LMFDB - Embedded newform 192.3.e.d.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(65,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.e (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:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	192.65
Dual form			192.3.e.d.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 2.82843i) q^{3} -5.65685i q^{5} +6.00000 q^{7} +(-7.00000 - 5.65685i) q^{9} +5.65685i q^{11} -10.0000 q^{13} +(-16.0000 - 5.65685i) q^{15} -22.6274i q^{17} +2.00000 q^{19} +(6.00000 - 16.9706i) q^{21} +11.3137i q^{23} -7.00000 q^{25} +(-23.0000 + 14.1421i) q^{27} -16.9706i q^{29} +22.0000 q^{31} +(16.0000 + 5.65685i) q^{33} -33.9411i q^{35} +6.00000 q^{37} +(-10.0000 + 28.2843i) q^{39} +33.9411i q^{41} +82.0000 q^{43} +(-32.0000 + 39.5980i) q^{45} -67.8823i q^{47} -13.0000 q^{49} +(-64.0000 - 22.6274i) q^{51} +62.2254i q^{53} +32.0000 q^{55} +(2.00000 - 5.65685i) q^{57} +73.5391i q^{59} +86.0000 q^{61} +(-42.0000 - 33.9411i) q^{63} +56.5685i q^{65} +2.00000 q^{67} +(32.0000 + 11.3137i) q^{69} +124.451i q^{71} +82.0000 q^{73} +(-7.00000 + 19.7990i) q^{75} +33.9411i q^{77} -10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391i q^{83} -128.000 q^{85} +(-48.0000 - 16.9706i) q^{87} -33.9411i q^{89} -60.0000 q^{91} +(22.0000 - 62.2254i) q^{93} -11.3137i q^{95} -94.0000 q^{97} +(32.0000 - 39.5980i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 12 q^{7} - 14 q^{9} - 20 q^{13} - 32 q^{15} + 4 q^{19} + 12 q^{21} - 14 q^{25} - 46 q^{27} + 44 q^{31} + 32 q^{33} + 12 q^{37} - 20 q^{39} + 164 q^{43} - 64 q^{45} - 26 q^{49} - 128 q^{51}+ \cdots + 64 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 12 * q^7 - 14 * q^9 - 20 * q^13 - 32 * q^15 + 4 * q^19 + 12 * q^21 - 14 * q^25 - 46 * q^27 + 44 * q^31 + 32 * q^33 + 12 * q^37 - 20 * q^39 + 164 * q^43 - 64 * q^45 - 26 * q^49 - 128 * q^51 + 64 * q^55 + 4 * q^57 + 172 * q^61 - 84 * q^63 + 4 * q^67 + 64 * q^69 + 164 * q^73 - 14 * q^75 - 20 * q^79 + 34 * q^81 - 256 * q^85 - 96 * q^87 - 120 * q^91 + 44 * q^93 - 188 * q^97 + 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.00000	−	2.82843i	0.333333	−	0.942809i
$3$	1.00000	−	2.82843i	0.333333	−	0.942809i
$4$		0			0
$5$		−	5.65685i		−	1.13137i	−0.824621	−	0.565685i	$-0.808612\pi$
$5$		−	5.65685i		−	1.13137i	0.824621	−	0.565685i	$-0.191388\pi$
$6$		0			0
$7$	6.00000			0.857143			0.428571	−	0.903508i	$-0.359017\pi$
$7$	6.00000			0.857143			0.428571	+	0.903508i	$0.359017\pi$
$8$		0			0
$9$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$10$		0			0
$11$			5.65685i			0.514259i	0.966377	+	0.257130i	$0.0827768\pi$
$11$			5.65685i			0.514259i	−0.966377	+	0.257130i	$0.917223\pi$
$12$		0			0
$13$	−10.0000			−0.769231			−0.384615	−	0.923077i	$-0.625666\pi$
$13$	−10.0000			−0.769231			−0.384615	+	0.923077i	$0.625666\pi$
$14$		0			0
$15$	−16.0000	−	5.65685i	−1.06667	−	0.377124i
$16$		0			0
$17$		−	22.6274i		−	1.33102i	−0.746387	−	0.665512i	$-0.768213\pi$
$17$		−	22.6274i		−	1.33102i	0.746387	−	0.665512i	$-0.231787\pi$
$18$		0			0
$19$	2.00000			0.105263			0.0526316	−	0.998614i	$-0.483239\pi$
$19$	2.00000			0.105263			0.0526316	+	0.998614i	$0.483239\pi$
$20$		0			0
$21$	6.00000	−	16.9706i	0.285714	−	0.808122i
$22$		0			0
$23$			11.3137i			0.491900i	0.969282	+	0.245950i	$0.0791000\pi$
$23$			11.3137i			0.491900i	−0.969282	+	0.245950i	$0.920900\pi$
$24$		0			0
$25$	−7.00000			−0.280000
$26$		0			0
$27$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$28$		0			0
$29$		−	16.9706i		−	0.585192i	−0.956236	−	0.292596i	$-0.905481\pi$
$29$		−	16.9706i		−	0.585192i	0.956236	−	0.292596i	$-0.0945191\pi$
$30$		0			0
$31$	22.0000			0.709677			0.354839	−	0.934928i	$-0.384536\pi$
$31$	22.0000			0.709677			0.354839	+	0.934928i	$0.384536\pi$
$32$		0			0
$33$	16.0000	+	5.65685i	0.484848	+	0.171420i
$34$		0			0
$35$		−	33.9411i		−	0.969746i
$36$		0			0
$37$	6.00000			0.162162			0.0810811	−	0.996708i	$-0.474163\pi$
$37$	6.00000			0.162162			0.0810811	+	0.996708i	$0.474163\pi$
$38$		0			0
$39$	−10.0000	+	28.2843i	−0.256410	+	0.725238i
$40$		0			0
$41$			33.9411i			0.827832i	0.910315	+	0.413916i	$0.135839\pi$
$41$			33.9411i			0.827832i	−0.910315	+	0.413916i	$0.864161\pi$
$42$		0			0
$43$	82.0000			1.90698			0.953488	−	0.301430i	$-0.0974639\pi$
$43$	82.0000			1.90698			0.953488	+	0.301430i	$0.0974639\pi$
$44$		0			0
$45$	−32.0000	+	39.5980i	−0.711111	+	0.879955i
$46$		0			0
$47$		−	67.8823i		−	1.44430i	−0.691735	−	0.722152i	$-0.743153\pi$
$47$		−	67.8823i		−	1.44430i	0.691735	−	0.722152i	$-0.256847\pi$
$48$		0			0
$49$	−13.0000			−0.265306
$50$		0			0
$51$	−64.0000	−	22.6274i	−1.25490	−	0.443675i
$52$		0			0
$53$			62.2254i			1.17406i	0.809564	+	0.587032i	$0.199704\pi$
$53$			62.2254i			1.17406i	−0.809564	+	0.587032i	$0.800296\pi$
$54$		0			0
$55$	32.0000			0.581818
$56$		0			0
$57$	2.00000	−	5.65685i	0.0350877	−	0.0992431i
$58$		0			0
$59$			73.5391i			1.24643i	0.782052	+	0.623213i	$0.214173\pi$
$59$			73.5391i			1.24643i	−0.782052	+	0.623213i	$0.785827\pi$
$60$		0			0
$61$	86.0000			1.40984			0.704918	−	0.709289i	$-0.250984\pi$
$61$	86.0000			1.40984			0.704918	+	0.709289i	$0.250984\pi$
$62$		0			0
$63$	−42.0000	−	33.9411i	−0.666667	−	0.538748i
$64$		0			0
$65$			56.5685i			0.870285i
$66$		0			0
$67$	2.00000			0.0298507			0.0149254	−	0.999889i	$-0.495249\pi$
$67$	2.00000			0.0298507			0.0149254	+	0.999889i	$0.495249\pi$
$68$		0			0
$69$	32.0000	+	11.3137i	0.463768	+	0.163967i
$70$		0			0
$71$			124.451i			1.75283i	0.481558	+	0.876414i	$0.340071\pi$
$71$			124.451i			1.75283i	−0.481558	+	0.876414i	$0.659929\pi$
$72$		0			0
$73$	82.0000			1.12329			0.561644	−	0.827379i	$-0.310169\pi$
$73$	82.0000			1.12329			0.561644	+	0.827379i	$0.310169\pi$
$74$		0			0
$75$	−7.00000	+	19.7990i	−0.0933333	+	0.263987i
$76$		0			0
$77$			33.9411i			0.440794i
$78$		0			0
$79$	−10.0000			−0.126582			−0.0632911	−	0.997995i	$-0.520160\pi$
$79$	−10.0000			−0.126582			−0.0632911	+	0.997995i	$0.520160\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$		−	73.5391i		−	0.886013i	−0.896518	−	0.443007i	$-0.853912\pi$
$83$		−	73.5391i		−	0.886013i	0.896518	−	0.443007i	$-0.146088\pi$
$84$		0			0
$85$	−128.000			−1.50588
$86$		0			0
$87$	−48.0000	−	16.9706i	−0.551724	−	0.195064i
$88$		0			0
$89$		−	33.9411i		−	0.381361i	−0.981652	−	0.190680i	$-0.938931\pi$
$89$		−	33.9411i		−	0.381361i	0.981652	−	0.190680i	$-0.0610694\pi$
$90$		0			0
$91$	−60.0000			−0.659341
$92$		0			0
$93$	22.0000	−	62.2254i	0.236559	−	0.669090i
$94$		0			0
$95$		−	11.3137i		−	0.119092i
$96$		0			0
$97$	−94.0000			−0.969072			−0.484536	−	0.874771i	$-0.661012\pi$
$97$	−94.0000			−0.969072			−0.484536	+	0.874771i	$0.661012\pi$
$98$		0			0
$99$	32.0000	−	39.5980i	0.323232	−	0.399980i
$100$		0			0
$101$		−	50.9117i		−	0.504076i	−0.967717	−	0.252038i	$-0.918899\pi$
$101$		−	50.9117i		−	0.504076i	0.967717	−	0.252038i	$-0.0811008\pi$
$102$		0			0
$103$	134.000			1.30097			0.650485	−	0.759519i	$-0.274565\pi$
$103$	134.000			1.30097			0.650485	+	0.759519i	$0.274565\pi$
$104$		0			0
$105$	−96.0000	−	33.9411i	−0.914286	−	0.323249i
$106$		0			0
$107$			50.9117i			0.475810i	0.971288	+	0.237905i	$0.0764607\pi$
$107$			50.9117i			0.475810i	−0.971288	+	0.237905i	$0.923539\pi$
$108$		0			0
$109$	−10.0000			−0.0917431			−0.0458716	−	0.998947i	$-0.514606\pi$
$109$	−10.0000			−0.0917431			−0.0458716	+	0.998947i	$0.514606\pi$
$110$		0			0
$111$	6.00000	−	16.9706i	0.0540541	−	0.152888i
$112$		0			0
$113$		−	67.8823i		−	0.600728i	−0.953825	−	0.300364i	$-0.902892\pi$
$113$		−	67.8823i		−	0.600728i	0.953825	−	0.300364i	$-0.0971081\pi$
$114$		0			0
$115$	64.0000			0.556522
$116$		0			0
$117$	70.0000	+	56.5685i	0.598291	+	0.483492i
$118$		0			0
$119$		−	135.765i		−	1.14088i
$120$		0			0
$121$	89.0000			0.735537
$122$		0			0
$123$	96.0000	+	33.9411i	0.780488	+	0.275944i
$124$		0			0
$125$		−	101.823i		−	0.814587i
$126$		0			0
$127$	−106.000			−0.834646			−0.417323	−	0.908758i	$-0.637032\pi$
$127$	−106.000			−0.834646			−0.417323	+	0.908758i	$0.637032\pi$
$128$		0			0
$129$	82.0000	−	231.931i	0.635659	−	1.79791i
$130$		0			0
$131$		−	5.65685i		−	0.0431821i	−0.999767	−	0.0215910i	$-0.993127\pi$
$131$		−	5.65685i		−	0.0431821i	0.999767	−	0.0215910i	$-0.00687318\pi$
$132$		0			0
$133$	12.0000			0.0902256
$134$		0			0
$135$	80.0000	+	130.108i	0.592593	+	0.963760i
$136$		0			0
$137$		−	101.823i		−	0.743236i	−0.928386	−	0.371618i	$-0.878803\pi$
$137$		−	101.823i		−	0.743236i	0.928386	−	0.371618i	$-0.121197\pi$
$138$		0			0
$139$	−78.0000			−0.561151			−0.280576	−	0.959832i	$-0.590525\pi$
$139$	−78.0000			−0.561151			−0.280576	+	0.959832i	$0.590525\pi$
$140$		0			0
$141$	−192.000	−	67.8823i	−1.36170	−	0.481434i
$142$		0			0
$143$		−	56.5685i		−	0.395584i
$144$		0			0
$145$	−96.0000			−0.662069
$146$		0			0
$147$	−13.0000	+	36.7696i	−0.0884354	+	0.250133i
$148$		0			0
$149$		−	164.049i		−	1.10100i	−0.834836	−	0.550499i	$-0.814437\pi$
$149$		−	164.049i		−	1.10100i	0.834836	−	0.550499i	$-0.185563\pi$
$150$		0			0
$151$	−218.000			−1.44371			−0.721854	−	0.692045i	$-0.756710\pi$
$151$	−218.000			−1.44371			−0.721854	+	0.692045i	$0.756710\pi$
$152$		0			0
$153$	−128.000	+	158.392i	−0.836601	+	1.03524i
$154$		0			0
$155$		−	124.451i		−	0.802908i
$156$		0			0
$157$	86.0000			0.547771			0.273885	−	0.961762i	$-0.411691\pi$
$157$	86.0000			0.547771			0.273885	+	0.961762i	$0.411691\pi$
$158$		0			0
$159$	176.000	+	62.2254i	1.10692	+	0.391355i
$160$		0			0
$161$			67.8823i			0.421629i
$162$		0			0
$163$	−222.000			−1.36196			−0.680982	−	0.732301i	$-0.738447\pi$
$163$	−222.000			−1.36196			−0.680982	+	0.732301i	$0.738447\pi$
$164$		0			0
$165$	32.0000	−	90.5097i	0.193939	−	0.548543i
$166$		0			0
$167$			169.706i			1.01620i	0.861298	+	0.508101i	$0.169652\pi$
$167$			169.706i			1.01620i	−0.861298	+	0.508101i	$0.830348\pi$
$168$		0			0
$169$	−69.0000			−0.408284
$170$		0			0
$171$	−14.0000	−	11.3137i	−0.0818713	−	0.0661620i
$172$		0			0
$173$			186.676i			1.07905i	0.841969	+	0.539527i	$0.181397\pi$
$173$			186.676i			1.07905i	−0.841969	+	0.539527i	$0.818603\pi$
$174$		0			0
$175$	−42.0000			−0.240000
$176$		0			0
$177$	208.000	+	73.5391i	1.17514	+	0.415475i
$178$		0			0
$179$			152.735i			0.853269i	0.904424	+	0.426634i	$0.140301\pi$
$179$			152.735i			0.853269i	−0.904424	+	0.426634i	$0.859699\pi$
$180$		0			0
$181$	−90.0000			−0.497238			−0.248619	−	0.968601i	$-0.579977\pi$
$181$	−90.0000			−0.497238			−0.248619	+	0.968601i	$0.579977\pi$
$182$		0			0
$183$	86.0000	−	243.245i	0.469945	−	1.32921i
$184$		0			0
$185$		−	33.9411i		−	0.183466i
$186$		0			0
$187$	128.000			0.684492
$188$		0			0
$189$	−138.000	+	84.8528i	−0.730159	+	0.448957i
$190$		0			0
$191$			271.529i			1.42162i	0.703385	+	0.710809i	$0.251671\pi$
$191$			271.529i			1.42162i	−0.703385	+	0.710809i	$0.748329\pi$
$192$		0			0
$193$	2.00000			0.0103627			0.00518135	−	0.999987i	$-0.498351\pi$
$193$	2.00000			0.0103627			0.00518135	+	0.999987i	$0.498351\pi$
$194$		0			0
$195$	160.000	+	56.5685i	0.820513	+	0.290095i
$196$		0			0
$197$			84.8528i			0.430725i	0.976534	+	0.215362i	$0.0690933\pi$
$197$			84.8528i			0.430725i	−0.976534	+	0.215362i	$0.930907\pi$
$198$		0			0
$199$	−250.000			−1.25628			−0.628141	−	0.778100i	$-0.716184\pi$
$199$	−250.000			−1.25628			−0.628141	+	0.778100i	$0.716184\pi$
$200$		0			0
$201$	2.00000	−	5.65685i	0.00995025	−	0.0281436i
$202$		0			0
$203$		−	101.823i		−	0.501593i
$204$		0			0
$205$	192.000			0.936585
$206$		0			0
$207$	64.0000	−	79.1960i	0.309179	−	0.382589i
$208$		0			0
$209$			11.3137i			0.0541326i
$210$		0			0
$211$	34.0000			0.161137			0.0805687	−	0.996749i	$-0.474326\pi$
$211$	34.0000			0.161137			0.0805687	+	0.996749i	$0.474326\pi$
$212$		0			0
$213$	352.000	+	124.451i	1.65258	+	0.584276i
$214$		0			0
$215$		−	463.862i		−	2.15750i
$216$		0			0
$217$	132.000			0.608295
$218$		0			0
$219$	82.0000	−	231.931i	0.374429	−	1.05905i
$220$		0			0
$221$			226.274i			1.02387i
$222$		0			0
$223$	278.000			1.24664			0.623318	−	0.781968i	$-0.285784\pi$
$223$	278.000			1.24664			0.623318	+	0.781968i	$0.285784\pi$
$224$		0			0
$225$	49.0000	+	39.5980i	0.217778	+	0.175991i
$226$		0			0
$227$			220.617i			0.971882i	0.873991	+	0.485941i	$0.161523\pi$
$227$			220.617i			0.971882i	−0.873991	+	0.485941i	$0.838477\pi$
$228$		0			0
$229$	−58.0000			−0.253275			−0.126638	−	0.991949i	$-0.540419\pi$
$229$	−58.0000			−0.253275			−0.126638	+	0.991949i	$0.540419\pi$
$230$		0			0
$231$	96.0000	+	33.9411i	0.415584	+	0.146931i
$232$		0			0
$233$			395.980i			1.69948i	0.527199	+	0.849742i	$0.323242\pi$
$233$			395.980i			1.69948i	−0.527199	+	0.849742i	$0.676758\pi$
$234$		0			0
$235$	−384.000			−1.63404
$236$		0			0
$237$	−10.0000	+	28.2843i	−0.0421941	+	0.119343i
$238$		0			0
$239$			22.6274i			0.0946754i	0.998879	+	0.0473377i	$0.0150737\pi$
$239$			22.6274i			0.0946754i	−0.998879	+	0.0473377i	$0.984926\pi$
$240$		0			0
$241$	−30.0000			−0.124481			−0.0622407	−	0.998061i	$-0.519825\pi$
$241$	−30.0000			−0.124481			−0.0622407	+	0.998061i	$0.519825\pi$
$242$		0			0
$243$	241.000	+	31.1127i	0.991770	+	0.128036i
$244$		0			0
$245$			73.5391i			0.300160i
$246$		0			0
$247$	−20.0000			−0.0809717
$248$		0			0
$249$	−208.000	−	73.5391i	−0.835341	−	0.295338i
$250$		0			0
$251$		−	107.480i		−	0.428208i	−0.976811	−	0.214104i	$-0.931317\pi$
$251$		−	107.480i		−	0.428208i	0.976811	−	0.214104i	$-0.0686832\pi$
$252$		0			0
$253$	−64.0000			−0.252964
$254$		0			0
$255$	−128.000	+	362.039i	−0.501961	+	1.41976i
$256$		0			0
$257$			181.019i			0.704355i	0.935933	+	0.352178i	$0.114559\pi$
$257$			181.019i			0.704355i	−0.935933	+	0.352178i	$0.885441\pi$
$258$		0			0
$259$	36.0000			0.138996
$260$		0			0
$261$	−96.0000	+	118.794i	−0.367816	+	0.455149i
$262$		0			0
$263$			214.960i			0.817340i	0.912682	+	0.408670i	$0.134007\pi$
$263$			214.960i			0.817340i	−0.912682	+	0.408670i	$0.865993\pi$
$264$		0			0
$265$	352.000			1.32830
$266$		0			0
$267$	−96.0000	−	33.9411i	−0.359551	−	0.127120i
$268$		0			0
$269$		−	401.637i		−	1.49307i	−0.665344	−	0.746537i	$-0.731715\pi$
$269$		−	401.637i		−	1.49307i	0.665344	−	0.746537i	$-0.268285\pi$
$270$		0			0
$271$	−266.000			−0.981550			−0.490775	−	0.871286i	$-0.663286\pi$
$271$	−266.000			−0.981550			−0.490775	+	0.871286i	$0.663286\pi$
$272$		0			0
$273$	−60.0000	+	169.706i	−0.219780	+	0.621632i
$274$		0			0
$275$		−	39.5980i		−	0.143993i
$276$		0			0
$277$	−346.000			−1.24910			−0.624549	−	0.780986i	$-0.714717\pi$
$277$	−346.000			−1.24910			−0.624549	+	0.780986i	$0.714717\pi$
$278$		0			0
$279$	−154.000	−	124.451i	−0.551971	−	0.446060i
$280$		0			0
$281$		−	124.451i		−	0.442885i	−0.975173	−	0.221443i	$-0.928923\pi$
$281$		−	124.451i		−	0.442885i	0.975173	−	0.221443i	$-0.0710766\pi$
$282$		0			0
$283$	−46.0000			−0.162544			−0.0812721	−	0.996692i	$-0.525898\pi$
$283$	−46.0000			−0.162544			−0.0812721	+	0.996692i	$0.525898\pi$
$284$		0			0
$285$	−32.0000	−	11.3137i	−0.112281	−	0.0396972i
$286$		0			0
$287$			203.647i			0.709571i
$288$		0			0
$289$	−223.000			−0.771626
$290$		0			0
$291$	−94.0000	+	265.872i	−0.323024	+	0.913650i
$292$		0			0
$293$			220.617i			0.752960i	0.926425	+	0.376480i	$0.122866\pi$
$293$			220.617i			0.752960i	−0.926425	+	0.376480i	$0.877134\pi$
$294$		0			0
$295$	416.000			1.41017
$296$		0			0
$297$	−80.0000	−	130.108i	−0.269360	−	0.438073i
$298$		0			0
$299$		−	113.137i		−	0.378385i
$300$		0			0
$301$	492.000			1.63455
$302$		0			0
$303$	−144.000	−	50.9117i	−0.475248	−	0.168025i
$304$		0			0
$305$		−	486.489i		−	1.59505i
$306$		0			0
$307$	−30.0000			−0.0977199			−0.0488599	−	0.998806i	$-0.515559\pi$
$307$	−30.0000			−0.0977199			−0.0488599	+	0.998806i	$0.515559\pi$
$308$		0			0
$309$	134.000	−	379.009i	0.433657	−	1.22657i
$310$		0			0
$311$		−	576.999i		−	1.85530i	−0.373447	−	0.927651i	$-0.621824\pi$
$311$		−	576.999i		−	1.85530i	0.373447	−	0.927651i	$-0.378176\pi$
$312$		0			0
$313$	210.000			0.670927			0.335463	−	0.942053i	$-0.391107\pi$
$313$	210.000			0.670927			0.335463	+	0.942053i	$0.391107\pi$
$314$		0			0
$315$	−192.000	+	237.588i	−0.609524	+	0.754247i
$316$		0			0
$317$		−	152.735i		−	0.481814i	−0.970548	−	0.240907i	$-0.922555\pi$
$317$		−	152.735i		−	0.481814i	0.970548	−	0.240907i	$-0.0774449\pi$
$318$		0			0
$319$	96.0000			0.300940
$320$		0			0
$321$	144.000	+	50.9117i	0.448598	+	0.158603i
$322$		0			0
$323$		−	45.2548i		−	0.140108i
$324$		0			0
$325$	70.0000			0.215385
$326$		0			0
$327$	−10.0000	+	28.2843i	−0.0305810	+	0.0864962i
$328$		0			0
$329$		−	407.294i		−	1.23797i
$330$		0			0
$331$	434.000			1.31118			0.655589	−	0.755118i	$-0.272420\pi$
$331$	434.000			1.31118			0.655589	+	0.755118i	$0.272420\pi$
$332$		0			0
$333$	−42.0000	−	33.9411i	−0.126126	−	0.101925i
$334$		0			0
$335$		−	11.3137i		−	0.0337723i
$336$		0			0
$337$	−510.000			−1.51335			−0.756677	−	0.653789i	$-0.773178\pi$
$337$	−510.000			−1.51335			−0.756677	+	0.653789i	$0.773178\pi$
$338$		0			0
$339$	−192.000	−	67.8823i	−0.566372	−	0.200243i
$340$		0			0
$341$			124.451i			0.364958i
$342$		0			0
$343$	−372.000			−1.08455
$344$		0			0
$345$	64.0000	−	181.019i	0.185507	−	0.524694i
$346$		0			0
$347$		−	152.735i		−	0.440159i	−0.975482	−	0.220079i	$-0.929368\pi$
$347$		−	152.735i		−	0.440159i	0.975482	−	0.220079i	$-0.0706316\pi$
$348$		0			0
$349$	−426.000			−1.22063			−0.610315	−	0.792159i	$-0.708957\pi$
$349$	−426.000			−1.22063			−0.610315	+	0.792159i	$0.708957\pi$
$350$		0			0
$351$	230.000	−	141.421i	0.655271	−	0.402910i
$352$		0			0
$353$			45.2548i			0.128201i	0.997943	+	0.0641003i	$0.0204178\pi$
$353$			45.2548i			0.128201i	−0.997943	+	0.0641003i	$0.979582\pi$
$354$		0			0
$355$	704.000			1.98310
$356$		0			0
$357$	−384.000	−	135.765i	−1.07563	−	0.380293i
$358$		0			0
$359$			441.235i			1.22907i	0.788891	+	0.614533i	$0.210655\pi$
$359$			441.235i			1.22907i	−0.788891	+	0.614533i	$0.789345\pi$
$360$		0			0
$361$	−357.000			−0.988920
$362$		0			0
$363$	89.0000	−	251.730i	0.245179	−	0.693471i
$364$		0			0
$365$		−	463.862i		−	1.27085i
$366$		0			0
$367$	566.000			1.54223			0.771117	−	0.636693i	$-0.219698\pi$
$367$	566.000			1.54223			0.771117	+	0.636693i	$0.219698\pi$
$368$		0			0
$369$	192.000	−	237.588i	0.520325	−	0.643870i
$370$		0			0
$371$			373.352i			1.00634i
$372$		0			0
$373$	−218.000			−0.584450			−0.292225	−	0.956350i	$-0.594396\pi$
$373$	−218.000			−0.584450			−0.292225	+	0.956350i	$0.594396\pi$
$374$		0			0
$375$	−288.000	−	101.823i	−0.768000	−	0.271529i
$376$		0			0
$377$			169.706i			0.450148i
$378$		0			0
$379$	−142.000			−0.374670			−0.187335	−	0.982296i	$-0.559985\pi$
$379$	−142.000			−0.374670			−0.187335	+	0.982296i	$0.559985\pi$
$380$		0			0
$381$	−106.000	+	299.813i	−0.278215	+	0.786911i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	192.000			0.498701
$386$		0			0
$387$	−574.000	−	463.862i	−1.48320	−	1.19861i
$388$		0			0
$389$		−	548.715i		−	1.41058i	−0.708920	−	0.705289i	$-0.750817\pi$
$389$		−	548.715i		−	1.41058i	0.708920	−	0.705289i	$-0.249183\pi$
$390$		0			0
$391$	256.000			0.654731
$392$		0			0
$393$	−16.0000	−	5.65685i	−0.0407125	−	0.0143940i
$394$		0			0
$395$			56.5685i			0.143211i
$396$		0			0
$397$	310.000			0.780856			0.390428	−	0.920633i	$-0.372327\pi$
$397$	310.000			0.780856			0.390428	+	0.920633i	$0.372327\pi$
$398$		0			0
$399$	12.0000	−	33.9411i	0.0300752	−	0.0850655i
$400$		0			0
$401$			339.411i			0.846412i	0.906033	+	0.423206i	$0.139095\pi$
$401$			339.411i			0.846412i	−0.906033	+	0.423206i	$0.860905\pi$
$402$		0			0
$403$	−220.000			−0.545906
$404$		0			0
$405$	448.000	−	96.1665i	1.10617	−	0.237448i
$406$		0			0
$407$			33.9411i			0.0833934i
$408$		0			0
$409$	−270.000			−0.660147			−0.330073	−	0.943955i	$-0.607073\pi$
$409$	−270.000			−0.660147			−0.330073	+	0.943955i	$0.607073\pi$
$410$		0			0
$411$	−288.000	−	101.823i	−0.700730	−	0.247745i
$412$		0			0
$413$			441.235i			1.06836i
$414$		0			0
$415$	−416.000			−1.00241
$416$		0			0
$417$	−78.0000	+	220.617i	−0.187050	+	0.529058i
$418$		0			0
$419$		−	50.9117i		−	0.121508i	−0.998153	−	0.0607538i	$-0.980650\pi$
$419$		−	50.9117i		−	0.121508i	0.998153	−	0.0607538i	$-0.0193505\pi$
$420$		0			0
$421$	454.000			1.07838			0.539192	−	0.842183i	$-0.318730\pi$
$421$	454.000			1.07838			0.539192	+	0.842183i	$0.318730\pi$
$422$		0			0
$423$	−384.000	+	475.176i	−0.907801	+	1.12335i
$424$		0			0
$425$			158.392i			0.372687i
$426$		0			0
$427$	516.000			1.20843
$428$		0			0
$429$	−160.000	−	56.5685i	−0.372960	−	0.131861i
$430$		0			0
$431$		−	248.902i		−	0.577498i	−0.957405	−	0.288749i	$-0.906761\pi$
$431$		−	248.902i		−	0.577498i	0.957405	−	0.288749i	$-0.0932393\pi$
$432$		0			0
$433$	706.000			1.63048			0.815242	−	0.579120i	$-0.196604\pi$
$433$	706.000			1.63048			0.815242	+	0.579120i	$0.196604\pi$
$434$		0			0
$435$	−96.0000	+	271.529i	−0.220690	+	0.624205i
$436$		0			0
$437$			22.6274i			0.0517790i
$438$		0			0
$439$	486.000			1.10706			0.553531	−	0.832829i	$-0.313280\pi$
$439$	486.000			1.10706			0.553531	+	0.832829i	$0.313280\pi$
$440$		0			0
$441$	91.0000	+	73.5391i	0.206349	+	0.166755i
$442$		0			0
$443$			707.107i			1.59618i	0.602540	+	0.798089i	$0.294156\pi$
$443$			707.107i			1.59618i	−0.602540	+	0.798089i	$0.705844\pi$
$444$		0			0
$445$	−192.000			−0.431461
$446$		0			0
$447$	−464.000	−	164.049i	−1.03803	−	0.366999i
$448$		0			0
$449$		−	724.077i		−	1.61264i	−0.591477	−	0.806322i	$-0.701455\pi$
$449$		−	724.077i		−	1.61264i	0.591477	−	0.806322i	$-0.298545\pi$
$450$		0			0
$451$	−192.000			−0.425721
$452$		0			0
$453$	−218.000	+	616.597i	−0.481236	+	1.36114i
$454$		0			0
$455$			339.411i			0.745959i
$456$		0			0
$457$	338.000			0.739606			0.369803	−	0.929110i	$-0.379425\pi$
$457$	338.000			0.739606			0.369803	+	0.929110i	$0.379425\pi$
$458$		0			0
$459$	320.000	+	520.431i	0.697168	+	1.13384i
$460$		0			0
$461$			774.989i			1.68110i	0.541731	+	0.840552i	$0.317769\pi$
$461$			774.989i			1.68110i	−0.541731	+	0.840552i	$0.682231\pi$
$462$		0			0
$463$	−74.0000			−0.159827			−0.0799136	−	0.996802i	$-0.525464\pi$
$463$	−74.0000			−0.159827			−0.0799136	+	0.996802i	$0.525464\pi$
$464$		0			0
$465$	−352.000	−	124.451i	−0.756989	−	0.267636i
$466$		0			0
$467$		−	797.616i		−	1.70796i	−0.520307	−	0.853979i	$-0.674183\pi$
$467$		−	797.616i		−	1.70796i	0.520307	−	0.853979i	$-0.325817\pi$
$468$		0			0
$469$	12.0000			0.0255864
$470$		0			0
$471$	86.0000	−	243.245i	0.182590	−	0.516443i
$472$		0			0
$473$			463.862i			0.980681i
$474$		0			0
$475$	−14.0000			−0.0294737
$476$		0			0
$477$	352.000	−	435.578i	0.737945	−	0.913161i
$478$		0			0
$479$		−	316.784i		−	0.661344i	−0.943746	−	0.330672i	$-0.892725\pi$
$479$		−	316.784i		−	0.661344i	0.943746	−	0.330672i	$-0.107275\pi$
$480$		0			0
$481$	−60.0000			−0.124740
$482$		0			0
$483$	192.000	+	67.8823i	0.397516	+	0.140543i
$484$		0			0
$485$			531.744i			1.09638i
$486$		0			0
$487$	134.000			0.275154			0.137577	−	0.990491i	$-0.456069\pi$
$487$	134.000			0.275154			0.137577	+	0.990491i	$0.456069\pi$
$488$		0			0
$489$	−222.000	+	627.911i	−0.453988	+	1.28407i
$490$		0			0
$491$			50.9117i			0.103690i	0.998655	+	0.0518449i	$0.0165101\pi$
$491$			50.9117i			0.103690i	−0.998655	+	0.0518449i	$0.983490\pi$
$492$		0			0
$493$	−384.000			−0.778905
$494$		0			0
$495$	−224.000	−	181.019i	−0.452525	−	0.365696i
$496$		0			0
$497$			746.705i			1.50242i
$498$		0			0
$499$	−30.0000			−0.0601202			−0.0300601	−	0.999548i	$-0.509570\pi$
$499$	−30.0000			−0.0601202			−0.0300601	+	0.999548i	$0.509570\pi$
$500$		0			0
$501$	480.000	+	169.706i	0.958084	+	0.338734i
$502$		0			0
$503$			237.588i			0.472342i	0.971712	+	0.236171i	$0.0758925\pi$
$503$			237.588i			0.472342i	−0.971712	+	0.236171i	$0.924107\pi$
$504$		0			0
$505$	−288.000			−0.570297
$506$		0			0
$507$	−69.0000	+	195.161i	−0.136095	+	0.384934i
$508$		0			0
$509$			118.794i			0.233387i	0.993168	+	0.116693i	$0.0372295\pi$
$509$			118.794i			0.233387i	−0.993168	+	0.116693i	$0.962770\pi$
$510$		0			0
$511$	492.000			0.962818
$512$		0			0
$513$	−46.0000	+	28.2843i	−0.0896686	+	0.0551350i
$514$		0			0
$515$		−	758.018i		−	1.47188i
$516$		0			0
$517$	384.000			0.742747
$518$		0			0
$519$	528.000	+	186.676i	1.01734	+	0.359684i
$520$		0			0
$521$			79.1960i			0.152008i	0.997108	+	0.0760038i	$0.0242161\pi$
$521$			79.1960i			0.152008i	−0.997108	+	0.0760038i	$0.975784\pi$
$522$		0			0
$523$	−494.000			−0.944551			−0.472275	−	0.881451i	$-0.656567\pi$
$523$	−494.000			−0.944551			−0.472275	+	0.881451i	$0.656567\pi$
$524$		0			0
$525$	−42.0000	+	118.794i	−0.0800000	+	0.226274i
$526$		0			0
$527$		−	497.803i		−	0.944598i
$528$		0			0
$529$	401.000			0.758034
$530$		0			0
$531$	416.000	−	514.774i	0.783427	−	0.969442i
$532$		0			0
$533$		−	339.411i		−	0.636794i
$534$		0			0
$535$	288.000			0.538318
$536$		0			0
$537$	432.000	+	152.735i	0.804469	+	0.284423i
$538$		0			0
$539$		−	73.5391i		−	0.136436i
$540$		0			0
$541$	−234.000			−0.432532			−0.216266	−	0.976334i	$-0.569388\pi$
$541$	−234.000			−0.432532			−0.216266	+	0.976334i	$0.569388\pi$
$542$		0			0
$543$	−90.0000	+	254.558i	−0.165746	+	0.468800i
$544$		0			0
$545$			56.5685i			0.103795i
$546$		0			0
$547$	290.000			0.530165			0.265082	−	0.964226i	$-0.414601\pi$
$547$	290.000			0.530165			0.265082	+	0.964226i	$0.414601\pi$
$548$		0			0
$549$	−602.000	−	486.489i	−1.09654	−	0.886137i
$550$		0			0
$551$		−	33.9411i		−	0.0615991i
$552$		0			0
$553$	−60.0000			−0.108499
$554$		0			0
$555$	−96.0000	−	33.9411i	−0.172973	−	0.0611552i
$556$		0			0
$557$		−	175.362i		−	0.314834i	−0.987532	−	0.157417i	$-0.949683\pi$
$557$		−	175.362i		−	0.314834i	0.987532	−	0.157417i	$-0.0503167\pi$
$558$		0			0
$559$	−820.000			−1.46691
$560$		0			0
$561$	128.000	−	362.039i	0.228164	−	0.645345i
$562$		0			0
$563$			243.245i			0.432051i	0.976388	+	0.216026i	$0.0693094\pi$
$563$			243.245i			0.432051i	−0.976388	+	0.216026i	$0.930691\pi$
$564$		0			0
$565$	−384.000			−0.679646
$566$		0			0
$567$	102.000	+	475.176i	0.179894	+	0.838052i
$568$		0			0
$569$		−	622.254i		−	1.09359i	−0.837266	−	0.546796i	$-0.815847\pi$
$569$		−	622.254i		−	1.09359i	0.837266	−	0.546796i	$-0.184153\pi$
$570$		0			0
$571$	402.000			0.704028			0.352014	−	0.935995i	$-0.385497\pi$
$571$	402.000			0.704028			0.352014	+	0.935995i	$0.385497\pi$
$572$		0			0
$573$	768.000	+	271.529i	1.34031	+	0.473873i
$574$		0			0
$575$		−	79.1960i		−	0.137732i
$576$		0			0
$577$	98.0000			0.169844			0.0849220	−	0.996388i	$-0.472936\pi$
$577$	98.0000			0.169844			0.0849220	+	0.996388i	$0.472936\pi$
$578$		0			0
$579$	2.00000	−	5.65685i	0.00345423	−	0.00977004i
$580$		0			0
$581$		−	441.235i		−	0.759440i
$582$		0			0
$583$	−352.000			−0.603774
$584$		0			0
$585$	320.000	−	395.980i	0.547009	−	0.676889i
$586$		0			0
$587$			548.715i			0.934778i	0.884052	+	0.467389i	$0.154805\pi$
$587$			548.715i			0.934778i	−0.884052	+	0.467389i	$0.845195\pi$
$588$		0			0
$589$	44.0000			0.0747029
$590$		0			0
$591$	240.000	+	84.8528i	0.406091	+	0.143575i
$592$		0			0
$593$			701.450i			1.18288i	0.806348	+	0.591442i	$0.201441\pi$
$593$			701.450i			1.18288i	−0.806348	+	0.591442i	$0.798559\pi$
$594$		0			0
$595$	−768.000			−1.29076
$596$		0			0
$597$	−250.000	+	707.107i	−0.418760	+	1.18443i
$598$		0			0
$599$			644.881i			1.07660i	0.842754	+	0.538298i	$0.180933\pi$
$599$			644.881i			1.07660i	−0.842754	+	0.538298i	$0.819067\pi$
$600$		0			0
$601$	−398.000			−0.662230			−0.331115	−	0.943590i	$-0.607425\pi$
$601$	−398.000			−0.662230			−0.331115	+	0.943590i	$0.607425\pi$
$602$		0			0
$603$	−14.0000	−	11.3137i	−0.0232172	−	0.0187624i
$604$		0			0
$605$		−	503.460i		−	0.832165i
$606$		0			0
$607$	−170.000			−0.280066			−0.140033	−	0.990147i	$-0.544721\pi$
$607$	−170.000			−0.280066			−0.140033	+	0.990147i	$0.544721\pi$
$608$		0			0
$609$	−288.000	−	101.823i	−0.472906	−	0.167198i
$610$		0			0
$611$			678.823i			1.11100i
$612$		0			0
$613$	1030.00			1.68026			0.840131	−	0.542384i	$-0.182478\pi$
$613$	1030.00			1.68026			0.840131	+	0.542384i	$0.182478\pi$
$614$		0			0
$615$	192.000	−	543.058i	0.312195	−	0.883021i
$616$		0			0
$617$		−	1052.17i		−	1.70531i	−0.522476	−	0.852654i	$-0.674992\pi$
$617$		−	1052.17i		−	1.70531i	0.522476	−	0.852654i	$-0.325008\pi$
$618$		0			0
$619$	−14.0000			−0.0226171			−0.0113086	−	0.999936i	$-0.503600\pi$
$619$	−14.0000			−0.0226171			−0.0113086	+	0.999936i	$0.503600\pi$
$620$		0			0
$621$	−160.000	−	260.215i	−0.257649	−	0.419026i
$622$		0			0
$623$		−	203.647i		−	0.326881i
$624$		0			0
$625$	−751.000			−1.20160
$626$		0			0
$627$	32.0000	+	11.3137i	0.0510367	+	0.0180442i
$628$		0			0
$629$		−	135.765i		−	0.215842i
$630$		0			0
$631$	−1114.00			−1.76545			−0.882726	−	0.469888i	$-0.844294\pi$
$631$	−1114.00			−1.76545			−0.882726	+	0.469888i	$0.844294\pi$
$632$		0			0
$633$	34.0000	−	96.1665i	0.0537125	−	0.151922i
$634$		0			0
$635$			599.627i			0.944294i
$636$		0			0
$637$	130.000			0.204082
$638$		0			0
$639$	704.000	−	871.156i	1.10172	−	1.36331i
$640$		0			0
$641$		−	452.548i		−	0.706004i	−0.935623	−	0.353002i	$-0.885161\pi$
$641$		−	452.548i		−	0.706004i	0.935623	−	0.353002i	$-0.114839\pi$
$642$		0			0
$643$	−798.000			−1.24106			−0.620529	−	0.784184i	$-0.713082\pi$
$643$	−798.000			−1.24106			−0.620529	+	0.784184i	$0.713082\pi$
$644$		0			0
$645$	−1312.00	−	463.862i	−2.03411	−	0.719166i
$646$		0			0
$647$		−	147.078i		−	0.227323i	−0.993520	−	0.113662i	$-0.963742\pi$
$647$		−	147.078i		−	0.227323i	0.993520	−	0.113662i	$-0.0362580\pi$
$648$		0			0
$649$	−416.000			−0.640986
$650$		0			0
$651$	132.000	−	373.352i	0.202765	−	0.573506i
$652$		0			0
$653$			322.441i			0.493784i	0.969043	+	0.246892i	$0.0794092\pi$
$653$			322.441i			0.493784i	−0.969043	+	0.246892i	$0.920591\pi$
$654$		0			0
$655$	−32.0000			−0.0488550
$656$		0			0
$657$	−574.000	−	463.862i	−0.873668	−	0.706031i
$658$		0			0
$659$		−	797.616i		−	1.21034i	−0.796095	−	0.605172i	$-0.793104\pi$
$659$		−	797.616i		−	1.21034i	0.796095	−	0.605172i	$-0.206896\pi$
$660$		0			0
$661$	−986.000			−1.49168			−0.745840	−	0.666126i	$-0.767951\pi$
$661$	−986.000			−1.49168			−0.745840	+	0.666126i	$0.767951\pi$
$662$		0			0
$663$	640.000	+	226.274i	0.965309	+	0.341288i
$664$		0			0
$665$		−	67.8823i		−	0.102079i
$666$		0			0
$667$	192.000			0.287856
$668$		0			0
$669$	278.000	−	786.303i	0.415546	−	1.17534i
$670$		0			0
$671$			486.489i			0.725022i
$672$		0			0
$673$	34.0000			0.0505201			0.0252600	−	0.999681i	$-0.491959\pi$
$673$	34.0000			0.0505201			0.0252600	+	0.999681i	$0.491959\pi$
$674$		0			0
$675$	161.000	−	98.9949i	0.238519	−	0.146659i
$676$		0			0
$677$			401.637i			0.593259i	0.954993	+	0.296630i	$0.0958627\pi$
$677$			401.637i			0.593259i	−0.954993	+	0.296630i	$0.904137\pi$
$678$		0			0
$679$	−564.000			−0.830633
$680$		0			0
$681$	624.000	+	220.617i	0.916300	+	0.323961i
$682$		0			0
$683$		−	130.108i		−	0.190494i	−0.995454	−	0.0952472i	$-0.969636\pi$
$683$		−	130.108i		−	0.190494i	0.995454	−	0.0952472i	$-0.0303641\pi$
$684$		0			0
$685$	−576.000			−0.840876
$686$		0			0
$687$	−58.0000	+	164.049i	−0.0844250	+	0.238790i
$688$		0			0
$689$		−	622.254i		−	0.903126i
$690$		0			0
$691$	578.000			0.836469			0.418234	−	0.908339i	$-0.362649\pi$
$691$	578.000			0.836469			0.418234	+	0.908339i	$0.362649\pi$
$692$		0			0
$693$	192.000	−	237.588i	0.277056	−	0.342840i
$694$		0			0
$695$			441.235i			0.634870i
$696$		0			0
$697$	768.000			1.10187
$698$		0			0
$699$	1120.00	+	395.980i	1.60229	+	0.566495i
$700$		0			0
$701$		−	1238.85i		−	1.76726i	−0.468183	−	0.883631i	$-0.655091\pi$
$701$		−	1238.85i		−	1.76726i	0.468183	−	0.883631i	$-0.344909\pi$
$702$		0			0
$703$	12.0000			0.0170697
$704$		0			0
$705$	−384.000	+	1086.12i	−0.544681	+	1.54059i
$706$		0			0
$707$		−	305.470i		−	0.432065i
$708$		0			0
$709$	1222.00			1.72355			0.861777	−	0.507287i	$-0.169352\pi$
$709$	1222.00			1.72355			0.861777	+	0.507287i	$0.169352\pi$
$710$		0			0
$711$	70.0000	+	56.5685i	0.0984529	+	0.0795619i
$712$		0			0
$713$			248.902i			0.349091i
$714$		0			0
$715$	−320.000			−0.447552
$716$		0			0
$717$	64.0000	+	22.6274i	0.0892608	+	0.0315585i
$718$		0			0
$719$			248.902i			0.346177i	0.984906	+	0.173089i	$0.0553747\pi$
$719$			248.902i			0.346177i	−0.984906	+	0.173089i	$0.944625\pi$
$720$		0			0
$721$	804.000			1.11512
$722$		0			0
$723$	−30.0000	+	84.8528i	−0.0414938	+	0.117362i
$724$		0			0
$725$			118.794i			0.163854i
$726$		0			0
$727$	870.000			1.19670			0.598349	−	0.801235i	$-0.295823\pi$
$727$	870.000			1.19670			0.598349	+	0.801235i	$0.295823\pi$
$728$		0			0
$729$	329.000	−	650.538i	0.451303	−	0.892371i
$730$		0			0
$731$		−	1855.45i		−	2.53823i
$732$		0			0
$733$	214.000			0.291951			0.145975	−	0.989288i	$-0.453368\pi$
$733$	214.000			0.291951			0.145975	+	0.989288i	$0.453368\pi$
$734$		0			0
$735$	208.000	+	73.5391i	0.282993	+	0.100053i
$736$		0			0
$737$			11.3137i			0.0153510i
$738$		0			0
$739$	−958.000			−1.29635			−0.648173	−	0.761493i	$-0.724467\pi$
$739$	−958.000			−1.29635			−0.648173	+	0.761493i	$0.724467\pi$
$740$		0			0
$741$	−20.0000	+	56.5685i	−0.0269906	+	0.0763408i
$742$		0			0
$743$		−	1006.92i		−	1.35521i	−0.735427	−	0.677604i	$-0.763018\pi$
$743$		−	1006.92i		−	1.35521i	0.735427	−	0.677604i	$-0.236982\pi$
$744$		0			0
$745$	−928.000			−1.24564
$746$		0			0
$747$	−416.000	+	514.774i	−0.556894	+	0.689121i
$748$		0			0
$749$			305.470i			0.407837i
$750$		0			0
$751$	630.000			0.838881			0.419441	−	0.907783i	$-0.362226\pi$
$751$	630.000			0.838881			0.419441	+	0.907783i	$0.362226\pi$
$752$		0			0
$753$	−304.000	−	107.480i	−0.403718	−	0.142736i
$754$		0			0
$755$			1233.19i			1.63337i
$756$		0			0
$757$	−602.000			−0.795244			−0.397622	−	0.917549i	$-0.630165\pi$
$757$	−602.000			−0.795244			−0.397622	+	0.917549i	$0.630165\pi$
$758$		0			0
$759$	−64.0000	+	181.019i	−0.0843215	+	0.238497i
$760$		0			0
$761$			1097.43i			1.44209i	0.692889	+	0.721044i	$0.256338\pi$
$761$			1097.43i			1.44209i	−0.692889	+	0.721044i	$0.743662\pi$
$762$		0			0
$763$	−60.0000			−0.0786370
$764$		0			0
$765$	896.000	+	724.077i	1.17124	+	0.946506i
$766$		0			0
$767$		−	735.391i		−	0.958789i
$768$		0			0
$769$	770.000			1.00130			0.500650	−	0.865650i	$-0.333094\pi$
$769$	770.000			1.00130			0.500650	+	0.865650i	$0.333094\pi$
$770$		0			0
$771$	512.000	+	181.019i	0.664073	+	0.234785i
$772$		0			0
$773$		−	186.676i		−	0.241496i	−0.992683	−	0.120748i	$-0.961471\pi$
$773$		−	186.676i		−	0.241496i	0.992683	−	0.120748i	$-0.0385293\pi$
$774$		0			0
$775$	−154.000			−0.198710
$776$		0			0
$777$	36.0000	−	101.823i	0.0463320	−	0.131047i
$778$		0			0
$779$			67.8823i			0.0871402i
$780$		0			0
$781$	−704.000			−0.901408
$782$		0			0
$783$	240.000	+	390.323i	0.306513	+	0.498497i
$784$		0			0
$785$		−	486.489i		−	0.619732i
$786$		0			0
$787$	514.000			0.653113			0.326557	−	0.945178i	$-0.394112\pi$
$787$	514.000			0.653113			0.326557	+	0.945178i	$0.394112\pi$
$788$		0			0
$789$	608.000	+	214.960i	0.770596	+	0.272447i
$790$		0			0
$791$		−	407.294i		−	0.514910i
$792$		0			0
$793$	−860.000

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 2.82843i) q^{3} -5.65685i q^{5} +6.00000 q^{7} +(-7.00000 - 5.65685i) q^{9} +5.65685i q^{11} -10.0000 q^{13} +(-16.0000 - 5.65685i) q^{15} -22.6274i q^{17} +2.00000 q^{19} +(6.00000 - 16.9706i) q^{21} +11.3137i q^{23} -7.00000 q^{25} +(-23.0000 + 14.1421i) q^{27} -16.9706i q^{29} +22.0000 q^{31} +(16.0000 + 5.65685i) q^{33} -33.9411i q^{35} +6.00000 q^{37} +(-10.0000 + 28.2843i) q^{39} +33.9411i q^{41} +82.0000 q^{43} +(-32.0000 + 39.5980i) q^{45} -67.8823i q^{47} -13.0000 q^{49} +(-64.0000 - 22.6274i) q^{51} +62.2254i q^{53} +32.0000 q^{55} +(2.00000 - 5.65685i) q^{57} +73.5391i q^{59} +86.0000 q^{61} +(-42.0000 - 33.9411i) q^{63} +56.5685i q^{65} +2.00000 q^{67} +(32.0000 + 11.3137i) q^{69} +124.451i q^{71} +82.0000 q^{73} +(-7.00000 + 19.7990i) q^{75} +33.9411i q^{77} -10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391i q^{83} -128.000 q^{85} +(-48.0000 - 16.9706i) q^{87} -33.9411i q^{89} -60.0000 q^{91} +(22.0000 - 62.2254i) q^{93} -11.3137i q^{95} -94.0000 q^{97} +(32.0000 - 39.5980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 12 q^{7} - 14 q^{9} - 20 q^{13} - 32 q^{15} + 4 q^{19} + 12 q^{21} - 14 q^{25} - 46 q^{27} + 44 q^{31} + 32 q^{33} + 12 q^{37} - 20 q^{39} + 164 q^{43} - 64 q^{45} - 26 q^{49} - 128 q^{51}+ \cdots + 64 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 12 * q^7 - 14 * q^9 - 20 * q^13 - 32 * q^15 + 4 * q^19 + 12 * q^21 - 14 * q^25 - 46 * q^27 + 44 * q^31 + 32 * q^33 + 12 * q^37 - 20 * q^39 + 164 * q^43 - 64 * q^45 - 26 * q^49 - 128 * q^51 + 64 * q^55 + 4 * q^57 + 172 * q^61 - 84 * q^63 + 4 * q^67 + 64 * q^69 + 164 * q^73 - 14 * q^75 - 20 * q^79 + 34 * q^81 - 256 * q^85 - 96 * q^87 - 120 * q^91 + 44 * q^93 - 188 * q^97 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more