LMFDB - Embedded newform 825.6.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$0$
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	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44737 q^{2} -9.00000 q^{3} -29.9051 q^{4} -13.0263 q^{6} -41.5023 q^{7} -89.5997 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+1.44737 q^{2} -9.00000 q^{3} -29.9051 q^{4} -13.0263 q^{6} -41.5023 q^{7} -89.5997 q^{8} +81.0000 q^{9} +121.000 q^{11} +269.146 q^{12} -434.580 q^{13} -60.0692 q^{14} +827.280 q^{16} -474.359 q^{17} +117.237 q^{18} -2586.54 q^{19} +373.520 q^{21} +175.132 q^{22} +3671.69 q^{23} +806.397 q^{24} -628.999 q^{26} -729.000 q^{27} +1241.13 q^{28} -4269.64 q^{29} -8415.46 q^{31} +4064.57 q^{32} -1089.00 q^{33} -686.573 q^{34} -2422.31 q^{36} -13940.3 q^{37} -3743.68 q^{38} +3911.22 q^{39} +1771.24 q^{41} +540.623 q^{42} +11481.3 q^{43} -3618.52 q^{44} +5314.30 q^{46} +11048.5 q^{47} -7445.52 q^{48} -15084.6 q^{49} +4269.23 q^{51} +12996.2 q^{52} -20924.0 q^{53} -1055.13 q^{54} +3718.59 q^{56} +23278.8 q^{57} -6179.75 q^{58} -33001.2 q^{59} -32348.2 q^{61} -12180.3 q^{62} -3361.68 q^{63} -20590.0 q^{64} -1576.19 q^{66} -28892.8 q^{67} +14185.7 q^{68} -33045.2 q^{69} -20476.5 q^{71} -7257.58 q^{72} +43333.2 q^{73} -20176.8 q^{74} +77350.7 q^{76} -5021.77 q^{77} +5660.99 q^{78} -95301.8 q^{79} +6561.00 q^{81} +2563.64 q^{82} +16682.8 q^{83} -11170.2 q^{84} +16617.7 q^{86} +38426.7 q^{87} -10841.6 q^{88} +143269. q^{89} +18036.1 q^{91} -109802. q^{92} +75739.1 q^{93} +15991.3 q^{94} -36581.1 q^{96} +128960. q^{97} -21833.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

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	$0.459166\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−29.9051	−0.934535
$5$	0	0
$5$	0	0
$6$	−13.0263	−0.147722
$7$	−41.5023	−0.320130	−0.160065	−	0.987106i	$-0.551170\pi$
$7$	−41.5023	−0.320130	−0.160065	+	0.987106i	$0.551170\pi$
$8$	−89.5997	−0.494973
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	269.146	0.539554
$13$	−434.580	−0.713201	−0.356600	−	0.934257i	$-0.616064\pi$
$13$	−434.580	−0.713201	−0.356600	+	0.934257i	$0.616064\pi$
$14$	−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	$0.563784\pi$
$18$	117.237	0.0852872
$19$	−2586.54	−1.64375	−0.821873	−	0.569671i	$-0.807071\pi$
$19$	−2586.54	−1.64375	−0.821873	+	0.569671i	$0.807071\pi$
$20$	0	0
$21$	373.520	0.184827
$22$	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	$0.242473\pi$
$24$	806.397	0.285773
$25$	0	0
$26$	−628.999	−0.182481
$27$	−729.000	−0.192450
$28$	1241.13	0.299173
$29$	−4269.64	−0.942749	−0.471374	−	0.881933i	$-0.656242\pi$
$29$	−4269.64	−0.942749	−0.471374	+	0.881933i	$0.656242\pi$
$30$	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
$33$	−1089.00	−0.174078
$34$	−686.573	−0.101857
$35$	0	0
$36$	−2422.31	−0.311512
$37$	−13940.3	−1.67405	−0.837024	−	0.547166i	$-0.815706\pi$
$37$	−13940.3	−1.67405	−0.837024	+	0.547166i	$0.815706\pi$
$38$	−3743.68	−0.420571
$39$	3911.22	0.411767
$40$	0	0
$41$	1771.24	0.164558	0.0822788	−	0.996609i	$-0.473780\pi$
$41$	1771.24	0.164558	0.0822788	+	0.996609i	$0.473780\pi$
$42$	540.623	0.0472902
$43$	11481.3	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$43$	11481.3	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$44$	−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	$0.381145\pi$
$48$	−7445.52	−0.466436
$49$	−15084.6	−0.897517
$50$	0	0
$51$	4269.23	0.229839
$52$	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	$0.670945\pi$
$54$	−1055.13	−0.0492406
$55$	0	0
$56$	3718.59	0.158456
$57$	23278.8	0.949017
$58$	−6179.75	−0.241213
$59$	−33001.2	−1.23424	−0.617121	−	0.786868i	$-0.711701\pi$
$59$	−33001.2	−1.23424	−0.617121	+	0.786868i	$0.711701\pi$
$60$	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
$63$	−3361.68	−0.106710
$64$	−20590.0	−0.628357
$65$	0	0
$66$	−1576.19	−0.0445398
$67$	−28892.8	−0.786325	−0.393163	−	0.919469i	$-0.628619\pi$
$67$	−28892.8	−0.786325	−0.393163	+	0.919469i	$0.628619\pi$
$68$	14185.7	0.372032
$69$	−33045.2	−0.835575
$70$	0	0
$71$	−20476.5	−0.482069	−0.241034	−	0.970517i	$-0.577487\pi$
$71$	−20476.5	−0.482069	−0.241034	+	0.970517i	$0.577487\pi$
$72$	−7257.58	−0.164991
$73$	43333.2	0.951729	0.475864	−	0.879519i	$-0.342135\pi$
$73$	43333.2	0.951729	0.475864	+	0.879519i	$0.342135\pi$
$74$	−20176.8	−0.428325
$75$	0	0
$76$	77350.7	1.53614
$77$	−5021.77	−0.0965229
$78$	5660.99	0.105355
$79$	−95301.8	−1.71804	−0.859020	−	0.511942i	$-0.828926\pi$
$79$	−95301.8	−1.71804	−0.859020	+	0.511942i	$0.828926\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	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	$0.457569\pi$
$84$	−11170.2	−0.172728
$85$	0	0
$86$	16617.7	0.242284
$87$	38426.7	0.544296
$88$	−10841.6	−0.149240
$89$	143269.	1.91724	0.958622	−	0.284681i	$-0.0918877\pi$
$89$	143269.	1.91724	0.958622	+	0.284681i	$0.0918877\pi$
$90$	0	0
$91$	18036.1	0.228317
$92$	−109802.	−1.35251
$93$	75739.1	0.908057
$94$	15991.3	0.186665
$95$	0	0
$96$	−36581.1	−0.405116
$97$	128960.	1.39163	0.695816	−	0.718220i	$-0.255043\pi$
$97$	128960.	1.39163	0.695816	+	0.718220i	$0.255043\pi$
$98$	−21833.0	−0.229640
$99$	9801.00	0.100504
$100$	0	0
$101$	147088.	1.43475	0.717373	−	0.696689i	$-0.245344\pi$
$101$	147088.	1.43475	0.717373	+	0.696689i	$0.245344\pi$
$102$	6179.16	0.0588070
$103$	−88889.0	−0.825572	−0.412786	−	0.910828i	$-0.635444\pi$
$103$	−88889.0	−0.825572	−0.412786	+	0.910828i	$0.635444\pi$
$104$	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	$0.712011\pi$
$108$	21800.8	0.179851
$109$	−111535.	−0.899174	−0.449587	−	0.893236i	$-0.648429\pi$
$109$	−111535.	−0.899174	−0.449587	+	0.893236i	$0.648429\pi$
$110$	0	0
$111$	125463.	0.966512
$112$	−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	$0.433104\pi$
$114$	33693.1	0.242817
$115$	0	0
$116$	127684.	0.881031
$117$	−35201.0	−0.237734
$118$	−47765.1	−0.315795
$119$	19687.0	0.127442
$120$	0	0
$121$	14641.0	0.0909091
$122$	−46819.9	−0.284794
$123$	−15941.2	−0.0950074
$124$	251665.	1.46984
$125$	0	0
$126$	−4865.61	−0.0273030
$127$	−79565.3	−0.437738	−0.218869	−	0.975754i	$-0.570237\pi$
$127$	−79565.3	−0.437738	−0.218869	+	0.975754i	$0.570237\pi$
$128$	−159868.	−0.862454
$129$	−103331.	−0.546712
$130$	0	0
$131$	73365.2	0.373518	0.186759	−	0.982406i	$-0.440202\pi$
$131$	73365.2	0.373518	0.186759	+	0.982406i	$0.440202\pi$
$132$	32566.7	0.162682
$133$	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	$0.403829\pi$
$138$	−47828.7	−0.213792
$139$	−366480.	−1.60884	−0.804420	−	0.594060i	$-0.797524\pi$
$139$	−366480.	−1.60884	−0.804420	+	0.594060i	$0.797524\pi$
$140$	0	0
$141$	−99436.5	−0.421209
$142$	−29637.0	−0.123343
$143$	−52584.2	−0.215038
$144$	67009.6	0.269297
$145$	0	0
$146$	62719.2	0.243511
$147$	135761.	0.518181
$148$	416886.	1.56446
$149$	476119.	1.75691	0.878455	−	0.477825i	$-0.158575\pi$
$149$	476119.	1.75691	0.878455	+	0.477825i	$0.158575\pi$
$150$	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
$153$	−38423.0	−0.132698
$154$	−7268.38	−0.0246965
$155$	0	0
$156$	−116966.	−0.384810
$157$	79283.9	0.256706	0.128353	−	0.991729i	$-0.459031\pi$
$157$	79283.9	0.256706	0.128353	+	0.991729i	$0.459031\pi$
$158$	−137937.	−0.439581
$159$	188316.	0.590738
$160$	0	0
$161$	−152383.	−0.463311
$162$	9496.21	0.0284291
$163$	303550.	0.894872	0.447436	−	0.894316i	$-0.352337\pi$
$163$	303550.	0.894872	0.447436	+	0.894316i	$0.352337\pi$
$164$	−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	$0.225245\pi$
$168$	−33467.3	−0.0914846
$169$	−182433.	−0.491345
$170$	0	0
$171$	−209509.	−0.547915
$172$	−343349.	−0.884941
$173$	350354.	0.890003	0.445001	−	0.895530i	$-0.353203\pi$
$173$	350354.	0.890003	0.445001	+	0.895530i	$0.353203\pi$
$174$	55617.8	0.139264
$175$	0	0
$176$	100101.	0.243588
$177$	297011.	0.712590
$178$	207364.	0.490549
$179$	185899.	0.433656	0.216828	−	0.976210i	$-0.430429\pi$
$179$	185899.	0.433656	0.216828	+	0.976210i	$0.430429\pi$
$180$	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
$183$	291134.	0.642636
$184$	−328982.	−0.716354
$185$	0	0
$186$	109623.	0.232337
$187$	−57397.4	−0.120030
$188$	−330407.	−0.681796
$189$	30255.2	0.0616091
$190$	0	0
$191$	715625.	1.41939	0.709695	−	0.704509i	$-0.248833\pi$
$191$	715625.	1.41939	0.709695	+	0.704509i	$0.248833\pi$
$192$	185310.	0.362782
$193$	−315860.	−0.610381	−0.305191	−	0.952291i	$-0.598720\pi$
$193$	−315860.	−0.610381	−0.305191	+	0.952291i	$0.598720\pi$
$194$	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	$0.540874\pi$
$198$	14185.7	0.0257151
$199$	183434.	0.328358	0.164179	−	0.986431i	$-0.447503\pi$
$199$	183434.	0.328358	0.164179	+	0.986431i	$0.447503\pi$
$200$	0	0
$201$	260035.	0.453985
$202$	212892.	0.367096
$203$	177200.	0.301802
$204$	−127672.	−0.214793
$205$	0	0
$206$	−128655.	−0.211232
$207$	297407.	0.482420
$208$	−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
$213$	184288.	0.278322
$214$	−211825.	−0.316186
$215$	0	0
$216$	65318.2	0.0952576
$217$	349261.	0.503501
$218$	−161432.	−0.230064
$219$	−389998.	−0.549481
$220$	0	0
$221$	206147.	0.283920
$222$	181591.	0.247293
$223$	1.01837e6	1.37133	0.685665	−	0.727917i	$-0.259511\pi$
$223$	1.01837e6	1.37133	0.685665	+	0.727917i	$0.259511\pi$
$224$	−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	$0.425015\pi$
$228$	−696156.	−0.886889
$229$	−1.38333e6	−1.74315	−0.871577	−	0.490258i	$-0.836902\pi$
$229$	−1.38333e6	−1.74315	−0.871577	+	0.490258i	$0.836902\pi$
$230$	0	0
$231$	45196.0	0.0557275
$232$	382558.	0.466635
$233$	−1.19085e6	−1.43704	−0.718519	−	0.695507i	$-0.755180\pi$
$233$	−1.19085e6	−1.43704	−0.718519	+	0.695507i	$0.755180\pi$
$234$	−50948.9	−0.0608269
$235$	0	0
$236$	986906.	1.15344
$237$	857716.	0.991911
$238$	28494.3	0.0326074
$239$	638385.	0.722916	0.361458	−	0.932388i	$-0.382279\pi$
$239$	638385.	0.722916	0.361458	+	0.932388i	$0.382279\pi$
$240$	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
$243$	−59049.0	−0.0641500
$244$	967378.	1.04021
$245$	0	0
$246$	−23072.8	−0.0243087
$247$	1.12406e6	1.17232
$248$	754023.	0.778494
$249$	−150145.	−0.153466
$250$	0	0
$251$	−285054.	−0.285590	−0.142795	−	0.989752i	$-0.545609\pi$
$251$	−285054.	−0.285590	−0.142795	+	0.989752i	$0.545609\pi$
$252$	100532.	0.0997243
$253$	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	$0.196675\pi$
$258$	−149559.	−0.139882
$259$	578554.	0.535913
$260$	0	0
$261$	−345841.	−0.314250
$262$	106187.	0.0955690
$263$	1.29514e6	1.15459	0.577296	−	0.816535i	$-0.304108\pi$
$263$	1.29514e6	1.15459	0.577296	+	0.816535i	$0.304108\pi$
$264$	97574.1	0.0861638
$265$	0	0
$266$	155371.	0.134638
$267$	−1.28942e6	−1.10692
$268$	864042.	0.734848
$269$	−1.96289e6	−1.65392	−0.826960	−	0.562261i	$-0.809932\pi$
$269$	−1.96289e6	−1.65392	−0.826960	+	0.562261i	$0.809932\pi$
$270$	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
$273$	−162325.	−0.131819
$274$	189226.	0.152266
$275$	0	0
$276$	988220.	0.780874
$277$	1.10131e6	0.862401	0.431201	−	0.902256i	$-0.358090\pi$
$277$	1.10131e6	0.862401	0.431201	+	0.902256i	$0.358090\pi$
$278$	−530433.	−0.411641
$279$	−681652.	−0.524267
$280$	0	0
$281$	1.21262e6	0.916135	0.458067	−	0.888917i	$-0.348542\pi$
$281$	1.21262e6	0.916135	0.458067	+	0.888917i	$0.348542\pi$
$282$	−143922.	−0.107771
$283$	−2.07749e6	−1.54196	−0.770979	−	0.636860i	$-0.780233\pi$
$283$	−2.07749e6	−1.54196	−0.770979	+	0.636860i	$0.780233\pi$
$284$	612351.	0.450510
$285$	0	0
$286$	−76108.9	−0.0550200
$287$	−73510.5	−0.0526799
$288$	329230.	0.233894
$289$	−1.19484e6	−0.841522
$290$	0	0
$291$	−1.16064e6	−0.803460
$292$	−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	$0.703225\pi$
$294$	196497.	0.132583
$295$	0	0
$296$	1.24905e6	0.828609
$297$	−88209.0	−0.0580259
$298$	689121.	0.449526
$299$	−1.59564e6	−1.03219
$300$	0	0
$301$	−476499.	−0.303142
$302$	335261.	0.211527
$303$	−1.32380e6	−0.828351
$304$	−2.13979e6	−1.32797
$305$	0	0
$306$	−55612.4	−0.0339522
$307$	−2.76532e6	−1.67456	−0.837278	−	0.546777i	$-0.815854\pi$
$307$	−2.76532e6	−1.67456	−0.837278	+	0.546777i	$0.815854\pi$
$308$	150177.	0.0902040
$309$	800001.	0.476644
$310$	0	0
$311$	138490.	0.0811925	0.0405962	−	0.999176i	$-0.487074\pi$
$311$	138490.	0.0811925	0.0405962	+	0.999176i	$0.487074\pi$
$312$	−350444.	−0.203813
$313$	−2.25033e6	−1.29833	−0.649165	−	0.760647i	$-0.724882\pi$
$313$	−2.25033e6	−1.29833	−0.649165	+	0.760647i	$0.724882\pi$
$314$	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	$0.473279\pi$
$318$	272563.	0.151147
$319$	−516626.	−0.284249
$320$	0	0
$321$	1.31716e6	0.713473
$322$	−220555.	−0.118544
$323$	1.22695e6	0.654363
$324$	−196207.	−0.103837
$325$	0	0
$326$	439350.	0.228963
$327$	1.00381e6	0.519139
$328$	−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
$333$	−1.12916e6	−0.558016
$334$	792796.	0.388862
$335$	0	0
$336$	309006.	0.149320
$337$	−119077.	−0.0571153	−0.0285576	−	0.999592i	$-0.509091\pi$
$337$	−119077.	−0.0571153	−0.0285576	+	0.999592i	$0.509091\pi$
$338$	−264048.	−0.125716
$339$	−509705.	−0.240890
$340$	0	0
$341$	−1.01827e6	−0.474217
$342$	−303238.	−0.140190
$343$	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	$0.323811\pi$
$348$	−1.14916e6	−0.508664
$349$	2.71913e6	1.19500	0.597499	−	0.801870i	$-0.296161\pi$
$349$	2.71913e6	1.19500	0.597499	+	0.801870i	$0.296161\pi$
$350$	0	0
$351$	316809.	0.137256
$352$	491813.	0.211565
$353$	787647.	0.336430	0.168215	−	0.985750i	$-0.446200\pi$
$353$	787647.	0.336430	0.168215	+	0.985750i	$0.446200\pi$
$354$	429885.	0.182324
$355$	0	0
$356$	−4.28448e6	−1.79173
$357$	−177183.	−0.0735784
$358$	269065.	0.110956
$359$	3.28528e6	1.34535	0.672676	−	0.739937i	$-0.265145\pi$
$359$	3.28528e6	1.34535	0.672676	+	0.739937i	$0.265145\pi$
$360$	0	0
$361$	4.21407e6	1.70190
$362$	358847.	0.143926
$363$	−131769.	−0.0524864
$364$	−539371.	−0.213370
$365$	0	0
$366$	421379.	0.164426
$367$	700644.	0.271539	0.135770	−	0.990740i	$-0.456649\pi$
$367$	700644.	0.271539	0.135770	+	0.990740i	$0.456649\pi$
$368$	3.03751e6	1.16923
$369$	143470.	0.0548525
$370$	0	0
$371$	868394.	0.327553
$372$	−2.26499e6	−0.848611
$373$	2.59078e6	0.964180	0.482090	−	0.876122i	$-0.339878\pi$
$373$	2.59078e6	0.964180	0.482090	+	0.876122i	$0.339878\pi$
$374$	−83075.4	−0.0307109
$375$	0	0
$376$	−989943.	−0.361111
$377$	1.85550e6	0.672369
$378$	43790.5	0.0157634
$379$	−672260.	−0.240402	−0.120201	−	0.992750i	$-0.538354\pi$
$379$	−672260.	−0.240402	−0.120201	+	0.992750i	$0.538354\pi$
$380$	0	0
$381$	716088.	0.252728
$382$	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	$0.638113\pi$
$384$	1.43881e6	0.497938
$385$	0	0
$386$	−457167.	−0.156173
$387$	929983.	0.315644
$388$	−3.85655e6	−1.30053
$389$	3.01046e6	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$389$	3.01046e6	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$390$	0	0
$391$	−1.74170e6	−0.576143
$392$	1.35157e6	0.444247
$393$	−660287.	−0.215651
$394$	−201921.	−0.0655300
$395$	0	0
$396$	−293100.	−0.0939243
$397$	−871291.	−0.277452	−0.138726	−	0.990331i	$-0.544301\pi$
$397$	−871291.	−0.277452	−0.138726	+	0.990331i	$0.544301\pi$
$398$	265498.	0.0840142
$399$	−966124.	−0.303809
$400$	0	0
$401$	−2.01901e6	−0.627014	−0.313507	−	0.949586i	$-0.601504\pi$
$401$	−2.01901e6	−0.627014	−0.313507	+	0.949586i	$0.601504\pi$
$402$	376367.	0.116157
$403$	3.65719e6	1.12172
$404$	−4.39869e6	−1.34082
$405$	0	0
$406$	256474.	0.0772196
$407$	−1.68678e6	−0.504744
$408$	−382522.	−0.113764
$409$	−346377.	−0.102386	−0.0511930	−	0.998689i	$-0.516302\pi$
$409$	−346377.	−0.102386	−0.0511930	+	0.998689i	$0.516302\pi$
$410$	0	0
$411$	−1.17664e6	−0.343588
$412$	2.65824e6	0.771526
$413$	1.36963e6	0.395118
$414$	430458.	0.123433
$415$	0	0
$416$	−1.76638e6	−0.500440
$417$	3.29832e6	0.928865
$418$	−452985.	−0.126807
$419$	−938584.	−0.261179	−0.130589	−	0.991437i	$-0.541687\pi$
$419$	−938584.	−0.261179	−0.130589	+	0.991437i	$0.541687\pi$
$420$	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
$423$	894929.	0.243185
$424$	1.87479e6	0.506450
$425$	0	0
$426$	266733.	0.0712120
$427$	1.34253e6	0.356330
$428$	4.37666e6	1.15487
$429$	473258.	0.124152
$430$	0	0
$431$	−5.27493e6	−1.36780	−0.683902	−	0.729574i	$-0.739718\pi$
$431$	−5.27493e6	−1.36780	−0.683902	+	0.729574i	$0.739718\pi$
$432$	−603087.	−0.155479
$433$	3.30374e6	0.846811	0.423405	−	0.905940i	$-0.360835\pi$
$433$	3.30374e6	0.846811	0.423405	+	0.905940i	$0.360835\pi$
$434$	505510.	0.128827
$435$	0	0
$436$	3.33546e6	0.840310
$437$	−9.49695e6	−2.37892
$438$	−564473.	−0.140591
$439$	5.92591e6	1.46755	0.733777	−	0.679391i	$-0.237756\pi$
$439$	5.92591e6	1.46755	0.733777	+	0.679391i	$0.237756\pi$
$440$	0	0
$441$	−1.22185e6	−0.299172
$442$	298371.	0.0726442
$443$	−2.36693e6	−0.573029	−0.286515	−	0.958076i	$-0.592497\pi$
$443$	−2.36693e6	−0.573029	−0.286515	+	0.958076i	$0.592497\pi$
$444$	−3.75198e6	−0.903239
$445$	0	0
$446$	1.47396e6	0.350871
$447$	−4.28507e6	−1.01435
$448$	854532.	0.201156
$449$	−3.35144e6	−0.784540	−0.392270	−	0.919850i	$-0.628310\pi$
$449$	−3.35144e6	−0.784540	−0.392270	+	0.919850i	$0.628310\pi$
$450$	0	0
$451$	214320.	0.0496160
$452$	−1.69364e6	−0.389920
$453$	−2.08471e6	−0.477310
$454$	524539.	0.119437
$455$	0	0
$456$	−2.08578e6	−0.469738
$457$	−4.04376e6	−0.905721	−0.452861	−	0.891581i	$-0.649596\pi$
$457$	−4.04376e6	−0.905721	−0.452861	+	0.891581i	$0.649596\pi$
$458$	−2.00219e6	−0.446006
$459$	345807.	0.0766130
$460$	0	0
$461$	−8.31607e6	−1.82249	−0.911247	−	0.411861i	$-0.864879\pi$
$461$	−8.31607e6	−1.82249	−0.911247	+	0.411861i	$0.864879\pi$
$462$	65415.4	0.0142585
$463$	5.54406e6	1.20192	0.600960	−	0.799279i	$-0.294785\pi$
$463$	5.54406e6	1.20192	0.600960	+	0.799279i	$0.294785\pi$
$464$	−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	$0.299820\pi$
$468$	1.05269e6	0.222170
$469$	1.19912e6	0.251726
$470$	0	0
$471$	−713555.	−0.148209
$472$	2.95690e6	0.610916
$473$	1.38923e6	0.285511
$474$	1.24143e6	0.253792
$475$	0	0
$476$	−588741.	−0.119099
$477$	−1.69484e6	−0.341063
$478$	923980.	0.184966
$479$	2.93070e6	0.583624	0.291812	−	0.956476i	$-0.405742\pi$
$479$	2.93070e6	0.583624	0.291812	+	0.956476i	$0.405742\pi$
$480$	0	0
$481$	6.05818e6	1.19393
$482$	436078.	0.0854961
$483$	1.37145e6	0.267493
$484$	−437841.	−0.0849577
$485$	0	0
$486$	−85465.9	−0.0164135
$487$	370838.	0.0708536	0.0354268	−	0.999372i	$-0.488721\pi$
$487$	370838.	0.0708536	0.0354268	+	0.999372i	$0.488721\pi$
$488$	2.89839e6	0.550944
$489$	−2.73195e6	−0.516655
$490$	0	0
$491$	592085.	0.110836	0.0554179	−	0.998463i	$-0.482351\pi$
$491$	592085.	0.110836	0.0554179	+	0.998463i	$0.482351\pi$
$492$	476722.	0.0887877
$493$	2.02534e6	0.375301
$494$	1.62693e6	0.299952
$495$	0	0
$496$	−6.96194e6	−1.27065
$497$	849819.	0.154325
$498$	−217315.	−0.0392661
$499$	−6.23154e6	−1.12032	−0.560162	−	0.828383i	$-0.689261\pi$
$499$	−6.23154e6	−1.12032	−0.560162	+	0.828383i	$0.689261\pi$
$500$	0	0
$501$	−4.92974e6	−0.877464
$502$	−412579.	−0.0730714
$503$	−6.58812e6	−1.16102	−0.580512	−	0.814251i	$-0.697148\pi$
$503$	−6.58812e6	−1.16102	−0.580512	+	0.814251i	$0.697148\pi$
$504$	301206.	0.0528186
$505$	0	0
$506$	643030.	0.111649
$507$	1.64190e6	0.283678
$508$	2.37941e6	0.409082
$509$	8.26967e6	1.41479	0.707397	−	0.706816i	$-0.249869\pi$
$509$	8.26967e6	1.41479	0.707397	+	0.706816i	$0.249869\pi$
$510$	0	0
$511$	−1.79842e6	−0.304677
$512$	5.73451e6	0.966765
$513$	1.88559e6	0.316339
$514$	2.49839e6	0.417112
$515$	0	0
$516$	3.09014e6	0.510921
$517$	1.33687e6	0.219969
$518$	837383.	0.137120
$519$	−3.15318e6	−0.513843
$520$	0	0
$521$	1.99187e6	0.321490	0.160745	−	0.986996i	$-0.448610\pi$
$521$	1.99187e6	0.321490	0.160745	+	0.986996i	$0.448610\pi$
$522$	−500560.	−0.0804044
$523$	−7.99327e6	−1.27782	−0.638911	−	0.769281i	$-0.720615\pi$
$523$	−7.99327e6	−1.27782	−0.638911	+	0.769281i	$0.720615\pi$
$524$	−2.19399e6	−0.349066
$525$	0	0
$526$	1.87455e6	0.295416
$527$	3.99194e6	0.626121
$528$	−900907.	−0.140636
$529$	7.04495e6	1.09456
$530$	0	0
$531$	−2.67310e6	−0.411414
$532$	−3.21023e6	−0.491764
$533$	−769746.	−0.117363
$534$	−1.86627e6	−0.283219
$535$	0	0
$536$	2.58878e6	0.389210
$537$	−1.67309e6	−0.250371
$538$	−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
$543$	−2.23137e6	−0.324767
$544$	−1.92806e6	−0.279334
$545$	0	0
$546$	−234944.	−0.0337274
$547$	−2.98240e6	−0.426184	−0.213092	−	0.977032i	$-0.568353\pi$
$547$	−2.98240e6	−0.426184	−0.213092	+	0.977032i	$0.568353\pi$
$548$	−3.90972e6	−0.556153
$549$	−2.62021e6	−0.371026
$550$	0	0
$551$	1.10436e7	1.54964
$552$	2.96084e6	0.413587
$553$	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	$0.808884\pi$
$558$	−986604.	−0.134140
$559$	−4.98954e6	−0.675353
$560$	0	0
$561$	516576.	0.0692991
$562$	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	$0.212058\pi$
$564$	2.97366e6	0.393635
$565$	0	0
$566$	−3.00690e6	−0.394528
$567$	−272296.	−0.0355700
$568$	1.83468e6	0.238611
$569$	−1.13220e6	−0.146603	−0.0733014	−	0.997310i	$-0.523353\pi$
$569$	−1.13220e6	−0.146603	−0.0733014	+	0.997310i	$0.523353\pi$
$570$	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
$573$	−6.44062e6	−0.819485
$574$	−106397.	−0.0134788
$575$	0	0
$576$	−1.66779e6	−0.209452
$577$	−3.70459e6	−0.463235	−0.231617	−	0.972807i	$-0.574402\pi$
$577$	−3.70459e6	−0.463235	−0.231617	+	0.972807i	$0.574402\pi$
$578$	−1.72938e6	−0.215313
$579$	2.84274e6	0.352404
$580$	0	0
$581$	−692373.	−0.0850941
$582$	−1.67987e6	−0.205574
$583$	−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	$0.632996\pi$
$588$	−4.05995e6	−0.484259
$589$	2.17669e7	2.58528
$590$	0	0
$591$	1.25558e6	0.147868
$592$	−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	$0.878291\pi$
$594$	−127671.	−0.0148466
$595$	0	0
$596$	−1.42384e7	−1.64189
$597$	−1.65091e6	−0.189578
$598$	−2.30949e6	−0.264097
$599$	4.74611e6	0.540469	0.270234	−	0.962795i	$-0.412899\pi$
$599$	4.74611e6	0.540469	0.270234	+	0.962795i	$0.412899\pi$
$600$	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
$603$	−2.34031e6	−0.262108
$604$	−6.92706e6	−0.772603
$605$	0	0
$606$	−1.91602e6	−0.211943
$607$	−1.28362e6	−0.141405	−0.0707025	−	0.997497i	$-0.522524\pi$
$607$	−1.28362e6	−0.141405	−0.0707025	+	0.997497i	$0.522524\pi$
$608$	−1.05132e7	−1.15339
$609$	−1.59480e6	−0.174246
$610$	0	0
$611$	−4.80146e6	−0.520320
$612$	1.14905e6	0.124011
$613$	−1.01402e7	−1.08992	−0.544960	−	0.838462i	$-0.683455\pi$
$613$	−1.01402e7	−1.08992	−0.544960	+	0.838462i	$0.683455\pi$
$614$	−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	$0.777582\pi$
$618$	1.15790e6	0.121955
$619$	1.03997e6	0.109092	0.0545462	−	0.998511i	$-0.482629\pi$
$619$	1.03997e6	0.109092	0.0545462	+	0.998511i	$0.482629\pi$
$620$	0	0
$621$	−2.67666e6	−0.278525
$622$	200446.	0.0207740
$623$	−5.94599e6	−0.613768
$624$	3.23567e6	0.332662
$625$	0	0
$626$	−3.25706e6	−0.332193
$627$	2.81674e6	0.286139
$628$	−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
$633$	3.07735e6	0.305258
$634$	434264.	0.0429073
$635$	0	0
$636$	−5.63161e6	−0.552065
$637$	6.55545e6	0.640109
$638$	−747750.	−0.0727285
$639$	−1.65859e6	−0.160690
$640$	0	0
$641$	5.13372e6	0.493500	0.246750	−	0.969079i	$-0.420637\pi$
$641$	5.13372e6	0.493500	0.246750	+	0.969079i	$0.420637\pi$
$642$	1.90643e6	0.182550
$643$	1.80636e7	1.72296	0.861482	−	0.507788i	$-0.169537\pi$
$643$	1.80636e7	1.72296	0.861482	+	0.507788i	$0.169537\pi$
$644$	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	$0.495536\pi$
$648$	−587864.	−0.0549970
$649$	−3.99315e6	−0.372138
$650$	0	0
$651$	−3.14335e6	−0.290696
$652$	−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	$0.522331\pi$
$654$	1.45289e6	0.132828
$655$	0	0
$656$	1.46531e6	0.132944
$657$	3.50999e6	0.317243
$658$	−663675.	−0.0597573
$659$	1.69648e7	1.52172	0.760862	−	0.648913i	$-0.224776\pi$
$659$	1.69648e7	1.52172	0.760862	+	0.648913i	$0.224776\pi$
$660$	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
$663$	−1.85532e6	−0.163921
$664$	−1.49477e6	−0.131569
$665$	0	0
$666$	−1.63432e6	−0.142775
$667$	−1.56768e7	−1.36440
$668$	−1.63805e7	−1.42032
$669$	−9.16530e6	−0.791738
$670$	0	0
$671$	−3.91414e6	−0.335606
$672$	1.51820e6	0.129690
$673$	2.29001e7	1.94895	0.974473	−	0.224505i	$-0.0720765\pi$
$673$	2.29001e7	1.94895	0.974473	+	0.224505i	$0.0720765\pi$
$674$	−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	$0.686523\pi$
$678$	−737732.	−0.0616346
$679$	−5.35212e6	−0.445504
$680$	0	0
$681$	−3.26167e6	−0.269508
$682$	−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	$0.165637\pi$
$684$	6.26540e6	0.512046
$685$	0	0
$686$	1.91570e6	0.155424
$687$	1.24499e7	1.00641
$688$	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
$693$	−406764.	−0.0321743
$694$	3.41315e6	0.269003
$695$	0	0
$696$	−3.44302e6	−0.269412
$697$	−840203.	−0.0655092
$698$	3.93560e6	0.305754
$699$	1.07177e7	0.829674
$700$	0	0
$701$	−5.70759e6	−0.438690	−0.219345	−	0.975647i	$-0.570392\pi$
$701$	−5.70759e6	−0.438690	−0.219345	+	0.975647i	$0.570392\pi$
$702$	458541.	0.0351184
$703$	3.60571e7	2.75171
$704$	−2.49139e6	−0.189457
$705$	0	0
$706$	1.14002e6	0.0860796
$707$	−6.10450e6	−0.459306
$708$	−8.88215e6	−0.665940
$709$	−6.05501e6	−0.452375	−0.226188	−	0.974084i	$-0.572626\pi$
$709$	−6.05501e6	−0.452375	−0.226188	+	0.974084i	$0.572626\pi$
$710$	0	0
$711$	−7.71945e6	−0.572680
$712$	−1.28369e7	−0.948985
$713$	−3.08989e7	−2.27625
$714$	−256449.	−0.0188259
$715$	0	0
$716$	−5.55934e6	−0.405266
$717$	−5.74546e6	−0.417376
$718$	4.75502e6	0.344224
$719$	−1.99372e6	−0.143828	−0.0719139	−	0.997411i	$-0.522911\pi$
$719$	−1.99372e6	−0.143828	−0.0719139	+	0.997411i	$0.522911\pi$
$720$	0	0
$721$	3.68909e6	0.264291
$722$	6.09933e6	0.435451
$723$	−2.71160e6	−0.192921
$724$	−7.41438e6	−0.525688
$725$	0	0
$726$	−190719.	−0.0134293
$727$	−1.66770e7	−1.17026	−0.585129	−	0.810940i	$-0.698956\pi$
$727$	−1.66770e7	−1.17026	−0.585129	+	0.810940i	$0.698956\pi$
$728$	−1.61603e6	−0.113011
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.44624e6	−0.376967
$732$	−8.70640e6	−0.600566
$733$	6.22120e6	0.427676	0.213838	−	0.976869i	$-0.431404\pi$
$733$	6.22120e6	0.427676	0.213838	+	0.976869i	$0.431404\pi$
$734$	1.01409e6	0.0694764
$735$	0	0
$736$	1.49238e7	1.01551
$737$	−3.49602e6	−0.237086
$738$	207655.	0.0140347
$739$	193628.	0.0130424	0.00652118	−	0.999979i	$-0.497924\pi$
$739$	193628.	0.0130424	0.00652118	+	0.999979i	$0.497924\pi$
$740$	0	0
$741$	−1.01165e7	−0.676840
$742$	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	$0.665436\pi$
$744$	−6.78620e6	−0.449464
$745$	0	0
$746$	3.74982e6	0.246697
$747$	1.35130e6	0.0886036
$748$	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
$753$	2.56548e6	0.164885
$754$	2.68560e6	0.172033
$755$	0	0
$756$	−904784.	−0.0575759
$757$	2.87783e7	1.82527	0.912633	−	0.408780i	$-0.134046\pi$
$757$	2.87783e7	1.82527	0.912633	+	0.408780i	$0.134046\pi$
$758$	−973009.	−0.0615098
$759$	−3.99847e6	−0.251935
$760$	0	0
$761$	1.92089e6	0.120238	0.0601189	−	0.998191i	$-0.480852\pi$
$761$	1.92089e6	0.120238	0.0601189	+	0.998191i	$0.480852\pi$
$762$	1.03645e6	0.0646635
$763$	4.62895e6	0.287853
$764$	−2.14008e7	−1.32647
$765$	0	0
$766$	−3.49363e6	−0.215132
$767$	1.43417e7	0.880262
$768$	−3.84743e6	−0.235379
$769$	1.51697e7	0.925043	0.462521	−	0.886608i	$-0.346945\pi$
$769$	1.51697e7	0.925043	0.462521	+	0.886608i	$0.346945\pi$
$770$	0	0
$771$	−1.55354e7	−0.941211
$772$	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	$0.384854\pi$
$774$	1.34603e6	0.0807612
$775$	0	0
$776$	−1.15548e7	−0.688821
$777$	−5.20699e6	−0.309410
$778$	4.35726e6	0.258086
$779$	−4.58138e6	−0.270491
$780$	0	0
$781$	−2.47765e6	−0.145349
$782$	−2.52088e6	−0.147413
$783$	3.11256e6	0.181432
$784$	−1.24791e7	−0.725095
$785$	0	0
$786$	−955680.	−0.0551768
$787$	1.07859e6	0.0620756	0.0310378	−	0.999518i	$-0.490119\pi$
$787$	1.07859e6	0.0620756	0.0310378	+	0.999518i	$0.490119\pi$
$788$	4.17202e6	0.239348
$789$	−1.16563e7	−0.666604
$790$	0	0
$791$	−2.35043e6	−0.133569
$792$	−878167.	−0.0497467
$793$	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	$0.569152\pi$
$798$	−1.39834e6	−0.0777331
$799$	−5.24095e6	−0.290431
$800$	0	0
$801$	1.16048e7	0.639082
$802$	−2.92225e6	−0.160429
$803$	5.24331e6	0.286957
$804$	−7.77637e6	−0.424265
$805$	0	0
$806$	5.29332e6	0.287006
$807$	1.76660e7	0.954891
$808$	−1.31791e7	−0.710161
$809$	−1.49228e6	−0.0801639	−0.0400819	−	0.999196i	$-0.512762\pi$
$809$	−1.49228e6	−0.0801639	−0.0400819	+	0.999196i	$0.512762\pi$
$810$	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
$813$	−3.92431e6	−0.208227
$814$	−2.44139e6	−0.129145
$815$	0	0
$816$	3.53184e6	0.185685
$817$	−2.96967e7	−1.55652
$818$	−501336.	−0.0261967
$819$	1.46092e6	0.0761057
$820$	0	0
$821$	−1.42454e7	−0.737594	−0.368797	−	0.929510i	$-0.620230\pi$
$821$	−1.42454e7	−0.737594	−0.368797	+	0.929510i	$0.620230\pi$
$822$	−1.70303e6	−0.0879109
$823$	1.91145e7	0.983702	0.491851	−	0.870679i	$-0.336320\pi$
$823$	1.91145e7	0.983702	0.491851	+	0.870679i	$0.336320\pi$
$824$	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	$0.260547\pi$
$828$	−8.89398e6	−0.450838
$829$	2.27066e7	1.14754	0.573768	−	0.819018i	$-0.305481\pi$
$829$	2.27066e7	1.14754	0.573768	+	0.819018i	$0.305481\pi$
$830$	0	0
$831$	−9.91177e6	−0.497908
$832$	8.94801e6	0.448145
$833$	7.15549e6	0.357295
$834$	4.77389e6	0.237661
$835$	0	0
$836$	9.35943e6	0.463163
$837$	6.13487e6	0.302686
$838$	−1.35848e6	−0.0668257
$839$	−1.54859e7	−0.759505	−0.379753	−	0.925088i	$-0.623991\pi$
$839$	−1.54859e7	−0.759505	−0.379753	+	0.925088i	$0.623991\pi$
$840$	0	0
$841$	−2.28136e6	−0.111225
$842$	7.06383e6	0.343368
$843$	−1.09136e7	−0.528931
$844$	1.02254e7	0.494110
$845$	0	0
$846$	1.29529e6	0.0622218
$847$	−607635.	−0.0291028
$848$	−1.73100e7	−0.826623
$849$	1.86974e7	0.890250
$850$	0	0
$851$	−5.11844e7	−2.42278
$852$	−5.51116e6	−0.260102
$853$	3.77200e6	0.177500	0.0887502	−	0.996054i	$-0.471713\pi$
$853$	3.77200e6	0.177500	0.0887502	+	0.996054i	$0.471713\pi$
$854$	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	$0.451684\pi$
$858$	684980.	0.0317658
$859$	−1.10411e7	−0.510538	−0.255269	−	0.966870i	$-0.582164\pi$
$859$	−1.10411e7	−0.510538	−0.255269	+	0.966870i	$0.582164\pi$
$860$	0	0
$861$	661594.	0.0304147
$862$	−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	$0.488653\pi$
$864$	−2.96307e6	−0.135039
$865$	0	0
$866$	4.78174e6	0.216666
$867$	1.07536e7	0.485853
$868$	−1.04447e7	−0.470539
$869$	−1.15315e7	−0.518009
$870$	0	0
$871$	1.25562e7	0.560808
$872$	9.99348e6	0.445067
$873$	1.04457e7	0.463878
$874$	−1.37456e7	−0.608676
$875$	0	0
$876$	1.16629e7	0.513509
$877$	−2.78476e7	−1.22261	−0.611307	−	0.791393i	$-0.709356\pi$
$877$	−2.78476e7	−1.22261	−0.611307	+	0.791393i	$0.709356\pi$
$878$	8.57700e6	0.375491
$879$	1.57635e7	0.688144
$880$	0	0
$881$	−1.52851e7	−0.663481	−0.331740	−	0.943371i	$-0.607636\pi$
$881$	−1.52851e7	−0.663481	−0.331740	+	0.943371i	$0.607636\pi$
$882$	−1.76847e6	−0.0765467
$883$	7.82254e6	0.337634	0.168817	−	0.985647i	$-0.446005\pi$
$883$	7.82254e6	0.337634	0.168817	+	0.985647i	$0.446005\pi$
$884$	−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	$0.670865\pi$
$888$	−1.12414e7	−0.478397
$889$	3.30214e6	0.140133
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.04544e7	−1.28156
$893$	−2.85774e7	−1.19920
$894$	−6.20209e6	−0.259534
$895$	0	0
$896$	6.63487e6	0.276098
$897$	1.43608e7	0.595933
$898$	−4.85078e6	−0.200734
$899$	3.59309e7	1.48276
$900$	0	0
$901$	9.92548e6	0.407324
$902$	310201.	0.0126948
$903$	4.28849e6	0.175019
$904$	−5.07438e6	−0.206520
$905$	0	0
$906$	−3.01735e6	−0.122125
$907$	5.48830e6	0.221523	0.110762	−	0.993847i	$-0.464671\pi$
$907$	5.48830e6	0.221523	0.110762	+	0.993847i	$0.464671\pi$
$908$	−1.08378e7	−0.436243
$909$	1.19142e7	0.478249
$910$	0	0
$911$	−918823.	−0.0366806	−0.0183403	−	0.999832i	$-0.505838\pi$
$911$	−918823.	−0.0366806	−0.0183403	+	0.999832i	$0.505838\pi$
$912$	1.92581e7	0.766702
$913$	2.01861e6	0.0801450
$914$	−5.85282e6	−0.231739
$915$	0	0
$916$	4.13685e7	1.62904
$917$	−3.04482e6	−0.119574
$918$	500512.	0.0196023
$919$	−6.92636e6	−0.270530	−0.135265	−	0.990809i	$-0.543189\pi$
$919$	−6.92636e6	−0.270530	−0.135265	+	0.990809i	$0.543189\pi$
$920$	0	0
$921$	2.48879e7	0.966805
$922$	−1.20365e7	−0.466306
$923$	8.89866e6	0.343812
$924$	−1.35159e6	−0.0520793
$925$	0	0
$926$	8.02432e6	0.307525
$927$	−7.20001e6	−0.275191
$928$	−1.73542e7	−0.661509
$929$	−2.62196e7	−0.996752	−0.498376	−	0.866961i	$-0.666070\pi$
$929$	−2.62196e7	−0.996752	−0.498376	+	0.866961i	$0.666070\pi$
$930$	0	0
$931$	3.90168e7	1.47529
$932$	3.56126e7	1.34296
$933$	−1.24641e6	−0.0468765
$934$	8.02527e6	0.301018
$935$	0	0
$936$	3.15400e6	0.117672
$937$	216082.	0.00804027	0.00402013	−	0.999992i	$-0.498720\pi$
$937$	216082.	0.00804027	0.00402013	+	0.999992i	$0.498720\pi$
$938$	1.73557e6	0.0644071
$939$	2.02530e7	0.749592
$940$	0	0
$941$	2.94074e7	1.08264	0.541318	−	0.840818i	$-0.317926\pi$
$941$	2.94074e7	1.08264	0.541318	+	0.840818i	$0.317926\pi$
$942$	−1.03278e6	−0.0379211
$943$	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	$0.226212\pi$
$948$	−2.56501e7	−0.926976
$949$	−1.88317e7	−0.678774
$950$	0	0
$951$	−2.70033e6	−0.0968200
$952$	−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	$0.421031\pi$
$954$	−2.45307e6	−0.0872648
$955$	0	0
$956$	−1.90910e7	−0.675590
$957$	4.64963e6	0.164111
$958$	4.24182e6	0.149327
$959$	−5.42590e6	−0.190513
$960$	0	0
$961$	4.21908e7	1.47370
$962$	8.76844e6	0.305481
$963$	−1.18545e7	−0.411924
$964$	−9.01009e6	−0.312275
$965$	0	0
$966$	1.98500e6	0.0684412
$967$	4.56284e7	1.56917	0.784584	−	0.620022i	$-0.212877\pi$
$967$	4.56284e7	1.56917	0.784584	+	0.620022i	$0.212877\pi$
$968$	−1.31183e6	−0.0449976
$969$	−1.10425e7	−0.377797
$970$	0	0
$971$	9.15569e6	0.311633	0.155816	−	0.987786i	$-0.450199\pi$
$971$	9.15569e6	0.311633	0.155816	+	0.987786i	$0.450199\pi$
$972$	1.76587e6	0.0599504
$973$	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	$0.439753\pi$
$978$	−3.95415e6	−0.132192
$979$	1.73356e7	0.578071
$980$	0	0
$981$	−9.03431e6	−0.299725
$982$	856967.	0.0283586
$983$	1.92294e7	0.634718	0.317359	−	0.948305i	$-0.397204\pi$
$983$	1.92294e7	0.634718	0.317359	+	0.948305i	$0.397204\pi$
$984$	1.42832e6	0.0470261
$985$	0	0
$986$	2.93142e6	0.0960252
$987$	4.12684e6	0.134842
$988$	−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
$993$	−5.74092e6	−0.184760
$994$	1.23000e6	0.0394858
$995$	0	0
$996$	4.49010e6	0.143419
$997$	−2.88621e7	−0.919583	−0.459791	−	0.888027i	$-0.652076\pi$
$997$	−2.88621e7	−0.919583	−0.459791	+	0.888027i	$0.652076\pi$
$998$	−9.01935e6	−0.286648
$999$	1.01625e7	0.322171

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.b.1.2	✓	3	5.4	even	2
495.6.a.d.1.2		3	15.14	odd	2
825.6.a.i.1.2		3	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44737 q^{2} -9.00000 q^{3} -29.9051 q^{4} -13.0263 q^{6} -41.5023 q^{7} -89.5997 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+1.44737 q^{2} -9.00000 q^{3} -29.9051 q^{4} -13.0263 q^{6} -41.5023 q^{7} -89.5997 q^{8} +81.0000 q^{9} +121.000 q^{11} +269.146 q^{12} -434.580 q^{13} -60.0692 q^{14} +827.280 q^{16} -474.359 q^{17} +117.237 q^{18} -2586.54 q^{19} +373.520 q^{21} +175.132 q^{22} +3671.69 q^{23} +806.397 q^{24} -628.999 q^{26} -729.000 q^{27} +1241.13 q^{28} -4269.64 q^{29} -8415.46 q^{31} +4064.57 q^{32} -1089.00 q^{33} -686.573 q^{34} -2422.31 q^{36} -13940.3 q^{37} -3743.68 q^{38} +3911.22 q^{39} +1771.24 q^{41} +540.623 q^{42} +11481.3 q^{43} -3618.52 q^{44} +5314.30 q^{46} +11048.5 q^{47} -7445.52 q^{48} -15084.6 q^{49} +4269.23 q^{51} +12996.2 q^{52} -20924.0 q^{53} -1055.13 q^{54} +3718.59 q^{56} +23278.8 q^{57} -6179.75 q^{58} -33001.2 q^{59} -32348.2 q^{61} -12180.3 q^{62} -3361.68 q^{63} -20590.0 q^{64} -1576.19 q^{66} -28892.8 q^{67} +14185.7 q^{68} -33045.2 q^{69} -20476.5 q^{71} -7257.58 q^{72} +43333.2 q^{73} -20176.8 q^{74} +77350.7 q^{76} -5021.77 q^{77} +5660.99 q^{78} -95301.8 q^{79} +6561.00 q^{81} +2563.64 q^{82} +16682.8 q^{83} -11170.2 q^{84} +16617.7 q^{86} +38426.7 q^{87} -10841.6 q^{88} +143269. q^{89} +18036.1 q^{91} -109802. q^{92} +75739.1 q^{93} +15991.3 q^{94} -36581.1 q^{96} +128960. q^{97} -21833.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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