LMFDB - Embedded newform 825.6.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.476452$ 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)

$f(q)$	$=$	$q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} -121.000 q^{11} -1.69545 q^{12} +69.9067 q^{13} -817.946 q^{14} -1017.94 q^{16} +500.495 q^{17} +456.854 q^{18} -670.685 q^{19} -1305.19 q^{21} -682.462 q^{22} -791.025 q^{23} -1633.93 q^{24} +394.287 q^{26} +729.000 q^{27} +27.3196 q^{28} +1545.91 q^{29} +2703.09 q^{31} +68.2013 q^{32} -1089.00 q^{33} +2822.88 q^{34} -15.2591 q^{36} +2667.13 q^{37} -3782.78 q^{38} +629.161 q^{39} +9622.09 q^{41} -7361.51 q^{42} +6681.57 q^{43} +22.7944 q^{44} -4461.52 q^{46} +1167.06 q^{47} -9161.43 q^{48} +4224.17 q^{49} +4504.46 q^{51} -13.1693 q^{52} -28872.9 q^{53} +4111.69 q^{54} +26328.4 q^{56} -6036.16 q^{57} +8719.22 q^{58} +23599.3 q^{59} +17601.3 q^{61} +15245.9 q^{62} -11746.7 q^{63} +32958.6 q^{64} -6142.15 q^{66} -16501.5 q^{67} -94.2851 q^{68} -7119.22 q^{69} +72059.4 q^{71} -14705.4 q^{72} +45480.8 q^{73} +15043.1 q^{74} +126.346 q^{76} +17547.6 q^{77} +3548.58 q^{78} -21688.7 q^{79} +6561.00 q^{81} +54270.3 q^{82} -6934.34 q^{83} +245.877 q^{84} +37685.2 q^{86} +13913.2 q^{87} +21967.3 q^{88} +42779.8 q^{89} -10138.0 q^{91} +149.016 q^{92} +24327.8 q^{93} +6582.41 q^{94} +613.812 q^{96} +20992.6 q^{97} +23825.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.64018	0.997052	0.498526	−	0.866875i	$-0.333875\pi$
$2$	5.64018	0.997052	0.498526	+	0.866875i	$0.333875\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−0.188384	−0.00588699
$5$	0	0
$5$	0	0
$6$	50.7616	0.575648
$7$	−145.021	−1.11863	−0.559315	−	0.828955i	$-0.688936\pi$
$7$	−145.021	−1.11863	−0.559315	+	0.828955i	$0.688936\pi$
$8$	−181.548	−1.00292
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−1.69545	−0.00339886
$13$	69.9067	0.114726	0.0573628	−	0.998353i	$-0.481731\pi$
$13$	69.9067	0.114726	0.0573628	+	0.998353i	$0.481731\pi$
$14$	−817.946	−1.11533
$15$	0	0
$16$	−1017.94	−0.994078
$17$	500.495	0.420027	0.210014	−	0.977698i	$-0.432649\pi$
$17$	500.495	0.420027	0.210014	+	0.977698i	$0.432649\pi$
$18$	456.854	0.332351
$19$	−670.685	−0.426221	−0.213110	−	0.977028i	$-0.568359\pi$
$19$	−670.685	−0.426221	−0.213110	+	0.977028i	$0.568359\pi$
$20$	0	0
$21$	−1305.19	−0.645842
$22$	−682.462	−0.300623
$23$	−791.025	−0.311796	−0.155898	−	0.987773i	$-0.549827\pi$
$23$	−791.025	−0.311796	−0.155898	+	0.987773i	$0.549827\pi$
$24$	−1633.93	−0.579037
$25$	0	0
$26$	394.287	0.114388
$27$	729.000	0.192450
$28$	27.3196	0.00658537
$29$	1545.91	0.341342	0.170671	−	0.985328i	$-0.445406\pi$
$29$	1545.91	0.341342	0.170671	+	0.985328i	$0.445406\pi$
$30$	0	0
$31$	2703.09	0.505191	0.252596	−	0.967572i	$-0.418716\pi$
$31$	2703.09	0.505191	0.252596	+	0.967572i	$0.418716\pi$
$32$	68.2013	0.0117738
$33$	−1089.00	−0.174078
$34$	2822.88	0.418789
$35$	0	0
$36$	−15.2591	−0.00196233
$37$	2667.13	0.320287	0.160144	−	0.987094i	$-0.448804\pi$
$37$	2667.13	0.320287	0.160144	+	0.987094i	$0.448804\pi$
$38$	−3782.78	−0.424964
$39$	629.161	0.0662369
$40$	0	0
$41$	9622.09	0.893942	0.446971	−	0.894548i	$-0.352503\pi$
$41$	9622.09	0.893942	0.446971	+	0.894548i	$0.352503\pi$
$42$	−7361.51	−0.643938
$43$	6681.57	0.551070	0.275535	−	0.961291i	$-0.411145\pi$
$43$	6681.57	0.551070	0.275535	+	0.961291i	$0.411145\pi$
$44$	22.7944	0.00177499
$45$	0	0
$46$	−4461.52	−0.310877
$47$	1167.06	0.0770632	0.0385316	−	0.999257i	$-0.487732\pi$
$47$	1167.06	0.0770632	0.0385316	+	0.999257i	$0.487732\pi$
$48$	−9161.43	−0.573931
$49$	4224.17	0.251334
$50$	0	0
$51$	4504.46	0.242503
$52$	−13.1693	−0.000675389 0
$53$	−28872.9	−1.41189	−0.705944	−	0.708268i	$-0.749477\pi$
$53$	−28872.9	−1.41189	−0.705944	+	0.708268i	$0.749477\pi$
$54$	4111.69	0.191883
$55$	0	0
$56$	26328.4	1.12190
$57$	−6036.16	−0.246079
$58$	8719.22	0.340336
$59$	23599.3	0.882610	0.441305	−	0.897357i	$-0.354516\pi$
$59$	23599.3	0.882610	0.441305	+	0.897357i	$0.354516\pi$
$60$	0	0
$61$	17601.3	0.605648	0.302824	−	0.953046i	$-0.402070\pi$
$61$	17601.3	0.605648	0.302824	+	0.953046i	$0.402070\pi$
$62$	15245.9	0.503702
$63$	−11746.7	−0.372877
$64$	32958.6	1.00582
$65$	0	0
$66$	−6142.15	−0.173565
$67$	−16501.5	−0.449092	−0.224546	−	0.974463i	$-0.572090\pi$
$67$	−16501.5	−0.449092	−0.224546	+	0.974463i	$0.572090\pi$
$68$	−94.2851	−0.00247270
$69$	−7119.22	−0.180016
$70$	0	0
$71$	72059.4	1.69646	0.848232	−	0.529625i	$-0.177667\pi$
$71$	72059.4	1.69646	0.848232	+	0.529625i	$0.177667\pi$
$72$	−14705.4	−0.334307
$73$	45480.8	0.998898	0.499449	−	0.866343i	$-0.333536\pi$
$73$	45480.8	0.998898	0.499449	+	0.866343i	$0.333536\pi$
$74$	15043.1	0.319343
$75$	0	0
$76$	126.346	0.00250916
$77$	17547.6	0.337280
$78$	3548.58	0.0660417
$79$	−21688.7	−0.390990	−0.195495	−	0.980705i	$-0.562631\pi$
$79$	−21688.7	−0.390990	−0.195495	+	0.980705i	$0.562631\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	54270.3	0.891307
$83$	−6934.34	−0.110487	−0.0552434	−	0.998473i	$-0.517593\pi$
$83$	−6934.34	−0.110487	−0.0552434	+	0.998473i	$0.517593\pi$
$84$	245.877	0.00380206
$85$	0	0
$86$	37685.2	0.549446
$87$	13913.2	0.197074
$88$	21967.3	0.302392
$89$	42779.8	0.572484	0.286242	−	0.958157i	$-0.407594\pi$
$89$	42779.8	0.572484	0.286242	+	0.958157i	$0.407594\pi$
$90$	0	0
$91$	−10138.0	−0.128336
$92$	149.016	0.00183554
$93$	24327.8	0.291672
$94$	6582.41	0.0768361
$95$	0	0
$96$	613.812	0.00679762
$97$	20992.6	0.226535	0.113268	−	0.993565i	$-0.463868\pi$
$97$	20992.6	0.226535	0.113268	+	0.993565i	$0.463868\pi$
$98$	23825.1	0.250593
$99$	−9801.00	−0.100504
$100$	0	0
$101$	189654.	1.84995	0.924973	−	0.380033i	$-0.124087\pi$
$101$	189654.	1.84995	0.924973	+	0.380033i	$0.124087\pi$
$102$	25405.9	0.241788
$103$	160899.	1.49438	0.747190	−	0.664611i	$-0.231403\pi$
$103$	160899.	1.49438	0.747190	+	0.664611i	$0.231403\pi$
$104$	−12691.4	−0.115061
$105$	0	0
$106$	−162848.	−1.40773
$107$	−113244.	−0.956216	−0.478108	−	0.878301i	$-0.658677\pi$
$107$	−113244.	−0.956216	−0.478108	+	0.878301i	$0.658677\pi$
$108$	−137.332	−0.00113295
$109$	−1768.32	−0.0142559	−0.00712797	−	0.999975i	$-0.502269\pi$
$109$	−1768.32	−0.0142559	−0.00712797	+	0.999975i	$0.502269\pi$
$110$	0	0
$111$	24004.1	0.184918
$112$	147622.	1.11201
$113$	77273.2	0.569289	0.284645	−	0.958633i	$-0.408124\pi$
$113$	77273.2	0.569289	0.284645	+	0.958633i	$0.408124\pi$
$114$	−34045.0	−0.245353
$115$	0	0
$116$	−291.224	−0.00200948
$117$	5662.45	0.0382419
$118$	133104.	0.880008
$119$	−72582.5	−0.469855
$120$	0	0
$121$	14641.0	0.0909091
$122$	99274.6	0.603863
$123$	86598.8	0.516118
$124$	−509.218	−0.00297406
$125$	0	0
$126$	−66253.6	−0.371778
$127$	−104707.	−0.576058	−0.288029	−	0.957622i	$-0.593000\pi$
$127$	−104707.	−0.576058	−0.288029	+	0.957622i	$0.593000\pi$
$128$	183710.	0.991079
$129$	60134.1	0.318161
$130$	0	0
$131$	101248.	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$131$	101248.	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$132$	205.150	0.00102479
$133$	97263.6	0.476783
$134$	−93071.2	−0.447768
$135$	0	0
$136$	−90864.0	−0.421255
$137$	−128311.	−0.584068	−0.292034	−	0.956408i	$-0.594332\pi$
$137$	−128311.	−0.584068	−0.292034	+	0.956408i	$0.594332\pi$
$138$	−40153.7	−0.179485
$139$	161754.	0.710100	0.355050	−	0.934847i	$-0.384464\pi$
$139$	161754.	0.710100	0.355050	+	0.934847i	$0.384464\pi$
$140$	0	0
$141$	10503.5	0.0444925
$142$	406428.	1.69146
$143$	−8458.72	−0.0345911
$144$	−82452.8	−0.331359
$145$	0	0
$146$	256520.	0.995954
$147$	38017.5	0.145108
$148$	−502.443	−0.00188553
$149$	−404421.	−1.49234	−0.746170	−	0.665755i	$-0.768110\pi$
$149$	−404421.	−1.49234	−0.746170	+	0.665755i	$0.768110\pi$
$150$	0	0
$151$	416180.	1.48538	0.742692	−	0.669633i	$-0.233549\pi$
$151$	416180.	1.48538	0.742692	+	0.669633i	$0.233549\pi$
$152$	121762.	0.427466
$153$	40540.1	0.140009
$154$	98971.5	0.336286
$155$	0	0
$156$	−118.524	−0.000389936 0
$157$	−332834.	−1.07765	−0.538827	−	0.842417i	$-0.681132\pi$
$157$	−332834.	−1.07765	−0.538827	+	0.842417i	$0.681132\pi$
$158$	−122328.	−0.389837
$159$	−259856.	−0.815154
$160$	0	0
$161$	114715.	0.348785
$162$	37005.2	0.110784
$163$	14764.9	0.0435272	0.0217636	−	0.999763i	$-0.493072\pi$
$163$	14764.9	0.0435272	0.0217636	+	0.999763i	$0.493072\pi$
$164$	−1812.64	−0.00526263
$165$	0	0
$166$	−39110.9	−0.110161
$167$	−169960.	−0.471579	−0.235790	−	0.971804i	$-0.575768\pi$
$167$	−169960.	−0.471579	−0.235790	+	0.971804i	$0.575768\pi$
$168$	236955.	0.647729
$169$	−366406.	−0.986838
$170$	0	0
$171$	−54325.5	−0.142074
$172$	−1258.70	−0.00324415
$173$	−725869.	−1.84392	−0.921962	−	0.387281i	$-0.873414\pi$
$173$	−725869.	−1.84392	−0.921962	+	0.387281i	$0.873414\pi$
$174$	78472.9	0.196493
$175$	0	0
$176$	123170.	0.299726
$177$	212394.	0.509575
$178$	241286.	0.570797
$179$	69311.4	0.161686	0.0808429	−	0.996727i	$-0.474239\pi$
$179$	69311.4	0.161686	0.0808429	+	0.996727i	$0.474239\pi$
$180$	0	0
$181$	790396.	1.79328	0.896640	−	0.442761i	$-0.146001\pi$
$181$	790396.	1.79328	0.896640	+	0.442761i	$0.146001\pi$
$182$	−57179.9	−0.127957
$183$	158412.	0.349671
$184$	143609.	0.312707
$185$	0	0
$186$	137213.	0.290813
$187$	−60559.9	−0.126643
$188$	−219.854	−0.000453670 0
$189$	−105721.	−0.215281
$190$	0	0
$191$	418979.	0.831015	0.415507	−	0.909590i	$-0.363604\pi$
$191$	418979.	0.831015	0.415507	+	0.909590i	$0.363604\pi$
$192$	296628.	0.580709
$193$	−265524.	−0.513110	−0.256555	−	0.966530i	$-0.582587\pi$
$193$	−265524.	−0.513110	−0.256555	+	0.966530i	$0.582587\pi$
$194$	118402.	0.225868
$195$	0	0
$196$	−795.765	−0.00147960
$197$	87754.1	0.161102	0.0805512	−	0.996750i	$-0.474332\pi$
$197$	87754.1	0.161102	0.0805512	+	0.996750i	$0.474332\pi$
$198$	−55279.4	−0.100208
$199$	380988.	0.681992	0.340996	−	0.940065i	$-0.389236\pi$
$199$	380988.	0.681992	0.340996	+	0.940065i	$0.389236\pi$
$200$	0	0
$201$	−148513.	−0.259284
$202$	1.06968e6	1.84449
$203$	−224190.	−0.381835
$204$	−848.566	−0.00142761
$205$	0	0
$206$	907501.	1.48997
$207$	−64073.0	−0.103932
$208$	−71160.6	−0.114046
$209$	81152.9	0.128510
$210$	0	0
$211$	624404.	0.965516	0.482758	−	0.875754i	$-0.339635\pi$
$211$	624404.	0.965516	0.482758	+	0.875754i	$0.339635\pi$
$212$	5439.18	0.00831177
$213$	648534.	0.979454
$214$	−638717.	−0.953398
$215$	0	0
$216$	−132349.	−0.193012
$217$	−392005.	−0.565123
$218$	−9973.67	−0.0142139
$219$	409328.	0.576714
$220$	0	0
$221$	34988.0	0.0481879
$222$	135388.	0.184373
$223$	658298.	0.886462	0.443231	−	0.896407i	$-0.353832\pi$
$223$	658298.	0.886462	0.443231	+	0.896407i	$0.353832\pi$
$224$	−9890.64	−0.0131706
$225$	0	0
$226$	435835.	0.567611
$227$	−219954.	−0.283313	−0.141657	−	0.989916i	$-0.545243\pi$
$227$	−219954.	−0.283313	−0.141657	+	0.989916i	$0.545243\pi$
$228$	1137.11	0.00144866
$229$	−313324.	−0.394825	−0.197412	−	0.980321i	$-0.563254\pi$
$229$	−313324.	−0.394825	−0.197412	+	0.980321i	$0.563254\pi$
$230$	0	0
$231$	157928.	0.194729
$232$	−280657.	−0.342339
$233$	1.01608e6	1.22614	0.613069	−	0.790030i	$-0.289935\pi$
$233$	1.01608e6	1.22614	0.613069	+	0.790030i	$0.289935\pi$
$234$	31937.2	0.0381292
$235$	0	0
$236$	−4445.72	−0.00519592
$237$	−195198.	−0.225738
$238$	−409378.	−0.468470
$239$	762420.	0.863375	0.431688	−	0.902023i	$-0.357918\pi$
$239$	762420.	0.863375	0.431688	+	0.902023i	$0.357918\pi$
$240$	0	0
$241$	−22909.2	−0.0254078	−0.0127039	−	0.999919i	$-0.504044\pi$
$241$	−22909.2	−0.0254078	−0.0127039	+	0.999919i	$0.504044\pi$
$242$	82577.9	0.0906411
$243$	59049.0	0.0641500
$244$	−3315.80	−0.00356545
$245$	0	0
$246$	488433.	0.514596
$247$	−46885.4	−0.0488985
$248$	−490741.	−0.506668
$249$	−62409.1	−0.0637895
$250$	0	0
$251$	−691159.	−0.692458	−0.346229	−	0.938150i	$-0.612538\pi$
$251$	−691159.	−0.692458	−0.346229	+	0.938150i	$0.612538\pi$
$252$	2212.89	0.00219512
$253$	95714.0	0.0940100
$254$	−590566.	−0.574360
$255$	0	0
$256$	−18518.2	−0.0176604
$257$	1.34845e6	1.27351	0.636753	−	0.771068i	$-0.280277\pi$
$257$	1.34845e6	1.27351	0.636753	+	0.771068i	$0.280277\pi$
$258$	339167.	0.317223
$259$	−386790.	−0.358283
$260$	0	0
$261$	125219.	0.113781
$262$	571057.	0.513957
$263$	−1.34541e6	−1.19940	−0.599700	−	0.800225i	$-0.704713\pi$
$263$	−1.34541e6	−1.19940	−0.599700	+	0.800225i	$0.704713\pi$
$264$	197706.	0.174586
$265$	0	0
$266$	548584.	0.475378
$267$	385018.	0.330524
$268$	3108.61	0.00264380
$269$	−2.03803e6	−1.71724	−0.858620	−	0.512613i	$-0.828678\pi$
$269$	−2.03803e6	−1.71724	−0.858620	+	0.512613i	$0.828678\pi$
$270$	0	0
$271$	−218053.	−0.180360	−0.0901799	−	0.995925i	$-0.528744\pi$
$271$	−218053.	−0.180360	−0.0901799	+	0.995925i	$0.528744\pi$
$272$	−509472.	−0.417540
$273$	−91241.7	−0.0740946
$274$	−723698.	−0.582346
$275$	0	0
$276$	1341.15	0.00105975
$277$	410179.	0.321199	0.160599	−	0.987020i	$-0.448657\pi$
$277$	410179.	0.321199	0.160599	+	0.987020i	$0.448657\pi$
$278$	912324.	0.708006
$279$	218950.	0.168397
$280$	0	0
$281$	−406589.	−0.307178	−0.153589	−	0.988135i	$-0.549083\pi$
$281$	−406589.	−0.307178	−0.153589	+	0.988135i	$0.549083\pi$
$282$	59241.7	0.0443613
$283$	−1.26416e6	−0.938291	−0.469146	−	0.883121i	$-0.655438\pi$
$283$	−1.26416e6	−0.938291	−0.469146	+	0.883121i	$0.655438\pi$
$284$	−13574.8	−0.00998707
$285$	0	0
$286$	−47708.7	−0.0344891
$287$	−1.39541e6	−0.999991
$288$	5524.30	0.00392461
$289$	−1.16936e6	−0.823577
$290$	0	0
$291$	188933.	0.130790
$292$	−8567.85	−0.00588050
$293$	−351099.	−0.238924	−0.119462	−	0.992839i	$-0.538117\pi$
$293$	−351099.	−0.238924	−0.119462	+	0.992839i	$0.538117\pi$
$294$	214426.	0.144680
$295$	0	0
$296$	−484212.	−0.321223
$297$	−88209.0	−0.0580259
$298$	−2.28101e6	−1.48794
$299$	−55298.0	−0.0357710
$300$	0	0
$301$	−968969.	−0.616444
$302$	2.34733e6	1.48101
$303$	1.70689e6	1.06807
$304$	682714.	0.423697
$305$	0	0
$306$	228654.	0.139596
$307$	1.46228e6	0.885493	0.442747	−	0.896647i	$-0.354004\pi$
$307$	1.46228e6	0.885493	0.442747	+	0.896647i	$0.354004\pi$
$308$	−3305.68	−0.00198556
$309$	1.44809e6	0.862781
$310$	0	0
$311$	2.20292e6	1.29151	0.645755	−	0.763545i	$-0.276543\pi$
$311$	2.20292e6	1.29151	0.645755	+	0.763545i	$0.276543\pi$
$312$	−114223.	−0.0664304
$313$	−647848.	−0.373777	−0.186888	−	0.982381i	$-0.559840\pi$
$313$	−647848.	−0.373777	−0.186888	+	0.982381i	$0.559840\pi$
$314$	−1.87725e6	−1.07448
$315$	0	0
$316$	4085.80	0.00230175
$317$	1.62933e6	0.910670	0.455335	−	0.890320i	$-0.349519\pi$
$317$	1.62933e6	0.910670	0.455335	+	0.890320i	$0.349519\pi$
$318$	−1.46563e6	−0.812751
$319$	−187055.	−0.102918
$320$	0	0
$321$	−1.01920e6	−0.552072
$322$	647016.	0.347756
$323$	−335675.	−0.179024
$324$	−1235.99	−0.000654110 0
$325$	0	0
$326$	83276.6	0.0433989
$327$	−15914.9	−0.00823067
$328$	−1.74687e6	−0.896554
$329$	−169248.	−0.0862053
$330$	0	0
$331$	799339.	0.401016	0.200508	−	0.979692i	$-0.435741\pi$
$331$	799339.	0.401016	0.200508	+	0.979692i	$0.435741\pi$
$332$	1306.32	0.000650434 0
$333$	216037.	0.106762
$334$	−958603.	−0.470189
$335$	0	0
$336$	1.32860e6	0.642017
$337$	3.76736e6	1.80702	0.903508	−	0.428570i	$-0.140983\pi$
$337$	3.76736e6	1.80702	0.903508	+	0.428570i	$0.140983\pi$
$338$	−2.06660e6	−0.983929
$339$	695459.	0.328679
$340$	0	0
$341$	−327074.	−0.152321
$342$	−306405.	−0.141655
$343$	1.82478e6	0.837481
$344$	−1.21303e6	−0.552681
$345$	0	0
$346$	−4.09403e6	−1.83849
$347$	−923519.	−0.411739	−0.205869	−	0.978579i	$-0.566002\pi$
$347$	−923519.	−0.411739	−0.205869	+	0.978579i	$0.566002\pi$
$348$	−2621.02	−0.00116017
$349$	2.61022e6	1.14713	0.573567	−	0.819159i	$-0.305559\pi$
$349$	2.61022e6	1.14713	0.573567	+	0.819159i	$0.305559\pi$
$350$	0	0
$351$	50962.0	0.0220790
$352$	−8252.36	−0.00354994
$353$	−1.25644e6	−0.536669	−0.268335	−	0.963326i	$-0.586473\pi$
$353$	−1.25644e6	−0.536669	−0.268335	+	0.963326i	$0.586473\pi$
$354$	1.19794e6	0.508073
$355$	0	0
$356$	−8059.01	−0.00337021
$357$	−653242.	−0.271271
$358$	390928.	0.161209
$359$	−2.38200e6	−0.975451	−0.487726	−	0.872997i	$-0.662173\pi$
$359$	−2.38200e6	−0.975451	−0.487726	+	0.872997i	$0.662173\pi$
$360$	0	0
$361$	−2.02628e6	−0.818336
$362$	4.45797e6	1.78799
$363$	131769.	0.0524864
$364$	1909.83	0.000755511 0
$365$	0	0
$366$	893471.	0.348641
$367$	3.46320e6	1.34219	0.671093	−	0.741373i	$-0.265825\pi$
$367$	3.46320e6	1.34219	0.671093	+	0.741373i	$0.265825\pi$
$368$	805213.	0.309950
$369$	779389.	0.297981
$370$	0	0
$371$	4.18718e6	1.57938
$372$	−4582.96	−0.00171707
$373$	−811517.	−0.302013	−0.151006	−	0.988533i	$-0.548251\pi$
$373$	−811517.	−0.302013	−0.151006	+	0.988533i	$0.548251\pi$
$374$	−341569.	−0.126270
$375$	0	0
$376$	−211877.	−0.0772884
$377$	108070.	0.0391607
$378$	−596283.	−0.214646
$379$	−522481.	−0.186841	−0.0934206	−	0.995627i	$-0.529780\pi$
$379$	−522481.	−0.186841	−0.0934206	+	0.995627i	$0.529780\pi$
$380$	0	0
$381$	−942362.	−0.332587
$382$	2.36312e6	0.828565
$383$	2.24960e6	0.783625	0.391813	−	0.920045i	$-0.371848\pi$
$383$	2.24960e6	0.783625	0.391813	+	0.920045i	$0.371848\pi$
$384$	1.65339e6	0.572200
$385$	0	0
$386$	−1.49760e6	−0.511598
$387$	541207.	0.183690
$388$	−3954.65	−0.00133361
$389$	−2.42055e6	−0.811035	−0.405518	−	0.914087i	$-0.632909\pi$
$389$	−2.42055e6	−0.811035	−0.405518	+	0.914087i	$0.632909\pi$
$390$	0	0
$391$	−395904.	−0.130963
$392$	−766891.	−0.252068
$393$	911233.	0.297610
$394$	494949.	0.160627
$395$	0	0
$396$	1846.35	0.000591665 0
$397$	−1.96075e6	−0.624375	−0.312187	−	0.950021i	$-0.601062\pi$
$397$	−1.96075e6	−0.624375	−0.312187	+	0.950021i	$0.601062\pi$
$398$	2.14884e6	0.679981
$399$	875372.	0.275271
$400$	0	0
$401$	−3.86313e6	−1.19972	−0.599858	−	0.800107i	$-0.704776\pi$
$401$	−3.86313e6	−1.19972	−0.599858	+	0.800107i	$0.704776\pi$
$402$	−837641.	−0.258519
$403$	188964.	0.0579584
$404$	−35727.8	−0.0108906
$405$	0	0
$406$	−1.26447e6	−0.380710
$407$	−322722.	−0.0965702
$408$	−817776.	−0.243212
$409$	1.75292e6	0.518149	0.259075	−	0.965857i	$-0.416582\pi$
$409$	1.75292e6	0.518149	0.259075	+	0.965857i	$0.416582\pi$
$410$	0	0
$411$	−1.15480e6	−0.337212
$412$	−30310.8	−0.00879740
$413$	−3.42240e6	−0.987314
$414$	−361383.	−0.103626
$415$	0	0
$416$	4767.73	0.00135076
$417$	1.45579e6	0.409976
$418$	457717.	0.128132
$419$	−4.28414e6	−1.19214	−0.596071	−	0.802932i	$-0.703272\pi$
$419$	−4.28414e6	−1.19214	−0.596071	+	0.802932i	$0.703272\pi$
$420$	0	0
$421$	5.74050e6	1.57850	0.789250	−	0.614072i	$-0.210470\pi$
$421$	5.74050e6	1.57850	0.789250	+	0.614072i	$0.210470\pi$
$422$	3.52175e6	0.962670
$423$	94531.6	0.0256877
$424$	5.24182e6	1.41601
$425$	0	0
$426$	3.65785e6	0.976567
$427$	−2.55257e6	−0.677497
$428$	21333.3	0.00562924
$429$	−76128.4	−0.0199712
$430$	0	0
$431$	5.29228e6	1.37230	0.686150	−	0.727460i	$-0.259299\pi$
$431$	5.29228e6	1.37230	0.686150	+	0.727460i	$0.259299\pi$
$432$	−742076.	−0.191310
$433$	2.37840e6	0.609628	0.304814	−	0.952412i	$-0.401406\pi$
$433$	2.37840e6	0.609628	0.304814	+	0.952412i	$0.401406\pi$
$434$	−2.21098e6	−0.563457
$435$	0	0
$436$	333.123	8.39245e−5 0
$437$	530528.	0.132894
$438$	2.30868e6	0.575014
$439$	6.44393e6	1.59584	0.797921	−	0.602762i	$-0.205933\pi$
$439$	6.44393e6	1.59584	0.797921	+	0.602762i	$0.205933\pi$
$440$	0	0
$441$	342158.	0.0837780
$442$	197339.	0.0480459
$443$	−2501.14	−0.000605520 0	−0.000302760	−	1.00000i	$-0.500096\pi$
$443$	−2501.14	−0.000605520 0	−0.000302760		1.00000i	$0.500096\pi$
$444$	−4521.99	−0.00108861
$445$	0	0
$446$	3.71292e6	0.883849
$447$	−3.63979e6	−0.861603
$448$	−4.77970e6	−1.12514
$449$	−5.04731e6	−1.18153	−0.590764	−	0.806844i	$-0.701174\pi$
$449$	−5.04731e6	−1.18153	−0.590764	+	0.806844i	$0.701174\pi$
$450$	0	0
$451$	−1.16427e6	−0.269534
$452$	−14557.0	−0.00335140
$453$	3.74562e6	0.857587
$454$	−1.24058e6	−0.282478
$455$	0	0
$456$	1.09585e6	0.246798
$457$	−4.62692e6	−1.03634	−0.518169	−	0.855278i	$-0.673386\pi$
$457$	−4.62692e6	−1.03634	−0.518169	+	0.855278i	$0.673386\pi$
$458$	−1.76720e6	−0.393661
$459$	364861.	0.0808343
$460$	0	0
$461$	1.43209e6	0.313846	0.156923	−	0.987611i	$-0.449842\pi$
$461$	1.43209e6	0.313846	0.156923	+	0.987611i	$0.449842\pi$
$462$	890743.	0.194155
$463$	−1.58115e6	−0.342785	−0.171393	−	0.985203i	$-0.554827\pi$
$463$	−1.58115e6	−0.342785	−0.171393	+	0.985203i	$0.554827\pi$
$464$	−1.57364e6	−0.339321
$465$	0	0
$466$	5.73089e6	1.22252
$467$	−8724.92	−0.00185127	−0.000925634	−	1.00000i	$-0.500295\pi$
$467$	−8724.92	−0.00185127	−0.000925634		1.00000i	$0.500295\pi$
$468$	−1066.71	−0.000225130 0
$469$	2.39306e6	0.502368
$470$	0	0
$471$	−2.99551e6	−0.622183
$472$	−4.28441e6	−0.885189
$473$	−808470.	−0.166154
$474$	−1.10095e6	−0.225073
$475$	0	0
$476$	13673.4	0.00276603
$477$	−2.33870e6	−0.470629
$478$	4.30018e6	0.860830
$479$	8.15482e6	1.62396	0.811981	−	0.583684i	$-0.198389\pi$
$479$	8.15482e6	1.62396	0.811981	+	0.583684i	$0.198389\pi$
$480$	0	0
$481$	186450.	0.0367452
$482$	−129212.	−0.0253329
$483$	1.03244e6	0.201371
$484$	−2758.13	−0.000535181 0
$485$	0	0
$486$	333047.	0.0639609
$487$	−9.00123e6	−1.71981	−0.859903	−	0.510457i	$-0.829476\pi$
$487$	−9.00123e6	−1.71981	−0.859903	+	0.510457i	$0.829476\pi$
$488$	−3.19549e6	−0.607418
$489$	132884.	0.0251305
$490$	0	0
$491$	6.03644e6	1.13000	0.564999	−	0.825092i	$-0.308877\pi$
$491$	6.03644e6	1.13000	0.564999	+	0.825092i	$0.308877\pi$
$492$	−16313.8	−0.00303838
$493$	773721.	0.143373
$494$	−264442.	−0.0487543
$495$	0	0
$496$	−2.75157e6	−0.502200
$497$	−1.04501e7	−1.89772
$498$	−351998.	−0.0636015
$499$	233452.	0.0419707	0.0209853	−	0.999780i	$-0.493320\pi$
$499$	233452.	0.0419707	0.0209853	+	0.999780i	$0.493320\pi$
$500$	0	0
$501$	−1.52964e6	−0.272267
$502$	−3.89826e6	−0.690417
$503$	7.79728e6	1.37412	0.687058	−	0.726603i	$-0.258902\pi$
$503$	7.79728e6	1.37412	0.687058	+	0.726603i	$0.258902\pi$
$504$	2.13260e6	0.373966
$505$	0	0
$506$	539844.	0.0937329
$507$	−3.29765e6	−0.569751
$508$	19725.1	0.00339125
$509$	5.32415e6	0.910869	0.455434	−	0.890269i	$-0.349484\pi$
$509$	5.32415e6	0.910869	0.455434	+	0.890269i	$0.349484\pi$
$510$	0	0
$511$	−6.59569e6	−1.11740
$512$	−5.98317e6	−1.00869
$513$	−488929.	−0.0820262
$514$	7.60548e6	1.26975
$515$	0	0
$516$	−11328.3	−0.00187301
$517$	−141214.	−0.0232354
$518$	−2.18157e6	−0.357227
$519$	−6.53282e6	−1.06459
$520$	0	0
$521$	−8.42076e6	−1.35912	−0.679559	−	0.733621i	$-0.737829\pi$
$521$	−8.42076e6	−1.35912	−0.679559	+	0.733621i	$0.737829\pi$
$522$	706257.	0.113445
$523$	6.81759e6	1.08988	0.544938	−	0.838476i	$-0.316553\pi$
$523$	6.81759e6	1.08988	0.544938	+	0.838476i	$0.316553\pi$
$524$	−19073.5	−0.00303460
$525$	0	0
$526$	−7.58833e6	−1.19586
$527$	1.35288e6	0.212194
$528$	1.10853e6	0.173047
$529$	−5.81062e6	−0.902783
$530$	0	0
$531$	1.91154e6	0.294203
$532$	−18322.9	−0.00280682
$533$	672649.	0.102558
$534$	2.17157e6	0.329550
$535$	0	0
$536$	2.99581e6	0.450404
$537$	623802.	0.0933493
$538$	−1.14949e7	−1.71218
$539$	−511125.	−0.0757801
$540$	0	0
$541$	−1.00313e7	−1.47354	−0.736770	−	0.676143i	$-0.763650\pi$
$541$	−1.00313e7	−1.47354	−0.736770	+	0.676143i	$0.763650\pi$
$542$	−1.22986e6	−0.179828
$543$	7.11356e6	1.03535
$544$	34134.4	0.00494533
$545$	0	0
$546$	−514619.	−0.0738762
$547$	−341056.	−0.0487368	−0.0243684	−	0.999703i	$-0.507757\pi$
$547$	−341056.	−0.0487368	−0.0243684	+	0.999703i	$0.507757\pi$
$548$	24171.7	0.00343840
$549$	1.42571e6	0.201883
$550$	0	0
$551$	−1.03682e6	−0.145487
$552$	1.29248e6	0.180541
$553$	3.14532e6	0.437373
$554$	2.31348e6	0.320252
$555$	0	0
$556$	−30471.9	−0.00418035
$557$	−8.70989e6	−1.18953	−0.594764	−	0.803901i	$-0.702754\pi$
$557$	−8.70989e6	−1.18953	−0.594764	+	0.803901i	$0.702754\pi$
$558$	1.23492e6	0.167901
$559$	467087.	0.0632219
$560$	0	0
$561$	−545039.	−0.0731174
$562$	−2.29324e6	−0.306272
$563$	1.05570e7	1.40369	0.701845	−	0.712330i	$-0.252360\pi$
$563$	1.05570e7	1.40369	0.701845	+	0.712330i	$0.252360\pi$
$564$	−1978.69	−0.000261927 0
$565$	0	0
$566$	−7.13011e6	−0.935525
$567$	−951485.	−0.124292
$568$	−1.30823e7	−1.70142
$569$	−2.88092e6	−0.373035	−0.186518	−	0.982452i	$-0.559720\pi$
$569$	−2.88092e6	−0.373035	−0.186518	+	0.982452i	$0.559720\pi$
$570$	0	0
$571$	−8.00484e6	−1.02745	−0.513727	−	0.857954i	$-0.671735\pi$
$571$	−8.00484e6	−1.02745	−0.513727	+	0.857954i	$0.671735\pi$
$572$	1593.48	0.000203637 0
$573$	3.77081e6	0.479786
$574$	−7.87035e6	−0.997043
$575$	0	0
$576$	2.66965e6	0.335272
$577$	1.42967e7	1.78770	0.893852	−	0.448361i	$-0.147992\pi$
$577$	1.42967e7	1.78770	0.893852	+	0.448361i	$0.147992\pi$
$578$	−6.59541e6	−0.821149
$579$	−2.38972e6	−0.296244
$580$	0	0
$581$	1.00563e6	0.123594
$582$	1.06562e6	0.130405
$583$	3.49362e6	0.425700
$584$	−8.25697e6	−1.00182
$585$	0	0
$586$	−1.98026e6	−0.238220
$587$	−9.83619e6	−1.17823	−0.589117	−	0.808047i	$-0.700524\pi$
$587$	−9.83619e6	−1.17823	−0.589117	+	0.808047i	$0.700524\pi$
$588$	−7161.88	−0.000854248 0
$589$	−1.81292e6	−0.215323
$590$	0	0
$591$	789787.	0.0930125
$592$	−2.71497e6	−0.318390
$593$	9.84429e6	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$593$	9.84429e6	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$594$	−497515.	−0.0578548
$595$	0	0
$596$	76186.3	0.00878540
$597$	3.42890e6	0.393748
$598$	−311890.	−0.0356656
$599$	−3.94699e6	−0.449468	−0.224734	−	0.974420i	$-0.572151\pi$
$599$	−3.94699e6	−0.449468	−0.224734	+	0.974420i	$0.572151\pi$
$600$	0	0
$601$	−7.63150e6	−0.861834	−0.430917	−	0.902392i	$-0.641810\pi$
$601$	−7.63150e6	−0.861834	−0.430917	+	0.902392i	$0.641810\pi$
$602$	−5.46516e6	−0.614627
$603$	−1.33662e6	−0.149697
$604$	−78401.5	−0.00874444
$605$	0	0
$606$	9.62715e6	1.06492
$607$	1.12292e7	1.23703	0.618513	−	0.785775i	$-0.287736\pi$
$607$	1.12292e7	1.23703	0.618513	+	0.785775i	$0.287736\pi$
$608$	−45741.6	−0.00501825
$609$	−2.01771e6	−0.220453
$610$	0	0
$611$	81585.1	0.00884113
$612$	−7637.10	−0.000824232 0
$613$	−2.62390e6	−0.282031	−0.141016	−	0.990007i	$-0.545037\pi$
$613$	−2.62390e6	−0.282031	−0.141016	+	0.990007i	$0.545037\pi$
$614$	8.24753e6	0.882883
$615$	0	0
$616$	−3.18573e6	−0.338265
$617$	3.59228e6	0.379889	0.189945	−	0.981795i	$-0.439169\pi$
$617$	3.59228e6	0.379889	0.189945	+	0.981795i	$0.439169\pi$
$618$	8.16750e6	0.860237
$619$	−1.19777e7	−1.25646	−0.628228	−	0.778029i	$-0.716219\pi$
$619$	−1.19777e7	−1.25646	−0.628228	+	0.778029i	$0.716219\pi$
$620$	0	0
$621$	−576657.	−0.0600052
$622$	1.24249e7	1.28770
$623$	−6.20398e6	−0.640398
$624$	−640445.	−0.0658447
$625$	0	0
$626$	−3.65398e6	−0.372675
$627$	730376.	0.0741955
$628$	62700.6	0.00634413
$629$	1.33488e6	0.134529
$630$	0	0
$631$	2.50844e6	0.250801	0.125401	−	0.992106i	$-0.459978\pi$
$631$	2.50844e6	0.250801	0.125401	+	0.992106i	$0.459978\pi$
$632$	3.93754e6	0.392132
$633$	5.61963e6	0.557441
$634$	9.18972e6	0.907986
$635$	0	0
$636$	48952.6	0.00479880
$637$	295298.	0.0288345
$638$	−1.05503e6	−0.102615
$639$	5.83681e6	0.565488
$640$	0	0
$641$	1.12431e7	1.08079	0.540395	−	0.841411i	$-0.318275\pi$
$641$	1.12431e7	1.08079	0.540395	+	0.841411i	$0.318275\pi$
$642$	−5.74845e6	−0.550444
$643$	−5.61271e6	−0.535360	−0.267680	−	0.963508i	$-0.586257\pi$
$643$	−5.61271e6	−0.535360	−0.267680	+	0.963508i	$0.586257\pi$
$644$	−21610.5	−0.00205329
$645$	0	0
$646$	−1.89326e6	−0.178497
$647$	−1.10739e7	−1.04002	−0.520008	−	0.854162i	$-0.674071\pi$
$647$	−1.10739e7	−1.04002	−0.520008	+	0.854162i	$0.674071\pi$
$648$	−1.19114e6	−0.111436
$649$	−2.85551e6	−0.266117
$650$	0	0
$651$	−3.52805e6	−0.326274
$652$	−2781.46	−0.000256244 0
$653$	7.83486e6	0.719032	0.359516	−	0.933139i	$-0.382942\pi$
$653$	7.83486e6	0.719032	0.359516	+	0.933139i	$0.382942\pi$
$654$	−89763.0	−0.00820640
$655$	0	0
$656$	−9.79467e6	−0.888649
$657$	3.68395e6	0.332966
$658$	−954589.	−0.0859511
$659$	1.02774e7	0.921868	0.460934	−	0.887434i	$-0.347514\pi$
$659$	1.02774e7	0.921868	0.460934	+	0.887434i	$0.347514\pi$
$660$	0	0
$661$	−1.42467e7	−1.26827	−0.634134	−	0.773223i	$-0.718643\pi$
$661$	−1.42467e7	−1.26827	−0.634134	+	0.773223i	$0.718643\pi$
$662$	4.50842e6	0.399833
$663$	314892.	0.0278213
$664$	1.25892e6	0.110810
$665$	0	0
$666$	1.21849e6	0.106448
$667$	−1.22285e6	−0.106429
$668$	32017.6	0.00277618
$669$	5.92468e6	0.511799
$670$	0	0
$671$	−2.12976e6	−0.182610
$672$	−89015.7	−0.00760403
$673$	3.85183e6	0.327815	0.163908	−	0.986476i	$-0.447590\pi$
$673$	3.85183e6	0.327815	0.163908	+	0.986476i	$0.447590\pi$
$674$	2.12486e7	1.80169
$675$	0	0
$676$	69024.9	0.00580951
$677$	−1.39430e7	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$677$	−1.39430e7	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$678$	3.92251e6	0.327710
$679$	−3.04437e6	−0.253409
$680$	0	0
$681$	−1.97959e6	−0.163571
$682$	−1.84475e6	−0.151872
$683$	1.24329e7	1.01981	0.509905	−	0.860231i	$-0.329681\pi$
$683$	1.24329e7	1.01981	0.509905	+	0.860231i	$0.329681\pi$
$684$	10234.0	0.000836386 0
$685$	0	0
$686$	1.02921e7	0.835012
$687$	−2.81991e6	−0.227952
$688$	−6.80141e6	−0.547807
$689$	−2.01841e6	−0.161980
$690$	0	0
$691$	1.46905e7	1.17042	0.585208	−	0.810883i	$-0.301013\pi$
$691$	1.46905e7	1.17042	0.585208	+	0.810883i	$0.301013\pi$
$692$	136742.	0.0108552
$693$	1.42135e6	0.112427
$694$	−5.20881e6	−0.410525
$695$	0	0
$696$	−2.52592e6	−0.197650
$697$	4.81581e6	0.375480
$698$	1.47221e7	1.14375
$699$	9.14474e6	0.707911
$700$	0	0
$701$	−3.55096e6	−0.272930	−0.136465	−	0.990645i	$-0.543574\pi$
$701$	−3.55096e6	−0.272930	−0.136465	+	0.990645i	$0.543574\pi$
$702$	287435.	0.0220139
$703$	−1.78880e6	−0.136513
$704$	−3.98799e6	−0.303265
$705$	0	0
$706$	−7.08657e6	−0.535087
$707$	−2.75039e7	−2.06941
$708$	−40011.5	−0.00299986
$709$	−8.56408e6	−0.639831	−0.319915	−	0.947446i	$-0.603654\pi$
$709$	−8.56408e6	−0.639831	−0.319915	+	0.947446i	$0.603654\pi$
$710$	0	0
$711$	−1.75678e6	−0.130330
$712$	−7.76659e6	−0.574157
$713$	−2.13821e6	−0.157517
$714$	−3.68440e6	−0.270472
$715$	0	0
$716$	−13057.1	−0.000951843 0
$717$	6.86178e6	0.498470
$718$	−1.34349e7	−0.972576
$719$	1.07318e7	0.774195	0.387097	−	0.922039i	$-0.373478\pi$
$719$	1.07318e7	0.774195	0.387097	+	0.922039i	$0.373478\pi$
$720$	0	0
$721$	−2.33338e7	−1.67166
$722$	−1.14286e7	−0.815924
$723$	−206182.	−0.0146692
$724$	−148898.	−0.0105570
$725$	0	0
$726$	743201.	0.0523317
$727$	−1.49237e7	−1.04722	−0.523611	−	0.851957i	$-0.675416\pi$
$727$	−1.49237e7	−1.04722	−0.523611	+	0.851957i	$0.675416\pi$
$728$	1.84053e6	0.128711
$729$	531441.	0.0370370
$730$	0	0
$731$	3.34409e6	0.231465
$732$	−29842.2	−0.00205851
$733$	−1.21777e7	−0.837155	−0.418577	−	0.908181i	$-0.637471\pi$
$733$	−1.21777e7	−0.837155	−0.418577	+	0.908181i	$0.637471\pi$
$734$	1.95331e7	1.33823
$735$	0	0
$736$	−53948.9	−0.00367103
$737$	1.99668e6	0.135406
$738$	4.39589e6	0.297102
$739$	4.10708e6	0.276644	0.138322	−	0.990387i	$-0.455829\pi$
$739$	4.10708e6	0.276644	0.138322	+	0.990387i	$0.455829\pi$
$740$	0	0
$741$	−421969.	−0.0282315
$742$	2.36164e7	1.57472
$743$	−2.73077e7	−1.81474	−0.907368	−	0.420337i	$-0.861912\pi$
$743$	−2.73077e7	−1.81474	−0.907368	+	0.420337i	$0.861912\pi$
$744$	−4.41667e6	−0.292525
$745$	0	0
$746$	−4.57710e6	−0.301123
$747$	−561682.	−0.0368289
$748$	11408.5	0.000745546 0
$749$	1.64228e7	1.06965
$750$	0	0
$751$	−1.54004e7	−0.996398	−0.498199	−	0.867063i	$-0.666005\pi$
$751$	−1.54004e7	−0.996398	−0.498199	+	0.867063i	$0.666005\pi$
$752$	−1.18799e6	−0.0766069
$753$	−6.22043e6	−0.399791
$754$	609532.	0.0390452
$755$	0	0
$756$	19916.0	0.00126735
$757$	−8.23327e6	−0.522195	−0.261097	−	0.965312i	$-0.584084\pi$
$757$	−8.23327e6	−0.522195	−0.261097	+	0.965312i	$0.584084\pi$
$758$	−2.94689e6	−0.186290
$759$	861426.	0.0542767
$760$	0	0
$761$	1.69020e7	1.05798	0.528989	−	0.848629i	$-0.322571\pi$
$761$	1.69020e7	1.05798	0.528989	+	0.848629i	$0.322571\pi$
$762$	−5.31509e6	−0.331607
$763$	256445.	0.0159471
$764$	−78928.8	−0.00489217
$765$	0	0
$766$	1.26881e7	0.781315
$767$	1.64975e6	0.101258
$768$	−166664.	−0.0101962
$769$	−1.72204e7	−1.05009	−0.525045	−	0.851075i	$-0.675951\pi$
$769$	−1.72204e7	−1.05009	−0.525045	+	0.851075i	$0.675951\pi$
$770$	0	0
$771$	1.21360e7	0.735259
$772$	50020.4	0.00302067
$773$	−1.86902e7	−1.12503	−0.562517	−	0.826786i	$-0.690167\pi$
$773$	−1.86902e7	−1.12503	−0.562517	+	0.826786i	$0.690167\pi$
$774$	3.05250e6	0.183149
$775$	0	0
$776$	−3.81116e6	−0.227197
$777$	−3.48111e6	−0.206855
$778$	−1.36523e7	−0.808644
$779$	−6.45339e6	−0.381017
$780$	0	0
$781$	−8.71919e6	−0.511503
$782$	−2.23297e6	−0.130577
$783$	1.12697e6	0.0656913
$784$	−4.29994e6	−0.249846
$785$	0	0
$786$	5.13952e6	0.296733
$787$	8.17563e6	0.470527	0.235264	−	0.971932i	$-0.424405\pi$
$787$	8.17563e6	0.470527	0.235264	+	0.971932i	$0.424405\pi$
$788$	−16531.4	−0.000948408 0
$789$	−1.21087e7	−0.692474
$790$	0	0
$791$	−1.12063e7	−0.636824
$792$	1.77935e6	0.100797
$793$	1.23045e6	0.0694834
$794$	−1.10590e7	−0.622534
$795$	0	0
$796$	−71772.0	−0.00401488
$797$	3.39844e7	1.89511	0.947554	−	0.319595i	$-0.103547\pi$
$797$	3.39844e7	1.89511	0.947554	+	0.319595i	$0.103547\pi$
$798$	4.93726e6	0.274460
$799$	584106.	0.0323687
$800$	0	0
$801$	3.46516e6	0.190828
$802$	−2.17887e7	−1.19618
$803$	−5.50318e6	−0.301179
$804$	27977.5	0.00152640
$805$	0	0
$806$	1.06579e6	0.0577876
$807$	−1.83423e7	−0.991449
$808$	−3.44314e7	−1.85535
$809$	2.50619e7	1.34630	0.673151	−	0.739505i	$-0.264940\pi$
$809$	2.50619e7	1.34630	0.673151	+	0.739505i	$0.264940\pi$
$810$	0	0
$811$	−1.73791e7	−0.927844	−0.463922	−	0.885876i	$-0.653558\pi$
$811$	−1.73791e7	−0.927844	−0.463922	+	0.885876i	$0.653558\pi$
$812$	42233.7	0.00224786
$813$	−1.96248e6	−0.104131
$814$	−1.82021e6	−0.0962855
$815$	0	0
$816$	−4.58525e6	−0.241067
$817$	−4.48123e6	−0.234878
$818$	9.88681e6	0.516622
$819$	−821175.	−0.0427786
$820$	0	0
$821$	−1.21448e7	−0.628827	−0.314413	−	0.949286i	$-0.601808\pi$
$821$	−1.21448e7	−0.628827	−0.314413	+	0.949286i	$0.601808\pi$
$822$	−6.51329e6	−0.336218
$823$	−9.33503e6	−0.480415	−0.240207	−	0.970722i	$-0.577215\pi$
$823$	−9.33503e6	−0.480415	−0.240207	+	0.970722i	$0.577215\pi$
$824$	−2.92110e7	−1.49875
$825$	0	0
$826$	−1.93029e7	−0.984404
$827$	1.15378e7	0.586622	0.293311	−	0.956017i	$-0.405243\pi$
$827$	1.15378e7	0.586622	0.293311	+	0.956017i	$0.405243\pi$
$828$	12070.3	0.000611847 0
$829$	−1.25927e7	−0.636404	−0.318202	−	0.948023i	$-0.603079\pi$
$829$	−1.25927e7	−0.636404	−0.318202	+	0.948023i	$0.603079\pi$
$830$	0	0
$831$	3.69161e6	0.185444
$832$	2.30403e6	0.115393
$833$	2.11418e6	0.105567
$834$	8.21092e6	0.408768
$835$	0	0
$836$	−15287.9	−0.000756539 0
$837$	1.97055e6	0.0972241
$838$	−2.41633e7	−1.18863
$839$	3.69853e7	1.81395	0.906973	−	0.421189i	$-0.138387\pi$
$839$	3.69853e7	1.81395	0.906973	+	0.421189i	$0.138387\pi$
$840$	0	0
$841$	−1.81213e7	−0.883486
$842$	3.23775e7	1.57385
$843$	−3.65930e6	−0.177349
$844$	−117627.	−0.00568398
$845$	0	0
$846$	533175.	0.0256120
$847$	−2.12326e6	−0.101694
$848$	2.93907e7	1.40353
$849$	−1.13775e7	−0.541723
$850$	0	0
$851$	−2.10976e6	−0.0998642
$852$	−122173.	−0.00576604
$853$	−3.49513e7	−1.64471	−0.822357	−	0.568972i	$-0.807341\pi$
$853$	−3.49513e7	−1.64471	−0.822357	+	0.568972i	$0.807341\pi$
$854$	−1.43969e7	−0.675500
$855$	0	0
$856$	2.05593e7	0.959010
$857$	2.24410e7	1.04373	0.521867	−	0.853027i	$-0.325236\pi$
$857$	2.24410e7	1.04373	0.521867	+	0.853027i	$0.325236\pi$
$858$	−429378.	−0.0199123
$859$	3.30887e7	1.53002	0.765010	−	0.644019i	$-0.222734\pi$
$859$	3.30887e7	1.53002	0.765010	+	0.644019i	$0.222734\pi$
$860$	0	0
$861$	−1.25587e7	−0.577345
$862$	2.98494e7	1.36826
$863$	−1.58950e6	−0.0726495	−0.0363248	−	0.999340i	$-0.511565\pi$
$863$	−1.58950e6	−0.0726495	−0.0363248	+	0.999340i	$0.511565\pi$
$864$	49718.7	0.00226587
$865$	0	0
$866$	1.34146e7	0.607830
$867$	−1.05243e7	−0.475492
$868$	73847.4	0.00332687
$869$	2.62433e6	0.117888
$870$	0	0
$871$	−1.15356e6	−0.0515224
$872$	321036.	0.0142976
$873$	1.70040e6	0.0755118
$874$	2.99228e6	0.132502
$875$	0	0
$876$	−77110.6	−0.00339511
$877$	1.78401e7	0.783246	0.391623	−	0.920126i	$-0.371914\pi$
$877$	1.78401e7	0.783246	0.391623	+	0.920126i	$0.371914\pi$
$878$	3.63449e7	1.59114
$879$	−3.15989e6	−0.137943
$880$	0	0
$881$	−1.53213e7	−0.665054	−0.332527	−	0.943094i	$-0.607901\pi$
$881$	−1.53213e7	−0.665054	−0.332527	+	0.943094i	$0.607901\pi$
$882$	1.92983e6	0.0835310
$883$	−1.92226e7	−0.829680	−0.414840	−	0.909894i	$-0.636162\pi$
$883$	−1.92226e7	−0.829680	−0.414840	+	0.909894i	$0.636162\pi$
$884$	−6591.17	−0.000283682 0
$885$	0	0
$886$	−14106.9	−0.000603735 0
$887$	1.78729e7	0.762756	0.381378	−	0.924419i	$-0.375450\pi$
$887$	1.78729e7	0.762756	0.381378	+	0.924419i	$0.375450\pi$
$888$	−4.35791e6	−0.185458
$889$	1.51847e7	0.644396
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−124013.	−0.00521860
$893$	−782727.	−0.0328459
$894$	−2.05291e7	−0.859063
$895$	0	0
$896$	−2.66419e7	−1.10865
$897$	−497682.	−0.0206524
$898$	−2.84677e7	−1.17804
$899$	4.17874e6	0.172443
$900$	0	0
$901$	−1.44507e7	−0.593032
$902$	−6.56670e6	−0.268739
$903$	−8.72072e6	−0.355904
$904$	−1.40288e7	−0.570953
$905$	0	0
$906$	2.11260e7	0.855059
$907$	−2.36602e7	−0.954993	−0.477497	−	0.878634i	$-0.658456\pi$
$907$	−2.36602e7	−0.954993	−0.477497	+	0.878634i	$0.658456\pi$
$908$	41435.7	0.00166786
$909$	1.53620e7	0.616649
$910$	0	0
$911$	2.62399e7	1.04753	0.523766	−	0.851862i	$-0.324527\pi$
$911$	2.62399e7	1.04753	0.523766	+	0.851862i	$0.324527\pi$
$912$	6.14443e6	0.244621
$913$	839056.	0.0333130
$914$	−2.60966e7	−1.03328
$915$	0	0
$916$	59025.0	0.00232433
$917$	−1.46831e7	−0.576627
$918$	2.05788e6	0.0805960
$919$	1.05336e7	0.411422	0.205711	−	0.978613i	$-0.434049\pi$
$919$	1.05336e7	0.411422	0.205711	+	0.978613i	$0.434049\pi$
$920$	0	0
$921$	1.31605e7	0.511240
$922$	8.07723e6	0.312921
$923$	5.03744e6	0.194628
$924$	−29751.1	−0.00114637
$925$	0	0
$926$	−8.91800e6	−0.341775
$927$	1.30328e7	0.498127
$928$	105433.	0.00401890
$929$	4.93218e7	1.87499	0.937497	−	0.347993i	$-0.113137\pi$
$929$	4.93218e7	1.87499	0.937497	+	0.347993i	$0.113137\pi$
$930$	0	0
$931$	−2.83309e6	−0.107124
$932$	−191413.	−0.00721826
$933$	1.98263e7	0.745653
$934$	−49210.1	−0.00184581
$935$	0	0
$936$	−1.02801e6	−0.0383536
$937$	−3.96100e6	−0.147386	−0.0736929	−	0.997281i	$-0.523478\pi$
$937$	−3.96100e6	−0.147386	−0.0736929	+	0.997281i	$0.523478\pi$
$938$	1.34973e7	0.500887
$939$	−5.83063e6	−0.215800
$940$	0	0
$941$	3.82868e7	1.40953	0.704766	−	0.709440i	$-0.251052\pi$
$941$	3.82868e7	1.40953	0.704766	+	0.709440i	$0.251052\pi$
$942$	−1.68952e7	−0.620349
$943$	−7.61131e6	−0.278728
$944$	−2.40226e7	−0.877383
$945$	0	0
$946$	−4.55991e6	−0.165664
$947$	1.34832e7	0.488562	0.244281	−	0.969705i	$-0.421448\pi$
$947$	1.34832e7	0.488562	0.244281	+	0.969705i	$0.421448\pi$
$948$	36772.2	0.00132892
$949$	3.17942e6	0.114599
$950$	0	0
$951$	1.46640e7	0.525776
$952$	1.31772e7	0.471228
$953$	9.20696e6	0.328385	0.164193	−	0.986428i	$-0.447498\pi$
$953$	9.20696e6	0.328385	0.164193	+	0.986428i	$0.447498\pi$
$954$	−1.31907e7	−0.469242
$955$	0	0
$956$	−143627.	−0.00508268
$957$	−1.68350e6	−0.0594200
$958$	4.59947e7	1.61917
$959$	1.86079e7	0.653356
$960$	0	0
$961$	−2.13225e7	−0.744782
$962$	1.05161e6	0.0366368
$963$	−9.17277e6	−0.318739
$964$	4315.71	0.000149575 0
$965$	0	0
$966$	5.82314e6	0.200777
$967$	−2.95285e6	−0.101549	−0.0507745	−	0.998710i	$-0.516169\pi$
$967$	−2.95285e6	−0.101549	−0.0507745	+	0.998710i	$0.516169\pi$
$968$	−2.65805e6	−0.0911747
$969$	−3.02107e6	−0.103360
$970$	0	0
$971$	−3.09624e7	−1.05387	−0.526935	−	0.849906i	$-0.676659\pi$
$971$	−3.09624e7	−1.05387	−0.526935	+	0.849906i	$0.676659\pi$
$972$	−11123.9	−0.000377651 0
$973$	−2.34578e7	−0.794339
$974$	−5.07686e7	−1.71474
$975$	0	0
$976$	−1.79170e7	−0.602062
$977$	2.85400e7	0.956573	0.478287	−	0.878204i	$-0.341258\pi$
$977$	2.85400e7	0.956573	0.478287	+	0.878204i	$0.341258\pi$
$978$	749489.	0.0250564
$979$	−5.17635e6	−0.172610
$980$	0	0
$981$	−143234.	−0.00475198
$982$	3.40466e7	1.12667
$983$	4.46869e7	1.47501	0.737507	−	0.675340i	$-0.236003\pi$
$983$	4.46869e7	1.47501	0.737507	+	0.675340i	$0.236003\pi$
$984$	−1.57219e7	−0.517626
$985$	0	0
$986$	4.36393e6	0.142950
$987$	−1.52323e6	−0.0497706
$988$	8832.44	0.000287865 0
$989$	−5.28529e6	−0.171822
$990$	0	0
$991$	5.85115e7	1.89259	0.946296	−	0.323302i	$-0.104793\pi$
$991$	5.85115e7	1.89259	0.946296	+	0.323302i	$0.104793\pi$
$992$	184354.	0.00594804
$993$	7.19405e6	0.231526
$994$	−5.89407e7	−1.89212
$995$	0	0
$996$	11756.9	0.000375528 0
$997$	4.32684e7	1.37858	0.689292	−	0.724483i	$-0.257922\pi$
$997$	4.32684e7	1.37858	0.689292	+	0.724483i	$0.257922\pi$
$998$	1.31671e6	0.0418469
$999$	1.94434e6	0.0616393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.h.1.3		3
5.4	even	2		165.6.a.d.1.1	✓	3
15.14	odd	2		495.6.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.1	✓	3	5.4	even	2
495.6.a.c.1.3		3	15.14	odd	2
825.6.a.h.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} -121.000 q^{11} -1.69545 q^{12} +69.9067 q^{13} -817.946 q^{14} -1017.94 q^{16} +500.495 q^{17} +456.854 q^{18} -670.685 q^{19} -1305.19 q^{21} -682.462 q^{22} -791.025 q^{23} -1633.93 q^{24} +394.287 q^{26} +729.000 q^{27} +27.3196 q^{28} +1545.91 q^{29} +2703.09 q^{31} +68.2013 q^{32} -1089.00 q^{33} +2822.88 q^{34} -15.2591 q^{36} +2667.13 q^{37} -3782.78 q^{38} +629.161 q^{39} +9622.09 q^{41} -7361.51 q^{42} +6681.57 q^{43} +22.7944 q^{44} -4461.52 q^{46} +1167.06 q^{47} -9161.43 q^{48} +4224.17 q^{49} +4504.46 q^{51} -13.1693 q^{52} -28872.9 q^{53} +4111.69 q^{54} +26328.4 q^{56} -6036.16 q^{57} +8719.22 q^{58} +23599.3 q^{59} +17601.3 q^{61} +15245.9 q^{62} -11746.7 q^{63} +32958.6 q^{64} -6142.15 q^{66} -16501.5 q^{67} -94.2851 q^{68} -7119.22 q^{69} +72059.4 q^{71} -14705.4 q^{72} +45480.8 q^{73} +15043.1 q^{74} +126.346 q^{76} +17547.6 q^{77} +3548.58 q^{78} -21688.7 q^{79} +6561.00 q^{81} +54270.3 q^{82} -6934.34 q^{83} +245.877 q^{84} +37685.2 q^{86} +13913.2 q^{87} +21967.3 q^{88} +42779.8 q^{89} -10138.0 q^{91} +149.016 q^{92} +24327.8 q^{93} +6582.41 q^{94} +613.812 q^{96} +20992.6 q^{97} +23825.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Introduction

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