LMFDB - Embedded newform 1536.2.j.j.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(385,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1536.385");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.j (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			385.2
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1536.385
Dual form			1536.2.j.j.1153.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(1.76537 + 1.76537i) q^{5} +0.917608i q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{3} +(1.76537 + 1.76537i) q^{5} +0.917608i q^{7} -1.00000i q^{9} +(-2.61313 - 2.61313i) q^{11} +(4.10973 - 4.10973i) q^{13} -2.49661 q^{15} +0.867091 q^{17} +(4.61313 - 4.61313i) q^{19} +(-0.648847 - 0.648847i) q^{21} -4.00000i q^{23} +1.23304i q^{25} +(0.707107 + 0.707107i) q^{27} +(5.46088 - 5.46088i) q^{29} -2.14386 q^{31} +3.69552 q^{33} +(-1.61991 + 1.61991i) q^{35} +(-1.75057 - 1.75057i) q^{37} +5.81204i q^{39} +9.58541i q^{41} +(6.77791 + 6.77791i) q^{43} +(1.76537 - 1.76537i) q^{45} +1.65685 q^{47} +6.15800 q^{49} +(-0.613126 + 0.613126i) q^{51} +(-2.39942 - 2.39942i) q^{53} -9.22625i q^{55} +6.52395i q^{57} +(-8.99321 - 8.99321i) q^{59} +(-0.0163888 + 0.0163888i) q^{61} +0.917608 q^{63} +14.5104 q^{65} +(-3.76696 + 3.76696i) q^{67} +(2.82843 + 2.82843i) q^{69} +11.8216i q^{71} +15.2809i q^{73} +(-0.871891 - 0.871891i) q^{75} +(2.39782 - 2.39782i) q^{77} -8.40968 q^{79} -1.00000 q^{81} +(1.38687 - 1.38687i) q^{83} +(1.53073 + 1.53073i) q^{85} +7.72286i q^{87} +2.79565i q^{89} +(3.77112 + 3.77112i) q^{91} +(1.51594 - 1.51594i) q^{93} +16.2877 q^{95} -2.26582 q^{97} +(-2.61313 + 2.61313i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} - 8 q^{13} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 32 q^{35} - 8 q^{37} + 16 q^{43} + 8 q^{45} - 32 q^{47} - 8 q^{49} + 16 q^{51} - 8 q^{53} - 32 q^{59} - 8 q^{61} + 16 q^{63} - 16 q^{65} - 32 q^{67} + 16 q^{75} - 48 q^{79} - 8 q^{81} + 32 q^{83} - 48 q^{91} + 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 8 * q^13 + 16 * q^19 + 8 * q^29 + 16 * q^31 - 32 * q^35 - 8 * q^37 + 16 * q^43 + 8 * q^45 - 32 * q^47 - 8 * q^49 + 16 * q^51 - 8 * q^53 - 32 * q^59 - 8 * q^61 + 16 * q^63 - 16 * q^65 - 32 * q^67 + 16 * q^75 - 48 * q^79 - 8 * q^81 + 32 * q^83 - 48 * q^91 + 64 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$	1.76537	+	1.76537i	0.789496	+	0.789496i	0.981411	−	0.191915i	$-0.0614699\pi$
$5$	1.76537	+	1.76537i	0.789496	+	0.789496i	−0.191915	+	0.981411i	$0.561470\pi$
$6$		0			0
$7$			0.917608i			0.346823i	0.984849	+	0.173412i	$0.0554791\pi$
$7$			0.917608i			0.346823i	−0.984849	+	0.173412i	$0.944521\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−2.61313	−	2.61313i	−0.787887	−	0.787887i	0.193260	−	0.981148i	$-0.438094\pi$
$11$	−2.61313	−	2.61313i	−0.787887	−	0.787887i	−0.981148	+	0.193260i	$0.938094\pi$
$12$		0			0
$13$	4.10973	−	4.10973i	1.13983	−	1.13983i	0.151355	−	0.988479i	$-0.451636\pi$
$13$	4.10973	−	4.10973i	1.13983	−	1.13983i	0.988479	−	0.151355i	$-0.0483637\pi$
$14$		0			0
$15$	−2.49661			−0.644621
$16$		0			0
$17$	0.867091			0.210300			0.105150	−	0.994456i	$-0.466468\pi$
$17$	0.867091			0.210300			0.105150	+	0.994456i	$0.466468\pi$
$18$		0			0
$19$	4.61313	−	4.61313i	1.05832	−	1.05832i	0.0601333	−	0.998190i	$-0.480847\pi$
$19$	4.61313	−	4.61313i	1.05832	−	1.05832i	0.998190	−	0.0601333i	$-0.0191526\pi$
$20$		0			0
$21$	−0.648847	−	0.648847i	−0.141590	−	0.141590i
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$			1.23304i			0.246608i
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$	5.46088	−	5.46088i	1.01406	−	1.01406i	0.0141612	−	0.999900i	$-0.495492\pi$
$29$	5.46088	−	5.46088i	1.01406	−	1.01406i	0.999900	−	0.0141612i	$-0.00450779\pi$
$30$		0			0
$31$	−2.14386			−0.385049			−0.192524	−	0.981292i	$-0.561667\pi$
$31$	−2.14386			−0.385049			−0.192524	+	0.981292i	$0.561667\pi$
$32$		0			0
$33$	3.69552			0.643307
$34$		0			0
$35$	−1.61991	+	1.61991i	−0.273816	+	0.273816i
$36$		0			0
$37$	−1.75057	−	1.75057i	−0.287792	−	0.287792i	0.548415	−	0.836207i	$-0.315232\pi$
$37$	−1.75057	−	1.75057i	−0.287792	−	0.287792i	−0.836207	+	0.548415i	$0.815232\pi$
$38$		0			0
$39$			5.81204i			0.930671i
$40$		0			0
$41$			9.58541i			1.49699i	0.663140	+	0.748495i	$0.269223\pi$
$41$			9.58541i			1.49699i	−0.663140	+	0.748495i	$0.730777\pi$
$42$		0			0
$43$	6.77791	+	6.77791i	1.03362	+	1.03362i	0.999415	+	0.0342069i	$0.0108905\pi$
$43$	6.77791	+	6.77791i	1.03362	+	1.03362i	0.0342069	+	0.999415i	$0.489109\pi$
$44$		0			0
$45$	1.76537	−	1.76537i	0.263165	−	0.263165i
$46$		0			0
$47$	1.65685			0.241677			0.120839	−	0.992672i	$-0.461442\pi$
$47$	1.65685			0.241677			0.120839	+	0.992672i	$0.461442\pi$
$48$		0			0
$49$	6.15800			0.879714
$50$		0			0
$51$	−0.613126	+	0.613126i	−0.0858548	+	0.0858548i
$52$		0			0
$53$	−2.39942	−	2.39942i	−0.329585	−	0.329585i	0.522843	−	0.852429i	$-0.324871\pi$
$53$	−2.39942	−	2.39942i	−0.329585	−	0.329585i	−0.852429	+	0.522843i	$0.824871\pi$
$54$		0			0
$55$		−	9.22625i		−	1.24407i
$56$		0			0
$57$			6.52395i			0.864118i
$58$		0			0
$59$	−8.99321	−	8.99321i	−1.17082	−	1.17082i	−0.982015	−	0.188801i	$-0.939540\pi$
$59$	−8.99321	−	8.99321i	−1.17082	−	1.17082i	−0.188801	−	0.982015i	$-0.560460\pi$
$60$		0			0
$61$	−0.0163888	+	0.0163888i	−0.00209837	+	0.00209837i	−0.708155	−	0.706057i	$-0.750472\pi$
$61$	−0.0163888	+	0.0163888i	−0.00209837	+	0.00209837i	0.706057	+	0.708155i	$0.250472\pi$
$62$		0			0
$63$	0.917608			0.115608
$64$		0			0
$65$	14.5104			1.79979
$66$		0			0
$67$	−3.76696	+	3.76696i	−0.460207	+	0.460207i	−0.898723	−	0.438516i	$-0.855504\pi$
$67$	−3.76696	+	3.76696i	−0.460207	+	0.460207i	0.438516	+	0.898723i	$0.355504\pi$
$68$		0			0
$69$	2.82843	+	2.82843i	0.340503	+	0.340503i
$70$		0			0
$71$			11.8216i			1.40297i	0.712684	+	0.701485i	$0.247479\pi$
$71$			11.8216i			1.40297i	−0.712684	+	0.701485i	$0.752521\pi$
$72$		0			0
$73$			15.2809i			1.78850i	0.447570	+	0.894249i	$0.352289\pi$
$73$			15.2809i			1.78850i	−0.447570	+	0.894249i	$0.647711\pi$
$74$		0			0
$75$	−0.871891	−	0.871891i	−0.100677	−	0.100677i
$76$		0			0
$77$	2.39782	−	2.39782i	0.273257	−	0.273257i
$78$		0			0
$79$	−8.40968			−0.946163			−0.473081	−	0.881019i	$-0.656858\pi$
$79$	−8.40968			−0.946163			−0.473081	+	0.881019i	$0.656858\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	1.38687	−	1.38687i	0.152229	−	0.152229i	−0.626884	−	0.779113i	$-0.715670\pi$
$83$	1.38687	−	1.38687i	0.152229	−	0.152229i	0.779113	+	0.626884i	$0.215670\pi$
$84$		0			0
$85$	1.53073	+	1.53073i	0.166031	+	0.166031i
$86$		0			0
$87$			7.72286i			0.827977i
$88$		0			0
$89$			2.79565i			0.296338i	0.988962	+	0.148169i	$0.0473380\pi$
$89$			2.79565i			0.296338i	−0.988962	+	0.148169i	$0.952662\pi$
$90$		0			0
$91$	3.77112	+	3.77112i	0.395321	+	0.395321i
$92$		0			0
$93$	1.51594	−	1.51594i	0.157195	−	0.157195i
$94$		0			0
$95$	16.2877			1.67108
$96$		0			0
$97$	−2.26582			−0.230059			−0.115029	−	0.993362i	$-0.536696\pi$
$97$	−2.26582			−0.230059			−0.115029	+	0.993362i	$0.536696\pi$
$98$		0			0
$99$	−2.61313	+	2.61313i	−0.262629	+	0.262629i
$100$		0			0
$101$	−7.06306	−	7.06306i	−0.702801	−	0.702801i	0.262210	−	0.965011i	$-0.415549\pi$
$101$	−7.06306	−	7.06306i	−0.702801	−	0.702801i	−0.965011	+	0.262210i	$0.915549\pi$
$102$		0			0
$103$		−	16.4180i		−	1.61771i	−0.588006	−	0.808857i	$-0.700087\pi$
$103$		−	16.4180i		−	1.61771i	0.588006	−	0.808857i	$-0.299913\pi$
$104$		0			0
$105$		−	2.29090i		−	0.223569i
$106$		0			0
$107$	8.66364	+	8.66364i	0.837546	+	0.837546i	0.988535	−	0.150989i	$-0.0482459\pi$
$107$	8.66364	+	8.66364i	0.837546	+	0.837546i	−0.150989	+	0.988535i	$0.548246\pi$
$108$		0			0
$109$	12.9382	−	12.9382i	1.23925	−	1.23925i	0.278943	−	0.960308i	$-0.410016\pi$
$109$	12.9382	−	12.9382i	1.23925	−	1.23925i	0.960308	−	0.278943i	$-0.0899841\pi$
$110$		0			0
$111$	2.47568			0.234981
$112$		0			0
$113$	−3.65685			−0.344008			−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685			−0.344008			−0.172004	+	0.985096i	$0.555024\pi$
$114$		0			0
$115$	7.06147	−	7.06147i	0.658485	−	0.658485i
$116$		0			0
$117$	−4.10973	−	4.10973i	−0.379945	−	0.379945i
$118$		0			0
$119$			0.795649i			0.0729371i
$120$		0			0
$121$			2.65685i			0.241532i
$122$		0			0
$123$	−6.77791	−	6.77791i	−0.611144	−	0.611144i
$124$		0			0
$125$	6.65007	−	6.65007i	0.594800	−	0.594800i
$126$		0			0
$127$	20.6382			1.83135			0.915673	−	0.401925i	$-0.131659\pi$
$127$	20.6382			1.83135			0.915673	+	0.401925i	$0.131659\pi$
$128$		0			0
$129$	−9.58541			−0.843949
$130$		0			0
$131$	−12.3842	+	12.3842i	−1.08202	+	1.08202i	−0.0856954	+	0.996321i	$0.527311\pi$
$131$	−12.3842	+	12.3842i	−1.08202	+	1.08202i	−0.996321	+	0.0856954i	$0.972689\pi$
$132$		0			0
$133$	4.23304	+	4.23304i	0.367051	+	0.367051i
$134$		0			0
$135$			2.49661i			0.214874i
$136$		0			0
$137$		−	8.25813i		−	0.705539i	−0.935710	−	0.352770i	$-0.885240\pi$
$137$		−	8.25813i		−	0.705539i	0.935710	−	0.352770i	$-0.114760\pi$
$138$		0			0
$139$	10.3978	+	10.3978i	0.881932	+	0.881932i	0.993731	−	0.111799i	$-0.0356612\pi$
$139$	10.3978	+	10.3978i	0.881932	+	0.881932i	−0.111799	+	0.993731i	$0.535661\pi$
$140$		0			0
$141$	−1.17157	+	1.17157i	−0.0986642	+	0.0986642i
$142$		0			0
$143$	−21.4785			−1.79612
$144$		0			0
$145$	19.2809			1.60119
$146$		0			0
$147$	−4.35436	+	4.35436i	−0.359142	+	0.359142i
$148$		0			0
$149$	11.4222	+	11.4222i	0.935745	+	0.935745i	0.998057	−	0.0623119i	$-0.0198474\pi$
$149$	11.4222	+	11.4222i	0.935745	+	0.935745i	−0.0623119	+	0.998057i	$0.519847\pi$
$150$		0			0
$151$		−	19.2691i		−	1.56810i	−0.620701	−	0.784048i	$-0.713152\pi$
$151$		−	19.2691i		−	1.56810i	0.620701	−	0.784048i	$-0.286848\pi$
$152$		0			0
$153$		−	0.867091i		−	0.0701002i
$154$		0			0
$155$	−3.78470	−	3.78470i	−0.303994	−	0.303994i
$156$		0			0
$157$	−0.578998	+	0.578998i	−0.0462091	+	0.0462091i	−0.729834	−	0.683625i	$-0.760403\pi$
$157$	−0.578998	+	0.578998i	−0.0462091	+	0.0462091i	0.683625	+	0.729834i	$0.260403\pi$
$158$		0			0
$159$	3.39329			0.269105
$160$		0			0
$161$	3.67043			0.289271
$162$		0			0
$163$	16.7225	−	16.7225i	1.30981	−	1.30981i	0.388253	−	0.921553i	$-0.373079\pi$
$163$	16.7225	−	16.7225i	1.30981	−	1.30981i	0.921553	−	0.388253i	$-0.126921\pi$
$164$		0			0
$165$	6.52395	+	6.52395i	0.507888	+	0.507888i
$166$		0			0
$167$			9.30358i			0.719933i	0.932965	+	0.359966i	$0.117212\pi$
$167$			9.30358i			0.719933i	−0.932965	+	0.359966i	$0.882788\pi$
$168$		0			0
$169$		−	20.7798i		−	1.59845i
$170$		0			0
$171$	−4.61313	−	4.61313i	−0.352775	−	0.352775i
$172$		0			0
$173$	−5.02440	+	5.02440i	−0.381998	+	0.381998i	−0.871821	−	0.489824i	$-0.837061\pi$
$173$	−5.02440	+	5.02440i	−0.381998	+	0.381998i	0.489824	+	0.871821i	$0.337061\pi$
$174$		0			0
$175$	−1.13145			−0.0855294
$176$		0			0
$177$	12.7183			0.955968
$178$		0			0
$179$	−1.88989	+	1.88989i	−0.141257	+	0.141257i	−0.774199	−	0.632942i	$-0.781847\pi$
$179$	−1.88989	+	1.88989i	−0.141257	+	0.141257i	0.632942	+	0.774199i	$0.281847\pi$
$180$		0			0
$181$	0.452877	+	0.452877i	0.0336621	+	0.0336621i	0.723737	−	0.690075i	$-0.242423\pi$
$181$	0.452877	+	0.452877i	0.0336621	+	0.0336621i	−0.690075	+	0.723737i	$0.742423\pi$
$182$		0			0
$183$		−	0.0231773i		−	0.00171332i
$184$		0			0
$185$		−	6.18080i		−	0.454421i
$186$		0			0
$187$	−2.26582	−	2.26582i	−0.165693	−	0.165693i
$188$		0			0
$189$	−0.648847	+	0.648847i	−0.0471967	+	0.0471967i
$190$		0			0
$191$	−14.6037			−1.05669			−0.528344	−	0.849031i	$-0.677187\pi$
$191$	−14.6037			−1.05669			−0.528344	+	0.849031i	$0.677187\pi$
$192$		0			0
$193$	−8.81485			−0.634507			−0.317253	−	0.948341i	$-0.602761\pi$
$193$	−8.81485			−0.634507			−0.317253	+	0.948341i	$0.602761\pi$
$194$		0			0
$195$	−10.2604	+	10.2604i	−0.734761	+	0.734761i
$196$		0			0
$197$	−2.64847	−	2.64847i	−0.188696	−	0.188696i	0.606436	−	0.795132i	$-0.292599\pi$
$197$	−2.64847	−	2.64847i	−0.188696	−	0.188696i	−0.795132	+	0.606436i	$0.792599\pi$
$198$		0			0
$199$			14.8267i			1.05104i	0.850782	+	0.525519i	$0.176129\pi$
$199$			14.8267i			1.05104i	−0.850782	+	0.525519i	$0.823871\pi$
$200$		0			0
$201$		−	5.32729i		−	0.375758i
$202$		0			0
$203$	5.01095	+	5.01095i	0.351700	+	0.351700i
$204$		0			0
$205$	−16.9218	+	16.9218i	−1.18187	+	1.18187i
$206$		0			0
$207$	−4.00000			−0.278019
$208$		0			0
$209$	−24.1094			−1.66768
$210$		0			0
$211$	8.05468	−	8.05468i	0.554507	−	0.554507i	−0.373231	−	0.927738i	$-0.621750\pi$
$211$	8.05468	−	8.05468i	0.554507	−	0.554507i	0.927738	+	0.373231i	$0.121750\pi$
$212$		0			0
$213$	−8.35916	−	8.35916i	−0.572760	−	0.572760i
$214$		0			0
$215$			23.9310i			1.63208i
$216$		0			0
$217$		−	1.96722i		−	0.133544i
$218$		0			0
$219$	−10.8052	−	10.8052i	−0.730151	−	0.730151i
$220$		0			0
$221$	3.56351	−	3.56351i	0.239708	−	0.239708i
$222$		0			0
$223$	4.78110			0.320166			0.160083	−	0.987104i	$-0.448824\pi$
$223$	4.78110			0.320166			0.160083	+	0.987104i	$0.448824\pi$
$224$		0			0
$225$	1.23304			0.0822027
$226$		0			0
$227$	−2.71416	+	2.71416i	−0.180145	+	0.180145i	−0.791419	−	0.611274i	$-0.790657\pi$
$227$	−2.71416	+	2.71416i	−0.180145	+	0.180145i	0.611274	+	0.791419i	$0.290657\pi$
$228$		0			0
$229$	−0.951736	−	0.951736i	−0.0628925	−	0.0628925i	0.674961	−	0.737853i	$-0.264160\pi$
$229$	−0.951736	−	0.951736i	−0.0628925	−	0.0628925i	−0.737853	+	0.674961i	$0.764160\pi$
$230$		0			0
$231$			3.39104i			0.223114i
$232$		0			0
$233$			17.6433i			1.15585i	0.816090	+	0.577925i	$0.196137\pi$
$233$			17.6433i			1.15585i	−0.816090	+	0.577925i	$0.803863\pi$
$234$		0			0
$235$	2.92496	+	2.92496i	0.190803	+	0.190803i
$236$		0			0
$237$	5.94654	−	5.94654i	0.386269	−	0.386269i
$238$		0			0
$239$	9.61500			0.621943			0.310971	−	0.950419i	$-0.399346\pi$
$239$	9.61500			0.621943			0.310971	+	0.950419i	$0.399346\pi$
$240$		0			0
$241$	−21.3583			−1.37581			−0.687903	−	0.725802i	$-0.741469\pi$
$241$	−21.3583			−1.37581			−0.687903	+	0.725802i	$0.741469\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$	10.8711	+	10.8711i	0.694531	+	0.694531i
$246$		0			0
$247$		−	37.9174i		−	2.41263i
$248$		0			0
$249$			1.96134i			0.124295i
$250$		0			0
$251$	3.04373	+	3.04373i	0.192118	+	0.192118i	0.796611	−	0.604492i	$-0.206624\pi$
$251$	3.04373	+	3.04373i	0.192118	+	0.192118i	−0.604492	+	0.796611i	$0.706624\pi$
$252$		0			0
$253$	−10.4525	+	10.4525i	−0.657143	+	0.657143i
$254$		0			0
$255$	−2.16478			−0.135564
$256$		0			0
$257$	−4.67271			−0.291476			−0.145738	−	0.989323i	$-0.546556\pi$
$257$	−4.67271			−0.291476			−0.145738	+	0.989323i	$0.546556\pi$
$258$		0			0
$259$	1.60634	−	1.60634i	0.0998130	−	0.0998130i
$260$		0			0
$261$	−5.46088	−	5.46088i	−0.338020	−	0.338020i
$262$		0			0
$263$			15.4502i			0.952701i	0.879255	+	0.476351i	$0.158041\pi$
$263$			15.4502i			0.952701i	−0.879255	+	0.476351i	$0.841959\pi$
$264$		0			0
$265$		−	8.47170i		−	0.520413i
$266$		0			0
$267$	−1.97682	−	1.97682i	−0.120980	−	0.120980i
$268$		0			0
$269$	−22.0942	+	22.0942i	−1.34711	+	1.34711i	−0.458318	+	0.888788i	$0.651548\pi$
$269$	−22.0942	+	22.0942i	−1.34711	+	1.34711i	−0.888788	+	0.458318i	$0.848452\pi$
$270$		0			0
$271$	−9.88817			−0.600664			−0.300332	−	0.953835i	$-0.597097\pi$
$271$	−9.88817			−0.600664			−0.300332	+	0.953835i	$0.597097\pi$
$272$		0			0
$273$	−5.33317			−0.322778
$274$		0			0
$275$	3.22209	−	3.22209i	0.194299	−	0.194299i
$276$		0			0
$277$	−16.2327	−	16.2327i	−0.975326	−	0.975326i	0.0243764	−	0.999703i	$-0.492240\pi$
$277$	−16.2327	−	16.2327i	−0.975326	−	0.975326i	−0.999703	+	0.0243764i	$0.992240\pi$
$278$		0			0
$279$			2.14386i			0.128350i
$280$		0			0
$281$		−	14.5754i		−	0.869498i	−0.900552	−	0.434749i	$-0.856837\pi$
$281$		−	14.5754i		−	0.869498i	0.900552	−	0.434749i	$-0.143163\pi$
$282$		0			0
$283$	3.33636	+	3.33636i	0.198326	+	0.198326i	0.799282	−	0.600956i	$-0.205213\pi$
$283$	3.33636	+	3.33636i	0.198326	+	0.198326i	−0.600956	+	0.799282i	$0.705213\pi$
$284$		0			0
$285$	−11.5172	+	11.5172i	−0.682217	+	0.682217i
$286$		0			0
$287$	−8.79565			−0.519191
$288$		0			0
$289$	−16.2482			−0.955774
$290$		0			0
$291$	1.60218	−	1.60218i	0.0939212	−	0.0939212i
$292$		0			0
$293$	−16.9143	−	16.9143i	−0.988143	−	0.988143i	0.0117871	−	0.999931i	$-0.496248\pi$
$293$	−16.9143	−	16.9143i	−0.988143	−	0.988143i	−0.999931	+	0.0117871i	$0.996248\pi$
$294$		0			0
$295$		−	31.7526i		−	1.84871i
$296$		0			0
$297$		−	3.69552i		−	0.214436i
$298$		0			0
$299$	−16.4389	−	16.4389i	−0.950688	−	0.950688i
$300$		0			0
$301$	−6.21946	+	6.21946i	−0.358484	+	0.358484i
$302$		0			0
$303$	9.98868			0.573834
$304$		0			0
$305$	−0.0578646			−0.00331332
$306$		0			0
$307$	14.6855	−	14.6855i	0.838148	−	0.838148i	−0.150467	−	0.988615i	$-0.548078\pi$
$307$	14.6855	−	14.6855i	0.838148	−	0.838148i	0.988615	+	0.150467i	$0.0480777\pi$
$308$		0			0
$309$	11.6093	+	11.6093i	0.660429	+	0.660429i
$310$		0			0
$311$		−	11.7798i		−	0.667971i	−0.942578	−	0.333985i	$-0.891606\pi$
$311$		−	11.7798i		−	0.667971i	0.942578	−	0.333985i	$-0.108394\pi$
$312$		0			0
$313$			11.1580i			0.630687i	0.948978	+	0.315344i	$0.102120\pi$
$313$			11.1580i			0.630687i	−0.948978	+	0.315344i	$0.897880\pi$
$314$		0			0
$315$	1.61991	+	1.61991i	0.0912718	+	0.0912718i
$316$		0			0
$317$	−11.2529	+	11.2529i	−0.632028	+	0.632028i	−0.948576	−	0.316549i	$-0.897476\pi$
$317$	−11.2529	+	11.2529i	−0.632028	+	0.632028i	0.316549	+	0.948576i	$0.397476\pi$
$318$		0			0
$319$	−28.5400			−1.59793
$320$		0			0
$321$	−12.2522			−0.683853
$322$		0			0
$323$	4.00000	−	4.00000i	0.222566	−	0.222566i
$324$		0			0
$325$	5.06746	+	5.06746i	0.281092	+	0.281092i
$326$		0			0
$327$			18.2973i			1.01184i
$328$		0			0
$329$			1.52034i			0.0838192i
$330$		0			0
$331$	1.93174	+	1.93174i	0.106178	+	0.106178i	0.758200	−	0.652022i	$-0.226079\pi$
$331$	1.93174	+	1.93174i	0.106178	+	0.106178i	−0.652022	+	0.758200i	$0.726079\pi$
$332$		0			0
$333$	−1.75057	+	1.75057i	−0.0959307	+	0.0959307i
$334$		0			0
$335$	−13.3001			−0.726664
$336$		0			0
$337$	1.49657			0.0815236			0.0407618	−	0.999169i	$-0.487022\pi$
$337$	1.49657			0.0815236			0.0407618	+	0.999169i	$0.487022\pi$
$338$		0			0
$339$	2.58579	−	2.58579i	0.140441	−	0.140441i
$340$		0			0
$341$	5.60218	+	5.60218i	0.303375	+	0.303375i
$342$		0			0
$343$			12.0739i			0.651928i
$344$		0			0
$345$			9.98642i			0.537651i
$346$		0			0
$347$	2.28356	+	2.28356i	0.122588	+	0.122588i	0.765739	−	0.643151i	$-0.222373\pi$
$347$	2.28356	+	2.28356i	0.122588	+	0.122588i	−0.643151	+	0.765739i	$0.722373\pi$
$348$		0			0
$349$	−5.31090	+	5.31090i	−0.284286	+	0.284286i	−0.834816	−	0.550530i	$-0.814426\pi$
$349$	−5.31090	+	5.31090i	−0.284286	+	0.284286i	0.550530	+	0.834816i	$0.314426\pi$
$350$		0			0
$351$	5.81204			0.310224
$352$		0			0
$353$	2.17491			0.115759			0.0578795	−	0.998324i	$-0.481566\pi$
$353$	2.17491			0.115759			0.0578795	+	0.998324i	$0.481566\pi$
$354$		0			0
$355$	−20.8695	+	20.8695i	−1.10764	+	1.10764i
$356$		0			0
$357$	−0.562609	−	0.562609i	−0.0297764	−	0.0297764i
$358$		0			0
$359$		−	12.6828i		−	0.669375i	−0.942329	−	0.334687i	$-0.891369\pi$
$359$		−	12.6828i		−	0.669375i	0.942329	−	0.334687i	$-0.108631\pi$
$360$		0			0
$361$		−	23.5619i		−	1.24010i
$362$		0			0
$363$	−1.87868	−	1.87868i	−0.0986051	−	0.0986051i
$364$		0			0
$365$	−26.9764	+	26.9764i	−1.41201	+	1.41201i
$366$		0			0
$367$	21.4576			1.12008			0.560038	−	0.828467i	$-0.310787\pi$
$367$	21.4576			1.12008			0.560038	+	0.828467i	$0.310787\pi$
$368$		0			0
$369$	9.58541			0.498997
$370$		0			0
$371$	2.20172	−	2.20172i	0.114308	−	0.114308i
$372$		0			0
$373$	0.359535	+	0.359535i	0.0186160	+	0.0186160i	0.716354	−	0.697738i	$-0.245810\pi$
$373$	0.359535	+	0.359535i	0.0186160	+	0.0186160i	−0.697738	+	0.716354i	$0.745810\pi$
$374$		0			0
$375$			9.40461i			0.485652i
$376$		0			0
$377$		−	44.8855i		−	2.31172i
$378$		0			0
$379$	14.0916	+	14.0916i	0.723838	+	0.723838i	0.969385	−	0.245547i	$-0.0789675\pi$
$379$	14.0916	+	14.0916i	0.723838	+	0.723838i	−0.245547	+	0.969385i	$0.578968\pi$
$380$		0			0
$381$	−14.5934	+	14.5934i	−0.747644	+	0.747644i
$382$		0			0
$383$	29.3858			1.50154			0.750772	−	0.660562i	$-0.229682\pi$
$383$	29.3858			1.50154			0.750772	+	0.660562i	$0.229682\pi$
$384$		0			0
$385$	8.46608			0.431471
$386$		0			0
$387$	6.77791	−	6.77791i	0.344541	−	0.344541i
$388$		0			0
$389$	0.709786	+	0.709786i	0.0359876	+	0.0359876i	0.724872	−	0.688884i	$-0.241899\pi$
$389$	0.709786	+	0.709786i	0.0359876	+	0.0359876i	−0.688884	+	0.724872i	$0.741899\pi$
$390$		0			0
$391$		−	3.46836i		−	0.175403i
$392$		0			0
$393$		−	17.5140i		−	0.883463i
$394$		0			0
$395$	−14.8462	−	14.8462i	−0.746992	−	0.746992i
$396$		0			0
$397$	4.96775	−	4.96775i	0.249324	−	0.249324i	−0.571369	−	0.820693i	$-0.693587\pi$
$397$	4.96775	−	4.96775i	0.249324	−	0.249324i	0.820693	+	0.571369i	$0.193587\pi$
$398$		0			0
$399$	−5.98642			−0.299696
$400$		0			0
$401$	−4.18080			−0.208779			−0.104390	−	0.994536i	$-0.533289\pi$
$401$	−4.18080			−0.208779			−0.104390	+	0.994536i	$0.533289\pi$
$402$		0			0
$403$	−8.81069	+	8.81069i	−0.438892	+	0.438892i
$404$		0			0
$405$	−1.76537	−	1.76537i	−0.0877218	−	0.0877218i
$406$		0			0
$407$			9.14892i			0.453495i
$408$		0			0
$409$			16.5818i			0.819918i	0.912104	+	0.409959i	$0.134457\pi$
$409$			16.5818i			0.819918i	−0.912104	+	0.409959i	$0.865543\pi$
$410$		0			0
$411$	5.83938	+	5.83938i	0.288035	+	0.288035i
$412$		0			0
$413$	8.25224	−	8.25224i	0.406066	−	0.406066i
$414$		0			0
$415$	4.89668			0.240369
$416$		0			0
$417$	−14.7047			−0.720094
$418$		0			0
$419$	−15.7383	+	15.7383i	−0.768868	+	0.768868i	−0.977907	−	0.209039i	$-0.932966\pi$
$419$	−15.7383	+	15.7383i	−0.768868	+	0.768868i	0.209039	+	0.977907i	$0.432966\pi$
$420$		0			0
$421$	20.0962	+	20.0962i	0.979427	+	0.979427i	0.999793	−	0.0203659i	$-0.00648313\pi$
$421$	20.0962	+	20.0962i	0.979427	+	0.979427i	−0.0203659	+	0.999793i	$0.506483\pi$
$422$		0			0
$423$		−	1.65685i		−	0.0805590i
$424$		0			0
$425$			1.06916i			0.0518618i
$426$		0			0
$427$	−0.0150385	−	0.0150385i	−0.000727765	−	0.000727765i
$428$		0			0
$429$	15.1876	−	15.1876i	0.733264	−	0.733264i
$430$		0			0
$431$	36.6138			1.76363			0.881813	−	0.471599i	$-0.156323\pi$
$431$	36.6138			1.76363			0.881813	+	0.471599i	$0.156323\pi$
$432$		0			0
$433$	−27.7526			−1.33371			−0.666853	−	0.745189i	$-0.732359\pi$
$433$	−27.7526			−1.33371			−0.666853	+	0.745189i	$0.732359\pi$
$434$		0			0
$435$	−13.6337	+	13.6337i	−0.653685	+	0.653685i
$436$		0			0
$437$	−18.4525	−	18.4525i	−0.882703	−	0.882703i
$438$		0			0
$439$			21.1461i			1.00925i	0.863339	+	0.504625i	$0.168369\pi$
$439$			21.1461i			1.00925i	−0.863339	+	0.504625i	$0.831631\pi$
$440$		0			0
$441$		−	6.15800i		−	0.293238i
$442$		0			0
$443$	−5.16666	−	5.16666i	−0.245476	−	0.245476i	0.573635	−	0.819111i	$-0.305533\pi$
$443$	−5.16666	−	5.16666i	−0.245476	−	0.245476i	−0.819111	+	0.573635i	$0.805533\pi$
$444$		0			0
$445$	−4.93535	+	4.93535i	−0.233958	+	0.233958i
$446$		0			0
$447$	−16.1535			−0.764032
$448$		0			0
$449$	−17.6356			−0.832275			−0.416137	−	0.909302i	$-0.636616\pi$
$449$	−17.6356			−0.832275			−0.416137	+	0.909302i	$0.636616\pi$
$450$		0			0
$451$	25.0479	−	25.0479i	1.17946	−	1.17946i
$452$		0			0
$453$	13.6253	+	13.6253i	0.640172	+	0.640172i
$454$		0			0
$455$			13.3148i			0.624209i
$456$		0			0
$457$		−	17.7070i		−	0.828300i	−0.910209	−	0.414150i	$-0.864079\pi$
$457$		−	17.7070i		−	0.828300i	0.910209	−	0.414150i	$-0.135921\pi$
$458$		0			0
$459$	0.613126	+	0.613126i	0.0286183	+	0.0286183i
$460$		0			0
$461$	26.1496	−	26.1496i	1.21791	−	1.21791i	0.249546	−	0.968363i	$-0.419719\pi$
$461$	26.1496	−	26.1496i	1.21791	−	1.21791i	0.968363	−	0.249546i	$-0.0802814\pi$
$462$		0			0
$463$	−10.9632			−0.509504			−0.254752	−	0.967006i	$-0.581994\pi$
$463$	−10.9632			−0.509504			−0.254752	+	0.967006i	$0.581994\pi$
$464$		0			0
$465$	5.35237			0.248210
$466$		0			0
$467$	−8.95627	+	8.95627i	−0.414447	+	0.414447i	−0.883284	−	0.468838i	$-0.844673\pi$
$467$	−8.95627	+	8.95627i	−0.414447	+	0.414447i	0.468838	+	0.883284i	$0.344673\pi$
$468$		0			0
$469$	−3.45659	−	3.45659i	−0.159611	−	0.159611i
$470$		0			0
$471$		−	0.818827i		−	0.0377295i
$472$		0			0
$473$		−	35.4231i		−	1.62875i
$474$		0			0
$475$	5.68817	+	5.68817i	0.260991	+	0.260991i
$476$		0			0
$477$	−2.39942	+	2.39942i	−0.109862	+	0.109862i
$478$		0			0
$479$	−13.3036			−0.607856			−0.303928	−	0.952695i	$-0.598298\pi$
$479$	−13.3036			−0.607856			−0.303928	+	0.952695i	$0.598298\pi$
$480$		0			0
$481$	−14.3888			−0.656071
$482$		0			0
$483$	−2.59539	+	2.59539i	−0.118094	+	0.118094i
$484$		0			0
$485$	−4.00000	−	4.00000i	−0.181631	−	0.181631i
$486$		0			0
$487$		−	5.44924i		−	0.246929i	−0.992349	−	0.123464i	$-0.960600\pi$
$487$		−	5.44924i		−	0.246929i	0.992349	−	0.123464i	$-0.0394005\pi$
$488$		0			0
$489$			23.6492i			1.06945i
$490$		0			0
$491$	15.4039	+	15.4039i	0.695167	+	0.695167i	0.963364	−	0.268197i	$-0.0864279\pi$
$491$	15.4039	+	15.4039i	0.695167	+	0.695167i	−0.268197	+	0.963364i	$0.586428\pi$
$492$		0			0
$493$	4.73508	−	4.73508i	0.213257	−	0.213257i
$494$		0			0
$495$	−9.22625			−0.414689
$496$		0			0
$497$	−10.8476			−0.486583
$498$		0			0
$499$	−14.2549	+	14.2549i	−0.638139	+	0.638139i	−0.950096	−	0.311957i	$-0.899015\pi$
$499$	−14.2549	+	14.2549i	−0.638139	+	0.638139i	0.311957	+	0.950096i	$0.399015\pi$
$500$		0			0
$501$	−6.57862	−	6.57862i	−0.293911	−	0.293911i
$502$		0			0
$503$		−	33.5879i		−	1.49761i	−0.662791	−	0.748804i	$-0.730628\pi$
$503$		−	33.5879i		−	1.49761i	0.662791	−	0.748804i	$-0.269372\pi$
$504$		0			0
$505$		−	24.9378i		−	1.10972i
$506$		0			0
$507$	14.6935	+	14.6935i	0.652563	+	0.652563i
$508$		0			0
$509$	7.91657	−	7.91657i	0.350896	−	0.350896i	−0.509547	−	0.860443i	$-0.670187\pi$
$509$	7.91657	−	7.91657i	0.350896	−	0.350896i	0.860443	+	0.509547i	$0.170187\pi$
$510$		0			0
$511$	−14.0219			−0.620292
$512$		0			0
$513$	6.52395			0.288039
$514$		0			0
$515$	28.9838	−	28.9838i	1.27718	−	1.27718i
$516$		0			0
$517$	−4.32957	−	4.32957i	−0.190414	−	0.190414i
$518$		0			0
$519$		−	7.10557i		−	0.311900i
$520$		0			0
$521$			35.2151i			1.54280i	0.636349	+	0.771401i	$0.280444\pi$
$521$			35.2151i			1.54280i	−0.636349	+	0.771401i	$0.719556\pi$
$522$		0			0
$523$	9.15309	+	9.15309i	0.400237	+	0.400237i	0.878316	−	0.478080i	$-0.158667\pi$
$523$	9.15309	+	9.15309i	0.400237	+	0.400237i	−0.478080	+	0.878316i	$0.658667\pi$
$524$		0			0
$525$	0.800054	−	0.800054i	0.0349172	−	0.0349172i
$526$		0			0
$527$	−1.85892			−0.0809759
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	−8.99321	+	8.99321i	−0.390272	+	0.390272i
$532$		0			0
$533$	39.3935	+	39.3935i	1.70632	+	1.70632i
$534$		0			0
$535$			30.5890i			1.32248i
$536$		0			0
$537$		−	2.67271i		−	0.115336i
$538$		0			0
$539$	−16.0916	−	16.0916i	−0.693115	−	0.693115i
$540$		0			0
$541$	−12.6278	+	12.6278i	−0.542911	+	0.542911i	−0.924381	−	0.381470i	$-0.875418\pi$
$541$	−12.6278	+	12.6278i	−0.542911	+	0.542911i	0.381470	+	0.924381i	$0.375418\pi$
$542$		0			0
$543$	−0.640465			−0.0274850
$544$		0			0
$545$	45.6812			1.95677
$546$		0			0
$547$	10.5494	−	10.5494i	0.451059	−	0.451059i	−0.444647	−	0.895706i	$-0.646671\pi$
$547$	10.5494	−	10.5494i	0.451059	−	0.451059i	0.895706	+	0.444647i	$0.146671\pi$
$548$		0			0
$549$	0.0163888	+	0.0163888i	0.000699458	+	0.000699458i
$550$		0			0
$551$		−	50.3835i		−	2.14641i
$552$		0			0
$553$		−	7.71679i		−	0.328151i
$554$		0			0
$555$	4.37049	+	4.37049i	0.185517	+	0.185517i
$556$		0			0
$557$	−17.2993	+	17.2993i	−0.732994	+	0.732994i	−0.971212	−	0.238218i	$-0.923437\pi$
$557$	−17.2993	+	17.2993i	−0.732994	+	0.732994i	0.238218	+	0.971212i	$0.423437\pi$
$558$		0			0
$559$	55.7108			2.35632
$560$		0			0
$561$	3.20435			0.135288
$562$		0			0
$563$	−28.5995	+	28.5995i	−1.20533	+	1.20533i	−0.232803	+	0.972524i	$0.574790\pi$
$563$	−28.5995	+	28.5995i	−1.20533	+	1.20533i	−0.972524	+	0.232803i	$0.925210\pi$
$564$		0			0
$565$	−6.45569	−	6.45569i	−0.271593	−	0.271593i
$566$		0			0
$567$		−	0.917608i		−	0.0385359i
$568$		0			0
$569$			42.1017i			1.76499i	0.470318	+	0.882497i	$0.344139\pi$
$569$			42.1017i			1.76499i	−0.470318	+	0.882497i	$0.655861\pi$
$570$		0			0
$571$	0.883854	+	0.883854i	0.0369881	+	0.0369881i	0.725359	−	0.688371i	$-0.241674\pi$
$571$	0.883854	+	0.883854i	0.0369881	+	0.0369881i	−0.688371	+	0.725359i	$0.741674\pi$
$572$		0			0
$573$	10.3264	−	10.3264i	0.431391	−	0.431391i
$574$		0			0
$575$	4.93216			0.205685
$576$		0			0
$577$	29.3767			1.22297			0.611484	−	0.791257i	$-0.290573\pi$
$577$	29.3767			1.22297			0.611484	+	0.791257i	$0.290573\pi$
$578$		0			0
$579$	6.23304	−	6.23304i	0.259036	−	0.259036i
$580$		0			0
$581$	1.27261	+	1.27261i	0.0527966	+	0.0527966i
$582$		0			0
$583$			12.5400i			0.519352i
$584$		0			0
$585$		−	14.5104i		−	0.599930i
$586$		0			0
$587$	−12.5208	−	12.5208i	−0.516787	−	0.516787i	0.399811	−	0.916598i	$-0.369076\pi$
$587$	−12.5208	−	12.5208i	−0.516787	−	0.516787i	−0.916598	+	0.399811i	$0.869076\pi$
$588$		0			0
$589$	−9.88989	+	9.88989i	−0.407506	+	0.407506i
$590$		0			0
$591$	3.74551			0.154070
$592$		0			0
$593$	−17.1207			−0.703061			−0.351530	−	0.936176i	$-0.614339\pi$
$593$	−17.1207			−0.703061			−0.351530	+	0.936176i	$0.614339\pi$
$594$		0			0
$595$	−1.40461	+	1.40461i	−0.0575835	+	0.0575835i
$596$		0			0
$597$	−10.4841	−	10.4841i	−0.429084	−	0.429084i
$598$		0			0
$599$			25.1908i			1.02927i	0.857410	+	0.514634i	$0.172072\pi$
$599$			25.1908i			1.02927i	−0.857410	+	0.514634i	$0.827928\pi$
$600$		0			0
$601$			22.5946i			0.921655i	0.887490	+	0.460827i	$0.152447\pi$
$601$			22.5946i			0.921655i	−0.887490	+	0.460827i	$0.847553\pi$
$602$		0			0
$603$	3.76696	+	3.76696i	0.153402	+	0.153402i
$604$		0			0
$605$	−4.69032	+	4.69032i	−0.190689	+	0.190689i
$606$		0			0
$607$	8.88408			0.360594			0.180297	−	0.983612i	$-0.442294\pi$
$607$	8.88408			0.360594			0.180297	+	0.983612i	$0.442294\pi$
$608$		0			0
$609$	−7.08655			−0.287162
$610$		0			0
$611$	6.80923	−	6.80923i	0.275472	−	0.275472i
$612$		0			0
$613$	11.3109	+	11.3109i	0.456843	+	0.456843i	0.897618	−	0.440775i	$-0.145296\pi$
$613$	11.3109	+	11.3109i	0.456843	+	0.456843i	−0.440775	+	0.897618i	$0.645296\pi$
$614$		0			0
$615$		−	23.9310i		−	0.964991i
$616$		0			0
$617$			2.99772i			0.120684i	0.998178	+	0.0603418i	$0.0192191\pi$
$617$			2.99772i			0.120684i	−0.998178	+	0.0603418i	$0.980781\pi$
$618$		0			0
$619$	−10.6264	−	10.6264i	−0.427109	−	0.427109i	0.460533	−	0.887643i	$-0.347658\pi$
$619$	−10.6264	−	10.6264i	−0.427109	−	0.427109i	−0.887643	+	0.460533i	$0.847658\pi$
$620$		0			0
$621$	2.82843	−	2.82843i	0.113501	−	0.113501i
$622$		0			0
$623$	−2.56531			−0.102777
$624$		0			0
$625$	29.6448			1.18579
$626$		0			0
$627$	17.0479	−	17.0479i	0.680827	−	0.680827i
$628$		0			0
$629$	−1.51790	−	1.51790i	−0.0605228	−	0.0605228i
$630$		0			0
$631$			19.3701i			0.771112i	0.922684	+	0.385556i	$0.125990\pi$
$631$			19.3701i			0.771112i	−0.922684	+	0.385556i	$0.874010\pi$
$632$		0			0
$633$			11.3910i			0.452753i
$634$		0			0
$635$	36.4340	+	36.4340i	1.44584	+	1.44584i
$636$		0			0
$637$	25.3077	−	25.3077i	1.00273	−	1.00273i
$638$		0			0
$639$	11.8216			0.467657
$640$		0			0
$641$	13.4489			0.531200			0.265600	−	0.964083i	$-0.414430\pi$
$641$	13.4489			0.531200			0.265600	+	0.964083i	$0.414430\pi$
$642$		0			0
$643$	1.73002	−	1.73002i	0.0682253	−	0.0682253i	−0.672171	−	0.740396i	$-0.734638\pi$
$643$	1.73002	−	1.73002i	0.0682253	−	0.0682253i	0.740396	+	0.672171i	$0.234638\pi$
$644$		0			0
$645$	−16.9218	−	16.9218i	−0.666294	−	0.666294i
$646$		0			0
$647$		−	16.1275i		−	0.634038i	−0.948419	−	0.317019i	$-0.897318\pi$
$647$		−	16.1275i		−	0.634038i	0.948419	−	0.317019i	$-0.102682\pi$
$648$		0			0
$649$			47.0008i			1.84494i
$650$		0			0
$651$	1.39104	+	1.39104i	0.0545190	+	0.0545190i
$652$		0			0
$653$	20.2405	−	20.2405i	0.792073	−	0.792073i	−0.189758	−	0.981831i	$-0.560770\pi$
$653$	20.2405	−	20.2405i	0.792073	−	0.792073i	0.981831	+	0.189758i	$0.0607704\pi$
$654$		0			0
$655$	−43.7255			−1.70850
$656$		0			0
$657$	15.2809			0.596166
$658$		0			0
$659$	−3.36006	+	3.36006i	−0.130889	+	0.130889i	−0.769516	−	0.638627i	$-0.779503\pi$
$659$	−3.36006	+	3.36006i	−0.130889	+	0.130889i	0.638627	+	0.769516i	$0.279503\pi$
$660$		0			0
$661$	−15.4210	−	15.4210i	−0.599807	−	0.599807i	0.340454	−	0.940261i	$-0.389419\pi$
$661$	−15.4210	−	15.4210i	−0.599807	−	0.599807i	−0.940261	+	0.340454i	$0.889419\pi$
$662$		0			0
$663$			5.03957i			0.195721i
$664$		0			0
$665$			14.9457i			0.579571i
$666$		0			0
$667$	−21.8435	−	21.8435i	−0.845785	−	0.845785i
$668$		0			0
$669$	−3.38075	+	3.38075i	−0.130707	+	0.130707i
$670$		0			0
$671$	0.0856521			0.00330656
$672$		0			0
$673$	6.34315			0.244510			0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315			0.244510			0.122255	+	0.992499i	$0.460987\pi$
$674$		0			0
$675$	−0.871891	+	0.871891i	−0.0335591	+	0.0335591i
$676$		0			0
$677$	−32.2680	−	32.2680i	−1.24016	−	1.24016i	−0.959933	−	0.280228i	$-0.909590\pi$
$677$	−32.2680	−	32.2680i	−1.24016	−	1.24016i	−0.280228	−	0.959933i	$-0.590410\pi$
$678$		0			0
$679$		−	2.07913i		−	0.0797898i
$680$		0			0
$681$		−	3.83840i		−	0.147088i
$682$		0			0
$683$	−26.8235	−	26.8235i	−1.02637	−	1.02637i	−0.999643	−	0.0267307i	$-0.991490\pi$
$683$	−26.8235	−	26.8235i	−1.02637	−	1.02637i	−0.0267307	−	0.999643i	$-0.508510\pi$
$684$		0			0
$685$	14.5786	−	14.5786i	0.557021	−	0.557021i
$686$		0			0
$687$	1.34596			0.0513515
$688$		0			0
$689$	−19.7219			−0.751345
$690$		0			0
$691$	−23.5973	+	23.5973i	−0.897682	+	0.897682i	−0.995231	−	0.0975485i	$-0.968900\pi$
$691$	−23.5973	+	23.5973i	−0.897682	+	0.897682i	0.0975485	+	0.995231i	$0.468900\pi$
$692$		0			0
$693$	−2.39782	−	2.39782i	−0.0910858	−	0.0910858i
$694$		0			0
$695$			36.7120i			1.39256i
$696$		0			0
$697$			8.31143i			0.314818i
$698$		0			0
$699$	−12.4757	−	12.4757i	−0.471874	−	0.471874i
$700$		0			0
$701$	16.5802	−	16.5802i	0.626226	−	0.626226i	−0.320890	−	0.947116i	$-0.603982\pi$
$701$	16.5802	−	16.5802i	0.626226	−	0.626226i	0.947116	+	0.320890i	$0.103982\pi$
$702$		0			0
$703$	−16.1512			−0.609154
$704$		0			0
$705$	−4.13651			−0.155790
$706$		0			0
$707$	6.48112	−	6.48112i	0.243748	−	0.243748i
$708$		0			0
$709$	−8.59576	−	8.59576i	−0.322821	−	0.322821i	0.527028	−	0.849848i	$-0.323306\pi$
$709$	−8.59576	−	8.59576i	−0.322821	−	0.322821i	−0.849848	+	0.527028i	$0.823306\pi$
$710$		0			0
$711$			8.40968i			0.315388i
$712$		0			0
$713$			8.57544i			0.321153i
$714$		0			0
$715$	−37.9174	−	37.9174i	−1.41803	−	1.41803i
$716$		0			0
$717$	−6.79884	+	6.79884i	−0.253907	+	0.253907i
$718$		0			0
$719$	4.06900			0.151748			0.0758741	−	0.997117i	$-0.475825\pi$
$719$	4.06900			0.151748			0.0758741	+	0.997117i	$0.475825\pi$
$720$		0			0
$721$	15.0653			0.561061
$722$		0			0
$723$	15.1026	−	15.1026i	0.561671	−	0.561671i
$724$		0			0
$725$	6.73349	+	6.73349i	0.250076	+	0.250076i
$726$		0			0
$727$			17.3211i			0.642402i	0.947011	+	0.321201i	$0.104087\pi$
$727$			17.3211i			0.642402i	−0.947011	+	0.321201i	$0.895913\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	5.87707	+	5.87707i	0.217371	+	0.217371i
$732$		0			0
$733$	15.1576	−	15.1576i	0.559860	−	0.559860i	−0.369408	−	0.929267i	$-0.620439\pi$
$733$	15.1576	−	15.1576i	0.559860	−	0.559860i	0.929267	+	0.369408i	$0.120439\pi$
$734$		0			0
$735$	−15.3741			−0.567082
$736$		0			0
$737$	19.6871			0.725183
$738$		0			0
$739$	−22.1003	+	22.1003i	−0.812972	+	0.812972i	−0.985078	−	0.172106i	$-0.944943\pi$
$739$	−22.1003	+	22.1003i	−0.812972	+	0.812972i	0.172106	+	0.985078i	$0.444943\pi$
$740$		0			0
$741$	26.8117	+	26.8117i	0.984951	+	0.984951i
$742$		0			0
$743$		−	17.7244i		−	0.650244i	−0.945672	−	0.325122i	$-0.894595\pi$
$743$		−	17.7244i		−	0.650244i	0.945672	−	0.325122i	$-0.105405\pi$
$744$		0			0
$745$			40.3288i			1.47753i
$746$		0			0
$747$	−1.38687	−	1.38687i	−0.0507431	−	0.0507431i
$748$		0			0
$749$	−7.94983	+	7.94983i	−0.290480	+	0.290480i
$750$		0			0
$751$	−7.38826			−0.269601			−0.134801	−	0.990873i	$-0.543039\pi$
$751$	−7.38826			−0.269601			−0.134801	+	0.990873i	$0.543039\pi$
$752$		0			0
$753$	−4.30448			−0.156864
$754$		0			0
$755$	34.0170	−	34.0170i	1.23801	−	1.23801i
$756$		0			0
$757$	−9.39066	−	9.39066i	−0.341309	−	0.341309i	0.515550	−	0.856859i	$-0.327588\pi$
$757$	−9.39066	−	9.39066i	−0.341309	−	0.341309i	−0.856859	+	0.515550i	$0.827588\pi$
$758$		0			0
$759$		−	14.7821i		−	0.536555i
$760$		0			0
$761$			28.1473i			1.02034i	0.860074	+	0.510169i	$0.170417\pi$
$761$			28.1473i			1.02034i	−0.860074	+	0.510169i	$0.829583\pi$
$762$		0			0
$763$	11.8722	+	11.8722i	0.429801	+	0.429801i
$764$		0			0
$765$	1.53073	−	1.53073i	0.0553438	−	0.0553438i
$766$		0			0
$767$	−73.9194			−2.66907
$768$		0			0
$769$	−11.9355			−0.430405			−0.215203	−	0.976569i	$-0.569041\pi$
$769$	−11.9355			−0.430405			−0.215203	+	0.976569i	$0.569041\pi$
$770$		0			0
$771$	3.30411	−	3.30411i	0.118995	−	0.118995i
$772$		0			0
$773$	10.3144	+	10.3144i	0.370983	+	0.370983i	0.867835	−	0.496852i	$-0.165511\pi$
$773$	10.3144	+	10.3144i	0.370983	+	0.370983i	−0.496852	+	0.867835i	$0.665511\pi$
$774$		0			0
$775$		−	2.64347i		−	0.0949561i
$776$		0			0
$777$			2.27170i			0.0814969i
$778$		0			0
$779$	44.2187	+	44.2187i	1.58430	+	1.58430i
$780$		0			0
$781$	30.8914	−	30.8914i	1.10538	−	1.10538i
$782$		0			0
$783$	7.72286			0.275992
$784$		0			0
$785$	−2.04429			−0.0729638
$786$		0			0
$787$	−27.3533	+	27.3533i	−0.975042	+	0.975042i	−0.999696	−	0.0246544i	$-0.992151\pi$
$787$	−27.3533	+	27.3533i	−0.975042	+	0.975042i	0.0246544	+	0.999696i	$0.492151\pi$
$788$		0			0
$789$	−10.9250	−	10.9250i	−0.388939	−	0.388939i
$790$		0			0
$791$		−	3.35556i		−	0.119310i
$792$		0			0
$793$			0.134707i			0.00478360i
$794$		0			0
$795$	5.99040	+	5.99040i	0.212458	+	0.212458i
$796$		0			0
$797$	20.7965	−	20.7965i	0.736650	−	0.736650i	−0.235278	−	0.971928i	$-0.575600\pi$
$797$	20.7965	−	20.7965i	0.736650	−	0.736650i	0.971928	+	0.235278i	$0.0756002\pi$
$798$		0			0
$799$	1.43664			0.0508248
$800$		0			0
$801$	2.79565			0.0987794
$802$		0			0
$803$	39.9310	−	39.9310i	1.40913	−	1.40913i
$804$		0			0
$805$	6.47966	+	6.47966i	0.228378	+	0.228378i
$806$		0			0
$807$		−	31.2459i		−	1.09991i
$808$		0			0
$809$		−	2.80334i		−	0.0985602i	−0.998785	−	0.0492801i	$-0.984307\pi$
$809$		−	2.80334i		−	0.0985602i	0.998785	−	0.0492801i	$-0.0156927\pi$
$810$		0			0
$811$	−36.1135	−	36.1135i	−1.26812	−	1.26812i	−0.947060	−	0.321058i	$-0.895962\pi$
$811$	−36.1135	−	36.1135i	−1.26812	−	1.26812i	−0.321058	−	0.947060i	$-0.604038\pi$
$812$		0			0
$813$	6.99199	−	6.99199i	0.245220	−	0.245220i
$814$		0			0
$815$	59.0426			2.06817
$816$		0			0
$817$	62.5347			2.18781
$818$		0			0
$819$	3.77112	−	3.77112i	0.131774	−	0.131774i
$820$		0			0
$821$	−12.3312	−	12.3312i	−0.430361	−	0.430361i	0.458390	−	0.888751i	$-0.348426\pi$
$821$	−12.3312	−	12.3312i	−0.430361	−	0.430361i	−0.888751	+	0.458390i	$0.848426\pi$
$822$		0			0
$823$		−	22.0665i		−	0.769191i	−0.923085	−	0.384595i	$-0.874341\pi$
$823$		−	22.0665i		−	0.769191i	0.923085	−	0.384595i	$-0.125659\pi$
$824$		0			0
$825$			4.55672i			0.158645i
$826$		0			0
$827$	1.24890	+	1.24890i	0.0434285	+	0.0434285i	0.728488	−	0.685059i	$-0.240224\pi$
$827$	1.24890	+	1.24890i	0.0434285	+	0.0434285i	−0.685059	+	0.728488i	$0.740224\pi$
$828$		0			0
$829$	−15.5927	+	15.5927i	−0.541558	+	0.541558i	−0.923985	−	0.382428i	$-0.875088\pi$
$829$	−15.5927	+	15.5927i	−0.541558	+	0.541558i	0.382428	+	0.923985i	$0.375088\pi$
$830$		0			0
$831$	22.9565			0.796351
$832$		0			0
$833$	5.33954			0.185004
$834$		0			0
$835$	−16.4242	+	16.4242i	−0.568384	+	0.568384i
$836$		0			0
$837$	−1.51594	−	1.51594i	−0.0523985	−	0.0523985i
$838$		0			0
$839$			22.1904i			0.766099i	0.923728	+	0.383050i	$0.125126\pi$
$839$			22.1904i			0.766099i	−0.923728	+	0.383050i	$0.874874\pi$
$840$		0			0
$841$		−	30.6425i		−	1.05664i
$842$		0			0
$843$	10.3064	+	10.3064i	0.354971	+	0.354971i
$844$		0			0
$845$	36.6840	−	36.6840i	1.26197	−	1.26197i
$846$		0			0
$847$	−2.43795			−0.0837690
$848$		0			0
$849$	−4.71832			−0.161932
$850$		0			0
$851$	−7.00228	+	7.00228i	−0.240035	+	0.240035i
$852$		0			0
$853$	26.5145	+	26.5145i	0.907839	+	0.907839i	0.996098	−	0.0882584i	$-0.0281301\pi$
$853$	26.5145	+	26.5145i	0.907839	+	0.907839i	−0.0882584	+	0.996098i	$0.528130\pi$
$854$		0			0
$855$		−	16.2877i		−	0.557028i
$856$		0			0
$857$		−	17.4761i		−	0.596971i	−0.954414	−	0.298485i	$-0.903519\pi$
$857$		−	17.4761i		−	0.596971i	0.954414	−	0.298485i	$-0.0964814\pi$
$858$		0			0
$859$	7.59727	+	7.59727i	0.259215	+	0.259215i	0.824735	−	0.565520i	$-0.191324\pi$
$859$	7.59727	+	7.59727i	0.259215	+	0.259215i	−0.565520	+	0.824735i	$0.691324\pi$
$860$		0			0
$861$	6.21946	−	6.21946i	0.211959	−	0.211959i
$862$		0			0
$863$	−23.1791			−0.789027			−0.394514	−	0.918890i	$-0.629087\pi$
$863$	−23.1791			−0.789027			−0.394514	+	0.918890i	$0.629087\pi$
$864$		0			0
$865$	−17.7398			−0.603171
$866$		0			0
$867$	11.4892	−	11.4892i	0.390193	−	0.390193i
$868$		0			0
$869$	21.9755	+	21.9755i	0.745469	+	0.745469i
$870$		0			0
$871$			30.9624i			1.04912i
$872$		0			0
$873$			2.26582i			0.0766863i
$874$		0			0
$875$	6.10215	+	6.10215i	0.206290	+	0.206290i
$876$		0			0
$877$	20.0602	−	20.0602i	0.677385	−	0.677385i	−0.282023	−	0.959408i	$-0.591005\pi$
$877$	20.0602	−	20.0602i	0.677385	−	0.677385i	0.959408	+	0.282023i	$0.0910055\pi$
$878$		0			0
$879$	23.9204			0.806816
$880$		0			0
$881$	58.0822			1.95684			0.978420	−	0.206628i	$-0.0662490\pi$
$881$	58.0822			1.95684			0.978420	+	0.206628i	$0.0662490\pi$
$882$		0			0
$883$	−3.93028	+	3.93028i	−0.132265	+	0.132265i	−0.770140	−	0.637875i	$-0.779814\pi$
$883$	−3.93028	+	3.93028i	−0.132265	+	0.132265i	0.637875	+	0.770140i	$0.279814\pi$
$884$		0			0
$885$	22.4525	+	22.4525i	0.754733	+	0.754733i
$886$		0			0
$887$		−	9.24359i		−	0.310369i	−0.987885	−	0.155185i	$-0.950403\pi$
$887$		−	9.24359i		−	0.310369i	0.987885	−	0.155185i	$-0.0495972\pi$
$888$		0			0
$889$			18.9378i			0.635153i
$890$		0			0
$891$	2.61313	+	2.61313i	0.0875430	+	0.0875430i
$892$		0			0
$893$	7.64328	−	7.64328i	0.255773	−	0.255773i
$894$		0			0
$895$	−6.67271			−0.223044
$896$		0			0
$897$	23.2482			0.776233
$898$		0			0
$899$	−11.7074	+	11.7074i	−0.390463	+	0.390463i
$900$		0			0
$901$	−2.08051	−	2.08051i	−0.0693119	−	0.0693119i
$902$		0			0
$903$		−	8.79565i		−	0.292701i
$904$		0			0
$905$			1.59899i			0.0531522i
$906$		0			0
$907$	−10.9563	−	10.9563i	−0.363797	−	0.363797i	0.501412	−	0.865209i	$-0.332814\pi$
$907$	−10.9563	−	10.9563i	−0.363797	−	0.363797i	−0.865209	+	0.501412i	$0.832814\pi$
$908$		0			0
$909$	−7.06306	+	7.06306i	−0.234267	+	0.234267i
$910$		0			0
$911$	−43.7108			−1.44820			−0.724101	−	0.689693i	$-0.757745\pi$
$911$	−43.7108			−1.44820			−0.724101	+	0.689693i	$0.757745\pi$
$912$		0			0
$913$	−7.24815			−0.239879
$914$		0			0
$915$	0.0409164	−	0.0409164i	0.00135266	−	0.00135266i
$916$		0			0
$917$	−11.3639	−	11.3639i	−0.375268	−	0.375268i
$918$		0			0
$919$			41.7955i			1.37871i	0.724426	+	0.689353i	$0.242105\pi$
$919$			41.7955i			1.37871i	−0.724426	+	0.689353i	$0.757895\pi$
$920$		0			0
$921$			20.7685i			0.684345i
$922$		0			0
$923$	48.5838	+	48.5838i	1.59915	+	1.59915i
$924$		0			0
$925$	2.15852	−	2.15852i	0.0709718	−	0.0709718i
$926$		0			0
$927$	−16.4180			−0.539238
$928$		0			0
$929$	24.1989			0.793942			0.396971	−	0.917831i	$-0.370061\pi$
$929$	24.1989			0.793942			0.396971	+	0.917831i	$0.370061\pi$
$930$		0			0
$931$	28.4076	−	28.4076i	0.931022	−	0.931022i
$932$		0			0
$933$	8.32957	+	8.32957i	0.272698	+	0.272698i
$934$		0			0
$935$		−	8.00000i		−	0.261628i
$936$		0			0
$937$		−	15.3183i		−	0.500426i	−0.968191	−	0.250213i	$-0.919499\pi$
$937$		−	15.3183i		−	0.500426i	0.968191	−	0.250213i	$-0.0805007\pi$
$938$		0			0
$939$	−7.88989	−	7.88989i	−0.257477	−	0.257477i
$940$		0			0
$941$	−24.9620	+	24.9620i	−0.813739	+	0.813739i	−0.985192	−	0.171453i	$-0.945154\pi$
$941$	−24.9620	+	24.9620i	−0.813739	+	0.813739i	0.171453	+	0.985192i	$0.445154\pi$
$942$		0			0
$943$	38.3417			1.24858
$944$		0			0
$945$	−2.29090			−0.0745231
$946$		0			0
$947$	10.0446	−	10.0446i	0.326404	−	0.326404i	−0.524813	−	0.851217i	$-0.675865\pi$
$947$	10.0446	−	10.0446i	0.326404	−	0.326404i	0.851217	+	0.524813i	$0.175865\pi$
$948$		0			0
$949$	62.8005	+	62.8005i	2.03859	+	2.03859i
$950$		0			0
$951$		−	15.9140i		−	0.516048i
$952$		0			0
$953$		−	19.6428i		−	0.636292i	−0.948042	−	0.318146i	$-0.896940\pi$
$953$		−	19.6428i		−	0.636292i	0.948042	−	0.318146i	$-0.103060\pi$
$954$		0			0
$955$	−25.7809	−	25.7809i	−0.834251	−	0.834251i
$956$		0			0
$957$	20.1808	−	20.1808i	0.652353	−	0.652353i
$958$		0			0
$959$	7.57772			0.244697
$960$		0			0
$961$	−26.4039			−0.851738
$962$		0			0
$963$	8.66364	−	8.66364i	0.279182	−	0.279182i
$964$		0			0
$965$	−15.5614	−	15.5614i	−0.500941	−	0.500941i
$966$		0			0
$967$			44.5337i			1.43211i	0.698045	+	0.716054i	$0.254054\pi$
$967$			44.5337i			1.43211i	−0.698045	+	0.716054i	$0.745946\pi$
$968$		0			0
$969$			5.65685i			0.181724i
$970$		0			0
$971$	0.991750	+	0.991750i	0.0318268	+	0.0318268i	0.722841	−	0.691014i	$-0.242836\pi$
$971$	0.991750	+	0.991750i	0.0318268	+	0.0318268i	−0.691014	+	0.722841i	$0.742836\pi$
$972$		0			0
$973$	−9.54113	+	9.54113i	−0.305874	+	0.305874i
$974$		0			0
$975$	−7.16648			−0.229511
$976$		0			0
$977$	37.0794			1.18628			0.593138	−	0.805101i	$-0.297889\pi$
$977$	37.0794			1.18628			0.593138	+	0.805101i	$0.297889\pi$
$978$		0			0
$979$	7.30538	−	7.30538i	0.233481	−	0.233481i
$980$		0			0
$981$	−12.9382	−	12.9382i	−0.413084	−	0.413084i
$982$		0			0
$983$		−	46.6894i		−	1.48916i	−0.667534	−	0.744580i	$-0.732650\pi$
$983$		−	46.6894i		−	1.48916i	0.667534	−	0.744580i	$-0.267350\pi$
$984$		0			0
$985$		−	9.35105i		−	0.297949i
$986$		0			0
$987$	−1.07504	−	1.07504i	−0.0342190	−	0.0342190i
$988$		0			0
$989$	27.1116	−	27.1116i	0.862100	−	0.862100i
$990$		0			0
$991$	−11.8780			−0.377318			−0.188659	−	0.982043i	$-0.560414\pi$
$991$	−11.8780			−0.377318			−0.188659	+	0.982043i	$0.560414\pi$
$992$		0			0
$993$	−2.73190			−0.0866942
$994$		0			0
$995$	−26.1746	+	26.1746i	−0.829790	+	0.829790i
$996$		0			0
$997$	−36.1695	−	36.1695i	−1.14550	−	1.14550i	−0.987427	−	0.158074i	$-0.949472\pi$
$997$	−36.1695	−	36.1695i	−1.14550	−	1.14550i	−0.158074	−	0.987427i	$-0.550528\pi$
$998$		0			0
$999$		−	2.47568i		−	0.0783271i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.j.j.385.2	yes	8
3.2	odd	2		4608.2.k.be.3457.2		8
4.3	odd	2		1536.2.j.i.385.4	yes	8
8.3	odd	2		1536.2.j.f.385.1	yes	8
8.5	even	2		1536.2.j.e.385.3	✓	8
12.11	even	2		4608.2.k.bc.3457.2		8
16.3	odd	4		1536.2.j.i.1153.4	yes	8
16.5	even	4		1536.2.j.e.1153.3	yes	8
16.11	odd	4		1536.2.j.f.1153.1	yes	8
16.13	even	4	inner	1536.2.j.j.1153.2	yes	8
24.5	odd	2		4608.2.k.bh.3457.3		8
24.11	even	2		4608.2.k.bj.3457.3		8
32.3	odd	8		3072.2.a.p.1.1		4
32.5	even	8		3072.2.d.e.1537.4		8
32.11	odd	8		3072.2.d.j.1537.1		8
32.13	even	8		3072.2.a.s.1.4		4
32.19	odd	8		3072.2.a.j.1.4		4
32.21	even	8		3072.2.d.e.1537.5		8
32.27	odd	8		3072.2.d.j.1537.8		8
32.29	even	8		3072.2.a.m.1.1		4
48.5	odd	4		4608.2.k.bh.1153.3		8
48.11	even	4		4608.2.k.bj.1153.3		8
48.29	odd	4		4608.2.k.be.1153.2		8
48.35	even	4		4608.2.k.bc.1153.2		8
96.29	odd	8		9216.2.a.bl.1.4		4
96.35	even	8		9216.2.a.z.1.4		4
96.77	odd	8		9216.2.a.bm.1.1		4
96.83	even	8		9216.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.3	✓	8	8.5	even	2
1536.2.j.e.1153.3	yes	8	16.5	even	4
1536.2.j.f.385.1	yes	8	8.3	odd	2
1536.2.j.f.1153.1	yes	8	16.11	odd	4
1536.2.j.i.385.4	yes	8	4.3	odd	2
1536.2.j.i.1153.4	yes	8	16.3	odd	4
1536.2.j.j.385.2	yes	8	1.1	even	1	trivial
1536.2.j.j.1153.2	yes	8	16.13	even	4	inner
3072.2.a.j.1.4		4	32.19	odd	8
3072.2.a.m.1.1		4	32.29	even	8
3072.2.a.p.1.1		4	32.3	odd	8
3072.2.a.s.1.4		4	32.13	even	8
3072.2.d.e.1537.4		8	32.5	even	8
3072.2.d.e.1537.5		8	32.21	even	8
3072.2.d.j.1537.1		8	32.11	odd	8
3072.2.d.j.1537.8		8	32.27	odd	8
4608.2.k.bc.1153.2		8	48.35	even	4
4608.2.k.bc.3457.2		8	12.11	even	2
4608.2.k.be.1153.2		8	48.29	odd	4
4608.2.k.be.3457.2		8	3.2	odd	2
4608.2.k.bh.1153.3		8	48.5	odd	4
4608.2.k.bh.3457.3		8	24.5	odd	2
4608.2.k.bj.1153.3		8	48.11	even	4
4608.2.k.bj.3457.3		8	24.11	even	2
9216.2.a.z.1.4		4	96.35	even	8
9216.2.a.ba.1.1		4	96.83	even	8
9216.2.a.bl.1.4		4	96.29	odd	8
9216.2.a.bm.1.1		4	96.77	odd	8

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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(1.76537 + 1.76537i) q^{5} +0.917608i q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} +(1.76537 + 1.76537i) q^{5} +0.917608i q^{7} -1.00000i q^{9} +(-2.61313 - 2.61313i) q^{11} +(4.10973 - 4.10973i) q^{13} -2.49661 q^{15} +0.867091 q^{17} +(4.61313 - 4.61313i) q^{19} +(-0.648847 - 0.648847i) q^{21} -4.00000i q^{23} +1.23304i q^{25} +(0.707107 + 0.707107i) q^{27} +(5.46088 - 5.46088i) q^{29} -2.14386 q^{31} +3.69552 q^{33} +(-1.61991 + 1.61991i) q^{35} +(-1.75057 - 1.75057i) q^{37} +5.81204i q^{39} +9.58541i q^{41} +(6.77791 + 6.77791i) q^{43} +(1.76537 - 1.76537i) q^{45} +1.65685 q^{47} +6.15800 q^{49} +(-0.613126 + 0.613126i) q^{51} +(-2.39942 - 2.39942i) q^{53} -9.22625i q^{55} +6.52395i q^{57} +(-8.99321 - 8.99321i) q^{59} +(-0.0163888 + 0.0163888i) q^{61} +0.917608 q^{63} +14.5104 q^{65} +(-3.76696 + 3.76696i) q^{67} +(2.82843 + 2.82843i) q^{69} +11.8216i q^{71} +15.2809i q^{73} +(-0.871891 - 0.871891i) q^{75} +(2.39782 - 2.39782i) q^{77} -8.40968 q^{79} -1.00000 q^{81} +(1.38687 - 1.38687i) q^{83} +(1.53073 + 1.53073i) q^{85} +7.72286i q^{87} +2.79565i q^{89} +(3.77112 + 3.77112i) q^{91} +(1.51594 - 1.51594i) q^{93} +16.2877 q^{95} -2.26582 q^{97} +(-2.61313 + 2.61313i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} - 8 q^{13} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 32 q^{35} - 8 q^{37} + 16 q^{43} + 8 q^{45} - 32 q^{47} - 8 q^{49} + 16 q^{51} - 8 q^{53} - 32 q^{59} - 8 q^{61} + 16 q^{63} - 16 q^{65} - 32 q^{67} + 16 q^{75} - 48 q^{79} - 8 q^{81} + 32 q^{83} - 48 q^{91} + 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 8 * q^13 + 16 * q^19 + 8 * q^29 + 16 * q^31 - 32 * q^35 - 8 * q^37 + 16 * q^43 + 8 * q^45 - 32 * q^47 - 8 * q^49 + 16 * q^51 - 8 * q^53 - 32 * q^59 - 8 * q^61 + 16 * q^63 - 16 * q^65 - 32 * q^67 + 16 * q^75 - 48 * q^79 - 8 * q^81 + 32 * q^83 - 48 * q^91 + 64 * q^95 - 32 * q^97

Introduction

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