LMFDB - Embedded newform 384.7.l.a.31.12

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			31.12
Character	$\chi$	$=$	384.31
Dual form			384.7.l.a.223.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.0227 - 11.0227i) q^{3} +(160.484 + 160.484i) q^{5} +53.5181 q^{7} +243.000i q^{9} +O(q^{10})$	\(q+(-11.0227 - 11.0227i) q^{3} +(160.484 + 160.484i) q^{5} +53.5181 q^{7} +243.000i q^{9} +(-122.360 + 122.360i) q^{11} +(-413.892 + 413.892i) q^{13} -3537.94i q^{15} -3293.73 q^{17} +(7932.29 + 7932.29i) q^{19} +(-589.914 - 589.914i) q^{21} +13647.7 q^{23} +35885.2i q^{25} +(2678.52 - 2678.52i) q^{27} +(-5980.28 + 5980.28i) q^{29} +23190.3i q^{31} +2697.48 q^{33} +(8588.80 + 8588.80i) q^{35} +(-62907.0 - 62907.0i) q^{37} +9124.43 q^{39} +90773.4i q^{41} +(101496. - 101496. i) q^{43} +(-38997.6 + 38997.6i) q^{45} -32613.3i q^{47} -114785. q^{49} +(36305.8 + 36305.8i) q^{51} +(28691.0 + 28691.0i) q^{53} -39273.7 q^{55} -174871. i q^{57} +(-135412. + 135412. i) q^{59} +(247804. - 247804. i) q^{61} +13004.9i q^{63} -132846. q^{65} +(-52377.3 - 52377.3i) q^{67} +(-150434. - 150434. i) q^{69} -381785. q^{71} +124029. i q^{73} +(395552. - 395552. i) q^{75} +(-6548.49 + 6548.49i) q^{77} +223099. i q^{79} -59049.0 q^{81} +(-568288. - 568288. i) q^{83} +(-528591. - 528591. i) q^{85} +131838. q^{87} +863372. i q^{89} +(-22150.7 + 22150.7i) q^{91} +(255620. - 255620. i) q^{93} +2.54601e6i q^{95} -1.32736e6 q^{97} +(-29733.5 - 29733.5i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	160.484	+	160.484i	1.28387	+	1.28387i	0.938446	+	0.345426i	$0.112266\pi$
$5$	160.484	+	160.484i	1.28387	+	1.28387i	0.345426	+	0.938446i	$0.387734\pi$
$6$		0			0
$7$	53.5181			0.156029			0.0780147	−	0.996952i	$-0.475142\pi$
$7$	53.5181			0.156029			0.0780147	+	0.996952i	$0.475142\pi$
$8$		0			0
$9$			243.000i			0.333333i
$10$		0			0
$11$	−122.360	+	122.360i	−0.0919310	+	0.0919310i	−0.751577	−	0.659646i	$-0.770706\pi$
$11$	−122.360	+	122.360i	−0.0919310	+	0.0919310i	0.659646	+	0.751577i	$0.270706\pi$
$12$		0			0
$13$	−413.892	+	413.892i	−0.188390	+	0.188390i	−0.795000	−	0.606610i	$-0.792529\pi$
$13$	−413.892	+	413.892i	−0.188390	+	0.188390i	0.606610	+	0.795000i	$0.292529\pi$
$14$		0			0
$15$		−	3537.94i		−	1.04828i
$16$		0			0
$17$	−3293.73			−0.670412			−0.335206	−	0.942145i	$-0.608806\pi$
$17$	−3293.73			−0.670412			−0.335206	+	0.942145i	$0.608806\pi$
$18$		0			0
$19$	7932.29	+	7932.29i	1.15648	+	1.15648i	0.985227	+	0.171252i	$0.0547811\pi$
$19$	7932.29	+	7932.29i	1.15648	+	1.15648i	0.171252	+	0.985227i	$0.445219\pi$
$20$		0			0
$21$	−589.914	−	589.914i	−0.0636988	−	0.0636988i
$22$		0			0
$23$	13647.7			1.12169			0.560847	−	0.827919i	$-0.310475\pi$
$23$	13647.7			1.12169			0.560847	+	0.827919i	$0.310475\pi$
$24$		0			0
$25$			35885.2i			2.29665i
$26$		0			0
$27$	2678.52	−	2678.52i	0.136083	−	0.136083i
$28$		0			0
$29$	−5980.28	+	5980.28i	−0.245204	+	0.245204i	−0.818999	−	0.573795i	$-0.805471\pi$
$29$	−5980.28	+	5980.28i	−0.245204	+	0.245204i	0.573795	+	0.818999i	$0.305471\pi$
$30$		0			0
$31$			23190.3i			0.778433i	0.921146	+	0.389216i	$0.127254\pi$
$31$			23190.3i			0.778433i	−0.921146	+	0.389216i	$0.872746\pi$
$32$		0			0
$33$	2697.48			0.0750614
$34$		0			0
$35$	8588.80	+	8588.80i	0.200322	+	0.200322i
$36$		0			0
$37$	−62907.0	−	62907.0i	−1.24192	−	1.24192i	−0.959203	−	0.282717i	$-0.908764\pi$
$37$	−62907.0	−	62907.0i	−1.24192	−	1.24192i	−0.282717	−	0.959203i	$-0.591236\pi$
$38$		0			0
$39$	9124.43			0.153820
$40$		0			0
$41$			90773.4i			1.31707i	0.752552	+	0.658533i	$0.228823\pi$
$41$			90773.4i			1.31707i	−0.752552	+	0.658533i	$0.771177\pi$
$42$		0			0
$43$	101496.	−	101496.i	1.27657	−	1.27657i	0.333989	−	0.942577i	$-0.391605\pi$
$43$	101496.	−	101496.i	1.27657	−	1.27657i	0.942577	−	0.333989i	$-0.108395\pi$
$44$		0			0
$45$	−38997.6	+	38997.6i	−0.427957	+	0.427957i
$46$		0			0
$47$		−	32613.3i		−	0.314124i	−0.987589	−	0.157062i	$-0.949798\pi$
$47$		−	32613.3i		−	0.314124i	0.987589	−	0.157062i	$-0.0502022\pi$
$48$		0			0
$49$	−114785.			−0.975655
$50$		0			0
$51$	36305.8	+	36305.8i	0.273694	+	0.273694i
$52$		0			0
$53$	28691.0	+	28691.0i	0.192716	+	0.192716i	0.796869	−	0.604153i	$-0.206488\pi$
$53$	28691.0	+	28691.0i	0.192716	+	0.192716i	−0.604153	+	0.796869i	$0.706488\pi$
$54$		0			0
$55$	−39273.7			−0.236055
$56$		0			0
$57$		−	174871.i		−	0.944261i
$58$		0			0
$59$	−135412.	+	135412.i	−0.659329	+	0.659329i	−0.955221	−	0.295892i	$-0.904383\pi$
$59$	−135412.	+	135412.i	−0.659329	+	0.659329i	0.295892	+	0.955221i	$0.404383\pi$
$60$		0			0
$61$	247804.	−	247804.i	1.09174	−	1.09174i	0.0963978	−	0.995343i	$-0.469268\pi$
$61$	247804.	−	247804.i	1.09174	−	1.09174i	0.995343	−	0.0963978i	$-0.0307321\pi$
$62$		0			0
$63$			13004.9i			0.0520098i
$64$		0			0
$65$	−132846.			−0.483737
$66$		0			0
$67$	−52377.3	−	52377.3i	−0.174148	−	0.174148i	0.614651	−	0.788799i	$-0.289297\pi$
$67$	−52377.3	−	52377.3i	−0.174148	−	0.174148i	−0.788799	+	0.614651i	$0.789297\pi$
$68$		0			0
$69$	−150434.	−	150434.i	−0.457930	−	0.457930i
$70$		0			0
$71$	−381785.			−1.06670			−0.533352	−	0.845893i	$-0.679068\pi$
$71$	−381785.			−1.06670			−0.533352	+	0.845893i	$0.679068\pi$
$72$		0			0
$73$			124029.i			0.318826i	0.987212	+	0.159413i	$0.0509602\pi$
$73$			124029.i			0.318826i	−0.987212	+	0.159413i	$0.949040\pi$
$74$		0			0
$75$	395552.	−	395552.i	0.937605	−	0.937605i
$76$		0			0
$77$	−6548.49	+	6548.49i	−0.0143440	+	0.0143440i
$78$		0			0
$79$			223099.i			0.452497i	0.974070	+	0.226249i	$0.0726462\pi$
$79$			223099.i			0.452497i	−0.974070	+	0.226249i	$0.927354\pi$
$80$		0			0
$81$	−59049.0			−0.111111
$82$		0			0
$83$	−568288.	−	568288.i	−0.993881	−	0.993881i	0.00610030	−	0.999981i	$-0.498058\pi$
$83$	−568288.	−	568288.i	−0.993881	−	0.993881i	−0.999981	+	0.00610030i	$0.998058\pi$
$84$		0			0
$85$	−528591.	−	528591.i	−0.860723	−	0.860723i
$86$		0			0
$87$	131838.			0.200208
$88$		0			0
$89$			863372.i			1.22470i	0.790589	+	0.612348i	$0.209775\pi$
$89$			863372.i			1.22470i	−0.790589	+	0.612348i	$0.790225\pi$
$90$		0			0
$91$	−22150.7	+	22150.7i	−0.0293944	+	0.0293944i
$92$		0			0
$93$	255620.	−	255620.i	0.317794	−	0.317794i
$94$		0			0
$95$			2.54601e6i			2.96954i
$96$		0			0
$97$	−1.32736e6			−1.45437			−0.727185	−	0.686441i	$-0.759172\pi$
$97$	−1.32736e6			−1.45437			−0.727185	+	0.686441i	$0.759172\pi$
$98$		0			0
$99$	−29733.5	−	29733.5i	−0.0306437	−	0.0306437i
$100$		0			0
$101$	909475.	+	909475.i	0.882727	+	0.882727i	0.993811	−	0.111084i	$-0.0354323\pi$
$101$	909475.	+	909475.i	0.882727	+	0.882727i	−0.111084	+	0.993811i	$0.535432\pi$
$102$		0			0
$103$	1.29950e6			1.18922			0.594611	−	0.804013i	$-0.297306\pi$
$103$	1.29950e6			1.18922			0.594611	+	0.804013i	$0.297306\pi$
$104$		0			0
$105$		−	189344.i		−	0.163562i
$106$		0			0
$107$	−253167.	+	253167.i	−0.206659	+	0.206659i	−0.802846	−	0.596187i	$-0.796682\pi$
$107$	−253167.	+	253167.i	−0.206659	+	0.206659i	0.596187	+	0.802846i	$0.296682\pi$
$108$		0			0
$109$	−643790.	+	643790.i	−0.497124	+	0.497124i	−0.910542	−	0.413418i	$-0.864335\pi$
$109$	−643790.	+	643790.i	−0.497124	+	0.497124i	0.413418	+	0.910542i	$0.364335\pi$
$110$		0			0
$111$			1.38681e6i			1.01402i
$112$		0			0
$113$	726527.			0.503520			0.251760	−	0.967790i	$-0.418991\pi$
$113$	726527.			0.503520			0.251760	+	0.967790i	$0.418991\pi$
$114$		0			0
$115$	2.19023e6	+	2.19023e6i	1.44011	+	1.44011i
$116$		0			0
$117$	−100576.	−	100576.i	−0.0627966	−	0.0627966i
$118$		0			0
$119$	−176274.			−0.104604
$120$		0			0
$121$			1.74162e6i			0.983097i
$122$		0			0
$123$	1.00057e6	−	1.00057e6i	0.537690	−	0.537690i
$124$		0			0
$125$	−3.25144e6	+	3.25144e6i	−1.66474	+	1.66474i
$126$		0			0
$127$		−	676922.i		−	0.330466i	−0.986255	−	0.165233i	$-0.947162\pi$
$127$		−	676922.i		−	0.330466i	0.986255	−	0.165233i	$-0.0528377\pi$
$128$		0			0
$129$	−2.23752e6			−1.04231
$130$		0			0
$131$	2.77209e6	+	2.77209e6i	1.23309	+	1.23309i	0.962772	+	0.270315i	$0.0871279\pi$
$131$	2.77209e6	+	2.77209e6i	1.23309	+	1.23309i	0.270315	+	0.962772i	$0.412872\pi$
$132$		0			0
$133$	424521.	+	424521.i	0.180445	+	0.180445i
$134$		0			0
$135$	859718.			0.349426
$136$		0			0
$137$			1.68453e6i			0.655114i	0.944831	+	0.327557i	$0.106225\pi$
$137$			1.68453e6i			0.655114i	−0.944831	+	0.327557i	$0.893775\pi$
$138$		0			0
$139$	−2.47206e6	+	2.47206e6i	−0.920480	+	0.920480i	−0.997063	−	0.0765831i	$-0.975599\pi$
$139$	−2.47206e6	+	2.47206e6i	−0.920480	+	0.920480i	0.0765831	+	0.997063i	$0.475599\pi$
$140$		0			0
$141$	−359486.	+	359486.i	−0.128240	+	0.128240i
$142$		0			0
$143$		−	101288.i		−	0.0346377i
$144$		0			0
$145$	−1.91948e6			−0.629621
$146$		0			0
$147$	1.26524e6	+	1.26524e6i	0.398309	+	0.398309i
$148$		0			0
$149$	−3.06263e6	−	3.06263e6i	−0.925840	−	0.925840i	0.0715935	−	0.997434i	$-0.477192\pi$
$149$	−3.06263e6	−	3.06263e6i	−0.925840	−	0.925840i	−0.997434	+	0.0715935i	$0.977192\pi$
$150$		0			0
$151$	−3698.15			−0.00107412			−0.000537060	−	1.00000i	$-0.500171\pi$
$151$	−3698.15			−0.00107412			−0.000537060		1.00000i	$0.500171\pi$
$152$		0			0
$153$		−	800377.i		−	0.223471i
$154$		0			0
$155$	−3.72167e6	+	3.72167e6i	−0.999408	+	0.999408i
$156$		0			0
$157$	2.39066e6	−	2.39066e6i	0.617758	−	0.617758i	−0.327198	−	0.944956i	$-0.606104\pi$
$157$	2.39066e6	−	2.39066e6i	0.617758	−	0.617758i	0.944956	+	0.327198i	$0.106104\pi$
$158$		0			0
$159$		−	632505.i		−	0.157352i
$160$		0			0
$161$	730397.			0.175017
$162$		0			0
$163$	547276.	+	547276.i	0.126370	+	0.126370i	0.767463	−	0.641093i	$-0.221519\pi$
$163$	547276.	+	547276.i	0.126370	+	0.126370i	−0.641093	+	0.767463i	$0.721519\pi$
$164$		0			0
$165$	432902.	+	432902.i	0.0963692	+	0.0963692i
$166$		0			0
$167$	4.43980e6			0.953265			0.476632	−	0.879103i	$-0.341857\pi$
$167$	4.43980e6			0.953265			0.476632	+	0.879103i	$0.341857\pi$
$168$		0			0
$169$			4.48420e6i			0.929019i
$170$		0			0
$171$	−1.92755e6	+	1.92755e6i	−0.385493	+	0.385493i
$172$		0			0
$173$	−5.45891e6	+	5.45891e6i	−1.05431	+	1.05431i	−0.0558712	+	0.998438i	$0.517794\pi$
$173$	−5.45891e6	+	5.45891e6i	−1.05431	+	1.05431i	−0.998438	+	0.0558712i	$0.982206\pi$
$174$		0			0
$175$			1.92051e6i			0.358346i
$176$		0			0
$177$	2.98522e6			0.538340
$178$		0			0
$179$	−278271.	−	278271.i	−0.0485187	−	0.0485187i	0.682431	−	0.730950i	$-0.260923\pi$
$179$	−278271.	−	278271.i	−0.0485187	−	0.0485187i	−0.730950	+	0.682431i	$0.760923\pi$
$180$		0			0
$181$	−2.95365e6	−	2.95365e6i	−0.498108	−	0.498108i	0.412740	−	0.910849i	$-0.364572\pi$
$181$	−2.95365e6	−	2.95365e6i	−0.498108	−	0.498108i	−0.910849	+	0.412740i	$0.864572\pi$
$182$		0			0
$183$	−5.46295e6			−0.891403
$184$		0			0
$185$		−	2.01911e7i		−	3.18893i
$186$		0			0
$187$	403022.	−	403022.i	0.0616316	−	0.0616316i
$188$		0			0
$189$	143349.	−	143349.i	0.0212329	−	0.0212329i
$190$		0			0
$191$		−	5.90746e6i		−	0.847814i	−0.905706	−	0.423907i	$-0.860658\pi$
$191$		−	5.90746e6i		−	0.847814i	0.905706	−	0.423907i	$-0.139342\pi$
$192$		0			0
$193$	−1.15329e7			−1.60423			−0.802115	−	0.597169i	$-0.796292\pi$
$193$	−1.15329e7			−1.60423			−0.802115	+	0.597169i	$0.796292\pi$
$194$		0			0
$195$	1.46432e6	+	1.46432e6i	0.197485	+	0.197485i
$196$		0			0
$197$	−1.68541e6	−	1.68541e6i	−0.220448	−	0.220448i	0.588239	−	0.808687i	$-0.299821\pi$
$197$	−1.68541e6	−	1.68541e6i	−0.220448	−	0.220448i	−0.808687	+	0.588239i	$0.799821\pi$
$198$		0			0
$199$	−1.03355e7			−1.31151			−0.655754	−	0.754975i	$-0.727649\pi$
$199$	−1.03355e7			−1.31151			−0.655754	+	0.754975i	$0.727649\pi$
$200$		0			0
$201$			1.15468e6i			0.142191i
$202$		0			0
$203$	−320053.	+	320053.i	−0.0382591	+	0.0382591i
$204$		0			0
$205$	−1.45677e7	+	1.45677e7i	−1.69094	+	1.69094i
$206$		0			0
$207$			3.31638e6i			0.373898i
$208$		0			0
$209$	−1.94119e6			−0.212633
$210$		0			0
$211$	−1.61899e6	−	1.61899e6i	−0.172344	−	0.172344i	0.615664	−	0.788009i	$-0.288888\pi$
$211$	−1.61899e6	−	1.61899e6i	−0.172344	−	0.172344i	−0.788009	+	0.615664i	$0.788888\pi$
$212$		0			0
$213$	4.20831e6	+	4.20831e6i	0.435480	+	0.435480i
$214$		0			0
$215$	3.25770e7			3.27790
$216$		0			0
$217$			1.24110e6i			0.121459i
$218$		0			0
$219$	1.36713e6	−	1.36713e6i	0.130160	−	0.130160i
$220$		0			0
$221$	1.36325e6	−	1.36325e6i	0.126299	−	0.126299i
$222$		0			0
$223$			1.42305e7i			1.28323i	0.767026	+	0.641615i	$0.221735\pi$
$223$			1.42305e7i			1.28323i	−0.767026	+	0.641615i	$0.778265\pi$
$224$		0			0
$225$	−8.72011e6			−0.765552
$226$		0			0
$227$	7.10217e6	+	7.10217e6i	0.607175	+	0.607175i	0.942207	−	0.335032i	$-0.108747\pi$
$227$	7.10217e6	+	7.10217e6i	0.607175	+	0.607175i	−0.335032	+	0.942207i	$0.608747\pi$
$228$		0			0
$229$	1.47594e7	+	1.47594e7i	1.22903	+	1.22903i	0.964332	+	0.264694i	$0.0852710\pi$
$229$	1.47594e7	+	1.47594e7i	1.22903	+	1.22903i	0.264694	+	0.964332i	$0.414729\pi$
$230$		0			0
$231$	144364.			0.0117118
$232$		0			0
$233$			1.15928e7i			0.916475i	0.888830	+	0.458237i	$0.151519\pi$
$233$			1.15928e7i			0.916475i	−0.888830	+	0.458237i	$0.848481\pi$
$234$		0			0
$235$	5.23391e6	−	5.23391e6i	0.403295	−	0.403295i
$236$		0			0
$237$	2.45915e6	−	2.45915e6i	0.184731	−	0.184731i
$238$		0			0
$239$		−	3.18157e6i		−	0.233049i	−0.993188	−	0.116525i	$-0.962825\pi$
$239$		−	3.18157e6i		−	0.233049i	0.993188	−	0.116525i	$-0.0371754\pi$
$240$		0			0
$241$	1.15359e7			0.824138			0.412069	−	0.911153i	$-0.364806\pi$
$241$	1.15359e7			0.824138			0.412069	+	0.911153i	$0.364806\pi$
$242$		0			0
$243$	650880.	+	650880.i	0.0453609	+	0.0453609i
$244$		0			0
$245$	−1.84211e7	−	1.84211e7i	−1.25262	−	1.25262i
$246$		0			0
$247$	−6.56623e6			−0.435738
$248$		0			0
$249$			1.25281e7i			0.811501i
$250$		0			0
$251$	−3.04103e6	+	3.04103e6i	−0.192309	+	0.192309i	−0.796693	−	0.604384i	$-0.793419\pi$
$251$	−3.04103e6	+	3.04103e6i	−0.192309	+	0.192309i	0.604384	+	0.796693i	$0.293419\pi$
$252$		0			0
$253$	−1.66993e6	+	1.66993e6i	−0.103119	+	0.103119i
$254$		0			0
$255$			1.16530e7i			0.702777i
$256$		0			0
$257$	−8.51882e6			−0.501857			−0.250929	−	0.968006i	$-0.580736\pi$
$257$	−8.51882e6			−0.501857			−0.250929	+	0.968006i	$0.580736\pi$
$258$		0			0
$259$	−3.36666e6	−	3.36666e6i	−0.193776	−	0.193776i
$260$		0			0
$261$	−1.45321e6	−	1.45321e6i	−0.0817347	−	0.0817347i
$262$		0			0
$263$	9.30372e6			0.511434			0.255717	−	0.966752i	$-0.417689\pi$
$263$	9.30372e6			0.511434			0.255717	+	0.966752i	$0.417689\pi$
$264$		0			0
$265$			9.20889e6i			0.494846i
$266$		0			0
$267$	9.51670e6	−	9.51670e6i	0.499980	−	0.499980i
$268$		0			0
$269$	2.28835e6	−	2.28835e6i	0.117562	−	0.117562i	−0.645878	−	0.763440i	$-0.723509\pi$
$269$	2.28835e6	−	2.28835e6i	0.117562	−	0.117562i	0.763440	+	0.645878i	$0.223509\pi$
$270$		0			0
$271$		−	9.66335e6i		−	0.485534i	−0.970085	−	0.242767i	$-0.921945\pi$
$271$		−	9.66335e6i		−	0.485534i	0.970085	−	0.242767i	$-0.0780551\pi$
$272$		0			0
$273$	488322.			0.0240004
$274$		0			0
$275$	−4.39092e6	−	4.39092e6i	−0.211134	−	0.211134i
$276$		0			0
$277$	6.67690e6	+	6.67690e6i	0.314149	+	0.314149i	0.846514	−	0.532366i	$-0.178697\pi$
$277$	6.67690e6	+	6.67690e6i	0.314149	+	0.314149i	−0.532366	+	0.846514i	$0.678697\pi$
$278$		0			0
$279$	−5.63524e6			−0.259478
$280$		0			0
$281$		−	4.14352e6i		−	0.186745i	−0.995631	−	0.0933727i	$-0.970235\pi$
$281$		−	4.14352e6i		−	0.186745i	0.995631	−	0.0933727i	$-0.0297648\pi$
$282$		0			0
$283$	−3.34427e6	+	3.34427e6i	−0.147551	+	0.147551i	−0.777023	−	0.629472i	$-0.783271\pi$
$283$	−3.34427e6	+	3.34427e6i	−0.147551	+	0.147551i	0.629472	+	0.777023i	$0.283271\pi$
$284$		0			0
$285$	2.80639e7	−	2.80639e7i	1.21231	−	1.21231i
$286$		0			0
$287$			4.85802e6i			0.205501i
$288$		0			0
$289$	−1.32889e7			−0.550548
$290$		0			0
$291$	1.46311e7	+	1.46311e7i	0.593744	+	0.593744i
$292$		0			0
$293$	1.53622e7	+	1.53622e7i	0.610730	+	0.610730i	0.943136	−	0.332406i	$-0.107860\pi$
$293$	1.53622e7	+	1.53622e7i	0.610730	+	0.610730i	−0.332406	+	0.943136i	$0.607860\pi$
$294$		0			0
$295$	−4.34630e7			−1.69299
$296$		0			0
$297$			655488.i			0.0250205i
$298$		0			0
$299$	−5.64866e6	+	5.64866e6i	−0.211316	+	0.211316i
$300$		0			0
$301$	5.43187e6	−	5.43187e6i	0.199182	−	0.199182i
$302$		0			0
$303$		−	2.00497e7i		−	0.720744i
$304$		0			0
$305$	7.95373e7			2.80331
$306$		0			0
$307$	−2.33452e7	−	2.33452e7i	−0.806832	−	0.806832i	0.177321	−	0.984153i	$-0.443257\pi$
$307$	−2.33452e7	−	2.33452e7i	−0.806832	−	0.806832i	−0.984153	+	0.177321i	$0.943257\pi$
$308$		0			0
$309$	−1.43240e7	−	1.43240e7i	−0.485498	−	0.485498i
$310$		0			0
$311$	−5.41001e7			−1.79853			−0.899263	−	0.437408i	$-0.855897\pi$
$311$	−5.41001e7			−1.79853			−0.899263	+	0.437408i	$0.855897\pi$
$312$		0			0
$313$		−	6.22133e6i		−	0.202885i	−0.994841	−	0.101443i	$-0.967654\pi$
$313$		−	6.22133e6i		−	0.202885i	0.994841	−	0.101443i	$-0.0323458\pi$
$314$		0			0
$315$	−2.08708e6	+	2.08708e6i	−0.0667740	+	0.0667740i
$316$		0			0
$317$	3.43285e7	−	3.43285e7i	1.07765	−	1.07765i	0.0809284	−	0.996720i	$-0.474211\pi$
$317$	3.43285e7	−	3.43285e7i	1.07765	−	1.07765i	0.996720	−	0.0809284i	$-0.0257885\pi$
$318$		0			0
$319$		−	1.46350e6i		−	0.0450837i
$320$		0			0
$321$	5.58116e6			0.168737
$322$		0			0
$323$	−2.61268e7	−	2.61268e7i	−0.775317	−	0.775317i
$324$		0			0
$325$	−1.48526e7	−	1.48526e7i	−0.432666	−	0.432666i
$326$		0			0
$327$	1.41926e7			0.405900
$328$		0			0
$329$		−	1.74540e6i		−	0.0490126i
$330$		0			0
$331$	3.16803e7	−	3.16803e7i	0.873585	−	0.873585i	−0.119276	−	0.992861i	$-0.538057\pi$
$331$	3.16803e7	−	3.16803e7i	0.873585	−	0.873585i	0.992861	+	0.119276i	$0.0380573\pi$
$332$		0			0
$333$	1.52864e7	−	1.52864e7i	0.413974	−	0.413974i
$334$		0			0
$335$		−	1.68114e7i		−	0.447168i
$336$		0			0
$337$	2.49344e6			0.0651491			0.0325746	−	0.999469i	$-0.489629\pi$
$337$	2.49344e6			0.0651491			0.0325746	+	0.999469i	$0.489629\pi$
$338$		0			0
$339$	−8.00829e6	−	8.00829e6i	−0.205561	−	0.205561i
$340$		0			0
$341$	−2.83757e6	−	2.83757e6i	−0.0715621	−	0.0715621i
$342$		0			0
$343$	−1.24394e7			−0.308260
$344$		0			0
$345$		−	4.82845e7i		−	1.17585i
$346$		0			0
$347$	6.81675e6	−	6.81675e6i	0.163151	−	0.163151i	−0.620810	−	0.783961i	$-0.713196\pi$
$347$	6.81675e6	−	6.81675e6i	0.163151	−	0.163151i	0.783961	+	0.620810i	$0.213196\pi$
$348$		0			0
$349$	4.27475e7	−	4.27475e7i	1.00562	−	1.00562i	0.00563622	−	0.999984i	$-0.498206\pi$
$349$	4.27475e7	−	4.27475e7i	1.00562	−	1.00562i	0.999984	−	0.00563622i	$-0.00179407\pi$
$350$		0			0
$351$			2.21724e6i			0.0512732i
$352$		0			0
$353$	3.37544e7			0.767374			0.383687	−	0.923463i	$-0.374654\pi$
$353$	3.37544e7			0.767374			0.383687	+	0.923463i	$0.374654\pi$
$354$		0			0
$355$	−6.12704e7	−	6.12704e7i	−1.36951	−	1.36951i
$356$		0			0
$357$	1.94302e6	+	1.94302e6i	0.0427044	+	0.0427044i
$358$		0			0
$359$	2.37154e7			0.512563			0.256281	−	0.966602i	$-0.417503\pi$
$359$	2.37154e7			0.512563			0.256281	+	0.966602i	$0.417503\pi$
$360$		0			0
$361$			7.87965e7i			1.67489i
$362$		0			0
$363$	1.91973e7	−	1.91973e7i	0.401348	−	0.401348i
$364$		0			0
$365$	−1.99046e7	+	1.99046e7i	−0.409332	+	0.409332i
$366$		0			0
$367$		−	1.87320e7i		−	0.378954i	−0.981885	−	0.189477i	$-0.939321\pi$
$367$		−	1.87320e7i		−	0.378954i	0.981885	−	0.189477i	$-0.0606793\pi$
$368$		0			0
$369$	−2.20579e7			−0.439022
$370$		0			0
$371$	1.53549e6	+	1.53549e6i	0.0300694	+	0.0300694i
$372$		0			0
$373$	1.84212e7	+	1.84212e7i	0.354969	+	0.354969i	0.861955	−	0.506985i	$-0.169240\pi$
$373$	1.84212e7	+	1.84212e7i	0.354969	+	0.354969i	−0.506985	+	0.861955i	$0.669240\pi$
$374$		0			0
$375$	7.16794e7			1.35925
$376$		0			0
$377$		−	4.95039e6i		−	0.0923879i
$378$		0			0
$379$	−7.09007e6	+	7.09007e6i	−0.130237	+	0.130237i	−0.769220	−	0.638984i	$-0.779355\pi$
$379$	−7.09007e6	+	7.09007e6i	−0.130237	+	0.130237i	0.638984	+	0.769220i	$0.279355\pi$
$380$		0			0
$381$	−7.46151e6	+	7.46151e6i	−0.134912	+	0.134912i
$382$		0			0
$383$			43105.7i			0.000767253i	1.00000		0.000383627i	$0.000122112\pi$
$383$			43105.7i			0.000767253i	−1.00000		0.000383627i	$0.999878\pi$
$384$		0			0
$385$	−2.10185e6			−0.0368316
$386$		0			0
$387$	2.46635e7	+	2.46635e7i	0.425522	+	0.425522i
$388$		0			0
$389$	2.56670e7	+	2.56670e7i	0.436039	+	0.436039i	0.890677	−	0.454637i	$-0.150231\pi$
$389$	2.56670e7	+	2.56670e7i	0.436039	+	0.436039i	−0.454637	+	0.890677i	$0.650231\pi$
$390$		0			0
$391$	−4.49517e7			−0.751997
$392$		0			0
$393$		−	6.11119e7i		−	1.00681i
$394$		0			0
$395$	−3.58038e7	+	3.58038e7i	−0.580948	+	0.580948i
$396$		0			0
$397$	−4.19642e7	+	4.19642e7i	−0.670667	+	0.670667i	−0.957870	−	0.287202i	$-0.907275\pi$
$397$	−4.19642e7	+	4.19642e7i	−0.670667	+	0.670667i	0.287202	+	0.957870i	$0.407275\pi$
$398$		0			0
$399$		−	9.35874e6i		−	0.147333i
$400$		0			0
$401$	−3.27072e7			−0.507236			−0.253618	−	0.967305i	$-0.581621\pi$
$401$	−3.27072e7			−0.507236			−0.253618	+	0.967305i	$0.581621\pi$
$402$		0			0
$403$	−9.59829e6	−	9.59829e6i	−0.146649	−	0.146649i
$404$		0			0
$405$	−9.47642e6	−	9.47642e6i	−0.142652	−	0.142652i
$406$		0			0
$407$	1.53946e7			0.228342
$408$		0			0
$409$		−	1.49793e7i		−	0.218938i	−0.993990	−	0.109469i	$-0.965085\pi$
$409$		−	1.49793e7i		−	0.218938i	0.993990	−	0.109469i	$-0.0349151\pi$
$410$		0			0
$411$	1.85681e7	−	1.85681e7i	0.267449	−	0.267449i
$412$		0			0
$413$	−7.24702e6	+	7.24702e6i	−0.102875	+	0.102875i
$414$		0			0
$415$		−	1.82402e8i		−	2.55203i
$416$		0			0
$417$	5.44975e7			0.751569
$418$		0			0
$419$	5.28358e7	+	5.28358e7i	0.718268	+	0.718268i	0.968250	−	0.249982i	$-0.0804248\pi$
$419$	5.28358e7	+	5.28358e7i	0.718268	+	0.718268i	−0.249982	+	0.968250i	$0.580425\pi$
$420$		0			0
$421$	3.22046e7	+	3.22046e7i	0.431591	+	0.431591i	0.889169	−	0.457578i	$-0.151283\pi$
$421$	3.22046e7	+	3.22046e7i	0.431591	+	0.431591i	−0.457578	+	0.889169i	$0.651283\pi$
$422$		0			0
$423$	7.92503e6			0.104708
$424$		0			0
$425$		−	1.18196e8i		−	1.53970i
$426$		0			0
$427$	1.32620e7	−	1.32620e7i	0.170344	−	0.170344i
$428$		0			0
$429$	−1.11647e6	+	1.11647e6i	−0.0141408	+	0.0141408i
$430$		0			0
$431$		−	2.33211e6i		−	0.0291285i	−0.999894	−	0.0145642i	$-0.995364\pi$
$431$		−	2.33211e6i		−	0.0291285i	0.999894	−	0.0145642i	$-0.00463611\pi$
$432$		0			0
$433$	−2.63406e7			−0.324460			−0.162230	−	0.986753i	$-0.551869\pi$
$433$	−2.63406e7			−0.324460			−0.162230	+	0.986753i	$0.551869\pi$
$434$		0			0
$435$	2.11579e7	+	2.11579e7i	0.257042	+	0.257042i
$436$		0			0
$437$	1.08257e8	+	1.08257e8i	1.29722	+	1.29722i
$438$		0			0
$439$	8.19570e7			0.968708			0.484354	−	0.874872i	$-0.339055\pi$
$439$	8.19570e7			0.968708			0.484354	+	0.874872i	$0.339055\pi$
$440$		0			0
$441$		−	2.78927e7i		−	0.325218i
$442$		0			0
$443$	6.85870e7	−	6.85870e7i	0.788916	−	0.788916i	−0.192401	−	0.981316i	$-0.561627\pi$
$443$	6.85870e7	−	6.85870e7i	0.788916	−	0.788916i	0.981316	+	0.192401i	$0.0616274\pi$
$444$		0			0
$445$	−1.38557e8	+	1.38557e8i	−1.57235	+	1.57235i
$446$		0			0
$447$			6.75170e7i			0.755945i
$448$		0			0
$449$	−1.60760e7			−0.177599			−0.0887994	−	0.996050i	$-0.528303\pi$
$449$	−1.60760e7			−0.177599			−0.0887994	+	0.996050i	$0.528303\pi$
$450$		0			0
$451$	−1.11071e7	−	1.11071e7i	−0.121079	−	0.121079i
$452$		0			0
$453$	40763.6	+	40763.6i	0.000438508	+	0.000438508i
$454$		0			0
$455$	−7.10968e6			−0.0754772
$456$		0			0
$457$		−	2.76233e7i		−	0.289419i	−0.989474	−	0.144709i	$-0.953775\pi$
$457$		−	2.76233e7i		−	0.289419i	0.989474	−	0.144709i	$-0.0462247\pi$
$458$		0			0
$459$	−8.82232e6	+	8.82232e6i	−0.0912315	+	0.0912315i
$460$		0			0
$461$	7.91602e7	−	7.91602e7i	0.807987	−	0.807987i	−0.176342	−	0.984329i	$-0.556427\pi$
$461$	7.91602e7	−	7.91602e7i	0.807987	−	0.807987i	0.984329	+	0.176342i	$0.0564265\pi$
$462$		0			0
$463$		−	3.84728e7i		−	0.387624i	−0.981039	−	0.193812i	$-0.937915\pi$
$463$		−	3.84728e7i		−	0.387624i	0.981039	−	0.193812i	$-0.0620852\pi$
$464$		0			0
$465$	8.20458e7			0.816013
$466$		0			0
$467$	−8.83440e7	−	8.83440e7i	−0.867414	−	0.867414i	0.124772	−	0.992185i	$-0.460180\pi$
$467$	−8.83440e7	−	8.83440e7i	−0.867414	−	0.867414i	−0.992185	+	0.124772i	$0.960180\pi$
$468$		0			0
$469$	−2.80313e6	−	2.80313e6i	−0.0271722	−	0.0271722i
$470$		0			0
$471$	−5.27030e7			−0.504397
$472$		0			0
$473$			2.48381e7i			0.234712i
$474$		0			0
$475$	−2.84652e8	+	2.84652e8i	−2.65603	+	2.65603i
$476$		0			0
$477$	−6.97191e6	+	6.97191e6i	−0.0642387	+	0.0642387i
$478$		0			0
$479$			4.42078e7i			0.402247i	0.979566	+	0.201123i	$0.0644592\pi$
$479$			4.42078e7i			0.402247i	−0.979566	+	0.201123i	$0.935541\pi$
$480$		0			0
$481$	5.20735e7			0.467930
$482$		0			0
$483$	−8.05095e6	−	8.05095e6i	−0.0714506	−	0.0714506i
$484$		0			0
$485$	−2.13021e8	−	2.13021e8i	−1.86723	−	1.86723i
$486$		0			0
$487$	−2.05189e8			−1.77651			−0.888254	−	0.459353i	$-0.848081\pi$
$487$	−2.05189e8			−1.77651			−0.888254	+	0.459353i	$0.848081\pi$
$488$		0			0
$489$		−	1.20649e7i		−	0.103181i
$490$		0			0
$491$	1.11072e8	−	1.11072e8i	0.938337	−	0.938337i	−0.0598694	−	0.998206i	$-0.519068\pi$
$491$	1.11072e8	−	1.11072e8i	0.938337	−	0.938337i	0.998206	+	0.0598694i	$0.0190684\pi$
$492$		0			0
$493$	1.96974e7	−	1.96974e7i	0.164388	−	0.164388i
$494$		0			0
$495$		−	9.54351e6i		−	0.0786851i
$496$		0			0
$497$	−2.04324e7			−0.166437
$498$		0			0
$499$	−3.75392e7	−	3.75392e7i	−0.302123	−	0.302123i	0.539721	−	0.841844i	$-0.318530\pi$
$499$	−3.75392e7	−	3.75392e7i	−0.302123	−	0.302123i	−0.841844	+	0.539721i	$0.818530\pi$
$500$		0			0
$501$	−4.89385e7	−	4.89385e7i	−0.389169	−	0.389169i
$502$		0			0
$503$	1.09161e8			0.857757			0.428878	−	0.903362i	$-0.358909\pi$
$503$	1.09161e8			0.857757			0.428878	+	0.903362i	$0.358909\pi$
$504$		0			0
$505$			2.91912e8i			2.26662i
$506$		0			0
$507$	4.94280e7	−	4.94280e7i	0.379270	−	0.379270i
$508$		0			0
$509$	1.66360e8	−	1.66360e8i	1.26152	−	1.26152i	0.311169	−	0.950355i	$-0.399279\pi$
$509$	1.66360e8	−	1.66360e8i	1.26152	−	1.26152i	0.950355	−	0.311169i	$-0.100721\pi$
$510$		0			0
$511$			6.63779e6i			0.0497463i
$512$		0			0
$513$	4.24935e7			0.314754
$514$		0			0
$515$	2.08548e8	+	2.08548e8i	1.52681	+	1.52681i
$516$		0			0
$517$	3.99057e6	+	3.99057e6i	0.0288777	+	0.0288777i
$518$		0			0
$519$	1.20344e8			0.860840
$520$		0			0
$521$		−	1.52012e8i		−	1.07489i	−0.843299	−	0.537445i	$-0.819390\pi$
$521$		−	1.52012e8i		−	1.07489i	0.843299	−	0.537445i	$-0.180610\pi$
$522$		0			0
$523$	−7.11802e7	+	7.11802e7i	−0.497570	+	0.497570i	−0.910681	−	0.413111i	$-0.864442\pi$
$523$	−7.11802e7	+	7.11802e7i	−0.497570	+	0.497570i	0.413111	+	0.910681i	$0.364442\pi$
$524$		0			0
$525$	2.11692e7	−	2.11692e7i	0.146294	−	0.146294i
$526$		0			0
$527$		−	7.63826e7i		−	0.521871i
$528$		0			0
$529$	3.82228e7			0.258199
$530$		0			0
$531$	−3.29052e7	−	3.29052e7i	−0.219776	−	0.219776i
$532$		0			0
$533$	−3.75704e7	−	3.75704e7i	−0.248122	−	0.248122i
$534$		0			0
$535$	−8.12584e7			−0.530648
$536$		0			0
$537$			6.13460e6i			0.0396153i
$538$		0			0
$539$	1.40451e7	−	1.40451e7i	0.0896929	−	0.0896929i
$540$		0			0
$541$	−1.59423e8	+	1.59423e8i	−1.00684	+	1.00684i	−0.00686056	+	0.999976i	$0.502184\pi$
$541$	−1.59423e8	+	1.59423e8i	−1.00684	+	1.00684i	−0.999976	+	0.00686056i	$0.997816\pi$
$542$		0			0
$543$			6.51145e7i			0.406704i
$544$		0			0
$545$	−2.06636e8			−1.27649
$546$		0			0
$547$	−5.89803e7	−	5.89803e7i	−0.360367	−	0.360367i	0.503581	−	0.863948i	$-0.332016\pi$
$547$	−5.89803e7	−	5.89803e7i	−0.360367	−	0.360367i	−0.863948	+	0.503581i	$0.832016\pi$
$548$		0			0
$549$	6.02165e7	+	6.02165e7i	0.363914	+	0.363914i
$550$		0			0
$551$	−9.48746e7			−0.567147
$552$		0			0
$553$			1.19398e7i			0.0706029i
$554$		0			0
$555$	−2.22561e8	+	2.22561e8i	−1.30188	+	1.30188i
$556$		0			0
$557$	6.92261e7	−	6.92261e7i	0.400594	−	0.400594i	−0.477849	−	0.878442i	$-0.658583\pi$
$557$	6.92261e7	−	6.92261e7i	0.400594	−	0.400594i	0.878442	+	0.477849i	$0.158583\pi$
$558$		0			0
$559$			8.40168e7i			0.480984i
$560$		0			0
$561$	−8.88478e6			−0.0503220
$562$		0			0
$563$	2.44399e8	+	2.44399e8i	1.36954	+	1.36954i	0.861094	+	0.508445i	$0.169780\pi$
$563$	2.44399e8	+	2.44399e8i	1.36954	+	1.36954i	0.508445	+	0.861094i	$0.330220\pi$
$564$		0			0
$565$	1.16596e8	+	1.16596e8i	0.646455	+	0.646455i
$566$		0			0
$567$	−3.16019e6			−0.0173366
$568$		0			0
$569$			1.51122e8i			0.820337i	0.912010	+	0.410168i	$0.134530\pi$
$569$			1.51122e8i			0.820337i	−0.912010	+	0.410168i	$0.865470\pi$
$570$		0			0
$571$	6.74457e7	−	6.74457e7i	0.362281	−	0.362281i	−0.502371	−	0.864652i	$-0.667539\pi$
$571$	6.74457e7	−	6.74457e7i	0.362281	−	0.362281i	0.864652	+	0.502371i	$0.167539\pi$
$572$		0			0
$573$	−6.51162e7	+	6.51162e7i	−0.346119	+	0.346119i
$574$		0			0
$575$			4.89749e8i			2.57615i
$576$		0			0
$577$	−1.55358e8			−0.808734			−0.404367	−	0.914597i	$-0.632508\pi$
$577$	−1.55358e8			−0.808734			−0.404367	+	0.914597i	$0.632508\pi$
$578$		0			0
$579$	1.27124e8	+	1.27124e8i	0.654924	+	0.654924i
$580$		0			0
$581$	−3.04137e7	−	3.04137e7i	−0.155075	−	0.155075i
$582$		0			0
$583$	−7.02127e6			−0.0354332
$584$		0			0
$585$		−	3.22816e7i		−	0.161246i
$586$		0			0
$587$	1.89580e8	−	1.89580e8i	0.937299	−	0.937299i	−0.0608477	−	0.998147i	$-0.519380\pi$
$587$	1.89580e8	−	1.89580e8i	0.937299	−	0.937299i	0.998147	+	0.0608477i	$0.0193804\pi$
$588$		0			0
$589$	−1.83952e8	+	1.83952e8i	−0.900241	+	0.900241i
$590$		0			0
$591$			3.71555e7i			0.179995i
$592$		0			0
$593$	−1.96574e8			−0.942673			−0.471337	−	0.881953i	$-0.656228\pi$
$593$	−1.96574e8			−0.942673			−0.471337	+	0.881953i	$0.656228\pi$
$594$		0			0
$595$	−2.82892e7	−	2.82892e7i	−0.134298	−	0.134298i
$596$		0			0
$597$	1.13925e8	+	1.13925e8i	0.535421	+	0.535421i
$598$		0			0
$599$	1.81636e8			0.845125			0.422563	−	0.906334i	$-0.361131\pi$
$599$	1.81636e8			0.845125			0.422563	+	0.906334i	$0.361131\pi$
$600$		0			0
$601$			4.06592e8i			1.87299i	0.350680	+	0.936496i	$0.385951\pi$
$601$			4.06592e8i			1.87299i	−0.350680	+	0.936496i	$0.614049\pi$
$602$		0			0
$603$	1.27277e7	−	1.27277e7i	0.0580493	−	0.0580493i
$604$		0			0
$605$	−2.79502e8	+	2.79502e8i	−1.26217	+	1.26217i
$606$		0			0
$607$			1.98245e8i			0.886411i	0.896420	+	0.443205i	$0.146159\pi$
$607$			1.98245e8i			0.886411i	−0.896420	+	0.443205i	$0.853841\pi$
$608$		0			0
$609$	7.05571e6			0.0312384
$610$		0			0
$611$	1.34984e7	+	1.34984e7i	0.0591777	+	0.0591777i
$612$		0			0
$613$	−1.87047e8	−	1.87047e8i	−0.812025	−	0.812025i	0.172912	−	0.984937i	$-0.444682\pi$
$613$	−1.87047e8	−	1.87047e8i	−0.812025	−	0.812025i	−0.984937	+	0.172912i	$0.944682\pi$
$614$		0			0
$615$	3.21151e8			1.38065
$616$		0			0
$617$		−	1.83619e7i		−	0.0781740i	−0.999236	−	0.0390870i	$-0.987555\pi$
$617$		−	1.83619e7i		−	0.0781740i	0.999236	−	0.0390870i	$-0.0124450\pi$
$618$		0			0
$619$	2.35104e8	−	2.35104e8i	0.991261	−	0.991261i	−0.00870092	−	0.999962i	$-0.502770\pi$
$619$	2.35104e8	−	2.35104e8i	0.991261	−	0.991261i	0.999962	+	0.00870092i	$0.00276962\pi$
$620$		0			0
$621$	3.65555e7	−	3.65555e7i	0.152643	−	0.152643i
$622$		0			0
$623$			4.62061e7i			0.191089i
$624$		0			0
$625$	−4.82902e8			−1.97797
$626$		0			0
$627$	2.13972e7	+	2.13972e7i	0.0868069	+	0.0868069i
$628$		0			0
$629$	2.07199e8	+	2.07199e8i	0.832598	+	0.832598i
$630$		0			0
$631$	6.18271e7			0.246088			0.123044	−	0.992401i	$-0.460734\pi$
$631$	6.18271e7			0.246088			0.123044	+	0.992401i	$0.460734\pi$
$632$		0			0
$633$			3.56913e7i			0.140719i
$634$		0			0
$635$	1.08635e8	−	1.08635e8i	0.424276	−	0.424276i
$636$		0			0
$637$	4.75086e7	−	4.75086e7i	0.183803	−	0.183803i
$638$		0			0
$639$		−	9.27738e7i		−	0.355568i
$640$		0			0
$641$	−1.39413e7			−0.0529334			−0.0264667	−	0.999650i	$-0.508426\pi$
$641$	−1.39413e7			−0.0529334			−0.0264667	+	0.999650i	$0.508426\pi$
$642$		0			0
$643$	−2.21654e8	−	2.21654e8i	−0.833762	−	0.833762i	0.154267	−	0.988029i	$-0.450698\pi$
$643$	−2.21654e8	−	2.21654e8i	−0.833762	−	0.833762i	−0.988029	+	0.154267i	$0.950698\pi$
$644$		0			0
$645$	−3.59086e8	−	3.59086e8i	−1.33820	−	1.33820i
$646$		0			0
$647$	1.71900e8			0.634692			0.317346	−	0.948310i	$-0.397208\pi$
$647$	1.71900e8			0.634692			0.317346	+	0.948310i	$0.397208\pi$
$648$		0			0
$649$		−	3.31382e7i		−	0.121226i
$650$		0			0
$651$	1.36803e7	−	1.36803e7i	0.0495852	−	0.0495852i
$652$		0			0
$653$	8.79241e7	−	8.79241e7i	0.315768	−	0.315768i	−0.531371	−	0.847139i	$-0.678323\pi$
$653$	8.79241e7	−	8.79241e7i	0.315768	−	0.315768i	0.847139	+	0.531371i	$0.178323\pi$
$654$		0			0
$655$			8.89753e8i			3.16625i
$656$		0			0
$657$	−3.01390e7			−0.106275
$658$		0			0
$659$	−1.77161e8	−	1.77161e8i	−0.619029	−	0.619029i	0.326253	−	0.945282i	$-0.394214\pi$
$659$	−1.77161e8	−	1.77161e8i	−0.619029	−	0.619029i	−0.945282	+	0.326253i	$0.894214\pi$
$660$		0			0
$661$	1.63208e8	+	1.63208e8i	0.565114	+	0.565114i	0.930756	−	0.365642i	$-0.119150\pi$
$661$	1.63208e8	+	1.63208e8i	0.565114	+	0.565114i	−0.365642	+	0.930756i	$0.619150\pi$
$662$		0			0
$663$	−3.00534e7			−0.103122
$664$		0			0
$665$			1.36258e8i			0.463336i
$666$		0			0
$667$	−8.16169e7	+	8.16169e7i	−0.275044	+	0.275044i
$668$		0			0
$669$	1.56858e8	−	1.56858e8i	0.523877	−	0.523877i
$670$		0			0
$671$			6.06428e7i			0.200730i
$672$		0			0
$673$	2.61585e8			0.858158			0.429079	−	0.903267i	$-0.358838\pi$
$673$	2.61585e8			0.858158			0.429079	+	0.903267i	$0.358838\pi$
$674$		0			0
$675$	9.61192e7	+	9.61192e7i	0.312535	+	0.312535i
$676$		0			0
$677$	−1.96092e8	−	1.96092e8i	−0.631966	−	0.631966i	0.316595	−	0.948561i	$-0.397460\pi$
$677$	−1.96092e8	−	1.96092e8i	−0.631966	−	0.631966i	−0.948561	+	0.316595i	$0.897460\pi$
$678$		0			0
$679$	−7.10381e7			−0.226925
$680$		0			0
$681$		−	1.56570e8i		−	0.495756i
$682$		0			0
$683$	−2.80121e8	+	2.80121e8i	−0.879193	+	0.879193i	−0.993451	−	0.114258i	$-0.963551\pi$
$683$	−2.80121e8	+	2.80121e8i	−0.879193	+	0.879193i	0.114258	+	0.993451i	$0.463551\pi$
$684$		0			0
$685$	−2.70340e8	+	2.70340e8i	−0.841083	+	0.841083i
$686$		0			0
$687$		−	3.25376e8i		−	1.00350i
$688$		0			0
$689$	−2.37500e7			−0.0726115
$690$		0			0
$691$	4.12201e8	+	4.12201e8i	1.24932	+	1.24932i	0.956019	+	0.293305i	$0.0947550\pi$
$691$	4.12201e8	+	4.12201e8i	1.24932	+	1.24932i	0.293305	+	0.956019i	$0.405245\pi$
$692$		0			0
$693$	−1.59128e6	−	1.59128e6i	−0.00478132	−	0.00478132i
$694$		0			0
$695$	−7.93452e8			−2.36356
$696$		0			0
$697$		−	2.98983e8i		−	0.882976i
$698$		0			0
$699$	1.27784e8	−	1.27784e8i	0.374149	−	0.374149i
$700$		0			0
$701$	2.49518e8	−	2.49518e8i	0.724348	−	0.724348i	−0.245140	−	0.969488i	$-0.578834\pi$
$701$	2.49518e8	−	2.49518e8i	0.724348	−	0.724348i	0.969488	+	0.245140i	$0.0788338\pi$
$702$		0			0
$703$		−	9.97993e8i		−	2.87251i
$704$		0			0
$705$	−1.15384e8			−0.329289
$706$		0			0
$707$	4.86734e7	+	4.86734e7i	0.137731	+	0.137731i
$708$		0			0
$709$	−1.19711e8	−	1.19711e8i	−0.335889	−	0.335889i	0.518928	−	0.854818i	$-0.326331\pi$
$709$	−1.19711e8	−	1.19711e8i	−0.335889	−	0.335889i	−0.854818	+	0.518928i	$0.826331\pi$
$710$		0			0
$711$	−5.42130e7			−0.150832
$712$		0			0
$713$			3.16493e8i			0.873164i
$714$		0			0
$715$	1.62551e7	−	1.62551e7i	0.0444704	−	0.0444704i
$716$		0			0
$717$	−3.50695e7	+	3.50695e7i	−0.0951420	+	0.0951420i
$718$		0			0
$719$		−	3.99472e8i		−	1.07473i	−0.843349	−	0.537366i	$-0.819420\pi$
$719$		−	3.99472e8i		−	1.07473i	0.843349	−	0.537366i	$-0.180580\pi$
$720$		0			0
$721$	6.95466e7			0.185554
$722$		0			0
$723$	−1.27157e8	−	1.27157e8i	−0.336453	−	0.336453i
$724$		0			0
$725$	−2.14604e8	−	2.14604e8i	−0.563149	−	0.563149i
$726$		0			0
$727$	3.64970e8			0.949847			0.474923	−	0.880027i	$-0.342476\pi$
$727$	3.64970e8			0.949847			0.474923	+	0.880027i	$0.342476\pi$
$728$		0			0
$729$		−	1.43489e7i		−	0.0370370i
$730$		0			0
$731$	−3.34301e8	+	3.34301e8i	−0.855825	+	0.855825i
$732$		0			0
$733$	1.06785e8	−	1.06785e8i	0.271143	−	0.271143i	−0.558417	−	0.829560i	$-0.688591\pi$
$733$	1.06785e8	−	1.06785e8i	0.271143	−	0.271143i	0.829560	+	0.558417i	$0.188591\pi$
$734$		0			0
$735$			4.06101e8i			1.02276i
$736$		0			0
$737$	1.28178e7			0.0320192
$738$		0			0
$739$	−5.28709e8	−	5.28709e8i	−1.31004	−	1.31004i	−0.921385	−	0.388650i	$-0.872942\pi$
$739$	−5.28709e8	−	5.28709e8i	−1.31004	−	1.31004i	−0.388650	−	0.921385i	$-0.627058\pi$
$740$		0			0
$741$	7.23776e7	+	7.23776e7i	0.177889	+	0.177889i
$742$		0			0
$743$	2.66673e8			0.650148			0.325074	−	0.945689i	$-0.394611\pi$
$743$	2.66673e8			0.650148			0.325074	+	0.945689i	$0.394611\pi$
$744$		0			0
$745$		−	9.83007e8i		−	2.37732i
$746$		0			0
$747$	1.38094e8	−	1.38094e8i	0.331294	−	0.331294i
$748$		0			0
$749$	−1.35490e7	+	1.35490e7i	−0.0322450	+	0.0322450i
$750$		0			0
$751$		−	5.88812e8i		−	1.39013i	−0.718945	−	0.695067i	$-0.755375\pi$
$751$		−	5.88812e8i		−	1.39013i	0.718945	−	0.695067i	$-0.244625\pi$
$752$		0			0
$753$	6.70408e7			0.157020
$754$		0			0
$755$	−593493.	−	593493.i	−0.00137903	−	0.00137903i
$756$		0			0
$757$	−1.33371e8	−	1.33371e8i	−0.307449	−	0.307449i	0.536470	−	0.843919i	$-0.319757\pi$
$757$	−1.33371e8	−	1.33371e8i	−0.307449	−	0.307449i	−0.843919	+	0.536470i	$0.819757\pi$
$758$		0			0
$759$	3.68143e7			0.0841959
$760$		0			0
$761$			1.09778e8i			0.249094i	0.992214	+	0.124547i	$0.0397477\pi$
$761$			1.09778e8i			0.249094i	−0.992214	+	0.124547i	$0.960252\pi$
$762$		0			0
$763$	−3.44544e7	+	3.44544e7i	−0.0775660	+	0.0775660i
$764$		0			0
$765$	1.28448e8	−	1.28448e8i	0.286908	−	0.286908i
$766$		0			0
$767$		−	1.12092e8i		−	0.248422i
$768$		0			0
$769$	−8.17993e7			−0.179875			−0.0899374	−	0.995947i	$-0.528667\pi$
$769$	−8.17993e7			−0.179875			−0.0899374	+	0.995947i	$0.528667\pi$
$770$		0			0
$771$	9.39004e7	+	9.39004e7i	0.204882	+	0.204882i
$772$		0			0
$773$	6.06930e8	+	6.06930e8i	1.31401	+	1.31401i	0.918430	+	0.395584i	$0.129458\pi$
$773$	6.06930e8	+	6.06930e8i	1.31401	+	1.31401i	0.395584	+	0.918430i	$0.370542\pi$
$774$		0			0
$775$	−8.32189e8			−1.78779
$776$		0			0
$777$			7.42195e7i			0.158218i
$778$		0			0
$779$	−7.20041e8	+	7.20041e8i	−1.52316	+	1.52316i
$780$		0			0
$781$	4.67153e7	−	4.67153e7i	0.0980632	−	0.0980632i
$782$		0			0
$783$			3.20366e7i			0.0667361i
$784$		0			0
$785$	7.67324e8			1.58624
$786$		0			0
$787$	3.37638e8	+	3.37638e8i	0.692671	+	0.692671i	0.962819	−	0.270148i	$-0.0870726\pi$
$787$	3.37638e8	+	3.37638e8i	0.692671	+	0.692671i	−0.270148	+	0.962819i	$0.587073\pi$
$788$		0			0
$789$	−1.02552e8	−	1.02552e8i	−0.208792	−	0.208792i
$790$		0			0
$791$	3.88823e7			0.0785639
$792$		0			0
$793$			2.05129e8i			0.411346i
$794$		0			0
$795$	1.01507e8	−	1.01507e8i	0.202020	−	0.202020i
$796$		0			0
$797$	−2.20099e8	+	2.20099e8i	−0.434753	+	0.434753i	−0.890242	−	0.455489i	$-0.849465\pi$
$797$	−2.20099e8	+	2.20099e8i	−0.434753	+	0.434753i	0.455489	+	0.890242i	$0.349465\pi$
$798$		0			0
$799$			1.07419e8i			0.210592i
$800$		0			0
$801$	−2.09799e8			−0.408232
$802$		0			0
$803$	−1.51762e7	−	1.51762e7i	−0.0293100	−	0.0293100i
$804$		0			0
$805$	1.17217e8	+	1.17217e8i	0.224700	+	0.224700i
$806$		0			0
$807$	−5.04477e7			−0.0959888
$808$		0			0
$809$		−	6.45967e7i		−	0.122001i	−0.998138	−	0.0610007i	$-0.980571\pi$
$809$		−	6.45967e7i		−	0.122001i	0.998138	−	0.0610007i	$-0.0194292\pi$
$810$		0			0
$811$	4.16966e8	−	4.16966e8i	0.781697	−	0.781697i	−0.198420	−	0.980117i	$-0.563581\pi$
$811$	4.16966e8	−	4.16966e8i	0.781697	−	0.781697i	0.980117	+	0.198420i	$0.0635810\pi$
$812$		0			0
$813$	−1.06516e8	+	1.06516e8i	−0.198219	+	0.198219i
$814$		0			0
$815$			1.75658e8i			0.324486i
$816$		0			0
$817$	1.61019e9			2.95264
$818$		0			0
$819$	−5.38263e6	−	5.38263e6i	−0.00979812	−	0.00979812i
$820$		0			0
$821$	2.63502e8	+	2.63502e8i	0.476162	+	0.476162i	0.903902	−	0.427740i	$-0.140690\pi$
$821$	2.63502e8	+	2.63502e8i	0.476162	+	0.476162i	−0.427740	+	0.903902i	$0.640690\pi$
$822$		0			0
$823$	6.87298e8			1.23295			0.616475	−	0.787375i	$-0.288560\pi$
$823$	6.87298e8			1.23295			0.616475	+	0.787375i	$0.288560\pi$
$824$		0			0
$825$			9.67997e7i			0.172390i
$826$		0			0
$827$	6.60675e8	−	6.60675e8i	1.16808	−	1.16808i	0.185417	−	0.982660i	$-0.440636\pi$
$827$	6.60675e8	−	6.60675e8i	1.16808	−	1.16808i	0.982660	−	0.185417i	$-0.0593637\pi$
$828$		0			0
$829$	1.09496e8	−	1.09496e8i	0.192192	−	0.192192i	−0.604451	−	0.796643i	$-0.706607\pi$
$829$	1.09496e8	−	1.09496e8i	0.192192	−	0.192192i	0.796643	+	0.604451i	$0.206607\pi$
$830$		0			0
$831$		−	1.47195e8i		−	0.256501i
$832$		0			0
$833$	3.78070e8			0.654090
$834$		0			0
$835$	7.12516e8	+	7.12516e8i	1.22387	+	1.22387i
$836$		0			0
$837$	6.21156e7	+	6.21156e7i	0.105931	+	0.105931i
$838$		0			0
$839$	7.60578e8			1.28783			0.643914	−	0.765098i	$-0.277310\pi$
$839$	7.60578e8			1.28783			0.643914	+	0.765098i	$0.277310\pi$
$840$		0			0
$841$			5.23296e8i			0.879750i
$842$		0			0
$843$	−4.56727e7	+	4.56727e7i	−0.0762385	+	0.0762385i
$844$		0			0
$845$	−7.19642e8	+	7.19642e8i	−1.19274	+	1.19274i
$846$		0			0
$847$			9.32081e7i			0.153392i
$848$		0			0
$849$	7.37258e7			0.120475
$850$		0			0
$851$	−8.58533e8	−	8.58533e8i	−1.39306	−	1.39306i
$852$		0			0
$853$	4.01377e8	+	4.01377e8i	0.646703	+	0.646703i	0.952195	−	0.305491i	$-0.0988207\pi$
$853$	4.01377e8	+	4.01377e8i	0.646703	+	0.646703i	−0.305491	+	0.952195i	$0.598821\pi$
$854$		0			0
$855$	−6.18681e8			−0.989847
$856$		0			0
$857$		−	7.49729e8i		−	1.19114i	−0.803305	−	0.595568i	$-0.796927\pi$
$857$		−	7.49729e8i		−	1.19114i	0.803305	−	0.595568i	$-0.203073\pi$
$858$		0			0
$859$	−6.75979e8	+	6.75979e8i	−1.06648	+	1.06648i	−0.0688566	+	0.997627i	$0.521935\pi$
$859$	−6.75979e8	+	6.75979e8i	−1.06648	+	1.06648i	−0.997627	+	0.0688566i	$0.978065\pi$
$860$		0			0
$861$	5.35486e7	−	5.35486e7i	0.0838954	−	0.0838954i
$862$		0			0
$863$		−	9.20482e8i		−	1.43213i	−0.698033	−	0.716066i	$-0.745941\pi$
$863$		−	9.20482e8i		−	1.43213i	0.698033	−	0.716066i	$-0.254059\pi$
$864$		0			0
$865$	−1.75214e9			−2.70720
$866$		0			0
$867$	1.46480e8	+	1.46480e8i	0.224760	+	0.224760i
$868$		0			0
$869$	−2.72984e7	−	2.72984e7i	−0.0415985	−	0.0415985i
$870$		0			0
$871$	4.33571e7			0.0656154
$872$		0			0
$873$		−	3.22550e8i		−	0.484790i
$874$		0			0
$875$	−1.74011e8	+	1.74011e8i	−0.259748	+	0.259748i
$876$		0			0
$877$	4.57604e8	−	4.57604e8i	0.678407	−	0.678407i	−0.281232	−	0.959640i	$-0.590743\pi$
$877$	4.57604e8	−	4.57604e8i	0.678407	−	0.678407i	0.959640	+	0.281232i	$0.0907431\pi$
$878$		0			0
$879$		−	3.38665e8i		−	0.498659i
$880$		0			0
$881$	6.68932e7			0.0978261			0.0489130	−	0.998803i	$-0.484424\pi$
$881$	6.68932e7			0.0978261			0.0489130	+	0.998803i	$0.484424\pi$
$882$		0			0
$883$	−1.34223e8	−	1.34223e8i	−0.194960	−	0.194960i	0.602875	−	0.797835i	$-0.294022\pi$
$883$	−1.34223e8	−	1.34223e8i	−0.194960	−	0.194960i	−0.797835	+	0.602875i	$0.794022\pi$
$884$		0			0
$885$	4.79080e8	+	4.79080e8i	0.691160	+	0.691160i
$886$		0			0
$887$	−2.54696e8			−0.364966			−0.182483	−	0.983209i	$-0.558413\pi$
$887$	−2.54696e8			−0.364966			−0.182483	+	0.983209i	$0.558413\pi$
$888$		0			0
$889$		−	3.62276e7i		−	0.0515625i
$890$		0			0
$891$	7.22525e6	−	7.22525e6i	0.0102146	−	0.0102146i
$892$		0			0
$893$	2.58698e8	−	2.58698e8i	0.363277	−	0.363277i
$894$		0			0
$895$		−	8.93161e7i		−	0.124584i
$896$		0			0
$897$	1.24527e8			0.172539
$898$		0			0
$899$	−1.38685e8	−	1.38685e8i	−0.190875	−	0.190875i
$900$		0			0
$901$	−9.45004e7	−	9.45004e7i	−0.129199	−	0.129199i
$902$		0			0
$903$	−1.19748e8			−0.162631
$904$		0			0
$905$		−	9.48029e8i		−	1.27902i
$906$		0			0
$907$	1.83019e8	−	1.83019e8i	0.245286	−	0.245286i	−0.573747	−	0.819033i	$-0.694511\pi$
$907$	1.83019e8	−	1.83019e8i	0.245286	−	0.245286i	0.819033	+	0.573747i	$0.194511\pi$
$908$		0			0
$909$	−2.21002e8	+	2.21002e8i	−0.294242	+	0.294242i
$910$		0			0
$911$		−	6.16653e8i		−	0.815616i	−0.913068	−	0.407808i	$-0.866293\pi$
$911$		−	6.16653e8i		−	0.815616i	0.913068	−	0.407808i	$-0.133707\pi$
$912$		0			0
$913$	1.39072e8			0.182737
$914$		0			0
$915$	−8.76716e8	−	8.76716e8i	−1.14445	−	1.14445i
$916$		0			0
$917$	1.48357e8	+	1.48357e8i	0.192398	+	0.192398i
$918$		0			0
$919$	3.48775e8			0.449364			0.224682	−	0.974432i	$-0.427866\pi$
$919$	3.48775e8			0.449364			0.224682	+	0.974432i	$0.427866\pi$
$920$		0			0
$921$			5.14655e8i			0.658776i
$922$		0			0
$923$	1.58018e8	−	1.58018e8i	0.200956	−	0.200956i
$924$		0			0
$925$	2.25743e9	−	2.25743e9i	2.85226	−	2.85226i
$926$		0			0
$927$			3.15778e8i			0.396408i
$928$		0			0
$929$	7.50608e8			0.936194			0.468097	−	0.883677i	$-0.344940\pi$
$929$	7.50608e8			0.936194			0.468097	+	0.883677i	$0.344940\pi$
$930$		0			0
$931$	−9.10506e8	−	9.10506e8i	−1.12832	−	1.12832i
$932$		0			0
$933$	5.96329e8	+	5.96329e8i	0.734245	+	0.734245i
$934$		0			0
$935$	1.29357e8			0.158254
$936$		0			0
$937$			2.76459e8i			0.336057i	0.985782	+	0.168028i	$0.0537400\pi$
$937$			2.76459e8i			0.336057i	−0.985782	+	0.168028i	$0.946260\pi$
$938$		0			0
$939$	−6.85758e7	+	6.85758e7i	−0.0828275	+	0.0828275i
$940$		0			0
$941$	−7.64592e8	+	7.64592e8i	−0.917616	+	0.917616i	−0.996856	−	0.0792394i	$-0.974751\pi$
$941$	−7.64592e8	+	7.64592e8i	−0.917616	+	0.917616i	0.0792394	+	0.996856i	$0.474751\pi$
$942$		0			0
$943$			1.23885e9i			1.47735i
$944$		0			0
$945$	4.60105e7			0.0545207
$946$		0			0
$947$	6.64677e8	+	6.64677e8i	0.782638	+	0.782638i	0.980275	−	0.197638i	$-0.0633269\pi$
$947$	6.64677e8	+	6.64677e8i	0.782638	+	0.782638i	−0.197638	+	0.980275i	$0.563327\pi$
$948$		0			0
$949$	−5.13346e7	−	5.13346e7i	−0.0600636	−	0.0600636i
$950$		0			0
$951$	−7.56786e8			−0.879896
$952$		0			0
$953$		−	4.52299e8i		−	0.522573i	−0.965261	−	0.261287i	$-0.915853\pi$
$953$		−	4.52299e8i		−	0.522573i	0.965261	−	0.261287i	$-0.0841468\pi$
$954$		0			0
$955$	9.48053e8	−	9.48053e8i	1.08849	−	1.08849i
$956$		0			0
$957$	−1.61317e7	+	1.61317e7i	−0.0184054	+	0.0184054i
$958$		0			0
$959$			9.01529e7i			0.102217i
$960$		0			0
$961$	3.49714e8			0.394042
$962$		0			0
$963$	−6.15195e7	−	6.15195e7i	−0.0688865	−	0.0688865i
$964$		0			0
$965$	−1.85085e9	−	1.85085e9i	−2.05963	−	2.05963i
$966$		0			0
$967$	−7.87797e8			−0.871235			−0.435617	−	0.900132i	$-0.643470\pi$
$967$	−7.87797e8			−0.871235			−0.435617	+	0.900132i	$0.643470\pi$
$968$		0			0
$969$			5.75977e8i			0.633044i
$970$		0			0
$971$	2.12327e8	−	2.12327e8i	0.231925	−	0.231925i	−0.581571	−	0.813496i	$-0.697562\pi$
$971$	2.12327e8	−	2.12327e8i	0.231925	−	0.231925i	0.813496	+	0.581571i	$0.197562\pi$
$972$		0			0
$973$	−1.32300e8	+	1.32300e8i	−0.143622	+	0.143622i
$974$		0			0
$975$			3.27432e8i			0.353271i
$976$		0			0
$977$	1.53376e9			1.64465			0.822323	−	0.569021i	$-0.192678\pi$
$977$	1.53376e9			1.64465			0.822323	+	0.569021i	$0.192678\pi$
$978$		0			0
$979$	−1.05642e8	−	1.05642e8i	−0.112587	−	0.112587i
$980$		0			0
$981$	−1.56441e8	−	1.56441e8i	−0.165708	−	0.165708i
$982$		0			0
$983$	−4.48887e8			−0.472581			−0.236291	−	0.971682i	$-0.575932\pi$
$983$	−4.48887e8			−0.472581			−0.236291	+	0.971682i	$0.575932\pi$
$984$		0			0
$985$		−	5.40962e8i		−	0.566055i
$986$		0			0
$987$	−1.92390e7	+	1.92390e7i	−0.0200093	+	0.0200093i
$988$		0			0
$989$	1.38518e9	−	1.38518e9i	1.43192	−	1.43192i
$990$		0			0
$991$		−	1.13972e9i		−	1.17106i	−0.810651	−	0.585529i	$-0.800887\pi$
$991$		−	1.13972e9i		−	1.17106i	0.810651	−	0.585529i	$-0.199113\pi$
$992$		0			0
$993$	−6.98405e8			−0.713280
$994$		0			0
$995$	−1.65868e9	−	1.65868e9i	−1.68381	−	1.68381i
$996$		0			0
$997$	−4.54437e8	−	4.54437e8i	−0.458552	−	0.458552i	0.439628	−	0.898180i	$-0.355110\pi$
$997$	−4.54437e8	−	4.54437e8i	−0.458552	−	0.458552i	−0.898180	+	0.439628i	$0.855110\pi$
$998$		0			0
$999$	−3.36995e8			−0.338008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.l.a.31.12		48
4.3	odd	2		384.7.l.b.31.13		48
8.3	odd	2		48.7.l.a.43.6	yes	48
8.5	even	2		192.7.l.a.79.13		48
16.3	odd	4	inner	384.7.l.a.223.12		48
16.5	even	4		48.7.l.a.19.6	✓	48
16.11	odd	4		192.7.l.a.175.13		48
16.13	even	4		384.7.l.b.223.13		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.l.a.19.6	✓	48	16.5	even	4
48.7.l.a.43.6	yes	48	8.3	odd	2
192.7.l.a.79.13		48	8.5	even	2
192.7.l.a.175.13		48	16.11	odd	4
384.7.l.a.31.12		48	1.1	even	1	trivial
384.7.l.a.223.12		48	16.3	odd	4	inner
384.7.l.b.31.13		48	4.3	odd	2
384.7.l.b.223.13		48	16.13	even	4

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)

\(f(q)\)	\(=\)	\(q+(-11.0227 - 11.0227i) q^{3} +(160.484 + 160.484i) q^{5} +53.5181 q^{7} +243.000i q^{9} +O(q^{10})\)	\(q+(-11.0227 - 11.0227i) q^{3} +(160.484 + 160.484i) q^{5} +53.5181 q^{7} +243.000i q^{9} +(-122.360 + 122.360i) q^{11} +(-413.892 + 413.892i) q^{13} -3537.94i q^{15} -3293.73 q^{17} +(7932.29 + 7932.29i) q^{19} +(-589.914 - 589.914i) q^{21} +13647.7 q^{23} +35885.2i q^{25} +(2678.52 - 2678.52i) q^{27} +(-5980.28 + 5980.28i) q^{29} +23190.3i q^{31} +2697.48 q^{33} +(8588.80 + 8588.80i) q^{35} +(-62907.0 - 62907.0i) q^{37} +9124.43 q^{39} +90773.4i q^{41} +(101496. - 101496. i) q^{43} +(-38997.6 + 38997.6i) q^{45} -32613.3i q^{47} -114785. q^{49} +(36305.8 + 36305.8i) q^{51} +(28691.0 + 28691.0i) q^{53} -39273.7 q^{55} -174871. i q^{57} +(-135412. + 135412. i) q^{59} +(247804. - 247804. i) q^{61} +13004.9i q^{63} -132846. q^{65} +(-52377.3 - 52377.3i) q^{67} +(-150434. - 150434. i) q^{69} -381785. q^{71} +124029. i q^{73} +(395552. - 395552. i) q^{75} +(-6548.49 + 6548.49i) q^{77} +223099. i q^{79} -59049.0 q^{81} +(-568288. - 568288. i) q^{83} +(-528591. - 528591. i) q^{85} +131838. q^{87} +863372. i q^{89} +(-22150.7 + 22150.7i) q^{91} +(255620. - 255620. i) q^{93} +2.54601e6i q^{95} -1.32736e6 q^{97} +(-29733.5 - 29733.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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