LMFDB - Embedded newform 825.6.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$10$
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			1.3
Root			$6.73090$ 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)

$f(q)$	$=$	$q-6.73090 q^{2} +9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} +2.41142 q^{7} +125.834 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-6.73090 q^{2} +9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} +2.41142 q^{7} +125.834 q^{8} +81.0000 q^{9} +121.000 q^{11} +119.745 q^{12} +334.983 q^{13} -16.2310 q^{14} -1272.74 q^{16} -1211.39 q^{17} -545.203 q^{18} -746.587 q^{19} +21.7028 q^{21} -814.439 q^{22} +3387.32 q^{23} +1132.51 q^{24} -2254.73 q^{26} +729.000 q^{27} +32.0839 q^{28} -2012.09 q^{29} +8689.78 q^{31} +4539.97 q^{32} +1089.00 q^{33} +8153.73 q^{34} +1077.70 q^{36} +1392.25 q^{37} +5025.20 q^{38} +3014.84 q^{39} +2339.11 q^{41} -146.079 q^{42} +10534.9 q^{43} +1609.90 q^{44} -22799.7 q^{46} -22537.9 q^{47} -11454.6 q^{48} -16801.2 q^{49} -10902.5 q^{51} +4456.94 q^{52} +5835.91 q^{53} -4906.83 q^{54} +303.439 q^{56} -6719.28 q^{57} +13543.2 q^{58} +27573.2 q^{59} +13969.9 q^{61} -58490.0 q^{62} +195.325 q^{63} +10169.5 q^{64} -7329.95 q^{66} +18128.3 q^{67} -16117.5 q^{68} +30485.8 q^{69} +986.398 q^{71} +10192.6 q^{72} -11598.8 q^{73} -9371.12 q^{74} -9933.34 q^{76} +291.782 q^{77} -20292.6 q^{78} -48553.4 q^{79} +6561.00 q^{81} -15744.3 q^{82} -34168.5 q^{83} +288.755 q^{84} -70909.2 q^{86} -18108.8 q^{87} +15225.9 q^{88} +124210. q^{89} +807.783 q^{91} +45068.2 q^{92} +78208.0 q^{93} +151700. q^{94} +40859.7 q^{96} -138028. q^{97} +113087. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−6.73090	−1.18987	−0.594933	−	0.803775i	$-0.702821\pi$
$2$	−6.73090	−1.18987	−0.594933	+	0.803775i	$0.702821\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	13.3050	0.415781
$5$	0	0
$5$	0	0
$6$	−60.5781	−0.686969
$7$	2.41142	0.0186006	0.00930031	−	0.999957i	$-0.497040\pi$
$7$	2.41142	0.0186006	0.00930031	+	0.999957i	$0.497040\pi$
$8$	125.834	0.695142
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	119.745	0.240051
$13$	334.983	0.549748	0.274874	−	0.961480i	$-0.411364\pi$
$13$	334.983	0.549748	0.274874	+	0.961480i	$0.411364\pi$
$14$	−16.2310	−0.0221322
$15$	0	0
$16$	−1272.74	−1.24291
$17$	−1211.39	−1.01663	−0.508313	−	0.861173i	$-0.669731\pi$
$17$	−1211.39	−1.01663	−0.508313	+	0.861173i	$0.669731\pi$
$18$	−545.203	−0.396622
$19$	−746.587	−0.474456	−0.237228	−	0.971454i	$-0.576239\pi$
$19$	−746.587	−0.474456	−0.237228	+	0.971454i	$0.576239\pi$
$20$	0	0
$21$	21.7028	0.0107391
$22$	−814.439	−0.358758
$23$	3387.32	1.33517	0.667584	−	0.744534i	$-0.267328\pi$
$23$	3387.32	1.33517	0.667584	+	0.744534i	$0.267328\pi$
$24$	1132.51	0.401341
$25$	0	0
$26$	−2254.73	−0.654127
$27$	729.000	0.192450
$28$	32.0839	0.00773379
$29$	−2012.09	−0.444275	−0.222137	−	0.975015i	$-0.571303\pi$
$29$	−2012.09	−0.444275	−0.222137	+	0.975015i	$0.571303\pi$
$30$	0	0
$31$	8689.78	1.62407	0.812035	−	0.583609i	$-0.198360\pi$
$31$	8689.78	1.62407	0.812035	+	0.583609i	$0.198360\pi$
$32$	4539.97	0.783751
$33$	1089.00	0.174078
$34$	8153.73	1.20965
$35$	0	0
$36$	1077.70	0.138594
$37$	1392.25	0.167192	0.0835958	−	0.996500i	$-0.473360\pi$
$37$	1392.25	0.167192	0.0835958	+	0.996500i	$0.473360\pi$
$38$	5025.20	0.564540
$39$	3014.84	0.317397
$40$	0	0
$41$	2339.11	0.217316	0.108658	−	0.994079i	$-0.465345\pi$
$41$	2339.11	0.217316	0.108658	+	0.994079i	$0.465345\pi$
$42$	−146.079	−0.0127781
$43$	10534.9	0.868877	0.434439	−	0.900701i	$-0.356947\pi$
$43$	10534.9	0.868877	0.434439	+	0.900701i	$0.356947\pi$
$44$	1609.90	0.125363
$45$	0	0
$46$	−22799.7	−1.58867
$47$	−22537.9	−1.48823	−0.744113	−	0.668054i	$-0.767127\pi$
$47$	−22537.9	−1.48823	−0.744113	+	0.668054i	$0.767127\pi$
$48$	−11454.6	−0.717593
$49$	−16801.2	−0.999654
$50$	0	0
$51$	−10902.5	−0.586949
$52$	4456.94	0.228575
$53$	5835.91	0.285377	0.142688	−	0.989768i	$-0.454425\pi$
$53$	5835.91	0.285377	0.142688	+	0.989768i	$0.454425\pi$
$54$	−4906.83	−0.228990
$55$	0	0
$56$	303.439	0.0129301
$57$	−6719.28	−0.273928
$58$	13543.2	0.528628
$59$	27573.2	1.03123	0.515617	−	0.856819i	$-0.327563\pi$
$59$	27573.2	1.03123	0.515617	+	0.856819i	$0.327563\pi$
$60$	0	0
$61$	13969.9	0.480693	0.240347	−	0.970687i	$-0.422739\pi$
$61$	13969.9	0.480693	0.240347	+	0.970687i	$0.422739\pi$
$62$	−58490.0	−1.93242
$63$	195.325	0.00620021
$64$	10169.5	0.310349
$65$	0	0
$66$	−7329.95	−0.207129
$67$	18128.3	0.493366	0.246683	−	0.969096i	$-0.420659\pi$
$67$	18128.3	0.493366	0.246683	+	0.969096i	$0.420659\pi$
$68$	−16117.5	−0.422694
$69$	30485.8	0.770860
$70$	0	0
$71$	986.398	0.0232224	0.0116112	−	0.999933i	$-0.496304\pi$
$71$	986.398	0.0232224	0.0116112	+	0.999933i	$0.496304\pi$
$72$	10192.6	0.231714
$73$	−11598.8	−0.254745	−0.127372	−	0.991855i	$-0.540654\pi$
$73$	−11598.8	−0.254745	−0.127372	+	0.991855i	$0.540654\pi$
$74$	−9371.12	−0.198936
$75$	0	0
$76$	−9933.34	−0.197270
$77$	291.782	0.00560830
$78$	−20292.6	−0.377660
$79$	−48553.4	−0.875289	−0.437645	−	0.899148i	$-0.644187\pi$
$79$	−48553.4	−0.875289	−0.437645	+	0.899148i	$0.644187\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−15744.3	−0.258577
$83$	−34168.5	−0.544416	−0.272208	−	0.962239i	$-0.587754\pi$
$83$	−34168.5	−0.544416	−0.272208	+	0.962239i	$0.587754\pi$
$84$	288.755	0.00446510
$85$	0	0
$86$	−70909.2	−1.03385
$87$	−18108.8	−0.256502
$88$	15225.9	0.209593
$89$	124210.	1.66219	0.831097	−	0.556128i	$-0.187713\pi$
$89$	124210.	1.66219	0.831097	+	0.556128i	$0.187713\pi$
$90$	0	0
$91$	807.783	0.0102257
$92$	45068.2	0.555138
$93$	78208.0	0.937657
$94$	151700.	1.77079
$95$	0	0
$96$	40859.7	0.452499
$97$	−138028.	−1.48949	−0.744744	−	0.667351i	$-0.767428\pi$
$97$	−138028.	−1.48949	−0.744744	+	0.667351i	$0.767428\pi$
$98$	113087.	1.18945
$99$	9801.00	0.100504
$100$	0	0
$101$	−5951.66	−0.0580543	−0.0290272	−	0.999579i	$-0.509241\pi$
$101$	−5951.66	−0.0580543	−0.0290272	+	0.999579i	$0.509241\pi$
$102$	73383.6	0.698390
$103$	−191198.	−1.77578	−0.887892	−	0.460052i	$-0.847831\pi$
$103$	−191198.	−1.77578	−0.887892	+	0.460052i	$0.847831\pi$
$104$	42152.3	0.382153
$105$	0	0
$106$	−39280.9	−0.339560
$107$	88404.1	0.746471	0.373236	−	0.927737i	$-0.378248\pi$
$107$	88404.1	0.746471	0.373236	+	0.927737i	$0.378248\pi$
$108$	9699.34	0.0800171
$109$	223100.	1.79859	0.899296	−	0.437341i	$-0.144080\pi$
$109$	223100.	1.79859	0.899296	+	0.437341i	$0.144080\pi$
$110$	0	0
$111$	12530.3	0.0965281
$112$	−3069.10	−0.0231188
$113$	12365.3	0.0910978	0.0455489	−	0.998962i	$-0.485496\pi$
$113$	12365.3	0.0910978	0.0455489	+	0.998962i	$0.485496\pi$
$114$	45226.8	0.325937
$115$	0	0
$116$	−26770.8	−0.184721
$117$	27133.6	0.183249
$118$	−185592.	−1.22703
$119$	−2921.16	−0.0189099
$120$	0	0
$121$	14641.0	0.0909091
$122$	−94029.9	−0.571961
$123$	21052.0	0.125467
$124$	115618.	0.675257
$125$	0	0
$126$	−1314.71	−0.00737741
$127$	−207000.	−1.13883	−0.569417	−	0.822049i	$-0.692831\pi$
$127$	−207000.	−1.13883	−0.569417	+	0.822049i	$0.692831\pi$
$128$	−213729.	−1.15302
$129$	94813.9	0.501647
$130$	0	0
$131$	−104541.	−0.532239	−0.266119	−	0.963940i	$-0.585742\pi$
$131$	−104541.	−0.532239	−0.266119	+	0.963940i	$0.585742\pi$
$132$	14489.1	0.0723782
$133$	−1800.33	−0.00882518
$134$	−122020.	−0.587040
$135$	0	0
$136$	−152434.	−0.706699
$137$	−159251.	−0.724907	−0.362453	−	0.932002i	$-0.618061\pi$
$137$	−159251.	−0.724907	−0.362453	+	0.932002i	$0.618061\pi$
$138$	−205197.	−0.917220
$139$	184879.	0.811614	0.405807	−	0.913959i	$-0.366990\pi$
$139$	184879.	0.811614	0.405807	+	0.913959i	$0.366990\pi$
$140$	0	0
$141$	−202841.	−0.859227
$142$	−6639.35	−0.0276315
$143$	40532.9	0.165755
$144$	−103092.	−0.414302
$145$	0	0
$146$	78070.3	0.303112
$147$	−151211.	−0.577151
$148$	18523.9	0.0695151
$149$	270575.	0.998442	0.499221	−	0.866475i	$-0.333620\pi$
$149$	270575.	0.998442	0.499221	+	0.866475i	$0.333620\pi$
$150$	0	0
$151$	−287926.	−1.02764	−0.513818	−	0.857899i	$-0.671769\pi$
$151$	−287926.	−1.02764	−0.513818	+	0.857899i	$0.671769\pi$
$152$	−93946.1	−0.329815
$153$	−98122.4	−0.338875
$154$	−1963.95	−0.00667312
$155$	0	0
$156$	40112.5	0.131968
$157$	400095.	1.29543	0.647715	−	0.761883i	$-0.275725\pi$
$157$	400095.	1.29543	0.647715	+	0.761883i	$0.275725\pi$
$158$	326808.	1.04148
$159$	52523.1	0.164762
$160$	0	0
$161$	8168.23	0.0248350
$162$	−44161.4	−0.132207
$163$	230209.	0.678661	0.339331	−	0.940667i	$-0.389799\pi$
$163$	230209.	0.678661	0.339331	+	0.940667i	$0.389799\pi$
$164$	31121.9	0.0903559
$165$	0	0
$166$	229985.	0.647782
$167$	−505911.	−1.40373	−0.701864	−	0.712311i	$-0.747649\pi$
$167$	−505911.	−1.40373	−0.701864	+	0.712311i	$0.747649\pi$
$168$	2730.95	0.00746518
$169$	−259080.	−0.697777
$170$	0	0
$171$	−60473.5	−0.158152
$172$	140167.	0.361263
$173$	−182355.	−0.463237	−0.231619	−	0.972807i	$-0.574402\pi$
$173$	−182355.	−0.463237	−0.231619	+	0.972807i	$0.574402\pi$
$174$	121888.	0.305203
$175$	0	0
$176$	−154001.	−0.374751
$177$	248159.	0.595383
$178$	−836045.	−1.97779
$179$	293039.	0.683586	0.341793	−	0.939775i	$-0.388966\pi$
$179$	293039.	0.683586	0.341793	+	0.939775i	$0.388966\pi$
$180$	0	0
$181$	234051.	0.531024	0.265512	−	0.964108i	$-0.414459\pi$
$181$	234051.	0.531024	0.265512	+	0.964108i	$0.414459\pi$
$182$	−5437.11	−0.0121672
$183$	125729.	0.277528
$184$	426240.	0.928132
$185$	0	0
$186$	−526410.	−1.11569
$187$	−146578.	−0.306524
$188$	−299867.	−0.618776
$189$	1757.92	0.00357969
$190$	0	0
$191$	560936.	1.11258	0.556288	−	0.830990i	$-0.312225\pi$
$191$	560936.	1.11258	0.556288	+	0.830990i	$0.312225\pi$
$192$	91525.5	0.179180
$193$	−490651.	−0.948155	−0.474078	−	0.880483i	$-0.657218\pi$
$193$	−490651.	−0.948155	−0.474078	+	0.880483i	$0.657218\pi$
$194$	929050.	1.77229
$195$	0	0
$196$	−223540.	−0.415637
$197$	1.01719e6	1.86740	0.933698	−	0.358061i	$-0.116562\pi$
$197$	1.01719e6	1.86740	0.933698	+	0.358061i	$0.116562\pi$
$198$	−65969.5	−0.119586
$199$	−104543.	−0.187138	−0.0935689	−	0.995613i	$-0.529828\pi$
$199$	−104543.	−0.187138	−0.0935689	+	0.995613i	$0.529828\pi$
$200$	0	0
$201$	163154.	0.284845
$202$	40060.0	0.0690769
$203$	−4851.98	−0.00826379
$204$	−145058.	−0.244042
$205$	0	0
$206$	1.28693e6	2.11295
$207$	274373.	0.445056
$208$	−426345.	−0.683286
$209$	−90337.0	−0.143054
$210$	0	0
$211$	29310.5	0.0453228	0.0226614	−	0.999743i	$-0.492786\pi$
$211$	29310.5	0.0453228	0.0226614	+	0.999743i	$0.492786\pi$
$212$	77646.7	0.118654
$213$	8877.59	0.0134074
$214$	−595039.	−0.888201
$215$	0	0
$216$	91733.1	0.133780
$217$	20954.7	0.0302087
$218$	−1.50166e6	−2.14008
$219$	−104389.	−0.147077
$220$	0	0
$221$	−405794.	−0.558888
$222$	−84340.1	−0.114855
$223$	−794924.	−1.07044	−0.535221	−	0.844712i	$-0.679772\pi$
$223$	−794924.	−1.07044	−0.535221	+	0.844712i	$0.679772\pi$
$224$	10947.8	0.0145782
$225$	0	0
$226$	−83229.4	−0.108394
$227$	−379790.	−0.489191	−0.244596	−	0.969625i	$-0.578655\pi$
$227$	−379790.	−0.489191	−0.244596	+	0.969625i	$0.578655\pi$
$228$	−89400.0	−0.113894
$229$	162001.	0.204140	0.102070	−	0.994777i	$-0.467453\pi$
$229$	162001.	0.204140	0.102070	+	0.994777i	$0.467453\pi$
$230$	0	0
$231$	2626.03	0.00323795
$232$	−253189.	−0.308834
$233$	463134.	0.558878	0.279439	−	0.960163i	$-0.409852\pi$
$233$	463134.	0.558878	0.279439	+	0.960163i	$0.409852\pi$
$234$	−182634.	−0.218042
$235$	0	0
$236$	366861.	0.428767
$237$	−436980.	−0.505348
$238$	19662.0	0.0225002
$239$	932706.	1.05621	0.528105	−	0.849179i	$-0.322903\pi$
$239$	932706.	1.05621	0.528105	+	0.849179i	$0.322903\pi$
$240$	0	0
$241$	414220.	0.459397	0.229699	−	0.973262i	$-0.426226\pi$
$241$	414220.	0.459397	0.229699	+	0.973262i	$0.426226\pi$
$242$	−98547.1	−0.108170
$243$	59049.0	0.0641500
$244$	185869.	0.199863
$245$	0	0
$246$	−141699.	−0.149289
$247$	−250094.	−0.260832
$248$	1.09347e6	1.12896
$249$	−307516.	−0.314318
$250$	0	0
$251$	1.90536e6	1.90894	0.954472	−	0.298300i	$-0.0964195\pi$
$251$	1.90536e6	1.90894	0.954472	+	0.298300i	$0.0964195\pi$
$252$	2598.80	0.00257793
$253$	409865.	0.402568
$254$	1.39329e6	1.35506
$255$	0	0
$256$	1.11316e6	1.06160
$257$	631671.	0.596565	0.298283	−	0.954478i	$-0.403586\pi$
$257$	631671.	0.596565	0.298283	+	0.954478i	$0.403586\pi$
$258$	−638183.	−0.596892
$259$	3357.31	0.00310987
$260$	0	0
$261$	−162979.	−0.148092
$262$	703652.	0.633293
$263$	961547.	0.857198	0.428599	−	0.903495i	$-0.359007\pi$
$263$	961547.	0.857198	0.428599	+	0.903495i	$0.359007\pi$
$264$	137033.	0.121009
$265$	0	0
$266$	12117.9	0.0105008
$267$	1.11789e6	0.959668
$268$	241197.	0.205132
$269$	720581.	0.607158	0.303579	−	0.952806i	$-0.401818\pi$
$269$	720581.	0.607158	0.303579	+	0.952806i	$0.401818\pi$
$270$	0	0
$271$	−632223.	−0.522934	−0.261467	−	0.965212i	$-0.584206\pi$
$271$	−632223.	−0.522934	−0.261467	+	0.965212i	$0.584206\pi$
$272$	1.54178e6	1.26357
$273$	7270.05	0.00590379
$274$	1.07191e6	0.862542
$275$	0	0
$276$	405614.	0.320509
$277$	2.37840e6	1.86246	0.931228	−	0.364437i	$-0.118738\pi$
$277$	2.37840e6	1.86246	0.931228	+	0.364437i	$0.118738\pi$
$278$	−1.24440e6	−0.965712
$279$	703872.	0.541356
$280$	0	0
$281$	2.11538e6	1.59817	0.799085	−	0.601218i	$-0.205317\pi$
$281$	2.11538e6	1.59817	0.799085	+	0.601218i	$0.205317\pi$
$282$	1.36530e6	1.02237
$283$	−40867.3	−0.0303326	−0.0151663	−	0.999885i	$-0.504828\pi$
$283$	−40867.3	−0.0303326	−0.0151663	+	0.999885i	$0.504828\pi$
$284$	13124.0	0.00965542
$285$	0	0
$286$	−272823.	−0.197227
$287$	5640.58	0.00404221
$288$	367738.	0.261250
$289$	47603.2	0.0335267
$290$	0	0
$291$	−1.24225e6	−0.859956
$292$	−154322.	−0.105918
$293$	837587.	0.569982	0.284991	−	0.958530i	$-0.408009\pi$
$293$	837587.	0.569982	0.284991	+	0.958530i	$0.408009\pi$
$294$	1.01778e6	0.686732
$295$	0	0
$296$	175193.	0.116222
$297$	88209.0	0.0580259
$298$	−1.82122e6	−1.18801
$299$	1.13469e6	0.734007
$300$	0	0
$301$	25404.0	0.0161617
$302$	1.93800e6	1.22275
$303$	−53564.9	−0.0335177
$304$	950209.	0.589705
$305$	0	0
$306$	660452.	0.403216
$307$	632213.	0.382840	0.191420	−	0.981508i	$-0.438691\pi$
$307$	632213.	0.382840	0.191420	+	0.981508i	$0.438691\pi$
$308$	3882.15	0.00233182
$309$	−1.72078e6	−1.02525
$310$	0	0
$311$	1.99159e6	1.16761	0.583806	−	0.811893i	$-0.301563\pi$
$311$	1.99159e6	1.16761	0.583806	+	0.811893i	$0.301563\pi$
$312$	379370.	0.220636
$313$	305651.	0.176346	0.0881728	−	0.996105i	$-0.471897\pi$
$313$	305651.	0.176346	0.0881728	+	0.996105i	$0.471897\pi$
$314$	−2.69300e6	−1.54139
$315$	0	0
$316$	−646002.	−0.363929
$317$	−770054.	−0.430401	−0.215200	−	0.976570i	$-0.569040\pi$
$317$	−770054.	−0.430401	−0.215200	+	0.976570i	$0.569040\pi$
$318$	−353528.	−0.196045
$319$	−243463.	−0.133954
$320$	0	0
$321$	795637.	0.430975
$322$	−54979.5	−0.0295503
$323$	904406.	0.482344
$324$	87294.1	0.0461979
$325$	0	0
$326$	−1.54951e6	−0.807516
$327$	2.00790e6	1.03842
$328$	294340.	0.151065
$329$	−54348.3	−0.0276819
$330$	0	0
$331$	965864.	0.484558	0.242279	−	0.970207i	$-0.422105\pi$
$331$	965864.	0.484558	0.242279	+	0.970207i	$0.422105\pi$
$332$	−454612.	−0.226358
$333$	112773.	0.0557305
$334$	3.40524e6	1.67025
$335$	0	0
$336$	−27621.9	−0.0133477
$337$	1.11434e6	0.534494	0.267247	−	0.963628i	$-0.413886\pi$
$337$	1.11434e6	0.534494	0.267247	+	0.963628i	$0.413886\pi$
$338$	1.74384e6	0.830261
$339$	111288.	0.0525953
$340$	0	0
$341$	1.05146e6	0.489675
$342$	407041.	0.188180
$343$	−81043.4	−0.0371948
$344$	1.32565e6	0.603993
$345$	0	0
$346$	1.22742e6	0.551190
$347$	2.70207e6	1.20468	0.602342	−	0.798238i	$-0.294234\pi$
$347$	2.70207e6	1.20468	0.602342	+	0.798238i	$0.294234\pi$
$348$	−240937.	−0.106649
$349$	−760601.	−0.334267	−0.167133	−	0.985934i	$-0.553451\pi$
$349$	−760601.	−0.334267	−0.167133	+	0.985934i	$0.553451\pi$
$350$	0	0
$351$	244202.	0.105799
$352$	549336.	0.236310
$353$	2.61113e6	1.11530	0.557650	−	0.830076i	$-0.311703\pi$
$353$	2.61113e6	1.11530	0.557650	+	0.830076i	$0.311703\pi$
$354$	−1.67033e6	−0.708426
$355$	0	0
$356$	1.65261e6	0.691109
$357$	−26290.5	−0.0109176
$358$	−1.97242e6	−0.813376
$359$	−1.34676e6	−0.551512	−0.275756	−	0.961228i	$-0.588928\pi$
$359$	−1.34676e6	−0.551512	−0.275756	+	0.961228i	$0.588928\pi$
$360$	0	0
$361$	−1.91871e6	−0.774891
$362$	−1.57537e6	−0.631847
$363$	131769.	0.0524864
$364$	10747.6	0.00425164
$365$	0	0
$366$	−846269.	−0.330222
$367$	1.80461e6	0.699388	0.349694	−	0.936864i	$-0.386286\pi$
$367$	1.80461e6	0.699388	0.349694	+	0.936864i	$0.386286\pi$
$368$	−4.31116e6	−1.65949
$369$	189468.	0.0724386
$370$	0	0
$371$	14072.8	0.00530818
$372$	1.04056e6	0.389860
$373$	−715758.	−0.266376	−0.133188	−	0.991091i	$-0.542521\pi$
$373$	−715758.	−0.266376	−0.133188	+	0.991091i	$0.542521\pi$
$374$	986601.	0.364723
$375$	0	0
$376$	−2.83604e6	−1.03453
$377$	−674014.	−0.244239
$378$	−11832.4	−0.00425935
$379$	4.22778e6	1.51187	0.755934	−	0.654648i	$-0.227183\pi$
$379$	4.22778e6	1.51187	0.755934	+	0.654648i	$0.227183\pi$
$380$	0	0
$381$	−1.86300e6	−0.657506
$382$	−3.77560e6	−1.32382
$383$	4.99412e6	1.73965	0.869825	−	0.493361i	$-0.164232\pi$
$383$	4.99412e6	1.73965	0.869825	+	0.493361i	$0.164232\pi$
$384$	−1.92356e6	−0.665699
$385$	0	0
$386$	3.30252e6	1.12818
$387$	853325.	0.289626
$388$	−1.83646e6	−0.619301
$389$	1.96218e6	0.657455	0.328727	−	0.944425i	$-0.393380\pi$
$389$	1.96218e6	0.657455	0.328727	+	0.944425i	$0.393380\pi$
$390$	0	0
$391$	−4.10335e6	−1.35737
$392$	−2.11416e6	−0.694902
$393$	−940865.	−0.307288
$394$	−6.84660e6	−2.22195
$395$	0	0
$396$	130402.	0.0417876
$397$	4.49019e6	1.42984	0.714921	−	0.699205i	$-0.246463\pi$
$397$	4.49019e6	1.42984	0.714921	+	0.699205i	$0.246463\pi$
$398$	703667.	0.222669
$399$	−16203.0	−0.00509522
$400$	0	0
$401$	4.74027e6	1.47212	0.736059	−	0.676918i	$-0.236685\pi$
$401$	4.74027e6	1.47212	0.736059	+	0.676918i	$0.236685\pi$
$402$	−1.09818e6	−0.338928
$403$	2.91093e6	0.892830
$404$	−79186.8	−0.0241379
$405$	0	0
$406$	32658.2	0.00983280
$407$	168463.	0.0504101
$408$	−1.37191e6	−0.408013
$409$	−1.77250e6	−0.523935	−0.261967	−	0.965077i	$-0.584371\pi$
$409$	−1.77250e6	−0.523935	−0.261967	+	0.965077i	$0.584371\pi$
$410$	0	0
$411$	−1.43326e6	−0.418525
$412$	−2.54389e6	−0.738338
$413$	66490.5	0.0191816
$414$	−1.84677e6	−0.529557
$415$	0	0
$416$	1.52081e6	0.430866
$417$	1.66391e6	0.468586
$418$	608049.	0.170215
$419$	−2.99507e6	−0.833435	−0.416718	−	0.909036i	$-0.636820\pi$
$419$	−2.99507e6	−0.833435	−0.416718	+	0.909036i	$0.636820\pi$
$420$	0	0
$421$	−5.49888e6	−1.51206	−0.756030	−	0.654537i	$-0.772864\pi$
$421$	−5.49888e6	−1.51206	−0.756030	+	0.654537i	$0.772864\pi$
$422$	−197286.	−0.0539281
$423$	−1.82557e6	−0.496075
$424$	734356.	0.198377
$425$	0	0
$426$	−59754.1	−0.0159531
$427$	33687.2	0.00894119
$428$	1.17622e6	0.310369
$429$	364796.	0.0956989
$430$	0	0
$431$	−597155.	−0.154844	−0.0774219	−	0.996998i	$-0.524669\pi$
$431$	−597155.	−0.154844	−0.0774219	+	0.996998i	$0.524669\pi$
$432$	−927825.	−0.239198
$433$	−1.07785e6	−0.276274	−0.138137	−	0.990413i	$-0.544111\pi$
$433$	−1.07785e6	−0.276274	−0.138137	+	0.990413i	$0.544111\pi$
$434$	−141044.	−0.0359443
$435$	0	0
$436$	2.96834e6	0.747820
$437$	−2.52893e6	−0.633479
$438$	702633.	0.175002
$439$	4.45787e6	1.10399	0.551996	−	0.833847i	$-0.313866\pi$
$439$	4.45787e6	1.10399	0.551996	+	0.833847i	$0.313866\pi$
$440$	0	0
$441$	−1.36090e6	−0.333218
$442$	2.73136e6	0.665002
$443$	2.35378e6	0.569845	0.284923	−	0.958551i	$-0.408032\pi$
$443$	2.35378e6	0.569845	0.284923	+	0.958551i	$0.408032\pi$
$444$	166715.	0.0401346
$445$	0	0
$446$	5.35055e6	1.27368
$447$	2.43518e6	0.576451
$448$	24522.9	0.00577268
$449$	1.43645e6	0.336260	0.168130	−	0.985765i	$-0.446227\pi$
$449$	1.43645e6	0.336260	0.168130	+	0.985765i	$0.446227\pi$
$450$	0	0
$451$	283033.	0.0655232
$452$	164520.	0.0378767
$453$	−2.59134e6	−0.593305
$454$	2.55633e6	0.582072
$455$	0	0
$456$	−845515.	−0.190419
$457$	−4.60085e6	−1.03050	−0.515250	−	0.857040i	$-0.672301\pi$
$457$	−4.60085e6	−1.03050	−0.515250	+	0.857040i	$0.672301\pi$
$458$	−1.09041e6	−0.242899
$459$	−883102.	−0.195650
$460$	0	0
$461$	−4.67111e6	−1.02369	−0.511844	−	0.859078i	$-0.671038\pi$
$461$	−4.67111e6	−1.02369	−0.511844	+	0.859078i	$0.671038\pi$
$462$	−17675.6	−0.00385273
$463$	−326607.	−0.0708065	−0.0354032	−	0.999373i	$-0.511272\pi$
$463$	−326607.	−0.0708065	−0.0354032	+	0.999373i	$0.511272\pi$
$464$	2.56086e6	0.552192
$465$	0	0
$466$	−3.11731e6	−0.664990
$467$	−1.63074e6	−0.346014	−0.173007	−	0.984921i	$-0.555348\pi$
$467$	−1.63074e6	−0.346014	−0.173007	+	0.984921i	$0.555348\pi$
$468$	361012.	0.0761917
$469$	43714.8	0.00917692
$470$	0	0
$471$	3.60085e6	0.747916
$472$	3.46965e6	0.716854
$473$	1.27472e6	0.261976
$474$	2.94127e6	0.601297
$475$	0	0
$476$	−38866.0	−0.00786236
$477$	472708.	0.0951256
$478$	−6.27795e6	−1.25675
$479$	−7.31820e6	−1.45736	−0.728678	−	0.684856i	$-0.759865\pi$
$479$	−7.31820e6	−1.45736	−0.728678	+	0.684856i	$0.759865\pi$
$480$	0	0
$481$	466381.	0.0919133
$482$	−2.78807e6	−0.546621
$483$	73514.1	0.0143385
$484$	194798.	0.0377983
$485$	0	0
$486$	−397453.	−0.0763299
$487$	−2.80682e6	−0.536281	−0.268141	−	0.963380i	$-0.586409\pi$
$487$	−2.80682e6	−0.536281	−0.268141	+	0.963380i	$0.586409\pi$
$488$	1.75789e6	0.334150
$489$	2.07188e6	0.391825
$490$	0	0
$491$	−2.39394e6	−0.448136	−0.224068	−	0.974574i	$-0.571934\pi$
$491$	−2.39394e6	−0.448136	−0.224068	+	0.974574i	$0.571934\pi$
$492$	280097.	0.0521670
$493$	2.43742e6	0.451661
$494$	1.68336e6	0.310355
$495$	0	0
$496$	−1.10598e7	−2.01857
$497$	2378.62	0.000431950 0
$498$	2.06986e6	0.373997
$499$	5.55266e6	0.998274	0.499137	−	0.866523i	$-0.333650\pi$
$499$	5.55266e6	0.998274	0.499137	+	0.866523i	$0.333650\pi$
$500$	0	0
$501$	−4.55320e6	−0.810443
$502$	−1.28248e7	−2.27139
$503$	3.38576e6	0.596673	0.298336	−	0.954461i	$-0.403568\pi$
$503$	3.38576e6	0.596673	0.298336	+	0.954461i	$0.403568\pi$
$504$	24578.5	0.00431002
$505$	0	0
$506$	−2.75876e6	−0.479002
$507$	−2.33172e6	−0.402862
$508$	−2.75413e6	−0.473506
$509$	4.81925e6	0.824489	0.412244	−	0.911073i	$-0.364745\pi$
$509$	4.81925e6	0.824489	0.412244	+	0.911073i	$0.364745\pi$
$510$	0	0
$511$	−27969.5	−0.00473841
$512$	−653266.	−0.110132
$513$	−544262.	−0.0913092
$514$	−4.25171e6	−0.709833
$515$	0	0
$516$	1.26150e6	0.208575
$517$	−2.72708e6	−0.448717
$518$	−22597.7	−0.00370032
$519$	−1.64120e6	−0.267450
$520$	0	0
$521$	6.32781e6	1.02131	0.510657	−	0.859785i	$-0.329402\pi$
$521$	6.32781e6	1.02131	0.510657	+	0.859785i	$0.329402\pi$
$522$	1.09700e6	0.176209
$523$	3.74833e6	0.599216	0.299608	−	0.954062i	$-0.403144\pi$
$523$	3.74833e6	0.599216	0.299608	+	0.954062i	$0.403144\pi$
$524$	−1.39091e6	−0.221295
$525$	0	0
$526$	−6.47208e6	−1.01995
$527$	−1.05267e7	−1.65107
$528$	−1.38601e6	−0.216362
$529$	5.03756e6	0.782674
$530$	0	0
$531$	2.23343e6	0.343744
$532$	−23953.4	−0.00366934
$533$	783562.	0.119469
$534$	−7.52441e6	−1.14188
$535$	0	0
$536$	2.28116e6	0.342960
$537$	2.63735e6	0.394669
$538$	−4.85016e6	−0.722437
$539$	−2.03294e6	−0.301407
$540$	0	0
$541$	4.24091e6	0.622968	0.311484	−	0.950251i	$-0.399174\pi$
$541$	4.24091e6	0.622968	0.311484	+	0.950251i	$0.399174\pi$
$542$	4.25543e6	0.622221
$543$	2.10646e6	0.306587
$544$	−5.49966e6	−0.796781
$545$	0	0
$546$	−48934.0	−0.00702472
$547$	−6.27240e6	−0.896325	−0.448162	−	0.893952i	$-0.647921\pi$
$547$	−6.27240e6	−0.896325	−0.448162	+	0.893952i	$0.647921\pi$
$548$	−2.11884e6	−0.301402
$549$	1.13156e6	0.160231
$550$	0	0
$551$	1.50220e6	0.210789
$552$	3.83616e6	0.535857
$553$	−117082.	−0.0162809
$554$	−1.60088e7	−2.21607
$555$	0	0
$556$	2.45981e6	0.337454
$557$	−7.08639e6	−0.967803	−0.483902	−	0.875122i	$-0.660781\pi$
$557$	−7.08639e6	−0.967803	−0.483902	+	0.875122i	$0.660781\pi$
$558$	−4.73769e6	−0.644142
$559$	3.52900e6	0.477664
$560$	0	0
$561$	−1.31920e6	−0.176972
$562$	−1.42384e7	−1.90161
$563$	−3.35539e6	−0.446140	−0.223070	−	0.974802i	$-0.571608\pi$
$563$	−3.35539e6	−0.446140	−0.223070	+	0.974802i	$0.571608\pi$
$564$	−2.69880e6	−0.357250
$565$	0	0
$566$	275074.	0.0360918
$567$	15821.3	0.00206674
$568$	124123.	0.0161428
$569$	7.59058e6	0.982865	0.491433	−	0.870916i	$-0.336473\pi$
$569$	7.59058e6	0.982865	0.491433	+	0.870916i	$0.336473\pi$
$570$	0	0
$571$	−1.00981e7	−1.29613	−0.648067	−	0.761584i	$-0.724422\pi$
$571$	−1.00981e7	−1.29613	−0.648067	+	0.761584i	$0.724422\pi$
$572$	539290.	0.0689180
$573$	5.04842e6	0.642346
$574$	−37966.2	−0.00480969
$575$	0	0
$576$	823730.	0.103450
$577$	−1.34784e7	−1.68538	−0.842692	−	0.538396i	$-0.819030\pi$
$577$	−1.34784e7	−1.68538	−0.842692	+	0.538396i	$0.819030\pi$
$578$	−320412.	−0.0398923
$579$	−4.41586e6	−0.547418
$580$	0	0
$581$	−82394.5	−0.0101265
$582$	8.36145e6	1.02323
$583$	706145.	0.0860443
$584$	−1.45952e6	−0.177084
$585$	0	0
$586$	−5.63771e6	−0.678202
$587$	3.66148e6	0.438593	0.219296	−	0.975658i	$-0.429624\pi$
$587$	3.66148e6	0.438593	0.219296	+	0.975658i	$0.429624\pi$
$588$	−2.01186e6	−0.239968
$589$	−6.48768e6	−0.770550
$590$	0	0
$591$	9.15471e6	1.07814
$592$	−1.77197e6	−0.207804
$593$	−4.20157e6	−0.490653	−0.245327	−	0.969440i	$-0.578895\pi$
$593$	−4.20157e6	−0.490653	−0.245327	+	0.969440i	$0.578895\pi$
$594$	−593726.	−0.0690430
$595$	0	0
$596$	3.60001e6	0.415133
$597$	−940886.	−0.108044
$598$	−7.63750e6	−0.873370
$599$	5.56521e6	0.633745	0.316873	−	0.948468i	$-0.397367\pi$
$599$	5.56521e6	0.633745	0.316873	+	0.948468i	$0.397367\pi$
$600$	0	0
$601$	−455310.	−0.0514187	−0.0257093	−	0.999669i	$-0.508184\pi$
$601$	−455310.	−0.0514187	−0.0257093	+	0.999669i	$0.508184\pi$
$602$	−170992.	−0.0192302
$603$	1.46839e6	0.164455
$604$	−3.83086e6	−0.427271
$605$	0	0
$606$	360540.	0.0398816
$607$	−8.22381e6	−0.905944	−0.452972	−	0.891525i	$-0.649636\pi$
$607$	−8.22381e6	−0.905944	−0.452972	+	0.891525i	$0.649636\pi$
$608$	−3.38948e6	−0.371856
$609$	−43667.8	−0.00477110
$610$	0	0
$611$	−7.54980e6	−0.818149
$612$	−1.30552e6	−0.140898
$613$	2.86125e6	0.307543	0.153771	−	0.988106i	$-0.450858\pi$
$613$	2.86125e6	0.307543	0.153771	+	0.988106i	$0.450858\pi$
$614$	−4.25536e6	−0.455528
$615$	0	0
$616$	36716.1	0.00389856
$617$	1.59938e7	1.69137	0.845685	−	0.533683i	$-0.179192\pi$
$617$	1.59938e7	1.69137	0.845685	+	0.533683i	$0.179192\pi$
$618$	1.15824e7	1.21991
$619$	6.22085e6	0.652564	0.326282	−	0.945273i	$-0.394204\pi$
$619$	6.22085e6	0.652564	0.326282	+	0.945273i	$0.394204\pi$
$620$	0	0
$621$	2.46935e6	0.256953
$622$	−1.34052e7	−1.38930
$623$	299522.	0.0309178
$624$	−3.83710e6	−0.394496
$625$	0	0
$626$	−2.05730e6	−0.209828
$627$	−813033.	−0.0825923
$628$	5.32326e6	0.538615
$629$	−1.68656e6	−0.169971
$630$	0	0
$631$	−9.11719e6	−0.911565	−0.455783	−	0.890091i	$-0.650641\pi$
$631$	−9.11719e6	−0.911565	−0.455783	+	0.890091i	$0.650641\pi$
$632$	−6.10967e6	−0.608450
$633$	263795.	0.0261672
$634$	5.18316e6	0.512119
$635$	0	0
$636$	698820.	0.0685051
$637$	−5.62811e6	−0.549558
$638$	1.63872e6	0.159387
$639$	79898.3	0.00774079
$640$	0	0
$641$	9.50107e6	0.913330	0.456665	−	0.889639i	$-0.349044\pi$
$641$	9.50107e6	0.913330	0.456665	+	0.889639i	$0.349044\pi$
$642$	−5.35535e6	−0.512803
$643$	5.32410e6	0.507830	0.253915	−	0.967227i	$-0.418282\pi$
$643$	5.32410e6	0.507830	0.253915	+	0.967227i	$0.418282\pi$
$644$	108678.	0.0103259
$645$	0	0
$646$	−6.08747e6	−0.573925
$647$	−2.02469e6	−0.190150	−0.0950752	−	0.995470i	$-0.530309\pi$
$647$	−2.02469e6	−0.190150	−0.0950752	+	0.995470i	$0.530309\pi$
$648$	825598.	0.0772380
$649$	3.33636e6	0.310929
$650$	0	0
$651$	188592.	0.0174410
$652$	3.06293e6	0.282175
$653$	1.14621e7	1.05192	0.525959	−	0.850510i	$-0.323707\pi$
$653$	1.14621e7	1.05192	0.525959	+	0.850510i	$0.323707\pi$
$654$	−1.35149e7	−1.23558
$655$	0	0
$656$	−2.97708e6	−0.270103
$657$	−939502.	−0.0849150
$658$	365813.	0.0329378
$659$	−8.95330e6	−0.803100	−0.401550	−	0.915837i	$-0.631528\pi$
$659$	−8.95330e6	−0.803100	−0.401550	+	0.915837i	$0.631528\pi$
$660$	0	0
$661$	−1.38575e7	−1.23362	−0.616812	−	0.787111i	$-0.711576\pi$
$661$	−1.38575e7	−1.23362	−0.616812	+	0.787111i	$0.711576\pi$
$662$	−6.50113e6	−0.576559
$663$	−3.65215e6	−0.322674
$664$	−4.29956e6	−0.378446
$665$	0	0
$666$	−759061.	−0.0663118
$667$	−6.81557e6	−0.593182
$668$	−6.73115e6	−0.583644
$669$	−7.15431e6	−0.618020
$670$	0	0
$671$	1.69036e6	0.144935
$672$	98529.9	0.00841676
$673$	9.94769e6	0.846612	0.423306	−	0.905987i	$-0.360870\pi$
$673$	9.94769e6	0.846612	0.423306	+	0.905987i	$0.360870\pi$
$674$	−7.50051e6	−0.635976
$675$	0	0
$676$	−3.44705e6	−0.290122
$677$	−1.32600e7	−1.11192	−0.555960	−	0.831209i	$-0.687649\pi$
$677$	−1.32600e7	−1.11192	−0.555960	+	0.831209i	$0.687649\pi$
$678$	−749065.	−0.0625814
$679$	−332842.	−0.0277054
$680$	0	0
$681$	−3.41811e6	−0.282435
$682$	−7.07729e6	−0.582648
$683$	2.05349e7	1.68438	0.842192	−	0.539178i	$-0.181265\pi$
$683$	2.05349e7	1.68438	0.842192	+	0.539178i	$0.181265\pi$
$684$	−804600.	−0.0657567
$685$	0	0
$686$	545495.	0.0442568
$687$	1.45801e6	0.117860
$688$	−1.34081e7	−1.07993
$689$	1.95493e6	0.156885
$690$	0	0
$691$	1.74373e7	1.38926	0.694630	−	0.719367i	$-0.255568\pi$
$691$	1.74373e7	1.38926	0.694630	+	0.719367i	$0.255568\pi$
$692$	−2.42624e6	−0.192605
$693$	23634.3	0.00186943
$694$	−1.81874e7	−1.43341
$695$	0	0
$696$	−2.27870e6	−0.178305
$697$	−2.83357e6	−0.220929
$698$	5.11953e6	0.397733
$699$	4.16821e6	0.322668
$700$	0	0
$701$	1.83721e6	0.141209	0.0706046	−	0.997504i	$-0.477507\pi$
$701$	1.83721e6	0.141209	0.0706046	+	0.997504i	$0.477507\pi$
$702$	−1.64370e6	−0.125887
$703$	−1.03944e6	−0.0793251
$704$	1.23051e6	0.0935736
$705$	0	0
$706$	−1.75752e7	−1.32706
$707$	−14351.9	−0.00107985
$708$	3.30175e6	0.247549
$709$	4.37773e6	0.327065	0.163532	−	0.986538i	$-0.447711\pi$
$709$	4.37773e6	0.327065	0.163532	+	0.986538i	$0.447711\pi$
$710$	0	0
$711$	−3.93282e6	−0.291763
$712$	1.56299e7	1.15546
$713$	2.94350e7	2.16841
$714$	176958.	0.0129905
$715$	0	0
$716$	3.89889e6	0.284222
$717$	8.39436e6	0.609803
$718$	9.06492e6	0.656225
$719$	−1.65258e7	−1.19217	−0.596087	−	0.802920i	$-0.703279\pi$
$719$	−1.65258e7	−1.19217	−0.596087	+	0.802920i	$0.703279\pi$
$720$	0	0
$721$	−461058.	−0.0330307
$722$	1.29146e7	0.922017
$723$	3.72798e6	0.265233
$724$	3.11405e6	0.220790
$725$	0	0
$726$	−886924.	−0.0624518
$727$	1.37064e7	0.961803	0.480902	−	0.876774i	$-0.340309\pi$
$727$	1.37064e7	0.961803	0.480902	+	0.876774i	$0.340309\pi$
$728$	101647.	0.00710829
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.27618e7	−0.883323
$732$	1.67282e6	0.115391
$733$	2.48655e6	0.170938	0.0854688	−	0.996341i	$-0.472761\pi$
$733$	2.48655e6	0.170938	0.0854688	+	0.996341i	$0.472761\pi$
$734$	−1.21466e7	−0.832178
$735$	0	0
$736$	1.53783e7	1.04644
$737$	2.19352e6	0.148756
$738$	−1.27529e6	−0.0861923
$739$	−2.27900e7	−1.53509	−0.767544	−	0.640996i	$-0.778521\pi$
$739$	−2.27900e7	−1.53509	−0.767544	+	0.640996i	$0.778521\pi$
$740$	0	0
$741$	−2.25084e6	−0.150591
$742$	−94722.6	−0.00631603
$743$	−1.24390e7	−0.826634	−0.413317	−	0.910587i	$-0.635630\pi$
$743$	−1.24390e7	−0.826634	−0.413317	+	0.910587i	$0.635630\pi$
$744$	9.84124e6	0.651805
$745$	0	0
$746$	4.81770e6	0.316951
$747$	−2.76765e6	−0.181472
$748$	−1.95022e6	−0.127447
$749$	213179.	0.0138848
$750$	0	0
$751$	2.85180e7	1.84510	0.922549	−	0.385880i	$-0.126102\pi$
$751$	2.85180e7	1.84510	0.922549	+	0.385880i	$0.126102\pi$
$752$	2.86848e7	1.84973
$753$	1.71483e7	1.10213
$754$	4.53672e6	0.290612
$755$	0	0
$756$	23389.2	0.00148837
$757$	−1.53660e6	−0.0974586	−0.0487293	−	0.998812i	$-0.515517\pi$
$757$	−1.53660e6	−0.0974586	−0.0487293	+	0.998812i	$0.515517\pi$
$758$	−2.84567e7	−1.79892
$759$	3.68879e6	0.232423
$760$	0	0
$761$	−3.69462e6	−0.231264	−0.115632	−	0.993292i	$-0.536889\pi$
$761$	−3.69462e6	−0.231264	−0.115632	+	0.993292i	$0.536889\pi$
$762$	1.25396e7	0.782344
$763$	537986.	0.0334549
$764$	7.46325e6	0.462588
$765$	0	0
$766$	−3.36149e7	−2.06995
$767$	9.23654e6	0.566919
$768$	1.00185e7	0.612913
$769$	−1.49715e6	−0.0912956	−0.0456478	−	0.998958i	$-0.514535\pi$
$769$	−1.49715e6	−0.0912956	−0.0456478	+	0.998958i	$0.514535\pi$
$770$	0	0
$771$	5.68504e6	0.344427
$772$	−6.52811e6	−0.394225
$773$	5.52587e6	0.332623	0.166311	−	0.986073i	$-0.446814\pi$
$773$	5.52587e6	0.332623	0.166311	+	0.986073i	$0.446814\pi$
$774$	−5.74365e6	−0.344616
$775$	0	0
$776$	−1.73686e7	−1.03541
$777$	30215.8	0.00179548
$778$	−1.32073e7	−0.782283
$779$	−1.74635e6	−0.103107
$780$	0	0
$781$	119354.	0.00700181
$782$	2.76192e7	1.61508
$783$	−1.46681e6	−0.0855007
$784$	2.13835e7	1.24248
$785$	0	0
$786$	6.33286e6	0.365632
$787$	6.04216e6	0.347741	0.173870	−	0.984769i	$-0.444373\pi$
$787$	6.04216e6	0.347741	0.173870	+	0.984769i	$0.444373\pi$
$788$	1.35337e7	0.776428
$789$	8.65393e6	0.494904
$790$	0	0
$791$	29817.9	0.00169448
$792$	1.23330e6	0.0698644
$793$	4.67967e6	0.264260
$794$	−3.02230e7	−1.70132
$795$	0	0
$796$	−1.39094e6	−0.0778084
$797$	−3.45685e6	−0.192768	−0.0963840	−	0.995344i	$-0.530728\pi$
$797$	−3.45685e6	−0.192768	−0.0963840	+	0.995344i	$0.530728\pi$
$798$	109061.	0.00606263
$799$	2.73021e7	1.51297
$800$	0	0
$801$	1.00610e7	0.554065
$802$	−3.19063e7	−1.75162
$803$	−1.40345e6	−0.0768085
$804$	2.17077e6	0.118433
$805$	0	0
$806$	−1.95931e7	−1.06235
$807$	6.48523e6	0.350543
$808$	−748922.	−0.0403560
$809$	7.36347e6	0.395559	0.197779	−	0.980247i	$-0.436627\pi$
$809$	7.36347e6	0.395559	0.197779	+	0.980247i	$0.436627\pi$
$810$	0	0
$811$	3.37010e7	1.79925	0.899623	−	0.436667i	$-0.143841\pi$
$811$	3.37010e7	1.79925	0.899623	+	0.436667i	$0.143841\pi$
$812$	−64555.6	−0.00343593
$813$	−5.69000e6	−0.301916
$814$	−1.13391e6	−0.0599813
$815$	0	0
$816$	1.38760e7	0.729523
$817$	−7.86520e6	−0.412244
$818$	1.19305e7	0.623412
$819$	65430.4	0.00340855
$820$	0	0
$821$	−2.10298e7	−1.08888	−0.544438	−	0.838801i	$-0.683257\pi$
$821$	−2.10298e7	−1.08888	−0.544438	+	0.838801i	$0.683257\pi$
$822$	9.64715e6	0.497989
$823$	−1.99288e7	−1.02561	−0.512803	−	0.858506i	$-0.671393\pi$
$823$	−1.99288e7	−1.02561	−0.512803	+	0.858506i	$0.671393\pi$
$824$	−2.40592e7	−1.23442
$825$	0	0
$826$	−447541.	−0.0228235
$827$	1.87978e7	0.955748	0.477874	−	0.878428i	$-0.341407\pi$
$827$	1.87978e7	0.955748	0.477874	+	0.878428i	$0.341407\pi$
$828$	3.65053e6	0.185046
$829$	7.27718e6	0.367771	0.183885	−	0.982948i	$-0.441133\pi$
$829$	7.27718e6	0.367771	0.183885	+	0.982948i	$0.441133\pi$
$830$	0	0
$831$	2.14056e7	1.07529
$832$	3.40661e6	0.170614
$833$	2.03527e7	1.01627
$834$	−1.11996e7	−0.557554
$835$	0	0
$836$	−1.20193e6	−0.0594792
$837$	6.33485e6	0.312552
$838$	2.01595e7	0.991677
$839$	3.52339e6	0.172805	0.0864025	−	0.996260i	$-0.472463\pi$
$839$	3.52339e6	0.172805	0.0864025	+	0.996260i	$0.472463\pi$
$840$	0	0
$841$	−1.64627e7	−0.802620
$842$	3.70124e7	1.79915
$843$	1.90384e7	0.922704
$844$	389976.	0.0188444
$845$	0	0
$846$	1.22877e7	0.590263
$847$	35305.6	0.00169097
$848$	−7.42757e6	−0.354697
$849$	−367806.	−0.0175125
$850$	0	0
$851$	4.71600e6	0.223229
$852$	118116.	0.00557456
$853$	−4.55380e6	−0.214290	−0.107145	−	0.994243i	$-0.534171\pi$
$853$	−4.55380e6	−0.214290	−0.107145	+	0.994243i	$0.534171\pi$
$854$	−226745.	−0.0106388
$855$	0	0
$856$	1.11243e7	0.518904
$857$	2.88776e7	1.34310	0.671552	−	0.740958i	$-0.265628\pi$
$857$	2.88776e7	1.34310	0.671552	+	0.740958i	$0.265628\pi$
$858$	−2.45541e6	−0.113869
$859$	1.25501e7	0.580316	0.290158	−	0.956979i	$-0.406292\pi$
$859$	1.25501e7	0.580316	0.290158	+	0.956979i	$0.406292\pi$
$860$	0	0
$861$	50765.2	0.00233377
$862$	4.01939e6	0.184243
$863$	−2.86046e7	−1.30740	−0.653701	−	0.756753i	$-0.726785\pi$
$863$	−2.86046e7	−1.30740	−0.653701	+	0.756753i	$0.726785\pi$
$864$	3.30964e6	0.150833
$865$	0	0
$866$	7.25492e6	0.328729
$867$	428428.	0.0193567
$868$	278802.	0.0125602
$869$	−5.87496e6	−0.263910
$870$	0	0
$871$	6.07266e6	0.271227
$872$	2.80736e7	1.25028
$873$	−1.11802e7	−0.496496
$874$	1.70219e7	0.753755
$875$	0	0
$876$	−1.38890e6	−0.0611519
$877$	−3.42227e7	−1.50250	−0.751251	−	0.660017i	$-0.770549\pi$
$877$	−3.42227e7	−1.50250	−0.751251	+	0.660017i	$0.770549\pi$
$878$	−3.00055e7	−1.31360
$879$	7.53828e6	0.329079
$880$	0	0
$881$	−5.25799e6	−0.228234	−0.114117	−	0.993467i	$-0.536404\pi$
$881$	−5.25799e6	−0.228234	−0.114117	+	0.993467i	$0.536404\pi$
$882$	9.16005e6	0.396485
$883$	−8.94488e6	−0.386076	−0.193038	−	0.981191i	$-0.561834\pi$
$883$	−8.94488e6	−0.386076	−0.193038	+	0.981191i	$0.561834\pi$
$884$	−5.39909e6	−0.232375
$885$	0	0
$886$	−1.58431e7	−0.678040
$887$	−4.14864e7	−1.77050	−0.885252	−	0.465112i	$-0.846014\pi$
$887$	−4.14864e7	−1.77050	−0.885252	+	0.465112i	$0.846014\pi$
$888$	1.57674e6	0.0671007
$889$	−499163.	−0.0211830
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.05765e7	−0.445070
$893$	1.68265e7	0.706098
$894$	−1.63909e7	−0.685899
$895$	0	0
$896$	−515390.	−0.0214470
$897$	1.02122e7	0.423779
$898$	−9.66861e6	−0.400104
$899$	−1.74846e7	−0.721533
$900$	0	0
$901$	−7.06954e6	−0.290121
$902$	−1.90506e6	−0.0779638
$903$	228636.	0.00933094
$904$	1.55597e6	0.0633259
$905$	0	0
$906$	1.74420e7	0.705954
$907$	−3.66068e7	−1.47755	−0.738777	−	0.673950i	$-0.764596\pi$
$907$	−3.66068e7	−1.47755	−0.738777	+	0.673950i	$0.764596\pi$
$908$	−5.05310e6	−0.203396
$909$	−482084.	−0.0193514
$910$	0	0
$911$	2.81370e7	1.12326	0.561631	−	0.827388i	$-0.310174\pi$
$911$	2.81370e7	1.12326	0.561631	+	0.827388i	$0.310174\pi$
$912$	8.55188e6	0.340467
$913$	−4.13439e6	−0.164147
$914$	3.09679e7	1.22616
$915$	0	0
$916$	2.15542e6	0.0848776
$917$	−252091.	−0.00989997
$918$	5.94407e6	0.232797
$919$	−9.39900e6	−0.367107	−0.183554	−	0.983010i	$-0.558760\pi$
$919$	−9.39900e6	−0.367107	−0.183554	+	0.983010i	$0.558760\pi$
$920$	0	0
$921$	5.68991e6	0.221033
$922$	3.14408e7	1.21805
$923$	330426.	0.0127665
$924$	34939.4	0.00134628
$925$	0	0
$926$	2.19836e6	0.0842502
$927$	−1.54870e7	−0.591928
$928$	−9.13481e6	−0.348201
$929$	−3.51723e7	−1.33709	−0.668546	−	0.743671i	$-0.733083\pi$
$929$	−3.51723e7	−1.33709	−0.668546	+	0.743671i	$0.733083\pi$
$930$	0	0
$931$	1.25435e7	0.474292
$932$	6.16200e6	0.232371
$933$	1.79243e7	0.674121
$934$	1.09764e7	0.411710
$935$	0	0
$936$	3.41433e6	0.127384
$937$	1.09773e7	0.408457	0.204229	−	0.978923i	$-0.434531\pi$
$937$	1.09773e7	0.408457	0.204229	+	0.978923i	$0.434531\pi$
$938$	−294240.	−0.0109193
$939$	2.75086e6	0.101813
$940$	0	0
$941$	2.86362e7	1.05424	0.527122	−	0.849790i	$-0.323271\pi$
$941$	2.86362e7	1.05424	0.527122	+	0.849790i	$0.323271\pi$
$942$	−2.42370e7	−0.889920
$943$	7.92331e6	0.290153
$944$	−3.50934e7	−1.28173
$945$	0	0
$946$	−8.58001e6	−0.311717
$947$	1.64152e7	0.594800	0.297400	−	0.954753i	$-0.403881\pi$
$947$	1.64152e7	0.594800	0.297400	+	0.954753i	$0.403881\pi$
$948$	−5.81402e6	−0.210114
$949$	−3.88539e6	−0.140046
$950$	0	0
$951$	−6.93049e6	−0.248492
$952$	−367582.	−0.0131450
$953$	2.15286e7	0.767862	0.383931	−	0.923362i	$-0.374570\pi$
$953$	2.15286e7	0.767862	0.383931	+	0.923362i	$0.374570\pi$
$954$	−3.18175e6	−0.113187
$955$	0	0
$956$	1.24097e7	0.439152
$957$	−2.19116e6	−0.0773383
$958$	4.92581e7	1.73406
$959$	−384022.	−0.0134837
$960$	0	0
$961$	4.68831e7	1.63760
$962$	−3.13916e6	−0.109365
$963$	7.16073e6	0.248824
$964$	5.51119e6	0.191009
$965$	0	0
$966$	−494816.	−0.0170609
$967$	−1.52551e7	−0.524624	−0.262312	−	0.964983i	$-0.584485\pi$
$967$	−1.52551e7	−0.524624	−0.262312	+	0.964983i	$0.584485\pi$
$968$	1.84234e6	0.0631947
$969$	8.13966e6	0.278482
$970$	0	0
$971$	−1.95235e7	−0.664521	−0.332261	−	0.943188i	$-0.607811\pi$
$971$	−1.95235e7	−0.664521	−0.332261	+	0.943188i	$0.607811\pi$
$972$	785647.	0.0266724
$973$	445820.	0.0150965
$974$	1.88924e7	0.638103
$975$	0	0
$976$	−1.77800e7	−0.597457
$977$	5.23977e7	1.75621	0.878104	−	0.478469i	$-0.158808\pi$
$977$	5.23977e7	1.75621	0.878104	+	0.478469i	$0.158808\pi$
$978$	−1.39456e7	−0.466219
$979$	1.50294e7	0.501170
$980$	0	0
$981$	1.80711e7	0.599531
$982$	1.61134e7	0.533222
$983$	−3.04687e7	−1.00570	−0.502851	−	0.864373i	$-0.667716\pi$
$983$	−3.04687e7	−1.00570	−0.502851	+	0.864373i	$0.667716\pi$
$984$	2.64906e6	0.0872177
$985$	0	0
$986$	−1.64060e7	−0.537416
$987$	−489134.	−0.0159822
$988$	−3.32750e6	−0.108449
$989$	3.56850e7	1.16010
$990$	0	0
$991$	2.04432e6	0.0661250	0.0330625	−	0.999453i	$-0.489474\pi$
$991$	2.04432e6	0.0661250	0.0330625	+	0.999453i	$0.489474\pi$
$992$	3.94513e7	1.27287
$993$	8.69277e6	0.279760
$994$	−16010.2	−0.000513963 0
$995$	0	0
$996$	−4.09150e6	−0.130688
$997$	−4.98989e7	−1.58984	−0.794919	−	0.606715i	$-0.792487\pi$
$997$	−4.98989e7	−1.58984	−0.794919	+	0.606715i	$0.792487\pi$
$998$	−3.73744e7	−1.18781
$999$	1.01495e6	0.0321760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.t.1.3	✓	10
5.4	even	2		825.6.a.u.1.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.3	✓	10	1.1	even	1	trivial
825.6.a.u.1.8	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)

\(f(q)\)	\(=\)	\(q-6.73090 q^{2} +9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} +2.41142 q^{7} +125.834 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-6.73090 q^{2} +9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} +2.41142 q^{7} +125.834 q^{8} +81.0000 q^{9} +121.000 q^{11} +119.745 q^{12} +334.983 q^{13} -16.2310 q^{14} -1272.74 q^{16} -1211.39 q^{17} -545.203 q^{18} -746.587 q^{19} +21.7028 q^{21} -814.439 q^{22} +3387.32 q^{23} +1132.51 q^{24} -2254.73 q^{26} +729.000 q^{27} +32.0839 q^{28} -2012.09 q^{29} +8689.78 q^{31} +4539.97 q^{32} +1089.00 q^{33} +8153.73 q^{34} +1077.70 q^{36} +1392.25 q^{37} +5025.20 q^{38} +3014.84 q^{39} +2339.11 q^{41} -146.079 q^{42} +10534.9 q^{43} +1609.90 q^{44} -22799.7 q^{46} -22537.9 q^{47} -11454.6 q^{48} -16801.2 q^{49} -10902.5 q^{51} +4456.94 q^{52} +5835.91 q^{53} -4906.83 q^{54} +303.439 q^{56} -6719.28 q^{57} +13543.2 q^{58} +27573.2 q^{59} +13969.9 q^{61} -58490.0 q^{62} +195.325 q^{63} +10169.5 q^{64} -7329.95 q^{66} +18128.3 q^{67} -16117.5 q^{68} +30485.8 q^{69} +986.398 q^{71} +10192.6 q^{72} -11598.8 q^{73} -9371.12 q^{74} -9933.34 q^{76} +291.782 q^{77} -20292.6 q^{78} -48553.4 q^{79} +6561.00 q^{81} -15744.3 q^{82} -34168.5 q^{83} +288.755 q^{84} -70909.2 q^{86} -18108.8 q^{87} +15225.9 q^{88} +124210. q^{89} +807.783 q^{91} +45068.2 q^{92} +78208.0 q^{93} +151700. q^{94} +40859.7 q^{96} -138028. q^{97} +113087. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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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