LMFDB - Embedded newform 720.6.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.a (trivial)

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:	yes
Analytic conductor:	$115.476350265$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2161}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 540 $ x^2 - x - 540
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$23.7433$ of defining polynomial
Character	$\chi$	$=$	720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0000 q^{5} +189.946 q^{7} +O(q^{10})$	$q-25.0000 q^{5} +189.946 q^{7} +615.892 q^{11} +863.839 q^{13} -615.946 q^{17} +2093.41 q^{19} +3909.30 q^{23} +625.000 q^{25} +2600.92 q^{29} -10440.3 q^{31} -4748.66 q^{35} +11070.3 q^{37} -5974.75 q^{41} +16887.1 q^{43} -13756.4 q^{47} +19272.6 q^{49} -30989.6 q^{53} -15397.3 q^{55} -20690.9 q^{59} +8319.05 q^{61} -21596.0 q^{65} -18744.0 q^{67} -38315.0 q^{71} +5252.99 q^{73} +116986. q^{77} +83490.7 q^{79} -70547.4 q^{83} +15398.7 q^{85} +21181.9 q^{89} +164083. q^{91} -52335.2 q^{95} +67392.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 8 q^{7}+O(q^{10}) $ 2 * q - 50 * q^5 + 8 * q^7	$ 2 q - 50 q^{5} + 8 q^{7} + 488 q^{11} + 612 q^{13} - 860 q^{17} + 96 q^{19} + 2984 q^{23} + 1250 q^{25} - 2236 q^{29} - 9352 q^{31} - 200 q^{35} + 10612 q^{37} - 17156 q^{41} - 440 q^{43} - 16728 q^{47} + 35570 q^{49} - 31484 q^{53} - 12200 q^{55} - 61464 q^{59} + 51596 q^{61} - 15300 q^{65} + 45816 q^{67} - 97456 q^{71} + 58852 q^{73} + 140256 q^{77} + 116776 q^{79} - 136632 q^{83} + 21500 q^{85} + 124924 q^{89} + 209904 q^{91} - 2400 q^{95} - 39260 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 8 * q^7 + 488 * q^11 + 612 * q^13 - 860 * q^17 + 96 * q^19 + 2984 * q^23 + 1250 * q^25 - 2236 * q^29 - 9352 * q^31 - 200 * q^35 + 10612 * q^37 - 17156 * q^41 - 440 * q^43 - 16728 * q^47 + 35570 * q^49 - 31484 * q^53 - 12200 * q^55 - 61464 * q^59 + 51596 * q^61 - 15300 * q^65 + 45816 * q^67 - 97456 * q^71 + 58852 * q^73 + 140256 * q^77 + 116776 * q^79 - 136632 * q^83 + 21500 * q^85 + 124924 * q^89 + 209904 * q^91 - 2400 * q^95 - 39260 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	189.946	1.46516	0.732581	−	0.680680i	$-0.238316\pi$
$7$	189.946	1.46516	0.732581	+	0.680680i	$0.238316\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	615.892	1.53470	0.767349	−	0.641229i	$-0.221575\pi$
$11$	615.892	1.53470	0.767349	+	0.641229i	$0.221575\pi$
$12$	0	0
$13$	863.839	1.41767	0.708834	−	0.705376i	$-0.249222\pi$
$13$	863.839	1.41767	0.708834	+	0.705376i	$0.249222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−615.946	−0.516917	−0.258458	−	0.966022i	$-0.583214\pi$
$17$	−615.946	−0.516917	−0.258458	+	0.966022i	$0.583214\pi$
$18$	0	0
$19$	2093.41	1.33036	0.665181	−	0.746682i	$-0.268354\pi$
$19$	2093.41	1.33036	0.665181	+	0.746682i	$0.268354\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3909.30	1.54092	0.770459	−	0.637490i	$-0.220027\pi$
$23$	3909.30	1.54092	0.770459	+	0.637490i	$0.220027\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2600.92	0.574292	0.287146	−	0.957887i	$-0.407294\pi$
$29$	2600.92	0.574292	0.287146	+	0.957887i	$0.407294\pi$
$30$	0	0
$31$	−10440.3	−1.95124	−0.975619	−	0.219472i	$-0.929566\pi$
$31$	−10440.3	−1.95124	−0.975619	+	0.219472i	$0.929566\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4748.66	−0.655240
$36$	0	0
$37$	11070.3	1.32940	0.664701	−	0.747109i	$-0.268559\pi$
$37$	11070.3	1.32940	0.664701	+	0.747109i	$0.268559\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5974.75	−0.555086	−0.277543	−	0.960713i	$-0.589520\pi$
$41$	−5974.75	−0.555086	−0.277543	+	0.960713i	$0.589520\pi$
$42$	0	0
$43$	16887.1	1.39278	0.696390	−	0.717663i	$-0.254788\pi$
$43$	16887.1	1.39278	0.696390	+	0.717663i	$0.254788\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13756.4	−0.908367	−0.454184	−	0.890908i	$-0.650069\pi$
$47$	−13756.4	−0.908367	−0.454184	+	0.890908i	$0.650069\pi$
$48$	0	0
$49$	19272.6	1.14670
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−30989.6	−1.51540	−0.757698	−	0.652605i	$-0.773676\pi$
$53$	−30989.6	−1.51540	−0.757698	+	0.652605i	$0.773676\pi$
$54$	0	0
$55$	−15397.3	−0.686338
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−20690.9	−0.773837	−0.386918	−	0.922114i	$-0.626461\pi$
$59$	−20690.9	−0.773837	−0.386918	+	0.922114i	$0.626461\pi$
$60$	0	0
$61$	8319.05	0.286253	0.143126	−	0.989704i	$-0.454284\pi$
$61$	8319.05	0.286253	0.143126	+	0.989704i	$0.454284\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21596.0	−0.634000
$66$	0	0
$67$	−18744.0	−0.510122	−0.255061	−	0.966925i	$-0.582096\pi$
$67$	−18744.0	−0.510122	−0.255061	+	0.966925i	$0.582096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−38315.0	−0.902034	−0.451017	−	0.892515i	$-0.648939\pi$
$71$	−38315.0	−0.902034	−0.451017	+	0.892515i	$0.648939\pi$
$72$	0	0
$73$	5252.99	0.115372	0.0576859	−	0.998335i	$-0.481628\pi$
$73$	5252.99	0.115372	0.0576859	+	0.998335i	$0.481628\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	116986.	2.24858
$78$	0	0
$79$	83490.7	1.50512	0.752559	−	0.658525i	$-0.228819\pi$
$79$	83490.7	1.50512	0.752559	+	0.658525i	$0.228819\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−70547.4	−1.12405	−0.562025	−	0.827120i	$-0.689977\pi$
$83$	−70547.4	−1.12405	−0.562025	+	0.827120i	$0.689977\pi$
$84$	0	0
$85$	15398.7	0.231172
$86$	0	0
$87$	0	0
$88$	0	0
$89$	21181.9	0.283459	0.141730	−	0.989905i	$-0.454734\pi$
$89$	21181.9	0.283459	0.141730	+	0.989905i	$0.454734\pi$
$90$	0	0
$91$	164083.	2.07711
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−52335.2	−0.594956
$96$	0	0
$97$	67392.8	0.727251	0.363626	−	0.931545i	$-0.381539\pi$
$97$	67392.8	0.727251	0.363626	+	0.931545i	$0.381539\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	74842.0	0.730032	0.365016	−	0.931001i	$-0.381064\pi$
$101$	74842.0	0.730032	0.365016	+	0.931001i	$0.381064\pi$
$102$	0	0
$103$	−170061.	−1.57947	−0.789736	−	0.613447i	$-0.789782\pi$
$103$	−170061.	−1.57947	−0.789736	+	0.613447i	$0.789782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2971.42	0.0250902	0.0125451	−	0.999921i	$-0.496007\pi$
$107$	2971.42	0.0250902	0.0125451	+	0.999921i	$0.496007\pi$
$108$	0	0
$109$	−111890.	−0.902037	−0.451018	−	0.892515i	$-0.648939\pi$
$109$	−111890.	−0.902037	−0.451018	+	0.892515i	$0.648939\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	154909.	1.14125	0.570625	−	0.821211i	$-0.306701\pi$
$113$	154909.	1.14125	0.570625	+	0.821211i	$0.306701\pi$
$114$	0	0
$115$	−97732.5	−0.689119
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−116997.	−0.757366
$120$	0	0
$121$	218273.	1.35530
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	133249.	0.733086	0.366543	−	0.930401i	$-0.380541\pi$
$127$	133249.	0.733086	0.366543	+	0.930401i	$0.380541\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−265229.	−1.35034	−0.675170	−	0.737662i	$-0.735930\pi$
$131$	−265229.	−1.35034	−0.675170	+	0.737662i	$0.735930\pi$
$132$	0	0
$133$	397635.	1.94920
$134$	0	0
$135$	0	0
$136$	0	0
$137$	264455.	1.20379	0.601895	−	0.798575i	$-0.294413\pi$
$137$	264455.	1.20379	0.601895	+	0.798575i	$0.294413\pi$
$138$	0	0
$139$	−65638.1	−0.288150	−0.144075	−	0.989567i	$-0.546021\pi$
$139$	−65638.1	−0.288150	−0.144075	+	0.989567i	$0.546021\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	532032.	2.17569
$144$	0	0
$145$	−65023.1	−0.256831
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−512954.	−1.89284	−0.946418	−	0.322945i	$-0.895327\pi$
$149$	−512954.	−1.89284	−0.946418	+	0.322945i	$0.895327\pi$
$150$	0	0
$151$	332828.	1.18789	0.593947	−	0.804504i	$-0.297569\pi$
$151$	332828.	1.18789	0.593947	+	0.804504i	$0.297569\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	261008.	0.872620
$156$	0	0
$157$	69539.9	0.225157	0.112578	−	0.993643i	$-0.464089\pi$
$157$	69539.9	0.225157	0.112578	+	0.993643i	$0.464089\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	742557.	2.25769
$162$	0	0
$163$	−185741.	−0.547568	−0.273784	−	0.961791i	$-0.588275\pi$
$163$	−185741.	−0.547568	−0.273784	+	0.961791i	$0.588275\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	304144.	0.843894	0.421947	−	0.906620i	$-0.361347\pi$
$167$	304144.	0.843894	0.421947	+	0.906620i	$0.361347\pi$
$168$	0	0
$169$	374924.	1.00978
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15586.0	0.0395932	0.0197966	−	0.999804i	$-0.493698\pi$
$173$	15586.0	0.0395932	0.0197966	+	0.999804i	$0.493698\pi$
$174$	0	0
$175$	118716.	0.293032
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−317898.	−0.741576	−0.370788	−	0.928718i	$-0.620912\pi$
$179$	−317898.	−0.741576	−0.370788	+	0.928718i	$0.620912\pi$
$180$	0	0
$181$	−129082.	−0.292867	−0.146433	−	0.989221i	$-0.546779\pi$
$181$	−129082.	−0.292867	−0.146433	+	0.989221i	$0.546779\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−276758.	−0.594527
$186$	0	0
$187$	−379357.	−0.793311
$188$	0	0
$189$	0	0
$190$	0	0
$191$	187911.	0.372707	0.186354	−	0.982483i	$-0.440333\pi$
$191$	187911.	0.372707	0.186354	+	0.982483i	$0.440333\pi$
$192$	0	0
$193$	403530.	0.779799	0.389900	−	0.920857i	$-0.372510\pi$
$193$	403530.	0.779799	0.389900	+	0.920857i	$0.372510\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	241295.	0.442978	0.221489	−	0.975163i	$-0.428908\pi$
$197$	241295.	0.442978	0.221489	+	0.975163i	$0.428908\pi$
$198$	0	0
$199$	−577766.	−1.03424	−0.517118	−	0.855914i	$-0.672995\pi$
$199$	−577766.	−1.03424	−0.517118	+	0.855914i	$0.672995\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	494036.	0.841431
$204$	0	0
$205$	149369.	0.248242
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.28931e6	2.04171
$210$	0	0
$211$	408610.	0.631833	0.315917	−	0.948787i	$-0.397688\pi$
$211$	408610.	0.631833	0.315917	+	0.948787i	$0.397688\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−422176.	−0.622870
$216$	0	0
$217$	−1.98310e6	−2.85888
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−532078.	−0.732816
$222$	0	0
$223$	384755.	0.518110	0.259055	−	0.965863i	$-0.416589\pi$
$223$	384755.	0.518110	0.259055	+	0.965863i	$0.416589\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	558251.	0.719059	0.359530	−	0.933134i	$-0.382937\pi$
$227$	558251.	0.719059	0.359530	+	0.933134i	$0.382937\pi$
$228$	0	0
$229$	−1.05791e6	−1.33309	−0.666544	−	0.745465i	$-0.732227\pi$
$229$	−1.05791e6	−1.33309	−0.666544	+	0.745465i	$0.732227\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−844056.	−1.01855	−0.509274	−	0.860604i	$-0.670086\pi$
$233$	−844056.	−1.01855	−0.509274	+	0.860604i	$0.670086\pi$
$234$	0	0
$235$	343911.	0.406234
$236$	0	0
$237$	0	0
$238$	0	0
$239$	449134.	0.508606	0.254303	−	0.967125i	$-0.418154\pi$
$239$	449134.	0.508606	0.254303	+	0.967125i	$0.418154\pi$
$240$	0	0
$241$	−227636.	−0.252464	−0.126232	−	0.992001i	$-0.540288\pi$
$241$	−227636.	−0.252464	−0.126232	+	0.992001i	$0.540288\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−481814.	−0.512819
$246$	0	0
$247$	1.80837e6	1.88601
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−308095.	−0.308674	−0.154337	−	0.988018i	$-0.549324\pi$
$251$	−308095.	−0.308674	−0.154337	+	0.988018i	$0.549324\pi$
$252$	0	0
$253$	2.40771e6	2.36485
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3488.47	0.00329460	0.00164730	−	0.999999i	$-0.499476\pi$
$257$	3488.47	0.00329460	0.00164730	+	0.999999i	$0.499476\pi$
$258$	0	0
$259$	2.10277e6	1.94779
$260$	0	0
$261$	0	0
$262$	0	0
$263$	987870.	0.880664	0.440332	−	0.897835i	$-0.354861\pi$
$263$	987870.	0.880664	0.440332	+	0.897835i	$0.354861\pi$
$264$	0	0
$265$	774740.	0.677706
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−643304.	−0.542045	−0.271023	−	0.962573i	$-0.587362\pi$
$269$	−643304.	−0.542045	−0.271023	+	0.962573i	$0.587362\pi$
$270$	0	0
$271$	1.20112e6	0.993490	0.496745	−	0.867896i	$-0.334528\pi$
$271$	1.20112e6	0.993490	0.496745	+	0.867896i	$0.334528\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	384933.	0.306940
$276$	0	0
$277$	2.05137e6	1.60637	0.803183	−	0.595733i	$-0.203138\pi$
$277$	2.05137e6	1.60637	0.803183	+	0.595733i	$0.203138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	574023.	0.433674	0.216837	−	0.976208i	$-0.430426\pi$
$281$	574023.	0.433674	0.216837	+	0.976208i	$0.430426\pi$
$282$	0	0
$283$	−578001.	−0.429005	−0.214503	−	0.976723i	$-0.568813\pi$
$283$	−578001.	−0.429005	−0.214503	+	0.976723i	$0.568813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.13488e6	−0.813291
$288$	0	0
$289$	−1.04047e6	−0.732797
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.38335e6	0.941376	0.470688	−	0.882300i	$-0.344006\pi$
$293$	1.38335e6	0.941376	0.470688	+	0.882300i	$0.344006\pi$
$294$	0	0
$295$	517273.	0.346070
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.37701e6	2.18451
$300$	0	0
$301$	3.20763e6	2.04065
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−207976.	−0.128016
$306$	0	0
$307$	−1.66221e6	−1.00656	−0.503281	−	0.864123i	$-0.667874\pi$
$307$	−1.66221e6	−1.00656	−0.503281	+	0.864123i	$0.667874\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−245097.	−0.143693	−0.0718467	−	0.997416i	$-0.522889\pi$
$311$	−245097.	−0.143693	−0.0718467	+	0.997416i	$0.522889\pi$
$312$	0	0
$313$	−834700.	−0.481581	−0.240791	−	0.970577i	$-0.577407\pi$
$313$	−834700.	−0.481581	−0.240791	+	0.970577i	$0.577407\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−94612.6	−0.0528811	−0.0264406	−	0.999650i	$-0.508417\pi$
$317$	−94612.6	−0.0528811	−0.0264406	+	0.999650i	$0.508417\pi$
$318$	0	0
$319$	1.60189e6	0.881365
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.28943e6	−0.687686
$324$	0	0
$325$	539899.	0.283533
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.61298e6	−1.33090
$330$	0	0
$331$	2.01733e6	1.01206	0.506031	−	0.862515i	$-0.331112\pi$
$331$	2.01733e6	1.01206	0.506031	+	0.862515i	$0.331112\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	468599.	0.228134
$336$	0	0
$337$	−312749.	−0.150010	−0.0750052	−	0.997183i	$-0.523897\pi$
$337$	−312749.	−0.150010	−0.0750052	+	0.997183i	$0.523897\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.43012e6	−2.99456
$342$	0	0
$343$	468326.	0.214938
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.78366e6	0.795223	0.397611	−	0.917554i	$-0.369839\pi$
$347$	1.78366e6	0.795223	0.397611	+	0.917554i	$0.369839\pi$
$348$	0	0
$349$	114997.	0.0505384	0.0252692	−	0.999681i	$-0.491956\pi$
$349$	114997.	0.0505384	0.0252692	+	0.999681i	$0.491956\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	123355.	0.0526892	0.0263446	−	0.999653i	$-0.491613\pi$
$353$	123355.	0.0526892	0.0263446	+	0.999653i	$0.491613\pi$
$354$	0	0
$355$	957875.	0.403402
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.32230e6	1.36051	0.680256	−	0.732975i	$-0.261869\pi$
$359$	3.32230e6	1.36051	0.680256	+	0.732975i	$0.261869\pi$
$360$	0	0
$361$	1.90626e6	0.769864
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−131325.	−0.0515958
$366$	0	0
$367$	−4.49581e6	−1.74238	−0.871190	−	0.490947i	$-0.836651\pi$
$367$	−4.49581e6	−1.74238	−0.871190	+	0.490947i	$0.836651\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.88636e6	−2.22030
$372$	0	0
$373$	−3.13103e6	−1.16524	−0.582620	−	0.812744i	$-0.697973\pi$
$373$	−3.13103e6	−1.16524	−0.582620	+	0.812744i	$0.697973\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.24678e6	0.814155
$378$	0	0
$379$	2.13639e6	0.763981	0.381991	−	0.924166i	$-0.375239\pi$
$379$	2.13639e6	0.763981	0.381991	+	0.924166i	$0.375239\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.79282e6	−1.66953	−0.834765	−	0.550607i	$-0.814396\pi$
$383$	−4.79282e6	−1.66953	−0.834765	+	0.550607i	$0.814396\pi$
$384$	0	0
$385$	−2.92466e6	−1.00560
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.07000e6	1.36371	0.681853	−	0.731489i	$-0.261174\pi$
$389$	4.07000e6	1.36371	0.681853	+	0.731489i	$0.261174\pi$
$390$	0	0
$391$	−2.40792e6	−0.796526
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.08727e6	−0.673109
$396$	0	0
$397$	4.05181e6	1.29025	0.645123	−	0.764078i	$-0.276806\pi$
$397$	4.05181e6	1.29025	0.645123	+	0.764078i	$0.276806\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	286296.	0.0889106	0.0444553	−	0.999011i	$-0.485845\pi$
$401$	286296.	0.0889106	0.0444553	+	0.999011i	$0.485845\pi$
$402$	0	0
$403$	−9.01876e6	−2.76621
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.81813e6	2.04023
$408$	0	0
$409$	−5.73483e6	−1.69517	−0.847583	−	0.530663i	$-0.821943\pi$
$409$	−5.73483e6	−1.69517	−0.847583	+	0.530663i	$0.821943\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.93016e6	−1.13380
$414$	0	0
$415$	1.76368e6	0.502690
$416$	0	0
$417$	0	0
$418$	0	0
$419$	185516.	0.0516234	0.0258117	−	0.999667i	$-0.491783\pi$
$419$	185516.	0.0516234	0.0258117	+	0.999667i	$0.491783\pi$
$420$	0	0
$421$	2.92989e6	0.805649	0.402824	−	0.915277i	$-0.368029\pi$
$421$	2.92989e6	0.805649	0.402824	+	0.915277i	$0.368029\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−384966.	−0.103383
$426$	0	0
$427$	1.58017e6	0.419406
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−507967.	−0.131717	−0.0658586	−	0.997829i	$-0.520979\pi$
$431$	−507967.	−0.131717	−0.0658586	+	0.997829i	$0.520979\pi$
$432$	0	0
$433$	−3.02418e6	−0.775155	−0.387577	−	0.921837i	$-0.626688\pi$
$433$	−3.02418e6	−0.775155	−0.387577	+	0.921837i	$0.626688\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.18376e6	2.04998
$438$	0	0
$439$	−703836.	−0.174305	−0.0871525	−	0.996195i	$-0.527777\pi$
$439$	−703836.	−0.174305	−0.0871525	+	0.996195i	$0.527777\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.41317e6	−1.79471	−0.897355	−	0.441309i	$-0.854514\pi$
$443$	−7.41317e6	−1.79471	−0.897355	+	0.441309i	$0.854514\pi$
$444$	0	0
$445$	−529548.	−0.126767
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.25222e6	0.995405	0.497703	−	0.867348i	$-0.334177\pi$
$449$	4.25222e6	0.995405	0.497703	+	0.867348i	$0.334177\pi$
$450$	0	0
$451$	−3.67981e6	−0.851890
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.10207e6	−0.928913
$456$	0	0
$457$	−5.49005e6	−1.22966	−0.614832	−	0.788658i	$-0.710776\pi$
$457$	−5.49005e6	−1.22966	−0.614832	+	0.788658i	$0.710776\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.28899e6	0.720792	0.360396	−	0.932799i	$-0.382642\pi$
$461$	3.28899e6	0.720792	0.360396	+	0.932799i	$0.382642\pi$
$462$	0	0
$463$	4.83662e6	1.04855	0.524276	−	0.851549i	$-0.324336\pi$
$463$	4.83662e6	1.04855	0.524276	+	0.851549i	$0.324336\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.87891e6	−0.823033	−0.411516	−	0.911402i	$-0.635001\pi$
$467$	−3.87891e6	−0.823033	−0.411516	+	0.911402i	$0.635001\pi$
$468$	0	0
$469$	−3.56034e6	−0.747412
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.04006e7	2.13750
$474$	0	0
$475$	1.30838e6	0.266072
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.60106e6	−0.916262	−0.458131	−	0.888885i	$-0.651481\pi$
$479$	−4.60106e6	−0.916262	−0.458131	+	0.888885i	$0.651481\pi$
$480$	0	0
$481$	9.56298e6	1.88465
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.68482e6	−0.325237
$486$	0	0
$487$	−3.25177e6	−0.621294	−0.310647	−	0.950525i	$-0.600546\pi$
$487$	−3.25177e6	−0.621294	−0.310647	+	0.950525i	$0.600546\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.01153e6	1.49972	0.749862	−	0.661594i	$-0.230120\pi$
$491$	8.01153e6	1.49972	0.749862	+	0.661594i	$0.230120\pi$
$492$	0	0
$493$	−1.60203e6	−0.296861
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.27779e6	−1.32163
$498$	0	0
$499$	−6.52202e6	−1.17255	−0.586274	−	0.810113i	$-0.699406\pi$
$499$	−6.52202e6	−1.17255	−0.586274	+	0.810113i	$0.699406\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.15794e6	−1.43767	−0.718837	−	0.695179i	$-0.755325\pi$
$503$	−8.15794e6	−1.43767	−0.718837	+	0.695179i	$0.755325\pi$
$504$	0	0
$505$	−1.87105e6	−0.326480
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.02637e6	−0.175594	−0.0877969	−	0.996138i	$-0.527983\pi$
$509$	−1.02637e6	−0.175594	−0.0877969	+	0.996138i	$0.527983\pi$
$510$	0	0
$511$	997786.	0.169038
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.25153e6	0.706361
$516$	0	0
$517$	−8.47249e6	−1.39407
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.55278e6	1.05762	0.528812	−	0.848739i	$-0.322638\pi$
$521$	6.55278e6	1.05762	0.528812	+	0.848739i	$0.322638\pi$
$522$	0	0
$523$	−6.66508e6	−1.06549	−0.532747	−	0.846274i	$-0.678840\pi$
$523$	−6.66508e6	−1.06549	−0.532747	+	0.846274i	$0.678840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.43068e6	1.00863
$528$	0	0
$529$	8.84629e6	1.37443
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.16122e6	−0.786927
$534$	0	0
$535$	−74285.4	−0.0112207
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.18698e7	1.75984
$540$	0	0
$541$	−9.07626e6	−1.33326	−0.666629	−	0.745390i	$-0.732263\pi$
$541$	−9.07626e6	−1.33326	−0.666629	+	0.745390i	$0.732263\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.79725e6	0.403403
$546$	0	0
$547$	−9.96038e6	−1.42334	−0.711668	−	0.702515i	$-0.752060\pi$
$547$	−9.96038e6	−1.42334	−0.711668	+	0.702515i	$0.752060\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.44480e6	0.764017
$552$	0	0
$553$	1.58588e7	2.20524
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.66650e6	0.500742	0.250371	−	0.968150i	$-0.419447\pi$
$557$	3.66650e6	0.500742	0.250371	+	0.968150i	$0.419447\pi$
$558$	0	0
$559$	1.45877e7	1.97450
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.05055e6	−0.937458	−0.468729	−	0.883342i	$-0.655288\pi$
$563$	−7.05055e6	−0.937458	−0.468729	+	0.883342i	$0.655288\pi$
$564$	0	0
$565$	−3.87272e6	−0.510382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.26209e7	−1.63422	−0.817109	−	0.576484i	$-0.804424\pi$
$569$	−1.26209e7	−1.63422	−0.817109	+	0.576484i	$0.804424\pi$
$570$	0	0
$571$	−6.96326e6	−0.893763	−0.446882	−	0.894593i	$-0.647465\pi$
$571$	−6.96326e6	−0.893763	−0.446882	+	0.894593i	$0.647465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.44331e6	0.308184
$576$	0	0
$577$	−9.74622e6	−1.21870	−0.609350	−	0.792901i	$-0.708570\pi$
$577$	−9.74622e6	−1.21870	−0.609350	+	0.792901i	$0.708570\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.34002e7	−1.64691
$582$	0	0
$583$	−1.90863e7	−2.32568
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.16376e7	−1.39402	−0.697009	−	0.717063i	$-0.745486\pi$
$587$	−1.16376e7	−1.39402	−0.697009	+	0.717063i	$0.745486\pi$
$588$	0	0
$589$	−2.18559e7	−2.59585
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.94011e6	1.16079	0.580396	−	0.814334i	$-0.302898\pi$
$593$	9.94011e6	1.16079	0.580396	+	0.814334i	$0.302898\pi$
$594$	0	0
$595$	2.92492e6	0.338705
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.17722e6	−0.703438	−0.351719	−	0.936106i	$-0.614403\pi$
$599$	−6.17722e6	−0.703438	−0.351719	+	0.936106i	$0.614403\pi$
$600$	0	0
$601$	1.63700e7	1.84868	0.924341	−	0.381567i	$-0.124615\pi$
$601$	1.63700e7	1.84868	0.924341	+	0.381567i	$0.124615\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.45681e6	−0.606109
$606$	0	0
$607$	−5.83969e6	−0.643307	−0.321653	−	0.946858i	$-0.604239\pi$
$607$	−5.83969e6	−0.643307	−0.321653	+	0.946858i	$0.604239\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.18833e7	−1.28776
$612$	0	0
$613$	−4.13369e6	−0.444310	−0.222155	−	0.975011i	$-0.571309\pi$
$613$	−4.13369e6	−0.444310	−0.222155	+	0.975011i	$0.571309\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.20271e7	−1.27188	−0.635941	−	0.771738i	$-0.719388\pi$
$617$	−1.20271e7	−1.27188	−0.635941	+	0.771738i	$0.719388\pi$
$618$	0	0
$619$	−2.85292e6	−0.299269	−0.149635	−	0.988741i	$-0.547810\pi$
$619$	−2.85292e6	−0.299269	−0.149635	+	0.988741i	$0.547810\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.02343e6	0.415314
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.81873e6	−0.687190
$630$	0	0
$631$	−2.43994e6	−0.243952	−0.121976	−	0.992533i	$-0.538923\pi$
$631$	−2.43994e6	−0.243952	−0.121976	+	0.992533i	$0.538923\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.33123e6	−0.327846
$636$	0	0
$637$	1.66484e7	1.62564
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.88121e6	0.661485	0.330742	−	0.943721i	$-0.392701\pi$
$641$	6.88121e6	0.661485	0.330742	+	0.943721i	$0.392701\pi$
$642$	0	0
$643$	−215429.	−0.0205483	−0.0102742	−	0.999947i	$-0.503270\pi$
$643$	−215429.	−0.0205483	−0.0102742	+	0.999947i	$0.503270\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	322210.	0.0302607	0.0151303	−	0.999886i	$-0.495184\pi$
$647$	322210.	0.0302607	0.0151303	+	0.999886i	$0.495184\pi$
$648$	0	0
$649$	−1.27434e7	−1.18761
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−44942.7	−0.00412455	−0.00206227	−	0.999998i	$-0.500656\pi$
$653$	−44942.7	−0.00412455	−0.00206227	+	0.999998i	$0.500656\pi$
$654$	0	0
$655$	6.63073e6	0.603891
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.34595e7	−1.20730	−0.603650	−	0.797250i	$-0.706287\pi$
$659$	−1.34595e7	−1.20730	−0.603650	+	0.797250i	$0.706287\pi$
$660$	0	0
$661$	9.08451e6	0.808719	0.404360	−	0.914600i	$-0.367494\pi$
$661$	9.08451e6	0.808719	0.404360	+	0.914600i	$0.367494\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.94088e6	−0.871707
$666$	0	0
$667$	1.01678e7	0.884937
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.12364e6	0.439311
$672$	0	0
$673$	−6.34175e6	−0.539724	−0.269862	−	0.962899i	$-0.586978\pi$
$673$	−6.34175e6	−0.539724	−0.269862	+	0.962899i	$0.586978\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.79666e7	1.50659	0.753293	−	0.657685i	$-0.228464\pi$
$677$	1.79666e7	1.50659	0.753293	+	0.657685i	$0.228464\pi$
$678$	0	0
$679$	1.28010e7	1.06554
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.67389e7	1.37301	0.686506	−	0.727124i	$-0.259143\pi$
$683$	1.67389e7	1.37301	0.686506	+	0.727124i	$0.259143\pi$
$684$	0	0
$685$	−6.61138e6	−0.538351
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.67700e7	−2.14833
$690$	0	0
$691$	4.32004e6	0.344185	0.172093	−	0.985081i	$-0.444947\pi$
$691$	4.32004e6	0.344185	0.172093	+	0.985081i	$0.444947\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.64095e6	0.128865
$696$	0	0
$697$	3.68013e6	0.286933
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.10699e7	1.61945	0.809726	−	0.586808i	$-0.199616\pi$
$701$	2.10699e7	1.61945	0.809726	+	0.586808i	$0.199616\pi$
$702$	0	0
$703$	2.31747e7	1.76859
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.42159e7	1.06961
$708$	0	0
$709$	−1.03966e7	−0.776744	−0.388372	−	0.921503i	$-0.626962\pi$
$709$	−1.03966e7	−0.776744	−0.388372	+	0.921503i	$0.626962\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.08144e7	−3.00670
$714$	0	0
$715$	−1.33008e7	−0.972999
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.12339e7	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$719$	2.12339e7	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$720$	0	0
$721$	−3.23024e7	−2.31418
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.62558e6	0.114858
$726$	0	0
$727$	2.32288e7	1.63001	0.815007	−	0.579451i	$-0.196733\pi$
$727$	2.32288e7	1.63001	0.815007	+	0.579451i	$0.196733\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.04015e7	−0.719951
$732$	0	0
$733$	2.18744e7	1.50375	0.751877	−	0.659303i	$-0.229149\pi$
$733$	2.18744e7	1.50375	0.751877	+	0.659303i	$0.229149\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.15443e7	−0.782884
$738$	0	0
$739$	1.43886e7	0.969186	0.484593	−	0.874740i	$-0.338968\pi$
$739$	1.43886e7	0.969186	0.484593	+	0.874740i	$0.338968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.49771e7	−0.995303	−0.497651	−	0.867377i	$-0.665804\pi$
$743$	−1.49771e7	−0.995303	−0.497651	+	0.867377i	$0.665804\pi$
$744$	0	0
$745$	1.28239e7	0.846502
$746$	0	0
$747$	0	0
$748$	0	0
$749$	564410.	0.0367612
$750$	0	0
$751$	2.81239e7	1.81960	0.909801	−	0.415045i	$-0.136234\pi$
$751$	2.81239e7	1.81960	0.909801	+	0.415045i	$0.136234\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.32070e6	−0.531242
$756$	0	0
$757$	1.85444e7	1.17618	0.588090	−	0.808795i	$-0.299880\pi$
$757$	1.85444e7	1.17618	0.588090	+	0.808795i	$0.299880\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.75700e7	1.09979	0.549896	−	0.835233i	$-0.314667\pi$
$761$	1.75700e7	1.09979	0.549896	+	0.835233i	$0.314667\pi$
$762$	0	0
$763$	−2.12530e7	−1.32163
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.78736e7	−1.09704
$768$	0	0
$769$	−646698.	−0.0394353	−0.0197177	−	0.999806i	$-0.506277\pi$
$769$	−646698.	−0.0394353	−0.0197177	+	0.999806i	$0.506277\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.72615e6	0.284485	0.142242	−	0.989832i	$-0.454569\pi$
$773$	4.72615e6	0.284485	0.142242	+	0.989832i	$0.454569\pi$
$774$	0	0
$775$	−6.52521e6	−0.390247
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.25076e7	−0.738466
$780$	0	0
$781$	−2.35979e7	−1.38435
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.73850e6	−0.100693
$786$	0	0
$787$	−2.78545e6	−0.160309	−0.0801547	−	0.996782i	$-0.525541\pi$
$787$	−2.78545e6	−0.160309	−0.0801547	+	0.996782i	$0.525541\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.94244e7	1.67211
$792$	0	0
$793$	7.18632e6	0.405811
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.44471e7	0.805628	0.402814	−	0.915282i	$-0.368032\pi$
$797$	1.44471e7	0.805628	0.402814	+	0.915282i	$0.368032\pi$
$798$	0	0
$799$	8.47323e6	0.469550
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.23528e6	0.177061
$804$	0	0
$805$	−1.85639e7	−1.00967
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.88445e6	0.262389	0.131194	−	0.991357i	$-0.458119\pi$
$809$	4.88445e6	0.262389	0.131194	+	0.991357i	$0.458119\pi$
$810$	0	0
$811$	−2.50815e7	−1.33907	−0.669533	−	0.742782i	$-0.733506\pi$
$811$	−2.50815e7	−1.33907	−0.669533	+	0.742782i	$0.733506\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.64352e6	0.244880
$816$	0	0
$817$	3.53515e7	1.85290
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.47256e6	0.231579	0.115789	−	0.993274i	$-0.463060\pi$
$821$	4.47256e6	0.231579	0.115789	+	0.993274i	$0.463060\pi$
$822$	0	0
$823$	−1.97977e7	−1.01886	−0.509431	−	0.860511i	$-0.670144\pi$
$823$	−1.97977e7	−1.01886	−0.509431	+	0.860511i	$0.670144\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.16308e6	0.211666	0.105833	−	0.994384i	$-0.466249\pi$
$827$	4.16308e6	0.211666	0.105833	+	0.994384i	$0.466249\pi$
$828$	0	0
$829$	4.64118e6	0.234554	0.117277	−	0.993099i	$-0.462584\pi$
$829$	4.64118e6	0.234554	0.117277	+	0.993099i	$0.462584\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.18709e7	−0.592748
$834$	0	0
$835$	−7.60360e6	−0.377401
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.03037e7	1.48624	0.743122	−	0.669156i	$-0.233344\pi$
$839$	3.03037e7	1.48624	0.743122	+	0.669156i	$0.233344\pi$
$840$	0	0
$841$	−1.37463e7	−0.670189
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.37311e6	−0.451587
$846$	0	0
$847$	4.14600e7	1.98573
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.32773e7	2.04850
$852$	0	0
$853$	3.59123e7	1.68994	0.844968	−	0.534817i	$-0.179620\pi$
$853$	3.59123e7	1.68994	0.844968	+	0.534817i	$0.179620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.25034e7	0.581537	0.290769	−	0.956793i	$-0.406089\pi$
$857$	1.25034e7	0.581537	0.290769	+	0.956793i	$0.406089\pi$
$858$	0	0
$859$	−9.41856e6	−0.435513	−0.217757	−	0.976003i	$-0.569874\pi$
$859$	−9.41856e6	−0.435513	−0.217757	+	0.976003i	$0.569874\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.22162e7	0.558351	0.279176	−	0.960240i	$-0.409939\pi$
$863$	1.22162e7	0.558351	0.279176	+	0.960240i	$0.409939\pi$
$864$	0	0
$865$	−389651.	−0.0177066
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.14213e7	2.30990
$870$	0	0
$871$	−1.61918e7	−0.723183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.96791e6	−0.131048
$876$	0	0
$877$	−3.74239e7	−1.64305	−0.821523	−	0.570175i	$-0.806875\pi$
$877$	−3.74239e7	−1.64305	−0.821523	+	0.570175i	$0.806875\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.81490e7	1.22187	0.610933	−	0.791683i	$-0.290795\pi$
$881$	2.81490e7	1.22187	0.610933	+	0.791683i	$0.290795\pi$
$882$	0	0
$883$	−3.20992e7	−1.38545	−0.692727	−	0.721200i	$-0.743591\pi$
$883$	−3.20992e7	−1.38545	−0.692727	+	0.721200i	$0.743591\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.27921e7	1.39946	0.699730	−	0.714408i	$-0.253304\pi$
$887$	3.27921e7	1.39946	0.699730	+	0.714408i	$0.253304\pi$
$888$	0	0
$889$	2.53102e7	1.07409
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.87978e7	−1.20846
$894$	0	0
$895$	7.94746e6	0.331643
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.71545e7	−1.12058
$900$	0	0
$901$	1.90879e7	0.783333
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.22706e6	0.130974
$906$	0	0
$907$	−3.05934e7	−1.23484	−0.617419	−	0.786634i	$-0.711822\pi$
$907$	−3.05934e7	−1.23484	−0.617419	+	0.786634i	$0.711822\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.08772e7	−0.434230	−0.217115	−	0.976146i	$-0.569665\pi$
$911$	−1.08772e7	−0.434230	−0.217115	+	0.976146i	$0.569665\pi$
$912$	0	0
$913$	−4.34496e7	−1.72508
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.03793e7	−1.97847
$918$	0	0
$919$	2.48621e6	0.0971065	0.0485532	−	0.998821i	$-0.484539\pi$
$919$	2.48621e6	0.0971065	0.0485532	+	0.998821i	$0.484539\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.30980e7	−1.27878
$924$	0	0
$925$	6.91896e6	0.265880
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.41958e7	−0.919815	−0.459908	−	0.887967i	$-0.652117\pi$
$929$	−2.41958e7	−0.919815	−0.459908	+	0.887967i	$0.652117\pi$
$930$	0	0
$931$	4.03454e7	1.52553
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.48392e6	0.354780
$936$	0	0
$937$	388651.	0.0144614	0.00723070	−	0.999974i	$-0.497698\pi$
$937$	388651.	0.0144614	0.00723070	+	0.999974i	$0.497698\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.13048e7	1.52064	0.760320	−	0.649549i	$-0.225042\pi$
$941$	4.13048e7	1.52064	0.760320	+	0.649549i	$0.225042\pi$
$942$	0	0
$943$	−2.33571e7	−0.855342
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.78183e7	−1.37033	−0.685167	−	0.728386i	$-0.740271\pi$
$947$	−3.78183e7	−1.37033	−0.685167	+	0.728386i	$0.740271\pi$
$948$	0	0
$949$	4.53774e6	0.163559
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.95133e6	−0.105265	−0.0526327	−	0.998614i	$-0.516761\pi$
$953$	−2.95133e6	−0.105265	−0.0526327	+	0.998614i	$0.516761\pi$
$954$	0	0
$955$	−4.69776e6	−0.166680
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.02323e7	1.76375
$960$	0	0
$961$	8.03714e7	2.80733
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.00883e7	−0.348737
$966$	0	0
$967$	2.89870e7	0.996866	0.498433	−	0.866928i	$-0.333909\pi$
$967$	2.89870e7	0.996866	0.498433	+	0.866928i	$0.333909\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.34180e7	0.797081	0.398541	−	0.917151i	$-0.369517\pi$
$971$	2.34180e7	0.797081	0.398541	+	0.917151i	$0.369517\pi$
$972$	0	0
$973$	−1.24677e7	−0.422187
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.05797e7	1.36010	0.680052	−	0.733164i	$-0.261957\pi$
$977$	4.05797e7	1.36010	0.680052	+	0.733164i	$0.261957\pi$
$978$	0	0
$979$	1.30458e7	0.435025
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.62683e7	−0.867057	−0.433529	−	0.901140i	$-0.642732\pi$
$983$	−2.62683e7	−0.867057	−0.433529	+	0.901140i	$0.642732\pi$
$984$	0	0
$985$	−6.03237e6	−0.198106
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.60166e7	2.14616
$990$	0	0
$991$	2.45268e7	0.793334	0.396667	−	0.917962i	$-0.370167\pi$
$991$	2.45268e7	0.793334	0.396667	+	0.917962i	$0.370167\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.44441e7	0.462524
$996$	0	0
$997$	−9.15082e6	−0.291556	−0.145778	−	0.989317i	$-0.546569\pi$
$997$	−9.15082e6	−0.291556	−0.145778	+	0.989317i	$0.546569\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.ba.1.2		2
3.2	odd	2		240.6.a.p.1.2		2
4.3	odd	2		360.6.a.k.1.1		2
12.11	even	2		120.6.a.h.1.1	✓	2
24.5	odd	2		960.6.a.bk.1.2		2
24.11	even	2		960.6.a.be.1.1		2
60.23	odd	4		600.6.f.m.49.4		4
60.47	odd	4		600.6.f.m.49.1		4
60.59	even	2		600.6.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.h.1.1	✓	2	12.11	even	2
240.6.a.p.1.2		2	3.2	odd	2
360.6.a.k.1.1		2	4.3	odd	2
600.6.a.l.1.2		2	60.59	even	2
600.6.f.m.49.1		4	60.47	odd	4
600.6.f.m.49.4		4	60.23	odd	4
720.6.a.ba.1.2		2	1.1	even	1	trivial
960.6.a.be.1.1		2	24.11	even	2
960.6.a.bk.1.2		2	24.5	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} +189.946 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} +189.946 q^{7} +615.892 q^{11} +863.839 q^{13} -615.946 q^{17} +2093.41 q^{19} +3909.30 q^{23} +625.000 q^{25} +2600.92 q^{29} -10440.3 q^{31} -4748.66 q^{35} +11070.3 q^{37} -5974.75 q^{41} +16887.1 q^{43} -13756.4 q^{47} +19272.6 q^{49} -30989.6 q^{53} -15397.3 q^{55} -20690.9 q^{59} +8319.05 q^{61} -21596.0 q^{65} -18744.0 q^{67} -38315.0 q^{71} +5252.99 q^{73} +116986. q^{77} +83490.7 q^{79} -70547.4 q^{83} +15398.7 q^{85} +21181.9 q^{89} +164083. q^{91} -52335.2 q^{95} +67392.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 8 q^{7}+O(q^{10}) \) 2 * q - 50 * q^5 + 8 * q^7	\( 2 q - 50 q^{5} + 8 q^{7} + 488 q^{11} + 612 q^{13} - 860 q^{17} + 96 q^{19} + 2984 q^{23} + 1250 q^{25} - 2236 q^{29} - 9352 q^{31} - 200 q^{35} + 10612 q^{37} - 17156 q^{41} - 440 q^{43} - 16728 q^{47} + 35570 q^{49} - 31484 q^{53} - 12200 q^{55} - 61464 q^{59} + 51596 q^{61} - 15300 q^{65} + 45816 q^{67} - 97456 q^{71} + 58852 q^{73} + 140256 q^{77} + 116776 q^{79} - 136632 q^{83} + 21500 q^{85} + 124924 q^{89} + 209904 q^{91} - 2400 q^{95} - 39260 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 8 * q^7 + 488 * q^11 + 612 * q^13 - 860 * q^17 + 96 * q^19 + 2984 * q^23 + 1250 * q^25 - 2236 * q^29 - 9352 * q^31 - 200 * q^35 + 10612 * q^37 - 17156 * q^41 - 440 * q^43 - 16728 * q^47 + 35570 * q^49 - 31484 * q^53 - 12200 * q^55 - 61464 * q^59 + 51596 * q^61 - 15300 * q^65 + 45816 * q^67 - 97456 * q^71 + 58852 * q^73 + 140256 * q^77 + 116776 * q^79 - 136632 * q^83 + 21500 * q^85 + 124924 * q^89 + 209904 * q^91 - 2400 * q^95 - 39260 * q^97

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