LMFDB - Embedded newform 825.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
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			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} -121.000 q^{11} +169.557 q^{12} +277.549 q^{13} +910.853 q^{14} -66.1983 q^{16} -704.489 q^{17} -293.845 q^{18} -2861.18 q^{19} +2259.73 q^{21} +438.954 q^{22} +1066.85 q^{23} -1659.89 q^{24} -1006.87 q^{26} -729.000 q^{27} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} -5661.67 q^{32} +1089.00 q^{33} +2555.69 q^{34} -1526.01 q^{36} +9042.34 q^{37} +10379.6 q^{38} -2497.94 q^{39} +18219.0 q^{41} -8197.68 q^{42} +4054.54 q^{43} +2279.60 q^{44} -3870.24 q^{46} -20750.8 q^{47} +595.785 q^{48} +46234.9 q^{49} +6340.40 q^{51} -5228.92 q^{52} +26485.9 q^{53} +2644.61 q^{54} -46307.4 q^{56} +25750.7 q^{57} +14283.9 q^{58} +4293.12 q^{59} -6831.76 q^{61} +2337.52 q^{62} -20337.6 q^{63} +22657.3 q^{64} -3950.59 q^{66} +56749.5 q^{67} +13272.3 q^{68} -9601.68 q^{69} +3187.09 q^{71} +14939.0 q^{72} +7397.14 q^{73} -32803.1 q^{74} +53903.7 q^{76} +30380.9 q^{77} +9061.82 q^{78} +24393.7 q^{79} +6561.00 q^{81} -66093.5 q^{82} -102795. q^{83} -42572.6 q^{84} -14708.7 q^{86} +35437.0 q^{87} -22316.3 q^{88} +49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} +5799.15 q^{93} +75278.1 q^{94} +50955.1 q^{96} +92279.5 q^{97} -167727. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * 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 $

$2$	−3.62772	−0.641296	−0.320648	−	0.947198i	$-0.603901\pi$
$2$	−3.62772	−0.641296	−0.320648	+	0.947198i	$0.603901\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−18.8397	−0.588739
$5$	0	0
$5$	0	0
$6$	32.6495	0.370252
$7$	−251.081	−1.93673	−0.968366	−	0.249534i	$-0.919722\pi$
$7$	−251.081	−1.93673	−0.968366	+	0.249534i	$0.919722\pi$
$8$	184.432	1.01885
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	169.557	0.339909
$13$	277.549	0.455492	0.227746	−	0.973721i	$-0.426864\pi$
$13$	277.549	0.455492	0.227746	+	0.973721i	$0.426864\pi$
$14$	910.853	1.24202
$15$	0	0
$16$	−66.1983	−0.0646468
$17$	−704.489	−0.591224	−0.295612	−	0.955308i	$-0.595523\pi$
$17$	−704.489	−0.591224	−0.295612	+	0.955308i	$0.595523\pi$
$18$	−293.845	−0.213765
$19$	−2861.18	−1.81828	−0.909142	−	0.416486i	$-0.863261\pi$
$19$	−2861.18	−1.81828	−0.909142	+	0.416486i	$0.863261\pi$
$20$	0	0
$21$	2259.73	1.11817
$22$	438.954	0.193358
$23$	1066.85	0.420518	0.210259	−	0.977646i	$-0.432569\pi$
$23$	1066.85	0.420518	0.210259	+	0.977646i	$0.432569\pi$
$24$	−1659.89	−0.588235
$25$	0	0
$26$	−1006.87	−0.292105
$27$	−729.000	−0.192450
$28$	4730.29	1.14023
$29$	−3937.44	−0.869399	−0.434700	−	0.900575i	$-0.643145\pi$
$29$	−3937.44	−0.869399	−0.434700	+	0.900575i	$0.643145\pi$
$30$	0	0
$31$	−644.350	−0.120425	−0.0602126	−	0.998186i	$-0.519178\pi$
$31$	−644.350	−0.120425	−0.0602126	+	0.998186i	$0.519178\pi$
$32$	−5661.67	−0.977395
$33$	1089.00	0.174078
$34$	2555.69	0.379149
$35$	0	0
$36$	−1526.01	−0.196246
$37$	9042.34	1.08587	0.542934	−	0.839776i	$-0.317314\pi$
$37$	9042.34	1.08587	0.542934	+	0.839776i	$0.317314\pi$
$38$	10379.6	1.16606
$39$	−2497.94	−0.262979
$40$	0	0
$41$	18219.0	1.69264	0.846322	−	0.532672i	$-0.178812\pi$
$41$	18219.0	1.69264	0.846322	+	0.532672i	$0.178812\pi$
$42$	−8197.68	−0.717080
$43$	4054.54	0.334403	0.167202	−	0.985923i	$-0.446527\pi$
$43$	4054.54	0.334403	0.167202	+	0.985923i	$0.446527\pi$
$44$	2279.60	0.177512
$45$	0	0
$46$	−3870.24	−0.269677
$47$	−20750.8	−1.37022	−0.685110	−	0.728439i	$-0.740246\pi$
$47$	−20750.8	−1.37022	−0.685110	+	0.728439i	$0.740246\pi$
$48$	595.785	0.0373238
$49$	46234.9	2.75093
$50$	0	0
$51$	6340.40	0.341343
$52$	−5228.92	−0.268166
$53$	26485.9	1.29517	0.647583	−	0.761995i	$-0.275780\pi$
$53$	26485.9	1.29517	0.647583	+	0.761995i	$0.275780\pi$
$54$	2644.61	0.123417
$55$	0	0
$56$	−46307.4	−1.97324
$57$	25750.7	1.04979
$58$	14283.9	0.557542
$59$	4293.12	0.160562	0.0802810	−	0.996772i	$-0.474418\pi$
$59$	4293.12	0.160562	0.0802810	+	0.996772i	$0.474418\pi$
$60$	0	0
$61$	−6831.76	−0.235076	−0.117538	−	0.993068i	$-0.537500\pi$
$61$	−6831.76	−0.235076	−0.117538	+	0.993068i	$0.537500\pi$
$62$	2337.52	0.0772282
$63$	−20337.6	−0.645577
$64$	22657.3	0.691446
$65$	0	0
$66$	−3950.59	−0.111635
$67$	56749.5	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$67$	56749.5	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$68$	13272.3	0.348077
$69$	−9601.68	−0.242786
$70$	0	0
$71$	3187.09	0.0750323	0.0375161	−	0.999296i	$-0.488055\pi$
$71$	3187.09	0.0750323	0.0375161	+	0.999296i	$0.488055\pi$
$72$	14939.0	0.339617
$73$	7397.14	0.162464	0.0812319	−	0.996695i	$-0.474115\pi$
$73$	7397.14	0.162464	0.0812319	+	0.996695i	$0.474115\pi$
$74$	−32803.1	−0.696362
$75$	0	0
$76$	53903.7	1.07050
$77$	30380.9	0.583947
$78$	9061.82	0.168647
$79$	24393.7	0.439754	0.219877	−	0.975528i	$-0.429434\pi$
$79$	24393.7	0.439754	0.219877	+	0.975528i	$0.429434\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−66093.5	−1.08549
$83$	−102795.	−1.63786	−0.818932	−	0.573890i	$-0.805434\pi$
$83$	−102795.	−1.63786	−0.818932	+	0.573890i	$0.805434\pi$
$84$	−42572.6	−0.658312
$85$	0	0
$86$	−14708.7	−0.214451
$87$	35437.0	0.501948
$88$	−22316.3	−0.307196
$89$	49599.4	0.663745	0.331873	−	0.943324i	$-0.392320\pi$
$89$	49599.4	0.663745	0.331873	+	0.943324i	$0.392320\pi$
$90$	0	0
$91$	−69687.4	−0.882166
$92$	−20099.1	−0.247576
$93$	5799.15	0.0695275
$94$	75278.1	0.878717
$95$	0	0
$96$	50955.1	0.564299
$97$	92279.5	0.995808	0.497904	−	0.867232i	$-0.334103\pi$
$97$	92279.5	0.995808	0.497904	+	0.867232i	$0.334103\pi$
$98$	−167727.	−1.76416
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−35546.1	−0.346728	−0.173364	−	0.984858i	$-0.555464\pi$
$101$	−35546.1	−0.346728	−0.173364	+	0.984858i	$0.555464\pi$
$102$	−23001.2	−0.218902
$103$	59876.8	0.556116	0.278058	−	0.960564i	$-0.410309\pi$
$103$	59876.8	0.556116	0.278058	+	0.960564i	$0.410309\pi$
$104$	51188.9	0.464079
$105$	0	0
$106$	−96083.5	−0.830586
$107$	−89253.8	−0.753646	−0.376823	−	0.926285i	$-0.622983\pi$
$107$	−89253.8	−0.753646	−0.376823	+	0.926285i	$0.622983\pi$
$108$	13734.1	0.113303
$109$	22796.0	0.183777	0.0918887	−	0.995769i	$-0.470710\pi$
$109$	22796.0	0.183777	0.0918887	+	0.995769i	$0.470710\pi$
$110$	0	0
$111$	−81381.1	−0.626926
$112$	16621.2	0.125203
$113$	166064.	1.22343	0.611717	−	0.791077i	$-0.290479\pi$
$113$	166064.	1.22343	0.611717	+	0.791077i	$0.290479\pi$
$114$	−93416.1	−0.673224
$115$	0	0
$116$	74180.1	0.511850
$117$	22481.5	0.151831
$118$	−15574.2	−0.102968
$119$	176884.	1.14504
$120$	0	0
$121$	14641.0	0.0909091
$122$	24783.7	0.150753
$123$	−163971.	−0.977248
$124$	12139.3	0.0708991
$125$	0	0
$126$	73779.1	0.414006
$127$	−159190.	−0.875802	−0.437901	−	0.899023i	$-0.644278\pi$
$127$	−159190.	−0.875802	−0.437901	+	0.899023i	$0.644278\pi$
$128$	98979.2	0.533973
$129$	−36490.9	−0.193068
$130$	0	0
$131$	232174.	1.18205	0.591025	−	0.806654i	$-0.298724\pi$
$131$	232174.	1.18205	0.591025	+	0.806654i	$0.298724\pi$
$132$	−20516.4	−0.102486
$133$	718390.	3.52153
$134$	−205871.	−0.990452
$135$	0	0
$136$	−129930.	−0.602369
$137$	−68205.4	−0.310468	−0.155234	−	0.987878i	$-0.549613\pi$
$137$	−68205.4	−0.310468	−0.155234	+	0.987878i	$0.549613\pi$
$138$	34832.2	0.155698
$139$	298162.	1.30893	0.654464	−	0.756093i	$-0.272894\pi$
$139$	298162.	1.30893	0.654464	+	0.756093i	$0.272894\pi$
$140$	0	0
$141$	186757.	0.791097
$142$	−11561.9	−0.0481179
$143$	−33583.4	−0.137336
$144$	−5362.06	−0.0215489
$145$	0	0
$146$	−26834.7	−0.104187
$147$	−416114.	−1.58825
$148$	−170355.	−0.639293
$149$	83131.5	0.306761	0.153380	−	0.988167i	$-0.450984\pi$
$149$	83131.5	0.306761	0.153380	+	0.988167i	$0.450984\pi$
$150$	0	0
$151$	167598.	0.598171	0.299085	−	0.954226i	$-0.403318\pi$
$151$	167598.	0.598171	0.299085	+	0.954226i	$0.403318\pi$
$152$	−527694.	−1.85256
$153$	−57063.6	−0.197075
$154$	−110213.	−0.374483
$155$	0	0
$156$	47060.3	0.154826
$157$	63259.7	0.204823	0.102411	−	0.994742i	$-0.467344\pi$
$157$	63259.7	0.204823	0.102411	+	0.994742i	$0.467344\pi$
$158$	−88493.4	−0.282012
$159$	−238373.	−0.747765
$160$	0	0
$161$	−267867.	−0.814431
$162$	−23801.5	−0.0712551
$163$	−237727.	−0.700825	−0.350412	−	0.936596i	$-0.613959\pi$
$163$	−237727.	−0.700825	−0.350412	+	0.936596i	$0.613959\pi$
$164$	−343240.	−0.996526
$165$	0	0
$166$	372912.	1.05036
$167$	−487880.	−1.35370	−0.676849	−	0.736122i	$-0.736655\pi$
$167$	−487880.	−1.35370	−0.676849	+	0.736122i	$0.736655\pi$
$168$	416767.	1.13925
$169$	−294260.	−0.792527
$170$	0	0
$171$	−231756.	−0.606095
$172$	−76386.1	−0.196876
$173$	−325942.	−0.827991	−0.413996	−	0.910279i	$-0.635867\pi$
$173$	−325942.	−0.827991	−0.413996	+	0.910279i	$0.635867\pi$
$174$	−128555.	−0.321897
$175$	0	0
$176$	8009.99	0.0194917
$177$	−38638.1	−0.0927005
$178$	−179933.	−0.425657
$179$	462906.	1.07984	0.539921	−	0.841716i	$-0.318454\pi$
$179$	462906.	1.07984	0.539921	+	0.841716i	$0.318454\pi$
$180$	0	0
$181$	23901.3	0.0542281	0.0271141	−	0.999632i	$-0.491368\pi$
$181$	23901.3	0.0542281	0.0271141	+	0.999632i	$0.491368\pi$
$182$	252806.	0.565730
$183$	61485.8	0.135721
$184$	196762.	0.428446
$185$	0	0
$186$	−21037.7	−0.0445877
$187$	85243.1	0.178261
$188$	390938.	0.806703
$189$	183038.	0.372724
$190$	0	0
$191$	−565986.	−1.12259	−0.561297	−	0.827615i	$-0.689697\pi$
$191$	−565986.	−1.12259	−0.561297	+	0.827615i	$0.689697\pi$
$192$	−203916.	−0.399207
$193$	−91762.3	−0.177325	−0.0886627	−	0.996062i	$-0.528259\pi$
$193$	−91762.3	−0.177325	−0.0886627	+	0.996062i	$0.528259\pi$
$194$	−334764.	−0.638608
$195$	0	0
$196$	−871049.	−1.61958
$197$	−485247.	−0.890836	−0.445418	−	0.895323i	$-0.646945\pi$
$197$	−485247.	−0.890836	−0.445418	+	0.895323i	$0.646945\pi$
$198$	35555.3	0.0644527
$199$	692632.	1.23985	0.619926	−	0.784660i	$-0.287162\pi$
$199$	692632.	1.23985	0.619926	+	0.784660i	$0.287162\pi$
$200$	0	0
$201$	−510745.	−0.891690
$202$	128951.	0.222355
$203$	988619.	1.68379
$204$	−119451.	−0.200962
$205$	0	0
$206$	−217216.	−0.356635
$207$	86415.1	0.140173
$208$	−18373.3	−0.0294461
$209$	346203.	0.548233
$210$	0	0
$211$	−441020.	−0.681949	−0.340975	−	0.940073i	$-0.610757\pi$
$211$	−441020.	−0.681949	−0.340975	+	0.940073i	$0.610757\pi$
$212$	−498986.	−0.762516
$213$	−28683.8	−0.0433199
$214$	323788.	0.483310
$215$	0	0
$216$	−134451.	−0.196078
$217$	161784.	0.233231
$218$	−82697.4	−0.117856
$219$	−66574.2	−0.0937985
$220$	0	0
$221$	−195530.	−0.269298
$222$	295228.	0.402045
$223$	−1.13133e6	−1.52345	−0.761726	−	0.647899i	$-0.775648\pi$
$223$	−1.13133e6	−1.52345	−0.761726	+	0.647899i	$0.775648\pi$
$224$	1.42154e6	1.89295
$225$	0	0
$226$	−602435.	−0.784583
$227$	820354.	1.05666	0.528332	−	0.849038i	$-0.322818\pi$
$227$	820354.	1.05666	0.528332	+	0.849038i	$0.322818\pi$
$228$	−485133.	−0.618051
$229$	−1.00301e6	−1.26392	−0.631958	−	0.775003i	$-0.717748\pi$
$229$	−1.00301e6	−1.26392	−0.631958	+	0.775003i	$0.717748\pi$
$230$	0	0
$231$	−273428.	−0.337142
$232$	−726191.	−0.885790
$233$	−734.569	−0.000886427 0	−0.000443214	−	1.00000i	$-0.500141\pi$
$233$	−734.569	−0.000886427 0	−0.000443214		1.00000i	$0.500141\pi$
$234$	−81556.4	−0.0973685
$235$	0	0
$236$	−80880.9	−0.0945291
$237$	−219543.	−0.253892
$238$	−641685.	−0.734311
$239$	−1.06403e6	−1.20492	−0.602460	−	0.798149i	$-0.705813\pi$
$239$	−1.06403e6	−1.20492	−0.602460	+	0.798149i	$0.705813\pi$
$240$	0	0
$241$	−661636.	−0.733798	−0.366899	−	0.930261i	$-0.619581\pi$
$241$	−661636.	−0.733798	−0.366899	+	0.930261i	$0.619581\pi$
$242$	−53113.4	−0.0582996
$243$	−59049.0	−0.0641500
$244$	128708.	0.138398
$245$	0	0
$246$	594841.	0.626705
$247$	−794118.	−0.828214
$248$	−118839.	−0.122696
$249$	925158.	0.945622
$250$	0	0
$251$	1.34864e6	1.35118	0.675590	−	0.737277i	$-0.263889\pi$
$251$	1.34864e6	1.35118	0.675590	+	0.737277i	$0.263889\pi$
$252$	383153.	0.380077
$253$	−129089.	−0.126791
$254$	577496.	0.561648
$255$	0	0
$256$	−1.08410e6	−1.03388
$257$	1.65015e6	1.55844	0.779220	−	0.626751i	$-0.215616\pi$
$257$	1.65015e6	1.55844	0.779220	+	0.626751i	$0.215616\pi$
$258$	132379.	0.123814
$259$	−2.27036e6	−2.10303
$260$	0	0
$261$	−318933.	−0.289800
$262$	−842262.	−0.758044
$263$	868163.	0.773948	0.386974	−	0.922091i	$-0.373520\pi$
$263$	868163.	0.773948	0.386974	+	0.922091i	$0.373520\pi$
$264$	200846.	0.177359
$265$	0	0
$266$	−2.60612e6	−2.25834
$267$	−446395.	−0.383213
$268$	−1.06914e6	−0.909280
$269$	271547.	0.228804	0.114402	−	0.993435i	$-0.463505\pi$
$269$	271547.	0.228804	0.114402	+	0.993435i	$0.463505\pi$
$270$	0	0
$271$	−752770.	−0.622643	−0.311322	−	0.950305i	$-0.600772\pi$
$271$	−752770.	−0.622643	−0.311322	+	0.950305i	$0.600772\pi$
$272$	46635.9	0.0382207
$273$	627186.	0.509319
$274$	247430.	0.199102
$275$	0	0
$276$	180892.	0.142938
$277$	−811445.	−0.635418	−0.317709	−	0.948188i	$-0.602914\pi$
$277$	−811445.	−0.635418	−0.317709	+	0.948188i	$0.602914\pi$
$278$	−1.08165e6	−0.839411
$279$	−52192.3	−0.0401417
$280$	0	0
$281$	−1.72395e6	−1.30244	−0.651221	−	0.758888i	$-0.725743\pi$
$281$	−1.72395e6	−1.30244	−0.651221	+	0.758888i	$0.725743\pi$
$282$	−677503.	−0.507327
$283$	−272979.	−0.202611	−0.101305	−	0.994855i	$-0.532302\pi$
$283$	−272979.	−0.202611	−0.101305	+	0.994855i	$0.532302\pi$
$284$	−60043.6	−0.0441744
$285$	0	0
$286$	121831.	0.0880731
$287$	−4.57446e6	−3.27820
$288$	−458596.	−0.325798
$289$	−923553.	−0.650455
$290$	0	0
$291$	−830515.	−0.574930
$292$	−139360.	−0.0956488
$293$	713441.	0.485500	0.242750	−	0.970089i	$-0.421951\pi$
$293$	713441.	0.485500	0.242750	+	0.970089i	$0.421951\pi$
$294$	1.50954e6	1.01854
$295$	0	0
$296$	1.66770e6	1.10634
$297$	88209.0	0.0580259
$298$	−301578.	−0.196725
$299$	296104.	0.191543
$300$	0	0
$301$	−1.01802e6	−0.647649
$302$	−607997.	−0.383605
$303$	319915.	0.200184
$304$	189405.	0.117546
$305$	0	0
$306$	207011.	0.126383
$307$	2.67734e6	1.62128	0.810640	−	0.585545i	$-0.199119\pi$
$307$	2.67734e6	1.62128	0.810640	+	0.585545i	$0.199119\pi$
$308$	−572365.	−0.343792
$309$	−538891.	−0.321074
$310$	0	0
$311$	−1.28078e6	−0.750886	−0.375443	−	0.926846i	$-0.622509\pi$
$311$	−1.28078e6	−0.750886	−0.375443	+	0.926846i	$0.622509\pi$
$312$	−460700.	−0.267936
$313$	−1.36808e6	−0.789313	−0.394656	−	0.918829i	$-0.629136\pi$
$313$	−1.36808e6	−0.789313	−0.394656	+	0.918829i	$0.629136\pi$
$314$	−229488.	−0.131352
$315$	0	0
$316$	−459569.	−0.258900
$317$	7686.28	0.00429604	0.00214802	−	0.999998i	$-0.499316\pi$
$317$	7686.28	0.00429604	0.00214802	+	0.999998i	$0.499316\pi$
$318$	864752.	0.479539
$319$	476431.	0.262134
$320$	0	0
$321$	803284.	0.435117
$322$	971746.	0.522292
$323$	2.01567e6	1.07501
$324$	−123607.	−0.0654155
$325$	0	0
$326$	862406.	0.449436
$327$	−205164.	−0.106104
$328$	3.36017e6	1.72455
$329$	5.21014e6	2.65375
$330$	0	0
$331$	958347.	0.480787	0.240393	−	0.970676i	$-0.422724\pi$
$331$	958347.	0.480787	0.240393	+	0.970676i	$0.422724\pi$
$332$	1.93663e6	0.964275
$333$	732430.	0.361956
$334$	1.76989e6	0.868121
$335$	0	0
$336$	−149590.	−0.0722862
$337$	−4.08768e6	−1.96066	−0.980330	−	0.197365i	$-0.936762\pi$
$337$	−4.08768e6	−1.96066	−0.980330	+	0.197365i	$0.936762\pi$
$338$	1.06749e6	0.508244
$339$	−1.49458e6	−0.706350
$340$	0	0
$341$	77966.3	0.0363096
$342$	840745.	0.388686
$343$	−7.38880e6	−3.39108
$344$	747787.	0.340707
$345$	0	0
$346$	1.18243e6	0.530988
$347$	−84250.2	−0.0375619	−0.0187809	−	0.999824i	$-0.505979\pi$
$347$	−84250.2	−0.0375619	−0.0187809	+	0.999824i	$0.505979\pi$
$348$	−667621.	−0.295517
$349$	1.11859e6	0.491597	0.245798	−	0.969321i	$-0.420950\pi$
$349$	1.11859e6	0.491597	0.245798	+	0.969321i	$0.420950\pi$
$350$	0	0
$351$	−202333.	−0.0876595
$352$	685063.	0.294696
$353$	3.73570e6	1.59564	0.797820	−	0.602896i	$-0.205987\pi$
$353$	3.73570e6	1.59564	0.797820	+	0.602896i	$0.205987\pi$
$354$	140168.	0.0594485
$355$	0	0
$356$	−934436.	−0.390773
$357$	−1.59196e6	−0.661090
$358$	−1.67929e6	−0.692499
$359$	1.45342e6	0.595189	0.297595	−	0.954692i	$-0.403816\pi$
$359$	1.45342e6	0.595189	0.297595	+	0.954692i	$0.403816\pi$
$360$	0	0
$361$	5.71027e6	2.30616
$362$	−86707.1	−0.0347763
$363$	−131769.	−0.0524864
$364$	1.31289e6	0.519366
$365$	0	0
$366$	−223053.	−0.0870374
$367$	−25363.4	−0.00982975	−0.00491487	−	0.999988i	$-0.501564\pi$
$367$	−25363.4	−0.00982975	−0.00491487	+	0.999988i	$0.501564\pi$
$368$	−70623.8	−0.0271851
$369$	1.47574e6	0.564214
$370$	0	0
$371$	−6.65013e6	−2.50839
$372$	−109254.	−0.0409336
$373$	1.72695e6	0.642699	0.321350	−	0.946961i	$-0.395864\pi$
$373$	1.72695e6	0.642699	0.321350	+	0.946961i	$0.395864\pi$
$374$	−309238.	−0.114318
$375$	0	0
$376$	−3.82711e6	−1.39605
$377$	−1.09283e6	−0.396005
$378$	−664012.	−0.239027
$379$	−4.29401e6	−1.53555	−0.767777	−	0.640718i	$-0.778637\pi$
$379$	−4.29401e6	−1.53555	−0.767777	+	0.640718i	$0.778637\pi$
$380$	0	0
$381$	1.43271e6	0.505644
$382$	2.05324e6	0.719915
$383$	−1.29297e6	−0.450392	−0.225196	−	0.974313i	$-0.572302\pi$
$383$	−1.29297e6	−0.450392	−0.225196	+	0.974313i	$0.572302\pi$
$384$	−890813.	−0.308289
$385$	0	0
$386$	332888.	0.113718
$387$	328418.	0.111468
$388$	−1.73851e6	−0.586272
$389$	1.55257e6	0.520209	0.260104	−	0.965581i	$-0.416243\pi$
$389$	1.55257e6	0.520209	0.260104	+	0.965581i	$0.416243\pi$
$390$	0	0
$391$	−751586.	−0.248620
$392$	8.52719e6	2.80279
$393$	−2.08957e6	−0.682456
$394$	1.76034e6	0.571290
$395$	0	0
$396$	184647.	0.0591705
$397$	−819569.	−0.260981	−0.130491	−	0.991450i	$-0.541655\pi$
$397$	−819569.	−0.260981	−0.130491	+	0.991450i	$0.541655\pi$
$398$	−2.51268e6	−0.795113
$399$	−6.46551e6	−2.03316
$400$	0	0
$401$	1.38956e6	0.431536	0.215768	−	0.976445i	$-0.430774\pi$
$401$	1.38956e6	0.431536	0.215768	+	0.976445i	$0.430774\pi$
$402$	1.85284e6	0.571838
$403$	−178839.	−0.0548528
$404$	669677.	0.204132
$405$	0	0
$406$	−3.58643e6	−1.07981
$407$	−1.09412e6	−0.327401
$408$	1.16937e6	0.347778
$409$	−3.53091e6	−1.04371	−0.521854	−	0.853035i	$-0.674759\pi$
$409$	−3.53091e6	−1.04371	−0.521854	+	0.853035i	$0.674759\pi$
$410$	0	0
$411$	613849.	0.179249
$412$	−1.12806e6	−0.327407
$413$	−1.07792e6	−0.310966
$414$	−313490.	−0.0898923
$415$	0	0
$416$	−1.57139e6	−0.445196
$417$	−2.68346e6	−0.755710
$418$	−1.25593e6	−0.351580
$419$	−6.69977e6	−1.86434	−0.932170	−	0.362021i	$-0.882087\pi$
$419$	−6.69977e6	−1.86434	−0.932170	+	0.362021i	$0.882087\pi$
$420$	0	0
$421$	−5.01124e6	−1.37797	−0.688986	−	0.724775i	$-0.741944\pi$
$421$	−5.01124e6	−1.37797	−0.688986	+	0.724775i	$0.741944\pi$
$422$	1.59990e6	0.437331
$423$	−1.68082e6	−0.456740
$424$	4.88485e6	1.31958
$425$	0	0
$426$	104057.	0.0277809
$427$	1.71533e6	0.455279
$428$	1.68151e6	0.443701
$429$	302251.	0.0792910
$430$	0	0
$431$	−1.14741e6	−0.297526	−0.148763	−	0.988873i	$-0.547529\pi$
$431$	−1.14741e6	−0.297526	−0.148763	+	0.988873i	$0.547529\pi$
$432$	48258.6	0.0124413
$433$	1.13393e6	0.290649	0.145324	−	0.989384i	$-0.453577\pi$
$433$	1.13393e6	0.290649	0.145324	+	0.989384i	$0.453577\pi$
$434$	−586908.	−0.149570
$435$	0	0
$436$	−429469.	−0.108197
$437$	−3.05246e6	−0.764622
$438$	241513.	0.0601526
$439$	7.44692e6	1.84423	0.922115	−	0.386915i	$-0.126459\pi$
$439$	7.44692e6	1.84423	0.922115	+	0.386915i	$0.126459\pi$
$440$	0	0
$441$	3.74503e6	0.916977
$442$	709328.	0.172700
$443$	−6.72521e6	−1.62816	−0.814079	−	0.580754i	$-0.802758\pi$
$443$	−6.72521e6	−1.62816	−0.814079	+	0.580754i	$0.802758\pi$
$444$	1.53319e6	0.369096
$445$	0	0
$446$	4.10416e6	0.976984
$447$	−748183.	−0.177108
$448$	−5.68883e6	−1.33915
$449$	2.17793e6	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$449$	2.17793e6	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$450$	0	0
$451$	−2.20450e6	−0.510351
$452$	−3.12860e6	−0.720284
$453$	−1.50838e6	−0.345354
$454$	−2.97601e6	−0.677634
$455$	0	0
$456$	4.74924e6	1.06958
$457$	3.72380e6	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$457$	3.72380e6	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$458$	3.63865e6	0.810544
$459$	513572.	0.113781
$460$	0	0
$461$	4.74202e6	1.03923	0.519615	−	0.854401i	$-0.326076\pi$
$461$	4.74202e6	1.03923	0.519615	+	0.854401i	$0.326076\pi$
$462$	991919.	0.216208
$463$	4.82803e6	1.04669	0.523344	−	0.852121i	$-0.324684\pi$
$463$	4.82803e6	1.04669	0.523344	+	0.852121i	$0.324684\pi$
$464$	260652.	0.0562039
$465$	0	0
$466$	2664.81	0.000568462 0
$467$	−7.56254e6	−1.60463	−0.802316	−	0.596900i	$-0.796399\pi$
$467$	−7.56254e6	−1.60463	−0.802316	+	0.596900i	$0.796399\pi$
$468$	−423543.	−0.0893887
$469$	−1.42487e7	−2.99119
$470$	0	0
$471$	−569337.	−0.118254
$472$	791788.	0.163589
$473$	−490599.	−0.100826
$474$	796441.	0.162820
$475$	0	0
$476$	−3.33243e6	−0.674131
$477$	2.14536e6	0.431722
$478$	3.85999e6	0.772710
$479$	−1.71897e6	−0.342319	−0.171159	−	0.985243i	$-0.554751\pi$
$479$	−1.71897e6	−0.342319	−0.171159	+	0.985243i	$0.554751\pi$
$480$	0	0
$481$	2.50969e6	0.494604
$482$	2.40023e6	0.470582
$483$	2.41080e6	0.470212
$484$	−275831.	−0.0535218
$485$	0	0
$486$	214213.	0.0411392
$487$	3.08125e6	0.588715	0.294357	−	0.955695i	$-0.404894\pi$
$487$	3.08125e6	0.588715	0.294357	+	0.955695i	$0.404894\pi$
$488$	−1.25999e6	−0.239508
$489$	2.13954e6	0.404621
$490$	0	0
$491$	−1.05909e6	−0.198258	−0.0991290	−	0.995075i	$-0.531606\pi$
$491$	−1.05909e6	−0.198258	−0.0991290	+	0.995075i	$0.531606\pi$
$492$	3.08916e6	0.575344
$493$	2.77388e6	0.514009
$494$	2.88084e6	0.531131
$495$	0	0
$496$	42654.9	0.00778510
$497$	−800218.	−0.145317
$498$	−3.35621e6	−0.606423
$499$	2.26415e6	0.407056	0.203528	−	0.979069i	$-0.434759\pi$
$499$	2.26415e6	0.407056	0.203528	+	0.979069i	$0.434759\pi$
$500$	0	0
$501$	4.39092e6	0.781558
$502$	−4.89250e6	−0.866507
$503$	7.65222e6	1.34855	0.674276	−	0.738480i	$-0.264456\pi$
$503$	7.65222e6	1.34855	0.674276	+	0.738480i	$0.264456\pi$
$504$	−3.75090e6	−0.657748
$505$	0	0
$506$	468299.	0.0813106
$507$	2.64834e6	0.457566
$508$	2.99908e6	0.515619
$509$	489107.	0.0836777	0.0418388	−	0.999124i	$-0.486678\pi$
$509$	489107.	0.0836777	0.0418388	+	0.999124i	$0.486678\pi$
$510$	0	0
$511$	−1.85728e6	−0.314649
$512$	765484.	0.129051
$513$	2.08580e6	0.349929
$514$	−5.98627e6	−0.999421
$515$	0	0
$516$	687475.	0.113667
$517$	2.51085e6	0.413137
$518$	8.23624e6	1.34867
$519$	2.93348e6	0.478041
$520$	0	0
$521$	1.36556e6	0.220402	0.110201	−	0.993909i	$-0.464851\pi$
$521$	1.36556e6	0.220402	0.110201	+	0.993909i	$0.464851\pi$
$522$	1.15700e6	0.185847
$523$	5.63994e6	0.901613	0.450807	−	0.892622i	$-0.351136\pi$
$523$	5.63994e6	0.901613	0.450807	+	0.892622i	$0.351136\pi$
$524$	−4.37408e6	−0.695919
$525$	0	0
$526$	−3.14945e6	−0.496330
$527$	453937.	0.0711982
$528$	−72089.9	−0.0112536
$529$	−5.29817e6	−0.823164
$530$	0	0
$531$	347742.	0.0535207
$532$	−1.35342e7	−2.07326
$533$	5.05667e6	0.770986
$534$	1.61939e6	0.245753
$535$	0	0
$536$	1.04664e7	1.57357
$537$	−4.16615e6	−0.623447
$538$	−985095.	−0.146731
$539$	−5.59442e6	−0.829437
$540$	0	0
$541$	8.24934e6	1.21179	0.605893	−	0.795546i	$-0.292816\pi$
$541$	8.24934e6	1.21179	0.605893	+	0.795546i	$0.292816\pi$
$542$	2.73084e6	0.399299
$543$	−215111.	−0.0313086
$544$	3.98859e6	0.577859
$545$	0	0
$546$	−2.27526e6	−0.326624
$547$	4.74537e6	0.678112	0.339056	−	0.940766i	$-0.389892\pi$
$547$	4.74537e6	0.678112	0.339056	+	0.940766i	$0.389892\pi$
$548$	1.28497e6	0.182785
$549$	−553372.	−0.0783586
$550$	0	0
$551$	1.12657e7	1.58082
$552$	−1.77086e6	−0.247363
$553$	−6.12480e6	−0.851685
$554$	2.94369e6	0.407491
$555$	0	0
$556$	−5.61728e6	−0.770617
$557$	−2.99690e6	−0.409293	−0.204647	−	0.978836i	$-0.565605\pi$
$557$	−2.99690e6	−0.409293	−0.204647	+	0.978836i	$0.565605\pi$
$558$	189339.	0.0257427
$559$	1.12533e6	0.152318
$560$	0	0
$561$	−767188.	−0.102919
$562$	6.25400e6	0.835251
$563$	3.89491e6	0.517876	0.258938	−	0.965894i	$-0.416627\pi$
$563$	3.89491e6	0.517876	0.258938	+	0.965894i	$0.416627\pi$
$564$	−3.51844e6	−0.465750
$565$	0	0
$566$	990290.	0.129934
$567$	−1.64735e6	−0.215192
$568$	587801.	0.0764468
$569$	2.08127e6	0.269493	0.134747	−	0.990880i	$-0.456978\pi$
$569$	2.08127e6	0.269493	0.134747	+	0.990880i	$0.456978\pi$
$570$	0	0
$571$	1.28958e7	1.65523	0.827615	−	0.561297i	$-0.189697\pi$
$571$	1.28958e7	1.65523	0.827615	+	0.561297i	$0.189697\pi$
$572$	632700.	0.0808551
$573$	5.09387e6	0.648129
$574$	1.65948e7	2.10229
$575$	0	0
$576$	1.83524e6	0.230482
$577$	−1.26163e6	−0.157759	−0.0788795	−	0.996884i	$-0.525134\pi$
$577$	−1.26163e6	−0.157759	−0.0788795	+	0.996884i	$0.525134\pi$
$578$	3.35039e6	0.417134
$579$	825861.	0.102379
$580$	0	0
$581$	2.58100e7	3.17210
$582$	3.01288e6	0.368701
$583$	−3.20480e6	−0.390508
$584$	1.36427e6	0.165527
$585$	0	0
$586$	−2.58816e6	−0.311349
$587$	1.25673e7	1.50538	0.752689	−	0.658376i	$-0.228756\pi$
$587$	1.25673e7	1.50538	0.752689	+	0.658376i	$0.228756\pi$
$588$	7.83945e6	0.935065
$589$	1.84360e6	0.218967
$590$	0	0
$591$	4.36723e6	0.514324
$592$	−598588.	−0.0701978
$593$	4.32620e6	0.505207	0.252604	−	0.967570i	$-0.418713\pi$
$593$	4.32620e6	0.505207	0.252604	+	0.967570i	$0.418713\pi$
$594$	−319997.	−0.0372118
$595$	0	0
$596$	−1.56617e6	−0.180602
$597$	−6.23369e6	−0.715829
$598$	−1.07418e6	−0.122836
$599$	1.45262e7	1.65419	0.827097	−	0.562060i	$-0.189991\pi$
$599$	1.45262e7	1.65419	0.827097	+	0.562060i	$0.189991\pi$
$600$	0	0
$601$	−380688.	−0.0429915	−0.0214958	−	0.999769i	$-0.506843\pi$
$601$	−380688.	−0.0429915	−0.0214958	+	0.999769i	$0.506843\pi$
$602$	3.69309e6	0.415335
$603$	4.59671e6	0.514818
$604$	−3.15748e6	−0.352167
$605$	0	0
$606$	−1.16056e6	−0.128377
$607$	4.99233e6	0.549961	0.274980	−	0.961450i	$-0.411329\pi$
$607$	4.99233e6	0.549961	0.274980	+	0.961450i	$0.411329\pi$
$608$	1.61991e7	1.77718
$609$	−8.89757e6	−0.972139
$610$	0	0
$611$	−5.75936e6	−0.624125
$612$	1.07506e6	0.116026
$613$	−2.99353e6	−0.321760	−0.160880	−	0.986974i	$-0.551433\pi$
$613$	−2.99353e6	−0.321760	−0.160880	+	0.986974i	$0.551433\pi$
$614$	−9.71265e6	−1.03972
$615$	0	0
$616$	5.60320e6	0.594955
$617$	−6.27781e6	−0.663889	−0.331945	−	0.943299i	$-0.607705\pi$
$617$	−6.27781e6	−0.663889	−0.331945	+	0.943299i	$0.607705\pi$
$618$	1.95494e6	0.205903
$619$	3.96477e6	0.415902	0.207951	−	0.978139i	$-0.433320\pi$
$619$	3.96477e6	0.415902	0.207951	+	0.978139i	$0.433320\pi$
$620$	0	0
$621$	−777736.	−0.0809288
$622$	4.64631e6	0.481540
$623$	−1.24535e7	−1.28550
$624$	165359.	0.0170007
$625$	0	0
$626$	4.96299e6	0.506183
$627$	−3.11583e6	−0.316523
$628$	−1.19179e6	−0.120587
$629$	−6.37023e6	−0.641990
$630$	0	0
$631$	−9.62587e6	−0.962425	−0.481212	−	0.876604i	$-0.659803\pi$
$631$	−9.62587e6	−0.962425	−0.481212	+	0.876604i	$0.659803\pi$
$632$	4.49898e6	0.448044
$633$	3.96918e6	0.393724
$634$	−27883.7	−0.00275503
$635$	0	0
$636$	4.49087e6	0.440239
$637$	1.28324e7	1.25303
$638$	−1.72836e6	−0.168105
$639$	258154.	0.0250108
$640$	0	0
$641$	−2.20090e6	−0.211571	−0.105785	−	0.994389i	$-0.533736\pi$
$641$	−2.20090e6	−0.211571	−0.105785	+	0.994389i	$0.533736\pi$
$642$	−2.91409e6	−0.279039
$643$	−2.00555e7	−1.91296	−0.956482	−	0.291791i	$-0.905749\pi$
$643$	−2.00555e7	−1.91296	−0.956482	+	0.291791i	$0.905749\pi$
$644$	5.04652e6	0.479488
$645$	0	0
$646$	−7.31229e6	−0.689401
$647$	−128900.	−0.0121057	−0.00605287	−	0.999982i	$-0.501927\pi$
$647$	−128900.	−0.0121057	−0.00605287	+	0.999982i	$0.501927\pi$
$648$	1.21006e6	0.113206
$649$	−519467.	−0.0484113
$650$	0	0
$651$	−1.45606e6	−0.134656
$652$	4.47869e6	0.412603
$653$	−3.03250e6	−0.278303	−0.139151	−	0.990271i	$-0.544437\pi$
$653$	−3.03250e6	−0.278303	−0.139151	+	0.990271i	$0.544437\pi$
$654$	744277.	0.0680441
$655$	0	0
$656$	−1.20607e6	−0.109424
$657$	599168.	0.0541546
$658$	−1.89009e7	−1.70184
$659$	−1.59132e7	−1.42740	−0.713698	−	0.700454i	$-0.752981\pi$
$659$	−1.59132e7	−1.42740	−0.713698	+	0.700454i	$0.752981\pi$
$660$	0	0
$661$	6.50892e6	0.579436	0.289718	−	0.957112i	$-0.406439\pi$
$661$	6.50892e6	0.579436	0.289718	+	0.957112i	$0.406439\pi$
$662$	−3.47661e6	−0.308327
$663$	1.75977e6	0.155479
$664$	−1.89587e7	−1.66874
$665$	0	0
$666$	−2.65705e6	−0.232121
$667$	−4.20067e6	−0.365598
$668$	9.19149e6	0.796975
$669$	1.01820e7	0.879565
$670$	0	0
$671$	826643.	0.0708780
$672$	−1.27939e7	−1.09290
$673$	5.66035e6	0.481732	0.240866	−	0.970558i	$-0.422569\pi$
$673$	5.66035e6	0.481732	0.240866	+	0.970558i	$0.422569\pi$
$674$	1.48290e7	1.25736
$675$	0	0
$676$	5.54375e6	0.466592
$677$	1.53601e7	1.28802	0.644012	−	0.765016i	$-0.277269\pi$
$677$	1.53601e7	1.28802	0.644012	+	0.765016i	$0.277269\pi$
$678$	5.42192e6	0.452979
$679$	−2.31697e7	−1.92861
$680$	0	0
$681$	−7.38319e6	−0.610065
$682$	−282840.	−0.0232852
$683$	−1.47069e6	−0.120634	−0.0603170	−	0.998179i	$-0.519211\pi$
$683$	−1.47069e6	−0.120634	−0.0603170	+	0.998179i	$0.519211\pi$
$684$	4.36620e6	0.356832
$685$	0	0
$686$	2.68045e7	2.17469
$687$	9.02712e6	0.729722
$688$	−268404.	−0.0216181
$689$	7.35114e6	0.589939
$690$	0	0
$691$	−3.65617e6	−0.291294	−0.145647	−	0.989337i	$-0.546526\pi$
$691$	−3.65617e6	−0.291294	−0.145647	+	0.989337i	$0.546526\pi$
$692$	6.14064e6	0.487471
$693$	2.46085e6	0.194649
$694$	305636.	0.0240883
$695$	0	0
$696$	6.53571e6	0.511411
$697$	−1.28351e7	−1.00073
$698$	−4.05794e6	−0.315259
$699$	6611.12	0.000511779 0
$700$	0	0
$701$	−5.21232e6	−0.400623	−0.200312	−	0.979732i	$-0.564195\pi$
$701$	−5.21232e6	−0.400623	−0.200312	+	0.979732i	$0.564195\pi$
$702$	734008.	0.0562157
$703$	−2.58718e7	−1.97442
$704$	−2.74153e6	−0.208479
$705$	0	0
$706$	−1.35521e7	−1.02328
$707$	8.92498e6	0.671519
$708$	727928.	0.0545764
$709$	−1.48388e7	−1.10862	−0.554311	−	0.832310i	$-0.687018\pi$
$709$	−1.48388e7	−1.10862	−0.554311	+	0.832310i	$0.687018\pi$
$710$	0	0
$711$	1.97589e6	0.146585
$712$	9.14772e6	0.676258
$713$	−687427.	−0.0506410
$714$	5.77517e6	0.423954
$715$	0	0
$716$	−8.72099e6	−0.635745
$717$	9.57625e6	0.695661
$718$	−5.27260e6	−0.381693
$719$	1.37497e7	0.991906	0.495953	−	0.868349i	$-0.334819\pi$
$719$	1.37497e7	0.991906	0.495953	+	0.868349i	$0.334819\pi$
$720$	0	0
$721$	−1.50339e7	−1.07705
$722$	−2.07153e7	−1.47893
$723$	5.95473e6	0.423659
$724$	−450292.	−0.0319262
$725$	0	0
$726$	478021.	0.0336593
$727$	8.20549e6	0.575796	0.287898	−	0.957661i	$-0.407044\pi$
$727$	8.20549e6	0.575796	0.287898	+	0.957661i	$0.407044\pi$
$728$	−1.28526e7	−0.898797
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.85638e6	−0.197707
$732$	−1.15837e6	−0.0799043
$733$	−1.20636e7	−0.829312	−0.414656	−	0.909978i	$-0.636098\pi$
$733$	−1.20636e7	−0.829312	−0.414656	+	0.909978i	$0.636098\pi$
$734$	92011.3	0.00630378
$735$	0	0
$736$	−6.04017e6	−0.411012
$737$	−6.86668e6	−0.465670
$738$	−5.35357e6	−0.361828
$739$	−1.02319e7	−0.689203	−0.344602	−	0.938749i	$-0.611986\pi$
$739$	−1.02319e7	−0.689203	−0.344602	+	0.938749i	$0.611986\pi$
$740$	0	0
$741$	7.14706e6	0.478170
$742$	2.41248e7	1.60862
$743$	−2.41980e7	−1.60808	−0.804039	−	0.594577i	$-0.797320\pi$
$743$	−2.41980e7	−1.60808	−0.804039	+	0.594577i	$0.797320\pi$
$744$	1.06955e6	0.0708383
$745$	0	0
$746$	−6.26489e6	−0.412161
$747$	−8.32642e6	−0.545955
$748$	−1.60595e6	−0.104949
$749$	2.24100e7	1.45961
$750$	0	0
$751$	1.10121e7	0.712476	0.356238	−	0.934395i	$-0.384059\pi$
$751$	1.10121e7	0.712476	0.356238	+	0.934395i	$0.384059\pi$
$752$	1.37367e6	0.0885803
$753$	−1.21378e7	−0.780104
$754$	3.96449e6	0.253956
$755$	0	0
$756$	−3.44838e6	−0.219437
$757$	2.08747e7	1.32398	0.661989	−	0.749513i	$-0.269712\pi$
$757$	2.08747e7	1.32398	0.661989	+	0.749513i	$0.269712\pi$
$758$	1.55775e7	0.984744
$759$	1.16180e6	0.0732028
$760$	0	0
$761$	2.85723e7	1.78848	0.894240	−	0.447587i	$-0.147716\pi$
$761$	2.85723e7	1.78848	0.894240	+	0.447587i	$0.147716\pi$
$762$	−5.19746e6	−0.324268
$763$	−5.72365e6	−0.355928
$764$	1.06630e7	0.660915
$765$	0	0
$766$	4.69052e6	0.288835
$767$	1.19155e6	0.0731347
$768$	9.75692e6	0.596911
$769$	−3.07246e6	−0.187357	−0.0936787	−	0.995602i	$-0.529863\pi$
$769$	−3.07246e6	−0.187357	−0.0936787	+	0.995602i	$0.529863\pi$
$770$	0	0
$771$	−1.48513e7	−0.899765
$772$	1.72877e6	0.104398
$773$	3.02076e7	1.81831	0.909153	−	0.416461i	$-0.136730\pi$
$773$	3.02076e7	1.81831	0.909153	+	0.416461i	$0.136730\pi$
$774$	−1.19141e6	−0.0714838
$775$	0	0
$776$	1.70193e7	1.01458
$777$	2.04333e7	1.21419
$778$	−5.63229e6	−0.333608
$779$	−5.21280e7	−3.07771
$780$	0	0
$781$	−385638.	−0.0226231
$782$	2.72654e6	0.159439
$783$	2.87040e6	0.167316
$784$	−3.06067e6	−0.177839
$785$	0	0
$786$	7.58036e6	0.437657
$787$	2.07854e7	1.19625	0.598126	−	0.801402i	$-0.295912\pi$
$787$	2.07854e7	1.19625	0.598126	+	0.801402i	$0.295912\pi$
$788$	9.14190e6	0.524470
$789$	−7.81346e6	−0.446839
$790$	0	0
$791$	−4.16957e7	−2.36946
$792$	−1.80762e6	−0.102399
$793$	−1.89615e6	−0.107075
$794$	2.97317e6	0.167366
$795$	0	0
$796$	−1.30490e7	−0.729950
$797$	7.95535e6	0.443622	0.221811	−	0.975090i	$-0.428803\pi$
$797$	7.95535e6	0.443622	0.221811	+	0.975090i	$0.428803\pi$
$798$	2.34551e7	1.30385
$799$	1.46187e7	0.810106
$800$	0	0
$801$	4.01755e6	0.221248
$802$	−5.04094e6	−0.276742
$803$	−895054.	−0.0489847
$804$	9.62226e6	0.524973
$805$	0	0
$806$	648776.	0.0351769
$807$	−2.44392e6	−0.132100
$808$	−6.55584e6	−0.353265
$809$	−3.04660e7	−1.63661	−0.818303	−	0.574787i	$-0.805085\pi$
$809$	−3.04660e7	−1.63661	−0.818303	+	0.574787i	$0.805085\pi$
$810$	0	0
$811$	1.28041e7	0.683594	0.341797	−	0.939774i	$-0.388964\pi$
$811$	1.28041e7	0.683594	0.341797	+	0.939774i	$0.388964\pi$
$812$	−1.86252e7	−0.991316
$813$	6.77493e6	0.359483
$814$	3.96917e6	0.209961
$815$	0	0
$816$	−419723.	−0.0220667
$817$	−1.16008e7	−0.608040
$818$	1.28092e7	0.669325
$819$	−5.64468e6	−0.294055
$820$	0	0
$821$	1.13635e7	0.588373	0.294186	−	0.955748i	$-0.404951\pi$
$821$	1.13635e7	0.588373	0.294186	+	0.955748i	$0.404951\pi$
$822$	−2.22687e6	−0.114952
$823$	−8.23741e6	−0.423927	−0.211964	−	0.977278i	$-0.567986\pi$
$823$	−8.23741e6	−0.423927	−0.211964	+	0.977278i	$0.567986\pi$
$824$	1.10432e7	0.566600
$825$	0	0
$826$	3.91040e6	0.199421
$827$	−2.21035e7	−1.12382	−0.561909	−	0.827199i	$-0.689933\pi$
$827$	−2.21035e7	−1.12382	−0.561909	+	0.827199i	$0.689933\pi$
$828$	−1.62803e6	−0.0825252
$829$	−5.57875e6	−0.281936	−0.140968	−	0.990014i	$-0.545021\pi$
$829$	−5.57875e6	−0.281936	−0.140968	+	0.990014i	$0.545021\pi$
$830$	0	0
$831$	7.30300e6	0.366859
$832$	6.28851e6	0.314948
$833$	−3.25720e7	−1.62641
$834$	9.73484e6	0.484634
$835$	0	0
$836$	−6.52235e6	−0.322766
$837$	469731.	0.0231758
$838$	2.43049e7	1.19559
$839$	−2.33576e7	−1.14558	−0.572788	−	0.819704i	$-0.694138\pi$
$839$	−2.33576e7	−1.14558	−0.572788	+	0.819704i	$0.694138\pi$
$840$	0	0
$841$	−5.00769e6	−0.244145
$842$	1.81794e7	0.883688
$843$	1.55155e7	0.751965
$844$	8.30866e6	0.401490
$845$	0	0
$846$	6.09753e6	0.292906
$847$	−3.67608e6	−0.176067
$848$	−1.75332e6	−0.0837284
$849$	2.45681e6	0.116977
$850$	0	0
$851$	9.64685e6	0.456627
$852$	540393.	0.0255041
$853$	−8.94855e6	−0.421095	−0.210548	−	0.977584i	$-0.567525\pi$
$853$	−8.94855e6	−0.421095	−0.210548	+	0.977584i	$0.567525\pi$
$854$	−6.22273e6	−0.291969
$855$	0	0
$856$	−1.64612e7	−0.767853
$857$	2.03190e6	0.0945039	0.0472520	−	0.998883i	$-0.484954\pi$
$857$	2.03190e6	0.0945039	0.0472520	+	0.998883i	$0.484954\pi$
$858$	−1.09648e6	−0.0508490
$859$	−1.92359e7	−0.889467	−0.444734	−	0.895663i	$-0.646702\pi$
$859$	−1.92359e7	−0.889467	−0.444734	+	0.895663i	$0.646702\pi$
$860$	0	0
$861$	4.11701e7	1.89267
$862$	4.16247e6	0.190802
$863$	−2.00380e7	−0.915856	−0.457928	−	0.888989i	$-0.651408\pi$
$863$	−2.00380e7	−0.915856	−0.457928	+	0.888989i	$0.651408\pi$
$864$	4.12736e6	0.188100
$865$	0	0
$866$	−4.11360e6	−0.186392
$867$	8.31197e6	0.375540
$868$	−3.04796e6	−0.137312
$869$	−2.95164e6	−0.132591
$870$	0	0
$871$	1.57507e7	0.703486
$872$	4.20431e6	0.187242
$873$	7.47464e6	0.331936
$874$	1.10735e7	0.490349
$875$	0	0
$876$	1.25424e6	0.0552229
$877$	−1.98487e7	−0.871430	−0.435715	−	0.900085i	$-0.643504\pi$
$877$	−1.98487e7	−0.871430	−0.435715	+	0.900085i	$0.643504\pi$
$878$	−2.70153e7	−1.18270
$879$	−6.42097e6	−0.280304
$880$	0	0
$881$	−2.93546e7	−1.27420	−0.637099	−	0.770782i	$-0.719866\pi$
$881$	−2.93546e7	−1.27420	−0.637099	+	0.770782i	$0.719866\pi$
$882$	−1.35859e7	−0.588054
$883$	−3.85199e7	−1.66258	−0.831291	−	0.555838i	$-0.812398\pi$
$883$	−3.85199e7	−1.66258	−0.831291	+	0.555838i	$0.812398\pi$
$884$	3.68372e6	0.158546
$885$	0	0
$886$	2.43972e7	1.04413
$887$	2.68797e6	0.114714	0.0573569	−	0.998354i	$-0.481733\pi$
$887$	2.68797e6	0.114714	0.0573569	+	0.998354i	$0.481733\pi$
$888$	−1.50093e7	−0.638745
$889$	3.99696e7	1.69619
$890$	0	0
$891$	−793881.	−0.0335013
$892$	2.13139e7	0.896916
$893$	5.93719e7	2.49145
$894$	2.71420e6	0.113579
$895$	0	0
$896$	−2.48519e7	−1.03416
$897$	−2.66493e6	−0.110587
$898$	−7.90093e6	−0.326955
$899$	2.53709e6	0.104698
$900$	0	0
$901$	−1.86590e7	−0.765733
$902$	7.99731e6	0.327286
$903$	9.16218e6	0.373921
$904$	3.06276e7	1.24650
$905$	0	0
$906$	5.47197e6	0.221474
$907$	7.78340e6	0.314160	0.157080	−	0.987586i	$-0.449792\pi$
$907$	7.78340e6	0.314160	0.157080	+	0.987586i	$0.449792\pi$
$908$	−1.54552e7	−0.622099
$909$	−2.87924e6	−0.115576
$910$	0	0
$911$	3.15361e7	1.25896	0.629481	−	0.777016i	$-0.283268\pi$
$911$	3.15361e7	1.25896	0.629481	+	0.777016i	$0.283268\pi$
$912$	−1.70465e6	−0.0678653
$913$	1.24382e7	0.493835
$914$	−1.35089e7	−0.534878
$915$	0	0
$916$	1.88964e7	0.744117
$917$	−5.82946e7	−2.28931
$918$	−1.86310e6	−0.0729673
$919$	4.21184e7	1.64507	0.822533	−	0.568717i	$-0.192560\pi$
$919$	4.21184e7	1.64507	0.822533	+	0.568717i	$0.192560\pi$
$920$	0	0
$921$	−2.40961e7	−0.936047
$922$	−1.72027e7	−0.666454
$923$	884572.	0.0341766
$924$	5.15128e6	0.198489
$925$	0	0
$926$	−1.75147e7	−0.671237
$927$	4.85002e6	0.185372
$928$	2.22925e7	0.849746
$929$	9.43595e6	0.358712	0.179356	−	0.983784i	$-0.442599\pi$
$929$	9.43595e6	0.358712	0.179356	+	0.983784i	$0.442599\pi$
$930$	0	0
$931$	−1.32287e8	−5.00197
$932$	13839.0	0.000521874 0
$933$	1.15270e7	0.433524
$934$	2.74348e7	1.02904
$935$	0	0
$936$	4.14630e6	0.154693
$937$	−2.17869e7	−0.810674	−0.405337	−	0.914167i	$-0.632846\pi$
$937$	−2.17869e7	−0.810674	−0.405337	+	0.914167i	$0.632846\pi$
$938$	5.16904e7	1.91824
$939$	1.23127e7	0.455710
$940$	0	0
$941$	2.18678e7	0.805065	0.402532	−	0.915406i	$-0.368130\pi$
$941$	2.18678e7	0.805065	0.402532	+	0.915406i	$0.368130\pi$
$942$	2.06540e6	0.0758361
$943$	1.94370e7	0.711787
$944$	−284197.	−0.0103798
$945$	0	0
$946$	1.77976e6	0.0646595
$947$	−2.42335e7	−0.878094	−0.439047	−	0.898464i	$-0.644684\pi$
$947$	−2.42335e7	−0.878094	−0.439047	+	0.898464i	$0.644684\pi$
$948$	4.13612e6	0.149476
$949$	2.05307e6	0.0740010
$950$	0	0
$951$	−69176.5	−0.00248032
$952$	3.26231e7	1.16663
$953$	−5.56092e7	−1.98342	−0.991710	−	0.128499i	$-0.958984\pi$
$953$	−5.56092e7	−1.98342	−0.991710	+	0.128499i	$0.958984\pi$
$954$	−7.78277e6	−0.276862
$955$	0	0
$956$	2.00459e7	0.709383
$957$	−4.28788e6	−0.151343
$958$	6.23595e6	0.219528
$959$	1.71251e7	0.601294
$960$	0	0
$961$	−2.82140e7	−0.985498
$962$	−9.10446e6	−0.317188
$963$	−7.22956e6	−0.251215
$964$	1.24650e7	0.432016
$965$	0	0
$966$	−8.74571e6	−0.301545
$967$	2.09122e7	0.719173	0.359586	−	0.933112i	$-0.382918\pi$
$967$	2.09122e7	0.719173	0.359586	+	0.933112i	$0.382918\pi$
$968$	2.70027e6	0.0926229
$969$	−1.81410e7	−0.620659
$970$	0	0
$971$	1.68184e7	0.572448	0.286224	−	0.958163i	$-0.407600\pi$
$971$	1.68184e7	0.572448	0.286224	+	0.958163i	$0.407600\pi$
$972$	1.11246e6	0.0377676
$973$	−7.48630e7	−2.53504
$974$	−1.11779e7	−0.377541
$975$	0	0
$976$	452251.	0.0151969
$977$	3.45472e7	1.15792	0.578958	−	0.815358i	$-0.303460\pi$
$977$	3.45472e7	1.15792	0.578958	+	0.815358i	$0.303460\pi$
$978$	−7.76166e6	−0.259482
$979$	−6.00153e6	−0.200127
$980$	0	0
$981$	1.84648e6	0.0612592
$982$	3.84210e6	0.127142
$983$	3.40232e7	1.12303	0.561516	−	0.827466i	$-0.310218\pi$
$983$	3.40232e7	1.12303	0.561516	+	0.827466i	$0.310218\pi$
$984$	−3.02415e7	−0.995671
$985$	0	0
$986$	−1.00629e7	−0.329632
$987$	−4.68913e7	−1.53214
$988$	1.49609e7	0.487602
$989$	4.32560e6	0.140623
$990$	0	0
$991$	−4.47458e7	−1.44733	−0.723665	−	0.690151i	$-0.757544\pi$
$991$	−4.47458e7	−1.44733	−0.723665	+	0.690151i	$0.757544\pi$
$992$	3.64810e6	0.117703
$993$	−8.62512e6	−0.277583
$994$	2.90297e6	0.0931915
$995$	0	0
$996$	−1.74297e7	−0.556725
$997$	5.86494e6	0.186864	0.0934320	−	0.995626i	$-0.470216\pi$
$997$	5.86494e6	0.186864	0.0934320	+	0.995626i	$0.470216\pi$
$998$	−8.21371e6	−0.261044
$999$	−6.59187e6	−0.208975

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} -121.000 q^{11} +169.557 q^{12} +277.549 q^{13} +910.853 q^{14} -66.1983 q^{16} -704.489 q^{17} -293.845 q^{18} -2861.18 q^{19} +2259.73 q^{21} +438.954 q^{22} +1066.85 q^{23} -1659.89 q^{24} -1006.87 q^{26} -729.000 q^{27} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} -5661.67 q^{32} +1089.00 q^{33} +2555.69 q^{34} -1526.01 q^{36} +9042.34 q^{37} +10379.6 q^{38} -2497.94 q^{39} +18219.0 q^{41} -8197.68 q^{42} +4054.54 q^{43} +2279.60 q^{44} -3870.24 q^{46} -20750.8 q^{47} +595.785 q^{48} +46234.9 q^{49} +6340.40 q^{51} -5228.92 q^{52} +26485.9 q^{53} +2644.61 q^{54} -46307.4 q^{56} +25750.7 q^{57} +14283.9 q^{58} +4293.12 q^{59} -6831.76 q^{61} +2337.52 q^{62} -20337.6 q^{63} +22657.3 q^{64} -3950.59 q^{66} +56749.5 q^{67} +13272.3 q^{68} -9601.68 q^{69} +3187.09 q^{71} +14939.0 q^{72} +7397.14 q^{73} -32803.1 q^{74} +53903.7 q^{76} +30380.9 q^{77} +9061.82 q^{78} +24393.7 q^{79} +6561.00 q^{81} -66093.5 q^{82} -102795. q^{83} -42572.6 q^{84} -14708.7 q^{86} +35437.0 q^{87} -22316.3 q^{88} +49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} +5799.15 q^{93} +75278.1 q^{94} +50955.1 q^{96} +92279.5 q^{97} -167727. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * q^98 - 19602 * q^99

Introduction

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