LMFDB - Embedded newform 825.6.a.s.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 $ x^9 - x^8 - 229x^7 + 267x^6 + 16434x^5 - 16568x^4 - 405504x^3 + 202288x^2 + 2184608*x + 1190400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$10.2706$ 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)

$f(q)$	$=$	$q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} -121.000 q^{11} -661.363 q^{12} -26.3756 q^{13} +1333.64 q^{14} +2024.50 q^{16} -1414.41 q^{17} +831.917 q^{18} +2336.09 q^{19} -1168.65 q^{21} -1242.74 q^{22} +939.766 q^{23} -3834.65 q^{24} -270.892 q^{26} -729.000 q^{27} +9542.04 q^{28} +6494.78 q^{29} +3501.81 q^{31} +7158.44 q^{32} +1089.00 q^{33} -14526.8 q^{34} +5952.27 q^{36} +15597.6 q^{37} +23993.0 q^{38} +237.380 q^{39} -12175.4 q^{41} -12002.8 q^{42} +10617.1 q^{43} -8891.66 q^{44} +9651.94 q^{46} -11000.2 q^{47} -18220.5 q^{48} +54.1649 q^{49} +12729.7 q^{51} -1938.20 q^{52} -2759.73 q^{53} -7487.25 q^{54} +55325.7 q^{56} -21024.8 q^{57} +66705.2 q^{58} -22393.5 q^{59} +34019.6 q^{61} +35965.6 q^{62} +10517.9 q^{63} +8737.42 q^{64} +11184.7 q^{66} -10536.4 q^{67} -103938. q^{68} -8457.89 q^{69} +62585.8 q^{71} +34511.9 q^{72} +5720.55 q^{73} +160196. q^{74} +171667. q^{76} -15711.9 q^{77} +2438.03 q^{78} -70532.7 q^{79} +6561.00 q^{81} -125048. q^{82} -25794.2 q^{83} -85878.3 q^{84} +109044. q^{86} -58453.0 q^{87} -51554.8 q^{88} +34850.4 q^{89} -3424.88 q^{91} +69058.4 q^{92} -31516.3 q^{93} -112979. q^{94} -64426.0 q^{96} +18576.6 q^{97} +556.305 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) $ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * 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 $

$2$	10.2706	1.81560	0.907799	−	0.419405i	$-0.137761\pi$
$2$	10.2706	1.81560	0.907799	+	0.419405i	$0.137761\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	73.4848	2.29640
$5$	0	0
$5$	0	0
$6$	−92.4352	−1.04824
$7$	129.851	1.00161	0.500805	−	0.865560i	$-0.333037\pi$
$7$	129.851	1.00161	0.500805	+	0.865560i	$0.333037\pi$
$8$	426.072	2.35374
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−661.363	−1.32583
$13$	−26.3756	−0.0432856	−0.0216428	−	0.999766i	$-0.506890\pi$
$13$	−26.3756	−0.0432856	−0.0216428	+	0.999766i	$0.506890\pi$
$14$	1333.64	1.81852
$15$	0	0
$16$	2024.50	1.97705
$17$	−1414.41	−1.18701	−0.593503	−	0.804831i	$-0.702256\pi$
$17$	−1414.41	−1.18701	−0.593503	+	0.804831i	$0.702256\pi$
$18$	831.917	0.605200
$19$	2336.09	1.48459	0.742294	−	0.670074i	$-0.233738\pi$
$19$	2336.09	1.48459	0.742294	+	0.670074i	$0.233738\pi$
$20$	0	0
$21$	−1168.65	−0.578280
$22$	−1242.74	−0.547424
$23$	939.766	0.370425	0.185212	−	0.982699i	$-0.440703\pi$
$23$	939.766	0.370425	0.185212	+	0.982699i	$0.440703\pi$
$24$	−3834.65	−1.35893
$25$	0	0
$26$	−270.892	−0.0785893
$27$	−729.000	−0.192450
$28$	9542.04	2.30010
$29$	6494.78	1.43407	0.717034	−	0.697038i	$-0.245499\pi$
$29$	6494.78	1.43407	0.717034	+	0.697038i	$0.245499\pi$
$30$	0	0
$31$	3501.81	0.654468	0.327234	−	0.944943i	$-0.393883\pi$
$31$	3501.81	0.654468	0.327234	+	0.944943i	$0.393883\pi$
$32$	7158.44	1.23579
$33$	1089.00	0.174078
$34$	−14526.8	−2.15513
$35$	0	0
$36$	5952.27	0.765466
$37$	15597.6	1.87306	0.936531	−	0.350585i	$-0.114017\pi$
$37$	15597.6	1.87306	0.936531	+	0.350585i	$0.114017\pi$
$38$	23993.0	2.69542
$39$	237.380	0.0249909
$40$	0	0
$41$	−12175.4	−1.13116	−0.565579	−	0.824694i	$-0.691347\pi$
$41$	−12175.4	−1.13116	−0.565579	+	0.824694i	$0.691347\pi$
$42$	−12002.8	−1.04992
$43$	10617.1	0.875658	0.437829	−	0.899058i	$-0.355748\pi$
$43$	10617.1	0.875658	0.437829	+	0.899058i	$0.355748\pi$
$44$	−8891.66	−0.692390
$45$	0	0
$46$	9651.94	0.672543
$47$	−11000.2	−0.726370	−0.363185	−	0.931717i	$-0.618311\pi$
$47$	−11000.2	−0.726370	−0.363185	+	0.931717i	$0.618311\pi$
$48$	−18220.5	−1.14145
$49$	54.1649	0.00322276
$50$	0	0
$51$	12729.7	0.685319
$52$	−1938.20	−0.0994010
$53$	−2759.73	−0.134951	−0.0674755	−	0.997721i	$-0.521494\pi$
$53$	−2759.73	−0.134951	−0.0674755	+	0.997721i	$0.521494\pi$
$54$	−7487.25	−0.349412
$55$	0	0
$56$	55325.7	2.35753
$57$	−21024.8	−0.857128
$58$	66705.2	2.60369
$59$	−22393.5	−0.837515	−0.418757	−	0.908098i	$-0.637534\pi$
$59$	−22393.5	−0.837515	−0.418757	+	0.908098i	$0.637534\pi$
$60$	0	0
$61$	34019.6	1.17059	0.585295	−	0.810820i	$-0.300979\pi$
$61$	34019.6	1.17059	0.585295	+	0.810820i	$0.300979\pi$
$62$	35965.6	1.18825
$63$	10517.9	0.333870
$64$	8737.42	0.266645
$65$	0	0
$66$	11184.7	0.316055
$67$	−10536.4	−0.286751	−0.143375	−	0.989668i	$-0.545796\pi$
$67$	−10536.4	−0.286751	−0.143375	+	0.989668i	$0.545796\pi$
$68$	−103938.	−2.72584
$69$	−8457.89	−0.213865
$70$	0	0
$71$	62585.8	1.47343	0.736716	−	0.676202i	$-0.236375\pi$
$71$	62585.8	1.47343	0.736716	+	0.676202i	$0.236375\pi$
$72$	34511.9	0.784580
$73$	5720.55	0.125641	0.0628204	−	0.998025i	$-0.479990\pi$
$73$	5720.55	0.125641	0.0628204	+	0.998025i	$0.479990\pi$
$74$	160196.	3.40073
$75$	0	0
$76$	171667.	3.40921
$77$	−15711.9	−0.301997
$78$	2438.03	0.0453735
$79$	−70532.7	−1.27152	−0.635759	−	0.771887i	$-0.719313\pi$
$79$	−70532.7	−1.27152	−0.635759	+	0.771887i	$0.719313\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−125048.	−2.05373
$83$	−25794.2	−0.410986	−0.205493	−	0.978659i	$-0.565880\pi$
$83$	−25794.2	−0.410986	−0.205493	+	0.978659i	$0.565880\pi$
$84$	−85878.3	−1.32796
$85$	0	0
$86$	109044.	1.58984
$87$	−58453.0	−0.827959
$88$	−51554.8	−0.709679
$89$	34850.4	0.466372	0.233186	−	0.972432i	$-0.425085\pi$
$89$	34850.4	0.466372	0.233186	+	0.972432i	$0.425085\pi$
$90$	0	0
$91$	−3424.88	−0.0433553
$92$	69058.4	0.850643
$93$	−31516.3	−0.377857
$94$	−112979.	−1.31880
$95$	0	0
$96$	−64426.0	−0.713482
$97$	18576.6	0.200464	0.100232	−	0.994964i	$-0.468042\pi$
$97$	18576.6	0.200464	0.100232	+	0.994964i	$0.468042\pi$
$98$	556.305	0.00585124
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−73280.9	−0.714805	−0.357403	−	0.933950i	$-0.616338\pi$
$101$	−73280.9	−0.714805	−0.357403	+	0.933950i	$0.616338\pi$
$102$	130741.	1.24426
$103$	151995.	1.41168	0.705842	−	0.708369i	$-0.250569\pi$
$103$	151995.	1.41168	0.705842	+	0.708369i	$0.250569\pi$
$104$	−11237.9	−0.101883
$105$	0	0
$106$	−28344.0	−0.245017
$107$	93690.7	0.791110	0.395555	−	0.918442i	$-0.370552\pi$
$107$	93690.7	0.791110	0.395555	+	0.918442i	$0.370552\pi$
$108$	−53570.4	−0.441942
$109$	139934.	1.12813	0.564063	−	0.825732i	$-0.309237\pi$
$109$	139934.	1.12813	0.564063	+	0.825732i	$0.309237\pi$
$110$	0	0
$111$	−140378.	−1.08141
$112$	262882.	1.98023
$113$	18255.0	0.134489	0.0672444	−	0.997737i	$-0.478579\pi$
$113$	18255.0	0.134489	0.0672444	+	0.997737i	$0.478579\pi$
$114$	−215937.	−1.55620
$115$	0	0
$116$	477268.	3.29319
$117$	−2136.42	−0.0144285
$118$	−229994.	−1.52059
$119$	−183662.	−1.18892
$120$	0	0
$121$	14641.0	0.0909091
$122$	349401.	2.12532
$123$	109578.	0.653074
$124$	257330.	1.50292
$125$	0	0
$126$	108025.	0.606174
$127$	174183.	0.958287	0.479143	−	0.877737i	$-0.340947\pi$
$127$	174183.	0.958287	0.479143	+	0.877737i	$0.340947\pi$
$128$	−139332.	−0.751667
$129$	−95553.8	−0.505561
$130$	0	0
$131$	106563.	0.542536	0.271268	−	0.962504i	$-0.412557\pi$
$131$	106563.	0.542536	0.271268	+	0.962504i	$0.412557\pi$
$132$	80024.9	0.399752
$133$	303343.	1.48698
$134$	−108215.	−0.520625
$135$	0	0
$136$	−602641.	−2.79391
$137$	31743.1	0.144493	0.0722466	−	0.997387i	$-0.476983\pi$
$137$	31743.1	0.144493	0.0722466	+	0.997387i	$0.476983\pi$
$138$	−86867.4	−0.388293
$139$	333919.	1.46590	0.732950	−	0.680283i	$-0.238143\pi$
$139$	333919.	1.46590	0.732950	+	0.680283i	$0.238143\pi$
$140$	0	0
$141$	99002.2	0.419370
$142$	642793.	2.67516
$143$	3191.44	0.0130511
$144$	163984.	0.659016
$145$	0	0
$146$	58753.3	0.228113
$147$	−487.484	−0.00186066
$148$	1.14618e6	4.30130
$149$	448303.	1.65427	0.827134	−	0.562005i	$-0.189970\pi$
$149$	448303.	1.65427	0.827134	+	0.562005i	$0.189970\pi$
$150$	0	0
$151$	−61013.9	−0.217764	−0.108882	−	0.994055i	$-0.534727\pi$
$151$	−61013.9	−0.217764	−0.108882	+	0.994055i	$0.534727\pi$
$152$	995345.	3.49434
$153$	−114567.	−0.395669
$154$	−161370.	−0.548305
$155$	0	0
$156$	17443.8	0.0573892
$157$	429707.	1.39131	0.695654	−	0.718377i	$-0.255115\pi$
$157$	429707.	1.39131	0.695654	+	0.718377i	$0.255115\pi$
$158$	−724411.	−2.30857
$159$	24837.5	0.0779140
$160$	0	0
$161$	122029.	0.371021
$162$	67385.3	0.201733
$163$	−261760.	−0.771675	−0.385837	−	0.922567i	$-0.626087\pi$
$163$	−261760.	−0.771675	−0.385837	+	0.922567i	$0.626087\pi$
$164$	−894705.	−2.59759
$165$	0	0
$166$	−264921.	−0.746185
$167$	−467471.	−1.29707	−0.648535	−	0.761185i	$-0.724618\pi$
$167$	−467471.	−1.29707	−0.648535	+	0.761185i	$0.724618\pi$
$168$	−497932.	−1.36112
$169$	−370597.	−0.998126
$170$	0	0
$171$	189224.	0.494863
$172$	780195.	2.01086
$173$	−231869.	−0.589015	−0.294508	−	0.955649i	$-0.595156\pi$
$173$	−231869.	−0.589015	−0.294508	+	0.955649i	$0.595156\pi$
$174$	−600346.	−1.50324
$175$	0	0
$176$	−244964.	−0.596103
$177$	201542.	0.483539
$178$	357933.	0.846744
$179$	−271629.	−0.633642	−0.316821	−	0.948485i	$-0.602616\pi$
$179$	−271629.	−0.633642	−0.316821	+	0.948485i	$0.602616\pi$
$180$	0	0
$181$	−14022.0	−0.0318137	−0.0159069	−	0.999873i	$-0.505064\pi$
$181$	−14022.0	−0.0318137	−0.0159069	+	0.999873i	$0.505064\pi$
$182$	−35175.5	−0.0787158
$183$	−306177.	−0.675840
$184$	400408.	0.871883
$185$	0	0
$186$	−323691.	−0.686037
$187$	171144.	0.357896
$188$	−808351.	−1.66804
$189$	−94661.0	−0.192760
$190$	0	0
$191$	92573.9	0.183614	0.0918068	−	0.995777i	$-0.470736\pi$
$191$	92573.9	0.183614	0.0918068	+	0.995777i	$0.470736\pi$
$192$	−78636.8	−0.153948
$193$	475271.	0.918434	0.459217	−	0.888324i	$-0.348130\pi$
$193$	475271.	0.918434	0.459217	+	0.888324i	$0.348130\pi$
$194$	190792.	0.363962
$195$	0	0
$196$	3980.30	0.00740074
$197$	−799351.	−1.46748	−0.733740	−	0.679431i	$-0.762227\pi$
$197$	−799351.	−1.46748	−0.733740	+	0.679431i	$0.762227\pi$
$198$	−100662.	−0.182475
$199$	−248215.	−0.444319	−0.222160	−	0.975010i	$-0.571311\pi$
$199$	−248215.	−0.444319	−0.222160	+	0.975010i	$0.571311\pi$
$200$	0	0
$201$	94827.5	0.165556
$202$	−752638.	−1.29780
$203$	843351.	1.43638
$204$	935439.	1.57377
$205$	0	0
$206$	1.56108e6	2.56305
$207$	76121.0	0.123475
$208$	−53397.3	−0.0855777
$209$	−282667.	−0.447620
$210$	0	0
$211$	−699006.	−1.08087	−0.540436	−	0.841385i	$-0.681741\pi$
$211$	−699006.	−1.08087	−0.540436	+	0.841385i	$0.681741\pi$
$212$	−202798.	−0.309902
$213$	−563272.	−0.850686
$214$	962258.	1.43634
$215$	0	0
$216$	−310607.	−0.452978
$217$	454712.	0.655522
$218$	1.43721e6	2.04822
$219$	−51484.9	−0.0725387
$220$	0	0
$221$	37305.9	0.0513803
$222$	−1.44176e6	−1.96341
$223$	−877162.	−1.18118	−0.590592	−	0.806970i	$-0.701106\pi$
$223$	−877162.	−1.18118	−0.590592	+	0.806970i	$0.701106\pi$
$224$	929528.	1.23778
$225$	0	0
$226$	187490.	0.244178
$227$	−1.16059e6	−1.49491	−0.747456	−	0.664311i	$-0.768725\pi$
$227$	−1.16059e6	−1.49491	−0.747456	+	0.664311i	$0.768725\pi$
$228$	−1.54501e6	−1.96831
$229$	80586.1	0.101548	0.0507740	−	0.998710i	$-0.483831\pi$
$229$	80586.1	0.101548	0.0507740	+	0.998710i	$0.483831\pi$
$230$	0	0
$231$	141407.	0.174358
$232$	2.76725e6	3.37542
$233$	178613.	0.215538	0.107769	−	0.994176i	$-0.465629\pi$
$233$	178613.	0.215538	0.107769	+	0.994176i	$0.465629\pi$
$234$	−21942.3	−0.0261964
$235$	0	0
$236$	−1.64558e6	−1.92327
$237$	634794.	0.734111
$238$	−1.88631e6	−2.15860
$239$	734357.	0.831596	0.415798	−	0.909457i	$-0.363502\pi$
$239$	734357.	0.831596	0.415798	+	0.909457i	$0.363502\pi$
$240$	0	0
$241$	1.75930e6	1.95118	0.975590	−	0.219598i	$-0.0704745\pi$
$241$	1.75930e6	1.95118	0.975590	+	0.219598i	$0.0704745\pi$
$242$	150372.	0.165054
$243$	−59049.0	−0.0641500
$244$	2.49992e6	2.68814
$245$	0	0
$246$	1.12543e6	1.18572
$247$	−61615.8	−0.0642613
$248$	1.49203e6	1.54045
$249$	232148.	0.237283
$250$	0	0
$251$	−57478.6	−0.0575867	−0.0287934	−	0.999585i	$-0.509166\pi$
$251$	−57478.6	−0.0575867	−0.0287934	+	0.999585i	$0.509166\pi$
$252$	772905.	0.766699
$253$	−113712.	−0.111687
$254$	1.78896e6	1.73986
$255$	0	0
$256$	−1.71062e6	−1.63137
$257$	−1.77829e6	−1.67946	−0.839730	−	0.543005i	$-0.817287\pi$
$257$	−1.77829e6	−1.67946	−0.839730	+	0.543005i	$0.817287\pi$
$258$	−981393.	−0.917896
$259$	2.02535e6	1.87608
$260$	0	0
$261$	526077.	0.478023
$262$	1.09447e6	0.985029
$263$	−2.11350e6	−1.88414	−0.942071	−	0.335414i	$-0.891124\pi$
$263$	−2.11350e6	−1.88414	−0.942071	+	0.335414i	$0.891124\pi$
$264$	463993.	0.409734
$265$	0	0
$266$	3.11551e6	2.69976
$267$	−313653.	−0.269260
$268$	−774264.	−0.658494
$269$	−1.40580e6	−1.18452	−0.592261	−	0.805746i	$-0.701765\pi$
$269$	−1.40580e6	−1.18452	−0.592261	+	0.805746i	$0.701765\pi$
$270$	0	0
$271$	−1.10334e6	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$271$	−1.10334e6	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$272$	−2.86347e6	−2.34677
$273$	30823.9	0.0250312
$274$	326020.	0.262342
$275$	0	0
$276$	−621526.	−0.491119
$277$	−1.01726e6	−0.796589	−0.398295	−	0.917258i	$-0.630398\pi$
$277$	−1.01726e6	−0.796589	−0.398295	+	0.917258i	$0.630398\pi$
$278$	3.42954e6	2.66148
$279$	283647.	0.218156
$280$	0	0
$281$	811048.	0.612747	0.306373	−	0.951911i	$-0.400884\pi$
$281$	811048.	0.612747	0.306373	+	0.951911i	$0.400884\pi$
$282$	1.01681e6	0.761407
$283$	848509.	0.629782	0.314891	−	0.949128i	$-0.398032\pi$
$283$	848509.	0.629782	0.314891	+	0.949128i	$0.398032\pi$
$284$	4.59910e6	3.38359
$285$	0	0
$286$	32778.0	0.0236956
$287$	−1.58098e6	−1.13298
$288$	579834.	0.411929
$289$	580701.	0.408985
$290$	0	0
$291$	−167189.	−0.115738
$292$	420373.	0.288521
$293$	2.16475e6	1.47312	0.736559	−	0.676373i	$-0.236449\pi$
$293$	2.16475e6	1.47312	0.736559	+	0.676373i	$0.236449\pi$
$294$	−5006.75	−0.00337822
$295$	0	0
$296$	6.64569e6	4.40870
$297$	88209.0	0.0580259
$298$	4.60433e6	3.00349
$299$	−24786.8	−0.0160341
$300$	0	0
$301$	1.37864e6	0.877068
$302$	−626648.	−0.395372
$303$	659528.	0.412693
$304$	4.72942e6	2.93510
$305$	0	0
$306$	−1.17667e6	−0.718376
$307$	−3.02969e6	−1.83464	−0.917322	−	0.398145i	$-0.869654\pi$
$307$	−3.02969e6	−1.83464	−0.917322	+	0.398145i	$0.869654\pi$
$308$	−1.15459e6	−0.693505
$309$	−1.36796e6	−0.815036
$310$	0	0
$311$	994885.	0.583273	0.291637	−	0.956529i	$-0.405800\pi$
$311$	994885.	0.583273	0.291637	+	0.956529i	$0.405800\pi$
$312$	101141.	0.0588222
$313$	−764574.	−0.441122	−0.220561	−	0.975373i	$-0.570789\pi$
$313$	−764574.	−0.441122	−0.220561	+	0.975373i	$0.570789\pi$
$314$	4.41334e6	2.52606
$315$	0	0
$316$	−5.18308e6	−2.91991
$317$	1.77055e6	0.989598	0.494799	−	0.869007i	$-0.335242\pi$
$317$	1.77055e6	0.989598	0.494799	+	0.869007i	$0.335242\pi$
$318$	255096.	0.141461
$319$	−785869.	−0.432388
$320$	0	0
$321$	−843216.	−0.456748
$322$	1.25331e6	0.673625
$323$	−3.30420e6	−1.76222
$324$	482134.	0.255155
$325$	0	0
$326$	−2.68843e6	−1.40105
$327$	−1.25941e6	−0.651324
$328$	−5.18760e6	−2.66245
$329$	−1.42839e6	−0.727540
$330$	0	0
$331$	−979878.	−0.491589	−0.245794	−	0.969322i	$-0.579049\pi$
$331$	−979878.	−0.491589	−0.245794	+	0.969322i	$0.579049\pi$
$332$	−1.89548e6	−0.943787
$333$	1.26340e6	0.624354
$334$	−4.80120e6	−2.35496
$335$	0	0
$336$	−2.36594e6	−1.14329
$337$	−1.82371e6	−0.874745	−0.437372	−	0.899280i	$-0.644091\pi$
$337$	−1.82371e6	−0.874745	−0.437372	+	0.899280i	$0.644091\pi$
$338$	−3.80625e6	−1.81220
$339$	−164295.	−0.0776472
$340$	0	0
$341$	−423719.	−0.197330
$342$	1.94344e6	0.898472
$343$	−2.17536e6	−0.998382
$344$	4.52365e6	2.06107
$345$	0	0
$346$	−2.38142e6	−1.06942
$347$	−1.87237e6	−0.834772	−0.417386	−	0.908729i	$-0.637054\pi$
$347$	−1.87237e6	−0.834772	−0.417386	+	0.908729i	$0.637054\pi$
$348$	−4.29541e6	−1.90132
$349$	−4.37905e6	−1.92449	−0.962247	−	0.272177i	$-0.912256\pi$
$349$	−4.37905e6	−1.92449	−0.962247	+	0.272177i	$0.912256\pi$
$350$	0	0
$351$	19227.8	0.00833031
$352$	−866172.	−0.372604
$353$	155327.	0.0663453	0.0331726	−	0.999450i	$-0.489439\pi$
$353$	155327.	0.0663453	0.0331726	+	0.999450i	$0.489439\pi$
$354$	2.06995e6	0.877913
$355$	0	0
$356$	2.56097e6	1.07098
$357$	1.65296e6	0.686422
$358$	−2.78979e6	−1.15044
$359$	−3.77688e6	−1.54667	−0.773334	−	0.633999i	$-0.781412\pi$
$359$	−3.77688e6	−1.54667	−0.773334	+	0.633999i	$0.781412\pi$
$360$	0	0
$361$	2.98123e6	1.20400
$362$	−144014.	−0.0577610
$363$	−131769.	−0.0524864
$364$	−251677.	−0.0995610
$365$	0	0
$366$	−3.14461e6	−1.22705
$367$	3.29301e6	1.27623	0.638113	−	0.769943i	$-0.279715\pi$
$367$	3.29301e6	1.27623	0.638113	+	0.769943i	$0.279715\pi$
$368$	1.90255e6	0.732348
$369$	−986206.	−0.377052
$370$	0	0
$371$	−358352.	−0.135168
$372$	−2.31597e6	−0.867711
$373$	−3.95094e6	−1.47037	−0.735187	−	0.677864i	$-0.762906\pi$
$373$	−3.95094e6	−1.47037	−0.735187	+	0.677864i	$0.762906\pi$
$374$	1.75774e6	0.649796
$375$	0	0
$376$	−4.68690e6	−1.70969
$377$	−171304.	−0.0620745
$378$	−972224.	−0.349975
$379$	2.04599e6	0.731654	0.365827	−	0.930683i	$-0.380786\pi$
$379$	2.04599e6	0.731654	0.365827	+	0.930683i	$0.380786\pi$
$380$	0	0
$381$	−1.56764e6	−0.553267
$382$	950787.	0.333369
$383$	−3.77153e6	−1.31378	−0.656888	−	0.753988i	$-0.728127\pi$
$383$	−3.77153e6	−1.31378	−0.656888	+	0.753988i	$0.728127\pi$
$384$	1.25399e6	0.433975
$385$	0	0
$386$	4.88131e6	1.66751
$387$	859984.	0.291886
$388$	1.36510e6	0.460345
$389$	3.09884e6	1.03830	0.519152	−	0.854682i	$-0.326248\pi$
$389$	3.09884e6	1.03830	0.519152	+	0.854682i	$0.326248\pi$
$390$	0	0
$391$	−1.32921e6	−0.439697
$392$	23078.2	0.00758554
$393$	−959069.	−0.313234
$394$	−8.20980e6	−2.66435
$395$	0	0
$396$	−720224.	−0.230797
$397$	−3.75693e6	−1.19635	−0.598173	−	0.801367i	$-0.704106\pi$
$397$	−3.75693e6	−1.19635	−0.598173	+	0.801367i	$0.704106\pi$
$398$	−2.54931e6	−0.806705
$399$	−2.73009e6	−0.858508
$400$	0	0
$401$	−3.88578e6	−1.20675	−0.603376	−	0.797457i	$-0.706178\pi$
$401$	−3.88578e6	−1.20675	−0.603376	+	0.797457i	$0.706178\pi$
$402$	973933.	0.300583
$403$	−92362.2	−0.0283290
$404$	−5.38503e6	−1.64148
$405$	0	0
$406$	8.66170e6	2.60788
$407$	−1.88730e6	−0.564749
$408$	5.42377e6	1.61306
$409$	1.74268e6	0.515120	0.257560	−	0.966262i	$-0.417081\pi$
$409$	1.74268e6	0.515120	0.257560	+	0.966262i	$0.417081\pi$
$410$	0	0
$411$	−285688.	−0.0834232
$412$	1.11693e7	3.24179
$413$	−2.90781e6	−0.838863
$414$	781807.	0.224181
$415$	0	0
$416$	−188808.	−0.0534918
$417$	−3.00527e6	−0.846337
$418$	−2.90316e6	−0.812699
$419$	2.81726e6	0.783956	0.391978	−	0.919974i	$-0.371791\pi$
$419$	2.81726e6	0.783956	0.391978	+	0.919974i	$0.371791\pi$
$420$	0	0
$421$	−3.92334e6	−1.07882	−0.539412	−	0.842042i	$-0.681353\pi$
$421$	−3.92334e6	−1.07882	−0.539412	+	0.842042i	$0.681353\pi$
$422$	−7.17919e6	−1.96243
$423$	−891020.	−0.242123
$424$	−1.17584e6	−0.317640
$425$	0	0
$426$	−5.78513e6	−1.54451
$427$	4.41747e6	1.17247
$428$	6.88484e6	1.81671
$429$	−28723.0	−0.00753505
$430$	0	0
$431$	3.54043e6	0.918044	0.459022	−	0.888425i	$-0.348200\pi$
$431$	3.54043e6	0.918044	0.459022	+	0.888425i	$0.348200\pi$
$432$	−1.47586e6	−0.380483
$433$	−3.93249e6	−1.00797	−0.503985	−	0.863712i	$-0.668133\pi$
$433$	−3.93249e6	−1.00797	−0.503985	+	0.863712i	$0.668133\pi$
$434$	4.67016e6	1.19016
$435$	0	0
$436$	1.02830e7	2.59063
$437$	2.19538e6	0.549928
$438$	−528780.	−0.131701
$439$	−6.90652e6	−1.71040	−0.855201	−	0.518296i	$-0.826566\pi$
$439$	−6.90652e6	−1.71040	−0.855201	+	0.518296i	$0.826566\pi$
$440$	0	0
$441$	4387.36	0.00107425
$442$	383153.	0.0932860
$443$	6.56130e6	1.58848	0.794238	−	0.607607i	$-0.207870\pi$
$443$	6.56130e6	1.58848	0.794238	+	0.607607i	$0.207870\pi$
$444$	−1.03156e7	−2.48336
$445$	0	0
$446$	−9.00896e6	−2.14456
$447$	−4.03472e6	−0.955092
$448$	1.13456e6	0.267074
$449$	−1.77881e6	−0.416403	−0.208201	−	0.978086i	$-0.566761\pi$
$449$	−1.77881e6	−0.416403	−0.208201	+	0.978086i	$0.566761\pi$
$450$	0	0
$451$	1.47322e6	0.341057
$452$	1.34147e6	0.308840
$453$	549125.	0.125726
$454$	−1.19200e7	−2.71416
$455$	0	0
$456$	−8.95811e6	−2.01746
$457$	1.32684e6	0.297186	0.148593	−	0.988898i	$-0.452526\pi$
$457$	1.32684e6	0.297186	0.148593	+	0.988898i	$0.452526\pi$
$458$	827666.	0.184370
$459$	1.03111e6	0.228440
$460$	0	0
$461$	5.96751e6	1.30780	0.653899	−	0.756582i	$-0.273132\pi$
$461$	5.96751e6	1.30780	0.653899	+	0.756582i	$0.273132\pi$
$462$	1.45233e6	0.316564
$463$	−2.85291e6	−0.618494	−0.309247	−	0.950982i	$-0.600077\pi$
$463$	−2.85291e6	−0.618494	−0.309247	+	0.950982i	$0.600077\pi$
$464$	1.31487e7	2.83522
$465$	0	0
$466$	1.83446e6	0.391330
$467$	4.25236e6	0.902274	0.451137	−	0.892455i	$-0.351019\pi$
$467$	4.25236e6	0.902274	0.451137	+	0.892455i	$0.351019\pi$
$468$	−156994.	−0.0331337
$469$	−1.36816e6	−0.287213
$470$	0	0
$471$	−3.86736e6	−0.803272
$472$	−9.54126e6	−1.97129
$473$	−1.28467e6	−0.264021
$474$	6.51970e6	1.33285
$475$	0	0
$476$	−1.34964e7	−2.73023
$477$	−223538.	−0.0449837
$478$	7.54227e6	1.50984
$479$	4.59659e6	0.915371	0.457686	−	0.889114i	$-0.348678\pi$
$479$	4.59659e6	0.915371	0.457686	+	0.889114i	$0.348678\pi$
$480$	0	0
$481$	−411394.	−0.0810766
$482$	1.80690e7	3.54256
$483$	−1.09826e6	−0.214209
$484$	1.07589e6	0.208764
$485$	0	0
$486$	−606467.	−0.116471
$487$	−4.45476e6	−0.851142	−0.425571	−	0.904925i	$-0.639927\pi$
$487$	−4.45476e6	−0.851142	−0.425571	+	0.904925i	$0.639927\pi$
$488$	1.44948e7	2.75526
$489$	2.35584e6	0.445527
$490$	0	0
$491$	9.96192e6	1.86483	0.932416	−	0.361388i	$-0.117697\pi$
$491$	9.96192e6	1.86483	0.932416	+	0.361388i	$0.117697\pi$
$492$	8.05235e6	1.49972
$493$	−9.18629e6	−1.70225
$494$	−632829.	−0.116673
$495$	0	0
$496$	7.08941e6	1.29392
$497$	8.12680e6	1.47580
$498$	2.38429e6	0.430810
$499$	1.01344e7	1.82199	0.910993	−	0.412422i	$-0.135317\pi$
$499$	1.01344e7	1.82199	0.910993	+	0.412422i	$0.135317\pi$
$500$	0	0
$501$	4.20724e6	0.748864
$502$	−590339.	−0.104554
$503$	−3.76588e6	−0.663662	−0.331831	−	0.943339i	$-0.607666\pi$
$503$	−3.76588e6	−0.663662	−0.331831	+	0.943339i	$0.607666\pi$
$504$	4.48139e6	0.785843
$505$	0	0
$506$	−1.16788e6	−0.202779
$507$	3.33538e6	0.576269
$508$	1.27998e7	2.20061
$509$	1.22083e6	0.208863	0.104432	−	0.994532i	$-0.466698\pi$
$509$	1.22083e6	0.208863	0.104432	+	0.994532i	$0.466698\pi$
$510$	0	0
$511$	742816.	0.125843
$512$	−1.31104e7	−2.21025
$513$	−1.70301e6	−0.285709
$514$	−1.82641e7	−3.04922
$515$	0	0
$516$	−7.02175e6	−1.16097
$517$	1.33103e6	0.219009
$518$	2.08015e7	3.40620
$519$	2.08682e6	0.340068
$520$	0	0
$521$	3.00864e6	0.485597	0.242798	−	0.970077i	$-0.421935\pi$
$521$	3.00864e6	0.485597	0.242798	+	0.970077i	$0.421935\pi$
$522$	5.40312e6	0.867897
$523$	−2.82585e6	−0.451747	−0.225874	−	0.974157i	$-0.572524\pi$
$523$	−2.82585e6	−0.451747	−0.225874	+	0.974157i	$0.572524\pi$
$524$	7.83077e6	1.24588
$525$	0	0
$526$	−2.17069e7	−3.42085
$527$	−4.95300e6	−0.776858
$528$	2.20468e6	0.344160
$529$	−5.55318e6	−0.862786
$530$	0	0
$531$	−1.81388e6	−0.279172
$532$	2.22911e7	3.41470
$533$	321133.	0.0489628
$534$	−3.22140e6	−0.488868
$535$	0	0
$536$	−4.48927e6	−0.674937
$537$	2.44467e6	0.365834
$538$	−1.44384e7	−2.15062
$539$	−6553.96	−0.000971699 0
$540$	0	0
$541$	9.06915e6	1.33221	0.666106	−	0.745857i	$-0.267960\pi$
$541$	9.06915e6	1.33221	0.666106	+	0.745857i	$0.267960\pi$
$542$	−1.13319e7	−1.65693
$543$	126198.	0.0183677
$544$	−1.01250e7	−1.46689
$545$	0	0
$546$	316580.	0.0454466
$547$	1.77907e6	0.254229	0.127114	−	0.991888i	$-0.459428\pi$
$547$	1.77907e6	0.254229	0.127114	+	0.991888i	$0.459428\pi$
$548$	2.33263e6	0.331814
$549$	2.75559e6	0.390197
$550$	0	0
$551$	1.51724e7	2.12900
$552$	−3.60367e6	−0.503382
$553$	−9.15871e6	−1.27357
$554$	−1.04479e7	−1.44629
$555$	0	0
$556$	2.45380e7	3.36629
$557$	−361115.	−0.0493182	−0.0246591	−	0.999696i	$-0.507850\pi$
$557$	−361115.	−0.0493182	−0.0246591	+	0.999696i	$0.507850\pi$
$558$	2.91322e6	0.396084
$559$	−280032.	−0.0379034
$560$	0	0
$561$	−1.54029e6	−0.206631
$562$	8.32993e6	1.11250
$563$	1.24161e7	1.65088	0.825440	−	0.564489i	$-0.190927\pi$
$563$	1.24161e7	1.65088	0.825440	+	0.564489i	$0.190927\pi$
$564$	7.27516e6	0.963041
$565$	0	0
$566$	8.71467e6	1.14343
$567$	851949.	0.111290
$568$	2.66661e7	3.46808
$569$	−1.47941e7	−1.91561	−0.957806	−	0.287417i	$-0.907204\pi$
$569$	−1.47941e7	−1.91561	−0.957806	+	0.287417i	$0.907204\pi$
$570$	0	0
$571$	7.97003e6	1.02299	0.511493	−	0.859287i	$-0.329093\pi$
$571$	7.97003e6	1.02299	0.511493	+	0.859287i	$0.329093\pi$
$572$	234522.	0.0299705
$573$	−833165.	−0.106009
$574$	−1.62376e7	−2.05703
$575$	0	0
$576$	707731.	0.0888817
$577$	−1.48026e7	−1.85096	−0.925480	−	0.378796i	$-0.876338\pi$
$577$	−1.48026e7	−1.85096	−0.925480	+	0.378796i	$0.876338\pi$
$578$	5.96413e6	0.742553
$579$	−4.27744e6	−0.530258
$580$	0	0
$581$	−3.34939e6	−0.411648
$582$	−1.71713e6	−0.210134
$583$	333927.	0.0406893
$584$	2.43737e6	0.295726
$585$	0	0
$586$	2.22332e7	2.67459
$587$	−1.07889e7	−1.29236	−0.646180	−	0.763185i	$-0.723635\pi$
$587$	−1.07889e7	−1.29236	−0.646180	+	0.763185i	$0.723635\pi$
$588$	−35822.7	−0.00427282
$589$	8.18056e6	0.971616
$590$	0	0
$591$	7.19416e6	0.847250
$592$	3.15772e7	3.70314
$593$	9.31223e6	1.08747	0.543734	−	0.839257i	$-0.317010\pi$
$593$	9.31223e6	1.08747	0.543734	+	0.839257i	$0.317010\pi$
$594$	905957.	0.105352
$595$	0	0
$596$	3.29434e7	3.79886
$597$	2.23393e6	0.256528
$598$	−254575.	−0.0291114
$599$	−1.19315e7	−1.35871	−0.679354	−	0.733811i	$-0.737740\pi$
$599$	−1.19315e7	−1.35871	−0.679354	+	0.733811i	$0.737740\pi$
$600$	0	0
$601$	−7.25405e6	−0.819208	−0.409604	−	0.912263i	$-0.634333\pi$
$601$	−7.25405e6	−0.819208	−0.409604	+	0.912263i	$0.634333\pi$
$602$	1.41594e7	1.59240
$603$	−853448.	−0.0955836
$604$	−4.48359e6	−0.500073
$605$	0	0
$606$	6.77374e6	0.749285
$607$	−1.37134e7	−1.51068	−0.755342	−	0.655331i	$-0.772529\pi$
$607$	−1.37134e7	−1.51068	−0.755342	+	0.655331i	$0.772529\pi$
$608$	1.67228e7	1.83464
$609$	−7.59016e6	−0.829292
$610$	0	0
$611$	290138.	0.0314414
$612$	−8.41895e6	−0.908614
$613$	−1.94436e6	−0.208990	−0.104495	−	0.994525i	$-0.533323\pi$
$613$	−1.94436e6	−0.208990	−0.104495	+	0.994525i	$0.533323\pi$
$614$	−3.11166e7	−3.33098
$615$	0	0
$616$	−6.69442e6	−0.710822
$617$	−1.02584e7	−1.08484	−0.542419	−	0.840108i	$-0.682491\pi$
$617$	−1.02584e7	−1.08484	−0.542419	+	0.840108i	$0.682491\pi$
$618$	−1.40497e7	−1.47978
$619$	4.58800e6	0.481279	0.240639	−	0.970615i	$-0.422643\pi$
$619$	4.58800e6	0.481279	0.240639	+	0.970615i	$0.422643\pi$
$620$	0	0
$621$	−685089.	−0.0712883
$622$	1.02180e7	1.05899
$623$	4.52534e6	0.467123
$624$	480575.	0.0494083
$625$	0	0
$626$	−7.85262e6	−0.800901
$627$	2.54401e6	0.258434
$628$	3.15769e7	3.19500
$629$	−2.20613e7	−2.22334
$630$	0	0
$631$	1.87262e7	1.87230	0.936152	−	0.351596i	$-0.114361\pi$
$631$	1.87262e7	1.87230	0.936152	+	0.351596i	$0.114361\pi$
$632$	−3.00520e7	−2.99282
$633$	6.29105e6	0.624042
$634$	1.81845e7	1.79671
$635$	0	0
$636$	1.82518e6	0.178922
$637$	−1428.63	−0.000139499 0
$638$	−8.07133e6	−0.785042
$639$	5.06945e6	0.491144
$640$	0	0
$641$	−4.84441e6	−0.465689	−0.232845	−	0.972514i	$-0.574803\pi$
$641$	−4.84441e6	−0.465689	−0.232845	+	0.972514i	$0.574803\pi$
$642$	−8.66032e6	−0.829271
$643$	−7.56977e6	−0.722030	−0.361015	−	0.932560i	$-0.617570\pi$
$643$	−7.56977e6	−0.722030	−0.361015	+	0.932560i	$0.617570\pi$
$644$	8.96728e6	0.852012
$645$	0	0
$646$	−3.39360e7	−3.19948
$647$	−1.35084e7	−1.26865	−0.634327	−	0.773065i	$-0.718723\pi$
$647$	−1.35084e7	−1.26865	−0.634327	+	0.773065i	$0.718723\pi$
$648$	2.79546e6	0.261527
$649$	2.70962e6	0.252520
$650$	0	0
$651$	−4.09241e6	−0.378466
$652$	−1.92354e7	−1.77207
$653$	−2.14348e6	−0.196714	−0.0983571	−	0.995151i	$-0.531359\pi$
$653$	−2.14348e6	−0.196714	−0.0983571	+	0.995151i	$0.531359\pi$
$654$	−1.29348e7	−1.18254
$655$	0	0
$656$	−2.46490e7	−2.23635
$657$	463364.	0.0418802
$658$	−1.46704e7	−1.32092
$659$	1.27830e7	1.14662	0.573311	−	0.819338i	$-0.305658\pi$
$659$	1.27830e7	1.14662	0.573311	+	0.819338i	$0.305658\pi$
$660$	0	0
$661$	1.67783e7	1.49363	0.746816	−	0.665030i	$-0.231581\pi$
$661$	1.67783e7	1.49363	0.746816	+	0.665030i	$0.231581\pi$
$662$	−1.00639e7	−0.892528
$663$	−335753.	−0.0296644
$664$	−1.09902e7	−0.967354
$665$	0	0
$666$	1.29759e7	1.13358
$667$	6.10357e6	0.531214
$668$	−3.43520e7	−2.97859
$669$	7.89446e6	0.681957
$670$	0	0
$671$	−4.11637e6	−0.352946
$672$	−8.36575e6	−0.714631
$673$	2.08834e7	1.77731	0.888657	−	0.458572i	$-0.151639\pi$
$673$	2.08834e7	1.77731	0.888657	+	0.458572i	$0.151639\pi$
$674$	−1.87306e7	−1.58819
$675$	0	0
$676$	−2.72333e7	−2.29210
$677$	1.10376e7	0.925555	0.462777	−	0.886475i	$-0.346853\pi$
$677$	1.10376e7	0.925555	0.462777	+	0.886475i	$0.346853\pi$
$678$	−1.68741e6	−0.140976
$679$	2.41218e6	0.200787
$680$	0	0
$681$	1.04453e7	0.863088
$682$	−4.35184e6	−0.358271
$683$	−6.12820e6	−0.502668	−0.251334	−	0.967900i	$-0.580869\pi$
$683$	−6.12820e6	−0.502668	−0.251334	+	0.967900i	$0.580869\pi$
$684$	1.39050e7	1.13640
$685$	0	0
$686$	−2.23423e7	−1.81266
$687$	−725275.	−0.0586288
$688$	2.14943e7	1.73122
$689$	72789.3	0.00584144
$690$	0	0
$691$	−1.18390e7	−0.943233	−0.471616	−	0.881804i	$-0.656329\pi$
$691$	−1.18390e7	−0.943233	−0.471616	+	0.881804i	$0.656329\pi$
$692$	−1.70388e7	−1.35261
$693$	−1.27267e6	−0.100666
$694$	−1.92303e7	−1.51561
$695$	0	0
$696$	−2.49052e7	−1.94880
$697$	1.72210e7	1.34269
$698$	−4.49754e7	−3.49411
$699$	−1.60752e6	−0.124441
$700$	0	0
$701$	−2.85962e6	−0.219793	−0.109896	−	0.993943i	$-0.535052\pi$
$701$	−2.85962e6	−0.219793	−0.109896	+	0.993943i	$0.535052\pi$
$702$	197480.	0.0151245
$703$	3.64373e7	2.78073
$704$	−1.05723e6	−0.0803965
$705$	0	0
$706$	1.59530e6	0.120456
$707$	−9.51557e6	−0.715956
$708$	1.48102e7	1.11040
$709$	−1.53361e7	−1.14577	−0.572887	−	0.819635i	$-0.694177\pi$
$709$	−1.53361e7	−1.14577	−0.572887	+	0.819635i	$0.694177\pi$
$710$	0	0
$711$	−5.71315e6	−0.423839
$712$	1.48488e7	1.09772
$713$	3.29088e6	0.242431
$714$	1.69768e7	1.24627
$715$	0	0
$716$	−1.99606e7	−1.45510
$717$	−6.60921e6	−0.480122
$718$	−3.87907e7	−2.80813
$719$	−4.90410e6	−0.353783	−0.176892	−	0.984230i	$-0.556604\pi$
$719$	−4.90410e6	−0.353783	−0.176892	+	0.984230i	$0.556604\pi$
$720$	0	0
$721$	1.97367e7	1.41396
$722$	3.06190e7	2.18599
$723$	−1.58337e7	−1.12651
$724$	−1.03041e6	−0.0730570
$725$	0	0
$726$	−1.35334e6	−0.0952942
$727$	1.07476e7	0.754180	0.377090	−	0.926177i	$-0.376925\pi$
$727$	1.07476e7	0.754180	0.377090	+	0.926177i	$0.376925\pi$
$728$	−1.45925e6	−0.102047
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.50169e7	−1.03941
$732$	−2.24993e7	−1.55200
$733$	−2.22318e7	−1.52832	−0.764161	−	0.645025i	$-0.776847\pi$
$733$	−2.22318e7	−1.52832	−0.764161	+	0.645025i	$0.776847\pi$
$734$	3.38211e7	2.31711
$735$	0	0
$736$	6.72726e6	0.457766
$737$	1.27490e6	0.0864587
$738$	−1.01289e7	−0.684576
$739$	1.16506e7	0.784758	0.392379	−	0.919804i	$-0.371652\pi$
$739$	1.16506e7	0.784758	0.392379	+	0.919804i	$0.371652\pi$
$740$	0	0
$741$	554542.	0.0371013
$742$	−3.68048e6	−0.245412
$743$	−8.41242e6	−0.559048	−0.279524	−	0.960139i	$-0.590177\pi$
$743$	−8.41242e6	−0.559048	−0.279524	+	0.960139i	$0.590177\pi$
$744$	−1.34282e7	−0.889378
$745$	0	0
$746$	−4.05784e7	−2.66961
$747$	−2.08933e6	−0.136995
$748$	1.25765e7	0.821872
$749$	1.21658e7	0.792384
$750$	0	0
$751$	4.28896e6	0.277493	0.138746	−	0.990328i	$-0.455693\pi$
$751$	4.28896e6	0.277493	0.138746	+	0.990328i	$0.455693\pi$
$752$	−2.22700e7	−1.43607
$753$	517308.	0.0332477
$754$	−1.75939e6	−0.112702
$755$	0	0
$756$	−6.95614e6	−0.442654
$757$	8.29492e6	0.526105	0.263053	−	0.964781i	$-0.415271\pi$
$757$	8.29492e6	0.526105	0.263053	+	0.964781i	$0.415271\pi$
$758$	2.10135e7	1.32839
$759$	1.02340e6	0.0644827
$760$	0	0
$761$	6.70114e6	0.419456	0.209728	−	0.977760i	$-0.432742\pi$
$761$	6.70114e6	0.419456	0.209728	+	0.977760i	$0.432742\pi$
$762$	−1.61006e7	−1.00451
$763$	1.81705e7	1.12994
$764$	6.80277e6	0.421650
$765$	0	0
$766$	−3.87358e7	−2.38529
$767$	590642.	0.0362523
$768$	1.53955e7	0.941872
$769$	−8.86059e6	−0.540314	−0.270157	−	0.962816i	$-0.587076\pi$
$769$	−8.86059e6	−0.540314	−0.270157	+	0.962816i	$0.587076\pi$
$770$	0	0
$771$	1.60046e7	0.969636
$772$	3.49252e7	2.10909
$773$	4.08253e6	0.245743	0.122871	−	0.992423i	$-0.460790\pi$
$773$	4.08253e6	0.245743	0.122871	+	0.992423i	$0.460790\pi$
$774$	8.83254e6	0.529948
$775$	0	0
$776$	7.91497e6	0.471840
$777$	−1.82282e7	−1.08315
$778$	3.18268e7	1.88514
$779$	−2.84428e7	−1.67930
$780$	0	0
$781$	−7.57288e6	−0.444256
$782$	−1.36518e7	−0.798313
$783$	−4.73470e6	−0.275986
$784$	109657.	0.00637156
$785$	0	0
$786$	−9.85019e6	−0.568707
$787$	−1.15210e7	−0.663063	−0.331531	−	0.943444i	$-0.607565\pi$
$787$	−1.15210e7	−0.663063	−0.331531	+	0.943444i	$0.607565\pi$
$788$	−5.87401e7	−3.36992
$789$	1.90215e7	1.08781
$790$	0	0
$791$	2.37042e6	0.134705
$792$	−4.17594e6	−0.236560
$793$	−897286.	−0.0506697
$794$	−3.85858e7	−2.17208
$795$	0	0
$796$	−1.82400e7	−1.02033
$797$	−1.71525e7	−0.956492	−0.478246	−	0.878226i	$-0.658727\pi$
$797$	−1.71525e7	−0.956492	−0.478246	+	0.878226i	$0.658727\pi$
$798$	−2.80396e7	−1.55871
$799$	1.55589e7	0.862206
$800$	0	0
$801$	2.82288e6	0.155457
$802$	−3.99092e7	−2.19098
$803$	−692186.	−0.0378821
$804$	6.96838e6	0.380182
$805$	0	0
$806$	−948614.	−0.0514342
$807$	1.26522e7	0.683884
$808$	−3.12230e7	−1.68247
$809$	−6.93330e6	−0.372450	−0.186225	−	0.982507i	$-0.559625\pi$
$809$	−6.93330e6	−0.372450	−0.186225	+	0.982507i	$0.559625\pi$
$810$	0	0
$811$	−2.74832e7	−1.46729	−0.733643	−	0.679535i	$-0.762182\pi$
$811$	−2.74832e7	−1.46729	−0.733643	+	0.679535i	$0.762182\pi$
$812$	6.19735e7	3.29849
$813$	9.93002e6	0.526895
$814$	−1.93837e7	−1.02536
$815$	0	0
$816$	2.57712e7	1.35491
$817$	2.48025e7	1.29999
$818$	1.78983e7	0.935252
$819$	−277415.	−0.0144518
$820$	0	0
$821$	−1.07753e7	−0.557921	−0.278961	−	0.960303i	$-0.589990\pi$
$821$	−1.07753e7	−0.557921	−0.278961	+	0.960303i	$0.589990\pi$
$822$	−2.93418e6	−0.151463
$823$	9.62133e6	0.495149	0.247574	−	0.968869i	$-0.420367\pi$
$823$	9.62133e6	0.495149	0.247574	+	0.968869i	$0.420367\pi$
$824$	6.47611e7	3.32274
$825$	0	0
$826$	−2.98649e7	−1.52304
$827$	1.14957e7	0.584482	0.292241	−	0.956345i	$-0.405599\pi$
$827$	1.14957e7	0.584482	0.292241	+	0.956345i	$0.405599\pi$
$828$	5.59373e6	0.283548
$829$	3.48689e7	1.76219	0.881094	−	0.472941i	$-0.156808\pi$
$829$	3.48689e7	1.76219	0.881094	+	0.472941i	$0.156808\pi$
$830$	0	0
$831$	9.15538e6	0.459911
$832$	−230454.	−0.0115419
$833$	−76611.5	−0.00382544
$834$	−3.08659e7	−1.53661
$835$	0	0
$836$	−2.07717e7	−1.02791
$837$	−2.55282e6	−0.125952
$838$	2.89349e7	1.42335
$839$	1.56070e7	0.765445	0.382722	−	0.923863i	$-0.374987\pi$
$839$	1.56070e7	0.765445	0.382722	+	0.923863i	$0.374987\pi$
$840$	0	0
$841$	2.16710e7	1.05655
$842$	−4.02950e7	−1.95871
$843$	−7.29944e6	−0.353769
$844$	−5.13663e7	−2.48211
$845$	0	0
$846$	−9.15129e6	−0.439599
$847$	1.90114e6	0.0910555
$848$	−5.58706e6	−0.266805
$849$	−7.63658e6	−0.363605
$850$	0	0
$851$	1.46580e7	0.693828
$852$	−4.13919e7	−1.95352
$853$	−5.56534e6	−0.261890	−0.130945	−	0.991390i	$-0.541801\pi$
$853$	−5.56534e6	−0.261890	−0.130945	+	0.991390i	$0.541801\pi$
$854$	4.53699e7	2.12874
$855$	0	0
$856$	3.99190e7	1.86207
$857$	−1.76227e7	−0.819633	−0.409817	−	0.912168i	$-0.634407\pi$
$857$	−1.76227e7	−0.819633	−0.409817	+	0.912168i	$0.634407\pi$
$858$	−295002.	−0.0136806
$859$	−7.71598e6	−0.356786	−0.178393	−	0.983959i	$-0.557090\pi$
$859$	−7.71598e6	−0.356786	−0.178393	+	0.983959i	$0.557090\pi$
$860$	0	0
$861$	1.42288e7	0.654126
$862$	3.63623e7	1.66680
$863$	−2.72946e7	−1.24753	−0.623764	−	0.781613i	$-0.714397\pi$
$863$	−2.72946e7	−1.24753	−0.623764	+	0.781613i	$0.714397\pi$
$864$	−5.21851e6	−0.237827
$865$	0	0
$866$	−4.03889e7	−1.83007
$867$	−5.22631e6	−0.236128
$868$	3.34144e7	1.50534
$869$	8.53445e6	0.383377
$870$	0	0
$871$	277903.	0.0124122
$872$	5.96221e7	2.65532
$873$	1.50470e6	0.0668213
$874$	2.25478e7	0.998449
$875$	0	0
$876$	−3.78336e6	−0.166578
$877$	263106.	0.0115513	0.00577567	−	0.999983i	$-0.498162\pi$
$877$	263106.	0.0115513	0.00577567	+	0.999983i	$0.498162\pi$
$878$	−7.09340e7	−3.10540
$879$	−1.94827e7	−0.850506
$880$	0	0
$881$	3.47050e7	1.50644	0.753222	−	0.657767i	$-0.228499\pi$
$881$	3.47050e7	1.50644	0.753222	+	0.657767i	$0.228499\pi$
$882$	45060.7	0.00195041
$883$	−1.77964e7	−0.768123	−0.384061	−	0.923308i	$-0.625475\pi$
$883$	−1.77964e7	−0.768123	−0.384061	+	0.923308i	$0.625475\pi$
$884$	2.74141e6	0.117990
$885$	0	0
$886$	6.73883e7	2.88403
$887$	−4.02082e7	−1.71595	−0.857976	−	0.513690i	$-0.828278\pi$
$887$	−4.02082e7	−1.71595	−0.857976	+	0.513690i	$0.828278\pi$
$888$	−5.98112e7	−2.54536
$889$	2.26177e7	0.959830
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−6.44581e7	−2.71247
$893$	−2.56976e7	−1.07836
$894$	−4.14389e7	−1.73406
$895$	0	0
$896$	−1.80923e7	−0.752877
$897$	223082.	0.00925726
$898$	−1.82694e7	−0.756021
$899$	2.27435e7	0.938552
$900$	0	0
$901$	3.90339e6	0.160188
$902$	1.51308e7	0.619222
$903$	−1.24077e7	−0.506375
$904$	7.77796e6	0.316552
$905$	0	0
$906$	5.63983e6	0.228268
$907$	−2.69086e7	−1.08611	−0.543054	−	0.839698i	$-0.682732\pi$
$907$	−2.69086e7	−1.08611	−0.543054	+	0.839698i	$0.682732\pi$
$908$	−8.52860e7	−3.43291
$909$	−5.93576e6	−0.238268
$910$	0	0
$911$	−1.43592e7	−0.573236	−0.286618	−	0.958045i	$-0.592531\pi$
$911$	−1.43592e7	−0.573236	−0.286618	+	0.958045i	$0.592531\pi$
$912$	−4.25647e7	−1.69458
$913$	3.12110e6	0.123917
$914$	1.36274e7	0.539571
$915$	0	0
$916$	5.92185e6	0.233195
$917$	1.38373e7	0.543410
$918$	1.05900e7	0.414755
$919$	−2.81706e6	−0.110029	−0.0550145	−	0.998486i	$-0.517521\pi$
$919$	−2.81706e6	−0.110029	−0.0550145	+	0.998486i	$0.517521\pi$
$920$	0	0
$921$	2.72672e7	1.05923
$922$	6.12897e7	2.37444
$923$	−1.65074e6	−0.0637784
$924$	1.03913e7	0.400395
$925$	0	0
$926$	−2.93010e7	−1.12294
$927$	1.23116e7	0.470561
$928$	4.64925e7	1.77220
$929$	−8.94528e6	−0.340059	−0.170030	−	0.985439i	$-0.554386\pi$
$929$	−8.94528e6	−0.340059	−0.170030	+	0.985439i	$0.554386\pi$
$930$	0	0
$931$	126534.	0.00478447
$932$	1.31253e7	0.494960
$933$	−8.95397e6	−0.336753
$934$	4.36742e7	1.63817
$935$	0	0
$936$	−910270.	−0.0339610
$937$	−3.68278e7	−1.37034	−0.685168	−	0.728385i	$-0.740271\pi$
$937$	−3.68278e7	−1.37034	−0.685168	+	0.728385i	$0.740271\pi$
$938$	−1.40518e7	−0.521463
$939$	6.88117e6	0.254682
$940$	0	0
$941$	3.22670e7	1.18791	0.593957	−	0.804497i	$-0.297565\pi$
$941$	3.22670e7	1.18791	0.593957	+	0.804497i	$0.297565\pi$
$942$	−3.97201e7	−1.45842
$943$	−1.14420e7	−0.419009
$944$	−4.53356e7	−1.65581
$945$	0	0
$946$	−1.31943e7	−0.479356
$947$	5.26498e7	1.90775	0.953877	−	0.300199i	$-0.0970531\pi$
$947$	5.26498e7	1.90775	0.953877	+	0.300199i	$0.0970531\pi$
$948$	4.66477e7	1.68581
$949$	−150883.	−0.00543843
$950$	0	0
$951$	−1.59349e7	−0.571345
$952$	−7.82533e7	−2.79840
$953$	4.80923e7	1.71531	0.857657	−	0.514222i	$-0.171919\pi$
$953$	4.80923e7	1.71531	0.857657	+	0.514222i	$0.171919\pi$
$954$	−2.29586e6	−0.0816723
$955$	0	0
$956$	5.39640e7	1.90968
$957$	7.07282e6	0.249639
$958$	4.72097e7	1.66195
$959$	4.12186e6	0.144726
$960$	0	0
$961$	−1.63665e7	−0.571671
$962$	−4.22526e6	−0.147203
$963$	7.58895e6	0.263703
$964$	1.29282e8	4.48069
$965$	0	0
$966$	−1.12798e7	−0.388918
$967$	−3.61508e7	−1.24323	−0.621615	−	0.783323i	$-0.713523\pi$
$967$	−3.61508e7	−1.24323	−0.621615	+	0.783323i	$0.713523\pi$
$968$	6.23813e6	0.213976
$969$	2.97378e7	1.01742
$970$	0	0
$971$	−3.40402e7	−1.15863	−0.579315	−	0.815104i	$-0.696680\pi$
$971$	−3.40402e7	−1.15863	−0.579315	+	0.815104i	$0.696680\pi$
$972$	−4.33920e6	−0.147314
$973$	4.33596e7	1.46826
$974$	−4.57530e7	−1.54533
$975$	0	0
$976$	6.88726e7	2.31431
$977$	−1.92360e7	−0.644730	−0.322365	−	0.946615i	$-0.604478\pi$
$977$	−1.92360e7	−0.644730	−0.322365	+	0.946615i	$0.604478\pi$
$978$	2.41958e7	0.808898
$979$	−4.21689e6	−0.140616
$980$	0	0
$981$	1.13347e7	0.376042
$982$	1.02315e8	3.38578
$983$	2.65797e7	0.877336	0.438668	−	0.898649i	$-0.355450\pi$
$983$	2.65797e7	0.877336	0.438668	+	0.898649i	$0.355450\pi$
$984$	4.66884e7	1.53717
$985$	0	0
$986$	−9.43485e7	−3.09060
$987$	1.28555e7	0.420045
$988$	−4.52782e6	−0.147570
$989$	9.97758e6	0.324365
$990$	0	0
$991$	5.56931e7	1.80143	0.900714	−	0.434412i	$-0.143044\pi$
$991$	5.56931e7	1.80143	0.900714	+	0.434412i	$0.143044\pi$
$992$	2.50675e7	0.808783
$993$	8.81890e6	0.283819
$994$	8.34670e7	2.67947
$995$	0	0
$996$	1.70593e7	0.544896
$997$	−2.24526e7	−0.715367	−0.357684	−	0.933843i	$-0.616433\pi$
$997$	−2.24526e7	−0.715367	−0.357684	+	0.933843i	$0.616433\pi$
$998$	1.04086e8	3.30800
$999$	−1.13706e7	−0.360471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.s.1.9	yes	9
5.4	even	2		825.6.a.r.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.1	✓	9	5.4	even	2
825.6.a.s.1.9	yes	9	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)

\(f(q)\)	\(=\)	\(q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} -121.000 q^{11} -661.363 q^{12} -26.3756 q^{13} +1333.64 q^{14} +2024.50 q^{16} -1414.41 q^{17} +831.917 q^{18} +2336.09 q^{19} -1168.65 q^{21} -1242.74 q^{22} +939.766 q^{23} -3834.65 q^{24} -270.892 q^{26} -729.000 q^{27} +9542.04 q^{28} +6494.78 q^{29} +3501.81 q^{31} +7158.44 q^{32} +1089.00 q^{33} -14526.8 q^{34} +5952.27 q^{36} +15597.6 q^{37} +23993.0 q^{38} +237.380 q^{39} -12175.4 q^{41} -12002.8 q^{42} +10617.1 q^{43} -8891.66 q^{44} +9651.94 q^{46} -11000.2 q^{47} -18220.5 q^{48} +54.1649 q^{49} +12729.7 q^{51} -1938.20 q^{52} -2759.73 q^{53} -7487.25 q^{54} +55325.7 q^{56} -21024.8 q^{57} +66705.2 q^{58} -22393.5 q^{59} +34019.6 q^{61} +35965.6 q^{62} +10517.9 q^{63} +8737.42 q^{64} +11184.7 q^{66} -10536.4 q^{67} -103938. q^{68} -8457.89 q^{69} +62585.8 q^{71} +34511.9 q^{72} +5720.55 q^{73} +160196. q^{74} +171667. q^{76} -15711.9 q^{77} +2438.03 q^{78} -70532.7 q^{79} +6561.00 q^{81} -125048. q^{82} -25794.2 q^{83} -85878.3 q^{84} +109044. q^{86} -58453.0 q^{87} -51554.8 q^{88} +34850.4 q^{89} -3424.88 q^{91} +69058.4 q^{92} -31516.3 q^{93} -112979. q^{94} -64426.0 q^{96} +18576.6 q^{97} +556.305 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * q^99

Introduction

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