LMFDB - Embedded newform 80.6.n.d.47.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,6,Mod(47,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("80.47");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	80.n (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.8307055850$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{65}\cdot 3^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.10
Root			$11.4741 + 7.80740i$ of defining polynomial
Character	$\chi$	$=$	80.47
Dual form			80.6.n.d.63.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(20.3843 + 20.3843i) q^{3} +(-46.4503 + 31.1026i) q^{5} +(76.9082 - 76.9082i) q^{7} +588.037i q^{9} +O(q^{10})$	\(q+(20.3843 + 20.3843i) q^{3} +(-46.4503 + 31.1026i) q^{5} +(76.9082 - 76.9082i) q^{7} +588.037i q^{9} +556.846i q^{11} +(141.317 - 141.317i) q^{13} +(-1580.86 - 312.851i) q^{15} +(-477.013 - 477.013i) q^{17} -1608.20 q^{19} +3135.44 q^{21} +(346.617 + 346.617i) q^{23} +(1190.25 - 2889.45i) q^{25} +(-7033.34 + 7033.34i) q^{27} +7486.67i q^{29} -7927.33i q^{31} +(-11350.9 + 11350.9i) q^{33} +(-1180.36 + 5964.46i) q^{35} +(3329.26 + 3329.26i) q^{37} +5761.27 q^{39} +18717.9 q^{41} +(8253.12 + 8253.12i) q^{43} +(-18289.5 - 27314.5i) q^{45} +(5098.15 - 5098.15i) q^{47} +4977.25i q^{49} -19447.1i q^{51} +(19488.7 - 19488.7i) q^{53} +(-17319.4 - 25865.6i) q^{55} +(-32782.1 - 32782.1i) q^{57} -108.893 q^{59} +14287.1 q^{61} +(45224.9 + 45224.9i) q^{63} +(-2168.88 + 10959.5i) q^{65} +(28986.5 - 28986.5i) q^{67} +14131.1i q^{69} +982.591i q^{71} +(-21571.6 + 21571.6i) q^{73} +(83161.8 - 34636.9i) q^{75} +(42826.0 + 42826.0i) q^{77} +9383.55 q^{79} -143846. q^{81} +(-9451.97 - 9451.97i) q^{83} +(36993.7 + 7321.02i) q^{85} +(-152610. + 152610. i) q^{87} -8489.66i q^{89} -21736.8i q^{91} +(161593. - 161593. i) q^{93} +(74701.5 - 50019.3i) q^{95} +(122282. + 122282. i) q^{97} -327446. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 44 q^{5}+O(q^{10}) $ 20 * q - 44 * q^5	$ 20 q - 44 q^{5} + 804 q^{13} - 2236 q^{17} - 4520 q^{21} + 948 q^{25} - 11096 q^{33} + 44260 q^{37} - 6760 q^{41} - 92816 q^{45} + 182452 q^{53} - 34288 q^{57} - 41080 q^{61} - 155772 q^{65} + 264372 q^{73} + 399304 q^{77} - 520220 q^{81} - 344796 q^{85} + 713496 q^{93} + 374772 q^{97}+O(q^{100}) $ 20 * q - 44 * q^5 + 804 * q^13 - 2236 * q^17 - 4520 * q^21 + 948 * q^25 - 11096 * q^33 + 44260 * q^37 - 6760 * q^41 - 92816 * q^45 + 182452 * q^53 - 34288 * q^57 - 41080 * q^61 - 155772 * q^65 + 264372 * q^73 + 399304 * q^77 - 520220 * q^81 - 344796 * q^85 + 713496 * q^93 + 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	20.3843	+	20.3843i	1.30765	+	1.30765i	0.923106	+	0.384546i	$0.125642\pi$
$3$	20.3843	+	20.3843i	1.30765	+	1.30765i	0.384546	+	0.923106i	$0.374358\pi$
$4$		0			0
$5$	−46.4503	+	31.1026i	−0.830928	+	0.556381i
$5$	−46.4503	+	31.1026i	−0.830928	+	0.556381i
$6$		0			0
$7$	76.9082	−	76.9082i	0.593236	−	0.593236i	−0.345268	−	0.938504i	$-0.612212\pi$
$7$	76.9082	−	76.9082i	0.593236	−	0.593236i	0.938504	+	0.345268i	$0.112212\pi$
$8$		0			0
$9$			588.037i			2.41991i
$10$		0			0
$11$			556.846i			1.38756i	0.720185	+	0.693782i	$0.244057\pi$
$11$			556.846i			1.38756i	−0.720185	+	0.693782i	$0.755943\pi$
$12$		0			0
$13$	141.317	−	141.317i	0.231918	−	0.231918i	−0.581575	−	0.813493i	$-0.697563\pi$
$13$	141.317	−	141.317i	0.231918	−	0.231918i	0.813493	+	0.581575i	$0.197563\pi$
$14$		0			0
$15$	−1580.86	−	312.851i	−1.81412	−	0.359012i
$16$		0			0
$17$	−477.013	−	477.013i	−0.400320	−	0.400320i	0.478026	−	0.878346i	$-0.341353\pi$
$17$	−477.013	−	477.013i	−0.400320	−	0.400320i	−0.878346	+	0.478026i	$0.841353\pi$
$18$		0			0
$19$	−1608.20			−1.02201			−0.511007	−	0.859577i	$-0.670727\pi$
$19$	−1608.20			−1.02201			−0.511007	+	0.859577i	$0.670727\pi$
$20$		0			0
$21$	3135.44			1.55149
$22$		0			0
$23$	346.617	+	346.617i	0.136625	+	0.136625i	0.772112	−	0.635487i	$-0.219201\pi$
$23$	346.617	+	346.617i	0.136625	+	0.136625i	−0.635487	+	0.772112i	$0.719201\pi$
$24$		0			0
$25$	1190.25	−	2889.45i	0.380881	−	0.924624i
$26$		0			0
$27$	−7033.34	+	7033.34i	−1.85674	+	1.85674i
$28$		0			0
$29$			7486.67i			1.65308i	0.562879	+	0.826539i	$0.309694\pi$
$29$			7486.67i			1.65308i	−0.562879	+	0.826539i	$0.690306\pi$
$30$		0			0
$31$		−	7927.33i		−	1.48157i	−0.671742	−	0.740786i	$-0.734453\pi$
$31$		−	7927.33i		−	1.48157i	0.671742	−	0.740786i	$-0.265547\pi$
$32$		0			0
$33$	−11350.9	+	11350.9i	−1.81445	+	1.81445i
$34$		0			0
$35$	−1180.36	+	5964.46i	−0.162871	+	0.823002i
$36$		0			0
$37$	3329.26	+	3329.26i	0.399801	+	0.399801i	0.878163	−	0.478362i	$-0.158769\pi$
$37$	3329.26	+	3329.26i	0.399801	+	0.399801i	−0.478362	+	0.878163i	$0.658769\pi$
$38$		0			0
$39$	5761.27			0.606536
$40$		0			0
$41$	18717.9			1.73899			0.869497	−	0.493938i	$-0.164443\pi$
$41$	18717.9			1.73899			0.869497	+	0.493938i	$0.164443\pi$
$42$		0			0
$43$	8253.12	+	8253.12i	0.680686	+	0.680686i	0.960155	−	0.279468i	$-0.0901582\pi$
$43$	8253.12	+	8253.12i	0.680686	+	0.680686i	−0.279468	+	0.960155i	$0.590158\pi$
$44$		0			0
$45$	−18289.5	−	27314.5i	−1.34639	−	2.01077i
$46$		0			0
$47$	5098.15	−	5098.15i	0.336642	−	0.336642i	−0.518460	−	0.855102i	$-0.673494\pi$
$47$	5098.15	−	5098.15i	0.336642	−	0.336642i	0.855102	+	0.518460i	$0.173494\pi$
$48$		0			0
$49$			4977.25i			0.296141i
$50$		0			0
$51$		−	19447.1i		−	1.04696i
$52$		0			0
$53$	19488.7	−	19488.7i	0.953001	−	0.953001i	−0.0459428	−	0.998944i	$-0.514629\pi$
$53$	19488.7	−	19488.7i	0.953001	−	0.953001i	0.998944	+	0.0459428i	$0.0146292\pi$
$54$		0			0
$55$	−17319.4	−	25865.6i	−0.772014	−	1.15297i
$56$		0			0
$57$	−32782.1	−	32782.1i	−1.33644	−	1.33644i
$58$		0			0
$59$	−108.893			−0.00407257			−0.00203628	−	0.999998i	$-0.500648\pi$
$59$	−108.893			−0.00407257			−0.00203628	+	0.999998i	$0.500648\pi$
$60$		0			0
$61$	14287.1			0.491608			0.245804	−	0.969320i	$-0.420948\pi$
$61$	14287.1			0.491608			0.245804	+	0.969320i	$0.420948\pi$
$62$		0			0
$63$	45224.9	+	45224.9i	1.43558	+	1.43558i
$64$		0			0
$65$	−2168.88	+	10959.5i	−0.0636724	+	0.321742i
$66$		0			0
$67$	28986.5	−	28986.5i	0.788875	−	0.788875i	−0.192435	−	0.981310i	$-0.561638\pi$
$67$	28986.5	−	28986.5i	0.788875	−	0.788875i	0.981310	+	0.192435i	$0.0616384\pi$
$68$		0			0
$69$			14131.1i			0.357316i
$70$		0			0
$71$			982.591i			0.0231327i	0.999933	+	0.0115664i	$0.00368177\pi$
$71$			982.591i			0.0231327i	−0.999933	+	0.0115664i	$0.996318\pi$
$72$		0			0
$73$	−21571.6	+	21571.6i	−0.473778	+	0.473778i	−0.903135	−	0.429357i	$-0.858740\pi$
$73$	−21571.6	+	21571.6i	−0.473778	+	0.473778i	0.429357	+	0.903135i	$0.358740\pi$
$74$		0			0
$75$	83161.8	−	34636.9i	1.70715	−	0.711026i
$76$		0			0
$77$	42826.0	+	42826.0i	0.823154	+	0.823154i
$78$		0			0
$79$	9383.55			0.169161			0.0845803	−	0.996417i	$-0.473045\pi$
$79$	9383.55			0.169161			0.0845803	+	0.996417i	$0.473045\pi$
$80$		0			0
$81$	−143846.			−2.43604
$82$		0			0
$83$	−9451.97	−	9451.97i	−0.150601	−	0.150601i	0.627786	−	0.778386i	$-0.283961\pi$
$83$	−9451.97	−	9451.97i	−0.150601	−	0.150601i	−0.778386	+	0.627786i	$0.783961\pi$
$84$		0			0
$85$	36993.7	+	7321.02i	0.555367	+	0.109907i
$86$		0			0
$87$	−152610.	+	152610.i	−2.16165	+	2.16165i
$88$		0			0
$89$		−	8489.66i		−	0.113610i	−0.998385	−	0.0568049i	$-0.981909\pi$
$89$		−	8489.66i		−	0.113610i	0.998385	−	0.0568049i	$-0.0180913\pi$
$90$		0			0
$91$		−	21736.8i		−	0.275165i
$92$		0			0
$93$	161593.	−	161593.i	1.93738	−	1.93738i
$94$		0			0
$95$	74701.5	−	50019.3i	0.849220	−	0.568629i
$96$		0			0
$97$	122282.	+	122282.i	1.31957	+	1.31957i	0.914116	+	0.405453i	$0.132886\pi$
$97$	122282.	+	122282.i	1.31957	+	1.31957i	0.405453	+	0.914116i	$0.367114\pi$
$98$		0			0
$99$	−327446.			−3.35778
$100$		0			0
$101$	−49322.0			−0.481102			−0.240551	−	0.970636i	$-0.577328\pi$
$101$	−49322.0			−0.481102			−0.240551	+	0.970636i	$0.577328\pi$
$102$		0			0
$103$	−63162.5	−	63162.5i	−0.586633	−	0.586633i	0.350085	−	0.936718i	$-0.386153\pi$
$103$	−63162.5	−	63162.5i	−0.586633	−	0.586633i	−0.936718	+	0.350085i	$0.886153\pi$
$104$		0			0
$105$	−145642.	+	97520.3i	−1.28918	+	0.863221i
$106$		0			0
$107$	25175.2	−	25175.2i	0.212575	−	0.212575i	−0.592785	−	0.805361i	$-0.701972\pi$
$107$	25175.2	−	25175.2i	0.212575	−	0.212575i	0.805361	+	0.592785i	$0.201972\pi$
$108$		0			0
$109$		−	79771.5i		−	0.643104i	−0.946892	−	0.321552i	$-0.895795\pi$
$109$		−	79771.5i		−	0.643104i	0.946892	−	0.321552i	$-0.104205\pi$
$110$		0			0
$111$			135729.i			1.04560i
$112$		0			0
$113$	93876.0	−	93876.0i	0.691606	−	0.691606i	−0.270980	−	0.962585i	$-0.587348\pi$
$113$	93876.0	−	93876.0i	0.691606	−	0.691606i	0.962585	+	0.270980i	$0.0873476\pi$
$114$		0			0
$115$	−26881.2	−	5319.76i	−0.189541	−	0.0375100i
$116$		0			0
$117$	83099.4	+	83099.4i	0.561220	+	0.561220i
$118$		0			0
$119$	−73372.4			−0.474969
$120$		0			0
$121$	−149026.			−0.925336
$122$		0			0
$123$	381551.	+	381551.i	2.27400	+	2.27400i
$124$		0			0
$125$	34581.8	+	171236.i	0.197958	+	0.980210i
$126$		0			0
$127$	5616.82	−	5616.82i	0.0309016	−	0.0309016i	−0.691487	−	0.722389i	$-0.743044\pi$
$127$	5616.82	−	5616.82i	0.0309016	−	0.0309016i	0.722389	+	0.691487i	$0.243044\pi$
$128$		0			0
$129$			336468.i			1.78020i
$130$		0			0
$131$		−	56715.0i		−	0.288749i	−0.989523	−	0.144374i	$-0.953883\pi$
$131$		−	56715.0i		−	0.288749i	0.989523	−	0.144374i	$-0.0461169\pi$
$132$		0			0
$133$	−123684.	+	123684.i	−0.606296	+	0.606296i
$134$		0			0
$135$	107945.	−	545456.i	0.509764	−	2.57588i
$136$		0			0
$137$	−95096.7	−	95096.7i	−0.432877	−	0.432877i	0.456729	−	0.889606i	$-0.349021\pi$
$137$	−95096.7	−	95096.7i	−0.432877	−	0.432877i	−0.889606	+	0.456729i	$0.849021\pi$
$138$		0			0
$139$	−276110.			−1.21212			−0.606059	−	0.795420i	$-0.707251\pi$
$139$	−276110.			−1.21212			−0.606059	+	0.795420i	$0.707251\pi$
$140$		0			0
$141$	207844.			0.880421
$142$		0			0
$143$	78691.5	+	78691.5i	0.321801	+	0.321801i
$144$		0			0
$145$	−232855.	−	347758.i	−0.919741	−	1.37359i
$146$		0			0
$147$	−101458.	+	101458.i	−0.387250	+	0.387250i
$148$		0			0
$149$		−	258844.i		−	0.955153i	−0.878590	−	0.477577i	$-0.841515\pi$
$149$		−	258844.i		−	0.955153i	0.878590	−	0.477577i	$-0.158485\pi$
$150$		0			0
$151$		−	399655.i		−	1.42640i	−0.700958	−	0.713202i	$-0.747244\pi$
$151$		−	399655.i		−	1.42640i	0.700958	−	0.713202i	$-0.252756\pi$
$152$		0			0
$153$	280501.	−	280501.i	0.968738	−	0.968738i
$154$		0			0
$155$	246561.	+	368226.i	0.824317	+	1.23108i
$156$		0			0
$157$	−174590.	−	174590.i	−0.565289	−	0.565289i	0.365516	−	0.930805i	$-0.380893\pi$
$157$	−174590.	−	174590.i	−0.565289	−	0.565289i	−0.930805	+	0.365516i	$0.880893\pi$
$158$		0			0
$159$	794527.			2.49239
$160$		0			0
$161$	53315.4			0.162102
$162$		0			0
$163$	1364.52	+	1364.52i	0.00402264	+	0.00402264i	0.709115	−	0.705093i	$-0.249095\pi$
$163$	1364.52	+	1364.52i	0.00402264	+	0.00402264i	−0.705093	+	0.709115i	$0.749095\pi$
$164$		0			0
$165$	174210.	−	880295.i	0.498152	−	2.51720i
$166$		0			0
$167$	−323133.	+	323133.i	−0.896581	+	0.896581i	−0.995132	−	0.0985507i	$-0.968579\pi$
$167$	−323133.	+	323133.i	−0.896581	+	0.896581i	0.0985507	+	0.995132i	$0.468579\pi$
$168$		0			0
$169$			331352.i			0.892428i
$170$		0			0
$171$		−	945684.i		−	2.47318i
$172$		0			0
$173$	−173917.	+	173917.i	−0.441800	+	0.441800i	−0.892617	−	0.450816i	$-0.851133\pi$
$173$	−173917.	+	173917.i	−0.441800	+	0.441800i	0.450816	+	0.892617i	$0.351133\pi$
$174$		0			0
$175$	−130682.	−	313763.i	−0.322568	−	0.774473i
$176$		0			0
$177$	−2219.70	−	2219.70i	−0.00532550	−	0.00532550i
$178$		0			0
$179$	60036.1			0.140049			0.0700245	−	0.997545i	$-0.477692\pi$
$179$	60036.1			0.140049			0.0700245	+	0.997545i	$0.477692\pi$
$180$		0			0
$181$	−71086.8			−0.161285			−0.0806423	−	0.996743i	$-0.525697\pi$
$181$	−71086.8			−0.161285			−0.0806423	+	0.996743i	$0.525697\pi$
$182$		0			0
$183$	291232.	+	291232.i	0.642852	+	0.642852i
$184$		0			0
$185$	−258194.	−	51096.3i	−0.554647	−	0.109764i
$186$		0			0
$187$	265622.	−	265622.i	0.555470	−	0.555470i
$188$		0			0
$189$			1.08184e6i			2.20298i
$190$		0			0
$191$			898374.i			1.78186i	0.454139	+	0.890931i	$0.349947\pi$
$191$			898374.i			1.78186i	−0.454139	+	0.890931i	$0.650053\pi$
$192$		0			0
$193$	−514879.	+	514879.i	−0.994974	+	0.994974i	−0.999987	−	0.00501311i	$-0.998404\pi$
$193$	−514879.	+	514879.i	−0.994974	+	0.994974i	0.00501311	+	0.999987i	$0.498404\pi$
$194$		0			0
$195$	−267613.	+	179191.i	−0.503988	+	0.337465i
$196$		0			0
$197$	320182.	+	320182.i	0.587802	+	0.587802i	0.937036	−	0.349234i	$-0.113558\pi$
$197$	320182.	+	320182.i	0.587802	+	0.587802i	−0.349234	+	0.937036i	$0.613558\pi$
$198$		0			0
$199$	−458886.			−0.821433			−0.410717	−	0.911763i	$-0.634721\pi$
$199$	−458886.			−0.821433			−0.410717	+	0.911763i	$0.634721\pi$
$200$		0			0
$201$	1.18174e6			2.06315
$202$		0			0
$203$	575786.	+	575786.i	0.980666	+	0.980666i
$204$		0			0
$205$	−869453.	+	582177.i	−1.44498	+	0.967542i
$206$		0			0
$207$	−203824.	+	203824.i	−0.330620	+	0.330620i
$208$		0			0
$209$		−	895521.i		−	1.41811i
$210$		0			0
$211$		−	561846.i		−	0.868783i	−0.900724	−	0.434391i	$-0.856963\pi$
$211$		−	561846.i		−	0.868783i	0.900724	−	0.434391i	$-0.143037\pi$
$212$		0			0
$213$	−20029.4	+	20029.4i	−0.0302495	+	0.0302495i
$214$		0			0
$215$	−640053.	−	126666.i	−0.944322	−	0.186880i
$216$		0			0
$217$	−609677.	−	609677.i	−0.878922	−	0.878922i
$218$		0			0
$219$	−879443.			−1.23907
$220$		0			0
$221$	−134820.			−0.185683
$222$		0			0
$223$	−821427.	−	821427.i	−1.10613	−	1.10613i	−0.993654	−	0.112478i	$-0.964121\pi$
$223$	−821427.	−	821427.i	−1.10613	−	1.10613i	−0.112478	−	0.993654i	$-0.535879\pi$
$224$		0			0
$225$	1.69910e6	+	699914.i	2.23750	+	0.921697i
$226$		0			0
$227$	276085.	−	276085.i	0.355613	−	0.355613i	−0.506580	−	0.862193i	$-0.669090\pi$
$227$	276085.	−	276085.i	0.355613	−	0.355613i	0.862193	+	0.506580i	$0.169090\pi$
$228$		0			0
$229$		−	503195.i		−	0.634085i	−0.948411	−	0.317043i	$-0.897310\pi$
$229$		−	503195.i		−	0.634085i	0.948411	−	0.317043i	$-0.102690\pi$
$230$		0			0
$231$			1.74596e6i			2.15280i
$232$		0			0
$233$	594823.	−	594823.i	0.717790	−	0.717790i	−0.250362	−	0.968152i	$-0.580550\pi$
$233$	594823.	−	594823.i	0.717790	−	0.717790i	0.968152	+	0.250362i	$0.0805496\pi$
$234$		0			0
$235$	−78244.7	+	395377.i	−0.0924241	+	0.467026i
$236$		0			0
$237$	191277.	+	191277.i	0.221203	+	0.221203i
$238$		0			0
$239$	−348307.			−0.394428			−0.197214	−	0.980360i	$-0.563189\pi$
$239$	−348307.			−0.394428			−0.197214	+	0.980360i	$0.563189\pi$
$240$		0			0
$241$	252895.			0.280477			0.140239	−	0.990118i	$-0.455213\pi$
$241$	252895.			0.280477			0.140239	+	0.990118i	$0.455213\pi$
$242$		0			0
$243$	−1.22309e6	−	1.22309e6i	−1.32875	−	1.32875i
$244$		0			0
$245$	−154805.	−	231194.i	−0.164767	−	0.246072i
$246$		0			0
$247$	−227266.	+	227266.i	−0.237024	+	0.237024i
$248$		0			0
$249$		−	385343.i		−	0.393867i
$250$		0			0
$251$			493537.i			0.494465i	0.968956	+	0.247232i	$0.0795211\pi$
$251$			493537.i			0.494465i	−0.968956	+	0.247232i	$0.920479\pi$
$252$		0			0
$253$	−193012.	+	193012.i	−0.189576	+	0.189576i
$254$		0			0
$255$	604856.	+	903323.i	0.582508	+	0.869947i
$256$		0			0
$257$	1.28934e6	+	1.28934e6i	1.21768	+	1.21768i	0.968441	+	0.249242i	$0.0801814\pi$
$257$	1.28934e6	+	1.28934e6i	1.21768	+	1.21768i	0.249242	+	0.968441i	$0.419819\pi$
$258$		0			0
$259$	512095.			0.474352
$260$		0			0
$261$	−4.40244e6			−4.00030
$262$		0			0
$263$	−562452.	−	562452.i	−0.501414	−	0.501414i	0.410463	−	0.911877i	$-0.365367\pi$
$263$	−562452.	−	562452.i	−0.501414	−	0.501414i	−0.911877	+	0.410463i	$0.865367\pi$
$264$		0			0
$265$	−299106.	+	1.51141e6i	−0.261644	+	1.32211i
$266$		0			0
$267$	173056.	−	173056.i	0.148562	−	0.148562i
$268$		0			0
$269$		−	305884.i		−	0.257737i	−0.991662	−	0.128868i	$-0.958866\pi$
$269$		−	305884.i		−	0.257737i	0.991662	−	0.128868i	$-0.0411345\pi$
$270$		0			0
$271$		−	1.98399e6i		−	1.64103i	−0.571626	−	0.820514i	$-0.693687\pi$
$271$		−	1.98399e6i		−	1.64103i	0.571626	−	0.820514i	$-0.306313\pi$
$272$		0			0
$273$	443089.	−	443089.i	0.359819	−	0.359819i
$274$		0			0
$275$	1.60898e6	+	662788.i	1.28298	+	0.528497i
$276$		0			0
$277$	669845.	+	669845.i	0.524536	+	0.524536i	0.918938	−	0.394402i	$-0.129048\pi$
$277$	669845.	+	669845.i	0.524536	+	0.524536i	−0.394402	+	0.918938i	$0.629048\pi$
$278$		0			0
$279$	4.66157e6			3.58526
$280$		0			0
$281$	374638.			0.283039			0.141520	−	0.989935i	$-0.454801\pi$
$281$	374638.			0.283039			0.141520	+	0.989935i	$0.454801\pi$
$282$		0			0
$283$	−894492.	−	894492.i	−0.663912	−	0.663912i	0.292388	−	0.956300i	$-0.405550\pi$
$283$	−894492.	−	894492.i	−0.663912	−	0.663912i	−0.956300	+	0.292388i	$0.905550\pi$
$284$		0			0
$285$	2.54234e6	+	503127.i	1.85405	+	0.366915i
$286$		0			0
$287$	1.43956e6	−	1.43956e6i	1.03163	−	1.03163i
$288$		0			0
$289$		−	964775.i		−	0.679488i
$290$		0			0
$291$			4.98525e6i			3.45107i
$292$		0			0
$293$	−788207.	+	788207.i	−0.536379	+	0.536379i	−0.922463	−	0.386085i	$-0.873827\pi$
$293$	−788207.	+	788207.i	−0.536379	+	0.536379i	0.386085	+	0.922463i	$0.373827\pi$
$294$		0			0
$295$	5058.09	−	3386.85i	0.00338401	−	0.00226590i
$296$		0			0
$297$	−3.91649e6	−	3.91649e6i	−2.57635	−	2.57635i
$298$		0			0
$299$	97965.5			0.0633717
$300$		0			0
$301$	1.26947e6			0.807616
$302$		0			0
$303$	−1.00539e6	−	1.00539e6i	−0.629114	−	0.629114i
$304$		0			0
$305$	−663638.	+	444365.i	−0.408490	+	0.273521i
$306$		0			0
$307$	2.03742e6	−	2.03742e6i	1.23377	−	1.23377i	0.271264	−	0.962505i	$-0.412558\pi$
$307$	2.03742e6	−	2.03742e6i	1.23377	−	1.23377i	0.962505	−	0.271264i	$-0.0874415\pi$
$308$		0			0
$309$		−	2.57504e6i		−	1.53422i
$310$		0			0
$311$		−	2.55652e6i		−	1.49881i	−0.662109	−	0.749407i	$-0.730339\pi$
$311$		−	2.55652e6i		−	1.49881i	0.662109	−	0.749407i	$-0.269661\pi$
$312$		0			0
$313$	985823.	−	985823.i	0.568772	−	0.568772i	−0.363012	−	0.931784i	$-0.618252\pi$
$313$	985823.	−	985823.i	0.568772	−	0.568772i	0.931784	+	0.363012i	$0.118252\pi$
$314$		0			0
$315$	−3.50732e6	−	694096.i	−1.99159	−	0.394133i
$316$		0			0
$317$	−113582.	−	113582.i	−0.0634836	−	0.0634836i	0.674652	−	0.738136i	$-0.264294\pi$
$317$	−113582.	−	113582.i	−0.0634836	−	0.0634836i	−0.738136	+	0.674652i	$0.764294\pi$
$318$		0			0
$319$	−4.16892e6			−2.29375
$320$		0			0
$321$	1.02636e6			0.555949
$322$		0			0
$323$	767133.	+	767133.i	0.409133	+	0.409133i
$324$		0			0
$325$	−240125.	−	576530.i	−0.126104	−	0.302770i
$326$		0			0
$327$	1.62608e6	−	1.62608e6i	0.840956	−	0.840956i
$328$		0			0
$329$		−	784180.i		−	0.399417i
$330$		0			0
$331$		−	327502.i		−	0.164303i	−0.996620	−	0.0821513i	$-0.973821\pi$
$331$		−	327502.i		−	0.164303i	0.996620	−	0.0821513i	$-0.0261791\pi$
$332$		0			0
$333$	−1.95773e6	+	1.95773e6i	−0.967480	+	0.967480i
$334$		0			0
$335$	−444874.	+	2.24798e6i	−0.216583	+	1.09441i
$336$		0			0
$337$	1.20574e6	+	1.20574e6i	0.578335	+	0.578335i	0.934444	−	0.356110i	$-0.115897\pi$
$337$	1.20574e6	+	1.20574e6i	0.578335	+	0.578335i	−0.356110	+	0.934444i	$0.615897\pi$
$338$		0			0
$339$	3.82719e6			1.80876
$340$		0			0
$341$	4.41430e6			2.05578
$342$		0			0
$343$	1.67539e6	+	1.67539e6i	0.768918	+	0.768918i
$344$		0			0
$345$	−439514.	−	656392.i	−0.198804	−	0.296904i
$346$		0			0
$347$	−1.84834e6	+	1.84834e6i	−0.824058	+	0.824058i	−0.986687	−	0.162629i	$-0.948002\pi$
$347$	−1.84834e6	+	1.84834e6i	−0.824058	+	0.824058i	0.162629	+	0.986687i	$0.448002\pi$
$348$		0			0
$349$			28570.5i			0.0125561i	0.999980	+	0.00627804i	$0.00199838\pi$
$349$			28570.5i			0.0125561i	−0.999980	+	0.00627804i	$0.998002\pi$
$350$		0			0
$351$			1.98785e6i			0.861225i
$352$		0			0
$353$	−1.77770e6	+	1.77770e6i	−0.759316	+	0.759316i	−0.976198	−	0.216882i	$-0.930411\pi$
$353$	−1.77770e6	+	1.77770e6i	−0.759316	+	0.759316i	0.216882	+	0.976198i	$0.430411\pi$
$354$		0			0
$355$	−30561.1	−	45641.6i	−0.0128706	−	0.0192216i
$356$		0			0
$357$	−1.49564e6	−	1.49564e6i	−0.621094	−	0.621094i
$358$		0			0
$359$	736136.			0.301454			0.150727	−	0.988575i	$-0.451838\pi$
$359$	736136.			0.301454			0.150727	+	0.988575i	$0.451838\pi$
$360$		0			0
$361$	110218.			0.0445130
$362$		0			0
$363$	−3.03779e6	−	3.03779e6i	−1.21002	−	1.21002i
$364$		0			0
$365$	331073.	−	1.67294e6i	0.130074	−	0.657276i
$366$		0			0
$367$	−477935.	+	477935.i	−0.185227	+	0.185227i	−0.793629	−	0.608402i	$-0.791811\pi$
$367$	−477935.	+	477935.i	−0.185227	+	0.185227i	0.608402	+	0.793629i	$0.291811\pi$
$368$		0			0
$369$			1.10068e7i			4.20820i
$370$		0			0
$371$		−	2.99769e6i		−	1.13071i
$372$		0			0
$373$	−3.22422e6	+	3.22422e6i	−1.19992	+	1.19992i	−0.225732	+	0.974189i	$0.572477\pi$
$373$	−3.22422e6	+	3.22422e6i	−1.19992	+	1.19992i	−0.974189	+	0.225732i	$0.927523\pi$
$374$		0			0
$375$	−2.78559e6	+	4.19544e6i	−1.02291	+	1.54063i
$376$		0			0
$377$	1.05799e6	+	1.05799e6i	0.383379	+	0.383379i
$378$		0			0
$379$	4.24297e6			1.51730			0.758652	−	0.651496i	$-0.225858\pi$
$379$	4.24297e6			1.51730			0.758652	+	0.651496i	$0.225858\pi$
$380$		0			0
$381$	228990.			0.0808171
$382$		0			0
$383$	−654083.	−	654083.i	−0.227843	−	0.227843i	0.583948	−	0.811791i	$-0.301507\pi$
$383$	−654083.	−	654083.i	−0.227843	−	0.227843i	−0.811791	+	0.583948i	$0.801507\pi$
$384$		0			0
$385$	−3.32128e6	−	657279.i	−1.14197	−	0.225994i
$386$		0			0
$387$	−4.85314e6	+	4.85314e6i	−1.64720	+	1.64720i
$388$		0			0
$389$		−	2.46468e6i		−	0.825823i	−0.910771	−	0.412911i	$-0.864512\pi$
$389$		−	2.46468e6i		−	0.825823i	0.910771	−	0.412911i	$-0.135488\pi$
$390$		0			0
$391$		−	330681.i		−	0.109388i
$392$		0			0
$393$	1.15609e6	−	1.15609e6i	0.377583	−	0.377583i
$394$		0			0
$395$	−435868.	+	291853.i	−0.140560	+	0.0941177i
$396$		0			0
$397$	4.06922e6	+	4.06922e6i	1.29579	+	1.29579i	0.931147	+	0.364645i	$0.118810\pi$
$397$	4.06922e6	+	4.06922e6i	1.29579	+	1.29579i	0.364645	+	0.931147i	$0.381190\pi$
$398$		0			0
$399$	−5.04242e6			−1.58565
$400$		0			0
$401$	3.14549e6			0.976849			0.488424	−	0.872606i	$-0.337572\pi$
$401$	3.14549e6			0.976849			0.488424	+	0.872606i	$0.337572\pi$
$402$		0			0
$403$	−1.12026e6	−	1.12026e6i	−0.343603	−	0.343603i
$404$		0			0
$405$	6.68168e6	−	4.47399e6i	2.02418	−	1.35537i
$406$		0			0
$407$	−1.85388e6	+	1.85388e6i	−0.554749	+	0.554749i
$408$		0			0
$409$		−	4.42634e6i		−	1.30839i	−0.756327	−	0.654194i	$-0.773008\pi$
$409$		−	4.42634e6i		−	1.30839i	0.756327	−	0.654194i	$-0.226992\pi$
$410$		0			0
$411$		−	3.87696e6i		−	1.13210i
$412$		0			0
$413$	−8374.74	+	8374.74i	−0.00241600	+	0.00241600i
$414$		0			0
$415$	733028.	+	145066.i	0.208930	+	0.0413470i
$416$		0			0
$417$	−5.62830e6	−	5.62830e6i	−1.58503	−	1.58503i
$418$		0			0
$419$	−2.19963e6			−0.612090			−0.306045	−	0.952017i	$-0.599006\pi$
$419$	−2.19963e6			−0.612090			−0.306045	+	0.952017i	$0.599006\pi$
$420$		0			0
$421$	1240.34			0.000341063			0.000170532	−	1.00000i	$-0.499946\pi$
$421$	1240.34			0.000341063			0.000170532		1.00000i	$0.499946\pi$
$422$		0			0
$423$	2.99791e6	+	2.99791e6i	0.814642	+	0.814642i
$424$		0			0
$425$	−1.94607e6	+	810538.i	−0.522620	+	0.217671i
$426$		0			0
$427$	1.09879e6	−	1.09879e6i	0.291640	−	0.291640i
$428$		0			0
$429$			3.20814e6i			0.841609i
$430$		0			0
$431$			1.29826e6i			0.336642i	0.985732	+	0.168321i	$0.0538344\pi$
$431$			1.29826e6i			0.336642i	−0.985732	+	0.168321i	$0.946166\pi$
$432$		0			0
$433$	665935.	−	665935.i	0.170691	−	0.170691i	−0.616592	−	0.787283i	$-0.711487\pi$
$433$	665935.	−	665935.i	0.170691	−	0.170691i	0.787283	+	0.616592i	$0.211487\pi$
$434$		0			0
$435$	2.34221e6	−	1.18354e7i	0.593475	−	2.99888i
$436$		0			0
$437$	−557431.	−	557431.i	−0.139633	−	0.139633i
$438$		0			0
$439$	−3.73469e6			−0.924898			−0.462449	−	0.886646i	$-0.653029\pi$
$439$	−3.73469e6			−0.924898			−0.462449	+	0.886646i	$0.653029\pi$
$440$		0			0
$441$	−2.92681e6			−0.716634
$442$		0			0
$443$	2.91756e6	+	2.91756e6i	0.706336	+	0.706336i	0.965763	−	0.259427i	$-0.0835337\pi$
$443$	2.91756e6	+	2.91756e6i	0.706336	+	0.706336i	−0.259427	+	0.965763i	$0.583534\pi$
$444$		0			0
$445$	264051.	+	394347.i	0.0632102	+	0.0944014i
$446$		0			0
$447$	5.27635e6	−	5.27635e6i	1.24901	−	1.24901i
$448$		0			0
$449$			992830.i			0.232412i	0.993225	+	0.116206i	$0.0370733\pi$
$449$			992830.i			0.232412i	−0.993225	+	0.116206i	$0.962927\pi$
$450$		0			0
$451$			1.04230e7i			2.41297i
$452$		0			0
$453$	8.14667e6	−	8.14667e6i	1.86524	−	1.86524i
$454$		0			0
$455$	676072.	+	1.00968e6i	0.153096	+	0.228642i
$456$		0			0
$457$	−4.19086e6	−	4.19086e6i	−0.938670	−	0.938670i	0.0595554	−	0.998225i	$-0.481032\pi$
$457$	−4.19086e6	−	4.19086e6i	−0.938670	−	0.938670i	−0.998225	+	0.0595554i	$0.981032\pi$
$458$		0			0
$459$	6.70998e6			1.48658
$460$		0			0
$461$	−5.83166e6			−1.27803			−0.639013	−	0.769196i	$-0.720657\pi$
$461$	−5.83166e6			−1.27803			−0.639013	+	0.769196i	$0.720657\pi$
$462$		0			0
$463$	4.17783e6	+	4.17783e6i	0.905728	+	0.905728i	0.995924	−	0.0901957i	$-0.0287492\pi$
$463$	4.17783e6	+	4.17783e6i	0.905728	+	0.905728i	−0.0901957	+	0.995924i	$0.528749\pi$
$464$		0			0
$465$	−2.48007e6	+	1.25320e7i	−0.531902	+	2.68774i
$466$		0			0
$467$	2.18280e6	−	2.18280e6i	0.463150	−	0.463150i	−0.436537	−	0.899686i	$-0.643795\pi$
$467$	2.18280e6	−	2.18280e6i	0.463150	−	0.463150i	0.899686	+	0.436537i	$0.143795\pi$
$468$		0			0
$469$		−	4.45859e6i		−	0.935978i
$470$		0			0
$471$		−	7.11779e6i		−	1.47840i
$472$		0			0
$473$	−4.59572e6	+	4.59572e6i	−0.944497	+	0.944497i
$474$		0			0
$475$	−1.91417e6	+	4.64682e6i	−0.389266	+	0.944979i
$476$		0			0
$477$	1.14601e7	+	1.14601e7i	2.30617	+	2.30617i
$478$		0			0
$479$	−1.70781e6			−0.340095			−0.170047	−	0.985436i	$-0.554392\pi$
$479$	−1.70781e6			−0.340095			−0.170047	+	0.985436i	$0.554392\pi$
$480$		0			0
$481$	940959.			0.185442
$482$		0			0
$483$	1.08680e6	+	1.08680e6i	0.211973	+	0.211973i
$484$		0			0
$485$	−9.48330e6	−	1.87674e6i	−1.83065	−	0.362284i
$486$		0			0
$487$	3.89197e6	−	3.89197e6i	0.743613	−	0.743613i	−0.229658	−	0.973271i	$-0.573761\pi$
$487$	3.89197e6	−	3.89197e6i	0.743613	−	0.743613i	0.973271	+	0.229658i	$0.0737608\pi$
$488$		0			0
$489$			55629.6i			0.0105204i
$490$		0			0
$491$		−	3.79408e6i		−	0.710236i	−0.934821	−	0.355118i	$-0.884441\pi$
$491$		−	3.79408e6i		−	0.710236i	0.934821	−	0.355118i	$-0.115559\pi$
$492$		0			0
$493$	3.57123e6	−	3.57123e6i	0.661761	−	0.661761i
$494$		0			0
$495$	1.52100e7	−	1.01844e7i	2.79007	−	1.86820i
$496$		0			0
$497$	75569.3	+	75569.3i	0.0137232	+	0.0137232i
$498$		0			0
$499$	−5.45318e6			−0.980388			−0.490194	−	0.871613i	$-0.663074\pi$
$499$	−5.45318e6			−0.980388			−0.490194	+	0.871613i	$0.663074\pi$
$500$		0			0
$501$	−1.31737e7			−2.34483
$502$		0			0
$503$	−5.05453e6	−	5.05453e6i	−0.890760	−	0.890760i	0.103834	−	0.994595i	$-0.466889\pi$
$503$	−5.05453e6	−	5.05453e6i	−0.890760	−	0.890760i	−0.994595	+	0.103834i	$0.966889\pi$
$504$		0			0
$505$	2.29102e6	−	1.53404e6i	0.399761	−	0.267676i
$506$		0			0
$507$	−6.75438e6	+	6.75438e6i	−1.16699	+	1.16699i
$508$		0			0
$509$		−	5.72886e6i		−	0.980108i	−0.871692	−	0.490054i	$-0.836977\pi$
$509$		−	5.72886e6i		−	0.980108i	0.871692	−	0.490054i	$-0.163023\pi$
$510$		0			0
$511$			3.31807e6i			0.562125i
$512$		0			0
$513$	1.13110e7	−	1.13110e7i	1.89762	−	1.89762i
$514$		0			0
$515$	4.89844e6	+	969396.i	0.813841	+	0.161058i
$516$		0			0
$517$	2.83889e6	+	2.83889e6i	0.467113	+	0.467113i
$518$		0			0
$519$	−7.09033e6			−1.15544
$520$		0			0
$521$	−6.55438e6			−1.05788			−0.528941	−	0.848659i	$-0.677411\pi$
$521$	−6.55438e6			−1.05788			−0.528941	+	0.848659i	$0.677411\pi$
$522$		0			0
$523$	7.16908e6	+	7.16908e6i	1.14606	+	1.14606i	0.987320	+	0.158745i	$0.0507447\pi$
$523$	7.16908e6	+	7.16908e6i	1.14606	+	1.14606i	0.158745	+	0.987320i	$0.449255\pi$
$524$		0			0
$525$	3.73197e6	−	9.05969e6i	0.590935	−	1.43455i
$526$		0			0
$527$	−3.78143e6	+	3.78143e6i	−0.593103	+	0.593103i
$528$		0			0
$529$		−	6.19606e6i		−	0.962667i
$530$		0			0
$531$		−	64033.0i		−	0.00985524i
$532$		0			0
$533$	2.64515e6	−	2.64515e6i	0.403304	−	0.403304i
$534$		0			0
$535$	−386380.	+	1.95241e6i	−0.0583619	+	0.294908i
$536$		0			0
$537$	1.22379e6	+	1.22379e6i	0.183135	+	0.183135i
$538$		0			0
$539$	−2.77156e6			−0.410915
$540$		0			0
$541$	47552.7			0.00698526			0.00349263	−	0.999994i	$-0.498888\pi$
$541$	47552.7			0.00698526			0.00349263	+	0.999994i	$0.498888\pi$
$542$		0			0
$543$	−1.44905e6	−	1.44905e6i	−0.210904	−	0.210904i
$544$		0			0
$545$	2.48110e6	+	3.70541e6i	0.357811	+	0.534373i
$546$		0			0
$547$	−4.55854e6	+	4.55854e6i	−0.651415	+	0.651415i	−0.953334	−	0.301919i	$-0.902373\pi$
$547$	−4.55854e6	+	4.55854e6i	−0.651415	+	0.651415i	0.301919	+	0.953334i	$0.402373\pi$
$548$		0			0
$549$			8.40134e6i			1.18965i
$550$		0			0
$551$		−	1.20401e7i		−	1.68947i
$552$		0			0
$553$	721672.	−	721672.i	0.100352	−	0.100352i
$554$		0			0
$555$	−4.22153e6	−	6.30465e6i	−0.581751	−	0.868818i
$556$		0			0
$557$	3.11785e6	+	3.11785e6i	0.425811	+	0.425811i	0.887199	−	0.461388i	$-0.152648\pi$
$557$	3.11785e6	+	3.11785e6i	0.425811	+	0.425811i	−0.461388	+	0.887199i	$0.652648\pi$
$558$		0			0
$559$	2.33261e6			0.315727
$560$		0			0
$561$	1.08290e7			1.45272
$562$		0			0
$563$	−8.10403e6	−	8.10403e6i	−1.07753	−	1.07753i	−0.996730	−	0.0808018i	$-0.974252\pi$
$563$	−8.10403e6	−	8.10403e6i	−1.07753	−	1.07753i	−0.0808018	−	0.996730i	$-0.525748\pi$
$564$		0			0
$565$	−1.44078e6	+	7.28035e6i	−0.189878	+	0.959470i
$566$		0			0
$567$	−1.10629e7	+	1.10629e7i	−1.44515	+	1.44515i
$568$		0			0
$569$			3.79477e6i			0.491366i	0.969350	+	0.245683i	$0.0790122\pi$
$569$			3.79477e6i			0.491366i	−0.969350	+	0.245683i	$0.920988\pi$
$570$		0			0
$571$			3.17124e6i			0.407042i	0.979071	+	0.203521i	$0.0652385\pi$
$571$			3.17124e6i			0.407042i	−0.979071	+	0.203521i	$0.934761\pi$
$572$		0			0
$573$	−1.83127e7	+	1.83127e7i	−2.33005	+	2.33005i
$574$		0			0
$575$	1.41410e6	−	588970.i	0.178365	−	0.0742889i
$576$		0			0
$577$	−407507.	−	407507.i	−0.0509560	−	0.0509560i	0.681170	−	0.732126i	$-0.261472\pi$
$577$	−407507.	−	407507.i	−0.0509560	−	0.0509560i	−0.732126	+	0.681170i	$0.761472\pi$
$578$		0			0
$579$	−2.09909e7			−2.60216
$580$		0			0
$581$	−1.45387e6			−0.178684
$582$		0			0
$583$	1.08522e7	+	1.08522e7i	1.32235	+	1.32235i
$584$		0			0
$585$	−6.44460e6	−	1.27538e6i	−0.778586	−	0.154081i
$586$		0			0
$587$	3.38451e6	−	3.38451e6i	0.405416	−	0.405416i	−0.474721	−	0.880136i	$-0.657451\pi$
$587$	3.38451e6	−	3.38451e6i	0.405416	−	0.405416i	0.880136	+	0.474721i	$0.157451\pi$
$588$		0			0
$589$			1.27488e7i			1.51419i
$590$		0			0
$591$			1.30533e7i			1.53728i
$592$		0			0
$593$	1.57715e6	−	1.57715e6i	0.184178	−	0.184178i	−0.608996	−	0.793174i	$-0.708427\pi$
$593$	1.57715e6	−	1.57715e6i	0.184178	−	0.184178i	0.793174	+	0.608996i	$0.208427\pi$
$594$		0			0
$595$	3.40817e6	−	2.28207e6i	0.394665	−	0.264264i
$596$		0			0
$597$	−9.35406e6	−	9.35406e6i	−1.07415	−	1.07415i
$598$		0			0
$599$	1.09992e7			1.25254			0.626272	−	0.779605i	$-0.284580\pi$
$599$	1.09992e7			1.25254			0.626272	+	0.779605i	$0.284580\pi$
$600$		0			0
$601$	1.44320e7			1.62982			0.814909	−	0.579589i	$-0.196787\pi$
$601$	1.44320e7			1.62982			0.814909	+	0.579589i	$0.196787\pi$
$602$		0			0
$603$	1.70451e7	+	1.70451e7i	1.90900	+	1.90900i
$604$		0			0
$605$	6.92231e6	−	4.63511e6i	0.768887	−	0.514839i
$606$		0			0
$607$	−4.21170e6	+	4.21170e6i	−0.463966	+	0.463966i	−0.899953	−	0.435987i	$-0.856399\pi$
$607$	−4.21170e6	+	4.21170e6i	−0.463966	+	0.463966i	0.435987	+	0.899953i	$0.356399\pi$
$608$		0			0
$609$			2.34740e7i			2.56474i
$610$		0			0
$611$		−	1.44091e6i		−	0.156147i
$612$		0			0
$613$	3.54916e6	−	3.54916e6i	0.381482	−	0.381482i	−0.490154	−	0.871636i	$-0.663059\pi$
$613$	3.54916e6	−	3.54916e6i	0.381482	−	0.381482i	0.871636	+	0.490154i	$0.163059\pi$
$614$		0			0
$615$	−2.95904e7	−	5.85591e6i	−3.15474	−	0.624320i
$616$		0			0
$617$	2.56234e6	+	2.56234e6i	0.270971	+	0.270971i	0.829491	−	0.558520i	$-0.188631\pi$
$617$	2.56234e6	+	2.56234e6i	0.270971	+	0.270971i	−0.558520	+	0.829491i	$0.688631\pi$
$618$		0			0
$619$	6.34177e6			0.665248			0.332624	−	0.943060i	$-0.392066\pi$
$619$	6.34177e6			0.665248			0.332624	+	0.943060i	$0.392066\pi$
$620$		0			0
$621$	−4.87575e6			−0.507356
$622$		0			0
$623$	−652925.	−	652925.i	−0.0673974	−	0.0673974i
$624$		0			0
$625$	−6.93222e6	−	6.87836e6i	−0.709859	−	0.704344i
$626$		0			0
$627$	1.82546e7	−	1.82546e7i	1.85440	−	1.85440i
$628$		0			0
$629$		−	3.17620e6i		−	0.320096i
$630$		0			0
$631$		−	4.54433e6i		−	0.454356i	−0.973853	−	0.227178i	$-0.927050\pi$
$631$		−	4.54433e6i		−	0.454356i	0.973853	−	0.227178i	$-0.0729500\pi$
$632$		0			0
$633$	1.14528e7	−	1.14528e7i	1.13607	−	1.13607i
$634$		0			0
$635$	−86205.0	+	435601.i	−0.00848395	+	0.0428701i
$636$		0			0
$637$	703367.	+	703367.i	0.0686805	+	0.0686805i
$638$		0			0
$639$	−577800.			−0.0559790
$640$		0			0
$641$	6.59831e6			0.634290			0.317145	−	0.948377i	$-0.397276\pi$
$641$	6.59831e6			0.634290			0.317145	+	0.948377i	$0.397276\pi$
$642$		0			0
$643$	−6.81404e6	−	6.81404e6i	−0.649946	−	0.649946i	0.303034	−	0.952980i	$-0.402000\pi$
$643$	−6.81404e6	−	6.81404e6i	−0.649946	−	0.649946i	−0.952980	+	0.303034i	$0.902000\pi$
$644$		0			0
$645$	−1.04650e7	−	1.56290e7i	−0.990470	−	1.47922i
$646$		0			0
$647$	−1.11169e7	+	1.11169e7i	−1.04405	+	1.04405i	−0.0450702	+	0.998984i	$0.514351\pi$
$647$	−1.11169e7	+	1.11169e7i	−1.04405	+	1.04405i	−0.998984	+	0.0450702i	$0.985649\pi$
$648$		0			0
$649$		−	60636.4i		−	0.00565095i
$650$		0			0
$651$		−	2.48556e7i		−	2.29865i
$652$		0			0
$653$	−1.27508e7	+	1.27508e7i	−1.17019	+	1.17019i	−0.188025	+	0.982164i	$0.560209\pi$
$653$	−1.27508e7	+	1.27508e7i	−1.17019	+	1.17019i	−0.982164	+	0.188025i	$0.939791\pi$
$654$		0			0
$655$	1.76399e6	+	2.63443e6i	0.160654	+	0.239929i
$656$		0			0
$657$	−1.26849e7	−	1.26849e7i	−1.14650	−	1.14650i
$658$		0			0
$659$	−9.10667e6			−0.816857			−0.408429	−	0.912790i	$-0.633923\pi$
$659$	−9.10667e6			−0.816857			−0.408429	+	0.912790i	$0.633923\pi$
$660$		0			0
$661$	1.89210e7			1.68438			0.842189	−	0.539182i	$-0.181266\pi$
$661$	1.89210e7			1.68438			0.842189	+	0.539182i	$0.181266\pi$
$662$		0			0
$663$	−2.74820e6	−	2.74820e6i	−0.242809	−	0.242809i
$664$		0			0
$665$	1.89826e6	−	9.59206e6i	0.166457	−	0.841119i
$666$		0			0
$667$	−2.59501e6	+	2.59501e6i	−0.225852	+	0.225852i
$668$		0			0
$669$		−	3.34884e7i		−	2.89287i
$670$		0			0
$671$			7.95570e6i			0.682138i
$672$		0			0
$673$	3.24765e6	−	3.24765e6i	0.276395	−	0.276395i	−0.555273	−	0.831668i	$-0.687386\pi$
$673$	3.24765e6	−	3.24765e6i	0.276395	−	0.276395i	0.831668	+	0.555273i	$0.187386\pi$
$674$		0			0
$675$	1.19510e7	+	2.86939e7i	1.00959	+	2.42399i
$676$		0			0
$677$	1.17838e7	+	1.17838e7i	0.988128	+	0.988128i	0.999930	−	0.0118027i	$-0.00375700\pi$
$677$	1.17838e7	+	1.17838e7i	0.988128	+	0.988128i	−0.0118027	+	0.999930i	$0.503757\pi$
$678$		0			0
$679$	1.88089e7			1.56563
$680$		0			0
$681$	1.12556e7			0.930036
$682$		0			0
$683$	−8.18028e6	−	8.18028e6i	−0.670991	−	0.670991i	0.286954	−	0.957944i	$-0.407357\pi$
$683$	−8.18028e6	−	8.18028e6i	−0.670991	−	0.670991i	−0.957944	+	0.286954i	$0.907357\pi$
$684$		0			0
$685$	7.37502e6	+	1.45951e6i	0.600533	+	0.118845i
$686$		0			0
$687$	1.02573e7	−	1.02573e7i	0.829163	−	0.829163i
$688$		0			0
$689$		−	5.50816e6i		−	0.442037i
$690$		0			0
$691$		−	1.23162e7i		−	0.981253i	−0.871370	−	0.490626i	$-0.836768\pi$
$691$		−	1.23162e7i		−	0.981253i	0.871370	−	0.490626i	$-0.163232\pi$
$692$		0			0
$693$	−2.51833e7	+	2.51833e7i	−1.99196	+	1.99196i
$694$		0			0
$695$	1.28254e7	−	8.58774e6i	1.00718	−	0.674399i
$696$		0			0
$697$	−8.92868e6	−	8.92868e6i	−0.696154	−	0.696154i
$698$		0			0
$699$	2.42501e7			1.87724
$700$		0			0
$701$	−2.10018e7			−1.61421			−0.807107	−	0.590405i	$-0.798968\pi$
$701$	−2.10018e7			−1.61421			−0.807107	+	0.590405i	$0.798968\pi$
$702$		0			0
$703$	−5.35413e6	−	5.35413e6i	−0.408602	−	0.408602i
$704$		0			0
$705$	−9.65443e6	+	6.46450e6i	−0.731566	+	0.489849i
$706$		0			0
$707$	−3.79327e6	+	3.79327e6i	−0.285407	+	0.285407i
$708$		0			0
$709$			1.87952e7i			1.40421i	0.712074	+	0.702105i	$0.247756\pi$
$709$			1.87952e7i			1.40421i	−0.712074	+	0.702105i	$0.752244\pi$
$710$		0			0
$711$			5.51788e6i			0.409353i
$712$		0			0
$713$	2.74775e6	−	2.74775e6i	0.202420	−	0.202420i
$714$		0			0
$715$	−6.10276e6	−	1.20773e6i	−0.446438	−	0.0883496i
$716$		0			0
$717$	−7.09999e6	−	7.09999e6i	−0.515774	−	0.515774i
$718$		0			0
$719$	2.33655e7			1.68559			0.842797	−	0.538232i	$-0.180908\pi$
$719$	2.33655e7			1.68559			0.842797	+	0.538232i	$0.180908\pi$
$720$		0			0
$721$	−9.71544e6			−0.696024
$722$		0			0
$723$	5.15508e6	+	5.15508e6i	0.366767	+	0.366767i
$724$		0			0
$725$	2.16324e7	+	8.91104e6i	1.52848	+	0.629627i
$726$		0			0
$727$	1.18759e7	−	1.18759e7i	0.833355	−	0.833355i	−0.154619	−	0.987974i	$-0.549415\pi$
$727$	1.18759e7	−	1.18759e7i	0.833355	−	0.833355i	0.987974	+	0.154619i	$0.0494151\pi$
$728$		0			0
$729$		−	1.49092e7i		−	1.03905i
$730$		0			0
$731$		−	7.87368e6i		−	0.544985i
$732$		0			0
$733$	8.08731e6	−	8.08731e6i	0.555961	−	0.555961i	−0.372194	−	0.928155i	$-0.621394\pi$
$733$	8.08731e6	−	8.08731e6i	0.555961	−	0.555961i	0.928155	+	0.372194i	$0.121394\pi$
$734$		0			0
$735$	1.55713e6	−	7.86833e6i	0.106318	−	0.537235i
$736$		0			0
$737$	1.61410e7	+	1.61410e7i	1.09461	+	1.09461i
$738$		0			0
$739$	−1.24230e7			−0.836786			−0.418393	−	0.908266i	$-0.637407\pi$
$739$	−1.24230e7			−0.836786			−0.418393	+	0.908266i	$0.637407\pi$
$740$		0			0
$741$	−9.26530e6			−0.619889
$742$		0			0
$743$	9.76492e6	+	9.76492e6i	0.648928	+	0.648928i	0.952734	−	0.303806i	$-0.0982575\pi$
$743$	9.76492e6	+	9.76492e6i	0.648928	+	0.648928i	−0.303806	+	0.952734i	$0.598257\pi$
$744$		0			0
$745$	8.05074e6	+	1.20234e7i	0.531429	+	0.793663i
$746$		0			0
$747$	5.55811e6	−	5.55811e6i	0.364440	−	0.364440i
$748$		0			0
$749$		−	3.87236e6i		−	0.252215i
$750$		0			0
$751$			2.15742e7i			1.39584i	0.716178	+	0.697918i	$0.245890\pi$
$751$			2.15742e7i			1.39584i	−0.716178	+	0.697918i	$0.754110\pi$
$752$		0			0
$753$	−1.00604e7	+	1.00604e7i	−0.646588	+	0.646588i
$754$		0			0
$755$	1.24303e7	+	1.85641e7i	0.793624	+	1.18524i
$756$		0			0
$757$	−1.09605e7	−	1.09605e7i	−0.695169	−	0.695169i	0.268196	−	0.963364i	$-0.413573\pi$
$757$	−1.09605e7	−	1.09605e7i	−0.695169	−	0.695169i	−0.963364	+	0.268196i	$0.913573\pi$
$758$		0			0
$759$	−7.86883e6			−0.495799
$760$		0			0
$761$	−3.94007e6			−0.246628			−0.123314	−	0.992368i	$-0.539352\pi$
$761$	−3.94007e6			−0.246628			−0.123314	+	0.992368i	$0.539352\pi$
$762$		0			0
$763$	−6.13508e6	−	6.13508e6i	−0.381513	−	0.381513i
$764$		0			0
$765$	−4.30503e6	+	2.17537e7i	−0.265964	+	1.34394i
$766$		0			0
$767$	−15388.3	+	15388.3i	−0.000944503	+	0.000944503i
$768$		0			0
$769$		−	1.05073e7i		−	0.640730i	−0.947294	−	0.320365i	$-0.896195\pi$
$769$		−	1.05073e7i		−	0.640730i	0.947294	−	0.320365i	$-0.103805\pi$
$770$		0			0
$771$			5.25645e7i			3.18461i
$772$		0			0
$773$	1.29271e7	−	1.29271e7i	0.778128	−	0.778128i	−0.201384	−	0.979512i	$-0.564544\pi$
$773$	1.29271e7	−	1.29271e7i	0.778128	−	0.778128i	0.979512	+	0.201384i	$0.0645439\pi$
$774$		0			0
$775$	−2.29056e7	−	9.43553e6i	−1.36990	−	0.564303i
$776$		0			0
$777$	1.04387e7	+	1.04387e7i	0.620288	+	0.620288i
$778$		0			0
$779$	−3.01022e7			−1.77728
$780$		0			0
$781$	−547151.			−0.0320981
$782$		0			0
$783$	−5.26563e7	−	5.26563e7i	−3.06934	−	3.06934i
$784$		0			0
$785$	1.35400e7	+	2.67955e6i	0.784230	+	0.155198i
$786$		0			0
$787$	2.08585e7	−	2.08585e7i	1.20046	−	1.20046i	0.226427	−	0.974028i	$-0.427295\pi$
$787$	2.08585e7	−	2.08585e7i	1.20046	−	1.20046i	0.974028	−	0.226427i	$-0.0727045\pi$
$788$		0			0
$789$		−	2.29304e7i		−	1.31135i
$790$		0			0
$791$		−	1.44397e7i		−	0.820571i
$792$		0			0
$793$	2.01900e6	−	2.01900e6i	0.114013	−	0.114013i
$794$		0			0
$795$	−3.69060e7	+	2.47119e7i	−2.07099	+	1.38672i
$796$		0			0
$797$	−2.06132e7	−	2.06132e7i	−1.14947	−	1.14947i	−0.986656	−	0.162817i	$-0.947942\pi$
$797$	−2.06132e7	−	2.06132e7i	−1.14947	−	1.14947i	−0.162817	−	0.986656i	$-0.552058\pi$
$798$		0			0
$799$	−4.86377e6			−0.269529
$800$		0			0
$801$	4.99224e6			0.274925
$802$		0			0
$803$	−1.20121e7	−	1.20121e7i	−0.657398	−	0.657398i
$804$		0			0
$805$	−2.47652e6	+	1.65825e6i	−0.134695	+	0.0901904i
$806$		0			0
$807$	6.23523e6	−	6.23523e6i	0.337030	−	0.337030i
$808$		0			0
$809$			1.76429e7i			0.947761i	0.880589	+	0.473881i	$0.157147\pi$
$809$			1.76429e7i			0.947761i	−0.880589	+	0.473881i	$0.842853\pi$
$810$		0			0
$811$			2.55639e7i			1.36482i	0.730970	+	0.682409i	$0.239068\pi$
$811$			2.55639e7i			1.36482i	−0.730970	+	0.682409i	$0.760932\pi$
$812$		0			0
$813$	4.04422e7	−	4.04422e7i	2.14589	−	2.14589i
$814$		0			0
$815$	−105823.	−	20942.2i	−0.00558064	−	0.00110440i
$816$		0			0
$817$	−1.32727e7	−	1.32727e7i	−0.695671	−	0.695671i
$818$		0			0
$819$	1.27821e7			0.665873
$820$		0			0
$821$	−1.26661e7			−0.655818			−0.327909	−	0.944709i	$-0.606344\pi$
$821$	−1.26661e7			−0.655818			−0.327909	+	0.944709i	$0.606344\pi$
$822$		0			0
$823$	4.72508e6	+	4.72508e6i	0.243170	+	0.243170i	0.818160	−	0.574990i	$-0.194994\pi$
$823$	4.72508e6	+	4.72508e6i	0.243170	+	0.243170i	−0.574990	+	0.818160i	$0.694994\pi$
$824$		0			0
$825$	1.92874e7	+	4.63083e7i	0.986595	+	2.36878i
$826$		0			0
$827$	−1.82843e7	+	1.82843e7i	−0.929637	+	0.929637i	−0.997682	−	0.0680450i	$-0.978324\pi$
$827$	−1.82843e7	+	1.82843e7i	−0.929637	+	0.929637i	0.0680450	+	0.997682i	$0.478324\pi$
$828$		0			0
$829$			2.47048e7i			1.24852i	0.781216	+	0.624260i	$0.214600\pi$
$829$			2.47048e7i			1.24852i	−0.781216	+	0.624260i	$0.785400\pi$
$830$		0			0
$831$			2.73086e7i			1.37182i
$832$		0			0
$833$	2.37421e6	−	2.37421e6i	0.118551	−	0.118551i
$834$		0			0
$835$	4.95933e6	−	2.50599e7i	0.246154	−	1.24383i
$836$		0			0
$837$	5.57556e7	+	5.57556e7i	2.75090	+	2.75090i
$838$		0			0
$839$	3.59284e7			1.76211			0.881054	−	0.473015i	$-0.156835\pi$
$839$	3.59284e7			1.76211			0.881054	+	0.473015i	$0.156835\pi$
$840$		0			0
$841$	−3.55390e7			−1.73267
$842$		0			0
$843$	7.63673e6	+	7.63673e6i	0.370117	+	0.370117i
$844$		0			0
$845$	−1.03059e7	−	1.53914e7i	−0.496530	−	0.741543i
$846$		0			0
$847$	−1.14614e7	+	1.14614e7i	−0.548943	+	0.548943i
$848$		0			0
$849$		−	3.64672e7i		−	1.73633i
$850$		0			0
$851$			2.30796e6i			0.109246i
$852$		0			0
$853$	−2.48308e7	+	2.48308e7i	−1.16847	+	1.16847i	−0.185905	+	0.982568i	$0.559522\pi$
$853$	−2.48308e7	+	2.48308e7i	−1.16847	+	1.16847i	−0.982568	+	0.185905i	$0.940478\pi$
$854$		0			0
$855$	2.94132e7	+	4.39273e7i	1.37603	+	2.05503i
$856$		0			0
$857$	−2.27271e7	−	2.27271e7i	−1.05704	−	1.05704i	−0.998271	−	0.0587713i	$-0.981282\pi$
$857$	−2.27271e7	−	2.27271e7i	−1.05704	−	1.05704i	−0.0587713	−	0.998271i	$-0.518718\pi$
$858$		0			0
$859$	−2.65352e7			−1.22698			−0.613492	−	0.789701i	$-0.710236\pi$
$859$	−2.65352e7			−1.22698			−0.613492	+	0.789701i	$0.710236\pi$
$860$		0			0
$861$	5.86889e7			2.69804
$862$		0			0
$863$	−2.09059e7	−	2.09059e7i	−0.955524	−	0.955524i	0.0435285	−	0.999052i	$-0.486140\pi$
$863$	−2.09059e7	−	2.09059e7i	−0.955524	−	0.955524i	−0.999052	+	0.0435285i	$0.986140\pi$
$864$		0			0
$865$	2.66921e6	−	1.34877e7i	0.121295	−	0.612913i
$866$		0			0
$867$	1.96662e7	−	1.96662e7i	0.888533	−	0.888533i
$868$		0			0
$869$			5.22519e6i			0.234721i
$870$		0			0
$871$		−	8.19253e6i		−	0.365909i
$872$		0			0
$873$	−7.19062e7	+	7.19062e7i	−3.19323	+	3.19323i
$874$		0			0
$875$	1.58291e7	+	1.05098e7i	0.698932	+	0.464061i
$876$		0			0
$877$	1.73327e7	+	1.73327e7i	0.760968	+	0.760968i	0.976497	−	0.215529i	$-0.0691477\pi$
$877$	1.73327e7	+	1.73327e7i	0.760968	+	0.760968i	−0.215529	+	0.976497i	$0.569148\pi$
$878$		0			0
$879$	−3.21341e7			−1.40279
$880$		0			0
$881$	−3.30949e7			−1.43655			−0.718276	−	0.695758i	$-0.755068\pi$
$881$	−3.30949e7			−1.43655			−0.718276	+	0.695758i	$0.755068\pi$
$882$		0			0
$883$	−7.21332e6	−	7.21332e6i	−0.311339	−	0.311339i	0.534089	−	0.845428i	$-0.320655\pi$
$883$	−7.21332e6	−	7.21332e6i	−0.311339	−	0.311339i	−0.845428	+	0.534089i	$0.820655\pi$
$884$		0			0
$885$	172144.	+	34067.1i	0.00738811	+	0.00146210i
$886$		0			0
$887$	−2.63581e7	+	2.63581e7i	−1.12488	+	1.12488i	−0.133879	+	0.990998i	$0.542743\pi$
$887$	−2.63581e7	+	2.63581e7i	−1.12488	+	1.12488i	−0.990998	+	0.133879i	$0.957257\pi$
$888$		0			0
$889$		−	863960.i		−	0.0366639i
$890$		0			0
$891$		−	8.01000e7i		−	3.38017i
$892$		0			0
$893$	−8.19887e6	+	8.19887e6i	−0.344053	+	0.344053i
$894$		0			0
$895$	−2.78869e6	+	1.86728e6i	−0.116371	+	0.0779205i
$896$		0			0
$897$	1.99696e6	+	1.99696e6i	0.0828681	+	0.0828681i
$898$		0			0
$899$	5.93493e7			2.44915
$900$		0			0
$901$	−1.85927e7			−0.763011
$902$		0			0
$903$	2.58771e7	+	2.58771e7i	1.05608	+	1.05608i
$904$		0			0
$905$	3.30200e6	−	2.21099e6i	0.134016	−	0.0897356i
$906$		0			0
$907$	−5.10817e6	+	5.10817e6i	−0.206180	+	0.206180i	−0.802642	−	0.596462i	$-0.796573\pi$
$907$	−5.10817e6	+	5.10817e6i	−0.206180	+	0.206180i	0.596462	+	0.802642i	$0.296573\pi$
$908$		0			0
$909$		−	2.90032e7i		−	1.16422i
$910$		0			0
$911$			1.09815e7i			0.438397i	0.975680	+	0.219198i	$0.0703442\pi$
$911$			1.09815e7i			0.438397i	−0.975680	+	0.219198i	$0.929656\pi$
$912$		0			0
$913$	5.26329e6	−	5.26329e6i	0.208968	−	0.208968i
$914$		0			0
$915$	−2.25859e7	−	4.46972e6i	−0.891834	−	0.176493i
$916$		0			0
$917$	−4.36185e6	−	4.36185e6i	−0.171296	−	0.171296i
$918$		0			0
$919$	7.43231e6			0.290292			0.145146	−	0.989410i	$-0.453635\pi$
$919$	7.43231e6			0.290292			0.145146	+	0.989410i	$0.453635\pi$
$920$		0			0
$921$	8.30625e7			3.22668
$922$		0			0
$923$	138856.	+	138856.i	0.00536490	+	0.00536490i
$924$		0			0
$925$	1.35824e7	−	5.65706e6i	0.521942	−	0.217389i
$926$		0			0
$927$	3.71419e7	−	3.71419e7i	1.41960	−	1.41960i
$928$		0			0
$929$			4.06455e6i			0.154516i	0.997011	+	0.0772580i	$0.0246165\pi$
$929$			4.06455e6i			0.154516i	−0.997011	+	0.0772580i	$0.975383\pi$
$930$		0			0
$931$		−	8.00442e6i		−	0.302661i
$932$		0			0
$933$	5.21128e7	−	5.21128e7i	1.95993	−	1.95993i
$934$		0			0
$935$	−4.07668e6	+	2.05998e7i	−0.152503	+	0.770608i
$936$		0			0
$937$	−2.62414e7	−	2.62414e7i	−0.976423	−	0.976423i	0.0233056	−	0.999728i	$-0.492581\pi$
$937$	−2.62414e7	−	2.62414e7i	−0.976423	−	0.976423i	−0.999728	+	0.0233056i	$0.992581\pi$
$938$		0			0
$939$	4.01906e7			1.48751
$940$		0			0
$941$	2.31934e7			0.853867			0.426934	−	0.904283i	$-0.359594\pi$
$941$	2.31934e7			0.853867			0.426934	+	0.904283i	$0.359594\pi$
$942$		0			0
$943$	6.48795e6	+	6.48795e6i	0.237590	+	0.237590i
$944$		0			0
$945$	−3.36482e7	−	5.02519e7i	−1.22569	−	1.83051i
$946$		0			0
$947$	−1.30147e7	+	1.30147e7i	−0.471585	+	0.471585i	−0.902427	−	0.430842i	$-0.858217\pi$
$947$	−1.30147e7	+	1.30147e7i	−0.471585	+	0.471585i	0.430842	+	0.902427i	$0.358217\pi$
$948$		0			0
$949$			6.09685e6i			0.219756i
$950$		0			0
$951$		−	4.63058e6i		−	0.166029i
$952$		0			0
$953$	2.46875e6	−	2.46875e6i	0.0880531	−	0.0880531i	−0.661708	−	0.749761i	$-0.730168\pi$
$953$	2.46875e6	−	2.46875e6i	0.0880531	−	0.0880531i	0.749761	+	0.661708i	$0.230168\pi$
$954$		0			0
$955$	−2.79418e7	−	4.17297e7i	−0.991393	−	1.48060i
$956$		0			0
$957$	−8.49804e7	−	8.49804e7i	−2.99943	−	2.99943i
$958$		0			0
$959$	−1.46274e7			−0.513596
$960$		0			0
$961$	−3.42134e7			−1.19505
$962$		0			0
$963$	1.48039e7	+	1.48039e7i	0.514413	+	0.514413i
$964$		0			0
$965$	7.90218e6	−	3.99303e7i	0.273167	−	1.38034i
$966$		0			0
$967$	2.59267e7	−	2.59267e7i	0.891624	−	0.891624i	−0.103052	−	0.994676i	$-0.532861\pi$
$967$	2.59267e7	−	2.59267e7i	0.891624	−	0.891624i	0.994676	+	0.103052i	$0.0328609\pi$
$968$		0			0
$969$			3.12749e7i			1.07001i
$970$		0			0
$971$			5.51016e6i			0.187550i	0.995593	+	0.0937748i	$0.0298934\pi$
$971$			5.51016e6i			0.187550i	−0.995593	+	0.0937748i	$0.970107\pi$
$972$		0			0
$973$	−2.12351e7	+	2.12351e7i	−0.719073	+	0.719073i
$974$		0			0
$975$	6.85738e6	−	1.66469e7i	0.231018	−	0.560818i
$976$		0			0
$977$	3.64191e7	+	3.64191e7i	1.22065	+	1.22065i	0.967399	+	0.253256i	$0.0815014\pi$
$977$	3.64191e7	+	3.64191e7i	1.22065	+	1.22065i	0.253256	+	0.967399i	$0.418499\pi$
$978$		0			0
$979$	4.72743e6			0.157641
$980$		0			0
$981$	4.69086e7			1.55625
$982$		0			0
$983$	1.26298e7	+	1.26298e7i	0.416881	+	0.416881i	0.884127	−	0.467246i	$-0.154754\pi$
$983$	1.26298e7	+	1.26298e7i	0.416881	+	0.416881i	−0.467246	+	0.884127i	$0.654754\pi$
$984$		0			0
$985$	−2.48310e7	−	4.91403e6i	−0.815462	−	0.161379i
$986$		0			0
$987$	1.59849e7	−	1.59849e7i	0.522298	−	0.522298i
$988$		0			0
$989$			5.72135e6i			0.185998i
$990$		0			0
$991$		−	3.90456e7i		−	1.26295i	−0.775395	−	0.631477i	$-0.782449\pi$
$991$		−	3.90456e7i		−	1.26295i	0.775395	−	0.631477i	$-0.217551\pi$
$992$		0			0
$993$	6.67590e6	−	6.67590e6i	0.214851	−	0.214851i
$994$		0			0
$995$	2.13154e7	−	1.42726e7i	0.682552	−	0.457030i
$996$		0			0
$997$	−2.55989e6	−	2.55989e6i	−0.0815612	−	0.0815612i	0.665149	−	0.746710i	$-0.268368\pi$
$997$	−2.55989e6	−	2.55989e6i	−0.0815612	−	0.0815612i	−0.746710	+	0.665149i	$0.768368\pi$
$998$		0			0
$999$	−4.68316e7			−1.48465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.n.d.47.10	yes	20
4.3	odd	2	inner	80.6.n.d.47.1	✓	20
5.2	odd	4		400.6.n.g.143.10		20
5.3	odd	4	inner	80.6.n.d.63.1	yes	20
5.4	even	2		400.6.n.g.207.1		20
20.3	even	4	inner	80.6.n.d.63.10	yes	20
20.7	even	4		400.6.n.g.143.1		20
20.19	odd	2		400.6.n.g.207.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.1	✓	20	4.3	odd	2	inner
80.6.n.d.47.10	yes	20	1.1	even	1	trivial
80.6.n.d.63.1	yes	20	5.3	odd	4	inner
80.6.n.d.63.10	yes	20	20.3	even	4	inner
400.6.n.g.143.1		20	20.7	even	4
400.6.n.g.143.10		20	5.2	odd	4
400.6.n.g.207.1		20	5.4	even	2
400.6.n.g.207.10		20	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(20.3843 + 20.3843i) q^{3} +(-46.4503 + 31.1026i) q^{5} +(76.9082 - 76.9082i) q^{7} +588.037i q^{9} +O(q^{10})\)	\(q+(20.3843 + 20.3843i) q^{3} +(-46.4503 + 31.1026i) q^{5} +(76.9082 - 76.9082i) q^{7} +588.037i q^{9} +556.846i q^{11} +(141.317 - 141.317i) q^{13} +(-1580.86 - 312.851i) q^{15} +(-477.013 - 477.013i) q^{17} -1608.20 q^{19} +3135.44 q^{21} +(346.617 + 346.617i) q^{23} +(1190.25 - 2889.45i) q^{25} +(-7033.34 + 7033.34i) q^{27} +7486.67i q^{29} -7927.33i q^{31} +(-11350.9 + 11350.9i) q^{33} +(-1180.36 + 5964.46i) q^{35} +(3329.26 + 3329.26i) q^{37} +5761.27 q^{39} +18717.9 q^{41} +(8253.12 + 8253.12i) q^{43} +(-18289.5 - 27314.5i) q^{45} +(5098.15 - 5098.15i) q^{47} +4977.25i q^{49} -19447.1i q^{51} +(19488.7 - 19488.7i) q^{53} +(-17319.4 - 25865.6i) q^{55} +(-32782.1 - 32782.1i) q^{57} -108.893 q^{59} +14287.1 q^{61} +(45224.9 + 45224.9i) q^{63} +(-2168.88 + 10959.5i) q^{65} +(28986.5 - 28986.5i) q^{67} +14131.1i q^{69} +982.591i q^{71} +(-21571.6 + 21571.6i) q^{73} +(83161.8 - 34636.9i) q^{75} +(42826.0 + 42826.0i) q^{77} +9383.55 q^{79} -143846. q^{81} +(-9451.97 - 9451.97i) q^{83} +(36993.7 + 7321.02i) q^{85} +(-152610. + 152610. i) q^{87} -8489.66i q^{89} -21736.8i q^{91} +(161593. - 161593. i) q^{93} +(74701.5 - 50019.3i) q^{95} +(122282. + 122282. i) q^{97} -327446. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 44 q^{5}+O(q^{10}) \) 20 * q - 44 * q^5	\( 20 q - 44 q^{5} + 804 q^{13} - 2236 q^{17} - 4520 q^{21} + 948 q^{25} - 11096 q^{33} + 44260 q^{37} - 6760 q^{41} - 92816 q^{45} + 182452 q^{53} - 34288 q^{57} - 41080 q^{61} - 155772 q^{65} + 264372 q^{73} + 399304 q^{77} - 520220 q^{81} - 344796 q^{85} + 713496 q^{93} + 374772 q^{97}+O(q^{100}) \) 20 * q - 44 * q^5 + 804 * q^13 - 2236 * q^17 - 4520 * q^21 + 948 * q^25 - 11096 * q^33 + 44260 * q^37 - 6760 * q^41 - 92816 * q^45 + 182452 * q^53 - 34288 * q^57 - 41080 * q^61 - 155772 * q^65 + 264372 * q^73 + 399304 * q^77 - 520220 * q^81 - 344796 * q^85 + 713496 * q^93 + 374772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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