LMFDB - Embedded newform 84.7.m.b.61.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,7,Mod(61,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("84.61");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	84.m (of order $6$, 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:	$19.3245430241$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + \cdots + 5192355364 $ x^8 - 2x^7 + 1061x^6 + 35442x^5 + 1155979x^4 + 17325616x^3 + 201523590x^2 + 1200774512*x + 5192355364
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.3
Root			$-5.94197 + 10.2918i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.7.m.b.73.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5000 - 7.79423i) q^{3} +(-0.0783677 - 0.0452456i) q^{5} +(-218.562 + 264.348i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})$	\(q+(13.5000 - 7.79423i) q^{3} +(-0.0783677 - 0.0452456i) q^{5} +(-218.562 + 264.348i) q^{7} +(121.500 - 210.444i) q^{9} +(-1139.94 - 1974.43i) q^{11} -11.1483i q^{13} -1.41062 q^{15} +(-4462.04 + 2576.16i) q^{17} +(-1501.28 - 866.766i) q^{19} +(-890.195 + 5272.22i) q^{21} +(1173.42 - 2032.43i) q^{23} +(-7812.50 - 13531.6i) q^{25} -3788.00i q^{27} -13880.2 q^{29} +(-35997.5 + 20783.2i) q^{31} +(-30778.3 - 17769.9i) q^{33} +(29.0887 - 10.8274i) q^{35} +(-1039.36 + 1800.23i) q^{37} +(-86.8924 - 150.502i) q^{39} -33805.2i q^{41} -105285. q^{43} +(-19.0433 + 10.9947i) q^{45} +(-86534.1 - 49960.5i) q^{47} +(-22110.6 - 115553. i) q^{49} +(-40158.4 + 69556.4i) q^{51} +(52238.9 + 90480.4i) q^{53} +206.309i q^{55} -27023.1 q^{57} +(194414. - 112245. i) q^{59} +(266700. + 153979. i) q^{61} +(29075.2 + 78113.3i) q^{63} +(-0.504411 + 0.873666i) q^{65} +(118138. + 204621. i) q^{67} -36583.8i q^{69} +586114. q^{71} +(-558139. + 322242. i) q^{73} +(-210937. - 121785. i) q^{75} +(771084. + 130195. i) q^{77} +(269093. - 466084. i) q^{79} +(-29524.5 - 51137.9i) q^{81} -602657. i q^{83} +466.240 q^{85} +(-187382. + 108185. i) q^{87} +(1.06824e6 + 616746. i) q^{89} +(2947.03 + 2436.59i) q^{91} +(-323978. + 561146. i) q^{93} +(78.4346 + 135.853i) q^{95} +339514. i q^{97} -554010. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9}+O(q^{10}) $ 8 * q + 108 * q^3 - 294 * q^5 + 232 * q^7 + 972 * q^9	$ 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9} + 378 q^{11} - 5292 q^{15} + 852 q^{17} + 3690 q^{19} + 3942 q^{21} + 15600 q^{23} + 3386 q^{25} - 68604 q^{29} + 23028 q^{31} + 10206 q^{33} + 93828 q^{35} + 15914 q^{37} + 25326 q^{39} - 170044 q^{43} - 71442 q^{45} + 102180 q^{47} + 157340 q^{49} + 7668 q^{51} + 196410 q^{53} + 66420 q^{57} - 662550 q^{59} - 23928 q^{61} + 50058 q^{63} + 14892 q^{65} + 774838 q^{67} - 721896 q^{71} - 1219050 q^{73} + 91422 q^{75} + 1584738 q^{77} - 493868 q^{79} - 236196 q^{81} - 1329816 q^{85} - 926154 q^{87} + 604260 q^{89} + 3831690 q^{91} + 207252 q^{93} + 448944 q^{95} + 183708 q^{99}+O(q^{100}) $ 8 * q + 108 * q^3 - 294 * q^5 + 232 * q^7 + 972 * q^9 + 378 * q^11 - 5292 * q^15 + 852 * q^17 + 3690 * q^19 + 3942 * q^21 + 15600 * q^23 + 3386 * q^25 - 68604 * q^29 + 23028 * q^31 + 10206 * q^33 + 93828 * q^35 + 15914 * q^37 + 25326 * q^39 - 170044 * q^43 - 71442 * q^45 + 102180 * q^47 + 157340 * q^49 + 7668 * q^51 + 196410 * q^53 + 66420 * q^57 - 662550 * q^59 - 23928 * q^61 + 50058 * q^63 + 14892 * q^65 + 774838 * q^67 - 721896 * q^71 - 1219050 * q^73 + 91422 * q^75 + 1584738 * q^77 - 493868 * q^79 - 236196 * q^81 - 1329816 * q^85 - 926154 * q^87 + 604260 * q^89 + 3831690 * q^91 + 207252 * q^93 + 448944 * q^95 + 183708 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	13.5000	−	7.79423i	0.500000	−	0.288675i
$3$	13.5000	−	7.79423i	0.500000	−	0.288675i
$4$		0			0
$5$	−0.0783677	−	0.0452456i	−0.000626941	−	0.000361965i	0.499686	−	0.866206i	$-0.333449\pi$
$5$	−0.0783677	−	0.0452456i	−0.000626941	−	0.000361965i	−0.500313	+	0.865844i	$0.666782\pi$
$6$		0			0
$7$	−218.562	+	264.348i	−0.637206	+	0.770693i
$7$	−218.562	+	264.348i	−0.637206	+	0.770693i
$8$		0			0
$9$	121.500	−	210.444i	0.166667	−	0.288675i
$10$		0			0
$11$	−1139.94	−	1974.43i	−0.856453	−	1.48342i	−0.875291	−	0.483597i	$-0.839330\pi$
$11$	−1139.94	−	1974.43i	−0.856453	−	1.48342i	0.0188378	−	0.999823i	$-0.494003\pi$
$12$		0			0
$13$		−	11.1483i		−	0.00507433i	−0.999997	−	0.00253716i	$-0.999192\pi$
$13$		−	11.1483i		−	0.00507433i	0.999997	−	0.00253716i	$-0.000807605\pi$
$14$		0			0
$15$	−1.41062			−0.000417961
$16$		0			0
$17$	−4462.04	+	2576.16i	−0.908211	+	0.524356i	−0.879855	−	0.475242i	$-0.842361\pi$
$17$	−4462.04	+	2576.16i	−0.908211	+	0.524356i	−0.0283562	+	0.999598i	$0.509027\pi$
$18$		0			0
$19$	−1501.28	−	866.766i	−0.218878	−	0.126369i	0.386553	−	0.922267i	$-0.373666\pi$
$19$	−1501.28	−	866.766i	−0.218878	−	0.126369i	−0.605430	+	0.795898i	$0.706999\pi$
$20$		0			0
$21$	−890.195	+	5272.22i	−0.0961230	+	0.569292i
$22$		0			0
$23$	1173.42	−	2032.43i	0.0964432	−	0.167045i	−0.813767	−	0.581192i	$-0.802587\pi$
$23$	1173.42	−	2032.43i	0.0964432	−	0.167045i	0.910210	+	0.414147i	$0.135920\pi$
$24$		0			0
$25$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$26$		0			0
$27$		−	3788.00i		−	0.192450i
$28$		0			0
$29$	−13880.2			−0.569116			−0.284558	−	0.958659i	$-0.591847\pi$
$29$	−13880.2			−0.569116			−0.284558	+	0.958659i	$0.591847\pi$
$30$		0			0
$31$	−35997.5	+	20783.2i	−1.20834	+	0.697633i	−0.962396	−	0.271652i	$-0.912430\pi$
$31$	−35997.5	+	20783.2i	−1.20834	+	0.697633i	−0.245940	+	0.969285i	$0.579097\pi$
$32$		0			0
$33$	−30778.3	−	17769.9i	−0.856453	−	0.494473i
$34$		0			0
$35$	29.0887	−	10.8274i	0.000678455	−	0.000252533i
$36$		0			0
$37$	−1039.36	+	1800.23i	−0.0205193	+	0.0355404i	−0.876103	−	0.482125i	$-0.839865\pi$
$37$	−1039.36	+	1800.23i	−0.0205193	+	0.0355404i	0.855583	+	0.517665i	$0.173199\pi$
$38$		0			0
$39$	−86.8924	−	150.502i	−0.00146483	−	0.00253716i
$40$		0			0
$41$		−	33805.2i		−	0.490492i	−0.969461	−	0.245246i	$-0.921131\pi$
$41$		−	33805.2i		−	0.490492i	0.969461	−	0.245246i	$-0.0788688\pi$
$42$		0			0
$43$	−105285.			−1.32423			−0.662114	−	0.749404i	$-0.730340\pi$
$43$	−105285.			−1.32423			−0.662114	+	0.749404i	$0.730340\pi$
$44$		0			0
$45$	−19.0433	+	10.9947i	−0.000208980	+	0.000120655i
$46$		0			0
$47$	−86534.1	−	49960.5i	−0.833477	−	0.481208i	0.0215644	−	0.999767i	$-0.493135\pi$
$47$	−86534.1	−	49960.5i	−0.833477	−	0.481208i	−0.855042	+	0.518559i	$0.826469\pi$
$48$		0			0
$49$	−22110.6	−	115553.i	−0.187937	−	0.982181i
$50$		0			0
$51$	−40158.4	+	69556.4i	−0.302737	+	0.524356i
$52$		0			0
$53$	52238.9	+	90480.4i	0.350886	+	0.607753i	0.986405	−	0.164333i	$-0.0525471\pi$
$53$	52238.9	+	90480.4i	0.350886	+	0.607753i	−0.635519	+	0.772085i	$0.719214\pi$
$54$		0			0
$55$			206.309i			0.00124002i
$56$		0			0
$57$	−27023.1			−0.145918
$58$		0			0
$59$	194414.	−	112245.i	0.946610	−	0.546526i	0.0545841	−	0.998509i	$-0.482617\pi$
$59$	194414.	−	112245.i	0.946610	−	0.546526i	0.892026	+	0.451983i	$0.149283\pi$
$60$		0			0
$61$	266700.	+	153979.i	1.17499	+	0.678380i	0.954850	−	0.297089i	$-0.0960158\pi$
$61$	266700.	+	153979.i	1.17499	+	0.678380i	0.220138	+	0.975469i	$0.429349\pi$
$62$		0			0
$63$	29075.2	+	78113.3i	0.116279	+	0.312394i
$64$		0			0
$65$	−0.504411	+	0.873666i	−1.83673e−6	+	3.18131e-6i
$66$		0			0
$67$	118138.	+	204621.i	0.392795	+	0.680341i	0.992817	−	0.119643i	$-0.0381749\pi$
$67$	118138.	+	204621.i	0.392795	+	0.680341i	−0.600022	+	0.799983i	$0.704842\pi$
$68$		0			0
$69$		−	36583.8i		−	0.111363i
$70$		0			0
$71$	586114.			1.63760			0.818799	−	0.574081i	$-0.194640\pi$
$71$	586114.			1.63760			0.818799	+	0.574081i	$0.194640\pi$
$72$		0			0
$73$	−558139.	+	322242.i	−1.43474	+	0.828348i	−0.997477	−	0.0709851i	$-0.977386\pi$
$73$	−558139.	+	322242.i	−1.43474	+	0.828348i	−0.437264	+	0.899333i	$0.644052\pi$
$74$		0			0
$75$	−210937.	−	121785.i	−0.500000	−	0.288675i
$76$		0			0
$77$	771084.	+	130195.i	1.68900	+	0.285182i
$78$		0			0
$79$	269093.	−	466084.i	0.545785	−	0.945328i	−0.452772	−	0.891626i	$-0.649565\pi$
$79$	269093.	−	466084.i	0.545785	−	0.945328i	0.998557	−	0.0537014i	$-0.0171019\pi$
$80$		0			0
$81$	−29524.5	−	51137.9i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	602657.i		−	1.05399i	−0.849869	−	0.526994i	$-0.823319\pi$
$83$		−	602657.i		−	1.05399i	0.849869	−	0.526994i	$-0.176681\pi$
$84$		0			0
$85$	466.240			0.000759194
$86$		0			0
$87$	−187382.	+	108185.i	−0.284558	+	0.164290i
$88$		0			0
$89$	1.06824e6	+	616746.i	1.51529	+	0.874856i	0.999839	+	0.0179396i	$0.00571067\pi$
$89$	1.06824e6	+	616746.i	1.51529	+	0.874856i	0.515456	+	0.856916i	$0.327623\pi$
$90$		0			0
$91$	2947.03	+	2436.59i	0.00391075	+	0.00323339i
$92$		0			0
$93$	−323978.	+	561146.i	−0.402779	+	0.697633i
$94$		0			0
$95$	78.4346	+	135.853i	9.14823e−5	+	0.000158452i
$96$		0			0
$97$			339514.i			0.372000i	0.982550	+	0.186000i	$0.0595524\pi$
$97$			339514.i			0.372000i	−0.982550	+	0.186000i	$0.940448\pi$
$98$		0			0
$99$	−554010.			−0.570969
$100$		0			0
$101$	479993.	−	277124.i	0.465877	−	0.268974i	−0.248635	−	0.968597i	$-0.579982\pi$
$101$	479993.	−	277124.i	0.465877	−	0.268974i	0.714512	+	0.699623i	$0.246649\pi$
$102$		0			0
$103$	−785695.	−	453621.i	−0.719022	−	0.415128i	0.0953705	−	0.995442i	$-0.469596\pi$
$103$	−785695.	−	453621.i	−0.719022	−	0.415128i	−0.814393	+	0.580314i	$0.802930\pi$
$104$		0			0
$105$	308.307	−	372.894i	0.000266327	−	0.000322120i
$106$		0			0
$107$	362326.	−	627566.i	0.295766	−	0.512281i	−0.679397	−	0.733771i	$-0.737759\pi$
$107$	362326.	−	627566.i	0.295766	−	0.512281i	0.975163	+	0.221490i	$0.0710919\pi$
$108$		0			0
$109$	−77877.9	−	134889.i	−0.0601360	−	0.104159i	0.834390	−	0.551174i	$-0.185820\pi$
$109$	−77877.9	−	134889.i	−0.0601360	−	0.104159i	−0.894526	+	0.447016i	$0.852487\pi$
$110$		0			0
$111$			32404.1i			0.0236936i
$112$		0			0
$113$	−225088.			−0.155997			−0.0779987	−	0.996953i	$-0.524853\pi$
$113$	−225088.			−0.155997			−0.0779987	+	0.996953i	$0.524853\pi$
$114$		0			0
$115$	−183.917	+	106.185i	−0.000120929	+	6.98181e-5i
$116$		0			0
$117$	−2346.09	−	1354.52i	−0.00146483	−	0.000845721i
$118$		0			0
$119$	294229.	−	1.74258e6i	0.174600	−	1.03408i
$120$		0			0
$121$	−1.71314e6	+	2.96725e6i	−0.967023	+	1.67493i
$122$		0			0
$123$	−263486.	−	456370.i	−0.141593	−	0.245246i
$124$		0			0
$125$			2827.85i			0.00144786i
$126$		0			0
$127$	−2.95637e6			−1.44327			−0.721635	−	0.692274i	$-0.756609\pi$
$127$	−2.95637e6			−1.44327			−0.721635	+	0.692274i	$0.756609\pi$
$128$		0			0
$129$	−1.42135e6	+	820618.i	−0.662114	+	0.382271i
$130$		0			0
$131$	289300.	+	167028.i	0.128687	+	0.0742976i	0.562962	−	0.826483i	$-0.309662\pi$
$131$	289300.	+	167028.i	0.128687	+	0.0742976i	−0.434274	+	0.900781i	$0.642995\pi$
$132$		0			0
$133$	557250.	−	207419.i	0.236862	−	0.0881644i
$134$		0			0
$135$	−171.390	+	296.856i	−6.96602e−5	+	0.000120655i
$136$		0			0
$137$	−116799.	−	202301.i	−0.0454231	−	0.0786751i	0.842420	−	0.538821i	$-0.181130\pi$
$137$	−116799.	−	202301.i	−0.0454231	−	0.0786751i	−0.887843	+	0.460146i	$0.847797\pi$
$138$		0			0
$139$		−	2.77347e6i		−	1.03271i	−0.856374	−	0.516355i	$-0.827288\pi$
$139$		−	2.77347e6i		−	1.03271i	0.856374	−	0.516355i	$-0.172712\pi$
$140$		0			0
$141$	−1.55761e6			−0.555652
$142$		0			0
$143$	−22011.6	+	12708.4i	−0.00752736	+	0.00434592i
$144$		0			0
$145$	1087.76	+	628.017i	0.000356803	+	0.000206000i
$146$		0			0
$147$	−1.19914e6	−	1.38763e6i	−0.377500	−	0.436838i
$148$		0			0
$149$	−399353.	+	691700.i	−0.120725	+	0.209102i	−0.920054	−	0.391792i	$-0.871855\pi$
$149$	−399353.	+	691700.i	−0.120725	+	0.209102i	0.799329	+	0.600894i	$0.205189\pi$
$150$		0			0
$151$	−2.96924e6	−	5.14288e6i	−0.862412	−	1.49374i	−0.869594	−	0.493768i	$-0.835619\pi$
$151$	−2.96924e6	−	5.14288e6i	−0.862412	−	1.49374i	0.00718135	−	0.999974i	$-0.497714\pi$
$152$		0			0
$153$			1.25201e6i			0.349571i
$154$		0			0
$155$	3761.39			0.00101007
$156$		0			0
$157$	−333838.	+	192741.i	−0.0862654	+	0.0498054i	−0.542512	−	0.840048i	$-0.682527\pi$
$157$	−333838.	+	192741.i	−0.0862654	+	0.0498054i	0.456247	+	0.889853i	$0.349193\pi$
$158$		0			0
$159$	1.41045e6	+	814324.i	0.350886	+	0.202584i
$160$		0			0
$161$	280803.	+	754404.i	0.0672860	+	0.180770i
$162$		0			0
$163$	−2.99408e6	+	5.18591e6i	−0.691355	+	1.19746i	0.280039	+	0.959989i	$0.409653\pi$
$163$	−2.99408e6	+	5.18591e6i	−0.691355	+	1.19746i	−0.971394	+	0.237474i	$0.923681\pi$
$164$		0			0
$165$	1608.02	+	2785.17i	0.000357964	+	0.000620011i
$166$		0			0
$167$		−	1.39878e6i		−	0.300332i	−0.988661	−	0.150166i	$-0.952019\pi$
$167$		−	1.39878e6i		−	0.300332i	0.988661	−	0.150166i	$-0.0479808\pi$
$168$		0			0
$169$	4.82668e6			0.999974
$170$		0			0
$171$	−364812.	+	210624.i	−0.0729592	+	0.0421230i
$172$		0			0
$173$	8.81139e6	+	5.08726e6i	1.70179	+	0.982530i	0.943949	+	0.330092i	$0.107080\pi$
$173$	8.81139e6	+	5.08726e6i	1.70179	+	0.982530i	0.757843	+	0.652437i	$0.226253\pi$
$174$		0			0
$175$	5.28457e6	+	892282.i	0.986043	+	0.166490i
$176$		0			0
$177$	1.74973e6	−	3.03061e6i	0.315537	−	0.546526i
$178$		0			0
$179$	−4.27964e6	−	7.41256e6i	−0.746188	−	1.29244i	−0.949638	−	0.313350i	$-0.898549\pi$
$179$	−4.27964e6	−	7.41256e6i	−0.746188	−	1.29244i	0.203450	−	0.979085i	$-0.434785\pi$
$180$		0			0
$181$			3.19101e6i			0.538137i	0.963121	+	0.269068i	$0.0867158\pi$
$181$			3.19101e6i			0.538137i	−0.963121	+	0.269068i	$0.913284\pi$
$182$		0			0
$183$	4.80060e6			0.783325
$184$		0			0
$185$	162.905	−	94.0531i	2.57288e−5	−	1.48545e-5i
$186$		0			0
$187$	1.01729e7	+	5.87333e6i	1.55568	+	0.898173i
$188$		0			0
$189$	1.00135e6	+	827911.i	0.148320	+	0.122630i
$190$		0			0
$191$	−4.67455e6	+	8.09655e6i	−0.670871	+	1.16198i	0.306786	+	0.951779i	$0.400746\pi$
$191$	−4.67455e6	+	8.09655e6i	−0.670871	+	1.16198i	−0.977657	+	0.210205i	$0.932587\pi$
$192$		0			0
$193$	−977142.	−	1.69246e6i	−0.135921	−	0.235422i	0.790028	−	0.613071i	$-0.210066\pi$
$193$	−977142.	−	1.69246e6i	−0.135921	−	0.235422i	−0.925949	+	0.377649i	$0.876733\pi$
$194$		0			0
$195$			15.7260i			2.12087e-6i
$196$		0			0
$197$	−6.09910e6			−0.797750			−0.398875	−	0.917005i	$-0.630599\pi$
$197$	−6.09910e6			−0.797750			−0.398875	+	0.917005i	$0.630599\pi$
$198$		0			0
$199$	8.27147e6	−	4.77554e6i	1.04960	−	0.605987i	0.127062	−	0.991895i	$-0.459445\pi$
$199$	8.27147e6	−	4.77554e6i	1.04960	−	0.605987i	0.922537	+	0.385908i	$0.126112\pi$
$200$		0			0
$201$	3.18973e6	+	1.84159e6i	0.392795	+	0.226780i
$202$		0			0
$203$	3.03368e6	−	3.66920e6i	0.362644	−	0.438614i
$204$		0			0
$205$	−1529.54	+	2649.24i	−0.000177541	+	0.000307510i
$206$		0			0
$207$	−285142.	−	493881.i	−0.0321477	−	0.0556815i
$208$		0			0
$209$			3.95224e6i			0.432917i
$210$		0			0
$211$	−3.50535e6			−0.373151			−0.186575	−	0.982441i	$-0.559739\pi$
$211$	−3.50535e6			−0.373151			−0.186575	+	0.982441i	$0.559739\pi$
$212$		0			0
$213$	7.91254e6	−	4.56831e6i	0.818799	−	0.472734i
$214$		0			0
$215$	8250.97	+	4763.70i	0.000830213	+	0.000479324i
$216$		0			0
$217$	2.37369e6	−	1.40583e7i	0.232298	−	1.37579i
$218$		0			0
$219$	−5.02325e6	+	8.70052e6i	−0.478247	+	0.828348i
$220$		0			0
$221$	28719.8	+	49744.2i	0.00266075	+	0.00460856i
$222$		0			0
$223$		−	3.12055e6i		−	0.281395i	−0.990053	−	0.140698i	$-0.955066\pi$
$223$		−	3.12055e6i		−	0.281395i	0.990053	−	0.140698i	$-0.0449345\pi$
$224$		0			0
$225$	−3.79687e6			−0.333333
$226$		0			0
$227$	−6.46876e6	+	3.73474e6i	−0.553023	+	0.319288i	−0.750340	−	0.661052i	$-0.770110\pi$
$227$	−6.46876e6	+	3.73474e6i	−0.553023	+	0.319288i	0.197317	+	0.980340i	$0.436777\pi$
$228$		0			0
$229$	7.96006e6	+	4.59574e6i	0.662841	+	0.382692i	0.793359	−	0.608754i	$-0.208330\pi$
$229$	7.96006e6	+	4.59574e6i	0.662841	+	0.382692i	−0.130517	+	0.991446i	$0.541664\pi$
$230$		0			0
$231$	1.14244e7	−	4.25237e6i	0.926824	−	0.344981i
$232$		0			0
$233$	−8.17134e6	+	1.41532e7i	−0.645990	+	1.11889i	0.338082	+	0.941117i	$0.390222\pi$
$233$	−8.17134e6	+	1.41532e7i	−0.645990	+	1.11889i	−0.984072	+	0.177771i	$0.943112\pi$
$234$		0			0
$235$	4520.99	+	7830.58i	0.000348361	+	0.000603379i
$236$		0			0
$237$		−	8.38950e6i		−	0.630219i
$238$		0			0
$239$	−2.11658e7			−1.55039			−0.775196	−	0.631721i	$-0.782349\pi$
$239$	−2.11658e7			−1.55039			−0.775196	+	0.631721i	$0.782349\pi$
$240$		0			0
$241$	−1.13158e7	+	6.53316e6i	−0.808412	+	0.466737i	−0.846404	−	0.532541i	$-0.821237\pi$
$241$	−1.13158e7	+	6.53316e6i	−0.808412	+	0.466737i	0.0379922	+	0.999278i	$0.487904\pi$
$242$		0			0
$243$	−797162.	−	460241.i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−3495.49	+	10056.0i	−0.000237690	+	0.000683796i
$246$		0			0
$247$	−9662.96	+	16736.7i	−0.000641238	+	0.00111066i
$248$		0			0
$249$	−4.69725e6	−	8.13587e6i	−0.304260	−	0.526994i
$250$		0			0
$251$		−	1.43596e7i		−	0.908077i	−0.890982	−	0.454038i	$-0.849983\pi$
$251$		−	1.43596e7i		−	0.908077i	0.890982	−	0.454038i	$-0.150017\pi$
$252$		0			0
$253$	−5.35053e6			−0.330396
$254$		0			0
$255$	6294.24	−	3633.98i	0.000379597	−	0.000219160i
$256$		0			0
$257$	2.51641e6	+	1.45285e6i	0.148246	+	0.0855896i	0.572288	−	0.820053i	$-0.306056\pi$
$257$	2.51641e6	+	1.45285e6i	0.148246	+	0.0855896i	−0.424042	+	0.905642i	$0.639389\pi$
$258$		0			0
$259$	−248722.	−	668214.i	−0.0143158	−	0.0384606i
$260$		0			0
$261$	−1.68644e6	+	2.92100e6i	−0.0948527	+	0.164290i
$262$		0			0
$263$	−1.34154e7	−	2.32361e7i	−0.737456	−	1.27731i	−0.953638	−	0.300957i	$-0.902694\pi$
$263$	−1.34154e7	−	2.32361e7i	−0.737456	−	1.27731i	0.216182	−	0.976353i	$-0.430640\pi$
$264$		0			0
$265$		−	9454.32i		−	0.000508034i
$266$		0			0
$267$	1.92282e7			1.01020
$268$		0			0
$269$	−1.58515e7	+	9.15186e6i	−0.814354	+	0.470168i	−0.848466	−	0.529250i	$-0.822473\pi$
$269$	−1.58515e7	+	9.15186e6i	−0.814354	+	0.470168i	0.0341115	+	0.999418i	$0.489140\pi$
$270$		0			0
$271$	2.11779e7	+	1.22270e7i	1.06408	+	0.614347i	0.926558	−	0.376152i	$-0.122753\pi$
$271$	2.11779e7	+	1.22270e7i	1.06408	+	0.614347i	0.137522	+	0.990499i	$0.456086\pi$
$272$		0			0
$273$	58776.2	+	9924.16i	0.00288878	+	0.000487760i
$274$		0			0
$275$	−1.78115e7	+	3.08505e7i	−0.856452	+	1.48342i
$276$		0			0
$277$	−1.53590e7	−	2.66025e7i	−0.722642	−	1.25165i	−0.959937	−	0.280215i	$-0.909594\pi$
$277$	−1.53590e7	−	2.66025e7i	−0.722642	−	1.25165i	0.237296	−	0.971438i	$-0.423739\pi$
$278$		0			0
$279$			1.01006e7i			0.465089i
$280$		0			0
$281$	−2.13967e7			−0.964333			−0.482167	−	0.876080i	$-0.660150\pi$
$281$	−2.13967e7			−0.964333			−0.482167	+	0.876080i	$0.660150\pi$
$282$		0			0
$283$	−2.02749e7	+	1.17057e7i	−0.894541	+	0.516464i	−0.875425	−	0.483354i	$-0.839419\pi$
$283$	−2.02749e7	+	1.17057e7i	−0.894541	+	0.516464i	−0.0191160	+	0.999817i	$0.506085\pi$
$284$		0			0
$285$	2117.74	+	1222.68i	9.14823e−5	+	5.28173e-5i
$286$		0			0
$287$	8.93634e6	+	7.38853e6i	0.378019	+	0.312545i
$288$		0			0
$289$	1.20443e6	−	2.08614e6i	0.0498986	−	0.0864270i
$290$		0			0
$291$	2.64625e6	+	4.58344e6i	0.107387	+	0.186000i
$292$		0			0
$293$		−	2.62759e7i		−	1.04461i	−0.852759	−	0.522305i	$-0.825072\pi$
$293$		−	2.62759e7i		−	1.04461i	0.852759	−	0.522305i	$-0.174928\pi$
$294$		0			0
$295$	−20314.4			−0.000791292
$296$		0			0
$297$	−7.47914e6	+	4.31808e6i	−0.285484	+	0.164824i
$298$		0			0
$299$	−22658.2	−	13081.7i	−0.000847639	−	0.000489385i
$300$		0			0
$301$	2.30113e7	−	2.78319e7i	0.843806	−	1.02057i
$302$		0			0
$303$	4.31994e6	−	7.48236e6i	0.155292	−	0.268974i
$304$		0			0
$305$	−13933.8	−	24134.0i	−0.000491099	−	0.000850609i
$306$		0			0
$307$		−	1.25570e7i		−	0.433981i	−0.976174	−	0.216990i	$-0.930376\pi$
$307$		−	1.25570e7i		−	0.433981i	0.976174	−	0.216990i	$-0.0696241\pi$
$308$		0			0
$309$	−1.41425e7			−0.479348
$310$		0			0
$311$	1.02129e6	−	589640.i	0.0339521	−	0.0196023i	−0.482928	−	0.875660i	$-0.660427\pi$
$311$	1.02129e6	−	589640.i	0.0339521	−	0.0196023i	0.516880	+	0.856058i	$0.327093\pi$
$312$		0			0
$313$	−7.70433e6	−	4.44810e6i	−0.251248	−	0.145058i	0.369088	−	0.929395i	$-0.379670\pi$
$313$	−7.70433e6	−	4.44810e6i	−0.251248	−	0.145058i	−0.620335	+	0.784337i	$0.713004\pi$
$314$		0			0
$315$	1255.73	−	7437.08i	4.01757e−5	−	0.000237942i
$316$		0			0
$317$	8.59927e6	−	1.48944e7i	0.269950	−	0.467568i	−0.698898	−	0.715221i	$-0.746326\pi$
$317$	8.59927e6	−	1.48944e7i	0.269950	−	0.467568i	0.968849	+	0.247653i	$0.0796594\pi$
$318$		0			0
$319$	1.58226e7	+	2.74055e7i	0.487421	+	0.844238i
$320$		0			0
$321$		−	1.12962e7i		−	0.341521i
$322$		0			0
$323$	8.93171e6			0.265050
$324$		0			0
$325$	−150855.	+	87096.0i	−0.00439449	+	0.00253716i
$326$		0			0
$327$	−2.10270e6	−	1.21400e6i	−0.0601360	−	0.0347196i
$328$		0			0
$329$	3.21200e7	−	1.19557e7i	0.901961	−	0.335727i
$330$		0			0
$331$	−2.93149e7	+	5.07749e7i	−0.808359	+	1.40012i	0.105642	+	0.994404i	$0.466310\pi$
$331$	−2.93149e7	+	5.07749e7i	−0.808359	+	1.40012i	−0.914000	+	0.405714i	$0.867023\pi$
$332$		0			0
$333$	252565.	+	437456.i	0.00683976	+	0.0118468i
$334$		0			0
$335$		−	21380.9i		−	0.000568712i
$336$		0			0
$337$	−5.26755e6			−0.137632			−0.0688159	−	0.997629i	$-0.521922\pi$
$337$	−5.26755e6			−0.137632			−0.0688159	+	0.997629i	$0.521922\pi$
$338$		0			0
$339$	−3.03869e6	+	1.75439e6i	−0.0779987	+	0.0450326i
$340$		0			0
$341$	8.20700e7	+	4.73831e7i	2.06977	+	1.19498i
$342$		0			0
$343$	3.53786e7	+	1.94105e7i	0.876715	+	0.481010i
$344$		0			0
$345$	−1655.25	+	2866.98i	−4.03095e−5	+	6.98181e-5i
$346$		0			0
$347$	3.35761e7	+	5.81555e7i	0.803604	+	1.39188i	0.917230	+	0.398359i	$0.130420\pi$
$347$	3.35761e7	+	5.81555e7i	0.803604	+	1.39188i	−0.113626	+	0.993524i	$0.536246\pi$
$348$		0			0
$349$			4.50914e6i			0.106076i	0.998592	+	0.0530380i	$0.0168905\pi$
$349$			4.50914e6i			0.106076i	−0.998592	+	0.0530380i	$0.983110\pi$
$350$		0			0
$351$	−42229.7			−0.000976555
$352$		0			0
$353$	5.90317e7	−	3.40820e7i	1.34203	−	0.774819i	0.354922	−	0.934896i	$-0.384507\pi$
$353$	5.90317e7	−	3.40820e7i	1.34203	−	0.774819i	0.987105	+	0.160077i	$0.0511741\pi$
$354$		0			0
$355$	−45932.4	−	26519.1i	−0.00102668	−	0.000592753i
$356$		0			0
$357$	−9.60999e6	−	2.58181e7i	−0.211212	−	0.567440i
$358$		0			0
$359$	2.92553e7	−	5.06717e7i	0.632298	−	1.09517i	−0.354783	−	0.934949i	$-0.615445\pi$
$359$	2.92553e7	−	5.06717e7i	0.632298	−	1.09517i	0.987081	−	0.160223i	$-0.0512214\pi$
$360$		0			0
$361$	−2.20204e7	−	3.81404e7i	−0.468062	−	0.810707i
$362$		0			0
$363$			5.34104e7i			1.11662i
$364$		0			0
$365$	58320.0			0.00119933
$366$		0			0
$367$	5.04590e7	−	2.91325e7i	1.02080	−	0.589359i	0.106464	−	0.994317i	$-0.466047\pi$
$367$	5.04590e7	−	2.91325e7i	1.02080	−	0.589359i	0.914335	+	0.404958i	$0.132714\pi$
$368$		0			0
$369$	−7.11411e6	−	4.10733e6i	−0.141593	−	0.0817487i
$370$		0			0
$371$	−3.53357e7	−	5.96631e6i	−0.691978	−	0.116838i
$372$		0			0
$373$	−2.16495e7	+	3.74981e7i	−0.417178	+	0.722574i	−0.995654	−	0.0931254i	$-0.970314\pi$
$373$	−2.16495e7	+	3.74981e7i	−0.417178	+	0.722574i	0.578476	+	0.815699i	$0.303648\pi$
$374$		0			0
$375$	22040.9	+	38176.0i	0.000417961	+	0.000723929i
$376$		0			0
$377$			154740.i			0.00288788i
$378$		0			0
$379$	3.56341e7			0.654558			0.327279	−	0.944928i	$-0.393868\pi$
$379$	3.56341e7			0.654558			0.327279	+	0.944928i	$0.393868\pi$
$380$		0			0
$381$	−3.99110e7	+	2.30426e7i	−0.721635	+	0.416636i
$382$		0			0
$383$	5.96797e6	+	3.44561e6i	0.106226	+	0.0613295i	0.552172	−	0.833730i	$-0.313799\pi$
$383$	5.96797e6	+	3.44561e6i	0.106226	+	0.0613295i	−0.445946	+	0.895060i	$0.647133\pi$
$384$		0			0
$385$	−54537.3	−	45091.2i	−0.000955678	−	0.000790150i
$386$		0			0
$387$	−1.27922e7	+	2.21567e7i	−0.220705	+	0.382271i
$388$		0			0
$389$	2.06591e6	+	3.57826e6i	0.0350964	+	0.0607887i	0.883040	−	0.469298i	$-0.155493\pi$
$389$	2.06591e6	+	3.57826e6i	0.0350964	+	0.0607887i	−0.847944	+	0.530086i	$0.822160\pi$
$390$		0			0
$391$			1.20917e7i			0.202282i
$392$		0			0
$393$	5.20741e6			0.0857914
$394$		0			0
$395$	−42176.5	+	24350.6i	−0.000684351	+	0.000395110i
$396$		0			0
$397$	−7.57634e7	−	4.37420e7i	−1.21084	−	0.699081i	−0.247900	−	0.968786i	$-0.579741\pi$
$397$	−7.57634e7	−	4.37420e7i	−1.21084	−	0.699081i	−0.962943	+	0.269705i	$0.913074\pi$
$398$		0			0
$399$	5.90621e6	−	7.14349e6i	0.0929801	−	0.112458i
$400$		0			0
$401$	3.54173e7	−	6.13446e7i	0.549266	−	0.951356i	−0.449060	−	0.893502i	$-0.648241\pi$
$401$	3.54173e7	−	6.13446e7i	0.549266	−	0.951356i	0.998325	−	0.0578539i	$-0.0184258\pi$
$402$		0			0
$403$	231697.	+	401311.i	0.00354002	+	0.00613149i
$404$		0			0
$405$			5343.41i			8.04366e-5i
$406$		0			0
$407$	4.73924e6			0.0702951
$408$		0			0
$409$	−4.74349e7	+	2.73866e7i	−0.693312	+	0.400284i	−0.804851	−	0.593476i	$-0.797755\pi$
$409$	−4.74349e7	+	2.73866e7i	−0.693312	+	0.400284i	0.111540	+	0.993760i	$0.464422\pi$
$410$		0			0
$411$	−3.15357e6	−	1.82071e6i	−0.0454231	−	0.0262250i
$412$		0			0
$413$	−1.28197e7	+	7.59253e7i	−0.181982	+	1.07780i
$414$		0			0
$415$	−27267.6	+	47228.8i	−0.000381507	+	0.000660789i
$416$		0			0
$417$	−2.16170e7	−	3.74418e7i	−0.298118	−	0.516355i
$418$		0			0
$419$		−	1.26910e8i		−	1.72525i	−0.505842	−	0.862626i	$-0.668818\pi$
$419$		−	1.26910e8i		−	1.72525i	0.505842	−	0.862626i	$-0.331182\pi$
$420$		0			0
$421$	−2.52286e7			−0.338102			−0.169051	−	0.985607i	$-0.554070\pi$
$421$	−2.52286e7			−0.338102			−0.169051	+	0.985607i	$0.554070\pi$
$422$		0			0
$423$	−2.10278e7	+	1.21404e7i	−0.277826	+	0.160403i
$424$		0			0
$425$	6.97194e7	+	4.02525e7i	0.908211	+	0.524356i
$426$		0			0
$427$	−9.89945e7	+	3.68476e7i	−1.27153	+	0.473288i
$428$		0			0
$429$	−198104.	+	343126.i	−0.00250912	+	0.00434592i
$430$		0			0
$431$	−7.00037e7	−	1.21250e8i	−0.874358	−	1.51443i	−0.857445	−	0.514576i	$-0.827949\pi$
$431$	−7.00037e7	−	1.21250e8i	−0.874358	−	1.51443i	−0.0169134	−	0.999857i	$-0.505384\pi$
$432$		0			0
$433$			7.55793e7i			0.930978i	0.885054	+	0.465489i	$0.154121\pi$
$433$			7.55793e7i			0.930978i	−0.885054	+	0.465489i	$0.845879\pi$
$434$		0			0
$435$	19579.6			0.000237868
$436$		0			0
$437$	−3.52328e6	+	2.03417e6i	−0.0422185	+	0.0243749i
$438$		0			0
$439$	−8.60392e7	−	4.96747e7i	−1.01696	−	0.587140i	−0.103736	−	0.994605i	$-0.533080\pi$
$439$	−8.60392e7	−	4.96747e7i	−1.01696	−	0.587140i	−0.913221	+	0.407465i	$0.866413\pi$
$440$		0			0
$441$	−2.70038e7	−	9.38660e6i	−0.314854	−	0.109444i
$442$		0			0
$443$	−3.18080e7	+	5.50930e7i	−0.365868	+	0.633703i	−0.988915	−	0.148482i	$-0.952561\pi$
$443$	−3.18080e7	+	5.50930e7i	−0.365868	+	0.633703i	0.623047	+	0.782185i	$0.285895\pi$
$444$		0			0
$445$	−55810.1	−	96665.9i	−0.000633334	−	0.00109697i
$446$		0			0
$447$			1.24506e7i			0.139402i
$448$		0			0
$449$	2.37071e6			0.0261903			0.0130951	−	0.999914i	$-0.495832\pi$
$449$	2.37071e6			0.0261903			0.0130951	+	0.999914i	$0.495832\pi$
$450$		0			0
$451$	−6.67461e7	+	3.85359e7i	−0.727606	+	0.420083i
$452$		0			0
$453$	−8.01696e7	−	4.62859e7i	−0.862412	−	0.497914i
$454$		0			0
$455$	−120.707	−	324.290i	−1.28144e−6	−	3.44270e-6i
$456$		0			0
$457$	8.95758e7	−	1.55150e8i	0.938517	−	1.62556i	0.170278	−	0.985396i	$-0.445533\pi$
$457$	8.95758e7	−	1.55150e8i	0.938517	−	1.62556i	0.768239	−	0.640163i	$-0.221133\pi$
$458$		0			0
$459$	9.75849e6	+	1.69022e7i	0.100912	+	0.174785i
$460$		0			0
$461$			6.44992e7i			0.658342i	0.944270	+	0.329171i	$0.106769\pi$
$461$			6.44992e7i			0.658342i	−0.944270	+	0.329171i	$0.893231\pi$
$462$		0			0
$463$	8.12336e7			0.818451			0.409226	−	0.912433i	$-0.365799\pi$
$463$	8.12336e7			0.818451			0.409226	+	0.912433i	$0.365799\pi$
$464$		0			0
$465$	50778.8	−	29317.1i	0.000505037	−	0.000291583i
$466$		0			0
$467$	−8.15323e7	−	4.70727e7i	−0.800533	−	0.462188i	0.0431244	−	0.999070i	$-0.486269\pi$
$467$	−8.15323e7	−	4.70727e7i	−0.800533	−	0.462188i	−0.843658	+	0.536882i	$0.819602\pi$
$468$		0			0
$469$	−7.99117e7	−	1.34928e7i	−0.774625	−	0.130793i
$470$		0			0
$471$	−3.00454e6	+	5.20402e6i	−0.0287551	+	0.0498054i
$472$		0			0
$473$	1.20019e8	+	2.07879e8i	1.13414	+	1.96438i
$474$		0			0
$475$			2.70864e7i			0.252738i
$476$		0			0
$477$	2.53881e7			0.233924
$478$		0			0
$479$	1.84049e8	−	1.06261e8i	1.67466	−	0.966865i	0.709686	−	0.704518i	$-0.248837\pi$
$479$	1.84049e8	−	1.06261e8i	1.67466	−	0.966865i	0.964974	−	0.262347i	$-0.0844964\pi$
$480$		0			0
$481$	20069.5	+	11587.1i	0.000180344	+	0.000104121i
$482$		0			0
$483$	9.67084e6	+	7.99581e6i	0.0858268	+	0.0709612i
$484$		0			0
$485$	15361.5	−	26606.9i	0.000134651	−	0.000233222i
$486$		0			0
$487$	1.39442e7	+	2.41521e7i	0.120728	+	0.209106i	0.920055	−	0.391790i	$-0.128144\pi$
$487$	1.39442e7	+	2.41521e7i	0.120728	+	0.209106i	−0.799327	+	0.600896i	$0.794811\pi$
$488$		0			0
$489$			9.33463e7i			0.798308i
$490$		0			0
$491$	−2.07655e8			−1.75427			−0.877137	−	0.480240i	$-0.840550\pi$
$491$	−2.07655e8			−1.75427			−0.877137	+	0.480240i	$0.840550\pi$
$492$		0			0
$493$	6.19339e7	−	3.57576e7i	0.516878	−	0.298420i
$494$		0			0
$495$	43416.5	+	25066.5i	0.000357964	+	0.000206670i
$496$		0			0
$497$	−1.28102e8	+	1.54938e8i	−1.04349	+	1.26209i
$498$		0			0
$499$	−4.96648e6	+	8.60219e6i	−0.0399712	+	0.0692321i	−0.885319	−	0.464984i	$-0.846060\pi$
$499$	−4.96648e6	+	8.60219e6i	−0.0399712	+	0.0692321i	0.845348	+	0.534216i	$0.179393\pi$
$500$		0			0
$501$	−1.09024e7	−	1.88836e7i	−0.0866983	−	0.150166i
$502$		0			0
$503$		−	1.17586e8i		−	0.923956i	−0.886891	−	0.461978i	$-0.847140\pi$
$503$		−	1.17586e8i		−	0.923956i	0.886891	−	0.461978i	$-0.152860\pi$
$504$		0			0
$505$	−50154.6			−0.000389437
$506$		0			0
$507$	6.51602e7	−	3.76203e7i	0.499987	−	0.288668i
$508$		0			0
$509$	−9.25834e7	−	5.34530e7i	−0.702069	−	0.405340i	0.106049	−	0.994361i	$-0.466180\pi$
$509$	−9.25834e7	−	5.34530e7i	−0.702069	−	0.405340i	−0.808117	+	0.589021i	$0.799513\pi$
$510$		0			0
$511$	3.68039e7	−	2.17972e8i	0.275823	−	1.63357i
$512$		0			0
$513$	−3.28330e6	+	5.68685e6i	−0.0243197	+	0.0421230i
$514$		0			0
$515$	41048.7	+	71098.5i	0.000300523	+	0.000520521i
$516$		0			0
$517$			2.27808e8i			1.64853i
$518$		0			0
$519$	1.58605e8			1.13453
$520$		0			0
$521$	7.88379e7	−	4.55171e7i	0.557471	−	0.321856i	−0.194659	−	0.980871i	$-0.562360\pi$
$521$	7.88379e7	−	4.55171e7i	0.557471	−	0.321856i	0.752130	+	0.659015i	$0.229027\pi$
$522$		0			0
$523$	4.65798e7	+	2.68929e7i	0.325606	+	0.187989i	0.653889	−	0.756591i	$-0.273136\pi$
$523$	4.65798e7	+	2.68929e7i	0.325606	+	0.187989i	−0.328282	+	0.944580i	$0.606470\pi$
$524$		0			0
$525$	7.82964e7	−	2.91434e7i	0.541083	−	0.201401i
$526$		0			0
$527$	1.07082e8	−	1.85471e8i	0.731616	−	1.26720i
$528$		0			0
$529$	7.12641e7	+	1.23433e8i	0.481397	+	0.833805i
$530$		0			0
$531$		−	5.45510e7i		−	0.364351i
$532$		0			0
$533$	−376871.			−0.00248892
$534$		0			0
$535$	−56789.2	+	32787.3i	−0.000370855	+	0.000214113i
$536$		0			0
$537$	−1.15550e8	−	6.67130e7i	−0.746188	−	0.430812i
$538$		0			0
$539$	−2.02946e8	+	1.75379e8i	−1.29603	+	1.11998i
$540$		0			0
$541$	8.51256e7	−	1.47442e8i	0.537612	−	0.931170i	−0.461421	−	0.887182i	$-0.652660\pi$
$541$	8.51256e7	−	1.47442e8i	0.537612	−	0.931170i	0.999032	−	0.0439889i	$-0.0140066\pi$
$542$		0			0
$543$	2.48715e7	+	4.30787e7i	0.155347	+	0.269068i
$544$		0			0
$545$			14094.5i			8.70685e-5i
$546$		0			0
$547$	2.69779e8			1.64834			0.824170	−	0.566343i	$-0.191642\pi$
$547$	2.69779e8			1.64834			0.824170	+	0.566343i	$0.191642\pi$
$548$		0			0
$549$	6.48081e7	−	3.74170e7i	0.391663	−	0.226127i
$550$		0			0
$551$	2.08381e7	+	1.20309e7i	0.124567	+	0.0719187i
$552$		0			0
$553$	6.43947e7	+	1.73002e8i	0.380780	+	1.02300i
$554$		0			0
$555$	1466.14	−	2539.43i	8.57625e−6	−	1.48545e-5i
$556$		0			0
$557$	8.73125e7	+	1.51230e8i	0.505255	+	0.875128i	0.999982	+	0.00607879i	$0.00193495\pi$
$557$	8.73125e7	+	1.51230e8i	0.505255	+	0.875128i	−0.494726	+	0.869049i	$0.664732\pi$
$558$		0			0
$559$			1.17375e6i			0.00671956i
$560$		0			0
$561$	1.83112e8			1.03712
$562$		0			0
$563$	−2.68901e8	+	1.55250e8i	−1.50684	+	0.869976i	−0.506874	+	0.862020i	$0.669199\pi$
$563$	−2.68901e8	+	1.55250e8i	−1.50684	+	0.869976i	−0.999968	+	0.00795588i	$0.997468\pi$
$564$		0			0
$565$	17639.6	+	10184.3i	9.78013e−5	+	5.64656e-5i
$566$		0			0
$567$	1.99711e7	+	3.37206e6i	0.109560	+	0.0184989i
$568$		0			0
$569$	3.21511e7	−	5.56873e7i	0.174525	−	0.302287i	−0.765472	−	0.643470i	$-0.777494\pi$
$569$	3.21511e7	−	5.56873e7i	0.174525	−	0.302287i	0.939997	+	0.341183i	$0.110828\pi$
$570$		0			0
$571$	2.92864e7	+	5.07255e7i	0.157310	+	0.272470i	0.933898	−	0.357540i	$-0.116384\pi$
$571$	2.92864e7	+	5.07255e7i	0.157310	+	0.272470i	−0.776588	+	0.630009i	$0.783051\pi$
$572$		0			0
$573$			1.45738e8i			0.774656i
$574$		0			0
$575$	−3.66695e7			−0.192886
$576$		0			0
$577$	1.85466e7	−	1.07079e7i	0.0965468	−	0.0557413i	−0.450949	−	0.892550i	$-0.648914\pi$
$577$	1.85466e7	−	1.07079e7i	0.0965468	−	0.0557413i	0.547496	+	0.836808i	$0.315581\pi$
$578$		0			0
$579$	−2.63828e7	−	1.52321e7i	−0.135921	−	0.0784739i
$580$		0			0
$581$	1.59311e8	+	1.31718e8i	0.812302	+	0.671608i
$582$		0			0
$583$	1.19098e8	−	2.06284e8i	0.601035	−	1.04102i
$584$		0			0
$585$	122.572	+	212.301i	6.12243e−7	+	1.06044e-6i
$586$		0			0
$587$		−	1.58381e8i		−	0.783050i	−0.920167	−	0.391525i	$-0.871948\pi$
$587$		−	1.58381e8i		−	0.783050i	0.920167	−	0.391525i	$-0.128052\pi$
$588$		0			0
$589$	7.20566e7			0.352637
$590$		0			0
$591$	−8.23378e7	+	4.75378e7i	−0.398875	+	0.230291i
$592$		0			0
$593$	−4.39682e6	−	2.53850e6i	−0.0210850	−	0.0121735i	0.489420	−	0.872048i	$-0.337208\pi$
$593$	−4.39682e6	−	2.53850e6i	−0.0210850	−	0.0121735i	−0.510505	+	0.859875i	$0.670542\pi$
$594$		0			0
$595$	−101902.	+	123250.i	−0.000483763	+	0.000585106i
$596$		0			0
$597$	7.44433e7	−	1.28939e8i	0.349866	−	0.605987i
$598$		0			0
$599$	1.46545e8	+	2.53823e8i	0.681850	+	1.18100i	0.974416	+	0.224754i	$0.0721578\pi$
$599$	1.46545e8	+	2.53823e8i	0.681850	+	1.18100i	−0.292565	+	0.956246i	$0.594509\pi$
$600$		0			0
$601$		−	2.90021e8i		−	1.33600i	−0.744162	−	0.667999i	$-0.767151\pi$
$601$		−	2.90021e8i		−	1.33600i	0.744162	−	0.667999i	$-0.232849\pi$
$602$		0			0
$603$	5.74151e7			0.261863
$604$		0			0
$605$	268510.	−	155024.i	0.00121253	−	0.000700056i
$606$		0			0
$607$	−2.64810e8	−	1.52888e8i	−1.18405	−	0.683609i	−0.227098	−	0.973872i	$-0.572924\pi$
$607$	−2.64810e8	−	1.52888e8i	−1.18405	−	0.683609i	−0.956947	+	0.290263i	$0.906257\pi$
$608$		0			0
$609$	1.23561e7	−	7.31793e7i	0.0547052	−	0.323994i
$610$		0			0
$611$	−556975.	+	964708.i	−0.00244181	+	0.00422934i
$612$		0			0
$613$	6.23753e6	+	1.08037e7i	0.0270789	+	0.0469020i	0.879247	−	0.476366i	$-0.158046\pi$
$613$	6.23753e6	+	1.08037e7i	0.0270789	+	0.0469020i	−0.852168	+	0.523268i	$0.824713\pi$
$614$		0			0
$615$			47686.2i			0.000205007i
$616$		0			0
$617$	−3.75554e8			−1.59888			−0.799442	−	0.600743i	$-0.794871\pi$
$617$	−3.75554e8			−1.59888			−0.799442	+	0.600743i	$0.794871\pi$
$618$		0			0
$619$	−7.09760e7	+	4.09780e7i	−0.299254	+	0.172774i	−0.642108	−	0.766615i	$-0.721940\pi$
$619$	−7.09760e7	+	4.09780e7i	−0.299254	+	0.172774i	0.342854	+	0.939389i	$0.388606\pi$
$620$		0			0
$621$	−7.69884e6	−	4.44493e6i	−0.0321477	−	0.0185605i
$622$		0			0
$623$	−3.96511e8	+	1.47589e8i	−1.63980	+	0.610364i
$624$		0			0
$625$	−1.22070e8	+	2.11432e8i	−0.499999	+	0.866024i
$626$		0			0
$627$	3.08046e7	+	5.33552e7i	0.124972	+	0.216458i
$628$		0			0
$629$		−	1.07103e7i		−	0.0430376i
$630$		0			0
$631$	3.38493e8			1.34729			0.673646	−	0.739054i	$-0.264727\pi$
$631$	3.38493e8			1.34729			0.673646	+	0.739054i	$0.264727\pi$
$632$		0			0
$633$	−4.73223e7	+	2.73215e7i	−0.186575	+	0.107719i
$634$		0			0
$635$	231684.	+	133763.i	0.000904846	+	0.000522413i
$636$		0			0
$637$	−1.28822e6	+	246495.i	−0.00498391	+	0.000953653i
$638$		0			0
$639$	7.12129e7	−	1.23344e8i	0.272933	−	0.472734i
$640$		0			0
$641$	−2.02548e8	−	3.50823e8i	−0.769048	−	1.33203i	−0.938080	−	0.346419i	$-0.887397\pi$
$641$	−2.02548e8	−	3.50823e8i	−0.769048	−	1.33203i	0.169032	−	0.985611i	$-0.445936\pi$
$642$		0			0
$643$			4.76136e8i			1.79101i	0.445050	+	0.895506i	$0.353186\pi$
$643$			4.76136e8i			1.79101i	−0.445050	+	0.895506i	$0.646814\pi$
$644$		0			0
$645$	148517.			0.000553475
$646$		0			0
$647$	1.52008e7	−	8.77616e6i	0.0561245	−	0.0324035i	−0.471675	−	0.881772i	$-0.656351\pi$
$647$	1.52008e7	−	8.77616e6i	0.0561245	−	0.0324035i	0.527800	+	0.849369i	$0.323017\pi$
$648$		0			0
$649$	−4.43240e8	−	2.55905e8i	−1.62145	−	0.936147i
$650$		0			0
$651$	−7.75286e7	−	2.08288e8i	−0.281008	−	0.754955i
$652$		0			0
$653$	2.28952e7	−	3.96557e7i	0.0822252	−	0.142418i	−0.821980	−	0.569516i	$-0.807131\pi$
$653$	2.28952e7	−	3.96557e7i	0.0822252	−	0.142418i	0.904205	+	0.427098i	$0.140464\pi$
$654$		0			0
$655$	−15114.5	−	26179.1i	−5.37862e−5	−	9.31604e-5i
$656$		0			0
$657$			1.56609e8i			0.552232i
$658$		0			0
$659$	−4.26685e8			−1.49091			−0.745455	−	0.666556i	$-0.767768\pi$
$659$	−4.26685e8			−1.49091			−0.745455	+	0.666556i	$0.767768\pi$
$660$		0			0
$661$	9.62322e7	−	5.55597e7i	0.333208	−	0.192378i	−0.324056	−	0.946038i	$-0.605047\pi$
$661$	9.62322e7	−	5.55597e7i	0.333208	−	0.192378i	0.657265	+	0.753660i	$0.271713\pi$
$662$		0			0
$663$	775435.	+	447698.i	0.00266075	+	0.00153619i
$664$		0			0
$665$	−53055.2	−	8958.19i	−0.000180411	−	3.04618e-5i
$666$		0			0
$667$	−1.62873e7	+	2.82105e7i	−0.0548874	+	0.0950678i
$668$		0			0
$669$	−2.43223e7	−	4.21274e7i	−0.0812317	−	0.140698i
$670$		0			0
$671$		−	7.02108e8i		−	2.32400i
$672$		0			0
$673$	−1.25013e8			−0.410119			−0.205060	−	0.978749i	$-0.565739\pi$
$673$	−1.25013e8			−0.410119			−0.205060	+	0.978749i	$0.565739\pi$
$674$		0			0
$675$	−5.12578e7	+	2.95937e7i	−0.166667	+	0.0962250i
$676$		0			0
$677$	4.17878e7	+	2.41262e7i	0.134674	+	0.0777539i	0.565823	−	0.824527i	$-0.308558\pi$
$677$	4.17878e7	+	2.41262e7i	0.134674	+	0.0777539i	−0.431149	+	0.902281i	$0.641892\pi$
$678$		0			0
$679$	−8.97498e7	−	7.42048e7i	−0.286698	−	0.237041i
$680$		0			0
$681$	−5.82188e7	+	1.00838e8i	−0.184341	+	0.319288i
$682$		0			0
$683$	2.23115e8	+	3.86446e8i	0.700271	+	1.21291i	0.968371	+	0.249514i	$0.0802709\pi$
$683$	2.23115e8	+	3.86446e8i	0.700271	+	1.21291i	−0.268100	+	0.963391i	$0.586396\pi$
$684$		0			0
$685$			21138.5i			6.57662e-5i
$686$		0			0
$687$	1.43281e8			0.441894
$688$		0			0
$689$	1.00870e6	−	582375.i	0.00308394	−	0.00178051i
$690$		0			0
$691$	−4.39974e8	−	2.54019e8i	−1.33350	−	0.769897i	−0.347666	−	0.937618i	$-0.613026\pi$
$691$	−4.39974e8	−	2.54019e8i	−1.33350	−	0.769897i	−0.985834	+	0.167722i	$0.946359\pi$
$692$		0			0
$693$	1.21085e8	−	1.46451e8i	0.363825	−	0.440042i
$694$		0			0
$695$	−125487.	+	217350.i	−0.000373805	+	0.000647449i
$696$		0			0
$697$	8.70877e7	+	1.50840e8i	0.257193	+	0.445471i
$698$		0			0
$699$			2.54757e8i			0.745925i
$700$		0			0
$701$	−5.62305e8			−1.63237			−0.816183	−	0.577793i	$-0.803914\pi$
$701$	−5.62305e8			−1.63237			−0.816183	+	0.577793i	$0.803914\pi$
$702$		0			0
$703$	3.12075e6	−	1.80177e6i	0.00898242	−	0.00518600i
$704$		0			0
$705$	122067.	+	70475.2i	0.000348361	+	0.000201126i
$706$		0			0
$707$	−3.16509e7	+	1.87454e8i	−0.0895630	+	0.530440i
$708$		0			0
$709$	5.94219e7	−	1.02922e8i	0.166728	−	0.288781i	−0.770540	−	0.637392i	$-0.780013\pi$
$709$	5.94219e7	−	1.02922e8i	0.166728	−	0.288781i	0.937267	+	0.348611i	$0.113347\pi$
$710$		0			0
$711$	−6.53897e7	−	1.13258e8i	−0.181928	−	0.315109i
$712$		0			0
$713$			9.75500e7i			0.269128i
$714$		0			0
$715$	2299.99			6.29228e−6
$716$		0			0
$717$	−2.85739e8	+	1.64971e8i	−0.775196	+	0.447560i
$718$		0			0
$719$	5.17751e8	+	2.98924e8i	1.39295	+	0.804219i	0.993641	−	0.112599i	$-0.0359175\pi$
$719$	5.17751e8	+	2.98924e8i	1.39295	+	0.804219i	0.399307	+	0.916817i	$0.369251\pi$
$720$		0			0
$721$	2.91637e8	−	1.08553e8i	0.778102	−	0.289624i
$722$		0			0
$723$	−1.01842e8	+	1.76395e8i	−0.269471	+	0.466737i
$724$		0			0
$725$	1.08439e8	+	1.87822e8i	0.284558	+	0.492869i
$726$		0			0
$727$		−	1.55915e8i		−	0.405775i	−0.979202	−	0.202887i	$-0.934967\pi$
$727$		−	1.55915e8i		−	0.405775i	0.979202	−	0.202887i	$-0.0650325\pi$
$728$		0			0
$729$	−1.43489e7			−0.0370370
$730$		0			0
$731$	4.69788e8	−	2.71232e8i	1.20268	−	0.694367i
$732$		0			0
$733$	2.33614e8	+	1.34877e8i	0.593180	+	0.342472i	0.766354	−	0.642419i	$-0.222069\pi$
$733$	2.33614e8	+	1.34877e8i	0.593180	+	0.342472i	−0.173174	+	0.984891i	$0.555402\pi$
$734$		0			0
$735$	31189.6	+	163001.i	7.85502e−5	+	0.000410513i
$736$		0			0
$737$	2.69341e8	−	4.66511e8i	0.672821	−	1.16536i
$738$		0			0
$739$	−2.68694e7	−	4.65391e7i	−0.0665770	−	0.115315i	0.830815	−	0.556548i	$-0.187874\pi$
$739$	−2.68694e7	−	4.65391e7i	−0.0665770	−	0.115315i	−0.897392	+	0.441233i	$0.854541\pi$
$740$		0			0
$741$			301261.i			0.000740438i
$742$		0			0
$743$	6.94445e8			1.69306			0.846528	−	0.532344i	$-0.178689\pi$
$743$	6.94445e8			1.69306			0.846528	+	0.532344i	$0.178689\pi$
$744$		0			0
$745$	62592.7	−	36137.9i	0.000151375	−	8.73966e-5i
$746$		0			0
$747$	−1.26826e8	−	7.32228e7i	−0.304260	−	0.175665i
$748$		0			0
$749$	8.67053e7	+	2.32942e8i	0.206348	+	0.554373i
$750$		0			0
$751$	−3.16023e8	+	5.47368e8i	−0.746104	+	1.29229i	0.203574	+	0.979060i	$0.434744\pi$
$751$	−3.16023e8	+	5.47368e8i	−0.746104	+	1.29229i	−0.949677	+	0.313230i	$0.898589\pi$
$752$		0			0
$753$	−1.11922e8	−	1.93855e8i	−0.262139	−	0.454038i
$754$		0			0
$755$			537381.i			0.00124865i
$756$		0			0
$757$	1.00073e8			0.230691			0.115346	−	0.993325i	$-0.463202\pi$
$757$	1.00073e8			0.230691			0.115346	+	0.993325i	$0.463202\pi$
$758$		0			0
$759$	−7.22322e7	+	4.17033e7i	−0.165198	+	0.0953772i
$760$		0			0
$761$	2.67004e8	+	1.54155e8i	0.605848	+	0.349787i	0.771339	−	0.636425i	$-0.219587\pi$
$761$	2.67004e8	+	1.54155e8i	0.605848	+	0.349787i	−0.165491	+	0.986211i	$0.552921\pi$
$762$		0			0
$763$	5.26786e7	+	8.89460e6i	0.118593	+	0.0200241i
$764$		0			0
$765$	56648.1	−	98117.5i	0.000126532	−	0.000219160i
$766$		0			0
$767$	−1.25134e6	−	2.16738e6i	−0.00277325	−	0.00480341i
$768$		0			0
$769$			6.37446e8i			1.40173i	0.713294	+	0.700865i	$0.247202\pi$
$769$			6.37446e8i			1.40173i	−0.713294	+	0.700865i	$0.752798\pi$
$770$		0			0
$771$	4.52953e7			0.0988304
$772$		0			0
$773$	4.94870e8	−	2.85713e8i	1.07140	−	0.618574i	0.142838	−	0.989746i	$-0.454377\pi$
$773$	4.94870e8	−	2.85713e8i	1.07140	−	0.618574i	0.928564	+	0.371172i	$0.121044\pi$
$774$		0			0
$775$	5.62461e8	+	3.24737e8i	1.20834	+	0.697633i
$776$		0			0
$777$	−8.56596e6	−	7.08230e6i	−0.0182605	−	0.0150977i
$778$		0			0
$779$	−2.93012e7	+	5.07512e7i	−0.0619831	+	0.107358i
$780$		0			0
$781$	−6.68134e8	−	1.15724e9i	−1.40253	−	2.42924i
$782$		0			0
$783$			5.25780e7i			0.109526i
$784$		0			0
$785$	34882.8			7.21111e−5
$786$		0			0
$787$	−7.50442e8	+	4.33268e8i	−1.53955	+	0.888857i	−0.540681	+	0.841228i	$0.681833\pi$
$787$	−7.50442e8	+	4.33268e8i	−1.53955	+	0.888857i	−0.998865	+	0.0476296i	$0.984833\pi$
$788$		0			0
$789$	−3.62215e8	−	2.09125e8i	−0.737456	−	0.425770i
$790$		0			0
$791$	4.91957e7	−	5.95016e7i	0.0994025	−	0.120226i
$792$		0			0
$793$	1.71661e6	−	2.97325e6i	0.00344232	−	0.00596227i
$794$		0			0
$795$	−73689.1	−	127633.i	−0.000146657	−	0.000254017i
$796$		0			0
$797$		−	7.87468e8i		−	1.55546i	−0.628601	−	0.777728i	$-0.716372\pi$
$797$		−	7.87468e8i		−	1.55546i	0.628601	−	0.777728i	$-0.283628\pi$
$798$		0			0
$799$	5.14825e8			1.00930
$800$		0			0
$801$	2.59581e8	−	1.49869e8i	0.505098	−	0.291619i
$802$		0			0
$803$	1.27249e9	+	7.34671e8i	2.45758	+	1.41888i
$804$		0			0
$805$	12127.6	−	71826.0i	2.32480e−5	−	0.000137687i
$806$		0			0
$807$	−1.42663e8	+	2.47100e8i	−0.271451	+	0.470168i
$808$		0			0
$809$	−3.24621e8	−	5.62259e8i	−0.613099	−	1.06192i	−0.990715	−	0.135956i	$-0.956589\pi$
$809$	−3.24621e8	−	5.62259e8i	−0.613099	−	1.06192i	0.377616	−	0.925962i	$-0.376744\pi$
$810$		0			0
$811$		−	1.94118e8i		−	0.363917i	−0.983306	−	0.181959i	$-0.941756\pi$
$811$		−	1.94118e8i		−	0.363917i	0.983306	−	0.181959i	$-0.0582437\pi$
$812$		0			0
$813$	3.81202e8			0.709387
$814$		0			0
$815$	469279.	−	270938.i	0.000866878	−	0.000500492i
$816$		0			0
$817$	1.58063e8	+	9.12577e7i	0.289844	+	0.167341i
$818$		0			0
$819$	870830.	−	324139.i	0.00158519	−	0.000590038i
$820$		0			0
$821$	3.47367e7	−	6.01658e7i	0.0627711	−	0.108723i	−0.832932	−	0.553375i	$-0.813340\pi$
$821$	3.47367e7	−	6.01658e7i	0.0627711	−	0.108723i	0.895703	+	0.444653i	$0.146673\pi$
$822$		0			0
$823$	2.87144e8	+	4.97347e8i	0.515109	+	0.892196i	0.999846	+	0.0175357i	$0.00558206\pi$
$823$	2.87144e8	+	4.97347e8i	0.515109	+	0.892196i	−0.484737	+	0.874660i	$0.661085\pi$
$824$		0			0
$825$			5.55309e8i			0.988946i
$826$		0			0
$827$	−1.02808e8			−0.181765			−0.0908826	−	0.995862i	$-0.528969\pi$
$827$	−1.02808e8			−0.181765			−0.0908826	+	0.995862i	$0.528969\pi$
$828$		0			0
$829$	3.09922e8	−	1.78934e8i	0.543988	−	0.314071i	−0.202706	−	0.979240i	$-0.564974\pi$
$829$	3.09922e8	−	1.78934e8i	0.543988	−	0.314071i	0.746693	+	0.665168i	$0.231640\pi$
$830$		0			0
$831$	−4.14692e8	−	2.39423e8i	−0.722642	−	0.417217i
$832$		0			0
$833$	3.96341e8	+	4.58640e8i	0.685699	+	0.793482i
$834$		0			0
$835$	−63288.8	+	109619.i	−0.000108710	+	0.000188290i
$836$		0			0
$837$	7.87266e7	+	1.36358e8i	0.134260	+	0.232544i
$838$		0			0
$839$		−	2.36873e8i		−	0.401079i	−0.979686	−	0.200539i	$-0.935731\pi$
$839$		−	2.36873e8i		−	0.401079i	0.979686	−	0.200539i	$-0.0642694\pi$
$840$		0			0
$841$	−4.02164e8			−0.676107
$842$		0			0
$843$	−2.88855e8	+	1.66771e8i	−0.482167	+	0.278379i
$844$		0			0
$845$	−378256.	−	218386.i	−0.000626925	−	0.000361955i
$846$		0			0
$847$	−4.09958e8	−	1.10139e9i	−0.674667	−	1.81256i
$848$		0			0
$849$	−1.82474e8	+	3.16055e8i	−0.298180	+	0.516464i
$850$		0			0
$851$	2.43923e6	+	4.22487e6i	0.00395789	+	0.00685526i
$852$		0			0
$853$			6.95204e8i			1.12012i	0.828451	+	0.560061i	$0.189222\pi$
$853$			6.95204e8i			1.12012i	−0.828451	+	0.560061i	$0.810778\pi$
$854$		0			0
$855$	38119.2			6.09882e−5
$856$		0			0
$857$	6.87613e8	−	3.96994e8i	1.09245	−	0.630727i	0.158223	−	0.987403i	$-0.449424\pi$
$857$	6.87613e8	−	3.96994e8i	1.09245	−	0.630727i	0.934228	+	0.356677i	$0.116090\pi$
$858$		0			0
$859$	6.84536e8	+	3.95217e8i	1.07998	+	0.623529i	0.930892	−	0.365295i	$-0.119032\pi$
$859$	6.84536e8	+	3.95217e8i	1.07998	+	0.623529i	0.149092	+	0.988823i	$0.452365\pi$
$860$		0			0
$861$	1.78228e8	+	3.00932e7i	0.279233	+	0.0471476i
$862$		0			0
$863$	−6.33504e8	+	1.09726e9i	−0.985637	+	1.70717i	−0.346567	+	0.938025i	$0.612653\pi$
$863$	−6.33504e8	+	1.09726e9i	−0.985637	+	1.70717i	−0.639070	+	0.769148i	$0.720681\pi$
$864$		0			0
$865$	−460352.	−	797354.i	−0.000711282	−	0.00123198i
$866$		0			0
$867$		−	3.75505e7i		−	0.0576180i
$868$		0			0
$869$	−1.22700e9			−1.86976
$870$		0			0
$871$	2.28118e6	−	1.31704e6i	0.00345227	−	0.00199317i
$872$		0			0
$873$	7.14488e7	+	4.12510e7i	0.107387	+	0.0620000i
$874$		0			0
$875$	−747536.	−	618059.i	−0.00111586	−	0.000922584i
$876$		0			0
$877$	−2.07029e8	+	3.58584e8i	−0.306924	+	0.531609i	−0.977688	−	0.210063i	$-0.932633\pi$
$877$	−2.07029e8	+	3.58584e8i	−0.306924	+	0.531609i	0.670764	+	0.741671i	$0.265967\pi$
$878$		0			0
$879$	−2.04800e8	−	3.54724e8i	−0.301553	−	0.522305i
$880$		0			0
$881$		−	1.11611e9i		−	1.63222i	−0.577897	−	0.816110i	$-0.696126\pi$
$881$		−	1.11611e9i		−	1.63222i	0.577897	−	0.816110i	$-0.303874\pi$
$882$		0			0
$883$	−7.77359e8			−1.12912			−0.564559	−	0.825392i	$-0.690954\pi$
$883$	−7.77359e8			−1.12912			−0.564559	+	0.825392i	$0.690954\pi$
$884$		0			0
$885$	−274244.	+	158335.i	−0.000395646	+	0.000228426i
$886$		0			0
$887$	2.79999e8	+	1.61657e8i	0.401223	+	0.231646i	0.687011	−	0.726647i	$-0.258922\pi$
$887$	2.79999e8	+	1.61657e8i	0.401223	+	0.231646i	−0.285789	+	0.958293i	$0.592256\pi$
$888$		0			0
$889$	6.46149e8	−	7.81510e8i	0.919661	−	1.11232i
$890$		0			0
$891$	−6.73122e7	+	1.16588e8i	−0.0951614	+	0.164824i
$892$		0			0
$893$	8.66081e7	+	1.50010e8i	0.121620	+	0.210652i
$894$		0			0
$895$			774540.i			0.00108038i
$896$		0			0
$897$	−407847.			−0.000565093
$898$		0			0
$899$	4.99652e8	−	2.88474e8i	0.687684	−	0.397034i
$900$		0			0
$901$	−4.66184e8	−	2.69152e8i	−0.637358	−	0.367979i
$902$		0			0
$903$	9.37245e7	−	5.55087e8i	0.127289	−	0.753872i
$904$		0			0
$905$	144379.	−	250072.i	0.000194787	−	0.000337380i
$906$		0			0
$907$	−5.41990e8	−	9.38754e8i	−0.726389	−	1.25814i	−0.958400	−	0.285430i	$-0.907864\pi$
$907$	−5.41990e8	−	9.38754e8i	−0.726389	−	1.25814i	0.232011	−	0.972713i	$-0.425470\pi$
$908$		0			0
$909$		−	1.34682e8i		−	0.179316i
$910$		0			0
$911$	−1.83298e8			−0.242439			−0.121219	−	0.992626i	$-0.538680\pi$
$911$	−1.83298e8			−0.242439			−0.121219	+	0.992626i	$0.538680\pi$
$912$		0			0
$913$	−1.18991e9	+	6.86992e8i	−1.56351	+	0.902692i
$914$		0			0
$915$	−376212.	−	217206.i	−0.000491099	−	0.000283536i
$916$		0			0
$917$	−1.07383e8	+	3.99701e7i	−0.139261	+	0.0518355i
$918$		0			0
$919$	−3.97916e8	+	6.89211e8i	−0.512679	+	0.887986i	0.487213	+	0.873283i	$0.338013\pi$
$919$	−3.97916e8	+	6.89211e8i	−0.512679	+	0.887986i	−0.999892	+	0.0147025i	$0.995320\pi$
$920$		0			0
$921$	−9.78721e7	−	1.69519e8i	−0.125280	−	0.216990i
$922$		0			0
$923$		−	6.53418e6i		−	0.00830971i
$924$		0			0
$925$	3.24801e7			0.0410385
$926$		0			0
$927$	−1.90924e8	+	1.10230e8i	−0.239674	+	0.138376i
$928$		0			0
$929$	−2.80851e8	−	1.62149e8i	−0.350290	−	0.202240i	0.314523	−	0.949250i	$-0.398156\pi$
$929$	−2.80851e8	−	1.62149e8i	−0.350290	−	0.202240i	−0.664813	+	0.747010i	$0.731489\pi$
$930$		0			0
$931$	−6.69628e7	+	1.92642e8i	−0.0829822	+	0.238727i
$932$		0			0
$933$	9.19158e6	−	1.59203e7i	0.0113174	−	0.0196023i
$934$		0			0
$935$	−531485.	−	920559.i	−0.000650214	−	0.00112620i
$936$		0			0
$937$			7.22586e8i			0.878356i	0.898400	+	0.439178i	$0.144730\pi$
$937$			7.22586e8i			0.878356i	−0.898400	+	0.439178i	$0.855270\pi$
$938$		0			0
$939$	−1.38678e8			−0.167498
$940$		0			0
$941$	−8.32202e8	+	4.80472e8i	−0.998757	+	0.576633i	−0.907880	−	0.419229i	$-0.862300\pi$
$941$	−8.32202e8	+	4.80472e8i	−0.998757	+	0.576633i	−0.0908770	+	0.995862i	$0.528967\pi$
$942$		0			0
$943$	−6.87068e7	−	3.96679e7i	−0.0819341	−	0.0473047i
$944$		0			0
$945$	−41014.0	−	110188.i	−4.86001e−5	−	0.000130569i
$946$		0			0
$947$	−1.20174e8	+	2.08148e8i	−0.141502	+	0.245088i	−0.928062	−	0.372425i	$-0.878526\pi$
$947$	−1.20174e8	+	2.08148e8i	−0.141502	+	0.245088i	0.786561	+	0.617513i	$0.211860\pi$
$948$		0			0
$949$	3.59244e6	+	6.22230e6i	0.00420331	+	0.00728035i
$950$		0			0
$951$		−	2.68099e8i		−	0.311712i
$952$		0			0
$953$	−1.26904e9			−1.46622			−0.733108	−	0.680113i	$-0.761931\pi$
$953$	−1.26904e9			−1.46622			−0.733108	+	0.680113i	$0.761931\pi$
$954$		0			0
$955$	732667.	−	423005.i	0.000841194	−	0.000485664i
$956$		0			0
$957$	4.27209e8	+	2.46649e8i	0.487421	+	0.281413i
$958$		0			0
$959$	7.90057e7	+	1.33398e7i	0.0895782	+	0.0151250i
$960$		0			0
$961$	4.20130e8	−	7.27686e8i	0.473384	−	0.819925i
$962$		0			0
$963$	−8.80451e7	−	1.52499e8i	−0.0985885	−	0.170760i
$964$		0			0
$965$			176846.i			0.000196794i
$966$		0			0
$967$	2.04679e8			0.226357			0.113178	−	0.993575i	$-0.463897\pi$
$967$	2.04679e8			0.226357			0.113178	+	0.993575i	$0.463897\pi$
$968$		0			0
$969$	1.20578e8	−	6.96158e7i	0.132525	−	0.0765132i
$970$		0			0
$971$	5.23119e8	+	3.02023e8i	0.571404	+	0.329900i	0.757710	−	0.652592i	$-0.226318\pi$
$971$	5.23119e8	+	3.02023e8i	0.571404	+	0.329900i	−0.186306	+	0.982492i	$0.559652\pi$
$972$		0			0
$973$	7.33160e8	+	6.06174e8i	0.795903	+	0.658049i
$974$		0			0
$975$	−1.35769e6	+	2.35159e6i	−0.00146483	+	0.00253716i
$976$		0			0
$977$	−7.57469e8	−	1.31197e9i	−0.812234	−	1.40683i	−0.911297	−	0.411749i	$-0.864918\pi$
$977$	−7.57469e8	−	1.31197e9i	−0.812234	−	1.40683i	0.0990637	−	0.995081i	$-0.468415\pi$
$978$		0			0
$979$		−	2.81221e9i		−	2.99709i
$980$		0			0
$981$	−3.78487e7			−0.0400907
$982$		0			0
$983$	−1.45203e9	+	8.38332e8i	−1.52868	+	0.882583i	−0.529260	+	0.848460i	$0.677530\pi$
$983$	−1.45203e9	+	8.38332e8i	−1.52868	+	0.882583i	−0.999418	+	0.0341231i	$0.989136\pi$
$984$		0			0
$985$	477972.	+	275957.i	0.000500143	+	0.000288757i
$986$		0			0
$987$	3.40435e8	−	4.11752e8i	0.354065	−	0.428237i
$988$		0			0
$989$	−1.23544e8	+	2.13985e8i	−0.127713	+	0.221205i
$990$		0			0
$991$	−3.22159e8	−	5.57995e8i	−0.331016	−	0.573336i	0.651695	−	0.758481i	$-0.274058\pi$
$991$	−3.22159e8	−	5.57995e8i	−0.331016	−	0.573336i	−0.982711	+	0.185144i	$0.940725\pi$
$992$		0			0
$993$			9.13947e8i			0.933412i
$994$		0			0
$995$	−864288.			−0.000877383
$996$		0			0
$997$	1.35199e9	−	7.80574e8i	1.36424	−	0.787642i	0.374051	−	0.927408i	$-0.377968\pi$
$997$	1.35199e9	−	7.80574e8i	1.36424	−	0.787642i	0.990185	+	0.139766i	$0.0446351\pi$
$998$		0			0
$999$	6.81926e6	+	3.93710e6i	0.00683976	+	0.00394893i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.7.m.b.61.3	✓	8
3.2	odd	2		252.7.z.e.145.2		8
4.3	odd	2		336.7.bh.a.145.3		8
7.2	even	3		588.7.d.a.97.6		8
7.3	odd	6	inner	84.7.m.b.73.3	yes	8
7.4	even	3		588.7.m.b.325.2		8
7.5	odd	6		588.7.d.a.97.3		8
7.6	odd	2		588.7.m.b.313.2		8
21.17	even	6		252.7.z.e.73.2		8
28.3	even	6		336.7.bh.a.241.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.7.m.b.61.3	✓	8	1.1	even	1	trivial
84.7.m.b.73.3	yes	8	7.3	odd	6	inner
252.7.z.e.73.2		8	21.17	even	6
252.7.z.e.145.2		8	3.2	odd	2
336.7.bh.a.145.3		8	4.3	odd	2
336.7.bh.a.241.3		8	28.3	even	6
588.7.d.a.97.3		8	7.5	odd	6
588.7.d.a.97.6		8	7.2	even	3
588.7.m.b.313.2		8	7.6	odd	2
588.7.m.b.325.2		8	7.4	even	3

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	84.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(13.5000 - 7.79423i) q^{3} +(-0.0783677 - 0.0452456i) q^{5} +(-218.562 + 264.348i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})\)	\(q+(13.5000 - 7.79423i) q^{3} +(-0.0783677 - 0.0452456i) q^{5} +(-218.562 + 264.348i) q^{7} +(121.500 - 210.444i) q^{9} +(-1139.94 - 1974.43i) q^{11} -11.1483i q^{13} -1.41062 q^{15} +(-4462.04 + 2576.16i) q^{17} +(-1501.28 - 866.766i) q^{19} +(-890.195 + 5272.22i) q^{21} +(1173.42 - 2032.43i) q^{23} +(-7812.50 - 13531.6i) q^{25} -3788.00i q^{27} -13880.2 q^{29} +(-35997.5 + 20783.2i) q^{31} +(-30778.3 - 17769.9i) q^{33} +(29.0887 - 10.8274i) q^{35} +(-1039.36 + 1800.23i) q^{37} +(-86.8924 - 150.502i) q^{39} -33805.2i q^{41} -105285. q^{43} +(-19.0433 + 10.9947i) q^{45} +(-86534.1 - 49960.5i) q^{47} +(-22110.6 - 115553. i) q^{49} +(-40158.4 + 69556.4i) q^{51} +(52238.9 + 90480.4i) q^{53} +206.309i q^{55} -27023.1 q^{57} +(194414. - 112245. i) q^{59} +(266700. + 153979. i) q^{61} +(29075.2 + 78113.3i) q^{63} +(-0.504411 + 0.873666i) q^{65} +(118138. + 204621. i) q^{67} -36583.8i q^{69} +586114. q^{71} +(-558139. + 322242. i) q^{73} +(-210937. - 121785. i) q^{75} +(771084. + 130195. i) q^{77} +(269093. - 466084. i) q^{79} +(-29524.5 - 51137.9i) q^{81} -602657. i q^{83} +466.240 q^{85} +(-187382. + 108185. i) q^{87} +(1.06824e6 + 616746. i) q^{89} +(2947.03 + 2436.59i) q^{91} +(-323978. + 561146. i) q^{93} +(78.4346 + 135.853i) q^{95} +339514. i q^{97} -554010. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9}+O(q^{10}) \) 8 * q + 108 * q^3 - 294 * q^5 + 232 * q^7 + 972 * q^9	\( 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9} + 378 q^{11} - 5292 q^{15} + 852 q^{17} + 3690 q^{19} + 3942 q^{21} + 15600 q^{23} + 3386 q^{25} - 68604 q^{29} + 23028 q^{31} + 10206 q^{33} + 93828 q^{35} + 15914 q^{37} + 25326 q^{39} - 170044 q^{43} - 71442 q^{45} + 102180 q^{47} + 157340 q^{49} + 7668 q^{51} + 196410 q^{53} + 66420 q^{57} - 662550 q^{59} - 23928 q^{61} + 50058 q^{63} + 14892 q^{65} + 774838 q^{67} - 721896 q^{71} - 1219050 q^{73} + 91422 q^{75} + 1584738 q^{77} - 493868 q^{79} - 236196 q^{81} - 1329816 q^{85} - 926154 q^{87} + 604260 q^{89} + 3831690 q^{91} + 207252 q^{93} + 448944 q^{95} + 183708 q^{99}+O(q^{100}) \) 8 * q + 108 * q^3 - 294 * q^5 + 232 * q^7 + 972 * q^9 + 378 * q^11 - 5292 * q^15 + 852 * q^17 + 3690 * q^19 + 3942 * q^21 + 15600 * q^23 + 3386 * q^25 - 68604 * q^29 + 23028 * q^31 + 10206 * q^33 + 93828 * q^35 + 15914 * q^37 + 25326 * q^39 - 170044 * q^43 - 71442 * q^45 + 102180 * q^47 + 157340 * q^49 + 7668 * q^51 + 196410 * q^53 + 66420 * q^57 - 662550 * q^59 - 23928 * q^61 + 50058 * q^63 + 14892 * q^65 + 774838 * q^67 - 721896 * q^71 - 1219050 * q^73 + 91422 * q^75 + 1584738 * q^77 - 493868 * q^79 - 236196 * q^81 - 1329816 * q^85 - 926154 * q^87 + 604260 * q^89 + 3831690 * q^91 + 207252 * q^93 + 448944 * q^95 + 183708 * q^99

Introduction

L-functions

Modular forms

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Database

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