LMFDB - Embedded newform 825.6.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 $ x^8 - x^7 - 176x^6 + 272x^5 + 9055x^4 - 15851x^3 - 118840x^2 + 149572x - 33248
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-4.15857$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.15857 q^{2} +9.00000 q^{3} -22.0234 q^{4} +28.4272 q^{6} +35.7175 q^{7} -170.637 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+3.15857 q^{2} +9.00000 q^{3} -22.0234 q^{4} +28.4272 q^{6} +35.7175 q^{7} -170.637 q^{8} +81.0000 q^{9} -121.000 q^{11} -198.211 q^{12} -128.844 q^{13} +112.816 q^{14} +165.780 q^{16} +233.062 q^{17} +255.844 q^{18} -533.777 q^{19} +321.457 q^{21} -382.187 q^{22} +4309.77 q^{23} -1535.73 q^{24} -406.964 q^{26} +729.000 q^{27} -786.621 q^{28} +7053.77 q^{29} -7484.11 q^{31} +5984.01 q^{32} -1089.00 q^{33} +736.144 q^{34} -1783.90 q^{36} -7778.03 q^{37} -1685.97 q^{38} -1159.60 q^{39} +5635.48 q^{41} +1015.35 q^{42} +1910.79 q^{43} +2664.83 q^{44} +13612.7 q^{46} -29248.2 q^{47} +1492.02 q^{48} -15531.3 q^{49} +2097.56 q^{51} +2837.59 q^{52} -7526.88 q^{53} +2302.60 q^{54} -6094.72 q^{56} -4803.99 q^{57} +22279.8 q^{58} -29371.8 q^{59} +42254.3 q^{61} -23639.1 q^{62} +2893.12 q^{63} +13596.0 q^{64} -3439.69 q^{66} -26453.6 q^{67} -5132.83 q^{68} +38787.9 q^{69} -24075.2 q^{71} -13821.6 q^{72} -75389.2 q^{73} -24567.5 q^{74} +11755.6 q^{76} -4321.82 q^{77} -3662.68 q^{78} +36440.5 q^{79} +6561.00 q^{81} +17800.1 q^{82} +44549.7 q^{83} -7079.59 q^{84} +6035.37 q^{86} +63483.9 q^{87} +20647.1 q^{88} -71509.6 q^{89} -4601.99 q^{91} -94915.8 q^{92} -67357.0 q^{93} -92382.5 q^{94} +53856.1 q^{96} -81704.7 q^{97} -49056.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) $ 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 - 968 * q^11 + 963 * q^12 - 382 * q^13 + 1048 * q^14 - 325 * q^16 - 288 * q^17 - 729 * q^18 - 988 * q^19 + 594 * q^21 + 1089 * q^22 - 5972 * q^23 - 1863 * q^24 - 14579 * q^26 + 5832 * q^27 - 5942 * q^28 + 1032 * q^29 - 4682 * q^31 - 3863 * q^32 - 8712 * q^33 + 2206 * q^34 + 8667 * q^36 + 17200 * q^37 - 11011 * q^38 - 3438 * q^39 - 13220 * q^41 + 9432 * q^42 - 22872 * q^43 - 12947 * q^44 + 9101 * q^46 - 6700 * q^47 - 2925 * q^48 - 43466 * q^49 - 2592 * q^51 - 5009 * q^52 - 6224 * q^53 - 6561 * q^54 + 20992 * q^56 - 8892 * q^57 - 33015 * q^58 - 77556 * q^59 + 11554 * q^61 - 12135 * q^62 + 5346 * q^63 - 149917 * q^64 + 9801 * q^66 - 20894 * q^67 - 91776 * q^68 - 53748 * q^69 - 21648 * q^71 - 16767 * q^72 - 64660 * q^73 - 179522 * q^74 + 24401 * q^76 - 7986 * q^77 - 131211 * q^78 - 22660 * q^79 + 52488 * q^81 + 56080 * q^82 - 100390 * q^83 - 53478 * q^84 + 47271 * q^86 + 9288 * q^87 + 25047 * q^88 - 25578 * q^89 + 73250 * q^91 - 95311 * q^92 - 42138 * q^93 - 170120 * q^94 - 34767 * q^96 - 142828 * q^97 - 303397 * q^98 - 78408 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.15857	0.558362	0.279181	−	0.960238i	$-0.409937\pi$
$2$	3.15857	0.558362	0.279181	+	0.960238i	$0.409937\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−22.0234	−0.688232
$5$	0	0
$5$	0	0
$6$	28.4272	0.322370
$7$	35.7175	0.275509	0.137755	−	0.990466i	$-0.456011\pi$
$7$	35.7175	0.275509	0.137755	+	0.990466i	$0.456011\pi$
$8$	−170.637	−0.942645
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−198.211	−0.397351
$13$	−128.844	−0.211450	−0.105725	−	0.994395i	$-0.533716\pi$
$13$	−128.844	−0.211450	−0.105725	+	0.994395i	$0.533716\pi$
$14$	112.816	0.153834
$15$	0	0
$16$	165.780	0.161895
$17$	233.062	0.195591	0.0977957	−	0.995207i	$-0.468821\pi$
$17$	233.062	0.195591	0.0977957	+	0.995207i	$0.468821\pi$
$18$	255.844	0.186121
$19$	−533.777	−0.339216	−0.169608	−	0.985512i	$-0.554250\pi$
$19$	−533.777	−0.339216	−0.169608	+	0.985512i	$0.554250\pi$
$20$	0	0
$21$	321.457	0.159065
$22$	−382.187	−0.168352
$23$	4309.77	1.69877	0.849384	−	0.527775i	$-0.176974\pi$
$23$	4309.77	1.69877	0.849384	+	0.527775i	$0.176974\pi$
$24$	−1535.73	−0.544236
$25$	0	0
$26$	−406.964	−0.118065
$27$	729.000	0.192450
$28$	−786.621	−0.189614
$29$	7053.77	1.55749	0.778747	−	0.627339i	$-0.215856\pi$
$29$	7053.77	1.55749	0.778747	+	0.627339i	$0.215856\pi$
$30$	0	0
$31$	−7484.11	−1.39874	−0.699368	−	0.714762i	$-0.746535\pi$
$31$	−7484.11	−1.39874	−0.699368	+	0.714762i	$0.746535\pi$
$32$	5984.01	1.03304
$33$	−1089.00	−0.174078
$34$	736.144	0.109211
$35$	0	0
$36$	−1783.90	−0.229411
$37$	−7778.03	−0.934039	−0.467020	−	0.884247i	$-0.654672\pi$
$37$	−7778.03	−0.934039	−0.467020	+	0.884247i	$0.654672\pi$
$38$	−1685.97	−0.189405
$39$	−1159.60	−0.122080
$40$	0	0
$41$	5635.48	0.523566	0.261783	−	0.965127i	$-0.415689\pi$
$41$	5635.48	0.523566	0.261783	+	0.965127i	$0.415689\pi$
$42$	1015.35	0.0888160
$43$	1910.79	0.157595	0.0787974	−	0.996891i	$-0.474892\pi$
$43$	1910.79	0.157595	0.0787974	+	0.996891i	$0.474892\pi$
$44$	2664.83	0.207510
$45$	0	0
$46$	13612.7	0.948528
$47$	−29248.2	−1.93132	−0.965660	−	0.259811i	$-0.916340\pi$
$47$	−29248.2	−1.93132	−0.965660	+	0.259811i	$0.916340\pi$
$48$	1492.02	0.0934701
$49$	−15531.3	−0.924095
$50$	0	0
$51$	2097.56	0.112925
$52$	2837.59	0.145526
$53$	−7526.88	−0.368066	−0.184033	−	0.982920i	$-0.558915\pi$
$53$	−7526.88	−0.368066	−0.184033	+	0.982920i	$0.558915\pi$
$54$	2302.60	0.107457
$55$	0	0
$56$	−6094.72	−0.259707
$57$	−4803.99	−0.195846
$58$	22279.8	0.869645
$59$	−29371.8	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$59$	−29371.8	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$60$	0	0
$61$	42254.3	1.45394	0.726970	−	0.686669i	$-0.240928\pi$
$61$	42254.3	1.45394	0.726970	+	0.686669i	$0.240928\pi$
$62$	−23639.1	−0.781001
$63$	2893.12	0.0918363
$64$	13596.0	0.414916
$65$	0	0
$66$	−3439.69	−0.0971983
$67$	−26453.6	−0.719942	−0.359971	−	0.932963i	$-0.617213\pi$
$67$	−26453.6	−0.719942	−0.359971	+	0.932963i	$0.617213\pi$
$68$	−5132.83	−0.134612
$69$	38787.9	0.980784
$70$	0	0
$71$	−24075.2	−0.566793	−0.283396	−	0.959003i	$-0.591461\pi$
$71$	−24075.2	−0.566793	−0.283396	+	0.959003i	$0.591461\pi$
$72$	−13821.6	−0.314215
$73$	−75389.2	−1.65578	−0.827889	−	0.560893i	$-0.810458\pi$
$73$	−75389.2	−1.65578	−0.827889	+	0.560893i	$0.810458\pi$
$74$	−24567.5	−0.521532
$75$	0	0
$76$	11755.6	0.233459
$77$	−4321.82	−0.0830691
$78$	−3662.68	−0.0681651
$79$	36440.5	0.656926	0.328463	−	0.944517i	$-0.393469\pi$
$79$	36440.5	0.656926	0.328463	+	0.944517i	$0.393469\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	17800.1	0.292339
$83$	44549.7	0.709822	0.354911	−	0.934900i	$-0.384511\pi$
$83$	44549.7	0.709822	0.354911	+	0.934900i	$0.384511\pi$
$84$	−7079.59	−0.109474
$85$	0	0
$86$	6035.37	0.0879950
$87$	63483.9	0.899219
$88$	20647.1	0.284218
$89$	−71509.6	−0.956950	−0.478475	−	0.878101i	$-0.658810\pi$
$89$	−71509.6	−0.956950	−0.478475	+	0.878101i	$0.658810\pi$
$90$	0	0
$91$	−4601.99	−0.0582562
$92$	−94915.8	−1.16915
$93$	−67357.0	−0.807561
$94$	−92382.5	−1.07838
$95$	0	0
$96$	53856.1	0.596426
$97$	−81704.7	−0.881694	−0.440847	−	0.897582i	$-0.645322\pi$
$97$	−81704.7	−0.881694	−0.440847	+	0.897582i	$0.645322\pi$
$98$	−49056.6	−0.515979
$99$	−9801.00	−0.100504
$100$	0	0
$101$	147465.	1.43842	0.719211	−	0.694792i	$-0.244504\pi$
$101$	147465.	1.43842	0.719211	+	0.694792i	$0.244504\pi$
$102$	6625.29	0.0630529
$103$	−69398.4	−0.644549	−0.322275	−	0.946646i	$-0.604447\pi$
$103$	−69398.4	−0.644549	−0.322275	+	0.946646i	$0.604447\pi$
$104$	21985.6	0.199322
$105$	0	0
$106$	−23774.2	−0.205514
$107$	−83958.7	−0.708935	−0.354467	−	0.935068i	$-0.615338\pi$
$107$	−83958.7	−0.708935	−0.354467	+	0.935068i	$0.615338\pi$
$108$	−16055.1	−0.132450
$109$	50656.1	0.408381	0.204190	−	0.978931i	$-0.434544\pi$
$109$	50656.1	0.408381	0.204190	+	0.978931i	$0.434544\pi$
$110$	0	0
$111$	−70002.2	−0.539268
$112$	5921.26	0.0446035
$113$	205869.	1.51669	0.758343	−	0.651855i	$-0.226009\pi$
$113$	205869.	1.51669	0.758343	+	0.651855i	$0.226009\pi$
$114$	−15173.8	−0.109353
$115$	0	0
$116$	−155348.	−1.07192
$117$	−10436.4	−0.0704832
$118$	−92773.0	−0.613361
$119$	8324.40	0.0538872
$120$	0	0
$121$	14641.0	0.0909091
$122$	133463.	0.811825
$123$	50719.4	0.302281
$124$	164826.	0.962655
$125$	0	0
$126$	9138.12	0.0512779
$127$	−132961.	−0.731501	−0.365751	−	0.930713i	$-0.619188\pi$
$127$	−132961.	−0.731501	−0.365751	+	0.930713i	$0.619188\pi$
$128$	−148545.	−0.801367
$129$	17197.1	0.0909874
$130$	0	0
$131$	−343760.	−1.75015	−0.875077	−	0.483983i	$-0.839190\pi$
$131$	−343760.	−1.75015	−0.875077	+	0.483983i	$0.839190\pi$
$132$	23983.5	0.119806
$133$	−19065.2	−0.0934570
$134$	−83555.6	−0.401988
$135$	0	0
$136$	−39769.0	−0.184373
$137$	−192965.	−0.878370	−0.439185	−	0.898397i	$-0.644733\pi$
$137$	−192965.	−0.878370	−0.439185	+	0.898397i	$0.644733\pi$
$138$	122514.	0.547633
$139$	−371990.	−1.63303	−0.816516	−	0.577323i	$-0.804097\pi$
$139$	−371990.	−1.63303	−0.816516	+	0.577323i	$0.804097\pi$
$140$	0	0
$141$	−263234.	−1.11505
$142$	−76043.3	−0.316476
$143$	15590.2	0.0637544
$144$	13428.2	0.0539650
$145$	0	0
$146$	−238122.	−0.924523
$147$	−139781.	−0.533526
$148$	171299.	0.642835
$149$	217037.	0.800883	0.400441	−	0.916322i	$-0.368857\pi$
$149$	217037.	0.800883	0.400441	+	0.916322i	$0.368857\pi$
$150$	0	0
$151$	−191640.	−0.683982	−0.341991	−	0.939703i	$-0.611101\pi$
$151$	−191640.	−0.683982	−0.341991	+	0.939703i	$0.611101\pi$
$152$	91082.1	0.319760
$153$	18878.0	0.0651971
$154$	−13650.8	−0.0463826
$155$	0	0
$156$	25538.3	0.0840196
$157$	−159174.	−0.515376	−0.257688	−	0.966228i	$-0.582961\pi$
$157$	−159174.	−0.515376	−0.257688	+	0.966228i	$0.582961\pi$
$158$	115100.	0.366802
$159$	−67741.9	−0.212503
$160$	0	0
$161$	153934.	0.468026
$162$	20723.4	0.0620402
$163$	195016.	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$163$	195016.	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$164$	−124113.	−0.360335
$165$	0	0
$166$	140713.	0.396338
$167$	−113805.	−0.315771	−0.157885	−	0.987457i	$-0.550468\pi$
$167$	−113805.	−0.315771	−0.157885	+	0.987457i	$0.550468\pi$
$168$	−54852.5	−0.149942
$169$	−354692.	−0.955289
$170$	0	0
$171$	−43235.9	−0.113072
$172$	−42082.2	−0.108462
$173$	−112967.	−0.286970	−0.143485	−	0.989653i	$-0.545831\pi$
$173$	−112967.	−0.286970	−0.143485	+	0.989653i	$0.545831\pi$
$174$	200519.	0.502090
$175$	0	0
$176$	−20059.4	−0.0488132
$177$	−264346.	−0.634220
$178$	−225868.	−0.534325
$179$	−370365.	−0.863966	−0.431983	−	0.901882i	$-0.642186\pi$
$179$	−370365.	−0.863966	−0.431983	+	0.901882i	$0.642186\pi$
$180$	0	0
$181$	−432563.	−0.981417	−0.490708	−	0.871324i	$-0.663262\pi$
$181$	−432563.	−0.981417	−0.490708	+	0.871324i	$0.663262\pi$
$182$	−14535.7	−0.0325281
$183$	380289.	0.839432
$184$	−735405.	−1.60133
$185$	0	0
$186$	−212752.	−0.450911
$187$	−28200.5	−0.0589730
$188$	644145.	1.32920
$189$	26038.1	0.0530217
$190$	0	0
$191$	309528.	0.613927	0.306964	−	0.951721i	$-0.400687\pi$
$191$	309528.	0.613927	0.306964	+	0.951721i	$0.400687\pi$
$192$	122364.	0.239552
$193$	−391542.	−0.756632	−0.378316	−	0.925677i	$-0.623497\pi$
$193$	−391542.	−0.756632	−0.378316	+	0.925677i	$0.623497\pi$
$194$	−258070.	−0.492304
$195$	0	0
$196$	342051.	0.635991
$197$	−964936.	−1.77147	−0.885733	−	0.464195i	$-0.846344\pi$
$197$	−964936.	−1.77147	−0.885733	+	0.464195i	$0.846344\pi$
$198$	−30957.2	−0.0561175
$199$	785779.	1.40659	0.703296	−	0.710898i	$-0.251711\pi$
$199$	785779.	1.40659	0.703296	+	0.710898i	$0.251711\pi$
$200$	0	0
$201$	−238082.	−0.415659
$202$	465779.	0.803160
$203$	251943.	0.429103
$204$	−46195.4	−0.0777184
$205$	0	0
$206$	−219200.	−0.359892
$207$	349091.	0.566256
$208$	−21359.9	−0.0342326
$209$	64587.0	0.102277
$210$	0	0
$211$	−1.19918e6	−1.85429	−0.927147	−	0.374697i	$-0.877747\pi$
$211$	−1.19918e6	−1.85429	−0.927147	+	0.374697i	$0.877747\pi$
$212$	165768.	0.253314
$213$	−216677.	−0.327238
$214$	−265190.	−0.395842
$215$	0	0
$216$	−124394.	−0.181412
$217$	−267314.	−0.385364
$218$	160001.	0.228024
$219$	−678503.	−0.955963
$220$	0	0
$221$	−30028.7	−0.0413577
$222$	−221107.	−0.301107
$223$	905470.	1.21930	0.609652	−	0.792669i	$-0.291309\pi$
$223$	905470.	1.21930	0.609652	+	0.792669i	$0.291309\pi$
$224$	213734.	0.284612
$225$	0	0
$226$	650254.	0.846860
$227$	351541.	0.452805	0.226403	−	0.974034i	$-0.427303\pi$
$227$	351541.	0.452805	0.226403	+	0.974034i	$0.427303\pi$
$228$	105800.	0.134788
$229$	−1.49170e6	−1.87972	−0.939858	−	0.341566i	$-0.889043\pi$
$229$	−1.49170e6	−1.87972	−0.939858	+	0.341566i	$0.889043\pi$
$230$	0	0
$231$	−38896.3	−0.0479600
$232$	−1.20363e6	−1.46816
$233$	1.10207e6	1.32990	0.664950	−	0.746887i	$-0.268453\pi$
$233$	1.10207e6	1.32990	0.664950	+	0.746887i	$0.268453\pi$
$234$	−32964.1	−0.0393551
$235$	0	0
$236$	646868.	0.756024
$237$	327964.	0.379276
$238$	26293.2	0.0300885
$239$	184453.	0.208877	0.104438	−	0.994531i	$-0.466696\pi$
$239$	184453.	0.208877	0.104438	+	0.994531i	$0.466696\pi$
$240$	0	0
$241$	1.34621e6	1.49303	0.746515	−	0.665368i	$-0.231725\pi$
$241$	1.34621e6	1.49303	0.746515	+	0.665368i	$0.231725\pi$
$242$	46244.7	0.0507602
$243$	59049.0	0.0641500
$244$	−930584.	−1.00065
$245$	0	0
$246$	160201.	0.168782
$247$	68774.1	0.0717270
$248$	1.27706e6	1.31851
$249$	400947.	0.409816
$250$	0	0
$251$	−873576.	−0.875219	−0.437609	−	0.899165i	$-0.644175\pi$
$251$	−873576.	−0.875219	−0.437609	+	0.899165i	$0.644175\pi$
$252$	−63716.3	−0.0632047
$253$	−521482.	−0.512198
$254$	−419967.	−0.408442
$255$	0	0
$256$	−904259.	−0.862369
$257$	−733907.	−0.693119	−0.346560	−	0.938028i	$-0.612650\pi$
$257$	−733907.	−0.693119	−0.346560	+	0.938028i	$0.612650\pi$
$258$	54318.4	0.0508039
$259$	−277812.	−0.257336
$260$	0	0
$261$	571355.	0.519164
$262$	−1.08579e6	−0.977220
$263$	−904254.	−0.806123	−0.403061	−	0.915173i	$-0.632054\pi$
$263$	−904254.	−0.806123	−0.403061	+	0.915173i	$0.632054\pi$
$264$	185824.	0.164093
$265$	0	0
$266$	−60218.7	−0.0521828
$267$	−643587.	−0.552496
$268$	582598.	0.495487
$269$	599284.	0.504954	0.252477	−	0.967603i	$-0.418755\pi$
$269$	599284.	0.504954	0.252477	+	0.967603i	$0.418755\pi$
$270$	0	0
$271$	−215385.	−0.178152	−0.0890762	−	0.996025i	$-0.528391\pi$
$271$	−215385.	−0.178152	−0.0890762	+	0.996025i	$0.528391\pi$
$272$	38637.2	0.0316653
$273$	−41417.9	−0.0336343
$274$	−609495.	−0.490449
$275$	0	0
$276$	−854242.	−0.675007
$277$	1.94463e6	1.52278	0.761390	−	0.648294i	$-0.224517\pi$
$277$	1.94463e6	1.52278	0.761390	+	0.648294i	$0.224517\pi$
$278$	−1.17496e6	−0.911823
$279$	−606213.	−0.466245
$280$	0	0
$281$	−592680.	−0.447770	−0.223885	−	0.974616i	$-0.571874\pi$
$281$	−592680.	−0.447770	−0.223885	+	0.974616i	$0.571874\pi$
$282$	−831442.	−0.622600
$283$	1.54936e6	1.14997	0.574985	−	0.818164i	$-0.305008\pi$
$283$	1.54936e6	1.14997	0.574985	+	0.818164i	$0.305008\pi$
$284$	530219.	0.390085
$285$	0	0
$286$	49242.6	0.0355980
$287$	201285.	0.144247
$288$	484705.	0.344347
$289$	−1.36554e6	−0.961744
$290$	0	0
$291$	−735343.	−0.509046
$292$	1.66033e6	1.13956
$293$	2.11300e6	1.43791	0.718953	−	0.695059i	$-0.244622\pi$
$293$	2.11300e6	1.43791	0.718953	+	0.695059i	$0.244622\pi$
$294$	−441510.	−0.297901
$295$	0	0
$296$	1.32722e6	0.880467
$297$	−88209.0	−0.0580259
$298$	685528.	0.447182
$299$	−555289.	−0.359204
$300$	0	0
$301$	68248.7	0.0434188
$302$	−605310.	−0.381910
$303$	1.32719e6	0.830473
$304$	−88489.8	−0.0549173
$305$	0	0
$306$	59627.7	0.0364036
$307$	−334984.	−0.202852	−0.101426	−	0.994843i	$-0.532340\pi$
$307$	−334984.	−0.202852	−0.101426	+	0.994843i	$0.532340\pi$
$308$	95181.2	0.0571708
$309$	−624585.	−0.372131
$310$	0	0
$311$	1.61240e6	0.945307	0.472654	−	0.881248i	$-0.343296\pi$
$311$	1.61240e6	0.945307	0.472654	+	0.881248i	$0.343296\pi$
$312$	197870.	0.115078
$313$	2.44560e6	1.41099	0.705495	−	0.708715i	$-0.250725\pi$
$313$	2.44560e6	1.41099	0.705495	+	0.708715i	$0.250725\pi$
$314$	−502764.	−0.287766
$315$	0	0
$316$	−802544.	−0.452117
$317$	−566437.	−0.316595	−0.158297	−	0.987391i	$-0.550600\pi$
$317$	−566437.	−0.316595	−0.158297	+	0.987391i	$0.550600\pi$
$318$	−213968.	−0.118653
$319$	−853506.	−0.469602
$320$	0	0
$321$	−755628.	−0.409304
$322$	486212.	0.261328
$323$	−124403.	−0.0663476
$324$	−144496.	−0.0764702
$325$	0	0
$326$	615972.	0.321009
$327$	455905.	0.235779
$328$	−961622.	−0.493537
$329$	−1.04467e6	−0.532096
$330$	0	0
$331$	606168.	0.304104	0.152052	−	0.988372i	$-0.451412\pi$
$331$	606168.	0.304104	0.152052	+	0.988372i	$0.451412\pi$
$332$	−981137.	−0.488522
$333$	−630020.	−0.311346
$334$	−359463.	−0.176314
$335$	0	0
$336$	53291.4	0.0257519
$337$	525210.	0.251917	0.125959	−	0.992035i	$-0.459799\pi$
$337$	525210.	0.251917	0.125959	+	0.992035i	$0.459799\pi$
$338$	−1.12032e6	−0.533397
$339$	1.85283e6	0.875659
$340$	0	0
$341$	905577.	0.421735
$342$	−136564.	−0.0631351
$343$	−1.15504e6	−0.530105
$344$	−326051.	−0.148556
$345$	0	0
$346$	−356814.	−0.160233
$347$	−3.30639e6	−1.47411	−0.737055	−	0.675833i	$-0.763784\pi$
$347$	−3.30639e6	−1.47411	−0.737055	+	0.675833i	$0.763784\pi$
$348$	−1.39813e6	−0.618871
$349$	−1.37274e6	−0.603290	−0.301645	−	0.953420i	$-0.597536\pi$
$349$	−1.37274e6	−0.603290	−0.301645	+	0.953420i	$0.597536\pi$
$350$	0	0
$351$	−93927.5	−0.0406935
$352$	−724065.	−0.311473
$353$	2.98834e6	1.27642	0.638209	−	0.769863i	$-0.279676\pi$
$353$	2.98834e6	1.27642	0.638209	+	0.769863i	$0.279676\pi$
$354$	−834957.	−0.354124
$355$	0	0
$356$	1.57489e6	0.658604
$357$	74919.6	0.0311118
$358$	−1.16982e6	−0.482406
$359$	−1.28200e6	−0.524993	−0.262497	−	0.964933i	$-0.584546\pi$
$359$	−1.28200e6	−0.524993	−0.262497	+	0.964933i	$0.584546\pi$
$360$	0	0
$361$	−2.19118e6	−0.884933
$362$	−1.36628e6	−0.547986
$363$	131769.	0.0524864
$364$	101352.	0.0400938
$365$	0	0
$366$	1.20117e6	0.468707
$367$	−1.79336e6	−0.695028	−0.347514	−	0.937675i	$-0.612974\pi$
$367$	−1.79336e6	−0.695028	−0.347514	+	0.937675i	$0.612974\pi$
$368$	714475.	0.275022
$369$	456474.	0.174522
$370$	0	0
$371$	−268841.	−0.101405
$372$	1.48343e6	0.555789
$373$	−4.37954e6	−1.62988	−0.814941	−	0.579543i	$-0.803231\pi$
$373$	−4.37954e6	−1.62988	−0.814941	+	0.579543i	$0.803231\pi$
$374$	−89073.4	−0.0329283
$375$	0	0
$376$	4.99082e6	1.82055
$377$	−908837.	−0.329331
$378$	82243.1	0.0296053
$379$	−2.33138e6	−0.833710	−0.416855	−	0.908973i	$-0.636868\pi$
$379$	−2.33138e6	−0.833710	−0.416855	+	0.908973i	$0.636868\pi$
$380$	0	0
$381$	−1.19665e6	−0.422332
$382$	977668.	0.342794
$383$	−4.16021e6	−1.44917	−0.724584	−	0.689187i	$-0.757968\pi$
$383$	−4.16021e6	−1.44917	−0.724584	+	0.689187i	$0.757968\pi$
$384$	−1.33690e6	−0.462670
$385$	0	0
$386$	−1.23671e6	−0.422474
$387$	154774.	0.0525316
$388$	1.79942e6	0.606810
$389$	−3.85296e6	−1.29098	−0.645491	−	0.763768i	$-0.723347\pi$
$389$	−3.85296e6	−1.29098	−0.645491	+	0.763768i	$0.723347\pi$
$390$	0	0
$391$	1.00444e6	0.332264
$392$	2.65021e6	0.871093
$393$	−3.09384e6	−1.01045
$394$	−3.04782e6	−0.989120
$395$	0	0
$396$	215852.	0.0691699
$397$	5.59555e6	1.78183	0.890915	−	0.454169i	$-0.150064\pi$
$397$	5.59555e6	1.78183	0.890915	+	0.454169i	$0.150064\pi$
$398$	2.48194e6	0.785387
$399$	−171587.	−0.0539574
$400$	0	0
$401$	−4.76784e6	−1.48068	−0.740339	−	0.672234i	$-0.765335\pi$
$401$	−4.76784e6	−1.48068	−0.740339	+	0.672234i	$0.765335\pi$
$402$	−752000.	−0.232088
$403$	964284.	0.295762
$404$	−3.24769e6	−0.989967
$405$	0	0
$406$	795780.	0.239595
$407$	941141.	0.281623
$408$	−357921.	−0.106448
$409$	3.34226e6	0.987944	0.493972	−	0.869478i	$-0.335545\pi$
$409$	3.34226e6	0.987944	0.493972	+	0.869478i	$0.335545\pi$
$410$	0	0
$411$	−1.73669e6	−0.507127
$412$	1.52839e6	0.443599
$413$	−1.04909e6	−0.302647
$414$	1.10263e6	0.316176
$415$	0	0
$416$	−771005.	−0.218436
$417$	−3.34791e6	−0.942831
$418$	204003.	0.0571078
$419$	234028.	0.0651227	0.0325614	−	0.999470i	$-0.489634\pi$
$419$	234028.	0.0651227	0.0325614	+	0.999470i	$0.489634\pi$
$420$	0	0
$421$	−4.52291e6	−1.24369	−0.621846	−	0.783139i	$-0.713617\pi$
$421$	−4.52291e6	−1.24369	−0.621846	+	0.783139i	$0.713617\pi$
$422$	−3.78770e6	−1.03537
$423$	−2.36910e6	−0.643773
$424$	1.28436e6	0.346955
$425$	0	0
$426$	−684390.	−0.182717
$427$	1.50922e6	0.400573
$428$	1.84906e6	0.487911
$429$	140311.	0.0368086
$430$	0	0
$431$	188933.	0.0489907	0.0244954	−	0.999700i	$-0.492202\pi$
$431$	188933.	0.0489907	0.0244954	+	0.999700i	$0.492202\pi$
$432$	120854.	0.0311567
$433$	2.85171e6	0.730945	0.365473	−	0.930822i	$-0.380907\pi$
$433$	2.85171e6	0.730945	0.365473	+	0.930822i	$0.380907\pi$
$434$	−844329.	−0.215173
$435$	0	0
$436$	−1.11562e6	−0.281061
$437$	−2.30045e6	−0.576249
$438$	−2.14310e6	−0.533774
$439$	1.71570e6	0.424894	0.212447	−	0.977173i	$-0.431857\pi$
$439$	1.71570e6	0.424894	0.212447	+	0.977173i	$0.431857\pi$
$440$	0	0
$441$	−1.25803e6	−0.308032
$442$	−94847.9	−0.0230926
$443$	5.89814e6	1.42793	0.713963	−	0.700183i	$-0.246898\pi$
$443$	5.89814e6	1.42793	0.713963	+	0.700183i	$0.246898\pi$
$444$	1.54169e6	0.371141
$445$	0	0
$446$	2.85999e6	0.680813
$447$	1.95334e6	0.462390
$448$	485613.	0.114313
$449$	4.24151e6	0.992898	0.496449	−	0.868066i	$-0.334637\pi$
$449$	4.24151e6	0.992898	0.496449	+	0.868066i	$0.334637\pi$
$450$	0	0
$451$	−681894.	−0.157861
$452$	−4.53395e6	−1.04383
$453$	−1.72476e6	−0.394897
$454$	1.11037e6	0.252829
$455$	0	0
$456$	819739.	0.184613
$457$	667519.	0.149511	0.0747555	−	0.997202i	$-0.476182\pi$
$457$	667519.	0.149511	0.0747555	+	0.997202i	$0.476182\pi$
$458$	−4.71164e6	−1.04956
$459$	169902.	0.0376416
$460$	0	0
$461$	2.73973e6	0.600419	0.300210	−	0.953873i	$-0.402943\pi$
$461$	2.73973e6	0.600419	0.300210	+	0.953873i	$0.402943\pi$
$462$	−122857.	−0.0267790
$463$	274239.	0.0594534	0.0297267	−	0.999558i	$-0.490536\pi$
$463$	274239.	0.0594534	0.0297267	+	0.999558i	$0.490536\pi$
$464$	1.16938e6	0.252150
$465$	0	0
$466$	3.48097e6	0.742566
$467$	624561.	0.132520	0.0662602	−	0.997802i	$-0.478893\pi$
$467$	624561.	0.132520	0.0662602	+	0.997802i	$0.478893\pi$
$468$	229845.	0.0485088
$469$	−944856.	−0.198351
$470$	0	0
$471$	−1.43257e6	−0.297552
$472$	5.01191e6	1.03550
$473$	−231206.	−0.0475166
$474$	1.03590e6	0.211774
$475$	0	0
$476$	−183332.	−0.0370869
$477$	−609677.	−0.122689
$478$	582607.	0.116629
$479$	5.44733e6	1.08479	0.542394	−	0.840124i	$-0.317518\pi$
$479$	5.44733e6	1.08479	0.542394	+	0.840124i	$0.317518\pi$
$480$	0	0
$481$	1.00215e6	0.197502
$482$	4.25209e6	0.833652
$483$	1.38541e6	0.270215
$484$	−322445.	−0.0625665
$485$	0	0
$486$	186511.	0.0358189
$487$	7.59155e6	1.45047	0.725234	−	0.688502i	$-0.241731\pi$
$487$	7.59155e6	1.45047	0.725234	+	0.688502i	$0.241731\pi$
$488$	−7.21014e6	−1.37055
$489$	1.75514e6	0.331925
$490$	0	0
$491$	−842332.	−0.157681	−0.0788406	−	0.996887i	$-0.525122\pi$
$491$	−842332.	−0.157681	−0.0788406	+	0.996887i	$0.525122\pi$
$492$	−1.11701e6	−0.208039
$493$	1.64397e6	0.304632
$494$	217228.	0.0400496
$495$	0	0
$496$	−1.24072e6	−0.226448
$497$	−859906.	−0.156157
$498$	1.26642e6	0.228826
$499$	−1.26955e6	−0.228244	−0.114122	−	0.993467i	$-0.536406\pi$
$499$	−1.26955e6	−0.228244	−0.114122	+	0.993467i	$0.536406\pi$
$500$	0	0
$501$	−1.02425e6	−0.182310
$502$	−2.75925e6	−0.488689
$503$	−2.13169e6	−0.375669	−0.187834	−	0.982201i	$-0.560147\pi$
$503$	−2.13169e6	−0.375669	−0.187834	+	0.982201i	$0.560147\pi$
$504$	−493672.	−0.0865690
$505$	0	0
$506$	−1.64714e6	−0.285992
$507$	−3.19223e6	−0.551536
$508$	2.92826e6	0.503442
$509$	4.53841e6	0.776443	0.388222	−	0.921566i	$-0.373090\pi$
$509$	4.53841e6	0.776443	0.388222	+	0.921566i	$0.373090\pi$
$510$	0	0
$511$	−2.69271e6	−0.456182
$512$	1.89726e6	0.319854
$513$	−389124.	−0.0652821
$514$	−2.31810e6	−0.387011
$515$	0	0
$516$	−378739.	−0.0626205
$517$	3.53903e6	0.582315
$518$	−877488.	−0.143687
$519$	−1.01670e6	−0.165682
$520$	0	0
$521$	768398.	0.124020	0.0620100	−	0.998076i	$-0.480249\pi$
$521$	768398.	0.124020	0.0620100	+	0.998076i	$0.480249\pi$
$522$	1.80467e6	0.289882
$523$	3.71601e6	0.594050	0.297025	−	0.954870i	$-0.404006\pi$
$523$	3.71601e6	0.594050	0.297025	+	0.954870i	$0.404006\pi$
$524$	7.57076e6	1.20451
$525$	0	0
$526$	−2.85615e6	−0.450108
$527$	−1.74426e6	−0.273581
$528$	−180535.	−0.0281823
$529$	1.21377e7	1.88581
$530$	0	0
$531$	−2.37912e6	−0.366167
$532$	419880.	0.0643201
$533$	−726100.	−0.110708
$534$	−2.03281e6	−0.308493
$535$	0	0
$536$	4.51396e6	0.678649
$537$	−3.33328e6	−0.498811
$538$	1.89288e6	0.281947
$539$	1.87928e6	0.278625
$540$	0	0
$541$	1.43075e6	0.210170	0.105085	−	0.994463i	$-0.466489\pi$
$541$	1.43075e6	0.210170	0.105085	+	0.994463i	$0.466489\pi$
$542$	−680308.	−0.0994736
$543$	−3.89307e6	−0.566621
$544$	1.39465e6	0.202054
$545$	0	0
$546$	−130822.	−0.0187801
$547$	2.44364e6	0.349196	0.174598	−	0.984640i	$-0.444137\pi$
$547$	2.44364e6	0.349196	0.174598	+	0.984640i	$0.444137\pi$
$548$	4.24975e6	0.604522
$549$	3.42260e6	0.484647
$550$	0	0
$551$	−3.76514e6	−0.528326
$552$	−6.61865e6	−0.924531
$553$	1.30156e6	0.180989
$554$	6.14225e6	0.850263
$555$	0	0
$556$	8.19250e6	1.12390
$557$	−6.30783e6	−0.861474	−0.430737	−	0.902477i	$-0.641746\pi$
$557$	−6.30783e6	−0.861474	−0.430737	+	0.902477i	$0.641746\pi$
$558$	−1.91477e6	−0.260334
$559$	−246194.	−0.0333234
$560$	0	0
$561$	−253805.	−0.0340481
$562$	−1.87202e6	−0.250018
$563$	−6.32773e6	−0.841350	−0.420675	−	0.907211i	$-0.638207\pi$
$563$	−6.32773e6	−0.841350	−0.420675	+	0.907211i	$0.638207\pi$
$564$	5.79730e6	0.767411
$565$	0	0
$566$	4.89377e6	0.642100
$567$	234342.	0.0306121
$568$	4.10812e6	0.534284
$569$	7.91435e6	1.02479	0.512395	−	0.858750i	$-0.328758\pi$
$569$	7.91435e6	1.02479	0.512395	+	0.858750i	$0.328758\pi$
$570$	0	0
$571$	1.92969e6	0.247683	0.123842	−	0.992302i	$-0.460479\pi$
$571$	1.92969e6	0.247683	0.123842	+	0.992302i	$0.460479\pi$
$572$	−343349.	−0.0438778
$573$	2.78576e6	0.354451
$574$	635774.	0.0805422
$575$	0	0
$576$	1.10127e6	0.138305
$577$	7.24847e6	0.906373	0.453186	−	0.891416i	$-0.350287\pi$
$577$	7.24847e6	0.906373	0.453186	+	0.891416i	$0.350287\pi$
$578$	−4.31315e6	−0.537001
$579$	−3.52387e6	−0.436842
$580$	0	0
$581$	1.59120e6	0.195562
$582$	−2.32263e6	−0.284232
$583$	910752.	0.110976
$584$	1.28642e7	1.56081
$585$	0	0
$586$	6.67406e6	0.802872
$587$	5.75930e6	0.689882	0.344941	−	0.938624i	$-0.387899\pi$
$587$	5.75930e6	0.689882	0.344941	+	0.938624i	$0.387899\pi$
$588$	3.07846e6	0.367190
$589$	3.99485e6	0.474473
$590$	0	0
$591$	−8.68442e6	−1.02276
$592$	−1.28944e6	−0.151216
$593$	1.82920e6	0.213612	0.106806	−	0.994280i	$-0.465938\pi$
$593$	1.82920e6	0.213612	0.106806	+	0.994280i	$0.465938\pi$
$594$	−278615.	−0.0323994
$595$	0	0
$596$	−4.77991e6	−0.551193
$597$	7.07201e6	0.812096
$598$	−1.75392e6	−0.200566
$599$	2.59180e6	0.295145	0.147572	−	0.989051i	$-0.452854\pi$
$599$	2.59180e6	0.295145	0.147572	+	0.989051i	$0.452854\pi$
$600$	0	0
$601$	3.37298e6	0.380914	0.190457	−	0.981696i	$-0.439003\pi$
$601$	3.37298e6	0.380914	0.190457	+	0.981696i	$0.439003\pi$
$602$	215568.	0.0242434
$603$	−2.14274e6	−0.239981
$604$	4.22058e6	0.470738
$605$	0	0
$606$	4.19201e6	0.463704
$607$	−2.37031e6	−0.261116	−0.130558	−	0.991441i	$-0.541677\pi$
$607$	−2.37031e6	−0.261116	−0.130558	+	0.991441i	$0.541677\pi$
$608$	−3.19413e6	−0.350424
$609$	2.26749e6	0.247743
$610$	0	0
$611$	3.76846e6	0.408376
$612$	−415759.	−0.0448707
$613$	−1.67511e7	−1.80049	−0.900246	−	0.435381i	$-0.856614\pi$
$613$	−1.67511e7	−1.80049	−0.900246	+	0.435381i	$0.856614\pi$
$614$	−1.05807e6	−0.113265
$615$	0	0
$616$	737461.	0.0783046
$617$	−8.03002e6	−0.849188	−0.424594	−	0.905384i	$-0.639583\pi$
$617$	−8.03002e6	−0.849188	−0.424594	+	0.905384i	$0.639583\pi$
$618$	−1.97280e6	−0.207784
$619$	−9.26345e6	−0.971731	−0.485866	−	0.874034i	$-0.661496\pi$
$619$	−9.26345e6	−0.971731	−0.485866	+	0.874034i	$0.661496\pi$
$620$	0	0
$621$	3.14182e6	0.326928
$622$	5.09290e6	0.527824
$623$	−2.55414e6	−0.263648
$624$	−192239.	−0.0197642
$625$	0	0
$626$	7.72459e6	0.787843
$627$	581283.	0.0590499
$628$	3.50556e6	0.354698
$629$	−1.81276e6	−0.182690
$630$	0	0
$631$	−5.10757e6	−0.510671	−0.255336	−	0.966853i	$-0.582186\pi$
$631$	−5.10757e6	−0.510671	−0.255336	+	0.966853i	$0.582186\pi$
$632$	−6.21809e6	−0.619248
$633$	−1.07926e7	−1.07058
$634$	−1.78913e6	−0.176774
$635$	0	0
$636$	1.49191e6	0.146251
$637$	2.00111e6	0.195399
$638$	−2.69586e6	−0.262208
$639$	−1.95009e6	−0.188931
$640$	0	0
$641$	−5.95583e6	−0.572528	−0.286264	−	0.958151i	$-0.592414\pi$
$641$	−5.95583e6	−0.572528	−0.286264	+	0.958151i	$0.592414\pi$
$642$	−2.38671e6	−0.228540
$643$	−9.47664e6	−0.903914	−0.451957	−	0.892040i	$-0.649274\pi$
$643$	−9.47664e6	−0.903914	−0.451957	+	0.892040i	$0.649274\pi$
$644$	−3.39015e6	−0.322110
$645$	0	0
$646$	−392937.	−0.0370460
$647$	619740.	0.0582035	0.0291017	−	0.999576i	$-0.490735\pi$
$647$	619740.	0.0582035	0.0291017	+	0.999576i	$0.490735\pi$
$648$	−1.11955e6	−0.104738
$649$	3.55399e6	0.331211
$650$	0	0
$651$	−2.40582e6	−0.222490
$652$	−4.29492e6	−0.395672
$653$	4.43748e6	0.407242	0.203621	−	0.979050i	$-0.434729\pi$
$653$	4.43748e6	0.407242	0.203621	+	0.979050i	$0.434729\pi$
$654$	1.44001e6	0.131650
$655$	0	0
$656$	934253.	0.0847628
$657$	−6.10652e6	−0.551926
$658$	−3.29967e6	−0.297102
$659$	7.33968e6	0.658360	0.329180	−	0.944267i	$-0.393228\pi$
$659$	7.33968e6	0.658360	0.329180	+	0.944267i	$0.393228\pi$
$660$	0	0
$661$	1.37197e7	1.22135	0.610676	−	0.791881i	$-0.290898\pi$
$661$	1.37197e7	1.22135	0.610676	+	0.791881i	$0.290898\pi$
$662$	1.91462e6	0.169800
$663$	−270259.	−0.0238779
$664$	−7.60182e6	−0.669110
$665$	0	0
$666$	−1.98996e6	−0.173844
$667$	3.04001e7	2.64582
$668$	2.50638e6	0.217323
$669$	8.14923e6	0.703966
$670$	0	0
$671$	−5.11277e6	−0.438379
$672$	1.92360e6	0.164321
$673$	6.01655e6	0.512047	0.256023	−	0.966671i	$-0.417588\pi$
$673$	6.01655e6	0.512047	0.256023	+	0.966671i	$0.417588\pi$
$674$	1.65891e6	0.140661
$675$	0	0
$676$	7.81153e6	0.657460
$677$	4.88047e6	0.409251	0.204626	−	0.978840i	$-0.434402\pi$
$677$	4.88047e6	0.409251	0.204626	+	0.978840i	$0.434402\pi$
$678$	5.85228e6	0.488935
$679$	−2.91829e6	−0.242915
$680$	0	0
$681$	3.16387e6	0.261427
$682$	2.86033e6	0.235481
$683$	7.00188e6	0.574332	0.287166	−	0.957881i	$-0.407287\pi$
$683$	7.00188e6	0.574332	0.287166	+	0.957881i	$0.407287\pi$
$684$	952203.	0.0778197
$685$	0	0
$686$	−3.64828e6	−0.295991
$687$	−1.34253e7	−1.08525
$688$	316772.	0.0255138
$689$	969795.	0.0778273
$690$	0	0
$691$	−4.38318e6	−0.349216	−0.174608	−	0.984638i	$-0.555866\pi$
$691$	−4.38318e6	−0.349216	−0.174608	+	0.984638i	$0.555866\pi$
$692$	2.48792e6	0.197502
$693$	−350067.	−0.0276897
$694$	−1.04435e7	−0.823087
$695$	0	0
$696$	−1.08327e7	−0.847644
$697$	1.31342e6	0.102405
$698$	−4.33591e6	−0.336854
$699$	9.91863e6	0.767819
$700$	0	0
$701$	−2.12876e7	−1.63618	−0.818091	−	0.575089i	$-0.804967\pi$
$701$	−2.12876e7	−1.63618	−0.818091	+	0.575089i	$0.804967\pi$
$702$	−296677.	−0.0227217
$703$	4.15173e6	0.316841
$704$	−1.64511e6	−0.125102
$705$	0	0
$706$	9.43888e6	0.712703
$707$	5.26709e6	0.396298
$708$	5.82181e6	0.436490
$709$	−1.39993e7	−1.04590	−0.522951	−	0.852363i	$-0.675169\pi$
$709$	−1.39993e7	−1.04590	−0.522951	+	0.852363i	$0.675169\pi$
$710$	0	0
$711$	2.95168e6	0.218975
$712$	1.22022e7	0.902064
$713$	−3.22547e7	−2.37613
$714$	236639.	0.0173716
$715$	0	0
$716$	8.15669e6	0.594609
$717$	1.66007e6	0.120595
$718$	−4.04931e6	−0.293136
$719$	2.71556e7	1.95901	0.979505	−	0.201419i	$-0.0645552\pi$
$719$	2.71556e7	1.95901	0.979505	+	0.201419i	$0.0645552\pi$
$720$	0	0
$721$	−2.47874e6	−0.177579
$722$	−6.92100e6	−0.494113
$723$	1.21158e7	0.862002
$724$	9.52653e6	0.675442
$725$	0	0
$726$	416202.	0.0293064
$727$	2.54023e7	1.78253	0.891265	−	0.453483i	$-0.149819\pi$
$727$	2.54023e7	1.78253	0.891265	+	0.453483i	$0.149819\pi$
$728$	785270.	0.0549149
$729$	531441.	0.0370370
$730$	0	0
$731$	445333.	0.0308242
$732$	−8.37526e6	−0.577724
$733$	1.99308e7	1.37014	0.685070	−	0.728477i	$-0.259772\pi$
$733$	1.99308e7	1.37014	0.685070	+	0.728477i	$0.259772\pi$
$734$	−5.66446e6	−0.388077
$735$	0	0
$736$	2.57897e7	1.75490
$737$	3.20088e6	0.217071
$738$	1.44181e6	0.0974465
$739$	1.48604e7	1.00096	0.500482	−	0.865747i	$-0.333156\pi$
$739$	1.48604e7	1.00096	0.500482	+	0.865747i	$0.333156\pi$
$740$	0	0
$741$	618967.	0.0414116
$742$	−849154.	−0.0566209
$743$	−1.33960e7	−0.890229	−0.445114	−	0.895474i	$-0.646837\pi$
$743$	−1.33960e7	−0.890229	−0.445114	+	0.895474i	$0.646837\pi$
$744$	1.14936e7	0.761243
$745$	0	0
$746$	−1.38331e7	−0.910065
$747$	3.60853e6	0.236607
$748$	621072.	0.0405871
$749$	−2.99879e6	−0.195318
$750$	0	0
$751$	1.90836e7	1.23470	0.617350	−	0.786689i	$-0.288206\pi$
$751$	1.90836e7	1.23470	0.617350	+	0.786689i	$0.288206\pi$
$752$	−4.84878e6	−0.312671
$753$	−7.86219e6	−0.505308
$754$	−2.87063e6	−0.183886
$755$	0	0
$756$	−573447.	−0.0364913
$757$	2.34286e7	1.48596	0.742978	−	0.669316i	$-0.233413\pi$
$757$	2.34286e7	1.48596	0.742978	+	0.669316i	$0.233413\pi$
$758$	−7.36383e6	−0.465512
$759$	−4.69334e6	−0.295718
$760$	0	0
$761$	8.73779e6	0.546940	0.273470	−	0.961881i	$-0.411829\pi$
$761$	8.73779e6	0.546940	0.273470	+	0.961881i	$0.411829\pi$
$762$	−3.77970e6	−0.235814
$763$	1.80931e6	0.112513
$764$	−6.81687e6	−0.422524
$765$	0	0
$766$	−1.31403e7	−0.809160
$767$	3.78439e6	0.232278
$768$	−8.13833e6	−0.497889
$769$	−2.18487e7	−1.33232	−0.666161	−	0.745808i	$-0.732064\pi$
$769$	−2.18487e7	−1.33232	−0.666161	+	0.745808i	$0.732064\pi$
$770$	0	0
$771$	−6.60516e6	−0.400173
$772$	8.62308e6	0.520738
$773$	−1.42932e7	−0.860363	−0.430182	−	0.902742i	$-0.641550\pi$
$773$	−1.42932e7	−0.860363	−0.430182	+	0.902742i	$0.641550\pi$
$774$	488865.	0.0293317
$775$	0	0
$776$	1.39418e7	0.831124
$777$	−2.50030e6	−0.148573
$778$	−1.21699e7	−0.720836
$779$	−3.00809e6	−0.177602
$780$	0	0
$781$	2.91310e6	0.170894
$782$	3.17261e6	0.185524
$783$	5.14220e6	0.299740
$784$	−2.57478e6	−0.149606
$785$	0	0
$786$	−9.77210e6	−0.564198
$787$	−9.06074e6	−0.521467	−0.260733	−	0.965411i	$-0.583964\pi$
$787$	−9.06074e6	−0.521467	−0.260733	+	0.965411i	$0.583964\pi$
$788$	2.12512e7	1.21918
$789$	−8.13829e6	−0.465415
$790$	0	0
$791$	7.35314e6	0.417861
$792$	1.67241e6	0.0947393
$793$	−5.44422e6	−0.307435
$794$	1.76739e7	0.994907
$795$	0	0
$796$	−1.73055e7	−0.968061
$797$	2.52752e7	1.40945	0.704723	−	0.709483i	$-0.251071\pi$
$797$	2.52752e7	1.40945	0.704723	+	0.709483i	$0.251071\pi$
$798$	−541969.	−0.0301278
$799$	−6.81664e6	−0.377749
$800$	0	0
$801$	−5.79228e6	−0.318983
$802$	−1.50596e7	−0.826754
$803$	9.12209e6	0.499236
$804$	5.24339e6	0.286070
$805$	0	0
$806$	3.04576e6	0.165142
$807$	5.39356e6	0.291536
$808$	−2.51630e7	−1.35592
$809$	−3.19699e7	−1.71739	−0.858696	−	0.512486i	$-0.828725\pi$
$809$	−3.19699e7	−1.71739	−0.858696	+	0.512486i	$0.828725\pi$
$810$	0	0
$811$	4.02925e6	0.215116	0.107558	−	0.994199i	$-0.465697\pi$
$811$	4.02925e6	0.215116	0.107558	+	0.994199i	$0.465697\pi$
$812$	−5.54864e6	−0.295323
$813$	−1.93846e6	−0.102856
$814$	2.97266e6	0.157248
$815$	0	0
$816$	347734.	0.0182819
$817$	−1.01994e6	−0.0534587
$818$	1.05568e7	0.551630
$819$	−372761.	−0.0194187
$820$	0	0
$821$	−4.71683e6	−0.244226	−0.122113	−	0.992516i	$-0.538967\pi$
$821$	−4.71683e6	−0.244226	−0.122113	+	0.992516i	$0.538967\pi$
$822$	−5.48545e6	−0.283161
$823$	−1.63939e6	−0.0843687	−0.0421844	−	0.999110i	$-0.513432\pi$
$823$	−1.63939e6	−0.0843687	−0.0421844	+	0.999110i	$0.513432\pi$
$824$	1.18419e7	0.607581
$825$	0	0
$826$	−3.31362e6	−0.168987
$827$	−1.32821e7	−0.675310	−0.337655	−	0.941270i	$-0.609634\pi$
$827$	−1.32821e7	−0.675310	−0.337655	+	0.941270i	$0.609634\pi$
$828$	−7.68818e6	−0.389715
$829$	−2.63551e7	−1.33192	−0.665961	−	0.745987i	$-0.731978\pi$
$829$	−2.63551e7	−1.33192	−0.665961	+	0.745987i	$0.731978\pi$
$830$	0	0
$831$	1.75017e7	0.879178
$832$	−1.75176e6	−0.0877337
$833$	−3.61975e6	−0.180745
$834$	−1.05746e7	−0.526441
$835$	0	0
$836$	−1.42243e6	−0.0703906
$837$	−5.45591e6	−0.269187
$838$	739194.	0.0363620
$839$	1.10047e7	0.539725	0.269863	−	0.962899i	$-0.413022\pi$
$839$	1.10047e7	0.539725	0.269863	+	0.962899i	$0.413022\pi$
$840$	0	0
$841$	2.92445e7	1.42578
$842$	−1.42859e7	−0.694431
$843$	−5.33412e6	−0.258520
$844$	2.64101e7	1.27618
$845$	0	0
$846$	−7.48298e6	−0.359458
$847$	522940.	0.0250463
$848$	−1.24781e6	−0.0595880
$849$	1.39443e7	0.663936
$850$	0	0
$851$	−3.35215e7	−1.58672
$852$	4.77197e6	0.225216
$853$	−21091.2	−0.000992496 0	−0.000496248	−	1.00000i	$-0.500158\pi$
$853$	−21091.2	−0.000992496 0	−0.000496248		1.00000i	$0.500158\pi$
$854$	4.76697e6	0.223665
$855$	0	0
$856$	1.43264e7	0.668273
$857$	1.05373e7	0.490093	0.245047	−	0.969511i	$-0.421197\pi$
$857$	1.05373e7	0.490093	0.245047	+	0.969511i	$0.421197\pi$
$858$	443184.	0.0205525
$859$	3.70966e6	0.171534	0.0857672	−	0.996315i	$-0.472666\pi$
$859$	3.70966e6	0.171534	0.0857672	+	0.996315i	$0.472666\pi$
$860$	0	0
$861$	1.81157e6	0.0832812
$862$	596758.	0.0273546
$863$	3.60754e6	0.164886	0.0824430	−	0.996596i	$-0.473728\pi$
$863$	3.60754e6	0.164886	0.0824430	+	0.996596i	$0.473728\pi$
$864$	4.36234e6	0.198809
$865$	0	0
$866$	9.00732e6	0.408132
$867$	−1.22899e7	−0.555263
$868$	5.88716e6	0.265220
$869$	−4.40930e6	−0.198071
$870$	0	0
$871$	3.40839e6	0.152231
$872$	−8.64380e6	−0.384958
$873$	−6.61808e6	−0.293898
$874$	−7.26615e6	−0.321755
$875$	0	0
$876$	1.49429e7	0.657924
$877$	3.50652e7	1.53949	0.769747	−	0.638350i	$-0.220383\pi$
$877$	3.50652e7	1.53949	0.769747	+	0.638350i	$0.220383\pi$
$878$	5.41917e6	0.237245
$879$	1.90170e7	0.830175
$880$	0	0
$881$	−3.29621e7	−1.43079	−0.715394	−	0.698721i	$-0.753753\pi$
$881$	−3.29621e7	−1.43079	−0.715394	+	0.698721i	$0.753753\pi$
$882$	−3.97359e6	−0.171993
$883$	−1.22078e7	−0.526907	−0.263454	−	0.964672i	$-0.584862\pi$
$883$	−1.22078e7	−0.526907	−0.263454	+	0.964672i	$0.584862\pi$
$884$	661335.	0.0284637
$885$	0	0
$886$	1.86297e7	0.797300
$887$	3.01046e7	1.28477	0.642383	−	0.766384i	$-0.277946\pi$
$887$	3.01046e7	1.28477	0.642383	+	0.766384i	$0.277946\pi$
$888$	1.19450e7	0.508338
$889$	−4.74903e6	−0.201535
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−1.99416e7	−0.839164
$893$	1.56120e7	0.655134
$894$	6.16976e6	0.258181
$895$	0	0
$896$	−5.30564e6	−0.220784
$897$	−4.99760e6	−0.207386
$898$	1.33971e7	0.554396
$899$	−5.27911e7	−2.17852
$900$	0	0
$901$	−1.75423e6	−0.0719904
$902$	−2.15381e6	−0.0881437
$903$	614238.	0.0250679
$904$	−3.51289e7	−1.42970
$905$	0	0
$906$	−5.44779e6	−0.220496
$907$	2.06605e7	0.833915	0.416958	−	0.908926i	$-0.363096\pi$
$907$	2.06605e7	0.833915	0.416958	+	0.908926i	$0.363096\pi$
$908$	−7.74214e6	−0.311635
$909$	1.19447e7	0.479474
$910$	0	0
$911$	−3.14271e7	−1.25461	−0.627304	−	0.778774i	$-0.715842\pi$
$911$	−3.14271e7	−1.25461	−0.627304	+	0.778774i	$0.715842\pi$
$912$	−796408.	−0.0317065
$913$	−5.39051e6	−0.214019
$914$	2.10841e6	0.0834813
$915$	0	0
$916$	3.28523e7	1.29368
$917$	−1.22782e7	−0.482183
$918$	536649.	0.0210176
$919$	−4.58231e7	−1.78976	−0.894882	−	0.446303i	$-0.852740\pi$
$919$	−4.58231e7	−1.78976	−0.894882	+	0.446303i	$0.852740\pi$
$920$	0	0
$921$	−3.01486e6	−0.117116
$922$	8.65362e6	0.335251
$923$	3.10195e6	0.119848
$924$	856631.	0.0330076
$925$	0	0
$926$	866204.	0.0331965
$927$	−5.62127e6	−0.214850
$928$	4.22098e7	1.60895
$929$	−4.07725e7	−1.54999	−0.774994	−	0.631969i	$-0.782247\pi$
$929$	−4.07725e7	−1.54999	−0.774994	+	0.631969i	$0.782247\pi$
$930$	0	0
$931$	8.29023e6	0.313467
$932$	−2.42713e7	−0.915280
$933$	1.45116e7	0.545773
$934$	1.97272e6	0.0739943
$935$	0	0
$936$	1.78083e6	0.0664406
$937$	−2.27957e7	−0.848211	−0.424105	−	0.905613i	$-0.639411\pi$
$937$	−2.27957e7	−0.848211	−0.424105	+	0.905613i	$0.639411\pi$
$938$	−2.98440e6	−0.110751
$939$	2.20104e7	0.814636
$940$	0	0
$941$	−2.13238e7	−0.785039	−0.392520	−	0.919744i	$-0.628396\pi$
$941$	−2.13238e7	−0.785039	−0.392520	+	0.919744i	$0.628396\pi$
$942$	−4.52487e6	−0.166142
$943$	2.42876e7	0.889418
$944$	−4.86927e6	−0.177842
$945$	0	0
$946$	−730280.	−0.0265315
$947$	−2.32964e7	−0.844139	−0.422070	−	0.906563i	$-0.638696\pi$
$947$	−2.32964e7	−0.844139	−0.422070	+	0.906563i	$0.638696\pi$
$948$	−7.22290e6	−0.261030
$949$	9.71346e6	0.350113
$950$	0	0
$951$	−5.09793e6	−0.182786
$952$	−1.42045e6	−0.0507964
$953$	−813207.	−0.0290047	−0.0145024	−	0.999895i	$-0.504616\pi$
$953$	−813207.	−0.0290047	−0.0145024	+	0.999895i	$0.504616\pi$
$954$	−1.92571e6	−0.0685046
$955$	0	0
$956$	−4.06228e6	−0.143756
$957$	−7.68155e6	−0.271125
$958$	1.72058e7	0.605704
$959$	−6.89223e6	−0.241999
$960$	0	0
$961$	2.73827e7	0.956462
$962$	3.16538e6	0.110278
$963$	−6.80065e6	−0.236312
$964$	−2.96480e7	−1.02755
$965$	0	0
$966$	4.37591e6	0.150878
$967$	3.59985e7	1.23799	0.618997	−	0.785393i	$-0.287539\pi$
$967$	3.59985e7	1.23799	0.618997	+	0.785393i	$0.287539\pi$
$968$	−2.49829e6	−0.0856950
$969$	−1.11963e6	−0.0383058
$970$	0	0
$971$	−6.11981e6	−0.208300	−0.104150	−	0.994562i	$-0.533212\pi$
$971$	−6.11981e6	−0.208300	−0.104150	+	0.994562i	$0.533212\pi$
$972$	−1.30046e6	−0.0441501
$973$	−1.32866e7	−0.449915
$974$	2.39785e7	0.809886
$975$	0	0
$976$	7.00494e6	0.235386
$977$	−1.31824e7	−0.441835	−0.220917	−	0.975293i	$-0.570905\pi$
$977$	−1.31824e7	−0.441835	−0.220917	+	0.975293i	$0.570905\pi$
$978$	5.54375e6	0.185334
$979$	8.65266e6	0.288531
$980$	0	0
$981$	4.10314e6	0.136127
$982$	−2.66057e6	−0.0880432
$983$	1.06411e7	0.351238	0.175619	−	0.984458i	$-0.443807\pi$
$983$	1.06411e7	0.351238	0.175619	+	0.984458i	$0.443807\pi$
$984$	−8.65459e6	−0.284944
$985$	0	0
$986$	5.19259e6	0.170095
$987$	−9.40204e6	−0.307206
$988$	−1.51464e6	−0.0493648
$989$	8.23506e6	0.267717
$990$	0	0
$991$	−5.48907e6	−0.177548	−0.0887738	−	0.996052i	$-0.528295\pi$
$991$	−5.48907e6	−0.177548	−0.0887738	+	0.996052i	$0.528295\pi$
$992$	−4.47850e7	−1.44495
$993$	5.45551e6	0.175575
$994$	−2.71608e6	−0.0871919
$995$	0	0
$996$	−8.83023e6	−0.282048
$997$	−1.37419e7	−0.437833	−0.218916	−	0.975744i	$-0.570252\pi$
$997$	−1.37419e7	−0.437833	−0.218916	+	0.975744i	$0.570252\pi$
$998$	−4.00998e6	−0.127443
$999$	−5.67018e6	−0.179756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.p.1.6	✓	8
5.4	even	2		825.6.a.q.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.6	✓	8	1.1	even	1	trivial
825.6.a.q.1.3	yes	8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.15857 q^{2} +9.00000 q^{3} -22.0234 q^{4} +28.4272 q^{6} +35.7175 q^{7} -170.637 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+3.15857 q^{2} +9.00000 q^{3} -22.0234 q^{4} +28.4272 q^{6} +35.7175 q^{7} -170.637 q^{8} +81.0000 q^{9} -121.000 q^{11} -198.211 q^{12} -128.844 q^{13} +112.816 q^{14} +165.780 q^{16} +233.062 q^{17} +255.844 q^{18} -533.777 q^{19} +321.457 q^{21} -382.187 q^{22} +4309.77 q^{23} -1535.73 q^{24} -406.964 q^{26} +729.000 q^{27} -786.621 q^{28} +7053.77 q^{29} -7484.11 q^{31} +5984.01 q^{32} -1089.00 q^{33} +736.144 q^{34} -1783.90 q^{36} -7778.03 q^{37} -1685.97 q^{38} -1159.60 q^{39} +5635.48 q^{41} +1015.35 q^{42} +1910.79 q^{43} +2664.83 q^{44} +13612.7 q^{46} -29248.2 q^{47} +1492.02 q^{48} -15531.3 q^{49} +2097.56 q^{51} +2837.59 q^{52} -7526.88 q^{53} +2302.60 q^{54} -6094.72 q^{56} -4803.99 q^{57} +22279.8 q^{58} -29371.8 q^{59} +42254.3 q^{61} -23639.1 q^{62} +2893.12 q^{63} +13596.0 q^{64} -3439.69 q^{66} -26453.6 q^{67} -5132.83 q^{68} +38787.9 q^{69} -24075.2 q^{71} -13821.6 q^{72} -75389.2 q^{73} -24567.5 q^{74} +11755.6 q^{76} -4321.82 q^{77} -3662.68 q^{78} +36440.5 q^{79} +6561.00 q^{81} +17800.1 q^{82} +44549.7 q^{83} -7079.59 q^{84} +6035.37 q^{86} +63483.9 q^{87} +20647.1 q^{88} -71509.6 q^{89} -4601.99 q^{91} -94915.8 q^{92} -67357.0 q^{93} -92382.5 q^{94} +53856.1 q^{96} -81704.7 q^{97} -49056.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 - 968 * q^11 + 963 * q^12 - 382 * q^13 + 1048 * q^14 - 325 * q^16 - 288 * q^17 - 729 * q^18 - 988 * q^19 + 594 * q^21 + 1089 * q^22 - 5972 * q^23 - 1863 * q^24 - 14579 * q^26 + 5832 * q^27 - 5942 * q^28 + 1032 * q^29 - 4682 * q^31 - 3863 * q^32 - 8712 * q^33 + 2206 * q^34 + 8667 * q^36 + 17200 * q^37 - 11011 * q^38 - 3438 * q^39 - 13220 * q^41 + 9432 * q^42 - 22872 * q^43 - 12947 * q^44 + 9101 * q^46 - 6700 * q^47 - 2925 * q^48 - 43466 * q^49 - 2592 * q^51 - 5009 * q^52 - 6224 * q^53 - 6561 * q^54 + 20992 * q^56 - 8892 * q^57 - 33015 * q^58 - 77556 * q^59 + 11554 * q^61 - 12135 * q^62 + 5346 * q^63 - 149917 * q^64 + 9801 * q^66 - 20894 * q^67 - 91776 * q^68 - 53748 * q^69 - 21648 * q^71 - 16767 * q^72 - 64660 * q^73 - 179522 * q^74 + 24401 * q^76 - 7986 * q^77 - 131211 * q^78 - 22660 * q^79 + 52488 * q^81 + 56080 * q^82 - 100390 * q^83 - 53478 * q^84 + 47271 * q^86 + 9288 * q^87 + 25047 * q^88 - 25578 * q^89 + 73250 * q^91 - 95311 * q^92 - 42138 * q^93 - 170120 * q^94 - 34767 * q^96 - 142828 * q^97 - 303397 * q^98 - 78408 * q^99

Introduction

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