LMFDB - Embedded newform 160.3.h.a.159.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(159,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("160.159");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.h (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:	$4.35968422976$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			159.4
Root			$0.273891i$ of defining polynomial
Character	$\chi$	$=$	160.159
Dual form			160.3.h.a.159.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.547781 q^{3} +(-3.75441 + 3.30219i) q^{5} -10.0566 q^{7} -8.69994 q^{9} +O(q^{10})$	$q+0.547781 q^{3} +(-3.75441 + 3.30219i) q^{5} -10.0566 q^{7} -8.69994 q^{9} -17.2087i q^{11} +4.41325i q^{13} +(-2.05659 + 1.80888i) q^{15} +27.0176i q^{17} -4.82650i q^{19} -5.50881 q^{21} -15.2653 q^{23} +(3.19112 - 24.7955i) q^{25} -9.69569 q^{27} -2.38225 q^{29} +38.0352i q^{31} -9.42663i q^{33} +(37.7565 - 33.2087i) q^{35} +16.5691i q^{37} +2.41749i q^{39} -13.3177 q^{41} +59.7918 q^{43} +(32.6631 - 28.7288i) q^{45} -62.4388 q^{47} +52.1351 q^{49} +14.7997i q^{51} -71.5952i q^{53} +(56.8265 + 64.6086i) q^{55} -2.64386i q^{57} +68.8265i q^{59} +40.9439 q^{61} +87.4917 q^{63} +(-14.5734 - 16.5691i) q^{65} -51.0080 q^{67} -8.36206 q^{69} -40.4527i q^{71} -35.8441i q^{73} +(1.74804 - 13.5825i) q^{75} +173.061i q^{77} -126.800i q^{79} +72.9883 q^{81} -75.1490 q^{83} +(-89.2172 - 101.435i) q^{85} -1.30495 q^{87} -106.523 q^{89} -44.3822i q^{91} +20.8350i q^{93} +(15.9380 + 18.1206i) q^{95} +85.4351i q^{97} +149.715i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 18 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 18 * q^9	$ 6 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 18 q^{9} + 36 q^{15} + 8 q^{21} + 68 q^{23} - 10 q^{25} - 184 q^{27} + 44 q^{29} + 108 q^{35} - 68 q^{41} + 76 q^{43} - 6 q^{45} - 268 q^{47} - 62 q^{49} + 288 q^{55} - 100 q^{61} + 172 q^{63} - 308 q^{67} - 184 q^{69} + 284 q^{75} + 238 q^{81} + 204 q^{83} - 32 q^{85} - 584 q^{87} + 76 q^{89} + 32 q^{95}+O(q^{100}) $ 6 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 18 * q^9 + 36 * q^15 + 8 * q^21 + 68 * q^23 - 10 * q^25 - 184 * q^27 + 44 * q^29 + 108 * q^35 - 68 * q^41 + 76 * q^43 - 6 * q^45 - 268 * q^47 - 62 * q^49 + 288 * q^55 - 100 * q^61 + 172 * q^63 - 308 * q^67 - 184 * q^69 + 284 * q^75 + 238 * q^81 + 204 * q^83 - 32 * q^85 - 584 * q^87 + 76 * q^89 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0.547781			0.182594			0.0912969	−	0.995824i	$-0.470899\pi$
$3$	0.547781			0.182594			0.0912969	+	0.995824i	$0.470899\pi$
$4$		0			0
$5$	−3.75441	+	3.30219i	−0.750881	+	0.660437i
$5$	−3.75441	+	3.30219i	−0.750881	+	0.660437i
$6$		0			0
$7$	−10.0566			−1.43666			−0.718328	−	0.695705i	$-0.755092\pi$
$7$	−10.0566			−1.43666			−0.718328	+	0.695705i	$0.755092\pi$
$8$		0			0
$9$	−8.69994			−0.966660
$10$		0			0
$11$		−	17.2087i		−	1.56443i	−0.623008	−	0.782216i	$-0.714089\pi$
$11$		−	17.2087i		−	1.56443i	0.623008	−	0.782216i	$-0.285911\pi$
$12$		0			0
$13$			4.41325i			0.339481i	0.985489	+	0.169740i	$0.0542929\pi$
$13$			4.41325i			0.339481i	−0.985489	+	0.169740i	$0.945707\pi$
$14$		0			0
$15$	−2.05659	+	1.80888i	−0.137106	+	0.120592i
$16$		0			0
$17$			27.0176i			1.58927i	0.607086	+	0.794636i	$0.292338\pi$
$17$			27.0176i			1.58927i	−0.607086	+	0.794636i	$0.707662\pi$
$18$		0			0
$19$		−	4.82650i		−	0.254026i	−0.991901	−	0.127013i	$-0.959461\pi$
$19$		−	4.82650i		−	0.254026i	0.991901	−	0.127013i	$-0.0405390\pi$
$20$		0			0
$21$	−5.50881			−0.262324
$22$		0			0
$23$	−15.2653			−0.663710			−0.331855	−	0.943330i	$-0.607675\pi$
$23$	−15.2653			−0.663710			−0.331855	+	0.943330i	$0.607675\pi$
$24$		0			0
$25$	3.19112	−	24.7955i	0.127645	−	0.991820i
$26$		0			0
$27$	−9.69569			−0.359100
$28$		0			0
$29$	−2.38225			−0.0821465			−0.0410733	−	0.999156i	$-0.513078\pi$
$29$	−2.38225			−0.0821465			−0.0410733	+	0.999156i	$0.513078\pi$
$30$		0			0
$31$			38.0352i			1.22694i	0.789717	+	0.613472i	$0.210228\pi$
$31$			38.0352i			1.22694i	−0.789717	+	0.613472i	$0.789772\pi$
$32$		0			0
$33$		−	9.42663i		−	0.285655i
$34$		0			0
$35$	37.7565	−	33.2087i	1.07876	−	0.948821i
$36$		0			0
$37$			16.5691i			0.447814i	0.974610	+	0.223907i	$0.0718812\pi$
$37$			16.5691i			0.447814i	−0.974610	+	0.223907i	$0.928119\pi$
$38$		0			0
$39$			2.41749i			0.0619870i
$40$		0			0
$41$	−13.3177			−0.324822			−0.162411	−	0.986723i	$-0.551927\pi$
$41$	−13.3177			−0.324822			−0.162411	+	0.986723i	$0.551927\pi$
$42$		0			0
$43$	59.7918			1.39051			0.695253	−	0.718765i	$-0.255292\pi$
$43$	59.7918			1.39051			0.695253	+	0.718765i	$0.255292\pi$
$44$		0			0
$45$	32.6631	−	28.7288i	0.725846	−	0.638418i
$46$		0			0
$47$	−62.4388			−1.32849			−0.664243	−	0.747517i	$-0.731246\pi$
$47$	−62.4388			−1.32849			−0.664243	+	0.747517i	$0.731246\pi$
$48$		0			0
$49$	52.1351			1.06398
$50$		0			0
$51$			14.7997i			0.290191i
$52$		0			0
$53$		−	71.5952i		−	1.35085i	−0.737427	−	0.675427i	$-0.763959\pi$
$53$		−	71.5952i		−	1.35085i	0.737427	−	0.675427i	$-0.236041\pi$
$54$		0			0
$55$	56.8265	+	64.6086i	1.03321	+	1.17470i
$56$		0			0
$57$		−	2.64386i		−	0.0463836i
$58$		0			0
$59$			68.8265i			1.16655i	0.812274	+	0.583275i	$0.198229\pi$
$59$			68.8265i			1.16655i	−0.812274	+	0.583275i	$0.801771\pi$
$60$		0			0
$61$	40.9439			0.671212			0.335606	−	0.942002i	$-0.391059\pi$
$61$	40.9439			0.671212			0.335606	+	0.942002i	$0.391059\pi$
$62$		0			0
$63$	87.4917			1.38876
$64$		0			0
$65$	−14.5734	−	16.5691i	−0.224206	−	0.254910i
$66$		0			0
$67$	−51.0080			−0.761313			−0.380656	−	0.924717i	$-0.624302\pi$
$67$	−51.0080			−0.761313			−0.380656	+	0.924717i	$0.624302\pi$
$68$		0			0
$69$	−8.36206			−0.121189
$70$		0			0
$71$		−	40.4527i		−	0.569757i	−0.958564	−	0.284878i	$-0.908047\pi$
$71$		−	40.4527i		−	0.569757i	0.958564	−	0.284878i	$-0.0919532\pi$
$72$		0			0
$73$		−	35.8441i		−	0.491015i	−0.969395	−	0.245508i	$-0.921045\pi$
$73$		−	35.8441i		−	0.491015i	0.969395	−	0.245508i	$-0.0789547\pi$
$74$		0			0
$75$	1.74804	−	13.5825i	0.0233072	−	0.181100i
$76$		0			0
$77$			173.061i			2.24755i
$78$		0			0
$79$		−	126.800i		−	1.60506i	−0.596612	−	0.802530i	$-0.703487\pi$
$79$		−	126.800i		−	1.60506i	0.596612	−	0.802530i	$-0.296513\pi$
$80$		0			0
$81$	72.9883			0.901090
$82$		0			0
$83$	−75.1490			−0.905409			−0.452705	−	0.891661i	$-0.649541\pi$
$83$	−75.1490			−0.905409			−0.452705	+	0.891661i	$0.649541\pi$
$84$		0			0
$85$	−89.2172	−	101.435i	−1.04961	−	1.19335i
$86$		0			0
$87$	−1.30495			−0.0149994
$88$		0			0
$89$	−106.523			−1.19689			−0.598445	−	0.801164i	$-0.704215\pi$
$89$	−106.523			−1.19689			−0.598445	+	0.801164i	$0.704215\pi$
$90$		0			0
$91$		−	44.3822i		−	0.487717i
$92$		0			0
$93$			20.8350i			0.224032i
$94$		0			0
$95$	15.9380	+	18.1206i	0.167768	+	0.190744i
$96$		0			0
$97$			85.4351i			0.880774i	0.897808	+	0.440387i	$0.145159\pi$
$97$			85.4351i			0.880774i	−0.897808	+	0.440387i	$0.854841\pi$
$98$		0			0
$99$			149.715i			1.51227i
$100$		0			0
$101$	−83.2877			−0.824631			−0.412315	−	0.911041i	$-0.635280\pi$
$101$	−83.2877			−0.824631			−0.412315	+	0.911041i	$0.635280\pi$
$102$		0			0
$103$	−47.2653			−0.458887			−0.229443	−	0.973322i	$-0.573691\pi$
$103$	−47.2653			−0.458887			−0.229443	+	0.973322i	$0.573691\pi$
$104$		0			0
$105$	20.6823	−	18.1911i	0.196974	−	0.173249i
$106$		0			0
$107$	−189.628			−1.77223			−0.886114	−	0.463467i	$-0.846605\pi$
$107$	−189.628			−1.77223			−0.886114	+	0.463467i	$0.846605\pi$
$108$		0			0
$109$	84.5617			0.775795			0.387898	−	0.921702i	$-0.373201\pi$
$109$	84.5617			0.775795			0.387898	+	0.921702i	$0.373201\pi$
$110$		0			0
$111$			9.07626i			0.0817681i
$112$		0			0
$113$			114.558i			1.01379i	0.862007	+	0.506896i	$0.169207\pi$
$113$			114.558i			1.01379i	−0.862007	+	0.506896i	$0.830793\pi$
$114$		0			0
$115$	57.3123	−	50.4090i	0.498368	−	0.438339i
$116$		0			0
$117$		−	38.3950i		−	0.328162i
$118$		0			0
$119$		−	271.705i		−	2.28324i
$120$		0			0
$121$	−175.141			−1.44745
$122$		0			0
$123$	−7.29518			−0.0593104
$124$		0			0
$125$	69.8986	+	103.630i	0.559189	+	0.829040i
$126$		0			0
$127$	−44.4121			−0.349701			−0.174851	−	0.984595i	$-0.555944\pi$
$127$	−44.4121			−0.349701			−0.174851	+	0.984595i	$0.555944\pi$
$128$		0			0
$129$	32.7528			0.253898
$130$		0			0
$131$		−	47.1200i		−	0.359695i	−0.983695	−	0.179847i	$-0.942440\pi$
$131$		−	47.1200i		−	0.359695i	0.983695	−	0.179847i	$-0.0575604\pi$
$132$		0			0
$133$			48.5381i			0.364948i
$134$		0			0
$135$	36.4016	−	32.0170i	0.269641	−	0.237163i
$136$		0			0
$137$			62.8617i			0.458845i	0.973327	+	0.229422i	$0.0736837\pi$
$137$			62.8617i			0.458845i	−0.973327	+	0.229422i	$0.926316\pi$
$138$		0			0
$139$			128.320i			0.923167i	0.887097	+	0.461584i	$0.152719\pi$
$139$			128.320i			0.923167i	−0.887097	+	0.461584i	$0.847281\pi$
$140$		0			0
$141$	−34.2028			−0.242573
$142$		0			0
$143$	75.9465			0.531094
$144$		0			0
$145$	8.94393	−	7.86663i	0.0616823	−	0.0542526i
$146$		0			0
$147$	28.5586			0.194276
$148$		0			0
$149$	−165.561			−1.11115			−0.555574	−	0.831467i	$-0.687501\pi$
$149$	−165.561			−1.11115			−0.555574	+	0.831467i	$0.687501\pi$
$150$		0			0
$151$			218.558i			1.44741i	0.690111	+	0.723704i	$0.257562\pi$
$151$			218.558i			1.44741i	−0.690111	+	0.723704i	$0.742438\pi$
$152$		0			0
$153$		−	235.052i		−	1.53628i
$154$		0			0
$155$	−125.599	−	142.800i	−0.810319	−	0.921289i
$156$		0			0
$157$		−	174.293i		−	1.11014i	−0.831802	−	0.555072i	$-0.812691\pi$
$157$		−	174.293i		−	1.11014i	0.831802	−	0.555072i	$-0.187309\pi$
$158$		0			0
$159$		−	39.2185i		−	0.246657i
$160$		0			0
$161$	153.517			0.953524
$162$		0			0
$163$	52.6353			0.322916			0.161458	−	0.986880i	$-0.448380\pi$
$163$	52.6353			0.322916			0.161458	+	0.986880i	$0.448380\pi$
$164$		0			0
$165$	31.1285	+	35.3914i	0.188657	+	0.214493i
$166$		0			0
$167$	76.6812			0.459169			0.229584	−	0.973289i	$-0.426263\pi$
$167$	76.6812			0.459169			0.229584	+	0.973289i	$0.426263\pi$
$168$		0			0
$169$	149.523			0.884753
$170$		0			0
$171$			41.9902i			0.245557i
$172$		0			0
$173$			9.72437i			0.0562102i	0.999605	+	0.0281051i	$0.00894731\pi$
$173$			9.72437i			0.0562102i	−0.999605	+	0.0281051i	$0.991053\pi$
$174$		0			0
$175$	−32.0918	+	249.358i	−0.183382	+	1.42490i
$176$		0			0
$177$			37.7019i			0.213005i
$178$		0			0
$179$			155.502i			0.868728i	0.900738	+	0.434364i	$0.143027\pi$
$179$			155.502i			0.868728i	−0.900738	+	0.434364i	$0.856973\pi$
$180$		0			0
$181$	250.346			1.38313			0.691563	−	0.722316i	$-0.256923\pi$
$181$	250.346			1.38313			0.691563	+	0.722316i	$0.256923\pi$
$182$		0			0
$183$	22.4283			0.122559
$184$		0			0
$185$	−54.7144	−	62.2072i	−0.295753	−	0.336255i
$186$		0			0
$187$	464.939			2.48631
$188$		0			0
$189$	97.5056			0.515903
$190$		0			0
$191$		−	111.128i		−	0.581825i	−0.956750	−	0.290912i	$-0.906041\pi$
$191$		−	111.128i		−	0.581825i	0.956750	−	0.290912i	$-0.0939588\pi$
$192$		0			0
$193$		−	10.2701i		−	0.0532130i	−0.999646	−	0.0266065i	$-0.991530\pi$
$193$		−	10.2701i		−	0.0532130i	0.999646	−	0.0266065i	$-0.00847011\pi$
$194$		0			0
$195$	−7.98302	−	9.07626i	−0.0409386	−	0.0465449i
$196$		0			0
$197$		−	163.213i		−	0.828492i	−0.910165	−	0.414246i	$-0.864045\pi$
$197$		−	163.213i		−	0.828492i	0.910165	−	0.414246i	$-0.135955\pi$
$198$		0			0
$199$			195.028i			0.980041i	0.871711	+	0.490021i	$0.163011\pi$
$199$			195.028i			0.980041i	−0.871711	+	0.490021i	$0.836989\pi$
$200$		0			0
$201$	−27.9412			−0.139011
$202$		0			0
$203$	23.9573			0.118016
$204$		0			0
$205$	50.0000	−	43.9775i	0.243902	−	0.214524i
$206$		0			0
$207$	132.807			0.641582
$208$		0			0
$209$	−83.0580			−0.397407
$210$		0			0
$211$		−	297.038i		−	1.40776i	−0.710317	−	0.703881i	$-0.751449\pi$
$211$		−	297.038i		−	1.40776i	0.710317	−	0.703881i	$-0.248551\pi$
$212$		0			0
$213$		−	22.1592i		−	0.104034i
$214$		0			0
$215$	−224.483	+	197.444i	−1.04411	+	0.918342i
$216$		0			0
$217$		−	382.505i		−	1.76270i
$218$		0			0
$219$		−	19.6347i		−	0.0896563i
$220$		0			0
$221$	−119.236			−0.539527
$222$		0			0
$223$	−405.335			−1.81764			−0.908822	−	0.417184i	$-0.863017\pi$
$223$	−405.335			−1.81764			−0.908822	+	0.417184i	$0.863017\pi$
$224$		0			0
$225$	−27.7626	+	215.719i	−0.123389	+	0.958752i
$226$		0			0
$227$	36.6013			0.161239			0.0806196	−	0.996745i	$-0.474310\pi$
$227$	36.6013			0.161239			0.0806196	+	0.996745i	$0.474310\pi$
$228$		0			0
$229$	−91.1467			−0.398021			−0.199010	−	0.979997i	$-0.563773\pi$
$229$	−91.1467			−0.398021			−0.199010	+	0.979997i	$0.563773\pi$
$230$		0			0
$231$			94.7997i			0.410388i
$232$		0			0
$233$			338.802i			1.45409i	0.686592	+	0.727043i	$0.259106\pi$
$233$			338.802i			1.45409i	−0.686592	+	0.727043i	$0.740894\pi$
$234$		0			0
$235$	234.421	−	206.185i	0.997535	−	0.877382i
$236$		0			0
$237$		−	69.4585i		−	0.293074i
$238$		0			0
$239$		−	30.7997i		−	0.128869i	−0.997922	−	0.0644346i	$-0.979476\pi$
$239$		−	30.7997i		−	0.128869i	0.997922	−	0.0644346i	$-0.0205244\pi$
$240$		0			0
$241$	240.282			0.997020			0.498510	−	0.866884i	$-0.333881\pi$
$241$	240.282			0.997020			0.498510	+	0.866884i	$0.333881\pi$
$242$		0			0
$243$	127.243			0.523633
$244$		0			0
$245$	−195.736	+	172.160i	−0.798923	+	0.702693i
$246$		0			0
$247$	21.3005			0.0862370
$248$		0			0
$249$	−41.1652			−0.165322
$250$		0			0
$251$			86.0268i			0.342736i	0.985207	+	0.171368i	$0.0548187\pi$
$251$			86.0268i			0.342736i	−0.985207	+	0.171368i	$0.945181\pi$
$252$		0			0
$253$			262.697i			1.03833i
$254$		0			0
$255$	−48.8715	−	55.5642i	−0.191653	−	0.217899i
$256$		0			0
$257$		−	355.963i		−	1.38507i	−0.721383	−	0.692536i	$-0.756493\pi$
$257$		−	355.963i		−	1.38507i	0.721383	−	0.692536i	$-0.243507\pi$
$258$		0			0
$259$		−	166.629i		−	0.643355i
$260$		0			0
$261$	20.7254			0.0794077
$262$		0			0
$263$	73.1254			0.278043			0.139022	−	0.990289i	$-0.455604\pi$
$263$	73.1254			0.278043			0.139022	+	0.990289i	$0.455604\pi$
$264$		0			0
$265$	236.421	+	268.798i	0.892154	+	1.01433i
$266$		0			0
$267$	−58.3514			−0.218545
$268$		0			0
$269$	268.002			0.996290			0.498145	−	0.867094i	$-0.334015\pi$
$269$	268.002			0.996290			0.498145	+	0.867094i	$0.334015\pi$
$270$		0			0
$271$			229.412i			0.846538i	0.906004	+	0.423269i	$0.139117\pi$
$271$			229.412i			0.846538i	−0.906004	+	0.423269i	$0.860883\pi$
$272$		0			0
$273$		−	24.3118i		−	0.0890541i
$274$		0			0
$275$	−426.699	−	54.9153i	−1.55163	−	0.199692i
$276$		0			0
$277$			126.327i			0.456053i	0.973655	+	0.228026i	$0.0732272\pi$
$277$			126.327i			0.456053i	−0.973655	+	0.228026i	$0.926773\pi$
$278$		0			0
$279$		−	330.904i		−	1.18604i
$280$		0			0
$281$	−458.746			−1.63255			−0.816275	−	0.577664i	$-0.803964\pi$
$281$	−458.746			−1.63255			−0.816275	+	0.577664i	$0.803964\pi$
$282$		0			0
$283$	−465.558			−1.64508			−0.822541	−	0.568706i	$-0.807444\pi$
$283$	−465.558			−1.64508			−0.822541	+	0.568706i	$0.807444\pi$
$284$		0			0
$285$	8.73053	+	9.92614i	0.0306335	+	0.0348286i
$286$		0			0
$287$	133.931			0.466657
$288$		0			0
$289$	−440.952			−1.52579
$290$		0			0
$291$			46.7997i			0.160824i
$292$		0			0
$293$		−	165.218i		−	0.563884i	−0.959431	−	0.281942i	$-0.909021\pi$
$293$		−	165.218i		−	0.563884i	0.959431	−	0.281942i	$-0.0909786\pi$
$294$		0			0
$295$	−227.278	−	258.403i	−0.770434	−	0.875941i
$296$		0			0
$297$			166.851i			0.561787i
$298$		0			0
$299$		−	67.3697i		−	0.225317i
$300$		0			0
$301$	−601.302			−1.99768
$302$		0			0
$303$	−45.6234			−0.150572
$304$		0			0
$305$	−153.720	+	135.205i	−0.504000	+	0.443293i
$306$		0			0
$307$	235.339			0.766577			0.383288	−	0.923629i	$-0.374792\pi$
$307$	235.339			0.766577			0.383288	+	0.923629i	$0.374792\pi$
$308$		0			0
$309$	−25.8911			−0.0837898
$310$		0			0
$311$		−	210.665i		−	0.677381i	−0.940898	−	0.338691i	$-0.890016\pi$
$311$		−	210.665i		−	0.677381i	0.940898	−	0.338691i	$-0.109984\pi$
$312$		0			0
$313$			318.738i			1.01833i	0.860668	+	0.509166i	$0.170046\pi$
$313$			318.738i			1.01833i	−0.860668	+	0.509166i	$0.829954\pi$
$314$		0			0
$315$	−328.479	+	288.914i	−1.04279	+	0.917187i
$316$		0			0
$317$			70.3950i			0.222066i	0.993817	+	0.111033i	$0.0354160\pi$
$317$			70.3950i			0.222066i	−0.993817	+	0.111033i	$0.964584\pi$
$318$		0			0
$319$			40.9955i			0.128513i
$320$		0			0
$321$	−103.875			−0.323598
$322$		0			0
$323$	130.401			0.403717
$324$		0			0
$325$	109.429	+	14.0832i	0.336704	+	0.0433330i
$326$		0			0
$327$	46.3213			0.141655
$328$		0			0
$329$	627.922			1.90858
$330$		0			0
$331$		−	280.807i		−	0.848359i	−0.905578	−	0.424180i	$-0.860562\pi$
$331$		−	280.807i		−	0.848359i	0.905578	−	0.424180i	$-0.139438\pi$
$332$		0			0
$333$		−	144.150i		−	0.432884i
$334$		0			0
$335$	191.505	−	168.438i	0.571656	−	0.502800i
$336$		0			0
$337$		−	120.969i		−	0.358959i	−0.983762	−	0.179480i	$-0.942559\pi$
$337$		−	120.969i		−	0.358959i	0.983762	−	0.179480i	$-0.0574414\pi$
$338$		0			0
$339$			62.7530i			0.185112i
$340$		0			0
$341$	654.539			1.91947
$342$		0			0
$343$	−31.5280			−0.0919182
$344$		0			0
$345$	31.3946	−	27.6131i	0.0909988	−	0.0800380i
$346$		0			0
$347$	214.333			0.617675			0.308838	−	0.951115i	$-0.400060\pi$
$347$	214.333			0.617675			0.308838	+	0.951115i	$0.400060\pi$
$348$		0			0
$349$	−418.041			−1.19782			−0.598912	−	0.800815i	$-0.704400\pi$
$349$	−418.041			−1.19782			−0.598912	+	0.800815i	$0.704400\pi$
$350$		0			0
$351$		−	42.7895i		−	0.121907i
$352$		0			0
$353$		−	364.929i		−	1.03379i	−0.856048	−	0.516897i	$-0.827087\pi$
$353$		−	364.929i		−	1.03379i	0.856048	−	0.516897i	$-0.172913\pi$
$354$		0			0
$355$	133.583	+	151.876i	0.376289	+	0.427820i
$356$		0			0
$357$		−	148.835i		−	0.416905i
$358$		0			0
$359$			242.169i			0.674567i	0.941403	+	0.337283i	$0.109508\pi$
$359$			242.169i			0.674567i	−0.941403	+	0.337283i	$0.890492\pi$
$360$		0			0
$361$	337.705			0.935471
$362$		0			0
$363$	−95.9389			−0.264295
$364$		0			0
$365$	118.364	+	134.573i	0.324285	+	0.368694i
$366$		0			0
$367$	224.564			0.611891			0.305945	−	0.952049i	$-0.401028\pi$
$367$	224.564			0.611891			0.305945	+	0.952049i	$0.401028\pi$
$368$		0			0
$369$	115.863			0.313992
$370$		0			0
$371$			720.004i			1.94071i
$372$		0			0
$373$			572.174i			1.53398i	0.641660	+	0.766989i	$0.278246\pi$
$373$			572.174i			1.53398i	−0.641660	+	0.766989i	$0.721754\pi$
$374$		0			0
$375$	38.2891	+	56.7666i	0.102104	+	0.151378i
$376$		0			0
$377$		−	10.5135i		−	0.0278872i
$378$		0			0
$379$			122.896i			0.324263i	0.986769	+	0.162132i	$0.0518369\pi$
$379$			122.896i			0.324263i	−0.986769	+	0.162132i	$0.948163\pi$
$380$		0			0
$381$	−24.3281			−0.0638533
$382$		0			0
$383$	289.717			0.756442			0.378221	−	0.925715i	$-0.376536\pi$
$383$	289.717			0.756442			0.378221	+	0.925715i	$0.376536\pi$
$384$		0			0
$385$	−571.481	−	649.743i	−1.48437	−	1.68764i
$386$		0			0
$387$	−520.185			−1.34415
$388$		0			0
$389$	111.590			0.286863			0.143432	−	0.989660i	$-0.454186\pi$
$389$	111.590			0.286863			0.143432	+	0.989660i	$0.454186\pi$
$390$		0			0
$391$		−	412.433i		−	1.05482i
$392$		0			0
$393$		−	25.8114i		−	0.0656780i
$394$		0			0
$395$	418.716	+	476.058i	1.06004	+	1.20521i
$396$		0			0
$397$		−	705.314i		−	1.77661i	−0.459253	−	0.888305i	$-0.651883\pi$
$397$		−	705.314i		−	1.77661i	0.459253	−	0.888305i	$-0.348117\pi$
$398$		0			0
$399$			26.5883i			0.0666373i
$400$		0			0
$401$	432.052			1.07744			0.538718	−	0.842486i	$-0.318909\pi$
$401$	432.052			1.07744			0.538718	+	0.842486i	$0.318909\pi$
$402$		0			0
$403$	−167.859			−0.416524
$404$		0			0
$405$	−274.028	+	241.021i	−0.676612	+	0.595114i
$406$		0			0
$407$	285.134			0.700575
$408$		0			0
$409$	−98.8102			−0.241590			−0.120795	−	0.992677i	$-0.538544\pi$
$409$	−98.8102			−0.241590			−0.120795	+	0.992677i	$0.538544\pi$
$410$		0			0
$411$			34.4345i			0.0837822i
$412$		0			0
$413$		−	692.160i		−	1.67593i
$414$		0			0
$415$	282.140	−	248.156i	0.679855	−	0.597966i
$416$		0			0
$417$			70.2914i			0.168565i
$418$		0			0
$419$		−	204.980i		−	0.489213i	−0.969622	−	0.244607i	$-0.921341\pi$
$419$		−	204.980i		−	0.489213i	0.969622	−	0.244607i	$-0.0786588\pi$
$420$		0			0
$421$	449.956			1.06878			0.534390	−	0.845238i	$-0.320541\pi$
$421$	449.956			1.06878			0.534390	+	0.845238i	$0.320541\pi$
$422$		0			0
$423$	543.214			1.28419
$424$		0			0
$425$	669.915	+	86.2166i	1.57627	+	0.202863i
$426$		0			0
$427$	−411.756			−0.964301
$428$		0			0
$429$	41.6021			0.0969745
$430$		0			0
$431$		−	314.588i		−	0.729903i	−0.931026	−	0.364952i	$-0.881086\pi$
$431$		−	314.588i		−	0.729903i	0.931026	−	0.364952i	$-0.118914\pi$
$432$		0			0
$433$			330.441i			0.763143i	0.924339	+	0.381571i	$0.124617\pi$
$433$			330.441i			0.763143i	−0.924339	+	0.381571i	$0.875383\pi$
$434$		0			0
$435$	4.89932	−	4.30919i	0.0112628	−	0.00990619i
$436$		0			0
$437$			73.6781i			0.168600i
$438$		0			0
$439$			664.815i			1.51439i	0.653191	+	0.757193i	$0.273430\pi$
$439$			664.815i			1.51439i	−0.653191	+	0.757193i	$0.726570\pi$
$440$		0			0
$441$	−453.572			−1.02851
$442$		0			0
$443$	312.860			0.706229			0.353115	−	0.935580i	$-0.385123\pi$
$443$	312.860			0.706229			0.353115	+	0.935580i	$0.385123\pi$
$444$		0			0
$445$	399.931	−	351.760i	0.898722	−	0.790471i
$446$		0			0
$447$	−90.6912			−0.202889
$448$		0			0
$449$	−246.330			−0.548620			−0.274310	−	0.961641i	$-0.588449\pi$
$449$	−246.330			−0.548620			−0.274310	+	0.961641i	$0.588449\pi$
$450$		0			0
$451$			229.181i			0.508161i
$452$		0			0
$453$			119.722i			0.264287i
$454$		0			0
$455$	146.558	+	166.629i	0.322107	+	0.366218i
$456$		0			0
$457$			707.094i			1.54725i	0.633642	+	0.773626i	$0.281559\pi$
$457$			707.094i			1.54725i	−0.633642	+	0.773626i	$0.718441\pi$
$458$		0			0
$459$		−	261.955i		−	0.570707i
$460$		0			0
$461$	−611.288			−1.32600			−0.663002	−	0.748618i	$-0.730718\pi$
$461$	−611.288			−1.32600			−0.663002	+	0.748618i	$0.730718\pi$
$462$		0			0
$463$	−334.063			−0.721519			−0.360760	−	0.932659i	$-0.617482\pi$
$463$	−334.063			−0.721519			−0.360760	+	0.932659i	$0.617482\pi$
$464$		0			0
$465$	−68.8010	−	78.2230i	−0.147959	−	0.168222i
$466$		0			0
$467$	128.864			0.275939			0.137970	−	0.990436i	$-0.455942\pi$
$467$	128.864			0.275939			0.137970	+	0.990436i	$0.455942\pi$
$468$		0			0
$469$	512.966			1.09374
$470$		0			0
$471$		−	95.4742i		−	0.202705i
$472$		0			0
$473$		−	1028.94i		−	2.17535i
$474$		0			0
$475$	−119.675	−	15.4020i	−0.251948	−	0.0324252i
$476$		0			0
$477$			622.874i			1.30582i
$478$		0			0
$479$			510.257i			1.06525i	0.846350	+	0.532627i	$0.178795\pi$
$479$			510.257i			1.06525i	−0.846350	+	0.532627i	$0.821205\pi$
$480$		0			0
$481$	−73.1237			−0.152024
$482$		0			0
$483$	84.0939			0.174107
$484$		0			0
$485$	−282.123	−	320.758i	−0.581696	−	0.661357i
$486$		0			0
$487$	−616.472			−1.26586			−0.632928	−	0.774211i	$-0.718147\pi$
$487$	−616.472			−1.26586			−0.632928	+	0.774211i	$0.718147\pi$
$488$		0			0
$489$	28.8326			0.0589624
$490$		0			0
$491$			603.190i			1.22849i	0.789114	+	0.614247i	$0.210540\pi$
$491$			603.190i			1.22849i	−0.789114	+	0.614247i	$0.789460\pi$
$492$		0			0
$493$		−	64.3627i		−	0.130553i
$494$		0			0
$495$	−494.387	−	562.091i	−0.998761	−	1.13554i
$496$		0			0
$497$			406.817i			0.818545i
$498$		0			0
$499$		−	867.976i		−	1.73943i	−0.493553	−	0.869716i	$-0.664302\pi$
$499$		−	867.976i		−	1.73943i	0.493553	−	0.869716i	$-0.335698\pi$
$500$		0			0
$501$	42.0045			0.0838413
$502$		0			0
$503$	−57.8408			−0.114992			−0.0574958	−	0.998346i	$-0.518312\pi$
$503$	−57.8408			−0.114992			−0.0574958	+	0.998346i	$0.518312\pi$
$504$		0			0
$505$	312.696	−	275.032i	0.619200	−	0.544617i
$506$		0			0
$507$	81.9060			0.161550
$508$		0			0
$509$	−168.498			−0.331038			−0.165519	−	0.986207i	$-0.552930\pi$
$509$	−168.498			−0.331038			−0.165519	+	0.986207i	$0.552930\pi$
$510$		0			0
$511$			360.470i			0.705420i
$512$		0			0
$513$			46.7962i			0.0912207i
$514$		0			0
$515$	177.453	−	156.079i	0.344569	−	0.303066i
$516$		0			0
$517$			1074.49i			2.07833i
$518$		0			0
$519$			5.32682i			0.0102636i
$520$		0			0
$521$	−87.1060			−0.167190			−0.0835951	−	0.996500i	$-0.526640\pi$
$521$	−87.1060			−0.167190			−0.0835951	+	0.996500i	$0.526640\pi$
$522$		0			0
$523$	243.469			0.465524			0.232762	−	0.972534i	$-0.425224\pi$
$523$	243.469			0.465524			0.232762	+	0.972534i	$0.425224\pi$
$524$		0			0
$525$	−17.5793	+	136.594i	−0.0334844	+	0.260179i
$526$		0			0
$527$	−1027.62			−1.94995
$528$		0			0
$529$	−295.969			−0.559488
$530$		0			0
$531$		−	598.786i		−	1.12766i
$532$		0			0
$533$		−	58.7743i		−	0.110271i
$534$		0			0
$535$	711.942	−	626.189i	1.33073	−	1.17045i
$536$		0			0
$537$			85.1812i			0.158624i
$538$		0			0
$539$		−	897.179i		−	1.66452i
$540$		0			0
$541$	−812.616			−1.50206			−0.751032	−	0.660266i	$-0.770443\pi$
$541$	−812.616			−1.50206			−0.751032	+	0.660266i	$0.770443\pi$
$542$		0			0
$543$	137.135			0.252550
$544$		0			0
$545$	−317.479	+	279.238i	−0.582530	+	0.512364i
$546$		0			0
$547$	−104.713			−0.191431			−0.0957155	−	0.995409i	$-0.530514\pi$
$547$	−104.713			−0.191431			−0.0957155	+	0.995409i	$0.530514\pi$
$548$		0			0
$549$	−356.210			−0.648833
$550$		0			0
$551$			11.4979i			0.0208674i
$552$		0			0
$553$			1275.17i			2.30592i
$554$		0			0
$555$	−29.9715	−	34.0759i	−0.0540027	−	0.0613981i
$556$		0			0
$557$			260.018i			0.466818i	0.972379	+	0.233409i	$0.0749882\pi$
$557$			260.018i			0.466818i	−0.972379	+	0.233409i	$0.925012\pi$
$558$		0			0
$559$			263.876i			0.472050i
$560$		0			0
$561$	254.685			0.453984
$562$		0			0
$563$	39.9073			0.0708833			0.0354416	−	0.999372i	$-0.488716\pi$
$563$	39.9073			0.0708833			0.0354416	+	0.999372i	$0.488716\pi$
$564$		0			0
$565$	−378.293	−	430.099i	−0.669546	−	0.761237i
$566$		0			0
$567$	−734.014			−1.29456
$568$		0			0
$569$	−211.588			−0.371860			−0.185930	−	0.982563i	$-0.559530\pi$
$569$	−211.588			−0.371860			−0.185930	+	0.982563i	$0.559530\pi$
$570$		0			0
$571$		−	556.938i		−	0.975372i	−0.873019	−	0.487686i	$-0.837841\pi$
$571$		−	556.938i		−	0.975372i	0.873019	−	0.487686i	$-0.162159\pi$
$572$		0			0
$573$		−	60.8741i		−	0.106237i
$574$		0			0
$575$	−48.7136	+	378.512i	−0.0847193	+	0.658281i
$576$		0			0
$577$		−	516.233i		−	0.894684i	−0.894363	−	0.447342i	$-0.852371\pi$
$577$		−	516.233i		−	0.894684i	0.894363	−	0.447342i	$-0.147629\pi$
$578$		0			0
$579$		−	5.62577i		−	0.00971636i
$580$		0			0
$581$	755.742			1.30076
$582$		0			0
$583$	−1232.06			−2.11332
$584$		0			0
$585$	126.787	+	144.150i	0.216731	+	0.246411i
$586$		0			0
$587$	−934.677			−1.59229			−0.796147	−	0.605103i	$-0.793132\pi$
$587$	−934.677			−1.59229			−0.796147	+	0.605103i	$0.793132\pi$
$588$		0			0
$589$	183.577			0.311676
$590$		0			0
$591$		−	89.4050i		−	0.151277i
$592$		0			0
$593$		−	634.665i		−	1.07026i	−0.844769	−	0.535131i	$-0.820262\pi$
$593$		−	634.665i		−	1.07026i	0.844769	−	0.535131i	$-0.179738\pi$
$594$		0			0
$595$	897.221	+	1020.09i	1.50794	+	1.71444i
$596$		0			0
$597$			106.833i			0.178949i
$598$		0			0
$599$			914.029i			1.52593i	0.646443	+	0.762963i	$0.276256\pi$
$599$			914.029i			1.52593i	−0.646443	+	0.762963i	$0.723744\pi$
$600$		0			0
$601$	−598.569			−0.995955			−0.497977	−	0.867190i	$-0.665924\pi$
$601$	−598.569			−0.995955			−0.497977	+	0.867190i	$0.665924\pi$
$602$		0			0
$603$	443.766			0.735930
$604$		0			0
$605$	657.550	−	578.348i	1.08686	−	0.955948i
$606$		0			0
$607$	201.830			0.332504			0.166252	−	0.986083i	$-0.446834\pi$
$607$	201.830			0.332504			0.166252	+	0.986083i	$0.446834\pi$
$608$		0			0
$609$	13.1234			0.0215490
$610$		0			0
$611$		−	275.558i		−	0.450995i
$612$		0			0
$613$		−	71.9205i		−	0.117325i	−0.998278	−	0.0586627i	$-0.981316\pi$
$613$		−	71.9205i		−	0.117325i	0.998278	−	0.0586627i	$-0.0186836\pi$
$614$		0			0
$615$	27.3891	−	24.0900i	0.0445350	−	0.0391708i
$616$		0			0
$617$			204.485i			0.331418i	0.986175	+	0.165709i	$0.0529912\pi$
$617$			204.485i			0.331418i	−0.986175	+	0.165709i	$0.947009\pi$
$618$		0			0
$619$		−	287.512i		−	0.464479i	−0.972659	−	0.232239i	$-0.925395\pi$
$619$		−	287.512i		−	0.464479i	0.972659	−	0.232239i	$-0.0746053\pi$
$620$		0			0
$621$	148.008			0.238338
$622$		0			0
$623$	1071.26			1.71952
$624$		0			0
$625$	−604.633	−	158.251i	−0.967414	−	0.253202i
$626$		0			0
$627$	−45.4976			−0.0725640
$628$		0			0
$629$	−447.658			−0.711699
$630$		0			0
$631$			274.203i			0.434554i	0.976110	+	0.217277i	$0.0697175\pi$
$631$			274.203i			0.434554i	−0.976110	+	0.217277i	$0.930283\pi$
$632$		0			0
$633$		−	162.712i		−	0.257049i
$634$		0			0
$635$	166.741	−	146.657i	0.262584	−	0.230956i
$636$		0			0
$637$			230.085i			0.361201i
$638$		0			0
$639$			351.936i			0.550761i
$640$		0			0
$641$	681.328			1.06291			0.531457	−	0.847085i	$-0.321645\pi$
$641$	681.328			1.06291			0.531457	+	0.847085i	$0.321645\pi$
$642$		0			0
$643$	−527.270			−0.820016			−0.410008	−	0.912082i	$-0.634474\pi$
$643$	−527.270			−0.820016			−0.410008	+	0.912082i	$0.634474\pi$
$644$		0			0
$645$	−122.967	+	108.156i	−0.190647	+	0.167684i
$646$		0			0
$647$	551.627			0.852592			0.426296	−	0.904584i	$-0.359818\pi$
$647$	551.627			0.852592			0.426296	+	0.904584i	$0.359818\pi$
$648$		0			0
$649$	1184.42			1.82499
$650$		0			0
$651$		−	209.529i		−	0.321857i
$652$		0			0
$653$			699.422i			1.07109i	0.844507	+	0.535545i	$0.179894\pi$
$653$			699.422i			1.07109i	−0.844507	+	0.535545i	$0.820106\pi$
$654$		0			0
$655$	155.599	+	176.908i	0.237556	+	0.270088i
$656$		0			0
$657$			311.842i			0.474645i
$658$		0			0
$659$			38.3257i			0.0581574i	0.999577	+	0.0290787i	$0.00925734\pi$
$659$			38.3257i			0.0581574i	−0.999577	+	0.0290787i	$0.990743\pi$
$660$		0			0
$661$	392.364			0.593591			0.296796	−	0.954941i	$-0.404082\pi$
$661$	392.364			0.593591			0.296796	+	0.954941i	$0.404082\pi$
$662$		0			0
$663$	−65.3150			−0.0985143
$664$		0			0
$665$	−160.282	−	182.232i	−0.241026	−	0.274033i
$666$		0			0
$667$	36.3658			0.0545215
$668$		0			0
$669$	−222.035			−0.331890
$670$		0			0
$671$		−	704.594i		−	1.05007i
$672$		0			0
$673$		−	427.290i		−	0.634904i	−0.948274	−	0.317452i	$-0.897173\pi$
$673$		−	427.290i		−	0.634904i	0.948274	−	0.317452i	$-0.102827\pi$
$674$		0			0
$675$	−30.9402	+	240.409i	−0.0458373	+	0.356162i
$676$		0			0
$677$		−	131.234i		−	0.193847i	−0.995292	−	0.0969233i	$-0.969100\pi$
$677$		−	131.234i		−	0.193847i	0.995292	−	0.0969233i	$-0.0309002\pi$
$678$		0			0
$679$		−	859.186i		−	1.26537i
$680$		0			0
$681$	20.0495			0.0294413
$682$		0			0
$683$	896.229			1.31219			0.656097	−	0.754676i	$-0.272206\pi$
$683$	896.229			1.31219			0.656097	+	0.754676i	$0.272206\pi$
$684$		0			0
$685$	−207.581	−	236.008i	−0.303038	−	0.344538i
$686$		0			0
$687$	−49.9285			−0.0726761
$688$		0			0
$689$	315.968			0.458589
$690$		0			0
$691$			520.465i			0.753206i	0.926375	+	0.376603i	$0.122908\pi$
$691$			520.465i			0.753206i	−0.926375	+	0.376603i	$0.877092\pi$
$692$		0			0
$693$		−	1505.62i		−	2.17262i
$694$		0			0
$695$	−423.737	−	481.766i	−0.609694	−	0.693189i
$696$		0			0
$697$		−	359.812i		−	0.516230i
$698$		0			0
$699$			185.589i			0.265507i
$700$		0			0
$701$	80.7527			0.115196			0.0575982	−	0.998340i	$-0.481656\pi$
$701$	80.7527			0.115196			0.0575982	+	0.998340i	$0.481656\pi$
$702$		0			0
$703$	79.9709			0.113757
$704$		0			0
$705$	128.411	−	112.944i	0.182144	−	0.160204i
$706$		0			0
$707$	837.591			1.18471
$708$		0			0
$709$	−174.828			−0.246584			−0.123292	−	0.992370i	$-0.539345\pi$
$709$	−174.828			−0.246584			−0.123292	+	0.992370i	$0.539345\pi$
$710$		0			0
$711$			1103.15i			1.55155i
$712$		0			0
$713$		−	580.621i		−	0.814335i
$714$		0			0
$715$	−285.134	+	250.789i	−0.398789	+	0.350755i
$716$		0			0
$717$		−	16.8715i		−	0.0235307i
$718$		0			0
$719$			889.905i			1.23770i	0.785510	+	0.618849i	$0.212401\pi$
$719$			889.905i			1.23770i	−0.785510	+	0.618849i	$0.787599\pi$
$720$		0			0
$721$	475.328			0.659263
$722$		0			0
$723$	131.622			0.182050
$724$		0			0
$725$	−7.60205	+	59.0690i	−0.0104856	+	0.0814745i
$726$		0			0
$727$	408.818			0.562335			0.281168	−	0.959659i	$-0.409278\pi$
$727$	408.818			0.562335			0.281168	+	0.959659i	$0.409278\pi$
$728$		0			0
$729$	−587.194			−0.805478
$730$		0			0
$731$			1615.43i			2.20989i
$732$		0			0
$733$			388.369i			0.529835i	0.964271	+	0.264917i	$0.0853446\pi$
$733$			388.369i			0.529835i	−0.964271	+	0.264917i	$0.914655\pi$
$734$		0			0
$735$	−107.221	+	94.3058i	−0.145878	+	0.128307i
$736$		0			0
$737$			877.783i			1.19102i
$738$		0			0
$739$			109.561i			0.148256i	0.997249	+	0.0741280i	$0.0236173\pi$
$739$			109.561i			0.148256i	−0.997249	+	0.0741280i	$0.976383\pi$
$740$		0			0
$741$	11.6680			0.0157463
$742$		0			0
$743$	732.717			0.986160			0.493080	−	0.869984i	$-0.335871\pi$
$743$	732.717			0.986160			0.493080	+	0.869984i	$0.335871\pi$
$744$		0			0
$745$	621.583	−	546.713i	0.834340	−	0.733844i
$746$		0			0
$747$	653.791			0.875222
$748$		0			0
$749$	1907.02			2.54608
$750$		0			0
$751$		−	420.809i		−	0.560332i	−0.959952	−	0.280166i	$-0.909610\pi$
$751$		−	420.809i		−	0.560332i	0.959952	−	0.280166i	$-0.0903895\pi$
$752$		0			0
$753$			47.1238i			0.0625814i
$754$		0			0
$755$	−721.721	−	820.557i	−0.955922	−	1.08683i
$756$		0			0
$757$		−	1306.99i		−	1.72654i	−0.504744	−	0.863269i	$-0.668413\pi$
$757$		−	1306.99i		−	1.72654i	0.504744	−	0.863269i	$-0.331587\pi$
$758$		0			0
$759$			143.901i			0.189592i
$760$		0			0
$761$	−1402.25			−1.84264			−0.921320	−	0.388804i	$-0.872888\pi$
$761$	−1402.25			−1.84264			−0.921320	+	0.388804i	$0.872888\pi$
$762$		0			0
$763$	−850.402			−1.11455
$764$		0			0
$765$	776.184	+	882.479i	1.01462	+	1.15357i
$766$		0			0
$767$	−303.748			−0.396021
$768$		0			0
$769$	−359.952			−0.468078			−0.234039	−	0.972227i	$-0.575194\pi$
$769$	−359.952			−0.468078			−0.234039	+	0.972227i	$0.575194\pi$
$770$		0			0
$771$		−	194.990i		−	0.252905i
$772$		0			0
$773$			437.539i			0.566027i	0.959116	+	0.283013i	$0.0913341\pi$
$773$			437.539i			0.566027i	−0.959116	+	0.283013i	$0.908666\pi$
$774$		0			0
$775$	943.103	+	121.375i	1.21691	+	0.156613i
$776$		0			0
$777$		−	91.2762i		−	0.117473i
$778$		0			0
$779$			64.2778i			0.0825132i
$780$		0			0
$781$	−696.141			−0.891346
$782$		0			0
$783$	23.0975			0.0294988
$784$		0			0
$785$	575.547	+	654.365i	0.733181	+	0.833586i
$786$		0			0
$787$	−438.946			−0.557746			−0.278873	−	0.960328i	$-0.589961\pi$
$787$	−438.946			−0.557746			−0.278873	+	0.960328i	$0.589961\pi$
$788$		0			0
$789$	40.0567			0.0507690
$790$		0			0
$791$		−	1152.07i		−	1.45647i
$792$		0			0
$793$			180.696i			0.227864i
$794$		0			0
$795$	129.507	+	147.242i	0.162902	+	0.185210i
$796$		0			0
$797$		−	982.414i		−	1.23264i	−0.787496	−	0.616320i	$-0.788623\pi$
$797$		−	982.414i		−	1.23264i	0.787496	−	0.616320i	$-0.211377\pi$
$798$		0			0
$799$		−	1686.95i		−	2.11133i
$800$		0			0
$801$	926.745			1.15699
$802$		0			0
$803$	−616.832			−0.768160
$804$		0			0
$805$	−576.366	+	506.943i	−0.715983	+	0.629743i
$806$		0			0
$807$	146.806			0.181916
$808$		0			0
$809$	−449.408			−0.555510			−0.277755	−	0.960652i	$-0.589590\pi$
$809$	−449.408			−0.555510			−0.277755	+	0.960652i	$0.589590\pi$
$810$		0			0
$811$			610.106i			0.752288i	0.926561	+	0.376144i	$0.122750\pi$
$811$			610.106i			0.752288i	−0.926561	+	0.376144i	$0.877250\pi$
$812$		0			0
$813$			125.667i			0.154572i
$814$		0			0
$815$	−197.614	+	173.811i	−0.242471	+	0.213266i
$816$		0			0
$817$		−	288.585i		−	0.353225i
$818$		0			0
$819$			386.123i			0.471456i
$820$		0			0
$821$	−1099.76			−1.33954			−0.669770	−	0.742568i	$-0.733607\pi$
$821$	−1099.76			−1.33954			−0.669770	+	0.742568i	$0.733607\pi$
$822$		0			0
$823$	236.092			0.286867			0.143434	−	0.989660i	$-0.454186\pi$
$823$	236.092			0.286867			0.143434	+	0.989660i	$0.454186\pi$
$824$		0			0
$825$	−233.738	−	30.0815i	−0.283319	−	0.0364625i
$826$		0			0
$827$	−1266.81			−1.53182			−0.765908	−	0.642950i	$-0.777710\pi$
$827$	−1266.81			−1.53182			−0.765908	+	0.642950i	$0.777710\pi$
$828$		0			0
$829$	−41.7642			−0.0503790			−0.0251895	−	0.999683i	$-0.508019\pi$
$829$	−41.7642			−0.0503790			−0.0251895	+	0.999683i	$0.508019\pi$
$830$		0			0
$831$			69.1993i			0.0832723i
$832$		0			0
$833$			1408.57i			1.69095i
$834$		0			0
$835$	−287.892	+	253.215i	−0.344781	+	0.303252i
$836$		0			0
$837$		−	368.778i		−	0.440595i
$838$		0			0
$839$		−	816.273i		−	0.972912i	−0.873705	−	0.486456i	$-0.838289\pi$
$839$		−	816.273i		−	0.972912i	0.873705	−	0.486456i	$-0.161711\pi$
$840$		0			0
$841$	−835.325			−0.993252
$842$		0			0
$843$	−251.293			−0.298093
$844$		0			0
$845$	−561.371	+	493.754i	−0.664344	+	0.584324i
$846$		0			0
$847$	1761.32			2.07948
$848$		0			0
$849$	−255.024			−0.300381
$850$		0			0
$851$		−	252.933i		−	0.297219i
$852$		0			0
$853$			1215.58i			1.42507i	0.701639	+	0.712533i	$0.252452\pi$
$853$			1215.58i			1.42507i	−0.701639	+	0.712533i	$0.747548\pi$
$854$		0			0
$855$	−138.660	−	157.648i	−0.162175	−	0.184384i
$856$		0			0
$857$		−	997.998i		−	1.16452i	−0.813001	−	0.582262i	$-0.802168\pi$
$857$		−	997.998i		−	1.16452i	0.813001	−	0.582262i	$-0.197832\pi$
$858$		0			0
$859$			1110.42i			1.29269i	0.763044	+	0.646347i	$0.223704\pi$
$859$			1110.42i			1.29269i	−0.763044	+	0.646347i	$0.776296\pi$
$860$		0			0
$861$	73.3646			0.0852086
$862$		0			0
$863$	−1372.33			−1.59019			−0.795094	−	0.606486i	$-0.792578\pi$
$863$	−1372.33			−1.59019			−0.795094	+	0.606486i	$0.792578\pi$
$864$		0			0
$865$	−32.1117	−	36.5092i	−0.0371233	−	0.0422072i
$866$		0			0
$867$	−241.545			−0.278599
$868$		0			0
$869$	−2182.06			−2.51101
$870$		0			0
$871$		−	225.111i		−	0.258451i
$872$		0			0
$873$		−	743.280i		−	0.851409i
$874$		0			0
$875$	−702.942	−	1042.17i	−0.803362	−	1.19105i
$876$		0			0
$877$			776.337i			0.885219i	0.896714	+	0.442609i	$0.145947\pi$
$877$			776.337i			0.885219i	−0.896714	+	0.442609i	$0.854053\pi$
$878$		0			0
$879$		−	90.5034i		−	0.102962i
$880$		0			0
$881$	1047.38			1.18885			0.594427	−	0.804150i	$-0.297379\pi$
$881$	1047.38			1.18885			0.594427	+	0.804150i	$0.297379\pi$
$882$		0			0
$883$	−1175.41			−1.33116			−0.665579	−	0.746327i	$-0.731815\pi$
$883$	−1175.41			−1.33116			−0.665579	+	0.746327i	$0.731815\pi$
$884$		0			0
$885$	−124.499	−	141.548i	−0.140676	−	0.159941i
$886$		0			0
$887$	109.768			0.123752			0.0618762	−	0.998084i	$-0.480292\pi$
$887$	109.768			0.123752			0.0618762	+	0.998084i	$0.480292\pi$
$888$		0			0
$889$	446.634			0.502401
$890$		0			0
$891$		−	1256.04i		−	1.40969i
$892$		0			0
$893$			301.361i			0.337470i
$894$		0			0
$895$	−513.497	−	583.818i	−0.573740	−	0.652311i
$896$		0			0
$897$		−	36.9039i		−	0.0411414i
$898$		0			0
$899$		−	90.6094i		−	0.100789i
$900$		0			0
$901$	1934.33			2.14687
$902$		0			0
$903$	−329.382			−0.364764
$904$		0			0
$905$	−939.899	+	826.688i	−1.03856	+	0.913468i
$906$		0			0
$907$	1639.15			1.80722			0.903609	−	0.428358i	$-0.140908\pi$
$907$	1639.15			1.80722			0.903609	+	0.428358i	$0.140908\pi$
$908$		0			0
$909$	724.598			0.797137
$910$		0			0
$911$			6.82023i			0.00748654i	0.999993	+	0.00374327i	$0.00119152\pi$
$911$			6.82023i			0.00748654i	−0.999993	+	0.00374327i	$0.998808\pi$
$912$		0			0
$913$			1293.22i			1.41645i
$914$		0			0
$915$	−84.2050	+	74.0625i	−0.0920273	+	0.0809426i
$916$		0			0
$917$			473.867i			0.516757i
$918$		0			0
$919$		−	72.4188i		−	0.0788017i	−0.999223	−	0.0394009i	$-0.987455\pi$
$919$		−	72.4188i		−	0.0788017i	0.999223	−	0.0394009i	$-0.0125449\pi$
$920$		0			0
$921$	128.914			0.139972
$922$		0			0
$923$	178.528			0.193421
$924$		0			0
$925$	410.840	+	52.8741i	0.444151	+	0.0571612i
$926$		0			0
$927$	411.205			0.443587
$928$		0			0
$929$	−368.537			−0.396703			−0.198352	−	0.980131i	$-0.563559\pi$
$929$	−368.537			−0.396703			−0.198352	+	0.980131i	$0.563559\pi$
$930$		0			0
$931$		−	251.630i		−	0.270279i
$932$		0			0
$933$		−	115.399i		−	0.123686i
$934$		0			0
$935$	−1745.57	+	1535.32i	−1.86692	+	1.64205i
$936$		0			0
$937$		−	590.172i		−	0.629852i	−0.949116	−	0.314926i	$-0.898020\pi$
$937$		−	590.172i		−	0.629852i	0.949116	−	0.314926i	$-0.101980\pi$
$938$		0			0
$939$			174.599i			0.185941i
$940$		0			0
$941$	−189.017			−0.200869			−0.100434	−	0.994944i	$-0.532023\pi$
$941$	−189.017			−0.200869			−0.100434	+	0.994944i	$0.532023\pi$
$942$		0			0
$943$	203.299			0.215588
$944$		0			0
$945$	−366.076	+	321.982i	−0.387382	+	0.340721i
$946$		0			0
$947$	664.319			0.701498			0.350749	−	0.936470i	$-0.385927\pi$
$947$	664.319			0.701498			0.350749	+	0.936470i	$0.385927\pi$
$948$		0			0
$949$	158.189			0.166690
$950$		0			0
$951$			38.5610i			0.0405479i
$952$		0			0
$953$		−	173.943i		−	0.182522i	−0.995827	−	0.0912610i	$-0.970910\pi$
$953$		−	173.943i		−	0.182522i	0.995827	−	0.0912610i	$-0.0290898\pi$
$954$		0			0
$955$	366.967	+	417.221i	0.384259	+	0.436881i
$956$		0			0
$957$			22.4566i			0.0234656i
$958$		0			0
$959$		−	632.175i		−	0.659202i
$960$		0			0
$961$	−485.680			−0.505390
$962$		0			0
$963$	1649.76			1.71314
$964$		0			0
$965$	33.9138	+	38.5581i	0.0351438	+	0.0399566i
$966$		0			0
$967$	255.364			0.264078			0.132039	−	0.991245i	$-0.457848\pi$
$967$	255.364			0.264078			0.132039	+	0.991245i	$0.457848\pi$
$968$		0			0
$969$	71.4309			0.0737161
$970$		0			0
$971$		−	1623.60i		−	1.67209i	−0.548659	−	0.836046i	$-0.684861\pi$
$971$		−	1623.60i		−	1.67209i	0.548659	−	0.836046i	$-0.315139\pi$
$972$		0			0
$973$		−	1290.46i		−	1.32627i
$974$		0			0
$975$	59.9430	+	7.71453i	0.0614800	+	0.00791233i
$976$		0			0
$977$			104.946i			0.107416i	0.998557	+	0.0537082i	$0.0171041\pi$
$977$			104.946i			0.107416i	−0.998557	+	0.0537082i	$0.982896\pi$
$978$		0			0
$979$			1833.13i			1.87245i
$980$		0			0
$981$	−735.681			−0.749930
$982$		0			0
$983$	−889.933			−0.905323			−0.452662	−	0.891682i	$-0.649525\pi$
$983$	−889.933			−0.905323			−0.452662	+	0.891682i	$0.649525\pi$
$984$		0			0
$985$	538.960	+	612.768i	0.547167	+	0.622099i
$986$		0			0
$987$	343.964			0.348494
$988$		0			0
$989$	−912.742			−0.922894
$990$		0			0
$991$			1509.57i			1.52328i	0.648001	+	0.761639i	$0.275605\pi$
$991$			1509.57i			1.52328i	−0.648001	+	0.761639i	$0.724395\pi$
$992$		0			0
$993$		−	153.821i		−	0.154905i
$994$		0			0
$995$	−644.020	−	732.215i	−0.647256	−	0.735894i
$996$		0			0
$997$		−	1599.93i		−	1.60474i	−0.596825	−	0.802371i	$-0.703571\pi$
$997$		−	1599.93i		−	1.60474i	0.596825	−	0.802371i	$-0.296429\pi$
$998$		0			0
$999$		−	160.649i		−	0.160810i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.h.a.159.4	yes	6
3.2	odd	2		1440.3.j.a.1279.5		6
4.3	odd	2		160.3.h.b.159.4	yes	6
5.2	odd	4		800.3.b.h.351.3		6
5.3	odd	4		800.3.b.i.351.4		6
5.4	even	2		160.3.h.b.159.3	yes	6
8.3	odd	2		320.3.h.f.319.3		6
8.5	even	2		320.3.h.g.319.3		6
12.11	even	2		1440.3.j.b.1279.5		6
15.14	odd	2		1440.3.j.b.1279.6		6
16.3	odd	4		1280.3.e.f.639.3		6
16.5	even	4		1280.3.e.i.639.3		6
16.11	odd	4		1280.3.e.h.639.4		6
16.13	even	4		1280.3.e.g.639.4		6
20.3	even	4		800.3.b.i.351.3		6
20.7	even	4		800.3.b.h.351.4		6
20.19	odd	2	inner	160.3.h.a.159.3	✓	6
40.3	even	4		1600.3.b.w.1151.4		6
40.13	odd	4		1600.3.b.w.1151.3		6
40.19	odd	2		320.3.h.g.319.4		6
40.27	even	4		1600.3.b.v.1151.3		6
40.29	even	2		320.3.h.f.319.4		6
40.37	odd	4		1600.3.b.v.1151.4		6
60.59	even	2		1440.3.j.a.1279.6		6
80.19	odd	4		1280.3.e.i.639.4		6
80.29	even	4		1280.3.e.h.639.3		6
80.59	odd	4		1280.3.e.g.639.3		6
80.69	even	4		1280.3.e.f.639.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.h.a.159.3	✓	6	20.19	odd	2	inner
160.3.h.a.159.4	yes	6	1.1	even	1	trivial
160.3.h.b.159.3	yes	6	5.4	even	2
160.3.h.b.159.4	yes	6	4.3	odd	2
320.3.h.f.319.3		6	8.3	odd	2
320.3.h.f.319.4		6	40.29	even	2
320.3.h.g.319.3		6	8.5	even	2
320.3.h.g.319.4		6	40.19	odd	2
800.3.b.h.351.3		6	5.2	odd	4
800.3.b.h.351.4		6	20.7	even	4
800.3.b.i.351.3		6	20.3	even	4
800.3.b.i.351.4		6	5.3	odd	4
1280.3.e.f.639.3		6	16.3	odd	4
1280.3.e.f.639.4		6	80.69	even	4
1280.3.e.g.639.3		6	80.59	odd	4
1280.3.e.g.639.4		6	16.13	even	4
1280.3.e.h.639.3		6	80.29	even	4
1280.3.e.h.639.4		6	16.11	odd	4
1280.3.e.i.639.3		6	16.5	even	4
1280.3.e.i.639.4		6	80.19	odd	4
1440.3.j.a.1279.5		6	3.2	odd	2
1440.3.j.a.1279.6		6	60.59	even	2
1440.3.j.b.1279.5		6	12.11	even	2
1440.3.j.b.1279.6		6	15.14	odd	2
1600.3.b.v.1151.3		6	40.27	even	4
1600.3.b.v.1151.4		6	40.37	odd	4
1600.3.b.w.1151.3		6	40.13	odd	4
1600.3.b.w.1151.4		6	40.3	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.547781 q^{3} +(-3.75441 + 3.30219i) q^{5} -10.0566 q^{7} -8.69994 q^{9} +O(q^{10})\)	\(q+0.547781 q^{3} +(-3.75441 + 3.30219i) q^{5} -10.0566 q^{7} -8.69994 q^{9} -17.2087i q^{11} +4.41325i q^{13} +(-2.05659 + 1.80888i) q^{15} +27.0176i q^{17} -4.82650i q^{19} -5.50881 q^{21} -15.2653 q^{23} +(3.19112 - 24.7955i) q^{25} -9.69569 q^{27} -2.38225 q^{29} +38.0352i q^{31} -9.42663i q^{33} +(37.7565 - 33.2087i) q^{35} +16.5691i q^{37} +2.41749i q^{39} -13.3177 q^{41} +59.7918 q^{43} +(32.6631 - 28.7288i) q^{45} -62.4388 q^{47} +52.1351 q^{49} +14.7997i q^{51} -71.5952i q^{53} +(56.8265 + 64.6086i) q^{55} -2.64386i q^{57} +68.8265i q^{59} +40.9439 q^{61} +87.4917 q^{63} +(-14.5734 - 16.5691i) q^{65} -51.0080 q^{67} -8.36206 q^{69} -40.4527i q^{71} -35.8441i q^{73} +(1.74804 - 13.5825i) q^{75} +173.061i q^{77} -126.800i q^{79} +72.9883 q^{81} -75.1490 q^{83} +(-89.2172 - 101.435i) q^{85} -1.30495 q^{87} -106.523 q^{89} -44.3822i q^{91} +20.8350i q^{93} +(15.9380 + 18.1206i) q^{95} +85.4351i q^{97} +149.715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 18 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 18 * q^9	\( 6 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 18 q^{9} + 36 q^{15} + 8 q^{21} + 68 q^{23} - 10 q^{25} - 184 q^{27} + 44 q^{29} + 108 q^{35} - 68 q^{41} + 76 q^{43} - 6 q^{45} - 268 q^{47} - 62 q^{49} + 288 q^{55} - 100 q^{61} + 172 q^{63} - 308 q^{67} - 184 q^{69} + 284 q^{75} + 238 q^{81} + 204 q^{83} - 32 q^{85} - 584 q^{87} + 76 q^{89} + 32 q^{95}+O(q^{100}) \) 6 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 18 * q^9 + 36 * q^15 + 8 * q^21 + 68 * q^23 - 10 * q^25 - 184 * q^27 + 44 * q^29 + 108 * q^35 - 68 * q^41 + 76 * q^43 - 6 * q^45 - 268 * q^47 - 62 * q^49 + 288 * q^55 - 100 * q^61 + 172 * q^63 - 308 * q^67 - 184 * q^69 + 284 * q^75 + 238 * q^81 + 204 * q^83 - 32 * q^85 - 584 * q^87 + 76 * q^89 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more