LMFDB - Embedded newform 1440.4.f.k.289.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,4,Mod(289,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.289"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-408] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$84.9627504083$
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:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-0.320221i$ of defining polynomial
Character	$\chi$	$=$	1440.289
Dual form			1440.4.f.k.289.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +65.2358 q^{11} +33.6697i q^{13} -73.3212i q^{17} +134.063 q^{19} -14.7007i q^{23} +(22.3212 + 122.991i) q^{25} -224.642 q^{29} -68.8271 q^{31} +(202.890 - 243.026i) q^{35} -196.312i q^{37} +143.147 q^{41} -15.0755i q^{43} -134.399i q^{47} -458.808 q^{49} +262.955i q^{53} +(-559.891 - 467.424i) q^{55} +119.698 q^{59} +16.5409 q^{61} +(241.248 - 288.973i) q^{65} +545.565i q^{67} -199.299 q^{71} -43.2667i q^{73} +1847.23i q^{77} +438.694 q^{79} +1220.89i q^{83} +(-525.358 + 629.285i) q^{85} +723.212 q^{89} -953.398 q^{91} +(-1150.60 - 960.581i) q^{95} -1136.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{5} - 408 q^{25} - 624 q^{29} + 192 q^{41} - 2424 q^{49} + 2112 q^{61} - 2176 q^{65} - 5376 q^{85} - 80 q^{89}+O(q^{100}) $ 8 * q - 32 * q^5 - 408 * q^25 - 624 * q^29 + 192 * q^41 - 2424 * q^49 + 2112 * q^61 - 2176 * q^65 - 5376 * q^85 - 80 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		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
$9$		0			0
$10$		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	$0.116939\pi$
$13$			33.6697i			0.718330i	−0.933274	+	0.359165i	$0.883061\pi$
$14$		0			0
$15$		0			0
$16$		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	$-0.199808\pi$
$19$	134.063			1.61874			0.809372	+	0.587297i	$0.199808\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	14.7007i		−	0.133274i	−0.997777	−	0.0666371i	$-0.978773\pi$
$23$		−	14.7007i		−	0.133274i	0.997777	−	0.0666371i	$-0.0212270\pi$
$24$		0			0
$25$	22.3212	+	122.991i	0.178570	+	0.983927i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−224.642			−1.43845			−0.719225	−	0.694777i	$-0.755503\pi$
$29$	−224.642			−1.43845			−0.719225	+	0.694777i	$0.755503\pi$
$30$		0			0
$31$	−68.8271			−0.398765			−0.199382	−	0.979922i	$-0.563894\pi$
$31$	−68.8271			−0.398765			−0.199382	+	0.979922i	$0.563894\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	202.890	−	243.026i	0.979847	−	1.17368i
$36$		0			0
$37$		−	196.312i		−	0.872257i	−0.899884	−	0.436129i	$-0.856349\pi$
$37$		−	196.312i		−	0.872257i	0.899884	−	0.436129i	$-0.143651\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	143.147			0.545263			0.272632	−	0.962118i	$-0.412106\pi$
$41$	143.147			0.545263			0.272632	+	0.962118i	$0.412106\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$		−	134.399i		−	0.417107i	−0.978011	−	0.208554i	$-0.933124\pi$
$47$		−	134.399i		−	0.417107i	0.978011	−	0.208554i	$-0.0668756\pi$
$48$		0			0
$49$	−458.808			−1.33763
$50$		0			0
$51$		0			0
$52$		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
$57$		0			0
$58$		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
$63$		0			0
$64$		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
$69$		0			0
$70$		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	$-0.988957\pi$
$73$		−	43.2667i		−	0.0693696i	0.999398	−	0.0346848i	$-0.0110427\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1847.23i			2.73391i
$78$		0			0
$79$	438.694			0.624772			0.312386	−	0.949955i	$-0.398872\pi$
$79$	438.694			0.624772			0.312386	+	0.949955i	$0.398872\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1220.89i			1.61458i	0.590154	+	0.807291i	$0.299067\pi$
$83$			1220.89i			1.61458i	−0.590154	+	0.807291i	$0.700933\pi$
$84$		0			0
$85$	−525.358	+	629.285i	−0.670389	+	0.803006i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	723.212			0.861352			0.430676	−	0.902507i	$-0.358275\pi$
$89$	723.212			0.861352			0.430676	+	0.902507i	$0.358275\pi$
$90$		0			0
$91$	−953.398			−1.09828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1150.60	−	960.581i	−1.24263	−	1.03741i
$96$		0			0
$97$		−	1136.00i		−	1.18911i	−0.804056	−	0.594553i	$-0.797329\pi$
$97$		−	1136.00i		−	1.18911i	0.804056	−	0.594553i	$-0.202671\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1175.21			1.15780			0.578901	−	0.815398i	$-0.303482\pi$
$101$	1175.21			1.15780			0.578901	+	0.815398i	$0.303482\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$		−	94.5981i		−	0.0854686i	−0.999086	−	0.0427343i	$-0.986393\pi$
$107$		−	94.5981i		−	0.0854686i	0.999086	−	0.0427343i	$-0.0136069\pi$
$108$		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
$111$		0			0
$112$		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
$117$		0			0
$118$		0			0
$119$	2076.18			1.59935
$120$		0			0
$121$	2924.71			2.19738
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		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
$135$		0			0
$136$		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	$-0.409326\pi$
$139$	921.077			0.562048			0.281024	+	0.959701i	$0.409326\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2196.47i			1.28446i
$144$		0			0
$145$	1928.01	+	1609.60i	1.10422	+	0.921860i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	265.259			0.145845			0.0729224	−	0.997338i	$-0.476767\pi$
$149$	265.259			0.145845			0.0729224	+	0.997338i	$0.476767\pi$
$150$		0			0
$151$	2507.69			1.35148			0.675738	−	0.737142i	$-0.263825\pi$
$151$	2507.69			1.35148			0.675738	+	0.737142i	$0.263825\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	590.713	+	493.156i	0.306111	+	0.255557i
$156$		0			0
$157$			14.8818i			0.00756495i	0.999993	+	0.00378248i	$0.00120400\pi$
$157$			14.8818i			0.00756495i	−0.999993	+	0.00378248i	$0.998796\pi$
$158$		0			0
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$		−	2688.78i		−	1.24589i	−0.782266	−	0.622945i	$-0.785936\pi$
$167$		−	2688.78i		−	1.24589i	0.782266	−	0.622945i	$-0.214064\pi$
$168$		0			0
$169$	1063.35			0.484002
$170$		0			0
$171$		0			0
$172$		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
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	−1406.61	+	1684.86i	−0.559004	+	0.669587i
$186$		0			0
$187$		−	4783.17i		−	1.87048i
$188$		0			0
$189$		0			0
$190$		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	$0.0503982\pi$
$193$			845.503i			0.315340i	−0.987492	+	0.157670i	$0.949602\pi$
$194$		0			0
$195$		0			0
$196$		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	$-0.189387\pi$
$199$	4649.70			1.65632			0.828162	+	0.560489i	$0.189387\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6361.02i		−	2.19929i
$204$		0			0
$205$	−1228.57	−	1025.67i	−0.418571	−	0.349443i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8745.70			2.89451
$210$		0			0
$211$	−3845.91			−1.25480			−0.627402	−	0.778696i	$-0.715882\pi$
$211$	−3845.91			−1.25480			−0.627402	+	0.778696i	$0.715882\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−108.018	+	129.386i	−0.0342640	+	0.0410422i
$216$		0			0
$217$		−	1948.92i		−	0.609684i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$			3435.16i			1.00440i	0.864750	+	0.502202i	$0.167477\pi$
$227$			3435.16i			1.00440i	−0.864750	+	0.502202i	$0.832523\pi$
$228$		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
$231$		0			0
$232$		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
$237$		0			0
$238$		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
$243$		0			0
$244$		0			0
$245$	3937.75	+	3287.43i	1.02683	+	0.857249i
$246$		0			0
$247$			4513.86i			1.16279i
$248$		0			0
$249$		0			0
$250$		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
$255$		0			0
$256$		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
$261$		0			0
$262$		0			0
$263$			4865.27i			1.14071i	0.821400	+	0.570353i	$0.193194\pi$
$263$			4865.27i			1.14071i	−0.821400	+	0.570353i	$0.806806\pi$
$264$		0			0
$265$	1884.11	−	2256.83i	0.436754	−	0.523154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6753.93			−1.53084			−0.765418	−	0.643534i	$-0.777468\pi$
$269$	−6753.93			−1.53084			−0.765418	+	0.643534i	$0.777468\pi$
$270$		0			0
$271$	4315.74			0.967390			0.483695	−	0.875237i	$-0.339294\pi$
$271$	4315.74			0.967390			0.483695	+	0.875237i	$0.339294\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1456.14	+	8023.41i	0.319304	+	1.75938i
$276$		0			0
$277$			5799.07i			1.25788i	0.777454	+	0.628939i	$0.216511\pi$
$277$			5799.07i			1.25788i	−0.777454	+	0.628939i	$0.783489\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	538.674			0.114358			0.0571790	−	0.998364i	$-0.481789\pi$
$281$	538.674			0.114358			0.0571790	+	0.998364i	$0.481789\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$			4053.38i			0.833670i
$288$		0			0
$289$	−463.000			−0.0942398
$290$		0			0
$291$		0			0
$292$		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
$297$		0			0
$298$		0			0
$299$	494.968			0.0957350
$300$		0			0
$301$	426.880			0.0817441
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		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	$0.116338\pi$
$313$			3958.27i			0.714808i	−0.933950	+	0.357404i	$0.883662\pi$
$314$		0			0
$315$		0			0
$316$		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
$321$		0			0
$322$		0			0
$323$		−	9829.65i		−	1.69330i
$324$		0			0
$325$	−4141.07	+	751.548i	−0.706785	+	0.128272i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3805.66			0.637728
$330$		0			0
$331$	6048.38			1.00438			0.502189	−	0.864758i	$-0.332528\pi$
$331$	6048.38			1.00438			0.502189	+	0.864758i	$0.332528\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3909.06	−	4682.35i	0.637536	−	0.763655i
$336$		0			0
$337$			8539.53i			1.38035i	0.723643	+	0.690175i	$0.242466\pi$
$337$			8539.53i			1.38035i	−0.723643	+	0.690175i	$0.757534\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4489.99			−0.713040
$342$		0			0
$343$		−	3279.23i		−	0.516215i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6610.79i			1.02273i	0.859365	+	0.511363i	$0.170859\pi$
$347$			6610.79i			1.02273i	−0.859365	+	0.511363i	$0.829141\pi$
$348$		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
$351$		0			0
$352$		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
$357$		0			0
$358$		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
$363$		0			0
$364$		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
$369$		0			0
$370$		0			0
$371$	−7445.88			−1.04197
$372$		0			0
$373$			5945.42i			0.825314i	0.910886	+	0.412657i	$0.135399\pi$
$373$			5945.42i			0.825314i	−0.910886	+	0.412657i	$0.864601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7563.64i		−	1.03328i
$378$		0			0
$379$	−12553.4			−1.70139			−0.850693	−	0.525663i	$-0.823817\pi$
$379$	−12553.4			−1.70139			−0.850693	+	0.525663i	$0.823817\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	7602.57i		−	1.01429i	−0.861861	−	0.507145i	$-0.830701\pi$
$383$		−	7602.57i		−	1.01429i	0.861861	−	0.507145i	$-0.169299\pi$
$384$		0			0
$385$	13235.7	−	15854.0i	1.75209	−	2.09869i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−484.765			−0.0631840			−0.0315920	−	0.999501i	$-0.510058\pi$
$389$	−484.765			−0.0631840			−0.0315920	+	0.999501i	$0.510058\pi$
$390$		0			0
$391$	−1077.87			−0.139413
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3765.13	−	3143.31i	−0.479605	−	0.400398i
$396$		0			0
$397$			9367.85i			1.18428i	0.805835	+	0.592140i	$0.201717\pi$
$397$			9367.85i			1.18428i	−0.805835	+	0.592140i	$0.798283\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6650.48			0.828203			0.414101	−	0.910231i	$-0.364096\pi$
$401$	6650.48			0.828203			0.414101	+	0.910231i	$0.364096\pi$
$402$		0			0
$403$		−	2317.39i		−	0.286445i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			3389.39i			0.403828i
$414$		0			0
$415$	8747.87	−	10478.4i	1.03474	−	1.23943i
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$	9017.84	−	1636.62i	1.02925	−	0.186794i
$426$		0			0
$427$			468.376i			0.0530827i
$428$		0			0
$429$		0			0
$430$		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	$-0.858021\pi$
$433$		−	7773.88i		−	0.862792i	0.902163	−	0.431396i	$-0.141979\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1970.82i		−	0.215737i
$438$		0			0
$439$	−3841.14			−0.417602			−0.208801	−	0.977958i	$-0.566956\pi$
$439$	−3841.14			−0.417602			−0.208801	+	0.977958i	$0.566956\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2955.34i			0.316958i	0.987362	+	0.158479i	$0.0506590\pi$
$443$			2955.34i			0.316958i	−0.987362	+	0.158479i	$0.949341\pi$
$444$		0			0
$445$	−6207.02	−	5181.92i	−0.661216	−	0.552015i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10786.9			−1.13377			−0.566886	−	0.823796i	$-0.691852\pi$
$449$	−10786.9			−1.13377			−0.566886	+	0.823796i	$0.691852\pi$
$450$		0			0
$451$	9338.31			0.974997
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8182.61	+	6831.24i	0.843092	+	0.703854i
$456$		0			0
$457$		−	15384.9i		−	1.57479i	−0.616451	−	0.787393i	$-0.711430\pi$
$457$		−	15384.9i		−	1.57479i	0.616451	−	0.787393i	$-0.288570\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14590.9			1.47411			0.737057	−	0.675830i	$-0.236215\pi$
$461$	14590.9			1.47411			0.737057	+	0.675830i	$0.236215\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$		−	4110.64i		−	0.407319i	−0.979042	−	0.203659i	$-0.934716\pi$
$467$		−	4110.64i		−	0.407319i	0.979042	−	0.203659i	$-0.0652835\pi$
$468$		0			0
$469$	−15448.3			−1.52098
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	983.461i		−	0.0956016i
$474$		0			0
$475$	2992.45	+	16488.5i	0.289059	+	1.59273i
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		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
$489$		0			0
$490$		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
$495$		0			0
$496$		0			0
$497$		−	5643.38i		−	0.509337i
$498$		0			0
$499$	−3587.97			−0.321883			−0.160941	−	0.986964i	$-0.551453\pi$
$499$	−3587.97			−0.321883			−0.160941	+	0.986964i	$0.551453\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18351.9i			1.62678i	0.581720	+	0.813389i	$0.302380\pi$
$503$			18351.9i			1.62678i	−0.581720	+	0.813389i	$0.697620\pi$
$504$		0			0
$505$	−10086.3	−	8420.57i	−0.888785	−	0.742001i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	992.521			0.0864297			0.0432149	−	0.999066i	$-0.486240\pi$
$509$	992.521			0.0864297			0.0432149	+	0.999066i	$0.486240\pi$
$510$		0			0
$511$	1225.15			0.106061
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12556.4	−	15040.3i	1.07437	−	1.28691i
$516$		0			0
$517$		−	8767.59i		−	0.745838i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11192.3			−0.941155			−0.470577	−	0.882359i	$-0.655954\pi$
$521$	−11192.3			−0.941155			−0.470577	+	0.882359i	$0.655954\pi$
$522$		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
$525$		0			0
$526$		0			0
$527$			5046.48i			0.417131i
$528$		0			0
$529$	11950.9			0.982238
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4819.72i			0.391679i
$534$		0			0
$535$	−677.810	+	811.895i	−0.0547743	+	0.0656099i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$	−30116.2			−2.32848
$552$		0			0
$553$			12422.2i			0.955233i
$554$		0			0
$555$		0			0
$556$		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
$561$		0			0
$562$		0			0
$563$			9751.95i			0.730010i	0.931006	+	0.365005i	$0.118933\pi$
$563$			9751.95i			0.730010i	−0.931006	+	0.365005i	$0.881067\pi$
$564$		0			0
$565$	−3792.15	+	4542.32i	−0.282366	+	0.338224i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5645.90			0.415973			0.207986	−	0.978132i	$-0.433309\pi$
$569$	5645.90			0.415973			0.207986	+	0.978132i	$0.433309\pi$
$570$		0			0
$571$	17188.1			1.25972			0.629860	−	0.776709i	$-0.283112\pi$
$571$	17188.1			1.25972			0.629860	+	0.776709i	$0.283112\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1808.05	−	328.138i	0.131132	−	0.0237987i
$576$		0			0
$577$		−	23294.0i		−	1.68066i	−0.542076	−	0.840330i	$-0.682361\pi$
$577$		−	23294.0i		−	1.68066i	0.542076	−	0.840330i	$-0.317639\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−34571.0			−2.46858
$582$		0			0
$583$			17154.0i			1.21861i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	15182.5i		−	1.06754i	−0.845629	−	0.533771i	$-0.820774\pi$
$587$		−	15182.5i		−	1.06754i	0.845629	−	0.533771i	$-0.179226\pi$
$588$		0			0
$589$	−9227.15			−0.645498
$590$		0			0
$591$		0			0
$592$		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
$597$		0			0
$598$		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
$603$		0			0
$604$		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
$609$		0			0
$610$		0			0
$611$	4525.16			0.299621
$612$		0			0
$613$			16492.5i			1.08667i	0.839518	+	0.543333i	$0.182838\pi$
$613$			16492.5i			1.08667i	−0.839518	+	0.543333i	$0.817162\pi$
$614$		0			0
$615$		0			0
$616$		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	$-0.702469\pi$
$619$	−18297.2			−1.18809			−0.594043	+	0.804434i	$0.702469\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			20478.6i			1.31695i
$624$		0			0
$625$	−14628.5	+	5490.61i	−0.936226	+	0.351399i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14393.8			−0.912433
$630$		0			0
$631$	−25681.4			−1.62022			−0.810112	−	0.586275i	$-0.800594\pi$
$631$	−25681.4			−1.62022			−0.810112	+	0.586275i	$0.800594\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5832.90	+	6986.78i	−0.364522	+	0.436633i
$636$		0			0
$637$		−	15447.9i		−	0.960861i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12994.7			−0.800720			−0.400360	−	0.916358i	$-0.631115\pi$
$641$	−12994.7			−0.800720			−0.400360	+	0.916358i	$0.631115\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$		−	8771.72i		−	0.533002i	−0.963835	−	0.266501i	$-0.914132\pi$
$647$		−	8771.72i		−	0.533002i	0.963835	−	0.266501i	$-0.0858675\pi$
$648$		0			0
$649$	7808.58			0.472286
$650$		0			0
$651$		0			0
$652$		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
$657$		0			0
$658$		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
$663$		0			0
$664$		0			0
$665$	27200.0	−	32580.8i	1.58612	−	1.89989i
$666$		0			0
$667$			3302.40i			0.191708i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1079.06			0.0620814
$672$		0			0
$673$			7649.89i			0.438160i	0.975707	+	0.219080i	$0.0703056\pi$
$673$			7649.89i			0.438160i	−0.975707	+	0.219080i	$0.929694\pi$
$674$		0			0
$675$		0			0
$676$		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
$681$		0			0
$682$		0			0
$683$			8827.56i			0.494549i	0.968945	+	0.247275i	$0.0795350\pi$
$683$			8827.56i			0.494549i	−0.968945	+	0.247275i	$0.920465\pi$
$684$		0			0
$685$	20828.9	−	24949.3i	1.16180	−	1.39163i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8853.60			−0.489543
$690$		0			0
$691$	−12913.1			−0.710908			−0.355454	−	0.934694i	$-0.615674\pi$
$691$	−12913.1			−0.710908			−0.355454	+	0.934694i	$0.615674\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7905.21	−	6599.65i	−0.431456	−	0.360200i
$696$		0			0
$697$		−	10495.7i		−	0.570378i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12297.2			0.662567			0.331283	−	0.943531i	$-0.392518\pi$
$701$	12297.2			0.662567			0.331283	+	0.943531i	$0.392518\pi$
$702$		0			0
$703$		−	26318.2i		−	1.41196i
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$			1011.81i			0.0531451i
$714$		0			0
$715$	15738.0	−	18851.4i	0.823174	−	0.986016i
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		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
$729$		0			0
$730$		0			0
$731$	−1105.35			−0.0559274
$732$		0			0
$733$			15221.6i			0.767017i	0.923537	+	0.383508i	$0.125284\pi$
$733$			15221.6i			0.767017i	−0.923537	+	0.383508i	$0.874716\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			35590.4i			1.77882i
$738$		0			0
$739$	9845.80			0.490099			0.245050	−	0.969511i	$-0.421196\pi$
$739$	9845.80			0.490099			0.245050	+	0.969511i	$0.421196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	35081.5i		−	1.73219i	−0.499881	−	0.866094i	$-0.666623\pi$
$743$		−	35081.5i		−	1.73219i	0.499881	−	0.866094i	$-0.333377\pi$
$744$		0			0
$745$	−2276.61	−	1900.62i	−0.111958	−	0.0934676i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2678.66			0.130676
$750$		0			0
$751$	20007.7			0.972158			0.486079	−	0.873915i	$-0.338427\pi$
$751$	20007.7			0.972158			0.486079	+	0.873915i	$0.338427\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21522.4	−	17968.0i	−1.03746	−	0.866122i
$756$		0			0
$757$		−	33478.7i		−	1.60741i	−0.595031	−	0.803703i	$-0.702860\pi$
$757$		−	33478.7i		−	1.60741i	0.595031	−	0.803703i	$-0.297140\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12728.1			−0.606298			−0.303149	−	0.952943i	$-0.598038\pi$
$761$	−12728.1			−0.606298			−0.303149	+	0.952943i	$0.598038\pi$
$762$		0			0
$763$			38217.8i			1.81334i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		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
$777$		0			0
$778$		0			0
$779$	19190.7			0.882642
$780$		0			0
$781$	−13001.4			−0.595681
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	14986.3			0.673643
$792$		0			0
$793$			556.928i			0.0249396i
$794$		0			0
$795$		0			0
$796$		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
$801$		0			0
$802$		0			0
$803$		−	2822.54i		−	0.124041i
$804$		0			0
$805$	−3572.65	−	2982.62i	−0.156422	−	0.130588i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15278.6			−0.663987			−0.331993	−	0.943282i	$-0.607721\pi$
$809$	−15278.6			−0.663987			−0.331993	+	0.943282i	$0.607721\pi$
$810$		0			0
$811$	27275.2			1.18096			0.590482	−	0.807051i	$-0.298938\pi$
$811$	27275.2			1.18096			0.590482	+	0.807051i	$0.298938\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8485.18	+	10163.7i	−0.364691	+	0.436835i
$816$		0			0
$817$		−	2021.06i		−	0.0865459i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11047.6			0.469627			0.234814	−	0.972040i	$-0.424552\pi$
$821$	11047.6			0.469627			0.234814	+	0.972040i	$0.424552\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$		−	7622.29i		−	0.320499i	−0.987077	−	0.160250i	$-0.948770\pi$
$827$		−	7622.29i		−	0.320499i	0.987077	−	0.160250i	$-0.0512299\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$			33640.3i			1.39924i
$834$		0			0
$835$	−19265.5	+	23076.6i	−0.798454	+	0.956406i
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$	−9126.29	−	7619.07i	−0.371543	−	0.310182i
$846$		0			0
$847$			82816.7i			3.35964i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2885.93			−0.116249
$852$		0			0
$853$			21077.3i			0.846040i	0.906120	+	0.423020i	$0.139030\pi$
$853$			21077.3i			0.846040i	−0.906120	+	0.423020i	$0.860970\pi$
$854$		0			0
$855$		0			0
$856$		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	$-0.329870\pi$
$859$	25649.1			1.01878			0.509392	+	0.860534i	$0.329870\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	31095.6i		−	1.22654i	−0.789872	−	0.613272i	$-0.789853\pi$
$863$		−	31095.6i		−	1.22654i	0.789872	−	0.613272i	$-0.210147\pi$
$864$		0			0
$865$	13905.5	−	16656.4i	0.546593	−	0.654721i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	28618.6			1.11717
$870$		0			0
$871$	−18369.0			−0.714593
$872$		0			0
$873$		0			0
$874$		0			0
$875$	34418.7	+	19529.0i	1.32979	+	0.754514i
$876$		0			0
$877$		−	26626.0i		−	1.02519i	−0.858630	−	0.512597i	$-0.828684\pi$
$877$		−	26626.0i		−	1.02519i	0.858630	−	0.512597i	$-0.171316\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18538.6			0.708945			0.354472	−	0.935066i	$-0.384660\pi$
$881$	18538.6			0.708945			0.354472	+	0.935066i	$0.384660\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$		−	19948.2i		−	0.755126i	−0.925984	−	0.377563i	$-0.876762\pi$
$887$		−	19948.2i		−	0.755126i	0.925984	−	0.377563i	$-0.123238\pi$
$888$		0			0
$889$	23051.3			0.869645
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	18017.8i		−	0.675190i
$894$		0			0
$895$	−13565.8	−	11325.4i	−0.506652	−	0.422978i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15461.5			0.573603
$900$		0			0
$901$	19280.1			0.712891
$902$		0			0
$903$		0			0
$904$		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
$909$		0			0
$910$		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
$915$		0			0
$916$		0			0
$917$		−	37656.4i		−	1.35608i
$918$		0			0
$919$	21804.9			0.782673			0.391336	−	0.920248i	$-0.372013\pi$
$919$	21804.9			0.782673			0.391336	+	0.920248i	$0.372013\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6710.33i		−	0.239299i
$924$		0			0
$925$	24144.6	−	4381.92i	0.858238	−	0.155759i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−49150.1			−1.73580			−0.867902	−	0.496735i	$-0.834532\pi$
$929$	−49150.1			−1.73580			−0.867902	+	0.496735i	$0.834532\pi$
$930$		0			0
$931$	−61509.1			−2.16528
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−34272.1	+	41051.9i	−1.19874	+	1.43587i
$936$		0			0
$937$		−	33165.6i		−	1.15632i	−0.815922	−	0.578161i	$-0.803770\pi$
$937$		−	33165.6i		−	1.15632i	0.815922	−	0.578161i	$-0.196230\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15184.8			0.526048			0.263024	−	0.964789i	$-0.415280\pi$
$941$	15184.8			0.526048			0.263024	+	0.964789i	$0.415280\pi$
$942$		0			0
$943$		−	2104.36i		−	0.0726696i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			44341.2i			1.52154i	0.649023	+	0.760769i	$0.275178\pi$
$947$			44341.2i			1.52154i	−0.649023	+	0.760769i	$0.724822\pi$
$948$		0			0
$949$	1456.78			0.0498303
$950$		0			0
$951$		0			0
$952$		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
$957$		0			0
$958$		0			0
$959$	−82314.5			−2.77171
$960$		0			0
$961$	−25053.8			−0.840987
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		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
$975$		0			0
$976$		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
$981$		0			0
$982$		0			0
$983$		−	31114.6i		−	1.00956i	−0.863247	−	0.504782i	$-0.831573\pi$
$983$		−	31114.6i		−	1.00956i	0.863247	−	0.504782i	$-0.168427\pi$
$984$		0			0
$985$	19586.9	−	23461.6i	0.633594	−	0.758933i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−221.620			−0.00712549
$990$		0			0
$991$	−44477.8			−1.42571			−0.712857	−	0.701309i	$-0.752599\pi$
$991$	−44477.8			−1.42571			−0.712857	+	0.701309i	$0.752599\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−39906.4	−	33315.8i	−1.27147	−	1.06149i
$996$		0			0
$997$			57489.4i			1.82619i	0.407752	+	0.913093i	$0.366313\pi$
$997$			57489.4i			1.82619i	−0.407752	+	0.913093i	$0.633687\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.f.k.289.2		8
3.2	odd	2		160.4.c.d.129.4	yes	8
4.3	odd	2	inner	1440.4.f.k.289.1		8
5.4	even	2	inner	1440.4.f.k.289.3		8
12.11	even	2		160.4.c.d.129.6	yes	8
15.2	even	4		800.4.a.z.1.2		4
15.8	even	4		800.4.a.y.1.3		4
15.14	odd	2		160.4.c.d.129.5	yes	8
20.19	odd	2	inner	1440.4.f.k.289.4		8
24.5	odd	2		320.4.c.j.129.5		8
24.11	even	2		320.4.c.j.129.3		8
60.23	odd	4		800.4.a.y.1.2		4
60.47	odd	4		800.4.a.z.1.3		4
60.59	even	2		160.4.c.d.129.3	✓	8
120.29	odd	2		320.4.c.j.129.4		8
120.53	even	4		1600.4.a.cv.1.2		4
120.59	even	2		320.4.c.j.129.6		8
120.77	even	4		1600.4.a.cu.1.3		4
120.83	odd	4		1600.4.a.cv.1.3		4
120.107	odd	4		1600.4.a.cu.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.d.129.3	✓	8	60.59	even	2
160.4.c.d.129.4	yes	8	3.2	odd	2
160.4.c.d.129.5	yes	8	15.14	odd	2
160.4.c.d.129.6	yes	8	12.11	even	2
320.4.c.j.129.3		8	24.11	even	2
320.4.c.j.129.4		8	120.29	odd	2
320.4.c.j.129.5		8	24.5	odd	2
320.4.c.j.129.6		8	120.59	even	2
800.4.a.y.1.2		4	60.23	odd	4
800.4.a.y.1.3		4	15.8	even	4
800.4.a.z.1.2		4	15.2	even	4
800.4.a.z.1.3		4	60.47	odd	4
1440.4.f.k.289.1		8	4.3	odd	2	inner
1440.4.f.k.289.2		8	1.1	even	1	trivial
1440.4.f.k.289.3		8	5.4	even	2	inner
1440.4.f.k.289.4		8	20.19	odd	2	inner
1600.4.a.cu.1.2		4	120.107	odd	4
1600.4.a.cu.1.3		4	120.77	even	4
1600.4.a.cv.1.2		4	120.53	even	4
1600.4.a.cv.1.3		4	120.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +65.2358 q^{11} +33.6697i q^{13} -73.3212i q^{17} +134.063 q^{19} -14.7007i q^{23} +(22.3212 + 122.991i) q^{25} -224.642 q^{29} -68.8271 q^{31} +(202.890 - 243.026i) q^{35} -196.312i q^{37} +143.147 q^{41} -15.0755i q^{43} -134.399i q^{47} -458.808 q^{49} +262.955i q^{53} +(-559.891 - 467.424i) q^{55} +119.698 q^{59} +16.5409 q^{61} +(241.248 - 288.973i) q^{65} +545.565i q^{67} -199.299 q^{71} -43.2667i q^{73} +1847.23i q^{77} +438.694 q^{79} +1220.89i q^{83} +(-525.358 + 629.285i) q^{85} +723.212 q^{89} -953.398 q^{91} +(-1150.60 - 960.581i) q^{95} -1136.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{5} - 408 q^{25} - 624 q^{29} + 192 q^{41} - 2424 q^{49} + 2112 q^{61} - 2176 q^{65} - 5376 q^{85} - 80 q^{89}+O(q^{100}) \) 8 * q - 32 * q^5 - 408 * q^25 - 624 * q^29 + 192 * q^41 - 2424 * q^49 + 2112 * q^61 - 2176 * q^65 - 5376 * q^85 - 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more