LMFDB - Embedded newform 825.6.a.r.1.4

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 $ x^9 - x^8 - 229x^7 + 267x^6 + 16434x^5 - 16568x^4 - 405504x^3 + 202288x^2 + 2184608*x + 1190400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.27244$ 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)

$f(q)$	$=$	$q-3.27244 q^{2} +9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} -115.628 q^{7} +174.392 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.27244 q^{2} +9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} -115.628 q^{7} +174.392 q^{8} +81.0000 q^{9} -121.000 q^{11} -191.620 q^{12} -883.035 q^{13} +378.385 q^{14} +110.630 q^{16} +519.889 q^{17} -265.068 q^{18} -1590.33 q^{19} -1040.65 q^{21} +395.965 q^{22} -1462.43 q^{23} +1569.53 q^{24} +2889.68 q^{26} +729.000 q^{27} +2461.85 q^{28} -7184.81 q^{29} +7836.58 q^{31} -5942.57 q^{32} -1089.00 q^{33} -1701.31 q^{34} -1724.58 q^{36} -10331.6 q^{37} +5204.27 q^{38} -7947.32 q^{39} -3161.57 q^{41} +3405.46 q^{42} +15968.3 q^{43} +2576.23 q^{44} +4785.73 q^{46} +10774.0 q^{47} +995.666 q^{48} -3437.22 q^{49} +4679.00 q^{51} +18800.8 q^{52} -2690.05 q^{53} -2385.61 q^{54} -20164.6 q^{56} -14313.0 q^{57} +23511.9 q^{58} -46938.2 q^{59} +2111.96 q^{61} -25644.7 q^{62} -9365.85 q^{63} +15906.6 q^{64} +3563.69 q^{66} -58553.4 q^{67} -11069.0 q^{68} -13161.9 q^{69} -57097.1 q^{71} +14125.8 q^{72} +29208.2 q^{73} +33809.6 q^{74} +33860.0 q^{76} +13991.0 q^{77} +26007.1 q^{78} +73679.8 q^{79} +6561.00 q^{81} +10346.0 q^{82} -39084.3 q^{83} +22156.6 q^{84} -52255.3 q^{86} -64663.3 q^{87} -21101.4 q^{88} +12826.5 q^{89} +102103. q^{91} +31136.9 q^{92} +70529.2 q^{93} -35257.2 q^{94} -53483.2 q^{96} -23693.2 q^{97} +11248.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9}+O(q^{10}) $ 9 * q - q^2 + 81 * q^3 + 171 * q^4 - 9 * q^6 + 57 * q^7 + 177 * q^8 + 729 * q^9	\( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9} - 1089 q^{11} + 1539 q^{12} - 723 q^{13} + 720 q^{14} + 4147 q^{16} + 2804 q^{17} - 81 q^{18} - 1601 q^{19} + 513 q^{21} + 121 q^{22} + 2392 q^{23} + 1593 q^{24} - 1987 q^{26} + 6561 q^{27} - 4436 q^{28} + 5966 q^{29} + 21575 q^{31} + 25493 q^{32} - 9801 q^{33} - 3098 q^{34} + 13851 q^{36} + 10228 q^{37} - 12765 q^{38} - 6507 q^{39} + 13304 q^{41} + 6480 q^{42} + 13829 q^{43} - 20691 q^{44} + 81283 q^{46} + 13998 q^{47} + 37323 q^{48} - 19468 q^{49} + 25236 q^{51} - 37131 q^{52} + 44166 q^{53} - 729 q^{54} + 79160 q^{56} - 14409 q^{57} + 7635 q^{58} + 11626 q^{59} + 49481 q^{61} + 94479 q^{62} + 4617 q^{63} + 109367 q^{64} + 1089 q^{66} - 26567 q^{67} + 119506 q^{68} + 21528 q^{69} + 78454 q^{71} + 14337 q^{72} + 100086 q^{73} + 162360 q^{74} + 194115 q^{76} - 6897 q^{77} - 17883 q^{78} - 8478 q^{79} + 59049 q^{81} + 52700 q^{82} + 157476 q^{83} - 39924 q^{84} + 251663 q^{86} + 53694 q^{87} - 21417 q^{88} - 65548 q^{89} - 106849 q^{91} + 350115 q^{92} + 194175 q^{93} - 35742 q^{94} + 229437 q^{96} - 116757 q^{97} + 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q - q^2 + 81 * q^3 + 171 * q^4 - 9 * q^6 + 57 * q^7 + 177 * q^8 + 729 * q^9 - 1089 * q^11 + 1539 * q^12 - 723 * q^13 + 720 * q^14 + 4147 * q^16 + 2804 * q^17 - 81 * q^18 - 1601 * q^19 + 513 * q^21 + 121 * q^22 + 2392 * q^23 + 1593 * q^24 - 1987 * q^26 + 6561 * q^27 - 4436 * q^28 + 5966 * q^29 + 21575 * q^31 + 25493 * q^32 - 9801 * q^33 - 3098 * q^34 + 13851 * q^36 + 10228 * q^37 - 12765 * q^38 - 6507 * q^39 + 13304 * q^41 + 6480 * q^42 + 13829 * q^43 - 20691 * q^44 + 81283 * q^46 + 13998 * q^47 + 37323 * q^48 - 19468 * q^49 + 25236 * q^51 - 37131 * q^52 + 44166 * q^53 - 729 * q^54 + 79160 * q^56 - 14409 * q^57 + 7635 * q^58 + 11626 * q^59 + 49481 * q^61 + 94479 * q^62 + 4617 * q^63 + 109367 * q^64 + 1089 * q^66 - 26567 * q^67 + 119506 * q^68 + 21528 * q^69 + 78454 * q^71 + 14337 * q^72 + 100086 * q^73 + 162360 * q^74 + 194115 * q^76 - 6897 * q^77 - 17883 * q^78 - 8478 * q^79 + 59049 * q^81 + 52700 * q^82 + 157476 * q^83 - 39924 * q^84 + 251663 * q^86 + 53694 * q^87 - 21417 * q^88 - 65548 * q^89 - 106849 * q^91 + 350115 * q^92 + 194175 * q^93 - 35742 * q^94 + 229437 * q^96 - 116757 * q^97 + 14949 * q^98 - 88209 * 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 $

$2$	−3.27244	−0.578491	−0.289245	−	0.957255i	$-0.593404\pi$
$2$	−3.27244	−0.578491	−0.289245	+	0.957255i	$0.593404\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−21.2911	−0.665348
$5$	0	0
$5$	0	0
$6$	−29.4519	−0.333992
$7$	−115.628	−0.891902	−0.445951	−	0.895057i	$-0.647134\pi$
$7$	−115.628	−0.891902	−0.445951	+	0.895057i	$0.647134\pi$
$8$	174.392	0.963389
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−191.620	−0.384139
$13$	−883.035	−1.44917	−0.724585	−	0.689185i	$-0.757969\pi$
$13$	−883.035	−1.44917	−0.724585	+	0.689185i	$0.757969\pi$
$14$	378.385	0.515957
$15$	0	0
$16$	110.630	0.108037
$17$	519.889	0.436303	0.218152	−	0.975915i	$-0.429997\pi$
$17$	519.889	0.436303	0.218152	+	0.975915i	$0.429997\pi$
$18$	−265.068	−0.192830
$19$	−1590.33	−1.01066	−0.505329	−	0.862927i	$-0.668629\pi$
$19$	−1590.33	−1.01066	−0.505329	+	0.862927i	$0.668629\pi$
$20$	0	0
$21$	−1040.65	−0.514940
$22$	395.965	0.174422
$23$	−1462.43	−0.576444	−0.288222	−	0.957564i	$-0.593064\pi$
$23$	−1462.43	−0.576444	−0.288222	+	0.957564i	$0.593064\pi$
$24$	1569.53	0.556213
$25$	0	0
$26$	2889.68	0.838332
$27$	729.000	0.192450
$28$	2461.85	0.593425
$29$	−7184.81	−1.58643	−0.793214	−	0.608943i	$-0.791594\pi$
$29$	−7184.81	−1.58643	−0.793214	+	0.608943i	$0.791594\pi$
$30$	0	0
$31$	7836.58	1.46461	0.732306	−	0.680976i	$-0.238444\pi$
$31$	7836.58	1.46461	0.732306	+	0.680976i	$0.238444\pi$
$32$	−5942.57	−1.02589
$33$	−1089.00	−0.174078
$34$	−1701.31	−0.252398
$35$	0	0
$36$	−1724.58	−0.221783
$37$	−10331.6	−1.24069	−0.620346	−	0.784328i	$-0.713008\pi$
$37$	−10331.6	−1.24069	−0.620346	+	0.784328i	$0.713008\pi$
$38$	5204.27	0.584656
$39$	−7947.32	−0.836679
$40$	0	0
$41$	−3161.57	−0.293726	−0.146863	−	0.989157i	$-0.546918\pi$
$41$	−3161.57	−0.293726	−0.146863	+	0.989157i	$0.546918\pi$
$42$	3405.46	0.297888
$43$	15968.3	1.31701	0.658503	−	0.752578i	$-0.271190\pi$
$43$	15968.3	1.31701	0.658503	+	0.752578i	$0.271190\pi$
$44$	2576.23	0.200610
$45$	0	0
$46$	4785.73	0.333467
$47$	10774.0	0.711429	0.355715	−	0.934595i	$-0.384238\pi$
$47$	10774.0	0.711429	0.355715	+	0.934595i	$0.384238\pi$
$48$	995.666	0.0623750
$49$	−3437.22	−0.204511
$50$	0	0
$51$	4679.00	0.251900
$52$	18800.8	0.964203
$53$	−2690.05	−0.131544	−0.0657719	−	0.997835i	$-0.520951\pi$
$53$	−2690.05	−0.131544	−0.0657719	+	0.997835i	$0.520951\pi$
$54$	−2385.61	−0.111331
$55$	0	0
$56$	−20164.6	−0.859248
$57$	−14313.0	−0.583504
$58$	23511.9	0.917734
$59$	−46938.2	−1.75548	−0.877742	−	0.479134i	$-0.840951\pi$
$59$	−46938.2	−1.75548	−0.877742	+	0.479134i	$0.840951\pi$
$60$	0	0
$61$	2111.96	0.0726710	0.0363355	−	0.999340i	$-0.488432\pi$
$61$	2111.96	0.0726710	0.0363355	+	0.999340i	$0.488432\pi$
$62$	−25644.7	−0.847264
$63$	−9365.85	−0.297301
$64$	15906.6	0.485430
$65$	0	0
$66$	3563.69	0.100702
$67$	−58553.4	−1.59355	−0.796774	−	0.604277i	$-0.793462\pi$
$67$	−58553.4	−1.59355	−0.796774	+	0.604277i	$0.793462\pi$
$68$	−11069.0	−0.290294
$69$	−13161.9	−0.332810
$70$	0	0
$71$	−57097.1	−1.34421	−0.672107	−	0.740454i	$-0.734611\pi$
$71$	−57097.1	−1.34421	−0.672107	+	0.740454i	$0.734611\pi$
$72$	14125.8	0.321130
$73$	29208.2	0.641502	0.320751	−	0.947164i	$-0.396065\pi$
$73$	29208.2	0.641502	0.320751	+	0.947164i	$0.396065\pi$
$74$	33809.6	0.717729
$75$	0	0
$76$	33860.0	0.672439
$77$	13991.0	0.268919
$78$	26007.1	0.484011
$79$	73679.8	1.32825	0.664126	−	0.747620i	$-0.268804\pi$
$79$	73679.8	1.32825	0.664126	+	0.747620i	$0.268804\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	10346.0	0.169918
$83$	−39084.3	−0.622741	−0.311370	−	0.950289i	$-0.600788\pi$
$83$	−39084.3	−0.622741	−0.311370	+	0.950289i	$0.600788\pi$
$84$	22156.6	0.342614
$85$	0	0
$86$	−52255.3	−0.761876
$87$	−64663.3	−0.915925
$88$	−21101.4	−0.290473
$89$	12826.5	0.171646	0.0858231	−	0.996310i	$-0.472648\pi$
$89$	12826.5	0.171646	0.0858231	+	0.996310i	$0.472648\pi$
$90$	0	0
$91$	102103.	1.29252
$92$	31136.9	0.383536
$93$	70529.2	0.845594
$94$	−35257.2	−0.411555
$95$	0	0
$96$	−53483.2	−0.592296
$97$	−23693.2	−0.255678	−0.127839	−	0.991795i	$-0.540804\pi$
$97$	−23693.2	−0.255678	−0.127839	+	0.991795i	$0.540804\pi$
$98$	11248.1	0.118308
$99$	−9801.00	−0.100504
$100$	0	0
$101$	114273.	1.11465	0.557325	−	0.830294i	$-0.311828\pi$
$101$	114273.	1.11465	0.557325	+	0.830294i	$0.311828\pi$
$102$	−15311.8	−0.145722
$103$	−134822.	−1.25218	−0.626092	−	0.779749i	$-0.715347\pi$
$103$	−134822.	−1.25218	−0.626092	+	0.779749i	$0.715347\pi$
$104$	−153994.	−1.39612
$105$	0	0
$106$	8803.02	0.0760969
$107$	29366.7	0.247968	0.123984	−	0.992284i	$-0.460433\pi$
$107$	29366.7	0.247968	0.123984	+	0.992284i	$0.460433\pi$
$108$	−15521.2	−0.128046
$109$	102755.	0.828392	0.414196	−	0.910188i	$-0.364063\pi$
$109$	102755.	0.828392	0.414196	+	0.910188i	$0.364063\pi$
$110$	0	0
$111$	−92984.5	−0.716314
$112$	−12791.8	−0.0963581
$113$	134021.	0.987362	0.493681	−	0.869643i	$-0.335651\pi$
$113$	134021.	0.987362	0.493681	+	0.869643i	$0.335651\pi$
$114$	46838.4	0.337551
$115$	0	0
$116$	152973.	1.05553
$117$	−71525.9	−0.483057
$118$	153603.	1.01553
$119$	−60113.6	−0.389140
$120$	0	0
$121$	14641.0	0.0909091
$122$	−6911.26	−0.0420395
$123$	−28454.1	−0.169583
$124$	−166850.	−0.974476
$125$	0	0
$126$	30649.2	0.171986
$127$	−138288.	−0.760809	−0.380405	−	0.924820i	$-0.624215\pi$
$127$	−138288.	−0.760809	−0.380405	+	0.924820i	$0.624215\pi$
$128$	138109.	0.745070
$129$	143715.	0.760374
$130$	0	0
$131$	241088.	1.22743	0.613716	−	0.789527i	$-0.289674\pi$
$131$	241088.	1.22743	0.613716	+	0.789527i	$0.289674\pi$
$132$	23186.1	0.115822
$133$	183887.	0.901407
$134$	191613.	0.921853
$135$	0	0
$136$	90664.5	0.420330
$137$	−249641.	−1.13636	−0.568179	−	0.822905i	$-0.692352\pi$
$137$	−249641.	−1.13636	−0.568179	+	0.822905i	$0.692352\pi$
$138$	43071.6	0.192528
$139$	−167292.	−0.734410	−0.367205	−	0.930140i	$-0.619685\pi$
$139$	−167292.	−0.734410	−0.367205	+	0.930140i	$0.619685\pi$
$140$	0	0
$141$	96965.8	0.410744
$142$	186847.	0.777616
$143$	106847.	0.436941
$144$	8960.99	0.0360122
$145$	0	0
$146$	−95582.1	−0.371103
$147$	−30935.0	−0.118075
$148$	219972.	0.825492
$149$	141666.	0.522758	0.261379	−	0.965236i	$-0.415823\pi$
$149$	141666.	0.522758	0.261379	+	0.965236i	$0.415823\pi$
$150$	0	0
$151$	365960.	1.30614	0.653072	−	0.757296i	$-0.273480\pi$
$151$	365960.	1.30614	0.653072	+	0.757296i	$0.273480\pi$
$152$	−277341.	−0.973656
$153$	42111.0	0.145434
$154$	−45784.6	−0.155567
$155$	0	0
$156$	169207.	0.556683
$157$	283151.	0.916789	0.458395	−	0.888749i	$-0.348425\pi$
$157$	283151.	0.916789	0.458395	+	0.888749i	$0.348425\pi$
$158$	−241113.	−0.768382
$159$	−24210.4	−0.0759468
$160$	0	0
$161$	169098.	0.514131
$162$	−21470.5	−0.0642768
$163$	−601651.	−1.77368	−0.886840	−	0.462076i	$-0.847105\pi$
$163$	−601651.	−1.77368	−0.886840	+	0.462076i	$0.847105\pi$
$164$	67313.4	0.195430
$165$	0	0
$166$	127901.	0.360250
$167$	−178294.	−0.494703	−0.247352	−	0.968926i	$-0.579560\pi$
$167$	−178294.	−0.494703	−0.247352	+	0.968926i	$0.579560\pi$
$168$	−181481.	−0.496087
$169$	408458.	1.10010
$170$	0	0
$171$	−128817.	−0.336886
$172$	−339984.	−0.876268
$173$	−760854.	−1.93280	−0.966398	−	0.257051i	$-0.917249\pi$
$173$	−760854.	−1.93280	−0.966398	+	0.257051i	$0.917249\pi$
$174$	211607.	0.529854
$175$	0	0
$176$	−13386.2	−0.0325743
$177$	−422444.	−1.01353
$178$	−41974.0	−0.0992957
$179$	707719.	1.65093	0.825464	−	0.564454i	$-0.190913\pi$
$179$	707719.	1.65093	0.825464	+	0.564454i	$0.190913\pi$
$180$	0	0
$181$	531826.	1.20663	0.603313	−	0.797504i	$-0.293847\pi$
$181$	531826.	1.20663	0.603313	+	0.797504i	$0.293847\pi$
$182$	−334127.	−0.747710
$183$	19007.6	0.0419566
$184$	−255037.	−0.555339
$185$	0	0
$186$	−230803.	−0.489168
$187$	−62906.6	−0.131550
$188$	−229390.	−0.473348
$189$	−84292.6	−0.171647
$190$	0	0
$191$	−519646.	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$191$	−519646.	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$192$	143159.	0.280263
$193$	624883.	1.20755	0.603775	−	0.797155i	$-0.293663\pi$
$193$	624883.	1.20755	0.603775	+	0.797155i	$0.293663\pi$
$194$	77534.5	0.147908
$195$	0	0
$196$	73182.4	0.136071
$197$	−34880.0	−0.0640340	−0.0320170	−	0.999487i	$-0.510193\pi$
$197$	−34880.0	−0.0640340	−0.0320170	+	0.999487i	$0.510193\pi$
$198$	32073.2	0.0581405
$199$	230348.	0.412336	0.206168	−	0.978517i	$-0.433901\pi$
$199$	230348.	0.412336	0.206168	+	0.978517i	$0.433901\pi$
$200$	0	0
$201$	−526981.	−0.920036
$202$	−373950.	−0.644815
$203$	830764.	1.41494
$204$	−99621.3	−0.167601
$205$	0	0
$206$	441197.	0.724377
$207$	−118457.	−0.192148
$208$	−97689.8	−0.156564
$209$	192430.	0.304725
$210$	0	0
$211$	−1.12880e6	−1.74546	−0.872730	−	0.488204i	$-0.837652\pi$
$211$	−1.12880e6	−1.74546	−0.872730	+	0.488204i	$0.837652\pi$
$212$	57274.2	0.0875224
$213$	−513874.	−0.776082
$214$	−96100.6	−0.143447
$215$	0	0
$216$	127132.	0.185404
$217$	−906126.	−1.30629
$218$	−336259.	−0.479217
$219$	262874.	0.370371
$220$	0	0
$221$	−459081.	−0.632278
$222$	304286.	0.414381
$223$	1.21861e6	1.64097	0.820487	−	0.571665i	$-0.193702\pi$
$223$	1.21861e6	1.64097	0.820487	+	0.571665i	$0.193702\pi$
$224$	687126.	0.914990
$225$	0	0
$226$	−438575.	−0.571180
$227$	834702.	1.07514	0.537572	−	0.843218i	$-0.319342\pi$
$227$	834702.	1.07514	0.537572	+	0.843218i	$0.319342\pi$
$228$	304740.	0.388233
$229$	781702.	0.985037	0.492519	−	0.870302i	$-0.336076\pi$
$229$	781702.	0.985037	0.492519	+	0.870302i	$0.336076\pi$
$230$	0	0
$231$	125919.	0.155260
$232$	−1.25297e6	−1.52835
$233$	207066.	0.249873	0.124936	−	0.992165i	$-0.460127\pi$
$233$	207066.	0.249873	0.124936	+	0.992165i	$0.460127\pi$
$234$	234064.	0.279444
$235$	0	0
$236$	999369.	1.16801
$237$	663118.	0.766867
$238$	196718.	0.225114
$239$	1.57438e6	1.78285	0.891424	−	0.453171i	$-0.149707\pi$
$239$	1.57438e6	1.78285	0.891424	+	0.453171i	$0.149707\pi$
$240$	0	0
$241$	1.02355e6	1.13519	0.567595	−	0.823308i	$-0.307874\pi$
$241$	1.02355e6	1.13519	0.567595	+	0.823308i	$0.307874\pi$
$242$	−47911.8	−0.0525901
$243$	59049.0	0.0641500
$244$	−44966.0	−0.0483515
$245$	0	0
$246$	93114.3	0.0981021
$247$	1.40432e6	1.46462
$248$	1.36664e6	1.41099
$249$	−351759.	−0.359539
$250$	0	0
$251$	572916.	0.573993	0.286996	−	0.957932i	$-0.407343\pi$
$251$	572916.	0.573993	0.286996	+	0.957932i	$0.407343\pi$
$252$	199410.	0.197808
$253$	176955.	0.173804
$254$	452540.	0.440121
$255$	0	0
$256$	−960963.	−0.916446
$257$	−607677.	−0.573905	−0.286953	−	0.957945i	$-0.592642\pi$
$257$	−607677.	−0.573905	−0.286953	+	0.957945i	$0.592642\pi$
$258$	−470298.	−0.439869
$259$	1.19462e6	1.10657
$260$	0	0
$261$	−581970.	−0.528809
$262$	−788946.	−0.710059
$263$	−1.39142e6	−1.24042	−0.620211	−	0.784435i	$-0.712953\pi$
$263$	−1.39142e6	−1.24042	−0.620211	+	0.784435i	$0.712953\pi$
$264$	−189913.	−0.167704
$265$	0	0
$266$	−601758.	−0.521456
$267$	115439.	0.0991000
$268$	1.24667e6	1.06026
$269$	221782.	0.186872	0.0934362	−	0.995625i	$-0.470215\pi$
$269$	221782.	0.186872	0.0934362	+	0.995625i	$0.470215\pi$
$270$	0	0
$271$	965446.	0.798554	0.399277	−	0.916830i	$-0.369261\pi$
$271$	965446.	0.798554	0.399277	+	0.916830i	$0.369261\pi$
$272$	57515.1	0.0471368
$273$	918930.	0.746236
$274$	816936.	0.657373
$275$	0	0
$276$	280232.	0.221435
$277$	−157129.	−0.123043	−0.0615217	−	0.998106i	$-0.519595\pi$
$277$	−157129.	−0.123043	−0.0615217	+	0.998106i	$0.519595\pi$
$278$	547453.	0.424850
$279$	634763.	0.488204
$280$	0	0
$281$	2.07218e6	1.56553	0.782765	−	0.622318i	$-0.213809\pi$
$281$	2.07218e6	1.56553	0.782765	+	0.622318i	$0.213809\pi$
$282$	−317315.	−0.237612
$283$	586163.	0.435063	0.217532	−	0.976053i	$-0.430199\pi$
$283$	586163.	0.435063	0.217532	+	0.976053i	$0.430199\pi$
$284$	1.21566e6	0.894371
$285$	0	0
$286$	−349651.	−0.252767
$287$	365565.	0.261975
$288$	−481348.	−0.341962
$289$	−1.14957e6	−0.809639
$290$	0	0
$291$	−213239.	−0.147616
$292$	−621877.	−0.426822
$293$	2.54116e6	1.72927	0.864635	−	0.502400i	$-0.167550\pi$
$293$	2.54116e6	1.72927	0.864635	+	0.502400i	$0.167550\pi$
$294$	101233.	0.0683051
$295$	0	0
$296$	−1.80175e6	−1.19527
$297$	−88209.0	−0.0580259
$298$	−463594.	−0.302411
$299$	1.29138e6	0.835366
$300$	0	0
$301$	−1.84638e6	−1.17464
$302$	−1.19758e6	−0.755593
$303$	1.02845e6	0.643543
$304$	−175938.	−0.109188
$305$	0	0
$306$	−137806.	−0.0841325
$307$	−1.45371e6	−0.880302	−0.440151	−	0.897924i	$-0.645075\pi$
$307$	−1.45371e6	−0.880302	−0.440151	+	0.897924i	$0.645075\pi$
$308$	−297884.	−0.178924
$309$	−1.21340e6	−0.722949
$310$	0	0
$311$	1.19477e6	0.700458	0.350229	−	0.936664i	$-0.386104\pi$
$311$	1.19477e6	0.700458	0.350229	+	0.936664i	$0.386104\pi$
$312$	−1.38595e6	−0.806047
$313$	2.43275e6	1.40358	0.701789	−	0.712385i	$-0.252385\pi$
$313$	2.43275e6	1.40358	0.701789	+	0.712385i	$0.252385\pi$
$314$	−926596.	−0.530354
$315$	0	0
$316$	−1.56873e6	−0.883751
$317$	931723.	0.520761	0.260381	−	0.965506i	$-0.416152\pi$
$317$	931723.	0.520761	0.260381	+	0.965506i	$0.416152\pi$
$318$	79227.2	0.0439346
$319$	869362.	0.478326
$320$	0	0
$321$	264300.	0.143164
$322$	−553363.	−0.297420
$323$	−826797.	−0.440953
$324$	−139691.	−0.0739276
$325$	0	0
$326$	1.96886e6	1.02606
$327$	924794.	0.478272
$328$	−551352.	−0.282972
$329$	−1.24577e6	−0.634525
$330$	0	0
$331$	2.23321e6	1.12036	0.560181	−	0.828370i	$-0.310731\pi$
$331$	2.23321e6	1.12036	0.560181	+	0.828370i	$0.310731\pi$
$332$	832150.	0.414339
$333$	−836861.	−0.413564
$334$	583455.	0.286181
$335$	0	0
$336$	−115127.	−0.0556324
$337$	−1.37340e6	−0.658751	−0.329376	−	0.944199i	$-0.606838\pi$
$337$	−1.37340e6	−0.658751	−0.329376	+	0.944199i	$0.606838\pi$
$338$	−1.33665e6	−0.636396
$339$	1.20619e6	0.570054
$340$	0	0
$341$	−948226.	−0.441597
$342$	421546.	0.194885
$343$	2.34079e6	1.07431
$344$	2.78475e6	1.26879
$345$	0	0
$346$	2.48985e6	1.11810
$347$	1.48507e6	0.662100	0.331050	−	0.943613i	$-0.392597\pi$
$347$	1.48507e6	0.662100	0.331050	+	0.943613i	$0.392597\pi$
$348$	1.37676e6	0.609409
$349$	−296745.	−0.130413	−0.0652063	−	0.997872i	$-0.520771\pi$
$349$	−296745.	−0.130413	−0.0652063	+	0.997872i	$0.520771\pi$
$350$	0	0
$351$	−643733.	−0.278893
$352$	719051.	0.309317
$353$	1.02630e6	0.438368	0.219184	−	0.975684i	$-0.429661\pi$
$353$	1.02630e6	0.438368	0.219184	+	0.975684i	$0.429661\pi$
$354$	1.38242e6	0.586317
$355$	0	0
$356$	−273092.	−0.114204
$357$	−541023.	−0.224670
$358$	−2.31597e6	−0.955047
$359$	3.46040e6	1.41707	0.708534	−	0.705677i	$-0.249357\pi$
$359$	3.46040e6	1.41707	0.708534	+	0.705677i	$0.249357\pi$
$360$	0	0
$361$	53060.4	0.0214290
$362$	−1.74037e6	−0.698022
$363$	131769.	0.0524864
$364$	−2.17390e6	−0.859975
$365$	0	0
$366$	−62201.3	−0.0242715
$367$	−2.54792e6	−0.987461	−0.493731	−	0.869615i	$-0.664367\pi$
$367$	−2.54792e6	−0.987461	−0.493731	+	0.869615i	$0.664367\pi$
$368$	−161789.	−0.0622771
$369$	−256087.	−0.0979087
$370$	0	0
$371$	311044.	0.117324
$372$	−1.50165e6	−0.562614
$373$	1.62873e6	0.606145	0.303073	−	0.952967i	$-0.401988\pi$
$373$	1.62873e6	0.606145	0.303073	+	0.952967i	$0.401988\pi$
$374$	205858.	0.0761007
$375$	0	0
$376$	1.87890e6	0.685383
$377$	6.34444e6	2.29901
$378$	275842.	0.0992960
$379$	−3.96243e6	−1.41698	−0.708490	−	0.705721i	$-0.750623\pi$
$379$	−3.96243e6	−1.41698	−0.708490	+	0.705721i	$0.750623\pi$
$380$	0	0
$381$	−1.24459e6	−0.439253
$382$	1.70051e6	0.596239
$383$	552568.	0.192481	0.0962407	−	0.995358i	$-0.469318\pi$
$383$	552568.	0.192481	0.0962407	+	0.995358i	$0.469318\pi$
$384$	1.24298e6	0.430167
$385$	0	0
$386$	−2.04489e6	−0.698557
$387$	1.29343e6	0.439002
$388$	504455.	0.170115
$389$	−4.13340e6	−1.38495	−0.692473	−	0.721443i	$-0.743479\pi$
$389$	−4.13340e6	−1.38495	−0.692473	+	0.721443i	$0.743479\pi$
$390$	0	0
$391$	−760304.	−0.251504
$392$	−599424.	−0.197024
$393$	2.16979e6	0.708659
$394$	114142.	0.0370431
$395$	0	0
$396$	208675.	0.0668700
$397$	−2.28281e6	−0.726933	−0.363466	−	0.931607i	$-0.618407\pi$
$397$	−2.28281e6	−0.726933	−0.363466	+	0.931607i	$0.618407\pi$
$398$	−753799.	−0.238533
$399$	1.65498e6	0.520428
$400$	0	0
$401$	3.83742e6	1.19173	0.595866	−	0.803084i	$-0.296809\pi$
$401$	3.83742e6	1.19173	0.595866	+	0.803084i	$0.296809\pi$
$402$	1.72451e6	0.532232
$403$	−6.91998e6	−2.12247
$404$	−2.43299e6	−0.741630
$405$	0	0
$406$	−2.71862e6	−0.818529
$407$	1.25013e6	0.374083
$408$	815981.	0.242678
$409$	4.06933e6	1.20286	0.601430	−	0.798926i	$-0.294598\pi$
$409$	4.06933e6	1.20286	0.601430	+	0.798926i	$0.294598\pi$
$410$	0	0
$411$	−2.24677e6	−0.656077
$412$	2.87052e6	0.833139
$413$	5.42736e6	1.56572
$414$	387644.	0.111156
$415$	0	0
$416$	5.24750e6	1.48669
$417$	−1.50563e6	−0.424012
$418$	−629716.	−0.176280
$419$	956542.	0.266176	0.133088	−	0.991104i	$-0.457511\pi$
$419$	956542.	0.266176	0.133088	+	0.991104i	$0.457511\pi$
$420$	0	0
$421$	3.40755e6	0.936994	0.468497	−	0.883465i	$-0.344796\pi$
$421$	3.40755e6	0.936994	0.468497	+	0.883465i	$0.344796\pi$
$422$	3.69392e6	1.00973
$423$	872693.	0.237143
$424$	−469123.	−0.126728
$425$	0	0
$426$	1.68162e6	0.448957
$427$	−244201.	−0.0648154
$428$	−625250.	−0.164985
$429$	961625.	0.252268
$430$	0	0
$431$	−1548.19	−0.000401450 0	−0.000200725	−	1.00000i	$-0.500064\pi$
$431$	−1548.19	−0.000401450 0	−0.000200725		1.00000i	$0.500064\pi$
$432$	80649.0	0.0207917
$433$	−4.72970e6	−1.21231	−0.606155	−	0.795347i	$-0.707289\pi$
$433$	−4.72970e6	−1.21231	−0.606155	+	0.795347i	$0.707289\pi$
$434$	2.96524e6	0.755676
$435$	0	0
$436$	−2.18777e6	−0.551169
$437$	2.32576e6	0.582587
$438$	−860239.	−0.214256
$439$	725064.	0.179562	0.0897811	−	0.995962i	$-0.471383\pi$
$439$	725064.	0.179562	0.0897811	+	0.995962i	$0.471383\pi$
$440$	0	0
$441$	−278415.	−0.0681704
$442$	1.50231e6	0.365767
$443$	−2.29072e6	−0.554578	−0.277289	−	0.960787i	$-0.589436\pi$
$443$	−2.29072e6	−0.554578	−0.277289	+	0.960787i	$0.589436\pi$
$444$	1.97975e6	0.476598
$445$	0	0
$446$	−3.98782e6	−0.949288
$447$	1.27500e6	0.301815
$448$	−1.83924e6	−0.432956
$449$	−3.83348e6	−0.897382	−0.448691	−	0.893687i	$-0.648110\pi$
$449$	−3.83348e6	−0.897382	−0.448691	+	0.893687i	$0.648110\pi$
$450$	0	0
$451$	382550.	0.0885618
$452$	−2.85346e6	−0.656940
$453$	3.29364e6	0.754103
$454$	−2.73151e6	−0.621961
$455$	0	0
$456$	−2.49607e6	−0.562141
$457$	−675446.	−0.151286	−0.0756432	−	0.997135i	$-0.524101\pi$
$457$	−675446.	−0.151286	−0.0756432	+	0.997135i	$0.524101\pi$
$458$	−2.55807e6	−0.569835
$459$	378999.	0.0839666
$460$	0	0
$461$	−987250.	−0.216359	−0.108179	−	0.994131i	$-0.534502\pi$
$461$	−987250.	−0.216359	−0.108179	+	0.994131i	$0.534502\pi$
$462$	−412061.	−0.0898166
$463$	3.27705e6	0.710445	0.355222	−	0.934782i	$-0.384405\pi$
$463$	3.27705e6	0.710445	0.355222	+	0.934782i	$0.384405\pi$
$464$	−794853.	−0.171392
$465$	0	0
$466$	−677610.	−0.144549
$467$	−1.97021e6	−0.418042	−0.209021	−	0.977911i	$-0.567028\pi$
$467$	−1.97021e6	−0.418042	−0.209021	+	0.977911i	$0.567028\pi$
$468$	1.52287e6	0.321401
$469$	6.77040e6	1.42129
$470$	0	0
$471$	2.54836e6	0.529309
$472$	−8.18565e6	−1.69121
$473$	−1.93217e6	−0.397092
$474$	−2.17001e6	−0.443626
$475$	0	0
$476$	1.27989e6	0.258913
$477$	−217894.	−0.0438479
$478$	−5.15205e6	−1.03136
$479$	6.21301e6	1.23727	0.618633	−	0.785680i	$-0.287687\pi$
$479$	6.21301e6	1.23727	0.618633	+	0.785680i	$0.287687\pi$
$480$	0	0
$481$	9.12318e6	1.79797
$482$	−3.34952e6	−0.656697
$483$	1.52188e6	0.296834
$484$	−311724.	−0.0604862
$485$	0	0
$486$	−193234.	−0.0371102
$487$	1.04014e6	0.198732	0.0993660	−	0.995051i	$-0.468319\pi$
$487$	1.04014e6	0.198732	0.0993660	+	0.995051i	$0.468319\pi$
$488$	368309.	0.0700104
$489$	−5.41486e6	−1.02403
$490$	0	0
$491$	−7.20841e6	−1.34939	−0.674693	−	0.738099i	$-0.735724\pi$
$491$	−7.20841e6	−1.34939	−0.674693	+	0.738099i	$0.735724\pi$
$492$	605820.	0.112832
$493$	−3.73531e6	−0.692164
$494$	−4.59555e6	−0.847267
$495$	0	0
$496$	866957.	0.158232
$497$	6.60201e6	1.19891
$498$	1.15111e6	0.207990
$499$	−2.91483e6	−0.524037	−0.262019	−	0.965063i	$-0.584388\pi$
$499$	−2.91483e6	−0.524037	−0.262019	+	0.965063i	$0.584388\pi$
$500$	0	0
$501$	−1.60464e6	−0.285617
$502$	−1.87483e6	−0.332050
$503$	−5.60301e6	−0.987419	−0.493710	−	0.869627i	$-0.664359\pi$
$503$	−5.60301e6	−0.987419	−0.493710	+	0.869627i	$0.664359\pi$
$504$	−1.63333e6	−0.286416
$505$	0	0
$506$	−579073.	−0.100544
$507$	3.67612e6	0.635141
$508$	2.94431e6	0.506203
$509$	−5.16044e6	−0.882861	−0.441430	−	0.897295i	$-0.645529\pi$
$509$	−5.16044e6	−0.882861	−0.441430	+	0.897295i	$0.645529\pi$
$510$	0	0
$511$	−3.37728e6	−0.572157
$512$	−1.27480e6	−0.214915
$513$	−1.15935e6	−0.194501
$514$	1.98859e6	0.331999
$515$	0	0
$516$	−3.05985e6	−0.505914
$517$	−1.30365e6	−0.214504
$518$	−3.90932e6	−0.640143
$519$	−6.84768e6	−1.11590
$520$	0	0
$521$	6.65504e6	1.07413	0.537064	−	0.843541i	$-0.319533\pi$
$521$	6.65504e6	1.07413	0.537064	+	0.843541i	$0.319533\pi$
$522$	1.90446e6	0.305911
$523$	6.22833e6	0.995674	0.497837	−	0.867271i	$-0.334128\pi$
$523$	6.22833e6	0.995674	0.497837	+	0.867271i	$0.334128\pi$
$524$	−5.13304e6	−0.816670
$525$	0	0
$526$	4.55334e6	0.717573
$527$	4.07415e6	0.639015
$528$	−120476.	−0.0188068
$529$	−4.29763e6	−0.667713
$530$	0	0
$531$	−3.80200e6	−0.585161
$532$	−3.91516e6	−0.599750
$533$	2.79177e6	0.425659
$534$	−377766.	−0.0573284
$535$	0	0
$536$	−1.02112e7	−1.53521
$537$	6.36947e6	0.953164
$538$	−725767.	−0.108104
$539$	415904.	0.0616625
$540$	0	0
$541$	−5.01609e6	−0.736838	−0.368419	−	0.929660i	$-0.620101\pi$
$541$	−5.01609e6	−0.736838	−0.368419	+	0.929660i	$0.620101\pi$
$542$	−3.15936e6	−0.461956
$543$	4.78643e6	0.696646
$544$	−3.08948e6	−0.447598
$545$	0	0
$546$	−3.00714e6	−0.431691
$547$	−7.65398e6	−1.09375	−0.546876	−	0.837213i	$-0.684183\pi$
$547$	−7.65398e6	−1.09375	−0.546876	+	0.837213i	$0.684183\pi$
$548$	5.31515e6	0.756074
$549$	171069.	0.0242237
$550$	0	0
$551$	1.14262e7	1.60334
$552$	−2.29533e6	−0.320625
$553$	−8.51943e6	−1.18467
$554$	514197.	0.0711795
$555$	0	0
$556$	3.56184e6	0.488639
$557$	1.43563e6	0.196067	0.0980335	−	0.995183i	$-0.468745\pi$
$557$	1.43563e6	0.196067	0.0980335	+	0.995183i	$0.468745\pi$
$558$	−2.07722e6	−0.282421
$559$	−1.41006e7	−1.90857
$560$	0	0
$561$	−566159.	−0.0759507
$562$	−6.78107e6	−0.905644
$563$	−2.98059e6	−0.396307	−0.198153	−	0.980171i	$-0.563494\pi$
$563$	−2.98059e6	−0.396307	−0.198153	+	0.980171i	$0.563494\pi$
$564$	−2.06451e6	−0.273288
$565$	0	0
$566$	−1.91818e6	−0.251680
$567$	−758634.	−0.0991002
$568$	−9.95728e6	−1.29500
$569$	−4.06173e6	−0.525934	−0.262967	−	0.964805i	$-0.584701\pi$
$569$	−4.06173e6	−0.525934	−0.262967	+	0.964805i	$0.584701\pi$
$570$	0	0
$571$	2.56327e6	0.329007	0.164503	−	0.986377i	$-0.447398\pi$
$571$	2.56327e6	0.329007	0.164503	+	0.986377i	$0.447398\pi$
$572$	−2.27490e6	−0.290718
$573$	−4.67681e6	−0.595064
$574$	−1.19629e6	−0.151550
$575$	0	0
$576$	1.28843e6	0.161810
$577$	−1.06121e7	−1.32698	−0.663489	−	0.748186i	$-0.730925\pi$
$577$	−1.06121e7	−1.32698	−0.663489	+	0.748186i	$0.730925\pi$
$578$	3.76190e6	0.468369
$579$	5.62394e6	0.697179
$580$	0	0
$581$	4.51923e6	0.555423
$582$	697810.	0.0853945
$583$	325496.	0.0396619
$584$	5.09368e6	0.618016
$585$	0	0
$586$	−8.31579e6	−1.00037
$587$	1.12956e7	1.35305	0.676526	−	0.736418i	$-0.263484\pi$
$587$	1.12956e7	1.35305	0.676526	+	0.736418i	$0.263484\pi$
$588$	658641.	0.0785607
$589$	−1.24628e7	−1.48022
$590$	0	0
$591$	−313920.	−0.0369700
$592$	−1.14298e6	−0.134040
$593$	−1.45361e7	−1.69751	−0.848753	−	0.528790i	$-0.822646\pi$
$593$	−1.45361e7	−1.69751	−0.848753	+	0.528790i	$0.822646\pi$
$594$	288659.	0.0335674
$595$	0	0
$596$	−3.01624e6	−0.347816
$597$	2.07313e6	0.238062
$598$	−4.22597e6	−0.483251
$599$	9.35623e6	1.06545	0.532726	−	0.846288i	$-0.321168\pi$
$599$	9.35623e6	1.06545	0.532726	+	0.846288i	$0.321168\pi$
$600$	0	0
$601$	−1.40395e6	−0.158549	−0.0792747	−	0.996853i	$-0.525260\pi$
$601$	−1.40395e6	−0.158549	−0.0792747	+	0.996853i	$0.525260\pi$
$602$	6.04217e6	0.679519
$603$	−4.74283e6	−0.531183
$604$	−7.79171e6	−0.869041
$605$	0	0
$606$	−3.36555e6	−0.372284
$607$	−9.82683e6	−1.08253	−0.541267	−	0.840851i	$-0.682055\pi$
$607$	−9.82683e6	−1.08253	−0.541267	+	0.840851i	$0.682055\pi$
$608$	9.45067e6	1.03682
$609$	7.47687e6	0.816915
$610$	0	0
$611$	−9.51381e6	−1.03098
$612$	−896592.	−0.0967646
$613$	−1.00133e6	−0.107629	−0.0538143	−	0.998551i	$-0.517138\pi$
$613$	−1.00133e6	−0.107629	−0.0538143	+	0.998551i	$0.517138\pi$
$614$	4.75718e6	0.509247
$615$	0	0
$616$	2.43991e6	0.259073
$617$	−1.27829e7	−1.35181	−0.675905	−	0.736988i	$-0.736247\pi$
$617$	−1.27829e7	−1.35181	−0.675905	+	0.736988i	$0.736247\pi$
$618$	3.97078e6	0.418219
$619$	−1.03109e7	−1.08161	−0.540803	−	0.841149i	$-0.681880\pi$
$619$	−1.03109e7	−1.08161	−0.540803	+	0.841149i	$0.681880\pi$
$620$	0	0
$621$	−1.06612e6	−0.110937
$622$	−3.90980e6	−0.405209
$623$	−1.48310e6	−0.153092
$624$	−879208.	−0.0903921
$625$	0	0
$626$	−7.96103e6	−0.811957
$627$	1.73187e6	0.175933
$628$	−6.02862e6	−0.609984
$629$	−5.37129e6	−0.541318
$630$	0	0
$631$	−104040.	−0.0104023	−0.00520114	−	0.999986i	$-0.501656\pi$
$631$	−104040.	−0.0104023	−0.00520114	+	0.999986i	$0.501656\pi$
$632$	1.28492e7	1.27962
$633$	−1.01592e7	−1.00774
$634$	−3.04901e6	−0.301256
$635$	0	0
$636$	515468.	0.0505311
$637$	3.03519e6	0.296372
$638$	−2.84493e6	−0.276707
$639$	−4.62487e6	−0.448071
$640$	0	0
$641$	−7.20754e6	−0.692855	−0.346427	−	0.938077i	$-0.612605\pi$
$641$	−7.20754e6	−0.692855	−0.346427	+	0.938077i	$0.612605\pi$
$642$	−864906.	−0.0828192
$643$	−1.77389e7	−1.69199	−0.845997	−	0.533188i	$-0.820994\pi$
$643$	−1.77389e7	−1.69199	−0.845997	+	0.533188i	$0.820994\pi$
$644$	−3.60029e6	−0.342076
$645$	0	0
$646$	2.70564e6	0.255087
$647$	1.23330e7	1.15826	0.579130	−	0.815235i	$-0.303392\pi$
$647$	1.23330e7	1.15826	0.579130	+	0.815235i	$0.303392\pi$
$648$	1.14419e6	0.107043
$649$	5.67953e6	0.529298
$650$	0	0
$651$	−8.15514e6	−0.754186
$652$	1.28098e7	1.18012
$653$	1.12222e6	0.102990	0.0514950	−	0.998673i	$-0.483601\pi$
$653$	1.12222e6	0.102990	0.0514950	+	0.998673i	$0.483601\pi$
$654$	−3.02633e6	−0.276676
$655$	0	0
$656$	−349763.	−0.0317332
$657$	2.36587e6	0.213834
$658$	4.07671e6	0.367067
$659$	−9.47379e6	−0.849787	−0.424894	−	0.905243i	$-0.639688\pi$
$659$	−9.47379e6	−0.849787	−0.424894	+	0.905243i	$0.639688\pi$
$660$	0	0
$661$	−1.25237e7	−1.11488	−0.557441	−	0.830216i	$-0.688217\pi$
$661$	−1.25237e7	−1.11488	−0.557441	+	0.830216i	$0.688217\pi$
$662$	−7.30803e6	−0.648120
$663$	−4.13172e6	−0.365046
$664$	−6.81599e6	−0.599941
$665$	0	0
$666$	2.73857e6	0.239243
$667$	1.05073e7	0.914487
$668$	3.79608e6	0.329150
$669$	1.09675e7	0.947417
$670$	0	0
$671$	−255547.	−0.0219111
$672$	6.18414e6	0.528270
$673$	−5.53368e6	−0.470952	−0.235476	−	0.971880i	$-0.575665\pi$
$673$	−5.53368e6	−0.470952	−0.235476	+	0.971880i	$0.575665\pi$
$674$	4.49436e6	0.381082
$675$	0	0
$676$	−8.69654e6	−0.731947
$677$	−4.42320e6	−0.370907	−0.185453	−	0.982653i	$-0.559375\pi$
$677$	−4.42320e6	−0.370907	−0.185453	+	0.982653i	$0.559375\pi$
$678$	−3.94718e6	−0.329771
$679$	2.73959e6	0.228040
$680$	0	0
$681$	7.51232e6	0.620735
$682$	3.10301e6	0.255460
$683$	−9.10450e6	−0.746800	−0.373400	−	0.927670i	$-0.621808\pi$
$683$	−9.10450e6	−0.746800	−0.373400	+	0.927670i	$0.621808\pi$
$684$	2.74266e6	0.224146
$685$	0	0
$686$	−7.66010e6	−0.621476
$687$	7.03532e6	0.568712
$688$	1.76657e6	0.142285
$689$	2.37541e6	0.190629
$690$	0	0
$691$	2.00055e7	1.59388	0.796939	−	0.604060i	$-0.206451\pi$
$691$	2.00055e7	1.59388	0.796939	+	0.604060i	$0.206451\pi$
$692$	1.61994e7	1.28598
$693$	1.13327e6	0.0896395
$694$	−4.85981e6	−0.383019
$695$	0	0
$696$	−1.12768e7	−0.882392
$697$	−1.64366e6	−0.128154
$698$	971080.	0.0754425
$699$	1.86359e6	0.144264
$700$	0	0
$701$	2.43992e7	1.87534	0.937672	−	0.347522i	$-0.112977\pi$
$701$	2.43992e7	1.87534	0.937672	+	0.347522i	$0.112977\pi$
$702$	2.10658e6	0.161337
$703$	1.64307e7	1.25391
$704$	−1.92469e6	−0.146363
$705$	0	0
$706$	−3.35851e6	−0.253592
$707$	−1.32131e7	−0.994158
$708$	8.99432e6	0.674350
$709$	−2.48282e6	−0.185494	−0.0927470	−	0.995690i	$-0.529565\pi$
$709$	−2.48282e6	−0.185494	−0.0927470	+	0.995690i	$0.529565\pi$
$710$	0	0
$711$	5.96806e6	0.442751
$712$	2.23684e6	0.165362
$713$	−1.14605e7	−0.844266
$714$	1.77046e6	0.129970
$715$	0	0
$716$	−1.50682e7	−1.09844
$717$	1.41694e7	1.02933
$718$	−1.13240e7	−0.819761
$719$	−1.24262e7	−0.896430	−0.448215	−	0.893926i	$-0.647940\pi$
$719$	−1.24262e7	−0.896430	−0.448215	+	0.893926i	$0.647940\pi$
$720$	0	0
$721$	1.55892e7	1.11683
$722$	−173637.	−0.0123965
$723$	9.21199e6	0.655402
$724$	−1.13232e7	−0.802827
$725$	0	0
$726$	−431206.	−0.0303629
$727$	789687.	0.0554139	0.0277070	−	0.999616i	$-0.491179\pi$
$727$	789687.	0.0554139	0.0277070	+	0.999616i	$0.491179\pi$
$728$	1.78060e7	1.24520
$729$	531441.	0.0370370
$730$	0	0
$731$	8.30176e6	0.574614
$732$	−404694.	−0.0279158
$733$	1.22120e7	0.839512	0.419756	−	0.907637i	$-0.362116\pi$
$733$	1.22120e7	0.839512	0.419756	+	0.907637i	$0.362116\pi$
$734$	8.33790e6	0.571237
$735$	0	0
$736$	8.69063e6	0.591366
$737$	7.08497e6	0.480473
$738$	838029.	0.0566393
$739$	−2.79083e7	−1.87985	−0.939923	−	0.341385i	$-0.889104\pi$
$739$	−2.79083e7	−1.87985	−0.939923	+	0.341385i	$0.889104\pi$
$740$	0	0
$741$	1.26389e7	0.845596
$742$	−1.01787e6	−0.0678709
$743$	2.25788e7	1.50047	0.750236	−	0.661170i	$-0.229940\pi$
$743$	2.25788e7	1.50047	0.750236	+	0.661170i	$0.229940\pi$
$744$	1.22997e7	0.814635
$745$	0	0
$746$	−5.32992e6	−0.350650
$747$	−3.16583e6	−0.207580
$748$	1.33935e6	0.0875268
$749$	−3.39560e6	−0.221163
$750$	0	0
$751$	3.67339e6	0.237666	0.118833	−	0.992914i	$-0.462085\pi$
$751$	3.67339e6	0.237666	0.118833	+	0.992914i	$0.462085\pi$
$752$	1.19192e6	0.0768604
$753$	5.15624e6	0.331395
$754$	−2.07618e7	−1.32995
$755$	0	0
$756$	1.79469e6	0.114205
$757$	−5.72422e6	−0.363058	−0.181529	−	0.983386i	$-0.558105\pi$
$757$	−5.72422e6	−0.363058	−0.181529	+	0.983386i	$0.558105\pi$
$758$	1.29668e7	0.819710
$759$	1.59259e6	0.100346
$760$	0	0
$761$	−1.00462e7	−0.628842	−0.314421	−	0.949284i	$-0.601810\pi$
$761$	−1.00462e7	−0.628842	−0.314421	+	0.949284i	$0.601810\pi$
$762$	4.07286e6	0.254104
$763$	−1.18813e7	−0.738845
$764$	1.10639e7	0.685762
$765$	0	0
$766$	−1.80825e6	−0.111349
$767$	4.14481e7	2.54400
$768$	−8.64867e6	−0.529110
$769$	−1.09550e7	−0.668033	−0.334017	−	0.942567i	$-0.608404\pi$
$769$	−1.09550e7	−0.668033	−0.334017	+	0.942567i	$0.608404\pi$
$770$	0	0
$771$	−5.46910e6	−0.331344
$772$	−1.33045e7	−0.803441
$773$	−1.11558e7	−0.671507	−0.335753	−	0.941950i	$-0.608991\pi$
$773$	−1.11558e7	−0.671507	−0.335753	+	0.941950i	$0.608991\pi$
$774$	−4.23268e6	−0.253959
$775$	0	0
$776$	−4.13190e6	−0.246318
$777$	1.07516e7	0.638881
$778$	1.35263e7	0.801179
$779$	5.02794e6	0.296857
$780$	0	0
$781$	6.90875e6	0.405296
$782$	2.48805e6	0.145493
$783$	−5.23773e6	−0.305308
$784$	−380258.	−0.0220947
$785$	0	0
$786$	−7.10052e6	−0.409953
$787$	2.62258e7	1.50936	0.754679	−	0.656094i	$-0.227793\pi$
$787$	2.62258e7	1.50936	0.754679	+	0.656094i	$0.227793\pi$
$788$	742634.	0.0426049
$789$	−1.25228e7	−0.716158
$790$	0	0
$791$	−1.54965e7	−0.880630
$792$	−1.70922e6	−0.0968242
$793$	−1.86494e6	−0.105313
$794$	7.47037e6	0.420524
$795$	0	0
$796$	−4.90437e6	−0.274347
$797$	2.91979e7	1.62819	0.814097	−	0.580729i	$-0.197232\pi$
$797$	2.91979e7	1.62819	0.814097	+	0.580729i	$0.197232\pi$
$798$	−5.41582e6	−0.301063
$799$	5.60128e6	0.310399
$800$	0	0
$801$	1.03895e6	0.0572154
$802$	−1.25577e7	−0.689406
$803$	−3.53420e6	−0.193420
$804$	1.12200e7	0.612144
$805$	0	0
$806$	2.26452e7	1.22783
$807$	1.99604e6	0.107891
$808$	1.99282e7	1.07384
$809$	6.94941e6	0.373316	0.186658	−	0.982425i	$-0.440234\pi$
$809$	6.94941e6	0.373316	0.186658	+	0.982425i	$0.440234\pi$
$810$	0	0
$811$	6.55556e6	0.349992	0.174996	−	0.984569i	$-0.444009\pi$
$811$	6.55556e6	0.349992	0.174996	+	0.984569i	$0.444009\pi$
$812$	−1.76879e7	−0.941427
$813$	8.68901e6	0.461046
$814$	−4.09096e6	−0.216403
$815$	0	0
$816$	517636.	0.0272144
$817$	−2.53949e7	−1.33104
$818$	−1.33166e7	−0.695843
$819$	8.27037e6	0.430839
$820$	0	0
$821$	−1.86984e7	−0.968160	−0.484080	−	0.875024i	$-0.660846\pi$
$821$	−1.86984e7	−0.968160	−0.484080	+	0.875024i	$0.660846\pi$
$822$	7.35242e6	0.379534
$823$	−7.32584e6	−0.377014	−0.188507	−	0.982072i	$-0.560365\pi$
$823$	−7.32584e6	−0.377014	−0.188507	+	0.982072i	$0.560365\pi$
$824$	−2.35119e7	−1.20634
$825$	0	0
$826$	−1.77607e7	−0.905754
$827$	−1.76737e7	−0.898596	−0.449298	−	0.893382i	$-0.648326\pi$
$827$	−1.76737e7	−0.898596	−0.449298	+	0.893382i	$0.648326\pi$
$828$	2.52209e6	0.127845
$829$	−8.48151e6	−0.428634	−0.214317	−	0.976764i	$-0.568753\pi$
$829$	−8.48151e6	−0.428634	−0.214317	+	0.976764i	$0.568753\pi$
$830$	0	0
$831$	−1.41417e6	−0.0710391
$832$	−1.40460e7	−0.703470
$833$	−1.78697e6	−0.0892289
$834$	4.92708e6	0.245287
$835$	0	0
$836$	−4.09706e6	−0.202748
$837$	5.71287e6	0.281865
$838$	−3.13022e6	−0.153980
$839$	3.35504e7	1.64548	0.822741	−	0.568417i	$-0.192444\pi$
$839$	3.35504e7	1.64548	0.822741	+	0.568417i	$0.192444\pi$
$840$	0	0
$841$	3.11104e7	1.51675
$842$	−1.11510e7	−0.542043
$843$	1.86496e7	0.903859
$844$	2.40334e7	1.16134
$845$	0	0
$846$	−2.85583e6	−0.137185
$847$	−1.69291e6	−0.0810820
$848$	−297599.	−0.0142116
$849$	5.27547e6	0.251184
$850$	0	0
$851$	1.51093e7	0.715189
$852$	1.09410e7	0.516365
$853$	−2.06919e7	−0.973704	−0.486852	−	0.873484i	$-0.661855\pi$
$853$	−2.06919e7	−0.973704	−0.486852	+	0.873484i	$0.661855\pi$
$854$	799133.	0.0374951
$855$	0	0
$856$	5.12131e6	0.238889
$857$	2.99048e7	1.39088	0.695440	−	0.718584i	$-0.255210\pi$
$857$	2.99048e7	1.39088	0.695440	+	0.718584i	$0.255210\pi$
$858$	−3.14686e6	−0.145935
$859$	2.64304e7	1.22214	0.611070	−	0.791577i	$-0.290739\pi$
$859$	2.64304e7	1.22214	0.611070	+	0.791577i	$0.290739\pi$
$860$	0	0
$861$	3.29008e6	0.151251
$862$	5066.36	0.000232235 0
$863$	1.40568e6	0.0642481	0.0321241	−	0.999484i	$-0.489773\pi$
$863$	1.40568e6	0.0642481	0.0321241	+	0.999484i	$0.489773\pi$
$864$	−4.33214e6	−0.197432
$865$	0	0
$866$	1.54776e7	0.701310
$867$	−1.03461e7	−0.467446
$868$	1.92925e7	0.869137
$869$	−8.91525e6	−0.400483
$870$	0	0
$871$	5.17047e7	2.30932
$872$	1.79196e7	0.798064
$873$	−1.91915e6	−0.0852261
$874$	−7.61090e6	−0.337021
$875$	0	0
$876$	−5.59689e6	−0.246426
$877$	−3.65903e7	−1.60645	−0.803225	−	0.595675i	$-0.796885\pi$
$877$	−3.65903e7	−1.60645	−0.803225	+	0.595675i	$0.796885\pi$
$878$	−2.37273e6	−0.103875
$879$	2.28704e7	0.998395
$880$	0	0
$881$	−4.07911e7	−1.77062	−0.885311	−	0.464999i	$-0.846055\pi$
$881$	−4.07911e7	−1.77062	−0.885311	+	0.464999i	$0.846055\pi$
$882$	911095.	0.0394360
$883$	−3.47123e7	−1.49824	−0.749120	−	0.662434i	$-0.769524\pi$
$883$	−3.47123e7	−1.49824	−0.749120	+	0.662434i	$0.769524\pi$
$884$	9.77435e6	0.420685
$885$	0	0
$886$	7.49623e6	0.320818
$887$	−1.02664e7	−0.438135	−0.219068	−	0.975710i	$-0.570302\pi$
$887$	−1.02664e7	−0.438135	−0.219068	+	0.975710i	$0.570302\pi$
$888$	−1.62158e7	−0.690088
$889$	1.59900e7	0.678567
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−2.59455e7	−1.09182
$893$	−1.71342e7	−0.719011
$894$	−4.17235e6	−0.174597
$895$	0	0
$896$	−1.59692e7	−0.664530
$897$	1.16224e7	0.482299
$898$	1.25448e7	0.519127
$899$	−5.63043e7	−2.32350
$900$	0	0
$901$	−1.39853e6	−0.0573930
$902$	−1.25187e6	−0.0512322
$903$	−1.66174e7	−0.678179
$904$	2.33722e7	0.951213
$905$	0	0
$906$	−1.07782e7	−0.436242
$907$	2.61646e7	1.05608	0.528039	−	0.849220i	$-0.322928\pi$
$907$	2.61646e7	1.05608	0.528039	+	0.849220i	$0.322928\pi$
$908$	−1.77718e7	−0.715345
$909$	9.25608e6	0.371550
$910$	0	0
$911$	−3.76509e7	−1.50307	−0.751536	−	0.659692i	$-0.770687\pi$
$911$	−3.76509e7	−1.50307	−0.751536	+	0.659692i	$0.770687\pi$
$912$	−1.58344e6	−0.0630398
$913$	4.72920e6	0.187763
$914$	2.21035e6	0.0875178
$915$	0	0
$916$	−1.66433e7	−0.655393
$917$	−2.78765e7	−1.09475
$918$	−1.24025e6	−0.0485739
$919$	2.89048e7	1.12897	0.564484	−	0.825444i	$-0.309075\pi$
$919$	2.89048e7	1.12897	0.564484	+	0.825444i	$0.309075\pi$
$920$	0	0
$921$	−1.30834e7	−0.508243
$922$	3.23072e6	0.125162
$923$	5.04188e7	1.94800
$924$	−2.68095e6	−0.103302
$925$	0	0
$926$	−1.07239e7	−0.410986
$927$	−1.09206e7	−0.417395
$928$	4.26963e7	1.62750
$929$	1.70718e7	0.648993	0.324496	−	0.945887i	$-0.394805\pi$
$929$	1.70718e7	0.648993	0.324496	+	0.945887i	$0.394805\pi$
$930$	0	0
$931$	5.46633e6	0.206691
$932$	−4.40867e6	−0.166252
$933$	1.07529e7	0.404410
$934$	6.44739e6	0.241834
$935$	0	0
$936$	−1.24735e7	−0.465372
$937$	−2.85410e7	−1.06199	−0.530994	−	0.847375i	$-0.678181\pi$
$937$	−2.85410e7	−1.06199	−0.530994	+	0.847375i	$0.678181\pi$
$938$	−2.21557e7	−0.822203
$939$	2.18948e7	0.810356
$940$	0	0
$941$	−1.24597e7	−0.458703	−0.229352	−	0.973344i	$-0.573661\pi$
$941$	−1.24597e7	−0.458703	−0.229352	+	0.973344i	$0.573661\pi$
$942$	−8.33936e6	−0.306200
$943$	4.62358e6	0.169317
$944$	−5.19276e6	−0.189657
$945$	0	0
$946$	6.32289e6	0.229714
$947$	1.05265e7	0.381424	0.190712	−	0.981646i	$-0.438920\pi$
$947$	1.05265e7	0.381424	0.190712	+	0.981646i	$0.438920\pi$
$948$	−1.41185e7	−0.510234
$949$	−2.57919e7	−0.929646
$950$	0	0
$951$	8.38551e6	0.300662
$952$	−1.04833e7	−0.374893
$953$	3.58903e7	1.28010	0.640051	−	0.768333i	$-0.278913\pi$
$953$	3.58903e7	1.28010	0.640051	+	0.768333i	$0.278913\pi$
$954$	713044.	0.0253656
$955$	0	0
$956$	−3.35203e7	−1.18621
$957$	7.82426e6	0.276162
$958$	−2.03317e7	−0.715747
$959$	2.88655e7	1.01352
$960$	0	0
$961$	3.27828e7	1.14509
$962$	−2.98550e7	−1.04011
$963$	2.37870e6	0.0826559
$964$	−2.17926e7	−0.755296
$965$	0	0
$966$	−4.98027e6	−0.171716
$967$	2.57707e7	0.886259	0.443129	−	0.896458i	$-0.353868\pi$
$967$	2.57707e7	0.886259	0.443129	+	0.896458i	$0.353868\pi$
$968$	2.55327e6	0.0875808
$969$	−7.44117e6	−0.254585
$970$	0	0
$971$	1.07132e7	0.364645	0.182322	−	0.983239i	$-0.441639\pi$
$971$	1.07132e7	0.364645	0.182322	+	0.983239i	$0.441639\pi$
$972$	−1.25722e6	−0.0426821
$973$	1.93436e7	0.655022
$974$	−3.40378e6	−0.114965
$975$	0	0
$976$	233645.	0.00785113
$977$	−6.88736e6	−0.230843	−0.115421	−	0.993317i	$-0.536822\pi$
$977$	−6.88736e6	−0.230843	−0.115421	+	0.993317i	$0.536822\pi$
$978$	1.77198e7	0.592395
$979$	−1.55201e6	−0.0517533
$980$	0	0
$981$	8.32314e6	0.276131
$982$	2.35891e7	0.780607
$983$	1.49277e7	0.492730	0.246365	−	0.969177i	$-0.420764\pi$
$983$	1.49277e7	0.492730	0.246365	+	0.969177i	$0.420764\pi$
$984$	−4.96217e6	−0.163374
$985$	0	0
$986$	1.22236e7	0.400410
$987$	−1.12119e7	−0.366343
$988$	−2.98996e7	−0.974480
$989$	−2.33526e7	−0.759180
$990$	0	0
$991$	3.68190e7	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$991$	3.68190e7	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$992$	−4.65694e7	−1.50253
$993$	2.00988e7	0.646842
$994$	−2.16047e7	−0.693557
$995$	0	0
$996$	7.48935e6	0.239219
$997$	2.71677e7	0.865595	0.432798	−	0.901491i	$-0.357526\pi$
$997$	2.71677e7	0.865595	0.432798	+	0.901491i	$0.357526\pi$
$998$	9.53861e6	0.303151
$999$	−7.53175e6	−0.238771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.r.1.4	✓	9
5.4	even	2		825.6.a.s.1.6	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.4	✓	9	1.1	even	1	trivial
825.6.a.s.1.6	yes	9	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-3.27244 q^{2} +9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} -115.628 q^{7} +174.392 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.27244 q^{2} +9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} -115.628 q^{7} +174.392 q^{8} +81.0000 q^{9} -121.000 q^{11} -191.620 q^{12} -883.035 q^{13} +378.385 q^{14} +110.630 q^{16} +519.889 q^{17} -265.068 q^{18} -1590.33 q^{19} -1040.65 q^{21} +395.965 q^{22} -1462.43 q^{23} +1569.53 q^{24} +2889.68 q^{26} +729.000 q^{27} +2461.85 q^{28} -7184.81 q^{29} +7836.58 q^{31} -5942.57 q^{32} -1089.00 q^{33} -1701.31 q^{34} -1724.58 q^{36} -10331.6 q^{37} +5204.27 q^{38} -7947.32 q^{39} -3161.57 q^{41} +3405.46 q^{42} +15968.3 q^{43} +2576.23 q^{44} +4785.73 q^{46} +10774.0 q^{47} +995.666 q^{48} -3437.22 q^{49} +4679.00 q^{51} +18800.8 q^{52} -2690.05 q^{53} -2385.61 q^{54} -20164.6 q^{56} -14313.0 q^{57} +23511.9 q^{58} -46938.2 q^{59} +2111.96 q^{61} -25644.7 q^{62} -9365.85 q^{63} +15906.6 q^{64} +3563.69 q^{66} -58553.4 q^{67} -11069.0 q^{68} -13161.9 q^{69} -57097.1 q^{71} +14125.8 q^{72} +29208.2 q^{73} +33809.6 q^{74} +33860.0 q^{76} +13991.0 q^{77} +26007.1 q^{78} +73679.8 q^{79} +6561.00 q^{81} +10346.0 q^{82} -39084.3 q^{83} +22156.6 q^{84} -52255.3 q^{86} -64663.3 q^{87} -21101.4 q^{88} +12826.5 q^{89} +102103. q^{91} +31136.9 q^{92} +70529.2 q^{93} -35257.2 q^{94} -53483.2 q^{96} -23693.2 q^{97} +11248.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9}+O(q^{10}) \) 9 * q - q^2 + 81 * q^3 + 171 * q^4 - 9 * q^6 + 57 * q^7 + 177 * q^8 + 729 * q^9	\( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9} - 1089 q^{11} + 1539 q^{12} - 723 q^{13} + 720 q^{14} + 4147 q^{16} + 2804 q^{17} - 81 q^{18} - 1601 q^{19} + 513 q^{21} + 121 q^{22} + 2392 q^{23} + 1593 q^{24} - 1987 q^{26} + 6561 q^{27} - 4436 q^{28} + 5966 q^{29} + 21575 q^{31} + 25493 q^{32} - 9801 q^{33} - 3098 q^{34} + 13851 q^{36} + 10228 q^{37} - 12765 q^{38} - 6507 q^{39} + 13304 q^{41} + 6480 q^{42} + 13829 q^{43} - 20691 q^{44} + 81283 q^{46} + 13998 q^{47} + 37323 q^{48} - 19468 q^{49} + 25236 q^{51} - 37131 q^{52} + 44166 q^{53} - 729 q^{54} + 79160 q^{56} - 14409 q^{57} + 7635 q^{58} + 11626 q^{59} + 49481 q^{61} + 94479 q^{62} + 4617 q^{63} + 109367 q^{64} + 1089 q^{66} - 26567 q^{67} + 119506 q^{68} + 21528 q^{69} + 78454 q^{71} + 14337 q^{72} + 100086 q^{73} + 162360 q^{74} + 194115 q^{76} - 6897 q^{77} - 17883 q^{78} - 8478 q^{79} + 59049 q^{81} + 52700 q^{82} + 157476 q^{83} - 39924 q^{84} + 251663 q^{86} + 53694 q^{87} - 21417 q^{88} - 65548 q^{89} - 106849 q^{91} + 350115 q^{92} + 194175 q^{93} - 35742 q^{94} + 229437 q^{96} - 116757 q^{97} + 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q - q^2 + 81 * q^3 + 171 * q^4 - 9 * q^6 + 57 * q^7 + 177 * q^8 + 729 * q^9 - 1089 * q^11 + 1539 * q^12 - 723 * q^13 + 720 * q^14 + 4147 * q^16 + 2804 * q^17 - 81 * q^18 - 1601 * q^19 + 513 * q^21 + 121 * q^22 + 2392 * q^23 + 1593 * q^24 - 1987 * q^26 + 6561 * q^27 - 4436 * q^28 + 5966 * q^29 + 21575 * q^31 + 25493 * q^32 - 9801 * q^33 - 3098 * q^34 + 13851 * q^36 + 10228 * q^37 - 12765 * q^38 - 6507 * q^39 + 13304 * q^41 + 6480 * q^42 + 13829 * q^43 - 20691 * q^44 + 81283 * q^46 + 13998 * q^47 + 37323 * q^48 - 19468 * q^49 + 25236 * q^51 - 37131 * q^52 + 44166 * q^53 - 729 * q^54 + 79160 * q^56 - 14409 * q^57 + 7635 * q^58 + 11626 * q^59 + 49481 * q^61 + 94479 * q^62 + 4617 * q^63 + 109367 * q^64 + 1089 * q^66 - 26567 * q^67 + 119506 * q^68 + 21528 * q^69 + 78454 * q^71 + 14337 * q^72 + 100086 * q^73 + 162360 * q^74 + 194115 * q^76 - 6897 * q^77 - 17883 * q^78 - 8478 * q^79 + 59049 * q^81 + 52700 * q^82 + 157476 * q^83 - 39924 * q^84 + 251663 * q^86 + 53694 * q^87 - 21417 * q^88 - 65548 * q^89 - 106849 * q^91 + 350115 * q^92 + 194175 * q^93 - 35742 * q^94 + 229437 * q^96 - 116757 * q^97 + 14949 * q^98 - 88209 * q^99

Introduction

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