LMFDB - Embedded newform 448.4.j.b.335.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not 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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.20
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.20488 + 1.20488i) q^{3} +(-11.4522 + 11.4522i) q^{5} +(-13.8299 + 12.3180i) q^{7} +24.0965i q^{9} +O(q^{10})$	\(q+(-1.20488 + 1.20488i) q^{3} +(-11.4522 + 11.4522i) q^{5} +(-13.8299 + 12.3180i) q^{7} +24.0965i q^{9} +(-33.6995 + 33.6995i) q^{11} +(52.3819 + 52.3819i) q^{13} -27.5971i q^{15} +18.7579i q^{17} +(88.4903 - 88.4903i) q^{19} +(1.82159 - 31.5050i) q^{21} -108.347 q^{23} -137.308i q^{25} +(-61.5650 - 61.5650i) q^{27} +(-65.1348 + 65.1348i) q^{29} -120.698 q^{31} -81.2075i q^{33} +(17.3140 - 299.452i) q^{35} +(235.103 + 235.103i) q^{37} -126.227 q^{39} -215.905 q^{41} +(65.1171 - 65.1171i) q^{43} +(-275.959 - 275.959i) q^{45} +124.833 q^{47} +(39.5316 - 340.714i) q^{49} +(-22.6010 - 22.6010i) q^{51} +(-264.870 - 264.870i) q^{53} -771.870i q^{55} +213.240i q^{57} +(374.504 + 374.504i) q^{59} +(579.243 + 579.243i) q^{61} +(-296.822 - 333.252i) q^{63} -1199.78 q^{65} +(-396.143 - 396.143i) q^{67} +(130.544 - 130.544i) q^{69} +50.2060 q^{71} +110.942 q^{73} +(165.439 + 165.439i) q^{75} +(50.9484 - 881.172i) q^{77} +621.257i q^{79} -502.250 q^{81} +(-98.7342 + 98.7342i) q^{83} +(-214.820 - 214.820i) q^{85} -156.959i q^{87} +618.581 q^{89} +(-1369.68 - 79.1932i) q^{91} +(145.426 - 145.426i) q^{93} +2026.82i q^{95} -1516.53i q^{97} +(-812.042 - 812.042i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7}+O(q^{10}) $ 88 * q + 4 * q^7	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.20488	+	1.20488i	−0.231879	+	0.231879i	−0.813476	−	0.581598i	$-0.802428\pi$
$3$	−1.20488	+	1.20488i	−0.231879	+	0.231879i	0.581598	+	0.813476i	$0.302428\pi$
$4$		0			0
$5$	−11.4522	+	11.4522i	−1.02432	+	1.02432i	−0.0246228	+	0.999697i	$0.507838\pi$
$5$	−11.4522	+	11.4522i	−1.02432	+	1.02432i	−0.999697	+	0.0246228i	$0.992162\pi$
$6$		0			0
$7$	−13.8299	+	12.3180i	−0.746744	+	0.665112i
$7$	−13.8299	+	12.3180i	−0.746744	+	0.665112i
$8$		0			0
$9$			24.0965i			0.892465i
$10$		0			0
$11$	−33.6995	+	33.6995i	−0.923708	+	0.923708i	−0.997289	−	0.0735816i	$-0.976557\pi$
$11$	−33.6995	+	33.6995i	−0.923708	+	0.923708i	0.0735816	+	0.997289i	$0.476557\pi$
$12$		0			0
$13$	52.3819	+	52.3819i	1.11755	+	1.11755i	0.992100	+	0.125448i	$0.0400368\pi$
$13$	52.3819	+	52.3819i	1.11755	+	1.11755i	0.125448	+	0.992100i	$0.459963\pi$
$14$		0			0
$15$		−	27.5971i		−	0.475036i
$16$		0			0
$17$			18.7579i			0.267615i	0.991007	+	0.133808i	$0.0427205\pi$
$17$			18.7579i			0.267615i	−0.991007	+	0.133808i	$0.957280\pi$
$18$		0			0
$19$	88.4903	−	88.4903i	1.06848	−	1.06848i	0.0710009	−	0.997476i	$-0.477381\pi$
$19$	88.4903	−	88.4903i	1.06848	−	1.06848i	0.997476	−	0.0710009i	$-0.0226193\pi$
$20$		0			0
$21$	1.82159	−	31.5050i	0.0189287	−	0.327379i
$22$		0			0
$23$	−108.347			−0.982254			−0.491127	−	0.871088i	$-0.663415\pi$
$23$	−108.347			−0.982254			−0.491127	+	0.871088i	$0.663415\pi$
$24$		0			0
$25$		−	137.308i		−	1.09846i
$26$		0			0
$27$	−61.5650	−	61.5650i	−0.438822	−	0.438822i
$28$		0			0
$29$	−65.1348	+	65.1348i	−0.417077	+	0.417077i	−0.884195	−	0.467118i	$-0.845292\pi$
$29$	−65.1348	+	65.1348i	−0.417077	+	0.417077i	0.467118	+	0.884195i	$0.345292\pi$
$30$		0			0
$31$	−120.698			−0.699288			−0.349644	−	0.936883i	$-0.613697\pi$
$31$	−120.698			−0.699288			−0.349644	+	0.936883i	$0.613697\pi$
$32$		0			0
$33$		−	81.2075i		−	0.428376i
$34$		0			0
$35$	17.3140	−	299.452i	0.0836171	−	1.44619i
$36$		0			0
$37$	235.103	+	235.103i	1.04461	+	1.04461i	0.998957	+	0.0456569i	$0.0145381\pi$
$37$	235.103	+	235.103i	1.04461	+	1.04461i	0.0456569	+	0.998957i	$0.485462\pi$
$38$		0			0
$39$	−126.227			−0.518271
$40$		0			0
$41$	−215.905			−0.822409			−0.411204	−	0.911543i	$-0.634892\pi$
$41$	−215.905			−0.822409			−0.411204	+	0.911543i	$0.634892\pi$
$42$		0			0
$43$	65.1171	−	65.1171i	0.230936	−	0.230936i	−0.582147	−	0.813083i	$-0.697787\pi$
$43$	65.1171	−	65.1171i	0.230936	−	0.230936i	0.813083	+	0.582147i	$0.197787\pi$
$44$		0			0
$45$	−275.959	−	275.959i	−0.914169	−	0.914169i
$46$		0			0
$47$	124.833			0.387419			0.193710	−	0.981059i	$-0.437948\pi$
$47$	124.833			0.387419			0.193710	+	0.981059i	$0.437948\pi$
$48$		0			0
$49$	39.5316	−	340.714i	0.115252	−	0.993336i
$50$		0			0
$51$	−22.6010	−	22.6010i	−0.0620543	−	0.0620543i
$52$		0			0
$53$	−264.870	−	264.870i	−0.686466	−	0.686466i	0.274983	−	0.961449i	$-0.411328\pi$
$53$	−264.870	−	264.870i	−0.686466	−	0.686466i	−0.961449	+	0.274983i	$0.911328\pi$
$54$		0			0
$55$		−	771.870i		−	1.89234i
$56$		0			0
$57$			213.240i			0.495514i
$58$		0			0
$59$	374.504	+	374.504i	0.826377	+	0.826377i	0.987014	−	0.160637i	$-0.0513548\pi$
$59$	374.504	+	374.504i	0.826377	+	0.826377i	−0.160637	+	0.987014i	$0.551355\pi$
$60$		0			0
$61$	579.243	+	579.243i	1.21581	+	1.21581i	0.969085	+	0.246727i	$0.0793550\pi$
$61$	579.243	+	579.243i	1.21581	+	1.21581i	0.246727	+	0.969085i	$0.420645\pi$
$62$		0			0
$63$	−296.822	−	333.252i	−0.593589	−	0.666442i
$64$		0			0
$65$	−1199.78			−2.28945
$66$		0			0
$67$	−396.143	−	396.143i	−0.722337	−	0.722337i	0.246744	−	0.969081i	$-0.420639\pi$
$67$	−396.143	−	396.143i	−0.722337	−	0.722337i	−0.969081	+	0.246744i	$0.920639\pi$
$68$		0			0
$69$	130.544	−	130.544i	0.227764	−	0.227764i
$70$		0			0
$71$	50.2060			0.0839205			0.0419603	−	0.999119i	$-0.486640\pi$
$71$	50.2060			0.0839205			0.0419603	+	0.999119i	$0.486640\pi$
$72$		0			0
$73$	110.942			0.177874			0.0889372	−	0.996037i	$-0.471653\pi$
$73$	110.942			0.177874			0.0889372	+	0.996037i	$0.471653\pi$
$74$		0			0
$75$	165.439	+	165.439i	0.254710	+	0.254710i
$76$		0			0
$77$	50.9484	−	881.172i	0.0754040	−	1.30414i
$78$		0			0
$79$			621.257i			0.884770i	0.896825	+	0.442385i	$0.145868\pi$
$79$			621.257i			0.884770i	−0.896825	+	0.442385i	$0.854132\pi$
$80$		0			0
$81$	−502.250			−0.688958
$82$		0			0
$83$	−98.7342	+	98.7342i	−0.130572	+	0.130572i	−0.769373	−	0.638800i	$-0.779431\pi$
$83$	−98.7342	+	98.7342i	−0.130572	+	0.130572i	0.638800	+	0.769373i	$0.279431\pi$
$84$		0			0
$85$	−214.820	−	214.820i	−0.274124	−	0.274124i
$86$		0			0
$87$		−	156.959i		−	0.193422i
$88$		0			0
$89$	618.581			0.736736			0.368368	−	0.929680i	$-0.379917\pi$
$89$	618.581			0.736736			0.368368	+	0.929680i	$0.379917\pi$
$90$		0			0
$91$	−1369.68	−	79.1932i	−1.57782	−	0.0912276i
$92$		0			0
$93$	145.426	−	145.426i	0.162150	−	0.162150i
$94$		0			0
$95$			2026.82i			2.18892i
$96$		0			0
$97$		−	1516.53i		−	1.58743i	−0.608290	−	0.793715i	$-0.708144\pi$
$97$		−	1516.53i		−	1.58743i	0.608290	−	0.793715i	$-0.291856\pi$
$98$		0			0
$99$	−812.042	−	812.042i	−0.824376	−	0.824376i
$100$		0			0
$101$	847.085	−	847.085i	0.834536	−	0.834536i	−0.153598	−	0.988133i	$-0.549086\pi$
$101$	847.085	−	847.085i	0.834536	−	0.834536i	0.988133	+	0.153598i	$0.0490861\pi$
$102$		0			0
$103$			1336.55i			1.27858i	0.768965	+	0.639291i	$0.220772\pi$
$103$			1336.55i			1.27858i	−0.768965	+	0.639291i	$0.779228\pi$
$104$		0			0
$105$	339.942	+	381.665i	0.315952	+	0.354730i
$106$		0			0
$107$	1180.45	−	1180.45i	1.06653	−	1.06653i	0.0689071	−	0.997623i	$-0.478049\pi$
$107$	1180.45	−	1180.45i	1.06653	−	1.06653i	0.997623	−	0.0689071i	$-0.0219512\pi$
$108$		0			0
$109$	224.180	−	224.180i	0.196996	−	0.196996i	−0.601715	−	0.798711i	$-0.705516\pi$
$109$	224.180	−	224.180i	0.196996	−	0.196996i	0.798711	+	0.601715i	$0.205516\pi$
$110$		0			0
$111$	−566.541			−0.484447
$112$		0			0
$113$	−1076.04			−0.895797			−0.447898	−	0.894084i	$-0.647827\pi$
$113$	−1076.04			−0.895797			−0.447898	+	0.894084i	$0.647827\pi$
$114$		0			0
$115$	1240.81	−	1240.81i	1.00614	−	1.00614i
$116$		0			0
$117$	−1262.22	+	1262.22i	−0.997372	+	0.997372i
$118$		0			0
$119$	−231.061	−	259.420i	−0.177994	−	0.199840i
$120$		0			0
$121$		−	940.313i		−	0.706471i
$122$		0			0
$123$	260.139	−	260.139i	0.190699	−	0.190699i
$124$		0			0
$125$	140.951	+	140.951i	0.100856	+	0.100856i
$126$		0			0
$127$		−	612.657i		−	0.428067i	−0.976826	−	0.214034i	$-0.931340\pi$
$127$		−	612.657i		−	0.428067i	0.976826	−	0.214034i	$-0.0686602\pi$
$128$		0			0
$129$			156.916i			0.107098i
$130$		0			0
$131$	−61.3158	+	61.3158i	−0.0408946	+	0.0408946i	−0.727258	−	0.686364i	$-0.759206\pi$
$131$	−61.3158	+	61.3158i	−0.0408946	+	0.0408946i	0.686364	+	0.727258i	$0.259206\pi$
$132$		0			0
$133$	−133.783	+	2313.84i	−0.0872218	+	1.50854i
$134$		0			0
$135$	1410.12			0.898988
$136$		0			0
$137$		−	945.912i		−	0.589888i	−0.955514	−	0.294944i	$-0.904699\pi$
$137$		−	945.912i		−	0.589888i	0.955514	−	0.294944i	$-0.0953011\pi$
$138$		0			0
$139$	1180.47	+	1180.47i	0.720330	+	0.720330i	0.968672	−	0.248343i	$-0.0798859\pi$
$139$	1180.47	+	1180.47i	0.720330	+	0.720330i	−0.248343	+	0.968672i	$0.579886\pi$
$140$		0			0
$141$	−150.408	+	150.408i	−0.0898343	+	0.0898343i
$142$		0			0
$143$	−3530.49			−2.06458
$144$		0			0
$145$		−	1491.88i		−	0.854440i
$146$		0			0
$147$	362.888	+	458.150i	0.203609	+	0.257058i
$148$		0			0
$149$	−428.127	−	428.127i	−0.235393	−	0.235393i	0.579546	−	0.814939i	$-0.303230\pi$
$149$	−428.127	−	428.127i	−0.235393	−	0.235393i	−0.814939	+	0.579546i	$0.803230\pi$
$150$		0			0
$151$	−790.239			−0.425886			−0.212943	−	0.977065i	$-0.568305\pi$
$151$	−790.239			−0.425886			−0.212943	+	0.977065i	$0.568305\pi$
$152$		0			0
$153$	−452.001			−0.238837
$154$		0			0
$155$	1382.26	−	1382.26i	0.716294	−	0.716294i
$156$		0			0
$157$	2185.95	+	2185.95i	1.11120	+	1.11120i	0.992989	+	0.118207i	$0.0377145\pi$
$157$	2185.95	+	2185.95i	1.11120	+	1.11120i	0.118207	+	0.992989i	$0.462285\pi$
$158$		0			0
$159$	638.272			0.318354
$160$		0			0
$161$	1498.42	−	1334.62i	0.733492	−	0.653309i
$162$		0			0
$163$	1895.52	+	1895.52i	0.910852	+	0.910852i	0.996339	−	0.0854877i	$-0.0272448\pi$
$163$	1895.52	+	1895.52i	0.910852	+	0.910852i	−0.0854877	+	0.996339i	$0.527245\pi$
$164$		0			0
$165$	930.008	+	930.008i	0.438794	+	0.438794i
$166$		0			0
$167$			2571.18i			1.19140i	0.803207	+	0.595700i	$0.203125\pi$
$167$			2571.18i			1.19140i	−0.803207	+	0.595700i	$0.796875\pi$
$168$		0			0
$169$			3290.73i			1.49783i
$170$		0			0
$171$	2132.31	+	2132.31i	0.953578	+	0.953578i
$172$		0			0
$173$	925.767	+	925.767i	0.406848	+	0.406848i	0.880638	−	0.473790i	$-0.157114\pi$
$173$	925.767	+	925.767i	0.406848	+	0.406848i	−0.473790	+	0.880638i	$0.657114\pi$
$174$		0			0
$175$	1691.36	+	1898.95i	0.730600	+	0.820269i
$176$		0			0
$177$	−902.462			−0.383238
$178$		0			0
$179$	−1984.79	−	1984.79i	−0.828770	−	0.828770i	0.158576	−	0.987347i	$-0.449310\pi$
$179$	−1984.79	−	1984.79i	−0.828770	−	0.828770i	−0.987347	+	0.158576i	$0.949310\pi$
$180$		0			0
$181$	−557.094	+	557.094i	−0.228776	+	0.228776i	−0.812181	−	0.583405i	$-0.801720\pi$
$181$	−557.094	+	557.094i	−0.228776	+	0.228776i	0.583405	+	0.812181i	$0.301720\pi$
$182$		0			0
$183$	−1395.83			−0.563842
$184$		0			0
$185$	−5384.92			−2.14004
$186$		0			0
$187$	−632.132	−	632.132i	−0.247198	−	0.247198i
$188$		0			0
$189$	1609.80	+	93.0767i	0.619554	+	0.0358219i
$190$		0			0
$191$			1559.36i			0.590741i	0.955383	+	0.295371i	$0.0954432\pi$
$191$			1559.36i			0.590741i	−0.955383	+	0.295371i	$0.904557\pi$
$192$		0			0
$193$	−352.570			−0.131495			−0.0657475	−	0.997836i	$-0.520943\pi$
$193$	−352.570			−0.131495			−0.0657475	+	0.997836i	$0.520943\pi$
$194$		0			0
$195$	1445.59	−	1445.59i	0.530875	−	0.530875i
$196$		0			0
$197$	2541.71	+	2541.71i	0.919234	+	0.919234i	0.996974	−	0.0777398i	$-0.0247704\pi$
$197$	2541.71	+	2541.71i	0.919234	+	0.919234i	−0.0777398	+	0.996974i	$0.524770\pi$
$198$		0			0
$199$		−	3310.03i		−	1.17910i	−0.807730	−	0.589552i	$-0.799304\pi$
$199$		−	3310.03i		−	1.17910i	0.807730	−	0.589552i	$-0.200696\pi$
$200$		0			0
$201$	954.607			0.334989
$202$		0			0
$203$	98.4736	−	1703.14i	0.0340468	−	0.588852i
$204$		0			0
$205$	2472.60	−	2472.60i	0.842409	−	0.842409i
$206$		0			0
$207$		−	2610.78i		−	0.876627i
$208$		0			0
$209$			5964.16i			1.97392i
$210$		0			0
$211$	−961.076	−	961.076i	−0.313570	−	0.313570i	0.532721	−	0.846291i	$-0.321169\pi$
$211$	−961.076	−	961.076i	−0.313570	−	0.313570i	−0.846291	+	0.532721i	$0.821169\pi$
$212$		0			0
$213$	−60.4921	+	60.4921i	−0.0194594	+	0.0194594i
$214$		0			0
$215$			1491.47i			0.473105i
$216$		0			0
$217$	1669.23	−	1486.76i	0.522189	−	0.465104i
$218$		0			0
$219$	−133.672	+	133.672i	−0.0412453	+	0.0412453i
$220$		0			0
$221$	−982.575	+	982.575i	−0.299073	+	0.299073i
$222$		0			0
$223$	−2441.56			−0.733180			−0.366590	−	0.930383i	$-0.619475\pi$
$223$	−2441.56			−0.733180			−0.366590	+	0.930383i	$0.619475\pi$
$224$		0			0
$225$	3308.64			0.980338
$226$		0			0
$227$	−2689.53	+	2689.53i	−0.786390	+	0.786390i	−0.980900	−	0.194511i	$-0.937688\pi$
$227$	−2689.53	+	2689.53i	−0.786390	+	0.786390i	0.194511	+	0.980900i	$0.437688\pi$
$228$		0			0
$229$	−1627.52	+	1627.52i	−0.469649	+	0.469649i	−0.901801	−	0.432152i	$-0.857755\pi$
$229$	−1627.52	+	1627.52i	−0.469649	+	0.469649i	0.432152	+	0.901801i	$0.357755\pi$
$230$		0			0
$231$	1000.32	+	1123.09i	0.284918	+	0.319887i
$232$		0			0
$233$			1374.80i			0.386550i	0.981145	+	0.193275i	$0.0619110\pi$
$233$			1374.80i			0.386550i	−0.981145	+	0.193275i	$0.938089\pi$
$234$		0			0
$235$	−1429.61	+	1429.61i	−0.396841	+	0.396841i
$236$		0			0
$237$	−748.538	−	748.538i	−0.205159	−	0.205159i
$238$		0			0
$239$			2751.60i			0.744712i	0.928090	+	0.372356i	$0.121450\pi$
$239$			2751.60i			0.744712i	−0.928090	+	0.372356i	$0.878550\pi$
$240$		0			0
$241$			3341.81i			0.893216i	0.894730	+	0.446608i	$0.147368\pi$
$241$			3341.81i			0.893216i	−0.894730	+	0.446608i	$0.852632\pi$
$242$		0			0
$243$	2267.41	−	2267.41i	0.598577	−	0.598577i
$244$		0			0
$245$	3449.22	+	4354.67i	0.899439	+	1.13555i
$246$		0			0
$247$	9270.58			2.38815
$248$		0			0
$249$		−	237.925i		−	0.0605538i
$250$		0			0
$251$	647.140	+	647.140i	0.162738	+	0.162738i	0.783778	−	0.621041i	$-0.213290\pi$
$251$	647.140	+	647.140i	0.162738	+	0.162738i	−0.621041	+	0.783778i	$0.713290\pi$
$252$		0			0
$253$	3651.23	−	3651.23i	0.907316	−	0.907316i
$254$		0			0
$255$	517.664			0.127127
$256$		0			0
$257$		−	3186.49i		−	0.773415i	−0.922202	−	0.386707i	$-0.873612\pi$
$257$		−	3186.49i		−	0.773415i	0.922202	−	0.386707i	$-0.126388\pi$
$258$		0			0
$259$	−6147.46	−	355.439i	−1.47484	−	0.0852738i
$260$		0			0
$261$	−1569.52	−	1569.52i	−0.372226	−	0.372226i
$262$		0			0
$263$	180.500			0.0423199			0.0211600	−	0.999776i	$-0.493264\pi$
$263$	180.500			0.0423199			0.0211600	+	0.999776i	$0.493264\pi$
$264$		0			0
$265$	6066.71			1.40632
$266$		0			0
$267$	−745.314	+	745.314i	−0.170833	+	0.170833i
$268$		0			0
$269$	−4036.50	−	4036.50i	−0.914906	−	0.914906i	0.0817469	−	0.996653i	$-0.473950\pi$
$269$	−4036.50	−	4036.50i	−0.914906	−	0.914906i	−0.996653	+	0.0817469i	$0.973950\pi$
$270$		0			0
$271$	−1603.39			−0.359405			−0.179703	−	0.983721i	$-0.557514\pi$
$271$	−1603.39			−0.359405			−0.179703	+	0.983721i	$0.557514\pi$
$272$		0			0
$273$	1745.71	−	1554.88i	0.387016	−	0.344708i
$274$		0			0
$275$	4627.20	+	4627.20i	1.01466	+	1.01466i
$276$		0			0
$277$	−4925.48	−	4925.48i	−1.06839	−	1.06839i	−0.997483	−	0.0709054i	$-0.977411\pi$
$277$	−4925.48	−	4925.48i	−1.06839	−	1.06839i	−0.0709054	−	0.997483i	$-0.522589\pi$
$278$		0			0
$279$		−	2908.39i		−	0.624089i
$280$		0			0
$281$		−	1665.96i		−	0.353676i	−0.984240	−	0.176838i	$-0.943413\pi$
$281$		−	1665.96i		−	0.353676i	0.984240	−	0.176838i	$-0.0565869\pi$
$282$		0			0
$283$	1185.83	+	1185.83i	0.249082	+	0.249082i	0.820594	−	0.571512i	$-0.193643\pi$
$283$	1185.83	+	1185.83i	0.249082	+	0.249082i	−0.571512	+	0.820594i	$0.693643\pi$
$284$		0			0
$285$	−2442.07	−	2442.07i	−0.507565	−	0.507565i
$286$		0			0
$287$	2985.95	−	2659.53i	0.614129	−	0.546994i
$288$		0			0
$289$	4561.14			0.928382
$290$		0			0
$291$	1827.24	+	1827.24i	0.368091	+	0.368091i
$292$		0			0
$293$	−148.821	+	148.821i	−0.0296732	+	0.0296732i	−0.721788	−	0.692115i	$-0.756679\pi$
$293$	−148.821	+	148.821i	−0.0296732	+	0.0296732i	0.692115	+	0.721788i	$0.256679\pi$
$294$		0			0
$295$	−8577.81			−1.69295
$296$		0			0
$297$	4149.42			0.810687
$298$		0			0
$299$	−5675.41	−	5675.41i	−1.09772	−	1.09772i
$300$		0			0
$301$	−98.4468	+	1702.68i	−0.0188518	+	0.326049i
$302$		0			0
$303$			2041.27i			0.387022i
$304$		0			0
$305$	−13267.3			−2.49076
$306$		0			0
$307$	−4046.03	+	4046.03i	−0.752180	+	0.752180i	−0.974886	−	0.222706i	$-0.928511\pi$
$307$	−4046.03	+	4046.03i	−0.752180	+	0.752180i	0.222706	+	0.974886i	$0.428511\pi$
$308$		0			0
$309$	−1610.38	−	1610.38i	−0.296476	−	0.296476i
$310$		0			0
$311$		−	522.260i		−	0.0952240i	−0.998866	−	0.0476120i	$-0.984839\pi$
$311$		−	522.260i		−	0.0952240i	0.998866	−	0.0476120i	$-0.0151611\pi$
$312$		0			0
$313$	−6047.35			−1.09207			−0.546033	−	0.837764i	$-0.683863\pi$
$313$	−6047.35			−1.09207			−0.546033	+	0.837764i	$0.683863\pi$
$314$		0			0
$315$	7215.77	+	417.208i	1.29067	+	0.0746253i
$316$		0			0
$317$	−6094.10	+	6094.10i	−1.07974	+	1.07974i	−0.0832118	+	0.996532i	$0.526518\pi$
$317$	−6094.10	+	6094.10i	−1.07974	+	1.07974i	−0.996532	+	0.0832118i	$0.973482\pi$
$318$		0			0
$319$		−	4390.02i		−	0.770514i
$320$		0			0
$321$			2844.60i			0.494611i
$322$		0			0
$323$	1659.89	+	1659.89i	0.285941	+	0.285941i
$324$		0			0
$325$	7192.44	−	7192.44i	1.22758	−	1.22758i
$326$		0			0
$327$			540.218i			0.0913581i
$328$		0			0
$329$	−1726.42	+	1537.69i	−0.289303	+	0.257677i
$330$		0			0
$331$	817.697	−	817.697i	0.135785	−	0.135785i	−0.635948	−	0.771732i	$-0.719391\pi$
$331$	817.697	−	817.697i	0.135785	−	0.135785i	0.771732	+	0.635948i	$0.219391\pi$
$332$		0			0
$333$	−5665.17	+	5665.17i	−0.932281	+	0.932281i
$334$		0			0
$335$	9073.45			1.47981
$336$		0			0
$337$	−1881.13			−0.304070			−0.152035	−	0.988375i	$-0.548583\pi$
$337$	−1881.13			−0.304070			−0.152035	+	0.988375i	$0.548583\pi$
$338$		0			0
$339$	1296.49	−	1296.49i	0.207716	−	0.207716i
$340$		0			0
$341$	4067.45	−	4067.45i	0.645937	−	0.645937i
$342$		0			0
$343$	3650.22	+	5198.99i	0.574616	+	0.818423i
$344$		0			0
$345$			2990.05i			0.466606i
$346$		0			0
$347$	−8789.76	+	8789.76i	−1.35982	+	1.35982i	−0.485698	+	0.874127i	$0.661434\pi$
$347$	−8789.76	+	8789.76i	−1.35982	+	1.35982i	−0.874127	+	0.485698i	$0.838566\pi$
$348$		0			0
$349$	2548.68	+	2548.68i	0.390910	+	0.390910i	0.875012	−	0.484102i	$-0.160853\pi$
$349$	2548.68	+	2548.68i	0.390910	+	0.390910i	−0.484102	+	0.875012i	$0.660853\pi$
$350$		0			0
$351$		−	6449.79i		−	0.980810i
$352$		0			0
$353$		−	3013.67i		−	0.454395i	−0.973849	−	0.227198i	$-0.927044\pi$
$353$		−	3013.67i		−	0.454395i	0.973849	−	0.227198i	$-0.0729563\pi$
$354$		0			0
$355$	−574.971	+	574.971i	−0.0859614	+	0.0859614i
$356$		0			0
$357$	590.969	+	34.1691i	0.0876117	+	0.00506561i
$358$		0			0
$359$	−4228.48			−0.621645			−0.310822	−	0.950468i	$-0.600604\pi$
$359$	−4228.48			−0.621645			−0.310822	+	0.950468i	$0.600604\pi$
$360$		0			0
$361$		−	8802.07i		−	1.28329i
$362$		0			0
$363$	1132.96	+	1132.96i	0.163816	+	0.163816i
$364$		0			0
$365$	−1270.54	+	1270.54i	−0.182200	+	0.182200i
$366$		0			0
$367$	2474.29			0.351926			0.175963	−	0.984397i	$-0.443696\pi$
$367$	2474.29			0.351926			0.175963	+	0.984397i	$0.443696\pi$
$368$		0			0
$369$		−	5202.57i		−	0.733971i
$370$		0			0
$371$	6925.81	+	400.442i	0.969191	+	0.0560375i
$372$		0			0
$373$	−5899.83	−	5899.83i	−0.818985	−	0.818985i	0.166976	−	0.985961i	$-0.446600\pi$
$373$	−5899.83	−	5899.83i	−0.818985	−	0.818985i	−0.985961	+	0.166976i	$0.946600\pi$
$374$		0			0
$375$	−339.656			−0.0467728
$376$		0			0
$377$	−6823.77			−0.932207
$378$		0			0
$379$	−1992.94	+	1992.94i	−0.270107	+	0.270107i	−0.829143	−	0.559036i	$-0.811171\pi$
$379$	−1992.94	+	1992.94i	−0.270107	+	0.270107i	0.559036	+	0.829143i	$0.311171\pi$
$380$		0			0
$381$	738.176	+	738.176i	0.0992596	+	0.0992596i
$382$		0			0
$383$	−3148.30			−0.420027			−0.210014	−	0.977698i	$-0.567351\pi$
$383$	−3148.30			−0.420027			−0.210014	+	0.977698i	$0.567351\pi$
$384$		0			0
$385$	9507.93	+	10674.9i	1.25862	+	1.41310i
$386$		0			0
$387$	1569.10	+	1569.10i	0.206102	+	0.206102i
$388$		0			0
$389$	−1839.33	−	1839.33i	−0.239737	−	0.239737i	0.577004	−	0.816741i	$-0.304222\pi$
$389$	−1839.33	−	1839.33i	−0.239737	−	0.239737i	−0.816741	+	0.577004i	$0.804222\pi$
$390$		0			0
$391$		−	2032.36i		−	0.262866i
$392$		0			0
$393$		−	147.756i		−	0.0189652i
$394$		0			0
$395$	−7114.78	−	7114.78i	−0.906288	−	0.906288i
$396$		0			0
$397$	230.201	+	230.201i	0.0291019	+	0.0291019i	0.721508	−	0.692406i	$-0.243449\pi$
$397$	230.201	+	230.201i	0.0291019	+	0.0291019i	−0.692406	+	0.721508i	$0.743449\pi$
$398$		0			0
$399$	−2626.70	−	2949.08i	−0.329572	−	0.370022i
$400$		0			0
$401$	13627.6			1.69708			0.848539	−	0.529133i	$-0.177483\pi$
$401$	13627.6			1.69708			0.848539	+	0.529133i	$0.177483\pi$
$402$		0			0
$403$	−6322.37	−	6322.37i	−0.781487	−	0.781487i
$404$		0			0
$405$	5751.89	−	5751.89i	0.705713	−	0.705713i
$406$		0			0
$407$	−15845.7			−1.92984
$408$		0			0
$409$	1750.53			0.211634			0.105817	−	0.994386i	$-0.466254\pi$
$409$	1750.53			0.211634			0.105817	+	0.994386i	$0.466254\pi$
$410$		0			0
$411$	1139.71	+	1139.71i	0.136783	+	0.136783i
$412$		0			0
$413$	−9792.50	−	566.191i	−1.16672	−	0.0674587i
$414$		0			0
$415$		−	2261.46i		−	0.267495i
$416$		0			0
$417$	−2844.63			−0.334058
$418$		0			0
$419$	−439.729	+	439.729i	−0.0512700	+	0.0512700i	−0.732277	−	0.681007i	$-0.761542\pi$
$419$	−439.729	+	439.729i	−0.0512700	+	0.0512700i	0.681007	+	0.732277i	$0.261542\pi$
$420$		0			0
$421$	−11783.4	−	11783.4i	−1.36411	−	1.36411i	−0.868607	−	0.495502i	$-0.834984\pi$
$421$	−11783.4	−	11783.4i	−1.36411	−	1.36411i	−0.495502	−	0.868607i	$-0.665016\pi$
$422$		0			0
$423$			3008.03i			0.345758i
$424$		0			0
$425$	2575.61			0.293965
$426$		0			0
$427$	−15146.0	−	875.725i	−1.71655	−	0.0992490i
$428$		0			0
$429$	4253.80	−	4253.80i	0.478731	−	0.478731i
$430$		0			0
$431$		−	5155.10i		−	0.576131i	−0.957611	−	0.288065i	$-0.906988\pi$
$431$		−	5155.10i		−	0.576131i	0.957611	−	0.288065i	$-0.0930120\pi$
$432$		0			0
$433$		−	15522.1i		−	1.72273i	−0.507985	−	0.861366i	$-0.669609\pi$
$433$		−	15522.1i		−	1.72273i	0.507985	−	0.861366i	$-0.330391\pi$
$434$		0			0
$435$	1797.53	+	1797.53i	0.198126	+	0.198126i
$436$		0			0
$437$	−9587.63	+	9587.63i	−1.04952	+	1.04952i
$438$		0			0
$439$		−	11301.8i		−	1.22871i	−0.789028	−	0.614357i	$-0.789415\pi$
$439$		−	11301.8i		−	1.22871i	0.789028	−	0.614357i	$-0.210585\pi$
$440$		0			0
$441$	8210.04	+	952.574i	0.886517	+	0.102859i
$442$		0			0
$443$	3621.82	−	3621.82i	0.388437	−	0.388437i	−0.485692	−	0.874130i	$-0.661433\pi$
$443$	3621.82	−	3621.82i	0.388437	−	0.388437i	0.874130	+	0.485692i	$0.161433\pi$
$444$		0			0
$445$	−7084.14	+	7084.14i	−0.754653	+	0.754653i
$446$		0			0
$447$	1031.68			0.109165
$448$		0			0
$449$	6744.34			0.708876			0.354438	−	0.935080i	$-0.384672\pi$
$449$	6744.34			0.708876			0.354438	+	0.935080i	$0.384672\pi$
$450$		0			0
$451$	7275.91	−	7275.91i	0.759665	−	0.759665i
$452$		0			0
$453$	952.141	−	952.141i	0.0987539	−	0.0987539i
$454$		0			0
$455$	16592.8	−	14778.9i	1.70963	−	1.52274i
$456$		0			0
$457$		−	10250.2i		−	1.04920i	−0.851348	−	0.524601i	$-0.824215\pi$
$457$		−	10250.2i		−	1.04920i	0.851348	−	0.524601i	$-0.175785\pi$
$458$		0			0
$459$	1154.83	−	1154.83i	0.117436	−	0.117436i
$460$		0			0
$461$	−9989.48	−	9989.48i	−1.00923	−	1.00923i	−0.999957	−	0.00927622i	$-0.997047\pi$
$461$	−9989.48	−	9989.48i	−1.00923	−	1.00923i	−0.00927622	−	0.999957i	$-0.502953\pi$
$462$		0			0
$463$		−	4481.11i		−	0.449794i	−0.974383	−	0.224897i	$-0.927795\pi$
$463$		−	4481.11i		−	0.449794i	0.974383	−	0.224897i	$-0.0722046\pi$
$464$		0			0
$465$			3330.90i			0.332187i
$466$		0			0
$467$	−3030.10	+	3030.10i	−0.300249	+	0.300249i	−0.841111	−	0.540862i	$-0.818098\pi$
$467$	−3030.10	+	3030.10i	−0.300249	+	0.300249i	0.540862	+	0.841111i	$0.318098\pi$
$468$		0			0
$469$	10358.3	+	598.906i	1.01984	+	0.0589657i
$470$		0			0
$471$	−5267.60			−0.515325
$472$		0			0
$473$			4388.83i			0.426635i
$474$		0			0
$475$	−12150.4	−	12150.4i	−1.17368	−	1.17368i
$476$		0			0
$477$	6382.45	−	6382.45i	0.612647	−	0.612647i
$478$		0			0
$479$	10364.7			0.988673			0.494336	−	0.869271i	$-0.335411\pi$
$479$	10364.7			0.988673			0.494336	+	0.869271i	$0.335411\pi$
$480$		0			0
$481$			24630.3i			2.33481i
$482$		0			0
$483$	−197.363	+	3413.47i	−0.0185928	+	0.321570i
$484$		0			0
$485$	17367.7	+	17367.7i	1.62604	+	1.62604i
$486$		0			0
$487$	−13176.2			−1.22602			−0.613009	−	0.790076i	$-0.710041\pi$
$487$	−13176.2			−1.22602			−0.613009	+	0.790076i	$0.710041\pi$
$488$		0			0
$489$	−4567.74			−0.422414
$490$		0			0
$491$	−646.906	+	646.906i	−0.0594592	+	0.0594592i	−0.736211	−	0.676752i	$-0.763387\pi$
$491$	−646.906	+	646.906i	−0.0594592	+	0.0594592i	0.676752	+	0.736211i	$0.263387\pi$
$492$		0			0
$493$	−1221.79	−	1221.79i	−0.111616	−	0.111616i
$494$		0			0
$495$	18599.4			1.68885
$496$		0			0
$497$	−694.343	+	618.440i	−0.0626671	+	0.0558165i
$498$		0			0
$499$	4359.07	+	4359.07i	0.391060	+	0.391060i	0.875065	−	0.484005i	$-0.160818\pi$
$499$	4359.07	+	4359.07i	0.391060	+	0.391060i	−0.484005	+	0.875065i	$0.660818\pi$
$500$		0			0
$501$	−3097.96	−	3097.96i	−0.276260	−	0.276260i
$502$		0			0
$503$		−	6447.91i		−	0.571567i	−0.958294	−	0.285783i	$-0.907746\pi$
$503$		−	6447.91i		−	0.571567i	0.958294	−	0.285783i	$-0.0922538\pi$
$504$		0			0
$505$			19402.0i			1.70966i
$506$		0			0
$507$	−3964.92	−	3964.92i	−0.347314	−	0.347314i
$508$		0			0
$509$	2047.71	+	2047.71i	0.178317	+	0.178317i	0.790622	−	0.612305i	$-0.209757\pi$
$509$	2047.71	+	2047.71i	0.178317	+	0.178317i	−0.612305	+	0.790622i	$0.709757\pi$
$510$		0			0
$511$	−1534.32	+	1366.59i	−0.132827	+	0.118306i
$512$		0			0
$513$	−10895.8			−0.937743
$514$		0			0
$515$	−15306.5	−	15306.5i	−1.30968	−	1.30968i
$516$		0			0
$517$	−4206.80	+	4206.80i	−0.357862	+	0.357862i
$518$		0			0
$519$	−2230.87			−0.188679
$520$		0			0
$521$	7528.32			0.633055			0.316528	−	0.948583i	$-0.397483\pi$
$521$	7528.32			0.633055			0.316528	+	0.948583i	$0.397483\pi$
$522$		0			0
$523$	15031.1	+	15031.1i	1.25672	+	1.25672i	0.952651	+	0.304065i	$0.0983441\pi$
$523$	15031.1	+	15031.1i	1.25672	+	1.25672i	0.304065	+	0.952651i	$0.401656\pi$
$524$		0			0
$525$	−4325.88	−	250.118i	−0.359613	−	0.0207924i
$526$		0			0
$527$		−	2264.03i		−	0.187140i
$528$		0			0
$529$	−427.990			−0.0351763
$530$		0			0
$531$	−9024.24	+	9024.24i	−0.737512	+	0.737512i
$532$		0			0
$533$	−11309.5	−	11309.5i	−0.919081	−	0.919081i
$534$		0			0
$535$			27037.7i			2.18494i
$536$		0			0
$537$	4782.85			0.384348
$538$		0			0
$539$	10149.7	+	12814.1i	0.811093	+	1.02401i
$540$		0			0
$541$	15537.6	−	15537.6i	1.23478	−	1.23478i	0.272668	−	0.962108i	$-0.412094\pi$
$541$	15537.6	−	15537.6i	1.23478	−	1.23478i	0.962108	−	0.272668i	$-0.0879062\pi$
$542$		0			0
$543$		−	1342.46i		−	0.106097i
$544$		0			0
$545$			5134.72i			0.403573i
$546$		0			0
$547$	5987.15	+	5987.15i	0.467992	+	0.467992i	0.901264	−	0.433271i	$-0.142641\pi$
$547$	5987.15	+	5987.15i	0.467992	+	0.467992i	−0.433271	+	0.901264i	$0.642641\pi$
$548$		0			0
$549$	−13957.8	+	13957.8i	−1.08507	+	1.08507i
$550$		0			0
$551$			11527.6i			0.891274i
$552$		0			0
$553$	−7652.67	−	8591.91i	−0.588471	−	0.660697i
$554$		0			0
$555$	6488.16	−	6488.16i	0.496229	−	0.496229i
$556$		0			0
$557$	−11088.1	+	11088.1i	−0.843477	+	0.843477i	−0.989309	−	0.145833i	$-0.953414\pi$
$557$	−11088.1	+	11088.1i	−0.843477	+	0.843477i	0.145833	+	0.989309i	$0.453414\pi$
$558$		0			0
$559$	6821.91			0.516165
$560$		0			0
$561$	1523.28			0.114640
$562$		0			0
$563$	−13271.2	+	13271.2i	−0.993452	+	0.993452i	−0.999979	−	0.00652692i	$-0.997922\pi$
$563$	−13271.2	+	13271.2i	−0.993452	+	0.993452i	0.00652692	+	0.999979i	$0.497922\pi$
$564$		0			0
$565$	12323.0	−	12323.0i	0.917582	−	0.917582i
$566$		0			0
$567$	6946.06	−	6186.74i	0.514475	−	0.458234i
$568$		0			0
$569$			21138.9i			1.55745i	0.627365	+	0.778725i	$0.284133\pi$
$569$			21138.9i			1.55745i	−0.627365	+	0.778725i	$0.715867\pi$
$570$		0			0
$571$	−11251.3	+	11251.3i	−0.824607	+	0.824607i	−0.986765	−	0.162158i	$-0.948155\pi$
$571$	−11251.3	+	11251.3i	−0.824607	+	0.824607i	0.162158	+	0.986765i	$0.448155\pi$
$572$		0			0
$573$	−1878.84	−	1878.84i	−0.136980	−	0.136980i
$574$		0			0
$575$			14876.8i			1.07897i
$576$		0			0
$577$		−	5345.41i		−	0.385671i	−0.981231	−	0.192835i	$-0.938232\pi$
$577$		−	5345.41i		−	0.385671i	0.981231	−	0.192835i	$-0.0617684\pi$
$578$		0			0
$579$	424.803	−	424.803i	0.0304909	−	0.0304909i
$580$		0			0
$581$	149.271	−	2581.70i	0.0106588	−	0.184349i
$582$		0			0
$583$	17852.0			1.26819
$584$		0			0
$585$		−	28910.6i		−	2.04326i
$586$		0			0
$587$	1153.38	+	1153.38i	0.0810992	+	0.0810992i	0.746493	−	0.665394i	$-0.231736\pi$
$587$	1153.38	+	1153.38i	0.0810992	+	0.0810992i	−0.665394	+	0.746493i	$0.731736\pi$
$588$		0			0
$589$	−10680.6	+	10680.6i	−0.747173	+	0.747173i
$590$		0			0
$591$	−6124.89			−0.426301
$592$		0			0
$593$			3106.70i			0.215138i	0.994198	+	0.107569i	$0.0343066\pi$
$593$			3106.70i			0.215138i	−0.994198	+	0.107569i	$0.965693\pi$
$594$		0			0
$595$	5617.10	+	324.774i	0.387023	+	0.0223772i
$596$		0			0
$597$	3988.18	+	3988.18i	0.273409	+	0.273409i
$598$		0			0
$599$	11406.6			0.778066			0.389033	−	0.921224i	$-0.372809\pi$
$599$	11406.6			0.778066			0.389033	+	0.921224i	$0.372809\pi$
$600$		0			0
$601$	8670.56			0.588485			0.294242	−	0.955731i	$-0.404933\pi$
$601$	8670.56			0.588485			0.294242	+	0.955731i	$0.404933\pi$
$602$		0			0
$603$	9545.68	−	9545.68i	0.644660	−	0.644660i
$604$		0			0
$605$	10768.7	+	10768.7i	0.723652	+	0.723652i
$606$		0			0
$607$	−26687.4			−1.78453			−0.892265	−	0.451512i	$-0.850885\pi$
$607$	−26687.4			−1.78453			−0.892265	+	0.451512i	$0.850885\pi$
$608$		0			0
$609$	1933.43	+	2170.72i	0.128648	+	0.144437i
$610$		0			0
$611$	6538.97	+	6538.97i	0.432960	+	0.432960i
$612$		0			0
$613$	6354.65	+	6354.65i	0.418698	+	0.418698i	0.884755	−	0.466057i	$-0.154326\pi$
$613$	6354.65	+	6354.65i	0.418698	+	0.418698i	−0.466057	+	0.884755i	$0.654326\pi$
$614$		0			0
$615$			5958.36i			0.390674i
$616$		0			0
$617$			25088.1i			1.63697i	0.574528	+	0.818485i	$0.305186\pi$
$617$			25088.1i			1.63697i	−0.574528	+	0.818485i	$0.694814\pi$
$618$		0			0
$619$	−630.701	−	630.701i	−0.0409532	−	0.0409532i	0.686334	−	0.727287i	$-0.259219\pi$
$619$	−630.701	−	630.701i	−0.0409532	−	0.0409532i	−0.727287	+	0.686334i	$0.759219\pi$
$620$		0			0
$621$	6670.37	+	6670.37i	0.431035	+	0.431035i
$622$		0			0
$623$	−8554.91	+	7619.71i	−0.550153	+	0.490012i
$624$		0			0
$625$	13935.1			0.891844
$626$		0			0
$627$	−7186.08	−	7186.08i	−0.457710	−	0.457710i
$628$		0			0
$629$	−4410.04	+	4410.04i	−0.279555	+	0.279555i
$630$		0			0
$631$	15726.7			0.992188			0.496094	−	0.868269i	$-0.334767\pi$
$631$	15726.7			0.992188			0.496094	+	0.868269i	$0.334767\pi$
$632$		0			0
$633$	2315.96			0.145420
$634$		0			0
$635$	7016.30	+	7016.30i	0.438478	+	0.438478i
$636$		0			0
$637$	19918.0	−	15776.5i	1.23890	−	0.981301i
$638$		0			0
$639$			1209.79i			0.0748961i
$640$		0			0
$641$	858.608			0.0529064			0.0264532	−	0.999650i	$-0.491579\pi$
$641$	858.608			0.0529064			0.0264532	+	0.999650i	$0.491579\pi$
$642$		0			0
$643$	19183.4	−	19183.4i	1.17654	−	1.17654i	0.195925	−	0.980619i	$-0.437229\pi$
$643$	19183.4	−	19183.4i	1.17654	−	1.17654i	0.980619	−	0.195925i	$-0.0627710\pi$
$644$		0			0
$645$	−1797.04	−	1797.04i	−0.109703	−	0.109703i
$646$		0			0
$647$		−	18568.8i		−	1.12831i	−0.825670	−	0.564154i	$-0.809203\pi$
$647$		−	18568.8i		−	1.12831i	0.825670	−	0.564154i	$-0.190797\pi$
$648$		0			0
$649$	−25241.2			−1.52666
$650$		0			0
$651$	−219.861	+	3802.58i	−0.0132366	+	0.228932i
$652$		0			0
$653$	−2088.28	+	2088.28i	−0.125147	+	0.125147i	−0.766906	−	0.641759i	$-0.778205\pi$
$653$	−2088.28	+	2088.28i	−0.125147	+	0.125147i	0.641759	+	0.766906i	$0.278205\pi$
$654$		0			0
$655$		−	1404.41i		−	0.0837782i
$656$		0			0
$657$			2673.33i			0.158747i
$658$		0			0
$659$	18978.8	+	18978.8i	1.12187	+	1.12187i	0.991461	+	0.130407i	$0.0416284\pi$
$659$	18978.8	+	18978.8i	1.12187	+	1.12187i	0.130407	+	0.991461i	$0.458372\pi$
$660$		0			0
$661$	14123.7	−	14123.7i	0.831088	−	0.831088i	−0.156578	−	0.987666i	$-0.550046\pi$
$661$	14123.7	−	14123.7i	0.831088	−	0.831088i	0.987666	+	0.156578i	$0.0500461\pi$
$662$		0			0
$663$		−	2367.76i		−	0.138697i
$664$		0			0
$665$	−24966.5	−	28030.8i	−1.45588	−	1.63457i
$666$		0			0
$667$	7057.14	−	7057.14i	0.409676	−	0.409676i
$668$		0			0
$669$	2941.78	−	2941.78i	0.170009	−	0.170009i
$670$		0			0
$671$	−39040.4			−2.24611
$672$		0			0
$673$	−16026.4			−0.917939			−0.458970	−	0.888452i	$-0.651781\pi$
$673$	−16026.4			−0.917939			−0.458970	+	0.888452i	$0.651781\pi$
$674$		0			0
$675$	−8453.35	+	8453.35i	−0.482029	+	0.482029i
$676$		0			0
$677$	−14013.4	+	14013.4i	−0.795535	+	0.795535i	−0.982388	−	0.186853i	$-0.940171\pi$
$677$	−14013.4	+	14013.4i	−0.795535	+	0.795535i	0.186853	+	0.982388i	$0.440171\pi$
$678$		0			0
$679$	18680.7	+	20973.5i	1.05582	+	1.18540i
$680$		0			0
$681$		−	6481.11i		−	0.364694i
$682$		0			0
$683$	−6313.42	+	6313.42i	−0.353699	+	0.353699i	−0.861484	−	0.507785i	$-0.830464\pi$
$683$	−6313.42	+	6313.42i	−0.353699	+	0.353699i	0.507785	+	0.861484i	$0.330464\pi$
$684$		0			0
$685$	10832.8	+	10832.8i	0.604234	+	0.604234i
$686$		0			0
$687$		−	3921.93i		−	0.217803i
$688$		0			0
$689$		−	27748.8i		−	1.53432i
$690$		0			0
$691$	5051.03	−	5051.03i	0.278076	−	0.278076i	−0.554265	−	0.832340i	$-0.687000\pi$
$691$	5051.03	−	5051.03i	0.278076	−	0.278076i	0.832340	+	0.554265i	$0.187000\pi$
$692$		0			0
$693$	21233.2	+	1227.68i	1.16390	+	0.0672954i
$694$		0			0
$695$	−27038.0			−1.47570
$696$		0			0
$697$		−	4049.94i		−	0.220089i
$698$		0			0
$699$	−1656.47	−	1656.47i	−0.0896328	−	0.0896328i
$700$		0			0
$701$	20637.4	−	20637.4i	1.11193	−	1.11193i	0.119041	−	0.992889i	$-0.462018\pi$
$701$	20637.4	−	20637.4i	1.11193	−	1.11193i	0.992889	−	0.119041i	$-0.0379820\pi$
$702$		0			0
$703$	41608.7			2.23229
$704$		0			0
$705$		−	3445.02i		−	0.184038i
$706$		0			0
$707$	−1280.66	+	22149.5i	−0.0681247	+	1.17824i
$708$		0			0
$709$	12489.6	+	12489.6i	0.661575	+	0.661575i	0.955751	−	0.294176i	$-0.0950452\pi$
$709$	12489.6	+	12489.6i	0.661575	+	0.661575i	−0.294176	+	0.955751i	$0.595045\pi$
$710$		0			0
$711$	−14970.1			−0.789626
$712$		0			0
$713$	13077.2			0.686878
$714$		0			0
$715$	40432.0	−	40432.0i	2.11479	−	2.11479i
$716$		0			0
$717$	−3315.34	−	3315.34i	−0.172683	−	0.172683i
$718$		0			0
$719$	−17897.5			−0.928325			−0.464162	−	0.885750i	$-0.653645\pi$
$719$	−17897.5			−0.928325			−0.464162	+	0.885750i	$0.653645\pi$
$720$		0			0
$721$	−16463.7	−	18484.3i	−0.850400	−	0.954773i
$722$		0			0
$723$	−4026.47	−	4026.47i	−0.207118	−	0.207118i
$724$		0			0
$725$	8943.51	+	8943.51i	0.458143	+	0.458143i
$726$		0			0
$727$			10369.4i			0.528995i	0.964386	+	0.264497i	$0.0852061\pi$
$727$			10369.4i			0.528995i	−0.964386	+	0.264497i	$0.914794\pi$
$728$		0			0
$729$		−	8096.86i		−	0.411363i
$730$		0			0
$731$	1221.46	+	1221.46i	0.0618021	+	0.0618021i
$732$		0			0
$733$	8525.15	+	8525.15i	0.429582	+	0.429582i	0.888486	−	0.458904i	$-0.151758\pi$
$733$	8525.15	+	8525.15i	0.429582	+	0.429582i	−0.458904	+	0.888486i	$0.651758\pi$
$734$		0			0
$735$	−9402.72	−	1090.96i	−0.471870	−	0.0547490i
$736$		0			0
$737$	26699.6			1.33446
$738$		0			0
$739$	−11399.0	−	11399.0i	−0.567417	−	0.567417i	0.363987	−	0.931404i	$-0.381415\pi$
$739$	−11399.0	−	11399.0i	−0.567417	−	0.567417i	−0.931404	+	0.363987i	$0.881415\pi$
$740$		0			0
$741$	−11169.9	+	11169.9i	−0.553761	+	0.553761i
$742$		0			0
$743$	−20533.6			−1.01387			−0.506934	−	0.861985i	$-0.669221\pi$
$743$	−20533.6			−1.01387			−0.506934	+	0.861985i	$0.669221\pi$
$744$		0			0
$745$	9806.03			0.482235
$746$		0			0
$747$	−2379.15	−	2379.15i	−0.116531	−	0.116531i
$748$		0			0
$749$	−1784.66	+	30866.4i	−0.0870629	+	1.50579i
$750$		0			0
$751$			9247.12i			0.449310i	0.974438	+	0.224655i	$0.0721256\pi$
$751$			9247.12i			0.449310i	−0.974438	+	0.224655i	$0.927874\pi$
$752$		0			0
$753$	−1559.45			−0.0754708
$754$		0			0
$755$	9050.01	−	9050.01i	0.436243	−	0.436243i
$756$		0			0
$757$	−14679.5	−	14679.5i	−0.704804	−	0.704804i	0.260633	−	0.965438i	$-0.416069\pi$
$757$	−14679.5	−	14679.5i	−0.704804	−	0.704804i	−0.965438	+	0.260633i	$0.916069\pi$
$758$		0			0
$759$			8798.57i			0.420774i
$760$		0			0
$761$	14262.2			0.679375			0.339688	−	0.940538i	$-0.389679\pi$
$761$	14262.2			0.679375			0.339688	+	0.940538i	$0.389679\pi$
$762$		0			0
$763$	−338.924	+	5861.83i	−0.0160811	+	0.278129i
$764$		0			0
$765$	5176.42	−	5176.42i	0.244646	−	0.244646i
$766$		0			0
$767$			39234.4i			1.84703i
$768$		0			0
$769$			7157.70i			0.335648i	0.985817	+	0.167824i	$0.0536740\pi$
$769$			7157.70i			0.335648i	−0.985817	+	0.167824i	$0.946326\pi$
$770$		0			0
$771$	3839.32	+	3839.32i	0.179338	+	0.179338i
$772$		0			0
$773$	−26152.7	+	26152.7i	−1.21688	+	1.21688i	−0.248157	+	0.968720i	$0.579825\pi$
$773$	−26152.7	+	26152.7i	−1.21688	+	1.21688i	−0.968720	+	0.248157i	$0.920175\pi$
$774$		0			0
$775$			16572.7i			0.768140i
$776$		0			0
$777$	7835.20	−	6978.67i	0.361758	−	0.322212i
$778$		0			0
$779$	−19105.5	+	19105.5i	−0.878725	+	0.878725i
$780$		0			0
$781$	−1691.92	+	1691.92i	−0.0775180	+	0.0775180i
$782$		0			0
$783$	8020.05			0.366045
$784$		0			0
$785$	−50068.0			−2.27644
$786$		0			0
$787$	16902.6	−	16902.6i	0.765579	−	0.765579i	−0.211745	−	0.977325i	$-0.567915\pi$
$787$	16902.6	−	16902.6i	0.765579	−	0.765579i	0.977325	+	0.211745i	$0.0679148\pi$
$788$		0			0
$789$	−217.481	+	217.481i	−0.00981309	+	0.00981309i
$790$		0			0
$791$	14881.5	−	13254.7i	0.668931	−	0.595805i
$792$		0			0
$793$			60683.7i			2.71746i
$794$		0			0
$795$	−7309.64	+	7309.64i	−0.326096	+	0.326096i
$796$		0			0
$797$	−5729.91	−	5729.91i	−0.254660	−	0.254660i	0.568218	−	0.822878i	$-0.307633\pi$
$797$	−5729.91	−	5729.91i	−0.254660	−	0.254660i	−0.822878	+	0.568218i	$0.807633\pi$
$798$		0			0
$799$			2341.60i			0.103679i
$800$		0			0
$801$			14905.7i			0.657511i
$802$		0			0
$803$	−3738.71	+	3738.71i	−0.164304	+	0.164304i
$804$		0			0
$805$	−1875.91	+	32444.7i	−0.0821333	+	1.42053i
$806$		0			0
$807$	9726.97			0.424295
$808$		0			0
$809$		−	13959.6i		−	0.606665i	−0.952885	−	0.303332i	$-0.901901\pi$
$809$		−	13959.6i		−	0.606665i	0.952885	−	0.303332i	$-0.0980992\pi$
$810$		0			0
$811$	18263.0	+	18263.0i	0.790754	+	0.790754i	0.981617	−	0.190862i	$-0.0611284\pi$
$811$	18263.0	+	18263.0i	0.790754	+	0.790754i	−0.190862	+	0.981617i	$0.561128\pi$
$812$		0			0
$813$	1931.89	−	1931.89i	0.0833385	−	0.0833385i
$814$		0			0
$815$	−43416.0			−1.86601
$816$		0			0
$817$		−	11524.5i		−	0.493500i
$818$		0			0
$819$	1908.28	−	33004.5i	0.0814174	−	1.40815i
$820$		0			0
$821$	−3597.99	−	3597.99i	−0.152949	−	0.152949i	0.626485	−	0.779434i	$-0.284493\pi$
$821$	−3597.99	−	3597.99i	−0.152949	−	0.152949i	−0.779434	+	0.626485i	$0.784493\pi$
$822$		0			0
$823$	44012.2			1.86412			0.932059	−	0.362306i	$-0.118010\pi$
$823$	44012.2			1.86412			0.932059	+	0.362306i	$0.118010\pi$
$824$		0			0
$825$	−11150.4			−0.470555
$826$		0			0
$827$	12058.7	−	12058.7i	0.507042	−	0.507042i	−0.406575	−	0.913617i	$-0.633277\pi$
$827$	12058.7	−	12058.7i	0.507042	−	0.507042i	0.913617	+	0.406575i	$0.133277\pi$
$828$		0			0
$829$	2012.30	+	2012.30i	0.0843064	+	0.0843064i	0.748002	−	0.663696i	$-0.231013\pi$
$829$	2012.30	+	2012.30i	0.0843064	+	0.0843064i	−0.663696	+	0.748002i	$0.731013\pi$
$830$		0			0
$831$	11869.2			0.495473
$832$		0			0
$833$	6391.09	+	741.530i	0.265832	+	0.0308433i
$834$		0			0
$835$	−29445.8	−	29445.8i	−1.22037	−	1.22037i
$836$		0			0
$837$	7430.75	+	7430.75i	0.306863	+	0.306863i
$838$		0			0
$839$			17518.3i			0.720856i	0.932787	+	0.360428i	$0.117369\pi$
$839$			17518.3i			0.720856i	−0.932787	+	0.360428i	$0.882631\pi$
$840$		0			0
$841$			15903.9i			0.652094i
$842$		0			0
$843$	2007.28	+	2007.28i	0.0820099	+	0.0820099i
$844$		0			0
$845$	−37686.2	−	37686.2i	−1.53425	−	1.53425i
$846$		0			0
$847$	11582.8	+	13004.4i	0.469882	+	0.527553i
$848$		0			0
$849$	−2857.56			−0.115514
$850$		0			0
$851$	−25472.7	−	25472.7i	−1.02608	−	1.02608i
$852$		0			0
$853$	10344.4	−	10344.4i	0.415222	−	0.415222i	−0.468331	−	0.883553i	$-0.655145\pi$
$853$	10344.4	−	10344.4i	0.415222	−	0.415222i	0.883553	+	0.468331i	$0.155145\pi$
$854$		0			0
$855$	−48839.5			−1.95354
$856$		0			0
$857$	−24026.6			−0.957683			−0.478841	−	0.877901i	$-0.658943\pi$
$857$	−24026.6			−0.957683			−0.478841	+	0.877901i	$0.658943\pi$
$858$		0			0
$859$	−5943.24	−	5943.24i	−0.236066	−	0.236066i	0.579153	−	0.815219i	$-0.303383\pi$
$859$	−5943.24	−	5943.24i	−0.236066	−	0.236066i	−0.815219	+	0.579153i	$0.803383\pi$
$860$		0			0
$861$	−393.290	+	6802.11i	−0.0155671	+	0.269240i
$862$		0			0
$863$		−	38562.8i		−	1.52108i	−0.649290	−	0.760541i	$-0.724934\pi$
$863$		−	38562.8i		−	1.52108i	0.649290	−	0.760541i	$-0.275066\pi$
$864$		0			0
$865$	−21204.2			−0.833485
$866$		0			0
$867$	−5495.61	+	5495.61i	−0.215272	+	0.215272i
$868$		0			0
$869$	−20936.0	−	20936.0i	−0.817269	−	0.817269i
$870$		0			0
$871$		−	41501.5i		−	1.61449i
$872$		0			0
$873$	36543.2			1.41673
$874$		0			0
$875$	−3685.57	−	213.095i	−0.142394	−	0.00823307i
$876$		0			0
$877$	7373.96	−	7373.96i	0.283924	−	0.283924i	−0.550748	−	0.834672i	$-0.685657\pi$
$877$	7373.96	−	7373.96i	0.283924	−	0.283924i	0.834672	+	0.550748i	$0.185657\pi$
$878$		0			0
$879$		−	358.623i		−	0.0137612i
$880$		0			0
$881$		−	22252.7i		−	0.850977i	−0.904964	−	0.425489i	$-0.860102\pi$
$881$		−	22252.7i		−	0.850977i	0.904964	−	0.425489i	$-0.139898\pi$
$882$		0			0
$883$	−13356.8	−	13356.8i	−0.509053	−	0.509053i	0.405183	−	0.914236i	$-0.367208\pi$
$883$	−13356.8	−	13356.8i	−0.509053	−	0.509053i	−0.914236	+	0.405183i	$0.867208\pi$
$884$		0			0
$885$	10335.2	−	10335.2i	0.392559	−	0.392559i
$886$		0			0
$887$		−	49543.5i		−	1.87543i	−0.347402	−	0.937716i	$-0.612936\pi$
$887$		−	49543.5i		−	1.87543i	0.347402	−	0.937716i	$-0.387064\pi$
$888$		0			0
$889$	7546.74	+	8472.98i	0.284713	+	0.319656i
$890$		0			0
$891$	16925.6	−	16925.6i	0.636395	−	0.636395i
$892$		0			0
$893$	11046.5	−	11046.5i	0.413949	−	0.413949i
$894$		0			0
$895$	45460.5			1.69785
$896$		0			0
$897$	13676.3			0.509074
$898$		0			0
$899$	7861.61	−	7861.61i	0.291657	−	0.291657i
$900$		0			0
$901$	4968.41	−	4968.41i	0.183709	−	0.183709i
$902$		0			0
$903$	−1932.90	−	2170.13i	−0.0712324	−	0.0799751i
$904$		0			0
$905$		−	12760.0i		−	0.468680i
$906$		0			0
$907$	−14623.7	+	14623.7i	−0.535360	+	0.535360i	−0.922162	−	0.386803i	$-0.873579\pi$
$907$	−14623.7	+	14623.7i	−0.535360	+	0.535360i	0.386803	+	0.922162i	$0.373579\pi$
$908$		0			0
$909$	20411.8	+	20411.8i	0.744793	+	0.744793i
$910$		0			0
$911$			39934.9i			1.45236i	0.687503	+	0.726181i	$0.258707\pi$
$911$			39934.9i			1.45236i	−0.687503	+	0.726181i	$0.741293\pi$
$912$		0			0
$913$		−	6654.59i		−	0.241221i
$914$		0			0
$915$	15985.4	−	15985.4i	0.577554	−	0.577554i
$916$		0			0
$917$	92.6999	−	1603.28i	0.00333830	−	0.0577372i
$918$		0			0
$919$	−29098.3			−1.04446			−0.522232	−	0.852803i	$-0.674901\pi$
$919$	−29098.3			−1.04446			−0.522232	+	0.852803i	$0.674901\pi$
$920$		0			0
$921$		−	9749.94i		−	0.348829i
$922$		0			0
$923$	2629.89	+	2629.89i	0.0937852	+	0.0937852i
$924$		0			0
$925$	32281.5	−	32281.5i	1.14747	−	1.14747i
$926$		0			0
$927$	−32206.2			−1.14109
$928$		0			0
$929$		−	35648.5i		−	1.25898i	−0.777009	−	0.629489i	$-0.783264\pi$
$929$		−	35648.5i		−	1.25898i	0.777009	−	0.629489i	$-0.216736\pi$
$930$		0			0
$931$	−26651.8	−	33648.1i	−0.938213	−	1.18450i
$932$		0			0
$933$	629.260	+	629.260i	0.0220804	+	0.0220804i
$934$		0			0
$935$	14478.7			0.506420
$936$		0			0
$937$	−19178.1			−0.668645			−0.334323	−	0.942459i	$-0.608508\pi$
$937$	−19178.1			−0.668645			−0.334323	+	0.942459i	$0.608508\pi$
$938$		0			0
$939$	7286.32	−	7286.32i	0.253227	−	0.253227i
$940$		0			0
$941$	2039.88	+	2039.88i	0.0706677	+	0.0706677i	0.741557	−	0.670890i	$-0.234088\pi$
$941$	2039.88	+	2039.88i	0.0706677	+	0.0706677i	−0.670890	+	0.741557i	$0.734088\pi$
$942$		0			0
$943$	23392.6			0.807815
$944$		0			0
$945$	−19501.7	+	17369.9i	−0.671314	+	0.597928i
$946$		0			0
$947$	346.475	+	346.475i	0.0118890	+	0.0118890i	0.713026	−	0.701137i	$-0.247324\pi$
$947$	346.475	+	346.475i	0.0118890	+	0.0118890i	−0.701137	+	0.713026i	$0.747324\pi$
$948$		0			0
$949$	5811.38	+	5811.38i	0.198783	+	0.198783i
$950$		0			0
$951$		−	14685.3i		−	0.500739i
$952$		0			0
$953$			35991.8i			1.22339i	0.791094	+	0.611694i	$0.209512\pi$
$953$			35991.8i			1.22339i	−0.791094	+	0.611694i	$0.790488\pi$
$954$		0			0
$955$	−17858.2	−	17858.2i	−0.605108	−	0.605108i
$956$		0			0
$957$	5289.44	+	5289.44i	0.178666	+	0.178666i
$958$		0			0
$959$	11651.8	+	13081.9i	0.392342	+	0.440496i
$960$		0			0
$961$	−15223.1			−0.510997
$962$		0			0
$963$	28444.8	+	28444.8i	0.951840	+	0.951840i
$964$		0			0
$965$	4037.71	−	4037.71i	0.134693	−	0.134693i
$966$		0			0
$967$	−33119.2			−1.10139			−0.550693	−	0.834708i	$-0.685636\pi$
$967$	−33119.2			−1.10139			−0.550693	+	0.834708i	$0.685636\pi$
$968$		0			0
$969$	−3999.93			−0.132607
$970$		0			0
$971$	−34423.3	−	34423.3i	−1.13769	−	1.13769i	−0.988863	−	0.148827i	$-0.952450\pi$
$971$	−34423.3	−	34423.3i	−1.13769	−	1.13769i	−0.148827	−	0.988863i	$-0.547550\pi$
$972$		0			0
$973$	−30866.7	−	1784.68i	−1.01700	−	0.0588019i
$974$		0			0
$975$			17332.0i			0.569301i
$976$		0			0
$977$	−20013.4			−0.655360			−0.327680	−	0.944789i	$-0.606267\pi$
$977$	−20013.4			−0.655360			−0.327680	+	0.944789i	$0.606267\pi$
$978$		0			0
$979$	−20845.9	+	20845.9i	−0.680528	+	0.680528i
$980$		0			0
$981$	5401.95	+	5401.95i	0.175812	+	0.175812i
$982$		0			0
$983$			15635.9i			0.507332i	0.967292	+	0.253666i	$0.0816364\pi$
$983$			15635.9i			0.507332i	−0.967292	+	0.253666i	$0.918364\pi$
$984$		0			0
$985$	−58216.5			−1.88318
$986$		0			0
$987$	227.393	−	3932.86i	0.00733334	−	0.126833i
$988$		0			0
$989$	−7055.22	+	7055.22i	−0.226838	+	0.226838i
$990$		0			0
$991$		−	54689.3i		−	1.75304i	−0.481364	−	0.876521i	$-0.659859\pi$
$991$		−	54689.3i		−	1.75304i	0.481364	−	0.876521i	$-0.340141\pi$
$992$		0			0
$993$			1970.45i			0.0629711i
$994$		0			0
$995$	37907.3	+	37907.3i	1.20778	+	1.20778i
$996$		0			0
$997$	24191.4	−	24191.4i	0.768455	−	0.768455i	−0.209380	−	0.977834i	$-0.567144\pi$
$997$	24191.4	−	24191.4i	0.768455	−	0.768455i	0.977834	+	0.209380i	$0.0671444\pi$
$998$		0			0
$999$		−	28948.3i		−	0.916800i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.20		88
4.3	odd	2		112.4.j.b.27.18	yes	88
7.6	odd	2	inner	448.4.j.b.335.25		88
16.3	odd	4	inner	448.4.j.b.111.25		88
16.13	even	4		112.4.j.b.83.17	yes	88
28.27	even	2		112.4.j.b.27.17	✓	88
112.13	odd	4		112.4.j.b.83.18	yes	88
112.83	even	4	inner	448.4.j.b.111.20		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.17	✓	88	28.27	even	2
112.4.j.b.27.18	yes	88	4.3	odd	2
112.4.j.b.83.17	yes	88	16.13	even	4
112.4.j.b.83.18	yes	88	112.13	odd	4
448.4.j.b.111.20		88	112.83	even	4	inner
448.4.j.b.111.25		88	16.3	odd	4	inner
448.4.j.b.335.20		88	1.1	even	1	trivial
448.4.j.b.335.25		88	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.20488 + 1.20488i) q^{3} +(-11.4522 + 11.4522i) q^{5} +(-13.8299 + 12.3180i) q^{7} +24.0965i q^{9} +O(q^{10})\)	\(q+(-1.20488 + 1.20488i) q^{3} +(-11.4522 + 11.4522i) q^{5} +(-13.8299 + 12.3180i) q^{7} +24.0965i q^{9} +(-33.6995 + 33.6995i) q^{11} +(52.3819 + 52.3819i) q^{13} -27.5971i q^{15} +18.7579i q^{17} +(88.4903 - 88.4903i) q^{19} +(1.82159 - 31.5050i) q^{21} -108.347 q^{23} -137.308i q^{25} +(-61.5650 - 61.5650i) q^{27} +(-65.1348 + 65.1348i) q^{29} -120.698 q^{31} -81.2075i q^{33} +(17.3140 - 299.452i) q^{35} +(235.103 + 235.103i) q^{37} -126.227 q^{39} -215.905 q^{41} +(65.1171 - 65.1171i) q^{43} +(-275.959 - 275.959i) q^{45} +124.833 q^{47} +(39.5316 - 340.714i) q^{49} +(-22.6010 - 22.6010i) q^{51} +(-264.870 - 264.870i) q^{53} -771.870i q^{55} +213.240i q^{57} +(374.504 + 374.504i) q^{59} +(579.243 + 579.243i) q^{61} +(-296.822 - 333.252i) q^{63} -1199.78 q^{65} +(-396.143 - 396.143i) q^{67} +(130.544 - 130.544i) q^{69} +50.2060 q^{71} +110.942 q^{73} +(165.439 + 165.439i) q^{75} +(50.9484 - 881.172i) q^{77} +621.257i q^{79} -502.250 q^{81} +(-98.7342 + 98.7342i) q^{83} +(-214.820 - 214.820i) q^{85} -156.959i q^{87} +618.581 q^{89} +(-1369.68 - 79.1932i) q^{91} +(145.426 - 145.426i) q^{93} +2026.82i q^{95} -1516.53i q^{97} +(-812.042 - 812.042i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7}+O(q^{10}) \) 88 * q + 4 * q^7	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

L-functions

Modular forms

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