LMFDB - Embedded newform 384.4.c.a.383.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.c (of order $2$, degree $1$, 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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 $ x^12 + 39x^10 + 549x^8 + 3500x^6 + 10236x^4 + 11952*x^2 + 4624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.4
Root			$-3.14286i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.4.c.a.383.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.12202 + 3.16369i) q^{3} +21.4043i q^{5} -20.9034i q^{7} +(6.98212 - 26.0816i) q^{9} +O(q^{10})$	\(q+(-4.12202 + 3.16369i) q^{3} +21.4043i q^{5} -20.9034i q^{7} +(6.98212 - 26.0816i) q^{9} -9.94564 q^{11} -67.8290 q^{13} +(-67.7167 - 88.2292i) q^{15} +7.97348i q^{17} -62.4014i q^{19} +(66.1320 + 86.1644i) q^{21} +101.816 q^{23} -333.146 q^{25} +(53.7337 + 129.598i) q^{27} -122.445i q^{29} +87.5072i q^{31} +(40.9961 - 31.4649i) q^{33} +447.424 q^{35} +106.136 q^{37} +(279.593 - 214.590i) q^{39} -90.3818i q^{41} -451.366i q^{43} +(558.260 + 149.448i) q^{45} +428.890 q^{47} -93.9530 q^{49} +(-25.2256 - 32.8669i) q^{51} +362.452i q^{53} -212.880i q^{55} +(197.419 + 257.220i) q^{57} -801.483 q^{59} +647.903 q^{61} +(-545.195 - 145.950i) q^{63} -1451.84i q^{65} -957.240i q^{67} +(-419.688 + 322.115i) q^{69} -224.090 q^{71} -108.474 q^{73} +(1373.24 - 1053.97i) q^{75} +207.898i q^{77} -615.569i q^{79} +(-631.500 - 364.210i) q^{81} -204.385 q^{83} -170.667 q^{85} +(387.378 + 504.721i) q^{87} +454.067i q^{89} +1417.86i q^{91} +(-276.846 - 360.707i) q^{93} +1335.66 q^{95} +740.490 q^{97} +(-69.4417 + 259.398i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{3}+O(q^{10}) $ 12 * q - 10 * q^3	$ 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) $ 12 * q - 10 * q^3 - 36 * q^11 - 84 * q^15 + 136 * q^21 + 120 * q^23 - 300 * q^25 + 266 * q^27 - 116 * q^33 - 432 * q^35 + 528 * q^37 - 620 * q^39 + 440 * q^45 + 1248 * q^47 - 948 * q^49 + 1072 * q^51 - 172 * q^57 - 2508 * q^59 + 624 * q^61 - 2744 * q^63 - 24 * q^69 + 2040 * q^71 - 216 * q^73 + 3894 * q^75 - 1076 * q^81 - 4572 * q^83 + 480 * q^85 - 4156 * q^87 - 112 * q^93 + 5448 * q^95 - 48 * q^97 + 6044 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.12202	+	3.16369i	−0.793283	+	0.608853i
$3$	−4.12202	+	3.16369i	−0.793283	+	0.608853i
$4$		0			0
$5$			21.4043i			1.91446i	0.289323	+	0.957232i	$0.406570\pi$
$5$			21.4043i			1.91446i	−0.289323	+	0.957232i	$0.593430\pi$
$6$		0			0
$7$		−	20.9034i		−	1.12868i	−0.825543	−	0.564339i	$-0.809131\pi$
$7$		−	20.9034i		−	1.12868i	0.825543	−	0.564339i	$-0.190869\pi$
$8$		0			0
$9$	6.98212	−	26.0816i	0.258597	−	0.965985i
$10$		0			0
$11$	−9.94564			−0.272611			−0.136306	−	0.990667i	$-0.543523\pi$
$11$	−9.94564			−0.272611			−0.136306	+	0.990667i	$0.543523\pi$
$12$		0			0
$13$	−67.8290			−1.44711			−0.723554	−	0.690268i	$-0.757493\pi$
$13$	−67.8290			−1.44711			−0.723554	+	0.690268i	$0.757493\pi$
$14$		0			0
$15$	−67.7167	−	88.2292i	−1.16563	−	1.51871i
$16$		0			0
$17$			7.97348i			0.113756i	0.998381	+	0.0568780i	$0.0181146\pi$
$17$			7.97348i			0.113756i	−0.998381	+	0.0568780i	$0.981885\pi$
$18$		0			0
$19$		−	62.4014i		−	0.753466i	−0.926322	−	0.376733i	$-0.877047\pi$
$19$		−	62.4014i		−	0.753466i	0.926322	−	0.376733i	$-0.122953\pi$
$20$		0			0
$21$	66.1320	+	86.1644i	0.687199	+	0.895362i
$22$		0			0
$23$	101.816			0.923049			0.461525	−	0.887127i	$-0.347303\pi$
$23$	101.816			0.923049			0.461525	+	0.887127i	$0.347303\pi$
$24$		0			0
$25$	−333.146			−2.66517
$26$		0			0
$27$	53.7337	+	129.598i	0.383002	+	0.923748i
$28$		0			0
$29$		−	122.445i		−	0.784051i	−0.919954	−	0.392025i	$-0.871775\pi$
$29$		−	122.445i		−	0.784051i	0.919954	−	0.392025i	$-0.128225\pi$
$30$		0			0
$31$			87.5072i			0.506992i	0.967336	+	0.253496i	$0.0815805\pi$
$31$			87.5072i			0.506992i	−0.967336	+	0.253496i	$0.918419\pi$
$32$		0			0
$33$	40.9961	−	31.4649i	0.216258	−	0.165980i
$34$		0			0
$35$	447.424			2.16081
$36$		0			0
$37$	106.136			0.471587			0.235794	−	0.971803i	$-0.424231\pi$
$37$	106.136			0.471587			0.235794	+	0.971803i	$0.424231\pi$
$38$		0			0
$39$	279.593	−	214.590i	1.14797	−	0.881075i
$40$		0			0
$41$		−	90.3818i		−	0.344275i	−0.985073	−	0.172137i	$-0.944933\pi$
$41$		−	90.3818i		−	0.344275i	0.985073	−	0.172137i	$-0.0550673\pi$
$42$		0			0
$43$		−	451.366i		−	1.60076i	−0.599494	−	0.800380i	$-0.704631\pi$
$43$		−	451.366i		−	1.60076i	0.599494	−	0.800380i	$-0.295369\pi$
$44$		0			0
$45$	558.260	+	149.448i	1.84934	+	0.495075i
$46$		0			0
$47$	428.890			1.33106			0.665532	−	0.746369i	$-0.268205\pi$
$47$	428.890			1.33106			0.665532	+	0.746369i	$0.268205\pi$
$48$		0			0
$49$	−93.9530			−0.273916
$50$		0			0
$51$	−25.2256	−	32.8669i	−0.0692607	−	0.0902408i
$52$		0			0
$53$			362.452i			0.939370i	0.882834	+	0.469685i	$0.155633\pi$
$53$			362.452i			0.939370i	−0.882834	+	0.469685i	$0.844367\pi$
$54$		0			0
$55$		−	212.880i		−	0.521904i
$56$		0			0
$57$	197.419	+	257.220i	0.458750	+	0.597712i
$58$		0			0
$59$	−801.483			−1.76855			−0.884273	−	0.466970i	$-0.845346\pi$
$59$	−801.483			−1.76855			−0.884273	+	0.466970i	$0.845346\pi$
$60$		0			0
$61$	647.903			1.35993			0.679963	−	0.733247i	$-0.261996\pi$
$61$	647.903			1.35993			0.679963	+	0.733247i	$0.261996\pi$
$62$		0			0
$63$	−545.195	−	145.950i	−1.09029	−	0.291873i
$64$		0			0
$65$		−	1451.84i		−	2.77043i
$66$		0			0
$67$		−	957.240i		−	1.74545i	−0.488208	−	0.872727i	$-0.662349\pi$
$67$		−	957.240i		−	1.74545i	0.488208	−	0.872727i	$-0.337651\pi$
$68$		0			0
$69$	−419.688	+	322.115i	−0.732240	+	0.562001i
$70$		0			0
$71$	−224.090			−0.374572			−0.187286	−	0.982305i	$-0.559969\pi$
$71$	−224.090			−0.374572			−0.187286	+	0.982305i	$0.559969\pi$
$72$		0			0
$73$	−108.474			−0.173916			−0.0869582	−	0.996212i	$-0.527715\pi$
$73$	−108.474			−0.173916			−0.0869582	+	0.996212i	$0.527715\pi$
$74$		0			0
$75$	1373.24	−	1053.97i	2.11423	−	1.62269i
$76$		0			0
$77$			207.898i			0.307690i
$78$		0			0
$79$		−	615.569i		−	0.876670i	−0.898812	−	0.438335i	$-0.855568\pi$
$79$		−	615.569i		−	0.876670i	0.898812	−	0.438335i	$-0.144432\pi$
$80$		0			0
$81$	−631.500	−	364.210i	−0.866255	−	0.499602i
$82$		0			0
$83$	−204.385			−0.270291			−0.135145	−	0.990826i	$-0.543150\pi$
$83$	−204.385			−0.270291			−0.135145	+	0.990826i	$0.543150\pi$
$84$		0			0
$85$	−170.667			−0.217782
$86$		0			0
$87$	387.378	+	504.721i	0.477371	+	0.621974i
$88$		0			0
$89$			454.067i			0.540798i	0.962748	+	0.270399i	$0.0871556\pi$
$89$			454.067i			0.540798i	−0.962748	+	0.270399i	$0.912844\pi$
$90$		0			0
$91$			1417.86i			1.63332i
$92$		0			0
$93$	−276.846	−	360.707i	−0.308684	−	0.402189i
$94$		0			0
$95$	1335.66			1.44248
$96$		0			0
$97$	740.490			0.775106			0.387553	−	0.921847i	$-0.373320\pi$
$97$	740.490			0.775106			0.387553	+	0.921847i	$0.373320\pi$
$98$		0			0
$99$	−69.4417	+	259.398i	−0.0704965	+	0.263338i
$100$		0			0
$101$			52.9454i			0.0521610i	0.999660	+	0.0260805i	$0.00830263\pi$
$101$			52.9454i			0.0521610i	−0.999660	+	0.0260805i	$0.991697\pi$
$102$		0			0
$103$		−	808.292i		−	0.773236i	−0.922240	−	0.386618i	$-0.873643\pi$
$103$		−	808.292i		−	0.773236i	0.922240	−	0.386618i	$-0.126357\pi$
$104$		0			0
$105$	−1844.29	+	1415.51i	−1.71414	+	1.31562i
$106$		0			0
$107$	728.059			0.657796			0.328898	−	0.944366i	$-0.393323\pi$
$107$	728.059			0.657796			0.328898	+	0.944366i	$0.393323\pi$
$108$		0			0
$109$	1488.85			1.30831			0.654157	−	0.756358i	$-0.273023\pi$
$109$	1488.85			1.30831			0.654157	+	0.756358i	$0.273023\pi$
$110$		0			0
$111$	−437.497	+	335.783i	−0.374102	+	0.287127i
$112$		0			0
$113$			38.0071i			0.0316408i	0.999875	+	0.0158204i	$0.00503600\pi$
$113$			38.0071i			0.0316408i	−0.999875	+	0.0158204i	$0.994964\pi$
$114$		0			0
$115$			2179.31i			1.76714i
$116$		0			0
$117$	−473.591	+	1769.09i	−0.374218	+	1.39788i
$118$		0			0
$119$	166.673			0.128394
$120$		0			0
$121$	−1232.08			−0.925683
$122$		0			0
$123$	285.940	+	372.556i	0.209613	+	0.273107i
$124$		0			0
$125$		−	4455.23i		−	3.18790i
$126$		0			0
$127$		−	226.671i		−	0.158376i	−0.996860	−	0.0791882i	$-0.974767\pi$
$127$		−	226.671i		−	0.158376i	0.996860	−	0.0791882i	$-0.0252328\pi$
$128$		0			0
$129$	1427.98	+	1860.54i	0.974626	+	1.26986i
$130$		0			0
$131$	−1580.10			−1.05385			−0.526925	−	0.849912i	$-0.676655\pi$
$131$	−1580.10			−1.05385			−0.526925	+	0.849912i	$0.676655\pi$
$132$		0			0
$133$	−1304.40			−0.850421
$134$		0			0
$135$	−2773.96	+	1150.13i	−1.76848	+	0.733243i
$136$		0			0
$137$		−	543.435i		−	0.338896i	−0.985539	−	0.169448i	$-0.945801\pi$
$137$		−	543.435i		−	0.338896i	0.985539	−	0.169448i	$-0.0541985\pi$
$138$		0			0
$139$		−	29.7527i		−	0.0181554i	−0.999959	−	0.00907768i	$-0.997110\pi$
$139$		−	29.7527i		−	0.0181554i	0.999959	−	0.00907768i	$-0.00288955\pi$
$140$		0			0
$141$	−1767.89	+	1356.87i	−1.05591	+	0.810422i
$142$		0			0
$143$	674.603			0.394498
$144$		0			0
$145$	2620.86			1.50104
$146$		0			0
$147$	387.276	−	297.238i	0.217293	−	0.166774i
$148$		0			0
$149$			489.426i			0.269096i	0.990907	+	0.134548i	$0.0429583\pi$
$149$			489.426i			0.269096i	−0.990907	+	0.134548i	$0.957042\pi$
$150$		0			0
$151$			396.852i			0.213877i	0.994266	+	0.106938i	$0.0341047\pi$
$151$			396.852i			0.213877i	−0.994266	+	0.106938i	$0.965895\pi$
$152$		0			0
$153$	207.961	+	55.6718i	0.109887	+	0.0294170i
$154$		0			0
$155$	−1873.04			−0.970618
$156$		0			0
$157$	−2542.51			−1.29245			−0.646224	−	0.763148i	$-0.723653\pi$
$157$	−2542.51			−1.29245			−0.646224	+	0.763148i	$0.723653\pi$
$158$		0			0
$159$	−1146.69	−	1494.03i	−0.571938	−	0.745187i
$160$		0			0
$161$		−	2128.31i		−	1.04183i
$162$		0			0
$163$		−	1807.94i		−	0.868765i	−0.900728	−	0.434383i	$-0.856967\pi$
$163$		−	1807.94i		−	0.868765i	0.900728	−	0.434383i	$-0.143033\pi$
$164$		0			0
$165$	673.486	+	877.496i	0.317763	+	0.414018i
$166$		0			0
$167$	25.5944			0.0118596			0.00592980	−	0.999982i	$-0.498112\pi$
$167$	25.5944			0.0118596			0.00592980	+	0.999982i	$0.498112\pi$
$168$		0			0
$169$	2403.78			1.09412
$170$		0			0
$171$	−1627.53	−	435.694i	−0.727837	−	0.194844i
$172$		0			0
$173$		−	677.278i		−	0.297644i	−0.988864	−	0.148822i	$-0.952452\pi$
$173$		−	677.278i		−	0.297644i	0.988864	−	0.148822i	$-0.0475482\pi$
$174$		0			0
$175$			6963.89i			3.00812i
$176$		0			0
$177$	3303.73	−	2535.64i	1.40296	−	1.07678i
$178$		0			0
$179$	−3240.41			−1.35307			−0.676535	−	0.736411i	$-0.736519\pi$
$179$	−3240.41			−1.35307			−0.676535	+	0.736411i	$0.736519\pi$
$180$		0			0
$181$	−2123.69			−0.872114			−0.436057	−	0.899919i	$-0.643625\pi$
$181$	−2123.69			−0.872114			−0.436057	+	0.899919i	$0.643625\pi$
$182$		0			0
$183$	−2670.67	+	2049.76i	−1.07881	+	0.827994i
$184$		0			0
$185$			2271.78i			0.902836i
$186$		0			0
$187$		−	79.3014i		−	0.0310112i
$188$		0			0
$189$	2709.05	−	1123.22i	1.04261	−	0.432286i
$190$		0			0
$191$	−3080.73			−1.16709			−0.583543	−	0.812082i	$-0.698334\pi$
$191$	−3080.73			−1.16709			−0.583543	+	0.812082i	$0.698334\pi$
$192$		0			0
$193$	−2614.09			−0.974955			−0.487478	−	0.873135i	$-0.662083\pi$
$193$	−2614.09			−0.974955			−0.487478	+	0.873135i	$0.662083\pi$
$194$		0			0
$195$	4593.16	+	5984.50i	1.68679	+	2.19774i
$196$		0			0
$197$		−	4223.51i		−	1.52748i	−0.645527	−	0.763738i	$-0.723362\pi$
$197$		−	4223.51i		−	1.52748i	0.645527	−	0.763738i	$-0.276638\pi$
$198$		0			0
$199$		−	783.071i		−	0.278947i	−0.990226	−	0.139474i	$-0.955459\pi$
$199$		−	783.071i		−	0.278947i	0.990226	−	0.139474i	$-0.0445410\pi$
$200$		0			0
$201$	3028.41	+	3945.76i	1.06272	+	1.38464i
$202$		0			0
$203$	−2559.52			−0.884941
$204$		0			0
$205$	1934.56			0.659101
$206$		0			0
$207$	710.893	−	2655.53i	0.238698	−	0.891652i
$208$		0			0
$209$			620.622i			0.205403i
$210$		0			0
$211$		−	3515.60i		−	1.14703i	−0.819195	−	0.573516i	$-0.805579\pi$
$211$		−	3515.60i		−	1.14703i	0.819195	−	0.573516i	$-0.194421\pi$
$212$		0			0
$213$	923.705	−	708.952i	0.297142	−	0.228059i
$214$		0			0
$215$	9661.19			3.06459
$216$		0			0
$217$	1829.20			0.572232
$218$		0			0
$219$	447.131	−	343.177i	0.137965	−	0.105889i
$220$		0			0
$221$		−	540.834i		−	0.164617i
$222$		0			0
$223$			1864.10i			0.559772i	0.960033	+	0.279886i	$0.0902967\pi$
$223$			1864.10i			0.559772i	−0.960033	+	0.279886i	$0.909703\pi$
$224$		0			0
$225$	−2326.07	+	8688.98i	−0.689205	+	2.57451i
$226$		0			0
$227$	−2871.51			−0.839600			−0.419800	−	0.907617i	$-0.637900\pi$
$227$	−2871.51			−0.839600			−0.419800	+	0.907617i	$0.637900\pi$
$228$		0			0
$229$	−3048.58			−0.879720			−0.439860	−	0.898066i	$-0.644972\pi$
$229$	−3048.58			−0.879720			−0.439860	+	0.898066i	$0.644972\pi$
$230$		0			0
$231$	−657.725	−	856.960i	−0.187338	−	0.244086i
$232$		0			0
$233$		−	2078.46i		−	0.584398i	−0.956358	−	0.292199i	$-0.905613\pi$
$233$		−	2078.46i		−	0.584398i	0.956358	−	0.292199i	$-0.0943870\pi$
$234$		0			0
$235$			9180.11i			2.54827i
$236$		0			0
$237$	1947.47	+	2537.39i	0.533763	+	0.695448i
$238$		0			0
$239$	−2778.79			−0.752072			−0.376036	−	0.926605i	$-0.622713\pi$
$239$	−2778.79			−0.752072			−0.376036	+	0.926605i	$0.622713\pi$
$240$		0			0
$241$	−2838.97			−0.758815			−0.379408	−	0.925230i	$-0.623872\pi$
$241$	−2838.97			−0.758815			−0.379408	+	0.925230i	$0.623872\pi$
$242$		0			0
$243$	3755.30	−	496.589i	0.991370	−	0.131095i
$244$		0			0
$245$		−	2011.00i		−	0.524401i
$246$		0			0
$247$			4232.63i			1.09035i
$248$		0			0
$249$	842.478	−	646.610i	0.214417	−	0.164567i
$250$		0			0
$251$	5339.24			1.34267			0.671334	−	0.741155i	$-0.265722\pi$
$251$	5339.24			1.34267			0.671334	+	0.741155i	$0.265722\pi$
$252$		0			0
$253$	−1012.63			−0.251634
$254$		0			0
$255$	703.494	−	539.938i	0.172763	−	0.132597i
$256$		0			0
$257$		−	6223.99i		−	1.51067i	−0.655339	−	0.755335i	$-0.727474\pi$
$257$		−	6223.99i		−	1.51067i	0.655339	−	0.755335i	$-0.272526\pi$
$258$		0			0
$259$		−	2218.61i		−	0.532270i
$260$		0			0
$261$	−3193.56	−	854.926i	−0.757381	−	0.202753i
$262$		0			0
$263$	−4779.81			−1.12067			−0.560334	−	0.828267i	$-0.689327\pi$
$263$	−4779.81			−1.12067			−0.560334	+	0.828267i	$0.689327\pi$
$264$		0			0
$265$	−7758.05			−1.79839
$266$		0			0
$267$	−1436.53	−	1871.67i	−0.329266	−	0.429006i
$268$		0			0
$269$			3916.03i			0.887601i	0.896126	+	0.443800i	$0.146370\pi$
$269$			3916.03i			0.887601i	−0.896126	+	0.443800i	$0.853630\pi$
$270$		0			0
$271$		−	6412.78i		−	1.43745i	−0.695295	−	0.718725i	$-0.744726\pi$
$271$		−	6412.78i		−	1.43745i	0.695295	−	0.718725i	$-0.255274\pi$
$272$		0			0
$273$	−4485.67	−	5844.45i	−0.994450	−	1.29568i
$274$		0			0
$275$	3313.35			0.726555
$276$		0			0
$277$	3745.22			0.812376			0.406188	−	0.913789i	$-0.366858\pi$
$277$	3745.22			0.812376			0.406188	+	0.913789i	$0.366858\pi$
$278$		0			0
$279$	2282.33	+	610.986i	0.489747	+	0.131107i
$280$		0			0
$281$		−	6656.43i		−	1.41313i	−0.707648	−	0.706565i	$-0.750244\pi$
$281$		−	6656.43i		−	1.41313i	0.707648	−	0.706565i	$-0.249756\pi$
$282$		0			0
$283$		−	6460.77i		−	1.35708i	−0.734564	−	0.678539i	$-0.762613\pi$
$283$		−	6460.77i		−	1.35708i	0.734564	−	0.678539i	$-0.237387\pi$
$284$		0			0
$285$	−5505.62	+	4225.62i	−1.14430	+	0.878260i
$286$		0			0
$287$	−1889.29			−0.388576
$288$		0			0
$289$	4849.42			0.987060
$290$		0			0
$291$	−3052.31	+	2342.68i	−0.614879	+	0.471925i
$292$		0			0
$293$			2620.84i			0.522564i	0.965263	+	0.261282i	$0.0841453\pi$
$293$			2620.84i			0.522564i	−0.965263	+	0.261282i	$0.915855\pi$
$294$		0			0
$295$		−	17155.2i		−	3.38582i
$296$		0			0
$297$	−534.416	−	1288.94i	−0.104411	−	0.251824i
$298$		0			0
$299$	−6906.09			−1.33575
$300$		0			0
$301$	−9435.09			−1.80674
$302$		0			0
$303$	−167.503	−	218.242i	−0.0317584	−	0.0413785i
$304$		0			0
$305$			13867.9i			2.60353i
$306$		0			0
$307$		−	952.063i		−	0.176994i	−0.996076	−	0.0884969i	$-0.971794\pi$
$307$		−	952.063i		−	0.176994i	0.996076	−	0.0884969i	$-0.0282063\pi$
$308$		0			0
$309$	2557.18	+	3331.80i	0.470787	+	0.613395i
$310$		0			0
$311$	464.965			0.0847773			0.0423887	−	0.999101i	$-0.486503\pi$
$311$	464.965			0.0847773			0.0423887	+	0.999101i	$0.486503\pi$
$312$		0			0
$313$	5656.16			1.02142			0.510711	−	0.859752i	$-0.329382\pi$
$313$	5656.16			1.02142			0.510711	+	0.859752i	$0.329382\pi$
$314$		0			0
$315$	3123.97	−	11669.5i	0.558780	−	2.08731i
$316$		0			0
$317$			4593.57i			0.813883i	0.913454	+	0.406941i	$0.133405\pi$
$317$			4593.57i			0.813883i	−0.913454	+	0.406941i	$0.866595\pi$
$318$		0			0
$319$			1217.79i			0.213741i
$320$		0			0
$321$	−3001.08	+	2303.35i	−0.521818	+	0.400501i
$322$		0			0
$323$	497.556			0.0857113
$324$		0			0
$325$	22597.0			3.85678
$326$		0			0
$327$	−6137.09	+	4710.27i	−1.03786	+	0.796571i
$328$		0			0
$329$		−	8965.26i		−	1.50234i
$330$		0			0
$331$			6709.02i			1.11408i	0.830485	+	0.557041i	$0.188063\pi$
$331$			6709.02i			1.11408i	−0.830485	+	0.557041i	$0.811937\pi$
$332$		0			0
$333$	741.058	−	2768.21i	0.121951	−	0.455546i
$334$		0			0
$335$	20489.1			3.34161
$336$		0			0
$337$	−3712.30			−0.600064			−0.300032	−	0.953929i	$-0.596997\pi$
$337$	−3712.30			−0.600064			−0.300032	+	0.953929i	$0.596997\pi$
$338$		0			0
$339$	−120.243	−	156.666i	−0.0192646	−	0.0251001i
$340$		0			0
$341$		−	870.316i		−	0.138212i
$342$		0			0
$343$		−	5205.93i		−	0.819516i
$344$		0			0
$345$	−6894.66	−	8983.16i	−1.07593	−	1.40185i
$346$		0			0
$347$	6331.12			0.979459			0.489729	−	0.871874i	$-0.337096\pi$
$347$	6331.12			0.979459			0.489729	+	0.871874i	$0.337096\pi$
$348$		0			0
$349$	8328.66			1.27743			0.638714	−	0.769444i	$-0.279467\pi$
$349$	8328.66			1.27743			0.638714	+	0.769444i	$0.279467\pi$
$350$		0			0
$351$	−3644.70	−	8790.52i	−0.554245	−	1.33676i
$352$		0			0
$353$		−	1494.95i		−	0.225406i	−0.993629	−	0.112703i	$-0.964049\pi$
$353$		−	1494.95i		−	0.225406i	0.993629	−	0.112703i	$-0.0359508\pi$
$354$		0			0
$355$		−	4796.51i		−	0.717104i
$356$		0			0
$357$	−687.030	+	527.302i	−0.101853	+	0.0781730i
$358$		0			0
$359$	−2701.95			−0.397223			−0.198612	−	0.980078i	$-0.563643\pi$
$359$	−2701.95			−0.397223			−0.198612	+	0.980078i	$0.563643\pi$
$360$		0			0
$361$	2965.07			0.432289
$362$		0			0
$363$	5078.68	−	3897.93i	0.734329	−	0.563605i
$364$		0			0
$365$		−	2321.81i		−	0.332956i
$366$		0			0
$367$			11089.3i			1.57727i	0.614861	+	0.788636i	$0.289212\pi$
$367$			11089.3i			1.57727i	−0.614861	+	0.788636i	$0.710788\pi$
$368$		0			0
$369$	−2357.30	−	631.057i	−0.332564	−	0.0890285i
$370$		0			0
$371$	7576.49			1.06025
$372$		0			0
$373$	1299.98			0.180456			0.0902282	−	0.995921i	$-0.471240\pi$
$373$	1299.98			0.180456			0.0902282	+	0.995921i	$0.471240\pi$
$374$		0			0
$375$	14095.0	+	18364.6i	1.94096	+	2.52891i
$376$		0			0
$377$			8305.33i			1.13461i
$378$		0			0
$379$			2953.18i			0.400250i	0.979770	+	0.200125i	$0.0641348\pi$
$379$			2953.18i			0.400250i	−0.979770	+	0.200125i	$0.935865\pi$
$380$		0			0
$381$	717.117	+	934.343i	0.0964279	+	0.125637i
$382$		0			0
$383$	2647.22			0.353176			0.176588	−	0.984285i	$-0.443494\pi$
$383$	2647.22			0.353176			0.176588	+	0.984285i	$0.443494\pi$
$384$		0			0
$385$	−4449.92			−0.589062
$386$		0			0
$387$	−11772.3	−	3151.49i	−1.54631	−	0.413952i
$388$		0			0
$389$		−	3585.34i		−	0.467310i	−0.972320	−	0.233655i	$-0.924931\pi$
$389$		−	3585.34i		−	0.467310i	0.972320	−	0.233655i	$-0.0750687\pi$
$390$		0			0
$391$			811.829i			0.105002i
$392$		0			0
$393$	6513.22	−	4998.96i	0.836001	−	0.641639i
$394$		0			0
$395$	13175.9			1.67835
$396$		0			0
$397$	6502.91			0.822094			0.411047	−	0.911614i	$-0.365163\pi$
$397$	6502.91			0.822094			0.411047	+	0.911614i	$0.365163\pi$
$398$		0			0
$399$	5376.77	−	4126.73i	0.674625	−	0.517781i
$400$		0			0
$401$			6673.49i			0.831068i	0.909578	+	0.415534i	$0.136405\pi$
$401$			6673.49i			0.831068i	−0.909578	+	0.415534i	$0.863595\pi$
$402$		0			0
$403$		−	5935.53i		−	0.733672i
$404$		0			0
$405$	7795.68	−	13516.8i	0.956470	−	1.65841i
$406$		0			0
$407$	−1055.59			−0.128560
$408$		0			0
$409$	−2660.34			−0.321627			−0.160814	−	0.986985i	$-0.551412\pi$
$409$	−2660.34			−0.321627			−0.160814	+	0.986985i	$0.551412\pi$
$410$		0			0
$411$	1719.26	+	2240.05i	0.206338	+	0.268841i
$412$		0			0
$413$			16753.7i			1.99612i
$414$		0			0
$415$		−	4374.72i		−	0.517461i
$416$		0			0
$417$	94.1285	+	122.641i	0.0110539	+	0.0144023i
$418$		0			0
$419$	8957.94			1.04445			0.522224	−	0.852808i	$-0.325102\pi$
$419$	8957.94			1.04445			0.522224	+	0.852808i	$0.325102\pi$
$420$		0			0
$421$	−12070.2			−1.39730			−0.698652	−	0.715461i	$-0.746217\pi$
$421$	−12070.2			−1.39730			−0.698652	+	0.715461i	$0.746217\pi$
$422$		0			0
$423$	2994.56	−	11186.1i	0.344209	−	1.28579i
$424$		0			0
$425$		−	2656.33i		−	0.303179i
$426$		0			0
$427$		−	13543.4i		−	1.53492i
$428$		0			0
$429$	−2780.73	+	2134.24i	−0.312948	+	0.240191i
$430$		0			0
$431$	−14197.2			−1.58667			−0.793334	−	0.608787i	$-0.791656\pi$
$431$	−14197.2			−1.58667			−0.793334	+	0.608787i	$0.791656\pi$
$432$		0			0
$433$	−7129.67			−0.791294			−0.395647	−	0.918403i	$-0.629480\pi$
$433$	−7129.67			−0.791294			−0.395647	+	0.918403i	$0.629480\pi$
$434$		0			0
$435$	−10803.2	+	8291.58i	−1.19075	+	0.913909i
$436$		0			0
$437$		−	6353.47i		−	0.695487i
$438$		0			0
$439$			5206.56i			0.566049i	0.959113	+	0.283025i	$0.0913378\pi$
$439$			5206.56i			0.566049i	−0.959113	+	0.283025i	$0.908662\pi$
$440$		0			0
$441$	−655.992	+	2450.45i	−0.0708338	+	0.264598i
$442$		0			0
$443$	−17034.3			−1.82692			−0.913459	−	0.406930i	$-0.866599\pi$
$443$	−17034.3			−1.82692			−0.913459	+	0.406930i	$0.866599\pi$
$444$		0			0
$445$	−9719.01			−1.03534
$446$		0			0
$447$	−1548.39	−	2017.42i	−0.163840	−	0.213470i
$448$		0			0
$449$		−	9312.78i		−	0.978836i	−0.872049	−	0.489418i	$-0.837209\pi$
$449$		−	9312.78i		−	0.978836i	0.872049	−	0.489418i	$-0.162791\pi$
$450$		0			0
$451$			898.905i			0.0938532i
$452$		0			0
$453$	−1255.52	−	1635.83i	−0.130219	−	0.169665i
$454$		0			0
$455$	−30348.4			−3.12693
$456$		0			0
$457$	13276.6			1.35898			0.679488	−	0.733687i	$-0.262202\pi$
$457$	13276.6			1.35898			0.679488	+	0.733687i	$0.262202\pi$
$458$		0			0
$459$	−1033.35	+	428.444i	−0.105082	+	0.0435688i
$460$		0			0
$461$			507.329i			0.0512552i	0.999672	+	0.0256276i	$0.00815842\pi$
$461$			507.329i			0.0512552i	−0.999672	+	0.0256276i	$0.991842\pi$
$462$		0			0
$463$			4468.90i			0.448569i	0.974524	+	0.224284i	$0.0720044\pi$
$463$			4468.90i			0.448569i	−0.974524	+	0.224284i	$0.927996\pi$
$464$		0			0
$465$	7720.69	−	5925.70i	0.769975	−	0.590963i
$466$		0			0
$467$	−12258.9			−1.21472			−0.607358	−	0.794428i	$-0.707771\pi$
$467$	−12258.9			−1.21472			−0.607358	+	0.794428i	$0.707771\pi$
$468$		0			0
$469$	−20009.6			−1.97006
$470$		0			0
$471$	10480.3	−	8043.71i	1.02528	−	0.786910i
$472$		0			0
$473$			4489.12i			0.436385i
$474$		0			0
$475$			20788.8i			2.00811i
$476$		0			0
$477$	9453.33	+	2530.68i	0.907418	+	0.242918i
$478$		0			0
$479$	−5781.56			−0.551495			−0.275747	−	0.961230i	$-0.588925\pi$
$479$	−5781.56			−0.551495			−0.275747	+	0.961230i	$0.588925\pi$
$480$		0			0
$481$	−7199.13			−0.682437
$482$		0			0
$483$	6733.30	+	8772.92i	0.634318	+	0.826463i
$484$		0			0
$485$			15849.7i			1.48391i
$486$		0			0
$487$		−	8699.77i		−	0.809495i	−0.914429	−	0.404747i	$-0.867359\pi$
$487$		−	8699.77i		−	0.809495i	0.914429	−	0.404747i	$-0.132641\pi$
$488$		0			0
$489$	5719.76	+	7452.36i	0.528950	+	0.689177i
$490$		0			0
$491$	−6084.95			−0.559287			−0.279643	−	0.960104i	$-0.590216\pi$
$491$	−6084.95			−0.559287			−0.279643	+	0.960104i	$0.590216\pi$
$492$		0			0
$493$	976.313			0.0891905
$494$		0			0
$495$	−5552.25	−	1486.35i	−0.504152	−	0.134963i
$496$		0			0
$497$			4684.25i			0.422772i
$498$		0			0
$499$		−	6970.12i		−	0.625301i	−0.949868	−	0.312651i	$-0.898783\pi$
$499$		−	6970.12i		−	0.625301i	0.949868	−	0.312651i	$-0.101217\pi$
$500$		0			0
$501$	−105.501	+	80.9728i	−0.00940803	+	0.00722075i
$502$		0			0
$503$	15721.2			1.39359			0.696795	−	0.717271i	$-0.254609\pi$
$503$	15721.2			1.39359			0.696795	+	0.717271i	$0.254609\pi$
$504$		0			0
$505$	−1133.26			−0.0998604
$506$		0			0
$507$	−9908.43	+	7604.82i	−0.867947	+	0.666157i
$508$		0			0
$509$			11249.3i			0.979602i	0.871834	+	0.489801i	$0.162931\pi$
$509$			11249.3i			0.979602i	−0.871834	+	0.489801i	$0.837069\pi$
$510$		0			0
$511$			2267.47i			0.196296i
$512$		0			0
$513$	8087.11	−	3353.05i	0.696013	−	0.288579i
$514$		0			0
$515$	17301.0			1.48033
$516$		0			0
$517$	−4265.58			−0.362863
$518$		0			0
$519$	2142.70	+	2791.75i	0.181222	+	0.236116i
$520$		0			0
$521$			12276.2i			1.03231i	0.856496	+	0.516153i	$0.172636\pi$
$521$			12276.2i			1.03231i	−0.856496	+	0.516153i	$0.827364\pi$
$522$		0			0
$523$			770.278i			0.0644013i	0.999481	+	0.0322007i	$0.0102516\pi$
$523$			770.278i			0.0644013i	−0.999481	+	0.0322007i	$0.989748\pi$
$524$		0			0
$525$	−22031.6	−	28705.3i	−1.83150	−	2.38629i
$526$		0			0
$527$	−697.737			−0.0576735
$528$		0			0
$529$	−1800.47			−0.147980
$530$		0			0
$531$	−5596.05	+	20904.0i	−0.457341	+	1.70839i
$532$		0			0
$533$			6130.51i			0.498202i
$534$		0			0
$535$			15583.6i			1.25933i
$536$		0			0
$537$	13357.0	−	10251.6i	1.07337	−	0.823820i
$538$		0			0
$539$	934.423			0.0746725
$540$		0			0
$541$	−14469.7			−1.14991			−0.574956	−	0.818184i	$-0.694981\pi$
$541$	−14469.7			−1.14991			−0.574956	+	0.818184i	$0.694981\pi$
$542$		0			0
$543$	8753.90	−	6718.70i	0.691834	−	0.530989i
$544$		0			0
$545$			31867.9i			2.50472i
$546$		0			0
$547$		−	1150.14i		−	0.0899021i	−0.998989	−	0.0449511i	$-0.985687\pi$
$547$		−	1150.14i		−	0.0899021i	0.998989	−	0.0449511i	$-0.0143132\pi$
$548$		0			0
$549$	4523.74	−	16898.3i	0.351673	−	1.31367i
$550$		0			0
$551$	−7640.74			−0.590756
$552$		0			0
$553$	−12867.5			−0.989479
$554$		0			0
$555$	−7187.21	−	9364.33i	−0.549694	−	0.716205i
$556$		0			0
$557$		−	24347.5i		−	1.85213i	−0.377361	−	0.926066i	$-0.623168\pi$
$557$		−	24347.5i		−	1.85213i	0.377361	−	0.926066i	$-0.376832\pi$
$558$		0			0
$559$			30615.7i			2.31647i
$560$		0			0
$561$	250.885	+	326.882i	0.0188812	+	0.0246007i
$562$		0			0
$563$	15515.2			1.16144			0.580718	−	0.814105i	$-0.302772\pi$
$563$	15515.2			1.16144			0.580718	+	0.814105i	$0.302772\pi$
$564$		0			0
$565$	−813.518			−0.0605752
$566$		0			0
$567$	−7613.23	+	13200.5i	−0.563890	+	0.977724i
$568$		0			0
$569$			10960.4i			0.807531i	0.914863	+	0.403765i	$0.132299\pi$
$569$			10960.4i			0.807531i	−0.914863	+	0.403765i	$0.867701\pi$
$570$		0			0
$571$			24367.3i			1.78588i	0.450174	+	0.892941i	$0.351362\pi$
$571$			24367.3i			1.78588i	−0.450174	+	0.892941i	$0.648638\pi$
$572$		0			0
$573$	12698.8	−	9746.46i	0.925830	−	0.710583i
$574$		0			0
$575$	−33919.7			−2.46008
$576$		0			0
$577$	17109.4			1.23445			0.617223	−	0.786788i	$-0.288258\pi$
$577$	17109.4			1.23445			0.617223	+	0.786788i	$0.288258\pi$
$578$		0			0
$579$	10775.3	−	8270.17i	0.773416	−	0.593604i
$580$		0			0
$581$			4272.34i			0.305071i
$582$		0			0
$583$		−	3604.82i		−	0.256083i
$584$		0			0
$585$	−37866.2	−	10136.9i	−2.67620	−	0.716426i
$586$		0			0
$587$	13267.9			0.932921			0.466460	−	0.884542i	$-0.345529\pi$
$587$	13267.9			0.932921			0.466460	+	0.884542i	$0.345529\pi$
$588$		0			0
$589$	5460.57			0.382002
$590$		0			0
$591$	13361.9	+	17409.4i	0.930007	+	1.21172i
$592$		0			0
$593$		−	16312.5i		−	1.12963i	−0.825216	−	0.564817i	$-0.808947\pi$
$593$		−	16312.5i		−	1.12963i	0.825216	−	0.564817i	$-0.191053\pi$
$594$		0			0
$595$			3567.53i			0.245806i
$596$		0			0
$597$	2477.40	+	3227.84i	0.169838	+	0.221284i
$598$		0			0
$599$	15038.9			1.02583			0.512915	−	0.858440i	$-0.328566\pi$
$599$	15038.9			1.02583			0.512915	+	0.858440i	$0.328566\pi$
$600$		0			0
$601$	−4408.70			−0.299226			−0.149613	−	0.988745i	$-0.547803\pi$
$601$	−4408.70			−0.299226			−0.149613	+	0.988745i	$0.547803\pi$
$602$		0			0
$603$	−24966.3	−	6683.57i	−1.68608	−	0.451370i
$604$		0			0
$605$		−	26372.0i		−	1.77219i
$606$		0			0
$607$			1271.09i			0.0849950i	0.999097	+	0.0424975i	$0.0135315\pi$
$607$			1271.09i			0.0849950i	−0.999097	+	0.0424975i	$0.986469\pi$
$608$		0			0
$609$	10550.4	−	8097.53i	0.702009	−	0.538799i
$610$		0			0
$611$	−29091.2			−1.92619
$612$		0			0
$613$	21306.8			1.40387			0.701937	−	0.712239i	$-0.252319\pi$
$613$	21306.8			1.40387			0.701937	+	0.712239i	$0.252319\pi$
$614$		0			0
$615$	−7974.31	+	6120.36i	−0.522854	+	0.401295i
$616$		0			0
$617$			26964.4i			1.75940i	0.475533	+	0.879698i	$0.342255\pi$
$617$			26964.4i			1.75940i	−0.475533	+	0.879698i	$0.657745\pi$
$618$		0			0
$619$		−	17633.6i		−	1.14500i	−0.819905	−	0.572500i	$-0.805974\pi$
$619$		−	17633.6i		−	1.14500i	0.819905	−	0.572500i	$-0.194026\pi$
$620$		0			0
$621$	5470.95	+	13195.2i	0.353529	+	0.852665i
$622$		0			0
$623$	9491.56			0.610387
$624$		0			0
$625$	53718.1			3.43796
$626$		0			0
$627$	−1963.46	−	2558.22i	−0.125060	−	0.162943i
$628$		0			0
$629$			846.277i			0.0536459i
$630$		0			0
$631$			168.115i			0.0106062i	0.999986	+	0.00530312i	$0.00168804\pi$
$631$			168.115i			0.0106062i	−0.999986	+	0.00530312i	$0.998312\pi$
$632$		0			0
$633$	11122.3	+	14491.4i	0.698373	+	0.909921i
$634$		0			0
$635$	4851.75			0.303206
$636$		0			0
$637$	6372.74			0.396385
$638$		0			0
$639$	−1564.63	+	5844.63i	−0.0968633	+	0.361831i
$640$		0			0
$641$			823.334i			0.0507328i	0.999678	+	0.0253664i	$0.00807525\pi$
$641$			823.334i			0.0507328i	−0.999678	+	0.0253664i	$0.991925\pi$
$642$		0			0
$643$			17860.7i			1.09543i	0.836666	+	0.547713i	$0.184501\pi$
$643$			17860.7i			1.09543i	−0.836666	+	0.547713i	$0.815499\pi$
$644$		0			0
$645$	−39823.6	+	30565.0i	−2.43109	+	1.86589i
$646$		0			0
$647$	2518.14			0.153011			0.0765057	−	0.997069i	$-0.475624\pi$
$647$	2518.14			0.153011			0.0765057	+	0.997069i	$0.475624\pi$
$648$		0			0
$649$	7971.26			0.482126
$650$		0			0
$651$	−7540.01	+	5787.03i	−0.453942	+	0.348405i
$652$		0			0
$653$			18937.3i			1.13488i	0.823416	+	0.567439i	$0.192066\pi$
$653$			18937.3i			1.13488i	−0.823416	+	0.567439i	$0.807934\pi$
$654$		0			0
$655$		−	33821.1i		−	2.01756i
$656$		0			0
$657$	−757.377	+	2829.17i	−0.0449743	+	0.168001i
$658$		0			0
$659$	−10400.8			−0.614807			−0.307404	−	0.951579i	$-0.599460\pi$
$659$	−10400.8			−0.614807			−0.307404	+	0.951579i	$0.599460\pi$
$660$		0			0
$661$	−3225.88			−0.189822			−0.0949108	−	0.995486i	$-0.530257\pi$
$661$	−3225.88			−0.189822			−0.0949108	+	0.995486i	$0.530257\pi$
$662$		0			0
$663$	1711.03	+	2229.33i	0.100228	+	0.130588i
$664$		0			0
$665$		−	27919.9i		−	1.62810i
$666$		0			0
$667$		−	12466.9i		−	0.723717i
$668$		0			0
$669$	−5897.43	−	7683.85i	−0.340819	−	0.444058i
$670$		0			0
$671$	−6443.81			−0.370731
$672$		0			0
$673$	1215.38			0.0696128			0.0348064	−	0.999394i	$-0.488919\pi$
$673$	1215.38			0.0696128			0.0348064	+	0.999394i	$0.488919\pi$
$674$		0			0
$675$	−17901.2	−	43175.1i	−1.02076	−	2.46194i
$676$		0			0
$677$		−	3901.92i		−	0.221511i	−0.993848	−	0.110755i	$-0.964673\pi$
$677$		−	3901.92i		−	0.221511i	0.993848	−	0.110755i	$-0.0353270\pi$
$678$		0			0
$679$		−	15478.8i		−	0.874846i
$680$		0			0
$681$	11836.4	−	9084.58i	0.666040	−	0.511192i
$682$		0			0
$683$	31547.8			1.76742			0.883708	−	0.468039i	$-0.155040\pi$
$683$	31547.8			1.76742			0.883708	+	0.468039i	$0.155040\pi$
$684$		0			0
$685$	11631.9			0.648805
$686$		0			0
$687$	12566.3	−	9644.76i	0.697867	−	0.535620i
$688$		0			0
$689$		−	24584.8i		−	1.35937i
$690$		0			0
$691$			2791.51i			0.153682i	0.997043	+	0.0768408i	$0.0244833\pi$
$691$			2791.51i			0.153682i	−0.997043	+	0.0768408i	$0.975517\pi$
$692$		0			0
$693$	5422.31	+	1451.57i	0.297224	+	0.0795679i
$694$		0			0
$695$	636.838			0.0347578
$696$		0			0
$697$	720.657			0.0391633
$698$		0			0
$699$	6575.62	+	8567.48i	0.355812	+	0.463593i
$700$		0			0
$701$			9830.35i			0.529653i	0.964296	+	0.264827i	$0.0853148\pi$
$701$			9830.35i			0.529653i	−0.964296	+	0.264827i	$0.914685\pi$
$702$		0			0
$703$		−	6623.06i		−	0.355325i
$704$		0			0
$705$	−29043.0	−	37840.6i	−1.55152	−	2.02150i
$706$		0			0
$707$	1106.74			0.0588730
$708$		0			0
$709$	−3029.84			−0.160491			−0.0802455	−	0.996775i	$-0.525570\pi$
$709$	−3029.84			−0.160491			−0.0802455	+	0.996775i	$0.525570\pi$
$710$		0			0
$711$	−16055.0	−	4297.98i	−0.846850	−	0.226704i
$712$		0			0
$713$			8909.65i			0.467979i
$714$		0			0
$715$			14439.4i			0.755251i
$716$		0			0
$717$	11454.2	−	8791.24i	0.596606	−	0.457901i
$718$		0			0
$719$	−28359.2			−1.47096			−0.735479	−	0.677548i	$-0.763043\pi$
$719$	−28359.2			−1.47096			−0.735479	+	0.677548i	$0.763043\pi$
$720$		0			0
$721$	−16896.1			−0.872735
$722$		0			0
$723$	11702.3	−	8981.64i	0.601955	−	0.462006i
$724$		0			0
$725$			40792.1i			2.08963i
$726$		0			0
$727$			24702.5i			1.26020i	0.776514	+	0.630100i	$0.216986\pi$
$727$			24702.5i			1.26020i	−0.776514	+	0.630100i	$0.783014\pi$
$728$		0			0
$729$	−13908.4	+	13927.6i	−0.706619	+	0.707594i
$730$		0			0
$731$	3598.96			0.182096
$732$		0			0
$733$	30507.1			1.53725			0.768627	−	0.639698i	$-0.220940\pi$
$733$	30507.1			1.53725			0.768627	+	0.639698i	$0.220940\pi$
$734$		0			0
$735$	6362.19	+	8289.40i	0.319283	+	0.415999i
$736$		0			0
$737$			9520.36i			0.475831i
$738$		0			0
$739$			4105.52i			0.204363i	0.994766	+	0.102181i	$0.0325822\pi$
$739$			4105.52i			0.204363i	−0.994766	+	0.102181i	$0.967418\pi$
$740$		0			0
$741$	−13390.7	−	17447.0i	−0.663860	−	0.864954i
$742$		0			0
$743$	−4140.87			−0.204460			−0.102230	−	0.994761i	$-0.532598\pi$
$743$	−4140.87			−0.204460			−0.102230	+	0.994761i	$0.532598\pi$
$744$		0			0
$745$	−10475.8			−0.515175
$746$		0			0
$747$	−1427.04	+	5330.68i	−0.0698964	+	0.261097i
$748$		0			0
$749$		−	15218.9i		−	0.742440i
$750$		0			0
$751$		−	22033.0i		−	1.07057i	−0.844672	−	0.535284i	$-0.820204\pi$
$751$		−	22033.0i		−	1.07057i	0.844672	−	0.535284i	$-0.179796\pi$
$752$		0			0
$753$	−22008.5	+	16891.7i	−1.06512	+	0.817487i
$754$		0			0
$755$	−8494.36			−0.409459
$756$		0			0
$757$	−37525.3			−1.80169			−0.900847	−	0.434137i	$-0.857053\pi$
$757$	−37525.3			−1.80169			−0.900847	+	0.434137i	$0.857053\pi$
$758$		0			0
$759$	4174.07	−	3203.64i	0.199617	−	0.153208i
$760$		0			0
$761$			32680.8i			1.55674i	0.627806	+	0.778369i	$0.283953\pi$
$761$			32680.8i			1.55674i	−0.627806	+	0.778369i	$0.716047\pi$
$762$		0			0
$763$		−	31122.1i		−	1.47667i
$764$		0			0
$765$	−1191.62	+	4451.27i	−0.0563177	+	0.210374i
$766$		0			0
$767$	54363.8			2.55928
$768$		0			0
$769$	−27457.2			−1.28756			−0.643779	−	0.765212i	$-0.722634\pi$
$769$	−27457.2			−1.28756			−0.643779	+	0.765212i	$0.722634\pi$
$770$		0			0
$771$	19690.8	+	25655.4i	0.919775	+	1.19839i
$772$		0			0
$773$			8851.25i			0.411847i	0.978568	+	0.205923i	$0.0660197\pi$
$773$			8851.25i			0.411847i	−0.978568	+	0.205923i	$0.933980\pi$
$774$		0			0
$775$		−	29152.7i		−	1.35122i
$776$		0			0
$777$	7019.01	+	9145.18i	0.324074	+	0.422241i
$778$		0			0
$779$	−5639.95			−0.259399
$780$		0			0
$781$	2228.72			0.102113
$782$		0			0
$783$	15868.7	−	6579.42i	0.724265	−	0.300293i
$784$		0			0
$785$		−	54420.8i		−	2.47434i
$786$		0			0
$787$			3892.16i			0.176290i	0.996108	+	0.0881452i	$0.0280939\pi$
$787$			3892.16i			0.176290i	−0.996108	+	0.0881452i	$0.971906\pi$
$788$		0			0
$789$	19702.5	−	15121.8i	0.889007	−	0.682321i
$790$		0			0
$791$	794.479			0.0357123
$792$		0			0
$793$	−43946.6			−1.96796
$794$		0			0
$795$	31978.8	−	24544.1i	1.42663	−	1.09495i
$796$		0			0
$797$		−	16462.8i		−	0.731672i	−0.930679	−	0.365836i	$-0.880783\pi$
$797$		−	16462.8i		−	0.731672i	0.930679	−	0.365836i	$-0.119217\pi$
$798$		0			0
$799$			3419.74i			0.151417i
$800$		0			0
$801$	11842.8	+	3170.35i	0.522403	+	0.139849i
$802$		0			0
$803$	1078.84			0.0474116
$804$		0			0
$805$	45555.0			1.99454
$806$		0			0
$807$	−12389.1	−	16142.0i	−0.540418	−	0.704119i
$808$		0			0
$809$			2287.89i			0.0994287i	0.998763	+	0.0497144i	$0.0158311\pi$
$809$			2287.89i			0.0994287i	−0.998763	+	0.0497144i	$0.984169\pi$
$810$		0			0
$811$		−	2892.28i		−	0.125230i	−0.998038	−	0.0626150i	$-0.980056\pi$
$811$		−	2892.28i		−	0.125230i	0.998038	−	0.0626150i	$-0.0199440\pi$
$812$		0			0
$813$	20288.1	+	26433.6i	0.875195	+	1.14031i
$814$		0			0
$815$	38697.8			1.66322
$816$		0			0
$817$	−28165.8			−1.20612
$818$		0			0
$819$	36980.0	+	9899.67i	1.57776	+	0.422372i
$820$		0			0
$821$		−	37351.7i		−	1.58780i	−0.608049	−	0.793899i	$-0.708048\pi$
$821$		−	37351.7i		−	1.58780i	0.608049	−	0.793899i	$-0.291952\pi$
$822$		0			0
$823$		−	25859.6i		−	1.09527i	−0.836717	−	0.547636i	$-0.815528\pi$
$823$		−	25859.6i		−	1.09527i	0.836717	−	0.547636i	$-0.184472\pi$
$824$		0			0
$825$	−13657.7	+	10482.4i	−0.576364	+	0.442365i
$826$		0			0
$827$	32182.0			1.35318			0.676589	−	0.736361i	$-0.263457\pi$
$827$	32182.0			1.35318			0.676589	+	0.736361i	$0.263457\pi$
$828$		0			0
$829$	−16710.7			−0.700105			−0.350053	−	0.936730i	$-0.613836\pi$
$829$	−16710.7			−0.700105			−0.350053	+	0.936730i	$0.613836\pi$
$830$		0			0
$831$	−15437.9	+	11848.7i	−0.644445	+	0.494617i
$832$		0			0
$833$		−	749.133i		−	0.0311595i
$834$		0			0
$835$			547.831i			0.0227048i
$836$		0			0
$837$	−11340.8	+	4702.08i	−0.468333	+	0.194179i
$838$		0			0
$839$	19182.4			0.789334			0.394667	−	0.918824i	$-0.370860\pi$
$839$	19182.4			0.789334			0.394667	+	0.918824i	$0.370860\pi$
$840$		0			0
$841$	9396.22			0.385265
$842$		0			0
$843$	21058.9	+	27438.0i	0.860388	+	1.12101i
$844$		0			0
$845$			51451.3i			2.09465i
$846$		0			0
$847$			25754.8i			1.04480i
$848$		0			0
$849$	20439.9	+	26631.4i	0.826260	+	1.07655i
$850$		0			0
$851$	10806.4			0.435298
$852$		0			0
$853$	−16306.8			−0.654552			−0.327276	−	0.944929i	$-0.606131\pi$
$853$	−16306.8			−0.654552			−0.327276	+	0.944929i	$0.606131\pi$
$854$		0			0
$855$	9325.75	−	34836.2i	0.373022	−	1.39342i
$856$		0			0
$857$		−	42374.8i		−	1.68903i	−0.535534	−	0.844514i	$-0.679890\pi$
$857$		−	42374.8i		−	1.68903i	0.535534	−	0.844514i	$-0.320110\pi$
$858$		0			0
$859$		−	21561.9i		−	0.856442i	−0.903674	−	0.428221i	$-0.859140\pi$
$859$		−	21561.9i		−	0.856442i	0.903674	−	0.428221i	$-0.140860\pi$
$860$		0			0
$861$	7787.69	−	5977.12i	0.308251	−	0.236585i
$862$		0			0
$863$	−16679.1			−0.657894			−0.328947	−	0.944348i	$-0.606694\pi$
$863$	−16679.1			−0.657894			−0.328947	+	0.944348i	$0.606694\pi$
$864$		0			0
$865$	14496.7			0.569829
$866$		0			0
$867$	−19989.4	+	15342.1i	−0.783018	+	0.600974i
$868$		0			0
$869$			6122.23i			0.238990i
$870$		0			0
$871$			64928.7i			2.52586i
$872$		0			0
$873$	5170.19	−	19313.2i	0.200440	−	0.748741i
$874$		0			0
$875$	−93129.6			−3.59812
$876$		0			0
$877$	14094.6			0.542690			0.271345	−	0.962482i	$-0.412532\pi$
$877$	14094.6			0.542690			0.271345	+	0.962482i	$0.412532\pi$
$878$		0			0
$879$	−8291.53	−	10803.2i	−0.318164	−	0.414541i
$880$		0			0
$881$			12553.3i			0.480057i	0.970766	+	0.240028i	$0.0771568\pi$
$881$			12553.3i			0.480057i	−0.970766	+	0.240028i	$0.922843\pi$
$882$		0			0
$883$		−	43192.3i		−	1.64613i	−0.567944	−	0.823067i	$-0.692261\pi$
$883$		−	43192.3i		−	1.64613i	0.567944	−	0.823067i	$-0.307739\pi$
$884$		0			0
$885$	54273.8	+	70714.2i	2.06146	+	2.68591i
$886$		0			0
$887$	−17930.0			−0.678725			−0.339362	−	0.940656i	$-0.610211\pi$
$887$	−17930.0			−0.678725			−0.339362	+	0.940656i	$0.610211\pi$
$888$		0			0
$889$	−4738.20			−0.178756
$890$		0			0
$891$	6280.67	+	3622.30i	0.236151	+	0.136197i
$892$		0			0
$893$		−	26763.3i		−	1.00291i
$894$		0			0
$895$		−	69358.8i		−	2.59040i
$896$		0			0
$897$	28467.1	−	21848.7i	1.05963	−	0.813276i
$898$		0			0
$899$	10714.8			0.397508
$900$		0			0
$901$	−2890.00			−0.106859
$902$		0			0
$903$	38891.6	−	29849.7i	1.43326	−	1.10004i
$904$		0			0
$905$		−	45456.2i		−	1.66963i
$906$		0			0
$907$			5289.85i			0.193657i	0.995301	+	0.0968283i	$0.0308698\pi$
$907$			5289.85i			0.193657i	−0.995301	+	0.0968283i	$0.969130\pi$
$908$		0			0
$909$	1380.90	+	369.671i	0.0503868	+	0.0134887i
$910$		0			0
$911$	4585.72			0.166774			0.0833872	−	0.996517i	$-0.473426\pi$
$911$	4585.72			0.166774			0.0833872	+	0.996517i	$0.473426\pi$
$912$		0			0
$913$	2032.74			0.0736843
$914$		0			0
$915$	−43873.9	−	57163.9i	−1.58516	−	2.06533i
$916$		0			0
$917$			33029.6i			1.18946i
$918$		0			0
$919$			43234.5i			1.55188i	0.630809	+	0.775939i	$0.282723\pi$
$919$			43234.5i			1.55188i	−0.630809	+	0.775939i	$0.717277\pi$
$920$		0			0
$921$	3012.03	+	3924.42i	0.107763	+	0.140406i
$922$		0			0
$923$	15199.8			0.542046
$924$		0			0
$925$	−35358.9			−1.25686
$926$		0			0
$927$	−21081.5	−	5643.59i	−0.746935	−	0.199957i
$928$		0			0
$929$		−	695.704i		−	0.0245697i	−0.999925	−	0.0122849i	$-0.996090\pi$
$929$		−	695.704i		−	0.0245697i	0.999925	−	0.0122849i	$-0.00391050\pi$
$930$		0			0
$931$			5862.80i			0.206386i
$932$		0			0
$933$	−1916.60	+	1471.01i	−0.0672525	+	0.0516169i
$934$		0			0
$935$	1697.39			0.0593698
$936$		0			0
$937$	−21092.2			−0.735381			−0.367691	−	0.929948i	$-0.619851\pi$
$937$	−21092.2			−0.735381			−0.367691	+	0.929948i	$0.619851\pi$
$938$		0			0
$939$	−23314.8	+	17894.3i	−0.810277	+	0.621896i
$940$		0			0
$941$		−	11392.6i		−	0.394676i	−0.980336	−	0.197338i	$-0.936770\pi$
$941$		−	11392.6i		−	0.394676i	0.980336	−	0.197338i	$-0.0632296\pi$
$942$		0			0
$943$		−	9202.33i		−	0.317783i
$944$		0			0
$945$	24041.7	+	57985.4i	0.827595	+	1.99605i
$946$		0			0
$947$	3540.41			0.121487			0.0607433	−	0.998153i	$-0.480653\pi$
$947$	3540.41			0.121487			0.0607433	+	0.998153i	$0.480653\pi$
$948$		0			0
$949$	7357.67			0.251676
$950$		0			0
$951$	−14532.6	−	18934.8i	−0.495535	−	0.645640i
$952$		0			0
$953$		−	29360.7i		−	0.997992i	−0.866604	−	0.498996i	$-0.833702\pi$
$953$		−	29360.7i		−	0.997992i	0.866604	−	0.498996i	$-0.166298\pi$
$954$		0			0
$955$		−	65940.9i		−	2.23434i
$956$		0			0
$957$	−3852.72	−	5019.77i	−0.130137	−	0.169557i
$958$		0			0
$959$	−11359.7			−0.382505
$960$		0			0
$961$	22133.5			0.742959
$962$		0			0
$963$	5083.40	−	18989.0i	0.170104	−	0.635421i
$964$		0			0
$965$		−	55952.9i		−	1.86652i
$966$		0			0
$967$			27094.7i			0.901043i	0.892766	+	0.450521i	$0.148762\pi$
$967$			27094.7i			0.901043i	−0.892766	+	0.450521i	$0.851238\pi$
$968$		0			0
$969$	−2050.94	+	1574.11i	−0.0679934	+	0.0521856i
$970$		0			0
$971$	−10806.5			−0.357154			−0.178577	−	0.983926i	$-0.557149\pi$
$971$	−10806.5			−0.357154			−0.178577	+	0.983926i	$0.557149\pi$
$972$		0			0
$973$	−621.934			−0.0204916
$974$		0			0
$975$	−93145.2	+	71489.9i	−3.05952	+	2.34821i
$976$		0			0
$977$			14327.6i			0.469172i	0.972095	+	0.234586i	$0.0753734\pi$
$977$			14327.6i			0.469172i	−0.972095	+	0.234586i	$0.924627\pi$
$978$		0			0
$979$		−	4515.99i		−	0.147428i
$980$		0			0
$981$	10395.4	−	38831.7i	0.338327	−	1.26381i
$982$		0			0
$983$	25691.9			0.833617			0.416808	−	0.908994i	$-0.363149\pi$
$983$	25691.9			0.833617			0.416808	+	0.908994i	$0.363149\pi$
$984$		0			0
$985$	90401.5			2.92430
$986$		0			0
$987$	28363.3	+	36955.0i	0.914706	+	1.19178i
$988$		0			0
$989$		−	45956.3i		−	1.47758i
$990$		0			0
$991$			49378.3i			1.58280i	0.611300	+	0.791399i	$0.290647\pi$
$991$			49378.3i			1.58280i	−0.611300	+	0.791399i	$0.709353\pi$
$992$		0			0
$993$	−21225.3	−	27654.7i	−0.678311	−	0.883782i
$994$		0			0
$995$	16761.1			0.534034
$996$		0			0
$997$	22705.1			0.721241			0.360621	−	0.932713i	$-0.382565\pi$
$997$	22705.1			0.721241			0.360621	+	0.932713i	$0.382565\pi$
$998$		0			0
$999$	5703.10	+	13755.1i	0.180619	+	0.435627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.c.a.383.4	yes	12
3.2	odd	2		384.4.c.d.383.10	yes	12
4.3	odd	2		384.4.c.d.383.9	yes	12
8.3	odd	2		384.4.c.b.383.4	yes	12
8.5	even	2		384.4.c.c.383.9	yes	12
12.11	even	2	inner	384.4.c.a.383.3	✓	12
16.3	odd	4		768.4.f.f.383.10		12
16.5	even	4		768.4.f.e.383.10		12
16.11	odd	4		768.4.f.g.383.3		12
16.13	even	4		768.4.f.h.383.3		12
24.5	odd	2		384.4.c.b.383.3	yes	12
24.11	even	2		384.4.c.c.383.10	yes	12
48.5	odd	4		768.4.f.f.383.9		12
48.11	even	4		768.4.f.h.383.4		12
48.29	odd	4		768.4.f.g.383.4		12
48.35	even	4		768.4.f.e.383.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.3	✓	12	12.11	even	2	inner
384.4.c.a.383.4	yes	12	1.1	even	1	trivial
384.4.c.b.383.3	yes	12	24.5	odd	2
384.4.c.b.383.4	yes	12	8.3	odd	2
384.4.c.c.383.9	yes	12	8.5	even	2
384.4.c.c.383.10	yes	12	24.11	even	2
384.4.c.d.383.9	yes	12	4.3	odd	2
384.4.c.d.383.10	yes	12	3.2	odd	2
768.4.f.e.383.9		12	48.35	even	4
768.4.f.e.383.10		12	16.5	even	4
768.4.f.f.383.9		12	48.5	odd	4
768.4.f.f.383.10		12	16.3	odd	4
768.4.f.g.383.3		12	16.11	odd	4
768.4.f.g.383.4		12	48.29	odd	4
768.4.f.h.383.3		12	16.13	even	4
768.4.f.h.383.4		12	48.11	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.12202 + 3.16369i) q^{3} +21.4043i q^{5} -20.9034i q^{7} +(6.98212 - 26.0816i) q^{9} +O(q^{10})\)	\(q+(-4.12202 + 3.16369i) q^{3} +21.4043i q^{5} -20.9034i q^{7} +(6.98212 - 26.0816i) q^{9} -9.94564 q^{11} -67.8290 q^{13} +(-67.7167 - 88.2292i) q^{15} +7.97348i q^{17} -62.4014i q^{19} +(66.1320 + 86.1644i) q^{21} +101.816 q^{23} -333.146 q^{25} +(53.7337 + 129.598i) q^{27} -122.445i q^{29} +87.5072i q^{31} +(40.9961 - 31.4649i) q^{33} +447.424 q^{35} +106.136 q^{37} +(279.593 - 214.590i) q^{39} -90.3818i q^{41} -451.366i q^{43} +(558.260 + 149.448i) q^{45} +428.890 q^{47} -93.9530 q^{49} +(-25.2256 - 32.8669i) q^{51} +362.452i q^{53} -212.880i q^{55} +(197.419 + 257.220i) q^{57} -801.483 q^{59} +647.903 q^{61} +(-545.195 - 145.950i) q^{63} -1451.84i q^{65} -957.240i q^{67} +(-419.688 + 322.115i) q^{69} -224.090 q^{71} -108.474 q^{73} +(1373.24 - 1053.97i) q^{75} +207.898i q^{77} -615.569i q^{79} +(-631.500 - 364.210i) q^{81} -204.385 q^{83} -170.667 q^{85} +(387.378 + 504.721i) q^{87} +454.067i q^{89} +1417.86i q^{91} +(-276.846 - 360.707i) q^{93} +1335.66 q^{95} +740.490 q^{97} +(-69.4417 + 259.398i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{3}+O(q^{10}) \) 12 * q - 10 * q^3	\( 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) \) 12 * q - 10 * q^3 - 36 * q^11 - 84 * q^15 + 136 * q^21 + 120 * q^23 - 300 * q^25 + 266 * q^27 - 116 * q^33 - 432 * q^35 + 528 * q^37 - 620 * q^39 + 440 * q^45 + 1248 * q^47 - 948 * q^49 + 1072 * q^51 - 172 * q^57 - 2508 * q^59 + 624 * q^61 - 2744 * q^63 - 24 * q^69 + 2040 * q^71 - 216 * q^73 + 3894 * q^75 - 1076 * q^81 - 4572 * q^83 + 480 * q^85 - 4156 * q^87 - 112 * q^93 + 5448 * q^95 - 48 * q^97 + 6044 * q^99

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