LMFDB - Embedded newform 825.6.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + \cdots - 7036608 $ x^10 - x^9 - 271x^8 + 309x^7 + 24456x^6 - 33410x^5 - 822204x^4 + 1367872x^3 + 7443872x^2 - 12856224x - 7036608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.65849$ 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)

$f(q)$	$=$	$q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.872 q^{12} -849.133 q^{13} +859.262 q^{14} -1166.52 q^{16} -608.382 q^{17} -620.338 q^{18} -2245.32 q^{19} +1009.78 q^{21} -926.677 q^{22} -1287.36 q^{23} -368.585 q^{24} +6503.08 q^{26} -729.000 q^{27} -2990.34 q^{28} +2679.38 q^{29} +3838.89 q^{31} +7623.29 q^{32} -1089.00 q^{33} +4659.28 q^{34} +2158.85 q^{36} -12016.2 q^{37} +17195.8 q^{38} +7642.20 q^{39} -618.409 q^{41} -7733.36 q^{42} -13349.9 q^{43} +3224.95 q^{44} +9859.21 q^{46} -6897.13 q^{47} +10498.7 q^{48} -4218.77 q^{49} +5475.43 q^{51} -22631.5 q^{52} +1982.49 q^{53} +5583.04 q^{54} -4594.92 q^{56} +20207.9 q^{57} -20520.0 q^{58} -38147.4 q^{59} +5604.46 q^{61} -29400.1 q^{62} -9087.98 q^{63} -21054.1 q^{64} +8340.10 q^{66} +35672.2 q^{67} -16214.9 q^{68} +11586.2 q^{69} +36035.5 q^{71} +3317.27 q^{72} -33184.8 q^{73} +92025.8 q^{74} -59843.4 q^{76} -13575.9 q^{77} -58527.7 q^{78} -31882.0 q^{79} +6561.00 q^{81} +4736.08 q^{82} -91005.1 q^{83} +26913.0 q^{84} +102240. q^{86} -24114.4 q^{87} +4955.42 q^{88} +50403.1 q^{89} +95270.4 q^{91} -34311.3 q^{92} -34550.0 q^{93} +52821.6 q^{94} -68609.6 q^{96} -149344. q^{97} +32309.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) $ 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−7.65849	−1.35384	−0.676921	−	0.736055i	$-0.736686\pi$
$2$	−7.65849	−1.35384	−0.676921	+	0.736055i	$0.736686\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	26.6525	0.832890
$5$	0	0
$5$	0	0
$6$	68.9264	0.781641
$7$	−112.197	−0.865440	−0.432720	−	0.901528i	$-0.642446\pi$
$7$	−112.197	−0.865440	−0.432720	+	0.901528i	$0.642446\pi$
$8$	40.9539	0.226241
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−239.872	−0.480869
$13$	−849.133	−1.39353	−0.696767	−	0.717298i	$-0.745379\pi$
$13$	−849.133	−1.39353	−0.696767	+	0.717298i	$0.745379\pi$
$14$	859.262	1.17167
$15$	0	0
$16$	−1166.52	−1.13918
$17$	−608.382	−0.510568	−0.255284	−	0.966866i	$-0.582169\pi$
$17$	−608.382	−0.510568	−0.255284	+	0.966866i	$0.582169\pi$
$18$	−620.338	−0.451281
$19$	−2245.32	−1.42690	−0.713452	−	0.700704i	$-0.752869\pi$
$19$	−2245.32	−1.42690	−0.713452	+	0.700704i	$0.752869\pi$
$20$	0	0
$21$	1009.78	0.499662
$22$	−926.677	−0.408199
$23$	−1287.36	−0.507434	−0.253717	−	0.967279i	$-0.581653\pi$
$23$	−1287.36	−0.507434	−0.253717	+	0.967279i	$0.581653\pi$
$24$	−368.585	−0.130620
$25$	0	0
$26$	6503.08	1.88662
$27$	−729.000	−0.192450
$28$	−2990.34	−0.720817
$29$	2679.38	0.591616	0.295808	−	0.955247i	$-0.404411\pi$
$29$	2679.38	0.591616	0.295808	+	0.955247i	$0.404411\pi$
$30$	0	0
$31$	3838.89	0.717467	0.358733	−	0.933440i	$-0.383209\pi$
$31$	3838.89	0.717467	0.358733	+	0.933440i	$0.383209\pi$
$32$	7623.29	1.31604
$33$	−1089.00	−0.174078
$34$	4659.28	0.691229
$35$	0	0
$36$	2158.85	0.277630
$37$	−12016.2	−1.44299	−0.721493	−	0.692422i	$-0.756544\pi$
$37$	−12016.2	−1.44299	−0.721493	+	0.692422i	$0.756544\pi$
$38$	17195.8	1.93180
$39$	7642.20	0.804557
$40$	0	0
$41$	−618.409	−0.0574534	−0.0287267	−	0.999587i	$-0.509145\pi$
$41$	−618.409	−0.0574534	−0.0287267	+	0.999587i	$0.509145\pi$
$42$	−7733.36	−0.676464
$43$	−13349.9	−1.10105	−0.550526	−	0.834818i	$-0.685573\pi$
$43$	−13349.9	−1.10105	−0.550526	+	0.834818i	$0.685573\pi$
$44$	3224.95	0.251126
$45$	0	0
$46$	9859.21	0.686986
$47$	−6897.13	−0.455433	−0.227716	−	0.973728i	$-0.573126\pi$
$47$	−6897.13	−0.455433	−0.227716	+	0.973728i	$0.573126\pi$
$48$	10498.7	0.657708
$49$	−4218.77	−0.251013
$50$	0	0
$51$	5475.43	0.294777
$52$	−22631.5	−1.16066
$53$	1982.49	0.0969438	0.0484719	−	0.998825i	$-0.484565\pi$
$53$	1982.49	0.0969438	0.0484719	+	0.998825i	$0.484565\pi$
$54$	5583.04	0.260547
$55$	0	0
$56$	−4594.92	−0.195798
$57$	20207.9	0.823824
$58$	−20520.0	−0.800954
$59$	−38147.4	−1.42671	−0.713353	−	0.700805i	$-0.752824\pi$
$59$	−38147.4	−1.42671	−0.713353	+	0.700805i	$0.752824\pi$
$60$	0	0
$61$	5604.46	0.192845	0.0964226	−	0.995340i	$-0.469260\pi$
$61$	5604.46	0.192845	0.0964226	+	0.995340i	$0.469260\pi$
$62$	−29400.1	−0.971337
$63$	−9087.98	−0.288480
$64$	−21054.1	−0.642521
$65$	0	0
$66$	8340.10	0.235674
$67$	35672.2	0.970830	0.485415	−	0.874284i	$-0.338669\pi$
$67$	35672.2	0.970830	0.485415	+	0.874284i	$0.338669\pi$
$68$	−16214.9	−0.425247
$69$	11586.2	0.292967
$70$	0	0
$71$	36035.5	0.848368	0.424184	−	0.905576i	$-0.360561\pi$
$71$	36035.5	0.848368	0.424184	+	0.905576i	$0.360561\pi$
$72$	3317.27	0.0754135
$73$	−33184.8	−0.728841	−0.364420	−	0.931235i	$-0.618733\pi$
$73$	−33184.8	−0.728841	−0.364420	+	0.931235i	$0.618733\pi$
$74$	92025.8	1.95358
$75$	0	0
$76$	−59843.4	−1.18845
$77$	−13575.9	−0.260940
$78$	−58527.7	−1.08924
$79$	−31882.0	−0.574749	−0.287374	−	0.957818i	$-0.592782\pi$
$79$	−31882.0	−0.574749	−0.287374	+	0.957818i	$0.592782\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	4736.08	0.0777829
$83$	−91005.1	−1.45001	−0.725004	−	0.688745i	$-0.758162\pi$
$83$	−91005.1	−1.45001	−0.725004	+	0.688745i	$0.758162\pi$
$84$	26913.0	0.416164
$85$	0	0
$86$	102240.	1.49065
$87$	−24114.4	−0.341569
$88$	4955.42	0.0682141
$89$	50403.1	0.674501	0.337250	−	0.941415i	$-0.390503\pi$
$89$	50403.1	0.674501	0.337250	+	0.941415i	$0.390503\pi$
$90$	0	0
$91$	95270.4	1.20602
$92$	−34311.3	−0.422637
$93$	−34550.0	−0.414230
$94$	52821.6	0.616584
$95$	0	0
$96$	−68609.6	−0.759814
$97$	−149344.	−1.61160	−0.805800	−	0.592188i	$-0.798264\pi$
$97$	−149344.	−1.61160	−0.805800	+	0.592188i	$0.798264\pi$
$98$	32309.4	0.339832
$99$	9801.00	0.100504
$100$	0	0
$101$	−55801.3	−0.544304	−0.272152	−	0.962254i	$-0.587735\pi$
$101$	−55801.3	−0.544304	−0.272152	+	0.962254i	$0.587735\pi$
$102$	−41933.6	−0.399081
$103$	41304.4	0.383622	0.191811	−	0.981432i	$-0.438564\pi$
$103$	41304.4	0.383622	0.191811	+	0.981432i	$0.438564\pi$
$104$	−34775.3	−0.315274
$105$	0	0
$106$	−15182.8	−0.131247
$107$	−133537.	−1.12756	−0.563782	−	0.825924i	$-0.690654\pi$
$107$	−133537.	−1.12756	−0.563782	+	0.825924i	$0.690654\pi$
$108$	−19429.7	−0.160290
$109$	−217266.	−1.75156	−0.875782	−	0.482706i	$-0.839654\pi$
$109$	−217266.	−1.75156	−0.875782	+	0.482706i	$0.839654\pi$
$110$	0	0
$111$	108146.	0.833108
$112$	130881.	0.985896
$113$	−144891.	−1.06744	−0.533722	−	0.845660i	$-0.679207\pi$
$113$	−144891.	−1.06744	−0.533722	+	0.845660i	$0.679207\pi$
$114$	−154762.	−1.11533
$115$	0	0
$116$	71412.2	0.492751
$117$	−68779.8	−0.464511
$118$	292151.	1.93154
$119$	68258.7	0.441866
$120$	0	0
$121$	14641.0	0.0909091
$122$	−42921.7	−0.261082
$123$	5565.68	0.0331707
$124$	102316.	0.597571
$125$	0	0
$126$	69600.2	0.390557
$127$	−263734.	−1.45096	−0.725482	−	0.688241i	$-0.758383\pi$
$127$	−263734.	−1.45096	−0.725482	+	0.688241i	$0.758383\pi$
$128$	−82702.5	−0.446163
$129$	120149.	0.635692
$130$	0	0
$131$	−343441.	−1.74853	−0.874266	−	0.485447i	$-0.838657\pi$
$131$	−343441.	−1.74853	−0.874266	+	0.485447i	$0.838657\pi$
$132$	−29024.6	−0.144988
$133$	251919.	1.23490
$134$	−273195.	−1.31435
$135$	0	0
$136$	−24915.6	−0.115511
$137$	−172845.	−0.786786	−0.393393	−	0.919370i	$-0.628699\pi$
$137$	−172845.	−0.786786	−0.393393	+	0.919370i	$0.628699\pi$
$138$	−88732.9	−0.396631
$139$	−79230.1	−0.347819	−0.173909	−	0.984762i	$-0.555640\pi$
$139$	−79230.1	−0.347819	−0.173909	+	0.984762i	$0.555640\pi$
$140$	0	0
$141$	62074.2	0.262944
$142$	−275977.	−1.14856
$143$	−102745.	−0.420166
$144$	−94488.5	−0.379728
$145$	0	0
$146$	254146.	0.986736
$147$	37969.0	0.144922
$148$	−320261.	−1.20185
$149$	−112367.	−0.414641	−0.207321	−	0.978273i	$-0.566474\pi$
$149$	−112367.	−0.414641	−0.207321	+	0.978273i	$0.566474\pi$
$150$	0	0
$151$	351417.	1.25424	0.627119	−	0.778923i	$-0.284234\pi$
$151$	351417.	1.25424	0.627119	+	0.778923i	$0.284234\pi$
$152$	−91954.8	−0.322824
$153$	−49278.9	−0.170189
$154$	103971.	0.353272
$155$	0	0
$156$	203684.	0.670107
$157$	−202204.	−0.654696	−0.327348	−	0.944904i	$-0.606155\pi$
$157$	−202204.	−0.654696	−0.327348	+	0.944904i	$0.606155\pi$
$158$	244168.	0.778120
$159$	−17842.4	−0.0559706
$160$	0	0
$161$	144438.	0.439154
$162$	−50247.4	−0.150427
$163$	317741.	0.936708	0.468354	−	0.883541i	$-0.344847\pi$
$163$	317741.	0.936708	0.468354	+	0.883541i	$0.344847\pi$
$164$	−16482.1	−0.0478524
$165$	0	0
$166$	696962.	1.96308
$167$	−590220.	−1.63766	−0.818828	−	0.574038i	$-0.805376\pi$
$167$	−590220.	−1.63766	−0.818828	+	0.574038i	$0.805376\pi$
$168$	41354.2	0.113044
$169$	349734.	0.941934
$170$	0	0
$171$	−181871.	−0.475635
$172$	−355809.	−0.917055
$173$	461802.	1.17311	0.586557	−	0.809908i	$-0.300483\pi$
$173$	461802.	1.17311	0.586557	+	0.809908i	$0.300483\pi$
$174$	184680.	0.462431
$175$	0	0
$176$	−141149.	−0.343477
$177$	343326.	0.823709
$178$	−386012.	−0.913168
$179$	60459.9	0.141038	0.0705188	−	0.997510i	$-0.477535\pi$
$179$	60459.9	0.141038	0.0705188	+	0.997510i	$0.477535\pi$
$180$	0	0
$181$	−472232.	−1.07142	−0.535709	−	0.844403i	$-0.679955\pi$
$181$	−472232.	−1.07142	−0.535709	+	0.844403i	$0.679955\pi$
$182$	−729627.	−1.63276
$183$	−50440.1	−0.111339
$184$	−52722.3	−0.114802
$185$	0	0
$186$	264601.	0.560802
$187$	−73614.2	−0.153942
$188$	−183826.	−0.379325
$189$	81791.8	0.166554
$190$	0	0
$191$	−85661.8	−0.169904	−0.0849521	−	0.996385i	$-0.527074\pi$
$191$	−85661.8	−0.169904	−0.0849521	+	0.996385i	$0.527074\pi$
$192$	189487.	0.370960
$193$	992251.	1.91747	0.958734	−	0.284304i	$-0.0917625\pi$
$193$	992251.	1.91747	0.958734	+	0.284304i	$0.0917625\pi$
$194$	1.14375e6	2.18185
$195$	0	0
$196$	−112441.	−0.209066
$197$	−854973.	−1.56959	−0.784796	−	0.619754i	$-0.787233\pi$
$197$	−854973.	−1.56959	−0.784796	+	0.619754i	$0.787233\pi$
$198$	−75060.9	−0.136066
$199$	77645.6	0.138990	0.0694951	−	0.997582i	$-0.477861\pi$
$199$	77645.6	0.138990	0.0694951	+	0.997582i	$0.477861\pi$
$200$	0	0
$201$	−321050.	−0.560509
$202$	427354.	0.736902
$203$	−300619.	−0.512008
$204$	145934.	0.245517
$205$	0	0
$206$	−316330.	−0.519364
$207$	−104276.	−0.169145
$208$	990534.	1.58749
$209$	−271684.	−0.430228
$210$	0	0
$211$	741653.	1.14682	0.573409	−	0.819269i	$-0.305621\pi$
$211$	741653.	1.14682	0.573409	+	0.819269i	$0.305621\pi$
$212$	52838.1	0.0807436
$213$	−324319.	−0.489806
$214$	1.02269e6	1.52654
$215$	0	0
$216$	−29855.4	−0.0435400
$217$	−430713.	−0.620925
$218$	1.66393e6	2.37134
$219$	298664.	0.420796
$220$	0	0
$221$	516597.	0.711494
$222$	−828232.	−1.12790
$223$	−278843.	−0.375490	−0.187745	−	0.982218i	$-0.560118\pi$
$223$	−278843.	−0.375490	−0.187745	+	0.982218i	$0.560118\pi$
$224$	−855313.	−1.13895
$225$	0	0
$226$	1.10965e6	1.44515
$227$	770180.	0.992036	0.496018	−	0.868312i	$-0.334795\pi$
$227$	770180.	0.992036	0.496018	+	0.868312i	$0.334795\pi$
$228$	538591.	0.686155
$229$	−1.03700e6	−1.30674	−0.653370	−	0.757039i	$-0.726645\pi$
$229$	−1.03700e6	−1.30674	−0.653370	+	0.757039i	$0.726645\pi$
$230$	0	0
$231$	122183.	0.150654
$232$	109731.	0.133847
$233$	1.31100e6	1.58203	0.791014	−	0.611799i	$-0.209554\pi$
$233$	1.31100e6	1.58203	0.791014	+	0.611799i	$0.209554\pi$
$234$	526749.	0.628875
$235$	0	0
$236$	−1.01672e6	−1.18829
$237$	286938.	0.331831
$238$	−522759.	−0.598217
$239$	173528.	0.196506	0.0982529	−	0.995161i	$-0.468675\pi$
$239$	173528.	0.196506	0.0982529	+	0.995161i	$0.468675\pi$
$240$	0	0
$241$	−496038.	−0.550138	−0.275069	−	0.961424i	$-0.588701\pi$
$241$	−496038.	−0.550138	−0.275069	+	0.961424i	$0.588701\pi$
$242$	−112128.	−0.123077
$243$	−59049.0	−0.0641500
$244$	149373.	0.160619
$245$	0	0
$246$	−42624.7	−0.0449080
$247$	1.90658e6	1.98844
$248$	157218.	0.162320
$249$	819046.	0.837162
$250$	0	0
$251$	831223.	0.832785	0.416393	−	0.909185i	$-0.363294\pi$
$251$	831223.	0.832785	0.416393	+	0.909185i	$0.363294\pi$
$252$	−242217.	−0.240272
$253$	−155770.	−0.152997
$254$	2.01980e6	1.96438
$255$	0	0
$256$	1.30711e6	1.24656
$257$	734210.	0.693406	0.346703	−	0.937975i	$-0.387301\pi$
$257$	734210.	0.693406	0.346703	+	0.937975i	$0.387301\pi$
$258$	−920162.	−0.860627
$259$	1.34818e6	1.24882
$260$	0	0
$261$	217030.	0.197205
$262$	2.63024e6	2.36724
$263$	89351.2	0.0796547	0.0398273	−	0.999207i	$-0.487319\pi$
$263$	89351.2	0.0796547	0.0398273	+	0.999207i	$0.487319\pi$
$264$	−44598.8	−0.0393834
$265$	0	0
$266$	−1.92932e6	−1.67186
$267$	−453628.	−0.389423
$268$	950753.	0.808594
$269$	−927745.	−0.781714	−0.390857	−	0.920451i	$-0.627821\pi$
$269$	−927745.	−0.781714	−0.390857	+	0.920451i	$0.627821\pi$
$270$	0	0
$271$	494541.	0.409052	0.204526	−	0.978861i	$-0.434435\pi$
$271$	494541.	0.409052	0.204526	+	0.978861i	$0.434435\pi$
$272$	709692.	0.581631
$273$	−857434.	−0.696296
$274$	1.32373e6	1.06518
$275$	0	0
$276$	308801.	0.244009
$277$	1.98776e6	1.55656	0.778278	−	0.627920i	$-0.216093\pi$
$277$	1.98776e6	1.55656	0.778278	+	0.627920i	$0.216093\pi$
$278$	606783.	0.470892
$279$	310950.	0.239156
$280$	0	0
$281$	−760827.	−0.574804	−0.287402	−	0.957810i	$-0.592792\pi$
$281$	−760827.	−0.574804	−0.287402	+	0.957810i	$0.592792\pi$
$282$	−475395.	−0.355985
$283$	−2.17752e6	−1.61620	−0.808102	−	0.589043i	$-0.799505\pi$
$283$	−2.17752e6	−1.61620	−0.808102	+	0.589043i	$0.799505\pi$
$284$	960435.	0.706598
$285$	0	0
$286$	786872.	0.568839
$287$	69383.8	0.0497225
$288$	617487.	0.438679
$289$	−1.04973e6	−0.739320
$290$	0	0
$291$	1.34409e6	0.930458
$292$	−884458.	−0.607044
$293$	573498.	0.390268	0.195134	−	0.980777i	$-0.437486\pi$
$293$	573498.	0.390268	0.195134	+	0.980777i	$0.437486\pi$
$294$	−290785.	−0.196202
$295$	0	0
$296$	−492109.	−0.326462
$297$	−88209.0	−0.0580259
$298$	860560.	0.561359
$299$	1.09314e6	0.707126
$300$	0	0
$301$	1.49782e6	0.952894
$302$	−2.69132e6	−1.69804
$303$	502212.	0.314254
$304$	2.61922e6	1.62551
$305$	0	0
$306$	377402.	0.230410
$307$	784034.	0.474776	0.237388	−	0.971415i	$-0.423709\pi$
$307$	784034.	0.474776	0.237388	+	0.971415i	$0.423709\pi$
$308$	−361831.	−0.217334
$309$	−371740.	−0.221484
$310$	0	0
$311$	−1.20531e6	−0.706639	−0.353320	−	0.935503i	$-0.614947\pi$
$311$	−1.20531e6	−0.706639	−0.353320	+	0.935503i	$0.614947\pi$
$312$	312978.	0.182023
$313$	1.46973e6	0.847962	0.423981	−	0.905671i	$-0.360632\pi$
$313$	1.46973e6	0.847962	0.423981	+	0.905671i	$0.360632\pi$
$314$	1.54858e6	0.886356
$315$	0	0
$316$	−849735.	−0.478703
$317$	−943173.	−0.527161	−0.263580	−	0.964637i	$-0.584903\pi$
$317$	−943173.	−0.527161	−0.263580	+	0.964637i	$0.584903\pi$
$318$	136646.	0.0757753
$319$	324205.	0.178379
$320$	0	0
$321$	1.20183e6	0.651000
$322$	−1.10618e6	−0.594545
$323$	1.36601e6	0.728532
$324$	174867.	0.0925433
$325$	0	0
$326$	−2.43342e6	−1.26816
$327$	1.95540e6	1.01127
$328$	−25326.2	−0.0129983
$329$	773840.	0.394150
$330$	0	0
$331$	−2.82790e6	−1.41871	−0.709357	−	0.704850i	$-0.751014\pi$
$331$	−2.82790e6	−1.41871	−0.709357	+	0.704850i	$0.751014\pi$
$332$	−2.42551e6	−1.20770
$333$	−973310.	−0.480995
$334$	4.52020e6	2.21713
$335$	0	0
$336$	−1.17793e6	−0.569207
$337$	−1.66721e6	−0.799678	−0.399839	−	0.916585i	$-0.630934\pi$
$337$	−1.66721e6	−0.799678	−0.399839	+	0.916585i	$0.630934\pi$
$338$	−2.67843e6	−1.27523
$339$	1.30402e6	0.616289
$340$	0	0
$341$	464506.	0.216324
$342$	1.39286e6	0.643935
$343$	2.35903e6	1.08268
$344$	−546732.	−0.249102
$345$	0	0
$346$	−3.53671e6	−1.58821
$347$	−2.99775e6	−1.33651	−0.668254	−	0.743933i	$-0.732958\pi$
$347$	−2.99775e6	−1.33651	−0.668254	+	0.743933i	$0.732958\pi$
$348$	−642710.	−0.284490
$349$	−46625.4	−0.0204908	−0.0102454	−	0.999948i	$-0.503261\pi$
$349$	−46625.4	−0.0204908	−0.0102454	+	0.999948i	$0.503261\pi$
$350$	0	0
$351$	619018.	0.268186
$352$	922418.	0.396800
$353$	1.70738e6	0.729280	0.364640	−	0.931149i	$-0.381192\pi$
$353$	1.70738e6	0.729280	0.364640	+	0.931149i	$0.381192\pi$
$354$	−2.62936e6	−1.11517
$355$	0	0
$356$	1.34337e6	0.561785
$357$	−614329.	−0.255112
$358$	−463032.	−0.190943
$359$	−2.51698e6	−1.03073	−0.515364	−	0.856971i	$-0.672343\pi$
$359$	−2.51698e6	−1.03073	−0.515364	+	0.856971i	$0.672343\pi$
$360$	0	0
$361$	2.56538e6	1.03606
$362$	3.61658e6	1.45053
$363$	−131769.	−0.0524864
$364$	2.53919e6	1.00448
$365$	0	0
$366$	386295.	0.150736
$367$	2.06581e6	0.800616	0.400308	−	0.916381i	$-0.368903\pi$
$367$	2.06581e6	0.800616	0.400308	+	0.916381i	$0.368903\pi$
$368$	1.50173e6	0.578061
$369$	−50091.1	−0.0191511
$370$	0	0
$371$	−222429.	−0.0838991
$372$	−920844.	−0.345008
$373$	3.01036e6	1.12033	0.560166	−	0.828380i	$-0.310737\pi$
$373$	3.01036e6	1.12033	0.560166	+	0.828380i	$0.310737\pi$
$374$	563773.	0.208413
$375$	0	0
$376$	−282465.	−0.103037
$377$	−2.27515e6	−0.824436
$378$	−626402.	−0.225488
$379$	510365.	0.182508	0.0912541	−	0.995828i	$-0.470912\pi$
$379$	510365.	0.182508	0.0912541	+	0.995828i	$0.470912\pi$
$380$	0	0
$381$	2.37361e6	0.837715
$382$	656040.	0.230024
$383$	3.20939e6	1.11796	0.558979	−	0.829182i	$-0.311193\pi$
$383$	3.20939e6	1.11796	0.558979	+	0.829182i	$0.311193\pi$
$384$	744322.	0.257592
$385$	0	0
$386$	−7.59914e6	−2.59595
$387$	−1.08134e6	−0.367017
$388$	−3.98038e6	−1.34229
$389$	−4.52943e6	−1.51764	−0.758822	−	0.651298i	$-0.774225\pi$
$389$	−4.52943e6	−1.51764	−0.758822	+	0.651298i	$0.774225\pi$
$390$	0	0
$391$	783204.	0.259080
$392$	−172775.	−0.0567893
$393$	3.09097e6	1.00952
$394$	6.54781e6	2.12498
$395$	0	0
$396$	261221.	0.0837086
$397$	2.35485e6	0.749870	0.374935	−	0.927051i	$-0.377665\pi$
$397$	2.35485e6	0.749870	0.374935	+	0.927051i	$0.377665\pi$
$398$	−594648.	−0.188171
$399$	−2.26727e6	−0.712970
$400$	0	0
$401$	3.24320e6	1.00719	0.503597	−	0.863939i	$-0.332010\pi$
$401$	3.24320e6	1.00719	0.503597	+	0.863939i	$0.332010\pi$
$402$	2.45876e6	0.758841
$403$	−3.25973e6	−0.999814
$404$	−1.48724e6	−0.453345
$405$	0	0
$406$	2.30229e6	0.693178
$407$	−1.45396e6	−0.435076
$408$	224240.	0.0666904
$409$	2.99435e6	0.885103	0.442551	−	0.896743i	$-0.354073\pi$
$409$	2.99435e6	0.885103	0.442551	+	0.896743i	$0.354073\pi$
$410$	0	0
$411$	1.55561e6	0.454251
$412$	1.10087e6	0.319515
$413$	4.28003e6	1.23473
$414$	798596.	0.228995
$415$	0	0
$416$	−6.47319e6	−1.83394
$417$	713071.	0.200813
$418$	2.08069e6	0.582461
$419$	1.09825e6	0.305608	0.152804	−	0.988257i	$-0.451170\pi$
$419$	1.09825e6	0.305608	0.152804	+	0.988257i	$0.451170\pi$
$420$	0	0
$421$	52153.9	0.0143411	0.00717053	−	0.999974i	$-0.497718\pi$
$421$	52153.9	0.0143411	0.00717053	+	0.999974i	$0.497718\pi$
$422$	−5.67994e6	−1.55261
$423$	−558668.	−0.151811
$424$	81190.5	0.0219326
$425$	0	0
$426$	2.48380e6	0.663120
$427$	−628805.	−0.166896
$428$	−3.55909e6	−0.939137
$429$	924706.	0.242583
$430$	0	0
$431$	380799.	0.0987423	0.0493711	−	0.998781i	$-0.484278\pi$
$431$	380799.	0.0987423	0.0493711	+	0.998781i	$0.484278\pi$
$432$	850396.	0.219236
$433$	−6.43191e6	−1.64862	−0.824309	−	0.566140i	$-0.808436\pi$
$433$	−6.43191e6	−1.64862	−0.824309	+	0.566140i	$0.808436\pi$
$434$	3.29861e6	0.840634
$435$	0	0
$436$	−5.79069e6	−1.45886
$437$	2.89053e6	0.724060
$438$	−2.28731e6	−0.569692
$439$	−2.43398e6	−0.602776	−0.301388	−	0.953502i	$-0.597450\pi$
$439$	−2.43398e6	−0.602776	−0.301388	+	0.953502i	$0.597450\pi$
$440$	0	0
$441$	−341721.	−0.0836710
$442$	−3.95635e6	−0.963250
$443$	4.37287e6	1.05866	0.529331	−	0.848415i	$-0.322443\pi$
$443$	4.37287e6	1.05866	0.529331	+	0.848415i	$0.322443\pi$
$444$	2.88235e6	0.693887
$445$	0	0
$446$	2.13552e6	0.508354
$447$	1.01130e6	0.239393
$448$	2.36222e6	0.556064
$449$	6.51066e6	1.52409	0.762043	−	0.647527i	$-0.224197\pi$
$449$	6.51066e6	1.52409	0.762043	+	0.647527i	$0.224197\pi$
$450$	0	0
$451$	−74827.4	−0.0173229
$452$	−3.86171e6	−0.889064
$453$	−3.16275e6	−0.724135
$454$	−5.89841e6	−1.34306
$455$	0	0
$456$	827593.	0.186382
$457$	3.02898e6	0.678431	0.339216	−	0.940709i	$-0.389838\pi$
$457$	3.02898e6	0.678431	0.339216	+	0.940709i	$0.389838\pi$
$458$	7.94184e6	1.76912
$459$	443510.	0.0982589
$460$	0	0
$461$	−6.31089e6	−1.38305	−0.691525	−	0.722352i	$-0.743061\pi$
$461$	−6.31089e6	−1.38305	−0.691525	+	0.722352i	$0.743061\pi$
$462$	−935736.	−0.203962
$463$	1.33303e6	0.288993	0.144497	−	0.989505i	$-0.453844\pi$
$463$	1.33303e6	0.288993	0.144497	+	0.989505i	$0.453844\pi$
$464$	−3.12556e6	−0.673959
$465$	0	0
$466$	−1.00403e7	−2.14182
$467$	2.53362e6	0.537587	0.268793	−	0.963198i	$-0.413375\pi$
$467$	2.53362e6	0.537587	0.268793	+	0.963198i	$0.413375\pi$
$468$	−1.83315e6	−0.386887
$469$	−4.00232e6	−0.840195
$470$	0	0
$471$	1.81983e6	0.377989
$472$	−1.56228e6	−0.322779
$473$	−1.61534e6	−0.331979
$474$	−2.19751e6	−0.449248
$475$	0	0
$476$	1.81927e6	0.368026
$477$	160581.	0.0323146
$478$	−1.32896e6	−0.266038
$479$	−5.07626e6	−1.01089	−0.505447	−	0.862858i	$-0.668672\pi$
$479$	−5.07626e6	−1.01089	−0.505447	+	0.862858i	$0.668672\pi$
$480$	0	0
$481$	1.02033e7	2.01085
$482$	3.79890e6	0.744801
$483$	−1.29994e6	−0.253546
$484$	390219.	0.0757173
$485$	0	0
$486$	452226.	0.0868491
$487$	4.42849e6	0.846123	0.423061	−	0.906101i	$-0.360955\pi$
$487$	4.42849e6	0.846123	0.423061	+	0.906101i	$0.360955\pi$
$488$	229524.	0.0436294
$489$	−2.85967e6	−0.540809
$490$	0	0
$491$	3.27064e6	0.612250	0.306125	−	0.951991i	$-0.400967\pi$
$491$	3.27064e6	0.612250	0.306125	+	0.951991i	$0.400967\pi$
$492$	148339.	0.0276276
$493$	−1.63009e6	−0.302060
$494$	−1.46015e7	−2.69203
$495$	0	0
$496$	−4.47816e6	−0.817327
$497$	−4.04308e6	−0.734212
$498$	−6.27265e6	−1.13339
$499$	−160223.	−0.0288053	−0.0144027	−	0.999896i	$-0.504585\pi$
$499$	−160223.	−0.0288053	−0.0144027	+	0.999896i	$0.504585\pi$
$500$	0	0
$501$	5.31198e6	0.945502
$502$	−6.36591e6	−1.12746
$503$	−4.82699e6	−0.850661	−0.425330	−	0.905038i	$-0.639842\pi$
$503$	−4.82699e6	−0.850661	−0.425330	+	0.905038i	$0.639842\pi$
$504$	−372188.	−0.0652659
$505$	0	0
$506$	1.19296e6	0.207134
$507$	−3.14760e6	−0.543826
$508$	−7.02917e6	−1.20849
$509$	−3.12997e6	−0.535483	−0.267741	−	0.963491i	$-0.586277\pi$
$509$	−3.12997e6	−0.535483	−0.267741	+	0.963491i	$0.586277\pi$
$510$	0	0
$511$	3.72325e6	0.630768
$512$	−7.36400e6	−1.24148
$513$	1.63684e6	0.274608
$514$	−5.62294e6	−0.938763
$515$	0	0
$516$	3.20228e6	0.529462
$517$	−834553.	−0.137318
$518$	−1.03250e7	−1.69070
$519$	−4.15622e6	−0.677298
$520$	0	0
$521$	7.30204e6	1.17856	0.589278	−	0.807930i	$-0.299412\pi$
$521$	7.30204e6	1.17856	0.589278	+	0.807930i	$0.299412\pi$
$522$	−1.66212e6	−0.266985
$523$	−3.34673e6	−0.535016	−0.267508	−	0.963556i	$-0.586200\pi$
$523$	−3.34673e6	−0.535016	−0.267508	+	0.963556i	$0.586200\pi$
$524$	−9.15355e6	−1.45634
$525$	0	0
$526$	−684296.	−0.107840
$527$	−2.33551e6	−0.366316
$528$	1.27035e6	0.198307
$529$	−4.77905e6	−0.742511
$530$	0	0
$531$	−3.08994e6	−0.475569
$532$	6.71427e6	1.02854
$533$	525111.	0.0800632
$534$	3.47411e6	0.527218
$535$	0	0
$536$	1.46092e6	0.219641
$537$	−544139.	−0.0814281
$538$	7.10513e6	1.05832
$539$	−510472.	−0.0756832
$540$	0	0
$541$	−4.74690e6	−0.697296	−0.348648	−	0.937254i	$-0.613359\pi$
$541$	−4.74690e6	−0.697296	−0.348648	+	0.937254i	$0.613359\pi$
$542$	−3.78744e6	−0.553793
$543$	4.25008e6	0.618583
$544$	−4.63787e6	−0.671926
$545$	0	0
$546$	6.56665e6	0.942675
$547$	9.86753e6	1.41007	0.705034	−	0.709173i	$-0.250932\pi$
$547$	9.86753e6	1.41007	0.705034	+	0.709173i	$0.250932\pi$
$548$	−4.60676e6	−0.655306
$549$	453961.	0.0642818
$550$	0	0
$551$	−6.01608e6	−0.844179
$552$	474501.	0.0662810
$553$	3.57708e6	0.497411
$554$	−1.52232e7	−2.10733
$555$	0	0
$556$	−2.11168e6	−0.289695
$557$	−799590.	−0.109202	−0.0546008	−	0.998508i	$-0.517389\pi$
$557$	−799590.	−0.109202	−0.0546008	+	0.998508i	$0.517389\pi$
$558$	−2.38141e6	−0.323779
$559$	1.13359e7	1.53435
$560$	0	0
$561$	662527.	0.0888785
$562$	5.82678e6	0.778194
$563$	1.95313e6	0.259693	0.129847	−	0.991534i	$-0.458552\pi$
$563$	1.95313e6	0.259693	0.129847	+	0.991534i	$0.458552\pi$
$564$	1.65443e6	0.219004
$565$	0	0
$566$	1.66765e7	2.18809
$567$	−736126.	−0.0961600
$568$	1.47579e6	0.191935
$569$	−1.55722e6	−0.201637	−0.100819	−	0.994905i	$-0.532146\pi$
$569$	−1.55722e6	−0.201637	−0.100819	+	0.994905i	$0.532146\pi$
$570$	0	0
$571$	1.77525e6	0.227861	0.113930	−	0.993489i	$-0.463656\pi$
$571$	1.77525e6	0.227861	0.113930	+	0.993489i	$0.463656\pi$
$572$	−2.73841e6	−0.349952
$573$	770957.	0.0980942
$574$	−531375.	−0.0673165
$575$	0	0
$576$	−1.70538e6	−0.214174
$577$	7.49718e6	0.937472	0.468736	−	0.883338i	$-0.344710\pi$
$577$	7.49718e6	0.937472	0.468736	+	0.883338i	$0.344710\pi$
$578$	8.03934e6	1.00092
$579$	−8.93026e6	−1.10705
$580$	0	0
$581$	1.02105e7	1.25490
$582$	−1.02937e7	−1.25969
$583$	239881.	0.0292297
$584$	−1.35905e6	−0.164893
$585$	0	0
$586$	−4.39213e6	−0.528361
$587$	8.46392e6	1.01386	0.506928	−	0.861988i	$-0.330781\pi$
$587$	8.46392e6	1.01386	0.506928	+	0.861988i	$0.330781\pi$
$588$	1.01197e6	0.120704
$589$	−8.61956e6	−1.02376
$590$	0	0
$591$	7.69476e6	0.906205
$592$	1.40172e7	1.64383
$593$	6.09896e6	0.712228	0.356114	−	0.934442i	$-0.384101\pi$
$593$	6.09896e6	0.712228	0.356114	+	0.934442i	$0.384101\pi$
$594$	675548.	0.0785579
$595$	0	0
$596$	−2.99486e6	−0.345351
$597$	−698811.	−0.0802461
$598$	−8.37178e6	−0.957337
$599$	−4.91865e6	−0.560118	−0.280059	−	0.959983i	$-0.590354\pi$
$599$	−4.91865e6	−0.560118	−0.280059	+	0.959983i	$0.590354\pi$
$600$	0	0
$601$	1.00491e7	1.13486	0.567431	−	0.823421i	$-0.307937\pi$
$601$	1.00491e7	1.13486	0.567431	+	0.823421i	$0.307937\pi$
$602$	−1.14711e7	−1.29007
$603$	2.88945e6	0.323610
$604$	9.36613e6	1.04464
$605$	0	0
$606$	−3.84619e6	−0.425450
$607$	−1.26381e7	−1.39222	−0.696112	−	0.717933i	$-0.745088\pi$
$607$	−1.26381e7	−1.39222	−0.696112	+	0.717933i	$0.745088\pi$
$608$	−1.71168e7	−1.87786
$609$	2.70557e6	0.295608
$610$	0	0
$611$	5.85658e6	0.634660
$612$	−1.31341e6	−0.141749
$613$	−1.14049e7	−1.22586	−0.612932	−	0.790136i	$-0.710010\pi$
$613$	−1.14049e7	−1.22586	−0.612932	+	0.790136i	$0.710010\pi$
$614$	−6.00452e6	−0.642772
$615$	0	0
$616$	−555985.	−0.0590352
$617$	−1.57880e7	−1.66961	−0.834804	−	0.550547i	$-0.814419\pi$
$617$	−1.57880e7	−1.66961	−0.834804	+	0.550547i	$0.814419\pi$
$618$	2.84697e6	0.299855
$619$	−3.22994e6	−0.338819	−0.169409	−	0.985546i	$-0.554186\pi$
$619$	−3.22994e6	−0.338819	−0.169409	+	0.985546i	$0.554186\pi$
$620$	0	0
$621$	938483.	0.0976557
$622$	9.23086e6	0.956679
$623$	−5.65509e6	−0.583740
$624$	−8.91481e6	−0.916538
$625$	0	0
$626$	−1.12559e7	−1.14801
$627$	2.44516e6	0.248392
$628$	−5.38923e6	−0.545290
$629$	7.31042e6	0.736742
$630$	0	0
$631$	−1.46803e7	−1.46778	−0.733890	−	0.679268i	$-0.762297\pi$
$631$	−1.46803e7	−1.46778	−0.733890	+	0.679268i	$0.762297\pi$
$632$	−1.30569e6	−0.130031
$633$	−6.67488e6	−0.662116
$634$	7.22328e6	0.713693
$635$	0	0
$636$	−475543.	−0.0466173
$637$	3.58230e6	0.349795
$638$	−2.48292e6	−0.241497
$639$	2.91887e6	0.282789
$640$	0	0
$641$	−3.89836e6	−0.374746	−0.187373	−	0.982289i	$-0.559997\pi$
$641$	−3.89836e6	−0.374746	−0.187373	+	0.982289i	$0.559997\pi$
$642$	−9.20421e6	−0.881351
$643$	−1.22406e7	−1.16754	−0.583772	−	0.811917i	$-0.698424\pi$
$643$	−1.22406e7	−1.16754	−0.583772	+	0.811917i	$0.698424\pi$
$644$	3.84963e6	0.365767
$645$	0	0
$646$	−1.04616e7	−0.986318
$647$	1.57395e7	1.47819	0.739095	−	0.673601i	$-0.235254\pi$
$647$	1.57395e7	1.47819	0.739095	+	0.673601i	$0.235254\pi$
$648$	268699.	0.0251378
$649$	−4.61583e6	−0.430168
$650$	0	0
$651$	3.87642e6	0.358491
$652$	8.46859e6	0.780175
$653$	−5.37711e6	−0.493476	−0.246738	−	0.969082i	$-0.579359\pi$
$653$	−5.37711e6	−0.493476	−0.246738	+	0.969082i	$0.579359\pi$
$654$	−1.49754e7	−1.36910
$655$	0	0
$656$	721389.	0.0654500
$657$	−2.68797e6	−0.242947
$658$	−5.92644e6	−0.533617
$659$	−5.58169e6	−0.500671	−0.250335	−	0.968159i	$-0.580541\pi$
$659$	−5.58169e6	−0.500671	−0.250335	+	0.968159i	$0.580541\pi$
$660$	0	0
$661$	2.97463e6	0.264807	0.132403	−	0.991196i	$-0.457731\pi$
$661$	2.97463e6	0.264807	0.132403	+	0.991196i	$0.457731\pi$
$662$	2.16575e7	1.92071
$663$	−4.64937e6	−0.410781
$664$	−3.72701e6	−0.328051
$665$	0	0
$666$	7.45409e6	0.651192
$667$	−3.44932e6	−0.300206
$668$	−1.57308e7	−1.36399
$669$	2.50959e6	0.216789
$670$	0	0
$671$	678139.	0.0581450
$672$	7.69781e6	0.657573
$673$	4.89911e6	0.416946	0.208473	−	0.978028i	$-0.433151\pi$
$673$	4.89911e6	0.416946	0.208473	+	0.978028i	$0.433151\pi$
$674$	1.27683e7	1.08264
$675$	0	0
$676$	9.32127e6	0.784528
$677$	−2.13572e7	−1.79091	−0.895453	−	0.445156i	$-0.853148\pi$
$677$	−2.13572e7	−1.79091	−0.895453	+	0.445156i	$0.853148\pi$
$678$	−9.98682e6	−0.834359
$679$	1.67559e7	1.39474
$680$	0	0
$681$	−6.93162e6	−0.572752
$682$	−3.55741e6	−0.292869
$683$	1.39735e7	1.14618	0.573089	−	0.819493i	$-0.305745\pi$
$683$	1.39735e7	1.14618	0.573089	+	0.819493i	$0.305745\pi$
$684$	−4.84732e6	−0.396152
$685$	0	0
$686$	−1.80666e7	−1.46577
$687$	9.33298e6	0.754446
$688$	1.55730e7	1.25430
$689$	−1.68339e6	−0.135094
$690$	0	0
$691$	1.97099e7	1.57033	0.785164	−	0.619288i	$-0.212579\pi$
$691$	1.97099e7	1.57033	0.785164	+	0.619288i	$0.212579\pi$
$692$	1.23082e7	0.977076
$693$	−1.09965e6	−0.0869800
$694$	2.29582e7	1.80942
$695$	0	0
$696$	−987580.	−0.0772768
$697$	376228.	0.0293339
$698$	357080.	0.0277413
$699$	−1.17990e7	−0.913384
$700$	0	0
$701$	−1.31790e7	−1.01295	−0.506473	−	0.862256i	$-0.669051\pi$
$701$	−1.31790e7	−1.01295	−0.506473	+	0.862256i	$0.669051\pi$
$702$	−4.74074e6	−0.363081
$703$	2.69802e7	2.05900
$704$	−2.54755e6	−0.193727
$705$	0	0
$706$	−1.30760e7	−0.987330
$707$	6.26076e6	0.471062
$708$	9.15050e6	0.686059
$709$	−2.28382e7	−1.70626	−0.853131	−	0.521697i	$-0.825299\pi$
$709$	−2.28382e7	−1.70626	−0.853131	+	0.521697i	$0.825299\pi$
$710$	0	0
$711$	−2.58244e6	−0.191583
$712$	2.06420e6	0.152599
$713$	−4.94203e6	−0.364067
$714$	4.70483e6	0.345381
$715$	0	0
$716$	1.61141e6	0.117469
$717$	−1.56175e6	−0.113453
$718$	1.92763e7	1.39544
$719$	−3.78947e6	−0.273373	−0.136687	−	0.990614i	$-0.543645\pi$
$719$	−3.78947e6	−0.273373	−0.136687	+	0.990614i	$0.543645\pi$
$720$	0	0
$721$	−4.63424e6	−0.332002
$722$	−1.96469e7	−1.40266
$723$	4.46434e6	0.317623
$724$	−1.25861e7	−0.892373
$725$	0	0
$726$	1.00915e6	0.0710583
$727$	2.52016e7	1.76845	0.884223	−	0.467065i	$-0.154688\pi$
$727$	2.52016e7	1.76845	0.884223	+	0.467065i	$0.154688\pi$
$728$	3.90169e6	0.272851
$729$	531441.	0.0370370
$730$	0	0
$731$	8.12185e6	0.562162
$732$	−1.34435e6	−0.0927334
$733$	−1.83693e7	−1.26279	−0.631397	−	0.775460i	$-0.717518\pi$
$733$	−1.83693e7	−1.26279	−0.631397	+	0.775460i	$0.717518\pi$
$734$	−1.58210e7	−1.08391
$735$	0	0
$736$	−9.81390e6	−0.667801
$737$	4.31634e6	0.292716
$738$	383622.	0.0259276
$739$	4.22050e6	0.284284	0.142142	−	0.989846i	$-0.454601\pi$
$739$	4.22050e6	0.284284	0.142142	+	0.989846i	$0.454601\pi$
$740$	0	0
$741$	−1.71592e7	−1.14803
$742$	1.70347e6	0.113586
$743$	6.24107e6	0.414750	0.207375	−	0.978261i	$-0.433508\pi$
$743$	6.24107e6	0.414750	0.207375	+	0.978261i	$0.433508\pi$
$744$	−1.41496e6	−0.0937155
$745$	0	0
$746$	−2.30548e7	−1.51675
$747$	−7.37141e6	−0.483336
$748$	−1.96200e6	−0.128217
$749$	1.49825e7	0.975840
$750$	0	0
$751$	−542004.	−0.0350673	−0.0175337	−	0.999846i	$-0.505581\pi$
$751$	−542004.	−0.0350673	−0.0175337	+	0.999846i	$0.505581\pi$
$752$	8.04568e6	0.518822
$753$	−7.48100e6	−0.480809
$754$	1.74242e7	1.11616
$755$	0	0
$756$	2.17995e6	0.138721
$757$	1.63709e7	1.03833	0.519164	−	0.854675i	$-0.326244\pi$
$757$	1.63709e7	1.03833	0.519164	+	0.854675i	$0.326244\pi$
$758$	−3.90862e6	−0.247087
$759$	1.40193e6	0.0883329
$760$	0	0
$761$	−1.17890e7	−0.737931	−0.368966	−	0.929443i	$-0.620288\pi$
$761$	−1.17890e7	−0.737931	−0.368966	+	0.929443i	$0.620288\pi$
$762$	−1.81782e7	−1.13413
$763$	2.43767e7	1.51587
$764$	−2.28310e6	−0.141511
$765$	0	0
$766$	−2.45791e7	−1.51354
$767$	3.23922e7	1.98816
$768$	−1.17640e7	−0.719699
$769$	−2.22979e7	−1.35972	−0.679858	−	0.733344i	$-0.737958\pi$
$769$	−2.22979e7	−1.35972	−0.679858	+	0.733344i	$0.737958\pi$
$770$	0	0
$771$	−6.60789e6	−0.400338
$772$	2.64459e7	1.59704
$773$	−2.38341e7	−1.43467	−0.717333	−	0.696730i	$-0.754637\pi$
$773$	−2.38341e7	−1.43467	−0.717333	+	0.696730i	$0.754637\pi$
$774$	8.28146e6	0.496883
$775$	0	0
$776$	−6.11620e6	−0.364609
$777$	−1.21336e7	−0.721005
$778$	3.46886e7	2.05465
$779$	1.38853e6	0.0819805
$780$	0	0
$781$	4.36029e6	0.255793
$782$	−5.99816e6	−0.350753
$783$	−1.95327e6	−0.113856
$784$	4.92130e6	0.285950
$785$	0	0
$786$	−2.36721e7	−1.36673
$787$	6.85248e6	0.394376	0.197188	−	0.980366i	$-0.436819\pi$
$787$	6.85248e6	0.394376	0.197188	+	0.980366i	$0.436819\pi$
$788$	−2.27872e7	−1.30730
$789$	−804161.	−0.0459886
$790$	0	0
$791$	1.62564e7	0.923810
$792$	401389.	0.0227380
$793$	−4.75893e6	−0.268736
$794$	−1.80346e7	−1.01521
$795$	0	0
$796$	2.06945e6	0.115764
$797$	4.96228e6	0.276717	0.138358	−	0.990382i	$-0.455817\pi$
$797$	4.96228e6	0.276717	0.138358	+	0.990382i	$0.455817\pi$
$798$	1.73639e7	0.965250
$799$	4.19609e6	0.232529
$800$	0	0
$801$	4.08265e6	0.224834
$802$	−2.48380e7	−1.36358
$803$	−4.01537e6	−0.219754
$804$	−8.55678e6	−0.466842
$805$	0	0
$806$	2.49646e7	1.35359
$807$	8.34971e6	0.451323
$808$	−2.28528e6	−0.123144
$809$	−1.48303e6	−0.0796669	−0.0398335	−	0.999206i	$-0.512683\pi$
$809$	−1.48303e6	−0.0796669	−0.0398335	+	0.999206i	$0.512683\pi$
$810$	0	0
$811$	−8.26617e6	−0.441319	−0.220659	−	0.975351i	$-0.570821\pi$
$811$	−8.26617e6	−0.441319	−0.220659	+	0.975351i	$0.570821\pi$
$812$	−8.01225e6	−0.426446
$813$	−4.45087e6	−0.236167
$814$	1.11351e7	0.589025
$815$	0	0
$816$	−6.38723e6	−0.335805
$817$	2.99749e7	1.57110
$818$	−2.29322e7	−1.19829
$819$	7.71690e6	0.402007
$820$	0	0
$821$	2.39516e7	1.24016	0.620078	−	0.784540i	$-0.287101\pi$
$821$	2.39516e7	1.24016	0.620078	+	0.784540i	$0.287101\pi$
$822$	−1.19136e7	−0.614984
$823$	1.11177e7	0.572156	0.286078	−	0.958206i	$-0.407648\pi$
$823$	1.11177e7	0.572156	0.286078	+	0.958206i	$0.407648\pi$
$824$	1.69158e6	0.0867908
$825$	0	0
$826$	−3.27786e7	−1.67163
$827$	1.84798e7	0.939580	0.469790	−	0.882778i	$-0.344330\pi$
$827$	1.84798e7	0.939580	0.469790	+	0.882778i	$0.344330\pi$
$828$	−2.77921e6	−0.140879
$829$	1.97075e7	0.995969	0.497984	−	0.867186i	$-0.334074\pi$
$829$	1.97075e7	0.995969	0.497984	+	0.867186i	$0.334074\pi$
$830$	0	0
$831$	−1.78898e7	−0.898678
$832$	1.78778e7	0.895374
$833$	2.56662e6	0.128159
$834$	−5.46105e6	−0.271870
$835$	0	0
$836$	−7.24106e6	−0.358333
$837$	−2.79855e6	−0.138077
$838$	−8.41090e6	−0.413745
$839$	3.06964e7	1.50550	0.752752	−	0.658304i	$-0.228726\pi$
$839$	3.06964e7	1.50550	0.752752	+	0.658304i	$0.228726\pi$
$840$	0	0
$841$	−1.33321e7	−0.649991
$842$	−399420.	−0.0194155
$843$	6.84744e6	0.331863
$844$	1.97669e7	0.955173
$845$	0	0
$846$	4.27855e6	0.205528
$847$	−1.64268e6	−0.0786764
$848$	−2.31262e6	−0.110437
$849$	1.95977e7	0.933116
$850$	0	0
$851$	1.54691e7	0.732220
$852$	−8.64392e6	−0.407954
$853$	1.45171e7	0.683137	0.341569	−	0.939857i	$-0.389042\pi$
$853$	1.45171e7	0.683137	0.341569	+	0.939857i	$0.389042\pi$
$854$	4.81570e6	0.225951
$855$	0	0
$856$	−5.46885e6	−0.255101
$857$	−2.63178e7	−1.22405	−0.612024	−	0.790840i	$-0.709644\pi$
$857$	−2.63178e7	−1.22405	−0.612024	+	0.790840i	$0.709644\pi$
$858$	−7.08185e6	−0.328419
$859$	−7.00715e6	−0.324010	−0.162005	−	0.986790i	$-0.551796\pi$
$859$	−7.00715e6	−0.324010	−0.162005	+	0.986790i	$0.551796\pi$
$860$	0	0
$861$	−624454.	−0.0287073
$862$	−2.91635e6	−0.133681
$863$	−1.80082e6	−0.0823084	−0.0411542	−	0.999153i	$-0.513103\pi$
$863$	−1.80082e6	−0.0823084	−0.0411542	+	0.999153i	$0.513103\pi$
$864$	−5.55738e6	−0.253271
$865$	0	0
$866$	4.92587e7	2.23197
$867$	9.44756e6	0.426847
$868$	−1.14796e7	−0.517162
$869$	−3.85772e6	−0.173293
$870$	0	0
$871$	−3.02904e7	−1.35288
$872$	−8.89790e6	−0.396275
$873$	−1.20968e7	−0.537200
$874$	−2.21371e7	−0.980263
$875$	0	0
$876$	7.96013e6	0.350477
$877$	4.22779e7	1.85615	0.928077	−	0.372388i	$-0.121461\pi$
$877$	4.22779e7	1.85615	0.928077	+	0.372388i	$0.121461\pi$
$878$	1.86406e7	0.816064
$879$	−5.16148e6	−0.225321
$880$	0	0
$881$	9.29766e6	0.403584	0.201792	−	0.979428i	$-0.435323\pi$
$881$	9.29766e6	0.403584	0.201792	+	0.979428i	$0.435323\pi$
$882$	2.61706e6	0.113277
$883$	3.50752e7	1.51390	0.756951	−	0.653472i	$-0.226688\pi$
$883$	3.50752e7	1.51390	0.756951	+	0.653472i	$0.226688\pi$
$884$	1.37686e7	0.592596
$885$	0	0
$886$	−3.34896e7	−1.43326
$887$	−3.66149e7	−1.56260	−0.781302	−	0.624153i	$-0.785444\pi$
$887$	−3.66149e7	−1.56260	−0.781302	+	0.624153i	$0.785444\pi$
$888$	4.42898e6	0.188483
$889$	2.95902e7	1.25572
$890$	0	0
$891$	793881.	0.0335013
$892$	−7.43186e6	−0.312742
$893$	1.54863e7	0.649859
$894$	−7.74504e6	−0.324101
$895$	0	0
$896$	9.27899e6	0.386128
$897$	−9.83823e6	−0.408259
$898$	−4.98618e7	−2.06337
$899$	1.02859e7	0.424464
$900$	0	0
$901$	−1.20611e6	−0.0494964
$902$	573065.	0.0234524
$903$	−1.34804e7	−0.550154
$904$	−5.93385e6	−0.241499
$905$	0	0
$906$	2.42219e7	0.980365
$907$	−4.65461e7	−1.87873	−0.939366	−	0.342916i	$-0.888585\pi$
$907$	−4.65461e7	−1.87873	−0.939366	+	0.342916i	$0.888585\pi$
$908$	2.05272e7	0.826257
$909$	−4.51991e6	−0.181435
$910$	0	0
$911$	3.05339e7	1.21895	0.609476	−	0.792805i	$-0.291380\pi$
$911$	3.05339e7	1.21895	0.609476	+	0.792805i	$0.291380\pi$
$912$	−2.35730e7	−0.938487
$913$	−1.10116e7	−0.437194
$914$	−2.31974e7	−0.918490
$915$	0	0
$916$	−2.76386e7	−1.08837
$917$	3.85331e7	1.51325
$918$	−3.39662e6	−0.133027
$919$	−2.39764e7	−0.936472	−0.468236	−	0.883603i	$-0.655110\pi$
$919$	−2.39764e7	−0.936472	−0.468236	+	0.883603i	$0.655110\pi$
$920$	0	0
$921$	−7.05630e6	−0.274112
$922$	4.83319e7	1.87243
$923$	−3.05989e7	−1.18223
$924$	3.25648e6	0.125478
$925$	0	0
$926$	−1.02090e7	−0.391251
$927$	3.34566e6	0.127874
$928$	2.04257e7	0.778587
$929$	3.60560e7	1.37069	0.685343	−	0.728221i	$-0.259652\pi$
$929$	3.60560e7	1.37069	0.685343	+	0.728221i	$0.259652\pi$
$930$	0	0
$931$	9.47251e6	0.358171
$932$	3.49415e7	1.31765
$933$	1.08478e7	0.407978
$934$	−1.94037e7	−0.727808
$935$	0	0
$936$	−2.81680e6	−0.105091
$937$	2.89459e7	1.07706	0.538528	−	0.842608i	$-0.318981\pi$
$937$	2.89459e7	1.07706	0.538528	+	0.842608i	$0.318981\pi$
$938$	3.06518e7	1.13749
$939$	−1.32276e7	−0.489571
$940$	0	0
$941$	−2.26496e7	−0.833849	−0.416924	−	0.908941i	$-0.636892\pi$
$941$	−2.26496e7	−0.833849	−0.416924	+	0.908941i	$0.636892\pi$
$942$	−1.39372e7	−0.511738
$943$	796113.	0.0291538
$944$	4.44998e7	1.62528
$945$	0	0
$946$	1.23711e7	0.449448
$947$	3.01070e7	1.09092	0.545460	−	0.838137i	$-0.316355\pi$
$947$	3.01070e7	1.09092	0.545460	+	0.838137i	$0.316355\pi$
$948$	7.64762e6	0.276379
$949$	2.81783e7	1.01566
$950$	0	0
$951$	8.48855e6	0.304356
$952$	2.79546e6	0.0999681
$953$	−3.69781e7	−1.31890	−0.659450	−	0.751748i	$-0.729211\pi$
$953$	−3.69781e7	−1.31890	−0.659450	+	0.751748i	$0.729211\pi$
$954$	−1.22981e6	−0.0437489
$955$	0	0
$956$	4.62496e6	0.163668
$957$	−2.91785e6	−0.102987
$958$	3.88765e7	1.36859
$959$	1.93928e7	0.680916
$960$	0	0
$961$	−1.38921e7	−0.485242
$962$	−7.81421e7	−2.72237
$963$	−1.08165e7	−0.375855
$964$	−1.32206e7	−0.458205
$965$	0	0
$966$	9.95559e6	0.343261
$967$	5.23212e6	0.179933	0.0899666	−	0.995945i	$-0.471324\pi$
$967$	5.23212e6	0.179933	0.0899666	+	0.995945i	$0.471324\pi$
$968$	599606.	0.0205673
$969$	−1.22941e7	−0.420618
$970$	0	0
$971$	5.23574e7	1.78209	0.891046	−	0.453913i	$-0.149972\pi$
$971$	5.23574e7	1.78209	0.891046	+	0.453913i	$0.149972\pi$
$972$	−1.57380e6	−0.0534299
$973$	8.88940e6	0.301017
$974$	−3.39156e7	−1.14552
$975$	0	0
$976$	−6.53774e6	−0.219686
$977$	3.76035e7	1.26035	0.630176	−	0.776452i	$-0.282983\pi$
$977$	3.76035e7	1.26035	0.630176	+	0.776452i	$0.282983\pi$
$978$	2.19007e7	0.732170
$979$	6.09878e6	0.203370
$980$	0	0
$981$	−1.75986e7	−0.583855
$982$	−2.50482e7	−0.828890
$983$	4.17452e6	0.137791	0.0688957	−	0.997624i	$-0.478052\pi$
$983$	4.17452e6	0.137791	0.0688957	+	0.997624i	$0.478052\pi$
$984$	227936.	0.00750457
$985$	0	0
$986$	1.24840e7	0.408942
$987$	−6.96456e6	−0.227562
$988$	5.08150e7	1.65615
$989$	1.71861e7	0.558711
$990$	0	0
$991$	−7.89607e6	−0.255404	−0.127702	−	0.991813i	$-0.540760\pi$
$991$	−7.89607e6	−0.255404	−0.127702	+	0.991813i	$0.540760\pi$
$992$	2.92650e7	0.944212
$993$	2.54511e7	0.819095
$994$	3.09639e7	0.994008
$995$	0	0
$996$	2.18296e7	0.697264
$997$	−9.45089e6	−0.301117	−0.150558	−	0.988601i	$-0.548107\pi$
$997$	−9.45089e6	−0.301117	−0.150558	+	0.988601i	$0.548107\pi$
$998$	1.22706e6	0.0389979
$999$	8.75979e6	0.277703

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.872 q^{12} -849.133 q^{13} +859.262 q^{14} -1166.52 q^{16} -608.382 q^{17} -620.338 q^{18} -2245.32 q^{19} +1009.78 q^{21} -926.677 q^{22} -1287.36 q^{23} -368.585 q^{24} +6503.08 q^{26} -729.000 q^{27} -2990.34 q^{28} +2679.38 q^{29} +3838.89 q^{31} +7623.29 q^{32} -1089.00 q^{33} +4659.28 q^{34} +2158.85 q^{36} -12016.2 q^{37} +17195.8 q^{38} +7642.20 q^{39} -618.409 q^{41} -7733.36 q^{42} -13349.9 q^{43} +3224.95 q^{44} +9859.21 q^{46} -6897.13 q^{47} +10498.7 q^{48} -4218.77 q^{49} +5475.43 q^{51} -22631.5 q^{52} +1982.49 q^{53} +5583.04 q^{54} -4594.92 q^{56} +20207.9 q^{57} -20520.0 q^{58} -38147.4 q^{59} +5604.46 q^{61} -29400.1 q^{62} -9087.98 q^{63} -21054.1 q^{64} +8340.10 q^{66} +35672.2 q^{67} -16214.9 q^{68} +11586.2 q^{69} +36035.5 q^{71} +3317.27 q^{72} -33184.8 q^{73} +92025.8 q^{74} -59843.4 q^{76} -13575.9 q^{77} -58527.7 q^{78} -31882.0 q^{79} +6561.00 q^{81} +4736.08 q^{82} -91005.1 q^{83} +26913.0 q^{84} +102240. q^{86} -24114.4 q^{87} +4955.42 q^{88} +50403.1 q^{89} +95270.4 q^{91} -34311.3 q^{92} -34550.0 q^{93} +52821.6 q^{94} -68609.6 q^{96} -149344. q^{97} +32309.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * q^99

Introduction

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