LMFDB - Embedded newform 20.6.c.a.9.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,6,Mod(9,20)] 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("20.9"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	20.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$3.20767639626$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 8 $ x^2 - x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.2
Root			$0.500000 - 2.78388i$ of defining polynomial
Character	$\chi$	$=$	20.9
Dual form			20.6.c.a.9.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.1355i q^{3} +(-5.00000 + 55.6776i) q^{5} +122.491i q^{7} +119.000 q^{9} -100.000 q^{11} -734.945i q^{13} +(-620.000 - 55.6776i) q^{15} -979.927i q^{17} +2244.00 q^{19} -1364.00 q^{21} +3418.61i q^{23} +(-3075.00 - 556.776i) q^{25} +4031.06i q^{27} +7854.00 q^{29} -2144.00 q^{31} -1113.55i q^{33} +(-6820.00 - 612.454i) q^{35} -10400.6i q^{37} +8184.00 q^{39} -7414.00 q^{41} -17761.2i q^{43} +(-595.000 + 6625.64i) q^{45} +9431.79i q^{47} +1803.00 q^{49} +10912.0 q^{51} +24253.2i q^{53} +(500.000 - 5567.76i) q^{55} +24988.1i q^{57} -25972.0 q^{59} -3058.00 q^{61} +14576.4i q^{63} +(40920.0 + 3674.72i) q^{65} -58784.5i q^{67} -38068.0 q^{69} +37608.0 q^{71} -24008.2i q^{73} +(6200.00 - 34241.8i) q^{75} -12249.1i q^{77} -79728.0 q^{79} -15971.0 q^{81} -16291.3i q^{83} +(54560.0 + 4899.63i) q^{85} +87458.4i q^{87} +826.000 q^{89} +90024.0 q^{91} -23874.6i q^{93} +(-11220.0 + 124941. i) q^{95} +37593.5i q^{97} -11900.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 238 q^{9} - 200 q^{11} - 1240 q^{15} + 4488 q^{19} - 2728 q^{21} - 6150 q^{25} + 15708 q^{29} - 4288 q^{31} - 13640 q^{35} + 16368 q^{39} - 14828 q^{41} - 1190 q^{45} + 3606 q^{49} + 21824 q^{51}+ \cdots - 23800 q^{99}+O(q^{100}) $ 2 * q - 10 * q^5 + 238 * q^9 - 200 * q^11 - 1240 * q^15 + 4488 * q^19 - 2728 * q^21 - 6150 * q^25 + 15708 * q^29 - 4288 * q^31 - 13640 * q^35 + 16368 * q^39 - 14828 * q^41 - 1190 * q^45 + 3606 * q^49 + 21824 * q^51 + 1000 * q^55 - 51944 * q^59 - 6116 * q^61 + 81840 * q^65 - 76136 * q^69 + 75216 * q^71 + 12400 * q^75 - 159456 * q^79 - 31942 * q^81 + 109120 * q^85 + 1652 * q^89 + 180048 * q^91 - 22440 * q^95 - 23800 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$-1$

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
$3$			11.1355i			0.714345i	0.934039	+	0.357172i	$0.116259\pi$
$3$			11.1355i			0.714345i	−0.934039	+	0.357172i	$0.883741\pi$
$4$		0			0
$5$	−5.00000	+	55.6776i	−0.0894427	+	0.995992i
$5$	−5.00000	+	55.6776i	−0.0894427	+	0.995992i
$6$		0			0
$7$			122.491i			0.944840i	0.881373	+	0.472420i	$0.156620\pi$
$7$			122.491i			0.944840i	−0.881373	+	0.472420i	$0.843380\pi$
$8$		0			0
$9$	119.000			0.489712
$10$		0			0
$11$	−100.000			−0.249183			−0.124591	−	0.992208i	$-0.539762\pi$
$11$	−100.000			−0.249183			−0.124591	+	0.992208i	$0.539762\pi$
$12$		0			0
$13$		−	734.945i		−	1.20614i	−0.797690	−	0.603068i	$-0.793945\pi$
$13$		−	734.945i		−	1.20614i	0.797690	−	0.603068i	$-0.206055\pi$
$14$		0			0
$15$	−620.000	−	55.6776i	−0.711481	−	0.0638929i
$16$		0			0
$17$		−	979.927i		−	0.822377i	−0.911550	−	0.411189i	$-0.865114\pi$
$17$		−	979.927i		−	0.822377i	0.911550	−	0.411189i	$-0.134886\pi$
$18$		0			0
$19$	2244.00			1.42606			0.713032	−	0.701132i	$-0.247322\pi$
$19$	2244.00			1.42606			0.713032	+	0.701132i	$0.247322\pi$
$20$		0			0
$21$	−1364.00			−0.674942
$22$		0			0
$23$			3418.61i			1.34750i	0.738958	+	0.673751i	$0.235318\pi$
$23$			3418.61i			1.34750i	−0.738958	+	0.673751i	$0.764682\pi$
$24$		0			0
$25$	−3075.00	−	556.776i	−0.984000	−	0.178168i
$26$		0			0
$27$			4031.06i			1.06417i
$28$		0			0
$29$	7854.00			1.73419			0.867093	−	0.498146i	$-0.165985\pi$
$29$	7854.00			1.73419			0.867093	+	0.498146i	$0.165985\pi$
$30$		0			0
$31$	−2144.00			−0.400701			−0.200351	−	0.979724i	$-0.564208\pi$
$31$	−2144.00			−0.400701			−0.200351	+	0.979724i	$0.564208\pi$
$32$		0			0
$33$		−	1113.55i		−	0.178002i
$34$		0			0
$35$	−6820.00	−	612.454i	−0.941053	−	0.0845091i
$36$		0			0
$37$		−	10400.6i		−	1.24897i	−0.781035	−	0.624487i	$-0.785308\pi$
$37$		−	10400.6i		−	1.24897i	0.781035	−	0.624487i	$-0.214692\pi$
$38$		0			0
$39$	8184.00			0.861597
$40$		0			0
$41$	−7414.00			−0.688800			−0.344400	−	0.938823i	$-0.611918\pi$
$41$	−7414.00			−0.688800			−0.344400	+	0.938823i	$0.611918\pi$
$42$		0			0
$43$		−	17761.2i		−	1.46487i	−0.680835	−	0.732437i	$-0.738383\pi$
$43$		−	17761.2i		−	1.46487i	0.680835	−	0.732437i	$-0.261617\pi$
$44$		0			0
$45$	−595.000	+	6625.64i	−0.0438012	+	0.487749i
$46$		0			0
$47$			9431.79i			0.622801i	0.950279	+	0.311401i	$0.100798\pi$
$47$			9431.79i			0.622801i	−0.950279	+	0.311401i	$0.899202\pi$
$48$		0			0
$49$	1803.00			0.107277
$50$		0			0
$51$	10912.0			0.587461
$52$		0			0
$53$			24253.2i			1.18598i	0.805208	+	0.592992i	$0.202054\pi$
$53$			24253.2i			1.18598i	−0.805208	+	0.592992i	$0.797946\pi$
$54$		0			0
$55$	500.000	−	5567.76i	0.0222876	−	0.248184i
$56$		0			0
$57$			24988.1i			1.01870i
$58$		0			0
$59$	−25972.0			−0.971349			−0.485675	−	0.874140i	$-0.661426\pi$
$59$	−25972.0			−0.971349			−0.485675	+	0.874140i	$0.661426\pi$
$60$		0			0
$61$	−3058.00			−0.105224			−0.0526118	−	0.998615i	$-0.516755\pi$
$61$	−3058.00			−0.105224			−0.0526118	+	0.998615i	$0.516755\pi$
$62$		0			0
$63$			14576.4i			0.462700i
$64$		0			0
$65$	40920.0	+	3674.72i	1.20130	+	0.107880i
$66$		0			0
$67$		−	58784.5i		−	1.59984i	−0.600109	−	0.799918i	$-0.704876\pi$
$67$		−	58784.5i		−	1.59984i	0.600109	−	0.799918i	$-0.295124\pi$
$68$		0			0
$69$	−38068.0			−0.962581
$70$		0			0
$71$	37608.0			0.885389			0.442695	−	0.896672i	$-0.354023\pi$
$71$	37608.0			0.885389			0.442695	+	0.896672i	$0.354023\pi$
$72$		0			0
$73$		−	24008.2i		−	0.527294i	−0.964619	−	0.263647i	$-0.915075\pi$
$73$		−	24008.2i		−	0.527294i	0.964619	−	0.263647i	$-0.0849253\pi$
$74$		0			0
$75$	6200.00	−	34241.8i	0.127274	−	0.702915i
$76$		0			0
$77$		−	12249.1i		−	0.235438i
$78$		0			0
$79$	−79728.0			−1.43729			−0.718643	−	0.695379i	$-0.755236\pi$
$79$	−79728.0			−1.43729			−0.718643	+	0.695379i	$0.755236\pi$
$80$		0			0
$81$	−15971.0			−0.270470
$82$		0			0
$83$		−	16291.3i		−	0.259573i	−0.991542	−	0.129787i	$-0.958571\pi$
$83$		−	16291.3i		−	0.259573i	0.991542	−	0.129787i	$-0.0414292\pi$
$84$		0			0
$85$	54560.0	+	4899.63i	0.819081	+	0.0735557i
$86$		0			0
$87$			87458.4i			1.23881i
$88$		0			0
$89$	826.000			0.0110536			0.00552682	−	0.999985i	$-0.498241\pi$
$89$	826.000			0.0110536			0.00552682	+	0.999985i	$0.498241\pi$
$90$		0			0
$91$	90024.0			1.13961
$92$		0			0
$93$		−	23874.6i		−	0.286239i
$94$		0			0
$95$	−11220.0	+	124941.i	−0.127551	+	1.42035i
$96$		0			0
$97$			37593.5i			0.405680i	0.979212	+	0.202840i	$0.0650172\pi$
$97$			37593.5i			0.405680i	−0.979212	+	0.202840i	$0.934983\pi$
$98$		0			0
$99$	−11900.0			−0.122028
$100$		0			0
$101$	−143594.			−1.40066			−0.700330	−	0.713819i	$-0.746964\pi$
$101$	−143594.			−1.40066			−0.700330	+	0.713819i	$0.746964\pi$
$102$		0			0
$103$			111834.i			1.03868i	0.854568	+	0.519339i	$0.173822\pi$
$103$			111834.i			1.03868i	−0.854568	+	0.519339i	$0.826178\pi$
$104$		0			0
$105$	6820.00	−	75944.3i	0.0603686	−	0.672236i
$106$		0			0
$107$		−	92235.6i		−	0.778824i	−0.921064	−	0.389412i	$-0.872678\pi$
$107$		−	92235.6i		−	0.778824i	0.921064	−	0.389412i	$-0.127322\pi$
$108$		0			0
$109$	106238.			0.856473			0.428236	−	0.903667i	$-0.359135\pi$
$109$	106238.			0.856473			0.428236	+	0.903667i	$0.359135\pi$
$110$		0			0
$111$	115816.			0.892198
$112$		0			0
$113$		−	113048.i		−	0.832849i	−0.909170	−	0.416425i	$-0.863283\pi$
$113$		−	113048.i		−	0.832849i	0.909170	−	0.416425i	$-0.136717\pi$
$114$		0			0
$115$	−190340.	−	17093.0i	−1.34210	−	0.120524i
$116$		0			0
$117$		−	87458.4i		−	0.590659i
$118$		0			0
$119$	120032.			0.777015
$120$		0			0
$121$	−151051.			−0.937908
$122$		0			0
$123$		−	82558.8i		−	0.492040i
$124$		0			0
$125$	46375.0	−	168425.i	0.265466	−	0.964120i
$126$		0			0
$127$			51568.6i			0.283711i	0.989887	+	0.141856i	$0.0453069\pi$
$127$			51568.6i			0.283711i	−0.989887	+	0.141856i	$0.954693\pi$
$128$		0			0
$129$	197780.			1.04642
$130$		0			0
$131$	−89100.0			−0.453628			−0.226814	−	0.973938i	$-0.572831\pi$
$131$	−89100.0			−0.453628			−0.226814	+	0.973938i	$0.572831\pi$
$132$		0			0
$133$			274869.i			1.34740i
$134$		0			0
$135$	−224440.	−	20155.3i	−1.05990	−	0.0951820i
$136$		0			0
$137$			38350.8i			0.174571i	0.996183	+	0.0872856i	$0.0278193\pi$
$137$			38350.8i			0.174571i	−0.996183	+	0.0872856i	$0.972181\pi$
$138$		0			0
$139$	134684.			0.591261			0.295630	−	0.955302i	$-0.404470\pi$
$139$	134684.			0.591261			0.295630	+	0.955302i	$0.404470\pi$
$140$		0			0
$141$	−105028.			−0.444895
$142$		0			0
$143$			73494.5i			0.300549i
$144$		0			0
$145$	−39270.0	+	437292.i	−0.155110	+	1.72724i
$146$		0			0
$147$			20077.4i			0.0766325i
$148$		0			0
$149$	248006.			0.915159			0.457579	−	0.889169i	$-0.348717\pi$
$149$	248006.			0.915159			0.457579	+	0.889169i	$0.348717\pi$
$150$		0			0
$151$	313720.			1.11970			0.559848	−	0.828596i	$-0.310860\pi$
$151$	313720.			1.11970			0.559848	+	0.828596i	$0.310860\pi$
$152$		0			0
$153$		−	116611.i		−	0.402728i
$154$		0			0
$155$	10720.0	−	119373.i	0.0358398	−	0.399095i
$156$		0			0
$157$			245583.i			0.795150i	0.917570	+	0.397575i	$0.130148\pi$
$157$			245583.i			0.795150i	−0.917570	+	0.397575i	$0.869852\pi$
$158$		0			0
$159$	−270072.			−0.847202
$160$		0			0
$161$	−418748.			−1.27317
$162$		0			0
$163$		−	397483.i		−	1.17179i	−0.810388	−	0.585894i	$-0.800743\pi$
$163$		−	397483.i		−	1.17179i	0.810388	−	0.585894i	$-0.199257\pi$
$164$		0			0
$165$	62000.0	+	5567.76i	0.177289	+	0.0159210i
$166$		0			0
$167$		−	189983.i		−	0.527138i	−0.964641	−	0.263569i	$-0.915100\pi$
$167$		−	189983.i		−	0.527138i	0.964641	−	0.263569i	$-0.0848996\pi$
$168$		0			0
$169$	−168851.			−0.454765
$170$		0			0
$171$	267036.			0.698360
$172$		0			0
$173$		−	81088.9i		−	0.205990i	−0.994682	−	0.102995i	$-0.967157\pi$
$173$		−	81088.9i		−	0.205990i	0.994682	−	0.102995i	$-0.0328426\pi$
$174$		0			0
$175$	68200.0	−	376659.i	0.168341	−	0.929723i
$176$		0			0
$177$		−	289212.i		−	0.693878i
$178$		0			0
$179$	−142108.			−0.331502			−0.165751	−	0.986168i	$-0.553005\pi$
$179$	−142108.			−0.331502			−0.165751	+	0.986168i	$0.553005\pi$
$180$		0			0
$181$	250790.			0.569002			0.284501	−	0.958676i	$-0.408172\pi$
$181$	250790.			0.569002			0.284501	+	0.958676i	$0.408172\pi$
$182$		0			0
$183$		−	34052.4i		−	0.0751659i
$184$		0			0
$185$	579080.	+	52002.9i	1.24397	+	0.111712i
$186$		0			0
$187$			97992.7i			0.204922i
$188$		0			0
$189$	−493768.			−1.00547
$190$		0			0
$191$	−209472.			−0.415473			−0.207736	−	0.978185i	$-0.566610\pi$
$191$	−209472.			−0.415473			−0.207736	+	0.978185i	$0.566610\pi$
$192$		0			0
$193$			356693.i			0.689289i	0.938733	+	0.344645i	$0.112001\pi$
$193$			356693.i			0.689289i	−0.938733	+	0.344645i	$0.887999\pi$
$194$		0			0
$195$	−40920.0	+	455666.i	−0.0770636	+	0.858144i
$196$		0			0
$197$		−	86478.5i		−	0.158761i	−0.996844	−	0.0793803i	$-0.974706\pi$
$197$		−	86478.5i		−	0.158761i	0.996844	−	0.0793803i	$-0.0252941\pi$
$198$		0			0
$199$	−749208.			−1.34113			−0.670563	−	0.741852i	$-0.733947\pi$
$199$	−749208.			−1.34113			−0.670563	+	0.741852i	$0.733947\pi$
$200$		0			0
$201$	654596.			1.14283
$202$		0			0
$203$			962043.i			1.63853i
$204$		0			0
$205$	37070.0	−	412794.i	0.0616081	−	0.686039i
$206$		0			0
$207$			406814.i			0.659888i
$208$		0			0
$209$	−224400.			−0.355351
$210$		0			0
$211$	287364.			0.444351			0.222176	−	0.975007i	$-0.428684\pi$
$211$	287364.			0.444351			0.222176	+	0.975007i	$0.428684\pi$
$212$		0			0
$213$			418785.i			0.632473i
$214$		0			0
$215$	988900.	+	88805.8i	1.45900	+	0.131022i
$216$		0			0
$217$		−	262620.i		−	0.378599i
$218$		0			0
$219$	267344.			0.376669
$220$		0			0
$221$	−720192.			−0.991899
$222$		0			0
$223$		−	1.18866e6i		−	1.60065i	−0.599567	−	0.800325i	$-0.704660\pi$
$223$		−	1.18866e6i		−	1.60065i	0.599567	−	0.800325i	$-0.295340\pi$
$224$		0			0
$225$	−365925.	−	66256.4i	−0.481877	−	0.0872512i
$226$		0			0
$227$			978334.i			1.26015i	0.776534	+	0.630075i	$0.216976\pi$
$227$			978334.i			1.26015i	−0.776534	+	0.630075i	$0.783024\pi$
$228$		0			0
$229$	−506474.			−0.638217			−0.319109	−	0.947718i	$-0.603383\pi$
$229$	−506474.			−0.638217			−0.319109	+	0.947718i	$0.603383\pi$
$230$		0			0
$231$	136400.			0.168184
$232$		0			0
$233$		−	1.55465e6i		−	1.87605i	−0.346571	−	0.938024i	$-0.612654\pi$
$233$		−	1.55465e6i		−	1.87605i	0.346571	−	0.938024i	$-0.387346\pi$
$234$		0			0
$235$	−525140.	−	47159.0i	−0.620305	−	0.0557051i
$236$		0			0
$237$		−	887813.i		−	1.02672i
$238$		0			0
$239$	374704.			0.424320			0.212160	−	0.977235i	$-0.431950\pi$
$239$	374704.			0.424320			0.212160	+	0.977235i	$0.431950\pi$
$240$		0			0
$241$	843634.			0.935646			0.467823	−	0.883822i	$-0.345039\pi$
$241$	843634.			0.935646			0.467823	+	0.883822i	$0.345039\pi$
$242$		0			0
$243$			801702.i			0.870959i
$244$		0			0
$245$	−9015.00	+	100387.i	−0.00959512	+	0.106847i
$246$		0			0
$247$		−	1.64922e6i		−	1.72003i
$248$		0			0
$249$	181412.			0.185425
$250$		0			0
$251$	−1.72050e6			−1.72373			−0.861867	−	0.507134i	$-0.830705\pi$
$251$	−1.72050e6			−1.72373			−0.861867	+	0.507134i	$0.830705\pi$
$252$		0			0
$253$		−	341861.i		−	0.335775i
$254$		0			0
$255$	−54560.0	+	607554.i	−0.0525441	+	0.585106i
$256$		0			0
$257$			1.55220e6i			1.46594i	0.680262	+	0.732969i	$0.261866\pi$
$257$			1.55220e6i			1.46594i	−0.680262	+	0.732969i	$0.738134\pi$
$258$		0			0
$259$	1.27398e6			1.18008
$260$		0			0
$261$	934626.			0.849252
$262$		0			0
$263$			407772.i			0.363520i	0.983343	+	0.181760i	$0.0581793\pi$
$263$			407772.i			0.363520i	−0.983343	+	0.181760i	$0.941821\pi$
$264$		0			0
$265$	−1.35036e6	−	121266.i	−1.18123	−	0.106078i
$266$		0			0
$267$			9197.95i			0.00789610i
$268$		0			0
$269$	1.82710e6			1.53951			0.769754	−	0.638340i	$-0.220379\pi$
$269$	1.82710e6			1.53951			0.769754	+	0.638340i	$0.220379\pi$
$270$		0			0
$271$	−616880.			−0.510243			−0.255122	−	0.966909i	$-0.582116\pi$
$271$	−616880.			−0.510243			−0.255122	+	0.966909i	$0.582116\pi$
$272$		0			0
$273$			1.00246e6i			0.814071i
$274$		0			0
$275$	307500.	+	55677.6i	0.245196	+	0.0443965i
$276$		0			0
$277$			1.83712e6i			1.43859i	0.694704	+	0.719296i	$0.255535\pi$
$277$			1.83712e6i			1.43859i	−0.694704	+	0.719296i	$0.744465\pi$
$278$		0			0
$279$	−255136.			−0.196228
$280$		0			0
$281$	−1.22093e6			−0.922415			−0.461208	−	0.887292i	$-0.652584\pi$
$281$	−1.22093e6			−0.922415			−0.461208	+	0.887292i	$0.652584\pi$
$282$		0			0
$283$			688766.i			0.511217i	0.966780	+	0.255609i	$0.0822758\pi$
$283$			688766.i			0.511217i	−0.966780	+	0.255609i	$0.917724\pi$
$284$		0			0
$285$	−1.39128e6	−	124941.i	−1.01462	−	0.0911154i
$286$		0			0
$287$		−	908147.i		−	0.650806i
$288$		0			0
$289$	459601.			0.323695
$290$		0			0
$291$	−418624.			−0.289796
$292$		0			0
$293$			856211.i			0.582655i	0.956623	+	0.291328i	$0.0940970\pi$
$293$			856211.i			0.582655i	−0.956623	+	0.291328i	$0.905903\pi$
$294$		0			0
$295$	129860.	−	1.44606e6i	0.0868801	−	0.967456i
$296$		0			0
$297$		−	403106.i		−	0.265172i
$298$		0			0
$299$	2.51249e6			1.62527
$300$		0			0
$301$	2.17558e6			1.38407
$302$		0			0
$303$		−	1.59900e6i		−	1.00055i
$304$		0			0
$305$	15290.0	−	170262.i	0.00941148	−	0.104802i
$306$		0			0
$307$		−	1.09617e6i		−	0.663792i	−0.943316	−	0.331896i	$-0.892312\pi$
$307$		−	1.09617e6i		−	0.663792i	0.943316	−	0.331896i	$-0.107688\pi$
$308$		0			0
$309$	−1.24533e6			−0.741974
$310$		0			0
$311$	−2.12465e6			−1.24562			−0.622811	−	0.782373i	$-0.714009\pi$
$311$	−2.12465e6			−1.24562			−0.622811	+	0.782373i	$0.714009\pi$
$312$		0			0
$313$			294824.i			0.170099i	0.996377	+	0.0850496i	$0.0271049\pi$
$313$			294824.i			0.170099i	−0.996377	+	0.0850496i	$0.972895\pi$
$314$		0			0
$315$	−811580.	−	72882.0i	−0.460845	−	0.0413851i
$316$		0			0
$317$		−	2.53153e6i		−	1.41493i	−0.706749	−	0.707465i	$-0.749839\pi$
$317$		−	2.53153e6i		−	1.41493i	0.706749	−	0.707465i	$-0.250161\pi$
$318$		0			0
$319$	−785400.			−0.432130
$320$		0			0
$321$	1.02709e6			0.556348
$322$		0			0
$323$		−	2.19896e6i		−	1.17276i
$324$		0			0
$325$	−409200.	+	2.25996e6i	−0.214895	+	1.18684i
$326$		0			0
$327$			1.18302e6i			0.611817i
$328$		0			0
$329$	−1.15531e6			−0.588448
$330$		0			0
$331$	−1.17021e6			−0.587076			−0.293538	−	0.955947i	$-0.594833\pi$
$331$	−1.17021e6			−0.587076			−0.293538	+	0.955947i	$0.594833\pi$
$332$		0			0
$333$		−	1.23767e6i		−	0.611637i
$334$		0			0
$335$	3.27298e6	+	293922.i	1.59342	+	0.143094i
$336$		0			0
$337$			1.86872e6i			0.896333i	0.893950	+	0.448167i	$0.147923\pi$
$337$			1.86872e6i			0.896333i	−0.893950	+	0.448167i	$0.852077\pi$
$338$		0			0
$339$	1.25885e6			0.594941
$340$		0			0
$341$	214400.			0.0998479
$342$		0			0
$343$			2.27955e6i			1.04620i
$344$		0			0
$345$	190340.	−	2.11954e6i	0.0860959	−	0.958723i
$346$		0			0
$347$			1.63342e6i			0.728237i	0.931353	+	0.364119i	$0.118630\pi$
$347$			1.63342e6i			0.728237i	−0.931353	+	0.364119i	$0.881370\pi$
$348$		0			0
$349$	−2.00629e6			−0.881719			−0.440859	−	0.897576i	$-0.645326\pi$
$349$	−2.00629e6			−0.881719			−0.440859	+	0.897576i	$0.645326\pi$
$350$		0			0
$351$	2.96261e6			1.28353
$352$		0			0
$353$			1.80859e6i			0.772508i	0.922392	+	0.386254i	$0.126231\pi$
$353$			1.80859e6i			0.772508i	−0.922392	+	0.386254i	$0.873769\pi$
$354$		0			0
$355$	−188040.	+	2.09392e6i	−0.0791916	+	0.881841i
$356$		0			0
$357$			1.33662e6i			0.555057i
$358$		0			0
$359$	−4.50674e6			−1.84555			−0.922777	−	0.385334i	$-0.874086\pi$
$359$	−4.50674e6			−1.84555			−0.922777	+	0.385334i	$0.874086\pi$
$360$		0			0
$361$	2.55944e6			1.03366
$362$		0			0
$363$		−	1.68203e6i		−	0.669989i
$364$		0			0
$365$	1.33672e6	+	120041.i	0.525180	+	0.0471626i
$366$		0			0
$367$			3.02796e6i			1.17351i	0.809766	+	0.586753i	$0.199594\pi$
$367$			3.02796e6i			1.17351i	−0.809766	+	0.586753i	$0.800406\pi$
$368$		0			0
$369$	−882266.			−0.337313
$370$		0			0
$371$	−2.97079e6			−1.12057
$372$		0			0
$373$		−	1.16342e6i		−	0.432976i	−0.976285	−	0.216488i	$-0.930540\pi$
$373$		−	1.16342e6i		−	0.432976i	0.976285	−	0.216488i	$-0.0694602\pi$
$374$		0			0
$375$	1.87550e6	+	516410.i	0.688714	+	0.189634i
$376$		0			0
$377$		−	5.77226e6i		−	2.09167i
$378$		0			0
$379$	−832052.			−0.297545			−0.148772	−	0.988871i	$-0.547532\pi$
$379$	−832052.			−0.297545			−0.148772	+	0.988871i	$0.547532\pi$
$380$		0			0
$381$	−574244.			−0.202667
$382$		0			0
$383$			2.86948e6i			0.999554i	0.866154	+	0.499777i	$0.166585\pi$
$383$			2.86948e6i			0.999554i	−0.866154	+	0.499777i	$0.833415\pi$
$384$		0			0
$385$	682000.	+	61245.4i	0.234494	+	0.0210582i
$386$		0			0
$387$		−	2.11358e6i		−	0.717366i
$388$		0			0
$389$	311926.			0.104515			0.0522574	−	0.998634i	$-0.483358\pi$
$389$	311926.			0.104515			0.0522574	+	0.998634i	$0.483358\pi$
$390$		0			0
$391$	3.34998e6			1.10816
$392$		0			0
$393$		−	992176.i		−	0.324046i
$394$		0			0
$395$	398640.	−	4.43907e6i	0.128555	−	1.43153i
$396$		0			0
$397$		−	2.95619e6i		−	0.941362i	−0.882304	−	0.470681i	$-0.844008\pi$
$397$		−	2.95619e6i		−	0.941362i	0.882304	−	0.470681i	$-0.155992\pi$
$398$		0			0
$399$	−3.06082e6			−0.962509
$400$		0			0
$401$	2770.00			0.000860238			0.000430119	−	1.00000i	$-0.499863\pi$
$401$	2770.00			0.000860238			0.000430119		1.00000i	$0.499863\pi$
$402$		0			0
$403$			1.57572e6i			0.483300i
$404$		0			0
$405$	79855.0	−	889228.i	0.0241916	−	0.269386i
$406$		0			0
$407$			1.04006e6i			0.311223i
$408$		0			0
$409$	−1.97985e6			−0.585225			−0.292613	−	0.956231i	$-0.594525\pi$
$409$	−1.97985e6			−0.585225			−0.292613	+	0.956231i	$0.594525\pi$
$410$		0			0
$411$	−427056.			−0.124704
$412$		0			0
$413$		−	3.18133e6i		−	0.917770i
$414$		0			0
$415$	907060.	+	81456.4i	0.258533	+	0.0232169i
$416$		0			0
$417$			1.49978e6i			0.422364i
$418$		0			0
$419$	5.10120e6			1.41951			0.709754	−	0.704450i	$-0.248806\pi$
$419$	5.10120e6			1.41951			0.709754	+	0.704450i	$0.248806\pi$
$420$		0			0
$421$	−2.43223e6			−0.668806			−0.334403	−	0.942430i	$-0.608535\pi$
$421$	−2.43223e6			−0.668806			−0.334403	+	0.942430i	$0.608535\pi$
$422$		0			0
$423$			1.12238e6i			0.304993i
$424$		0			0
$425$	−545600.	+	3.01327e6i	−0.146522	+	0.809219i
$426$		0			0
$427$		−	374577.i		−	0.0994194i
$428$		0			0
$429$	−818400.			−0.214695
$430$		0			0
$431$	918896.			0.238272			0.119136	−	0.992878i	$-0.461988\pi$
$431$	918896.			0.238272			0.119136	+	0.992878i	$0.461988\pi$
$432$		0			0
$433$			2.05455e6i			0.526619i	0.964711	+	0.263310i	$0.0848141\pi$
$433$			2.05455e6i			0.526619i	−0.964711	+	0.263310i	$0.915186\pi$
$434$		0			0
$435$	−4.86948e6	−	437292.i	−1.23384	−	0.110802i
$436$		0			0
$437$			7.67135e6i			1.92162i
$438$		0			0
$439$	676632.			0.167568			0.0837840	−	0.996484i	$-0.473299\pi$
$439$	676632.			0.167568			0.0837840	+	0.996484i	$0.473299\pi$
$440$		0			0
$441$	214557.			0.0525347
$442$		0			0
$443$		−	2.53092e6i		−	0.612729i	−0.951914	−	0.306365i	$-0.900887\pi$
$443$		−	2.53092e6i		−	0.612729i	0.951914	−	0.306365i	$-0.0991126\pi$
$444$		0			0
$445$	−4130.00	+	45989.7i	−0.000988667	+	0.0110093i
$446$		0			0
$447$			2.76168e6i			0.653739i
$448$		0			0
$449$	5.17619e6			1.21170			0.605849	−	0.795579i	$-0.292833\pi$
$449$	5.17619e6			1.21170			0.605849	+	0.795579i	$0.292833\pi$
$450$		0			0
$451$	741400.			0.171637
$452$		0			0
$453$			3.49344e6i			0.799848i
$454$		0			0
$455$	−450120.	+	5.01232e6i	−0.101929	+	1.13504i
$456$		0			0
$457$		−	3.11274e6i		−	0.697191i	−0.937273	−	0.348596i	$-0.886659\pi$
$457$		−	3.11274e6i		−	0.697191i	0.937273	−	0.348596i	$-0.113341\pi$
$458$		0			0
$459$	3.95014e6			0.875147
$460$		0			0
$461$	−2.64957e6			−0.580662			−0.290331	−	0.956926i	$-0.593765\pi$
$461$	−2.64957e6			−0.580662			−0.290331	+	0.956926i	$0.593765\pi$
$462$		0			0
$463$		−	2.59165e6i		−	0.561854i	−0.959729	−	0.280927i	$-0.909358\pi$
$463$		−	2.59165e6i		−	0.561854i	0.959729	−	0.280927i	$-0.0906420\pi$
$464$		0			0
$465$	1.32928e6	+	119373.i	0.285091	+	0.0256020i
$466$		0			0
$467$		−	6.62135e6i		−	1.40493i	−0.711719	−	0.702465i	$-0.752083\pi$
$467$		−	6.62135e6i		−	1.40493i	0.711719	−	0.702465i	$-0.247917\pi$
$468$		0			0
$469$	7.20056e6			1.51159
$470$		0			0
$471$	−2.73470e6			−0.568011
$472$		0			0
$473$			1.77612e6i			0.365022i
$474$		0			0
$475$	−6.90030e6	−	1.24941e6i	−1.40325	−	0.254080i
$476$		0			0
$477$			2.88613e6i			0.580791i
$478$		0			0
$479$	−6.89322e6			−1.37272			−0.686362	−	0.727260i	$-0.740793\pi$
$479$	−6.89322e6			−1.37272			−0.686362	+	0.727260i	$0.740793\pi$
$480$		0			0
$481$	−7.64386e6			−1.50643
$482$		0			0
$483$		−	4.66298e6i		−	0.909485i
$484$		0			0
$485$	−2.09312e6	−	187968.i	−0.404054	−	0.0362852i
$486$		0			0
$487$		−	5.65370e6i		−	1.08021i	−0.841596	−	0.540107i	$-0.818384\pi$
$487$		−	5.65370e6i		−	1.08021i	0.841596	−	0.540107i	$-0.181616\pi$
$488$		0			0
$489$	4.42618e6			0.837061
$490$		0			0
$491$	5.88390e6			1.10144			0.550721	−	0.834689i	$-0.314353\pi$
$491$	5.88390e6			1.10144			0.550721	+	0.834689i	$0.314353\pi$
$492$		0			0
$493$		−	7.69634e6i		−	1.42616i
$494$		0			0
$495$	59500.0	−	662564.i	0.0109145	−	0.121539i
$496$		0			0
$497$			4.60663e6i			0.836552i
$498$		0			0
$499$	6.72080e6			1.20829			0.604143	−	0.796876i	$-0.293515\pi$
$499$	6.72080e6			1.20829			0.604143	+	0.796876i	$0.293515\pi$
$500$		0			0
$501$	2.11556e6			0.376558
$502$		0			0
$503$		−	469262.i		−	0.0826981i	−0.999145	−	0.0413491i	$-0.986834\pi$
$503$		−	469262.i		−	0.0826981i	0.999145	−	0.0413491i	$-0.0131656\pi$
$504$		0			0
$505$	717970.	−	7.99498e6i	0.125279	−	1.39505i
$506$		0			0
$507$		−	1.88025e6i		−	0.324859i
$508$		0			0
$509$	294414.			0.0503691			0.0251845	−	0.999683i	$-0.491983\pi$
$509$	294414.			0.0503691			0.0251845	+	0.999683i	$0.491983\pi$
$510$		0			0
$511$	2.94078e6			0.498208
$512$		0			0
$513$			9.04570e6i			1.51757i
$514$		0			0
$515$	−6.22666e6	−	559171.i	−1.03452	−	0.0929023i
$516$		0			0
$517$		−	943179.i		−	0.155191i
$518$		0			0
$519$	902968.			0.147148
$520$		0			0
$521$	−7.10025e6			−1.14599			−0.572993	−	0.819560i	$-0.694218\pi$
$521$	−7.10025e6			−1.14599			−0.572993	+	0.819560i	$0.694218\pi$
$522$		0			0
$523$			5.96567e6i			0.953685i	0.878989	+	0.476843i	$0.158219\pi$
$523$			5.96567e6i			0.953685i	−0.878989	+	0.476843i	$0.841781\pi$
$524$		0			0
$525$	4.19430e6	+	759443.i	0.664142	+	0.120253i
$526$		0			0
$527$			2.10096e6i			0.329528i
$528$		0			0
$529$	−5.25053e6			−0.815763
$530$		0			0
$531$	−3.09067e6			−0.475681
$532$		0			0
$533$			5.44888e6i			0.830786i
$534$		0			0
$535$	5.13546e6	+	461178.i	0.775702	+	0.0696601i
$536$		0			0
$537$		−	1.58245e6i		−	0.236807i
$538$		0			0
$539$	−180300.			−0.0267315
$540$		0			0
$541$	−2.72367e6			−0.400093			−0.200046	−	0.979786i	$-0.564109\pi$
$541$	−2.72367e6			−0.400093			−0.200046	+	0.979786i	$0.564109\pi$
$542$		0			0
$543$			2.79268e6i			0.406463i
$544$		0			0
$545$	−531190.	+	5.91508e6i	−0.0766053	+	0.853040i
$546$		0			0
$547$			9.22148e6i			1.31775i	0.752254	+	0.658874i	$0.228967\pi$
$547$			9.22148e6i			1.31775i	−0.752254	+	0.658874i	$0.771033\pi$
$548$		0			0
$549$	−363902.			−0.0515292
$550$		0			0
$551$	1.76244e7			2.47306
$552$		0			0
$553$		−	9.76595e6i		−	1.35801i
$554$		0			0
$555$	−579080.	+	6.44836e6i	−0.0798006	+	0.888622i
$556$		0			0
$557$		−	3.42852e6i		−	0.468240i	−0.972208	−	0.234120i	$-0.924779\pi$
$557$		−	3.42852e6i		−	0.468240i	0.972208	−	0.234120i	$-0.0752209\pi$
$558$		0			0
$559$	−1.30535e7			−1.76684
$560$		0			0
$561$	−1.09120e6			−0.146385
$562$		0			0
$563$			5.77899e6i			0.768389i	0.923252	+	0.384195i	$0.125521\pi$
$563$			5.77899e6i			0.768389i	−0.923252	+	0.384195i	$0.874479\pi$
$564$		0			0
$565$	6.29424e6	+	565239.i	0.829511	+	0.0744923i
$566$		0			0
$567$		−	1.95630e6i		−	0.255551i
$568$		0			0
$569$	3.89257e6			0.504029			0.252015	−	0.967723i	$-0.418907\pi$
$569$	3.89257e6			0.504029			0.252015	+	0.967723i	$0.418907\pi$
$570$		0			0
$571$	−5.06277e6			−0.649828			−0.324914	−	0.945744i	$-0.605335\pi$
$571$	−5.06277e6			−0.649828			−0.324914	+	0.945744i	$0.605335\pi$
$572$		0			0
$573$		−	2.33258e6i		−	0.296791i
$574$		0			0
$575$	1.90340e6	−	1.05122e7i	0.240082	−	1.32594i
$576$		0			0
$577$		−	3.30075e6i		−	0.412737i	−0.978474	−	0.206368i	$-0.933836\pi$
$577$		−	3.30075e6i		−	0.412737i	0.978474	−	0.206368i	$-0.0661645\pi$
$578$		0			0
$579$	−3.97197e6			−0.492390
$580$		0			0
$581$	1.99553e6			0.245255
$582$		0			0
$583$		−	2.42532e6i		−	0.295527i
$584$		0			0
$585$	4.86948e6	+	437292.i	0.588292	+	0.0528302i
$586$		0			0
$587$			5.16997e6i			0.619288i	0.950853	+	0.309644i	$0.100210\pi$
$587$			5.16997e6i			0.619288i	−0.950853	+	0.309644i	$0.899790\pi$
$588$		0			0
$589$	−4.81114e6			−0.571425
$590$		0			0
$591$	962984.			0.113410
$592$		0			0
$593$		−	1.58484e7i		−	1.85075i	−0.379055	−	0.925374i	$-0.623751\pi$
$593$		−	1.58484e7i		−	1.85075i	0.379055	−	0.925374i	$-0.376249\pi$
$594$		0			0
$595$	−600160.	+	6.68310e6i	−0.0694984	+	0.773901i
$596$		0			0
$597$		−	8.34283e6i		−	0.958026i
$598$		0			0
$599$	−1.66146e7			−1.89201			−0.946004	−	0.324156i	$-0.894920\pi$
$599$	−1.66146e7			−1.89201			−0.946004	+	0.324156i	$0.894920\pi$
$600$		0			0
$601$	7.88249e6			0.890179			0.445089	−	0.895486i	$-0.353172\pi$
$601$	7.88249e6			0.890179			0.445089	+	0.895486i	$0.353172\pi$
$602$		0			0
$603$		−	6.99535e6i		−	0.783459i
$604$		0			0
$605$	755255.	−	8.41016e6i	0.0838890	−	0.934149i
$606$		0			0
$607$			782594.i			0.0862114i	0.999071	+	0.0431057i	$0.0137252\pi$
$607$			782594.i			0.0862114i	−0.999071	+	0.0431057i	$0.986275\pi$
$608$		0			0
$609$	−1.07129e7			−1.17047
$610$		0			0
$611$	6.93185e6			0.751183
$612$		0			0
$613$			2.41233e6i			0.259290i	0.991560	+	0.129645i	$0.0413838\pi$
$613$			2.41233e6i			0.259290i	−0.991560	+	0.129645i	$0.958616\pi$
$614$		0			0
$615$	4.59668e6	+	412794.i	0.490068	+	0.0440094i
$616$		0			0
$617$			9.66355e6i			1.02194i	0.859600	+	0.510968i	$0.170713\pi$
$617$			9.66355e6i			1.02194i	−0.859600	+	0.510968i	$0.829287\pi$
$618$		0			0
$619$	1.80036e7			1.88857			0.944283	−	0.329134i	$-0.106757\pi$
$619$	1.80036e7			1.88857			0.944283	+	0.329134i	$0.106757\pi$
$620$		0			0
$621$	−1.37806e7			−1.43397
$622$		0			0
$623$			101177.i			0.0104439i
$624$		0			0
$625$	9.14562e6	+	3.42418e6i	0.936512	+	0.350636i
$626$		0			0
$627$		−	2.49881e6i		−	0.253843i
$628$		0			0
$629$	−1.01918e7			−1.02713
$630$		0			0
$631$	−4.80081e6			−0.480000			−0.240000	−	0.970773i	$-0.577147\pi$
$631$	−4.80081e6			−0.480000			−0.240000	+	0.970773i	$0.577147\pi$
$632$		0			0
$633$			3.19995e6i			0.317420i
$634$		0			0
$635$	−2.87122e6	−	257843.i	−0.282574	−	0.0253759i
$636$		0			0
$637$		−	1.32511e6i		−	0.129390i
$638$		0			0
$639$	4.47535e6			0.433586
$640$		0			0
$641$	1.44950e7			1.39340			0.696698	−	0.717365i	$-0.254652\pi$
$641$	1.44950e7			1.39340			0.696698	+	0.717365i	$0.254652\pi$
$642$		0			0
$643$			1.82430e7i			1.74008i	0.492979	+	0.870041i	$0.335908\pi$
$643$			1.82430e7i			1.74008i	−0.492979	+	0.870041i	$0.664092\pi$
$644$		0			0
$645$	−988900.	+	1.10119e7i	−0.0935951	+	1.04223i
$646$		0			0
$647$		−	9.64592e6i		−	0.905905i	−0.891535	−	0.452953i	$-0.850371\pi$
$647$		−	9.64592e6i		−	0.905905i	0.891535	−	0.452953i	$-0.149629\pi$
$648$		0			0
$649$	2.59720e6			0.242044
$650$		0			0
$651$	2.92442e6			0.270450
$652$		0			0
$653$			1.92807e7i			1.76945i	0.466111	+	0.884726i	$0.345655\pi$
$653$			1.92807e7i			1.76945i	−0.466111	+	0.884726i	$0.654345\pi$
$654$		0			0
$655$	445500.	−	4.96088e6i	0.0405737	−	0.451809i
$656$		0			0
$657$		−	2.85698e6i		−	0.258222i
$658$		0			0
$659$	−9.70592e6			−0.870609			−0.435304	−	0.900283i	$-0.643359\pi$
$659$	−9.70592e6			−0.870609			−0.435304	+	0.900283i	$0.643359\pi$
$660$		0			0
$661$	−4.28396e6			−0.381366			−0.190683	−	0.981652i	$-0.561070\pi$
$661$	−4.28396e6			−0.381366			−0.190683	+	0.981652i	$0.561070\pi$
$662$		0			0
$663$		−	8.01972e6i		−	0.708558i
$664$		0			0
$665$	−1.53041e7	−	1.37435e6i	−1.34200	−	0.120515i
$666$		0			0
$667$			2.68497e7i			2.33682i
$668$		0			0
$669$	1.32364e7			1.14342
$670$		0			0
$671$	305800.			0.0262199
$672$		0			0
$673$			1.30585e7i			1.11136i	0.831395	+	0.555681i	$0.187542\pi$
$673$			1.30585e7i			1.11136i	−0.831395	+	0.555681i	$0.812458\pi$
$674$		0			0
$675$	2.24440e6	−	1.23955e7i	0.189601	−	1.04714i
$676$		0			0
$677$		−	8.42565e6i		−	0.706532i	−0.935523	−	0.353266i	$-0.885071\pi$
$677$		−	8.42565e6i		−	0.706532i	0.935523	−	0.353266i	$-0.114929\pi$
$678$		0			0
$679$	−4.60486e6			−0.383303
$680$		0			0
$681$	−1.08943e7			−0.900182
$682$		0			0
$683$		−	1.64100e7i		−	1.34603i	−0.739627	−	0.673017i	$-0.764998\pi$
$683$		−	1.64100e7i		−	1.34603i	0.739627	−	0.673017i	$-0.235002\pi$
$684$		0			0
$685$	−2.13528e6	−	191754.i	−0.173872	−	0.0156141i
$686$		0			0
$687$		−	5.63986e6i		−	0.455907i
$688$		0			0
$689$	1.78248e7			1.43046
$690$		0			0
$691$	−1.12139e7			−0.893428			−0.446714	−	0.894677i	$-0.647406\pi$
$691$	−1.12139e7			−0.893428			−0.446714	+	0.894677i	$0.647406\pi$
$692$		0			0
$693$		−	1.45764e6i		−	0.115297i
$694$		0			0
$695$	−673420.	+	7.49889e6i	−0.0528840	+	0.588891i
$696$		0			0
$697$			7.26518e6i			0.566453i
$698$		0			0
$699$	1.73119e7			1.34014
$700$		0			0
$701$	2.04707e7			1.57339			0.786696	−	0.617340i	$-0.211790\pi$
$701$	2.04707e7			1.57339			0.786696	+	0.617340i	$0.211790\pi$
$702$		0			0
$703$		−	2.33389e7i		−	1.78112i
$704$		0			0
$705$	525140.	−	5.84771e6i	0.0397926	−	0.443112i
$706$		0			0
$707$		−	1.75889e7i		−	1.32340i
$708$		0			0
$709$	81654.0			0.00610045			0.00305023	−	0.999995i	$-0.499029\pi$
$709$	81654.0			0.00610045			0.00305023	+	0.999995i	$0.499029\pi$
$710$		0			0
$711$	−9.48763e6			−0.703856
$712$		0			0
$713$		−	7.32949e6i		−	0.539946i
$714$		0			0
$715$	−4.09200e6	−	367472.i	−0.299344	−	0.0268819i
$716$		0			0
$717$			4.17253e6i			0.303111i
$718$		0			0
$719$	8.61006e6			0.621132			0.310566	−	0.950552i	$-0.399481\pi$
$719$	8.61006e6			0.621132			0.310566	+	0.950552i	$0.399481\pi$
$720$		0			0
$721$	−1.36987e7			−0.981386
$722$		0			0
$723$			9.39431e6i			0.668373i
$724$		0			0
$725$	−2.41510e7	−	4.37292e6i	−1.70644	−	0.308977i
$726$		0			0
$727$		−	1.17682e7i		−	0.825796i	−0.910777	−	0.412898i	$-0.864517\pi$
$727$		−	1.17682e7i		−	0.825796i	0.910777	−	0.412898i	$-0.135483\pi$
$728$		0			0
$729$	−1.28083e7			−0.892635
$730$		0			0
$731$	−1.74046e7			−1.20468
$732$		0			0
$733$			3.93759e6i			0.270689i	0.990799	+	0.135344i	$0.0432141\pi$
$733$			3.93759e6i			0.270689i	−0.990799	+	0.135344i	$0.956786\pi$
$734$		0			0
$735$	−1.11786e6	−	100387.i	−0.0763254	−	0.00685422i
$736$		0			0
$737$			5.87845e6i			0.398652i
$738$		0			0
$739$	2.30602e7			1.55329			0.776643	−	0.629941i	$-0.216921\pi$
$739$	2.30602e7			1.55329			0.776643	+	0.629941i	$0.216921\pi$
$740$		0			0
$741$	1.83649e7			1.22869
$742$		0			0
$743$			1.72675e6i			0.114751i	0.998353	+	0.0573757i	$0.0182733\pi$
$743$			1.72675e6i			0.114751i	−0.998353	+	0.0573757i	$0.981727\pi$
$744$		0			0
$745$	−1.24003e6	+	1.38084e7i	−0.0818543	+	0.911491i
$746$		0			0
$747$		−	1.93866e6i		−	0.127116i
$748$		0			0
$749$	1.12980e7			0.735864
$750$		0			0
$751$	−2.58030e6			−0.166944			−0.0834720	−	0.996510i	$-0.526601\pi$
$751$	−2.58030e6			−0.166944			−0.0834720	+	0.996510i	$0.526601\pi$
$752$		0			0
$753$		−	1.91587e7i		−	1.23134i
$754$		0			0
$755$	−1.56860e6	+	1.74672e7i	−0.100149	+	1.11521i
$756$		0			0
$757$			5.75878e6i			0.365251i	0.983183	+	0.182625i	$0.0584595\pi$
$757$			5.75878e6i			0.365251i	−0.983183	+	0.182625i	$0.941540\pi$
$758$		0			0
$759$	3.80680e6			0.239859
$760$		0			0
$761$	1.40499e7			0.879450			0.439725	−	0.898133i	$-0.355076\pi$
$761$	1.40499e7			0.879450			0.439725	+	0.898133i	$0.355076\pi$
$762$		0			0
$763$			1.30132e7i			0.809230i
$764$		0			0
$765$	6.49264e6	+	583056.i	0.401114	+	0.0360211i
$766$		0			0
$767$			1.90880e7i			1.17158i
$768$		0			0
$769$	5.59898e6			0.341423			0.170712	−	0.985321i	$-0.445393\pi$
$769$	5.59898e6			0.341423			0.170712	+	0.985321i	$0.445393\pi$
$770$		0			0
$771$	−1.72846e7			−1.04719
$772$		0			0
$773$		−	6.34625e6i		−	0.382004i	−0.981590	−	0.191002i	$-0.938826\pi$
$773$		−	6.34625e6i		−	0.382004i	0.981590	−	0.191002i	$-0.0611738\pi$
$774$		0			0
$775$	6.59280e6	+	1.19373e6i	0.394290	+	0.0713923i
$776$		0			0
$777$			1.41864e7i			0.842984i
$778$		0			0
$779$	−1.66370e7			−0.982272
$780$		0			0
$781$	−3.76080e6			−0.220624
$782$		0			0
$783$			3.16600e7i			1.84547i
$784$		0			0
$785$	−1.36735e7	−	1.22791e6i	−0.791963	−	0.0711204i
$786$		0			0
$787$		−	1.73688e7i		−	0.999617i	−0.866136	−	0.499809i	$-0.833404\pi$
$787$		−	1.73688e7i		−	0.999617i	0.866136	−	0.499809i	$-0.166596\pi$
$788$		0			0
$789$	−4.54076e6			−0.259678
$790$		0			0
$791$	1.38473e7			0.786909
$792$		0			0
$793$			2.24746e6i			0.126914i
$794$		0			0
$795$	1.35036e6	−	1.50370e7i	0.0757760	−	0.843806i
$796$		0			0
$797$			8.06932e6i			0.449978i	0.974361	+	0.224989i	$0.0722346\pi$
$797$			8.06932e6i			0.449978i	−0.974361	+	0.224989i	$0.927765\pi$
$798$		0			0
$799$	9.24246e6			0.512178
$800$		0			0
$801$	98294.0			0.00541310
$802$		0			0
$803$			2.40082e6i			0.131393i
$804$		0			0
$805$	2.09374e6	−	2.33149e7i	0.113876	−	1.26807i
$806$		0			0
$807$			2.03457e7i			1.09974i
$808$		0			0
$809$	−1.94554e7			−1.04513			−0.522564	−	0.852600i	$-0.675024\pi$
$809$	−1.94554e7			−1.04513			−0.522564	+	0.852600i	$0.675024\pi$
$810$		0			0
$811$	−2.85204e6			−0.152266			−0.0761330	−	0.997098i	$-0.524257\pi$
$811$	−2.85204e6			−0.152266			−0.0761330	+	0.997098i	$0.524257\pi$
$812$		0			0
$813$		−	6.86928e6i		−	0.364490i
$814$		0			0
$815$	2.21309e7	+	1.98741e6i	1.16709	+	0.104808i
$816$		0			0
$817$		−	3.98561e7i		−	2.08900i
$818$		0			0
$819$	1.07129e7			0.558079
$820$		0			0
$821$	−5.48431e6			−0.283965			−0.141982	−	0.989869i	$-0.545348\pi$
$821$	−5.48431e6			−0.283965			−0.141982	+	0.989869i	$0.545348\pi$
$822$		0			0
$823$		−	6.57652e6i		−	0.338452i	−0.985577	−	0.169226i	$-0.945873\pi$
$823$		−	6.57652e6i		−	0.338452i	0.985577	−	0.169226i	$-0.0541267\pi$
$824$		0			0
$825$	−620000.	+	3.42418e6i	−0.0317144	+	0.175154i
$826$		0			0
$827$		−	4.19715e6i		−	0.213398i	−0.994291	−	0.106699i	$-0.965972\pi$
$827$		−	4.19715e6i		−	0.213398i	0.994291	−	0.106699i	$-0.0340281\pi$
$828$		0			0
$829$	−2.78889e7			−1.40943			−0.704717	−	0.709488i	$-0.748926\pi$
$829$	−2.78889e7			−1.40943			−0.704717	+	0.709488i	$0.748926\pi$
$830$		0			0
$831$	−2.04573e7			−1.02765
$832$		0			0
$833$		−	1.76681e6i		−	0.0882220i
$834$		0			0
$835$	1.05778e7	+	949916.i	0.525025	+	0.0471486i
$836$		0			0
$837$		−	8.64260e6i		−	0.426413i
$838$		0			0
$839$	−1.27752e7			−0.626560			−0.313280	−	0.949661i	$-0.601428\pi$
$839$	−1.27752e7			−0.626560			−0.313280	+	0.949661i	$0.601428\pi$
$840$		0			0
$841$	4.11742e7			2.00740
$842$		0			0
$843$		−	1.35957e7i		−	0.658922i
$844$		0			0
$845$	844255.	−	9.40123e6i	0.0406754	−	0.452942i
$846$		0			0
$847$		−	1.85024e7i		−	0.886173i
$848$		0			0
$849$	−7.66977e6			−0.365185
$850$		0			0
$851$	3.55555e7			1.68300
$852$		0			0
$853$		−	5.37122e6i		−	0.252755i	−0.991982	−	0.126378i	$-0.959665\pi$
$853$		−	5.37122e6i		−	0.252755i	0.991982	−	0.126378i	$-0.0403351\pi$
$854$		0			0
$855$	−1.33518e6	+	1.48679e7i	−0.0624632	+	0.695561i
$856$		0			0
$857$		−	1.61585e7i		−	0.751535i	−0.926714	−	0.375767i	$-0.877379\pi$
$857$		−	1.61585e7i		−	0.751535i	0.926714	−	0.375767i	$-0.122621\pi$
$858$		0			0
$859$	−3.34643e7			−1.54739			−0.773693	−	0.633561i	$-0.781593\pi$
$859$	−3.34643e7			−1.54739			−0.773693	+	0.633561i	$0.781593\pi$
$860$		0			0
$861$	1.01127e7			0.464899
$862$		0			0
$863$			2.51788e7i			1.15082i	0.817864	+	0.575412i	$0.195158\pi$
$863$			2.51788e7i			1.15082i	−0.817864	+	0.575412i	$0.804842\pi$
$864$		0			0
$865$	4.51484e6	+	405445.i	0.205164	+	0.0184243i
$866$		0			0
$867$			5.11790e6i			0.231230i
$868$		0			0
$869$	7.97280e6			0.358147
$870$		0			0
$871$	−4.32033e7			−1.92962
$872$		0			0
$873$			4.47363e6i			0.198666i
$874$		0			0
$875$	2.06305e7	+	5.68051e6i	0.910940	+	0.250823i
$876$		0			0
$877$			2.46327e7i			1.08146i	0.841195	+	0.540732i	$0.181853\pi$
$877$			2.46327e7i			1.08146i	−0.841195	+	0.540732i	$0.818147\pi$
$878$		0			0
$879$	−9.53436e6			−0.416217
$880$		0			0
$881$	−1.10564e7			−0.479926			−0.239963	−	0.970782i	$-0.577135\pi$
$881$	−1.10564e7			−0.479926			−0.239963	+	0.970782i	$0.577135\pi$
$882$		0			0
$883$		−	3.74671e7i		−	1.61714i	−0.588398	−	0.808572i	$-0.700241\pi$
$883$		−	3.74671e7i		−	1.61714i	0.588398	−	0.808572i	$-0.299759\pi$
$884$		0			0
$885$	1.61026e7	+	1.44606e6i	0.691097	+	0.0620623i
$886$		0			0
$887$			2.96341e7i			1.26469i	0.774689	+	0.632343i	$0.217906\pi$
$887$			2.96341e7i			1.26469i	−0.774689	+	0.632343i	$0.782094\pi$
$888$		0			0
$889$	−6.31668e6			−0.268062
$890$		0			0
$891$	1.59710e6			0.0673966
$892$		0			0
$893$			2.11649e7i			0.888154i
$894$		0			0
$895$	710540.	−	7.91224e6i	0.0296504	−	0.330173i
$896$		0			0
$897$			2.79779e7i			1.16100i
$898$		0			0
$899$	−1.68390e7			−0.694891
$900$		0			0
$901$	2.37663e7			0.975327
$902$		0			0
$903$			2.42262e7i			0.988705i
$904$		0			0
$905$	−1.25395e6	+	1.39634e7i	−0.0508931	+	0.566721i
$906$		0			0
$907$		−	1.94430e6i		−	0.0784773i	−0.999230	−	0.0392387i	$-0.987507\pi$
$907$		−	1.94430e6i		−	0.0784773i	0.999230	−	0.0392387i	$-0.0124933\pi$
$908$		0			0
$909$	−1.70877e7			−0.685920
$910$		0			0
$911$	1.28064e7			0.511245			0.255623	−	0.966777i	$-0.417720\pi$
$911$	1.28064e7			0.511245			0.255623	+	0.966777i	$0.417720\pi$
$912$		0			0
$913$			1.62913e6i			0.0646812i
$914$		0			0
$915$	1.89596e6	+	170262.i	0.0748646	+	0.00672304i
$916$		0			0
$917$		−	1.09139e7i		−	0.428606i
$918$		0			0
$919$	3.58404e7			1.39986			0.699929	−	0.714213i	$-0.253215\pi$
$919$	3.58404e7			1.39986			0.699929	+	0.714213i	$0.253215\pi$
$920$		0			0
$921$	1.22064e7			0.474176
$922$		0			0
$923$		−	2.76398e7i		−	1.06790i
$924$		0			0
$925$	−5.79080e6	+	3.19818e7i	−0.222528	+	1.22899i
$926$		0			0
$927$			1.33083e7i			0.508653i
$928$		0			0
$929$	8.47792e6			0.322292			0.161146	−	0.986931i	$-0.448481\pi$
$929$	8.47792e6			0.322292			0.161146	+	0.986931i	$0.448481\pi$
$930$		0			0
$931$	4.04593e6			0.152983
$932$		0			0
$933$		−	2.36591e7i		−	0.889803i
$934$		0			0
$935$	−5.45600e6	−	489963.i	−0.204101	−	0.0183288i
$936$		0			0
$937$			1.60566e7i			0.597454i	0.954339	+	0.298727i	$0.0965619\pi$
$937$			1.60566e7i			0.597454i	−0.954339	+	0.298727i	$0.903438\pi$
$938$		0			0
$939$	−3.28302e6			−0.121509
$940$		0			0
$941$	−4.24675e7			−1.56344			−0.781722	−	0.623627i	$-0.785658\pi$
$941$	−4.24675e7			−1.56344			−0.781722	+	0.623627i	$0.785658\pi$
$942$		0			0
$943$		−	2.53456e7i		−	0.928159i
$944$		0			0
$945$	2.46884e6	−	2.74918e7i	0.0899318	−	1.00144i
$946$		0			0
$947$		−	2.78877e7i		−	1.01050i	−0.862972	−	0.505252i	$-0.831400\pi$
$947$		−	2.78877e7i		−	1.01050i	0.862972	−	0.505252i	$-0.168600\pi$
$948$		0			0
$949$	−1.76447e7			−0.635988
$950$		0			0
$951$	2.81899e7			1.01075
$952$		0			0
$953$			4.77895e7i			1.70451i	0.523123	+	0.852257i	$0.324767\pi$
$953$			4.77895e7i			1.70451i	−0.523123	+	0.852257i	$0.675233\pi$
$954$		0			0
$955$	1.04736e6	−	1.16629e7i	0.0371610	−	0.413808i
$956$		0			0
$957$		−	8.74584e6i		−	0.308690i
$958$		0			0
$959$	−4.69762e6			−0.164942
$960$		0			0
$961$	−2.40324e7			−0.839439
$962$		0			0
$963$		−	1.09760e7i		−	0.381399i
$964$		0			0
$965$	−1.98598e7	−	1.78347e6i	−0.686527	−	0.0616519i
$966$		0			0
$967$		−	4.51588e7i		−	1.55302i	−0.630107	−	0.776509i	$-0.716989\pi$
$967$		−	4.51588e7i		−	1.55302i	0.630107	−	0.776509i	$-0.283011\pi$
$968$		0			0
$969$	2.44865e7			0.837756
$970$		0			0
$971$	−2.70901e7			−0.922067			−0.461033	−	0.887383i	$-0.652521\pi$
$971$	−2.70901e7			−0.922067			−0.461033	+	0.887383i	$0.652521\pi$
$972$		0			0
$973$			1.64976e7i			0.558647i
$974$		0			0
$975$	−2.51658e7	−	4.55666e6i	−0.847811	−	0.153509i
$976$		0			0
$977$		−	4.34630e7i		−	1.45675i	−0.685181	−	0.728373i	$-0.740277\pi$
$977$		−	4.34630e7i		−	1.45675i	0.685181	−	0.728373i	$-0.259723\pi$
$978$		0			0
$979$	−82600.0			−0.00275438
$980$		0			0
$981$	1.26423e7			0.419425
$982$		0			0
$983$			6.56037e6i			0.216543i	0.994121	+	0.108272i	$0.0345316\pi$
$983$			6.56037e6i			0.216543i	−0.994121	+	0.108272i	$0.965468\pi$
$984$		0			0
$985$	4.81492e6	+	432393.i	0.158124	+	0.0142000i
$986$		0			0
$987$		−	1.28650e7i		−	0.420355i
$988$		0			0
$989$	6.07185e7			1.97392
$990$		0			0
$991$	1.72942e7			0.559391			0.279696	−	0.960089i	$-0.409766\pi$
$991$	1.72942e7			0.559391			0.279696	+	0.960089i	$0.409766\pi$
$992$		0			0
$993$		−	1.30309e7i		−	0.419375i
$994$		0			0
$995$	3.74604e6	−	4.17141e7i	0.119954	−	1.33575i
$996$		0			0
$997$			2.24107e7i			0.714031i	0.934098	+	0.357015i	$0.116206\pi$
$997$			2.24107e7i			0.714031i	−0.934098	+	0.357015i	$0.883794\pi$
$998$		0			0
$999$	4.19254e7			1.32912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.c.a.9.2	yes	2
3.2	odd	2		180.6.d.b.109.1		2
4.3	odd	2		80.6.c.b.49.1		2
5.2	odd	4		100.6.a.d.1.2		2
5.3	odd	4		100.6.a.d.1.1		2
5.4	even	2	inner	20.6.c.a.9.1	✓	2
8.3	odd	2		320.6.c.d.129.2		2
8.5	even	2		320.6.c.e.129.1		2
12.11	even	2		720.6.f.d.289.1		2
15.2	even	4		900.6.a.q.1.1		2
15.8	even	4		900.6.a.q.1.2		2
15.14	odd	2		180.6.d.b.109.2		2
20.3	even	4		400.6.a.s.1.2		2
20.7	even	4		400.6.a.s.1.1		2
20.19	odd	2		80.6.c.b.49.2		2
40.19	odd	2		320.6.c.d.129.1		2
40.29	even	2		320.6.c.e.129.2		2
60.59	even	2		720.6.f.d.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.c.a.9.1	✓	2	5.4	even	2	inner
20.6.c.a.9.2	yes	2	1.1	even	1	trivial
80.6.c.b.49.1		2	4.3	odd	2
80.6.c.b.49.2		2	20.19	odd	2
100.6.a.d.1.1		2	5.3	odd	4
100.6.a.d.1.2		2	5.2	odd	4
180.6.d.b.109.1		2	3.2	odd	2
180.6.d.b.109.2		2	15.14	odd	2
320.6.c.d.129.1		2	40.19	odd	2
320.6.c.d.129.2		2	8.3	odd	2
320.6.c.e.129.1		2	8.5	even	2
320.6.c.e.129.2		2	40.29	even	2
400.6.a.s.1.1		2	20.7	even	4
400.6.a.s.1.2		2	20.3	even	4
720.6.f.d.289.1		2	12.11	even	2
720.6.f.d.289.2		2	60.59	even	2
900.6.a.q.1.1		2	15.2	even	4
900.6.a.q.1.2		2	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	20.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.1355i q^{3} +(-5.00000 + 55.6776i) q^{5} +122.491i q^{7} +119.000 q^{9} -100.000 q^{11} -734.945i q^{13} +(-620.000 - 55.6776i) q^{15} -979.927i q^{17} +2244.00 q^{19} -1364.00 q^{21} +3418.61i q^{23} +(-3075.00 - 556.776i) q^{25} +4031.06i q^{27} +7854.00 q^{29} -2144.00 q^{31} -1113.55i q^{33} +(-6820.00 - 612.454i) q^{35} -10400.6i q^{37} +8184.00 q^{39} -7414.00 q^{41} -17761.2i q^{43} +(-595.000 + 6625.64i) q^{45} +9431.79i q^{47} +1803.00 q^{49} +10912.0 q^{51} +24253.2i q^{53} +(500.000 - 5567.76i) q^{55} +24988.1i q^{57} -25972.0 q^{59} -3058.00 q^{61} +14576.4i q^{63} +(40920.0 + 3674.72i) q^{65} -58784.5i q^{67} -38068.0 q^{69} +37608.0 q^{71} -24008.2i q^{73} +(6200.00 - 34241.8i) q^{75} -12249.1i q^{77} -79728.0 q^{79} -15971.0 q^{81} -16291.3i q^{83} +(54560.0 + 4899.63i) q^{85} +87458.4i q^{87} +826.000 q^{89} +90024.0 q^{91} -23874.6i q^{93} +(-11220.0 + 124941. i) q^{95} +37593.5i q^{97} -11900.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 238 q^{9} - 200 q^{11} - 1240 q^{15} + 4488 q^{19} - 2728 q^{21} - 6150 q^{25} + 15708 q^{29} - 4288 q^{31} - 13640 q^{35} + 16368 q^{39} - 14828 q^{41} - 1190 q^{45} + 3606 q^{49} + 21824 q^{51}+ \cdots - 23800 q^{99}+O(q^{100}) \) 2 * q - 10 * q^5 + 238 * q^9 - 200 * q^11 - 1240 * q^15 + 4488 * q^19 - 2728 * q^21 - 6150 * q^25 + 15708 * q^29 - 4288 * q^31 - 13640 * q^35 + 16368 * q^39 - 14828 * q^41 - 1190 * q^45 + 3606 * q^49 + 21824 * q^51 + 1000 * q^55 - 51944 * q^59 - 6116 * q^61 + 81840 * q^65 - 76136 * q^69 + 75216 * q^71 + 12400 * q^75 - 159456 * q^79 - 31942 * q^81 + 109120 * q^85 + 1652 * q^89 + 180048 * q^91 - 22440 * q^95 - 23800 * q^99

Introduction

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