LMFDB - Embedded newform 520.2.d.b.209.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [520,2,Mod(209,520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("520.209"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] 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:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.5
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	520.209
Dual form			520.2.d.b.209.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17009i q^{3} +(0.539189 + 2.17009i) q^{5} +0.630898i q^{7} +1.63090 q^{9} -0.539189 q^{11} +1.00000i q^{13} +(-2.53919 + 0.630898i) q^{15} +3.07838i q^{17} -0.879362 q^{19} -0.738205 q^{21} -2.58864i q^{23} +(-4.41855 + 2.34017i) q^{25} +5.41855i q^{27} +2.29072 q^{29} -7.95774 q^{31} -0.630898i q^{33} +(-1.36910 + 0.340173i) q^{35} +4.78765i q^{37} -1.17009 q^{39} +3.41855 q^{41} +3.17009i q^{43} +(0.879362 + 3.53919i) q^{45} -8.94441i q^{47} +6.60197 q^{49} -3.60197 q^{51} +0.496928i q^{53} +(-0.290725 - 1.17009i) q^{55} -1.02893i q^{57} +8.29791 q^{59} +4.04945 q^{61} +1.02893i q^{63} +(-2.17009 + 0.539189i) q^{65} +3.36910i q^{67} +3.02893 q^{69} -9.06278 q^{71} -12.2062i q^{73} +(-2.73820 - 5.17009i) q^{75} -0.340173i q^{77} +5.65983 q^{79} -1.44748 q^{81} +1.02893i q^{83} +(-6.68035 + 1.65983i) q^{85} +2.68035i q^{87} +16.8371 q^{89} -0.630898 q^{91} -9.31124i q^{93} +(-0.474142 - 1.90829i) q^{95} -5.23513i q^{97} -0.879362 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} - 12 q^{15} + 20 q^{19} - 20 q^{21} + 2 q^{25} + 28 q^{29} - 16 q^{31} - 16 q^{35} + 4 q^{39} - 8 q^{41} - 20 q^{45} + 2 q^{49} + 16 q^{51} - 16 q^{55} - 4 q^{59} - 12 q^{61} - 2 q^{65} + 48 q^{69}+ \cdots + 20 q^{99}+O(q^{100}) $ 6 * q + 2 * q^9 - 12 * q^15 + 20 * q^19 - 20 * q^21 + 2 * q^25 + 28 * q^29 - 16 * q^31 - 16 * q^35 + 4 * q^39 - 8 * q^41 - 20 * q^45 + 2 * q^49 + 16 * q^51 - 16 * q^55 - 4 * q^59 - 12 * q^61 - 2 * q^65 + 48 * q^69 - 20 * q^71 - 32 * q^75 + 56 * q^79 - 10 * q^81 + 4 * q^85 + 44 * q^89 + 4 * q^91 - 8 * q^95 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.17009i			0.675550i	0.941227	+	0.337775i	$0.109674\pi$
$3$			1.17009i			0.675550i	−0.941227	+	0.337775i	$0.890326\pi$
$4$		0			0
$5$	0.539189	+	2.17009i	0.241133	+	0.970492i
$5$	0.539189	+	2.17009i	0.241133	+	0.970492i
$6$		0			0
$7$			0.630898i			0.238457i	0.992867	+	0.119228i	$0.0380421\pi$
$7$			0.630898i			0.238457i	−0.992867	+	0.119228i	$0.961958\pi$
$8$		0			0
$9$	1.63090			0.543633
$10$		0			0
$11$	−0.539189			−0.162572			−0.0812858	−	0.996691i	$-0.525903\pi$
$11$	−0.539189			−0.162572			−0.0812858	+	0.996691i	$0.525903\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	−2.53919	+	0.630898i	−0.655616	+	0.162897i
$16$		0			0
$17$			3.07838i			0.746616i	0.927707	+	0.373308i	$0.121777\pi$
$17$			3.07838i			0.746616i	−0.927707	+	0.373308i	$0.878223\pi$
$18$		0			0
$19$	−0.879362			−0.201739			−0.100870	−	0.994900i	$-0.532163\pi$
$19$	−0.879362			−0.201739			−0.100870	+	0.994900i	$0.532163\pi$
$20$		0			0
$21$	−0.738205			−0.161089
$22$		0			0
$23$		−	2.58864i		−	0.539768i	−0.962893	−	0.269884i	$-0.913015\pi$
$23$		−	2.58864i		−	0.539768i	0.962893	−	0.269884i	$-0.0869854\pi$
$24$		0			0
$25$	−4.41855	+	2.34017i	−0.883710	+	0.468035i
$26$		0			0
$27$			5.41855i			1.04280i
$28$		0			0
$29$	2.29072			0.425377			0.212688	−	0.977120i	$-0.431778\pi$
$29$	2.29072			0.425377			0.212688	+	0.977120i	$0.431778\pi$
$30$		0			0
$31$	−7.95774			−1.42925			−0.714626	−	0.699507i	$-0.753403\pi$
$31$	−7.95774			−1.42925			−0.714626	+	0.699507i	$0.753403\pi$
$32$		0			0
$33$		−	0.630898i		−	0.109825i
$34$		0			0
$35$	−1.36910	+	0.340173i	−0.231421	+	0.0574997i
$36$		0			0
$37$			4.78765i			0.787085i	0.919306	+	0.393543i	$0.128751\pi$
$37$			4.78765i			0.787085i	−0.919306	+	0.393543i	$0.871249\pi$
$38$		0			0
$39$	−1.17009			−0.187364
$40$		0			0
$41$	3.41855			0.533888			0.266944	−	0.963712i	$-0.413986\pi$
$41$	3.41855			0.533888			0.266944	+	0.963712i	$0.413986\pi$
$42$		0			0
$43$			3.17009i			0.483434i	0.970347	+	0.241717i	$0.0777105\pi$
$43$			3.17009i			0.483434i	−0.970347	+	0.241717i	$0.922289\pi$
$44$		0			0
$45$	0.879362	+	3.53919i	0.131088	+	0.527591i
$46$		0			0
$47$		−	8.94441i		−	1.30468i	−0.757928	−	0.652338i	$-0.773788\pi$
$47$		−	8.94441i		−	1.30468i	0.757928	−	0.652338i	$-0.226212\pi$
$48$		0			0
$49$	6.60197			0.943138
$50$		0			0
$51$	−3.60197			−0.504376
$52$		0			0
$53$			0.496928i			0.0682584i	0.999417	+	0.0341292i	$0.0108658\pi$
$53$			0.496928i			0.0682584i	−0.999417	+	0.0341292i	$0.989134\pi$
$54$		0			0
$55$	−0.290725	−	1.17009i	−0.0392013	−	0.157774i
$56$		0			0
$57$		−	1.02893i		−	0.136285i
$58$		0			0
$59$	8.29791			1.08030			0.540148	−	0.841570i	$-0.318368\pi$
$59$	8.29791			1.08030			0.540148	+	0.841570i	$0.318368\pi$
$60$		0			0
$61$	4.04945			0.518479			0.259239	−	0.965813i	$-0.416528\pi$
$61$	4.04945			0.518479			0.259239	+	0.965813i	$0.416528\pi$
$62$		0			0
$63$			1.02893i			0.129633i
$64$		0			0
$65$	−2.17009	+	0.539189i	−0.269166	+	0.0668781i
$66$		0			0
$67$			3.36910i			0.411601i	0.978594	+	0.205801i	$0.0659799\pi$
$67$			3.36910i			0.411601i	−0.978594	+	0.205801i	$0.934020\pi$
$68$		0			0
$69$	3.02893			0.364640
$70$		0			0
$71$	−9.06278			−1.07555			−0.537777	−	0.843087i	$-0.680736\pi$
$71$	−9.06278			−1.07555			−0.537777	+	0.843087i	$0.680736\pi$
$72$		0			0
$73$		−	12.2062i		−	1.42863i	−0.699825	−	0.714314i	$-0.746739\pi$
$73$		−	12.2062i		−	1.42863i	0.699825	−	0.714314i	$-0.253261\pi$
$74$		0			0
$75$	−2.73820	−	5.17009i	−0.316181	−	0.596990i
$76$		0			0
$77$		−	0.340173i		−	0.0387663i
$78$		0			0
$79$	5.65983			0.636780			0.318390	−	0.947960i	$-0.396858\pi$
$79$	5.65983			0.636780			0.318390	+	0.947960i	$0.396858\pi$
$80$		0			0
$81$	−1.44748			−0.160831
$82$		0			0
$83$			1.02893i			0.112940i	0.998404	+	0.0564698i	$0.0179845\pi$
$83$			1.02893i			0.112940i	−0.998404	+	0.0564698i	$0.982016\pi$
$84$		0			0
$85$	−6.68035	+	1.65983i	−0.724585	+	0.180034i
$86$		0			0
$87$			2.68035i			0.287363i
$88$		0			0
$89$	16.8371			1.78473			0.892365	−	0.451315i	$-0.149045\pi$
$89$	16.8371			1.78473			0.892365	+	0.451315i	$0.149045\pi$
$90$		0			0
$91$	−0.630898			−0.0661360
$92$		0			0
$93$		−	9.31124i		−	0.965531i
$94$		0			0
$95$	−0.474142	−	1.90829i	−0.0486460	−	0.195787i
$96$		0			0
$97$		−	5.23513i		−	0.531547i	−0.964035	−	0.265774i	$-0.914373\pi$
$97$		−	5.23513i		−	0.531547i	0.964035	−	0.265774i	$-0.0856274\pi$
$98$		0			0
$99$	−0.879362			−0.0883792
$100$		0			0
$101$	5.23513			0.520915			0.260458	−	0.965485i	$-0.416127\pi$
$101$	5.23513			0.520915			0.260458	+	0.965485i	$0.416127\pi$
$102$		0			0
$103$			3.69368i			0.363949i	0.983303	+	0.181974i	$0.0582488\pi$
$103$			3.69368i			0.363949i	−0.983303	+	0.181974i	$0.941751\pi$
$104$		0			0
$105$	−0.398032	−	1.60197i	−0.0388439	−	0.156336i
$106$		0			0
$107$		−	15.8504i		−	1.53232i	−0.642650	−	0.766160i	$-0.722165\pi$
$107$		−	15.8504i		−	1.53232i	0.642650	−	0.766160i	$-0.277835\pi$
$108$		0			0
$109$	3.81658			0.365562			0.182781	−	0.983154i	$-0.441490\pi$
$109$	3.81658			0.365562			0.182781	+	0.983154i	$0.441490\pi$
$110$		0			0
$111$	−5.60197			−0.531715
$112$		0			0
$113$		−	11.4452i		−	1.07668i	−0.842729	−	0.538338i	$-0.819053\pi$
$113$		−	11.4452i		−	1.07668i	0.842729	−	0.538338i	$-0.180947\pi$
$114$		0			0
$115$	5.61757	−	1.39576i	0.523841	−	0.130156i
$116$		0			0
$117$			1.63090i			0.150777i
$118$		0			0
$119$	−1.94214			−0.178036
$120$		0			0
$121$	−10.7093			−0.973570
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$	−7.46081	−	8.32684i	−0.667315	−	0.744775i
$126$		0			0
$127$		−	5.35350i		−	0.475047i	−0.971382	−	0.237523i	$-0.923664\pi$
$127$		−	5.35350i		−	0.475047i	0.971382	−	0.237523i	$-0.0763356\pi$
$128$		0			0
$129$	−3.70928			−0.326583
$130$		0			0
$131$	18.9360			1.65445			0.827223	−	0.561874i	$-0.189919\pi$
$131$	18.9360			1.65445			0.827223	+	0.561874i	$0.189919\pi$
$132$		0			0
$133$		−	0.554787i		−	0.0481062i
$134$		0			0
$135$	−11.7587	+	2.92162i	−1.01203	+	0.251453i
$136$		0			0
$137$		−	6.09890i		−	0.521064i	−0.965465	−	0.260532i	$-0.916102\pi$
$137$		−	6.09890i		−	0.521064i	0.965465	−	0.260532i	$-0.0838979\pi$
$138$		0			0
$139$	21.5441			1.82735			0.913674	−	0.406448i	$-0.133233\pi$
$139$	21.5441			1.82735			0.913674	+	0.406448i	$0.133233\pi$
$140$		0			0
$141$	10.4657			0.881374
$142$		0			0
$143$		−	0.539189i		−	0.0450892i
$144$		0			0
$145$	1.23513	+	4.97107i	0.102572	+	0.412825i
$146$		0			0
$147$			7.72487i			0.637137i
$148$		0			0
$149$	−8.28231			−0.678514			−0.339257	−	0.940694i	$-0.610176\pi$
$149$	−8.28231			−0.678514			−0.339257	+	0.940694i	$0.610176\pi$
$150$		0			0
$151$	−7.61757			−0.619909			−0.309954	−	0.950751i	$-0.600314\pi$
$151$	−7.61757			−0.619909			−0.309954	+	0.950751i	$0.600314\pi$
$152$		0			0
$153$			5.02052i			0.405885i
$154$		0			0
$155$	−4.29072	−	17.2690i	−0.344639	−	1.38708i
$156$		0			0
$157$			3.47641i			0.277448i	0.990331	+	0.138724i	$0.0443000\pi$
$157$			3.47641i			0.277448i	−0.990331	+	0.138724i	$0.955700\pi$
$158$		0			0
$159$	−0.581449			−0.0461119
$160$		0			0
$161$	1.63317			0.128711
$162$		0			0
$163$			9.12783i			0.714947i	0.933923	+	0.357473i	$0.116362\pi$
$163$			9.12783i			0.714947i	−0.933923	+	0.357473i	$0.883638\pi$
$164$		0			0
$165$	1.36910	−	0.340173i	0.106584	−	0.0264824i
$166$		0			0
$167$		−	12.8143i		−	0.991601i	−0.868436	−	0.495801i	$-0.834875\pi$
$167$		−	12.8143i		−	0.991601i	0.868436	−	0.495801i	$-0.165125\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−1.43415			−0.109672
$172$		0			0
$173$		−	4.18342i		−	0.318059i	−0.987274	−	0.159030i	$-0.949163\pi$
$173$		−	4.18342i		−	0.318059i	0.987274	−	0.159030i	$-0.0508366\pi$
$174$		0			0
$175$	−1.47641	−	2.78765i	−0.111606	−	0.210727i
$176$		0			0
$177$			9.70928i			0.729794i
$178$		0			0
$179$	13.7587			1.02838			0.514188	−	0.857678i	$-0.328093\pi$
$179$	13.7587			1.02838			0.514188	+	0.857678i	$0.328093\pi$
$180$		0			0
$181$	−21.8082			−1.62099			−0.810494	−	0.585746i	$-0.800801\pi$
$181$	−21.8082			−1.62099			−0.810494	+	0.585746i	$0.800801\pi$
$182$		0			0
$183$			4.73820i			0.350258i
$184$		0			0
$185$	−10.3896	+	2.58145i	−0.763860	+	0.189792i
$186$		0			0
$187$		−	1.65983i		−	0.121379i
$188$		0			0
$189$	−3.41855			−0.248663
$190$		0			0
$191$	−2.83710			−0.205285			−0.102643	−	0.994718i	$-0.532730\pi$
$191$	−2.83710			−0.205285			−0.102643	+	0.994718i	$0.532730\pi$
$192$		0			0
$193$			21.6163i			1.55598i	0.628278	+	0.777989i	$0.283760\pi$
$193$			21.6163i			1.55598i	−0.628278	+	0.777989i	$0.716240\pi$
$194$		0			0
$195$	−0.630898	−	2.53919i	−0.0451795	−	0.181835i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	14.4391			1.02356			0.511779	−	0.859117i	$-0.328987\pi$
$199$	14.4391			1.02356			0.511779	+	0.859117i	$0.328987\pi$
$200$		0			0
$201$	−3.94214			−0.278057
$202$		0			0
$203$			1.44521i			0.101434i
$204$		0			0
$205$	1.84324	+	7.41855i	0.128738	+	0.518134i
$206$		0			0
$207$		−	4.22180i		−	0.293436i
$208$		0			0
$209$	0.474142			0.0327971
$210$		0			0
$211$	4.28231			0.294807			0.147403	−	0.989076i	$-0.452908\pi$
$211$	4.28231			0.294807			0.147403	+	0.989076i	$0.452908\pi$
$212$		0			0
$213$		−	10.6042i		−	0.726590i
$214$		0			0
$215$	−6.87936	+	1.70928i	−0.469169	+	0.116572i
$216$		0			0
$217$		−	5.02052i		−	0.340815i
$218$		0			0
$219$	14.2823			0.965109
$220$		0			0
$221$	−3.07838			−0.207074
$222$		0			0
$223$			22.8865i			1.53260i	0.642485	+	0.766298i	$0.277904\pi$
$223$			22.8865i			1.53260i	−0.642485	+	0.766298i	$0.722096\pi$
$224$		0			0
$225$	−7.20620	+	3.81658i	−0.480414	+	0.254439i
$226$		0			0
$227$		−	5.89269i		−	0.391112i	−0.980693	−	0.195556i	$-0.937349\pi$
$227$		−	5.89269i		−	0.391112i	0.980693	−	0.195556i	$-0.0626511\pi$
$228$		0			0
$229$	−17.1194			−1.13128			−0.565641	−	0.824651i	$-0.691371\pi$
$229$	−17.1194			−1.13128			−0.565641	+	0.824651i	$0.691371\pi$
$230$		0			0
$231$	0.398032			0.0261886
$232$		0			0
$233$		−	13.8888i		−	0.909887i	−0.890520	−	0.454943i	$-0.849660\pi$
$233$		−	13.8888i		−	0.909887i	0.890520	−	0.454943i	$-0.150340\pi$
$234$		0			0
$235$	19.4101	−	4.82273i	1.26618	−	0.314600i
$236$		0			0
$237$			6.62249i			0.430177i
$238$		0			0
$239$	−20.9516			−1.35525			−0.677623	−	0.735409i	$-0.736990\pi$
$239$	−20.9516			−1.35525			−0.677623	+	0.735409i	$0.736990\pi$
$240$		0			0
$241$	−8.68035			−0.559150			−0.279575	−	0.960124i	$-0.590194\pi$
$241$	−8.68035			−0.559150			−0.279575	+	0.960124i	$0.590194\pi$
$242$		0			0
$243$			14.5620i			0.934151i
$244$		0			0
$245$	3.55971	+	14.3268i	0.227421	+	0.915308i
$246$		0			0
$247$		−	0.879362i		−	0.0559525i
$248$		0			0
$249$	−1.20394			−0.0762964
$250$		0			0
$251$	8.58145			0.541656			0.270828	−	0.962628i	$-0.412702\pi$
$251$	8.58145			0.541656			0.270828	+	0.962628i	$0.412702\pi$
$252$		0			0
$253$			1.39576i			0.0877510i
$254$		0			0
$255$	−1.94214	−	7.81658i	−0.121622	−	0.489493i
$256$		0			0
$257$			14.2557i			0.889243i	0.895719	+	0.444622i	$0.146662\pi$
$257$			14.2557i			0.889243i	−0.895719	+	0.444622i	$0.853338\pi$
$258$		0			0
$259$	−3.02052			−0.187686
$260$		0			0
$261$	3.73594			0.231249
$262$		0			0
$263$			10.8299i			0.667801i	0.942608	+	0.333901i	$0.108365\pi$
$263$			10.8299i			0.667801i	−0.942608	+	0.333901i	$0.891635\pi$
$264$		0			0
$265$	−1.07838	+	0.267938i	−0.0662442	+	0.0164593i
$266$		0			0
$267$			19.7009i			1.20567i
$268$		0			0
$269$	21.4329			1.30679			0.653394	−	0.757018i	$-0.273344\pi$
$269$	21.4329			1.30679			0.653394	+	0.757018i	$0.273344\pi$
$270$		0			0
$271$	21.4752			1.30452			0.652262	−	0.757993i	$-0.273820\pi$
$271$	21.4752			1.30452			0.652262	+	0.757993i	$0.273820\pi$
$272$		0			0
$273$		−	0.738205i		−	0.0446782i
$274$		0			0
$275$	2.38243	−	1.26180i	0.143666	−	0.0760891i
$276$		0			0
$277$			3.28846i			0.197584i	0.995108	+	0.0987921i	$0.0314979\pi$
$277$			3.28846i			0.197584i	−0.995108	+	0.0987921i	$0.968502\pi$
$278$		0			0
$279$	−12.9783			−0.776988
$280$		0			0
$281$	−28.1978			−1.68214			−0.841070	−	0.540927i	$-0.818074\pi$
$281$	−28.1978			−1.68214			−0.841070	+	0.540927i	$0.818074\pi$
$282$		0			0
$283$			12.1639i			0.723071i	0.932358	+	0.361536i	$0.117747\pi$
$283$			12.1639i			0.723071i	−0.932358	+	0.361536i	$0.882253\pi$
$284$		0			0
$285$	2.23287	−	0.554787i	0.132264	−	0.0328628i
$286$		0			0
$287$			2.15676i			0.127309i
$288$		0			0
$289$	7.52359			0.442564
$290$		0			0
$291$	6.12556			0.359087
$292$		0			0
$293$		−	9.36910i		−	0.547349i	−0.961822	−	0.273674i	$-0.911761\pi$
$293$		−	9.36910i		−	0.547349i	0.961822	−	0.273674i	$-0.0882391\pi$
$294$		0			0
$295$	4.47414	+	18.0072i	0.260495	+	1.04842i
$296$		0			0
$297$		−	2.92162i		−	0.169530i
$298$		0			0
$299$	2.58864			0.149705
$300$		0			0
$301$	−2.00000			−0.115278
$302$		0			0
$303$			6.12556i			0.351904i
$304$		0			0
$305$	2.18342	+	8.78765i	0.125022	+	0.503180i
$306$		0			0
$307$			31.0700i			1.77326i	0.462483	+	0.886628i	$0.346959\pi$
$307$			31.0700i			1.77326i	−0.462483	+	0.886628i	$0.653041\pi$
$308$		0			0
$309$	−4.32192			−0.245866
$310$		0			0
$311$	−3.68649			−0.209042			−0.104521	−	0.994523i	$-0.533331\pi$
$311$	−3.68649			−0.209042			−0.104521	+	0.994523i	$0.533331\pi$
$312$		0			0
$313$			2.49693i			0.141135i	0.997507	+	0.0705674i	$0.0224810\pi$
$313$			2.49693i			0.141135i	−0.997507	+	0.0705674i	$0.977519\pi$
$314$		0			0
$315$	−2.23287	+	0.554787i	−0.125808	+	0.0312587i
$316$		0			0
$317$		−	27.5090i		−	1.54506i	−0.634977	−	0.772531i	$-0.718991\pi$
$317$		−	27.5090i		−	1.54506i	0.634977	−	0.772531i	$-0.281009\pi$
$318$		0			0
$319$	−1.23513			−0.0691542
$320$		0			0
$321$	18.5464			1.03516
$322$		0			0
$323$		−	2.70701i		−	0.150622i
$324$		0			0
$325$	−2.34017	−	4.41855i	−0.129809	−	0.245097i
$326$		0			0
$327$			4.46573i			0.246956i
$328$		0			0
$329$	5.64301			0.311109
$330$		0			0
$331$	−9.30406			−0.511397			−0.255699	−	0.966757i	$-0.582305\pi$
$331$	−9.30406			−0.511397			−0.255699	+	0.966757i	$0.582305\pi$
$332$		0			0
$333$			7.80817i			0.427885i
$334$		0			0
$335$	−7.31124	+	1.81658i	−0.399456	+	0.0992505i
$336$		0			0
$337$			21.0205i			1.14506i	0.819884	+	0.572530i	$0.194038\pi$
$337$			21.0205i			1.14506i	−0.819884	+	0.572530i	$0.805962\pi$
$338$		0			0
$339$	13.3919			0.727348
$340$		0			0
$341$	4.29072			0.232356
$342$		0			0
$343$			8.58145i			0.463355i
$344$		0			0
$345$	1.63317	+	6.57304i	0.0879266	+	0.353881i
$346$		0			0
$347$			11.9083i			0.639271i	0.947541	+	0.319635i	$0.103560\pi$
$347$			11.9083i			0.639271i	−0.947541	+	0.319635i	$0.896440\pi$
$348$		0			0
$349$	−4.07223			−0.217982			−0.108991	−	0.994043i	$-0.534762\pi$
$349$	−4.07223			−0.217982			−0.108991	+	0.994043i	$0.534762\pi$
$350$		0			0
$351$	−5.41855			−0.289221
$352$		0			0
$353$		−	26.5152i		−	1.41126i	−0.708580	−	0.705630i	$-0.750664\pi$
$353$		−	26.5152i		−	1.41126i	0.708580	−	0.705630i	$-0.249336\pi$
$354$		0			0
$355$	−4.88655	−	19.6670i	−0.259351	−	1.04382i
$356$		0			0
$357$		−	2.27247i		−	0.120272i
$358$		0			0
$359$	−28.4501			−1.50154			−0.750770	−	0.660563i	$-0.770317\pi$
$359$	−28.4501			−1.50154			−0.750770	+	0.660563i	$0.770317\pi$
$360$		0			0
$361$	−18.2267			−0.959301
$362$		0			0
$363$		−	12.5308i		−	0.657695i
$364$		0			0
$365$	26.4885	−	6.58145i	1.38647	−	0.344489i
$366$		0			0
$367$		−	36.3740i		−	1.89871i	−0.314208	−	0.949354i	$-0.601739\pi$
$367$		−	36.3740i		−	1.89871i	0.314208	−	0.949354i	$-0.398261\pi$
$368$		0			0
$369$	5.57531			0.290239
$370$		0			0
$371$	−0.313511			−0.0162767
$372$		0			0
$373$		−	23.3607i		−	1.20957i	−0.796388	−	0.604785i	$-0.793259\pi$
$373$		−	23.3607i		−	1.20957i	0.796388	−	0.604785i	$-0.206741\pi$
$374$		0			0
$375$	9.74313	−	8.72979i	0.503133	−	0.450805i
$376$		0			0
$377$			2.29072i			0.117978i
$378$		0			0
$379$	−30.0300			−1.54254			−0.771268	−	0.636510i	$-0.780377\pi$
$379$	−30.0300			−1.54254			−0.771268	+	0.636510i	$0.780377\pi$
$380$		0			0
$381$	6.26406			0.320918
$382$		0			0
$383$		−	2.02279i		−	0.103360i	−0.998664	−	0.0516798i	$-0.983542\pi$
$383$		−	2.02279i		−	0.103360i	0.998664	−	0.0516798i	$-0.0164575\pi$
$384$		0			0
$385$	0.738205	−	0.183417i	0.0376224	−	0.00934782i
$386$		0			0
$387$			5.17009i			0.262810i
$388$		0			0
$389$	−35.6742			−1.80875			−0.904377	−	0.426735i	$-0.859664\pi$
$389$	−35.6742			−1.80875			−0.904377	+	0.426735i	$0.859664\pi$
$390$		0			0
$391$	7.96880			0.403000
$392$		0			0
$393$			22.1568i			1.11766i
$394$		0			0
$395$	3.05172	+	12.2823i	0.153548	+	0.617990i
$396$		0			0
$397$		−	9.05559i		−	0.454487i	−0.973838	−	0.227244i	$-0.927029\pi$
$397$		−	9.05559i		−	0.454487i	0.973838	−	0.227244i	$-0.0729713\pi$
$398$		0			0
$399$	0.649149			0.0324981
$400$		0			0
$401$	27.9421			1.39536			0.697682	−	0.716408i	$-0.254215\pi$
$401$	27.9421			1.39536			0.697682	+	0.716408i	$0.254215\pi$
$402$		0			0
$403$		−	7.95774i		−	0.396403i
$404$		0			0
$405$	−0.780465	−	3.14116i	−0.0387816	−	0.156085i
$406$		0			0
$407$		−	2.58145i		−	0.127958i
$408$		0			0
$409$	−32.0989			−1.58719			−0.793594	−	0.608447i	$-0.791793\pi$
$409$	−32.0989			−1.58719			−0.793594	+	0.608447i	$0.791793\pi$
$410$		0			0
$411$	7.13624			0.352005
$412$		0			0
$413$			5.23513i			0.257604i
$414$		0			0
$415$	−2.23287	+	0.554787i	−0.109607	+	0.0272334i
$416$		0			0
$417$			25.2085i			1.23446i
$418$		0			0
$419$	−18.3090			−0.894452			−0.447226	−	0.894421i	$-0.647588\pi$
$419$	−18.3090			−0.894452			−0.447226	+	0.894421i	$0.647588\pi$
$420$		0			0
$421$	−27.0205			−1.31690			−0.658450	−	0.752625i	$-0.728787\pi$
$421$	−27.0205			−1.31690			−0.658450	+	0.752625i	$0.728787\pi$
$422$		0			0
$423$		−	14.5874i		−	0.709264i
$424$		0			0
$425$	−7.20394	−	13.6020i	−0.349442	−	0.659792i
$426$		0			0
$427$			2.55479i			0.123635i
$428$		0			0
$429$	0.630898			0.0304600
$430$		0			0
$431$	14.7636			0.711140			0.355570	−	0.934650i	$-0.384287\pi$
$431$	14.7636			0.711140			0.355570	+	0.934650i	$0.384287\pi$
$432$		0			0
$433$		−	26.7214i		−	1.28415i	−0.766643	−	0.642074i	$-0.778074\pi$
$433$		−	26.7214i		−	1.28415i	0.766643	−	0.642074i	$-0.221926\pi$
$434$		0			0
$435$	−5.81658	+	1.44521i	−0.278884	+	0.0692926i
$436$		0			0
$437$			2.27635i			0.108893i
$438$		0			0
$439$	23.6475			1.12864			0.564318	−	0.825558i	$-0.309139\pi$
$439$	23.6475			1.12864			0.564318	+	0.825558i	$0.309139\pi$
$440$		0			0
$441$	10.7671			0.512721
$442$		0			0
$443$		−	25.4524i		−	1.20928i	−0.796499	−	0.604640i	$-0.793317\pi$
$443$		−	25.4524i		−	1.20928i	0.796499	−	0.604640i	$-0.206683\pi$
$444$		0			0
$445$	9.07838	+	36.5380i	0.430356	+	1.73207i
$446$		0			0
$447$		−	9.69102i		−	0.458370i
$448$		0			0
$449$	28.2967			1.33540			0.667702	−	0.744429i	$-0.267278\pi$
$449$	28.2967			1.33540			0.667702	+	0.744429i	$0.267278\pi$
$450$		0			0
$451$	−1.84324			−0.0867950
$452$		0			0
$453$		−	8.91321i		−	0.418779i
$454$		0			0
$455$	−0.340173	−	1.36910i	−0.0159476	−	0.0641845i
$456$		0			0
$457$		−	21.9155i		−	1.02516i	−0.858639	−	0.512581i	$-0.828689\pi$
$457$		−	21.9155i		−	1.02516i	0.858639	−	0.512581i	$-0.171311\pi$
$458$		0			0
$459$	−16.6803			−0.778572
$460$		0			0
$461$	20.3402			0.947336			0.473668	−	0.880703i	$-0.342930\pi$
$461$	20.3402			0.947336			0.473668	+	0.880703i	$0.342930\pi$
$462$		0			0
$463$		−	11.7503i		−	0.546083i	−0.962002	−	0.273042i	$-0.911970\pi$
$463$		−	11.7503i		−	0.546083i	0.962002	−	0.273042i	$-0.0880297\pi$
$464$		0			0
$465$	20.2062	−	5.02052i	0.937040	−	0.232821i
$466$		0			0
$467$		−	19.1578i		−	0.886517i	−0.896394	−	0.443259i	$-0.853822\pi$
$467$		−	19.1578i		−	0.886517i	0.896394	−	0.443259i	$-0.146178\pi$
$468$		0			0
$469$	−2.12556			−0.0981492
$470$		0			0
$471$	−4.06770			−0.187430
$472$		0			0
$473$		−	1.70928i		−	0.0785926i
$474$		0			0
$475$	3.88550	−	2.05786i	0.178279	−	0.0944210i
$476$		0			0
$477$			0.810439i			0.0371075i
$478$		0			0
$479$	−1.72875			−0.0789886			−0.0394943	−	0.999220i	$-0.512575\pi$
$479$	−1.72875			−0.0789886			−0.0394943	+	0.999220i	$0.512575\pi$
$480$		0			0
$481$	−4.78765			−0.218298
$482$		0			0
$483$			1.91094i			0.0869510i
$484$		0			0
$485$	11.3607	−	2.82273i	0.515862	−	0.128173i
$486$		0			0
$487$		−	42.4885i		−	1.92534i	−0.270680	−	0.962669i	$-0.587249\pi$
$487$		−	42.4885i		−	1.92534i	0.270680	−	0.962669i	$-0.412751\pi$
$488$		0			0
$489$	−10.6803			−0.482982
$490$		0			0
$491$	−0.398032			−0.0179629			−0.00898146	−	0.999960i	$-0.502859\pi$
$491$	−0.398032			−0.0179629			−0.00898146	+	0.999960i	$0.502859\pi$
$492$		0			0
$493$			7.05172i			0.317593i
$494$		0			0
$495$	−0.474142	−	1.90829i	−0.0213111	−	0.0857713i
$496$		0			0
$497$		−	5.71769i		−	0.256473i
$498$		0			0
$499$	−8.56585			−0.383460			−0.191730	−	0.981448i	$-0.561410\pi$
$499$	−8.56585			−0.383460			−0.191730	+	0.981448i	$0.561410\pi$
$500$		0			0
$501$	14.9939			0.669876
$502$		0			0
$503$			35.7347i			1.59333i	0.604420	+	0.796666i	$0.293405\pi$
$503$			35.7347i			1.59333i	−0.604420	+	0.796666i	$0.706595\pi$
$504$		0			0
$505$	2.82273	+	11.3607i	0.125610	+	0.505544i
$506$		0			0
$507$		−	1.17009i		−	0.0519654i
$508$		0			0
$509$	6.13009			0.271712			0.135856	−	0.990729i	$-0.456622\pi$
$509$	6.13009			0.271712			0.135856	+	0.990729i	$0.456622\pi$
$510$		0			0
$511$	7.70086			0.340666
$512$		0			0
$513$		−	4.76487i		−	0.210374i
$514$		0			0
$515$	−8.01560	+	1.99159i	−0.353209	+	0.0877599i
$516$		0			0
$517$			4.82273i			0.212103i
$518$		0			0
$519$	4.89496			0.214865
$520$		0			0
$521$	24.7442			1.08406			0.542031	−	0.840359i	$-0.317656\pi$
$521$	24.7442			1.08406			0.542031	+	0.840359i	$0.317656\pi$
$522$		0			0
$523$		−	23.7081i		−	1.03668i	−0.855174	−	0.518340i	$-0.826550\pi$
$523$		−	23.7081i		−	1.03668i	0.855174	−	0.518340i	$-0.173450\pi$
$524$		0			0
$525$	3.26180	−	1.72753i	0.142356	−	0.0753955i
$526$		0			0
$527$		−	24.4969i		−	1.06710i
$528$		0			0
$529$	16.2990			0.708650
$530$		0			0
$531$	13.5330			0.587284
$532$		0			0
$533$			3.41855i			0.148074i
$534$		0			0
$535$	34.3968	−	8.54638i	1.48710	−	0.369492i
$536$		0			0
$537$			16.0989i			0.694719i
$538$		0			0
$539$	−3.55971			−0.153327
$540$		0			0
$541$	17.6475			0.758727			0.379364	−	0.925248i	$-0.376143\pi$
$541$	17.6475			0.758727			0.379364	+	0.925248i	$0.376143\pi$
$542$		0			0
$543$		−	25.5174i		−	1.09506i
$544$		0			0
$545$	2.05786	+	8.28231i	0.0881490	+	0.354775i
$546$		0			0
$547$			24.8299i			1.06165i	0.847481	+	0.530825i	$0.178118\pi$
$547$			24.8299i			1.06165i	−0.847481	+	0.530825i	$0.821882\pi$
$548$		0			0
$549$	6.60424			0.281862
$550$		0			0
$551$	−2.01438			−0.0858153
$552$		0			0
$553$			3.57077i			0.151845i
$554$		0			0
$555$	−3.02052	−	12.1568i	−0.128214	−	0.516026i
$556$		0			0
$557$			18.0950i			0.766711i	0.923601	+	0.383355i	$0.125232\pi$
$557$			18.0950i			0.766711i	−0.923601	+	0.383355i	$0.874768\pi$
$558$		0			0
$559$	−3.17009			−0.134080
$560$		0			0
$561$	1.94214			0.0819973
$562$		0			0
$563$		−	6.67316i		−	0.281240i	−0.990064	−	0.140620i	$-0.955090\pi$
$563$		−	6.67316i		−	0.281240i	0.990064	−	0.140620i	$-0.0449096\pi$
$564$		0			0
$565$	24.8371	−	6.17113i	1.04490	−	0.259621i
$566$		0			0
$567$		−	0.913212i		−	0.0383513i
$568$		0			0
$569$	−10.7877			−0.452242			−0.226121	−	0.974099i	$-0.572604\pi$
$569$	−10.7877			−0.452242			−0.226121	+	0.974099i	$0.572604\pi$
$570$		0			0
$571$	−22.2245			−0.930065			−0.465032	−	0.885294i	$-0.653957\pi$
$571$	−22.2245			−0.930065			−0.465032	+	0.885294i	$0.653957\pi$
$572$		0			0
$573$		−	3.31965i		−	0.138681i
$574$		0			0
$575$	6.05786	+	11.4380i	0.252630	+	0.476999i
$576$		0			0
$577$		−	34.6309i		−	1.44170i	−0.693089	−	0.720852i	$-0.743751\pi$
$577$		−	34.6309i		−	1.44170i	0.693089	−	0.720852i	$-0.256249\pi$
$578$		0			0
$579$	−25.2930			−1.05114
$580$		0			0
$581$	−0.649149			−0.0269312
$582$		0			0
$583$		−	0.267938i		−	0.0110969i
$584$		0			0
$585$	−3.53919	+	0.879362i	−0.146327	+	0.0363571i
$586$		0			0
$587$			23.8660i			0.985057i	0.870296	+	0.492528i	$0.163927\pi$
$587$			23.8660i			0.985057i	−0.870296	+	0.492528i	$0.836073\pi$
$588$		0			0
$589$	6.99773			0.288337
$590$		0			0
$591$	−2.34017			−0.0962619
$592$		0			0
$593$		−	2.39803i		−	0.0984754i	−0.998787	−	0.0492377i	$-0.984321\pi$
$593$		−	2.39803i		−	0.0984754i	0.998787	−	0.0492377i	$-0.0156792\pi$
$594$		0			0
$595$	−1.04718	−	4.21461i	−0.0429302	−	0.172782i
$596$		0			0
$597$			16.8950i			0.691465i
$598$		0			0
$599$	−40.8248			−1.66806			−0.834028	−	0.551722i	$-0.813971\pi$
$599$	−40.8248			−1.66806			−0.834028	+	0.551722i	$0.813971\pi$
$600$		0			0
$601$	−20.4703			−0.835000			−0.417500	−	0.908677i	$-0.637094\pi$
$601$	−20.4703			−0.835000			−0.417500	+	0.908677i	$0.637094\pi$
$602$		0			0
$603$			5.49466i			0.223760i
$604$		0			0
$605$	−5.77432	−	23.2401i	−0.234760	−	0.944843i
$606$		0			0
$607$			37.1578i			1.50819i	0.656766	+	0.754094i	$0.271924\pi$
$607$			37.1578i			1.50819i	−0.656766	+	0.754094i	$0.728076\pi$
$608$		0			0
$609$	−1.69102			−0.0685237
$610$		0			0
$611$	8.94441			0.361852
$612$		0			0
$613$		−	20.1711i		−	0.814704i	−0.913271	−	0.407352i	$-0.866452\pi$
$613$		−	20.1711i		−	0.814704i	0.913271	−	0.407352i	$-0.133548\pi$
$614$		0			0
$615$	−8.68035	+	2.15676i	−0.350025	+	0.0869688i
$616$		0			0
$617$		−	14.4657i		−	0.582368i	−0.956667	−	0.291184i	$-0.905951\pi$
$617$		−	14.4657i		−	0.582368i	0.956667	−	0.291184i	$-0.0940493\pi$
$618$		0			0
$619$	−19.5330			−0.785099			−0.392550	−	0.919731i	$-0.628407\pi$
$619$	−19.5330			−0.785099			−0.392550	+	0.919731i	$0.628407\pi$
$620$		0			0
$621$	14.0267			0.562871
$622$		0			0
$623$			10.6225i			0.425581i
$624$		0			0
$625$	14.0472	−	20.6803i	0.561887	−	0.827214i
$626$		0			0
$627$			0.554787i			0.0221561i
$628$		0			0
$629$	−14.7382			−0.587651
$630$		0			0
$631$	−22.9783			−0.914750			−0.457375	−	0.889274i	$-0.651210\pi$
$631$	−22.9783			−0.914750			−0.457375	+	0.889274i	$0.651210\pi$
$632$		0			0
$633$			5.01068i			0.199157i
$634$		0			0
$635$	11.6176	−	2.88655i	0.461029	−	0.114549i
$636$		0			0
$637$			6.60197i			0.261580i
$638$		0			0
$639$	−14.7805			−0.584706
$640$		0			0
$641$	5.00614			0.197731			0.0988654	−	0.995101i	$-0.468479\pi$
$641$	5.00614			0.197731			0.0988654	+	0.995101i	$0.468479\pi$
$642$		0			0
$643$			3.95055i			0.155795i	0.996961	+	0.0778973i	$0.0248206\pi$
$643$			3.95055i			0.155795i	−0.996961	+	0.0778973i	$0.975179\pi$
$644$		0			0
$645$	−2.00000	−	8.04945i	−0.0787499	−	0.316947i
$646$		0			0
$647$		−	32.8710i		−	1.29229i	−0.763214	−	0.646145i	$-0.776380\pi$
$647$		−	32.8710i		−	1.29229i	0.763214	−	0.646145i	$-0.223620\pi$
$648$		0			0
$649$	−4.47414			−0.175625
$650$		0			0
$651$	5.87444			0.230238
$652$		0			0
$653$			24.4703i			0.957596i	0.877925	+	0.478798i	$0.158927\pi$
$653$			24.4703i			0.957596i	−0.877925	+	0.478798i	$0.841073\pi$
$654$		0			0
$655$	10.2101	+	41.0928i	0.398941	+	1.60563i
$656$		0			0
$657$		−	19.9071i		−	0.776649i
$658$		0			0
$659$	−21.0928			−0.821657			−0.410829	−	0.911713i	$-0.634761\pi$
$659$	−21.0928			−0.821657			−0.410829	+	0.911713i	$0.634761\pi$
$660$		0			0
$661$	−28.5958			−1.11225			−0.556124	−	0.831099i	$-0.687712\pi$
$661$	−28.5958			−1.11225			−0.556124	+	0.831099i	$0.687712\pi$
$662$		0			0
$663$		−	3.60197i		−	0.139889i
$664$		0			0
$665$	1.20394	−	0.299135i	0.0466867	−	0.0116000i
$666$		0			0
$667$		−	5.92986i		−	0.229605i
$668$		0			0
$669$	−26.7792			−1.03535
$670$		0			0
$671$	−2.18342			−0.0842899
$672$		0			0
$673$		−	1.02052i		−	0.0393381i	−0.999807	−	0.0196691i	$-0.993739\pi$
$673$		−	1.02052i		−	0.0393381i	0.999807	−	0.0196691i	$-0.00626126\pi$
$674$		0			0
$675$	−12.6803	−	23.9421i	−0.488067	−	0.921533i
$676$		0			0
$677$			33.3295i			1.28096i	0.767976	+	0.640478i	$0.221264\pi$
$677$			33.3295i			1.28096i	−0.767976	+	0.640478i	$0.778736\pi$
$678$		0			0
$679$	3.30283			0.126751
$680$		0			0
$681$	6.89496			0.264215
$682$		0			0
$683$		−	1.52586i		−	0.0583853i	−0.999574	−	0.0291927i	$-0.990706\pi$
$683$		−	1.52586i		−	0.0583853i	0.999574	−	0.0291927i	$-0.00929363\pi$
$684$		0			0
$685$	13.2351	−	3.28846i	0.505688	−	0.125645i
$686$		0			0
$687$		−	20.0312i		−	0.764238i
$688$		0			0
$689$	−0.496928			−0.0189315
$690$		0			0
$691$	−21.2339			−0.807776			−0.403888	−	0.914808i	$-0.632341\pi$
$691$	−21.2339			−0.807776			−0.403888	+	0.914808i	$0.632341\pi$
$692$		0			0
$693$		−	0.554787i		−	0.0210746i
$694$		0			0
$695$	11.6163	+	46.7526i	0.440633	+	1.77343i
$696$		0			0
$697$			10.5236i			0.398609i
$698$		0			0
$699$	16.2511			0.614674
$700$		0			0
$701$	0.974946			0.0368232			0.0184116	−	0.999830i	$-0.494139\pi$
$701$	0.974946			0.0368232			0.0184116	+	0.999830i	$0.494139\pi$
$702$		0			0
$703$		−	4.21008i		−	0.158786i
$704$		0			0
$705$	5.64301	+	22.7115i	0.212528	+	0.855366i
$706$		0			0
$707$			3.30283i			0.124216i
$708$		0			0
$709$	−4.14238			−0.155570			−0.0777852	−	0.996970i	$-0.524785\pi$
$709$	−4.14238			−0.155570			−0.0777852	+	0.996970i	$0.524785\pi$
$710$		0			0
$711$	9.23060			0.346174
$712$		0			0
$713$			20.5997i			0.771465i
$714$		0			0
$715$	1.17009	−	0.290725i	0.0437588	−	0.0108725i
$716$		0			0
$717$		−	24.5152i		−	0.915536i
$718$		0			0
$719$	9.26180			0.345407			0.172703	−	0.984974i	$-0.444750\pi$
$719$	9.26180			0.345407			0.172703	+	0.984974i	$0.444750\pi$
$720$		0			0
$721$	−2.33033			−0.0867861
$722$		0			0
$723$		−	10.1568i		−	0.377734i
$724$		0			0
$725$	−10.1217	+	5.36069i	−0.375910	+	0.199091i
$726$		0			0
$727$			38.9165i			1.44333i	0.692240	+	0.721667i	$0.256624\pi$
$727$			38.9165i			1.44333i	−0.692240	+	0.721667i	$0.743376\pi$
$728$		0			0
$729$	−21.3812			−0.791897
$730$		0			0
$731$	−9.75872			−0.360939
$732$		0			0
$733$			51.1917i			1.89081i	0.325903	+	0.945403i	$0.394332\pi$
$733$			51.1917i			1.89081i	−0.325903	+	0.945403i	$0.605668\pi$
$734$		0			0
$735$	−16.7636	+	4.16517i	−0.618336	+	0.153634i
$736$		0			0
$737$		−	1.81658i		−	0.0669147i
$738$		0			0
$739$	10.8260			0.398242			0.199121	−	0.979975i	$-0.436191\pi$
$739$	10.8260			0.398242			0.199121	+	0.979975i	$0.436191\pi$
$740$		0			0
$741$	1.02893			0.0377987
$742$		0			0
$743$			43.3667i			1.59097i	0.605974	+	0.795484i	$0.292783\pi$
$743$			43.3667i			1.59097i	−0.605974	+	0.795484i	$0.707217\pi$
$744$		0			0
$745$	−4.46573	−	17.9733i	−0.163612	−	0.658492i
$746$		0			0
$747$			1.67808i			0.0613977i
$748$		0			0
$749$	10.0000			0.365392
$750$		0			0
$751$	23.9299			0.873213			0.436606	−	0.899653i	$-0.356180\pi$
$751$	23.9299			0.873213			0.436606	+	0.899653i	$0.356180\pi$
$752$		0			0
$753$			10.0410i			0.365916i
$754$		0			0
$755$	−4.10731	−	16.5308i	−0.149480	−	0.601617i
$756$		0			0
$757$		−	25.9733i		−	0.944017i	−0.881594	−	0.472009i	$-0.843529\pi$
$757$		−	25.9733i		−	0.944017i	0.881594	−	0.472009i	$-0.156471\pi$
$758$		0			0
$759$	−1.63317			−0.0592801
$760$		0			0
$761$	17.7854			0.644720			0.322360	−	0.946617i	$-0.395524\pi$
$761$	17.7854			0.644720			0.322360	+	0.946617i	$0.395524\pi$
$762$		0			0
$763$			2.40787i			0.0871708i
$764$		0			0
$765$	−10.8950	+	2.70701i	−0.393908	+	0.0978721i
$766$		0			0
$767$			8.29791i			0.299620i
$768$		0			0
$769$	13.9011			0.501287			0.250643	−	0.968080i	$-0.419358\pi$
$769$	13.9011			0.501287			0.250643	+	0.968080i	$0.419358\pi$
$770$		0			0
$771$	−16.6803			−0.600728
$772$		0			0
$773$			10.7298i			0.385924i	0.981206	+	0.192962i	$0.0618094\pi$
$773$			10.7298i			0.385924i	−0.981206	+	0.192962i	$0.938191\pi$
$774$		0			0
$775$	35.1617	−	18.6225i	1.26304	−	0.668939i
$776$		0			0
$777$		−	3.53427i		−	0.126791i
$778$		0			0
$779$	−3.00614			−0.107706
$780$		0			0
$781$	4.88655			0.174854
$782$		0			0
$783$			12.4124i			0.443583i
$784$		0			0
$785$	−7.54411	+	1.87444i	−0.269261	+	0.0669017i
$786$		0			0
$787$			9.76260i			0.347999i	0.984746	+	0.174000i	$0.0556691\pi$
$787$			9.76260i			0.347999i	−0.984746	+	0.174000i	$0.944331\pi$
$788$		0			0
$789$	−12.6719			−0.451133
$790$		0			0
$791$	7.22076			0.256741
$792$		0			0
$793$			4.04945i			0.143800i
$794$		0			0
$795$	−0.313511	−	1.26180i	−0.0111191	−	0.0447513i
$796$		0			0
$797$			42.7792i			1.51532i	0.652650	+	0.757659i	$0.273657\pi$
$797$			42.7792i			1.51532i	−0.652650	+	0.757659i	$0.726343\pi$
$798$		0			0
$799$	27.5343			0.974092
$800$		0			0
$801$	27.4596			0.970237
$802$		0			0
$803$			6.58145i			0.232254i
$804$		0			0
$805$	0.880584	+	3.54411i	0.0310365	+	0.124913i
$806$		0			0
$807$			25.0784i			0.882801i
$808$		0			0
$809$	−23.1155			−0.812699			−0.406350	−	0.913718i	$-0.633198\pi$
$809$	−23.1155			−0.812699			−0.406350	+	0.913718i	$0.633198\pi$
$810$		0			0
$811$	7.59090			0.266553			0.133276	−	0.991079i	$-0.457450\pi$
$811$	7.59090			0.266553			0.133276	+	0.991079i	$0.457450\pi$
$812$		0			0
$813$			25.1278i			0.881271i
$814$		0			0
$815$	−19.8082	+	4.92162i	−0.693850	+	0.172397i
$816$		0			0
$817$		−	2.78765i		−	0.0975276i
$818$		0			0
$819$	−1.02893			−0.0359537
$820$		0			0
$821$	14.0722			0.491124			0.245562	−	0.969381i	$-0.421027\pi$
$821$	14.0722			0.491124			0.245562	+	0.969381i	$0.421027\pi$
$822$		0			0
$823$			8.10608i			0.282560i	0.989970	+	0.141280i	$0.0451218\pi$
$823$			8.10608i			0.282560i	−0.989970	+	0.141280i	$0.954878\pi$
$824$		0			0
$825$	1.47641	+	2.78765i	0.0514020	+	0.0970536i
$826$		0			0
$827$			50.1750i			1.74476i	0.488832	+	0.872378i	$0.337423\pi$
$827$			50.1750i			1.74476i	−0.488832	+	0.872378i	$0.662577\pi$
$828$		0			0
$829$	−21.2123			−0.736735			−0.368368	−	0.929680i	$-0.620083\pi$
$829$	−21.2123			−0.736735			−0.368368	+	0.929680i	$0.620083\pi$
$830$		0			0
$831$	−3.84778			−0.133478
$832$		0			0
$833$			20.3234i			0.704162i
$834$		0			0
$835$	27.8082	−	6.90934i	0.962341	−	0.239107i
$836$		0			0
$837$		−	43.1194i		−	1.49043i
$838$		0			0
$839$	46.4645			1.60413			0.802066	−	0.597235i	$-0.203734\pi$
$839$	46.4645			1.60413			0.802066	+	0.597235i	$0.203734\pi$
$840$		0			0
$841$	−23.7526			−0.819055
$842$		0			0
$843$		−	32.9939i		−	1.13637i
$844$		0			0
$845$	−0.539189	−	2.17009i	−0.0185487	−	0.0746532i
$846$		0			0
$847$		−	6.75646i		−	0.232155i
$848$		0			0
$849$	−14.2329			−0.488471
$850$		0			0
$851$	12.3935			0.424844
$852$		0			0
$853$			3.99159i			0.136669i	0.997662	+	0.0683347i	$0.0217686\pi$
$853$			3.99159i			0.136669i	−0.997662	+	0.0683347i	$0.978231\pi$
$854$		0			0
$855$	−0.773277	−	3.11223i	−0.0264455	−	0.106436i
$856$		0			0
$857$			31.8843i			1.08915i	0.838713	+	0.544573i	$0.183308\pi$
$857$			31.8843i			1.08915i	−0.838713	+	0.544573i	$0.816692\pi$
$858$		0			0
$859$	−26.8059			−0.914606			−0.457303	−	0.889311i	$-0.651184\pi$
$859$	−26.8059			−0.914606			−0.457303	+	0.889311i	$0.651184\pi$
$860$		0			0
$861$	−2.52359			−0.0860037
$862$		0			0
$863$			41.5259i			1.41356i	0.707435	+	0.706778i	$0.249852\pi$
$863$			41.5259i			1.41356i	−0.707435	+	0.706778i	$0.750148\pi$
$864$		0			0
$865$	9.07838	−	2.25565i	0.308674	−	0.0766945i
$866$		0			0
$867$			8.80325i			0.298974i
$868$		0			0
$869$	−3.05172			−0.103522
$870$		0			0
$871$	−3.36910			−0.114158
$872$		0			0
$873$		−	8.53797i		−	0.288966i
$874$		0			0
$875$	5.25338	−	4.70701i	0.177597	−	0.159126i
$876$		0			0
$877$		−	9.98562i		−	0.337191i	−0.985685	−	0.168595i	$-0.946077\pi$
$877$		−	9.98562i		−	0.337191i	0.985685	−	0.168595i	$-0.0539231\pi$
$878$		0			0
$879$	10.9627			0.369761
$880$		0			0
$881$	8.41628			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$881$	8.41628			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$882$		0			0
$883$		−	30.4585i		−	1.02501i	−0.858684	−	0.512506i	$-0.828717\pi$
$883$		−	30.4585i		−	1.02501i	0.858684	−	0.512506i	$-0.171283\pi$
$884$		0			0
$885$	−21.0700	+	5.23513i	−0.708259	+	0.175977i
$886$		0			0
$887$		−	54.7454i		−	1.83817i	−0.394059	−	0.919085i	$-0.628929\pi$
$887$		−	54.7454i		−	1.83817i	0.394059	−	0.919085i	$-0.371071\pi$
$888$		0			0
$889$	3.37751			0.113278
$890$		0			0
$891$	0.780465			0.0261466
$892$		0			0
$893$			7.86537i			0.263205i
$894$		0			0
$895$	7.41855	+	29.8576i	0.247975	+	0.998030i
$896$		0			0
$897$			3.02893i			0.101133i
$898$		0			0
$899$	−18.2290			−0.607971
$900$		0			0
$901$	−1.52973			−0.0509628
$902$		0			0
$903$		−	2.34017i		−	0.0778761i
$904$		0			0
$905$	−11.7587	−	47.3256i	−0.390873	−	1.57316i
$906$		0			0
$907$			30.4007i			1.00944i	0.863284	+	0.504719i	$0.168404\pi$
$907$			30.4007i			1.00944i	−0.863284	+	0.504719i	$0.831596\pi$
$908$		0			0
$909$	8.53797			0.283186
$910$		0			0
$911$	−10.7526			−0.356249			−0.178124	−	0.984008i	$-0.557003\pi$
$911$	−10.7526			−0.356249			−0.178124	+	0.984008i	$0.557003\pi$
$912$		0			0
$913$		−	0.554787i		−	0.0183608i
$914$		0			0
$915$	−10.2823	+	2.55479i	−0.339923	+	0.0844587i
$916$		0			0
$917$			11.9467i			0.394514i
$918$		0			0
$919$	37.3607			1.23242			0.616208	−	0.787584i	$-0.288668\pi$
$919$	37.3607			1.23242			0.616208	+	0.787584i	$0.288668\pi$
$920$		0			0
$921$	−36.3545			−1.19792
$922$		0			0
$923$		−	9.06278i		−	0.298305i
$924$		0			0
$925$	−11.2039	−	21.1545i	−0.368383	−	0.695555i
$926$		0			0
$927$			6.02401i			0.197854i
$928$		0			0
$929$	−33.6742			−1.10481			−0.552407	−	0.833574i	$-0.686291\pi$
$929$	−33.6742			−1.10481			−0.552407	+	0.833574i	$0.686291\pi$
$930$		0			0
$931$	−5.80552			−0.190268
$932$		0			0
$933$		−	4.31351i		−	0.141218i
$934$		0			0
$935$	3.60197	−	0.894960i	0.117797	−	0.0292683i
$936$		0			0
$937$		−	25.0394i		−	0.818003i	−0.912534	−	0.409001i	$-0.865877\pi$
$937$		−	25.0394i		−	0.818003i	0.912534	−	0.409001i	$-0.134123\pi$
$938$		0			0
$939$	−2.92162			−0.0953435
$940$		0			0
$941$	7.07838			0.230749			0.115374	−	0.993322i	$-0.463193\pi$
$941$	7.07838			0.230749			0.115374	+	0.993322i	$0.463193\pi$
$942$		0			0
$943$		−	8.84939i		−	0.288176i
$944$		0			0
$945$	−1.84324	−	7.41855i	−0.0599607	−	0.241325i
$946$		0			0
$947$		−	24.1795i		−	0.785730i	−0.919596	−	0.392865i	$-0.871484\pi$
$947$		−	24.1795i		−	0.785730i	0.919596	−	0.392865i	$-0.128516\pi$
$948$		0			0
$949$	12.2062			0.396230
$950$		0			0
$951$	32.1880			1.04377
$952$		0			0
$953$			21.6742i			0.702096i	0.936357	+	0.351048i	$0.114175\pi$
$953$			21.6742i			0.702096i	−0.936357	+	0.351048i	$0.885825\pi$
$954$		0			0
$955$	−1.52973	−	6.15676i	−0.0495010	−	0.199228i
$956$		0			0
$957$		−	1.44521i		−	0.0467171i
$958$		0			0
$959$	3.84778			0.124251
$960$		0			0
$961$	32.3256			1.04276
$962$		0			0
$963$		−	25.8504i		−	0.833019i
$964$		0			0
$965$	−46.9093	+	11.6553i	−1.51006	+	0.375197i
$966$		0			0
$967$			1.14220i			0.0367307i	0.999831	+	0.0183654i	$0.00584621\pi$
$967$			1.14220i			0.0367307i	−0.999831	+	0.0183654i	$0.994154\pi$
$968$		0			0
$969$	3.16743			0.101753
$970$		0			0
$971$	−41.9565			−1.34645			−0.673224	−	0.739438i	$-0.735091\pi$
$971$	−41.9565			−1.34645			−0.673224	+	0.739438i	$0.735091\pi$
$972$		0			0
$973$			13.5921i			0.435744i
$974$		0			0
$975$	5.17009	−	2.73820i	0.165575	−	0.0876927i
$976$		0			0
$977$			21.0023i			0.671922i	0.941876	+	0.335961i	$0.109061\pi$
$977$			21.0023i			0.671922i	−0.941876	+	0.335961i	$0.890939\pi$
$978$		0			0
$979$	−9.07838			−0.290146
$980$		0			0
$981$	6.22446			0.198732
$982$		0			0
$983$		−	44.1171i		−	1.40712i	−0.710637	−	0.703559i	$-0.751593\pi$
$983$		−	44.1171i		−	1.40712i	0.710637	−	0.703559i	$-0.248407\pi$
$984$		0			0
$985$	−4.34017	+	1.07838i	−0.138289	+	0.0343600i
$986$		0			0
$987$			6.60281i			0.210170i
$988$		0			0
$989$	8.20620			0.260942
$990$		0			0
$991$	48.9549			1.55510			0.777552	−	0.628819i	$-0.216461\pi$
$991$	48.9549			1.55510			0.777552	+	0.628819i	$0.216461\pi$
$992$		0			0
$993$		−	10.8865i		−	0.345474i
$994$		0			0
$995$	7.78539	+	31.3340i	0.246813	+	0.993356i
$996$		0			0
$997$		−	52.9914i		−	1.67825i	−0.543935	−	0.839127i	$-0.683066\pi$
$997$		−	52.9914i		−	1.67825i	0.543935	−	0.839127i	$-0.316934\pi$
$998$		0			0
$999$	−25.9421			−0.820773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.d.b.209.5	yes	6
3.2	odd	2		4680.2.l.d.2809.3		6
4.3	odd	2		1040.2.d.e.209.2		6
5.2	odd	4		2600.2.a.x.1.3		3
5.3	odd	4		2600.2.a.y.1.1		3
5.4	even	2	inner	520.2.d.b.209.2	✓	6
15.14	odd	2		4680.2.l.d.2809.4		6
20.3	even	4		5200.2.a.cc.1.3		3
20.7	even	4		5200.2.a.ch.1.1		3
20.19	odd	2		1040.2.d.e.209.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.d.b.209.2	✓	6	5.4	even	2	inner
520.2.d.b.209.5	yes	6	1.1	even	1	trivial
1040.2.d.e.209.2		6	4.3	odd	2
1040.2.d.e.209.5		6	20.19	odd	2
2600.2.a.x.1.3		3	5.2	odd	4
2600.2.a.y.1.1		3	5.3	odd	4
4680.2.l.d.2809.3		6	3.2	odd	2
4680.2.l.d.2809.4		6	15.14	odd	2
5200.2.a.cc.1.3		3	20.3	even	4
5200.2.a.ch.1.1		3	20.7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.17009i q^{3} +(0.539189 + 2.17009i) q^{5} +0.630898i q^{7} +1.63090 q^{9} -0.539189 q^{11} +1.00000i q^{13} +(-2.53919 + 0.630898i) q^{15} +3.07838i q^{17} -0.879362 q^{19} -0.738205 q^{21} -2.58864i q^{23} +(-4.41855 + 2.34017i) q^{25} +5.41855i q^{27} +2.29072 q^{29} -7.95774 q^{31} -0.630898i q^{33} +(-1.36910 + 0.340173i) q^{35} +4.78765i q^{37} -1.17009 q^{39} +3.41855 q^{41} +3.17009i q^{43} +(0.879362 + 3.53919i) q^{45} -8.94441i q^{47} +6.60197 q^{49} -3.60197 q^{51} +0.496928i q^{53} +(-0.290725 - 1.17009i) q^{55} -1.02893i q^{57} +8.29791 q^{59} +4.04945 q^{61} +1.02893i q^{63} +(-2.17009 + 0.539189i) q^{65} +3.36910i q^{67} +3.02893 q^{69} -9.06278 q^{71} -12.2062i q^{73} +(-2.73820 - 5.17009i) q^{75} -0.340173i q^{77} +5.65983 q^{79} -1.44748 q^{81} +1.02893i q^{83} +(-6.68035 + 1.65983i) q^{85} +2.68035i q^{87} +16.8371 q^{89} -0.630898 q^{91} -9.31124i q^{93} +(-0.474142 - 1.90829i) q^{95} -5.23513i q^{97} -0.879362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} - 12 q^{15} + 20 q^{19} - 20 q^{21} + 2 q^{25} + 28 q^{29} - 16 q^{31} - 16 q^{35} + 4 q^{39} - 8 q^{41} - 20 q^{45} + 2 q^{49} + 16 q^{51} - 16 q^{55} - 4 q^{59} - 12 q^{61} - 2 q^{65} + 48 q^{69}+ \cdots + 20 q^{99}+O(q^{100}) \) 6 * q + 2 * q^9 - 12 * q^15 + 20 * q^19 - 20 * q^21 + 2 * q^25 + 28 * q^29 - 16 * q^31 - 16 * q^35 + 4 * q^39 - 8 * q^41 - 20 * q^45 + 2 * q^49 + 16 * q^51 - 16 * q^55 - 4 * q^59 - 12 * q^61 - 2 * q^65 + 48 * q^69 - 20 * q^71 - 32 * q^75 + 56 * q^79 - 10 * q^81 + 4 * q^85 + 44 * q^89 + 4 * q^91 - 8 * q^95 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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