LMFDB - Embedded newform 1620.3.b.a.809.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.15
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.78224 - 4.15441i) q^{5} +0.702000i q^{7} +O(q^{10})$	$q+(2.78224 - 4.15441i) q^{5} +0.702000i q^{7} -0.462564i q^{11} -14.1065i q^{13} +14.3982 q^{17} +5.37885 q^{19} -41.6369 q^{23} +(-9.51832 - 23.1171i) q^{25} -9.34749i q^{29} +21.8753 q^{31} +(2.91640 + 1.95313i) q^{35} -34.0899i q^{37} -22.3653i q^{41} +55.9804i q^{43} -7.64552 q^{47} +48.5072 q^{49} -36.2290 q^{53} +(-1.92168 - 1.28696i) q^{55} -35.0730i q^{59} -58.8745 q^{61} +(-58.6043 - 39.2477i) q^{65} -61.8925i q^{67} -43.6717i q^{71} +50.7423i q^{73} +0.324719 q^{77} -96.1799 q^{79} +26.0085 q^{83} +(40.0592 - 59.8162i) q^{85} -125.106i q^{89} +9.90277 q^{91} +(14.9652 - 22.3460i) q^{95} +117.788i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) $ 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	2.78224	−	4.15441i	0.556447	−	0.830883i
$5$	2.78224	−	4.15441i	0.556447	−	0.830883i
$6$		0			0
$7$			0.702000i			0.100286i	0.998742	+	0.0501428i	$0.0159677\pi$
$7$			0.702000i			0.100286i	−0.998742	+	0.0501428i	$0.984032\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.462564i		−	0.0420512i	−0.999779	−	0.0210256i	$-0.993307\pi$
$11$		−	0.462564i		−	0.0420512i	0.999779	−	0.0210256i	$-0.00669316\pi$
$12$		0			0
$13$		−	14.1065i		−	1.08512i	−0.840018	−	0.542558i	$-0.817456\pi$
$13$		−	14.1065i		−	1.08512i	0.840018	−	0.542558i	$-0.182544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.3982			0.846954			0.423477	−	0.905907i	$-0.360809\pi$
$17$	14.3982			0.846954			0.423477	+	0.905907i	$0.360809\pi$
$18$		0			0
$19$	5.37885			0.283098			0.141549	−	0.989931i	$-0.454792\pi$
$19$	5.37885			0.283098			0.141549	+	0.989931i	$0.454792\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−41.6369			−1.81030			−0.905149	−	0.425094i	$-0.860241\pi$
$23$	−41.6369			−1.81030			−0.905149	+	0.425094i	$0.860241\pi$
$24$		0			0
$25$	−9.51832	−	23.1171i	−0.380733	−	0.924685i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.34749i		−	0.322327i	−0.986928	−	0.161164i	$-0.948475\pi$
$29$		−	9.34749i		−	0.322327i	0.986928	−	0.161164i	$-0.0515247\pi$
$30$		0			0
$31$	21.8753			0.705654			0.352827	−	0.935689i	$-0.385220\pi$
$31$	21.8753			0.705654			0.352827	+	0.935689i	$0.385220\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.91640	+	1.95313i	0.0833256	+	0.0558037i
$36$		0			0
$37$		−	34.0899i		−	0.921348i	−0.887569	−	0.460674i	$-0.847608\pi$
$37$		−	34.0899i		−	0.921348i	0.887569	−	0.460674i	$-0.152392\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	22.3653i		−	0.545496i	−0.962086	−	0.272748i	$-0.912067\pi$
$41$		−	22.3653i		−	0.545496i	0.962086	−	0.272748i	$-0.0879325\pi$
$42$		0			0
$43$			55.9804i			1.30187i	0.759133	+	0.650935i	$0.225623\pi$
$43$			55.9804i			1.30187i	−0.759133	+	0.650935i	$0.774377\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.64552			−0.162671			−0.0813354	−	0.996687i	$-0.525918\pi$
$47$	−7.64552			−0.162671			−0.0813354	+	0.996687i	$0.525918\pi$
$48$		0			0
$49$	48.5072			0.989943
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−36.2290			−0.683566			−0.341783	−	0.939779i	$-0.611031\pi$
$53$	−36.2290			−0.683566			−0.341783	+	0.939779i	$0.611031\pi$
$54$		0			0
$55$	−1.92168	−	1.28696i	−0.0349397	−	0.0233993i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	35.0730i		−	0.594457i	−0.954806	−	0.297229i	$-0.903938\pi$
$59$		−	35.0730i		−	0.594457i	0.954806	−	0.297229i	$-0.0960624\pi$
$60$		0			0
$61$	−58.8745			−0.965155			−0.482578	−	0.875853i	$-0.660299\pi$
$61$	−58.8745			−0.965155			−0.482578	+	0.875853i	$0.660299\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−58.6043	−	39.2477i	−0.901605	−	0.603810i
$66$		0			0
$67$		−	61.8925i		−	0.923769i	−0.886940	−	0.461884i	$-0.847174\pi$
$67$		−	61.8925i		−	0.923769i	0.886940	−	0.461884i	$-0.152826\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	43.6717i		−	0.615095i	−0.951533	−	0.307547i	$-0.900492\pi$
$71$		−	43.6717i		−	0.615095i	0.951533	−	0.307547i	$-0.0995083\pi$
$72$		0			0
$73$			50.7423i			0.695101i	0.937661	+	0.347550i	$0.112986\pi$
$73$			50.7423i			0.695101i	−0.937661	+	0.347550i	$0.887014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.324719			0.00421714
$78$		0			0
$79$	−96.1799			−1.21747			−0.608734	−	0.793375i	$-0.708322\pi$
$79$	−96.1799			−1.21747			−0.608734	+	0.793375i	$0.708322\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	26.0085			0.313355			0.156678	−	0.987650i	$-0.449922\pi$
$83$	26.0085			0.313355			0.156678	+	0.987650i	$0.449922\pi$
$84$		0			0
$85$	40.0592	−	59.8162i	0.471285	−	0.703720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	125.106i		−	1.40568i	−0.711347	−	0.702841i	$-0.751914\pi$
$89$		−	125.106i		−	1.40568i	0.711347	−	0.702841i	$-0.248086\pi$
$90$		0			0
$91$	9.90277			0.108822
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.9652	−	22.3460i	0.157529	−	0.235221i
$96$		0			0
$97$			117.788i			1.21431i	0.794582	+	0.607157i	$0.207690\pi$
$97$			117.788i			1.21431i	−0.794582	+	0.607157i	$0.792310\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	129.551i		−	1.28268i	−0.767256	−	0.641341i	$-0.778379\pi$
$101$		−	129.551i		−	1.28268i	0.767256	−	0.641341i	$-0.221621\pi$
$102$		0			0
$103$			35.6070i			0.345699i	0.984948	+	0.172850i	$0.0552974\pi$
$103$			35.6070i			0.345699i	−0.984948	+	0.172850i	$0.944703\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−132.242			−1.23591			−0.617953	−	0.786215i	$-0.712038\pi$
$107$	−132.242			−1.23591			−0.617953	+	0.786215i	$0.712038\pi$
$108$		0			0
$109$	16.9302			0.155323			0.0776616	−	0.996980i	$-0.475255\pi$
$109$	16.9302			0.155323			0.0776616	+	0.996980i	$0.475255\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	33.9494			0.300437			0.150219	−	0.988653i	$-0.452002\pi$
$113$	33.9494			0.300437			0.150219	+	0.988653i	$0.452002\pi$
$114$		0			0
$115$	−115.844	+	172.977i	−1.00734	+	1.50415i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.1075i			0.0849373i
$120$		0			0
$121$	120.786			0.998232
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−122.520	−	24.7742i	−0.980163	−	0.198194i
$126$		0			0
$127$			82.5012i			0.649616i	0.945780	+	0.324808i	$0.105300\pi$
$127$			82.5012i			0.649616i	−0.945780	+	0.324808i	$0.894700\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	68.2493i		−	0.520987i	−0.965476	−	0.260494i	$-0.916115\pi$
$131$		−	68.2493i		−	0.520987i	0.965476	−	0.260494i	$-0.0838853\pi$
$132$		0			0
$133$			3.77595i			0.0283906i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−170.934			−1.24769			−0.623847	−	0.781547i	$-0.714431\pi$
$137$	−170.934			−1.24769			−0.623847	+	0.781547i	$0.714431\pi$
$138$		0			0
$139$	217.137			1.56214			0.781070	−	0.624444i	$-0.214674\pi$
$139$	217.137			1.56214			0.781070	+	0.624444i	$0.214674\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.52516			−0.0456305
$144$		0			0
$145$	−38.8334	−	26.0069i	−0.267816	−	0.179358i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	59.9620i		−	0.402429i	−0.979547	−	0.201215i	$-0.935511\pi$
$149$		−	59.9620i		−	0.402429i	0.979547	−	0.201215i	$-0.0644889\pi$
$150$		0			0
$151$	−279.060			−1.84808			−0.924039	−	0.382297i	$-0.875133\pi$
$151$	−279.060			−1.84808			−0.924039	+	0.382297i	$0.875133\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	60.8622	−	90.8789i	0.392659	−	0.586316i
$156$		0			0
$157$		−	82.8343i		−	0.527607i	−0.964576	−	0.263804i	$-0.915023\pi$
$157$		−	82.8343i		−	0.527607i	0.964576	−	0.263804i	$-0.0849771\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	29.2291i		−	0.181547i
$162$		0			0
$163$		−	172.317i		−	1.05716i	−0.848884	−	0.528579i	$-0.822725\pi$
$163$		−	172.317i		−	1.05716i	0.848884	−	0.528579i	$-0.177275\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−214.557			−1.28477			−0.642386	−	0.766381i	$-0.722055\pi$
$167$	−214.557			−1.28477			−0.642386	+	0.766381i	$0.722055\pi$
$168$		0			0
$169$	−29.9939			−0.177479
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−180.493			−1.04331			−0.521655	−	0.853157i	$-0.674685\pi$
$173$	−180.493			−1.04331			−0.521655	+	0.853157i	$0.674685\pi$
$174$		0			0
$175$	16.2282	−	6.68186i	0.0927326	−	0.0381821i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	215.571i		−	1.20431i	−0.798381	−	0.602153i	$-0.794310\pi$
$179$		−	215.571i		−	1.20431i	0.798381	−	0.602153i	$-0.205690\pi$
$180$		0			0
$181$	−88.6029			−0.489519			−0.244759	−	0.969584i	$-0.578709\pi$
$181$	−88.6029			−0.489519			−0.244759	+	0.969584i	$0.578709\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−141.624	−	94.8461i	−0.765533	−	0.512682i
$186$		0			0
$187$		−	6.66009i		−	0.0356155i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	66.7132i		−	0.349284i	−0.984632	−	0.174642i	$-0.944123\pi$
$191$		−	66.7132i		−	0.349284i	0.984632	−	0.174642i	$-0.0558767\pi$
$192$		0			0
$193$		−	365.878i		−	1.89574i	−0.318655	−	0.947871i	$-0.603231\pi$
$193$		−	365.878i		−	1.89574i	0.318655	−	0.947871i	$-0.396769\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	77.3887			0.392836			0.196418	−	0.980520i	$-0.437069\pi$
$197$	77.3887			0.392836			0.196418	+	0.980520i	$0.437069\pi$
$198$		0			0
$199$	259.440			1.30372			0.651860	−	0.758339i	$-0.273989\pi$
$199$	259.440			1.30372			0.651860	+	0.758339i	$0.273989\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.56193			0.0323248
$204$		0			0
$205$	−92.9149	−	62.2256i	−0.453243	−	0.303540i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.48806i		−	0.0119046i
$210$		0			0
$211$	68.7221			0.325697			0.162849	−	0.986651i	$-0.447932\pi$
$211$	68.7221			0.325697			0.162849	+	0.986651i	$0.447932\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	232.566	+	155.751i	1.08170	+	0.724422i
$216$		0			0
$217$			15.3564i			0.0707669i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	203.109i		−	0.919044i
$222$		0			0
$223$			233.182i			1.04566i	0.852437	+	0.522830i	$0.175124\pi$
$223$			233.182i			1.04566i	−0.852437	+	0.522830i	$0.824876\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	263.951			1.16278			0.581390	−	0.813625i	$-0.302509\pi$
$227$	263.951			1.16278			0.581390	+	0.813625i	$0.302509\pi$
$228$		0			0
$229$	252.110			1.10092			0.550459	−	0.834862i	$-0.314453\pi$
$229$	252.110			1.10092			0.550459	+	0.834862i	$0.314453\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−97.4784			−0.418362			−0.209181	−	0.977877i	$-0.567080\pi$
$233$	−97.4784			−0.418362			−0.209181	+	0.977877i	$0.567080\pi$
$234$		0			0
$235$	−21.2717	+	31.7627i	−0.0905177	+	0.135160i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	64.8703i		−	0.271424i	−0.990748	−	0.135712i	$-0.956668\pi$
$239$		−	64.8703i		−	0.271424i	0.990748	−	0.135712i	$-0.0433321\pi$
$240$		0			0
$241$	70.3914			0.292080			0.146040	−	0.989279i	$-0.453347\pi$
$241$	70.3914			0.292080			0.146040	+	0.989279i	$0.453347\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	134.958	−	201.519i	0.550851	−	0.822527i
$246$		0			0
$247$		−	75.8769i		−	0.307194i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			153.769i			0.612627i	0.951931	+	0.306313i	$0.0990956\pi$
$251$			153.769i			0.612627i	−0.951931	+	0.306313i	$0.900904\pi$
$252$		0			0
$253$			19.2597i			0.0761253i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−52.2311			−0.203234			−0.101617	−	0.994824i	$-0.532402\pi$
$257$	−52.2311			−0.203234			−0.101617	+	0.994824i	$0.532402\pi$
$258$		0			0
$259$	23.9311			0.0923980
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−320.902			−1.22016			−0.610079	−	0.792341i	$-0.708862\pi$
$263$	−320.902			−1.22016			−0.610079	+	0.792341i	$0.708862\pi$
$264$		0			0
$265$	−100.798	+	150.510i	−0.380368	+	0.567963i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	364.325i		−	1.35437i	−0.735814	−	0.677183i	$-0.763200\pi$
$269$		−	364.325i		−	1.35437i	0.735814	−	0.677183i	$-0.236800\pi$
$270$		0			0
$271$	120.782			0.445689			0.222845	−	0.974854i	$-0.428466\pi$
$271$	120.782			0.445689			0.222845	+	0.974854i	$0.428466\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.6931	+	4.40283i	−0.0388841	+	0.0160103i
$276$		0			0
$277$		−	42.8183i		−	0.154579i	−0.997009	−	0.0772894i	$-0.975373\pi$
$277$		−	42.8183i		−	0.154579i	0.997009	−	0.0772894i	$-0.0246265\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	171.250i		−	0.609432i	−0.952443	−	0.304716i	$-0.901438\pi$
$281$		−	171.250i		−	0.609432i	0.952443	−	0.304716i	$-0.0985615\pi$
$282$		0			0
$283$		−	98.8967i		−	0.349458i	−0.984617	−	0.174729i	$-0.944095\pi$
$283$		−	98.8967i		−	0.349458i	0.984617	−	0.174729i	$-0.0559050\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.7005			0.0547054
$288$		0			0
$289$	−81.6913			−0.282669
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−275.287			−0.939545			−0.469772	−	0.882788i	$-0.655664\pi$
$293$	−275.287			−0.939545			−0.469772	+	0.882788i	$0.655664\pi$
$294$		0			0
$295$	−145.708	−	97.5813i	−0.493925	−	0.330784i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			587.351i			1.96439i
$300$		0			0
$301$	−39.2982			−0.130559
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−163.803	+	244.589i	−0.537058	+	0.801931i
$306$		0			0
$307$		−	397.193i		−	1.29379i	−0.762580	−	0.646893i	$-0.776068\pi$
$307$		−	397.193i		−	1.29379i	0.762580	−	0.646893i	$-0.223932\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			342.087i			1.09996i	0.835179	+	0.549979i	$0.185364\pi$
$311$			342.087i			1.09996i	−0.835179	+	0.549979i	$0.814636\pi$
$312$		0			0
$313$		−	194.755i		−	0.622222i	−0.950374	−	0.311111i	$-0.899299\pi$
$313$		−	194.755i		−	0.622222i	0.950374	−	0.311111i	$-0.100701\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−130.611			−0.412023			−0.206012	−	0.978550i	$-0.566049\pi$
$317$	−130.611			−0.412023			−0.206012	+	0.978550i	$0.566049\pi$
$318$		0			0
$319$	−4.32381			−0.0135543
$320$		0			0
$321$		0			0
$322$		0			0
$323$	77.4459			0.239771
$324$		0			0
$325$	−326.102	+	134.270i	−1.00339	+	0.413140i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	5.36716i		−	0.0163135i
$330$		0			0
$331$	−154.500			−0.466766			−0.233383	−	0.972385i	$-0.574980\pi$
$331$	−154.500			−0.466766			−0.233383	+	0.972385i	$0.574980\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−257.127	−	172.200i	−0.767544	−	0.514029i
$336$		0			0
$337$			331.705i			0.984289i	0.870513	+	0.492145i	$0.163787\pi$
$337$			331.705i			0.984289i	−0.870513	+	0.492145i	$0.836213\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	10.1187i		−	0.0296736i
$342$		0			0
$343$			68.4500i			0.199563i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	639.366			1.84255			0.921277	−	0.388907i	$-0.127147\pi$
$347$	639.366			1.84255			0.921277	+	0.388907i	$0.127147\pi$
$348$		0			0
$349$	581.904			1.66735			0.833673	−	0.552259i	$-0.186234\pi$
$349$	581.904			1.66735			0.833673	+	0.552259i	$0.186234\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	588.797			1.66798			0.833990	−	0.551780i	$-0.186051\pi$
$353$	588.797			1.66798			0.833990	+	0.551780i	$0.186051\pi$
$354$		0			0
$355$	−181.431	−	121.505i	−0.511072	−	0.342268i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	138.855i		−	0.386781i	−0.981122	−	0.193391i	$-0.938052\pi$
$359$		−	138.855i		−	0.386781i	0.981122	−	0.193391i	$-0.0619485\pi$
$360$		0			0
$361$	−332.068			−0.919856
$362$		0			0
$363$		0			0
$364$		0			0
$365$	210.805	+	141.177i	0.577547	+	0.386787i
$366$		0			0
$367$			427.942i			1.16606i	0.812452	+	0.583028i	$0.198132\pi$
$367$			427.942i			1.16606i	−0.812452	+	0.583028i	$0.801868\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	25.4327i		−	0.0685518i
$372$		0			0
$373$			718.271i			1.92566i	0.270110	+	0.962830i	$0.412940\pi$
$373$			718.271i			1.92566i	−0.270110	+	0.962830i	$0.587060\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−131.861			−0.349763
$378$		0			0
$379$	512.901			1.35330			0.676650	−	0.736304i	$-0.263431\pi$
$379$	512.901			1.35330			0.676650	+	0.736304i	$0.263431\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	443.406			1.15772			0.578859	−	0.815428i	$-0.303498\pi$
$383$	443.406			1.15772			0.578859	+	0.815428i	$0.303498\pi$
$384$		0			0
$385$	0.903446	−	1.34902i	0.00234661	−	0.00350395i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			470.259i			1.20889i	0.796646	+	0.604446i	$0.206606\pi$
$389$			470.259i			1.20889i	−0.796646	+	0.604446i	$0.793394\pi$
$390$		0			0
$391$	−599.497			−1.53324
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−267.595	+	399.571i	−0.677456	+	1.01157i
$396$		0			0
$397$			692.805i			1.74510i	0.488524	+	0.872551i	$0.337536\pi$
$397$			692.805i			1.74510i	−0.488524	+	0.872551i	$0.662464\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	208.112i		−	0.518983i	−0.965745	−	0.259491i	$-0.916445\pi$
$401$		−	208.112i		−	0.518983i	0.965745	−	0.259491i	$-0.0835549\pi$
$402$		0			0
$403$		−	308.584i		−	0.765717i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.7687			−0.0387438
$408$		0			0
$409$	−27.8433			−0.0680765			−0.0340382	−	0.999421i	$-0.510837\pi$
$409$	−27.8433			−0.0680765			−0.0340382	+	0.999421i	$0.510837\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.6212			0.0596156
$414$		0			0
$415$	72.3617	−	108.050i	0.174366	−	0.260361i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			756.881i			1.80640i	0.429221	+	0.903199i	$0.358788\pi$
$419$			756.881i			1.80640i	−0.429221	+	0.903199i	$0.641212\pi$
$420$		0			0
$421$	414.751			0.985158			0.492579	−	0.870268i	$-0.336054\pi$
$421$	414.751			0.985158			0.492579	+	0.870268i	$0.336054\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−137.047	−	332.845i	−0.322463	−	0.783166i
$426$		0			0
$427$		−	41.3299i		−	0.0967912i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			592.001i			1.37355i	0.726869	+	0.686776i	$0.240975\pi$
$431$			592.001i			1.37355i	−0.726869	+	0.686776i	$0.759025\pi$
$432$		0			0
$433$			515.795i			1.19121i	0.803277	+	0.595606i	$0.203088\pi$
$433$			515.795i			1.19121i	−0.803277	+	0.595606i	$0.796912\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−223.959			−0.512491
$438$		0			0
$439$	463.708			1.05628			0.528141	−	0.849157i	$-0.322889\pi$
$439$	463.708			1.05628			0.528141	+	0.849157i	$0.322889\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	615.676			1.38979			0.694894	−	0.719112i	$-0.255451\pi$
$443$	615.676			1.38979			0.694894	+	0.719112i	$0.255451\pi$
$444$		0			0
$445$	−519.741	−	348.074i	−1.16796	−	0.782188i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	856.832i		−	1.90831i	−0.299310	−	0.954156i	$-0.596756\pi$
$449$		−	856.832i		−	1.90831i	0.299310	−	0.954156i	$-0.403244\pi$
$450$		0			0
$451$	−10.3454			−0.0229388
$452$		0			0
$453$		0			0
$454$		0			0
$455$	27.5518	−	41.1402i	0.0605535	−	0.0904181i
$456$		0			0
$457$		−	661.209i		−	1.44685i	−0.690404	−	0.723424i	$-0.742567\pi$
$457$		−	661.209i		−	1.44685i	0.690404	−	0.723424i	$-0.257433\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			136.414i			0.295908i	0.988994	+	0.147954i	$0.0472688\pi$
$461$			136.414i			0.295908i	−0.988994	+	0.147954i	$0.952731\pi$
$462$		0			0
$463$			559.962i			1.20942i	0.796445	+	0.604711i	$0.206711\pi$
$463$			559.962i			1.20942i	−0.796445	+	0.604711i	$0.793289\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	398.411			0.853129			0.426565	−	0.904457i	$-0.359724\pi$
$467$	398.411			0.853129			0.426565	+	0.904457i	$0.359724\pi$
$468$		0			0
$469$	43.4485			0.0926407
$470$		0			0
$471$		0			0
$472$		0			0
$473$	25.8945			0.0547452
$474$		0			0
$475$	−51.1977	−	124.344i	−0.107785	−	0.261776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			633.729i			1.32302i	0.749934	+	0.661512i	$0.230085\pi$
$479$			633.729i			1.32302i	−0.749934	+	0.661512i	$0.769915\pi$
$480$		0			0
$481$	−480.890			−0.999771
$482$		0			0
$483$		0			0
$484$		0			0
$485$	489.342	+	327.715i	1.00895	+	0.675701i
$486$		0			0
$487$		−	673.549i		−	1.38306i	−0.722349	−	0.691528i	$-0.756938\pi$
$487$		−	673.549i		−	1.38306i	0.722349	−	0.691528i	$-0.243062\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			525.821i			1.07092i	0.844561	+	0.535459i	$0.179861\pi$
$491$			525.821i			1.07092i	−0.844561	+	0.535459i	$0.820139\pi$
$492$		0			0
$493$		−	134.587i		−	0.272996i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	30.6575			0.0616852
$498$		0			0
$499$	229.252			0.459424			0.229712	−	0.973259i	$-0.426222\pi$
$499$	229.252			0.459424			0.229712	+	0.973259i	$0.426222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	331.323			0.658694			0.329347	−	0.944209i	$-0.393171\pi$
$503$	331.323			0.658694			0.329347	+	0.944209i	$0.393171\pi$
$504$		0			0
$505$	−538.208	−	360.441i	−1.06576	−	0.713744i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	136.587i		−	0.268345i	−0.990958	−	0.134172i	$-0.957162\pi$
$509$		−	136.587i		−	0.268345i	0.990958	−	0.134172i	$-0.0428375\pi$
$510$		0			0
$511$	−35.6211			−0.0697086
$512$		0			0
$513$		0			0
$514$		0			0
$515$	147.926	+	99.0671i	0.287236	+	0.192363i
$516$		0			0
$517$			3.53654i			0.00684051i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			811.351i			1.55730i	0.627461	+	0.778648i	$0.284094\pi$
$521$			811.351i			1.55730i	−0.627461	+	0.778648i	$0.715906\pi$
$522$		0			0
$523$		−	254.507i		−	0.486629i	−0.969947	−	0.243315i	$-0.921765\pi$
$523$		−	254.507i		−	0.486629i	0.969947	−	0.243315i	$-0.0782347\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	314.965			0.597656
$528$		0			0
$529$	1204.63			2.27718
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−315.497			−0.591927
$534$		0			0
$535$	−367.928	+	549.388i	−0.687716	+	1.02689i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	22.4377i		−	0.0416283i
$540$		0			0
$541$	−499.608			−0.923489			−0.461745	−	0.887013i	$-0.652776\pi$
$541$	−499.608			−0.923489			−0.461745	+	0.887013i	$0.652776\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	47.1039	−	70.3352i	0.0864292	−	0.129055i
$546$		0			0
$547$		−	701.196i		−	1.28189i	−0.767585	−	0.640947i	$-0.778542\pi$
$547$		−	701.196i		−	1.28189i	0.767585	−	0.640947i	$-0.221458\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	50.2788i		−	0.0912501i
$552$		0			0
$553$		−	67.5183i		−	0.122095i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−344.509			−0.618508			−0.309254	−	0.950979i	$-0.600079\pi$
$557$	−344.509			−0.618508			−0.309254	+	0.950979i	$0.600079\pi$
$558$		0			0
$559$	789.689			1.41268
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−382.276			−0.678998			−0.339499	−	0.940606i	$-0.610258\pi$
$563$	−382.276			−0.678998			−0.339499	+	0.940606i	$0.610258\pi$
$564$		0			0
$565$	94.4552	−	141.040i	0.167177	−	0.249628i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1122.74i			1.97319i	0.163201	+	0.986593i	$0.447818\pi$
$569$			1122.74i			1.97319i	−0.163201	+	0.986593i	$0.552182\pi$
$570$		0			0
$571$	−269.669			−0.472275			−0.236137	−	0.971720i	$-0.575882\pi$
$571$	−269.669			−0.472275			−0.236137	+	0.971720i	$0.575882\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	396.313	+	962.525i	0.689240	+	1.67396i
$576$		0			0
$577$			440.053i			0.762657i	0.924440	+	0.381329i	$0.124533\pi$
$577$			440.053i			0.762657i	−0.924440	+	0.381329i	$0.875467\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			18.2579i			0.0314250i
$582$		0			0
$583$			16.7582i			0.0287448i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	186.549			0.317801			0.158901	−	0.987295i	$-0.449205\pi$
$587$	186.549			0.317801			0.158901	+	0.987295i	$0.449205\pi$
$588$		0			0
$589$	117.664			0.199769
$590$		0			0
$591$		0			0
$592$		0			0
$593$	161.121			0.271705			0.135853	−	0.990729i	$-0.456623\pi$
$593$	161.121			0.271705			0.135853	+	0.990729i	$0.456623\pi$
$594$		0			0
$595$	41.9909	+	28.1216i	0.0705730	+	0.0472632i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			58.9378i			0.0983937i	0.998789	+	0.0491969i	$0.0156662\pi$
$599$			58.9378i			0.0983937i	−0.998789	+	0.0491969i	$0.984334\pi$
$600$		0			0
$601$	762.960			1.26948			0.634742	−	0.772724i	$-0.281106\pi$
$601$	762.960			1.26948			0.634742	+	0.772724i	$0.281106\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	336.055	−	501.795i	0.555463	−	0.829414i
$606$		0			0
$607$			718.517i			1.18372i	0.806041	+	0.591859i	$0.201606\pi$
$607$			718.517i			1.18372i	−0.806041	+	0.591859i	$0.798394\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			107.852i			0.176517i
$612$		0			0
$613$		−	614.827i		−	1.00298i	−0.865163	−	0.501490i	$-0.832785\pi$
$613$		−	614.827i		−	1.00298i	0.865163	−	0.501490i	$-0.167215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−605.662			−0.981624			−0.490812	−	0.871266i	$-0.663300\pi$
$617$	−605.662			−0.981624			−0.490812	+	0.871266i	$0.663300\pi$
$618$		0			0
$619$	407.471			0.658273			0.329136	−	0.944282i	$-0.393242\pi$
$619$	407.471			0.658273			0.329136	+	0.944282i	$0.393242\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	87.8242			0.140970
$624$		0			0
$625$	−443.803	+	440.073i	−0.710085	+	0.704116i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	490.834i		−	0.780340i
$630$		0			0
$631$	192.314			0.304776			0.152388	−	0.988321i	$-0.451304\pi$
$631$	192.314			0.304776			0.152388	+	0.988321i	$0.451304\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	342.744	+	229.538i	0.539755	+	0.361477i
$636$		0			0
$637$		−	684.268i		−	1.07420i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	807.956i		−	1.26046i	−0.776408	−	0.630231i	$-0.782960\pi$
$641$		−	807.956i		−	1.26046i	0.776408	−	0.630231i	$-0.217040\pi$
$642$		0			0
$643$		−	1234.20i		−	1.91944i	−0.280950	−	0.959722i	$-0.590650\pi$
$643$		−	1234.20i		−	1.91944i	0.280950	−	0.959722i	$-0.409350\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	818.693			1.26537			0.632684	−	0.774410i	$-0.281953\pi$
$647$	818.693			1.26537			0.632684	+	0.774410i	$0.281953\pi$
$648$		0			0
$649$	−16.2235			−0.0249977
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−177.215			−0.271386			−0.135693	−	0.990751i	$-0.543326\pi$
$653$	−177.215			−0.271386			−0.135693	+	0.990751i	$0.543326\pi$
$654$		0			0
$655$	−283.536	−	189.886i	−0.432880	−	0.289902i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1247.10i			1.89241i	0.323564	+	0.946206i	$0.395119\pi$
$659$			1247.10i			1.89241i	−0.323564	+	0.946206i	$0.604881\pi$
$660$		0			0
$661$	−318.753			−0.482229			−0.241114	−	0.970497i	$-0.577513\pi$
$661$	−318.753			−0.482229			−0.241114	+	0.970497i	$0.577513\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.6869	+	10.5056i	0.0235893	+	0.0157979i
$666$		0			0
$667$			389.200i			0.583509i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			27.2332i			0.0405860i
$672$		0			0
$673$		−	63.7076i		−	0.0946622i	−0.998879	−	0.0473311i	$-0.984928\pi$
$673$		−	63.7076i		−	0.0946622i	0.998879	−	0.0473311i	$-0.0150716\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	687.627			1.01570			0.507849	−	0.861446i	$-0.330441\pi$
$677$	687.627			1.01570			0.507849	+	0.861446i	$0.330441\pi$
$678$		0			0
$679$	−82.6874			−0.121778
$680$		0			0
$681$		0			0
$682$		0			0
$683$	419.303			0.613914			0.306957	−	0.951723i	$-0.400689\pi$
$683$	419.303			0.613914			0.306957	+	0.951723i	$0.400689\pi$
$684$		0			0
$685$	−475.579	+	710.131i	−0.694276	+	1.03669i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			511.065i			0.741749i
$690$		0			0
$691$	−614.804			−0.889731			−0.444865	−	0.895597i	$-0.646748\pi$
$691$	−614.804			−0.889731			−0.444865	+	0.895597i	$0.646748\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	604.128	−	902.079i	0.869248	−	1.29796i
$696$		0			0
$697$		−	322.021i		−	0.462010i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			528.543i			0.753984i	0.926217	+	0.376992i	$0.123042\pi$
$701$			528.543i			0.753984i	−0.926217	+	0.376992i	$0.876958\pi$
$702$		0			0
$703$		−	183.365i		−	0.260831i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	90.9446			0.128635
$708$		0			0
$709$	748.620			1.05588			0.527941	−	0.849281i	$-0.322964\pi$
$709$	748.620			1.05588			0.527941	+	0.849281i	$0.322964\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−910.817			−1.27744
$714$		0			0
$715$	−18.1545	+	27.1082i	−0.0253910	+	0.0379136i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1273.40i		−	1.77107i	−0.464576	−	0.885533i	$-0.653793\pi$
$719$		−	1273.40i		−	1.77107i	0.464576	−	0.885533i	$-0.346207\pi$
$720$		0			0
$721$	−24.9961			−0.0346687
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−216.087	+	88.9724i	−0.298051	+	0.122721i
$726$		0			0
$727$			1098.74i			1.51134i	0.654953	+	0.755670i	$0.272689\pi$
$727$			1098.74i			1.51134i	−0.654953	+	0.755670i	$0.727311\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			806.018i			1.10262i
$732$		0			0
$733$			293.909i			0.400967i	0.979697	+	0.200483i	$0.0642513\pi$
$733$			293.909i			0.400967i	−0.979697	+	0.200483i	$0.935749\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−28.6292			−0.0388456
$738$		0			0
$739$	604.344			0.817786			0.408893	−	0.912582i	$-0.365915\pi$
$739$	604.344			0.817786			0.408893	+	0.912582i	$0.365915\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1336.87			1.79929			0.899645	−	0.436622i	$-0.143825\pi$
$743$	1336.87			1.79929			0.899645	+	0.436622i	$0.143825\pi$
$744$		0			0
$745$	−249.107	−	166.828i	−0.334372	−	0.223931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	92.8337i		−	0.123944i
$750$		0			0
$751$	−717.255			−0.955067			−0.477533	−	0.878614i	$-0.658469\pi$
$751$	−717.255			−0.955067			−0.477533	+	0.878614i	$0.658469\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−776.411	+	1159.33i	−1.02836	+	1.53554i
$756$		0			0
$757$		−	343.933i		−	0.454337i	−0.973855	−	0.227168i	$-0.927053\pi$
$757$		−	343.933i		−	0.454337i	0.973855	−	0.227168i	$-0.0729468\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	637.818i		−	0.838132i	−0.907956	−	0.419066i	$-0.862358\pi$
$761$		−	637.818i		−	0.838132i	0.907956	−	0.419066i	$-0.137642\pi$
$762$		0			0
$763$			11.8850i			0.0155767i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−494.758			−0.645056
$768$		0			0
$769$	538.198			0.699867			0.349934	−	0.936774i	$-0.386204\pi$
$769$	538.198			0.699867			0.349934	+	0.936774i	$0.386204\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1107.17			−1.43231			−0.716154	−	0.697942i	$-0.754099\pi$
$773$	−1107.17			−1.43231			−0.716154	+	0.697942i	$0.754099\pi$
$774$		0			0
$775$	−208.216	−	505.693i	−0.268666	−	0.652507i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	120.300i		−	0.154429i
$780$		0			0
$781$	−20.2010			−0.0258655
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−344.128	−	230.465i	−0.438380	−	0.293586i
$786$		0			0
$787$			707.513i			0.899000i	0.893280	+	0.449500i	$0.148398\pi$
$787$			707.513i			0.899000i	−0.893280	+	0.449500i	$0.851602\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			23.8325i			0.0301295i
$792$		0			0
$793$			830.514i			1.04731i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1209.26			1.51726			0.758630	−	0.651522i	$-0.225869\pi$
$797$	1209.26			1.51726			0.758630	+	0.651522i	$0.225869\pi$
$798$		0			0
$799$	−110.082			−0.137775
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.4716			0.0292298
$804$		0			0
$805$	−121.430	−	81.3222i	−0.150844	−	0.101021i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1090.45i		−	1.34789i	−0.738779	−	0.673947i	$-0.764597\pi$
$809$		−	1090.45i		−	1.34789i	0.738779	−	0.673947i	$-0.235403\pi$
$810$		0			0
$811$	−497.042			−0.612876			−0.306438	−	0.951891i	$-0.599137\pi$
$811$	−497.042			−0.612876			−0.306438	+	0.951891i	$0.599137\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−715.876	−	479.426i	−0.878375	−	0.588253i
$816$		0			0
$817$			301.110i			0.368556i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	650.456i		−	0.792273i	−0.918192	−	0.396136i	$-0.870351\pi$
$821$		−	650.456i		−	0.792273i	0.918192	−	0.396136i	$-0.129649\pi$
$822$		0			0
$823$			1064.92i			1.29395i	0.762509	+	0.646977i	$0.223967\pi$
$823$			1064.92i			1.29395i	−0.762509	+	0.646977i	$0.776033\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−718.833			−0.869206			−0.434603	−	0.900622i	$-0.643111\pi$
$827$	−718.833			−0.869206			−0.434603	+	0.900622i	$0.643111\pi$
$828$		0			0
$829$	209.803			0.253080			0.126540	−	0.991962i	$-0.459613\pi$
$829$	209.803			0.253080			0.126540	+	0.991962i	$0.459613\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	698.417			0.838436
$834$		0			0
$835$	−596.948	+	891.359i	−0.714908	+	1.06750i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1071.72i			1.27738i	0.769463	+	0.638691i	$0.220524\pi$
$839$			1071.72i			1.27738i	−0.769463	+	0.638691i	$0.779476\pi$
$840$		0			0
$841$	753.624			0.896105
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−83.4501	+	124.607i	−0.0987576	+	0.147464i
$846$		0			0
$847$			84.7917i			0.100108i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1419.40i			1.66792i
$852$		0			0
$853$		−	1119.66i		−	1.31262i	−0.754492	−	0.656310i	$-0.772116\pi$
$853$		−	1119.66i		−	1.31262i	0.754492	−	0.656310i	$-0.227884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−417.845			−0.487567			−0.243784	−	0.969830i	$-0.578389\pi$
$857$	−417.845			−0.487567			−0.243784	+	0.969830i	$0.578389\pi$
$858$		0			0
$859$	1481.77			1.72499			0.862497	−	0.506063i	$-0.168900\pi$
$859$	1481.77			1.72499			0.862497	+	0.506063i	$0.168900\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−511.566			−0.592776			−0.296388	−	0.955068i	$-0.595782\pi$
$863$	−511.566			−0.592776			−0.296388	+	0.955068i	$0.595782\pi$
$864$		0			0
$865$	−502.173	+	749.841i	−0.580547	+	0.866869i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			44.4893i			0.0511960i
$870$		0			0
$871$	−873.088			−1.00240
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.3915	−	86.0092i	0.0198760	−	0.0982963i
$876$		0			0
$877$		−	737.224i		−	0.840620i	−0.907381	−	0.420310i	$-0.861921\pi$
$877$		−	737.224i		−	0.840620i	0.907381	−	0.420310i	$-0.138079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	455.461i		−	0.516981i	−0.966014	−	0.258491i	$-0.916775\pi$
$881$		−	455.461i		−	0.516981i	0.966014	−	0.258491i	$-0.0832251\pi$
$882$		0			0
$883$		−	35.2571i		−	0.0399288i	−0.999801	−	0.0199644i	$-0.993645\pi$
$883$		−	35.2571i		−	0.0399288i	0.999801	−	0.0199644i	$-0.00635528\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1745.92			−1.96834			−0.984169	−	0.177232i	$-0.943286\pi$
$887$	−1745.92			−1.96834			−0.984169	+	0.177232i	$0.943286\pi$
$888$		0			0
$889$	−57.9158			−0.0651472
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−41.1242			−0.0460517
$894$		0			0
$895$	−895.570	−	599.769i	−1.00064	−	0.670132i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	204.479i		−	0.227451i
$900$		0			0
$901$	−521.633			−0.578949
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−246.514	+	368.093i	−0.272391	+	0.406733i
$906$		0			0
$907$			976.102i			1.07619i	0.842885	+	0.538094i	$0.180855\pi$
$907$			976.102i			1.07619i	−0.842885	+	0.538094i	$0.819145\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1352.10i		−	1.48420i	−0.670290	−	0.742099i	$-0.733830\pi$
$911$		−	1352.10i		−	1.48420i	0.670290	−	0.742099i	$-0.266170\pi$
$912$		0			0
$913$		−	12.0306i		−	0.0131770i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	47.9110			0.0522476
$918$		0			0
$919$	1058.71			1.15202			0.576011	−	0.817442i	$-0.304609\pi$
$919$	1058.71			1.15202			0.576011	+	0.817442i	$0.304609\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−616.056			−0.667450
$924$		0			0
$925$	−788.060	+	324.479i	−0.851957	+	0.350788i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	95.6546i		−	0.102965i	−0.998674	−	0.0514826i	$-0.983605\pi$
$929$		−	95.6546i		−	0.102965i	0.998674	−	0.0514826i	$-0.0163947\pi$
$930$		0			0
$931$	260.913			0.280250
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−27.6688	−	18.5299i	−0.0295923	−	0.0198181i
$936$		0			0
$937$		−	1488.98i		−	1.58910i	−0.607201	−	0.794549i	$-0.707708\pi$
$937$		−	1488.98i		−	1.58910i	0.607201	−	0.794549i	$-0.292292\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1480.85i		−	1.57370i	−0.617145	−	0.786850i	$-0.711711\pi$
$941$		−	1480.85i		−	1.57370i	0.617145	−	0.786850i	$-0.288289\pi$
$942$		0			0
$943$			931.222i			0.987510i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1077.27			1.13756			0.568781	−	0.822489i	$-0.307415\pi$
$947$	1077.27			1.13756			0.568781	+	0.822489i	$0.307415\pi$
$948$		0			0
$949$	715.798			0.754265
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1197.06			1.25609			0.628046	−	0.778176i	$-0.283855\pi$
$953$	1197.06			1.25609			0.628046	+	0.778176i	$0.283855\pi$
$954$		0			0
$955$	−277.154	−	185.612i	−0.290214	−	0.194358i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	119.996i		−	0.125126i
$960$		0			0
$961$	−482.473			−0.502053
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1520.01	−	1017.96i	−1.57514	−	1.05488i
$966$		0			0
$967$		−	1155.33i		−	1.19475i	−0.801960	−	0.597377i	$-0.796210\pi$
$967$		−	1155.33i		−	1.19475i	0.801960	−	0.597377i	$-0.203790\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			261.219i			0.269021i	0.990912	+	0.134510i	$0.0429461\pi$
$971$			261.219i			0.269021i	−0.990912	+	0.134510i	$0.957054\pi$
$972$		0			0
$973$			152.430i			0.156660i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	705.097			0.721696			0.360848	−	0.932625i	$-0.382487\pi$
$977$	705.097			0.721696			0.360848	+	0.932625i	$0.382487\pi$
$978$		0			0
$979$	−57.8694			−0.0591107
$980$		0			0
$981$		0			0
$982$		0			0
$983$	825.915			0.840198			0.420099	−	0.907478i	$-0.361995\pi$
$983$	825.915			0.840198			0.420099	+	0.907478i	$0.361995\pi$
$984$		0			0
$985$	215.314	−	321.505i	0.218592	−	0.326401i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	2330.85i		−	2.35677i
$990$		0			0
$991$	556.033			0.561083			0.280542	−	0.959842i	$-0.409486\pi$
$991$	556.033			0.561083			0.280542	+	0.959842i	$0.409486\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	721.824	−	1077.82i	0.725451	−	1.08324i
$996$		0			0
$997$			514.787i			0.516336i	0.966100	+	0.258168i	$0.0831188\pi$
$997$			514.787i			0.516336i	−0.966100	+	0.258168i	$0.916881\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.a.809.15	yes	24
3.2	odd	2	inner	1620.3.b.a.809.10	yes	24
5.4	even	2	inner	1620.3.b.a.809.9	✓	24
9.2	odd	6		1620.3.t.f.1349.9		48
9.4	even	3		1620.3.t.f.269.1		48
9.5	odd	6		1620.3.t.f.269.24		48
9.7	even	3		1620.3.t.f.1349.16		48
15.14	odd	2	inner	1620.3.b.a.809.16	yes	24
45.4	even	6		1620.3.t.f.269.9		48
45.14	odd	6		1620.3.t.f.269.16		48
45.29	odd	6		1620.3.t.f.1349.1		48
45.34	even	6		1620.3.t.f.1349.24		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.9	✓	24	5.4	even	2	inner
1620.3.b.a.809.10	yes	24	3.2	odd	2	inner
1620.3.b.a.809.15	yes	24	1.1	even	1	trivial
1620.3.b.a.809.16	yes	24	15.14	odd	2	inner
1620.3.t.f.269.1		48	9.4	even	3
1620.3.t.f.269.9		48	45.4	even	6
1620.3.t.f.269.16		48	45.14	odd	6
1620.3.t.f.269.24		48	9.5	odd	6
1620.3.t.f.1349.1		48	45.29	odd	6
1620.3.t.f.1349.9		48	9.2	odd	6
1620.3.t.f.1349.16		48	9.7	even	3
1620.3.t.f.1349.24		48	45.34	even	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.78224 - 4.15441i) q^{5} +0.702000i q^{7} +O(q^{10})\)	\(q+(2.78224 - 4.15441i) q^{5} +0.702000i q^{7} -0.462564i q^{11} -14.1065i q^{13} +14.3982 q^{17} +5.37885 q^{19} -41.6369 q^{23} +(-9.51832 - 23.1171i) q^{25} -9.34749i q^{29} +21.8753 q^{31} +(2.91640 + 1.95313i) q^{35} -34.0899i q^{37} -22.3653i q^{41} +55.9804i q^{43} -7.64552 q^{47} +48.5072 q^{49} -36.2290 q^{53} +(-1.92168 - 1.28696i) q^{55} -35.0730i q^{59} -58.8745 q^{61} +(-58.6043 - 39.2477i) q^{65} -61.8925i q^{67} -43.6717i q^{71} +50.7423i q^{73} +0.324719 q^{77} -96.1799 q^{79} +26.0085 q^{83} +(40.0592 - 59.8162i) q^{85} -125.106i q^{89} +9.90277 q^{91} +(14.9652 - 22.3460i) q^{95} +117.788i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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