LMFDB - Embedded newform 160.4.c.d.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

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([0, 0, 1]))

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

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.359712057600.22
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 $ x^8 + 17x^6 + 82x^4 + 96*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$-0.320221i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.d.129.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +O(q^{10})$	$q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +65.2358 q^{11} -33.6697i q^{13} +(-30.8223 - 36.9196i) q^{15} -73.3212i q^{17} -134.063 q^{19} +121.808 q^{21} +14.7007i q^{23} +(22.3212 - 122.991i) q^{25} -152.690i q^{27} +224.642 q^{29} +68.8271 q^{31} -280.624i q^{33} +(202.890 + 243.026i) q^{35} +196.312i q^{37} -144.837 q^{39} -143.147 q^{41} -15.0755i q^{43} +(72.9129 - 60.8712i) q^{45} +134.399i q^{47} -458.808 q^{49} -315.405 q^{51} +262.955i q^{53} +(559.891 - 467.424i) q^{55} +576.697i q^{57} +119.698 q^{59} +16.5409 q^{61} +240.559i q^{63} +(-241.248 - 288.973i) q^{65} +545.565i q^{67} +63.2379 q^{69} -199.299 q^{71} +43.2667i q^{73} +(-529.069 - 96.0190i) q^{75} +1847.23i q^{77} -438.694 q^{79} -427.450 q^{81} -1220.89i q^{83} +(-525.358 - 629.285i) q^{85} -966.342i q^{87} -723.212 q^{89} +953.398 q^{91} -296.073i q^{93} +(-1150.60 + 960.581i) q^{95} +1136.00i q^{97} +554.208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{5} - 152 q^{9}+O(q^{10}) $ 8 * q + 32 * q^5 - 152 * q^9	$ 8 q + 32 q^{5} - 152 q^{9} - 272 q^{21} - 408 q^{25} + 624 q^{29} - 192 q^{41} + 400 q^{45} - 2424 q^{49} + 2112 q^{61} + 2176 q^{65} + 3952 q^{69} - 1000 q^{81} - 5376 q^{85} + 80 q^{89}+O(q^{100}) $ 8 * q + 32 * q^5 - 152 * q^9 - 272 * q^21 - 408 * q^25 + 624 * q^29 - 192 * q^41 + 400 * q^45 - 2424 * q^49 + 2112 * q^61 + 2176 * q^65 + 3952 * q^69 - 1000 * q^81 - 5376 * q^85 + 80 * q^89

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
$3$		−	4.30169i		−	0.827861i	−0.910309	−	0.413930i	$-0.864156\pi$
$3$		−	4.30169i		−	0.827861i	0.910309	−	0.413930i	$-0.135844\pi$
$4$		0			0
$5$	8.58258	−	7.16515i	0.767649	−	0.640871i
$5$	8.58258	−	7.16515i	0.767649	−	0.640871i
$6$		0			0
$7$			28.3162i			1.52893i	0.644664	+	0.764466i	$0.276997\pi$
$7$			28.3162i			1.52893i	−0.644664	+	0.764466i	$0.723003\pi$
$8$		0			0
$9$	8.49545			0.314646
$10$		0			0
$11$	65.2358			1.78812			0.894061	−	0.447946i	$-0.147844\pi$
$11$	65.2358			1.78812			0.894061	+	0.447946i	$0.147844\pi$
$12$		0			0
$13$		−	33.6697i		−	0.718330i	−0.933274	−	0.359165i	$-0.883061\pi$
$13$		−	33.6697i		−	0.718330i	0.933274	−	0.359165i	$-0.116939\pi$
$14$		0			0
$15$	−30.8223	−	36.9196i	−0.530552	−	0.635506i
$16$		0			0
$17$		−	73.3212i		−	1.04606i	−0.852315	−	0.523030i	$-0.824802\pi$
$17$		−	73.3212i		−	1.04606i	0.852315	−	0.523030i	$-0.175198\pi$
$18$		0			0
$19$	−134.063			−1.61874			−0.809372	−	0.587297i	$-0.800192\pi$
$19$	−134.063			−1.61874			−0.809372	+	0.587297i	$0.800192\pi$
$20$		0			0
$21$	121.808			1.26574
$22$		0			0
$23$			14.7007i			0.133274i	0.997777	+	0.0666371i	$0.0212270\pi$
$23$			14.7007i			0.133274i	−0.997777	+	0.0666371i	$0.978773\pi$
$24$		0			0
$25$	22.3212	−	122.991i	0.178570	−	0.983927i
$26$		0			0
$27$		−	152.690i		−	1.08834i
$28$		0			0
$29$	224.642			1.43845			0.719225	−	0.694777i	$-0.244497\pi$
$29$	224.642			1.43845			0.719225	+	0.694777i	$0.244497\pi$
$30$		0			0
$31$	68.8271			0.398765			0.199382	−	0.979922i	$-0.436106\pi$
$31$	68.8271			0.398765			0.199382	+	0.979922i	$0.436106\pi$
$32$		0			0
$33$		−	280.624i		−	1.48032i
$34$		0			0
$35$	202.890	+	243.026i	0.979847	+	1.17368i
$36$		0			0
$37$			196.312i			0.872257i	0.899884	+	0.436129i	$0.143651\pi$
$37$			196.312i			0.872257i	−0.899884	+	0.436129i	$0.856349\pi$
$38$		0			0
$39$	−144.837			−0.594678
$40$		0			0
$41$	−143.147			−0.545263			−0.272632	−	0.962118i	$-0.587894\pi$
$41$	−143.147			−0.545263			−0.272632	+	0.962118i	$0.587894\pi$
$42$		0			0
$43$		−	15.0755i		−	0.0534648i	−0.999643	−	0.0267324i	$-0.991490\pi$
$43$		−	15.0755i		−	0.0534648i	0.999643	−	0.0267324i	$-0.00851021\pi$
$44$		0			0
$45$	72.9129	−	60.8712i	0.241538	−	0.201648i
$46$		0			0
$47$			134.399i			0.417107i	0.978011	+	0.208554i	$0.0668756\pi$
$47$			134.399i			0.417107i	−0.978011	+	0.208554i	$0.933124\pi$
$48$		0			0
$49$	−458.808			−1.33763
$50$		0			0
$51$	−315.405			−0.865991
$52$		0			0
$53$			262.955i			0.681502i	0.940154	+	0.340751i	$0.110681\pi$
$53$			262.955i			0.681502i	−0.940154	+	0.340751i	$0.889319\pi$
$54$		0			0
$55$	559.891	−	467.424i	1.37265	−	1.14595i
$56$		0			0
$57$			576.697i			1.34009i
$58$		0			0
$59$	119.698			0.264124			0.132062	−	0.991241i	$-0.457840\pi$
$59$	119.698			0.264124			0.132062	+	0.991241i	$0.457840\pi$
$60$		0			0
$61$	16.5409			0.0347188			0.0173594	−	0.999849i	$-0.494474\pi$
$61$	16.5409			0.0347188			0.0173594	+	0.999849i	$0.494474\pi$
$62$		0			0
$63$			240.559i			0.481073i
$64$		0			0
$65$	−241.248	−	288.973i	−0.460357	−	0.551425i
$66$		0			0
$67$			545.565i			0.994797i	0.867522	+	0.497399i	$0.165711\pi$
$67$			545.565i			0.994797i	−0.867522	+	0.497399i	$0.834289\pi$
$68$		0			0
$69$	63.2379			0.110333
$70$		0			0
$71$	−199.299			−0.333132			−0.166566	−	0.986030i	$-0.553268\pi$
$71$	−199.299			−0.333132			−0.166566	+	0.986030i	$0.553268\pi$
$72$		0			0
$73$			43.2667i			0.0693696i	0.999398	+	0.0346848i	$0.0110427\pi$
$73$			43.2667i			0.0693696i	−0.999398	+	0.0346848i	$0.988957\pi$
$74$		0			0
$75$	−529.069	−	96.0190i	−0.814555	−	0.147831i
$76$		0			0
$77$			1847.23i			2.73391i
$78$		0			0
$79$	−438.694			−0.624772			−0.312386	−	0.949955i	$-0.601128\pi$
$79$	−438.694			−0.624772			−0.312386	+	0.949955i	$0.601128\pi$
$80$		0			0
$81$	−427.450			−0.586351
$82$		0			0
$83$		−	1220.89i		−	1.61458i	−0.590154	−	0.807291i	$-0.700933\pi$
$83$		−	1220.89i		−	1.61458i	0.590154	−	0.807291i	$-0.299067\pi$
$84$		0			0
$85$	−525.358	−	629.285i	−0.670389	−	0.803006i
$86$		0			0
$87$		−	966.342i		−	1.19084i
$88$		0			0
$89$	−723.212			−0.861352			−0.430676	−	0.902507i	$-0.641725\pi$
$89$	−723.212			−0.861352			−0.430676	+	0.902507i	$0.641725\pi$
$90$		0			0
$91$	953.398			1.09828
$92$		0			0
$93$		−	296.073i		−	0.330122i
$94$		0			0
$95$	−1150.60	+	960.581i	−1.24263	+	1.03741i
$96$		0			0
$97$			1136.00i			1.18911i	0.804056	+	0.594553i	$0.202671\pi$
$97$			1136.00i			1.18911i	−0.804056	+	0.594553i	$0.797329\pi$
$98$		0			0
$99$	554.208			0.562626
$100$		0			0
$101$	−1175.21			−1.15780			−0.578901	−	0.815398i	$-0.696518\pi$
$101$	−1175.21			−1.15780			−0.578901	+	0.815398i	$0.696518\pi$
$102$		0			0
$103$			1752.43i			1.67643i	0.545344	+	0.838213i	$0.316399\pi$
$103$			1752.43i			1.67643i	−0.545344	+	0.838213i	$0.683601\pi$
$104$		0			0
$105$	1045.42	−	872.770i	0.971646	−	0.811177i
$106$		0			0
$107$			94.5981i			0.0854686i	0.999086	+	0.0427343i	$0.0136069\pi$
$107$			94.5981i			0.0854686i	−0.999086	+	0.0427343i	$0.986393\pi$
$108$		0			0
$109$	1349.68			1.18602			0.593009	−	0.805196i	$-0.297940\pi$
$109$	1349.68			1.18602			0.593009	+	0.805196i	$0.297940\pi$
$110$		0			0
$111$	844.474			0.722108
$112$		0			0
$113$		−	529.248i		−	0.440597i	−0.975432	−	0.220299i	$-0.929297\pi$
$113$		−	529.248i		−	0.440597i	0.975432	−	0.220299i	$-0.0707032\pi$
$114$		0			0
$115$	105.333	+	126.170i	0.0854116	+	0.102308i
$116$		0			0
$117$		−	286.039i		−	0.226020i
$118$		0			0
$119$	2076.18			1.59935
$120$		0			0
$121$	2924.71			2.19738
$122$		0			0
$123$			615.774i			0.451402i
$124$		0			0
$125$	−689.675	−	1215.51i	−0.493491	−	0.869751i
$126$		0			0
$127$		−	814.066i		−	0.568793i	−0.958707	−	0.284396i	$-0.908207\pi$
$127$		−	814.066i		−	0.568793i	0.958707	−	0.284396i	$-0.0917931\pi$
$128$		0			0
$129$	−64.8500			−0.0442614
$130$		0			0
$131$	−1329.85			−0.886946			−0.443473	−	0.896288i	$-0.646254\pi$
$131$	−1329.85			−0.886946			−0.443473	+	0.896288i	$0.646254\pi$
$132$		0			0
$133$		−	3796.15i		−	2.47495i
$134$		0			0
$135$	−1094.05	−	1310.48i	−0.697488	−	0.835466i
$136$		0			0
$137$			2906.98i			1.81284i	0.422373	+	0.906422i	$0.361197\pi$
$137$			2906.98i			1.81284i	−0.422373	+	0.906422i	$0.638803\pi$
$138$		0			0
$139$	−921.077			−0.562048			−0.281024	−	0.959701i	$-0.590674\pi$
$139$	−921.077			−0.562048			−0.281024	+	0.959701i	$0.590674\pi$
$140$		0			0
$141$	578.141			0.345307
$142$		0			0
$143$		−	2196.47i		−	1.28446i
$144$		0			0
$145$	1928.01	−	1609.60i	1.10422	−	0.921860i
$146$		0			0
$147$			1973.65i			1.10737i
$148$		0			0
$149$	−265.259			−0.145845			−0.0729224	−	0.997338i	$-0.523233\pi$
$149$	−265.259			−0.145845			−0.0729224	+	0.997338i	$0.523233\pi$
$150$		0			0
$151$	−2507.69			−1.35148			−0.675738	−	0.737142i	$-0.736175\pi$
$151$	−2507.69			−1.35148			−0.675738	+	0.737142i	$0.736175\pi$
$152$		0			0
$153$		−	622.897i		−	0.329139i
$154$		0			0
$155$	590.713	−	493.156i	0.306111	−	0.255557i
$156$		0			0
$157$		−	14.8818i		−	0.00756495i	−0.999993	−	0.00378248i	$-0.998796\pi$
$157$		−	14.8818i		−	0.00756495i	0.999993	−	0.00378248i	$-0.00120400\pi$
$158$		0			0
$159$	1131.15			0.564188
$160$		0			0
$161$	−416.268			−0.203767
$162$		0			0
$163$		−	1184.23i		−	0.569055i	−0.958668	−	0.284528i	$-0.908163\pi$
$163$		−	1184.23i		−	0.569055i	0.958668	−	0.284528i	$-0.0918368\pi$
$164$		0			0
$165$	−2010.72	−	2408.48i	−0.948691	−	1.13636i
$166$		0			0
$167$			2688.78i			1.24589i	0.782266	+	0.622945i	$0.214064\pi$
$167$			2688.78i			1.24589i	−0.782266	+	0.622945i	$0.785936\pi$
$168$		0			0
$169$	1063.35			0.484002
$170$		0			0
$171$	−1138.92			−0.509332
$172$		0			0
$173$			1940.72i			0.852891i	0.904513	+	0.426445i	$0.140234\pi$
$173$			1940.72i			0.852891i	−0.904513	+	0.426445i	$0.859766\pi$
$174$		0			0
$175$	3482.64	+	632.052i	1.50436	+	0.273021i
$176$		0			0
$177$		−	514.903i		−	0.218658i
$178$		0			0
$179$	1580.62			0.660005			0.330002	−	0.943980i	$-0.392950\pi$
$179$	1580.62			0.660005			0.330002	+	0.943980i	$0.392950\pi$
$180$		0			0
$181$	−2561.93			−1.05208			−0.526040	−	0.850460i	$-0.676324\pi$
$181$	−2561.93			−1.05208			−0.526040	+	0.850460i	$0.676324\pi$
$182$		0			0
$183$		−	71.1539i		−	0.0287423i
$184$		0			0
$185$	1406.61	+	1684.86i	0.559004	+	0.669587i
$186$		0			0
$187$		−	4783.17i		−	1.87048i
$188$		0			0
$189$	4323.62			1.66400
$190$		0			0
$191$	−137.654			−0.0521482			−0.0260741	−	0.999660i	$-0.508301\pi$
$191$	−137.654			−0.0521482			−0.0260741	+	0.999660i	$0.508301\pi$
$192$		0			0
$193$		−	845.503i		−	0.315340i	−0.987492	−	0.157670i	$-0.949602\pi$
$193$		−	845.503i		−	0.315340i	0.987492	−	0.157670i	$-0.0503982\pi$
$194$		0			0
$195$	−1243.07	+	1037.78i	−0.456504	+	0.381111i
$196$		0			0
$197$			2733.63i			0.988646i	0.869278	+	0.494323i	$0.164584\pi$
$197$			2733.63i			0.988646i	−0.869278	+	0.494323i	$0.835416\pi$
$198$		0			0
$199$	−4649.70			−1.65632			−0.828162	−	0.560489i	$-0.810613\pi$
$199$	−4649.70			−1.65632			−0.828162	+	0.560489i	$0.810613\pi$
$200$		0			0
$201$	2346.85			0.823553
$202$		0			0
$203$			6361.02i			2.19929i
$204$		0			0
$205$	−1228.57	+	1025.67i	−0.418571	+	0.349443i
$206$		0			0
$207$			124.889i			0.0419343i
$208$		0			0
$209$	−8745.70			−2.89451
$210$		0			0
$211$	3845.91			1.25480			0.627402	−	0.778696i	$-0.284118\pi$
$211$	3845.91			1.25480			0.627402	+	0.778696i	$0.284118\pi$
$212$		0			0
$213$			857.321i			0.275787i
$214$		0			0
$215$	−108.018	−	129.386i	−0.0342640	−	0.0410422i
$216$		0			0
$217$			1948.92i			0.609684i
$218$		0			0
$219$	186.120			0.0574284
$220$		0			0
$221$	−2468.70			−0.751416
$222$		0			0
$223$		−	1460.74i		−	0.438648i	−0.975652	−	0.219324i	$-0.929615\pi$
$223$		−	1460.74i		−	0.438648i	0.975652	−	0.219324i	$-0.0703851\pi$
$224$		0			0
$225$	189.629	−	1044.86i	0.0561863	−	0.309589i
$226$		0			0
$227$		−	3435.16i		−	1.00440i	−0.864750	−	0.502202i	$-0.832523\pi$
$227$		−	3435.16i		−	1.00440i	0.864750	−	0.502202i	$-0.167477\pi$
$228$		0			0
$229$	595.358			0.171801			0.0859003	−	0.996304i	$-0.472623\pi$
$229$	595.358			0.171801			0.0859003	+	0.996304i	$0.472623\pi$
$230$		0			0
$231$	7946.21			2.26330
$232$		0			0
$233$			5432.62i			1.52748i	0.645523	+	0.763740i	$0.276639\pi$
$233$			5432.62i			1.52748i	−0.645523	+	0.763740i	$0.723361\pi$
$234$		0			0
$235$	962.986	+	1153.49i	0.267312	+	0.320192i
$236$		0			0
$237$			1887.13i			0.517224i
$238$		0			0
$239$	−3931.51			−1.06405			−0.532026	−	0.846728i	$-0.678569\pi$
$239$	−3931.51			−1.06405			−0.532026	+	0.846728i	$0.678569\pi$
$240$		0			0
$241$	−2587.88			−0.691701			−0.345851	−	0.938290i	$-0.612410\pi$
$241$	−2587.88			−0.691701			−0.345851	+	0.938290i	$0.612410\pi$
$242$		0			0
$243$		−	2283.89i		−	0.602927i
$244$		0			0
$245$	−3937.75	+	3287.43i	−1.02683	+	0.857249i
$246$		0			0
$247$			4513.86i			1.16279i
$248$		0			0
$249$	−5251.90			−1.33665
$250$		0			0
$251$	−2231.79			−0.561232			−0.280616	−	0.959820i	$-0.590539\pi$
$251$	−2231.79			−0.561232			−0.280616	+	0.959820i	$0.590539\pi$
$252$		0			0
$253$			959.012i			0.238311i
$254$		0			0
$255$	−2706.99	+	2259.93i	−0.664777	+	0.554988i
$256$		0			0
$257$		−	2477.07i		−	0.601226i	−0.953746	−	0.300613i	$-0.902809\pi$
$257$		−	2477.07i		−	0.601226i	0.953746	−	0.300613i	$-0.0971913\pi$
$258$		0			0
$259$	−5558.81			−1.33362
$260$		0			0
$261$	1908.44			0.452603
$262$		0			0
$263$		−	4865.27i		−	1.14071i	−0.821400	−	0.570353i	$-0.806806\pi$
$263$		−	4865.27i		−	1.14071i	0.821400	−	0.570353i	$-0.193194\pi$
$264$		0			0
$265$	1884.11	+	2256.83i	0.436754	+	0.523154i
$266$		0			0
$267$			3111.04i			0.713080i
$268$		0			0
$269$	6753.93			1.53084			0.765418	−	0.643534i	$-0.222532\pi$
$269$	6753.93			1.53084			0.765418	+	0.643534i	$0.222532\pi$
$270$		0			0
$271$	−4315.74			−0.967390			−0.483695	−	0.875237i	$-0.660706\pi$
$271$	−4315.74			−0.967390			−0.483695	+	0.875237i	$0.660706\pi$
$272$		0			0
$273$		−	4101.22i		−	0.909221i
$274$		0			0
$275$	1456.14	−	8023.41i	0.319304	−	1.75938i
$276$		0			0
$277$		−	5799.07i		−	1.25788i	−0.777454	−	0.628939i	$-0.783489\pi$
$277$		−	5799.07i		−	1.25788i	0.777454	−	0.628939i	$-0.216511\pi$
$278$		0			0
$279$	584.717			0.125470
$280$		0			0
$281$	−538.674			−0.114358			−0.0571790	−	0.998364i	$-0.518211\pi$
$281$	−538.674			−0.114358			−0.0571790	+	0.998364i	$0.518211\pi$
$282$		0			0
$283$		−	1648.50i		−	0.346265i	−0.984899	−	0.173133i	$-0.944611\pi$
$283$		−	1648.50i		−	0.346265i	0.984899	−	0.173133i	$-0.0553889\pi$
$284$		0			0
$285$	4132.12	+	4949.55i	0.858827	+	1.02872i
$286$		0			0
$287$		−	4053.38i		−	0.833670i
$288$		0			0
$289$	−463.000			−0.0942398
$290$		0			0
$291$	4886.72			0.984415
$292$		0			0
$293$			4963.73i			0.989707i	0.868976	+	0.494854i	$0.164778\pi$
$293$			4963.73i			0.989707i	−0.868976	+	0.494854i	$0.835222\pi$
$294$		0			0
$295$	1027.32	−	857.653i	0.202755	−	0.169269i
$296$		0			0
$297$		−	9960.88i		−	1.94609i
$298$		0			0
$299$	494.968			0.0957350
$300$		0			0
$301$	426.880			0.0817441
$302$		0			0
$303$			5055.40i			0.958499i
$304$		0			0
$305$	141.964	−	118.518i	0.0266518	−	0.0222503i
$306$		0			0
$307$			9906.61i			1.84169i	0.389925	+	0.920847i	$0.372501\pi$
$307$			9906.61i			1.84169i	−0.389925	+	0.920847i	$0.627499\pi$
$308$		0			0
$309$	7538.40			1.38785
$310$		0			0
$311$	−3477.27			−0.634012			−0.317006	−	0.948424i	$-0.602677\pi$
$311$	−3477.27			−0.634012			−0.317006	+	0.948424i	$0.602677\pi$
$312$		0			0
$313$		−	3958.27i		−	0.714808i	−0.933950	−	0.357404i	$-0.883662\pi$
$313$		−	3958.27i		−	0.714808i	0.933950	−	0.357404i	$-0.116338\pi$
$314$		0			0
$315$	1723.64	+	2064.62i	0.308305	+	0.369295i
$316$		0			0
$317$		−	3296.86i		−	0.584132i	−0.956398	−	0.292066i	$-0.905657\pi$
$317$		−	3296.86i		−	0.584132i	0.956398	−	0.292066i	$-0.0943428\pi$
$318$		0			0
$319$	14654.7			2.57212
$320$		0			0
$321$	406.932			0.0707561
$322$		0			0
$323$			9829.65i			1.69330i
$324$		0			0
$325$	−4141.07	−	751.548i	−0.706785	−	0.128272i
$326$		0			0
$327$		−	5805.91i		−	0.981858i
$328$		0			0
$329$	−3805.66			−0.637728
$330$		0			0
$331$	−6048.38			−1.00438			−0.502189	−	0.864758i	$-0.667472\pi$
$331$	−6048.38			−1.00438			−0.502189	+	0.864758i	$0.667472\pi$
$332$		0			0
$333$			1667.76i			0.274453i
$334$		0			0
$335$	3909.06	+	4682.35i	0.637536	+	0.763655i
$336$		0			0
$337$		−	8539.53i		−	1.38035i	−0.723643	−	0.690175i	$-0.757534\pi$
$337$		−	8539.53i		−	1.38035i	0.723643	−	0.690175i	$-0.242466\pi$
$338$		0			0
$339$	−2276.66			−0.364753
$340$		0			0
$341$	4489.99			0.713040
$342$		0			0
$343$		−	3279.23i		−	0.516215i
$344$		0			0
$345$	542.744	−	453.109i	0.0846967	−	0.0707089i
$346$		0			0
$347$		−	6610.79i		−	1.02273i	−0.859365	−	0.511363i	$-0.829141\pi$
$347$		−	6610.79i		−	1.02273i	0.859365	−	0.511363i	$-0.170859\pi$
$348$		0			0
$349$	2491.07			0.382074			0.191037	−	0.981583i	$-0.438815\pi$
$349$	2491.07			0.382074			0.191037	+	0.981583i	$0.438815\pi$
$350$		0			0
$351$	−5141.04			−0.781791
$352$		0			0
$353$			3383.35i			0.510134i	0.966923	+	0.255067i	$0.0820975\pi$
$353$			3383.35i			0.510134i	−0.966923	+	0.255067i	$0.917902\pi$
$354$		0			0
$355$	−1710.50	+	1428.00i	−0.255729	+	0.213495i
$356$		0			0
$357$		−	8931.08i		−	1.32404i
$358$		0			0
$359$	1303.53			0.191637			0.0958185	−	0.995399i	$-0.469453\pi$
$359$	1303.53			0.191637			0.0958185	+	0.995399i	$0.469453\pi$
$360$		0			0
$361$	11113.8			1.62033
$362$		0			0
$363$		−	12581.2i		−	1.81912i
$364$		0			0
$365$	310.012	+	371.339i	0.0444569	+	0.0532515i
$366$		0			0
$367$		−	6672.42i		−	0.949040i	−0.880245	−	0.474520i	$-0.842622\pi$
$367$		−	6672.42i		−	0.949040i	0.880245	−	0.474520i	$-0.157378\pi$
$368$		0			0
$369$	−1216.10			−0.171565
$370$		0			0
$371$	−7445.88			−1.04197
$372$		0			0
$373$		−	5945.42i		−	0.825314i	−0.910886	−	0.412657i	$-0.864601\pi$
$373$		−	5945.42i		−	0.825314i	0.910886	−	0.412657i	$-0.135399\pi$
$374$		0			0
$375$	−5228.76	+	2966.77i	−0.720033	+	0.408542i
$376$		0			0
$377$		−	7563.64i		−	1.03328i
$378$		0			0
$379$	12553.4			1.70139			0.850693	−	0.525663i	$-0.176183\pi$
$379$	12553.4			1.70139			0.850693	+	0.525663i	$0.176183\pi$
$380$		0			0
$381$	−3501.86			−0.470881
$382$		0			0
$383$			7602.57i			1.01429i	0.861861	+	0.507145i	$0.169299\pi$
$383$			7602.57i			1.01429i	−0.861861	+	0.507145i	$0.830701\pi$
$384$		0			0
$385$	13235.7	+	15854.0i	1.75209	+	2.09869i
$386$		0			0
$387$		−	128.073i		−	0.0168225i
$388$		0			0
$389$	484.765			0.0631840			0.0315920	−	0.999501i	$-0.489942\pi$
$389$	484.765			0.0631840			0.0315920	+	0.999501i	$0.489942\pi$
$390$		0			0
$391$	1077.87			0.139413
$392$		0			0
$393$			5720.62i			0.734268i
$394$		0			0
$395$	−3765.13	+	3143.31i	−0.479605	+	0.400398i
$396$		0			0
$397$		−	9367.85i		−	1.18428i	−0.805835	−	0.592140i	$-0.798283\pi$
$397$		−	9367.85i		−	1.18428i	0.805835	−	0.592140i	$-0.201717\pi$
$398$		0			0
$399$	−16329.9			−2.04891
$400$		0			0
$401$	−6650.48			−0.828203			−0.414101	−	0.910231i	$-0.635904\pi$
$401$	−6650.48			−0.828203			−0.414101	+	0.910231i	$0.635904\pi$
$402$		0			0
$403$		−	2317.39i		−	0.286445i
$404$		0			0
$405$	−3668.62	+	3062.74i	−0.450112	+	0.375775i
$406$		0			0
$407$			12806.6i			1.55970i
$408$		0			0
$409$	1615.31			0.195286			0.0976432	−	0.995221i	$-0.468870\pi$
$409$	1615.31			0.195286			0.0976432	+	0.995221i	$0.468870\pi$
$410$		0			0
$411$	12504.9			1.50078
$412$		0			0
$413$			3389.39i			0.403828i
$414$		0			0
$415$	−8747.87	−	10478.4i	−1.03474	−	1.23943i
$416$		0			0
$417$			3962.19i			0.465298i
$418$		0			0
$419$	10181.0			1.18705			0.593526	−	0.804815i	$-0.297736\pi$
$419$	10181.0			1.18705			0.593526	+	0.804815i	$0.297736\pi$
$420$		0			0
$421$	−4145.13			−0.479860			−0.239930	−	0.970790i	$-0.577124\pi$
$421$	−4145.13			−0.479860			−0.239930	+	0.970790i	$0.577124\pi$
$422$		0			0
$423$			1141.78i			0.131241i
$424$		0			0
$425$	−9017.84	−	1636.62i	−1.02925	−	0.186794i
$426$		0			0
$427$			468.376i			0.0530827i
$428$		0			0
$429$	−9448.53			−1.06336
$430$		0			0
$431$	14283.0			1.59627			0.798133	−	0.602482i	$-0.205821\pi$
$431$	14283.0			1.59627			0.798133	+	0.602482i	$0.205821\pi$
$432$		0			0
$433$			7773.88i			0.862792i	0.902163	+	0.431396i	$0.141979\pi$
$433$			7773.88i			0.862792i	−0.902163	+	0.431396i	$0.858021\pi$
$434$		0			0
$435$	−6923.99	−	8293.71i	−0.763172	−	0.914144i
$436$		0			0
$437$		−	1970.82i		−	0.215737i
$438$		0			0
$439$	3841.14			0.417602			0.208801	−	0.977958i	$-0.433044\pi$
$439$	3841.14			0.417602			0.208801	+	0.977958i	$0.433044\pi$
$440$		0			0
$441$	−3897.78			−0.420881
$442$		0			0
$443$		−	2955.34i		−	0.316958i	−0.987362	−	0.158479i	$-0.949341\pi$
$443$		−	2955.34i		−	0.316958i	0.987362	−	0.158479i	$-0.0506590\pi$
$444$		0			0
$445$	−6207.02	+	5181.92i	−0.661216	+	0.552015i
$446$		0			0
$447$			1141.06i			0.120739i
$448$		0			0
$449$	10786.9			1.13377			0.566886	−	0.823796i	$-0.308148\pi$
$449$	10786.9			1.13377			0.566886	+	0.823796i	$0.308148\pi$
$450$		0			0
$451$	−9338.31			−0.974997
$452$		0			0
$453$			10787.3i			1.11883i
$454$		0			0
$455$	8182.61	−	6831.24i	0.843092	−	0.703854i
$456$		0			0
$457$			15384.9i			1.57479i	0.616451	+	0.787393i	$0.288570\pi$
$457$			15384.9i			1.57479i	−0.616451	+	0.787393i	$0.711430\pi$
$458$		0			0
$459$	−11195.5			−1.13847
$460$		0			0
$461$	−14590.9			−1.47411			−0.737057	−	0.675830i	$-0.763785\pi$
$461$	−14590.9			−1.47411			−0.737057	+	0.675830i	$0.763785\pi$
$462$		0			0
$463$			8828.72i			0.886188i	0.896475	+	0.443094i	$0.146119\pi$
$463$			8828.72i			0.886188i	−0.896475	+	0.443094i	$0.853881\pi$
$464$		0			0
$465$	−2121.41	−	2541.07i	−0.211565	−	0.253418i
$466$		0			0
$467$			4110.64i			0.407319i	0.979042	+	0.203659i	$0.0652835\pi$
$467$			4110.64i			0.407319i	−0.979042	+	0.203659i	$0.934716\pi$
$468$		0			0
$469$	−15448.3			−1.52098
$470$		0			0
$471$	−64.0169			−0.00626273
$472$		0			0
$473$		−	983.461i		−	0.0956016i
$474$		0			0
$475$	−2992.45	+	16488.5i	−0.289059	+	1.59273i
$476$		0			0
$477$			2233.92i			0.214432i
$478$		0			0
$479$	−1852.96			−0.176751			−0.0883757	−	0.996087i	$-0.528168\pi$
$479$	−1852.96			−0.176751			−0.0883757	+	0.996087i	$0.528168\pi$
$480$		0			0
$481$	6609.77			0.626569
$482$		0			0
$483$			1790.66i			0.168691i
$484$		0			0
$485$	8139.61	+	9749.81i	0.762063	+	0.912816i
$486$		0			0
$487$			5285.78i			0.491831i	0.969291	+	0.245915i	$0.0790885\pi$
$487$			5285.78i			0.491831i	−0.969291	+	0.245915i	$0.920912\pi$
$488$		0			0
$489$	−5094.19			−0.471099
$490$		0			0
$491$	12149.4			1.11669			0.558347	−	0.829608i	$-0.311436\pi$
$491$	12149.4			1.11669			0.558347	+	0.829608i	$0.311436\pi$
$492$		0			0
$493$		−	16471.1i		−	1.50470i
$494$		0			0
$495$	4756.53	−	3970.98i	0.431899	−	0.360570i
$496$		0			0
$497$		−	5643.38i		−	0.509337i
$498$		0			0
$499$	3587.97			0.321883			0.160941	−	0.986964i	$-0.448547\pi$
$499$	3587.97			0.321883			0.160941	+	0.986964i	$0.448547\pi$
$500$		0			0
$501$	11566.3			1.03142
$502$		0			0
$503$		−	18351.9i		−	1.62678i	−0.581720	−	0.813389i	$-0.697620\pi$
$503$		−	18351.9i		−	1.62678i	0.581720	−	0.813389i	$-0.302380\pi$
$504$		0			0
$505$	−10086.3	+	8420.57i	−0.888785	+	0.742001i
$506$		0			0
$507$		−	4574.21i		−	0.400686i
$508$		0			0
$509$	−992.521			−0.0864297			−0.0432149	−	0.999066i	$-0.513760\pi$
$509$	−992.521			−0.0864297			−0.0432149	+	0.999066i	$0.513760\pi$
$510$		0			0
$511$	−1225.15			−0.106061
$512$		0			0
$513$			20470.1i			1.76175i
$514$		0			0
$515$	12556.4	+	15040.3i	1.07437	+	1.28691i
$516$		0			0
$517$			8767.59i			0.745838i
$518$		0			0
$519$	8348.37			0.706075
$520$		0			0
$521$	11192.3			0.941155			0.470577	−	0.882359i	$-0.344046\pi$
$521$	11192.3			0.941155			0.470577	+	0.882359i	$0.344046\pi$
$522$		0			0
$523$			4959.46i			0.414650i	0.978272	+	0.207325i	$0.0664758\pi$
$523$			4959.46i			0.414650i	−0.978272	+	0.207325i	$0.933524\pi$
$524$		0			0
$525$	2718.89	−	14981.2i	0.226023	−	1.24540i
$526$		0			0
$527$		−	5046.48i		−	0.417131i
$528$		0			0
$529$	11950.9			0.982238
$530$		0			0
$531$	1016.89			0.0831057
$532$		0			0
$533$			4819.72i			0.391679i
$534$		0			0
$535$	677.810	+	811.895i	0.0547743	+	0.0656099i
$536$		0			0
$537$		−	6799.33i		−	0.546392i
$538$		0			0
$539$	−29930.7			−2.39185
$540$		0			0
$541$	−18555.9			−1.47464			−0.737319	−	0.675545i	$-0.763908\pi$
$541$	−18555.9			−1.47464			−0.737319	+	0.675545i	$0.763908\pi$
$542$		0			0
$543$			11020.6i			0.870976i
$544$		0			0
$545$	11583.7	−	9670.66i	0.910445	−	0.760084i
$546$		0			0
$547$		−	24604.8i		−	1.92326i	−0.274345	−	0.961631i	$-0.588461\pi$
$547$		−	24604.8i		−	1.92326i	0.274345	−	0.961631i	$-0.411539\pi$
$548$		0			0
$549$	140.523			0.0109241
$550$		0			0
$551$	−30116.2			−2.32848
$552$		0			0
$553$		−	12422.2i		−	0.955233i
$554$		0			0
$555$	7247.76	−	6050.78i	0.554325	−	0.462777i
$556$		0			0
$557$		−	13010.2i		−	0.989694i	−0.868980	−	0.494847i	$-0.835224\pi$
$557$		−	13010.2i		−	0.989694i	0.868980	−	0.494847i	$-0.164776\pi$
$558$		0			0
$559$	−507.587			−0.0384054
$560$		0			0
$561$	−20575.7			−1.54850
$562$		0			0
$563$		−	9751.95i		−	0.730010i	−0.931006	−	0.365005i	$-0.881067\pi$
$563$		−	9751.95i		−	0.730010i	0.931006	−	0.365005i	$-0.118933\pi$
$564$		0			0
$565$	−3792.15	−	4542.32i	−0.282366	−	0.338224i
$566$		0			0
$567$		−	12103.8i		−	0.896491i
$568$		0			0
$569$	−5645.90			−0.415973			−0.207986	−	0.978132i	$-0.566691\pi$
$569$	−5645.90			−0.415973			−0.207986	+	0.978132i	$0.566691\pi$
$570$		0			0
$571$	−17188.1			−1.25972			−0.629860	−	0.776709i	$-0.716888\pi$
$571$	−17188.1			−1.25972			−0.629860	+	0.776709i	$0.716888\pi$
$572$		0			0
$573$			592.145i			0.0431714i
$574$		0			0
$575$	1808.05	+	328.138i	0.131132	+	0.0237987i
$576$		0			0
$577$			23294.0i			1.68066i	0.542076	+	0.840330i	$0.317639\pi$
$577$			23294.0i			1.68066i	−0.542076	+	0.840330i	$0.682361\pi$
$578$		0			0
$579$	−3637.09			−0.261058
$580$		0			0
$581$	34571.0			2.46858
$582$		0			0
$583$			17154.0i			1.21861i
$584$		0			0
$585$	−2049.52	−	2454.95i	−0.144850	−	0.173504i
$586$		0			0
$587$			15182.5i			1.06754i	0.845629	+	0.533771i	$0.179226\pi$
$587$			15182.5i			1.06754i	−0.845629	+	0.533771i	$0.820774\pi$
$588$		0			0
$589$	−9227.15			−0.645498
$590$		0			0
$591$	11759.2			0.818461
$592$		0			0
$593$		−	26734.6i		−	1.85137i	−0.378301	−	0.925683i	$-0.623492\pi$
$593$		−	26734.6i		−	1.85137i	0.378301	−	0.925683i	$-0.376508\pi$
$594$		0			0
$595$	17819.0	−	14876.1i	1.22774	−	1.02498i
$596$		0			0
$597$			20001.6i			1.37121i
$598$		0			0
$599$	−2868.56			−0.195670			−0.0978350	−	0.995203i	$-0.531192\pi$
$599$	−2868.56			−0.195670			−0.0978350	+	0.995203i	$0.531192\pi$
$600$		0			0
$601$	−4242.60			−0.287952			−0.143976	−	0.989581i	$-0.545989\pi$
$601$	−4242.60			−0.287952			−0.143976	+	0.989581i	$0.545989\pi$
$602$		0			0
$603$			4634.82i			0.313009i
$604$		0			0
$605$	25101.5	−	20956.0i	1.68681	−	1.40823i
$606$		0			0
$607$			2058.87i			0.137672i	0.997628	+	0.0688361i	$0.0219285\pi$
$607$			2058.87i			0.137672i	−0.997628	+	0.0688361i	$0.978071\pi$
$608$		0			0
$609$	27363.1			1.82071
$610$		0			0
$611$	4525.16			0.299621
$612$		0			0
$613$		−	16492.5i		−	1.08667i	−0.839518	−	0.543333i	$-0.817162\pi$
$613$		−	16492.5i		−	1.08667i	0.839518	−	0.543333i	$-0.182838\pi$
$614$		0			0
$615$	4412.11	+	5284.93i	0.289290	+	0.346518i
$616$		0			0
$617$		−	12562.1i		−	0.819662i	−0.912161	−	0.409831i	$-0.865588\pi$
$617$		−	12562.1i		−	0.819662i	0.912161	−	0.409831i	$-0.134412\pi$
$618$		0			0
$619$	18297.2			1.18809			0.594043	−	0.804434i	$-0.297531\pi$
$619$	18297.2			1.18809			0.594043	+	0.804434i	$0.297531\pi$
$620$		0			0
$621$	2244.66			0.145048
$622$		0			0
$623$		−	20478.6i		−	1.31695i
$624$		0			0
$625$	−14628.5	−	5490.61i	−0.936226	−	0.351399i
$626$		0			0
$627$			37621.3i			2.39625i
$628$		0			0
$629$	14393.8			0.912433
$630$		0			0
$631$	25681.4			1.62022			0.810112	−	0.586275i	$-0.199406\pi$
$631$	25681.4			1.62022			0.810112	+	0.586275i	$0.199406\pi$
$632$		0			0
$633$		−	16543.9i		−	1.03880i
$634$		0			0
$635$	−5832.90	−	6986.78i	−0.364522	−	0.436633i
$636$		0			0
$637$			15447.9i			0.960861i
$638$		0			0
$639$	−1693.13			−0.104819
$640$		0			0
$641$	12994.7			0.800720			0.400360	−	0.916358i	$-0.368885\pi$
$641$	12994.7			0.800720			0.400360	+	0.916358i	$0.368885\pi$
$642$		0			0
$643$		−	4625.71i		−	0.283701i	−0.989888	−	0.141851i	$-0.954695\pi$
$643$		−	4625.71i		−	0.283701i	0.989888	−	0.141851i	$-0.0453053\pi$
$644$		0			0
$645$	−556.580	+	464.660i	−0.0339773	+	0.0283659i
$646$		0			0
$647$			8771.72i			0.533002i	0.963835	+	0.266501i	$0.0858675\pi$
$647$			8771.72i			0.533002i	−0.963835	+	0.266501i	$0.914132\pi$
$648$		0			0
$649$	7808.58			0.472286
$650$		0			0
$651$	8383.66			0.504733
$652$		0			0
$653$			2239.49i			0.134209i	0.997746	+	0.0671043i	$0.0213760\pi$
$653$			2239.49i			0.134209i	−0.997746	+	0.0671043i	$0.978624\pi$
$654$		0			0
$655$	−11413.6	+	9528.61i	−0.680863	+	0.568418i
$656$		0			0
$657$			367.570i			0.0218269i
$658$		0			0
$659$	−4539.62			−0.268344			−0.134172	−	0.990958i	$-0.542837\pi$
$659$	−4539.62			−0.268344			−0.134172	+	0.990958i	$0.542837\pi$
$660$		0			0
$661$	28228.4			1.66105			0.830526	−	0.556980i	$-0.188040\pi$
$661$	28228.4			1.66105			0.830526	+	0.556980i	$0.188040\pi$
$662$		0			0
$663$			10619.6i			0.622068i
$664$		0			0
$665$	−27200.0	−	32580.8i	−1.58612	−	1.89989i
$666$		0			0
$667$			3302.40i			0.191708i
$668$		0			0
$669$	−6283.65			−0.363139
$670$		0			0
$671$	1079.06			0.0620814
$672$		0			0
$673$		−	7649.89i		−	0.438160i	−0.975707	−	0.219080i	$-0.929694\pi$
$673$		−	7649.89i		−	0.438160i	0.975707	−	0.219080i	$-0.0703056\pi$
$674$		0			0
$675$	−18779.5	−	3408.24i	−1.07085	−	0.194345i
$676$		0			0
$677$			12087.9i			0.686228i	0.939294	+	0.343114i	$0.111482\pi$
$677$			12087.9i			0.686228i	−0.939294	+	0.343114i	$0.888518\pi$
$678$		0			0
$679$	−32167.2			−1.81806
$680$		0			0
$681$	−14777.0			−0.831506
$682$		0			0
$683$		−	8827.56i		−	0.494549i	−0.968945	−	0.247275i	$-0.920465\pi$
$683$		−	8827.56i		−	0.494549i	0.968945	−	0.247275i	$-0.0795350\pi$
$684$		0			0
$685$	20828.9	+	24949.3i	1.16180	+	1.39163i
$686$		0			0
$687$		−	2561.04i		−	0.142227i
$688$		0			0
$689$	8853.60			0.489543
$690$		0			0
$691$	12913.1			0.710908			0.355454	−	0.934694i	$-0.384326\pi$
$691$	12913.1			0.710908			0.355454	+	0.934694i	$0.384326\pi$
$692$		0			0
$693$			15693.1i			0.860217i
$694$		0			0
$695$	−7905.21	+	6599.65i	−0.431456	+	0.360200i
$696$		0			0
$697$			10495.7i			0.570378i
$698$		0			0
$699$	23369.5			1.26454
$700$		0			0
$701$	−12297.2			−0.662567			−0.331283	−	0.943531i	$-0.607482\pi$
$701$	−12297.2			−0.662567			−0.331283	+	0.943531i	$0.607482\pi$
$702$		0			0
$703$		−	26318.2i		−	1.41196i
$704$		0			0
$705$	4961.94	−	4142.47i	0.265074	−	0.221297i
$706$		0			0
$707$		−	33277.5i		−	1.77020i
$708$		0			0
$709$	−34308.2			−1.81731			−0.908653	−	0.417552i	$-0.862888\pi$
$709$	−34308.2			−1.81731			−0.908653	+	0.417552i	$0.862888\pi$
$710$		0			0
$711$	−3726.91			−0.196582
$712$		0			0
$713$			1011.81i			0.0531451i
$714$		0			0
$715$	−15738.0	−	18851.4i	−0.823174	−	0.986016i
$716$		0			0
$717$			16912.1i			0.880887i
$718$		0			0
$719$	21631.4			1.12200			0.560999	−	0.827817i	$-0.310417\pi$
$719$	21631.4			1.12200			0.560999	+	0.827817i	$0.310417\pi$
$720$		0			0
$721$	−49622.1			−2.56314
$722$		0			0
$723$			11132.3i			0.572632i
$724$		0			0
$725$	5014.29	−	27629.0i	0.256864	−	1.41533i
$726$		0			0
$727$		−	12049.8i		−	0.614723i	−0.951593	−	0.307361i	$-0.900554\pi$
$727$		−	12049.8i		−	0.614723i	0.951593	−	0.307361i	$-0.0994460\pi$
$728$		0			0
$729$	−21365.7			−1.08549
$730$		0			0
$731$	−1105.35			−0.0559274
$732$		0			0
$733$		−	15221.6i		−	0.767017i	−0.923537	−	0.383508i	$-0.874716\pi$
$733$		−	15221.6i		−	0.767017i	0.923537	−	0.383508i	$-0.125284\pi$
$734$		0			0
$735$	14141.5	+	16939.0i	0.709683	+	0.850073i
$736$		0			0
$737$			35590.4i			1.77882i
$738$		0			0
$739$	−9845.80			−0.490099			−0.245050	−	0.969511i	$-0.578804\pi$
$739$	−9845.80			−0.490099			−0.245050	+	0.969511i	$0.578804\pi$
$740$		0			0
$741$	19417.2			0.962630
$742$		0			0
$743$			35081.5i			1.73219i	0.499881	+	0.866094i	$0.333377\pi$
$743$			35081.5i			1.73219i	−0.499881	+	0.866094i	$0.666623\pi$
$744$		0			0
$745$	−2276.61	+	1900.62i	−0.111958	+	0.0934676i
$746$		0			0
$747$		−	10372.0i		−	0.508022i
$748$		0			0
$749$	−2678.66			−0.130676
$750$		0			0
$751$	−20007.7			−0.972158			−0.486079	−	0.873915i	$-0.661573\pi$
$751$	−20007.7			−0.972158			−0.486079	+	0.873915i	$0.661573\pi$
$752$		0			0
$753$			9600.47i			0.464622i
$754$		0			0
$755$	−21522.4	+	17968.0i	−1.03746	+	0.866122i
$756$		0			0
$757$			33478.7i			1.60741i	0.595031	+	0.803703i	$0.297140\pi$
$757$			33478.7i			1.60741i	−0.595031	+	0.803703i	$0.702860\pi$
$758$		0			0
$759$	4125.37			0.197288
$760$		0			0
$761$	12728.1			0.606298			0.303149	−	0.952943i	$-0.401962\pi$
$761$	12728.1			0.606298			0.303149	+	0.952943i	$0.401962\pi$
$762$		0			0
$763$			38217.8i			1.81334i
$764$		0			0
$765$	−4463.15	−	5346.06i	−0.210935	−	0.252663i
$766$		0			0
$767$		−	4030.19i		−	0.189728i
$768$		0			0
$769$	20061.4			0.940747			0.470373	−	0.882468i	$-0.344119\pi$
$769$	20061.4			0.940747			0.470373	+	0.882468i	$0.344119\pi$
$770$		0			0
$771$	−10655.6			−0.497732
$772$		0			0
$773$			672.445i			0.0312887i	0.999878	+	0.0156444i	$0.00497996\pi$
$773$			672.445i			0.0312887i	−0.999878	+	0.0156444i	$0.995020\pi$
$774$		0			0
$775$	1536.30	−	8465.10i	0.0712073	−	0.392355i
$776$		0			0
$777$			23912.3i			1.10405i
$778$		0			0
$779$	19190.7			0.882642
$780$		0			0
$781$	−13001.4			−0.595681
$782$		0			0
$783$		−	34300.8i		−	1.56553i
$784$		0			0
$785$	−106.630	−	127.724i	−0.00484816	−	0.00580723i
$786$		0			0
$787$			7928.23i			0.359099i	0.983749	+	0.179550i	$0.0574640\pi$
$787$			7928.23i			0.359099i	−0.983749	+	0.179550i	$0.942536\pi$
$788$		0			0
$789$	−20928.9			−0.944346
$790$		0			0
$791$	14986.3			0.673643
$792$		0			0
$793$		−	556.928i		−	0.0249396i
$794$		0			0
$795$	9708.17	−	8104.86i	0.433099	−	0.361572i
$796$		0			0
$797$			3196.35i			0.142059i	0.997474	+	0.0710293i	$0.0226284\pi$
$797$			3196.35i			0.142059i	−0.997474	+	0.0710293i	$0.977372\pi$
$798$		0			0
$799$	9854.26			0.436319
$800$		0			0
$801$	−6144.02			−0.271021
$802$		0			0
$803$			2822.54i			0.124041i
$804$		0			0
$805$	−3572.65	+	2982.62i	−0.156422	+	0.130588i
$806$		0			0
$807$		−	29053.3i		−	1.26732i
$808$		0			0
$809$	15278.6			0.663987			0.331993	−	0.943282i	$-0.392279\pi$
$809$	15278.6			0.663987			0.331993	+	0.943282i	$0.392279\pi$
$810$		0			0
$811$	−27275.2			−1.18096			−0.590482	−	0.807051i	$-0.701062\pi$
$811$	−27275.2			−1.18096			−0.590482	+	0.807051i	$0.701062\pi$
$812$		0			0
$813$			18565.0i			0.800864i
$814$		0			0
$815$	−8485.18	−	10163.7i	−0.364691	−	0.436835i
$816$		0			0
$817$			2021.06i			0.0865459i
$818$		0			0
$819$	8099.55			0.345569
$820$		0			0
$821$	−11047.6			−0.469627			−0.234814	−	0.972040i	$-0.575448\pi$
$821$	−11047.6			−0.469627			−0.234814	+	0.972040i	$0.575448\pi$
$822$		0			0
$823$		−	46589.2i		−	1.97327i	−0.162959	−	0.986633i	$-0.552104\pi$
$823$		−	46589.2i		−	1.97327i	0.162959	−	0.986633i	$-0.447896\pi$
$824$		0			0
$825$	−34514.2	−	6263.87i	−1.45652	−	0.264339i
$826$		0			0
$827$			7622.29i			0.320499i	0.987077	+	0.160250i	$0.0512299\pi$
$827$			7622.29i			0.320499i	−0.987077	+	0.160250i	$0.948770\pi$
$828$		0			0
$829$	637.729			0.0267180			0.0133590	−	0.999911i	$-0.495748\pi$
$829$	637.729			0.0267180			0.0133590	+	0.999911i	$0.495748\pi$
$830$		0			0
$831$	−24945.8			−1.04135
$832$		0			0
$833$			33640.3i			1.39924i
$834$		0			0
$835$	19265.5	+	23076.6i	0.798454	+	0.956406i
$836$		0			0
$837$		−	10509.2i		−	0.433993i
$838$		0			0
$839$	38563.3			1.58683			0.793417	−	0.608679i	$-0.208300\pi$
$839$	38563.3			1.58683			0.793417	+	0.608679i	$0.208300\pi$
$840$		0			0
$841$	26075.2			1.06914
$842$		0			0
$843$			2317.21i			0.0946725i
$844$		0			0
$845$	9126.29	−	7619.07i	0.371543	−	0.310182i
$846$		0			0
$847$			82816.7i			3.35964i
$848$		0			0
$849$	−7091.33			−0.286660
$850$		0			0
$851$	−2885.93			−0.116249
$852$		0			0
$853$		−	21077.3i		−	0.846040i	−0.906120	−	0.423020i	$-0.860970\pi$
$853$		−	21077.3i		−	0.846040i	0.906120	−	0.423020i	$-0.139030\pi$
$854$		0			0
$855$	−9774.91	+	8160.57i	−0.390988	+	0.326416i
$856$		0			0
$857$		−	24424.6i		−	0.973546i	−0.873528	−	0.486773i	$-0.838174\pi$
$857$		−	24424.6i		−	0.973546i	0.873528	−	0.486773i	$-0.161826\pi$
$858$		0			0
$859$	−25649.1			−1.01878			−0.509392	−	0.860534i	$-0.670130\pi$
$859$	−25649.1			−1.01878			−0.509392	+	0.860534i	$0.670130\pi$
$860$		0			0
$861$	−17436.4			−0.690163
$862$		0			0
$863$			31095.6i			1.22654i	0.789872	+	0.613272i	$0.210147\pi$
$863$			31095.6i			1.22654i	−0.789872	+	0.613272i	$0.789853\pi$
$864$		0			0
$865$	13905.5	+	16656.4i	0.546593	+	0.654721i
$866$		0			0
$867$			1991.68i			0.0780174i
$868$		0			0
$869$	−28618.6			−1.11717
$870$		0			0
$871$	18369.0			0.714593
$872$		0			0
$873$			9650.84i			0.374148i
$874$		0			0
$875$	34418.7	−	19529.0i	1.32979	−	0.754514i
$876$		0			0
$877$			26626.0i			1.02519i	0.858630	+	0.512597i	$0.171316\pi$
$877$			26626.0i			1.02519i	−0.858630	+	0.512597i	$0.828684\pi$
$878$		0			0
$879$	21352.4			0.819340
$880$		0			0
$881$	−18538.6			−0.708945			−0.354472	−	0.935066i	$-0.615340\pi$
$881$	−18538.6			−0.708945			−0.354472	+	0.935066i	$0.615340\pi$
$882$		0			0
$883$		−	16994.0i		−	0.647672i	−0.946113	−	0.323836i	$-0.895027\pi$
$883$		−	16994.0i		−	0.647672i	0.946113	−	0.323836i	$-0.104973\pi$
$884$		0			0
$885$	−3689.36	−	4419.19i	−0.140132	−	0.167853i
$886$		0			0
$887$			19948.2i			0.755126i	0.925984	+	0.377563i	$0.123238\pi$
$887$			19948.2i			0.755126i	−0.925984	+	0.377563i	$0.876762\pi$
$888$		0			0
$889$	23051.3			0.869645
$890$		0			0
$891$	−27885.0			−1.04847
$892$		0			0
$893$		−	18017.8i		−	0.675190i
$894$		0			0
$895$	13565.8	−	11325.4i	0.506652	−	0.422978i
$896$		0			0
$897$		−	2129.20i		−	0.0792552i
$898$		0			0
$899$	15461.5			0.573603
$900$		0			0
$901$	19280.1			0.712891
$902$		0			0
$903$		−	1836.31i		−	0.0676727i
$904$		0			0
$905$	−21987.9	+	18356.6i	−0.807628	+	0.674247i
$906$		0			0
$907$		−	31481.3i		−	1.15250i	−0.817273	−	0.576251i	$-0.804515\pi$
$907$		−	31481.3i		−	1.15250i	0.817273	−	0.576251i	$-0.195485\pi$
$908$		0			0
$909$	−9983.96			−0.364298
$910$		0			0
$911$	−29685.8			−1.07962			−0.539811	−	0.841787i	$-0.681504\pi$
$911$	−29685.8			−1.07962			−0.539811	+	0.841787i	$0.681504\pi$
$912$		0			0
$913$		−	79645.8i		−	2.88707i
$914$		0			0
$915$	−509.828	−	610.684i	−0.0184201	−	0.0220640i
$916$		0			0
$917$		−	37656.4i		−	1.35608i
$918$		0			0
$919$	−21804.9			−0.782673			−0.391336	−	0.920248i	$-0.627987\pi$
$919$	−21804.9			−0.782673			−0.391336	+	0.920248i	$0.627987\pi$
$920$		0			0
$921$	42615.2			1.52467
$922$		0			0
$923$			6710.33i			0.239299i
$924$		0			0
$925$	24144.6	+	4381.92i	0.858238	+	0.155759i
$926$		0			0
$927$			14887.7i			0.527481i
$928$		0			0
$929$	49150.1			1.73580			0.867902	−	0.496735i	$-0.165468\pi$
$929$	49150.1			1.73580			0.867902	+	0.496735i	$0.165468\pi$
$930$		0			0
$931$	61509.1			2.16528
$932$		0			0
$933$			14958.1i			0.524873i
$934$		0			0
$935$	−34272.1	−	41051.9i	−1.19874	−	1.43587i
$936$		0			0
$937$			33165.6i			1.15632i	0.815922	+	0.578161i	$0.196230\pi$
$937$			33165.6i			1.15632i	−0.815922	+	0.578161i	$0.803770\pi$
$938$		0			0
$939$	−17027.3			−0.591761
$940$		0			0
$941$	−15184.8			−0.526048			−0.263024	−	0.964789i	$-0.584720\pi$
$941$	−15184.8			−0.526048			−0.263024	+	0.964789i	$0.584720\pi$
$942$		0			0
$943$		−	2104.36i		−	0.0726696i
$944$		0			0
$945$	37107.8	−	30979.4i	1.27737	−	1.06641i
$946$		0			0
$947$		−	44341.2i		−	1.52154i	−0.649023	−	0.760769i	$-0.724822\pi$
$947$		−	44341.2i		−	1.52154i	0.649023	−	0.760769i	$-0.275178\pi$
$948$		0			0
$949$	1456.78			0.0498303
$950$		0			0
$951$	−14182.1			−0.483580
$952$		0			0
$953$			31056.8i			1.05564i	0.849356	+	0.527821i	$0.176991\pi$
$953$			31056.8i			1.05564i	−0.849356	+	0.527821i	$0.823009\pi$
$954$		0			0
$955$	−1181.43	+	986.313i	−0.0400315	+	0.0334202i
$956$		0			0
$957$		−	63040.1i		−	2.12936i
$958$		0			0
$959$	−82314.5			−2.77171
$960$		0			0
$961$	−25053.8			−0.840987
$962$		0			0
$963$			803.654i			0.0268924i
$964$		0			0
$965$	−6058.16	−	7256.59i	−0.202092	−	0.242070i
$966$		0			0
$967$		−	40508.8i		−	1.34713i	−0.739128	−	0.673564i	$-0.764762\pi$
$967$		−	40508.8i		−	1.34713i	0.739128	−	0.673564i	$-0.235238\pi$
$968$		0			0
$969$	42284.1			1.40182
$970$		0			0
$971$	36986.9			1.22242			0.611208	−	0.791470i	$-0.290684\pi$
$971$	36986.9			1.22242			0.611208	+	0.791470i	$0.290684\pi$
$972$		0			0
$973$		−	26081.4i		−	0.859333i
$974$		0			0
$975$	−3232.93	+	17813.6i	−0.106191	+	0.585119i
$976$		0			0
$977$		−	28979.9i		−	0.948975i	−0.880262	−	0.474488i	$-0.842633\pi$
$977$		−	28979.9i		−	0.948975i	0.880262	−	0.474488i	$-0.157367\pi$
$978$		0			0
$979$	−47179.3			−1.54020
$980$		0			0
$981$	11466.1			0.373176
$982$		0			0
$983$			31114.6i			1.00956i	0.863247	+	0.504782i	$0.168427\pi$
$983$			31114.6i			1.00956i	−0.863247	+	0.504782i	$0.831573\pi$
$984$		0			0
$985$	19586.9	+	23461.6i	0.633594	+	0.758933i
$986$		0			0
$987$			16370.8i			0.527950i
$988$		0			0
$989$	221.620			0.00712549
$990$		0			0
$991$	44477.8			1.42571			0.712857	−	0.701309i	$-0.247401\pi$
$991$	44477.8			1.42571			0.712857	+	0.701309i	$0.247401\pi$
$992$		0			0
$993$			26018.3i			0.831485i
$994$		0			0
$995$	−39906.4	+	33315.8i	−1.27147	+	1.06149i
$996$		0			0
$997$		−	57489.4i		−	1.82619i	−0.407752	−	0.913093i	$-0.633687\pi$
$997$		−	57489.4i		−	1.82619i	0.407752	−	0.913093i	$-0.366313\pi$
$998$		0			0
$999$	29975.0			0.949316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.c.d.129.3	✓	8
3.2	odd	2		1440.4.f.k.289.4		8
4.3	odd	2	inner	160.4.c.d.129.5	yes	8
5.2	odd	4		800.4.a.y.1.2		4
5.3	odd	4		800.4.a.z.1.3		4
5.4	even	2	inner	160.4.c.d.129.6	yes	8
8.3	odd	2		320.4.c.j.129.4		8
8.5	even	2		320.4.c.j.129.6		8
12.11	even	2		1440.4.f.k.289.3		8
15.14	odd	2		1440.4.f.k.289.1		8
20.3	even	4		800.4.a.z.1.2		4
20.7	even	4		800.4.a.y.1.3		4
20.19	odd	2	inner	160.4.c.d.129.4	yes	8
40.3	even	4		1600.4.a.cu.1.3		4
40.13	odd	4		1600.4.a.cu.1.2		4
40.19	odd	2		320.4.c.j.129.5		8
40.27	even	4		1600.4.a.cv.1.2		4
40.29	even	2		320.4.c.j.129.3		8
40.37	odd	4		1600.4.a.cv.1.3		4
60.59	even	2		1440.4.f.k.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.d.129.3	✓	8	1.1	even	1	trivial
160.4.c.d.129.4	yes	8	20.19	odd	2	inner
160.4.c.d.129.5	yes	8	4.3	odd	2	inner
160.4.c.d.129.6	yes	8	5.4	even	2	inner
320.4.c.j.129.3		8	40.29	even	2
320.4.c.j.129.4		8	8.3	odd	2
320.4.c.j.129.5		8	40.19	odd	2
320.4.c.j.129.6		8	8.5	even	2
800.4.a.y.1.2		4	5.2	odd	4
800.4.a.y.1.3		4	20.7	even	4
800.4.a.z.1.2		4	20.3	even	4
800.4.a.z.1.3		4	5.3	odd	4
1440.4.f.k.289.1		8	15.14	odd	2
1440.4.f.k.289.2		8	60.59	even	2
1440.4.f.k.289.3		8	12.11	even	2
1440.4.f.k.289.4		8	3.2	odd	2
1600.4.a.cu.1.2		4	40.13	odd	4
1600.4.a.cu.1.3		4	40.3	even	4
1600.4.a.cv.1.2		4	40.27	even	4
1600.4.a.cv.1.3		4	40.37	odd	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +O(q^{10})\)	\(q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +65.2358 q^{11} -33.6697i q^{13} +(-30.8223 - 36.9196i) q^{15} -73.3212i q^{17} -134.063 q^{19} +121.808 q^{21} +14.7007i q^{23} +(22.3212 - 122.991i) q^{25} -152.690i q^{27} +224.642 q^{29} +68.8271 q^{31} -280.624i q^{33} +(202.890 + 243.026i) q^{35} +196.312i q^{37} -144.837 q^{39} -143.147 q^{41} -15.0755i q^{43} +(72.9129 - 60.8712i) q^{45} +134.399i q^{47} -458.808 q^{49} -315.405 q^{51} +262.955i q^{53} +(559.891 - 467.424i) q^{55} +576.697i q^{57} +119.698 q^{59} +16.5409 q^{61} +240.559i q^{63} +(-241.248 - 288.973i) q^{65} +545.565i q^{67} +63.2379 q^{69} -199.299 q^{71} +43.2667i q^{73} +(-529.069 - 96.0190i) q^{75} +1847.23i q^{77} -438.694 q^{79} -427.450 q^{81} -1220.89i q^{83} +(-525.358 - 629.285i) q^{85} -966.342i q^{87} -723.212 q^{89} +953.398 q^{91} -296.073i q^{93} +(-1150.60 + 960.581i) q^{95} +1136.00i q^{97} +554.208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{5} - 152 q^{9}+O(q^{10}) \) 8 * q + 32 * q^5 - 152 * q^9	\( 8 q + 32 q^{5} - 152 q^{9} - 272 q^{21} - 408 q^{25} + 624 q^{29} - 192 q^{41} + 400 q^{45} - 2424 q^{49} + 2112 q^{61} + 2176 q^{65} + 3952 q^{69} - 1000 q^{81} - 5376 q^{85} + 80 q^{89}+O(q^{100}) \) 8 * q + 32 * q^5 - 152 * q^9 - 272 * q^21 - 408 * q^25 + 624 * q^29 - 192 * q^41 + 400 * q^45 - 2424 * q^49 + 2112 * q^61 + 2176 * q^65 + 3952 * q^69 - 1000 * q^81 - 5376 * q^85 + 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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