LMFDB - Embedded newform 384.7.l.b.31.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,7,Mod(31,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("384.31");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	384.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$88.3407681100$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.21
Character	$\chi$	$=$	384.31
Dual form			384.7.l.b.223.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.0227 + 11.0227i) q^{3} +(15.5658 + 15.5658i) q^{5} +10.2991 q^{7} +243.000i q^{9} +O(q^{10})$	\(q+(11.0227 + 11.0227i) q^{3} +(15.5658 + 15.5658i) q^{5} +10.2991 q^{7} +243.000i q^{9} +(-1309.74 + 1309.74i) q^{11} +(-1999.54 + 1999.54i) q^{13} +343.155i q^{15} -1800.59 q^{17} +(-1703.06 - 1703.06i) q^{19} +(113.524 + 113.524i) q^{21} +5649.07 q^{23} -15140.4i q^{25} +(-2678.52 + 2678.52i) q^{27} +(11764.6 - 11764.6i) q^{29} -32237.2i q^{31} -28873.7 q^{33} +(160.314 + 160.314i) q^{35} +(49520.9 + 49520.9i) q^{37} -44080.7 q^{39} -87389.3i q^{41} +(-7129.74 + 7129.74i) q^{43} +(-3782.49 + 3782.49i) q^{45} +79423.3i q^{47} -117543. q^{49} +(-19847.3 - 19847.3i) q^{51} +(-58057.5 - 58057.5i) q^{53} -40774.3 q^{55} -37544.8i q^{57} +(-9265.40 + 9265.40i) q^{59} +(-31233.4 + 31233.4i) q^{61} +2502.69i q^{63} -62249.0 q^{65} +(-140680. - 140680. i) q^{67} +(62268.0 + 62268.0i) q^{69} +445186. q^{71} -674960. i q^{73} +(166888. - 166888. i) q^{75} +(-13489.2 + 13489.2i) q^{77} +171535. i q^{79} -59049.0 q^{81} +(-709618. - 709618. i) q^{83} +(-28027.6 - 28027.6i) q^{85} +259355. q^{87} +549423. i q^{89} +(-20593.5 + 20593.5i) q^{91} +(355342. - 355342. i) q^{93} -53019.2i q^{95} -675921. q^{97} +(-318267. - 318267. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 2720 q^{11} + 3936 q^{19} + 26240 q^{23} - 66400 q^{29} - 162336 q^{35} + 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} - 443680 q^{53} + 232704 q^{55} + 886144 q^{59} + 326496 q^{61} - 372832 q^{65} + 962112 q^{67} - 541728 q^{69} + 534016 q^{71} + 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} + 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) $ 48 * q + 2720 * q^11 + 3936 * q^19 + 26240 * q^23 - 66400 * q^29 - 162336 * q^35 + 7200 * q^37 - 340704 * q^43 + 806736 * q^49 - 80352 * q^51 - 443680 * q^53 + 232704 * q^55 + 886144 * q^59 + 326496 * q^61 - 372832 * q^65 + 962112 * q^67 - 541728 * q^69 + 534016 * q^71 + 1073088 * q^75 + 932960 * q^77 - 2834352 * q^81 + 2497760 * q^83 + 372000 * q^85 - 775008 * q^91 + 660960 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	11.0227	+	11.0227i	0.408248	+	0.408248i
$3$	11.0227	+	11.0227i	0.408248	+	0.408248i
$4$		0			0
$5$	15.5658	+	15.5658i	0.124526	+	0.124526i	0.766623	−	0.642097i	$-0.221935\pi$
$5$	15.5658	+	15.5658i	0.124526	+	0.124526i	−0.642097	+	0.766623i	$0.721935\pi$
$6$		0			0
$7$	10.2991			0.0300266			0.0150133	−	0.999887i	$-0.495221\pi$
$7$	10.2991			0.0300266			0.0150133	+	0.999887i	$0.495221\pi$
$8$		0			0
$9$			243.000i			0.333333i
$10$		0			0
$11$	−1309.74	+	1309.74i	−0.984027	+	0.984027i	−0.999874	−	0.0158475i	$-0.994955\pi$
$11$	−1309.74	+	1309.74i	−0.984027	+	0.984027i	0.0158475	+	0.999874i	$0.494955\pi$
$12$		0			0
$13$	−1999.54	+	1999.54i	−0.910123	+	0.910123i	−0.996281	−	0.0861580i	$-0.972541\pi$
$13$	−1999.54	+	1999.54i	−0.910123	+	0.910123i	0.0861580	+	0.996281i	$0.472541\pi$
$14$		0			0
$15$			343.155i			0.101675i
$16$		0			0
$17$	−1800.59			−0.366495			−0.183247	−	0.983067i	$-0.558661\pi$
$17$	−1800.59			−0.366495			−0.183247	+	0.983067i	$0.558661\pi$
$18$		0			0
$19$	−1703.06	−	1703.06i	−0.248296	−	0.248296i	0.571975	−	0.820271i	$-0.306177\pi$
$19$	−1703.06	−	1703.06i	−0.248296	−	0.248296i	−0.820271	+	0.571975i	$0.806177\pi$
$20$		0			0
$21$	113.524	+	113.524i	0.0122583	+	0.0122583i
$22$		0			0
$23$	5649.07			0.464295			0.232147	−	0.972681i	$-0.425425\pi$
$23$	5649.07			0.464295			0.232147	+	0.972681i	$0.425425\pi$
$24$		0			0
$25$		−	15140.4i		−	0.968986i
$26$		0			0
$27$	−2678.52	+	2678.52i	−0.136083	+	0.136083i
$28$		0			0
$29$	11764.6	−	11764.6i	0.482373	−	0.482373i	−0.423516	−	0.905889i	$-0.639204\pi$
$29$	11764.6	−	11764.6i	0.482373	−	0.482373i	0.905889	+	0.423516i	$0.139204\pi$
$30$		0			0
$31$		−	32237.2i		−	1.08211i	−0.840986	−	0.541057i	$-0.818024\pi$
$31$		−	32237.2i		−	1.08211i	0.840986	−	0.541057i	$-0.181976\pi$
$32$		0			0
$33$	−28873.7			−0.803455
$34$		0			0
$35$	160.314	+	160.314i	0.00373911	+	0.00373911i
$36$		0			0
$37$	49520.9	+	49520.9i	0.977651	+	0.977651i	0.999756	−	0.0221048i	$-0.00703675\pi$
$37$	49520.9	+	49520.9i	0.977651	+	0.977651i	−0.0221048	+	0.999756i	$0.507037\pi$
$38$		0			0
$39$	−44080.7			−0.743113
$40$		0			0
$41$		−	87389.3i		−	1.26796i	−0.773348	−	0.633982i	$-0.781420\pi$
$41$		−	87389.3i		−	1.26796i	0.773348	−	0.633982i	$-0.218580\pi$
$42$		0			0
$43$	−7129.74	+	7129.74i	−0.0896744	+	0.0896744i	−0.750521	−	0.660847i	$-0.770197\pi$
$43$	−7129.74	+	7129.74i	−0.0896744	+	0.0896744i	0.660847	+	0.750521i	$0.270197\pi$
$44$		0			0
$45$	−3782.49	+	3782.49i	−0.0415088	+	0.0415088i
$46$		0			0
$47$			79423.3i			0.764987i	0.923958	+	0.382494i	$0.124935\pi$
$47$			79423.3i			0.764987i	−0.923958	+	0.382494i	$0.875065\pi$
$48$		0			0
$49$	−117543.			−0.999098
$50$		0			0
$51$	−19847.3	−	19847.3i	−0.149621	−	0.149621i
$52$		0			0
$53$	−58057.5	−	58057.5i	−0.389970	−	0.389970i	0.484707	−	0.874677i	$-0.338926\pi$
$53$	−58057.5	−	58057.5i	−0.389970	−	0.389970i	−0.874677	+	0.484707i	$0.838926\pi$
$54$		0			0
$55$	−40774.3			−0.245075
$56$		0			0
$57$		−	37544.8i		−	0.202733i
$58$		0			0
$59$	−9265.40	+	9265.40i	−0.0451137	+	0.0451137i	−0.729304	−	0.684190i	$-0.760156\pi$
$59$	−9265.40	+	9265.40i	−0.0451137	+	0.0451137i	0.684190	+	0.729304i	$0.260156\pi$
$60$		0			0
$61$	−31233.4	+	31233.4i	−0.137604	+	0.137604i	−0.772553	−	0.634950i	$-0.781021\pi$
$61$	−31233.4	+	31233.4i	−0.137604	+	0.137604i	0.634950	+	0.772553i	$0.281021\pi$
$62$		0			0
$63$			2502.69i			0.0100089i
$64$		0			0
$65$	−62249.0			−0.226669
$66$		0			0
$67$	−140680.	−	140680.i	−0.467742	−	0.467742i	0.433440	−	0.901182i	$-0.357300\pi$
$67$	−140680.	−	140680.i	−0.467742	−	0.467742i	−0.901182	+	0.433440i	$0.857300\pi$
$68$		0			0
$69$	62268.0	+	62268.0i	0.189547	+	0.189547i
$70$		0			0
$71$	445186.			1.24385			0.621923	−	0.783079i	$-0.286352\pi$
$71$	445186.			1.24385			0.621923	+	0.783079i	$0.286352\pi$
$72$		0			0
$73$		−	674960.i		−	1.73504i	−0.497402	−	0.867520i	$-0.665713\pi$
$73$		−	674960.i		−	1.73504i	0.497402	−	0.867520i	$-0.334287\pi$
$74$		0			0
$75$	166888.	−	166888.i	0.395587	−	0.395587i
$76$		0			0
$77$	−13489.2	+	13489.2i	−0.0295470	+	0.0295470i
$78$		0			0
$79$			171535.i			0.347914i	0.984753	+	0.173957i	$0.0556553\pi$
$79$			171535.i			0.347914i	−0.984753	+	0.173957i	$0.944345\pi$
$80$		0			0
$81$	−59049.0			−0.111111
$82$		0			0
$83$	−709618.	−	709618.i	−1.24105	−	1.24105i	−0.959565	−	0.281488i	$-0.909172\pi$
$83$	−709618.	−	709618.i	−1.24105	−	1.24105i	−0.281488	−	0.959565i	$-0.590828\pi$
$84$		0			0
$85$	−28027.6	−	28027.6i	−0.0456383	−	0.0456383i
$86$		0			0
$87$	259355.			0.393856
$88$		0			0
$89$			549423.i			0.779358i	0.920951	+	0.389679i	$0.127414\pi$
$89$			549423.i			0.779358i	−0.920951	+	0.389679i	$0.872586\pi$
$90$		0			0
$91$	−20593.5	+	20593.5i	−0.0273279	+	0.0273279i
$92$		0			0
$93$	355342.	−	355342.i	0.441771	−	0.441771i
$94$		0			0
$95$		−	53019.2i		−	0.0618389i
$96$		0			0
$97$	−675921.			−0.740595			−0.370297	−	0.928913i	$-0.620744\pi$
$97$	−675921.			−0.740595			−0.370297	+	0.928913i	$0.620744\pi$
$98$		0			0
$99$	−318267.	−	318267.i	−0.328009	−	0.328009i
$100$		0			0
$101$	−196185.	−	196185.i	−0.190415	−	0.190415i	0.605460	−	0.795875i	$-0.292989\pi$
$101$	−196185.	−	196185.i	−0.190415	−	0.190415i	−0.795875	+	0.605460i	$0.792989\pi$
$102$		0			0
$103$	1.82223e6			1.66759			0.833797	−	0.552071i	$-0.186162\pi$
$103$	1.82223e6			1.66759			0.833797	+	0.552071i	$0.186162\pi$
$104$		0			0
$105$			3534.19i			0.00305297i
$106$		0			0
$107$	1.18982e6	−	1.18982e6i	0.971248	−	0.971248i	−0.0283501	−	0.999598i	$-0.509025\pi$
$107$	1.18982e6	−	1.18982e6i	0.971248	−	0.971248i	0.999598	+	0.0283501i	$0.00902534\pi$
$108$		0			0
$109$	−895661.	+	895661.i	−0.691615	+	0.691615i	−0.962587	−	0.270972i	$-0.912655\pi$
$109$	−895661.	+	895661.i	−0.691615	+	0.691615i	0.270972	+	0.962587i	$0.412655\pi$
$110$		0			0
$111$			1.09171e6i			0.798249i
$112$		0			0
$113$	−2.27050e6			−1.57357			−0.786784	−	0.617228i	$-0.788256\pi$
$113$	−2.27050e6			−1.57357			−0.786784	+	0.617228i	$0.788256\pi$
$114$		0			0
$115$	87932.4	+	87932.4i	0.0578170	+	0.0578170i
$116$		0			0
$117$	−485889.	−	485889.i	−0.303374	−	0.303374i
$118$		0			0
$119$	−18544.5			−0.0110046
$120$		0			0
$121$		−	1.65928e6i		−	0.936618i
$122$		0			0
$123$	963266.	−	963266.i	0.517644	−	0.517644i
$124$		0			0
$125$	478888.	−	478888.i	0.245191	−	0.245191i
$126$		0			0
$127$		−	3.63397e6i		−	1.77407i	−0.461705	−	0.887034i	$-0.652762\pi$
$127$		−	3.63397e6i		−	1.77407i	0.461705	−	0.887034i	$-0.347238\pi$
$128$		0			0
$129$	−157178.			−0.0732188
$130$		0			0
$131$	1.91691e6	+	1.91691e6i	0.852685	+	0.852685i	0.990463	−	0.137778i	$-0.0439961\pi$
$131$	1.91691e6	+	1.91691e6i	0.852685	+	0.852685i	−0.137778	+	0.990463i	$0.543996\pi$
$132$		0			0
$133$	−17540.1	−	17540.1i	−0.00745550	−	0.00745550i
$134$		0			0
$135$	−83386.6			−0.0338918
$136$		0			0
$137$		−	16843.3i		−	0.00655037i	−0.999995	−	0.00327519i	$-0.998957\pi$
$137$		−	16843.3i		−	0.00655037i	0.999995	−	0.00327519i	$-0.00104253\pi$
$138$		0			0
$139$	1.61360e6	−	1.61360e6i	0.600831	−	0.600831i	−0.339702	−	0.940533i	$-0.610326\pi$
$139$	1.61360e6	−	1.61360e6i	0.600831	−	0.600831i	0.940533	+	0.339702i	$0.110326\pi$
$140$		0			0
$141$	−875459.	+	875459.i	−0.312305	+	0.312305i
$142$		0			0
$143$		−	5.23776e6i		−	1.79117i
$144$		0			0
$145$	366251.			0.120136
$146$		0			0
$147$	−1.29564e6	−	1.29564e6i	−0.407880	−	0.407880i
$148$		0			0
$149$	−3.25465e6	−	3.25465e6i	−0.983889	−	0.983889i	0.0159835	−	0.999872i	$-0.494912\pi$
$149$	−3.25465e6	−	3.25465e6i	−0.983889	−	0.983889i	−0.999872	+	0.0159835i	$0.994912\pi$
$150$		0			0
$151$	−534226.			−0.155165			−0.0775826	−	0.996986i	$-0.524720\pi$
$151$	−534226.			−0.155165			−0.0775826	+	0.996986i	$0.524720\pi$
$152$		0			0
$153$		−	437543.i		−	0.122165i
$154$		0			0
$155$	501799.	−	501799.i	0.134752	−	0.134752i
$156$		0			0
$157$	−3.24398e6	+	3.24398e6i	−0.838260	+	0.838260i	−0.988630	−	0.150370i	$-0.951954\pi$
$157$	−3.24398e6	+	3.24398e6i	−0.838260	+	0.838260i	0.150370	+	0.988630i	$0.451954\pi$
$158$		0			0
$159$		−	1.27990e6i		−	0.318409i
$160$		0			0
$161$	58180.5			0.0139412
$162$		0			0
$163$	1.38258e6	+	1.38258e6i	0.319247	+	0.319247i	0.848478	−	0.529231i	$-0.177520\pi$
$163$	1.38258e6	+	1.38258e6i	0.319247	+	0.319247i	−0.529231	+	0.848478i	$0.677520\pi$
$164$		0			0
$165$	−449443.	−	449443.i	−0.100051	−	0.100051i
$166$		0			0
$167$	758120.			0.162775			0.0813877	−	0.996683i	$-0.474065\pi$
$167$	758120.			0.162775			0.0813877	+	0.996683i	$0.474065\pi$
$168$		0			0
$169$		−	3.16952e6i		−	0.656650i
$170$		0			0
$171$	413845.	−	413845.i	0.0827655	−	0.0827655i
$172$		0			0
$173$	3.74301e6	−	3.74301e6i	0.722908	−	0.722908i	−0.246289	−	0.969196i	$-0.579211\pi$
$173$	3.74301e6	−	3.74301e6i	0.722908	−	0.722908i	0.969196	+	0.246289i	$0.0792112\pi$
$174$		0			0
$175$		−	155933.i		−	0.0290954i
$176$		0			0
$177$	−204260.			−0.0368352
$178$		0			0
$179$	4.64678e6	+	4.64678e6i	0.810201	+	0.810201i	0.984664	−	0.174463i	$-0.0558188\pi$
$179$	4.64678e6	+	4.64678e6i	0.810201	+	0.810201i	−0.174463	+	0.984664i	$0.555819\pi$
$180$		0			0
$181$	−6.37439e6	−	6.37439e6i	−1.07499	−	1.07499i	−0.996951	−	0.0780350i	$-0.975135\pi$
$181$	−6.37439e6	−	6.37439e6i	−1.07499	−	1.07499i	−0.0780350	−	0.996951i	$-0.524865\pi$
$182$		0			0
$183$	−688553.			−0.112353
$184$		0			0
$185$			1.54167e6i			0.243487i
$186$		0			0
$187$	2.35830e6	−	2.35830e6i	0.360640	−	0.360640i
$188$		0			0
$189$	−27586.4	+	27586.4i	−0.00408610	+	0.00408610i
$190$		0			0
$191$		−	8.82548e6i		−	1.26660i	−0.773908	−	0.633298i	$-0.781701\pi$
$191$		−	8.82548e6i		−	1.26660i	0.773908	−	0.633298i	$-0.218299\pi$
$192$		0			0
$193$	−1.15481e7			−1.60635			−0.803175	−	0.595743i	$-0.796858\pi$
$193$	−1.15481e7			−1.60635			−0.803175	+	0.595743i	$0.796858\pi$
$194$		0			0
$195$	−686152.	−	686152.i	−0.0925372	−	0.0925372i
$196$		0			0
$197$	−1.01337e7	−	1.01337e7i	−1.32547	−	1.32547i	−0.909275	−	0.416195i	$-0.863363\pi$
$197$	−1.01337e7	−	1.01337e7i	−1.32547	−	1.32547i	−0.416195	−	0.909275i	$-0.636637\pi$
$198$		0			0
$199$	−1.23678e6			−0.156940			−0.0784702	−	0.996916i	$-0.525004\pi$
$199$	−1.23678e6			−0.156940			−0.0784702	+	0.996916i	$0.525004\pi$
$200$		0			0
$201$		−	3.10134e6i		−	0.381910i
$202$		0			0
$203$	121165.	−	121165.i	0.0144840	−	0.0144840i
$204$		0			0
$205$	1.36028e6	−	1.36028e6i	0.157895	−	0.157895i
$206$		0			0
$207$			1.37272e6i			0.154765i
$208$		0			0
$209$	4.46114e6			0.488661
$210$		0			0
$211$	−1.18103e6	−	1.18103e6i	−0.125722	−	0.125722i	0.641446	−	0.767168i	$-0.278335\pi$
$211$	−1.18103e6	−	1.18103e6i	−0.125722	−	0.125722i	−0.767168	+	0.641446i	$0.778335\pi$
$212$		0			0
$213$	4.90715e6	+	4.90715e6i	0.507798	+	0.507798i
$214$		0			0
$215$	−221960.			−0.0223337
$216$		0			0
$217$		−	332016.i		−	0.0324922i
$218$		0			0
$219$	7.43989e6	−	7.43989e6i	0.708327	−	0.708327i
$220$		0			0
$221$	3.60035e6	−	3.60035e6i	0.333555	−	0.333555i
$222$		0			0
$223$			1.20834e7i			1.08962i	0.838561	+	0.544808i	$0.183398\pi$
$223$			1.20834e7i			1.08962i	−0.838561	+	0.544808i	$0.816602\pi$
$224$		0			0
$225$	3.67912e6			0.322995
$226$		0			0
$227$	−1.40935e7	−	1.40935e7i	−1.20488	−	1.20488i	−0.972666	−	0.232210i	$-0.925404\pi$
$227$	−1.40935e7	−	1.40935e7i	−1.20488	−	1.20488i	−0.232210	−	0.972666i	$-0.574596\pi$
$228$		0			0
$229$	−4.56390e6	−	4.56390e6i	−0.380040	−	0.380040i	0.491076	−	0.871117i	$-0.336604\pi$
$229$	−4.56390e6	−	4.56390e6i	−0.380040	−	0.380040i	−0.871117	+	0.491076i	$0.836604\pi$
$230$		0			0
$231$	−297374.			−0.0241250
$232$		0			0
$233$			1.88250e7i			1.48822i	0.668056	+	0.744111i	$0.267127\pi$
$233$			1.88250e7i			1.48822i	−0.668056	+	0.744111i	$0.732873\pi$
$234$		0			0
$235$	−1.23629e6	+	1.23629e6i	−0.0952612	+	0.0952612i
$236$		0			0
$237$	−1.89078e6	+	1.89078e6i	−0.142035	+	0.142035i
$238$		0			0
$239$		−	1.87271e7i		−	1.37176i	−0.727716	−	0.685879i	$-0.759418\pi$
$239$		−	1.87271e7i		−	1.37176i	0.727716	−	0.685879i	$-0.240582\pi$
$240$		0			0
$241$	3.72390e6			0.266040			0.133020	−	0.991113i	$-0.457533\pi$
$241$	3.72390e6			0.266040			0.133020	+	0.991113i	$0.457533\pi$
$242$		0			0
$243$	−650880.	−	650880.i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	−1.82965e6	−	1.82965e6i	−0.124414	−	0.124414i
$246$		0			0
$247$	6.81070e6			0.451961
$248$		0			0
$249$		−	1.56438e7i		−	1.01332i
$250$		0			0
$251$	−1.06030e7	+	1.06030e7i	−0.670513	+	0.670513i	−0.957834	−	0.287321i	$-0.907235\pi$
$251$	−1.06030e7	+	1.06030e7i	−0.670513	+	0.670513i	0.287321	+	0.957834i	$0.407235\pi$
$252$		0			0
$253$	−7.39881e6	+	7.39881e6i	−0.456878	+	0.456878i
$254$		0			0
$255$		−	617880.i		−	0.0372635i
$256$		0			0
$257$	1.38806e6			0.0817730			0.0408865	−	0.999164i	$-0.486982\pi$
$257$	1.38806e6			0.0817730			0.0408865	+	0.999164i	$0.486982\pi$
$258$		0			0
$259$	510023.	+	510023.i	0.0293555	+	0.0293555i
$260$		0			0
$261$	2.85880e6	+	2.85880e6i	0.160791	+	0.160791i
$262$		0			0
$263$	7.75571e6			0.426338			0.213169	−	0.977015i	$-0.431621\pi$
$263$	7.75571e6			0.426338			0.213169	+	0.977015i	$0.431621\pi$
$264$		0			0
$265$		−	1.80742e6i		−	0.0971231i
$266$		0			0
$267$	−6.05613e6	+	6.05613e6i	−0.318171	+	0.318171i
$268$		0			0
$269$	−7.01455e6	+	7.01455e6i	−0.360365	+	0.360365i	−0.863947	−	0.503582i	$-0.832015\pi$
$269$	−7.01455e6	+	7.01455e6i	−0.360365	+	0.360365i	0.503582	+	0.863947i	$0.332015\pi$
$270$		0			0
$271$			3.58375e7i			1.80065i	0.435216	+	0.900326i	$0.356672\pi$
$271$			3.58375e7i			1.80065i	−0.435216	+	0.900326i	$0.643328\pi$
$272$		0			0
$273$	−453993.			−0.0223132
$274$		0			0
$275$	1.98300e7	+	1.98300e7i	0.953509	+	0.953509i
$276$		0			0
$277$	1.43034e7	+	1.43034e7i	0.672976	+	0.672976i	0.958401	−	0.285425i	$-0.0921347\pi$
$277$	1.43034e7	+	1.43034e7i	0.672976	+	0.672976i	−0.285425	+	0.958401i	$0.592135\pi$
$278$		0			0
$279$	7.83365e6			0.360705
$280$		0			0
$281$		−	1.53217e7i		−	0.690539i	−0.938504	−	0.345270i	$-0.887787\pi$
$281$		−	1.53217e7i		−	0.690539i	0.938504	−	0.345270i	$-0.112213\pi$
$282$		0			0
$283$	2.25052e7	−	2.25052e7i	0.992942	−	0.992942i	−0.00703372	−	0.999975i	$-0.502239\pi$
$283$	2.25052e7	−	2.25052e7i	0.992942	−	0.992942i	0.999975	+	0.00703372i	$0.00223892\pi$
$284$		0			0
$285$	584414.	−	584414.i	0.0252456	−	0.0252456i
$286$		0			0
$287$		−	900033.i		−	0.0380726i
$288$		0			0
$289$	−2.08955e7			−0.865682
$290$		0			0
$291$	−7.45047e6	−	7.45047e6i	−0.302346	−	0.302346i
$292$		0			0
$293$	2.66381e7	+	2.66381e7i	1.05901	+	1.05901i	0.998146	+	0.0608658i	$0.0193862\pi$
$293$	2.66381e7	+	2.66381e7i	1.05901	+	1.05901i	0.0608658	+	0.998146i	$0.480614\pi$
$294$		0			0
$295$	−288447.			−0.0112357
$296$		0			0
$297$		−	7.01632e6i		−	0.267818i
$298$		0			0
$299$	−1.12956e7	+	1.12956e7i	−0.422565	+	0.422565i
$300$		0			0
$301$	−73430.1	+	73430.1i	−0.00269262	+	0.00269262i
$302$		0			0
$303$		−	4.32498e6i		−	0.155473i
$304$		0			0
$305$	−972346.			−0.0342706
$306$		0			0
$307$	−2.12747e6	−	2.12747e6i	−0.0735273	−	0.0735273i	0.669387	−	0.742914i	$-0.266557\pi$
$307$	−2.12747e6	−	2.12747e6i	−0.0735273	−	0.0735273i	−0.742914	+	0.669387i	$0.766557\pi$
$308$		0			0
$309$	2.00858e7	+	2.00858e7i	0.680792	+	0.680792i
$310$		0			0
$311$	−2.09965e7			−0.698018			−0.349009	−	0.937119i	$-0.613482\pi$
$311$	−2.09965e7			−0.698018			−0.349009	+	0.937119i	$0.613482\pi$
$312$		0			0
$313$			3.90976e7i			1.27502i	0.770442	+	0.637511i	$0.220036\pi$
$313$			3.90976e7i			1.27502i	−0.770442	+	0.637511i	$0.779964\pi$
$314$		0			0
$315$	−38956.4	+	38956.4i	−0.00124637	+	0.00124637i
$316$		0			0
$317$	3.72393e7	−	3.72393e7i	1.16902	−	1.16902i	0.186585	−	0.982439i	$-0.440258\pi$
$317$	3.72393e7	−	3.72393e7i	1.16902	−	1.16902i	0.982439	−	0.186585i	$-0.0597421\pi$
$318$		0			0
$319$			3.08171e7i			0.949336i
$320$		0			0
$321$	2.62301e7			0.793021
$322$		0			0
$323$	3.06652e6	+	3.06652e6i	0.0909992	+	0.0909992i
$324$		0			0
$325$	3.02739e7	+	3.02739e7i	0.881897	+	0.881897i
$326$		0			0
$327$	−1.97452e7			−0.564701
$328$		0			0
$329$			817991.i			0.0229700i
$330$		0			0
$331$	−4.89023e7	+	4.89023e7i	−1.34848	+	1.34848i	−0.461170	+	0.887312i	$0.652570\pi$
$331$	−4.89023e7	+	4.89023e7i	−1.34848	+	1.34848i	−0.887312	+	0.461170i	$0.847430\pi$
$332$		0			0
$333$	−1.20336e7	+	1.20336e7i	−0.325884	+	0.325884i
$334$		0			0
$335$		−	4.37958e6i		−	0.116493i
$336$		0			0
$337$	−2.08256e7			−0.544136			−0.272068	−	0.962278i	$-0.587708\pi$
$337$	−2.08256e7			−0.544136			−0.272068	+	0.962278i	$0.587708\pi$
$338$		0			0
$339$	−2.50270e7	−	2.50270e7i	−0.642407	−	0.642407i
$340$		0			0
$341$	4.22224e7	+	4.22224e7i	1.06483	+	1.06483i
$342$		0			0
$343$	−2.42227e6			−0.0600261
$344$		0			0
$345$			1.93850e6i			0.0472073i
$346$		0			0
$347$	−1.19435e6	+	1.19435e6i	−0.0285854	+	0.0285854i	−0.721255	−	0.692670i	$-0.756434\pi$
$347$	−1.19435e6	+	1.19435e6i	−0.0285854	+	0.0285854i	0.692670	+	0.721255i	$0.256434\pi$
$348$		0			0
$349$	−2.69344e7	+	2.69344e7i	−0.633622	+	0.633622i	−0.948975	−	0.315352i	$-0.897877\pi$
$349$	−2.69344e7	+	2.69344e7i	−0.633622	+	0.633622i	0.315352	+	0.948975i	$0.397877\pi$
$350$		0			0
$351$		−	1.07116e7i		−	0.247704i
$352$		0			0
$353$	4.91191e7			1.11667			0.558337	−	0.829615i	$-0.311440\pi$
$353$	4.91191e7			1.11667			0.558337	+	0.829615i	$0.311440\pi$
$354$		0			0
$355$	6.92968e6	+	6.92968e6i	0.154892	+	0.154892i
$356$		0			0
$357$	−204410.	−	204410.i	−0.00449260	−	0.00449260i
$358$		0			0
$359$	3.41281e7			0.737613			0.368806	−	0.929506i	$-0.379767\pi$
$359$	3.41281e7			0.737613			0.368806	+	0.929506i	$0.379767\pi$
$360$		0			0
$361$		−	4.12450e7i		−	0.876698i
$362$		0			0
$363$	1.82897e7	−	1.82897e7i	0.382373	−	0.382373i
$364$		0			0
$365$	1.05063e7	−	1.05063e7i	0.216058	−	0.216058i
$366$		0			0
$367$			9.40637e7i			1.90293i	0.307751	+	0.951467i	$0.400424\pi$
$367$			9.40637e7i			1.90293i	−0.307751	+	0.951467i	$0.599576\pi$
$368$		0			0
$369$	2.12356e7			0.422654
$370$		0			0
$371$	−597942.	−	597942.i	−0.0117095	−	0.0117095i
$372$		0			0
$373$	−6.53429e7	−	6.53429e7i	−1.25913	−	1.25913i	−0.951507	−	0.307626i	$-0.900465\pi$
$373$	−6.53429e7	−	6.53429e7i	−1.25913	−	1.25913i	−0.307626	−	0.951507i	$-0.599535\pi$
$374$		0			0
$375$	1.05573e7			0.200198
$376$		0			0
$377$			4.70476e7i			0.878038i
$378$		0			0
$379$	−7.30308e7	+	7.30308e7i	−1.34149	+	1.34149i	−0.446918	+	0.894575i	$0.647478\pi$
$379$	−7.30308e7	+	7.30308e7i	−1.34149	+	1.34149i	−0.894575	+	0.446918i	$0.852522\pi$
$380$		0			0
$381$	4.00562e7	−	4.00562e7i	0.724260	−	0.724260i
$382$		0			0
$383$			8.34668e6i			0.148565i	0.997237	+	0.0742827i	$0.0236667\pi$
$383$			8.34668e6i			0.148565i	−0.997237	+	0.0742827i	$0.976333\pi$
$384$		0			0
$385$	−419940.			−0.00735877
$386$		0			0
$387$	−1.73253e6	−	1.73253e6i	−0.0298915	−	0.0298915i
$388$		0			0
$389$	−2.74528e7	−	2.74528e7i	−0.466377	−	0.466377i	0.434361	−	0.900739i	$-0.356974\pi$
$389$	−2.74528e7	−	2.74528e7i	−0.466377	−	0.466377i	−0.900739	+	0.434361i	$0.856974\pi$
$390$		0			0
$391$	−1.01716e7			−0.170161
$392$		0			0
$393$			4.22591e7i			0.696214i
$394$		0			0
$395$	−2.67008e6	+	2.67008e6i	−0.0433244	+	0.0433244i
$396$		0			0
$397$	−3.93307e7	+	3.93307e7i	−0.628579	+	0.628579i	−0.947710	−	0.319131i	$-0.896609\pi$
$397$	−3.93307e7	+	3.93307e7i	−0.628579	+	0.628579i	0.319131	+	0.947710i	$0.396609\pi$
$398$		0			0
$399$		−	386678.i		−	0.00608739i
$400$		0			0
$401$	5.95571e7			0.923635			0.461817	−	0.886975i	$-0.347198\pi$
$401$	5.95571e7			0.923635			0.461817	+	0.886975i	$0.347198\pi$
$402$		0			0
$403$	6.44597e7	+	6.44597e7i	0.984857	+	0.984857i
$404$		0			0
$405$	−919145.	−	919145.i	−0.0138363	−	0.0138363i
$406$		0			0
$407$	−1.29719e8			−1.92407
$408$		0			0
$409$			4.56064e7i			0.666586i	0.942823	+	0.333293i	$0.108160\pi$
$409$			4.56064e7i			0.666586i	−0.942823	+	0.333293i	$0.891840\pi$
$410$		0			0
$411$	185659.	−	185659.i	0.00267418	−	0.00267418i
$412$		0			0
$413$	−95425.5	+	95425.5i	−0.00135461	+	0.00135461i
$414$		0			0
$415$		−	2.20915e7i		−	0.309088i
$416$		0			0
$417$	3.55725e7			0.490576
$418$		0			0
$419$	−6.90997e7	−	6.90997e7i	−0.939364	−	0.939364i	0.0589000	−	0.998264i	$-0.481241\pi$
$419$	−6.90997e7	−	6.90997e7i	−0.939364	−	0.939364i	−0.998264	+	0.0589000i	$0.981241\pi$
$420$		0			0
$421$	3.73557e6	+	3.73557e6i	0.0500623	+	0.0500623i	0.731695	−	0.681632i	$-0.238730\pi$
$421$	3.73557e6	+	3.73557e6i	0.0500623	+	0.0500623i	−0.681632	+	0.731695i	$0.738730\pi$
$422$		0			0
$423$	−1.92999e7			−0.254996
$424$		0			0
$425$			2.72616e7i			0.355128i
$426$		0			0
$427$	−321677.	+	321677.i	−0.00413177	+	0.00413177i
$428$		0			0
$429$	5.77343e7	−	5.77343e7i	0.731243	−	0.731243i
$430$		0			0
$431$			5.26388e7i			0.657467i	0.944423	+	0.328734i	$0.106622\pi$
$431$			5.26388e7i			0.657467i	−0.944423	+	0.328734i	$0.893378\pi$
$432$		0			0
$433$	4.50617e7			0.555065			0.277532	−	0.960716i	$-0.410483\pi$
$433$	4.50617e7			0.555065			0.277532	+	0.960716i	$0.410483\pi$
$434$		0			0
$435$	4.03707e6	+	4.03707e6i	0.0490455	+	0.0490455i
$436$		0			0
$437$	−9.62073e6	−	9.62073e6i	−0.115283	−	0.115283i
$438$		0			0
$439$	−6.13040e7			−0.724595			−0.362298	−	0.932062i	$-0.618008\pi$
$439$	−6.13040e7			−0.724595			−0.362298	+	0.932062i	$0.618008\pi$
$440$		0			0
$441$		−	2.85629e7i		−	0.333033i
$442$		0			0
$443$	2.20176e7	−	2.20176e7i	0.253255	−	0.253255i	−0.569049	−	0.822304i	$-0.692688\pi$
$443$	2.20176e7	−	2.20176e7i	0.253255	−	0.253255i	0.822304	+	0.569049i	$0.192688\pi$
$444$		0			0
$445$	−8.55221e6	+	8.55221e6i	−0.0970506	+	0.0970506i
$446$		0			0
$447$		−	7.17502e7i		−	0.803342i
$448$		0			0
$449$	5.18241e7			0.572523			0.286262	−	0.958151i	$-0.407587\pi$
$449$	5.18241e7			0.572523			0.286262	+	0.958151i	$0.407587\pi$
$450$		0			0
$451$	1.14457e8	+	1.14457e8i	1.24771	+	1.24771i
$452$		0			0
$453$	−5.88862e6	−	5.88862e6i	−0.0633459	−	0.0633459i
$454$		0			0
$455$	−641110.			−0.00680610
$456$		0			0
$457$			3.31182e7i			0.346991i	0.984835	+	0.173495i	$0.0555061\pi$
$457$			3.31182e7i			0.346991i	−0.984835	+	0.173495i	$0.944494\pi$
$458$		0			0
$459$	4.82290e6	−	4.82290e6i	0.0498736	−	0.0498736i
$460$		0			0
$461$	−1.62828e7	+	1.62828e7i	−0.166199	+	0.166199i	−0.785306	−	0.619108i	$-0.787494\pi$
$461$	−1.62828e7	+	1.62828e7i	−0.166199	+	0.166199i	0.619108	+	0.785306i	$0.287494\pi$
$462$		0			0
$463$		−	4.56709e7i		−	0.460147i	−0.973173	−	0.230074i	$-0.926103\pi$
$463$		−	4.56709e7i		−	0.460147i	0.973173	−	0.230074i	$-0.0738967\pi$
$464$		0			0
$465$	1.10624e7			0.110024
$466$		0			0
$467$	−5.27710e7	−	5.27710e7i	−0.518137	−	0.518137i	0.398870	−	0.917007i	$-0.369402\pi$
$467$	−5.27710e7	−	5.27710e7i	−0.518137	−	0.518137i	−0.917007	+	0.398870i	$0.869402\pi$
$468$		0			0
$469$	−1.44888e6	−	1.44888e6i	−0.0140447	−	0.0140447i
$470$		0			0
$471$	−7.15148e7			−0.684437
$472$		0			0
$473$		−	1.86762e7i		−	0.176484i
$474$		0			0
$475$	−2.57851e7	+	2.57851e7i	−0.240596	+	0.240596i
$476$		0			0
$477$	1.41080e7	−	1.41080e7i	0.129990	−	0.129990i
$478$		0			0
$479$		−	8.65283e7i		−	0.787320i	−0.919256	−	0.393660i	$-0.871209\pi$
$479$		−	8.65283e7i		−	0.787320i	0.919256	−	0.393660i	$-0.128791\pi$
$480$		0			0
$481$	−1.98038e8			−1.77957
$482$		0			0
$483$	641306.	+	641306.i	0.00569147	+	0.00569147i
$484$		0			0
$485$	−1.05213e7	−	1.05213e7i	−0.0922236	−	0.0922236i
$486$		0			0
$487$	1.04559e8			0.905263			0.452632	−	0.891698i	$-0.350485\pi$
$487$	1.04559e8			0.905263			0.452632	+	0.891698i	$0.350485\pi$
$488$		0			0
$489$			3.04795e7i			0.260664i
$490$		0			0
$491$	5.07335e7	−	5.07335e7i	0.428598	−	0.428598i	−0.459553	−	0.888151i	$-0.651990\pi$
$491$	5.07335e7	−	5.07335e7i	0.428598	−	0.428598i	0.888151	+	0.459553i	$0.151990\pi$
$492$		0			0
$493$	−2.11832e7	+	2.11832e7i	−0.176787	+	0.176787i
$494$		0			0
$495$		−	9.90816e6i		−	0.0816916i
$496$		0			0
$497$	4.58503e6			0.0373485
$498$		0			0
$499$	1.06158e8	+	1.06158e8i	0.854379	+	0.854379i	0.990669	−	0.136290i	$-0.0435180\pi$
$499$	1.06158e8	+	1.06158e8i	0.854379	+	0.854379i	−0.136290	+	0.990669i	$0.543518\pi$
$500$		0			0
$501$	8.35654e6	+	8.35654e6i	0.0664528	+	0.0664528i
$502$		0			0
$503$	1.26794e8			0.996310			0.498155	−	0.867088i	$-0.334011\pi$
$503$	1.26794e8			0.996310			0.498155	+	0.867088i	$0.334011\pi$
$504$		0			0
$505$		−	6.10755e6i		−	0.0474234i
$506$		0			0
$507$	3.49367e7	−	3.49367e7i	0.268076	−	0.268076i
$508$		0			0
$509$	−241668.	+	241668.i	−0.00183259	+	0.00183259i	−0.708022	−	0.706190i	$-0.750412\pi$
$509$	−241668.	+	241668.i	−0.00183259	+	0.00183259i	0.706190	+	0.708022i	$0.250412\pi$
$510$		0			0
$511$		−	6.95150e6i		−	0.0520974i
$512$		0			0
$513$	9.12338e6			0.0675777
$514$		0			0
$515$	2.83644e7	+	2.83644e7i	0.207660	+	0.207660i
$516$		0			0
$517$	−1.04024e8	−	1.04024e8i	−0.752768	−	0.752768i
$518$		0			0
$519$	8.25162e7			0.590252
$520$		0			0
$521$		−	2.33312e8i		−	1.64977i	−0.565301	−	0.824885i	$-0.691240\pi$
$521$		−	2.33312e8i		−	1.64977i	0.565301	−	0.824885i	$-0.308760\pi$
$522$		0			0
$523$	−1.17256e8	+	1.17256e8i	−0.819652	+	0.819652i	−0.986057	−	0.166405i	$-0.946784\pi$
$523$	−1.17256e8	+	1.17256e8i	−0.819652	+	0.819652i	0.166405	+	0.986057i	$0.446784\pi$
$524$		0			0
$525$	1.71880e6	−	1.71880e6i	0.0118781	−	0.0118781i
$526$		0			0
$527$			5.80460e7i			0.396589i
$528$		0			0
$529$	−1.16124e8			−0.784431
$530$		0			0
$531$	−2.25149e6	−	2.25149e6i	−0.0150379	−	0.0150379i
$532$		0			0
$533$	1.74738e8	+	1.74738e8i	1.15400	+	1.15400i
$534$		0			0
$535$	3.70410e7			0.241892
$536$		0			0
$537$			1.02440e8i			0.661527i
$538$		0			0
$539$	1.53951e8	−	1.53951e8i	0.983140	−	0.983140i
$540$		0			0
$541$	−1.79653e8	+	1.79653e8i	−1.13460	+	1.13460i	−0.145197	+	0.989403i	$0.546382\pi$
$541$	−1.79653e8	+	1.79653e8i	−1.13460	+	1.13460i	−0.989403	+	0.145197i	$0.953618\pi$
$542$		0			0
$543$		−	1.40526e8i		−	0.877722i
$544$		0			0
$545$	−2.78834e7			−0.172249
$546$		0			0
$547$	−3.98185e7	−	3.98185e7i	−0.243289	−	0.243289i	0.574920	−	0.818209i	$-0.305033\pi$
$547$	−3.98185e7	−	3.98185e7i	−0.243289	−	0.243289i	−0.818209	+	0.574920i	$0.805033\pi$
$548$		0			0
$549$	−7.58971e6	−	7.58971e6i	−0.0458679	−	0.0458679i
$550$		0			0
$551$	−4.00717e7			−0.239543
$552$		0			0
$553$			1.76666e6i			0.0104467i
$554$		0			0
$555$	−1.69933e7	+	1.69933e7i	−0.0994031	+	0.0994031i
$556$		0			0
$557$	3.27259e7	−	3.27259e7i	0.189377	−	0.189377i	−0.606050	−	0.795427i	$-0.707247\pi$
$557$	3.27259e7	−	3.27259e7i	0.189377	−	0.189377i	0.795427	+	0.606050i	$0.207247\pi$
$558$		0			0
$559$		−	2.85124e7i		−	0.163230i
$560$		0			0
$561$	5.19897e7			0.294462
$562$		0			0
$563$	−1.04641e8	−	1.04641e8i	−0.586376	−	0.586376i	0.350272	−	0.936648i	$-0.386089\pi$
$563$	−1.04641e8	−	1.04641e8i	−0.586376	−	0.586376i	−0.936648	+	0.350272i	$0.886089\pi$
$564$		0			0
$565$	−3.53421e7	−	3.53421e7i	−0.195951	−	0.195951i
$566$		0			0
$567$	−608153.			−0.00333629
$568$		0			0
$569$		−	1.76056e8i		−	0.955684i	−0.878446	−	0.477842i	$-0.841419\pi$
$569$		−	1.76056e8i		−	0.955684i	0.878446	−	0.477842i	$-0.158581\pi$
$570$		0			0
$571$	1.10539e8	−	1.10539e8i	0.593754	−	0.593754i	−0.344889	−	0.938643i	$-0.612083\pi$
$571$	1.10539e8	−	1.10539e8i	0.593754	−	0.593754i	0.938643	+	0.344889i	$0.112083\pi$
$572$		0			0
$573$	9.72807e7	−	9.72807e7i	0.517086	−	0.517086i
$574$		0			0
$575$		−	8.55293e7i		−	0.449895i
$576$		0			0
$577$	−2.03195e8			−1.05775			−0.528877	−	0.848698i	$-0.677387\pi$
$577$	−2.03195e8			−1.05775			−0.528877	+	0.848698i	$0.677387\pi$
$578$		0			0
$579$	−1.27292e8	−	1.27292e8i	−0.655790	−	0.655790i
$580$		0			0
$581$	−7.30844e6	−	7.30844e6i	−0.0372646	−	0.0372646i
$582$		0			0
$583$	1.52081e8			0.767482
$584$		0			0
$585$		−	1.51265e7i		−	0.0755563i
$586$		0			0
$587$	2.05167e8	−	2.05167e8i	1.01436	−	1.01436i	0.0144688	−	0.999895i	$-0.495394\pi$
$587$	2.05167e8	−	2.05167e8i	1.01436	−	1.01436i	0.999895	−	0.0144688i	$-0.00460572\pi$
$588$		0			0
$589$	−5.49021e7	+	5.49021e7i	−0.268685	+	0.268685i
$590$		0			0
$591$		−	2.23402e8i		−	1.08224i
$592$		0			0
$593$	3.11010e7			0.149146			0.0745728	−	0.997216i	$-0.476241\pi$
$593$	3.11010e7			0.149146			0.0745728	+	0.997216i	$0.476241\pi$
$594$		0			0
$595$	−288660.	−	288660.i	−0.00137036	−	0.00137036i
$596$		0			0
$597$	−1.36327e7	−	1.36327e7i	−0.0640706	−	0.0640706i
$598$		0			0
$599$	1.95480e8			0.909541			0.454770	−	0.890609i	$-0.349721\pi$
$599$	1.95480e8			0.909541			0.454770	+	0.890609i	$0.349721\pi$
$600$		0			0
$601$			6.46859e7i			0.297979i	0.988839	+	0.148990i	$0.0476021\pi$
$601$			6.46859e7i			0.297979i	−0.988839	+	0.148990i	$0.952398\pi$
$602$		0			0
$603$	3.41851e7	−	3.41851e7i	0.155914	−	0.155914i
$604$		0			0
$605$	2.58280e7	−	2.58280e7i	0.116634	−	0.116634i
$606$		0			0
$607$			3.99248e8i			1.78516i	0.450893	+	0.892578i	$0.351106\pi$
$607$			3.99248e8i			1.78516i	−0.450893	+	0.892578i	$0.648894\pi$
$608$		0			0
$609$	2.67113e6			0.0118262
$610$		0			0
$611$	−1.58810e8	−	1.58810e8i	−0.696233	−	0.696233i
$612$		0			0
$613$	−2.95990e8	−	2.95990e8i	−1.28498	−	1.28498i	−0.937800	−	0.347177i	$-0.887140\pi$
$613$	−2.95990e8	−	2.95990e8i	−1.28498	−	1.28498i	−0.347177	−	0.937800i	$-0.612860\pi$
$614$		0			0
$615$	2.99880e7			0.128921
$616$		0			0
$617$		−	1.85247e8i		−	0.788671i	−0.918967	−	0.394335i	$-0.870975\pi$
$617$		−	1.85247e8i		−	0.788671i	0.918967	−	0.394335i	$-0.129025\pi$
$618$		0			0
$619$	5.32171e7	−	5.32171e7i	0.224378	−	0.224378i	−0.585961	−	0.810339i	$-0.699283\pi$
$619$	5.32171e7	−	5.32171e7i	0.224378	−	0.224378i	0.810339	+	0.585961i	$0.199283\pi$
$620$		0			0
$621$	−1.51311e7	+	1.51311e7i	−0.0631825	+	0.0631825i
$622$		0			0
$623$			5.65858e6i			0.0234015i
$624$		0			0
$625$	−2.21660e8			−0.907921
$626$		0			0
$627$	4.91739e7	+	4.91739e7i	0.199495	+	0.199495i
$628$		0			0
$629$	−8.91668e7	−	8.91668e7i	−0.358304	−	0.358304i
$630$		0			0
$631$	9.30000e7			0.370165			0.185082	−	0.982723i	$-0.440745\pi$
$631$	9.30000e7			0.370165			0.185082	+	0.982723i	$0.440745\pi$
$632$		0			0
$633$		−	2.60362e7i		−	0.102652i
$634$		0			0
$635$	5.65657e7	−	5.65657e7i	0.220918	−	0.220918i
$636$		0			0
$637$	2.35032e8	−	2.35032e8i	0.909303	−	0.909303i
$638$		0			0
$639$			1.08180e8i			0.414615i
$640$		0			0
$641$	−4.74473e8			−1.80151			−0.900756	−	0.434326i	$-0.856986\pi$
$641$	−4.74473e8			−1.80151			−0.900756	+	0.434326i	$0.856986\pi$
$642$		0			0
$643$	−1.92710e8	−	1.92710e8i	−0.724890	−	0.724890i	0.244707	−	0.969597i	$-0.421308\pi$
$643$	−1.92710e8	−	1.92710e8i	−0.724890	−	0.724890i	−0.969597	+	0.244707i	$0.921308\pi$
$644$		0			0
$645$	−2.44660e6	−	2.44660e6i	−0.00911768	−	0.00911768i
$646$		0			0
$647$	−1.21114e8			−0.447179			−0.223589	−	0.974683i	$-0.571777\pi$
$647$	−1.21114e8			−0.447179			−0.223589	+	0.974683i	$0.571777\pi$
$648$		0			0
$649$		−	2.42705e7i		−	0.0887861i
$650$		0			0
$651$	3.65971e6	−	3.65971e6i	0.0132649	−	0.0132649i
$652$		0			0
$653$	−2.00270e8	+	2.00270e8i	−0.719244	+	0.719244i	−0.968450	−	0.249207i	$-0.919830\pi$
$653$	−2.00270e8	+	2.00270e8i	−0.719244	+	0.719244i	0.249207	+	0.968450i	$0.419830\pi$
$654$		0			0
$655$			5.96766e7i			0.212364i
$656$		0			0
$657$	1.64015e8			0.578347
$658$		0			0
$659$	−5.29275e7	−	5.29275e7i	−0.184938	−	0.184938i	0.608566	−	0.793503i	$-0.291745\pi$
$659$	−5.29275e7	−	5.29275e7i	−0.184938	−	0.184938i	−0.793503	+	0.608566i	$0.791745\pi$
$660$		0			0
$661$	−9.26348e6	−	9.26348e6i	−0.0320752	−	0.0320752i	0.690887	−	0.722963i	$-0.257220\pi$
$661$	−9.26348e6	−	9.26348e6i	−0.0320752	−	0.0320752i	−0.722963	+	0.690887i	$0.757220\pi$
$662$		0			0
$663$	7.93712e7			0.272347
$664$		0			0
$665$		−	546051.i		−	0.00185681i
$666$		0			0
$667$	6.64590e7	−	6.64590e7i	0.223963	−	0.223963i
$668$		0			0
$669$	−1.33191e8	+	1.33191e8i	−0.444834	+	0.444834i
$670$		0			0
$671$		−	8.18152e7i		−	0.270811i
$672$		0			0
$673$	−1.11352e8			−0.365304			−0.182652	−	0.983178i	$-0.558468\pi$
$673$	−1.11352e8			−0.365304			−0.182652	+	0.983178i	$0.558468\pi$
$674$		0			0
$675$	4.05538e7	+	4.05538e7i	0.131862	+	0.131862i
$676$		0			0
$677$	1.72222e8	+	1.72222e8i	0.555039	+	0.555039i	0.927891	−	0.372852i	$-0.121620\pi$
$677$	1.72222e8	+	1.72222e8i	0.555039	+	0.555039i	−0.372852	+	0.927891i	$0.621620\pi$
$678$		0			0
$679$	−6.96139e6			−0.0222375
$680$		0			0
$681$		−	3.10698e8i		−	0.983777i
$682$		0			0
$683$	−2.99546e8	+	2.99546e8i	−0.940160	+	0.940160i	−0.998308	−	0.0581483i	$-0.981480\pi$
$683$	−2.99546e8	+	2.99546e8i	−0.940160	+	0.940160i	0.0581483	+	0.998308i	$0.481480\pi$
$684$		0			0
$685$	262180.	−	262180.i	0.000815695	−	0.000815695i
$686$		0			0
$687$		−	1.00613e8i		−	0.310302i
$688$		0			0
$689$	2.32177e8			0.709841
$690$		0			0
$691$	1.31649e7	+	1.31649e7i	0.0399008	+	0.0399008i	0.726776	−	0.686875i	$-0.241018\pi$
$691$	1.31649e7	+	1.31649e7i	0.0399008	+	0.0399008i	−0.686875	+	0.726776i	$0.741018\pi$
$692$		0			0
$693$	−3.27787e6	−	3.27787e6i	−0.00984900	−	0.00984900i
$694$		0			0
$695$	5.02341e7			0.149639
$696$		0			0
$697$			1.57352e8i			0.464702i
$698$		0			0
$699$	−2.07503e8	+	2.07503e8i	−0.607564	+	0.607564i
$700$		0			0
$701$	−5.88418e7	+	5.88418e7i	−0.170817	+	0.170817i	−0.787338	−	0.616521i	$-0.788542\pi$
$701$	−5.88418e7	+	5.88418e7i	−0.170817	+	0.170817i	0.616521	+	0.787338i	$0.288542\pi$
$702$		0			0
$703$		−	1.68675e8i		−	0.485494i
$704$		0			0
$705$	−2.72545e7			−0.0777804
$706$		0			0
$707$	−2.02053e6	−	2.02053e6i	−0.00571752	−	0.00571752i
$708$		0			0
$709$	−1.03453e8	−	1.03453e8i	−0.290272	−	0.290272i	0.546915	−	0.837188i	$-0.315802\pi$
$709$	−1.03453e8	−	1.03453e8i	−0.290272	−	0.290272i	−0.837188	+	0.546915i	$0.815802\pi$
$710$		0			0
$711$	−4.16830e7			−0.115971
$712$		0			0
$713$		−	1.82111e8i		−	0.502419i
$714$		0			0
$715$	8.15299e7	−	8.15299e7i	0.223048	−	0.223048i
$716$		0			0
$717$	2.06424e8	−	2.06424e8i	0.560018	−	0.560018i
$718$		0			0
$719$		−	2.43205e8i		−	0.654314i	−0.944970	−	0.327157i	$-0.893909\pi$
$719$		−	2.43205e8i		−	0.654314i	0.944970	−	0.327157i	$-0.106091\pi$
$720$		0			0
$721$	1.87673e7			0.0500722
$722$		0			0
$723$	4.10475e7	+	4.10475e7i	0.108610	+	0.108610i
$724$		0			0
$725$	−1.78121e8	−	1.78121e8i	−0.467413	−	0.467413i
$726$		0			0
$727$	4.48848e7			0.116814			0.0584071	−	0.998293i	$-0.481398\pi$
$727$	4.48848e7			0.116814			0.0584071	+	0.998293i	$0.481398\pi$
$728$		0			0
$729$		−	1.43489e7i		−	0.0370370i
$730$		0			0
$731$	1.28377e7	−	1.28377e7i	0.0328652	−	0.0328652i
$732$		0			0
$733$	2.55826e8	−	2.55826e8i	0.649579	−	0.649579i	−0.303312	−	0.952891i	$-0.598092\pi$
$733$	2.55826e8	−	2.55826e8i	0.649579	−	0.649579i	0.952891	+	0.303312i	$0.0980925\pi$
$734$		0			0
$735$		−	4.03354e7i		−	0.101584i
$736$		0			0
$737$	3.68507e8			0.920542
$738$		0			0
$739$	2.81372e8	+	2.81372e8i	0.697184	+	0.697184i	0.963802	−	0.266619i	$-0.0859064\pi$
$739$	2.81372e8	+	2.81372e8i	0.697184	+	0.697184i	−0.266619	+	0.963802i	$0.585906\pi$
$740$		0			0
$741$	7.50723e7	+	7.50723e7i	0.184512	+	0.184512i
$742$		0			0
$743$	−1.35968e8			−0.331489			−0.165744	−	0.986169i	$-0.553003\pi$
$743$	−1.35968e8			−0.331489			−0.165744	+	0.986169i	$0.553003\pi$
$744$		0			0
$745$		−	1.01323e8i		−	0.245040i
$746$		0			0
$747$	1.72437e8	−	1.72437e8i	0.413684	−	0.413684i
$748$		0			0
$749$	1.22541e7	−	1.22541e7i	0.0291633	−	0.0291633i
$750$		0			0
$751$			3.87723e7i			0.0915382i	0.998952	+	0.0457691i	$0.0145738\pi$
$751$			3.87723e7i			0.0915382i	−0.998952	+	0.0457691i	$0.985426\pi$
$752$		0			0
$753$	−2.33747e8			−0.547471
$754$		0			0
$755$	−8.31566e6	−	8.31566e6i	−0.0193222	−	0.0193222i
$756$		0			0
$757$	4.20208e8	+	4.20208e8i	0.968672	+	0.968672i	0.999524	−	0.0308517i	$-0.00982196\pi$
$757$	4.20208e8	+	4.20208e8i	0.968672	+	0.968672i	−0.0308517	+	0.999524i	$0.509822\pi$
$758$		0			0
$759$	−1.63110e8			−0.373040
$760$		0			0
$761$			5.17271e8i			1.17372i	0.809689	+	0.586860i	$0.199636\pi$
$761$			5.17271e8i			1.17372i	−0.809689	+	0.586860i	$0.800364\pi$
$762$		0			0
$763$	−9.22453e6	+	9.22453e6i	−0.0207669	+	0.0207669i
$764$		0			0
$765$	6.81071e6	−	6.81071e6i	0.0152128	−	0.0152128i
$766$		0			0
$767$		−	3.70531e7i		−	0.0821180i
$768$		0			0
$769$	5.94158e8			1.30654			0.653270	−	0.757125i	$-0.273397\pi$
$769$	5.94158e8			1.30654			0.653270	+	0.757125i	$0.273397\pi$
$770$		0			0
$771$	1.53002e7	+	1.53002e7i	0.0333837	+	0.0333837i
$772$		0			0
$773$	−2.95585e8	−	2.95585e8i	−0.639948	−	0.639948i	0.310595	−	0.950542i	$-0.399472\pi$
$773$	−2.95585e8	−	2.95585e8i	−0.639948	−	0.639948i	−0.950542	+	0.310595i	$0.899472\pi$
$774$		0			0
$775$	−4.88085e8			−1.04855
$776$		0			0
$777$			1.12437e7i			0.0239687i
$778$		0			0
$779$	−1.48830e8	+	1.48830e8i	−0.314831	+	0.314831i
$780$		0			0
$781$	−5.83078e8	+	5.83078e8i	−1.22398	+	1.22398i
$782$		0			0
$783$			6.30233e7i			0.131285i
$784$		0			0
$785$	−1.00990e8			−0.208771
$786$		0			0
$787$	1.97521e8	+	1.97521e8i	0.405218	+	0.405218i	0.880067	−	0.474849i	$-0.157497\pi$
$787$	1.97521e8	+	1.97521e8i	0.405218	+	0.405218i	−0.474849	+	0.880067i	$0.657497\pi$
$788$		0			0
$789$	8.54889e7	+	8.54889e7i	0.174052	+	0.174052i
$790$		0			0
$791$	−2.33841e7			−0.0472489
$792$		0			0
$793$		−	1.24905e8i		−	0.250472i
$794$		0			0
$795$	1.99227e7	−	1.99227e7i	0.0396503	−	0.0396503i
$796$		0			0
$797$	4.31312e8	−	4.31312e8i	0.851954	−	0.851954i	−0.138419	−	0.990374i	$-0.544202\pi$
$797$	4.31312e8	−	4.31312e8i	0.851954	−	0.851954i	0.990374	+	0.138419i	$0.0442022\pi$
$798$		0			0
$799$		−	1.43009e8i		−	0.280364i
$800$		0			0
$801$	−1.33510e8			−0.259786
$802$		0			0
$803$	8.84022e8	+	8.84022e8i	1.70733	+	1.70733i
$804$		0			0
$805$	905627.	+	905627.i	0.00173605	+	0.00173605i
$806$		0			0
$807$	−1.54639e8			−0.294237
$808$		0			0
$809$			5.32321e8i			1.00538i	0.864468	+	0.502688i	$0.167655\pi$
$809$			5.32321e8i			1.00538i	−0.864468	+	0.502688i	$0.832345\pi$
$810$		0			0
$811$	−9.63456e7	+	9.63456e7i	−0.180622	+	0.180622i	−0.791627	−	0.611005i	$-0.790765\pi$
$811$	−9.63456e7	+	9.63456e7i	−0.180622	+	0.180622i	0.611005	+	0.791627i	$0.290765\pi$
$812$		0			0
$813$	−3.95026e8	+	3.95026e8i	−0.735113	+	0.735113i
$814$		0			0
$815$			4.30418e7i			0.0795093i
$816$		0			0
$817$	2.42848e7			0.0445316
$818$		0			0
$819$	−5.00423e6	−	5.00423e6i	−0.00910931	−	0.00910931i
$820$		0			0
$821$	2.51953e8	+	2.51953e8i	0.455293	+	0.455293i	0.897107	−	0.441814i	$-0.145665\pi$
$821$	2.51953e8	+	2.51953e8i	0.455293	+	0.455293i	−0.441814	+	0.897107i	$0.645665\pi$
$822$		0			0
$823$	−8.42435e8			−1.51125			−0.755626	−	0.655004i	$-0.772667\pi$
$823$	−8.42435e8			−1.51125			−0.755626	+	0.655004i	$0.772667\pi$
$824$		0			0
$825$			4.37160e8i			0.778537i
$826$		0			0
$827$	−1.58922e8	+	1.58922e8i	−0.280975	+	0.280975i	−0.833498	−	0.552523i	$-0.813665\pi$
$827$	−1.58922e8	+	1.58922e8i	−0.280975	+	0.280975i	0.552523	+	0.833498i	$0.313665\pi$
$828$		0			0
$829$	1.73710e8	−	1.73710e8i	0.304902	−	0.304902i	−0.538026	−	0.842928i	$-0.680830\pi$
$829$	1.73710e8	−	1.73710e8i	0.304902	−	0.304902i	0.842928	+	0.538026i	$0.180830\pi$
$830$		0			0
$831$			3.15324e8i			0.549483i
$832$		0			0
$833$	2.11646e8			0.366164
$834$		0			0
$835$	1.18008e7	+	1.18008e7i	0.0202698	+	0.0202698i
$836$		0			0
$837$	8.63480e7	+	8.63480e7i	0.147257	+	0.147257i
$838$		0			0
$839$	1.40758e8			0.238335			0.119168	−	0.992874i	$-0.461977\pi$
$839$	1.40758e8			0.238335			0.119168	+	0.992874i	$0.461977\pi$
$840$		0			0
$841$			3.18012e8i			0.534633i
$842$		0			0
$843$	1.68887e8	−	1.68887e8i	0.281911	−	0.281911i
$844$		0			0
$845$	4.93362e7	−	4.93362e7i	0.0817702	−	0.0817702i
$846$		0			0
$847$		−	1.70891e7i		−	0.0281235i
$848$		0			0
$849$	4.96136e8			0.810733
$850$		0			0
$851$	2.79747e8	+	2.79747e8i	0.453918	+	0.453918i
$852$		0			0
$853$	2.21132e8	+	2.21132e8i	0.356291	+	0.356291i	0.862444	−	0.506153i	$-0.168933\pi$
$853$	2.21132e8	+	2.21132e8i	0.356291	+	0.356291i	−0.506153	+	0.862444i	$0.668933\pi$
$854$		0			0
$855$	1.28837e7			0.0206130
$856$		0			0
$857$			1.00862e9i			1.60245i	0.598365	+	0.801224i	$0.295817\pi$
$857$			1.00862e9i			1.60245i	−0.598365	+	0.801224i	$0.704183\pi$
$858$		0			0
$859$	−6.56398e8	+	6.56398e8i	−1.03559	+	1.03559i	−0.0362465	+	0.999343i	$0.511540\pi$
$859$	−6.56398e8	+	6.56398e8i	−1.03559	+	1.03559i	−0.999343	+	0.0362465i	$0.988460\pi$
$860$		0			0
$861$	9.92080e6	−	9.92080e6i	0.0155431	−	0.0155431i
$862$		0			0
$863$			1.27673e8i			0.198641i	0.995056	+	0.0993204i	$0.0316669\pi$
$863$			1.27673e8i			0.198641i	−0.995056	+	0.0993204i	$0.968333\pi$
$864$		0			0
$865$	1.16526e8			0.180042
$866$		0			0
$867$	−2.30324e8	−	2.30324e8i	−0.353413	−	0.353413i
$868$		0			0
$869$	−2.24666e8	−	2.24666e8i	−0.342356	−	0.342356i
$870$		0			0
$871$	5.62589e8			0.851407
$872$		0			0
$873$		−	1.64249e8i		−	0.246865i
$874$		0			0
$875$	4.93213e6	−	4.93213e6i	0.00736225	−	0.00736225i
$876$		0			0
$877$	−2.52901e8	+	2.52901e8i	−0.374931	+	0.374931i	−0.869269	−	0.494339i	$-0.835410\pi$
$877$	−2.52901e8	+	2.52901e8i	−0.374931	+	0.374931i	0.494339	+	0.869269i	$0.335410\pi$
$878$		0			0
$879$			5.87248e8i			0.864679i
$880$		0			0
$881$	1.24837e9			1.82564			0.912819	−	0.408365i	$-0.133901\pi$
$881$	1.24837e9			1.82564			0.912819	+	0.408365i	$0.133901\pi$
$882$		0			0
$883$	−9.14023e8	−	9.14023e8i	−1.32762	−	1.32762i	−0.907438	−	0.420185i	$-0.861965\pi$
$883$	−9.14023e8	−	9.14023e8i	−1.32762	−	1.32762i	−0.420185	−	0.907438i	$-0.638035\pi$
$884$		0			0
$885$	−3.17946e6	−	3.17946e6i	−0.00458695	−	0.00458695i
$886$		0			0
$887$	−2.29976e8			−0.329542			−0.164771	−	0.986332i	$-0.552689\pi$
$887$	−2.29976e8			−0.329542			−0.164771	+	0.986332i	$0.552689\pi$
$888$		0			0
$889$		−	3.74267e7i		−	0.0532692i
$890$		0			0
$891$	7.73388e7	−	7.73388e7i	0.109336	−	0.109336i
$892$		0			0
$893$	1.35263e8	−	1.35263e8i	0.189944	−	0.189944i
$894$		0			0
$895$			1.44662e8i			0.201783i
$896$		0			0
$897$	−2.49015e8			−0.345023
$898$		0			0
$899$	−3.79258e8	−	3.79258e8i	−0.521982	−	0.521982i
$900$		0			0
$901$	1.04538e8	+	1.04538e8i	0.142922	+	0.142922i
$902$		0			0
$903$	−1.61880e6			−0.00219851
$904$		0			0
$905$		−	1.98445e8i		−	0.267728i
$906$		0			0
$907$	4.51748e8	−	4.51748e8i	0.605444	−	0.605444i	−0.336308	−	0.941752i	$-0.609178\pi$
$907$	4.51748e8	−	4.51748e8i	0.605444	−	0.605444i	0.941752	+	0.336308i	$0.109178\pi$
$908$		0			0
$909$	4.76729e7	−	4.76729e7i	0.0634717	−	0.0634717i
$910$		0			0
$911$			5.67087e8i			0.750058i	0.927013	+	0.375029i	$0.122367\pi$
$911$			5.67087e8i			0.750058i	−0.927013	+	0.375029i	$0.877633\pi$
$912$		0			0
$913$	1.85883e9			2.44246
$914$		0			0
$915$	−1.07179e7	−	1.07179e7i	−0.0139909	−	0.0139909i
$916$		0			0
$917$	1.97425e7	+	1.97425e7i	0.0256032	+	0.0256032i
$918$		0			0
$919$	8.38744e8			1.08065			0.540323	−	0.841458i	$-0.318302\pi$
$919$	8.38744e8			1.08065			0.540323	+	0.841458i	$0.318302\pi$
$920$		0			0
$921$		−	4.69010e7i		−	0.0600348i
$922$		0			0
$923$	−8.90168e8	+	8.90168e8i	−1.13205	+	1.13205i
$924$		0			0
$925$	7.49768e8	−	7.49768e8i	0.947330	−	0.947330i
$926$		0			0
$927$			4.42801e8i			0.555865i
$928$		0			0
$929$	−1.36976e9			−1.70842			−0.854212	−	0.519924i	$-0.825960\pi$
$929$	−1.36976e9			−1.70842			−0.854212	+	0.519924i	$0.825960\pi$
$930$		0			0
$931$	2.00183e8	+	2.00183e8i	0.248072	+	0.248072i
$932$		0			0
$933$	−2.31439e8	−	2.31439e8i	−0.284964	−	0.284964i
$934$		0			0
$935$	7.34177e7			0.0898186
$936$		0			0
$937$		−	6.89998e8i		−	0.838743i	−0.907815	−	0.419371i	$-0.862251\pi$
$937$		−	6.89998e8i		−	0.838743i	0.907815	−	0.419371i	$-0.137749\pi$
$938$		0			0
$939$	−4.30962e8	+	4.30962e8i	−0.520525	+	0.520525i
$940$		0			0
$941$	1.06002e8	−	1.06002e8i	0.127218	−	0.127218i	−0.640631	−	0.767849i	$-0.721327\pi$
$941$	1.06002e8	−	1.06002e8i	0.127218	−	0.127218i	0.767849	+	0.640631i	$0.221327\pi$
$942$		0			0
$943$		−	4.93668e8i		−	0.588708i
$944$		0			0
$945$	−858809.			−0.00101766
$946$		0			0
$947$	1.85082e8	+	1.85082e8i	0.217928	+	0.217928i	0.807625	−	0.589696i	$-0.200753\pi$
$947$	1.85082e8	+	1.85082e8i	0.217928	+	0.217928i	−0.589696	+	0.807625i	$0.700753\pi$
$948$		0			0
$949$	1.34961e9	+	1.34961e9i	1.57910	+	1.57910i
$950$		0			0
$951$	8.20955e8			0.954504
$952$		0			0
$953$			7.03272e8i			0.812540i	0.913753	+	0.406270i	$0.133171\pi$
$953$			7.03272e8i			0.812540i	−0.913753	+	0.406270i	$0.866829\pi$
$954$		0			0
$955$	1.37376e8	−	1.37376e8i	0.157725	−	0.157725i
$956$		0			0
$957$	−3.39688e8	+	3.39688e8i	−0.387565	+	0.387565i
$958$		0			0
$959$		−	173471.i		−	0.000196685i
$960$		0			0
$961$	−1.51736e8			−0.170970
$962$		0			0
$963$	2.89126e8	+	2.89126e8i	0.323749	+	0.323749i
$964$		0			0
$965$	−1.79756e8	−	1.79756e8i	−0.200033	−	0.200033i
$966$		0			0
$967$	−5.58827e8			−0.618013			−0.309007	−	0.951060i	$-0.599997\pi$
$967$	−5.58827e8			−0.618013			−0.309007	+	0.951060i	$0.599997\pi$
$968$		0			0
$969$			6.76026e7i			0.0743006i
$970$		0			0
$971$	−2.38035e8	+	2.38035e8i	−0.260006	+	0.260006i	−0.825056	−	0.565051i	$-0.808857\pi$
$971$	−2.38035e8	+	2.38035e8i	−0.260006	+	0.260006i	0.565051	+	0.825056i	$0.308857\pi$
$972$		0			0
$973$	1.66187e7	−	1.66187e7i	0.0180409	−	0.0180409i
$974$		0			0
$975$			6.67400e8i			0.720066i
$976$		0			0
$977$	−9.76599e8			−1.04721			−0.523603	−	0.851962i	$-0.675413\pi$
$977$	−9.76599e8			−1.04721			−0.523603	+	0.851962i	$0.675413\pi$
$978$		0			0
$979$	−7.19601e8	−	7.19601e8i	−0.766909	−	0.766909i
$980$		0			0
$981$	−2.17646e8	−	2.17646e8i	−0.230538	−	0.230538i
$982$		0			0
$983$	−1.57271e9			−1.65573			−0.827863	−	0.560930i	$-0.810444\pi$
$983$	−1.57271e9			−1.65573			−0.827863	+	0.560930i	$0.810444\pi$
$984$		0			0
$985$		−	3.15479e8i		−	0.330112i
$986$		0			0
$987$	−9.01647e6	+	9.01647e6i	−0.00937745	+	0.00937745i
$988$		0			0
$989$	−4.02764e7	+	4.02764e7i	−0.0416353	+	0.0416353i
$990$		0			0
$991$			9.47817e8i			0.973875i	0.873437	+	0.486938i	$0.161886\pi$
$991$			9.47817e8i			0.973875i	−0.873437	+	0.486938i	$0.838114\pi$
$992$		0			0
$993$	−1.07807e9			−1.10103
$994$		0			0
$995$	−1.92515e7	−	1.92515e7i	−0.0195432	−	0.0195432i
$996$		0			0
$997$	5.42772e8	+	5.42772e8i	0.547686	+	0.547686i	0.925771	−	0.378085i	$-0.123417\pi$
$997$	5.42772e8	+	5.42772e8i	0.547686	+	0.547686i	−0.378085	+	0.925771i	$0.623417\pi$
$998$		0			0
$999$	−2.65285e8			−0.266083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.l.b.31.21		48
4.3	odd	2		384.7.l.a.31.4		48
8.3	odd	2		192.7.l.a.79.21		48
8.5	even	2		48.7.l.a.43.15	yes	48
16.3	odd	4	inner	384.7.l.b.223.21		48
16.5	even	4		192.7.l.a.175.21		48
16.11	odd	4		48.7.l.a.19.15	✓	48
16.13	even	4		384.7.l.a.223.4		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.l.a.19.15	✓	48	16.11	odd	4
48.7.l.a.43.15	yes	48	8.5	even	2
192.7.l.a.79.21		48	8.3	odd	2
192.7.l.a.175.21		48	16.5	even	4
384.7.l.a.31.4		48	4.3	odd	2
384.7.l.a.223.4		48	16.13	even	4
384.7.l.b.31.21		48	1.1	even	1	trivial
384.7.l.b.223.21		48	16.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	384.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(11.0227 + 11.0227i) q^{3} +(15.5658 + 15.5658i) q^{5} +10.2991 q^{7} +243.000i q^{9} +O(q^{10})\)	\(q+(11.0227 + 11.0227i) q^{3} +(15.5658 + 15.5658i) q^{5} +10.2991 q^{7} +243.000i q^{9} +(-1309.74 + 1309.74i) q^{11} +(-1999.54 + 1999.54i) q^{13} +343.155i q^{15} -1800.59 q^{17} +(-1703.06 - 1703.06i) q^{19} +(113.524 + 113.524i) q^{21} +5649.07 q^{23} -15140.4i q^{25} +(-2678.52 + 2678.52i) q^{27} +(11764.6 - 11764.6i) q^{29} -32237.2i q^{31} -28873.7 q^{33} +(160.314 + 160.314i) q^{35} +(49520.9 + 49520.9i) q^{37} -44080.7 q^{39} -87389.3i q^{41} +(-7129.74 + 7129.74i) q^{43} +(-3782.49 + 3782.49i) q^{45} +79423.3i q^{47} -117543. q^{49} +(-19847.3 - 19847.3i) q^{51} +(-58057.5 - 58057.5i) q^{53} -40774.3 q^{55} -37544.8i q^{57} +(-9265.40 + 9265.40i) q^{59} +(-31233.4 + 31233.4i) q^{61} +2502.69i q^{63} -62249.0 q^{65} +(-140680. - 140680. i) q^{67} +(62268.0 + 62268.0i) q^{69} +445186. q^{71} -674960. i q^{73} +(166888. - 166888. i) q^{75} +(-13489.2 + 13489.2i) q^{77} +171535. i q^{79} -59049.0 q^{81} +(-709618. - 709618. i) q^{83} +(-28027.6 - 28027.6i) q^{85} +259355. q^{87} +549423. i q^{89} +(-20593.5 + 20593.5i) q^{91} +(355342. - 355342. i) q^{93} -53019.2i q^{95} -675921. q^{97} +(-318267. - 318267. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 2720 q^{11} + 3936 q^{19} + 26240 q^{23} - 66400 q^{29} - 162336 q^{35} + 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} - 443680 q^{53} + 232704 q^{55} + 886144 q^{59} + 326496 q^{61} - 372832 q^{65} + 962112 q^{67} - 541728 q^{69} + 534016 q^{71} + 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} + 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) \) 48 * q + 2720 * q^11 + 3936 * q^19 + 26240 * q^23 - 66400 * q^29 - 162336 * q^35 + 7200 * q^37 - 340704 * q^43 + 806736 * q^49 - 80352 * q^51 - 443680 * q^53 + 232704 * q^55 + 886144 * q^59 + 326496 * q^61 - 372832 * q^65 + 962112 * q^67 - 541728 * q^69 + 534016 * q^71 + 1073088 * q^75 + 932960 * q^77 - 2834352 * q^81 + 2497760 * q^83 + 372000 * q^85 - 775008 * q^91 + 660960 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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