LMFDB - Embedded newform 825.6.a.e.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.15207$ 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)

$f(q)$	$=$	$q-4.15207 q^{2} +9.00000 q^{3} -14.7603 q^{4} -37.3686 q^{6} +76.4793 q^{7} +194.152 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.15207 q^{2} +9.00000 q^{3} -14.7603 q^{4} -37.3686 q^{6} +76.4793 q^{7} +194.152 q^{8} +81.0000 q^{9} -121.000 q^{11} -132.843 q^{12} -169.779 q^{13} -317.547 q^{14} -333.802 q^{16} +0.875959 q^{17} -336.317 q^{18} -817.825 q^{19} +688.314 q^{21} +502.400 q^{22} -749.657 q^{23} +1747.37 q^{24} +704.932 q^{26} +729.000 q^{27} -1128.86 q^{28} +6040.65 q^{29} -1475.69 q^{31} -4826.90 q^{32} -1089.00 q^{33} -3.63704 q^{34} -1195.59 q^{36} +15862.7 q^{37} +3395.66 q^{38} -1528.01 q^{39} -7626.27 q^{41} -2857.93 q^{42} +18279.5 q^{43} +1786.00 q^{44} +3112.62 q^{46} +12878.1 q^{47} -3004.22 q^{48} -10957.9 q^{49} +7.88363 q^{51} +2505.99 q^{52} -21763.5 q^{53} -3026.86 q^{54} +14848.6 q^{56} -7360.42 q^{57} -25081.2 q^{58} -1168.45 q^{59} +14026.3 q^{61} +6127.16 q^{62} +6194.83 q^{63} +30723.3 q^{64} +4521.60 q^{66} -36954.6 q^{67} -12.9295 q^{68} -6746.91 q^{69} -37513.1 q^{71} +15726.3 q^{72} -80452.0 q^{73} -65862.9 q^{74} +12071.4 q^{76} -9254.00 q^{77} +6344.39 q^{78} -62139.9 q^{79} +6561.00 q^{81} +31664.8 q^{82} +11905.8 q^{83} -10159.7 q^{84} -75897.7 q^{86} +54365.9 q^{87} -23492.4 q^{88} +146348. q^{89} -12984.5 q^{91} +11065.2 q^{92} -13281.2 q^{93} -53470.8 q^{94} -43442.1 q^{96} +138129. q^{97} +45498.0 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q + 5 * q^2 + 18 * q^3 + 37 * q^4 + 45 * q^6 + 286 * q^7 + 375 * q^8 + 162 * q^9	\( 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9} - 242 q^{11} + 333 q^{12} + 166 q^{13} + 1600 q^{14} - 335 q^{16} + 800 q^{17} + 405 q^{18} - 1476 q^{19} + 2574 q^{21} - 605 q^{22} + 3370 q^{23} + 3375 q^{24} + 3778 q^{26} + 1458 q^{27} + 9716 q^{28} + 6600 q^{29} - 7528 q^{31} - 10625 q^{32} - 2178 q^{33} + 7310 q^{34} + 2997 q^{36} + 29916 q^{37} - 2628 q^{38} + 1494 q^{39} - 5780 q^{41} + 14400 q^{42} + 16656 q^{43} - 4477 q^{44} + 40816 q^{46} - 7850 q^{47} - 3015 q^{48} + 16134 q^{49} + 7200 q^{51} + 19886 q^{52} - 14178 q^{53} + 3645 q^{54} + 52740 q^{56} - 13284 q^{57} - 19962 q^{58} + 17300 q^{59} - 2946 q^{61} - 49264 q^{62} + 23166 q^{63} - 22303 q^{64} - 5445 q^{66} - 31336 q^{67} + 41350 q^{68} + 30330 q^{69} - 33810 q^{71} + 30375 q^{72} - 60644 q^{73} + 62754 q^{74} - 21996 q^{76} - 34606 q^{77} + 34002 q^{78} + 1870 q^{79} + 13122 q^{81} + 48562 q^{82} + 58296 q^{83} + 87444 q^{84} - 90756 q^{86} + 59400 q^{87} - 45375 q^{88} + 92388 q^{89} + 57368 q^{91} + 224300 q^{92} - 67752 q^{93} - 243176 q^{94} - 95625 q^{96} - 7120 q^{97} + 293445 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 18 * q^3 + 37 * q^4 + 45 * q^6 + 286 * q^7 + 375 * q^8 + 162 * q^9 - 242 * q^11 + 333 * q^12 + 166 * q^13 + 1600 * q^14 - 335 * q^16 + 800 * q^17 + 405 * q^18 - 1476 * q^19 + 2574 * q^21 - 605 * q^22 + 3370 * q^23 + 3375 * q^24 + 3778 * q^26 + 1458 * q^27 + 9716 * q^28 + 6600 * q^29 - 7528 * q^31 - 10625 * q^32 - 2178 * q^33 + 7310 * q^34 + 2997 * q^36 + 29916 * q^37 - 2628 * q^38 + 1494 * q^39 - 5780 * q^41 + 14400 * q^42 + 16656 * q^43 - 4477 * q^44 + 40816 * q^46 - 7850 * q^47 - 3015 * q^48 + 16134 * q^49 + 7200 * q^51 + 19886 * q^52 - 14178 * q^53 + 3645 * q^54 + 52740 * q^56 - 13284 * q^57 - 19962 * q^58 + 17300 * q^59 - 2946 * q^61 - 49264 * q^62 + 23166 * q^63 - 22303 * q^64 - 5445 * q^66 - 31336 * q^67 + 41350 * q^68 + 30330 * q^69 - 33810 * q^71 + 30375 * q^72 - 60644 * q^73 + 62754 * q^74 - 21996 * q^76 - 34606 * q^77 + 34002 * q^78 + 1870 * q^79 + 13122 * q^81 + 48562 * q^82 + 58296 * q^83 + 87444 * q^84 - 90756 * q^86 + 59400 * q^87 - 45375 * q^88 + 92388 * q^89 + 57368 * q^91 + 224300 * q^92 - 67752 * q^93 - 243176 * q^94 - 95625 * q^96 - 7120 * q^97 + 293445 * q^98 - 19602 * 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 $

$2$	−4.15207	−0.733989	−0.366994	−	0.930223i	$-0.619613\pi$
$2$	−4.15207	−0.733989	−0.366994	+	0.930223i	$0.619613\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−14.7603	−0.461261
$5$	0	0
$5$	0	0
$6$	−37.3686	−0.423769
$7$	76.4793	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$7$	76.4793	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$8$	194.152	1.07255
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−132.843	−0.266309
$13$	−169.779	−0.278628	−0.139314	−	0.990248i	$-0.544490\pi$
$13$	−169.779	−0.278628	−0.139314	+	0.990248i	$0.544490\pi$
$14$	−317.547	−0.433000
$15$	0	0
$16$	−333.802	−0.325978
$17$	0.875959	0.000735126 0	0.000367563	−	1.00000i	$-0.499883\pi$
$17$	0.875959	0.000735126 0	0.000367563		1.00000i	$0.499883\pi$
$18$	−336.317	−0.244663
$19$	−817.825	−0.519728	−0.259864	−	0.965645i	$-0.583678\pi$
$19$	−817.825	−0.519728	−0.259864	+	0.965645i	$0.583678\pi$
$20$	0	0
$21$	688.314	0.340595
$22$	502.400	0.221306
$23$	−749.657	−0.295490	−0.147745	−	0.989025i	$-0.547201\pi$
$23$	−749.657	−0.295490	−0.147745	+	0.989025i	$0.547201\pi$
$24$	1747.37	0.619236
$25$	0	0
$26$	704.932	0.204510
$27$	729.000	0.192450
$28$	−1128.86	−0.272110
$29$	6040.65	1.33379	0.666897	−	0.745150i	$-0.267622\pi$
$29$	6040.65	1.33379	0.666897	+	0.745150i	$0.267622\pi$
$30$	0	0
$31$	−1475.69	−0.275798	−0.137899	−	0.990446i	$-0.544035\pi$
$31$	−1475.69	−0.275798	−0.137899	+	0.990446i	$0.544035\pi$
$32$	−4826.90	−0.833284
$33$	−1089.00	−0.174078
$34$	−3.63704	−0.000539574 0
$35$	0	0
$36$	−1195.59	−0.153754
$37$	15862.7	1.90490	0.952450	−	0.304694i	$-0.0985543\pi$
$37$	15862.7	1.90490	0.952450	+	0.304694i	$0.0985543\pi$
$38$	3395.66	0.381475
$39$	−1528.01	−0.160866
$40$	0	0
$41$	−7626.27	−0.708521	−0.354260	−	0.935147i	$-0.615267\pi$
$41$	−7626.27	−0.708521	−0.354260	+	0.935147i	$0.615267\pi$
$42$	−2857.93	−0.249993
$43$	18279.5	1.50762	0.753812	−	0.657090i	$-0.228213\pi$
$43$	18279.5	1.50762	0.753812	+	0.657090i	$0.228213\pi$
$44$	1786.00	0.139075
$45$	0	0
$46$	3112.62	0.216886
$47$	12878.1	0.850370	0.425185	−	0.905106i	$-0.360209\pi$
$47$	12878.1	0.850370	0.425185	+	0.905106i	$0.360209\pi$
$48$	−3004.22	−0.188204
$49$	−10957.9	−0.651985
$50$	0	0
$51$	7.88363	0.000424425 0
$52$	2505.99	0.128520
$53$	−21763.5	−1.06424	−0.532118	−	0.846670i	$-0.678604\pi$
$53$	−21763.5	−1.06424	−0.532118	+	0.846670i	$0.678604\pi$
$54$	−3026.86	−0.141256
$55$	0	0
$56$	14848.6	0.632726
$57$	−7360.42	−0.300065
$58$	−25081.2	−0.978990
$59$	−1168.45	−0.0436999	−0.0218500	−	0.999761i	$-0.506956\pi$
$59$	−1168.45	−0.0436999	−0.0218500	+	0.999761i	$0.506956\pi$
$60$	0	0
$61$	14026.3	0.482635	0.241318	−	0.970446i	$-0.422420\pi$
$61$	14026.3	0.482635	0.241318	+	0.970446i	$0.422420\pi$
$62$	6127.16	0.202432
$63$	6194.83	0.196643
$64$	30723.3	0.937600
$65$	0	0
$66$	4521.60	0.127771
$67$	−36954.6	−1.00573	−0.502865	−	0.864365i	$-0.667721\pi$
$67$	−36954.6	−1.00573	−0.502865	+	0.864365i	$0.667721\pi$
$68$	−12.9295	−0.000339084 0
$69$	−6746.91	−0.170601
$70$	0	0
$71$	−37513.1	−0.883155	−0.441578	−	0.897223i	$-0.645581\pi$
$71$	−37513.1	−0.883155	−0.441578	+	0.897223i	$0.645581\pi$
$72$	15726.3	0.357516
$73$	−80452.0	−1.76697	−0.883486	−	0.468458i	$-0.844810\pi$
$73$	−80452.0	−1.76697	−0.883486	+	0.468458i	$0.844810\pi$
$74$	−65862.9	−1.39818
$75$	0	0
$76$	12071.4	0.239730
$77$	−9254.00	−0.177870
$78$	6344.39	0.118074
$79$	−62139.9	−1.12022	−0.560109	−	0.828419i	$-0.689241\pi$
$79$	−62139.9	−1.12022	−0.560109	+	0.828419i	$0.689241\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	31664.8	0.520046
$83$	11905.8	0.189699	0.0948495	−	0.995492i	$-0.469763\pi$
$83$	11905.8	0.189699	0.0948495	+	0.995492i	$0.469763\pi$
$84$	−10159.7	−0.157103
$85$	0	0
$86$	−75897.7	−1.10658
$87$	54365.9	0.770066
$88$	−23492.4	−0.323386
$89$	146348.	1.95844	0.979220	−	0.202800i	$-0.0650041\pi$
$89$	146348.	1.95844	0.979220	+	0.202800i	$0.0650041\pi$
$90$	0	0
$91$	−12984.5	−0.164370
$92$	11065.2	0.136298
$93$	−13281.2	−0.159232
$94$	−53470.8	−0.624162
$95$	0	0
$96$	−43442.1	−0.481097
$97$	138129.	1.49058	0.745291	−	0.666740i	$-0.232311\pi$
$97$	138129.	1.49058	0.745291	+	0.666740i	$0.232311\pi$
$98$	45498.0	0.478550
$99$	−9801.00	−0.100504
$100$	0	0
$101$	36615.1	0.357155	0.178577	−	0.983926i	$-0.442850\pi$
$101$	36615.1	0.357155	0.178577	+	0.983926i	$0.442850\pi$
$102$	−32.7334	−0.000311523 0
$103$	−45198.5	−0.419789	−0.209895	−	0.977724i	$-0.567312\pi$
$103$	−45198.5	−0.419789	−0.209895	+	0.977724i	$0.567312\pi$
$104$	−32962.9	−0.298842
$105$	0	0
$106$	90363.4	0.781138
$107$	28086.8	0.237161	0.118580	−	0.992944i	$-0.462166\pi$
$107$	28086.8	0.237161	0.118580	+	0.992944i	$0.462166\pi$
$108$	−10760.3	−0.0887696
$109$	−33852.9	−0.272916	−0.136458	−	0.990646i	$-0.543572\pi$
$109$	−33852.9	−0.272916	−0.136458	+	0.990646i	$0.543572\pi$
$110$	0	0
$111$	142764.	1.09979
$112$	−25528.9	−0.192304
$113$	−168559.	−1.24181	−0.620904	−	0.783886i	$-0.713234\pi$
$113$	−168559.	−1.24181	−0.620904	+	0.783886i	$0.713234\pi$
$114$	30561.0	0.220244
$115$	0	0
$116$	−89162.1	−0.615227
$117$	−13752.1	−0.0928759
$118$	4851.49	0.0320752
$119$	66.9928	0.000433671 0
$120$	0	0
$121$	14641.0	0.0909091
$122$	−58238.2	−0.354249
$123$	−68636.4	−0.409065
$124$	21781.7	0.127215
$125$	0	0
$126$	−25721.3	−0.144333
$127$	336859.	1.85327	0.926636	−	0.375960i	$-0.122687\pi$
$127$	336859.	1.85327	0.926636	+	0.375960i	$0.122687\pi$
$128$	26895.7	0.145097
$129$	164515.	0.870427
$130$	0	0
$131$	−83896.5	−0.427135	−0.213568	−	0.976928i	$-0.568508\pi$
$131$	−83896.5	−0.427135	−0.213568	+	0.976928i	$0.568508\pi$
$132$	16074.0	0.0802952
$133$	−62546.7	−0.306602
$134$	153438.	0.738195
$135$	0	0
$136$	170.069	0.000788458 0
$137$	196507.	0.894492	0.447246	−	0.894411i	$-0.352405\pi$
$137$	196507.	0.894492	0.447246	+	0.894411i	$0.352405\pi$
$138$	28013.6	0.125219
$139$	263232.	1.15559	0.577793	−	0.816183i	$-0.303914\pi$
$139$	263232.	1.15559	0.577793	+	0.816183i	$0.303914\pi$
$140$	0	0
$141$	115903.	0.490961
$142$	155757.	0.648226
$143$	20543.2	0.0840094
$144$	−27037.9	−0.108659
$145$	0	0
$146$	334042.	1.29694
$147$	−98621.2	−0.376424
$148$	−234139.	−0.878655
$149$	−337412.	−1.24507	−0.622536	−	0.782591i	$-0.713898\pi$
$149$	−337412.	−1.24507	−0.622536	+	0.782591i	$0.713898\pi$
$150$	0	0
$151$	−157583.	−0.562428	−0.281214	−	0.959645i	$-0.590737\pi$
$151$	−157583.	−0.562428	−0.281214	+	0.959645i	$0.590737\pi$
$152$	−158782.	−0.557434
$153$	70.9527	0.000245042 0
$154$	38423.2	0.130555
$155$	0	0
$156$	22553.9	0.0742011
$157$	−5369.59	−0.0173857	−0.00869285	−	0.999962i	$-0.502767\pi$
$157$	−5369.59	−0.0173857	−0.00869285	+	0.999962i	$0.502767\pi$
$158$	258009.	0.822228
$159$	−195871.	−0.614437
$160$	0	0
$161$	−57333.2	−0.174318
$162$	−27241.7	−0.0815543
$163$	503102.	1.48316	0.741579	−	0.670866i	$-0.234077\pi$
$163$	503102.	1.48316	0.741579	+	0.670866i	$0.234077\pi$
$164$	112566.	0.326813
$165$	0	0
$166$	−49433.9	−0.139237
$167$	584523.	1.62185	0.810924	−	0.585151i	$-0.198965\pi$
$167$	584523.	1.62185	0.810924	+	0.585151i	$0.198965\pi$
$168$	133638.	0.365305
$169$	−342468.	−0.922367
$170$	0	0
$171$	−66243.8	−0.173243
$172$	−269811.	−0.695407
$173$	55544.0	0.141098	0.0705491	−	0.997508i	$-0.477525\pi$
$173$	55544.0	0.141098	0.0705491	+	0.997508i	$0.477525\pi$
$174$	−225731.	−0.565220
$175$	0	0
$176$	40390.0	0.0982861
$177$	−10516.1	−0.0252302
$178$	−607645.	−1.43747
$179$	215136.	0.501858	0.250929	−	0.968006i	$-0.419264\pi$
$179$	215136.	0.501858	0.250929	+	0.968006i	$0.419264\pi$
$180$	0	0
$181$	682113.	1.54761	0.773803	−	0.633427i	$-0.218352\pi$
$181$	682113.	1.54761	0.773803	+	0.633427i	$0.218352\pi$
$182$	53912.7	0.120646
$183$	126237.	0.278650
$184$	−145547.	−0.316927
$185$	0	0
$186$	55144.4	0.116874
$187$	−105.991	−0.000221649 0
$188$	−190085.	−0.392242
$189$	55753.4	0.113532
$190$	0	0
$191$	−89697.4	−0.177908	−0.0889542	−	0.996036i	$-0.528352\pi$
$191$	−89697.4	−0.177908	−0.0889542	+	0.996036i	$0.528352\pi$
$192$	276509.	0.541323
$193$	−657062.	−1.26973	−0.634867	−	0.772621i	$-0.718945\pi$
$193$	−657062.	−1.26973	−0.634867	+	0.772621i	$0.718945\pi$
$194$	−573521.	−1.09407
$195$	0	0
$196$	161742.	0.300735
$197$	−720419.	−1.32257	−0.661286	−	0.750134i	$-0.729989\pi$
$197$	−720419.	−1.32257	−0.661286	+	0.750134i	$0.729989\pi$
$198$	40694.4	0.0737686
$199$	140848.	0.252127	0.126063	−	0.992022i	$-0.459766\pi$
$199$	140848.	0.252127	0.126063	+	0.992022i	$0.459766\pi$
$200$	0	0
$201$	−332592.	−0.580659
$202$	−152028.	−0.262148
$203$	461985.	0.786842
$204$	−116.365	−0.000195770 0
$205$	0	0
$206$	187667.	0.308121
$207$	−60722.2	−0.0984967
$208$	56672.4	0.0908266
$209$	98956.8	0.156704
$210$	0	0
$211$	−111929.	−0.173075	−0.0865377	−	0.996249i	$-0.527580\pi$
$211$	−111929.	−0.173075	−0.0865377	+	0.996249i	$0.527580\pi$
$212$	321236.	0.490890
$213$	−337618.	−0.509890
$214$	−116618.	−0.174073
$215$	0	0
$216$	141537.	0.206412
$217$	−112860.	−0.162701
$218$	140559.	0.200317
$219$	−724068.	−1.02016
$220$	0	0
$221$	−148.719	−0.000204826 0
$222$	−592766.	−0.807237
$223$	906826.	1.22113	0.610565	−	0.791966i	$-0.290942\pi$
$223$	906826.	1.22113	0.610565	+	0.791966i	$0.290942\pi$
$224$	−369158.	−0.491578
$225$	0	0
$226$	699867.	0.911474
$227$	1.35715e6	1.74809	0.874046	−	0.485844i	$-0.161488\pi$
$227$	1.35715e6	1.74809	0.874046	+	0.485844i	$0.161488\pi$
$228$	108642.	0.138408
$229$	−344956.	−0.434685	−0.217342	−	0.976095i	$-0.569739\pi$
$229$	−344956.	−0.434685	−0.217342	+	0.976095i	$0.569739\pi$
$230$	0	0
$231$	−83286.0	−0.102693
$232$	1.17281e6	1.43056
$233$	1.18163e6	1.42591	0.712955	−	0.701210i	$-0.247357\pi$
$233$	1.18163e6	1.42591	0.712955	+	0.701210i	$0.247357\pi$
$234$	57099.5	0.0681699
$235$	0	0
$236$	17246.7	0.0201570
$237$	−559259.	−0.646759
$238$	−278.158	−0.000318310 0
$239$	−723612.	−0.819429	−0.409714	−	0.912214i	$-0.634372\pi$
$239$	−723612.	−0.819429	−0.409714	+	0.912214i	$0.634372\pi$
$240$	0	0
$241$	394322.	0.437329	0.218664	−	0.975800i	$-0.429830\pi$
$241$	394322.	0.437329	0.218664	+	0.975800i	$0.429830\pi$
$242$	−60790.4	−0.0667262
$243$	59049.0	0.0641500
$244$	−207033.	−0.222621
$245$	0	0
$246$	284983.	0.300249
$247$	138849.	0.144811
$248$	−286508.	−0.295806
$249$	107153.	0.109523
$250$	0	0
$251$	1.49884e6	1.50165	0.750827	−	0.660499i	$-0.229655\pi$
$251$	1.49884e6	1.50165	0.750827	+	0.660499i	$0.229655\pi$
$252$	−91437.7	−0.0907035
$253$	90708.5	0.0890936
$254$	−1.39866e6	−1.36028
$255$	0	0
$256$	−1.09482e6	−1.04410
$257$	512138.	0.483676	0.241838	−	0.970317i	$-0.422250\pi$
$257$	512138.	0.483676	0.241838	+	0.970317i	$0.422250\pi$
$258$	−683079.	−0.638884
$259$	1.21317e6	1.12375
$260$	0	0
$261$	489293.	0.444598
$262$	348344.	0.313513
$263$	−154350.	−0.137600	−0.0687998	−	0.997630i	$-0.521917\pi$
$263$	−154350.	−0.137600	−0.0687998	+	0.997630i	$0.521917\pi$
$264$	−211432.	−0.186707
$265$	0	0
$266$	259698.	0.225043
$267$	1.31713e6	1.13071
$268$	545463.	0.463904
$269$	2.27104e6	1.91357	0.956783	−	0.290802i	$-0.0939221\pi$
$269$	2.27104e6	1.91357	0.956783	+	0.290802i	$0.0939221\pi$
$270$	0	0
$271$	304916.	0.252207	0.126103	−	0.992017i	$-0.459753\pi$
$271$	304916.	0.252207	0.126103	+	0.992017i	$0.459753\pi$
$272$	−292.397	−0.000239635 0
$273$	−116861.	−0.0948993
$274$	−815910.	−0.656547
$275$	0	0
$276$	99586.7	0.0786916
$277$	884418.	0.692561	0.346280	−	0.938131i	$-0.387445\pi$
$277$	884418.	0.692561	0.346280	+	0.938131i	$0.387445\pi$
$278$	−1.09296e6	−0.848187
$279$	−119531.	−0.0919325
$280$	0	0
$281$	2.03032e6	1.53391	0.766955	−	0.641701i	$-0.221771\pi$
$281$	2.03032e6	1.53391	0.766955	+	0.641701i	$0.221771\pi$
$282$	−481237.	−0.360360
$283$	−1.32361e6	−0.982415	−0.491207	−	0.871043i	$-0.663444\pi$
$283$	−1.32361e6	−0.982415	−0.491207	+	0.871043i	$0.663444\pi$
$284$	553706.	0.407365
$285$	0	0
$286$	−85296.8	−0.0616620
$287$	−583252.	−0.417976
$288$	−390979.	−0.277761
$289$	−1.41986e6	−0.999999
$290$	0	0
$291$	1.24316e6	0.860587
$292$	1.18750e6	0.815034
$293$	2.50642e6	1.70563	0.852816	−	0.522211i	$-0.174893\pi$
$293$	2.50642e6	1.70563	0.852816	+	0.522211i	$0.174893\pi$
$294$	409482.	0.276291
$295$	0	0
$296$	3.07977e6	2.04310
$297$	−88209.0	−0.0580259
$298$	1.40096e6	0.913869
$299$	127276.	0.0823317
$300$	0	0
$301$	1.39800e6	0.889389
$302$	654295.	0.412816
$303$	329536.	0.206203
$304$	272991.	0.169420
$305$	0	0
$306$	−294.600	−0.000179858 0
$307$	−871198.	−0.527559	−0.263779	−	0.964583i	$-0.584969\pi$
$307$	−871198.	−0.527559	−0.263779	+	0.964583i	$0.584969\pi$
$308$	136592.	0.0820444
$309$	−406787.	−0.242365
$310$	0	0
$311$	1.26077e6	0.739156	0.369578	−	0.929200i	$-0.379502\pi$
$311$	1.26077e6	0.739156	0.369578	+	0.929200i	$0.379502\pi$
$312$	−296666.	−0.172536
$313$	−326948.	−0.188633	−0.0943164	−	0.995542i	$-0.530067\pi$
$313$	−326948.	−0.188633	−0.0943164	+	0.995542i	$0.530067\pi$
$314$	22294.9	0.0127609
$315$	0	0
$316$	917206.	0.516713
$317$	969525.	0.541890	0.270945	−	0.962595i	$-0.412664\pi$
$317$	969525.	0.541890	0.270945	+	0.962595i	$0.412664\pi$
$318$	813270.	0.450990
$319$	−730919.	−0.402154
$320$	0	0
$321$	252781.	0.136925
$322$	238051.	0.127947
$323$	−716.381	−0.000382065 0
$324$	−96842.6	−0.0512512
$325$	0	0
$326$	−2.08891e6	−1.08862
$327$	−304676.	−0.157568
$328$	−1.48066e6	−0.759923
$329$	984910.	0.501657
$330$	0	0
$331$	2.06283e6	1.03489	0.517444	−	0.855717i	$-0.326884\pi$
$331$	2.06283e6	1.03489	0.517444	+	0.855717i	$0.326884\pi$
$332$	−175734.	−0.0875006
$333$	1.28488e6	0.634967
$334$	−2.42698e6	−1.19042
$335$	0	0
$336$	−229760.	−0.111027
$337$	2.86366e6	1.37356	0.686778	−	0.726867i	$-0.259024\pi$
$337$	2.86366e6	1.37356	0.686778	+	0.726867i	$0.259024\pi$
$338$	1.42195e6	0.677007
$339$	−1.51703e6	−0.716959
$340$	0	0
$341$	178558.	0.0831561
$342$	275049.	0.127158
$343$	−2.12344e6	−0.974552
$344$	3.54900e6	1.61700
$345$	0	0
$346$	−230622.	−0.103565
$347$	2.33015e6	1.03887	0.519434	−	0.854511i	$-0.326143\pi$
$347$	2.33015e6	1.03887	0.519434	+	0.854511i	$0.326143\pi$
$348$	−802458.	−0.355201
$349$	949067.	0.417094	0.208547	−	0.978012i	$-0.433127\pi$
$349$	949067.	0.417094	0.208547	+	0.978012i	$0.433127\pi$
$350$	0	0
$351$	−123769.	−0.0536219
$352$	584055.	0.251245
$353$	2.05584e6	0.878119	0.439059	−	0.898458i	$-0.355312\pi$
$353$	2.05584e6	0.878119	0.439059	+	0.898458i	$0.355312\pi$
$354$	43663.4	0.0185187
$355$	0	0
$356$	−2.16014e6	−0.903351
$357$	602.935	0.000250380 0
$358$	−893259.	−0.368358
$359$	−1.11901e6	−0.458244	−0.229122	−	0.973398i	$-0.573585\pi$
$359$	−1.11901e6	−0.458244	−0.229122	+	0.973398i	$0.573585\pi$
$360$	0	0
$361$	−1.80726e6	−0.729883
$362$	−2.83218e6	−1.13592
$363$	131769.	0.0524864
$364$	191656.	0.0758175
$365$	0	0
$366$	−524144.	−0.204526
$367$	−4.10134e6	−1.58950	−0.794749	−	0.606938i	$-0.792398\pi$
$367$	−4.10134e6	−1.58950	−0.794749	+	0.606938i	$0.792398\pi$
$368$	250237.	0.0963233
$369$	−617728.	−0.236174
$370$	0	0
$371$	−1.66445e6	−0.627823
$372$	196035.	0.0734474
$373$	3.54538e6	1.31945	0.659723	−	0.751509i	$-0.270674\pi$
$373$	3.54538e6	1.31945	0.659723	+	0.751509i	$0.270674\pi$
$374$	440.082	0.000162688 0
$375$	0	0
$376$	2.50031e6	0.912063
$377$	−1.02557e6	−0.371632
$378$	−231492.	−0.0833310
$379$	3.23295e6	1.15612	0.578058	−	0.815996i	$-0.303811\pi$
$379$	3.23295e6	1.15612	0.578058	+	0.815996i	$0.303811\pi$
$380$	0	0
$381$	3.03173e6	1.06999
$382$	372430.	0.130583
$383$	3.72375e6	1.29713	0.648565	−	0.761159i	$-0.275369\pi$
$383$	3.72375e6	1.29713	0.648565	+	0.761159i	$0.275369\pi$
$384$	242061.	0.0837717
$385$	0	0
$386$	2.72817e6	0.931971
$387$	1.48064e6	0.502541
$388$	−2.03883e6	−0.687546
$389$	−1.00909e6	−0.338108	−0.169054	−	0.985607i	$-0.554071\pi$
$389$	−1.00909e6	−0.338108	−0.169054	+	0.985607i	$0.554071\pi$
$390$	0	0
$391$	−656.669	−0.000217222 0
$392$	−2.12750e6	−0.699286
$393$	−755068.	−0.246607
$394$	2.99123e6	0.970753
$395$	0	0
$396$	144666.	0.0463584
$397$	366919.	0.116841	0.0584203	−	0.998292i	$-0.481394\pi$
$397$	366919.	0.116841	0.0584203	+	0.998292i	$0.481394\pi$
$398$	−584811.	−0.185058
$399$	−562920.	−0.177017
$400$	0	0
$401$	1.21086e6	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$401$	1.21086e6	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$402$	1.38094e6	0.426197
$403$	250540.	0.0768449
$404$	−540451.	−0.164741
$405$	0	0
$406$	−1.91819e6	−0.577533
$407$	−1.91938e6	−0.574349
$408$	1530.62	0.000455216 0
$409$	314741.	0.0930346	0.0465173	−	0.998917i	$-0.485188\pi$
$409$	314741.	0.0930346	0.0465173	+	0.998917i	$0.485188\pi$
$410$	0	0
$411$	1.76856e6	0.516435
$412$	667145.	0.193632
$413$	−89362.4	−0.0257798
$414$	252123.	0.0722954
$415$	0	0
$416$	819504.	0.232176
$417$	2.36909e6	0.667178
$418$	−410875.	−0.115019
$419$	5.98987e6	1.66679	0.833397	−	0.552674i	$-0.186393\pi$
$419$	5.98987e6	1.66679	0.833397	+	0.552674i	$0.186393\pi$
$420$	0	0
$421$	−3.19815e6	−0.879414	−0.439707	−	0.898141i	$-0.644918\pi$
$421$	−3.19815e6	−0.879414	−0.439707	+	0.898141i	$0.644918\pi$
$422$	464735.	0.127035
$423$	1.04313e6	0.283457
$424$	−4.22542e6	−1.14145
$425$	0	0
$426$	1.40181e6	0.374254
$427$	1.07272e6	0.284720
$428$	−414571.	−0.109393
$429$	184889.	0.0485029
$430$	0	0
$431$	−5.24498e6	−1.36004	−0.680019	−	0.733195i	$-0.738028\pi$
$431$	−5.24498e6	−1.36004	−0.680019	+	0.733195i	$0.738028\pi$
$432$	−243341.	−0.0627345
$433$	−5.44629e6	−1.39599	−0.697993	−	0.716105i	$-0.745923\pi$
$433$	−5.44629e6	−1.39599	−0.697993	+	0.716105i	$0.745923\pi$
$434$	468601.	0.119421
$435$	0	0
$436$	499680.	0.125885
$437$	613088.	0.153574
$438$	3.00638e6	0.748787
$439$	−6.38391e6	−1.58098	−0.790488	−	0.612477i	$-0.790173\pi$
$439$	−6.38391e6	−1.58098	−0.790488	+	0.612477i	$0.790173\pi$
$440$	0	0
$441$	−887591.	−0.217328
$442$	617.492	0.000150340 0
$443$	−2.26340e6	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$443$	−2.26340e6	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$444$	−2.10725e6	−0.507292
$445$	0	0
$446$	−3.76520e6	−0.896296
$447$	−3.03671e6	−0.718843
$448$	2.34969e6	0.553116
$449$	4.05102e6	0.948306	0.474153	−	0.880442i	$-0.342754\pi$
$449$	4.05102e6	0.948306	0.474153	+	0.880442i	$0.342754\pi$
$450$	0	0
$451$	922779.	0.213627
$452$	2.48798e6	0.572797
$453$	−1.41825e6	−0.324718
$454$	−5.63499e6	−1.28308
$455$	0	0
$456$	−1.42904e6	−0.321835
$457$	5.07731e6	1.13722	0.568608	−	0.822609i	$-0.307482\pi$
$457$	5.07731e6	1.13722	0.568608	+	0.822609i	$0.307482\pi$
$458$	1.43228e6	0.319054
$459$	638.574	0.000141475 0
$460$	0	0
$461$	−3.81197e6	−0.835405	−0.417702	−	0.908584i	$-0.637165\pi$
$461$	−3.81197e6	−0.835405	−0.417702	+	0.908584i	$0.637165\pi$
$462$	345809.	0.0753757
$463$	−6.00414e6	−1.30166	−0.650832	−	0.759222i	$-0.725580\pi$
$463$	−6.00414e6	−1.30166	−0.650832	+	0.759222i	$0.725580\pi$
$464$	−2.01638e6	−0.434788
$465$	0	0
$466$	−4.90621e6	−1.04660
$467$	3.29877e6	0.699939	0.349969	−	0.936761i	$-0.386192\pi$
$467$	3.29877e6	0.699939	0.349969	+	0.936761i	$0.386192\pi$
$468$	202985.	0.0428400
$469$	−2.82626e6	−0.593309
$470$	0	0
$471$	−48326.3	−0.0100376
$472$	−226857.	−0.0468703
$473$	−2.21182e6	−0.454566
$474$	2.32208e6	0.474714
$475$	0	0
$476$	−988.836	−0.000200035 0
$477$	−1.76284e6	−0.354746
$478$	3.00449e6	0.601452
$479$	5.86911e6	1.16878	0.584391	−	0.811472i	$-0.301333\pi$
$479$	5.86911e6	1.16878	0.584391	+	0.811472i	$0.301333\pi$
$480$	0	0
$481$	−2.69314e6	−0.530758
$482$	−1.63725e6	−0.320995
$483$	−515999.	−0.100642
$484$	−216106.	−0.0419328
$485$	0	0
$486$	−245175.	−0.0470854
$487$	−981038.	−0.187441	−0.0937203	−	0.995599i	$-0.529876\pi$
$487$	−981038.	−0.187441	−0.0937203	+	0.995599i	$0.529876\pi$
$488$	2.72324e6	0.517650
$489$	4.52792e6	0.856301
$490$	0	0
$491$	5.94395e6	1.11268	0.556342	−	0.830954i	$-0.312205\pi$
$491$	5.94395e6	1.11268	0.556342	+	0.830954i	$0.312205\pi$
$492$	1.01310e6	0.188685
$493$	5291.36	0.000980506 0
$494$	−576511.	−0.106289
$495$	0	0
$496$	492587.	0.0899040
$497$	−2.86898e6	−0.520998
$498$	−444905.	−0.0803884
$499$	6.59714e6	1.18605	0.593027	−	0.805183i	$-0.297933\pi$
$499$	6.59714e6	1.18605	0.593027	+	0.805183i	$0.297933\pi$
$500$	0	0
$501$	5.26070e6	0.936374
$502$	−6.22327e6	−1.10220
$503$	7.55854e6	1.33204	0.666021	−	0.745933i	$-0.267996\pi$
$503$	7.55854e6	1.33204	0.666021	+	0.745933i	$0.267996\pi$
$504$	1.20274e6	0.210909
$505$	0	0
$506$	−376628.	−0.0653937
$507$	−3.08221e6	−0.532529
$508$	−4.97216e6	−0.854841
$509$	4.67934e6	0.800553	0.400276	−	0.916394i	$-0.368914\pi$
$509$	4.67934e6	0.800553	0.400276	+	0.916394i	$0.368914\pi$
$510$	0	0
$511$	−6.15291e6	−1.04239
$512$	3.68509e6	0.621260
$513$	−596194.	−0.100022
$514$	−2.12643e6	−0.355013
$515$	0	0
$516$	−2.42830e6	−0.401494
$517$	−1.55825e6	−0.256396
$518$	−5.03715e6	−0.824823
$519$	499896.	0.0814631
$520$	0	0
$521$	557794.	0.0900284	0.0450142	−	0.998986i	$-0.485667\pi$
$521$	557794.	0.0900284	0.0450142	+	0.998986i	$0.485667\pi$
$522$	−2.03158e6	−0.326330
$523$	−3.87401e6	−0.619308	−0.309654	−	0.950849i	$-0.600213\pi$
$523$	−3.87401e6	−0.619308	−0.309654	+	0.950849i	$0.600213\pi$
$524$	1.23834e6	0.197021
$525$	0	0
$526$	640871.	0.100997
$527$	−1292.64	−0.000202746 0
$528$	363510.	0.0567455
$529$	−5.87436e6	−0.912686
$530$	0	0
$531$	−94644.6	−0.0145666
$532$	923210.	0.141423
$533$	1.29478e6	0.197414
$534$	−5.46880e6	−0.829926
$535$	0	0
$536$	−7.17481e6	−1.07870
$537$	1.93622e6	0.289748
$538$	−9.42950e6	−1.40454
$539$	1.32591e6	0.196581
$540$	0	0
$541$	−5.42985e6	−0.797618	−0.398809	−	0.917034i	$-0.630576\pi$
$541$	−5.42985e6	−0.797618	−0.398809	+	0.917034i	$0.630576\pi$
$542$	−1.26603e6	−0.185117
$543$	6.13902e6	0.893510
$544$	−4228.17	−0.000612569 0
$545$	0	0
$546$	485215.	0.0696550
$547$	853295.	0.121936	0.0609679	−	0.998140i	$-0.480581\pi$
$547$	853295.	0.121936	0.0609679	+	0.998140i	$0.480581\pi$
$548$	−2.90051e6	−0.412594
$549$	1.13613e6	0.160878
$550$	0	0
$551$	−4.94019e6	−0.693210
$552$	−1.30993e6	−0.182978
$553$	−4.75242e6	−0.660848
$554$	−3.67216e6	−0.508332
$555$	0	0
$556$	−3.88540e6	−0.533026
$557$	−1.08605e6	−0.148324	−0.0741618	−	0.997246i	$-0.523628\pi$
$557$	−1.08605e6	−0.148324	−0.0741618	+	0.997246i	$0.523628\pi$
$558$	496300.	0.0674775
$559$	−3.10347e6	−0.420066
$560$	0	0
$561$	−953.919	−0.000127969 0
$562$	−8.43005e6	−1.12587
$563$	−6.05610e6	−0.805235	−0.402617	−	0.915368i	$-0.631899\pi$
$563$	−6.05610e6	−0.805235	−0.402617	+	0.915368i	$0.631899\pi$
$564$	−1.71077e6	−0.226461
$565$	0	0
$566$	5.49573e6	0.721081
$567$	501781.	0.0655475
$568$	−7.28325e6	−0.947227
$569$	4.03513e6	0.522489	0.261245	−	0.965273i	$-0.415867\pi$
$569$	4.03513e6	0.522489	0.261245	+	0.965273i	$0.415867\pi$
$570$	0	0
$571$	597325.	0.0766691	0.0383345	−	0.999265i	$-0.487795\pi$
$571$	597325.	0.0766691	0.0383345	+	0.999265i	$0.487795\pi$
$572$	−303225.	−0.0387502
$573$	−807277.	−0.102715
$574$	2.42170e6	0.306790
$575$	0	0
$576$	2.48858e6	0.312533
$577$	−4.08910e6	−0.511315	−0.255657	−	0.966767i	$-0.582292\pi$
$577$	−4.08910e6	−0.511315	−0.255657	+	0.966767i	$0.582292\pi$
$578$	5.89534e6	0.733988
$579$	−5.91356e6	−0.733082
$580$	0	0
$581$	910551.	0.111909
$582$	−5.16169e6	−0.631662
$583$	2.63338e6	0.320879
$584$	−1.56199e7	−1.89516
$585$	0	0
$586$	−1.04068e7	−1.25191
$587$	−1.56181e6	−0.187082	−0.0935409	−	0.995615i	$-0.529819\pi$
$587$	−1.56181e6	−0.187082	−0.0935409	+	0.995615i	$0.529819\pi$
$588$	1.45568e6	0.173629
$589$	1.20685e6	0.143340
$590$	0	0
$591$	−6.48377e6	−0.763587
$592$	−5.29499e6	−0.620956
$593$	8.87674e6	1.03661	0.518307	−	0.855195i	$-0.326563\pi$
$593$	8.87674e6	1.03661	0.518307	+	0.855195i	$0.326563\pi$
$594$	366250.	0.0425903
$595$	0	0
$596$	4.98031e6	0.574303
$597$	1.26763e6	0.145565
$598$	−528457.	−0.0604306
$599$	−7.17157e6	−0.816671	−0.408336	−	0.912832i	$-0.633891\pi$
$599$	−7.17157e6	−0.816671	−0.408336	+	0.912832i	$0.633891\pi$
$600$	0	0
$601$	−4.79599e6	−0.541617	−0.270809	−	0.962633i	$-0.587291\pi$
$601$	−4.79599e6	−0.541617	−0.270809	+	0.962633i	$0.587291\pi$
$602$	−5.80460e6	−0.652802
$603$	−2.99332e6	−0.335244
$604$	2.32598e6	0.259426
$605$	0	0
$606$	−1.36825e6	−0.151351
$607$	8.49023e6	0.935293	0.467646	−	0.883916i	$-0.345102\pi$
$607$	8.49023e6	0.935293	0.467646	+	0.883916i	$0.345102\pi$
$608$	3.94756e6	0.433081
$609$	4.15786e6	0.454284
$610$	0	0
$611$	−2.18643e6	−0.236937
$612$	−1047.29	−0.000113028 0
$613$	−6.52329e6	−0.701158	−0.350579	−	0.936533i	$-0.614015\pi$
$613$	−6.52329e6	−0.701158	−0.350579	+	0.936533i	$0.614015\pi$
$614$	3.61727e6	0.387222
$615$	0	0
$616$	−1.79668e6	−0.190774
$617$	−1.20215e7	−1.27129	−0.635647	−	0.771980i	$-0.719267\pi$
$617$	−1.20215e7	−1.27129	−0.635647	+	0.771980i	$0.719267\pi$
$618$	1.68901e6	0.177893
$619$	1.34686e7	1.41285	0.706424	−	0.707789i	$-0.250307\pi$
$619$	1.34686e7	1.41285	0.706424	+	0.707789i	$0.250307\pi$
$620$	0	0
$621$	−546500.	−0.0568671
$622$	−5.23482e6	−0.542532
$623$	1.11926e7	1.15534
$624$	510051.	0.0524388
$625$	0	0
$626$	1.35751e6	0.138454
$627$	890611.	0.0904731
$628$	79257.0	0.00801934
$629$	13895.1	0.00140034
$630$	0	0
$631$	−1.08509e7	−1.08490	−0.542452	−	0.840087i	$-0.682504\pi$
$631$	−1.08509e7	−1.08490	−0.542452	+	0.840087i	$0.682504\pi$
$632$	−1.20646e7	−1.20149
$633$	−1.00736e6	−0.0999251
$634$	−4.02554e6	−0.397741
$635$	0	0
$636$	2.89112e6	0.283416
$637$	1.86042e6	0.181661
$638$	3.03482e6	0.295177
$639$	−3.03856e6	−0.294385
$640$	0	0
$641$	−9.31338e6	−0.895287	−0.447643	−	0.894212i	$-0.647737\pi$
$641$	−9.31338e6	−0.895287	−0.447643	+	0.894212i	$0.647737\pi$
$642$	−1.04956e6	−0.100501
$643$	−1.35760e7	−1.29493	−0.647464	−	0.762096i	$-0.724170\pi$
$643$	−1.35760e7	−1.29493	−0.647464	+	0.762096i	$0.724170\pi$
$644$	846258.	0.0804059
$645$	0	0
$646$	2974.46	0.000280432 0
$647$	1.18628e7	1.11411	0.557053	−	0.830477i	$-0.311932\pi$
$647$	1.18628e7	1.11411	0.557053	+	0.830477i	$0.311932\pi$
$648$	1.27383e6	0.119172
$649$	141383.	0.0131760
$650$	0	0
$651$	−1.01574e6	−0.0939353
$652$	−7.42596e6	−0.684122
$653$	−1.22670e7	−1.12579	−0.562893	−	0.826530i	$-0.690312\pi$
$653$	−1.22670e7	−1.12579	−0.562893	+	0.826530i	$0.690312\pi$
$654$	1.26503e6	0.115653
$655$	0	0
$656$	2.54566e6	0.230962
$657$	−6.51661e6	−0.588991
$658$	−4.08941e6	−0.368211
$659$	4.60998e6	0.413509	0.206755	−	0.978393i	$-0.433710\pi$
$659$	4.60998e6	0.413509	0.206755	+	0.978393i	$0.433710\pi$
$660$	0	0
$661$	−1.04708e7	−0.932130	−0.466065	−	0.884750i	$-0.654329\pi$
$661$	−1.04708e7	−0.932130	−0.466065	+	0.884750i	$0.654329\pi$
$662$	−8.56500e6	−0.759596
$663$	−1338.47	−0.000118257 0
$664$	2.31154e6	0.203461
$665$	0	0
$666$	−5.33490e6	−0.466058
$667$	−4.52841e6	−0.394123
$668$	−8.62775e6	−0.748094
$669$	8.16143e6	0.705020
$670$	0	0
$671$	−1.69718e6	−0.145520
$672$	−3.32242e6	−0.283813
$673$	5.32923e6	0.453552	0.226776	−	0.973947i	$-0.427181\pi$
$673$	5.32923e6	0.453552	0.226776	+	0.973947i	$0.427181\pi$
$674$	−1.18901e7	−1.00818
$675$	0	0
$676$	5.05495e6	0.425451
$677$	−1.75693e6	−0.147327	−0.0736634	−	0.997283i	$-0.523469\pi$
$677$	−1.75693e6	−0.147327	−0.0736634	+	0.997283i	$0.523469\pi$
$678$	6.29880e6	0.526240
$679$	1.05640e7	0.879335
$680$	0	0
$681$	1.22144e7	1.00926
$682$	−741386.	−0.0610357
$683$	1.42170e7	1.16615	0.583077	−	0.812417i	$-0.301848\pi$
$683$	1.42170e7	1.16615	0.583077	+	0.812417i	$0.301848\pi$
$684$	977781.	0.0799100
$685$	0	0
$686$	8.81667e6	0.715310
$687$	−3.10460e6	−0.250965
$688$	−6.10173e6	−0.491453
$689$	3.69497e6	0.296526
$690$	0	0
$691$	−2.40296e7	−1.91448	−0.957240	−	0.289294i	$-0.906579\pi$
$691$	−2.40296e7	−1.91448	−0.957240	+	0.289294i	$0.906579\pi$
$692$	−819848.	−0.0650831
$693$	−749574.	−0.0592900
$694$	−9.67494e6	−0.762517
$695$	0	0
$696$	1.05552e7	0.825934
$697$	−6680.30	−0.000520852 0
$698$	−3.94059e6	−0.306142
$699$	1.06347e7	0.823249
$700$	0	0
$701$	−5.04194e6	−0.387528	−0.193764	−	0.981048i	$-0.562070\pi$
$701$	−5.04194e6	−0.387528	−0.193764	+	0.981048i	$0.562070\pi$
$702$	513895.	0.0393579
$703$	−1.29729e7	−0.990030
$704$	−3.71751e6	−0.282697
$705$	0	0
$706$	−8.53600e6	−0.644529
$707$	2.80030e6	0.210696
$708$	155221.	0.0116377
$709$	1.87351e7	1.39972	0.699860	−	0.714280i	$-0.253246\pi$
$709$	1.87351e7	1.39972	0.699860	+	0.714280i	$0.253246\pi$
$710$	0	0
$711$	−5.03333e6	−0.373406
$712$	2.84137e7	2.10052
$713$	1.10626e6	0.0814955
$714$	−2503.43	−0.000183776 0
$715$	0	0
$716$	−3.17548e6	−0.231487
$717$	−6.51251e6	−0.473097
$718$	4.64619e6	0.336346
$719$	9.84863e6	0.710483	0.355242	−	0.934775i	$-0.384399\pi$
$719$	9.84863e6	0.710483	0.355242	+	0.934775i	$0.384399\pi$
$720$	0	0
$721$	−3.45675e6	−0.247645
$722$	7.50387e6	0.535726
$723$	3.54890e6	0.252492
$724$	−1.00682e7	−0.713849
$725$	0	0
$726$	−547114.	−0.0385244
$727$	1.76700e7	1.23994	0.619968	−	0.784627i	$-0.287146\pi$
$727$	1.76700e7	1.23994	0.619968	+	0.784627i	$0.287146\pi$
$728$	−2.52098e6	−0.176295
$729$	531441.	0.0370370
$730$	0	0
$731$	16012.1	0.00110829
$732$	−1.86330e6	−0.128530
$733$	1.72881e7	1.18847	0.594235	−	0.804291i	$-0.297455\pi$
$733$	1.72881e7	1.18847	0.594235	+	0.804291i	$0.297455\pi$
$734$	1.70290e7	1.16667
$735$	0	0
$736$	3.61852e6	0.246227
$737$	4.47151e6	0.303239
$738$	2.56485e6	0.173349
$739$	−5.26063e6	−0.354345	−0.177172	−	0.984180i	$-0.556695\pi$
$739$	−5.26063e6	−0.354345	−0.177172	+	0.984180i	$0.556695\pi$
$740$	0	0
$741$	1.24964e6	0.0836065
$742$	6.91093e6	0.460815
$743$	−5.40803e6	−0.359391	−0.179695	−	0.983722i	$-0.557511\pi$
$743$	−5.40803e6	−0.359391	−0.179695	+	0.983722i	$0.557511\pi$
$744$	−2.57857e6	−0.170784
$745$	0	0
$746$	−1.47207e7	−0.968458
$747$	964373.	0.0632330
$748$	1564.46	0.000102238 0
$749$	2.14806e6	0.139908
$750$	0	0
$751$	−1.50076e7	−0.970980	−0.485490	−	0.874242i	$-0.661359\pi$
$751$	−1.50076e7	−0.970980	−0.485490	+	0.874242i	$0.661359\pi$
$752$	−4.29874e6	−0.277202
$753$	1.34895e7	0.866980
$754$	4.25825e6	0.272774
$755$	0	0
$756$	−822939.	−0.0523677
$757$	−5.65482e6	−0.358657	−0.179328	−	0.983789i	$-0.557392\pi$
$757$	−5.65482e6	−0.358657	−0.179328	+	0.983789i	$0.557392\pi$
$758$	−1.34234e7	−0.848576
$759$	816376.	0.0514382
$760$	0	0
$761$	1.90760e7	1.19406	0.597030	−	0.802219i	$-0.296347\pi$
$761$	1.90760e7	1.19406	0.597030	+	0.802219i	$0.296347\pi$
$762$	−1.25880e7	−0.785358
$763$	−2.58904e6	−0.161001
$764$	1.32396e6	0.0820621
$765$	0	0
$766$	−1.54613e7	−0.952079
$767$	198378.	0.0121760
$768$	−9.85336e6	−0.602811
$769$	2.03557e7	1.24128	0.620639	−	0.784096i	$-0.286873\pi$
$769$	2.03557e7	1.24128	0.620639	+	0.784096i	$0.286873\pi$
$770$	0	0
$771$	4.60925e6	0.279250
$772$	9.69845e6	0.585678
$773$	9.29224e6	0.559334	0.279667	−	0.960097i	$-0.409776\pi$
$773$	9.29224e6	0.559334	0.279667	+	0.960097i	$0.409776\pi$
$774$	−6.14771e6	−0.368860
$775$	0	0
$776$	2.68180e7	1.59872
$777$	1.09185e7	0.648800
$778$	4.18981e6	0.248168
$779$	6.23695e6	0.368238
$780$	0	0
$781$	4.53909e6	0.266281
$782$	2726.53	0.000159439 0
$783$	4.40364e6	0.256689
$784$	3.65777e6	0.212533
$785$	0	0
$786$	3.13509e6	0.181007
$787$	−1.28703e7	−0.740716	−0.370358	−	0.928889i	$-0.620765\pi$
$787$	−1.28703e7	−0.740716	−0.370358	+	0.928889i	$0.620765\pi$
$788$	1.06336e7	0.610050
$789$	−1.38915e6	−0.0794431
$790$	0	0
$791$	−1.28912e7	−0.732578
$792$	−1.90288e6	−0.107795
$793$	−2.38137e6	−0.134476
$794$	−1.52347e6	−0.0857597
$795$	0	0
$796$	−2.07897e6	−0.116296
$797$	−8.82772e6	−0.492269	−0.246135	−	0.969236i	$-0.579161\pi$
$797$	−8.82772e6	−0.492269	−0.246135	+	0.969236i	$0.579161\pi$
$798$	2.33728e6	0.129928
$799$	11280.7	0.000625129 0
$800$	0	0
$801$	1.18541e7	0.652813
$802$	−5.02756e6	−0.276008
$803$	9.73469e6	0.532762
$804$	4.90916e6	0.267835
$805$	0	0
$806$	−1.04026e6	−0.0564033
$807$	2.04393e7	1.10480
$808$	7.10890e6	0.383066
$809$	640426.	0.0344031	0.0172016	−	0.999852i	$-0.494524\pi$
$809$	640426.	0.0344031	0.0172016	+	0.999852i	$0.494524\pi$
$810$	0	0
$811$	1.30571e7	0.697099	0.348550	−	0.937290i	$-0.386674\pi$
$811$	1.30571e7	0.697099	0.348550	+	0.937290i	$0.386674\pi$
$812$	−6.81905e6	−0.362939
$813$	2.74424e6	0.145612
$814$	7.96941e6	0.421566
$815$	0	0
$816$	−2631.57	−0.000138353 0
$817$	−1.49494e7	−0.783555
$818$	−1.30682e6	−0.0682864
$819$	−1.05175e6	−0.0547901
$820$	0	0
$821$	3.01122e7	1.55914	0.779570	−	0.626315i	$-0.215438\pi$
$821$	3.01122e7	1.55914	0.779570	+	0.626315i	$0.215438\pi$
$822$	−7.34319e6	−0.379057
$823$	−1.95327e7	−1.00523	−0.502613	−	0.864512i	$-0.667628\pi$
$823$	−1.95327e7	−1.00523	−0.502613	+	0.864512i	$0.667628\pi$
$824$	−8.77539e6	−0.450244
$825$	0	0
$826$	371039.	0.0189221
$827$	−6.83832e6	−0.347685	−0.173842	−	0.984773i	$-0.555618\pi$
$827$	−6.83832e6	−0.347685	−0.173842	+	0.984773i	$0.555618\pi$
$828$	896280.	0.0454326
$829$	7.34516e6	0.371206	0.185603	−	0.982625i	$-0.440576\pi$
$829$	7.34516e6	0.371206	0.185603	+	0.982625i	$0.440576\pi$
$830$	0	0
$831$	7.95976e6	0.399850
$832$	−5.21615e6	−0.261241
$833$	−9598.68	−0.000479291 0
$834$	−9.83662e6	−0.489701
$835$	0	0
$836$	−1.46064e6	−0.0722813
$837$	−1.07578e6	−0.0530773
$838$	−2.48703e7	−1.22341
$839$	−3.88085e7	−1.90336	−0.951682	−	0.307084i	$-0.900647\pi$
$839$	−3.88085e7	−1.90336	−0.951682	+	0.307084i	$0.900647\pi$
$840$	0	0
$841$	1.59783e7	0.779007
$842$	1.32789e7	0.645480
$843$	1.82729e7	0.885603
$844$	1.65211e6	0.0798328
$845$	0	0
$846$	−4.33114e6	−0.208054
$847$	1.11973e6	0.0536298
$848$	7.26468e6	0.346918
$849$	−1.19125e7	−0.567197
$850$	0	0
$851$	−1.18916e7	−0.562879
$852$	4.98335e6	0.235192
$853$	1.11803e7	0.526115	0.263058	−	0.964780i	$-0.415269\pi$
$853$	1.11803e7	0.526115	0.263058	+	0.964780i	$0.415269\pi$
$854$	−4.45402e6	−0.208981
$855$	0	0
$856$	5.45311e6	0.254367
$857$	2.30926e7	1.07404	0.537020	−	0.843569i	$-0.319550\pi$
$857$	2.30926e7	1.07404	0.537020	+	0.843569i	$0.319550\pi$
$858$	−767671.	−0.0356006
$859$	−1.28419e7	−0.593809	−0.296904	−	0.954907i	$-0.595954\pi$
$859$	−1.28419e7	−0.593809	−0.296904	+	0.954907i	$0.595954\pi$
$860$	0	0
$861$	−5.24927e6	−0.241319
$862$	2.17775e7	0.998252
$863$	2.98309e7	1.36345	0.681725	−	0.731608i	$-0.261230\pi$
$863$	2.98309e7	1.36345	0.681725	+	0.731608i	$0.261230\pi$
$864$	−3.51881e6	−0.160366
$865$	0	0
$866$	2.26134e7	1.02464
$867$	−1.27787e7	−0.577350
$868$	1.66585e6	0.0750474
$869$	7.51893e6	0.337759
$870$	0	0
$871$	6.27410e6	0.280225
$872$	−6.57260e6	−0.292716
$873$	1.11885e7	0.496860
$874$	−2.54558e6	−0.112722
$875$	0	0
$876$	1.06875e7	0.470560
$877$	3.85241e7	1.69135	0.845674	−	0.533700i	$-0.179199\pi$
$877$	3.85241e7	1.69135	0.845674	+	0.533700i	$0.179199\pi$
$878$	2.65064e7	1.16042
$879$	2.25578e7	0.984747
$880$	0	0
$881$	−3.12023e7	−1.35440	−0.677200	−	0.735799i	$-0.736807\pi$
$881$	−3.12023e7	−1.35440	−0.677200	+	0.735799i	$0.736807\pi$
$882$	3.68534e6	0.159517
$883$	1.08832e7	0.469738	0.234869	−	0.972027i	$-0.424534\pi$
$883$	1.08832e7	0.469738	0.234869	+	0.972027i	$0.424534\pi$
$884$	2195.14	9.44783e−5 0
$885$	0	0
$886$	9.39779e6	0.402199
$887$	−4.34842e7	−1.85576	−0.927882	−	0.372875i	$-0.878372\pi$
$887$	−4.34842e7	−1.85576	−0.927882	+	0.372875i	$0.878372\pi$
$888$	2.77180e7	1.17958
$889$	2.57628e7	1.09330
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−1.33851e7	−0.563259
$893$	−1.05320e7	−0.441961
$894$	1.26086e7	0.527623
$895$	0	0
$896$	2.05697e6	0.0855967
$897$	1.14548e6	0.0475342
$898$	−1.68201e7	−0.696046
$899$	−8.91412e6	−0.367857
$900$	0	0
$901$	−19063.9	−0.000782348 0
$902$	−3.83144e6	−0.156800
$903$	1.25820e7	0.513489
$904$	−3.27260e7	−1.33190
$905$	0	0
$906$	5.88866e6	0.238339
$907$	9.01861e6	0.364017	0.182008	−	0.983297i	$-0.441740\pi$
$907$	9.01861e6	0.364017	0.182008	+	0.983297i	$0.441740\pi$
$908$	−2.00320e7	−0.806326
$909$	2.96582e6	0.119052
$910$	0	0
$911$	−6.69826e6	−0.267403	−0.133701	−	0.991022i	$-0.542686\pi$
$911$	−6.69826e6	−0.267403	−0.133701	+	0.991022i	$0.542686\pi$
$912$	2.45692e6	0.0978147
$913$	−1.44061e6	−0.0571964
$914$	−2.10813e7	−0.834704
$915$	0	0
$916$	5.09166e6	0.200503
$917$	−6.41635e6	−0.251979
$918$	−2651.40	−0.000103841 0
$919$	−1.01836e7	−0.397753	−0.198877	−	0.980025i	$-0.563729\pi$
$919$	−1.01836e7	−0.397753	−0.198877	+	0.980025i	$0.563729\pi$
$920$	0	0
$921$	−7.84078e6	−0.304586
$922$	1.58275e7	0.613178
$923$	6.36892e6	0.246072
$924$	1.22933e6	0.0473684
$925$	0	0
$926$	2.49296e7	0.955406
$927$	−3.66108e6	−0.139930
$928$	−2.91576e7	−1.11143
$929$	7.05588e6	0.268233	0.134116	−	0.990966i	$-0.457180\pi$
$929$	7.05588e6	0.268233	0.134116	+	0.990966i	$0.457180\pi$
$930$	0	0
$931$	8.96165e6	0.338855
$932$	−1.74413e7	−0.657716
$933$	1.13470e7	0.426752
$934$	−1.36967e7	−0.513747
$935$	0	0
$936$	−2.66999e6	−0.0996140
$937$	−1.62634e6	−0.0605150	−0.0302575	−	0.999542i	$-0.509633\pi$
$937$	−1.62634e6	−0.0605150	−0.0302575	+	0.999542i	$0.509633\pi$
$938$	1.17348e7	0.435482
$939$	−2.94253e6	−0.108907
$940$	0	0
$941$	−2.73619e7	−1.00733	−0.503665	−	0.863899i	$-0.668015\pi$
$941$	−2.73619e7	−1.00733	−0.503665	+	0.863899i	$0.668015\pi$
$942$	200654.	0.00736752
$943$	5.71709e6	0.209361
$944$	390031.	0.0142452
$945$	0	0
$946$	9.18362e6	0.333646
$947$	−4.28329e7	−1.55204	−0.776019	−	0.630709i	$-0.782764\pi$
$947$	−4.28329e7	−1.55204	−0.776019	+	0.630709i	$0.782764\pi$
$948$	8.25485e6	0.298324
$949$	1.36590e7	0.492327
$950$	0	0
$951$	8.72573e6	0.312860
$952$	13006.8	0.000465133 0
$953$	−2.80611e7	−1.00086	−0.500429	−	0.865777i	$-0.666824\pi$
$953$	−2.80611e7	−1.00086	−0.500429	+	0.865777i	$0.666824\pi$
$954$	7.31943e6	0.260379
$955$	0	0
$956$	1.06808e7	0.377970
$957$	−6.57827e6	−0.232184
$958$	−2.43690e7	−0.857873
$959$	1.50287e7	0.527686
$960$	0	0
$961$	−2.64515e7	−0.923936
$962$	1.11821e7	0.389571
$963$	2.27503e6	0.0790536
$964$	−5.82032e6	−0.201723
$965$	0	0
$966$	2.14246e6	0.0738704
$967$	3.20738e7	1.10302	0.551512	−	0.834167i	$-0.314051\pi$
$967$	3.20738e7	1.10302	0.551512	+	0.834167i	$0.314051\pi$
$968$	2.84258e6	0.0975044
$969$	−6447.43	−0.000220586 0
$970$	0	0
$971$	4.92583e7	1.67661	0.838304	−	0.545203i	$-0.183548\pi$
$971$	4.92583e7	1.67661	0.838304	+	0.545203i	$0.183548\pi$
$972$	−871583.	−0.0295899
$973$	2.01318e7	0.681712
$974$	4.07334e6	0.137579
$975$	0	0
$976$	−4.68201e6	−0.157329
$977$	1.26363e7	0.423530	0.211765	−	0.977321i	$-0.432079\pi$
$977$	1.26363e7	0.423530	0.211765	+	0.977321i	$0.432079\pi$
$978$	−1.88002e7	−0.628516
$979$	−1.77081e7	−0.590492
$980$	0	0
$981$	−2.74208e6	−0.0909720
$982$	−2.46797e7	−0.816697
$983$	−6.52333e6	−0.215321	−0.107660	−	0.994188i	$-0.534336\pi$
$983$	−6.52333e6	−0.215321	−0.107660	+	0.994188i	$0.534336\pi$
$984$	−1.33259e7	−0.438742
$985$	0	0
$986$	−21970.1	−0.000719681 0
$987$	8.86419e6	0.289632
$988$	−2.04946e6	−0.0667955
$989$	−1.37033e7	−0.445488
$990$	0	0
$991$	2.28897e7	0.740382	0.370191	−	0.928956i	$-0.379292\pi$
$991$	2.28897e7	0.740382	0.370191	+	0.928956i	$0.379292\pi$
$992$	7.12300e6	0.229818
$993$	1.85655e7	0.597492
$994$	1.19122e7	0.382407
$995$	0	0
$996$	−1.58161e6	−0.0505185
$997$	1.82397e7	0.581138	0.290569	−	0.956854i	$-0.406155\pi$
$997$	1.82397e7	0.581138	0.290569	+	0.956854i	$0.406155\pi$
$998$	−2.73918e7	−0.870550
$999$	1.15639e7	0.366598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.e.1.1		2
5.4	even	2		33.6.a.c.1.2	✓	2
15.14	odd	2		99.6.a.f.1.1		2
20.19	odd	2		528.6.a.s.1.1		2
55.54	odd	2		363.6.a.j.1.1		2
165.164	even	2		1089.6.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.c.1.2	✓	2	5.4	even	2
99.6.a.f.1.1		2	15.14	odd	2
363.6.a.j.1.1		2	55.54	odd	2
528.6.a.s.1.1		2	20.19	odd	2
825.6.a.e.1.1		2	1.1	even	1	trivial
1089.6.a.j.1.2		2	165.164	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)

\(f(q)\)	\(=\)	\(q-4.15207 q^{2} +9.00000 q^{3} -14.7603 q^{4} -37.3686 q^{6} +76.4793 q^{7} +194.152 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.15207 q^{2} +9.00000 q^{3} -14.7603 q^{4} -37.3686 q^{6} +76.4793 q^{7} +194.152 q^{8} +81.0000 q^{9} -121.000 q^{11} -132.843 q^{12} -169.779 q^{13} -317.547 q^{14} -333.802 q^{16} +0.875959 q^{17} -336.317 q^{18} -817.825 q^{19} +688.314 q^{21} +502.400 q^{22} -749.657 q^{23} +1747.37 q^{24} +704.932 q^{26} +729.000 q^{27} -1128.86 q^{28} +6040.65 q^{29} -1475.69 q^{31} -4826.90 q^{32} -1089.00 q^{33} -3.63704 q^{34} -1195.59 q^{36} +15862.7 q^{37} +3395.66 q^{38} -1528.01 q^{39} -7626.27 q^{41} -2857.93 q^{42} +18279.5 q^{43} +1786.00 q^{44} +3112.62 q^{46} +12878.1 q^{47} -3004.22 q^{48} -10957.9 q^{49} +7.88363 q^{51} +2505.99 q^{52} -21763.5 q^{53} -3026.86 q^{54} +14848.6 q^{56} -7360.42 q^{57} -25081.2 q^{58} -1168.45 q^{59} +14026.3 q^{61} +6127.16 q^{62} +6194.83 q^{63} +30723.3 q^{64} +4521.60 q^{66} -36954.6 q^{67} -12.9295 q^{68} -6746.91 q^{69} -37513.1 q^{71} +15726.3 q^{72} -80452.0 q^{73} -65862.9 q^{74} +12071.4 q^{76} -9254.00 q^{77} +6344.39 q^{78} -62139.9 q^{79} +6561.00 q^{81} +31664.8 q^{82} +11905.8 q^{83} -10159.7 q^{84} -75897.7 q^{86} +54365.9 q^{87} -23492.4 q^{88} +146348. q^{89} -12984.5 q^{91} +11065.2 q^{92} -13281.2 q^{93} -53470.8 q^{94} -43442.1 q^{96} +138129. q^{97} +45498.0 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q + 5 * q^2 + 18 * q^3 + 37 * q^4 + 45 * q^6 + 286 * q^7 + 375 * q^8 + 162 * q^9	\( 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9} - 242 q^{11} + 333 q^{12} + 166 q^{13} + 1600 q^{14} - 335 q^{16} + 800 q^{17} + 405 q^{18} - 1476 q^{19} + 2574 q^{21} - 605 q^{22} + 3370 q^{23} + 3375 q^{24} + 3778 q^{26} + 1458 q^{27} + 9716 q^{28} + 6600 q^{29} - 7528 q^{31} - 10625 q^{32} - 2178 q^{33} + 7310 q^{34} + 2997 q^{36} + 29916 q^{37} - 2628 q^{38} + 1494 q^{39} - 5780 q^{41} + 14400 q^{42} + 16656 q^{43} - 4477 q^{44} + 40816 q^{46} - 7850 q^{47} - 3015 q^{48} + 16134 q^{49} + 7200 q^{51} + 19886 q^{52} - 14178 q^{53} + 3645 q^{54} + 52740 q^{56} - 13284 q^{57} - 19962 q^{58} + 17300 q^{59} - 2946 q^{61} - 49264 q^{62} + 23166 q^{63} - 22303 q^{64} - 5445 q^{66} - 31336 q^{67} + 41350 q^{68} + 30330 q^{69} - 33810 q^{71} + 30375 q^{72} - 60644 q^{73} + 62754 q^{74} - 21996 q^{76} - 34606 q^{77} + 34002 q^{78} + 1870 q^{79} + 13122 q^{81} + 48562 q^{82} + 58296 q^{83} + 87444 q^{84} - 90756 q^{86} + 59400 q^{87} - 45375 q^{88} + 92388 q^{89} + 57368 q^{91} + 224300 q^{92} - 67752 q^{93} - 243176 q^{94} - 95625 q^{96} - 7120 q^{97} + 293445 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 18 * q^3 + 37 * q^4 + 45 * q^6 + 286 * q^7 + 375 * q^8 + 162 * q^9 - 242 * q^11 + 333 * q^12 + 166 * q^13 + 1600 * q^14 - 335 * q^16 + 800 * q^17 + 405 * q^18 - 1476 * q^19 + 2574 * q^21 - 605 * q^22 + 3370 * q^23 + 3375 * q^24 + 3778 * q^26 + 1458 * q^27 + 9716 * q^28 + 6600 * q^29 - 7528 * q^31 - 10625 * q^32 - 2178 * q^33 + 7310 * q^34 + 2997 * q^36 + 29916 * q^37 - 2628 * q^38 + 1494 * q^39 - 5780 * q^41 + 14400 * q^42 + 16656 * q^43 - 4477 * q^44 + 40816 * q^46 - 7850 * q^47 - 3015 * q^48 + 16134 * q^49 + 7200 * q^51 + 19886 * q^52 - 14178 * q^53 + 3645 * q^54 + 52740 * q^56 - 13284 * q^57 - 19962 * q^58 + 17300 * q^59 - 2946 * q^61 - 49264 * q^62 + 23166 * q^63 - 22303 * q^64 - 5445 * q^66 - 31336 * q^67 + 41350 * q^68 + 30330 * q^69 - 33810 * q^71 + 30375 * q^72 - 60644 * q^73 + 62754 * q^74 - 21996 * q^76 - 34606 * q^77 + 34002 * q^78 + 1870 * q^79 + 13122 * q^81 + 48562 * q^82 + 58296 * q^83 + 87444 * q^84 - 90756 * q^86 + 59400 * q^87 - 45375 * q^88 + 92388 * q^89 + 57368 * q^91 + 224300 * q^92 - 67752 * q^93 - 243176 * q^94 - 95625 * q^96 - 7120 * q^97 + 293445 * q^98 - 19602 * q^99

Introduction

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