LMFDB - Embedded newform 825.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{313}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 78 $ x^2 - x - 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.34590$ 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)

$f(q)$	$=$	$q-9.34590 q^{2} +9.00000 q^{3} +55.3459 q^{4} -84.1131 q^{6} -8.69181 q^{7} -218.189 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.34590 q^{2} +9.00000 q^{3} +55.3459 q^{4} -84.1131 q^{6} -8.69181 q^{7} -218.189 q^{8} +81.0000 q^{9} +121.000 q^{11} +498.113 q^{12} +970.666 q^{13} +81.2328 q^{14} +268.100 q^{16} +424.616 q^{17} -757.018 q^{18} -1430.62 q^{19} -78.2263 q^{21} -1130.85 q^{22} -2852.99 q^{23} -1963.70 q^{24} -9071.75 q^{26} +729.000 q^{27} -481.056 q^{28} -7467.66 q^{29} +10346.3 q^{31} +4476.40 q^{32} +1089.00 q^{33} -3968.42 q^{34} +4483.02 q^{36} -167.311 q^{37} +13370.4 q^{38} +8735.99 q^{39} +5682.18 q^{41} +731.095 q^{42} -21148.9 q^{43} +6696.85 q^{44} +26663.8 q^{46} +9785.11 q^{47} +2412.90 q^{48} -16731.5 q^{49} +3821.55 q^{51} +53722.4 q^{52} -25639.5 q^{53} -6813.16 q^{54} +1896.45 q^{56} -12875.5 q^{57} +69792.0 q^{58} -23411.5 q^{59} +18591.8 q^{61} -96695.6 q^{62} -704.036 q^{63} -50415.2 q^{64} -10177.7 q^{66} -39477.7 q^{67} +23500.8 q^{68} -25676.9 q^{69} +3283.01 q^{71} -17673.3 q^{72} -29536.7 q^{73} +1563.67 q^{74} -79178.8 q^{76} -1051.71 q^{77} -81645.7 q^{78} -10280.8 q^{79} +6561.00 q^{81} -53105.1 q^{82} +38360.3 q^{83} -4329.50 q^{84} +197656. q^{86} -67208.9 q^{87} -26400.8 q^{88} -2319.99 q^{89} -8436.84 q^{91} -157902. q^{92} +93116.8 q^{93} -91450.7 q^{94} +40287.6 q^{96} +81868.3 q^{97} +156371. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9} + 242 q^{11} + 837 q^{12} + 66 q^{13} + 304 q^{14} - 543 q^{16} + 920 q^{17} - 81 q^{18} - 2932 q^{19} + 162 q^{21} - 121 q^{22} - 5246 q^{23} - 1539 q^{24} - 16622 q^{26} + 1458 q^{27} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} - 3803 q^{32} + 2178 q^{33} + 166 q^{34} + 7533 q^{36} - 5996 q^{37} + 840 q^{38} + 594 q^{39} + 24244 q^{41} + 2736 q^{42} - 20360 q^{43} + 11253 q^{44} + 6692 q^{46} + 5806 q^{47} - 4887 q^{48} - 32826 q^{49} + 8280 q^{51} + 19658 q^{52} - 40770 q^{53} - 729 q^{54} + 3156 q^{56} - 26388 q^{57} + 26958 q^{58} + 18212 q^{59} - 11398 q^{61} - 100120 q^{62} + 1458 q^{63} - 93559 q^{64} - 1089 q^{66} - 65368 q^{67} + 42154 q^{68} - 47214 q^{69} + 61446 q^{71} - 13851 q^{72} - 53412 q^{73} - 47082 q^{74} - 135712 q^{76} + 2178 q^{77} - 149598 q^{78} + 17122 q^{79} + 13122 q^{81} + 101810 q^{82} + 14304 q^{83} + 4716 q^{84} + 204240 q^{86} - 113400 q^{87} - 20691 q^{88} - 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 89424 q^{93} - 124660 q^{94} - 34227 q^{96} + 183056 q^{97} + 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9 + 242 * q^11 + 837 * q^12 + 66 * q^13 + 304 * q^14 - 543 * q^16 + 920 * q^17 - 81 * q^18 - 2932 * q^19 + 162 * q^21 - 121 * q^22 - 5246 * q^23 - 1539 * q^24 - 16622 * q^26 + 1458 * q^27 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 - 3803 * q^32 + 2178 * q^33 + 166 * q^34 + 7533 * q^36 - 5996 * q^37 + 840 * q^38 + 594 * q^39 + 24244 * q^41 + 2736 * q^42 - 20360 * q^43 + 11253 * q^44 + 6692 * q^46 + 5806 * q^47 - 4887 * q^48 - 32826 * q^49 + 8280 * q^51 + 19658 * q^52 - 40770 * q^53 - 729 * q^54 + 3156 * q^56 - 26388 * q^57 + 26958 * q^58 + 18212 * q^59 - 11398 * q^61 - 100120 * q^62 + 1458 * q^63 - 93559 * q^64 - 1089 * q^66 - 65368 * q^67 + 42154 * q^68 - 47214 * q^69 + 61446 * q^71 - 13851 * q^72 - 53412 * q^73 - 47082 * q^74 - 135712 * q^76 + 2178 * q^77 - 149598 * q^78 + 17122 * q^79 + 13122 * q^81 + 101810 * q^82 + 14304 * q^83 + 4716 * q^84 + 204240 * q^86 - 113400 * q^87 - 20691 * q^88 - 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 89424 * q^93 - 124660 * q^94 - 34227 * q^96 + 183056 * q^97 + 22047 * q^98 + 19602 * 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 $

$2$	−9.34590	−1.65214	−0.826069	−	0.563569i	$-0.809428\pi$
$2$	−9.34590	−1.65214	−0.826069	+	0.563569i	$0.809428\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	55.3459	1.72956
$5$	0	0
$5$	0	0
$6$	−84.1131	−0.953862
$7$	−8.69181	−0.0670448	−0.0335224	−	0.999438i	$-0.510673\pi$
$7$	−8.69181	−0.0670448	−0.0335224	+	0.999438i	$0.510673\pi$
$8$	−218.189	−1.20533
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	498.113	0.998562
$13$	970.666	1.59298	0.796492	−	0.604649i	$-0.206687\pi$
$13$	970.666	1.59298	0.796492	+	0.604649i	$0.206687\pi$
$14$	81.2328	0.110767
$15$	0	0
$16$	268.100	0.261816
$17$	424.616	0.356348	0.178174	−	0.983999i	$-0.442981\pi$
$17$	424.616	0.356348	0.178174	+	0.983999i	$0.442981\pi$
$18$	−757.018	−0.550713
$19$	−1430.62	−0.909158	−0.454579	−	0.890707i	$-0.650210\pi$
$19$	−1430.62	−0.909158	−0.454579	+	0.890707i	$0.650210\pi$
$20$	0	0
$21$	−78.2263	−0.0387083
$22$	−1130.85	−0.498138
$23$	−2852.99	−1.12456	−0.562278	−	0.826948i	$-0.690075\pi$
$23$	−2852.99	−1.12456	−0.562278	+	0.826948i	$0.690075\pi$
$24$	−1963.70	−0.695899
$25$	0	0
$26$	−9071.75	−2.63183
$27$	729.000	0.192450
$28$	−481.056	−0.115958
$29$	−7467.66	−1.64888	−0.824441	−	0.565948i	$-0.808510\pi$
$29$	−7467.66	−1.64888	−0.824441	+	0.565948i	$0.808510\pi$
$30$	0	0
$31$	10346.3	1.93366	0.966832	−	0.255411i	$-0.0822109\pi$
$31$	10346.3	1.93366	0.966832	+	0.255411i	$0.0822109\pi$
$32$	4476.40	0.772776
$33$	1089.00	0.174078
$34$	−3968.42	−0.588736
$35$	0	0
$36$	4483.02	0.576520
$37$	−167.311	−0.0200919	−0.0100459	−	0.999950i	$-0.503198\pi$
$37$	−167.311	−0.0200919	−0.0100459	+	0.999950i	$0.503198\pi$
$38$	13370.4	1.50205
$39$	8735.99	0.919710
$40$	0	0
$41$	5682.18	0.527905	0.263952	−	0.964536i	$-0.414974\pi$
$41$	5682.18	0.527905	0.263952	+	0.964536i	$0.414974\pi$
$42$	731.095	0.0639515
$43$	−21148.9	−1.74428	−0.872142	−	0.489253i	$-0.837269\pi$
$43$	−21148.9	−1.74428	−0.872142	+	0.489253i	$0.837269\pi$
$44$	6696.85	0.521482
$45$	0	0
$46$	26663.8	1.85792
$47$	9785.11	0.646132	0.323066	−	0.946376i	$-0.395286\pi$
$47$	9785.11	0.646132	0.323066	+	0.946376i	$0.395286\pi$
$48$	2412.90	0.151160
$49$	−16731.5	−0.995505
$50$	0	0
$51$	3821.55	0.205738
$52$	53722.4	2.75516
$53$	−25639.5	−1.25377	−0.626887	−	0.779110i	$-0.715671\pi$
$53$	−25639.5	−1.25377	−0.626887	+	0.779110i	$0.715671\pi$
$54$	−6813.16	−0.317954
$55$	0	0
$56$	1896.45	0.0808113
$57$	−12875.5	−0.524902
$58$	69792.0	2.72418
$59$	−23411.5	−0.875588	−0.437794	−	0.899075i	$-0.644240\pi$
$59$	−23411.5	−0.875588	−0.437794	+	0.899075i	$0.644240\pi$
$60$	0	0
$61$	18591.8	0.639732	0.319866	−	0.947463i	$-0.396362\pi$
$61$	18591.8	0.639732	0.319866	+	0.947463i	$0.396362\pi$
$62$	−96695.6	−3.19468
$63$	−704.036	−0.0223483
$64$	−50415.2	−1.53855
$65$	0	0
$66$	−10177.7	−0.287600
$67$	−39477.7	−1.07440	−0.537198	−	0.843456i	$-0.680517\pi$
$67$	−39477.7	−1.07440	−0.537198	+	0.843456i	$0.680517\pi$
$68$	23500.8	0.616325
$69$	−25676.9	−0.649263
$70$	0	0
$71$	3283.01	0.0772905	0.0386453	−	0.999253i	$-0.487696\pi$
$71$	3283.01	0.0772905	0.0386453	+	0.999253i	$0.487696\pi$
$72$	−17673.3	−0.401778
$73$	−29536.7	−0.648716	−0.324358	−	0.945934i	$-0.605148\pi$
$73$	−29536.7	−0.648716	−0.324358	+	0.945934i	$0.605148\pi$
$74$	1563.67	0.0331945
$75$	0	0
$76$	−79178.8	−1.57244
$77$	−1051.71	−0.0202148
$78$	−81645.7	−1.51949
$79$	−10280.8	−0.185335	−0.0926676	−	0.995697i	$-0.529539\pi$
$79$	−10280.8	−0.185335	−0.0926676	+	0.995697i	$0.529539\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−53105.1	−0.872171
$83$	38360.3	0.611206	0.305603	−	0.952159i	$-0.401142\pi$
$83$	38360.3	0.611206	0.305603	+	0.952159i	$0.401142\pi$
$84$	−4329.50	−0.0669483
$85$	0	0
$86$	197656.	2.88180
$87$	−67208.9	−0.951982
$88$	−26400.8	−0.363422
$89$	−2319.99	−0.0310464	−0.0155232	−	0.999880i	$-0.504941\pi$
$89$	−2319.99	−0.0310464	−0.0155232	+	0.999880i	$0.504941\pi$
$90$	0	0
$91$	−8436.84	−0.106801
$92$	−157902.	−1.94499
$93$	93116.8	1.11640
$94$	−91450.7	−1.06750
$95$	0	0
$96$	40287.6	0.446162
$97$	81868.3	0.883459	0.441729	−	0.897148i	$-0.354365\pi$
$97$	81868.3	0.883459	0.441729	+	0.897148i	$0.354365\pi$
$98$	156371.	1.64471
$99$	9801.00	0.100504
$100$	0	0
$101$	−180138.	−1.75712	−0.878560	−	0.477631i	$-0.841496\pi$
$101$	−180138.	−1.75712	−0.878560	+	0.477631i	$0.841496\pi$
$102$	−35715.8	−0.339907
$103$	109911.	1.02082	0.510410	−	0.859931i	$-0.329494\pi$
$103$	109911.	1.02082	0.510410	+	0.859931i	$0.329494\pi$
$104$	−211788.	−1.92008
$105$	0	0
$106$	239624.	2.07141
$107$	−65683.5	−0.554621	−0.277311	−	0.960780i	$-0.589443\pi$
$107$	−65683.5	−0.554621	−0.277311	+	0.960780i	$0.589443\pi$
$108$	40347.2	0.332854
$109$	146333.	1.17971	0.589854	−	0.807510i	$-0.299185\pi$
$109$	146333.	1.17971	0.589854	+	0.807510i	$0.299185\pi$
$110$	0	0
$111$	−1505.80	−0.0116000
$112$	−2330.27	−0.0175534
$113$	−261872.	−1.92927	−0.964636	−	0.263585i	$-0.915095\pi$
$113$	−261872.	−1.92927	−0.964636	+	0.263585i	$0.915095\pi$
$114$	120334.	0.867211
$115$	0	0
$116$	−413304.	−2.85184
$117$	78623.9	0.530995
$118$	218802.	1.44659
$119$	−3690.68	−0.0238913
$120$	0	0
$121$	14641.0	0.0909091
$122$	−173758.	−1.05693
$123$	51139.6	0.304786
$124$	572626.	3.34439
$125$	0	0
$126$	6579.85	0.0369224
$127$	22466.6	0.123603	0.0618015	−	0.998088i	$-0.480315\pi$
$127$	22466.6	0.123603	0.0618015	+	0.998088i	$0.480315\pi$
$128$	327931.	1.76912
$129$	−190340.	−1.00706
$130$	0	0
$131$	119595.	0.608886	0.304443	−	0.952531i	$-0.401530\pi$
$131$	119595.	0.608886	0.304443	+	0.952531i	$0.401530\pi$
$132$	60271.7	0.301078
$133$	12434.6	0.0609543
$134$	368954.	1.77505
$135$	0	0
$136$	−92646.4	−0.429518
$137$	−176443.	−0.803160	−0.401580	−	0.915824i	$-0.631539\pi$
$137$	−176443.	−0.803160	−0.401580	+	0.915824i	$0.631539\pi$
$138$	239974.	1.07267
$139$	288218.	1.26527	0.632636	−	0.774450i	$-0.281973\pi$
$139$	288218.	1.26527	0.632636	+	0.774450i	$0.281973\pi$
$140$	0	0
$141$	88066.0	0.373044
$142$	−30682.7	−0.127695
$143$	117451.	0.480303
$144$	21716.1	0.0872722
$145$	0	0
$146$	276047.	1.07177
$147$	−150583.	−0.574755
$148$	−9259.98	−0.0347501
$149$	−133635.	−0.493123	−0.246561	−	0.969127i	$-0.579301\pi$
$149$	−133635.	−0.493123	−0.246561	+	0.969127i	$0.579301\pi$
$150$	0	0
$151$	406122.	1.44949	0.724744	−	0.689019i	$-0.241958\pi$
$151$	406122.	1.44949	0.724744	+	0.689019i	$0.241958\pi$
$152$	312144.	1.09584
$153$	34393.9	0.118783
$154$	9829.17	0.0333976
$155$	0	0
$156$	483501.	1.59069
$157$	158746.	0.513988	0.256994	−	0.966413i	$-0.417268\pi$
$157$	158746.	0.513988	0.256994	+	0.966413i	$0.417268\pi$
$158$	96083.1	0.306199
$159$	−230755.	−0.723867
$160$	0	0
$161$	24797.7	0.0753956
$162$	−61318.5	−0.183571
$163$	−174460.	−0.514311	−0.257155	−	0.966370i	$-0.582785\pi$
$163$	−174460.	−0.514311	−0.257155	+	0.966370i	$0.582785\pi$
$164$	314486.	0.913043
$165$	0	0
$166$	−358512.	−1.00980
$167$	−105879.	−0.293777	−0.146889	−	0.989153i	$-0.546926\pi$
$167$	−105879.	−0.293777	−0.146889	+	0.989153i	$0.546926\pi$
$168$	17068.1	0.0466564
$169$	570899.	1.53760
$170$	0	0
$171$	−115880.	−0.303053
$172$	−1.17051e6	−3.01684
$173$	432459.	1.09857	0.549287	−	0.835634i	$-0.314899\pi$
$173$	432459.	1.09857	0.549287	+	0.835634i	$0.314899\pi$
$174$	628128.	1.57281
$175$	0	0
$176$	32440.1	0.0789406
$177$	−210704.	−0.505521
$178$	21682.4	0.0512929
$179$	−624448.	−1.45668	−0.728340	−	0.685216i	$-0.759708\pi$
$179$	−624448.	−1.45668	−0.728340	+	0.685216i	$0.759708\pi$
$180$	0	0
$181$	10602.4	0.0240551	0.0120275	−	0.999928i	$-0.496171\pi$
$181$	10602.4	0.0240551	0.0120275	+	0.999928i	$0.496171\pi$
$182$	78849.9	0.176450
$183$	167327.	0.369349
$184$	622490.	1.35546
$185$	0	0
$186$	−870260.	−1.84445
$187$	51378.6	0.107443
$188$	541566.	1.11752
$189$	−6336.33	−0.0129028
$190$	0	0
$191$	441955.	0.876586	0.438293	−	0.898832i	$-0.355583\pi$
$191$	441955.	0.876586	0.438293	+	0.898832i	$0.355583\pi$
$192$	−453736.	−0.888282
$193$	444354.	0.858690	0.429345	−	0.903141i	$-0.358745\pi$
$193$	444354.	0.858690	0.429345	+	0.903141i	$0.358745\pi$
$194$	−765133.	−1.45960
$195$	0	0
$196$	−926017.	−1.72179
$197$	416576.	0.764766	0.382383	−	0.924004i	$-0.375103\pi$
$197$	416576.	0.764766	0.382383	+	0.924004i	$0.375103\pi$
$198$	−91599.2	−0.166046
$199$	−502973.	−0.900351	−0.450176	−	0.892940i	$-0.648639\pi$
$199$	−502973.	−0.900351	−0.450176	+	0.892940i	$0.648639\pi$
$200$	0	0
$201$	−355299.	−0.620303
$202$	1.68355e6	2.90301
$203$	64907.4	0.110549
$204$	211507.	0.355836
$205$	0	0
$206$	−1.02722e6	−1.68653
$207$	−231092.	−0.374852
$208$	260236.	0.417069
$209$	−173105.	−0.274121
$210$	0	0
$211$	−897392.	−1.38764	−0.693819	−	0.720149i	$-0.744073\pi$
$211$	−897392.	−1.38764	−0.693819	+	0.720149i	$0.744073\pi$
$212$	−1.41904e6	−2.16848
$213$	29547.1	0.0446237
$214$	613871.	0.916311
$215$	0	0
$216$	−159059.	−0.231966
$217$	−89928.1	−0.129642
$218$	−1.36761e6	−1.94904
$219$	−265830.	−0.374536
$220$	0	0
$221$	412161.	0.567657
$222$	14073.1	0.0191649
$223$	−117328.	−0.157994	−0.0789971	−	0.996875i	$-0.525172\pi$
$223$	−117328.	−0.157994	−0.0789971	+	0.996875i	$0.525172\pi$
$224$	−38908.0	−0.0518106
$225$	0	0
$226$	2.44743e6	3.18742
$227$	−75051.3	−0.0966704	−0.0483352	−	0.998831i	$-0.515392\pi$
$227$	−75051.3	−0.0966704	−0.0483352	+	0.998831i	$0.515392\pi$
$228$	−712609.	−0.907850
$229$	634950.	0.800112	0.400056	−	0.916491i	$-0.368991\pi$
$229$	634950.	0.800112	0.400056	+	0.916491i	$0.368991\pi$
$230$	0	0
$231$	−9465.38	−0.0116710
$232$	1.62936e6	1.98745
$233$	375281.	0.452863	0.226431	−	0.974027i	$-0.427294\pi$
$233$	375281.	0.452863	0.226431	+	0.974027i	$0.427294\pi$
$234$	−734812.	−0.877276
$235$	0	0
$236$	−1.29573e6	−1.51438
$237$	−92527.0	−0.107003
$238$	34492.8	0.0394717
$239$	−859529.	−0.973343	−0.486671	−	0.873585i	$-0.661789\pi$
$239$	−859529.	−0.973343	−0.486671	+	0.873585i	$0.661789\pi$
$240$	0	0
$241$	−552284.	−0.612520	−0.306260	−	0.951948i	$-0.599078\pi$
$241$	−552284.	−0.612520	−0.306260	+	0.951948i	$0.599078\pi$
$242$	−136833.	−0.150194
$243$	59049.0	0.0641500
$244$	1.02898e6	1.10645
$245$	0	0
$246$	−477946.	−0.503548
$247$	−1.38865e6	−1.44827
$248$	−2.25745e6	−2.33071
$249$	345243.	0.352880
$250$	0	0
$251$	198740.	0.199114	0.0995569	−	0.995032i	$-0.468257\pi$
$251$	198740.	0.199114	0.0995569	+	0.995032i	$0.468257\pi$
$252$	−38965.5	−0.0386526
$253$	−345212.	−0.339066
$254$	−209971.	−0.204209
$255$	0	0
$256$	−1.45152e6	−1.38428
$257$	76585.7	0.0723294	0.0361647	−	0.999346i	$-0.488486\pi$
$257$	76585.7	0.0723294	0.0361647	+	0.999346i	$0.488486\pi$
$258$	1.77890e6	1.66381
$259$	1454.24	0.00134705
$260$	0	0
$261$	−604880.	−0.549627
$262$	−1.11773e6	−1.00596
$263$	−2.10812e6	−1.87935	−0.939673	−	0.342073i	$-0.888871\pi$
$263$	−2.10812e6	−1.87935	−0.939673	+	0.342073i	$0.888871\pi$
$264$	−237607.	−0.209822
$265$	0	0
$266$	−116213.	−0.100705
$267$	−20879.9	−0.0179246
$268$	−2.18493e6	−1.85823
$269$	−1.97320e6	−1.66261	−0.831306	−	0.555816i	$-0.812406\pi$
$269$	−1.97320e6	−1.66261	−0.831306	+	0.555816i	$0.812406\pi$
$270$	0	0
$271$	−926426.	−0.766280	−0.383140	−	0.923690i	$-0.625157\pi$
$271$	−926426.	−0.766280	−0.383140	+	0.923690i	$0.625157\pi$
$272$	113840.	0.0932978
$273$	−75931.5	−0.0616617
$274$	1.64902e6	1.32693
$275$	0	0
$276$	−1.42111e6	−1.12294
$277$	1.60640e6	1.25792	0.628962	−	0.777436i	$-0.283480\pi$
$277$	1.60640e6	1.25792	0.628962	+	0.777436i	$0.283480\pi$
$278$	−2.69365e6	−2.09040
$279$	838051.	0.644555
$280$	0	0
$281$	−814868.	−0.615632	−0.307816	−	0.951446i	$-0.599598\pi$
$281$	−814868.	−0.615632	−0.307816	+	0.951446i	$0.599598\pi$
$282$	−823056.	−0.616321
$283$	61465.6	0.0456211	0.0228106	−	0.999740i	$-0.492739\pi$
$283$	61465.6	0.0456211	0.0228106	+	0.999740i	$0.492739\pi$
$284$	181701.	0.133679
$285$	0	0
$286$	−1.09768e6	−0.793526
$287$	−49388.4	−0.0353933
$288$	362588.	0.257592
$289$	−1.23956e6	−0.873016
$290$	0	0
$291$	736814.	0.510065
$292$	−1.63473e6	−1.12199
$293$	298524.	0.203147	0.101573	−	0.994828i	$-0.467612\pi$
$293$	298524.	0.203147	0.101573	+	0.994828i	$0.467612\pi$
$294$	1.40733e6	0.949575
$295$	0	0
$296$	36505.4	0.0242174
$297$	88209.0	0.0580259
$298$	1.24894e6	0.814707
$299$	−2.76930e6	−1.79140
$300$	0	0
$301$	183822.	0.116945
$302$	−3.79558e6	−2.39475
$303$	−1.62124e6	−1.01447
$304$	−383548.	−0.238032
$305$	0	0
$306$	−321442.	−0.196245
$307$	1.35385e6	0.819834	0.409917	−	0.912123i	$-0.365558\pi$
$307$	1.35385e6	0.819834	0.409917	+	0.912123i	$0.365558\pi$
$308$	−58207.8	−0.0349626
$309$	989201.	0.589370
$310$	0	0
$311$	−2.68320e6	−1.57309	−0.786544	−	0.617535i	$-0.788132\pi$
$311$	−2.68320e6	−1.57309	−0.786544	+	0.617535i	$0.788132\pi$
$312$	−1.90609e6	−1.10856
$313$	−2.85968e6	−1.64989	−0.824947	−	0.565210i	$-0.808795\pi$
$313$	−2.85968e6	−1.64989	−0.824947	+	0.565210i	$0.808795\pi$
$314$	−1.48362e6	−0.849180
$315$	0	0
$316$	−568999.	−0.320548
$317$	111110.	0.0621020	0.0310510	−	0.999518i	$-0.490115\pi$
$317$	111110.	0.0621020	0.0310510	+	0.999518i	$0.490115\pi$
$318$	2.15662e6	1.19593
$319$	−903587.	−0.497157
$320$	0	0
$321$	−591151.	−0.320211
$322$	−231757.	−0.124564
$323$	−607463.	−0.323977
$324$	363124.	0.192173
$325$	0	0
$326$	1.63048e6	0.849712
$327$	1.31699e6	0.681105
$328$	−1.23979e6	−0.636301
$329$	−85050.3	−0.0433198
$330$	0	0
$331$	−1.36346e6	−0.684028	−0.342014	−	0.939695i	$-0.611109\pi$
$331$	−1.36346e6	−0.684028	−0.342014	+	0.939695i	$0.611109\pi$
$332$	2.12309e6	1.05712
$333$	−13552.2	−0.00669729
$334$	989534.	0.485361
$335$	0	0
$336$	−20972.5	−0.0101345
$337$	621625.	0.298163	0.149082	−	0.988825i	$-0.452368\pi$
$337$	621625.	0.298163	0.149082	+	0.988825i	$0.452368\pi$
$338$	−5.33557e6	−2.54032
$339$	−2.35685e6	−1.11387
$340$	0	0
$341$	1.25190e6	0.583022
$342$	1.08300e6	0.500685
$343$	291510.	0.133788
$344$	4.61445e6	2.10244
$345$	0	0
$346$	−4.04172e6	−1.81500
$347$	−125580.	−0.0559883	−0.0279942	−	0.999608i	$-0.508912\pi$
$347$	−125580.	−0.0559883	−0.0279942	+	0.999608i	$0.508912\pi$
$348$	−3.71974e6	−1.64651
$349$	3.24565e6	1.42639	0.713195	−	0.700966i	$-0.247247\pi$
$349$	3.24565e6	1.42639	0.713195	+	0.700966i	$0.247247\pi$
$350$	0	0
$351$	707615.	0.306570
$352$	541644.	0.233001
$353$	−1.67773e6	−0.716614	−0.358307	−	0.933604i	$-0.616646\pi$
$353$	−1.67773e6	−0.716614	−0.358307	+	0.933604i	$0.616646\pi$
$354$	1.96922e6	0.835191
$355$	0	0
$356$	−128402.	−0.0536966
$357$	−33216.1	−0.0137936
$358$	5.83603e6	2.40663
$359$	−453973.	−0.185906	−0.0929531	−	0.995670i	$-0.529631\pi$
$359$	−453973.	−0.185906	−0.0929531	+	0.995670i	$0.529631\pi$
$360$	0	0
$361$	−429436.	−0.173432
$362$	−99088.7	−0.0397423
$363$	131769.	0.0524864
$364$	−466944.	−0.184719
$365$	0	0
$366$	−1.56382e6	−0.610216
$367$	−4.87840e6	−1.89066	−0.945328	−	0.326121i	$-0.894258\pi$
$367$	−4.87840e6	−1.89066	−0.945328	+	0.326121i	$0.894258\pi$
$368$	−764888.	−0.294427
$369$	460257.	0.175968
$370$	0	0
$371$	222853.	0.0840590
$372$	5.15363e6	1.93088
$373$	605344.	0.225284	0.112642	−	0.993636i	$-0.464069\pi$
$373$	605344.	0.225284	0.112642	+	0.993636i	$0.464069\pi$
$374$	−480179.	−0.177511
$375$	0	0
$376$	−2.13500e6	−0.778804
$377$	−7.24860e6	−2.62664
$378$	59218.7	0.0213172
$379$	−3.90340e6	−1.39587	−0.697936	−	0.716160i	$-0.745898\pi$
$379$	−3.90340e6	−1.39587	−0.697936	+	0.716160i	$0.745898\pi$
$380$	0	0
$381$	202200.	0.0713622
$382$	−4.13047e6	−1.44824
$383$	−2.80018e6	−0.975413	−0.487707	−	0.873008i	$-0.662166\pi$
$383$	−2.80018e6	−0.975413	−0.487707	+	0.873008i	$0.662166\pi$
$384$	2.95138e6	1.02140
$385$	0	0
$386$	−4.15289e6	−1.41867
$387$	−1.71306e6	−0.581428
$388$	4.53107e6	1.52799
$389$	3.71874e6	1.24601	0.623006	−	0.782217i	$-0.285911\pi$
$389$	3.71874e6	1.24601	0.623006	+	0.782217i	$0.285911\pi$
$390$	0	0
$391$	−1.21143e6	−0.400733
$392$	3.65061e6	1.19991
$393$	1.07636e6	0.351540
$394$	−3.89328e6	−1.26350
$395$	0	0
$396$	542445.	0.173827
$397$	−3.85343e6	−1.22708	−0.613538	−	0.789665i	$-0.710254\pi$
$397$	−3.85343e6	−1.22708	−0.613538	+	0.789665i	$0.710254\pi$
$398$	4.70074e6	1.48750
$399$	111912.	0.0351920
$400$	0	0
$401$	1.08196e6	0.336009	0.168004	−	0.985786i	$-0.446268\pi$
$401$	1.08196e6	0.336009	0.168004	+	0.985786i	$0.446268\pi$
$402$	3.32059e6	1.02483
$403$	1.00428e7	3.08030
$404$	−9.96989e6	−3.03905
$405$	0	0
$406$	−606619.	−0.182642
$407$	−20244.6	−0.00605793
$408$	−833818.	−0.247982
$409$	1.98167e6	0.585765	0.292883	−	0.956148i	$-0.405385\pi$
$409$	1.98167e6	0.585765	0.292883	+	0.956148i	$0.405385\pi$
$410$	0	0
$411$	−1.58798e6	−0.463705
$412$	6.08313e6	1.76557
$413$	203489.	0.0587036
$414$	2.15977e6	0.619307
$415$	0	0
$416$	4.34508e6	1.23102
$417$	2.59396e6	0.730505
$418$	1.61782e6	0.452886
$419$	−997473.	−0.277566	−0.138783	−	0.990323i	$-0.544319\pi$
$419$	−997473.	−0.277566	−0.138783	+	0.990323i	$0.544319\pi$
$420$	0	0
$421$	5.36616e6	1.47556	0.737782	−	0.675039i	$-0.235873\pi$
$421$	5.36616e6	1.47556	0.737782	+	0.675039i	$0.235873\pi$
$422$	8.38694e6	2.29257
$423$	792594.	0.215377
$424$	5.59424e6	1.51122
$425$	0	0
$426$	−276144.	−0.0737245
$427$	−161597.	−0.0428907
$428$	−3.63531e6	−0.959250
$429$	1.05705e6	0.277303
$430$	0	0
$431$	−3.37897e6	−0.876175	−0.438088	−	0.898932i	$-0.644344\pi$
$431$	−3.37897e6	−0.876175	−0.438088	+	0.898932i	$0.644344\pi$
$432$	195445.	0.0503866
$433$	−3.06239e6	−0.784948	−0.392474	−	0.919763i	$-0.628381\pi$
$433$	−3.06239e6	−0.784948	−0.392474	+	0.919763i	$0.628381\pi$
$434$	840459.	0.214187
$435$	0	0
$436$	8.09891e6	2.04038
$437$	4.08154e6	1.02240
$438$	2.48442e6	0.618786
$439$	2.56020e6	0.634035	0.317018	−	0.948420i	$-0.397319\pi$
$439$	2.56020e6	0.634035	0.317018	+	0.948420i	$0.397319\pi$
$440$	0	0
$441$	−1.35525e6	−0.331835
$442$	−3.85201e6	−0.937847
$443$	1.10849e6	0.268363	0.134181	−	0.990957i	$-0.457160\pi$
$443$	1.10849e6	0.268363	0.134181	+	0.990957i	$0.457160\pi$
$444$	−83339.8	−0.0200630
$445$	0	0
$446$	1.09654e6	0.261028
$447$	−1.20272e6	−0.284705
$448$	438199.	0.103152
$449$	−5.98494e6	−1.40102	−0.700509	−	0.713643i	$-0.747044\pi$
$449$	−5.98494e6	−1.40102	−0.700509	+	0.713643i	$0.747044\pi$
$450$	0	0
$451$	687544.	0.159169
$452$	−1.44936e7	−3.33679
$453$	3.65510e6	0.836862
$454$	701422.	0.159713
$455$	0	0
$456$	2.80930e6	0.632682
$457$	−3.91747e6	−0.877436	−0.438718	−	0.898625i	$-0.644567\pi$
$457$	−3.91747e6	−0.877436	−0.438718	+	0.898625i	$0.644567\pi$
$458$	−5.93418e6	−1.32190
$459$	309545.	0.0685792
$460$	0	0
$461$	−2.16827e6	−0.475184	−0.237592	−	0.971365i	$-0.576358\pi$
$461$	−2.16827e6	−0.475184	−0.237592	+	0.971365i	$0.576358\pi$
$462$	88462.5	0.0192821
$463$	5.01423e6	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$463$	5.01423e6	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$464$	−2.00208e6	−0.431704
$465$	0	0
$466$	−3.50734e6	−0.748191
$467$	2.57098e6	0.545514	0.272757	−	0.962083i	$-0.412064\pi$
$467$	2.57098e6	0.545514	0.272757	+	0.962083i	$0.412064\pi$
$468$	4.35151e6	0.918387
$469$	343132.	0.0720326
$470$	0	0
$471$	1.42871e6	0.296751
$472$	5.10813e6	1.05538
$473$	−2.55902e6	−0.525921
$474$	864748.	0.176784
$475$	0	0
$476$	−204264.	−0.0413214
$477$	−2.07680e6	−0.417925
$478$	8.03307e6	1.60810
$479$	5.16179e6	1.02793	0.513963	−	0.857813i	$-0.328177\pi$
$479$	5.16179e6	1.02793	0.513963	+	0.857813i	$0.328177\pi$
$480$	0	0
$481$	−162403.	−0.0320060
$482$	5.16160e6	1.01197
$483$	223179.	0.0435297
$484$	810319.	0.157233
$485$	0	0
$486$	−551866.	−0.105985
$487$	−5.85746e6	−1.11915	−0.559573	−	0.828781i	$-0.689035\pi$
$487$	−5.85746e6	−1.11915	−0.559573	+	0.828781i	$0.689035\pi$
$488$	−4.05653e6	−0.771090
$489$	−1.57014e6	−0.296937
$490$	0	0
$491$	956557.	0.179064	0.0895318	−	0.995984i	$-0.471463\pi$
$491$	956557.	0.179064	0.0895318	+	0.995984i	$0.471463\pi$
$492$	2.83037e6	0.527145
$493$	−3.17089e6	−0.587576
$494$	1.29782e7	2.39275
$495$	0	0
$496$	2.77385e6	0.506265
$497$	−28535.3	−0.00518192
$498$	−3.22661e6	−0.583006
$499$	−1.47579e6	−0.265322	−0.132661	−	0.991161i	$-0.542352\pi$
$499$	−1.47579e6	−0.265322	−0.132661	+	0.991161i	$0.542352\pi$
$500$	0	0
$501$	−952911.	−0.169612
$502$	−1.85741e6	−0.328963
$503$	−991048.	−0.174652	−0.0873262	−	0.996180i	$-0.527832\pi$
$503$	−991048.	−0.174652	−0.0873262	+	0.996180i	$0.527832\pi$
$504$	153613.	0.0269371
$505$	0	0
$506$	3.22632e6	0.560185
$507$	5.13809e6	0.887732
$508$	1.24344e6	0.213779
$509$	−8.24580e6	−1.41071	−0.705356	−	0.708854i	$-0.749213\pi$
$509$	−8.24580e6	−1.41071	−0.705356	+	0.708854i	$0.749213\pi$
$510$	0	0
$511$	256727.	0.0434930
$512$	3.07201e6	0.517901
$513$	−1.04292e6	−0.174967
$514$	−715762.	−0.119498
$515$	0	0
$516$	−1.05346e7	−1.74177
$517$	1.18400e6	0.194816
$518$	−13591.1	−0.00222552
$519$	3.89213e6	0.634262
$520$	0	0
$521$	−1.18607e6	−0.191432	−0.0957161	−	0.995409i	$-0.530514\pi$
$521$	−1.18607e6	−0.191432	−0.0957161	+	0.995409i	$0.530514\pi$
$522$	5.65315e6	0.908060
$523$	−2.80341e6	−0.448159	−0.224079	−	0.974571i	$-0.571938\pi$
$523$	−2.80341e6	−0.448159	−0.224079	+	0.974571i	$0.571938\pi$
$524$	6.61911e6	1.05310
$525$	0	0
$526$	1.97023e7	3.10494
$527$	4.39321e6	0.689058
$528$	291961.	0.0455764
$529$	1.70323e6	0.264627
$530$	0	0
$531$	−1.89633e6	−0.291863
$532$	688206.	0.105424
$533$	5.51550e6	0.840944
$534$	195142.	0.0296140
$535$	0	0
$536$	8.61357e6	1.29500
$537$	−5.62003e6	−0.841014
$538$	1.84413e7	2.74686
$539$	−2.02451e6	−0.300156
$540$	0	0
$541$	5.30698e6	0.779569	0.389785	−	0.920906i	$-0.372549\pi$
$541$	5.30698e6	0.779569	0.389785	+	0.920906i	$0.372549\pi$
$542$	8.65829e6	1.26600
$543$	95421.3	0.0138882
$544$	1.90075e6	0.275377
$545$	0	0
$546$	709649.	0.101874
$547$	−7.62305e6	−1.08933	−0.544666	−	0.838653i	$-0.683344\pi$
$547$	−7.62305e6	−1.08933	−0.544666	+	0.838653i	$0.683344\pi$
$548$	−9.76538e6	−1.38911
$549$	1.50594e6	0.213244
$550$	0	0
$551$	1.06834e7	1.49909
$552$	5.60241e6	0.782578
$553$	89358.5	0.0124258
$554$	−1.50133e7	−2.07826
$555$	0	0
$556$	1.59517e7	2.18836
$557$	−7.81748e6	−1.06765	−0.533825	−	0.845595i	$-0.679246\pi$
$557$	−7.81748e6	−1.06765	−0.533825	+	0.845595i	$0.679246\pi$
$558$	−7.83234e6	−1.06489
$559$	−2.05285e7	−2.77861
$560$	0	0
$561$	462407.	0.0620322
$562$	7.61567e6	1.01711
$563$	−4.46693e6	−0.593933	−0.296967	−	0.954888i	$-0.595975\pi$
$563$	−4.46693e6	−0.593933	−0.296967	+	0.954888i	$0.595975\pi$
$564$	4.87409e6	0.645203
$565$	0	0
$566$	−574452.	−0.0753724
$567$	−57026.9	−0.00744942
$568$	−716315.	−0.0931608
$569$	−1.03786e7	−1.34387	−0.671933	−	0.740612i	$-0.734536\pi$
$569$	−1.03786e7	−1.34387	−0.671933	+	0.740612i	$0.734536\pi$
$570$	0	0
$571$	5.85319e6	0.751280	0.375640	−	0.926766i	$-0.377423\pi$
$571$	5.85319e6	0.751280	0.375640	+	0.926766i	$0.377423\pi$
$572$	6.50041e6	0.830712
$573$	3.97760e6	0.506097
$574$	461579.	0.0584745
$575$	0	0
$576$	−4.08363e6	−0.512850
$577$	−1.01944e7	−1.27475	−0.637373	−	0.770555i	$-0.719979\pi$
$577$	−1.01944e7	−1.27475	−0.637373	+	0.770555i	$0.719979\pi$
$578$	1.15848e7	1.44234
$579$	3.99919e6	0.495765
$580$	0	0
$581$	−333421.	−0.0409781
$582$	−6.88620e6	−0.842698
$583$	−3.10238e6	−0.378027
$584$	6.44457e6	0.781919
$585$	0	0
$586$	−2.78997e6	−0.335626
$587$	3.10970e6	0.372497	0.186249	−	0.982503i	$-0.440367\pi$
$587$	3.10970e6	0.372497	0.186249	+	0.982503i	$0.440367\pi$
$588$	−8.33416e6	−0.994073
$589$	−1.48016e7	−1.75801
$590$	0	0
$591$	3.74918e6	0.441538
$592$	−44856.1	−0.00526038
$593$	1.59890e7	1.86717	0.933587	−	0.358352i	$-0.116661\pi$
$593$	1.59890e7	1.86717	0.933587	+	0.358352i	$0.116661\pi$
$594$	−824393.	−0.0958668
$595$	0	0
$596$	−7.39616e6	−0.852885
$597$	−4.52676e6	−0.519818
$598$	2.58816e7	2.95964
$599$	−4.21384e6	−0.479856	−0.239928	−	0.970791i	$-0.577124\pi$
$599$	−4.21384e6	−0.479856	−0.239928	+	0.970791i	$0.577124\pi$
$600$	0	0
$601$	4.68871e6	0.529502	0.264751	−	0.964317i	$-0.414710\pi$
$601$	4.68871e6	0.529502	0.264751	+	0.964317i	$0.414710\pi$
$602$	−1.71799e6	−0.193209
$603$	−3.19769e6	−0.358132
$604$	2.24772e7	2.50697
$605$	0	0
$606$	1.51520e7	1.67605
$607$	−4.29265e6	−0.472883	−0.236441	−	0.971646i	$-0.575981\pi$
$607$	−4.29265e6	−0.472883	−0.236441	+	0.971646i	$0.575981\pi$
$608$	−6.40401e6	−0.702575
$609$	584167.	0.0638254
$610$	0	0
$611$	9.49807e6	1.02928
$612$	1.90356e6	0.205442
$613$	−1.67420e7	−1.79951	−0.899757	−	0.436392i	$-0.856256\pi$
$613$	−1.67420e7	−1.79951	−0.899757	+	0.436392i	$0.856256\pi$
$614$	−1.26530e7	−1.35448
$615$	0	0
$616$	229471.	0.0243655
$617$	2.10611e6	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$617$	2.10611e6	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$618$	−9.24497e6	−0.973721
$619$	2.52452e6	0.264821	0.132410	−	0.991195i	$-0.457728\pi$
$619$	2.52452e6	0.264821	0.132410	+	0.991195i	$0.457728\pi$
$620$	0	0
$621$	−2.07983e6	−0.216421
$622$	2.50770e7	2.59896
$623$	20164.9	0.00208150
$624$	2.34212e6	0.240795
$625$	0	0
$626$	2.67263e7	2.72585
$627$	−1.55794e6	−0.158264
$628$	8.78594e6	0.888974
$629$	−71043.0	−0.00715970
$630$	0	0
$631$	1.11540e6	0.111521	0.0557607	−	0.998444i	$-0.482242\pi$
$631$	1.11540e6	0.111521	0.0557607	+	0.998444i	$0.482242\pi$
$632$	2.24315e6	0.223391
$633$	−8.07653e6	−0.801153
$634$	−1.03843e6	−0.102601
$635$	0	0
$636$	−1.27714e7	−1.25197
$637$	−1.62406e7	−1.58582
$638$	8.44483e6	0.821371
$639$	265924.	0.0257635
$640$	0	0
$641$	9.80630e6	0.942671	0.471336	−	0.881954i	$-0.343772\pi$
$641$	9.80630e6	0.942671	0.471336	+	0.881954i	$0.343772\pi$
$642$	5.52484e6	0.529032
$643$	5.97068e6	0.569504	0.284752	−	0.958601i	$-0.408089\pi$
$643$	5.97068e6	0.569504	0.284752	+	0.958601i	$0.408089\pi$
$644$	1.37245e6	0.130401
$645$	0	0
$646$	5.67729e6	0.535254
$647$	−8.55002e6	−0.802984	−0.401492	−	0.915863i	$-0.631508\pi$
$647$	−8.55002e6	−0.802984	−0.401492	+	0.915863i	$0.631508\pi$
$648$	−1.43154e6	−0.133926
$649$	−2.83280e6	−0.264000
$650$	0	0
$651$	−809353.	−0.0748489
$652$	−9.65562e6	−0.889531
$653$	4.56279e6	0.418743	0.209371	−	0.977836i	$-0.432858\pi$
$653$	4.56279e6	0.418743	0.209371	+	0.977836i	$0.432858\pi$
$654$	−1.23085e7	−1.12528
$655$	0	0
$656$	1.52339e6	0.138214
$657$	−2.39247e6	−0.216239
$658$	794872.	0.0715702
$659$	−1.97288e6	−0.176965	−0.0884825	−	0.996078i	$-0.528202\pi$
$659$	−1.97288e6	−0.176965	−0.0884825	+	0.996078i	$0.528202\pi$
$660$	0	0
$661$	−1.78964e7	−1.59317	−0.796585	−	0.604527i	$-0.793362\pi$
$661$	−1.78964e7	−1.59317	−0.796585	+	0.604527i	$0.793362\pi$
$662$	1.27428e7	1.13011
$663$	3.70945e6	0.327737
$664$	−8.36979e6	−0.736706
$665$	0	0
$666$	126657.	0.0110648
$667$	2.13052e7	1.85426
$668$	−5.85997e6	−0.508106
$669$	−1.05596e6	−0.0912180
$670$	0	0
$671$	2.24961e6	0.192886
$672$	−350172.	−0.0299129
$673$	7.40111e6	0.629882	0.314941	−	0.949111i	$-0.398015\pi$
$673$	7.40111e6	0.629882	0.314941	+	0.949111i	$0.398015\pi$
$674$	−5.80965e6	−0.492607
$675$	0	0
$676$	3.15969e7	2.65937
$677$	6.94001e6	0.581954	0.290977	−	0.956730i	$-0.406020\pi$
$677$	6.94001e6	0.581954	0.290977	+	0.956730i	$0.406020\pi$
$678$	2.20269e7	1.84026
$679$	−711583.	−0.0592313
$680$	0	0
$681$	−675461.	−0.0558127
$682$	−1.17002e7	−0.963233
$683$	−5.97233e6	−0.489882	−0.244941	−	0.969538i	$-0.578769\pi$
$683$	−5.97233e6	−0.489882	−0.244941	+	0.969538i	$0.578769\pi$
$684$	−6.41348e6	−0.524147
$685$	0	0
$686$	−2.72442e6	−0.221037
$687$	5.71455e6	0.461945
$688$	−5.67003e6	−0.456682
$689$	−2.48874e7	−1.99724
$690$	0	0
$691$	−2.05711e7	−1.63893	−0.819467	−	0.573126i	$-0.805731\pi$
$691$	−2.05711e7	−1.63893	−0.819467	+	0.573126i	$0.805731\pi$
$692$	2.39348e7	1.90005
$693$	−85188.4	−0.00673825
$694$	1.17366e6	0.0925004
$695$	0	0
$696$	1.46642e7	1.14746
$697$	2.41275e6	0.188118
$698$	−3.03335e7	−2.35659
$699$	3.37753e6	0.261460
$700$	0	0
$701$	9.61010e6	0.738640	0.369320	−	0.929302i	$-0.379591\pi$
$701$	9.61010e6	0.738640	0.369320	+	0.929302i	$0.379591\pi$
$702$	−6.61330e6	−0.506496
$703$	239358.	0.0182667
$704$	−6.10024e6	−0.463890
$705$	0	0
$706$	1.56799e7	1.18394
$707$	1.56572e6	0.117806
$708$	−1.16616e7	−0.874329
$709$	2.13666e6	0.159632	0.0798161	−	0.996810i	$-0.474567\pi$
$709$	2.13666e6	0.159632	0.0798161	+	0.996810i	$0.474567\pi$
$710$	0	0
$711$	−832743.	−0.0617784
$712$	506195.	0.0374212
$713$	−2.95180e7	−2.17451
$714$	310435.	0.0227890
$715$	0	0
$716$	−3.45607e7	−2.51941
$717$	−7.73576e6	−0.561960
$718$	4.24279e6	0.307143
$719$	1.57303e7	1.13479	0.567395	−	0.823446i	$-0.307951\pi$
$719$	1.57303e7	1.13479	0.567395	+	0.823446i	$0.307951\pi$
$720$	0	0
$721$	−955327.	−0.0684406
$722$	4.01346e6	0.286534
$723$	−4.97056e6	−0.353639
$724$	586797.	0.0416046
$725$	0	0
$726$	−1.23150e6	−0.0867147
$727$	−1.39695e7	−0.980271	−0.490136	−	0.871646i	$-0.663053\pi$
$727$	−1.39695e7	−0.980271	−0.490136	+	0.871646i	$0.663053\pi$
$728$	1.84082e6	0.128731
$729$	531441.	0.0370370
$730$	0	0
$731$	−8.98018e6	−0.621572
$732$	9.26084e6	0.638812
$733$	−2.03921e6	−0.140185	−0.0700927	−	0.997540i	$-0.522329\pi$
$733$	−2.03921e6	−0.140185	−0.0700927	+	0.997540i	$0.522329\pi$
$734$	4.55931e7	3.12362
$735$	0	0
$736$	−1.27711e7	−0.869030
$737$	−4.77680e6	−0.323943
$738$	−4.30152e6	−0.290724
$739$	−1.13456e7	−0.764216	−0.382108	−	0.924118i	$-0.624802\pi$
$739$	−1.13456e7	−0.764216	−0.382108	+	0.924118i	$0.624802\pi$
$740$	0	0
$741$	−1.24979e7	−0.836161
$742$	−2.08277e6	−0.138877
$743$	1.17404e7	0.780206	0.390103	−	0.920771i	$-0.372439\pi$
$743$	1.17404e7	0.780206	0.390103	+	0.920771i	$0.372439\pi$
$744$	−2.03170e7	−1.34564
$745$	0	0
$746$	−5.65749e6	−0.372200
$747$	3.10719e6	0.203735
$748$	2.84359e6	0.185829
$749$	570908.	0.0371845
$750$	0	0
$751$	2.23069e7	1.44324	0.721620	−	0.692289i	$-0.243398\pi$
$751$	2.23069e7	1.44324	0.721620	+	0.692289i	$0.243398\pi$
$752$	2.62339e6	0.169168
$753$	1.78866e6	0.114958
$754$	6.77447e7	4.33957
$755$	0	0
$756$	−350690.	−0.0223161
$757$	2.86550e7	1.81744	0.908721	−	0.417403i	$-0.137060\pi$
$757$	2.86550e7	1.81744	0.908721	+	0.417403i	$0.137060\pi$
$758$	3.64808e7	2.30617
$759$	−3.10691e6	−0.195760
$760$	0	0
$761$	−1.61728e7	−1.01233	−0.506167	−	0.862435i	$-0.668938\pi$
$761$	−1.61728e7	−1.01233	−0.506167	+	0.862435i	$0.668938\pi$
$762$	−1.88974e6	−0.117900
$763$	−1.27189e6	−0.0790933
$764$	2.44604e7	1.51611
$765$	0	0
$766$	2.61702e7	1.61152
$767$	−2.27248e7	−1.39480
$768$	−1.30637e7	−0.799214
$769$	1.31452e7	0.801588	0.400794	−	0.916168i	$-0.368734\pi$
$769$	1.31452e7	0.801588	0.400794	+	0.916168i	$0.368734\pi$
$770$	0	0
$771$	689271.	0.0417594
$772$	2.45932e7	1.48515
$773$	2.45719e7	1.47907	0.739537	−	0.673116i	$-0.235045\pi$
$773$	2.45719e7	1.47907	0.739537	+	0.673116i	$0.235045\pi$
$774$	1.60101e7	0.960599
$775$	0	0
$776$	−1.78627e7	−1.06486
$777$	13088.1	0.000777722 0
$778$	−3.47550e7	−2.05858
$779$	−8.12902e6	−0.479949
$780$	0	0
$781$	397244.	0.0233040
$782$	1.13219e7	0.662067
$783$	−5.44392e6	−0.317327
$784$	−4.48570e6	−0.260640
$785$	0	0
$786$	−1.00595e7	−0.580793
$787$	2.62366e7	1.50998	0.754989	−	0.655737i	$-0.227642\pi$
$787$	2.62366e7	1.50998	0.754989	+	0.655737i	$0.227642\pi$
$788$	2.30558e7	1.32271
$789$	−1.89731e7	−1.08504
$790$	0	0
$791$	2.27614e6	0.129348
$792$	−2.13847e6	−0.121141
$793$	1.80465e7	1.01908
$794$	3.60138e7	2.02730
$795$	0	0
$796$	−2.78375e7	−1.55721
$797$	333892.	0.0186192	0.00930959	−	0.999957i	$-0.497037\pi$
$797$	333892.	0.0186192	0.00930959	+	0.999957i	$0.497037\pi$
$798$	−1.04592e6	−0.0581420
$799$	4.15492e6	0.230248
$800$	0	0
$801$	−187919.	−0.0103488
$802$	−1.01119e7	−0.555133
$803$	−3.57394e6	−0.195595
$804$	−1.96643e7	−1.07285
$805$	0	0
$806$	−9.38591e7	−5.08907
$807$	−1.77588e7	−0.959909
$808$	3.93040e7	2.11792
$809$	2.96083e6	0.159053	0.0795265	−	0.996833i	$-0.474659\pi$
$809$	2.96083e6	0.159053	0.0795265	+	0.996833i	$0.474659\pi$
$810$	0	0
$811$	−2.24739e7	−1.19985	−0.599925	−	0.800056i	$-0.704803\pi$
$811$	−2.24739e7	−1.19985	−0.599925	+	0.800056i	$0.704803\pi$
$812$	3.59236e6	0.191201
$813$	−8.33783e6	−0.442412
$814$	189204.	0.0100085
$815$	0	0
$816$	1.02456e6	0.0538655
$817$	3.02560e7	1.58583
$818$	−1.85205e7	−0.967765
$819$	−683384.	−0.0356004
$820$	0	0
$821$	2.07336e7	1.07353	0.536767	−	0.843730i	$-0.319645\pi$
$821$	2.07336e7	1.07353	0.536767	+	0.843730i	$0.319645\pi$
$822$	1.48411e7	0.766104
$823$	−2.99008e7	−1.53880	−0.769401	−	0.638766i	$-0.779445\pi$
$823$	−2.99008e7	−1.53880	−0.769401	+	0.638766i	$0.779445\pi$
$824$	−2.39814e7	−1.23043
$825$	0	0
$826$	−1.90178e6	−0.0969865
$827$	3.09628e7	1.57426	0.787130	−	0.616787i	$-0.211566\pi$
$827$	3.09628e7	1.57426	0.787130	+	0.616787i	$0.211566\pi$
$828$	−1.27900e7	−0.648329
$829$	−2.79498e7	−1.41251	−0.706255	−	0.707957i	$-0.749617\pi$
$829$	−2.79498e7	−1.41251	−0.706255	+	0.707957i	$0.749617\pi$
$830$	0	0
$831$	1.44576e7	0.726263
$832$	−4.89363e7	−2.45088
$833$	−7.10445e6	−0.354746
$834$	−2.42429e7	−1.20689
$835$	0	0
$836$	−9.58063e6	−0.474109
$837$	7.54246e6	0.372134
$838$	9.32228e6	0.458577
$839$	−5.17408e6	−0.253763	−0.126882	−	0.991918i	$-0.540497\pi$
$839$	−5.17408e6	−0.253763	−0.126882	+	0.991918i	$0.540497\pi$
$840$	0	0
$841$	3.52548e7	1.71881
$842$	−5.01516e7	−2.43784
$843$	−7.33381e6	−0.355435
$844$	−4.96670e7	−2.40000
$845$	0	0
$846$	−7.40751e6	−0.355833
$847$	−127257.	−0.00609498
$848$	−6.87394e6	−0.328259
$849$	553190.	0.0263394
$850$	0	0
$851$	477337.	0.0225944
$852$	1.63531e6	0.0771793
$853$	−4.50230e6	−0.211866	−0.105933	−	0.994373i	$-0.533783\pi$
$853$	−4.50230e6	−0.211866	−0.105933	+	0.994373i	$0.533783\pi$
$854$	1.51027e6	0.0708613
$855$	0	0
$856$	1.43314e7	0.668503
$857$	−2.97267e7	−1.38259	−0.691296	−	0.722572i	$-0.742960\pi$
$857$	−2.97267e7	−1.38259	−0.691296	+	0.722572i	$0.742960\pi$
$858$	−9.87913e6	−0.458143
$859$	2.65435e7	1.22737	0.613685	−	0.789551i	$-0.289686\pi$
$859$	2.65435e7	1.22737	0.613685	+	0.789551i	$0.289686\pi$
$860$	0	0
$861$	−444496.	−0.0204343
$862$	3.15795e7	1.44756
$863$	1.61893e7	0.739949	0.369974	−	0.929042i	$-0.379366\pi$
$863$	1.61893e7	0.739949	0.369974	+	0.929042i	$0.379366\pi$
$864$	3.26329e6	0.148721
$865$	0	0
$866$	2.86208e7	1.29684
$867$	−1.11560e7	−0.504036
$868$	−4.97715e6	−0.224224
$869$	−1.24397e6	−0.0558807
$870$	0	0
$871$	−3.83196e7	−1.71150
$872$	−3.19281e7	−1.42194
$873$	6.63133e6	0.294486
$874$	−3.81457e7	−1.68914
$875$	0	0
$876$	−1.47126e7	−0.647783
$877$	1.71684e6	0.0753757	0.0376879	−	0.999290i	$-0.488001\pi$
$877$	1.71684e6	0.0753757	0.0376879	+	0.999290i	$0.488001\pi$
$878$	−2.39274e7	−1.04751
$879$	2.68671e6	0.117287
$880$	0	0
$881$	1.59472e6	0.0692221	0.0346110	−	0.999401i	$-0.488981\pi$
$881$	1.59472e6	0.0692221	0.0346110	+	0.999401i	$0.488981\pi$
$882$	1.26660e7	0.548237
$883$	−3.27757e7	−1.41465	−0.707327	−	0.706887i	$-0.750099\pi$
$883$	−3.27757e7	−1.41465	−0.707327	+	0.706887i	$0.750099\pi$
$884$	2.28114e7	0.981796
$885$	0	0
$886$	−1.03598e7	−0.443372
$887$	−2.30536e7	−0.983853	−0.491926	−	0.870637i	$-0.663707\pi$
$887$	−2.30536e7	−0.983853	−0.491926	+	0.870637i	$0.663707\pi$
$888$	328548.	0.0139819
$889$	−195276.	−0.00828693
$890$	0	0
$891$	793881.	0.0335013
$892$	−6.49365e6	−0.273260
$893$	−1.39987e7	−0.587436
$894$	1.12405e7	0.470371
$895$	0	0
$896$	−2.85031e6	−0.118610
$897$	−2.49237e7	−1.03427
$898$	5.59347e7	2.31468
$899$	−7.72627e7	−3.18838
$900$	0	0
$901$	−1.08869e7	−0.446780
$902$	−6.42572e6	−0.262970
$903$	1.65440e6	0.0675183
$904$	5.71376e7	2.32542
$905$	0	0
$906$	−3.41602e7	−1.38261
$907$	2.13981e7	0.863689	0.431845	−	0.901948i	$-0.357863\pi$
$907$	2.13981e7	0.863689	0.431845	+	0.901948i	$0.357863\pi$
$908$	−4.15378e6	−0.167197
$909$	−1.45912e7	−0.585707
$910$	0	0
$911$	−2.14014e7	−0.854369	−0.427184	−	0.904165i	$-0.640494\pi$
$911$	−2.14014e7	−0.854369	−0.427184	+	0.904165i	$0.640494\pi$
$912$	−3.45194e6	−0.137428
$913$	4.64160e6	0.184285
$914$	3.66123e7	1.44964
$915$	0	0
$916$	3.51419e7	1.38384
$917$	−1.03950e6	−0.0408226
$918$	−2.89298e6	−0.113302
$919$	5.88304e6	0.229781	0.114890	−	0.993378i	$-0.463348\pi$
$919$	5.88304e6	0.229781	0.114890	+	0.993378i	$0.463348\pi$
$920$	0	0
$921$	1.21847e7	0.473331
$922$	2.02645e7	0.785069
$923$	3.18670e6	0.123123
$924$	−523870.	−0.0201857
$925$	0	0
$926$	−4.68625e7	−1.79597
$927$	8.90281e6	0.340273
$928$	−3.34282e7	−1.27422
$929$	1.32356e7	0.503158	0.251579	−	0.967837i	$-0.419050\pi$
$929$	1.32356e7	0.503158	0.251579	+	0.967837i	$0.419050\pi$
$930$	0	0
$931$	2.39363e7	0.905071
$932$	2.07703e7	0.783253
$933$	−2.41488e7	−0.908222
$934$	−2.40281e7	−0.901265
$935$	0	0
$936$	−1.71548e7	−0.640025
$937$	−4.19890e7	−1.56238	−0.781189	−	0.624294i	$-0.785387\pi$
$937$	−4.19890e7	−1.56238	−0.781189	+	0.624294i	$0.785387\pi$
$938$	−3.20688e6	−0.119008
$939$	−2.57371e7	−0.952567
$940$	0	0
$941$	−4.99313e7	−1.83823	−0.919113	−	0.393993i	$-0.871093\pi$
$941$	−4.99313e7	−1.83823	−0.919113	+	0.393993i	$0.871093\pi$
$942$	−1.33526e7	−0.490274
$943$	−1.62112e7	−0.593659
$944$	−6.27664e6	−0.229243
$945$	0	0
$946$	2.39163e7	0.868894
$947$	1.72916e7	0.626557	0.313278	−	0.949661i	$-0.398573\pi$
$947$	1.72916e7	0.626557	0.313278	+	0.949661i	$0.398573\pi$
$948$	−5.12099e6	−0.185069
$949$	−2.86703e7	−1.03339
$950$	0	0
$951$	999992.	0.0358546
$952$	805265.	0.0287969
$953$	1.17729e7	0.419905	0.209953	−	0.977712i	$-0.432669\pi$
$953$	1.17729e7	0.419905	0.209953	+	0.977712i	$0.432669\pi$
$954$	1.94095e7	0.690469
$955$	0	0
$956$	−4.75714e7	−1.68345
$957$	−8.13228e6	−0.287033
$958$	−4.82416e7	−1.69827
$959$	1.53360e6	0.0538477
$960$	0	0
$961$	7.84170e7	2.73906
$962$	1.51780e6	0.0528783
$963$	−5.32036e6	−0.184874
$964$	−3.05667e7	−1.05939
$965$	0	0
$966$	−2.08581e6	−0.0719170
$967$	−4.31614e7	−1.48433	−0.742164	−	0.670218i	$-0.766200\pi$
$967$	−4.31614e7	−1.48433	−0.742164	+	0.670218i	$0.766200\pi$
$968$	−3.19450e6	−0.109576
$969$	−5.46717e6	−0.187048
$970$	0	0
$971$	1.06735e7	0.363293	0.181647	−	0.983364i	$-0.441857\pi$
$971$	1.06735e7	0.363293	0.181647	+	0.983364i	$0.441857\pi$
$972$	3.26812e6	0.110951
$973$	−2.50513e6	−0.0848298
$974$	5.47432e7	1.84898
$975$	0	0
$976$	4.98448e6	0.167492
$977$	3.55410e7	1.19122	0.595612	−	0.803272i	$-0.296909\pi$
$977$	3.55410e7	1.19122	0.595612	+	0.803272i	$0.296909\pi$
$978$	1.46743e7	0.490582
$979$	−280719.	−0.00936084
$980$	0	0
$981$	1.18529e7	0.393236
$982$	−8.93989e6	−0.295838
$983$	1.30764e6	0.0431623	0.0215811	−	0.999767i	$-0.493130\pi$
$983$	1.30764e6	0.0431623	0.0215811	+	0.999767i	$0.493130\pi$
$984$	−1.11581e7	−0.367368
$985$	0	0
$986$	2.96348e7	0.970756
$987$	−765453.	−0.0250107
$988$	−7.68561e7	−2.50487
$989$	6.03377e7	1.96154
$990$	0	0
$991$	−2.32452e7	−0.751882	−0.375941	−	0.926644i	$-0.622680\pi$
$991$	−2.32452e7	−0.751882	−0.375941	+	0.926644i	$0.622680\pi$
$992$	4.63142e7	1.49429
$993$	−1.22712e7	−0.394924
$994$	266688.	0.00856125
$995$	0	0
$996$	1.91078e7	0.610326
$997$	−6.20462e6	−0.197686	−0.0988432	−	0.995103i	$-0.531514\pi$
$997$	−6.20462e6	−0.197686	−0.0988432	+	0.995103i	$0.531514\pi$
$998$	1.37926e7	0.438348
$999$	−121970.	−0.00386668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.d.1.1		2
5.4	even	2		33.6.a.d.1.2	✓	2
15.14	odd	2		99.6.a.e.1.1		2
20.19	odd	2		528.6.a.q.1.2		2
55.54	odd	2		363.6.a.g.1.1		2
165.164	even	2		1089.6.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.d.1.2	✓	2	5.4	even	2
99.6.a.e.1.1		2	15.14	odd	2
363.6.a.g.1.1		2	55.54	odd	2
528.6.a.q.1.2		2	20.19	odd	2
825.6.a.d.1.1		2	1.1	even	1	trivial
1089.6.a.o.1.2		2	165.164	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)

\(f(q)\)	\(=\)	\(q-9.34590 q^{2} +9.00000 q^{3} +55.3459 q^{4} -84.1131 q^{6} -8.69181 q^{7} -218.189 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.34590 q^{2} +9.00000 q^{3} +55.3459 q^{4} -84.1131 q^{6} -8.69181 q^{7} -218.189 q^{8} +81.0000 q^{9} +121.000 q^{11} +498.113 q^{12} +970.666 q^{13} +81.2328 q^{14} +268.100 q^{16} +424.616 q^{17} -757.018 q^{18} -1430.62 q^{19} -78.2263 q^{21} -1130.85 q^{22} -2852.99 q^{23} -1963.70 q^{24} -9071.75 q^{26} +729.000 q^{27} -481.056 q^{28} -7467.66 q^{29} +10346.3 q^{31} +4476.40 q^{32} +1089.00 q^{33} -3968.42 q^{34} +4483.02 q^{36} -167.311 q^{37} +13370.4 q^{38} +8735.99 q^{39} +5682.18 q^{41} +731.095 q^{42} -21148.9 q^{43} +6696.85 q^{44} +26663.8 q^{46} +9785.11 q^{47} +2412.90 q^{48} -16731.5 q^{49} +3821.55 q^{51} +53722.4 q^{52} -25639.5 q^{53} -6813.16 q^{54} +1896.45 q^{56} -12875.5 q^{57} +69792.0 q^{58} -23411.5 q^{59} +18591.8 q^{61} -96695.6 q^{62} -704.036 q^{63} -50415.2 q^{64} -10177.7 q^{66} -39477.7 q^{67} +23500.8 q^{68} -25676.9 q^{69} +3283.01 q^{71} -17673.3 q^{72} -29536.7 q^{73} +1563.67 q^{74} -79178.8 q^{76} -1051.71 q^{77} -81645.7 q^{78} -10280.8 q^{79} +6561.00 q^{81} -53105.1 q^{82} +38360.3 q^{83} -4329.50 q^{84} +197656. q^{86} -67208.9 q^{87} -26400.8 q^{88} -2319.99 q^{89} -8436.84 q^{91} -157902. q^{92} +93116.8 q^{93} -91450.7 q^{94} +40287.6 q^{96} +81868.3 q^{97} +156371. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9} + 242 q^{11} + 837 q^{12} + 66 q^{13} + 304 q^{14} - 543 q^{16} + 920 q^{17} - 81 q^{18} - 2932 q^{19} + 162 q^{21} - 121 q^{22} - 5246 q^{23} - 1539 q^{24} - 16622 q^{26} + 1458 q^{27} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} - 3803 q^{32} + 2178 q^{33} + 166 q^{34} + 7533 q^{36} - 5996 q^{37} + 840 q^{38} + 594 q^{39} + 24244 q^{41} + 2736 q^{42} - 20360 q^{43} + 11253 q^{44} + 6692 q^{46} + 5806 q^{47} - 4887 q^{48} - 32826 q^{49} + 8280 q^{51} + 19658 q^{52} - 40770 q^{53} - 729 q^{54} + 3156 q^{56} - 26388 q^{57} + 26958 q^{58} + 18212 q^{59} - 11398 q^{61} - 100120 q^{62} + 1458 q^{63} - 93559 q^{64} - 1089 q^{66} - 65368 q^{67} + 42154 q^{68} - 47214 q^{69} + 61446 q^{71} - 13851 q^{72} - 53412 q^{73} - 47082 q^{74} - 135712 q^{76} + 2178 q^{77} - 149598 q^{78} + 17122 q^{79} + 13122 q^{81} + 101810 q^{82} + 14304 q^{83} + 4716 q^{84} + 204240 q^{86} - 113400 q^{87} - 20691 q^{88} - 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 89424 q^{93} - 124660 q^{94} - 34227 q^{96} + 183056 q^{97} + 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9 + 242 * q^11 + 837 * q^12 + 66 * q^13 + 304 * q^14 - 543 * q^16 + 920 * q^17 - 81 * q^18 - 2932 * q^19 + 162 * q^21 - 121 * q^22 - 5246 * q^23 - 1539 * q^24 - 16622 * q^26 + 1458 * q^27 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 - 3803 * q^32 + 2178 * q^33 + 166 * q^34 + 7533 * q^36 - 5996 * q^37 + 840 * q^38 + 594 * q^39 + 24244 * q^41 + 2736 * q^42 - 20360 * q^43 + 11253 * q^44 + 6692 * q^46 + 5806 * q^47 - 4887 * q^48 - 32826 * q^49 + 8280 * q^51 + 19658 * q^52 - 40770 * q^53 - 729 * q^54 + 3156 * q^56 - 26388 * q^57 + 26958 * q^58 + 18212 * q^59 - 11398 * q^61 - 100120 * q^62 + 1458 * q^63 - 93559 * q^64 - 1089 * q^66 - 65368 * q^67 + 42154 * q^68 - 47214 * q^69 + 61446 * q^71 - 13851 * q^72 - 53412 * q^73 - 47082 * q^74 - 135712 * q^76 + 2178 * q^77 - 149598 * q^78 + 17122 * q^79 + 13122 * q^81 + 101810 * q^82 + 14304 * q^83 + 4716 * q^84 + 204240 * q^86 - 113400 * q^87 - 20691 * q^88 - 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 89424 * q^93 - 124660 * q^94 - 34227 * q^96 + 183056 * q^97 + 22047 * q^98 + 19602 * q^99

Introduction

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