LMFDB - Embedded newform 480.2.h.b.191.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,2,Mod(191,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	480.191
Dual form			480.2.h.b.191.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.70711 - 0.292893i) q^{3} +1.00000i q^{5} -0.585786i q^{7} +(2.82843 + 1.00000i) q^{9} -2.82843 q^{11} -2.00000 q^{13} +(0.292893 - 1.70711i) q^{15} +3.65685i q^{17} +2.82843i q^{19} +(-0.171573 + 1.00000i) q^{21} -4.58579 q^{23} -1.00000 q^{25} +(-4.53553 - 2.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(4.82843 + 0.828427i) q^{33} +0.585786 q^{35} -11.6569 q^{37} +(3.41421 + 0.585786i) q^{39} -2.00000i q^{41} +11.8995i q^{43} +(-1.00000 + 2.82843i) q^{45} -2.24264 q^{47} +6.65685 q^{49} +(1.07107 - 6.24264i) q^{51} -7.65685i q^{53} -2.82843i q^{55} +(0.828427 - 4.82843i) q^{57} +9.65685 q^{59} -5.65685 q^{61} +(0.585786 - 1.65685i) q^{63} -2.00000i q^{65} -13.0711i q^{67} +(7.82843 + 1.34315i) q^{69} -6.82843 q^{71} +4.34315 q^{73} +(1.70711 + 0.292893i) q^{75} +1.65685i q^{77} +12.4853i q^{79} +(7.00000 + 5.65685i) q^{81} +9.07107 q^{83} -3.65685 q^{85} +(2.34315 - 13.6569i) q^{87} -15.3137i q^{89} +1.17157i q^{91} +(1.65685 - 9.65685i) q^{93} -2.82843 q^{95} -14.9706 q^{97} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{13} + 4 q^{15} - 12 q^{21} - 24 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 4 q^{45} + 8 q^{47} + 4 q^{49} - 24 q^{51} - 8 q^{57} + 16 q^{59} + 8 q^{63}+ \cdots - 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^13 + 4 * q^15 - 12 * q^21 - 24 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 4 * q^45 + 8 * q^47 + 4 * q^49 - 24 * q^51 - 8 * q^57 + 16 * q^59 + 8 * q^63 + 20 * q^69 - 16 * q^71 + 40 * q^73 + 4 * q^75 + 28 * q^81 + 8 * q^83 + 8 * q^85 + 32 * q^87 - 16 * q^93 + 8 * q^97 - 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	0.585786i		−	0.221406i	−0.993854	−	0.110703i	$-0.964690\pi$
$7$		−	0.585786i		−	0.221406i	0.993854	−	0.110703i	$-0.0353103\pi$
$8$		0			0
$9$	2.82843	+	1.00000i	0.942809	+	0.333333i
$10$		0			0
$11$	−2.82843			−0.852803			−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843			−0.852803			−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	0.292893	−	1.70711i	0.0756247	−	0.440773i
$16$		0			0
$17$			3.65685i			0.886917i	0.896295	+	0.443459i	$0.146249\pi$
$17$			3.65685i			0.886917i	−0.896295	+	0.443459i	$0.853751\pi$
$18$		0			0
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$	−0.171573	+	1.00000i	−0.0374403	+	0.218218i
$22$		0			0
$23$	−4.58579			−0.956203			−0.478101	−	0.878305i	$-0.658675\pi$
$23$	−4.58579			−0.956203			−0.478101	+	0.878305i	$0.658675\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.53553	−	2.53553i	−0.872864	−	0.487964i
$28$		0			0
$29$			8.00000i			1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$			8.00000i			1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$		0			0
$31$			5.65685i			1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$31$			5.65685i			1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$32$		0			0
$33$	4.82843	+	0.828427i	0.840521	+	0.144211i
$34$		0			0
$35$	0.585786			0.0990160
$36$		0			0
$37$	−11.6569			−1.91638			−0.958188	−	0.286141i	$-0.907627\pi$
$37$	−11.6569			−1.91638			−0.958188	+	0.286141i	$0.907627\pi$
$38$		0			0
$39$	3.41421	+	0.585786i	0.546712	+	0.0938009i
$40$		0			0
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$			11.8995i			1.81466i	0.420424	+	0.907328i	$0.361882\pi$
$43$			11.8995i			1.81466i	−0.420424	+	0.907328i	$0.638118\pi$
$44$		0			0
$45$	−1.00000	+	2.82843i	−0.149071	+	0.421637i
$46$		0			0
$47$	−2.24264			−0.327123			−0.163561	−	0.986533i	$-0.552298\pi$
$47$	−2.24264			−0.327123			−0.163561	+	0.986533i	$0.552298\pi$
$48$		0			0
$49$	6.65685			0.950979
$50$		0			0
$51$	1.07107	−	6.24264i	0.149979	−	0.874145i
$52$		0			0
$53$		−	7.65685i		−	1.05175i	−0.850562	−	0.525875i	$-0.823738\pi$
$53$		−	7.65685i		−	1.05175i	0.850562	−	0.525875i	$-0.176262\pi$
$54$		0			0
$55$		−	2.82843i		−	0.381385i
$56$		0			0
$57$	0.828427	−	4.82843i	0.109728	−	0.639541i
$58$		0			0
$59$	9.65685			1.25722			0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685			1.25722			0.628608	+	0.777723i	$0.283625\pi$
$60$		0			0
$61$	−5.65685			−0.724286			−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685			−0.724286			−0.362143	+	0.932123i	$0.617955\pi$
$62$		0			0
$63$	0.585786	−	1.65685i	0.0738022	−	0.208744i
$64$		0			0
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$		−	13.0711i		−	1.59689i	−0.602071	−	0.798443i	$-0.705658\pi$
$67$		−	13.0711i		−	1.59689i	0.602071	−	0.798443i	$-0.294342\pi$
$68$		0			0
$69$	7.82843	+	1.34315i	0.942432	+	0.161696i
$70$		0			0
$71$	−6.82843			−0.810385			−0.405193	−	0.914231i	$-0.632796\pi$
$71$	−6.82843			−0.810385			−0.405193	+	0.914231i	$0.632796\pi$
$72$		0			0
$73$	4.34315			0.508327			0.254163	−	0.967161i	$-0.418200\pi$
$73$	4.34315			0.508327			0.254163	+	0.967161i	$0.418200\pi$
$74$		0			0
$75$	1.70711	+	0.292893i	0.197120	+	0.0338204i
$76$		0			0
$77$			1.65685i			0.188816i
$78$		0			0
$79$			12.4853i			1.40470i	0.711830	+	0.702352i	$0.247867\pi$
$79$			12.4853i			1.40470i	−0.711830	+	0.702352i	$0.752133\pi$
$80$		0			0
$81$	7.00000	+	5.65685i	0.777778	+	0.628539i
$82$		0			0
$83$	9.07107			0.995679			0.497840	−	0.867269i	$-0.334127\pi$
$83$	9.07107			0.995679			0.497840	+	0.867269i	$0.334127\pi$
$84$		0			0
$85$	−3.65685			−0.396642
$86$		0			0
$87$	2.34315	−	13.6569i	0.251212	−	1.46417i
$88$		0			0
$89$		−	15.3137i		−	1.62325i	−0.584179	−	0.811625i	$-0.698583\pi$
$89$		−	15.3137i		−	1.62325i	0.584179	−	0.811625i	$-0.301417\pi$
$90$		0			0
$91$			1.17157i			0.122814i
$92$		0			0
$93$	1.65685	−	9.65685i	0.171808	−	1.00137i
$94$		0			0
$95$	−2.82843			−0.290191
$96$		0			0
$97$	−14.9706			−1.52003			−0.760015	−	0.649905i	$-0.774809\pi$
$97$	−14.9706			−1.52003			−0.760015	+	0.649905i	$0.774809\pi$
$98$		0			0
$99$	−8.00000	−	2.82843i	−0.804030	−	0.284268i
$100$		0			0
$101$		−	15.3137i		−	1.52377i	−0.647712	−	0.761885i	$-0.724274\pi$
$101$		−	15.3137i		−	1.52377i	0.647712	−	0.761885i	$-0.275726\pi$
$102$		0			0
$103$		−	6.24264i		−	0.615106i	−0.951531	−	0.307553i	$-0.900490\pi$
$103$		−	6.24264i		−	0.615106i	0.951531	−	0.307553i	$-0.0995101\pi$
$104$		0			0
$105$	−1.00000	−	0.171573i	−0.0975900	−	0.0167438i
$106$		0			0
$107$	12.5858			1.21671			0.608357	−	0.793664i	$-0.291829\pi$
$107$	12.5858			1.21671			0.608357	+	0.793664i	$0.291829\pi$
$108$		0			0
$109$	6.34315			0.607563			0.303782	−	0.952742i	$-0.401751\pi$
$109$	6.34315			0.607563			0.303782	+	0.952742i	$0.401751\pi$
$110$		0			0
$111$	19.8995	+	3.41421i	1.88878	+	0.324063i
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$		−	4.58579i		−	0.427627i
$116$		0			0
$117$	−5.65685	−	2.00000i	−0.522976	−	0.184900i
$118$		0			0
$119$	2.14214			0.196369
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	−0.585786	+	3.41421i	−0.0528186	+	0.307849i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			2.92893i			0.259901i	0.991521	+	0.129950i	$0.0414818\pi$
$127$			2.92893i			0.259901i	−0.991521	+	0.129950i	$0.958518\pi$
$128$		0			0
$129$	3.48528	−	20.3137i	0.306862	−	1.78852i
$130$		0			0
$131$	5.17157			0.451842			0.225921	−	0.974146i	$-0.427461\pi$
$131$	5.17157			0.451842			0.225921	+	0.974146i	$0.427461\pi$
$132$		0			0
$133$	1.65685			0.143667
$134$		0			0
$135$	2.53553	−	4.53553i	0.218224	−	0.390357i
$136$		0			0
$137$			5.31371i			0.453981i	0.973897	+	0.226990i	$0.0728886\pi$
$137$			5.31371i			0.453981i	−0.973897	+	0.226990i	$0.927111\pi$
$138$		0			0
$139$			18.8284i			1.59701i	0.601991	+	0.798503i	$0.294374\pi$
$139$			18.8284i			1.59701i	−0.601991	+	0.798503i	$0.705626\pi$
$140$		0			0
$141$	3.82843	+	0.656854i	0.322412	+	0.0553171i
$142$		0			0
$143$	5.65685			0.473050
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$	−11.3640	−	1.94975i	−0.937284	−	0.160812i
$148$		0			0
$149$			9.31371i			0.763009i	0.924367	+	0.381504i	$0.124594\pi$
$149$			9.31371i			0.763009i	−0.924367	+	0.381504i	$0.875406\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−3.65685	+	10.3431i	−0.295639	+	0.836194i
$154$		0			0
$155$	−5.65685			−0.454369
$156$		0			0
$157$	3.65685			0.291849			0.145924	−	0.989296i	$-0.453384\pi$
$157$	3.65685			0.291849			0.145924	+	0.989296i	$0.453384\pi$
$158$		0			0
$159$	−2.24264	+	13.0711i	−0.177853	+	1.03660i
$160$		0			0
$161$			2.68629i			0.211709i
$162$		0			0
$163$			1.75736i			0.137647i	0.997629	+	0.0688235i	$0.0219245\pi$
$163$			1.75736i			0.137647i	−0.997629	+	0.0688235i	$0.978075\pi$
$164$		0			0
$165$	−0.828427	+	4.82843i	−0.0644930	+	0.375893i
$166$		0			0
$167$	6.72792			0.520622			0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792			0.520622			0.260311	+	0.965525i	$0.416175\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−2.82843	+	8.00000i	−0.216295	+	0.611775i
$172$		0			0
$173$		−	2.00000i		−	0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$		−	2.00000i		−	0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$		0			0
$175$			0.585786i			0.0442813i
$176$		0			0
$177$	−16.4853	−	2.82843i	−1.23911	−	0.212598i
$178$		0			0
$179$	−6.34315			−0.474109			−0.237054	−	0.971496i	$-0.576182\pi$
$179$	−6.34315			−0.474109			−0.237054	+	0.971496i	$0.576182\pi$
$180$		0			0
$181$	13.3137			0.989600			0.494800	−	0.869007i	$-0.335241\pi$
$181$	13.3137			0.989600			0.494800	+	0.869007i	$0.335241\pi$
$182$		0			0
$183$	9.65685	+	1.65685i	0.713855	+	0.122478i
$184$		0			0
$185$		−	11.6569i		−	0.857029i
$186$		0			0
$187$		−	10.3431i		−	0.756366i
$188$		0			0
$189$	−1.48528	+	2.65685i	−0.108038	+	0.193258i
$190$		0			0
$191$	−10.1421			−0.733859			−0.366930	−	0.930249i	$-0.619591\pi$
$191$	−10.1421			−0.733859			−0.366930	+	0.930249i	$0.619591\pi$
$192$		0			0
$193$	3.65685			0.263226			0.131613	−	0.991301i	$-0.457984\pi$
$193$	3.65685			0.263226			0.131613	+	0.991301i	$0.457984\pi$
$194$		0			0
$195$	−0.585786	+	3.41421i	−0.0419490	+	0.244497i
$196$		0			0
$197$			10.9706i			0.781620i	0.920471	+	0.390810i	$0.127805\pi$
$197$			10.9706i			0.781620i	−0.920471	+	0.390810i	$0.872195\pi$
$198$		0			0
$199$		−	17.1716i		−	1.21726i	−0.793454	−	0.608630i	$-0.791719\pi$
$199$		−	17.1716i		−	1.21726i	0.793454	−	0.608630i	$-0.208281\pi$
$200$		0			0
$201$	−3.82843	+	22.3137i	−0.270036	+	1.57389i
$202$		0			0
$203$	4.68629			0.328913
$204$		0			0
$205$	2.00000			0.139686
$206$		0			0
$207$	−12.9706	−	4.58579i	−0.901516	−	0.318734i
$208$		0			0
$209$		−	8.00000i		−	0.553372i
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$		0			0
$213$	11.6569	+	2.00000i	0.798714	+	0.137038i
$214$		0			0
$215$	−11.8995			−0.811539
$216$		0			0
$217$	3.31371			0.224949
$218$		0			0
$219$	−7.41421	−	1.27208i	−0.501006	−	0.0859591i
$220$		0			0
$221$		−	7.31371i		−	0.491973i
$222$		0			0
$223$			11.8995i			0.796849i	0.917201	+	0.398425i	$0.130443\pi$
$223$			11.8995i			0.796849i	−0.917201	+	0.398425i	$0.869557\pi$
$224$		0			0
$225$	−2.82843	−	1.00000i	−0.188562	−	0.0666667i
$226$		0			0
$227$	−4.58579			−0.304369			−0.152185	−	0.988352i	$-0.548631\pi$
$227$	−4.58579			−0.304369			−0.152185	+	0.988352i	$0.548631\pi$
$228$		0			0
$229$	25.3137			1.67278			0.836388	−	0.548137i	$-0.184663\pi$
$229$	25.3137			1.67278			0.836388	+	0.548137i	$0.184663\pi$
$230$		0			0
$231$	0.485281	−	2.82843i	0.0319292	−	0.186097i
$232$		0			0
$233$			26.9706i			1.76690i	0.468525	+	0.883450i	$0.344786\pi$
$233$			26.9706i			1.76690i	−0.468525	+	0.883450i	$0.655214\pi$
$234$		0			0
$235$		−	2.24264i		−	0.146294i
$236$		0			0
$237$	3.65685	−	21.3137i	0.237538	−	1.38447i
$238$		0			0
$239$	−13.6569			−0.883388			−0.441694	−	0.897166i	$-0.645622\pi$
$239$	−13.6569			−0.883388			−0.441694	+	0.897166i	$0.645622\pi$
$240$		0			0
$241$	−1.65685			−0.106727			−0.0533637	−	0.998575i	$-0.516994\pi$
$241$	−1.65685			−0.106727			−0.0533637	+	0.998575i	$0.516994\pi$
$242$		0			0
$243$	−10.2929	−	11.7071i	−0.660289	−	0.751011i
$244$		0			0
$245$			6.65685i			0.425291i
$246$		0			0
$247$		−	5.65685i		−	0.359937i
$248$		0			0
$249$	−15.4853	−	2.65685i	−0.981340	−	0.168371i
$250$		0			0
$251$	−16.4853			−1.04054			−0.520271	−	0.854001i	$-0.674169\pi$
$251$	−16.4853			−1.04054			−0.520271	+	0.854001i	$0.674169\pi$
$252$		0			0
$253$	12.9706			0.815452
$254$		0			0
$255$	6.24264	+	1.07107i	0.390929	+	0.0670729i
$256$		0			0
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$			6.82843i			0.424298i
$260$		0			0
$261$	−8.00000	+	22.6274i	−0.495188	+	1.40060i
$262$		0			0
$263$	−7.89949			−0.487104			−0.243552	−	0.969888i	$-0.578313\pi$
$263$	−7.89949			−0.487104			−0.243552	+	0.969888i	$0.578313\pi$
$264$		0			0
$265$	7.65685			0.470357
$266$		0			0
$267$	−4.48528	+	26.1421i	−0.274495	+	1.59987i
$268$		0			0
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$		−	13.6569i		−	0.829595i	−0.909914	−	0.414797i	$-0.863852\pi$
$271$		−	13.6569i		−	0.829595i	0.909914	−	0.414797i	$-0.136148\pi$
$272$		0			0
$273$	0.343146	−	2.00000i	0.0207681	−	0.121046i
$274$		0			0
$275$	2.82843			0.170561
$276$		0			0
$277$	0.343146			0.0206176			0.0103088	−	0.999947i	$-0.496719\pi$
$277$	0.343146			0.0206176			0.0103088	+	0.999947i	$0.496719\pi$
$278$		0			0
$279$	−5.65685	+	16.0000i	−0.338667	+	0.957895i
$280$		0			0
$281$		−	9.31371i		−	0.555609i	−0.960638	−	0.277805i	$-0.910393\pi$
$281$		−	9.31371i		−	0.555609i	0.960638	−	0.277805i	$-0.0896068\pi$
$282$		0			0
$283$		−	17.5563i		−	1.04362i	−0.853063	−	0.521808i	$-0.825258\pi$
$283$		−	17.5563i		−	1.04362i	0.853063	−	0.521808i	$-0.174742\pi$
$284$		0			0
$285$	4.82843	+	0.828427i	0.286011	+	0.0490718i
$286$		0			0
$287$	−1.17157			−0.0691558
$288$		0			0
$289$	3.62742			0.213377
$290$		0			0
$291$	25.5563	+	4.38478i	1.49814	+	0.257040i
$292$		0			0
$293$		−	22.0000i		−	1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$		−	22.0000i		−	1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$		0			0
$295$			9.65685i			0.562244i
$296$		0			0
$297$	12.8284	+	7.17157i	0.744381	+	0.416137i
$298$		0			0
$299$	9.17157			0.530406
$300$		0			0
$301$	6.97056			0.401777
$302$		0			0
$303$	−4.48528	+	26.1421i	−0.257673	+	1.50183i
$304$		0			0
$305$		−	5.65685i		−	0.323911i
$306$		0			0
$307$		−	11.8995i		−	0.679140i	−0.940581	−	0.339570i	$-0.889718\pi$
$307$		−	11.8995i		−	0.679140i	0.940581	−	0.339570i	$-0.110282\pi$
$308$		0			0
$309$	−1.82843	+	10.6569i	−0.104016	+	0.606247i
$310$		0			0
$311$	−28.4853			−1.61525			−0.807626	−	0.589695i	$-0.799248\pi$
$311$	−28.4853			−1.61525			−0.807626	+	0.589695i	$0.799248\pi$
$312$		0			0
$313$	−5.31371			−0.300349			−0.150174	−	0.988660i	$-0.547983\pi$
$313$	−5.31371			−0.300349			−0.150174	+	0.988660i	$0.547983\pi$
$314$		0			0
$315$	1.65685	+	0.585786i	0.0933532	+	0.0330053i
$316$		0			0
$317$			11.6569i			0.654714i	0.944901	+	0.327357i	$0.106158\pi$
$317$			11.6569i			0.654714i	−0.944901	+	0.327357i	$0.893842\pi$
$318$		0			0
$319$		−	22.6274i		−	1.26689i
$320$		0			0
$321$	−21.4853	−	3.68629i	−1.19919	−	0.205749i
$322$		0			0
$323$	−10.3431			−0.575508
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$	−10.8284	−	1.85786i	−0.598813	−	0.102740i
$328$		0			0
$329$			1.31371i			0.0724271i
$330$		0			0
$331$			31.3137i			1.72116i	0.509318	+	0.860579i	$0.329898\pi$
$331$			31.3137i			1.72116i	−0.509318	+	0.860579i	$0.670102\pi$
$332$		0			0
$333$	−32.9706	−	11.6569i	−1.80678	−	0.638792i
$334$		0			0
$335$	13.0711			0.714149
$336$		0			0
$337$	13.3137			0.725244			0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137			0.725244			0.362622	+	0.931936i	$0.381882\pi$
$338$		0			0
$339$	2.92893	−	17.0711i	0.159078	−	0.927173i
$340$		0			0
$341$		−	16.0000i		−	0.866449i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$	−1.34315	+	7.82843i	−0.0723125	+	0.421468i
$346$		0			0
$347$	31.6985			1.70166			0.850832	−	0.525438i	$-0.176099\pi$
$347$	31.6985			1.70166			0.850832	+	0.525438i	$0.176099\pi$
$348$		0			0
$349$	−28.6274			−1.53239			−0.766195	−	0.642608i	$-0.777852\pi$
$349$	−28.6274			−1.53239			−0.766195	+	0.642608i	$0.777852\pi$
$350$		0			0
$351$	9.07107	+	5.07107i	0.484178	+	0.270674i
$352$		0			0
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$		0			0
$355$		−	6.82843i		−	0.362415i
$356$		0			0
$357$	−3.65685	−	0.627417i	−0.193541	−	0.0332064i
$358$		0			0
$359$	18.3431			0.968114			0.484057	−	0.875036i	$-0.339163\pi$
$359$	18.3431			0.968114			0.484057	+	0.875036i	$0.339163\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$	5.12132	+	0.878680i	0.268800	+	0.0461187i
$364$		0			0
$365$			4.34315i			0.227331i
$366$		0			0
$367$			29.0711i			1.51750i	0.651383	+	0.758749i	$0.274189\pi$
$367$			29.0711i			1.51750i	−0.651383	+	0.758749i	$0.725811\pi$
$368$		0			0
$369$	2.00000	−	5.65685i	0.104116	−	0.294484i
$370$		0			0
$371$	−4.48528			−0.232864
$372$		0			0
$373$	−8.34315			−0.431992			−0.215996	−	0.976394i	$-0.569300\pi$
$373$	−8.34315			−0.431992			−0.215996	+	0.976394i	$0.569300\pi$
$374$		0			0
$375$	−0.292893	+	1.70711i	−0.0151249	+	0.0881546i
$376$		0			0
$377$		−	16.0000i		−	0.824042i
$378$		0			0
$379$		−	14.1421i		−	0.726433i	−0.931705	−	0.363216i	$-0.881679\pi$
$379$		−	14.1421i		−	0.726433i	0.931705	−	0.363216i	$-0.118321\pi$
$380$		0			0
$381$	0.857864	−	5.00000i	0.0439497	−	0.256158i
$382$		0			0
$383$	−3.41421			−0.174458			−0.0872291	−	0.996188i	$-0.527801\pi$
$383$	−3.41421			−0.174458			−0.0872291	+	0.996188i	$0.527801\pi$
$384$		0			0
$385$	−1.65685			−0.0844411
$386$		0			0
$387$	−11.8995	+	33.6569i	−0.604885	+	1.71087i
$388$		0			0
$389$			13.3137i			0.675032i	0.941320	+	0.337516i	$0.109587\pi$
$389$			13.3137i			0.675032i	−0.941320	+	0.337516i	$0.890413\pi$
$390$		0			0
$391$		−	16.7696i		−	0.848073i
$392$		0			0
$393$	−8.82843	−	1.51472i	−0.445335	−	0.0764074i
$394$		0			0
$395$	−12.4853			−0.628203
$396$		0			0
$397$	24.6274			1.23601			0.618007	−	0.786172i	$-0.287940\pi$
$397$	24.6274			1.23601			0.618007	+	0.786172i	$0.287940\pi$
$398$		0			0
$399$	−2.82843	−	0.485281i	−0.141598	−	0.0242945i
$400$		0			0
$401$		−	8.00000i		−	0.399501i	−0.979847	−	0.199750i	$-0.935987\pi$
$401$		−	8.00000i		−	0.399501i	0.979847	−	0.199750i	$-0.0640132\pi$
$402$		0			0
$403$		−	11.3137i		−	0.563576i
$404$		0			0
$405$	−5.65685	+	7.00000i	−0.281091	+	0.347833i
$406$		0			0
$407$	32.9706			1.63429
$408$		0			0
$409$	−5.65685			−0.279713			−0.139857	−	0.990172i	$-0.544664\pi$
$409$	−5.65685			−0.279713			−0.139857	+	0.990172i	$0.544664\pi$
$410$		0			0
$411$	1.55635	−	9.07107i	0.0767690	−	0.447443i
$412$		0			0
$413$		−	5.65685i		−	0.278356i
$414$		0			0
$415$			9.07107i			0.445281i
$416$		0			0
$417$	5.51472	−	32.1421i	0.270057	−	1.57401i
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	−16.9706			−0.827095			−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706			−0.827095			−0.413547	+	0.910483i	$0.635710\pi$
$422$		0			0
$423$	−6.34315	−	2.24264i	−0.308414	−	0.109041i
$424$		0			0
$425$		−	3.65685i		−	0.177383i
$426$		0			0
$427$			3.31371i			0.160362i
$428$		0			0
$429$	−9.65685	−	1.65685i	−0.466237	−	0.0799937i
$430$		0			0
$431$	28.4853			1.37209			0.686044	−	0.727560i	$-0.259346\pi$
$431$	28.4853			1.37209			0.686044	+	0.727560i	$0.259346\pi$
$432$		0			0
$433$	4.34315			0.208718			0.104359	−	0.994540i	$-0.466721\pi$
$433$	4.34315			0.208718			0.104359	+	0.994540i	$0.466721\pi$
$434$		0			0
$435$	13.6569	+	2.34315i	0.654796	+	0.112345i
$436$		0			0
$437$		−	12.9706i		−	0.620466i
$438$		0			0
$439$		−	22.8284i		−	1.08954i	−0.838585	−	0.544771i	$-0.816617\pi$
$439$		−	22.8284i		−	1.08954i	0.838585	−	0.544771i	$-0.183383\pi$
$440$		0			0
$441$	18.8284	+	6.65685i	0.896592	+	0.316993i
$442$		0			0
$443$	−16.1005			−0.764958			−0.382479	−	0.923964i	$-0.624930\pi$
$443$	−16.1005			−0.764958			−0.382479	+	0.923964i	$0.624930\pi$
$444$		0			0
$445$	15.3137			0.725939
$446$		0			0
$447$	2.72792	−	15.8995i	0.129026	−	0.752020i
$448$		0			0
$449$			2.00000i			0.0943858i	0.998886	+	0.0471929i	$0.0150276\pi$
$449$			2.00000i			0.0943858i	−0.998886	+	0.0471929i	$0.984972\pi$
$450$		0			0
$451$			5.65685i			0.266371i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.17157			−0.0549242
$456$		0			0
$457$	17.3137			0.809901			0.404951	−	0.914339i	$-0.367289\pi$
$457$	17.3137			0.809901			0.404951	+	0.914339i	$0.367289\pi$
$458$		0			0
$459$	9.27208	−	16.5858i	0.432784	−	0.774158i
$460$		0			0
$461$		−	18.6274i		−	0.867565i	−0.901018	−	0.433783i	$-0.857179\pi$
$461$		−	18.6274i		−	0.867565i	0.901018	−	0.433783i	$-0.142821\pi$
$462$		0			0
$463$		−	23.4142i		−	1.08815i	−0.839036	−	0.544075i	$-0.816881\pi$
$463$		−	23.4142i		−	1.08815i	0.839036	−	0.544075i	$-0.183119\pi$
$464$		0			0
$465$	9.65685	+	1.65685i	0.447826	+	0.0768348i
$466$		0			0
$467$	−16.1005			−0.745043			−0.372521	−	0.928024i	$-0.621507\pi$
$467$	−16.1005			−0.745043			−0.372521	+	0.928024i	$0.621507\pi$
$468$		0			0
$469$	−7.65685			−0.353561
$470$		0			0
$471$	−6.24264	−	1.07107i	−0.287646	−	0.0493522i
$472$		0			0
$473$		−	33.6569i		−	1.54754i
$474$		0			0
$475$		−	2.82843i		−	0.129777i
$476$		0			0
$477$	7.65685	−	21.6569i	0.350583	−	0.991599i
$478$		0			0
$479$	32.9706			1.50646			0.753232	−	0.657755i	$-0.228494\pi$
$479$	32.9706			1.50646			0.753232	+	0.657755i	$0.228494\pi$
$480$		0			0
$481$	23.3137			1.06301
$482$		0			0
$483$	0.786797	−	4.58579i	0.0358005	−	0.208661i
$484$		0			0
$485$		−	14.9706i		−	0.679778i
$486$		0			0
$487$			27.8995i			1.26425i	0.774868	+	0.632123i	$0.217816\pi$
$487$			27.8995i			1.26425i	−0.774868	+	0.632123i	$0.782184\pi$
$488$		0			0
$489$	0.514719	−	3.00000i	0.0232764	−	0.135665i
$490$		0			0
$491$	−23.1127			−1.04306			−0.521531	−	0.853232i	$-0.674639\pi$
$491$	−23.1127			−1.04306			−0.521531	+	0.853232i	$0.674639\pi$
$492$		0			0
$493$	−29.2548			−1.31757
$494$		0			0
$495$	2.82843	−	8.00000i	0.127128	−	0.359573i
$496$		0			0
$497$			4.00000i			0.179425i
$498$		0			0
$499$		−	17.4558i		−	0.781431i	−0.920512	−	0.390715i	$-0.872228\pi$
$499$		−	17.4558i		−	0.781431i	0.920512	−	0.390715i	$-0.127772\pi$
$500$		0			0
$501$	−11.4853	−	1.97056i	−0.513125	−	0.0880383i
$502$		0			0
$503$	43.2132			1.92678			0.963391	−	0.268101i	$-0.0863962\pi$
$503$	43.2132			1.92678			0.963391	+	0.268101i	$0.0863962\pi$
$504$		0			0
$505$	15.3137			0.681451
$506$		0			0
$507$	15.3640	+	2.63604i	0.682337	+	0.117071i
$508$		0			0
$509$			22.6274i			1.00294i	0.865174	+	0.501471i	$0.167208\pi$
$509$			22.6274i			1.00294i	−0.865174	+	0.501471i	$0.832792\pi$
$510$		0			0
$511$		−	2.54416i		−	0.112547i
$512$		0			0
$513$	7.17157	−	12.8284i	0.316633	−	0.566389i
$514$		0			0
$515$	6.24264			0.275084
$516$		0			0
$517$	6.34315			0.278971
$518$		0			0
$519$	−0.585786	+	3.41421i	−0.0257132	+	0.149867i
$520$		0			0
$521$		−	16.0000i		−	0.700973i	−0.936568	−	0.350486i	$-0.886016\pi$
$521$		−	16.0000i		−	0.700973i	0.936568	−	0.350486i	$-0.113984\pi$
$522$		0			0
$523$			8.58579i			0.375430i	0.982224	+	0.187715i	$0.0601082\pi$
$523$			8.58579i			0.375430i	−0.982224	+	0.187715i	$0.939892\pi$
$524$		0			0
$525$	0.171573	−	1.00000i	0.00748805	−	0.0436436i
$526$		0			0
$527$	−20.6863			−0.901109
$528$		0			0
$529$	−1.97056			−0.0856766
$530$		0			0
$531$	27.3137	+	9.65685i	1.18531	+	0.419072i
$532$		0			0
$533$			4.00000i			0.173259i
$534$		0			0
$535$			12.5858i			0.544131i
$536$		0			0
$537$	10.8284	+	1.85786i	0.467281	+	0.0801728i
$538$		0			0
$539$	−18.8284			−0.810998
$540$		0			0
$541$	−37.3137			−1.60424			−0.802121	−	0.597162i	$-0.796295\pi$
$541$	−37.3137			−1.60424			−0.802121	+	0.597162i	$0.796295\pi$
$542$		0			0
$543$	−22.7279	−	3.89949i	−0.975348	−	0.167343i
$544$		0			0
$545$			6.34315i			0.271711i
$546$		0			0
$547$			30.0416i			1.28449i	0.766501	+	0.642244i	$0.221996\pi$
$547$			30.0416i			1.28449i	−0.766501	+	0.642244i	$0.778004\pi$
$548$		0			0
$549$	−16.0000	−	5.65685i	−0.682863	−	0.241429i
$550$		0			0
$551$	−22.6274			−0.963960
$552$		0			0
$553$	7.31371			0.311011
$554$		0			0
$555$	−3.41421	+	19.8995i	−0.144925	+	0.844687i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$		−	23.7990i		−	1.00659i
$560$		0			0
$561$	−3.02944	+	17.6569i	−0.127903	+	0.745473i
$562$		0			0
$563$	−5.55635			−0.234172			−0.117086	−	0.993122i	$-0.537355\pi$
$563$	−5.55635			−0.234172			−0.117086	+	0.993122i	$0.537355\pi$
$564$		0			0
$565$	−10.0000			−0.420703
$566$		0			0
$567$	3.31371	−	4.10051i	0.139163	−	0.172205i
$568$		0			0
$569$			16.6274i			0.697058i	0.937298	+	0.348529i	$0.113319\pi$
$569$			16.6274i			0.697058i	−0.937298	+	0.348529i	$0.886681\pi$
$570$		0			0
$571$		−	40.2843i		−	1.68584i	−0.538036	−	0.842922i	$-0.680833\pi$
$571$		−	40.2843i		−	1.68584i	0.538036	−	0.842922i	$-0.319167\pi$
$572$		0			0
$573$	17.3137	+	2.97056i	0.723291	+	0.124097i
$574$		0			0
$575$	4.58579			0.191241
$576$		0			0
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$		0			0
$579$	−6.24264	−	1.07107i	−0.259435	−	0.0445121i
$580$		0			0
$581$		−	5.31371i		−	0.220450i
$582$		0			0
$583$			21.6569i			0.896935i
$584$		0			0
$585$	2.00000	−	5.65685i	0.0826898	−	0.233882i
$586$		0			0
$587$	6.72792			0.277691			0.138846	−	0.990314i	$-0.455661\pi$
$587$	6.72792			0.277691			0.138846	+	0.990314i	$0.455661\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	3.21320	−	18.7279i	0.132174	−	0.770364i
$592$		0			0
$593$			12.6274i			0.518546i	0.965804	+	0.259273i	$0.0834829\pi$
$593$			12.6274i			0.518546i	−0.965804	+	0.259273i	$0.916517\pi$
$594$		0			0
$595$			2.14214i			0.0878190i
$596$		0			0
$597$	−5.02944	+	29.3137i	−0.205841	+	1.19973i
$598$		0			0
$599$	−14.6274			−0.597660			−0.298830	−	0.954306i	$-0.596596\pi$
$599$	−14.6274			−0.597660			−0.298830	+	0.954306i	$0.596596\pi$
$600$		0			0
$601$	−28.9706			−1.18173			−0.590867	−	0.806769i	$-0.701214\pi$
$601$	−28.9706			−1.18173			−0.590867	+	0.806769i	$0.701214\pi$
$602$		0			0
$603$	13.0711	−	36.9706i	0.532295	−	1.50556i
$604$		0			0
$605$		−	3.00000i		−	0.121967i
$606$		0			0
$607$			26.9289i			1.09301i	0.837455	+	0.546506i	$0.184042\pi$
$607$			26.9289i			1.09301i	−0.837455	+	0.546506i	$0.815958\pi$
$608$		0			0
$609$	−8.00000	−	1.37258i	−0.324176	−	0.0556199i
$610$		0			0
$611$	4.48528			0.181455
$612$		0			0
$613$	−14.6863			−0.593174			−0.296587	−	0.955006i	$-0.595848\pi$
$613$	−14.6863			−0.593174			−0.296587	+	0.955006i	$0.595848\pi$
$614$		0			0
$615$	−3.41421	−	0.585786i	−0.137674	−	0.0236212i
$616$		0			0
$617$		−	7.65685i		−	0.308253i	−0.988051	−	0.154127i	$-0.950744\pi$
$617$		−	7.65685i		−	0.308253i	0.988051	−	0.154127i	$-0.0492564\pi$
$618$		0			0
$619$			33.4558i			1.34470i	0.740232	+	0.672352i	$0.234716\pi$
$619$			33.4558i			1.34470i	−0.740232	+	0.672352i	$0.765284\pi$
$620$		0			0
$621$	20.7990	+	11.6274i	0.834635	+	0.466592i
$622$		0			0
$623$	−8.97056			−0.359398
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−2.34315	+	13.6569i	−0.0935762	+	0.545402i
$628$		0			0
$629$		−	42.6274i		−	1.69967i
$630$		0			0
$631$			19.3137i			0.768867i	0.923153	+	0.384433i	$0.125603\pi$
$631$			19.3137i			0.768867i	−0.923153	+	0.384433i	$0.874397\pi$
$632$		0			0
$633$	−3.51472	+	20.4853i	−0.139698	+	0.814217i
$634$		0			0
$635$	−2.92893			−0.116231
$636$		0			0
$637$	−13.3137			−0.527508
$638$		0			0
$639$	−19.3137	−	6.82843i	−0.764038	−	0.270128i
$640$		0			0
$641$			17.3137i			0.683850i	0.939727	+	0.341925i	$0.111079\pi$
$641$			17.3137i			0.683850i	−0.939727	+	0.341925i	$0.888921\pi$
$642$		0			0
$643$			35.8995i			1.41574i	0.706344	+	0.707869i	$0.250343\pi$
$643$			35.8995i			1.41574i	−0.706344	+	0.707869i	$0.749657\pi$
$644$		0			0
$645$	20.3137	+	3.48528i	0.799851	+	0.137233i
$646$		0			0
$647$	42.2426			1.66073			0.830365	−	0.557220i	$-0.188132\pi$
$647$	42.2426			1.66073			0.830365	+	0.557220i	$0.188132\pi$
$648$		0			0
$649$	−27.3137			−1.07216
$650$		0			0
$651$	−5.65685	−	0.970563i	−0.221710	−	0.0380394i
$652$		0			0
$653$			43.6569i			1.70842i	0.519924	+	0.854212i	$0.325960\pi$
$653$			43.6569i			1.70842i	−0.519924	+	0.854212i	$0.674040\pi$
$654$		0			0
$655$			5.17157i			0.202070i
$656$		0			0
$657$	12.2843	+	4.34315i	0.479255	+	0.169442i
$658$		0			0
$659$	−33.6569			−1.31108			−0.655542	−	0.755158i	$-0.727560\pi$
$659$	−33.6569			−1.31108			−0.655542	+	0.755158i	$0.727560\pi$
$660$		0			0
$661$	21.6569			0.842354			0.421177	−	0.906978i	$-0.361617\pi$
$661$	21.6569			0.842354			0.421177	+	0.906978i	$0.361617\pi$
$662$		0			0
$663$	−2.14214	+	12.4853i	−0.0831937	+	0.484888i
$664$		0			0
$665$			1.65685i			0.0642501i
$666$		0			0
$667$		−	36.6863i		−	1.42050i
$668$		0			0
$669$	3.48528	−	20.3137i	0.134749	−	0.785373i
$670$		0			0
$671$	16.0000			0.617673
$672$		0			0
$673$	−35.6569			−1.37447			−0.687235	−	0.726435i	$-0.741176\pi$
$673$	−35.6569			−1.37447			−0.687235	+	0.726435i	$0.741176\pi$
$674$		0			0
$675$	4.53553	+	2.53553i	0.174573	+	0.0975927i
$676$		0			0
$677$		−	11.6569i		−	0.448009i	−0.974588	−	0.224005i	$-0.928087\pi$
$677$		−	11.6569i		−	0.448009i	0.974588	−	0.224005i	$-0.0719131\pi$
$678$		0			0
$679$			8.76955i			0.336545i
$680$		0			0
$681$	7.82843	+	1.34315i	0.299986	+	0.0514695i
$682$		0			0
$683$	21.5563			0.824831			0.412415	−	0.910996i	$-0.364685\pi$
$683$	21.5563			0.824831			0.412415	+	0.910996i	$0.364685\pi$
$684$		0			0
$685$	−5.31371			−0.203026
$686$		0			0
$687$	−43.2132	−	7.41421i	−1.64869	−	0.282870i
$688$		0			0
$689$			15.3137i			0.583406i
$690$		0			0
$691$			16.2843i			0.619483i	0.950821	+	0.309741i	$0.100242\pi$
$691$			16.2843i			0.619483i	−0.950821	+	0.309741i	$0.899758\pi$
$692$		0			0
$693$	−1.65685	+	4.68629i	−0.0629387	+	0.178017i
$694$		0			0
$695$	−18.8284			−0.714203
$696$		0			0
$697$	7.31371			0.277026
$698$		0			0
$699$	7.89949	−	46.0416i	0.298786	−	1.74145i
$700$		0			0
$701$			51.9411i			1.96179i	0.194540	+	0.980895i	$0.437679\pi$
$701$			51.9411i			1.96179i	−0.194540	+	0.980895i	$0.562321\pi$
$702$		0			0
$703$		−	32.9706i		−	1.24351i
$704$		0			0
$705$	−0.656854	+	3.82843i	−0.0247386	+	0.144187i
$706$		0			0
$707$	−8.97056			−0.337373
$708$		0			0
$709$	−25.3137			−0.950676			−0.475338	−	0.879803i	$-0.657674\pi$
$709$	−25.3137			−0.950676			−0.475338	+	0.879803i	$0.657674\pi$
$710$		0			0
$711$	−12.4853	+	35.3137i	−0.468235	+	1.32437i
$712$		0			0
$713$		−	25.9411i		−	0.971503i
$714$		0			0
$715$			5.65685i			0.211554i
$716$		0			0
$717$	23.3137	+	4.00000i	0.870666	+	0.149383i
$718$		0			0
$719$	−44.2843			−1.65152			−0.825762	−	0.564018i	$-0.809255\pi$
$719$	−44.2843			−1.65152			−0.825762	+	0.564018i	$0.809255\pi$
$720$		0			0
$721$	−3.65685			−0.136188
$722$		0			0
$723$	2.82843	+	0.485281i	0.105190	+	0.0180478i
$724$		0			0
$725$		−	8.00000i		−	0.297113i
$726$		0			0
$727$			48.1838i			1.78704i	0.449026	+	0.893518i	$0.351771\pi$
$727$			48.1838i			1.78704i	−0.449026	+	0.893518i	$0.648229\pi$
$728$		0			0
$729$	14.1421	+	23.0000i	0.523783	+	0.851852i
$730$		0			0
$731$	−43.5147			−1.60945
$732$		0			0
$733$	42.9706			1.58715			0.793577	−	0.608470i	$-0.208216\pi$
$733$	42.9706			1.58715			0.793577	+	0.608470i	$0.208216\pi$
$734$		0			0
$735$	1.94975	−	11.3640i	0.0719175	−	0.419166i
$736$		0			0
$737$			36.9706i			1.36183i
$738$		0			0
$739$			32.4853i			1.19499i	0.801872	+	0.597495i	$0.203837\pi$
$739$			32.4853i			1.19499i	−0.801872	+	0.597495i	$0.796163\pi$
$740$		0			0
$741$	−1.65685	+	9.65685i	−0.0608661	+	0.354753i
$742$		0			0
$743$	−30.7279			−1.12730			−0.563649	−	0.826014i	$-0.690603\pi$
$743$	−30.7279			−1.12730			−0.563649	+	0.826014i	$0.690603\pi$
$744$		0			0
$745$	−9.31371			−0.341228
$746$		0			0
$747$	25.6569	+	9.07107i	0.938735	+	0.331893i
$748$		0			0
$749$		−	7.37258i		−	0.269388i
$750$		0			0
$751$			6.62742i			0.241838i	0.992662	+	0.120919i	$0.0385841\pi$
$751$			6.62742i			0.241838i	−0.992662	+	0.120919i	$0.961416\pi$
$752$		0			0
$753$	28.1421	+	4.82843i	1.02556	+	0.175958i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	27.6569			1.00521			0.502603	−	0.864517i	$-0.332376\pi$
$757$	27.6569			1.00521			0.502603	+	0.864517i	$0.332376\pi$
$758$		0			0
$759$	−22.1421	−	3.79899i	−0.803709	−	0.137895i
$760$		0			0
$761$		−	11.3137i		−	0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$		−	3.71573i		−	0.134518i
$764$		0			0
$765$	−10.3431	−	3.65685i	−0.373957	−	0.132214i
$766$		0			0
$767$	−19.3137			−0.697378
$768$		0			0
$769$	2.68629			0.0968701			0.0484351	−	0.998826i	$-0.484577\pi$
$769$	2.68629			0.0968701			0.0484351	+	0.998826i	$0.484577\pi$
$770$		0			0
$771$	6.44365	−	37.5563i	0.232062	−	1.35256i
$772$		0			0
$773$			30.2843i			1.08925i	0.838680	+	0.544625i	$0.183328\pi$
$773$			30.2843i			1.08925i	−0.838680	+	0.544625i	$0.816672\pi$
$774$		0			0
$775$		−	5.65685i		−	0.203200i
$776$		0			0
$777$	2.00000	−	11.6569i	0.0717496	−	0.418187i
$778$		0			0
$779$	5.65685			0.202678
$780$		0			0
$781$	19.3137			0.691099
$782$		0			0
$783$	20.2843	−	36.2843i	0.724901	−	1.29669i
$784$		0			0
$785$			3.65685i			0.130519i
$786$		0			0
$787$		−	9.55635i		−	0.340647i	−0.985388	−	0.170324i	$-0.945519\pi$
$787$		−	9.55635i		−	0.340647i	0.985388	−	0.170324i	$-0.0544813\pi$
$788$		0			0
$789$	13.4853	+	2.31371i	0.480089	+	0.0823702i
$790$		0			0
$791$	5.85786			0.208282
$792$		0			0
$793$	11.3137			0.401762
$794$		0			0
$795$	−13.0711	−	2.24264i	−0.463583	−	0.0795383i
$796$		0			0
$797$		−	2.97056i		−	0.105223i	−0.998615	−	0.0526114i	$-0.983246\pi$
$797$		−	2.97056i		−	0.105223i	0.998615	−	0.0526114i	$-0.0167545\pi$
$798$		0			0
$799$		−	8.20101i		−	0.290131i
$800$		0			0
$801$	15.3137	−	43.3137i	0.541083	−	1.53041i
$802$		0			0
$803$	−12.2843			−0.433503
$804$		0			0
$805$	−2.68629			−0.0946793
$806$		0			0
$807$	5.27208	−	30.7279i	0.185586	−	1.08167i
$808$		0			0
$809$		−	18.6274i		−	0.654905i	−0.944868	−	0.327453i	$-0.893810\pi$
$809$		−	18.6274i		−	0.654905i	0.944868	−	0.327453i	$-0.106190\pi$
$810$		0			0
$811$		−	2.62742i		−	0.0922611i	−0.998935	−	0.0461305i	$-0.985311\pi$
$811$		−	2.62742i		−	0.0922611i	0.998935	−	0.0461305i	$-0.0146890\pi$
$812$		0			0
$813$	−4.00000	+	23.3137i	−0.140286	+	0.817647i
$814$		0			0
$815$	−1.75736			−0.0615576
$816$		0			0
$817$	−33.6569			−1.17750
$818$		0			0
$819$	−1.17157	+	3.31371i	−0.0409381	+	0.115790i
$820$		0			0
$821$			16.6274i			0.580301i	0.956981	+	0.290150i	$0.0937053\pi$
$821$			16.6274i			0.580301i	−0.956981	+	0.290150i	$0.906295\pi$
$822$		0			0
$823$		−	35.8995i		−	1.25138i	−0.780073	−	0.625689i	$-0.784818\pi$
$823$		−	35.8995i		−	1.25138i	0.780073	−	0.625689i	$-0.215182\pi$
$824$		0			0
$825$	−4.82843	−	0.828427i	−0.168104	−	0.0288421i
$826$		0			0
$827$	−27.4142			−0.953286			−0.476643	−	0.879097i	$-0.658146\pi$
$827$	−27.4142			−0.953286			−0.476643	+	0.879097i	$0.658146\pi$
$828$		0			0
$829$	20.9706			0.728338			0.364169	−	0.931333i	$-0.381353\pi$
$829$	20.9706			0.728338			0.364169	+	0.931333i	$0.381353\pi$
$830$		0			0
$831$	−0.585786	−	0.100505i	−0.0203207	−	0.00348648i
$832$		0			0
$833$			24.3431i			0.843440i
$834$		0			0
$835$			6.72792i			0.232829i
$836$		0			0
$837$	14.3431	−	25.6569i	0.495772	−	0.886831i
$838$		0			0
$839$	49.9411			1.72416			0.862080	−	0.506773i	$-0.169162\pi$
$839$	49.9411			1.72416			0.862080	+	0.506773i	$0.169162\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$		0			0
$843$	−2.72792	+	15.8995i	−0.0939546	+	0.547608i
$844$		0			0
$845$		−	9.00000i		−	0.309609i
$846$		0			0
$847$			1.75736i			0.0603836i
$848$		0			0
$849$	−5.14214	+	29.9706i	−0.176478	+	1.02859i
$850$		0			0
$851$	53.4558			1.83244
$852$		0			0
$853$	−19.9411			−0.682771			−0.341386	−	0.939923i	$-0.610896\pi$
$853$	−19.9411			−0.682771			−0.341386	+	0.939923i	$0.610896\pi$
$854$		0			0
$855$	−8.00000	−	2.82843i	−0.273594	−	0.0967302i
$856$		0			0
$857$			10.0000i			0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$			10.0000i			0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$		0			0
$859$		−	24.4853i		−	0.835427i	−0.908579	−	0.417714i	$-0.862832\pi$
$859$		−	24.4853i		−	0.835427i	0.908579	−	0.417714i	$-0.137168\pi$
$860$		0			0
$861$	2.00000	+	0.343146i	0.0681598	+	0.0116944i
$862$		0			0
$863$	13.5563			0.461463			0.230732	−	0.973017i	$-0.425888\pi$
$863$	13.5563			0.461463			0.230732	+	0.973017i	$0.425888\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$	−6.19239	−	1.06245i	−0.210305	−	0.0360826i
$868$		0			0
$869$		−	35.3137i		−	1.19794i
$870$		0			0
$871$			26.1421i			0.885792i
$872$		0			0
$873$	−42.3431	−	14.9706i	−1.43310	−	0.506677i
$874$		0			0
$875$	−0.585786			−0.0198032
$876$		0			0
$877$	−12.3431			−0.416798			−0.208399	−	0.978044i	$-0.566825\pi$
$877$	−12.3431			−0.416798			−0.208399	+	0.978044i	$0.566825\pi$
$878$		0			0
$879$	−6.44365	+	37.5563i	−0.217339	+	1.26674i
$880$		0			0
$881$		−	16.6274i		−	0.560192i	−0.959972	−	0.280096i	$-0.909634\pi$
$881$		−	16.6274i		−	0.560192i	0.959972	−	0.280096i	$-0.0903663\pi$
$882$		0			0
$883$			26.7279i			0.899466i	0.893163	+	0.449733i	$0.148481\pi$
$883$			26.7279i			0.899466i	−0.893163	+	0.449733i	$0.851519\pi$
$884$		0			0
$885$	2.82843	−	16.4853i	0.0950765	−	0.554147i
$886$		0			0
$887$	7.89949			0.265239			0.132620	−	0.991167i	$-0.457661\pi$
$887$	7.89949			0.265239			0.132620	+	0.991167i	$0.457661\pi$
$888$		0			0
$889$	1.71573			0.0575437
$890$		0			0
$891$	−19.7990	−	16.0000i	−0.663291	−	0.536020i
$892$		0			0
$893$		−	6.34315i		−	0.212265i
$894$		0			0
$895$		−	6.34315i		−	0.212028i
$896$		0			0
$897$	−15.6569	−	2.68629i	−0.522767	−	0.0896927i
$898$		0			0
$899$	−45.2548			−1.50933
$900$		0			0
$901$	28.0000			0.932815
$902$		0			0
$903$	−11.8995	−	2.04163i	−0.395990	−	0.0679412i
$904$		0			0
$905$			13.3137i			0.442563i
$906$		0			0
$907$		−	13.0711i		−	0.434018i	−0.976170	−	0.217009i	$-0.930370\pi$
$907$		−	13.0711i		−	0.434018i	0.976170	−	0.217009i	$-0.0696300\pi$
$908$		0			0
$909$	15.3137	−	43.3137i	0.507924	−	1.43663i
$910$		0			0
$911$	51.1127			1.69344			0.846720	−	0.532039i	$-0.178574\pi$
$911$	51.1127			1.69344			0.846720	+	0.532039i	$0.178574\pi$
$912$		0			0
$913$	−25.6569			−0.849118
$914$		0			0
$915$	−1.65685	+	9.65685i	−0.0547739	+	0.319246i
$916$		0			0
$917$		−	3.02944i		−	0.100041i
$918$		0			0
$919$			27.5147i			0.907627i	0.891097	+	0.453813i	$0.149937\pi$
$919$			27.5147i			0.907627i	−0.891097	+	0.453813i	$0.850063\pi$
$920$		0			0
$921$	−3.48528	+	20.3137i	−0.114844	+	0.669359i
$922$		0			0
$923$	13.6569			0.449521
$924$		0			0
$925$	11.6569			0.383275
$926$		0			0
$927$	6.24264	−	17.6569i	0.205035	−	0.579927i
$928$		0			0
$929$			46.0000i			1.50921i	0.656179	+	0.754606i	$0.272172\pi$
$929$			46.0000i			1.50921i	−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$			18.8284i			0.617077i
$932$		0			0
$933$	48.6274	+	8.34315i	1.59199	+	0.273142i
$934$		0			0
$935$	10.3431			0.338257
$936$		0			0
$937$	−41.5980			−1.35895			−0.679473	−	0.733700i	$-0.737792\pi$
$937$	−41.5980			−1.35895			−0.679473	+	0.733700i	$0.737792\pi$
$938$		0			0
$939$	9.07107	+	1.55635i	0.296023	+	0.0507895i
$940$		0			0
$941$		−	12.0000i		−	0.391189i	−0.980685	−	0.195594i	$-0.937336\pi$
$941$		−	12.0000i		−	0.391189i	0.980685	−	0.195594i	$-0.0626636\pi$
$942$		0			0
$943$			9.17157i			0.298668i
$944$		0			0
$945$	−2.65685	−	1.48528i	−0.0864275	−	0.0483162i
$946$		0			0
$947$	−19.2132			−0.624345			−0.312173	−	0.950025i	$-0.601057\pi$
$947$	−19.2132			−0.624345			−0.312173	+	0.950025i	$0.601057\pi$
$948$		0			0
$949$	−8.68629			−0.281969
$950$		0			0
$951$	3.41421	−	19.8995i	0.110713	−	0.645285i
$952$		0			0
$953$		−	2.68629i		−	0.0870175i	−0.999053	−	0.0435088i	$-0.986146\pi$
$953$		−	2.68629i		−	0.0870175i	0.999053	−	0.0435088i	$-0.0138536\pi$
$954$		0			0
$955$		−	10.1421i		−	0.328192i
$956$		0			0
$957$	−6.62742	+	38.6274i	−0.214234	+	1.24865i
$958$		0			0
$959$	3.11270			0.100514
$960$		0			0
$961$	−1.00000			−0.0322581
$962$		0			0
$963$	35.5980	+	12.5858i	1.14713	+	0.405571i
$964$		0			0
$965$			3.65685i			0.117718i
$966$		0			0
$967$		−	8.58579i		−	0.276100i	−0.990425	−	0.138050i	$-0.955916\pi$
$967$		−	8.58579i		−	0.276100i	0.990425	−	0.138050i	$-0.0440835\pi$
$968$		0			0
$969$	17.6569	+	3.02944i	0.567220	+	0.0973195i
$970$		0			0
$971$	−15.5147			−0.497891			−0.248946	−	0.968517i	$-0.580084\pi$
$971$	−15.5147			−0.497891			−0.248946	+	0.968517i	$0.580084\pi$
$972$		0			0
$973$	11.0294			0.353588
$974$		0			0
$975$	−3.41421	−	0.585786i	−0.109342	−	0.0187602i
$976$		0			0
$977$		−	54.9706i		−	1.75866i	−0.476209	−	0.879332i	$-0.657989\pi$
$977$		−	54.9706i		−	1.75866i	0.476209	−	0.879332i	$-0.342011\pi$
$978$		0			0
$979$			43.3137i			1.38431i
$980$		0			0
$981$	17.9411	+	6.34315i	0.572816	+	0.202521i
$982$		0			0
$983$	30.9289			0.986480			0.493240	−	0.869893i	$-0.335812\pi$
$983$	30.9289			0.986480			0.493240	+	0.869893i	$0.335812\pi$
$984$		0			0
$985$	−10.9706			−0.349551
$986$		0			0
$987$	0.384776	−	2.24264i	0.0122476	−	0.0713840i
$988$		0			0
$989$		−	54.5685i		−	1.73518i
$990$		0			0
$991$		−	28.2843i		−	0.898479i	−0.893411	−	0.449240i	$-0.851695\pi$
$991$		−	28.2843i		−	0.898479i	0.893411	−	0.449240i	$-0.148305\pi$
$992$		0			0
$993$	9.17157	−	53.4558i	0.291051	−	1.69637i
$994$		0			0
$995$	17.1716			0.544375
$996$		0			0
$997$	−18.0000			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$998$		0			0
$999$	52.8701	+	29.5563i	1.67273	+	0.935122i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.h.b.191.1	✓	4
3.2	odd	2		480.2.h.d.191.3	yes	4
4.3	odd	2		480.2.h.d.191.4	yes	4
5.2	odd	4		2400.2.o.c.2399.4		4
5.3	odd	4		2400.2.o.i.2399.1		4
5.4	even	2		2400.2.h.e.1151.4		4
8.3	odd	2		960.2.h.b.191.1		4
8.5	even	2		960.2.h.f.191.4		4
12.11	even	2	inner	480.2.h.b.191.2	yes	4
15.2	even	4		2400.2.o.b.2399.3		4
15.8	even	4		2400.2.o.j.2399.2		4
15.14	odd	2		2400.2.h.b.1151.2		4
20.3	even	4		2400.2.o.b.2399.4		4
20.7	even	4		2400.2.o.j.2399.1		4
20.19	odd	2		2400.2.h.b.1151.1		4
24.5	odd	2		960.2.h.b.191.2		4
24.11	even	2		960.2.h.f.191.3		4
60.23	odd	4		2400.2.o.c.2399.3		4
60.47	odd	4		2400.2.o.i.2399.2		4
60.59	even	2		2400.2.h.e.1151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.h.b.191.1	✓	4	1.1	even	1	trivial
480.2.h.b.191.2	yes	4	12.11	even	2	inner
480.2.h.d.191.3	yes	4	3.2	odd	2
480.2.h.d.191.4	yes	4	4.3	odd	2
960.2.h.b.191.1		4	8.3	odd	2
960.2.h.b.191.2		4	24.5	odd	2
960.2.h.f.191.3		4	24.11	even	2
960.2.h.f.191.4		4	8.5	even	2
2400.2.h.b.1151.1		4	20.19	odd	2
2400.2.h.b.1151.2		4	15.14	odd	2
2400.2.h.e.1151.3		4	60.59	even	2
2400.2.h.e.1151.4		4	5.4	even	2
2400.2.o.b.2399.3		4	15.2	even	4
2400.2.o.b.2399.4		4	20.3	even	4
2400.2.o.c.2399.3		4	60.23	odd	4
2400.2.o.c.2399.4		4	5.2	odd	4
2400.2.o.i.2399.1		4	5.3	odd	4
2400.2.o.i.2399.2		4	60.47	odd	4
2400.2.o.j.2399.1		4	20.7	even	4
2400.2.o.j.2399.2		4	15.8	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 - 0.292893i) q^{3} +1.00000i q^{5} -0.585786i q^{7} +(2.82843 + 1.00000i) q^{9} -2.82843 q^{11} -2.00000 q^{13} +(0.292893 - 1.70711i) q^{15} +3.65685i q^{17} +2.82843i q^{19} +(-0.171573 + 1.00000i) q^{21} -4.58579 q^{23} -1.00000 q^{25} +(-4.53553 - 2.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(4.82843 + 0.828427i) q^{33} +0.585786 q^{35} -11.6569 q^{37} +(3.41421 + 0.585786i) q^{39} -2.00000i q^{41} +11.8995i q^{43} +(-1.00000 + 2.82843i) q^{45} -2.24264 q^{47} +6.65685 q^{49} +(1.07107 - 6.24264i) q^{51} -7.65685i q^{53} -2.82843i q^{55} +(0.828427 - 4.82843i) q^{57} +9.65685 q^{59} -5.65685 q^{61} +(0.585786 - 1.65685i) q^{63} -2.00000i q^{65} -13.0711i q^{67} +(7.82843 + 1.34315i) q^{69} -6.82843 q^{71} +4.34315 q^{73} +(1.70711 + 0.292893i) q^{75} +1.65685i q^{77} +12.4853i q^{79} +(7.00000 + 5.65685i) q^{81} +9.07107 q^{83} -3.65685 q^{85} +(2.34315 - 13.6569i) q^{87} -15.3137i q^{89} +1.17157i q^{91} +(1.65685 - 9.65685i) q^{93} -2.82843 q^{95} -14.9706 q^{97} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{13} + 4 q^{15} - 12 q^{21} - 24 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 4 q^{45} + 8 q^{47} + 4 q^{49} - 24 q^{51} - 8 q^{57} + 16 q^{59} + 8 q^{63}+ \cdots - 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^13 + 4 * q^15 - 12 * q^21 - 24 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 4 * q^45 + 8 * q^47 + 4 * q^49 - 24 * q^51 - 8 * q^57 + 16 * q^59 + 8 * q^63 + 20 * q^69 - 16 * q^71 + 40 * q^73 + 4 * q^75 + 28 * q^81 + 8 * q^83 + 8 * q^85 + 32 * q^87 - 16 * q^93 + 8 * q^97 - 32 * q^99

Introduction

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