LMFDB - Embedded newform 1280.3.h.n.1279.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, 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("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 $ x^16 + 19x^14 + 301x^12 + 1102x^10 + 3238x^8 + 1102x^6 + 301x^4 + 19*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{34} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.13
Root			$-0.940543 - 1.62907i$ of defining polynomial
Character	$\chi$	$=$	1280.1279
Dual form			1280.3.h.n.1279.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.13399 q^{3} +(-4.14474 - 2.79662i) q^{5} +6.19337 q^{7} +17.3578 q^{9} +O(q^{10})$	$q+5.13399 q^{3} +(-4.14474 - 2.79662i) q^{5} +6.19337 q^{7} +17.3578 q^{9} +20.0431i q^{11} +15.8612i q^{13} +(-21.2791 - 14.3578i) q^{15} +6.98882i q^{17} +10.3923i q^{19} +31.7967 q^{21} -22.3871 q^{23} +(9.35782 + 23.1826i) q^{25} +42.9089 q^{27} +4.20563 q^{29} -20.7156i q^{31} +102.901i q^{33} +(-25.6699 - 17.3205i) q^{35} +35.4786i q^{37} +81.4313i q^{39} +37.0735 q^{41} -23.8329 q^{43} +(-71.9437 - 48.5432i) q^{45} -48.7515 q^{47} -10.6422 q^{49} +35.8805i q^{51} -77.4691i q^{53} +(56.0529 - 83.0735i) q^{55} +53.3540i q^{57} +0.497984i q^{59} +60.7490 q^{61} +107.503 q^{63} +(44.3578 - 65.7407i) q^{65} +82.5209 q^{67} -114.935 q^{69} -28.7156i q^{71} +10.1706i q^{73} +(48.0429 + 119.019i) q^{75} +124.134i q^{77} +87.5782i q^{79} +64.0735 q^{81} +103.057 q^{83} +(19.5451 - 28.9669i) q^{85} +21.5917 q^{87} +49.2844 q^{89} +98.2344i q^{91} -106.354i q^{93} +(29.0633 - 43.0735i) q^{95} -84.4911i q^{97} +347.904i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 96 q^{9}+O(q^{10}) $ 16 * q + 96 * q^9	$ 16 q + 96 q^{9} - 32 q^{25} + 48 q^{41} - 352 q^{49} + 528 q^{65} + 480 q^{81} + 1152 q^{89}+O(q^{100}) $ 16 * q + 96 * q^9 - 32 * q^25 + 48 * q^41 - 352 * q^49 + 528 * q^65 + 480 * q^81 + 1152 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	5.13399			1.71133			0.855664	−	0.517531i	$-0.173149\pi$
$3$	5.13399			1.71133			0.855664	+	0.517531i	$0.173149\pi$
$4$		0			0
$5$	−4.14474	−	2.79662i	−0.828949	−	0.559324i
$5$	−4.14474	−	2.79662i	−0.828949	−	0.559324i
$6$		0			0
$7$	6.19337			0.884767			0.442383	−	0.896826i	$-0.354133\pi$
$7$	6.19337			0.884767			0.442383	+	0.896826i	$0.354133\pi$
$8$		0			0
$9$	17.3578			1.92865
$10$		0			0
$11$			20.0431i			1.82210i	0.412298	+	0.911049i	$0.364726\pi$
$11$			20.0431i			1.82210i	−0.412298	+	0.911049i	$0.635274\pi$
$12$		0			0
$13$			15.8612i			1.22009i	0.792365	+	0.610047i	$0.208849\pi$
$13$			15.8612i			1.22009i	−0.792365	+	0.610047i	$0.791151\pi$
$14$		0			0
$15$	−21.2791	−	14.3578i	−1.41860	−	0.957188i
$16$		0			0
$17$			6.98882i			0.411107i	0.978646	+	0.205554i	$0.0658995\pi$
$17$			6.98882i			0.411107i	−0.978646	+	0.205554i	$0.934100\pi$
$18$		0			0
$19$			10.3923i			0.546963i	0.961877	+	0.273482i	$0.0881753\pi$
$19$			10.3923i			0.546963i	−0.961877	+	0.273482i	$0.911825\pi$
$20$		0			0
$21$	31.7967			1.51413
$22$		0			0
$23$	−22.3871			−0.973353			−0.486676	−	0.873582i	$-0.661791\pi$
$23$	−22.3871			−0.973353			−0.486676	+	0.873582i	$0.661791\pi$
$24$		0			0
$25$	9.35782	+	23.1826i	0.374313	+	0.927303i
$26$		0			0
$27$	42.9089			1.58922
$28$		0			0
$29$	4.20563			0.145022			0.0725109	−	0.997368i	$-0.476899\pi$
$29$	4.20563			0.145022			0.0725109	+	0.997368i	$0.476899\pi$
$30$		0			0
$31$		−	20.7156i		−	0.668246i	−0.942529	−	0.334123i	$-0.891560\pi$
$31$		−	20.7156i		−	0.668246i	0.942529	−	0.334123i	$-0.108440\pi$
$32$		0			0
$33$			102.901i			3.11821i
$34$		0			0
$35$	−25.6699	−	17.3205i	−0.733427	−	0.494872i
$36$		0			0
$37$			35.4786i			0.958882i	0.877574	+	0.479441i	$0.159161\pi$
$37$			35.4786i			0.958882i	−0.877574	+	0.479441i	$0.840839\pi$
$38$		0			0
$39$			81.4313i			2.08798i
$40$		0			0
$41$	37.0735			0.904230			0.452115	−	0.891960i	$-0.350670\pi$
$41$	37.0735			0.904230			0.452115	+	0.891960i	$0.350670\pi$
$42$		0			0
$43$	−23.8329			−0.554254			−0.277127	−	0.960833i	$-0.589382\pi$
$43$	−23.8329			−0.554254			−0.277127	+	0.960833i	$0.589382\pi$
$44$		0			0
$45$	−71.9437	−	48.5432i	−1.59875	−	1.07874i
$46$		0			0
$47$	−48.7515			−1.03727			−0.518633	−	0.854997i	$-0.673559\pi$
$47$	−48.7515			−1.03727			−0.518633	+	0.854997i	$0.673559\pi$
$48$		0			0
$49$	−10.6422			−0.217187
$50$		0			0
$51$			35.8805i			0.703540i
$52$		0			0
$53$		−	77.4691i		−	1.46168i	−0.682549	−	0.730840i	$-0.739128\pi$
$53$		−	77.4691i		−	1.46168i	0.682549	−	0.730840i	$-0.260872\pi$
$54$		0			0
$55$	56.0529	−	83.0735i	1.01914	−	1.51043i
$56$		0			0
$57$			53.3540i			0.936034i
$58$		0			0
$59$			0.497984i			0.00844041i	0.999991	+	0.00422020i	$0.00134334\pi$
$59$			0.497984i			0.00844041i	−0.999991	+	0.00422020i	$0.998657\pi$
$60$		0			0
$61$	60.7490			0.995885			0.497943	−	0.867210i	$-0.334089\pi$
$61$	60.7490			0.995885			0.497943	+	0.867210i	$0.334089\pi$
$62$		0			0
$63$	107.503			1.70640
$64$		0			0
$65$	44.3578	−	65.7407i	0.682428	−	1.01140i
$66$		0			0
$67$	82.5209			1.23165			0.615827	−	0.787881i	$-0.288822\pi$
$67$	82.5209			1.23165			0.615827	+	0.787881i	$0.288822\pi$
$68$		0			0
$69$	−114.935			−1.66573
$70$		0			0
$71$		−	28.7156i		−	0.404446i	−0.979340	−	0.202223i	$-0.935183\pi$
$71$		−	28.7156i		−	0.404446i	0.979340	−	0.202223i	$-0.0648165\pi$
$72$		0			0
$73$			10.1706i			0.139324i	0.997571	+	0.0696620i	$0.0221921\pi$
$73$			10.1706i			0.139324i	−0.997571	+	0.0696620i	$0.977808\pi$
$74$		0			0
$75$	48.0429	+	119.019i	0.640572	+	1.58692i
$76$		0			0
$77$			124.134i			1.61213i
$78$		0			0
$79$			87.5782i			1.10858i	0.832322	+	0.554292i	$0.187011\pi$
$79$			87.5782i			1.10858i	−0.832322	+	0.554292i	$0.812989\pi$
$80$		0			0
$81$	64.0735			0.791030
$82$		0			0
$83$	103.057			1.24165			0.620824	−	0.783950i	$-0.286798\pi$
$83$	103.057			1.24165			0.620824	+	0.783950i	$0.286798\pi$
$84$		0			0
$85$	19.5451	−	28.9669i	0.229942	−	0.340787i
$86$		0			0
$87$	21.5917			0.248180
$88$		0			0
$89$	49.2844			0.553757			0.276878	−	0.960905i	$-0.410700\pi$
$89$	49.2844			0.553757			0.276878	+	0.960905i	$0.410700\pi$
$90$		0			0
$91$			98.2344i			1.07950i
$92$		0			0
$93$		−	106.354i		−	1.14359i
$94$		0			0
$95$	29.0633	−	43.0735i	0.305930	−	0.453405i
$96$		0			0
$97$		−	84.4911i		−	0.871042i	−0.900179	−	0.435521i	$-0.856564\pi$
$97$		−	84.4911i		−	0.871042i	0.900179	−	0.435521i	$-0.143436\pi$
$98$		0			0
$99$			347.904i			3.51418i
$100$		0			0
$101$	92.7892			0.918705			0.459353	−	0.888254i	$-0.348081\pi$
$101$	92.7892			0.918705			0.459353	+	0.888254i	$0.348081\pi$
$102$		0			0
$103$	−42.7284			−0.414839			−0.207419	−	0.978252i	$-0.566506\pi$
$103$	−42.7284			−0.414839			−0.207419	+	0.978252i	$0.566506\pi$
$104$		0			0
$105$	−131.789	−	88.9233i	−1.25513	−	0.846888i
$106$		0			0
$107$	29.3440			0.274243			0.137121	−	0.990554i	$-0.456215\pi$
$107$	29.3440			0.274243			0.137121	+	0.990554i	$0.456215\pi$
$108$		0			0
$109$	163.433			1.49938			0.749691	−	0.661789i	$-0.230202\pi$
$109$	163.433			1.49938			0.749691	+	0.661789i	$0.230202\pi$
$110$		0			0
$111$			182.147i			1.64096i
$112$		0			0
$113$		−	120.686i		−	1.06801i	−0.845480	−	0.534007i	$-0.820686\pi$
$113$		−	120.686i		−	1.06801i	0.845480	−	0.534007i	$-0.179314\pi$
$114$		0			0
$115$	92.7888	+	62.6083i	0.806860	+	0.544420i
$116$		0			0
$117$			275.316i			2.35313i
$118$		0			0
$119$			43.2844i			0.363734i
$120$		0			0
$121$	−280.725			−2.32004
$122$		0			0
$123$	190.335			1.54744
$124$		0			0
$125$	26.0471	−	122.256i	0.208377	−	0.978049i
$126$		0			0
$127$	227.564			1.79184			0.895920	−	0.444215i	$-0.146517\pi$
$127$	227.564			1.79184			0.895920	+	0.444215i	$0.146517\pi$
$128$		0			0
$129$	−122.358			−0.948510
$130$		0			0
$131$		−	34.1430i		−	0.260634i	−0.991472	−	0.130317i	$-0.958401\pi$
$131$		−	34.1430i		−	0.260634i	0.991472	−	0.130317i	$-0.0415995\pi$
$132$		0			0
$133$			64.3634i			0.483935i
$134$		0			0
$135$	−177.847	−	120.000i	−1.31738	−	0.888889i
$136$		0			0
$137$		−	216.598i		−	1.58100i	−0.612459	−	0.790502i	$-0.709820\pi$
$137$		−	216.598i		−	1.58100i	0.612459	−	0.790502i	$-0.290180\pi$
$138$		0			0
$139$		−	113.076i		−	0.813495i	−0.913541	−	0.406748i	$-0.866663\pi$
$139$		−	113.076i		−	0.813495i	0.913541	−	0.406748i	$-0.133337\pi$
$140$		0			0
$141$	−250.290			−1.77510
$142$		0			0
$143$	−317.908			−2.22313
$144$		0			0
$145$	−17.4313	−	11.7616i	−0.120216	−	0.0811142i
$146$		0			0
$147$	−54.6368			−0.371679
$148$		0			0
$149$	−171.844			−1.15331			−0.576657	−	0.816986i	$-0.695643\pi$
$149$	−171.844			−1.15331			−0.576657	+	0.816986i	$0.695643\pi$
$150$		0			0
$151$			30.5687i			0.202442i	0.994864	+	0.101221i	$0.0322749\pi$
$151$			30.5687i			0.202442i	−0.994864	+	0.101221i	$0.967725\pi$
$152$		0			0
$153$			121.311i			0.792881i
$154$		0			0
$155$	−57.9338	+	85.8610i	−0.373766	+	0.553942i
$156$		0			0
$157$			265.212i			1.68925i	0.535357	+	0.844626i	$0.320177\pi$
$157$			265.212i			1.68925i	−0.535357	+	0.844626i	$0.679823\pi$
$158$		0			0
$159$		−	397.725i		−	2.50142i
$160$		0			0
$161$	−138.652			−0.861190
$162$		0			0
$163$	−69.6617			−0.427372			−0.213686	−	0.976902i	$-0.568547\pi$
$163$	−69.6617			−0.427372			−0.213686	+	0.976902i	$0.568547\pi$
$164$		0			0
$165$	287.775	−	426.498i	1.74409	−	2.58484i
$166$		0			0
$167$	−101.140			−0.605627			−0.302814	−	0.953050i	$-0.597926\pi$
$167$	−101.140			−0.605627			−0.302814	+	0.953050i	$0.597926\pi$
$168$		0			0
$169$	−82.5782			−0.488628
$170$		0			0
$171$			180.388i			1.05490i
$172$		0			0
$173$		−	65.5284i		−	0.378777i	−0.981902	−	0.189388i	$-0.939349\pi$
$173$		−	65.5284i		−	0.378777i	0.981902	−	0.189388i	$-0.0606505\pi$
$174$		0			0
$175$	57.9564	+	143.578i	0.331179	+	0.820447i
$176$		0			0
$177$			2.55664i			0.0144443i
$178$		0			0
$179$		−	47.2688i		−	0.264072i	−0.991245	−	0.132036i	$-0.957849\pi$
$179$		−	47.2688i		−	0.264072i	0.991245	−	0.132036i	$-0.0421514\pi$
$180$		0			0
$181$	−255.613			−1.41223			−0.706113	−	0.708099i	$-0.749553\pi$
$181$	−255.613			−1.41223			−0.706113	+	0.708099i	$0.749553\pi$
$182$		0			0
$183$	311.885			1.70429
$184$		0			0
$185$	99.2204	−	147.050i	0.536326	−	0.794865i
$186$		0			0
$187$	−140.078			−0.749078
$188$		0			0
$189$	265.751			1.40609
$190$		0			0
$191$			67.7062i			0.354483i	0.984167	+	0.177241i	$0.0567173\pi$
$191$			67.7062i			0.354483i	−0.984167	+	0.177241i	$0.943283\pi$
$192$		0			0
$193$		−	327.113i		−	1.69488i	−0.530888	−	0.847442i	$-0.678142\pi$
$193$		−	327.113i		−	1.69488i	0.530888	−	0.847442i	$-0.321858\pi$
$194$		0			0
$195$	227.732	−	337.512i	1.16786	−	1.73083i
$196$		0			0
$197$			120.542i			0.611890i	0.952049	+	0.305945i	$0.0989723\pi$
$197$			120.542i			0.611890i	−0.952049	+	0.305945i	$0.901028\pi$
$198$		0			0
$199$		−	175.578i		−	0.882302i	−0.897433	−	0.441151i	$-0.854570\pi$
$199$		−	175.578i		−	0.882302i	0.897433	−	0.441151i	$-0.145430\pi$
$200$		0			0
$201$	423.661			2.10777
$202$		0			0
$203$	26.0470			0.128310
$204$		0			0
$205$	−153.660	−	103.680i	−0.749561	−	0.505758i
$206$		0			0
$207$	−388.591			−1.87725
$208$		0			0
$209$	−208.294			−0.996621
$210$		0			0
$211$			237.053i			1.12347i	0.827316	+	0.561737i	$0.189867\pi$
$211$			237.053i			1.12347i	−0.827316	+	0.561737i	$0.810133\pi$
$212$		0			0
$213$		−	147.426i		−	0.692139i
$214$		0			0
$215$	98.7813	+	66.6516i	0.459448	+	0.310008i
$216$		0			0
$217$		−	128.300i		−	0.591242i
$218$		0			0
$219$			52.2160i			0.238429i
$220$		0			0
$221$	−110.851			−0.501589
$222$		0			0
$223$	−13.8074			−0.0619165			−0.0309582	−	0.999521i	$-0.509856\pi$
$223$	−13.8074			−0.0619165			−0.0309582	+	0.999521i	$0.509856\pi$
$224$		0			0
$225$	162.431	+	402.399i	0.721917	+	1.78844i
$226$		0			0
$227$	−2.21412			−0.00975382			−0.00487691	−	0.999988i	$-0.501552\pi$
$227$	−2.21412			−0.00975382			−0.00487691	+	0.999988i	$0.501552\pi$
$228$		0			0
$229$	241.756			1.05571			0.527853	−	0.849336i	$-0.322997\pi$
$229$	241.756			1.05571			0.527853	+	0.849336i	$0.322997\pi$
$230$		0			0
$231$			637.303i			2.75889i
$232$		0			0
$233$		−	30.5119i		−	0.130953i	−0.997854	−	0.0654763i	$-0.979143\pi$
$233$		−	30.5119i		−	0.130953i	0.997854	−	0.0654763i	$-0.0208567\pi$
$234$		0			0
$235$	202.062	+	136.339i	0.859840	+	0.580168i
$236$		0			0
$237$			449.625i			1.89715i
$238$		0			0
$239$		−	426.441i		−	1.78427i	−0.451768	−	0.892135i	$-0.649207\pi$
$239$		−	426.441i		−	1.78427i	0.451768	−	0.892135i	$-0.350793\pi$
$240$		0			0
$241$	147.936			0.613842			0.306921	−	0.951735i	$-0.400701\pi$
$241$	147.936			0.613842			0.306921	+	0.951735i	$0.400701\pi$
$242$		0			0
$243$	−57.2280			−0.235506
$244$		0			0
$245$	44.1091	+	29.7622i	0.180037	+	0.121478i
$246$		0			0
$247$	−164.835			−0.667347
$248$		0			0
$249$	529.092			2.12487
$250$		0			0
$251$			97.7364i			0.389388i	0.980864	+	0.194694i	$0.0623714\pi$
$251$			97.7364i			0.389388i	−0.980864	+	0.194694i	$0.937629\pi$
$252$		0			0
$253$		−	448.707i		−	1.77354i
$254$		0			0
$255$	100.344	−	148.716i	0.393507	−	0.583199i
$256$		0			0
$257$			354.443i			1.37915i	0.724212	+	0.689577i	$0.242204\pi$
$257$			354.443i			1.37915i	−0.724212	+	0.689577i	$0.757796\pi$
$258$		0			0
$259$			219.732i			0.848387i
$260$		0			0
$261$	73.0006			0.279696
$262$		0			0
$263$	−44.6039			−0.169597			−0.0847984	−	0.996398i	$-0.527025\pi$
$263$	−44.6039			−0.169597			−0.0847984	+	0.996398i	$0.527025\pi$
$264$		0			0
$265$	−216.652	+	321.089i	−0.817553	+	1.21166i
$266$		0			0
$267$	253.025			0.947660
$268$		0			0
$269$	−314.879			−1.17055			−0.585277	−	0.810834i	$-0.699014\pi$
$269$	−314.879			−1.17055			−0.585277	+	0.810834i	$0.699014\pi$
$270$		0			0
$271$			449.009i			1.65686i	0.560092	+	0.828431i	$0.310766\pi$
$271$			449.009i			1.65686i	−0.560092	+	0.828431i	$0.689234\pi$
$272$		0			0
$273$			504.334i			1.84738i
$274$		0			0
$275$	−464.650	+	187.559i	−1.68964	+	0.682034i
$276$		0			0
$277$			330.330i			1.19253i	0.802789	+	0.596264i	$0.203349\pi$
$277$			330.330i			1.19253i	−0.802789	+	0.596264i	$0.796651\pi$
$278$		0			0
$279$		−	359.578i		−	1.28881i
$280$		0			0
$281$	−307.936			−1.09586			−0.547929	−	0.836525i	$-0.684584\pi$
$281$	−307.936			−1.09586			−0.547929	+	0.836525i	$0.684584\pi$
$282$		0			0
$283$	182.609			0.645263			0.322631	−	0.946525i	$-0.395433\pi$
$283$	182.609			0.645263			0.322631	+	0.946525i	$0.395433\pi$
$284$		0			0
$285$	149.211	−	221.139i	0.523547	−	0.775925i
$286$		0			0
$287$	229.610			0.800033
$288$		0			0
$289$	240.156			0.830991
$290$		0			0
$291$		−	433.776i		−	1.49064i
$292$		0			0
$293$		−	259.373i		−	0.885231i	−0.896711	−	0.442616i	$-0.854051\pi$
$293$		−	259.373i		−	0.885231i	0.896711	−	0.442616i	$-0.145949\pi$
$294$		0			0
$295$	1.39267	−	2.06402i	0.00472092	−	0.00699667i
$296$		0			0
$297$			860.027i			2.89571i
$298$		0			0
$299$		−	355.087i		−	1.18758i
$300$		0			0
$301$	−147.606			−0.490385
$302$		0			0
$303$	476.379			1.57221
$304$		0			0
$305$	−251.789	−	169.892i	−0.825538	−	0.557023i
$306$		0			0
$307$	209.411			0.682119			0.341060	−	0.940042i	$-0.389214\pi$
$307$	209.411			0.682119			0.341060	+	0.940042i	$0.389214\pi$
$308$		0			0
$309$	−219.367			−0.709926
$310$		0			0
$311$			149.137i			0.479542i	0.970830	+	0.239771i	$0.0770723\pi$
$311$			149.137i			0.479542i	−0.970830	+	0.239771i	$0.922928\pi$
$312$		0			0
$313$		−	296.601i		−	0.947606i	−0.880631	−	0.473803i	$-0.842881\pi$
$313$		−	296.601i		−	0.947606i	0.880631	−	0.473803i	$-0.157119\pi$
$314$		0			0
$315$	−445.574	−	300.646i	−1.41452	−	0.954432i
$316$		0			0
$317$		−	23.0450i		−	0.0726971i	−0.999339	−	0.0363486i	$-0.988427\pi$
$317$		−	23.0450i		−	0.0726971i	0.999339	−	0.0363486i	$-0.0115727\pi$
$318$		0			0
$319$			84.2938i			0.264244i
$320$		0			0
$321$	150.652			0.469320
$322$		0			0
$323$	−72.6300			−0.224861
$324$		0			0
$325$	−367.704	+	148.426i	−1.13140	+	0.456696i
$326$		0			0
$327$	839.060			2.56593
$328$		0			0
$329$	−301.936			−0.917739
$330$		0			0
$331$		−	419.665i		−	1.26787i	−0.773386	−	0.633935i	$-0.781439\pi$
$331$		−	419.665i		−	1.26787i	0.773386	−	0.633935i	$-0.218561\pi$
$332$		0			0
$333$			615.832i			1.84934i
$334$		0			0
$335$	−342.028	−	230.780i	−1.02098	−	0.688894i
$336$		0			0
$337$			52.0477i			0.154444i	0.997014	+	0.0772221i	$0.0246050\pi$
$337$			52.0477i			0.154444i	−0.997014	+	0.0772221i	$0.975395\pi$
$338$		0			0
$339$		−	619.598i		−	1.82772i
$340$		0			0
$341$	415.205			1.21761
$342$		0			0
$343$	−369.386			−1.07693
$344$		0			0
$345$	476.377	+	321.430i	1.38080	+	0.931681i
$346$		0			0
$347$	−329.706			−0.950163			−0.475081	−	0.879942i	$-0.657581\pi$
$347$	−329.706			−0.950163			−0.475081	+	0.879942i	$0.657581\pi$
$348$		0			0
$349$	517.423			1.48259			0.741294	−	0.671180i	$-0.234212\pi$
$349$	517.423			1.48259			0.741294	+	0.671180i	$0.234212\pi$
$350$		0			0
$351$			680.588i			1.93900i
$352$		0			0
$353$			44.4337i			0.125874i	0.998017	+	0.0629372i	$0.0200468\pi$
$353$			44.4337i			0.125874i	−0.998017	+	0.0629372i	$0.979953\pi$
$354$		0			0
$355$	−80.3068	+	119.019i	−0.226216	+	0.335265i
$356$		0			0
$357$			222.221i			0.622469i
$358$		0			0
$359$		−	53.1375i		−	0.148015i	−0.997258	−	0.0740076i	$-0.976421\pi$
$359$		−	53.1375i		−	0.148015i	0.997258	−	0.0740076i	$-0.0235789\pi$
$360$		0			0
$361$	253.000			0.700831
$362$		0			0
$363$	−1441.24			−3.97035
$364$		0			0
$365$	28.4434	−	42.1547i	0.0779273	−	0.115492i
$366$		0			0
$367$	374.954			1.02167			0.510837	−	0.859678i	$-0.329336\pi$
$367$	374.954			1.02167			0.510837	+	0.859678i	$0.329336\pi$
$368$		0			0
$369$	643.514			1.74394
$370$		0			0
$371$		−	479.794i		−	1.29325i
$372$		0			0
$373$		−	24.6208i		−	0.0660076i	−0.999455	−	0.0330038i	$-0.989493\pi$
$373$		−	24.6208i		−	0.0660076i	0.999455	−	0.0330038i	$-0.0105073\pi$
$374$		0			0
$375$	133.725	−	627.661i	0.356601	−	1.67376i
$376$		0			0
$377$			66.7064i			0.176940i
$378$		0			0
$379$			170.472i			0.449793i	0.974383	+	0.224897i	$0.0722044\pi$
$379$			170.472i			0.449793i	−0.974383	+	0.224897i	$0.927796\pi$
$380$		0			0
$381$	1168.31			3.06643
$382$		0			0
$383$	136.084			0.355310			0.177655	−	0.984093i	$-0.443149\pi$
$383$	136.084			0.355310			0.177655	+	0.984093i	$0.443149\pi$
$384$		0			0
$385$	347.156	−	514.505i	0.901705	−	1.33638i
$386$		0			0
$387$	−413.687			−1.06896
$388$		0			0
$389$	−60.5055			−0.155541			−0.0777705	−	0.996971i	$-0.524780\pi$
$389$	−60.5055			−0.155541			−0.0777705	+	0.996971i	$0.524780\pi$
$390$		0			0
$391$		−	156.460i		−	0.400152i
$392$		0			0
$393$		−	175.290i		−	0.446030i
$394$		0			0
$395$	244.923	−	362.989i	0.620058	−	0.918960i
$396$		0			0
$397$		−	124.216i		−	0.312888i	−0.987687	−	0.156444i	$-0.949997\pi$
$397$		−	124.216i		−	0.312888i	0.987687	−	0.156444i	$-0.0500030\pi$
$398$		0			0
$399$			330.441i			0.828172i
$400$		0			0
$401$	46.7156			0.116498			0.0582489	−	0.998302i	$-0.481448\pi$
$401$	46.7156			0.116498			0.0582489	+	0.998302i	$0.481448\pi$
$402$		0			0
$403$	328.575			0.815323
$404$		0			0
$405$	−265.568	−	179.189i	−0.655724	−	0.442442i
$406$		0			0
$407$	−711.101			−1.74718
$408$		0			0
$409$	−582.377			−1.42390			−0.711952	−	0.702228i	$-0.752189\pi$
$409$	−582.377			−1.42390			−0.711952	+	0.702228i	$0.752189\pi$
$410$		0			0
$411$		−	1112.01i		−	2.70562i
$412$		0			0
$413$			3.08420i			0.00746779i
$414$		0			0
$415$	−427.144	−	288.211i	−1.02926	−	0.694484i
$416$		0			0
$417$		−	580.530i		−	1.39216i
$418$		0			0
$419$		−	479.296i		−	1.14391i	−0.820287	−	0.571953i	$-0.806186\pi$
$419$		−	479.296i		−	1.14391i	0.820287	−	0.571953i	$-0.193814\pi$
$420$		0			0
$421$	−116.905			−0.277685			−0.138842	−	0.990314i	$-0.544338\pi$
$421$	−116.905			−0.277685			−0.138842	+	0.990314i	$0.544338\pi$
$422$		0			0
$423$	−846.220			−2.00052
$424$		0			0
$425$	−162.019	+	65.4001i	−0.381221	+	0.153883i
$426$		0			0
$427$	376.241			0.881126
$428$		0			0
$429$	−1632.13			−3.80451
$430$		0			0
$431$		−	204.460i		−	0.474384i	−0.971463	−	0.237192i	$-0.923773\pi$
$431$		−	204.460i		−	0.474384i	0.971463	−	0.237192i	$-0.0762271\pi$
$432$		0			0
$433$		−	403.364i		−	0.931558i	−0.884901	−	0.465779i	$-0.845774\pi$
$433$		−	403.364i		−	0.931558i	0.884901	−	0.465779i	$-0.154226\pi$
$434$		0			0
$435$	−89.4919	−	60.3837i	−0.205728	−	0.138813i
$436$		0			0
$437$		−	232.654i		−	0.532388i
$438$		0			0
$439$		−	114.275i		−	0.260307i	−0.991494	−	0.130154i	$-0.958453\pi$
$439$		−	114.275i		−	0.260307i	0.991494	−	0.130154i	$-0.0415471\pi$
$440$		0			0
$441$	−184.725			−0.418878
$442$		0			0
$443$	750.268			1.69361			0.846803	−	0.531906i	$-0.178524\pi$
$443$	750.268			1.69361			0.846803	+	0.531906i	$0.178524\pi$
$444$		0			0
$445$	−204.271	−	137.830i	−0.459036	−	0.309730i
$446$		0			0
$447$	−882.244			−1.97370
$448$		0			0
$449$	−554.377			−1.23469			−0.617346	−	0.786692i	$-0.711792\pi$
$449$	−554.377			−1.23469			−0.617346	+	0.786692i	$0.711792\pi$
$450$		0			0
$451$			743.066i			1.64760i
$452$		0			0
$453$			156.939i			0.346445i
$454$		0			0
$455$	274.724	−	407.156i	0.603790	−	0.894849i
$456$		0			0
$457$		−	870.823i		−	1.90552i	−0.303724	−	0.952760i	$-0.598230\pi$
$457$		−	870.823i		−	1.90552i	0.303724	−	0.952760i	$-0.401770\pi$
$458$		0			0
$459$			299.883i			0.653340i
$460$		0			0
$461$	37.3636			0.0810490			0.0405245	−	0.999179i	$-0.487097\pi$
$461$	37.3636			0.0810490			0.0405245	+	0.999179i	$0.487097\pi$
$462$		0			0
$463$	228.189			0.492849			0.246424	−	0.969162i	$-0.420744\pi$
$463$	228.189			0.492849			0.246424	+	0.969162i	$0.420744\pi$
$464$		0			0
$465$	−297.431	+	440.809i	−0.639637	+	0.947977i
$466$		0			0
$467$	−414.441			−0.887455			−0.443727	−	0.896162i	$-0.646344\pi$
$467$	−414.441			−0.887455			−0.443727	+	0.896162i	$0.646344\pi$
$468$		0			0
$469$	511.082			1.08973
$470$		0			0
$471$			1361.60i			2.89086i
$472$		0			0
$473$		−	477.685i		−	1.00990i
$474$		0			0
$475$	−240.920	+	97.2493i	−0.507201	+	0.204735i
$476$		0			0
$477$		−	1344.69i		−	2.81906i
$478$		0			0
$479$			871.744i			1.81992i	0.414691	+	0.909962i	$0.363890\pi$
$479$			871.744i			1.81992i	−0.414691	+	0.909962i	$0.636110\pi$
$480$		0			0
$481$	−562.735			−1.16993
$482$		0			0
$483$	−711.836			−1.47378
$484$		0			0
$485$	−236.290	+	350.194i	−0.487195	+	0.722049i
$486$		0			0
$487$	−316.431			−0.649756			−0.324878	−	0.945756i	$-0.605323\pi$
$487$	−316.431			−0.649756			−0.324878	+	0.945756i	$0.605323\pi$
$488$		0			0
$489$	−357.642			−0.731375
$490$		0			0
$491$			372.142i			0.757927i	0.925412	+	0.378963i	$0.123719\pi$
$491$			372.142i			0.757927i	−0.925412	+	0.378963i	$0.876281\pi$
$492$		0			0
$493$			29.3924i			0.0596195i
$494$		0			0
$495$	972.956	−	1441.97i	1.96557	−	2.91308i
$496$		0			0
$497$		−	177.847i		−	0.357840i
$498$		0			0
$499$		−	198.184i		−	0.397163i	−0.980084	−	0.198582i	$-0.936367\pi$
$499$		−	198.184i		−	0.397163i	0.980084	−	0.198582i	$-0.0636335\pi$
$500$		0			0
$501$	−519.250			−1.03643
$502$		0			0
$503$	638.202			1.26879			0.634395	−	0.773009i	$-0.281249\pi$
$503$	638.202			1.26879			0.634395	+	0.773009i	$0.281249\pi$
$504$		0			0
$505$	−384.588	−	259.496i	−0.761560	−	0.513854i
$506$		0			0
$507$	−423.955			−0.836204
$508$		0			0
$509$	58.6571			0.115240			0.0576199	−	0.998339i	$-0.481649\pi$
$509$	58.6571			0.115240			0.0576199	+	0.998339i	$0.481649\pi$
$510$		0			0
$511$			62.9906i			0.123269i
$512$		0			0
$513$			445.923i			0.869245i
$514$		0			0
$515$	177.098	+	119.495i	0.343880	+	0.232029i
$516$		0			0
$517$		−	977.130i		−	1.89000i
$518$		0			0
$519$		−	336.422i		−	0.648212i
$520$		0			0
$521$	−171.431			−0.329043			−0.164521	−	0.986374i	$-0.552608\pi$
$521$	−171.431			−0.329043			−0.164521	+	0.986374i	$0.552608\pi$
$522$		0			0
$523$	−407.751			−0.779638			−0.389819	−	0.920892i	$-0.627462\pi$
$523$	−407.751			−0.779638			−0.389819	+	0.920892i	$0.627462\pi$
$524$		0			0
$525$	297.547	+	737.128i	0.566757	+	1.40405i
$526$		0			0
$527$	144.778			0.274721
$528$		0			0
$529$	−27.8174			−0.0525848
$530$		0			0
$531$			8.64391i			0.0162786i
$532$		0			0
$533$			588.030i			1.10325i
$534$		0			0
$535$	−121.623	−	82.0640i	−0.227333	−	0.153391i
$536$		0			0
$537$		−	242.677i		−	0.451913i
$538$		0			0
$539$		−	213.302i		−	0.395737i
$540$		0			0
$541$	−144.762			−0.267582			−0.133791	−	0.991010i	$-0.542715\pi$
$541$	−144.762			−0.267582			−0.133791	+	0.991010i	$0.542715\pi$
$542$		0			0
$543$	−1312.31			−2.41678
$544$		0			0
$545$	−677.386	−	457.059i	−1.24291	−	0.838640i
$546$		0			0
$547$	570.627			1.04319			0.521597	−	0.853192i	$-0.325337\pi$
$547$	570.627			1.04319			0.521597	+	0.853192i	$0.325337\pi$
$548$		0			0
$549$	1054.47			1.92071
$550$		0			0
$551$			43.7062i			0.0793216i
$552$		0			0
$553$			542.404i			0.980839i
$554$		0			0
$555$	509.396	−	754.952i	0.917831	−	1.36027i
$556$		0			0
$557$			824.208i			1.47973i	0.672757	+	0.739864i	$0.265110\pi$
$557$			824.208i			1.47973i	−0.672757	+	0.739864i	$0.734890\pi$
$558$		0			0
$559$		−	378.019i		−	0.676241i
$560$		0			0
$561$	−719.156			−1.28192
$562$		0			0
$563$	216.333			0.384251			0.192125	−	0.981370i	$-0.438462\pi$
$563$	216.333			0.384251			0.192125	+	0.981370i	$0.438462\pi$
$564$		0			0
$565$	−337.512	+	500.211i	−0.597366	+	0.885329i
$566$		0			0
$567$	396.831			0.699877
$568$		0			0
$569$	468.230			0.822899			0.411450	−	0.911432i	$-0.365023\pi$
$569$	468.230			0.822899			0.411450	+	0.911432i	$0.365023\pi$
$570$		0			0
$571$			759.434i			1.33001i	0.746840	+	0.665004i	$0.231570\pi$
$571$			759.434i			1.33001i	−0.746840	+	0.665004i	$0.768430\pi$
$572$		0			0
$573$			347.603i			0.606636i
$574$		0			0
$575$	−209.494	−	518.991i	−0.364338	−	0.902592i
$576$		0			0
$577$		−	29.2057i		−	0.0506164i	−0.999680	−	0.0253082i	$-0.991943\pi$
$577$		−	29.2057i		−	0.0506164i	0.999680	−	0.0253082i	$-0.00805671\pi$
$578$		0			0
$579$		−	1679.39i		−	2.90050i
$580$		0			0
$581$	638.269			1.09857
$582$		0			0
$583$	1552.72			2.66332
$584$		0			0
$585$	769.955	−	1141.11i	1.31616	−	1.95062i
$586$		0			0
$587$	−1014.74			−1.72869			−0.864345	−	0.502900i	$-0.832266\pi$
$587$	−1014.74			−1.72869			−0.864345	+	0.502900i	$0.832266\pi$
$588$		0			0
$589$	215.283			0.365506
$590$		0			0
$591$			618.863i			1.04714i
$592$		0			0
$593$		−	835.823i		−	1.40948i	−0.709465	−	0.704741i	$-0.751063\pi$
$593$		−	835.823i		−	1.40948i	0.709465	−	0.704741i	$-0.248937\pi$
$594$		0			0
$595$	121.050	−	179.403i	0.203445	−	0.301517i
$596$		0			0
$597$		−	901.416i		−	1.50991i
$598$		0			0
$599$			851.616i			1.42173i	0.703329	+	0.710865i	$0.251696\pi$
$599$			851.616i			1.42173i	−0.703329	+	0.710865i	$0.748304\pi$
$600$		0			0
$601$	618.377			1.02891			0.514456	−	0.857516i	$-0.327994\pi$
$601$	618.377			1.02891			0.514456	+	0.857516i	$0.327994\pi$
$602$		0			0
$603$	1432.38			2.37543
$604$		0			0
$605$	1163.53	+	785.082i	1.92320	+	1.29766i
$606$		0			0
$607$	182.393			0.300483			0.150241	−	0.988649i	$-0.451995\pi$
$607$	182.393			0.300483			0.150241	+	0.988649i	$0.451995\pi$
$608$		0			0
$609$	133.725			0.219581
$610$		0			0
$611$		−	773.258i		−	1.26556i
$612$		0			0
$613$		−	638.630i		−	1.04181i	−0.853614	−	0.520906i	$-0.825594\pi$
$613$		−	638.630i		−	1.04181i	0.853614	−	0.520906i	$-0.174406\pi$
$614$		0			0
$615$	−788.888	−	532.294i	−1.28275	−	0.865518i
$616$		0			0
$617$		−	416.661i		−	0.675301i	−0.941271	−	0.337651i	$-0.890368\pi$
$617$		−	416.661i		−	0.675301i	0.941271	−	0.337651i	$-0.109632\pi$
$618$		0			0
$619$			1090.24i			1.76129i	0.473775	+	0.880646i	$0.342891\pi$
$619$			1090.24i			1.76129i	−0.473775	+	0.880646i	$0.657109\pi$
$620$		0			0
$621$	−960.607			−1.54687
$622$		0			0
$623$	305.236			0.489946
$624$		0			0
$625$	−449.863	+	433.876i	−0.719780	+	0.694202i
$626$		0			0
$627$	−1069.38			−1.70555
$628$		0			0
$629$	−247.954			−0.394204
$630$		0			0
$631$			574.569i			0.910569i	0.890346	+	0.455284i	$0.150462\pi$
$631$			574.569i			0.910569i	−0.890346	+	0.455284i	$0.849538\pi$
$632$		0			0
$633$			1217.03i			1.92263i
$634$		0			0
$635$	−943.194	−	636.410i	−1.48534	−	1.00222i
$636$		0			0
$637$		−	168.798i		−	0.264989i
$638$		0			0
$639$		−	498.441i		−	0.780032i
$640$		0			0
$641$	893.367			1.39371			0.696854	−	0.717213i	$-0.254582\pi$
$641$	893.367			1.39371			0.696854	+	0.717213i	$0.254582\pi$
$642$		0			0
$643$	−675.801			−1.05101			−0.525506	−	0.850790i	$-0.676124\pi$
$643$	−675.801			−1.05101			−0.525506	+	0.850790i	$0.676124\pi$
$644$		0			0
$645$	507.142	+	342.188i	0.786267	+	0.530525i
$646$		0			0
$647$	737.921			1.14053			0.570263	−	0.821462i	$-0.306841\pi$
$647$	737.921			1.14053			0.570263	+	0.821462i	$0.306841\pi$
$648$		0			0
$649$	−9.98113			−0.0153792
$650$		0			0
$651$		−	658.688i		−	1.01181i
$652$		0			0
$653$		−	771.853i		−	1.18201i	−0.806668	−	0.591005i	$-0.798731\pi$
$653$		−	771.853i		−	1.18201i	0.806668	−	0.591005i	$-0.201269\pi$
$654$		0			0
$655$	−95.4851	+	141.514i	−0.145779	+	0.216052i
$656$		0			0
$657$			176.540i			0.268707i
$658$		0			0
$659$			814.838i			1.23648i	0.785991	+	0.618238i	$0.212153\pi$
$659$			814.838i			1.23648i	−0.785991	+	0.618238i	$0.787847\pi$
$660$		0			0
$661$	−1038.13			−1.57055			−0.785276	−	0.619146i	$-0.787479\pi$
$661$	−1038.13			−1.57055			−0.785276	+	0.619146i	$0.787479\pi$
$662$		0			0
$663$	−569.109			−0.858384
$664$		0			0
$665$	180.000	−	266.770i	0.270677	−	0.401158i
$666$		0			0
$667$	−94.1519			−0.141157
$668$		0			0
$669$	−70.8869			−0.105959
$670$		0			0
$671$			1217.60i			1.81460i
$672$		0			0
$673$			367.795i			0.546501i	0.961943	+	0.273250i	$0.0880988\pi$
$673$			367.795i			0.546501i	−0.961943	+	0.273250i	$0.911901\pi$
$674$		0			0
$675$	401.534	+	994.739i	0.594865	+	1.47369i
$676$		0			0
$677$		−	463.978i		−	0.685344i	−0.939455	−	0.342672i	$-0.888668\pi$
$677$		−	463.978i		−	0.685344i	0.939455	−	0.342672i	$-0.111332\pi$
$678$		0			0
$679$		−	523.284i		−	0.770669i
$680$		0			0
$681$	−11.3673			−0.0166920
$682$		0			0
$683$	−632.563			−0.926154			−0.463077	−	0.886318i	$-0.653255\pi$
$683$	−632.563			−0.926154			−0.463077	+	0.886318i	$0.653255\pi$
$684$		0			0
$685$	−605.742	+	897.742i	−0.884294	+	1.31057i
$686$		0			0
$687$	1241.17			1.80666
$688$		0			0
$689$	1228.75			1.78339
$690$		0			0
$691$			436.001i			0.630970i	0.948930	+	0.315485i	$0.102167\pi$
$691$			436.001i			0.630970i	−0.948930	+	0.315485i	$0.897833\pi$
$692$		0			0
$693$			2154.70i			3.10923i
$694$		0			0
$695$	−316.230	+	468.670i	−0.455008	+	0.674346i
$696$		0			0
$697$			259.100i			0.371736i
$698$		0			0
$699$		−	156.648i		−	0.224103i
$700$		0			0
$701$	−364.085			−0.519380			−0.259690	−	0.965692i	$-0.583620\pi$
$701$	−364.085			−0.519380			−0.259690	+	0.965692i	$0.583620\pi$
$702$		0			0
$703$	−368.705			−0.524474
$704$		0			0
$705$	1037.39	+	699.965i	1.47147	+	0.992858i
$706$		0			0
$707$	574.678			0.812840
$708$		0			0
$709$	789.593			1.11367			0.556836	−	0.830622i	$-0.312015\pi$
$709$	789.593			1.11367			0.556836	+	0.830622i	$0.312015\pi$
$710$		0			0
$711$			1520.17i			2.13807i
$712$		0			0
$713$			463.763i			0.650439i
$714$		0			0
$715$	1317.65	+	889.067i	1.84286	+	1.24345i
$716$		0			0
$717$		−	2189.34i		−	3.05347i
$718$		0			0
$719$		−	168.422i		−	0.234245i	−0.993118	−	0.117122i	$-0.962633\pi$
$719$		−	168.422i		−	0.234245i	0.993118	−	0.117122i	$-0.0373669\pi$
$720$		0			0
$721$	−264.633			−0.367036
$722$		0			0
$723$	759.501			1.05049
$724$		0			0
$725$	39.3555	+	97.4973i	0.0542835	+	0.134479i
$726$		0			0
$727$	−382.568			−0.526229			−0.263114	−	0.964765i	$-0.584750\pi$
$727$	−382.568			−0.526229			−0.263114	+	0.964765i	$0.584750\pi$
$728$		0			0
$729$	−870.469			−1.19406
$730$		0			0
$731$		−	166.564i		−	0.227858i
$732$		0			0
$733$			74.5492i			0.101704i	0.998706	+	0.0508521i	$0.0161937\pi$
$733$			74.5492i			0.101704i	−0.998706	+	0.0508521i	$0.983806\pi$
$734$		0			0
$735$	226.456	+	152.799i	0.308103	+	0.207889i
$736$		0			0
$737$			1653.97i			2.24420i
$738$		0			0
$739$			682.206i			0.923148i	0.887102	+	0.461574i	$0.152715\pi$
$739$			682.206i			0.923148i	−0.887102	+	0.461574i	$0.847285\pi$
$740$		0			0
$741$	−846.259			−1.14205
$742$		0			0
$743$	359.274			0.483545			0.241772	−	0.970333i	$-0.422271\pi$
$743$	359.274			0.483545			0.241772	+	0.970333i	$0.422271\pi$
$744$		0			0
$745$	712.249	+	480.582i	0.956038	+	0.645077i
$746$		0			0
$747$	1788.84			2.39470
$748$		0			0
$749$	181.738			0.242641
$750$		0			0
$751$		−	532.038i		−	0.708439i	−0.935162	−	0.354220i	$-0.884747\pi$
$751$		−	532.038i		−	0.708439i	0.935162	−	0.354220i	$-0.115253\pi$
$752$		0			0
$753$			501.777i			0.666371i
$754$		0			0
$755$	85.4892	−	126.700i	0.113231	−	0.167814i
$756$		0			0
$757$		−	444.854i		−	0.587653i	−0.955859	−	0.293827i	$-0.905071\pi$
$757$		−	444.854i		−	0.587653i	0.955859	−	0.293827i	$-0.0949288\pi$
$758$		0			0
$759$		−	2303.65i		−	3.03512i
$760$		0			0
$761$	244.735			0.321596			0.160798	−	0.986987i	$-0.448593\pi$
$761$	244.735			0.321596			0.160798	+	0.986987i	$0.448593\pi$
$762$		0			0
$763$	1012.20			1.32660
$764$		0			0
$765$	339.260	−	502.802i	0.443477	−	0.657258i
$766$		0			0
$767$	−7.89863			−0.0102981
$768$		0			0
$769$	−1131.73			−1.47168			−0.735842	−	0.677153i	$-0.763213\pi$
$769$	−1131.73			−1.47168			−0.735842	+	0.677153i	$0.763213\pi$
$770$		0			0
$771$			1819.70i			2.36019i
$772$		0			0
$773$		−	294.508i		−	0.380994i	−0.981688	−	0.190497i	$-0.938990\pi$
$773$		−	294.508i		−	0.380994i	0.981688	−	0.190497i	$-0.0610099\pi$
$774$		0			0
$775$	480.241	−	193.853i	0.619666	−	0.250133i
$776$		0			0
$777$			1128.10i			1.45187i
$778$		0			0
$779$			385.279i			0.494581i
$780$		0			0
$781$	575.550			0.736939
$782$		0			0
$783$	180.459			0.230471
$784$		0			0
$785$	741.699	−	1099.24i	0.944839	−	1.40030i
$786$		0			0
$787$	1017.00			1.29225			0.646127	−	0.763230i	$-0.276388\pi$
$787$	1017.00			1.29225			0.646127	+	0.763230i	$0.276388\pi$
$788$		0			0
$789$	−228.996			−0.290236
$790$		0			0
$791$		−	747.450i		−	0.944943i
$792$		0			0
$793$			963.553i			1.21507i
$794$		0			0
$795$	−1112.29	+	1648.47i	−1.39910	+	2.07355i
$796$		0			0
$797$		−	1105.61i		−	1.38722i	−0.720353	−	0.693608i	$-0.756020\pi$
$797$		−	1105.61i		−	1.38722i	0.720353	−	0.693608i	$-0.243980\pi$
$798$		0			0
$799$		−	340.716i		−	0.426428i
$800$		0			0
$801$	855.469			1.06800
$802$		0			0
$803$	−203.851			−0.253862
$804$		0			0
$805$	574.676	+	387.756i	0.713883	+	0.481685i
$806$		0			0
$807$	−1616.58			−2.00320
$808$		0			0
$809$	594.881			0.735329			0.367665	−	0.929958i	$-0.380157\pi$
$809$	594.881			0.735329			0.367665	+	0.929958i	$0.380157\pi$
$810$		0			0
$811$			45.3204i			0.0558822i	0.999610	+	0.0279411i	$0.00889508\pi$
$811$			45.3204i			0.0558822i	−0.999610	+	0.0279411i	$0.991105\pi$
$812$		0			0
$813$			2305.21i			2.83543i
$814$		0			0
$815$	288.730	+	194.817i	0.354270	+	0.239040i
$816$		0			0
$817$		−	247.679i		−	0.303156i
$818$		0			0
$819$			1705.13i			2.08197i
$820$		0			0
$821$	−310.208			−0.377842			−0.188921	−	0.981992i	$-0.560499\pi$
$821$	−310.208			−0.377842			−0.188921	+	0.981992i	$0.560499\pi$
$822$		0			0
$823$	−660.022			−0.801971			−0.400985	−	0.916084i	$-0.631332\pi$
$823$	−660.022			−0.801971			−0.400985	+	0.916084i	$0.631332\pi$
$824$		0			0
$825$	−2385.51	+	962.928i	−2.89152	+	1.16719i
$826$		0			0
$827$	168.339			0.203553			0.101777	−	0.994807i	$-0.467547\pi$
$827$	168.339			0.203553			0.101777	+	0.994807i	$0.467547\pi$
$828$		0			0
$829$	−597.939			−0.721278			−0.360639	−	0.932706i	$-0.617441\pi$
$829$	−597.939			−0.721278			−0.360639	+	0.932706i	$0.617441\pi$
$830$		0			0
$831$			1695.91i			2.04081i
$832$		0			0
$833$		−	74.3764i		−	0.0892873i
$834$		0			0
$835$	419.198	+	282.849i	0.502034	+	0.338742i
$836$		0			0
$837$		−	888.885i		−	1.06199i
$838$		0			0
$839$		−	1113.51i		−	1.32718i	−0.748095	−	0.663592i	$-0.769031\pi$
$839$		−	1113.51i		−	1.32718i	0.748095	−	0.663592i	$-0.230969\pi$
$840$		0			0
$841$	−823.313			−0.978969
$842$		0			0
$843$	−1580.94			−1.87537
$844$		0			0
$845$	342.265	+	230.940i	0.405048	+	0.273302i
$846$		0			0
$847$	−1738.63			−2.05270
$848$		0			0
$849$	937.514			1.10426
$850$		0			0
$851$		−	794.264i		−	0.933331i
$852$		0			0
$853$		−	889.393i		−	1.04266i	−0.853354	−	0.521332i	$-0.825435\pi$
$853$		−	889.393i		−	1.04266i	0.853354	−	0.521332i	$-0.174565\pi$
$854$		0			0
$855$	504.476	−	747.661i	0.590031	−	0.874457i
$856$		0			0
$857$			1246.97i			1.45504i	0.686087	+	0.727520i	$0.259327\pi$
$857$			1246.97i			1.45504i	−0.686087	+	0.727520i	$0.740673\pi$
$858$		0			0
$859$			977.927i			1.13845i	0.822182	+	0.569224i	$0.192756\pi$
$859$			977.927i			1.13845i	−0.822182	+	0.569224i	$0.807244\pi$
$860$		0			0
$861$	1178.81			1.36912
$862$		0			0
$863$	1366.41			1.58332			0.791661	−	0.610961i	$-0.209217\pi$
$863$	1366.41			1.58332			0.791661	+	0.610961i	$0.209217\pi$
$864$		0			0
$865$	−183.258	+	271.598i	−0.211859	+	0.313987i
$866$		0			0
$867$	1232.96			1.42210
$868$		0			0
$869$	−1755.34			−2.01995
$870$		0			0
$871$			1308.88i			1.50273i
$872$		0			0
$873$		−	1466.58i		−	1.67993i
$874$		0			0
$875$	161.319	−	757.177i	0.184365	−	0.865345i
$876$		0			0
$877$			87.0797i			0.0992927i	0.998767	+	0.0496464i	$0.0158094\pi$
$877$			87.0797i			0.0992927i	−0.998767	+	0.0496464i	$0.984191\pi$
$878$		0			0
$879$		−	1331.62i		−	1.51492i
$880$		0			0
$881$	154.927			0.175853			0.0879265	−	0.996127i	$-0.471976\pi$
$881$	154.927			0.175853			0.0879265	+	0.996127i	$0.471976\pi$
$882$		0			0
$883$	1604.11			1.81666			0.908332	−	0.418249i	$-0.137356\pi$
$883$	1604.11			1.81666			0.908332	+	0.418249i	$0.137356\pi$
$884$		0			0
$885$	7.14996	−	10.5966i	0.00807905	−	0.0119736i
$886$		0			0
$887$	196.767			0.221834			0.110917	−	0.993830i	$-0.464621\pi$
$887$	196.767			0.221834			0.110917	+	0.993830i	$0.464621\pi$
$888$		0			0
$889$	1409.39			1.58536
$890$		0			0
$891$			1284.23i			1.44133i
$892$		0			0
$893$		−	506.640i		−	0.567346i
$894$		0			0
$895$	−132.193	+	195.917i	−0.147702	+	0.218902i
$896$		0			0
$897$		−	1823.01i		−	2.03234i
$898$		0			0
$899$		−	87.1223i		−	0.0969102i
$900$		0			0
$901$	541.418			0.600907
$902$		0			0
$903$	−757.807			−0.839210
$904$		0			0
$905$	1059.45	+	714.852i	1.17066	+	0.789892i
$906$		0			0
$907$	1079.04			1.18968			0.594838	−	0.803845i	$-0.297216\pi$
$907$	1079.04			1.18968			0.594838	+	0.803845i	$0.297216\pi$
$908$		0			0
$909$	1610.62			1.77186
$910$		0			0
$911$			1815.91i			1.99331i	0.0816953	+	0.996657i	$0.473967\pi$
$911$			1815.91i			1.99331i	−0.0816953	+	0.996657i	$0.526033\pi$
$912$		0			0
$913$			2065.58i			2.26241i
$914$		0			0
$915$	−1292.68	−	872.223i	−1.41277	−	0.953249i
$916$		0			0
$917$		−	211.460i		−	0.230600i
$918$		0			0
$919$			437.303i			0.475847i	0.971284	+	0.237923i	$0.0764667\pi$
$919$			437.303i			0.475847i	−0.971284	+	0.237923i	$0.923533\pi$
$920$		0			0
$921$	1075.11			1.16733
$922$		0			0
$923$	455.465			0.493461
$924$		0			0
$925$	−822.486	+	332.003i	−0.889174	+	0.358922i
$926$		0			0
$927$	−741.672			−0.800077
$928$		0			0
$929$	−164.320			−0.176878			−0.0884392	−	0.996082i	$-0.528188\pi$
$929$	−164.320			−0.176878			−0.0884392	+	0.996082i	$0.528188\pi$
$930$		0			0
$931$		−	110.597i		−	0.118794i
$932$		0			0
$933$			765.670i			0.820654i
$934$		0			0
$935$	580.586	+	391.744i	0.620947	+	0.418977i
$936$		0			0
$937$		−	1238.62i		−	1.32190i	−0.750431	−	0.660949i	$-0.770154\pi$
$937$		−	1238.62i		−	1.32190i	0.750431	−	0.660949i	$-0.229846\pi$
$938$		0			0
$939$		−	1522.74i		−	1.62167i
$940$		0			0
$941$	−1846.13			−1.96188			−0.980942	−	0.194299i	$-0.937757\pi$
$941$	−1846.13			−1.96188			−0.980942	+	0.194299i	$0.937757\pi$
$942$		0			0
$943$	−829.967			−0.880135
$944$		0			0
$945$	−1101.47	−	743.204i	−1.16558	−	0.786460i
$946$		0			0
$947$	−466.767			−0.492890			−0.246445	−	0.969157i	$-0.579263\pi$
$947$	−466.767			−0.492890			−0.246445	+	0.969157i	$0.579263\pi$
$948$		0			0
$949$	−161.319			−0.169988
$950$		0			0
$951$		−	118.313i		−	0.124409i
$952$		0			0
$953$		−	1316.75i		−	1.38168i	−0.723006	−	0.690842i	$-0.757240\pi$
$953$		−	1316.75i		−	1.38168i	0.723006	−	0.690842i	$-0.242760\pi$
$954$		0			0
$955$	189.349	−	280.625i	0.198271	−	0.293848i
$956$		0			0
$957$			432.763i			0.452208i
$958$		0			0
$959$		−	1341.47i		−	1.39882i
$960$		0			0
$961$	531.863			0.553447
$962$		0			0
$963$	509.347			0.528917
$964$		0			0
$965$	−914.810	+	1355.80i	−0.947990	+	1.40497i
$966$		0			0
$967$	572.176			0.591702			0.295851	−	0.955234i	$-0.404397\pi$
$967$	572.176			0.591702			0.295851	+	0.955234i	$0.404397\pi$
$968$		0			0
$969$	−372.881			−0.384811
$970$		0			0
$971$			881.863i			0.908201i	0.890951	+	0.454100i	$0.150039\pi$
$971$			881.863i			0.908201i	−0.890951	+	0.454100i	$0.849961\pi$
$972$		0			0
$973$		−	700.320i		−	0.719754i
$974$		0			0
$975$	−1887.79	+	762.019i	−1.93619	+	0.781558i
$976$		0			0
$977$		−	512.461i		−	0.524525i	−0.964997	−	0.262263i	$-0.915531\pi$
$977$		−	512.461i		−	0.524525i	0.964997	−	0.262263i	$-0.0844687\pi$
$978$		0			0
$979$			987.811i			1.00900i
$980$		0			0
$981$	2836.83			2.89178
$982$		0			0
$983$	−1665.68			−1.69448			−0.847241	−	0.531209i	$-0.821738\pi$
$983$	−1665.68			−1.69448			−0.847241	+	0.531209i	$0.821738\pi$
$984$		0			0
$985$	337.111	−	499.617i	0.342245	−	0.507225i
$986$		0			0
$987$	−1550.14			−1.57055
$988$		0			0
$989$	533.550			0.539484
$990$		0			0
$991$		−	1696.66i		−	1.71207i	−0.516916	−	0.856036i	$-0.672920\pi$
$991$		−	1696.66i		−	1.71207i	0.516916	−	0.856036i	$-0.327080\pi$
$992$		0			0
$993$		−	2154.56i		−	2.16974i
$994$		0			0
$995$	−491.026	+	727.727i	−0.493493	+	0.731384i
$996$		0			0
$997$			1405.45i			1.40968i	0.709367	+	0.704840i	$0.248981\pi$
$997$			1405.45i			1.40968i	−0.709367	+	0.704840i	$0.751019\pi$
$998$		0			0
$999$			1522.35i			1.52387i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.h.n.1279.13		16
4.3	odd	2	inner	1280.3.h.n.1279.1		16
5.4	even	2	inner	1280.3.h.n.1279.2		16
8.3	odd	2	inner	1280.3.h.n.1279.16		16
8.5	even	2	inner	1280.3.h.n.1279.4		16
16.3	odd	4		320.3.e.b.159.2	yes	16
16.5	even	4		320.3.e.b.159.3	yes	16
16.11	odd	4		320.3.e.b.159.15	yes	16
16.13	even	4		320.3.e.b.159.14	yes	16
20.19	odd	2	inner	1280.3.h.n.1279.14		16
40.19	odd	2	inner	1280.3.h.n.1279.3		16
40.29	even	2	inner	1280.3.h.n.1279.15		16
80.3	even	4		1600.3.g.k.351.13		16
80.13	odd	4		1600.3.g.k.351.4		16
80.19	odd	4		320.3.e.b.159.16	yes	16
80.27	even	4		1600.3.g.k.351.16		16
80.29	even	4		320.3.e.b.159.4	yes	16
80.37	odd	4		1600.3.g.k.351.1		16
80.43	even	4		1600.3.g.k.351.2		16
80.53	odd	4		1600.3.g.k.351.15		16
80.59	odd	4		320.3.e.b.159.1	✓	16
80.67	even	4		1600.3.g.k.351.3		16
80.69	even	4		320.3.e.b.159.13	yes	16
80.77	odd	4		1600.3.g.k.351.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.b.159.1	✓	16	80.59	odd	4
320.3.e.b.159.2	yes	16	16.3	odd	4
320.3.e.b.159.3	yes	16	16.5	even	4
320.3.e.b.159.4	yes	16	80.29	even	4
320.3.e.b.159.13	yes	16	80.69	even	4
320.3.e.b.159.14	yes	16	16.13	even	4
320.3.e.b.159.15	yes	16	16.11	odd	4
320.3.e.b.159.16	yes	16	80.19	odd	4
1280.3.h.n.1279.1		16	4.3	odd	2	inner
1280.3.h.n.1279.2		16	5.4	even	2	inner
1280.3.h.n.1279.3		16	40.19	odd	2	inner
1280.3.h.n.1279.4		16	8.5	even	2	inner
1280.3.h.n.1279.13		16	1.1	even	1	trivial
1280.3.h.n.1279.14		16	20.19	odd	2	inner
1280.3.h.n.1279.15		16	40.29	even	2	inner
1280.3.h.n.1279.16		16	8.3	odd	2	inner
1600.3.g.k.351.1		16	80.37	odd	4
1600.3.g.k.351.2		16	80.43	even	4
1600.3.g.k.351.3		16	80.67	even	4
1600.3.g.k.351.4		16	80.13	odd	4
1600.3.g.k.351.13		16	80.3	even	4
1600.3.g.k.351.14		16	80.77	odd	4
1600.3.g.k.351.15		16	80.53	odd	4
1600.3.g.k.351.16		16	80.27	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.13399 q^{3} +(-4.14474 - 2.79662i) q^{5} +6.19337 q^{7} +17.3578 q^{9} +O(q^{10})\)	\(q+5.13399 q^{3} +(-4.14474 - 2.79662i) q^{5} +6.19337 q^{7} +17.3578 q^{9} +20.0431i q^{11} +15.8612i q^{13} +(-21.2791 - 14.3578i) q^{15} +6.98882i q^{17} +10.3923i q^{19} +31.7967 q^{21} -22.3871 q^{23} +(9.35782 + 23.1826i) q^{25} +42.9089 q^{27} +4.20563 q^{29} -20.7156i q^{31} +102.901i q^{33} +(-25.6699 - 17.3205i) q^{35} +35.4786i q^{37} +81.4313i q^{39} +37.0735 q^{41} -23.8329 q^{43} +(-71.9437 - 48.5432i) q^{45} -48.7515 q^{47} -10.6422 q^{49} +35.8805i q^{51} -77.4691i q^{53} +(56.0529 - 83.0735i) q^{55} +53.3540i q^{57} +0.497984i q^{59} +60.7490 q^{61} +107.503 q^{63} +(44.3578 - 65.7407i) q^{65} +82.5209 q^{67} -114.935 q^{69} -28.7156i q^{71} +10.1706i q^{73} +(48.0429 + 119.019i) q^{75} +124.134i q^{77} +87.5782i q^{79} +64.0735 q^{81} +103.057 q^{83} +(19.5451 - 28.9669i) q^{85} +21.5917 q^{87} +49.2844 q^{89} +98.2344i q^{91} -106.354i q^{93} +(29.0633 - 43.0735i) q^{95} -84.4911i q^{97} +347.904i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 96 q^{9}+O(q^{10}) \) 16 * q + 96 * q^9	\( 16 q + 96 q^{9} - 32 q^{25} + 48 q^{41} - 352 q^{49} + 528 q^{65} + 480 q^{81} + 1152 q^{89}+O(q^{100}) \) 16 * q + 96 * q^9 - 32 * q^25 + 48 * q^41 - 352 * q^49 + 528 * q^65 + 480 * q^81 + 1152 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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