LMFDB - Embedded newform 880.3.n.a.639.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,3,Mod(639,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.639"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	880.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.9782632637$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} - 35 x^{18} + 600 x^{17} + 281 x^{16} - 17140 x^{15} + 16590 x^{14} + \cdots + 10140467375 $ x^20 - 10x^19 - 35x^18 + 600x^17 + 281x^16 - 17140x^15 + 16590x^14 + 237880x^13 - 363739x^12 - 1891870x^11 + 3816735x^10 + 6526640x^9 + 1425331x^8 - 80078900x^7 + 158315250x^6 - 131267320x^5 + 731692651x^4 - 1369163130x^3 + 2865489185x^2 - 2184739000*x + 10140467375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20}\cdot 5^{4}\cdot 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.2
Root			$5.71178 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	880.639
Dual form			880.3.n.a.639.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.21178 q^{3} +(2.33138 + 4.42320i) q^{5} -8.94099 q^{7} +18.1626 q^{9} +3.31662i q^{11} -4.60640i q^{13} +(-12.1507 - 23.0527i) q^{15} +32.7204i q^{17} +17.4499i q^{19} +46.5985 q^{21} +23.6253 q^{23} +(-14.1293 + 20.6243i) q^{25} -47.7537 q^{27} +7.05234 q^{29} +40.2409i q^{31} -17.2855i q^{33} +(-20.8449 - 39.5478i) q^{35} -54.9230i q^{37} +24.0075i q^{39} -13.5959 q^{41} +14.7714 q^{43} +(42.3441 + 80.3369i) q^{45} +59.2416 q^{47} +30.9413 q^{49} -170.531i q^{51} +13.6744i q^{53} +(-14.6701 + 7.73232i) q^{55} -90.9448i q^{57} +15.0250i q^{59} -109.420 q^{61} -162.392 q^{63} +(20.3750 - 10.7393i) q^{65} -129.380 q^{67} -123.130 q^{69} -44.6372i q^{71} +96.7596i q^{73} +(73.6389 - 107.489i) q^{75} -29.6539i q^{77} -46.4190i q^{79} +85.4178 q^{81} -129.339 q^{83} +(-144.729 + 76.2837i) q^{85} -36.7552 q^{87} +14.9683 q^{89} +41.1858i q^{91} -209.727i q^{93} +(-77.1841 + 40.6823i) q^{95} -168.151i q^{97} +60.2387i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 10 q^{5} + 40 q^{9} + 184 q^{21} - 18 q^{25} + 16 q^{29} + 144 q^{41} + 16 q^{45} + 36 q^{49} - 384 q^{61} - 32 q^{65} - 244 q^{69} + 236 q^{81} - 348 q^{85} - 100 q^{89}+O(q^{100}) $ 20 * q - 10 * q^5 + 40 * q^9 + 184 * q^21 - 18 * q^25 + 16 * q^29 + 144 * q^41 + 16 * q^45 + 36 * q^49 - 384 * q^61 - 32 * q^65 - 244 * q^69 + 236 * q^81 - 348 * q^85 - 100 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−5.21178			−1.73726			−0.868630	−	0.495462i	$-0.834999\pi$
$3$	−5.21178			−1.73726			−0.868630	+	0.495462i	$0.834999\pi$
$4$		0			0
$5$	2.33138	+	4.42320i	0.466276	+	0.884639i
$5$	2.33138	+	4.42320i	0.466276	+	0.884639i
$6$		0			0
$7$	−8.94099			−1.27728			−0.638642	−	0.769504i	$-0.720504\pi$
$7$	−8.94099			−1.27728			−0.638642	+	0.769504i	$0.720504\pi$
$8$		0			0
$9$	18.1626			2.01807
$10$		0			0
$11$			3.31662i			0.301511i
$11$			3.31662i			0.301511i
$12$		0			0
$13$		−	4.60640i		−	0.354339i	−0.984180	−	0.177169i	$-0.943306\pi$
$13$		−	4.60640i		−	0.354339i	0.984180	−	0.177169i	$-0.0566940\pi$
$14$		0			0
$15$	−12.1507	−	23.0527i	−0.810043	−	1.53685i
$16$		0			0
$17$			32.7204i			1.92473i	0.271766	+	0.962363i	$0.412392\pi$
$17$			32.7204i			1.92473i	−0.271766	+	0.962363i	$0.587608\pi$
$18$		0			0
$19$			17.4499i			0.918414i	0.888329	+	0.459207i	$0.151866\pi$
$19$			17.4499i			0.918414i	−0.888329	+	0.459207i	$0.848134\pi$
$20$		0			0
$21$	46.5985			2.21897
$22$		0			0
$23$	23.6253			1.02719			0.513593	−	0.858034i	$-0.328314\pi$
$23$	23.6253			1.02719			0.513593	+	0.858034i	$0.328314\pi$
$24$		0			0
$25$	−14.1293	+	20.6243i	−0.565172	+	0.824973i
$26$		0			0
$27$	−47.7537			−1.76865
$28$		0			0
$29$	7.05234			0.243184			0.121592	−	0.992580i	$-0.461200\pi$
$29$	7.05234			0.243184			0.121592	+	0.992580i	$0.461200\pi$
$30$		0			0
$31$			40.2409i			1.29809i	0.760749	+	0.649047i	$0.224832\pi$
$31$			40.2409i			1.29809i	−0.760749	+	0.649047i	$0.775168\pi$
$32$		0			0
$33$		−	17.2855i		−	0.523804i
$34$		0			0
$35$	−20.8449	−	39.5478i	−0.595568	−	1.12994i
$36$		0			0
$37$		−	54.9230i		−	1.48441i	−0.670176	−	0.742203i	$-0.733781\pi$
$37$		−	54.9230i		−	1.48441i	0.670176	−	0.742203i	$-0.266219\pi$
$38$		0			0
$39$			24.0075i			0.615578i
$40$		0			0
$41$	−13.5959			−0.331606			−0.165803	−	0.986159i	$-0.553022\pi$
$41$	−13.5959			−0.331606			−0.165803	+	0.986159i	$0.553022\pi$
$42$		0			0
$43$	14.7714			0.343521			0.171760	−	0.985139i	$-0.445054\pi$
$43$	14.7714			0.343521			0.171760	+	0.985139i	$0.445054\pi$
$44$		0			0
$45$	42.3441	+	80.3369i	0.940979	+	1.78526i
$46$		0			0
$47$	59.2416			1.26046			0.630229	−	0.776409i	$-0.282961\pi$
$47$	59.2416			1.26046			0.630229	+	0.776409i	$0.282961\pi$
$48$		0			0
$49$	30.9413			0.631456
$50$		0			0
$51$		−	170.531i		−	3.34375i
$52$		0			0
$53$			13.6744i			0.258007i	0.991644	+	0.129003i	$0.0411778\pi$
$53$			13.6744i			0.258007i	−0.991644	+	0.129003i	$0.958822\pi$
$54$		0			0
$55$	−14.6701	+	7.73232i	−0.266729	+	0.140588i
$56$		0			0
$57$		−	90.9448i		−	1.59552i
$58$		0			0
$59$			15.0250i			0.254661i	0.991860	+	0.127331i	$0.0406409\pi$
$59$			15.0250i			0.254661i	−0.991860	+	0.127331i	$0.959359\pi$
$60$		0			0
$61$	−109.420			−1.79378			−0.896889	−	0.442256i	$-0.854178\pi$
$61$	−109.420			−1.79378			−0.896889	+	0.442256i	$0.854178\pi$
$62$		0			0
$63$	−162.392			−2.57765
$64$		0			0
$65$	20.3750	−	10.7393i	0.313462	−	0.165220i
$66$		0			0
$67$	−129.380			−1.93105			−0.965523	−	0.260319i	$-0.916172\pi$
$67$	−129.380			−1.93105			−0.965523	+	0.260319i	$0.916172\pi$
$68$		0			0
$69$	−123.130			−1.78449
$70$		0			0
$71$		−	44.6372i		−	0.628693i	−0.949308	−	0.314346i	$-0.898215\pi$
$71$		−	44.6372i		−	0.628693i	0.949308	−	0.314346i	$-0.101785\pi$
$72$		0			0
$73$			96.7596i			1.32547i	0.748852	+	0.662737i	$0.230605\pi$
$73$			96.7596i			1.32547i	−0.748852	+	0.662737i	$0.769395\pi$
$74$		0			0
$75$	73.6389	−	107.489i	0.981851	−	1.43319i
$76$		0			0
$77$		−	29.6539i		−	0.385116i
$78$		0			0
$79$		−	46.4190i		−	0.587582i	−0.955870	−	0.293791i	$-0.905083\pi$
$79$		−	46.4190i		−	0.587582i	0.955870	−	0.293791i	$-0.0949169\pi$
$80$		0			0
$81$	85.4178			1.05454
$82$		0			0
$83$	−129.339			−1.55830			−0.779148	−	0.626840i	$-0.784348\pi$
$83$	−129.339			−1.55830			−0.779148	+	0.626840i	$0.784348\pi$
$84$		0			0
$85$	−144.729	+	76.2837i	−1.70269	+	0.897455i
$86$		0			0
$87$	−36.7552			−0.422474
$88$		0			0
$89$	14.9683			0.168184			0.0840919	−	0.996458i	$-0.473201\pi$
$89$	14.9683			0.168184			0.0840919	+	0.996458i	$0.473201\pi$
$90$		0			0
$91$			41.1858i			0.452591i
$92$		0			0
$93$		−	209.727i		−	2.25513i
$94$		0			0
$95$	−77.1841	+	40.6823i	−0.812465	+	0.428235i
$96$		0			0
$97$		−	168.151i		−	1.73352i	−0.498728	−	0.866759i	$-0.666199\pi$
$97$		−	168.151i		−	1.73352i	0.498728	−	0.866759i	$-0.333801\pi$
$98$		0			0
$99$			60.2387i			0.608471i
$100$		0			0
$101$	40.0084			0.396123			0.198062	−	0.980190i	$-0.436535\pi$
$101$	40.0084			0.396123			0.198062	+	0.980190i	$0.436535\pi$
$102$		0			0
$103$	−43.3528			−0.420901			−0.210451	−	0.977604i	$-0.567493\pi$
$103$	−43.3528			−0.420901			−0.210451	+	0.977604i	$0.567493\pi$
$104$		0			0
$105$	108.639	+	206.114i	1.03466	+	1.96299i
$106$		0			0
$107$	98.8535			0.923864			0.461932	−	0.886915i	$-0.347156\pi$
$107$	98.8535			0.923864			0.461932	+	0.886915i	$0.347156\pi$
$108$		0			0
$109$	−93.2143			−0.855177			−0.427589	−	0.903973i	$-0.640637\pi$
$109$	−93.2143			−0.855177			−0.427589	+	0.903973i	$0.640637\pi$
$110$		0			0
$111$			286.246i			2.57880i
$112$		0			0
$113$			11.7332i			0.103834i	0.998651	+	0.0519169i	$0.0165331\pi$
$113$			11.7332i			0.103834i	−0.998651	+	0.0519169i	$0.983467\pi$
$114$		0			0
$115$	55.0796	+	104.499i	0.478953	+	0.908689i
$116$		0			0
$117$		−	83.6644i		−	0.715080i
$118$		0			0
$119$		−	292.552i		−	2.45842i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$		0			0
$123$	70.8586			0.576086
$124$		0			0
$125$	−124.166	−	14.4135i	−0.993330	−	0.115308i
$126$		0			0
$127$	−18.3713			−0.144656			−0.0723281	−	0.997381i	$-0.523043\pi$
$127$	−18.3713			−0.144656			−0.0723281	+	0.997381i	$0.523043\pi$
$128$		0			0
$129$	−76.9853			−0.596785
$130$		0			0
$131$			17.1165i			0.130660i	0.997864	+	0.0653300i	$0.0208100\pi$
$131$			17.1165i			0.130660i	−0.997864	+	0.0653300i	$0.979190\pi$
$132$		0			0
$133$		−	156.019i		−	1.17308i
$134$		0			0
$135$	−111.332	−	211.224i	−0.824682	−	1.56462i
$136$		0			0
$137$		−	204.809i		−	1.49496i	−0.664285	−	0.747479i	$-0.731264\pi$
$137$		−	204.809i		−	1.49496i	0.664285	−	0.747479i	$-0.268736\pi$
$138$		0			0
$139$		−	154.632i		−	1.11246i	−0.831029	−	0.556229i	$-0.812248\pi$
$139$		−	154.632i		−	1.11246i	0.831029	−	0.556229i	$-0.187752\pi$
$140$		0			0
$141$	−308.754			−2.18974
$142$		0			0
$143$	15.2777			0.106837
$144$		0			0
$145$	16.4417	+	31.1939i	0.113391	+	0.215130i
$146$		0			0
$147$	−161.259			−1.09700
$148$		0			0
$149$	109.984			0.738151			0.369075	−	0.929399i	$-0.379674\pi$
$149$	109.984			0.738151			0.369075	+	0.929399i	$0.379674\pi$
$150$		0			0
$151$			157.308i			1.04178i	0.853625	+	0.520888i	$0.174399\pi$
$151$			157.308i			1.04178i	−0.853625	+	0.520888i	$0.825601\pi$
$152$		0			0
$153$			594.288i			3.88424i
$154$		0			0
$155$	−177.993	+	93.8169i	−1.14834	+	0.605270i
$156$		0			0
$157$		−	31.9788i		−	0.203687i	−0.994800	−	0.101843i	$-0.967526\pi$
$157$		−	31.9788i		−	0.203687i	0.994800	−	0.101843i	$-0.0324740\pi$
$158$		0			0
$159$		−	71.2677i		−	0.448225i
$160$		0			0
$161$	−211.234			−1.31201
$162$		0			0
$163$	−164.309			−1.00803			−0.504014	−	0.863695i	$-0.668144\pi$
$163$	−164.309			−1.00803			−0.504014	+	0.863695i	$0.668144\pi$
$164$		0			0
$165$	76.4572	−	40.2991i	0.463377	−	0.244237i
$166$		0			0
$167$	−73.3521			−0.439234			−0.219617	−	0.975586i	$-0.570481\pi$
$167$	−73.3521			−0.439234			−0.219617	+	0.975586i	$0.570481\pi$
$168$		0			0
$169$	147.781			0.874444
$170$		0			0
$171$			316.936i			1.85342i
$172$		0			0
$173$			252.766i			1.46108i	0.682872	+	0.730538i	$0.260731\pi$
$173$			252.766i			1.46108i	−0.682872	+	0.730538i	$0.739269\pi$
$174$		0			0
$175$	126.330	−	184.402i	0.721886	−	1.05372i
$176$		0			0
$177$		−	78.3071i		−	0.442413i
$178$		0			0
$179$		−	198.094i		−	1.10667i	−0.832958	−	0.553336i	$-0.813355\pi$
$179$		−	198.094i		−	1.10667i	0.832958	−	0.553336i	$-0.186645\pi$
$180$		0			0
$181$	162.434			0.897427			0.448713	−	0.893676i	$-0.351882\pi$
$181$	162.434			0.897427			0.448713	+	0.893676i	$0.351882\pi$
$182$		0			0
$183$	570.275			3.11626
$184$		0			0
$185$	242.935	−	128.046i	1.31316	−	0.692143i
$186$		0			0
$187$	−108.521			−0.580327
$188$		0			0
$189$	426.965			2.25907
$190$		0			0
$191$			339.870i			1.77943i	0.456520	+	0.889713i	$0.349096\pi$
$191$			339.870i			1.77943i	−0.456520	+	0.889713i	$0.650904\pi$
$192$		0			0
$193$			12.7864i			0.0662509i	0.999451	+	0.0331255i	$0.0105461\pi$
$193$			12.7864i			0.0662509i	−0.999451	+	0.0331255i	$0.989454\pi$
$194$		0			0
$195$	−106.190	+	55.9708i	−0.544564	+	0.287030i
$196$		0			0
$197$		−	164.109i		−	0.833041i	−0.909126	−	0.416520i	$-0.863249\pi$
$197$		−	164.109i		−	0.833041i	0.909126	−	0.416520i	$-0.136751\pi$
$198$		0			0
$199$			7.06858i			0.0355205i	0.999842	+	0.0177602i	$0.00565356\pi$
$199$			7.06858i			0.0355205i	−0.999842	+	0.0177602i	$0.994346\pi$
$200$		0			0
$201$	674.300			3.35473
$202$		0			0
$203$	−63.0549			−0.310615
$204$		0			0
$205$	−31.6971	−	60.1371i	−0.154620	−	0.293352i
$206$		0			0
$207$	429.098			2.07294
$208$		0			0
$209$	−57.8746			−0.276912
$210$		0			0
$211$		−	322.289i		−	1.52743i	−0.645551	−	0.763717i	$-0.723372\pi$
$211$		−	322.289i		−	1.52743i	0.645551	−	0.763717i	$-0.276628\pi$
$212$		0			0
$213$			232.639i			1.09220i
$214$		0			0
$215$	34.4378	+	65.3368i	0.160176	+	0.303892i
$216$		0			0
$217$		−	359.794i		−	1.65803i
$218$		0			0
$219$		−	504.290i		−	2.30269i
$220$		0			0
$221$	150.723			0.682005
$222$		0			0
$223$	223.474			1.00213			0.501063	−	0.865411i	$-0.332943\pi$
$223$	223.474			1.00213			0.501063	+	0.865411i	$0.332943\pi$
$224$		0			0
$225$	−256.626	+	374.592i	−1.14056	+	1.66485i
$226$		0			0
$227$	207.360			0.913481			0.456741	−	0.889600i	$-0.349017\pi$
$227$	207.360			0.913481			0.456741	+	0.889600i	$0.349017\pi$
$228$		0			0
$229$	−341.681			−1.49206			−0.746028	−	0.665915i	$-0.768042\pi$
$229$	−341.681			−1.49206			−0.746028	+	0.665915i	$0.768042\pi$
$230$		0			0
$231$			154.550i			0.669046i
$232$		0			0
$233$			284.842i			1.22250i	0.791439	+	0.611248i	$0.209332\pi$
$233$			284.842i			1.22250i	−0.791439	+	0.611248i	$0.790668\pi$
$234$		0			0
$235$	138.115	+	262.037i	0.587722	+	1.11505i
$236$		0			0
$237$			241.925i			1.02078i
$238$		0			0
$239$			284.588i			1.19075i	0.803449	+	0.595373i	$0.202996\pi$
$239$			284.588i			1.19075i	−0.803449	+	0.595373i	$0.797004\pi$
$240$		0			0
$241$	59.1427			0.245405			0.122703	−	0.992443i	$-0.460844\pi$
$241$	59.1427			0.245405			0.122703	+	0.992443i	$0.460844\pi$
$242$		0			0
$243$	−15.3957			−0.0633567
$244$		0			0
$245$	72.1361	+	136.860i	0.294433	+	0.558610i
$246$		0			0
$247$	80.3811			0.325429
$248$		0			0
$249$	674.084			2.70716
$250$		0			0
$251$		−	265.930i		−	1.05948i	−0.848160	−	0.529741i	$-0.822289\pi$
$251$		−	265.930i		−	1.05948i	0.848160	−	0.529741i	$-0.177711\pi$
$252$		0			0
$253$			78.3562i			0.309708i
$254$		0			0
$255$	754.293	−	397.574i	2.95801	−	1.55911i
$256$		0			0
$257$		−	7.52232i		−	0.0292697i	−0.999893	−	0.0146349i	$-0.995341\pi$
$257$		−	7.52232i		−	0.0292697i	0.999893	−	0.0146349i	$-0.00465859\pi$
$258$		0			0
$259$			491.066i			1.89601i
$260$		0			0
$261$	128.089			0.490763
$262$		0			0
$263$	150.331			0.571602			0.285801	−	0.958289i	$-0.407740\pi$
$263$	150.331			0.571602			0.285801	+	0.958289i	$0.407740\pi$
$264$		0			0
$265$	−60.4843	+	31.8801i	−0.228243	+	0.120302i
$266$		0			0
$267$	−78.0117			−0.292179
$268$		0			0
$269$	−458.467			−1.70434			−0.852169	−	0.523267i	$-0.824713\pi$
$269$	−458.467			−1.70434			−0.852169	+	0.523267i	$0.824713\pi$
$270$		0			0
$271$			308.200i			1.13727i	0.822590	+	0.568635i	$0.192528\pi$
$271$			308.200i			1.13727i	−0.822590	+	0.568635i	$0.807472\pi$
$272$		0			0
$273$		−	214.651i		−	0.786268i
$274$		0			0
$275$	−68.4031	−	46.8616i	−0.248739	−	0.170406i
$276$		0			0
$277$			234.559i			0.846783i	0.905947	+	0.423391i	$0.139161\pi$
$277$			234.559i			0.846783i	−0.905947	+	0.423391i	$0.860839\pi$
$278$		0			0
$279$			730.881i			2.61965i
$280$		0			0
$281$	−425.745			−1.51511			−0.757553	−	0.652774i	$-0.773605\pi$
$281$	−425.745			−1.51511			−0.757553	+	0.652774i	$0.773605\pi$
$282$		0			0
$283$	120.687			0.426455			0.213228	−	0.977003i	$-0.431602\pi$
$283$	120.687			0.426455			0.213228	+	0.977003i	$0.431602\pi$
$284$		0			0
$285$	402.267	−	212.027i	1.41146	−	0.743955i
$286$		0			0
$287$	121.560			0.423556
$288$		0			0
$289$	−781.622			−2.70457
$290$		0			0
$291$			876.367i			3.01157i
$292$		0			0
$293$		−	479.969i		−	1.63812i	−0.573708	−	0.819060i	$-0.694495\pi$
$293$		−	479.969i		−	1.63812i	0.573708	−	0.819060i	$-0.305505\pi$
$294$		0			0
$295$	−66.4586	+	35.0291i	−0.225283	+	0.118743i
$296$		0			0
$297$		−	158.381i		−	0.533269i
$298$		0			0
$299$		−	108.828i		−	0.363972i
$300$		0			0
$301$	−132.071			−0.438774
$302$		0			0
$303$	−208.515			−0.688169
$304$		0			0
$305$	−255.101	−	483.988i	−0.836396	−	1.58685i
$306$		0			0
$307$	65.1335			0.212161			0.106081	−	0.994358i	$-0.466170\pi$
$307$	65.1335			0.212161			0.106081	+	0.994358i	$0.466170\pi$
$308$		0			0
$309$	225.945			0.731215
$310$		0			0
$311$			83.2282i			0.267615i	0.991007	+	0.133807i	$0.0427203\pi$
$311$			83.2282i			0.267615i	−0.991007	+	0.133807i	$0.957280\pi$
$312$		0			0
$313$			192.654i			0.615508i	0.951466	+	0.307754i	$0.0995775\pi$
$313$			192.654i			0.615508i	−0.951466	+	0.307754i	$0.900423\pi$
$314$		0			0
$315$	−378.598	−	718.292i	−1.20190	−	2.28029i
$316$		0			0
$317$			232.076i			0.732100i	0.930595	+	0.366050i	$0.119290\pi$
$317$			232.076i			0.732100i	−0.930595	+	0.366050i	$0.880710\pi$
$318$		0			0
$319$			23.3900i			0.0733228i
$320$		0			0
$321$	−515.203			−1.60499
$322$		0			0
$323$	−570.966			−1.76770
$324$		0			0
$325$	95.0039	+	65.0853i	0.292320	+	0.200262i
$326$		0			0
$327$	485.813			1.48567
$328$		0			0
$329$	−529.678			−1.60996
$330$		0			0
$331$			97.7582i			0.295342i	0.989037	+	0.147671i	$0.0471777\pi$
$331$			97.7582i			0.295342i	−0.989037	+	0.147671i	$0.952822\pi$
$332$		0			0
$333$		−	997.547i		−	2.99564i
$334$		0			0
$335$	−301.634	−	572.273i	−0.900401	−	1.70828i
$336$		0			0
$337$		−	422.995i		−	1.25518i	−0.778545	−	0.627589i	$-0.784042\pi$
$337$		−	422.995i		−	1.25518i	0.778545	−	0.627589i	$-0.215958\pi$
$338$		0			0
$339$		−	61.1509i		−	0.180386i
$340$		0			0
$341$	−133.464			−0.391390
$342$		0			0
$343$	161.462			0.470736
$344$		0			0
$345$	−287.063	−	544.627i	−0.832066	−	1.57863i
$346$		0			0
$347$	−440.881			−1.27055			−0.635276	−	0.772286i	$-0.719113\pi$
$347$	−440.881			−1.27055			−0.635276	+	0.772286i	$0.719113\pi$
$348$		0			0
$349$	436.709			1.25131			0.625657	−	0.780098i	$-0.284831\pi$
$349$	436.709			1.25131			0.625657	+	0.780098i	$0.284831\pi$
$350$		0			0
$351$			219.973i			0.626702i
$352$		0			0
$353$		−	270.780i		−	0.767082i	−0.923524	−	0.383541i	$-0.874705\pi$
$353$		−	270.780i		−	0.767082i	0.923524	−	0.383541i	$-0.125295\pi$
$354$		0			0
$355$	197.439	−	104.066i	0.556166	−	0.293145i
$356$		0			0
$357$			1524.72i			4.27092i
$358$		0			0
$359$		−	482.341i		−	1.34357i	−0.740747	−	0.671785i	$-0.765528\pi$
$359$		−	482.341i		−	1.34357i	0.740747	−	0.671785i	$-0.234472\pi$
$360$		0			0
$361$	56.5023			0.156516
$362$		0			0
$363$	57.3296			0.157933
$364$		0			0
$365$	−427.987	+	225.584i	−1.17257	+	0.618037i
$366$		0			0
$367$	−116.977			−0.318739			−0.159370	−	0.987219i	$-0.550946\pi$
$367$	−116.977			−0.318739			−0.159370	+	0.987219i	$0.550946\pi$
$368$		0			0
$369$	−246.937			−0.669205
$370$		0			0
$371$		−	122.262i		−	0.329548i
$372$		0			0
$373$			24.6905i			0.0661944i	0.999452	+	0.0330972i	$0.0105371\pi$
$373$			24.6905i			0.0661944i	−0.999452	+	0.0330972i	$0.989463\pi$
$374$		0			0
$375$	647.127	+	75.1201i	1.72567	+	0.200320i
$376$		0			0
$377$		−	32.4859i		−	0.0861695i
$378$		0			0
$379$		−	73.5825i		−	0.194149i	−0.995277	−	0.0970746i	$-0.969051\pi$
$379$		−	73.5825i		−	0.194149i	0.995277	−	0.0970746i	$-0.0309485\pi$
$380$		0			0
$381$	95.7474			0.251305
$382$		0			0
$383$	6.13890			0.0160285			0.00801423	−	0.999968i	$-0.497449\pi$
$383$	6.13890			0.0160285			0.00801423	+	0.999968i	$0.497449\pi$
$384$		0			0
$385$	131.165	−	69.1346i	0.340688	−	0.179570i
$386$		0			0
$387$	268.288			0.693250
$388$		0			0
$389$	503.766			1.29503			0.647514	−	0.762053i	$-0.275809\pi$
$389$	503.766			1.29503			0.647514	+	0.762053i	$0.275809\pi$
$390$		0			0
$391$			773.028i			1.97705i
$392$		0			0
$393$		−	89.2072i		−	0.226990i
$394$		0			0
$395$	205.320	−	108.220i	0.519798	−	0.273976i
$396$		0			0
$397$		−	567.375i		−	1.42916i	−0.699555	−	0.714578i	$-0.746619\pi$
$397$		−	567.375i		−	1.42916i	0.699555	−	0.714578i	$-0.253381\pi$
$398$		0			0
$399$			813.137i			2.03794i
$400$		0			0
$401$	−516.351			−1.28766			−0.643830	−	0.765169i	$-0.722656\pi$
$401$	−516.351			−1.28766			−0.643830	+	0.765169i	$0.722656\pi$
$402$		0			0
$403$	185.366			0.459965
$404$		0			0
$405$	199.142	+	377.820i	0.491708	+	0.932888i
$406$		0			0
$407$	182.159			0.447565
$408$		0			0
$409$	−393.925			−0.963141			−0.481571	−	0.876407i	$-0.659933\pi$
$409$	−393.925			−0.963141			−0.481571	+	0.876407i	$0.659933\pi$
$410$		0			0
$411$			1067.42i			2.59713i
$412$		0			0
$413$		−	134.339i		−	0.325275i
$414$		0			0
$415$	−301.538	−	572.090i	−0.726597	−	1.37853i
$416$		0			0
$417$			805.906i			1.93263i
$418$		0			0
$419$			107.398i			0.256319i	0.991754	+	0.128159i	$0.0409069\pi$
$419$			107.398i			0.256319i	−0.991754	+	0.128159i	$0.959093\pi$
$420$		0			0
$421$	92.3044			0.219250			0.109625	−	0.993973i	$-0.465035\pi$
$421$	92.3044			0.219250			0.109625	+	0.993973i	$0.465035\pi$
$422$		0			0
$423$	1075.98			2.54370
$424$		0			0
$425$	−674.835	−	462.316i	−1.58785	−	1.08780i
$426$		0			0
$427$	978.327			2.29116
$428$		0			0
$429$	−79.6240			−0.185604
$430$		0			0
$431$		−	443.446i		−	1.02888i	−0.857527	−	0.514439i	$-0.828000\pi$
$431$		−	443.446i		−	1.02888i	0.857527	−	0.514439i	$-0.172000\pi$
$432$		0			0
$433$			468.756i			1.08258i	0.840837	+	0.541289i	$0.182063\pi$
$433$			468.756i			1.08258i	−0.840837	+	0.541289i	$0.817937\pi$
$434$		0			0
$435$	−85.6905	−	162.576i	−0.196990	−	0.373737i
$436$		0			0
$437$			412.258i			0.943382i
$438$		0			0
$439$			67.3501i			0.153417i	0.997054	+	0.0767085i	$0.0244411\pi$
$439$			67.3501i			0.153417i	−0.997054	+	0.0767085i	$0.975559\pi$
$440$		0			0
$441$	561.976			1.27432
$442$		0			0
$443$	−338.549			−0.764219			−0.382110	−	0.924117i	$-0.624802\pi$
$443$	−338.549			−0.764219			−0.382110	+	0.924117i	$0.624802\pi$
$444$		0			0
$445$	34.8969	+	66.2079i	0.0784201	+	0.148782i
$446$		0			0
$447$	−573.215			−1.28236
$448$		0			0
$449$	132.750			0.295657			0.147828	−	0.989013i	$-0.452772\pi$
$449$	132.750			0.295657			0.147828	+	0.989013i	$0.452772\pi$
$450$		0			0
$451$		−	45.0924i		−	0.0999831i
$452$		0			0
$453$		−	819.856i		−	1.80984i
$454$		0			0
$455$	−182.173	+	96.0198i	−0.400380	+	0.211033i
$456$		0			0
$457$			529.101i			1.15777i	0.815409	+	0.578885i	$0.196512\pi$
$457$			529.101i			1.15777i	−0.815409	+	0.578885i	$0.803488\pi$
$458$		0			0
$459$		−	1562.52i		−	3.40418i
$460$		0			0
$461$	491.865			1.06695			0.533477	−	0.845815i	$-0.320885\pi$
$461$	491.865			1.06695			0.533477	+	0.845815i	$0.320885\pi$
$462$		0			0
$463$	−714.900			−1.54406			−0.772031	−	0.635585i	$-0.780759\pi$
$463$	−714.900			−1.54406			−0.772031	+	0.635585i	$0.780759\pi$
$464$		0			0
$465$	927.662	−	488.953i	1.99497	−	1.05151i
$466$		0			0
$467$	−606.408			−1.29852			−0.649260	−	0.760567i	$-0.724921\pi$
$467$	−606.408			−1.29852			−0.649260	+	0.760567i	$0.724921\pi$
$468$		0			0
$469$	1156.79			2.46649
$470$		0			0
$471$			166.666i			0.353856i
$472$		0			0
$473$			48.9912i			0.103575i
$474$		0			0
$475$	−359.892	−	246.555i	−0.757666	−	0.519062i
$476$		0			0
$477$			248.362i			0.520676i
$478$		0			0
$479$			176.822i			0.369148i	0.982819	+	0.184574i	$0.0590906\pi$
$479$			176.822i			0.369148i	−0.982819	+	0.184574i	$0.940909\pi$
$480$		0			0
$481$	−252.997			−0.525982
$482$		0			0
$483$	1100.90			2.27930
$484$		0			0
$485$	743.766	−	392.025i	1.53354	−	0.808298i
$486$		0			0
$487$	479.170			0.983922			0.491961	−	0.870617i	$-0.336280\pi$
$487$	479.170			0.983922			0.491961	+	0.870617i	$0.336280\pi$
$488$		0			0
$489$	856.340			1.75121
$490$		0			0
$491$			397.341i			0.809248i	0.914483	+	0.404624i	$0.132598\pi$
$491$			397.341i			0.809248i	−0.914483	+	0.404624i	$0.867402\pi$
$492$		0			0
$493$			230.755i			0.468063i
$494$		0			0
$495$	−266.447	+	140.439i	−0.538278	+	0.283716i
$496$		0			0
$497$			399.101i			0.803019i
$498$		0			0
$499$		−	251.648i		−	0.504305i	−0.967687	−	0.252153i	$-0.918862\pi$
$499$		−	251.648i		−	0.504305i	0.967687	−	0.252153i	$-0.0811385\pi$
$500$		0			0
$501$	382.295			0.763063
$502$		0			0
$503$	−341.999			−0.679919			−0.339959	−	0.940440i	$-0.610413\pi$
$503$	−341.999			−0.679919			−0.339959	+	0.940440i	$0.610413\pi$
$504$		0			0
$505$	93.2749	+	176.965i	0.184703	+	0.350426i
$506$		0			0
$507$	−770.202			−1.51914
$508$		0			0
$509$	−348.704			−0.685077			−0.342538	−	0.939504i	$-0.611287\pi$
$509$	−348.704			−0.685077			−0.342538	+	0.939504i	$0.611287\pi$
$510$		0			0
$511$		−	865.127i		−	1.69301i
$512$		0			0
$513$		−	833.295i		−	1.62436i
$514$		0			0
$515$	−101.072	−	191.758i	−0.196256	−	0.372346i
$516$		0			0
$517$			196.482i			0.380043i
$518$		0			0
$519$		−	1317.36i		−	2.53827i
$520$		0			0
$521$	−415.301			−0.797123			−0.398562	−	0.917142i	$-0.630491\pi$
$521$	−415.301			−0.797123			−0.398562	+	0.917142i	$0.630491\pi$
$522$		0			0
$523$	744.535			1.42359			0.711793	−	0.702389i	$-0.247883\pi$
$523$	744.535			1.42359			0.711793	+	0.702389i	$0.247883\pi$
$524$		0			0
$525$	−658.404	+	961.062i	−1.25410	+	1.83059i
$526$		0			0
$527$	−1316.70			−2.49848
$528$		0			0
$529$	29.1544			0.0551122
$530$		0			0
$531$			272.894i			0.513925i
$532$		0			0
$533$			62.6280i			0.117501i
$534$		0			0
$535$	230.465	+	437.248i	0.430776	+	0.817287i
$536$		0			0
$537$			1032.42i			1.92258i
$538$		0			0
$539$			102.621i			0.190391i
$540$		0			0
$541$	−257.151			−0.475326			−0.237663	−	0.971348i	$-0.576381\pi$
$541$	−257.151			−0.475326			−0.237663	+	0.971348i	$0.576381\pi$
$542$		0			0
$543$	−846.571			−1.55906
$544$		0			0
$545$	−217.318	−	412.305i	−0.398749	−	0.756523i
$546$		0			0
$547$	551.325			1.00791			0.503953	−	0.863731i	$-0.331878\pi$
$547$	551.325			1.00791			0.503953	+	0.863731i	$0.331878\pi$
$548$		0			0
$549$	−1987.36			−3.61997
$550$		0			0
$551$			123.062i			0.223344i
$552$		0			0
$553$			415.032i			0.750509i
$554$		0			0
$555$	−1266.12	+	667.350i	−2.28130	+	1.20243i
$556$		0			0
$557$			841.665i			1.51107i	0.655109	+	0.755534i	$0.272623\pi$
$557$			841.665i			1.51107i	−0.655109	+	0.755534i	$0.727377\pi$
$558$		0			0
$559$		−	68.0430i		−	0.121723i
$560$		0			0
$561$	565.588			1.00818
$562$		0			0
$563$	−575.688			−1.02254			−0.511269	−	0.859421i	$-0.670824\pi$
$563$	−575.688			−1.02254			−0.511269	+	0.859421i	$0.670824\pi$
$564$		0			0
$565$	−51.8983	+	27.3546i	−0.0918554	+	0.0484152i
$566$		0			0
$567$	−763.720			−1.34695
$568$		0			0
$569$	735.995			1.29349			0.646744	−	0.762707i	$-0.276130\pi$
$569$	735.995			1.29349			0.646744	+	0.762707i	$0.276130\pi$
$570$		0			0
$571$		−	632.991i		−	1.10857i	−0.832328	−	0.554283i	$-0.812993\pi$
$571$		−	632.991i		−	1.10857i	0.832328	−	0.554283i	$-0.187007\pi$
$572$		0			0
$573$		−	1771.33i		−	3.09133i
$574$		0			0
$575$	−333.809	+	487.256i	−0.580538	+	0.847401i
$576$		0			0
$577$		−	860.484i		−	1.49131i	−0.666334	−	0.745654i	$-0.732137\pi$
$577$		−	860.484i		−	1.49131i	0.666334	−	0.745654i	$-0.267863\pi$
$578$		0			0
$579$		−	66.6400i		−	0.115095i
$580$		0			0
$581$	1156.41			1.99039
$582$		0			0
$583$	−45.3527			−0.0777919
$584$		0			0
$585$	370.064	−	195.054i	0.632588	−	0.333425i
$586$		0			0
$587$	−645.463			−1.09960			−0.549798	−	0.835298i	$-0.685295\pi$
$587$	−645.463			−1.09960			−0.549798	+	0.835298i	$0.685295\pi$
$588$		0			0
$589$	−702.198			−1.19219
$590$		0			0
$591$			855.300i			1.44721i
$592$		0			0
$593$			159.924i			0.269686i	0.990867	+	0.134843i	$0.0430530\pi$
$593$			159.924i			0.269686i	−0.990867	+	0.134843i	$0.956947\pi$
$594$		0			0
$595$	1294.02	−	682.052i	2.17482	−	1.14631i
$596$		0			0
$597$		−	36.8399i		−	0.0617083i
$598$		0			0
$599$			45.3949i			0.0757844i	0.999282	+	0.0378922i	$0.0120644\pi$
$599$			45.3949i			0.0757844i	−0.999282	+	0.0378922i	$0.987936\pi$
$600$		0			0
$601$	455.715			0.758262			0.379131	−	0.925343i	$-0.376223\pi$
$601$	455.715			0.758262			0.379131	+	0.925343i	$0.376223\pi$
$602$		0			0
$603$	−2349.88			−3.89699
$604$		0			0
$605$	−25.6452	−	48.6551i	−0.0423888	−	0.0804217i
$606$		0			0
$607$	−57.0950			−0.0940609			−0.0470305	−	0.998893i	$-0.514976\pi$
$607$	−57.0950			−0.0940609			−0.0470305	+	0.998893i	$0.514976\pi$
$608$		0			0
$609$	328.628			0.539619
$610$		0			0
$611$		−	272.890i		−	0.446629i
$612$		0			0
$613$			756.252i			1.23369i	0.787085	+	0.616845i	$0.211589\pi$
$613$			756.252i			1.23369i	−0.787085	+	0.616845i	$0.788411\pi$
$614$		0			0
$615$	165.199	+	313.422i	0.268616	+	0.509628i
$616$		0			0
$617$		−	392.159i		−	0.635590i	−0.948159	−	0.317795i	$-0.897058\pi$
$617$		−	392.159i		−	0.635590i	0.948159	−	0.317795i	$-0.102942\pi$
$618$		0			0
$619$		−	1017.10i		−	1.64314i	−0.570109	−	0.821569i	$-0.693099\pi$
$619$		−	1017.10i		−	1.64314i	0.570109	−	0.821569i	$-0.306901\pi$
$620$		0			0
$621$	−1128.19			−1.81674
$622$		0			0
$623$	−133.832			−0.214818
$624$		0			0
$625$	−225.725	−	582.815i	−0.361160	−	0.932504i
$626$		0			0
$627$	301.630			0.481068
$628$		0			0
$629$	1797.10			2.85707
$630$		0			0
$631$			203.151i			0.321952i	0.986958	+	0.160976i	$0.0514641\pi$
$631$			203.151i			0.321952i	−0.986958	+	0.160976i	$0.948536\pi$
$632$		0			0
$633$			1679.70i			2.65355i
$634$		0			0
$635$	−42.8306	−	81.2600i	−0.0674498	−	0.127969i
$636$		0			0
$637$		−	142.528i		−	0.223749i
$638$		0			0
$639$		−	810.729i		−	1.26875i
$640$		0			0
$641$	239.790			0.374087			0.187043	−	0.982352i	$-0.440109\pi$
$641$	239.790			0.374087			0.187043	+	0.982352i	$0.440109\pi$
$642$		0			0
$643$	−44.3768			−0.0690153			−0.0345076	−	0.999404i	$-0.510986\pi$
$643$	−44.3768			−0.0690153			−0.0345076	+	0.999404i	$0.510986\pi$
$644$		0			0
$645$	−179.482	−	340.521i	−0.278267	−	0.527939i
$646$		0			0
$647$	234.286			0.362111			0.181055	−	0.983473i	$-0.442049\pi$
$647$	234.286			0.362111			0.181055	+	0.983473i	$0.442049\pi$
$648$		0			0
$649$	−49.8323			−0.0767833
$650$		0			0
$651$			1875.16i			2.88044i
$652$		0			0
$653$		−	710.306i		−	1.08776i	−0.839163	−	0.543879i	$-0.816955\pi$
$653$		−	710.306i		−	1.08776i	0.839163	−	0.543879i	$-0.183045\pi$
$654$		0			0
$655$	−75.7094	+	39.9050i	−0.115587	+	0.0609237i
$656$		0			0
$657$			1757.41i			2.67490i
$658$		0			0
$659$		−	355.382i		−	0.539275i	−0.962962	−	0.269637i	$-0.913096\pi$
$659$		−	355.382i		−	0.539275i	0.962962	−	0.269637i	$-0.0869038\pi$
$660$		0			0
$661$	−442.779			−0.669862			−0.334931	−	0.942243i	$-0.608713\pi$
$661$	−442.779			−0.669862			−0.334931	+	0.942243i	$0.608713\pi$
$662$		0			0
$663$	−785.535			−1.18482
$664$		0			0
$665$	690.103	−	363.740i	1.03775	−	0.546978i
$666$		0			0
$667$	166.614			0.249795
$668$		0			0
$669$	−1164.70			−1.74095
$670$		0			0
$671$		−	362.906i		−	0.540844i
$672$		0			0
$673$		−	179.454i		−	0.266648i	−0.991072	−	0.133324i	$-0.957435\pi$
$673$		−	179.454i		−	0.266648i	0.991072	−	0.133324i	$-0.0425651\pi$
$674$		0			0
$675$	674.727	−	984.887i	0.999595	−	1.45909i
$676$		0			0
$677$			573.195i			0.846669i	0.905973	+	0.423335i	$0.139141\pi$
$677$			573.195i			0.846669i	−0.905973	+	0.423335i	$0.860859\pi$
$678$		0			0
$679$			1503.44i			2.21419i
$680$		0			0
$681$	−1080.72			−1.58695
$682$		0			0
$683$	−410.490			−0.601010			−0.300505	−	0.953780i	$-0.597155\pi$
$683$	−410.490			−0.601010			−0.300505	+	0.953780i	$0.597155\pi$
$684$		0			0
$685$	905.911	−	477.489i	1.32250	−	0.697064i
$686$		0			0
$687$	1780.76			2.59209
$688$		0			0
$689$	62.9895			0.0914217
$690$		0			0
$691$			1283.81i			1.85789i	0.370212	+	0.928947i	$0.379285\pi$
$691$			1283.81i			1.85789i	−0.370212	+	0.928947i	$0.620715\pi$
$692$		0			0
$693$		−	538.593i		−	0.777191i
$694$		0			0
$695$	683.966	−	360.505i	0.984123	−	0.518713i
$696$		0			0
$697$		−	444.861i		−	0.638252i
$698$		0			0
$699$		−	1484.53i		−	2.12379i
$700$		0			0
$701$	−653.254			−0.931889			−0.465944	−	0.884814i	$-0.654285\pi$
$701$	−653.254			−0.931889			−0.465944	+	0.884814i	$0.654285\pi$
$702$		0			0
$703$	958.399			1.36330
$704$		0			0
$705$	−719.823	−	1365.68i	−1.02103	−	1.93713i
$706$		0			0
$707$	−357.715			−0.505962
$708$		0			0
$709$	175.725			0.247850			0.123925	−	0.992292i	$-0.460452\pi$
$709$	175.725			0.247850			0.123925	+	0.992292i	$0.460452\pi$
$710$		0			0
$711$		−	843.091i		−	1.18578i
$712$		0			0
$713$			950.703i			1.33338i
$714$		0			0
$715$	35.6182	+	67.5763i	0.0498156	+	0.0945123i
$716$		0			0
$717$		−	1483.21i		−	2.06863i
$718$		0			0
$719$		−	695.911i		−	0.967887i	−0.875099	−	0.483943i	$-0.839204\pi$
$719$		−	695.911i		−	0.967887i	0.875099	−	0.483943i	$-0.160796\pi$
$720$		0			0
$721$	387.617			0.537611
$722$		0			0
$723$	−308.239			−0.426333
$724$		0			0
$725$	−99.6447	+	145.450i	−0.137441	+	0.200620i
$726$		0			0
$727$	−187.681			−0.258158			−0.129079	−	0.991634i	$-0.541202\pi$
$727$	−187.681			−0.258158			−0.129079	+	0.991634i	$0.541202\pi$
$728$		0			0
$729$	−688.521			−0.944474
$730$		0			0
$731$			483.325i			0.661184i
$732$		0			0
$733$			156.326i			0.213269i	0.994298	+	0.106635i	$0.0340075\pi$
$733$			156.326i			0.213269i	−0.994298	+	0.106635i	$0.965993\pi$
$734$		0			0
$735$	−375.957	−	713.282i	−0.511507	−	0.970451i
$736$		0			0
$737$		−	429.105i		−	0.582232i
$738$		0			0
$739$			934.196i			1.26414i	0.774913	+	0.632068i	$0.217794\pi$
$739$			934.196i			1.26414i	−0.774913	+	0.632068i	$0.782206\pi$
$740$		0			0
$741$	−418.928			−0.565355
$742$		0			0
$743$	949.248			1.27759			0.638794	−	0.769378i	$-0.279434\pi$
$743$	949.248			1.27759			0.638794	+	0.769378i	$0.279434\pi$
$744$		0			0
$745$	256.416	+	486.483i	0.344182	+	0.652997i
$746$		0			0
$747$	−2349.13			−3.14475
$748$		0			0
$749$	−883.848			−1.18004
$750$		0			0
$751$		−	643.029i		−	0.856230i	−0.903724	−	0.428115i	$-0.859178\pi$
$751$		−	643.029i		−	0.856230i	0.903724	−	0.428115i	$-0.140822\pi$
$752$		0			0
$753$			1385.97i			1.84059i
$754$		0			0
$755$	−695.805	+	366.746i	−0.921596	+	0.485756i
$756$		0			0
$757$		−	932.399i		−	1.23170i	−0.787862	−	0.615851i	$-0.788812\pi$
$757$		−	932.399i		−	1.23170i	0.787862	−	0.615851i	$-0.211188\pi$
$758$		0			0
$759$		−	408.375i		−	0.538044i
$760$		0			0
$761$	542.027			0.712257			0.356128	−	0.934437i	$-0.384097\pi$
$761$	542.027			0.712257			0.356128	+	0.934437i	$0.384097\pi$
$762$		0			0
$763$	833.429			1.09230
$764$		0			0
$765$	−2628.65	+	1385.51i	−3.43615	+	1.81113i
$766$		0			0
$767$	69.2112			0.0902363
$768$		0			0
$769$	746.509			0.970753			0.485377	−	0.874305i	$-0.338682\pi$
$769$	746.509			0.970753			0.485377	+	0.874305i	$0.338682\pi$
$770$		0			0
$771$			39.2047i			0.0508491i
$772$		0			0
$773$			978.971i			1.26646i	0.773965	+	0.633228i	$0.218271\pi$
$773$			978.971i			1.26646i	−0.773965	+	0.633228i	$0.781729\pi$
$774$		0			0
$775$	−829.941	−	568.576i	−1.07089	−	0.733647i
$776$		0			0
$777$		−	2559.33i		−	3.29386i
$778$		0			0
$779$		−	237.246i		−	0.304552i
$780$		0			0
$781$	148.045			0.189558
$782$		0			0
$783$	−336.775			−0.430109
$784$		0			0
$785$	141.448	−	74.5548i	0.180189	−	0.0949742i
$786$		0			0
$787$	−1215.01			−1.54385			−0.771924	−	0.635715i	$-0.780705\pi$
$787$	−1215.01			−1.54385			−0.771924	+	0.635715i	$0.780705\pi$
$788$		0			0
$789$	−783.494			−0.993021
$790$		0			0
$791$		−	104.907i		−	0.132625i
$792$		0			0
$793$			504.034i			0.635604i
$794$		0			0
$795$	315.231	−	166.152i	0.396517	−	0.208997i
$796$		0			0
$797$			70.2191i			0.0881043i	0.999029	+	0.0440521i	$0.0140268\pi$
$797$			70.2191i			0.0881043i	−0.999029	+	0.0440521i	$0.985973\pi$
$798$		0			0
$799$			1938.40i			2.42604i
$800$		0			0
$801$	271.865			0.339407
$802$		0			0
$803$	−320.915			−0.399645
$804$		0			0
$805$	−492.466	−	934.327i	−0.611759	−	1.16065i
$806$		0			0
$807$	2389.43			2.96088
$808$		0			0
$809$	−1177.27			−1.45521			−0.727607	−	0.685995i	$-0.759367\pi$
$809$	−1177.27			−1.45521			−0.727607	+	0.685995i	$0.759367\pi$
$810$		0			0
$811$		−	1061.68i		−	1.30910i	−0.756019	−	0.654550i	$-0.772858\pi$
$811$		−	1061.68i		−	1.30910i	0.756019	−	0.654550i	$-0.227142\pi$
$812$		0			0
$813$		−	1606.27i		−	1.97573i
$814$		0			0
$815$	−383.066	−	726.769i	−0.470020	−	0.891741i
$816$		0			0
$817$			257.759i			0.315494i
$818$		0			0
$819$			748.043i			0.913361i
$820$		0			0
$821$	322.247			0.392505			0.196253	−	0.980553i	$-0.437123\pi$
$821$	322.247			0.392505			0.196253	+	0.980553i	$0.437123\pi$
$822$		0			0
$823$	−751.577			−0.913216			−0.456608	−	0.889668i	$-0.650936\pi$
$823$	−751.577			−0.913216			−0.456608	+	0.889668i	$0.650936\pi$
$824$		0			0
$825$	356.502	+	244.232i	0.432124	+	0.296039i
$826$		0			0
$827$	−1601.64			−1.93669			−0.968344	−	0.249620i	$-0.919694\pi$
$827$	−1601.64			−1.93669			−0.968344	+	0.249620i	$0.919694\pi$
$828$		0			0
$829$	1249.91			1.50773			0.753866	−	0.657028i	$-0.228187\pi$
$829$	1249.91			1.50773			0.753866	+	0.657028i	$0.228187\pi$
$830$		0			0
$831$		−	1222.47i		−	1.47108i
$832$		0			0
$833$			1012.41i			1.21538i
$834$		0			0
$835$	−171.012	−	324.450i	−0.204804	−	0.388563i
$836$		0			0
$837$		−	1921.65i		−	2.29588i
$838$		0			0
$839$		−	1039.99i		−	1.23956i	−0.784778	−	0.619778i	$-0.787223\pi$
$839$		−	1039.99i		−	1.23956i	0.784778	−	0.619778i	$-0.212777\pi$
$840$		0			0
$841$	−791.265			−0.940861
$842$		0			0
$843$	2218.89			2.63213
$844$		0			0
$845$	344.534	+	653.665i	0.407733	+	0.773568i
$846$		0			0
$847$	98.3509			0.116117
$848$		0			0
$849$	−628.993			−0.740864
$850$		0			0
$851$		−	1297.57i		−	1.52476i
$852$		0			0
$853$			1161.51i			1.36167i	0.732436	+	0.680836i	$0.238383\pi$
$853$			1161.51i			1.36167i	−0.732436	+	0.680836i	$0.761617\pi$
$854$		0			0
$855$	−1401.87	+	738.898i	−1.63961	+	0.864208i
$856$		0			0
$857$		−	18.6298i		−	0.0217384i	−0.999941	−	0.0108692i	$-0.996540\pi$
$857$		−	18.6298i		−	0.0217384i	0.999941	−	0.0108692i	$-0.00345984\pi$
$858$		0			0
$859$			1310.76i			1.52591i	0.646449	+	0.762957i	$0.276253\pi$
$859$			1310.76i			1.52591i	−0.646449	+	0.762957i	$0.723747\pi$
$860$		0			0
$861$	−633.546			−0.735826
$862$		0			0
$863$	−332.009			−0.384715			−0.192357	−	0.981325i	$-0.561613\pi$
$863$	−332.009			−0.384715			−0.192357	+	0.981325i	$0.561613\pi$
$864$		0			0
$865$	−1118.03	+	589.294i	−1.29252	+	0.681265i
$866$		0			0
$867$	4073.64			4.69855
$868$		0			0
$869$	153.954			0.177163
$870$		0			0
$871$			595.976i			0.684244i
$872$		0			0
$873$		−	3054.07i		−	3.49836i
$874$		0			0
$875$	1110.17	+	128.871i	1.26876	+	0.147281i
$876$		0			0
$877$			182.631i			0.208245i	0.994564	+	0.104123i	$0.0332035\pi$
$877$			182.631i			0.208245i	−0.994564	+	0.104123i	$0.966797\pi$
$878$		0			0
$879$			2501.49i			2.84584i
$880$		0			0
$881$	1470.26			1.66885			0.834426	−	0.551120i	$-0.185799\pi$
$881$	1470.26			1.66885			0.834426	+	0.551120i	$0.185799\pi$
$882$		0			0
$883$	1041.14			1.17910			0.589548	−	0.807733i	$-0.299306\pi$
$883$	1041.14			1.17910			0.589548	+	0.807733i	$0.299306\pi$
$884$		0			0
$885$	346.367	−	182.564i	0.391376	−	0.206287i
$886$		0			0
$887$	−1402.58			−1.58127			−0.790633	−	0.612290i	$-0.790249\pi$
$887$	−1402.58			−1.58127			−0.790633	+	0.612290i	$0.790249\pi$
$888$		0			0
$889$	164.258			0.184767
$890$		0			0
$891$			283.299i			0.317956i
$892$		0			0
$893$			1033.76i			1.15762i
$894$		0			0
$895$	876.209	−	461.833i	0.979005	−	0.516015i
$896$		0			0
$897$			567.185i			0.632313i
$898$		0			0
$899$			283.792i			0.315676i
$900$		0			0
$901$	−447.430			−0.496592
$902$		0			0
$903$	688.325			0.762264
$904$		0			0
$905$	378.696	+	718.478i	0.418449	+	0.793899i
$906$		0			0
$907$	699.199			0.770892			0.385446	−	0.922730i	$-0.374048\pi$
$907$	699.199			0.770892			0.385446	+	0.922730i	$0.374048\pi$
$908$		0			0
$909$	726.659			0.799405
$910$		0			0
$911$		−	1223.02i		−	1.34251i	−0.741228	−	0.671253i	$-0.765756\pi$
$911$		−	1223.02i		−	1.34251i	0.741228	−	0.671253i	$-0.234244\pi$
$912$		0			0
$913$		−	428.967i		−	0.469844i
$914$		0			0
$915$	1329.53	+	2522.44i	1.45304	+	2.75676i
$916$		0			0
$917$		−	153.038i		−	0.166890i
$918$		0			0
$919$			803.727i			0.874566i	0.899324	+	0.437283i	$0.144059\pi$
$919$			803.727i			0.874566i	−0.899324	+	0.437283i	$0.855941\pi$
$920$		0			0
$921$	−339.462			−0.368579
$922$		0			0
$923$	−205.617			−0.222770
$924$		0			0
$925$	1132.75	+	776.024i	1.22459	+	0.838945i
$926$		0			0
$927$	−787.402			−0.849409
$928$		0			0
$929$	−180.570			−0.194371			−0.0971853	−	0.995266i	$-0.530984\pi$
$929$	−180.570			−0.194371			−0.0971853	+	0.995266i	$0.530984\pi$
$930$		0			0
$931$			539.922i			0.579938i
$932$		0			0
$933$		−	433.767i		−	0.464916i
$934$		0			0
$935$	−253.004	−	480.010i	−0.270593	−	0.513380i
$936$		0			0
$937$		−	159.144i		−	0.169844i	−0.996388	−	0.0849222i	$-0.972936\pi$
$937$		−	159.144i		−	0.169844i	0.996388	−	0.0849222i	$-0.0270642\pi$
$938$		0			0
$939$		−	1004.07i		−	1.06930i
$940$		0			0
$941$	1060.32			1.12680			0.563401	−	0.826183i	$-0.309493\pi$
$941$	1060.32			1.12680			0.563401	+	0.826183i	$0.309493\pi$
$942$		0			0
$943$	−321.206			−0.340622
$944$		0			0
$945$	995.419	+	1888.55i	1.05335	+	1.99847i
$946$		0			0
$947$	42.3814			0.0447533			0.0223767	−	0.999750i	$-0.492877\pi$
$947$	42.3814			0.0447533			0.0223767	+	0.999750i	$0.492877\pi$
$948$		0			0
$949$	445.713			0.469666
$950$		0			0
$951$		−	1209.53i		−	1.27185i
$952$		0			0
$953$			500.390i			0.525069i	0.964923	+	0.262534i	$0.0845583\pi$
$953$			500.390i			0.525069i	−0.964923	+	0.262534i	$0.915442\pi$
$954$		0			0
$955$	−1503.31	+	792.368i	−1.57415	+	0.829705i
$956$		0			0
$957$		−	121.903i		−	0.127381i
$958$		0			0
$959$			1831.20i			1.90949i
$960$		0			0
$961$	−658.330			−0.685047
$962$		0			0
$963$	1795.44			1.86442
$964$		0			0
$965$	−56.5568	+	29.8100i	−0.0586081	+	0.0308912i
$966$		0			0
$967$	−390.757			−0.404092			−0.202046	−	0.979376i	$-0.564759\pi$
$967$	−390.757			−0.404092			−0.202046	+	0.979376i	$0.564759\pi$
$968$		0			0
$969$	2975.75			3.07095
$970$		0			0
$971$			461.787i			0.475579i	0.971317	+	0.237790i	$0.0764229\pi$
$971$			461.787i			0.475579i	−0.971317	+	0.237790i	$0.923577\pi$
$972$		0			0
$973$			1382.56i			1.42092i
$974$		0			0
$975$	−495.139	−	339.210i	−0.507835	−	0.347908i
$976$		0			0
$977$			901.857i			0.923088i	0.887117	+	0.461544i	$0.152704\pi$
$977$			901.857i			0.923088i	−0.887117	+	0.461544i	$0.847296\pi$
$978$		0			0
$979$			49.6444i			0.0507093i
$980$		0			0
$981$	−1693.02			−1.72581
$982$		0			0
$983$	−98.1581			−0.0998556			−0.0499278	−	0.998753i	$-0.515899\pi$
$983$	−98.1581			−0.0998556			−0.0499278	+	0.998753i	$0.515899\pi$
$984$		0			0
$985$	725.886	−	382.601i	0.736941	−	0.388427i
$986$		0			0
$987$	2760.57			2.79693
$988$		0			0
$989$	348.979			0.352860
$990$		0			0
$991$		−	427.551i		−	0.431434i	−0.976456	−	0.215717i	$-0.930791\pi$
$991$		−	427.551i		−	0.431434i	0.976456	−	0.215717i	$-0.0692088\pi$
$992$		0			0
$993$		−	509.494i		−	0.513086i
$994$		0			0
$995$	−31.2657	+	16.4796i	−0.0314228	+	0.0165624i
$996$		0			0
$997$		−	724.591i		−	0.726772i	−0.931639	−	0.363386i	$-0.881621\pi$
$997$		−	724.591i		−	0.726772i	0.931639	−	0.363386i	$-0.118379\pi$
$998$		0			0
$999$			2622.77i			2.62540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.n.a.639.2	yes	20
4.3	odd	2	inner	880.3.n.a.639.20	yes	20
5.4	even	2	inner	880.3.n.a.639.19	yes	20
20.19	odd	2	inner	880.3.n.a.639.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.3.n.a.639.1	✓	20	20.19	odd	2	inner
880.3.n.a.639.2	yes	20	1.1	even	1	trivial
880.3.n.a.639.19	yes	20	5.4	even	2	inner
880.3.n.a.639.20	yes	20	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.21178 q^{3} +(2.33138 + 4.42320i) q^{5} -8.94099 q^{7} +18.1626 q^{9} +3.31662i q^{11} -4.60640i q^{13} +(-12.1507 - 23.0527i) q^{15} +32.7204i q^{17} +17.4499i q^{19} +46.5985 q^{21} +23.6253 q^{23} +(-14.1293 + 20.6243i) q^{25} -47.7537 q^{27} +7.05234 q^{29} +40.2409i q^{31} -17.2855i q^{33} +(-20.8449 - 39.5478i) q^{35} -54.9230i q^{37} +24.0075i q^{39} -13.5959 q^{41} +14.7714 q^{43} +(42.3441 + 80.3369i) q^{45} +59.2416 q^{47} +30.9413 q^{49} -170.531i q^{51} +13.6744i q^{53} +(-14.6701 + 7.73232i) q^{55} -90.9448i q^{57} +15.0250i q^{59} -109.420 q^{61} -162.392 q^{63} +(20.3750 - 10.7393i) q^{65} -129.380 q^{67} -123.130 q^{69} -44.6372i q^{71} +96.7596i q^{73} +(73.6389 - 107.489i) q^{75} -29.6539i q^{77} -46.4190i q^{79} +85.4178 q^{81} -129.339 q^{83} +(-144.729 + 76.2837i) q^{85} -36.7552 q^{87} +14.9683 q^{89} +41.1858i q^{91} -209.727i q^{93} +(-77.1841 + 40.6823i) q^{95} -168.151i q^{97} +60.2387i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 10 q^{5} + 40 q^{9} + 184 q^{21} - 18 q^{25} + 16 q^{29} + 144 q^{41} + 16 q^{45} + 36 q^{49} - 384 q^{61} - 32 q^{65} - 244 q^{69} + 236 q^{81} - 348 q^{85} - 100 q^{89}+O(q^{100}) \) 20 * q - 10 * q^5 + 40 * q^9 + 184 * q^21 - 18 * q^25 + 16 * q^29 + 144 * q^41 + 16 * q^45 + 36 * q^49 - 384 * q^61 - 32 * q^65 - 244 * q^69 + 236 * q^81 - 348 * q^85 - 100 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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