LMFDB - Embedded newform 825.6.a.g.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 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			$0.305203$ 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)

$f(q)$	$=$	$q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +121.000 q^{11} -283.655 q^{12} +674.655 q^{13} +57.7851 q^{14} +977.890 q^{16} -1927.66 q^{17} -56.2785 q^{18} -149.751 q^{19} -748.514 q^{21} -84.0704 q^{22} +1355.59 q^{23} +397.184 q^{24} -468.748 q^{26} +729.000 q^{27} +2621.24 q^{28} -7320.99 q^{29} -4215.76 q^{31} -2091.65 q^{32} +1089.00 q^{33} +1339.33 q^{34} -2552.90 q^{36} +13420.1 q^{37} +104.046 q^{38} +6071.90 q^{39} +2865.39 q^{41} +520.065 q^{42} +22078.1 q^{43} -3813.59 q^{44} -941.861 q^{46} +14556.4 q^{47} +8801.01 q^{48} -9890.04 q^{49} -17348.9 q^{51} -21263.3 q^{52} -13349.7 q^{53} -506.507 q^{54} -3670.35 q^{56} -1347.76 q^{57} +5086.60 q^{58} +45803.3 q^{59} +18996.5 q^{61} +2929.10 q^{62} -6736.63 q^{63} -29839.2 q^{64} -756.634 q^{66} -6651.05 q^{67} +60754.5 q^{68} +12200.3 q^{69} -61028.7 q^{71} +3574.66 q^{72} +17353.4 q^{73} -9324.27 q^{74} +4719.73 q^{76} -10063.4 q^{77} -4218.74 q^{78} +61676.5 q^{79} +6561.00 q^{81} -1990.87 q^{82} +65230.8 q^{83} +23591.1 q^{84} -15339.8 q^{86} -65888.9 q^{87} +5339.92 q^{88} -109563. q^{89} -56109.9 q^{91} -42724.5 q^{92} -37941.8 q^{93} -10113.8 q^{94} -18824.8 q^{96} -83736.1 q^{97} +6871.57 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22} + 640 q^{23} + 216 q^{24} + 2132 q^{26} + 2187 q^{27} + 580 q^{28} - 4538 q^{29} - 14968 q^{31} + 2496 q^{32} + 3267 q^{33} - 13704 q^{34} - 3402 q^{36} + 6190 q^{37} + 11668 q^{38} - 2610 q^{39} - 8926 q^{41} + 8244 q^{42} + 33592 q^{43} - 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 5310 q^{48} - 14693 q^{49} - 3906 q^{51} - 18780 q^{52} + 22934 q^{53} - 1458 q^{54} - 40012 q^{56} - 25704 q^{57} + 32304 q^{58} - 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 5508 q^{63} + 35474 q^{64} - 2178 q^{66} - 16868 q^{67} + 71288 q^{68} + 5760 q^{69} + 4856 q^{71} + 1944 q^{72} - 1910 q^{73} + 29404 q^{74} + 6116 q^{76} + 8228 q^{77} + 19188 q^{78} - 36844 q^{79} + 19683 q^{81} - 84000 q^{82} + 48796 q^{83} + 5220 q^{84} - 83492 q^{86} - 40842 q^{87} + 2904 q^{88} - 188978 q^{89} - 93208 q^{91} + 6976 q^{92} - 134712 q^{93} + 70472 q^{94} + 22464 q^{96} - 247526 q^{97} + 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 + 363 * q^11 - 378 * q^12 - 290 * q^13 + 916 * q^14 - 590 * q^16 - 434 * q^17 - 162 * q^18 - 2856 * q^19 + 612 * q^21 - 242 * q^22 + 640 * q^23 + 216 * q^24 + 2132 * q^26 + 2187 * q^27 + 580 * q^28 - 4538 * q^29 - 14968 * q^31 + 2496 * q^32 + 3267 * q^33 - 13704 * q^34 - 3402 * q^36 + 6190 * q^37 + 11668 * q^38 - 2610 * q^39 - 8926 * q^41 + 8244 * q^42 + 33592 * q^43 - 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 5310 * q^48 - 14693 * q^49 - 3906 * q^51 - 18780 * q^52 + 22934 * q^53 - 1458 * q^54 - 40012 * q^56 - 25704 * q^57 + 32304 * q^58 - 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 5508 * q^63 + 35474 * q^64 - 2178 * q^66 - 16868 * q^67 + 71288 * q^68 + 5760 * q^69 + 4856 * q^71 + 1944 * q^72 - 1910 * q^73 + 29404 * q^74 + 6116 * q^76 + 8228 * q^77 + 19188 * q^78 - 36844 * q^79 + 19683 * q^81 - 84000 * q^82 + 48796 * q^83 + 5220 * q^84 - 83492 * q^86 - 40842 * q^87 + 2904 * q^88 - 188978 * q^89 - 93208 * q^91 + 6976 * q^92 - 134712 * q^93 + 70472 * q^94 + 22464 * q^96 - 247526 * q^97 + 154654 * 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 $

$2$	−0.694797	−0.122824	−0.0614120	−	0.998113i	$-0.519560\pi$
$2$	−0.694797	−0.122824	−0.0614120	+	0.998113i	$0.519560\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.5173	−0.984914
$5$	0	0
$5$	0	0
$6$	−6.25317	−0.0709124
$7$	−83.1683	−0.641523	−0.320762	−	0.947160i	$-0.603939\pi$
$7$	−83.1683	−0.641523	−0.320762	+	0.947160i	$0.603939\pi$
$8$	44.1316	0.243795
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−283.655	−0.568641
$13$	674.655	1.10719	0.553597	−	0.832785i	$-0.313255\pi$
$13$	674.655	1.10719	0.553597	+	0.832785i	$0.313255\pi$
$14$	57.7851	0.0787944
$15$	0	0
$16$	977.890	0.954970
$17$	−1927.66	−1.61774	−0.808868	−	0.587991i	$-0.799919\pi$
$17$	−1927.66	−1.61774	−0.808868	+	0.587991i	$0.799919\pi$
$18$	−56.2785	−0.0409413
$19$	−149.751	−0.0951666	−0.0475833	−	0.998867i	$-0.515152\pi$
$19$	−149.751	−0.0951666	−0.0475833	+	0.998867i	$0.515152\pi$
$20$	0	0
$21$	−748.514	−0.370384
$22$	−84.0704	−0.0370328
$23$	1355.59	0.534330	0.267165	−	0.963651i	$-0.413913\pi$
$23$	1355.59	0.534330	0.267165	+	0.963651i	$0.413913\pi$
$24$	397.184	0.140755
$25$	0	0
$26$	−468.748	−0.135990
$27$	729.000	0.192450
$28$	2621.24	0.631846
$29$	−7320.99	−1.61650	−0.808248	−	0.588842i	$-0.799584\pi$
$29$	−7320.99	−1.61650	−0.808248	+	0.588842i	$0.799584\pi$
$30$	0	0
$31$	−4215.76	−0.787901	−0.393951	−	0.919132i	$-0.628892\pi$
$31$	−4215.76	−0.787901	−0.393951	+	0.919132i	$0.628892\pi$
$32$	−2091.65	−0.361088
$33$	1089.00	0.174078
$34$	1339.33	0.198697
$35$	0	0
$36$	−2552.90	−0.328305
$37$	13420.1	1.61158	0.805792	−	0.592199i	$-0.201740\pi$
$37$	13420.1	1.61158	0.805792	+	0.592199i	$0.201740\pi$
$38$	104.046	0.0116887
$39$	6071.90	0.639238
$40$	0	0
$41$	2865.39	0.266210	0.133105	−	0.991102i	$-0.457505\pi$
$41$	2865.39	0.266210	0.133105	+	0.991102i	$0.457505\pi$
$42$	520.065	0.0454920
$43$	22078.1	1.82092	0.910458	−	0.413601i	$-0.135729\pi$
$43$	22078.1	1.82092	0.910458	+	0.413601i	$0.135729\pi$
$44$	−3813.59	−0.296963
$45$	0	0
$46$	−941.861	−0.0656285
$47$	14556.4	0.961192	0.480596	−	0.876942i	$-0.340420\pi$
$47$	14556.4	0.961192	0.480596	+	0.876942i	$0.340420\pi$
$48$	8801.01	0.551352
$49$	−9890.04	−0.588448
$50$	0	0
$51$	−17348.9	−0.934000
$52$	−21263.3	−1.09049
$53$	−13349.7	−0.652805	−0.326402	−	0.945231i	$-0.605836\pi$
$53$	−13349.7	−0.652805	−0.326402	+	0.945231i	$0.605836\pi$
$54$	−506.507	−0.0236375
$55$	0	0
$56$	−3670.35	−0.156400
$57$	−1347.76	−0.0549445
$58$	5086.60	0.198544
$59$	45803.3	1.71304	0.856518	−	0.516117i	$-0.172623\pi$
$59$	45803.3	1.71304	0.856518	+	0.516117i	$0.172623\pi$
$60$	0	0
$61$	18996.5	0.653655	0.326827	−	0.945084i	$-0.394020\pi$
$61$	18996.5	0.653655	0.326827	+	0.945084i	$0.394020\pi$
$62$	2929.10	0.0967731
$63$	−6736.63	−0.213841
$64$	−29839.2	−0.910620
$65$	0	0
$66$	−756.634	−0.0213809
$67$	−6651.05	−0.181010	−0.0905052	−	0.995896i	$-0.528848\pi$
$67$	−6651.05	−0.181010	−0.0905052	+	0.995896i	$0.528848\pi$
$68$	60754.5	1.59333
$69$	12200.3	0.308495
$70$	0	0
$71$	−61028.7	−1.43677	−0.718386	−	0.695644i	$-0.755119\pi$
$71$	−61028.7	−1.43677	−0.718386	+	0.695644i	$0.755119\pi$
$72$	3574.66	0.0812650
$73$	17353.4	0.381134	0.190567	−	0.981674i	$-0.438967\pi$
$73$	17353.4	0.381134	0.190567	+	0.981674i	$0.438967\pi$
$74$	−9324.27	−0.197941
$75$	0	0
$76$	4719.73	0.0937309
$77$	−10063.4	−0.193427
$78$	−4218.74	−0.0785138
$79$	61676.5	1.11187	0.555933	−	0.831227i	$-0.312361\pi$
$79$	61676.5	1.11187	0.555933	+	0.831227i	$0.312361\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−1990.87	−0.0326969
$83$	65230.8	1.03934	0.519670	−	0.854367i	$-0.326055\pi$
$83$	65230.8	1.03934	0.519670	+	0.854367i	$0.326055\pi$
$84$	23591.1	0.364796
$85$	0	0
$86$	−15339.8	−0.223652
$87$	−65888.9	−0.933284
$88$	5339.92	0.0735069
$89$	−109563.	−1.46618	−0.733091	−	0.680131i	$-0.761923\pi$
$89$	−109563.	−1.46618	−0.733091	+	0.680131i	$0.761923\pi$
$90$	0	0
$91$	−56109.9	−0.710291
$92$	−42724.5	−0.526269
$93$	−37941.8	−0.454895
$94$	−10113.8	−0.118057
$95$	0	0
$96$	−18824.8	−0.208474
$97$	−83736.1	−0.903615	−0.451808	−	0.892115i	$-0.649221\pi$
$97$	−83736.1	−0.903615	−0.451808	+	0.892115i	$0.649221\pi$
$98$	6871.57	0.0722754
$99$	9801.00	0.100504
$100$	0	0
$101$	−129877.	−1.26686	−0.633428	−	0.773801i	$-0.718353\pi$
$101$	−129877.	−1.26686	−0.633428	+	0.773801i	$0.718353\pi$
$102$	12054.0	0.114718
$103$	−201851.	−1.87472	−0.937362	−	0.348358i	$-0.886739\pi$
$103$	−201851.	−1.87472	−0.937362	+	0.348358i	$0.886739\pi$
$104$	29773.6	0.269928
$105$	0	0
$106$	9275.36	0.0801800
$107$	132285.	1.11699	0.558496	−	0.829507i	$-0.311378\pi$
$107$	132285.	1.11699	0.558496	+	0.829507i	$0.311378\pi$
$108$	−22976.1	−0.189547
$109$	−144044.	−1.16126	−0.580629	−	0.814168i	$-0.697193\pi$
$109$	−144044.	−1.16126	−0.580629	+	0.814168i	$0.697193\pi$
$110$	0	0
$111$	120781.	0.930448
$112$	−81329.4	−0.612636
$113$	1095.62	0.00807165	0.00403582	−	0.999992i	$-0.498715\pi$
$113$	1095.62	0.00807165	0.00403582	+	0.999992i	$0.498715\pi$
$114$	936.416	0.00674849
$115$	0	0
$116$	230737.	1.59211
$117$	54647.1	0.369065
$118$	−31824.0	−0.210402
$119$	160320.	1.03782
$120$	0	0
$121$	14641.0	0.0909091
$122$	−13198.7	−0.0802844
$123$	25788.5	0.153696
$124$	132869.	0.776015
$125$	0	0
$126$	4680.59	0.0262648
$127$	−315758.	−1.73718	−0.868591	−	0.495529i	$-0.834974\pi$
$127$	−315758.	−1.73718	−0.868591	+	0.495529i	$0.834974\pi$
$128$	87664.9	0.472934
$129$	198703.	1.05131
$130$	0	0
$131$	−187001.	−0.952064	−0.476032	−	0.879428i	$-0.657925\pi$
$131$	−187001.	−0.952064	−0.476032	+	0.879428i	$0.657925\pi$
$132$	−34322.3	−0.171452
$133$	12454.5	0.0610516
$134$	4621.13	0.0222324
$135$	0	0
$136$	−85070.6	−0.394396
$137$	−232301.	−1.05743	−0.528713	−	0.848801i	$-0.677325\pi$
$137$	−232301.	−1.05743	−0.528713	+	0.848801i	$0.677325\pi$
$138$	−8476.75	−0.0378906
$139$	−311665.	−1.36820	−0.684102	−	0.729386i	$-0.739806\pi$
$139$	−311665.	−1.36820	−0.684102	+	0.729386i	$0.739806\pi$
$140$	0	0
$141$	131008.	0.554944
$142$	42402.5	0.176470
$143$	81633.3	0.333831
$144$	79209.1	0.318323
$145$	0	0
$146$	−12057.1	−0.0468124
$147$	−89010.4	−0.339740
$148$	−422966.	−1.58727
$149$	30354.5	0.112010	0.0560050	−	0.998430i	$-0.482164\pi$
$149$	30354.5	0.112010	0.0560050	+	0.998430i	$0.482164\pi$
$150$	0	0
$151$	131015.	0.467604	0.233802	−	0.972284i	$-0.424883\pi$
$151$	131015.	0.467604	0.233802	+	0.972284i	$0.424883\pi$
$152$	−6608.73	−0.0232011
$153$	−156140.	−0.539245
$154$	6991.99	0.0237574
$155$	0	0
$156$	−191370.	−0.629595
$157$	−326922.	−1.05851	−0.529255	−	0.848463i	$-0.677528\pi$
$157$	−326922.	−1.05851	−0.529255	+	0.848463i	$0.677528\pi$
$158$	−42852.7	−0.136564
$159$	−120148.	−0.376897
$160$	0	0
$161$	−112742.	−0.342785
$162$	−4558.56	−0.0136471
$163$	501854.	1.47948	0.739739	−	0.672894i	$-0.234949\pi$
$163$	501854.	1.47948	0.739739	+	0.672894i	$0.234949\pi$
$164$	−90309.3	−0.262194
$165$	0	0
$166$	−45322.2	−0.127656
$167$	590462.	1.63833	0.819164	−	0.573559i	$-0.194438\pi$
$167$	590462.	1.63833	0.819164	+	0.573559i	$0.194438\pi$
$168$	−33033.1	−0.0902977
$169$	83866.7	0.225877
$170$	0	0
$171$	−12129.8	−0.0317222
$172$	−695840.	−1.79345
$173$	−570934.	−1.45034	−0.725172	−	0.688568i	$-0.758240\pi$
$173$	−570934.	−1.45034	−0.725172	+	0.688568i	$0.758240\pi$
$174$	45779.4	0.114630
$175$	0	0
$176$	118325.	0.287934
$177$	412230.	0.989022
$178$	76123.9	0.180082
$179$	−464402.	−1.08333	−0.541666	−	0.840594i	$-0.682206\pi$
$179$	−464402.	−1.08333	−0.541666	+	0.840594i	$0.682206\pi$
$180$	0	0
$181$	−409792.	−0.929751	−0.464876	−	0.885376i	$-0.653901\pi$
$181$	−409792.	−0.929751	−0.464876	+	0.885376i	$0.653901\pi$
$182$	38985.0	0.0872407
$183$	170968.	0.377388
$184$	59824.4	0.130267
$185$	0	0
$186$	26361.9	0.0558720
$187$	−233247.	−0.487766
$188$	−458778.	−0.946691
$189$	−60629.7	−0.123461
$190$	0	0
$191$	−591029.	−1.17226	−0.586132	−	0.810216i	$-0.699350\pi$
$191$	−591029.	−1.17226	−0.586132	+	0.810216i	$0.699350\pi$
$192$	−268553.	−0.525747
$193$	310907.	0.600811	0.300405	−	0.953812i	$-0.402878\pi$
$193$	310907.	0.600811	0.300405	+	0.953812i	$0.402878\pi$
$194$	58179.6	0.110986
$195$	0	0
$196$	311707.	0.579571
$197$	−161613.	−0.296695	−0.148348	−	0.988935i	$-0.547395\pi$
$197$	−161613.	−0.296695	−0.148348	+	0.988935i	$0.547395\pi$
$198$	−6809.70	−0.0123443
$199$	435622.	0.779790	0.389895	−	0.920859i	$-0.372511\pi$
$199$	435622.	0.779790	0.389895	+	0.920859i	$0.372511\pi$
$200$	0	0
$201$	−59859.5	−0.104506
$202$	90237.8	0.155600
$203$	608874.	1.03702
$204$	546790.	0.919910
$205$	0	0
$206$	140245.	0.230261
$207$	109803.	0.178110
$208$	659738.	1.05734
$209$	−18119.8	−0.0286938
$210$	0	0
$211$	−539434.	−0.834127	−0.417063	−	0.908877i	$-0.636941\pi$
$211$	−539434.	−0.834127	−0.417063	+	0.908877i	$0.636941\pi$
$212$	420747.	0.642957
$213$	−549258.	−0.829521
$214$	−91911.1	−0.137193
$215$	0	0
$216$	32171.9	0.0469184
$217$	350618.	0.505457
$218$	100081.	0.142630
$219$	156181.	0.220048
$220$	0	0
$221$	−1.30050e6	−1.79115
$222$	−83918.5	−0.114281
$223$	−452834.	−0.609786	−0.304893	−	0.952387i	$-0.598621\pi$
$223$	−452834.	−0.609786	−0.304893	+	0.952387i	$0.598621\pi$
$224$	173959.	0.231646
$225$	0	0
$226$	−761.230	−0.000991391 0
$227$	40608.6	0.0523062	0.0261531	−	0.999658i	$-0.491674\pi$
$227$	40608.6	0.0523062	0.0261531	+	0.999658i	$0.491674\pi$
$228$	42477.5	0.0541156
$229$	988410.	1.24551	0.622757	−	0.782415i	$-0.286013\pi$
$229$	988410.	1.24551	0.622757	+	0.782415i	$0.286013\pi$
$230$	0	0
$231$	−90570.2	−0.111675
$232$	−323087.	−0.394093
$233$	47673.4	0.0575289	0.0287645	−	0.999586i	$-0.490843\pi$
$233$	47673.4	0.0575289	0.0287645	+	0.999586i	$0.490843\pi$
$234$	−37968.6	−0.0453299
$235$	0	0
$236$	−1.44359e6	−1.68719
$237$	555089.	0.641936
$238$	−111390.	−0.127469
$239$	−165174.	−0.187045	−0.0935224	−	0.995617i	$-0.529813\pi$
$239$	−165174.	−0.187045	−0.0935224	+	0.995617i	$0.529813\pi$
$240$	0	0
$241$	−1.13302e6	−1.25659	−0.628294	−	0.777976i	$-0.716247\pi$
$241$	−1.13302e6	−1.25659	−0.628294	+	0.777976i	$0.716247\pi$
$242$	−10172.5	−0.0111658
$243$	59049.0	0.0641500
$244$	−598717.	−0.643794
$245$	0	0
$246$	−17917.8	−0.0188776
$247$	−101030.	−0.105368
$248$	−186048.	−0.192086
$249$	587078.	0.600063
$250$	0	0
$251$	−1.08030e6	−1.08233	−0.541165	−	0.840917i	$-0.682016\pi$
$251$	−1.08030e6	−1.08233	−0.541165	+	0.840917i	$0.682016\pi$
$252$	212320.	0.210615
$253$	164027.	0.161106
$254$	219388.	0.213368
$255$	0	0
$256$	893945.	0.852533
$257$	−301460.	−0.284707	−0.142353	−	0.989816i	$-0.545467\pi$
$257$	−301460.	−0.284707	−0.142353	+	0.989816i	$0.545467\pi$
$258$	−138058.	−0.129126
$259$	−1.11613e6	−1.03387
$260$	0	0
$261$	−593000.	−0.538832
$262$	129928.	0.116936
$263$	347755.	0.310016	0.155008	−	0.987913i	$-0.450460\pi$
$263$	347755.	0.310016	0.155008	+	0.987913i	$0.450460\pi$
$264$	48059.3	0.0424392
$265$	0	0
$266$	−8653.34	−0.00749860
$267$	−986065.	−0.846501
$268$	209623.	0.178280
$269$	980953.	0.826547	0.413274	−	0.910607i	$-0.364385\pi$
$269$	980953.	0.826547	0.413274	+	0.910607i	$0.364385\pi$
$270$	0	0
$271$	9630.73	0.00796592	0.00398296	−	0.999992i	$-0.498732\pi$
$271$	9630.73	0.00796592	0.00398296	+	0.999992i	$0.498732\pi$
$272$	−1.88504e6	−1.54489
$273$	−504989.	−0.410086
$274$	161402.	0.129877
$275$	0	0
$276$	−384521.	−0.303842
$277$	−324368.	−0.254003	−0.127001	−	0.991903i	$-0.540535\pi$
$277$	−324368.	−0.254003	−0.127001	+	0.991903i	$0.540535\pi$
$278$	216544.	0.168048
$279$	−341477.	−0.262634
$280$	0	0
$281$	866856.	0.654909	0.327455	−	0.944867i	$-0.393809\pi$
$281$	866856.	0.654909	0.327455	+	0.944867i	$0.393809\pi$
$282$	−91023.8	−0.0681604
$283$	575157.	0.426895	0.213447	−	0.976955i	$-0.431531\pi$
$283$	575157.	0.426895	0.213447	+	0.976955i	$0.431531\pi$
$284$	1.92346e6	1.41510
$285$	0	0
$286$	−56718.6	−0.0410025
$287$	−238310.	−0.170780
$288$	−169423.	−0.120363
$289$	2.29601e6	1.61707
$290$	0	0
$291$	−753625.	−0.521702
$292$	−546932.	−0.375384
$293$	−1.96073e6	−1.33429	−0.667144	−	0.744929i	$-0.732483\pi$
$293$	−1.96073e6	−1.33429	−0.667144	+	0.744929i	$0.732483\pi$
$294$	61844.1	0.0417282
$295$	0	0
$296$	592252.	0.392896
$297$	88209.0	0.0580259
$298$	−21090.2	−0.0137575
$299$	914557.	0.591606
$300$	0	0
$301$	−1.83620e6	−1.16816
$302$	−91028.8	−0.0574330
$303$	−1.16889e6	−0.731420
$304$	−146440.	−0.0908813
$305$	0	0
$306$	108486.	0.0662322
$307$	−1.60549e6	−0.972215	−0.486108	−	0.873899i	$-0.661584\pi$
$307$	−1.60549e6	−0.972215	−0.486108	+	0.873899i	$0.661584\pi$
$308$	317170.	0.190509
$309$	−1.81666e6	−1.08237
$310$	0	0
$311$	6622.52	0.00388260	0.00194130	−	0.999998i	$-0.499382\pi$
$311$	6622.52	0.00388260	0.00194130	+	0.999998i	$0.499382\pi$
$312$	267963.	0.155843
$313$	1.08764e6	0.627516	0.313758	−	0.949503i	$-0.398412\pi$
$313$	1.08764e6	0.627516	0.313758	+	0.949503i	$0.398412\pi$
$314$	227144.	0.130010
$315$	0	0
$316$	−1.94387e6	−1.09509
$317$	456183.	0.254971	0.127486	−	0.991840i	$-0.459309\pi$
$317$	456183.	0.254971	0.127486	+	0.991840i	$0.459309\pi$
$318$	83478.2	0.0462920
$319$	−885839.	−0.487392
$320$	0	0
$321$	1.19056e6	0.644896
$322$	78332.9	0.0421022
$323$	288668.	0.153954
$324$	−206785.	−0.109435
$325$	0	0
$326$	−348687.	−0.181715
$327$	−1.29640e6	−0.670452
$328$	126454.	0.0649006
$329$	−1.21063e6	−0.616627
$330$	0	0
$331$	1.73077e6	0.868297	0.434149	−	0.900841i	$-0.357049\pi$
$331$	1.73077e6	0.868297	0.434149	+	0.900841i	$0.357049\pi$
$332$	−2.05590e6	−1.02366
$333$	1.08703e6	0.537194
$334$	−410251.	−0.201226
$335$	0	0
$336$	−731965.	−0.353706
$337$	−3.01707e6	−1.44714	−0.723571	−	0.690250i	$-0.757501\pi$
$337$	−3.01707e6	−1.44714	−0.723571	+	0.690250i	$0.757501\pi$
$338$	−58270.3	−0.0277431
$339$	9860.54	0.00466017
$340$	0	0
$341$	−510107.	−0.237561
$342$	8427.75	0.00389624
$343$	2.22035e6	1.01903
$344$	974341.	0.443930
$345$	0	0
$346$	396683.	0.178137
$347$	−1.26947e6	−0.565978	−0.282989	−	0.959123i	$-0.591326\pi$
$347$	−1.26947e6	−0.565978	−0.282989	+	0.959123i	$0.591326\pi$
$348$	2.07664e6	0.919205
$349$	3.26420e6	1.43454	0.717270	−	0.696795i	$-0.245391\pi$
$349$	3.26420e6	1.43454	0.717270	+	0.696795i	$0.245391\pi$
$350$	0	0
$351$	491824.	0.213079
$352$	−253089.	−0.108872
$353$	−1.70741e6	−0.729290	−0.364645	−	0.931147i	$-0.618810\pi$
$353$	−1.70741e6	−0.729290	−0.364645	+	0.931147i	$0.618810\pi$
$354$	−286416.	−0.121476
$355$	0	0
$356$	3.45312e6	1.44406
$357$	1.44288e6	0.599183
$358$	322665.	0.133059
$359$	257688.	0.105526	0.0527628	−	0.998607i	$-0.483197\pi$
$359$	257688.	0.105526	0.0527628	+	0.998607i	$0.483197\pi$
$360$	0	0
$361$	−2.45367e6	−0.990943
$362$	284722.	0.114196
$363$	131769.	0.0524864
$364$	1.76843e6	0.699575
$365$	0	0
$366$	−118788.	−0.0463522
$367$	683489.	0.264891	0.132445	−	0.991190i	$-0.457717\pi$
$367$	683489.	0.264891	0.132445	+	0.991190i	$0.457717\pi$
$368$	1.32562e6	0.510269
$369$	232097.	0.0887367
$370$	0	0
$371$	1.11027e6	0.418790
$372$	1.19582e6	0.448033
$373$	−3.58176e6	−1.33298	−0.666492	−	0.745512i	$-0.732205\pi$
$373$	−3.58176e6	−1.33298	−0.666492	+	0.745512i	$0.732205\pi$
$374$	162059.	0.0599093
$375$	0	0
$376$	642398.	0.234334
$377$	−4.93914e6	−1.78977
$378$	42125.3	0.0151640
$379$	−4.87955e6	−1.74494	−0.872472	−	0.488665i	$-0.837484\pi$
$379$	−4.87955e6	−1.74494	−0.872472	+	0.488665i	$0.837484\pi$
$380$	0	0
$381$	−2.84182e6	−1.00296
$382$	410645.	0.143982
$383$	−4.97559e6	−1.73319	−0.866597	−	0.499008i	$-0.833698\pi$
$383$	−4.97559e6	−1.73319	−0.866597	+	0.499008i	$0.833698\pi$
$384$	788984.	0.273049
$385$	0	0
$386$	−216017.	−0.0737939
$387$	1.78832e6	0.606972
$388$	2.63913e6	0.889983
$389$	−4.09866e6	−1.37331	−0.686654	−	0.726984i	$-0.740921\pi$
$389$	−4.09866e6	−1.37331	−0.686654	+	0.726984i	$0.740921\pi$
$390$	0	0
$391$	−2.61312e6	−0.864404
$392$	−436463.	−0.143461
$393$	−1.68301e6	−0.549674
$394$	112288.	0.0364413
$395$	0	0
$396$	−308901.	−0.0989876
$397$	−1.61215e6	−0.513369	−0.256684	−	0.966495i	$-0.582630\pi$
$397$	−1.61215e6	−0.513369	−0.256684	+	0.966495i	$0.582630\pi$
$398$	−302669.	−0.0957768
$399$	112090.	0.0352482
$400$	0	0
$401$	−648362.	−0.201352	−0.100676	−	0.994919i	$-0.532101\pi$
$401$	−648362.	−0.201352	−0.100676	+	0.994919i	$0.532101\pi$
$402$	41590.2	0.0128359
$403$	−2.84419e6	−0.872359
$404$	4.09335e6	1.24775
$405$	0	0
$406$	−423044.	−0.127371
$407$	1.62384e6	0.485911
$408$	−765635.	−0.227704
$409$	1.88327e6	0.556677	0.278338	−	0.960483i	$-0.410216\pi$
$409$	1.88327e6	0.556677	0.278338	+	0.960483i	$0.410216\pi$
$410$	0	0
$411$	−2.09071e6	−0.610505
$412$	6.36178e6	1.84644
$413$	−3.80938e6	−1.09895
$414$	−76290.7	−0.0218762
$415$	0	0
$416$	−1.41114e6	−0.399794
$417$	−2.80499e6	−0.789933
$418$	12589.6	0.00352429
$419$	−5.10510e6	−1.42059	−0.710296	−	0.703903i	$-0.751439\pi$
$419$	−5.10510e6	−1.42059	−0.710296	+	0.703903i	$0.751439\pi$
$420$	0	0
$421$	5.48811e6	1.50910	0.754550	−	0.656243i	$-0.227855\pi$
$421$	5.48811e6	1.50910	0.754550	+	0.656243i	$0.227855\pi$
$422$	374797.	0.102451
$423$	1.17907e6	0.320397
$424$	−589145.	−0.159150
$425$	0	0
$426$	381623.	0.101885
$427$	−1.57990e6	−0.419335
$428$	−4.16925e6	−1.10014
$429$	734700.	0.192738
$430$	0	0
$431$	−3.59589e6	−0.932424	−0.466212	−	0.884673i	$-0.654382\pi$
$431$	−3.59589e6	−0.932424	−0.466212	+	0.884673i	$0.654382\pi$
$432$	712882.	0.183784
$433$	5.03603e6	1.29083	0.645414	−	0.763833i	$-0.276685\pi$
$433$	5.03603e6	1.29083	0.645414	+	0.763833i	$0.276685\pi$
$434$	−243608.	−0.0620822
$435$	0	0
$436$	4.53987e6	1.14374
$437$	−203001.	−0.0508503
$438$	−108514.	−0.0270271
$439$	−3.25961e6	−0.807244	−0.403622	−	0.914926i	$-0.632249\pi$
$439$	−3.25961e6	−0.807244	−0.403622	+	0.914926i	$0.632249\pi$
$440$	0	0
$441$	−801093.	−0.196149
$442$	903586.	0.219996
$443$	1.35658e6	0.328424	0.164212	−	0.986425i	$-0.447492\pi$
$443$	1.35658e6	0.328424	0.164212	+	0.986425i	$0.447492\pi$
$444$	−3.80669e6	−0.916412
$445$	0	0
$446$	314628.	0.0748962
$447$	273190.	0.0646690
$448$	2.48167e6	0.584184
$449$	−6.80906e6	−1.59394	−0.796969	−	0.604020i	$-0.793565\pi$
$449$	−6.80906e6	−1.59394	−0.796969	+	0.604020i	$0.793565\pi$
$450$	0	0
$451$	346712.	0.0802653
$452$	−34530.8	−0.00794988
$453$	1.17913e6	0.269972
$454$	−28214.7	−0.00642445
$455$	0	0
$456$	−59478.6	−0.0133952
$457$	3.03977e6	0.680848	0.340424	−	0.940272i	$-0.389429\pi$
$457$	3.03977e6	0.680848	0.340424	+	0.940272i	$0.389429\pi$
$458$	−686744.	−0.152979
$459$	−1.40526e6	−0.311333
$460$	0	0
$461$	3.27858e6	0.718511	0.359256	−	0.933239i	$-0.383031\pi$
$461$	3.27858e6	0.718511	0.359256	+	0.933239i	$0.383031\pi$
$462$	62927.9	0.0137163
$463$	−5.51970e6	−1.19664	−0.598319	−	0.801258i	$-0.704164\pi$
$463$	−5.51970e6	−1.19664	−0.598319	+	0.801258i	$0.704164\pi$
$464$	−7.15912e6	−1.54371
$465$	0	0
$466$	−33123.3	−0.00706593
$467$	4.07534e6	0.864711	0.432356	−	0.901703i	$-0.357683\pi$
$467$	4.07534e6	0.864711	0.432356	+	0.901703i	$0.357683\pi$
$468$	−1.72233e6	−0.363497
$469$	553157.	0.116122
$470$	0	0
$471$	−2.94230e6	−0.611131
$472$	2.02137e6	0.417630
$473$	2.67145e6	0.549027
$474$	−385674.	−0.0788450
$475$	0	0
$476$	−5.05284e6	−1.02216
$477$	−1.08133e6	−0.217602
$478$	114762.	0.0229736
$479$	4.77799e6	0.951495	0.475748	−	0.879582i	$-0.342178\pi$
$479$	4.77799e6	0.951495	0.475748	+	0.879582i	$0.342178\pi$
$480$	0	0
$481$	9.05397e6	1.78433
$482$	787216.	0.154339
$483$	−1.01468e6	−0.197907
$484$	−461444.	−0.0895377
$485$	0	0
$486$	−41027.1	−0.00787916
$487$	−1.76923e6	−0.338035	−0.169017	−	0.985613i	$-0.554059\pi$
$487$	−1.76923e6	−0.338035	−0.169017	+	0.985613i	$0.554059\pi$
$488$	838345.	0.159358
$489$	4.51669e6	0.854177
$490$	0	0
$491$	−1.04139e6	−0.194943	−0.0974716	−	0.995238i	$-0.531076\pi$
$491$	−1.04139e6	−0.194943	−0.0974716	+	0.995238i	$0.531076\pi$
$492$	−812784.	−0.151378
$493$	1.41124e7	2.61506
$494$	70195.3	0.0129417
$495$	0	0
$496$	−4.12255e6	−0.752422
$497$	5.07565e6	0.921723
$498$	−407900.	−0.0737021
$499$	2.03068e6	0.365082	0.182541	−	0.983198i	$-0.441568\pi$
$499$	2.03068e6	0.365082	0.182541	+	0.983198i	$0.441568\pi$
$500$	0	0
$501$	5.31416e6	0.945890
$502$	750588.	0.132936
$503$	5.06008e6	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$503$	5.06008e6	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$504$	−297298.	−0.0521334
$505$	0	0
$506$	−113965.	−0.0197877
$507$	754800.	0.130410
$508$	9.95183e6	1.71098
$509$	5.64302e6	0.965421	0.482711	−	0.875780i	$-0.339652\pi$
$509$	5.64302e6	0.965421	0.482711	+	0.875780i	$0.339652\pi$
$510$	0	0
$511$	−1.44325e6	−0.244506
$512$	−3.42639e6	−0.577645
$513$	−109168.	−0.0183148
$514$	209454.	0.0349688
$515$	0	0
$516$	−6.26256e6	−1.03545
$517$	1.76133e6	0.289810
$518$	775484.	0.126984
$519$	−5.13841e6	−0.837356
$520$	0	0
$521$	5.37673e6	0.867809	0.433905	−	0.900959i	$-0.357136\pi$
$521$	5.37673e6	0.867809	0.433905	+	0.900959i	$0.357136\pi$
$522$	412014.	0.0661814
$523$	−8.42403e6	−1.34668	−0.673342	−	0.739331i	$-0.735142\pi$
$523$	−8.42403e6	−1.34668	−0.673342	+	0.739331i	$0.735142\pi$
$524$	5.89376e6	0.937701
$525$	0	0
$526$	−241619.	−0.0380773
$527$	8.12654e6	1.27462
$528$	1.06492e6	0.166239
$529$	−4.59871e6	−0.714492
$530$	0	0
$531$	3.71007e6	0.571012
$532$	−392532.	−0.0601306
$533$	1.93315e6	0.294746
$534$	685115.	0.103970
$535$	0	0
$536$	−293522.	−0.0441294
$537$	−4.17962e6	−0.625462
$538$	−681563.	−0.101520
$539$	−1.19669e6	−0.177424
$540$	0	0
$541$	8.85130e6	1.30021	0.650106	−	0.759844i	$-0.274725\pi$
$541$	8.85130e6	1.30021	0.650106	+	0.759844i	$0.274725\pi$
$542$	−6691.40	−0.000978405 0
$543$	−3.68813e6	−0.536792
$544$	4.03198e6	0.584145
$545$	0	0
$546$	350865.	0.0503684
$547$	−3.13327e6	−0.447744	−0.223872	−	0.974619i	$-0.571870\pi$
$547$	−3.13327e6	−0.447744	−0.223872	+	0.974619i	$0.571870\pi$
$548$	7.32150e6	1.04147
$549$	1.53871e6	0.217885
$550$	0	0
$551$	1.09632e6	0.153836
$552$	538420.	0.0752096
$553$	−5.12953e6	−0.713288
$554$	225370.	0.0311976
$555$	0	0
$556$	9.82283e6	1.34756
$557$	5.46691e6	0.746627	0.373313	−	0.927705i	$-0.378222\pi$
$557$	5.46691e6	0.746627	0.373313	+	0.927705i	$0.378222\pi$
$558$	237257.	0.0322577
$559$	1.48951e7	2.01611
$560$	0	0
$561$	−2.09922e6	−0.281612
$562$	−602289.	−0.0804385
$563$	−5.19043e6	−0.690132	−0.345066	−	0.938578i	$-0.612143\pi$
$563$	−5.19043e6	−0.690132	−0.345066	+	0.938578i	$0.612143\pi$
$564$	−4.12901e6	−0.546573
$565$	0	0
$566$	−399618.	−0.0524329
$567$	−545667.	−0.0712804
$568$	−2.69329e6	−0.350278
$569$	1.24058e7	1.60636	0.803180	−	0.595737i	$-0.203140\pi$
$569$	1.24058e7	1.60636	0.803180	+	0.595737i	$0.203140\pi$
$570$	0	0
$571$	8.79234e6	1.12853	0.564266	−	0.825593i	$-0.309159\pi$
$571$	8.79234e6	1.12853	0.564266	+	0.825593i	$0.309159\pi$
$572$	−2.57286e6	−0.328795
$573$	−5.31926e6	−0.676807
$574$	165577.	0.0209759
$575$	0	0
$576$	−2.41698e6	−0.303540
$577$	1.14336e7	1.42969	0.714845	−	0.699283i	$-0.246497\pi$
$577$	1.14336e7	1.42969	0.714845	+	0.699283i	$0.246497\pi$
$578$	−1.59526e6	−0.198615
$579$	2.79817e6	0.346878
$580$	0	0
$581$	−5.42514e6	−0.666761
$582$	523616.	0.0640775
$583$	−1.61532e6	−0.196828
$584$	765834.	0.0929185
$585$	0	0
$586$	1.36231e6	0.163882
$587$	2.32244e6	0.278195	0.139097	−	0.990279i	$-0.455580\pi$
$587$	2.32244e6	0.278195	0.139097	+	0.990279i	$0.455580\pi$
$588$	2.80536e6	0.334615
$589$	631313.	0.0749819
$590$	0	0
$591$	−1.45452e6	−0.171297
$592$	1.31234e7	1.53901
$593$	−7.98435e6	−0.932401	−0.466201	−	0.884679i	$-0.654377\pi$
$593$	−7.98435e6	−0.932401	−0.466201	+	0.884679i	$0.654377\pi$
$594$	−61287.3	−0.00712697
$595$	0	0
$596$	−956690.	−0.110320
$597$	3.92060e6	0.450212
$598$	−635431.	−0.0726634
$599$	−139225.	−0.0158544	−0.00792721	−	0.999969i	$-0.502523\pi$
$599$	−139225.	−0.0158544	−0.00792721	+	0.999969i	$0.502523\pi$
$600$	0	0
$601$	1.55602e7	1.75723	0.878616	−	0.477529i	$-0.158467\pi$
$601$	1.55602e7	1.75723	0.878616	+	0.477529i	$0.158467\pi$
$602$	1.27578e6	0.143478
$603$	−538735.	−0.0603368
$604$	−4.12923e6	−0.460550
$605$	0	0
$606$	812141.	0.0898358
$607$	6.00544e6	0.661566	0.330783	−	0.943707i	$-0.392687\pi$
$607$	6.00544e6	0.661566	0.330783	+	0.943707i	$0.392687\pi$
$608$	313225.	0.0343635
$609$	5.47986e6	0.598724
$610$	0	0
$611$	9.82057e6	1.06423
$612$	4.92111e6	0.531110
$613$	2.32775e6	0.250199	0.125099	−	0.992144i	$-0.460075\pi$
$613$	2.32775e6	0.250199	0.125099	+	0.992144i	$0.460075\pi$
$614$	1.11549e6	0.119411
$615$	0	0
$616$	−444112.	−0.0471564
$617$	−1.30284e7	−1.37777	−0.688886	−	0.724870i	$-0.741900\pi$
$617$	−1.30284e7	−1.37777	−0.688886	+	0.724870i	$0.741900\pi$
$618$	1.26221e6	0.132941
$619$	−2.53159e6	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$619$	−2.53159e6	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$620$	0	0
$621$	988226.	0.102832
$622$	−4601.30	−0.000476876 0
$623$	9.11214e6	0.940590
$624$	5.93765e6	0.610454
$625$	0	0
$626$	−755689.	−0.0770739
$627$	−163078.	−0.0165664
$628$	1.03037e7	1.04254
$629$	−2.58694e7	−2.60712
$630$	0	0
$631$	−2.17733e6	−0.217696	−0.108848	−	0.994058i	$-0.534716\pi$
$631$	−2.17733e6	−0.217696	−0.108848	+	0.994058i	$0.534716\pi$
$632$	2.72188e6	0.271067
$633$	−4.85491e6	−0.481583
$634$	−316955.	−0.0313166
$635$	0	0
$636$	3.78673e6	0.371211
$637$	−6.67237e6	−0.651525
$638$	615478.	0.0598634
$639$	−4.94332e6	−0.478924
$640$	0	0
$641$	−2.58180e6	−0.248187	−0.124093	−	0.992271i	$-0.539602\pi$
$641$	−2.58180e6	−0.248187	−0.124093	+	0.992271i	$0.539602\pi$
$642$	−827200.	−0.0792087
$643$	1.67813e7	1.60066	0.800329	−	0.599561i	$-0.204658\pi$
$643$	1.67813e7	1.60066	0.800329	+	0.599561i	$0.204658\pi$
$644$	3.55333e6	0.337614
$645$	0	0
$646$	−200566.	−0.0189093
$647$	1.01691e7	0.955037	0.477518	−	0.878622i	$-0.341536\pi$
$647$	1.01691e7	0.955037	0.477518	+	0.878622i	$0.341536\pi$
$648$	289547.	0.0270883
$649$	5.54220e6	0.516500
$650$	0	0
$651$	3.15556e6	0.291826
$652$	−1.58171e7	−1.45716
$653$	8.99282e6	0.825302	0.412651	−	0.910889i	$-0.364603\pi$
$653$	8.99282e6	0.825302	0.412651	+	0.910889i	$0.364603\pi$
$654$	900731.	0.0823476
$655$	0	0
$656$	2.80204e6	0.254223
$657$	1.40563e6	0.127045
$658$	841144.	0.0757365
$659$	−1.03480e7	−0.928203	−0.464102	−	0.885782i	$-0.653623\pi$
$659$	−1.03480e7	−0.928203	−0.464102	+	0.885782i	$0.653623\pi$
$660$	0	0
$661$	1.48595e7	1.32282	0.661409	−	0.750025i	$-0.269959\pi$
$661$	1.48595e7	1.32282	0.661409	+	0.750025i	$0.269959\pi$
$662$	−1.20253e6	−0.106648
$663$	−1.17045e7	−1.03412
$664$	2.87874e6	0.253386
$665$	0	0
$666$	−755266.	−0.0659803
$667$	−9.92427e6	−0.863742
$668$	−1.86098e7	−1.61361
$669$	−4.07551e6	−0.352060
$670$	0	0
$671$	2.29857e6	0.197084
$672$	1.56563e6	0.133741
$673$	−217545.	−0.0185145	−0.00925725	−	0.999957i	$-0.502947\pi$
$673$	−217545.	−0.0185145	−0.00925725	+	0.999957i	$0.502947\pi$
$674$	2.09625e6	0.177744
$675$	0	0
$676$	−2.64325e6	−0.222470
$677$	2.64314e6	0.221640	0.110820	−	0.993840i	$-0.464652\pi$
$677$	2.64314e6	0.221640	0.110820	+	0.993840i	$0.464652\pi$
$678$	−6851.07	−0.000572380 0
$679$	6.96419e6	0.579690
$680$	0	0
$681$	365477.	0.0301990
$682$	354421.	0.0291782
$683$	−4.66785e6	−0.382882	−0.191441	−	0.981504i	$-0.561316\pi$
$683$	−4.66785e6	−0.382882	−0.191441	+	0.981504i	$0.561316\pi$
$684$	382298.	0.0312436
$685$	0	0
$686$	−1.54269e6	−0.125161
$687$	8.89569e6	0.719098
$688$	2.15899e7	1.73892
$689$	−9.00647e6	−0.722781
$690$	0	0
$691$	3.11273e6	0.247997	0.123998	−	0.992282i	$-0.460428\pi$
$691$	3.11273e6	0.247997	0.123998	+	0.992282i	$0.460428\pi$
$692$	1.79943e7	1.42846
$693$	−815132.	−0.0644755
$694$	882026.	0.0695156
$695$	0	0
$696$	−2.90778e6	−0.227530
$697$	−5.52349e6	−0.430657
$698$	−2.26795e6	−0.176196
$699$	429061.	0.0332143
$700$	0	0
$701$	1.16786e7	0.897629	0.448815	−	0.893625i	$-0.351846\pi$
$701$	1.16786e7	0.897629	0.448815	+	0.893625i	$0.351846\pi$
$702$	−341718.	−0.0261713
$703$	−2.00967e6	−0.153369
$704$	−3.61054e6	−0.274562
$705$	0	0
$706$	1.18630e6	0.0895743
$707$	1.08016e7	0.812718
$708$	−1.29923e7	−0.974102
$709$	−1.96571e7	−1.46860	−0.734302	−	0.678823i	$-0.762490\pi$
$709$	−1.96571e7	−1.46860	−0.734302	+	0.678823i	$0.762490\pi$
$710$	0	0
$711$	4.99580e6	0.370622
$712$	−4.83518e6	−0.357448
$713$	−5.71485e6	−0.420999
$714$	−1.00251e6	−0.0735940
$715$	0	0
$716$	1.46367e7	1.06699
$717$	−1.48656e6	−0.107990
$718$	−179041.	−0.0129611
$719$	−1.89013e7	−1.36354	−0.681772	−	0.731565i	$-0.738790\pi$
$719$	−1.89013e7	−1.36354	−0.681772	+	0.731565i	$0.738790\pi$
$720$	0	0
$721$	1.67876e7	1.20268
$722$	1.70480e6	0.121712
$723$	−1.01971e7	−0.725492
$724$	1.29155e7	0.915725
$725$	0	0
$726$	−91552.7	−0.00644658
$727$	−6.78994e6	−0.476463	−0.238232	−	0.971208i	$-0.576568\pi$
$727$	−6.78994e6	−0.476463	−0.238232	+	0.971208i	$0.576568\pi$
$728$	−2.47622e6	−0.173165
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.25590e7	−2.94576
$732$	−5.38845e6	−0.371695
$733$	3.60300e6	0.247688	0.123844	−	0.992302i	$-0.460478\pi$
$733$	3.60300e6	0.247688	0.123844	+	0.992302i	$0.460478\pi$
$734$	−474886.	−0.0325349
$735$	0	0
$736$	−2.83542e6	−0.192940
$737$	−804778.	−0.0545767
$738$	−161260.	−0.0108990
$739$	−7.58466e6	−0.510887	−0.255443	−	0.966824i	$-0.582221\pi$
$739$	−7.58466e6	−0.510887	−0.255443	+	0.966824i	$0.582221\pi$
$740$	0	0
$741$	−909270.	−0.0608341
$742$	−771416.	−0.0514374
$743$	−1.17473e7	−0.780669	−0.390335	−	0.920673i	$-0.627641\pi$
$743$	−1.17473e7	−0.780669	−0.390335	+	0.920673i	$0.627641\pi$
$744$	−1.67443e6	−0.110901
$745$	0	0
$746$	2.48860e6	0.163722
$747$	5.28370e6	0.346447
$748$	7.35129e6	0.480407
$749$	−1.10019e7	−0.716577
$750$	0	0
$751$	4.05057e6	0.262070	0.131035	−	0.991378i	$-0.458170\pi$
$751$	4.05057e6	0.262070	0.131035	+	0.991378i	$0.458170\pi$
$752$	1.42346e7	0.917910
$753$	−9.72269e6	−0.624883
$754$	3.43170e6	0.219827
$755$	0	0
$756$	1.91088e6	0.121599
$757$	−2.12290e7	−1.34645	−0.673225	−	0.739438i	$-0.735091\pi$
$757$	−2.12290e7	−1.34645	−0.673225	+	0.739438i	$0.735091\pi$
$758$	3.39029e6	0.214321
$759$	1.47624e6	0.0930149
$760$	0	0
$761$	−1.06438e7	−0.666247	−0.333123	−	0.942883i	$-0.608103\pi$
$761$	−1.06438e7	−0.666247	−0.333123	+	0.942883i	$0.608103\pi$
$762$	1.97449e6	0.123188
$763$	1.19799e7	0.744974
$764$	1.86276e7	1.15458
$765$	0	0
$766$	3.45702e6	0.212878
$767$	3.09014e7	1.89666
$768$	8.04551e6	0.492210
$769$	3.17109e6	0.193372	0.0966859	−	0.995315i	$-0.469176\pi$
$769$	3.17109e6	0.193372	0.0966859	+	0.995315i	$0.469176\pi$
$770$	0	0
$771$	−2.71314e6	−0.164375
$772$	−9.79894e6	−0.591747
$773$	−1.80501e7	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$773$	−1.80501e7	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$774$	−1.24252e6	−0.0745507
$775$	0	0
$776$	−3.69541e6	−0.220297
$777$	−1.00452e7	−0.596904
$778$	2.84774e6	0.168675
$779$	−429094.	−0.0253343
$780$	0	0
$781$	−7.38447e6	−0.433203
$782$	1.81559e6	0.106169
$783$	−5.33700e6	−0.311095
$784$	−9.67137e6	−0.561950
$785$	0	0
$786$	1.16935e6	0.0675132
$787$	2.58223e7	1.48614	0.743068	−	0.669216i	$-0.233370\pi$
$787$	2.58223e7	1.48614	0.743068	+	0.669216i	$0.233370\pi$
$788$	5.09360e6	0.292219
$789$	3.12979e6	0.178988
$790$	0	0
$791$	−91120.5	−0.00517815
$792$	432534.	0.0245023
$793$	1.28161e7	0.723722
$794$	1.12012e6	0.0630540
$795$	0	0
$796$	−1.37296e7	−0.768026
$797$	−1.50658e7	−0.840129	−0.420065	−	0.907494i	$-0.637993\pi$
$797$	−1.50658e7	−0.840129	−0.420065	+	0.907494i	$0.637993\pi$
$798$	−77880.1	−0.00432932
$799$	−2.80598e7	−1.55495
$800$	0	0
$801$	−8.87458e6	−0.488727
$802$	450480.	0.0247309
$803$	2.09976e6	0.114916
$804$	1.88661e6	0.102930
$805$	0	0
$806$	1.97613e6	0.107147
$807$	8.82858e6	0.477207
$808$	−5.73166e6	−0.308853
$809$	−2.08560e7	−1.12037	−0.560183	−	0.828369i	$-0.689269\pi$
$809$	−2.08560e7	−1.12037	−0.560183	+	0.828369i	$0.689269\pi$
$810$	0	0
$811$	8.46233e6	0.451791	0.225896	−	0.974152i	$-0.427469\pi$
$811$	8.46233e6	0.451791	0.225896	+	0.974152i	$0.427469\pi$
$812$	−1.91900e7	−1.02138
$813$	86676.5	0.00459912
$814$	−1.12824e6	−0.0596814
$815$	0	0
$816$	−1.69653e7	−0.891943
$817$	−3.30620e6	−0.173290
$818$	−1.30849e6	−0.0683732
$819$	−4.54490e6	−0.236764
$820$	0	0
$821$	8.67359e6	0.449098	0.224549	−	0.974463i	$-0.427909\pi$
$821$	8.67359e6	0.449098	0.224549	+	0.974463i	$0.427909\pi$
$822$	1.45262e6	0.0749846
$823$	1.24194e7	0.639148	0.319574	−	0.947561i	$-0.396460\pi$
$823$	1.24194e7	0.639148	0.319574	+	0.947561i	$0.396460\pi$
$824$	−8.90799e6	−0.457048
$825$	0	0
$826$	2.64675e6	0.134978
$827$	2.38473e7	1.21248	0.606242	−	0.795280i	$-0.292676\pi$
$827$	2.38473e7	1.21248	0.606242	+	0.795280i	$0.292676\pi$
$828$	−3.46069e6	−0.175423
$829$	−3.26873e7	−1.65193	−0.825967	−	0.563719i	$-0.809370\pi$
$829$	−3.26873e7	−1.65193	−0.825967	+	0.563719i	$0.809370\pi$
$830$	0	0
$831$	−2.91931e6	−0.146649
$832$	−2.01312e7	−1.00823
$833$	1.90646e7	0.951953
$834$	1.94890e6	0.0970227
$835$	0	0
$836$	571087.	0.0282609
$837$	−3.07329e6	−0.151632
$838$	3.54701e6	0.174483
$839$	8.09423e6	0.396982	0.198491	−	0.980103i	$-0.436396\pi$
$839$	8.09423e6	0.396982	0.198491	+	0.980103i	$0.436396\pi$
$840$	0	0
$841$	3.30857e7	1.61306
$842$	−3.81313e6	−0.185354
$843$	7.80171e6	0.378112
$844$	1.70015e7	0.821544
$845$	0	0
$846$	−819214.	−0.0393524
$847$	−1.21767e6	−0.0583203
$848$	−1.30546e7	−0.623409
$849$	5.17642e6	0.246468
$850$	0	0
$851$	1.81922e7	0.861117
$852$	1.73111e7	0.817007
$853$	−3.67970e6	−0.173157	−0.0865785	−	0.996245i	$-0.527593\pi$
$853$	−3.67970e6	−0.173157	−0.0865785	+	0.996245i	$0.527593\pi$
$854$	1.09771e6	0.0515043
$855$	0	0
$856$	5.83794e6	0.272317
$857$	−2.17589e7	−1.01201	−0.506004	−	0.862531i	$-0.668878\pi$
$857$	−2.17589e7	−1.01201	−0.506004	+	0.862531i	$0.668878\pi$
$858$	−510467.	−0.0236728
$859$	−3.76305e7	−1.74003	−0.870015	−	0.493025i	$-0.835891\pi$
$859$	−3.76305e7	−1.74003	−0.870015	+	0.493025i	$0.835891\pi$
$860$	0	0
$861$	−2.14479e6	−0.0985998
$862$	2.49841e6	0.114524
$863$	2.50264e7	1.14386	0.571928	−	0.820304i	$-0.306196\pi$
$863$	2.50264e7	1.14386	0.571928	+	0.820304i	$0.306196\pi$
$864$	−1.52481e6	−0.0694914
$865$	0	0
$866$	−3.49902e6	−0.158545
$867$	2.06641e7	0.933615
$868$	−1.10505e7	−0.497832
$869$	7.46286e6	0.335240
$870$	0	0
$871$	−4.48717e6	−0.200414
$872$	−6.35689e6	−0.283109
$873$	−6.78263e6	−0.301205
$874$	141044.	0.00624564
$875$	0	0
$876$	−4.92239e6	−0.216728
$877$	−1.12677e7	−0.494692	−0.247346	−	0.968927i	$-0.579558\pi$
$877$	−1.12677e7	−0.494692	−0.247346	+	0.968927i	$0.579558\pi$
$878$	2.26477e6	0.0991488
$879$	−1.76466e7	−0.770351
$880$	0	0
$881$	−4.36223e7	−1.89352	−0.946758	−	0.321947i	$-0.895663\pi$
$881$	−4.36223e7	−1.89352	−0.946758	+	0.321947i	$0.895663\pi$
$882$	556597.	0.0240918
$883$	3.14744e7	1.35849	0.679244	−	0.733912i	$-0.262308\pi$
$883$	3.14744e7	1.35849	0.679244	+	0.733912i	$0.262308\pi$
$884$	4.09883e7	1.76413
$885$	0	0
$886$	−942545.	−0.0403383
$887$	−2.33777e7	−0.997684	−0.498842	−	0.866693i	$-0.666241\pi$
$887$	−2.33777e7	−0.997684	−0.498842	+	0.866693i	$0.666241\pi$
$888$	5.33027e6	0.226839
$889$	2.62611e7	1.11444
$890$	0	0
$891$	793881.	0.0335013
$892$	1.42721e7	0.600587
$893$	−2.17983e6	−0.0914733
$894$	−189812.	−0.00794290
$895$	0	0
$896$	−7.29093e6	−0.303398
$897$	8.23101e6	0.341564
$898$	4.73091e6	0.195774
$899$	3.08635e7	1.27364
$900$	0	0
$901$	2.57337e7	1.05607
$902$	−240895.	−0.00985850
$903$	−1.65258e7	−0.674438
$904$	48351.3	0.00196783
$905$	0	0
$906$	−819259.	−0.0331590
$907$	−2.58345e7	−1.04275	−0.521377	−	0.853326i	$-0.674582\pi$
$907$	−2.58345e7	−1.04275	−0.521377	+	0.853326i	$0.674582\pi$
$908$	−1.27987e6	−0.0515171
$909$	−1.05200e7	−0.422285
$910$	0	0
$911$	1.24995e7	0.498996	0.249498	−	0.968375i	$-0.419734\pi$
$911$	1.24995e7	0.498996	0.249498	+	0.968375i	$0.419734\pi$
$912$	−1.31796e6	−0.0524703
$913$	7.89293e6	0.313373
$914$	−2.11202e6	−0.0836244
$915$	0	0
$916$	−3.11520e7	−1.22672
$917$	1.55526e7	0.610771
$918$	976372.	0.0382392
$919$	1.24766e7	0.487313	0.243657	−	0.969862i	$-0.421653\pi$
$919$	1.24766e7	0.487313	0.243657	+	0.969862i	$0.421653\pi$
$920$	0	0
$921$	−1.44494e7	−0.561309
$922$	−2.27795e6	−0.0882503
$923$	−4.11733e7	−1.59079
$924$	2.85453e6	0.109990
$925$	0	0
$926$	3.83507e6	0.146976
$927$	−1.63499e7	−0.624908
$928$	1.53129e7	0.583697
$929$	−1.63852e6	−0.0622891	−0.0311445	−	0.999515i	$-0.509915\pi$
$929$	−1.63852e6	−0.0622891	−0.0311445	+	0.999515i	$0.509915\pi$
$930$	0	0
$931$	1.48104e6	0.0560006
$932$	−1.50253e6	−0.0566611
$933$	59602.7	0.00224162
$934$	−2.83153e6	−0.106207
$935$	0	0
$936$	2.41166e6	0.0899760
$937$	−1.03137e7	−0.383766	−0.191883	−	0.981418i	$-0.561459\pi$
$937$	−1.03137e7	−0.383766	−0.191883	+	0.981418i	$0.561459\pi$
$938$	−384331.	−0.0142626
$939$	9.78876e6	0.362296
$940$	0	0
$941$	−1.30200e7	−0.479332	−0.239666	−	0.970855i	$-0.577038\pi$
$941$	−1.30200e7	−0.479332	−0.239666	+	0.970855i	$0.577038\pi$
$942$	2.04430e6	0.0750615
$943$	3.88430e6	0.142244
$944$	4.47906e7	1.63590
$945$	0	0
$946$	−1.85611e6	−0.0674336
$947$	−4.07950e7	−1.47820	−0.739098	−	0.673598i	$-0.764748\pi$
$947$	−4.07950e7	−1.47820	−0.739098	+	0.673598i	$0.764748\pi$
$948$	−1.74949e7	−0.632252
$949$	1.17076e7	0.421989
$950$	0	0
$951$	4.10565e6	0.147208
$952$	7.07517e6	0.253014
$953$	−4.68439e7	−1.67078	−0.835392	−	0.549654i	$-0.814760\pi$
$953$	−4.68439e7	−1.67078	−0.835392	+	0.549654i	$0.814760\pi$
$954$	751304.	0.0267267
$955$	0	0
$956$	5.20582e6	0.184223
$957$	−7.97255e6	−0.281396
$958$	−3.31973e6	−0.116866
$959$	1.93201e7	0.678364
$960$	0	0
$961$	−1.08565e7	−0.379212
$962$	−6.29067e6	−0.219159
$963$	1.07151e7	0.372331
$964$	3.57095e7	1.23763
$965$	0	0
$966$	704996.	0.0243077
$967$	−1.63484e7	−0.562223	−0.281111	−	0.959675i	$-0.590703\pi$
$967$	−1.63484e7	−0.562223	−0.281111	+	0.959675i	$0.590703\pi$
$968$	646131.	0.0221632
$969$	2.59801e6	0.0888856
$970$	0	0
$971$	−1.62139e6	−0.0551875	−0.0275937	−	0.999619i	$-0.508784\pi$
$971$	−1.62139e6	−0.0551875	−0.0275937	+	0.999619i	$0.508784\pi$
$972$	−1.86106e6	−0.0631823
$973$	2.59206e7	0.877735
$974$	1.22925e6	0.0415187
$975$	0	0
$976$	1.85765e7	0.624221
$977$	2.33840e7	0.783759	0.391880	−	0.920017i	$-0.371825\pi$
$977$	2.33840e7	0.783759	0.391880	+	0.920017i	$0.371825\pi$
$978$	−3.13818e6	−0.104913
$979$	−1.32571e7	−0.442070
$980$	0	0
$981$	−1.16676e7	−0.387086
$982$	723552.	0.0239437
$983$	−1.96781e7	−0.649529	−0.324764	−	0.945795i	$-0.605285\pi$
$983$	−1.96781e7	−0.649529	−0.324764	+	0.945795i	$0.605285\pi$
$984$	1.13809e6	0.0374704
$985$	0	0
$986$	−9.80522e6	−0.321192
$987$	−1.08957e7	−0.356010
$988$	3.18419e6	0.103778
$989$	2.99289e7	0.972970
$990$	0	0
$991$	1.56015e7	0.504642	0.252321	−	0.967644i	$-0.418806\pi$
$991$	1.56015e7	0.504642	0.252321	+	0.967644i	$0.418806\pi$
$992$	8.81788e6	0.284502
$993$	1.55769e7	0.501312
$994$	−3.52654e6	−0.113210
$995$	0	0
$996$	−1.85031e7	−0.591011
$997$	5.71913e6	0.182218	0.0911091	−	0.995841i	$-0.470959\pi$
$997$	5.71913e6	0.182218	0.0911091	+	0.995841i	$0.470959\pi$
$998$	−1.41091e6	−0.0448408
$999$	9.78328e6	0.310149

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.2	✓	3	5.4	even	2
495.6.a.b.1.2		3	15.14	odd	2
825.6.a.g.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)

\(f(q)\)	\(=\)	\(q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +121.000 q^{11} -283.655 q^{12} +674.655 q^{13} +57.7851 q^{14} +977.890 q^{16} -1927.66 q^{17} -56.2785 q^{18} -149.751 q^{19} -748.514 q^{21} -84.0704 q^{22} +1355.59 q^{23} +397.184 q^{24} -468.748 q^{26} +729.000 q^{27} +2621.24 q^{28} -7320.99 q^{29} -4215.76 q^{31} -2091.65 q^{32} +1089.00 q^{33} +1339.33 q^{34} -2552.90 q^{36} +13420.1 q^{37} +104.046 q^{38} +6071.90 q^{39} +2865.39 q^{41} +520.065 q^{42} +22078.1 q^{43} -3813.59 q^{44} -941.861 q^{46} +14556.4 q^{47} +8801.01 q^{48} -9890.04 q^{49} -17348.9 q^{51} -21263.3 q^{52} -13349.7 q^{53} -506.507 q^{54} -3670.35 q^{56} -1347.76 q^{57} +5086.60 q^{58} +45803.3 q^{59} +18996.5 q^{61} +2929.10 q^{62} -6736.63 q^{63} -29839.2 q^{64} -756.634 q^{66} -6651.05 q^{67} +60754.5 q^{68} +12200.3 q^{69} -61028.7 q^{71} +3574.66 q^{72} +17353.4 q^{73} -9324.27 q^{74} +4719.73 q^{76} -10063.4 q^{77} -4218.74 q^{78} +61676.5 q^{79} +6561.00 q^{81} -1990.87 q^{82} +65230.8 q^{83} +23591.1 q^{84} -15339.8 q^{86} -65888.9 q^{87} +5339.92 q^{88} -109563. q^{89} -56109.9 q^{91} -42724.5 q^{92} -37941.8 q^{93} -10113.8 q^{94} -18824.8 q^{96} -83736.1 q^{97} +6871.57 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22} + 640 q^{23} + 216 q^{24} + 2132 q^{26} + 2187 q^{27} + 580 q^{28} - 4538 q^{29} - 14968 q^{31} + 2496 q^{32} + 3267 q^{33} - 13704 q^{34} - 3402 q^{36} + 6190 q^{37} + 11668 q^{38} - 2610 q^{39} - 8926 q^{41} + 8244 q^{42} + 33592 q^{43} - 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 5310 q^{48} - 14693 q^{49} - 3906 q^{51} - 18780 q^{52} + 22934 q^{53} - 1458 q^{54} - 40012 q^{56} - 25704 q^{57} + 32304 q^{58} - 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 5508 q^{63} + 35474 q^{64} - 2178 q^{66} - 16868 q^{67} + 71288 q^{68} + 5760 q^{69} + 4856 q^{71} + 1944 q^{72} - 1910 q^{73} + 29404 q^{74} + 6116 q^{76} + 8228 q^{77} + 19188 q^{78} - 36844 q^{79} + 19683 q^{81} - 84000 q^{82} + 48796 q^{83} + 5220 q^{84} - 83492 q^{86} - 40842 q^{87} + 2904 q^{88} - 188978 q^{89} - 93208 q^{91} + 6976 q^{92} - 134712 q^{93} + 70472 q^{94} + 22464 q^{96} - 247526 q^{97} + 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 + 363 * q^11 - 378 * q^12 - 290 * q^13 + 916 * q^14 - 590 * q^16 - 434 * q^17 - 162 * q^18 - 2856 * q^19 + 612 * q^21 - 242 * q^22 + 640 * q^23 + 216 * q^24 + 2132 * q^26 + 2187 * q^27 + 580 * q^28 - 4538 * q^29 - 14968 * q^31 + 2496 * q^32 + 3267 * q^33 - 13704 * q^34 - 3402 * q^36 + 6190 * q^37 + 11668 * q^38 - 2610 * q^39 - 8926 * q^41 + 8244 * q^42 + 33592 * q^43 - 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 5310 * q^48 - 14693 * q^49 - 3906 * q^51 - 18780 * q^52 + 22934 * q^53 - 1458 * q^54 - 40012 * q^56 - 25704 * q^57 + 32304 * q^58 - 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 5508 * q^63 + 35474 * q^64 - 2178 * q^66 - 16868 * q^67 + 71288 * q^68 + 5760 * q^69 + 4856 * q^71 + 1944 * q^72 - 1910 * q^73 + 29404 * q^74 + 6116 * q^76 + 8228 * q^77 + 19188 * q^78 - 36844 * q^79 + 19683 * q^81 - 84000 * q^82 + 48796 * q^83 + 5220 * q^84 - 83492 * q^86 - 40842 * q^87 + 2904 * q^88 - 188978 * q^89 - 93208 * q^91 + 6976 * q^92 - 134712 * q^93 + 70472 * q^94 + 22464 * q^96 - 247526 * q^97 + 154654 * q^98 + 29403 * q^99

Introduction

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