LMFDB - Embedded newform 8085.2.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 13x^{6} + 53x^{4} - 72x^{2} - 6x + 24 $ x^8 - 13x^6 + 53x^4 - 72x^2 - 6x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56369$ of defining polynomial
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56369 q^{2} -1.00000 q^{3} +4.57249 q^{4} +1.00000 q^{5} +2.56369 q^{6} -6.59506 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.56369 q^{2} -1.00000 q^{3} +4.57249 q^{4} +1.00000 q^{5} +2.56369 q^{6} -6.59506 q^{8} +1.00000 q^{9} -2.56369 q^{10} -1.00000 q^{11} -4.57249 q^{12} +1.89800 q^{13} -1.00000 q^{15} +7.76268 q^{16} -7.14498 q^{17} -2.56369 q^{18} -5.78820 q^{19} +4.57249 q^{20} +2.56369 q^{22} +1.58845 q^{23} +6.59506 q^{24} +1.00000 q^{25} -4.86588 q^{26} -1.00000 q^{27} -5.41638 q^{29} +2.56369 q^{30} -2.70144 q^{31} -6.71096 q^{32} +1.00000 q^{33} +18.3175 q^{34} +4.57249 q^{36} +10.4849 q^{37} +14.8391 q^{38} -1.89800 q^{39} -6.59506 q^{40} -8.49189 q^{41} +7.05730 q^{43} -4.57249 q^{44} +1.00000 q^{45} -4.07229 q^{46} +13.6294 q^{47} -7.76268 q^{48} -2.56369 q^{50} +7.14498 q^{51} +8.67858 q^{52} +8.59824 q^{53} +2.56369 q^{54} -1.00000 q^{55} +5.78820 q^{57} +13.8859 q^{58} -4.77930 q^{59} -4.57249 q^{60} +7.36219 q^{61} +6.92564 q^{62} +1.67945 q^{64} +1.89800 q^{65} -2.56369 q^{66} +3.21249 q^{67} -32.6703 q^{68} -1.58845 q^{69} -6.06777 q^{71} -6.59506 q^{72} -11.0940 q^{73} -26.8801 q^{74} -1.00000 q^{75} -26.4665 q^{76} +4.86588 q^{78} +8.62986 q^{79} +7.76268 q^{80} +1.00000 q^{81} +21.7705 q^{82} +6.27078 q^{83} -7.14498 q^{85} -18.0927 q^{86} +5.41638 q^{87} +6.59506 q^{88} +13.6075 q^{89} -2.56369 q^{90} +7.26318 q^{92} +2.70144 q^{93} -34.9416 q^{94} -5.78820 q^{95} +6.71096 q^{96} +15.5039 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 10 q^{4} + 8 q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 10 * q^4 + 8 * q^5 + 8 * q^9	$ 8 q - 8 q^{3} + 10 q^{4} + 8 q^{5} + 8 q^{9} - 8 q^{11} - 10 q^{12} - 4 q^{13} - 8 q^{15} + 2 q^{16} - 4 q^{17} - 9 q^{19} + 10 q^{20} - 5 q^{23} + 8 q^{25} - 32 q^{26} - 8 q^{27} - 5 q^{29} - 5 q^{31} + 8 q^{33} + 10 q^{36} + 7 q^{37} + 8 q^{38} + 4 q^{39} - 9 q^{41} + 14 q^{43} - 10 q^{44} + 8 q^{45} + 18 q^{46} + 5 q^{47} - 2 q^{48} + 4 q^{51} - 8 q^{52} - q^{53} - 8 q^{55} + 9 q^{57} + 10 q^{58} - 16 q^{59} - 10 q^{60} - 26 q^{61} + 16 q^{62} - 8 q^{64} - 4 q^{65} + 3 q^{67} - 88 q^{68} + 5 q^{69} - 30 q^{71} - 15 q^{73} - 18 q^{74} - 8 q^{75} - 22 q^{76} + 32 q^{78} + 11 q^{79} + 2 q^{80} + 8 q^{81} - 42 q^{82} - 12 q^{83} - 4 q^{85} - 48 q^{86} + 5 q^{87} + 28 q^{92} + 5 q^{93} - 24 q^{94} - 9 q^{95} - 44 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 10 * q^4 + 8 * q^5 + 8 * q^9 - 8 * q^11 - 10 * q^12 - 4 * q^13 - 8 * q^15 + 2 * q^16 - 4 * q^17 - 9 * q^19 + 10 * q^20 - 5 * q^23 + 8 * q^25 - 32 * q^26 - 8 * q^27 - 5 * q^29 - 5 * q^31 + 8 * q^33 + 10 * q^36 + 7 * q^37 + 8 * q^38 + 4 * q^39 - 9 * q^41 + 14 * q^43 - 10 * q^44 + 8 * q^45 + 18 * q^46 + 5 * q^47 - 2 * q^48 + 4 * q^51 - 8 * q^52 - q^53 - 8 * q^55 + 9 * q^57 + 10 * q^58 - 16 * q^59 - 10 * q^60 - 26 * q^61 + 16 * q^62 - 8 * q^64 - 4 * q^65 + 3 * q^67 - 88 * q^68 + 5 * q^69 - 30 * q^71 - 15 * q^73 - 18 * q^74 - 8 * q^75 - 22 * q^76 + 32 * q^78 + 11 * q^79 + 2 * q^80 + 8 * q^81 - 42 * q^82 - 12 * q^83 - 4 * q^85 - 48 * q^86 + 5 * q^87 + 28 * q^92 + 5 * q^93 - 24 * q^94 - 9 * q^95 - 44 * q^97 - 8 * 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 $

$2$	−2.56369	−1.81280	−0.906400	−	0.422420i	$-0.861181\pi$
$2$	−2.56369	−1.81280	−0.906400	+	0.422420i	$0.861181\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.57249	2.28624
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.56369	1.04662
$7$	0	0
$7$	0	0
$8$	−6.59506	−2.33170
$9$	1.00000	0.333333
$10$	−2.56369	−0.810709
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.57249	−1.31996
$13$	1.89800	0.526411	0.263205	−	0.964740i	$-0.415220\pi$
$13$	1.89800	0.526411	0.263205	+	0.964740i	$0.415220\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	7.76268	1.94067
$17$	−7.14498	−1.73291	−0.866456	−	0.499254i	$-0.833608\pi$
$17$	−7.14498	−1.73291	−0.866456	+	0.499254i	$0.833608\pi$
$18$	−2.56369	−0.604267
$19$	−5.78820	−1.32790	−0.663952	−	0.747775i	$-0.731122\pi$
$19$	−5.78820	−1.32790	−0.663952	+	0.747775i	$0.731122\pi$
$20$	4.57249	1.02244
$21$	0	0
$22$	2.56369	0.546580
$23$	1.58845	0.331215	0.165608	−	0.986192i	$-0.447041\pi$
$23$	1.58845	0.331215	0.165608	+	0.986192i	$0.447041\pi$
$24$	6.59506	1.34621
$25$	1.00000	0.200000
$26$	−4.86588	−0.954277
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.41638	−1.00580	−0.502898	−	0.864346i	$-0.667733\pi$
$29$	−5.41638	−1.00580	−0.502898	+	0.864346i	$0.667733\pi$
$30$	2.56369	0.468063
$31$	−2.70144	−0.485193	−0.242596	−	0.970127i	$-0.577999\pi$
$31$	−2.70144	−0.485193	−0.242596	+	0.970127i	$0.577999\pi$
$32$	−6.71096	−1.18634
$33$	1.00000	0.174078
$34$	18.3175	3.14142
$35$	0	0
$36$	4.57249	0.762082
$37$	10.4849	1.72371	0.861857	−	0.507151i	$-0.169301\pi$
$37$	10.4849	1.72371	0.861857	+	0.507151i	$0.169301\pi$
$38$	14.8391	2.40722
$39$	−1.89800	−0.303923
$40$	−6.59506	−1.04277
$41$	−8.49189	−1.32621	−0.663105	−	0.748526i	$-0.730762\pi$
$41$	−8.49189	−1.32621	−0.663105	+	0.748526i	$0.730762\pi$
$42$	0	0
$43$	7.05730	1.07623	0.538114	−	0.842872i	$-0.319137\pi$
$43$	7.05730	1.07623	0.538114	+	0.842872i	$0.319137\pi$
$44$	−4.57249	−0.689329
$45$	1.00000	0.149071
$46$	−4.07229	−0.600427
$47$	13.6294	1.98806	0.994029	−	0.109115i	$-0.0348017\pi$
$47$	13.6294	1.98806	0.994029	+	0.109115i	$0.0348017\pi$
$48$	−7.76268	−1.12045
$49$	0	0
$50$	−2.56369	−0.362560
$51$	7.14498	1.00050
$52$	8.67858	1.20350
$53$	8.59824	1.18106	0.590530	−	0.807016i	$-0.298919\pi$
$53$	8.59824	1.18106	0.590530	+	0.807016i	$0.298919\pi$
$54$	2.56369	0.348874
$55$	−1.00000	−0.134840
$56$	0	0
$57$	5.78820	0.766666
$58$	13.8859	1.82331
$59$	−4.77930	−0.622211	−0.311106	−	0.950375i	$-0.600699\pi$
$59$	−4.77930	−0.622211	−0.311106	+	0.950375i	$0.600699\pi$
$60$	−4.57249	−0.590306
$61$	7.36219	0.942631	0.471316	−	0.881965i	$-0.343779\pi$
$61$	7.36219	0.942631	0.471316	+	0.881965i	$0.343779\pi$
$62$	6.92564	0.879557
$63$	0	0
$64$	1.67945	0.209931
$65$	1.89800	0.235418
$66$	−2.56369	−0.315568
$67$	3.21249	0.392468	0.196234	−	0.980557i	$-0.437129\pi$
$67$	3.21249	0.392468	0.196234	+	0.980557i	$0.437129\pi$
$68$	−32.6703	−3.96186
$69$	−1.58845	−0.191227
$70$	0	0
$71$	−6.06777	−0.720112	−0.360056	−	0.932931i	$-0.617242\pi$
$71$	−6.06777	−0.720112	−0.360056	+	0.932931i	$0.617242\pi$
$72$	−6.59506	−0.777235
$73$	−11.0940	−1.29845	−0.649226	−	0.760595i	$-0.724907\pi$
$73$	−11.0940	−1.29845	−0.649226	+	0.760595i	$0.724907\pi$
$74$	−26.8801	−3.12475
$75$	−1.00000	−0.115470
$76$	−26.4665	−3.03591
$77$	0	0
$78$	4.86588	0.550952
$79$	8.62986	0.970935	0.485467	−	0.874255i	$-0.338649\pi$
$79$	8.62986	0.970935	0.485467	+	0.874255i	$0.338649\pi$
$80$	7.76268	0.867894
$81$	1.00000	0.111111
$82$	21.7705	2.40415
$83$	6.27078	0.688307	0.344154	−	0.938913i	$-0.388166\pi$
$83$	6.27078	0.688307	0.344154	+	0.938913i	$0.388166\pi$
$84$	0	0
$85$	−7.14498	−0.774982
$86$	−18.0927	−1.95099
$87$	5.41638	0.580697
$88$	6.59506	0.703035
$89$	13.6075	1.44239	0.721194	−	0.692734i	$-0.243594\pi$
$89$	13.6075	1.44239	0.721194	+	0.692734i	$0.243594\pi$
$90$	−2.56369	−0.270236
$91$	0	0
$92$	7.26318	0.757239
$93$	2.70144	0.280126
$94$	−34.9416	−3.60395
$95$	−5.78820	−0.593857
$96$	6.71096	0.684935
$97$	15.5039	1.57419	0.787094	−	0.616834i	$-0.211585\pi$
$97$	15.5039	1.57419	0.787094	+	0.616834i	$0.211585\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.57249	0.457249
$101$	11.6021	1.15445	0.577227	−	0.816584i	$-0.304135\pi$
$101$	11.6021	1.15445	0.577227	+	0.816584i	$0.304135\pi$
$102$	−18.3175	−1.81370
$103$	−2.00794	−0.197848	−0.0989240	−	0.995095i	$-0.531540\pi$
$103$	−2.00794	−0.197848	−0.0989240	+	0.995095i	$0.531540\pi$
$104$	−12.5174	−1.22743
$105$	0	0
$106$	−22.0432	−2.14102
$107$	−7.69415	−0.743821	−0.371911	−	0.928269i	$-0.621297\pi$
$107$	−7.69415	−0.743821	−0.371911	+	0.928269i	$0.621297\pi$
$108$	−4.57249	−0.439988
$109$	−5.01535	−0.480383	−0.240192	−	0.970725i	$-0.577210\pi$
$109$	−5.01535	−0.480383	−0.240192	+	0.970725i	$0.577210\pi$
$110$	2.56369	0.244438
$111$	−10.4849	−0.995187
$112$	0	0
$113$	−3.23524	−0.304345	−0.152173	−	0.988354i	$-0.548627\pi$
$113$	−3.23524	−0.304345	−0.152173	+	0.988354i	$0.548627\pi$
$114$	−14.8391	−1.38981
$115$	1.58845	0.148124
$116$	−24.7663	−2.29950
$117$	1.89800	0.175470
$118$	12.2526	1.12794
$119$	0	0
$120$	6.59506	0.602044
$121$	1.00000	0.0909091
$122$	−18.8743	−1.70880
$123$	8.49189	0.765688
$124$	−12.3523	−1.10927
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.4611	1.46069	0.730345	−	0.683079i	$-0.239360\pi$
$127$	16.4611	1.46069	0.730345	+	0.683079i	$0.239360\pi$
$128$	9.11634	0.805778
$129$	−7.05730	−0.621361
$130$	−4.86588	−0.426766
$131$	−9.03945	−0.789780	−0.394890	−	0.918728i	$-0.629217\pi$
$131$	−9.03945	−0.789780	−0.394890	+	0.918728i	$0.629217\pi$
$132$	4.57249	0.397984
$133$	0	0
$134$	−8.23581	−0.711465
$135$	−1.00000	−0.0860663
$136$	47.1215	4.04064
$137$	−11.1221	−0.950227	−0.475114	−	0.879925i	$-0.657593\pi$
$137$	−11.1221	−0.950227	−0.475114	+	0.879925i	$0.657593\pi$
$138$	4.07229	0.346657
$139$	9.55890	0.810775	0.405388	−	0.914145i	$-0.367137\pi$
$139$	9.55890	0.810775	0.405388	+	0.914145i	$0.367137\pi$
$140$	0	0
$141$	−13.6294	−1.14781
$142$	15.5559	1.30542
$143$	−1.89800	−0.158719
$144$	7.76268	0.646890
$145$	−5.41638	−0.449806
$146$	28.4415	2.35384
$147$	0	0
$148$	47.9423	3.94083
$149$	8.19135	0.671062	0.335531	−	0.942029i	$-0.391084\pi$
$149$	8.19135	0.671062	0.335531	+	0.942029i	$0.391084\pi$
$150$	2.56369	0.209324
$151$	−8.81039	−0.716979	−0.358490	−	0.933534i	$-0.616708\pi$
$151$	−8.81039	−0.716979	−0.358490	+	0.933534i	$0.616708\pi$
$152$	38.1735	3.09628
$153$	−7.14498	−0.577637
$154$	0	0
$155$	−2.70144	−0.216985
$156$	−8.67858	−0.694843
$157$	−16.0037	−1.27723	−0.638615	−	0.769526i	$-0.720492\pi$
$157$	−16.0037	−1.27723	−0.638615	+	0.769526i	$0.720492\pi$
$158$	−22.1243	−1.76011
$159$	−8.59824	−0.681885
$160$	−6.71096	−0.530548
$161$	0	0
$162$	−2.56369	−0.201422
$163$	−10.2364	−0.801779	−0.400889	−	0.916126i	$-0.631299\pi$
$163$	−10.2364	−0.801779	−0.400889	+	0.916126i	$0.631299\pi$
$164$	−38.8291	−3.03204
$165$	1.00000	0.0778499
$166$	−16.0763	−1.24776
$167$	−1.57975	−0.122245	−0.0611224	−	0.998130i	$-0.519468\pi$
$167$	−1.57975	−0.122245	−0.0611224	+	0.998130i	$0.519468\pi$
$168$	0	0
$169$	−9.39760	−0.722892
$170$	18.3175	1.40489
$171$	−5.78820	−0.442635
$172$	32.2694	2.46052
$173$	−20.3783	−1.54933	−0.774667	−	0.632369i	$-0.782083\pi$
$173$	−20.3783	−1.54933	−0.774667	+	0.632369i	$0.782083\pi$
$174$	−13.8859	−1.05269
$175$	0	0
$176$	−7.76268	−0.585134
$177$	4.77930	0.359234
$178$	−34.8852	−2.61476
$179$	−14.9470	−1.11719	−0.558595	−	0.829441i	$-0.688659\pi$
$179$	−14.9470	−1.11719	−0.558595	+	0.829441i	$0.688659\pi$
$180$	4.57249	0.340813
$181$	7.19546	0.534834	0.267417	−	0.963581i	$-0.413830\pi$
$181$	7.19546	0.534834	0.267417	+	0.963581i	$0.413830\pi$
$182$	0	0
$183$	−7.36219	−0.544228
$184$	−10.4759	−0.772296
$185$	10.4849	0.770869
$186$	−6.92564	−0.507813
$187$	7.14498	0.522493
$188$	62.3205	4.54519
$189$	0	0
$190$	14.8391	1.07654
$191$	−13.8760	−1.00403	−0.502015	−	0.864859i	$-0.667408\pi$
$191$	−13.8760	−1.00403	−0.502015	+	0.864859i	$0.667408\pi$
$192$	−1.67945	−0.121204
$193$	11.7602	0.846519	0.423260	−	0.906008i	$-0.360886\pi$
$193$	11.7602	0.846519	0.423260	+	0.906008i	$0.360886\pi$
$194$	−39.7473	−2.85369
$195$	−1.89800	−0.135919
$196$	0	0
$197$	−1.28355	−0.0914492	−0.0457246	−	0.998954i	$-0.514560\pi$
$197$	−1.28355	−0.0914492	−0.0457246	+	0.998954i	$0.514560\pi$
$198$	2.56369	0.182193
$199$	−18.8234	−1.33436	−0.667179	−	0.744897i	$-0.732498\pi$
$199$	−18.8234	−1.33436	−0.667179	+	0.744897i	$0.732498\pi$
$200$	−6.59506	−0.466341
$201$	−3.21249	−0.226591
$202$	−29.7442	−2.09279
$203$	0	0
$204$	32.6703	2.28738
$205$	−8.49189	−0.593099
$206$	5.14773	0.358659
$207$	1.58845	0.110405
$208$	14.7336	1.02159
$209$	5.78820	0.400378
$210$	0	0
$211$	15.6903	1.08017	0.540084	−	0.841611i	$-0.318393\pi$
$211$	15.6903	1.08017	0.540084	+	0.841611i	$0.318393\pi$
$212$	39.3154	2.70019
$213$	6.06777	0.415757
$214$	19.7254	1.34840
$215$	7.05730	0.481304
$216$	6.59506	0.448737
$217$	0	0
$218$	12.8578	0.870839
$219$	11.0940	0.749662
$220$	−4.57249	−0.308277
$221$	−13.5612	−0.912223
$222$	26.8801	1.80408
$223$	−17.0892	−1.14438	−0.572188	−	0.820123i	$-0.693905\pi$
$223$	−17.0892	−1.14438	−0.572188	+	0.820123i	$0.693905\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	8.29413	0.551717
$227$	−10.4108	−0.690988	−0.345494	−	0.938421i	$-0.612289\pi$
$227$	−10.4108	−0.690988	−0.345494	+	0.938421i	$0.612289\pi$
$228$	26.4665	1.75278
$229$	7.12793	0.471027	0.235514	−	0.971871i	$-0.424323\pi$
$229$	7.12793	0.471027	0.235514	+	0.971871i	$0.424323\pi$
$230$	−4.07229	−0.268519
$231$	0	0
$232$	35.7213	2.34522
$233$	−17.2588	−1.13066	−0.565329	−	0.824865i	$-0.691251\pi$
$233$	−17.2588	−1.13066	−0.565329	+	0.824865i	$0.691251\pi$
$234$	−4.86588	−0.318092
$235$	13.6294	0.889087
$236$	−21.8533	−1.42253
$237$	−8.62986	−0.560569
$238$	0	0
$239$	−12.8102	−0.828624	−0.414312	−	0.910135i	$-0.635978\pi$
$239$	−12.8102	−0.828624	−0.414312	+	0.910135i	$0.635978\pi$
$240$	−7.76268	−0.501079
$241$	−12.3733	−0.797034	−0.398517	−	0.917161i	$-0.630475\pi$
$241$	−12.3733	−0.797034	−0.398517	+	0.917161i	$0.630475\pi$
$242$	−2.56369	−0.164800
$243$	−1.00000	−0.0641500
$244$	33.6635	2.15509
$245$	0	0
$246$	−21.7705	−1.38804
$247$	−10.9860	−0.699022
$248$	17.8161	1.13133
$249$	−6.27078	−0.397394
$250$	−2.56369	−0.162142
$251$	−25.3091	−1.59750	−0.798748	−	0.601665i	$-0.794504\pi$
$251$	−25.3091	−1.59750	−0.798748	+	0.601665i	$0.794504\pi$
$252$	0	0
$253$	−1.58845	−0.0998652
$254$	−42.2012	−2.64794
$255$	7.14498	0.447436
$256$	−26.7303	−1.67065
$257$	−25.5099	−1.59127	−0.795633	−	0.605779i	$-0.792862\pi$
$257$	−25.5099	−1.59127	−0.795633	+	0.605779i	$0.792862\pi$
$258$	18.0927	1.12640
$259$	0	0
$260$	8.67858	0.538223
$261$	−5.41638	−0.335265
$262$	23.1743	1.43171
$263$	12.1733	0.750638	0.375319	−	0.926896i	$-0.377533\pi$
$263$	12.1733	0.750638	0.375319	+	0.926896i	$0.377533\pi$
$264$	−6.59506	−0.405898
$265$	8.59824	0.528186
$266$	0	0
$267$	−13.6075	−0.832763
$268$	14.6891	0.897277
$269$	11.7041	0.713610	0.356805	−	0.934179i	$-0.383866\pi$
$269$	11.7041	0.713610	0.356805	+	0.934179i	$0.383866\pi$
$270$	2.56369	0.156021
$271$	3.91385	0.237749	0.118875	−	0.992909i	$-0.462071\pi$
$271$	3.91385	0.237749	0.118875	+	0.992909i	$0.462071\pi$
$272$	−55.4642	−3.36301
$273$	0	0
$274$	28.5136	1.72257
$275$	−1.00000	−0.0603023
$276$	−7.26318	−0.437192
$277$	−0.784251	−0.0471211	−0.0235605	−	0.999722i	$-0.507500\pi$
$277$	−0.784251	−0.0471211	−0.0235605	+	0.999722i	$0.507500\pi$
$278$	−24.5060	−1.46977
$279$	−2.70144	−0.161731
$280$	0	0
$281$	6.75257	0.402825	0.201412	−	0.979507i	$-0.435447\pi$
$281$	6.75257	0.402825	0.201412	+	0.979507i	$0.435447\pi$
$282$	34.9416	2.08074
$283$	−6.12759	−0.364247	−0.182124	−	0.983276i	$-0.558297\pi$
$283$	−6.12759	−0.364247	−0.182124	+	0.983276i	$0.558297\pi$
$284$	−27.7448	−1.64635
$285$	5.78820	0.342863
$286$	4.86588	0.287725
$287$	0	0
$288$	−6.71096	−0.395447
$289$	34.0507	2.00298
$290$	13.8859	0.815408
$291$	−15.5039	−0.908857
$292$	−50.7271	−2.96858
$293$	−13.2684	−0.775145	−0.387573	−	0.921839i	$-0.626686\pi$
$293$	−13.2684	−0.775145	−0.387573	+	0.921839i	$0.626686\pi$
$294$	0	0
$295$	−4.77930	−0.278261
$296$	−69.1488	−4.01919
$297$	1.00000	0.0580259
$298$	−21.0001	−1.21650
$299$	3.01488	0.174355
$300$	−4.57249	−0.263993
$301$	0	0
$302$	22.5871	1.29974
$303$	−11.6021	−0.666524
$304$	−44.9319	−2.57702
$305$	7.36219	0.421558
$306$	18.3175	1.04714
$307$	−8.30013	−0.473713	−0.236857	−	0.971545i	$-0.576117\pi$
$307$	−8.30013	−0.473713	−0.236857	+	0.971545i	$0.576117\pi$
$308$	0	0
$309$	2.00794	0.114228
$310$	6.92564	0.393350
$311$	−3.51739	−0.199453	−0.0997263	−	0.995015i	$-0.531797\pi$
$311$	−3.51739	−0.199453	−0.0997263	+	0.995015i	$0.531797\pi$
$312$	12.5174	0.708659
$313$	29.3143	1.65694	0.828470	−	0.560033i	$-0.189212\pi$
$313$	29.3143	1.65694	0.828470	+	0.560033i	$0.189212\pi$
$314$	41.0283	2.31536
$315$	0	0
$316$	39.4599	2.21979
$317$	11.6461	0.654108	0.327054	−	0.945006i	$-0.393944\pi$
$317$	11.6461	0.654108	0.327054	+	0.945006i	$0.393944\pi$
$318$	22.0432	1.23612
$319$	5.41638	0.303259
$320$	1.67945	0.0938842
$321$	7.69415	0.429445
$322$	0	0
$323$	41.3565	2.30114
$324$	4.57249	0.254027
$325$	1.89800	0.105282
$326$	26.2430	1.45346
$327$	5.01535	0.277349
$328$	56.0045	3.09233
$329$	0	0
$330$	−2.56369	−0.141126
$331$	4.77246	0.262318	0.131159	−	0.991361i	$-0.458130\pi$
$331$	4.77246	0.262318	0.131159	+	0.991361i	$0.458130\pi$
$332$	28.6731	1.57364
$333$	10.4849	0.574571
$334$	4.04998	0.221605
$335$	3.21249	0.175517
$336$	0	0
$337$	10.0268	0.546192	0.273096	−	0.961987i	$-0.411952\pi$
$337$	10.0268	0.546192	0.273096	+	0.961987i	$0.411952\pi$
$338$	24.0925	1.31046
$339$	3.23524	0.175714
$340$	−32.6703	−1.77180
$341$	2.70144	0.146291
$342$	14.8391	0.802408
$343$	0	0
$344$	−46.5433	−2.50945
$345$	−1.58845	−0.0855194
$346$	52.2436	2.80863
$347$	11.3061	0.606946	0.303473	−	0.952840i	$-0.401854\pi$
$347$	11.3061	0.606946	0.303473	+	0.952840i	$0.401854\pi$
$348$	24.7663	1.32762
$349$	8.89490	0.476133	0.238066	−	0.971249i	$-0.423486\pi$
$349$	8.89490	0.476133	0.238066	+	0.971249i	$0.423486\pi$
$350$	0	0
$351$	−1.89800	−0.101308
$352$	6.71096	0.357696
$353$	−26.6236	−1.41703	−0.708516	−	0.705695i	$-0.750635\pi$
$353$	−26.6236	−1.41703	−0.708516	+	0.705695i	$0.750635\pi$
$354$	−12.2526	−0.651219
$355$	−6.06777	−0.322044
$356$	62.2199	3.29765
$357$	0	0
$358$	38.3193	2.02524
$359$	15.0877	0.796296	0.398148	−	0.917321i	$-0.369653\pi$
$359$	15.0877	0.796296	0.398148	+	0.917321i	$0.369653\pi$
$360$	−6.59506	−0.347590
$361$	14.5032	0.763328
$362$	−18.4469	−0.969547
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−11.0940	−0.580686
$366$	18.8743	0.986577
$367$	−23.4009	−1.22152	−0.610759	−	0.791816i	$-0.709136\pi$
$367$	−23.4009	−1.22152	−0.610759	+	0.791816i	$0.709136\pi$
$368$	12.3306	0.642779
$369$	−8.49189	−0.442070
$370$	−26.8801	−1.39743
$371$	0	0
$372$	12.3523	0.640437
$373$	−19.6316	−1.01649	−0.508244	−	0.861213i	$-0.669705\pi$
$373$	−19.6316	−1.01649	−0.508244	+	0.861213i	$0.669705\pi$
$374$	−18.3175	−0.947175
$375$	−1.00000	−0.0516398
$376$	−89.8869	−4.63556
$377$	−10.2803	−0.529462
$378$	0	0
$379$	29.1037	1.49496	0.747478	−	0.664286i	$-0.231264\pi$
$379$	29.1037	1.49496	0.747478	+	0.664286i	$0.231264\pi$
$380$	−26.4665	−1.35770
$381$	−16.4611	−0.843329
$382$	35.5736	1.82011
$383$	21.6663	1.10710	0.553548	−	0.832817i	$-0.313274\pi$
$383$	21.6663	1.10710	0.553548	+	0.832817i	$0.313274\pi$
$384$	−9.11634	−0.465216
$385$	0	0
$386$	−30.1495	−1.53457
$387$	7.05730	0.358743
$388$	70.8916	3.59898
$389$	33.7596	1.71168	0.855841	−	0.517239i	$-0.173040\pi$
$389$	33.7596	1.71168	0.855841	+	0.517239i	$0.173040\pi$
$390$	4.86588	0.246393
$391$	−11.3495	−0.573967
$392$	0	0
$393$	9.03945	0.455980
$394$	3.29062	0.165779
$395$	8.62986	0.434215
$396$	−4.57249	−0.229776
$397$	6.34401	0.318397	0.159198	−	0.987247i	$-0.449109\pi$
$397$	6.34401	0.318397	0.159198	+	0.987247i	$0.449109\pi$
$398$	48.2574	2.41892
$399$	0	0
$400$	7.76268	0.388134
$401$	8.67229	0.433074	0.216537	−	0.976274i	$-0.430524\pi$
$401$	8.67229	0.433074	0.216537	+	0.976274i	$0.430524\pi$
$402$	8.23581	0.410765
$403$	−5.12733	−0.255410
$404$	53.0505	2.63936
$405$	1.00000	0.0496904
$406$	0	0
$407$	−10.4849	−0.519719
$408$	−47.1215	−2.33286
$409$	13.6059	0.672767	0.336384	−	0.941725i	$-0.390796\pi$
$409$	13.6059	0.672767	0.336384	+	0.941725i	$0.390796\pi$
$410$	21.7705	1.07517
$411$	11.1221	0.548614
$412$	−9.18128	−0.452329
$413$	0	0
$414$	−4.07229	−0.200142
$415$	6.27078	0.307820
$416$	−12.7374	−0.624503
$417$	−9.55890	−0.468101
$418$	−14.8391	−0.725805
$419$	−3.73618	−0.182524	−0.0912622	−	0.995827i	$-0.529090\pi$
$419$	−3.73618	−0.182524	−0.0912622	+	0.995827i	$0.529090\pi$
$420$	0	0
$421$	−31.4235	−1.53149	−0.765745	−	0.643145i	$-0.777629\pi$
$421$	−31.4235	−1.53149	−0.765745	+	0.643145i	$0.777629\pi$
$422$	−40.2251	−1.95813
$423$	13.6294	0.662686
$424$	−56.7059	−2.75388
$425$	−7.14498	−0.346582
$426$	−15.5559	−0.753684
$427$	0	0
$428$	−35.1814	−1.70056
$429$	1.89800	0.0916363
$430$	−18.0927	−0.872508
$431$	−37.8027	−1.82089	−0.910445	−	0.413630i	$-0.864261\pi$
$431$	−37.8027	−1.82089	−0.910445	+	0.413630i	$0.864261\pi$
$432$	−7.76268	−0.373482
$433$	−17.9561	−0.862917	−0.431458	−	0.902133i	$-0.642001\pi$
$433$	−17.9561	−0.862917	−0.431458	+	0.902133i	$0.642001\pi$
$434$	0	0
$435$	5.41638	0.259696
$436$	−22.9326	−1.09827
$437$	−9.19428	−0.439822
$438$	−28.4415	−1.35899
$439$	−20.0266	−0.955816	−0.477908	−	0.878410i	$-0.658605\pi$
$439$	−20.0266	−0.955816	−0.477908	+	0.878410i	$0.658605\pi$
$440$	6.59506	0.314407
$441$	0	0
$442$	34.7666	1.65368
$443$	−39.9672	−1.89890	−0.949451	−	0.313916i	$-0.898359\pi$
$443$	−39.9672	−1.89890	−0.949451	+	0.313916i	$0.898359\pi$
$444$	−47.9423	−2.27524
$445$	13.6075	0.645055
$446$	43.8113	2.07452
$447$	−8.19135	−0.387438
$448$	0	0
$449$	−19.3186	−0.911703	−0.455852	−	0.890056i	$-0.650665\pi$
$449$	−19.3186	−0.911703	−0.455852	+	0.890056i	$0.650665\pi$
$450$	−2.56369	−0.120853
$451$	8.49189	0.399868
$452$	−14.7931	−0.695808
$453$	8.81039	0.413948
$454$	26.6900	1.25262
$455$	0	0
$456$	−38.1735	−1.78764
$457$	29.0468	1.35875	0.679377	−	0.733789i	$-0.262250\pi$
$457$	29.0468	1.35875	0.679377	+	0.733789i	$0.262250\pi$
$458$	−18.2738	−0.853878
$459$	7.14498	0.333499
$460$	7.26318	0.338648
$461$	−21.9266	−1.02122	−0.510611	−	0.859812i	$-0.670581\pi$
$461$	−21.9266	−1.02122	−0.510611	+	0.859812i	$0.670581\pi$
$462$	0	0
$463$	−21.2759	−0.988773	−0.494387	−	0.869242i	$-0.664607\pi$
$463$	−21.2759	−0.988773	−0.494387	+	0.869242i	$0.664607\pi$
$464$	−42.0456	−1.95192
$465$	2.70144	0.125276
$466$	44.2460	2.04966
$467$	23.3521	1.08061	0.540303	−	0.841470i	$-0.318310\pi$
$467$	23.3521	1.08061	0.540303	+	0.841470i	$0.318310\pi$
$468$	8.67858	0.401168
$469$	0	0
$470$	−34.9416	−1.61174
$471$	16.0037	0.737409
$472$	31.5197	1.45081
$473$	−7.05730	−0.324495
$474$	22.1243	1.01620
$475$	−5.78820	−0.265581
$476$	0	0
$477$	8.59824	0.393686
$478$	32.8414	1.50213
$479$	−15.5638	−0.711126	−0.355563	−	0.934652i	$-0.615711\pi$
$479$	−15.5638	−0.711126	−0.355563	+	0.934652i	$0.615711\pi$
$480$	6.71096	0.306312
$481$	19.9004	0.907381
$482$	31.7212	1.44486
$483$	0	0
$484$	4.57249	0.207840
$485$	15.5039	0.703998
$486$	2.56369	0.116291
$487$	−16.4860	−0.747054	−0.373527	−	0.927619i	$-0.621852\pi$
$487$	−16.4860	−0.747054	−0.373527	+	0.927619i	$0.621852\pi$
$488$	−48.5540	−2.19794
$489$	10.2364	0.462907
$490$	0	0
$491$	22.3868	1.01030	0.505150	−	0.863032i	$-0.331437\pi$
$491$	22.3868	1.01030	0.505150	+	0.863032i	$0.331437\pi$
$492$	38.8291	1.75055
$493$	38.6999	1.74296
$494$	28.1647	1.26719
$495$	−1.00000	−0.0449467
$496$	−20.9704	−0.941599
$497$	0	0
$498$	16.0763	0.720397
$499$	−11.7656	−0.526702	−0.263351	−	0.964700i	$-0.584828\pi$
$499$	−11.7656	−0.526702	−0.263351	+	0.964700i	$0.584828\pi$
$500$	4.57249	0.204488
$501$	1.57975	0.0705780
$502$	64.8846	2.89594
$503$	−15.8132	−0.705075	−0.352538	−	0.935798i	$-0.614681\pi$
$503$	−15.8132	−0.705075	−0.352538	+	0.935798i	$0.614681\pi$
$504$	0	0
$505$	11.6021	0.516287
$506$	4.07229	0.181036
$507$	9.39760	0.417362
$508$	75.2683	3.33949
$509$	25.5576	1.13282	0.566411	−	0.824123i	$-0.308331\pi$
$509$	25.5576	1.13282	0.566411	+	0.824123i	$0.308331\pi$
$510$	−18.3175	−0.811112
$511$	0	0
$512$	50.2955	2.22277
$513$	5.78820	0.255555
$514$	65.3995	2.88465
$515$	−2.00794	−0.0884803
$516$	−32.2694	−1.42058
$517$	−13.6294	−0.599422
$518$	0	0
$519$	20.3783	0.894509
$520$	−12.5174	−0.548925
$521$	−3.43167	−0.150344	−0.0751722	−	0.997171i	$-0.523951\pi$
$521$	−3.43167	−0.150344	−0.0751722	+	0.997171i	$0.523951\pi$
$522$	13.8859	0.607769
$523$	−12.1294	−0.530380	−0.265190	−	0.964196i	$-0.585435\pi$
$523$	−12.1294	−0.530380	−0.265190	+	0.964196i	$0.585435\pi$
$524$	−41.3328	−1.80563
$525$	0	0
$526$	−31.2085	−1.36076
$527$	19.3017	0.840796
$528$	7.76268	0.337827
$529$	−20.4768	−0.890296
$530$	−22.0432	−0.957495
$531$	−4.77930	−0.207404
$532$	0	0
$533$	−16.1176	−0.698131
$534$	34.8852	1.50963
$535$	−7.69415	−0.332647
$536$	−21.1865	−0.915118
$537$	14.9470	0.645009
$538$	−30.0056	−1.29363
$539$	0	0
$540$	−4.57249	−0.196769
$541$	−29.3063	−1.25998	−0.629988	−	0.776605i	$-0.716940\pi$
$541$	−29.3063	−1.25998	−0.629988	+	0.776605i	$0.716940\pi$
$542$	−10.0339	−0.430992
$543$	−7.19546	−0.308787
$544$	47.9497	2.05583
$545$	−5.01535	−0.214834
$546$	0	0
$547$	−13.7688	−0.588712	−0.294356	−	0.955696i	$-0.595105\pi$
$547$	−13.7688	−0.588712	−0.294356	+	0.955696i	$0.595105\pi$
$548$	−50.8558	−2.17245
$549$	7.36219	0.314210
$550$	2.56369	0.109316
$551$	31.3511	1.33560
$552$	10.4759	0.445885
$553$	0	0
$554$	2.01057	0.0854211
$555$	−10.4849	−0.445061
$556$	43.7080	1.85363
$557$	−26.2899	−1.11394	−0.556969	−	0.830533i	$-0.688036\pi$
$557$	−26.2899	−1.11394	−0.556969	+	0.830533i	$0.688036\pi$
$558$	6.92564	0.293186
$559$	13.3948	0.566538
$560$	0	0
$561$	−7.14498	−0.301661
$562$	−17.3115	−0.730241
$563$	−9.66380	−0.407281	−0.203640	−	0.979046i	$-0.565277\pi$
$563$	−9.66380	−0.407281	−0.203640	+	0.979046i	$0.565277\pi$
$564$	−62.3205	−2.62417
$565$	−3.23524	−0.136107
$566$	15.7092	0.660308
$567$	0	0
$568$	40.0173	1.67909
$569$	17.2830	0.724540	0.362270	−	0.932073i	$-0.382002\pi$
$569$	17.2830	0.724540	0.362270	+	0.932073i	$0.382002\pi$
$570$	−14.8391	−0.621543
$571$	5.49607	0.230003	0.115002	−	0.993365i	$-0.463313\pi$
$571$	5.49607	0.230003	0.115002	+	0.993365i	$0.463313\pi$
$572$	−8.67858	−0.362870
$573$	13.8760	0.579677
$574$	0	0
$575$	1.58845	0.0662431
$576$	1.67945	0.0699771
$577$	−30.1932	−1.25696	−0.628481	−	0.777825i	$-0.716323\pi$
$577$	−30.1932	−1.25696	−0.628481	+	0.777825i	$0.716323\pi$
$578$	−87.2954	−3.63101
$579$	−11.7602	−0.488738
$580$	−24.7663	−1.02837
$581$	0	0
$582$	39.7473	1.64758
$583$	−8.59824	−0.356103
$584$	73.1654	3.02761
$585$	1.89800	0.0784726
$586$	34.0159	1.40518
$587$	0.946272	0.0390568	0.0195284	−	0.999809i	$-0.493784\pi$
$587$	0.946272	0.0390568	0.0195284	+	0.999809i	$0.493784\pi$
$588$	0	0
$589$	15.6365	0.644289
$590$	12.2526	0.504432
$591$	1.28355	0.0527982
$592$	81.3913	3.34516
$593$	−23.9011	−0.981500	−0.490750	−	0.871301i	$-0.663277\pi$
$593$	−23.9011	−0.981500	−0.490750	+	0.871301i	$0.663277\pi$
$594$	−2.56369	−0.105189
$595$	0	0
$596$	37.4549	1.53421
$597$	18.8234	0.770392
$598$	−7.72922	−0.316071
$599$	−24.8429	−1.01505	−0.507526	−	0.861637i	$-0.669440\pi$
$599$	−24.8429	−1.01505	−0.507526	+	0.861637i	$0.669440\pi$
$600$	6.59506	0.269242
$601$	32.2357	1.31492	0.657462	−	0.753488i	$-0.271630\pi$
$601$	32.2357	1.31492	0.657462	+	0.753488i	$0.271630\pi$
$602$	0	0
$603$	3.21249	0.130823
$604$	−40.2854	−1.63919
$605$	1.00000	0.0406558
$606$	29.7442	1.20827
$607$	−14.1837	−0.575699	−0.287850	−	0.957676i	$-0.592940\pi$
$607$	−14.1837	−0.575699	−0.287850	+	0.957676i	$0.592940\pi$
$608$	38.8444	1.57535
$609$	0	0
$610$	−18.8743	−0.764200
$611$	25.8687	1.04653
$612$	−32.6703	−1.32062
$613$	30.3024	1.22390	0.611950	−	0.790896i	$-0.290385\pi$
$613$	30.3024	1.22390	0.611950	+	0.790896i	$0.290385\pi$
$614$	21.2789	0.858748
$615$	8.49189	0.342426
$616$	0	0
$617$	41.4017	1.66677	0.833384	−	0.552694i	$-0.186400\pi$
$617$	41.4017	1.66677	0.833384	+	0.552694i	$0.186400\pi$
$618$	−5.14773	−0.207072
$619$	−18.0036	−0.723625	−0.361813	−	0.932251i	$-0.617842\pi$
$619$	−18.0036	−0.723625	−0.361813	+	0.932251i	$0.617842\pi$
$620$	−12.3523	−0.496080
$621$	−1.58845	−0.0637424
$622$	9.01748	0.361568
$623$	0	0
$624$	−14.7336	−0.589815
$625$	1.00000	0.0400000
$626$	−75.1526	−3.00370
$627$	−5.78820	−0.231158
$628$	−73.1765	−2.92006
$629$	−74.9147	−2.98704
$630$	0	0
$631$	−24.2616	−0.965838	−0.482919	−	0.875665i	$-0.660424\pi$
$631$	−24.2616	−0.965838	−0.482919	+	0.875665i	$0.660424\pi$
$632$	−56.9144	−2.26393
$633$	−15.6903	−0.623635
$634$	−29.8569	−1.18577
$635$	16.4611	0.653240
$636$	−39.3154	−1.55896
$637$	0	0
$638$	−13.8859	−0.549748
$639$	−6.06777	−0.240037
$640$	9.11634	0.360355
$641$	35.1794	1.38950	0.694750	−	0.719251i	$-0.255515\pi$
$641$	35.1794	1.38950	0.694750	+	0.719251i	$0.255515\pi$
$642$	−19.7254	−0.778499
$643$	2.91111	0.114803	0.0574015	−	0.998351i	$-0.481718\pi$
$643$	2.91111	0.114803	0.0574015	+	0.998351i	$0.481718\pi$
$644$	0	0
$645$	−7.05730	−0.277881
$646$	−106.025	−4.17151
$647$	29.8357	1.17296	0.586482	−	0.809963i	$-0.300513\pi$
$647$	29.8357	1.17296	0.586482	+	0.809963i	$0.300513\pi$
$648$	−6.59506	−0.259078
$649$	4.77930	0.187604
$650$	−4.86588	−0.190855
$651$	0	0
$652$	−46.8060	−1.83306
$653$	−2.09860	−0.0821245	−0.0410622	−	0.999157i	$-0.513074\pi$
$653$	−2.09860	−0.0821245	−0.0410622	+	0.999157i	$0.513074\pi$
$654$	−12.8578	−0.502779
$655$	−9.03945	−0.353200
$656$	−65.9198	−2.57374
$657$	−11.0940	−0.432818
$658$	0	0
$659$	25.9807	1.01206	0.506032	−	0.862515i	$-0.331112\pi$
$659$	25.9807	1.01206	0.506032	+	0.862515i	$0.331112\pi$
$660$	4.57249	0.177984
$661$	43.8009	1.70366	0.851829	−	0.523821i	$-0.175494\pi$
$661$	43.8009	1.70366	0.851829	+	0.523821i	$0.175494\pi$
$662$	−12.2351	−0.475531
$663$	13.5612	0.526672
$664$	−41.3561	−1.60493
$665$	0	0
$666$	−26.8801	−1.04158
$667$	−8.60366	−0.333135
$668$	−7.22339	−0.279481
$669$	17.0892	0.660706
$670$	−8.23581	−0.318177
$671$	−7.36219	−0.284214
$672$	0	0
$673$	−17.9117	−0.690444	−0.345222	−	0.938521i	$-0.612196\pi$
$673$	−17.9117	−0.690444	−0.345222	+	0.938521i	$0.612196\pi$
$674$	−25.7055	−0.990137
$675$	−1.00000	−0.0384900
$676$	−42.9704	−1.65271
$677$	−30.1787	−1.15986	−0.579931	−	0.814666i	$-0.696920\pi$
$677$	−30.1787	−1.15986	−0.579931	+	0.814666i	$0.696920\pi$
$678$	−8.29413	−0.318534
$679$	0	0
$680$	47.1215	1.80703
$681$	10.4108	0.398942
$682$	−6.92564	−0.265196
$683$	0.00239243	9.15439e−5 0	4.57719e−5	−	1.00000i	$-0.499985\pi$
$683$	0.00239243	9.15439e−5 0	4.57719e−5		1.00000i	$0.499985\pi$
$684$	−26.4665	−1.01197
$685$	−11.1221	−0.424954
$686$	0	0
$687$	−7.12793	−0.271948
$688$	54.7836	2.08860
$689$	16.3195	0.621722
$690$	4.07229	0.155030
$691$	−30.1192	−1.14579	−0.572893	−	0.819630i	$-0.694179\pi$
$691$	−30.1192	−1.14579	−0.572893	+	0.819630i	$0.694179\pi$
$692$	−93.1796	−3.54216
$693$	0	0
$694$	−28.9854	−1.10027
$695$	9.55890	0.362590
$696$	−35.7213	−1.35401
$697$	60.6744	2.29821
$698$	−22.8037	−0.863134
$699$	17.2588	0.652786
$700$	0	0
$701$	−11.3960	−0.430419	−0.215210	−	0.976568i	$-0.569043\pi$
$701$	−11.3960	−0.430419	−0.215210	+	0.976568i	$0.569043\pi$
$702$	4.86588	0.183651
$703$	−60.6889	−2.28893
$704$	−1.67945	−0.0632967
$705$	−13.6294	−0.513314
$706$	68.2546	2.56879
$707$	0	0
$708$	21.8533	0.821296
$709$	−29.0546	−1.09117	−0.545584	−	0.838056i	$-0.683692\pi$
$709$	−29.0546	−1.09117	−0.545584	+	0.838056i	$0.683692\pi$
$710$	15.5559	0.583801
$711$	8.62986	0.323645
$712$	−89.7419	−3.36322
$713$	−4.29111	−0.160703
$714$	0	0
$715$	−1.89800	−0.0709812
$716$	−68.3449	−2.55417
$717$	12.8102	0.478406
$718$	−38.6800	−1.44353
$719$	−7.06722	−0.263563	−0.131781	−	0.991279i	$-0.542070\pi$
$719$	−7.06722	−0.263563	−0.131781	+	0.991279i	$0.542070\pi$
$720$	7.76268	0.289298
$721$	0	0
$722$	−37.1818	−1.38376
$723$	12.3733	0.460168
$724$	32.9011	1.22276
$725$	−5.41638	−0.201159
$726$	2.56369	0.0951473
$727$	−20.4353	−0.757903	−0.378951	−	0.925417i	$-0.623715\pi$
$727$	−20.4353	−0.757903	−0.378951	+	0.925417i	$0.623715\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	28.4415	1.05267
$731$	−50.4243	−1.86501
$732$	−33.6635	−1.24424
$733$	−21.4552	−0.792465	−0.396232	−	0.918150i	$-0.629683\pi$
$733$	−21.4552	−0.792465	−0.396232	+	0.918150i	$0.629683\pi$
$734$	59.9927	2.21437
$735$	0	0
$736$	−10.6600	−0.392935
$737$	−3.21249	−0.118333
$738$	21.7705	0.801385
$739$	−19.5787	−0.720215	−0.360108	−	0.932911i	$-0.617260\pi$
$739$	−19.5787	−0.720215	−0.360108	+	0.932911i	$0.617260\pi$
$740$	47.9423	1.76239
$741$	10.9860	0.403581
$742$	0	0
$743$	−32.5938	−1.19575	−0.597875	−	0.801589i	$-0.703988\pi$
$743$	−32.5938	−1.19575	−0.597875	+	0.801589i	$0.703988\pi$
$744$	−17.8161	−0.653171
$745$	8.19135	0.300108
$746$	50.3294	1.84269
$747$	6.27078	0.229436
$748$	32.6703	1.19455
$749$	0	0
$750$	2.56369	0.0936126
$751$	−52.1890	−1.90440	−0.952202	−	0.305470i	$-0.901187\pi$
$751$	−52.1890	−1.90440	−0.952202	+	0.305470i	$0.901187\pi$
$752$	105.801	3.85816
$753$	25.3091	0.922315
$754$	26.3554	0.959808
$755$	−8.81039	−0.320643
$756$	0	0
$757$	32.6845	1.18794	0.593969	−	0.804488i	$-0.297560\pi$
$757$	32.6845	1.18794	0.593969	+	0.804488i	$0.297560\pi$
$758$	−74.6128	−2.71006
$759$	1.58845	0.0576572
$760$	38.1735	1.38470
$761$	−20.7136	−0.750868	−0.375434	−	0.926849i	$-0.622506\pi$
$761$	−20.7136	−0.750868	−0.375434	+	0.926849i	$0.622506\pi$
$762$	42.2012	1.52879
$763$	0	0
$764$	−63.4477	−2.29546
$765$	−7.14498	−0.258327
$766$	−55.5456	−2.00694
$767$	−9.07110	−0.327539
$768$	26.7303	0.964548
$769$	−6.07692	−0.219139	−0.109570	−	0.993979i	$-0.534947\pi$
$769$	−6.07692	−0.219139	−0.109570	+	0.993979i	$0.534947\pi$
$770$	0	0
$771$	25.5099	0.918718
$772$	53.7735	1.93535
$773$	48.3108	1.73762	0.868809	−	0.495147i	$-0.164886\pi$
$773$	48.3108	1.73762	0.868809	+	0.495147i	$0.164886\pi$
$774$	−18.0927	−0.650329
$775$	−2.70144	−0.0970385
$776$	−102.249	−3.67054
$777$	0	0
$778$	−86.5491	−3.10294
$779$	49.1527	1.76108
$780$	−8.67858	−0.310743
$781$	6.06777	0.217122
$782$	29.0965	1.04049
$783$	5.41638	0.193566
$784$	0	0
$785$	−16.0037	−0.571195
$786$	−23.1743	−0.826600
$787$	32.1142	1.14475	0.572374	−	0.819993i	$-0.306023\pi$
$787$	32.1142	1.14475	0.572374	+	0.819993i	$0.306023\pi$
$788$	−5.86902	−0.209075
$789$	−12.1733	−0.433381
$790$	−22.1243	−0.787146
$791$	0	0
$792$	6.59506	0.234345
$793$	13.9734	0.496211
$794$	−16.2640	−0.577189
$795$	−8.59824	−0.304948
$796$	−86.0700	−3.05067
$797$	−27.9461	−0.989903	−0.494952	−	0.868921i	$-0.664814\pi$
$797$	−27.9461	−0.989903	−0.494952	+	0.868921i	$0.664814\pi$
$798$	0	0
$799$	−97.3821	−3.44513
$800$	−6.71096	−0.237268
$801$	13.6075	0.480796
$802$	−22.2330	−0.785076
$803$	11.0940	0.391498
$804$	−14.6891	−0.518043
$805$	0	0
$806$	13.1449	0.463008
$807$	−11.7041	−0.412003
$808$	−76.5166	−2.69184
$809$	8.37249	0.294361	0.147181	−	0.989110i	$-0.452980\pi$
$809$	8.37249	0.294361	0.147181	+	0.989110i	$0.452980\pi$
$810$	−2.56369	−0.0900788
$811$	−37.6282	−1.32130	−0.660652	−	0.750693i	$-0.729720\pi$
$811$	−37.6282	−1.32130	−0.660652	+	0.750693i	$0.729720\pi$
$812$	0	0
$813$	−3.91385	−0.137265
$814$	26.8801	0.942148
$815$	−10.2364	−0.358566
$816$	55.4642	1.94163
$817$	−40.8491	−1.42913
$818$	−34.8812	−1.21959
$819$	0	0
$820$	−38.8291	−1.35597
$821$	14.1973	0.495488	0.247744	−	0.968826i	$-0.420311\pi$
$821$	14.1973	0.495488	0.247744	+	0.968826i	$0.420311\pi$
$822$	−28.5136	−0.994527
$823$	−2.76253	−0.0962957	−0.0481478	−	0.998840i	$-0.515332\pi$
$823$	−2.76253	−0.0962957	−0.0481478	+	0.998840i	$0.515332\pi$
$824$	13.2425	0.461323
$825$	1.00000	0.0348155
$826$	0	0
$827$	4.46450	0.155246	0.0776230	−	0.996983i	$-0.475267\pi$
$827$	4.46450	0.155246	0.0776230	+	0.996983i	$0.475267\pi$
$828$	7.26318	0.252413
$829$	−8.14580	−0.282915	−0.141458	−	0.989944i	$-0.545179\pi$
$829$	−8.14580	−0.282915	−0.141458	+	0.989944i	$0.545179\pi$
$830$	−16.0763	−0.558017
$831$	0.784251	0.0272054
$832$	3.18760	0.110510
$833$	0	0
$834$	24.5060	0.848574
$835$	−1.57975	−0.0546695
$836$	26.4665	0.915362
$837$	2.70144	0.0933754
$838$	9.57840	0.330880
$839$	7.52439	0.259771	0.129885	−	0.991529i	$-0.458539\pi$
$839$	7.52439	0.259771	0.129885	+	0.991529i	$0.458539\pi$
$840$	0	0
$841$	0.337168	0.0116265
$842$	80.5601	2.77628
$843$	−6.75257	−0.232571
$844$	71.7439	2.46953
$845$	−9.39760	−0.323287
$846$	−34.9416	−1.20132
$847$	0	0
$848$	66.7454	2.29205
$849$	6.12759	0.210298
$850$	18.3175	0.628285
$851$	16.6548	0.570921
$852$	27.7448	0.950522
$853$	38.2038	1.30807	0.654037	−	0.756462i	$-0.273074\pi$
$853$	38.2038	1.30807	0.654037	+	0.756462i	$0.273074\pi$
$854$	0	0
$855$	−5.78820	−0.197952
$856$	50.7433	1.73437
$857$	−39.3821	−1.34527	−0.672634	−	0.739976i	$-0.734837\pi$
$857$	−39.3821	−1.34527	−0.672634	+	0.739976i	$0.734837\pi$
$858$	−4.86588	−0.166118
$859$	−46.8527	−1.59859	−0.799296	−	0.600937i	$-0.794794\pi$
$859$	−46.8527	−1.59859	−0.799296	+	0.600937i	$0.794794\pi$
$860$	32.2694	1.10038
$861$	0	0
$862$	96.9142	3.30091
$863$	44.0908	1.50087	0.750433	−	0.660946i	$-0.229845\pi$
$863$	44.0908	1.50087	0.750433	+	0.660946i	$0.229845\pi$
$864$	6.71096	0.228312
$865$	−20.3783	−0.692883
$866$	46.0339	1.56430
$867$	−34.0507	−1.15642
$868$	0	0
$869$	−8.62986	−0.292748
$870$	−13.8859	−0.470776
$871$	6.09730	0.206599
$872$	33.0765	1.12011
$873$	15.5039	0.524729
$874$	23.5712	0.797309
$875$	0	0
$876$	50.7271	1.71391
$877$	28.7938	0.972297	0.486148	−	0.873876i	$-0.338401\pi$
$877$	28.7938	0.972297	0.486148	+	0.873876i	$0.338401\pi$
$878$	51.3418	1.73270
$879$	13.2684	0.447530
$880$	−7.76268	−0.261680
$881$	−9.53677	−0.321302	−0.160651	−	0.987011i	$-0.551359\pi$
$881$	−9.53677	−0.321302	−0.160651	+	0.987011i	$0.551359\pi$
$882$	0	0
$883$	−29.2161	−0.983200	−0.491600	−	0.870821i	$-0.663588\pi$
$883$	−29.2161	−0.983200	−0.491600	+	0.870821i	$0.663588\pi$
$884$	−62.0083	−2.08556
$885$	4.77930	0.160654
$886$	102.464	3.44233
$887$	−28.6339	−0.961431	−0.480716	−	0.876876i	$-0.659623\pi$
$887$	−28.6339	−0.961431	−0.480716	+	0.876876i	$0.659623\pi$
$888$	69.1488	2.32048
$889$	0	0
$890$	−34.8852	−1.16936
$891$	−1.00000	−0.0335013
$892$	−78.1401	−2.61632
$893$	−78.8899	−2.63995
$894$	21.0001	0.702347
$895$	−14.9470	−0.499622
$896$	0	0
$897$	−3.01488	−0.100664
$898$	49.5270	1.65274
$899$	14.6320	0.488005
$900$	4.57249	0.152416
$901$	−61.4342	−2.04667
$902$	−21.7705	−0.724880
$903$	0	0
$904$	21.3366	0.709644
$905$	7.19546	0.239185
$906$	−22.5871	−0.750405
$907$	4.39324	0.145875	0.0729376	−	0.997337i	$-0.476763\pi$
$907$	4.39324	0.145875	0.0729376	+	0.997337i	$0.476763\pi$
$908$	−47.6032	−1.57977
$909$	11.6021	0.384818
$910$	0	0
$911$	−14.2517	−0.472181	−0.236090	−	0.971731i	$-0.575866\pi$
$911$	−14.2517	−0.472181	−0.236090	+	0.971731i	$0.575866\pi$
$912$	44.9319	1.48784
$913$	−6.27078	−0.207532
$914$	−74.4670	−2.46315
$915$	−7.36219	−0.243386
$916$	32.5924	1.07688
$917$	0	0
$918$	−18.3175	−0.604567
$919$	−24.7488	−0.816388	−0.408194	−	0.912895i	$-0.633841\pi$
$919$	−24.7488	−0.816388	−0.408194	+	0.912895i	$0.633841\pi$
$920$	−10.4759	−0.345381
$921$	8.30013	0.273499
$922$	56.2128	1.85127
$923$	−11.5166	−0.379075
$924$	0	0
$925$	10.4849	0.344743
$926$	54.5446	1.79245
$927$	−2.00794	−0.0659494
$928$	36.3491	1.19322
$929$	−55.6621	−1.82622	−0.913108	−	0.407719i	$-0.866324\pi$
$929$	−55.6621	−1.82622	−0.913108	+	0.407719i	$0.866324\pi$
$930$	−6.92564	−0.227101
$931$	0	0
$932$	−78.9155	−2.58496
$933$	3.51739	0.115154
$934$	−59.8675	−1.95892
$935$	7.14498	0.233666
$936$	−12.5174	−0.409145
$937$	28.1402	0.919300	0.459650	−	0.888100i	$-0.347975\pi$
$937$	28.1402	0.919300	0.459650	+	0.888100i	$0.347975\pi$
$938$	0	0
$939$	−29.3143	−0.956635
$940$	62.3205	2.03267
$941$	−25.3456	−0.826242	−0.413121	−	0.910676i	$-0.635561\pi$
$941$	−25.3456	−0.826242	−0.413121	+	0.910676i	$0.635561\pi$
$942$	−41.0283	−1.33678
$943$	−13.4890	−0.439261
$944$	−37.1001	−1.20751
$945$	0	0
$946$	18.0927	0.588245
$947$	−1.93638	−0.0629239	−0.0314620	−	0.999505i	$-0.510016\pi$
$947$	−1.93638	−0.0629239	−0.0314620	+	0.999505i	$0.510016\pi$
$948$	−39.4599	−1.28160
$949$	−21.0564	−0.683519
$950$	14.8391	0.481445
$951$	−11.6461	−0.377649
$952$	0	0
$953$	34.5110	1.11792	0.558960	−	0.829194i	$-0.311200\pi$
$953$	34.5110	1.11792	0.558960	+	0.829194i	$0.311200\pi$
$954$	−22.0432	−0.713675
$955$	−13.8760	−0.449016
$956$	−58.5746	−1.89444
$957$	−5.41638	−0.175087
$958$	39.9006	1.28913
$959$	0	0
$960$	−1.67945	−0.0542040
$961$	−23.7022	−0.764588
$962$	−51.0185	−1.64490
$963$	−7.69415	−0.247940
$964$	−56.5767	−1.82221
$965$	11.7602	0.378575
$966$	0	0
$967$	26.9952	0.868109	0.434054	−	0.900887i	$-0.357083\pi$
$967$	26.9952	0.868109	0.434054	+	0.900887i	$0.357083\pi$
$968$	−6.59506	−0.211973
$969$	−41.3565	−1.32856
$970$	−39.7473	−1.27621
$971$	−28.4811	−0.914003	−0.457002	−	0.889466i	$-0.651077\pi$
$971$	−28.4811	−0.914003	−0.457002	+	0.889466i	$0.651077\pi$
$972$	−4.57249	−0.146663
$973$	0	0
$974$	42.2650	1.35426
$975$	−1.89800	−0.0607846
$976$	57.1503	1.82934
$977$	58.2655	1.86408	0.932039	−	0.362359i	$-0.118028\pi$
$977$	58.2655	1.86408	0.932039	+	0.362359i	$0.118028\pi$
$978$	−26.2430	−0.839158
$979$	−13.6075	−0.434896
$980$	0	0
$981$	−5.01535	−0.160128
$982$	−57.3926	−1.83147
$983$	−19.7266	−0.629181	−0.314591	−	0.949227i	$-0.601867\pi$
$983$	−19.7266	−0.629181	−0.314591	+	0.949227i	$0.601867\pi$
$984$	−56.0045	−1.78536
$985$	−1.28355	−0.0408973
$986$	−99.2145	−3.15963
$987$	0	0
$988$	−50.2334	−1.59814
$989$	11.2102	0.356463
$990$	2.56369	0.0814793
$991$	19.7951	0.628811	0.314405	−	0.949289i	$-0.398195\pi$
$991$	19.7951	0.628811	0.314405	+	0.949289i	$0.398195\pi$
$992$	18.1293	0.575604
$993$	−4.77246	−0.151450
$994$	0	0
$995$	−18.8234	−0.596743
$996$	−28.6731	−0.908541
$997$	6.15140	0.194817	0.0974084	−	0.995244i	$-0.468945\pi$
$997$	6.15140	0.194817	0.0974084	+	0.995244i	$0.468945\pi$
$998$	30.1634	0.954805
$999$	−10.4849	−0.331729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.ce.1.1		8
7.3	odd	6		1155.2.q.j.331.8	✓	16
7.5	odd	6		1155.2.q.j.991.8	yes	16
7.6	odd	2		8085.2.a.cf.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.q.j.331.8	✓	16	7.3	odd	6
1155.2.q.j.991.8	yes	16	7.5	odd	6
8085.2.a.ce.1.1		8	1.1	even	1	trivial
8085.2.a.cf.1.1		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56369 q^{2} -1.00000 q^{3} +4.57249 q^{4} +1.00000 q^{5} +2.56369 q^{6} -6.59506 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.56369 q^{2} -1.00000 q^{3} +4.57249 q^{4} +1.00000 q^{5} +2.56369 q^{6} -6.59506 q^{8} +1.00000 q^{9} -2.56369 q^{10} -1.00000 q^{11} -4.57249 q^{12} +1.89800 q^{13} -1.00000 q^{15} +7.76268 q^{16} -7.14498 q^{17} -2.56369 q^{18} -5.78820 q^{19} +4.57249 q^{20} +2.56369 q^{22} +1.58845 q^{23} +6.59506 q^{24} +1.00000 q^{25} -4.86588 q^{26} -1.00000 q^{27} -5.41638 q^{29} +2.56369 q^{30} -2.70144 q^{31} -6.71096 q^{32} +1.00000 q^{33} +18.3175 q^{34} +4.57249 q^{36} +10.4849 q^{37} +14.8391 q^{38} -1.89800 q^{39} -6.59506 q^{40} -8.49189 q^{41} +7.05730 q^{43} -4.57249 q^{44} +1.00000 q^{45} -4.07229 q^{46} +13.6294 q^{47} -7.76268 q^{48} -2.56369 q^{50} +7.14498 q^{51} +8.67858 q^{52} +8.59824 q^{53} +2.56369 q^{54} -1.00000 q^{55} +5.78820 q^{57} +13.8859 q^{58} -4.77930 q^{59} -4.57249 q^{60} +7.36219 q^{61} +6.92564 q^{62} +1.67945 q^{64} +1.89800 q^{65} -2.56369 q^{66} +3.21249 q^{67} -32.6703 q^{68} -1.58845 q^{69} -6.06777 q^{71} -6.59506 q^{72} -11.0940 q^{73} -26.8801 q^{74} -1.00000 q^{75} -26.4665 q^{76} +4.86588 q^{78} +8.62986 q^{79} +7.76268 q^{80} +1.00000 q^{81} +21.7705 q^{82} +6.27078 q^{83} -7.14498 q^{85} -18.0927 q^{86} +5.41638 q^{87} +6.59506 q^{88} +13.6075 q^{89} -2.56369 q^{90} +7.26318 q^{92} +2.70144 q^{93} -34.9416 q^{94} -5.78820 q^{95} +6.71096 q^{96} +15.5039 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 10 q^{4} + 8 q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 10 * q^4 + 8 * q^5 + 8 * q^9	\( 8 q - 8 q^{3} + 10 q^{4} + 8 q^{5} + 8 q^{9} - 8 q^{11} - 10 q^{12} - 4 q^{13} - 8 q^{15} + 2 q^{16} - 4 q^{17} - 9 q^{19} + 10 q^{20} - 5 q^{23} + 8 q^{25} - 32 q^{26} - 8 q^{27} - 5 q^{29} - 5 q^{31} + 8 q^{33} + 10 q^{36} + 7 q^{37} + 8 q^{38} + 4 q^{39} - 9 q^{41} + 14 q^{43} - 10 q^{44} + 8 q^{45} + 18 q^{46} + 5 q^{47} - 2 q^{48} + 4 q^{51} - 8 q^{52} - q^{53} - 8 q^{55} + 9 q^{57} + 10 q^{58} - 16 q^{59} - 10 q^{60} - 26 q^{61} + 16 q^{62} - 8 q^{64} - 4 q^{65} + 3 q^{67} - 88 q^{68} + 5 q^{69} - 30 q^{71} - 15 q^{73} - 18 q^{74} - 8 q^{75} - 22 q^{76} + 32 q^{78} + 11 q^{79} + 2 q^{80} + 8 q^{81} - 42 q^{82} - 12 q^{83} - 4 q^{85} - 48 q^{86} + 5 q^{87} + 28 q^{92} + 5 q^{93} - 24 q^{94} - 9 q^{95} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 10 * q^4 + 8 * q^5 + 8 * q^9 - 8 * q^11 - 10 * q^12 - 4 * q^13 - 8 * q^15 + 2 * q^16 - 4 * q^17 - 9 * q^19 + 10 * q^20 - 5 * q^23 + 8 * q^25 - 32 * q^26 - 8 * q^27 - 5 * q^29 - 5 * q^31 + 8 * q^33 + 10 * q^36 + 7 * q^37 + 8 * q^38 + 4 * q^39 - 9 * q^41 + 14 * q^43 - 10 * q^44 + 8 * q^45 + 18 * q^46 + 5 * q^47 - 2 * q^48 + 4 * q^51 - 8 * q^52 - q^53 - 8 * q^55 + 9 * q^57 + 10 * q^58 - 16 * q^59 - 10 * q^60 - 26 * q^61 + 16 * q^62 - 8 * q^64 - 4 * q^65 + 3 * q^67 - 88 * q^68 + 5 * q^69 - 30 * q^71 - 15 * q^73 - 18 * q^74 - 8 * q^75 - 22 * q^76 + 32 * q^78 + 11 * q^79 + 2 * q^80 + 8 * q^81 - 42 * q^82 - 12 * q^83 - 4 * q^85 - 48 * q^86 + 5 * q^87 + 28 * q^92 + 5 * q^93 - 24 * q^94 - 9 * q^95 - 44 * q^97 - 8 * q^99

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