LMFDB - Embedded newform 825.6.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
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			1.2
Root			$9.15502$ 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)

$f(q)$	$=$	$q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +121.000 q^{11} +466.329 q^{12} -60.5015 q^{13} -2075.04 q^{14} +2.66672 q^{16} +1984.50 q^{17} -741.556 q^{18} -1094.38 q^{19} +2039.90 q^{21} -1107.76 q^{22} -4636.35 q^{23} -1632.61 q^{24} +553.893 q^{26} +729.000 q^{27} +11744.0 q^{28} +8565.55 q^{29} -409.809 q^{31} +5780.41 q^{32} +1089.00 q^{33} -18168.1 q^{34} +4196.96 q^{36} -10085.7 q^{37} +10019.1 q^{38} -544.514 q^{39} +9021.70 q^{41} -18675.3 q^{42} +10712.1 q^{43} +6269.53 q^{44} +42445.8 q^{46} -17017.3 q^{47} +24.0005 q^{48} +34565.8 q^{49} +17860.5 q^{51} -3134.85 q^{52} +28394.2 q^{53} -6674.01 q^{54} -41115.5 q^{56} -9849.44 q^{57} -78417.8 q^{58} +30227.5 q^{59} -36839.5 q^{61} +3751.81 q^{62} +18359.1 q^{63} -53005.0 q^{64} -9969.81 q^{66} -11315.1 q^{67} +102825. q^{68} -41727.1 q^{69} +31917.4 q^{71} -14693.4 q^{72} -20881.1 q^{73} +92334.8 q^{74} -56704.7 q^{76} +27425.3 q^{77} +4985.03 q^{78} +77091.6 q^{79} +6561.00 q^{81} -82593.8 q^{82} +99048.9 q^{83} +105696. q^{84} -98069.0 q^{86} +77090.0 q^{87} -21949.5 q^{88} +130505. q^{89} -13713.0 q^{91} -240229. q^{92} -3688.28 q^{93} +155794. q^{94} +52023.6 q^{96} +74343.6 q^{97} -316450. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.15502	−1.61839	−0.809197	−	0.587538i	$-0.800097\pi$
$2$	−9.15502	−1.61839	−0.809197	+	0.587538i	$0.800097\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	51.8143	1.61920
$5$	0	0
$5$	0	0
$6$	−82.3952	−0.934380
$7$	226.656	1.74832	0.874161	−	0.485636i	$-0.161412\pi$
$7$	226.656	1.74832	0.874161	+	0.485636i	$0.161412\pi$
$8$	−181.401	−1.00211
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	466.329	0.934844
$13$	−60.5015	−0.0992906	−0.0496453	−	0.998767i	$-0.515809\pi$
$13$	−60.5015	−0.0992906	−0.0496453	+	0.998767i	$0.515809\pi$
$14$	−2075.04	−2.82947
$15$	0	0
$16$	2.66672	0.00260422
$17$	1984.50	1.66544	0.832718	−	0.553697i	$-0.186783\pi$
$17$	1984.50	1.66544	0.832718	+	0.553697i	$0.186783\pi$
$18$	−741.556	−0.539465
$19$	−1094.38	−0.695481	−0.347740	−	0.937591i	$-0.613051\pi$
$19$	−1094.38	−0.695481	−0.347740	+	0.937591i	$0.613051\pi$
$20$	0	0
$21$	2039.90	1.00939
$22$	−1107.76	−0.487964
$23$	−4636.35	−1.82750	−0.913748	−	0.406282i	$-0.866825\pi$
$23$	−4636.35	−1.82750	−0.913748	+	0.406282i	$0.866825\pi$
$24$	−1632.61	−0.578566
$25$	0	0
$26$	553.893	0.160691
$27$	729.000	0.192450
$28$	11744.0	2.83088
$29$	8565.55	1.89130	0.945650	−	0.325186i	$-0.105427\pi$
$29$	8565.55	1.89130	0.945650	+	0.325186i	$0.105427\pi$
$30$	0	0
$31$	−409.809	−0.0765908	−0.0382954	−	0.999266i	$-0.512193\pi$
$31$	−409.809	−0.0765908	−0.0382954	+	0.999266i	$0.512193\pi$
$32$	5780.41	0.997892
$33$	1089.00	0.174078
$34$	−18168.1	−2.69533
$35$	0	0
$36$	4196.96	0.539733
$37$	−10085.7	−1.21116	−0.605581	−	0.795784i	$-0.707059\pi$
$37$	−10085.7	−1.21116	−0.605581	+	0.795784i	$0.707059\pi$
$38$	10019.1	1.12556
$39$	−544.514	−0.0573254
$40$	0	0
$41$	9021.70	0.838163	0.419082	−	0.907949i	$-0.362352\pi$
$41$	9021.70	0.838163	0.419082	+	0.907949i	$0.362352\pi$
$42$	−18675.3	−1.63360
$43$	10712.1	0.883490	0.441745	−	0.897141i	$-0.354360\pi$
$43$	10712.1	0.883490	0.441745	+	0.897141i	$0.354360\pi$
$44$	6269.53	0.488207
$45$	0	0
$46$	42445.8	2.95761
$47$	−17017.3	−1.12369	−0.561844	−	0.827243i	$-0.689908\pi$
$47$	−17017.3	−1.12369	−0.561844	+	0.827243i	$0.689908\pi$
$48$	24.0005	0.00150355
$49$	34565.8	2.05663
$50$	0	0
$51$	17860.5	0.961540
$52$	−3134.85	−0.160771
$53$	28394.2	1.38848	0.694241	−	0.719742i	$-0.255740\pi$
$53$	28394.2	1.38848	0.694241	+	0.719742i	$0.255740\pi$
$54$	−6674.01	−0.311460
$55$	0	0
$56$	−41115.5	−1.75200
$57$	−9849.44	−0.401536
$58$	−78417.8	−3.06087
$59$	30227.5	1.13050	0.565252	−	0.824918i	$-0.308779\pi$
$59$	30227.5	1.13050	0.565252	+	0.824918i	$0.308779\pi$
$60$	0	0
$61$	−36839.5	−1.26762	−0.633810	−	0.773489i	$-0.718510\pi$
$61$	−36839.5	−1.26762	−0.633810	+	0.773489i	$0.718510\pi$
$62$	3751.81	0.123954
$63$	18359.1	0.582774
$64$	−53005.0	−1.61759
$65$	0	0
$66$	−9969.81	−0.281726
$67$	−11315.1	−0.307945	−0.153972	−	0.988075i	$-0.549207\pi$
$67$	−11315.1	−0.307945	−0.153972	+	0.988075i	$0.549207\pi$
$68$	102825.	2.69667
$69$	−41727.1	−1.05510
$70$	0	0
$71$	31917.4	0.751417	0.375709	−	0.926738i	$-0.377399\pi$
$71$	31917.4	0.751417	0.375709	+	0.926738i	$0.377399\pi$
$72$	−14693.4	−0.334035
$73$	−20881.1	−0.458612	−0.229306	−	0.973354i	$-0.573646\pi$
$73$	−20881.1	−0.458612	−0.229306	+	0.973354i	$0.573646\pi$
$74$	92334.8	1.96014
$75$	0	0
$76$	−56704.7	−1.12612
$77$	27425.3	0.527139
$78$	4985.03	0.0927751
$79$	77091.6	1.38976	0.694879	−	0.719126i	$-0.255458\pi$
$79$	77091.6	1.38976	0.694879	+	0.719126i	$0.255458\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−82593.8	−1.35648
$83$	99048.9	1.57817	0.789086	−	0.614283i	$-0.210555\pi$
$83$	99048.9	1.57817	0.789086	+	0.614283i	$0.210555\pi$
$84$	105696.	1.63441
$85$	0	0
$86$	−98069.0	−1.42983
$87$	77090.0	1.09194
$88$	−21949.5	−0.302146
$89$	130505.	1.74643	0.873217	−	0.487332i	$-0.162030\pi$
$89$	130505.	1.74643	0.873217	+	0.487332i	$0.162030\pi$
$90$	0	0
$91$	−13713.0	−0.173592
$92$	−240229.	−2.95908
$93$	−3688.28	−0.0442197
$94$	155794.	1.81857
$95$	0	0
$96$	52023.6	0.576133
$97$	74343.6	0.802259	0.401129	−	0.916021i	$-0.368618\pi$
$97$	74343.6	0.802259	0.401129	+	0.916021i	$0.368618\pi$
$98$	−316450.	−3.32844
$99$	9801.00	0.100504
$100$	0	0
$101$	7288.41	0.0710934	0.0355467	−	0.999368i	$-0.488683\pi$
$101$	7288.41	0.0710934	0.0355467	+	0.999368i	$0.488683\pi$
$102$	−163513.	−1.55615
$103$	55373.1	0.514287	0.257144	−	0.966373i	$-0.417219\pi$
$103$	55373.1	0.514287	0.257144	+	0.966373i	$0.417219\pi$
$104$	10975.0	0.0994997
$105$	0	0
$106$	−259950.	−2.24711
$107$	−227505.	−1.92102	−0.960510	−	0.278245i	$-0.910247\pi$
$107$	−227505.	−1.92102	−0.960510	+	0.278245i	$0.910247\pi$
$108$	37772.7	0.311615
$109$	−116049.	−0.935566	−0.467783	−	0.883843i	$-0.654947\pi$
$109$	−116049.	−0.935566	−0.467783	+	0.883843i	$0.654947\pi$
$110$	0	0
$111$	−90771.3	−0.699264
$112$	604.428	0.00455302
$113$	−213268.	−1.57119	−0.785597	−	0.618738i	$-0.787644\pi$
$113$	−213268.	−1.57119	−0.785597	+	0.618738i	$0.787644\pi$
$114$	90171.8	0.649843
$115$	0	0
$116$	443819.	3.06239
$117$	−4900.62	−0.0330969
$118$	−276733.	−1.82960
$119$	449797.	2.91172
$120$	0	0
$121$	14641.0	0.0909091
$122$	337266.	2.05151
$123$	81195.3	0.483914
$124$	−21234.0	−0.124016
$125$	0	0
$126$	−168078.	−0.943158
$127$	−76087.7	−0.418606	−0.209303	−	0.977851i	$-0.567119\pi$
$127$	−76087.7	−0.418606	−0.209303	+	0.977851i	$0.567119\pi$
$128$	300289.	1.62000
$129$	96408.5	0.510083
$130$	0	0
$131$	147751.	0.752233	0.376116	−	0.926572i	$-0.377259\pi$
$131$	147751.	0.752233	0.376116	+	0.926572i	$0.377259\pi$
$132$	56425.8	0.281866
$133$	−248048.	−1.21592
$134$	103590.	0.498376
$135$	0	0
$136$	−359989.	−1.66894
$137$	−231518.	−1.05386	−0.526931	−	0.849908i	$-0.676657\pi$
$137$	−231518.	−1.05386	−0.526931	+	0.849908i	$0.676657\pi$
$138$	382012.	1.70758
$139$	−121702.	−0.534270	−0.267135	−	0.963659i	$-0.586077\pi$
$139$	−121702.	−0.534270	−0.267135	+	0.963659i	$0.586077\pi$
$140$	0	0
$141$	−153156.	−0.648762
$142$	−292204.	−1.21609
$143$	−7320.69	−0.0299372
$144$	216.005	0.000868074 0
$145$	0	0
$146$	191167.	0.742215
$147$	311092.	1.18740
$148$	−522584.	−1.96111
$149$	−301130.	−1.11119	−0.555594	−	0.831454i	$-0.687509\pi$
$149$	−301130.	−1.11119	−0.555594	+	0.831454i	$0.687509\pi$
$150$	0	0
$151$	259623.	0.926619	0.463310	−	0.886196i	$-0.346662\pi$
$151$	259623.	0.926619	0.463310	+	0.886196i	$0.346662\pi$
$152$	198522.	0.696946
$153$	160744.	0.555146
$154$	−251079.	−0.853118
$155$	0	0
$156$	−28213.6	−0.0928212
$157$	−14890.4	−0.0482122	−0.0241061	−	0.999709i	$-0.507674\pi$
$157$	−14890.4	−0.0482122	−0.0241061	+	0.999709i	$0.507674\pi$
$158$	−705775.	−2.24918
$159$	255548.	0.801641
$160$	0	0
$161$	−1.05085e6	−3.19505
$162$	−60066.1	−0.179822
$163$	−200766.	−0.591862	−0.295931	−	0.955209i	$-0.595630\pi$
$163$	−200766.	−0.591862	−0.295931	+	0.955209i	$0.595630\pi$
$164$	467453.	1.35715
$165$	0	0
$166$	−906794.	−2.55410
$167$	89683.6	0.248841	0.124420	−	0.992230i	$-0.460293\pi$
$167$	89683.6	0.248841	0.124420	+	0.992230i	$0.460293\pi$
$168$	−370039.	−1.01152
$169$	−367633.	−0.990141
$170$	0	0
$171$	−88645.0	−0.231827
$172$	555038.	1.43054
$173$	413325.	1.04997	0.524984	−	0.851112i	$-0.324071\pi$
$173$	413325.	1.04997	0.524984	+	0.851112i	$0.324071\pi$
$174$	−705760.	−1.76719
$175$	0	0
$176$	322.674	0.000785203 0
$177$	272047.	0.652697
$178$	−1.19478e6	−2.82642
$179$	36462.3	0.0850572	0.0425286	−	0.999095i	$-0.486459\pi$
$179$	36462.3	0.0850572	0.0425286	+	0.999095i	$0.486459\pi$
$180$	0	0
$181$	143714.	0.326064	0.163032	−	0.986621i	$-0.447873\pi$
$181$	143714.	0.326064	0.163032	+	0.986621i	$0.447873\pi$
$182$	125543.	0.280940
$183$	−331556.	−0.731861
$184$	841036.	1.83134
$185$	0	0
$186$	33766.2	0.0715650
$187$	240124.	0.502148
$188$	−881740.	−1.81947
$189$	165232.	0.336465
$190$	0	0
$191$	144930.	0.287458	0.143729	−	0.989617i	$-0.454091\pi$
$191$	144930.	0.287458	0.143729	+	0.989617i	$0.454091\pi$
$192$	−477045.	−0.933913
$193$	177102.	0.342239	0.171120	−	0.985250i	$-0.445262\pi$
$193$	177102.	0.342239	0.171120	+	0.985250i	$0.445262\pi$
$194$	−680617.	−1.29837
$195$	0	0
$196$	1.79100e6	3.33009
$197$	−1.01364e6	−1.86087	−0.930437	−	0.366453i	$-0.880572\pi$
$197$	−1.01364e6	−1.86087	−0.930437	+	0.366453i	$0.880572\pi$
$198$	−89728.3	−0.162655
$199$	235430.	0.421434	0.210717	−	0.977547i	$-0.432420\pi$
$199$	235430.	0.421434	0.210717	+	0.977547i	$0.432420\pi$
$200$	0	0
$201$	−101836.	−0.177792
$202$	−66725.5	−0.115057
$203$	1.94143e6	3.30660
$204$	925429.	1.55692
$205$	0	0
$206$	−506942.	−0.832320
$207$	−375544.	−0.609165
$208$	−161.341	−0.000258575 0
$209$	−132420.	−0.209695
$210$	0	0
$211$	−919022.	−1.42108	−0.710542	−	0.703655i	$-0.751550\pi$
$211$	−919022.	−1.42108	−0.710542	+	0.703655i	$0.751550\pi$
$212$	1.47123e6	2.24823
$213$	287256.	0.433831
$214$	2.08281e6	3.10897
$215$	0	0
$216$	−132241.	−0.192855
$217$	−92885.4	−0.133905
$218$	1.06243e6	1.51411
$219$	−187930.	−0.264780
$220$	0	0
$221$	−120065.	−0.165362
$222$	831013.	1.13168
$223$	1.28777e6	1.73411	0.867053	−	0.498216i	$-0.166011\pi$
$223$	1.28777e6	1.73411	0.867053	+	0.498216i	$0.166011\pi$
$224$	1.31016e6	1.74464
$225$	0	0
$226$	1.95247e6	2.54281
$227$	−545866.	−0.703107	−0.351554	−	0.936168i	$-0.614346\pi$
$227$	−545866.	−0.703107	−0.351554	+	0.936168i	$0.614346\pi$
$228$	−510342.	−0.650166
$229$	−542716.	−0.683886	−0.341943	−	0.939721i	$-0.611085\pi$
$229$	−542716.	−0.683886	−0.341943	+	0.939721i	$0.611085\pi$
$230$	0	0
$231$	246828.	0.304344
$232$	−1.55380e6	−1.89528
$233$	126501.	0.152653	0.0763265	−	0.997083i	$-0.475681\pi$
$233$	126501.	0.152653	0.0763265	+	0.997083i	$0.475681\pi$
$234$	44865.3	0.0535637
$235$	0	0
$236$	1.56622e6	1.83051
$237$	693825.	0.802378
$238$	−4.11790e6	−4.71231
$239$	1.36942e6	1.55075	0.775377	−	0.631499i	$-0.217560\pi$
$239$	1.36942e6	1.55075	0.775377	+	0.631499i	$0.217560\pi$
$240$	0	0
$241$	1.28969e6	1.43036	0.715178	−	0.698942i	$-0.246346\pi$
$241$	1.28969e6	1.43036	0.715178	+	0.698942i	$0.246346\pi$
$242$	−134039.	−0.147127
$243$	59049.0	0.0641500
$244$	−1.90881e6	−2.05253
$245$	0	0
$246$	−743344.	−0.783163
$247$	66211.8	0.0690547
$248$	74339.5	0.0767522
$249$	891440.	0.911158
$250$	0	0
$251$	316251.	0.316845	0.158423	−	0.987371i	$-0.449359\pi$
$251$	316251.	0.316845	0.158423	+	0.987371i	$0.449359\pi$
$252$	951265.	0.943626
$253$	−560998.	−0.551011
$254$	696585.	0.677469
$255$	0	0
$256$	−1.05299e6	−1.00421
$257$	−100900.	−0.0952922	−0.0476461	−	0.998864i	$-0.515172\pi$
$257$	−100900.	−0.0952922	−0.0476461	+	0.998864i	$0.515172\pi$
$258$	−882621.	−0.825515
$259$	−2.28598e6	−2.11750
$260$	0	0
$261$	693810.	0.630433
$262$	−1.35266e6	−1.21741
$263$	−346997.	−0.309340	−0.154670	−	0.987966i	$-0.549431\pi$
$263$	−346997.	−0.309340	−0.154670	+	0.987966i	$0.549431\pi$
$264$	−197545.	−0.174444
$265$	0	0
$266$	2.27088e6	1.96784
$267$	1.17454e6	1.00830
$268$	−586287.	−0.498624
$269$	1.12964e6	0.951831	0.475915	−	0.879491i	$-0.342117\pi$
$269$	1.12964e6	0.951831	0.475915	+	0.879491i	$0.342117\pi$
$270$	0	0
$271$	756638.	0.625842	0.312921	−	0.949779i	$-0.398692\pi$
$271$	756638.	0.625842	0.312921	+	0.949779i	$0.398692\pi$
$272$	5292.11	0.00433717
$273$	−123417.	−0.100223
$274$	2.11955e6	1.70556
$275$	0	0
$276$	−2.16206e6	−1.70842
$277$	1.61681e6	1.26607	0.633036	−	0.774122i	$-0.281808\pi$
$277$	1.61681e6	1.26607	0.633036	+	0.774122i	$0.281808\pi$
$278$	1.11419e6	0.864660
$279$	−33194.5	−0.0255303
$280$	0	0
$281$	1.28139e6	0.968091	0.484046	−	0.875043i	$-0.339167\pi$
$281$	1.28139e6	0.968091	0.484046	+	0.875043i	$0.339167\pi$
$282$	1.40214e6	1.04995
$283$	−1.11420e6	−0.826986	−0.413493	−	0.910507i	$-0.635691\pi$
$283$	−1.11420e6	−0.826986	−0.413493	+	0.910507i	$0.635691\pi$
$284$	1.65378e6	1.21669
$285$	0	0
$286$	67021.0	0.0484502
$287$	2.04482e6	1.46538
$288$	468213.	0.332631
$289$	2.51837e6	1.77368
$290$	0	0
$291$	669093.	0.463184
$292$	−1.08194e6	−0.742584
$293$	1.03577e6	0.704843	0.352421	−	0.935841i	$-0.385358\pi$
$293$	1.03577e6	0.704843	0.352421	+	0.935841i	$0.385358\pi$
$294$	−2.84805e6	−1.92167
$295$	0	0
$296$	1.82955e6	1.21371
$297$	88209.0	0.0580259
$298$	2.75685e6	1.79834
$299$	280506.	0.181453
$300$	0	0
$301$	2.42795e6	1.54462
$302$	−2.37686e6	−1.49963
$303$	65595.7	0.0410458
$304$	−2918.42	−0.00181119
$305$	0	0
$306$	−1.47162e6	−0.898444
$307$	2.52316e6	1.52791	0.763957	−	0.645267i	$-0.223254\pi$
$307$	2.52316e6	1.52791	0.763957	+	0.645267i	$0.223254\pi$
$308$	1.42103e6	0.853542
$309$	498358.	0.296924
$310$	0	0
$311$	1.75014e6	1.02606	0.513028	−	0.858372i	$-0.328524\pi$
$311$	1.75014e6	1.02606	0.513028	+	0.858372i	$0.328524\pi$
$312$	98775.1	0.0574462
$313$	840872.	0.485142	0.242571	−	0.970134i	$-0.422009\pi$
$313$	840872.	0.485142	0.242571	+	0.970134i	$0.422009\pi$
$314$	136322.	0.0780263
$315$	0	0
$316$	3.99445e6	2.25029
$317$	1.34573e6	0.752157	0.376079	−	0.926588i	$-0.377272\pi$
$317$	1.34573e6	0.752157	0.376079	+	0.926588i	$0.377272\pi$
$318$	−2.33955e6	−1.29737
$319$	1.03643e6	0.570248
$320$	0	0
$321$	−2.04755e6	−1.10910
$322$	9.62059e6	5.17085
$323$	−2.17180e6	−1.15828
$324$	339954.	0.179911
$325$	0	0
$326$	1.83801e6	0.957865
$327$	−1.04444e6	−0.540149
$328$	−1.63654e6	−0.839928
$329$	−3.85707e6	−1.96457
$330$	0	0
$331$	−1.71783e6	−0.861808	−0.430904	−	0.902398i	$-0.641805\pi$
$331$	−1.71783e6	−0.861808	−0.430904	+	0.902398i	$0.641805\pi$
$332$	5.13215e6	2.55537
$333$	−816942.	−0.403720
$334$	−821055.	−0.402723
$335$	0	0
$336$	5439.85	0.00262869
$337$	−1.88053e6	−0.901996	−0.450998	−	0.892525i	$-0.648932\pi$
$337$	−1.88053e6	−0.901996	−0.450998	+	0.892525i	$0.648932\pi$
$338$	3.36568e6	1.60244
$339$	−1.91941e6	−0.907130
$340$	0	0
$341$	−49586.8	−0.0230930
$342$	811546.	0.375187
$343$	4.02512e6	1.84733
$344$	−1.94317e6	−0.885350
$345$	0	0
$346$	−3.78399e6	−1.69926
$347$	777998.	0.346860	0.173430	−	0.984846i	$-0.444515\pi$
$347$	777998.	0.346860	0.173430	+	0.984846i	$0.444515\pi$
$348$	3.99437e6	1.76807
$349$	−36672.3	−0.0161166	−0.00805831	−	0.999968i	$-0.502565\pi$
$349$	−36672.3	−0.0161166	−0.00805831	+	0.999968i	$0.502565\pi$
$350$	0	0
$351$	−44105.6	−0.0191085
$352$	699429.	0.300876
$353$	−2.53998e6	−1.08491	−0.542455	−	0.840085i	$-0.682505\pi$
$353$	−2.53998e6	−1.08491	−0.542455	+	0.840085i	$0.682505\pi$
$354$	−2.49060e6	−1.05632
$355$	0	0
$356$	6.76203e6	2.82782
$357$	4.04818e6	1.68108
$358$	−333813.	−0.137656
$359$	460599.	0.188620	0.0943098	−	0.995543i	$-0.469936\pi$
$359$	460599.	0.188620	0.0943098	+	0.995543i	$0.469936\pi$
$360$	0	0
$361$	−1.27843e6	−0.516306
$362$	−1.31570e6	−0.527699
$363$	131769.	0.0524864
$364$	−710531.	−0.281080
$365$	0	0
$366$	3.03540e6	1.18444
$367$	−918849.	−0.356106	−0.178053	−	0.984021i	$-0.556980\pi$
$367$	−918849.	−0.356106	−0.178053	+	0.984021i	$0.556980\pi$
$368$	−12363.9	−0.00475921
$369$	730757.	0.279388
$370$	0	0
$371$	6.43571e6	2.42751
$372$	−191106.	−0.0716005
$373$	476713.	0.177413	0.0887064	−	0.996058i	$-0.471727\pi$
$373$	476713.	0.177413	0.0887064	+	0.996058i	$0.471727\pi$
$374$	−2.19834e6	−0.812673
$375$	0	0
$376$	3.08695e6	1.12606
$377$	−518229.	−0.187788
$378$	−1.51270e6	−0.544532
$379$	340866.	0.121895	0.0609474	−	0.998141i	$-0.480588\pi$
$379$	340866.	0.121895	0.0609474	+	0.998141i	$0.480588\pi$
$380$	0	0
$381$	−684790.	−0.241682
$382$	−1.32684e6	−0.465221
$383$	322302.	0.112271	0.0561353	−	0.998423i	$-0.482122\pi$
$383$	322302.	0.112271	0.0561353	+	0.998423i	$0.482122\pi$
$384$	2.70260e6	0.935307
$385$	0	0
$386$	−1.62137e6	−0.553878
$387$	867676.	0.294497
$388$	3.85207e6	1.29902
$389$	2.91721e6	0.977447	0.488724	−	0.872439i	$-0.337463\pi$
$389$	2.91721e6	0.977447	0.488724	+	0.872439i	$0.337463\pi$
$390$	0	0
$391$	−9.20082e6	−3.04358
$392$	−6.27025e6	−2.06096
$393$	1.32976e6	0.434302
$394$	9.27986e6	3.01163
$395$	0	0
$396$	507832.	0.162736
$397$	2.07740e6	0.661520	0.330760	−	0.943715i	$-0.392695\pi$
$397$	2.07740e6	0.661520	0.330760	+	0.943715i	$0.392695\pi$
$398$	−2.15537e6	−0.682047
$399$	−2.23243e6	−0.702014
$400$	0	0
$401$	−2.82127e6	−0.876161	−0.438080	−	0.898936i	$-0.644341\pi$
$401$	−2.82127e6	−0.876161	−0.438080	+	0.898936i	$0.644341\pi$
$402$	932313.	0.287738
$403$	24794.1	0.00760475
$404$	377644.	0.115114
$405$	0	0
$406$	−1.77738e7	−5.35138
$407$	−1.22037e6	−0.365179
$408$	−3.23990e6	−0.963566
$409$	−2.31242e6	−0.683530	−0.341765	−	0.939785i	$-0.611025\pi$
$409$	−2.31242e6	−0.683530	−0.341765	+	0.939785i	$0.611025\pi$
$410$	0	0
$411$	−2.08366e6	−0.608447
$412$	2.86912e6	0.832733
$413$	6.85123e6	1.97649
$414$	3.43811e6	0.985869
$415$	0	0
$416$	−349723.	−0.0990812
$417$	−1.09532e6	−0.308461
$418$	1.21231e6	0.339370
$419$	5.50334e6	1.53141	0.765705	−	0.643192i	$-0.222390\pi$
$419$	5.50334e6	1.53141	0.765705	+	0.643192i	$0.222390\pi$
$420$	0	0
$421$	−4.12010e6	−1.13293	−0.566464	−	0.824086i	$-0.691689\pi$
$421$	−4.12010e6	−1.13293	−0.566464	+	0.824086i	$0.691689\pi$
$422$	8.41366e6	2.29987
$423$	−1.37840e6	−0.374563
$424$	−5.15073e6	−1.39141
$425$	0	0
$426$	−2.62984e6	−0.702109
$427$	−8.34988e6	−2.21621
$428$	−1.17880e7	−3.11051
$429$	−65886.2	−0.0172843
$430$	0	0
$431$	721675.	0.187132	0.0935661	−	0.995613i	$-0.470173\pi$
$431$	721675.	0.187132	0.0935661	+	0.995613i	$0.470173\pi$
$432$	1944.04	0.000501183 0
$433$	−1.55374e6	−0.398253	−0.199126	−	0.979974i	$-0.563810\pi$
$433$	−1.55374e6	−0.398253	−0.199126	+	0.979974i	$0.563810\pi$
$434$	850368.	0.216712
$435$	0	0
$436$	−6.01299e6	−1.51487
$437$	5.07394e6	1.27099
$438$	1.72050e6	0.428518
$439$	−6.59983e6	−1.63445	−0.817224	−	0.576320i	$-0.804488\pi$
$439$	−6.59983e6	−1.63445	−0.817224	+	0.576320i	$0.804488\pi$
$440$	0	0
$441$	2.79983e6	0.685543
$442$	1.09920e6	0.267621
$443$	−1.41708e6	−0.343071	−0.171536	−	0.985178i	$-0.554873\pi$
$443$	−1.41708e6	−0.343071	−0.171536	+	0.985178i	$0.554873\pi$
$444$	−4.70326e6	−1.13225
$445$	0	0
$446$	−1.17895e7	−2.80647
$447$	−2.71017e6	−0.641545
$448$	−1.20139e7	−2.82806
$449$	−3.90859e6	−0.914964	−0.457482	−	0.889219i	$-0.651249\pi$
$449$	−3.90859e6	−0.914964	−0.457482	+	0.889219i	$0.651249\pi$
$450$	0	0
$451$	1.09163e6	0.252716
$452$	−1.10504e7	−2.54408
$453$	2.33661e6	0.534984
$454$	4.99741e6	1.13790
$455$	0	0
$456$	1.78669e6	0.402382
$457$	4.85837e6	1.08818	0.544090	−	0.839027i	$-0.316875\pi$
$457$	4.85837e6	1.08818	0.544090	+	0.839027i	$0.316875\pi$
$458$	4.96857e6	1.10680
$459$	1.44670e6	0.320513
$460$	0	0
$461$	905464.	0.198435	0.0992176	−	0.995066i	$-0.468366\pi$
$461$	905464.	0.198435	0.0992176	+	0.995066i	$0.468366\pi$
$462$	−2.25971e6	−0.492548
$463$	−3.54245e6	−0.767981	−0.383991	−	0.923337i	$-0.625451\pi$
$463$	−3.54245e6	−0.767981	−0.383991	+	0.923337i	$0.625451\pi$
$464$	22842.0	0.00492537
$465$	0	0
$466$	−1.15812e6	−0.247052
$467$	−7.14306e6	−1.51563	−0.757813	−	0.652472i	$-0.773732\pi$
$467$	−7.14306e6	−1.51563	−0.757813	+	0.652472i	$0.773732\pi$
$468$	−253923.	−0.0535904
$469$	−2.56464e6	−0.538387
$470$	0	0
$471$	−134013.	−0.0278353
$472$	−5.48329e6	−1.13289
$473$	1.29616e6	0.266382
$474$	−6.35198e6	−1.29856
$475$	0	0
$476$	2.33060e7	4.71465
$477$	2.29993e6	0.462828
$478$	−1.25371e7	−2.50973
$479$	4.49284e6	0.894711	0.447355	−	0.894356i	$-0.352366\pi$
$479$	4.49284e6	0.894711	0.447355	+	0.894356i	$0.352366\pi$
$480$	0	0
$481$	610201.	0.120257
$482$	−1.18072e7	−2.31488
$483$	−9.45768e6	−1.84466
$484$	758614.	0.147200
$485$	0	0
$486$	−540595.	−0.103820
$487$	−2.83972e6	−0.542567	−0.271283	−	0.962500i	$-0.587448\pi$
$487$	−2.83972e6	−0.542567	−0.271283	+	0.962500i	$0.587448\pi$
$488$	6.68271e6	1.27029
$489$	−1.80689e6	−0.341712
$490$	0	0
$491$	−604654.	−0.113189	−0.0565944	−	0.998397i	$-0.518024\pi$
$491$	−604654.	−0.113189	−0.0565944	+	0.998397i	$0.518024\pi$
$492$	4.20708e6	0.783552
$493$	1.69983e7	3.14984
$494$	−606170.	−0.111758
$495$	0	0
$496$	−1092.85	−0.000199460 0
$497$	7.23425e6	1.31372
$498$	−8.16115e6	−1.47461
$499$	979877.	0.176165	0.0880826	−	0.996113i	$-0.471926\pi$
$499$	979877.	0.176165	0.0880826	+	0.996113i	$0.471926\pi$
$500$	0	0
$501$	807153.	0.143668
$502$	−2.89528e6	−0.512781
$503$	6.83985e6	1.20539	0.602693	−	0.797973i	$-0.294094\pi$
$503$	6.83985e6	1.20539	0.602693	+	0.797973i	$0.294094\pi$
$504$	−3.33035e6	−0.584001
$505$	0	0
$506$	5.13595e6	0.891752
$507$	−3.30869e6	−0.571658
$508$	−3.94244e6	−0.677806
$509$	3.42217e6	0.585473	0.292737	−	0.956193i	$-0.405434\pi$
$509$	3.42217e6	0.585473	0.292737	+	0.956193i	$0.405434\pi$
$510$	0	0
$511$	−4.73281e6	−0.801802
$512$	30894.6	0.00520844
$513$	−797805.	−0.133845
$514$	923739.	0.154220
$515$	0	0
$516$	4.99534e6	0.825925
$517$	−2.05909e6	−0.338805
$518$	2.09282e7	3.42695
$519$	3.71992e6	0.606199
$520$	0	0
$521$	3.25613e6	0.525542	0.262771	−	0.964858i	$-0.415364\pi$
$521$	3.25613e6	0.525542	0.262771	+	0.964858i	$0.415364\pi$
$522$	−6.35184e6	−1.02029
$523$	−2.77887e6	−0.444237	−0.222118	−	0.975020i	$-0.571297\pi$
$523$	−2.77887e6	−0.444237	−0.222118	+	0.975020i	$0.571297\pi$
$524$	7.65562e6	1.21801
$525$	0	0
$526$	3.17676e6	0.500634
$527$	−813264.	−0.127557
$528$	2904.06	0.000453337 0
$529$	1.50594e7	2.33974
$530$	0	0
$531$	2.44843e6	0.376835
$532$	−1.28524e7	−1.96882
$533$	−545826.	−0.0832217
$534$	−1.07530e7	−1.63183
$535$	0	0
$536$	2.05257e6	0.308594
$537$	328161.	0.0491078
$538$	−1.03419e7	−1.54044
$539$	4.18246e6	0.620097
$540$	0	0
$541$	−882770.	−0.129675	−0.0648373	−	0.997896i	$-0.520653\pi$
$541$	−882770.	−0.129675	−0.0648373	+	0.997896i	$0.520653\pi$
$542$	−6.92703e6	−1.01286
$543$	1.29342e6	0.188253
$544$	1.14712e7	1.66193
$545$	0	0
$546$	1.12989e6	0.162201
$547$	6.88538e6	0.983920	0.491960	−	0.870618i	$-0.336281\pi$
$547$	6.88538e6	0.983920	0.491960	+	0.870618i	$0.336281\pi$
$548$	−1.19960e7	−1.70641
$549$	−2.98400e6	−0.422540
$550$	0	0
$551$	−9.37399e6	−1.31536
$552$	7.56932e6	1.05733
$553$	1.74732e7	2.42975
$554$	−1.48019e7	−2.04900
$555$	0	0
$556$	−6.30591e6	−0.865090
$557$	1.75983e6	0.240344	0.120172	−	0.992753i	$-0.461655\pi$
$557$	1.75983e6	0.240344	0.120172	+	0.992753i	$0.461655\pi$
$558$	303896.	0.0413180
$559$	−648096.	−0.0877222
$560$	0	0
$561$	2.16112e6	0.289915
$562$	−1.17312e7	−1.56675
$563$	1.06993e7	1.42260	0.711301	−	0.702888i	$-0.248107\pi$
$563$	1.06993e7	1.42260	0.711301	+	0.702888i	$0.248107\pi$
$564$	−7.93566e6	−1.05047
$565$	0	0
$566$	1.02006e7	1.33839
$567$	1.48709e6	0.194258
$568$	−5.78983e6	−0.753000
$569$	−1.02640e6	−0.132904	−0.0664518	−	0.997790i	$-0.521168\pi$
$569$	−1.02640e6	−0.132904	−0.0664518	+	0.997790i	$0.521168\pi$
$570$	0	0
$571$	−1.40704e7	−1.80599	−0.902995	−	0.429650i	$-0.858637\pi$
$571$	−1.40704e7	−1.80599	−0.902995	+	0.429650i	$0.858637\pi$
$572$	−379316.	−0.0484743
$573$	1.30437e6	0.165964
$574$	−1.87203e7	−2.37156
$575$	0	0
$576$	−4.29341e6	−0.539195
$577$	1.54276e6	0.192912	0.0964562	−	0.995337i	$-0.469249\pi$
$577$	1.54276e6	0.192912	0.0964562	+	0.995337i	$0.469249\pi$
$578$	−2.30557e7	−2.87051
$579$	1.59392e6	0.197592
$580$	0	0
$581$	2.24500e7	2.75915
$582$	−6.12555e6	−0.749614
$583$	3.43570e6	0.418643
$584$	3.78784e6	0.459578
$585$	0	0
$586$	−9.48245e6	−1.14071
$587$	−3.85126e6	−0.461325	−0.230663	−	0.973034i	$-0.574089\pi$
$587$	−3.85126e6	−0.461325	−0.230663	+	0.973034i	$0.574089\pi$
$588$	1.61190e7	1.92263
$589$	448487.	0.0532675
$590$	0	0
$591$	−9.12273e6	−1.07438
$592$	−26895.8	−0.00315413
$593$	−304507.	−0.0355599	−0.0177799	−	0.999842i	$-0.505660\pi$
$593$	−304507.	−0.0355599	−0.0177799	+	0.999842i	$0.505660\pi$
$594$	−807555.	−0.0939087
$595$	0	0
$596$	−1.56028e7	−1.79923
$597$	2.11887e6	0.243315
$598$	−2.56804e6	−0.293662
$599$	5.46626e6	0.622476	0.311238	−	0.950332i	$-0.399256\pi$
$599$	5.46626e6	0.622476	0.311238	+	0.950332i	$0.399256\pi$
$600$	0	0
$601$	−4.68185e6	−0.528726	−0.264363	−	0.964423i	$-0.585162\pi$
$601$	−4.68185e6	−0.528726	−0.264363	+	0.964423i	$0.585162\pi$
$602$	−2.22279e7	−2.49981
$603$	−916527.	−0.102648
$604$	1.34522e7	1.50038
$605$	0	0
$606$	−600530.	−0.0664283
$607$	−6.36108e6	−0.700744	−0.350372	−	0.936611i	$-0.613945\pi$
$607$	−6.36108e6	−0.700744	−0.350372	+	0.936611i	$0.613945\pi$
$608$	−6.32597e6	−0.694014
$609$	1.74729e7	1.90907
$610$	0	0
$611$	1.02957e6	0.111572
$612$	8.32886e6	0.898891
$613$	8.61598e6	0.926091	0.463045	−	0.886335i	$-0.346757\pi$
$613$	8.61598e6	0.926091	0.463045	+	0.886335i	$0.346757\pi$
$614$	−2.30996e7	−2.47277
$615$	0	0
$616$	−4.97497e6	−0.528249
$617$	1.46873e6	0.155321	0.0776605	−	0.996980i	$-0.475255\pi$
$617$	1.46873e6	0.155321	0.0776605	+	0.996980i	$0.475255\pi$
$618$	−4.56248e6	−0.480540
$619$	1.01690e7	1.06672	0.533362	−	0.845887i	$-0.320928\pi$
$619$	1.01690e7	1.06672	0.533362	+	0.845887i	$0.320928\pi$
$620$	0	0
$621$	−3.37990e6	−0.351702
$622$	−1.60225e7	−1.66056
$623$	2.95797e7	3.05333
$624$	−1452.07	−0.000149288 0
$625$	0	0
$626$	−7.69819e6	−0.785151
$627$	−1.19178e6	−0.121068
$628$	−771535.	−0.0780650
$629$	−2.00150e7	−2.01711
$630$	0	0
$631$	5.17219e6	0.517131	0.258566	−	0.965994i	$-0.416750\pi$
$631$	5.17219e6	0.517131	0.258566	+	0.965994i	$0.416750\pi$
$632$	−1.39845e7	−1.39269
$633$	−8.27120e6	−0.820463
$634$	−1.23201e7	−1.21729
$635$	0	0
$636$	1.32411e7	1.29802
$637$	−2.09128e6	−0.204204
$638$	−9.48855e6	−0.922887
$639$	2.58531e6	0.250472
$640$	0	0
$641$	−1.09775e7	−1.05526	−0.527629	−	0.849475i	$-0.676919\pi$
$641$	−1.09775e7	−1.05526	−0.527629	+	0.849475i	$0.676919\pi$
$642$	1.87453e7	1.79496
$643$	−348856.	−0.0332750	−0.0166375	−	0.999862i	$-0.505296\pi$
$643$	−348856.	−0.0332750	−0.0166375	+	0.999862i	$0.505296\pi$
$644$	−5.44493e7	−5.17342
$645$	0	0
$646$	1.98829e7	1.87455
$647$	−1.90506e7	−1.78915	−0.894577	−	0.446915i	$-0.852523\pi$
$647$	−1.90506e7	−1.78915	−0.894577	+	0.446915i	$0.852523\pi$
$648$	−1.19017e6	−0.111345
$649$	3.65753e6	0.340860
$650$	0	0
$651$	−835969.	−0.0773103
$652$	−1.04025e7	−0.958342
$653$	3.51919e6	0.322968	0.161484	−	0.986875i	$-0.448372\pi$
$653$	3.51919e6	0.322968	0.161484	+	0.986875i	$0.448372\pi$
$654$	9.56186e6	0.874174
$655$	0	0
$656$	24058.4	0.00218276
$657$	−1.69137e6	−0.152871
$658$	3.53115e7	3.17945
$659$	1.68730e7	1.51349	0.756743	−	0.653712i	$-0.226789\pi$
$659$	1.68730e7	1.51349	0.756743	+	0.653712i	$0.226789\pi$
$660$	0	0
$661$	1.61331e6	0.143620	0.0718098	−	0.997418i	$-0.477123\pi$
$661$	1.61331e6	0.143620	0.0718098	+	0.997418i	$0.477123\pi$
$662$	1.57268e7	1.39474
$663$	−1.08059e6	−0.0954719
$664$	−1.79675e7	−1.58150
$665$	0	0
$666$	7.47912e6	0.653378
$667$	−3.97129e7	−3.45634
$668$	4.64690e6	0.402923
$669$	1.15899e7	1.00119
$670$	0	0
$671$	−4.45758e6	−0.382202
$672$	1.17915e7	1.00727
$673$	−1.03328e7	−0.879387	−0.439693	−	0.898148i	$-0.644913\pi$
$673$	−1.03328e7	−0.879387	−0.439693	+	0.898148i	$0.644913\pi$
$674$	1.72162e7	1.45978
$675$	0	0
$676$	−1.90486e7	−1.60323
$677$	1.19467e7	1.00179	0.500895	−	0.865508i	$-0.333004\pi$
$677$	1.19467e7	1.00179	0.500895	+	0.865508i	$0.333004\pi$
$678$	1.75723e7	1.46809
$679$	1.68504e7	1.40261
$680$	0	0
$681$	−4.91279e6	−0.405939
$682$	453968.	0.0373736
$683$	−1.23358e7	−1.01185	−0.505926	−	0.862577i	$-0.668849\pi$
$683$	−1.23358e7	−1.01185	−0.505926	+	0.862577i	$0.668849\pi$
$684$	−4.59308e6	−0.375374
$685$	0	0
$686$	−3.68501e7	−2.98970
$687$	−4.88444e6	−0.394842
$688$	28566.1	0.00230080
$689$	−1.71789e6	−0.137863
$690$	0	0
$691$	2.00464e7	1.59714	0.798568	−	0.601904i	$-0.205591\pi$
$691$	2.00464e7	1.59714	0.798568	+	0.601904i	$0.205591\pi$
$692$	2.14161e7	1.70011
$693$	2.22145e6	0.175713
$694$	−7.12259e6	−0.561357
$695$	0	0
$696$	−1.39842e7	−1.09424
$697$	1.79035e7	1.39591
$698$	335735.	0.0260830
$699$	1.13851e6	0.0881342
$700$	0	0
$701$	1.38095e7	1.06141	0.530706	−	0.847556i	$-0.321927\pi$
$701$	1.38095e7	1.06141	0.530706	+	0.847556i	$0.321927\pi$
$702$	403788.	0.0309250
$703$	1.10376e7	0.842339
$704$	−6.41361e6	−0.487720
$705$	0	0
$706$	2.32536e7	1.75581
$707$	1.65196e6	0.124294
$708$	1.40960e7	1.05685
$709$	−9.12564e6	−0.681785	−0.340893	−	0.940102i	$-0.610729\pi$
$709$	−9.12564e6	−0.681785	−0.340893	+	0.940102i	$0.610729\pi$
$710$	0	0
$711$	6.24442e6	0.463253
$712$	−2.36737e7	−1.75011
$713$	1.90001e6	0.139969
$714$	−3.70611e7	−2.72065
$715$	0	0
$716$	1.88927e6	0.137724
$717$	1.23248e7	0.895328
$718$	−4.21679e6	−0.305261
$719$	1.20353e7	0.868232	0.434116	−	0.900857i	$-0.357061\pi$
$719$	1.20353e7	0.868232	0.434116	+	0.900857i	$0.357061\pi$
$720$	0	0
$721$	1.25506e7	0.899140
$722$	1.17040e7	0.835587
$723$	1.16073e7	0.825817
$724$	7.44644e6	0.527961
$725$	0	0
$726$	−1.20635e6	−0.0849436
$727$	2.37157e7	1.66418	0.832089	−	0.554642i	$-0.187145\pi$
$727$	2.37157e7	1.66418	0.832089	+	0.554642i	$0.187145\pi$
$728$	2.48755e6	0.173957
$729$	531441.	0.0370370
$730$	0	0
$731$	2.12580e7	1.47140
$732$	−1.71793e7	−1.18503
$733$	−2.43268e7	−1.67234	−0.836172	−	0.548467i	$-0.815212\pi$
$733$	−2.43268e7	−1.67234	−0.836172	+	0.548467i	$0.815212\pi$
$734$	8.41208e6	0.576320
$735$	0	0
$736$	−2.68000e7	−1.82364
$737$	−1.36913e6	−0.0928489
$738$	−6.69010e6	−0.452159
$739$	2.33006e7	1.56948	0.784739	−	0.619826i	$-0.212797\pi$
$739$	2.33006e7	1.56948	0.784739	+	0.619826i	$0.212797\pi$
$740$	0	0
$741$	595906.	0.0398687
$742$	−5.89191e7	−3.92867
$743$	5.15145e6	0.342340	0.171170	−	0.985242i	$-0.445245\pi$
$743$	5.15145e6	0.342340	0.171170	+	0.985242i	$0.445245\pi$
$744$	669056.	0.0443129
$745$	0	0
$746$	−4.36431e6	−0.287124
$747$	8.02296e6	0.526057
$748$	1.24419e7	0.813077
$749$	−5.15653e7	−3.35856
$750$	0	0
$751$	−1.59164e7	−1.02978	−0.514891	−	0.857256i	$-0.672168\pi$
$751$	−1.59164e7	−1.02978	−0.514891	+	0.857256i	$0.672168\pi$
$752$	−45380.5	−0.00292634
$753$	2.84626e6	0.182931
$754$	4.74440e6	0.303915
$755$	0	0
$756$	8.56138e6	0.544803
$757$	−4.92277e6	−0.312227	−0.156113	−	0.987739i	$-0.549897\pi$
$757$	−4.92277e6	−0.312227	−0.156113	+	0.987739i	$0.549897\pi$
$758$	−3.12063e6	−0.197274
$759$	−5.04898e6	−0.318126
$760$	0	0
$761$	−2.61608e7	−1.63753	−0.818765	−	0.574129i	$-0.805341\pi$
$761$	−2.61608e7	−1.63753	−0.818765	+	0.574129i	$0.805341\pi$
$762$	6.26926e6	0.391137
$763$	−2.63031e7	−1.63567
$764$	7.50945e6	0.465452
$765$	0	0
$766$	−2.95068e6	−0.181698
$767$	−1.82881e6	−0.112248
$768$	−9.47691e6	−0.579781
$769$	2.56323e6	0.156305	0.0781524	−	0.996941i	$-0.475098\pi$
$769$	2.56323e6	0.156305	0.0781524	+	0.996941i	$0.475098\pi$
$770$	0	0
$771$	−908098.	−0.0550170
$772$	9.17642e6	0.554153
$773$	5.47140e6	0.329344	0.164672	−	0.986348i	$-0.447343\pi$
$773$	5.47140e6	0.329344	0.164672	+	0.986348i	$0.447343\pi$
$774$	−7.94359e6	−0.476611
$775$	0	0
$776$	−1.34860e7	−0.803948
$777$	−2.05738e7	−1.22254
$778$	−2.67071e7	−1.58189
$779$	−9.87319e6	−0.582926
$780$	0	0
$781$	3.86200e6	0.226561
$782$	8.42336e7	4.92571
$783$	6.24429e6	0.363981
$784$	92177.4	0.00535592
$785$	0	0
$786$	−1.21740e7	−0.702871
$787$	1.29522e7	0.745429	0.372714	−	0.927946i	$-0.378427\pi$
$787$	1.29522e7	0.745429	0.372714	+	0.927946i	$0.378427\pi$
$788$	−5.25209e7	−3.01312
$789$	−3.12297e6	−0.178597
$790$	0	0
$791$	−4.83384e7	−2.74695
$792$	−1.77791e6	−0.100715
$793$	2.22885e6	0.125863
$794$	−1.90186e7	−1.07060
$795$	0	0
$796$	1.21987e7	0.682386
$797$	8.84349e6	0.493149	0.246574	−	0.969124i	$-0.420695\pi$
$797$	8.84349e6	0.493149	0.246574	+	0.969124i	$0.420695\pi$
$798$	2.04379e7	1.13614
$799$	−3.37708e7	−1.87143
$800$	0	0
$801$	1.05709e7	0.582145
$802$	2.58288e7	1.41797
$803$	−2.52661e6	−0.138277
$804$	−5.27658e6	−0.287881
$805$	0	0
$806$	−226990.	−0.0123075
$807$	1.01668e7	0.549540
$808$	−1.32212e6	−0.0712431
$809$	2.89181e7	1.55345	0.776727	−	0.629838i	$-0.216879\pi$
$809$	2.89181e7	1.55345	0.776727	+	0.629838i	$0.216879\pi$
$810$	0	0
$811$	3.10413e7	1.65725	0.828624	−	0.559805i	$-0.189124\pi$
$811$	3.10413e7	1.65725	0.828624	+	0.559805i	$0.189124\pi$
$812$	1.00594e8	5.35404
$813$	6.80974e6	0.361330
$814$	1.11725e7	0.591003
$815$	0	0
$816$	47629.0	0.00250407
$817$	−1.17231e7	−0.614450
$818$	2.11702e7	1.10622
$819$	−1.11075e6	−0.0578640
$820$	0	0
$821$	−2.95235e7	−1.52866	−0.764329	−	0.644827i	$-0.776929\pi$
$821$	−2.95235e7	−1.52866	−0.764329	+	0.644827i	$0.776929\pi$
$822$	1.90760e7	0.984707
$823$	2.12030e7	1.09118	0.545592	−	0.838051i	$-0.316305\pi$
$823$	2.12030e7	1.09118	0.545592	+	0.838051i	$0.316305\pi$
$824$	−1.00447e7	−0.515371
$825$	0	0
$826$	−6.27231e7	−3.19873
$827$	−1.92124e7	−0.976829	−0.488414	−	0.872612i	$-0.662424\pi$
$827$	−1.92124e7	−0.976829	−0.488414	+	0.872612i	$0.662424\pi$
$828$	−1.94586e7	−0.986359
$829$	−4.31098e6	−0.217866	−0.108933	−	0.994049i	$-0.534743\pi$
$829$	−4.31098e6	−0.217866	−0.108933	+	0.994049i	$0.534743\pi$
$830$	0	0
$831$	1.45513e7	0.730968
$832$	3.20689e6	0.160611
$833$	6.85957e7	3.42519
$834$	1.00277e7	0.499212
$835$	0	0
$836$	−6.86127e6	−0.339538
$837$	−298751.	−0.0147399
$838$	−5.03832e7	−2.47842
$839$	−1.03289e7	−0.506583	−0.253291	−	0.967390i	$-0.581513\pi$
$839$	−1.03289e7	−0.506583	−0.253291	+	0.967390i	$0.581513\pi$
$840$	0	0
$841$	5.28576e7	2.57702
$842$	3.77196e7	1.83352
$843$	1.15325e7	0.558928
$844$	−4.76185e7	−2.30102
$845$	0	0
$846$	1.26193e7	0.606190
$847$	3.31846e6	0.158938
$848$	75719.6	0.00361592
$849$	−1.00278e7	−0.477461
$850$	0	0
$851$	4.67608e7	2.21339
$852$	1.48840e7	0.702458
$853$	−3.79900e6	−0.178771	−0.0893855	−	0.995997i	$-0.528490\pi$
$853$	−3.79900e6	−0.178771	−0.0893855	+	0.995997i	$0.528490\pi$
$854$	7.64433e7	3.58670
$855$	0	0
$856$	4.12696e7	1.92507
$857$	−3.34577e7	−1.55612	−0.778062	−	0.628188i	$-0.783797\pi$
$857$	−3.34577e7	−1.55612	−0.778062	+	0.628188i	$0.783797\pi$
$858$	603189.	0.0279728
$859$	−7.44861e6	−0.344423	−0.172212	−	0.985060i	$-0.555091\pi$
$859$	−7.44861e6	−0.344423	−0.172212	+	0.985060i	$0.555091\pi$
$860$	0	0
$861$	1.84034e7	0.846037
$862$	−6.60695e6	−0.302854
$863$	−3.47712e7	−1.58925	−0.794626	−	0.607099i	$-0.792333\pi$
$863$	−3.47712e7	−1.58925	−0.794626	+	0.607099i	$0.792333\pi$
$864$	4.21392e6	0.192044
$865$	0	0
$866$	1.42245e7	0.644529
$867$	2.26653e7	1.02403
$868$	−4.81280e6	−0.216819
$869$	9.32809e6	0.419028
$870$	0	0
$871$	684584.	0.0305760
$872$	2.10513e7	0.937536
$873$	6.02183e6	0.267420
$874$	−4.64520e7	−2.05696
$875$	0	0
$876$	−9.73745e6	−0.428731
$877$	−3.96598e7	−1.74121	−0.870605	−	0.491982i	$-0.836273\pi$
$877$	−3.96598e7	−1.74121	−0.870605	+	0.491982i	$0.836273\pi$
$878$	6.04215e7	2.64518
$879$	9.32189e6	0.406941
$880$	0	0
$881$	−4.15251e7	−1.80248	−0.901240	−	0.433320i	$-0.857342\pi$
$881$	−4.15251e7	−1.80248	−0.901240	+	0.433320i	$0.857342\pi$
$882$	−2.56325e7	−1.10948
$883$	−1.44451e7	−0.623473	−0.311737	−	0.950169i	$-0.600911\pi$
$883$	−1.44451e7	−0.623473	−0.311737	+	0.950169i	$0.600911\pi$
$884$	−6.22109e6	−0.267754
$885$	0	0
$886$	1.29734e7	0.555224
$887$	1.93342e7	0.825122	0.412561	−	0.910930i	$-0.364634\pi$
$887$	1.93342e7	0.825122	0.412561	+	0.910930i	$0.364634\pi$
$888$	1.64660e7	0.700737
$889$	−1.72457e7	−0.731858
$890$	0	0
$891$	793881.	0.0335013
$892$	6.67249e7	2.80786
$893$	1.86234e7	0.781504
$894$	2.48116e7	1.03827
$895$	0	0
$896$	6.80622e7	2.83228
$897$	2.52455e6	0.104762
$898$	3.57832e7	1.48077
$899$	−3.51024e6	−0.144856
$900$	0	0
$901$	5.63483e7	2.31243
$902$	−9.99385e6	−0.408993
$903$	2.18515e7	0.891789
$904$	3.86870e7	1.57450
$905$	0	0
$906$	−2.13917e7	−0.865815
$907$	−1.72690e7	−0.697025	−0.348513	−	0.937304i	$-0.613313\pi$
$907$	−1.72690e7	−0.697025	−0.348513	+	0.937304i	$0.613313\pi$
$908$	−2.82837e7	−1.13847
$909$	590361.	0.0236978
$910$	0	0
$911$	4.42672e6	0.176720	0.0883601	−	0.996089i	$-0.471837\pi$
$911$	4.42672e6	0.176720	0.0883601	+	0.996089i	$0.471837\pi$
$912$	−26265.7	−0.00104569
$913$	1.19849e7	0.475837
$914$	−4.44785e7	−1.76110
$915$	0	0
$916$	−2.81205e7	−1.10735
$917$	3.34886e7	1.31514
$918$	−1.32445e7	−0.518717
$919$	1.62676e7	0.635382	0.317691	−	0.948194i	$-0.397093\pi$
$919$	1.62676e7	0.635382	0.317691	+	0.948194i	$0.397093\pi$
$920$	0	0
$921$	2.27085e7	0.882142
$922$	−8.28954e6	−0.321146
$923$	−1.93105e6	−0.0746087
$924$	1.27892e7	0.492793
$925$	0	0
$926$	3.24312e7	1.24290
$927$	4.48522e6	0.171429
$928$	4.95124e7	1.88731
$929$	2.99303e7	1.13782	0.568908	−	0.822401i	$-0.307366\pi$
$929$	2.99303e7	1.13782	0.568908	+	0.822401i	$0.307366\pi$
$930$	0	0
$931$	−3.78282e7	−1.43035
$932$	6.55458e6	0.247175
$933$	1.57512e7	0.592394
$934$	6.53948e7	2.45288
$935$	0	0
$936$	888976.	0.0331666
$937$	−1.50548e7	−0.560179	−0.280089	−	0.959974i	$-0.590364\pi$
$937$	−1.50548e7	−0.560179	−0.280089	+	0.959974i	$0.590364\pi$
$938$	2.34793e7	0.871322
$939$	7.56784e6	0.280097
$940$	0	0
$941$	−1.96722e7	−0.724233	−0.362117	−	0.932133i	$-0.617946\pi$
$941$	−1.96722e7	−0.724233	−0.362117	+	0.932133i	$0.617946\pi$
$942$	1.22689e6	0.0450485
$943$	−4.18277e7	−1.53174
$944$	80608.4	0.00294409
$945$	0	0
$946$	−1.18664e7	−0.431111
$947$	−1.92584e7	−0.697824	−0.348912	−	0.937155i	$-0.613449\pi$
$947$	−1.92584e7	−0.697824	−0.348912	+	0.937155i	$0.613449\pi$
$948$	3.59501e7	1.29921
$949$	1.26334e6	0.0455359
$950$	0	0
$951$	1.21115e7	0.434258
$952$	−8.15935e7	−2.91785
$953$	−1.49922e7	−0.534729	−0.267364	−	0.963596i	$-0.586153\pi$
$953$	−1.49922e7	−0.534729	−0.267364	+	0.963596i	$0.586153\pi$
$954$	−2.10559e7	−0.749037
$955$	0	0
$956$	7.09557e7	2.51098
$957$	9.32789e6	0.329233
$958$	−4.11321e7	−1.44799
$959$	−5.24749e7	−1.84249
$960$	0	0
$961$	−2.84612e7	−0.994134
$962$	−5.58640e6	−0.194623
$963$	−1.84279e7	−0.640340
$964$	6.68247e7	2.31603
$965$	0	0
$966$	8.65853e7	2.98539
$967$	2.53960e7	0.873370	0.436685	−	0.899614i	$-0.356152\pi$
$967$	2.53960e7	0.873370	0.436685	+	0.899614i	$0.356152\pi$
$968$	−2.65589e6	−0.0911006
$969$	−1.95462e7	−0.668733
$970$	0	0
$971$	3.80328e7	1.29453	0.647263	−	0.762267i	$-0.275914\pi$
$971$	3.80328e7	1.29453	0.647263	+	0.762267i	$0.275914\pi$
$972$	3.05958e6	0.103872
$973$	−2.75845e7	−0.934077
$974$	2.59977e7	0.878086
$975$	0	0
$976$	−98240.8	−0.00330117
$977$	−7.60558e6	−0.254915	−0.127458	−	0.991844i	$-0.540682\pi$
$977$	−7.60558e6	−0.254915	−0.127458	+	0.991844i	$0.540682\pi$
$978$	1.65421e7	0.553024
$979$	1.57911e7	0.526570
$980$	0	0
$981$	−9.39995e6	−0.311855
$982$	5.53562e6	0.183184
$983$	−3.96257e7	−1.30796	−0.653979	−	0.756513i	$-0.726901\pi$
$983$	−3.96257e7	−1.30796	−0.653979	+	0.756513i	$0.726901\pi$
$984$	−1.47289e7	−0.484933
$985$	0	0
$986$	−1.55620e8	−5.09768
$987$	−3.47136e7	−1.13424
$988$	3.43072e6	0.111813
$989$	−4.96648e7	−1.61457
$990$	0	0
$991$	5.43979e7	1.75954	0.879768	−	0.475403i	$-0.157698\pi$
$991$	5.43979e7	1.75954	0.879768	+	0.475403i	$0.157698\pi$
$992$	−2.36886e6	−0.0764294
$993$	−1.54605e7	−0.497565
$994$	−6.62297e7	−2.12612
$995$	0	0
$996$	4.61894e7	1.47535
$997$	−4.80608e7	−1.53127	−0.765637	−	0.643273i	$-0.777576\pi$
$997$	−4.80608e7	−1.53127	−0.765637	+	0.643273i	$0.777576\pi$
$998$	−8.97079e6	−0.285105
$999$	−7.35248e6	−0.233088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.t.1.2	✓	10
5.4	even	2		825.6.a.u.1.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.2	✓	10	1.1	even	1	trivial
825.6.a.u.1.9	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)

\(f(q)\)	\(=\)	\(q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +121.000 q^{11} +466.329 q^{12} -60.5015 q^{13} -2075.04 q^{14} +2.66672 q^{16} +1984.50 q^{17} -741.556 q^{18} -1094.38 q^{19} +2039.90 q^{21} -1107.76 q^{22} -4636.35 q^{23} -1632.61 q^{24} +553.893 q^{26} +729.000 q^{27} +11744.0 q^{28} +8565.55 q^{29} -409.809 q^{31} +5780.41 q^{32} +1089.00 q^{33} -18168.1 q^{34} +4196.96 q^{36} -10085.7 q^{37} +10019.1 q^{38} -544.514 q^{39} +9021.70 q^{41} -18675.3 q^{42} +10712.1 q^{43} +6269.53 q^{44} +42445.8 q^{46} -17017.3 q^{47} +24.0005 q^{48} +34565.8 q^{49} +17860.5 q^{51} -3134.85 q^{52} +28394.2 q^{53} -6674.01 q^{54} -41115.5 q^{56} -9849.44 q^{57} -78417.8 q^{58} +30227.5 q^{59} -36839.5 q^{61} +3751.81 q^{62} +18359.1 q^{63} -53005.0 q^{64} -9969.81 q^{66} -11315.1 q^{67} +102825. q^{68} -41727.1 q^{69} +31917.4 q^{71} -14693.4 q^{72} -20881.1 q^{73} +92334.8 q^{74} -56704.7 q^{76} +27425.3 q^{77} +4985.03 q^{78} +77091.6 q^{79} +6561.00 q^{81} -82593.8 q^{82} +99048.9 q^{83} +105696. q^{84} -98069.0 q^{86} +77090.0 q^{87} -21949.5 q^{88} +130505. q^{89} -13713.0 q^{91} -240229. q^{92} -3688.28 q^{93} +155794. q^{94} +52023.6 q^{96} +74343.6 q^{97} -316450. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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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