LMFDB - Embedded newform 180.6.d.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,6,Mod(109,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("180.109");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	180.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$28.8690875663$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 2.78388i$ of defining polynomial
Character	$\chi$	$=$	180.109
Dual form			180.6.d.b.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.00000 - 55.6776i) q^{5} +122.491i q^{7} +O(q^{10})$	$q+(5.00000 - 55.6776i) q^{5} +122.491i q^{7} +100.000 q^{11} -734.945i q^{13} +979.927i q^{17} +2244.00 q^{19} -3418.61i q^{23} +(-3075.00 - 556.776i) q^{25} -7854.00 q^{29} -2144.00 q^{31} +(6820.00 + 612.454i) q^{35} -10400.6i q^{37} +7414.00 q^{41} -17761.2i q^{43} -9431.79i q^{47} +1803.00 q^{49} -24253.2i q^{53} +(500.000 - 5567.76i) q^{55} +25972.0 q^{59} -3058.00 q^{61} +(-40920.0 - 3674.72i) q^{65} -58784.5i q^{67} -37608.0 q^{71} -24008.2i q^{73} +12249.1i q^{77} -79728.0 q^{79} +16291.3i q^{83} +(54560.0 + 4899.63i) q^{85} -826.000 q^{89} +90024.0 q^{91} +(11220.0 - 124941. i) q^{95} +37593.5i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5}+O(q^{10}) $ 2 * q + 10 * q^5	$ 2 q + 10 q^{5} + 200 q^{11} + 4488 q^{19} - 6150 q^{25} - 15708 q^{29} - 4288 q^{31} + 13640 q^{35} + 14828 q^{41} + 3606 q^{49} + 1000 q^{55} + 51944 q^{59} - 6116 q^{61} - 81840 q^{65} - 75216 q^{71} - 159456 q^{79} + 109120 q^{85} - 1652 q^{89} + 180048 q^{91} + 22440 q^{95}+O(q^{100}) $ 2 * q + 10 * q^5 + 200 * q^11 + 4488 * q^19 - 6150 * q^25 - 15708 * q^29 - 4288 * q^31 + 13640 * q^35 + 14828 * q^41 + 3606 * q^49 + 1000 * q^55 + 51944 * q^59 - 6116 * q^61 - 81840 * q^65 - 75216 * q^71 - 159456 * q^79 + 109120 * q^85 - 1652 * q^89 + 180048 * q^91 + 22440 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		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
$9$		0			0
$10$		0			0
$11$	100.000			0.249183			0.124591	−	0.992208i	$-0.460238\pi$
$11$	100.000			0.249183			0.124591	+	0.992208i	$0.460238\pi$
$12$		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
$15$		0			0
$16$		0			0
$17$			979.927i			0.822377i	0.911550	+	0.411189i	$0.134886\pi$
$17$			979.927i			0.822377i	−0.911550	+	0.411189i	$0.865114\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$		−	3418.61i		−	1.34750i	−0.738958	−	0.673751i	$-0.764682\pi$
$23$		−	3418.61i		−	1.34750i	0.738958	−	0.673751i	$-0.235318\pi$
$24$		0			0
$25$	−3075.00	−	556.776i	−0.984000	−	0.178168i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7854.00			−1.73419			−0.867093	−	0.498146i	$-0.834015\pi$
$29$	−7854.00			−1.73419			−0.867093	+	0.498146i	$0.834015\pi$
$30$		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
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	7414.00			0.688800			0.344400	−	0.938823i	$-0.388082\pi$
$41$	7414.00			0.688800			0.344400	+	0.938823i	$0.388082\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$		−	9431.79i		−	0.622801i	−0.950279	−	0.311401i	$-0.899202\pi$
$47$		−	9431.79i		−	0.622801i	0.950279	−	0.311401i	$-0.100798\pi$
$48$		0			0
$49$	1803.00			0.107277
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	24253.2i		−	1.18598i	−0.805208	−	0.592992i	$-0.797946\pi$
$53$		−	24253.2i		−	1.18598i	0.805208	−	0.592992i	$-0.202054\pi$
$54$		0			0
$55$	500.000	−	5567.76i	0.0222876	−	0.248184i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	25972.0			0.971349			0.485675	−	0.874140i	$-0.338574\pi$
$59$	25972.0			0.971349			0.485675	+	0.874140i	$0.338574\pi$
$60$		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
$63$		0			0
$64$		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
$69$		0			0
$70$		0			0
$71$	−37608.0			−0.885389			−0.442695	−	0.896672i	$-0.645977\pi$
$71$	−37608.0			−0.885389			−0.442695	+	0.896672i	$0.645977\pi$
$72$		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
$75$		0			0
$76$		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
$81$		0			0
$82$		0			0
$83$			16291.3i			0.259573i	0.991542	+	0.129787i	$0.0414292\pi$
$83$			16291.3i			0.259573i	−0.991542	+	0.129787i	$0.958571\pi$
$84$		0			0
$85$	54560.0	+	4899.63i	0.819081	+	0.0735557i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−826.000			−0.0110536			−0.00552682	−	0.999985i	$-0.501759\pi$
$89$	−826.000			−0.0110536			−0.00552682	+	0.999985i	$0.501759\pi$
$90$		0			0
$91$	90024.0			1.13961
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$	143594.			1.40066			0.700330	−	0.713819i	$-0.253036\pi$
$101$	143594.			1.40066			0.700330	+	0.713819i	$0.253036\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$			92235.6i			0.778824i	0.921064	+	0.389412i	$0.127322\pi$
$107$			92235.6i			0.778824i	−0.921064	+	0.389412i	$0.872678\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$			113048.i			0.832849i	0.909170	+	0.416425i	$0.136717\pi$
$113$			113048.i			0.832849i	−0.909170	+	0.416425i	$0.863283\pi$
$114$		0			0
$115$	−190340.	−	17093.0i	−1.34210	−	0.120524i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−120032.			−0.777015
$120$		0			0
$121$	−151051.			−0.937908
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	89100.0			0.453628			0.226814	−	0.973938i	$-0.427169\pi$
$131$	89100.0			0.453628			0.226814	+	0.973938i	$0.427169\pi$
$132$		0			0
$133$			274869.i			1.34740i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	38350.8i		−	0.174571i	−0.996183	−	0.0872856i	$-0.972181\pi$
$137$		−	38350.8i		−	0.174571i	0.996183	−	0.0872856i	$-0.0278193\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$		−	73494.5i		−	0.300549i
$144$		0			0
$145$	−39270.0	+	437292.i	−0.155110	+	1.72724i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−248006.			−0.915159			−0.457579	−	0.889169i	$-0.651283\pi$
$149$	−248006.			−0.915159			−0.457579	+	0.889169i	$0.651283\pi$
$150$		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
$153$		0			0
$154$		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
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$			189983.i			0.527138i	0.964641	+	0.263569i	$0.0848996\pi$
$167$			189983.i			0.527138i	−0.964641	+	0.263569i	$0.915100\pi$
$168$		0			0
$169$	−168851.			−0.454765
$170$		0			0
$171$		0			0
$172$		0			0
$173$			81088.9i			0.205990i	0.994682	+	0.102995i	$0.0328426\pi$
$173$			81088.9i			0.205990i	−0.994682	+	0.102995i	$0.967157\pi$
$174$		0			0
$175$	68200.0	−	376659.i	0.168341	−	0.929723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	142108.			0.331502			0.165751	−	0.986168i	$-0.446995\pi$
$179$	142108.			0.331502			0.165751	+	0.986168i	$0.446995\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	−579080.	−	52002.9i	−1.24397	−	0.111712i
$186$		0			0
$187$			97992.7i			0.204922i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	209472.			0.415473			0.207736	−	0.978185i	$-0.433390\pi$
$191$	209472.			0.415473			0.207736	+	0.978185i	$0.433390\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$			86478.5i			0.158761i	0.996844	+	0.0793803i	$0.0252941\pi$
$197$			86478.5i			0.158761i	−0.996844	+	0.0793803i	$0.974706\pi$
$198$		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
$201$		0			0
$202$		0			0
$203$		−	962043.i		−	1.63853i
$204$		0			0
$205$	37070.0	−	412794.i	0.0616081	−	0.686039i
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	−988900.	−	88805.8i	−1.45900	−	0.131022i
$216$		0			0
$217$		−	262620.i		−	0.378599i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$		−	978334.i		−	1.26015i	−0.776534	−	0.630075i	$-0.783024\pi$
$227$		−	978334.i		−	1.26015i	0.776534	−	0.630075i	$-0.216976\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$			1.55465e6i			1.87605i	0.346571	+	0.938024i	$0.387346\pi$
$233$			1.55465e6i			1.87605i	−0.346571	+	0.938024i	$0.612654\pi$
$234$		0			0
$235$	−525140.	−	47159.0i	−0.620305	−	0.0557051i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−374704.			−0.424320			−0.212160	−	0.977235i	$-0.568050\pi$
$239$	−374704.			−0.424320			−0.212160	+	0.977235i	$0.568050\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	9015.00	−	100387.i	0.00959512	−	0.106847i
$246$		0			0
$247$		−	1.64922e6i		−	1.72003i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.72050e6			1.72373			0.861867	−	0.507134i	$-0.169295\pi$
$251$	1.72050e6			1.72373			0.861867	+	0.507134i	$0.169295\pi$
$252$		0			0
$253$		−	341861.i		−	0.335775i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.55220e6i		−	1.46594i	−0.680262	−	0.732969i	$-0.738134\pi$
$257$		−	1.55220e6i		−	1.46594i	0.680262	−	0.732969i	$-0.261866\pi$
$258$		0			0
$259$	1.27398e6			1.18008
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	407772.i		−	0.363520i	−0.983343	−	0.181760i	$-0.941821\pi$
$263$		−	407772.i		−	0.363520i	0.983343	−	0.181760i	$-0.0581793\pi$
$264$		0			0
$265$	−1.35036e6	−	121266.i	−1.18123	−	0.106078i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.82710e6			−1.53951			−0.769754	−	0.638340i	$-0.779621\pi$
$269$	−1.82710e6			−1.53951			−0.769754	+	0.638340i	$0.779621\pi$
$270$		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
$273$		0			0
$274$		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
$279$		0			0
$280$		0			0
$281$	1.22093e6			0.922415			0.461208	−	0.887292i	$-0.347416\pi$
$281$	1.22093e6			0.922415			0.461208	+	0.887292i	$0.347416\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$			908147.i			0.650806i
$288$		0			0
$289$	459601.			0.323695
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	856211.i		−	0.582655i	−0.956623	−	0.291328i	$-0.905903\pi$
$293$		−	856211.i		−	0.582655i	0.956623	−	0.291328i	$-0.0940970\pi$
$294$		0			0
$295$	129860.	−	1.44606e6i	0.0868801	−	0.967456i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.51249e6			−1.62527
$300$		0			0
$301$	2.17558e6			1.38407
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	2.12465e6			1.24562			0.622811	−	0.782373i	$-0.285991\pi$
$311$	2.12465e6			1.24562			0.622811	+	0.782373i	$0.285991\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$			2.53153e6i			1.41493i	0.706749	+	0.707465i	$0.250161\pi$
$317$			2.53153e6i			1.41493i	−0.706749	+	0.707465i	$0.749839\pi$
$318$		0			0
$319$	−785400.			−0.432130
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.19896e6i			1.17276i
$324$		0			0
$325$	−409200.	+	2.25996e6i	−0.214895	+	1.18684i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$	−214400.			−0.0998479
$342$		0			0
$343$			2.27955e6i			1.04620i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1.63342e6i		−	0.728237i	−0.931353	−	0.364119i	$-0.881370\pi$
$347$		−	1.63342e6i		−	0.728237i	0.931353	−	0.364119i	$-0.118630\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$		−	1.80859e6i		−	0.772508i	−0.922392	−	0.386254i	$-0.873769\pi$
$353$		−	1.80859e6i		−	0.772508i	0.922392	−	0.386254i	$-0.126231\pi$
$354$		0			0
$355$	−188040.	+	2.09392e6i	−0.0791916	+	0.881841i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.50674e6			1.84555			0.922777	−	0.385334i	$-0.125914\pi$
$359$	4.50674e6			1.84555			0.922777	+	0.385334i	$0.125914\pi$
$360$		0			0
$361$	2.55944e6			1.03366
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		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
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$		−	2.86948e6i		−	0.999554i	−0.866154	−	0.499777i	$-0.833415\pi$
$383$		−	2.86948e6i		−	0.999554i	0.866154	−	0.499777i	$-0.166585\pi$
$384$		0			0
$385$	682000.	+	61245.4i	0.234494	+	0.0210582i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−311926.			−0.104515			−0.0522574	−	0.998634i	$-0.516642\pi$
$389$	−311926.			−0.104515			−0.0522574	+	0.998634i	$0.516642\pi$
$390$		0			0
$391$	3.34998e6			1.10816
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	−2770.00			−0.000860238			−0.000430119	−	1.00000i	$-0.500137\pi$
$401$	−2770.00			−0.000860238			−0.000430119		1.00000i	$0.500137\pi$
$402$		0			0
$403$			1.57572e6i			0.483300i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			3.18133e6i			0.917770i
$414$		0			0
$415$	907060.	+	81456.4i	0.258533	+	0.0232169i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.10120e6			−1.41951			−0.709754	−	0.704450i	$-0.751194\pi$
$419$	−5.10120e6			−1.41951			−0.709754	+	0.704450i	$0.751194\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$	545600.	−	3.01327e6i	0.146522	−	0.809219i
$426$		0			0
$427$		−	374577.i		−	0.0994194i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−918896.			−0.238272			−0.119136	−	0.992878i	$-0.538012\pi$
$431$	−918896.			−0.238272			−0.119136	+	0.992878i	$0.538012\pi$
$432$		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
$435$		0			0
$436$		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
$441$		0			0
$442$		0			0
$443$			2.53092e6i			0.612729i	0.951914	+	0.306365i	$0.0991126\pi$
$443$			2.53092e6i			0.612729i	−0.951914	+	0.306365i	$0.900887\pi$
$444$		0			0
$445$	−4130.00	+	45989.7i	−0.000988667	+	0.0110093i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.17619e6			−1.21170			−0.605849	−	0.795579i	$-0.707167\pi$
$449$	−5.17619e6			−1.21170			−0.605849	+	0.795579i	$0.707167\pi$
$450$		0			0
$451$	741400.			0.171637
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	2.64957e6			0.580662			0.290331	−	0.956926i	$-0.406235\pi$
$461$	2.64957e6			0.580662			0.290331	+	0.956926i	$0.406235\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$			6.62135e6i			1.40493i	0.711719	+	0.702465i	$0.247917\pi$
$467$			6.62135e6i			1.40493i	−0.711719	+	0.702465i	$0.752083\pi$
$468$		0			0
$469$	7.20056e6			1.51159
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.77612e6i		−	0.365022i
$474$		0			0
$475$	−6.90030e6	−	1.24941e6i	−1.40325	−	0.254080i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.89322e6			1.37272			0.686362	−	0.727260i	$-0.259207\pi$
$479$	6.89322e6			1.37272			0.686362	+	0.727260i	$0.259207\pi$
$480$		0			0
$481$	−7.64386e6			−1.50643
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	−5.88390e6			−1.10144			−0.550721	−	0.834689i	$-0.685647\pi$
$491$	−5.88390e6			−1.10144			−0.550721	+	0.834689i	$0.685647\pi$
$492$		0			0
$493$		−	7.69634e6i		−	1.42616i
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		0			0
$503$			469262.i			0.0826981i	0.999145	+	0.0413491i	$0.0131656\pi$
$503$			469262.i			0.0826981i	−0.999145	+	0.0413491i	$0.986834\pi$
$504$		0			0
$505$	717970.	−	7.99498e6i	0.125279	−	1.39505i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−294414.			−0.0503691			−0.0251845	−	0.999683i	$-0.508017\pi$
$509$	−294414.			−0.0503691			−0.0251845	+	0.999683i	$0.508017\pi$
$510$		0			0
$511$	2.94078e6			0.498208
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.22666e6	+	559171.i	1.03452	+	0.0929023i
$516$		0			0
$517$		−	943179.i		−	0.155191i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.10025e6			1.14599			0.572993	−	0.819560i	$-0.305782\pi$
$521$	7.10025e6			1.14599			0.572993	+	0.819560i	$0.305782\pi$
$522$		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
$525$		0			0
$526$		0			0
$527$		−	2.10096e6i		−	0.329528i
$528$		0			0
$529$	−5.25053e6			−0.815763
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.44888e6i		−	0.830786i
$534$		0			0
$535$	5.13546e6	+	461178.i	0.775702	+	0.0696601i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$	−1.76244e7			−2.47306
$552$		0			0
$553$		−	9.76595e6i		−	1.35801i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.42852e6i			0.468240i	0.972208	+	0.234120i	$0.0752209\pi$
$557$			3.42852e6i			0.468240i	−0.972208	+	0.234120i	$0.924779\pi$
$558$		0			0
$559$	−1.30535e7			−1.76684
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	5.77899e6i		−	0.768389i	−0.923252	−	0.384195i	$-0.874479\pi$
$563$		−	5.77899e6i		−	0.768389i	0.923252	−	0.384195i	$-0.125521\pi$
$564$		0			0
$565$	6.29424e6	+	565239.i	0.829511	+	0.0744923i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.89257e6			−0.504029			−0.252015	−	0.967723i	$-0.581093\pi$
$569$	−3.89257e6			−0.504029			−0.252015	+	0.967723i	$0.581093\pi$
$570$		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
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	−1.99553e6			−0.245255
$582$		0			0
$583$		−	2.42532e6i		−	0.295527i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	5.16997e6i		−	0.619288i	−0.950853	−	0.309644i	$-0.899790\pi$
$587$		−	5.16997e6i		−	0.619288i	0.950853	−	0.309644i	$-0.100210\pi$
$588$		0			0
$589$	−4.81114e6			−0.571425
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.58484e7i			1.85075i	0.379055	+	0.925374i	$0.376249\pi$
$593$			1.58484e7i			1.85075i	−0.379055	+	0.925374i	$0.623751\pi$
$594$		0			0
$595$	−600160.	+	6.68310e6i	−0.0694984	+	0.773901i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.66146e7			1.89201			0.946004	−	0.324156i	$-0.105080\pi$
$599$	1.66146e7			1.89201			0.946004	+	0.324156i	$0.105080\pi$
$600$		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
$603$		0			0
$604$		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
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$		−	9.66355e6i		−	1.02194i	−0.859600	−	0.510968i	$-0.829287\pi$
$617$		−	9.66355e6i		−	1.02194i	0.859600	−	0.510968i	$-0.170713\pi$
$618$		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
$621$		0			0
$622$		0			0
$623$		−	101177.i		−	0.0104439i
$624$		0			0
$625$	9.14562e6	+	3.42418e6i	0.936512	+	0.350636i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	2.87122e6	+	257843.i	0.282574	+	0.0253759i
$636$		0			0
$637$		−	1.32511e6i		−	0.129390i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.44950e7			−1.39340			−0.696698	−	0.717365i	$-0.745348\pi$
$641$	−1.44950e7			−1.39340			−0.696698	+	0.717365i	$0.745348\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$			9.64592e6i			0.905905i	0.891535	+	0.452953i	$0.149629\pi$
$647$			9.64592e6i			0.905905i	−0.891535	+	0.452953i	$0.850371\pi$
$648$		0			0
$649$	2.59720e6			0.242044
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.92807e7i		−	1.76945i	−0.466111	−	0.884726i	$-0.654345\pi$
$653$		−	1.92807e7i		−	1.76945i	0.466111	−	0.884726i	$-0.345655\pi$
$654$		0			0
$655$	445500.	−	4.96088e6i	0.0405737	−	0.451809i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.70592e6			0.870609			0.435304	−	0.900283i	$-0.356641\pi$
$659$	9.70592e6			0.870609			0.435304	+	0.900283i	$0.356641\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	1.53041e7	+	1.37435e6i	1.34200	+	0.120515i
$666$		0			0
$667$			2.68497e7i			2.33682i
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$			8.42565e6i			0.706532i	0.935523	+	0.353266i	$0.114929\pi$
$677$			8.42565e6i			0.706532i	−0.935523	+	0.353266i	$0.885071\pi$
$678$		0			0
$679$	−4.60486e6			−0.383303
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.64100e7i			1.34603i	0.739627	+	0.673017i	$0.235002\pi$
$683$			1.64100e7i			1.34603i	−0.739627	+	0.673017i	$0.764998\pi$
$684$		0			0
$685$	−2.13528e6	−	191754.i	−0.173872	−	0.0156141i
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	673420.	−	7.49889e6i	0.0528840	−	0.588891i
$696$		0			0
$697$			7.26518e6i			0.566453i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.04707e7			−1.57339			−0.786696	−	0.617340i	$-0.788210\pi$
$701$	−2.04707e7			−1.57339			−0.786696	+	0.617340i	$0.788210\pi$
$702$		0			0
$703$		−	2.33389e7i		−	1.78112i
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$			7.32949e6i			0.539946i
$714$		0			0
$715$	−4.09200e6	−	367472.i	−0.299344	−	0.0268819i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.61006e6			−0.621132			−0.310566	−	0.950552i	$-0.600519\pi$
$719$	−8.61006e6			−0.621132			−0.310566	+	0.950552i	$0.600519\pi$
$720$		0			0
$721$	−1.36987e7			−0.981386
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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
$735$		0			0
$736$		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
$741$		0			0
$742$		0			0
$743$		−	1.72675e6i		−	0.114751i	−0.998353	−	0.0573757i	$-0.981727\pi$
$743$		−	1.72675e6i		−	0.114751i	0.998353	−	0.0573757i	$-0.0182733\pi$
$744$		0			0
$745$	−1.24003e6	+	1.38084e7i	−0.0818543	+	0.911491i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−1.40499e7			−0.879450			−0.439725	−	0.898133i	$-0.644924\pi$
$761$	−1.40499e7			−0.879450			−0.439725	+	0.898133i	$0.644924\pi$
$762$		0			0
$763$			1.30132e7i			0.809230i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$			6.34625e6i			0.382004i	0.981590	+	0.191002i	$0.0611738\pi$
$773$			6.34625e6i			0.382004i	−0.981590	+	0.191002i	$0.938826\pi$
$774$		0			0
$775$	6.59280e6	+	1.19373e6i	0.394290	+	0.0713923i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.66370e7			0.982272
$780$		0			0
$781$	−3.76080e6			−0.220624
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	−1.38473e7			−0.786909
$792$		0			0
$793$			2.24746e6i			0.126914i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	8.06932e6i		−	0.449978i	−0.974361	−	0.224989i	$-0.927765\pi$
$797$		−	8.06932e6i		−	0.449978i	0.974361	−	0.224989i	$-0.0722346\pi$
$798$		0			0
$799$	9.24246e6			0.512178
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	2.40082e6i		−	0.131393i
$804$		0			0
$805$	2.09374e6	−	2.33149e7i	0.113876	−	1.26807i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.94554e7			1.04513			0.522564	−	0.852600i	$-0.324976\pi$
$809$	1.94554e7			1.04513			0.522564	+	0.852600i	$0.324976\pi$
$810$		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
$813$		0			0
$814$		0			0
$815$	−2.21309e7	−	1.98741e6i	−1.16709	−	0.104808i
$816$		0			0
$817$		−	3.98561e7i		−	2.08900i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.48431e6			0.283965			0.141982	−	0.989869i	$-0.454652\pi$
$821$	5.48431e6			0.283965			0.141982	+	0.989869i	$0.454652\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$			4.19715e6i			0.213398i	0.994291	+	0.106699i	$0.0340281\pi$
$827$			4.19715e6i			0.213398i	−0.994291	+	0.106699i	$0.965972\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$			1.76681e6i			0.0882220i
$834$		0			0
$835$	1.05778e7	+	949916.i	0.525025	+	0.0471486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.27752e7			0.626560			0.313280	−	0.949661i	$-0.398572\pi$
$839$	1.27752e7			0.626560			0.313280	+	0.949661i	$0.398572\pi$
$840$		0			0
$841$	4.11742e7			2.00740
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−844255.	+	9.40123e6i	−0.0406754	+	0.452942i
$846$		0			0
$847$		−	1.85024e7i		−	0.886173i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$			1.61585e7i			0.751535i	0.926714	+	0.375767i	$0.122621\pi$
$857$			1.61585e7i			0.751535i	−0.926714	+	0.375767i	$0.877379\pi$
$858$		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
$861$		0			0
$862$		0			0
$863$		−	2.51788e7i		−	1.15082i	−0.817864	−	0.575412i	$-0.804842\pi$
$863$		−	2.51788e7i		−	1.15082i	0.817864	−	0.575412i	$-0.195158\pi$
$864$		0			0
$865$	4.51484e6	+	405445.i	0.205164	+	0.0184243i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.97280e6			−0.358147
$870$		0			0
$871$	−4.32033e7			−1.92962
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$	1.10564e7			0.479926			0.239963	−	0.970782i	$-0.422865\pi$
$881$	1.10564e7			0.479926			0.239963	+	0.970782i	$0.422865\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$		−	2.96341e7i		−	1.26469i	−0.774689	−	0.632343i	$-0.782094\pi$
$887$		−	2.96341e7i		−	1.26469i	0.774689	−	0.632343i	$-0.217906\pi$
$888$		0			0
$889$	−6.31668e6			−0.268062
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.11649e7i		−	0.888154i
$894$		0			0
$895$	710540.	−	7.91224e6i	0.0296504	−	0.330173i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.68390e7			0.694891
$900$		0			0
$901$	2.37663e7			0.975327
$902$		0			0
$903$		0			0
$904$		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
$909$		0			0
$910$		0			0
$911$	−1.28064e7			−0.511245			−0.255623	−	0.966777i	$-0.582280\pi$
$911$	−1.28064e7			−0.511245			−0.255623	+	0.966777i	$0.582280\pi$
$912$		0			0
$913$			1.62913e6i			0.0646812i
$914$		0			0
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$			2.76398e7i			1.06790i
$924$		0			0
$925$	−5.79080e6	+	3.19818e7i	−0.222528	+	1.22899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.47792e6			−0.322292			−0.161146	−	0.986931i	$-0.551519\pi$
$929$	−8.47792e6			−0.322292			−0.161146	+	0.986931i	$0.551519\pi$
$930$		0			0
$931$	4.04593e6			0.152983
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	4.24675e7			1.56344			0.781722	−	0.623627i	$-0.214342\pi$
$941$	4.24675e7			1.56344			0.781722	+	0.623627i	$0.214342\pi$
$942$		0			0
$943$		−	2.53456e7i		−	0.928159i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.78877e7i			1.01050i	0.862972	+	0.505252i	$0.168600\pi$
$947$			2.78877e7i			1.01050i	−0.862972	+	0.505252i	$0.831400\pi$
$948$		0			0
$949$	−1.76447e7			−0.635988
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	4.77895e7i		−	1.70451i	−0.523123	−	0.852257i	$-0.675233\pi$
$953$		−	4.77895e7i		−	1.70451i	0.523123	−	0.852257i	$-0.324767\pi$
$954$		0			0
$955$	1.04736e6	−	1.16629e7i	0.0371610	−	0.413808i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.69762e6			0.164942
$960$		0			0
$961$	−2.40324e7			−0.839439
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	2.70901e7			0.922067			0.461033	−	0.887383i	$-0.347479\pi$
$971$	2.70901e7			0.922067			0.461033	+	0.887383i	$0.347479\pi$
$972$		0			0
$973$			1.64976e7i			0.558647i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.34630e7i			1.45675i	0.685181	+	0.728373i	$0.259723\pi$
$977$			4.34630e7i			1.45675i	−0.685181	+	0.728373i	$0.740277\pi$
$978$		0			0
$979$	−82600.0			−0.00275438
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	6.56037e6i		−	0.216543i	−0.994121	−	0.108272i	$-0.965468\pi$
$983$		−	6.56037e6i		−	0.216543i	0.994121	−	0.108272i	$-0.0345316\pi$
$984$		0			0
$985$	4.81492e6	+	432393.i	0.158124	+	0.0142000i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		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
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.c.a.9.1	✓	2	15.14	odd	2
20.6.c.a.9.2	yes	2	3.2	odd	2
80.6.c.b.49.1		2	12.11	even	2
80.6.c.b.49.2		2	60.59	even	2
100.6.a.d.1.1		2	15.8	even	4
100.6.a.d.1.2		2	15.2	even	4
180.6.d.b.109.1		2	1.1	even	1	trivial
180.6.d.b.109.2		2	5.4	even	2	inner
320.6.c.d.129.1		2	120.59	even	2
320.6.c.d.129.2		2	24.11	even	2
320.6.c.e.129.1		2	24.5	odd	2
320.6.c.e.129.2		2	120.29	odd	2
400.6.a.s.1.1		2	60.47	odd	4
400.6.a.s.1.2		2	60.23	odd	4
720.6.f.d.289.1		2	4.3	odd	2
720.6.f.d.289.2		2	20.19	odd	2
900.6.a.q.1.1		2	5.2	odd	4
900.6.a.q.1.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	180.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(5.00000 - 55.6776i) q^{5} +122.491i q^{7} +O(q^{10})\)	\(q+(5.00000 - 55.6776i) q^{5} +122.491i q^{7} +100.000 q^{11} -734.945i q^{13} +979.927i q^{17} +2244.00 q^{19} -3418.61i q^{23} +(-3075.00 - 556.776i) q^{25} -7854.00 q^{29} -2144.00 q^{31} +(6820.00 + 612.454i) q^{35} -10400.6i q^{37} +7414.00 q^{41} -17761.2i q^{43} -9431.79i q^{47} +1803.00 q^{49} -24253.2i q^{53} +(500.000 - 5567.76i) q^{55} +25972.0 q^{59} -3058.00 q^{61} +(-40920.0 - 3674.72i) q^{65} -58784.5i q^{67} -37608.0 q^{71} -24008.2i q^{73} +12249.1i q^{77} -79728.0 q^{79} +16291.3i q^{83} +(54560.0 + 4899.63i) q^{85} -826.000 q^{89} +90024.0 q^{91} +(11220.0 - 124941. i) q^{95} +37593.5i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5}+O(q^{10}) \) 2 * q + 10 * q^5	\( 2 q + 10 q^{5} + 200 q^{11} + 4488 q^{19} - 6150 q^{25} - 15708 q^{29} - 4288 q^{31} + 13640 q^{35} + 14828 q^{41} + 3606 q^{49} + 1000 q^{55} + 51944 q^{59} - 6116 q^{61} - 81840 q^{65} - 75216 q^{71} - 159456 q^{79} + 109120 q^{85} - 1652 q^{89} + 180048 q^{91} + 22440 q^{95}+O(q^{100}) \) 2 * q + 10 * q^5 + 200 * q^11 + 4488 * q^19 - 6150 * q^25 - 15708 * q^29 - 4288 * q^31 + 13640 * q^35 + 14828 * q^41 + 3606 * q^49 + 1000 * q^55 + 51944 * q^59 - 6116 * q^61 - 81840 * q^65 - 75216 * q^71 - 159456 * q^79 + 109120 * q^85 - 1652 * q^89 + 180048 * q^91 + 22440 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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