LMFDB - Embedded newform 225.4.f.b.143.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,4,Mod(107,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.107"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	225.143
Dual form			225.4.f.b.107.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.12132 + 2.12132i) q^{2} +1.00000i q^{4} +(9.00000 - 9.00000i) q^{7} +(14.8492 - 14.8492i) q^{8} -38.1838i q^{11} +(-63.0000 - 63.0000i) q^{13} +38.1838 q^{14} +71.0000 q^{16} +(29.6985 + 29.6985i) q^{17} +70.0000i q^{19} +(81.0000 - 81.0000i) q^{22} +(72.1249 - 72.1249i) q^{23} -267.286i q^{26} +(9.00000 + 9.00000i) q^{28} +229.103 q^{29} +196.000 q^{31} +(31.8198 + 31.8198i) q^{32} +126.000i q^{34} +(-207.000 + 207.000i) q^{37} +(-148.492 + 148.492i) q^{38} -267.286i q^{41} +(144.000 + 144.000i) q^{43} +38.1838 q^{44} +306.000 q^{46} +(-356.382 - 356.382i) q^{47} +181.000i q^{49} +(63.0000 - 63.0000i) q^{52} +(-224.860 + 224.860i) q^{53} -267.286i q^{56} +(486.000 + 486.000i) q^{58} +267.286 q^{59} -322.000 q^{61} +(415.779 + 415.779i) q^{62} -433.000i q^{64} +(-378.000 + 378.000i) q^{67} +(-29.6985 + 29.6985i) q^{68} +840.043i q^{71} +(-378.000 - 378.000i) q^{73} -878.227 q^{74} -70.0000 q^{76} +(-343.654 - 343.654i) q^{77} +488.000i q^{79} +(567.000 - 567.000i) q^{82} +(772.161 - 772.161i) q^{83} +610.940i q^{86} +(-567.000 - 567.000i) q^{88} +267.286 q^{89} -1134.00 q^{91} +(72.1249 + 72.1249i) q^{92} -1512.00i q^{94} +(252.000 - 252.000i) q^{97} +(-383.959 + 383.959i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{7} - 252 q^{13} + 284 q^{16} + 324 q^{22} + 36 q^{28} + 784 q^{31} - 828 q^{37} + 576 q^{43} + 1224 q^{46} + 252 q^{52} + 1944 q^{58} - 1288 q^{61} - 1512 q^{67} - 1512 q^{73} - 280 q^{76} + 2268 q^{82}+ \cdots + 1008 q^{97}+O(q^{100}) $ 4 * q + 36 * q^7 - 252 * q^13 + 284 * q^16 + 324 * q^22 + 36 * q^28 + 784 * q^31 - 828 * q^37 + 576 * q^43 + 1224 * q^46 + 252 * q^52 + 1944 * q^58 - 1288 * q^61 - 1512 * q^67 - 1512 * q^73 - 280 * q^76 + 2268 * q^82 - 2268 * q^88 - 4536 * q^91 + 1008 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.12132	+	2.12132i	0.750000	+	0.750000i	0.974479	−	0.224479i	$-0.0720680\pi$
$2$	2.12132	+	2.12132i	0.750000	+	0.750000i	−0.224479	+	0.974479i	$0.572068\pi$
$3$		0			0
$3$		0			0
$4$			1.00000i			0.125000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	9.00000	−	9.00000i	0.485954	−	0.485954i	−0.421073	−	0.907027i	$-0.638346\pi$
$7$	9.00000	−	9.00000i	0.485954	−	0.485954i	0.907027	+	0.421073i	$0.138346\pi$
$8$	14.8492	−	14.8492i	0.656250	−	0.656250i
$9$		0			0
$10$		0			0
$11$		−	38.1838i		−	1.04662i	−0.852142	−	0.523311i	$-0.824697\pi$
$11$		−	38.1838i		−	1.04662i	0.852142	−	0.523311i	$-0.175303\pi$
$12$		0			0
$13$	−63.0000	−	63.0000i	−1.34408	−	1.34408i	−0.891953	−	0.452128i	$-0.850665\pi$
$13$	−63.0000	−	63.0000i	−1.34408	−	1.34408i	−0.452128	−	0.891953i	$-0.649335\pi$
$14$	38.1838			0.728931
$15$		0			0
$16$	71.0000			1.10938
$17$	29.6985	+	29.6985i	0.423702	+	0.423702i	0.886476	−	0.462774i	$-0.153146\pi$
$17$	29.6985	+	29.6985i	0.423702	+	0.423702i	−0.462774	+	0.886476i	$0.653146\pi$
$18$		0			0
$19$			70.0000i			0.845216i	0.906313	+	0.422608i	$0.138885\pi$
$19$			70.0000i			0.845216i	−0.906313	+	0.422608i	$0.861115\pi$
$20$		0			0
$21$		0			0
$22$	81.0000	−	81.0000i	0.784966	−	0.784966i
$23$	72.1249	−	72.1249i	0.653873	−	0.653873i	−0.300050	−	0.953923i	$-0.597003\pi$
$23$	72.1249	−	72.1249i	0.653873	−	0.653873i	0.953923	+	0.300050i	$0.0970035\pi$
$24$		0			0
$25$		0			0
$26$		−	267.286i		−	2.01612i
$27$		0			0
$28$	9.00000	+	9.00000i	0.0607443	+	0.0607443i
$29$	229.103			1.46701			0.733505	−	0.679684i	$-0.237883\pi$
$29$	229.103			1.46701			0.733505	+	0.679684i	$0.237883\pi$
$30$		0			0
$31$	196.000			1.13557			0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000			1.13557			0.567785	+	0.823177i	$0.307801\pi$
$32$	31.8198	+	31.8198i	0.175781	+	0.175781i
$33$		0			0
$34$			126.000i			0.635554i
$35$		0			0
$36$		0			0
$37$	−207.000	+	207.000i	−0.919746	+	0.919746i	−0.997011	−	0.0772649i	$-0.975381\pi$
$37$	−207.000	+	207.000i	−0.919746	+	0.919746i	0.0772649	+	0.997011i	$0.475381\pi$
$38$	−148.492	+	148.492i	−0.633912	+	0.633912i
$39$		0			0
$40$		0			0
$41$		−	267.286i		−	1.01812i	−0.860730	−	0.509062i	$-0.829992\pi$
$41$		−	267.286i		−	1.01812i	0.860730	−	0.509062i	$-0.170008\pi$
$42$		0			0
$43$	144.000	+	144.000i	0.510693	+	0.510693i	0.914739	−	0.404046i	$-0.132396\pi$
$43$	144.000	+	144.000i	0.510693	+	0.510693i	−0.404046	+	0.914739i	$0.632396\pi$
$44$	38.1838			0.130828
$45$		0			0
$46$	306.000			0.980810
$47$	−356.382	−	356.382i	−1.10603	−	1.10603i	−0.993667	−	0.112368i	$-0.964156\pi$
$47$	−356.382	−	356.382i	−1.10603	−	1.10603i	−0.112368	−	0.993667i	$-0.535844\pi$
$48$		0			0
$49$			181.000i			0.527697i
$50$		0			0
$51$		0			0
$52$	63.0000	−	63.0000i	0.168010	−	0.168010i
$53$	−224.860	+	224.860i	−0.582772	+	0.582772i	−0.935664	−	0.352892i	$-0.885198\pi$
$53$	−224.860	+	224.860i	−0.582772	+	0.582772i	0.352892	+	0.935664i	$0.385198\pi$
$54$		0			0
$55$		0			0
$56$		−	267.286i		−	0.637815i
$57$		0			0
$58$	486.000	+	486.000i	1.10026	+	1.10026i
$59$	267.286			0.589792			0.294896	−	0.955529i	$-0.404715\pi$
$59$	267.286			0.589792			0.294896	+	0.955529i	$0.404715\pi$
$60$		0			0
$61$	−322.000			−0.675867			−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000			−0.675867			−0.337933	+	0.941170i	$0.609728\pi$
$62$	415.779	+	415.779i	0.851677	+	0.851677i
$63$		0			0
$64$		−	433.000i		−	0.845703i
$65$		0			0
$66$		0			0
$67$	−378.000	+	378.000i	−0.689254	+	0.689254i	−0.962067	−	0.272813i	$-0.912046\pi$
$67$	−378.000	+	378.000i	−0.689254	+	0.689254i	0.272813	+	0.962067i	$0.412046\pi$
$68$	−29.6985	+	29.6985i	−0.0529628	+	0.0529628i
$69$		0			0
$70$		0			0
$71$			840.043i			1.40415i	0.712102	+	0.702076i	$0.247743\pi$
$71$			840.043i			1.40415i	−0.712102	+	0.702076i	$0.752257\pi$
$72$		0			0
$73$	−378.000	−	378.000i	−0.606049	−	0.606049i	0.335862	−	0.941911i	$-0.390972\pi$
$73$	−378.000	−	378.000i	−0.606049	−	0.606049i	−0.941911	+	0.335862i	$0.890972\pi$
$74$	−878.227			−1.37962
$75$		0			0
$76$	−70.0000			−0.105652
$77$	−343.654	−	343.654i	−0.508610	−	0.508610i
$78$		0			0
$79$			488.000i			0.694991i	0.937682	+	0.347496i	$0.112968\pi$
$79$			488.000i			0.694991i	−0.937682	+	0.347496i	$0.887032\pi$
$80$		0			0
$81$		0			0
$82$	567.000	−	567.000i	0.763594	−	0.763594i
$83$	772.161	−	772.161i	1.02115	−	1.02115i	0.0213809	−	0.999771i	$-0.493194\pi$
$83$	772.161	−	772.161i	1.02115	−	1.02115i	0.999771	−	0.0213809i	$-0.00680626\pi$
$84$		0			0
$85$		0			0
$86$			610.940i			0.766039i
$87$		0			0
$88$	−567.000	−	567.000i	−0.686845	−	0.686845i
$89$	267.286			0.318340			0.159170	−	0.987251i	$-0.449118\pi$
$89$	267.286			0.318340			0.159170	+	0.987251i	$0.449118\pi$
$90$		0			0
$91$	−1134.00			−1.30632
$92$	72.1249	+	72.1249i	0.0817341	+	0.0817341i
$93$		0			0
$94$		−	1512.00i		−	1.65905i
$95$		0			0
$96$		0			0
$97$	252.000	−	252.000i	0.263781	−	0.263781i	−0.562807	−	0.826588i	$-0.690279\pi$
$97$	252.000	−	252.000i	0.263781	−	0.263781i	0.826588	+	0.562807i	$0.190279\pi$
$98$	−383.959	+	383.959i	−0.395773	+	0.395773i
$99$		0			0
$100$		0			0
$101$			1603.72i			1.57996i	0.613133	+	0.789980i	$0.289909\pi$
$101$			1603.72i			1.57996i	−0.613133	+	0.789980i	$0.710091\pi$
$102$		0			0
$103$	−945.000	−	945.000i	−0.904016	−	0.904016i	0.0917650	−	0.995781i	$-0.470749\pi$
$103$	−945.000	−	945.000i	−0.904016	−	0.904016i	−0.995781	+	0.0917650i	$0.970749\pi$
$104$	−1871.00			−1.76411
$105$		0			0
$106$	−954.000			−0.874157
$107$	1323.70	+	1323.70i	1.19596	+	1.19596i	0.975367	+	0.220589i	$0.0707980\pi$
$107$	1323.70	+	1323.70i	1.19596	+	1.19596i	0.220589	+	0.975367i	$0.429202\pi$
$108$		0			0
$109$		−	70.0000i		−	0.0615118i	−0.999527	−	0.0307559i	$-0.990209\pi$
$109$		−	70.0000i		−	0.0615118i	0.999527	−	0.0307559i	$-0.00979145\pi$
$110$		0			0
$111$		0			0
$112$	639.000	−	639.000i	0.539106	−	0.539106i
$113$	521.845	−	521.845i	0.434434	−	0.434434i	−0.455700	−	0.890134i	$-0.650611\pi$
$113$	521.845	−	521.845i	0.434434	−	0.434434i	0.890134	+	0.455700i	$0.150611\pi$
$114$		0			0
$115$		0			0
$116$			229.103i			0.183376i
$117$		0			0
$118$	567.000	+	567.000i	0.442344	+	0.442344i
$119$	534.573			0.411800
$120$		0			0
$121$	−127.000			−0.0954170
$122$	−683.065	−	683.065i	−0.506900	−	0.506900i
$123$		0			0
$124$			196.000i			0.141946i
$125$		0			0
$126$		0			0
$127$	−315.000	+	315.000i	−0.220092	+	0.220092i	−0.808537	−	0.588445i	$-0.799740\pi$
$127$	−315.000	+	315.000i	−0.220092	+	0.220092i	0.588445	+	0.808537i	$0.299740\pi$
$128$	1173.09	−	1173.09i	0.810059	−	0.810059i
$129$		0			0
$130$		0			0
$131$			1336.43i			0.891333i	0.895199	+	0.445666i	$0.147033\pi$
$131$			1336.43i			0.891333i	−0.895199	+	0.445666i	$0.852967\pi$
$132$		0			0
$133$	630.000	+	630.000i	0.410736	+	0.410736i
$134$	−1603.72			−1.03388
$135$		0			0
$136$	882.000			0.556109
$137$	−1294.01	−	1294.01i	−0.806966	−	0.806966i	0.177208	−	0.984174i	$-0.443294\pi$
$137$	−1294.01	−	1294.01i	−0.806966	−	0.806966i	−0.984174	+	0.177208i	$0.943294\pi$
$138$		0			0
$139$		−	308.000i		−	0.187944i	−0.995575	−	0.0939720i	$-0.970044\pi$
$139$		−	308.000i		−	0.187944i	0.995575	−	0.0939720i	$-0.0299564\pi$
$140$		0			0
$141$		0			0
$142$	−1782.00	+	1782.00i	−1.05311	+	1.05311i
$143$	−2405.58	+	2405.58i	−1.40674	+	1.40674i
$144$		0			0
$145$		0			0
$146$		−	1603.72i		−	0.909073i
$147$		0			0
$148$	−207.000	−	207.000i	−0.114968	−	0.114968i
$149$	2443.76			1.34363			0.671814	−	0.740719i	$-0.265515\pi$
$149$	2443.76			1.34363			0.671814	+	0.740719i	$0.265515\pi$
$150$		0			0
$151$	−2072.00			−1.11667			−0.558334	−	0.829616i	$-0.688559\pi$
$151$	−2072.00			−1.11667			−0.558334	+	0.829616i	$0.688559\pi$
$152$	1039.45	+	1039.45i	0.554673	+	0.554673i
$153$		0			0
$154$		−	1458.00i		−	0.762916i
$155$		0			0
$156$		0			0
$157$	315.000	−	315.000i	0.160126	−	0.160126i	−0.622497	−	0.782622i	$-0.713882\pi$
$157$	315.000	−	315.000i	0.160126	−	0.160126i	0.782622	+	0.622497i	$0.213882\pi$
$158$	−1035.20	+	1035.20i	−0.521243	+	0.521243i
$159$		0			0
$160$		0			0
$161$		−	1298.25i		−	0.635505i
$162$		0			0
$163$	2394.00	+	2394.00i	1.15038	+	1.15038i	0.986476	+	0.163908i	$0.0524101\pi$
$163$	2394.00	+	2394.00i	1.15038	+	1.15038i	0.163908	+	0.986476i	$0.447590\pi$
$164$	267.286			0.127266
$165$		0			0
$166$	3276.00			1.53173
$167$	−742.462	−	742.462i	−0.344033	−	0.344033i	0.513848	−	0.857881i	$-0.328219\pi$
$167$	−742.462	−	742.462i	−0.344033	−	0.344033i	−0.857881	+	0.513848i	$0.828219\pi$
$168$		0			0
$169$			5741.00i			2.61311i
$170$		0			0
$171$		0			0
$172$	−144.000	+	144.000i	−0.0638366	+	0.0638366i
$173$	−326.683	+	326.683i	−0.143568	+	0.143568i	−0.775238	−	0.631670i	$-0.782370\pi$
$173$	−326.683	+	326.683i	−0.143568	+	0.143568i	0.631670	+	0.775238i	$0.282370\pi$
$174$		0			0
$175$		0			0
$176$		−	2711.05i		−	1.16110i
$177$		0			0
$178$	567.000	+	567.000i	0.238755	+	0.238755i
$179$	1565.53			0.653707			0.326853	−	0.945075i	$-0.394012\pi$
$179$	1565.53			0.653707			0.326853	+	0.945075i	$0.394012\pi$
$180$		0			0
$181$	3094.00			1.27058			0.635291	−	0.772273i	$-0.280880\pi$
$181$	3094.00			1.27058			0.635291	+	0.772273i	$0.280880\pi$
$182$	−2405.58	−	2405.58i	−0.979743	−	0.979743i
$183$		0			0
$184$		−	2142.00i		−	0.858208i
$185$		0			0
$186$		0			0
$187$	1134.00	−	1134.00i	0.443456	−	0.443456i
$188$	356.382	−	356.382i	0.138254	−	0.138254i
$189$		0			0
$190$		0			0
$191$		−	840.043i		−	0.318238i	−0.987259	−	0.159119i	$-0.949135\pi$
$191$		−	840.043i		−	0.318238i	0.987259	−	0.159119i	$-0.0508653\pi$
$192$		0			0
$193$	1242.00	+	1242.00i	0.463218	+	0.463218i	0.899709	−	0.436491i	$-0.143779\pi$
$193$	1242.00	+	1242.00i	0.463218	+	0.463218i	−0.436491	+	0.899709i	$0.643779\pi$
$194$	1069.15			0.395671
$195$		0			0
$196$	−181.000			−0.0659621
$197$	3725.04	+	3725.04i	1.34720	+	1.34720i	0.888691	+	0.458507i	$0.151616\pi$
$197$	3725.04	+	3725.04i	1.34720	+	1.34720i	0.458507	+	0.888691i	$0.348384\pi$
$198$		0			0
$199$		−	700.000i		−	0.249355i	−0.992197	−	0.124678i	$-0.960210\pi$
$199$		−	700.000i		−	0.249355i	0.992197	−	0.124678i	$-0.0397897\pi$
$200$		0			0
$201$		0			0
$202$	−3402.00	+	3402.00i	−1.18497	+	1.18497i
$203$	2061.92	−	2061.92i	0.712900	−	0.712900i
$204$		0			0
$205$		0			0
$206$		−	4009.30i		−	1.35602i
$207$		0			0
$208$	−4473.00	−	4473.00i	−1.49109	−	1.49109i
$209$	2672.86			0.884621
$210$		0			0
$211$	1316.00			0.429371			0.214685	−	0.976683i	$-0.431127\pi$
$211$	1316.00			0.429371			0.214685	+	0.976683i	$0.431127\pi$
$212$	−224.860	−	224.860i	−0.0728464	−	0.0728464i
$213$		0			0
$214$			5616.00i			1.79393i
$215$		0			0
$216$		0			0
$217$	1764.00	−	1764.00i	0.551835	−	0.551835i
$218$	148.492	−	148.492i	0.0461338	−	0.0461338i
$219$		0			0
$220$		0			0
$221$		−	3742.01i		−	1.13898i
$222$		0			0
$223$	−2205.00	−	2205.00i	−0.662142	−	0.662142i	0.293742	−	0.955885i	$-0.405099\pi$
$223$	−2205.00	−	2205.00i	−0.662142	−	0.662142i	−0.955885	+	0.293742i	$0.905099\pi$
$224$	572.756			0.170843
$225$		0			0
$226$	2214.00			0.651651
$227$	−653.367	−	653.367i	−0.191037	−	0.191037i	0.605107	−	0.796144i	$-0.293130\pi$
$227$	−653.367	−	653.367i	−0.191037	−	0.191037i	−0.796144	+	0.605107i	$0.793130\pi$
$228$		0			0
$229$		−	3206.00i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	3206.00i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$		0			0
$232$	3402.00	−	3402.00i	0.962725	−	0.962725i
$233$	−12.7279	+	12.7279i	−0.00357869	+	0.00357869i	−0.708894	−	0.705315i	$-0.750805\pi$
$233$	−12.7279	+	12.7279i	−0.00357869	+	0.00357869i	0.705315	+	0.708894i	$0.250805\pi$
$234$		0			0
$235$		0			0
$236$			267.286i			0.0737240i
$237$		0			0
$238$	1134.00	+	1134.00i	0.308850	+	0.308850i
$239$	−1298.25			−0.351367			−0.175683	−	0.984447i	$-0.556214\pi$
$239$	−1298.25			−0.351367			−0.175683	+	0.984447i	$0.556214\pi$
$240$		0			0
$241$	−700.000			−0.187099			−0.0935497	−	0.995615i	$-0.529821\pi$
$241$	−700.000			−0.187099			−0.0935497	+	0.995615i	$0.529821\pi$
$242$	−269.408	−	269.408i	−0.0715627	−	0.0715627i
$243$		0			0
$244$		−	322.000i		−	0.0844834i
$245$		0			0
$246$		0			0
$247$	4410.00	−	4410.00i	1.13604	−	1.13604i
$248$	2910.45	−	2910.45i	0.745217	−	0.745217i
$249$		0			0
$250$		0			0
$251$			5078.44i			1.27709i	0.769587	+	0.638543i	$0.220462\pi$
$251$			5078.44i			1.27709i	−0.769587	+	0.638543i	$0.779538\pi$
$252$		0			0
$253$	−2754.00	−	2754.00i	−0.684358	−	0.684358i
$254$	−1336.43			−0.330139
$255$		0			0
$256$	1513.00			0.369385
$257$	2880.75	+	2880.75i	0.699208	+	0.699208i	0.964240	−	0.265032i	$-0.0853824\pi$
$257$	2880.75	+	2880.75i	0.699208	+	0.699208i	−0.265032	+	0.964240i	$0.585382\pi$
$258$		0			0
$259$			3726.00i			0.893909i
$260$		0			0
$261$		0			0
$262$	−2835.00	+	2835.00i	−0.668500	+	0.668500i
$263$	−1412.80	+	1412.80i	−0.331243	+	0.331243i	−0.853058	−	0.521815i	$-0.825255\pi$
$263$	−1412.80	+	1412.80i	−0.331243	+	0.331243i	0.521815	+	0.853058i	$0.325255\pi$
$264$		0			0
$265$		0			0
$266$			2672.86i			0.616104i
$267$		0			0
$268$	−378.000	−	378.000i	−0.0861568	−	0.0861568i
$269$	1603.72			0.363496			0.181748	−	0.983345i	$-0.441824\pi$
$269$	1603.72			0.363496			0.181748	+	0.983345i	$0.441824\pi$
$270$		0			0
$271$	−2716.00			−0.608802			−0.304401	−	0.952544i	$-0.598456\pi$
$271$	−2716.00			−0.608802			−0.304401	+	0.952544i	$0.598456\pi$
$272$	2108.59	+	2108.59i	0.470045	+	0.470045i
$273$		0			0
$274$		−	5490.00i		−	1.21045i
$275$		0			0
$276$		0			0
$277$	−3843.00	+	3843.00i	−0.833587	+	0.833587i	−0.988005	−	0.154419i	$-0.950649\pi$
$277$	−3843.00	+	3843.00i	−0.833587	+	0.833587i	0.154419	+	0.988005i	$0.450649\pi$
$278$	653.367	−	653.367i	0.140958	−	0.140958i
$279$		0			0
$280$		0			0
$281$		−	3780.19i		−	0.802517i	−0.915965	−	0.401259i	$-0.868573\pi$
$281$		−	3780.19i		−	0.802517i	0.915965	−	0.401259i	$-0.131427\pi$
$282$		0			0
$283$	−126.000	−	126.000i	−0.0264662	−	0.0264662i	0.693750	−	0.720216i	$-0.255957\pi$
$283$	−126.000	−	126.000i	−0.0264662	−	0.0264662i	−0.720216	+	0.693750i	$0.755957\pi$
$284$	−840.043			−0.175519
$285$		0			0
$286$	−10206.0			−2.11012
$287$	−2405.58	−	2405.58i	−0.494762	−	0.494762i
$288$		0			0
$289$		−	3149.00i		−	0.640953i
$290$		0			0
$291$		0			0
$292$	378.000	−	378.000i	0.0757561	−	0.0757561i
$293$	−772.161	+	772.161i	−0.153959	+	0.153959i	−0.779884	−	0.625924i	$-0.784722\pi$
$293$	−772.161	+	772.161i	−0.153959	+	0.153959i	0.625924	+	0.779884i	$0.284722\pi$
$294$		0			0
$295$		0			0
$296$			6147.59i			1.20717i
$297$		0			0
$298$	5184.00	+	5184.00i	1.00772	+	1.00772i
$299$	−9087.74			−1.75772
$300$		0			0
$301$	2592.00			0.496347
$302$	−4395.38	−	4395.38i	−0.837501	−	0.837501i
$303$		0			0
$304$			4970.00i			0.937661i
$305$		0			0
$306$		0			0
$307$	−3024.00	+	3024.00i	−0.562178	+	0.562178i	−0.929926	−	0.367747i	$-0.880129\pi$
$307$	−3024.00	+	3024.00i	−0.562178	+	0.562178i	0.367747	+	0.929926i	$0.380129\pi$
$308$	343.654	−	343.654i	0.0635763	−	0.0635763i
$309$		0			0
$310$		0			0
$311$			10156.9i			1.85191i	0.377634	+	0.925955i	$0.376738\pi$
$311$			10156.9i			1.85191i	−0.377634	+	0.925955i	$0.623262\pi$
$312$		0			0
$313$	5418.00	+	5418.00i	0.978414	+	0.978414i	0.999772	−	0.0213583i	$-0.00679906\pi$
$313$	5418.00	+	5418.00i	0.978414	+	0.978414i	−0.0213583	+	0.999772i	$0.506799\pi$
$314$	1336.43			0.240189
$315$		0			0
$316$	−488.000			−0.0868739
$317$	−1977.07	−	1977.07i	−0.350294	−	0.350294i	0.509925	−	0.860219i	$-0.329673\pi$
$317$	−1977.07	−	1977.07i	−0.350294	−	0.350294i	−0.860219	+	0.509925i	$0.829673\pi$
$318$		0			0
$319$		−	8748.00i		−	1.53540i
$320$		0			0
$321$		0			0
$322$	2754.00	−	2754.00i	0.476629	−	0.476629i
$323$	−2078.89	+	2078.89i	−0.358120	+	0.358120i
$324$		0			0
$325$		0			0
$326$			10156.9i			1.72558i
$327$		0			0
$328$	−3969.00	−	3969.00i	−0.668144	−	0.668144i
$329$	−6414.87			−1.07496
$330$		0			0
$331$	−6622.00			−1.09963			−0.549816	−	0.835286i	$-0.685302\pi$
$331$	−6622.00			−1.09963			−0.549816	+	0.835286i	$0.685302\pi$
$332$	772.161	+	772.161i	0.127644	+	0.127644i
$333$		0			0
$334$		−	3150.00i		−	0.516049i
$335$		0			0
$336$		0			0
$337$	−2790.00	+	2790.00i	−0.450982	+	0.450982i	−0.895680	−	0.444698i	$-0.853311\pi$
$337$	−2790.00	+	2790.00i	−0.450982	+	0.450982i	0.444698	+	0.895680i	$0.353311\pi$
$338$	−12178.5	+	12178.5i	−1.95983	+	1.95983i
$339$		0			0
$340$		0			0
$341$		−	7484.02i		−	1.18851i
$342$		0			0
$343$	4716.00	+	4716.00i	0.742391	+	0.742391i
$344$	4276.58			0.670284
$345$		0			0
$346$	−1386.00			−0.215352
$347$	1858.28	+	1858.28i	0.287486	+	0.287486i	0.836085	−	0.548600i	$-0.184839\pi$
$347$	1858.28	+	1858.28i	0.287486	+	0.287486i	−0.548600	+	0.836085i	$0.684839\pi$
$348$		0			0
$349$			5614.00i			0.861062i	0.902576	+	0.430531i	$0.141674\pi$
$349$			5614.00i			0.861062i	−0.902576	+	0.430531i	$0.858326\pi$
$350$		0			0
$351$		0			0
$352$	1215.00	−	1215.00i	0.183976	−	0.183976i
$353$	1514.62	−	1514.62i	0.228372	−	0.228372i	−0.583640	−	0.812012i	$-0.698372\pi$
$353$	1514.62	−	1514.62i	0.228372	−	0.228372i	0.812012	+	0.583640i	$0.198372\pi$
$354$		0			0
$355$		0			0
$356$			267.286i			0.0397926i
$357$		0			0
$358$	3321.00	+	3321.00i	0.490280	+	0.490280i
$359$	−11455.1			−1.68406			−0.842032	−	0.539428i	$-0.818640\pi$
$359$	−11455.1			−1.68406			−0.842032	+	0.539428i	$0.818640\pi$
$360$		0			0
$361$	1959.00			0.285610
$362$	6563.37	+	6563.37i	0.952936	+	0.952936i
$363$		0			0
$364$		−	1134.00i		−	0.163291i
$365$		0			0
$366$		0			0
$367$	−5481.00	+	5481.00i	−0.779580	+	0.779580i	−0.979759	−	0.200179i	$-0.935848\pi$
$367$	−5481.00	+	5481.00i	−0.779580	+	0.779580i	0.200179	+	0.979759i	$0.435848\pi$
$368$	5120.87	−	5120.87i	0.725390	−	0.725390i
$369$		0			0
$370$		0			0
$371$			4047.48i			0.566401i
$372$		0			0
$373$	−2709.00	−	2709.00i	−0.376050	−	0.376050i	0.493625	−	0.869675i	$-0.335672\pi$
$373$	−2709.00	−	2709.00i	−0.376050	−	0.376050i	−0.869675	+	0.493625i	$0.835672\pi$
$374$	4811.15			0.665184
$375$		0			0
$376$	−10584.0			−1.45167
$377$	−14433.5	−	14433.5i	−1.97178	−	1.97178i
$378$		0			0
$379$		−	13916.0i		−	1.88606i	−0.332707	−	0.943030i	$-0.607962\pi$
$379$		−	13916.0i		−	1.88606i	0.332707	−	0.943030i	$-0.392038\pi$
$380$		0			0
$381$		0			0
$382$	1782.00	−	1782.00i	0.238678	−	0.238678i
$383$	6890.05	−	6890.05i	0.919230	−	0.919230i	−0.0777435	−	0.996973i	$-0.524772\pi$
$383$	6890.05	−	6890.05i	0.919230	−	0.919230i	0.996973	+	0.0777435i	$0.0247715\pi$
$384$		0			0
$385$		0			0
$386$			5269.36i			0.694827i
$387$		0			0
$388$	252.000	+	252.000i	0.0329726	+	0.0329726i
$389$	6720.34			0.875925			0.437963	−	0.898993i	$-0.355700\pi$
$389$	6720.34			0.875925			0.437963	+	0.898993i	$0.355700\pi$
$390$		0			0
$391$	4284.00			0.554095
$392$	2687.71	+	2687.71i	0.346301	+	0.346301i
$393$		0			0
$394$			15804.0i			2.02080i
$395$		0			0
$396$		0			0
$397$	5607.00	−	5607.00i	0.708834	−	0.708834i	−0.257456	−	0.966290i	$-0.582884\pi$
$397$	5607.00	−	5607.00i	0.708834	−	0.708834i	0.966290	+	0.257456i	$0.0828841\pi$
$398$	1484.92	−	1484.92i	0.187016	−	0.187016i
$399$		0			0
$400$		0			0
$401$		−	8056.77i		−	1.00333i	−0.865061	−	0.501666i	$-0.832721\pi$
$401$		−	8056.77i		−	1.00333i	0.865061	−	0.501666i	$-0.167279\pi$
$402$		0			0
$403$	−12348.0	−	12348.0i	−1.52630	−	1.52630i
$404$	−1603.72			−0.197495
$405$		0			0
$406$	8748.00			1.06935
$407$	7904.04	+	7904.04i	0.962626	+	0.962626i
$408$		0			0
$409$		−	1694.00i		−	0.204799i	−0.994743	−	0.102400i	$-0.967348\pi$
$409$		−	1694.00i		−	0.204799i	0.994743	−	0.102400i	$-0.0326521\pi$
$410$		0			0
$411$		0			0
$412$	945.000	−	945.000i	0.113002	−	0.113002i
$413$	2405.58	−	2405.58i	0.286612	−	0.286612i
$414$		0			0
$415$		0			0
$416$		−	4009.30i		−	0.472529i
$417$		0			0
$418$	5670.00	+	5670.00i	0.663466	+	0.663466i
$419$	267.286			0.0311642			0.0155821	−	0.999879i	$-0.495040\pi$
$419$	267.286			0.0311642			0.0155821	+	0.999879i	$0.495040\pi$
$420$		0			0
$421$	12850.0			1.48758			0.743789	−	0.668414i	$-0.233027\pi$
$421$	12850.0			1.48758			0.743789	+	0.668414i	$0.233027\pi$
$422$	2791.66	+	2791.66i	0.322028	+	0.322028i
$423$		0			0
$424$			6678.00i			0.764888i
$425$		0			0
$426$		0			0
$427$	−2898.00	+	2898.00i	−0.328440	+	0.328440i
$428$	−1323.70	+	1323.70i	−0.149494	+	0.149494i
$429$		0			0
$430$		0			0
$431$		−	7178.55i		−	0.802270i	−0.916019	−	0.401135i	$-0.868616\pi$
$431$		−	7178.55i		−	0.802270i	0.916019	−	0.401135i	$-0.131384\pi$
$432$		0			0
$433$	−11214.0	−	11214.0i	−1.24460	−	1.24460i	−0.958073	−	0.286524i	$-0.907500\pi$
$433$	−11214.0	−	11214.0i	−1.24460	−	1.24460i	−0.286524	−	0.958073i	$-0.592500\pi$
$434$	7484.02			0.827752
$435$		0			0
$436$	70.0000			0.00768897
$437$	5048.74	+	5048.74i	0.552664	+	0.552664i
$438$		0			0
$439$			2324.00i			0.252662i	0.991988	+	0.126331i	$0.0403201\pi$
$439$			2324.00i			0.252662i	−0.991988	+	0.126331i	$0.959680\pi$
$440$		0			0
$441$		0			0
$442$	7938.00	−	7938.00i	0.854236	−	0.854236i
$443$	−5685.14	+	5685.14i	−0.609727	+	0.609727i	−0.942875	−	0.333148i	$-0.891889\pi$
$443$	−5685.14	+	5685.14i	−0.609727	+	0.609727i	0.333148	+	0.942875i	$0.391889\pi$
$444$		0			0
$445$		0			0
$446$		−	9355.02i		−	0.993213i
$447$		0			0
$448$	−3897.00	−	3897.00i	−0.410973	−	0.410973i
$449$	3703.83			0.389297			0.194648	−	0.980873i	$-0.437643\pi$
$449$	3703.83			0.389297			0.194648	+	0.980873i	$0.437643\pi$
$450$		0			0
$451$	−10206.0			−1.06559
$452$	521.845	+	521.845i	0.0543042	+	0.0543042i
$453$		0			0
$454$		−	2772.00i		−	0.286556i
$455$		0			0
$456$		0			0
$457$	7200.00	−	7200.00i	0.736984	−	0.736984i	−0.235009	−	0.971993i	$-0.575512\pi$
$457$	7200.00	−	7200.00i	0.736984	−	0.736984i	0.971993	+	0.235009i	$0.0755120\pi$
$458$	6800.95	−	6800.95i	0.693860	−	0.693860i
$459$		0			0
$460$		0			0
$461$		−	13898.9i		−	1.40420i	−0.712079	−	0.702100i	$-0.752246\pi$
$461$		−	13898.9i		−	1.40420i	0.712079	−	0.702100i	$-0.247754\pi$
$462$		0			0
$463$	4095.00	+	4095.00i	0.411038	+	0.411038i	0.882100	−	0.471062i	$-0.156129\pi$
$463$	4095.00	+	4095.00i	0.411038	+	0.411038i	−0.471062	+	0.882100i	$0.656129\pi$
$464$	16266.3			1.62746
$465$		0			0
$466$	−54.0000			−0.00536803
$467$	−9919.29	−	9919.29i	−0.982891	−	0.982891i	0.0169649	−	0.999856i	$-0.494600\pi$
$467$	−9919.29	−	9919.29i	−0.982891	−	0.982891i	−0.999856	+	0.0169649i	$0.994600\pi$
$468$		0			0
$469$			6804.00i			0.669892i
$470$		0			0
$471$		0			0
$472$	3969.00	−	3969.00i	0.387051	−	0.387051i
$473$	5498.46	−	5498.46i	0.534502	−	0.534502i
$474$		0			0
$475$		0			0
$476$			534.573i			0.0514750i
$477$		0			0
$478$	−2754.00	−	2754.00i	−0.263525	−	0.263525i
$479$	−15502.6			−1.47877			−0.739387	−	0.673281i	$-0.764884\pi$
$479$	−15502.6			−1.47877			−0.739387	+	0.673281i	$0.764884\pi$
$480$		0			0
$481$	26082.0			2.47243
$482$	−1484.92	−	1484.92i	−0.140325	−	0.140325i
$483$		0			0
$484$		−	127.000i		−	0.0119271i
$485$		0			0
$486$		0			0
$487$	−6975.00	+	6975.00i	−0.649009	+	0.649009i	−0.952754	−	0.303744i	$-0.901763\pi$
$487$	−6975.00	+	6975.00i	−0.649009	+	0.649009i	0.303744	+	0.952754i	$0.401763\pi$
$488$	−4781.46	+	4781.46i	−0.443538	+	0.443538i
$489$		0			0
$490$		0			0
$491$		−	2100.11i		−	0.193028i	−0.995332	−	0.0965138i	$-0.969231\pi$
$491$		−	2100.11i		−	0.193028i	0.995332	−	0.0965138i	$-0.0307692\pi$
$492$		0			0
$493$	6804.00	+	6804.00i	0.621576	+	0.621576i
$494$	18710.0			1.70406
$495$		0			0
$496$	13916.0			1.25977
$497$	7560.39	+	7560.39i	0.682353	+	0.682353i
$498$		0			0
$499$		−	574.000i		−	0.0514945i	−0.999668	−	0.0257473i	$-0.991803\pi$
$499$		−	574.000i		−	0.0514945i	0.999668	−	0.0257473i	$-0.00819651\pi$
$500$		0			0
$501$		0			0
$502$	−10773.0	+	10773.0i	−0.957814	+	0.957814i
$503$	8077.99	−	8077.99i	0.716063	−	0.716063i	−0.251734	−	0.967797i	$-0.581001\pi$
$503$	8077.99	−	8077.99i	0.716063	−	0.716063i	0.967797	+	0.251734i	$0.0810007\pi$
$504$		0			0
$505$		0			0
$506$		−	11684.2i		−	1.02654i
$507$		0			0
$508$	−315.000	−	315.000i	−0.0275115	−	0.0275115i
$509$	−4811.15			−0.418960			−0.209480	−	0.977813i	$-0.567177\pi$
$509$	−4811.15			−0.418960			−0.209480	+	0.977813i	$0.567177\pi$
$510$		0			0
$511$	−6804.00			−0.589024
$512$	−6175.16	−	6175.16i	−0.533020	−	0.533020i
$513$		0			0
$514$			12222.0i			1.04881i
$515$		0			0
$516$		0			0
$517$	−13608.0	+	13608.0i	−1.15760	+	1.15760i
$518$	−7904.04	+	7904.04i	−0.670432	+	0.670432i
$519$		0			0
$520$		0			0
$521$		−	14166.2i		−	1.19123i	−0.803270	−	0.595616i	$-0.796908\pi$
$521$		−	14166.2i		−	1.19123i	0.803270	−	0.595616i	$-0.203092\pi$
$522$		0			0
$523$	8694.00	+	8694.00i	0.726887	+	0.726887i	0.969998	−	0.243111i	$-0.0781680\pi$
$523$	8694.00	+	8694.00i	0.726887	+	0.726887i	−0.243111	+	0.969998i	$0.578168\pi$
$524$	−1336.43			−0.111417
$525$		0			0
$526$	−5994.00			−0.496865
$527$	5820.90	+	5820.90i	0.481143	+	0.481143i
$528$		0			0
$529$			1763.00i			0.144900i
$530$		0			0
$531$		0			0
$532$	−630.000	+	630.000i	−0.0513420	+	0.0513420i
$533$	−16839.0	+	16839.0i	−1.36844	+	1.36844i
$534$		0			0
$535$		0			0
$536$			11226.0i			0.904647i
$537$		0			0
$538$	3402.00	+	3402.00i	0.272622	+	0.272622i
$539$	6911.26			0.552299
$540$		0			0
$541$	−646.000			−0.0513377			−0.0256689	−	0.999671i	$-0.508172\pi$
$541$	−646.000			−0.0513377			−0.0256689	+	0.999671i	$0.508172\pi$
$542$	−5761.51	−	5761.51i	−0.456601	−	0.456601i
$543$		0			0
$544$			1890.00i			0.148958i
$545$		0			0
$546$		0			0
$547$	11340.0	−	11340.0i	0.886405	−	0.886405i	−0.107771	−	0.994176i	$-0.534371\pi$
$547$	11340.0	−	11340.0i	0.886405	−	0.886405i	0.994176	+	0.107771i	$0.0343713\pi$
$548$	1294.01	−	1294.01i	0.100871	−	0.100871i
$549$		0			0
$550$		0			0
$551$			16037.2i			1.23994i
$552$		0			0
$553$	4392.00	+	4392.00i	0.337734	+	0.337734i
$554$	−16304.5			−1.25038
$555$		0			0
$556$	308.000			0.0234930
$557$	−5867.57	−	5867.57i	−0.446350	−	0.446350i	0.447789	−	0.894139i	$-0.352212\pi$
$557$	−5867.57	−	5867.57i	−0.446350	−	0.446350i	−0.894139	+	0.447789i	$0.852212\pi$
$558$		0			0
$559$		−	18144.0i		−	1.37283i
$560$		0			0
$561$		0			0
$562$	8019.00	−	8019.00i	0.601888	−	0.601888i
$563$	−2672.86	+	2672.86i	−0.200085	+	0.200085i	−0.800036	−	0.599952i	$-0.795187\pi$
$563$	−2672.86	+	2672.86i	−0.200085	+	0.200085i	0.599952	+	0.800036i	$0.295187\pi$
$564$		0			0
$565$		0			0
$566$		−	534.573i		−	0.0396992i
$567$		0			0
$568$	12474.0	+	12474.0i	0.921474	+	0.921474i
$569$	20886.5			1.53885			0.769427	−	0.638734i	$-0.220542\pi$
$569$	20886.5			1.53885			0.769427	+	0.638734i	$0.220542\pi$
$570$		0			0
$571$	−12238.0			−0.896925			−0.448463	−	0.893802i	$-0.648028\pi$
$571$	−12238.0			−0.896925			−0.448463	+	0.893802i	$0.648028\pi$
$572$	−2405.58	−	2405.58i	−0.175843	−	0.175843i
$573$		0			0
$574$		−	10206.0i		−	0.742143i
$575$		0			0
$576$		0			0
$577$	−4536.00	+	4536.00i	−0.327272	+	0.327272i	−0.851548	−	0.524276i	$-0.824336\pi$
$577$	−4536.00	+	4536.00i	−0.327272	+	0.327272i	0.524276	+	0.851548i	$0.324336\pi$
$578$	6680.04	−	6680.04i	0.480714	−	0.480714i
$579$		0			0
$580$		0			0
$581$		−	13898.9i		−	0.992467i
$582$		0			0
$583$	8586.00	+	8586.00i	0.609941	+	0.609941i
$584$	−11226.0			−0.795439
$585$		0			0
$586$	−3276.00			−0.230939
$587$	−9147.13	−	9147.13i	−0.643173	−	0.643173i	0.308161	−	0.951334i	$-0.400286\pi$
$587$	−9147.13	−	9147.13i	−0.643173	−	0.643173i	−0.951334	+	0.308161i	$0.900286\pi$
$588$		0			0
$589$			13720.0i			0.959801i
$590$		0			0
$591$		0			0
$592$	−14697.0	+	14697.0i	−1.02034	+	1.02034i
$593$	−14285.0	+	14285.0i	−0.989230	+	0.989230i	−0.999943	−	0.0107125i	$-0.996590\pi$
$593$	−14285.0	+	14285.0i	−0.989230	+	0.989230i	0.0107125	+	0.999943i	$0.496590\pi$
$594$		0			0
$595$		0			0
$596$			2443.76i			0.167954i
$597$		0			0
$598$	−19278.0	−	19278.0i	−1.31829	−	1.31829i
$599$	−7789.49			−0.531335			−0.265668	−	0.964065i	$-0.585592\pi$
$599$	−7789.49			−0.531335			−0.265668	+	0.964065i	$0.585592\pi$
$600$		0			0
$601$	6748.00			0.457998			0.228999	−	0.973427i	$-0.426455\pi$
$601$	6748.00			0.457998			0.228999	+	0.973427i	$0.426455\pi$
$602$	5498.46	+	5498.46i	0.372260	+	0.372260i
$603$		0			0
$604$		−	2072.00i		−	0.139584i
$605$		0			0
$606$		0			0
$607$	13671.0	−	13671.0i	0.914150	−	0.914150i	−0.0824460	−	0.996596i	$-0.526273\pi$
$607$	13671.0	−	13671.0i	0.914150	−	0.914150i	0.996596	+	0.0824460i	$0.0262732\pi$
$608$	−2227.39	+	2227.39i	−0.148573	+	0.148573i
$609$		0			0
$610$		0			0
$611$			44904.1i			2.97320i
$612$		0			0
$613$	−3213.00	−	3213.00i	−0.211700	−	0.211700i	0.593290	−	0.804989i	$-0.297829\pi$
$613$	−3213.00	−	3213.00i	−0.211700	−	0.211700i	−0.804989	+	0.593290i	$0.797829\pi$
$614$	−12829.7			−0.843268
$615$		0			0
$616$	−10206.0			−0.667551
$617$	9906.57	+	9906.57i	0.646391	+	0.646391i	0.952119	−	0.305728i	$-0.0988998\pi$
$617$	9906.57	+	9906.57i	0.646391	+	0.646391i	−0.305728	+	0.952119i	$0.598900\pi$
$618$		0			0
$619$			2828.00i			0.183630i	0.995776	+	0.0918150i	$0.0292668\pi$
$619$			2828.00i			0.183630i	−0.995776	+	0.0918150i	$0.970733\pi$
$620$		0			0
$621$		0			0
$622$	−21546.0	+	21546.0i	−1.38893	+	1.38893i
$623$	2405.58	−	2405.58i	0.154699	−	0.154699i
$624$		0			0
$625$		0			0
$626$			22986.6i			1.46762i
$627$		0			0
$628$	315.000	+	315.000i	0.0200157	+	0.0200157i
$629$	−12295.2			−0.779397
$630$		0			0
$631$	−12220.0			−0.770952			−0.385476	−	0.922718i	$-0.625963\pi$
$631$	−12220.0			−0.770952			−0.385476	+	0.922718i	$0.625963\pi$
$632$	7246.43	+	7246.43i	0.456088	+	0.456088i
$633$		0			0
$634$		−	8388.00i		−	0.525442i
$635$		0			0
$636$		0			0
$637$	11403.0	−	11403.0i	0.709267	−	0.709267i
$638$	18557.3	−	18557.3i	1.15155	−	1.15155i
$639$		0			0
$640$		0			0
$641$			2176.47i			0.134112i	0.997749	+	0.0670558i	$0.0213606\pi$
$641$			2176.47i			0.134112i	−0.997749	+	0.0670558i	$0.978639\pi$
$642$		0			0
$643$	−4914.00	−	4914.00i	−0.301383	−	0.301383i	0.540172	−	0.841555i	$-0.318359\pi$
$643$	−4914.00	−	4914.00i	−0.301383	−	0.301383i	−0.841555	+	0.540172i	$0.818359\pi$
$644$	1298.25			0.0794381
$645$		0			0
$646$	−8820.00			−0.537180
$647$	2375.88	+	2375.88i	0.144367	+	0.144367i	0.775596	−	0.631229i	$-0.217449\pi$
$647$	2375.88	+	2375.88i	0.144367	+	0.144367i	−0.631229	+	0.775596i	$0.717449\pi$
$648$		0			0
$649$		−	10206.0i		−	0.617289i
$650$		0			0
$651$		0			0
$652$	−2394.00	+	2394.00i	−0.143798	+	0.143798i
$653$	21603.5	−	21603.5i	1.29466	−	1.29466i	0.362784	−	0.931873i	$-0.381826\pi$
$653$	21603.5	−	21603.5i	1.29466	−	1.29466i	0.931873	−	0.362784i	$-0.118174\pi$
$654$		0			0
$655$		0			0
$656$		−	18977.3i		−	1.12948i
$657$		0			0
$658$	−13608.0	−	13608.0i	−0.806224	−	0.806224i
$659$	−14395.3			−0.850926			−0.425463	−	0.904976i	$-0.639889\pi$
$659$	−14395.3			−0.850926			−0.425463	+	0.904976i	$0.639889\pi$
$660$		0			0
$661$	−22610.0			−1.33045			−0.665225	−	0.746643i	$-0.731664\pi$
$661$	−22610.0			−1.33045			−0.665225	+	0.746643i	$0.731664\pi$
$662$	−14047.4	−	14047.4i	−0.824724	−	0.824724i
$663$		0			0
$664$		−	22932.0i		−	1.34026i
$665$		0			0
$666$		0			0
$667$	16524.0	−	16524.0i	0.959238	−	0.959238i
$668$	742.462	−	742.462i	0.0430041	−	0.0430041i
$669$		0			0
$670$		0			0
$671$			12295.2i			0.707377i
$672$		0			0
$673$	−1278.00	−	1278.00i	−0.0731995	−	0.0731995i	0.669559	−	0.742759i	$-0.266483\pi$
$673$	−1278.00	−	1278.00i	−0.0731995	−	0.0731995i	−0.742759	+	0.669559i	$0.766483\pi$
$674$	−11837.0			−0.676473
$675$		0			0
$676$	−5741.00			−0.326639
$677$	7840.40	+	7840.40i	0.445098	+	0.445098i	0.893721	−	0.448623i	$-0.148086\pi$
$677$	7840.40	+	7840.40i	0.445098	+	0.445098i	−0.448623	+	0.893721i	$0.648086\pi$
$678$		0			0
$679$		−	4536.00i		−	0.256371i
$680$		0			0
$681$		0			0
$682$	15876.0	−	15876.0i	0.891383	−	0.891383i
$683$	−5125.11	+	5125.11i	−0.287126	+	0.287126i	−0.835943	−	0.548817i	$-0.815078\pi$
$683$	−5125.11	+	5125.11i	−0.287126	+	0.287126i	0.548817	+	0.835943i	$0.315078\pi$
$684$		0			0
$685$		0			0
$686$			20008.3i			1.11359i
$687$		0			0
$688$	10224.0	+	10224.0i	0.566550	+	0.566550i
$689$	28332.4			1.56658
$690$		0			0
$691$	12418.0			0.683651			0.341826	−	0.939763i	$-0.388955\pi$
$691$	12418.0			0.683651			0.341826	+	0.939763i	$0.388955\pi$
$692$	−326.683	−	326.683i	−0.0179460	−	0.0179460i
$693$		0			0
$694$			7884.00i			0.431228i
$695$		0			0
$696$		0			0
$697$	7938.00	−	7938.00i	0.431382	−	0.431382i
$698$	−11909.1	+	11909.1i	−0.645796	+	0.645796i
$699$		0			0
$700$		0			0
$701$			7254.92i			0.390891i	0.980715	+	0.195445i	$0.0626152\pi$
$701$			7254.92i			0.390891i	−0.980715	+	0.195445i	$0.937385\pi$
$702$		0			0
$703$	−14490.0	−	14490.0i	−0.777384	−	0.777384i
$704$	−16533.6			−0.885131
$705$		0			0
$706$	6426.00			0.342558
$707$	14433.5	+	14433.5i	0.767788	+	0.767788i
$708$		0			0
$709$			2086.00i			0.110496i	0.998473	+	0.0552478i	$0.0175949\pi$
$709$			2086.00i			0.110496i	−0.998473	+	0.0552478i	$0.982405\pi$
$710$		0			0
$711$		0			0
$712$	3969.00	−	3969.00i	0.208911	−	0.208911i
$713$	14136.5	−	14136.5i	0.742518	−	0.742518i
$714$		0			0
$715$		0			0
$716$			1565.53i			0.0817134i
$717$		0			0
$718$	−24300.0	−	24300.0i	−1.26305	−	1.26305i
$719$	−10691.5			−0.554554			−0.277277	−	0.960790i	$-0.589432\pi$
$719$	−10691.5			−0.554554			−0.277277	+	0.960790i	$0.589432\pi$
$720$		0			0
$721$	−17010.0			−0.878621
$722$	4155.67	+	4155.67i	0.214208	+	0.214208i
$723$		0			0
$724$			3094.00i			0.158823i
$725$		0			0
$726$		0			0
$727$	−14931.0	+	14931.0i	−0.761706	+	0.761706i	−0.976631	−	0.214925i	$-0.931049\pi$
$727$	−14931.0	+	14931.0i	−0.761706	+	0.761706i	0.214925	+	0.976631i	$0.431049\pi$
$728$	−16839.0	+	16839.0i	−0.857275	+	0.857275i
$729$		0			0
$730$		0			0
$731$			8553.16i			0.432764i
$732$		0			0
$733$	8757.00	+	8757.00i	0.441265	+	0.441265i	0.892437	−	0.451172i	$-0.148994\pi$
$733$	8757.00	+	8757.00i	0.441265	+	0.441265i	−0.451172	+	0.892437i	$0.648994\pi$
$734$	−23253.9			−1.16937
$735$		0			0
$736$	4590.00			0.229877
$737$	14433.5	+	14433.5i	0.721389	+	0.721389i
$738$		0			0
$739$			33010.0i			1.64316i	0.570096	+	0.821578i	$0.306906\pi$
$739$			33010.0i			1.64316i	−0.570096	+	0.821578i	$0.693094\pi$
$740$		0			0
$741$		0			0
$742$	−8586.00	+	8586.00i	−0.424801	+	0.424801i
$743$	−4734.79	+	4734.79i	−0.233785	+	0.233785i	−0.814271	−	0.580485i	$-0.802863\pi$
$743$	−4734.79	+	4734.79i	−0.233785	+	0.233785i	0.580485	+	0.814271i	$0.302863\pi$
$744$		0			0
$745$		0			0
$746$		−	11493.3i		−	0.564075i
$747$		0			0
$748$	1134.00	+	1134.00i	0.0554320	+	0.0554320i
$749$	23826.7			1.16236
$750$		0			0
$751$	−18704.0			−0.908813			−0.454407	−	0.890794i	$-0.650149\pi$
$751$	−18704.0			−0.908813			−0.454407	+	0.890794i	$0.650149\pi$
$752$	−25303.1	−	25303.1i	−1.22701	−	1.22701i
$753$		0			0
$754$		−	61236.0i		−	2.95767i
$755$		0			0
$756$		0			0
$757$	2187.00	−	2187.00i	0.105004	−	0.105004i	−0.652653	−	0.757657i	$-0.726344\pi$
$757$	2187.00	−	2187.00i	0.105004	−	0.105004i	0.757657	+	0.652653i	$0.226344\pi$
$758$	29520.3	−	29520.3i	1.41455	−	1.41455i
$759$		0			0
$760$		0			0
$761$		−	4543.87i		−	0.216446i	−0.994127	−	0.108223i	$-0.965484\pi$
$761$		−	4543.87i		−	0.216446i	0.994127	−	0.108223i	$-0.0345160\pi$
$762$		0			0
$763$	−630.000	−	630.000i	−0.0298919	−	0.0298919i
$764$	840.043			0.0397797
$765$		0			0
$766$	29232.0			1.37884
$767$	−16839.0	−	16839.0i	−0.792728	−	0.792728i
$768$		0			0
$769$			35336.0i			1.65702i	0.559974	+	0.828510i	$0.310811\pi$
$769$			35336.0i			1.65702i	−0.559974	+	0.828510i	$0.689189\pi$
$770$		0			0
$771$		0			0
$772$	−1242.00	+	1242.00i	−0.0579023	+	0.0579023i
$773$	−9919.29	+	9919.29i	−0.461542	+	0.461542i	−0.899161	−	0.437618i	$-0.855822\pi$
$773$	−9919.29	+	9919.29i	−0.461542	+	0.461542i	0.437618	+	0.899161i	$0.355822\pi$
$774$		0			0
$775$		0			0
$776$		−	7484.02i		−	0.346212i
$777$		0			0
$778$	14256.0	+	14256.0i	0.656944	+	0.656944i
$779$	18710.0			0.860535
$780$		0			0
$781$	32076.0			1.46962
$782$	9087.74	+	9087.74i	0.415571	+	0.415571i
$783$		0			0
$784$			12851.0i			0.585414i
$785$		0			0
$786$		0			0
$787$	−19278.0	+	19278.0i	−0.873172	+	0.873172i	−0.992817	−	0.119645i	$-0.961824\pi$
$787$	−19278.0	+	19278.0i	−0.873172	+	0.873172i	0.119645	+	0.992817i	$0.461824\pi$
$788$	−3725.04	+	3725.04i	−0.168400	+	0.168400i
$789$		0			0
$790$		0			0
$791$		−	9393.21i		−	0.422230i
$792$		0			0
$793$	20286.0	+	20286.0i	0.908420	+	0.908420i
$794$	23788.5			1.06325
$795$		0			0
$796$	700.000			0.0311694
$797$	−24441.9	−	24441.9i	−1.08629	−	1.08629i	−0.995907	−	0.0903853i	$-0.971190\pi$
$797$	−24441.9	−	24441.9i	−1.08629	−	1.08629i	−0.0903853	−	0.995907i	$-0.528810\pi$
$798$		0			0
$799$		−	21168.0i		−	0.937259i
$800$		0			0
$801$		0			0
$802$	17091.0	−	17091.0i	0.752499	−	0.752499i
$803$	−14433.5	+	14433.5i	−0.634304	+	0.634304i
$804$		0			0
$805$		0			0
$806$		−	52388.1i		−	2.28945i
$807$		0			0
$808$	23814.0	+	23814.0i	1.03685	+	1.03685i
$809$	28370.5			1.23295			0.616474	−	0.787375i	$-0.288561\pi$
$809$	28370.5			1.23295			0.616474	+	0.787375i	$0.288561\pi$
$810$		0			0
$811$	−37100.0			−1.60636			−0.803180	−	0.595737i	$-0.796860\pi$
$811$	−37100.0			−1.60636			−0.803180	+	0.595737i	$0.796860\pi$
$812$	2061.92	+	2061.92i	0.0891125	+	0.0891125i
$813$		0			0
$814$			33534.0i			1.44394i
$815$		0			0
$816$		0			0
$817$	−10080.0	+	10080.0i	−0.431646	+	0.431646i
$818$	3593.52	−	3593.52i	0.153599	−	0.153599i
$819$		0			0
$820$		0			0
$821$		−	12600.6i		−	0.535646i	−0.963468	−	0.267823i	$-0.913696\pi$
$821$		−	12600.6i		−	0.535646i	0.963468	−	0.267823i	$-0.0863043\pi$
$822$		0			0
$823$	1575.00	+	1575.00i	0.0667084	+	0.0667084i	0.739674	−	0.672965i	$-0.234980\pi$
$823$	1575.00	+	1575.00i	0.0667084	+	0.0667084i	−0.672965	+	0.739674i	$0.734980\pi$
$824$	−28065.1			−1.18652
$825$		0			0
$826$	10206.0			0.429918
$827$	23181.8	+	23181.8i	0.974740	+	0.974740i	0.999689	−	0.0249490i	$-0.00794233\pi$
$827$	23181.8	+	23181.8i	0.974740	+	0.974740i	−0.0249490	+	0.999689i	$0.507942\pi$
$828$		0			0
$829$		−	1190.00i		−	0.0498557i	−0.999689	−	0.0249279i	$-0.992064\pi$
$829$		−	1190.00i		−	0.0498557i	0.999689	−	0.0249279i	$-0.00793561\pi$
$830$		0			0
$831$		0			0
$832$	−27279.0	+	27279.0i	−1.13669	+	1.13669i
$833$	−5375.43	+	5375.43i	−0.223586	+	0.223586i
$834$		0			0
$835$		0			0
$836$			2672.86i			0.110578i
$837$		0			0
$838$	567.000	+	567.000i	0.0233731	+	0.0233731i
$839$	9622.31			0.395946			0.197973	−	0.980207i	$-0.436564\pi$
$839$	9622.31			0.395946			0.197973	+	0.980207i	$0.436564\pi$
$840$		0			0
$841$	28099.0			1.15212
$842$	27259.0	+	27259.0i	1.11568	+	1.11568i
$843$		0			0
$844$			1316.00i			0.0536713i
$845$		0			0
$846$		0			0
$847$	−1143.00	+	1143.00i	−0.0463683	+	0.0463683i
$848$	−15965.1	+	15965.1i	−0.646512	+	0.646512i
$849$		0			0
$850$		0			0
$851$			29859.7i			1.20279i
$852$		0			0
$853$	−3969.00	−	3969.00i	−0.159315	−	0.159315i	0.622948	−	0.782263i	$-0.285935\pi$
$853$	−3969.00	−	3969.00i	−0.159315	−	0.159315i	−0.782263	+	0.622948i	$0.785935\pi$
$854$	−12295.2			−0.492661
$855$		0			0
$856$	39312.0			1.56969
$857$	−33292.0	−	33292.0i	−1.32699	−	1.32699i	−0.907980	−	0.419013i	$-0.862376\pi$
$857$	−33292.0	−	33292.0i	−1.32699	−	1.32699i	−0.419013	−	0.907980i	$-0.637624\pi$
$858$		0			0
$859$			13412.0i			0.532726i	0.963873	+	0.266363i	$0.0858220\pi$
$859$			13412.0i			0.532726i	−0.963873	+	0.266363i	$0.914178\pi$
$860$		0			0
$861$		0			0
$862$	15228.0	−	15228.0i	0.601703	−	0.601703i
$863$	−33572.0	+	33572.0i	−1.32422	+	1.32422i	−0.413902	+	0.910322i	$0.635834\pi$
$863$	−33572.0	+	33572.0i	−1.32422	+	1.32422i	−0.910322	+	0.413902i	$0.864166\pi$
$864$		0			0
$865$		0			0
$866$		−	47577.0i		−	1.86690i
$867$		0			0
$868$	1764.00	+	1764.00i	0.0689793	+	0.0689793i
$869$	18633.7			0.727393
$870$		0			0
$871$	47628.0			1.85283
$872$	−1039.45	−	1039.45i	−0.0403671	−	0.0403671i
$873$		0			0
$874$			21420.0i			0.828996i
$875$		0			0
$876$		0			0
$877$	15939.0	−	15939.0i	0.613708	−	0.613708i	−0.330202	−	0.943910i	$-0.607117\pi$
$877$	15939.0	−	15939.0i	0.613708	−	0.613708i	0.943910	+	0.330202i	$0.107117\pi$
$878$	−4929.95	+	4929.95i	−0.189496	+	0.189496i
$879$		0			0
$880$		0			0
$881$		−	30737.9i		−	1.17547i	−0.809054	−	0.587734i	$-0.800020\pi$
$881$		−	30737.9i		−	1.17547i	0.809054	−	0.587734i	$-0.199980\pi$
$882$		0			0
$883$	11574.0	+	11574.0i	0.441105	+	0.441105i	0.892383	−	0.451278i	$-0.149032\pi$
$883$	11574.0	+	11574.0i	0.441105	+	0.441105i	−0.451278	+	0.892383i	$0.649032\pi$
$884$	3742.01			0.142373
$885$		0			0
$886$	−24120.0			−0.914591
$887$	−13512.8	−	13512.8i	−0.511517	−	0.511517i	0.403474	−	0.914991i	$-0.367803\pi$
$887$	−13512.8	−	13512.8i	−0.511517	−	0.511517i	−0.914991	+	0.403474i	$0.867803\pi$
$888$		0			0
$889$			5670.00i			0.213910i
$890$		0			0
$891$		0			0
$892$	2205.00	−	2205.00i	0.0827678	−	0.0827678i
$893$	24946.7	−	24946.7i	0.934838	−	0.934838i
$894$		0			0
$895$		0			0
$896$		−	21115.6i		−	0.787303i
$897$		0			0
$898$	7857.00	+	7857.00i	0.291973	+	0.291973i
$899$	44904.1			1.66589
$900$		0			0
$901$	−13356.0			−0.493843
$902$	−21650.2	−	21650.2i	−0.799194	−	0.799194i
$903$		0			0
$904$		−	15498.0i		−	0.570194i
$905$		0			0
$906$		0			0
$907$	−30600.0	+	30600.0i	−1.12024	+	1.12024i	−0.128533	+	0.991705i	$0.541027\pi$
$907$	−30600.0	+	30600.0i	−1.12024	+	1.12024i	−0.991705	+	0.128533i	$0.958973\pi$
$908$	653.367	−	653.367i	0.0238797	−	0.0238797i
$909$		0			0
$910$		0			0
$911$		−	15731.7i		−	0.572135i	−0.958209	−	0.286067i	$-0.907652\pi$
$911$		−	15731.7i		−	0.572135i	0.958209	−	0.286067i	$-0.0923481\pi$
$912$		0			0
$913$	−29484.0	−	29484.0i	−1.06876	−	1.06876i
$914$	30547.0			1.10548
$915$		0			0
$916$	3206.00			0.115643
$917$	12027.9	+	12027.9i	0.433147	+	0.433147i
$918$		0			0
$919$			3724.00i			0.133671i	0.997764	+	0.0668354i	$0.0212902\pi$
$919$			3724.00i			0.133671i	−0.997764	+	0.0668354i	$0.978710\pi$
$920$		0			0
$921$		0			0
$922$	29484.0	−	29484.0i	1.05315	−	1.05315i
$923$	52922.7	−	52922.7i	1.88729	−	1.88729i
$924$		0			0
$925$		0			0
$926$			17373.6i			0.616558i
$927$		0			0
$928$	7290.00	+	7290.00i	0.257873	+	0.257873i
$929$	−51586.3			−1.82184			−0.910921	−	0.412582i	$-0.864627\pi$
$929$	−51586.3			−1.82184			−0.910921	+	0.412582i	$0.864627\pi$
$930$		0			0
$931$	−12670.0			−0.446018
$932$	−12.7279	−	12.7279i	−0.000447336	−	0.000447336i
$933$		0			0
$934$		−	42084.0i		−	1.47434i
$935$		0			0
$936$		0			0
$937$	−30744.0	+	30744.0i	−1.07189	+	1.07189i	−0.0746847	+	0.997207i	$0.523795\pi$
$937$	−30744.0	+	30744.0i	−1.07189	+	1.07189i	−0.997207	+	0.0746847i	$0.976205\pi$
$938$	−14433.5	+	14433.5i	−0.502419	+	0.502419i
$939$		0			0
$940$		0			0
$941$			1069.15i			0.0370384i	0.999829	+	0.0185192i	$0.00589518\pi$
$941$			1069.15i			0.0370384i	−0.999829	+	0.0185192i	$0.994105\pi$
$942$		0			0
$943$	−19278.0	−	19278.0i	−0.665724	−	0.665724i
$944$	18977.3			0.654300
$945$		0			0
$946$	23328.0			0.801753
$947$	−2986.82	−	2986.82i	−0.102491	−	0.102491i	0.654002	−	0.756493i	$-0.273089\pi$
$947$	−2986.82	−	2986.82i	−0.102491	−	0.102491i	−0.756493	+	0.654002i	$0.773089\pi$
$948$		0			0
$949$			47628.0i			1.62916i
$950$		0			0
$951$		0			0
$952$	7938.00	−	7938.00i	0.270244	−	0.270244i
$953$	−3220.16	+	3220.16i	−0.109456	+	0.109456i	−0.759714	−	0.650258i	$-0.774661\pi$
$953$	−3220.16	+	3220.16i	−0.109456	+	0.109456i	0.650258	+	0.759714i	$0.274661\pi$
$954$		0			0
$955$		0			0
$956$		−	1298.25i		−	0.0439209i
$957$		0			0
$958$	−32886.0	−	32886.0i	−1.10908	−	1.10908i
$959$	−23292.1			−0.784297
$960$		0			0
$961$	8625.00			0.289517
$962$	55328.3	+	55328.3i	1.85432	+	1.85432i
$963$		0			0
$964$		−	700.000i		−	0.0233874i
$965$		0			0
$966$		0			0
$967$	30051.0	−	30051.0i	0.999354	−	0.999354i	−0.000646108	−	1.00000i	$-0.500206\pi$
$967$	30051.0	−	30051.0i	0.999354	−	0.999354i	1.00000		0.000646108i	$0.000205663\pi$
$968$	−1885.85	+	1885.85i	−0.0626174	+	0.0626174i
$969$		0			0
$970$		0			0
$971$		−	10958.7i		−	0.362186i	−0.983466	−	0.181093i	$-0.942036\pi$
$971$		−	10958.7i		−	0.362186i	0.983466	−	0.181093i	$-0.0579635\pi$
$972$		0			0
$973$	−2772.00	−	2772.00i	−0.0913322	−	0.0913322i
$974$	−29592.4			−0.973514
$975$		0			0
$976$	−22862.0			−0.749790
$977$	19053.7	+	19053.7i	0.623932	+	0.623932i	0.946535	−	0.322602i	$-0.104558\pi$
$977$	19053.7	+	19053.7i	0.623932	+	0.623932i	−0.322602	+	0.946535i	$0.604558\pi$
$978$		0			0
$979$		−	10206.0i		−	0.333182i
$980$		0			0
$981$		0			0
$982$	4455.00	−	4455.00i	0.144771	−	0.144771i
$983$	20551.4	−	20551.4i	0.666822	−	0.666822i	−0.290157	−	0.956979i	$-0.593707\pi$
$983$	20551.4	−	20551.4i	0.666822	−	0.666822i	0.956979	+	0.290157i	$0.0937075\pi$
$984$		0			0
$985$		0			0
$986$			28866.9i			0.932363i
$987$		0			0
$988$	4410.00	+	4410.00i	0.142005	+	0.142005i
$989$	20772.0			0.667857
$990$		0			0
$991$	−40516.0			−1.29872			−0.649361	−	0.760480i	$-0.724963\pi$
$991$	−40516.0			−1.29872			−0.649361	+	0.760480i	$0.724963\pi$
$992$	6236.68	+	6236.68i	0.199612	+	0.199612i
$993$		0			0
$994$			32076.0i			1.02353i
$995$		0			0
$996$		0			0
$997$	945.000	−	945.000i	0.0300185	−	0.0300185i	−0.691938	−	0.721957i	$-0.743243\pi$
$997$	945.000	−	945.000i	0.0300185	−	0.0300185i	0.721957	+	0.691938i	$0.243243\pi$
$998$	1217.64	−	1217.64i	0.0386209	−	0.0386209i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.f.b.143.2	yes	4
3.2	odd	2	inner	225.4.f.b.143.1	yes	4
5.2	odd	4	inner	225.4.f.b.107.1	yes	4
5.3	odd	4		225.4.f.a.107.2	yes	4
5.4	even	2		225.4.f.a.143.1	yes	4
15.2	even	4	inner	225.4.f.b.107.2	yes	4
15.8	even	4		225.4.f.a.107.1	✓	4
15.14	odd	2		225.4.f.a.143.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.f.a.107.1	✓	4	15.8	even	4
225.4.f.a.107.2	yes	4	5.3	odd	4
225.4.f.a.143.1	yes	4	5.4	even	2
225.4.f.a.143.2	yes	4	15.14	odd	2
225.4.f.b.107.1	yes	4	5.2	odd	4	inner
225.4.f.b.107.2	yes	4	15.2	even	4	inner
225.4.f.b.143.1	yes	4	3.2	odd	2	inner
225.4.f.b.143.2	yes	4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 2.12132i) q^{2} +1.00000i q^{4} +(9.00000 - 9.00000i) q^{7} +(14.8492 - 14.8492i) q^{8} -38.1838i q^{11} +(-63.0000 - 63.0000i) q^{13} +38.1838 q^{14} +71.0000 q^{16} +(29.6985 + 29.6985i) q^{17} +70.0000i q^{19} +(81.0000 - 81.0000i) q^{22} +(72.1249 - 72.1249i) q^{23} -267.286i q^{26} +(9.00000 + 9.00000i) q^{28} +229.103 q^{29} +196.000 q^{31} +(31.8198 + 31.8198i) q^{32} +126.000i q^{34} +(-207.000 + 207.000i) q^{37} +(-148.492 + 148.492i) q^{38} -267.286i q^{41} +(144.000 + 144.000i) q^{43} +38.1838 q^{44} +306.000 q^{46} +(-356.382 - 356.382i) q^{47} +181.000i q^{49} +(63.0000 - 63.0000i) q^{52} +(-224.860 + 224.860i) q^{53} -267.286i q^{56} +(486.000 + 486.000i) q^{58} +267.286 q^{59} -322.000 q^{61} +(415.779 + 415.779i) q^{62} -433.000i q^{64} +(-378.000 + 378.000i) q^{67} +(-29.6985 + 29.6985i) q^{68} +840.043i q^{71} +(-378.000 - 378.000i) q^{73} -878.227 q^{74} -70.0000 q^{76} +(-343.654 - 343.654i) q^{77} +488.000i q^{79} +(567.000 - 567.000i) q^{82} +(772.161 - 772.161i) q^{83} +610.940i q^{86} +(-567.000 - 567.000i) q^{88} +267.286 q^{89} -1134.00 q^{91} +(72.1249 + 72.1249i) q^{92} -1512.00i q^{94} +(252.000 - 252.000i) q^{97} +(-383.959 + 383.959i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{7} - 252 q^{13} + 284 q^{16} + 324 q^{22} + 36 q^{28} + 784 q^{31} - 828 q^{37} + 576 q^{43} + 1224 q^{46} + 252 q^{52} + 1944 q^{58} - 1288 q^{61} - 1512 q^{67} - 1512 q^{73} - 280 q^{76} + 2268 q^{82}+ \cdots + 1008 q^{97}+O(q^{100}) \) 4 * q + 36 * q^7 - 252 * q^13 + 284 * q^16 + 324 * q^22 + 36 * q^28 + 784 * q^31 - 828 * q^37 + 576 * q^43 + 1224 * q^46 + 252 * q^52 + 1944 * q^58 - 1288 * q^61 - 1512 * q^67 - 1512 * q^73 - 280 * q^76 + 2268 * q^82 - 2268 * q^88 - 4536 * q^91 + 1008 * q^97

Introduction

L-functions

Modular forms

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Database

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