LMFDB - Embedded newform 825.6.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$8$
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			1.2
Root			$7.96197$ 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)

$f(q)$	$=$	$q-8.96197 q^{2} +9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} -191.454 q^{7} -146.231 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-8.96197 q^{2} +9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} -191.454 q^{7} -146.231 q^{8} +81.0000 q^{9} -121.000 q^{11} +434.852 q^{12} +257.481 q^{13} +1715.80 q^{14} -235.621 q^{16} +1247.35 q^{17} -725.919 q^{18} -2954.73 q^{19} -1723.09 q^{21} +1084.40 q^{22} -374.592 q^{23} -1316.08 q^{24} -2307.53 q^{26} +729.000 q^{27} -9250.46 q^{28} +4573.39 q^{29} +2523.66 q^{31} +6791.02 q^{32} -1089.00 q^{33} -11178.7 q^{34} +3913.66 q^{36} +11378.4 q^{37} +26480.2 q^{38} +2317.32 q^{39} -7607.66 q^{41} +15442.2 q^{42} -853.530 q^{43} -5846.34 q^{44} +3357.08 q^{46} +18520.4 q^{47} -2120.59 q^{48} +19847.7 q^{49} +11226.2 q^{51} +12440.6 q^{52} -15523.8 q^{53} -6533.27 q^{54} +27996.5 q^{56} -26592.6 q^{57} -40986.5 q^{58} -13932.5 q^{59} +21313.0 q^{61} -22617.0 q^{62} -15507.8 q^{63} -53321.0 q^{64} +9759.58 q^{66} +55365.4 q^{67} +60268.1 q^{68} -3371.33 q^{69} -72039.8 q^{71} -11844.7 q^{72} -57248.2 q^{73} -101973. q^{74} -142763. q^{76} +23165.9 q^{77} -20767.8 q^{78} +75155.7 q^{79} +6561.00 q^{81} +68179.6 q^{82} -102915. q^{83} -83254.1 q^{84} +7649.30 q^{86} +41160.5 q^{87} +17694.0 q^{88} +11387.8 q^{89} -49295.7 q^{91} -18099.1 q^{92} +22712.9 q^{93} -165980. q^{94} +61119.2 q^{96} -23382.2 q^{97} -177874. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.96197	−1.58427	−0.792133	−	0.610348i	$-0.791030\pi$
$2$	−8.96197	−1.58427	−0.792133	+	0.610348i	$0.791030\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	48.3168	1.50990
$5$	0	0
$5$	0	0
$6$	−80.6577	−0.914677
$7$	−191.454	−1.47679	−0.738396	−	0.674367i	$-0.764417\pi$
$7$	−191.454	−1.47679	−0.738396	+	0.674367i	$0.764417\pi$
$8$	−146.231	−0.807820
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	434.852	0.871742
$13$	257.481	0.422558	0.211279	−	0.977426i	$-0.432237\pi$
$13$	257.481	0.422558	0.211279	+	0.977426i	$0.432237\pi$
$14$	1715.80	2.33963
$15$	0	0
$16$	−235.621	−0.230099
$17$	1247.35	1.04681	0.523404	−	0.852085i	$-0.324662\pi$
$17$	1247.35	1.04681	0.523404	+	0.852085i	$0.324662\pi$
$18$	−725.919	−0.528089
$19$	−2954.73	−1.87774	−0.938868	−	0.344278i	$-0.888124\pi$
$19$	−2954.73	−1.87774	−0.938868	+	0.344278i	$0.888124\pi$
$20$	0	0
$21$	−1723.09	−0.852627
$22$	1084.40	0.477674
$23$	−374.592	−0.147652	−0.0738260	−	0.997271i	$-0.523521\pi$
$23$	−374.592	−0.147652	−0.0738260	+	0.997271i	$0.523521\pi$
$24$	−1316.08	−0.466395
$25$	0	0
$26$	−2307.53	−0.669444
$27$	729.000	0.192450
$28$	−9250.46	−2.22981
$29$	4573.39	1.00982	0.504909	−	0.863173i	$-0.331526\pi$
$29$	4573.39	1.00982	0.504909	+	0.863173i	$0.331526\pi$
$30$	0	0
$31$	2523.66	0.471657	0.235829	−	0.971795i	$-0.424220\pi$
$31$	2523.66	0.471657	0.235829	+	0.971795i	$0.424220\pi$
$32$	6791.02	1.17236
$33$	−1089.00	−0.174078
$34$	−11178.7	−1.65842
$35$	0	0
$36$	3913.66	0.503300
$37$	11378.4	1.36640	0.683199	−	0.730232i	$-0.260588\pi$
$37$	11378.4	1.36640	0.683199	+	0.730232i	$0.260588\pi$
$38$	26480.2	2.97483
$39$	2317.32	0.243964
$40$	0	0
$41$	−7607.66	−0.706791	−0.353396	−	0.935474i	$-0.614973\pi$
$41$	−7607.66	−0.706791	−0.353396	+	0.935474i	$0.614973\pi$
$42$	15442.2	1.35079
$43$	−853.530	−0.0703959	−0.0351980	−	0.999380i	$-0.511206\pi$
$43$	−853.530	−0.0703959	−0.0351980	+	0.999380i	$0.511206\pi$
$44$	−5846.34	−0.455252
$45$	0	0
$46$	3357.08	0.233920
$47$	18520.4	1.22294	0.611472	−	0.791266i	$-0.290578\pi$
$47$	18520.4	1.22294	0.611472	+	0.791266i	$0.290578\pi$
$48$	−2120.59	−0.132848
$49$	19847.7	1.18092
$50$	0	0
$51$	11226.2	0.604375
$52$	12440.6	0.638020
$53$	−15523.8	−0.759116	−0.379558	−	0.925168i	$-0.623924\pi$
$53$	−15523.8	−0.759116	−0.379558	+	0.925168i	$0.623924\pi$
$54$	−6533.27	−0.304892
$55$	0	0
$56$	27996.5	1.19298
$57$	−26592.6	−1.08411
$58$	−40986.5	−1.59982
$59$	−13932.5	−0.521074	−0.260537	−	0.965464i	$-0.583900\pi$
$59$	−13932.5	−0.521074	−0.260537	+	0.965464i	$0.583900\pi$
$60$	0	0
$61$	21313.0	0.733366	0.366683	−	0.930346i	$-0.380493\pi$
$61$	21313.0	0.733366	0.366683	+	0.930346i	$0.380493\pi$
$62$	−22617.0	−0.747231
$63$	−15507.8	−0.492264
$64$	−53321.0	−1.62723
$65$	0	0
$66$	9759.58	0.275785
$67$	55365.4	1.50679	0.753393	−	0.657571i	$-0.228416\pi$
$67$	55365.4	1.50679	0.753393	+	0.657571i	$0.228416\pi$
$68$	60268.1	1.58058
$69$	−3371.33	−0.0852469
$70$	0	0
$71$	−72039.8	−1.69600	−0.848002	−	0.529993i	$-0.822194\pi$
$71$	−72039.8	−1.69600	−0.848002	+	0.529993i	$0.822194\pi$
$72$	−11844.7	−0.269273
$73$	−57248.2	−1.25735	−0.628673	−	0.777670i	$-0.716402\pi$
$73$	−57248.2	−1.25735	−0.628673	+	0.777670i	$0.716402\pi$
$74$	−101973.	−2.16474
$75$	0	0
$76$	−142763.	−2.83520
$77$	23165.9	0.445270
$78$	−20767.8	−0.386504
$79$	75155.7	1.35486	0.677430	−	0.735587i	$-0.263094\pi$
$79$	75155.7	1.35486	0.677430	+	0.735587i	$0.263094\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	68179.6	1.11975
$83$	−102915.	−1.63978	−0.819889	−	0.572523i	$-0.805965\pi$
$83$	−102915.	−1.63978	−0.819889	+	0.572523i	$0.805965\pi$
$84$	−83254.1	−1.28738
$85$	0	0
$86$	7649.30	0.111526
$87$	41160.5	0.583018
$88$	17694.0	0.243567
$89$	11387.8	0.152393	0.0761965	−	0.997093i	$-0.475722\pi$
$89$	11387.8	0.152393	0.0761965	+	0.997093i	$0.475722\pi$
$90$	0	0
$91$	−49295.7	−0.624030
$92$	−18099.1	−0.222940
$93$	22712.9	0.272312
$94$	−165980.	−1.93747
$95$	0	0
$96$	61119.2	0.676861
$97$	−23382.2	−0.252323	−0.126161	−	0.992010i	$-0.540266\pi$
$97$	−23382.2	−0.252323	−0.126161	+	0.992010i	$0.540266\pi$
$98$	−177874.	−1.87089
$99$	−9801.00	−0.100504
$100$	0	0
$101$	71052.3	0.693066	0.346533	−	0.938038i	$-0.387359\pi$
$101$	71052.3	0.693066	0.346533	+	0.938038i	$0.387359\pi$
$102$	−100609.	−0.957491
$103$	133153.	1.23668	0.618339	−	0.785912i	$-0.287806\pi$
$103$	133153.	1.23668	0.618339	+	0.785912i	$0.287806\pi$
$104$	−37651.6	−0.341351
$105$	0	0
$106$	139124.	1.20264
$107$	22312.0	0.188400	0.0941998	−	0.995553i	$-0.469971\pi$
$107$	22312.0	0.188400	0.0941998	+	0.995553i	$0.469971\pi$
$108$	35223.0	0.290581
$109$	−212605.	−1.71399	−0.856993	−	0.515328i	$-0.827670\pi$
$109$	−212605.	−1.71399	−0.856993	+	0.515328i	$0.827670\pi$
$110$	0	0
$111$	102406.	0.788891
$112$	45110.7	0.339809
$113$	246202.	1.81383	0.906914	−	0.421316i	$-0.138432\pi$
$113$	246202.	1.81383	0.906914	+	0.421316i	$0.138432\pi$
$114$	238322.	1.71752
$115$	0	0
$116$	220972.	1.52472
$117$	20855.9	0.140853
$118$	124863.	0.825520
$119$	−238811.	−1.54592
$120$	0	0
$121$	14641.0	0.0909091
$122$	−191007.	−1.16185
$123$	−68468.9	−0.408066
$124$	121935.	0.712156
$125$	0	0
$126$	138980.	0.779878
$127$	119855.	0.659399	0.329699	−	0.944086i	$-0.393053\pi$
$127$	119855.	0.659399	0.329699	+	0.944086i	$0.393053\pi$
$128$	260549.	1.40561
$129$	−7681.77	−0.0406431
$130$	0	0
$131$	−89009.9	−0.453169	−0.226585	−	0.973992i	$-0.572756\pi$
$131$	−89009.9	−0.453169	−0.226585	+	0.973992i	$0.572756\pi$
$132$	−52617.0	−0.262840
$133$	565696.	2.77303
$134$	−496183.	−2.38715
$135$	0	0
$136$	−182402.	−0.845632
$137$	91980.3	0.418691	0.209345	−	0.977842i	$-0.432867\pi$
$137$	91980.3	0.418691	0.209345	+	0.977842i	$0.432867\pi$
$138$	30213.7	0.135054
$139$	233520.	1.02515	0.512574	−	0.858643i	$-0.328692\pi$
$139$	233520.	1.02515	0.512574	+	0.858643i	$0.328692\pi$
$140$	0	0
$141$	166684.	0.706067
$142$	645619.	2.68692
$143$	−31155.1	−0.127406
$144$	−19085.3	−0.0766997
$145$	0	0
$146$	513057.	1.99197
$147$	178629.	0.681802
$148$	549769.	2.06313
$149$	−226695.	−0.836520	−0.418260	−	0.908327i	$-0.637360\pi$
$149$	−226695.	−0.836520	−0.418260	+	0.908327i	$0.637360\pi$
$150$	0	0
$151$	−461783.	−1.64814	−0.824072	−	0.566485i	$-0.808303\pi$
$151$	−461783.	−1.64814	−0.824072	+	0.566485i	$0.808303\pi$
$152$	432074.	1.51687
$153$	101036.	0.348936
$154$	−207612.	−0.705426
$155$	0	0
$156$	111966.	0.368361
$157$	−9453.41	−0.0306083	−0.0153042	−	0.999883i	$-0.504872\pi$
$157$	−9453.41	−0.0306083	−0.0153042	+	0.999883i	$0.504872\pi$
$158$	−673543.	−2.14646
$159$	−139714.	−0.438276
$160$	0	0
$161$	71717.2	0.218051
$162$	−58799.5	−0.176030
$163$	−183701.	−0.541554	−0.270777	−	0.962642i	$-0.587281\pi$
$163$	−183701.	−0.541554	−0.270777	+	0.962642i	$0.587281\pi$
$164$	−367578.	−1.06719
$165$	0	0
$166$	922324.	2.59785
$167$	−339882.	−0.943054	−0.471527	−	0.881852i	$-0.656297\pi$
$167$	−339882.	−0.943054	−0.471527	+	0.881852i	$0.656297\pi$
$168$	251969.	0.688769
$169$	−304997.	−0.821445
$170$	0	0
$171$	−239333.	−0.625912
$172$	−41239.9	−0.106291
$173$	−12098.7	−0.0307342	−0.0153671	−	0.999882i	$-0.504892\pi$
$173$	−12098.7	−0.0307342	−0.0153671	+	0.999882i	$0.504892\pi$
$174$	−368879.	−0.923657
$175$	0	0
$176$	28510.2	0.0693775
$177$	−125393.	−0.300842
$178$	−102057.	−0.241431
$179$	366015.	0.853820	0.426910	−	0.904294i	$-0.359602\pi$
$179$	366015.	0.853820	0.426910	+	0.904294i	$0.359602\pi$
$180$	0	0
$181$	−187033.	−0.424347	−0.212174	−	0.977232i	$-0.568054\pi$
$181$	−187033.	−0.424347	−0.212174	+	0.977232i	$0.568054\pi$
$182$	441786.	0.988630
$183$	191817.	0.423409
$184$	54777.0	0.119276
$185$	0	0
$186$	−203553.	−0.431414
$187$	−150930.	−0.315624
$188$	894849.	1.84653
$189$	−139570.	−0.284209
$190$	0	0
$191$	75921.1	0.150584	0.0752920	−	0.997162i	$-0.476011\pi$
$191$	75921.1	0.150584	0.0752920	+	0.997162i	$0.476011\pi$
$192$	−479889.	−0.939481
$193$	897793.	1.73493	0.867467	−	0.497495i	$-0.165747\pi$
$193$	897793.	1.73493	0.867467	+	0.497495i	$0.165747\pi$
$194$	209551.	0.399747
$195$	0	0
$196$	958976.	1.78307
$197$	731669.	1.34323	0.671613	−	0.740902i	$-0.265602\pi$
$197$	731669.	1.34323	0.671613	+	0.740902i	$0.265602\pi$
$198$	87836.2	0.159225
$199$	60771.5	0.108785	0.0543923	−	0.998520i	$-0.482678\pi$
$199$	60771.5	0.108785	0.0543923	+	0.998520i	$0.482678\pi$
$200$	0	0
$201$	498289.	0.869943
$202$	−636768.	−1.09800
$203$	−875593.	−1.49129
$204$	542413.	0.912546
$205$	0	0
$206$	−1.19331e6	−1.95923
$207$	−30342.0	−0.0492173
$208$	−60667.9	−0.0972301
$209$	357523.	0.566159
$210$	0	0
$211$	52810.1	0.0816603	0.0408301	−	0.999166i	$-0.487000\pi$
$211$	52810.1	0.0816603	0.0408301	+	0.999166i	$0.487000\pi$
$212$	−750061.	−1.14619
$213$	−648358.	−0.979188
$214$	−199960.	−0.298475
$215$	0	0
$216$	−106602.	−0.155465
$217$	−483165.	−0.696540
$218$	1.90536e6	2.71541
$219$	−515234.	−0.725929
$220$	0	0
$221$	321169.	0.442337
$222$	−917757.	−1.24981
$223$	274535.	0.369688	0.184844	−	0.982768i	$-0.440822\pi$
$223$	274535.	0.369688	0.184844	+	0.982768i	$0.440822\pi$
$224$	−1.30017e6	−1.73133
$225$	0	0
$226$	−2.20646e6	−2.87359
$227$	−459993.	−0.592498	−0.296249	−	0.955111i	$-0.595736\pi$
$227$	−459993.	−0.592498	−0.296249	+	0.955111i	$0.595736\pi$
$228$	−1.28487e6	−1.63690
$229$	708694.	0.893038	0.446519	−	0.894774i	$-0.352664\pi$
$229$	708694.	0.893038	0.446519	+	0.894774i	$0.352664\pi$
$230$	0	0
$231$	208493.	0.257077
$232$	−668771.	−0.815751
$233$	−972539.	−1.17359	−0.586796	−	0.809735i	$-0.699611\pi$
$233$	−972539.	−1.17359	−0.586796	+	0.809735i	$0.699611\pi$
$234$	−186910.	−0.223148
$235$	0	0
$236$	−673175.	−0.786770
$237$	676402.	0.782229
$238$	2.14021e6	2.44915
$239$	506701.	0.573795	0.286898	−	0.957961i	$-0.407376\pi$
$239$	506701.	0.573795	0.286898	+	0.957961i	$0.407376\pi$
$240$	0	0
$241$	256791.	0.284798	0.142399	−	0.989809i	$-0.454518\pi$
$241$	256791.	0.284798	0.142399	+	0.989809i	$0.454518\pi$
$242$	−131212.	−0.144024
$243$	59049.0	0.0641500
$244$	1.02978e6	1.10731
$245$	0	0
$246$	613616.	0.646486
$247$	−760786.	−0.793451
$248$	−369038.	−0.381014
$249$	−926238.	−0.946726
$250$	0	0
$251$	−1.58330e6	−1.58628	−0.793138	−	0.609042i	$-0.791554\pi$
$251$	−1.58330e6	−1.58628	−0.793138	+	0.609042i	$0.791554\pi$
$252$	−749287.	−0.743270
$253$	45325.6	0.0445187
$254$	−1.07414e6	−1.04466
$255$	0	0
$256$	−628755.	−0.599628
$257$	−955020.	−0.901945	−0.450972	−	0.892538i	$-0.648923\pi$
$257$	−955020.	−0.901945	−0.450972	+	0.892538i	$0.648923\pi$
$258$	68843.7	0.0643895
$259$	−2.17844e6	−2.01789
$260$	0	0
$261$	370444.	0.336606
$262$	797704.	0.717941
$263$	−1.31065e6	−1.16841	−0.584207	−	0.811605i	$-0.698594\pi$
$263$	−1.31065e6	−1.16841	−0.584207	+	0.811605i	$0.698594\pi$
$264$	159246.	0.140623
$265$	0	0
$266$	−5.06975e6	−4.39321
$267$	102490.	0.0879842
$268$	2.67508e6	2.27510
$269$	1.14655e6	0.966076	0.483038	−	0.875599i	$-0.339533\pi$
$269$	1.14655e6	0.966076	0.483038	+	0.875599i	$0.339533\pi$
$270$	0	0
$271$	−1.21682e6	−1.00648	−0.503239	−	0.864147i	$-0.667859\pi$
$271$	−1.21682e6	−1.00648	−0.503239	+	0.864147i	$0.667859\pi$
$272$	−293903.	−0.240869
$273$	−443661.	−0.360284
$274$	−824325.	−0.663318
$275$	0	0
$276$	−162892.	−0.128714
$277$	772459.	0.604889	0.302445	−	0.953167i	$-0.402197\pi$
$277$	772459.	0.604889	0.302445	+	0.953167i	$0.402197\pi$
$278$	−2.09279e6	−1.62411
$279$	204417.	0.157219
$280$	0	0
$281$	−2.06441e6	−1.55966	−0.779830	−	0.625991i	$-0.784695\pi$
$281$	−2.06441e6	−1.55966	−0.779830	+	0.625991i	$0.784695\pi$
$282$	−1.49382e6	−1.11860
$283$	−2.22733e6	−1.65317	−0.826586	−	0.562810i	$-0.809720\pi$
$283$	−2.22733e6	−1.65317	−0.826586	+	0.562810i	$0.809720\pi$
$284$	−3.48074e6	−2.56080
$285$	0	0
$286$	279211.	0.201845
$287$	1.45652e6	1.04378
$288$	550073.	0.390786
$289$	136031.	0.0958063
$290$	0	0
$291$	−210440.	−0.145679
$292$	−2.76605e6	−1.89847
$293$	−2.69919e6	−1.83681	−0.918405	−	0.395641i	$-0.870522\pi$
$293$	−2.69919e6	−1.83681	−0.918405	+	0.395641i	$0.870522\pi$
$294$	−1.60087e6	−1.08016
$295$	0	0
$296$	−1.66388e6	−1.10380
$297$	−88209.0	−0.0580259
$298$	2.03163e6	1.32527
$299$	−96450.2	−0.0623914
$300$	0	0
$301$	163412.	0.103960
$302$	4.13848e6	2.61110
$303$	639471.	0.400142
$304$	696199.	0.432065
$305$	0	0
$306$	−905477.	−0.552808
$307$	−1.10821e6	−0.671086	−0.335543	−	0.942025i	$-0.608920\pi$
$307$	−1.10821e6	−0.671086	−0.335543	+	0.942025i	$0.608920\pi$
$308$	1.11931e6	0.672313
$309$	1.19837e6	0.713996
$310$	0	0
$311$	−2.05197e6	−1.20301	−0.601505	−	0.798869i	$-0.705432\pi$
$311$	−2.05197e6	−1.20301	−0.601505	+	0.798869i	$0.705432\pi$
$312$	−338865.	−0.197079
$313$	2.73410e6	1.57744	0.788721	−	0.614751i	$-0.210743\pi$
$313$	2.73410e6	1.57744	0.788721	+	0.614751i	$0.210743\pi$
$314$	84721.2	0.0484917
$315$	0	0
$316$	3.63129e6	2.04570
$317$	−2.45349e6	−1.37131	−0.685656	−	0.727926i	$-0.740485\pi$
$317$	−2.45349e6	−1.37131	−0.685656	+	0.727926i	$0.740485\pi$
$318$	1.25211e6	0.694346
$319$	−553380.	−0.304471
$320$	0	0
$321$	200808.	0.108773
$322$	−642727.	−0.345451
$323$	−3.68559e6	−1.96563
$324$	317007.	0.167767
$325$	0	0
$326$	1.64632e6	0.857966
$327$	−1.91345e6	−0.989570
$328$	1.11248e6	0.570960
$329$	−3.54581e6	−1.80604
$330$	0	0
$331$	1.15167e6	0.577773	0.288886	−	0.957363i	$-0.406715\pi$
$331$	1.15167e6	0.577773	0.288886	+	0.957363i	$0.406715\pi$
$332$	−4.97255e6	−2.47590
$333$	921652.	0.455466
$334$	3.04601e6	1.49405
$335$	0	0
$336$	405996.	0.196189
$337$	−2.40826e6	−1.15512	−0.577562	−	0.816347i	$-0.695996\pi$
$337$	−2.40826e6	−1.15512	−0.577562	+	0.816347i	$0.695996\pi$
$338$	2.73337e6	1.30139
$339$	2.21582e6	1.04721
$340$	0	0
$341$	−305363.	−0.142210
$342$	2.14490e6	0.991611
$343$	−582147.	−0.267176
$344$	124813.	0.0568672
$345$	0	0
$346$	108428.	0.0486912
$347$	−2.02361e6	−0.902202	−0.451101	−	0.892473i	$-0.648969\pi$
$347$	−2.02361e6	−0.902202	−0.451101	+	0.892473i	$0.648969\pi$
$348$	1.98874e6	0.880300
$349$	1.64732e6	0.723960	0.361980	−	0.932186i	$-0.382101\pi$
$349$	1.64732e6	0.723960	0.361980	+	0.932186i	$0.382101\pi$
$350$	0	0
$351$	187703.	0.0813213
$352$	−821714.	−0.353479
$353$	−2.08981e6	−0.892626	−0.446313	−	0.894877i	$-0.647263\pi$
$353$	−2.08981e6	−0.892626	−0.446313	+	0.894877i	$0.647263\pi$
$354$	1.12376e6	0.476614
$355$	0	0
$356$	550223.	0.230099
$357$	−2.14930e6	−0.892536
$358$	−3.28021e6	−1.35268
$359$	887838.	0.363578	0.181789	−	0.983338i	$-0.441811\pi$
$359$	887838.	0.363578	0.181789	+	0.983338i	$0.441811\pi$
$360$	0	0
$361$	6.25435e6	2.52589
$362$	1.67618e6	0.672279
$363$	131769.	0.0524864
$364$	−2.38181e6	−0.942224
$365$	0	0
$366$	−1.71906e6	−0.670793
$367$	−1.96673e6	−0.762218	−0.381109	−	0.924530i	$-0.624458\pi$
$367$	−1.96673e6	−0.762218	−0.381109	+	0.924530i	$0.624458\pi$
$368$	88261.9	0.0339746
$369$	−616220.	−0.235597
$370$	0	0
$371$	2.97209e6	1.12106
$372$	1.09742e6	0.411164
$373$	−4.81788e6	−1.79302	−0.896508	−	0.443028i	$-0.853904\pi$
$373$	−4.81788e6	−1.79302	−0.896508	+	0.443028i	$0.853904\pi$
$374$	1.35263e6	0.500033
$375$	0	0
$376$	−2.70826e6	−0.987919
$377$	1.17756e6	0.426706
$378$	1.25082e6	0.450263
$379$	−2.14286e6	−0.766296	−0.383148	−	0.923687i	$-0.625160\pi$
$379$	−2.14286e6	−0.766296	−0.383148	+	0.923687i	$0.625160\pi$
$380$	0	0
$381$	1.07870e6	0.380704
$382$	−680402.	−0.238565
$383$	−939889.	−0.327401	−0.163700	−	0.986510i	$-0.552343\pi$
$383$	−939889.	−0.327401	−0.163700	+	0.986510i	$0.552343\pi$
$384$	2.34494e6	0.811528
$385$	0	0
$386$	−8.04599e6	−2.74860
$387$	−69135.9	−0.0234653
$388$	−1.12976e6	−0.380982
$389$	−3.30060e6	−1.10591	−0.552953	−	0.833212i	$-0.686499\pi$
$389$	−3.30060e6	−1.10591	−0.552953	+	0.833212i	$0.686499\pi$
$390$	0	0
$391$	−467248.	−0.154563
$392$	−2.90234e6	−0.953968
$393$	−801089.	−0.261637
$394$	−6.55719e6	−2.12803
$395$	0	0
$396$	−473553.	−0.151751
$397$	4.66064e6	1.48412	0.742061	−	0.670333i	$-0.233849\pi$
$397$	4.66064e6	1.48412	0.742061	+	0.670333i	$0.233849\pi$
$398$	−544632.	−0.172344
$399$	5.09126e6	1.60101
$400$	0	0
$401$	−1.19103e6	−0.369879	−0.184940	−	0.982750i	$-0.559209\pi$
$401$	−1.19103e6	−0.369879	−0.184940	+	0.982750i	$0.559209\pi$
$402$	−4.46565e6	−1.37822
$403$	649793.	0.199302
$404$	3.43302e6	1.04646
$405$	0	0
$406$	7.84704e6	2.36260
$407$	−1.37679e6	−0.411985
$408$	−1.64161e6	−0.488226
$409$	1.65037e6	0.487834	0.243917	−	0.969796i	$-0.421568\pi$
$409$	1.65037e6	0.487834	0.243917	+	0.969796i	$0.421568\pi$
$410$	0	0
$411$	827823.	0.241731
$412$	6.43351e6	1.86726
$413$	2.66743e6	0.769518
$414$	271924.	0.0779733
$415$	0	0
$416$	1.74856e6	0.495389
$417$	2.10168e6	0.591869
$418$	−3.20411e6	−0.896946
$419$	−1.54582e6	−0.430155	−0.215078	−	0.976597i	$-0.569000\pi$
$419$	−1.54582e6	−0.430155	−0.215078	+	0.976597i	$0.569000\pi$
$420$	0	0
$421$	1.55879e6	0.428629	0.214314	−	0.976765i	$-0.431248\pi$
$421$	1.55879e6	0.428629	0.214314	+	0.976765i	$0.431248\pi$
$422$	−473282.	−0.129372
$423$	1.50016e6	0.407648
$424$	2.27006e6	0.613229
$425$	0	0
$426$	5.81057e6	1.55130
$427$	−4.08047e6	−1.08303
$428$	1.07805e6	0.284465
$429$	−280396.	−0.0735578
$430$	0	0
$431$	2.66308e6	0.690543	0.345272	−	0.938503i	$-0.387787\pi$
$431$	2.66308e6	0.690543	0.345272	+	0.938503i	$0.387787\pi$
$432$	−171768.	−0.0442826
$433$	2.04309e6	0.523682	0.261841	−	0.965111i	$-0.415670\pi$
$433$	2.04309e6	0.523682	0.261841	+	0.965111i	$0.415670\pi$
$434$	4.33011e6	1.10351
$435$	0	0
$436$	−1.02724e7	−2.58795
$437$	1.10682e6	0.277251
$438$	4.61751e6	1.15007
$439$	−4.40979e6	−1.09209	−0.546043	−	0.837757i	$-0.683866\pi$
$439$	−4.40979e6	−1.09209	−0.546043	+	0.837757i	$0.683866\pi$
$440$	0	0
$441$	1.60766e6	0.393639
$442$	−2.87831e6	−0.700779
$443$	2.98414e6	0.722453	0.361227	−	0.932478i	$-0.382358\pi$
$443$	2.98414e6	0.722453	0.361227	+	0.932478i	$0.382358\pi$
$444$	4.94792e6	1.19115
$445$	0	0
$446$	−2.46037e6	−0.585684
$447$	−2.04025e6	−0.482965
$448$	1.02085e7	2.40308
$449$	−718293.	−0.168146	−0.0840729	−	0.996460i	$-0.526793\pi$
$449$	−718293.	−0.168146	−0.0840729	+	0.996460i	$0.526793\pi$
$450$	0	0
$451$	920526.	0.213106
$452$	1.18957e7	2.73870
$453$	−4.15604e6	−0.951557
$454$	4.12244e6	0.938674
$455$	0	0
$456$	3.88866e6	0.875767
$457$	−2.69166e6	−0.602879	−0.301439	−	0.953485i	$-0.597467\pi$
$457$	−2.69166e6	−0.602879	−0.301439	+	0.953485i	$0.597467\pi$
$458$	−6.35129e6	−1.41481
$459$	909320.	0.201458
$460$	0	0
$461$	1.09475e6	0.239918	0.119959	−	0.992779i	$-0.461724\pi$
$461$	1.09475e6	0.239918	0.119959	+	0.992779i	$0.461724\pi$
$462$	−1.86851e6	−0.407278
$463$	−8.57985e6	−1.86006	−0.930031	−	0.367482i	$-0.880220\pi$
$463$	−8.57985e6	−1.86006	−0.930031	+	0.367482i	$0.880220\pi$
$464$	−1.07759e6	−0.232358
$465$	0	0
$466$	8.71586e6	1.85928
$467$	−6.54827e6	−1.38942	−0.694711	−	0.719289i	$-0.744468\pi$
$467$	−6.54827e6	−1.38942	−0.694711	+	0.719289i	$0.744468\pi$
$468$	1.00769e6	0.212673
$469$	−1.05999e7	−2.22521
$470$	0	0
$471$	−85080.7	−0.0176717
$472$	2.03736e6	0.420934
$473$	103277.	0.0212252
$474$	−6.06189e6	−1.23926
$475$	0	0
$476$	−1.15386e7	−2.33418
$477$	−1.25743e6	−0.253039
$478$	−4.54104e6	−0.909045
$479$	−2.02247e6	−0.402756	−0.201378	−	0.979514i	$-0.564542\pi$
$479$	−2.02247e6	−0.402756	−0.201378	+	0.979514i	$0.564542\pi$
$480$	0	0
$481$	2.92972e6	0.577382
$482$	−2.30135e6	−0.451196
$483$	645455.	0.125892
$484$	707407.	0.137264
$485$	0	0
$486$	−529195.	−0.101631
$487$	−4.51310e6	−0.862289	−0.431144	−	0.902283i	$-0.641890\pi$
$487$	−4.51310e6	−0.862289	−0.431144	+	0.902283i	$0.641890\pi$
$488$	−3.11663e6	−0.592428
$489$	−1.65331e6	−0.312666
$490$	0	0
$491$	−4.42836e6	−0.828971	−0.414485	−	0.910056i	$-0.636038\pi$
$491$	−4.42836e6	−0.828971	−0.414485	+	0.910056i	$0.636038\pi$
$492$	−3.30820e6	−0.616140
$493$	5.70463e6	1.05708
$494$	6.81814e6	1.25704
$495$	0	0
$496$	−594629.	−0.108528
$497$	1.37923e7	2.50465
$498$	8.30092e6	1.49987
$499$	−1.20046e6	−0.215822	−0.107911	−	0.994161i	$-0.534416\pi$
$499$	−1.20046e6	−0.215822	−0.107911	+	0.994161i	$0.534416\pi$
$500$	0	0
$501$	−3.05894e6	−0.544473
$502$	1.41895e7	2.51308
$503$	3.10866e6	0.547840	0.273920	−	0.961753i	$-0.411680\pi$
$503$	3.10866e6	0.547840	0.273920	+	0.961753i	$0.411680\pi$
$504$	2.26772e6	0.397661
$505$	0	0
$506$	−406207.	−0.0705295
$507$	−2.74497e6	−0.474262
$508$	5.79103e6	0.995627
$509$	9.30479e6	1.59189	0.795943	−	0.605371i	$-0.206975\pi$
$509$	9.30479e6	1.59189	0.795943	+	0.605371i	$0.206975\pi$
$510$	0	0
$511$	1.09604e7	1.85684
$512$	−2.70268e6	−0.455637
$513$	−2.15400e6	−0.361370
$514$	8.55886e6	1.42892
$515$	0	0
$516$	−371159.	−0.0613671
$517$	−2.24097e6	−0.368732
$518$	1.95231e7	3.19687
$519$	−108888.	−0.0177444
$520$	0	0
$521$	−1.64699e6	−0.265826	−0.132913	−	0.991128i	$-0.542433\pi$
$521$	−1.64699e6	−0.265826	−0.132913	+	0.991128i	$0.542433\pi$
$522$	−3.31991e6	−0.533273
$523$	321424.	0.0513835	0.0256917	−	0.999670i	$-0.491821\pi$
$523$	321424.	0.0513835	0.0256917	+	0.999670i	$0.491821\pi$
$524$	−4.30068e6	−0.684241
$525$	0	0
$526$	1.17460e7	1.85108
$527$	3.14789e6	0.493735
$528$	256592.	0.0400551
$529$	−6.29602e6	−0.978199
$530$	0	0
$531$	−1.12853e6	−0.173691
$532$	2.73326e7	4.18700
$533$	−1.95882e6	−0.298660
$534$	−918515.	−0.139390
$535$	0	0
$536$	−8.09614e6	−1.21721
$537$	3.29413e6	0.492953
$538$	−1.02753e7	−1.53052
$539$	−2.40157e6	−0.356060
$540$	0	0
$541$	−7.18074e6	−1.05481	−0.527407	−	0.849613i	$-0.676836\pi$
$541$	−7.18074e6	−1.05481	−0.527407	+	0.849613i	$0.676836\pi$
$542$	1.09051e7	1.59453
$543$	−1.68329e6	−0.244997
$544$	8.47080e6	1.22723
$545$	0	0
$546$	3.97608e6	0.570786
$547$	1.18114e6	0.168785	0.0843927	−	0.996433i	$-0.473105\pi$
$547$	1.18114e6	0.168785	0.0843927	+	0.996433i	$0.473105\pi$
$548$	4.44420e6	0.632182
$549$	1.72636e6	0.244455
$550$	0	0
$551$	−1.35131e7	−1.89617
$552$	492993.	0.0688641
$553$	−1.43889e7	−2.00085
$554$	−6.92275e6	−0.958306
$555$	0	0
$556$	1.12829e7	1.54787
$557$	8.98105e6	1.22656	0.613280	−	0.789865i	$-0.289850\pi$
$557$	8.98105e6	1.22656	0.613280	+	0.789865i	$0.289850\pi$
$558$	−1.83197e6	−0.249077
$559$	−219767.	−0.0297463
$560$	0	0
$561$	−1.35837e6	−0.182226
$562$	1.85012e7	2.47092
$563$	−8.32751e6	−1.10725	−0.553623	−	0.832768i	$-0.686755\pi$
$563$	−8.32751e6	−1.10725	−0.553623	+	0.832768i	$0.686755\pi$
$564$	8.05364e6	1.06609
$565$	0	0
$566$	1.99612e7	2.61907
$567$	−1.25613e6	−0.164088
$568$	1.05345e7	1.37007
$569$	−1.16305e7	−1.50597	−0.752985	−	0.658038i	$-0.771387\pi$
$569$	−1.16305e7	−1.50597	−0.752985	+	0.658038i	$0.771387\pi$
$570$	0	0
$571$	−9.03556e6	−1.15975	−0.579876	−	0.814705i	$-0.696899\pi$
$571$	−9.03556e6	−1.15975	−0.579876	+	0.814705i	$0.696899\pi$
$572$	−1.50532e6	−0.192370
$573$	683290.	0.0869397
$574$	−1.30533e7	−1.65363
$575$	0	0
$576$	−4.31901e6	−0.542410
$577$	1.69135e6	0.211492	0.105746	−	0.994393i	$-0.466277\pi$
$577$	1.69135e6	0.211492	0.105746	+	0.994393i	$0.466277\pi$
$578$	−1.21911e6	−0.151783
$579$	8.08013e6	1.00166
$580$	0	0
$581$	1.97036e7	2.42161
$582$	1.88596e6	0.230794
$583$	1.87838e6	0.228882
$584$	8.37147e6	1.01571
$585$	0	0
$586$	2.41900e7	2.91000
$587$	1.42543e7	1.70746	0.853731	−	0.520714i	$-0.174334\pi$
$587$	1.42543e7	1.70746	0.853731	+	0.520714i	$0.174334\pi$
$588$	8.63079e6	1.02945
$589$	−7.45675e6	−0.885648
$590$	0	0
$591$	6.58502e6	0.775512
$592$	−2.68100e6	−0.314407
$593$	−8.74561e6	−1.02130	−0.510650	−	0.859789i	$-0.670595\pi$
$593$	−8.74561e6	−1.02130	−0.510650	+	0.859789i	$0.670595\pi$
$594$	790526.	0.0919285
$595$	0	0
$596$	−1.09532e7	−1.26306
$597$	546944.	0.0628068
$598$	864383.	0.0988447
$599$	−2.04663e6	−0.233062	−0.116531	−	0.993187i	$-0.537177\pi$
$599$	−2.04663e6	−0.233062	−0.116531	+	0.993187i	$0.537177\pi$
$600$	0	0
$601$	−1.07049e7	−1.20891	−0.604456	−	0.796639i	$-0.706609\pi$
$601$	−1.07049e7	−1.20891	−0.604456	+	0.796639i	$0.706609\pi$
$602$	−1.46449e6	−0.164701
$603$	4.48460e6	0.502262
$604$	−2.23119e7	−2.48854
$605$	0	0
$606$	−5.73091e6	−0.633931
$607$	5.61449e6	0.618499	0.309249	−	0.950981i	$-0.399922\pi$
$607$	5.61449e6	0.618499	0.309249	+	0.950981i	$0.399922\pi$
$608$	−2.00657e7	−2.20138
$609$	−7.88034e6	−0.860997
$610$	0	0
$611$	4.76865e6	0.516765
$612$	4.88172e6	0.526859
$613$	−1.04084e7	−1.11875	−0.559373	−	0.828916i	$-0.688958\pi$
$613$	−1.04084e7	−1.11875	−0.559373	+	0.828916i	$0.688958\pi$
$614$	9.93178e6	1.06318
$615$	0	0
$616$	−3.38758e6	−0.359698
$617$	9.38303e6	0.992271	0.496136	−	0.868245i	$-0.334752\pi$
$617$	9.38303e6	0.992271	0.496136	+	0.868245i	$0.334752\pi$
$618$	−1.07398e7	−1.13116
$619$	−1.43312e7	−1.50334	−0.751668	−	0.659542i	$-0.770750\pi$
$619$	−1.43312e7	−1.50334	−0.751668	+	0.659542i	$0.770750\pi$
$620$	0	0
$621$	−273078.	−0.0284156
$622$	1.83896e7	1.90589
$623$	−2.18024e6	−0.225053
$624$	−546011.	−0.0561358
$625$	0	0
$626$	−2.45029e7	−2.49909
$627$	3.21771e6	0.326872
$628$	−456759.	−0.0462155
$629$	1.41929e7	1.43036
$630$	0	0
$631$	−1.34563e7	−1.34541	−0.672703	−	0.739912i	$-0.734867\pi$
$631$	−1.34563e7	−1.34541	−0.672703	+	0.739912i	$0.734867\pi$
$632$	−1.09901e7	−1.09448
$633$	475291.	0.0471466
$634$	2.19881e7	2.17252
$635$	0	0
$636$	−6.75055e6	−0.661754
$637$	5.11039e6	0.499005
$638$	4.95937e6	0.482364
$639$	−5.83523e6	−0.565335
$640$	0	0
$641$	2.02992e6	0.195134	0.0975670	−	0.995229i	$-0.468894\pi$
$641$	2.02992e6	0.195134	0.0975670	+	0.995229i	$0.468894\pi$
$642$	−1.79964e6	−0.172325
$643$	6.50526e6	0.620494	0.310247	−	0.950656i	$-0.399588\pi$
$643$	6.50526e6	0.620494	0.310247	+	0.950656i	$0.399588\pi$
$644$	3.46515e6	0.329236
$645$	0	0
$646$	3.30302e7	3.11408
$647$	1.62702e7	1.52804	0.764018	−	0.645195i	$-0.223224\pi$
$647$	1.62702e7	1.52804	0.764018	+	0.645195i	$0.223224\pi$
$648$	−959422.	−0.0897578
$649$	1.68583e6	0.157110
$650$	0	0
$651$	−4.34849e6	−0.402148
$652$	−8.87584e6	−0.817693
$653$	−1.40371e7	−1.28823	−0.644116	−	0.764927i	$-0.722775\pi$
$653$	−1.40371e7	−1.28823	−0.644116	+	0.764927i	$0.722775\pi$
$654$	1.71482e7	1.56774
$655$	0	0
$656$	1.79253e6	0.162632
$657$	−4.63711e6	−0.419115
$658$	3.17775e7	2.86124
$659$	1.08544e7	0.973626	0.486813	−	0.873506i	$-0.338159\pi$
$659$	1.08544e7	0.973626	0.486813	+	0.873506i	$0.338159\pi$
$660$	0	0
$661$	7.93752e6	0.706612	0.353306	−	0.935508i	$-0.385057\pi$
$661$	7.93752e6	0.706612	0.353306	+	0.935508i	$0.385057\pi$
$662$	−1.03212e7	−0.915346
$663$	2.89052e6	0.255383
$664$	1.50494e7	1.32465
$665$	0	0
$666$	−8.25981e6	−0.721580
$667$	−1.71315e6	−0.149101
$668$	−1.64220e7	−1.42392
$669$	2.47081e6	0.213439
$670$	0	0
$671$	−2.57888e6	−0.221118
$672$	−1.17015e7	−0.999584
$673$	1.11210e7	0.946471	0.473236	−	0.880936i	$-0.343086\pi$
$673$	1.11210e7	0.946471	0.473236	+	0.880936i	$0.343086\pi$
$674$	2.15827e7	1.83002
$675$	0	0
$676$	−1.47365e7	−1.24030
$677$	−6.96129e6	−0.583738	−0.291869	−	0.956458i	$-0.594277\pi$
$677$	−6.96129e6	−0.583738	−0.291869	+	0.956458i	$0.594277\pi$
$678$	−1.98581e7	−1.65907
$679$	4.47662e6	0.372628
$680$	0	0
$681$	−4.13994e6	−0.342079
$682$	2.73665e6	0.225299
$683$	−1.53699e7	−1.26072	−0.630360	−	0.776303i	$-0.717093\pi$
$683$	−1.53699e7	−1.26072	−0.630360	+	0.776303i	$0.717093\pi$
$684$	−1.15638e7	−0.945065
$685$	0	0
$686$	5.21718e6	0.423278
$687$	6.37824e6	0.515596
$688$	201110.	0.0161980
$689$	−3.99708e6	−0.320770
$690$	0	0
$691$	8.12461e6	0.647302	0.323651	−	0.946176i	$-0.395090\pi$
$691$	8.12461e6	0.647302	0.323651	+	0.946176i	$0.395090\pi$
$692$	−584569.	−0.0464056
$693$	1.87644e6	0.148423
$694$	1.81356e7	1.42933
$695$	0	0
$696$	−6.01894e6	−0.470974
$697$	−9.48943e6	−0.739875
$698$	−1.47632e7	−1.14695
$699$	−8.75285e6	−0.677574
$700$	0	0
$701$	1.69865e7	1.30560	0.652799	−	0.757532i	$-0.273595\pi$
$701$	1.69865e7	1.30560	0.652799	+	0.757532i	$0.273595\pi$
$702$	−1.68219e6	−0.128835
$703$	−3.36202e7	−2.56574
$704$	6.45185e6	0.490628
$705$	0	0
$706$	1.87288e7	1.41416
$707$	−1.36032e7	−1.02351
$708$	−6.05857e6	−0.454242
$709$	9.79250e6	0.731607	0.365804	−	0.930692i	$-0.380794\pi$
$709$	9.79250e6	0.731607	0.365804	+	0.930692i	$0.380794\pi$
$710$	0	0
$711$	6.08761e6	0.451620
$712$	−1.66525e6	−0.123106
$713$	−945343.	−0.0696411
$714$	1.92619e7	1.41402
$715$	0	0
$716$	1.76847e7	1.28918
$717$	4.56031e6	0.331281
$718$	−7.95678e6	−0.576005
$719$	−2.91539e6	−0.210317	−0.105159	−	0.994455i	$-0.533535\pi$
$719$	−2.91539e6	−0.210317	−0.105159	+	0.994455i	$0.533535\pi$
$720$	0	0
$721$	−2.54926e7	−1.82632
$722$	−5.60513e7	−4.00168
$723$	2.31112e6	0.164428
$724$	−9.03683e6	−0.640722
$725$	0	0
$726$	−1.18091e6	−0.0831524
$727$	−3.44448e6	−0.241706	−0.120853	−	0.992670i	$-0.538563\pi$
$727$	−3.44448e6	−0.241706	−0.120853	+	0.992670i	$0.538563\pi$
$728$	7.20856e6	0.504104
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.06465e6	−0.0736910
$732$	9.26801e6	0.639306
$733$	2.28891e7	1.57351	0.786755	−	0.617265i	$-0.211759\pi$
$733$	2.28891e7	1.57351	0.786755	+	0.617265i	$0.211759\pi$
$734$	1.76258e7	1.20756
$735$	0	0
$736$	−2.54386e6	−0.173101
$737$	−6.69922e6	−0.454313
$738$	5.52254e6	0.373249
$739$	1.84115e7	1.24016	0.620080	−	0.784539i	$-0.287100\pi$
$739$	1.84115e7	1.24016	0.620080	+	0.784539i	$0.287100\pi$
$740$	0	0
$741$	−6.84708e6	−0.458099
$742$	−2.66358e7	−1.77605
$743$	−9.95080e6	−0.661281	−0.330640	−	0.943757i	$-0.607265\pi$
$743$	−9.95080e6	−0.661281	−0.330640	+	0.943757i	$0.607265\pi$
$744$	−3.32134e6	−0.219979
$745$	0	0
$746$	4.31777e7	2.84062
$747$	−8.33614e6	−0.546593
$748$	−7.29244e6	−0.476562
$749$	−4.27173e6	−0.278227
$750$	0	0
$751$	484044.	0.0313173	0.0156587	−	0.999877i	$-0.495015\pi$
$751$	484044.	0.0313173	0.0156587	+	0.999877i	$0.495015\pi$
$752$	−4.36381e6	−0.281398
$753$	−1.42497e7	−0.915837
$754$	−1.05532e7	−0.676016
$755$	0	0
$756$	−6.74358e6	−0.429127
$757$	2.43101e7	1.54187	0.770935	−	0.636914i	$-0.219789\pi$
$757$	2.43101e7	1.54187	0.770935	+	0.636914i	$0.219789\pi$
$758$	1.92043e7	1.21402
$759$	407931.	0.0257029
$760$	0	0
$761$	−2.78776e6	−0.174499	−0.0872495	−	0.996186i	$-0.527808\pi$
$761$	−2.78776e6	−0.174499	−0.0872495	+	0.996186i	$0.527808\pi$
$762$	−9.66725e6	−0.603137
$763$	4.07041e7	2.53120
$764$	3.66827e6	0.227367
$765$	0	0
$766$	8.42325e6	0.518690
$767$	−3.58735e6	−0.220184
$768$	−5.65880e6	−0.346195
$769$	1.43510e7	0.875115	0.437558	−	0.899190i	$-0.355844\pi$
$769$	1.43510e7	0.875115	0.437558	+	0.899190i	$0.355844\pi$
$770$	0	0
$771$	−8.59518e6	−0.520738
$772$	4.33785e7	2.61958
$773$	7.14561e6	0.430121	0.215060	−	0.976601i	$-0.431005\pi$
$773$	7.14561e6	0.430121	0.215060	+	0.976601i	$0.431005\pi$
$774$	619594.	0.0371753
$775$	0	0
$776$	3.41921e6	0.203831
$777$	−1.96060e7	−1.16503
$778$	2.95798e7	1.75205
$779$	2.24786e7	1.32717
$780$	0	0
$781$	8.71682e6	0.511364
$782$	4.18747e6	0.244869
$783$	3.33400e6	0.194339
$784$	−4.67653e6	−0.271728
$785$	0	0
$786$	7.17934e6	0.414503
$787$	7.00732e6	0.403288	0.201644	−	0.979459i	$-0.435372\pi$
$787$	7.00732e6	0.403288	0.201644	+	0.979459i	$0.435372\pi$
$788$	3.53519e7	2.02814
$789$	−1.17958e7	−0.674584
$790$	0	0
$791$	−4.71364e7	−2.67865
$792$	1.43321e6	0.0811890
$793$	5.48769e6	0.309889
$794$	−4.17685e7	−2.35124
$795$	0	0
$796$	2.93629e6	0.164254
$797$	−3.12519e7	−1.74273	−0.871367	−	0.490633i	$-0.836766\pi$
$797$	−3.12519e7	−1.74273	−0.871367	+	0.490633i	$0.836766\pi$
$798$	−4.56277e7	−2.53642
$799$	2.31015e7	1.28019
$800$	0	0
$801$	922413.	0.0507977
$802$	1.06739e7	0.585988
$803$	6.92703e6	0.379104
$804$	2.40757e7	1.31353
$805$	0	0
$806$	−5.82343e6	−0.315748
$807$	1.03189e7	0.557764
$808$	−1.03900e7	−0.559873
$809$	−1.13855e7	−0.611620	−0.305810	−	0.952093i	$-0.598927\pi$
$809$	−1.13855e7	−0.611620	−0.305810	+	0.952093i	$0.598927\pi$
$810$	0	0
$811$	857075.	0.0457580	0.0228790	−	0.999738i	$-0.492717\pi$
$811$	857075.	0.0457580	0.0228790	+	0.999738i	$0.492717\pi$
$812$	−4.23059e7	−2.25170
$813$	−1.09514e7	−0.581091
$814$	1.23387e7	0.652694
$815$	0	0
$816$	−2.64513e6	−0.139066
$817$	2.52195e6	0.132185
$818$	−1.47905e7	−0.772859
$819$	−3.99295e6	−0.208010
$820$	0	0
$821$	1.28434e7	0.665001	0.332501	−	0.943103i	$-0.392108\pi$
$821$	1.28434e7	0.665001	0.332501	+	0.943103i	$0.392108\pi$
$822$	−7.41892e6	−0.382967
$823$	−2.50854e7	−1.29099	−0.645493	−	0.763767i	$-0.723348\pi$
$823$	−2.50854e7	−1.29099	−0.645493	+	0.763767i	$0.723348\pi$
$824$	−1.94710e7	−0.999012
$825$	0	0
$826$	−2.39055e7	−1.21912
$827$	1.78039e7	0.905217	0.452608	−	0.891709i	$-0.350494\pi$
$827$	1.78039e7	0.905217	0.452608	+	0.891709i	$0.350494\pi$
$828$	−1.46603e6	−0.0743133
$829$	−9.98789e6	−0.504763	−0.252381	−	0.967628i	$-0.581214\pi$
$829$	−9.98789e6	−0.504763	−0.252381	+	0.967628i	$0.581214\pi$
$830$	0	0
$831$	6.95213e6	0.349233
$832$	−1.37291e7	−0.687598
$833$	2.47570e7	1.23619
$834$	−1.88352e7	−0.937679
$835$	0	0
$836$	1.72744e7	0.854844
$837$	1.83975e6	0.0907705
$838$	1.38536e7	0.681480
$839$	1.59655e7	0.783028	0.391514	−	0.920172i	$-0.371951\pi$
$839$	1.59655e7	0.783028	0.391514	+	0.920172i	$0.371951\pi$
$840$	0	0
$841$	404715.	0.0197314
$842$	−1.39698e7	−0.679062
$843$	−1.85797e7	−0.900471
$844$	2.55162e6	0.123299
$845$	0	0
$846$	−1.34443e7	−0.645823
$847$	−2.80308e6	−0.134254
$848$	3.65774e6	0.174672
$849$	−2.00460e7	−0.954460
$850$	0	0
$851$	−4.26227e6	−0.201751
$852$	−3.13266e7	−1.47848
$853$	−2.94930e7	−1.38786	−0.693932	−	0.720041i	$-0.744123\pi$
$853$	−2.94930e7	−1.38786	−0.693932	+	0.720041i	$0.744123\pi$
$854$	3.65690e7	1.71581
$855$	0	0
$856$	−3.26271e6	−0.152193
$857$	−2.53036e6	−0.117688	−0.0588438	−	0.998267i	$-0.518741\pi$
$857$	−2.53036e6	−0.117688	−0.0588438	+	0.998267i	$0.518741\pi$
$858$	2.51290e6	0.116535
$859$	−3.52718e7	−1.63097	−0.815484	−	0.578780i	$-0.803529\pi$
$859$	−3.52718e7	−1.63097	−0.815484	+	0.578780i	$0.803529\pi$
$860$	0	0
$861$	1.31086e7	0.602629
$862$	−2.38664e7	−1.09400
$863$	−2.36369e7	−1.08035	−0.540174	−	0.841553i	$-0.681642\pi$
$863$	−2.36369e7	−1.08035	−0.540174	+	0.841553i	$0.681642\pi$
$864$	4.95066e6	0.225620
$865$	0	0
$866$	−1.83101e7	−0.829651
$867$	1.22428e6	0.0553138
$868$	−2.33450e7	−1.05171
$869$	−9.09384e6	−0.408506
$870$	0	0
$871$	1.42555e7	0.636704
$872$	3.10895e7	1.38459
$873$	−1.89396e6	−0.0841076
$874$	−9.91929e6	−0.439240
$875$	0	0
$876$	−2.48945e7	−1.09608
$877$	3.68580e7	1.61820	0.809102	−	0.587668i	$-0.199954\pi$
$877$	3.68580e7	1.61820	0.809102	+	0.587668i	$0.199954\pi$
$878$	3.95204e7	1.73015
$879$	−2.42927e7	−1.06048
$880$	0	0
$881$	8.29757e6	0.360173	0.180087	−	0.983651i	$-0.442362\pi$
$881$	8.29757e6	0.360173	0.180087	+	0.983651i	$0.442362\pi$
$882$	−1.44078e7	−0.623629
$883$	1.91280e7	0.825598	0.412799	−	0.910822i	$-0.364551\pi$
$883$	1.91280e7	0.825598	0.412799	+	0.910822i	$0.364551\pi$
$884$	1.55179e7	0.667885
$885$	0	0
$886$	−2.67438e7	−1.14456
$887$	−3.13587e7	−1.33829	−0.669144	−	0.743133i	$-0.733339\pi$
$887$	−3.13587e7	−1.33829	−0.669144	+	0.743133i	$0.733339\pi$
$888$	−1.49749e7	−0.637282
$889$	−2.29468e7	−0.973795
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.32647e7	0.558192
$893$	−5.47230e7	−2.29637
$894$	1.82847e7	0.765145
$895$	0	0
$896$	−4.98831e7	−2.07579
$897$	−868052.	−0.0360217
$898$	6.43732e6	0.266388
$899$	1.15417e7	0.476288
$900$	0	0
$901$	−1.93636e7	−0.794649
$902$	−8.24973e6	−0.337616
$903$	1.47071e6	0.0600214
$904$	−3.60024e7	−1.46525
$905$	0	0
$906$	3.72463e7	1.50752
$907$	−2.21583e7	−0.894371	−0.447185	−	0.894441i	$-0.647574\pi$
$907$	−2.21583e7	−0.894371	−0.447185	+	0.894441i	$0.647574\pi$
$908$	−2.22254e7	−0.894613
$909$	5.75523e6	0.231022
$910$	0	0
$911$	2.70378e7	1.07938	0.539691	−	0.841863i	$-0.318541\pi$
$911$	2.70378e7	1.07938	0.539691	+	0.841863i	$0.318541\pi$
$912$	6.26579e6	0.249453
$913$	1.24528e7	0.494412
$914$	2.41226e7	0.955121
$915$	0	0
$916$	3.42419e7	1.34840
$917$	1.70413e7	0.669237
$918$	−8.14930e6	−0.319164
$919$	−4.73521e7	−1.84948	−0.924741	−	0.380597i	$-0.875718\pi$
$919$	−4.73521e7	−1.84948	−0.924741	+	0.380597i	$0.875718\pi$
$920$	0	0
$921$	−9.97393e6	−0.387452
$922$	−9.81111e6	−0.380094
$923$	−1.85489e7	−0.716659
$924$	1.00737e7	0.388160
$925$	0	0
$926$	7.68923e7	2.94683
$927$	1.07854e7	0.412226
$928$	3.10580e7	1.18387
$929$	−1.66983e7	−0.634793	−0.317397	−	0.948293i	$-0.602809\pi$
$929$	−1.66983e7	−0.634793	−0.317397	+	0.948293i	$0.602809\pi$
$930$	0	0
$931$	−5.86446e7	−2.21745
$932$	−4.69900e7	−1.77201
$933$	−1.84677e7	−0.694558
$934$	5.86854e7	2.20122
$935$	0	0
$936$	−3.04978e6	−0.113784
$937$	1.68775e7	0.628000	0.314000	−	0.949423i	$-0.398331\pi$
$937$	1.68775e7	0.628000	0.314000	+	0.949423i	$0.398331\pi$
$938$	9.49963e7	3.52533
$939$	2.46069e7	0.910737
$940$	0	0
$941$	1.00920e7	0.371538	0.185769	−	0.982593i	$-0.440522\pi$
$941$	1.00920e7	0.371538	0.185769	+	0.982593i	$0.440522\pi$
$942$	762490.	0.0279967
$943$	2.84977e6	0.104359
$944$	3.28280e6	0.119899
$945$	0	0
$946$	−925566.	−0.0336263
$947$	−4.25599e7	−1.54215	−0.771073	−	0.636747i	$-0.780280\pi$
$947$	−4.25599e7	−1.54215	−0.771073	+	0.636747i	$0.780280\pi$
$948$	3.26816e7	1.18109
$949$	−1.47403e7	−0.531301
$950$	0	0
$951$	−2.20814e7	−0.791727
$952$	3.49215e7	1.24882
$953$	1.84695e7	0.658754	0.329377	−	0.944199i	$-0.393161\pi$
$953$	1.84695e7	0.658754	0.329377	+	0.944199i	$0.393161\pi$
$954$	1.12690e7	0.400881
$955$	0	0
$956$	2.44822e7	0.866374
$957$	−4.98042e6	−0.175787
$958$	1.81253e7	0.638074
$959$	−1.76100e7	−0.618320
$960$	0	0
$961$	−2.22603e7	−0.777539
$962$	−2.62561e7	−0.914728
$963$	1.80728e6	0.0627998
$964$	1.24073e7	0.430017
$965$	0	0
$966$	−5.78454e6	−0.199446
$967$	3.43843e7	1.18248	0.591241	−	0.806495i	$-0.298638\pi$
$967$	3.43843e7	1.18248	0.591241	+	0.806495i	$0.298638\pi$
$968$	−2.14097e6	−0.0734382
$969$	−3.31704e7	−1.13486
$970$	0	0
$971$	−2.08153e7	−0.708492	−0.354246	−	0.935152i	$-0.615262\pi$
$971$	−2.08153e7	−0.708492	−0.354246	+	0.935152i	$0.615262\pi$
$972$	2.85306e6	0.0968602
$973$	−4.47083e7	−1.51393
$974$	4.04463e7	1.36610
$975$	0	0
$976$	−5.02181e6	−0.168747
$977$	−1.72220e7	−0.577228	−0.288614	−	0.957446i	$-0.593194\pi$
$977$	−1.72220e7	−0.577228	−0.288614	+	0.957446i	$0.593194\pi$
$978$	1.48169e7	0.495347
$979$	−1.37793e6	−0.0459482
$980$	0	0
$981$	−1.72210e7	−0.571329
$982$	3.96868e7	1.31331
$983$	3.85176e7	1.27138	0.635690	−	0.771945i	$-0.280716\pi$
$983$	3.85176e7	1.27138	0.635690	+	0.771945i	$0.280716\pi$
$984$	1.00123e7	0.329644
$985$	0	0
$986$	−5.11247e7	−1.67470
$987$	−3.19123e7	−1.04271
$988$	−3.67588e7	−1.19803
$989$	319726.	0.0103941
$990$	0	0
$991$	−1.18485e7	−0.383249	−0.191624	−	0.981468i	$-0.561376\pi$
$991$	−1.18485e7	−0.383249	−0.191624	+	0.981468i	$0.561376\pi$
$992$	1.71382e7	0.552952
$993$	1.03650e7	0.333577
$994$	−1.23606e8	−3.96803
$995$	0	0
$996$	−4.47529e7	−1.42946
$997$	−2.06293e7	−0.657273	−0.328636	−	0.944457i	$-0.606589\pi$
$997$	−2.06293e7	−0.657273	−0.328636	+	0.944457i	$0.606589\pi$
$998$	1.07584e7	0.341919
$999$	8.29487e6	0.262964

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.p.1.2	✓	8
5.4	even	2		825.6.a.q.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.2	✓	8	1.1	even	1	trivial
825.6.a.q.1.7	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)

\(f(q)\)	\(=\)	\(q-8.96197 q^{2} +9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} -191.454 q^{7} -146.231 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-8.96197 q^{2} +9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} -191.454 q^{7} -146.231 q^{8} +81.0000 q^{9} -121.000 q^{11} +434.852 q^{12} +257.481 q^{13} +1715.80 q^{14} -235.621 q^{16} +1247.35 q^{17} -725.919 q^{18} -2954.73 q^{19} -1723.09 q^{21} +1084.40 q^{22} -374.592 q^{23} -1316.08 q^{24} -2307.53 q^{26} +729.000 q^{27} -9250.46 q^{28} +4573.39 q^{29} +2523.66 q^{31} +6791.02 q^{32} -1089.00 q^{33} -11178.7 q^{34} +3913.66 q^{36} +11378.4 q^{37} +26480.2 q^{38} +2317.32 q^{39} -7607.66 q^{41} +15442.2 q^{42} -853.530 q^{43} -5846.34 q^{44} +3357.08 q^{46} +18520.4 q^{47} -2120.59 q^{48} +19847.7 q^{49} +11226.2 q^{51} +12440.6 q^{52} -15523.8 q^{53} -6533.27 q^{54} +27996.5 q^{56} -26592.6 q^{57} -40986.5 q^{58} -13932.5 q^{59} +21313.0 q^{61} -22617.0 q^{62} -15507.8 q^{63} -53321.0 q^{64} +9759.58 q^{66} +55365.4 q^{67} +60268.1 q^{68} -3371.33 q^{69} -72039.8 q^{71} -11844.7 q^{72} -57248.2 q^{73} -101973. q^{74} -142763. q^{76} +23165.9 q^{77} -20767.8 q^{78} +75155.7 q^{79} +6561.00 q^{81} +68179.6 q^{82} -102915. q^{83} -83254.1 q^{84} +7649.30 q^{86} +41160.5 q^{87} +17694.0 q^{88} +11387.8 q^{89} -49295.7 q^{91} -18099.1 q^{92} +22712.9 q^{93} -165980. q^{94} +61119.2 q^{96} -23382.2 q^{97} -177874. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) \) 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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