LMFDB - Embedded newform 825.6.a.t.1.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + \cdots - 7036608 $ x^10 - x^9 - 271x^8 + 309x^7 + 24456x^6 - 33410x^5 - 822204x^4 + 1367872x^3 + 7443872x^2 - 12856224x - 7036608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.444648$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +121.000 q^{11} -286.221 q^{12} -893.005 q^{13} +91.5105 q^{14} +1005.06 q^{16} +836.291 q^{17} +36.0165 q^{18} +816.231 q^{19} +1852.24 q^{21} +53.8024 q^{22} +3125.01 q^{23} -255.326 q^{24} -397.073 q^{26} +729.000 q^{27} -6545.05 q^{28} -7307.37 q^{29} +152.387 q^{31} +1354.72 q^{32} +1089.00 q^{33} +371.855 q^{34} -2575.99 q^{36} +8734.38 q^{37} +362.935 q^{38} -8037.05 q^{39} -9608.76 q^{41} +823.594 q^{42} +9208.31 q^{43} -3848.08 q^{44} +1389.53 q^{46} +11806.9 q^{47} +9045.53 q^{48} +25548.5 q^{49} +7526.62 q^{51} +28399.6 q^{52} +12879.4 q^{53} +324.148 q^{54} -5838.58 q^{56} +7346.08 q^{57} -3249.21 q^{58} -517.608 q^{59} -50201.8 q^{61} +67.7587 q^{62} +16670.2 q^{63} -31559.5 q^{64} +484.221 q^{66} +4050.14 q^{67} -26596.0 q^{68} +28125.1 q^{69} +16467.7 q^{71} -2297.93 q^{72} -55131.2 q^{73} +3883.72 q^{74} -25958.0 q^{76} +24902.3 q^{77} -3573.66 q^{78} +25009.6 q^{79} +6561.00 q^{81} -4272.51 q^{82} -76147.7 q^{83} -58905.5 q^{84} +4094.45 q^{86} -65766.3 q^{87} -3432.72 q^{88} +104461. q^{89} -183784. q^{91} -99382.6 q^{92} +1371.49 q^{93} +5249.89 q^{94} +12192.5 q^{96} +39347.8 q^{97} +11360.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) $ 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 + 1210 * q^11 + 2007 * q^12 - 1102 * q^13 + 684 * q^14 + 7019 * q^16 + 1106 * q^17 - 81 * q^18 + 2586 * q^19 + 1692 * q^21 - 121 * q^22 + 2206 * q^23 + 1593 * q^24 + 12001 * q^26 + 7290 * q^27 + 14452 * q^28 + 5824 * q^29 - 4586 * q^31 - 10627 * q^32 + 10890 * q^33 + 7426 * q^34 + 18063 * q^36 - 18362 * q^37 + 44001 * q^38 - 9918 * q^39 - 5474 * q^41 + 6156 * q^42 + 20496 * q^43 + 26983 * q^44 + 19981 * q^46 - 14970 * q^47 + 63171 * q^48 + 68582 * q^49 + 9954 * q^51 - 58603 * q^52 + 61980 * q^53 - 729 * q^54 + 7132 * q^56 + 23274 * q^57 + 7161 * q^58 + 61190 * q^59 + 8230 * q^61 - 4509 * q^62 + 15228 * q^63 + 152223 * q^64 - 1089 * q^66 - 11930 * q^67 + 105598 * q^68 + 19854 * q^69 + 59822 * q^71 + 14337 * q^72 - 20680 * q^73 + 132564 * q^74 + 68165 * q^76 + 22748 * q^77 + 108009 * q^78 + 234494 * q^79 + 65610 * q^81 + 151948 * q^82 + 185478 * q^83 + 130068 * q^84 - 17825 * q^86 + 52416 * q^87 + 21417 * q^88 + 181834 * q^89 + 206274 * q^91 + 98373 * q^92 - 41274 * q^93 - 64998 * q^94 - 95643 * q^96 + 304358 * q^97 + 153453 * q^98 + 98010 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.444648	0.0786034	0.0393017	−	0.999227i	$-0.487487\pi$
$2$	0.444648	0.0786034	0.0393017	+	0.999227i	$0.487487\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.8023	−0.993822
$5$	0	0
$5$	0	0
$6$	4.00183	0.0453817
$7$	205.804	1.58748	0.793742	−	0.608254i	$-0.208130\pi$
$7$	205.804	1.58748	0.793742	+	0.608254i	$0.208130\pi$
$8$	−28.3695	−0.156721
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−286.221	−0.573783
$13$	−893.005	−1.46553	−0.732767	−	0.680480i	$-0.761771\pi$
$13$	−893.005	−1.46553	−0.732767	+	0.680480i	$0.761771\pi$
$14$	91.5105	0.124782
$15$	0	0
$16$	1005.06	0.981503
$17$	836.291	0.701835	0.350918	−	0.936406i	$-0.385870\pi$
$17$	836.291	0.701835	0.350918	+	0.936406i	$0.385870\pi$
$18$	36.0165	0.0262011
$19$	816.231	0.518715	0.259358	−	0.965781i	$-0.416489\pi$
$19$	816.231	0.518715	0.259358	+	0.965781i	$0.416489\pi$
$20$	0	0
$21$	1852.24	0.916535
$22$	53.8024	0.0236998
$23$	3125.01	1.23178	0.615889	−	0.787833i	$-0.288797\pi$
$23$	3125.01	1.23178	0.615889	+	0.787833i	$0.288797\pi$
$24$	−255.326	−0.0904830
$25$	0	0
$26$	−397.073	−0.115196
$27$	729.000	0.192450
$28$	−6545.05	−1.57768
$29$	−7307.37	−1.61349	−0.806745	−	0.590900i	$-0.798773\pi$
$29$	−7307.37	−1.61349	−0.806745	+	0.590900i	$0.798773\pi$
$30$	0	0
$31$	152.387	0.0284803	0.0142401	−	0.999899i	$-0.495467\pi$
$31$	152.387	0.0284803	0.0142401	+	0.999899i	$0.495467\pi$
$32$	1354.72	0.233871
$33$	1089.00	0.174078
$34$	371.855	0.0551666
$35$	0	0
$36$	−2575.99	−0.331274
$37$	8734.38	1.04888	0.524442	−	0.851446i	$-0.324274\pi$
$37$	8734.38	1.04888	0.524442	+	0.851446i	$0.324274\pi$
$38$	362.935	0.0407728
$39$	−8037.05	−0.846126
$40$	0	0
$41$	−9608.76	−0.892704	−0.446352	−	0.894857i	$-0.647277\pi$
$41$	−9608.76	−0.892704	−0.446352	+	0.894857i	$0.647277\pi$
$42$	823.594	0.0720427
$43$	9208.31	0.759467	0.379733	−	0.925096i	$-0.376016\pi$
$43$	9208.31	0.759467	0.379733	+	0.925096i	$0.376016\pi$
$44$	−3848.08	−0.299648
$45$	0	0
$46$	1389.53	0.0968219
$47$	11806.9	0.779632	0.389816	−	0.920893i	$-0.372539\pi$
$47$	11806.9	0.779632	0.389816	+	0.920893i	$0.372539\pi$
$48$	9045.53	0.566671
$49$	25548.5	1.52011
$50$	0	0
$51$	7526.62	0.405205
$52$	28399.6	1.45648
$53$	12879.4	0.629807	0.314904	−	0.949124i	$-0.398028\pi$
$53$	12879.4	0.629807	0.314904	+	0.949124i	$0.398028\pi$
$54$	324.148	0.0151272
$55$	0	0
$56$	−5838.58	−0.248792
$57$	7346.08	0.299480
$58$	−3249.21	−0.126826
$59$	−517.608	−0.0193585	−0.00967924	−	0.999953i	$-0.503081\pi$
$59$	−517.608	−0.0193585	−0.00967924	+	0.999953i	$0.503081\pi$
$60$	0	0
$61$	−50201.8	−1.72741	−0.863704	−	0.503999i	$-0.831861\pi$
$61$	−50201.8	−1.72741	−0.863704	+	0.503999i	$0.831861\pi$
$62$	67.7587	0.00223865
$63$	16670.2	0.529162
$64$	−31559.5	−0.963120
$65$	0	0
$66$	484.221	0.0136831
$67$	4050.14	0.110226	0.0551129	−	0.998480i	$-0.482448\pi$
$67$	4050.14	0.110226	0.0551129	+	0.998480i	$0.482448\pi$
$68$	−26596.0	−0.697499
$69$	28125.1	0.711167
$70$	0	0
$71$	16467.7	0.387692	0.193846	−	0.981032i	$-0.437904\pi$
$71$	16467.7	0.387692	0.193846	+	0.981032i	$0.437904\pi$
$72$	−2297.93	−0.0522404
$73$	−55131.2	−1.21085	−0.605425	−	0.795903i	$-0.706997\pi$
$73$	−55131.2	−1.21085	−0.605425	+	0.795903i	$0.706997\pi$
$74$	3883.72	0.0824458
$75$	0	0
$76$	−25958.0	−0.515510
$77$	24902.3	0.478645
$78$	−3573.66	−0.0665084
$79$	25009.6	0.450858	0.225429	−	0.974260i	$-0.427622\pi$
$79$	25009.6	0.450858	0.225429	+	0.974260i	$0.427622\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−4272.51	−0.0701696
$83$	−76147.7	−1.21328	−0.606641	−	0.794976i	$-0.707483\pi$
$83$	−76147.7	−1.21328	−0.606641	+	0.794976i	$0.707483\pi$
$84$	−58905.5	−0.910872
$85$	0	0
$86$	4094.45	0.0596966
$87$	−65766.3	−0.931548
$88$	−3432.72	−0.0472532
$89$	104461.	1.39790	0.698952	−	0.715169i	$-0.253650\pi$
$89$	104461.	1.39790	0.698952	+	0.715169i	$0.253650\pi$
$90$	0	0
$91$	−183784.	−2.32651
$92$	−99382.6	−1.22417
$93$	1371.49	0.0164431
$94$	5249.89	0.0612817
$95$	0	0
$96$	12192.5	0.135025
$97$	39347.8	0.424611	0.212306	−	0.977203i	$-0.431903\pi$
$97$	39347.8	0.424611	0.212306	+	0.977203i	$0.431903\pi$
$98$	11360.1	0.119486
$99$	9801.00	0.100504
$100$	0	0
$101$	−96213.1	−0.938493	−0.469246	−	0.883067i	$-0.655474\pi$
$101$	−96213.1	−0.938493	−0.469246	+	0.883067i	$0.655474\pi$
$102$	3346.70	0.0318505
$103$	54269.8	0.504040	0.252020	−	0.967722i	$-0.418905\pi$
$103$	54269.8	0.504040	0.252020	+	0.967722i	$0.418905\pi$
$104$	25334.2	0.229680
$105$	0	0
$106$	5726.82	0.0495050
$107$	−46180.6	−0.389942	−0.194971	−	0.980809i	$-0.562461\pi$
$107$	−46180.6	−0.389942	−0.194971	+	0.980809i	$0.562461\pi$
$108$	−23183.9	−0.191261
$109$	162090.	1.30675	0.653373	−	0.757036i	$-0.273353\pi$
$109$	162090.	1.30675	0.653373	+	0.757036i	$0.273353\pi$
$110$	0	0
$111$	78609.4	0.605574
$112$	206846.	1.55812
$113$	252815.	1.86255	0.931274	−	0.364321i	$-0.118699\pi$
$113$	252815.	1.86255	0.931274	+	0.364321i	$0.118699\pi$
$114$	3266.42	0.0235402
$115$	0	0
$116$	232391.	1.60352
$117$	−72333.4	−0.488511
$118$	−230.153	−0.00152164
$119$	172112.	1.11415
$120$	0	0
$121$	14641.0	0.0909091
$122$	−22322.1	−0.135780
$123$	−86478.8	−0.515403
$124$	−4846.26	−0.0283043
$125$	0	0
$126$	7412.35	0.0415939
$127$	171792.	0.945135	0.472568	−	0.881294i	$-0.343327\pi$
$127$	171792.	0.945135	0.472568	+	0.881294i	$0.343327\pi$
$128$	−57384.0	−0.309575
$129$	82874.8	0.438478
$130$	0	0
$131$	140420.	0.714911	0.357455	−	0.933930i	$-0.383644\pi$
$131$	140420.	0.714911	0.357455	+	0.933930i	$0.383644\pi$
$132$	−34632.7	−0.173002
$133$	167984.	0.823453
$134$	1800.89	0.00866411
$135$	0	0
$136$	−23725.2	−0.109992
$137$	194429.	0.885034	0.442517	−	0.896760i	$-0.354086\pi$
$137$	194429.	0.885034	0.442517	+	0.896760i	$0.354086\pi$
$138$	12505.8	0.0559001
$139$	−246396.	−1.08168	−0.540838	−	0.841127i	$-0.681893\pi$
$139$	−246396.	−1.08168	−0.540838	+	0.841127i	$0.681893\pi$
$140$	0	0
$141$	106262.	0.450121
$142$	7322.32	0.0304739
$143$	−108054.	−0.441875
$144$	81409.8	0.327168
$145$	0	0
$146$	−24514.0	−0.0951768
$147$	229936.	0.877635
$148$	−277773.	−1.04240
$149$	−329064.	−1.21427	−0.607135	−	0.794599i	$-0.707681\pi$
$149$	−329064.	−1.21427	−0.607135	+	0.794599i	$0.707681\pi$
$150$	0	0
$151$	125794.	0.448971	0.224486	−	0.974477i	$-0.427930\pi$
$151$	125794.	0.448971	0.224486	+	0.974477i	$0.427930\pi$
$152$	−23156.1	−0.0812936
$153$	67739.6	0.233945
$154$	11072.8	0.0376231
$155$	0	0
$156$	255597.	0.840898
$157$	21521.1	0.0696810	0.0348405	−	0.999393i	$-0.488908\pi$
$157$	21521.1	0.0696810	0.0348405	+	0.999393i	$0.488908\pi$
$158$	11120.5	0.0354389
$159$	115915.	0.363619
$160$	0	0
$161$	643142.	1.95543
$162$	2917.33	0.00873371
$163$	383784.	1.13140	0.565702	−	0.824610i	$-0.308605\pi$
$163$	383784.	1.13140	0.565702	+	0.824610i	$0.308605\pi$
$164$	305580.	0.887189
$165$	0	0
$166$	−33858.9	−0.0953681
$167$	512066.	1.42081	0.710403	−	0.703795i	$-0.248513\pi$
$167$	512066.	1.42081	0.710403	+	0.703795i	$0.248513\pi$
$168$	−52547.2	−0.143640
$169$	426166.	1.14779
$170$	0	0
$171$	66114.7	0.172905
$172$	−292845.	−0.754774
$173$	−260098.	−0.660726	−0.330363	−	0.943854i	$-0.607171\pi$
$173$	−260098.	−0.660726	−0.330363	+	0.943854i	$0.607171\pi$
$174$	−29242.8	−0.0732228
$175$	0	0
$176$	121612.	0.295934
$177$	−4658.47	−0.0111766
$178$	46448.2	0.109880
$179$	112495.	0.262421	0.131211	−	0.991354i	$-0.458114\pi$
$179$	112495.	0.262421	0.131211	+	0.991354i	$0.458114\pi$
$180$	0	0
$181$	56459.6	0.128098	0.0640488	−	0.997947i	$-0.479599\pi$
$181$	56459.6	0.128098	0.0640488	+	0.997947i	$0.479599\pi$
$182$	−81719.4	−0.182872
$183$	−451817.	−0.997320
$184$	−88655.2	−0.193046
$185$	0	0
$186$	609.828	0.00129248
$187$	101191.	0.211611
$188$	−375485.	−0.774815
$189$	150031.	0.305512
$190$	0	0
$191$	356386.	0.706865	0.353433	−	0.935460i	$-0.385014\pi$
$191$	356386.	0.706865	0.353433	+	0.935460i	$0.385014\pi$
$192$	−284036.	−0.556057
$193$	607666.	1.17428	0.587140	−	0.809486i	$-0.300254\pi$
$193$	607666.	1.17428	0.587140	+	0.809486i	$0.300254\pi$
$194$	17495.9	0.0333759
$195$	0	0
$196$	−812500.	−1.51072
$197$	−91261.2	−0.167541	−0.0837704	−	0.996485i	$-0.526696\pi$
$197$	−91261.2	−0.167541	−0.0837704	+	0.996485i	$0.526696\pi$
$198$	4357.99	0.00789994
$199$	840148.	1.50392	0.751958	−	0.659212i	$-0.229110\pi$
$199$	840148.	1.50392	0.751958	+	0.659212i	$0.229110\pi$
$200$	0	0
$201$	36451.3	0.0636389
$202$	−42781.0	−0.0737687
$203$	−1.50389e6	−2.56139
$204$	−239364.	−0.402701
$205$	0	0
$206$	24130.9	0.0396192
$207$	253126.	0.410593
$208$	−897523.	−1.43843
$209$	98763.9	0.156399
$210$	0	0
$211$	945258.	1.46165	0.730826	−	0.682563i	$-0.239135\pi$
$211$	945258.	1.46165	0.730826	+	0.682563i	$0.239135\pi$
$212$	−409596.	−0.625916
$213$	148209.	0.223834
$214$	−20534.1	−0.0306508
$215$	0	0
$216$	−20681.4	−0.0301610
$217$	31362.0	0.0452120
$218$	72073.2	0.102715
$219$	−496181.	−0.699084
$220$	0	0
$221$	−746813.	−1.02856
$222$	34953.5	0.0476001
$223$	−951062.	−1.28070	−0.640349	−	0.768084i	$-0.721210\pi$
$223$	−951062.	−1.28070	−0.640349	+	0.768084i	$0.721210\pi$
$224$	278808.	0.371266
$225$	0	0
$226$	112414.	0.146402
$227$	1.13419e6	1.46090	0.730448	−	0.682968i	$-0.239311\pi$
$227$	1.13419e6	1.46090	0.730448	+	0.682968i	$0.239311\pi$
$228$	−233622.	−0.297630
$229$	1.57670e6	1.98682	0.993412	−	0.114595i	$-0.0365569\pi$
$229$	1.57670e6	1.98682	0.993412	+	0.114595i	$0.0365569\pi$
$230$	0	0
$231$	224121.	0.276346
$232$	207307.	0.252868
$233$	1.06285e6	1.28258	0.641289	−	0.767299i	$-0.278400\pi$
$233$	1.06285e6	1.28258	0.641289	+	0.767299i	$0.278400\pi$
$234$	−32162.9	−0.0383986
$235$	0	0
$236$	16461.1	0.0192389
$237$	225087.	0.260303
$238$	76529.4	0.0875762
$239$	−62094.6	−0.0703168	−0.0351584	−	0.999382i	$-0.511194\pi$
$239$	−62094.6	−0.0703168	−0.0351584	+	0.999382i	$0.511194\pi$
$240$	0	0
$241$	90536.8	0.100411	0.0502056	−	0.998739i	$-0.484012\pi$
$241$	90536.8	0.100411	0.0502056	+	0.998739i	$0.484012\pi$
$242$	6510.09	0.00714576
$243$	59049.0	0.0641500
$244$	1.59653e6	1.71674
$245$	0	0
$246$	−38452.6	−0.0405124
$247$	−728899.	−0.760194
$248$	−4323.16	−0.00446346
$249$	−685330.	−0.700489
$250$	0	0
$251$	−1.35882e6	−1.36138	−0.680689	−	0.732572i	$-0.738320\pi$
$251$	−1.35882e6	−1.36138	−0.680689	+	0.732572i	$0.738320\pi$
$252$	−530149.	−0.525892
$253$	378127.	0.371395
$254$	76387.0	0.0742908
$255$	0	0
$256$	984389.	0.938786
$257$	648294.	0.612265	0.306132	−	0.951989i	$-0.400965\pi$
$257$	648294.	0.612265	0.306132	+	0.951989i	$0.400965\pi$
$258$	36850.1	0.0344659
$259$	1.79757e6	1.66509
$260$	0	0
$261$	−591897.	−0.537830
$262$	62437.6	0.0561944
$263$	−1.78919e6	−1.59502	−0.797510	−	0.603306i	$-0.793850\pi$
$263$	−1.78919e6	−1.59502	−0.797510	+	0.603306i	$0.793850\pi$
$264$	−30894.4	−0.0272816
$265$	0	0
$266$	74693.7	0.0647261
$267$	940145.	0.807080
$268$	−128804.	−0.109545
$269$	1.39327e6	1.17397	0.586984	−	0.809599i	$-0.300315\pi$
$269$	1.39327e6	1.17397	0.586984	+	0.809599i	$0.300315\pi$
$270$	0	0
$271$	−694086.	−0.574104	−0.287052	−	0.957915i	$-0.592675\pi$
$271$	−694086.	−0.574104	−0.287052	+	0.957915i	$0.592675\pi$
$272$	840522.	0.688853
$273$	−1.65406e6	−1.34321
$274$	86452.4	0.0695666
$275$	0	0
$276$	−894443.	−0.706773
$277$	1.92718e6	1.50911	0.754557	−	0.656234i	$-0.227852\pi$
$277$	1.92718e6	1.50911	0.754557	+	0.656234i	$0.227852\pi$
$278$	−109559.	−0.0850233
$279$	12343.4	0.00949343
$280$	0	0
$281$	473885.	0.358020	0.179010	−	0.983847i	$-0.442711\pi$
$281$	473885.	0.358020	0.179010	+	0.983847i	$0.442711\pi$
$282$	47249.0	0.0353810
$283$	568257.	0.421773	0.210887	−	0.977511i	$-0.432365\pi$
$283$	568257.	0.421773	0.210887	+	0.977511i	$0.432365\pi$
$284$	−523710.	−0.385297
$285$	0	0
$286$	−48045.8	−0.0347329
$287$	−1.97752e6	−1.41715
$288$	109733.	0.0779568
$289$	−720474.	−0.507427
$290$	0	0
$291$	354130.	0.245149
$292$	1.75330e6	1.20337
$293$	−1.76731e6	−1.20266	−0.601331	−	0.799000i	$-0.705363\pi$
$293$	−1.76731e6	−1.20266	−0.601331	+	0.799000i	$0.705363\pi$
$294$	102241.	0.0689851
$295$	0	0
$296$	−247790.	−0.164382
$297$	88209.0	0.0580259
$298$	−146318.	−0.0954457
$299$	−2.79065e6	−1.80521
$300$	0	0
$301$	1.89511e6	1.20564
$302$	55934.1	0.0352906
$303$	−865918.	−0.541839
$304$	820360.	0.509120
$305$	0	0
$306$	30120.3	0.0183889
$307$	−2.67297e6	−1.61863	−0.809316	−	0.587374i	$-0.800162\pi$
$307$	−2.67297e6	−1.61863	−0.809316	+	0.587374i	$0.800162\pi$
$308$	−791951.	−0.475687
$309$	488428.	0.291008
$310$	0	0
$311$	2.23621e6	1.31102	0.655512	−	0.755185i	$-0.272453\pi$
$311$	2.23621e6	1.31102	0.655512	+	0.755185i	$0.272453\pi$
$312$	228007.	0.132606
$313$	−1.55507e6	−0.897199	−0.448600	−	0.893733i	$-0.648077\pi$
$313$	−1.55507e6	−0.897199	−0.448600	+	0.893733i	$0.648077\pi$
$314$	9569.29	0.00547716
$315$	0	0
$316$	−795363.	−0.448072
$317$	1.64671e6	0.920383	0.460192	−	0.887820i	$-0.347781\pi$
$317$	1.64671e6	0.920383	0.460192	+	0.887820i	$0.347781\pi$
$318$	51541.4	0.0285817
$319$	−884192.	−0.486485
$320$	0	0
$321$	−415626.	−0.225133
$322$	285972.	0.153703
$323$	682607.	0.364053
$324$	−208655.	−0.110425
$325$	0	0
$326$	170649.	0.0889322
$327$	1.45881e6	0.754450
$328$	272596.	0.139906
$329$	2.42990e6	1.23765
$330$	0	0
$331$	249065.	0.124952	0.0624760	−	0.998046i	$-0.480100\pi$
$331$	249065.	0.124952	0.0624760	+	0.998046i	$0.480100\pi$
$332$	2.42167e6	1.20579
$333$	707484.	0.349628
$334$	227689.	0.111680
$335$	0	0
$336$	1.86161e6	0.899581
$337$	712382.	0.341695	0.170847	−	0.985298i	$-0.445350\pi$
$337$	712382.	0.341695	0.170847	+	0.985298i	$0.445350\pi$
$338$	189494.	0.0902200
$339$	2.27534e6	1.07534
$340$	0	0
$341$	18438.9	0.00858713
$342$	29397.8	0.0135909
$343$	1.79903e6	0.825664
$344$	−261236.	−0.119024
$345$	0	0
$346$	−115652.	−0.0519353
$347$	−1.05963e6	−0.472421	−0.236210	−	0.971702i	$-0.575905\pi$
$347$	−1.05963e6	−0.472421	−0.236210	+	0.971702i	$0.575905\pi$
$348$	2.09152e6	0.925793
$349$	−2.96255e6	−1.30197	−0.650986	−	0.759090i	$-0.725644\pi$
$349$	−2.96255e6	−1.30197	−0.650986	+	0.759090i	$0.725644\pi$
$350$	0	0
$351$	−651001.	−0.282042
$352$	163921.	0.0705146
$353$	2.45696e6	1.04945	0.524725	−	0.851272i	$-0.324168\pi$
$353$	2.45696e6	1.04945	0.524725	+	0.851272i	$0.324168\pi$
$354$	−2071.38	−0.000878520 0
$355$	0	0
$356$	−3.32209e6	−1.38927
$357$	1.54901e6	0.643257
$358$	50020.5	0.0206272
$359$	3.72854e6	1.52687	0.763436	−	0.645884i	$-0.223511\pi$
$359$	3.72854e6	1.52687	0.763436	+	0.645884i	$0.223511\pi$
$360$	0	0
$361$	−1.80987e6	−0.730935
$362$	25104.6	0.0100689
$363$	131769.	0.0524864
$364$	5.84477e6	2.31214
$365$	0	0
$366$	−200899.	−0.0783927
$367$	−141287.	−0.0547566	−0.0273783	−	0.999625i	$-0.508716\pi$
$367$	−141287.	−0.0547566	−0.0273783	+	0.999625i	$0.508716\pi$
$368$	3.14082e6	1.20899
$369$	−778309.	−0.297568
$370$	0	0
$371$	2.65065e6	0.999809
$372$	−43616.4	−0.0163415
$373$	−826181.	−0.307470	−0.153735	−	0.988112i	$-0.549130\pi$
$373$	−826181.	−0.307470	−0.153735	+	0.988112i	$0.549130\pi$
$374$	44994.5	0.0166334
$375$	0	0
$376$	−334955.	−0.122185
$377$	6.52552e6	2.36462
$378$	66711.1	0.0240142
$379$	−1.51195e6	−0.540678	−0.270339	−	0.962765i	$-0.587136\pi$
$379$	−1.51195e6	−0.540678	−0.270339	+	0.962765i	$0.587136\pi$
$380$	0	0
$381$	1.54613e6	0.545674
$382$	158466.	0.0555620
$383$	−3.81853e6	−1.33015	−0.665074	−	0.746778i	$-0.731600\pi$
$383$	−3.81853e6	−1.33015	−0.665074	+	0.746778i	$0.731600\pi$
$384$	−516456.	−0.178733
$385$	0	0
$386$	270197.	0.0923023
$387$	745873.	0.253156
$388$	−1.25135e6	−0.421988
$389$	3.87738e6	1.29916	0.649582	−	0.760292i	$-0.274944\pi$
$389$	3.87738e6	1.29916	0.649582	+	0.760292i	$0.274944\pi$
$390$	0	0
$391$	2.61342e6	0.864505
$392$	−724798.	−0.238233
$393$	1.26378e6	0.412754
$394$	−40579.1	−0.0131693
$395$	0	0
$396$	−311694.	−0.0998828
$397$	−4.70336e6	−1.49772	−0.748862	−	0.662725i	$-0.769400\pi$
$397$	−4.70336e6	−1.49772	−0.748862	+	0.662725i	$0.769400\pi$
$398$	373570.	0.118213
$399$	1.51186e6	0.475421
$400$	0	0
$401$	−2.21418e6	−0.687625	−0.343812	−	0.939038i	$-0.611718\pi$
$401$	−2.21418e6	−0.687625	−0.343812	+	0.939038i	$0.611718\pi$
$402$	16208.0	0.00500223
$403$	−136083.	−0.0417388
$404$	3.05980e6	0.932694
$405$	0	0
$406$	−668701.	−0.201334
$407$	1.05686e6	0.316251
$408$	−213527.	−0.0635041
$409$	−2.24429e6	−0.663394	−0.331697	−	0.943386i	$-0.607621\pi$
$409$	−2.24429e6	−0.663394	−0.331697	+	0.943386i	$0.607621\pi$
$410$	0	0
$411$	1.74986e6	0.510974
$412$	−1.72590e6	−0.500926
$413$	−106526.	−0.0307313
$414$	112552.	0.0322740
$415$	0	0
$416$	−1.20977e6	−0.342745
$417$	−2.21757e6	−0.624505
$418$	43915.2	0.0122935
$419$	−1.31098e6	−0.364806	−0.182403	−	0.983224i	$-0.558388\pi$
$419$	−1.31098e6	−0.364806	−0.182403	+	0.983224i	$0.558388\pi$
$420$	0	0
$421$	898192.	0.246981	0.123491	−	0.992346i	$-0.460591\pi$
$421$	898192.	0.246981	0.123491	+	0.992346i	$0.460591\pi$
$422$	420307.	0.114891
$423$	956356.	0.259877
$424$	−365384.	−0.0987040
$425$	0	0
$426$	65900.9	0.0175941
$427$	−1.03318e7	−2.74223
$428$	1.46865e6	0.387533
$429$	−972483.	−0.255117
$430$	0	0
$431$	6.21890e6	1.61258	0.806288	−	0.591523i	$-0.201473\pi$
$431$	6.21890e6	1.61258	0.806288	+	0.591523i	$0.201473\pi$
$432$	732688.	0.188890
$433$	−1.00093e6	−0.256557	−0.128279	−	0.991738i	$-0.540945\pi$
$433$	−1.00093e6	−0.256557	−0.128279	+	0.991738i	$0.540945\pi$
$434$	13945.0	0.00355382
$435$	0	0
$436$	−5.15485e6	−1.29867
$437$	2.55073e6	0.638942
$438$	−220626.	−0.0549504
$439$	4.86510e6	1.20484	0.602422	−	0.798178i	$-0.294203\pi$
$439$	4.86510e6	1.20484	0.602422	+	0.798178i	$0.294203\pi$
$440$	0	0
$441$	2.06943e6	0.506703
$442$	−332069.	−0.0808485
$443$	7.54601e6	1.82687	0.913435	−	0.406984i	$-0.133420\pi$
$443$	7.54601e6	1.82687	0.913435	+	0.406984i	$0.133420\pi$
$444$	−2.49996e6	−0.601832
$445$	0	0
$446$	−422888.	−0.100667
$447$	−2.96158e6	−0.701059
$448$	−6.49509e6	−1.52894
$449$	−2.70457e6	−0.633115	−0.316557	−	0.948573i	$-0.602527\pi$
$449$	−2.70457e6	−0.633115	−0.316557	+	0.948573i	$0.602527\pi$
$450$	0	0
$451$	−1.16266e6	−0.269160
$452$	−8.04011e6	−1.85104
$453$	1.13215e6	0.259214
$454$	504313.	0.114831
$455$	0	0
$456$	−208405.	−0.0469349
$457$	−3.69700e6	−0.828055	−0.414027	−	0.910264i	$-0.635878\pi$
$457$	−3.69700e6	−0.828055	−0.414027	+	0.910264i	$0.635878\pi$
$458$	701075.	0.156171
$459$	609656.	0.135068
$460$	0	0
$461$	8.68496e6	1.90334	0.951669	−	0.307127i	$-0.0993676\pi$
$461$	8.68496e6	1.90334	0.951669	+	0.307127i	$0.0993676\pi$
$462$	99654.9	0.0217217
$463$	−2.75398e6	−0.597047	−0.298523	−	0.954402i	$-0.596494\pi$
$463$	−2.75398e6	−0.597047	−0.298523	+	0.954402i	$0.596494\pi$
$464$	−7.34433e6	−1.58364
$465$	0	0
$466$	472596.	0.100815
$467$	−7.31776e6	−1.55269	−0.776347	−	0.630306i	$-0.782929\pi$
$467$	−7.31776e6	−1.55269	−0.776347	+	0.630306i	$0.782929\pi$
$468$	2.30037e6	0.485493
$469$	833537.	0.174982
$470$	0	0
$471$	193690.	0.0402304
$472$	14684.3	0.00303388
$473$	1.11421e6	0.228988
$474$	100084.	0.0204607
$475$	0	0
$476$	−5.47357e6	−1.10727
$477$	1.04324e6	0.209936
$478$	−27610.2	−0.00552713
$479$	−230118.	−0.0458260	−0.0229130	−	0.999737i	$-0.507294\pi$
$479$	−230118.	−0.0458260	−0.0229130	+	0.999737i	$0.507294\pi$
$480$	0	0
$481$	−7.79985e6	−1.53718
$482$	40257.0	0.00789267
$483$	5.78827e6	1.12897
$484$	−465617.	−0.0903474
$485$	0	0
$486$	26256.0	0.00504241
$487$	8.18489e6	1.56383	0.781917	−	0.623383i	$-0.214242\pi$
$487$	8.18489e6	1.56383	0.781917	+	0.623383i	$0.214242\pi$
$488$	1.42420e6	0.270721
$489$	3.45405e6	0.653216
$490$	0	0
$491$	−1.27094e6	−0.237915	−0.118957	−	0.992899i	$-0.537955\pi$
$491$	−1.27094e6	−0.237915	−0.118957	+	0.992899i	$0.537955\pi$
$492$	2.75022e6	0.512219
$493$	−6.11109e6	−1.13240
$494$	−324103.	−0.0597538
$495$	0	0
$496$	153158.	0.0279535
$497$	3.38912e6	0.615455
$498$	−304730.	−0.0550608
$499$	−9.67837e6	−1.74001	−0.870003	−	0.493046i	$-0.835883\pi$
$499$	−9.67837e6	−1.74001	−0.870003	+	0.493046i	$0.835883\pi$
$500$	0	0
$501$	4.60859e6	0.820303
$502$	−604198.	−0.107009
$503$	−6.99048e6	−1.23193	−0.615966	−	0.787773i	$-0.711234\pi$
$503$	−6.99048e6	−1.23193	−0.615966	+	0.787773i	$0.711234\pi$
$504$	−472925.	−0.0829308
$505$	0	0
$506$	168133.	0.0291929
$507$	3.83549e6	0.662676
$508$	−5.46338e6	−0.939296
$509$	−6.03866e6	−1.03311	−0.516555	−	0.856254i	$-0.672786\pi$
$509$	−6.03866e6	−1.03311	−0.516555	+	0.856254i	$0.672786\pi$
$510$	0	0
$511$	−1.13462e7	−1.92221
$512$	2.27399e6	0.383367
$513$	595032.	0.0998268
$514$	288262.	0.0481261
$515$	0	0
$516$	−2.63561e6	−0.435769
$517$	1.42863e6	0.235068
$518$	799287.	0.130882
$519$	−2.34088e6	−0.381470
$520$	0	0
$521$	6.75056e6	1.08955	0.544773	−	0.838584i	$-0.316616\pi$
$521$	6.75056e6	1.08955	0.544773	+	0.838584i	$0.316616\pi$
$522$	−263186.	−0.0422752
$523$	−3.61437e6	−0.577801	−0.288900	−	0.957359i	$-0.593290\pi$
$523$	−3.61437e6	−0.577801	−0.288900	+	0.957359i	$0.593290\pi$
$524$	−4.46569e6	−0.710494
$525$	0	0
$526$	−795557.	−0.125374
$527$	127440.	0.0199885
$528$	1.09451e6	0.170858
$529$	3.32937e6	0.517276
$530$	0	0
$531$	−41926.3	−0.00645282
$532$	−5.34227e6	−0.818365
$533$	8.58067e6	1.30829
$534$	418033.	0.0634392
$535$	0	0
$536$	−114901.	−0.0172747
$537$	1.01245e6	0.151509
$538$	619516.	0.0922778
$539$	3.09136e6	0.458330
$540$	0	0
$541$	7.17998e6	1.05470	0.527351	−	0.849647i	$-0.323185\pi$
$541$	7.17998e6	1.05470	0.527351	+	0.849647i	$0.323185\pi$
$542$	−308624.	−0.0451265
$543$	508136.	0.0739572
$544$	1.13294e6	0.164139
$545$	0	0
$546$	−735474.	−0.105581
$547$	−5.53996e6	−0.791659	−0.395830	−	0.918324i	$-0.629543\pi$
$547$	−5.53996e6	−0.791659	−0.395830	+	0.918324i	$0.629543\pi$
$548$	−6.18329e6	−0.879565
$549$	−4.06635e6	−0.575803
$550$	0	0
$551$	−5.96450e6	−0.836941
$552$	−797897.	−0.111455
$553$	5.14709e6	0.715730
$554$	856915.	0.118621
$555$	0	0
$556$	7.83596e6	1.07499
$557$	1.07664e7	1.47039	0.735197	−	0.677854i	$-0.237090\pi$
$557$	1.07664e7	1.47039	0.735197	+	0.677854i	$0.237090\pi$
$558$	5488.45	0.000746216 0
$559$	−8.22307e6	−1.11302
$560$	0	0
$561$	910721.	0.122174
$562$	210712.	0.0281416
$563$	6.29087e6	0.836449	0.418224	−	0.908344i	$-0.362653\pi$
$563$	6.29087e6	0.836449	0.418224	+	0.908344i	$0.362653\pi$
$564$	−3.37937e6	−0.447340
$565$	0	0
$566$	252674.	0.0331528
$567$	1.35028e6	0.176387
$568$	−467181.	−0.0607595
$569$	−1.25297e7	−1.62241	−0.811205	−	0.584761i	$-0.801188\pi$
$569$	−1.25297e7	−1.62241	−0.811205	+	0.584761i	$0.801188\pi$
$570$	0	0
$571$	−300370.	−0.0385538	−0.0192769	−	0.999814i	$-0.506136\pi$
$571$	−300370.	−0.0385538	−0.0192769	+	0.999814i	$0.506136\pi$
$572$	3.43635e6	0.439145
$573$	3.20747e6	0.408109
$574$	−879302.	−0.111393
$575$	0	0
$576$	−2.55632e6	−0.321040
$577$	4.91623e6	0.614742	0.307371	−	0.951590i	$-0.400551\pi$
$577$	4.91623e6	0.614742	0.307371	+	0.951590i	$0.400551\pi$
$578$	−320357.	−0.0398855
$579$	5.46899e6	0.677971
$580$	0	0
$581$	−1.56715e7	−1.92607
$582$	157463.	0.0192696
$583$	1.55841e6	0.189894
$584$	1.56405e6	0.189766
$585$	0	0
$586$	−785830.	−0.0945333
$587$	−1.27769e7	−1.53048	−0.765242	−	0.643742i	$-0.777381\pi$
$587$	−1.27769e7	−1.53048	−0.765242	+	0.643742i	$0.777381\pi$
$588$	−7.31250e6	−0.872213
$589$	124383.	0.0147732
$590$	0	0
$591$	−821351.	−0.0967298
$592$	8.77856e6	1.02948
$593$	2.84036e6	0.331694	0.165847	−	0.986152i	$-0.446964\pi$
$593$	2.84036e6	0.331694	0.165847	+	0.986152i	$0.446964\pi$
$594$	39221.9	0.00456103
$595$	0	0
$596$	1.04650e7	1.20677
$597$	7.56134e6	0.868286
$598$	−1.24086e6	−0.141896
$599$	−1.61252e7	−1.83628	−0.918140	−	0.396257i	$-0.870309\pi$
$599$	−1.61252e7	−1.83628	−0.918140	+	0.396257i	$0.870309\pi$
$600$	0	0
$601$	−6.84876e6	−0.773438	−0.386719	−	0.922198i	$-0.626392\pi$
$601$	−6.84876e6	−0.773438	−0.386719	+	0.922198i	$0.626392\pi$
$602$	842657.	0.0947675
$603$	328061.	0.0367419
$604$	−4.00055e6	−0.446197
$605$	0	0
$606$	−385029.	−0.0425904
$607$	−1.39670e7	−1.53862	−0.769312	−	0.638874i	$-0.779401\pi$
$607$	−1.39670e7	−1.53862	−0.769312	+	0.638874i	$0.779401\pi$
$608$	1.10577e6	0.121312
$609$	−1.35350e7	−1.47882
$610$	0	0
$611$	−1.05436e7	−1.14258
$612$	−2.15427e6	−0.232500
$613$	7.54369e6	0.810835	0.405418	−	0.914132i	$-0.367126\pi$
$613$	7.54369e6	0.810835	0.405418	+	0.914132i	$0.367126\pi$
$614$	−1.18853e6	−0.127230
$615$	0	0
$616$	−706468.	−0.0750137
$617$	3.93986e6	0.416647	0.208323	−	0.978060i	$-0.433199\pi$
$617$	3.93986e6	0.416647	0.208323	+	0.978060i	$0.433199\pi$
$618$	217178.	0.0228742
$619$	−5.93266e6	−0.622333	−0.311166	−	0.950355i	$-0.600720\pi$
$619$	−5.93266e6	−0.622333	−0.311166	+	0.950355i	$0.600720\pi$
$620$	0	0
$621$	2.27813e6	0.237056
$622$	994324.	0.103051
$623$	2.14984e7	2.21915
$624$	−8.07771e6	−0.830475
$625$	0	0
$626$	−691458.	−0.0705229
$627$	888875.	0.0902967
$628$	−684419.	−0.0692505
$629$	7.30448e6	0.736144
$630$	0	0
$631$	−4.19698e6	−0.419627	−0.209813	−	0.977741i	$-0.567286\pi$
$631$	−4.19698e6	−0.419627	−0.209813	+	0.977741i	$0.567286\pi$
$632$	−709512.	−0.0706589
$633$	8.50732e6	0.843886
$634$	732205.	0.0723452
$635$	0	0
$636$	−3.68636e6	−0.361373
$637$	−2.28149e7	−2.22777
$638$	−393154.	−0.0382394
$639$	1.33388e6	0.129231
$640$	0	0
$641$	1.95566e7	1.87995	0.939977	−	0.341236i	$-0.110846\pi$
$641$	1.95566e7	1.87995	0.939977	+	0.341236i	$0.110846\pi$
$642$	−184807.	−0.0176962
$643$	−1.89046e7	−1.80318	−0.901591	−	0.432590i	$-0.857600\pi$
$643$	−1.89046e7	−1.80318	−0.901591	+	0.432590i	$0.857600\pi$
$644$	−2.04534e7	−1.94335
$645$	0	0
$646$	303520.	0.0286158
$647$	1.35066e7	1.26849	0.634244	−	0.773133i	$-0.281311\pi$
$647$	1.35066e7	1.26849	0.634244	+	0.773133i	$0.281311\pi$
$648$	−186133.	−0.0174135
$649$	−62630.6	−0.00583680
$650$	0	0
$651$	282258.	0.0261032
$652$	−1.22052e7	−1.12441
$653$	−1.11108e7	−1.01968	−0.509840	−	0.860269i	$-0.670295\pi$
$653$	−1.11108e7	−1.01968	−0.509840	+	0.860269i	$0.670295\pi$
$654$	648658.	0.0593023
$655$	0	0
$656$	−9.65736e6	−0.876192
$657$	−4.46563e6	−0.403616
$658$	1.08045e6	0.0972838
$659$	1.35954e7	1.21949	0.609745	−	0.792597i	$-0.291272\pi$
$659$	1.35954e7	1.21949	0.609745	+	0.792597i	$0.291272\pi$
$660$	0	0
$661$	−856154.	−0.0762163	−0.0381082	−	0.999274i	$-0.512133\pi$
$661$	−856154.	−0.0762163	−0.0381082	+	0.999274i	$0.512133\pi$
$662$	110746.	0.00982164
$663$	−6.72131e6	−0.593841
$664$	2.16028e6	0.190147
$665$	0	0
$666$	314581.	0.0274819
$667$	−2.28356e7	−1.98746
$668$	−1.62849e7	−1.41203
$669$	−8.55956e6	−0.739411
$670$	0	0
$671$	−6.07442e6	−0.520833
$672$	2.50927e6	0.214350
$673$	1.21181e7	1.03133	0.515665	−	0.856790i	$-0.327545\pi$
$673$	1.21181e7	1.03133	0.515665	+	0.856790i	$0.327545\pi$
$674$	316759.	0.0268583
$675$	0	0
$676$	−1.35530e7	−1.14070
$677$	−8.70607e6	−0.730046	−0.365023	−	0.930998i	$-0.618939\pi$
$677$	−8.70607e6	−0.730046	−0.365023	+	0.930998i	$0.618939\pi$
$678$	1.01172e6	0.0845255
$679$	8.09796e6	0.674064
$680$	0	0
$681$	1.02077e7	0.843449
$682$	8198.80	0.000674977 0
$683$	3.49312e6	0.286525	0.143262	−	0.989685i	$-0.454241\pi$
$683$	3.49312e6	0.286525	0.143262	+	0.989685i	$0.454241\pi$
$684$	−2.10260e6	−0.171837
$685$	0	0
$686$	799936.	0.0649000
$687$	1.41903e7	1.14709
$688$	9.25489e6	0.745419
$689$	−1.15014e7	−0.923003
$690$	0	0
$691$	−2.98454e6	−0.237784	−0.118892	−	0.992907i	$-0.537934\pi$
$691$	−2.98454e6	−0.237784	−0.118892	+	0.992907i	$0.537934\pi$
$692$	8.27170e6	0.656643
$693$	2.01709e6	0.159548
$694$	−471160.	−0.0371339
$695$	0	0
$696$	1.86576e6	0.145993
$697$	−8.03572e6	−0.626531
$698$	−1.31729e6	−0.102339
$699$	9.56569e6	0.740497
$700$	0	0
$701$	−3.32551e6	−0.255601	−0.127801	−	0.991800i	$-0.540792\pi$
$701$	−3.32551e6	−0.255601	−0.127801	+	0.991800i	$0.540792\pi$
$702$	−289466.	−0.0221695
$703$	7.12927e6	0.544072
$704$	−3.81870e6	−0.290392
$705$	0	0
$706$	1.09248e6	0.0824903
$707$	−1.98011e7	−1.48984
$708$	148150.	0.0111076
$709$	1.53451e7	1.14645	0.573225	−	0.819398i	$-0.305692\pi$
$709$	1.53451e7	1.14645	0.573225	+	0.819398i	$0.305692\pi$
$710$	0	0
$711$	2.02578e6	0.150286
$712$	−2.96350e6	−0.219081
$713$	476212.	0.0350814
$714$	688765.	0.0505621
$715$	0	0
$716$	−3.57759e6	−0.260800
$717$	−558851.	−0.0405974
$718$	1.65789e6	0.120017
$719$	2.09235e7	1.50943	0.754714	−	0.656054i	$-0.227776\pi$
$719$	2.09235e7	1.50943	0.754714	+	0.656054i	$0.227776\pi$
$720$	0	0
$721$	1.11690e7	0.800156
$722$	−804753.	−0.0574539
$723$	814831.	0.0579725
$724$	−1.79554e6	−0.127306
$725$	0	0
$726$	58590.8	0.00412561
$727$	−1.75630e7	−1.23243	−0.616217	−	0.787576i	$-0.711336\pi$
$727$	−1.75630e7	−1.23243	−0.616217	+	0.787576i	$0.711336\pi$
$728$	5.21388e6	0.364614
$729$	531441.	0.0370370
$730$	0	0
$731$	7.70083e6	0.533021
$732$	1.43688e7	0.991158
$733$	1.52765e7	1.05018	0.525090	−	0.851047i	$-0.324032\pi$
$733$	1.52765e7	1.05018	0.525090	+	0.851047i	$0.324032\pi$
$734$	−62822.8	−0.00430405
$735$	0	0
$736$	4.23353e6	0.288076
$737$	490067.	0.0332343
$738$	−346074.	−0.0233899
$739$	2.22683e7	1.49995	0.749974	−	0.661467i	$-0.230066\pi$
$739$	2.22683e7	1.49995	0.749974	+	0.661467i	$0.230066\pi$
$740$	0	0
$741$	−6.56009e6	−0.438898
$742$	1.17860e6	0.0785884
$743$	−840214.	−0.0558365	−0.0279182	−	0.999610i	$-0.508888\pi$
$743$	−840214.	−0.0558365	−0.0279182	+	0.999610i	$0.508888\pi$
$744$	−38908.4	−0.00257698
$745$	0	0
$746$	−367360.	−0.0241682
$747$	−6.16797e6	−0.404427
$748$	−3.21811e6	−0.210304
$749$	−9.50418e6	−0.619028
$750$	0	0
$751$	−2.63571e7	−1.70529	−0.852644	−	0.522493i	$-0.825002\pi$
$751$	−2.63571e7	−1.70529	−0.852644	+	0.522493i	$0.825002\pi$
$752$	1.18666e7	0.765211
$753$	−1.22294e7	−0.785992
$754$	2.90156e6	0.185867
$755$	0	0
$756$	−4.77134e6	−0.303624
$757$	−1.71589e7	−1.08830	−0.544150	−	0.838988i	$-0.683148\pi$
$757$	−1.71589e7	−1.08830	−0.544150	+	0.838988i	$0.683148\pi$
$758$	−672284.	−0.0424991
$759$	3.40314e6	0.214425
$760$	0	0
$761$	−2.23433e7	−1.39857	−0.699287	−	0.714841i	$-0.746499\pi$
$761$	−2.23433e7	−1.39857	−0.699287	+	0.714841i	$0.746499\pi$
$762$	687483.	0.0428918
$763$	3.33589e7	2.07444
$764$	−1.13339e7	−0.702498
$765$	0	0
$766$	−1.69790e6	−0.104554
$767$	462227.	0.0283705
$768$	8.85950e6	0.542008
$769$	−8.69614e6	−0.530287	−0.265143	−	0.964209i	$-0.585419\pi$
$769$	−8.69614e6	−0.530287	−0.265143	+	0.964209i	$0.585419\pi$
$770$	0	0
$771$	5.83464e6	0.353491
$772$	−1.93252e7	−1.16702
$773$	5.79015e6	0.348531	0.174265	−	0.984699i	$-0.444245\pi$
$773$	5.79015e6	0.348531	0.174265	+	0.984699i	$0.444245\pi$
$774$	331651.	0.0198989
$775$	0	0
$776$	−1.11628e6	−0.0665455
$777$	1.61782e7	0.961339
$778$	1.72407e6	0.102119
$779$	−7.84296e6	−0.463059
$780$	0	0
$781$	1.99259e6	0.116894
$782$	1.16205e6	0.0679530
$783$	−5.32707e6	−0.310516
$784$	2.56777e7	1.49199
$785$	0	0
$786$	561938.	0.0324438
$787$	−3.16605e7	−1.82213	−0.911067	−	0.412258i	$-0.864740\pi$
$787$	−3.16605e7	−1.82213	−0.911067	+	0.412258i	$0.864740\pi$
$788$	2.90232e6	0.166506
$789$	−1.61027e7	−0.920885
$790$	0	0
$791$	5.20305e7	2.95677
$792$	−278050.	−0.0157511
$793$	4.48305e7	2.53157
$794$	−2.09134e6	−0.117726
$795$	0	0
$796$	−2.67186e7	−1.49462
$797$	−1.21137e7	−0.675511	−0.337755	−	0.941234i	$-0.609668\pi$
$797$	−1.21137e7	−0.675511	−0.337755	+	0.941234i	$0.609668\pi$
$798$	672243.	0.0373697
$799$	9.87397e6	0.547173
$800$	0	0
$801$	8.46131e6	0.465968
$802$	−984529.	−0.0540496
$803$	−6.67087e6	−0.365085
$804$	−1.15923e6	−0.0632457
$805$	0	0
$806$	−60508.9	−0.00328081
$807$	1.25395e7	0.677790
$808$	2.72952e6	0.147082
$809$	−1.66584e7	−0.894875	−0.447438	−	0.894315i	$-0.647663\pi$
$809$	−1.66584e7	−0.894875	−0.447438	+	0.894315i	$0.647663\pi$
$810$	0	0
$811$	−2.63057e7	−1.40442	−0.702211	−	0.711969i	$-0.747804\pi$
$811$	−2.63057e7	−1.40442	−0.702211	+	0.711969i	$0.747804\pi$
$812$	4.78271e7	2.54556
$813$	−6.24678e6	−0.331459
$814$	469930.	0.0248584
$815$	0	0
$816$	7.56470e6	0.397710
$817$	7.51610e6	0.393947
$818$	−997920.	−0.0521450
$819$	−1.48865e7	−0.775504
$820$	0	0
$821$	2.92543e6	0.151472	0.0757358	−	0.997128i	$-0.475869\pi$
$821$	2.92543e6	0.151472	0.0757358	+	0.997128i	$0.475869\pi$
$822$	778072.	0.0401643
$823$	−2.99240e7	−1.54000	−0.770000	−	0.638044i	$-0.779744\pi$
$823$	−2.99240e7	−1.54000	−0.770000	+	0.638044i	$0.779744\pi$
$824$	−1.53961e6	−0.0789937
$825$	0	0
$826$	−47366.6	−0.00241558
$827$	1.24304e6	0.0632008	0.0316004	−	0.999501i	$-0.489940\pi$
$827$	1.24304e6	0.0632008	0.0316004	+	0.999501i	$0.489940\pi$
$828$	−8.04999e6	−0.408056
$829$	7.42673e6	0.375328	0.187664	−	0.982233i	$-0.439908\pi$
$829$	7.42673e6	0.375328	0.187664	+	0.982233i	$0.439908\pi$
$830$	0	0
$831$	1.73446e7	0.871288
$832$	2.81828e7	1.41148
$833$	2.13660e7	1.06687
$834$	−986035.	−0.0490882
$835$	0	0
$836$	−3.14092e6	−0.155432
$837$	111090.	0.00548103
$838$	−582926.	−0.0286750
$839$	−9.30289e6	−0.456260	−0.228130	−	0.973631i	$-0.573261\pi$
$839$	−9.30289e6	−0.456260	−0.228130	+	0.973631i	$0.573261\pi$
$840$	0	0
$841$	3.28865e7	1.60335
$842$	399379.	0.0194136
$843$	4.26497e6	0.206703
$844$	−3.00614e7	−1.45262
$845$	0	0
$846$	425241.	0.0204272
$847$	3.01318e6	0.144317
$848$	1.29446e7	0.618157
$849$	5.11431e6	0.243511
$850$	0	0
$851$	2.72950e7	1.29199
$852$	−4.71339e6	−0.222451
$853$	−3.06103e7	−1.44044	−0.720219	−	0.693746i	$-0.755959\pi$
$853$	−3.06103e7	−1.44044	−0.720219	+	0.693746i	$0.755959\pi$
$854$	−4.59399e6	−0.215549
$855$	0	0
$856$	1.31012e6	0.0611122
$857$	8.04257e6	0.374061	0.187031	−	0.982354i	$-0.440114\pi$
$857$	8.04257e6	0.374061	0.187031	+	0.982354i	$0.440114\pi$
$858$	−432412.	−0.0200530
$859$	2.02293e7	0.935403	0.467702	−	0.883886i	$-0.345082\pi$
$859$	2.02293e7	0.935403	0.467702	+	0.883886i	$0.345082\pi$
$860$	0	0
$861$	−1.77977e7	−0.818194
$862$	2.76522e6	0.126754
$863$	2.06034e7	0.941700	0.470850	−	0.882213i	$-0.343947\pi$
$863$	2.06034e7	0.941700	0.470850	+	0.882213i	$0.343947\pi$
$864$	987593.	0.0450084
$865$	0	0
$866$	−445062.	−0.0201663
$867$	−6.48427e6	−0.292963
$868$	−997382.	−0.0449327
$869$	3.02616e6	0.135939
$870$	0	0
$871$	−3.61680e6	−0.161540
$872$	−4.59843e6	−0.204795
$873$	3.18717e6	0.141537
$874$	1.13418e6	0.0502230
$875$	0	0
$876$	1.57797e7	0.694765
$877$	2.65867e7	1.16726	0.583628	−	0.812021i	$-0.301633\pi$
$877$	2.65867e7	1.16726	0.583628	+	0.812021i	$0.301633\pi$
$878$	2.16326e6	0.0947047
$879$	−1.59058e7	−0.694357
$880$	0	0
$881$	4.01120e7	1.74114	0.870572	−	0.492041i	$-0.163749\pi$
$881$	4.01120e7	1.74114	0.870572	+	0.492041i	$0.163749\pi$
$882$	920166.	0.0398285
$883$	−3.61504e7	−1.56031	−0.780155	−	0.625586i	$-0.784860\pi$
$883$	−3.61504e7	−1.56031	−0.780155	+	0.625586i	$0.784860\pi$
$884$	2.37504e7	1.02221
$885$	0	0
$886$	3.35531e6	0.143598
$887$	−2.25484e7	−0.962290	−0.481145	−	0.876641i	$-0.659779\pi$
$887$	−2.25484e7	−0.962290	−0.481145	+	0.876641i	$0.659779\pi$
$888$	−2.23011e6	−0.0949062
$889$	3.53556e7	1.50039
$890$	0	0
$891$	793881.	0.0335013
$892$	3.02459e7	1.27278
$893$	9.63712e6	0.404407
$894$	−1.31686e6	−0.0551056
$895$	0	0
$896$	−1.18099e7	−0.491446
$897$	−2.51159e7	−1.04224
$898$	−1.20258e6	−0.0497650
$899$	−1.11355e6	−0.0459526
$900$	0	0
$901$	1.07710e7	0.442021
$902$	−516974.	−0.0211569
$903$	1.70560e7	0.696078
$904$	−7.17226e6	−0.291900
$905$	0	0
$906$	503407.	0.0203751
$907$	1.61295e7	0.651032	0.325516	−	0.945537i	$-0.394462\pi$
$907$	1.61295e7	0.651032	0.325516	+	0.945537i	$0.394462\pi$
$908$	−3.60697e7	−1.45187
$909$	−7.79326e6	−0.312831
$910$	0	0
$911$	−1.72749e7	−0.689637	−0.344819	−	0.938669i	$-0.612060\pi$
$911$	−1.72749e7	−0.689637	−0.344819	+	0.938669i	$0.612060\pi$
$912$	7.38324e6	0.293941
$913$	−9.21388e6	−0.365818
$914$	−1.64386e6	−0.0650879
$915$	0	0
$916$	−5.01426e7	−1.97455
$917$	2.88991e7	1.13491
$918$	271082.	0.0106168
$919$	966289.	0.0377414	0.0188707	−	0.999822i	$-0.493993\pi$
$919$	966289.	0.0377414	0.0188707	+	0.999822i	$0.493993\pi$
$920$	0	0
$921$	−2.40567e7	−0.934517
$922$	3.86175e6	0.149609
$923$	−1.47057e7	−0.568176
$924$	−7.12756e6	−0.274638
$925$	0	0
$926$	−1.22455e6	−0.0469299
$927$	4.39585e6	0.168013
$928$	−9.89946e6	−0.377348
$929$	−1.54446e7	−0.587133	−0.293567	−	0.955939i	$-0.594842\pi$
$929$	−1.54446e7	−0.587133	−0.293567	+	0.955939i	$0.594842\pi$
$930$	0	0
$931$	2.08534e7	0.788503
$932$	−3.38012e7	−1.27465
$933$	2.01259e7	0.756920
$934$	−3.25382e6	−0.122047
$935$	0	0
$936$	2.05207e6	0.0765600
$937$	−3.01699e7	−1.12260	−0.561300	−	0.827612i	$-0.689699\pi$
$937$	−3.01699e7	−1.12260	−0.561300	+	0.827612i	$0.689699\pi$
$938$	370630.	0.0137542
$939$	−1.39956e7	−0.517998
$940$	0	0
$941$	−1.74463e7	−0.642286	−0.321143	−	0.947031i	$-0.604067\pi$
$941$	−1.74463e7	−0.642286	−0.321143	+	0.947031i	$0.604067\pi$
$942$	86123.6	0.00316224
$943$	−3.00275e7	−1.09961
$944$	−520227.	−0.0190004
$945$	0	0
$946$	495429.	0.0179992
$947$	3.47258e7	1.25828	0.629140	−	0.777292i	$-0.283407\pi$
$947$	3.47258e7	1.25828	0.629140	+	0.777292i	$0.283407\pi$
$948$	−7.15827e6	−0.258694
$949$	4.92324e7	1.77454
$950$	0	0
$951$	1.48204e7	0.531383
$952$	−4.88275e6	−0.174611
$953$	−5.30473e7	−1.89204	−0.946021	−	0.324104i	$-0.894937\pi$
$953$	−5.30473e7	−1.89204	−0.946021	+	0.324104i	$0.894937\pi$
$954$	463872.	0.0165017
$955$	0	0
$956$	1.97475e6	0.0698823
$957$	−7.95772e6	−0.280872
$958$	−102322.	−0.00360208
$959$	4.00144e7	1.40498
$960$	0	0
$961$	−2.86059e7	−0.999189
$962$	−3.46818e6	−0.120827
$963$	−3.74063e6	−0.129981
$964$	−2.87928e6	−0.0997909
$965$	0	0
$966$	2.57374e6	0.0887406
$967$	8.74388e6	0.300703	0.150352	−	0.988633i	$-0.451959\pi$
$967$	8.74388e6	0.300703	0.150352	+	0.988633i	$0.451959\pi$
$968$	−415359.	−0.0142474
$969$	6.14346e6	0.210186
$970$	0	0
$971$	2.90606e7	0.989136	0.494568	−	0.869139i	$-0.335326\pi$
$971$	2.90606e7	0.989136	0.494568	+	0.869139i	$0.335326\pi$
$972$	−1.87789e6	−0.0637537
$973$	−5.07094e7	−1.71714
$974$	3.63939e6	0.122923
$975$	0	0
$976$	−5.04558e7	−1.69546
$977$	4.53644e6	0.152047	0.0760236	−	0.997106i	$-0.475778\pi$
$977$	4.53644e6	0.152047	0.0760236	+	0.997106i	$0.475778\pi$
$978$	1.53584e6	0.0513450
$979$	1.26397e7	0.421484
$980$	0	0
$981$	1.31293e7	0.435582
$982$	−565121.	−0.0187009
$983$	2.56053e7	0.845174	0.422587	−	0.906322i	$-0.361122\pi$
$983$	2.56053e7	0.845174	0.422587	+	0.906322i	$0.361122\pi$
$984$	2.45336e6	0.0807745
$985$	0	0
$986$	−2.71728e6	−0.0890107
$987$	2.18691e7	0.714560
$988$	2.31806e7	0.755498
$989$	2.87761e7	0.935494
$990$	0	0
$991$	−7.43675e6	−0.240547	−0.120273	−	0.992741i	$-0.538377\pi$
$991$	−7.43675e6	−0.240547	−0.120273	+	0.992741i	$0.538377\pi$
$992$	206442.	0.00666070
$993$	2.24159e6	0.0721410
$994$	1.50697e6	0.0483769
$995$	0	0
$996$	2.17951e7	0.696161
$997$	−2.77690e7	−0.884752	−0.442376	−	0.896830i	$-0.645864\pi$
$997$	−2.77690e7	−0.884752	−0.442376	+	0.896830i	$0.645864\pi$
$998$	−4.30347e6	−0.136770
$999$	6.36736e6	0.201858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.t.1.6	✓	10
5.4	even	2		825.6.a.u.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.6	✓	10	1.1	even	1	trivial
825.6.a.u.1.5	yes	10	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)

\(f(q)\)	\(=\)	\(q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +121.000 q^{11} -286.221 q^{12} -893.005 q^{13} +91.5105 q^{14} +1005.06 q^{16} +836.291 q^{17} +36.0165 q^{18} +816.231 q^{19} +1852.24 q^{21} +53.8024 q^{22} +3125.01 q^{23} -255.326 q^{24} -397.073 q^{26} +729.000 q^{27} -6545.05 q^{28} -7307.37 q^{29} +152.387 q^{31} +1354.72 q^{32} +1089.00 q^{33} +371.855 q^{34} -2575.99 q^{36} +8734.38 q^{37} +362.935 q^{38} -8037.05 q^{39} -9608.76 q^{41} +823.594 q^{42} +9208.31 q^{43} -3848.08 q^{44} +1389.53 q^{46} +11806.9 q^{47} +9045.53 q^{48} +25548.5 q^{49} +7526.62 q^{51} +28399.6 q^{52} +12879.4 q^{53} +324.148 q^{54} -5838.58 q^{56} +7346.08 q^{57} -3249.21 q^{58} -517.608 q^{59} -50201.8 q^{61} +67.7587 q^{62} +16670.2 q^{63} -31559.5 q^{64} +484.221 q^{66} +4050.14 q^{67} -26596.0 q^{68} +28125.1 q^{69} +16467.7 q^{71} -2297.93 q^{72} -55131.2 q^{73} +3883.72 q^{74} -25958.0 q^{76} +24902.3 q^{77} -3573.66 q^{78} +25009.6 q^{79} +6561.00 q^{81} -4272.51 q^{82} -76147.7 q^{83} -58905.5 q^{84} +4094.45 q^{86} -65766.3 q^{87} -3432.72 q^{88} +104461. q^{89} -183784. q^{91} -99382.6 q^{92} +1371.49 q^{93} +5249.89 q^{94} +12192.5 q^{96} +39347.8 q^{97} +11360.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 + 1210 * q^11 + 2007 * q^12 - 1102 * q^13 + 684 * q^14 + 7019 * q^16 + 1106 * q^17 - 81 * q^18 + 2586 * q^19 + 1692 * q^21 - 121 * q^22 + 2206 * q^23 + 1593 * q^24 + 12001 * q^26 + 7290 * q^27 + 14452 * q^28 + 5824 * q^29 - 4586 * q^31 - 10627 * q^32 + 10890 * q^33 + 7426 * q^34 + 18063 * q^36 - 18362 * q^37 + 44001 * q^38 - 9918 * q^39 - 5474 * q^41 + 6156 * q^42 + 20496 * q^43 + 26983 * q^44 + 19981 * q^46 - 14970 * q^47 + 63171 * q^48 + 68582 * q^49 + 9954 * q^51 - 58603 * q^52 + 61980 * q^53 - 729 * q^54 + 7132 * q^56 + 23274 * q^57 + 7161 * q^58 + 61190 * q^59 + 8230 * q^61 - 4509 * q^62 + 15228 * q^63 + 152223 * q^64 - 1089 * q^66 - 11930 * q^67 + 105598 * q^68 + 19854 * q^69 + 59822 * q^71 + 14337 * q^72 - 20680 * q^73 + 132564 * q^74 + 68165 * q^76 + 22748 * q^77 + 108009 * q^78 + 234494 * q^79 + 65610 * q^81 + 151948 * q^82 + 185478 * q^83 + 130068 * q^84 - 17825 * q^86 + 52416 * q^87 + 21417 * q^88 + 181834 * q^89 + 206274 * q^91 + 98373 * q^92 - 41274 * q^93 - 64998 * q^94 - 95643 * q^96 + 304358 * q^97 + 153453 * q^98 + 98010 * q^99

Introduction

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