LMFDB - Embedded newform 320.3.e.b.159.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(159,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.159");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.e (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:	$8.71936845953$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			159.15
Root			$-0.265804 + 0.460386i$ of defining polynomial
Character	$\chi$	$=$	320.159
Dual form			320.3.e.b.159.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.13399i q^{3} +(2.79662 - 4.14474i) q^{5} +6.19337 q^{7} -17.3578 q^{9} +O(q^{10})$	$q+5.13399i q^{3} +(2.79662 - 4.14474i) q^{5} +6.19337 q^{7} -17.3578 q^{9} +20.0431 q^{11} +15.8612 q^{13} +(21.2791 + 14.3578i) q^{15} +6.98882i q^{17} -10.3923 q^{19} +31.7967i q^{21} -22.3871 q^{23} +(-9.35782 - 23.1826i) q^{25} -42.9089i q^{27} -4.20563i q^{29} +20.7156i q^{31} +102.901i q^{33} +(17.3205 - 25.6699i) q^{35} -35.4786 q^{37} +81.4313i q^{39} -37.0735 q^{41} +23.8329i q^{43} +(-48.5432 + 71.9437i) q^{45} +48.7515 q^{47} -10.6422 q^{49} -35.8805 q^{51} +77.4691 q^{53} +(56.0529 - 83.0735i) q^{55} -53.3540i q^{57} +0.497984 q^{59} -60.7490i q^{61} -107.503 q^{63} +(44.3578 - 65.7407i) q^{65} +82.5209i q^{67} -114.935i q^{69} -28.7156i q^{71} -10.1706i q^{73} +(119.019 - 48.0429i) q^{75} +124.134 q^{77} -87.5782i q^{79} +64.0735 q^{81} +103.057i q^{83} +(28.9669 + 19.5451i) q^{85} +21.5917 q^{87} -49.2844 q^{89} +98.2344 q^{91} -106.354 q^{93} +(-29.0633 + 43.0735i) q^{95} -84.4911i q^{97} -347.904 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			5.13399i			1.71133i	0.517531	+	0.855664i	$0.326851\pi$
$3$			5.13399i			1.71133i	−0.517531	+	0.855664i	$0.673149\pi$
$4$		0			0
$5$	2.79662	−	4.14474i	0.559324	−	0.828949i
$5$	2.79662	−	4.14474i	0.559324	−	0.828949i
$6$		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
$11$	20.0431			1.82210			0.911049	−	0.412298i	$-0.135274\pi$
$11$	20.0431			1.82210			0.911049	+	0.412298i	$0.135274\pi$
$12$		0			0
$13$	15.8612			1.22009			0.610047	−	0.792365i	$-0.291151\pi$
$13$	15.8612			1.22009			0.610047	+	0.792365i	$0.291151\pi$
$14$		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
$19$	−10.3923			−0.546963			−0.273482	−	0.961877i	$-0.588175\pi$
$19$	−10.3923			−0.546963			−0.273482	+	0.961877i	$0.588175\pi$
$20$		0			0
$21$			31.7967i			1.51413i
$22$		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
$27$		−	42.9089i		−	1.58922i
$28$		0			0
$29$		−	4.20563i		−	0.145022i	−0.997368	−	0.0725109i	$-0.976899\pi$
$29$		−	4.20563i		−	0.145022i	0.997368	−	0.0725109i	$-0.0231012\pi$
$30$		0			0
$31$			20.7156i			0.668246i	0.942529	+	0.334123i	$0.108440\pi$
$31$			20.7156i			0.668246i	−0.942529	+	0.334123i	$0.891560\pi$
$32$		0			0
$33$			102.901i			3.11821i
$34$		0			0
$35$	17.3205	−	25.6699i	0.494872	−	0.733427i
$36$		0			0
$37$	−35.4786			−0.958882			−0.479441	−	0.877574i	$-0.659161\pi$
$37$	−35.4786			−0.958882			−0.479441	+	0.877574i	$0.659161\pi$
$38$		0			0
$39$			81.4313i			2.08798i
$40$		0			0
$41$	−37.0735			−0.904230			−0.452115	−	0.891960i	$-0.649330\pi$
$41$	−37.0735			−0.904230			−0.452115	+	0.891960i	$0.649330\pi$
$42$		0			0
$43$			23.8329i			0.554254i	0.960833	+	0.277127i	$0.0893822\pi$
$43$			23.8329i			0.554254i	−0.960833	+	0.277127i	$0.910618\pi$
$44$		0			0
$45$	−48.5432	+	71.9437i	−1.07874	+	1.59875i
$46$		0			0
$47$	48.7515			1.03727			0.518633	−	0.854997i	$-0.326441\pi$
$47$	48.7515			1.03727			0.518633	+	0.854997i	$0.326441\pi$
$48$		0			0
$49$	−10.6422			−0.217187
$50$		0			0
$51$	−35.8805			−0.703540
$52$		0			0
$53$	77.4691			1.46168			0.730840	−	0.682549i	$-0.239128\pi$
$53$	77.4691			1.46168			0.730840	+	0.682549i	$0.239128\pi$
$54$		0			0
$55$	56.0529	−	83.0735i	1.01914	−	1.51043i
$56$		0			0
$57$		−	53.3540i		−	0.936034i
$58$		0			0
$59$	0.497984			0.00844041			0.00422020	−	0.999991i	$-0.498657\pi$
$59$	0.497984			0.00844041			0.00422020	+	0.999991i	$0.498657\pi$
$60$		0			0
$61$		−	60.7490i		−	0.995885i	−0.867210	−	0.497943i	$-0.834089\pi$
$61$		−	60.7490i		−	0.995885i	0.867210	−	0.497943i	$-0.165911\pi$
$62$		0			0
$63$	−107.503			−1.70640
$64$		0			0
$65$	44.3578	−	65.7407i	0.682428	−	1.01140i
$66$		0			0
$67$			82.5209i			1.23165i	0.787881	+	0.615827i	$0.211178\pi$
$67$			82.5209i			1.23165i	−0.787881	+	0.615827i	$0.788822\pi$
$68$		0			0
$69$		−	114.935i		−	1.66573i
$70$		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	$-0.977808\pi$
$73$		−	10.1706i		−	0.139324i	0.997571	−	0.0696620i	$-0.0221921\pi$
$74$		0			0
$75$	119.019	−	48.0429i	1.58692	−	0.640572i
$76$		0			0
$77$	124.134			1.61213
$78$		0			0
$79$		−	87.5782i		−	1.10858i	−0.832322	−	0.554292i	$-0.812989\pi$
$79$		−	87.5782i		−	1.10858i	0.832322	−	0.554292i	$-0.187011\pi$
$80$		0			0
$81$	64.0735			0.791030
$82$		0			0
$83$			103.057i			1.24165i	0.783950	+	0.620824i	$0.213202\pi$
$83$			103.057i			1.24165i	−0.783950	+	0.620824i	$0.786798\pi$
$84$		0			0
$85$	28.9669	+	19.5451i	0.340787	+	0.229942i
$86$		0			0
$87$	21.5917			0.248180
$88$		0			0
$89$	−49.2844			−0.553757			−0.276878	−	0.960905i	$-0.589300\pi$
$89$	−49.2844			−0.553757			−0.276878	+	0.960905i	$0.589300\pi$
$90$		0			0
$91$	98.2344			1.07950
$92$		0			0
$93$	−106.354			−1.14359
$94$		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
$99$	−347.904			−3.51418
$100$		0			0
$101$			92.7892i			0.918705i	0.888254	+	0.459353i	$0.151919\pi$
$101$			92.7892i			0.918705i	−0.888254	+	0.459353i	$0.848081\pi$
$102$		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
$107$		−	29.3440i		−	0.274243i	−0.990554	−	0.137121i	$-0.956215\pi$
$107$		−	29.3440i		−	0.274243i	0.990554	−	0.137121i	$-0.0437851\pi$
$108$		0			0
$109$		−	163.433i		−	1.49938i	−0.661789	−	0.749691i	$-0.730202\pi$
$109$		−	163.433i		−	1.49938i	0.661789	−	0.749691i	$-0.269798\pi$
$110$		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
$115$	−62.6083	+	92.7888i	−0.544420	+	0.806860i
$116$		0			0
$117$	−275.316			−2.35313
$118$		0			0
$119$			43.2844i			0.363734i
$120$		0			0
$121$	280.725			2.32004
$122$		0			0
$123$		−	190.335i		−	1.54744i
$124$		0			0
$125$	−122.256	−	26.0471i	−0.978049	−	0.208377i
$126$		0			0
$127$	−227.564			−1.79184			−0.895920	−	0.444215i	$-0.853483\pi$
$127$	−227.564			−1.79184			−0.895920	+	0.444215i	$0.853483\pi$
$128$		0			0
$129$	−122.358			−0.948510
$130$		0			0
$131$	34.1430			0.260634			0.130317	−	0.991472i	$-0.458401\pi$
$131$	34.1430			0.260634			0.130317	+	0.991472i	$0.458401\pi$
$132$		0			0
$133$	−64.3634			−0.483935
$134$		0			0
$135$	−177.847	−	120.000i	−1.31738	−	0.888889i
$136$		0			0
$137$			216.598i			1.58100i	0.612459	+	0.790502i	$0.290180\pi$
$137$			216.598i			1.58100i	−0.612459	+	0.790502i	$0.709820\pi$
$138$		0			0
$139$	−113.076			−0.813495			−0.406748	−	0.913541i	$-0.633337\pi$
$139$	−113.076			−0.813495			−0.406748	+	0.913541i	$0.633337\pi$
$140$		0			0
$141$			250.290i			1.77510i
$142$		0			0
$143$	317.908			2.22313
$144$		0			0
$145$	−17.4313	−	11.7616i	−0.120216	−	0.0811142i
$146$		0			0
$147$		−	54.6368i		−	0.371679i
$148$		0			0
$149$		−	171.844i		−	1.15331i	−0.816986	−	0.576657i	$-0.804357\pi$
$149$		−	171.844i		−	1.15331i	0.816986	−	0.576657i	$-0.195643\pi$
$150$		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
$155$	85.8610	+	57.9338i	0.553942	+	0.373766i
$156$		0			0
$157$	265.212			1.68925			0.844626	−	0.535357i	$-0.179823\pi$
$157$	265.212			1.68925			0.844626	+	0.535357i	$0.179823\pi$
$158$		0			0
$159$			397.725i			2.50142i
$160$		0			0
$161$	−138.652			−0.861190
$162$		0			0
$163$		−	69.6617i		−	0.427372i	−0.976902	−	0.213686i	$-0.931453\pi$
$163$		−	69.6617i		−	0.427372i	0.976902	−	0.213686i	$-0.0685470\pi$
$164$		0			0
$165$	426.498	+	287.775i	2.58484	+	1.74409i
$166$		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
$171$	180.388			1.05490
$172$		0			0
$173$	−65.5284			−0.378777			−0.189388	−	0.981902i	$-0.560651\pi$
$173$	−65.5284			−0.378777			−0.189388	+	0.981902i	$0.560651\pi$
$174$		0			0
$175$	−57.9564	−	143.578i	−0.331179	−	0.820447i
$176$		0			0
$177$			2.55664i			0.0144443i
$178$		0			0
$179$	47.2688			0.264072			0.132036	−	0.991245i	$-0.457849\pi$
$179$	47.2688			0.264072			0.132036	+	0.991245i	$0.457849\pi$
$180$		0			0
$181$		−	255.613i		−	1.41223i	−0.708099	−	0.706113i	$-0.750447\pi$
$181$		−	255.613i		−	1.41223i	0.708099	−	0.706113i	$-0.249553\pi$
$182$		0			0
$183$	311.885			1.70429
$184$		0			0
$185$	−99.2204	+	147.050i	−0.536326	+	0.794865i
$186$		0			0
$187$			140.078i			0.749078i
$188$		0			0
$189$		−	265.751i		−	1.40609i
$190$		0			0
$191$		−	67.7062i		−	0.354483i	−0.984167	−	0.177241i	$-0.943283\pi$
$191$		−	67.7062i		−	0.354483i	0.984167	−	0.177241i	$-0.0567173\pi$
$192$		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
$195$	337.512	+	227.732i	1.73083	+	1.16786i
$196$		0			0
$197$	−120.542			−0.611890			−0.305945	−	0.952049i	$-0.598972\pi$
$197$	−120.542			−0.611890			−0.305945	+	0.952049i	$0.598972\pi$
$198$		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
$203$		−	26.0470i		−	0.128310i
$204$		0			0
$205$	−103.680	+	153.660i	−0.505758	+	0.749561i
$206$		0			0
$207$	388.591			1.87725
$208$		0			0
$209$	−208.294			−0.996621
$210$		0			0
$211$	−237.053			−1.12347			−0.561737	−	0.827316i	$-0.689867\pi$
$211$	−237.053			−1.12347			−0.561737	+	0.827316i	$0.689867\pi$
$212$		0			0
$213$	147.426			0.692139
$214$		0			0
$215$	98.7813	+	66.6516i	0.459448	+	0.310008i
$216$		0			0
$217$			128.300i			0.591242i
$218$		0			0
$219$	52.2160			0.238429
$220$		0			0
$221$			110.851i			0.501589i
$222$		0			0
$223$	13.8074			0.0619165			0.0309582	−	0.999521i	$-0.490144\pi$
$223$	13.8074			0.0619165			0.0309582	+	0.999521i	$0.490144\pi$
$224$		0			0
$225$	162.431	+	402.399i	0.721917	+	1.78844i
$226$		0			0
$227$		−	2.21412i		−	0.00975382i	−0.999988	−	0.00487691i	$-0.998448\pi$
$227$		−	2.21412i		−	0.00975382i	0.999988	−	0.00487691i	$-0.00155238\pi$
$228$		0			0
$229$			241.756i			1.05571i	0.849336	+	0.527853i	$0.177003\pi$
$229$			241.756i			1.05571i	−0.849336	+	0.527853i	$0.822997\pi$
$230$		0			0
$231$			637.303i			2.75889i
$232$		0			0
$233$			30.5119i			0.130953i	0.997854	+	0.0654763i	$0.0208567\pi$
$233$			30.5119i			0.130953i	−0.997854	+	0.0654763i	$0.979143\pi$
$234$		0			0
$235$	136.339	−	202.062i	0.580168	−	0.859840i
$236$		0			0
$237$	449.625			1.89715
$238$		0			0
$239$			426.441i			1.78427i	0.451768	+	0.892135i	$0.350793\pi$
$239$			426.441i			1.78427i	−0.451768	+	0.892135i	$0.649207\pi$
$240$		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
$243$		−	57.2280i		−	0.235506i
$244$		0			0
$245$	−29.7622	+	44.1091i	−0.121478	+	0.180037i
$246$		0			0
$247$	−164.835			−0.667347
$248$		0			0
$249$	−529.092			−2.12487
$250$		0			0
$251$	97.7364			0.389388			0.194694	−	0.980864i	$-0.437629\pi$
$251$	97.7364			0.389388			0.194694	+	0.980864i	$0.437629\pi$
$252$		0			0
$253$	−448.707			−1.77354
$254$		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
$259$	−219.732			−0.848387
$260$		0			0
$261$			73.0006i			0.279696i
$262$		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
$267$		−	253.025i		−	0.947660i
$268$		0			0
$269$			314.879i			1.17055i	0.810834	+	0.585277i	$0.199014\pi$
$269$			314.879i			1.17055i	−0.810834	+	0.585277i	$0.800986\pi$
$270$		0			0
$271$		−	449.009i		−	1.65686i	−0.560092	−	0.828431i	$-0.689234\pi$
$271$		−	449.009i		−	1.65686i	0.560092	−	0.828431i	$-0.310766\pi$
$272$		0			0
$273$			504.334i			1.84738i
$274$		0			0
$275$	−187.559	−	464.650i	−0.682034	−	1.68964i
$276$		0			0
$277$	−330.330			−1.19253			−0.596264	−	0.802789i	$-0.703349\pi$
$277$	−330.330			−1.19253			−0.596264	+	0.802789i	$0.703349\pi$
$278$		0			0
$279$		−	359.578i		−	1.28881i
$280$		0			0
$281$	307.936			1.09586			0.547929	−	0.836525i	$-0.315416\pi$
$281$	307.936			1.09586			0.547929	+	0.836525i	$0.315416\pi$
$282$		0			0
$283$		−	182.609i		−	0.645263i	−0.946525	−	0.322631i	$-0.895433\pi$
$283$		−	182.609i		−	0.645263i	0.946525	−	0.322631i	$-0.104567\pi$
$284$		0			0
$285$	−221.139	−	149.211i	−0.775925	−	0.523547i
$286$		0			0
$287$	−229.610			−0.800033
$288$		0			0
$289$	240.156			0.830991
$290$		0			0
$291$	433.776			1.49064
$292$		0			0
$293$	259.373			0.885231			0.442616	−	0.896711i	$-0.354051\pi$
$293$	259.373			0.885231			0.442616	+	0.896711i	$0.354051\pi$
$294$		0			0
$295$	1.39267	−	2.06402i	0.00472092	−	0.00699667i
$296$		0			0
$297$		−	860.027i		−	2.89571i
$298$		0			0
$299$	−355.087			−1.18758
$300$		0			0
$301$			147.606i			0.490385i
$302$		0			0
$303$	−476.379			−1.57221
$304$		0			0
$305$	−251.789	−	169.892i	−0.825538	−	0.557023i
$306$		0			0
$307$			209.411i			0.682119i	0.940042	+	0.341060i	$0.110786\pi$
$307$			209.411i			0.682119i	−0.940042	+	0.341060i	$0.889214\pi$
$308$		0			0
$309$		−	219.367i		−	0.709926i
$310$		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	$0.157119\pi$
$313$			296.601i			0.947606i	−0.880631	+	0.473803i	$0.842881\pi$
$314$		0			0
$315$	−300.646	+	445.574i	−0.954432	+	1.41452i
$316$		0			0
$317$	−23.0450			−0.0726971			−0.0363486	−	0.999339i	$-0.511573\pi$
$317$	−23.0450			−0.0726971			−0.0363486	+	0.999339i	$0.511573\pi$
$318$		0			0
$319$		−	84.2938i		−	0.264244i
$320$		0			0
$321$	150.652			0.469320
$322$		0			0
$323$		−	72.6300i		−	0.224861i
$324$		0			0
$325$	−148.426	−	367.704i	−0.456696	−	1.13140i
$326$		0			0
$327$	839.060			2.56593
$328$		0			0
$329$	301.936			0.917739
$330$		0			0
$331$	−419.665			−1.26787			−0.633935	−	0.773386i	$-0.718561\pi$
$331$	−419.665			−1.26787			−0.633935	+	0.773386i	$0.718561\pi$
$332$		0			0
$333$	615.832			1.84934
$334$		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
$339$	619.598			1.82772
$340$		0			0
$341$			415.205i			1.21761i
$342$		0			0
$343$	−369.386			−1.07693
$344$		0			0
$345$	−476.377	−	321.430i	−1.38080	−	0.931681i
$346$		0			0
$347$			329.706i			0.950163i	0.879942	+	0.475081i	$0.157581\pi$
$347$			329.706i			0.950163i	−0.879942	+	0.475081i	$0.842419\pi$
$348$		0			0
$349$		−	517.423i		−	1.48259i	−0.671180	−	0.741294i	$-0.734212\pi$
$349$		−	517.423i		−	1.48259i	0.671180	−	0.741294i	$-0.265788\pi$
$350$		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
$355$	−119.019	−	80.3068i	−0.335265	−	0.226216i
$356$		0			0
$357$	−222.221			−0.622469
$358$		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
$363$			1441.24i			3.97035i
$364$		0			0
$365$	−42.1547	−	28.4434i	−0.115492	−	0.0779273i
$366$		0			0
$367$	−374.954			−1.02167			−0.510837	−	0.859678i	$-0.670664\pi$
$367$	−374.954			−1.02167			−0.510837	+	0.859678i	$0.670664\pi$
$368$		0			0
$369$	643.514			1.74394
$370$		0			0
$371$	479.794			1.29325
$372$		0			0
$373$	24.6208			0.0660076			0.0330038	−	0.999455i	$-0.489493\pi$
$373$	24.6208			0.0660076			0.0330038	+	0.999455i	$0.489493\pi$
$374$		0			0
$375$	133.725	−	627.661i	0.356601	−	1.67376i
$376$		0			0
$377$		−	66.7064i		−	0.176940i
$378$		0			0
$379$	170.472			0.449793			0.224897	−	0.974383i	$-0.427796\pi$
$379$	170.472			0.449793			0.224897	+	0.974383i	$0.427796\pi$
$380$		0			0
$381$		−	1168.31i		−	3.06643i
$382$		0			0
$383$	−136.084			−0.355310			−0.177655	−	0.984093i	$-0.556851\pi$
$383$	−136.084			−0.355310			−0.177655	+	0.984093i	$0.556851\pi$
$384$		0			0
$385$	347.156	−	514.505i	0.901705	−	1.33638i
$386$		0			0
$387$		−	413.687i		−	1.06896i
$388$		0			0
$389$		−	60.5055i		−	0.155541i	−0.996971	−	0.0777705i	$-0.975220\pi$
$389$		−	60.5055i		−	0.155541i	0.996971	−	0.0777705i	$-0.0247801\pi$
$390$		0			0
$391$		−	156.460i		−	0.400152i
$392$		0			0
$393$			175.290i			0.446030i
$394$		0			0
$395$	−362.989	−	244.923i	−0.918960	−	0.620058i
$396$		0			0
$397$	−124.216			−0.312888			−0.156444	−	0.987687i	$-0.550003\pi$
$397$	−124.216			−0.312888			−0.156444	+	0.987687i	$0.550003\pi$
$398$		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
$403$			328.575i			0.815323i
$404$		0			0
$405$	179.189	−	265.568i	0.442442	−	0.655724i
$406$		0			0
$407$	−711.101			−1.74718
$408$		0			0
$409$	582.377			1.42390			0.711952	−	0.702228i	$-0.247811\pi$
$409$	582.377			1.42390			0.711952	+	0.702228i	$0.247811\pi$
$410$		0			0
$411$	−1112.01			−2.70562
$412$		0			0
$413$	3.08420			0.00746779
$414$		0			0
$415$	427.144	+	288.211i	1.02926	+	0.694484i
$416$		0			0
$417$		−	580.530i		−	1.39216i
$418$		0			0
$419$	479.296			1.14391			0.571953	−	0.820287i	$-0.306186\pi$
$419$	479.296			1.14391			0.571953	+	0.820287i	$0.306186\pi$
$420$		0			0
$421$		−	116.905i		−	0.277685i	−0.990314	−	0.138842i	$-0.955662\pi$
$421$		−	116.905i		−	0.277685i	0.990314	−	0.138842i	$-0.0443381\pi$
$422$		0			0
$423$	−846.220			−2.00052
$424$		0			0
$425$	162.019	−	65.4001i	0.381221	−	0.153883i
$426$		0			0
$427$		−	376.241i		−	0.881126i
$428$		0			0
$429$			1632.13i			3.80451i
$430$		0			0
$431$			204.460i			0.474384i	0.971463	+	0.237192i	$0.0762271\pi$
$431$			204.460i			0.474384i	−0.971463	+	0.237192i	$0.923773\pi$
$432$		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
$435$	60.3837	−	89.4919i	0.138813	−	0.205728i
$436$		0			0
$437$	232.654			0.532388
$438$		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
$443$		−	750.268i		−	1.69361i	−0.531906	−	0.846803i	$-0.678524\pi$
$443$		−	750.268i		−	1.69361i	0.531906	−	0.846803i	$-0.321476\pi$
$444$		0			0
$445$	−137.830	+	204.271i	−0.309730	+	0.459036i
$446$		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
$451$	−743.066			−1.64760
$452$		0			0
$453$	−156.939			−0.346445
$454$		0			0
$455$	274.724	−	407.156i	0.603790	−	0.894849i
$456$		0			0
$457$			870.823i			1.90552i	0.303724	+	0.952760i	$0.401770\pi$
$457$			870.823i			1.90552i	−0.303724	+	0.952760i	$0.598230\pi$
$458$		0			0
$459$	299.883			0.653340
$460$		0			0
$461$		−	37.3636i		−	0.0810490i	−0.999179	−	0.0405245i	$-0.987097\pi$
$461$		−	37.3636i		−	0.0810490i	0.999179	−	0.0405245i	$-0.0129029\pi$
$462$		0			0
$463$	−228.189			−0.492849			−0.246424	−	0.969162i	$-0.579256\pi$
$463$	−228.189			−0.492849			−0.246424	+	0.969162i	$0.579256\pi$
$464$		0			0
$465$	−297.431	+	440.809i	−0.639637	+	0.947977i
$466$		0			0
$467$		−	414.441i		−	0.887455i	−0.896162	−	0.443727i	$-0.853656\pi$
$467$		−	414.441i		−	0.887455i	0.896162	−	0.443727i	$-0.146344\pi$
$468$		0			0
$469$			511.082i			1.08973i
$470$		0			0
$471$			1361.60i			2.89086i
$472$		0			0
$473$			477.685i			1.00990i
$474$		0			0
$475$	97.2493	+	240.920i	0.204735	+	0.507201i
$476$		0			0
$477$	−1344.69			−2.81906
$478$		0			0
$479$		−	871.744i		−	1.81992i	−0.414691	−	0.909962i	$-0.636110\pi$
$479$		−	871.744i		−	1.81992i	0.414691	−	0.909962i	$-0.363890\pi$
$480$		0			0
$481$	−562.735			−1.16993
$482$		0			0
$483$		−	711.836i		−	1.47378i
$484$		0			0
$485$	−350.194	−	236.290i	−0.722049	−	0.487195i
$486$		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
$491$	372.142			0.757927			0.378963	−	0.925412i	$-0.376281\pi$
$491$	372.142			0.757927			0.378963	+	0.925412i	$0.376281\pi$
$492$		0			0
$493$	29.3924			0.0596195
$494$		0			0
$495$	−972.956	+	1441.97i	−1.96557	+	2.91308i
$496$		0			0
$497$		−	177.847i		−	0.357840i
$498$		0			0
$499$	198.184			0.397163			0.198582	−	0.980084i	$-0.436367\pi$
$499$	198.184			0.397163			0.198582	+	0.980084i	$0.436367\pi$
$500$		0			0
$501$		−	519.250i		−	1.03643i
$502$		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
$507$			423.955i			0.836204i
$508$		0			0
$509$		−	58.6571i		−	0.115240i	−0.998339	−	0.0576199i	$-0.981649\pi$
$509$		−	58.6571i		−	0.115240i	0.998339	−	0.0576199i	$-0.0183512\pi$
$510$		0			0
$511$		−	62.9906i		−	0.123269i
$512$		0			0
$513$			445.923i			0.869245i
$514$		0			0
$515$	−119.495	+	177.098i	−0.232029	+	0.343880i
$516$		0			0
$517$	977.130			1.89000
$518$		0			0
$519$		−	336.422i		−	0.648212i
$520$		0			0
$521$	171.431			0.329043			0.164521	−	0.986374i	$-0.447392\pi$
$521$	171.431			0.329043			0.164521	+	0.986374i	$0.447392\pi$
$522$		0			0
$523$			407.751i			0.779638i	0.920892	+	0.389819i	$0.127462\pi$
$523$			407.751i			0.779638i	−0.920892	+	0.389819i	$0.872538\pi$
$524$		0			0
$525$	737.128	−	297.547i	1.40405	−	0.566757i
$526$		0			0
$527$	−144.778			−0.274721
$528$		0			0
$529$	−27.8174			−0.0525848
$530$		0			0
$531$	−8.64391			−0.0162786
$532$		0			0
$533$	−588.030			−1.10325
$534$		0			0
$535$	−121.623	−	82.0640i	−0.227333	−	0.153391i
$536$		0			0
$537$			242.677i			0.451913i
$538$		0			0
$539$	−213.302			−0.395737
$540$		0			0
$541$			144.762i			0.267582i	0.991010	+	0.133791i	$0.0427150\pi$
$541$			144.762i			0.267582i	−0.991010	+	0.133791i	$0.957285\pi$
$542$		0			0
$543$	1312.31			2.41678
$544$		0			0
$545$	−677.386	−	457.059i	−1.24291	−	0.838640i
$546$		0			0
$547$			570.627i			1.04319i	0.853192	+	0.521597i	$0.174663\pi$
$547$			570.627i			1.04319i	−0.853192	+	0.521597i	$0.825337\pi$
$548$		0			0
$549$			1054.47i			1.92071i
$550$		0			0
$551$			43.7062i			0.0793216i
$552$		0			0
$553$		−	542.404i		−	0.980839i
$554$		0			0
$555$	−754.952	−	509.396i	−1.36027	−	0.917831i
$556$		0			0
$557$	824.208			1.47973			0.739864	−	0.672757i	$-0.234890\pi$
$557$	824.208			1.47973			0.739864	+	0.672757i	$0.234890\pi$
$558$		0			0
$559$			378.019i			0.676241i
$560$		0			0
$561$	−719.156			−1.28192
$562$		0			0
$563$			216.333i			0.384251i	0.981370	+	0.192125i	$0.0615380\pi$
$563$			216.333i			0.384251i	−0.981370	+	0.192125i	$0.938462\pi$
$564$		0			0
$565$	−500.211	−	337.512i	−0.885329	−	0.597366i
$566$		0			0
$567$	396.831			0.699877
$568$		0			0
$569$	−468.230			−0.822899			−0.411450	−	0.911432i	$-0.634977\pi$
$569$	−468.230			−0.822899			−0.411450	+	0.911432i	$0.634977\pi$
$570$		0			0
$571$	759.434			1.33001			0.665004	−	0.746840i	$-0.268430\pi$
$571$	759.434			1.33001			0.665004	+	0.746840i	$0.268430\pi$
$572$		0			0
$573$	347.603			0.606636
$574$		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
$579$	1679.39			2.90050
$580$		0			0
$581$			638.269i			1.09857i
$582$		0			0
$583$	1552.72			2.66332
$584$		0			0
$585$	−769.955	+	1141.11i	−1.31616	+	1.95062i
$586$		0			0
$587$			1014.74i			1.72869i	0.502900	+	0.864345i	$0.332266\pi$
$587$			1014.74i			1.72869i	−0.502900	+	0.864345i	$0.667734\pi$
$588$		0			0
$589$		−	215.283i		−	0.365506i
$590$		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
$595$	179.403	+	121.050i	0.301517	+	0.203445i
$596$		0			0
$597$	901.416			1.50991
$598$		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	$-0.672006\pi$
$601$	−618.377			−1.02891			−0.514456	+	0.857516i	$0.672006\pi$
$602$		0			0
$603$		−	1432.38i		−	2.37543i
$604$		0			0
$605$	785.082	−	1163.53i	1.29766	−	1.92320i
$606$		0			0
$607$	−182.393			−0.300483			−0.150241	−	0.988649i	$-0.548005\pi$
$607$	−182.393			−0.300483			−0.150241	+	0.988649i	$0.548005\pi$
$608$		0			0
$609$	133.725			0.219581
$610$		0			0
$611$	773.258			1.26556
$612$		0			0
$613$	638.630			1.04181			0.520906	−	0.853614i	$-0.325594\pi$
$613$	638.630			1.04181			0.520906	+	0.853614i	$0.325594\pi$
$614$		0			0
$615$	−788.888	−	532.294i	−1.28275	−	0.865518i
$616$		0			0
$617$			416.661i			0.675301i	0.941271	+	0.337651i	$0.109632\pi$
$617$			416.661i			0.675301i	−0.941271	+	0.337651i	$0.890368\pi$
$618$		0			0
$619$	1090.24			1.76129			0.880646	−	0.473775i	$-0.157109\pi$
$619$	1090.24			1.76129			0.880646	+	0.473775i	$0.157109\pi$
$620$		0			0
$621$			960.607i			1.54687i
$622$		0			0
$623$	−305.236			−0.489946
$624$		0			0
$625$	−449.863	+	433.876i	−0.719780	+	0.694202i
$626$		0			0
$627$		−	1069.38i		−	1.70555i
$628$		0			0
$629$		−	247.954i		−	0.394204i
$630$		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
$635$	−636.410	+	943.194i	−1.00222	+	1.48534i
$636$		0			0
$637$	−168.798			−0.264989
$638$		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
$643$		−	675.801i		−	1.05101i	−0.850790	−	0.525506i	$-0.823876\pi$
$643$		−	675.801i		−	1.05101i	0.850790	−	0.525506i	$-0.176124\pi$
$644$		0			0
$645$	−342.188	+	507.142i	−0.530525	+	0.786267i
$646$		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
$651$	−658.688			−1.01181
$652$		0			0
$653$	−771.853			−1.18201			−0.591005	−	0.806668i	$-0.701269\pi$
$653$	−771.853			−1.18201			−0.591005	+	0.806668i	$0.701269\pi$
$654$		0			0
$655$	95.4851	−	141.514i	0.145779	−	0.216052i
$656$		0			0
$657$			176.540i			0.268707i
$658$		0			0
$659$	−814.838			−1.23648			−0.618238	−	0.785991i	$-0.712153\pi$
$659$	−814.838			−1.23648			−0.618238	+	0.785991i	$0.712153\pi$
$660$		0			0
$661$		−	1038.13i		−	1.57055i	−0.619146	−	0.785276i	$-0.712521\pi$
$661$		−	1038.13i		−	1.57055i	0.619146	−	0.785276i	$-0.287479\pi$
$662$		0			0
$663$	−569.109			−0.858384
$664$		0			0
$665$	−180.000	+	266.770i	−0.270677	+	0.401158i
$666$		0			0
$667$			94.1519i			0.141157i
$668$		0			0
$669$			70.8869i			0.105959i
$670$		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
$675$	−994.739	+	401.534i	−1.47369	+	0.594865i
$676$		0			0
$677$	463.978			0.685344			0.342672	−	0.939455i	$-0.388668\pi$
$677$	463.978			0.685344			0.342672	+	0.939455i	$0.388668\pi$
$678$		0			0
$679$		−	523.284i		−	0.770669i
$680$		0			0
$681$	11.3673			0.0166920
$682$		0			0
$683$			632.563i			0.926154i	0.886318	+	0.463077i	$0.153255\pi$
$683$			632.563i			0.926154i	−0.886318	+	0.463077i	$0.846745\pi$
$684$		0			0
$685$	897.742	+	605.742i	1.31057	+	0.884294i
$686$		0			0
$687$	−1241.17			−1.80666
$688$		0			0
$689$	1228.75			1.78339
$690$		0			0
$691$	−436.001			−0.630970			−0.315485	−	0.948930i	$-0.602167\pi$
$691$	−436.001			−0.630970			−0.315485	+	0.948930i	$0.602167\pi$
$692$		0			0
$693$	−2154.70			−3.10923
$694$		0			0
$695$	−316.230	+	468.670i	−0.455008	+	0.674346i
$696$		0			0
$697$		−	259.100i		−	0.371736i
$698$		0			0
$699$	−156.648			−0.224103
$700$		0			0
$701$			364.085i			0.519380i	0.965692	+	0.259690i	$0.0836203\pi$
$701$			364.085i			0.519380i	−0.965692	+	0.259690i	$0.916380\pi$
$702$		0			0
$703$	368.705			0.524474
$704$		0			0
$705$	1037.39	+	699.965i	1.47147	+	0.992858i
$706$		0			0
$707$			574.678i			0.812840i
$708$		0			0
$709$			789.593i			1.11367i	0.830622	+	0.556836i	$0.187985\pi$
$709$			789.593i			1.11367i	−0.830622	+	0.556836i	$0.812015\pi$
$710$		0			0
$711$			1520.17i			2.13807i
$712$		0			0
$713$		−	463.763i		−	0.650439i
$714$		0			0
$715$	889.067	−	1317.65i	1.24345	−	1.84286i
$716$		0			0
$717$	−2189.34			−3.05347
$718$		0			0
$719$			168.422i			0.234245i	0.993118	+	0.117122i	$0.0373669\pi$
$719$			168.422i			0.234245i	−0.993118	+	0.117122i	$0.962633\pi$
$720$		0			0
$721$	−264.633			−0.367036
$722$		0			0
$723$			759.501i			1.05049i
$724$		0			0
$725$	−97.4973	+	39.3555i	−0.134479	+	0.0542835i
$726$		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
$731$	−166.564			−0.227858
$732$		0			0
$733$	74.5492			0.101704			0.0508521	−	0.998706i	$-0.483806\pi$
$733$	74.5492			0.101704			0.0508521	+	0.998706i	$0.483806\pi$
$734$		0			0
$735$	−226.456	−	152.799i	−0.308103	−	0.207889i
$736$		0			0
$737$			1653.97i			2.24420i
$738$		0			0
$739$	−682.206			−0.923148			−0.461574	−	0.887102i	$-0.652715\pi$
$739$	−682.206			−0.923148			−0.461574	+	0.887102i	$0.652715\pi$
$740$		0			0
$741$		−	846.259i		−	1.14205i
$742$		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
$747$		−	1788.84i		−	2.39470i
$748$		0			0
$749$		−	181.738i		−	0.242641i
$750$		0			0
$751$			532.038i			0.708439i	0.935162	+	0.354220i	$0.115253\pi$
$751$			532.038i			0.708439i	−0.935162	+	0.354220i	$0.884747\pi$
$752$		0			0
$753$			501.777i			0.666371i
$754$		0			0
$755$	126.700	+	85.4892i	0.167814	+	0.113231i
$756$		0			0
$757$	444.854			0.587653			0.293827	−	0.955859i	$-0.405071\pi$
$757$	444.854			0.587653			0.293827	+	0.955859i	$0.405071\pi$
$758$		0			0
$759$		−	2303.65i		−	3.03512i
$760$		0			0
$761$	−244.735			−0.321596			−0.160798	−	0.986987i	$-0.551407\pi$
$761$	−244.735			−0.321596			−0.160798	+	0.986987i	$0.551407\pi$
$762$		0			0
$763$		−	1012.20i		−	1.32660i
$764$		0			0
$765$	−502.802	−	339.260i	−0.657258	−	0.443477i
$766$		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
$771$	−1819.70			−2.36019
$772$		0			0
$773$	294.508			0.380994			0.190497	−	0.981688i	$-0.438990\pi$
$773$	294.508			0.380994			0.190497	+	0.981688i	$0.438990\pi$
$774$		0			0
$775$	480.241	−	193.853i	0.619666	−	0.250133i
$776$		0			0
$777$		−	1128.10i		−	1.45187i
$778$		0			0
$779$	385.279			0.494581
$780$		0			0
$781$		−	575.550i		−	0.736939i
$782$		0			0
$783$	−180.459			−0.230471
$784$		0			0
$785$	741.699	−	1099.24i	0.944839	−	1.40030i
$786$		0			0
$787$			1017.00i			1.29225i	0.763230	+	0.646127i	$0.223612\pi$
$787$			1017.00i			1.29225i	−0.763230	+	0.646127i	$0.776388\pi$
$788$		0			0
$789$		−	228.996i		−	0.290236i
$790$		0			0
$791$		−	747.450i		−	0.944943i
$792$		0			0
$793$		−	963.553i		−	1.21507i
$794$		0			0
$795$	1648.47	+	1112.29i	2.07355	+	1.39910i
$796$		0			0
$797$	−1105.61			−1.38722			−0.693608	−	0.720353i	$-0.743980\pi$
$797$	−1105.61			−1.38722			−0.693608	+	0.720353i	$0.743980\pi$
$798$		0			0
$799$			340.716i			0.426428i
$800$		0			0
$801$	855.469			1.06800
$802$		0			0
$803$		−	203.851i		−	0.253862i
$804$		0			0
$805$	−387.756	+	574.676i	−0.481685	+	0.713883i
$806$		0			0
$807$	−1616.58			−2.00320
$808$		0			0
$809$	−594.881			−0.735329			−0.367665	−	0.929958i	$-0.619843\pi$
$809$	−594.881			−0.735329			−0.367665	+	0.929958i	$0.619843\pi$
$810$		0			0
$811$	45.3204			0.0558822			0.0279411	−	0.999610i	$-0.491105\pi$
$811$	45.3204			0.0558822			0.0279411	+	0.999610i	$0.491105\pi$
$812$		0			0
$813$	2305.21			2.83543
$814$		0			0
$815$	−288.730	−	194.817i	−0.354270	−	0.239040i
$816$		0			0
$817$		−	247.679i		−	0.303156i
$818$		0			0
$819$	−1705.13			−2.08197
$820$		0			0
$821$		−	310.208i		−	0.377842i	−0.981992	−	0.188921i	$-0.939501\pi$
$821$		−	310.208i		−	0.377842i	0.981992	−	0.188921i	$-0.0604989\pi$
$822$		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
$827$		−	168.339i		−	0.203553i	−0.994807	−	0.101777i	$-0.967547\pi$
$827$		−	168.339i		−	0.203553i	0.994807	−	0.101777i	$-0.0324527\pi$
$828$		0			0
$829$			597.939i			0.721278i	0.932706	+	0.360639i	$0.117441\pi$
$829$			597.939i			0.721278i	−0.932706	+	0.360639i	$0.882559\pi$
$830$		0			0
$831$		−	1695.91i		−	2.04081i
$832$		0			0
$833$		−	74.3764i		−	0.0892873i
$834$		0			0
$835$	−282.849	+	419.198i	−0.338742	+	0.502034i
$836$		0			0
$837$	888.885			1.06199
$838$		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
$843$			1580.94i			1.87537i
$844$		0			0
$845$	230.940	−	342.265i	0.273302	−	0.405048i
$846$		0			0
$847$	1738.63			2.05270
$848$		0			0
$849$	937.514			1.10426
$850$		0			0
$851$	794.264			0.933331
$852$		0			0
$853$	889.393			1.04266			0.521332	−	0.853354i	$-0.325435\pi$
$853$	889.393			1.04266			0.521332	+	0.853354i	$0.325435\pi$
$854$		0			0
$855$	504.476	−	747.661i	0.590031	−	0.874457i
$856$		0			0
$857$		−	1246.97i		−	1.45504i	−0.686087	−	0.727520i	$-0.740673\pi$
$857$		−	1246.97i		−	1.45504i	0.686087	−	0.727520i	$-0.259327\pi$
$858$		0			0
$859$	977.927			1.13845			0.569224	−	0.822182i	$-0.307244\pi$
$859$	977.927			1.13845			0.569224	+	0.822182i	$0.307244\pi$
$860$		0			0
$861$		−	1178.81i		−	1.36912i
$862$		0			0
$863$	−1366.41			−1.58332			−0.791661	−	0.610961i	$-0.790783\pi$
$863$	−1366.41			−1.58332			−0.791661	+	0.610961i	$0.790783\pi$
$864$		0			0
$865$	−183.258	+	271.598i	−0.211859	+	0.313987i
$866$		0			0
$867$			1232.96i			1.42210i
$868$		0			0
$869$		−	1755.34i		−	2.01995i
$870$		0			0
$871$			1308.88i			1.50273i
$872$		0			0
$873$			1466.58i			1.67993i
$874$		0			0
$875$	−757.177	−	161.319i	−0.865345	−	0.184365i
$876$		0			0
$877$	87.0797			0.0992927			0.0496464	−	0.998767i	$-0.484191\pi$
$877$	87.0797			0.0992927			0.0496464	+	0.998767i	$0.484191\pi$
$878$		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
$883$			1604.11i			1.81666i	0.418249	+	0.908332i	$0.362644\pi$
$883$			1604.11i			1.81666i	−0.418249	+	0.908332i	$0.637356\pi$
$884$		0			0
$885$	10.5966	+	7.14996i	0.0119736	+	0.00807905i
$886$		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
$891$	1284.23			1.44133
$892$		0			0
$893$	−506.640			−0.567346
$894$		0			0
$895$	132.193	−	195.917i	0.147702	−	0.218902i
$896$		0			0
$897$		−	1823.01i		−	2.03234i
$898$		0			0
$899$	87.1223			0.0969102
$900$		0			0
$901$			541.418i			0.600907i
$902$		0			0
$903$	−757.807			−0.839210
$904$		0			0
$905$	−1059.45	−	714.852i	−1.17066	−	0.789892i
$906$		0			0
$907$		−	1079.04i		−	1.18968i	−0.803845	−	0.594838i	$-0.797216\pi$
$907$		−	1079.04i		−	1.18968i	0.803845	−	0.594838i	$-0.202784\pi$
$908$		0			0
$909$		−	1610.62i		−	1.77186i
$910$		0			0
$911$		−	1815.91i		−	1.99331i	−0.0816953	−	0.996657i	$-0.526033\pi$
$911$		−	1815.91i		−	1.99331i	0.0816953	−	0.996657i	$-0.473967\pi$
$912$		0			0
$913$			2065.58i			2.26241i
$914$		0			0
$915$	872.223	−	1292.68i	0.953249	−	1.41277i
$916$		0			0
$917$	211.460			0.230600
$918$		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
$923$		−	455.465i		−	0.493461i
$924$		0			0
$925$	332.003	+	822.486i	0.358922	+	0.889174i
$926$		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
$931$	110.597			0.118794
$932$		0			0
$933$	−765.670			−0.820654
$934$		0			0
$935$	580.586	+	391.744i	0.620947	+	0.418977i
$936$		0			0
$937$			1238.62i			1.32190i	0.750431	+	0.660949i	$0.229846\pi$
$937$			1238.62i			1.32190i	−0.750431	+	0.660949i	$0.770154\pi$
$938$		0			0
$939$	−1522.74			−1.62167
$940$		0			0
$941$			1846.13i			1.96188i	0.194299	+	0.980942i	$0.437757\pi$
$941$			1846.13i			1.96188i	−0.194299	+	0.980942i	$0.562243\pi$
$942$		0			0
$943$	829.967			0.880135
$944$		0			0
$945$	−1101.47	−	743.204i	−1.16558	−	0.786460i
$946$		0			0
$947$		−	466.767i		−	0.492890i	−0.969157	−	0.246445i	$-0.920737\pi$
$947$		−	466.767i		−	0.492890i	0.969157	−	0.246445i	$-0.0792625\pi$
$948$		0			0
$949$		−	161.319i		−	0.169988i
$950$		0			0
$951$		−	118.313i		−	0.124409i
$952$		0			0
$953$			1316.75i			1.38168i	0.723006	+	0.690842i	$0.242760\pi$
$953$			1316.75i			1.38168i	−0.723006	+	0.690842i	$0.757240\pi$
$954$		0			0
$955$	−280.625	−	189.349i	−0.293848	−	0.198271i
$956$		0			0
$957$	432.763			0.452208
$958$		0			0
$959$			1341.47i			1.39882i
$960$		0			0
$961$	531.863			0.553447
$962$		0			0
$963$			509.347i			0.528917i
$964$		0			0
$965$	−1355.80	−	914.810i	−1.40497	−	0.947990i
$966$		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
$971$	881.863			0.908201			0.454100	−	0.890951i	$-0.349961\pi$
$971$	881.863			0.908201			0.454100	+	0.890951i	$0.349961\pi$
$972$		0			0
$973$	−700.320			−0.719754
$974$		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
$979$	−987.811			−1.00900
$980$		0			0
$981$			2836.83i			2.89178i
$982$		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
$987$			1550.14i			1.57055i
$988$		0			0
$989$		−	533.550i		−	0.539484i
$990$		0			0
$991$			1696.66i			1.71207i	0.516916	+	0.856036i	$0.327080\pi$
$991$			1696.66i			1.71207i	−0.516916	+	0.856036i	$0.672920\pi$
$992$		0			0
$993$		−	2154.56i		−	2.16974i
$994$		0			0
$995$	−727.727	−	491.026i	−0.731384	−	0.493493i
$996$		0			0
$997$	−1405.45			−1.40968			−0.704840	−	0.709367i	$-0.748981\pi$
$997$	−1405.45			−1.40968			−0.704840	+	0.709367i	$0.748981\pi$
$998$		0			0
$999$			1522.35i			1.52387i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.b.159.1	✓	16	5.4	even	2	inner
320.3.e.b.159.2	yes	16	8.5	even	2	inner
320.3.e.b.159.3	yes	16	4.3	odd	2	inner
320.3.e.b.159.4	yes	16	40.19	odd	2	inner
320.3.e.b.159.13	yes	16	20.19	odd	2	inner
320.3.e.b.159.14	yes	16	8.3	odd	2	inner
320.3.e.b.159.15	yes	16	1.1	even	1	trivial
320.3.e.b.159.16	yes	16	40.29	even	2	inner
1280.3.h.n.1279.1		16	16.13	even	4
1280.3.h.n.1279.2		16	80.19	odd	4
1280.3.h.n.1279.3		16	80.69	even	4
1280.3.h.n.1279.4		16	16.11	odd	4
1280.3.h.n.1279.13		16	16.3	odd	4
1280.3.h.n.1279.14		16	80.29	even	4
1280.3.h.n.1279.15		16	80.59	odd	4
1280.3.h.n.1279.16		16	16.5	even	4
1600.3.g.k.351.1		16	20.7	even	4
1600.3.g.k.351.2		16	5.3	odd	4
1600.3.g.k.351.3		16	40.37	odd	4
1600.3.g.k.351.4		16	40.3	even	4
1600.3.g.k.351.13		16	40.13	odd	4
1600.3.g.k.351.14		16	40.27	even	4
1600.3.g.k.351.15		16	20.3	even	4
1600.3.g.k.351.16		16	5.2	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.e (of order \(2\), degree \(1\), minimal)

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