LMFDB - Embedded newform 171.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,12,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("171.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	171.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:	$131.386683876$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 10491x^{5} + 5390x^{4} + 33206195x^{3} + 155482410x^{2} - 32794886585x - 417193412918 $ x^7 - 2x^6 - 10491x^5 + 5390x^4 + 33206195x^3 + 155482410x^2 - 32794886585x - 417193412918
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-76.9480$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-75.9480 q^{2} +3720.10 q^{4} +559.147 q^{5} -42737.5 q^{7} -126992. q^{8} +O(q^{10})$	\(q-75.9480 q^{2} +3720.10 q^{4} +559.147 q^{5} -42737.5 q^{7} -126992. q^{8} -42466.1 q^{10} +760245. q^{11} -1.08264e6 q^{13} +3.24583e6 q^{14} +2.02605e6 q^{16} +147825. q^{17} +2.47610e6 q^{19} +2.08008e6 q^{20} -5.77391e7 q^{22} -4.63915e7 q^{23} -4.85155e7 q^{25} +8.22242e7 q^{26} -1.58988e8 q^{28} +6.40882e7 q^{29} +2.79716e8 q^{31} +1.06206e8 q^{32} -1.12270e7 q^{34} -2.38966e7 q^{35} -2.37124e8 q^{37} -1.88055e8 q^{38} -7.10073e7 q^{40} +9.81332e8 q^{41} +4.72722e7 q^{43} +2.82818e9 q^{44} +3.52334e9 q^{46} -1.85661e9 q^{47} -1.50830e8 q^{49} +3.68465e9 q^{50} -4.02752e9 q^{52} +2.12327e9 q^{53} +4.25088e8 q^{55} +5.42734e9 q^{56} -4.86737e9 q^{58} +2.01394e9 q^{59} +1.20487e10 q^{61} -2.12439e10 q^{62} -1.22155e10 q^{64} -6.05353e8 q^{65} -2.14288e10 q^{67} +5.49924e8 q^{68} +1.81490e9 q^{70} -5.76489e9 q^{71} -2.64202e10 q^{73} +1.80091e10 q^{74} +9.21133e9 q^{76} -3.24910e10 q^{77} -1.83608e10 q^{79} +1.13286e9 q^{80} -7.45302e10 q^{82} +2.66752e10 q^{83} +8.26560e7 q^{85} -3.59023e9 q^{86} -9.65452e10 q^{88} -2.35606e10 q^{89} +4.62693e10 q^{91} -1.72581e11 q^{92} +1.41006e11 q^{94} +1.38450e9 q^{95} -1.06270e10 q^{97} +1.14552e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 9 q^{2} + 6661 q^{4} + 14307 q^{5} - 2209 q^{7} + 72891 q^{8}+O(q^{10}) $ 7 * q + 9 * q^2 + 6661 * q^4 + 14307 * q^5 - 2209 * q^7 + 72891 * q^8	\( 7 q + 9 q^{2} + 6661 q^{4} + 14307 q^{5} - 2209 q^{7} + 72891 q^{8} + 610116 q^{10} + 1399461 q^{11} - 2639296 q^{13} + 6202863 q^{14} - 11910239 q^{16} + 9760773 q^{17} + 17332693 q^{19} + 60822132 q^{20} - 96770562 q^{22} + 117151884 q^{23} - 40644056 q^{25} + 140205807 q^{26} - 95042353 q^{28} + 380102178 q^{29} + 108083372 q^{31} - 639362133 q^{32} + 632902677 q^{34} - 258176013 q^{35} + 470630222 q^{37} + 22284891 q^{38} + 1263596868 q^{40} + 572622810 q^{41} + 1170836039 q^{43} - 1857175194 q^{44} + 6333771957 q^{46} + 3485769735 q^{47} + 2791682862 q^{49} + 6699338667 q^{50} - 4968372073 q^{52} + 9143305584 q^{53} - 18483973467 q^{55} + 11872570905 q^{56} + 10101698853 q^{58} + 847227714 q^{59} - 8144938567 q^{61} - 21626318880 q^{62} - 14662293047 q^{64} + 9033654252 q^{65} - 5797397824 q^{67} + 6895758945 q^{68} + 63346638156 q^{70} + 4538589186 q^{71} - 1815379657 q^{73} - 52966894338 q^{74} + 16493295439 q^{76} - 77952325659 q^{77} - 54907201480 q^{79} - 77197994292 q^{80} + 66047089392 q^{82} + 68609936724 q^{83} + 50328221961 q^{85} - 11262589308 q^{86} - 118799781006 q^{88} + 105124973892 q^{89} - 53584340240 q^{91} - 22337690499 q^{92} + 203078543976 q^{94} + 35425548393 q^{95} - 386940894664 q^{97} + 357308952786 q^{98}+O(q^{100}) \) 7 * q + 9 * q^2 + 6661 * q^4 + 14307 * q^5 - 2209 * q^7 + 72891 * q^8 + 610116 * q^10 + 1399461 * q^11 - 2639296 * q^13 + 6202863 * q^14 - 11910239 * q^16 + 9760773 * q^17 + 17332693 * q^19 + 60822132 * q^20 - 96770562 * q^22 + 117151884 * q^23 - 40644056 * q^25 + 140205807 * q^26 - 95042353 * q^28 + 380102178 * q^29 + 108083372 * q^31 - 639362133 * q^32 + 632902677 * q^34 - 258176013 * q^35 + 470630222 * q^37 + 22284891 * q^38 + 1263596868 * q^40 + 572622810 * q^41 + 1170836039 * q^43 - 1857175194 * q^44 + 6333771957 * q^46 + 3485769735 * q^47 + 2791682862 * q^49 + 6699338667 * q^50 - 4968372073 * q^52 + 9143305584 * q^53 - 18483973467 * q^55 + 11872570905 * q^56 + 10101698853 * q^58 + 847227714 * q^59 - 8144938567 * q^61 - 21626318880 * q^62 - 14662293047 * q^64 + 9033654252 * q^65 - 5797397824 * q^67 + 6895758945 * q^68 + 63346638156 * q^70 + 4538589186 * q^71 - 1815379657 * q^73 - 52966894338 * q^74 + 16493295439 * q^76 - 77952325659 * q^77 - 54907201480 * q^79 - 77197994292 * q^80 + 66047089392 * q^82 + 68609936724 * q^83 + 50328221961 * q^85 - 11262589308 * q^86 - 118799781006 * q^88 + 105124973892 * q^89 - 53584340240 * q^91 - 22337690499 * q^92 + 203078543976 * q^94 + 35425548393 * q^95 - 386940894664 * q^97 + 357308952786 * q^98

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 $

$2$	−75.9480	−1.67823	−0.839115	−	0.543955i	$-0.816926\pi$
$2$	−75.9480	−1.67823	−0.839115	+	0.543955i	$0.816926\pi$
$3$	0	0
$3$	0	0
$4$	3720.10	1.81645
$5$	559.147	0.0800186	0.0400093	−	0.999199i	$-0.487261\pi$
$5$	559.147	0.0800186	0.0400093	+	0.999199i	$0.487261\pi$
$6$	0	0
$7$	−42737.5	−0.961104	−0.480552	−	0.876966i	$-0.659564\pi$
$7$	−42737.5	−0.961104	−0.480552	+	0.876966i	$0.659564\pi$
$8$	−126992.	−1.37020
$9$	0	0
$10$	−42466.1	−0.134289
$11$	760245.	1.42329	0.711645	−	0.702539i	$-0.247950\pi$
$11$	760245.	1.42329	0.711645	+	0.702539i	$0.247950\pi$
$12$	0	0
$13$	−1.08264e6	−0.808714	−0.404357	−	0.914601i	$-0.632505\pi$
$13$	−1.08264e6	−0.808714	−0.404357	+	0.914601i	$0.632505\pi$
$14$	3.24583e6	1.61295
$15$	0	0
$16$	2.02605e6	0.483049
$17$	147825.	0.0252510	0.0126255	−	0.999920i	$-0.495981\pi$
$17$	147825.	0.0252510	0.0126255	+	0.999920i	$0.495981\pi$
$18$	0	0
$19$	2.47610e6	0.229416
$19$	2.47610e6	0.229416
$20$	2.08008e6	0.145350
$21$	0	0
$22$	−5.77391e7	−2.38861
$23$	−4.63915e7	−1.50292	−0.751459	−	0.659780i	$-0.770649\pi$
$23$	−4.63915e7	−1.50292	−0.751459	+	0.659780i	$0.770649\pi$
$24$	0	0
$25$	−4.85155e7	−0.993597
$26$	8.22242e7	1.35721
$27$	0	0
$28$	−1.58988e8	−1.74580
$29$	6.40882e7	0.580215	0.290108	−	0.956994i	$-0.406309\pi$
$29$	6.40882e7	0.580215	0.290108	+	0.956994i	$0.406309\pi$
$30$	0	0
$31$	2.79716e8	1.75480	0.877402	−	0.479757i	$-0.159275\pi$
$31$	2.79716e8	1.75480	0.877402	+	0.479757i	$0.159275\pi$
$32$	1.06206e8	0.559529
$33$	0	0
$34$	−1.12270e7	−0.0423770
$35$	−2.38966e7	−0.0769061
$36$	0	0
$37$	−2.37124e8	−0.562167	−0.281084	−	0.959683i	$-0.590694\pi$
$37$	−2.37124e8	−0.562167	−0.281084	+	0.959683i	$0.590694\pi$
$38$	−1.88055e8	−0.385012
$39$	0	0
$40$	−7.10073e7	−0.109641
$41$	9.81332e8	1.32283	0.661416	−	0.750019i	$-0.269956\pi$
$41$	9.81332e8	1.32283	0.661416	+	0.750019i	$0.269956\pi$
$42$	0	0
$43$	4.72722e7	0.0490376	0.0245188	−	0.999699i	$-0.492195\pi$
$43$	4.72722e7	0.0490376	0.0245188	+	0.999699i	$0.492195\pi$
$44$	2.82818e9	2.58534
$45$	0	0
$46$	3.52334e9	2.52224
$47$	−1.85661e9	−1.18082	−0.590409	−	0.807104i	$-0.701034\pi$
$47$	−1.85661e9	−1.18082	−0.590409	+	0.807104i	$0.701034\pi$
$48$	0	0
$49$	−1.50830e8	−0.0762796
$50$	3.68465e9	1.66748
$51$	0	0
$52$	−4.02752e9	−1.46899
$53$	2.12327e9	0.697409	0.348704	−	0.937233i	$-0.386622\pi$
$53$	2.12327e9	0.697409	0.348704	+	0.937233i	$0.386622\pi$
$54$	0	0
$55$	4.25088e8	0.113890
$56$	5.42734e9	1.31690
$57$	0	0
$58$	−4.86737e9	−0.973734
$59$	2.01394e9	0.366743	0.183371	−	0.983044i	$-0.441299\pi$
$59$	2.01394e9	0.366743	0.183371	+	0.983044i	$0.441299\pi$
$60$	0	0
$61$	1.20487e10	1.82653	0.913263	−	0.407371i	$-0.133554\pi$
$61$	1.20487e10	1.82653	0.913263	+	0.407371i	$0.133554\pi$
$62$	−2.12439e10	−2.94496
$63$	0	0
$64$	−1.22155e10	−1.42207
$65$	−6.05353e8	−0.0647121
$66$	0	0
$67$	−2.14288e10	−1.93904	−0.969520	−	0.245012i	$-0.921208\pi$
$67$	−2.14288e10	−1.93904	−0.969520	+	0.245012i	$0.921208\pi$
$68$	5.49924e8	0.0458673
$69$	0	0
$70$	1.81490e9	0.129066
$71$	−5.76489e9	−0.379201	−0.189601	−	0.981861i	$-0.560719\pi$
$71$	−5.76489e9	−0.379201	−0.189601	+	0.981861i	$0.560719\pi$
$72$	0	0
$73$	−2.64202e10	−1.49163	−0.745814	−	0.666154i	$-0.767939\pi$
$73$	−2.64202e10	−1.49163	−0.745814	+	0.666154i	$0.767939\pi$
$74$	1.80091e10	0.943445
$75$	0	0
$76$	9.21133e9	0.416723
$77$	−3.24910e10	−1.36793
$78$	0	0
$79$	−1.83608e10	−0.671342	−0.335671	−	0.941979i	$-0.608963\pi$
$79$	−1.83608e10	−0.671342	−0.335671	+	0.941979i	$0.608963\pi$
$80$	1.13286e9	0.0386529
$81$	0	0
$82$	−7.45302e10	−2.22002
$83$	2.66752e10	0.743324	0.371662	−	0.928368i	$-0.378788\pi$
$83$	2.66752e10	0.743324	0.371662	+	0.928368i	$0.378788\pi$
$84$	0	0
$85$	8.26560e7	0.00202055
$86$	−3.59023e9	−0.0822964
$87$	0	0
$88$	−9.65452e10	−1.95019
$89$	−2.35606e10	−0.447241	−0.223620	−	0.974676i	$-0.571788\pi$
$89$	−2.35606e10	−0.447241	−0.223620	+	0.974676i	$0.571788\pi$
$90$	0	0
$91$	4.62693e10	0.777258
$92$	−1.72581e11	−2.72998
$93$	0	0
$94$	1.41006e11	1.98168
$95$	1.38450e9	0.0183575
$96$	0	0
$97$	−1.06270e10	−0.125651	−0.0628257	−	0.998025i	$-0.520011\pi$
$97$	−1.06270e10	−0.125651	−0.0628257	+	0.998025i	$0.520011\pi$
$98$	1.14552e10	0.128015
$99$	0	0
$100$	−1.80482e11	−1.80482
$101$	6.85020e10	0.648539	0.324269	−	0.945965i	$-0.394882\pi$
$101$	6.85020e10	0.648539	0.324269	+	0.945965i	$0.394882\pi$
$102$	0	0
$103$	2.94639e9	0.0250430	0.0125215	−	0.999922i	$-0.496014\pi$
$103$	2.94639e9	0.0250430	0.0125215	+	0.999922i	$0.496014\pi$
$104$	1.37487e11	1.10810
$105$	0	0
$106$	−1.61258e11	−1.17041
$107$	−7.18708e10	−0.495384	−0.247692	−	0.968839i	$-0.579672\pi$
$107$	−7.18708e10	−0.495384	−0.247692	+	0.968839i	$0.579672\pi$
$108$	0	0
$109$	−2.37162e11	−1.47639	−0.738194	−	0.674589i	$-0.764321\pi$
$109$	−2.37162e11	−1.47639	−0.738194	+	0.674589i	$0.764321\pi$
$110$	−3.22846e10	−0.191133
$111$	0	0
$112$	−8.65886e10	−0.464260
$113$	−2.89617e11	−1.47874	−0.739371	−	0.673298i	$-0.764877\pi$
$113$	−2.89617e11	−1.47874	−0.739371	+	0.673298i	$0.764877\pi$
$114$	0	0
$115$	−2.59396e10	−0.120261
$116$	2.38414e11	1.05393
$117$	0	0
$118$	−1.52955e11	−0.615478
$119$	−6.31768e9	−0.0242689
$120$	0	0
$121$	2.92660e11	1.02576
$122$	−9.15074e11	−3.06533
$123$	0	0
$124$	1.04057e12	3.18752
$125$	−5.44294e10	−0.159525
$126$	0	0
$127$	6.97694e11	1.87389	0.936945	−	0.349476i	$-0.113640\pi$
$127$	6.97694e11	1.87389	0.936945	+	0.349476i	$0.113640\pi$
$128$	7.10230e11	1.82702
$129$	0	0
$130$	4.59754e10	0.108602
$131$	3.48233e8	0.000788638 0	0.000394319	−	1.00000i	$-0.499874\pi$
$131$	3.48233e8	0.000788638 0	0.000394319		1.00000i	$0.499874\pi$
$132$	0	0
$133$	−1.05822e11	−0.220492
$134$	1.62748e12	3.25415
$135$	0	0
$136$	−1.87727e10	−0.0345989
$137$	3.68529e11	0.652391	0.326196	−	0.945302i	$-0.394233\pi$
$137$	3.68529e11	0.652391	0.326196	+	0.945302i	$0.394233\pi$
$138$	0	0
$139$	−5.82223e11	−0.951718	−0.475859	−	0.879522i	$-0.657863\pi$
$139$	−5.82223e11	−0.951718	−0.475859	+	0.879522i	$0.657863\pi$
$140$	−8.88975e10	−0.139696
$141$	0	0
$142$	4.37831e11	0.636387
$143$	−8.23070e11	−1.15103
$144$	0	0
$145$	3.58347e10	0.0464280
$146$	2.00656e12	2.50329
$147$	0	0
$148$	−8.82123e11	−1.02115
$149$	−8.65612e11	−0.965603	−0.482801	−	0.875730i	$-0.660381\pi$
$149$	−8.65612e11	−0.965603	−0.482801	+	0.875730i	$0.660381\pi$
$150$	0	0
$151$	−1.29180e12	−1.33913	−0.669563	−	0.742755i	$-0.733519\pi$
$151$	−1.29180e12	−1.33913	−0.669563	+	0.742755i	$0.733519\pi$
$152$	−3.14446e11	−0.314344
$153$	0	0
$154$	2.46762e12	2.29570
$155$	1.56402e11	0.140417
$156$	0	0
$157$	3.77137e11	0.315537	0.157769	−	0.987476i	$-0.449570\pi$
$157$	3.77137e11	0.315537	0.157769	+	0.987476i	$0.449570\pi$
$158$	1.39447e12	1.12667
$159$	0	0
$160$	5.93845e10	0.0447727
$161$	1.98266e12	1.44446
$162$	0	0
$163$	1.43201e12	0.974795	0.487397	−	0.873180i	$-0.337946\pi$
$163$	1.43201e12	0.974795	0.487397	+	0.873180i	$0.337946\pi$
$164$	3.65065e12	2.40286
$165$	0	0
$166$	−2.02593e12	−1.24747
$167$	5.48607e11	0.326829	0.163415	−	0.986557i	$-0.447749\pi$
$167$	5.48607e11	0.326829	0.163415	+	0.986557i	$0.447749\pi$
$168$	0	0
$169$	−6.20056e11	−0.345982
$170$	−6.27755e9	−0.00339095
$171$	0	0
$172$	1.75857e11	0.0890746
$173$	−2.36053e12	−1.15813	−0.579064	−	0.815282i	$-0.696582\pi$
$173$	−2.36053e12	−1.15813	−0.579064	+	0.815282i	$0.696582\pi$
$174$	0	0
$175$	2.07343e12	0.954950
$176$	1.54030e12	0.687519
$177$	0	0
$178$	1.78938e12	0.750572
$179$	1.03168e12	0.419615	0.209808	−	0.977743i	$-0.432716\pi$
$179$	1.03168e12	0.419615	0.209808	+	0.977743i	$0.432716\pi$
$180$	0	0
$181$	9.06364e11	0.346793	0.173397	−	0.984852i	$-0.444526\pi$
$181$	9.06364e11	0.346793	0.173397	+	0.984852i	$0.444526\pi$
$182$	−3.51406e12	−1.30442
$183$	0	0
$184$	5.89136e12	2.05929
$185$	−1.32587e11	−0.0449838
$186$	0	0
$187$	1.12383e11	0.0359396
$188$	−6.90678e12	−2.14490
$189$	0	0
$190$	−1.05150e11	−0.0308081
$191$	5.75476e12	1.63811	0.819056	−	0.573713i	$-0.194498\pi$
$191$	5.75476e12	1.63811	0.819056	+	0.573713i	$0.194498\pi$
$192$	0	0
$193$	6.35485e12	1.70821	0.854103	−	0.520104i	$-0.174107\pi$
$193$	6.35485e12	1.70821	0.854103	+	0.520104i	$0.174107\pi$
$194$	8.07101e11	0.210872
$195$	0	0
$196$	−5.61101e11	−0.138558
$197$	3.53349e12	0.848476	0.424238	−	0.905551i	$-0.360542\pi$
$197$	3.53349e12	0.848476	0.424238	+	0.905551i	$0.360542\pi$
$198$	0	0
$199$	3.79492e12	0.862008	0.431004	−	0.902350i	$-0.358160\pi$
$199$	3.79492e12	0.862008	0.431004	+	0.902350i	$0.358160\pi$
$200$	6.16109e12	1.36142
$201$	0	0
$202$	−5.20259e12	−1.08840
$203$	−2.73897e12	−0.557647
$204$	0	0
$205$	5.48709e11	0.105851
$206$	−2.23773e11	−0.0420278
$207$	0	0
$208$	−2.19348e12	−0.390648
$209$	1.88244e12	0.326525
$210$	0	0
$211$	1.53937e12	0.253390	0.126695	−	0.991942i	$-0.459563\pi$
$211$	1.53937e12	0.253390	0.126695	+	0.991942i	$0.459563\pi$
$212$	7.89876e12	1.26681
$213$	0	0
$214$	5.45844e12	0.831367
$215$	2.64321e10	0.00392392
$216$	0	0
$217$	−1.19544e13	−1.68655
$218$	1.80120e13	2.47772
$219$	0	0
$220$	1.58137e12	0.206875
$221$	−1.60041e11	−0.0204209
$222$	0	0
$223$	−5.61894e12	−0.682304	−0.341152	−	0.940008i	$-0.610817\pi$
$223$	−5.61894e12	−0.682304	−0.341152	+	0.940008i	$0.610817\pi$
$224$	−4.53896e12	−0.537765
$225$	0	0
$226$	2.19958e13	2.48167
$227$	8.78670e12	0.967573	0.483786	−	0.875186i	$-0.339261\pi$
$227$	8.78670e12	0.967573	0.483786	+	0.875186i	$0.339261\pi$
$228$	0	0
$229$	1.78937e12	0.187761	0.0938806	−	0.995583i	$-0.470073\pi$
$229$	1.78937e12	0.187761	0.0938806	+	0.995583i	$0.470073\pi$
$230$	1.97006e12	0.201826
$231$	0	0
$232$	−8.13871e12	−0.795008
$233$	1.49446e13	1.42570	0.712850	−	0.701317i	$-0.247404\pi$
$233$	1.49446e13	1.42570	0.712850	+	0.701317i	$0.247404\pi$
$234$	0	0
$235$	−1.03812e12	−0.0944874
$236$	7.49206e12	0.666171
$237$	0	0
$238$	4.79815e11	0.0407287
$239$	7.59815e12	0.630259	0.315130	−	0.949049i	$-0.397952\pi$
$239$	7.59815e12	0.630259	0.315130	+	0.949049i	$0.397952\pi$
$240$	0	0
$241$	−1.75241e13	−1.38849	−0.694243	−	0.719741i	$-0.744261\pi$
$241$	−1.75241e13	−1.38849	−0.694243	+	0.719741i	$0.744261\pi$
$242$	−2.22270e13	−1.72145
$243$	0	0
$244$	4.48223e13	3.31780
$245$	−8.43359e10	−0.00610378
$246$	0	0
$247$	−2.68072e12	−0.185532
$248$	−3.55218e13	−2.40442
$249$	0	0
$250$	4.13380e12	0.267719
$251$	2.73332e13	1.73175	0.865876	−	0.500259i	$-0.166762\pi$
$251$	2.73332e13	1.73175	0.865876	+	0.500259i	$0.166762\pi$
$252$	0	0
$253$	−3.52689e13	−2.13909
$254$	−5.29884e13	−3.14482
$255$	0	0
$256$	−2.89233e13	−1.64410
$257$	−1.03532e13	−0.576025	−0.288013	−	0.957627i	$-0.592995\pi$
$257$	−1.03532e13	−0.576025	−0.288013	+	0.957627i	$0.592995\pi$
$258$	0	0
$259$	1.01341e13	0.540301
$260$	−2.25197e12	−0.117547
$261$	0	0
$262$	−2.64476e10	−0.00132352
$263$	1.68167e11	0.00824109	0.00412055	−	0.999992i	$-0.498688\pi$
$263$	1.68167e11	0.00824109	0.00412055	+	0.999992i	$0.498688\pi$
$264$	0	0
$265$	1.18722e12	0.0558057
$266$	8.03700e12	0.370037
$267$	0	0
$268$	−7.97173e13	−3.52218
$269$	−7.11871e12	−0.308151	−0.154075	−	0.988059i	$-0.549240\pi$
$269$	−7.11871e12	−0.308151	−0.154075	+	0.988059i	$0.549240\pi$
$270$	0	0
$271$	4.18480e13	1.73918	0.869588	−	0.493778i	$-0.164385\pi$
$271$	4.18480e13	1.73918	0.869588	+	0.493778i	$0.164385\pi$
$272$	2.99502e11	0.0121975
$273$	0	0
$274$	−2.79890e13	−1.09486
$275$	−3.68836e13	−1.41418
$276$	0	0
$277$	−4.34333e12	−0.160024	−0.0800118	−	0.996794i	$-0.525496\pi$
$277$	−4.34333e12	−0.160024	−0.0800118	+	0.996794i	$0.525496\pi$
$278$	4.42187e13	1.59720
$279$	0	0
$280$	3.03468e12	0.105376
$281$	−2.12282e13	−0.722816	−0.361408	−	0.932408i	$-0.617704\pi$
$281$	−2.12282e13	−0.722816	−0.361408	+	0.932408i	$0.617704\pi$
$282$	0	0
$283$	−1.29126e13	−0.422851	−0.211425	−	0.977394i	$-0.567811\pi$
$283$	−1.29126e13	−0.422851	−0.211425	+	0.977394i	$0.567811\pi$
$284$	−2.14459e13	−0.688801
$285$	0	0
$286$	6.25105e13	1.93170
$287$	−4.19397e13	−1.27138
$288$	0	0
$289$	−3.42500e13	−0.999362
$290$	−2.72157e12	−0.0779168
$291$	0	0
$292$	−9.82857e13	−2.70947
$293$	−1.15382e13	−0.312153	−0.156076	−	0.987745i	$-0.549885\pi$
$293$	−1.15382e13	−0.312153	−0.156076	+	0.987745i	$0.549885\pi$
$294$	0	0
$295$	1.12609e12	0.0293462
$296$	3.01129e13	0.770279
$297$	0	0
$298$	6.57415e13	1.62050
$299$	5.02252e13	1.21543
$300$	0	0
$301$	−2.02030e12	−0.0471303
$302$	9.81095e13	2.24736
$303$	0	0
$304$	5.01671e12	0.110819
$305$	6.73699e12	0.146156
$306$	0	0
$307$	8.77982e13	1.83749	0.918744	−	0.394854i	$-0.129205\pi$
$307$	8.77982e13	1.83749	0.918744	+	0.394854i	$0.129205\pi$
$308$	−1.20870e14	−2.48478
$309$	0	0
$310$	−1.18785e13	−0.235652
$311$	−5.44838e13	−1.06190	−0.530952	−	0.847402i	$-0.678165\pi$
$311$	−5.44838e13	−1.06190	−0.530952	+	0.847402i	$0.678165\pi$
$312$	0	0
$313$	1.17147e13	0.220413	0.110206	−	0.993909i	$-0.464849\pi$
$313$	1.17147e13	0.220413	0.110206	+	0.993909i	$0.464849\pi$
$314$	−2.86428e13	−0.529544
$315$	0	0
$316$	−6.83041e13	−1.21946
$317$	4.61261e13	0.809320	0.404660	−	0.914467i	$-0.367390\pi$
$317$	4.61261e13	0.809320	0.404660	+	0.914467i	$0.367390\pi$
$318$	0	0
$319$	4.87227e13	0.825815
$320$	−6.83023e12	−0.113792
$321$	0	0
$322$	−1.50579e14	−2.42413
$323$	3.66030e11	0.00579299
$324$	0	0
$325$	5.25247e13	0.803535
$326$	−1.08758e14	−1.63593
$327$	0	0
$328$	−1.24622e14	−1.81254
$329$	7.93471e13	1.13489
$330$	0	0
$331$	−2.72631e12	−0.0377157	−0.0188578	−	0.999822i	$-0.506003\pi$
$331$	−2.72631e12	−0.0377157	−0.0188578	+	0.999822i	$0.506003\pi$
$332$	9.92343e13	1.35021
$333$	0	0
$334$	−4.16656e13	−0.548494
$335$	−1.19819e13	−0.155159
$336$	0	0
$337$	6.31846e13	0.791857	0.395929	−	0.918281i	$-0.370423\pi$
$337$	6.31846e13	0.791857	0.395929	+	0.918281i	$0.370423\pi$
$338$	4.70920e13	0.580638
$339$	0	0
$340$	3.07488e11	0.00367024
$341$	2.12653e14	2.49760
$342$	0	0
$343$	9.09522e13	1.03442
$344$	−6.00321e12	−0.0671912
$345$	0	0
$346$	1.79278e14	1.94360
$347$	−4.91646e13	−0.524614	−0.262307	−	0.964984i	$-0.584483\pi$
$347$	−4.91646e13	−0.524614	−0.262307	+	0.964984i	$0.584483\pi$
$348$	0	0
$349$	6.23394e13	0.644500	0.322250	−	0.946655i	$-0.395561\pi$
$349$	6.23394e13	0.644500	0.322250	+	0.946655i	$0.395561\pi$
$350$	−1.57473e14	−1.60262
$351$	0	0
$352$	8.07422e13	0.796372
$353$	1.18251e14	1.14827	0.574135	−	0.818761i	$-0.305338\pi$
$353$	1.18251e14	1.14827	0.574135	+	0.818761i	$0.305338\pi$
$354$	0	0
$355$	−3.22342e12	−0.0303431
$356$	−8.76477e13	−0.812392
$357$	0	0
$358$	−7.83536e13	−0.704211
$359$	7.39160e13	0.654213	0.327106	−	0.944987i	$-0.393926\pi$
$359$	7.39160e13	0.654213	0.327106	+	0.944987i	$0.393926\pi$
$360$	0	0
$361$	6.13107e12	0.0526316
$362$	−6.88365e13	−0.581998
$363$	0	0
$364$	1.72126e14	1.41185
$365$	−1.47728e13	−0.119358
$366$	0	0
$367$	2.23518e14	1.75247	0.876233	−	0.481887i	$-0.160048\pi$
$367$	2.23518e14	1.75247	0.876233	+	0.481887i	$0.160048\pi$
$368$	−9.39916e13	−0.725983
$369$	0	0
$370$	1.00697e13	0.0754932
$371$	−9.07432e13	−0.670282
$372$	0	0
$373$	1.48916e14	1.06793	0.533966	−	0.845506i	$-0.320701\pi$
$373$	1.48916e14	1.06793	0.533966	+	0.845506i	$0.320701\pi$
$374$	−8.53529e12	−0.0603149
$375$	0	0
$376$	2.35776e14	1.61795
$377$	−6.93843e13	−0.469228
$378$	0	0
$379$	−3.63324e13	−0.238659	−0.119330	−	0.992855i	$-0.538075\pi$
$379$	−3.63324e13	−0.238659	−0.119330	+	0.992855i	$0.538075\pi$
$380$	5.15048e12	0.0333456
$381$	0	0
$382$	−4.37062e14	−2.74913
$383$	5.78932e13	0.358950	0.179475	−	0.983763i	$-0.442560\pi$
$383$	5.78932e13	0.358950	0.179475	+	0.983763i	$0.442560\pi$
$384$	0	0
$385$	−1.81672e13	−0.109460
$386$	−4.82638e14	−2.86676
$387$	0	0
$388$	−3.95335e13	−0.228240
$389$	−2.33229e13	−0.132758	−0.0663789	−	0.997794i	$-0.521145\pi$
$389$	−2.33229e13	−0.132758	−0.0663789	+	0.997794i	$0.521145\pi$
$390$	0	0
$391$	−6.85783e12	−0.0379502
$392$	1.91542e13	0.104518
$393$	0	0
$394$	−2.68361e14	−1.42394
$395$	−1.02664e13	−0.0537198
$396$	0	0
$397$	1.46796e14	0.747081	0.373540	−	0.927614i	$-0.378144\pi$
$397$	1.46796e14	0.747081	0.373540	+	0.927614i	$0.378144\pi$
$398$	−2.88217e14	−1.44665
$399$	0	0
$400$	−9.82950e13	−0.479956
$401$	−2.02235e12	−0.00974010	−0.00487005	−	0.999988i	$-0.501550\pi$
$401$	−2.02235e12	−0.00974010	−0.00487005	+	0.999988i	$0.501550\pi$
$402$	0	0
$403$	−3.02831e14	−1.41913
$404$	2.54834e14	1.17804
$405$	0	0
$406$	2.08019e14	0.935859
$407$	−1.80272e14	−0.800127
$408$	0	0
$409$	−4.39137e13	−0.189724	−0.0948619	−	0.995490i	$-0.530241\pi$
$409$	−4.39137e13	−0.189724	−0.0948619	+	0.995490i	$0.530241\pi$
$410$	−4.16733e13	−0.177643
$411$	0	0
$412$	1.09609e13	0.0454894
$413$	−8.60710e13	−0.352478
$414$	0	0
$415$	1.49154e13	0.0594797
$416$	−1.14982e14	−0.452498
$417$	0	0
$418$	−1.42968e14	−0.547984
$419$	2.82908e14	1.07021	0.535104	−	0.844786i	$-0.320272\pi$
$419$	2.82908e14	1.07021	0.535104	+	0.844786i	$0.320272\pi$
$420$	0	0
$421$	−2.07413e14	−0.764336	−0.382168	−	0.924093i	$-0.624822\pi$
$421$	−2.07413e14	−0.764336	−0.382168	+	0.924093i	$0.624822\pi$
$422$	−1.16912e14	−0.425246
$423$	0	0
$424$	−2.69639e14	−0.955587
$425$	−7.17181e12	−0.0250894
$426$	0	0
$427$	−5.14931e14	−1.75548
$428$	−2.67366e14	−0.899841
$429$	0	0
$430$	−2.00747e12	−0.00658524
$431$	−4.80227e14	−1.55533	−0.777664	−	0.628680i	$-0.783595\pi$
$431$	−4.80227e14	−1.55533	−0.777664	+	0.628680i	$0.783595\pi$
$432$	0	0
$433$	1.91065e14	0.603250	0.301625	−	0.953427i	$-0.402471\pi$
$433$	1.91065e14	0.603250	0.301625	+	0.953427i	$0.402471\pi$
$434$	9.07911e14	2.83041
$435$	0	0
$436$	−8.82267e14	−2.68179
$437$	−1.14870e14	−0.344793
$438$	0	0
$439$	2.19249e14	0.641774	0.320887	−	0.947118i	$-0.396019\pi$
$439$	2.19249e14	0.641774	0.320887	+	0.947118i	$0.396019\pi$
$440$	−5.39830e13	−0.156051
$441$	0	0
$442$	1.21548e13	0.0342709
$443$	5.02480e14	1.39926	0.699629	−	0.714506i	$-0.253348\pi$
$443$	5.02480e14	1.39926	0.699629	+	0.714506i	$0.253348\pi$
$444$	0	0
$445$	−1.31738e13	−0.0357876
$446$	4.26748e14	1.14506
$447$	0	0
$448$	5.22059e14	1.36675
$449$	5.06653e14	1.31026	0.655128	−	0.755518i	$-0.272615\pi$
$449$	5.06653e14	1.31026	0.655128	+	0.755518i	$0.272615\pi$
$450$	0	0
$451$	7.46053e14	1.88278
$452$	−1.07740e15	−2.68607
$453$	0	0
$454$	−6.67332e14	−1.62381
$455$	2.58713e13	0.0621950
$456$	0	0
$457$	5.36557e14	1.25915	0.629574	−	0.776941i	$-0.283230\pi$
$457$	5.36557e14	1.25915	0.629574	+	0.776941i	$0.283230\pi$
$458$	−1.35899e14	−0.315106
$459$	0	0
$460$	−9.64980e13	−0.218449
$461$	4.03273e14	0.902077	0.451039	−	0.892504i	$-0.351054\pi$
$461$	4.03273e14	0.902077	0.451039	+	0.892504i	$0.351054\pi$
$462$	0	0
$463$	−7.11827e14	−1.55482	−0.777408	−	0.628997i	$-0.783466\pi$
$463$	−7.11827e14	−1.55482	−0.777408	+	0.628997i	$0.783466\pi$
$464$	1.29846e14	0.280272
$465$	0	0
$466$	−1.13502e15	−2.39265
$467$	8.38289e14	1.74643	0.873215	−	0.487336i	$-0.162031\pi$
$467$	8.38289e14	1.74643	0.873215	+	0.487336i	$0.162031\pi$
$468$	0	0
$469$	9.15815e14	1.86362
$470$	7.88431e13	0.158572
$471$	0	0
$472$	−2.55755e14	−0.502509
$473$	3.59385e13	0.0697948
$474$	0	0
$475$	−1.20129e14	−0.227947
$476$	−2.35024e13	−0.0440833
$477$	0	0
$478$	−5.77064e14	−1.05772
$479$	−5.73586e14	−1.03933	−0.519665	−	0.854370i	$-0.673943\pi$
$479$	−5.73586e14	−1.03933	−0.519665	+	0.854370i	$0.673943\pi$
$480$	0	0
$481$	2.56719e14	0.454632
$482$	1.33092e15	2.33020
$483$	0	0
$484$	1.08872e15	1.86324
$485$	−5.94207e12	−0.0100544
$486$	0	0
$487$	5.05893e14	0.836853	0.418427	−	0.908251i	$-0.362582\pi$
$487$	5.05893e14	0.836853	0.418427	+	0.908251i	$0.362582\pi$
$488$	−1.53009e15	−2.50270
$489$	0	0
$490$	6.40514e12	0.0102435
$491$	−6.80267e14	−1.07580	−0.537899	−	0.843009i	$-0.680782\pi$
$491$	−6.80267e14	−1.07580	−0.537899	+	0.843009i	$0.680782\pi$
$492$	0	0
$493$	9.47385e12	0.0146510
$494$	2.03595e14	0.311365
$495$	0	0
$496$	5.66720e14	0.847656
$497$	2.46377e14	0.364452
$498$	0	0
$499$	−3.65905e13	−0.0529439	−0.0264719	−	0.999650i	$-0.508427\pi$
$499$	−3.65905e13	−0.0529439	−0.0264719	+	0.999650i	$0.508427\pi$
$500$	−2.02482e14	−0.289769
$501$	0	0
$502$	−2.07590e15	−2.90628
$503$	−7.50275e14	−1.03895	−0.519477	−	0.854484i	$-0.673873\pi$
$503$	−7.50275e14	−1.03895	−0.519477	+	0.854484i	$0.673873\pi$
$504$	0	0
$505$	3.83027e13	0.0518951
$506$	2.67860e15	3.58988
$507$	0	0
$508$	2.59549e15	3.40384
$509$	9.81460e14	1.27328	0.636641	−	0.771160i	$-0.280323\pi$
$509$	9.81460e14	1.27328	0.636641	+	0.771160i	$0.280323\pi$
$510$	0	0
$511$	1.12913e15	1.43361
$512$	7.42115e14	0.932151
$513$	0	0
$514$	7.86303e14	0.966703
$515$	1.64747e12	0.00200390
$516$	0	0
$517$	−1.41148e15	−1.68065
$518$	−7.69663e14	−0.906749
$519$	0	0
$520$	7.68752e13	0.0886682
$521$	1.10841e15	1.26500	0.632502	−	0.774558i	$-0.282028\pi$
$521$	1.10841e15	1.26500	0.632502	+	0.774558i	$0.282028\pi$
$522$	0	0
$523$	−4.97440e14	−0.555881	−0.277940	−	0.960598i	$-0.589652\pi$
$523$	−4.97440e14	−0.555881	−0.277940	+	0.960598i	$0.589652\pi$
$524$	1.29546e12	0.00143252
$525$	0	0
$526$	−1.27720e13	−0.0138304
$527$	4.13491e13	0.0443106
$528$	0	0
$529$	1.19936e15	1.25876
$530$	−9.01668e13	−0.0936547
$531$	0	0
$532$	−3.93669e14	−0.400514
$533$	−1.06243e15	−1.06979
$534$	0	0
$535$	−4.01863e13	−0.0396399
$536$	2.72130e15	2.65686
$537$	0	0
$538$	5.40651e14	0.517148
$539$	−1.14667e14	−0.108568
$540$	0	0
$541$	−5.29002e14	−0.490764	−0.245382	−	0.969426i	$-0.578913\pi$
$541$	−5.29002e14	−0.490764	−0.245382	+	0.969426i	$0.578913\pi$
$542$	−3.17827e15	−2.91874
$543$	0	0
$544$	1.56999e13	0.0141287
$545$	−1.32609e14	−0.118138
$546$	0	0
$547$	−2.95428e14	−0.257942	−0.128971	−	0.991648i	$-0.541167\pi$
$547$	−2.95428e14	−0.257942	−0.128971	+	0.991648i	$0.541167\pi$
$548$	1.37096e15	1.18504
$549$	0	0
$550$	2.80124e15	2.37331
$551$	1.58689e14	0.133110
$552$	0	0
$553$	7.84697e14	0.645229
$554$	3.29867e14	0.268556
$555$	0	0
$556$	−2.16593e15	−1.72875
$557$	−4.35634e14	−0.344285	−0.172142	−	0.985072i	$-0.555069\pi$
$557$	−4.35634e14	−0.344285	−0.172142	+	0.985072i	$0.555069\pi$
$558$	0	0
$559$	−5.11787e13	−0.0396574
$560$	−4.84157e13	−0.0371494
$561$	0	0
$562$	1.61224e15	1.21305
$563$	−2.30001e15	−1.71369	−0.856847	−	0.515570i	$-0.827580\pi$
$563$	−2.30001e15	−1.71369	−0.856847	+	0.515570i	$0.827580\pi$
$564$	0	0
$565$	−1.61938e14	−0.118327
$566$	9.80683e14	0.709640
$567$	0	0
$568$	7.32096e14	0.519580
$569$	9.08105e14	0.638291	0.319145	−	0.947706i	$-0.396604\pi$
$569$	9.08105e14	0.638291	0.319145	+	0.947706i	$0.396604\pi$
$570$	0	0
$571$	−1.56459e15	−1.07870	−0.539351	−	0.842081i	$-0.681331\pi$
$571$	−1.56459e15	−1.07870	−0.539351	+	0.842081i	$0.681331\pi$
$572$	−3.06190e15	−2.09080
$573$	0	0
$574$	3.18524e15	2.13367
$575$	2.25070e15	1.49329
$576$	0	0
$577$	1.87429e14	0.122003	0.0610015	−	0.998138i	$-0.480571\pi$
$577$	1.87429e14	0.122003	0.0610015	+	0.998138i	$0.480571\pi$
$578$	2.60122e15	1.67716
$579$	0	0
$580$	1.33309e14	0.0843343
$581$	−1.14003e15	−0.714412
$582$	0	0
$583$	1.61420e15	0.992616
$584$	3.35516e15	2.04382
$585$	0	0
$586$	8.76306e14	0.523864
$587$	1.82614e15	1.08149	0.540746	−	0.841186i	$-0.318142\pi$
$587$	1.82614e15	1.08149	0.540746	+	0.841186i	$0.318142\pi$
$588$	0	0
$589$	6.92605e14	0.402579
$590$	−8.55243e13	−0.0492497
$591$	0	0
$592$	−4.80426e14	−0.271554
$593$	1.26725e15	0.709678	0.354839	−	0.934927i	$-0.384536\pi$
$593$	1.26725e15	0.709678	0.354839	+	0.934927i	$0.384536\pi$
$594$	0	0
$595$	−3.53251e12	−0.00194196
$596$	−3.22016e15	−1.75397
$597$	0	0
$598$	−3.81450e15	−2.03977
$599$	−2.10895e15	−1.11742	−0.558712	−	0.829361i	$-0.688704\pi$
$599$	−2.10895e15	−1.11742	−0.558712	+	0.829361i	$0.688704\pi$
$600$	0	0
$601$	6.37611e14	0.331701	0.165850	−	0.986151i	$-0.446963\pi$
$601$	6.37611e14	0.331701	0.165850	+	0.986151i	$0.446963\pi$
$602$	1.53438e14	0.0790954
$603$	0	0
$604$	−4.80562e15	−2.43246
$605$	1.63640e14	0.0820796
$606$	0	0
$607$	9.44930e14	0.465438	0.232719	−	0.972544i	$-0.425238\pi$
$607$	9.44930e14	0.465438	0.232719	+	0.972544i	$0.425238\pi$
$608$	2.62975e14	0.128365
$609$	0	0
$610$	−5.11660e14	−0.245283
$611$	2.01004e15	0.954944
$612$	0	0
$613$	1.58352e14	0.0738910	0.0369455	−	0.999317i	$-0.488237\pi$
$613$	1.58352e14	0.0738910	0.0369455	+	0.999317i	$0.488237\pi$
$614$	−6.66809e15	−3.08373
$615$	0	0
$616$	4.12611e15	1.87433
$617$	−3.98602e15	−1.79462	−0.897308	−	0.441405i	$-0.854480\pi$
$617$	−3.98602e15	−1.79462	−0.897308	+	0.441405i	$0.854480\pi$
$618$	0	0
$619$	−3.48649e15	−1.54202	−0.771010	−	0.636823i	$-0.780248\pi$
$619$	−3.48649e15	−1.54202	−0.771010	+	0.636823i	$0.780248\pi$
$620$	5.81832e14	0.255061
$621$	0	0
$622$	4.13793e15	1.78212
$623$	1.00692e15	0.429845
$624$	0	0
$625$	2.33849e15	0.980832
$626$	−8.89707e14	−0.369903
$627$	0	0
$628$	1.40299e15	0.573159
$629$	−3.50529e13	−0.0141953
$630$	0	0
$631$	−2.15442e15	−0.857371	−0.428686	−	0.903454i	$-0.641023\pi$
$631$	−2.15442e15	−0.857371	−0.428686	+	0.903454i	$0.641023\pi$
$632$	2.33169e15	0.919870
$633$	0	0
$634$	−3.50318e15	−1.35823
$635$	3.90113e14	0.149946
$636$	0	0
$637$	1.63294e14	0.0616883
$638$	−3.70039e15	−1.38591
$639$	0	0
$640$	3.97123e14	0.146196
$641$	2.90418e14	0.106000	0.0529998	−	0.998595i	$-0.483122\pi$
$641$	2.90418e14	0.106000	0.0529998	+	0.998595i	$0.483122\pi$
$642$	0	0
$643$	−1.83731e15	−0.659206	−0.329603	−	0.944120i	$-0.606915\pi$
$643$	−1.83731e15	−0.659206	−0.329603	+	0.944120i	$0.606915\pi$
$644$	7.37568e15	2.62379
$645$	0	0
$646$	−2.77992e13	−0.00972196
$647$	4.64352e15	1.61018	0.805089	−	0.593154i	$-0.202118\pi$
$647$	4.64352e15	1.61018	0.805089	+	0.593154i	$0.202118\pi$
$648$	0	0
$649$	1.53109e15	0.521981
$650$	−3.98914e15	−1.34852
$651$	0	0
$652$	5.32720e15	1.77067
$653$	2.90253e15	0.956652	0.478326	−	0.878182i	$-0.341244\pi$
$653$	2.90253e15	0.956652	0.478326	+	0.878182i	$0.341244\pi$
$654$	0	0
$655$	1.94713e11	6.31057e−5 0
$656$	1.98823e15	0.638993
$657$	0	0
$658$	−6.02625e15	−1.90460
$659$	−3.52056e15	−1.10342	−0.551711	−	0.834036i	$-0.686025\pi$
$659$	−3.52056e15	−1.10342	−0.551711	+	0.834036i	$0.686025\pi$
$660$	0	0
$661$	−1.40798e15	−0.433998	−0.216999	−	0.976172i	$-0.569627\pi$
$661$	−1.40798e15	−0.433998	−0.216999	+	0.976172i	$0.569627\pi$
$662$	2.07058e14	0.0632956
$663$	0	0
$664$	−3.38755e15	−1.01850
$665$	−5.91702e13	−0.0176435
$666$	0	0
$667$	−2.97315e15	−0.872015
$668$	2.04087e15	0.593670
$669$	0	0
$670$	9.09998e14	0.260393
$671$	9.15995e15	2.59968
$672$	0	0
$673$	−2.28201e15	−0.637139	−0.318569	−	0.947900i	$-0.603202\pi$
$673$	−2.28201e15	−0.637139	−0.318569	+	0.947900i	$0.603202\pi$
$674$	−4.79874e15	−1.32892
$675$	0	0
$676$	−2.30667e15	−0.628461
$677$	9.49590e14	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$677$	9.49590e14	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$678$	0	0
$679$	4.54173e14	0.120764
$680$	−1.04967e13	−0.00276855
$681$	0	0
$682$	−1.61506e16	−4.19154
$683$	4.27936e15	1.10170	0.550852	−	0.834603i	$-0.314303\pi$
$683$	4.27936e15	1.10170	0.550852	+	0.834603i	$0.314303\pi$
$684$	0	0
$685$	2.06062e14	0.0522034
$686$	−6.90763e15	−1.73599
$687$	0	0
$688$	9.57761e13	0.0236876
$689$	−2.29873e15	−0.564004
$690$	0	0
$691$	−3.20692e15	−0.774388	−0.387194	−	0.921998i	$-0.626556\pi$
$691$	−3.20692e15	−0.774388	−0.387194	+	0.921998i	$0.626556\pi$
$692$	−8.78141e15	−2.10369
$693$	0	0
$694$	3.73395e15	0.880423
$695$	−3.25548e14	−0.0761551
$696$	0	0
$697$	1.45066e14	0.0334029
$698$	−4.73455e15	−1.08162
$699$	0	0
$700$	7.71337e15	1.73462
$701$	8.97609e14	0.200280	0.100140	−	0.994973i	$-0.468071\pi$
$701$	8.97609e14	0.200280	0.100140	+	0.994973i	$0.468071\pi$
$702$	0	0
$703$	−5.87142e14	−0.128970
$704$	−9.28674e15	−2.02401
$705$	0	0
$706$	−8.98093e15	−1.92706
$707$	−2.92761e15	−0.623313
$708$	0	0
$709$	−2.08099e15	−0.436230	−0.218115	−	0.975923i	$-0.569991\pi$
$709$	−2.08099e15	−0.436230	−0.218115	+	0.975923i	$0.569991\pi$
$710$	2.44812e14	0.0509227
$711$	0	0
$712$	2.99202e15	0.612807
$713$	−1.29765e16	−2.63732
$714$	0	0
$715$	−4.60217e14	−0.0921041
$716$	3.83793e15	0.762212
$717$	0	0
$718$	−5.61377e15	−1.09792
$719$	8.09021e15	1.57019	0.785093	−	0.619378i	$-0.212615\pi$
$719$	8.09021e15	1.57019	0.785093	+	0.619378i	$0.212615\pi$
$720$	0	0
$721$	−1.25922e14	−0.0240689
$722$	−4.65642e14	−0.0883279
$723$	0	0
$724$	3.37176e15	0.629933
$725$	−3.10927e15	−0.576500
$726$	0	0
$727$	3.88682e15	0.709832	0.354916	−	0.934898i	$-0.384510\pi$
$727$	3.88682e15	0.709832	0.354916	+	0.934898i	$0.384510\pi$
$728$	−5.87584e15	−1.06499
$729$	0	0
$730$	1.12196e15	0.200310
$731$	6.98803e12	0.00123825
$732$	0	0
$733$	−1.76691e15	−0.308420	−0.154210	−	0.988038i	$-0.549283\pi$
$733$	−1.76691e15	−0.308420	−0.154210	+	0.988038i	$0.549283\pi$
$734$	−1.69758e16	−2.94104
$735$	0	0
$736$	−4.92703e15	−0.840925
$737$	−1.62911e16	−2.75982
$738$	0	0
$739$	−1.10711e16	−1.84777	−0.923884	−	0.382673i	$-0.875004\pi$
$739$	−1.10711e16	−1.84777	−0.923884	+	0.382673i	$0.875004\pi$
$740$	−4.93236e14	−0.0817110
$741$	0	0
$742$	6.89177e15	1.12489
$743$	1.07932e15	0.174868	0.0874341	−	0.996170i	$-0.472133\pi$
$743$	1.07932e15	0.174868	0.0874341	+	0.996170i	$0.472133\pi$
$744$	0	0
$745$	−4.84004e14	−0.0772662
$746$	−1.13099e16	−1.79224
$747$	0	0
$748$	4.18077e14	0.0652826
$749$	3.07158e15	0.476115
$750$	0	0
$751$	8.16253e15	1.24682	0.623412	−	0.781893i	$-0.285746\pi$
$751$	8.16253e15	1.24682	0.623412	+	0.781893i	$0.285746\pi$
$752$	−3.76160e15	−0.570393
$753$	0	0
$754$	5.26960e15	0.787472
$755$	−7.22305e14	−0.107155
$756$	0	0
$757$	−1.51608e15	−0.221663	−0.110832	−	0.993839i	$-0.535351\pi$
$757$	−1.51608e15	−0.221663	−0.110832	+	0.993839i	$0.535351\pi$
$758$	2.75937e15	0.400525
$759$	0	0
$760$	−1.75821e14	−0.0251534
$761$	7.78799e15	1.10614	0.553069	−	0.833135i	$-0.313456\pi$
$761$	7.78799e15	1.10614	0.553069	+	0.833135i	$0.313456\pi$
$762$	0	0
$763$	1.01357e16	1.41896
$764$	2.14083e16	2.97555
$765$	0	0
$766$	−4.39687e15	−0.602401
$767$	−2.18037e15	−0.296590
$768$	0	0
$769$	−1.58020e15	−0.211893	−0.105947	−	0.994372i	$-0.533787\pi$
$769$	−1.58020e15	−0.211893	−0.105947	+	0.994372i	$0.533787\pi$
$770$	1.37976e15	0.183699
$771$	0	0
$772$	2.36407e16	3.10288
$773$	7.85872e15	1.02415	0.512076	−	0.858940i	$-0.328877\pi$
$773$	7.85872e15	1.02415	0.512076	+	0.858940i	$0.328877\pi$
$774$	0	0
$775$	−1.35706e16	−1.74357
$776$	1.34955e15	0.172167
$777$	0	0
$778$	1.77133e15	0.222798
$779$	2.42988e15	0.303479
$780$	0	0
$781$	−4.38272e15	−0.539714
$782$	5.20838e14	0.0636892
$783$	0	0
$784$	−3.05589e14	−0.0368468
$785$	2.10875e14	0.0252489
$786$	0	0
$787$	8.58324e15	1.01342	0.506711	−	0.862116i	$-0.330861\pi$
$787$	8.58324e15	1.01342	0.506711	+	0.862116i	$0.330861\pi$
$788$	1.31449e16	1.54122
$789$	0	0
$790$	7.79713e14	0.0901542
$791$	1.23775e16	1.42122
$792$	0	0
$793$	−1.30444e16	−1.47714
$794$	−1.11489e16	−1.25377
$795$	0	0
$796$	1.41175e16	1.56580
$797$	1.71049e16	1.88408	0.942040	−	0.335501i	$-0.108906\pi$
$797$	1.71049e16	1.88408	0.942040	+	0.335501i	$0.108906\pi$
$798$	0	0
$799$	−2.74454e14	−0.0298169
$800$	−5.15261e15	−0.555946
$801$	0	0
$802$	1.53594e14	0.0163461
$803$	−2.00858e16	−2.12302
$804$	0	0
$805$	1.10860e15	0.115584
$806$	2.29994e16	2.38163
$807$	0	0
$808$	−8.69923e15	−0.888625
$809$	1.08992e16	1.10580	0.552899	−	0.833248i	$-0.313521\pi$
$809$	1.08992e16	1.10580	0.552899	+	0.833248i	$0.313521\pi$
$810$	0	0
$811$	7.63422e15	0.764100	0.382050	−	0.924142i	$-0.375218\pi$
$811$	7.63422e15	0.764100	0.382050	+	0.924142i	$0.375218\pi$
$812$	−1.01892e16	−1.01294
$813$	0	0
$814$	1.36913e16	1.34280
$815$	8.00702e14	0.0780017
$816$	0	0
$817$	1.17051e14	0.0112500
$818$	3.33516e15	0.318400
$819$	0	0
$820$	2.04125e15	0.192274
$821$	9.27585e15	0.867893	0.433947	−	0.900939i	$-0.357121\pi$
$821$	9.27585e15	0.867893	0.433947	+	0.900939i	$0.357121\pi$
$822$	0	0
$823$	1.13215e16	1.04521	0.522606	−	0.852575i	$-0.324960\pi$
$823$	1.13215e16	1.04521	0.522606	+	0.852575i	$0.324960\pi$
$824$	−3.74169e14	−0.0343138
$825$	0	0
$826$	6.53692e15	0.591538
$827$	−8.48407e15	−0.762648	−0.381324	−	0.924441i	$-0.624532\pi$
$827$	−8.48407e15	−0.762648	−0.381324	+	0.924441i	$0.624532\pi$
$828$	0	0
$829$	6.02526e15	0.534473	0.267236	−	0.963631i	$-0.413890\pi$
$829$	6.02526e15	0.534473	0.267236	+	0.963631i	$0.413890\pi$
$830$	−1.13279e15	−0.0998206
$831$	0	0
$832$	1.32249e16	1.15004
$833$	−2.22964e13	−0.00192614
$834$	0	0
$835$	3.06752e14	0.0261524
$836$	7.00286e15	0.593118
$837$	0	0
$838$	−2.14863e16	−1.79605
$839$	8.10476e15	0.673053	0.336526	−	0.941674i	$-0.390748\pi$
$839$	8.10476e15	0.673053	0.336526	+	0.941674i	$0.390748\pi$
$840$	0	0
$841$	−8.09321e15	−0.663350
$842$	1.57526e16	1.28273
$843$	0	0
$844$	5.72659e15	0.460270
$845$	−3.46702e14	−0.0276850
$846$	0	0
$847$	−1.25076e16	−0.985859
$848$	4.30186e15	0.336883
$849$	0	0
$850$	5.44685e14	0.0421057
$851$	1.10005e16	0.844891
$852$	0	0
$853$	2.11690e15	0.160502	0.0802512	−	0.996775i	$-0.474428\pi$
$853$	2.11690e15	0.160502	0.0802512	+	0.996775i	$0.474428\pi$
$854$	3.91080e16	2.94610
$855$	0	0
$856$	9.12704e15	0.678772
$857$	−1.63925e16	−1.21130	−0.605648	−	0.795733i	$-0.707086\pi$
$857$	−1.63925e16	−1.21130	−0.605648	+	0.795733i	$0.707086\pi$
$858$	0	0
$859$	1.09497e16	0.798801	0.399400	−	0.916777i	$-0.369218\pi$
$859$	1.09497e16	0.798801	0.399400	+	0.916777i	$0.369218\pi$
$860$	9.83300e13	0.00712762
$861$	0	0
$862$	3.64723e16	2.61020
$863$	1.28096e16	0.910914	0.455457	−	0.890258i	$-0.349476\pi$
$863$	1.28096e16	0.910914	0.455457	+	0.890258i	$0.349476\pi$
$864$	0	0
$865$	−1.31989e15	−0.0926718
$866$	−1.45110e16	−1.01239
$867$	0	0
$868$	−4.44715e16	−3.06354
$869$	−1.39587e16	−0.955515
$870$	0	0
$871$	2.31996e16	1.56813
$872$	3.01178e16	2.02294
$873$	0	0
$874$	8.72414e15	0.578641
$875$	2.32618e15	0.153320
$876$	0	0
$877$	2.83216e16	1.84340	0.921702	−	0.387898i	$-0.126799\pi$
$877$	2.83216e16	1.84340	0.921702	+	0.387898i	$0.126799\pi$
$878$	−1.66515e16	−1.07704
$879$	0	0
$880$	8.61252e14	0.0550143
$881$	1.34936e16	0.856567	0.428284	−	0.903644i	$-0.359118\pi$
$881$	1.34936e16	0.856567	0.428284	+	0.903644i	$0.359118\pi$
$882$	0	0
$883$	5.14902e15	0.322805	0.161403	−	0.986889i	$-0.448398\pi$
$883$	5.14902e15	0.322805	0.161403	+	0.986889i	$0.448398\pi$
$884$	−5.95368e14	−0.0370935
$885$	0	0
$886$	−3.81623e16	−2.34828
$887$	3.44513e15	0.210681	0.105341	−	0.994436i	$-0.466407\pi$
$887$	3.44513e15	0.210681	0.105341	+	0.994436i	$0.466407\pi$
$888$	0	0
$889$	−2.98177e16	−1.80100
$890$	1.00053e15	0.0600597
$891$	0	0
$892$	−2.09030e16	−1.23937
$893$	−4.59716e15	−0.270898
$894$	0	0
$895$	5.76858e14	0.0335770
$896$	−3.03535e16	−1.75596
$897$	0	0
$898$	−3.84793e16	−2.19891
$899$	1.79265e16	1.01816
$900$	0	0
$901$	3.13872e14	0.0176103
$902$	−5.66612e16	−3.15973
$903$	0	0
$904$	3.67791e16	2.02617
$905$	5.06790e14	0.0277499
$906$	0	0
$907$	1.19358e16	0.645671	0.322835	−	0.946455i	$-0.395364\pi$
$907$	1.19358e16	0.645671	0.322835	+	0.946455i	$0.395364\pi$
$908$	3.26874e16	1.75755
$909$	0	0
$910$	−1.96487e15	−0.104378
$911$	−2.50826e16	−1.32441	−0.662204	−	0.749323i	$-0.730379\pi$
$911$	−2.50826e16	−1.32441	−0.662204	+	0.749323i	$0.730379\pi$
$912$	0	0
$913$	2.02797e16	1.05797
$914$	−4.07504e16	−2.11314
$915$	0	0
$916$	6.65664e15	0.341059
$917$	−1.48826e13	−0.000757963 0
$918$	0	0
$919$	−2.52670e16	−1.27151	−0.635754	−	0.771892i	$-0.719311\pi$
$919$	−2.52670e16	−1.27151	−0.635754	+	0.771892i	$0.719311\pi$
$920$	3.29414e15	0.164781
$921$	0	0
$922$	−3.06277e16	−1.51389
$923$	6.24128e15	0.306665
$924$	0	0
$925$	1.15042e16	0.558568
$926$	5.40618e16	2.60934
$927$	0	0
$928$	6.80652e15	0.324647
$929$	5.68538e15	0.269571	0.134785	−	0.990875i	$-0.456966\pi$
$929$	5.68538e15	0.269571	0.134785	+	0.990875i	$0.456966\pi$
$930$	0	0
$931$	−3.73469e14	−0.0174997
$932$	5.55955e16	2.58972
$933$	0	0
$934$	−6.36664e16	−2.93091
$935$	6.28388e13	0.00287583
$936$	0	0
$937$	3.25735e16	1.47332	0.736660	−	0.676263i	$-0.236402\pi$
$937$	3.25735e16	1.47332	0.736660	+	0.676263i	$0.236402\pi$
$938$	−6.95543e16	−3.12758
$939$	0	0
$940$	−3.86190e15	−0.171632
$941$	−1.39335e16	−0.615628	−0.307814	−	0.951447i	$-0.599597\pi$
$941$	−1.39335e16	−0.615628	−0.307814	+	0.951447i	$0.599597\pi$
$942$	0	0
$943$	−4.55254e16	−1.98811
$944$	4.08036e15	0.177155
$945$	0	0
$946$	−2.72945e15	−0.117132
$947$	−6.46550e15	−0.275853	−0.137926	−	0.990442i	$-0.544044\pi$
$947$	−6.46550e15	−0.275853	−0.137926	+	0.990442i	$0.544044\pi$
$948$	0	0
$949$	2.86035e16	1.20630
$950$	9.12357e15	0.382547
$951$	0	0
$952$	8.02297e14	0.0332531
$953$	−2.42277e16	−0.998392	−0.499196	−	0.866489i	$-0.666371\pi$
$953$	−2.42277e16	−0.998392	−0.499196	+	0.866489i	$0.666371\pi$
$954$	0	0
$955$	3.21776e15	0.131079
$956$	2.82658e16	1.14484
$957$	0	0
$958$	4.35627e16	1.74423
$959$	−1.57500e16	−0.627016
$960$	0	0
$961$	5.28327e16	2.07933
$962$	−1.94973e16	−0.762977
$963$	0	0
$964$	−6.51912e16	−2.52212
$965$	3.55330e15	0.136688
$966$	0	0
$967$	1.89676e16	0.721385	0.360692	−	0.932685i	$-0.382540\pi$
$967$	1.89676e16	0.721385	0.360692	+	0.932685i	$0.382540\pi$
$968$	−3.71656e16	−1.40549
$969$	0	0
$970$	4.51288e14	0.0168737
$971$	3.65394e15	0.135849	0.0679243	−	0.997690i	$-0.478362\pi$
$971$	3.65394e15	0.135849	0.0679243	+	0.997690i	$0.478362\pi$
$972$	0	0
$973$	2.48828e16	0.914699
$974$	−3.84215e16	−1.40443
$975$	0	0
$976$	2.44113e16	0.882301
$977$	3.65869e15	0.131494	0.0657469	−	0.997836i	$-0.479057\pi$
$977$	3.65869e15	0.131494	0.0657469	+	0.997836i	$0.479057\pi$
$978$	0	0
$979$	−1.79118e16	−0.636553
$980$	−3.13738e14	−0.0110872
$981$	0	0
$982$	5.16649e16	1.80544
$983$	3.22084e16	1.11925	0.559623	−	0.828747i	$-0.310946\pi$
$983$	3.22084e16	1.11925	0.559623	+	0.828747i	$0.310946\pi$
$984$	0	0
$985$	1.97574e15	0.0678938
$986$	−7.19520e14	−0.0245878
$987$	0	0
$988$	−9.97253e15	−0.337009
$989$	−2.19303e15	−0.0736995
$990$	0	0
$991$	1.65787e16	0.550991	0.275496	−	0.961302i	$-0.411158\pi$
$991$	1.65787e16	0.550991	0.275496	+	0.961302i	$0.411158\pi$
$992$	2.97074e16	0.981863
$993$	0	0
$994$	−1.87118e16	−0.611634
$995$	2.12192e15	0.0689766
$996$	0	0
$997$	−2.48578e16	−0.799171	−0.399585	−	0.916696i	$-0.630846\pi$
$997$	−2.48578e16	−0.799171	−0.399585	+	0.916696i	$0.630846\pi$
$998$	2.77898e15	0.0888520
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.12.a.a.1.1		7
3.2	odd	2		19.12.a.a.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.12.a.a.1.7	✓	7	3.2	odd	2
171.12.a.a.1.1		7	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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q-75.9480 q^{2} +3720.10 q^{4} +559.147 q^{5} -42737.5 q^{7} -126992. q^{8} +O(q^{10})\)	\(q-75.9480 q^{2} +3720.10 q^{4} +559.147 q^{5} -42737.5 q^{7} -126992. q^{8} -42466.1 q^{10} +760245. q^{11} -1.08264e6 q^{13} +3.24583e6 q^{14} +2.02605e6 q^{16} +147825. q^{17} +2.47610e6 q^{19} +2.08008e6 q^{20} -5.77391e7 q^{22} -4.63915e7 q^{23} -4.85155e7 q^{25} +8.22242e7 q^{26} -1.58988e8 q^{28} +6.40882e7 q^{29} +2.79716e8 q^{31} +1.06206e8 q^{32} -1.12270e7 q^{34} -2.38966e7 q^{35} -2.37124e8 q^{37} -1.88055e8 q^{38} -7.10073e7 q^{40} +9.81332e8 q^{41} +4.72722e7 q^{43} +2.82818e9 q^{44} +3.52334e9 q^{46} -1.85661e9 q^{47} -1.50830e8 q^{49} +3.68465e9 q^{50} -4.02752e9 q^{52} +2.12327e9 q^{53} +4.25088e8 q^{55} +5.42734e9 q^{56} -4.86737e9 q^{58} +2.01394e9 q^{59} +1.20487e10 q^{61} -2.12439e10 q^{62} -1.22155e10 q^{64} -6.05353e8 q^{65} -2.14288e10 q^{67} +5.49924e8 q^{68} +1.81490e9 q^{70} -5.76489e9 q^{71} -2.64202e10 q^{73} +1.80091e10 q^{74} +9.21133e9 q^{76} -3.24910e10 q^{77} -1.83608e10 q^{79} +1.13286e9 q^{80} -7.45302e10 q^{82} +2.66752e10 q^{83} +8.26560e7 q^{85} -3.59023e9 q^{86} -9.65452e10 q^{88} -2.35606e10 q^{89} +4.62693e10 q^{91} -1.72581e11 q^{92} +1.41006e11 q^{94} +1.38450e9 q^{95} -1.06270e10 q^{97} +1.14552e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 9 q^{2} + 6661 q^{4} + 14307 q^{5} - 2209 q^{7} + 72891 q^{8}+O(q^{10}) \) 7 * q + 9 * q^2 + 6661 * q^4 + 14307 * q^5 - 2209 * q^7 + 72891 * q^8	\( 7 q + 9 q^{2} + 6661 q^{4} + 14307 q^{5} - 2209 q^{7} + 72891 q^{8} + 610116 q^{10} + 1399461 q^{11} - 2639296 q^{13} + 6202863 q^{14} - 11910239 q^{16} + 9760773 q^{17} + 17332693 q^{19} + 60822132 q^{20} - 96770562 q^{22} + 117151884 q^{23} - 40644056 q^{25} + 140205807 q^{26} - 95042353 q^{28} + 380102178 q^{29} + 108083372 q^{31} - 639362133 q^{32} + 632902677 q^{34} - 258176013 q^{35} + 470630222 q^{37} + 22284891 q^{38} + 1263596868 q^{40} + 572622810 q^{41} + 1170836039 q^{43} - 1857175194 q^{44} + 6333771957 q^{46} + 3485769735 q^{47} + 2791682862 q^{49} + 6699338667 q^{50} - 4968372073 q^{52} + 9143305584 q^{53} - 18483973467 q^{55} + 11872570905 q^{56} + 10101698853 q^{58} + 847227714 q^{59} - 8144938567 q^{61} - 21626318880 q^{62} - 14662293047 q^{64} + 9033654252 q^{65} - 5797397824 q^{67} + 6895758945 q^{68} + 63346638156 q^{70} + 4538589186 q^{71} - 1815379657 q^{73} - 52966894338 q^{74} + 16493295439 q^{76} - 77952325659 q^{77} - 54907201480 q^{79} - 77197994292 q^{80} + 66047089392 q^{82} + 68609936724 q^{83} + 50328221961 q^{85} - 11262589308 q^{86} - 118799781006 q^{88} + 105124973892 q^{89} - 53584340240 q^{91} - 22337690499 q^{92} + 203078543976 q^{94} + 35425548393 q^{95} - 386940894664 q^{97} + 357308952786 q^{98}+O(q^{100}) \) 7 * q + 9 * q^2 + 6661 * q^4 + 14307 * q^5 - 2209 * q^7 + 72891 * q^8 + 610116 * q^10 + 1399461 * q^11 - 2639296 * q^13 + 6202863 * q^14 - 11910239 * q^16 + 9760773 * q^17 + 17332693 * q^19 + 60822132 * q^20 - 96770562 * q^22 + 117151884 * q^23 - 40644056 * q^25 + 140205807 * q^26 - 95042353 * q^28 + 380102178 * q^29 + 108083372 * q^31 - 639362133 * q^32 + 632902677 * q^34 - 258176013 * q^35 + 470630222 * q^37 + 22284891 * q^38 + 1263596868 * q^40 + 572622810 * q^41 + 1170836039 * q^43 - 1857175194 * q^44 + 6333771957 * q^46 + 3485769735 * q^47 + 2791682862 * q^49 + 6699338667 * q^50 - 4968372073 * q^52 + 9143305584 * q^53 - 18483973467 * q^55 + 11872570905 * q^56 + 10101698853 * q^58 + 847227714 * q^59 - 8144938567 * q^61 - 21626318880 * q^62 - 14662293047 * q^64 + 9033654252 * q^65 - 5797397824 * q^67 + 6895758945 * q^68 + 63346638156 * q^70 + 4538589186 * q^71 - 1815379657 * q^73 - 52966894338 * q^74 + 16493295439 * q^76 - 77952325659 * q^77 - 54907201480 * q^79 - 77197994292 * q^80 + 66047089392 * q^82 + 68609936724 * q^83 + 50328221961 * q^85 - 11262589308 * q^86 - 118799781006 * q^88 + 105124973892 * q^89 - 53584340240 * q^91 - 22337690499 * q^92 + 203078543976 * q^94 + 35425548393 * q^95 - 386940894664 * q^97 + 357308952786 * q^98

Introduction

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