LMFDB - Embedded newform 729.6.a.e.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,6,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	729.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:	$116.919804644$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.08001 q^{2} -6.19352 q^{4} +16.0034 q^{5} -124.463 q^{7} +194.023 q^{8} -81.2975 q^{10} -569.839 q^{11} +506.151 q^{13} +632.275 q^{14} -787.448 q^{16} +802.220 q^{17} +1985.08 q^{19} -99.1175 q^{20} +2894.78 q^{22} -2626.14 q^{23} -2868.89 q^{25} -2571.25 q^{26} +770.867 q^{28} -2863.57 q^{29} -1061.51 q^{31} -2208.51 q^{32} -4075.28 q^{34} -1991.84 q^{35} -10064.4 q^{37} -10084.2 q^{38} +3105.04 q^{40} +8260.52 q^{41} +16683.2 q^{43} +3529.31 q^{44} +13340.8 q^{46} +17511.0 q^{47} -1315.86 q^{49} +14574.0 q^{50} -3134.85 q^{52} -15083.4 q^{53} -9119.36 q^{55} -24148.8 q^{56} +14547.0 q^{58} -7859.28 q^{59} -34078.8 q^{61} +5392.50 q^{62} +36417.6 q^{64} +8100.14 q^{65} -62234.1 q^{67} -4968.57 q^{68} +10118.6 q^{70} -10390.9 q^{71} +22248.0 q^{73} +51127.0 q^{74} -12294.6 q^{76} +70924.1 q^{77} -85664.5 q^{79} -12601.9 q^{80} -41963.5 q^{82} +32676.2 q^{83} +12838.3 q^{85} -84750.7 q^{86} -110562. q^{88} +116390. q^{89} -62997.2 q^{91} +16265.0 q^{92} -88955.9 q^{94} +31768.1 q^{95} -163216. q^{97} +6684.57 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28}+ \cdots + 463410 q^{98}+O(q^{100}) $ 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * q^98

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 $

$2$	−5.08001	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$2$	−5.08001	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$3$	0	0
$3$	0	0
$4$	−6.19352	−0.193548
$5$	16.0034	0.286278	0.143139	−	0.989703i	$-0.454280\pi$
$5$	16.0034	0.286278	0.143139	+	0.989703i	$0.454280\pi$
$6$	0	0
$7$	−124.463	−0.960056	−0.480028	−	0.877253i	$-0.659374\pi$
$7$	−124.463	−0.960056	−0.480028	+	0.877253i	$0.659374\pi$
$8$	194.023	1.07184
$9$	0	0
$10$	−81.2975	−0.257085
$11$	−569.839	−1.41994	−0.709970	−	0.704232i	$-0.751292\pi$
$11$	−569.839	−1.41994	−0.709970	+	0.704232i	$0.751292\pi$
$12$	0	0
$13$	506.151	0.830656	0.415328	−	0.909672i	$-0.363667\pi$
$13$	506.151	0.830656	0.415328	+	0.909672i	$0.363667\pi$
$14$	632.275	0.862156
$15$	0	0
$16$	−787.448	−0.768992
$17$	802.220	0.673242	0.336621	−	0.941640i	$-0.390716\pi$
$17$	802.220	0.673242	0.336621	+	0.941640i	$0.390716\pi$
$18$	0	0
$19$	1985.08	1.26152	0.630760	−	0.775978i	$-0.282743\pi$
$19$	1985.08	1.26152	0.630760	+	0.775978i	$0.282743\pi$
$20$	−99.1175	−0.0554084
$21$	0	0
$22$	2894.78	1.27514
$23$	−2626.14	−1.03514	−0.517568	−	0.855642i	$-0.673163\pi$
$23$	−2626.14	−1.03514	−0.517568	+	0.855642i	$0.673163\pi$
$24$	0	0
$25$	−2868.89	−0.918045
$26$	−2571.25	−0.745952
$27$	0	0
$28$	770.867	0.185817
$29$	−2863.57	−0.632285	−0.316143	−	0.948712i	$-0.602388\pi$
$29$	−2863.57	−0.632285	−0.316143	+	0.948712i	$0.602388\pi$
$30$	0	0
$31$	−1061.51	−0.198391	−0.0991955	−	0.995068i	$-0.531627\pi$
$31$	−1061.51	−0.198391	−0.0991955	+	0.995068i	$0.531627\pi$
$32$	−2208.51	−0.381263
$33$	0	0
$34$	−4075.28	−0.604589
$35$	−1991.84	−0.274843
$36$	0	0
$37$	−10064.4	−1.20860	−0.604299	−	0.796758i	$-0.706547\pi$
$37$	−10064.4	−1.20860	−0.604299	+	0.796758i	$0.706547\pi$
$38$	−10084.2	−1.13288
$39$	0	0
$40$	3105.04	0.306843
$41$	8260.52	0.767446	0.383723	−	0.923448i	$-0.374642\pi$
$41$	8260.52	0.767446	0.383723	+	0.923448i	$0.374642\pi$
$42$	0	0
$43$	16683.2	1.37597	0.687983	−	0.725727i	$-0.258496\pi$
$43$	16683.2	1.37597	0.687983	+	0.725727i	$0.258496\pi$
$44$	3529.31	0.274826
$45$	0	0
$46$	13340.8	0.929581
$47$	17511.0	1.15629	0.578143	−	0.815935i	$-0.303777\pi$
$47$	17511.0	1.15629	0.578143	+	0.815935i	$0.303777\pi$
$48$	0	0
$49$	−1315.86	−0.0782923
$50$	14574.0	0.824429
$51$	0	0
$52$	−3134.85	−0.160772
$53$	−15083.4	−0.737583	−0.368791	−	0.929512i	$-0.620228\pi$
$53$	−15083.4	−0.737583	−0.368791	+	0.929512i	$0.620228\pi$
$54$	0	0
$55$	−9119.36	−0.406497
$56$	−24148.8	−1.02902
$57$	0	0
$58$	14547.0	0.567809
$59$	−7859.28	−0.293936	−0.146968	−	0.989141i	$-0.546951\pi$
$59$	−7859.28	−0.293936	−0.146968	+	0.989141i	$0.546951\pi$
$60$	0	0
$61$	−34078.8	−1.17263	−0.586314	−	0.810084i	$-0.699422\pi$
$61$	−34078.8	−1.17263	−0.586314	+	0.810084i	$0.699422\pi$
$62$	5392.50	0.178160
$63$	0	0
$64$	36417.6	1.11138
$65$	8100.14	0.237798
$66$	0	0
$67$	−62234.1	−1.69372	−0.846859	−	0.531817i	$-0.821509\pi$
$67$	−62234.1	−1.69372	−0.846859	+	0.531817i	$0.821509\pi$
$68$	−4968.57	−0.130304
$69$	0	0
$70$	10118.6	0.246816
$71$	−10390.9	−0.244629	−0.122315	−	0.992491i	$-0.539032\pi$
$71$	−10390.9	−0.244629	−0.122315	+	0.992491i	$0.539032\pi$
$72$	0	0
$73$	22248.0	0.488635	0.244317	−	0.969695i	$-0.421436\pi$
$73$	22248.0	0.488635	0.244317	+	0.969695i	$0.421436\pi$
$74$	51127.0	1.08535
$75$	0	0
$76$	−12294.6	−0.244164
$77$	70924.1	1.36322
$78$	0	0
$79$	−85664.5	−1.54431	−0.772153	−	0.635437i	$-0.780820\pi$
$79$	−85664.5	−1.54431	−0.772153	+	0.635437i	$0.780820\pi$
$80$	−12601.9	−0.220145
$81$	0	0
$82$	−41963.5	−0.689187
$83$	32676.2	0.520639	0.260319	−	0.965523i	$-0.416172\pi$
$83$	32676.2	0.520639	0.260319	+	0.965523i	$0.416172\pi$
$84$	0	0
$85$	12838.3	0.192734
$86$	−84750.7	−1.23565
$87$	0	0
$88$	−110562.	−1.52195
$89$	116390.	1.55754	0.778772	−	0.627308i	$-0.215843\pi$
$89$	116390.	1.55754	0.778772	+	0.627308i	$0.215843\pi$
$90$	0	0
$91$	−62997.2	−0.797477
$92$	16265.0	0.200348
$93$	0	0
$94$	−88955.9	−1.03838
$95$	31768.1	0.361145
$96$	0	0
$97$	−163216.	−1.76130	−0.880648	−	0.473771i	$-0.842892\pi$
$97$	−163216.	−1.76130	−0.880648	+	0.473771i	$0.842892\pi$
$98$	6684.57	0.0703086
$99$	0	0
$100$	17768.5	0.177685
$101$	−17987.3	−0.175454	−0.0877268	−	0.996145i	$-0.527960\pi$
$101$	−17987.3	−0.175454	−0.0877268	+	0.996145i	$0.527960\pi$
$102$	0	0
$103$	146320.	1.35898	0.679488	−	0.733687i	$-0.262202\pi$
$103$	146320.	1.35898	0.679488	+	0.733687i	$0.262202\pi$
$104$	98205.0	0.890329
$105$	0	0
$106$	76624.0	0.662369
$107$	−200073.	−1.68938	−0.844692	−	0.535253i	$-0.820216\pi$
$107$	−200073.	−1.68938	−0.844692	+	0.535253i	$0.820216\pi$
$108$	0	0
$109$	−117299.	−0.945645	−0.472822	−	0.881158i	$-0.656765\pi$
$109$	−117299.	−0.945645	−0.472822	+	0.881158i	$0.656765\pi$
$110$	46326.4	0.365046
$111$	0	0
$112$	98008.4	0.738275
$113$	219278.	1.61547	0.807736	−	0.589544i	$-0.200692\pi$
$113$	219278.	1.61547	0.807736	+	0.589544i	$0.200692\pi$
$114$	0	0
$115$	−42027.2	−0.296337
$116$	17735.6	0.122377
$117$	0	0
$118$	39925.2	0.263962
$119$	−99847.0	−0.646350
$120$	0	0
$121$	163665.	1.01623
$122$	173121.	1.05305
$123$	0	0
$124$	6574.52	0.0383981
$125$	−95922.7	−0.549094
$126$	0	0
$127$	70990.5	0.390563	0.195282	−	0.980747i	$-0.437438\pi$
$127$	70990.5	0.390563	0.195282	+	0.980747i	$0.437438\pi$
$128$	−114329.	−0.616783
$129$	0	0
$130$	−41148.8	−0.213549
$131$	−105374.	−0.536480	−0.268240	−	0.963352i	$-0.586442\pi$
$131$	−105374.	−0.536480	−0.268240	+	0.963352i	$0.586442\pi$
$132$	0	0
$133$	−247070.	−1.21113
$134$	316150.	1.52100
$135$	0	0
$136$	155649.	0.721606
$137$	−211495.	−0.962717	−0.481358	−	0.876524i	$-0.659856\pi$
$137$	−211495.	−0.962717	−0.481358	+	0.876524i	$0.659856\pi$
$138$	0	0
$139$	−118100.	−0.518459	−0.259230	−	0.965816i	$-0.583469\pi$
$139$	−118100.	−0.518459	−0.259230	+	0.965816i	$0.583469\pi$
$140$	12336.5	0.0531951
$141$	0	0
$142$	52786.0	0.219684
$143$	−288424.	−1.17948
$144$	0	0
$145$	−45826.9	−0.181009
$146$	−113020.	−0.438807
$147$	0	0
$148$	62333.8	0.233921
$149$	68356.6	0.252241	0.126120	−	0.992015i	$-0.459747\pi$
$149$	68356.6	0.252241	0.126120	+	0.992015i	$0.459747\pi$
$150$	0	0
$151$	96700.2	0.345132	0.172566	−	0.984998i	$-0.444794\pi$
$151$	96700.2	0.345132	0.172566	+	0.984998i	$0.444794\pi$
$152$	385152.	1.35214
$153$	0	0
$154$	−360295.	−1.22421
$155$	−16987.9	−0.0567949
$156$	0	0
$157$	−249547.	−0.807986	−0.403993	−	0.914762i	$-0.632378\pi$
$157$	−249547.	−0.807986	−0.403993	+	0.914762i	$0.632378\pi$
$158$	435176.	1.38683
$159$	0	0
$160$	−35343.7	−0.109147
$161$	326858.	0.993789
$162$	0	0
$163$	434926.	1.28217	0.641086	−	0.767469i	$-0.278484\pi$
$163$	434926.	1.28217	0.641086	+	0.767469i	$0.278484\pi$
$164$	−51161.7	−0.148537
$165$	0	0
$166$	−165995.	−0.467548
$167$	649997.	1.80352	0.901758	−	0.432242i	$-0.142277\pi$
$167$	649997.	1.80352	0.901758	+	0.432242i	$0.142277\pi$
$168$	0	0
$169$	−115105.	−0.310010
$170$	−65218.5	−0.173081
$171$	0	0
$172$	−103328.	−0.266315
$173$	637423.	1.61924	0.809622	−	0.586952i	$-0.199672\pi$
$173$	637423.	1.61924	0.809622	+	0.586952i	$0.199672\pi$
$174$	0	0
$175$	357072.	0.881375
$176$	448718.	1.09192
$177$	0	0
$178$	−591261.	−1.39872
$179$	−130748.	−0.305002	−0.152501	−	0.988303i	$-0.548733\pi$
$179$	−130748.	−0.305002	−0.152501	+	0.988303i	$0.548733\pi$
$180$	0	0
$181$	39307.8	0.0891831	0.0445916	−	0.999005i	$-0.485801\pi$
$181$	39307.8	0.0891831	0.0445916	+	0.999005i	$0.485801\pi$
$182$	320026.	0.716156
$183$	0	0
$184$	−509532.	−1.10950
$185$	−161064.	−0.345995
$186$	0	0
$187$	−457136.	−0.955963
$188$	−108455.	−0.223797
$189$	0	0
$190$	−161382.	−0.324318
$191$	483071.	0.958136	0.479068	−	0.877778i	$-0.340975\pi$
$191$	483071.	0.958136	0.479068	+	0.877778i	$0.340975\pi$
$192$	0	0
$193$	−466979.	−0.902410	−0.451205	−	0.892420i	$-0.649006\pi$
$193$	−466979.	−0.902410	−0.451205	+	0.892420i	$0.649006\pi$
$194$	829137.	1.58169
$195$	0	0
$196$	8149.80	0.0151533
$197$	−458069.	−0.840941	−0.420471	−	0.907306i	$-0.638135\pi$
$197$	−458069.	−0.840941	−0.420471	+	0.907306i	$0.638135\pi$
$198$	0	0
$199$	684326.	1.22498	0.612492	−	0.790477i	$-0.290167\pi$
$199$	684326.	1.22498	0.612492	+	0.790477i	$0.290167\pi$
$200$	−556632.	−0.983995
$201$	0	0
$202$	91375.6	0.157562
$203$	356410.	0.607029
$204$	0	0
$205$	132197.	0.219703
$206$	−743308.	−1.22040
$207$	0	0
$208$	−398567.	−0.638768
$209$	−1.13118e6	−1.79128
$210$	0	0
$211$	541424.	0.837205	0.418602	−	0.908170i	$-0.362520\pi$
$211$	541424.	0.837205	0.418602	+	0.908170i	$0.362520\pi$
$212$	93419.6	0.142757
$213$	0	0
$214$	1.01637e6	1.51711
$215$	266988.	0.393908
$216$	0	0
$217$	132120.	0.190466
$218$	595880.	0.849214
$219$	0	0
$220$	56481.0	0.0786766
$221$	406044.	0.559233
$222$	0	0
$223$	1.29203e6	1.73985	0.869923	−	0.493188i	$-0.164168\pi$
$223$	1.29203e6	1.73985	0.869923	+	0.493188i	$0.164168\pi$
$224$	274879.	0.366033
$225$	0	0
$226$	−1.11394e6	−1.45074
$227$	960510.	1.23719	0.618596	−	0.785709i	$-0.287702\pi$
$227$	960510.	1.23719	0.618596	+	0.785709i	$0.287702\pi$
$228$	0	0
$229$	−130323.	−0.164222	−0.0821112	−	0.996623i	$-0.526166\pi$
$229$	−130323.	−0.164222	−0.0821112	+	0.996623i	$0.526166\pi$
$230$	213498.	0.266118
$231$	0	0
$232$	−555600.	−0.677707
$233$	593425.	0.716104	0.358052	−	0.933702i	$-0.383441\pi$
$233$	593425.	0.716104	0.358052	+	0.933702i	$0.383441\pi$
$234$	0	0
$235$	280235.	0.331019
$236$	48676.6	0.0568906
$237$	0	0
$238$	507224.	0.580440
$239$	1.23978e6	1.40395	0.701974	−	0.712202i	$-0.252302\pi$
$239$	1.23978e6	1.40395	0.701974	+	0.712202i	$0.252302\pi$
$240$	0	0
$241$	1.04508e6	1.15906	0.579532	−	0.814949i	$-0.303235\pi$
$241$	1.04508e6	1.15906	0.579532	+	0.814949i	$0.303235\pi$
$242$	−831420.	−0.912603
$243$	0	0
$244$	211068.	0.226959
$245$	−21058.2	−0.0224133
$246$	0	0
$247$	1.00475e6	1.04789
$248$	−205959.	−0.212643
$249$	0	0
$250$	487288.	0.493101
$251$	670594.	0.671854	0.335927	−	0.941888i	$-0.390950\pi$
$251$	670594.	0.671854	0.335927	+	0.941888i	$0.390950\pi$
$252$	0	0
$253$	1.49647e6	1.46983
$254$	−360632.	−0.350736
$255$	0	0
$256$	−584569.	−0.557488
$257$	1.10596e6	1.04449	0.522247	−	0.852795i	$-0.325094\pi$
$257$	1.10596e6	1.04449	0.522247	+	0.852795i	$0.325094\pi$
$258$	0	0
$259$	1.25264e6	1.16032
$260$	−50168.4	−0.0460253
$261$	0	0
$262$	535298.	0.481773
$263$	521342.	0.464765	0.232382	−	0.972624i	$-0.425348\pi$
$263$	521342.	0.464765	0.232382	+	0.972624i	$0.425348\pi$
$264$	0	0
$265$	−241387.	−0.211154
$266$	1.25512e6	1.08763
$267$	0	0
$268$	385448.	0.327815
$269$	752426.	0.633991	0.316995	−	0.948427i	$-0.397326\pi$
$269$	752426.	0.633991	0.316995	+	0.948427i	$0.397326\pi$
$270$	0	0
$271$	−1.05481e6	−0.872475	−0.436237	−	0.899832i	$-0.643689\pi$
$271$	−1.05481e6	−0.872475	−0.436237	+	0.899832i	$0.643689\pi$
$272$	−631706.	−0.517718
$273$	0	0
$274$	1.07440e6	0.864545
$275$	1.63480e6	1.30357
$276$	0	0
$277$	−70666.7	−0.0553370	−0.0276685	−	0.999617i	$-0.508808\pi$
$277$	−70666.7	−0.0553370	−0.0276685	+	0.999617i	$0.508808\pi$
$278$	599951.	0.465590
$279$	0	0
$280$	−386463.	−0.294587
$281$	−850903.	−0.642857	−0.321429	−	0.946934i	$-0.604163\pi$
$281$	−850903.	−0.642857	−0.321429	+	0.946934i	$0.604163\pi$
$282$	0	0
$283$	−9066.40	−0.00672928	−0.00336464	−	0.999994i	$-0.501071\pi$
$283$	−9066.40	−0.00672928	−0.00336464	+	0.999994i	$0.501071\pi$
$284$	64356.5	0.0473474
$285$	0	0
$286$	1.46520e6	1.05921
$287$	−1.02813e6	−0.736791
$288$	0	0
$289$	−776300.	−0.546745
$290$	232801.	0.162551
$291$	0	0
$292$	−137794.	−0.0945740
$293$	1.56759e6	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$293$	1.56759e6	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$294$	0	0
$295$	−125775.	−0.0841473
$296$	−1.95272e6	−1.29542
$297$	0	0
$298$	−347252.	−0.226519
$299$	−1.32922e6	−0.859843
$300$	0	0
$301$	−2.07645e6	−1.32100
$302$	−491238.	−0.309938
$303$	0	0
$304$	−1.56315e6	−0.970098
$305$	−545378.	−0.335697
$306$	0	0
$307$	−117066.	−0.0708899	−0.0354450	−	0.999372i	$-0.511285\pi$
$307$	−117066.	−0.0708899	−0.0354450	+	0.999372i	$0.511285\pi$
$308$	−439270.	−0.263848
$309$	0	0
$310$	86298.5	0.0510034
$311$	1.09944e6	0.644568	0.322284	−	0.946643i	$-0.395549\pi$
$311$	1.09944e6	0.644568	0.322284	+	0.946643i	$0.395549\pi$
$312$	0	0
$313$	2.54078e6	1.46591	0.732954	−	0.680278i	$-0.238141\pi$
$313$	2.54078e6	1.46591	0.732954	+	0.680278i	$0.238141\pi$
$314$	1.26770e6	0.725593
$315$	0	0
$316$	530565.	0.298897
$317$	432967.	0.241995	0.120998	−	0.992653i	$-0.461391\pi$
$317$	432967.	0.241995	0.120998	+	0.992653i	$0.461391\pi$
$318$	0	0
$319$	1.63177e6	0.897808
$320$	582805.	0.318162
$321$	0	0
$322$	−1.66044e6	−0.892450
$323$	1.59247e6	0.849308
$324$	0	0
$325$	−1.45209e6	−0.762580
$326$	−2.20943e6	−1.15143
$327$	0	0
$328$	1.60273e6	0.822577
$329$	−2.17948e6	−1.11010
$330$	0	0
$331$	−685734.	−0.344021	−0.172011	−	0.985095i	$-0.555026\pi$
$331$	−685734.	−0.344021	−0.172011	+	0.985095i	$0.555026\pi$
$332$	−202381.	−0.100768
$333$	0	0
$334$	−3.30199e6	−1.61961
$335$	−995958.	−0.484874
$336$	0	0
$337$	−2.32900e6	−1.11711	−0.558553	−	0.829469i	$-0.688643\pi$
$337$	−2.32900e6	−1.11711	−0.558553	+	0.829469i	$0.688643\pi$
$338$	584732.	0.278397
$339$	0	0
$340$	−79514.0	−0.0373032
$341$	604892.	0.281703
$342$	0	0
$343$	2.25563e6	1.03522
$344$	3.23693e6	1.47481
$345$	0	0
$346$	−3.23811e6	−1.45412
$347$	−1.99993e6	−0.891643	−0.445822	−	0.895122i	$-0.647088\pi$
$347$	−1.99993e6	−0.891643	−0.445822	+	0.895122i	$0.647088\pi$
$348$	0	0
$349$	−749766.	−0.329505	−0.164753	−	0.986335i	$-0.552683\pi$
$349$	−749766.	−0.329505	−0.164753	+	0.986335i	$0.552683\pi$
$350$	−1.81393e6	−0.791498
$351$	0	0
$352$	1.25849e6	0.541370
$353$	1.84975e6	0.790088	0.395044	−	0.918662i	$-0.370729\pi$
$353$	1.84975e6	0.790088	0.395044	+	0.918662i	$0.370729\pi$
$354$	0	0
$355$	−166290.	−0.0700320
$356$	−720863.	−0.301459
$357$	0	0
$358$	664201.	0.273900
$359$	−255855.	−0.104775	−0.0523875	−	0.998627i	$-0.516683\pi$
$359$	−255855.	−0.104775	−0.0523875	+	0.998627i	$0.516683\pi$
$360$	0	0
$361$	1.46445e6	0.591432
$362$	−199684.	−0.0800888
$363$	0	0
$364$	390175.	0.154350
$365$	356044.	0.139885
$366$	0	0
$367$	−866048.	−0.335642	−0.167821	−	0.985817i	$-0.553673\pi$
$367$	−866048.	−0.335642	−0.167821	+	0.985817i	$0.553673\pi$
$368$	2.06795e6	0.796012
$369$	0	0
$370$	818207.	0.310713
$371$	1.87734e6	0.708121
$372$	0	0
$373$	928422.	0.345520	0.172760	−	0.984964i	$-0.444731\pi$
$373$	928422.	0.345520	0.172760	+	0.984964i	$0.444731\pi$
$374$	2.32225e6	0.858481
$375$	0	0
$376$	3.39754e6	1.23935
$377$	−1.44940e6	−0.525212
$378$	0	0
$379$	1.89728e6	0.678475	0.339238	−	0.940701i	$-0.389831\pi$
$379$	1.89728e6	0.678475	0.339238	+	0.940701i	$0.389831\pi$
$380$	−196756.	−0.0698988
$381$	0	0
$382$	−2.45400e6	−0.860432
$383$	3.48210e6	1.21295	0.606477	−	0.795101i	$-0.292582\pi$
$383$	3.48210e6	1.21295	0.606477	+	0.795101i	$0.292582\pi$
$384$	0	0
$385$	1.13503e6	0.390260
$386$	2.37226e6	0.810389
$387$	0	0
$388$	1.01088e6	0.340895
$389$	2.34168e6	0.784609	0.392304	−	0.919835i	$-0.371678\pi$
$389$	2.34168e6	0.784609	0.392304	+	0.919835i	$0.371678\pi$
$390$	0	0
$391$	−2.10674e6	−0.696898
$392$	−255307.	−0.0839167
$393$	0	0
$394$	2.32700e6	0.755188
$395$	−1.37093e6	−0.442100
$396$	0	0
$397$	2.13641e6	0.680314	0.340157	−	0.940369i	$-0.389520\pi$
$397$	2.13641e6	0.680314	0.340157	+	0.940369i	$0.389520\pi$
$398$	−3.47638e6	−1.10007
$399$	0	0
$400$	2.25910e6	0.705969
$401$	−5.13546e6	−1.59485	−0.797423	−	0.603421i	$-0.793804\pi$
$401$	−5.13546e6	−1.59485	−0.797423	+	0.603421i	$0.793804\pi$
$402$	0	0
$403$	−537286.	−0.164795
$404$	111405.	0.0339586
$405$	0	0
$406$	−1.81057e6	−0.545129
$407$	5.73506e6	1.71614
$408$	0	0
$409$	166729.	0.0492835	0.0246418	−	0.999696i	$-0.492155\pi$
$409$	166729.	0.0492835	0.0246418	+	0.999696i	$0.492155\pi$
$410$	−671559.	−0.197299
$411$	0	0
$412$	−906238.	−0.263026
$413$	978192.	0.282195
$414$	0	0
$415$	522931.	0.149047
$416$	−1.11784e6	−0.316698
$417$	0	0
$418$	5.74638e6	1.60862
$419$	1.18915e6	0.330903	0.165452	−	0.986218i	$-0.447092\pi$
$419$	1.18915e6	0.330903	0.165452	+	0.986218i	$0.447092\pi$
$420$	0	0
$421$	−5.88992e6	−1.61959	−0.809793	−	0.586716i	$-0.800420\pi$
$421$	−5.88992e6	−1.61959	−0.809793	+	0.586716i	$0.800420\pi$
$422$	−2.75044e6	−0.751832
$423$	0	0
$424$	−2.92654e6	−0.790569
$425$	−2.30148e6	−0.618066
$426$	0	0
$427$	4.24157e6	1.12579
$428$	1.23915e6	0.326976
$429$	0	0
$430$	−1.35630e6	−0.353740
$431$	5.29916e6	1.37409	0.687043	−	0.726617i	$-0.258909\pi$
$431$	5.29916e6	1.37409	0.687043	+	0.726617i	$0.258909\pi$
$432$	0	0
$433$	−702037.	−0.179945	−0.0899726	−	0.995944i	$-0.528678\pi$
$433$	−702037.	−0.179945	−0.0899726	+	0.995944i	$0.528678\pi$
$434$	−671169.	−0.171044
$435$	0	0
$436$	726494.	0.183027
$437$	−5.21309e6	−1.30585
$438$	0	0
$439$	537299.	0.133062	0.0665311	−	0.997784i	$-0.478807\pi$
$439$	537299.	0.133062	0.0665311	+	0.997784i	$0.478807\pi$
$440$	−1.76937e6	−0.435699
$441$	0	0
$442$	−2.06271e6	−0.502206
$443$	3.74855e6	0.907515	0.453758	−	0.891125i	$-0.350083\pi$
$443$	3.74855e6	0.907515	0.453758	+	0.891125i	$0.350083\pi$
$444$	0	0
$445$	1.86264e6	0.445890
$446$	−6.56353e6	−1.56243
$447$	0	0
$448$	−4.53265e6	−1.06698
$449$	688744.	0.161229	0.0806143	−	0.996745i	$-0.474312\pi$
$449$	688744.	0.161229	0.0806143	+	0.996745i	$0.474312\pi$
$450$	0	0
$451$	−4.70716e6	−1.08973
$452$	−1.35811e6	−0.312671
$453$	0	0
$454$	−4.87940e6	−1.11103
$455$	−1.00817e6	−0.228300
$456$	0	0
$457$	−3.41882e6	−0.765748	−0.382874	−	0.923801i	$-0.625066\pi$
$457$	−3.41882e6	−0.765748	−0.382874	+	0.923801i	$0.625066\pi$
$458$	662042.	0.147476
$459$	0	0
$460$	260296.	0.0573553
$461$	−1.27283e6	−0.278944	−0.139472	−	0.990226i	$-0.544541\pi$
$461$	−1.27283e6	−0.278944	−0.139472	+	0.990226i	$0.544541\pi$
$462$	0	0
$463$	6.56896e6	1.42411	0.712056	−	0.702123i	$-0.247764\pi$
$463$	6.56896e6	1.42411	0.712056	+	0.702123i	$0.247764\pi$
$464$	2.25491e6	0.486222
$465$	0	0
$466$	−3.01461e6	−0.643081
$467$	−160043.	−0.0339581	−0.0169791	−	0.999856i	$-0.505405\pi$
$467$	−160043.	−0.0339581	−0.0169791	+	0.999856i	$0.505405\pi$
$468$	0	0
$469$	7.74586e6	1.62606
$470$	−1.42360e6	−0.297264
$471$	0	0
$472$	−1.52488e6	−0.315052
$473$	−9.50672e6	−1.95379
$474$	0	0
$475$	−5.69498e6	−1.15813
$476$	618405.	0.125099
$477$	0	0
$478$	−6.29811e6	−1.26078
$479$	−3.22750e6	−0.642729	−0.321364	−	0.946956i	$-0.604141\pi$
$479$	−3.22750e6	−0.642729	−0.321364	+	0.946956i	$0.604141\pi$
$480$	0	0
$481$	−5.09408e6	−1.00393
$482$	−5.30902e6	−1.04087
$483$	0	0
$484$	−1.01366e6	−0.196689
$485$	−2.61201e6	−0.504220
$486$	0	0
$487$	8.61882e6	1.64674	0.823371	−	0.567504i	$-0.192091\pi$
$487$	8.61882e6	1.64674	0.823371	+	0.567504i	$0.192091\pi$
$488$	−6.61209e6	−1.25687
$489$	0	0
$490$	106976.	0.0201278
$491$	1.39734e6	0.261576	0.130788	−	0.991410i	$-0.458249\pi$
$491$	1.39734e6	0.261576	0.130788	+	0.991410i	$0.458249\pi$
$492$	0	0
$493$	−2.29721e6	−0.425681
$494$	−5.10414e6	−0.941033
$495$	0	0
$496$	835887.	0.152561
$497$	1.29329e6	0.234858
$498$	0	0
$499$	237068.	0.0426208	0.0213104	−	0.999773i	$-0.493216\pi$
$499$	237068.	0.0426208	0.0213104	+	0.999773i	$0.493216\pi$
$500$	594100.	0.106276
$501$	0	0
$502$	−3.40662e6	−0.603343
$503$	−6.60991e6	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$503$	−6.60991e6	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$504$	0	0
$505$	−287858.	−0.0502285
$506$	−7.60210e6	−1.31995
$507$	0	0
$508$	−439681.	−0.0755925
$509$	2.59325e6	0.443660	0.221830	−	0.975085i	$-0.428797\pi$
$509$	2.59325e6	0.443660	0.221830	+	0.975085i	$0.428797\pi$
$510$	0	0
$511$	−2.76906e6	−0.469117
$512$	6.62815e6	1.11742
$513$	0	0
$514$	−5.61827e6	−0.937983
$515$	2.34162e6	0.389044
$516$	0	0
$517$	−9.97843e6	−1.64186
$518$	−6.36344e6	−1.04200
$519$	0	0
$520$	1.57162e6	0.254881
$521$	−7.93434e6	−1.28061	−0.640304	−	0.768121i	$-0.721192\pi$
$521$	−7.93434e6	−1.28061	−0.640304	+	0.768121i	$0.721192\pi$
$522$	0	0
$523$	−1.63391e6	−0.261201	−0.130600	−	0.991435i	$-0.541691\pi$
$523$	−1.63391e6	−0.261201	−0.130600	+	0.991435i	$0.541691\pi$
$524$	652633.	0.103834
$525$	0	0
$526$	−2.64842e6	−0.417371
$527$	−851568.	−0.133565
$528$	0	0
$529$	460253.	0.0715084
$530$	1.22625e6	0.189622
$531$	0	0
$532$	1.53023e6	0.234411
$533$	4.18107e6	0.637484
$534$	0	0
$535$	−3.20185e6	−0.483633
$536$	−1.20749e7	−1.81539
$537$	0	0
$538$	−3.82233e6	−0.569341
$539$	749827.	0.111170
$540$	0	0
$541$	4.48902e6	0.659414	0.329707	−	0.944083i	$-0.393050\pi$
$541$	4.48902e6	0.659414	0.329707	+	0.944083i	$0.393050\pi$
$542$	5.35847e6	0.783506
$543$	0	0
$544$	−1.77171e6	−0.256682
$545$	−1.87718e6	−0.270717
$546$	0	0
$547$	−1.04617e6	−0.149497	−0.0747487	−	0.997202i	$-0.523815\pi$
$547$	−1.04617e6	−0.149497	−0.0747487	+	0.997202i	$0.523815\pi$
$548$	1.30990e6	0.186331
$549$	0	0
$550$	−8.30482e6	−1.17064
$551$	−5.68442e6	−0.797641
$552$	0	0
$553$	1.06621e7	1.48262
$554$	358987.	0.0496941
$555$	0	0
$556$	731458.	0.100346
$557$	4.12965e6	0.563995	0.281998	−	0.959415i	$-0.409003\pi$
$557$	4.12965e6	0.563995	0.281998	+	0.959415i	$0.409003\pi$
$558$	0	0
$559$	8.44420e6	1.14295
$560$	1.56847e6	0.211352
$561$	0	0
$562$	4.32260e6	0.577303
$563$	−1.35498e7	−1.80161	−0.900806	−	0.434221i	$-0.857024\pi$
$563$	−1.35498e7	−1.80161	−0.900806	+	0.434221i	$0.857024\pi$
$564$	0	0
$565$	3.50920e6	0.462474
$566$	46057.4	0.00604308
$567$	0	0
$568$	−2.01608e6	−0.262203
$569$	4.47889e6	0.579949	0.289975	−	0.957034i	$-0.406353\pi$
$569$	4.47889e6	0.579949	0.289975	+	0.957034i	$0.406353\pi$
$570$	0	0
$571$	−9.06059e6	−1.16296	−0.581482	−	0.813559i	$-0.697527\pi$
$571$	−9.06059e6	−1.16296	−0.581482	+	0.813559i	$0.697527\pi$
$572$	1.78636e6	0.228286
$573$	0	0
$574$	5.22292e6	0.661658
$575$	7.53410e6	0.950302
$576$	0	0
$577$	−567132.	−0.0709160	−0.0354580	−	0.999371i	$-0.511289\pi$
$577$	−567132.	−0.0709160	−0.0354580	+	0.999371i	$0.511289\pi$
$578$	3.94361e6	0.490992
$579$	0	0
$580$	283830.	0.0350339
$581$	−4.06699e6	−0.499843
$582$	0	0
$583$	8.59513e6	1.04732
$584$	4.31664e6	0.523737
$585$	0	0
$586$	−7.96337e6	−0.957972
$587$	6.96933e6	0.834826	0.417413	−	0.908717i	$-0.362937\pi$
$587$	6.96933e6	0.834826	0.417413	+	0.908717i	$0.362937\pi$
$588$	0	0
$589$	−2.10719e6	−0.250274
$590$	638939.	0.0755665
$591$	0	0
$592$	7.92515e6	0.929402
$593$	−8.17118e6	−0.954219	−0.477109	−	0.878844i	$-0.658315\pi$
$593$	−8.17118e6	−0.954219	−0.477109	+	0.878844i	$0.658315\pi$
$594$	0	0
$595$	−1.59789e6	−0.185036
$596$	−423368.	−0.0488206
$597$	0	0
$598$	6.75245e6	0.772162
$599$	−5.80504e6	−0.661056	−0.330528	−	0.943796i	$-0.607227\pi$
$599$	−5.80504e6	−0.661056	−0.330528	+	0.943796i	$0.607227\pi$
$600$	0	0
$601$	−2.82872e6	−0.319451	−0.159725	−	0.987162i	$-0.551061\pi$
$601$	−2.82872e6	−0.319451	−0.159725	+	0.987162i	$0.551061\pi$
$602$	1.05484e7	1.18630
$603$	0	0
$604$	−598915.	−0.0667995
$605$	2.61920e6	0.290924
$606$	0	0
$607$	−1.02918e7	−1.13376	−0.566878	−	0.823802i	$-0.691849\pi$
$607$	−1.02918e7	−1.13376	−0.566878	+	0.823802i	$0.691849\pi$
$608$	−4.38407e6	−0.480970
$609$	0	0
$610$	2.77052e6	0.301465
$611$	8.86319e6	0.960477
$612$	0	0
$613$	1.09558e7	1.17759	0.588795	−	0.808282i	$-0.299602\pi$
$613$	1.09558e7	1.17759	0.588795	+	0.808282i	$0.299602\pi$
$614$	594696.	0.0636611
$615$	0	0
$616$	1.37609e7	1.46115
$617$	−1.14615e6	−0.121207	−0.0606037	−	0.998162i	$-0.519303\pi$
$617$	−1.14615e6	−0.121207	−0.0606037	+	0.998162i	$0.519303\pi$
$618$	0	0
$619$	−1.10949e6	−0.116385	−0.0581926	−	0.998305i	$-0.518534\pi$
$619$	−1.10949e6	−0.116385	−0.0581926	+	0.998305i	$0.518534\pi$
$620$	105215.	0.0109925
$621$	0	0
$622$	−5.58514e6	−0.578840
$623$	−1.44863e7	−1.49533
$624$	0	0
$625$	7.43019e6	0.760852
$626$	−1.29072e7	−1.31643
$627$	0	0
$628$	1.54558e6	0.156384
$629$	−8.07383e6	−0.813679
$630$	0	0
$631$	−1.05600e7	−1.05582	−0.527912	−	0.849299i	$-0.677025\pi$
$631$	−1.05600e7	−1.05582	−0.527912	+	0.849299i	$0.677025\pi$
$632$	−1.66209e7	−1.65525
$633$	0	0
$634$	−2.19948e6	−0.217318
$635$	1.13609e6	0.111810
$636$	0	0
$637$	−666023.	−0.0650340
$638$	−8.28942e6	−0.806255
$639$	0	0
$640$	−1.82966e6	−0.176571
$641$	1.60094e7	1.53897	0.769483	−	0.638668i	$-0.220514\pi$
$641$	1.60094e7	1.53897	0.769483	+	0.638668i	$0.220514\pi$
$642$	0	0
$643$	−7.66694e6	−0.731298	−0.365649	−	0.930753i	$-0.619153\pi$
$643$	−7.66694e6	−0.731298	−0.365649	+	0.930753i	$0.619153\pi$
$644$	−2.02440e6	−0.192346
$645$	0	0
$646$	−8.08977e6	−0.762702
$647$	−1.47237e7	−1.38279	−0.691397	−	0.722475i	$-0.743004\pi$
$647$	−1.47237e7	−1.38279	−0.691397	+	0.722475i	$0.743004\pi$
$648$	0	0
$649$	4.47852e6	0.417371
$650$	7.37663e6	0.684817
$651$	0	0
$652$	−2.69372e6	−0.248161
$653$	−1.29579e7	−1.18919	−0.594595	−	0.804025i	$-0.702688\pi$
$653$	−1.29579e7	−1.18919	−0.594595	+	0.804025i	$0.702688\pi$
$654$	0	0
$655$	−1.68634e6	−0.153582
$656$	−6.50473e6	−0.590159
$657$	0	0
$658$	1.10718e7	0.996900
$659$	1.98288e7	1.77862	0.889311	−	0.457302i	$-0.151184\pi$
$659$	1.98288e7	1.77862	0.889311	+	0.457302i	$0.151184\pi$
$660$	0	0
$661$	6.67103e6	0.593867	0.296934	−	0.954898i	$-0.404036\pi$
$661$	6.67103e6	0.593867	0.296934	+	0.954898i	$0.404036\pi$
$662$	3.48353e6	0.308941
$663$	0	0
$664$	6.33995e6	0.558041
$665$	−3.95396e6	−0.346720
$666$	0	0
$667$	7.52013e6	0.654502
$668$	−4.02577e6	−0.349066
$669$	0	0
$670$	5.05947e6	0.435430
$671$	1.94194e7	1.66506
$672$	0	0
$673$	1.60379e6	0.136493	0.0682463	−	0.997669i	$-0.478260\pi$
$673$	1.60379e6	0.136493	0.0682463	+	0.997669i	$0.478260\pi$
$674$	1.18313e7	1.00319
$675$	0	0
$676$	712903.	0.0600017
$677$	5.42092e6	0.454571	0.227285	−	0.973828i	$-0.427015\pi$
$677$	5.42092e6	0.454571	0.227285	+	0.973828i	$0.427015\pi$
$678$	0	0
$679$	2.03144e7	1.69094
$680$	2.49092e6	0.206580
$681$	0	0
$682$	−3.07286e6	−0.252977
$683$	8.07067e6	0.661999	0.331000	−	0.943631i	$-0.392614\pi$
$683$	8.07067e6	0.661999	0.331000	+	0.943631i	$0.392614\pi$
$684$	0	0
$685$	−3.38464e6	−0.275604
$686$	−1.14586e7	−0.929656
$687$	0	0
$688$	−1.31371e7	−1.05811
$689$	−7.63449e6	−0.612678
$690$	0	0
$691$	1.58046e7	1.25918	0.629590	−	0.776928i	$-0.283223\pi$
$691$	1.58046e7	1.25918	0.629590	+	0.776928i	$0.283223\pi$
$692$	−3.94789e6	−0.313401
$693$	0	0
$694$	1.01597e7	0.800719
$695$	−1.89001e6	−0.148423
$696$	0	0
$697$	6.62675e6	0.516677
$698$	3.80882e6	0.295905
$699$	0	0
$700$	−2.21153e6	−0.170588
$701$	2.34265e6	0.180058	0.0900291	−	0.995939i	$-0.471304\pi$
$701$	2.34265e6	0.180058	0.0900291	+	0.995939i	$0.471304\pi$
$702$	0	0
$703$	−1.99786e7	−1.52467
$704$	−2.07521e7	−1.57809
$705$	0	0
$706$	−9.39672e6	−0.709520
$707$	2.23876e6	0.168445
$708$	0	0
$709$	−9.68102e6	−0.723278	−0.361639	−	0.932318i	$-0.617783\pi$
$709$	−9.68102e6	−0.723278	−0.361639	+	0.932318i	$0.617783\pi$
$710$	844757.	0.0628906
$711$	0	0
$712$	2.25824e7	1.66943
$713$	2.78768e6	0.205362
$714$	0	0
$715$	−4.61577e6	−0.337660
$716$	809791.	0.0590324
$717$	0	0
$718$	1.29974e6	0.0940908
$719$	2.52286e6	0.182000	0.0909998	−	0.995851i	$-0.470994\pi$
$719$	2.52286e6	0.182000	0.0909998	+	0.995851i	$0.470994\pi$
$720$	0	0
$721$	−1.82115e7	−1.30469
$722$	−7.43939e6	−0.531122
$723$	0	0
$724$	−243454.	−0.0172612
$725$	8.21527e6	0.580466
$726$	0	0
$727$	2.62482e7	1.84189	0.920946	−	0.389691i	$-0.127418\pi$
$727$	2.62482e7	1.84189	0.920946	+	0.389691i	$0.127418\pi$
$728$	−1.22229e7	−0.854766
$729$	0	0
$730$	−1.80871e6	−0.125621
$731$	1.33836e7	0.926358
$732$	0	0
$733$	1.46222e7	1.00520	0.502602	−	0.864518i	$-0.332376\pi$
$733$	1.46222e7	1.00520	0.502602	+	0.864518i	$0.332376\pi$
$734$	4.39953e6	0.301416
$735$	0	0
$736$	5.79985e6	0.394659
$737$	3.54634e7	2.40498
$738$	0	0
$739$	1.37426e7	0.925671	0.462835	−	0.886444i	$-0.346832\pi$
$739$	1.37426e7	0.925671	0.462835	+	0.886444i	$0.346832\pi$
$740$	997554.	0.0669664
$741$	0	0
$742$	−9.53688e6	−0.635912
$743$	−1.64137e7	−1.09077	−0.545387	−	0.838184i	$-0.683617\pi$
$743$	−1.64137e7	−1.09077	−0.545387	+	0.838184i	$0.683617\pi$
$744$	0	0
$745$	1.09394e6	0.0722109
$746$	−4.71639e6	−0.310286
$747$	0	0
$748$	2.83128e6	0.185024
$749$	2.49017e7	1.62190
$750$	0	0
$751$	−9.97932e6	−0.645656	−0.322828	−	0.946458i	$-0.604634\pi$
$751$	−9.97932e6	−0.645656	−0.322828	+	0.946458i	$0.604634\pi$
$752$	−1.37890e7	−0.889175
$753$	0	0
$754$	7.36296e6	0.471654
$755$	1.54753e6	0.0988036
$756$	0	0
$757$	1.33937e7	0.849495	0.424747	−	0.905312i	$-0.360363\pi$
$757$	1.33937e7	0.849495	0.424747	+	0.905312i	$0.360363\pi$
$758$	−9.63821e6	−0.609289
$759$	0	0
$760$	6.16375e6	0.387089
$761$	−1.43357e7	−0.897341	−0.448670	−	0.893697i	$-0.648102\pi$
$761$	−1.43357e7	−0.897341	−0.448670	+	0.893697i	$0.648102\pi$
$762$	0	0
$763$	1.45994e7	0.907872
$764$	−2.99191e6	−0.185445
$765$	0	0
$766$	−1.76891e7	−1.08927
$767$	−3.97798e6	−0.244160
$768$	0	0
$769$	2.75117e6	0.167765	0.0838826	−	0.996476i	$-0.473268\pi$
$769$	2.75117e6	0.167765	0.0838826	+	0.996476i	$0.473268\pi$
$770$	−5.76595e6	−0.350464
$771$	0	0
$772$	2.89224e6	0.174659
$773$	1.11095e7	0.668722	0.334361	−	0.942445i	$-0.391480\pi$
$773$	1.11095e7	0.668722	0.334361	+	0.942445i	$0.391480\pi$
$774$	0	0
$775$	3.04537e6	0.182132
$776$	−3.16676e7	−1.88782
$777$	0	0
$778$	−1.18957e7	−0.704600
$779$	1.63978e7	0.968148
$780$	0	0
$781$	5.92115e6	0.347359
$782$	1.07023e7	0.625833
$783$	0	0
$784$	1.03617e6	0.0602061
$785$	−3.99361e6	−0.231308
$786$	0	0
$787$	−1.09250e7	−0.628757	−0.314379	−	0.949298i	$-0.601796\pi$
$787$	−1.09250e7	−0.628757	−0.314379	+	0.949298i	$0.601796\pi$
$788$	2.83706e6	0.162762
$789$	0	0
$790$	6.96431e6	0.397018
$791$	−2.72921e7	−1.55094
$792$	0	0
$793$	−1.72490e7	−0.974051
$794$	−1.08530e7	−0.610940
$795$	0	0
$796$	−4.23839e6	−0.237093
$797$	2.23804e7	1.24802	0.624011	−	0.781416i	$-0.285502\pi$
$797$	2.23804e7	1.24802	0.624011	+	0.781416i	$0.285502\pi$
$798$	0	0
$799$	1.40477e7	0.778461
$800$	6.33597e6	0.350016
$801$	0	0
$802$	2.60882e7	1.43221
$803$	−1.26778e7	−0.693832
$804$	0	0
$805$	5.23084e6	0.284500
$806$	2.72942e6	0.147990
$807$	0	0
$808$	−3.48996e6	−0.188058
$809$	−1.69242e7	−0.909153	−0.454576	−	0.890708i	$-0.650209\pi$
$809$	−1.69242e7	−0.909153	−0.454576	+	0.890708i	$0.650209\pi$
$810$	0	0
$811$	2.67183e6	0.142645	0.0713226	−	0.997453i	$-0.477278\pi$
$811$	2.67183e6	0.142645	0.0713226	+	0.997453i	$0.477278\pi$
$812$	−2.20743e6	−0.117489
$813$	0	0
$814$	−2.91341e7	−1.54114
$815$	6.96030e6	0.367057
$816$	0	0
$817$	3.31174e7	1.73581
$818$	−846983.	−0.0442579
$819$	0	0
$820$	−818762.	−0.0425229
$821$	−5.35751e6	−0.277399	−0.138700	−	0.990335i	$-0.544292\pi$
$821$	−5.35751e6	−0.277399	−0.138700	+	0.990335i	$0.544292\pi$
$822$	0	0
$823$	357088.	0.0183770	0.00918851	−	0.999958i	$-0.497075\pi$
$823$	357088.	0.0183770	0.00918851	+	0.999958i	$0.497075\pi$
$824$	2.83896e7	1.45660
$825$	0	0
$826$	−4.96922e6	−0.253419
$827$	−1.09197e7	−0.555198	−0.277599	−	0.960697i	$-0.589539\pi$
$827$	−1.09197e7	−0.555198	−0.277599	+	0.960697i	$0.589539\pi$
$828$	0	0
$829$	−2.62577e7	−1.32700	−0.663499	−	0.748177i	$-0.730929\pi$
$829$	−2.62577e7	−1.32700	−0.663499	+	0.748177i	$0.730929\pi$
$830$	−2.65649e6	−0.133849
$831$	0	0
$832$	1.84328e7	0.923171
$833$	−1.05561e6	−0.0527097
$834$	0	0
$835$	1.04022e7	0.516307
$836$	7.00596e6	0.346699
$837$	0	0
$838$	−6.04088e6	−0.297160
$839$	1.74920e7	0.857896	0.428948	−	0.903329i	$-0.358884\pi$
$839$	1.74920e7	0.857896	0.428948	+	0.903329i	$0.358884\pi$
$840$	0	0
$841$	−1.23111e7	−0.600215
$842$	2.99208e7	1.45443
$843$	0	0
$844$	−3.35332e6	−0.162039
$845$	−1.84207e6	−0.0887490
$846$	0	0
$847$	−2.03703e7	−0.975639
$848$	1.18774e7	0.567195
$849$	0	0
$850$	1.16915e7	0.555040
$851$	2.64304e7	1.25106
$852$	0	0
$853$	−3.14997e7	−1.48229	−0.741147	−	0.671343i	$-0.765718\pi$
$853$	−3.14997e7	−1.48229	−0.741147	+	0.671343i	$0.765718\pi$
$854$	−2.15472e7	−1.01099
$855$	0	0
$856$	−3.88188e7	−1.81075
$857$	1.44326e7	0.671265	0.335632	−	0.941993i	$-0.391050\pi$
$857$	1.44326e7	0.671265	0.335632	+	0.941993i	$0.391050\pi$
$858$	0	0
$859$	106278.	0.00491427	0.00245714	−	0.999997i	$-0.499218\pi$
$859$	106278.	0.00491427	0.00245714	+	0.999997i	$0.499218\pi$
$860$	−1.65359e6	−0.0762400
$861$	0	0
$862$	−2.69198e7	−1.23397
$863$	1.69735e7	0.775792	0.387896	−	0.921703i	$-0.373202\pi$
$863$	1.69735e7	0.775792	0.387896	+	0.921703i	$0.373202\pi$
$864$	0	0
$865$	1.02009e7	0.463553
$866$	3.56635e6	0.161596
$867$	0	0
$868$	−818287.	−0.0368643
$869$	4.88150e7	2.19282
$870$	0	0
$871$	−3.14998e7	−1.40690
$872$	−2.27587e7	−1.01358
$873$	0	0
$874$	2.64826e7	1.17268
$875$	1.19389e7	0.527161
$876$	0	0
$877$	2.08946e7	0.917350	0.458675	−	0.888604i	$-0.348324\pi$
$877$	2.08946e7	0.917350	0.458675	+	0.888604i	$0.348324\pi$
$878$	−2.72948e6	−0.119493
$879$	0	0
$880$	7.18102e6	0.312593
$881$	3.16804e7	1.37515	0.687576	−	0.726113i	$-0.258675\pi$
$881$	3.16804e7	1.37515	0.687576	+	0.726113i	$0.258675\pi$
$882$	0	0
$883$	2.61037e7	1.12668	0.563339	−	0.826226i	$-0.309516\pi$
$883$	2.61037e7	1.12668	0.563339	+	0.826226i	$0.309516\pi$
$884$	−2.51484e6	−0.108238
$885$	0	0
$886$	−1.90427e7	−0.814973
$887$	−3.74614e7	−1.59873	−0.799365	−	0.600846i	$-0.794831\pi$
$887$	−3.74614e7	−1.59873	−0.799365	+	0.600846i	$0.794831\pi$
$888$	0	0
$889$	−8.83572e6	−0.374962
$890$	−9.46220e6	−0.400421
$891$	0	0
$892$	−8.00222e6	−0.336743
$893$	3.47607e7	1.45868
$894$	0	0
$895$	−2.09242e6	−0.0873153
$896$	1.42298e7	0.592146
$897$	0	0
$898$	−3.49882e6	−0.144788
$899$	3.03972e6	0.125440
$900$	0	0
$901$	−1.21002e7	−0.496572
$902$	2.39124e7	0.978605
$903$	0	0
$904$	4.25451e7	1.73153
$905$	629059.	0.0255311
$906$	0	0
$907$	−2.38271e7	−0.961728	−0.480864	−	0.876795i	$-0.659677\pi$
$907$	−2.38271e7	−0.961728	−0.480864	+	0.876795i	$0.659677\pi$
$908$	−5.94894e6	−0.239456
$909$	0	0
$910$	5.12152e6	0.205019
$911$	4.26255e7	1.70166	0.850832	−	0.525438i	$-0.176098\pi$
$911$	4.26255e7	1.70166	0.850832	+	0.525438i	$0.176098\pi$
$912$	0	0
$913$	−1.86202e7	−0.739276
$914$	1.73676e7	0.687663
$915$	0	0
$916$	807158.	0.0317848
$917$	1.31152e7	0.515051
$918$	0	0
$919$	8.22683e6	0.321325	0.160662	−	0.987009i	$-0.448637\pi$
$919$	8.22683e6	0.321325	0.160662	+	0.987009i	$0.448637\pi$
$920$	−8.15425e6	−0.317625
$921$	0	0
$922$	6.46597e6	0.250499
$923$	−5.25938e6	−0.203203
$924$	0	0
$925$	2.88735e7	1.10955
$926$	−3.33703e7	−1.27889
$927$	0	0
$928$	6.32422e6	0.241067
$929$	−2.09820e7	−0.797640	−0.398820	−	0.917029i	$-0.630580\pi$
$929$	−2.09820e7	−0.797640	−0.398820	+	0.917029i	$0.630580\pi$
$930$	0	0
$931$	−2.61209e6	−0.0987673
$932$	−3.67539e6	−0.138600
$933$	0	0
$934$	813019.	0.0304953
$935$	−7.31574e6	−0.273671
$936$	0	0
$937$	−1.32127e7	−0.491636	−0.245818	−	0.969316i	$-0.579057\pi$
$937$	−1.32127e7	−0.491636	−0.245818	+	0.969316i	$0.579057\pi$
$938$	−3.93490e7	−1.46025
$939$	0	0
$940$	−1.73564e6	−0.0640680
$941$	2.57409e7	0.947655	0.473827	−	0.880618i	$-0.342872\pi$
$941$	2.57409e7	0.947655	0.473827	+	0.880618i	$0.342872\pi$
$942$	0	0
$943$	−2.16933e7	−0.794411
$944$	6.18877e6	0.226034
$945$	0	0
$946$	4.82942e7	1.75456
$947$	−8.68204e6	−0.314592	−0.157296	−	0.987552i	$-0.550278\pi$
$947$	−8.68204e6	−0.314592	−0.157296	+	0.987552i	$0.550278\pi$
$948$	0	0
$949$	1.12608e7	0.405887
$950$	2.89305e7	1.04003
$951$	0	0
$952$	−1.93727e7	−0.692782
$953$	1.11979e7	0.399396	0.199698	−	0.979858i	$-0.436004\pi$
$953$	1.11979e7	0.399396	0.199698	+	0.979858i	$0.436004\pi$
$954$	0	0
$955$	7.73078e6	0.274293
$956$	−7.67863e6	−0.271731
$957$	0	0
$958$	1.63957e7	0.577188
$959$	2.63234e7	0.924262
$960$	0	0
$961$	−2.75023e7	−0.960641
$962$	2.58780e7	0.901556
$963$	0	0
$964$	−6.47274e6	−0.224334
$965$	−7.47326e6	−0.258340
$966$	0	0
$967$	2.32908e7	0.800972	0.400486	−	0.916303i	$-0.368841\pi$
$967$	2.32908e7	0.800972	0.400486	+	0.916303i	$0.368841\pi$
$968$	3.17548e7	1.08923
$969$	0	0
$970$	1.32690e7	0.452803
$971$	4.05544e7	1.38035	0.690175	−	0.723642i	$-0.257533\pi$
$971$	4.05544e7	1.38035	0.690175	+	0.723642i	$0.257533\pi$
$972$	0	0
$973$	1.46992e7	0.497750
$974$	−4.37837e7	−1.47882
$975$	0	0
$976$	2.68353e7	0.901741
$977$	−1.64638e7	−0.551816	−0.275908	−	0.961184i	$-0.588979\pi$
$977$	−1.64638e7	−0.551816	−0.275908	+	0.961184i	$0.588979\pi$
$978$	0	0
$979$	−6.63234e7	−2.21162
$980$	130425.	0.00433805
$981$	0	0
$982$	−7.09849e6	−0.234902
$983$	4.87657e7	1.60965	0.804824	−	0.593514i	$-0.202260\pi$
$983$	4.87657e7	1.60965	0.804824	+	0.593514i	$0.202260\pi$
$984$	0	0
$985$	−7.33067e6	−0.240743
$986$	1.16699e7	0.382273
$987$	0	0
$988$	−6.22294e6	−0.202816
$989$	−4.38123e7	−1.42431
$990$	0	0
$991$	−2.21394e7	−0.716114	−0.358057	−	0.933700i	$-0.616561\pi$
$991$	−2.21394e7	−0.716114	−0.358057	+	0.933700i	$0.616561\pi$
$992$	2.34436e6	0.0756390
$993$	0	0
$994$	−6.56993e6	−0.210909
$995$	1.09515e7	0.350686
$996$	0	0
$997$	−3.62667e7	−1.15550	−0.577751	−	0.816213i	$-0.696069\pi$
$997$	−3.62667e7	−1.15550	−0.577751	+	0.816213i	$0.696069\pi$
$998$	−1.20431e6	−0.0382746
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.e.1.12		42
3.2	odd	2		729.6.a.c.1.31		42
27.2	odd	18		27.6.e.a.4.5	✓	84
27.13	even	9		81.6.e.a.19.10		84
27.14	odd	18		27.6.e.a.7.5	yes	84
27.25	even	9		81.6.e.a.64.10		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.5	✓	84	27.2	odd	18
27.6.e.a.7.5	yes	84	27.14	odd	18
81.6.e.a.19.10		84	27.13	even	9
81.6.e.a.64.10		84	27.25	even	9
729.6.a.c.1.31		42	3.2	odd	2
729.6.a.e.1.12		42	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-5.08001 q^{2} -6.19352 q^{4} +16.0034 q^{5} -124.463 q^{7} +194.023 q^{8} -81.2975 q^{10} -569.839 q^{11} +506.151 q^{13} +632.275 q^{14} -787.448 q^{16} +802.220 q^{17} +1985.08 q^{19} -99.1175 q^{20} +2894.78 q^{22} -2626.14 q^{23} -2868.89 q^{25} -2571.25 q^{26} +770.867 q^{28} -2863.57 q^{29} -1061.51 q^{31} -2208.51 q^{32} -4075.28 q^{34} -1991.84 q^{35} -10064.4 q^{37} -10084.2 q^{38} +3105.04 q^{40} +8260.52 q^{41} +16683.2 q^{43} +3529.31 q^{44} +13340.8 q^{46} +17511.0 q^{47} -1315.86 q^{49} +14574.0 q^{50} -3134.85 q^{52} -15083.4 q^{53} -9119.36 q^{55} -24148.8 q^{56} +14547.0 q^{58} -7859.28 q^{59} -34078.8 q^{61} +5392.50 q^{62} +36417.6 q^{64} +8100.14 q^{65} -62234.1 q^{67} -4968.57 q^{68} +10118.6 q^{70} -10390.9 q^{71} +22248.0 q^{73} +51127.0 q^{74} -12294.6 q^{76} +70924.1 q^{77} -85664.5 q^{79} -12601.9 q^{80} -41963.5 q^{82} +32676.2 q^{83} +12838.3 q^{85} -84750.7 q^{86} -110562. q^{88} +116390. q^{89} -62997.2 q^{91} +16265.0 q^{92} -88955.9 q^{94} +31768.1 q^{95} -163216. q^{97} +6684.57 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28}+ \cdots + 463410 q^{98}+O(q^{100}) \) 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * q^98

Introduction

L-functions

Modular forms

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Database

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