LMFDB - Embedded newform 825.6.a.f.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.307532.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 76x + 168 $ x^3 - x^2 - 76*x + 168
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.91848$ 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)

$f(q)$	$=$	$q-9.91848 q^{2} +9.00000 q^{3} +66.3762 q^{4} -89.2663 q^{6} -92.6461 q^{7} -340.959 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.91848 q^{2} +9.00000 q^{3} +66.3762 q^{4} -89.2663 q^{6} -92.6461 q^{7} -340.959 q^{8} +81.0000 q^{9} +121.000 q^{11} +597.386 q^{12} -800.249 q^{13} +918.908 q^{14} +1257.76 q^{16} +117.742 q^{17} -803.397 q^{18} +831.422 q^{19} -833.814 q^{21} -1200.14 q^{22} -2952.23 q^{23} -3068.63 q^{24} +7937.25 q^{26} +729.000 q^{27} -6149.49 q^{28} +5765.87 q^{29} -61.7803 q^{31} -1564.36 q^{32} +1089.00 q^{33} -1167.83 q^{34} +5376.47 q^{36} +10236.6 q^{37} -8246.44 q^{38} -7202.24 q^{39} +9599.21 q^{41} +8270.17 q^{42} +17473.2 q^{43} +8031.52 q^{44} +29281.7 q^{46} -18748.3 q^{47} +11319.8 q^{48} -8223.71 q^{49} +1059.68 q^{51} -53117.5 q^{52} +9703.45 q^{53} -7230.57 q^{54} +31588.5 q^{56} +7482.79 q^{57} -57188.6 q^{58} -24401.4 q^{59} +33910.3 q^{61} +612.766 q^{62} -7504.33 q^{63} -24732.2 q^{64} -10801.2 q^{66} +4989.24 q^{67} +7815.29 q^{68} -26570.1 q^{69} -62961.6 q^{71} -27617.7 q^{72} +54232.5 q^{73} -101531. q^{74} +55186.6 q^{76} -11210.2 q^{77} +71435.2 q^{78} -56996.4 q^{79} +6561.00 q^{81} -95209.6 q^{82} -49685.9 q^{83} -55345.4 q^{84} -173308. q^{86} +51892.8 q^{87} -41256.1 q^{88} -87990.1 q^{89} +74139.9 q^{91} -195958. q^{92} -556.022 q^{93} +185954. q^{94} -14079.3 q^{96} +44817.9 q^{97} +81566.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22} - 1220 q^{23} - 2079 q^{24} + 17062 q^{26} + 2187 q^{27} - 11120 q^{28} + 4402 q^{29} - 10688 q^{31} - 12439 q^{32} + 3267 q^{33} - 4094 q^{34} + 5913 q^{36} + 8190 q^{37} - 13792 q^{38} + 810 q^{39} + 5974 q^{41} - 7056 q^{42} - 18868 q^{43} + 8833 q^{44} + 46220 q^{46} - 55500 q^{47} - 3735 q^{48} + 1907 q^{49} - 17406 q^{51} - 27330 q^{52} - 9206 q^{53} - 5103 q^{54} + 73248 q^{56} + 18756 q^{57} - 15366 q^{58} - 59196 q^{59} + 79902 q^{61} - 64616 q^{62} - 7452 q^{63} + 2129 q^{64} - 7623 q^{66} - 4468 q^{67} + 1218 q^{68} - 10980 q^{69} - 75164 q^{71} - 18711 q^{72} + 61290 q^{73} - 56766 q^{74} + 37816 q^{76} - 11132 q^{77} + 153558 q^{78} - 83564 q^{79} + 19683 q^{81} - 147410 q^{82} - 74764 q^{83} - 100080 q^{84} - 253432 q^{86} + 39618 q^{87} - 27951 q^{88} + 37342 q^{89} - 126488 q^{91} - 148164 q^{92} - 96192 q^{93} + 59252 q^{94} - 111951 q^{96} - 33486 q^{97} + 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9 + 363 * q^11 + 657 * q^12 + 90 * q^13 - 784 * q^14 - 415 * q^16 - 1934 * q^17 - 567 * q^18 + 2084 * q^19 - 828 * q^21 - 847 * q^22 - 1220 * q^23 - 2079 * q^24 + 17062 * q^26 + 2187 * q^27 - 11120 * q^28 + 4402 * q^29 - 10688 * q^31 - 12439 * q^32 + 3267 * q^33 - 4094 * q^34 + 5913 * q^36 + 8190 * q^37 - 13792 * q^38 + 810 * q^39 + 5974 * q^41 - 7056 * q^42 - 18868 * q^43 + 8833 * q^44 + 46220 * q^46 - 55500 * q^47 - 3735 * q^48 + 1907 * q^49 - 17406 * q^51 - 27330 * q^52 - 9206 * q^53 - 5103 * q^54 + 73248 * q^56 + 18756 * q^57 - 15366 * q^58 - 59196 * q^59 + 79902 * q^61 - 64616 * q^62 - 7452 * q^63 + 2129 * q^64 - 7623 * q^66 - 4468 * q^67 + 1218 * q^68 - 10980 * q^69 - 75164 * q^71 - 18711 * q^72 + 61290 * q^73 - 56766 * q^74 + 37816 * q^76 - 11132 * q^77 + 153558 * q^78 - 83564 * q^79 + 19683 * q^81 - 147410 * q^82 - 74764 * q^83 - 100080 * q^84 - 253432 * q^86 + 39618 * q^87 - 27951 * q^88 + 37342 * q^89 - 126488 * q^91 - 148164 * q^92 - 96192 * q^93 + 59252 * q^94 - 111951 * q^96 - 33486 * q^97 + 95249 * q^98 + 29403 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−9.91848	−1.75336	−0.876678	−	0.481078i	$-0.840245\pi$
$2$	−9.91848	−1.75336	−0.876678	+	0.481078i	$0.840245\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	66.3762	2.07426
$5$	0	0
$5$	0	0
$6$	−89.2663	−1.01230
$7$	−92.6461	−0.714631	−0.357315	−	0.933984i	$-0.616308\pi$
$7$	−92.6461	−0.714631	−0.357315	+	0.933984i	$0.616308\pi$
$8$	−340.959	−1.88355
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	597.386	1.19757
$13$	−800.249	−1.31331	−0.656654	−	0.754192i	$-0.728029\pi$
$13$	−800.249	−1.31331	−0.656654	+	0.754192i	$0.728029\pi$
$14$	918.908	1.25300
$15$	0	0
$16$	1257.76	1.22828
$17$	117.742	0.0988122	0.0494061	−	0.998779i	$-0.484267\pi$
$17$	117.742	0.0988122	0.0494061	+	0.998779i	$0.484267\pi$
$18$	−803.397	−0.584452
$19$	831.422	0.528369	0.264185	−	0.964472i	$-0.414897\pi$
$19$	831.422	0.528369	0.264185	+	0.964472i	$0.414897\pi$
$20$	0	0
$21$	−833.814	−0.412592
$22$	−1200.14	−0.528657
$23$	−2952.23	−1.16367	−0.581837	−	0.813306i	$-0.697666\pi$
$23$	−2952.23	−1.16367	−0.581837	+	0.813306i	$0.697666\pi$
$24$	−3068.63	−1.08747
$25$	0	0
$26$	7937.25	2.30270
$27$	729.000	0.192450
$28$	−6149.49	−1.48233
$29$	5765.87	1.27312	0.636560	−	0.771227i	$-0.280357\pi$
$29$	5765.87	1.27312	0.636560	+	0.771227i	$0.280357\pi$
$30$	0	0
$31$	−61.7803	−0.0115464	−0.00577319	−	0.999983i	$-0.501838\pi$
$31$	−61.7803	−0.0115464	−0.00577319	+	0.999983i	$0.501838\pi$
$32$	−1564.36	−0.270061
$33$	1089.00	0.174078
$34$	−1167.83	−0.173253
$35$	0	0
$36$	5376.47	0.691419
$37$	10236.6	1.22928	0.614640	−	0.788808i	$-0.289301\pi$
$37$	10236.6	1.22928	0.614640	+	0.788808i	$0.289301\pi$
$38$	−8246.44	−0.926419
$39$	−7202.24	−0.758239
$40$	0	0
$41$	9599.21	0.891818	0.445909	−	0.895078i	$-0.352881\pi$
$41$	9599.21	0.891818	0.445909	+	0.895078i	$0.352881\pi$
$42$	8270.17	0.723421
$43$	17473.2	1.44113	0.720563	−	0.693389i	$-0.243883\pi$
$43$	17473.2	1.44113	0.720563	+	0.693389i	$0.243883\pi$
$44$	8031.52	0.625412
$45$	0	0
$46$	29281.7	2.04033
$47$	−18748.3	−1.23799	−0.618995	−	0.785395i	$-0.712460\pi$
$47$	−18748.3	−1.23799	−0.618995	+	0.785395i	$0.712460\pi$
$48$	11319.8	0.709148
$49$	−8223.71	−0.489303
$50$	0	0
$51$	1059.68	0.0570493
$52$	−53117.5	−2.72414
$53$	9703.45	0.474500	0.237250	−	0.971449i	$-0.423754\pi$
$53$	9703.45	0.474500	0.237250	+	0.971449i	$0.423754\pi$
$54$	−7230.57	−0.337433
$55$	0	0
$56$	31588.5	1.34604
$57$	7482.79	0.305054
$58$	−57188.6	−2.23223
$59$	−24401.4	−0.912609	−0.456305	−	0.889824i	$-0.650827\pi$
$59$	−24401.4	−0.912609	−0.456305	+	0.889824i	$0.650827\pi$
$60$	0	0
$61$	33910.3	1.16683	0.583413	−	0.812175i	$-0.301717\pi$
$61$	33910.3	1.16683	0.583413	+	0.812175i	$0.301717\pi$
$62$	612.766	0.0202449
$63$	−7504.33	−0.238210
$64$	−24732.2	−0.754768
$65$	0	0
$66$	−10801.2	−0.305220
$67$	4989.24	0.135784	0.0678918	−	0.997693i	$-0.478373\pi$
$67$	4989.24	0.135784	0.0678918	+	0.997693i	$0.478373\pi$
$68$	7815.29	0.204962
$69$	−26570.1	−0.671847
$70$	0	0
$71$	−62961.6	−1.48228	−0.741140	−	0.671351i	$-0.765714\pi$
$71$	−62961.6	−1.48228	−0.741140	+	0.671351i	$0.765714\pi$
$72$	−27617.7	−0.627851
$73$	54232.5	1.19111	0.595556	−	0.803314i	$-0.296932\pi$
$73$	54232.5	1.19111	0.595556	+	0.803314i	$0.296932\pi$
$74$	−101531.	−2.15537
$75$	0	0
$76$	55186.6	1.09597
$77$	−11210.2	−0.215469
$78$	71435.2	1.32946
$79$	−56996.4	−1.02750	−0.513748	−	0.857941i	$-0.671743\pi$
$79$	−56996.4	−1.02750	−0.513748	+	0.857941i	$0.671743\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−95209.6	−1.56367
$83$	−49685.9	−0.791659	−0.395830	−	0.918324i	$-0.629543\pi$
$83$	−49685.9	−0.791659	−0.395830	+	0.918324i	$0.629543\pi$
$84$	−55345.4	−0.855822
$85$	0	0
$86$	−173308.	−2.52681
$87$	51892.8	0.735037
$88$	−41256.1	−0.567912
$89$	−87990.1	−1.17749	−0.588747	−	0.808317i	$-0.700379\pi$
$89$	−87990.1	−1.17749	−0.588747	+	0.808317i	$0.700379\pi$
$90$	0	0
$91$	74139.9	0.938531
$92$	−195958.	−2.41376
$93$	−556.022	−0.00666630
$94$	185954.	2.17064
$95$	0	0
$96$	−14079.3	−0.155920
$97$	44817.9	0.483640	0.241820	−	0.970321i	$-0.422256\pi$
$97$	44817.9	0.483640	0.241820	+	0.970321i	$0.422256\pi$
$98$	81566.7	0.857921
$99$	9801.00	0.100504
$100$	0	0
$101$	82871.7	0.808356	0.404178	−	0.914680i	$-0.367558\pi$
$101$	82871.7	0.808356	0.404178	+	0.914680i	$0.367558\pi$
$102$	−10510.4	−0.100028
$103$	−153697.	−1.42749	−0.713744	−	0.700406i	$-0.753002\pi$
$103$	−153697.	−1.42749	−0.713744	+	0.700406i	$0.753002\pi$
$104$	272852.	2.47368
$105$	0	0
$106$	−96243.5	−0.831968
$107$	−41662.3	−0.351790	−0.175895	−	0.984409i	$-0.556282\pi$
$107$	−41662.3	−0.351790	−0.175895	+	0.984409i	$0.556282\pi$
$108$	48388.2	0.399191
$109$	39745.0	0.320417	0.160209	−	0.987083i	$-0.448783\pi$
$109$	39745.0	0.320417	0.160209	+	0.987083i	$0.448783\pi$
$110$	0	0
$111$	92129.3	0.709725
$112$	−116526.	−0.877768
$113$	191157.	1.40829	0.704147	−	0.710054i	$-0.251329\pi$
$113$	191157.	1.40829	0.704147	+	0.710054i	$0.251329\pi$
$114$	−74217.9	−0.534868
$115$	0	0
$116$	382716.	2.64078
$117$	−64820.1	−0.437769
$118$	242025.	1.60013
$119$	−10908.4	−0.0706143
$120$	0	0
$121$	14641.0	0.0909091
$122$	−336338.	−2.04586
$123$	86392.9	0.514891
$124$	−4100.74	−0.0239501
$125$	0	0
$126$	74431.5	0.417667
$127$	212879.	1.17118	0.585590	−	0.810608i	$-0.300863\pi$
$127$	212879.	1.17118	0.585590	+	0.810608i	$0.300863\pi$
$128$	295366.	1.59344
$129$	157259.	0.832035
$130$	0	0
$131$	−146592.	−0.746333	−0.373166	−	0.927764i	$-0.621728\pi$
$131$	−146592.	−0.746333	−0.373166	+	0.927764i	$0.621728\pi$
$132$	72283.7	0.361082
$133$	−77027.9	−0.377589
$134$	−49485.7	−0.238077
$135$	0	0
$136$	−40145.4	−0.186118
$137$	−335075.	−1.52525	−0.762623	−	0.646843i	$-0.776089\pi$
$137$	−335075.	−1.52525	−0.762623	+	0.646843i	$0.776089\pi$
$138$	263535.	1.17799
$139$	30053.6	0.131935	0.0659674	−	0.997822i	$-0.478987\pi$
$139$	30053.6	0.131935	0.0659674	+	0.997822i	$0.478987\pi$
$140$	0	0
$141$	−168735.	−0.714754
$142$	624483.	2.59896
$143$	−96830.1	−0.395977
$144$	101879.	0.409427
$145$	0	0
$146$	−537904.	−2.08844
$147$	−74013.4	−0.282499
$148$	679466.	2.54984
$149$	240256.	0.886559	0.443280	−	0.896383i	$-0.353815\pi$
$149$	240256.	0.886559	0.443280	+	0.896383i	$0.353815\pi$
$150$	0	0
$151$	37997.8	0.135617	0.0678087	−	0.997698i	$-0.478399\pi$
$151$	37997.8	0.135617	0.0678087	+	0.997698i	$0.478399\pi$
$152$	−283481.	−0.995211
$153$	9537.13	0.0329374
$154$	111188.	0.377794
$155$	0	0
$156$	−478057.	−1.57278
$157$	260511.	0.843483	0.421742	−	0.906716i	$-0.361419\pi$
$157$	260511.	0.843483	0.421742	+	0.906716i	$0.361419\pi$
$158$	565318.	1.80157
$159$	87331.1	0.273953
$160$	0	0
$161$	273513.	0.831597
$162$	−65075.1	−0.194817
$163$	364741.	1.07527	0.537633	−	0.843179i	$-0.319319\pi$
$163$	364741.	1.07527	0.537633	+	0.843179i	$0.319319\pi$
$164$	637159.	1.84986
$165$	0	0
$166$	492809.	1.38806
$167$	−70995.9	−0.196989	−0.0984945	−	0.995138i	$-0.531403\pi$
$167$	−70995.9	−0.196989	−0.0984945	+	0.995138i	$0.531403\pi$
$168$	284297.	0.777139
$169$	269105.	0.724778
$170$	0	0
$171$	67345.2	0.176123
$172$	1.15981e6	2.98927
$173$	−295710.	−0.751192	−0.375596	−	0.926783i	$-0.622562\pi$
$173$	−295710.	−0.751192	−0.375596	+	0.926783i	$0.622562\pi$
$174$	−514698.	−1.28878
$175$	0	0
$176$	152189.	0.370341
$177$	−219613.	−0.526895
$178$	872727.	2.06457
$179$	−567691.	−1.32428	−0.662139	−	0.749381i	$-0.730351\pi$
$179$	−567691.	−1.32428	−0.662139	+	0.749381i	$0.730351\pi$
$180$	0	0
$181$	−361118.	−0.819319	−0.409660	−	0.912239i	$-0.634352\pi$
$181$	−361118.	−0.819319	−0.409660	+	0.912239i	$0.634352\pi$
$182$	−735355.	−1.64558
$183$	305192.	0.673668
$184$	1.00659e6	2.19184
$185$	0	0
$186$	5514.90	0.0116884
$187$	14246.8	0.0297930
$188$	−1.24444e6	−2.56791
$189$	−67539.0	−0.137531
$190$	0	0
$191$	334871.	0.664193	0.332096	−	0.943245i	$-0.392244\pi$
$191$	334871.	0.664193	0.332096	+	0.943245i	$0.392244\pi$
$192$	−222590.	−0.435765
$193$	286645.	0.553926	0.276963	−	0.960881i	$-0.410672\pi$
$193$	286645.	0.553926	0.276963	+	0.960881i	$0.410672\pi$
$194$	−444525.	−0.847992
$195$	0	0
$196$	−545858.	−1.01494
$197$	−402554.	−0.739024	−0.369512	−	0.929226i	$-0.620475\pi$
$197$	−402554.	−0.739024	−0.369512	+	0.929226i	$0.620475\pi$
$198$	−97211.0	−0.176219
$199$	506857.	0.907304	0.453652	−	0.891179i	$-0.350121\pi$
$199$	506857.	0.907304	0.453652	+	0.891179i	$0.350121\pi$
$200$	0	0
$201$	44903.2	0.0783948
$202$	−821961.	−1.41734
$203$	−534185.	−0.909812
$204$	70337.6	0.118335
$205$	0	0
$206$	1.52444e6	2.50289
$207$	−239131.	−0.387891
$208$	−1.00652e6	−1.61311
$209$	100602.	0.159309
$210$	0	0
$211$	166764.	0.257867	0.128934	−	0.991653i	$-0.458845\pi$
$211$	166764.	0.257867	0.128934	+	0.991653i	$0.458845\pi$
$212$	644078.	0.984235
$213$	−566655.	−0.855794
$214$	413227.	0.616813
$215$	0	0
$216$	−248559.	−0.362490
$217$	5723.70	0.00825140
$218$	−394209.	−0.561805
$219$	488093.	0.687689
$220$	0	0
$221$	−94223.2	−0.129771
$222$	−913782.	−1.24440
$223$	−556053.	−0.748780	−0.374390	−	0.927271i	$-0.622148\pi$
$223$	−556053.	−0.748780	−0.374390	+	0.927271i	$0.622148\pi$
$224$	144932.	0.192994
$225$	0	0
$226$	−1.89598e6	−2.46924
$227$	−507849.	−0.654139	−0.327069	−	0.945000i	$-0.606061\pi$
$227$	−507849.	−0.654139	−0.327069	+	0.945000i	$0.606061\pi$
$228$	496679.	0.632760
$229$	−627193.	−0.790337	−0.395169	−	0.918609i	$-0.629314\pi$
$229$	−627193.	−0.790337	−0.395169	+	0.918609i	$0.629314\pi$
$230$	0	0
$231$	−100892.	−0.124401
$232$	−1.96593e6	−2.39799
$233$	−955870.	−1.15348	−0.576738	−	0.816929i	$-0.695675\pi$
$233$	−955870.	−1.15348	−0.576738	+	0.816929i	$0.695675\pi$
$234$	642917.	0.767565
$235$	0	0
$236$	−1.61967e6	−1.89298
$237$	−512968.	−0.593225
$238$	108194.	0.123812
$239$	−509255.	−0.576687	−0.288344	−	0.957527i	$-0.593105\pi$
$239$	−509255.	−0.576687	−0.288344	+	0.957527i	$0.593105\pi$
$240$	0	0
$241$	−666125.	−0.738776	−0.369388	−	0.929275i	$-0.620433\pi$
$241$	−666125.	−0.738776	−0.369388	+	0.929275i	$0.620433\pi$
$242$	−145216.	−0.159396
$243$	59049.0	0.0641500
$244$	2.25083e6	2.42030
$245$	0	0
$246$	−856886.	−0.902787
$247$	−665344.	−0.693911
$248$	21064.6	0.0217482
$249$	−447173.	−0.457065
$250$	0	0
$251$	505010.	0.505960	0.252980	−	0.967472i	$-0.418589\pi$
$251$	505010.	0.505960	0.252980	+	0.967472i	$0.418589\pi$
$252$	−498109.	−0.494109
$253$	−357220.	−0.350861
$254$	−2.11144e6	−2.05349
$255$	0	0
$256$	−2.13815e6	−2.03909
$257$	2.02669e6	1.91405	0.957027	−	0.290000i	$-0.0936552\pi$
$257$	2.02669e6	1.91405	0.957027	+	0.290000i	$0.0936552\pi$
$258$	−1.55977e6	−1.45885
$259$	−948379.	−0.878482
$260$	0	0
$261$	467035.	0.424374
$262$	1.45397e6	1.30859
$263$	−1.57349e6	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$263$	−1.57349e6	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$264$	−371305.	−0.327884
$265$	0	0
$266$	764000.	0.662048
$267$	−791911.	−0.679826
$268$	331167.	0.281650
$269$	−1.55958e6	−1.31409	−0.657046	−	0.753850i	$-0.728194\pi$
$269$	−1.55958e6	−1.31409	−0.657046	+	0.753850i	$0.728194\pi$
$270$	0	0
$271$	1.35274e6	1.11890	0.559451	−	0.828864i	$-0.311012\pi$
$271$	1.35274e6	1.11890	0.559451	+	0.828864i	$0.311012\pi$
$272$	148092.	0.121369
$273$	667259.	0.541861
$274$	3.32343e6	2.67430
$275$	0	0
$276$	−1.76362e6	−1.39358
$277$	−2.47205e6	−1.93579	−0.967895	−	0.251353i	$-0.919124\pi$
$277$	−2.47205e6	−1.93579	−0.967895	+	0.251353i	$0.919124\pi$
$278$	−298086.	−0.231328
$279$	−5004.20	−0.00384879
$280$	0	0
$281$	769611.	0.581441	0.290721	−	0.956808i	$-0.406105\pi$
$281$	769611.	0.581441	0.290721	+	0.956808i	$0.406105\pi$
$282$	1.67359e6	1.25322
$283$	−2.52887e6	−1.87698	−0.938490	−	0.345305i	$-0.887775\pi$
$283$	−2.52887e6	−1.87698	−0.938490	+	0.345305i	$0.887775\pi$
$284$	−4.17915e6	−3.07463
$285$	0	0
$286$	960407.	0.694289
$287$	−889329.	−0.637320
$288$	−126713.	−0.0900204
$289$	−1.40599e6	−0.990236
$290$	0	0
$291$	403361.	0.279229
$292$	3.59975e6	2.47067
$293$	−2.06542e6	−1.40553	−0.702764	−	0.711423i	$-0.748051\pi$
$293$	−2.06542e6	−1.40553	−0.702764	+	0.711423i	$0.748051\pi$
$294$	734100.	0.495321
$295$	0	0
$296$	−3.49026e6	−2.31541
$297$	88209.0	0.0580259
$298$	−2.38297e6	−1.55445
$299$	2.36252e6	1.52826
$300$	0	0
$301$	−1.61883e6	−1.02987
$302$	−376880.	−0.237786
$303$	745845.	0.466704
$304$	1.04573e6	0.648986
$305$	0	0
$306$	−94593.9	−0.0577510
$307$	2.17085e6	1.31457	0.657286	−	0.753641i	$-0.271704\pi$
$307$	2.17085e6	1.31457	0.657286	+	0.753641i	$0.271704\pi$
$308$	−744088.	−0.446939
$309$	−1.38327e6	−0.824161
$310$	0	0
$311$	−198576.	−0.116420	−0.0582098	−	0.998304i	$-0.518539\pi$
$311$	−198576.	−0.116420	−0.0582098	+	0.998304i	$0.518539\pi$
$312$	2.45567e6	1.42818
$313$	483762.	0.279107	0.139554	−	0.990215i	$-0.455433\pi$
$313$	483762.	0.279107	0.139554	+	0.990215i	$0.455433\pi$
$314$	−2.58387e6	−1.47893
$315$	0	0
$316$	−3.78321e6	−2.13129
$317$	1.68099e6	0.939542	0.469771	−	0.882788i	$-0.344336\pi$
$317$	1.68099e6	0.939542	0.469771	+	0.882788i	$0.344336\pi$
$318$	−866191.	−0.480337
$319$	697670.	0.383860
$320$	0	0
$321$	−374961.	−0.203106
$322$	−2.71283e6	−1.45809
$323$	97893.6	0.0522093
$324$	435494.	0.230473
$325$	0	0
$326$	−3.61768e6	−1.88532
$327$	357705.	0.184993
$328$	−3.27294e6	−1.67979
$329$	1.73696e6	0.884706
$330$	0	0
$331$	−1.18881e6	−0.596407	−0.298204	−	0.954502i	$-0.596387\pi$
$331$	−1.18881e6	−0.596407	−0.298204	+	0.954502i	$0.596387\pi$
$332$	−3.29796e6	−1.64210
$333$	829164.	0.409760
$334$	704171.	0.345392
$335$	0	0
$336$	−1.04874e6	−0.506779
$337$	−1.78561e6	−0.856470	−0.428235	−	0.903667i	$-0.640865\pi$
$337$	−1.78561e6	−0.856470	−0.428235	+	0.903667i	$0.640865\pi$
$338$	−2.66911e6	−1.27079
$339$	1.72041e6	0.813079
$340$	0	0
$341$	−7475.41	−0.00348136
$342$	−667961.	−0.308806
$343$	2.31900e6	1.06430
$344$	−5.95766e6	−2.71444
$345$	0	0
$346$	2.93299e6	1.31711
$347$	−2.71689e6	−1.21129	−0.605645	−	0.795735i	$-0.707085\pi$
$347$	−2.71689e6	−1.21129	−0.605645	+	0.795735i	$0.707085\pi$
$348$	3.44445e6	1.52465
$349$	−1.87050e6	−0.822044	−0.411022	−	0.911626i	$-0.634828\pi$
$349$	−1.87050e6	−0.822044	−0.411022	+	0.911626i	$0.634828\pi$
$350$	0	0
$351$	−583381.	−0.252746
$352$	−189288.	−0.0814265
$353$	4.27952e6	1.82793	0.913963	−	0.405798i	$-0.133006\pi$
$353$	4.27952e6	1.82793	0.913963	+	0.405798i	$0.133006\pi$
$354$	2.17822e6	0.923835
$355$	0	0
$356$	−5.84045e6	−2.44242
$357$	−98175.3	−0.0407692
$358$	5.63063e6	2.32193
$359$	−1.39681e6	−0.572007	−0.286003	−	0.958229i	$-0.592327\pi$
$359$	−1.39681e6	−0.572007	−0.286003	+	0.958229i	$0.592327\pi$
$360$	0	0
$361$	−1.78484e6	−0.720826
$362$	3.58174e6	1.43656
$363$	131769.	0.0524864
$364$	4.92112e6	1.94675
$365$	0	0
$366$	−3.02704e6	−1.18118
$367$	5.05674e6	1.95977	0.979885	−	0.199561i	$-0.0639515\pi$
$367$	5.05674e6	1.95977	0.979885	+	0.199561i	$0.0639515\pi$
$368$	−3.71320e6	−1.42932
$369$	777536.	0.297273
$370$	0	0
$371$	−898987.	−0.339093
$372$	−36906.6	−0.0138276
$373$	4.68117e6	1.74214	0.871069	−	0.491161i	$-0.163427\pi$
$373$	4.68117e6	1.74214	0.871069	+	0.491161i	$0.163427\pi$
$374$	−141307.	−0.0522377
$375$	0	0
$376$	6.39241e6	2.33182
$377$	−4.61413e6	−1.67200
$378$	669884.	0.241140
$379$	−3.76641e6	−1.34688	−0.673442	−	0.739240i	$-0.735185\pi$
$379$	−3.76641e6	−1.34688	−0.673442	+	0.739240i	$0.735185\pi$
$380$	0	0
$381$	1.91591e6	0.676181
$382$	−3.32141e6	−1.16457
$383$	−4.17276e6	−1.45354	−0.726770	−	0.686881i	$-0.758979\pi$
$383$	−4.17276e6	−1.45354	−0.726770	+	0.686881i	$0.758979\pi$
$384$	2.65829e6	0.919972
$385$	0	0
$386$	−2.84308e6	−0.971228
$387$	1.41533e6	0.480376
$388$	2.97484e6	1.00319
$389$	−3.73222e6	−1.25053	−0.625265	−	0.780413i	$-0.715009\pi$
$389$	−3.73222e6	−1.25053	−0.625265	+	0.780413i	$0.715009\pi$
$390$	0	0
$391$	−347603.	−0.114985
$392$	2.80395e6	0.921627
$393$	−1.31933e6	−0.430896
$394$	3.99272e6	1.29577
$395$	0	0
$396$	650553.	0.208471
$397$	−5.71418e6	−1.81961	−0.909804	−	0.415038i	$-0.863768\pi$
$397$	−5.71418e6	−1.81961	−0.909804	+	0.415038i	$0.863768\pi$
$398$	−5.02725e6	−1.59083
$399$	−693251.	−0.218001
$400$	0	0
$401$	3.60785e6	1.12044	0.560219	−	0.828345i	$-0.310717\pi$
$401$	3.60785e6	1.12044	0.560219	+	0.828345i	$0.310717\pi$
$402$	−445371.	−0.137454
$403$	49439.6	0.0151639
$404$	5.50070e6	1.67674
$405$	0	0
$406$	5.29830e6	1.59522
$407$	1.23863e6	0.370642
$408$	−361308.	−0.107455
$409$	−4.59452e6	−1.35810	−0.679050	−	0.734092i	$-0.737608\pi$
$409$	−4.59452e6	−1.35810	−0.679050	+	0.734092i	$0.737608\pi$
$410$	0	0
$411$	−3.01567e6	−0.880601
$412$	−1.02018e7	−2.96098
$413$	2.26069e6	0.652179
$414$	2.37181e6	0.680111
$415$	0	0
$416$	1.25188e6	0.354673
$417$	270482.	0.0761725
$418$	−997819.	−0.279326
$419$	1.92383e6	0.535343	0.267671	−	0.963510i	$-0.413746\pi$
$419$	1.92383e6	0.535343	0.267671	+	0.963510i	$0.413746\pi$
$420$	0	0
$421$	−4.28936e6	−1.17947	−0.589736	−	0.807596i	$-0.700768\pi$
$421$	−4.28936e6	−1.17947	−0.589736	+	0.807596i	$0.700768\pi$
$422$	−1.65404e6	−0.452133
$423$	−1.51861e6	−0.412663
$424$	−3.30848e6	−0.893746
$425$	0	0
$426$	5.62035e6	1.50051
$427$	−3.14165e6	−0.833851
$428$	−2.76538e6	−0.729703
$429$	−871471.	−0.228618
$430$	0	0
$431$	477444.	0.123802	0.0619012	−	0.998082i	$-0.480284\pi$
$431$	477444.	0.123802	0.0619012	+	0.998082i	$0.480284\pi$
$432$	916907.	0.236383
$433$	6.24919e6	1.60178	0.800892	−	0.598808i	$-0.204359\pi$
$433$	6.24919e6	1.60178	0.800892	+	0.598808i	$0.204359\pi$
$434$	−56770.4	−0.0144676
$435$	0	0
$436$	2.63812e6	0.664627
$437$	−2.45455e6	−0.614849
$438$	−4.84113e6	−1.20576
$439$	5.45632e6	1.35126	0.675629	−	0.737241i	$-0.263872\pi$
$439$	5.45632e6	1.35126	0.675629	+	0.737241i	$0.263872\pi$
$440$	0	0
$441$	−666120.	−0.163101
$442$	934551.	0.227534
$443$	3.75783e6	0.909762	0.454881	−	0.890552i	$-0.349682\pi$
$443$	3.75783e6	0.909762	0.454881	+	0.890552i	$0.349682\pi$
$444$	6.11519e6	1.47215
$445$	0	0
$446$	5.51520e6	1.31288
$447$	2.16230e6	0.511855
$448$	2.29134e6	0.539380
$449$	−7.17310e6	−1.67916	−0.839578	−	0.543239i	$-0.817198\pi$
$449$	−7.17310e6	−1.67916	−0.839578	+	0.543239i	$0.817198\pi$
$450$	0	0
$451$	1.16150e6	0.268893
$452$	1.26883e7	2.92116
$453$	341980.	0.0782988
$454$	5.03709e6	1.14694
$455$	0	0
$456$	−2.55133e6	−0.574585
$457$	3.21240e6	0.719513	0.359757	−	0.933046i	$-0.382860\pi$
$457$	3.21240e6	0.719513	0.359757	+	0.933046i	$0.382860\pi$
$458$	6.22080e6	1.38574
$459$	85834.2	0.0190164
$460$	0	0
$461$	−5.17808e6	−1.13479	−0.567396	−	0.823445i	$-0.692049\pi$
$461$	−5.17808e6	−1.13479	−0.567396	+	0.823445i	$0.692049\pi$
$462$	1.00069e6	0.218120
$463$	−2.95142e6	−0.639851	−0.319926	−	0.947443i	$-0.603658\pi$
$463$	−2.95142e6	−0.639851	−0.319926	+	0.947443i	$0.603658\pi$
$464$	7.25208e6	1.56375
$465$	0	0
$466$	9.48077e6	2.02245
$467$	−7.26608e6	−1.54173	−0.770865	−	0.636999i	$-0.780175\pi$
$467$	−7.26608e6	−1.54173	−0.770865	+	0.636999i	$0.780175\pi$
$468$	−4.30251e6	−0.908045
$469$	−462234.	−0.0970352
$470$	0	0
$471$	2.34460e6	0.486985
$472$	8.31989e6	1.71895
$473$	2.11426e6	0.434516
$474$	5.08786e6	1.04013
$475$	0	0
$476$	−724056.	−0.146472
$477$	785980.	0.158167
$478$	5.05103e6	1.01114
$479$	−4.09584e6	−0.815651	−0.407826	−	0.913060i	$-0.633713\pi$
$479$	−4.09584e6	−0.815651	−0.407826	+	0.913060i	$0.633713\pi$
$480$	0	0
$481$	−8.19181e6	−1.61442
$482$	6.60694e6	1.29534
$483$	2.46162e6	0.480123
$484$	971814.	0.188569
$485$	0	0
$486$	−585676.	−0.112478
$487$	−218477.	−0.0417429	−0.0208715	−	0.999782i	$-0.506644\pi$
$487$	−218477.	−0.0417429	−0.0208715	+	0.999782i	$0.506644\pi$
$488$	−1.15620e7	−2.19778
$489$	3.28267e6	0.620805
$490$	0	0
$491$	−3.49848e6	−0.654901	−0.327451	−	0.944868i	$-0.606190\pi$
$491$	−3.49848e6	−0.654901	−0.327451	+	0.944868i	$0.606190\pi$
$492$	5.73443e6	1.06802
$493$	678887.	0.125800
$494$	6.59920e6	1.21667
$495$	0	0
$496$	−77704.7	−0.0141822
$497$	5.83315e6	1.05928
$498$	4.43528e6	0.801397
$499$	−3.62715e6	−0.652100	−0.326050	−	0.945353i	$-0.605718\pi$
$499$	−3.62715e6	−0.652100	−0.326050	+	0.945353i	$0.605718\pi$
$500$	0	0
$501$	−638963.	−0.113732
$502$	−5.00893e6	−0.887127
$503$	−6.00845e6	−1.05887	−0.529435	−	0.848351i	$-0.677596\pi$
$503$	−6.00845e6	−1.05887	−0.529435	+	0.848351i	$0.677596\pi$
$504$	2.55867e6	0.448682
$505$	0	0
$506$	3.54308e6	0.615184
$507$	2.42194e6	0.418451
$508$	1.41301e7	2.42933
$509$	4.28786e6	0.733578	0.366789	−	0.930304i	$-0.380457\pi$
$509$	4.28786e6	0.733578	0.366789	+	0.930304i	$0.380457\pi$
$510$	0	0
$511$	−5.02443e6	−0.851205
$512$	1.17554e7	1.98182
$513$	606106.	0.101685
$514$	−2.01017e7	−3.35602
$515$	0	0
$516$	1.04383e7	1.72585
$517$	−2.26854e6	−0.373268
$518$	9.40648e6	1.54029
$519$	−2.66139e6	−0.433701
$520$	0	0
$521$	1.06243e7	1.71477	0.857386	−	0.514673i	$-0.172087\pi$
$521$	1.06243e7	1.71477	0.857386	+	0.514673i	$0.172087\pi$
$522$	−4.63228e6	−0.744078
$523$	−123402.	−0.0197273	−0.00986367	−	0.999951i	$-0.503140\pi$
$523$	−123402.	−0.0197273	−0.00986367	+	0.999951i	$0.503140\pi$
$524$	−9.73023e6	−1.54809
$525$	0	0
$526$	1.56066e7	2.45949
$527$	−7274.16	−0.00114092
$528$	1.36970e6	0.213816
$529$	2.27934e6	0.354136
$530$	0	0
$531$	−1.97651e6	−0.304203
$532$	−5.11282e6	−0.783216
$533$	−7.68176e6	−1.17123
$534$	7.85455e6	1.19198
$535$	0	0
$536$	−1.70113e6	−0.255756
$537$	−5.10922e6	−0.764573
$538$	1.54686e7	2.30407
$539$	−995069.	−0.147530
$540$	0	0
$541$	−5.60908e6	−0.823946	−0.411973	−	0.911196i	$-0.635160\pi$
$541$	−5.60908e6	−0.823946	−0.411973	+	0.911196i	$0.635160\pi$
$542$	−1.34171e7	−1.96183
$543$	−3.25006e6	−0.473034
$544$	−184192.	−0.0266853
$545$	0	0
$546$	−6.61819e6	−0.950075
$547$	−2.72729e6	−0.389729	−0.194864	−	0.980830i	$-0.562427\pi$
$547$	−2.72729e6	−0.389729	−0.194864	+	0.980830i	$0.562427\pi$
$548$	−2.22410e7	−3.16375
$549$	2.74673e6	0.388942
$550$	0	0
$551$	4.79387e6	0.672678
$552$	9.05933e6	1.26546
$553$	5.28050e6	0.734280
$554$	2.45190e7	3.39413
$555$	0	0
$556$	1.99484e6	0.273666
$557$	−3.50042e6	−0.478059	−0.239030	−	0.971012i	$-0.576829\pi$
$557$	−3.50042e6	−0.478059	−0.239030	+	0.971012i	$0.576829\pi$
$558$	49634.1	0.00674830
$559$	−1.39829e7	−1.89264
$560$	0	0
$561$	128221.	0.0172010
$562$	−7.63337e6	−1.01947
$563$	−1.03320e7	−1.37377	−0.686886	−	0.726766i	$-0.741023\pi$
$563$	−1.03320e7	−1.37377	−0.686886	+	0.726766i	$0.741023\pi$
$564$	−1.12000e7	−1.48258
$565$	0	0
$566$	2.50825e7	3.29101
$567$	−607851.	−0.0794034
$568$	2.14674e7	2.79195
$569$	−1.31943e7	−1.70846	−0.854229	−	0.519897i	$-0.825970\pi$
$569$	−1.31943e7	−1.70846	−0.854229	+	0.519897i	$0.825970\pi$
$570$	0	0
$571$	−5.98575e6	−0.768295	−0.384148	−	0.923272i	$-0.625505\pi$
$571$	−5.98575e6	−0.768295	−0.384148	+	0.923272i	$0.625505\pi$
$572$	−6.42721e6	−0.821358
$573$	3.01384e6	0.383472
$574$	8.82079e6	1.11745
$575$	0	0
$576$	−2.00331e6	−0.251589
$577$	−7.74177e6	−0.968056	−0.484028	−	0.875052i	$-0.660827\pi$
$577$	−7.74177e6	−0.968056	−0.484028	+	0.875052i	$0.660827\pi$
$578$	1.39453e7	1.73624
$579$	2.57981e6	0.319809
$580$	0	0
$581$	4.60321e6	0.565744
$582$	−4.00072e6	−0.489589
$583$	1.17412e6	0.143067
$584$	−1.84911e7	−2.24352
$585$	0	0
$586$	2.04858e7	2.46439
$587$	−1.19577e7	−1.43236	−0.716181	−	0.697915i	$-0.754111\pi$
$587$	−1.19577e7	−1.43236	−0.716181	+	0.697915i	$0.754111\pi$
$588$	−4.91273e6	−0.585975
$589$	−51365.5	−0.00610075
$590$	0	0
$591$	−3.62298e6	−0.426676
$592$	1.28752e7	1.50990
$593$	1.08502e7	1.26707	0.633536	−	0.773713i	$-0.281603\pi$
$593$	1.08502e7	1.26707	0.633536	+	0.773713i	$0.281603\pi$
$594$	−874899.	−0.101740
$595$	0	0
$596$	1.59472e7	1.83895
$597$	4.56171e6	0.523832
$598$	−2.34326e7	−2.67959
$599$	−1.23306e7	−1.40416	−0.702082	−	0.712096i	$-0.747746\pi$
$599$	−1.23306e7	−1.40416	−0.702082	+	0.712096i	$0.747746\pi$
$600$	0	0
$601$	1.17382e7	1.32561	0.662807	−	0.748790i	$-0.269365\pi$
$601$	1.17382e7	1.32561	0.662807	+	0.748790i	$0.269365\pi$
$602$	1.60563e7	1.80574
$603$	404129.	0.0452612
$604$	2.52215e6	0.281305
$605$	0	0
$606$	−7.39765e6	−0.818299
$607$	−9.20559e6	−1.01410	−0.507049	−	0.861917i	$-0.669264\pi$
$607$	−9.20559e6	−1.01410	−0.507049	+	0.861917i	$0.669264\pi$
$608$	−1.30064e6	−0.142692
$609$	−4.80766e6	−0.525280
$610$	0	0
$611$	1.50033e7	1.62586
$612$	633039.	0.0683206
$613$	−5.83164e6	−0.626815	−0.313408	−	0.949619i	$-0.601471\pi$
$613$	−5.83164e6	−0.626815	−0.313408	+	0.949619i	$0.601471\pi$
$614$	−2.15316e7	−2.30491
$615$	0	0
$616$	3.82221e6	0.405848
$617$	9.28462e6	0.981863	0.490932	−	0.871198i	$-0.336656\pi$
$617$	9.28462e6	0.981863	0.490932	+	0.871198i	$0.336656\pi$
$618$	1.37200e7	1.44505
$619$	2.47650e6	0.259784	0.129892	−	0.991528i	$-0.458537\pi$
$619$	2.47650e6	0.259784	0.129892	+	0.991528i	$0.458537\pi$
$620$	0	0
$621$	−2.15218e6	−0.223949
$622$	1.96957e6	0.204125
$623$	8.15193e6	0.841474
$624$	−9.05869e6	−0.931330
$625$	0	0
$626$	−4.79819e6	−0.489374
$627$	905418.	0.0919772
$628$	1.72917e7	1.74960
$629$	1.20528e6	0.121468
$630$	0	0
$631$	8.58315e6	0.858170	0.429085	−	0.903264i	$-0.358836\pi$
$631$	8.58315e6	0.858170	0.429085	+	0.903264i	$0.358836\pi$
$632$	1.94335e7	1.93534
$633$	1.50087e6	0.148880
$634$	−1.66728e7	−1.64735
$635$	0	0
$636$	5.79670e6	0.568248
$637$	6.58101e6	0.642605
$638$	−6.91982e6	−0.673044
$639$	−5.09989e6	−0.494093
$640$	0	0
$641$	4.59033e6	0.441264	0.220632	−	0.975357i	$-0.429188\pi$
$641$	4.59033e6	0.441264	0.220632	+	0.975357i	$0.429188\pi$
$642$	3.71904e6	0.356117
$643$	−9.70391e6	−0.925592	−0.462796	−	0.886465i	$-0.653154\pi$
$643$	−9.70391e6	−0.925592	−0.462796	+	0.886465i	$0.653154\pi$
$644$	1.81547e7	1.72494
$645$	0	0
$646$	−970955.	−0.0915415
$647$	9.01887e6	0.847015	0.423508	−	0.905892i	$-0.360799\pi$
$647$	9.01887e6	0.847015	0.423508	+	0.905892i	$0.360799\pi$
$648$	−2.23703e6	−0.209284
$649$	−2.95257e6	−0.275162
$650$	0	0
$651$	51513.3	0.00476395
$652$	2.42101e7	2.23038
$653$	−1.94337e6	−0.178349	−0.0891747	−	0.996016i	$-0.528423\pi$
$653$	−1.94337e6	−0.178349	−0.0891747	+	0.996016i	$0.528423\pi$
$654$	−3.54789e6	−0.324358
$655$	0	0
$656$	1.20735e7	1.09540
$657$	4.39283e6	0.397037
$658$	−1.72279e7	−1.55120
$659$	−6.36990e6	−0.571372	−0.285686	−	0.958323i	$-0.592221\pi$
$659$	−6.36990e6	−0.571372	−0.285686	+	0.958323i	$0.592221\pi$
$660$	0	0
$661$	−7.88417e6	−0.701863	−0.350931	−	0.936401i	$-0.614135\pi$
$661$	−7.88417e6	−0.701863	−0.350931	+	0.936401i	$0.614135\pi$
$662$	1.17912e7	1.04571
$663$	−848009.	−0.0749232
$664$	1.69409e7	1.49113
$665$	0	0
$666$	−8.22404e6	−0.718455
$667$	−1.70222e7	−1.48150
$668$	−4.71244e6	−0.408606
$669$	−5.00448e6	−0.432308
$670$	0	0
$671$	4.10314e6	0.351812
$672$	1.30439e6	0.111425
$673$	4.51895e6	0.384592	0.192296	−	0.981337i	$-0.438407\pi$
$673$	4.51895e6	0.384592	0.192296	+	0.981337i	$0.438407\pi$
$674$	1.77106e7	1.50170
$675$	0	0
$676$	1.78622e7	1.50337
$677$	−1.17659e7	−0.986624	−0.493312	−	0.869852i	$-0.664214\pi$
$677$	−1.17659e7	−0.986624	−0.493312	+	0.869852i	$0.664214\pi$
$678$	−1.70639e7	−1.42562
$679$	−4.15220e6	−0.345624
$680$	0	0
$681$	−4.57064e6	−0.377667
$682$	74144.7	0.00610407
$683$	1.27636e7	1.04694	0.523470	−	0.852044i	$-0.324637\pi$
$683$	1.27636e7	1.04694	0.523470	+	0.852044i	$0.324637\pi$
$684$	4.47011e6	0.365324
$685$	0	0
$686$	−2.30009e7	−1.86610
$687$	−5.64474e6	−0.456302
$688$	2.19771e7	1.77011
$689$	−7.76518e6	−0.623165
$690$	0	0
$691$	1.94008e7	1.54570	0.772849	−	0.634590i	$-0.218831\pi$
$691$	1.94008e7	1.54570	0.772849	+	0.634590i	$0.218831\pi$
$692$	−1.96281e7	−1.55816
$693$	−908024.	−0.0718231
$694$	2.69474e7	2.12382
$695$	0	0
$696$	−1.76933e7	−1.38448
$697$	1.13023e6	0.0881225
$698$	1.85525e7	1.44133
$699$	−8.60283e6	−0.665960
$700$	0	0
$701$	1.81026e7	1.39138	0.695690	−	0.718342i	$-0.255099\pi$
$701$	1.81026e7	1.39138	0.695690	+	0.718342i	$0.255099\pi$
$702$	5.78625e6	0.443154
$703$	8.51092e6	0.649514
$704$	−2.99260e6	−0.227571
$705$	0	0
$706$	−4.24463e7	−3.20500
$707$	−7.67773e6	−0.577676
$708$	−1.45771e7	−1.09292
$709$	1.55411e7	1.16109	0.580547	−	0.814227i	$-0.302839\pi$
$709$	1.55411e7	1.16109	0.580547	+	0.814227i	$0.302839\pi$
$710$	0	0
$711$	−4.61671e6	−0.342499
$712$	3.00010e7	2.21787
$713$	182390.	0.0134362
$714$	973750.	0.0714828
$715$	0	0
$716$	−3.76812e7	−2.74689
$717$	−4.58329e6	−0.332951
$718$	1.38542e7	1.00293
$719$	8.88792e6	0.641177	0.320589	−	0.947219i	$-0.396119\pi$
$719$	8.88792e6	0.641177	0.320589	+	0.947219i	$0.396119\pi$
$720$	0	0
$721$	1.42394e7	1.02013
$722$	1.77029e7	1.26386
$723$	−5.99512e6	−0.426533
$724$	−2.39697e7	−1.69948
$725$	0	0
$726$	−1.30695e6	−0.0920273
$727$	5.73697e6	0.402575	0.201287	−	0.979532i	$-0.435487\pi$
$727$	5.73697e6	0.402575	0.201287	+	0.979532i	$0.435487\pi$
$728$	−2.52787e7	−1.76777
$729$	531441.	0.0370370
$730$	0	0
$731$	2.05734e6	0.142401
$732$	2.02575e7	1.39736
$733$	−1.77002e6	−0.121680	−0.0608399	−	0.998148i	$-0.519378\pi$
$733$	−1.77002e6	−0.121680	−0.0608399	+	0.998148i	$0.519378\pi$
$734$	−5.01551e7	−3.43618
$735$	0	0
$736$	4.61836e6	0.314263
$737$	603698.	0.0409403
$738$	−7.71198e6	−0.521224
$739$	1.53957e7	1.03702	0.518512	−	0.855070i	$-0.326486\pi$
$739$	1.53957e7	1.03702	0.518512	+	0.855070i	$0.326486\pi$
$740$	0	0
$741$	−5.98810e6	−0.400630
$742$	8.91658e6	0.594550
$743$	−1.65861e7	−1.10223	−0.551115	−	0.834429i	$-0.685798\pi$
$743$	−1.65861e7	−1.10223	−0.551115	+	0.834429i	$0.685798\pi$
$744$	189581.	0.0125563
$745$	0	0
$746$	−4.64301e7	−3.05459
$747$	−4.02456e6	−0.263886
$748$	945650.	0.0617983
$749$	3.85985e6	0.251400
$750$	0	0
$751$	9.40157e6	0.608276	0.304138	−	0.952628i	$-0.401632\pi$
$751$	9.40157e6	0.608276	0.304138	+	0.952628i	$0.401632\pi$
$752$	−2.35808e7	−1.52060
$753$	4.54509e6	0.292116
$754$	4.57651e7	2.93161
$755$	0	0
$756$	−4.48298e6	−0.285274
$757$	−1.22686e7	−0.778134	−0.389067	−	0.921209i	$-0.627203\pi$
$757$	−1.22686e7	−0.778134	−0.389067	+	0.921209i	$0.627203\pi$
$758$	3.73571e7	2.36157
$759$	−3.21498e6	−0.202570
$760$	0	0
$761$	−2.14104e7	−1.34018	−0.670090	−	0.742280i	$-0.733744\pi$
$761$	−2.14104e7	−1.34018	−0.670090	+	0.742280i	$0.733744\pi$
$762$	−1.90029e7	−1.18559
$763$	−3.68221e6	−0.228980
$764$	2.22275e7	1.37771
$765$	0	0
$766$	4.13875e7	2.54857
$767$	1.95272e7	1.19854
$768$	−1.92433e7	−1.17727
$769$	8.34064e6	0.508609	0.254304	−	0.967124i	$-0.418154\pi$
$769$	8.34064e6	0.508609	0.254304	+	0.967124i	$0.418154\pi$
$770$	0	0
$771$	1.82402e7	1.10508
$772$	1.90264e7	1.14898
$773$	−3.00417e7	−1.80832	−0.904162	−	0.427191i	$-0.859503\pi$
$773$	−3.00417e7	−1.80832	−0.904162	+	0.427191i	$0.859503\pi$
$774$	−1.40379e7	−0.842269
$775$	0	0
$776$	−1.52811e7	−0.910960
$777$	−8.53541e6	−0.507192
$778$	3.70180e7	2.19262
$779$	7.98099e6	0.471209
$780$	0	0
$781$	−7.61836e6	−0.446924
$782$	3.44769e6	0.201610
$783$	4.20332e6	0.245012
$784$	−1.03434e7	−0.601001
$785$	0	0
$786$	1.30857e7	0.755513
$787$	1.55411e7	0.894427	0.447213	−	0.894427i	$-0.352416\pi$
$787$	1.55411e7	0.894427	0.447213	+	0.894427i	$0.352416\pi$
$788$	−2.67200e7	−1.53292
$789$	−1.41614e7	−0.809868
$790$	0	0
$791$	−1.77099e7	−1.00641
$792$	−3.34174e6	−0.189304
$793$	−2.71366e7	−1.53240
$794$	5.66760e7	3.19042
$795$	0	0
$796$	3.36432e7	1.88198
$797$	2.13571e7	1.19096	0.595478	−	0.803371i	$-0.296963\pi$
$797$	2.13571e7	1.19096	0.595478	+	0.803371i	$0.296963\pi$
$798$	6.87600e6	0.382233
$799$	−2.20747e6	−0.122328
$800$	0	0
$801$	−7.12720e6	−0.392498
$802$	−3.57844e7	−1.96453
$803$	6.56213e6	0.359134
$804$	2.98050e6	0.162611
$805$	0	0
$806$	−490365.	−0.0265878
$807$	−1.40362e7	−0.758692
$808$	−2.82559e7	−1.52258
$809$	9.58699e6	0.515004	0.257502	−	0.966278i	$-0.417101\pi$
$809$	9.58699e6	0.515004	0.257502	+	0.966278i	$0.417101\pi$
$810$	0	0
$811$	2.27762e6	0.121599	0.0607995	−	0.998150i	$-0.480635\pi$
$811$	2.27762e6	0.121599	0.0607995	+	0.998150i	$0.480635\pi$
$812$	−3.54572e7	−1.88718
$813$	1.21747e7	0.645998
$814$	−1.22853e7	−0.649867
$815$	0	0
$816$	1.33282e6	0.0700725
$817$	1.45276e7	0.761447
$818$	4.55706e7	2.38123
$819$	6.00533e6	0.312844
$820$	0	0
$821$	2.31851e7	1.20047	0.600234	−	0.799824i	$-0.295074\pi$
$821$	2.31851e7	1.20047	0.600234	+	0.799824i	$0.295074\pi$
$822$	2.99109e7	1.54401
$823$	−3.30629e7	−1.70153	−0.850767	−	0.525542i	$-0.823862\pi$
$823$	−3.30629e7	−1.70153	−0.850767	+	0.525542i	$0.823862\pi$
$824$	5.24045e7	2.68875
$825$	0	0
$826$	−2.24226e7	−1.14350
$827$	−2.84571e7	−1.44686	−0.723431	−	0.690396i	$-0.757436\pi$
$827$	−2.84571e7	−1.44686	−0.723431	+	0.690396i	$0.757436\pi$
$828$	−1.58726e7	−0.804585
$829$	−3.34334e7	−1.68964	−0.844821	−	0.535050i	$-0.820293\pi$
$829$	−3.34334e7	−1.68964	−0.844821	+	0.535050i	$0.820293\pi$
$830$	0	0
$831$	−2.22485e7	−1.11763
$832$	1.97919e7	0.991243
$833$	−968279.	−0.0483491
$834$	−2.68277e6	−0.133558
$835$	0	0
$836$	6.67758e6	0.330448
$837$	−45037.8	−0.00222210
$838$	−1.90815e7	−0.938646
$839$	−1.79019e7	−0.877999	−0.438999	−	0.898487i	$-0.644667\pi$
$839$	−1.79019e7	−0.877999	−0.438999	+	0.898487i	$0.644667\pi$
$840$	0	0
$841$	1.27341e7	0.620837
$842$	4.25440e7	2.06803
$843$	6.92650e6	0.335695
$844$	1.10691e7	0.534882
$845$	0	0
$846$	1.50623e7	0.723545
$847$	−1.35643e6	−0.0649665
$848$	1.22046e7	0.582820
$849$	−2.27598e7	−1.08368
$850$	0	0
$851$	−3.02208e7	−1.43048
$852$	−3.76124e7	−1.77514
$853$	2.79631e7	1.31587	0.657934	−	0.753076i	$-0.271431\pi$
$853$	2.79631e7	1.31587	0.657934	+	0.753076i	$0.271431\pi$
$854$	3.11604e7	1.46204
$855$	0	0
$856$	1.42052e7	0.662615
$857$	−9.24668e6	−0.430065	−0.215032	−	0.976607i	$-0.568986\pi$
$857$	−9.24668e6	−0.430065	−0.215032	+	0.976607i	$0.568986\pi$
$858$	8.64366e6	0.400848
$859$	1.46860e7	0.679081	0.339540	−	0.940591i	$-0.389728\pi$
$859$	1.46860e7	0.679081	0.339540	+	0.940591i	$0.389728\pi$
$860$	0	0
$861$	−8.00396e6	−0.367957
$862$	−4.73551e6	−0.217070
$863$	−1.14915e7	−0.525231	−0.262615	−	0.964901i	$-0.584585\pi$
$863$	−1.14915e7	−0.525231	−0.262615	+	0.964901i	$0.584585\pi$
$864$	−1.14042e6	−0.0519733
$865$	0	0
$866$	−6.19825e7	−2.80850
$867$	−1.26539e7	−0.571713
$868$	379917.	0.0171155
$869$	−6.89657e6	−0.309802
$870$	0	0
$871$	−3.99263e6	−0.178326
$872$	−1.35514e7	−0.603523
$873$	3.63025e6	0.161213
$874$	2.43454e7	1.07805
$875$	0	0
$876$	3.23977e7	1.42644
$877$	−2.92652e7	−1.28485	−0.642426	−	0.766348i	$-0.722072\pi$
$877$	−2.92652e7	−1.28485	−0.642426	+	0.766348i	$0.722072\pi$
$878$	−5.41184e7	−2.36924
$879$	−1.85888e7	−0.811481
$880$	0	0
$881$	−6.92950e6	−0.300789	−0.150395	−	0.988626i	$-0.548054\pi$
$881$	−6.92950e6	−0.300789	−0.150395	+	0.988626i	$0.548054\pi$
$882$	6.60690e6	0.285974
$883$	−4.40887e7	−1.90294	−0.951470	−	0.307740i	$-0.900427\pi$
$883$	−4.40887e7	−1.90294	−0.951470	+	0.307740i	$0.900427\pi$
$884$	−6.25418e6	−0.269178
$885$	0	0
$886$	−3.72719e7	−1.59514
$887$	−4.26047e7	−1.81823	−0.909115	−	0.416545i	$-0.863241\pi$
$887$	−4.26047e7	−1.81823	−0.909115	+	0.416545i	$0.863241\pi$
$888$	−3.14123e7	−1.33680
$889$	−1.97224e7	−0.836961
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.69087e7	−1.55316
$893$	−1.55877e7	−0.654115
$894$	−2.14467e7	−0.897464
$895$	0	0
$896$	−2.73645e7	−1.13872
$897$	2.12627e7	0.882342
$898$	7.11462e7	2.94416
$899$	−356217.	−0.0146999
$900$	0	0
$901$	1.14251e6	0.0468864
$902$	−1.15204e7	−0.471465
$903$	−1.45694e7	−0.594598
$904$	−6.51767e7	−2.65260
$905$	0	0
$906$	−3.39192e6	−0.137286
$907$	−2.03750e7	−0.822392	−0.411196	−	0.911547i	$-0.634889\pi$
$907$	−2.03750e7	−0.822392	−0.411196	+	0.911547i	$0.634889\pi$
$908$	−3.37091e7	−1.35685
$909$	6.71260e6	0.269452
$910$	0	0
$911$	1.25015e7	0.499077	0.249538	−	0.968365i	$-0.419721\pi$
$911$	1.25015e7	0.499077	0.249538	+	0.968365i	$0.419721\pi$
$912$	9.41156e6	0.374692
$913$	−6.01200e6	−0.238694
$914$	−3.18621e7	−1.26156
$915$	0	0
$916$	−4.16307e7	−1.63936
$917$	1.35812e7	0.533353
$918$	−851345.	−0.0333425
$919$	2.12364e7	0.829455	0.414728	−	0.909946i	$-0.363877\pi$
$919$	2.12364e7	0.829455	0.414728	+	0.909946i	$0.363877\pi$
$920$	0	0
$921$	1.95377e7	0.758969
$922$	5.13586e7	1.98969
$923$	5.03850e7	1.94669
$924$	−6.69680e6	−0.258040
$925$	0	0
$926$	2.92736e7	1.12189
$927$	−1.24495e7	−0.475829
$928$	−9.01990e6	−0.343821
$929$	2.38747e7	0.907608	0.453804	−	0.891102i	$-0.350067\pi$
$929$	2.38747e7	0.907608	0.453804	+	0.891102i	$0.350067\pi$
$930$	0	0
$931$	−6.83737e6	−0.258532
$932$	−6.34470e7	−2.39261
$933$	−1.78719e6	−0.0672149
$934$	7.20685e7	2.70320
$935$	0	0
$936$	2.21010e7	0.824561
$937$	1.33814e7	0.497914	0.248957	−	0.968515i	$-0.419912\pi$
$937$	1.33814e7	0.497914	0.248957	+	0.968515i	$0.419912\pi$
$938$	4.58465e6	0.170137
$939$	4.35386e6	0.161143
$940$	0	0
$941$	5.91644e6	0.217814	0.108907	−	0.994052i	$-0.465265\pi$
$941$	5.91644e6	0.217814	0.108907	+	0.994052i	$0.465265\pi$
$942$	−2.32548e7	−0.853858
$943$	−2.83391e7	−1.03778
$944$	−3.06911e7	−1.12094
$945$	0	0
$946$	−2.09703e7	−0.761861
$947$	4.78035e7	1.73215	0.866074	−	0.499916i	$-0.166636\pi$
$947$	4.78035e7	1.73215	0.866074	+	0.499916i	$0.166636\pi$
$948$	−3.40489e7	−1.23050
$949$	−4.33995e7	−1.56430
$950$	0	0
$951$	1.51289e7	0.542445
$952$	3.71931e6	0.133006
$953$	3.75680e7	1.33994	0.669971	−	0.742387i	$-0.266306\pi$
$953$	3.75680e7	1.33994	0.669971	+	0.742387i	$0.266306\pi$
$954$	−7.79572e6	−0.277323
$955$	0	0
$956$	−3.38024e7	−1.19620
$957$	6.27903e6	0.221622
$958$	4.06245e7	1.43013
$959$	3.10433e7	1.08999
$960$	0	0
$961$	−2.86253e7	−0.999867
$962$	8.12503e7	2.83066
$963$	−3.37465e6	−0.117263
$964$	−4.42148e7	−1.53241
$965$	0	0
$966$	−2.44155e7	−0.841826
$967$	2.07684e7	0.714228	0.357114	−	0.934061i	$-0.383761\pi$
$967$	2.07684e7	0.714228	0.357114	+	0.934061i	$0.383761\pi$
$968$	−4.99199e6	−0.171232
$969$	881042.	0.0301431
$970$	0	0
$971$	1.39451e7	0.474650	0.237325	−	0.971430i	$-0.423729\pi$
$971$	1.39451e7	0.474650	0.237325	+	0.971430i	$0.423729\pi$
$972$	3.91945e6	0.133064
$973$	−2.78434e6	−0.0942846
$974$	2.16696e6	0.0731902
$975$	0	0
$976$	4.26510e7	1.43319
$977$	−2.07797e7	−0.696471	−0.348236	−	0.937407i	$-0.613219\pi$
$977$	−2.07797e7	−0.696471	−0.348236	+	0.937407i	$0.613219\pi$
$978$	−3.25591e7	−1.08849
$979$	−1.06468e7	−0.355028
$980$	0	0
$981$	3.21934e6	0.106806
$982$	3.46996e7	1.14828
$983$	−1.16598e7	−0.384864	−0.192432	−	0.981310i	$-0.561637\pi$
$983$	−1.16598e7	−0.384864	−0.192432	+	0.981310i	$0.561637\pi$
$984$	−2.94565e7	−0.969824
$985$	0	0
$986$	−6.73353e6	−0.220572
$987$	1.56326e7	0.510785
$988$	−4.41630e7	−1.43935
$989$	−5.15851e7	−1.67700
$990$	0	0
$991$	−2.35287e7	−0.761050	−0.380525	−	0.924771i	$-0.624257\pi$
$991$	−2.35287e7	−0.761050	−0.380525	+	0.924771i	$0.624257\pi$
$992$	96646.7	0.00311823
$993$	−1.06993e7	−0.344336
$994$	−5.78559e7	−1.85730
$995$	0	0
$996$	−2.96817e7	−0.948069
$997$	4.85989e7	1.54842	0.774210	−	0.632929i	$-0.218147\pi$
$997$	4.85989e7	1.54842	0.774210	+	0.632929i	$0.218147\pi$
$998$	3.59758e7	1.14336
$999$	7.46247e6	0.236575

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-9.91848 q^{2} +9.00000 q^{3} +66.3762 q^{4} -89.2663 q^{6} -92.6461 q^{7} -340.959 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.91848 q^{2} +9.00000 q^{3} +66.3762 q^{4} -89.2663 q^{6} -92.6461 q^{7} -340.959 q^{8} +81.0000 q^{9} +121.000 q^{11} +597.386 q^{12} -800.249 q^{13} +918.908 q^{14} +1257.76 q^{16} +117.742 q^{17} -803.397 q^{18} +831.422 q^{19} -833.814 q^{21} -1200.14 q^{22} -2952.23 q^{23} -3068.63 q^{24} +7937.25 q^{26} +729.000 q^{27} -6149.49 q^{28} +5765.87 q^{29} -61.7803 q^{31} -1564.36 q^{32} +1089.00 q^{33} -1167.83 q^{34} +5376.47 q^{36} +10236.6 q^{37} -8246.44 q^{38} -7202.24 q^{39} +9599.21 q^{41} +8270.17 q^{42} +17473.2 q^{43} +8031.52 q^{44} +29281.7 q^{46} -18748.3 q^{47} +11319.8 q^{48} -8223.71 q^{49} +1059.68 q^{51} -53117.5 q^{52} +9703.45 q^{53} -7230.57 q^{54} +31588.5 q^{56} +7482.79 q^{57} -57188.6 q^{58} -24401.4 q^{59} +33910.3 q^{61} +612.766 q^{62} -7504.33 q^{63} -24732.2 q^{64} -10801.2 q^{66} +4989.24 q^{67} +7815.29 q^{68} -26570.1 q^{69} -62961.6 q^{71} -27617.7 q^{72} +54232.5 q^{73} -101531. q^{74} +55186.6 q^{76} -11210.2 q^{77} +71435.2 q^{78} -56996.4 q^{79} +6561.00 q^{81} -95209.6 q^{82} -49685.9 q^{83} -55345.4 q^{84} -173308. q^{86} +51892.8 q^{87} -41256.1 q^{88} -87990.1 q^{89} +74139.9 q^{91} -195958. q^{92} -556.022 q^{93} +185954. q^{94} -14079.3 q^{96} +44817.9 q^{97} +81566.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22} - 1220 q^{23} - 2079 q^{24} + 17062 q^{26} + 2187 q^{27} - 11120 q^{28} + 4402 q^{29} - 10688 q^{31} - 12439 q^{32} + 3267 q^{33} - 4094 q^{34} + 5913 q^{36} + 8190 q^{37} - 13792 q^{38} + 810 q^{39} + 5974 q^{41} - 7056 q^{42} - 18868 q^{43} + 8833 q^{44} + 46220 q^{46} - 55500 q^{47} - 3735 q^{48} + 1907 q^{49} - 17406 q^{51} - 27330 q^{52} - 9206 q^{53} - 5103 q^{54} + 73248 q^{56} + 18756 q^{57} - 15366 q^{58} - 59196 q^{59} + 79902 q^{61} - 64616 q^{62} - 7452 q^{63} + 2129 q^{64} - 7623 q^{66} - 4468 q^{67} + 1218 q^{68} - 10980 q^{69} - 75164 q^{71} - 18711 q^{72} + 61290 q^{73} - 56766 q^{74} + 37816 q^{76} - 11132 q^{77} + 153558 q^{78} - 83564 q^{79} + 19683 q^{81} - 147410 q^{82} - 74764 q^{83} - 100080 q^{84} - 253432 q^{86} + 39618 q^{87} - 27951 q^{88} + 37342 q^{89} - 126488 q^{91} - 148164 q^{92} - 96192 q^{93} + 59252 q^{94} - 111951 q^{96} - 33486 q^{97} + 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9 + 363 * q^11 + 657 * q^12 + 90 * q^13 - 784 * q^14 - 415 * q^16 - 1934 * q^17 - 567 * q^18 + 2084 * q^19 - 828 * q^21 - 847 * q^22 - 1220 * q^23 - 2079 * q^24 + 17062 * q^26 + 2187 * q^27 - 11120 * q^28 + 4402 * q^29 - 10688 * q^31 - 12439 * q^32 + 3267 * q^33 - 4094 * q^34 + 5913 * q^36 + 8190 * q^37 - 13792 * q^38 + 810 * q^39 + 5974 * q^41 - 7056 * q^42 - 18868 * q^43 + 8833 * q^44 + 46220 * q^46 - 55500 * q^47 - 3735 * q^48 + 1907 * q^49 - 17406 * q^51 - 27330 * q^52 - 9206 * q^53 - 5103 * q^54 + 73248 * q^56 + 18756 * q^57 - 15366 * q^58 - 59196 * q^59 + 79902 * q^61 - 64616 * q^62 - 7452 * q^63 + 2129 * q^64 - 7623 * q^66 - 4468 * q^67 + 1218 * q^68 - 10980 * q^69 - 75164 * q^71 - 18711 * q^72 + 61290 * q^73 - 56766 * q^74 + 37816 * q^76 - 11132 * q^77 + 153558 * q^78 - 83564 * q^79 + 19683 * q^81 - 147410 * q^82 - 74764 * q^83 - 100080 * q^84 - 253432 * q^86 + 39618 * q^87 - 27951 * q^88 + 37342 * q^89 - 126488 * q^91 - 148164 * q^92 - 96192 * q^93 + 59252 * q^94 - 111951 * q^96 - 33486 * q^97 + 95249 * q^98 + 29403 * q^99

Introduction

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