LMFDB - Embedded newform 1089.6.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 129x^{3} - 11x^{2} + 3044x - 528 $ x^5 - x^4 - 129x^3 - 11x^2 + 3044*x - 528
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 121)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.11644$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.11644 q^{2} +18.6437 q^{4} -72.6055 q^{5} -4.73719 q^{7} -95.0493 q^{8} +O(q^{10})$	$q+7.11644 q^{2} +18.6437 q^{4} -72.6055 q^{5} -4.73719 q^{7} -95.0493 q^{8} -516.693 q^{10} +1011.28 q^{13} -33.7119 q^{14} -1273.01 q^{16} +263.786 q^{17} -376.262 q^{19} -1353.64 q^{20} +3187.09 q^{23} +2146.57 q^{25} +7196.72 q^{26} -88.3188 q^{28} +3736.64 q^{29} -485.467 q^{31} -6017.73 q^{32} +1877.22 q^{34} +343.947 q^{35} +7422.17 q^{37} -2677.64 q^{38} +6901.10 q^{40} -6063.29 q^{41} +5438.15 q^{43} +22680.8 q^{46} -22852.2 q^{47} -16784.6 q^{49} +15275.9 q^{50} +18854.0 q^{52} +8134.05 q^{53} +450.267 q^{56} +26591.6 q^{58} -33812.2 q^{59} -46243.8 q^{61} -3454.80 q^{62} -2088.44 q^{64} -73424.6 q^{65} +20298.5 q^{67} +4917.96 q^{68} +2447.67 q^{70} +25538.6 q^{71} -2458.86 q^{73} +52819.4 q^{74} -7014.91 q^{76} +22225.6 q^{79} +92427.6 q^{80} -43149.0 q^{82} -35249.5 q^{83} -19152.4 q^{85} +38700.3 q^{86} +52932.4 q^{89} -4790.63 q^{91} +59419.2 q^{92} -162626. q^{94} +27318.7 q^{95} -107925. q^{97} -119446. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 102 q^{4} - 29 q^{5} + 102 q^{7} + 102 q^{8}+O(q^{10}) $ 5 * q + 4 * q^2 + 102 * q^4 - 29 * q^5 + 102 * q^7 + 102 * q^8	$ 5 q + 4 q^{2} + 102 q^{4} - 29 q^{5} + 102 q^{7} + 102 q^{8} - 632 q^{10} + 621 q^{13} - 2636 q^{14} + 1506 q^{16} + 169 q^{17} - 4162 q^{19} + 446 q^{20} - 7498 q^{23} + 10140 q^{25} + 7696 q^{26} + 14584 q^{28} + 17739 q^{29} + 11670 q^{31} - 5242 q^{32} + 17692 q^{34} - 15590 q^{35} - 8887 q^{37} - 39684 q^{38} - 34770 q^{40} + 14089 q^{41} - 19552 q^{43} + 50876 q^{46} - 53914 q^{47} - 3903 q^{49} - 38296 q^{50} - 57494 q^{52} + 21723 q^{53} - 29304 q^{56} - 28992 q^{58} - 71896 q^{59} - 71986 q^{61} + 34708 q^{62} - 60022 q^{64} - 36773 q^{65} + 24058 q^{67} + 169910 q^{68} - 67964 q^{70} - 27660 q^{71} + 102438 q^{73} - 48792 q^{74} - 84928 q^{76} - 106562 q^{79} + 253490 q^{80} - 70532 q^{82} + 111906 q^{83} + 246799 q^{85} + 76248 q^{86} - 111123 q^{89} - 195274 q^{91} - 262832 q^{92} + 68828 q^{94} - 129486 q^{95} - 1553 q^{97} - 436908 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 102 * q^4 - 29 * q^5 + 102 * q^7 + 102 * q^8 - 632 * q^10 + 621 * q^13 - 2636 * q^14 + 1506 * q^16 + 169 * q^17 - 4162 * q^19 + 446 * q^20 - 7498 * q^23 + 10140 * q^25 + 7696 * q^26 + 14584 * q^28 + 17739 * q^29 + 11670 * q^31 - 5242 * q^32 + 17692 * q^34 - 15590 * q^35 - 8887 * q^37 - 39684 * q^38 - 34770 * q^40 + 14089 * q^41 - 19552 * q^43 + 50876 * q^46 - 53914 * q^47 - 3903 * q^49 - 38296 * q^50 - 57494 * q^52 + 21723 * q^53 - 29304 * q^56 - 28992 * q^58 - 71896 * q^59 - 71986 * q^61 + 34708 * q^62 - 60022 * q^64 - 36773 * q^65 + 24058 * q^67 + 169910 * q^68 - 67964 * q^70 - 27660 * q^71 + 102438 * q^73 - 48792 * q^74 - 84928 * q^76 - 106562 * q^79 + 253490 * q^80 - 70532 * q^82 + 111906 * q^83 + 246799 * q^85 + 76248 * q^86 - 111123 * q^89 - 195274 * q^91 - 262832 * q^92 + 68828 * q^94 - 129486 * q^95 - 1553 * q^97 - 436908 * q^98

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 $

$2$	7.11644	1.25802	0.629010	−	0.777397i	$-0.283460\pi$
$2$	7.11644	1.25802	0.629010	+	0.777397i	$0.283460\pi$
$3$	0	0
$3$	0	0
$4$	18.6437	0.582616
$5$	−72.6055	−1.29881	−0.649404	−	0.760444i	$-0.724981\pi$
$5$	−72.6055	−1.29881	−0.649404	+	0.760444i	$0.724981\pi$
$6$	0	0
$7$	−4.73719	−0.0365406	−0.0182703	−	0.999833i	$-0.505816\pi$
$7$	−4.73719	−0.0365406	−0.0182703	+	0.999833i	$0.505816\pi$
$8$	−95.0493	−0.525078
$9$	0	0
$10$	−516.693	−1.63393
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1011.28	1.65964	0.829819	−	0.558033i	$-0.188444\pi$
$13$	1011.28	1.65964	0.829819	+	0.558033i	$0.188444\pi$
$14$	−33.7119	−0.0459689
$15$	0	0
$16$	−1273.01	−1.24317
$17$	263.786	0.221376	0.110688	−	0.993855i	$-0.464695\pi$
$17$	263.786	0.221376	0.110688	+	0.993855i	$0.464695\pi$
$18$	0	0
$19$	−376.262	−0.239114	−0.119557	−	0.992827i	$-0.538147\pi$
$19$	−376.262	−0.239114	−0.119557	+	0.992827i	$0.538147\pi$
$20$	−1353.64	−0.756706
$21$	0	0
$22$	0	0
$23$	3187.09	1.25625	0.628124	−	0.778113i	$-0.283823\pi$
$23$	3187.09	1.25625	0.628124	+	0.778113i	$0.283823\pi$
$24$	0	0
$25$	2146.57	0.686901
$26$	7196.72	2.08786
$27$	0	0
$28$	−88.3188	−0.0212891
$29$	3736.64	0.825062	0.412531	−	0.910944i	$-0.364645\pi$
$29$	3736.64	0.825062	0.412531	+	0.910944i	$0.364645\pi$
$30$	0	0
$31$	−485.467	−0.0907309	−0.0453655	−	0.998970i	$-0.514445\pi$
$31$	−485.467	−0.0907309	−0.0453655	+	0.998970i	$0.514445\pi$
$32$	−6017.73	−1.03886
$33$	0	0
$34$	1877.22	0.278495
$35$	343.947	0.0474592
$36$	0	0
$37$	7422.17	0.891305	0.445653	−	0.895206i	$-0.352972\pi$
$37$	7422.17	0.891305	0.445653	+	0.895206i	$0.352972\pi$
$38$	−2677.64	−0.300811
$39$	0	0
$40$	6901.10	0.681975
$41$	−6063.29	−0.563311	−0.281656	−	0.959516i	$-0.590884\pi$
$41$	−6063.29	−0.563311	−0.281656	+	0.959516i	$0.590884\pi$
$42$	0	0
$43$	5438.15	0.448519	0.224259	−	0.974530i	$-0.428004\pi$
$43$	5438.15	0.448519	0.224259	+	0.974530i	$0.428004\pi$
$44$	0	0
$45$	0	0
$46$	22680.8	1.58039
$47$	−22852.2	−1.50898	−0.754490	−	0.656311i	$-0.772116\pi$
$47$	−22852.2	−1.50898	−0.754490	+	0.656311i	$0.772116\pi$
$48$	0	0
$49$	−16784.6	−0.998665
$50$	15275.9	0.864135
$51$	0	0
$52$	18854.0	0.966931
$53$	8134.05	0.397757	0.198878	−	0.980024i	$-0.436270\pi$
$53$	8134.05	0.397757	0.198878	+	0.980024i	$0.436270\pi$
$54$	0	0
$55$	0	0
$56$	450.267	0.0191867
$57$	0	0
$58$	26591.6	1.03795
$59$	−33812.2	−1.26457	−0.632286	−	0.774735i	$-0.717883\pi$
$59$	−33812.2	−1.26457	−0.632286	+	0.774735i	$0.717883\pi$
$60$	0	0
$61$	−46243.8	−1.59121	−0.795607	−	0.605813i	$-0.792848\pi$
$61$	−46243.8	−1.59121	−0.795607	+	0.605813i	$0.792848\pi$
$62$	−3454.80	−0.114141
$63$	0	0
$64$	−2088.44	−0.0637341
$65$	−73424.6	−2.15555
$66$	0	0
$67$	20298.5	0.552431	0.276215	−	0.961096i	$-0.410920\pi$
$67$	20298.5	0.552431	0.276215	+	0.961096i	$0.410920\pi$
$68$	4917.96	0.128977
$69$	0	0
$70$	2447.67	0.0597047
$71$	25538.6	0.601244	0.300622	−	0.953743i	$-0.402806\pi$
$71$	25538.6	0.601244	0.300622	+	0.953743i	$0.402806\pi$
$72$	0	0
$73$	−2458.86	−0.0540041	−0.0270021	−	0.999635i	$-0.508596\pi$
$73$	−2458.86	−0.0540041	−0.0270021	+	0.999635i	$0.508596\pi$
$74$	52819.4	1.12128
$75$	0	0
$76$	−7014.91	−0.139312
$77$	0	0
$78$	0	0
$79$	22225.6	0.400670	0.200335	−	0.979728i	$-0.435797\pi$
$79$	22225.6	0.400670	0.200335	+	0.979728i	$0.435797\pi$
$80$	92427.6	1.61464
$81$	0	0
$82$	−43149.0	−0.708657
$83$	−35249.5	−0.561639	−0.280820	−	0.959761i	$-0.590606\pi$
$83$	−35249.5	−0.561639	−0.280820	+	0.959761i	$0.590606\pi$
$84$	0	0
$85$	−19152.4	−0.287525
$86$	38700.3	0.564246
$87$	0	0
$88$	0	0
$89$	52932.4	0.708347	0.354174	−	0.935180i	$-0.384762\pi$
$89$	52932.4	0.708347	0.354174	+	0.935180i	$0.384762\pi$
$90$	0	0
$91$	−4790.63	−0.0606442
$92$	59419.2	0.731910
$93$	0	0
$94$	−162626.	−1.89833
$95$	27318.7	0.310564
$96$	0	0
$97$	−107925.	−1.16464	−0.582321	−	0.812959i	$-0.697855\pi$
$97$	−107925.	−1.16464	−0.582321	+	0.812959i	$0.697855\pi$
$98$	−119446.	−1.25634
$99$	0	0
$100$	40019.9	0.400199
$101$	−123480.	−1.20447	−0.602233	−	0.798320i	$-0.705722\pi$
$101$	−123480.	−1.20447	−0.602233	+	0.798320i	$0.705722\pi$
$102$	0	0
$103$	27863.1	0.258784	0.129392	−	0.991594i	$-0.458697\pi$
$103$	27863.1	0.258784	0.129392	+	0.991594i	$0.458697\pi$
$104$	−96121.5	−0.871439
$105$	0	0
$106$	57885.5	0.500386
$107$	−199746.	−1.68662	−0.843311	−	0.537426i	$-0.819397\pi$
$107$	−199746.	−1.68662	−0.843311	+	0.537426i	$0.819397\pi$
$108$	0	0
$109$	−177872.	−1.43398	−0.716989	−	0.697085i	$-0.754480\pi$
$109$	−177872.	−1.43398	−0.716989	+	0.697085i	$0.754480\pi$
$110$	0	0
$111$	0	0
$112$	6030.50	0.0454264
$113$	139195.	1.02548	0.512739	−	0.858544i	$-0.328631\pi$
$113$	139195.	1.02548	0.512739	+	0.858544i	$0.328631\pi$
$114$	0	0
$115$	−231401.	−1.63162
$116$	69664.9	0.480694
$117$	0	0
$118$	−240622.	−1.59086
$119$	−1249.61	−0.00808921
$120$	0	0
$121$	0	0
$122$	−329091.	−2.00178
$123$	0	0
$124$	−9050.90	−0.0528613
$125$	71039.8	0.406655
$126$	0	0
$127$	69910.1	0.384619	0.192309	−	0.981334i	$-0.438402\pi$
$127$	69910.1	0.384619	0.192309	+	0.981334i	$0.438402\pi$
$128$	177705.	0.958682
$129$	0	0
$130$	−522521.	−2.71173
$131$	97750.5	0.497669	0.248834	−	0.968546i	$-0.419952\pi$
$131$	97750.5	0.497669	0.248834	+	0.968546i	$0.419952\pi$
$132$	0	0
$133$	1782.42	0.00873739
$134$	144453.	0.694969
$135$	0	0
$136$	−25072.7	−0.116240
$137$	−310828.	−1.41488	−0.707438	−	0.706775i	$-0.750149\pi$
$137$	−310828.	−1.41488	−0.707438	+	0.706775i	$0.750149\pi$
$138$	0	0
$139$	−346449.	−1.52091	−0.760453	−	0.649393i	$-0.775023\pi$
$139$	−346449.	−1.52091	−0.760453	+	0.649393i	$0.775023\pi$
$140$	6412.44	0.0276505
$141$	0	0
$142$	181744.	0.756377
$143$	0	0
$144$	0	0
$145$	−271301.	−1.07160
$146$	−17498.3	−0.0679383
$147$	0	0
$148$	138377.	0.519288
$149$	473122.	1.74585	0.872925	−	0.487854i	$-0.162220\pi$
$149$	473122.	1.74585	0.872925	+	0.487854i	$0.162220\pi$
$150$	0	0
$151$	−457431.	−1.63261	−0.816307	−	0.577618i	$-0.803982\pi$
$151$	−457431.	−1.63261	−0.816307	+	0.577618i	$0.803982\pi$
$152$	35763.4	0.125554
$153$	0	0
$154$	0	0
$155$	35247.6	0.117842
$156$	0	0
$157$	90143.9	0.291869	0.145934	−	0.989294i	$-0.453381\pi$
$157$	90143.9	0.291869	0.145934	+	0.989294i	$0.453381\pi$
$158$	158167.	0.504051
$159$	0	0
$160$	436920.	1.34928
$161$	−15097.9	−0.0459041
$162$	0	0
$163$	−287935.	−0.848841	−0.424420	−	0.905465i	$-0.639522\pi$
$163$	−287935.	−0.848841	−0.424420	+	0.905465i	$0.639522\pi$
$164$	−113042.	−0.328194
$165$	0	0
$166$	−250851.	−0.706554
$167$	−383590.	−1.06433	−0.532165	−	0.846641i	$-0.678621\pi$
$167$	−383590.	−1.06433	−0.532165	+	0.846641i	$0.678621\pi$
$168$	0	0
$169$	651395.	1.75440
$170$	−136297.	−0.361712
$171$	0	0
$172$	101387.	0.261314
$173$	244427.	0.620919	0.310459	−	0.950587i	$-0.399517\pi$
$173$	244427.	0.620919	0.310459	+	0.950587i	$0.399517\pi$
$174$	0	0
$175$	−10168.7	−0.0250998
$176$	0	0
$177$	0	0
$178$	376690.	0.891116
$179$	366727.	0.855482	0.427741	−	0.903901i	$-0.359310\pi$
$179$	366727.	0.855482	0.427741	+	0.903901i	$0.359310\pi$
$180$	0	0
$181$	123794.	0.280869	0.140434	−	0.990090i	$-0.455150\pi$
$181$	123794.	0.280869	0.140434	+	0.990090i	$0.455150\pi$
$182$	−34092.2	−0.0762916
$183$	0	0
$184$	−302931.	−0.659628
$185$	−538890.	−1.15763
$186$	0	0
$187$	0	0
$188$	−426050.	−0.879156
$189$	0	0
$190$	194412.	0.390695
$191$	−25034.6	−0.0496543	−0.0248272	−	0.999692i	$-0.507904\pi$
$191$	−25034.6	−0.0496543	−0.0248272	+	0.999692i	$0.507904\pi$
$192$	0	0
$193$	−934246.	−1.80538	−0.902689	−	0.430294i	$-0.858410\pi$
$193$	−934246.	−1.80538	−0.902689	+	0.430294i	$0.858410\pi$
$194$	−768041.	−1.46514
$195$	0	0
$196$	−312926.	−0.581838
$197$	128005.	0.234997	0.117498	−	0.993073i	$-0.462513\pi$
$197$	128005.	0.234997	0.117498	+	0.993073i	$0.462513\pi$
$198$	0	0
$199$	−429035.	−0.767998	−0.383999	−	0.923333i	$-0.625453\pi$
$199$	−429035.	−0.767998	−0.383999	+	0.923333i	$0.625453\pi$
$200$	−204029.	−0.360677
$201$	0	0
$202$	−878741.	−1.51524
$203$	−17701.2	−0.0301483
$204$	0	0
$205$	440228.	0.731633
$206$	198286.	0.325555
$207$	0	0
$208$	−1.28737e6	−2.06322
$209$	0	0
$210$	0	0
$211$	−829092.	−1.28203	−0.641013	−	0.767530i	$-0.721485\pi$
$211$	−829092.	−1.28203	−0.641013	+	0.767530i	$0.721485\pi$
$212$	151649.	0.231739
$213$	0	0
$214$	−1.42148e6	−2.12180
$215$	−394840.	−0.582539
$216$	0	0
$217$	2299.75	0.00331537
$218$	−1.26582e6	−1.80397
$219$	0	0
$220$	0	0
$221$	266762.	0.367404
$222$	0	0
$223$	1.39839e6	1.88307	0.941535	−	0.336915i	$-0.109384\pi$
$223$	1.39839e6	1.88307	0.941535	+	0.336915i	$0.109384\pi$
$224$	28507.1	0.0379606
$225$	0	0
$226$	990570.	1.29007
$227$	−927669.	−1.19489	−0.597446	−	0.801909i	$-0.703818\pi$
$227$	−927669.	−1.19489	−0.597446	+	0.801909i	$0.703818\pi$
$228$	0	0
$229$	−433530.	−0.546299	−0.273150	−	0.961972i	$-0.588065\pi$
$229$	−433530.	−0.546299	−0.273150	+	0.961972i	$0.588065\pi$
$230$	−1.64675e6	−2.05262
$231$	0	0
$232$	−355165.	−0.433222
$233$	−304539.	−0.367497	−0.183748	−	0.982973i	$-0.558823\pi$
$233$	−304539.	−0.367497	−0.183748	+	0.982973i	$0.558823\pi$
$234$	0	0
$235$	1.65920e6	1.95988
$236$	−630385.	−0.736759
$237$	0	0
$238$	−8892.76	−0.0101764
$239$	−254240.	−0.287905	−0.143953	−	0.989585i	$-0.545981\pi$
$239$	−254240.	−0.287905	−0.143953	+	0.989585i	$0.545981\pi$
$240$	0	0
$241$	1.56307e6	1.73355	0.866774	−	0.498701i	$-0.166189\pi$
$241$	1.56307e6	1.73355	0.866774	+	0.498701i	$0.166189\pi$
$242$	0	0
$243$	0	0
$244$	−862155.	−0.927066
$245$	1.21865e6	1.29707
$246$	0	0
$247$	−380506.	−0.396843
$248$	46143.3	0.0476408
$249$	0	0
$250$	505550.	0.511581
$251$	−1.22758e6	−1.22989	−0.614944	−	0.788571i	$-0.710821\pi$
$251$	−1.22758e6	−1.22989	−0.614944	+	0.788571i	$0.710821\pi$
$252$	0	0
$253$	0	0
$254$	497511.	0.483858
$255$	0	0
$256$	1.33146e6	1.26978
$257$	110731.	0.104577	0.0522883	−	0.998632i	$-0.483349\pi$
$257$	110731.	0.104577	0.0522883	+	0.998632i	$0.483349\pi$
$258$	0	0
$259$	−35160.2	−0.0325688
$260$	−1.36891e6	−1.25586
$261$	0	0
$262$	695635.	0.626078
$263$	832925.	0.742534	0.371267	−	0.928526i	$-0.378923\pi$
$263$	832925.	0.742534	0.371267	+	0.928526i	$0.378923\pi$
$264$	0	0
$265$	−590577.	−0.516609
$266$	12684.5	0.0109918
$267$	0	0
$268$	378440.	0.321855
$269$	−873299.	−0.735838	−0.367919	−	0.929858i	$-0.619930\pi$
$269$	−873299.	−0.735838	−0.367919	+	0.929858i	$0.619930\pi$
$270$	0	0
$271$	−948707.	−0.784709	−0.392355	−	0.919814i	$-0.628339\pi$
$271$	−948707.	−0.784709	−0.392355	+	0.919814i	$0.628339\pi$
$272$	−335803.	−0.275209
$273$	0	0
$274$	−2.21199e6	−1.77994
$275$	0	0
$276$	0	0
$277$	871060.	0.682101	0.341050	−	0.940045i	$-0.389217\pi$
$277$	871060.	0.682101	0.341050	+	0.940045i	$0.389217\pi$
$278$	−2.46548e6	−1.91333
$279$	0	0
$280$	−32691.9	−0.0249198
$281$	1.16277e6	0.878469	0.439234	−	0.898373i	$-0.355250\pi$
$281$	1.16277e6	0.878469	0.439234	+	0.898373i	$0.355250\pi$
$282$	0	0
$283$	2.03304e6	1.50897	0.754483	−	0.656319i	$-0.227888\pi$
$283$	2.03304e6	1.50897	0.754483	+	0.656319i	$0.227888\pi$
$284$	476134.	0.350294
$285$	0	0
$286$	0	0
$287$	28723.0	0.0205838
$288$	0	0
$289$	−1.35027e6	−0.950993
$290$	−1.93070e6	−1.34809
$291$	0	0
$292$	−45842.3	−0.0314636
$293$	548226.	0.373070	0.186535	−	0.982448i	$-0.440274\pi$
$293$	548226.	0.373070	0.186535	+	0.982448i	$0.440274\pi$
$294$	0	0
$295$	2.45495e6	1.64244
$296$	−705472.	−0.468005
$297$	0	0
$298$	3.36694e6	2.19632
$299$	3.22305e6	2.08492
$300$	0	0
$301$	−25761.6	−0.0163892
$302$	−3.25528e6	−2.05386
$303$	0	0
$304$	478985.	0.297261
$305$	3.35755e6	2.06668
$306$	0	0
$307$	−1.34393e6	−0.813824	−0.406912	−	0.913467i	$-0.633394\pi$
$307$	−1.34393e6	−0.813824	−0.406912	+	0.913467i	$0.633394\pi$
$308$	0	0
$309$	0	0
$310$	250837.	0.148248
$311$	242814.	0.142355	0.0711774	−	0.997464i	$-0.477324\pi$
$311$	242814.	0.142355	0.0711774	+	0.997464i	$0.477324\pi$
$312$	0	0
$313$	−2.16100e6	−1.24679	−0.623396	−	0.781906i	$-0.714248\pi$
$313$	−2.16100e6	−1.24679	−0.623396	+	0.781906i	$0.714248\pi$
$314$	641504.	0.367177
$315$	0	0
$316$	414368.	0.233436
$317$	−1.03346e6	−0.577626	−0.288813	−	0.957386i	$-0.593261\pi$
$317$	−1.03346e6	−0.577626	−0.288813	+	0.957386i	$0.593261\pi$
$318$	0	0
$319$	0	0
$320$	151632.	0.0827784
$321$	0	0
$322$	−107443.	−0.0577483
$323$	−99252.7	−0.0529342
$324$	0	0
$325$	2.17078e6	1.14001
$326$	−2.04908e6	−1.06786
$327$	0	0
$328$	576311.	0.295782
$329$	108255.	0.0551391
$330$	0	0
$331$	−1.56898e6	−0.787134	−0.393567	−	0.919296i	$-0.628759\pi$
$331$	−1.56898e6	−0.787134	−0.393567	+	0.919296i	$0.628759\pi$
$332$	−657181.	−0.327220
$333$	0	0
$334$	−2.72979e6	−1.33895
$335$	−1.47379e6	−0.717501
$336$	0	0
$337$	−416215.	−0.199638	−0.0998189	−	0.995006i	$-0.531826\pi$
$337$	−416215.	−0.199638	−0.0998189	+	0.995006i	$0.531826\pi$
$338$	4.63561e6	2.20707
$339$	0	0
$340$	−357071.	−0.167516
$341$	0	0
$342$	0	0
$343$	159130.	0.0730325
$344$	−516893.	−0.235507
$345$	0	0
$346$	1.73945e6	0.781128
$347$	1.11364e6	0.496503	0.248251	−	0.968696i	$-0.420144\pi$
$347$	1.11364e6	0.496503	0.248251	+	0.968696i	$0.420144\pi$
$348$	0	0
$349$	2.68720e6	1.18096	0.590481	−	0.807052i	$-0.298938\pi$
$349$	2.68720e6	1.18096	0.590481	+	0.807052i	$0.298938\pi$
$350$	−72364.9	−0.0315761
$351$	0	0
$352$	0	0
$353$	−493757.	−0.210900	−0.105450	−	0.994425i	$-0.533628\pi$
$353$	−493757.	−0.210900	−0.105450	+	0.994425i	$0.533628\pi$
$354$	0	0
$355$	−1.85424e6	−0.780900
$356$	986855.	0.412694
$357$	0	0
$358$	2.60979e6	1.07621
$359$	−2.80267e6	−1.14772	−0.573860	−	0.818954i	$-0.694555\pi$
$359$	−2.80267e6	−1.14772	−0.573860	+	0.818954i	$0.694555\pi$
$360$	0	0
$361$	−2.33453e6	−0.942824
$362$	880974.	0.353339
$363$	0	0
$364$	−89315.1	−0.0353323
$365$	178527.	0.0701410
$366$	0	0
$367$	−1.36407e6	−0.528654	−0.264327	−	0.964433i	$-0.585150\pi$
$367$	−1.36407e6	−0.528654	−0.264327	+	0.964433i	$0.585150\pi$
$368$	−4.05721e6	−1.56174
$369$	0	0
$370$	−3.83498e6	−1.45633
$371$	−38532.6	−0.0145343
$372$	0	0
$373$	−8875.48	−0.00330308	−0.00165154	−	0.999999i	$-0.500526\pi$
$373$	−8875.48	−0.00330308	−0.00165154	+	0.999999i	$0.500526\pi$
$374$	0	0
$375$	0	0
$376$	2.17209e6	0.792333
$377$	3.77879e6	1.36930
$378$	0	0
$379$	3.14218e6	1.12366	0.561828	−	0.827254i	$-0.310098\pi$
$379$	3.14218e6	1.12366	0.561828	+	0.827254i	$0.310098\pi$
$380$	509321.	0.180939
$381$	0	0
$382$	−178157.	−0.0624661
$383$	3.26896e6	1.13871	0.569354	−	0.822093i	$-0.307194\pi$
$383$	3.26896e6	1.13871	0.569354	+	0.822093i	$0.307194\pi$
$384$	0	0
$385$	0	0
$386$	−6.64851e6	−2.27120
$387$	0	0
$388$	−2.01212e6	−0.678538
$389$	−4.27076e6	−1.43097	−0.715486	−	0.698627i	$-0.753795\pi$
$389$	−4.27076e6	−1.43097	−0.715486	+	0.698627i	$0.753795\pi$
$390$	0	0
$391$	840713.	0.278103
$392$	1.59536e6	0.524377
$393$	0	0
$394$	910940.	0.295631
$395$	−1.61370e6	−0.520393
$396$	0	0
$397$	−1.74435e6	−0.555465	−0.277732	−	0.960658i	$-0.589583\pi$
$397$	−1.74435e6	−0.555465	−0.277732	+	0.960658i	$0.589583\pi$
$398$	−3.05320e6	−0.966157
$399$	0	0
$400$	−2.73260e6	−0.853938
$401$	3.19770e6	0.993062	0.496531	−	0.868019i	$-0.334607\pi$
$401$	3.19770e6	0.993062	0.496531	+	0.868019i	$0.334607\pi$
$402$	0	0
$403$	−490943.	−0.150580
$404$	−2.30213e6	−0.701741
$405$	0	0
$406$	−125970.	−0.0379272
$407$	0	0
$408$	0	0
$409$	−2.74709e6	−0.812017	−0.406009	−	0.913869i	$-0.633080\pi$
$409$	−2.74709e6	−0.812017	−0.406009	+	0.913869i	$0.633080\pi$
$410$	3.13286e6	0.920409
$411$	0	0
$412$	519472.	0.150771
$413$	160175.	0.0462082
$414$	0	0
$415$	2.55931e6	0.729461
$416$	−6.08561e6	−1.72413
$417$	0	0
$418$	0	0
$419$	−3.90357e6	−1.08624	−0.543122	−	0.839654i	$-0.682758\pi$
$419$	−3.90357e6	−1.08624	−0.543122	+	0.839654i	$0.682758\pi$
$420$	0	0
$421$	1.00618e6	0.276676	0.138338	−	0.990385i	$-0.455824\pi$
$421$	1.00618e6	0.276676	0.138338	+	0.990385i	$0.455824\pi$
$422$	−5.90018e6	−1.61281
$423$	0	0
$424$	−773136.	−0.208853
$425$	566235.	0.152063
$426$	0	0
$427$	219066.	0.0581440
$428$	−3.72400e6	−0.982652
$429$	0	0
$430$	−2.80986e6	−0.732846
$431$	3.16110e6	0.819681	0.409840	−	0.912157i	$-0.365584\pi$
$431$	3.16110e6	0.819681	0.409840	+	0.912157i	$0.365584\pi$
$432$	0	0
$433$	−5.35215e6	−1.37186	−0.685928	−	0.727670i	$-0.740603\pi$
$433$	−5.35215e6	−1.37186	−0.685928	+	0.727670i	$0.740603\pi$
$434$	16366.0	0.00417080
$435$	0	0
$436$	−3.31620e6	−0.835458
$437$	−1.19918e6	−0.300387
$438$	0	0
$439$	7.69017e6	1.90447	0.952236	−	0.305364i	$-0.0987780\pi$
$439$	7.69017e6	1.90447	0.952236	+	0.305364i	$0.0987780\pi$
$440$	0	0
$441$	0	0
$442$	1.89840e6	0.462201
$443$	−4.64182e6	−1.12378	−0.561888	−	0.827214i	$-0.689925\pi$
$443$	−4.64182e6	−1.12378	−0.561888	+	0.827214i	$0.689925\pi$
$444$	0	0
$445$	−3.84318e6	−0.920007
$446$	9.95156e6	2.36894
$447$	0	0
$448$	9893.35	0.00232889
$449$	2.31530e6	0.541990	0.270995	−	0.962581i	$-0.412647\pi$
$449$	2.31530e6	0.541990	0.270995	+	0.962581i	$0.412647\pi$
$450$	0	0
$451$	0	0
$452$	2.59510e6	0.597460
$453$	0	0
$454$	−6.60170e6	−1.50320
$455$	347826.	0.0787651
$456$	0	0
$457$	−532842.	−0.119346	−0.0596730	−	0.998218i	$-0.519006\pi$
$457$	−532842.	−0.119346	−0.0596730	+	0.998218i	$0.519006\pi$
$458$	−3.08519e6	−0.687256
$459$	0	0
$460$	−4.31417e6	−0.950610
$461$	1.58678e6	0.347748	0.173874	−	0.984768i	$-0.444371\pi$
$461$	1.58678e6	0.347748	0.173874	+	0.984768i	$0.444371\pi$
$462$	0	0
$463$	−2.45369e6	−0.531946	−0.265973	−	0.963981i	$-0.585693\pi$
$463$	−2.45369e6	−0.531946	−0.265973	+	0.963981i	$0.585693\pi$
$464$	−4.75679e6	−1.02570
$465$	0	0
$466$	−2.16723e6	−0.462318
$467$	83334.2	0.0176820	0.00884099	−	0.999961i	$-0.497186\pi$
$467$	83334.2	0.0176820	0.00884099	+	0.999961i	$0.497186\pi$
$468$	0	0
$469$	−96158.1	−0.0201862
$470$	1.18076e7	2.46556
$471$	0	0
$472$	3.21383e6	0.663999
$473$	0	0
$474$	0	0
$475$	−807670.	−0.164248
$476$	−23297.3	−0.00471290
$477$	0	0
$478$	−1.80928e6	−0.362190
$479$	−435027.	−0.0866318	−0.0433159	−	0.999061i	$-0.513792\pi$
$479$	−435027.	−0.0866318	−0.0433159	+	0.999061i	$0.513792\pi$
$480$	0	0
$481$	7.50589e6	1.47924
$482$	1.11235e7	2.18084
$483$	0	0
$484$	0	0
$485$	7.83594e6	1.51264
$486$	0	0
$487$	−3.61785e6	−0.691239	−0.345619	−	0.938375i	$-0.612331\pi$
$487$	−3.61785e6	−0.691239	−0.345619	+	0.938375i	$0.612331\pi$
$488$	4.39544e6	0.835512
$489$	0	0
$490$	8.67246e6	1.63174
$491$	−6.94158e6	−1.29944	−0.649718	−	0.760176i	$-0.725113\pi$
$491$	−6.94158e6	−1.29944	−0.649718	+	0.760176i	$0.725113\pi$
$492$	0	0
$493$	985676.	0.182649
$494$	−2.70785e6	−0.499237
$495$	0	0
$496$	618005.	0.112794
$497$	−120981.	−0.0219698
$498$	0	0
$499$	3.49859e6	0.628987	0.314493	−	0.949260i	$-0.398165\pi$
$499$	3.49859e6	0.628987	0.314493	+	0.949260i	$0.398165\pi$
$500$	1.32444e6	0.236924
$501$	0	0
$502$	−8.73600e6	−1.54722
$503$	905951.	0.159656	0.0798279	−	0.996809i	$-0.474563\pi$
$503$	905951.	0.159656	0.0798279	+	0.996809i	$0.474563\pi$
$504$	0	0
$505$	8.96536e6	1.56437
$506$	0	0
$507$	0	0
$508$	1.30338e6	0.224085
$509$	−811390.	−0.138815	−0.0694073	−	0.997588i	$-0.522111\pi$
$509$	−811390.	−0.138815	−0.0694073	+	0.997588i	$0.522111\pi$
$510$	0	0
$511$	11648.1	0.00197334
$512$	3.78867e6	0.638722
$513$	0	0
$514$	788007.	0.131560
$515$	−2.02302e6	−0.336110
$516$	0	0
$517$	0	0
$518$	−250216.	−0.0409723
$519$	0	0
$520$	6.97895e6	1.13183
$521$	4.71269e6	0.760632	0.380316	−	0.924857i	$-0.375815\pi$
$521$	4.71269e6	0.760632	0.380316	+	0.924857i	$0.375815\pi$
$522$	0	0
$523$	−9.52412e6	−1.52255	−0.761274	−	0.648431i	$-0.775426\pi$
$523$	−9.52412e6	−1.52255	−0.761274	+	0.648431i	$0.775426\pi$
$524$	1.82243e6	0.289950
$525$	0	0
$526$	5.92746e6	0.934123
$527$	−128060.	−0.0200856
$528$	0	0
$529$	3.72123e6	0.578159
$530$	−4.20281e6	−0.649905
$531$	0	0
$532$	33231.0	0.00509054
$533$	−6.13168e6	−0.934893
$534$	0	0
$535$	1.45026e7	2.19060
$536$	−1.92936e6	−0.290069
$537$	0	0
$538$	−6.21478e6	−0.925699
$539$	0	0
$540$	0	0
$541$	9.18972e6	1.34992	0.674962	−	0.737853i	$-0.264160\pi$
$541$	9.18972e6	1.34992	0.674962	+	0.737853i	$0.264160\pi$
$542$	−6.75141e6	−0.987180
$543$	0	0
$544$	−1.58740e6	−0.229979
$545$	1.29145e7	1.86246
$546$	0	0
$547$	−3.21881e6	−0.459968	−0.229984	−	0.973194i	$-0.573867\pi$
$547$	−3.21881e6	−0.459968	−0.229984	+	0.973194i	$0.573867\pi$
$548$	−5.79498e6	−0.824329
$549$	0	0
$550$	0	0
$551$	−1.40596e6	−0.197284
$552$	0	0
$553$	−105287.	−0.0146407
$554$	6.19885e6	0.858097
$555$	0	0
$556$	−6.45909e6	−0.886103
$557$	−7.40741e6	−1.01165	−0.505823	−	0.862637i	$-0.668811\pi$
$557$	−7.40741e6	−1.01165	−0.505823	+	0.862637i	$0.668811\pi$
$558$	0	0
$559$	5.49950e6	0.744378
$560$	−437848.	−0.0590001
$561$	0	0
$562$	8.27475e6	1.10513
$563$	−5.19774e6	−0.691105	−0.345552	−	0.938399i	$-0.612308\pi$
$563$	−5.19774e6	−0.691105	−0.345552	+	0.938399i	$0.612308\pi$
$564$	0	0
$565$	−1.01063e7	−1.33190
$566$	1.44680e7	1.89831
$567$	0	0
$568$	−2.42742e6	−0.315700
$569$	−4.68496e6	−0.606632	−0.303316	−	0.952890i	$-0.598094\pi$
$569$	−4.68496e6	−0.606632	−0.303316	+	0.952890i	$0.598094\pi$
$570$	0	0
$571$	3.49234e6	0.448256	0.224128	−	0.974560i	$-0.428047\pi$
$571$	3.49234e6	0.448256	0.224128	+	0.974560i	$0.428047\pi$
$572$	0	0
$573$	0	0
$574$	204405.	0.0258948
$575$	6.84131e6	0.862918
$576$	0	0
$577$	−1.22300e7	−1.52928	−0.764642	−	0.644456i	$-0.777084\pi$
$577$	−1.22300e7	−1.52928	−0.764642	+	0.644456i	$0.777084\pi$
$578$	−9.60914e6	−1.19637
$579$	0	0
$580$	−5.05806e6	−0.624329
$581$	166984.	0.0205227
$582$	0	0
$583$	0	0
$584$	233713.	0.0283564
$585$	0	0
$586$	3.90142e6	0.469330
$587$	−1.76423e6	−0.211330	−0.105665	−	0.994402i	$-0.533697\pi$
$587$	−1.76423e6	−0.211330	−0.105665	+	0.994402i	$0.533697\pi$
$588$	0	0
$589$	182663.	0.0216951
$590$	1.74705e7	2.06622
$591$	0	0
$592$	−9.44850e6	−1.10805
$593$	6.25365e6	0.730293	0.365146	−	0.930950i	$-0.381019\pi$
$593$	6.25365e6	0.730293	0.365146	+	0.930950i	$0.381019\pi$
$594$	0	0
$595$	90728.4	0.0105063
$596$	8.82074e6	1.01716
$597$	0	0
$598$	2.29366e7	2.62287
$599$	−9.62733e6	−1.09632	−0.548162	−	0.836372i	$-0.684672\pi$
$599$	−9.62733e6	−1.09632	−0.548162	+	0.836372i	$0.684672\pi$
$600$	0	0
$601$	−750549.	−0.0847604	−0.0423802	−	0.999102i	$-0.513494\pi$
$601$	−750549.	−0.0847604	−0.0423802	+	0.999102i	$0.513494\pi$
$602$	−183331.	−0.0206179
$603$	0	0
$604$	−8.52821e6	−0.951186
$605$	0	0
$606$	0	0
$607$	−1.74412e6	−0.192134	−0.0960669	−	0.995375i	$-0.530626\pi$
$607$	−1.74412e6	−0.192134	−0.0960669	+	0.995375i	$0.530626\pi$
$608$	2.26424e6	0.248407
$609$	0	0
$610$	2.38938e7	2.59993
$611$	−2.31100e7	−2.50436
$612$	0	0
$613$	−2.21034e6	−0.237579	−0.118789	−	0.992919i	$-0.537901\pi$
$613$	−2.21034e6	−0.237579	−0.118789	+	0.992919i	$0.537901\pi$
$614$	−9.56399e6	−1.02381
$615$	0	0
$616$	0	0
$617$	−5.47689e6	−0.579190	−0.289595	−	0.957149i	$-0.593521\pi$
$617$	−5.47689e6	−0.579190	−0.289595	+	0.957149i	$0.593521\pi$
$618$	0	0
$619$	9.71832e6	1.01945	0.509724	−	0.860338i	$-0.329748\pi$
$619$	9.71832e6	1.01945	0.509724	+	0.860338i	$0.329748\pi$
$620$	657146.	0.0686566
$621$	0	0
$622$	1.72797e6	0.179085
$623$	−250751.	−0.0258835
$624$	0	0
$625$	−1.18659e7	−1.21507
$626$	−1.53786e7	−1.56849
$627$	0	0
$628$	1.68062e6	0.170047
$629$	1.95787e6	0.197313
$630$	0	0
$631$	1.12066e7	1.12047	0.560236	−	0.828333i	$-0.310711\pi$
$631$	1.12066e7	1.12047	0.560236	+	0.828333i	$0.310711\pi$
$632$	−2.11253e6	−0.210383
$633$	0	0
$634$	−7.35457e6	−0.726665
$635$	−5.07586e6	−0.499546
$636$	0	0
$637$	−1.69739e7	−1.65742
$638$	0	0
$639$	0	0
$640$	−1.29024e7	−1.24514
$641$	1.34005e7	1.28818	0.644090	−	0.764950i	$-0.277236\pi$
$641$	1.34005e7	1.28818	0.644090	+	0.764950i	$0.277236\pi$
$642$	0	0
$643$	−1.22269e7	−1.16625	−0.583123	−	0.812384i	$-0.698169\pi$
$643$	−1.22269e7	−1.16625	−0.583123	+	0.812384i	$0.698169\pi$
$644$	−281480.	−0.0267444
$645$	0	0
$646$	−706326.	−0.0665923
$647$	6.86830e6	0.645043	0.322521	−	0.946562i	$-0.395470\pi$
$647$	6.86830e6	0.645043	0.322521	+	0.946562i	$0.395470\pi$
$648$	0	0
$649$	0	0
$650$	1.54482e7	1.43415
$651$	0	0
$652$	−5.36818e6	−0.494548
$653$	1.35580e6	0.124427	0.0622133	−	0.998063i	$-0.480184\pi$
$653$	1.35580e6	0.124427	0.0622133	+	0.998063i	$0.480184\pi$
$654$	0	0
$655$	−7.09723e6	−0.646376
$656$	7.71863e6	0.700294
$657$	0	0
$658$	770393.	0.0693661
$659$	1.46979e7	1.31838	0.659190	−	0.751976i	$-0.270899\pi$
$659$	1.46979e7	1.31838	0.659190	+	0.751976i	$0.270899\pi$
$660$	0	0
$661$	−8.83401e6	−0.786419	−0.393210	−	0.919449i	$-0.628635\pi$
$661$	−8.83401e6	−0.786419	−0.393210	+	0.919449i	$0.628635\pi$
$662$	−1.11656e7	−0.990231
$663$	0	0
$664$	3.35044e6	0.294905
$665$	−129414.	−0.0113482
$666$	0	0
$667$	1.19090e7	1.03648
$668$	−7.15154e6	−0.620095
$669$	0	0
$670$	−1.04881e7	−0.902631
$671$	0	0
$672$	0	0
$673$	−6.27278e6	−0.533854	−0.266927	−	0.963717i	$-0.586008\pi$
$673$	−6.27278e6	−0.533854	−0.266927	+	0.963717i	$0.586008\pi$
$674$	−2.96197e6	−0.251148
$675$	0	0
$676$	1.21444e7	1.02214
$677$	−2.29559e7	−1.92497	−0.962483	−	0.271341i	$-0.912533\pi$
$677$	−2.29559e7	−1.92497	−0.962483	+	0.271341i	$0.912533\pi$
$678$	0	0
$679$	511261.	0.0425567
$680$	1.82042e6	0.150973
$681$	0	0
$682$	0	0
$683$	−1.26518e7	−1.03777	−0.518885	−	0.854844i	$-0.673653\pi$
$683$	−1.26518e7	−1.03777	−0.518885	+	0.854844i	$0.673653\pi$
$684$	0	0
$685$	2.25678e7	1.83765
$686$	1.13244e6	0.0918763
$687$	0	0
$688$	−6.92283e6	−0.557587
$689$	8.22581e6	0.660132
$690$	0	0
$691$	2.48572e6	0.198042	0.0990209	−	0.995085i	$-0.468429\pi$
$691$	2.48572e6	0.198042	0.0990209	+	0.995085i	$0.468429\pi$
$692$	4.55703e6	0.361757
$693$	0	0
$694$	7.92516e6	0.624610
$695$	2.51541e7	1.97536
$696$	0	0
$697$	−1.59941e6	−0.124704
$698$	1.91233e7	1.48567
$699$	0	0
$700$	−189582.	−0.0146235
$701$	2.46959e6	0.189814	0.0949072	−	0.995486i	$-0.469745\pi$
$701$	2.46959e6	0.189814	0.0949072	+	0.995486i	$0.469745\pi$
$702$	0	0
$703$	−2.79268e6	−0.213124
$704$	0	0
$705$	0	0
$706$	−3.51379e6	−0.265316
$707$	584951.	0.0440120
$708$	0	0
$709$	−2.82363e6	−0.210956	−0.105478	−	0.994422i	$-0.533637\pi$
$709$	−2.82363e6	−0.210956	−0.105478	+	0.994422i	$0.533637\pi$
$710$	−1.31956e7	−0.982388
$711$	0	0
$712$	−5.03118e6	−0.371938
$713$	−1.54723e6	−0.113981
$714$	0	0
$715$	0	0
$716$	6.83716e6	0.498417
$717$	0	0
$718$	−1.99450e7	−1.44386
$719$	−1.55911e7	−1.12475	−0.562373	−	0.826884i	$-0.690111\pi$
$719$	−1.55911e7	−1.12475	−0.562373	+	0.826884i	$0.690111\pi$
$720$	0	0
$721$	−131993.	−0.00945612
$722$	−1.66135e7	−1.18609
$723$	0	0
$724$	2.30798e6	0.163639
$725$	8.02095e6	0.566736
$726$	0	0
$727$	1.38292e7	0.970423	0.485212	−	0.874397i	$-0.338743\pi$
$727$	1.38292e7	0.970423	0.485212	+	0.874397i	$0.338743\pi$
$728$	455346.	0.0318429
$729$	0	0
$730$	1.27048e6	0.0882388
$731$	1.43451e6	0.0992912
$732$	0	0
$733$	2.68190e7	1.84367	0.921834	−	0.387586i	$-0.126691\pi$
$733$	2.68190e7	1.84367	0.921834	+	0.387586i	$0.126691\pi$
$734$	−9.70731e6	−0.665057
$735$	0	0
$736$	−1.91791e7	−1.30507
$737$	0	0
$738$	0	0
$739$	1.27952e7	0.861858	0.430929	−	0.902386i	$-0.358186\pi$
$739$	1.27952e7	0.861858	0.430929	+	0.902386i	$0.358186\pi$
$740$	−1.00469e7	−0.674456
$741$	0	0
$742$	−274215.	−0.0182844
$743$	1.27337e7	0.846218	0.423109	−	0.906079i	$-0.360939\pi$
$743$	1.27337e7	0.846218	0.423109	+	0.906079i	$0.360939\pi$
$744$	0	0
$745$	−3.43512e7	−2.26752
$746$	−63161.8	−0.00415535
$747$	0	0
$748$	0	0
$749$	946233.	0.0616302
$750$	0	0
$751$	−9.66088e6	−0.625053	−0.312527	−	0.949909i	$-0.601175\pi$
$751$	−9.66088e6	−0.625053	−0.312527	+	0.949909i	$0.601175\pi$
$752$	2.90911e7	1.87593
$753$	0	0
$754$	2.68916e7	1.72261
$755$	3.32120e7	2.12045
$756$	0	0
$757$	1.59887e7	1.01408	0.507041	−	0.861922i	$-0.330739\pi$
$757$	1.59887e7	1.01408	0.507041	+	0.861922i	$0.330739\pi$
$758$	2.23611e7	1.41358
$759$	0	0
$760$	−2.59662e6	−0.163070
$761$	2.24607e7	1.40592	0.702961	−	0.711228i	$-0.251861\pi$
$761$	2.24607e7	1.40592	0.702961	+	0.711228i	$0.251861\pi$
$762$	0	0
$763$	842616.	0.0523985
$764$	−466737.	−0.0289294
$765$	0	0
$766$	2.32633e7	1.43252
$767$	−3.41936e7	−2.09873
$768$	0	0
$769$	9.50203e6	0.579430	0.289715	−	0.957113i	$-0.406440\pi$
$769$	9.50203e6	0.579430	0.289715	+	0.957113i	$0.406440\pi$
$770$	0	0
$771$	0	0
$772$	−1.74178e7	−1.05184
$773$	−1.40655e7	−0.846656	−0.423328	−	0.905976i	$-0.639138\pi$
$773$	−1.40655e7	−0.846656	−0.423328	+	0.905976i	$0.639138\pi$
$774$	0	0
$775$	−1.04209e6	−0.0623232
$776$	1.02582e7	0.611528
$777$	0	0
$778$	−3.03926e7	−1.80019
$779$	2.28138e6	0.134696
$780$	0	0
$781$	0	0
$782$	5.98288e6	0.349859
$783$	0	0
$784$	2.13669e7	1.24151
$785$	−6.54495e6	−0.379081
$786$	0	0
$787$	−2.06203e7	−1.18675	−0.593374	−	0.804927i	$-0.702205\pi$
$787$	−2.06203e7	−1.18675	−0.593374	+	0.804927i	$0.702205\pi$
$788$	2.38649e6	0.136913
$789$	0	0
$790$	−1.14838e7	−0.654665
$791$	−659392.	−0.0374716
$792$	0	0
$793$	−4.67654e7	−2.64084
$794$	−1.24135e7	−0.698786
$795$	0	0
$796$	−7.99880e6	−0.447448
$797$	−1.62525e7	−0.906304	−0.453152	−	0.891433i	$-0.649701\pi$
$797$	−1.62525e7	−0.906304	−0.453152	+	0.891433i	$0.649701\pi$
$798$	0	0
$799$	−6.02811e6	−0.334052
$800$	−1.29174e7	−0.713595
$801$	0	0
$802$	2.27562e7	1.24929
$803$	0	0
$804$	0	0
$805$	1.09619e6	0.0596206
$806$	−3.49377e6	−0.189433
$807$	0	0
$808$	1.17367e7	0.632439
$809$	−2.28295e7	−1.22638	−0.613191	−	0.789934i	$-0.710115\pi$
$809$	−2.28295e7	−1.22638	−0.613191	+	0.789934i	$0.710115\pi$
$810$	0	0
$811$	−8.35950e6	−0.446301	−0.223151	−	0.974784i	$-0.571634\pi$
$811$	−8.35950e6	−0.446301	−0.223151	+	0.974784i	$0.571634\pi$
$812$	−330016.	−0.0175649
$813$	0	0
$814$	0	0
$815$	2.09057e7	1.10248
$816$	0	0
$817$	−2.04617e6	−0.107247
$818$	−1.95495e7	−1.02153
$819$	0	0
$820$	8.20749e6	0.426261
$821$	−1.64897e6	−0.0853799	−0.0426900	−	0.999088i	$-0.513593\pi$
$821$	−1.64897e6	−0.0853799	−0.0426900	+	0.999088i	$0.513593\pi$
$822$	0	0
$823$	−1.64213e7	−0.845101	−0.422551	−	0.906339i	$-0.638865\pi$
$823$	−1.64213e7	−0.845101	−0.422551	+	0.906339i	$0.638865\pi$
$824$	−2.64837e6	−0.135882
$825$	0	0
$826$	1.13988e6	0.0581309
$827$	−2.02615e7	−1.03017	−0.515084	−	0.857139i	$-0.672239\pi$
$827$	−2.02615e7	−1.03017	−0.515084	+	0.857139i	$0.672239\pi$
$828$	0	0
$829$	2.63116e7	1.32972	0.664862	−	0.746967i	$-0.268490\pi$
$829$	2.63116e7	1.32972	0.664862	+	0.746967i	$0.268490\pi$
$830$	1.82132e7	0.917678
$831$	0	0
$832$	−2.11200e6	−0.105776
$833$	−4.42754e6	−0.221080
$834$	0	0
$835$	2.78508e7	1.38236
$836$	0	0
$837$	0	0
$838$	−2.77795e7	−1.36652
$839$	−3.01813e7	−1.48024	−0.740122	−	0.672472i	$-0.765232\pi$
$839$	−3.01813e7	−1.48024	−0.740122	+	0.672472i	$0.765232\pi$
$840$	0	0
$841$	−6.54864e6	−0.319272
$842$	7.16043e6	0.348064
$843$	0	0
$844$	−1.54573e7	−0.746928
$845$	−4.72949e7	−2.27862
$846$	0	0
$847$	0	0
$848$	−1.03547e7	−0.494481
$849$	0	0
$850$	4.02958e6	0.191299
$851$	2.36551e7	1.11970
$852$	0	0
$853$	1.44365e7	0.679343	0.339671	−	0.940544i	$-0.389684\pi$
$853$	1.44365e7	0.679343	0.339671	+	0.940544i	$0.389684\pi$
$854$	1.55897e6	0.0731463
$855$	0	0
$856$	1.89857e7	0.885608
$857$	−1.60984e7	−0.748740	−0.374370	−	0.927279i	$-0.622141\pi$
$857$	−1.60984e7	−0.748740	−0.374370	+	0.927279i	$0.622141\pi$
$858$	0	0
$859$	−8.70793e6	−0.402654	−0.201327	−	0.979524i	$-0.564525\pi$
$859$	−8.70793e6	−0.402654	−0.201327	+	0.979524i	$0.564525\pi$
$860$	−7.36128e6	−0.339397
$861$	0	0
$862$	2.24958e7	1.03118
$863$	3.95132e7	1.80599	0.902995	−	0.429652i	$-0.141364\pi$
$863$	3.95132e7	1.80599	0.902995	+	0.429652i	$0.141364\pi$
$864$	0	0
$865$	−1.77468e7	−0.806454
$866$	−3.80882e7	−1.72582
$867$	0	0
$868$	42875.9	0.00193158
$869$	0	0
$870$	0	0
$871$	2.05275e7	0.916835
$872$	1.69066e7	0.752950
$873$	0	0
$874$	−8.53390e6	−0.377893
$875$	−336529.	−0.0148594
$876$	0	0
$877$	2.77083e7	1.21650	0.608248	−	0.793747i	$-0.291873\pi$
$877$	2.77083e7	1.21650	0.608248	+	0.793747i	$0.291873\pi$
$878$	5.47266e7	2.39586
$879$	0	0
$880$	0	0
$881$	−1.20665e7	−0.523771	−0.261886	−	0.965099i	$-0.584344\pi$
$881$	−1.20665e7	−0.523771	−0.261886	+	0.965099i	$0.584344\pi$
$882$	0	0
$883$	1.86562e7	0.805234	0.402617	−	0.915369i	$-0.368101\pi$
$883$	1.86562e7	0.805234	0.402617	+	0.915369i	$0.368101\pi$
$884$	4.97343e6	0.214055
$885$	0	0
$886$	−3.30333e7	−1.41373
$887$	4.53116e7	1.93375	0.966874	−	0.255254i	$-0.0821590\pi$
$887$	4.53116e7	1.93375	0.966874	+	0.255254i	$0.0821590\pi$
$888$	0	0
$889$	−331177.	−0.0140542
$890$	−2.73498e7	−1.15739
$891$	0	0
$892$	2.60712e7	1.09711
$893$	8.59841e6	0.360819
$894$	0	0
$895$	−2.66264e7	−1.11111
$896$	−841823.	−0.0350309
$897$	0	0
$898$	1.64767e7	0.681835
$899$	−1.81402e6	−0.0748587
$900$	0	0
$901$	2.14565e6	0.0880537
$902$	0	0
$903$	0	0
$904$	−1.32304e7	−0.538456
$905$	−8.98814e6	−0.364795
$906$	0	0
$907$	3.84187e6	0.155069	0.0775344	−	0.996990i	$-0.475295\pi$
$907$	3.84187e6	0.155069	0.0775344	+	0.996990i	$0.475295\pi$
$908$	−1.72952e7	−0.696163
$909$	0	0
$910$	2.47529e6	0.0990882
$911$	−1.79804e7	−0.717800	−0.358900	−	0.933376i	$-0.616848\pi$
$911$	−1.79804e7	−0.717800	−0.358900	+	0.933376i	$0.616848\pi$
$912$	0	0
$913$	0	0
$914$	−3.79194e6	−0.150140
$915$	0	0
$916$	−8.08261e6	−0.318283
$917$	−463063.	−0.0181851
$918$	0	0
$919$	−2.07958e7	−0.812243	−0.406121	−	0.913819i	$-0.633119\pi$
$919$	−2.07958e7	−0.812243	−0.406121	+	0.913819i	$0.633119\pi$
$920$	2.19945e7	0.856730
$921$	0	0
$922$	1.12922e7	0.437475
$923$	2.58267e7	0.997847
$924$	0	0
$925$	1.59322e7	0.612238
$926$	−1.74615e7	−0.669199
$927$	0	0
$928$	−2.24861e7	−0.857125
$929$	1.49177e7	0.567103	0.283551	−	0.958957i	$-0.408487\pi$
$929$	1.49177e7	0.567103	0.283551	+	0.958957i	$0.408487\pi$
$930$	0	0
$931$	6.31538e6	0.238795
$932$	−5.67774e6	−0.214109
$933$	0	0
$934$	593043.	0.0222443
$935$	0	0
$936$	0	0
$937$	3.73278e7	1.38894	0.694470	−	0.719522i	$-0.255639\pi$
$937$	3.73278e7	1.38894	0.694470	+	0.719522i	$0.255639\pi$
$938$	−684303.	−0.0253946
$939$	0	0
$940$	3.09336e7	1.14185
$941$	−2.15371e7	−0.792892	−0.396446	−	0.918058i	$-0.629757\pi$
$941$	−2.15371e7	−0.792892	−0.396446	+	0.918058i	$0.629757\pi$
$942$	0	0
$943$	−1.93243e7	−0.707659
$944$	4.30433e7	1.57208
$945$	0	0
$946$	0	0
$947$	2.60682e7	0.944572	0.472286	−	0.881445i	$-0.343429\pi$
$947$	2.60682e7	0.944572	0.472286	+	0.881445i	$0.343429\pi$
$948$	0	0
$949$	−2.48660e6	−0.0896273
$950$	−5.74773e6	−0.206627
$951$	0	0
$952$	118774.	0.00424747
$953$	−5.49000e7	−1.95812	−0.979062	−	0.203561i	$-0.934748\pi$
$953$	−5.49000e7	−1.95812	−0.979062	+	0.203561i	$0.934748\pi$
$954$	0	0
$955$	1.81765e6	0.0644914
$956$	−4.73998e6	−0.167738
$957$	0	0
$958$	−3.09584e6	−0.108985
$959$	1.47245e6	0.0517005
$960$	0	0
$961$	−2.83935e7	−0.991768
$962$	5.34152e7	1.86092
$963$	0	0
$964$	2.91414e7	1.00999
$965$	6.78315e7	2.34484
$966$	0	0
$967$	2.40709e7	0.827800	0.413900	−	0.910322i	$-0.364166\pi$
$967$	2.40709e7	0.827800	0.413900	+	0.910322i	$0.364166\pi$
$968$	0	0
$969$	0	0
$970$	5.57640e7	1.90294
$971$	3.52357e7	1.19932	0.599660	−	0.800255i	$-0.295302\pi$
$971$	3.52357e7	1.19932	0.599660	+	0.800255i	$0.295302\pi$
$972$	0	0
$973$	1.64120e6	0.0555748
$974$	−2.57462e7	−0.869593
$975$	0	0
$976$	5.88688e7	1.97816
$977$	1.91075e7	0.640425	0.320213	−	0.947346i	$-0.396246\pi$
$977$	1.91075e7	0.640425	0.320213	+	0.947346i	$0.396246\pi$
$978$	0	0
$979$	0	0
$980$	2.27202e7	0.755695
$981$	0	0
$982$	−4.93993e7	−1.63472
$983$	6.11595e6	0.201874	0.100937	−	0.994893i	$-0.467816\pi$
$983$	6.11595e6	0.201874	0.100937	+	0.994893i	$0.467816\pi$
$984$	0	0
$985$	−9.29388e6	−0.305215
$986$	7.01450e6	0.229776
$987$	0	0
$988$	−7.09404e6	−0.231207
$989$	1.73319e7	0.563451
$990$	0	0
$991$	2.24387e7	0.725794	0.362897	−	0.931829i	$-0.381788\pi$
$991$	2.24387e7	0.725794	0.362897	+	0.931829i	$0.381788\pi$
$992$	2.92141e6	0.0942569
$993$	0	0
$994$	−860955.	−0.0276385
$995$	3.11503e7	0.997482
$996$	0	0
$997$	−6.04252e7	−1.92522	−0.962609	−	0.270895i	$-0.912680\pi$
$997$	−6.04252e7	−1.92522	−0.962609	+	0.270895i	$0.912680\pi$
$998$	2.48975e7	0.791278
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.y.1.4		5
3.2	odd	2		121.6.a.e.1.2	✓	5
11.10	odd	2		1089.6.a.v.1.2		5
33.32	even	2		121.6.a.f.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.6.a.e.1.2	✓	5	3.2	odd	2
121.6.a.f.1.4	yes	5	33.32	even	2
1089.6.a.v.1.2		5	11.10	odd	2
1089.6.a.y.1.4		5	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+7.11644 q^{2} +18.6437 q^{4} -72.6055 q^{5} -4.73719 q^{7} -95.0493 q^{8} +O(q^{10})\)	\(q+7.11644 q^{2} +18.6437 q^{4} -72.6055 q^{5} -4.73719 q^{7} -95.0493 q^{8} -516.693 q^{10} +1011.28 q^{13} -33.7119 q^{14} -1273.01 q^{16} +263.786 q^{17} -376.262 q^{19} -1353.64 q^{20} +3187.09 q^{23} +2146.57 q^{25} +7196.72 q^{26} -88.3188 q^{28} +3736.64 q^{29} -485.467 q^{31} -6017.73 q^{32} +1877.22 q^{34} +343.947 q^{35} +7422.17 q^{37} -2677.64 q^{38} +6901.10 q^{40} -6063.29 q^{41} +5438.15 q^{43} +22680.8 q^{46} -22852.2 q^{47} -16784.6 q^{49} +15275.9 q^{50} +18854.0 q^{52} +8134.05 q^{53} +450.267 q^{56} +26591.6 q^{58} -33812.2 q^{59} -46243.8 q^{61} -3454.80 q^{62} -2088.44 q^{64} -73424.6 q^{65} +20298.5 q^{67} +4917.96 q^{68} +2447.67 q^{70} +25538.6 q^{71} -2458.86 q^{73} +52819.4 q^{74} -7014.91 q^{76} +22225.6 q^{79} +92427.6 q^{80} -43149.0 q^{82} -35249.5 q^{83} -19152.4 q^{85} +38700.3 q^{86} +52932.4 q^{89} -4790.63 q^{91} +59419.2 q^{92} -162626. q^{94} +27318.7 q^{95} -107925. q^{97} -119446. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 102 q^{4} - 29 q^{5} + 102 q^{7} + 102 q^{8}+O(q^{10}) \) 5 * q + 4 * q^2 + 102 * q^4 - 29 * q^5 + 102 * q^7 + 102 * q^8	\( 5 q + 4 q^{2} + 102 q^{4} - 29 q^{5} + 102 q^{7} + 102 q^{8} - 632 q^{10} + 621 q^{13} - 2636 q^{14} + 1506 q^{16} + 169 q^{17} - 4162 q^{19} + 446 q^{20} - 7498 q^{23} + 10140 q^{25} + 7696 q^{26} + 14584 q^{28} + 17739 q^{29} + 11670 q^{31} - 5242 q^{32} + 17692 q^{34} - 15590 q^{35} - 8887 q^{37} - 39684 q^{38} - 34770 q^{40} + 14089 q^{41} - 19552 q^{43} + 50876 q^{46} - 53914 q^{47} - 3903 q^{49} - 38296 q^{50} - 57494 q^{52} + 21723 q^{53} - 29304 q^{56} - 28992 q^{58} - 71896 q^{59} - 71986 q^{61} + 34708 q^{62} - 60022 q^{64} - 36773 q^{65} + 24058 q^{67} + 169910 q^{68} - 67964 q^{70} - 27660 q^{71} + 102438 q^{73} - 48792 q^{74} - 84928 q^{76} - 106562 q^{79} + 253490 q^{80} - 70532 q^{82} + 111906 q^{83} + 246799 q^{85} + 76248 q^{86} - 111123 q^{89} - 195274 q^{91} - 262832 q^{92} + 68828 q^{94} - 129486 q^{95} - 1553 q^{97} - 436908 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 102 * q^4 - 29 * q^5 + 102 * q^7 + 102 * q^8 - 632 * q^10 + 621 * q^13 - 2636 * q^14 + 1506 * q^16 + 169 * q^17 - 4162 * q^19 + 446 * q^20 - 7498 * q^23 + 10140 * q^25 + 7696 * q^26 + 14584 * q^28 + 17739 * q^29 + 11670 * q^31 - 5242 * q^32 + 17692 * q^34 - 15590 * q^35 - 8887 * q^37 - 39684 * q^38 - 34770 * q^40 + 14089 * q^41 - 19552 * q^43 + 50876 * q^46 - 53914 * q^47 - 3903 * q^49 - 38296 * q^50 - 57494 * q^52 + 21723 * q^53 - 29304 * q^56 - 28992 * q^58 - 71896 * q^59 - 71986 * q^61 + 34708 * q^62 - 60022 * q^64 - 36773 * q^65 + 24058 * q^67 + 169910 * q^68 - 67964 * q^70 - 27660 * q^71 + 102438 * q^73 - 48792 * q^74 - 84928 * q^76 - 106562 * q^79 + 253490 * q^80 - 70532 * q^82 + 111906 * q^83 + 246799 * q^85 + 76248 * q^86 - 111123 * q^89 - 195274 * q^91 - 262832 * q^92 + 68828 * q^94 - 129486 * q^95 - 1553 * q^97 - 436908 * q^98

Introduction

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