LMFDB - Embedded newform 495.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.723686$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44737 q^{2} -29.9051 q^{4} +25.0000 q^{5} +41.5023 q^{7} -89.5997 q^{8} +O(q^{10})$	\(q+1.44737 q^{2} -29.9051 q^{4} +25.0000 q^{5} +41.5023 q^{7} -89.5997 q^{8} +36.1843 q^{10} -121.000 q^{11} +434.580 q^{13} +60.0692 q^{14} +827.280 q^{16} -474.359 q^{17} -2586.54 q^{19} -747.628 q^{20} -175.132 q^{22} +3671.69 q^{23} +625.000 q^{25} +628.999 q^{26} -1241.13 q^{28} +4269.64 q^{29} -8415.46 q^{31} +4064.57 q^{32} -686.573 q^{34} +1037.56 q^{35} +13940.3 q^{37} -3743.68 q^{38} -2239.99 q^{40} -1771.24 q^{41} -11481.3 q^{43} +3618.52 q^{44} +5314.30 q^{46} +11048.5 q^{47} -15084.6 q^{49} +904.607 q^{50} -12996.2 q^{52} -20924.0 q^{53} -3025.00 q^{55} -3718.59 q^{56} +6179.75 q^{58} +33001.2 q^{59} -32348.2 q^{61} -12180.3 q^{62} -20590.0 q^{64} +10864.5 q^{65} +28892.8 q^{67} +14185.7 q^{68} +1501.73 q^{70} +20476.5 q^{71} -43333.2 q^{73} +20176.8 q^{74} +77350.7 q^{76} -5021.77 q^{77} -95301.8 q^{79} +20682.0 q^{80} -2563.64 q^{82} +16682.8 q^{83} -11859.0 q^{85} -16617.7 q^{86} +10841.6 q^{88} -143269. q^{89} +18036.1 q^{91} -109802. q^{92} +15991.3 q^{94} -64663.4 q^{95} -128960. q^{97} -21833.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * 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 $

$2$	1.44737	0.255862	0.127931	−	0.991783i	$-0.459166\pi$
$2$	1.44737	0.255862	0.127931	+	0.991783i	$0.459166\pi$
$3$	0	0
$3$	0	0
$4$	−29.9051	−0.934535
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	41.5023	0.320130	0.160065	−	0.987106i	$-0.448830\pi$
$7$	41.5023	0.320130	0.160065	+	0.987106i	$0.448830\pi$
$8$	−89.5997	−0.494973
$9$	0	0
$10$	36.1843	0.114425
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	434.580	0.713201	0.356600	−	0.934257i	$-0.383936\pi$
$13$	434.580	0.713201	0.356600	+	0.934257i	$0.383936\pi$
$14$	60.0692	0.0819090
$15$	0	0
$16$	827.280	0.807890
$17$	−474.359	−0.398093	−0.199046	−	0.979990i	$-0.563784\pi$
$17$	−474.359	−0.398093	−0.199046	+	0.979990i	$0.563784\pi$
$18$	0	0
$19$	−2586.54	−1.64375	−0.821873	−	0.569671i	$-0.807071\pi$
$19$	−2586.54	−1.64375	−0.821873	+	0.569671i	$0.807071\pi$
$20$	−747.628	−0.417937
$21$	0	0
$22$	−175.132	−0.0771452
$23$	3671.69	1.44726	0.723629	−	0.690189i	$-0.242473\pi$
$23$	3671.69	1.44726	0.723629	+	0.690189i	$0.242473\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	628.999	0.182481
$27$	0	0
$28$	−1241.13	−0.299173
$29$	4269.64	0.942749	0.471374	−	0.881933i	$-0.343758\pi$
$29$	4269.64	0.942749	0.471374	+	0.881933i	$0.343758\pi$
$30$	0	0
$31$	−8415.46	−1.57280	−0.786400	−	0.617717i	$-0.788058\pi$
$31$	−8415.46	−1.57280	−0.786400	+	0.617717i	$0.788058\pi$
$32$	4064.57	0.701681
$33$	0	0
$34$	−686.573	−0.101857
$35$	1037.56	0.143167
$36$	0	0
$37$	13940.3	1.67405	0.837024	−	0.547166i	$-0.184294\pi$
$37$	13940.3	1.67405	0.837024	+	0.547166i	$0.184294\pi$
$38$	−3743.68	−0.420571
$39$	0	0
$40$	−2239.99	−0.221359
$41$	−1771.24	−0.164558	−0.0822788	−	0.996609i	$-0.526220\pi$
$41$	−1771.24	−0.164558	−0.0822788	+	0.996609i	$0.526220\pi$
$42$	0	0
$43$	−11481.3	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$43$	−11481.3	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$44$	3618.52	0.281773
$45$	0	0
$46$	5314.30	0.370298
$47$	11048.5	0.729556	0.364778	−	0.931094i	$-0.381145\pi$
$47$	11048.5	0.729556	0.364778	+	0.931094i	$0.381145\pi$
$48$	0	0
$49$	−15084.6	−0.897517
$50$	904.607	0.0511723
$51$	0	0
$52$	−12996.2	−0.666511
$53$	−20924.0	−1.02319	−0.511594	−	0.859227i	$-0.670945\pi$
$53$	−20924.0	−1.02319	−0.511594	+	0.859227i	$0.670945\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−3718.59	−0.158456
$57$	0	0
$58$	6179.75	0.241213
$59$	33001.2	1.23424	0.617121	−	0.786868i	$-0.288299\pi$
$59$	33001.2	1.23424	0.617121	+	0.786868i	$0.288299\pi$
$60$	0	0
$61$	−32348.2	−1.11308	−0.556539	−	0.830821i	$-0.687871\pi$
$61$	−32348.2	−1.11308	−0.556539	+	0.830821i	$0.687871\pi$
$62$	−12180.3	−0.402419
$63$	0	0
$64$	−20590.0	−0.628357
$65$	10864.5	0.318953
$66$	0	0
$67$	28892.8	0.786325	0.393163	−	0.919469i	$-0.371381\pi$
$67$	28892.8	0.786325	0.393163	+	0.919469i	$0.371381\pi$
$68$	14185.7	0.372032
$69$	0	0
$70$	1501.73	0.0366308
$71$	20476.5	0.482069	0.241034	−	0.970517i	$-0.422513\pi$
$71$	20476.5	0.482069	0.241034	+	0.970517i	$0.422513\pi$
$72$	0	0
$73$	−43333.2	−0.951729	−0.475864	−	0.879519i	$-0.657865\pi$
$73$	−43333.2	−0.951729	−0.475864	+	0.879519i	$0.657865\pi$
$74$	20176.8	0.428325
$75$	0	0
$76$	77350.7	1.53614
$77$	−5021.77	−0.0965229
$78$	0	0
$79$	−95301.8	−1.71804	−0.859020	−	0.511942i	$-0.828926\pi$
$79$	−95301.8	−1.71804	−0.859020	+	0.511942i	$0.828926\pi$
$80$	20682.0	0.361299
$81$	0	0
$82$	−2563.64	−0.0421040
$83$	16682.8	0.265811	0.132905	−	0.991129i	$-0.457569\pi$
$83$	16682.8	0.265811	0.132905	+	0.991129i	$0.457569\pi$
$84$	0	0
$85$	−11859.0	−0.178033
$86$	−16617.7	−0.242284
$87$	0	0
$88$	10841.6	0.149240
$89$	−143269.	−1.91724	−0.958622	−	0.284681i	$-0.908112\pi$
$89$	−143269.	−1.91724	−0.958622	+	0.284681i	$0.908112\pi$
$90$	0	0
$91$	18036.1	0.228317
$92$	−109802.	−1.35251
$93$	0	0
$94$	15991.3	0.186665
$95$	−64663.4	−0.735105
$96$	0	0
$97$	−128960.	−1.39163	−0.695816	−	0.718220i	$-0.744957\pi$
$97$	−128960.	−1.39163	−0.695816	+	0.718220i	$0.744957\pi$
$98$	−21833.0	−0.229640
$99$	0	0
$100$	−18690.7	−0.186907
$101$	−147088.	−1.43475	−0.717373	−	0.696689i	$-0.754656\pi$
$101$	−147088.	−1.43475	−0.717373	+	0.696689i	$0.754656\pi$
$102$	0	0
$103$	88889.0	0.825572	0.412786	−	0.910828i	$-0.364556\pi$
$103$	88889.0	0.825572	0.412786	+	0.910828i	$0.364556\pi$
$104$	−38938.3	−0.353015
$105$	0	0
$106$	−30284.8	−0.261794
$107$	−146352.	−1.23577	−0.617886	−	0.786268i	$-0.712011\pi$
$107$	−146352.	−1.23577	−0.617886	+	0.786268i	$0.712011\pi$
$108$	0	0
$109$	−111535.	−0.899174	−0.449587	−	0.893236i	$-0.648429\pi$
$109$	−111535.	−0.899174	−0.449587	+	0.893236i	$0.648429\pi$
$110$	−4378.30	−0.0345004
$111$	0	0
$112$	34334.0	0.258630
$113$	56633.9	0.417234	0.208617	−	0.977997i	$-0.433104\pi$
$113$	56633.9	0.417234	0.208617	+	0.977997i	$0.433104\pi$
$114$	0	0
$115$	91792.2	0.647234
$116$	−127684.	−0.881031
$117$	0	0
$118$	47765.1	0.315795
$119$	−19687.0	−0.127442
$120$	0	0
$121$	14641.0	0.0909091
$122$	−46819.9	−0.284794
$123$	0	0
$124$	251665.	1.46984
$125$	15625.0	0.0894427
$126$	0	0
$127$	79565.3	0.437738	0.218869	−	0.975754i	$-0.429763\pi$
$127$	79565.3	0.437738	0.218869	+	0.975754i	$0.429763\pi$
$128$	−159868.	−0.862454
$129$	0	0
$130$	15725.0	0.0816078
$131$	−73365.2	−0.373518	−0.186759	−	0.982406i	$-0.559798\pi$
$131$	−73365.2	−0.373518	−0.186759	+	0.982406i	$0.559798\pi$
$132$	0	0
$133$	−107347.	−0.526213
$134$	41818.6	0.201190
$135$	0	0
$136$	42502.4	0.197045
$137$	130737.	0.595112	0.297556	−	0.954704i	$-0.403829\pi$
$137$	130737.	0.595112	0.297556	+	0.954704i	$0.403829\pi$
$138$	0	0
$139$	−366480.	−1.60884	−0.804420	−	0.594060i	$-0.797524\pi$
$139$	−366480.	−1.60884	−0.804420	+	0.594060i	$0.797524\pi$
$140$	−31028.3	−0.133794
$141$	0	0
$142$	29637.0	0.123343
$143$	−52584.2	−0.215038
$144$	0	0
$145$	106741.	0.421610
$146$	−62719.2	−0.243511
$147$	0	0
$148$	−416886.	−1.56446
$149$	−476119.	−1.75691	−0.878455	−	0.477825i	$-0.841425\pi$
$149$	−476119.	−1.75691	−0.878455	+	0.477825i	$0.841425\pi$
$150$	0	0
$151$	231635.	0.826725	0.413362	−	0.910567i	$-0.364354\pi$
$151$	231635.	0.826725	0.413362	+	0.910567i	$0.364354\pi$
$152$	231753.	0.813610
$153$	0	0
$154$	−7268.38	−0.0246965
$155$	−210386.	−0.703378
$156$	0	0
$157$	−79283.9	−0.256706	−0.128353	−	0.991729i	$-0.540969\pi$
$157$	−79283.9	−0.256706	−0.128353	+	0.991729i	$0.540969\pi$
$158$	−137937.	−0.439581
$159$	0	0
$160$	101614.	0.313801
$161$	152383.	0.463311
$162$	0	0
$163$	−303550.	−0.894872	−0.447436	−	0.894316i	$-0.647663\pi$
$163$	−303550.	−0.894872	−0.447436	+	0.894316i	$0.647663\pi$
$164$	52969.1	0.153785
$165$	0	0
$166$	24146.2	0.0680108
$167$	547748.	1.51981	0.759906	−	0.650033i	$-0.225245\pi$
$167$	547748.	1.51981	0.759906	+	0.650033i	$0.225245\pi$
$168$	0	0
$169$	−182433.	−0.491345
$170$	−17164.3	−0.0455517
$171$	0	0
$172$	343349.	0.884941
$173$	350354.	0.890003	0.445001	−	0.895530i	$-0.353203\pi$
$173$	350354.	0.890003	0.445001	+	0.895530i	$0.353203\pi$
$174$	0	0
$175$	25938.9	0.0640261
$176$	−100101.	−0.243588
$177$	0	0
$178$	−207364.	−0.490549
$179$	−185899.	−0.433656	−0.216828	−	0.976210i	$-0.569571\pi$
$179$	−185899.	−0.433656	−0.216828	+	0.976210i	$0.569571\pi$
$180$	0	0
$181$	247930.	0.562513	0.281257	−	0.959633i	$-0.409249\pi$
$181$	247930.	0.562513	0.281257	+	0.959633i	$0.409249\pi$
$182$	26104.9	0.0584176
$183$	0	0
$184$	−328982.	−0.716354
$185$	348508.	0.748657
$186$	0	0
$187$	57397.4	0.120030
$188$	−330407.	−0.681796
$189$	0	0
$190$	−93592.0	−0.188085
$191$	−715625.	−1.41939	−0.709695	−	0.704509i	$-0.751167\pi$
$191$	−715625.	−1.41939	−0.709695	+	0.704509i	$0.751167\pi$
$192$	0	0
$193$	315860.	0.610381	0.305191	−	0.952291i	$-0.401280\pi$
$193$	315860.	0.610381	0.305191	+	0.952291i	$0.401280\pi$
$194$	−186653.	−0.356065
$195$	0	0
$196$	451106.	0.838761
$197$	−139508.	−0.256115	−0.128058	−	0.991767i	$-0.540874\pi$
$197$	−139508.	−0.256115	−0.128058	+	0.991767i	$0.540874\pi$
$198$	0	0
$199$	183434.	0.328358	0.164179	−	0.986431i	$-0.447503\pi$
$199$	183434.	0.328358	0.164179	+	0.986431i	$0.447503\pi$
$200$	−55999.8	−0.0989946
$201$	0	0
$202$	−212892.	−0.367096
$203$	177200.	0.301802
$204$	0	0
$205$	−44281.0	−0.0735924
$206$	128655.	0.211232
$207$	0	0
$208$	359519.	0.576188
$209$	312971.	0.495608
$210$	0	0
$211$	−341928.	−0.528723	−0.264361	−	0.964424i	$-0.585161\pi$
$211$	−341928.	−0.528723	−0.264361	+	0.964424i	$0.585161\pi$
$212$	625735.	0.956204
$213$	0	0
$214$	−211825.	−0.316186
$215$	−287032.	−0.423481
$216$	0	0
$217$	−349261.	−0.503501
$218$	−161432.	−0.230064
$219$	0	0
$220$	90463.0	0.126013
$221$	−206147.	−0.283920
$222$	0	0
$223$	−1.01837e6	−1.37133	−0.685665	−	0.727917i	$-0.740489\pi$
$223$	−1.01837e6	−1.37133	−0.685665	+	0.727917i	$0.740489\pi$
$224$	168689.	0.224629
$225$	0	0
$226$	81970.3	0.106754
$227$	362408.	0.466802	0.233401	−	0.972381i	$-0.425015\pi$
$227$	362408.	0.466802	0.233401	+	0.972381i	$0.425015\pi$
$228$	0	0
$229$	−1.38333e6	−1.74315	−0.871577	−	0.490258i	$-0.836902\pi$
$229$	−1.38333e6	−1.74315	−0.871577	+	0.490258i	$0.836902\pi$
$230$	132857.	0.165602
$231$	0	0
$232$	−382558.	−0.466635
$233$	−1.19085e6	−1.43704	−0.718519	−	0.695507i	$-0.755180\pi$
$233$	−1.19085e6	−1.43704	−0.718519	+	0.695507i	$0.755180\pi$
$234$	0	0
$235$	276213.	0.326267
$236$	−986906.	−1.15344
$237$	0	0
$238$	−28494.3	−0.0326074
$239$	−638385.	−0.722916	−0.361458	−	0.932388i	$-0.617721\pi$
$239$	−638385.	−0.722916	−0.361458	+	0.932388i	$0.617721\pi$
$240$	0	0
$241$	301289.	0.334150	0.167075	−	0.985944i	$-0.446568\pi$
$241$	301289.	0.334150	0.167075	+	0.985944i	$0.446568\pi$
$242$	21191.0	0.0232601
$243$	0	0
$244$	967378.	1.04021
$245$	−377114.	−0.401382
$246$	0	0
$247$	−1.12406e6	−1.17232
$248$	754023.	0.778494
$249$	0	0
$250$	22615.2	0.0228850
$251$	285054.	0.285590	0.142795	−	0.989752i	$-0.454391\pi$
$251$	285054.	0.285590	0.142795	+	0.989752i	$0.454391\pi$
$252$	0	0
$253$	−444274.	−0.436365
$254$	115161.	0.112000
$255$	0	0
$256$	427492.	0.407688
$257$	1.72616e6	1.63022	0.815112	−	0.579303i	$-0.196675\pi$
$257$	1.72616e6	1.63022	0.815112	+	0.579303i	$0.196675\pi$
$258$	0	0
$259$	578554.	0.535913
$260$	−324904.	−0.298073
$261$	0	0
$262$	−106187.	−0.0955690
$263$	1.29514e6	1.15459	0.577296	−	0.816535i	$-0.304108\pi$
$263$	1.29514e6	1.15459	0.577296	+	0.816535i	$0.304108\pi$
$264$	0	0
$265$	−523100.	−0.457583
$266$	−155371.	−0.134638
$267$	0	0
$268$	−864042.	−0.734848
$269$	1.96289e6	1.65392	0.826960	−	0.562261i	$-0.190068\pi$
$269$	1.96289e6	1.65392	0.826960	+	0.562261i	$0.190068\pi$
$270$	0	0
$271$	436034.	0.360659	0.180330	−	0.983606i	$-0.442284\pi$
$271$	436034.	0.360659	0.180330	+	0.983606i	$0.442284\pi$
$272$	−392427.	−0.321615
$273$	0	0
$274$	189226.	0.152266
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−1.10131e6	−0.862401	−0.431201	−	0.902256i	$-0.641910\pi$
$277$	−1.10131e6	−0.862401	−0.431201	+	0.902256i	$0.641910\pi$
$278$	−530433.	−0.411641
$279$	0	0
$280$	−92964.8	−0.0708636
$281$	−1.21262e6	−0.916135	−0.458067	−	0.888917i	$-0.651458\pi$
$281$	−1.21262e6	−0.916135	−0.458067	+	0.888917i	$0.651458\pi$
$282$	0	0
$283$	2.07749e6	1.54196	0.770979	−	0.636860i	$-0.219767\pi$
$283$	2.07749e6	1.54196	0.770979	+	0.636860i	$0.219767\pi$
$284$	−612351.	−0.450510
$285$	0	0
$286$	−76108.9	−0.0550200
$287$	−73510.5	−0.0526799
$288$	0	0
$289$	−1.19484e6	−0.841522
$290$	154494.	0.107874
$291$	0	0
$292$	1.29588e6	0.889424
$293$	−1.75150e6	−1.19190	−0.595950	−	0.803021i	$-0.703225\pi$
$293$	−1.75150e6	−1.19190	−0.595950	+	0.803021i	$0.703225\pi$
$294$	0	0
$295$	825031.	0.551970
$296$	−1.24905e6	−0.828609
$297$	0	0
$298$	−689121.	−0.449526
$299$	1.59564e6	1.03219
$300$	0	0
$301$	−476499.	−0.303142
$302$	335261.	0.211527
$303$	0	0
$304$	−2.13979e6	−1.32797
$305$	−808706.	−0.497784
$306$	0	0
$307$	2.76532e6	1.67456	0.837278	−	0.546777i	$-0.184146\pi$
$307$	2.76532e6	1.67456	0.837278	+	0.546777i	$0.184146\pi$
$308$	150177.	0.0902040
$309$	0	0
$310$	−304507.	−0.179967
$311$	−138490.	−0.0811925	−0.0405962	−	0.999176i	$-0.512926\pi$
$311$	−138490.	−0.0811925	−0.0405962	+	0.999176i	$0.512926\pi$
$312$	0	0
$313$	2.25033e6	1.29833	0.649165	−	0.760647i	$-0.275118\pi$
$313$	2.25033e6	1.29833	0.649165	+	0.760647i	$0.275118\pi$
$314$	−114753.	−0.0656812
$315$	0	0
$316$	2.85001e6	1.60557
$317$	300036.	0.167697	0.0838486	−	0.996479i	$-0.473279\pi$
$317$	300036.	0.167697	0.0838486	+	0.996479i	$0.473279\pi$
$318$	0	0
$319$	−516626.	−0.284249
$320$	−514750.	−0.281010
$321$	0	0
$322$	220555.	0.118544
$323$	1.22695e6	0.654363
$324$	0	0
$325$	271613.	0.142640
$326$	−439350.	−0.228963
$327$	0	0
$328$	158703.	0.0814516
$329$	458538.	0.233553
$330$	0	0
$331$	637880.	0.320014	0.160007	−	0.987116i	$-0.448848\pi$
$331$	637880.	0.320014	0.160007	+	0.987116i	$0.448848\pi$
$332$	−498900.	−0.248410
$333$	0	0
$334$	792796.	0.388862
$335$	722319.	0.351655
$336$	0	0
$337$	119077.	0.0571153	0.0285576	−	0.999592i	$-0.490909\pi$
$337$	119077.	0.0571153	0.0285576	+	0.999592i	$0.490909\pi$
$338$	−264048.	−0.125716
$339$	0	0
$340$	354644.	0.166378
$341$	1.01827e6	0.474217
$342$	0	0
$343$	−1.32357e6	−0.607453
$344$	1.02872e6	0.468706
$345$	0	0
$346$	507092.	0.227718
$347$	2.35817e6	1.05136	0.525681	−	0.850682i	$-0.323811\pi$
$347$	2.35817e6	1.05136	0.525681	+	0.850682i	$0.323811\pi$
$348$	0	0
$349$	2.71913e6	1.19500	0.597499	−	0.801870i	$-0.296161\pi$
$349$	2.71913e6	1.19500	0.597499	+	0.801870i	$0.296161\pi$
$350$	37543.3	0.0163818
$351$	0	0
$352$	−491813.	−0.211565
$353$	787647.	0.336430	0.168215	−	0.985750i	$-0.446200\pi$
$353$	787647.	0.336430	0.168215	+	0.985750i	$0.446200\pi$
$354$	0	0
$355$	511911.	0.215588
$356$	4.28448e6	1.79173
$357$	0	0
$358$	−269065.	−0.110956
$359$	−3.28528e6	−1.34535	−0.672676	−	0.739937i	$-0.734855\pi$
$359$	−3.28528e6	−1.34535	−0.672676	+	0.739937i	$0.734855\pi$
$360$	0	0
$361$	4.21407e6	1.70190
$362$	358847.	0.143926
$363$	0	0
$364$	−539371.	−0.213370
$365$	−1.08333e6	−0.425626
$366$	0	0
$367$	−700644.	−0.271539	−0.135770	−	0.990740i	$-0.543351\pi$
$367$	−700644.	−0.271539	−0.135770	+	0.990740i	$0.543351\pi$
$368$	3.03751e6	1.16923
$369$	0	0
$370$	504420.	0.191553
$371$	−868394.	−0.327553
$372$	0	0
$373$	−2.59078e6	−0.964180	−0.482090	−	0.876122i	$-0.660122\pi$
$373$	−2.59078e6	−0.964180	−0.482090	+	0.876122i	$0.660122\pi$
$374$	83075.4	0.0307109
$375$	0	0
$376$	−989943.	−0.361111
$377$	1.85550e6	0.672369
$378$	0	0
$379$	−672260.	−0.240402	−0.120201	−	0.992750i	$-0.538354\pi$
$379$	−672260.	−0.240402	−0.120201	+	0.992750i	$0.538354\pi$
$380$	1.93377e6	0.686982
$381$	0	0
$382$	−1.03577e6	−0.363168
$383$	−2.41377e6	−0.840813	−0.420407	−	0.907336i	$-0.638113\pi$
$383$	−2.41377e6	−0.840813	−0.420407	+	0.907336i	$0.638113\pi$
$384$	0	0
$385$	−125544.	−0.0431664
$386$	457167.	0.156173
$387$	0	0
$388$	3.85655e6	1.30053
$389$	−3.01046e6	−1.00869	−0.504347	−	0.863501i	$-0.668267\pi$
$389$	−3.01046e6	−1.00869	−0.504347	+	0.863501i	$0.668267\pi$
$390$	0	0
$391$	−1.74170e6	−0.576143
$392$	1.35157e6	0.444247
$393$	0	0
$394$	−201921.	−0.0655300
$395$	−2.38255e6	−0.768331
$396$	0	0
$397$	871291.	0.277452	0.138726	−	0.990331i	$-0.455699\pi$
$397$	871291.	0.277452	0.138726	+	0.990331i	$0.455699\pi$
$398$	265498.	0.0840142
$399$	0	0
$400$	517050.	0.161578
$401$	2.01901e6	0.627014	0.313507	−	0.949586i	$-0.398496\pi$
$401$	2.01901e6	0.627014	0.313507	+	0.949586i	$0.398496\pi$
$402$	0	0
$403$	−3.65719e6	−1.12172
$404$	4.39869e6	1.34082
$405$	0	0
$406$	256474.	0.0772196
$407$	−1.68678e6	−0.504744
$408$	0	0
$409$	−346377.	−0.102386	−0.0511930	−	0.998689i	$-0.516302\pi$
$409$	−346377.	−0.102386	−0.0511930	+	0.998689i	$0.516302\pi$
$410$	−64091.1	−0.0188295
$411$	0	0
$412$	−2.65824e6	−0.771526
$413$	1.36963e6	0.395118
$414$	0	0
$415$	417069.	0.118874
$416$	1.76638e6	0.500440
$417$	0	0
$418$	452985.	0.126807
$419$	938584.	0.261179	0.130589	−	0.991437i	$-0.458313\pi$
$419$	938584.	0.261179	0.130589	+	0.991437i	$0.458313\pi$
$420$	0	0
$421$	4.88046e6	1.34201	0.671004	−	0.741454i	$-0.265863\pi$
$421$	4.88046e6	1.34201	0.671004	+	0.741454i	$0.265863\pi$
$422$	−494897.	−0.135280
$423$	0	0
$424$	1.87479e6	0.506450
$425$	−296474.	−0.0796186
$426$	0	0
$427$	−1.34253e6	−0.356330
$428$	4.37666e6	1.15487
$429$	0	0
$430$	−415442.	−0.108353
$431$	5.27493e6	1.36780	0.683902	−	0.729574i	$-0.260282\pi$
$431$	5.27493e6	1.36780	0.683902	+	0.729574i	$0.260282\pi$
$432$	0	0
$433$	−3.30374e6	−0.846811	−0.423405	−	0.905940i	$-0.639165\pi$
$433$	−3.30374e6	−0.846811	−0.423405	+	0.905940i	$0.639165\pi$
$434$	−505510.	−0.128827
$435$	0	0
$436$	3.33546e6	0.840310
$437$	−9.49695e6	−2.37892
$438$	0	0
$439$	5.92591e6	1.46755	0.733777	−	0.679391i	$-0.237756\pi$
$439$	5.92591e6	1.46755	0.733777	+	0.679391i	$0.237756\pi$
$440$	271039.	0.0667422
$441$	0	0
$442$	−298371.	−0.0726442
$443$	−2.36693e6	−0.573029	−0.286515	−	0.958076i	$-0.592497\pi$
$443$	−2.36693e6	−0.573029	−0.286515	+	0.958076i	$0.592497\pi$
$444$	0	0
$445$	−3.58173e6	−0.857418
$446$	−1.47396e6	−0.350871
$447$	0	0
$448$	−854532.	−0.201156
$449$	3.35144e6	0.784540	0.392270	−	0.919850i	$-0.371690\pi$
$449$	3.35144e6	0.784540	0.392270	+	0.919850i	$0.371690\pi$
$450$	0	0
$451$	214320.	0.0496160
$452$	−1.69364e6	−0.389920
$453$	0	0
$454$	524539.	0.119437
$455$	450902.	0.102107
$456$	0	0
$457$	4.04376e6	0.905721	0.452861	−	0.891581i	$-0.350404\pi$
$457$	4.04376e6	0.905721	0.452861	+	0.891581i	$0.350404\pi$
$458$	−2.00219e6	−0.446006
$459$	0	0
$460$	−2.74506e6	−0.604862
$461$	8.31607e6	1.82249	0.911247	−	0.411861i	$-0.135121\pi$
$461$	8.31607e6	1.82249	0.911247	+	0.411861i	$0.135121\pi$
$462$	0	0
$463$	−5.54406e6	−1.20192	−0.600960	−	0.799279i	$-0.705215\pi$
$463$	−5.54406e6	−1.20192	−0.600960	+	0.799279i	$0.705215\pi$
$464$	3.53218e6	0.761637
$465$	0	0
$466$	−1.72361e6	−0.367683
$467$	5.54472e6	1.17649	0.588244	−	0.808684i	$-0.299820\pi$
$467$	5.54472e6	1.17649	0.588244	+	0.808684i	$0.299820\pi$
$468$	0	0
$469$	1.19912e6	0.251726
$470$	399782.	0.0834793
$471$	0	0
$472$	−2.95690e6	−0.610916
$473$	1.38923e6	0.285511
$474$	0	0
$475$	−1.61659e6	−0.328749
$476$	588741.	0.119099
$477$	0	0
$478$	−923980.	−0.184966
$479$	−2.93070e6	−0.583624	−0.291812	−	0.956476i	$-0.594258\pi$
$479$	−2.93070e6	−0.583624	−0.291812	+	0.956476i	$0.594258\pi$
$480$	0	0
$481$	6.05818e6	1.19393
$482$	436078.	0.0854961
$483$	0	0
$484$	−437841.	−0.0849577
$485$	−3.22399e6	−0.622357
$486$	0	0
$487$	−370838.	−0.0708536	−0.0354268	−	0.999372i	$-0.511279\pi$
$487$	−370838.	−0.0708536	−0.0354268	+	0.999372i	$0.511279\pi$
$488$	2.89839e6	0.550944
$489$	0	0
$490$	−545824.	−0.102698
$491$	−592085.	−0.110836	−0.0554179	−	0.998463i	$-0.517649\pi$
$491$	−592085.	−0.110836	−0.0554179	+	0.998463i	$0.517649\pi$
$492$	0	0
$493$	−2.02534e6	−0.375301
$494$	−1.62693e6	−0.299952
$495$	0	0
$496$	−6.96194e6	−1.27065
$497$	849819.	0.154325
$498$	0	0
$499$	−6.23154e6	−1.12032	−0.560162	−	0.828383i	$-0.689261\pi$
$499$	−6.23154e6	−1.12032	−0.560162	+	0.828383i	$0.689261\pi$
$500$	−467267.	−0.0835873
$501$	0	0
$502$	412579.	0.0730714
$503$	−6.58812e6	−1.16102	−0.580512	−	0.814251i	$-0.697148\pi$
$503$	−6.58812e6	−1.16102	−0.580512	+	0.814251i	$0.697148\pi$
$504$	0	0
$505$	−3.67721e6	−0.641638
$506$	−643030.	−0.111649
$507$	0	0
$508$	−2.37941e6	−0.409082
$509$	−8.26967e6	−1.41479	−0.707397	−	0.706816i	$-0.750131\pi$
$509$	−8.26967e6	−1.41479	−0.707397	+	0.706816i	$0.750131\pi$
$510$	0	0
$511$	−1.79842e6	−0.304677
$512$	5.73451e6	0.966765
$513$	0	0
$514$	2.49839e6	0.417112
$515$	2.22222e6	0.369207
$516$	0	0
$517$	−1.33687e6	−0.219969
$518$	837383.	0.137120
$519$	0	0
$520$	−973457.	−0.157873
$521$	−1.99187e6	−0.321490	−0.160745	−	0.986996i	$-0.551390\pi$
$521$	−1.99187e6	−0.321490	−0.160745	+	0.986996i	$0.551390\pi$
$522$	0	0
$523$	7.99327e6	1.27782	0.638911	−	0.769281i	$-0.279385\pi$
$523$	7.99327e6	1.27782	0.638911	+	0.769281i	$0.279385\pi$
$524$	2.19399e6	0.349066
$525$	0	0
$526$	1.87455e6	0.295416
$527$	3.99194e6	0.626121
$528$	0	0
$529$	7.04495e6	1.09456
$530$	−757121.	−0.117078
$531$	0	0
$532$	3.21023e6	0.491764
$533$	−769746.	−0.117363
$534$	0	0
$535$	−3.65879e6	−0.552654
$536$	−2.58878e6	−0.389210
$537$	0	0
$538$	2.84103e6	0.423175
$539$	1.82523e6	0.270611
$540$	0	0
$541$	−8.83361e6	−1.29761	−0.648806	−	0.760954i	$-0.724731\pi$
$541$	−8.83361e6	−1.29761	−0.648806	+	0.760954i	$0.724731\pi$
$542$	631103.	0.0922789
$543$	0	0
$544$	−1.92806e6	−0.279334
$545$	−2.78837e6	−0.402123
$546$	0	0
$547$	2.98240e6	0.426184	0.213092	−	0.977032i	$-0.431647\pi$
$547$	2.98240e6	0.426184	0.213092	+	0.977032i	$0.431647\pi$
$548$	−3.90972e6	−0.556153
$549$	0	0
$550$	−109457.	−0.0154290
$551$	−1.10436e7	−1.54964
$552$	0	0
$553$	−3.95524e6	−0.549997
$554$	−1.59400e6	−0.220655
$555$	0	0
$556$	1.09596e7	1.50352
$557$	−1.20831e7	−1.65021	−0.825105	−	0.564979i	$-0.808884\pi$
$557$	−1.20831e7	−1.65021	−0.825105	+	0.564979i	$0.808884\pi$
$558$	0	0
$559$	−4.98954e6	−0.675353
$560$	858350.	0.115663
$561$	0	0
$562$	−1.75511e6	−0.234404
$563$	1.18255e7	1.57235	0.786176	−	0.618003i	$-0.212058\pi$
$563$	1.18255e7	1.57235	0.786176	+	0.618003i	$0.212058\pi$
$564$	0	0
$565$	1.41585e6	0.186593
$566$	3.00690e6	0.394528
$567$	0	0
$568$	−1.83468e6	−0.238611
$569$	1.13220e6	0.146603	0.0733014	−	0.997310i	$-0.476647\pi$
$569$	1.13220e6	0.146603	0.0733014	+	0.997310i	$0.476647\pi$
$570$	0	0
$571$	−2.92831e6	−0.375860	−0.187930	−	0.982182i	$-0.560178\pi$
$571$	−2.92831e6	−0.375860	−0.187930	+	0.982182i	$0.560178\pi$
$572$	1.57254e6	0.200961
$573$	0	0
$574$	−106397.	−0.0134788
$575$	2.29480e6	0.289452
$576$	0	0
$577$	3.70459e6	0.463235	0.231617	−	0.972807i	$-0.425598\pi$
$577$	3.70459e6	0.463235	0.231617	+	0.972807i	$0.425598\pi$
$578$	−1.72938e6	−0.215313
$579$	0	0
$580$	−3.19210e6	−0.394009
$581$	692373.	0.0850941
$582$	0	0
$583$	2.53181e6	0.308503
$584$	3.88264e6	0.471080
$585$	0	0
$586$	−2.53507e6	−0.304962
$587$	−6.77491e6	−0.811537	−0.405769	−	0.913976i	$-0.632996\pi$
$587$	−6.77491e6	−0.811537	−0.405769	+	0.913976i	$0.632996\pi$
$588$	0	0
$589$	2.17669e7	2.58528
$590$	1.19413e6	0.141228
$591$	0	0
$592$	1.15325e7	1.35245
$593$	−1.58897e7	−1.85557	−0.927787	−	0.373111i	$-0.878291\pi$
$593$	−1.58897e7	−1.85557	−0.927787	+	0.373111i	$0.878291\pi$
$594$	0	0
$595$	−492174.	−0.0569936
$596$	1.42384e7	1.64189
$597$	0	0
$598$	2.30949e6	0.264097
$599$	−4.74611e6	−0.540469	−0.270234	−	0.962795i	$-0.587101\pi$
$599$	−4.74611e6	−0.540469	−0.270234	+	0.962795i	$0.587101\pi$
$600$	0	0
$601$	−2.55803e6	−0.288882	−0.144441	−	0.989513i	$-0.546138\pi$
$601$	−2.55803e6	−0.288882	−0.144441	+	0.989513i	$0.546138\pi$
$602$	−689671.	−0.0775623
$603$	0	0
$604$	−6.92706e6	−0.772603
$605$	366025.	0.0406558
$606$	0	0
$607$	1.28362e6	0.141405	0.0707025	−	0.997497i	$-0.477476\pi$
$607$	1.28362e6	0.141405	0.0707025	+	0.997497i	$0.477476\pi$
$608$	−1.05132e7	−1.15339
$609$	0	0
$610$	−1.17050e6	−0.127364
$611$	4.80146e6	0.520320
$612$	0	0
$613$	1.01402e7	1.08992	0.544960	−	0.838462i	$-0.316545\pi$
$613$	1.01402e7	1.08992	0.544960	+	0.838462i	$0.316545\pi$
$614$	4.00245e6	0.428455
$615$	0	0
$616$	449950.	0.0477763
$617$	−1.44801e7	−1.53130	−0.765649	−	0.643259i	$-0.777582\pi$
$617$	−1.44801e7	−1.53130	−0.765649	+	0.643259i	$0.777582\pi$
$618$	0	0
$619$	1.03997e6	0.109092	0.0545462	−	0.998511i	$-0.482629\pi$
$619$	1.03997e6	0.109092	0.0545462	+	0.998511i	$0.482629\pi$
$620$	6.29163e6	0.657331
$621$	0	0
$622$	−200446.	−0.0207740
$623$	−5.94599e6	−0.613768
$624$	0	0
$625$	390625.	0.0400000
$626$	3.25706e6	0.332193
$627$	0	0
$628$	2.37099e6	0.239901
$629$	−6.61270e6	−0.666426
$630$	0	0
$631$	1.25757e7	1.25736	0.628681	−	0.777664i	$-0.283595\pi$
$631$	1.25757e7	1.25736	0.628681	+	0.777664i	$0.283595\pi$
$632$	8.53902e6	0.850384
$633$	0	0
$634$	434264.	0.0429073
$635$	1.98913e6	0.195763
$636$	0	0
$637$	−6.55545e6	−0.640109
$638$	−747750.	−0.0727285
$639$	0	0
$640$	−3.99669e6	−0.385701
$641$	−5.13372e6	−0.493500	−0.246750	−	0.969079i	$-0.579363\pi$
$641$	−5.13372e6	−0.493500	−0.246750	+	0.969079i	$0.579363\pi$
$642$	0	0
$643$	−1.80636e7	−1.72296	−0.861482	−	0.507788i	$-0.830463\pi$
$643$	−1.80636e7	−1.72296	−0.861482	+	0.507788i	$0.830463\pi$
$644$	−4.55704e6	−0.432981
$645$	0	0
$646$	1.77585e6	0.167426
$647$	298642.	0.0280472	0.0140236	−	0.999902i	$-0.495536\pi$
$647$	298642.	0.0280472	0.0140236	+	0.999902i	$0.495536\pi$
$648$	0	0
$649$	−3.99315e6	−0.372138
$650$	393125.	0.0364961
$651$	0	0
$652$	9.07770e6	0.836289
$653$	−1.52761e6	−0.140194	−0.0700971	−	0.997540i	$-0.522331\pi$
$653$	−1.52761e6	−0.140194	−0.0700971	+	0.997540i	$0.522331\pi$
$654$	0	0
$655$	−1.83413e6	−0.167042
$656$	−1.46531e6	−0.132944
$657$	0	0
$658$	663675.	0.0597573
$659$	−1.69648e7	−1.52172	−0.760862	−	0.648913i	$-0.775224\pi$
$659$	−1.69648e7	−1.52172	−0.760862	+	0.648913i	$0.775224\pi$
$660$	0	0
$661$	−7.24635e6	−0.645083	−0.322542	−	0.946555i	$-0.604537\pi$
$661$	−7.24635e6	−0.645083	−0.322542	+	0.946555i	$0.604537\pi$
$662$	923249.	0.0818793
$663$	0	0
$664$	−1.49477e6	−0.131569
$665$	−2.68368e6	−0.235330
$666$	0	0
$667$	1.56768e7	1.36440
$668$	−1.63805e7	−1.42032
$669$	0	0
$670$	1.04546e6	0.0899751
$671$	3.91414e6	0.335606
$672$	0	0
$673$	−2.29001e7	−1.94895	−0.974473	−	0.224505i	$-0.927923\pi$
$673$	−2.29001e7	−1.94895	−0.974473	+	0.224505i	$0.927923\pi$
$674$	172348.	0.0146136
$675$	0	0
$676$	5.45568e6	0.459179
$677$	−1.31898e7	−1.10603	−0.553015	−	0.833171i	$-0.686523\pi$
$677$	−1.31898e7	−1.10603	−0.553015	+	0.833171i	$0.686523\pi$
$678$	0	0
$679$	−5.35212e6	−0.445504
$680$	1.06256e6	0.0881213
$681$	0	0
$682$	1.47382e6	0.121334
$683$	2.11554e7	1.73528	0.867638	−	0.497196i	$-0.165637\pi$
$683$	2.11554e7	1.73528	0.867638	+	0.497196i	$0.165637\pi$
$684$	0	0
$685$	3.26844e6	0.266142
$686$	−1.91570e6	−0.155424
$687$	0	0
$688$	−9.49822e6	−0.765017
$689$	−9.09316e6	−0.729738
$690$	0	0
$691$	−1.40736e7	−1.12127	−0.560636	−	0.828063i	$-0.689443\pi$
$691$	−1.40736e7	−1.12127	−0.560636	+	0.828063i	$0.689443\pi$
$692$	−1.04774e7	−0.831739
$693$	0	0
$694$	3.41315e6	0.269003
$695$	−9.16200e6	−0.719495
$696$	0	0
$697$	840203.	0.0655092
$698$	3.93560e6	0.305754
$699$	0	0
$700$	−775706.	−0.0598346
$701$	5.70759e6	0.438690	0.219345	−	0.975647i	$-0.429608\pi$
$701$	5.70759e6	0.438690	0.219345	+	0.975647i	$0.429608\pi$
$702$	0	0
$703$	−3.60571e7	−2.75171
$704$	2.49139e6	0.189457
$705$	0	0
$706$	1.14002e6	0.0860796
$707$	−6.10450e6	−0.459306
$708$	0	0
$709$	−6.05501e6	−0.452375	−0.226188	−	0.974084i	$-0.572626\pi$
$709$	−6.05501e6	−0.452375	−0.226188	+	0.974084i	$0.572626\pi$
$710$	740926.	0.0551606
$711$	0	0
$712$	1.28369e7	0.948985
$713$	−3.08989e7	−2.27625
$714$	0	0
$715$	−1.31461e6	−0.0961679
$716$	5.55934e6	0.405266
$717$	0	0
$718$	−4.75502e6	−0.344224
$719$	1.99372e6	0.143828	0.0719139	−	0.997411i	$-0.477089\pi$
$719$	1.99372e6	0.143828	0.0719139	+	0.997411i	$0.477089\pi$
$720$	0	0
$721$	3.68909e6	0.264291
$722$	6.09933e6	0.435451
$723$	0	0
$724$	−7.41438e6	−0.525688
$725$	2.66852e6	0.188550
$726$	0	0
$727$	1.66770e7	1.17026	0.585129	−	0.810940i	$-0.301044\pi$
$727$	1.66770e7	1.17026	0.585129	+	0.810940i	$0.301044\pi$
$728$	−1.61603e6	−0.113011
$729$	0	0
$730$	−1.56798e6	−0.108901
$731$	5.44624e6	0.376967
$732$	0	0
$733$	−6.22120e6	−0.427676	−0.213838	−	0.976869i	$-0.568596\pi$
$733$	−6.22120e6	−0.427676	−0.213838	+	0.976869i	$0.568596\pi$
$734$	−1.01409e6	−0.0694764
$735$	0	0
$736$	1.49238e7	1.01551
$737$	−3.49602e6	−0.237086
$738$	0	0
$739$	193628.	0.0130424	0.00652118	−	0.999979i	$-0.497924\pi$
$739$	193628.	0.0130424	0.00652118	+	0.999979i	$0.497924\pi$
$740$	−1.04222e7	−0.699646
$741$	0	0
$742$	−1.25689e6	−0.0838083
$743$	−1.49469e7	−0.993295	−0.496648	−	0.867952i	$-0.665436\pi$
$743$	−1.49469e7	−0.993295	−0.496648	+	0.867952i	$0.665436\pi$
$744$	0	0
$745$	−1.19030e7	−0.785714
$746$	−3.74982e6	−0.246697
$747$	0	0
$748$	−1.71648e6	−0.112172
$749$	−6.07393e6	−0.395608
$750$	0	0
$751$	−3.06276e6	−0.198159	−0.0990793	−	0.995080i	$-0.531590\pi$
$751$	−3.06276e6	−0.198159	−0.0990793	+	0.995080i	$0.531590\pi$
$752$	9.14020e6	0.589401
$753$	0	0
$754$	2.68560e6	0.172033
$755$	5.79086e6	0.369723
$756$	0	0
$757$	−2.87783e7	−1.82527	−0.912633	−	0.408780i	$-0.865954\pi$
$757$	−2.87783e7	−1.82527	−0.912633	+	0.408780i	$0.865954\pi$
$758$	−973009.	−0.0615098
$759$	0	0
$760$	5.79382e6	0.363857
$761$	−1.92089e6	−0.120238	−0.0601189	−	0.998191i	$-0.519148\pi$
$761$	−1.92089e6	−0.120238	−0.0601189	+	0.998191i	$0.519148\pi$
$762$	0	0
$763$	−4.62895e6	−0.287853
$764$	2.14008e7	1.32647
$765$	0	0
$766$	−3.49363e6	−0.215132
$767$	1.43417e7	0.880262
$768$	0	0
$769$	1.51697e7	0.925043	0.462521	−	0.886608i	$-0.346945\pi$
$769$	1.51697e7	0.925043	0.462521	+	0.886608i	$0.346945\pi$
$770$	−181709.	−0.0110446
$771$	0	0
$772$	−9.44583e6	−0.570423
$773$	1.17588e7	0.707809	0.353904	−	0.935282i	$-0.384854\pi$
$773$	1.17588e7	0.707809	0.353904	+	0.935282i	$0.384854\pi$
$774$	0	0
$775$	−5.25966e6	−0.314560
$776$	1.15548e7	0.688821
$777$	0	0
$778$	−4.35726e6	−0.258086
$779$	4.58138e6	0.270491
$780$	0	0
$781$	−2.47765e6	−0.145349
$782$	−2.52088e6	−0.147413
$783$	0	0
$784$	−1.24791e7	−0.725095
$785$	−1.98210e6	−0.114802
$786$	0	0
$787$	−1.07859e6	−0.0620756	−0.0310378	−	0.999518i	$-0.509881\pi$
$787$	−1.07859e6	−0.0620756	−0.0310378	+	0.999518i	$0.509881\pi$
$788$	4.17202e6	0.239348
$789$	0	0
$790$	−3.44843e6	−0.196586
$791$	2.35043e6	0.133569
$792$	0	0
$793$	−1.40579e7	−0.793849
$794$	1.26108e6	0.0709892
$795$	0	0
$796$	−5.48562e6	−0.306862
$797$	−7.73047e6	−0.431082	−0.215541	−	0.976495i	$-0.569152\pi$
$797$	−7.73047e6	−0.431082	−0.215541	+	0.976495i	$0.569152\pi$
$798$	0	0
$799$	−5.24095e6	−0.290431
$800$	2.54036e6	0.140336
$801$	0	0
$802$	2.92225e6	0.160429
$803$	5.24331e6	0.286957
$804$	0	0
$805$	3.80958e6	0.207199
$806$	−5.29332e6	−0.287006
$807$	0	0
$808$	1.31791e7	0.710161
$809$	1.49228e6	0.0801639	0.0400819	−	0.999196i	$-0.487238\pi$
$809$	1.49228e6	0.0801639	0.0400819	+	0.999196i	$0.487238\pi$
$810$	0	0
$811$	−7.38116e6	−0.394069	−0.197035	−	0.980397i	$-0.563131\pi$
$811$	−7.38116e6	−0.394069	−0.197035	+	0.980397i	$0.563131\pi$
$812$	−5.29917e6	−0.282045
$813$	0	0
$814$	−2.44139e6	−0.129145
$815$	−7.58875e6	−0.400199
$816$	0	0
$817$	2.96967e7	1.55652
$818$	−501336.	−0.0261967
$819$	0	0
$820$	1.32423e6	0.0687746
$821$	1.42454e7	0.737594	0.368797	−	0.929510i	$-0.379770\pi$
$821$	1.42454e7	0.737594	0.368797	+	0.929510i	$0.379770\pi$
$822$	0	0
$823$	−1.91145e7	−0.983702	−0.491851	−	0.870679i	$-0.663680\pi$
$823$	−1.91145e7	−0.983702	−0.491851	+	0.870679i	$0.663680\pi$
$824$	−7.96443e6	−0.408636
$825$	0	0
$826$	1.98236e6	0.101096
$827$	2.68783e7	1.36659	0.683294	−	0.730143i	$-0.260547\pi$
$827$	2.68783e7	1.36659	0.683294	+	0.730143i	$0.260547\pi$
$828$	0	0
$829$	2.27066e7	1.14754	0.573768	−	0.819018i	$-0.305481\pi$
$829$	2.27066e7	1.14754	0.573768	+	0.819018i	$0.305481\pi$
$830$	603654.	0.0304154
$831$	0	0
$832$	−8.94801e6	−0.448145
$833$	7.15549e6	0.357295
$834$	0	0
$835$	1.36937e7	0.679681
$836$	−9.35943e6	−0.463163
$837$	0	0
$838$	1.35848e6	0.0668257
$839$	1.54859e7	0.759505	0.379753	−	0.925088i	$-0.376009\pi$
$839$	1.54859e7	0.759505	0.379753	+	0.925088i	$0.376009\pi$
$840$	0	0
$841$	−2.28136e6	−0.111225
$842$	7.06383e6	0.343368
$843$	0	0
$844$	1.02254e7	0.494110
$845$	−4.56082e6	−0.219736
$846$	0	0
$847$	607635.	0.0291028
$848$	−1.73100e7	−0.826623
$849$	0	0
$850$	−429108.	−0.0203713
$851$	5.11844e7	2.42278
$852$	0	0
$853$	−3.77200e6	−0.177500	−0.0887502	−	0.996054i	$-0.528287\pi$
$853$	−3.77200e6	−0.177500	−0.0887502	+	0.996054i	$0.528287\pi$
$854$	−1.94313e6	−0.0911712
$855$	0	0
$856$	1.31131e7	0.611674
$857$	6.50207e6	0.302412	0.151206	−	0.988502i	$-0.451684\pi$
$857$	6.50207e6	0.302412	0.151206	+	0.988502i	$0.451684\pi$
$858$	0	0
$859$	−1.10411e7	−0.510538	−0.255269	−	0.966870i	$-0.582164\pi$
$859$	−1.10411e7	−0.510538	−0.255269	+	0.966870i	$0.582164\pi$
$860$	8.58372e6	0.395758
$861$	0	0
$862$	7.63479e6	0.349968
$863$	1.55955e6	0.0712808	0.0356404	−	0.999365i	$-0.488653\pi$
$863$	1.55955e6	0.0712808	0.0356404	+	0.999365i	$0.488653\pi$
$864$	0	0
$865$	8.75884e6	0.398021
$866$	−4.78174e6	−0.216666
$867$	0	0
$868$	1.04447e7	0.470539
$869$	1.15315e7	0.518009
$870$	0	0
$871$	1.25562e7	0.560808
$872$	9.99348e6	0.445067
$873$	0	0
$874$	−1.37456e7	−0.608676
$875$	648473.	0.0286333
$876$	0	0
$877$	2.78476e7	1.22261	0.611307	−	0.791393i	$-0.290644\pi$
$877$	2.78476e7	1.22261	0.611307	+	0.791393i	$0.290644\pi$
$878$	8.57700e6	0.375491
$879$	0	0
$880$	−2.50252e6	−0.108936
$881$	1.52851e7	0.663481	0.331740	−	0.943371i	$-0.392364\pi$
$881$	1.52851e7	0.663481	0.331740	+	0.943371i	$0.392364\pi$
$882$	0	0
$883$	−7.82254e6	−0.337634	−0.168817	−	0.985647i	$-0.553995\pi$
$883$	−7.82254e6	−0.337634	−0.168817	+	0.985647i	$0.553995\pi$
$884$	6.16485e6	0.265333
$885$	0	0
$886$	−3.42583e6	−0.146616
$887$	−2.39652e7	−1.02276	−0.511379	−	0.859355i	$-0.670865\pi$
$887$	−2.39652e7	−1.02276	−0.511379	+	0.859355i	$0.670865\pi$
$888$	0	0
$889$	3.30214e6	0.140133
$890$	−5.18409e6	−0.219380
$891$	0	0
$892$	3.04544e7	1.28156
$893$	−2.85774e7	−1.19920
$894$	0	0
$895$	−4.64748e6	−0.193937
$896$	−6.63487e6	−0.276098
$897$	0	0
$898$	4.85078e6	0.200734
$899$	−3.59309e7	−1.48276
$900$	0	0
$901$	9.92548e6	0.407324
$902$	310201.	0.0126948
$903$	0	0
$904$	−5.07438e6	−0.206520
$905$	6.19825e6	0.251564
$906$	0	0
$907$	−5.48830e6	−0.221523	−0.110762	−	0.993847i	$-0.535329\pi$
$907$	−5.48830e6	−0.221523	−0.110762	+	0.993847i	$0.535329\pi$
$908$	−1.08378e7	−0.436243
$909$	0	0
$910$	652623.	0.0261251
$911$	918823.	0.0366806	0.0183403	−	0.999832i	$-0.494162\pi$
$911$	918823.	0.0366806	0.0183403	+	0.999832i	$0.494162\pi$
$912$	0	0
$913$	−2.01861e6	−0.0801450
$914$	5.85282e6	0.231739
$915$	0	0
$916$	4.13685e7	1.62904
$917$	−3.04482e6	−0.119574
$918$	0	0
$919$	−6.92636e6	−0.270530	−0.135265	−	0.990809i	$-0.543189\pi$
$919$	−6.92636e6	−0.270530	−0.135265	+	0.990809i	$0.543189\pi$
$920$	−8.22455e6	−0.320363
$921$	0	0
$922$	1.20365e7	0.466306
$923$	8.89866e6	0.343812
$924$	0	0
$925$	8.71269e6	0.334810
$926$	−8.02432e6	−0.307525
$927$	0	0
$928$	1.73542e7	0.661509
$929$	2.62196e7	0.996752	0.498376	−	0.866961i	$-0.333930\pi$
$929$	2.62196e7	0.996752	0.498376	+	0.866961i	$0.333930\pi$
$930$	0	0
$931$	3.90168e7	1.47529
$932$	3.56126e7	1.34296
$933$	0	0
$934$	8.02527e6	0.301018
$935$	1.43493e6	0.0536788
$936$	0	0
$937$	−216082.	−0.00804027	−0.00402013	−	0.999992i	$-0.501280\pi$
$937$	−216082.	−0.00804027	−0.00402013	+	0.999992i	$0.501280\pi$
$938$	1.73557e6	0.0644071
$939$	0	0
$940$	−8.26017e6	−0.304908
$941$	−2.94074e7	−1.08264	−0.541318	−	0.840818i	$-0.682074\pi$
$941$	−2.94074e7	−1.08264	−0.541318	+	0.840818i	$0.682074\pi$
$942$	0	0
$943$	−6.50344e6	−0.238157
$944$	2.73012e7	0.997132
$945$	0	0
$946$	2.01074e6	0.0730513
$947$	4.18343e7	1.51585	0.757927	−	0.652339i	$-0.226212\pi$
$947$	4.18343e7	1.51585	0.757927	+	0.652339i	$0.226212\pi$
$948$	0	0
$949$	−1.88317e7	−0.678774
$950$	−2.33980e6	−0.0841143
$951$	0	0
$952$	1.76395e6	0.0630802
$953$	1.37691e7	0.491104	0.245552	−	0.969383i	$-0.421031\pi$
$953$	1.37691e7	0.491104	0.245552	+	0.969383i	$0.421031\pi$
$954$	0	0
$955$	−1.78906e7	−0.634771
$956$	1.90910e7	0.675590
$957$	0	0
$958$	−4.24182e6	−0.149327
$959$	5.42590e6	0.190513
$960$	0	0
$961$	4.21908e7	1.47370
$962$	8.76844e6	0.305481
$963$	0	0
$964$	−9.01009e6	−0.312275
$965$	7.89650e6	0.272971
$966$	0	0
$967$	−4.56284e7	−1.56917	−0.784584	−	0.620022i	$-0.787123\pi$
$967$	−4.56284e7	−1.56917	−0.784584	+	0.620022i	$0.787123\pi$
$968$	−1.31183e6	−0.0449976
$969$	0	0
$970$	−4.66632e6	−0.159237
$971$	−9.15569e6	−0.311633	−0.155816	−	0.987786i	$-0.549801\pi$
$971$	−9.15569e6	−0.311633	−0.155816	+	0.987786i	$0.549801\pi$
$972$	0	0
$973$	−1.52097e7	−0.515039
$974$	−536741.	−0.0181287
$975$	0	0
$976$	−2.67610e7	−0.899246
$977$	1.12269e7	0.376289	0.188145	−	0.982141i	$-0.439753\pi$
$977$	1.12269e7	0.376289	0.188145	+	0.982141i	$0.439753\pi$
$978$	0	0
$979$	1.73356e7	0.578071
$980$	1.12776e7	0.375105
$981$	0	0
$982$	−856967.	−0.0283586
$983$	1.92294e7	0.634718	0.317359	−	0.948305i	$-0.397204\pi$
$983$	1.92294e7	0.634718	0.317359	+	0.948305i	$0.397204\pi$
$984$	0	0
$985$	−3.48771e6	−0.114538
$986$	−2.93142e6	−0.0960252
$987$	0	0
$988$	3.36151e7	1.09557
$989$	−4.21556e7	−1.37046
$990$	0	0
$991$	4.87732e7	1.57760	0.788801	−	0.614649i	$-0.210702\pi$
$991$	4.87732e7	1.57760	0.788801	+	0.614649i	$0.210702\pi$
$992$	−3.42052e7	−1.10360
$993$	0	0
$994$	1.23000e6	0.0394858
$995$	4.58586e6	0.146846
$996$	0	0
$997$	2.88621e7	0.919583	0.459791	−	0.888027i	$-0.347924\pi$
$997$	2.88621e7	0.919583	0.459791	+	0.888027i	$0.347924\pi$
$998$	−9.01935e6	−0.286648
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.d.1.2		3
3.2	odd	2		165.6.a.b.1.2	✓	3
15.14	odd	2		825.6.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.b.1.2	✓	3	3.2	odd	2
495.6.a.d.1.2		3	1.1	even	1	trivial
825.6.a.i.1.2		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44737 q^{2} -29.9051 q^{4} +25.0000 q^{5} +41.5023 q^{7} -89.5997 q^{8} +O(q^{10})\)	\(q+1.44737 q^{2} -29.9051 q^{4} +25.0000 q^{5} +41.5023 q^{7} -89.5997 q^{8} +36.1843 q^{10} -121.000 q^{11} +434.580 q^{13} +60.0692 q^{14} +827.280 q^{16} -474.359 q^{17} -2586.54 q^{19} -747.628 q^{20} -175.132 q^{22} +3671.69 q^{23} +625.000 q^{25} +628.999 q^{26} -1241.13 q^{28} +4269.64 q^{29} -8415.46 q^{31} +4064.57 q^{32} -686.573 q^{34} +1037.56 q^{35} +13940.3 q^{37} -3743.68 q^{38} -2239.99 q^{40} -1771.24 q^{41} -11481.3 q^{43} +3618.52 q^{44} +5314.30 q^{46} +11048.5 q^{47} -15084.6 q^{49} +904.607 q^{50} -12996.2 q^{52} -20924.0 q^{53} -3025.00 q^{55} -3718.59 q^{56} +6179.75 q^{58} +33001.2 q^{59} -32348.2 q^{61} -12180.3 q^{62} -20590.0 q^{64} +10864.5 q^{65} +28892.8 q^{67} +14185.7 q^{68} +1501.73 q^{70} +20476.5 q^{71} -43333.2 q^{73} +20176.8 q^{74} +77350.7 q^{76} -5021.77 q^{77} -95301.8 q^{79} +20682.0 q^{80} -2563.64 q^{82} +16682.8 q^{83} -11859.0 q^{85} -16617.7 q^{86} +10841.6 q^{88} -143269. q^{89} +18036.1 q^{91} -109802. q^{92} +15991.3 q^{94} -64663.4 q^{95} -128960. q^{97} -21833.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * q^98

Introduction

L-functions

Modular forms

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Database

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