LMFDB - Embedded newform 240.3.c.c.209.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,3,Mod(209,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("240.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	240.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.53952634465$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 16x^{2} + 81 $ x^4 + 16*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.3
Root			$0.707107 - 2.91548i$ of defining polynomial
Character	$\chi$	$=$	240.209
Dual form			240.3.c.c.209.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 2.91548i) q^{3} +(-2.82843 - 4.12311i) q^{5} +5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} +O(q^{10})$	$q+(0.707107 - 2.91548i) q^{3} +(-2.82843 - 4.12311i) q^{5} +5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} -16.4924i q^{11} +(-14.0208 + 5.33074i) q^{15} -11.3137 q^{17} -12.0000 q^{19} +(17.0000 + 4.12311i) q^{21} -24.0416 q^{23} +(-9.00000 + 23.3238i) q^{25} +(-17.6777 + 20.4083i) q^{27} +32.0000 q^{31} +(-48.0833 - 11.6619i) q^{33} +(24.0416 - 16.4924i) q^{35} -23.3238i q^{37} -57.7235i q^{41} -40.8167i q^{43} +(5.62742 + 44.6467i) q^{45} +35.3553 q^{47} +15.0000 q^{49} +(-8.00000 + 32.9848i) q^{51} +67.8823 q^{53} +(-68.0000 + 46.6476i) q^{55} +(-8.48528 + 34.9857i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +(24.0416 - 46.6476i) q^{63} -5.83095i q^{67} +(-17.0000 + 70.0928i) q^{69} -116.619i q^{73} +(61.6360 + 42.7317i) q^{75} +96.1665 q^{77} +72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} +43.8406 q^{83} +(32.0000 + 46.6476i) q^{85} -65.9697i q^{89} +(22.6274 - 93.2952i) q^{93} +(33.9411 + 49.4773i) q^{95} +163.267i q^{97} +(-68.0000 + 131.939i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{9}+O(q^{10}) $ 4 * q - 32 * q^9	$ 4 q - 32 q^{9} - 8 q^{15} - 48 q^{19} + 68 q^{21} - 36 q^{25} + 128 q^{31} - 68 q^{45} + 60 q^{49} - 32 q^{51} - 272 q^{55} - 64 q^{61} - 68 q^{69} + 272 q^{75} + 288 q^{79} + 188 q^{81} + 128 q^{85} - 272 q^{99}+O(q^{100}) $ 4 * q - 32 * q^9 - 8 * q^15 - 48 * q^19 + 68 * q^21 - 36 * q^25 + 128 * q^31 - 68 * q^45 + 60 * q^49 - 32 * q^51 - 272 * q^55 - 64 * q^61 - 68 * q^69 + 272 * q^75 + 288 * q^79 + 188 * q^81 + 128 * q^85 - 272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0.707107	−	2.91548i	0.235702	−	0.971825i
$3$	0.707107	−	2.91548i	0.235702	−	0.971825i
$4$		0			0
$5$	−2.82843	−	4.12311i	−0.565685	−	0.824621i
$5$	−2.82843	−	4.12311i	−0.565685	−	0.824621i
$6$		0			0
$7$			5.83095i			0.832993i	0.909137	+	0.416497i	$0.136742\pi$
$7$			5.83095i			0.832993i	−0.909137	+	0.416497i	$0.863258\pi$
$8$		0			0
$9$	−8.00000	−	4.12311i	−0.888889	−	0.458123i
$10$		0			0
$11$		−	16.4924i		−	1.49931i	−0.661828	−	0.749656i	$-0.730219\pi$
$11$		−	16.4924i		−	1.49931i	0.661828	−	0.749656i	$-0.269781\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−14.0208	+	5.33074i	−0.934721	+	0.355382i
$16$		0			0
$17$	−11.3137			−0.665512			−0.332756	−	0.943013i	$-0.607979\pi$
$17$	−11.3137			−0.665512			−0.332756	+	0.943013i	$0.607979\pi$
$18$		0			0
$19$	−12.0000			−0.631579			−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000			−0.631579			−0.315789	+	0.948829i	$0.602269\pi$
$20$		0			0
$21$	17.0000	+	4.12311i	0.809524	+	0.196338i
$22$		0			0
$23$	−24.0416			−1.04529			−0.522644	−	0.852551i	$-0.675054\pi$
$23$	−24.0416			−1.04529			−0.522644	+	0.852551i	$0.675054\pi$
$24$		0			0
$25$	−9.00000	+	23.3238i	−0.360000	+	0.932952i
$26$		0			0
$27$	−17.6777	+	20.4083i	−0.654729	+	0.755864i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	32.0000			1.03226			0.516129	−	0.856511i	$-0.327372\pi$
$31$	32.0000			1.03226			0.516129	+	0.856511i	$0.327372\pi$
$32$		0			0
$33$	−48.0833	−	11.6619i	−1.45707	−	0.353391i
$34$		0			0
$35$	24.0416	−	16.4924i	0.686904	−	0.471212i
$36$		0			0
$37$		−	23.3238i		−	0.630373i	−0.949030	−	0.315187i	$-0.897933\pi$
$37$		−	23.3238i		−	0.630373i	0.949030	−	0.315187i	$-0.102067\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	57.7235i		−	1.40789i	−0.710255	−	0.703945i	$-0.751420\pi$
$41$		−	57.7235i		−	1.40789i	0.710255	−	0.703945i	$-0.248580\pi$
$42$		0			0
$43$		−	40.8167i		−	0.949225i	−0.880195	−	0.474612i	$-0.842588\pi$
$43$		−	40.8167i		−	0.949225i	0.880195	−	0.474612i	$-0.157412\pi$
$44$		0			0
$45$	5.62742	+	44.6467i	0.125054	+	0.992150i
$46$		0			0
$47$	35.3553			0.752241			0.376121	−	0.926571i	$-0.377258\pi$
$47$	35.3553			0.752241			0.376121	+	0.926571i	$0.377258\pi$
$48$		0			0
$49$	15.0000			0.306122
$50$		0			0
$51$	−8.00000	+	32.9848i	−0.156863	+	0.646762i
$52$		0			0
$53$	67.8823			1.28080			0.640399	−	0.768043i	$-0.278769\pi$
$53$	67.8823			1.28080			0.640399	+	0.768043i	$0.278769\pi$
$54$		0			0
$55$	−68.0000	+	46.6476i	−1.23636	+	0.848138i
$56$		0			0
$57$	−8.48528	+	34.9857i	−0.148865	+	0.613784i
$58$		0			0
$59$		−	16.4924i		−	0.279533i	−0.990185	−	0.139766i	$-0.955365\pi$
$59$		−	16.4924i		−	0.279533i	0.990185	−	0.139766i	$-0.0446351\pi$
$60$		0			0
$61$	−16.0000			−0.262295			−0.131148	−	0.991363i	$-0.541866\pi$
$61$	−16.0000			−0.262295			−0.131148	+	0.991363i	$0.541866\pi$
$62$		0			0
$63$	24.0416	−	46.6476i	0.381613	−	0.740438i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	5.83095i		−	0.0870291i	−0.999053	−	0.0435146i	$-0.986145\pi$
$67$		−	5.83095i		−	0.0870291i	0.999053	−	0.0435146i	$-0.0138555\pi$
$68$		0			0
$69$	−17.0000	+	70.0928i	−0.246377	+	1.01584i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	116.619i		−	1.59752i	−0.601649	−	0.798761i	$-0.705489\pi$
$73$		−	116.619i		−	1.59752i	0.601649	−	0.798761i	$-0.294511\pi$
$74$		0			0
$75$	61.6360	+	42.7317i	0.821814	+	0.569756i
$76$		0			0
$77$	96.1665			1.24892
$78$		0			0
$79$	72.0000			0.911392			0.455696	−	0.890135i	$-0.349390\pi$
$79$	72.0000			0.911392			0.455696	+	0.890135i	$0.349390\pi$
$80$		0			0
$81$	47.0000	+	65.9697i	0.580247	+	0.814441i
$82$		0			0
$83$	43.8406			0.528200			0.264100	−	0.964495i	$-0.414925\pi$
$83$	43.8406			0.528200			0.264100	+	0.964495i	$0.414925\pi$
$84$		0			0
$85$	32.0000	+	46.6476i	0.376471	+	0.548795i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	65.9697i		−	0.741232i	−0.928786	−	0.370616i	$-0.879147\pi$
$89$		−	65.9697i		−	0.741232i	0.928786	−	0.370616i	$-0.120853\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	22.6274	−	93.2952i	0.243306	−	1.00317i
$94$		0			0
$95$	33.9411	+	49.4773i	0.357275	+	0.520813i
$96$		0			0
$97$			163.267i			1.68316i	0.540131	+	0.841581i	$0.318375\pi$
$97$			163.267i			1.68316i	−0.540131	+	0.841581i	$0.681625\pi$
$98$		0			0
$99$	−68.0000	+	131.939i	−0.686869	+	1.33272i
$100$		0			0
$101$			131.939i			1.30633i	0.757215	+	0.653165i	$0.226559\pi$
$101$			131.939i			1.30633i	−0.757215	+	0.653165i	$0.773441\pi$
$102$		0			0
$103$		−	99.1262i		−	0.962390i	−0.876614	−	0.481195i	$-0.840203\pi$
$103$		−	99.1262i		−	0.962390i	0.876614	−	0.481195i	$-0.159797\pi$
$104$		0			0
$105$	−31.0833	−	81.7547i	−0.296031	−	0.778616i
$106$		0			0
$107$	55.1543			0.515461			0.257731	−	0.966217i	$-0.417025\pi$
$107$	55.1543			0.515461			0.257731	+	0.966217i	$0.417025\pi$
$108$		0			0
$109$	80.0000			0.733945			0.366972	−	0.930232i	$-0.380394\pi$
$109$	80.0000			0.733945			0.366972	+	0.930232i	$0.380394\pi$
$110$		0			0
$111$	−68.0000	−	16.4924i	−0.612613	−	0.148580i
$112$		0			0
$113$	−152.735			−1.35164			−0.675819	−	0.737068i	$-0.736210\pi$
$113$	−152.735			−1.35164			−0.675819	+	0.737068i	$0.736210\pi$
$114$		0			0
$115$	68.0000	+	99.1262i	0.591304	+	0.861967i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	65.9697i		−	0.554367i
$120$		0			0
$121$	−151.000			−1.24793
$122$		0			0
$123$	−168.291	−	40.8167i	−1.36822	−	0.331843i
$124$		0			0
$125$	121.622	−	28.8617i	0.972979	−	0.230894i
$126$		0			0
$127$			40.8167i			0.321391i	0.987004	+	0.160696i	$0.0513737\pi$
$127$			40.8167i			0.321391i	−0.987004	+	0.160696i	$0.948626\pi$
$128$		0			0
$129$	−119.000	−	28.8617i	−0.922481	−	0.223734i
$130$		0			0
$131$		−	49.4773i		−	0.377689i	−0.982007	−	0.188845i	$-0.939526\pi$
$131$		−	49.4773i		−	0.377689i	0.982007	−	0.188845i	$-0.0604742\pi$
$132$		0			0
$133$		−	69.9714i		−	0.526101i
$134$		0			0
$135$	134.146	+	15.1634i	0.993672	+	0.112322i
$136$		0			0
$137$	−50.9117			−0.371618			−0.185809	−	0.982586i	$-0.559491\pi$
$137$	−50.9117			−0.371618			−0.185809	+	0.982586i	$0.559491\pi$
$138$		0			0
$139$	−44.0000			−0.316547			−0.158273	−	0.987395i	$-0.550593\pi$
$139$	−44.0000			−0.316547			−0.158273	+	0.987395i	$0.550593\pi$
$140$		0			0
$141$	25.0000	−	103.078i	0.177305	−	0.731047i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.6066	−	43.7321i	0.0721538	−	0.297498i
$148$		0			0
$149$		−	8.24621i		−	0.0553437i	−0.999617	−	0.0276718i	$-0.991191\pi$
$149$		−	8.24621i		−	0.0553437i	0.999617	−	0.0276718i	$-0.00880935\pi$
$150$		0			0
$151$	−136.000			−0.900662			−0.450331	−	0.892862i	$-0.648694\pi$
$151$	−136.000			−0.900662			−0.450331	+	0.892862i	$0.648694\pi$
$152$		0			0
$153$	90.5097	+	46.6476i	0.591566	+	0.304886i
$154$		0			0
$155$	−90.5097	−	131.939i	−0.583933	−	0.851222i
$156$		0			0
$157$		−	116.619i		−	0.742796i	−0.928474	−	0.371398i	$-0.878878\pi$
$157$		−	116.619i		−	0.742796i	0.928474	−	0.371398i	$-0.121122\pi$
$158$		0			0
$159$	48.0000	−	197.909i	0.301887	−	1.24471i
$160$		0			0
$161$		−	140.186i		−	0.870718i
$162$		0			0
$163$		−	99.1262i		−	0.608136i	−0.952650	−	0.304068i	$-0.901655\pi$
$163$		−	99.1262i		−	0.608136i	0.952650	−	0.304068i	$-0.0983450\pi$
$164$		0			0
$165$	87.9167	+	231.237i	0.532829	+	1.40144i
$166$		0			0
$167$	−292.742			−1.75295			−0.876474	−	0.481450i	$-0.840110\pi$
$167$	−292.742			−1.75295			−0.876474	+	0.481450i	$0.840110\pi$
$168$		0			0
$169$	169.000			1.00000
$170$		0			0
$171$	96.0000	+	49.4773i	0.561404	+	0.289341i
$172$		0			0
$173$	−164.049			−0.948259			−0.474129	−	0.880455i	$-0.657237\pi$
$173$	−164.049			−0.948259			−0.474129	+	0.880455i	$0.657237\pi$
$174$		0			0
$175$	−136.000	−	52.4786i	−0.777143	−	0.299878i
$176$		0			0
$177$	−48.0833	−	11.6619i	−0.271657	−	0.0658865i
$178$		0			0
$179$		−	16.4924i		−	0.0921364i	−0.998938	−	0.0460682i	$-0.985331\pi$
$179$		−	16.4924i		−	0.0921364i	0.998938	−	0.0460682i	$-0.0146692\pi$
$180$		0			0
$181$	−82.0000			−0.453039			−0.226519	−	0.974007i	$-0.572735\pi$
$181$	−82.0000			−0.453039			−0.226519	+	0.974007i	$0.572735\pi$
$182$		0			0
$183$	−11.3137	+	46.6476i	−0.0618235	+	0.254905i
$184$		0			0
$185$	−96.1665	+	65.9697i	−0.519819	+	0.356593i
$186$		0			0
$187$			186.590i			0.997810i
$188$		0			0
$189$	−119.000	−	103.078i	−0.629630	−	0.545384i
$190$		0			0
$191$			296.864i			1.55426i	0.629340	+	0.777130i	$0.283325\pi$
$191$			296.864i			1.55426i	−0.629340	+	0.777130i	$0.716675\pi$
$192$		0			0
$193$		−	116.619i		−	0.604244i	−0.953269	−	0.302122i	$-0.902305\pi$
$193$		−	116.619i		−	0.604244i	0.953269	−	0.302122i	$-0.0976950\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−192.333			−0.976310			−0.488155	−	0.872757i	$-0.662330\pi$
$197$	−192.333			−0.976310			−0.488155	+	0.872757i	$0.662330\pi$
$198$		0			0
$199$	312.000			1.56784			0.783920	−	0.620862i	$-0.213217\pi$
$199$	312.000			1.56784			0.783920	+	0.620862i	$0.213217\pi$
$200$		0			0
$201$	−17.0000	−	4.12311i	−0.0845771	−	0.0205130i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−238.000	+	163.267i	−1.16098	+	0.796423i
$206$		0			0
$207$	192.333	+	99.1262i	0.929145	+	0.478870i
$208$		0			0
$209$			197.909i			0.946933i
$210$		0			0
$211$	12.0000			0.0568720			0.0284360	−	0.999596i	$-0.490947\pi$
$211$	12.0000			0.0568720			0.0284360	+	0.999596i	$0.490947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−168.291	+	115.447i	−0.782751	+	0.536963i
$216$		0			0
$217$			186.590i			0.859864i
$218$		0			0
$219$	−340.000	−	82.4621i	−1.55251	−	0.376539i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	40.8167i		−	0.183034i	−0.995803	−	0.0915172i	$-0.970828\pi$
$223$		−	40.8167i		−	0.183034i	0.995803	−	0.0915172i	$-0.0291716\pi$
$224$		0			0
$225$	168.167	−	149.483i	0.747407	−	0.664367i
$226$		0			0
$227$	159.806			0.703992			0.351996	−	0.936002i	$-0.385503\pi$
$227$	159.806			0.703992			0.351996	+	0.936002i	$0.385503\pi$
$228$		0			0
$229$	82.0000			0.358079			0.179039	−	0.983842i	$-0.442701\pi$
$229$	82.0000			0.358079			0.179039	+	0.983842i	$0.442701\pi$
$230$		0			0
$231$	68.0000	−	280.371i	0.294372	−	1.21373i
$232$		0			0
$233$	192.333			0.825464			0.412732	−	0.910853i	$-0.364575\pi$
$233$	192.333			0.825464			0.412732	+	0.910853i	$0.364575\pi$
$234$		0			0
$235$	−100.000	−	145.774i	−0.425532	−	0.620314i
$236$		0			0
$237$	50.9117	−	209.914i	0.214817	−	0.885714i
$238$		0			0
$239$			461.788i			1.93217i	0.258231	+	0.966083i	$0.416861\pi$
$239$			461.788i			1.93217i	−0.258231	+	0.966083i	$0.583139\pi$
$240$		0			0
$241$	304.000			1.26141			0.630705	−	0.776022i	$-0.282766\pi$
$241$	304.000			1.26141			0.630705	+	0.776022i	$0.282766\pi$
$242$		0			0
$243$	225.567	−	90.3798i	0.928260	−	0.371933i
$244$		0			0
$245$	−42.4264	−	61.8466i	−0.173169	−	0.252435i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	31.0000	−	127.816i	0.124498	−	0.513318i
$250$		0			0
$251$		−	346.341i		−	1.37984i	−0.723884	−	0.689922i	$-0.757645\pi$
$251$		−	346.341i		−	1.37984i	0.723884	−	0.689922i	$-0.242355\pi$
$252$		0			0
$253$			396.505i			1.56721i
$254$		0			0
$255$	158.627	−	60.3104i	0.622068	−	0.236511i
$256$		0			0
$257$	390.323			1.51877			0.759383	−	0.650644i	$-0.225501\pi$
$257$	390.323			1.51877			0.759383	+	0.650644i	$0.225501\pi$
$258$		0			0
$259$	136.000			0.525097
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−295.571			−1.12384			−0.561921	−	0.827191i	$-0.689938\pi$
$263$	−295.571			−1.12384			−0.561921	+	0.827191i	$0.689938\pi$
$264$		0			0
$265$	−192.000	−	279.886i	−0.724528	−	1.05617i
$266$		0			0
$267$	−192.333	−	46.6476i	−0.720348	−	0.174710i
$268$		0			0
$269$			74.2159i			0.275896i	0.990440	+	0.137948i	$0.0440506\pi$
$269$			74.2159i			0.275896i	−0.990440	+	0.137948i	$0.955949\pi$
$270$		0			0
$271$	−40.0000			−0.147601			−0.0738007	−	0.997273i	$-0.523513\pi$
$271$	−40.0000			−0.147601			−0.0738007	+	0.997273i	$0.523513\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	384.666	+	148.432i	1.39879	+	0.539752i
$276$		0			0
$277$		−	443.152i		−	1.59983i	−0.600115	−	0.799914i	$-0.704878\pi$
$277$		−	443.152i		−	1.59983i	0.600115	−	0.799914i	$-0.295122\pi$
$278$		0			0
$279$	−256.000	−	131.939i	−0.917563	−	0.472901i
$280$		0			0
$281$			519.511i			1.84879i	0.381431	+	0.924397i	$0.375431\pi$
$281$			519.511i			1.84879i	−0.381431	+	0.924397i	$0.624569\pi$
$282$		0			0
$283$			320.702i			1.13322i	0.823985	+	0.566612i	$0.191746\pi$
$283$			320.702i			1.13322i	−0.823985	+	0.566612i	$0.808254\pi$
$284$		0			0
$285$	168.250	−	63.9688i	0.590350	−	0.224452i
$286$		0			0
$287$	336.583			1.17276
$288$		0			0
$289$	−161.000			−0.557093
$290$		0			0
$291$	476.000	+	115.447i	1.63574	+	0.396725i
$292$		0			0
$293$	84.8528			0.289600			0.144800	−	0.989461i	$-0.453746\pi$
$293$	84.8528			0.289600			0.144800	+	0.989461i	$0.453746\pi$
$294$		0			0
$295$	−68.0000	+	46.6476i	−0.230508	+	0.158128i
$296$		0			0
$297$	336.583	+	291.548i	1.13328	+	0.981642i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	238.000			0.790698
$302$		0			0
$303$	384.666	+	93.2952i	1.26953	+	0.307905i
$304$		0			0
$305$	45.2548	+	65.9697i	0.148377	+	0.216294i
$306$		0			0
$307$		−	367.350i		−	1.19658i	−0.801280	−	0.598290i	$-0.795847\pi$
$307$		−	367.350i		−	1.19658i	0.801280	−	0.598290i	$-0.204153\pi$
$308$		0			0
$309$	−289.000	−	70.0928i	−0.935275	−	0.226838i
$310$		0			0
$311$			98.9545i			0.318182i	0.987264	+	0.159091i	$0.0508563\pi$
$311$			98.9545i			0.318182i	−0.987264	+	0.159091i	$0.949144\pi$
$312$		0			0
$313$		−	186.590i		−	0.596136i	−0.954545	−	0.298068i	$-0.903658\pi$
$313$		−	186.590i		−	0.596136i	0.954545	−	0.298068i	$-0.0963422\pi$
$314$		0			0
$315$	−260.333	+	32.8132i	−0.826454	+	0.104169i
$316$		0			0
$317$	−520.431			−1.64174			−0.820868	−	0.571117i	$-0.806510\pi$
$317$	−520.431			−1.64174			−0.820868	+	0.571117i	$0.806510\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	39.0000	−	160.801i	0.121495	−	0.500938i
$322$		0			0
$323$	135.765			0.420324
$324$		0			0
$325$		0			0
$326$		0			0
$327$	56.5685	−	233.238i	0.172992	−	0.713266i
$328$		0			0
$329$			206.155i			0.626612i
$330$		0			0
$331$	−292.000			−0.882175			−0.441088	−	0.897464i	$-0.645407\pi$
$331$	−292.000			−0.882175			−0.441088	+	0.897464i	$0.645407\pi$
$332$		0			0
$333$	−96.1665	+	186.590i	−0.288788	+	0.560332i
$334$		0			0
$335$	−24.0416	+	16.4924i	−0.0717661	+	0.0492311i
$336$		0			0
$337$			326.533i			0.968942i	0.874807	+	0.484471i	$0.160988\pi$
$337$			326.533i			0.968942i	−0.874807	+	0.484471i	$0.839012\pi$
$338$		0			0
$339$	−108.000	+	445.295i	−0.318584	+	1.31356i
$340$		0			0
$341$		−	527.758i		−	1.54768i
$342$		0			0
$343$			373.181i			1.08799i
$344$		0			0
$345$	337.083	−	128.160i	0.977053	−	0.371477i
$346$		0			0
$347$	−394.566			−1.13708			−0.568538	−	0.822657i	$-0.692491\pi$
$347$	−394.566			−1.13708			−0.568538	+	0.822657i	$0.692491\pi$
$348$		0			0
$349$	254.000			0.727794			0.363897	−	0.931439i	$-0.381446\pi$
$349$	254.000			0.727794			0.363897	+	0.931439i	$0.381446\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	345.068			0.977530			0.488765	−	0.872415i	$-0.337448\pi$
$353$	345.068			0.977530			0.488765	+	0.872415i	$0.337448\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−192.333	−	46.6476i	−0.538748	−	0.130666i
$358$		0			0
$359$		−	395.818i		−	1.10256i	−0.834321	−	0.551279i	$-0.814140\pi$
$359$		−	395.818i		−	1.10256i	0.834321	−	0.551279i	$-0.185860\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$		0			0
$363$	−106.773	+	440.237i	−0.294141	+	1.21277i
$364$		0			0
$365$	−480.833	+	329.848i	−1.31735	+	0.903694i
$366$		0			0
$367$		−	413.998i		−	1.12806i	−0.825755	−	0.564029i	$-0.809250\pi$
$367$		−	413.998i		−	1.12806i	0.825755	−	0.564029i	$-0.190750\pi$
$368$		0			0
$369$	−238.000	+	461.788i	−0.644986	+	1.25146i
$370$		0			0
$371$			395.818i			1.06690i
$372$		0			0
$373$			629.743i			1.68832i	0.536092	+	0.844159i	$0.319900\pi$
$373$			629.743i			1.68832i	−0.536092	+	0.844159i	$0.680100\pi$
$374$		0			0
$375$	1.85429	−	374.995i	0.00494478	−	0.999988i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	572.000			1.50923			0.754617	−	0.656165i	$-0.227822\pi$
$379$	572.000			1.50923			0.754617	+	0.656165i	$0.227822\pi$
$380$		0			0
$381$	119.000	+	28.8617i	0.312336	+	0.0757526i
$382$		0			0
$383$	193.747			0.505868			0.252934	−	0.967484i	$-0.418605\pi$
$383$	193.747			0.505868			0.252934	+	0.967484i	$0.418605\pi$
$384$		0			0
$385$	−272.000	−	396.505i	−0.706494	−	1.02988i
$386$		0			0
$387$	−168.291	+	326.533i	−0.434862	+	0.843755i
$388$		0			0
$389$		−	387.572i		−	0.996329i	−0.867083	−	0.498164i	$-0.834008\pi$
$389$		−	387.572i		−	0.996329i	0.867083	−	0.498164i	$-0.165992\pi$
$390$		0			0
$391$	272.000			0.695652
$392$		0			0
$393$	−144.250	−	34.9857i	−0.367048	−	0.0890222i
$394$		0			0
$395$	−203.647	−	296.864i	−0.515561	−	0.751553i
$396$		0			0
$397$		−	513.124i		−	1.29250i	−0.763124	−	0.646252i	$-0.776336\pi$
$397$		−	513.124i		−	1.29250i	0.763124	−	0.646252i	$-0.223664\pi$
$398$		0			0
$399$	−204.000	−	49.4773i	−0.511278	−	0.124003i
$400$		0			0
$401$		−	65.9697i		−	0.164513i	−0.996611	−	0.0822565i	$-0.973787\pi$
$401$		−	65.9697i		−	0.164513i	0.996611	−	0.0822565i	$-0.0262127\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	139.064	−	380.376i	0.343368	−	0.939201i
$406$		0			0
$407$	−384.666			−0.945126
$408$		0			0
$409$	640.000			1.56479			0.782396	−	0.622781i	$-0.213997\pi$
$409$	640.000			1.56479			0.782396	+	0.622781i	$0.213997\pi$
$410$		0			0
$411$	−36.0000	+	148.432i	−0.0875912	+	0.361148i
$412$		0			0
$413$	96.1665			0.232849
$414$		0			0
$415$	−124.000	−	180.760i	−0.298795	−	0.435565i
$416$		0			0
$417$	−31.1127	+	128.281i	−0.0746108	+	0.307628i
$418$		0			0
$419$		−	577.235i		−	1.37765i	−0.724928	−	0.688824i	$-0.758127\pi$
$419$		−	577.235i		−	1.37765i	0.724928	−	0.688824i	$-0.241873\pi$
$420$		0			0
$421$	−656.000			−1.55819			−0.779097	−	0.626903i	$-0.784322\pi$
$421$	−656.000			−1.55819			−0.779097	+	0.626903i	$0.784322\pi$
$422$		0			0
$423$	−282.843	−	145.774i	−0.668659	−	0.344619i
$424$		0			0
$425$	101.823	−	263.879i	0.239584	−	0.620891i
$426$		0			0
$427$		−	93.2952i		−	0.218490i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	362.833i		−	0.841841i	−0.907098	−	0.420920i	$-0.861707\pi$
$431$		−	362.833i		−	0.841841i	0.907098	−	0.420920i	$-0.138293\pi$
$432$		0			0
$433$			163.267i			0.377059i	0.982068	+	0.188530i	$0.0603722\pi$
$433$			163.267i			0.377059i	−0.982068	+	0.188530i	$0.939628\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	288.500			0.660182
$438$		0			0
$439$	−432.000			−0.984055			−0.492027	−	0.870580i	$-0.663744\pi$
$439$	−432.000			−0.984055			−0.492027	+	0.870580i	$0.663744\pi$
$440$		0			0
$441$	−120.000	−	61.8466i	−0.272109	−	0.140242i
$442$		0			0
$443$	−123.037			−0.277735			−0.138867	−	0.990311i	$-0.544346\pi$
$443$	−123.037			−0.277735			−0.138867	+	0.990311i	$0.544346\pi$
$444$		0			0
$445$	−272.000	+	186.590i	−0.611236	+	0.419304i
$446$		0			0
$447$	−24.0416	−	5.83095i	−0.0537844	−	0.0130446i
$448$		0			0
$449$		−	865.852i		−	1.92840i	−0.265174	−	0.964201i	$-0.585429\pi$
$449$		−	865.852i		−	1.92840i	0.265174	−	0.964201i	$-0.414571\pi$
$450$		0			0
$451$	−952.000			−2.11086
$452$		0			0
$453$	−96.1665	+	396.505i	−0.212288	+	0.875286i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			466.476i			1.02074i	0.859956	+	0.510368i	$0.170491\pi$
$457$			466.476i			1.02074i	−0.859956	+	0.510368i	$0.829509\pi$
$458$		0			0
$459$	200.000	−	230.894i	0.435730	−	0.503037i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			612.250i			1.32235i	0.750230	+	0.661177i	$0.229943\pi$
$463$			612.250i			1.32235i	−0.750230	+	0.661177i	$0.770057\pi$
$464$		0			0
$465$	−448.666	+	170.584i	−0.964873	+	0.366846i
$466$		0			0
$467$	767.918			1.64436			0.822182	−	0.569225i	$-0.192757\pi$
$467$	767.918			1.64436			0.822182	+	0.569225i	$0.192757\pi$
$468$		0			0
$469$	34.0000			0.0724947
$470$		0			0
$471$	−340.000	−	82.4621i	−0.721868	−	0.175079i
$472$		0			0
$473$	−673.166			−1.42318
$474$		0			0
$475$	108.000	−	279.886i	0.227368	−	0.589233i
$476$		0			0
$477$	−543.058	−	279.886i	−1.13849	−	0.586762i
$478$		0			0
$479$			560.742i			1.17065i	0.810798	+	0.585326i	$0.199033\pi$
$479$			560.742i			1.17065i	−0.810798	+	0.585326i	$0.800967\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−408.708	−	99.1262i	−0.846186	−	0.205230i
$484$		0			0
$485$	673.166	−	461.788i	1.38797	−	0.952140i
$486$		0			0
$487$			647.236i			1.32903i	0.747277	+	0.664513i	$0.231361\pi$
$487$			647.236i			1.32903i	−0.747277	+	0.664513i	$0.768639\pi$
$488$		0			0
$489$	−289.000	−	70.0928i	−0.591002	−	0.143339i
$490$		0			0
$491$		−	346.341i		−	0.705379i	−0.935740	−	0.352689i	$-0.885267\pi$
$491$		−	346.341i		−	0.705379i	0.935740	−	0.352689i	$-0.114733\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	736.333	−	92.8097i	1.48754	−	0.187494i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	660.000			1.32265			0.661323	−	0.750102i	$-0.269995\pi$
$499$	660.000			1.32265			0.661323	+	0.750102i	$0.269995\pi$
$500$		0			0
$501$	−207.000	+	853.483i	−0.413174	+	1.70356i
$502$		0			0
$503$	182.434			0.362691			0.181345	−	0.983419i	$-0.441955\pi$
$503$	182.434			0.362691			0.181345	+	0.983419i	$0.441955\pi$
$504$		0			0
$505$	544.000	−	373.181i	1.07723	−	0.738972i
$506$		0			0
$507$	119.501	−	492.715i	0.235702	−	0.971825i
$508$		0			0
$509$			395.818i			0.777639i	0.921314	+	0.388819i	$0.127117\pi$
$509$			395.818i			0.777639i	−0.921314	+	0.388819i	$0.872883\pi$
$510$		0			0
$511$	680.000			1.33072
$512$		0			0
$513$	212.132	−	244.900i	0.413513	−	0.477388i
$514$		0			0
$515$	−408.708	+	280.371i	−0.793607	+	0.544410i
$516$		0			0
$517$		−	583.095i		−	1.12784i
$518$		0			0
$519$	−116.000	+	478.280i	−0.223507	+	0.921542i
$520$		0			0
$521$			131.939i			0.253243i	0.991951	+	0.126621i	$0.0404133\pi$
$521$			131.939i			0.253243i	−0.991951	+	0.126621i	$0.959587\pi$
$522$		0			0
$523$			145.774i			0.278726i	0.990241	+	0.139363i	$0.0445055\pi$
$523$			145.774i			0.278726i	−0.990241	+	0.139363i	$0.955494\pi$
$524$		0			0
$525$	−249.167	+	359.397i	−0.474603	+	0.684565i
$526$		0			0
$527$	−362.039			−0.686980
$528$		0			0
$529$	49.0000			0.0926276
$530$		0			0
$531$	−68.0000	+	131.939i	−0.128060	+	0.248473i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−156.000	−	227.407i	−0.291589	−	0.425060i
$536$		0			0
$537$	−48.0833	−	11.6619i	−0.0895405	−	0.0217168i
$538$		0			0
$539$		−	247.386i		−	0.458973i
$540$		0			0
$541$	418.000			0.772643			0.386322	−	0.922364i	$-0.373745\pi$
$541$	418.000			0.772643			0.386322	+	0.922364i	$0.373745\pi$
$542$		0			0
$543$	−57.9828	+	239.069i	−0.106782	+	0.440274i
$544$		0			0
$545$	−226.274	−	329.848i	−0.415182	−	0.605227i
$546$		0			0
$547$			285.717i			0.522334i	0.965294	+	0.261167i	$0.0841073\pi$
$547$			285.717i			0.522334i	−0.965294	+	0.261167i	$0.915893\pi$
$548$		0			0
$549$	128.000	+	65.9697i	0.233151	+	0.120163i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			419.829i			0.759184i
$554$		0			0
$555$	124.333	+	327.019i	0.224024	+	0.589223i
$556$		0			0
$557$	424.264			0.761695			0.380847	−	0.924638i	$-0.375632\pi$
$557$	424.264			0.761695			0.380847	+	0.924638i	$0.375632\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	544.000	+	131.939i	0.969697	+	0.235186i
$562$		0			0
$563$	813.173			1.44436			0.722178	−	0.691707i	$-0.243141\pi$
$563$	813.173			1.44436			0.722178	+	0.691707i	$0.243141\pi$
$564$		0			0
$565$	432.000	+	629.743i	0.764602	+	1.11459i
$566$		0			0
$567$	−384.666	+	274.055i	−0.678423	+	0.483342i
$568$		0			0
$569$			453.542i			0.797085i	0.917150	+	0.398543i	$0.130484\pi$
$569$			453.542i			0.797085i	−0.917150	+	0.398543i	$0.869516\pi$
$570$		0			0
$571$	−220.000			−0.385289			−0.192644	−	0.981269i	$-0.561706\pi$
$571$	−220.000			−0.385289			−0.192644	+	0.981269i	$0.561706\pi$
$572$		0			0
$573$	865.499	+	209.914i	1.51047	+	0.366343i
$574$		0			0
$575$	216.375	−	560.742i	0.376304	−	0.975204i
$576$		0			0
$577$			46.6476i			0.0808451i	0.999183	+	0.0404225i	$0.0128704\pi$
$577$			46.6476i			0.0808451i	−0.999183	+	0.0404225i	$0.987130\pi$
$578$		0			0
$579$	−340.000	−	82.4621i	−0.587219	−	0.142422i
$580$		0			0
$581$			255.633i			0.439987i
$582$		0			0
$583$		−	1119.54i		−	1.92031i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−55.1543			−0.0939597			−0.0469798	−	0.998896i	$-0.514960\pi$
$587$	−55.1543			−0.0939597			−0.0469798	+	0.998896i	$0.514960\pi$
$588$		0			0
$589$	−384.000			−0.651952
$590$		0			0
$591$	−136.000	+	560.742i	−0.230118	+	0.948803i
$592$		0			0
$593$	390.323			0.658217			0.329109	−	0.944292i	$-0.393252\pi$
$593$	390.323			0.658217			0.329109	+	0.944292i	$0.393252\pi$
$594$		0			0
$595$	−272.000	+	186.590i	−0.457143	+	0.313597i
$596$		0			0
$597$	220.617	−	909.628i	0.369543	−	1.52367i
$598$		0			0
$599$			98.9545i			0.165200i	0.996583	+	0.0825998i	$0.0263223\pi$
$599$			98.9545i			0.165200i	−0.996583	+	0.0825998i	$0.973678\pi$
$600$		0			0
$601$	−880.000			−1.46423			−0.732113	−	0.681183i	$-0.761466\pi$
$601$	−880.000			−1.46423			−0.732113	+	0.681183i	$0.761466\pi$
$602$		0			0
$603$	−24.0416	+	46.6476i	−0.0398700	+	0.0773592i
$604$		0			0
$605$	427.092	+	622.589i	0.705938	+	1.02907i
$606$		0			0
$607$		−	425.659i		−	0.701251i	−0.936516	−	0.350626i	$-0.885969\pi$
$607$		−	425.659i		−	0.701251i	0.936516	−	0.350626i	$-0.114031\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	606.419i		−	0.989264i	−0.869102	−	0.494632i	$-0.835303\pi$
$613$		−	606.419i		−	0.989264i	0.869102	−	0.494632i	$-0.164697\pi$
$614$		0			0
$615$	307.709	+	809.330i	0.500339	+	1.31598i
$616$		0			0
$617$	−113.137			−0.183366			−0.0916832	−	0.995788i	$-0.529225\pi$
$617$	−113.137			−0.183366			−0.0916832	+	0.995788i	$0.529225\pi$
$618$		0			0
$619$	−52.0000			−0.0840065			−0.0420032	−	0.999117i	$-0.513374\pi$
$619$	−52.0000			−0.0840065			−0.0420032	+	0.999117i	$0.513374\pi$
$620$		0			0
$621$	425.000	−	490.650i	0.684380	−	0.790096i
$622$		0			0
$623$	384.666			0.617442
$624$		0			0
$625$	−463.000	−	419.829i	−0.740800	−	0.671726i
$626$		0			0
$627$	576.999	+	139.943i	0.920254	+	0.223194i
$628$		0			0
$629$			263.879i			0.419521i
$630$		0			0
$631$	544.000			0.862124			0.431062	−	0.902322i	$-0.358139\pi$
$631$	544.000			0.862124			0.431062	+	0.902322i	$0.358139\pi$
$632$		0			0
$633$	8.48528	−	34.9857i	0.0134049	−	0.0552697i
$634$		0			0
$635$	168.291	−	115.447i	0.265026	−	0.181806i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	849.360i		−	1.32505i	−0.749038	−	0.662527i	$-0.769484\pi$
$641$		−	849.360i		−	1.32505i	0.749038	−	0.662527i	$-0.230516\pi$
$642$		0			0
$643$		−	367.350i		−	0.571306i	−0.958333	−	0.285653i	$-0.907789\pi$
$643$		−	367.350i		−	0.571306i	0.958333	−	0.285653i	$-0.0922105\pi$
$644$		0			0
$645$	217.583	+	572.283i	0.337338	+	0.887260i
$646$		0			0
$647$	971.565			1.50165			0.750823	−	0.660504i	$-0.229657\pi$
$647$	971.565			1.50165			0.750823	+	0.660504i	$0.229657\pi$
$648$		0			0
$649$	−272.000			−0.419106
$650$		0			0
$651$	544.000	+	131.939i	0.835637	+	0.202672i
$652$		0			0
$653$	350.725			0.537098			0.268549	−	0.963266i	$-0.413456\pi$
$653$	350.725			0.537098			0.268549	+	0.963266i	$0.413456\pi$
$654$		0			0
$655$	−204.000	+	139.943i	−0.311450	+	0.213653i
$656$		0			0
$657$	−480.833	+	932.952i	−0.731861	+	1.42002i
$658$		0			0
$659$			577.235i			0.875925i	0.898993	+	0.437963i	$0.144300\pi$
$659$			577.235i			0.875925i	−0.898993	+	0.437963i	$0.855700\pi$
$660$		0			0
$661$	80.0000			0.121029			0.0605144	−	0.998167i	$-0.480726\pi$
$661$	80.0000			0.121029			0.0605144	+	0.998167i	$0.480726\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−288.500	+	197.909i	−0.433834	+	0.297608i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−119.000	−	28.8617i	−0.177877	−	0.0431416i
$670$		0			0
$671$			263.879i			0.393262i
$672$		0			0
$673$		−	489.800i		−	0.727786i	−0.931441	−	0.363893i	$-0.881447\pi$
$673$		−	489.800i		−	0.727786i	0.931441	−	0.363893i	$-0.118553\pi$
$674$		0			0
$675$	−316.901	−	595.986i	−0.469483	−	0.882942i
$676$		0			0
$677$	192.333			0.284096			0.142048	−	0.989860i	$-0.454631\pi$
$677$	192.333			0.284096			0.142048	+	0.989860i	$0.454631\pi$
$678$		0			0
$679$	−952.000			−1.40206
$680$		0			0
$681$	113.000	−	465.911i	0.165932	−	0.684157i
$682$		0			0
$683$	−236.174			−0.345789			−0.172894	−	0.984940i	$-0.555312\pi$
$683$	−236.174			−0.345789			−0.172894	+	0.984940i	$0.555312\pi$
$684$		0			0
$685$	144.000	+	209.914i	0.210219	+	0.306444i
$686$		0			0
$687$	57.9828	−	239.069i	0.0843999	−	0.347990i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	548.000			0.793054			0.396527	−	0.918023i	$-0.370215\pi$
$691$	548.000			0.793054			0.396527	+	0.918023i	$0.370215\pi$
$692$		0			0
$693$	−769.332	−	396.505i	−1.11015	−	0.572157i
$694$		0			0
$695$	124.451	+	181.417i	0.179066	+	0.261031i
$696$		0			0
$697$			653.067i			0.936968i
$698$		0			0
$699$	136.000	−	560.742i	0.194564	−	0.802207i
$700$		0			0
$701$		−	57.7235i		−	0.0823445i	−0.999152	−	0.0411722i	$-0.986891\pi$
$701$		−	57.7235i		−	0.0823445i	0.999152	−	0.0411722i	$-0.0131092\pi$
$702$		0			0
$703$			279.886i			0.398130i
$704$		0			0
$705$	−495.711	+	188.470i	−0.703136	+	0.267333i
$706$		0			0
$707$	−769.332			−1.08816
$708$		0			0
$709$	1230.00			1.73484			0.867419	−	0.497579i	$-0.165777\pi$
$709$	1230.00			1.73484			0.867419	+	0.497579i	$0.165777\pi$
$710$		0			0
$711$	−576.000	−	296.864i	−0.810127	−	0.417530i
$712$		0			0
$713$	−769.332			−1.07901
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1346.33	+	326.533i	1.87773	+	0.455416i
$718$		0			0
$719$		−	626.712i		−	0.871644i	−0.900033	−	0.435822i	$-0.856458\pi$
$719$		−	626.712i		−	0.871644i	0.900033	−	0.435822i	$-0.143542\pi$
$720$		0			0
$721$	578.000			0.801664
$722$		0			0
$723$	214.960	−	886.305i	0.297317	−	1.22587i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			367.350i			0.505296i	0.967558	+	0.252648i	$0.0813014\pi$
$727$			367.350i			0.505296i	−0.967558	+	0.252648i	$0.918699\pi$
$728$		0			0
$729$	−104.000	−	721.543i	−0.142661	−	0.989772i
$730$		0			0
$731$			461.788i			0.631721i
$732$		0			0
$733$		−	1002.92i		−	1.36825i	−0.729367	−	0.684123i	$-0.760185\pi$
$733$		−	1002.92i		−	1.36825i	0.729367	−	0.684123i	$-0.239815\pi$
$734$		0			0
$735$	−210.312	+	79.9610i	−0.286139	+	0.108791i
$736$		0			0
$737$	−96.1665			−0.130484
$738$		0			0
$739$	340.000			0.460081			0.230041	−	0.973181i	$-0.426114\pi$
$739$	340.000			0.460081			0.230041	+	0.973181i	$0.426114\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1243.09			−1.67307			−0.836537	−	0.547911i	$-0.815423\pi$
$743$	−1243.09			−1.67307			−0.836537	+	0.547911i	$0.815423\pi$
$744$		0			0
$745$	−34.0000	+	23.3238i	−0.0456376	+	0.0313071i
$746$		0			0
$747$	−350.725	−	180.760i	−0.469511	−	0.241981i
$748$		0			0
$749$			321.602i			0.429375i
$750$		0			0
$751$	520.000			0.692410			0.346205	−	0.938159i	$-0.387470\pi$
$751$	520.000			0.692410			0.346205	+	0.938159i	$0.387470\pi$
$752$		0			0
$753$	−1009.75	−	244.900i	−1.34097	−	0.325232i
$754$		0			0
$755$	384.666	+	560.742i	0.509492	+	0.742705i
$756$		0			0
$757$			816.333i			1.07838i	0.842184	+	0.539190i	$0.181269\pi$
$757$			816.333i			1.07838i	−0.842184	+	0.539190i	$0.818731\pi$
$758$		0			0
$759$	1156.00	+	280.371i	1.52306	+	0.369395i
$760$		0			0
$761$			395.818i			0.520129i	0.965591	+	0.260064i	$0.0837438\pi$
$761$			395.818i			0.520129i	−0.965591	+	0.260064i	$0.916256\pi$
$762$		0			0
$763$			466.476i			0.611371i
$764$		0			0
$765$	−63.6670	−	505.120i	−0.0832248	−	0.660288i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−306.000			−0.397919			−0.198960	−	0.980008i	$-0.563756\pi$
$769$	−306.000			−0.397919			−0.198960	+	0.980008i	$0.563756\pi$
$770$		0			0
$771$	276.000	−	1137.98i	0.357977	−	1.47598i
$772$		0			0
$773$	305.470			0.395175			0.197587	−	0.980285i	$-0.436689\pi$
$773$	305.470			0.395175			0.197587	+	0.980285i	$0.436689\pi$
$774$		0			0
$775$	−288.000	+	746.362i	−0.371613	+	0.963048i
$776$		0			0
$777$	96.1665	−	396.505i	0.123766	−	0.510302i
$778$		0			0
$779$			692.682i			0.889194i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−480.833	+	329.848i	−0.612526	+	0.420189i
$786$		0			0
$787$			565.602i			0.718681i	0.933206	+	0.359341i	$0.116998\pi$
$787$			565.602i			0.718681i	−0.933206	+	0.359341i	$0.883002\pi$
$788$		0			0
$789$	−209.000	+	861.729i	−0.264892	+	1.09218i
$790$		0			0
$791$		−	890.591i		−	1.12590i
$792$		0			0
$793$		0			0
$794$		0			0

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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 2.91548i) q^{3} +(-2.82843 - 4.12311i) q^{5} +5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} +O(q^{10})\)	\(q+(0.707107 - 2.91548i) q^{3} +(-2.82843 - 4.12311i) q^{5} +5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} -16.4924i q^{11} +(-14.0208 + 5.33074i) q^{15} -11.3137 q^{17} -12.0000 q^{19} +(17.0000 + 4.12311i) q^{21} -24.0416 q^{23} +(-9.00000 + 23.3238i) q^{25} +(-17.6777 + 20.4083i) q^{27} +32.0000 q^{31} +(-48.0833 - 11.6619i) q^{33} +(24.0416 - 16.4924i) q^{35} -23.3238i q^{37} -57.7235i q^{41} -40.8167i q^{43} +(5.62742 + 44.6467i) q^{45} +35.3553 q^{47} +15.0000 q^{49} +(-8.00000 + 32.9848i) q^{51} +67.8823 q^{53} +(-68.0000 + 46.6476i) q^{55} +(-8.48528 + 34.9857i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +(24.0416 - 46.6476i) q^{63} -5.83095i q^{67} +(-17.0000 + 70.0928i) q^{69} -116.619i q^{73} +(61.6360 + 42.7317i) q^{75} +96.1665 q^{77} +72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} +43.8406 q^{83} +(32.0000 + 46.6476i) q^{85} -65.9697i q^{89} +(22.6274 - 93.2952i) q^{93} +(33.9411 + 49.4773i) q^{95} +163.267i q^{97} +(-68.0000 + 131.939i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{9}+O(q^{10}) \) 4 * q - 32 * q^9	\( 4 q - 32 q^{9} - 8 q^{15} - 48 q^{19} + 68 q^{21} - 36 q^{25} + 128 q^{31} - 68 q^{45} + 60 q^{49} - 32 q^{51} - 272 q^{55} - 64 q^{61} - 68 q^{69} + 272 q^{75} + 288 q^{79} + 188 q^{81} + 128 q^{85} - 272 q^{99}+O(q^{100}) \) 4 * q - 32 * q^9 - 8 * q^15 - 48 * q^19 + 68 * q^21 - 36 * q^25 + 128 * q^31 - 68 * q^45 + 60 * q^49 - 32 * q^51 - 272 * q^55 - 64 * q^61 - 68 * q^69 + 272 * q^75 + 288 * q^79 + 188 * q^81 + 128 * q^85 - 272 * q^99

Introduction

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