LMFDB - Embedded newform 448.4.p.g.383.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(255,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("448.255");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.p (of order $6$, 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:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			383.2
Root			$2.68858 + 4.65676i$ of defining polynomial
Character	$\chi$	$=$	448.383
Dual form			448.4.p.g.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.38417 + 4.12950i) q^{3} +(-14.6315 - 8.44749i) q^{5} +(-8.89981 + 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +O(q^{10})$	\(q+(2.38417 + 4.12950i) q^{3} +(-14.6315 - 8.44749i) q^{5} +(-8.89981 + 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +(-35.2840 + 20.3712i) q^{11} -56.7321i q^{13} -80.5609i q^{15} +(106.377 - 61.4170i) q^{17} +(-37.7473 + 65.3803i) q^{19} +(-88.2889 + 1.97120i) q^{21} +(91.9414 + 53.0824i) q^{23} +(80.2201 + 138.945i) q^{25} +149.072 q^{27} +146.703 q^{29} +(21.2956 + 36.8850i) q^{31} +(-168.246 - 97.1369i) q^{33} +(267.419 - 162.459i) q^{35} +(40.4571 - 70.0738i) q^{37} +(234.275 - 135.259i) q^{39} -53.8052i q^{41} -341.166i q^{43} +(-62.3733 + 36.0112i) q^{45} +(2.06393 - 3.57482i) q^{47} +(-184.587 - 289.096i) q^{49} +(507.244 + 292.857i) q^{51} +(139.849 + 242.225i) q^{53} +688.342 q^{55} -359.984 q^{57} +(-87.4155 - 151.408i) q^{59} +(404.509 + 233.543i) q^{61} +(40.9918 + 67.4754i) q^{63} +(-479.244 + 830.075i) q^{65} +(652.335 - 376.626i) q^{67} +506.229i q^{69} +669.583i q^{71} +(723.802 - 417.887i) q^{73} +(-382.516 + 662.538i) q^{75} +(-16.8426 - 754.372i) q^{77} +(-958.008 - 553.106i) q^{79} +(297.864 + 515.915i) q^{81} -552.905 q^{83} -2075.28 q^{85} +(349.765 + 605.811i) q^{87} +(106.062 + 61.2347i) q^{89} +(921.427 + 504.906i) q^{91} +(-101.545 + 175.880i) q^{93} +(1104.60 - 637.740i) q^{95} -291.203i q^{97} +173.683i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9}+O(q^{10}) $ 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9	$ 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} + 15 q^{21} - 15 q^{23} + 306 q^{25} - 362 q^{27} + 348 q^{29} + 205 q^{31} - 471 q^{33} + 1185 q^{35} + 249 q^{37} + 288 q^{39} - 2118 q^{45} + 75 q^{47} + 702 q^{49} + 2505 q^{51} + 645 q^{53} + 918 q^{55} - 6 q^{57} - 321 q^{59} + 1707 q^{61} - 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} - 555 q^{77} - 3447 q^{79} + 225 q^{81} - 24 q^{83} - 3786 q^{85} + 3642 q^{87} - 2607 q^{89} + 2448 q^{91} + 2991 q^{93} + 2085 q^{95}+O(q^{100}) $ 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 + 15 * q^21 - 15 * q^23 + 306 * q^25 - 362 * q^27 + 348 * q^29 + 205 * q^31 - 471 * q^33 + 1185 * q^35 + 249 * q^37 + 288 * q^39 - 2118 * q^45 + 75 * q^47 + 702 * q^49 + 2505 * q^51 + 645 * q^53 + 918 * q^55 - 6 * q^57 - 321 * q^59 + 1707 * q^61 - 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 - 555 * q^77 - 3447 * q^79 + 225 * q^81 - 24 * q^83 - 3786 * q^85 + 3642 * q^87 - 2607 * q^89 + 2448 * q^91 + 2991 * q^93 + 2085 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	2.38417	+	4.12950i	0.458834	+	0.794723i	0.998900	−	0.0468996i	$-0.0149341\pi$
$3$	2.38417	+	4.12950i	0.458834	+	0.794723i	−0.540066	+	0.841623i	$0.681601\pi$
$4$		0			0
$5$	−14.6315	−	8.44749i	−1.30868	−	0.755566i	−0.326803	−	0.945092i	$-0.605971\pi$
$5$	−14.6315	−	8.44749i	−1.30868	−	0.755566i	−0.981876	+	0.189526i	$0.939305\pi$
$6$		0			0
$7$	−8.89981	+	16.2417i	−0.480545	+	0.876970i
$7$	−8.89981	+	16.2417i	−0.480545	+	0.876970i
$8$		0			0
$9$	2.13148	−	3.69182i	0.0789436	−	0.136734i
$10$		0			0
$11$	−35.2840	+	20.3712i	−0.967138	+	0.558378i	−0.898363	−	0.439255i	$-0.855243\pi$
$11$	−35.2840	+	20.3712i	−0.967138	+	0.558378i	−0.0687757	+	0.997632i	$0.521909\pi$
$12$		0			0
$13$		−	56.7321i		−	1.21036i	−0.796089	−	0.605179i	$-0.793101\pi$
$13$		−	56.7321i		−	1.21036i	0.796089	−	0.605179i	$-0.206899\pi$
$14$		0			0
$15$		−	80.5609i		−	1.38672i
$16$		0			0
$17$	106.377	−	61.4170i	1.51767	−	0.876225i	0.517882	−	0.855452i	$-0.326721\pi$
$17$	106.377	−	61.4170i	1.51767	−	0.876225i	0.999784	−	0.0207724i	$-0.00661253\pi$
$18$		0			0
$19$	−37.7473	+	65.3803i	−0.455780	+	0.789435i	−0.998733	−	0.0503288i	$-0.983973\pi$
$19$	−37.7473	+	65.3803i	−0.455780	+	0.789435i	0.542952	+	0.839763i	$0.317306\pi$
$20$		0			0
$21$	−88.2889	+	1.97120i	−0.917438	+	0.0204833i
$22$		0			0
$23$	91.9414	+	53.0824i	0.833526	+	0.481236i	0.855058	−	0.518532i	$-0.173521\pi$
$23$	91.9414	+	53.0824i	0.833526	+	0.481236i	−0.0215324	+	0.999768i	$0.506854\pi$
$24$		0			0
$25$	80.2201	+	138.945i	0.641760	+	1.11156i
$26$		0			0
$27$	149.072			1.06255
$28$		0			0
$29$	146.703			0.939382			0.469691	−	0.882831i	$-0.344365\pi$
$29$	146.703			0.939382			0.469691	+	0.882831i	$0.344365\pi$
$30$		0			0
$31$	21.2956	+	36.8850i	0.123381	+	0.213702i	0.921099	−	0.389329i	$-0.127293\pi$
$31$	21.2956	+	36.8850i	0.123381	+	0.213702i	−0.797718	+	0.603030i	$0.793960\pi$
$32$		0			0
$33$	−168.246	−	97.1369i	−0.887511	−	0.512405i
$34$		0			0
$35$	267.419	−	162.459i	1.29149	−	0.784589i
$36$		0			0
$37$	40.4571	−	70.0738i	0.179760	−	0.311353i	−0.762039	−	0.647532i	$-0.775801\pi$
$37$	40.4571	−	70.0738i	0.179760	−	0.311353i	0.941798	+	0.336179i	$0.109135\pi$
$38$		0			0
$39$	234.275	−	135.259i	0.961900	−	0.555353i
$40$		0			0
$41$		−	53.8052i		−	0.204950i	−0.994736	−	0.102475i	$-0.967324\pi$
$41$		−	53.8052i		−	0.204950i	0.994736	−	0.102475i	$-0.0326762\pi$
$42$		0			0
$43$		−	341.166i		−	1.20994i	−0.796249	−	0.604970i	$-0.793185\pi$
$43$		−	341.166i		−	1.20994i	0.796249	−	0.604970i	$-0.206815\pi$
$44$		0			0
$45$	−62.3733	+	36.0112i	−0.206624	+	0.119294i
$46$		0			0
$47$	2.06393	−	3.57482i	0.00640542	−	0.0110945i	−0.862805	−	0.505537i	$-0.831294\pi$
$47$	2.06393	−	3.57482i	0.00640542	−	0.0110945i	0.869210	+	0.494442i	$0.164628\pi$
$48$		0			0
$49$	−184.587	−	289.096i	−0.538153	−	0.842847i
$50$		0			0
$51$	507.244	+	292.857i	1.39271	+	0.804083i
$52$		0			0
$53$	139.849	+	242.225i	0.362447	+	0.627777i	0.988363	−	0.152114i	$-0.0486080\pi$
$53$	139.849	+	242.225i	0.362447	+	0.627777i	−0.625916	+	0.779891i	$0.715275\pi$
$54$		0			0
$55$	688.342			1.68756
$56$		0			0
$57$	−359.984			−0.836509
$58$		0			0
$59$	−87.4155	−	151.408i	−0.192890	−	0.334096i	0.753317	−	0.657658i	$-0.228453\pi$
$59$	−87.4155	−	151.408i	−0.192890	−	0.334096i	−0.946207	+	0.323562i	$0.895119\pi$
$60$		0			0
$61$	404.509	+	233.543i	0.849050	+	0.490199i	0.860330	−	0.509737i	$-0.170257\pi$
$61$	404.509	+	233.543i	0.849050	+	0.490199i	−0.0112801	+	0.999936i	$0.503591\pi$
$62$		0			0
$63$	40.9918	+	67.4754i	0.0819759	+	0.134938i
$64$		0			0
$65$	−479.244	+	830.075i	−0.914506	+	1.58397i
$66$		0			0
$67$	652.335	−	376.626i	1.18948	−	0.686748i	0.231294	−	0.972884i	$-0.425704\pi$
$67$	652.335	−	376.626i	1.18948	−	0.686748i	0.958189	+	0.286136i	$0.0923708\pi$
$68$		0			0
$69$			506.229i			0.883230i
$70$		0			0
$71$			669.583i			1.11922i	0.828755	+	0.559611i	$0.189050\pi$
$71$			669.583i			1.11922i	−0.828755	+	0.559611i	$0.810950\pi$
$72$		0			0
$73$	723.802	−	417.887i	1.16047	−	0.670000i	0.209057	−	0.977904i	$-0.432961\pi$
$73$	723.802	−	417.887i	1.16047	−	0.670000i	0.951418	+	0.307904i	$0.0996274\pi$
$74$		0			0
$75$	−382.516	+	662.538i	−0.588922	+	1.02004i
$76$		0			0
$77$	−16.8426	−	754.372i	−0.0249272	−	1.11648i
$78$		0			0
$79$	−958.008	−	553.106i	−1.36436	−	0.787713i	−0.374159	−	0.927365i	$-0.622068\pi$
$79$	−958.008	−	553.106i	−1.36436	−	0.787713i	−0.990201	+	0.139652i	$0.955402\pi$
$80$		0			0
$81$	297.864	+	515.915i	0.408592	+	0.707703i
$82$		0			0
$83$	−552.905			−0.731195			−0.365597	−	0.930773i	$-0.619135\pi$
$83$	−552.905			−0.731195			−0.365597	+	0.930773i	$0.619135\pi$
$84$		0			0
$85$	−2075.28			−2.64818
$86$		0			0
$87$	349.765	+	605.811i	0.431020	+	0.746548i
$88$		0			0
$89$	106.062	+	61.2347i	0.126320	+	0.0729310i	0.561829	−	0.827254i	$-0.310098\pi$
$89$	106.062	+	61.2347i	0.126320	+	0.0729310i	−0.435508	+	0.900185i	$0.643431\pi$
$90$		0			0
$91$	921.427	+	504.906i	1.06145	+	0.581632i
$92$		0			0
$93$	−101.545	+	175.880i	−0.113222	+	0.196107i
$94$		0			0
$95$	1104.60	−	637.740i	1.19294	−	0.688744i
$96$		0			0
$97$		−	291.203i		−	0.304816i	−0.988318	−	0.152408i	$-0.951297\pi$
$97$		−	291.203i		−	0.304816i	0.988318	−	0.152408i	$-0.0487028\pi$
$98$		0			0
$99$			173.683i			0.176321i
$100$		0			0
$101$	1168.30	−	674.520i	1.15099	−	0.664527i	0.201865	−	0.979413i	$-0.435300\pi$
$101$	1168.30	−	674.520i	1.15099	−	0.664527i	0.949129	+	0.314887i	$0.101966\pi$
$102$		0			0
$103$	989.035	−	1713.06i	0.946141	−	1.63876i	0.192690	−	0.981260i	$-0.438279\pi$
$103$	989.035	−	1713.06i	0.946141	−	1.63876i	0.753451	−	0.657504i	$-0.228388\pi$
$104$		0			0
$105$	1308.45	+	716.977i	1.21611	+	0.666379i
$106$		0			0
$107$	732.931	+	423.158i	0.662198	+	0.382320i	0.793114	−	0.609073i	$-0.208459\pi$
$107$	732.931	+	423.158i	0.662198	+	0.382320i	−0.130916	+	0.991393i	$0.541792\pi$
$108$		0			0
$109$	463.538	+	802.872i	0.407330	+	0.705516i	0.994590	−	0.103883i	$-0.0331267\pi$
$109$	463.538	+	802.872i	0.407330	+	0.705516i	−0.587260	+	0.809398i	$0.699793\pi$
$110$		0			0
$111$	385.826			0.329919
$112$		0			0
$113$	−599.053			−0.498709			−0.249355	−	0.968412i	$-0.580218\pi$
$113$	−599.053			−0.498709			−0.249355	+	0.968412i	$0.580218\pi$
$114$		0			0
$115$	−896.825	−	1553.35i	−0.727212	−	1.25957i
$116$		0			0
$117$	−209.445	−	120.923i	−0.165497	−	0.0955500i
$118$		0			0
$119$	50.7787	+	2274.35i	0.0391166	+	1.75201i
$120$		0			0
$121$	164.473	−	284.876i	0.123571	−	0.214031i
$122$		0			0
$123$	222.189	−	128.281i	0.162879	−	0.0940380i
$124$		0			0
$125$		−	598.760i		−	0.428438i
$126$		0			0
$127$		−	1833.90i		−	1.28136i	−0.767810	−	0.640678i	$-0.778653\pi$
$127$		−	1833.90i		−	1.28136i	0.767810	−	0.640678i	$-0.221347\pi$
$128$		0			0
$129$	1408.85	−	813.398i	0.961567	−	0.555161i
$130$		0			0
$131$	−689.574	+	1194.38i	−0.459911	+	0.796589i	−0.998956	−	0.0456878i	$-0.985452\pi$
$131$	−689.574	+	1194.38i	−0.459911	+	0.796589i	0.539045	+	0.842277i	$0.318785\pi$
$132$		0			0
$133$	−725.943	−	1194.95i	−0.473288	−	0.779064i
$134$		0			0
$135$	−2181.15	−	1259.29i	−1.39054	−	0.802831i
$136$		0			0
$137$	−736.714	−	1276.03i	−0.459429	−	0.795754i	0.539502	−	0.841984i	$-0.318612\pi$
$137$	−736.714	−	1276.03i	−0.459429	−	0.795754i	−0.998931	+	0.0462305i	$0.985279\pi$
$138$		0			0
$139$	1929.87			1.17762			0.588811	−	0.808271i	$-0.299596\pi$
$139$	1929.87			1.17762			0.588811	+	0.808271i	$0.299596\pi$
$140$		0			0
$141$	19.6830			0.0117561
$142$		0			0
$143$	1155.70	+	2001.74i	0.675837	+	1.17058i
$144$		0			0
$145$	−2146.48	−	1239.27i	−1.22935	−	0.709765i
$146$		0			0
$147$	753.739	−	1451.51i	0.422907	−	0.814409i
$148$		0			0
$149$	1153.42	−	1997.78i	0.634172	−	1.09842i	−0.352517	−	0.935805i	$-0.614674\pi$
$149$	1153.42	−	1997.78i	0.634172	−	1.09842i	0.986690	−	0.162614i	$-0.0519924\pi$
$150$		0			0
$151$	−887.906	+	512.633i	−0.478522	+	0.276275i	−0.719800	−	0.694181i	$-0.755767\pi$
$151$	−887.906	+	512.633i	−0.478522	+	0.276275i	0.241278	+	0.970456i	$0.422433\pi$
$152$		0			0
$153$		−	523.636i		−	0.276689i
$154$		0			0
$155$		−	719.577i		−	0.372889i
$156$		0			0
$157$	−66.6966	+	38.5073i	−0.0339043	+	0.0195746i	−0.516856	−	0.856072i	$-0.672898\pi$
$157$	−66.6966	+	38.5073i	−0.0339043	+	0.0195746i	0.482952	+	0.875647i	$0.339565\pi$
$158$		0			0
$159$	−666.846	+	1155.01i	−0.332606	+	0.576090i
$160$		0			0
$161$	−1680.41	+	1020.86i	−0.822577	+	0.499722i
$162$		0			0
$163$	−3366.71	−	1943.77i	−1.61780	−	0.934036i	−0.987489	−	0.157691i	$-0.949595\pi$
$163$	−3366.71	−	1943.77i	−1.61780	−	0.934036i	−0.630308	−	0.776345i	$-0.717072\pi$
$164$		0			0
$165$	1641.12	+	2842.51i	0.774311	+	1.34115i
$166$		0			0
$167$	117.863			0.0546141			0.0273070	−	0.999627i	$-0.491307\pi$
$167$	117.863			0.0546141			0.0273070	+	0.999627i	$0.491307\pi$
$168$		0			0
$169$	−1021.54			−0.464969
$170$		0			0
$171$	160.915	+	278.713i	0.0719618	+	0.124642i
$172$		0			0
$173$	2480.48	+	1432.11i	1.09010	+	0.629370i	0.933603	−	0.358308i	$-0.116646\pi$
$173$	2480.48	+	1432.11i	1.09010	+	0.629370i	0.156498	+	0.987678i	$0.449980\pi$
$174$		0			0
$175$	−2970.65	+	66.3248i	−1.28320	+	0.0286496i
$176$		0			0
$177$	416.826	−	721.965i	0.177009	−	0.306589i
$178$		0			0
$179$	−1195.93	+	690.471i	−0.499375	+	0.288314i	−0.728455	−	0.685093i	$-0.759761\pi$
$179$	−1195.93	+	690.471i	−0.499375	+	0.288314i	0.229081	+	0.973407i	$0.426428\pi$
$180$		0			0
$181$			204.817i			0.0841103i	0.999115	+	0.0420551i	$0.0133905\pi$
$181$			204.817i			0.0841103i	−0.999115	+	0.0420551i	$0.986609\pi$
$182$		0			0
$183$			2227.23i			0.899680i
$184$		0			0
$185$	−1183.89	+	683.522i	−0.470495	+	0.271641i
$186$		0			0
$187$	−2502.28	+	4334.08i	−0.978528	+	1.69486i
$188$		0			0
$189$	−1326.72	+	2421.19i	−0.510605	+	0.931829i
$190$		0			0
$191$	−3609.38	−	2083.88i	−1.36736	−	0.789445i	−0.376769	−	0.926307i	$-0.622965\pi$
$191$	−3609.38	−	2083.88i	−1.36736	−	0.789445i	−0.990590	+	0.136862i	$0.956298\pi$
$192$		0			0
$193$	2392.97	+	4144.74i	0.892485	+	1.54583i	0.836887	+	0.547376i	$0.184373\pi$
$193$	2392.97	+	4144.74i	0.892485	+	1.54583i	0.0555978	+	0.998453i	$0.482294\pi$
$194$		0			0
$195$	−4570.39			−1.67842
$196$		0			0
$197$	377.368			0.136479			0.0682395	−	0.997669i	$-0.478262\pi$
$197$	377.368			0.136479			0.0682395	+	0.997669i	$0.478262\pi$
$198$		0			0
$199$	−809.764	−	1402.55i	−0.288455	−	0.499620i	0.684986	−	0.728556i	$-0.259808\pi$
$199$	−809.764	−	1402.55i	−0.288455	−	0.499620i	−0.973441	+	0.228937i	$0.926475\pi$
$200$		0			0
$201$	3110.55	+	1795.88i	1.09155	+	0.630206i
$202$		0			0
$203$	−1305.63	+	2382.71i	−0.451415	+	0.823810i
$204$		0			0
$205$	−454.519	+	787.249i	−0.154853	+	0.268214i
$206$		0			0
$207$	391.942	−	226.288i	0.131603	−	0.0759810i
$208$		0			0
$209$		−	3075.83i		−	1.01799i
$210$		0			0
$211$		−	3010.54i		−	0.982247i	−0.871090	−	0.491123i	$-0.836586\pi$
$211$		−	3010.54i		−	0.982247i	0.871090	−	0.491123i	$-0.163414\pi$
$212$		0			0
$213$	−2765.04	+	1596.40i	−0.889472	+	0.513537i
$214$		0			0
$215$	−2882.00	+	4991.77i	−0.914189	+	1.58342i
$216$		0			0
$217$	−788.603	+	17.6069i	−0.246700	+	0.00550798i
$218$		0			0
$219$	3451.33	+	1992.63i	1.06493	+	0.614837i
$220$		0			0
$221$	−3484.32	−	6035.02i	−1.06055	−	1.83692i
$222$		0			0
$223$	−879.692			−0.264164			−0.132082	−	0.991239i	$-0.542166\pi$
$223$	−879.692			−0.264164			−0.132082	+	0.991239i	$0.542166\pi$
$224$		0			0
$225$	683.948			0.202651
$226$		0			0
$227$	−712.315	−	1233.77i	−0.208273	−	0.360740i	0.742897	−	0.669405i	$-0.233451\pi$
$227$	−712.315	−	1233.77i	−0.208273	−	0.360740i	−0.951171	+	0.308665i	$0.900118\pi$
$228$		0			0
$229$	−206.746	−	119.365i	−0.0596602	−	0.0344448i	0.469873	−	0.882734i	$-0.344300\pi$
$229$	−206.746	−	119.365i	−0.0596602	−	0.0344448i	−0.529533	+	0.848289i	$0.677633\pi$
$230$		0			0
$231$	3075.03	−	1868.10i	0.875852	−	0.532087i
$232$		0			0
$233$	−2920.86	+	5059.08i	−0.821253	+	1.42245i	0.0834970	+	0.996508i	$0.473391\pi$
$233$	−2920.86	+	5059.08i	−0.821253	+	1.42245i	−0.904750	+	0.425943i	$0.859942\pi$
$234$		0			0
$235$	−60.3966	+	34.8700i	−0.0167653	+	0.00967943i
$236$		0			0
$237$		−	5274.80i		−	1.44572i
$238$		0			0
$239$		−	2101.91i		−	0.568876i	−0.958694	−	0.284438i	$-0.908193\pi$
$239$		−	2101.91i		−	0.568876i	0.958694	−	0.284438i	$-0.0918070\pi$
$240$		0			0
$241$	1883.76	−	1087.59i	0.503501	−	0.290697i	−0.226657	−	0.973975i	$-0.572780\pi$
$241$	1883.76	−	1087.59i	0.503501	−	0.290697i	0.730158	+	0.683278i	$0.239446\pi$
$242$		0			0
$243$	592.161	−	1025.65i	0.156326	−	0.270764i
$244$		0			0
$245$	258.636	+	5789.20i	0.0674435	+	1.50963i
$246$		0			0
$247$	3709.16	+	2141.49i	0.955499	+	0.551658i
$248$		0			0
$249$	−1318.22	−	2283.22i	−0.335497	−	0.581097i
$250$		0			0
$251$	−2069.98			−0.520542			−0.260271	−	0.965536i	$-0.583812\pi$
$251$	−2069.98			−0.520542			−0.260271	+	0.965536i	$0.583812\pi$
$252$		0			0
$253$	−4325.41			−1.07485
$254$		0			0
$255$	−4947.81	−	8569.87i	−1.21508	−	2.10457i
$256$		0			0
$257$	4498.76	+	2597.36i	1.09192	+	0.630423i	0.934088	−	0.357042i	$-0.116215\pi$
$257$	4498.76	+	2597.36i	1.09192	+	0.630423i	0.157836	+	0.987465i	$0.449548\pi$
$258$		0			0
$259$	778.057	+	1280.74i	0.186665	+	0.307263i
$260$		0			0
$261$	312.694	−	541.602i	0.0741581	−	0.128446i
$262$		0			0
$263$	1580.24	−	912.352i	0.370501	−	0.213909i	−0.303176	−	0.952934i	$-0.598047\pi$
$263$	1580.24	−	912.352i	0.370501	−	0.213909i	0.673677	+	0.739026i	$0.264714\pi$
$264$		0			0
$265$		−	4725.48i		−	1.09541i
$266$		0			0
$267$			583.975i			0.133853i
$268$		0			0
$269$	−2162.50	+	1248.52i	−0.490148	+	0.282987i	−0.724636	−	0.689132i	$-0.757992\pi$
$269$	−2162.50	+	1248.52i	−0.490148	+	0.282987i	0.234488	+	0.972119i	$0.424659\pi$
$270$		0			0
$271$	961.292	−	1665.01i	0.215477	−	0.373217i	−0.737943	−	0.674863i	$-0.764203\pi$
$271$	961.292	−	1665.01i	0.215477	−	0.373217i	0.953420	+	0.301646i	$0.0975360\pi$
$272$		0			0
$273$	111.830	+	5008.82i	0.0247922	+	1.11043i
$274$		0			0
$275$	−5660.97	−	3268.36i	−1.24134	−	0.716689i
$276$		0			0
$277$	−942.717	−	1632.83i	−0.204485	−	0.354179i	0.745483	−	0.666524i	$-0.232219\pi$
$277$	−942.717	−	1632.83i	−0.204485	−	0.354179i	−0.949969	+	0.312345i	$0.898885\pi$
$278$		0			0
$279$	181.564			0.0389604
$280$		0			0
$281$	−2666.13			−0.566006			−0.283003	−	0.959119i	$-0.591331\pi$
$281$	−2666.13			−0.566006			−0.283003	+	0.959119i	$0.591331\pi$
$282$		0			0
$283$	1873.94	+	3245.75i	0.393618	+	0.681766i	0.992924	−	0.118754i	$-0.0378899\pi$
$283$	1873.94	+	3245.75i	0.393618	+	0.681766i	−0.599306	+	0.800520i	$0.704557\pi$
$284$		0			0
$285$	5267.09	+	3040.96i	1.09472	+	0.632038i
$286$		0			0
$287$	873.889	+	478.856i	0.179735	+	0.0984878i
$288$		0			0
$289$	5087.61	−	8811.99i	1.03554	−	1.79361i
$290$		0			0
$291$	1202.52	−	694.276i	0.242244	−	0.139860i
$292$		0			0
$293$		−	6486.41i		−	1.29331i	−0.762782	−	0.646655i	$-0.776167\pi$
$293$		−	6486.41i		−	1.29331i	0.762782	−	0.646655i	$-0.223833\pi$
$294$		0			0
$295$			2953.76i			0.582965i
$296$		0			0
$297$	−5259.87	+	3036.78i	−1.02764	+	0.593307i
$298$		0			0
$299$	3011.48	−	5216.03i	0.582469	−	1.00887i
$300$		0			0
$301$	5541.13	+	3036.32i	1.06108	+	0.581430i
$302$		0			0
$303$	5570.86	+	3216.34i	1.05623	+	0.609814i
$304$		0			0
$305$	−3945.71	−	6834.17i	−0.740756	−	1.28303i
$306$		0			0
$307$	8884.08			1.65160			0.825800	−	0.563963i	$-0.190724\pi$
$307$	8884.08			1.65160			0.825800	+	0.563963i	$0.190724\pi$
$308$		0			0
$309$	9432.11			1.73648
$310$		0			0
$311$	3500.88	+	6063.71i	0.638318	+	1.10560i	0.985802	+	0.167913i	$0.0537029\pi$
$311$	3500.88	+	6063.71i	0.638318	+	1.10560i	−0.347484	+	0.937686i	$0.612964\pi$
$312$		0			0
$313$	4140.54	+	2390.54i	0.747722	+	0.431697i	0.824870	−	0.565322i	$-0.191248\pi$
$313$	4140.54	+	2390.54i	0.747722	+	0.431697i	−0.0771483	+	0.997020i	$0.524581\pi$
$314$		0			0
$315$	−29.7735	−	1333.54i	−0.00532555	−	0.238529i
$316$		0			0
$317$	−1801.72	+	3120.66i	−0.319226	+	0.552915i	−0.980327	−	0.197382i	$-0.936756\pi$
$317$	−1801.72	+	3120.66i	−0.319226	+	0.552915i	0.661101	+	0.750297i	$0.270089\pi$
$318$		0			0
$319$	−5176.27	+	2988.52i	−0.908512	+	0.524530i
$320$		0			0
$321$			4035.52i			0.701685i
$322$		0			0
$323$			9273.31i			1.59746i
$324$		0			0
$325$	7882.66	−	4551.06i	1.34539	−	0.776760i
$326$		0			0
$327$	−2210.31	+	3828.37i	−0.373793	+	0.647428i
$328$		0			0
$329$	39.6927	+	65.3370i	0.00665146	+	0.0109488i
$330$		0			0
$331$	−2861.42	−	1652.04i	−0.475159	−	0.274333i	0.243238	−	0.969967i	$-0.421790\pi$
$331$	−2861.42	−	1652.04i	−0.475159	−	0.274333i	−0.718397	+	0.695633i	$0.755124\pi$
$332$		0			0
$333$	−172.467	−	298.721i	−0.0283817	−	0.0491586i
$334$		0			0
$335$	−12726.2			−2.07554
$336$		0			0
$337$	−6650.35			−1.07498			−0.537489	−	0.843271i	$-0.680627\pi$
$337$	−6650.35			−1.07498			−0.537489	+	0.843271i	$0.680627\pi$
$338$		0			0
$339$	−1428.24	−	2473.79i	−0.228825	−	0.396336i
$340$		0			0
$341$	−1502.79	−	867.634i	−0.238652	−	0.137786i
$342$		0			0
$343$	6338.21	−	425.098i	0.997758	−	0.0669188i
$344$		0			0
$345$	4276.37	−	7406.88i	0.667339	−	1.15586i
$346$		0			0
$347$	203.832	−	117.683i	0.0315339	−	0.0182061i	−0.484150	−	0.874985i	$-0.660871\pi$
$347$	203.832	−	117.683i	0.0315339	−	0.0182061i	0.515684	+	0.856779i	$0.327538\pi$
$348$		0			0
$349$			2984.02i			0.457681i	0.973464	+	0.228841i	$0.0734935\pi$
$349$			2984.02i			0.457681i	−0.973464	+	0.228841i	$0.926507\pi$
$350$		0			0
$351$		−	8457.19i		−	1.28607i
$352$		0			0
$353$	−2118.54	+	1223.14i	−0.319429	+	0.184422i	−0.651138	−	0.758959i	$-0.725708\pi$
$353$	−2118.54	+	1223.14i	−0.319429	+	0.184422i	0.331709	+	0.943382i	$0.392375\pi$
$354$		0			0
$355$	5656.29	−	9796.98i	0.845647	−	1.46470i
$356$		0			0
$357$	−9270.88	+	5632.13i	−1.37442	+	0.834969i
$358$		0			0
$359$	4931.22	+	2847.04i	0.724957	+	0.418554i	0.816574	−	0.577240i	$-0.195870\pi$
$359$	4931.22	+	2847.04i	0.724957	+	0.418554i	−0.0916173	+	0.995794i	$0.529204\pi$
$360$		0			0
$361$	579.782	+	1004.21i	0.0845286	+	0.146408i
$362$		0			0
$363$	1568.53			0.226794
$364$		0			0
$365$	−14120.4			−2.02492
$366$		0			0
$367$	−168.161	−	291.263i	−0.0239180	−	0.0414272i	0.853819	−	0.520571i	$-0.174281\pi$
$367$	−168.161	−	291.263i	−0.0239180	−	0.0414272i	−0.877737	+	0.479143i	$0.840947\pi$
$368$		0			0
$369$	−198.639	−	114.684i	−0.0280237	−	0.0161795i
$370$		0			0
$371$	−5178.78	+	115.625i	−0.724714	+	0.0161804i
$372$		0			0
$373$	4908.42	−	8501.64i	0.681363	−	1.18016i	−0.293202	−	0.956051i	$-0.594721\pi$
$373$	4908.42	−	8501.64i	0.681363	−	1.18016i	0.974565	−	0.224105i	$-0.0719459\pi$
$374$		0			0
$375$	2472.58	−	1427.54i	0.340489	−	0.196582i
$376$		0			0
$377$		−	8322.78i		−	1.13699i
$378$		0			0
$379$			3978.71i			0.539242i	0.962967	+	0.269621i	$0.0868984\pi$
$379$			3978.71i			0.539242i	−0.962967	+	0.269621i	$0.913102\pi$
$380$		0			0
$381$	7573.09	−	4372.33i	1.01832	−	0.587929i
$382$		0			0
$383$	1862.99	−	3226.80i	0.248550	−	0.430501i	−0.714574	−	0.699560i	$-0.753379\pi$
$383$	1862.99	−	3226.80i	0.248550	−	0.430501i	0.963124	+	0.269059i	$0.0867127\pi$
$384$		0			0
$385$	−6126.12	+	11179.9i	−0.810950	+	1.47994i
$386$		0			0
$387$	−1259.53	−	727.188i	−0.165440	−	0.0955169i
$388$		0			0
$389$	6346.89	+	10993.1i	0.827250	+	1.43284i	0.900187	+	0.435503i	$0.143429\pi$
$389$	6346.89	+	10993.1i	0.827250	+	1.43284i	−0.0729372	+	0.997337i	$0.523237\pi$
$390$		0			0
$391$	13040.6			1.68669
$392$		0			0
$393$	−6576.24			−0.844091
$394$		0			0
$395$	9344.72	+	16185.5i	1.19034	+	2.06173i
$396$		0			0
$397$	−1794.63	−	1036.13i	−0.226877	−	0.130987i	0.382254	−	0.924057i	$-0.375148\pi$
$397$	−1794.63	−	1036.13i	−0.226877	−	0.130987i	−0.609130	+	0.793070i	$0.708481\pi$
$398$		0			0
$399$	3203.79	−	5846.76i	0.401980	−	0.733594i
$400$		0			0
$401$	4928.69	−	8536.75i	0.613784	−	1.06310i	−0.376813	−	0.926289i	$-0.622980\pi$
$401$	4928.69	−	8536.75i	0.613784	−	1.06310i	0.990597	−	0.136815i	$-0.0436866\pi$
$402$		0			0
$403$	2092.57	−	1208.14i	0.258656	−	0.149335i
$404$		0			0
$405$		−	10064.8i		−	1.23487i
$406$		0			0
$407$			3296.64i			0.401495i
$408$		0			0
$409$	1190.33	−	687.236i	0.143907	−	0.0830847i	−0.426318	−	0.904573i	$-0.640189\pi$
$409$	1190.33	−	687.236i	0.143907	−	0.0830847i	0.570225	+	0.821489i	$0.306856\pi$
$410$		0			0
$411$	3512.90	−	6084.52i	0.421603	−	0.730237i
$412$		0			0
$413$	3237.11	−	72.2738i	0.385684	−	0.00861104i
$414$		0			0
$415$	8089.81	+	4670.66i	0.956899	+	0.552466i
$416$		0			0
$417$	4601.14	+	7969.40i	0.540332	+	0.935883i
$418$		0			0
$419$	11066.7			1.29032			0.645158	−	0.764049i	$-0.276791\pi$
$419$	11066.7			1.29032			0.645158	+	0.764049i	$0.276791\pi$
$420$		0			0
$421$	−15288.1			−1.76982			−0.884910	−	0.465761i	$-0.845781\pi$
$421$	−15288.1			−1.76982			−0.884910	+	0.465761i	$0.845781\pi$
$422$		0			0
$423$	−8.79842	−	15.2393i	−0.00101133	−	0.00175168i
$424$		0			0
$425$	17067.2	+	9853.76i	1.94796	+	1.12465i
$426$		0			0
$427$	−7393.20	+	4491.43i	−0.837897	+	0.509029i
$428$		0			0
$429$	−5510.78	+	9544.95i	−0.620194	+	1.07421i
$430$		0			0
$431$	2265.08	−	1307.74i	0.253143	−	0.146152i	−0.368059	−	0.929802i	$-0.619978\pi$
$431$	2265.08	−	1307.74i	0.253143	−	0.146152i	0.621203	+	0.783650i	$0.286644\pi$
$432$		0			0
$433$			3152.43i			0.349875i	0.984580	+	0.174938i	$0.0559723\pi$
$433$			3152.43i			0.349875i	−0.984580	+	0.174938i	$0.944028\pi$
$434$		0			0
$435$		−	11818.5i		−	1.30266i
$436$		0			0
$437$	−6941.08	+	4007.43i	−0.759810	+	0.438676i
$438$		0			0
$439$	3470.23	−	6010.62i	0.377278	−	0.653465i	−0.613387	−	0.789783i	$-0.710193\pi$
$439$	3470.23	−	6010.62i	0.377278	−	0.653465i	0.990665	+	0.136317i	$0.0435267\pi$
$440$		0			0
$441$	−1460.74	+	65.2592i	−0.157730	+	0.00704667i
$442$		0			0
$443$	10854.6	+	6266.93i	1.16415	+	0.672123i	0.952295	−	0.305177i	$-0.0987158\pi$
$443$	10854.6	+	6266.93i	1.16415	+	0.672123i	0.211856	+	0.977301i	$0.432049\pi$
$444$		0			0
$445$	−1034.56	−	1791.91i	−0.110208	−	0.190887i
$446$		0			0
$447$	10999.8			1.16392
$448$		0			0
$449$	8637.25			0.907833			0.453917	−	0.891044i	$-0.350026\pi$
$449$	8637.25			0.907833			0.453917	+	0.891044i	$0.350026\pi$
$450$		0			0
$451$	1096.08	+	1898.46i	0.114440	+	0.198215i
$452$		0			0
$453$	−4233.84	−	2444.41i	−0.439124	−	0.253528i
$454$		0			0
$455$	−9216.66	−	15171.3i	−0.949634	−	1.56316i
$456$		0			0
$457$	−8248.34	+	14286.5i	−0.844291	+	1.46236i	0.0419444	+	0.999120i	$0.486645\pi$
$457$	−8248.34	+	14286.5i	−0.844291	+	1.46236i	−0.886235	+	0.463235i	$0.846689\pi$
$458$		0			0
$459$	15857.9	−	9155.58i	1.61260	−	0.931037i
$460$		0			0
$461$		−	6423.89i		−	0.649003i	−0.945885	−	0.324501i	$-0.894803\pi$
$461$		−	6423.89i		−	0.649003i	0.945885	−	0.324501i	$-0.105197\pi$
$462$		0			0
$463$		−	3390.22i		−	0.340295i	−0.985419	−	0.170148i	$-0.945576\pi$
$463$		−	3390.22i		−	0.340295i	0.985419	−	0.170148i	$-0.0544245\pi$
$464$		0			0
$465$	2971.49	−	1715.59i	0.296343	−	0.171094i
$466$		0			0
$467$	−580.262	+	1005.04i	−0.0574975	+	0.0995886i	−0.893341	−	0.449379i	$-0.851645\pi$
$467$	−580.262	+	1005.04i	−0.0574975	+	0.0995886i	0.835844	+	0.548967i	$0.184979\pi$
$468$		0			0
$469$	311.388	+	13946.9i	0.0306580	+	1.37315i
$470$		0			0
$471$	−318.032	−	183.616i	−0.0311128	−	0.0179630i
$472$		0			0
$473$	6949.97	+	12037.7i	0.675603	+	1.17018i
$474$		0			0
$475$	−12112.4			−1.17001
$476$		0			0
$477$	1192.34			0.114451
$478$		0			0
$479$	2042.46	+	3537.65i	0.194828	+	0.337452i	0.946844	−	0.321693i	$-0.104252\pi$
$479$	2042.46	+	3537.65i	0.194828	+	0.337452i	−0.752016	+	0.659145i	$0.770919\pi$
$480$		0			0
$481$	−3975.43	−	2295.22i	−0.376849	−	0.217574i
$482$		0			0
$483$	−8222.03	−	4505.35i	−0.774566	−	0.424431i
$484$		0			0
$485$	−2459.93	+	4260.72i	−0.230309	+	0.398906i
$486$		0			0
$487$	−6340.08	+	3660.44i	−0.589931	+	0.340597i	−0.765070	−	0.643947i	$-0.777296\pi$
$487$	−6340.08	+	3660.44i	−0.589931	+	0.340597i	0.175139	+	0.984544i	$0.443962\pi$
$488$		0			0
$489$		−	18537.1i		−	1.71427i
$490$		0			0
$491$		−	32.0507i		−	0.00294588i	−0.999999	−	0.00147294i	$-0.999531\pi$
$491$		−	32.0507i		−	0.00294588i	0.999999	−	0.00147294i	$-0.000468851\pi$
$492$		0			0
$493$	15605.9	−	9010.07i	1.42567	−	0.823110i
$494$		0			0
$495$	1467.19	−	2541.24i	0.133222	−	0.230748i
$496$		0			0
$497$	−10875.2	−	5959.16i	−0.981525	−	0.537837i
$498$		0			0
$499$	6513.95	+	3760.83i	0.584378	+	0.337391i	0.762871	−	0.646550i	$-0.223789\pi$
$499$	6513.95	+	3760.83i	0.584378	+	0.337391i	−0.178493	+	0.983941i	$0.557122\pi$
$500$		0			0
$501$	281.006	+	486.717i	0.0250588	+	0.0434031i
$502$		0			0
$503$	−4599.64			−0.407729			−0.203865	−	0.978999i	$-0.565350\pi$
$503$	−4599.64			−0.407729			−0.203865	+	0.978999i	$0.565350\pi$
$504$		0			0
$505$	−22792.0			−2.00838
$506$		0			0
$507$	−2435.51	−	4218.43i	−0.213343	−	0.369521i
$508$		0			0
$509$	18006.2	+	10395.9i	1.56800	+	0.905286i	0.996402	+	0.0847525i	$0.0270100\pi$
$509$	18006.2	+	10395.9i	1.56800	+	0.905286i	0.571599	+	0.820533i	$0.306323\pi$
$510$		0			0
$511$	345.503	+	15474.9i	0.0299103	+	1.33967i
$512$		0			0
$513$	−5627.08	+	9746.39i	−0.484292	+	0.838818i
$514$		0			0
$515$	−28942.1	+	16709.7i	−2.47639	+	1.42974i
$516$		0			0
$517$			168.179i			0.0143066i
$518$		0			0
$519$			13657.5i			1.15511i
$520$		0			0
$521$	16707.9	−	9646.31i	1.40496	−	0.811156i	0.410067	−	0.912055i	$-0.365505\pi$
$521$	16707.9	−	9646.31i	1.40496	−	0.811156i	0.994897	+	0.100899i	$0.0321718\pi$
$522$		0			0
$523$	8344.02	−	14452.3i	0.697626	−	1.20832i	−0.271661	−	0.962393i	$-0.587573\pi$
$523$	8344.02	−	14452.3i	0.697626	−	1.20832i	0.969287	−	0.245931i	$-0.0790936\pi$
$524$		0			0
$525$	−7356.43	−	12109.2i	−0.611544	−	1.00664i
$526$		0			0
$527$	4530.74	+	2615.82i	0.374501	+	0.216218i
$528$		0			0
$529$	−448.024	−	776.000i	−0.0368229	−	0.0637791i
$530$		0			0
$531$	−745.296			−0.0609098
$532$		0			0
$533$	−3052.48			−0.248063
$534$		0			0
$535$	−7149.24	−	12382.9i	−0.577736	−	1.00067i
$536$		0			0
$537$	−5702.60	−	3292.40i	−0.458260	−	0.264576i
$538$		0			0
$539$	12402.2	+	6440.22i	0.991096	+	0.514657i
$540$		0			0
$541$	1525.67	−	2642.54i	0.121245	−	0.210003i	−0.799014	−	0.601313i	$-0.794645\pi$
$541$	1525.67	−	2642.54i	0.121245	−	0.210003i	0.920259	+	0.391310i	$0.127978\pi$
$542$		0			0
$543$	−845.794	+	488.320i	−0.0668444	+	0.0385926i
$544$		0			0
$545$		−	15662.9i		−	1.23106i
$546$		0			0
$547$			12022.3i			0.939740i	0.882736	+	0.469870i	$0.155699\pi$
$547$			12022.3i			0.939740i	−0.882736	+	0.469870i	$0.844301\pi$
$548$		0			0
$549$	1724.40	−	995.584i	0.134054	−	0.0773962i
$550$		0			0
$551$	−5537.65	+	9591.48i	−0.428152	+	0.741581i
$552$		0			0
$553$	17509.5	−	10637.2i	1.34644	−	0.817971i
$554$		0			0
$555$	−5645.21	−	3259.26i	−0.431758	−	0.249276i
$556$		0			0
$557$	5913.89	+	10243.2i	0.449873	+	0.779204i	0.998377	−	0.0569444i	$-0.0181358\pi$
$557$	5913.89	+	10243.2i	0.449873	+	0.779204i	−0.548504	+	0.836148i	$0.684802\pi$
$558$		0			0
$559$	−19355.1			−1.46446
$560$		0			0
$561$	−23863.4			−1.79593
$562$		0			0
$563$	−6841.32	−	11849.5i	−0.512127	−	0.887030i	−0.999901	−	0.0140600i	$-0.995524\pi$
$563$	−6841.32	−	11849.5i	−0.512127	−	0.887030i	0.487774	−	0.872970i	$-0.337809\pi$
$564$		0			0
$565$	8765.03	+	5060.49i	0.652650	+	0.376808i
$566$		0			0
$567$	−11030.3	+	246.269i	−0.816981	+	0.0182405i
$568$		0			0
$569$	−816.737	+	1414.63i	−0.0601747	+	0.104226i	−0.894543	−	0.446981i	$-0.852499\pi$
$569$	−816.737	+	1414.63i	−0.0601747	+	0.104226i	0.834369	+	0.551207i	$0.185832\pi$
$570$		0			0
$571$	−12962.1	+	7483.67i	−0.949995	+	0.548480i	−0.893079	−	0.449899i	$-0.851460\pi$
$571$	−12962.1	+	7483.67i	−0.949995	+	0.548480i	−0.0569155	+	0.998379i	$0.518127\pi$
$572$		0			0
$573$		−	19873.3i		−	1.44890i
$574$		0			0
$575$			17033.1i			1.23535i
$576$		0			0
$577$	11197.4	−	6464.81i	0.807891	−	0.466436i	−0.0383317	−	0.999265i	$-0.512204\pi$
$577$	11197.4	−	6464.81i	0.807891	−	0.466436i	0.846223	+	0.532829i	$0.178871\pi$
$578$		0			0
$579$	−11410.5	+	19763.5i	−0.819004	+	1.41856i
$580$		0			0
$581$	4920.75	−	8980.12i	0.351372	−	0.641236i
$582$		0			0
$583$	−9868.84	−	5697.78i	−0.701073	−	0.404765i
$584$		0			0
$585$	2042.99	+	3538.57i	0.144389	+	0.250089i
$586$		0			0
$587$	−7486.36			−0.526397			−0.263199	−	0.964742i	$-0.584777\pi$
$587$	−7486.36			−0.526397			−0.263199	+	0.964742i	$0.584777\pi$
$588$		0			0
$589$	−3215.40			−0.224938
$590$		0			0
$591$	899.710	+	1558.34i	0.0626212	+	0.108463i
$592$		0			0
$593$	2294.33	+	1324.63i	0.158882	+	0.0917304i	0.577333	−	0.816509i	$-0.304094\pi$
$593$	2294.33	+	1324.63i	0.158882	+	0.0917304i	−0.418451	+	0.908239i	$0.637427\pi$
$594$		0			0
$595$	18469.6	−	33706.1i	1.27257	−	2.32238i
$596$		0			0
$597$	3861.23	−	6687.84i	0.264706	−	0.458484i
$598$		0			0
$599$	−15164.6	+	8755.28i	−1.03440	+	0.597213i	−0.918243	−	0.396017i	$-0.870392\pi$
$599$	−15164.6	+	8755.28i	−1.03440	+	0.597213i	−0.116161	+	0.993230i	$0.537059\pi$
$600$		0			0
$601$			12124.7i			0.822922i	0.911427	+	0.411461i	$0.134981\pi$
$601$			12124.7i			0.822922i	−0.911427	+	0.411461i	$0.865019\pi$
$602$		0			0
$603$		−	3211.07i		−	0.216857i
$604$		0			0
$605$	−4812.96	+	2778.77i	−0.323429	+	0.186732i
$606$		0			0
$607$	5706.27	−	9883.55i	0.381566	−	0.660891i	−0.609721	−	0.792616i	$-0.708718\pi$
$607$	5706.27	−	9883.55i	0.381566	−	0.660891i	0.991286	+	0.131725i	$0.0420517\pi$
$608$		0			0
$609$	−12952.2	+	289.180i	−0.861825	+	0.0192417i
$610$		0			0
$611$	−202.807	−	117.091i	−0.0134283	−	0.00775285i
$612$		0			0
$613$	3379.38	+	5853.26i	0.222662	+	0.385662i	0.955615	−	0.294617i	$-0.0951920\pi$
$613$	3379.38	+	5853.26i	0.222662	+	0.385662i	−0.732953	+	0.680279i	$0.761859\pi$
$614$		0			0
$615$	−4334.60			−0.284208
$616$		0			0
$617$	−9991.14			−0.651909			−0.325955	−	0.945385i	$-0.605686\pi$
$617$	−9991.14			−0.651909			−0.325955	+	0.945385i	$0.605686\pi$
$618$		0			0
$619$	−10332.1	−	17895.6i	−0.670889	−	1.16201i	−0.977652	−	0.210228i	$-0.932579\pi$
$619$	−10332.1	−	17895.6i	−0.670889	−	1.16201i	0.306764	−	0.951786i	$-0.400754\pi$
$620$		0			0
$621$	13705.9	+	7913.11i	0.885667	+	0.511340i
$622$		0			0
$623$	−1938.48	+	1177.64i	−0.124661	+	0.0757325i
$624$		0			0
$625$	4969.49	−	8607.41i	0.318047	−	0.550874i
$626$		0			0
$627$	12701.7	−	7333.31i	0.809020	−	0.467088i
$628$		0			0
$629$		−	9939.02i		−	0.630039i
$630$		0			0
$631$		−	28295.8i		−	1.78516i	−0.450887	−	0.892581i	$-0.648892\pi$
$631$		−	28295.8i		−	1.78516i	0.450887	−	0.892581i	$-0.351108\pi$
$632$		0			0
$633$	12432.0	−	7177.63i	0.780614	−	0.450688i
$634$		0			0
$635$	−15491.8	+	26832.6i	−0.968149	+	1.67688i
$636$		0			0
$637$	−16401.1	+	10472.0i	−1.02015	+	0.651359i
$638$		0			0
$639$	2471.98	+	1427.20i	0.153036	+	0.0883554i
$640$		0			0
$641$	8207.85	+	14216.4i	0.505758	+	0.875998i	0.999978	+	0.00666106i	$0.00212030\pi$
$641$	8207.85	+	14216.4i	0.505758	+	0.875998i	−0.494220	+	0.869337i	$0.664546\pi$
$642$		0			0
$643$	−14139.9			−0.867220			−0.433610	−	0.901101i	$-0.642760\pi$
$643$	−14139.9			−0.867220			−0.433610	+	0.901101i	$0.642760\pi$
$644$		0			0
$645$	−27484.7			−1.67784
$646$		0			0
$647$	10974.5	+	19008.5i	0.666853	+	1.15502i	0.978779	+	0.204917i	$0.0656925\pi$
$647$	10974.5	+	19008.5i	0.666853	+	1.15502i	−0.311926	+	0.950106i	$0.600974\pi$
$648$		0			0
$649$	6168.73	+	3561.52i	0.373103	+	0.215411i
$650$		0			0
$651$	−1952.87	−	3214.56i	−0.117571	−	0.193531i
$652$		0			0
$653$	2938.88	−	5090.29i	0.176121	−	0.305051i	−0.764427	−	0.644710i	$-0.776978\pi$
$653$	2938.88	−	5090.29i	0.176121	−	0.305051i	0.940549	+	0.339659i	$0.110312\pi$
$654$		0			0
$655$	20179.0	−	11650.3i	1.20375	−	0.694987i
$656$		0			0
$657$		−	3562.87i		−	0.211569i
$658$		0			0
$659$			11002.0i			0.650344i	0.945655	+	0.325172i	$0.105422\pi$
$659$			11002.0i			0.650344i	−0.945655	+	0.325172i	$0.894578\pi$
$660$		0			0
$661$	−3196.72	+	1845.63i	−0.188106	+	0.108603i	−0.591095	−	0.806602i	$-0.701304\pi$
$661$	−3196.72	+	1845.63i	−0.188106	+	0.108603i	0.402990	+	0.915205i	$0.367971\pi$
$662$		0			0
$663$	16614.4	−	28777.0i	0.973229	−	1.68568i
$664$		0			0
$665$	527.274	+	23616.3i	0.0307471	+	1.37715i
$666$		0			0
$667$	13488.1	+	7787.35i	0.782999	+	0.452065i
$668$		0			0
$669$	−2097.33	−	3632.69i	−0.121207	−	0.209937i
$670$		0			0
$671$	−19030.2			−1.09487
$672$		0			0
$673$	6819.19			0.390580			0.195290	−	0.980746i	$-0.437435\pi$
$673$	6819.19			0.390580			0.195290	+	0.980746i	$0.437435\pi$
$674$		0			0
$675$	11958.6	+	20712.9i	0.681906	+	1.18110i
$676$		0			0
$677$	−13683.4	−	7900.10i	−0.776802	−	0.448487i	0.0584939	−	0.998288i	$-0.481370\pi$
$677$	−13683.4	−	7900.10i	−0.776802	−	0.448487i	−0.835296	+	0.549801i	$0.814704\pi$
$678$		0			0
$679$	4729.63	+	2591.65i	0.267314	+	0.146478i
$680$		0			0
$681$	3396.56	−	5883.02i	0.191126	−	0.331039i
$682$		0			0
$683$	−18649.6	+	10767.3i	−1.04481	+	0.603222i	−0.921192	−	0.389108i	$-0.872783\pi$
$683$	−18649.6	+	10767.3i	−1.04481	+	0.603222i	−0.123619	+	0.992330i	$0.539450\pi$
$684$		0			0
$685$			24893.5i			1.38851i
$686$		0			0
$687$		−	1138.35i		−	0.0632177i
$688$		0			0
$689$	13742.0	−	7933.92i	0.759835	−	0.438691i
$690$		0			0
$691$	2019.27	−	3497.48i	0.111167	−	0.192548i	−0.805074	−	0.593175i	$-0.797874\pi$
$691$	2019.27	−	3497.48i	0.111167	−	0.192548i	0.916241	+	0.400627i	$0.131208\pi$
$692$		0			0
$693$	−2820.91	−	1545.75i	−0.154628	−	0.0847302i
$694$		0			0
$695$	−28236.8	−	16302.6i	−1.54113	−	0.889771i
$696$		0			0
$697$	−3304.56	−	5723.66i	−0.179582	−	0.311046i
$698$		0			0
$699$	−27855.3			−1.50727
$700$		0			0
$701$	−11823.9			−0.637066			−0.318533	−	0.947912i	$-0.603190\pi$
$701$	−11823.9			−0.637066			−0.318533	+	0.947912i	$0.603190\pi$
$702$		0			0
$703$	3054.29	+	5290.19i	0.163862	+	0.283817i
$704$		0			0
$705$	−287.991	−	166.272i	−0.0153849	−	0.00888249i
$706$		0			0
$707$	557.683	+	24978.3i	0.0296659	+	1.32872i
$708$		0			0
$709$	13143.3	−	22764.8i	0.696200	−	1.20585i	−0.273575	−	0.961851i	$-0.588206\pi$
$709$	13143.3	−	22764.8i	0.696200	−	1.20585i	0.969775	−	0.244003i	$-0.0784606\pi$
$710$		0			0
$711$	−4083.94	+	2357.87i	−0.215415	+	0.124370i
$712$		0			0
$713$			4521.68i			0.237501i
$714$		0			0
$715$		−	39051.1i		−	2.04256i
$716$		0			0
$717$	8679.85	−	5011.32i	0.452099	−	0.261020i
$718$		0			0
$719$	−9614.97	+	16653.6i	−0.498717	+	0.863804i	−0.999999	−	0.00148052i	$-0.999529\pi$
$719$	−9614.97	+	16653.6i	−0.498717	+	0.863804i	0.501282	+	0.865284i	$0.332862\pi$
$720$		0			0
$721$	19020.8	+	31309.5i	0.982484	+	1.61724i
$722$		0			0
$723$	8982.42	+	5186.00i	0.462047	+	0.266763i
$724$		0			0
$725$	11768.5	+	20383.7i	0.602858	+	1.04418i
$726$		0			0
$727$	33771.2			1.72284			0.861419	−	0.507895i	$-0.169576\pi$
$727$	33771.2			1.72284			0.861419	+	0.507895i	$0.169576\pi$
$728$		0			0
$729$	21731.9			1.10409
$730$		0			0
$731$	−20953.4	−	36292.4i	−1.06018	−	1.83628i
$732$		0			0
$733$	−1835.93	−	1059.97i	−0.0925125	−	0.0534121i	0.453030	−	0.891495i	$-0.350343\pi$
$733$	−1835.93	−	1059.97i	−0.0925125	−	0.0534121i	−0.545543	+	0.838083i	$0.683676\pi$
$734$		0			0
$735$	−23289.9	+	14870.5i	−1.16879	+	0.746266i
$736$		0			0
$737$	−15344.6	+	26577.7i	−0.766930	+	1.32836i
$738$		0			0
$739$	9238.36	−	5333.77i	0.459863	−	0.265502i	−0.252124	−	0.967695i	$-0.581129\pi$
$739$	9238.36	−	5333.77i	0.459863	−	0.265502i	0.711987	+	0.702193i	$0.247796\pi$
$740$		0			0
$741$			20422.7i			1.01248i
$742$		0			0
$743$			26684.2i			1.31756i	0.752335	+	0.658781i	$0.228928\pi$
$743$			26684.2i			1.31756i	−0.752335	+	0.658781i	$0.771072\pi$
$744$		0			0
$745$	−33752.4	+	19487.0i	−1.65986	+	0.958318i
$746$		0			0
$747$	−1178.50	+	2041.23i	−0.0577231	+	0.0999794i
$748$		0			0
$749$	−13395.8	+	8138.03i	−0.653499	+	0.397006i
$750$		0			0
$751$	−30415.3	−	17560.3i	−1.47785	−	0.853240i	−0.478168	−	0.878268i	$-0.658699\pi$
$751$	−30415.3	−	17560.3i	−1.47785	−	0.853240i	−0.999687	+	0.0250284i	$0.992032\pi$
$752$		0			0
$753$	−4935.19	−	8547.99i	−0.238842	−	0.413687i
$754$		0			0
$755$	17321.8			0.834975
$756$		0			0
$757$	−10060.1			−0.483011			−0.241506	−	0.970399i	$-0.577641\pi$
$757$	−10060.1			−0.483011			−0.241506	+	0.970399i	$0.577641\pi$
$758$		0			0
$759$	−10312.5	−	17861.8i	−0.493176	−	0.854205i
$760$		0			0
$761$	−31480.5	−	18175.3i	−1.49956	−	0.865774i	−0.499565	−	0.866276i	$-0.666507\pi$
$761$	−31480.5	−	18175.3i	−1.49956	−	0.865774i	−1.00000		0.000502093i	$0.999840\pi$
$762$		0			0
$763$	−17165.4	+	383.247i	−0.814456	+	0.0181841i
$764$		0			0
$765$	−4423.41	+	7661.56i	−0.209057	+	0.362097i
$766$		0			0
$767$	−8589.70	+	4959.27i	−0.404376	+	0.233466i
$768$		0			0
$769$		−	17664.9i		−	0.828366i	−0.910194	−	0.414183i	$-0.864067\pi$
$769$		−	17664.9i		−	0.828366i	0.910194	−	0.414183i	$-0.135933\pi$
$770$		0			0
$771$			24770.2i			1.15704i
$772$		0			0
$773$	6848.89	−	3954.21i	0.318677	−	0.183988i	−0.332126	−	0.943235i	$-0.607766\pi$
$773$	6848.89	−	3954.21i	0.318677	−	0.183988i	0.650803	+	0.759247i	$0.274432\pi$
$774$		0			0
$775$	−3416.67	+	5917.84i	−0.158362	+	0.274290i
$776$		0			0
$777$	−3433.78	+	6266.48i	−0.158541	+	0.289329i
$778$		0			0
$779$	3517.80	+	2031.00i	0.161795	+	0.0934123i
$780$		0			0
$781$	−13640.2	−	23625.5i	−0.624949	−	1.08244i
$782$		0			0
$783$	21869.4			0.998145
$784$		0			0
$785$	1301.16			0.0591598
$786$		0			0
$787$	5940.56	+	10289.4i	0.269070	+	0.466043i	0.968622	−	0.248539i	$-0.0799503\pi$
$787$	5940.56	+	10289.4i	0.269070	+	0.466043i	−0.699552	+	0.714582i	$0.746617\pi$
$788$		0			0
$789$	7535.12	+	4350.40i	0.339997	+	0.196297i
$790$		0			0
$791$	5331.46	−	9729.65i	0.239652	−	0.437353i
$792$		0			0
$793$	13249.4	−	22948.7i	0.593317	−	1.02766i
$794$		0			0
$795$	19513.9	−	11266.3i	0.870549	−	0.502611i
$796$		0			0
$797$		−	10203.4i		−	0.453480i	−0.973955	−	0.226740i	$-0.927193\pi$
$797$		−	10203.4i		−	0.453480i	0.973955	−	0.226740i	$-0.0728068\pi$
$798$		0			0
$799$		−	507.041i		−	0.0224503i
$800$		0			0
$801$	452.135	−	261.040i	0.0199443	−	0.0115149i
$802$		0			0
$803$	−17025.7	+	29489.5i	−0.748226	+	1.29597i
$804$		0			0
$805$	33210.6	−	741.482i	1.45406	−	0.0324643i
$806$		0			0
$807$	−10311.5	−	5953.36i	−0.449793	−	0.259688i
$808$		0			0
$809$	−10471.1	−	18136.4i	−0.455059	−	0.788185i	0.543633	−	0.839323i	$-0.317048\pi$
$809$	−10471.1	−	18136.4i	−0.455059	−	0.788185i	−0.998692	+	0.0511383i	$0.983715\pi$
$810$		0			0
$811$	−5434.74			−0.235314			−0.117657	−	0.993054i	$-0.537538\pi$
$811$	−5434.74			−0.235314			−0.117657	+	0.993054i	$0.537538\pi$
$812$		0			0
$813$	9167.53			0.395473
$814$		0			0
$815$	32839.9	+	56880.4i	1.41145	+	2.44471i
$816$		0			0
$817$	22305.5	+	12878.1i	0.955168	+	0.551467i
$818$		0			0
$819$	3828.02	−	2325.55i	0.163323	−	0.0992203i
$820$		0			0
$821$	−1066.43	+	1847.11i	−0.0453334	+	0.0785197i	−0.887802	−	0.460226i	$-0.847768\pi$
$821$	−1066.43	+	1847.11i	−0.0453334	+	0.0785197i	0.842468	+	0.538746i	$0.181102\pi$
$822$		0			0
$823$	−16591.2	+	9578.95i	−0.702714	+	0.405712i	−0.808358	−	0.588692i	$-0.799643\pi$
$823$	−16591.2	+	9578.95i	−0.702714	+	0.405712i	0.105643	+	0.994404i	$0.466310\pi$
$824$		0			0
$825$		−	31169.3i		−	1.31536i
$826$		0			0
$827$		−	27208.2i		−	1.14404i	−0.820239	−	0.572021i	$-0.806160\pi$
$827$		−	27208.2i		−	1.14404i	0.820239	−	0.572021i	$-0.193840\pi$
$828$		0			0
$829$	−14900.9	+	8603.02i	−0.624281	+	0.360429i	−0.778534	−	0.627603i	$-0.784036\pi$
$829$	−14900.9	+	8603.02i	−0.624281	+	0.360429i	0.154253	+	0.988031i	$0.450703\pi$
$830$		0			0
$831$	4495.19	−	7785.91i	0.187649	−	0.325018i
$832$		0			0
$833$	−37391.3	−	19416.6i	−1.55526	−	0.807617i
$834$		0			0
$835$	−1724.52	−	995.650i	−0.0714723	−	0.0412645i
$836$		0			0
$837$	3174.58	+	5498.54i	0.131099	+	0.227070i
$838$		0			0
$839$	9526.10			0.391988			0.195994	−	0.980605i	$-0.437207\pi$
$839$	9526.10			0.391988			0.195994	+	0.980605i	$0.437207\pi$
$840$		0			0
$841$	−2867.21			−0.117562
$842$		0			0
$843$	−6356.49	−	11009.8i	−0.259703	−	0.449818i
$844$		0			0
$845$	14946.6	+	8629.41i	0.608495	+	0.351315i
$846$		0			0
$847$	3163.09	+	5206.66i	0.128318	+	0.211220i
$848$		0			0
$849$	−8935.56	+	15476.8i	−0.361210	+	0.625635i
$850$		0			0
$851$	7439.36	−	4295.12i	0.299669	−	0.173014i
$852$		0			0
$853$		−	8447.62i		−	0.339087i	−0.985523	−	0.169543i	$-0.945771\pi$
$853$		−	8447.62i		−	0.339087i	0.985523	−	0.169543i	$-0.0542293\pi$
$854$		0			0
$855$		−	5437.31i		−	0.217488i
$856$		0			0
$857$	−22211.6	+	12823.9i	−0.885337	+	0.511150i	−0.872414	−	0.488767i	$-0.837447\pi$
$857$	−22211.6	+	12823.9i	−0.885337	+	0.511150i	−0.0129226	+	0.999916i	$0.504114\pi$
$858$		0			0
$859$	−3738.66	+	6475.54i	−0.148500	+	0.257209i	−0.930673	−	0.365852i	$-0.880778\pi$
$859$	−3738.66	+	6475.54i	−0.148500	+	0.257209i	0.782173	+	0.623061i	$0.214111\pi$
$860$		0			0
$861$	106.061	+	4750.40i	0.00419807	+	0.188029i
$862$		0			0
$863$	20094.6	+	11601.6i	0.792616	+	0.457617i	0.840883	−	0.541218i	$-0.182036\pi$
$863$	20094.6	+	11601.6i	0.792616	+	0.457617i	−0.0482670	+	0.998834i	$0.515370\pi$
$864$		0			0
$865$	−24195.4	−	41907.7i	−0.951062	−	1.64729i
$866$		0			0
$867$	48518.9			1.90056
$868$		0			0
$869$	45069.8			1.75937
$870$		0			0
$871$	−21366.8	−	37008.3i	−0.831212	−	1.43970i
$872$		0			0
$873$	−1075.07	−	620.691i	−0.0416788	−	0.0240633i
$874$		0			0
$875$	9724.89	+	5328.85i	0.375727	+	0.205883i
$876$		0			0
$877$	6141.61	−	10637.6i	0.236474	−	0.409584i	−0.723226	−	0.690611i	$-0.757342\pi$
$877$	6141.61	−	10637.6i	0.236474	−	0.409584i	0.959700	+	0.281027i	$0.0906750\pi$
$878$		0			0
$879$	26785.6	−	15464.7i	1.02782	−	0.593414i
$880$		0			0
$881$			44697.8i			1.70931i	0.519193	+	0.854657i	$0.326233\pi$
$881$			44697.8i			1.70931i	−0.519193	+	0.854657i	$0.673767\pi$
$882$		0			0
$883$		−	9474.88i		−	0.361104i	−0.983565	−	0.180552i	$-0.942212\pi$
$883$		−	9474.88i		−	0.361104i	0.983565	−	0.180552i	$-0.0577884\pi$
$884$		0			0
$885$	−12197.6	+	7042.27i	−0.463296	+	0.267484i
$886$		0			0
$887$	−7502.29	+	12994.3i	−0.283993	+	0.491891i	−0.972365	−	0.233468i	$-0.924993\pi$
$887$	−7502.29	+	12994.3i	−0.283993	+	0.491891i	0.688371	+	0.725359i	$0.258326\pi$
$888$		0			0
$889$	29785.7	+	16321.4i	1.12371	+	0.615749i
$890$		0			0
$891$	−21019.6	−	12135.7i	−0.790330	−	0.456298i
$892$		0			0
$893$	155.815	+	269.880i	0.00583892	+	0.0101133i
$894$		0			0
$895$	23331.0			0.871361
$896$		0			0
$897$	28719.5			1.06902
$898$		0			0
$899$	3124.13	+	5411.15i	0.115902	+	0.200747i
$900$		0			0
$901$	29753.5	+	17178.2i	1.10015	+	0.635170i
$902$		0			0
$903$	672.506	+	30121.2i	0.0247836	+	1.11004i
$904$		0			0
$905$	1730.19	−	2996.78i	0.0635509	−	0.110073i
$906$		0			0
$907$	24244.0	−	13997.3i	0.887549	−	0.512427i	0.0144093	−	0.999896i	$-0.495413\pi$
$907$	24244.0	−	13997.3i	0.887549	−	0.512427i	0.873140	+	0.487469i	$0.162080\pi$
$908$		0			0
$909$		−	5750.89i		−	0.209840i
$910$		0			0
$911$		−	27683.1i		−	1.00679i	−0.864057	−	0.503393i	$-0.832085\pi$
$911$		−	27683.1i		−	1.00679i	0.864057	−	0.503393i	$-0.167915\pi$
$912$		0			0
$913$	19508.7	−	11263.3i	0.707167	−	0.408283i
$914$		0			0
$915$	18814.5	−	32587.6i	0.679767	−	1.17739i
$916$		0			0
$917$	−13261.7	−	21829.6i	−0.477577	−	0.786125i
$918$		0			0
$919$	−15.0992	−	8.71755i	−0.000541978	−	0.000312911i	0.499729	−	0.866182i	$-0.333433\pi$
$919$	−15.0992	−	8.71755i	−0.000541978	−	0.000312911i	−0.500271	+	0.865869i	$0.666766\pi$
$920$		0			0
$921$	21181.2	+	36686.8i	0.757810	+	1.31256i
$922$		0			0
$923$	37986.9			1.35466
$924$		0			0
$925$	12981.9			0.461451
$926$		0			0
$927$	−4216.21	−	7302.69i	−0.149383	−	0.258740i
$928$		0			0
$929$	36888.5	+	21297.6i	1.30277	+	0.752155i	0.980878	−	0.194621i	$-0.0623478\pi$
$929$	36888.5	+	21297.6i	1.30277	+	0.752155i	0.321892	+	0.946776i	$0.395681\pi$
$930$		0			0
$931$	25868.8	−	1155.71i	0.910652	−	0.0406839i
$932$		0			0
$933$	−16693.4	+	28913.8i	−0.585764	+	1.01457i
$934$		0			0
$935$	73224.1	−	42275.9i	2.56116	−	1.47869i
$936$		0			0
$937$		−	26316.4i		−	0.917522i	−0.888560	−	0.458761i	$-0.848293\pi$
$937$		−	26316.4i		−	0.917522i	0.888560	−	0.458761i	$-0.151707\pi$
$938$		0			0
$939$			22797.8i			0.792309i
$940$		0			0
$941$	−25507.0	+	14726.5i	−0.883640	+	0.510170i	−0.871857	−	0.489761i	$-0.837084\pi$
$941$	−25507.0	+	14726.5i	−0.883640	+	0.510170i	−0.0117829	+	0.999931i	$0.503751\pi$
$942$		0			0
$943$	2856.11	−	4946.92i	0.0986295	−	0.170831i
$944$		0			0
$945$	39864.8	−	24218.2i	1.37228	−	0.833669i
$946$		0			0
$947$	−18190.8	−	10502.5i	−0.624205	−	0.360385i	0.154299	−	0.988024i	$-0.450688\pi$
$947$	−18190.8	−	10502.5i	−0.624205	−	0.360385i	−0.778504	+	0.627639i	$0.784021\pi$
$948$		0			0
$949$	−23707.6	−	41062.8i	−0.810940	−	1.40459i
$950$		0			0
$951$	−17182.4			−0.585885
$952$		0			0
$953$	10707.4			0.363952			0.181976	−	0.983303i	$-0.441751\pi$
$953$	10707.4			0.363952			0.181976	+	0.983303i	$0.441751\pi$
$954$		0			0
$955$	35207.0	+	60980.4i	1.19296	+	2.06626i
$956$		0			0
$957$	−24682.2	−	14250.3i	−0.833712	−	0.481344i
$958$		0			0
$959$	27281.5	−	609.104i	0.918628	−	0.0205099i
$960$		0			0
$961$	13988.5	−	24228.8i	0.469554	−	0.813292i
$962$		0			0
$963$	3124.45	−	1803.90i	0.104552	−	0.0603634i
$964$		0			0
$965$		−	80858.2i		−	2.69733i
$966$		0			0
$967$		−	20353.4i		−	0.676859i	−0.940992	−	0.338429i	$-0.890104\pi$
$967$		−	20353.4i		−	0.676859i	0.940992	−	0.338429i	$-0.109896\pi$
$968$		0			0
$969$	−38294.2	+	22109.1i	−1.26954	+	0.732970i
$970$		0			0
$971$	27664.6	−	47916.6i	0.914316	−	1.58364i	0.106416	−	0.994322i	$-0.466062\pi$
$971$	27664.6	−	47916.6i	0.914316	−	1.58364i	0.807900	−	0.589320i	$-0.200604\pi$
$972$		0			0
$973$	−17175.5	+	31344.4i	−0.565900	+	1.03274i
$974$		0			0
$975$	37587.2	+	21701.0i	1.23462	+	0.712808i
$976$		0			0
$977$	−14962.5	−	25915.8i	−0.489963	−	0.848640i	0.509971	−	0.860192i	$-0.329656\pi$
$977$	−14962.5	−	25915.8i	−0.489963	−	0.848640i	−0.999933	+	0.0115516i	$0.996323\pi$
$978$		0			0
$979$	−4989.70			−0.162892
$980$		0			0
$981$	3952.08			0.128624
$982$		0			0
$983$	2281.10	+	3950.97i	0.0740139	+	0.128196i	0.900657	−	0.434531i	$-0.143086\pi$
$983$	2281.10	+	3950.97i	0.0740139	+	0.128196i	−0.826643	+	0.562727i	$0.809752\pi$
$984$		0			0
$985$	−5521.45	−	3187.81i	−0.178607	−	0.103119i
$986$		0			0
$987$	−175.175	+	319.686i	−0.00564932	+	0.0103097i
$988$		0			0
$989$	18109.9	−	31367.3i	0.582267	−	1.00852i
$990$		0			0
$991$	−1060.34	+	612.186i	−0.0339886	+	0.0196233i	−0.516898	−	0.856047i	$-0.672913\pi$
$991$	−1060.34	+	612.186i	−0.0339886	+	0.0196233i	0.482909	+	0.875670i	$0.339580\pi$
$992$		0			0
$993$		−	15755.0i		−	0.503493i
$994$		0			0
$995$			27361.9i			0.871789i
$996$		0			0
$997$	−39922.0	+	23049.0i	−1.26815	+	0.732165i	−0.974637	−	0.223791i	$-0.928157\pi$
$997$	−39922.0	+	23049.0i	−1.26815	+	0.732165i	−0.293510	+	0.955956i	$0.594823\pi$
$998$		0			0
$999$	6031.03	−	10446.1i	0.191004	−	0.330829i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.p.g.383.2		6
4.3	odd	2		448.4.p.f.383.2		6
7.3	odd	6		448.4.p.f.255.2		6
8.3	odd	2		112.4.p.g.47.2	yes	6
8.5	even	2		112.4.p.f.47.2	yes	6
28.3	even	6	inner	448.4.p.g.255.2		6
56.3	even	6		112.4.p.f.31.2	✓	6
56.5	odd	6		784.4.f.g.783.4		6
56.19	even	6		784.4.f.h.783.4		6
56.37	even	6		784.4.f.h.783.3		6
56.45	odd	6		112.4.p.g.31.2	yes	6
56.51	odd	6		784.4.f.g.783.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.2	✓	6	56.3	even	6
112.4.p.f.47.2	yes	6	8.5	even	2
112.4.p.g.31.2	yes	6	56.45	odd	6
112.4.p.g.47.2	yes	6	8.3	odd	2
448.4.p.f.255.2		6	7.3	odd	6
448.4.p.f.383.2		6	4.3	odd	2
448.4.p.g.255.2		6	28.3	even	6	inner
448.4.p.g.383.2		6	1.1	even	1	trivial
784.4.f.g.783.3		6	56.51	odd	6
784.4.f.g.783.4		6	56.5	odd	6
784.4.f.h.783.3		6	56.37	even	6
784.4.f.h.783.4		6	56.19	even	6

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.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.38417 + 4.12950i) q^{3} +(-14.6315 - 8.44749i) q^{5} +(-8.89981 + 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +O(q^{10})\)	\(q+(2.38417 + 4.12950i) q^{3} +(-14.6315 - 8.44749i) q^{5} +(-8.89981 + 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +(-35.2840 + 20.3712i) q^{11} -56.7321i q^{13} -80.5609i q^{15} +(106.377 - 61.4170i) q^{17} +(-37.7473 + 65.3803i) q^{19} +(-88.2889 + 1.97120i) q^{21} +(91.9414 + 53.0824i) q^{23} +(80.2201 + 138.945i) q^{25} +149.072 q^{27} +146.703 q^{29} +(21.2956 + 36.8850i) q^{31} +(-168.246 - 97.1369i) q^{33} +(267.419 - 162.459i) q^{35} +(40.4571 - 70.0738i) q^{37} +(234.275 - 135.259i) q^{39} -53.8052i q^{41} -341.166i q^{43} +(-62.3733 + 36.0112i) q^{45} +(2.06393 - 3.57482i) q^{47} +(-184.587 - 289.096i) q^{49} +(507.244 + 292.857i) q^{51} +(139.849 + 242.225i) q^{53} +688.342 q^{55} -359.984 q^{57} +(-87.4155 - 151.408i) q^{59} +(404.509 + 233.543i) q^{61} +(40.9918 + 67.4754i) q^{63} +(-479.244 + 830.075i) q^{65} +(652.335 - 376.626i) q^{67} +506.229i q^{69} +669.583i q^{71} +(723.802 - 417.887i) q^{73} +(-382.516 + 662.538i) q^{75} +(-16.8426 - 754.372i) q^{77} +(-958.008 - 553.106i) q^{79} +(297.864 + 515.915i) q^{81} -552.905 q^{83} -2075.28 q^{85} +(349.765 + 605.811i) q^{87} +(106.062 + 61.2347i) q^{89} +(921.427 + 504.906i) q^{91} +(-101.545 + 175.880i) q^{93} +(1104.60 - 637.740i) q^{95} -291.203i q^{97} +173.683i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9}+O(q^{10}) \) 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9	\( 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} + 15 q^{21} - 15 q^{23} + 306 q^{25} - 362 q^{27} + 348 q^{29} + 205 q^{31} - 471 q^{33} + 1185 q^{35} + 249 q^{37} + 288 q^{39} - 2118 q^{45} + 75 q^{47} + 702 q^{49} + 2505 q^{51} + 645 q^{53} + 918 q^{55} - 6 q^{57} - 321 q^{59} + 1707 q^{61} - 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} - 555 q^{77} - 3447 q^{79} + 225 q^{81} - 24 q^{83} - 3786 q^{85} + 3642 q^{87} - 2607 q^{89} + 2448 q^{91} + 2991 q^{93} + 2085 q^{95}+O(q^{100}) \) 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 + 15 * q^21 - 15 * q^23 + 306 * q^25 - 362 * q^27 + 348 * q^29 + 205 * q^31 - 471 * q^33 + 1185 * q^35 + 249 * q^37 + 288 * q^39 - 2118 * q^45 + 75 * q^47 + 702 * q^49 + 2505 * q^51 + 645 * q^53 + 918 * q^55 - 6 * q^57 - 321 * q^59 + 1707 * q^61 - 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 - 555 * q^77 - 3447 * q^79 + 225 * q^81 - 24 * q^83 - 3786 * q^85 + 3642 * q^87 - 2607 * q^89 + 2448 * q^91 + 2991 * q^93 + 2085 * q^95

Introduction

L-functions

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Database

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