LMFDB - Embedded newform 825.6.a.r.1.9

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			1.9
Root			$-11.1064$ 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)

$f(q)$	$=$	$q+11.1064 q^{2} +9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} +76.1461 q^{7} +659.202 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+11.1064 q^{2} +9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} +76.1461 q^{7} +659.202 q^{8} +81.0000 q^{9} -121.000 q^{11} +822.178 q^{12} -470.783 q^{13} +845.713 q^{14} +4398.09 q^{16} -494.756 q^{17} +899.622 q^{18} -44.8076 q^{19} +685.315 q^{21} -1343.88 q^{22} +4242.60 q^{23} +5932.82 q^{24} -5228.73 q^{26} +729.000 q^{27} +6956.19 q^{28} -786.920 q^{29} +5594.34 q^{31} +27752.7 q^{32} -1089.00 q^{33} -5494.98 q^{34} +7399.60 q^{36} +6453.89 q^{37} -497.653 q^{38} -4237.05 q^{39} +7703.22 q^{41} +7611.42 q^{42} -2425.31 q^{43} -11053.7 q^{44} +47120.2 q^{46} +18816.2 q^{47} +39582.8 q^{48} -11008.8 q^{49} -4452.80 q^{51} -43007.5 q^{52} -34459.8 q^{53} +8096.60 q^{54} +50195.7 q^{56} -403.268 q^{57} -8739.88 q^{58} +27937.1 q^{59} +14661.0 q^{61} +62133.2 q^{62} +6167.84 q^{63} +167495. q^{64} -12094.9 q^{66} -47205.0 q^{67} -45197.5 q^{68} +38183.4 q^{69} -39501.5 q^{71} +53395.4 q^{72} +26993.8 q^{73} +71679.8 q^{74} -4093.31 q^{76} -9213.68 q^{77} -47058.6 q^{78} -12293.2 q^{79} +6561.00 q^{81} +85555.4 q^{82} -70308.6 q^{83} +62605.7 q^{84} -26936.6 q^{86} -7082.28 q^{87} -79763.4 q^{88} -143142. q^{89} -35848.3 q^{91} +387574. q^{92} +50349.1 q^{93} +208981. q^{94} +249774. q^{96} +91322.4 q^{97} -122268. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	11.1064	1.96336	0.981680	−	0.190536i	$-0.0610225\pi$
$2$	11.1064	1.96336	0.981680	+	0.190536i	$0.0610225\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	91.3531	2.85478
$5$	0	0
$5$	0	0
$6$	99.9580	1.13355
$7$	76.1461	0.587358	0.293679	−	0.955904i	$-0.405120\pi$
$7$	76.1461	0.587358	0.293679	+	0.955904i	$0.405120\pi$
$8$	659.202	3.64161
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	822.178	1.64821
$13$	−470.783	−0.772614	−0.386307	−	0.922370i	$-0.626250\pi$
$13$	−470.783	−0.772614	−0.386307	+	0.922370i	$0.626250\pi$
$14$	845.713	1.15320
$15$	0	0
$16$	4398.09	4.29501
$17$	−494.756	−0.415211	−0.207605	−	0.978213i	$-0.566567\pi$
$17$	−494.756	−0.415211	−0.207605	+	0.978213i	$0.566567\pi$
$18$	899.622	0.654454
$19$	−44.8076	−0.0284752	−0.0142376	−	0.999899i	$-0.504532\pi$
$19$	−44.8076	−0.0284752	−0.0142376	+	0.999899i	$0.504532\pi$
$20$	0	0
$21$	685.315	0.339111
$22$	−1343.88	−0.591975
$23$	4242.60	1.67229	0.836146	−	0.548507i	$-0.184804\pi$
$23$	4242.60	1.67229	0.836146	+	0.548507i	$0.184804\pi$
$24$	5932.82	2.10249
$25$	0	0
$26$	−5228.73	−1.51692
$27$	729.000	0.192450
$28$	6956.19	1.67678
$29$	−786.920	−0.173754	−0.0868772	−	0.996219i	$-0.527689\pi$
$29$	−786.920	−0.173754	−0.0868772	+	0.996219i	$0.527689\pi$
$30$	0	0
$31$	5594.34	1.04555	0.522775	−	0.852471i	$-0.324897\pi$
$31$	5594.34	1.04555	0.522775	+	0.852471i	$0.324897\pi$
$32$	27752.7	4.79104
$33$	−1089.00	−0.174078
$34$	−5494.98	−0.815209
$35$	0	0
$36$	7399.60	0.951595
$37$	6453.89	0.775028	0.387514	−	0.921864i	$-0.373334\pi$
$37$	6453.89	0.775028	0.387514	+	0.921864i	$0.373334\pi$
$38$	−497.653	−0.0559072
$39$	−4237.05	−0.446069
$40$	0	0
$41$	7703.22	0.715670	0.357835	−	0.933785i	$-0.383515\pi$
$41$	7703.22	0.715670	0.357835	+	0.933785i	$0.383515\pi$
$42$	7611.42	0.665798
$43$	−2425.31	−0.200031	−0.100015	−	0.994986i	$-0.531889\pi$
$43$	−2425.31	−0.200031	−0.100015	+	0.994986i	$0.531889\pi$
$44$	−11053.7	−0.860750
$45$	0	0
$46$	47120.2	3.28331
$47$	18816.2	1.24247	0.621237	−	0.783622i	$-0.286630\pi$
$47$	18816.2	1.24247	0.621237	+	0.783622i	$0.286630\pi$
$48$	39582.8	2.47973
$49$	−11008.8	−0.655011
$50$	0	0
$51$	−4452.80	−0.239722
$52$	−43007.5	−2.20565
$53$	−34459.8	−1.68509	−0.842545	−	0.538626i	$-0.818944\pi$
$53$	−34459.8	−1.68509	−0.842545	+	0.538626i	$0.818944\pi$
$54$	8096.60	0.377849
$55$	0	0
$56$	50195.7	2.13893
$57$	−403.268	−0.0164402
$58$	−8739.88	−0.341142
$59$	27937.1	1.04485	0.522423	−	0.852687i	$-0.325028\pi$
$59$	27937.1	1.04485	0.522423	+	0.852687i	$0.325028\pi$
$60$	0	0
$61$	14661.0	0.504475	0.252237	−	0.967665i	$-0.418834\pi$
$61$	14661.0	0.504475	0.252237	+	0.967665i	$0.418834\pi$
$62$	62133.2	2.05279
$63$	6167.84	0.195786
$64$	167495.	5.11154
$65$	0	0
$66$	−12094.9	−0.341777
$67$	−47205.0	−1.28470	−0.642349	−	0.766412i	$-0.722040\pi$
$67$	−47205.0	−1.28470	−0.642349	+	0.766412i	$0.722040\pi$
$68$	−45197.5	−1.18534
$69$	38183.4	0.965498
$70$	0	0
$71$	−39501.5	−0.929967	−0.464983	−	0.885319i	$-0.653940\pi$
$71$	−39501.5	−0.929967	−0.464983	+	0.885319i	$0.653940\pi$
$72$	53395.4	1.21387
$73$	26993.8	0.592867	0.296433	−	0.955054i	$-0.404203\pi$
$73$	26993.8	0.592867	0.296433	+	0.955054i	$0.404203\pi$
$74$	71679.8	1.52166
$75$	0	0
$76$	−4093.31	−0.0812907
$77$	−9213.68	−0.177095
$78$	−47058.6	−0.875795
$79$	−12293.2	−0.221614	−0.110807	−	0.993842i	$-0.535344\pi$
$79$	−12293.2	−0.221614	−0.110807	+	0.993842i	$0.535344\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	85555.4	1.40512
$83$	−70308.6	−1.12025	−0.560123	−	0.828409i	$-0.689246\pi$
$83$	−70308.6	−1.12025	−0.560123	+	0.828409i	$0.689246\pi$
$84$	62605.7	0.968090
$85$	0	0
$86$	−26936.6	−0.392733
$87$	−7082.28	−0.100317
$88$	−79763.4	−1.09799
$89$	−143142.	−1.91554	−0.957769	−	0.287539i	$-0.907163\pi$
$89$	−143142.	−1.91554	−0.957769	+	0.287539i	$0.907163\pi$
$90$	0	0
$91$	−35848.3	−0.453801
$92$	387574.	4.77403
$93$	50349.1	0.603648
$94$	208981.	2.43943
$95$	0	0
$96$	249774.	2.76611
$97$	91322.4	0.985480	0.492740	−	0.870177i	$-0.335995\pi$
$97$	91322.4	0.985480	0.492740	+	0.870177i	$0.335995\pi$
$98$	−122268.	−1.28602
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−141045.	−1.37580	−0.687899	−	0.725806i	$-0.741467\pi$
$101$	−141045.	−1.37580	−0.687899	+	0.725806i	$0.741467\pi$
$102$	−49454.8	−0.470661
$103$	−139369.	−1.29442	−0.647208	−	0.762314i	$-0.724063\pi$
$103$	−139369.	−1.29442	−0.647208	+	0.762314i	$0.724063\pi$
$104$	−310341.	−2.81356
$105$	0	0
$106$	−382726.	−3.30844
$107$	−189660.	−1.60146	−0.800729	−	0.599027i	$-0.795554\pi$
$107$	−189660.	−1.60146	−0.800729	+	0.599027i	$0.795554\pi$
$108$	66596.4	0.549404
$109$	−61287.5	−0.494089	−0.247045	−	0.969004i	$-0.579459\pi$
$109$	−61287.5	−0.494089	−0.247045	+	0.969004i	$0.579459\pi$
$110$	0	0
$111$	58085.0	0.447462
$112$	334898.	2.52271
$113$	243824.	1.79631	0.898153	−	0.439683i	$-0.144909\pi$
$113$	243824.	1.79631	0.898153	+	0.439683i	$0.144909\pi$
$114$	−4478.88	−0.0322780
$115$	0	0
$116$	−71887.6	−0.496031
$117$	−38133.5	−0.257538
$118$	310282.	2.05141
$119$	−37673.8	−0.243877
$120$	0	0
$121$	14641.0	0.0909091
$122$	162832.	0.990466
$123$	69329.0	0.413192
$124$	511060.	2.98482
$125$	0	0
$126$	68502.7	0.384398
$127$	101881.	0.560512	0.280256	−	0.959925i	$-0.409581\pi$
$127$	101881.	0.560512	0.280256	+	0.959925i	$0.409581\pi$
$128$	972186.	5.24474
$129$	−21827.8	−0.115488
$130$	0	0
$131$	−235642.	−1.19971	−0.599853	−	0.800110i	$-0.704774\pi$
$131$	−235642.	−1.19971	−0.599853	+	0.800110i	$0.704774\pi$
$132$	−99483.5	−0.496954
$133$	−3411.92	−0.0167252
$134$	−524280.	−2.52233
$135$	0	0
$136$	−326144.	−1.51204
$137$	180612.	0.822139	0.411070	−	0.911604i	$-0.365155\pi$
$137$	180612.	0.822139	0.411070	+	0.911604i	$0.365155\pi$
$138$	424081.	1.89562
$139$	307919.	1.35176	0.675879	−	0.737013i	$-0.263764\pi$
$139$	307919.	1.35176	0.675879	+	0.737013i	$0.263764\pi$
$140$	0	0
$141$	169346.	0.717343
$142$	−438721.	−1.82586
$143$	56964.8	0.232952
$144$	356245.	1.43167
$145$	0	0
$146$	299805.	1.16401
$147$	−99078.9	−0.378171
$148$	589583.	2.21254
$149$	325175.	1.19992	0.599959	−	0.800030i	$-0.295183\pi$
$149$	325175.	1.19992	0.599959	+	0.800030i	$0.295183\pi$
$150$	0	0
$151$	44535.7	0.158952	0.0794761	−	0.996837i	$-0.474675\pi$
$151$	44535.7	0.158952	0.0794761	+	0.996837i	$0.474675\pi$
$152$	−29537.2	−0.103696
$153$	−40075.2	−0.138404
$154$	−102331.	−0.347701
$155$	0	0
$156$	−387068.	−1.27343
$157$	−354519.	−1.14786	−0.573932	−	0.818903i	$-0.694583\pi$
$157$	−354519.	−1.14786	−0.573932	+	0.818903i	$0.694583\pi$
$158$	−136534.	−0.435108
$159$	−310138.	−0.972887
$160$	0	0
$161$	323057.	0.982234
$162$	72869.4	0.218151
$163$	−97011.8	−0.285993	−0.142996	−	0.989723i	$-0.545674\pi$
$163$	−97011.8	−0.285993	−0.142996	+	0.989723i	$0.545674\pi$
$164$	703713.	2.04308
$165$	0	0
$166$	−780879.	−2.19945
$167$	385119.	1.06857	0.534287	−	0.845303i	$-0.320580\pi$
$167$	385119.	1.06857	0.534287	+	0.845303i	$0.320580\pi$
$168$	451761.	1.23491
$169$	−149656.	−0.403067
$170$	0	0
$171$	−3629.41	−0.00949175
$172$	−221560.	−0.571045
$173$	544249.	1.38255	0.691277	−	0.722590i	$-0.257048\pi$
$173$	544249.	1.38255	0.691277	+	0.722590i	$0.257048\pi$
$174$	−78659.0	−0.196959
$175$	0	0
$176$	−532169.	−1.29499
$177$	251434.	0.603242
$178$	−1.58979e6	−3.76089
$179$	−265639.	−0.619668	−0.309834	−	0.950791i	$-0.600273\pi$
$179$	−265639.	−0.619668	−0.309834	+	0.950791i	$0.600273\pi$
$180$	0	0
$181$	−748969.	−1.69929	−0.849645	−	0.527355i	$-0.823184\pi$
$181$	−748969.	−1.69929	−0.849645	+	0.527355i	$0.823184\pi$
$182$	−398148.	−0.890975
$183$	131949.	0.291259
$184$	2.79673e6	6.08984
$185$	0	0
$186$	559199.	1.18518
$187$	59865.5	0.125191
$188$	1.71892e6	3.54700
$189$	55510.5	0.113037
$190$	0	0
$191$	−964220.	−1.91246	−0.956231	−	0.292613i	$-0.905475\pi$
$191$	−964220.	−1.91246	−0.956231	+	0.292613i	$0.905475\pi$
$192$	1.50745e6	2.95115
$193$	−167314.	−0.323325	−0.161662	−	0.986846i	$-0.551686\pi$
$193$	−167314.	−0.323325	−0.161662	+	0.986846i	$0.551686\pi$
$194$	1.01427e6	1.93485
$195$	0	0
$196$	−1.00568e6	−1.86991
$197$	−99359.5	−0.182408	−0.0912040	−	0.995832i	$-0.529072\pi$
$197$	−99359.5	−0.182408	−0.0912040	+	0.995832i	$0.529072\pi$
$198$	−108854.	−0.197325
$199$	−56809.9	−0.101693	−0.0508466	−	0.998706i	$-0.516192\pi$
$199$	−56809.9	−0.101693	−0.0508466	+	0.998706i	$0.516192\pi$
$200$	0	0
$201$	−424845.	−0.741721
$202$	−1.56651e6	−2.70119
$203$	−59920.9	−0.102056
$204$	−406778.	−0.684355
$205$	0	0
$206$	−1.54790e6	−2.54140
$207$	343650.	0.557431
$208$	−2.07055e6	−3.31839
$209$	5421.72	0.00858561
$210$	0	0
$211$	529807.	0.819241	0.409620	−	0.912256i	$-0.365661\pi$
$211$	529807.	0.819241	0.409620	+	0.912256i	$0.365661\pi$
$212$	−3.14801e6	−4.81057
$213$	−355513.	−0.536916
$214$	−2.10645e6	−3.14424
$215$	0	0
$216$	480558.	0.700828
$217$	425987.	0.614112
$218$	−680686.	−0.970076
$219$	242944.	0.342292
$220$	0	0
$221$	232923.	0.320798
$222$	645118.	0.878530
$223$	−757914.	−1.02061	−0.510303	−	0.859995i	$-0.670467\pi$
$223$	−757914.	−1.02061	−0.510303	+	0.859995i	$0.670467\pi$
$224$	2.11326e6	2.81406
$225$	0	0
$226$	2.70802e6	3.52680
$227$	−794647.	−1.02355	−0.511776	−	0.859119i	$-0.671012\pi$
$227$	−794647.	−1.02355	−0.511776	+	0.859119i	$0.671012\pi$
$228$	−36839.8	−0.0469332
$229$	−191327.	−0.241095	−0.120548	−	0.992708i	$-0.538465\pi$
$229$	−191327.	−0.241095	−0.120548	+	0.992708i	$0.538465\pi$
$230$	0	0
$231$	−82923.2	−0.102246
$232$	−518739.	−0.632746
$233$	845860.	1.02072	0.510362	−	0.859960i	$-0.329511\pi$
$233$	845860.	1.02072	0.510362	+	0.859960i	$0.329511\pi$
$234$	−423527.	−0.505640
$235$	0	0
$236$	2.55214e6	2.98281
$237$	−110639.	−0.127949
$238$	−418422.	−0.478819
$239$	1.41892e6	1.60681	0.803403	−	0.595435i	$-0.203020\pi$
$239$	1.41892e6	1.60681	0.803403	+	0.595435i	$0.203020\pi$
$240$	0	0
$241$	−902862.	−1.00133	−0.500667	−	0.865640i	$-0.666912\pi$
$241$	−902862.	−1.00133	−0.500667	+	0.865640i	$0.666912\pi$
$242$	162609.	0.178487
$243$	59049.0	0.0641500
$244$	1.33933e6	1.44017
$245$	0	0
$246$	769999.	0.811245
$247$	21094.7	0.0220004
$248$	3.68780e6	3.80748
$249$	−632778.	−0.646774
$250$	0	0
$251$	−58009.8	−0.0581188	−0.0290594	−	0.999578i	$-0.509251\pi$
$251$	−58009.8	−0.0581188	−0.0290594	+	0.999578i	$0.509251\pi$
$252$	563451.	0.558927
$253$	−513354.	−0.504215
$254$	1.13154e6	1.10049
$255$	0	0
$256$	5.43769e6	5.18579
$257$	661274.	0.624523	0.312262	−	0.949996i	$-0.398913\pi$
$257$	661274.	0.624523	0.312262	+	0.949996i	$0.398913\pi$
$258$	−242430.	−0.226744
$259$	491439.	0.455219
$260$	0	0
$261$	−63740.5	−0.0579181
$262$	−2.61715e6	−2.35546
$263$	−1.09488e6	−0.976058	−0.488029	−	0.872827i	$-0.662284\pi$
$263$	−1.09488e6	−0.976058	−0.488029	+	0.872827i	$0.662284\pi$
$264$	−717871.	−0.633923
$265$	0	0
$266$	−37894.3	−0.0328375
$267$	−1.28827e6	−1.10594
$268$	−4.31232e6	−3.66754
$269$	−206823.	−0.174268	−0.0871341	−	0.996197i	$-0.527771\pi$
$269$	−206823.	−0.174268	−0.0871341	+	0.996197i	$0.527771\pi$
$270$	0	0
$271$	−298237.	−0.246683	−0.123341	−	0.992364i	$-0.539361\pi$
$271$	−298237.	−0.246683	−0.123341	+	0.992364i	$0.539361\pi$
$272$	−2.17598e6	−1.78334
$273$	−322635.	−0.262002
$274$	2.00596e6	1.61416
$275$	0	0
$276$	3.48817e6	2.75629
$277$	1.71987e6	1.34678	0.673390	−	0.739287i	$-0.264837\pi$
$277$	1.71987e6	1.34678	0.673390	+	0.739287i	$0.264837\pi$
$278$	3.41988e6	2.65399
$279$	453141.	0.348516
$280$	0	0
$281$	2.04137e6	1.54225	0.771126	−	0.636683i	$-0.219694\pi$
$281$	2.04137e6	1.54225	0.771126	+	0.636683i	$0.219694\pi$
$282$	1.88083e6	1.40840
$283$	−1.62392e6	−1.20531	−0.602656	−	0.798001i	$-0.705891\pi$
$283$	−1.62392e6	−1.20531	−0.602656	+	0.798001i	$0.705891\pi$
$284$	−3.60858e6	−2.65485
$285$	0	0
$286$	632676.	0.457369
$287$	586571.	0.420354
$288$	2.24797e6	1.59701
$289$	−1.17507e6	−0.827600
$290$	0	0
$291$	821901.	0.568967
$292$	2.46597e6	1.69251
$293$	−2.37278e6	−1.61469	−0.807344	−	0.590082i	$-0.799096\pi$
$293$	−2.37278e6	−1.61469	−0.807344	+	0.590082i	$0.799096\pi$
$294$	−1.10041e6	−0.742485
$295$	0	0
$296$	4.25442e6	2.82235
$297$	−88209.0	−0.0580259
$298$	3.61154e6	2.35587
$299$	−1.99734e6	−1.29204
$300$	0	0
$301$	−184678.	−0.117490
$302$	494634.	0.312080
$303$	−1.26941e6	−0.794318
$304$	−197068.	−0.122301
$305$	0	0
$306$	−445093.	−0.271736
$307$	−1.29238e6	−0.782608	−0.391304	−	0.920261i	$-0.627976\pi$
$307$	−1.29238e6	−0.782608	−0.391304	+	0.920261i	$0.627976\pi$
$308$	−841699.	−0.505568
$309$	−1.25432e6	−0.747331
$310$	0	0
$311$	−1.55279e6	−0.910357	−0.455179	−	0.890400i	$-0.650425\pi$
$311$	−1.55279e6	−0.910357	−0.455179	+	0.890400i	$0.650425\pi$
$312$	−2.79307e6	−1.62441
$313$	2.14996e6	1.24042	0.620210	−	0.784436i	$-0.287047\pi$
$313$	2.14996e6	1.24042	0.620210	+	0.784436i	$0.287047\pi$
$314$	−3.93745e6	−2.25367
$315$	0	0
$316$	−1.12302e6	−0.632660
$317$	−1.34306e6	−0.750669	−0.375335	−	0.926889i	$-0.622472\pi$
$317$	−1.34306e6	−0.750669	−0.375335	+	0.926889i	$0.622472\pi$
$318$	−3.44453e6	−1.91013
$319$	95217.3	0.0523889
$320$	0	0
$321$	−1.70694e6	−0.924602
$322$	3.58802e6	1.92848
$323$	22168.8	0.0118232
$324$	599368.	0.317198
$325$	0	0
$326$	−1.07746e6	−0.561507
$327$	−551587.	−0.285263
$328$	5.07798e6	2.60619
$329$	1.43278e6	0.729777
$330$	0	0
$331$	1.24921e6	0.626706	0.313353	−	0.949637i	$-0.398548\pi$
$331$	1.24921e6	0.626706	0.313353	+	0.949637i	$0.398548\pi$
$332$	−6.42291e6	−3.19806
$333$	522765.	0.258343
$334$	4.27731e6	2.09799
$335$	0	0
$336$	3.01408e6	1.45649
$337$	3.36237e6	1.61277	0.806383	−	0.591394i	$-0.201422\pi$
$337$	3.36237e6	1.61277	0.806383	+	0.591394i	$0.201422\pi$
$338$	−1.66215e6	−0.791366
$339$	2.19442e6	1.03710
$340$	0	0
$341$	−676915.	−0.315245
$342$	−40309.9	−0.0186357
$343$	−2.11806e6	−0.972084
$344$	−1.59877e6	−0.728435
$345$	0	0
$346$	6.04467e6	2.71445
$347$	3.03255e6	1.35202	0.676012	−	0.736890i	$-0.263707\pi$
$347$	3.03255e6	1.35202	0.676012	+	0.736890i	$0.263707\pi$
$348$	−646988.	−0.286384
$349$	690808.	0.303595	0.151797	−	0.988412i	$-0.451494\pi$
$349$	690808.	0.303595	0.151797	+	0.988412i	$0.451494\pi$
$350$	0	0
$351$	−343201.	−0.148690
$352$	−3.35808e6	−1.44455
$353$	−3.16374e6	−1.35134	−0.675669	−	0.737205i	$-0.736145\pi$
$353$	−3.16374e6	−1.35134	−0.675669	+	0.737205i	$0.736145\pi$
$354$	2.79254e6	1.18438
$355$	0	0
$356$	−1.30764e7	−5.46845
$357$	−339064.	−0.140803
$358$	−2.95030e6	−1.21663
$359$	−392579.	−0.160765	−0.0803825	−	0.996764i	$-0.525614\pi$
$359$	−392579.	−0.160765	−0.0803825	+	0.996764i	$0.525614\pi$
$360$	0	0
$361$	−2.47409e6	−0.999189
$362$	−8.31839e6	−3.33632
$363$	131769.	0.0524864
$364$	−3.27486e6	−1.29550
$365$	0	0
$366$	1.46549e6	0.571846
$367$	−2.92576e6	−1.13390	−0.566949	−	0.823753i	$-0.691876\pi$
$367$	−2.92576e6	−1.13390	−0.566949	+	0.823753i	$0.691876\pi$
$368$	1.86593e7	7.18251
$369$	623961.	0.238557
$370$	0	0
$371$	−2.62398e6	−0.989751
$372$	4.59954e6	1.72329
$373$	−3.29066e6	−1.22465	−0.612323	−	0.790607i	$-0.709765\pi$
$373$	−3.29066e6	−1.22465	−0.612323	+	0.790607i	$0.709765\pi$
$374$	664893.	0.245795
$375$	0	0
$376$	1.24037e7	4.52461
$377$	370469.	0.134245
$378$	616525.	0.221933
$379$	2.02091e6	0.722683	0.361342	−	0.932434i	$-0.382319\pi$
$379$	2.02091e6	0.722683	0.361342	+	0.932434i	$0.382319\pi$
$380$	0	0
$381$	916932.	0.323612
$382$	−1.07091e7	−3.75485
$383$	−1.49093e6	−0.519349	−0.259674	−	0.965696i	$-0.583615\pi$
$383$	−1.49093e6	−0.519349	−0.259674	+	0.965696i	$0.583615\pi$
$384$	8.74967e6	3.02805
$385$	0	0
$386$	−1.85826e6	−0.634803
$387$	−196450.	−0.0666769
$388$	8.34258e6	2.81333
$389$	4.98939e6	1.67176	0.835878	−	0.548915i	$-0.184959\pi$
$389$	4.98939e6	1.67176	0.835878	+	0.548915i	$0.184959\pi$
$390$	0	0
$391$	−2.09905e6	−0.694354
$392$	−7.25700e6	−2.38529
$393$	−2.12078e6	−0.692651
$394$	−1.10353e6	−0.358133
$395$	0	0
$396$	−895352.	−0.286917
$397$	−2.73054e6	−0.869504	−0.434752	−	0.900550i	$-0.643164\pi$
$397$	−2.73054e6	−0.869504	−0.434752	+	0.900550i	$0.643164\pi$
$398$	−630956.	−0.199660
$399$	−30707.3	−0.00965628
$400$	0	0
$401$	−2.44533e6	−0.759409	−0.379705	−	0.925108i	$-0.623974\pi$
$401$	−2.44533e6	−0.759409	−0.379705	+	0.925108i	$0.623974\pi$
$402$	−4.71852e6	−1.45627
$403$	−2.63372e6	−0.807807
$404$	−1.28849e7	−3.92761
$405$	0	0
$406$	−665509.	−0.200373
$407$	−780921.	−0.233680
$408$	−2.93530e6	−0.872975
$409$	5.04690e6	1.49182	0.745909	−	0.666047i	$-0.232015\pi$
$409$	5.04690e6	1.49182	0.745909	+	0.666047i	$0.232015\pi$
$410$	0	0
$411$	1.62551e6	0.474662
$412$	−1.27318e7	−3.69528
$413$	2.12731e6	0.613698
$414$	3.81673e6	1.09444
$415$	0	0
$416$	−1.30655e7	−3.70163
$417$	2.77127e6	0.780438
$418$	60216.0	0.0168566
$419$	−30652.8	−0.00852971	−0.00426486	−	0.999991i	$-0.501358\pi$
$419$	−30652.8	−0.00852971	−0.00426486	+	0.999991i	$0.501358\pi$
$420$	0	0
$421$	2.31067e6	0.635379	0.317689	−	0.948195i	$-0.397093\pi$
$421$	2.31067e6	0.635379	0.317689	+	0.948195i	$0.397093\pi$
$422$	5.88427e6	1.60847
$423$	1.52411e6	0.414158
$424$	−2.27160e7	−6.13644
$425$	0	0
$426$	−3.94849e6	−1.05416
$427$	1.11638e6	0.296307
$428$	−1.73260e7	−4.57182
$429$	512683.	0.134495
$430$	0	0
$431$	7.38790e6	1.91570	0.957851	−	0.287266i	$-0.0927464\pi$
$431$	7.38790e6	1.91570	0.957851	+	0.287266i	$0.0927464\pi$
$432$	3.20621e6	0.826575
$433$	−3.90981e6	−1.00216	−0.501079	−	0.865402i	$-0.667063\pi$
$433$	−3.90981e6	−1.00216	−0.501079	+	0.865402i	$0.667063\pi$
$434$	4.73120e6	1.20572
$435$	0	0
$436$	−5.59880e6	−1.41052
$437$	−190100.	−0.0476189
$438$	2.69825e6	0.672042
$439$	2.57832e6	0.638522	0.319261	−	0.947667i	$-0.396565\pi$
$439$	2.57832e6	0.638522	0.319261	+	0.947667i	$0.396565\pi$
$440$	0	0
$441$	−891710.	−0.218337
$442$	2.58695e6	0.629842
$443$	−300253.	−0.0726905	−0.0363452	−	0.999339i	$-0.511572\pi$
$443$	−300253.	−0.0726905	−0.0363452	+	0.999339i	$0.511572\pi$
$444$	5.30625e6	1.27741
$445$	0	0
$446$	−8.41774e6	−2.00382
$447$	2.92658e6	0.692773
$448$	1.27541e7	3.00230
$449$	−3.40803e6	−0.797788	−0.398894	−	0.916997i	$-0.630606\pi$
$449$	−3.40803e6	−0.797788	−0.398894	+	0.916997i	$0.630606\pi$
$450$	0	0
$451$	−932090.	−0.215783
$452$	2.22741e7	5.12807
$453$	400822.	0.0917710
$454$	−8.82571e6	−2.00960
$455$	0	0
$456$	−265835.	−0.0598688
$457$	3.76619e6	0.843552	0.421776	−	0.906700i	$-0.361407\pi$
$457$	3.76619e6	0.843552	0.421776	+	0.906700i	$0.361407\pi$
$458$	−2.12497e6	−0.473356
$459$	−360677.	−0.0799074
$460$	0	0
$461$	−8.18562e6	−1.79390	−0.896952	−	0.442128i	$-0.854224\pi$
$461$	−8.18562e6	−1.79390	−0.896952	+	0.442128i	$0.854224\pi$
$462$	−920981.	−0.200746
$463$	950861.	0.206141	0.103071	−	0.994674i	$-0.467133\pi$
$463$	950861.	0.206141	0.103071	+	0.994674i	$0.467133\pi$
$464$	−3.46095e6	−0.746277
$465$	0	0
$466$	9.39449e6	2.00405
$467$	−3.61259e6	−0.766526	−0.383263	−	0.923639i	$-0.625200\pi$
$467$	−3.61259e6	−0.766526	−0.383263	+	0.923639i	$0.625200\pi$
$468$	−3.48361e6	−0.735216
$469$	−3.59448e6	−0.754577
$470$	0	0
$471$	−3.19067e6	−0.662720
$472$	1.84162e7	3.80492
$473$	293463.	0.0603116
$474$	−1.22880e6	−0.251210
$475$	0	0
$476$	−3.44162e6	−0.696218
$477$	−2.79124e6	−0.561697
$478$	1.57592e7	3.15474
$479$	8.09916e6	1.61288	0.806439	−	0.591318i	$-0.201392\pi$
$479$	8.09916e6	1.61288	0.806439	+	0.591318i	$0.201392\pi$
$480$	0	0
$481$	−3.03838e6	−0.598797
$482$	−1.00276e7	−1.96598
$483$	2.90752e6	0.567093
$484$	1.33750e6	0.259526
$485$	0	0
$486$	655824.	0.125950
$487$	2.22319e6	0.424771	0.212386	−	0.977186i	$-0.431877\pi$
$487$	2.22319e6	0.424771	0.212386	+	0.977186i	$0.431877\pi$
$488$	9.66457e6	1.83710
$489$	−873106.	−0.165118
$490$	0	0
$491$	1.80556e6	0.337993	0.168996	−	0.985617i	$-0.445947\pi$
$491$	1.80556e6	0.337993	0.168996	+	0.985617i	$0.445947\pi$
$492$	6.33342e6	1.17958
$493$	389333.	0.0721447
$494$	234287.	0.0431947
$495$	0	0
$496$	2.46044e7	4.49065
$497$	−3.00788e6	−0.546223
$498$	−7.02791e6	−1.26985
$499$	−3.16604e6	−0.569201	−0.284600	−	0.958646i	$-0.591861\pi$
$499$	−3.16604e6	−0.569201	−0.284600	+	0.958646i	$0.591861\pi$
$500$	0	0
$501$	3.46607e6	0.616941
$502$	−644282.	−0.114108
$503$	−4.86714e6	−0.857737	−0.428869	−	0.903367i	$-0.641088\pi$
$503$	−4.86714e6	−0.857737	−0.428869	+	0.903367i	$0.641088\pi$
$504$	4.06585e6	0.712976
$505$	0	0
$506$	−5.70154e6	−0.989956
$507$	−1.34690e6	−0.232711
$508$	9.30717e6	1.60014
$509$	4.22678e6	0.723128	0.361564	−	0.932347i	$-0.382243\pi$
$509$	4.22678e6	0.723128	0.361564	+	0.932347i	$0.382243\pi$
$510$	0	0
$511$	2.05547e6	0.348225
$512$	2.92835e7	4.93683
$513$	−32664.7	−0.00548006
$514$	7.34441e6	1.22616
$515$	0	0
$516$	−1.99404e6	−0.329693
$517$	−2.27676e6	−0.374620
$518$	5.45814e6	0.893758
$519$	4.89824e6	0.798218
$520$	0	0
$521$	1.11140e7	1.79380	0.896901	−	0.442232i	$-0.145813\pi$
$521$	1.11140e7	1.79380	0.896901	+	0.442232i	$0.145813\pi$
$522$	−707931.	−0.113714
$523$	532067.	0.0850574	0.0425287	−	0.999095i	$-0.486459\pi$
$523$	532067.	0.0850574	0.0425287	+	0.999095i	$0.486459\pi$
$524$	−2.15267e7	−3.42490
$525$	0	0
$526$	−1.21602e7	−1.91635
$527$	−2.76783e6	−0.434124
$528$	−4.78952e6	−0.747665
$529$	1.15633e7	1.79656
$530$	0	0
$531$	2.26291e6	0.348282
$532$	−311690.	−0.0477467
$533$	−3.62655e6	−0.552937
$534$	−1.43081e7	−2.17135
$535$	0	0
$536$	−3.11176e7	−4.67837
$537$	−2.39075e6	−0.357765
$538$	−2.29707e6	−0.342151
$539$	1.33206e6	0.197493
$540$	0	0
$541$	−9.77951e6	−1.43656	−0.718280	−	0.695754i	$-0.755070\pi$
$541$	−9.77951e6	−1.43656	−0.718280	+	0.695754i	$0.755070\pi$
$542$	−3.31235e6	−0.484327
$543$	−6.74072e6	−0.981086
$544$	−1.37308e7	−1.98929
$545$	0	0
$546$	−3.58333e6	−0.514405
$547$	8.61940e6	1.23171	0.615855	−	0.787859i	$-0.288811\pi$
$547$	8.61940e6	1.23171	0.615855	+	0.787859i	$0.288811\pi$
$548$	1.64995e7	2.34703
$549$	1.18754e6	0.168158
$550$	0	0
$551$	35260.0	0.00494770
$552$	2.51706e7	3.51597
$553$	−936079.	−0.130167
$554$	1.91017e7	2.64422
$555$	0	0
$556$	2.81293e7	3.85898
$557$	−1.37932e7	−1.88376	−0.941881	−	0.335946i	$-0.890944\pi$
$557$	−1.37932e7	−1.88376	−0.941881	+	0.335946i	$0.890944\pi$
$558$	5.03279e6	0.684263
$559$	1.14180e6	0.154547
$560$	0	0
$561$	538789.	0.0722789
$562$	2.26723e7	3.02800
$563$	−8.21998e6	−1.09295	−0.546474	−	0.837476i	$-0.684030\pi$
$563$	−8.21998e6	−1.09295	−0.546474	+	0.837476i	$0.684030\pi$
$564$	1.54703e7	2.04786
$565$	0	0
$566$	−1.80360e7	−2.36646
$567$	499595.	0.0652620
$568$	−2.60394e7	−3.38658
$569$	8.25501e6	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$569$	8.25501e6	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$570$	0	0
$571$	9.47816e6	1.21656	0.608280	−	0.793722i	$-0.291860\pi$
$571$	9.47816e6	1.21656	0.608280	+	0.793722i	$0.291860\pi$
$572$	5.20391e6	0.665028
$573$	−8.67798e6	−1.10416
$574$	6.51472e6	0.825307
$575$	0	0
$576$	1.35671e7	1.70385
$577$	5.03255e6	0.629287	0.314644	−	0.949210i	$-0.398115\pi$
$577$	5.03255e6	0.629287	0.314644	+	0.949210i	$0.398115\pi$
$578$	−1.30509e7	−1.62488
$579$	−1.50583e6	−0.186672
$580$	0	0
$581$	−5.35373e6	−0.657985
$582$	9.12840e6	1.11709
$583$	4.16964e6	0.508074
$584$	1.77944e7	2.15899
$585$	0	0
$586$	−2.63531e7	−3.17021
$587$	2.93358e6	0.351401	0.175700	−	0.984444i	$-0.443781\pi$
$587$	2.93358e6	0.351401	0.175700	+	0.984444i	$0.443781\pi$
$588$	−9.05116e6	−1.07960
$589$	−250669.	−0.0297723
$590$	0	0
$591$	−894236.	−0.105313
$592$	2.83848e7	3.32875
$593$	−1.08060e7	−1.26191	−0.630953	−	0.775821i	$-0.717336\pi$
$593$	−1.08060e7	−1.26191	−0.630953	+	0.775821i	$0.717336\pi$
$594$	−979688.	−0.113926
$595$	0	0
$596$	2.97058e7	3.42551
$597$	−511289.	−0.0587125
$598$	−2.21834e7	−2.53673
$599$	3.28617e6	0.374217	0.187108	−	0.982339i	$-0.440088\pi$
$599$	3.28617e6	0.374217	0.187108	+	0.982339i	$0.440088\pi$
$600$	0	0
$601$	6.11530e6	0.690608	0.345304	−	0.938491i	$-0.387776\pi$
$601$	6.11530e6	0.690608	0.345304	+	0.938491i	$0.387776\pi$
$602$	−2.05112e6	−0.230675
$603$	−3.82360e6	−0.428233
$604$	4.06848e6	0.453774
$605$	0	0
$606$	−1.40986e7	−1.55953
$607$	7.27336e6	0.801241	0.400621	−	0.916244i	$-0.368795\pi$
$607$	7.27336e6	0.801241	0.400621	+	0.916244i	$0.368795\pi$
$608$	−1.24353e6	−0.136426
$609$	−539288.	−0.0589220
$610$	0	0
$611$	−8.85836e6	−0.959954
$612$	−3.66100e6	−0.395113
$613$	7.10635e6	0.763827	0.381914	−	0.924198i	$-0.375265\pi$
$613$	7.10635e6	0.763827	0.381914	+	0.924198i	$0.375265\pi$
$614$	−1.43538e7	−1.53654
$615$	0	0
$616$	−6.07368e6	−0.644911
$617$	5.11598e6	0.541023	0.270512	−	0.962717i	$-0.412807\pi$
$617$	5.11598e6	0.541023	0.270512	+	0.962717i	$0.412807\pi$
$618$	−1.39311e7	−1.46728
$619$	−2.88443e6	−0.302576	−0.151288	−	0.988490i	$-0.548342\pi$
$619$	−2.88443e6	−0.302576	−0.151288	+	0.988490i	$0.548342\pi$
$620$	0	0
$621$	3.09285e6	0.321833
$622$	−1.72460e7	−1.78736
$623$	−1.08997e7	−1.12511
$624$	−1.86349e7	−1.91587
$625$	0	0
$626$	2.38784e7	2.43539
$627$	48795.5	0.00495690
$628$	−3.23864e7	−3.27690
$629$	−3.19310e6	−0.321800
$630$	0	0
$631$	−8.37374e6	−0.837233	−0.418616	−	0.908163i	$-0.637485\pi$
$631$	−8.37374e6	−0.837233	−0.418616	+	0.908163i	$0.637485\pi$
$632$	−8.10370e6	−0.807031
$633$	4.76826e6	0.472989
$634$	−1.49167e7	−1.47383
$635$	0	0
$636$	−2.83321e7	−2.77738
$637$	5.18274e6	0.506071
$638$	1.05753e6	0.102858
$639$	−3.19962e6	−0.309989
$640$	0	0
$641$	1.14615e7	1.10179	0.550893	−	0.834576i	$-0.314287\pi$
$641$	1.14615e7	1.10179	0.550893	+	0.834576i	$0.314287\pi$
$642$	−1.89580e7	−1.81533
$643$	−1.69107e7	−1.61300	−0.806498	−	0.591237i	$-0.798640\pi$
$643$	−1.69107e7	−1.61300	−0.806498	+	0.591237i	$0.798640\pi$
$644$	2.95123e7	2.80407
$645$	0	0
$646$	246217.	0.0232133
$647$	−1.21273e7	−1.13895	−0.569473	−	0.822010i	$-0.692853\pi$
$647$	−1.21273e7	−1.13895	−0.569473	+	0.822010i	$0.692853\pi$
$648$	4.32502e6	0.404623
$649$	−3.38039e6	−0.315033
$650$	0	0
$651$	3.83389e6	0.354558
$652$	−8.86233e6	−0.816448
$653$	1.09620e7	1.00602	0.503010	−	0.864281i	$-0.332226\pi$
$653$	1.09620e7	1.00602	0.503010	+	0.864281i	$0.332226\pi$
$654$	−6.12617e6	−0.560073
$655$	0	0
$656$	3.38795e7	3.07381
$657$	2.18650e6	0.197622
$658$	1.59131e7	1.43282
$659$	2.65796e6	0.238416	0.119208	−	0.992869i	$-0.461964\pi$
$659$	2.65796e6	0.238416	0.119208	+	0.992869i	$0.461964\pi$
$660$	0	0
$661$	8.86632e6	0.789295	0.394648	−	0.918833i	$-0.370867\pi$
$661$	8.86632e6	0.789295	0.394648	+	0.918833i	$0.370867\pi$
$662$	1.38742e7	1.23045
$663$	2.09631e6	0.185213
$664$	−4.63476e7	−4.07950
$665$	0	0
$666$	5.80606e6	0.507220
$667$	−3.33858e6	−0.290568
$668$	3.51819e7	3.05055
$669$	−6.82123e6	−0.589247
$670$	0	0
$671$	−1.77398e6	−0.152105
$672$	1.90193e7	1.62470
$673$	−1.56559e7	−1.33242	−0.666209	−	0.745765i	$-0.732084\pi$
$673$	−1.56559e7	−1.33242	−0.666209	+	0.745765i	$0.732084\pi$
$674$	3.73440e7	3.16644
$675$	0	0
$676$	−1.36715e7	−1.15067
$677$	2.51293e6	0.210721	0.105361	−	0.994434i	$-0.466400\pi$
$677$	2.51293e6	0.210721	0.105361	+	0.994434i	$0.466400\pi$
$678$	2.43722e7	2.03620
$679$	6.95385e6	0.578830
$680$	0	0
$681$	−7.15183e6	−0.590948
$682$	−7.51812e6	−0.618940
$683$	1.40231e7	1.15025	0.575125	−	0.818066i	$-0.304953\pi$
$683$	1.40231e7	1.15025	0.575125	+	0.818066i	$0.304953\pi$
$684$	−331558.	−0.0270969
$685$	0	0
$686$	−2.35242e7	−1.90855
$687$	−1.72195e6	−0.139196
$688$	−1.06668e7	−0.859135
$689$	1.62231e7	1.30192
$690$	0	0
$691$	−2.03284e7	−1.61960	−0.809799	−	0.586707i	$-0.800424\pi$
$691$	−2.03284e7	−1.61960	−0.809799	+	0.586707i	$0.800424\pi$
$692$	4.97188e7	3.94690
$693$	−746308.	−0.0590317
$694$	3.36809e7	2.65451
$695$	0	0
$696$	−4.66865e6	−0.365316
$697$	−3.81122e6	−0.297154
$698$	7.67243e6	0.596066
$699$	7.61274e6	0.589315
$700$	0	0
$701$	8.73851e6	0.671649	0.335825	−	0.941925i	$-0.390985\pi$
$701$	8.73851e6	0.671649	0.335825	+	0.941925i	$0.390985\pi$
$702$	−3.81174e6	−0.291932
$703$	−289183.	−0.0220691
$704$	−2.02669e7	−1.54119
$705$	0	0
$706$	−3.51379e7	−2.65316
$707$	−1.07401e7	−0.808086
$708$	2.29693e7	1.72212
$709$	6.30404e6	0.470981	0.235491	−	0.971877i	$-0.424330\pi$
$709$	6.30404e6	0.470981	0.235491	+	0.971877i	$0.424330\pi$
$710$	0	0
$711$	−995748.	−0.0738713
$712$	−9.43592e7	−6.97564
$713$	2.37345e7	1.74846
$714$	−3.76579e6	−0.276446
$715$	0	0
$716$	−2.42669e7	−1.76902
$717$	1.27703e7	0.927690
$718$	−4.36016e6	−0.315640
$719$	6.08432e6	0.438924	0.219462	−	0.975621i	$-0.429570\pi$
$719$	6.08432e6	0.438924	0.219462	+	0.975621i	$0.429570\pi$
$720$	0	0
$721$	−1.06124e7	−0.760285
$722$	−2.74784e7	−1.96177
$723$	−8.12575e6	−0.578120
$724$	−6.84207e7	−4.85111
$725$	0	0
$726$	1.46349e6	0.103050
$727$	−2.10256e7	−1.47541	−0.737704	−	0.675125i	$-0.764090\pi$
$727$	−2.10256e7	−1.47541	−0.737704	+	0.675125i	$0.764090\pi$
$728$	−2.36313e7	−1.65257
$729$	531441.	0.0370370
$730$	0	0
$731$	1.19994e6	0.0830550
$732$	1.20540e7	0.831480
$733$	9.82965e6	0.675738	0.337869	−	0.941193i	$-0.390294\pi$
$733$	9.82965e6	0.675738	0.337869	+	0.941193i	$0.390294\pi$
$734$	−3.24948e7	−2.22625
$735$	0	0
$736$	1.17743e8	8.01202
$737$	5.71180e6	0.387351
$738$	6.92999e6	0.468373
$739$	−1.51937e7	−1.02342	−0.511709	−	0.859159i	$-0.670987\pi$
$739$	−1.51937e7	−1.02342	−0.511709	+	0.859159i	$0.670987\pi$
$740$	0	0
$741$	189852.	0.0127019
$742$	−2.91431e7	−1.94324
$743$	4.92706e6	0.327428	0.163714	−	0.986508i	$-0.447653\pi$
$743$	4.92706e6	0.327428	0.163714	+	0.986508i	$0.447653\pi$
$744$	3.31902e7	2.19825
$745$	0	0
$746$	−3.65475e7	−2.40442
$747$	−5.69500e6	−0.373415
$748$	5.46890e6	0.357393
$749$	−1.44419e7	−0.940629
$750$	0	0
$751$	−1.07515e6	−0.0695617	−0.0347809	−	0.999395i	$-0.511073\pi$
$751$	−1.07515e6	−0.0695617	−0.0347809	+	0.999395i	$0.511073\pi$
$752$	8.27554e7	5.33644
$753$	−522088.	−0.0335549
$754$	4.11459e6	0.263572
$755$	0	0
$756$	5.07106e6	0.322697
$757$	1.80869e7	1.14716	0.573580	−	0.819150i	$-0.305554\pi$
$757$	1.80869e7	1.14716	0.573580	+	0.819150i	$0.305554\pi$
$758$	2.24451e7	1.41889
$759$	−4.62019e6	−0.291109
$760$	0	0
$761$	540730.	0.0338469	0.0169234	−	0.999857i	$-0.494613\pi$
$761$	540730.	0.0338469	0.0169234	+	0.999857i	$0.494613\pi$
$762$	1.01838e7	0.635367
$763$	−4.66681e6	−0.290207
$764$	−8.80845e7	−5.45967
$765$	0	0
$766$	−1.65589e7	−1.01967
$767$	−1.31523e7	−0.807262
$768$	4.89392e7	2.99402
$769$	8.60213e6	0.524554	0.262277	−	0.964993i	$-0.415527\pi$
$769$	8.60213e6	0.524554	0.262277	+	0.964993i	$0.415527\pi$
$770$	0	0
$771$	5.95147e6	0.360569
$772$	−1.52847e7	−0.923023
$773$	−1.45721e7	−0.877150	−0.438575	−	0.898695i	$-0.644517\pi$
$773$	−1.45721e7	−0.877150	−0.438575	+	0.898695i	$0.644517\pi$
$774$	−2.18187e6	−0.130911
$775$	0	0
$776$	6.01999e7	3.58874
$777$	4.42295e6	0.262821
$778$	5.54143e7	3.28226
$779$	−345163.	−0.0203789
$780$	0	0
$781$	4.77968e6	0.280395
$782$	−2.33130e7	−1.36327
$783$	−573665.	−0.0334390
$784$	−4.84176e7	−2.81328
$785$	0	0
$786$	−2.35543e7	−1.35992
$787$	−9.86427e6	−0.567712	−0.283856	−	0.958867i	$-0.591614\pi$
$787$	−9.86427e6	−0.567712	−0.283856	+	0.958867i	$0.591614\pi$
$788$	−9.07680e6	−0.520736
$789$	−9.85388e6	−0.563527
$790$	0	0
$791$	1.85663e7	1.05507
$792$	−6.46084e6	−0.365996
$793$	−6.90216e6	−0.389764
$794$	−3.03266e7	−1.70715
$795$	0	0
$796$	−5.18976e6	−0.290312
$797$	−1.18967e7	−0.663410	−0.331705	−	0.943383i	$-0.607624\pi$
$797$	−1.18967e7	−0.663410	−0.331705	+	0.943383i	$0.607624\pi$
$798$	−341049.	−0.0189588
$799$	−9.30943e6	−0.515889
$800$	0	0
$801$	−1.15945e7	−0.638513
$802$	−2.71589e7	−1.49099
$803$	−3.26625e6	−0.178756
$804$	−3.88109e7	−2.11745
$805$	0	0
$806$	−2.92513e7	−1.58602
$807$	−1.86141e6	−0.100614
$808$	−9.29773e7	−5.01012
$809$	1.09831e7	0.590005	0.295002	−	0.955497i	$-0.404680\pi$
$809$	1.09831e7	0.590005	0.295002	+	0.955497i	$0.404680\pi$
$810$	0	0
$811$	−3.01879e6	−0.161169	−0.0805845	−	0.996748i	$-0.525679\pi$
$811$	−3.01879e6	−0.161169	−0.0805845	+	0.996748i	$0.525679\pi$
$812$	−5.47396e6	−0.291348
$813$	−2.68413e6	−0.142422
$814$	−8.67325e6	−0.458797
$815$	0	0
$816$	−1.95838e7	−1.02961
$817$	108672.	0.00569593
$818$	5.60531e7	2.92898
$819$	−2.90372e6	−0.151267
$820$	0	0
$821$	−5.39706e6	−0.279447	−0.139723	−	0.990191i	$-0.544621\pi$
$821$	−5.39706e6	−0.279447	−0.139723	+	0.990191i	$0.544621\pi$
$822$	1.80536e7	0.931933
$823$	−2.28710e7	−1.17702	−0.588512	−	0.808488i	$-0.700286\pi$
$823$	−2.28710e7	−1.17702	−0.588512	+	0.808488i	$0.700286\pi$
$824$	−9.18725e7	−4.71376
$825$	0	0
$826$	2.36268e7	1.20491
$827$	7.63569e6	0.388226	0.194113	−	0.980979i	$-0.437817\pi$
$827$	7.63569e6	0.388226	0.194113	+	0.980979i	$0.437817\pi$
$828$	3.13935e7	1.59134
$829$	−7.03185e6	−0.355372	−0.177686	−	0.984087i	$-0.556861\pi$
$829$	−7.03185e6	−0.355372	−0.177686	+	0.984087i	$0.556861\pi$
$830$	0	0
$831$	1.54789e7	0.777564
$832$	−7.88538e7	−3.94925
$833$	5.44665e6	0.271968
$834$	3.07789e7	1.53228
$835$	0	0
$836$	495291.	0.0245101
$837$	4.07827e6	0.201216
$838$	−340443.	−0.0167469
$839$	6.66791e6	0.327028	0.163514	−	0.986541i	$-0.447717\pi$
$839$	6.66791e6	0.327028	0.163514	+	0.986541i	$0.447717\pi$
$840$	0	0
$841$	−1.98919e7	−0.969809
$842$	2.56633e7	1.24748
$843$	1.83723e7	0.890420
$844$	4.83995e7	2.33876
$845$	0	0
$846$	1.69275e7	0.813142
$847$	1.11486e6	0.0533962
$848$	−1.51557e8	−7.23748
$849$	−1.46153e7	−0.695888
$850$	0	0
$851$	2.73812e7	1.29607
$852$	−3.24772e7	−1.53278
$853$	−3.51845e7	−1.65569	−0.827845	−	0.560956i	$-0.810434\pi$
$853$	−3.51845e7	−1.65569	−0.827845	+	0.560956i	$0.810434\pi$
$854$	1.23990e7	0.581758
$855$	0	0
$856$	−1.25024e8	−5.83189
$857$	1.17588e7	0.546905	0.273452	−	0.961886i	$-0.411834\pi$
$857$	1.17588e7	0.546905	0.273452	+	0.961886i	$0.411834\pi$
$858$	5.69409e6	0.264062
$859$	3.68112e7	1.70215	0.851074	−	0.525045i	$-0.175952\pi$
$859$	3.68112e7	1.70215	0.851074	+	0.525045i	$0.175952\pi$
$860$	0	0
$861$	5.27914e6	0.242692
$862$	8.20533e7	3.76121
$863$	−3.57608e7	−1.63448	−0.817242	−	0.576295i	$-0.804498\pi$
$863$	−3.57608e7	−1.63448	−0.817242	+	0.576295i	$0.804498\pi$
$864$	2.02317e7	0.922037
$865$	0	0
$866$	−4.34241e7	−1.96760
$867$	−1.05757e7	−0.477815
$868$	3.89153e7	1.75316
$869$	1.48748e6	0.0668191
$870$	0	0
$871$	2.22233e7	0.992576
$872$	−4.04008e7	−1.79928
$873$	7.39711e6	0.328493
$874$	−2.11134e6	−0.0934931
$875$	0	0
$876$	2.21937e7	0.977169
$877$	−4.23836e7	−1.86080	−0.930398	−	0.366550i	$-0.880539\pi$
$877$	−4.23836e7	−1.86080	−0.930398	+	0.366550i	$0.880539\pi$
$878$	2.86360e7	1.25365
$879$	−2.13550e7	−0.932240
$880$	0	0
$881$	1.28486e7	0.557721	0.278861	−	0.960332i	$-0.410043\pi$
$881$	1.28486e7	0.557721	0.278861	+	0.960332i	$0.410043\pi$
$882$	−9.90373e6	−0.428674
$883$	−1.50474e7	−0.649469	−0.324735	−	0.945805i	$-0.605275\pi$
$883$	−1.50474e7	−0.649469	−0.324735	+	0.945805i	$0.605275\pi$
$884$	2.12782e7	0.915809
$885$	0	0
$886$	−3.33474e6	−0.142718
$887$	1.40467e7	0.599466	0.299733	−	0.954023i	$-0.403102\pi$
$887$	1.40467e7	0.599466	0.299733	+	0.954023i	$0.403102\pi$
$888$	3.82897e7	1.62948
$889$	7.75787e6	0.329221
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−6.92378e7	−2.91361
$893$	−843109.	−0.0353798
$894$	3.25039e7	1.36016
$895$	0	0
$896$	7.40282e7	3.08054
$897$	−1.79761e7	−0.745958
$898$	−3.78511e7	−1.56634
$899$	−4.40230e6	−0.181669
$900$	0	0
$901$	1.70492e7	0.699668
$902$	−1.03522e7	−0.423659
$903$	−1.66211e6	−0.0678327
$904$	1.60729e8	6.54145
$905$	0	0
$906$	4.45170e6	0.180180
$907$	8.25845e6	0.333335	0.166667	−	0.986013i	$-0.446699\pi$
$907$	8.25845e6	0.333335	0.166667	+	0.986013i	$0.446699\pi$
$908$	−7.25935e7	−2.92202
$909$	−1.14247e7	−0.458600
$910$	0	0
$911$	2.01230e7	0.803335	0.401667	−	0.915786i	$-0.368431\pi$
$911$	2.01230e7	0.803335	0.401667	+	0.915786i	$0.368431\pi$
$912$	−1.77361e6	−0.0706108
$913$	8.50735e6	0.337767
$914$	4.18290e7	1.65620
$915$	0	0
$916$	−1.74783e7	−0.688274
$917$	−1.79433e7	−0.704657
$918$	−4.00584e6	−0.156887
$919$	1.70445e7	0.665725	0.332862	−	0.942975i	$-0.391986\pi$
$919$	1.70445e7	0.665725	0.332862	+	0.942975i	$0.391986\pi$
$920$	0	0
$921$	−1.16314e7	−0.451839
$922$	−9.09131e7	−3.52208
$923$	1.85966e7	0.718506
$924$	−7.57529e6	−0.291890
$925$	0	0
$926$	1.05607e7	0.404729
$927$	−1.12889e7	−0.431472
$928$	−2.18392e7	−0.832465
$929$	1.53002e7	0.581644	0.290822	−	0.956777i	$-0.406071\pi$
$929$	1.53002e7	0.581644	0.290822	+	0.956777i	$0.406071\pi$
$930$	0	0
$931$	493276.	0.0186516
$932$	7.72719e7	2.91395
$933$	−1.39751e7	−0.525595
$934$	−4.01231e7	−1.50497
$935$	0	0
$936$	−2.51377e7	−0.937854
$937$	2.49555e7	0.928574	0.464287	−	0.885685i	$-0.346311\pi$
$937$	2.49555e7	0.928574	0.464287	+	0.885685i	$0.346311\pi$
$938$	−3.99219e7	−1.48151
$939$	1.93496e7	0.716157
$940$	0	0
$941$	2.67710e7	0.985576	0.492788	−	0.870149i	$-0.335978\pi$
$941$	2.67710e7	0.985576	0.492788	+	0.870149i	$0.335978\pi$
$942$	−3.54370e7	−1.30116
$943$	3.26817e7	1.19681
$944$	1.22870e8	4.48762
$945$	0	0
$946$	3.25933e6	0.118413
$947$	3.32255e7	1.20392	0.601958	−	0.798528i	$-0.294388\pi$
$947$	3.32255e7	1.20392	0.601958	+	0.798528i	$0.294388\pi$
$948$	−1.01072e7	−0.365266
$949$	−1.27082e7	−0.458057
$950$	0	0
$951$	−1.20876e7	−0.433399
$952$	−2.48346e7	−0.888107
$953$	−2.27164e7	−0.810229	−0.405114	−	0.914266i	$-0.632768\pi$
$953$	−2.27164e7	−0.810229	−0.405114	+	0.914266i	$0.632768\pi$
$954$	−3.10008e7	−1.10281
$955$	0	0
$956$	1.29623e8	4.58709
$957$	856956.	0.0302467
$958$	8.99529e7	3.16666
$959$	1.37529e7	0.482890
$960$	0	0
$961$	2.66748e6	0.0931736
$962$	−3.37456e7	−1.17566
$963$	−1.53624e7	−0.533819
$964$	−8.24792e7	−2.85859
$965$	0	0
$966$	3.22922e7	1.11341
$967$	4.69627e7	1.61505	0.807526	−	0.589832i	$-0.200806\pi$
$967$	4.69627e7	1.61505	0.807526	+	0.589832i	$0.200806\pi$
$968$	9.65138e6	0.331056
$969$	199519.	0.00682615
$970$	0	0
$971$	−261687.	−0.00890706	−0.00445353	−	0.999990i	$-0.501418\pi$
$971$	−261687.	−0.00890706	−0.00445353	+	0.999990i	$0.501418\pi$
$972$	5.39431e6	0.183135
$973$	2.34468e7	0.793966
$974$	2.46918e7	0.833979
$975$	0	0
$976$	6.44805e7	2.16672
$977$	3.02629e7	1.01432	0.507159	−	0.861853i	$-0.330696\pi$
$977$	3.02629e7	1.01432	0.507159	+	0.861853i	$0.330696\pi$
$978$	−9.69710e6	−0.324186
$979$	1.73201e7	0.577556
$980$	0	0
$981$	−4.96429e6	−0.164696
$982$	2.00533e7	0.663601
$983$	2.18315e7	0.720610	0.360305	−	0.932835i	$-0.382673\pi$
$983$	2.18315e7	0.720610	0.360305	+	0.932835i	$0.382673\pi$
$984$	4.57018e7	1.50469
$985$	0	0
$986$	4.32411e6	0.141646
$987$	1.28950e7	0.421337
$988$	1.92706e6	0.0628064
$989$	−1.02896e7	−0.334510
$990$	0	0
$991$	−5.46573e7	−1.76792	−0.883962	−	0.467558i	$-0.845134\pi$
$991$	−5.46573e7	−1.76792	−0.883962	+	0.467558i	$0.845134\pi$
$992$	1.55258e8	5.00927
$993$	1.12428e7	0.361829
$994$	−3.34069e7	−1.07243
$995$	0	0
$996$	−5.78062e7	−1.84640
$997$	1.96397e7	0.625745	0.312872	−	0.949795i	$-0.398709\pi$
$997$	1.96397e7	0.625745	0.312872	+	0.949795i	$0.398709\pi$
$998$	−3.51635e7	−1.11755
$999$	4.70488e6	0.149154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.r.1.9	✓	9
5.4	even	2		825.6.a.s.1.1	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.9	✓	9	1.1	even	1	trivial
825.6.a.s.1.1	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)

\(f(q)\)	\(=\)	\(q+11.1064 q^{2} +9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} +76.1461 q^{7} +659.202 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+11.1064 q^{2} +9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} +76.1461 q^{7} +659.202 q^{8} +81.0000 q^{9} -121.000 q^{11} +822.178 q^{12} -470.783 q^{13} +845.713 q^{14} +4398.09 q^{16} -494.756 q^{17} +899.622 q^{18} -44.8076 q^{19} +685.315 q^{21} -1343.88 q^{22} +4242.60 q^{23} +5932.82 q^{24} -5228.73 q^{26} +729.000 q^{27} +6956.19 q^{28} -786.920 q^{29} +5594.34 q^{31} +27752.7 q^{32} -1089.00 q^{33} -5494.98 q^{34} +7399.60 q^{36} +6453.89 q^{37} -497.653 q^{38} -4237.05 q^{39} +7703.22 q^{41} +7611.42 q^{42} -2425.31 q^{43} -11053.7 q^{44} +47120.2 q^{46} +18816.2 q^{47} +39582.8 q^{48} -11008.8 q^{49} -4452.80 q^{51} -43007.5 q^{52} -34459.8 q^{53} +8096.60 q^{54} +50195.7 q^{56} -403.268 q^{57} -8739.88 q^{58} +27937.1 q^{59} +14661.0 q^{61} +62133.2 q^{62} +6167.84 q^{63} +167495. q^{64} -12094.9 q^{66} -47205.0 q^{67} -45197.5 q^{68} +38183.4 q^{69} -39501.5 q^{71} +53395.4 q^{72} +26993.8 q^{73} +71679.8 q^{74} -4093.31 q^{76} -9213.68 q^{77} -47058.6 q^{78} -12293.2 q^{79} +6561.00 q^{81} +85555.4 q^{82} -70308.6 q^{83} +62605.7 q^{84} -26936.6 q^{86} -7082.28 q^{87} -79763.4 q^{88} -143142. q^{89} -35848.3 q^{91} +387574. q^{92} +50349.1 q^{93} +208981. q^{94} +249774. q^{96} +91322.4 q^{97} -122268. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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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