LMFDB - Embedded newform 825.6.a.q.1.5

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			1.5
Root			$0.894159$ 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)

$f(q)$	$=$	$q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} -121.000 q^{11} +255.709 q^{12} -536.985 q^{13} -363.654 q^{14} +692.440 q^{16} +1324.89 q^{17} +153.427 q^{18} -912.564 q^{19} +1727.88 q^{21} -229.193 q^{22} +4124.56 q^{23} +1029.87 q^{24} -1017.14 q^{26} -729.000 q^{27} +5454.77 q^{28} +340.490 q^{29} +4697.47 q^{31} +4973.36 q^{32} +1089.00 q^{33} +2509.55 q^{34} -2301.38 q^{36} +8279.99 q^{37} -1728.54 q^{38} +4832.86 q^{39} -17495.0 q^{41} +3272.89 q^{42} -5947.60 q^{43} +3437.87 q^{44} +7812.57 q^{46} -628.580 q^{47} -6231.96 q^{48} +20052.0 q^{49} -11924.0 q^{51} +15256.9 q^{52} +16952.3 q^{53} -1380.84 q^{54} +21969.1 q^{56} +8213.08 q^{57} +644.942 q^{58} -28406.8 q^{59} +51510.2 q^{61} +8897.76 q^{62} -15550.9 q^{63} -12737.7 q^{64} +2062.74 q^{66} -15258.0 q^{67} -37643.0 q^{68} -37121.0 q^{69} +38398.8 q^{71} -9268.85 q^{72} -40463.3 q^{73} +15683.6 q^{74} +25927.9 q^{76} +23230.4 q^{77} +9154.22 q^{78} -27358.8 q^{79} +6561.00 q^{81} -33138.2 q^{82} +51818.0 q^{83} -49092.9 q^{84} -11265.7 q^{86} -3064.41 q^{87} +13846.1 q^{88} +20215.2 q^{89} +103094. q^{91} -117188. q^{92} -42277.2 q^{93} -1190.63 q^{94} -44760.2 q^{96} +166901. q^{97} +37981.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.89416	0.334843	0.167422	−	0.985885i	$-0.446456\pi$
$2$	1.89416	0.334843	0.167422	+	0.985885i	$0.446456\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−28.4122	−0.887880
$5$	0	0
$5$	0	0
$6$	−17.0474	−0.193322
$7$	−191.987	−1.48090	−0.740452	−	0.672109i	$-0.765388\pi$
$7$	−191.987	−1.48090	−0.740452	+	0.672109i	$0.765388\pi$
$8$	−114.430	−0.632144
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	255.709	0.512618
$13$	−536.985	−0.881259	−0.440630	−	0.897689i	$-0.645245\pi$
$13$	−536.985	−0.881259	−0.440630	+	0.897689i	$0.645245\pi$
$14$	−363.654	−0.495871
$15$	0	0
$16$	692.440	0.676211
$17$	1324.89	1.11188	0.555940	−	0.831223i	$-0.312359\pi$
$17$	1324.89	1.11188	0.555940	+	0.831223i	$0.312359\pi$
$18$	153.427	0.111614
$19$	−912.564	−0.579935	−0.289968	−	0.957036i	$-0.593645\pi$
$19$	−912.564	−0.579935	−0.289968	+	0.957036i	$0.593645\pi$
$20$	0	0
$21$	1727.88	0.855000
$22$	−229.193	−0.100959
$23$	4124.56	1.62577	0.812883	−	0.582427i	$-0.197897\pi$
$23$	4124.56	1.62577	0.812883	+	0.582427i	$0.197897\pi$
$24$	1029.87	0.364968
$25$	0	0
$26$	−1017.14	−0.295084
$27$	−729.000	−0.192450
$28$	5454.77	1.31486
$29$	340.490	0.0751811	0.0375906	−	0.999293i	$-0.488032\pi$
$29$	340.490	0.0751811	0.0375906	+	0.999293i	$0.488032\pi$
$30$	0	0
$31$	4697.47	0.877930	0.438965	−	0.898504i	$-0.355345\pi$
$31$	4697.47	0.877930	0.438965	+	0.898504i	$0.355345\pi$
$32$	4973.36	0.858568
$33$	1089.00	0.174078
$34$	2509.55	0.372305
$35$	0	0
$36$	−2301.38	−0.295960
$37$	8279.99	0.994318	0.497159	−	0.867659i	$-0.334377\pi$
$37$	8279.99	0.994318	0.497159	+	0.867659i	$0.334377\pi$
$38$	−1728.54	−0.194187
$39$	4832.86	0.508795
$40$	0	0
$41$	−17495.0	−1.62537	−0.812687	−	0.582701i	$-0.801996\pi$
$41$	−17495.0	−1.62537	−0.812687	+	0.582701i	$0.801996\pi$
$42$	3272.89	0.286291
$43$	−5947.60	−0.490535	−0.245268	−	0.969455i	$-0.578876\pi$
$43$	−5947.60	−0.490535	−0.245268	+	0.969455i	$0.578876\pi$
$44$	3437.87	0.267706
$45$	0	0
$46$	7812.57	0.544377
$47$	−628.580	−0.0415065	−0.0207532	−	0.999785i	$-0.506606\pi$
$47$	−628.580	−0.0415065	−0.0207532	+	0.999785i	$0.506606\pi$
$48$	−6231.96	−0.390411
$49$	20052.0	1.19308
$50$	0	0
$51$	−11924.0	−0.641944
$52$	15256.9	0.782453
$53$	16952.3	0.828969	0.414485	−	0.910056i	$-0.363962\pi$
$53$	16952.3	0.828969	0.414485	+	0.910056i	$0.363962\pi$
$54$	−1380.84	−0.0644406
$55$	0	0
$56$	21969.1	0.936144
$57$	8213.08	0.334826
$58$	644.942	0.0251739
$59$	−28406.8	−1.06241	−0.531204	−	0.847244i	$-0.678260\pi$
$59$	−28406.8	−1.06241	−0.531204	+	0.847244i	$0.678260\pi$
$60$	0	0
$61$	51510.2	1.77243	0.886215	−	0.463275i	$-0.153326\pi$
$61$	51510.2	1.77243	0.886215	+	0.463275i	$0.153326\pi$
$62$	8897.76	0.293969
$63$	−15550.9	−0.493635
$64$	−12737.7	−0.388725
$65$	0	0
$66$	2062.74	0.0582887
$67$	−15258.0	−0.415251	−0.207625	−	0.978208i	$-0.566573\pi$
$67$	−15258.0	−0.415251	−0.207625	+	0.978208i	$0.566573\pi$
$68$	−37643.0	−0.987215
$69$	−37121.0	−0.938636
$70$	0	0
$71$	38398.8	0.904007	0.452004	−	0.892016i	$-0.350709\pi$
$71$	38398.8	0.904007	0.452004	+	0.892016i	$0.350709\pi$
$72$	−9268.85	−0.210715
$73$	−40463.3	−0.888699	−0.444349	−	0.895854i	$-0.646565\pi$
$73$	−40463.3	−0.888699	−0.444349	+	0.895854i	$0.646565\pi$
$74$	15683.6	0.332941
$75$	0	0
$76$	25927.9	0.514913
$77$	23230.4	0.446509
$78$	9154.22	0.170367
$79$	−27358.8	−0.493207	−0.246604	−	0.969116i	$-0.579315\pi$
$79$	−27358.8	−0.493207	−0.246604	+	0.969116i	$0.579315\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−33138.2	−0.544245
$83$	51818.0	0.825631	0.412815	−	0.910815i	$-0.364545\pi$
$83$	51818.0	0.825631	0.412815	+	0.910815i	$0.364545\pi$
$84$	−49092.9	−0.759138
$85$	0	0
$86$	−11265.7	−0.164252
$87$	−3064.41	−0.0434058
$88$	13846.1	0.190599
$89$	20215.2	0.270523	0.135261	−	0.990810i	$-0.456813\pi$
$89$	20215.2	0.270523	0.135261	+	0.990810i	$0.456813\pi$
$90$	0	0
$91$	103094.	1.30506
$92$	−117188.	−1.44348
$93$	−42277.2	−0.506873
$94$	−1190.63	−0.0138982
$95$	0	0
$96$	−44760.2	−0.495695
$97$	166901.	1.80106	0.900531	−	0.434791i	$-0.143178\pi$
$97$	166901.	1.80106	0.900531	+	0.434791i	$0.143178\pi$
$98$	37981.7	0.399493
$99$	−9801.00	−0.100504
$100$	0	0
$101$	18494.4	0.180400	0.0901999	−	0.995924i	$-0.471249\pi$
$101$	18494.4	0.180400	0.0901999	+	0.995924i	$0.471249\pi$
$102$	−22586.0	−0.214951
$103$	−88991.8	−0.826527	−0.413263	−	0.910612i	$-0.635611\pi$
$103$	−88991.8	−0.826527	−0.413263	+	0.910612i	$0.635611\pi$
$104$	61447.3	0.557083
$105$	0	0
$106$	32110.3	0.277575
$107$	−112300.	−0.948248	−0.474124	−	0.880458i	$-0.657235\pi$
$107$	−112300.	−0.948248	−0.474124	+	0.880458i	$0.657235\pi$
$108$	20712.5	0.170873
$109$	−194142.	−1.56514	−0.782569	−	0.622564i	$-0.786091\pi$
$109$	−194142.	−1.56514	−0.782569	+	0.622564i	$0.786091\pi$
$110$	0	0
$111$	−74519.9	−0.574070
$112$	−132940.	−1.00140
$113$	4270.92	0.0314648	0.0157324	−	0.999876i	$-0.494992\pi$
$113$	4270.92	0.0314648	0.0157324	+	0.999876i	$0.494992\pi$
$114$	15556.9	0.112114
$115$	0	0
$116$	−9674.05	−0.0667518
$117$	−43495.8	−0.293753
$118$	−53806.9	−0.355740
$119$	−254362.	−1.64659
$120$	0	0
$121$	14641.0	0.0909091
$122$	97568.6	0.593486
$123$	157455.	0.938410
$124$	−133465.	−0.779496
$125$	0	0
$126$	−29456.0	−0.165290
$127$	338246.	1.86090	0.930452	−	0.366414i	$-0.119415\pi$
$127$	338246.	1.86090	0.930452	+	0.366414i	$0.119415\pi$
$128$	−183275.	−0.988730
$129$	53528.4	0.283211
$130$	0	0
$131$	−113323.	−0.576950	−0.288475	−	0.957487i	$-0.593148\pi$
$131$	−113323.	−0.576950	−0.288475	+	0.957487i	$0.593148\pi$
$132$	−30940.8	−0.154560
$133$	175201.	0.858828
$134$	−28901.1	−0.139044
$135$	0	0
$136$	−151608.	−0.702868
$137$	−28035.0	−0.127614	−0.0638072	−	0.997962i	$-0.520324\pi$
$137$	−28035.0	−0.127614	−0.0638072	+	0.997962i	$0.520324\pi$
$138$	−70313.1	−0.314296
$139$	−59383.8	−0.260694	−0.130347	−	0.991468i	$-0.541609\pi$
$139$	−59383.8	−0.260694	−0.130347	+	0.991468i	$0.541609\pi$
$140$	0	0
$141$	5657.22	0.0239638
$142$	72733.5	0.302701
$143$	64975.2	0.265710
$144$	56087.6	0.225404
$145$	0	0
$146$	−76644.0	−0.297575
$147$	−180468.	−0.688823
$148$	−235252.	−0.882835
$149$	−226515.	−0.835855	−0.417928	−	0.908480i	$-0.637243\pi$
$149$	−226515.	−0.835855	−0.417928	+	0.908480i	$0.637243\pi$
$150$	0	0
$151$	231802.	0.827323	0.413661	−	0.910431i	$-0.364250\pi$
$151$	231802.	0.827323	0.413661	+	0.910431i	$0.364250\pi$
$152$	104425.	0.366602
$153$	107316.	0.370626
$154$	44002.1	0.149511
$155$	0	0
$156$	−137312.	−0.451749
$157$	196398.	0.635897	0.317949	−	0.948108i	$-0.397006\pi$
$157$	196398.	0.635897	0.317949	+	0.948108i	$0.397006\pi$
$158$	−51822.0	−0.165147
$159$	−152570.	−0.478606
$160$	0	0
$161$	−791862.	−2.40760
$162$	12427.6	0.0372048
$163$	604868.	1.78317	0.891583	−	0.452857i	$-0.149595\pi$
$163$	604868.	1.78317	0.891583	+	0.452857i	$0.149595\pi$
$164$	497069.	1.44314
$165$	0	0
$166$	98151.6	0.276457
$167$	−140409.	−0.389587	−0.194793	−	0.980844i	$-0.562404\pi$
$167$	−140409.	−0.389587	−0.194793	+	0.980844i	$0.562404\pi$
$168$	−197722.	−0.540483
$169$	−82940.2	−0.223382
$170$	0	0
$171$	−73917.7	−0.193312
$172$	168984.	0.435537
$173$	−352914.	−0.896506	−0.448253	−	0.893907i	$-0.647954\pi$
$173$	−352914.	−0.896506	−0.448253	+	0.893907i	$0.647954\pi$
$174$	−5804.47	−0.0145342
$175$	0	0
$176$	−83785.2	−0.203885
$177$	255661.	0.613382
$178$	38290.9	0.0905827
$179$	−784162.	−1.82925	−0.914625	−	0.404303i	$-0.867514\pi$
$179$	−784162.	−1.82925	−0.914625	+	0.404303i	$0.867514\pi$
$180$	0	0
$181$	−2532.32	−0.00574542	−0.00287271	−	0.999996i	$-0.500914\pi$
$181$	−2532.32	−0.00574542	−0.00287271	+	0.999996i	$0.500914\pi$
$182$	195277.	0.436991
$183$	−463592.	−1.02331
$184$	−471974.	−1.02772
$185$	0	0
$186$	−80079.8	−0.169723
$187$	−160312.	−0.335244
$188$	17859.3	0.0368527
$189$	139959.	0.285000
$190$	0	0
$191$	185241.	0.367411	0.183706	−	0.982981i	$-0.441191\pi$
$191$	185241.	0.367411	0.183706	+	0.982981i	$0.441191\pi$
$192$	114640.	0.224431
$193$	−523822.	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$193$	−523822.	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$194$	316136.	0.603074
$195$	0	0
$196$	−569721.	−1.05931
$197$	166997.	0.306580	0.153290	−	0.988181i	$-0.451013\pi$
$197$	166997.	0.306580	0.153290	+	0.988181i	$0.451013\pi$
$198$	−18564.7	−0.0336530
$199$	−715314.	−1.28045	−0.640227	−	0.768186i	$-0.721160\pi$
$199$	−715314.	−1.28045	−0.640227	+	0.768186i	$0.721160\pi$
$200$	0	0
$201$	137322.	0.239745
$202$	35031.3	0.0604056
$203$	−65369.6	−0.111336
$204$	338787.	0.569969
$205$	0	0
$206$	−168565.	−0.276757
$207$	334089.	0.541922
$208$	−371830.	−0.595917
$209$	110420.	0.174857
$210$	0	0
$211$	917870.	1.41930	0.709651	−	0.704553i	$-0.248852\pi$
$211$	917870.	1.41930	0.709651	+	0.704553i	$0.248852\pi$
$212$	−481651.	−0.736025
$213$	−345589.	−0.521929
$214$	−212715.	−0.317515
$215$	0	0
$216$	83419.7	0.121656
$217$	−901854.	−1.30013
$218$	−367735.	−0.524076
$219$	364170.	0.513090
$220$	0	0
$221$	−711446.	−0.979854
$222$	−141153.	−0.192223
$223$	65289.2	0.0879182	0.0439591	−	0.999033i	$-0.486003\pi$
$223$	65289.2	0.0879182	0.0439591	+	0.999033i	$0.486003\pi$
$224$	−954821.	−1.27146
$225$	0	0
$226$	8089.80	0.0105358
$227$	−558110.	−0.718879	−0.359439	−	0.933168i	$-0.617032\pi$
$227$	−558110.	−0.718879	−0.359439	+	0.933168i	$0.617032\pi$
$228$	−233351.	−0.297285
$229$	980050.	1.23498	0.617490	−	0.786579i	$-0.288150\pi$
$229$	980050.	1.23498	0.617490	+	0.786579i	$0.288150\pi$
$230$	0	0
$231$	−209074.	−0.257792
$232$	−38962.3	−0.0475253
$233$	−798554.	−0.963640	−0.481820	−	0.876270i	$-0.660024\pi$
$233$	−798554.	−0.963640	−0.481820	+	0.876270i	$0.660024\pi$
$234$	−82387.9	−0.0983612
$235$	0	0
$236$	807097.	0.943292
$237$	246229.	0.284753
$238$	−481802.	−0.551348
$239$	−1.31819e6	−1.49274	−0.746369	−	0.665532i	$-0.768205\pi$
$239$	−1.31819e6	−1.49274	−0.746369	+	0.665532i	$0.768205\pi$
$240$	0	0
$241$	−1.07597e6	−1.19332	−0.596660	−	0.802494i	$-0.703506\pi$
$241$	−1.07597e6	−1.19332	−0.596660	+	0.802494i	$0.703506\pi$
$242$	27732.4	0.0304403
$243$	−59049.0	−0.0641500
$244$	−1.46352e6	−1.57370
$245$	0	0
$246$	298244.	0.314220
$247$	490033.	0.511073
$248$	−537533.	−0.554978
$249$	−466362.	−0.476678
$250$	0	0
$251$	−814105.	−0.815635	−0.407818	−	0.913063i	$-0.633710\pi$
$251$	−814105.	−0.815635	−0.407818	+	0.913063i	$0.633710\pi$
$252$	441836.	0.438288
$253$	−499072.	−0.490187
$254$	640693.	0.623111
$255$	0	0
$256$	60456.1	0.0576554
$257$	1.89491e6	1.78960	0.894801	−	0.446466i	$-0.147318\pi$
$257$	1.89491e6	1.78960	0.894801	+	0.446466i	$0.147318\pi$
$258$	101391.	0.0948312
$259$	−1.58965e6	−1.47249
$260$	0	0
$261$	27579.7	0.0250604
$262$	−214651.	−0.193188
$263$	−1.39679e6	−1.24521	−0.622603	−	0.782538i	$-0.713925\pi$
$263$	−1.39679e6	−1.24521	−0.622603	+	0.782538i	$0.713925\pi$
$264$	−124615.	−0.110042
$265$	0	0
$266$	331858.	0.287573
$267$	−181937.	−0.156186
$268$	433513.	0.368693
$269$	1.06051e6	0.893585	0.446793	−	0.894638i	$-0.352566\pi$
$269$	1.06051e6	0.893585	0.446793	+	0.894638i	$0.352566\pi$
$270$	0	0
$271$	−710526.	−0.587702	−0.293851	−	0.955851i	$-0.594937\pi$
$271$	−710526.	−0.587702	−0.293851	+	0.955851i	$0.594937\pi$
$272$	917407.	0.751865
$273$	−927847.	−0.753477
$274$	−53102.8	−0.0427308
$275$	0	0
$276$	1.05469e6	0.833396
$277$	−678042.	−0.530955	−0.265477	−	0.964117i	$-0.585530\pi$
$277$	−678042.	−0.530955	−0.265477	+	0.964117i	$0.585530\pi$
$278$	−112482.	−0.0872917
$279$	380495.	0.292643
$280$	0	0
$281$	527403.	0.398453	0.199226	−	0.979953i	$-0.436157\pi$
$281$	527403.	0.398453	0.199226	+	0.979953i	$0.436157\pi$
$282$	10715.7	0.00802410
$283$	563399.	0.418167	0.209084	−	0.977898i	$-0.432952\pi$
$283$	563399.	0.418167	0.209084	+	0.977898i	$0.432952\pi$
$284$	−1.09099e6	−0.802650
$285$	0	0
$286$	123073.	0.0889711
$287$	3.35880e6	2.40702
$288$	402842.	0.286189
$289$	335478.	0.236276
$290$	0	0
$291$	−1.50211e6	−1.03984
$292$	1.14965e6	0.789058
$293$	1.31300e6	0.893505	0.446753	−	0.894657i	$-0.352580\pi$
$293$	1.31300e6	0.893505	0.446753	+	0.894657i	$0.352580\pi$
$294$	−341836.	−0.230648
$295$	0	0
$296$	−947481.	−0.628552
$297$	88209.0	0.0580259
$298$	−429055.	−0.279880
$299$	−2.21483e6	−1.43272
$300$	0	0
$301$	1.14186e6	0.726436
$302$	439070.	0.277023
$303$	−166449.	−0.104154
$304$	−631896.	−0.392159
$305$	0	0
$306$	203274.	0.124102
$307$	2.59454e6	1.57114	0.785570	−	0.618773i	$-0.212370\pi$
$307$	2.59454e6	1.57114	0.785570	+	0.618773i	$0.212370\pi$
$308$	−660027.	−0.396447
$309$	800926.	0.477196
$310$	0	0
$311$	−68979.7	−0.0404408	−0.0202204	−	0.999796i	$-0.506437\pi$
$311$	−68979.7	−0.0404408	−0.0202204	+	0.999796i	$0.506437\pi$
$312$	−553026.	−0.321632
$313$	2.10415e6	1.21399	0.606996	−	0.794705i	$-0.292374\pi$
$313$	2.10415e6	1.21399	0.606996	+	0.794705i	$0.292374\pi$
$314$	372008.	0.212926
$315$	0	0
$316$	777323.	0.437909
$317$	3.32293e6	1.85726	0.928632	−	0.371003i	$-0.120986\pi$
$317$	3.32293e6	1.85726	0.928632	+	0.371003i	$0.120986\pi$
$318$	−288993.	−0.160258
$319$	−41199.2	−0.0226680
$320$	0	0
$321$	1.01070e6	0.547472
$322$	−1.49991e6	−0.806169
$323$	−1.20905e6	−0.644818
$324$	−186412.	−0.0986533
$325$	0	0
$326$	1.14572e6	0.597081
$327$	1.74728e6	0.903633
$328$	2.00195e6	1.02747
$329$	120679.	0.0614671
$330$	0	0
$331$	−2.65460e6	−1.33177	−0.665886	−	0.746054i	$-0.731946\pi$
$331$	−2.65460e6	−1.33177	−0.665886	+	0.746054i	$0.731946\pi$
$332$	−1.47226e6	−0.733061
$333$	670679.	0.331439
$334$	−265957.	−0.130451
$335$	0	0
$336$	1.19646e6	0.578160
$337$	1.56818e6	0.752179	0.376089	−	0.926583i	$-0.377269\pi$
$337$	1.56818e6	0.752179	0.376089	+	0.926583i	$0.377269\pi$
$338$	−157102.	−0.0747980
$339$	−38438.3	−0.0181662
$340$	0	0
$341$	−568394.	−0.264706
$342$	−140012.	−0.0647291
$343$	−623002.	−0.285926
$344$	680585.	0.310089
$345$	0	0
$346$	−668475.	−0.300189
$347$	−3.47302e6	−1.54840	−0.774202	−	0.632939i	$-0.781848\pi$
$347$	−3.47302e6	−1.54840	−0.774202	+	0.632939i	$0.781848\pi$
$348$	87066.4	0.0385392
$349$	1.54641e6	0.679611	0.339805	−	0.940496i	$-0.389639\pi$
$349$	1.54641e6	0.679611	0.339805	+	0.940496i	$0.389639\pi$
$350$	0	0
$351$	391462.	0.169598
$352$	−601777.	−0.258868
$353$	1.08681e6	0.464214	0.232107	−	0.972690i	$-0.425438\pi$
$353$	1.08681e6	0.464214	0.232107	+	0.972690i	$0.425438\pi$
$354$	484262.	0.205387
$355$	0	0
$356$	−574358.	−0.240192
$357$	2.28926e6	0.950657
$358$	−1.48533e6	−0.612512
$359$	−1.38446e6	−0.566948	−0.283474	−	0.958980i	$-0.591487\pi$
$359$	−1.38446e6	−0.566948	−0.283474	+	0.958980i	$0.591487\pi$
$360$	0	0
$361$	−1.64333e6	−0.663675
$362$	−4796.62	−0.00192382
$363$	−131769.	−0.0524864
$364$	−2.92913e6	−1.15874
$365$	0	0
$366$	−878117.	−0.342649
$367$	−3.09462e6	−1.19934	−0.599669	−	0.800248i	$-0.704701\pi$
$367$	−3.09462e6	−1.19934	−0.599669	+	0.800248i	$0.704701\pi$
$368$	2.85601e6	1.09936
$369$	−1.41709e6	−0.541791
$370$	0	0
$371$	−3.25462e6	−1.22762
$372$	1.20119e6	0.450043
$373$	−3.81239e6	−1.41881	−0.709407	−	0.704799i	$-0.751037\pi$
$373$	−3.81239e6	−1.41881	−0.709407	+	0.704799i	$0.751037\pi$
$374$	−303656.	−0.112254
$375$	0	0
$376$	71928.5	0.0262380
$377$	−182838.	−0.0662541
$378$	265104.	0.0954303
$379$	−1.79432e6	−0.641656	−0.320828	−	0.947138i	$-0.603961\pi$
$379$	−1.79432e6	−0.641656	−0.320828	+	0.947138i	$0.603961\pi$
$380$	0	0
$381$	−3.04422e6	−1.07439
$382$	350875.	0.123025
$383$	1.79578e6	0.625541	0.312771	−	0.949829i	$-0.398743\pi$
$383$	1.79578e6	0.625541	0.312771	+	0.949829i	$0.398743\pi$
$384$	1.64947e6	0.570844
$385$	0	0
$386$	−992202.	−0.338947
$387$	−481755.	−0.163512
$388$	−4.74201e6	−1.59913
$389$	3.20505e6	1.07389	0.536946	−	0.843617i	$-0.319578\pi$
$389$	3.20505e6	1.07389	0.536946	+	0.843617i	$0.319578\pi$
$390$	0	0
$391$	5.46459e6	1.80765
$392$	−2.29456e6	−0.754195
$393$	1.01990e6	0.333102
$394$	316319.	0.102656
$395$	0	0
$396$	278468.	0.0892353
$397$	5.06822e6	1.61391	0.806954	−	0.590614i	$-0.201114\pi$
$397$	5.06822e6	1.61391	0.806954	+	0.590614i	$0.201114\pi$
$398$	−1.35492e6	−0.428751
$399$	−1.57680e6	−0.495845
$400$	0	0
$401$	−5.04243e6	−1.56595	−0.782977	−	0.622051i	$-0.786300\pi$
$401$	−5.04243e6	−1.56595	−0.782977	+	0.622051i	$0.786300\pi$
$402$	260110.	0.0802770
$403$	−2.52247e6	−0.773684
$404$	−525465.	−0.160173
$405$	0	0
$406$	−123820.	−0.0372801
$407$	−1.00188e6	−0.299798
$408$	1.36447e6	0.405801
$409$	−6.11825e6	−1.80850	−0.904252	−	0.427000i	$-0.859570\pi$
$409$	−6.11825e6	−1.80850	−0.904252	+	0.427000i	$0.859570\pi$
$410$	0	0
$411$	252315.	0.0736782
$412$	2.52845e6	0.733857
$413$	5.45373e6	1.57333
$414$	632818.	0.181459
$415$	0	0
$416$	−2.67062e6	−0.756621
$417$	534454.	0.150512
$418$	209154.	0.0585497
$419$	6.32243e6	1.75934	0.879669	−	0.475586i	$-0.157764\pi$
$419$	6.32243e6	1.75934	0.879669	+	0.475586i	$0.157764\pi$
$420$	0	0
$421$	1.04093e6	0.286232	0.143116	−	0.989706i	$-0.454288\pi$
$421$	1.04093e6	0.286232	0.143116	+	0.989706i	$0.454288\pi$
$422$	1.73859e6	0.475244
$423$	−50914.9	−0.0138355
$424$	−1.93985e6	−0.524028
$425$	0	0
$426$	−654601.	−0.174764
$427$	−9.88930e6	−2.62480
$428$	3.19070e6	0.841931
$429$	−584777.	−0.153408
$430$	0	0
$431$	2.82045e6	0.731349	0.365674	−	0.930743i	$-0.380838\pi$
$431$	2.82045e6	0.731349	0.365674	+	0.930743i	$0.380838\pi$
$432$	−504789.	−0.130137
$433$	−2.15739e6	−0.552980	−0.276490	−	0.961017i	$-0.589171\pi$
$433$	−2.15739e6	−0.552980	−0.276490	+	0.961017i	$0.589171\pi$
$434$	−1.70825e6	−0.435340
$435$	0	0
$436$	5.51599e6	1.38966
$437$	−3.76392e6	−0.942839
$438$	689796.	0.171805
$439$	4.60164e6	1.13960	0.569798	−	0.821785i	$-0.307021\pi$
$439$	4.60164e6	1.13960	0.569798	+	0.821785i	$0.307021\pi$
$440$	0	0
$441$	1.62421e6	0.397692
$442$	−1.34759e6	−0.328097
$443$	5.03529e6	1.21903	0.609516	−	0.792774i	$-0.291364\pi$
$443$	5.03529e6	1.21903	0.609516	+	0.792774i	$0.291364\pi$
$444$	2.11727e6	0.509705
$445$	0	0
$446$	123668.	0.0294388
$447$	2.03863e6	0.482581
$448$	2.44548e6	0.575664
$449$	−6.46043e6	−1.51233	−0.756163	−	0.654383i	$-0.772928\pi$
$449$	−6.46043e6	−1.51233	−0.756163	+	0.654383i	$0.772928\pi$
$450$	0	0
$451$	2.11689e6	0.490068
$452$	−121346.	−0.0279370
$453$	−2.08622e6	−0.477655
$454$	−1.05715e6	−0.240712
$455$	0	0
$456$	−939825.	−0.211658
$457$	−4.15028e6	−0.929580	−0.464790	−	0.885421i	$-0.653870\pi$
$457$	−4.15028e6	−0.929580	−0.464790	+	0.885421i	$0.653870\pi$
$458$	1.85637e6	0.413524
$459$	−965845.	−0.213981
$460$	0	0
$461$	2.82145e6	0.618330	0.309165	−	0.951008i	$-0.399950\pi$
$461$	2.82145e6	0.618330	0.309165	+	0.951008i	$0.399950\pi$
$462$	−396019.	−0.0863200
$463$	−2.16321e6	−0.468971	−0.234486	−	0.972120i	$-0.575341\pi$
$463$	−2.16321e6	−0.468971	−0.234486	+	0.972120i	$0.575341\pi$
$464$	235769.	0.0508383
$465$	0	0
$466$	−1.51259e6	−0.322668
$467$	1.87682e6	0.398226	0.199113	−	0.979977i	$-0.436194\pi$
$467$	1.87682e6	0.398226	0.199113	+	0.979977i	$0.436194\pi$
$468$	1.23581e6	0.260818
$469$	2.92934e6	0.614946
$470$	0	0
$471$	−1.76758e6	−0.367135
$472$	3.25059e6	0.671595
$473$	719659.	0.147902
$474$	466398.	0.0953478
$475$	0	0
$476$	7.22697e6	1.46197
$477$	1.37313e6	0.276323
$478$	−2.49686e6	−0.499833
$479$	−4.84619e6	−0.965076	−0.482538	−	0.875875i	$-0.660285\pi$
$479$	−4.84619e6	−0.965076	−0.482538	+	0.875875i	$0.660285\pi$
$480$	0	0
$481$	−4.44623e6	−0.876252
$482$	−2.03806e6	−0.399575
$483$	7.12676e6	1.39003
$484$	−415982.	−0.0807164
$485$	0	0
$486$	−111848.	−0.0214802
$487$	−9.13800e6	−1.74594	−0.872969	−	0.487776i	$-0.837808\pi$
$487$	−9.13800e6	−1.74594	−0.872969	+	0.487776i	$0.837808\pi$
$488$	−5.89433e6	−1.12043
$489$	−5.44382e6	−1.02951
$490$	0	0
$491$	−5.14270e6	−0.962692	−0.481346	−	0.876531i	$-0.659852\pi$
$491$	−5.14270e6	−0.962692	−0.481346	+	0.876531i	$0.659852\pi$
$492$	−4.47362e6	−0.833195
$493$	451111.	0.0835923
$494$	928201.	0.171129
$495$	0	0
$496$	3.25272e6	0.593666
$497$	−7.37207e6	−1.33875
$498$	−883365.	−0.159612
$499$	2.18975e6	0.393679	0.196840	−	0.980436i	$-0.436932\pi$
$499$	2.18975e6	0.393679	0.196840	+	0.980436i	$0.436932\pi$
$500$	0	0
$501$	1.26368e6	0.224928
$502$	−1.54204e6	−0.273110
$503$	−2.77987e6	−0.489897	−0.244948	−	0.969536i	$-0.578771\pi$
$503$	−2.77987e6	−0.489897	−0.244948	+	0.969536i	$0.578771\pi$
$504$	1.77950e6	0.312048
$505$	0	0
$506$	−945321.	−0.164136
$507$	746462.	0.128970
$508$	−9.61031e6	−1.65226
$509$	885490.	0.151492	0.0757459	−	0.997127i	$-0.475866\pi$
$509$	885490.	0.151492	0.0757459	+	0.997127i	$0.475866\pi$
$510$	0	0
$511$	7.76843e6	1.31608
$512$	5.97931e6	1.00804
$513$	665259.	0.111609
$514$	3.58927e6	0.599236
$515$	0	0
$516$	−1.52086e6	−0.251457
$517$	76058.1	0.0125147
$518$	−3.01105e6	−0.493053
$519$	3.17622e6	0.517598
$520$	0	0
$521$	−3.25668e6	−0.525631	−0.262816	−	0.964846i	$-0.584651\pi$
$521$	−3.25668e6	−0.525631	−0.262816	+	0.964846i	$0.584651\pi$
$522$	52240.3	0.00839130
$523$	−2.14727e6	−0.343268	−0.171634	−	0.985161i	$-0.554905\pi$
$523$	−2.14727e6	−0.343268	−0.171634	+	0.985161i	$0.554905\pi$
$524$	3.21974e6	0.512263
$525$	0	0
$526$	−2.64574e6	−0.416949
$527$	6.22363e6	0.976152
$528$	754067.	0.117713
$529$	1.05756e7	1.64311
$530$	0	0
$531$	−2.30095e6	−0.354136
$532$	−4.97783e6	−0.762536
$533$	9.39453e6	1.43238
$534$	−344618.	−0.0522979
$535$	0	0
$536$	1.74598e6	0.262498
$537$	7.05746e6	1.05612
$538$	2.00878e6	0.299211
$539$	−2.42629e6	−0.359726
$540$	0	0
$541$	2.92326e6	0.429412	0.214706	−	0.976679i	$-0.431121\pi$
$541$	2.92326e6	0.429412	0.214706	+	0.976679i	$0.431121\pi$
$542$	−1.34585e6	−0.196788
$543$	22790.9	0.00331712
$544$	6.58916e6	0.954625
$545$	0	0
$546$	−1.75749e6	−0.252297
$547$	−9.05734e6	−1.29429	−0.647146	−	0.762366i	$-0.724038\pi$
$547$	−9.05734e6	−1.29429	−0.647146	+	0.762366i	$0.724038\pi$
$548$	796535.	0.113306
$549$	4.17233e6	0.590810
$550$	0	0
$551$	−310719.	−0.0436002
$552$	4.24777e6	0.593353
$553$	5.25254e6	0.730393
$554$	−1.28432e6	−0.177787
$555$	0	0
$556$	1.68722e6	0.231465
$557$	9.32565e6	1.27362	0.636812	−	0.771019i	$-0.280253\pi$
$557$	9.32565e6	1.27362	0.636812	+	0.771019i	$0.280253\pi$
$558$	720718.	0.0979896
$559$	3.19377e6	0.432289
$560$	0	0
$561$	1.44281e6	0.193553
$562$	998985.	0.133419
$563$	−1.27073e6	−0.168960	−0.0844798	−	0.996425i	$-0.526923\pi$
$563$	−1.27073e6	−0.168960	−0.0844798	+	0.996425i	$0.526923\pi$
$564$	−160734.	−0.0212769
$565$	0	0
$566$	1.06717e6	0.140020
$567$	−1.25963e6	−0.164545
$568$	−4.39399e6	−0.571463
$569$	7.32601e6	0.948608	0.474304	−	0.880361i	$-0.342700\pi$
$569$	7.32601e6	0.948608	0.474304	+	0.880361i	$0.342700\pi$
$570$	0	0
$571$	1.10876e7	1.42314	0.711569	−	0.702617i	$-0.247985\pi$
$571$	1.10876e7	1.42314	0.711569	+	0.702617i	$0.247985\pi$
$572$	−1.84609e6	−0.235918
$573$	−1.66717e6	−0.212125
$574$	6.36211e6	0.805975
$575$	0	0
$576$	−1.03176e6	−0.129575
$577$	−3.23610e6	−0.404653	−0.202327	−	0.979318i	$-0.564850\pi$
$577$	−3.23610e6	−0.404653	−0.202327	+	0.979318i	$0.564850\pi$
$578$	635448.	0.0791153
$579$	4.71440e6	0.584426
$580$	0	0
$581$	−9.94839e6	−1.22268
$582$	−2.84523e6	−0.348185
$583$	−2.05123e6	−0.249944
$584$	4.63023e6	0.561785
$585$	0	0
$586$	2.48704e6	0.299184
$587$	9.64747e6	1.15563	0.577814	−	0.816168i	$-0.303906\pi$
$587$	9.64747e6	1.15563	0.577814	+	0.816168i	$0.303906\pi$
$588$	5.12749e6	0.611592
$589$	−4.28674e6	−0.509142
$590$	0	0
$591$	−1.50298e6	−0.177004
$592$	5.73339e6	0.672369
$593$	120468.	0.0140681	0.00703406	−	0.999975i	$-0.497761\pi$
$593$	120468.	0.0140681	0.00703406	+	0.999975i	$0.497761\pi$
$594$	167082.	0.0194296
$595$	0	0
$596$	6.43578e6	0.742139
$597$	6.43783e6	0.739270
$598$	−4.19523e6	−0.479737
$599$	1.39130e7	1.58436	0.792178	−	0.610290i	$-0.208947\pi$
$599$	1.39130e7	1.58436	0.792178	+	0.610290i	$0.208947\pi$
$600$	0	0
$601$	6.27530e6	0.708677	0.354339	−	0.935117i	$-0.384706\pi$
$601$	6.27530e6	0.708677	0.354339	+	0.935117i	$0.384706\pi$
$602$	2.16287e6	0.243242
$603$	−1.23590e6	−0.138417
$604$	−6.58600e6	−0.734563
$605$	0	0
$606$	−315281.	−0.0348752
$607$	−7.85752e6	−0.865593	−0.432796	−	0.901492i	$-0.642473\pi$
$607$	−7.85752e6	−0.865593	−0.432796	+	0.901492i	$0.642473\pi$
$608$	−4.53851e6	−0.497914
$609$	588326.	0.0642799
$610$	0	0
$611$	337538.	0.0365779
$612$	−3.04908e6	−0.329072
$613$	1.19554e7	1.28503	0.642514	−	0.766274i	$-0.277891\pi$
$613$	1.19554e7	1.28503	0.642514	+	0.766274i	$0.277891\pi$
$614$	4.91447e6	0.526085
$615$	0	0
$616$	−2.65826e6	−0.282258
$617$	−8.19632e6	−0.866774	−0.433387	−	0.901208i	$-0.642682\pi$
$617$	−8.19632e6	−0.866774	−0.433387	+	0.901208i	$0.642682\pi$
$618$	1.51708e6	0.159786
$619$	−3.47785e6	−0.364824	−0.182412	−	0.983222i	$-0.558390\pi$
$619$	−3.47785e6	−0.364824	−0.182412	+	0.983222i	$0.558390\pi$
$620$	0	0
$621$	−3.00680e6	−0.312879
$622$	−130658.	−0.0135413
$623$	−3.88106e6	−0.400618
$624$	3.34647e6	0.344053
$625$	0	0
$626$	3.98560e6	0.406497
$627$	−993783.	−0.100954
$628$	−5.58008e6	−0.564601
$629$	1.09701e7	1.10556
$630$	0	0
$631$	1.07592e6	0.107574	0.0537868	−	0.998552i	$-0.482871\pi$
$631$	1.07592e6	0.107574	0.0537868	+	0.998552i	$0.482871\pi$
$632$	3.13068e6	0.311778
$633$	−8.26083e6	−0.819435
$634$	6.29416e6	0.621892
$635$	0	0
$636$	4.33486e6	0.424944
$637$	−1.07676e7	−1.05141
$638$	−78037.9	−0.00759021
$639$	3.11030e6	0.301336
$640$	0	0
$641$	1.30837e7	1.25772	0.628862	−	0.777517i	$-0.283521\pi$
$641$	1.30837e7	1.25772	0.628862	+	0.777517i	$0.283521\pi$
$642$	1.91443e6	0.183317
$643$	7.84966e6	0.748727	0.374363	−	0.927282i	$-0.377861\pi$
$643$	7.84966e6	0.748727	0.374363	+	0.927282i	$0.377861\pi$
$644$	2.24985e7	2.13766
$645$	0	0
$646$	−2.29013e6	−0.215913
$647$	−2.02801e7	−1.90463	−0.952313	−	0.305123i	$-0.901302\pi$
$647$	−2.02801e7	−1.90463	−0.952313	+	0.305123i	$0.901302\pi$
$648$	−750777.	−0.0702382
$649$	3.43722e6	0.320328
$650$	0	0
$651$	8.11668e6	0.750630
$652$	−1.71856e7	−1.58324
$653$	1.04023e7	0.954655	0.477328	−	0.878725i	$-0.341606\pi$
$653$	1.04023e7	0.954655	0.477328	+	0.878725i	$0.341606\pi$
$654$	3.30962e6	0.302575
$655$	0	0
$656$	−1.21142e7	−1.09910
$657$	−3.27753e6	−0.296233
$658$	228585.	0.0205818
$659$	−1.05552e7	−0.946785	−0.473392	−	0.880852i	$-0.656971\pi$
$659$	−1.05552e7	−0.946785	−0.473392	+	0.880852i	$0.656971\pi$
$660$	0	0
$661$	−4.20388e6	−0.374237	−0.187119	−	0.982337i	$-0.559915\pi$
$661$	−4.20388e6	−0.374237	−0.187119	+	0.982337i	$0.559915\pi$
$662$	−5.02824e6	−0.445935
$663$	6.40302e6	0.565719
$664$	−5.92955e6	−0.521917
$665$	0	0
$666$	1.27037e6	0.110980
$667$	1.40437e6	0.122227
$668$	3.98933e6	0.345906
$669$	−587602.	−0.0507596
$670$	0	0
$671$	−6.23274e6	−0.534407
$672$	8.59339e6	0.734076
$673$	1.41220e7	1.20187	0.600935	−	0.799298i	$-0.294795\pi$
$673$	1.41220e7	1.20187	0.600935	+	0.799298i	$0.294795\pi$
$674$	2.97038e6	0.251862
$675$	0	0
$676$	2.35651e6	0.198336
$677$	6.70088e6	0.561901	0.280951	−	0.959722i	$-0.409350\pi$
$677$	6.70088e6	0.561901	0.280951	+	0.959722i	$0.409350\pi$
$678$	−72808.2	−0.00608283
$679$	−3.20428e7	−2.66720
$680$	0	0
$681$	5.02299e6	0.415045
$682$	−1.07663e6	−0.0886350
$683$	−7.21441e6	−0.591765	−0.295882	−	0.955224i	$-0.595614\pi$
$683$	−7.21441e6	−0.591765	−0.295882	+	0.955224i	$0.595614\pi$
$684$	2.10016e6	0.171638
$685$	0	0
$686$	−1.18007e6	−0.0957405
$687$	−8.82045e6	−0.713016
$688$	−4.11835e6	−0.331705
$689$	−9.10312e6	−0.730537
$690$	0	0
$691$	−7.64521e6	−0.609108	−0.304554	−	0.952495i	$-0.598507\pi$
$691$	−7.64521e6	−0.609108	−0.304554	+	0.952495i	$0.598507\pi$
$692$	1.00270e7	0.795990
$693$	1.88166e6	0.148836
$694$	−6.57846e6	−0.518472
$695$	0	0
$696$	350661.	0.0274387
$697$	−2.31789e7	−1.80722
$698$	2.92914e6	0.227563
$699$	7.18699e6	0.556358
$700$	0	0
$701$	6.16637e6	0.473952	0.236976	−	0.971515i	$-0.423844\pi$
$701$	6.16637e6	0.473952	0.236976	+	0.971515i	$0.423844\pi$
$702$	741491.	0.0567889
$703$	−7.55602e6	−0.576640
$704$	1.54127e6	0.117205
$705$	0	0
$706$	2.05860e6	0.155439
$707$	−3.55068e6	−0.267155
$708$	−7.26388e6	−0.544610
$709$	−1.96108e7	−1.46514	−0.732571	−	0.680691i	$-0.761680\pi$
$709$	−1.96108e7	−1.46514	−0.732571	+	0.680691i	$0.761680\pi$
$710$	0	0
$711$	−2.21606e6	−0.164402
$712$	−2.31323e6	−0.171009
$713$	1.93750e7	1.42731
$714$	4.33622e6	0.318321
$715$	0	0
$716$	2.22797e7	1.62415
$717$	1.18637e7	0.861833
$718$	−2.62238e6	−0.189839
$719$	−8.44447e6	−0.609187	−0.304593	−	0.952483i	$-0.598521\pi$
$719$	−8.44447e6	−0.609187	−0.304593	+	0.952483i	$0.598521\pi$
$720$	0	0
$721$	1.70853e7	1.22401
$722$	−3.11272e6	−0.222227
$723$	9.68372e6	0.688964
$724$	71948.7	0.00510125
$725$	0	0
$726$	−249591.	−0.0175747
$727$	−2.39210e7	−1.67858	−0.839291	−	0.543682i	$-0.817030\pi$
$727$	−2.39210e7	−1.67858	−0.839291	+	0.543682i	$0.817030\pi$
$728$	−1.17971e7	−0.824986
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.87991e6	−0.545416
$732$	1.31717e7	0.908579
$733$	1.40807e7	0.967973	0.483986	−	0.875076i	$-0.339188\pi$
$733$	1.40807e7	0.967973	0.483986	+	0.875076i	$0.339188\pi$
$734$	−5.86170e6	−0.401590
$735$	0	0
$736$	2.05129e7	1.39583
$737$	1.84622e6	0.125203
$738$	−2.68420e6	−0.181415
$739$	−1.14610e7	−0.771987	−0.385993	−	0.922502i	$-0.626141\pi$
$739$	−1.14610e7	−0.771987	−0.385993	+	0.922502i	$0.626141\pi$
$740$	0	0
$741$	−4.41030e6	−0.295068
$742$	−6.16476e6	−0.411061
$743$	2.82841e7	1.87962	0.939810	−	0.341698i	$-0.111002\pi$
$743$	2.82841e7	1.87962	0.939810	+	0.341698i	$0.111002\pi$
$744$	4.83780e6	0.320417
$745$	0	0
$746$	−7.22128e6	−0.475081
$747$	4.19726e6	0.275210
$748$	4.55480e6	0.297657
$749$	2.15602e7	1.40426
$750$	0	0
$751$	−1.65142e7	−1.06846	−0.534228	−	0.845340i	$-0.679398\pi$
$751$	−1.65142e7	−1.06846	−0.534228	+	0.845340i	$0.679398\pi$
$752$	−435254.	−0.0280671
$753$	7.32694e6	0.470907
$754$	−346324.	−0.0221847
$755$	0	0
$756$	−3.97652e6	−0.253046
$757$	−1.68113e7	−1.06626	−0.533128	−	0.846034i	$-0.678984\pi$
$757$	−1.68113e7	−1.06626	−0.533128	+	0.846034i	$0.678984\pi$
$758$	−3.39873e6	−0.214854
$759$	4.49164e6	0.283009
$760$	0	0
$761$	391721.	0.0245197	0.0122598	−	0.999925i	$-0.496097\pi$
$761$	391721.	0.0245197	0.0122598	+	0.999925i	$0.496097\pi$
$762$	−5.76623e6	−0.359753
$763$	3.72727e7	2.31782
$764$	−5.26308e6	−0.326217
$765$	0	0
$766$	3.40149e6	0.209458
$767$	1.52540e7	0.936258
$768$	−544105.	−0.0332874
$769$	2.68596e7	1.63789	0.818944	−	0.573874i	$-0.194560\pi$
$769$	2.68596e7	1.63789	0.818944	+	0.573874i	$0.194560\pi$
$770$	0	0
$771$	−1.70542e7	−1.03323
$772$	1.48829e7	0.898762
$773$	−1.65258e7	−0.994748	−0.497374	−	0.867536i	$-0.665702\pi$
$773$	−1.65258e7	−0.994748	−0.497374	+	0.867536i	$0.665702\pi$
$774$	−912521.	−0.0547508
$775$	0	0
$776$	−1.90985e7	−1.13853
$777$	1.43069e7	0.850142
$778$	6.07087e6	0.359585
$779$	1.59653e7	0.942611
$780$	0	0
$781$	−4.64626e6	−0.272568
$782$	1.03508e7	0.605281
$783$	−248217.	−0.0144686
$784$	1.38848e7	0.806771
$785$	0	0
$786$	1.93186e6	0.111537
$787$	368901.	0.0212311	0.0106155	−	0.999944i	$-0.496621\pi$
$787$	368901.	0.0212311	0.0106155	+	0.999944i	$0.496621\pi$
$788$	−4.74475e6	−0.272206
$789$	1.25711e7	0.718921
$790$	0	0
$791$	−819961.	−0.0465963
$792$	1.12153e6	0.0635328
$793$	−2.76602e7	−1.56197
$794$	9.60001e6	0.540406
$795$	0	0
$796$	2.03236e7	1.13689
$797$	−1.03717e7	−0.578370	−0.289185	−	0.957273i	$-0.593384\pi$
$797$	−1.03717e7	−0.578370	−0.289185	+	0.957273i	$0.593384\pi$
$798$	−2.98672e6	−0.166030
$799$	−832799.	−0.0461502
$800$	0	0
$801$	1.63743e6	0.0901742
$802$	−9.55117e6	−0.524349
$803$	4.89606e6	0.267953
$804$	−3.90161e6	−0.212865
$805$	0	0
$806$	−4.77796e6	−0.259063
$807$	−9.54463e6	−0.515912
$808$	−2.11631e6	−0.114039
$809$	−2.51123e7	−1.34901	−0.674505	−	0.738271i	$-0.735643\pi$
$809$	−2.51123e7	−1.34901	−0.674505	+	0.738271i	$0.735643\pi$
$810$	0	0
$811$	−1.14510e7	−0.611350	−0.305675	−	0.952136i	$-0.598882\pi$
$811$	−1.14510e7	−0.611350	−0.305675	+	0.952136i	$0.598882\pi$
$812$	1.85729e6	0.0988530
$813$	6.39474e6	0.339310
$814$	−1.89772e6	−0.100385
$815$	0	0
$816$	−8.25666e6	−0.434089
$817$	5.42756e6	0.284479
$818$	−1.15889e7	−0.605565
$819$	8.35063e6	0.435020
$820$	0	0
$821$	−1.48065e7	−0.766646	−0.383323	−	0.923614i	$-0.625220\pi$
$821$	−1.48065e7	−0.766646	−0.383323	+	0.923614i	$0.625220\pi$
$822$	477925.	0.0246706
$823$	2.56659e6	0.132086	0.0660430	−	0.997817i	$-0.478963\pi$
$823$	2.56659e6	0.132086	0.0660430	+	0.997817i	$0.478963\pi$
$824$	1.01834e7	0.522484
$825$	0	0
$826$	1.03302e7	0.526817
$827$	3.42288e7	1.74032	0.870159	−	0.492772i	$-0.164016\pi$
$827$	3.42288e7	1.74032	0.870159	+	0.492772i	$0.164016\pi$
$828$	−9.49220e6	−0.481162
$829$	7.49591e6	0.378824	0.189412	−	0.981898i	$-0.439342\pi$
$829$	7.49591e6	0.378824	0.189412	+	0.981898i	$0.439342\pi$
$830$	0	0
$831$	6.10238e6	0.306547
$832$	6.83998e6	0.342568
$833$	2.65667e7	1.32656
$834$	1.01234e6	0.0503979
$835$	0	0
$836$	−3.13728e6	−0.155252
$837$	−3.42446e6	−0.168958
$838$	1.19757e7	0.589102
$839$	2.91814e7	1.43120	0.715601	−	0.698509i	$-0.246153\pi$
$839$	2.91814e7	1.43120	0.715601	+	0.698509i	$0.246153\pi$
$840$	0	0
$841$	−2.03952e7	−0.994348
$842$	1.97169e6	0.0958427
$843$	−4.74663e6	−0.230047
$844$	−2.60787e7	−1.26017
$845$	0	0
$846$	−96441.0	−0.00463272
$847$	−2.81088e6	−0.134628
$848$	1.17384e7	0.560558
$849$	−5.07059e6	−0.241429
$850$	0	0
$851$	3.41513e7	1.61653
$852$	9.81894e6	0.463410
$853$	−265247.	−0.0124818	−0.00624091	−	0.999981i	$-0.501987\pi$
$853$	−265247.	−0.0124818	−0.00624091	+	0.999981i	$0.501987\pi$
$854$	−1.87319e7	−0.878895
$855$	0	0
$856$	1.28506e7	0.599429
$857$	2.09223e7	0.973102	0.486551	−	0.873652i	$-0.338255\pi$
$857$	2.09223e7	0.973102	0.486551	+	0.873652i	$0.338255\pi$
$858$	−1.10766e6	−0.0513675
$859$	−3.16166e7	−1.46195	−0.730976	−	0.682404i	$-0.760935\pi$
$859$	−3.16166e7	−1.46195	−0.730976	+	0.682404i	$0.760935\pi$
$860$	0	0
$861$	−3.02292e7	−1.38969
$862$	5.34237e6	0.244887
$863$	−3.04614e7	−1.39227	−0.696133	−	0.717913i	$-0.745098\pi$
$863$	−3.04614e7	−1.39227	−0.696133	+	0.717913i	$0.745098\pi$
$864$	−3.62558e6	−0.165232
$865$	0	0
$866$	−4.08645e6	−0.185162
$867$	−3.01930e6	−0.136414
$868$	2.56236e7	1.15436
$869$	3.31042e6	0.148708
$870$	0	0
$871$	8.19331e6	0.365944
$872$	2.22157e7	0.989393
$873$	1.35190e7	0.600354
$874$	−7.12947e6	−0.315703
$875$	0	0
$876$	−1.03469e7	−0.455563
$877$	−3.21784e6	−0.141275	−0.0706376	−	0.997502i	$-0.522503\pi$
$877$	−3.21784e6	−0.141275	−0.0706376	+	0.997502i	$0.522503\pi$
$878$	8.71623e6	0.381586
$879$	−1.18170e7	−0.515866
$880$	0	0
$881$	−2.71105e7	−1.17679	−0.588394	−	0.808574i	$-0.700240\pi$
$881$	−2.71105e7	−1.17679	−0.588394	+	0.808574i	$0.700240\pi$
$882$	3.07652e6	0.133164
$883$	−7.92568e6	−0.342086	−0.171043	−	0.985264i	$-0.554714\pi$
$883$	−7.92568e6	−0.342086	−0.171043	+	0.985264i	$0.554714\pi$
$884$	2.02137e7	0.869993
$885$	0	0
$886$	9.53764e6	0.408185
$887$	−2.02221e7	−0.863011	−0.431506	−	0.902110i	$-0.642017\pi$
$887$	−2.02221e7	−0.863011	−0.431506	+	0.902110i	$0.642017\pi$
$888$	8.52733e6	0.362895
$889$	−6.49389e7	−2.75582
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−1.85501e6	−0.0780608
$893$	573619.	0.0240711
$894$	3.86150e6	0.161589
$895$	0	0
$896$	3.51864e7	1.46421
$897$	1.99334e7	0.827182
$898$	−1.22371e7	−0.506392
$899$	1.59944e6	0.0660038
$900$	0	0
$901$	2.24599e7	0.921714
$902$	4.00973e6	0.164096
$903$	−1.02768e7	−0.419408
$904$	−488722.	−0.0198903
$905$	0	0
$906$	−3.95163e6	−0.159940
$907$	−7.88692e6	−0.318338	−0.159169	−	0.987251i	$-0.550882\pi$
$907$	−7.88692e6	−0.318338	−0.159169	+	0.987251i	$0.550882\pi$
$908$	1.58571e7	0.638278
$909$	1.49804e6	0.0601332
$910$	0	0
$911$	−6.12818e6	−0.244644	−0.122322	−	0.992490i	$-0.539034\pi$
$911$	−6.12818e6	−0.244644	−0.122322	+	0.992490i	$0.539034\pi$
$912$	5.68706e6	0.226413
$913$	−6.26998e6	−0.248937
$914$	−7.86129e6	−0.311264
$915$	0	0
$916$	−2.78453e7	−1.09651
$917$	2.17565e7	0.854408
$918$	−1.82946e6	−0.0716502
$919$	2.38050e7	0.929779	0.464890	−	0.885369i	$-0.346094\pi$
$919$	2.38050e7	0.929779	0.464890	+	0.885369i	$0.346094\pi$
$920$	0	0
$921$	−2.33509e7	−0.907098
$922$	5.34428e6	0.207044
$923$	−2.06196e7	−0.796665
$924$	5.94024e6	0.228889
$925$	0	0
$926$	−4.09746e6	−0.157032
$927$	−7.20834e6	−0.275509
$928$	1.69338e6	0.0645481
$929$	3.89856e7	1.48206	0.741028	−	0.671474i	$-0.234339\pi$
$929$	3.89856e7	1.48206	0.741028	+	0.671474i	$0.234339\pi$
$930$	0	0
$931$	−1.82988e7	−0.691907
$932$	2.26887e7	0.855596
$933$	620817.	0.0233485
$934$	3.55499e6	0.133343
$935$	0	0
$936$	4.97723e6	0.185694
$937$	1.66079e7	0.617967	0.308983	−	0.951067i	$-0.400011\pi$
$937$	1.66079e7	0.617967	0.308983	+	0.951067i	$0.400011\pi$
$938$	5.54863e6	0.205911
$939$	−1.89374e7	−0.700899
$940$	0	0
$941$	−2.15988e7	−0.795161	−0.397581	−	0.917567i	$-0.630150\pi$
$941$	−2.15988e7	−0.795161	−0.397581	+	0.917567i	$0.630150\pi$
$942$	−3.34807e6	−0.122933
$943$	−7.21589e7	−2.64248
$944$	−1.96700e7	−0.718412
$945$	0	0
$946$	1.36315e6	0.0495240
$947$	−4.65688e7	−1.68741	−0.843705	−	0.536808i	$-0.819630\pi$
$947$	−4.65688e7	−1.68741	−0.843705	+	0.536808i	$0.819630\pi$
$948$	−6.99591e6	−0.252827
$949$	2.17282e7	0.783174
$950$	0	0
$951$	−2.99064e7	−1.07229
$952$	2.91067e7	1.04088
$953$	−2.71746e7	−0.969240	−0.484620	−	0.874725i	$-0.661042\pi$
$953$	−2.71746e7	−0.969240	−0.484620	+	0.874725i	$0.661042\pi$
$954$	2.60094e6	0.0925249
$955$	0	0
$956$	3.74527e7	1.32537
$957$	370793.	0.0130874
$958$	−9.17946e6	−0.323149
$959$	5.38236e6	0.188985
$960$	0	0
$961$	−6.56292e6	−0.229239
$962$	−8.42186e6	−0.293407
$963$	−9.09634e6	−0.316083
$964$	3.05706e7	1.05953
$965$	0	0
$966$	1.34992e7	0.465442
$967$	2.23871e6	0.0769897	0.0384948	−	0.999259i	$-0.487744\pi$
$967$	2.23871e6	0.0769897	0.0384948	+	0.999259i	$0.487744\pi$
$968$	−1.67537e6	−0.0574676
$969$	1.08814e7	0.372286
$970$	0	0
$971$	−1.25051e7	−0.425638	−0.212819	−	0.977092i	$-0.568264\pi$
$971$	−1.25051e7	−0.425638	−0.212819	+	0.977092i	$0.568264\pi$
$972$	1.67771e6	0.0569575
$973$	1.14009e7	0.386063
$974$	−1.73088e7	−0.584616
$975$	0	0
$976$	3.56677e7	1.19854
$977$	−5.79995e7	−1.94396	−0.971982	−	0.235056i	$-0.924473\pi$
$977$	−5.79995e7	−1.94396	−0.971982	+	0.235056i	$0.924473\pi$
$978$	−1.03115e7	−0.344725
$979$	−2.44604e6	−0.0815656
$980$	0	0
$981$	−1.57255e7	−0.521713
$982$	−9.74109e6	−0.322351
$983$	−3.40518e7	−1.12397	−0.561987	−	0.827146i	$-0.689963\pi$
$983$	−3.40518e7	−1.12397	−0.561987	+	0.827146i	$0.689963\pi$
$984$	−1.80176e7	−0.593210
$985$	0	0
$986$	854477.	0.0279903
$987$	−1.08611e6	−0.0354880
$988$	−1.39229e7	−0.453772
$989$	−2.45312e7	−0.797495
$990$	0	0
$991$	2.08858e7	0.675565	0.337782	−	0.941224i	$-0.390323\pi$
$991$	2.08858e7	0.675565	0.337782	+	0.941224i	$0.390323\pi$
$992$	2.33622e7	0.753763
$993$	2.38914e7	0.768899
$994$	−1.39639e7	−0.448271
$995$	0	0
$996$	1.32504e7	0.423233
$997$	−3.35610e7	−1.06929	−0.534646	−	0.845076i	$-0.679555\pi$
$997$	−3.35610e7	−1.06929	−0.534646	+	0.845076i	$0.679555\pi$
$998$	4.14773e6	0.131821
$999$	−6.03611e6	−0.191357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.q.1.5	yes	8
5.4	even	2		825.6.a.p.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.4	✓	8	5.4	even	2
825.6.a.q.1.5	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} -121.000 q^{11} +255.709 q^{12} -536.985 q^{13} -363.654 q^{14} +692.440 q^{16} +1324.89 q^{17} +153.427 q^{18} -912.564 q^{19} +1727.88 q^{21} -229.193 q^{22} +4124.56 q^{23} +1029.87 q^{24} -1017.14 q^{26} -729.000 q^{27} +5454.77 q^{28} +340.490 q^{29} +4697.47 q^{31} +4973.36 q^{32} +1089.00 q^{33} +2509.55 q^{34} -2301.38 q^{36} +8279.99 q^{37} -1728.54 q^{38} +4832.86 q^{39} -17495.0 q^{41} +3272.89 q^{42} -5947.60 q^{43} +3437.87 q^{44} +7812.57 q^{46} -628.580 q^{47} -6231.96 q^{48} +20052.0 q^{49} -11924.0 q^{51} +15256.9 q^{52} +16952.3 q^{53} -1380.84 q^{54} +21969.1 q^{56} +8213.08 q^{57} +644.942 q^{58} -28406.8 q^{59} +51510.2 q^{61} +8897.76 q^{62} -15550.9 q^{63} -12737.7 q^{64} +2062.74 q^{66} -15258.0 q^{67} -37643.0 q^{68} -37121.0 q^{69} +38398.8 q^{71} -9268.85 q^{72} -40463.3 q^{73} +15683.6 q^{74} +25927.9 q^{76} +23230.4 q^{77} +9154.22 q^{78} -27358.8 q^{79} +6561.00 q^{81} -33138.2 q^{82} +51818.0 q^{83} -49092.9 q^{84} -11265.7 q^{86} -3064.41 q^{87} +13846.1 q^{88} +20215.2 q^{89} +103094. q^{91} -117188. q^{92} -42277.2 q^{93} -1190.63 q^{94} -44760.2 q^{96} +166901. q^{97} +37981.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) 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

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