0.566895			0.283448	+	0.958988i	$0.408522\pi$
$572$	−95.5244	+	95.5244i	−0.00698265	+	0.00698265i
$573$		0			0
$574$		−	129.888i		−	0.00944497i
$575$		0			0
$576$		0			0
$577$	−10178.7	−	10178.7i	−0.734391	−	0.734391i	0.237096	−	0.971486i	$-0.423804\pi$
$577$	−10178.7	−	10178.7i	−0.734391	−	0.734391i	−0.971486	+	0.237096i	$0.923804\pi$
$578$	−3588.12	−	3588.12i	−0.258211	−	0.258211i
$579$		0			0
$580$		0			0
$581$		−	31.6035i		−	0.00225669i
$582$		0			0
$583$	−1035.09	+	1035.09i	−0.0735319	+	0.0735319i
$584$	−9584.70			−0.679140
$585$		0			0
$586$	10697.8			0.754135
$587$	−7041.14	+	7041.14i	−0.495092	+	0.495092i	−0.909906	−	0.414814i	$-0.863847\pi$
$587$	−7041.14	+	7041.14i	−0.495092	+	0.495092i	0.414814	+	0.909906i	$0.363847\pi$
$588$		0			0
$589$			22257.6i			1.55706i
$590$		0			0
$591$		0			0
$592$	−2715.71	−	2715.71i	−0.188539	−	0.188539i
$593$	−15972.6	−	15972.6i	−1.10610	−	1.10610i	−0.993659	−	0.112437i	$-0.964134\pi$
$593$	−15972.6	−	15972.6i	−1.10610	−	1.10610i	−0.112437	−	0.993659i	$-0.535866\pi$
$594$		0			0
$595$		0			0
$596$		−	7550.17i		−	0.518904i
$597$		0			0
$598$	1781.27	−	1781.27i	0.121808	−	0.121808i
$599$	26647.8			1.81769			0.908847	−	0.417131i	$-0.136964\pi$
$599$	26647.8			1.81769			0.908847	+	0.417131i	$0.136964\pi$
$600$		0			0
$601$	23514.8			1.59599			0.797993	−	0.602666i	$-0.205895\pi$
$601$	23514.8			1.59599			0.797993	+	0.602666i	$0.205895\pi$
$602$	70.9126	−	70.9126i	0.00480097	−	0.00480097i
$603$		0			0
$604$		−	3212.84i		−	0.216438i
$605$		0			0
$606$		0			0
$607$	−1843.31	−	1843.31i	−0.123258	−	0.123258i	0.642787	−	0.766045i	$-0.277778\pi$
$607$	−1843.31	−	1843.31i	−0.123258	−	0.123258i	−0.766045	+	0.642787i	$0.777778\pi$
$608$	−2499.98	−	2499.98i	−0.166756	−	0.166756i
$609$		0			0
$610$		0			0
$611$		−	1199.48i		−	0.0794200i
$612$		0			0
$613$	14527.5	−	14527.5i	0.957197	−	0.957197i	−0.0419235	−	0.999121i	$-0.513349\pi$
$613$	14527.5	−	14527.5i	0.957197	−	0.957197i	0.999121	+	0.0419235i	$0.0133486\pi$
$614$	−9488.47			−0.623654
$615$		0			0
$616$	−5.93877			−0.000388441
$617$	13639.0	−	13639.0i	0.889928	−	0.889928i	−0.104588	−	0.994516i	$-0.533352\pi$
$617$	13639.0	−	13639.0i	0.889928	−	0.889928i	0.994516	+	0.104588i	$0.0333524\pi$
$618$		0			0
$619$			11981.4i			0.777985i	0.921241	+	0.388993i	$0.127177\pi$
$619$			11981.4i			0.777985i	−0.921241	+	0.388993i	$0.872823\pi$
$620$		0			0
$621$		0			0
$622$	−6049.19	−	6049.19i	−0.389953	−	0.389953i
$623$	−152.205	−	152.205i	−0.00978805	−	0.00978805i
$624$		0			0
$625$		0			0
$626$		−	6537.53i		−	0.417400i
$627$		0			0
$628$	4265.82	−	4265.82i	0.271059	−	0.271059i
$629$	−11700.0			−0.741668
$630$		0			0
$631$	−373.658			−0.0235739			−0.0117869	−	0.999931i	$-0.503752\pi$
$631$	−373.658			−0.0235739			−0.0117869	+	0.999931i	$0.503752\pi$
$632$	2471.18	−	2471.18i	0.155535	−	0.155535i
$633$		0			0
$634$			11886.2i			0.744577i
$635$		0			0
$636$		0			0
$637$	1930.52	+	1930.52i	0.120078	+	0.120078i
$638$	−318.328	−	318.328i	−0.0197535	−	0.0197535i
$639$		0			0
$640$		0			0
$641$		−	6251.83i		−	0.385230i	−0.981274	−	0.192615i	$-0.938303\pi$
$641$		−	6251.83i		−	0.385230i	0.981274	−	0.192615i	$-0.0616969\pi$
$642$		0			0
$643$	3502.38	−	3502.38i	0.214806	−	0.214806i	−0.591499	−	0.806305i	$-0.701464\pi$
$643$	3502.38	−	3502.38i	0.214806	−	0.214806i	0.806305	+	0.591499i	$0.201464\pi$
$644$	110.742			0.00677614
$645$		0			0
$646$	−10770.6			−0.655979
$647$	−13374.2	+	13374.2i	−0.812665	+	0.812665i	−0.985033	−	0.172368i	$-0.944858\pi$
$647$	−13374.2	+	13374.2i	−0.812665	+	0.812665i	0.172368	+	0.985033i	$0.444858\pi$
$648$		0			0
$649$		−	242.908i		−	0.0146918i
$650$		0			0
$651$		0			0
$652$	−8338.15	−	8338.15i	−0.500839	−	0.500839i
$653$	−20892.8	−	20892.8i	−1.25207	−	1.25207i	−0.954793	−	0.297272i	$-0.903923\pi$
$653$	−20892.8	−	20892.8i	−1.25207	−	1.25207i	−0.297272	−	0.954793i	$-0.596077\pi$
$654$		0			0
$655$		0			0
$656$		−	5938.66i		−	0.353454i
$657$		0			0
$658$	37.2858	−	37.2858i	0.00220905	−	0.00220905i
$659$	8493.05			0.502037			0.251018	−	0.967982i	$-0.419235\pi$
$659$	8493.05			0.502037			0.251018	+	0.967982i	$0.419235\pi$
$660$		0			0
$661$	27756.2			1.63327			0.816635	−	0.577155i	$-0.195837\pi$
$661$	27756.2			1.63327			0.816635	+	0.577155i	$0.195837\pi$
$662$	16448.7	−	16448.7i	0.965703	−	0.965703i
$663$		0			0
$664$		−	1444.96i		−	0.0844507i
$665$		0			0
$666$		0			0
$667$	5935.94	+	5935.94i	0.344588	+	0.344588i
$668$	2938.89	+	2938.89i	0.170223	+	0.170223i
$669$		0			0
$670$		0			0
$671$		−	2587.36i		−	0.148858i
$672$		0			0
$673$	−13533.3	+	13533.3i	−0.775143	+	0.775143i	−0.979001	−	0.203858i	$-0.934652\pi$
$673$	−13533.3	+	13533.3i	−0.775143	+	0.775143i	0.203858	+	0.979001i	$0.434652\pi$
$674$	−3071.66			−0.175543
$675$		0			0
$676$	8534.53			0.485579
$677$	−16217.2	+	16217.2i	−0.920647	+	0.920647i	−0.997075	−	0.0764277i	$-0.975649\pi$
$677$	−16217.2	+	16217.2i	−0.920647	+	0.920647i	0.0764277	+	0.997075i	$0.475649\pi$
$678$		0			0
$679$		−	50.1276i		−	0.00283317i
$680$		0			0
$681$		0			0
$682$	1208.72	+	1208.72i	0.0678658	+	0.0678658i
$683$	1055.14	+	1055.14i	0.0591125	+	0.0591125i	0.736045	−	0.676933i	$-0.236691\pi$
$683$	1055.14	+	1055.14i	0.0591125	+	0.0591125i	−0.676933	+	0.736045i	$0.736691\pi$
$684$		0			0
$685$		0			0
$686$			240.052i			0.0133604i
$687$		0			0
$688$	3242.22	−	3242.22i	0.179664	−	0.179664i
$689$	−2746.57			−0.151866
$690$		0			0
$691$	32208.2			1.77316			0.886582	−	0.462571i	$-0.153073\pi$
$691$	32208.2			1.77316			0.886582	+	0.462571i	$0.153073\pi$
$692$	−6255.67	+	6255.67i	−0.343648	+	0.343648i
$693$		0			0
$694$			17401.7i			0.951816i
$695$		0			0
$696$		0			0
$697$	−12792.6	−	12792.6i	−0.695201	−	0.695201i
$698$	−6384.19	−	6384.19i	−0.346197	−	0.346197i
$699$		0			0
$700$		0			0
$701$			7585.34i			0.408694i	0.978899	+	0.204347i	$0.0655071\pi$
$701$			7585.34i			0.408694i	−0.978899	+	0.204347i	$0.934493\pi$
$702$		0			0
$703$	18752.8	−	18752.8i	1.00608	−	1.00608i
$704$	−271.529			−0.0145364
$705$		0			0
$706$	−613.087			−0.0326825
$707$	−117.852	+	117.852i	−0.00626914	+	0.00626914i
$708$		0			0
$709$			1207.97i			0.0639862i	0.999488	+	0.0319931i	$0.0101855\pi$
$709$			1207.97i			0.0639862i	−0.999488	+	0.0319931i	$0.989815\pi$
$710$		0			0
$711$		0			0
$712$	−6959.02	−	6959.02i	−0.366293	−	0.366293i
$713$	−22539.4	−	22539.4i	−1.18388	−	1.18388i
$714$		0			0
$715$		0			0
$716$		−	10649.6i		−	0.555860i
$717$		0			0
$718$	5298.83	−	5298.83i	0.275418	−	0.275418i
$719$	−20470.8			−1.06180			−0.530898	−	0.847436i	$-0.678145\pi$
$719$	−20470.8			−1.06180			−0.530898	+	0.847436i	$0.678145\pi$
$720$		0			0
$721$	−213.459			−0.0110258
$722$	7563.03	−	7563.03i	0.389843	−	0.389843i
$723$		0			0
$724$		−	12913.8i		−	0.662897i
$725$		0			0
$726$		0			0
$727$	−3850.51	−	3850.51i	−0.196434	−	0.196434i	0.602036	−	0.798469i	$-0.294357\pi$
$727$	−3850.51	−	3850.51i	−0.196434	−	0.196434i	−0.798469	+	0.602036i	$0.794357\pi$
$728$	−7.87913	−	7.87913i	−0.000401126	−	0.000401126i
$729$		0			0
$730$		0			0
$731$		−	13968.3i		−	0.706754i
$732$		0			0
$733$	−18950.5	+	18950.5i	−0.954914	+	0.954914i	−0.999027	−	0.0441122i	$-0.985954\pi$
$733$	−18950.5	+	18950.5i	−0.954914	+	0.954914i	0.0441122	+	0.999027i	$0.485954\pi$
$734$	12703.1			0.638801
$735$		0			0
$736$	5063.27			0.253579
$737$	−1446.00	+	1446.00i	−0.0722715	+	0.0722715i
$738$		0			0
$739$		−	8217.11i		−	0.409028i	−0.978864	−	0.204514i	$-0.934439\pi$
$739$		−	8217.11i		−	0.409028i	0.978864	−	0.204514i	$-0.0655613\pi$
$740$		0			0
$741$		0			0
$742$	−85.3774	−	85.3774i	−0.00422413	−	0.00422413i
$743$	6551.87	+	6551.87i	0.323506	+	0.323506i	0.850110	−	0.526605i	$-0.176535\pi$
$743$	6551.87	+	6551.87i	0.323506	+	0.323506i	−0.526605	+	0.850110i	$0.676535\pi$
$744$		0			0
$745$		0			0
$746$		−	23500.4i		−	1.15337i
$747$		0			0
$748$	−584.908	+	584.908i	−0.0285914	+	0.0285914i
$749$	−147.215			−0.00718175
$750$		0			0
$751$	−12001.4			−0.583138			−0.291569	−	0.956550i	$-0.594177\pi$
$751$	−12001.4			−0.583138			−0.291569	+	0.956550i	$0.594177\pi$
$752$	1704.76	−	1704.76i	0.0826679	−	0.0826679i
$753$		0			0
$754$		−	844.669i		−	0.0407971i
$755$		0			0
$756$		0			0
$757$	15831.8	+	15831.8i	0.760127	+	0.760127i	0.976345	−	0.216218i	$-0.0693722\pi$
$757$	15831.8	+	15831.8i	0.760127	+	0.760127i	−0.216218	+	0.976345i	$0.569372\pi$
$758$	16917.8	+	16917.8i	0.810664	+	0.810664i
$759$		0			0
$760$		0			0
$761$			38246.7i			1.82187i	0.412553	+	0.910933i	$0.364637\pi$
$761$			38246.7i			1.82187i	−0.412553	+	0.910933i	$0.635363\pi$
$762$		0			0
$763$	−5.70268	+	5.70268i	−0.000270578	+	0.000270578i
$764$	−11605.6			−0.549576
$765$		0			0
$766$	−27846.0			−1.31347
$767$	322.273	−	322.273i	0.0151716	−	0.0151716i
$768$		0			0
$769$			24315.9i			1.14025i	0.821557	+	0.570126i	$0.193106\pi$
$769$			24315.9i			1.14025i	−0.821557	+	0.570126i	$0.806894\pi$
$770$		0			0
$771$		0			0
$772$	−4293.52	−	4293.52i	−0.200165	−	0.200165i
$773$	5549.98	+	5549.98i	0.258239	+	0.258239i	0.824338	−	0.566098i	$-0.191548\pi$
$773$	5549.98	+	5549.98i	0.258239	+	0.258239i	−0.566098	+	0.824338i	$0.691548\pi$
$774$		0			0
$775$		0			0
$776$		−	2291.91i		−	0.106024i
$777$		0			0
$778$	19566.9	−	19566.9i	0.901683	−	0.901683i
$779$	41008.2			1.88610
$780$		0			0
$781$	−4202.36			−0.192538
$782$	10906.9	−	10906.9i	0.498760	−	0.498760i
$783$		0			0
$784$			5487.51i			0.249978i
$785$		0			0
$786$		0			0
$787$	12044.5	+	12044.5i	0.545542	+	0.545542i	0.925148	−	0.379606i	$-0.123941\pi$
$787$	12044.5	+	12044.5i	0.545542	+	0.545542i	−0.379606	+	0.925148i	$0.623941\pi$
$788$	6759.89	+	6759.89i	0.305598	+	0.305598i
$789$		0			0
$790$		0			0
$791$		−	282.093i		−	0.0126802i
$792$		0			0
$793$	3432.72	−	3432.72i	0.153719	−	0.153719i
$794$	3538.11			0.158139
$795$		0			0
$796$	−4887.29			−0.217620
$797$	446.462	−	446.462i	0.0198425	−	0.0198425i	−0.697116	−	0.716958i	$-0.745534\pi$
$797$	446.462	−	446.462i	0.0198425	−	0.0198425i	0.716958	+	0.697116i	$0.245534\pi$
$798$		0			0
$799$		−	7344.55i		−	0.325196i
$800$		0			0
$801$		0			0
$802$	−13067.7	−	13067.7i	−0.575356	−	0.575356i
$803$	−3594.26	−	3594.26i	−0.157956	−	0.157956i
$804$		0			0
$805$		0			0
$806$			3207.30i			0.140164i
$807$		0			0
$808$	−5388.37	+	5388.37i	−0.234607	+	0.234607i
$809$	5823.00			0.253060			0.126530	−	0.991963i	$-0.459616\pi$
$809$	5823.00			0.253060			0.126530	+	0.991963i	$0.459616\pi$
$810$		0			0
$811$	9681.83			0.419205			0.209602	−	0.977787i	$-0.432783\pi$
$811$	9681.83			0.419205			0.209602	+	0.977787i	$0.432783\pi$
$812$	26.2566	−	26.2566i	0.00113476	−	0.00113476i
$813$		0			0
$814$		−	2036.79i		−	0.0877018i
$815$		0			0
$816$		0			0
$817$	22388.5	+	22388.5i	0.958721	+	0.958721i
$818$	5544.86	+	5544.86i	0.237007	+	0.237007i
$819$		0			0
$820$		0			0
$821$			613.797i			0.0260922i	0.999915	+	0.0130461i	$0.00415282\pi$
$821$			613.797i			0.0260922i	−0.999915	+	0.0130461i	$0.995847\pi$
$822$		0			0
$823$	4884.02	−	4884.02i	0.206861	−	0.206861i	−0.596071	−	0.802932i	$-0.703272\pi$
$823$	4884.02	−	4884.02i	0.206861	−	0.206861i	0.802932	+	0.596071i	$0.203272\pi$
$824$	−9759.66			−0.412614
$825$		0			0
$826$	20.0358			0.000843987
$827$	−31687.2	+	31687.2i	−1.33237	+	1.33237i	−0.429126	+	0.903245i	$0.641178\pi$
$827$	−31687.2	+	31687.2i	−1.33237	+	1.33237i	−0.903245	+	0.429126i	$0.858822\pi$
$828$		0			0
$829$		−	31717.5i		−	1.32882i	−0.747367	−	0.664411i	$-0.768682\pi$
$829$		−	31717.5i		−	1.32882i	0.747367	−	0.664411i	$-0.231318\pi$
$830$		0			0
$831$		0			0
$832$	−360.245	−	360.245i	−0.0150111	−	0.0150111i
$833$	11820.8	+	11820.8i	0.491676	+	0.491676i
$834$		0			0
$835$		0			0
$836$		−	1874.99i		−	0.0775691i
$837$		0			0
$838$	−16842.5	+	16842.5i	−0.694288	+	0.694288i
$839$	1336.26			0.0549854			0.0274927	−	0.999622i	$-0.491248\pi$
$839$	1336.26			0.0549854			0.0274927	+	0.999622i	$0.491248\pi$
$840$		0			0
$841$	−21574.2			−0.884587
$842$	14998.8	−	14998.8i	0.613886	−	0.613886i
$843$		0			0
$844$		−	18744.0i		−	0.764448i
$845$		0			0
$846$		0			0
$847$	162.450	+	162.450i	0.00659014	+	0.00659014i
$848$	−3903.58	−	3903.58i	−0.158077	−	0.158077i
$849$		0			0
$850$		0			0
$851$			37980.4i			1.52991i
$852$		0			0
$853$	−7849.59	+	7849.59i	−0.315082	+	0.315082i	−0.846875	−	0.531793i	$-0.821519\pi$
$853$	−7849.59	+	7849.59i	−0.315082	+	0.315082i	0.531793	+	0.846875i	$0.321519\pi$
$854$	213.413			0.00855134
$855$		0			0
$856$	−6730.90			−0.268759
$857$	−5652.60	+	5652.60i	−0.225308	+	0.225308i	−0.810729	−	0.585421i	$-0.800929\pi$
$857$	−5652.60	+	5652.60i	−0.225308	+	0.225308i	0.585421	+	0.810729i	$0.300929\pi$
$858$		0			0
$859$		−	22449.7i		−	0.891706i	−0.895106	−	0.445853i	$-0.852900\pi$
$859$		−	22449.7i		−	0.891706i	0.895106	−	0.445853i	$-0.147100\pi$
$860$		0			0
$861$		0			0
$862$	24124.9	+	24124.9i	0.953245	+	0.953245i
$863$	19455.8	+	19455.8i	0.767420	+	0.767420i	0.977652	−	0.210231i	$-0.0674217\pi$
$863$	19455.8	+	19455.8i	0.767420	+	0.767420i	−0.210231	+	0.977652i	$0.567422\pi$
$864$		0			0
$865$		0			0
$866$		−	11347.0i		−	0.445250i
$867$		0			0
$868$	−99.6991	+	99.6991i	−0.00389863	+	0.00389863i
$869$	1853.38			0.0723496
$870$		0			0
$871$	−3836.89			−0.149263
$872$	−260.735	+	260.735i	−0.0101257	+	0.0101257i
$873$		0			0
$874$			34963.3i			1.35315i
$875$		0			0
$876$		0			0
$877$	−15660.5	−	15660.5i	−0.602983	−	0.602983i	0.338120	−	0.941103i	$-0.390209\pi$
$877$	−15660.5	−	15660.5i	−0.602983	−	0.602983i	−0.941103	+	0.338120i	$0.890209\pi$
$878$	9370.82	+	9370.82i	0.360193	+	0.360193i
$879$		0			0
$880$		0			0
$881$			23.8106i			0.000910554i	1.00000		0.000455277i	$0.000144919\pi$
$881$			23.8106i			0.000910554i	−1.00000		0.000455277i	$0.999855\pi$
$882$		0			0
$883$	−19225.0	+	19225.0i	−0.732698	+	0.732698i	−0.971153	−	0.238456i	$-0.923359\pi$
$883$	−19225.0	+	19225.0i	−0.732698	+	0.732698i	0.238456	+	0.971153i	$0.423359\pi$
$884$	−1552.03			−0.0590501
$885$		0			0
$886$	−35385.9			−1.34178
$887$	13803.0	−	13803.0i	0.522502	−	0.522502i	−0.395824	−	0.918326i	$-0.629541\pi$
$887$	13803.0	−	13803.0i	0.522502	−	0.522502i	0.918326	+	0.395824i	$0.129541\pi$
$888$		0			0
$889$			155.847i			0.00587958i
$890$		0			0
$891$		0			0
$892$	−14983.4	−	14983.4i	−0.562425	−	0.562425i
$893$	11771.9	+	11771.9i	0.441132	+	0.441132i
$894$		0			0
$895$		0			0
$896$		−	22.3965i		−	0.000835061i
$897$		0			0
$898$	14078.9	−	14078.9i	0.523183	−	0.523183i
$899$	−10688.1			−0.396515
$900$		0			0
$901$	−16817.6			−0.621837
$902$	2227.00	−	2227.00i	0.0822072	−	0.0822072i
$903$		0			0
$904$		−	12897.7i		−	0.474525i
$905$		0			0
$906$		0			0
$907$	28217.1	+	28217.1i	1.03300	+	1.03300i	0.999436	+	0.0335664i	$0.0106865\pi$
$907$	28217.1	+	28217.1i	1.03300	+	1.03300i	0.0335664	+	0.999436i	$0.489313\pi$
$908$	13066.6	+	13066.6i	0.477568	+	0.477568i
$909$		0			0
$910$		0			0
$911$			24262.0i			0.882368i	0.897417	+	0.441184i	$0.145441\pi$
$911$			24262.0i			0.882368i	−0.897417	+	0.441184i	$0.854559\pi$
$912$		0			0
$913$	541.860	−	541.860i	0.0196418	−	0.0196418i
$914$	13283.3			0.480713
$915$		0			0
$916$	4207.63			0.151773
$917$	−92.1258	+	92.1258i	−0.00331762	+	0.00331762i
$918$		0			0
$919$		−	23354.5i		−	0.838295i	−0.907918	−	0.419148i	$-0.862329\pi$
$919$		−	23354.5i		−	0.838295i	0.907918	−	0.419148i	$-0.137671\pi$
$920$		0			0
$921$		0			0
$922$	−2590.26	−	2590.26i	−0.0925222	−	0.0925222i
$923$	−5575.38	−	5575.38i	−0.198825	−	0.198825i
$924$		0			0
$925$		0			0
$926$			23762.0i			0.843269i
$927$		0			0
$928$	1200.49	−	1200.49i	0.0424655	−	0.0424655i
$929$	2534.31			0.0895026			0.0447513	−	0.998998i	$-0.485750\pi$
$929$	2534.31			0.0895026			0.0447513	+	0.998998i	$0.485750\pi$
$930$		0			0
$931$	−37892.9			−1.33393
$932$	−3889.25	+	3889.25i	−0.136692	+	0.136692i
$933$		0			0
$934$		−	35126.9i		−	1.23061i
$935$		0			0
$936$		0			0
$937$	21606.6	+	21606.6i	0.753316	+	0.753316i	0.975097	−	0.221780i	$-0.0711867\pi$
$937$	21606.6	+	21606.6i	0.753316	+	0.753316i	−0.221780	+	0.975097i	$0.571187\pi$
$938$	−119.270	−	119.270i	−0.00415172	−	0.00415172i
$939$		0			0
$940$		0			0
$941$			2386.44i			0.0826735i	0.999145	+	0.0413368i	$0.0131616\pi$
$941$			2386.44i			0.0826735i	−0.999145	+	0.0413368i	$0.986838\pi$
$942$		0			0
$943$	−41527.3	+	41527.3i	−1.43406	+	1.43406i
$944$	916.064			0.0315841
$945$		0			0
$946$	2431.67			0.0835733
$947$	−26386.0	+	26386.0i	−0.905417	+	0.905417i	−0.995898	−	0.0904815i	$-0.971159\pi$
$947$	−26386.0	+	26386.0i	−0.905417	+	0.905417i	0.0904815	+	0.995898i	$0.471159\pi$
$948$		0			0
$949$		−	9537.21i		−	0.326229i
$950$		0			0
$951$		0			0
$952$	−48.2449	−	48.2449i	−0.00164247	−	0.00164247i
$953$	7162.80	+	7162.80i	0.243469	+	0.243469i	0.818284	−	0.574815i	$-0.194926\pi$
$953$	7162.80	+	7162.80i	0.243469	+	0.243469i	−0.574815	+	0.818284i	$0.694926\pi$
$954$		0			0
$955$		0			0
$956$		−	22084.2i		−	0.747126i
$957$		0			0
$958$	−9532.87	+	9532.87i	−0.321496	+	0.321496i
$959$	−155.744			−0.00524425
$960$		0			0
$961$	10792.7			0.362282
$962$	2702.26	−	2702.26i	0.0905658	−	0.0905658i
$963$		0			0
$964$			462.939i			0.0154671i
$965$		0			0
$966$		0			0
$967$	−11164.0	−	11164.0i	−0.371262	−	0.371262i	0.496675	−	0.867937i	$-0.334554\pi$
$967$	−11164.0	−	11164.0i	−0.371262	−	0.371262i	−0.867937	+	0.496675i	$0.834554\pi$
$968$	7427.45	+	7427.45i	0.246619	+	0.246619i
$969$		0			0
$970$		0			0
$971$			39205.7i			1.29575i	0.761747	+	0.647874i	$0.224342\pi$
$971$			39205.7i			1.29575i	−0.761747	+	0.647874i	$0.775658\pi$
$972$		0			0
$973$	−28.4263	+	28.4263i	−0.000936593	+	0.000936593i
$974$	−20640.1			−0.679005
$975$		0			0
$976$	9757.55			0.320012
$977$	3864.86	−	3864.86i	0.126559	−	0.126559i	−0.640990	−	0.767549i	$-0.721476\pi$
$977$	3864.86	−	3864.86i	0.126559	−	0.126559i	0.767549	+	0.640990i	$0.221476\pi$
$978$		0			0
$979$		−	5219.27i		−	0.170387i
$980$		0			0
$981$		0			0
$982$	−12260.9	−	12260.9i	−0.398434	−	0.398434i
$983$	21793.2	+	21793.2i	0.707116	+	0.707116i	0.965928	−	0.258812i	$-0.0833310\pi$
$983$	21793.2	+	21793.2i	0.707116	+	0.707116i	−0.258812	+	0.965928i	$0.583331\pi$
$984$		0			0
$985$		0			0
$986$		−	5172.02i		−	0.167049i
$987$		0			0
$988$	2487.60	−	2487.60i	0.0801022	−	0.0801022i
$989$	−45343.9			−1.45789
$990$		0			0
$991$	−33283.2			−1.06688			−0.533439	−	0.845839i	$-0.679101\pi$
$991$	−33283.2			−1.06688			−0.533439	+	0.845839i	$0.679101\pi$
$992$	−4558.39	+	4558.39i	−0.145896	+	0.145896i
$993$		0			0
$994$		−	346.623i		−	0.0110606i
$995$		0			0
$996$		0			0
$997$	−36019.7	−	36019.7i	−1.14419	−	1.14419i	−0.987676	−	0.156511i	$-0.949975\pi$
$997$	−36019.7	−	36019.7i	−1.14419	−	1.14419i	−0.156511	−	0.987676i	$-0.550025\pi$
$998$	8710.52	+	8710.52i	0.276279	+	0.276279i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.f.d.107.2	✓	8
3.2	odd	2	inner	450.4.f.d.107.4	yes	8
5.2	odd	4		450.4.f.f.143.1	yes	8
5.3	odd	4	inner	450.4.f.d.143.4	yes	8
5.4	even	2		450.4.f.f.107.3	yes	8
15.2	even	4		450.4.f.f.143.3	yes	8
15.8	even	4	inner	450.4.f.d.143.2	yes	8
15.14	odd	2		450.4.f.f.107.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.4.f.d.107.2	✓	8	1.1	even	1	trivial
450.4.f.d.107.4	yes	8	3.2	odd	2	inner
450.4.f.d.143.2	yes	8	15.8	even	4	inner
450.4.f.d.143.4	yes	8	5.3	odd	4	inner
450.4.f.f.107.1	yes	8	15.14	odd	2
450.4.f.f.107.3	yes	8	5.4	even	2
450.4.f.f.143.1	yes	8	5.2	odd	4
450.4.f.f.143.3	yes	8	15.2	even	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.41421i) q^{2} -4.00000i q^{4} +(0.123724 + 0.123724i) q^{7} +(5.65685 + 5.65685i) q^{8} +O(q^{10})\)	\(q+(-1.41421 + 1.41421i) q^{2} -4.00000i q^{4} +(0.123724 + 0.123724i) q^{7} +(5.65685 + 5.65685i) q^{8} +4.24264i q^{11} +(-5.62883 + 5.62883i) q^{13} -0.349945 q^{14} -16.0000 q^{16} +(-34.4660 + 34.4660i) q^{17} -110.485i q^{19} +(-6.00000 - 6.00000i) q^{22} +(111.883 + 111.883i) q^{23} -15.9207i q^{26} +(0.494897 - 0.494897i) q^{28} +53.0547 q^{29} -201.454 q^{31} +(22.6274 - 22.6274i) q^{32} -97.4847i q^{34} +(169.732 + 169.732i) q^{37} +(156.249 + 156.249i) q^{38} +371.166i q^{41} +(-202.639 + 202.639i) q^{43} +16.9706 q^{44} -316.454 q^{46} +(-106.548 + 106.548i) q^{47} -342.969i q^{49} +(22.5153 + 22.5153i) q^{52} +(243.973 + 243.973i) q^{53} +1.39978i q^{56} +(-75.0306 + 75.0306i) q^{58} -57.2540 q^{59} -609.847 q^{61} +(284.899 - 284.899i) q^{62} +64.0000i q^{64} +(340.825 + 340.825i) q^{67} +(137.864 + 137.864i) q^{68} +990.505i q^{71} +(-847.176 + 847.176i) q^{73} -480.075 q^{74} -441.939 q^{76} +(-0.524918 + 0.524918i) q^{77} -436.847i q^{79} +(-524.908 - 524.908i) q^{82} +(-127.718 - 127.718i) q^{83} -573.150i q^{86} +(-24.0000 + 24.0000i) q^{88} -1230.19 q^{89} -1.39285 q^{91} +(447.534 - 447.534i) q^{92} -301.362i q^{94} +(-202.578 - 202.578i) q^{97} +(485.032 + 485.032i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 48 q^{7}+O(q^{10}) \) 8 * q - 48 * q^7	\( 8 q - 48 q^{7} - 192 q^{13} - 128 q^{16} - 48 q^{22} - 192 q^{28} + 152 q^{31} + 672 q^{37} - 2160 q^{43} - 768 q^{46} + 768 q^{52} - 1776 q^{58} + 1000 q^{61} + 816 q^{67} - 3936 q^{73} - 1184 q^{76} - 672 q^{82} - 192 q^{88} + 4104 q^{91} + 192 q^{97}+O(q^{100}) \) 8 * q - 48 * q^7 - 192 * q^13 - 128 * q^16 - 48 * q^22 - 192 * q^28 + 152 * q^31 + 672 * q^37 - 2160 * q^43 - 768 * q^46 + 768 * q^52 - 1776 * q^58 + 1000 * q^61 + 816 * q^67 - 3936 * q^73 - 1184 * q^76 - 672 * q^82 - 192 * q^88 + 4104 * q^91 + 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more