LMFDB - Embedded newform 8619.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.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:	$68.8230615021$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} -1.00000 q^{3} +4.56155 q^{4} -3.56155 q^{5} -2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.56155 q^{2} -1.00000 q^{3} +4.56155 q^{4} -3.56155 q^{5} -2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} -9.12311 q^{10} -1.56155 q^{11} -4.56155 q^{12} +3.56155 q^{15} +7.68466 q^{16} +1.00000 q^{17} +2.56155 q^{18} +4.68466 q^{19} -16.2462 q^{20} -4.00000 q^{22} -2.43845 q^{23} -6.56155 q^{24} +7.68466 q^{25} -1.00000 q^{27} -8.24621 q^{29} +9.12311 q^{30} -3.12311 q^{31} +6.56155 q^{32} +1.56155 q^{33} +2.56155 q^{34} +4.56155 q^{36} +5.12311 q^{37} +12.0000 q^{38} -23.3693 q^{40} +3.56155 q^{41} +4.68466 q^{43} -7.12311 q^{44} -3.56155 q^{45} -6.24621 q^{46} +11.1231 q^{47} -7.68466 q^{48} -7.00000 q^{49} +19.6847 q^{50} -1.00000 q^{51} +12.2462 q^{53} -2.56155 q^{54} +5.56155 q^{55} -4.68466 q^{57} -21.1231 q^{58} -7.12311 q^{59} +16.2462 q^{60} +9.12311 q^{61} -8.00000 q^{62} +1.43845 q^{64} +4.00000 q^{66} -4.00000 q^{67} +4.56155 q^{68} +2.43845 q^{69} +6.24621 q^{71} +6.56155 q^{72} +12.2462 q^{73} +13.1231 q^{74} -7.68466 q^{75} +21.3693 q^{76} -9.36932 q^{79} -27.3693 q^{80} +1.00000 q^{81} +9.12311 q^{82} +0.876894 q^{83} -3.56155 q^{85} +12.0000 q^{86} +8.24621 q^{87} -10.2462 q^{88} +1.12311 q^{89} -9.12311 q^{90} -11.1231 q^{92} +3.12311 q^{93} +28.4924 q^{94} -16.6847 q^{95} -6.56155 q^{96} +2.87689 q^{97} -17.9309 q^{98} -1.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 - q^6 + 9 * q^8 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9} - 10 q^{10} + q^{11} - 5 q^{12} + 3 q^{15} + 3 q^{16} + 2 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 8 q^{22} - 9 q^{23} - 9 q^{24} + 3 q^{25} - 2 q^{27} + 10 q^{30} + 2 q^{31} + 9 q^{32} - q^{33} + q^{34} + 5 q^{36} + 2 q^{37} + 24 q^{38} - 22 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{44} - 3 q^{45} + 4 q^{46} + 14 q^{47} - 3 q^{48} - 14 q^{49} + 27 q^{50} - 2 q^{51} + 8 q^{53} - q^{54} + 7 q^{55} + 3 q^{57} - 34 q^{58} - 6 q^{59} + 16 q^{60} + 10 q^{61} - 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} + 5 q^{68} + 9 q^{69} - 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 3 q^{75} + 18 q^{76} + 6 q^{79} - 30 q^{80} + 2 q^{81} + 10 q^{82} + 10 q^{83} - 3 q^{85} + 24 q^{86} - 4 q^{88} - 6 q^{89} - 10 q^{90} - 14 q^{92} - 2 q^{93} + 24 q^{94} - 21 q^{95} - 9 q^{96} + 14 q^{97} - 7 q^{98} + q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 - q^6 + 9 * q^8 + 2 * q^9 - 10 * q^10 + q^11 - 5 * q^12 + 3 * q^15 + 3 * q^16 + 2 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 8 * q^22 - 9 * q^23 - 9 * q^24 + 3 * q^25 - 2 * q^27 + 10 * q^30 + 2 * q^31 + 9 * q^32 - q^33 + q^34 + 5 * q^36 + 2 * q^37 + 24 * q^38 - 22 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^44 - 3 * q^45 + 4 * q^46 + 14 * q^47 - 3 * q^48 - 14 * q^49 + 27 * q^50 - 2 * q^51 + 8 * q^53 - q^54 + 7 * q^55 + 3 * q^57 - 34 * q^58 - 6 * q^59 + 16 * q^60 + 10 * q^61 - 16 * q^62 + 7 * q^64 + 8 * q^66 - 8 * q^67 + 5 * q^68 + 9 * q^69 - 4 * q^71 + 9 * q^72 + 8 * q^73 + 18 * q^74 - 3 * q^75 + 18 * q^76 + 6 * q^79 - 30 * q^80 + 2 * q^81 + 10 * q^82 + 10 * q^83 - 3 * q^85 + 24 * q^86 - 4 * q^88 - 6 * q^89 - 10 * q^90 - 14 * q^92 - 2 * q^93 + 24 * q^94 - 21 * q^95 - 9 * q^96 + 14 * q^97 - 7 * q^98 + 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 $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.56155	2.28078
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	−2.56155	−1.04575
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	6.56155	2.31986
$9$	1.00000	0.333333
$10$	−9.12311	−2.88498
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	−4.56155	−1.31681
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.56155	0.919589
$16$	7.68466	1.92116
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.56155	0.603764
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	−16.2462	−3.63276
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	−6.56155	−1.33937
$25$	7.68466	1.53693
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	9.12311	1.66564
$31$	−3.12311	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$31$	−3.12311	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$32$	6.56155	1.15993
$33$	1.56155	0.271831
$34$	2.56155	0.439303
$35$	0	0
$36$	4.56155	0.760259
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	12.0000	1.94666
$39$	0	0
$40$	−23.3693	−3.69501
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	0	0
$43$	4.68466	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$44$	−7.12311	−1.07385
$45$	−3.56155	−0.530925
$46$	−6.24621	−0.920954
$47$	11.1231	1.62247	0.811236	−	0.584719i	$-0.198795\pi$
$47$	11.1231	1.62247	0.811236	+	0.584719i	$0.198795\pi$
$48$	−7.68466	−1.10918
$49$	−7.00000	−1.00000
$50$	19.6847	2.78383
$51$	−1.00000	−0.140028
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	−2.56155	−0.348583
$55$	5.56155	0.749920
$56$	0	0
$57$	−4.68466	−0.620498
$58$	−21.1231	−2.77360
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	16.2462	2.09738
$61$	9.12311	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$61$	9.12311	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.43845	0.179806
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.56155	0.553170
$69$	2.43845	0.293555
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	6.56155	0.773286
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	13.1231	1.52553
$75$	−7.68466	−0.887348
$76$	21.3693	2.45123
$77$	0	0
$78$	0	0
$79$	−9.36932	−1.05413	−0.527065	−	0.849825i	$-0.676708\pi$
$79$	−9.36932	−1.05413	−0.527065	+	0.849825i	$0.676708\pi$
$80$	−27.3693	−3.05998
$81$	1.00000	0.111111
$82$	9.12311	1.00748
$83$	0.876894	0.0962517	0.0481258	−	0.998841i	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	0	0
$85$	−3.56155	−0.386305
$86$	12.0000	1.29399
$87$	8.24621	0.884087
$88$	−10.2462	−1.09225
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	−9.12311	−0.961660
$91$	0	0
$92$	−11.1231	−1.15966
$93$	3.12311	0.323851
$94$	28.4924	2.93877
$95$	−16.6847	−1.71181
$96$	−6.56155	−0.669686
$97$	2.87689	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$97$	2.87689	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$98$	−17.9309	−1.81129
$99$	−1.56155	−0.156942
$100$	35.0540	3.50540
$101$	10.8769	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$102$	−2.56155	−0.253632
$103$	16.6847	1.64399	0.821994	−	0.569496i	$-0.192862\pi$
$103$	16.6847	1.64399	0.821994	+	0.569496i	$0.192862\pi$
$104$	0	0
$105$	0	0
$106$	31.3693	3.04686
$107$	−4.68466	−0.452883	−0.226442	−	0.974025i	$-0.572709\pi$
$107$	−4.68466	−0.452883	−0.226442	+	0.974025i	$0.572709\pi$
$108$	−4.56155	−0.438936
$109$	6.87689	0.658687	0.329344	−	0.944210i	$-0.393173\pi$
$109$	6.87689	0.658687	0.329344	+	0.944210i	$0.393173\pi$
$110$	14.2462	1.35832
$111$	−5.12311	−0.486264
$112$	0	0
$113$	−0.438447	−0.0412456	−0.0206228	−	0.999787i	$-0.506565\pi$
$113$	−0.438447	−0.0412456	−0.0206228	+	0.999787i	$0.506565\pi$
$114$	−12.0000	−1.12390
$115$	8.68466	0.809849
$116$	−37.6155	−3.49251
$117$	0	0
$118$	−18.2462	−1.67970
$119$	0	0
$120$	23.3693	2.13332
$121$	−8.56155	−0.778323
$122$	23.3693	2.11576
$123$	−3.56155	−0.321134
$124$	−14.2462	−1.27935
$125$	−9.56155	−0.855211
$126$	0	0
$127$	19.8078	1.75765	0.878827	−	0.477140i	$-0.158326\pi$
$127$	19.8078	1.75765	0.878827	+	0.477140i	$0.158326\pi$
$128$	−9.43845	−0.834249
$129$	−4.68466	−0.412461
$130$	0	0
$131$	14.4384	1.26149	0.630746	−	0.775989i	$-0.282749\pi$
$131$	14.4384	1.26149	0.630746	+	0.775989i	$0.282749\pi$
$132$	7.12311	0.619987
$133$	0	0
$134$	−10.2462	−0.885138
$135$	3.56155	0.306530
$136$	6.56155	0.562649
$137$	−0.246211	−0.0210352	−0.0105176	−	0.999945i	$-0.503348\pi$
$137$	−0.246211	−0.0210352	−0.0105176	+	0.999945i	$0.503348\pi$
$138$	6.24621	0.531713
$139$	−0.876894	−0.0743772	−0.0371886	−	0.999308i	$-0.511840\pi$
$139$	−0.876894	−0.0743772	−0.0371886	+	0.999308i	$0.511840\pi$
$140$	0	0
$141$	−11.1231	−0.936734
$142$	16.0000	1.34269
$143$	0	0
$144$	7.68466	0.640388
$145$	29.3693	2.43899
$146$	31.3693	2.59614
$147$	7.00000	0.577350
$148$	23.3693	1.92095
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	−19.6847	−1.60725
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	30.7386	2.49323
$153$	1.00000	0.0808452
$154$	0	0
$155$	11.1231	0.893429
$156$	0	0
$157$	6.68466	0.533494	0.266747	−	0.963767i	$-0.414051\pi$
$157$	6.68466	0.533494	0.266747	+	0.963767i	$0.414051\pi$
$158$	−24.0000	−1.90934
$159$	−12.2462	−0.971188
$160$	−23.3693	−1.84751
$161$	0	0
$162$	2.56155	0.201255
$163$	15.1231	1.18453	0.592267	−	0.805742i	$-0.298233\pi$
$163$	15.1231	1.18453	0.592267	+	0.805742i	$0.298233\pi$
$164$	16.2462	1.26862
$165$	−5.56155	−0.432966
$166$	2.24621	0.174340
$167$	−19.8078	−1.53277	−0.766385	−	0.642381i	$-0.777947\pi$
$167$	−19.8078	−1.53277	−0.766385	+	0.642381i	$0.777947\pi$
$168$	0	0
$169$	0	0
$170$	−9.12311	−0.699710
$171$	4.68466	0.358245
$172$	21.3693	1.62940
$173$	1.80776	0.137442	0.0687209	−	0.997636i	$-0.478108\pi$
$173$	1.80776	0.137442	0.0687209	+	0.997636i	$0.478108\pi$
$174$	21.1231	1.60134
$175$	0	0
$176$	−12.0000	−0.904534
$177$	7.12311	0.535405
$178$	2.87689	0.215632
$179$	−0.876894	−0.0655422	−0.0327711	−	0.999463i	$-0.510433\pi$
$179$	−0.876894	−0.0655422	−0.0327711	+	0.999463i	$0.510433\pi$
$180$	−16.2462	−1.21092
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−9.12311	−0.674399
$184$	−16.0000	−1.17954
$185$	−18.2462	−1.34149
$186$	8.00000	0.586588
$187$	−1.56155	−0.114192
$188$	50.7386	3.70050
$189$	0	0
$190$	−42.7386	−3.10059
$191$	−4.87689	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$191$	−4.87689	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$192$	−1.43845	−0.103811
$193$	7.75379	0.558130	0.279065	−	0.960272i	$-0.409976\pi$
$193$	7.75379	0.558130	0.279065	+	0.960272i	$0.409976\pi$
$194$	7.36932	0.529086
$195$	0	0
$196$	−31.9309	−2.28078
$197$	8.93087	0.636298	0.318149	−	0.948041i	$-0.396939\pi$
$197$	8.93087	0.636298	0.318149	+	0.948041i	$0.396939\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	50.4233	3.56547
$201$	4.00000	0.282138
$202$	27.8617	1.96035
$203$	0	0
$204$	−4.56155	−0.319373
$205$	−12.6847	−0.885935
$206$	42.7386	2.97774
$207$	−2.43845	−0.169484
$208$	0	0
$209$	−7.31534	−0.506013
$210$	0	0
$211$	13.3693	0.920382	0.460191	−	0.887820i	$-0.347781\pi$
$211$	13.3693	0.920382	0.460191	+	0.887820i	$0.347781\pi$
$212$	55.8617	3.83660
$213$	−6.24621	−0.427983
$214$	−12.0000	−0.820303
$215$	−16.6847	−1.13788
$216$	−6.56155	−0.446457
$217$	0	0
$218$	17.6155	1.19307
$219$	−12.2462	−0.827522
$220$	25.3693	1.71040
$221$	0	0
$222$	−13.1231	−0.880765
$223$	14.9309	0.999845	0.499922	−	0.866070i	$-0.333362\pi$
$223$	14.9309	0.999845	0.499922	+	0.866070i	$0.333362\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	−1.12311	−0.0747079
$227$	14.0540	0.932795	0.466398	−	0.884575i	$-0.345552\pi$
$227$	14.0540	0.932795	0.466398	+	0.884575i	$0.345552\pi$
$228$	−21.3693	−1.41522
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	22.2462	1.46687
$231$	0	0
$232$	−54.1080	−3.55236
$233$	−3.56155	−0.233325	−0.116663	−	0.993172i	$-0.537220\pi$
$233$	−3.56155	−0.233325	−0.116663	+	0.993172i	$0.537220\pi$
$234$	0	0
$235$	−39.6155	−2.58423
$236$	−32.4924	−2.11508
$237$	9.36932	0.608603
$238$	0	0
$239$	6.24621	0.404034	0.202017	−	0.979382i	$-0.435250\pi$
$239$	6.24621	0.404034	0.202017	+	0.979382i	$0.435250\pi$
$240$	27.3693	1.76668
$241$	−3.36932	−0.217037	−0.108518	−	0.994094i	$-0.534611\pi$
$241$	−3.36932	−0.217037	−0.108518	+	0.994094i	$0.534611\pi$
$242$	−21.9309	−1.40977
$243$	−1.00000	−0.0641500
$244$	41.6155	2.66416
$245$	24.9309	1.59277
$246$	−9.12311	−0.581668
$247$	0	0
$248$	−20.4924	−1.30127
$249$	−0.876894	−0.0555709
$250$	−24.4924	−1.54904
$251$	8.49242	0.536037	0.268018	−	0.963414i	$-0.413631\pi$
$251$	8.49242	0.536037	0.268018	+	0.963414i	$0.413631\pi$
$252$	0	0
$253$	3.80776	0.239392
$254$	50.7386	3.18363
$255$	3.56155	0.223033
$256$	−27.0540	−1.69087
$257$	−15.3693	−0.958712	−0.479356	−	0.877621i	$-0.659130\pi$
$257$	−15.3693	−0.958712	−0.479356	+	0.877621i	$0.659130\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	0	0
$261$	−8.24621	−0.510428
$262$	36.9848	2.28493
$263$	−20.4924	−1.26362	−0.631808	−	0.775125i	$-0.717687\pi$
$263$	−20.4924	−1.26362	−0.631808	+	0.775125i	$0.717687\pi$
$264$	10.2462	0.630611
$265$	−43.6155	−2.67928
$266$	0	0
$267$	−1.12311	−0.0687329
$268$	−18.2462	−1.11456
$269$	16.4384	1.00227	0.501135	−	0.865369i	$-0.332916\pi$
$269$	16.4384	1.00227	0.501135	+	0.865369i	$0.332916\pi$
$270$	9.12311	0.555215
$271$	−19.8078	−1.20324	−0.601618	−	0.798784i	$-0.705477\pi$
$271$	−19.8078	−1.20324	−0.601618	+	0.798784i	$0.705477\pi$
$272$	7.68466	0.465951
$273$	0	0
$274$	−0.630683	−0.0381010
$275$	−12.0000	−0.723627
$276$	11.1231	0.669532
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−2.24621	−0.134719
$279$	−3.12311	−0.186975
$280$	0	0
$281$	10.8769	0.648861	0.324431	−	0.945910i	$-0.394827\pi$
$281$	10.8769	0.648861	0.324431	+	0.945910i	$0.394827\pi$
$282$	−28.4924	−1.69670
$283$	21.3693	1.27027	0.635137	−	0.772399i	$-0.280944\pi$
$283$	21.3693	1.27027	0.635137	+	0.772399i	$0.280944\pi$
$284$	28.4924	1.69071
$285$	16.6847	0.988314
$286$	0	0
$287$	0	0
$288$	6.56155	0.386643
$289$	1.00000	0.0588235
$290$	75.2311	4.41772
$291$	−2.87689	−0.168647
$292$	55.8617	3.26906
$293$	−1.12311	−0.0656125	−0.0328063	−	0.999462i	$-0.510444\pi$
$293$	−1.12311	−0.0656125	−0.0328063	+	0.999462i	$0.510444\pi$
$294$	17.9309	1.04575
$295$	25.3693	1.47706
$296$	33.6155	1.95386
$297$	1.56155	0.0906105
$298$	−31.3693	−1.81718
$299$	0	0
$300$	−35.0540	−2.02384
$301$	0	0
$302$	−20.4924	−1.17921
$303$	−10.8769	−0.624861
$304$	36.0000	2.06474
$305$	−32.4924	−1.86051
$306$	2.56155	0.146434
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	0	0
$309$	−16.6847	−0.949157
$310$	28.4924	1.61826
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	33.6155	1.90006	0.950031	−	0.312156i	$-0.101051\pi$
$313$	33.6155	1.90006	0.950031	+	0.312156i	$0.101051\pi$
$314$	17.1231	0.966313
$315$	0	0
$316$	−42.7386	−2.40424
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−31.3693	−1.75910
$319$	12.8769	0.720968
$320$	−5.12311	−0.286390
$321$	4.68466	0.261472
$322$	0	0
$323$	4.68466	0.260661
$324$	4.56155	0.253420
$325$	0	0
$326$	38.7386	2.14553
$327$	−6.87689	−0.380293
$328$	23.3693	1.29035
$329$	0	0
$330$	−14.2462	−0.784228
$331$	34.9309	1.91997	0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309	1.91997	0.959987	+	0.280044i	$0.0903491\pi$
$332$	4.00000	0.219529
$333$	5.12311	0.280744
$334$	−50.7386	−2.77629
$335$	14.2462	0.778354
$336$	0	0
$337$	−16.7386	−0.911811	−0.455906	−	0.890028i	$-0.650685\pi$
$337$	−16.7386	−0.911811	−0.455906	+	0.890028i	$0.650685\pi$
$338$	0	0
$339$	0.438447	0.0238132
$340$	−16.2462	−0.881075
$341$	4.87689	0.264099
$342$	12.0000	0.648886
$343$	0	0
$344$	30.7386	1.65732
$345$	−8.68466	−0.467566
$346$	4.63068	0.248947
$347$	−8.49242	−0.455897	−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242	−0.455897	−0.227949	+	0.973673i	$0.573202\pi$
$348$	37.6155	2.01640
$349$	−11.5616	−0.618876	−0.309438	−	0.950920i	$-0.600141\pi$
$349$	−11.5616	−0.618876	−0.309438	+	0.950920i	$0.600141\pi$
$350$	0	0
$351$	0	0
$352$	−10.2462	−0.546125
$353$	10.4924	0.558455	0.279228	−	0.960225i	$-0.409922\pi$
$353$	10.4924	0.558455	0.279228	+	0.960225i	$0.409922\pi$
$354$	18.2462	0.969775
$355$	−22.2462	−1.18071
$356$	5.12311	0.271524
$357$	0	0
$358$	−2.24621	−0.118716
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	−23.3693	−1.23167
$361$	2.94602	0.155054
$362$	15.3693	0.807793
$363$	8.56155	0.449365
$364$	0	0
$365$	−43.6155	−2.28294
$366$	−23.3693	−1.22153
$367$	−1.75379	−0.0915470	−0.0457735	−	0.998952i	$-0.514575\pi$
$367$	−1.75379	−0.0915470	−0.0457735	+	0.998952i	$0.514575\pi$
$368$	−18.7386	−0.976819
$369$	3.56155	0.185407
$370$	−46.7386	−2.42983
$371$	0	0
$372$	14.2462	0.738632
$373$	−0.246211	−0.0127483	−0.00637417	−	0.999980i	$-0.502029\pi$
$373$	−0.246211	−0.0127483	−0.00637417	+	0.999980i	$0.502029\pi$
$374$	−4.00000	−0.206835
$375$	9.56155	0.493756
$376$	72.9848	3.76391
$377$	0	0
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	−76.1080	−3.90426
$381$	−19.8078	−1.01478
$382$	−12.4924	−0.639168
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	9.43845	0.481654
$385$	0	0
$386$	19.8617	1.01094
$387$	4.68466	0.238135
$388$	13.1231	0.666225
$389$	35.8617	1.81826	0.909131	−	0.416510i	$-0.136747\pi$
$389$	35.8617	1.81826	0.909131	+	0.416510i	$0.136747\pi$
$390$	0	0
$391$	−2.43845	−0.123318
$392$	−45.9309	−2.31986
$393$	−14.4384	−0.728323
$394$	22.8769	1.15252
$395$	33.3693	1.67899
$396$	−7.12311	−0.357950
$397$	19.3693	0.972118	0.486059	−	0.873926i	$-0.338434\pi$
$397$	19.3693	0.972118	0.486059	+	0.873926i	$0.338434\pi$
$398$	40.9848	2.05438
$399$	0	0
$400$	59.0540	2.95270
$401$	−39.1771	−1.95641	−0.978205	−	0.207641i	$-0.933421\pi$
$401$	−39.1771	−1.95641	−0.978205	+	0.207641i	$0.933421\pi$
$402$	10.2462	0.511035
$403$	0	0
$404$	49.6155	2.46846
$405$	−3.56155	−0.176975
$406$	0	0
$407$	−8.00000	−0.396545
$408$	−6.56155	−0.324845
$409$	14.6847	0.726110	0.363055	−	0.931768i	$-0.381734\pi$
$409$	14.6847	0.726110	0.363055	+	0.931768i	$0.381734\pi$
$410$	−32.4924	−1.60469
$411$	0.246211	0.0121447
$412$	76.1080	3.74957
$413$	0	0
$414$	−6.24621	−0.306985
$415$	−3.12311	−0.153307
$416$	0	0
$417$	0.876894	0.0429417
$418$	−18.7386	−0.916537
$419$	−0.492423	−0.0240564	−0.0120282	−	0.999928i	$-0.503829\pi$
$419$	−0.492423	−0.0240564	−0.0120282	+	0.999928i	$0.503829\pi$
$420$	0	0
$421$	−24.4384	−1.19106	−0.595529	−	0.803334i	$-0.703057\pi$
$421$	−24.4384	−1.19106	−0.595529	+	0.803334i	$0.703057\pi$
$422$	34.2462	1.66708
$423$	11.1231	0.540824
$424$	80.3542	3.90234
$425$	7.68466	0.372761
$426$	−16.0000	−0.775203
$427$	0	0
$428$	−21.3693	−1.03292
$429$	0	0
$430$	−42.7386	−2.06104
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−7.68466	−0.369728
$433$	26.6847	1.28238	0.641191	−	0.767381i	$-0.278440\pi$
$433$	26.6847	1.28238	0.641191	+	0.767381i	$0.278440\pi$
$434$	0	0
$435$	−29.3693	−1.40815
$436$	31.3693	1.50232
$437$	−11.4233	−0.546450
$438$	−31.3693	−1.49888
$439$	−22.2462	−1.06175	−0.530877	−	0.847449i	$-0.678137\pi$
$439$	−22.2462	−1.06175	−0.530877	+	0.847449i	$0.678137\pi$
$440$	36.4924	1.73971
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−31.1231	−1.47870	−0.739352	−	0.673319i	$-0.764868\pi$
$443$	−31.1231	−1.47870	−0.739352	+	0.673319i	$0.764868\pi$
$444$	−23.3693	−1.10906
$445$	−4.00000	−0.189618
$446$	38.2462	1.81101
$447$	12.2462	0.579226
$448$	0	0
$449$	−36.7386	−1.73380	−0.866902	−	0.498479i	$-0.833892\pi$
$449$	−36.7386	−1.73380	−0.866902	+	0.498479i	$0.833892\pi$
$450$	19.6847	0.927944
$451$	−5.56155	−0.261883
$452$	−2.00000	−0.0940721
$453$	8.00000	0.375873
$454$	36.0000	1.68956
$455$	0	0
$456$	−30.7386	−1.43947
$457$	−13.8078	−0.645900	−0.322950	−	0.946416i	$-0.604675\pi$
$457$	−13.8078	−0.645900	−0.322950	+	0.946416i	$0.604675\pi$
$458$	−15.3693	−0.718161
$459$	−1.00000	−0.0466760
$460$	39.6155	1.84708
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	40.9848	1.90473	0.952364	−	0.304965i	$-0.0986447\pi$
$463$	40.9848	1.90473	0.952364	+	0.304965i	$0.0986447\pi$
$464$	−63.3693	−2.94185
$465$	−11.1231	−0.515822
$466$	−9.12311	−0.422620
$467$	21.3693	0.988854	0.494427	−	0.869219i	$-0.335378\pi$
$467$	21.3693	0.988854	0.494427	+	0.869219i	$0.335378\pi$
$468$	0	0
$469$	0	0
$470$	−101.477	−4.68080
$471$	−6.68466	−0.308013
$472$	−46.7386	−2.15132
$473$	−7.31534	−0.336360
$474$	24.0000	1.10236
$475$	36.0000	1.65179
$476$	0	0
$477$	12.2462	0.560715
$478$	16.0000	0.731823
$479$	−24.3002	−1.11030	−0.555152	−	0.831749i	$-0.687340\pi$
$479$	−24.3002	−1.11030	−0.555152	+	0.831749i	$0.687340\pi$
$480$	23.3693	1.06666
$481$	0	0
$482$	−8.63068	−0.393117
$483$	0	0
$484$	−39.0540	−1.77518
$485$	−10.2462	−0.465256
$486$	−2.56155	−0.116194
$487$	−17.3693	−0.787079	−0.393539	−	0.919308i	$-0.628750\pi$
$487$	−17.3693	−0.787079	−0.393539	+	0.919308i	$0.628750\pi$
$488$	59.8617	2.70981
$489$	−15.1231	−0.683890
$490$	63.8617	2.88498
$491$	−21.3693	−0.964384	−0.482192	−	0.876066i	$-0.660159\pi$
$491$	−21.3693	−0.964384	−0.482192	+	0.876066i	$0.660159\pi$
$492$	−16.2462	−0.732436
$493$	−8.24621	−0.371391
$494$	0	0
$495$	5.56155	0.249973
$496$	−24.0000	−1.07763
$497$	0	0
$498$	−2.24621	−0.100655
$499$	−13.3693	−0.598493	−0.299246	−	0.954176i	$-0.596735\pi$
$499$	−13.3693	−0.598493	−0.299246	+	0.954176i	$0.596735\pi$
$500$	−43.6155	−1.95055
$501$	19.8078	0.884946
$502$	21.7538	0.970919
$503$	−29.5616	−1.31808	−0.659042	−	0.752106i	$-0.729038\pi$
$503$	−29.5616	−1.31808	−0.659042	+	0.752106i	$0.729038\pi$
$504$	0	0
$505$	−38.7386	−1.72385
$506$	9.75379	0.433609
$507$	0	0
$508$	90.3542	4.00882
$509$	−25.1231	−1.11356	−0.556781	−	0.830659i	$-0.687964\pi$
$509$	−25.1231	−1.11356	−0.556781	+	0.830659i	$0.687964\pi$
$510$	9.12311	0.403978
$511$	0	0
$512$	−50.4233	−2.22842
$513$	−4.68466	−0.206833
$514$	−39.3693	−1.73651
$515$	−59.4233	−2.61850
$516$	−21.3693	−0.940732
$517$	−17.3693	−0.763902
$518$	0	0
$519$	−1.80776	−0.0793520
$520$	0	0
$521$	−35.5616	−1.55798	−0.778990	−	0.627036i	$-0.784268\pi$
$521$	−35.5616	−1.55798	−0.778990	+	0.627036i	$0.784268\pi$
$522$	−21.1231	−0.924533
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	65.8617	2.87718
$525$	0	0
$526$	−52.4924	−2.28878
$527$	−3.12311	−0.136045
$528$	12.0000	0.522233
$529$	−17.0540	−0.741477
$530$	−111.723	−4.85296
$531$	−7.12311	−0.309116
$532$	0	0
$533$	0	0
$534$	−2.87689	−0.124495
$535$	16.6847	0.721341
$536$	−26.2462	−1.13366
$537$	0.876894	0.0378408
$538$	42.1080	1.81540
$539$	10.9309	0.470826
$540$	16.2462	0.699126
$541$	−34.1080	−1.46642	−0.733208	−	0.680005i	$-0.761978\pi$
$541$	−34.1080	−1.46642	−0.733208	+	0.680005i	$0.761978\pi$
$542$	−50.7386	−2.17941
$543$	−6.00000	−0.257485
$544$	6.56155	0.281324
$545$	−24.4924	−1.04914
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−1.12311	−0.0479767
$549$	9.12311	0.389365
$550$	−30.7386	−1.31070
$551$	−38.6307	−1.64572
$552$	16.0000	0.681005
$553$	0	0
$554$	15.3693	0.652980
$555$	18.2462	0.774509
$556$	−4.00000	−0.169638
$557$	−26.4924	−1.12252	−0.561260	−	0.827640i	$-0.689683\pi$
$557$	−26.4924	−1.12252	−0.561260	+	0.827640i	$0.689683\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	0	0
$561$	1.56155	0.0659288
$562$	27.8617	1.17528
$563$	31.1231	1.31168	0.655841	−	0.754899i	$-0.272314\pi$
$563$	31.1231	1.31168	0.655841	+	0.754899i	$0.272314\pi$
$564$	−50.7386	−2.13648
$565$	1.56155	0.0656950
$566$	54.7386	2.30084
$567$	0	0
$568$	40.9848	1.71969
$569$	21.1231	0.885527	0.442763	−	0.896639i	$-0.353998\pi$
$569$	21.1231	0.885527	0.442763	+	0.896639i	$0.353998\pi$
$570$	42.7386	1.79012
$571$	−30.7386	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$571$	−30.7386	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$572$	0	0
$573$	4.87689	0.203735
$574$	0	0
$575$	−18.7386	−0.781455
$576$	1.43845	0.0599353
$577$	3.94602	0.164275	0.0821376	−	0.996621i	$-0.473825\pi$
$577$	3.94602	0.164275	0.0821376	+	0.996621i	$0.473825\pi$
$578$	2.56155	0.106547
$579$	−7.75379	−0.322236
$580$	133.970	5.56279
$581$	0	0
$582$	−7.36932	−0.305468
$583$	−19.1231	−0.791998
$584$	80.3542	3.32508
$585$	0	0
$586$	−2.87689	−0.118843
$587$	28.9848	1.19633	0.598166	−	0.801372i	$-0.295896\pi$
$587$	28.9848	1.19633	0.598166	+	0.801372i	$0.295896\pi$
$588$	31.9309	1.31681
$589$	−14.6307	−0.602847
$590$	64.9848	2.67538
$591$	−8.93087	−0.367367
$592$	39.3693	1.61807
$593$	−27.7538	−1.13971	−0.569856	−	0.821745i	$-0.693001\pi$
$593$	−27.7538	−1.13971	−0.569856	+	0.821745i	$0.693001\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−55.8617	−2.28819
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−0.384472	−0.0157091	−0.00785455	−	0.999969i	$-0.502500\pi$
$599$	−0.384472	−0.0157091	−0.00785455	+	0.999969i	$0.502500\pi$
$600$	−50.4233	−2.05852
$601$	−30.9848	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$601$	−30.9848	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−36.4924	−1.48486
$605$	30.4924	1.23969
$606$	−27.8617	−1.13181
$607$	−9.36932	−0.380289	−0.190144	−	0.981756i	$-0.560896\pi$
$607$	−9.36932	−0.380289	−0.190144	+	0.981756i	$0.560896\pi$
$608$	30.7386	1.24662
$609$	0	0
$610$	−83.2311	−3.36993
$611$	0	0
$612$	4.56155	0.184390
$613$	−14.6847	−0.593108	−0.296554	−	0.955016i	$-0.595837\pi$
$613$	−14.6847	−0.593108	−0.296554	+	0.955016i	$0.595837\pi$
$614$	−83.2311	−3.35893
$615$	12.6847	0.511495
$616$	0	0
$617$	44.2462	1.78129	0.890643	−	0.454704i	$-0.150255\pi$
$617$	44.2462	1.78129	0.890643	+	0.454704i	$0.150255\pi$
$618$	−42.7386	−1.71920
$619$	−5.36932	−0.215811	−0.107906	−	0.994161i	$-0.534414\pi$
$619$	−5.36932	−0.215811	−0.107906	+	0.994161i	$0.534414\pi$
$620$	50.7386	2.03771
$621$	2.43845	0.0978515
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	86.1080	3.44157
$627$	7.31534	0.292147
$628$	30.4924	1.21678
$629$	5.12311	0.204272
$630$	0	0
$631$	0.684658	0.0272558	0.0136279	−	0.999907i	$-0.495662\pi$
$631$	0.684658	0.0272558	0.0136279	+	0.999907i	$0.495662\pi$
$632$	−61.4773	−2.44543
$633$	−13.3693	−0.531383
$634$	46.1080	1.83118
$635$	−70.5464	−2.79955
$636$	−55.8617	−2.21506
$637$	0	0
$638$	32.9848	1.30588
$639$	6.24621	0.247096
$640$	33.6155	1.32877
$641$	−28.9309	−1.14270	−0.571350	−	0.820706i	$-0.693580\pi$
$641$	−28.9309	−1.14270	−0.571350	+	0.820706i	$0.693580\pi$
$642$	12.0000	0.473602
$643$	13.7538	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$643$	13.7538	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$644$	0	0
$645$	16.6847	0.656958
$646$	12.0000	0.472134
$647$	−9.36932	−0.368346	−0.184173	−	0.982894i	$-0.558961\pi$
$647$	−9.36932	−0.368346	−0.184173	+	0.982894i	$0.558961\pi$
$648$	6.56155	0.257762
$649$	11.1231	0.436620
$650$	0	0
$651$	0	0
$652$	68.9848	2.70166
$653$	−32.9309	−1.28868	−0.644342	−	0.764737i	$-0.722869\pi$
$653$	−32.9309	−1.28868	−0.644342	+	0.764737i	$0.722869\pi$
$654$	−17.6155	−0.688822
$655$	−51.4233	−2.00927
$656$	27.3693	1.06859
$657$	12.2462	0.477770
$658$	0	0
$659$	−9.86174	−0.384159	−0.192079	−	0.981379i	$-0.561523\pi$
$659$	−9.86174	−0.384159	−0.192079	+	0.981379i	$0.561523\pi$
$660$	−25.3693	−0.987499
$661$	−13.3153	−0.517907	−0.258953	−	0.965890i	$-0.583378\pi$
$661$	−13.3153	−0.517907	−0.258953	+	0.965890i	$0.583378\pi$
$662$	89.4773	3.47763
$663$	0	0
$664$	5.75379	0.223290
$665$	0	0
$666$	13.1231	0.508510
$667$	20.1080	0.778583
$668$	−90.3542	−3.49591
$669$	−14.9309	−0.577261
$670$	36.4924	1.40983
$671$	−14.2462	−0.549969
$672$	0	0
$673$	−0.738634	−0.0284722	−0.0142361	−	0.999899i	$-0.504532\pi$
$673$	−0.738634	−0.0284722	−0.0142361	+	0.999899i	$0.504532\pi$
$674$	−42.8769	−1.65156
$675$	−7.68466	−0.295783
$676$	0	0
$677$	−1.31534	−0.0505527	−0.0252763	−	0.999681i	$-0.508047\pi$
$677$	−1.31534	−0.0505527	−0.0252763	+	0.999681i	$0.508047\pi$
$678$	1.12311	0.0431326
$679$	0	0
$680$	−23.3693	−0.896172
$681$	−14.0540	−0.538550
$682$	12.4924	0.478360
$683$	9.56155	0.365863	0.182931	−	0.983126i	$-0.441441\pi$
$683$	9.56155	0.365863	0.182931	+	0.983126i	$0.441441\pi$
$684$	21.3693	0.817076
$685$	0.876894	0.0335044
$686$	0	0
$687$	6.00000	0.228914
$688$	36.0000	1.37249
$689$	0	0
$690$	−22.2462	−0.846899
$691$	28.9848	1.10264	0.551318	−	0.834295i	$-0.314125\pi$
$691$	28.9848	1.10264	0.551318	+	0.834295i	$0.314125\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	−21.7538	−0.825763
$695$	3.12311	0.118466
$696$	54.1080	2.05096
$697$	3.56155	0.134903
$698$	−29.6155	−1.12096
$699$	3.56155	0.134710
$700$	0	0
$701$	15.3693	0.580491	0.290246	−	0.956952i	$-0.406263\pi$
$701$	15.3693	0.580491	0.290246	+	0.956952i	$0.406263\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−2.24621	−0.0846573
$705$	39.6155	1.49201
$706$	26.8769	1.01153
$707$	0	0
$708$	32.4924	1.22114
$709$	44.7386	1.68019	0.840097	−	0.542436i	$-0.182498\pi$
$709$	44.7386	1.68019	0.840097	+	0.542436i	$0.182498\pi$
$710$	−56.9848	−2.13860
$711$	−9.36932	−0.351377
$712$	7.36932	0.276177
$713$	7.61553	0.285204
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−6.24621	−0.233269
$718$	−36.4924	−1.36189
$719$	−11.8078	−0.440355	−0.220178	−	0.975460i	$-0.570664\pi$
$719$	−11.8078	−0.440355	−0.220178	+	0.975460i	$0.570664\pi$
$720$	−27.3693	−1.01999
$721$	0	0
$722$	7.54640	0.280848
$723$	3.36932	0.125306
$724$	27.3693	1.01717
$725$	−63.3693	−2.35348
$726$	21.9309	0.813931
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−111.723	−4.13507
$731$	4.68466	0.173268
$732$	−41.6155	−1.53815
$733$	11.7538	0.434136	0.217068	−	0.976156i	$-0.430351\pi$
$733$	11.7538	0.434136	0.217068	+	0.976156i	$0.430351\pi$
$734$	−4.49242	−0.165818
$735$	−24.9309	−0.919589
$736$	−16.0000	−0.589768
$737$	6.24621	0.230082
$738$	9.12311	0.335826
$739$	20.6847	0.760897	0.380449	−	0.924802i	$-0.375770\pi$
$739$	20.6847	0.760897	0.380449	+	0.924802i	$0.375770\pi$
$740$	−83.2311	−3.05963
$741$	0	0
$742$	0	0
$743$	−28.4924	−1.04529	−0.522643	−	0.852552i	$-0.675054\pi$
$743$	−28.4924	−1.04529	−0.522643	+	0.852552i	$0.675054\pi$
$744$	20.4924	0.751289
$745$	43.6155	1.59795
$746$	−0.630683	−0.0230909
$747$	0.876894	0.0320839
$748$	−7.12311	−0.260447
$749$	0	0
$750$	24.4924	0.894337
$751$	−25.3693	−0.925740	−0.462870	−	0.886426i	$-0.653180\pi$
$751$	−25.3693	−0.925740	−0.462870	+	0.886426i	$0.653180\pi$
$752$	85.4773	3.11704
$753$	−8.49242	−0.309481
$754$	0	0
$755$	28.4924	1.03695
$756$	0	0
$757$	16.0540	0.583492	0.291746	−	0.956496i	$-0.405764\pi$
$757$	16.0540	0.583492	0.291746	+	0.956496i	$0.405764\pi$
$758$	30.7386	1.11648
$759$	−3.80776	−0.138213
$760$	−109.477	−3.97116
$761$	15.7538	0.571074	0.285537	−	0.958368i	$-0.407828\pi$
$761$	15.7538	0.571074	0.285537	+	0.958368i	$0.407828\pi$
$762$	−50.7386	−1.83807
$763$	0	0
$764$	−22.2462	−0.804840
$765$	−3.56155	−0.128768
$766$	−16.0000	−0.578103
$767$	0	0
$768$	27.0540	0.976226
$769$	−40.5464	−1.46214	−0.731070	−	0.682302i	$-0.760979\pi$
$769$	−40.5464	−1.46214	−0.731070	+	0.682302i	$0.760979\pi$
$770$	0	0
$771$	15.3693	0.553512
$772$	35.3693	1.27297
$773$	8.63068	0.310424	0.155212	−	0.987881i	$-0.450394\pi$
$773$	8.63068	0.310424	0.155212	+	0.987881i	$0.450394\pi$
$774$	12.0000	0.431331
$775$	−24.0000	−0.862105
$776$	18.8769	0.677641
$777$	0	0
$778$	91.8617	3.29340
$779$	16.6847	0.597790
$780$	0	0
$781$	−9.75379	−0.349018
$782$	−6.24621	−0.223364
$783$	8.24621	0.294696
$784$	−53.7926	−1.92116
$785$	−23.8078	−0.849736
$786$	−36.9848	−1.31921
$787$	10.2462	0.365238	0.182619	−	0.983184i	$-0.441543\pi$
$787$	10.2462	0.365238	0.182619	+	0.983184i	$0.441543\pi$
$788$	40.7386	1.45125
$789$	20.4924	0.729550
$790$	85.4773	3.04114
$791$	0	0
$792$	−10.2462	−0.364083
$793$	0	0
$794$	49.6155	1.76079
$795$	43.6155	1.54688
$796$	72.9848	2.58688
$797$	−9.61553	−0.340599	−0.170300	−	0.985392i	$-0.554474\pi$
$797$	−9.61553	−0.340599	−0.170300	+	0.985392i	$0.554474\pi$
$798$	0	0
$799$	11.1231	0.393507
$800$	50.4233	1.78273
$801$	1.12311	0.0396830
$802$	−100.354	−3.54363
$803$	−19.1231	−0.674840
$804$	18.2462	0.643494
$805$	0	0
$806$	0	0
$807$	−16.4384	−0.578661
$808$	71.3693	2.51076
$809$	15.9460	0.560632	0.280316	−	0.959908i	$-0.409561\pi$
$809$	15.9460	0.560632	0.280316	+	0.959908i	$0.409561\pi$
$810$	−9.12311	−0.320553
$811$	45.3693	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$811$	45.3693	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$812$	0	0
$813$	19.8078	0.694689
$814$	−20.4924	−0.718259
$815$	−53.8617	−1.88669
$816$	−7.68466	−0.269017
$817$	21.9460	0.767794
$818$	37.6155	1.31520
$819$	0	0
$820$	−57.8617	−2.02062
$821$	12.4384	0.434105	0.217052	−	0.976160i	$-0.430356\pi$
$821$	12.4384	0.434105	0.217052	+	0.976160i	$0.430356\pi$
$822$	0.630683	0.0219976
$823$	3.50758	0.122266	0.0611332	−	0.998130i	$-0.480529\pi$
$823$	3.50758	0.122266	0.0611332	+	0.998130i	$0.480529\pi$
$824$	109.477	3.81382
$825$	12.0000	0.417786
$826$	0	0
$827$	−47.4233	−1.64907	−0.824535	−	0.565811i	$-0.808563\pi$
$827$	−47.4233	−1.64907	−0.824535	+	0.565811i	$0.808563\pi$
$828$	−11.1231	−0.386555
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	−8.00000	−0.277684
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−7.00000	−0.242536
$834$	2.24621	0.0777799
$835$	70.5464	2.44136
$836$	−33.3693	−1.15410
$837$	3.12311	0.107950
$838$	−1.26137	−0.0435732
$839$	26.0540	0.899483	0.449742	−	0.893159i	$-0.351516\pi$
$839$	26.0540	0.899483	0.449742	+	0.893159i	$0.351516\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	−62.6004	−2.15735
$843$	−10.8769	−0.374620
$844$	60.9848	2.09918
$845$	0	0
$846$	28.4924	0.979590
$847$	0	0
$848$	94.1080	3.23168
$849$	−21.3693	−0.733393
$850$	19.6847	0.675178
$851$	−12.4924	−0.428235
$852$	−28.4924	−0.976134
$853$	28.7386	0.983992	0.491996	−	0.870597i	$-0.336267\pi$
$853$	28.7386	0.983992	0.491996	+	0.870597i	$0.336267\pi$
$854$	0	0
$855$	−16.6847	−0.570603
$856$	−30.7386	−1.05062
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	−76.1080	−2.59526
$861$	0	0
$862$	61.4773	2.09392
$863$	9.75379	0.332023	0.166011	−	0.986124i	$-0.446911\pi$
$863$	9.75379	0.332023	0.166011	+	0.986124i	$0.446911\pi$
$864$	−6.56155	−0.223229
$865$	−6.43845	−0.218914
$866$	68.3542	2.32277
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	14.6307	0.496312
$870$	−75.2311	−2.55057
$871$	0	0
$872$	45.1231	1.52806
$873$	2.87689	0.0973681
$874$	−29.2614	−0.989780
$875$	0	0
$876$	−55.8617	−1.88739
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−56.9848	−1.92315
$879$	1.12311	0.0378814
$880$	42.7386	1.44072
$881$	40.2462	1.35593	0.677965	−	0.735095i	$-0.262862\pi$
$881$	40.2462	1.35593	0.677965	+	0.735095i	$0.262862\pi$
$882$	−17.9309	−0.603764
$883$	−23.4233	−0.788257	−0.394128	−	0.919055i	$-0.628953\pi$
$883$	−23.4233	−0.788257	−0.394128	+	0.919055i	$0.628953\pi$
$884$	0	0
$885$	−25.3693	−0.852780
$886$	−79.7235	−2.67836
$887$	18.4384	0.619102	0.309551	−	0.950883i	$-0.399821\pi$
$887$	18.4384	0.619102	0.309551	+	0.950883i	$0.399821\pi$
$888$	−33.6155	−1.12806
$889$	0	0
$890$	−10.2462	−0.343454
$891$	−1.56155	−0.0523140
$892$	68.1080	2.28042
$893$	52.1080	1.74373
$894$	31.3693	1.04915
$895$	3.12311	0.104394
$896$	0	0
$897$	0	0
$898$	−94.1080	−3.14042
$899$	25.7538	0.858937
$900$	35.0540	1.16847
$901$	12.2462	0.407980
$902$	−14.2462	−0.474347
$903$	0	0
$904$	−2.87689	−0.0956841
$905$	−21.3693	−0.710340
$906$	20.4924	0.680815
$907$	−9.86174	−0.327454	−0.163727	−	0.986506i	$-0.552352\pi$
$907$	−9.86174	−0.327454	−0.163727	+	0.986506i	$0.552352\pi$
$908$	64.1080	2.12750
$909$	10.8769	0.360764
$910$	0	0
$911$	24.3002	0.805101	0.402551	−	0.915398i	$-0.368124\pi$
$911$	24.3002	0.805101	0.402551	+	0.915398i	$0.368124\pi$
$912$	−36.0000	−1.19208
$913$	−1.36932	−0.0453178
$914$	−35.3693	−1.16991
$915$	32.4924	1.07417
$916$	−27.3693	−0.904308
$917$	0	0
$918$	−2.56155	−0.0845438
$919$	−16.6847	−0.550376	−0.275188	−	0.961390i	$-0.588740\pi$
$919$	−16.6847	−0.550376	−0.275188	+	0.961390i	$0.588740\pi$
$920$	56.9848	1.87873
$921$	32.4924	1.07066
$922$	21.1231	0.695652
$923$	0	0
$924$	0	0
$925$	39.3693	1.29446
$926$	104.985	3.45002
$927$	16.6847	0.547996
$928$	−54.1080	−1.77618
$929$	−3.06913	−0.100695	−0.0503474	−	0.998732i	$-0.516033\pi$
$929$	−3.06913	−0.100695	−0.0503474	+	0.998732i	$0.516033\pi$
$930$	−28.4924	−0.934303
$931$	−32.7926	−1.07473
$932$	−16.2462	−0.532162
$933$	0	0
$934$	54.7386	1.79110
$935$	5.56155	0.181882
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−33.6155	−1.09700
$940$	−180.708	−5.89406
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−17.1231	−0.557901
$943$	−8.68466	−0.282811
$944$	−54.7386	−1.78159
$945$	0	0
$946$	−18.7386	−0.609246
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	42.7386	1.38809
$949$	0	0
$950$	92.2159	2.99188
$951$	−18.0000	−0.583690
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	31.3693	1.01562
$955$	17.3693	0.562058
$956$	28.4924	0.921511
$957$	−12.8769	−0.416251
$958$	−62.2462	−2.01108
$959$	0	0
$960$	5.12311	0.165348
$961$	−21.2462	−0.685362
$962$	0	0
$963$	−4.68466	−0.150961
$964$	−15.3693	−0.495012
$965$	−27.6155	−0.888975
$966$	0	0
$967$	−42.4384	−1.36473	−0.682364	−	0.731012i	$-0.739048\pi$
$967$	−42.4384	−1.36473	−0.682364	+	0.731012i	$0.739048\pi$
$968$	−56.1771	−1.80560
$969$	−4.68466	−0.150493
$970$	−26.2462	−0.842715
$971$	−43.6155	−1.39969	−0.699844	−	0.714295i	$-0.746747\pi$
$971$	−43.6155	−1.39969	−0.699844	+	0.714295i	$0.746747\pi$
$972$	−4.56155	−0.146312
$973$	0	0
$974$	−44.4924	−1.42563
$975$	0	0
$976$	70.1080	2.24410
$977$	−8.24621	−0.263820	−0.131910	−	0.991262i	$-0.542111\pi$
$977$	−8.24621	−0.263820	−0.131910	+	0.991262i	$0.542111\pi$
$978$	−38.7386	−1.23872
$979$	−1.75379	−0.0560513
$980$	113.723	3.63276
$981$	6.87689	0.219562
$982$	−54.7386	−1.74678
$983$	−30.9309	−0.986542	−0.493271	−	0.869876i	$-0.664199\pi$
$983$	−30.9309	−0.986542	−0.493271	+	0.869876i	$0.664199\pi$
$984$	−23.3693	−0.744987
$985$	−31.8078	−1.01348
$986$	−21.1231	−0.672697
$987$	0	0
$988$	0	0
$989$	−11.4233	−0.363240
$990$	14.2462	0.452774
$991$	42.7386	1.35764	0.678819	−	0.734306i	$-0.262492\pi$
$991$	42.7386	1.35764	0.678819	+	0.734306i	$0.262492\pi$
$992$	−20.4924	−0.650635
$993$	−34.9309	−1.10850
$994$	0	0
$995$	−56.9848	−1.80654
$996$	−4.00000	−0.126745
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−34.2462	−1.08404
$999$	−5.12311	−0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.q.1.2		2
13.12	even	2		51.2.a.b.1.1	✓	2
39.38	odd	2		153.2.a.e.1.2		2
52.51	odd	2		816.2.a.m.1.2		2
65.12	odd	4		1275.2.b.d.1174.1		4
65.38	odd	4		1275.2.b.d.1174.4		4
65.64	even	2		1275.2.a.n.1.2		2
91.90	odd	2		2499.2.a.o.1.1		2
104.51	odd	2		3264.2.a.bg.1.1		2
104.77	even	2		3264.2.a.bl.1.1		2
143.142	odd	2		6171.2.a.p.1.2		2
156.155	even	2		2448.2.a.v.1.1		2
195.194	odd	2		3825.2.a.s.1.1		2
221.12	odd	16		867.2.h.j.688.3		16
221.25	even	8		867.2.e.f.829.4		8
221.38	even	4		867.2.d.c.577.4		4
221.64	even	4		867.2.d.c.577.3		4
221.77	even	8		867.2.e.f.829.3		8
221.90	odd	16		867.2.h.j.688.4		16
221.116	odd	16		867.2.h.j.757.2		16
221.129	odd	16		867.2.h.j.712.3		16
221.142	odd	16		867.2.h.j.733.1		16
221.155	even	8		867.2.e.f.616.1		8
221.168	even	8		867.2.e.f.616.2		8
221.181	odd	16		867.2.h.j.733.2		16
221.194	odd	16		867.2.h.j.712.4		16
221.207	odd	16		867.2.h.j.757.1		16
221.220	even	2		867.2.a.f.1.1		2
273.272	even	2		7497.2.a.v.1.2		2
312.77	odd	2		9792.2.a.cy.1.2		2
312.155	even	2		9792.2.a.cz.1.2		2
663.662	odd	2		2601.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.1	✓	2	13.12	even	2
153.2.a.e.1.2		2	39.38	odd	2
816.2.a.m.1.2		2	52.51	odd	2
867.2.a.f.1.1		2	221.220	even	2
867.2.d.c.577.3		4	221.64	even	4
867.2.d.c.577.4		4	221.38	even	4
867.2.e.f.616.1		8	221.155	even	8
867.2.e.f.616.2		8	221.168	even	8
867.2.e.f.829.3		8	221.77	even	8
867.2.e.f.829.4		8	221.25	even	8
867.2.h.j.688.3		16	221.12	odd	16
867.2.h.j.688.4		16	221.90	odd	16
867.2.h.j.712.3		16	221.129	odd	16
867.2.h.j.712.4		16	221.194	odd	16
867.2.h.j.733.1		16	221.142	odd	16
867.2.h.j.733.2		16	221.181	odd	16
867.2.h.j.757.1		16	221.207	odd	16
867.2.h.j.757.2		16	221.116	odd	16
1275.2.a.n.1.2		2	65.64	even	2
1275.2.b.d.1174.1		4	65.12	odd	4
1275.2.b.d.1174.4		4	65.38	odd	4
2448.2.a.v.1.1		2	156.155	even	2
2499.2.a.o.1.1		2	91.90	odd	2
2601.2.a.t.1.2		2	663.662	odd	2
3264.2.a.bg.1.1		2	104.51	odd	2
3264.2.a.bl.1.1		2	104.77	even	2
3825.2.a.s.1.1		2	195.194	odd	2
6171.2.a.p.1.2		2	143.142	odd	2
7497.2.a.v.1.2		2	273.272	even	2
8619.2.a.q.1.2		2	1.1	even	1	trivial
9792.2.a.cy.1.2		2	312.77	odd	2
9792.2.a.cz.1.2		2	312.155	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} -1.00000 q^{3} +4.56155 q^{4} -3.56155 q^{5} -2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.56155 q^{2} -1.00000 q^{3} +4.56155 q^{4} -3.56155 q^{5} -2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} -9.12311 q^{10} -1.56155 q^{11} -4.56155 q^{12} +3.56155 q^{15} +7.68466 q^{16} +1.00000 q^{17} +2.56155 q^{18} +4.68466 q^{19} -16.2462 q^{20} -4.00000 q^{22} -2.43845 q^{23} -6.56155 q^{24} +7.68466 q^{25} -1.00000 q^{27} -8.24621 q^{29} +9.12311 q^{30} -3.12311 q^{31} +6.56155 q^{32} +1.56155 q^{33} +2.56155 q^{34} +4.56155 q^{36} +5.12311 q^{37} +12.0000 q^{38} -23.3693 q^{40} +3.56155 q^{41} +4.68466 q^{43} -7.12311 q^{44} -3.56155 q^{45} -6.24621 q^{46} +11.1231 q^{47} -7.68466 q^{48} -7.00000 q^{49} +19.6847 q^{50} -1.00000 q^{51} +12.2462 q^{53} -2.56155 q^{54} +5.56155 q^{55} -4.68466 q^{57} -21.1231 q^{58} -7.12311 q^{59} +16.2462 q^{60} +9.12311 q^{61} -8.00000 q^{62} +1.43845 q^{64} +4.00000 q^{66} -4.00000 q^{67} +4.56155 q^{68} +2.43845 q^{69} +6.24621 q^{71} +6.56155 q^{72} +12.2462 q^{73} +13.1231 q^{74} -7.68466 q^{75} +21.3693 q^{76} -9.36932 q^{79} -27.3693 q^{80} +1.00000 q^{81} +9.12311 q^{82} +0.876894 q^{83} -3.56155 q^{85} +12.0000 q^{86} +8.24621 q^{87} -10.2462 q^{88} +1.12311 q^{89} -9.12311 q^{90} -11.1231 q^{92} +3.12311 q^{93} +28.4924 q^{94} -16.6847 q^{95} -6.56155 q^{96} +2.87689 q^{97} -17.9309 q^{98} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 - q^6 + 9 * q^8 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9} - 10 q^{10} + q^{11} - 5 q^{12} + 3 q^{15} + 3 q^{16} + 2 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 8 q^{22} - 9 q^{23} - 9 q^{24} + 3 q^{25} - 2 q^{27} + 10 q^{30} + 2 q^{31} + 9 q^{32} - q^{33} + q^{34} + 5 q^{36} + 2 q^{37} + 24 q^{38} - 22 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{44} - 3 q^{45} + 4 q^{46} + 14 q^{47} - 3 q^{48} - 14 q^{49} + 27 q^{50} - 2 q^{51} + 8 q^{53} - q^{54} + 7 q^{55} + 3 q^{57} - 34 q^{58} - 6 q^{59} + 16 q^{60} + 10 q^{61} - 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} + 5 q^{68} + 9 q^{69} - 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 3 q^{75} + 18 q^{76} + 6 q^{79} - 30 q^{80} + 2 q^{81} + 10 q^{82} + 10 q^{83} - 3 q^{85} + 24 q^{86} - 4 q^{88} - 6 q^{89} - 10 q^{90} - 14 q^{92} - 2 q^{93} + 24 q^{94} - 21 q^{95} - 9 q^{96} + 14 q^{97} - 7 q^{98} + q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 - q^6 + 9 * q^8 + 2 * q^9 - 10 * q^10 + q^11 - 5 * q^12 + 3 * q^15 + 3 * q^16 + 2 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 8 * q^22 - 9 * q^23 - 9 * q^24 + 3 * q^25 - 2 * q^27 + 10 * q^30 + 2 * q^31 + 9 * q^32 - q^33 + q^34 + 5 * q^36 + 2 * q^37 + 24 * q^38 - 22 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^44 - 3 * q^45 + 4 * q^46 + 14 * q^47 - 3 * q^48 - 14 * q^49 + 27 * q^50 - 2 * q^51 + 8 * q^53 - q^54 + 7 * q^55 + 3 * q^57 - 34 * q^58 - 6 * q^59 + 16 * q^60 + 10 * q^61 - 16 * q^62 + 7 * q^64 + 8 * q^66 - 8 * q^67 + 5 * q^68 + 9 * q^69 - 4 * q^71 + 9 * q^72 + 8 * q^73 + 18 * q^74 - 3 * q^75 + 18 * q^76 + 6 * q^79 - 30 * q^80 + 2 * q^81 + 10 * q^82 + 10 * q^83 - 3 * q^85 + 24 * q^86 - 4 * q^88 - 6 * q^89 - 10 * q^90 - 14 * q^92 - 2 * q^93 + 24 * q^94 - 21 * q^95 - 9 * q^96 + 14 * q^97 - 7 * q^98 + q^99

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