LMFDB - Embedded newform 825.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.76300$ 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)

$f(q)$	$=$	$q-7.52601 q^{2} -9.00000 q^{3} +24.6408 q^{4} +67.7341 q^{6} +234.126 q^{7} +55.3856 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-7.52601 q^{2} -9.00000 q^{3} +24.6408 q^{4} +67.7341 q^{6} +234.126 q^{7} +55.3856 q^{8} +81.0000 q^{9} +121.000 q^{11} -221.767 q^{12} -236.184 q^{13} -1762.04 q^{14} -1205.34 q^{16} +608.015 q^{17} -609.607 q^{18} -1799.81 q^{19} -2107.14 q^{21} -910.647 q^{22} +4773.40 q^{23} -498.470 q^{24} +1777.52 q^{26} -729.000 q^{27} +5769.05 q^{28} +2804.99 q^{29} +10258.3 q^{31} +7299.04 q^{32} -1089.00 q^{33} -4575.93 q^{34} +1995.90 q^{36} +7629.84 q^{37} +13545.4 q^{38} +2125.66 q^{39} +8355.48 q^{41} +15858.3 q^{42} +11967.9 q^{43} +2981.53 q^{44} -35924.7 q^{46} -8385.48 q^{47} +10848.0 q^{48} +38008.1 q^{49} -5472.14 q^{51} -5819.76 q^{52} -3453.00 q^{53} +5486.46 q^{54} +12967.2 q^{56} +16198.3 q^{57} -21110.4 q^{58} -51032.4 q^{59} +9988.24 q^{61} -77204.1 q^{62} +18964.2 q^{63} -16361.8 q^{64} +8195.82 q^{66} +36567.6 q^{67} +14982.0 q^{68} -42960.6 q^{69} -30522.3 q^{71} +4486.23 q^{72} -83879.5 q^{73} -57422.2 q^{74} -44348.7 q^{76} +28329.3 q^{77} -15997.7 q^{78} +103505. q^{79} +6561.00 q^{81} -62883.4 q^{82} +4341.54 q^{83} -51921.5 q^{84} -90070.3 q^{86} -25244.9 q^{87} +6701.65 q^{88} -45464.1 q^{89} -55296.9 q^{91} +117620. q^{92} -92324.8 q^{93} +63109.2 q^{94} -65691.3 q^{96} +183170. q^{97} -286049. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−7.52601	−1.33042	−0.665211	−	0.746655i	$-0.731659\pi$
$2$	−7.52601	−1.33042	−0.665211	+	0.746655i	$0.731659\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	24.6408	0.770024
$5$	0	0
$5$	0	0
$6$	67.7341	0.768120
$7$	234.126	1.80595	0.902973	−	0.429696i	$-0.141379\pi$
$7$	234.126	1.80595	0.902973	+	0.429696i	$0.141379\pi$
$8$	55.3856	0.305965
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−221.767	−0.444574
$13$	−236.184	−0.387608	−0.193804	−	0.981040i	$-0.562083\pi$
$13$	−236.184	−0.387608	−0.193804	+	0.981040i	$0.562083\pi$
$14$	−1762.04	−2.40267
$15$	0	0
$16$	−1205.34	−1.17709
$17$	608.015	0.510261	0.255130	−	0.966907i	$-0.417882\pi$
$17$	608.015	0.510261	0.255130	+	0.966907i	$0.417882\pi$
$18$	−609.607	−0.443474
$19$	−1799.81	−1.14378	−0.571890	−	0.820331i	$-0.693789\pi$
$19$	−1799.81	−1.14378	−0.571890	+	0.820331i	$0.693789\pi$
$20$	0	0
$21$	−2107.14	−1.04266
$22$	−910.647	−0.401137
$23$	4773.40	1.88152	0.940759	−	0.339075i	$-0.110114\pi$
$23$	4773.40	1.88152	0.940759	+	0.339075i	$0.110114\pi$
$24$	−498.470	−0.176649
$25$	0	0
$26$	1777.52	0.515682
$27$	−729.000	−0.192450
$28$	5769.05	1.39062
$29$	2804.99	0.619351	0.309676	−	0.950842i	$-0.399780\pi$
$29$	2804.99	0.619351	0.309676	+	0.950842i	$0.399780\pi$
$30$	0	0
$31$	10258.3	1.91722	0.958610	−	0.284724i	$-0.0919019\pi$
$31$	10258.3	1.91722	0.958610	+	0.284724i	$0.0919019\pi$
$32$	7299.04	1.26006
$33$	−1089.00	−0.174078
$34$	−4575.93	−0.678862
$35$	0	0
$36$	1995.90	0.256675
$37$	7629.84	0.916244	0.458122	−	0.888889i	$-0.348522\pi$
$37$	7629.84	0.916244	0.458122	+	0.888889i	$0.348522\pi$
$38$	13545.4	1.52171
$39$	2125.66	0.223785
$40$	0	0
$41$	8355.48	0.776268	0.388134	−	0.921603i	$-0.373120\pi$
$41$	8355.48	0.776268	0.388134	+	0.921603i	$0.373120\pi$
$42$	15858.3	1.38718
$43$	11967.9	0.987065	0.493533	−	0.869727i	$-0.335705\pi$
$43$	11967.9	0.987065	0.493533	+	0.869727i	$0.335705\pi$
$44$	2981.53	0.232171
$45$	0	0
$46$	−35924.7	−2.50321
$47$	−8385.48	−0.553711	−0.276856	−	0.960912i	$-0.589292\pi$
$47$	−8385.48	−0.553711	−0.276856	+	0.960912i	$0.589292\pi$
$48$	10848.0	0.679591
$49$	38008.1	2.26144
$50$	0	0
$51$	−5472.14	−0.294599
$52$	−5819.76	−0.298467
$53$	−3453.00	−0.168852	−0.0844261	−	0.996430i	$-0.526906\pi$
$53$	−3453.00	−0.168852	−0.0844261	+	0.996430i	$0.526906\pi$
$54$	5486.46	0.256040
$55$	0	0
$56$	12967.2	0.552556
$57$	16198.3	0.660361
$58$	−21110.4	−0.823999
$59$	−51032.4	−1.90860	−0.954302	−	0.298842i	$-0.903400\pi$
$59$	−51032.4	−1.90860	−0.954302	+	0.298842i	$0.903400\pi$
$60$	0	0
$61$	9988.24	0.343688	0.171844	−	0.985124i	$-0.445028\pi$
$61$	9988.24	0.343688	0.171844	+	0.985124i	$0.445028\pi$
$62$	−77204.1	−2.55071
$63$	18964.2	0.601982
$64$	−16361.8	−0.499323
$65$	0	0
$66$	8195.82	0.231597
$67$	36567.6	0.995198	0.497599	−	0.867407i	$-0.334215\pi$
$67$	36567.6	0.995198	0.497599	+	0.867407i	$0.334215\pi$
$68$	14982.0	0.392913
$69$	−42960.6	−1.08630
$70$	0	0
$71$	−30522.3	−0.718573	−0.359286	−	0.933227i	$-0.616980\pi$
$71$	−30522.3	−0.718573	−0.359286	+	0.933227i	$0.616980\pi$
$72$	4486.23	0.101988
$73$	−83879.5	−1.84225	−0.921125	−	0.389267i	$-0.872728\pi$
$73$	−83879.5	−1.84225	−0.921125	+	0.389267i	$0.872728\pi$
$74$	−57422.2	−1.21899
$75$	0	0
$76$	−44348.7	−0.880738
$77$	28329.3	0.544513
$78$	−15997.7	−0.297729
$79$	103505.	1.86592	0.932961	−	0.359978i	$-0.117216\pi$
$79$	103505.	1.86592	0.932961	+	0.359978i	$0.117216\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−62883.4	−1.03276
$83$	4341.54	0.0691749	0.0345875	−	0.999402i	$-0.488988\pi$
$83$	4341.54	0.0691749	0.0345875	+	0.999402i	$0.488988\pi$
$84$	−51921.5	−0.802876
$85$	0	0
$86$	−90070.3	−1.31321
$87$	−25244.9	−0.357583
$88$	6701.65	0.0922519
$89$	−45464.1	−0.608406	−0.304203	−	0.952607i	$-0.598390\pi$
$89$	−45464.1	−0.608406	−0.304203	+	0.952607i	$0.598390\pi$
$90$	0	0
$91$	−55296.9	−0.699999
$92$	117620.	1.44881
$93$	−92324.8	−1.10691
$94$	63109.2	0.736670
$95$	0	0
$96$	−65691.3	−0.727495
$97$	183170.	1.97663	0.988314	−	0.152432i	$-0.0487106\pi$
$97$	183170.	1.97663	0.988314	+	0.152432i	$0.0487106\pi$
$98$	−286049.	−3.00868
$99$	9801.00	0.100504
$100$	0	0
$101$	−28833.3	−0.281249	−0.140625	−	0.990063i	$-0.544911\pi$
$101$	−28833.3	−0.281249	−0.140625	+	0.990063i	$0.544911\pi$
$102$	41183.3	0.391941
$103$	59444.6	0.552102	0.276051	−	0.961143i	$-0.410974\pi$
$103$	59444.6	0.552102	0.276051	+	0.961143i	$0.410974\pi$
$104$	−13081.2	−0.118594
$105$	0	0
$106$	25987.3	0.224645
$107$	−13840.6	−0.116868	−0.0584342	−	0.998291i	$-0.518611\pi$
$107$	−13840.6	−0.116868	−0.0584342	+	0.998291i	$0.518611\pi$
$108$	−17963.1	−0.148191
$109$	−13278.1	−0.107046	−0.0535231	−	0.998567i	$-0.517045\pi$
$109$	−13278.1	−0.107046	−0.0535231	+	0.998567i	$0.517045\pi$
$110$	0	0
$111$	−68668.5	−0.528993
$112$	−282201.	−2.12576
$113$	−130086.	−0.958370	−0.479185	−	0.877714i	$-0.659068\pi$
$113$	−130086.	−0.958370	−0.479185	+	0.877714i	$0.659068\pi$
$114$	−121908.	−0.878560
$115$	0	0
$116$	69117.2	0.476915
$117$	−19130.9	−0.129203
$118$	384070.	2.53925
$119$	142352.	0.921504
$120$	0	0
$121$	14641.0	0.0909091
$122$	−75171.5	−0.457250
$123$	−75199.3	−0.448178
$124$	252773.	1.47631
$125$	0	0
$126$	−142725.	−0.800891
$127$	274904.	1.51242	0.756209	−	0.654330i	$-0.227049\pi$
$127$	274904.	1.51242	0.756209	+	0.654330i	$0.227049\pi$
$128$	−110430.	−0.595748
$129$	−107711.	−0.569882
$130$	0	0
$131$	−119374.	−0.607761	−0.303881	−	0.952710i	$-0.598282\pi$
$131$	−119374.	−0.607761	−0.303881	+	0.952710i	$0.598282\pi$
$132$	−26833.8	−0.134044
$133$	−421382.	−2.06560
$134$	−275208.	−1.32403
$135$	0	0
$136$	33675.3	0.156122
$137$	132278.	0.602123	0.301061	−	0.953605i	$-0.402659\pi$
$137$	132278.	0.602123	0.301061	+	0.953605i	$0.402659\pi$
$138$	323322.	1.44523
$139$	−102326.	−0.449209	−0.224604	−	0.974450i	$-0.572109\pi$
$139$	−102326.	−0.449209	−0.224604	+	0.974450i	$0.572109\pi$
$140$	0	0
$141$	75469.3	0.319685
$142$	229711.	0.956005
$143$	−28578.3	−0.116868
$144$	−97632.3	−0.392362
$145$	0	0
$146$	631278.	2.45097
$147$	−342073.	−1.30565
$148$	188005.	0.705530
$149$	371696.	1.37158	0.685791	−	0.727798i	$-0.259456\pi$
$149$	371696.	1.37158	0.685791	+	0.727798i	$0.259456\pi$
$150$	0	0
$151$	−111725.	−0.398757	−0.199379	−	0.979923i	$-0.563892\pi$
$151$	−111725.	−0.398757	−0.199379	+	0.979923i	$0.563892\pi$
$152$	−99683.4	−0.349956
$153$	49249.2	0.170087
$154$	−213206.	−0.724433
$155$	0	0
$156$	52377.8	0.172320
$157$	244726.	0.792375	0.396187	−	0.918170i	$-0.370333\pi$
$157$	244726.	0.792375	0.396187	+	0.918170i	$0.370333\pi$
$158$	−778979.	−2.48246
$159$	31077.0	0.0974869
$160$	0	0
$161$	1.11758e6	3.39792
$162$	−49378.1	−0.147825
$163$	−435623.	−1.28423	−0.642114	−	0.766610i	$-0.721942\pi$
$163$	−435623.	−1.28423	−0.642114	+	0.766610i	$0.721942\pi$
$164$	205885.	0.597745
$165$	0	0
$166$	−32674.5	−0.0920319
$167$	−288486.	−0.800450	−0.400225	−	0.916417i	$-0.631068\pi$
$167$	−288486.	−0.800450	−0.400225	+	0.916417i	$0.631068\pi$
$168$	−116705.	−0.319019
$169$	−315510.	−0.849760
$170$	0	0
$171$	−145784.	−0.381260
$172$	294898.	0.760064
$173$	−539633.	−1.37083	−0.685414	−	0.728154i	$-0.740379\pi$
$173$	−539633.	−1.37083	−0.685414	+	0.728154i	$0.740379\pi$
$174$	189994.	0.475736
$175$	0	0
$176$	−145846.	−0.354905
$177$	459292.	1.10193
$178$	342163.	0.809438
$179$	−378874.	−0.883816	−0.441908	−	0.897060i	$-0.645698\pi$
$179$	−378874.	−0.883816	−0.441908	+	0.897060i	$0.645698\pi$
$180$	0	0
$181$	−621633.	−1.41038	−0.705192	−	0.709016i	$-0.749139\pi$
$181$	−621633.	−1.41038	−0.705192	+	0.709016i	$0.749139\pi$
$182$	416165.	0.931294
$183$	−89894.1	−0.198428
$184$	264378.	0.575679
$185$	0	0
$186$	694837.	1.47265
$187$	73569.8	0.153849
$188$	−206625.	−0.426371
$189$	−170678.	−0.347555
$190$	0	0
$191$	8509.61	0.0168782	0.00843911	−	0.999964i	$-0.497314\pi$
$191$	8509.61	0.0168782	0.00843911	+	0.999964i	$0.497314\pi$
$192$	147256.	0.288284
$193$	564588.	1.09103	0.545517	−	0.838100i	$-0.316333\pi$
$193$	564588.	1.09103	0.545517	+	0.838100i	$0.316333\pi$
$194$	−1.37854e6	−2.62975
$195$	0	0
$196$	936549.	1.74137
$197$	432994.	0.794908	0.397454	−	0.917622i	$-0.369894\pi$
$197$	432994.	0.794908	0.397454	+	0.917622i	$0.369894\pi$
$198$	−73762.4	−0.133712
$199$	−614871.	−1.10066	−0.550328	−	0.834949i	$-0.685497\pi$
$199$	−614871.	−1.10066	−0.550328	+	0.834949i	$0.685497\pi$
$200$	0	0
$201$	−329108.	−0.574578
$202$	217000.	0.374180
$203$	656723.	1.11852
$204$	−134838.	−0.226849
$205$	0	0
$206$	−447380.	−0.734528
$207$	386646.	0.627173
$208$	284682.	0.456248
$209$	−217777.	−0.344862
$210$	0	0
$211$	72348.2	0.111872	0.0559360	−	0.998434i	$-0.482186\pi$
$211$	72348.2	0.111872	0.0559360	+	0.998434i	$0.482186\pi$
$212$	−85084.6	−0.130020
$213$	274700.	0.414868
$214$	104165.	0.155484
$215$	0	0
$216$	−40376.1	−0.0588830
$217$	2.40174e6	3.46240
$218$	99931.4	0.142417
$219$	754915.	1.06362
$220$	0	0
$221$	−143604.	−0.197781
$222$	516800.	0.703785
$223$	−370371.	−0.498740	−0.249370	−	0.968408i	$-0.580224\pi$
$223$	−370371.	−0.498740	−0.249370	+	0.968408i	$0.580224\pi$
$224$	1.70890e6	2.27560
$225$	0	0
$226$	979025.	1.27504
$227$	−325481.	−0.419239	−0.209619	−	0.977783i	$-0.567222\pi$
$227$	−325481.	−0.419239	−0.209619	+	0.977783i	$0.567222\pi$
$228$	399138.	0.508494
$229$	−57345.0	−0.0722614	−0.0361307	−	0.999347i	$-0.511503\pi$
$229$	−57345.0	−0.0722614	−0.0361307	+	0.999347i	$0.511503\pi$
$230$	0	0
$231$	−254963.	−0.314375
$232$	155356.	0.189500
$233$	833891.	1.00628	0.503141	−	0.864205i	$-0.332178\pi$
$233$	833891.	1.00628	0.503141	+	0.864205i	$0.332178\pi$
$234$	143979.	0.171894
$235$	0	0
$236$	−1.25748e6	−1.46967
$237$	−931545.	−1.07729
$238$	−1.07134e6	−1.22599
$239$	−813986.	−0.921769	−0.460885	−	0.887460i	$-0.652468\pi$
$239$	−813986.	−0.921769	−0.460885	+	0.887460i	$0.652468\pi$
$240$	0	0
$241$	−1.18456e6	−1.31376	−0.656878	−	0.753997i	$-0.728123\pi$
$241$	−1.18456e6	−1.31376	−0.656878	+	0.753997i	$0.728123\pi$
$242$	−110188.	−0.120948
$243$	−59049.0	−0.0641500
$244$	246118.	0.264648
$245$	0	0
$246$	565950.	0.596267
$247$	425086.	0.443338
$248$	568163.	0.586602
$249$	−39073.9	−0.0399382
$250$	0	0
$251$	−601749.	−0.602880	−0.301440	−	0.953485i	$-0.597467\pi$
$251$	−601749.	−0.602880	−0.301440	+	0.953485i	$0.597467\pi$
$252$	467293.	0.463541
$253$	577582.	0.567299
$254$	−2.06893e6	−2.01216
$255$	0	0
$256$	1.35468e6	1.29192
$257$	−904570.	−0.854298	−0.427149	−	0.904181i	$-0.640482\pi$
$257$	−904570.	−0.854298	−0.427149	+	0.904181i	$0.640482\pi$
$258$	810633.	0.758184
$259$	1.78634e6	1.65469
$260$	0	0
$261$	227204.	0.206450
$262$	898413.	0.808580
$263$	−1.47174e6	−1.31202	−0.656011	−	0.754751i	$-0.727758\pi$
$263$	−1.47174e6	−1.31202	−0.656011	+	0.754751i	$0.727758\pi$
$264$	−60314.9	−0.0532617
$265$	0	0
$266$	3.17133e6	2.74813
$267$	409177.	0.351264
$268$	901054.	0.766327
$269$	897427.	0.756168	0.378084	−	0.925771i	$-0.376583\pi$
$269$	897427.	0.756168	0.378084	+	0.925771i	$0.376583\pi$
$270$	0	0
$271$	134225.	0.111022	0.0555111	−	0.998458i	$-0.482321\pi$
$271$	134225.	0.111022	0.0555111	+	0.998458i	$0.482321\pi$
$272$	−732863.	−0.600621
$273$	497672.	0.404145
$274$	−995523.	−0.801078
$275$	0	0
$276$	−1.05858e6	−0.836474
$277$	2.01306e6	1.57637	0.788184	−	0.615439i	$-0.211021\pi$
$277$	2.01306e6	1.57637	0.788184	+	0.615439i	$0.211021\pi$
$278$	770105.	0.597637
$279$	830923.	0.639073
$280$	0	0
$281$	1.45204e6	1.09702	0.548509	−	0.836145i	$-0.315196\pi$
$281$	1.45204e6	1.09702	0.548509	+	0.836145i	$0.315196\pi$
$282$	−567983.	−0.425316
$283$	2.24416e6	1.66566	0.832832	−	0.553525i	$-0.186718\pi$
$283$	2.24416e6	1.66566	0.832832	+	0.553525i	$0.186718\pi$
$284$	−752092.	−0.553318
$285$	0	0
$286$	215080.	0.155484
$287$	1.95624e6	1.40190
$288$	591222.	0.420019
$289$	−1.05017e6	−0.739634
$290$	0	0
$291$	−1.64853e6	−1.14121
$292$	−2.06686e6	−1.41858
$293$	−148693.	−0.101187	−0.0505933	−	0.998719i	$-0.516111\pi$
$293$	−148693.	−0.101187	−0.0505933	+	0.998719i	$0.516111\pi$
$294$	2.57444e6	1.73706
$295$	0	0
$296$	422583.	0.280338
$297$	−88209.0	−0.0580259
$298$	−2.79739e6	−1.82478
$299$	−1.12740e6	−0.729291
$300$	0	0
$301$	2.80199e6	1.78259
$302$	840845.	0.530516
$303$	259500.	0.162379
$304$	2.16938e6	1.34633
$305$	0	0
$306$	−370650.	−0.226287
$307$	−2.02304e6	−1.22507	−0.612533	−	0.790445i	$-0.709849\pi$
$307$	−2.02304e6	−1.22507	−0.612533	+	0.790445i	$0.709849\pi$
$308$	698055.	0.419289
$309$	−535001.	−0.318756
$310$	0	0
$311$	1.87652e6	1.10015	0.550074	−	0.835116i	$-0.314599\pi$
$311$	1.87652e6	1.10015	0.550074	+	0.835116i	$0.314599\pi$
$312$	117731.	0.0684705
$313$	89494.6	0.0516340	0.0258170	−	0.999667i	$-0.491781\pi$
$313$	89494.6	0.0516340	0.0258170	+	0.999667i	$0.491781\pi$
$314$	−1.84181e6	−1.05419
$315$	0	0
$316$	2.55044e6	1.43680
$317$	−260115.	−0.145384	−0.0726920	−	0.997354i	$-0.523159\pi$
$317$	−260115.	−0.145384	−0.0726920	+	0.997354i	$0.523159\pi$
$318$	−233886.	−0.129699
$319$	339404.	0.186741
$320$	0	0
$321$	124566.	0.0674740
$322$	−8.41090e6	−4.52067
$323$	−1.09431e6	−0.583626
$324$	161668.	0.0855582
$325$	0	0
$326$	3.27850e6	1.70856
$327$	119503.	0.0618031
$328$	462773.	0.237511
$329$	−1.96326e6	−0.999973
$330$	0	0
$331$	2.19339e6	1.10039	0.550193	−	0.835037i	$-0.314554\pi$
$331$	2.19339e6	1.10039	0.550193	+	0.835037i	$0.314554\pi$
$332$	106979.	0.0532664
$333$	618017.	0.305415
$334$	2.17115e6	1.06494
$335$	0	0
$336$	2.53981e6	1.22731
$337$	−933037.	−0.447532	−0.223766	−	0.974643i	$-0.571835\pi$
$337$	−933037.	−0.447532	−0.223766	+	0.974643i	$0.571835\pi$
$338$	2.37453e6	1.13054
$339$	1.17077e6	0.553315
$340$	0	0
$341$	1.24126e6	0.578063
$342$	1.09717e6	0.507237
$343$	4.96373e6	2.27810
$344$	662848.	0.302007
$345$	0	0
$346$	4.06128e6	1.82378
$347$	1.72984e6	0.771229	0.385615	−	0.922660i	$-0.373989\pi$
$347$	1.72984e6	0.771229	0.385615	+	0.922660i	$0.373989\pi$
$348$	−622055.	−0.275347
$349$	1.01770e6	0.447258	0.223629	−	0.974674i	$-0.428210\pi$
$349$	1.01770e6	0.447258	0.223629	+	0.974674i	$0.428210\pi$
$350$	0	0
$351$	172178.	0.0745951
$352$	883183.	0.379922
$353$	3.29128e6	1.40581	0.702907	−	0.711282i	$-0.251885\pi$
$353$	3.29128e6	1.40581	0.702907	+	0.711282i	$0.251885\pi$
$354$	−3.45663e6	−1.46604
$355$	0	0
$356$	−1.12027e6	−0.468488
$357$	−1.28117e6	−0.532030
$358$	2.85140e6	1.17585
$359$	2.32128e6	0.950584	0.475292	−	0.879828i	$-0.342342\pi$
$359$	2.32128e6	0.950584	0.475292	+	0.879828i	$0.342342\pi$
$360$	0	0
$361$	763211.	0.308231
$362$	4.67841e6	1.87641
$363$	−131769.	−0.0524864
$364$	−1.36256e6	−0.539016
$365$	0	0
$366$	676544.	0.263993
$367$	1.81399e6	0.703024	0.351512	−	0.936183i	$-0.385668\pi$
$367$	1.81399e6	0.703024	0.351512	+	0.936183i	$0.385668\pi$
$368$	−5.75356e6	−2.21471
$369$	676794.	0.258756
$370$	0	0
$371$	−808438.	−0.304938
$372$	−2.27496e6	−0.852345
$373$	160496.	0.0597298	0.0298649	−	0.999554i	$-0.490492\pi$
$373$	160496.	0.0597298	0.0298649	+	0.999554i	$0.490492\pi$
$374$	−553687.	−0.204685
$375$	0	0
$376$	−464435.	−0.169416
$377$	−662495.	−0.240065
$378$	1.28452e6	0.462395
$379$	−622442.	−0.222587	−0.111294	−	0.993788i	$-0.535499\pi$
$379$	−622442.	−0.222587	−0.111294	+	0.993788i	$0.535499\pi$
$380$	0	0
$381$	−2.47414e6	−0.873195
$382$	−64043.4	−0.0224552
$383$	−1.49075e6	−0.519289	−0.259644	−	0.965704i	$-0.583605\pi$
$383$	−1.49075e6	−0.519289	−0.259644	+	0.965704i	$0.583605\pi$
$384$	993871.	0.343955
$385$	0	0
$386$	−4.24909e6	−1.45154
$387$	969398.	0.329022
$388$	4.51345e6	1.52205
$389$	−1.66085e6	−0.556490	−0.278245	−	0.960510i	$-0.589753\pi$
$389$	−1.66085e6	−0.556490	−0.278245	+	0.960510i	$0.589753\pi$
$390$	0	0
$391$	2.90230e6	0.960065
$392$	2.10510e6	0.691923
$393$	1.07437e6	0.350891
$394$	−3.25872e6	−1.05756
$395$	0	0
$396$	241504.	0.0773903
$397$	−2.46357e6	−0.784493	−0.392247	−	0.919860i	$-0.628302\pi$
$397$	−2.46357e6	−0.784493	−0.392247	+	0.919860i	$0.628302\pi$
$398$	4.62752e6	1.46434
$399$	3.79244e6	1.19258
$400$	0	0
$401$	−664739.	−0.206438	−0.103219	−	0.994659i	$-0.532914\pi$
$401$	−664739.	−0.206438	−0.103219	+	0.994659i	$0.532914\pi$
$402$	2.47687e6	0.764431
$403$	−2.42285e6	−0.743129
$404$	−710475.	−0.216569
$405$	0	0
$406$	−4.94250e6	−1.48810
$407$	923210.	0.276258
$408$	−303077.	−0.0901370
$409$	−111122.	−0.0328468	−0.0164234	−	0.999865i	$-0.505228\pi$
$409$	−111122.	−0.0328468	−0.0164234	+	0.999865i	$0.505228\pi$
$410$	0	0
$411$	−1.19050e6	−0.347636
$412$	1.46476e6	0.425132
$413$	−1.19480e7	−3.44684
$414$	−2.90990e6	−0.834405
$415$	0	0
$416$	−1.72392e6	−0.488408
$417$	920932.	0.259351
$418$	1.63899e6	0.458813
$419$	−5.34446e6	−1.48720	−0.743598	−	0.668627i	$-0.766882\pi$
$419$	−5.34446e6	−1.48720	−0.743598	+	0.668627i	$0.766882\pi$
$420$	0	0
$421$	−483590.	−0.132976	−0.0664878	−	0.997787i	$-0.521179\pi$
$421$	−483590.	−0.132976	−0.0664878	+	0.997787i	$0.521179\pi$
$422$	−544493.	−0.148837
$423$	−679224.	−0.184570
$424$	−191246.	−0.0516629
$425$	0	0
$426$	−2.06740e6	−0.551950
$427$	2.33851e6	0.620682
$428$	−341044.	−0.0899915
$429$	257205.	0.0674738
$430$	0	0
$431$	−241542.	−0.0626325	−0.0313162	−	0.999510i	$-0.509970\pi$
$431$	−241542.	−0.0626325	−0.0313162	+	0.999510i	$0.509970\pi$
$432$	878691.	0.226530
$433$	1.03998e6	0.266567	0.133284	−	0.991078i	$-0.457448\pi$
$433$	1.03998e6	0.266567	0.133284	+	0.991078i	$0.457448\pi$
$434$	−1.80755e7	−4.60645
$435$	0	0
$436$	−327184.	−0.0824281
$437$	−8.59121e6	−2.15204
$438$	−5.68150e6	−1.41507
$439$	−3.32986e6	−0.824642	−0.412321	−	0.911039i	$-0.635282\pi$
$439$	−3.32986e6	−0.824642	−0.412321	+	0.911039i	$0.635282\pi$
$440$	0	0
$441$	3.07866e6	0.753815
$442$	1.08076e6	0.263132
$443$	−5.12280e6	−1.24022	−0.620109	−	0.784515i	$-0.712912\pi$
$443$	−5.12280e6	−1.24022	−0.620109	+	0.784515i	$0.712912\pi$
$444$	−1.69205e6	−0.407338
$445$	0	0
$446$	2.78741e6	0.663535
$447$	−3.34526e6	−0.791884
$448$	−3.83073e6	−0.901750
$449$	5.82328e6	1.36318	0.681588	−	0.731736i	$-0.261290\pi$
$449$	5.82328e6	1.36318	0.681588	+	0.731736i	$0.261290\pi$
$450$	0	0
$451$	1.01101e6	0.234054
$452$	−3.20541e6	−0.737968
$453$	1.00553e6	0.230223
$454$	2.44957e6	0.557765
$455$	0	0
$456$	897151.	0.202047
$457$	5.32427e6	1.19253	0.596266	−	0.802787i	$-0.296651\pi$
$457$	5.32427e6	1.19253	0.596266	+	0.802787i	$0.296651\pi$
$458$	431579.	0.0961382
$459$	−443243.	−0.0981997
$460$	0	0
$461$	673694.	0.147642	0.0738211	−	0.997272i	$-0.476481\pi$
$461$	673694.	0.147642	0.0738211	+	0.997272i	$0.476481\pi$
$462$	1.91886e6	0.418252
$463$	1.92230e6	0.416742	0.208371	−	0.978050i	$-0.433184\pi$
$463$	1.92230e6	0.416742	0.208371	+	0.978050i	$0.433184\pi$
$464$	−3.38096e6	−0.729030
$465$	0	0
$466$	−6.27587e6	−1.33878
$467$	7.94668e6	1.68614	0.843070	−	0.537804i	$-0.180746\pi$
$467$	7.94668e6	1.68614	0.843070	+	0.537804i	$0.180746\pi$
$468$	−471401.	−0.0994891
$469$	8.56144e6	1.79727
$470$	0	0
$471$	−2.20253e6	−0.457478
$472$	−2.82646e6	−0.583966
$473$	1.44811e6	0.297611
$474$	7.01081e6	1.43325
$475$	0	0
$476$	3.50767e6	0.709580
$477$	−279693.	−0.0562841
$478$	6.12606e6	1.22634
$479$	−1.80345e6	−0.359141	−0.179570	−	0.983745i	$-0.557471\pi$
$479$	−1.80345e6	−0.359141	−0.179570	+	0.983745i	$0.557471\pi$
$480$	0	0
$481$	−1.80205e6	−0.355143
$482$	8.91501e6	1.74785
$483$	−1.00582e7	−1.96179
$484$	360766.	0.0700022
$485$	0	0
$486$	444403.	0.0853466
$487$	−3.04965e6	−0.582678	−0.291339	−	0.956620i	$-0.594101\pi$
$487$	−3.04965e6	−0.582678	−0.291339	+	0.956620i	$0.594101\pi$
$488$	553204.	0.105156
$489$	3.92061e6	0.741449
$490$	0	0
$491$	−7.53319e6	−1.41018	−0.705091	−	0.709117i	$-0.749094\pi$
$491$	−7.53319e6	−1.41018	−0.705091	+	0.709117i	$0.749094\pi$
$492$	−1.85297e6	−0.345108
$493$	1.70548e6	0.316031
$494$	−3.19920e6	−0.589826
$495$	0	0
$496$	−1.23647e7	−2.25673
$497$	−7.14606e6	−1.29770
$498$	294070.	0.0531346
$499$	6.01840e6	1.08201	0.541003	−	0.841021i	$-0.318045\pi$
$499$	6.01840e6	1.08201	0.541003	+	0.841021i	$0.318045\pi$
$500$	0	0
$501$	2.59638e6	0.462140
$502$	4.52876e6	0.802085
$503$	−8.91752e6	−1.57153	−0.785767	−	0.618522i	$-0.787732\pi$
$503$	−8.91752e6	−1.57153	−0.785767	+	0.618522i	$0.787732\pi$
$504$	1.05034e6	0.184185
$505$	0	0
$506$	−4.34688e6	−0.754748
$507$	2.83959e6	0.490609
$508$	6.77385e6	1.16460
$509$	9.71294e6	1.66171	0.830857	−	0.556486i	$-0.187851\pi$
$509$	9.71294e6	1.66171	0.830857	+	0.556486i	$0.187851\pi$
$510$	0	0
$511$	−1.96384e7	−3.32701
$512$	−6.66153e6	−1.12305
$513$	1.31206e6	0.220120
$514$	6.80780e6	1.13658
$515$	0	0
$516$	−2.65408e6	−0.438823
$517$	−1.01464e6	−0.166950
$518$	−1.34440e7	−2.20143
$519$	4.85669e6	0.791448
$520$	0	0
$521$	−591081.	−0.0954009	−0.0477004	−	0.998862i	$-0.515189\pi$
$521$	−591081.	−0.0954009	−0.0477004	+	0.998862i	$0.515189\pi$
$522$	−1.70994e6	−0.274666
$523$	−2.83152e6	−0.452653	−0.226327	−	0.974051i	$-0.572672\pi$
$523$	−2.83152e6	−0.452653	−0.226327	+	0.974051i	$0.572672\pi$
$524$	−2.94148e6	−0.467991
$525$	0	0
$526$	1.10763e7	1.74554
$527$	6.23721e6	0.978282
$528$	1.31261e6	0.204905
$529$	1.63490e7	2.54011
$530$	0	0
$531$	−4.13363e6	−0.636202
$532$	−1.03832e7	−1.59057
$533$	−1.97343e6	−0.300887
$534$	−3.07947e6	−0.467329
$535$	0	0
$536$	2.02532e6	0.304496
$537$	3.40986e6	0.510271
$538$	−6.75404e6	−1.00602
$539$	4.59898e6	0.681851
$540$	0	0
$541$	9.93914e6	1.46001	0.730004	−	0.683442i	$-0.239518\pi$
$541$	9.93914e6	1.46001	0.730004	+	0.683442i	$0.239518\pi$
$542$	−1.01018e6	−0.147706
$543$	5.59469e6	0.814286
$544$	4.43793e6	0.642958
$545$	0	0
$546$	−3.74548e6	−0.537683
$547$	4.34846e6	0.621394	0.310697	−	0.950509i	$-0.399437\pi$
$547$	4.34846e6	0.621394	0.310697	+	0.950509i	$0.399437\pi$
$548$	3.25942e6	0.463649
$549$	809047.	0.114563
$550$	0	0
$551$	−5.04845e6	−0.708401
$552$	−2.37940e6	−0.332368
$553$	2.42332e7	3.36976
$554$	−1.51503e7	−2.09724
$555$	0	0
$556$	−2.52139e6	−0.345902
$557$	−1.07234e7	−1.46452	−0.732261	−	0.681024i	$-0.761535\pi$
$557$	−1.07234e7	−1.46452	−0.732261	+	0.681024i	$0.761535\pi$
$558$	−6.25354e6	−0.850237
$559$	−2.82662e6	−0.382594
$560$	0	0
$561$	−662129.	−0.0888250
$562$	−1.09281e7	−1.45950
$563$	−1.67966e6	−0.223332	−0.111666	−	0.993746i	$-0.535619\pi$
$563$	−1.67966e6	−0.223332	−0.111666	+	0.993746i	$0.535619\pi$
$564$	1.85962e6	0.246165
$565$	0	0
$566$	−1.68896e7	−2.21604
$567$	1.53610e6	0.200661
$568$	−1.69049e6	−0.219858
$569$	1.04510e7	1.35325	0.676626	−	0.736327i	$-0.263441\pi$
$569$	1.04510e7	1.35325	0.676626	+	0.736327i	$0.263441\pi$
$570$	0	0
$571$	670155.	0.0860171	0.0430085	−	0.999075i	$-0.486306\pi$
$571$	670155.	0.0860171	0.0430085	+	0.999075i	$0.486306\pi$
$572$	−704191.	−0.0899913
$573$	−76586.5	−0.00974464
$574$	−1.47226e7	−1.86512
$575$	0	0
$576$	−1.32531e6	−0.166441
$577$	7.87543e6	0.984769	0.492385	−	0.870378i	$-0.336125\pi$
$577$	7.87543e6	0.984769	0.492385	+	0.870378i	$0.336125\pi$
$578$	7.90362e6	0.984026
$579$	−5.08129e6	−0.629909
$580$	0	0
$581$	1.01647e6	0.124926
$582$	1.24068e7	1.51829
$583$	−417813.	−0.0509109
$584$	−4.64571e6	−0.563664
$585$	0	0
$586$	1.11907e6	0.134621
$587$	9.13414e6	1.09414	0.547069	−	0.837087i	$-0.315743\pi$
$587$	9.13414e6	1.09414	0.547069	+	0.837087i	$0.315743\pi$
$588$	−8.42894e6	−1.00538
$589$	−1.84630e7	−2.19288
$590$	0	0
$591$	−3.89695e6	−0.458940
$592$	−9.19652e6	−1.07850
$593$	−1.25096e7	−1.46085	−0.730425	−	0.682993i	$-0.760678\pi$
$593$	−1.25096e7	−1.46085	−0.730425	+	0.682993i	$0.760678\pi$
$594$	663862.	0.0771989
$595$	0	0
$596$	9.15887e6	1.05615
$597$	5.53384e6	0.635464
$598$	8.48484e6	0.970265
$599$	1.59560e7	1.81700	0.908502	−	0.417881i	$-0.137227\pi$
$599$	1.59560e7	1.81700	0.908502	+	0.417881i	$0.137227\pi$
$600$	0	0
$601$	763693.	0.0862448	0.0431224	−	0.999070i	$-0.486269\pi$
$601$	763693.	0.0862448	0.0431224	+	0.999070i	$0.486269\pi$
$602$	−2.10878e7	−2.37159
$603$	2.96198e6	0.331733
$604$	−2.75300e6	−0.307053
$605$	0	0
$606$	−1.95300e6	−0.216033
$607$	8.51753e6	0.938301	0.469150	−	0.883118i	$-0.344560\pi$
$607$	8.51753e6	0.938301	0.469150	+	0.883118i	$0.344560\pi$
$608$	−1.31369e7	−1.44123
$609$	−5.91050e6	−0.645775
$610$	0	0
$611$	1.98052e6	0.214623
$612$	1.21354e6	0.130971
$613$	−9.58182e6	−1.02990	−0.514952	−	0.857219i	$-0.672190\pi$
$613$	−9.58182e6	−1.02990	−0.514952	+	0.857219i	$0.672190\pi$
$614$	1.52254e7	1.62985
$615$	0	0
$616$	1.56903e6	0.166602
$617$	1.12974e7	1.19472	0.597362	−	0.801972i	$-0.296216\pi$
$617$	1.12974e7	1.19472	0.597362	+	0.801972i	$0.296216\pi$
$618$	4.02642e6	0.424080
$619$	−4.12945e6	−0.433177	−0.216588	−	0.976263i	$-0.569493\pi$
$619$	−4.12945e6	−0.433177	−0.216588	+	0.976263i	$0.569493\pi$
$620$	0	0
$621$	−3.47981e6	−0.362098
$622$	−1.41227e7	−1.46366
$623$	−1.06443e7	−1.09875
$624$	−2.56213e6	−0.263415
$625$	0	0
$626$	−673537.	−0.0686951
$627$	1.95999e6	0.199106
$628$	6.03023e6	0.610148
$629$	4.63906e6	0.467523
$630$	0	0
$631$	1.29753e7	1.29731	0.648656	−	0.761082i	$-0.275332\pi$
$631$	1.29753e7	1.29731	0.648656	+	0.761082i	$0.275332\pi$
$632$	5.73268e6	0.570907
$633$	−651133.	−0.0645893
$634$	1.95762e6	0.193422
$635$	0	0
$636$	765761.	0.0750672
$637$	−8.97691e6	−0.876553
$638$	−2.55436e6	−0.248445
$639$	−2.47230e6	−0.239524
$640$	0	0
$641$	−130086.	−0.0125051	−0.00625255	−	0.999980i	$-0.501990\pi$
$641$	−130086.	−0.0125051	−0.00625255	+	0.999980i	$0.501990\pi$
$642$	−937483.	−0.0897689
$643$	6.47507e6	0.617614	0.308807	−	0.951125i	$-0.400070\pi$
$643$	6.47507e6	0.617614	0.308807	+	0.951125i	$0.400070\pi$
$644$	2.75380e7	2.61648
$645$	0	0
$646$	8.23579e6	0.776469
$647$	5.42540e6	0.509531	0.254766	−	0.967003i	$-0.418002\pi$
$647$	5.42540e6	0.509531	0.254766	+	0.967003i	$0.418002\pi$
$648$	363385.	0.0339961
$649$	−6.17492e6	−0.575466
$650$	0	0
$651$	−2.16157e7	−1.99902
$652$	−1.07341e7	−0.988886
$653$	−9.15697e6	−0.840367	−0.420183	−	0.907439i	$-0.638034\pi$
$653$	−9.15697e6	−0.840367	−0.420183	+	0.907439i	$0.638034\pi$
$654$	−899382.	−0.0822243
$655$	0	0
$656$	−1.00712e7	−0.913735
$657$	−6.79424e6	−0.614083
$658$	1.47755e7	1.33039
$659$	−4.88080e6	−0.437802	−0.218901	−	0.975747i	$-0.570247\pi$
$659$	−4.88080e6	−0.437802	−0.218901	+	0.975747i	$0.570247\pi$
$660$	0	0
$661$	1.75690e7	1.56402	0.782010	−	0.623266i	$-0.214194\pi$
$661$	1.75690e7	1.56402	0.782010	+	0.623266i	$0.214194\pi$
$662$	−1.65074e7	−1.46398
$663$	1.29243e6	0.114189
$664$	240459.	0.0211651
$665$	0	0
$666$	−4.65120e6	−0.406330
$667$	1.33894e7	1.16532
$668$	−7.10853e6	−0.616366
$669$	3.33333e6	0.287948
$670$	0	0
$671$	1.20858e6	0.103626
$672$	−1.53801e7	−1.31382
$673$	−1.60675e7	−1.36745	−0.683725	−	0.729740i	$-0.739641\pi$
$673$	−1.60675e7	−1.36745	−0.683725	+	0.729740i	$0.739641\pi$
$674$	7.02205e6	0.595407
$675$	0	0
$676$	−7.77441e6	−0.654336
$677$	1.51931e7	1.27401	0.637007	−	0.770858i	$-0.280172\pi$
$677$	1.51931e7	1.27401	0.637007	+	0.770858i	$0.280172\pi$
$678$	−8.81123e6	−0.736143
$679$	4.28849e7	3.56968
$680$	0	0
$681$	2.92933e6	0.242048
$682$	−9.34170e6	−0.769068
$683$	9.75866e6	0.800457	0.400229	−	0.916415i	$-0.368931\pi$
$683$	9.75866e6	0.800457	0.400229	+	0.916415i	$0.368931\pi$
$684$	−3.59224e6	−0.293579
$685$	0	0
$686$	−3.73571e7	−3.03084
$687$	516105.	0.0417202
$688$	−1.44253e7	−1.16186
$689$	815544.	0.0654484
$690$	0	0
$691$	−1.74840e7	−1.39298	−0.696490	−	0.717567i	$-0.745256\pi$
$691$	−1.74840e7	−1.39298	−0.696490	+	0.717567i	$0.745256\pi$
$692$	−1.32970e7	−1.05557
$693$	2.29467e6	0.181504
$694$	−1.30188e7	−1.02606
$695$	0	0
$696$	−1.39821e6	−0.109408
$697$	5.08026e6	0.396099
$698$	−7.65925e6	−0.595042
$699$	−7.50502e6	−0.580977
$700$	0	0
$701$	4.31053e6	0.331311	0.165656	−	0.986184i	$-0.447026\pi$
$701$	4.31053e6	0.331311	0.165656	+	0.986184i	$0.447026\pi$
$702$	−1.29581e6	−0.0992431
$703$	−1.37322e7	−1.04798
$704$	−1.97978e6	−0.150551
$705$	0	0
$706$	−2.47702e7	−1.87033
$707$	−6.75064e6	−0.507921
$708$	1.13173e7	0.848516
$709$	−1.44521e6	−0.107973	−0.0539866	−	0.998542i	$-0.517193\pi$
$709$	−1.44521e6	−0.107973	−0.0539866	+	0.998542i	$0.517193\pi$
$710$	0	0
$711$	8.38390e6	0.621974
$712$	−2.51806e6	−0.186151
$713$	4.89671e7	3.60728
$714$	9.64210e6	0.707825
$715$	0	0
$716$	−9.33574e6	−0.680559
$717$	7.32587e6	0.532184
$718$	−1.74699e7	−1.26468
$719$	−4.22567e6	−0.304841	−0.152420	−	0.988316i	$-0.548707\pi$
$719$	−4.22567e6	−0.304841	−0.152420	+	0.988316i	$0.548707\pi$
$720$	0	0
$721$	1.39175e7	0.997066
$722$	−5.74393e6	−0.410078
$723$	1.06610e7	0.758498
$724$	−1.53175e7	−1.08603
$725$	0	0
$726$	991694.	0.0698291
$727$	−1.62930e7	−1.14331	−0.571657	−	0.820493i	$-0.693699\pi$
$727$	−1.62930e7	−1.14331	−0.571657	+	0.820493i	$0.693699\pi$
$728$	−3.06265e6	−0.214175
$729$	531441.	0.0370370
$730$	0	0
$731$	7.27665e6	0.503661
$732$	−2.21506e6	−0.152795
$733$	3.00376e6	0.206493	0.103247	−	0.994656i	$-0.467077\pi$
$733$	3.00376e6	0.206493	0.103247	+	0.994656i	$0.467077\pi$
$734$	−1.36521e7	−0.935319
$735$	0	0
$736$	3.48412e7	2.37082
$737$	4.42468e6	0.300064
$738$	−5.09355e6	−0.344255
$739$	5.58694e6	0.376325	0.188163	−	0.982138i	$-0.439747\pi$
$739$	5.58694e6	0.376325	0.188163	+	0.982138i	$0.439747\pi$
$740$	0	0
$741$	−3.82578e6	−0.255961
$742$	6.08431e6	0.405697
$743$	8.33428e6	0.553855	0.276927	−	0.960891i	$-0.410684\pi$
$743$	8.33428e6	0.553855	0.276927	+	0.960891i	$0.410684\pi$
$744$	−5.11346e6	−0.338675
$745$	0	0
$746$	−1.20789e6	−0.0794659
$747$	351665.	0.0230583
$748$	1.81282e6	0.118468
$749$	−3.24046e6	−0.211058
$750$	0	0
$751$	−1.55555e7	−1.00643	−0.503215	−	0.864162i	$-0.667849\pi$
$751$	−1.55555e7	−1.00643	−0.503215	+	0.864162i	$0.667849\pi$
$752$	1.01073e7	0.651766
$753$	5.41574e6	0.348073
$754$	4.98594e6	0.319388
$755$	0	0
$756$	−4.20564e6	−0.267625
$757$	−2.42999e7	−1.54122	−0.770611	−	0.637306i	$-0.780049\pi$
$757$	−2.42999e7	−1.54122	−0.770611	+	0.637306i	$0.780049\pi$
$758$	4.68450e6	0.296135
$759$	−5.19824e6	−0.327530
$760$	0	0
$761$	1.84687e7	1.15605	0.578023	−	0.816021i	$-0.303825\pi$
$761$	1.84687e7	1.15605	0.578023	+	0.816021i	$0.303825\pi$
$762$	1.86204e7	1.16172
$763$	−3.10876e6	−0.193320
$764$	209684.	0.0129966
$765$	0	0
$766$	1.12194e7	0.690873
$767$	1.20530e7	0.739790
$768$	−1.21921e7	−0.745890
$769$	−1.51725e6	−0.0925210	−0.0462605	−	0.998929i	$-0.514730\pi$
$769$	−1.51725e6	−0.0925210	−0.0462605	+	0.998929i	$0.514730\pi$
$770$	0	0
$771$	8.14113e6	0.493229
$772$	1.39119e7	0.840122
$773$	−3.28907e6	−0.197981	−0.0989907	−	0.995088i	$-0.531561\pi$
$773$	−3.28907e6	−0.197981	−0.0989907	+	0.995088i	$0.531561\pi$
$774$	−7.29569e6	−0.437738
$775$	0	0
$776$	1.01450e7	0.604779
$777$	−1.60771e7	−0.955334
$778$	1.24996e7	0.740367
$779$	−1.50383e7	−0.887879
$780$	0	0
$781$	−3.69319e6	−0.216658
$782$	−2.18427e7	−1.27729
$783$	−2.04484e6	−0.119194
$784$	−4.58126e7	−2.66192
$785$	0	0
$786$	−8.08572e6	−0.466834
$787$	1.46769e7	0.844692	0.422346	−	0.906435i	$-0.361207\pi$
$787$	1.46769e7	0.844692	0.422346	+	0.906435i	$0.361207\pi$
$788$	1.06693e7	0.612098
$789$	1.32457e7	0.757497
$790$	0	0
$791$	−3.04564e7	−1.73076
$792$	542834.	0.0307506
$793$	−2.35906e6	−0.133216
$794$	1.85409e7	1.04371
$795$	0	0
$796$	−1.51509e7	−0.847531
$797$	−4.95422e6	−0.276267	−0.138134	−	0.990414i	$-0.544110\pi$
$797$	−4.95422e6	−0.276267	−0.138134	+	0.990414i	$0.544110\pi$
$798$	−2.85419e7	−1.58663
$799$	−5.09850e6	−0.282537
$800$	0	0
$801$	−3.68259e6	−0.202802
$802$	5.00283e6	0.274650
$803$	−1.01494e7	−0.555459
$804$	−8.10949e6	−0.442439
$805$	0	0
$806$	1.82344e7	0.988675
$807$	−8.07685e6	−0.436574
$808$	−1.59695e6	−0.0860524
$809$	1.90776e7	1.02483	0.512414	−	0.858738i	$-0.328751\pi$
$809$	1.90776e7	1.02483	0.512414	+	0.858738i	$0.328751\pi$
$810$	0	0
$811$	2.54611e7	1.35933	0.679665	−	0.733523i	$-0.262125\pi$
$811$	2.54611e7	1.35933	0.679665	+	0.733523i	$0.262125\pi$
$812$	1.61822e7	0.861284
$813$	−1.20802e6	−0.0640987
$814$	−6.94809e6	−0.367540
$815$	0	0
$816$	6.59577e6	0.346769
$817$	−2.15399e7	−1.12899
$818$	836308.	0.0437002
$819$	−4.47905e6	−0.233333
$820$	0	0
$821$	−3.75556e7	−1.94454	−0.972269	−	0.233867i	$-0.924862\pi$
$821$	−3.75556e7	−1.94454	−0.972269	+	0.233867i	$0.924862\pi$
$822$	8.95970e6	0.462502
$823$	3.42815e7	1.76425	0.882126	−	0.471014i	$-0.156112\pi$
$823$	3.42815e7	1.76425	0.882126	+	0.471014i	$0.156112\pi$
$824$	3.29237e6	0.168924
$825$	0	0
$826$	8.99209e7	4.58575
$827$	−2.92800e7	−1.48870	−0.744351	−	0.667789i	$-0.767241\pi$
$827$	−2.92800e7	−1.48870	−0.744351	+	0.667789i	$0.767241\pi$
$828$	9.52725e6	0.482938
$829$	−7.40301e6	−0.374130	−0.187065	−	0.982348i	$-0.559897\pi$
$829$	−7.40301e6	−0.374130	−0.187065	+	0.982348i	$0.559897\pi$
$830$	0	0
$831$	−1.81176e7	−0.910117
$832$	3.86440e6	0.193541
$833$	2.31095e7	1.15393
$834$	−6.93094e6	−0.345046
$835$	0	0
$836$	−5.36619e6	−0.265552
$837$	−7.47831e6	−0.368969
$838$	4.02224e7	1.97860
$839$	−3.44573e7	−1.68996	−0.844981	−	0.534797i	$-0.820388\pi$
$839$	−3.44573e7	−1.68996	−0.844981	+	0.534797i	$0.820388\pi$
$840$	0	0
$841$	−1.26432e7	−0.616404
$842$	3.63950e6	0.176914
$843$	−1.30684e7	−0.633364
$844$	1.78271e6	0.0861441
$845$	0	0
$846$	5.11184e6	0.245557
$847$	3.42784e6	0.164177
$848$	4.16203e6	0.198754
$849$	−2.01974e7	−0.961672
$850$	0	0
$851$	3.64203e7	1.72393
$852$	6.76883e6	0.319459
$853$	1.70125e7	0.800561	0.400281	−	0.916393i	$-0.368913\pi$
$853$	1.70125e7	0.800561	0.400281	+	0.916393i	$0.368913\pi$
$854$	−1.75996e7	−0.825769
$855$	0	0
$856$	−766572.	−0.0357576
$857$	−2.11650e7	−0.984387	−0.492194	−	0.870486i	$-0.663805\pi$
$857$	−2.11650e7	−0.984387	−0.492194	+	0.870486i	$0.663805\pi$
$858$	−1.93572e6	−0.0897687
$859$	−3.17802e7	−1.46952	−0.734758	−	0.678330i	$-0.762704\pi$
$859$	−3.17802e7	−1.46952	−0.734758	+	0.678330i	$0.762704\pi$
$860$	0	0
$861$	−1.76061e7	−0.809386
$862$	1.81785e6	0.0833277
$863$	3.53199e7	1.61433	0.807166	−	0.590325i	$-0.201000\pi$
$863$	3.53199e7	1.61433	0.807166	+	0.590325i	$0.201000\pi$
$864$	−5.32100e6	−0.242498
$865$	0	0
$866$	−7.82692e6	−0.354647
$867$	9.45157e6	0.427028
$868$	5.91807e7	2.66613
$869$	1.25241e7	0.562597
$870$	0	0
$871$	−8.63669e6	−0.385746
$872$	−735417.	−0.0327524
$873$	1.48368e7	0.658876
$874$	6.46575e7	2.86313
$875$	0	0
$876$	1.86017e7	0.819016
$877$	−256700.	−0.0112701	−0.00563503	−	0.999984i	$-0.501794\pi$
$877$	−256700.	−0.0112701	−0.00563503	+	0.999984i	$0.501794\pi$
$878$	2.50606e7	1.09712
$879$	1.33824e6	0.0584201
$880$	0	0
$881$	3.37169e7	1.46355	0.731776	−	0.681545i	$-0.238692\pi$
$881$	3.37169e7	1.46355	0.731776	+	0.681545i	$0.238692\pi$
$882$	−2.31700e7	−1.00289
$883$	2.45722e7	1.06058	0.530289	−	0.847817i	$-0.322083\pi$
$883$	2.45722e7	1.06058	0.530289	+	0.847817i	$0.322083\pi$
$884$	−3.53850e6	−0.152296
$885$	0	0
$886$	3.85542e7	1.65001
$887$	−220190.	−0.00939697	−0.00469849	−	0.999989i	$-0.501496\pi$
$887$	−220190.	−0.00939697	−0.00469849	+	0.999989i	$0.501496\pi$
$888$	−3.80325e6	−0.161853
$889$	6.43623e7	2.73135
$890$	0	0
$891$	793881.	0.0335013
$892$	−9.12622e6	−0.384042
$893$	1.50923e7	0.633323
$894$	2.51765e7	1.05354
$895$	0	0
$896$	−2.58546e7	−1.07589
$897$	1.01466e7	0.421056
$898$	−4.38260e7	−1.81360
$899$	2.87745e7	1.18743
$900$	0	0
$901$	−2.09948e6	−0.0861587
$902$	−7.60889e6	−0.311390
$903$	−2.52179e7	−1.02918
$904$	−7.20487e6	−0.293228
$905$	0	0
$906$	−7.56760e6	−0.306293
$907$	6.02862e6	0.243332	0.121666	−	0.992571i	$-0.461176\pi$
$907$	6.02862e6	0.243332	0.121666	+	0.992571i	$0.461176\pi$
$908$	−8.02011e6	−0.322824
$909$	−2.33550e6	−0.0937497
$910$	0	0
$911$	1.68076e7	0.670979	0.335490	−	0.942044i	$-0.391098\pi$
$911$	1.68076e7	0.670979	0.335490	+	0.942044i	$0.391098\pi$
$912$	−1.95244e7	−0.777303
$913$	525327.	0.0208570
$914$	−4.00705e7	−1.58657
$915$	0	0
$916$	−1.41302e6	−0.0556430
$917$	−2.79487e7	−1.09758
$918$	3.33585e6	0.130647
$919$	4.57189e7	1.78569	0.892846	−	0.450362i	$-0.148705\pi$
$919$	4.57189e7	1.78569	0.892846	+	0.450362i	$0.148705\pi$
$920$	0	0
$921$	1.82074e7	0.707292
$922$	−5.07022e6	−0.196426
$923$	7.20887e6	0.278524
$924$	−6.28250e6	−0.242076
$925$	0	0
$926$	−1.44672e7	−0.554443
$927$	4.81501e6	0.184034
$928$	2.04737e7	0.780418
$929$	−2.84053e7	−1.07984	−0.539921	−	0.841716i	$-0.681546\pi$
$929$	−2.84053e7	−1.07984	−0.539921	+	0.841716i	$0.681546\pi$
$930$	0	0
$931$	−6.84073e7	−2.58659
$932$	2.05477e7	0.774861
$933$	−1.68886e7	−0.635171
$934$	−5.98068e7	−2.24328
$935$	0	0
$936$	−1.05958e6	−0.0395315
$937$	8.99373e6	0.334650	0.167325	−	0.985902i	$-0.446487\pi$
$937$	8.99373e6	0.334650	0.167325	+	0.985902i	$0.446487\pi$
$938$	−6.44334e7	−2.39113
$939$	−805452.	−0.0298109
$940$	0	0
$941$	−2.96562e7	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$941$	−2.96562e7	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$942$	1.65763e7	0.608639
$943$	3.98841e7	1.46056
$944$	6.15113e7	2.24659
$945$	0	0
$946$	−1.08985e7	−0.395949
$947$	1.62357e7	0.588297	0.294148	−	0.955760i	$-0.404964\pi$
$947$	1.62357e7	0.588297	0.294148	+	0.955760i	$0.404964\pi$
$948$	−2.29540e7	−0.829540
$949$	1.98110e7	0.714070
$950$	0	0
$951$	2.34103e6	0.0839375
$952$	7.88427e6	0.281948
$953$	−2.13990e7	−0.763241	−0.381621	−	0.924319i	$-0.624634\pi$
$953$	−2.13990e7	−0.763241	−0.381621	+	0.924319i	$0.624634\pi$
$954$	2.10497e6	0.0748816
$955$	0	0
$956$	−2.00572e7	−0.709785
$957$	−3.05464e6	−0.107815
$958$	1.35728e7	0.477809
$959$	3.09697e7	1.08740
$960$	0	0
$961$	7.66039e7	2.67573
$962$	1.35622e7	0.472490
$963$	−1.12109e6	−0.0389561
$964$	−2.91885e7	−1.01162
$965$	0	0
$966$	7.56981e7	2.61001
$967$	6.31736e6	0.217255	0.108627	−	0.994083i	$-0.465354\pi$
$967$	6.31736e6	0.217255	0.108627	+	0.994083i	$0.465354\pi$
$968$	810900.	0.0278150
$969$	9.84880e6	0.336956
$970$	0	0
$971$	4.14293e7	1.41013	0.705065	−	0.709142i	$-0.250918\pi$
$971$	4.14293e7	1.41013	0.705065	+	0.709142i	$0.250918\pi$
$972$	−1.45501e6	−0.0493971
$973$	−2.39572e7	−0.811247
$974$	2.29517e7	0.775207
$975$	0	0
$976$	−1.20392e7	−0.404550
$977$	333761.	0.0111866	0.00559331	−	0.999984i	$-0.498220\pi$
$977$	333761.	0.0111866	0.00559331	+	0.999984i	$0.498220\pi$
$978$	−2.95065e7	−0.986440
$979$	−5.50116e6	−0.183441
$980$	0	0
$981$	−1.07553e6	−0.0356820
$982$	5.66948e7	1.87614
$983$	3.43832e7	1.13491	0.567457	−	0.823403i	$-0.307927\pi$
$983$	3.43832e7	1.13491	0.567457	+	0.823403i	$0.307927\pi$
$984$	−4.16496e6	−0.137127
$985$	0	0
$986$	−1.28354e7	−0.420454
$987$	1.76693e7	0.577335
$988$	1.04745e7	0.341381
$989$	5.71275e7	1.85718
$990$	0	0
$991$	4.26710e6	0.138022	0.0690111	−	0.997616i	$-0.478016\pi$
$991$	4.26710e6	0.138022	0.0690111	+	0.997616i	$0.478016\pi$
$992$	7.48758e7	2.41581
$993$	−1.97405e7	−0.635308
$994$	5.37813e7	1.72649
$995$	0	0
$996$	−962811.	−0.0307534
$997$	−2.61381e7	−0.832790	−0.416395	−	0.909184i	$-0.636707\pi$
$997$	−2.61381e7	−0.832790	−0.416395	+	0.909184i	$0.636707\pi$
$998$	−4.52945e7	−1.43953
$999$	−5.56215e6	−0.176331

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.52601 q^{2} -9.00000 q^{3} +24.6408 q^{4} +67.7341 q^{6} +234.126 q^{7} +55.3856 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-7.52601 q^{2} -9.00000 q^{3} +24.6408 q^{4} +67.7341 q^{6} +234.126 q^{7} +55.3856 q^{8} +81.0000 q^{9} +121.000 q^{11} -221.767 q^{12} -236.184 q^{13} -1762.04 q^{14} -1205.34 q^{16} +608.015 q^{17} -609.607 q^{18} -1799.81 q^{19} -2107.14 q^{21} -910.647 q^{22} +4773.40 q^{23} -498.470 q^{24} +1777.52 q^{26} -729.000 q^{27} +5769.05 q^{28} +2804.99 q^{29} +10258.3 q^{31} +7299.04 q^{32} -1089.00 q^{33} -4575.93 q^{34} +1995.90 q^{36} +7629.84 q^{37} +13545.4 q^{38} +2125.66 q^{39} +8355.48 q^{41} +15858.3 q^{42} +11967.9 q^{43} +2981.53 q^{44} -35924.7 q^{46} -8385.48 q^{47} +10848.0 q^{48} +38008.1 q^{49} -5472.14 q^{51} -5819.76 q^{52} -3453.00 q^{53} +5486.46 q^{54} +12967.2 q^{56} +16198.3 q^{57} -21110.4 q^{58} -51032.4 q^{59} +9988.24 q^{61} -77204.1 q^{62} +18964.2 q^{63} -16361.8 q^{64} +8195.82 q^{66} +36567.6 q^{67} +14982.0 q^{68} -42960.6 q^{69} -30522.3 q^{71} +4486.23 q^{72} -83879.5 q^{73} -57422.2 q^{74} -44348.7 q^{76} +28329.3 q^{77} -15997.7 q^{78} +103505. q^{79} +6561.00 q^{81} -62883.4 q^{82} +4341.54 q^{83} -51921.5 q^{84} -90070.3 q^{86} -25244.9 q^{87} +6701.65 q^{88} -45464.1 q^{89} -55296.9 q^{91} +117620. q^{92} -92324.8 q^{93} +63109.2 q^{94} -65691.3 q^{96} +183170. q^{97} -286049. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Introduction

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