LMFDB - Embedded newform 4114.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4114,2,Mod(1,4114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.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:	$32.8504553916$
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} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 $ x^8 - 3x^7 - 12x^6 + 28x^5 + 51x^4 - 80x^3 - 92x^2 + 67*x + 59
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 374)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.21054$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.21054 q^{3} +1.00000 q^{4} -2.95515 q^{5} +2.21054 q^{6} -3.36619 q^{7} -1.00000 q^{8} +1.88651 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.21054 q^{3} +1.00000 q^{4} -2.95515 q^{5} +2.21054 q^{6} -3.36619 q^{7} -1.00000 q^{8} +1.88651 q^{9} +2.95515 q^{10} -2.21054 q^{12} -3.45114 q^{13} +3.36619 q^{14} +6.53249 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.88651 q^{18} +0.923274 q^{19} -2.95515 q^{20} +7.44112 q^{21} +1.70547 q^{23} +2.21054 q^{24} +3.73291 q^{25} +3.45114 q^{26} +2.46142 q^{27} -3.36619 q^{28} -8.92413 q^{29} -6.53249 q^{30} -9.67978 q^{31} -1.00000 q^{32} -1.00000 q^{34} +9.94760 q^{35} +1.88651 q^{36} -2.34213 q^{37} -0.923274 q^{38} +7.62891 q^{39} +2.95515 q^{40} +5.29728 q^{41} -7.44112 q^{42} +11.7968 q^{43} -5.57492 q^{45} -1.70547 q^{46} +6.50633 q^{47} -2.21054 q^{48} +4.33125 q^{49} -3.73291 q^{50} -2.21054 q^{51} -3.45114 q^{52} +11.0366 q^{53} -2.46142 q^{54} +3.36619 q^{56} -2.04094 q^{57} +8.92413 q^{58} -1.87553 q^{59} +6.53249 q^{60} +8.16280 q^{61} +9.67978 q^{62} -6.35035 q^{63} +1.00000 q^{64} +10.1987 q^{65} +12.2524 q^{67} +1.00000 q^{68} -3.77002 q^{69} -9.94760 q^{70} -0.707032 q^{71} -1.88651 q^{72} +10.4546 q^{73} +2.34213 q^{74} -8.25177 q^{75} +0.923274 q^{76} -7.62891 q^{78} -13.3003 q^{79} -2.95515 q^{80} -11.1006 q^{81} -5.29728 q^{82} -7.09197 q^{83} +7.44112 q^{84} -2.95515 q^{85} -11.7968 q^{86} +19.7272 q^{87} +8.42835 q^{89} +5.57492 q^{90} +11.6172 q^{91} +1.70547 q^{92} +21.3976 q^{93} -6.50633 q^{94} -2.72841 q^{95} +2.21054 q^{96} +2.47057 q^{97} -4.33125 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.21054	−1.27626	−0.638129	−	0.769929i	$-0.720291\pi$
$3$	−2.21054	−1.27626	−0.638129	+	0.769929i	$0.720291\pi$
$4$	1.00000	0.500000
$5$	−2.95515	−1.32158	−0.660792	−	0.750569i	$-0.729779\pi$
$5$	−2.95515	−1.32158	−0.660792	+	0.750569i	$0.729779\pi$
$6$	2.21054	0.902451
$7$	−3.36619	−1.27230	−0.636150	−	0.771565i	$-0.719474\pi$
$7$	−3.36619	−1.27230	−0.636150	+	0.771565i	$0.719474\pi$
$8$	−1.00000	−0.353553
$9$	1.88651	0.628836
$10$	2.95515	0.934501
$11$	0	0
$11$	0	0
$12$	−2.21054	−0.638129
$13$	−3.45114	−0.957175	−0.478588	−	0.878040i	$-0.658851\pi$
$13$	−3.45114	−0.957175	−0.478588	+	0.878040i	$0.658851\pi$
$14$	3.36619	0.899653
$15$	6.53249	1.68668
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.88651	−0.444654
$19$	0.923274	0.211814	0.105907	−	0.994376i	$-0.466225\pi$
$19$	0.923274	0.211814	0.105907	+	0.994376i	$0.466225\pi$
$20$	−2.95515	−0.660792
$21$	7.44112	1.62379
$22$	0	0
$23$	1.70547	0.355616	0.177808	−	0.984065i	$-0.443099\pi$
$23$	1.70547	0.355616	0.177808	+	0.984065i	$0.443099\pi$
$24$	2.21054	0.451226
$25$	3.73291	0.746583
$26$	3.45114	0.676825
$27$	2.46142	0.473701
$28$	−3.36619	−0.636150
$29$	−8.92413	−1.65717	−0.828585	−	0.559863i	$-0.810854\pi$
$29$	−8.92413	−1.65717	−0.828585	+	0.559863i	$0.810854\pi$
$30$	−6.53249	−1.19266
$31$	−9.67978	−1.73854	−0.869270	−	0.494337i	$-0.835411\pi$
$31$	−9.67978	−1.73854	−0.869270	+	0.494337i	$0.835411\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	9.94760	1.68145
$36$	1.88651	0.314418
$37$	−2.34213	−0.385044	−0.192522	−	0.981293i	$-0.561667\pi$
$37$	−2.34213	−0.385044	−0.192522	+	0.981293i	$0.561667\pi$
$38$	−0.923274	−0.149775
$39$	7.62891	1.22160
$40$	2.95515	0.467250
$41$	5.29728	0.827296	0.413648	−	0.910437i	$-0.364254\pi$
$41$	5.29728	0.827296	0.413648	+	0.910437i	$0.364254\pi$
$42$	−7.44112	−1.14819
$43$	11.7968	1.79900	0.899498	−	0.436926i	$-0.143933\pi$
$43$	11.7968	1.79900	0.899498	+	0.436926i	$0.143933\pi$
$44$	0	0
$45$	−5.57492	−0.831059
$46$	−1.70547	−0.251458
$47$	6.50633	0.949046	0.474523	−	0.880243i	$-0.342621\pi$
$47$	6.50633	0.949046	0.474523	+	0.880243i	$0.342621\pi$
$48$	−2.21054	−0.319065
$49$	4.33125	0.618750
$50$	−3.73291	−0.527914
$51$	−2.21054	−0.309538
$52$	−3.45114	−0.478588
$53$	11.0366	1.51599	0.757997	−	0.652258i	$-0.226178\pi$
$53$	11.0366	1.51599	0.757997	+	0.652258i	$0.226178\pi$
$54$	−2.46142	−0.334957
$55$	0	0
$56$	3.36619	0.449826
$57$	−2.04094	−0.270329
$58$	8.92413	1.17180
$59$	−1.87553	−0.244174	−0.122087	−	0.992519i	$-0.538959\pi$
$59$	−1.87553	−0.244174	−0.122087	+	0.992519i	$0.538959\pi$
$60$	6.53249	0.843341
$61$	8.16280	1.04514	0.522570	−	0.852597i	$-0.324973\pi$
$61$	8.16280	1.04514	0.522570	+	0.852597i	$0.324973\pi$
$62$	9.67978	1.22933
$63$	−6.35035	−0.800069
$64$	1.00000	0.125000
$65$	10.1987	1.26499
$66$	0	0
$67$	12.2524	1.49687	0.748436	−	0.663207i	$-0.230805\pi$
$67$	12.2524	1.49687	0.748436	+	0.663207i	$0.230805\pi$
$68$	1.00000	0.121268
$69$	−3.77002	−0.453857
$70$	−9.94760	−1.18897
$71$	−0.707032	−0.0839093	−0.0419546	−	0.999120i	$-0.513358\pi$
$71$	−0.707032	−0.0839093	−0.0419546	+	0.999120i	$0.513358\pi$
$72$	−1.88651	−0.222327
$73$	10.4546	1.22362	0.611808	−	0.791007i	$-0.290443\pi$
$73$	10.4546	1.22362	0.611808	+	0.791007i	$0.290443\pi$
$74$	2.34213	0.272267
$75$	−8.25177	−0.952833
$76$	0.923274	0.105907
$77$	0	0
$78$	−7.62891	−0.863804
$79$	−13.3003	−1.49641	−0.748203	−	0.663470i	$-0.769083\pi$
$79$	−13.3003	−1.49641	−0.748203	+	0.663470i	$0.769083\pi$
$80$	−2.95515	−0.330396
$81$	−11.1006	−1.23340
$82$	−5.29728	−0.584987
$83$	−7.09197	−0.778445	−0.389222	−	0.921144i	$-0.627256\pi$
$83$	−7.09197	−0.778445	−0.389222	+	0.921144i	$0.627256\pi$
$84$	7.44112	0.811893
$85$	−2.95515	−0.320531
$86$	−11.7968	−1.27208
$87$	19.7272	2.11498
$88$	0	0
$89$	8.42835	0.893404	0.446702	−	0.894683i	$-0.352598\pi$
$89$	8.42835	0.893404	0.446702	+	0.894683i	$0.352598\pi$
$90$	5.57492	0.587648
$91$	11.6172	1.21781
$92$	1.70547	0.177808
$93$	21.3976	2.21883
$94$	−6.50633	−0.671077
$95$	−2.72841	−0.279929
$96$	2.21054	0.225613
$97$	2.47057	0.250848	0.125424	−	0.992103i	$-0.459971\pi$
$97$	2.47057	0.250848	0.125424	+	0.992103i	$0.459971\pi$
$98$	−4.33125	−0.437522
$99$	0	0
$100$	3.73291	0.373291
$101$	−5.38506	−0.535833	−0.267917	−	0.963442i	$-0.586335\pi$
$101$	−5.38506	−0.535833	−0.267917	+	0.963442i	$0.586335\pi$
$102$	2.21054	0.218877
$103$	11.4455	1.12776	0.563880	−	0.825856i	$-0.309308\pi$
$103$	11.4455	1.12776	0.563880	+	0.825856i	$0.309308\pi$
$104$	3.45114	0.338413
$105$	−21.9896	−2.14597
$106$	−11.0366	−1.07197
$107$	−8.92851	−0.863152	−0.431576	−	0.902077i	$-0.642042\pi$
$107$	−8.92851	−0.863152	−0.431576	+	0.902077i	$0.642042\pi$
$108$	2.46142	0.236851
$109$	−0.920094	−0.0881290	−0.0440645	−	0.999029i	$-0.514031\pi$
$109$	−0.920094	−0.0881290	−0.0440645	+	0.999029i	$0.514031\pi$
$110$	0	0
$111$	5.17739	0.491416
$112$	−3.36619	−0.318075
$113$	0.671866	0.0632038	0.0316019	−	0.999501i	$-0.489939\pi$
$113$	0.671866	0.0632038	0.0316019	+	0.999501i	$0.489939\pi$
$114$	2.04094	0.191151
$115$	−5.03993	−0.469976
$116$	−8.92413	−0.828585
$117$	−6.51061	−0.601906
$118$	1.87553	0.172657
$119$	−3.36619	−0.308578
$120$	−6.53249	−0.596332
$121$	0	0
$122$	−8.16280	−0.739025
$123$	−11.7099	−1.05584
$124$	−9.67978	−0.869270
$125$	3.74443	0.334912
$126$	6.35035	0.565734
$127$	−15.0801	−1.33814	−0.669069	−	0.743200i	$-0.733307\pi$
$127$	−15.0801	−1.33814	−0.669069	+	0.743200i	$0.733307\pi$
$128$	−1.00000	−0.0883883
$129$	−26.0774	−2.29598
$130$	−10.1987	−0.894481
$131$	−5.47424	−0.478286	−0.239143	−	0.970984i	$-0.576866\pi$
$131$	−5.47424	−0.478286	−0.239143	+	0.970984i	$0.576866\pi$
$132$	0	0
$133$	−3.10792	−0.269491
$134$	−12.2524	−1.05845
$135$	−7.27388	−0.626036
$136$	−1.00000	−0.0857493
$137$	−13.7391	−1.17381	−0.586903	−	0.809657i	$-0.699653\pi$
$137$	−13.7391	−1.17381	−0.586903	+	0.809657i	$0.699653\pi$
$138$	3.77002	0.320926
$139$	17.6093	1.49360	0.746799	−	0.665050i	$-0.231590\pi$
$139$	17.6093	1.49360	0.746799	+	0.665050i	$0.231590\pi$
$140$	9.94760	0.840726
$141$	−14.3825	−1.21123
$142$	0.707032	0.0593328
$143$	0	0
$144$	1.88651	0.157209
$145$	26.3722	2.19009
$146$	−10.4546	−0.865227
$147$	−9.57442	−0.789685
$148$	−2.34213	−0.192522
$149$	−6.01770	−0.492989	−0.246495	−	0.969144i	$-0.579279\pi$
$149$	−6.01770	−0.492989	−0.246495	+	0.969144i	$0.579279\pi$
$150$	8.25177	0.673754
$151$	−1.14380	−0.0930807	−0.0465404	−	0.998916i	$-0.514820\pi$
$151$	−1.14380	−0.0930807	−0.0465404	+	0.998916i	$0.514820\pi$
$152$	−0.923274	−0.0748874
$153$	1.88651	0.152515
$154$	0	0
$155$	28.6052	2.29763
$156$	7.62891	0.610802
$157$	21.5097	1.71666	0.858330	−	0.513098i	$-0.171502\pi$
$157$	21.5097	1.71666	0.858330	+	0.513098i	$0.171502\pi$
$158$	13.3003	1.05812
$159$	−24.3969	−1.93480
$160$	2.95515	0.233625
$161$	−5.74095	−0.452450
$162$	11.1006	0.872146
$163$	7.97671	0.624784	0.312392	−	0.949953i	$-0.398870\pi$
$163$	7.97671	0.624784	0.312392	+	0.949953i	$0.398870\pi$
$164$	5.29728	0.413648
$165$	0	0
$166$	7.09197	0.550444
$167$	11.1818	0.865273	0.432637	−	0.901568i	$-0.357583\pi$
$167$	11.1818	0.865273	0.432637	+	0.901568i	$0.357583\pi$
$168$	−7.44112	−0.574095
$169$	−1.08960	−0.0838156
$170$	2.95515	0.226650
$171$	1.74176	0.133196
$172$	11.7968	0.899498
$173$	−13.0500	−0.992170	−0.496085	−	0.868274i	$-0.665230\pi$
$173$	−13.0500	−0.992170	−0.496085	+	0.868274i	$0.665230\pi$
$174$	−19.7272	−1.49551
$175$	−12.5657	−0.949878
$176$	0	0
$177$	4.14595	0.311629
$178$	−8.42835	−0.631732
$179$	9.65513	0.721658	0.360829	−	0.932632i	$-0.382494\pi$
$179$	9.65513	0.721658	0.360829	+	0.932632i	$0.382494\pi$
$180$	−5.57492	−0.415530
$181$	16.0310	1.19158	0.595789	−	0.803141i	$-0.296840\pi$
$181$	16.0310	1.19158	0.595789	+	0.803141i	$0.296840\pi$
$182$	−11.6172	−0.861125
$183$	−18.0442	−1.33387
$184$	−1.70547	−0.125729
$185$	6.92135	0.508868
$186$	−21.3976	−1.56895
$187$	0	0
$188$	6.50633	0.474523
$189$	−8.28562	−0.602690
$190$	2.72841	0.197940
$191$	−6.77033	−0.489884	−0.244942	−	0.969538i	$-0.578769\pi$
$191$	−6.77033	−0.489884	−0.244942	+	0.969538i	$0.578769\pi$
$192$	−2.21054	−0.159532
$193$	12.9312	0.930806	0.465403	−	0.885099i	$-0.345909\pi$
$193$	12.9312	0.930806	0.465403	+	0.885099i	$0.345909\pi$
$194$	−2.47057	−0.177376
$195$	−22.5446	−1.61445
$196$	4.33125	0.309375
$197$	−2.14463	−0.152798	−0.0763992	−	0.997077i	$-0.524342\pi$
$197$	−2.14463	−0.152798	−0.0763992	+	0.997077i	$0.524342\pi$
$198$	0	0
$199$	−7.94362	−0.563109	−0.281554	−	0.959545i	$-0.590850\pi$
$199$	−7.94362	−0.563109	−0.281554	+	0.959545i	$0.590850\pi$
$200$	−3.73291	−0.263957
$201$	−27.0846	−1.91040
$202$	5.38506	0.378891
$203$	30.0403	2.10842
$204$	−2.21054	−0.154769
$205$	−15.6543	−1.09334
$206$	−11.4455	−0.797447
$207$	3.21739	0.223624
$208$	−3.45114	−0.239294
$209$	0	0
$210$	21.9896	1.51743
$211$	−6.65928	−0.458444	−0.229222	−	0.973374i	$-0.573618\pi$
$211$	−6.65928	−0.458444	−0.229222	+	0.973374i	$0.573618\pi$
$212$	11.0366	0.757997
$213$	1.56293	0.107090
$214$	8.92851	0.610341
$215$	−34.8613	−2.37752
$216$	−2.46142	−0.167479
$217$	32.5840	2.21195
$218$	0.920094	0.0623166
$219$	−23.1103	−1.56165
$220$	0	0
$221$	−3.45114	−0.232149
$222$	−5.17739	−0.347483
$223$	−18.2556	−1.22248	−0.611242	−	0.791444i	$-0.709330\pi$
$223$	−18.2556	−1.22248	−0.611242	+	0.791444i	$0.709330\pi$
$224$	3.36619	0.224913
$225$	7.04217	0.469478
$226$	−0.671866	−0.0446919
$227$	−25.7298	−1.70775	−0.853873	−	0.520482i	$-0.825752\pi$
$227$	−25.7298	−1.70775	−0.853873	+	0.520482i	$0.825752\pi$
$228$	−2.04094	−0.135165
$229$	23.1569	1.53025	0.765127	−	0.643879i	$-0.222676\pi$
$229$	23.1569	1.53025	0.765127	+	0.643879i	$0.222676\pi$
$230$	5.03993	0.332323
$231$	0	0
$232$	8.92413	0.585898
$233$	−5.89280	−0.386050	−0.193025	−	0.981194i	$-0.561830\pi$
$233$	−5.89280	−0.386050	−0.193025	+	0.981194i	$0.561830\pi$
$234$	6.51061	0.425612
$235$	−19.2272	−1.25424
$236$	−1.87553	−0.122087
$237$	29.4010	1.90980
$238$	3.36619	0.218198
$239$	−27.3521	−1.76926	−0.884631	−	0.466292i	$-0.845590\pi$
$239$	−27.3521	−1.76926	−0.884631	+	0.466292i	$0.845590\pi$
$240$	6.53249	0.421671
$241$	7.91912	0.510116	0.255058	−	0.966926i	$-0.417905\pi$
$241$	7.91912	0.510116	0.255058	+	0.966926i	$0.417905\pi$
$242$	0	0
$243$	17.1541	1.10044
$244$	8.16280	0.522570
$245$	−12.7995	−0.817729
$246$	11.7099	0.746595
$247$	−3.18635	−0.202743
$248$	9.67978	0.614667
$249$	15.6771	0.993497
$250$	−3.74443	−0.236819
$251$	−13.4587	−0.849505	−0.424753	−	0.905309i	$-0.639639\pi$
$251$	−13.4587	−0.849505	−0.424753	+	0.905309i	$0.639639\pi$
$252$	−6.35035	−0.400034
$253$	0	0
$254$	15.0801	0.946207
$255$	6.53249	0.409081
$256$	1.00000	0.0625000
$257$	16.5938	1.03509	0.517547	−	0.855655i	$-0.326845\pi$
$257$	16.5938	1.03509	0.517547	+	0.855655i	$0.326845\pi$
$258$	26.0774	1.62351
$259$	7.88406	0.489892
$260$	10.1987	0.632493
$261$	−16.8354	−1.04209
$262$	5.47424	0.338199
$263$	−4.62340	−0.285091	−0.142545	−	0.989788i	$-0.545529\pi$
$263$	−4.62340	−0.285091	−0.142545	+	0.989788i	$0.545529\pi$
$264$	0	0
$265$	−32.6148	−2.00351
$266$	3.10792	0.190559
$267$	−18.6313	−1.14021
$268$	12.2524	0.748436
$269$	−0.674059	−0.0410981	−0.0205490	−	0.999789i	$-0.506541\pi$
$269$	−0.674059	−0.0410981	−0.0205490	+	0.999789i	$0.506541\pi$
$270$	7.27388	0.442674
$271$	−17.5092	−1.06361	−0.531803	−	0.846868i	$-0.678485\pi$
$271$	−17.5092	−1.06361	−0.531803	+	0.846868i	$0.678485\pi$
$272$	1.00000	0.0606339
$273$	−25.6804	−1.55425
$274$	13.7391	0.830006
$275$	0	0
$276$	−3.77002	−0.226929
$277$	−22.8868	−1.37514	−0.687569	−	0.726119i	$-0.741322\pi$
$277$	−22.8868	−1.37514	−0.687569	+	0.726119i	$0.741322\pi$
$278$	−17.6093	−1.05613
$279$	−18.2610	−1.09326
$280$	−9.94760	−0.594483
$281$	−2.37406	−0.141625	−0.0708124	−	0.997490i	$-0.522559\pi$
$281$	−2.37406	−0.141625	−0.0708124	+	0.997490i	$0.522559\pi$
$282$	14.3825	0.856468
$283$	−9.10583	−0.541285	−0.270643	−	0.962680i	$-0.587236\pi$
$283$	−9.10583	−0.541285	−0.270643	+	0.962680i	$0.587236\pi$
$284$	−0.707032	−0.0419546
$285$	6.03128	0.357262
$286$	0	0
$287$	−17.8317	−1.05257
$288$	−1.88651	−0.111164
$289$	1.00000	0.0588235
$290$	−26.3722	−1.54863
$291$	−5.46130	−0.320147
$292$	10.4546	0.611808
$293$	−4.79170	−0.279934	−0.139967	−	0.990156i	$-0.544700\pi$
$293$	−4.79170	−0.279934	−0.139967	+	0.990156i	$0.544700\pi$
$294$	9.57442	0.558391
$295$	5.54248	0.322696
$296$	2.34213	0.136134
$297$	0	0
$298$	6.01770	0.348596
$299$	−5.88583	−0.340386
$300$	−8.25177	−0.476416
$301$	−39.7103	−2.28886
$302$	1.14380	0.0658180
$303$	11.9039	0.683862
$304$	0.923274	0.0529534
$305$	−24.1223	−1.38124
$306$	−1.88651	−0.107844
$307$	16.9664	0.968325	0.484163	−	0.874978i	$-0.339124\pi$
$307$	16.9664	0.968325	0.484163	+	0.874978i	$0.339124\pi$
$308$	0	0
$309$	−25.3008	−1.43931
$310$	−28.6052	−1.62467
$311$	−20.7871	−1.17873	−0.589363	−	0.807868i	$-0.700621\pi$
$311$	−20.7871	−1.17873	−0.589363	+	0.807868i	$0.700621\pi$
$312$	−7.62891	−0.431902
$313$	20.1169	1.13707	0.568536	−	0.822658i	$-0.307510\pi$
$313$	20.1169	1.13707	0.568536	+	0.822658i	$0.307510\pi$
$314$	−21.5097	−1.21386
$315$	18.7662	1.05736
$316$	−13.3003	−0.748203
$317$	24.1714	1.35760	0.678801	−	0.734322i	$-0.262500\pi$
$317$	24.1714	1.35760	0.678801	+	0.734322i	$0.262500\pi$
$318$	24.3969	1.36811
$319$	0	0
$320$	−2.95515	−0.165198
$321$	19.7369	1.10161
$322$	5.74095	0.319930
$323$	0.923274	0.0513724
$324$	−11.1006	−0.616701
$325$	−12.8828	−0.714611
$326$	−7.97671	−0.441789
$327$	2.03391	0.112475
$328$	−5.29728	−0.292493
$329$	−21.9016	−1.20747
$330$	0	0
$331$	−8.71943	−0.479263	−0.239632	−	0.970864i	$-0.577027\pi$
$331$	−8.71943	−0.479263	−0.239632	+	0.970864i	$0.577027\pi$
$332$	−7.09197	−0.389222
$333$	−4.41845	−0.242130
$334$	−11.1818	−0.611841
$335$	−36.2078	−1.97824
$336$	7.44112	0.405946
$337$	−17.3983	−0.947745	−0.473872	−	0.880594i	$-0.657144\pi$
$337$	−17.3983	−0.947745	−0.473872	+	0.880594i	$0.657144\pi$
$338$	1.08960	0.0592666
$339$	−1.48519	−0.0806644
$340$	−2.95515	−0.160266
$341$	0	0
$342$	−1.74176	−0.0941838
$343$	8.98353	0.485065
$344$	−11.7968	−0.636041
$345$	11.1410	0.599810
$346$	13.0500	0.701570
$347$	26.4403	1.41939	0.709695	−	0.704509i	$-0.248833\pi$
$347$	26.4403	1.41939	0.709695	+	0.704509i	$0.248833\pi$
$348$	19.7272	1.05749
$349$	13.6611	0.731262	0.365631	−	0.930760i	$-0.380853\pi$
$349$	13.6611	0.731262	0.365631	+	0.930760i	$0.380853\pi$
$350$	12.5657	0.671665
$351$	−8.49473	−0.453415
$352$	0	0
$353$	−2.93540	−0.156236	−0.0781179	−	0.996944i	$-0.524891\pi$
$353$	−2.93540	−0.156236	−0.0781179	+	0.996944i	$0.524891\pi$
$354$	−4.14595	−0.220355
$355$	2.08939	0.110893
$356$	8.42835	0.446702
$357$	7.44112	0.393826
$358$	−9.65513	−0.510289
$359$	−25.6578	−1.35417	−0.677083	−	0.735907i	$-0.736756\pi$
$359$	−25.6578	−1.35417	−0.677083	+	0.735907i	$0.736756\pi$
$360$	5.57492	0.293824
$361$	−18.1476	−0.955135
$362$	−16.0310	−0.842573
$363$	0	0
$364$	11.6172	0.608907
$365$	−30.8948	−1.61711
$366$	18.0442	0.943187
$367$	−6.86945	−0.358582	−0.179291	−	0.983796i	$-0.557380\pi$
$367$	−6.86945	−0.358582	−0.179291	+	0.983796i	$0.557380\pi$
$368$	1.70547	0.0889039
$369$	9.99337	0.520234
$370$	−6.92135	−0.359824
$371$	−37.1513	−1.92880
$372$	21.3976	1.10941
$373$	−3.66847	−0.189946	−0.0949731	−	0.995480i	$-0.530276\pi$
$373$	−3.66847	−0.189946	−0.0949731	+	0.995480i	$0.530276\pi$
$374$	0	0
$375$	−8.27723	−0.427434
$376$	−6.50633	−0.335538
$377$	30.7985	1.58620
$378$	8.28562	0.426166
$379$	−12.3629	−0.635037	−0.317519	−	0.948252i	$-0.602850\pi$
$379$	−12.3629	−0.635037	−0.317519	+	0.948252i	$0.602850\pi$
$380$	−2.72841	−0.139965
$381$	33.3351	1.70781
$382$	6.77033	0.346400
$383$	30.6948	1.56843	0.784216	−	0.620488i	$-0.213066\pi$
$383$	30.6948	1.56843	0.784216	+	0.620488i	$0.213066\pi$
$384$	2.21054	0.112806
$385$	0	0
$386$	−12.9312	−0.658179
$387$	22.2548	1.13127
$388$	2.47057	0.125424
$389$	6.78457	0.343991	0.171996	−	0.985098i	$-0.444979\pi$
$389$	6.78457	0.343991	0.171996	+	0.985098i	$0.444979\pi$
$390$	22.5446	1.14159
$391$	1.70547	0.0862494
$392$	−4.33125	−0.218761
$393$	12.1010	0.610417
$394$	2.14463	0.108045
$395$	39.3045	1.97762
$396$	0	0
$397$	−22.4616	−1.12731	−0.563657	−	0.826009i	$-0.690606\pi$
$397$	−22.4616	−1.12731	−0.563657	+	0.826009i	$0.690606\pi$
$398$	7.94362	0.398178
$399$	6.87019	0.343940
$400$	3.73291	0.186646
$401$	−8.51103	−0.425021	−0.212510	−	0.977159i	$-0.568164\pi$
$401$	−8.51103	−0.425021	−0.212510	+	0.977159i	$0.568164\pi$
$402$	27.0846	1.35085
$403$	33.4063	1.66409
$404$	−5.38506	−0.267917
$405$	32.8040	1.63004
$406$	−30.0403	−1.49088
$407$	0	0
$408$	2.21054	0.109438
$409$	12.7824	0.632048	0.316024	−	0.948751i	$-0.397652\pi$
$409$	12.7824	0.632048	0.316024	+	0.948751i	$0.397652\pi$
$410$	15.6543	0.773109
$411$	30.3708	1.49808
$412$	11.4455	0.563880
$413$	6.31341	0.310662
$414$	−3.21739	−0.158126
$415$	20.9578	1.02878
$416$	3.45114	0.169206
$417$	−38.9260	−1.90622
$418$	0	0
$419$	0.115102	0.00562310	0.00281155	−	0.999996i	$-0.499105\pi$
$419$	0.115102	0.00562310	0.00281155	+	0.999996i	$0.499105\pi$
$420$	−21.9896	−1.07298
$421$	34.9143	1.70162	0.850809	−	0.525475i	$-0.176113\pi$
$421$	34.9143	1.70162	0.850809	+	0.525475i	$0.176113\pi$
$422$	6.65928	0.324169
$423$	12.2742	0.596794
$424$	−11.0366	−0.535985
$425$	3.73291	0.181073
$426$	−1.56293	−0.0757240
$427$	−27.4775	−1.32973
$428$	−8.92851	−0.431576
$429$	0	0
$430$	34.8613	1.68116
$431$	−20.4544	−0.985255	−0.492627	−	0.870240i	$-0.663963\pi$
$431$	−20.4544	−0.985255	−0.492627	+	0.870240i	$0.663963\pi$
$432$	2.46142	0.118425
$433$	35.8900	1.72476	0.862382	−	0.506259i	$-0.168972\pi$
$433$	35.8900	1.72476	0.862382	+	0.506259i	$0.168972\pi$
$434$	−32.5840	−1.56408
$435$	−58.2968	−2.79512
$436$	−0.920094	−0.0440645
$437$	1.57462	0.0753242
$438$	23.1103	1.10425
$439$	23.7099	1.13161	0.565807	−	0.824538i	$-0.308565\pi$
$439$	23.7099	1.13161	0.565807	+	0.824538i	$0.308565\pi$
$440$	0	0
$441$	8.17093	0.389092
$442$	3.45114	0.164154
$443$	−17.5178	−0.832297	−0.416149	−	0.909297i	$-0.636620\pi$
$443$	−17.5178	−0.832297	−0.416149	+	0.909297i	$0.636620\pi$
$444$	5.17739	0.245708
$445$	−24.9070	−1.18071
$446$	18.2556	0.864427
$447$	13.3024	0.629182
$448$	−3.36619	−0.159038
$449$	3.05885	0.144356	0.0721781	−	0.997392i	$-0.477005\pi$
$449$	3.05885	0.144356	0.0721781	+	0.997392i	$0.477005\pi$
$450$	−7.04217	−0.331971
$451$	0	0
$452$	0.671866	0.0316019
$453$	2.52841	0.118795
$454$	25.7298	1.20756
$455$	−34.3306	−1.60944
$456$	2.04094	0.0955757
$457$	5.05814	0.236610	0.118305	−	0.992977i	$-0.462254\pi$
$457$	5.05814	0.236610	0.118305	+	0.992977i	$0.462254\pi$
$458$	−23.1569	−1.08205
$459$	2.46142	0.114889
$460$	−5.03993	−0.234988
$461$	−3.15889	−0.147124	−0.0735622	−	0.997291i	$-0.523437\pi$
$461$	−3.15889	−0.147124	−0.0735622	+	0.997291i	$0.523437\pi$
$462$	0	0
$463$	−20.9933	−0.975642	−0.487821	−	0.872944i	$-0.662208\pi$
$463$	−20.9933	−0.975642	−0.487821	+	0.872944i	$0.662208\pi$
$464$	−8.92413	−0.414292
$465$	−63.2331	−2.93237
$466$	5.89280	0.272979
$467$	−25.8521	−1.19629	−0.598146	−	0.801387i	$-0.704096\pi$
$467$	−25.8521	−1.19629	−0.598146	+	0.801387i	$0.704096\pi$
$468$	−6.51061	−0.300953
$469$	−41.2440	−1.90447
$470$	19.2272	0.886884
$471$	−47.5481	−2.19090
$472$	1.87553	0.0863284
$473$	0	0
$474$	−29.4010	−1.35043
$475$	3.44650	0.158136
$476$	−3.36619	−0.154289
$477$	20.8206	0.953312
$478$	27.3521	1.25106
$479$	5.92727	0.270824	0.135412	−	0.990789i	$-0.456764\pi$
$479$	5.92727	0.270824	0.135412	+	0.990789i	$0.456764\pi$
$480$	−6.53249	−0.298166
$481$	8.08303	0.368555
$482$	−7.91912	−0.360706
$483$	12.6906	0.577443
$484$	0	0
$485$	−7.30090	−0.331517
$486$	−17.1541	−0.778127
$487$	−21.8394	−0.989640	−0.494820	−	0.868996i	$-0.664766\pi$
$487$	−21.8394	−0.989640	−0.494820	+	0.868996i	$0.664766\pi$
$488$	−8.16280	−0.369513
$489$	−17.6329	−0.797386
$490$	12.7995	0.578222
$491$	40.6640	1.83514	0.917569	−	0.397576i	$-0.130149\pi$
$491$	40.6640	1.83514	0.917569	+	0.397576i	$0.130149\pi$
$492$	−11.7099	−0.527922
$493$	−8.92413	−0.401923
$494$	3.18635	0.143361
$495$	0	0
$496$	−9.67978	−0.434635
$497$	2.38001	0.106758
$498$	−15.6771	−0.702508
$499$	31.3093	1.40160	0.700798	−	0.713360i	$-0.252828\pi$
$499$	31.3093	1.40160	0.700798	+	0.713360i	$0.252828\pi$
$500$	3.74443	0.167456
$501$	−24.7179	−1.10431
$502$	13.4587	0.600691
$503$	38.7304	1.72690	0.863451	−	0.504432i	$-0.168298\pi$
$503$	38.7304	1.72690	0.863451	+	0.504432i	$0.168298\pi$
$504$	6.35035	0.282867
$505$	15.9137	0.708148
$506$	0	0
$507$	2.40862	0.106970
$508$	−15.0801	−0.669069
$509$	15.8344	0.701846	0.350923	−	0.936404i	$-0.385868\pi$
$509$	15.8344	0.701846	0.350923	+	0.936404i	$0.385868\pi$
$510$	−6.53249	−0.289264
$511$	−35.1921	−1.55681
$512$	−1.00000	−0.0441942
$513$	2.27257	0.100336
$514$	−16.5938	−0.731921
$515$	−33.8232	−1.49043
$516$	−26.0774	−1.14799
$517$	0	0
$518$	−7.88406	−0.346406
$519$	28.8475	1.26627
$520$	−10.1987	−0.447240
$521$	−23.7133	−1.03890	−0.519450	−	0.854501i	$-0.673863\pi$
$521$	−23.7133	−1.03890	−0.519450	+	0.854501i	$0.673863\pi$
$522$	16.8354	0.736868
$523$	−16.4660	−0.720009	−0.360005	−	0.932951i	$-0.617225\pi$
$523$	−16.4660	−0.720009	−0.360005	+	0.932951i	$0.617225\pi$
$524$	−5.47424	−0.239143
$525$	27.7771	1.21229
$526$	4.62340	0.201590
$527$	−9.67978	−0.421658
$528$	0	0
$529$	−20.0914	−0.873538
$530$	32.6148	1.41670
$531$	−3.53821	−0.153545
$532$	−3.10792	−0.134745
$533$	−18.2817	−0.791868
$534$	18.6313	0.806253
$535$	26.3851	1.14073
$536$	−12.2524	−0.529224
$537$	−21.3431	−0.921022
$538$	0.674059	0.0290607
$539$	0	0
$540$	−7.27388	−0.313018
$541$	−7.42138	−0.319070	−0.159535	−	0.987192i	$-0.551000\pi$
$541$	−7.42138	−0.319070	−0.159535	+	0.987192i	$0.551000\pi$
$542$	17.5092	0.752083
$543$	−35.4373	−1.52076
$544$	−1.00000	−0.0428746
$545$	2.71902	0.116470
$546$	25.6804	1.09902
$547$	38.4637	1.64459	0.822295	−	0.569062i	$-0.192693\pi$
$547$	38.4637	1.64459	0.822295	+	0.569062i	$0.192693\pi$
$548$	−13.7391	−0.586903
$549$	15.3992	0.657221
$550$	0	0
$551$	−8.23942	−0.351011
$552$	3.77002	0.160463
$553$	44.7715	1.90388
$554$	22.8868	0.972369
$555$	−15.3000	−0.649447
$556$	17.6093	0.746799
$557$	−29.0110	−1.22924	−0.614618	−	0.788825i	$-0.710690\pi$
$557$	−29.0110	−1.22924	−0.614618	+	0.788825i	$0.710690\pi$
$558$	18.2610	0.773049
$559$	−40.7125	−1.72195
$560$	9.94760	0.420363
$561$	0	0
$562$	2.37406	0.100144
$563$	−13.0051	−0.548101	−0.274051	−	0.961715i	$-0.588364\pi$
$563$	−13.0051	−0.548101	−0.274051	+	0.961715i	$0.588364\pi$
$564$	−14.3825	−0.605614
$565$	−1.98547	−0.0835291
$566$	9.10583	0.382747
$567$	37.3668	1.56926
$568$	0.707032	0.0296664
$569$	−29.2224	−1.22507	−0.612533	−	0.790445i	$-0.709849\pi$
$569$	−29.2224	−1.22507	−0.612533	+	0.790445i	$0.709849\pi$
$570$	−6.03128	−0.252623
$571$	−14.2276	−0.595407	−0.297704	−	0.954658i	$-0.596221\pi$
$571$	−14.2276	−0.595407	−0.297704	+	0.954658i	$0.596221\pi$
$572$	0	0
$573$	14.9661	0.625218
$574$	17.8317	0.744279
$575$	6.36638	0.265496
$576$	1.88651	0.0786045
$577$	30.0427	1.25069	0.625346	−	0.780348i	$-0.284958\pi$
$577$	30.0427	1.25069	0.625346	+	0.780348i	$0.284958\pi$
$578$	−1.00000	−0.0415945
$579$	−28.5849	−1.18795
$580$	26.3722	1.09504
$581$	23.8729	0.990416
$582$	5.46130	0.226378
$583$	0	0
$584$	−10.4546	−0.432613
$585$	19.2398	0.795469
$586$	4.79170	0.197943
$587$	−0.00793311	−0.000327434 0	−0.000163717	−	1.00000i	$-0.500052\pi$
$587$	−0.00793311	−0.000327434 0	−0.000163717		1.00000i	$0.500052\pi$
$588$	−9.57442	−0.394842
$589$	−8.93710	−0.368247
$590$	−5.54248	−0.228180
$591$	4.74080	0.195010
$592$	−2.34213	−0.0962610
$593$	−46.0087	−1.88935	−0.944676	−	0.328005i	$-0.893624\pi$
$593$	−46.0087	−1.88935	−0.944676	+	0.328005i	$0.893624\pi$
$594$	0	0
$595$	9.94760	0.407812
$596$	−6.01770	−0.246495
$597$	17.5597	0.718672
$598$	5.88583	0.240690
$599$	33.7585	1.37934	0.689668	−	0.724126i	$-0.257757\pi$
$599$	33.7585	1.37934	0.689668	+	0.724126i	$0.257757\pi$
$600$	8.25177	0.336877
$601$	22.2289	0.906736	0.453368	−	0.891323i	$-0.350222\pi$
$601$	22.2289	0.906736	0.453368	+	0.891323i	$0.350222\pi$
$602$	39.7103	1.61847
$603$	23.1143	0.941288
$604$	−1.14380	−0.0465404
$605$	0	0
$606$	−11.9039	−0.483563
$607$	20.5167	0.832745	0.416373	−	0.909194i	$-0.363301\pi$
$607$	20.5167	0.832745	0.416373	+	0.909194i	$0.363301\pi$
$608$	−0.923274	−0.0374437
$609$	−66.4055	−2.69089
$610$	24.1223	0.976683
$611$	−22.4543	−0.908403
$612$	1.88651	0.0762576
$613$	−18.7372	−0.756789	−0.378394	−	0.925644i	$-0.623524\pi$
$613$	−18.7372	−0.756789	−0.378394	+	0.925644i	$0.623524\pi$
$614$	−16.9664	−0.684709
$615$	34.6045	1.39539
$616$	0	0
$617$	6.48202	0.260956	0.130478	−	0.991451i	$-0.458349\pi$
$617$	6.48202	0.260956	0.130478	+	0.991451i	$0.458349\pi$
$618$	25.3008	1.01775
$619$	−48.1609	−1.93575	−0.967874	−	0.251435i	$-0.919098\pi$
$619$	−48.1609	−1.93575	−0.967874	+	0.251435i	$0.919098\pi$
$620$	28.6052	1.14881
$621$	4.19789	0.168456
$622$	20.7871	0.833485
$623$	−28.3715	−1.13668
$624$	7.62891	0.305401
$625$	−29.7299	−1.18920
$626$	−20.1169	−0.804032
$627$	0	0
$628$	21.5097	0.858330
$629$	−2.34213	−0.0933869
$630$	−18.7662	−0.747665
$631$	−23.2195	−0.924355	−0.462177	−	0.886788i	$-0.652932\pi$
$631$	−23.2195	−0.924355	−0.462177	+	0.886788i	$0.652932\pi$
$632$	13.3003	0.529059
$633$	14.7206	0.585093
$634$	−24.1714	−0.959970
$635$	44.5638	1.76846
$636$	−24.3969	−0.967400
$637$	−14.9478	−0.592252
$638$	0	0
$639$	−1.33382	−0.0527652
$640$	2.95515	0.116813
$641$	−19.1409	−0.756022	−0.378011	−	0.925801i	$-0.623392\pi$
$641$	−19.1409	−0.756022	−0.378011	+	0.925801i	$0.623392\pi$
$642$	−19.7369	−0.778952
$643$	−26.0092	−1.02570	−0.512852	−	0.858477i	$-0.671411\pi$
$643$	−26.0092	−1.02570	−0.512852	+	0.858477i	$0.671411\pi$
$644$	−5.74095	−0.226225
$645$	77.0625	3.03433
$646$	−0.923274	−0.0363257
$647$	−15.9795	−0.628219	−0.314109	−	0.949387i	$-0.601706\pi$
$647$	−15.9795	−0.628219	−0.314109	+	0.949387i	$0.601706\pi$
$648$	11.1006	0.436073
$649$	0	0
$650$	12.8828	0.505306
$651$	−72.0284	−2.82302
$652$	7.97671	0.312392
$653$	−32.7344	−1.28099	−0.640497	−	0.767960i	$-0.721272\pi$
$653$	−32.7344	−1.28099	−0.640497	+	0.767960i	$0.721272\pi$
$654$	−2.03391	−0.0795321
$655$	16.1772	0.632095
$656$	5.29728	0.206824
$657$	19.7226	0.769453
$658$	21.9016	0.853812
$659$	17.6528	0.687655	0.343827	−	0.939033i	$-0.388276\pi$
$659$	17.6528	0.687655	0.343827	+	0.939033i	$0.388276\pi$
$660$	0	0
$661$	51.1477	1.98942	0.994708	−	0.102738i	$-0.0327605\pi$
$661$	51.1477	1.98942	0.994708	+	0.102738i	$0.0327605\pi$
$662$	8.71943	0.338890
$663$	7.62891	0.296282
$664$	7.09197	0.275222
$665$	9.18437	0.356154
$666$	4.41845	0.171211
$667$	−15.2199	−0.589315
$668$	11.1818	0.432637
$669$	40.3548	1.56021
$670$	36.2078	1.39883
$671$	0	0
$672$	−7.44112	−0.287047
$673$	47.3404	1.82484	0.912418	−	0.409259i	$-0.134213\pi$
$673$	47.3404	1.82484	0.912418	+	0.409259i	$0.134213\pi$
$674$	17.3983	0.670157
$675$	9.18828	0.353657
$676$	−1.08960	−0.0419078
$677$	42.4338	1.63086	0.815432	−	0.578853i	$-0.196499\pi$
$677$	42.4338	1.63086	0.815432	+	0.578853i	$0.196499\pi$
$678$	1.48519	0.0570384
$679$	−8.31640	−0.319154
$680$	2.95515	0.113325
$681$	56.8768	2.17952
$682$	0	0
$683$	31.2394	1.19534	0.597671	−	0.801742i	$-0.296093\pi$
$683$	31.2394	1.19534	0.597671	+	0.801742i	$0.296093\pi$
$684$	1.74176	0.0665980
$685$	40.6010	1.55128
$686$	−8.98353	−0.342993
$687$	−51.1895	−1.95300
$688$	11.7968	0.449749
$689$	−38.0889	−1.45107
$690$	−11.1410	−0.424130
$691$	9.32703	0.354817	0.177408	−	0.984137i	$-0.443229\pi$
$691$	9.32703	0.354817	0.177408	+	0.984137i	$0.443229\pi$
$692$	−13.0500	−0.496085
$693$	0	0
$694$	−26.4403	−1.00366
$695$	−52.0380	−1.97391
$696$	−19.7272	−0.747757
$697$	5.29728	0.200649
$698$	−13.6611	−0.517080
$699$	13.0263	0.492700
$700$	−12.5657	−0.474939
$701$	44.3273	1.67422	0.837109	−	0.547036i	$-0.184244\pi$
$701$	44.3273	1.67422	0.837109	+	0.547036i	$0.184244\pi$
$702$	8.49473	0.320613
$703$	−2.16243	−0.0815576
$704$	0	0
$705$	42.5026	1.60074
$706$	2.93540	0.110475
$707$	18.1271	0.681741
$708$	4.14595	0.155814
$709$	1.03132	0.0387322	0.0193661	−	0.999812i	$-0.493835\pi$
$709$	1.03132	0.0387322	0.0193661	+	0.999812i	$0.493835\pi$
$710$	−2.08939	−0.0784133
$711$	−25.0912	−0.940994
$712$	−8.42835	−0.315866
$713$	−16.5086	−0.618252
$714$	−7.44112	−0.278477
$715$	0	0
$716$	9.65513	0.360829
$717$	60.4631	2.25804
$718$	25.6578	0.957540
$719$	50.1991	1.87211	0.936055	−	0.351854i	$-0.114449\pi$
$719$	50.1991	1.87211	0.936055	+	0.351854i	$0.114449\pi$
$720$	−5.57492	−0.207765
$721$	−38.5278	−1.43485
$722$	18.1476	0.675382
$723$	−17.5056	−0.651039
$724$	16.0310	0.595789
$725$	−33.3130	−1.23721
$726$	0	0
$727$	−43.7880	−1.62401	−0.812003	−	0.583653i	$-0.801623\pi$
$727$	−43.7880	−1.62401	−0.812003	+	0.583653i	$0.801623\pi$
$728$	−11.6172	−0.430563
$729$	−4.61813	−0.171042
$730$	30.8948	1.14347
$731$	11.7968	0.436320
$732$	−18.0442	−0.666934
$733$	11.9058	0.439753	0.219876	−	0.975528i	$-0.429435\pi$
$733$	11.9058	0.439753	0.219876	+	0.975528i	$0.429435\pi$
$734$	6.86945	0.253556
$735$	28.2938	1.04363
$736$	−1.70547	−0.0628645
$737$	0	0
$738$	−9.99337	−0.367861
$739$	41.7213	1.53474	0.767372	−	0.641203i	$-0.221564\pi$
$739$	41.7213	1.53474	0.767372	+	0.641203i	$0.221564\pi$
$740$	6.92135	0.254434
$741$	7.04358	0.258752
$742$	37.1513	1.36387
$743$	−35.6670	−1.30850	−0.654248	−	0.756280i	$-0.727015\pi$
$743$	−35.6670	−1.30850	−0.654248	+	0.756280i	$0.727015\pi$
$744$	−21.3976	−0.784474
$745$	17.7832	0.651527
$746$	3.66847	0.134312
$747$	−13.3791	−0.489514
$748$	0	0
$749$	30.0551	1.09819
$750$	8.27723	0.302242
$751$	40.0805	1.46256	0.731280	−	0.682078i	$-0.238924\pi$
$751$	40.0805	1.46256	0.731280	+	0.682078i	$0.238924\pi$
$752$	6.50633	0.237262
$753$	29.7511	1.08419
$754$	−30.7985	−1.12161
$755$	3.38009	0.123014
$756$	−8.28562	−0.301345
$757$	48.4245	1.76002	0.880010	−	0.474956i	$-0.157536\pi$
$757$	48.4245	1.76002	0.880010	+	0.474956i	$0.157536\pi$
$758$	12.3629	0.449039
$759$	0	0
$760$	2.72841	0.0989700
$761$	4.84161	0.175508	0.0877541	−	0.996142i	$-0.472031\pi$
$761$	4.84161	0.175508	0.0877541	+	0.996142i	$0.472031\pi$
$762$	−33.3351	−1.20760
$763$	3.09721	0.112127
$764$	−6.77033	−0.244942
$765$	−5.57492	−0.201561
$766$	−30.6948	−1.10905
$767$	6.47274	0.233717
$768$	−2.21054	−0.0797662
$769$	−9.23759	−0.333116	−0.166558	−	0.986032i	$-0.553265\pi$
$769$	−9.23759	−0.333116	−0.166558	+	0.986032i	$0.553265\pi$
$770$	0	0
$771$	−36.6813	−1.32105
$772$	12.9312	0.465403
$773$	−14.7277	−0.529717	−0.264858	−	0.964287i	$-0.585325\pi$
$773$	−14.7277	−0.529717	−0.264858	+	0.964287i	$0.585325\pi$
$774$	−22.2548	−0.799931
$775$	−36.1338	−1.29796
$776$	−2.47057	−0.0886882
$777$	−17.4281	−0.625229
$778$	−6.78457	−0.243238
$779$	4.89084	0.175233
$780$	−22.5446	−0.807225
$781$	0	0
$782$	−1.70547	−0.0609876
$783$	−21.9661	−0.785003
$784$	4.33125	0.154687
$785$	−63.5644	−2.26871
$786$	−12.1010	−0.431630
$787$	−16.7457	−0.596920	−0.298460	−	0.954422i	$-0.596473\pi$
$787$	−16.7457	−0.596920	−0.298460	+	0.954422i	$0.596473\pi$
$788$	−2.14463	−0.0763992
$789$	10.2202	0.363850
$790$	−39.3045	−1.39839
$791$	−2.26163	−0.0804143
$792$	0	0
$793$	−28.1710	−1.00038
$794$	22.4616	0.797132
$795$	72.0965	2.55700
$796$	−7.94362	−0.281554
$797$	9.73443	0.344811	0.172406	−	0.985026i	$-0.444846\pi$
$797$	9.73443	0.344811	0.172406	+	0.985026i	$0.444846\pi$
$798$	−6.87019	−0.243202
$799$	6.50633	0.230178
$800$	−3.73291	−0.131978
$801$	15.9002	0.561804
$802$	8.51103	0.300535
$803$	0	0
$804$	−27.0846	−0.955198
$805$	16.9654	0.597950
$806$	−33.4063	−1.17669
$807$	1.49004	0.0524518
$808$	5.38506	0.189446
$809$	9.06349	0.318655	0.159328	−	0.987226i	$-0.449067\pi$
$809$	9.06349	0.318655	0.159328	+	0.987226i	$0.449067\pi$
$810$	−32.8040	−1.15261
$811$	12.3454	0.433505	0.216752	−	0.976227i	$-0.430454\pi$
$811$	12.3454	0.433505	0.216752	+	0.976227i	$0.430454\pi$
$812$	30.0403	1.05421
$813$	38.7048	1.35744
$814$	0	0
$815$	−23.5724	−0.825704
$816$	−2.21054	−0.0773845
$817$	10.8917	0.381052
$818$	−12.7824	−0.446926
$819$	21.9160	0.765806
$820$	−15.6543	−0.546671
$821$	−35.0666	−1.22383	−0.611916	−	0.790923i	$-0.709601\pi$
$821$	−35.0666	−1.22383	−0.611916	+	0.790923i	$0.709601\pi$
$822$	−30.3708	−1.05930
$823$	−11.2794	−0.393176	−0.196588	−	0.980486i	$-0.562986\pi$
$823$	−11.2794	−0.393176	−0.196588	+	0.980486i	$0.562986\pi$
$824$	−11.4455	−0.398724
$825$	0	0
$826$	−6.31341	−0.219671
$827$	−40.4720	−1.40735	−0.703674	−	0.710523i	$-0.748459\pi$
$827$	−40.4720	−1.40735	−0.703674	+	0.710523i	$0.748459\pi$
$828$	3.21739	0.111812
$829$	15.6624	0.543979	0.271989	−	0.962300i	$-0.412318\pi$
$829$	15.6624	0.543979	0.271989	+	0.962300i	$0.412318\pi$
$830$	−20.9578	−0.727457
$831$	50.5924	1.75503
$832$	−3.45114	−0.119647
$833$	4.33125	0.150069
$834$	38.9260	1.34790
$835$	−33.0439	−1.14353
$836$	0	0
$837$	−23.8260	−0.823549
$838$	−0.115102	−0.00397613
$839$	−1.48269	−0.0511881	−0.0255940	−	0.999672i	$-0.508148\pi$
$839$	−1.48269	−0.0511881	−0.0255940	+	0.999672i	$0.508148\pi$
$840$	21.9896	0.758714
$841$	50.6401	1.74621
$842$	−34.9143	−1.20323
$843$	5.24797	0.180750
$844$	−6.65928	−0.229222
$845$	3.21994	0.110769
$846$	−12.2742	−0.421997
$847$	0	0
$848$	11.0366	0.378998
$849$	20.1288	0.690820
$850$	−3.73291	−0.128038
$851$	−3.99444	−0.136928
$852$	1.56293	0.0535450
$853$	−1.41669	−0.0485065	−0.0242532	−	0.999706i	$-0.507721\pi$
$853$	−1.41669	−0.0485065	−0.0242532	+	0.999706i	$0.507721\pi$
$854$	27.4775	0.940262
$855$	−5.14718	−0.176030
$856$	8.92851	0.305170
$857$	−44.0250	−1.50386	−0.751932	−	0.659240i	$-0.770878\pi$
$857$	−44.0250	−1.50386	−0.751932	+	0.659240i	$0.770878\pi$
$858$	0	0
$859$	−28.4553	−0.970882	−0.485441	−	0.874269i	$-0.661341\pi$
$859$	−28.4553	−0.970882	−0.485441	+	0.874269i	$0.661341\pi$
$860$	−34.8613	−1.18876
$861$	39.4177	1.34335
$862$	20.4544	0.696680
$863$	8.89103	0.302654	0.151327	−	0.988484i	$-0.451645\pi$
$863$	8.89103	0.302654	0.151327	+	0.988484i	$0.451645\pi$
$864$	−2.46142	−0.0837393
$865$	38.5646	1.31124
$866$	−35.8900	−1.21959
$867$	−2.21054	−0.0750740
$868$	32.5840	1.10597
$869$	0	0
$870$	58.2968	1.97645
$871$	−42.2849	−1.43277
$872$	0.920094	0.0311583
$873$	4.66075	0.157742
$874$	−1.57462	−0.0532623
$875$	−12.6045	−0.426109
$876$	−23.1103	−0.780825
$877$	−0.0866002	−0.00292428	−0.00146214	−	0.999999i	$-0.500465\pi$
$877$	−0.0866002	−0.00292428	−0.00146214	+	0.999999i	$0.500465\pi$
$878$	−23.7099	−0.800172
$879$	10.5923	0.357268
$880$	0	0
$881$	−10.4299	−0.351392	−0.175696	−	0.984444i	$-0.556218\pi$
$881$	−10.4299	−0.351392	−0.175696	+	0.984444i	$0.556218\pi$
$882$	−8.17093	−0.275130
$883$	17.2303	0.579844	0.289922	−	0.957050i	$-0.406371\pi$
$883$	17.2303	0.579844	0.289922	+	0.957050i	$0.406371\pi$
$884$	−3.45114	−0.116075
$885$	−12.2519	−0.411843
$886$	17.5178	0.588523
$887$	17.3988	0.584194	0.292097	−	0.956389i	$-0.405647\pi$
$887$	17.3988	0.584194	0.292097	+	0.956389i	$0.405647\pi$
$888$	−5.17739	−0.173742
$889$	50.7624	1.70251
$890$	24.9070	0.834886
$891$	0	0
$892$	−18.2556	−0.611242
$893$	6.00713	0.201021
$894$	−13.3024	−0.444899
$895$	−28.5324	−0.953731
$896$	3.36619	0.112457
$897$	13.0109	0.434421
$898$	−3.05885	−0.102075
$899$	86.3837	2.88106
$900$	7.04217	0.234739
$901$	11.0366	0.367682
$902$	0	0
$903$	87.7814	2.92118
$904$	−0.671866	−0.0223459
$905$	−47.3741	−1.57477
$906$	−2.52841	−0.0840008
$907$	−33.3697	−1.10802	−0.554012	−	0.832509i	$-0.686904\pi$
$907$	−33.3697	−1.10802	−0.554012	+	0.832509i	$0.686904\pi$
$908$	−25.7298	−0.853873
$909$	−10.1590	−0.336951
$910$	34.3306	1.13805
$911$	48.9738	1.62257	0.811287	−	0.584648i	$-0.198767\pi$
$911$	48.9738	1.62257	0.811287	+	0.584648i	$0.198767\pi$
$912$	−2.04094	−0.0675823
$913$	0	0
$914$	−5.05814	−0.167308
$915$	53.3234	1.76282
$916$	23.1569	0.765127
$917$	18.4273	0.608524
$918$	−2.46142	−0.0812391
$919$	7.98210	0.263305	0.131653	−	0.991296i	$-0.457972\pi$
$919$	7.98210	0.263305	0.131653	+	0.991296i	$0.457972\pi$
$920$	5.03993	0.166161
$921$	−37.5051	−1.23583
$922$	3.15889	0.104033
$923$	2.44007	0.0803159
$924$	0	0
$925$	−8.74298	−0.287467
$926$	20.9933	0.689883
$927$	21.5921	0.709177
$928$	8.92413	0.292949
$929$	−60.5701	−1.98724	−0.993620	−	0.112782i	$-0.964024\pi$
$929$	−60.5701	−1.98724	−0.993620	+	0.112782i	$0.964024\pi$
$930$	63.2331	2.07350
$931$	3.99893	0.131060
$932$	−5.89280	−0.193025
$933$	45.9507	1.50436
$934$	25.8521	0.845906
$935$	0	0
$936$	6.51061	0.212806
$937$	41.6556	1.36083	0.680415	−	0.732827i	$-0.261800\pi$
$937$	41.6556	1.36083	0.680415	+	0.732827i	$0.261800\pi$
$938$	41.2440	1.34667
$939$	−44.4692	−1.45120
$940$	−19.2272	−0.627122
$941$	8.41344	0.274270	0.137135	−	0.990552i	$-0.456211\pi$
$941$	8.41344	0.274270	0.137135	+	0.990552i	$0.456211\pi$
$942$	47.5481	1.54920
$943$	9.03437	0.294200
$944$	−1.87553	−0.0610434
$945$	24.4853	0.796506
$946$	0	0
$947$	−15.0936	−0.490475	−0.245238	−	0.969463i	$-0.578866\pi$
$947$	−15.0936	−0.490475	−0.245238	+	0.969463i	$0.578866\pi$
$948$	29.4010	0.954900
$949$	−36.0802	−1.17121
$950$	−3.44650	−0.111819
$951$	−53.4320	−1.73265
$952$	3.36619	0.109099
$953$	22.3271	0.723247	0.361624	−	0.932324i	$-0.382223\pi$
$953$	22.3271	0.723247	0.361624	+	0.932324i	$0.382223\pi$
$954$	−20.8206	−0.674093
$955$	20.0073	0.647422
$956$	−27.3521	−0.884631
$957$	0	0
$958$	−5.92727	−0.191501
$959$	46.2483	1.49343
$960$	6.53249	0.210835
$961$	62.6982	2.02252
$962$	−8.08303	−0.260607
$963$	−16.8437	−0.542781
$964$	7.91912	0.255058
$965$	−38.2136	−1.23014
$966$	−12.6906	−0.408314
$967$	−15.0608	−0.484322	−0.242161	−	0.970236i	$-0.577856\pi$
$967$	−15.0608	−0.484322	−0.242161	+	0.970236i	$0.577856\pi$
$968$	0	0
$969$	−2.04094	−0.0655644
$970$	7.30090	0.234418
$971$	−18.3530	−0.588974	−0.294487	−	0.955655i	$-0.595149\pi$
$971$	−18.3530	−0.588974	−0.294487	+	0.955655i	$0.595149\pi$
$972$	17.1541	0.550219
$973$	−59.2761	−1.90031
$974$	21.8394	0.699781
$975$	28.4781	0.912028
$976$	8.16280	0.261285
$977$	−1.81095	−0.0579373	−0.0289686	−	0.999580i	$-0.509222\pi$
$977$	−1.81095	−0.0579373	−0.0289686	+	0.999580i	$0.509222\pi$
$978$	17.6329	0.563837
$979$	0	0
$980$	−12.7995	−0.408865
$981$	−1.73576	−0.0554187
$982$	−40.6640	−1.29764
$983$	−21.5481	−0.687278	−0.343639	−	0.939102i	$-0.611660\pi$
$983$	−21.5481	−0.687278	−0.343639	+	0.939102i	$0.611660\pi$
$984$	11.7099	0.373297
$985$	6.33770	0.201936
$986$	8.92413	0.284202
$987$	48.4144	1.54105
$988$	−3.18635	−0.101371
$989$	20.1191	0.639751
$990$	0	0
$991$	4.80748	0.152715	0.0763574	−	0.997081i	$-0.475671\pi$
$991$	4.80748	0.152715	0.0763574	+	0.997081i	$0.475671\pi$
$992$	9.67978	0.307333
$993$	19.2747	0.611664
$994$	−2.38001	−0.0754892
$995$	23.4746	0.744195
$996$	15.6771	0.496748
$997$	−10.3763	−0.328620	−0.164310	−	0.986409i	$-0.552540\pi$
$997$	−10.3763	−0.328620	−0.164310	+	0.986409i	$0.552540\pi$
$998$	−31.3093	−0.991077
$999$	−5.76498	−0.182396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bg.1.3		8
11.2	odd	10		374.2.g.f.103.2	yes	16
11.6	odd	10		374.2.g.f.69.2	✓	16
11.10	odd	2		4114.2.a.bi.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.g.f.69.2	✓	16	11.6	odd	10
374.2.g.f.103.2	yes	16	11.2	odd	10
4114.2.a.bg.1.3		8	1.1	even	1	trivial
4114.2.a.bi.1.3		8	11.10	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:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.21054 q^{3} +1.00000 q^{4} -2.95515 q^{5} +2.21054 q^{6} -3.36619 q^{7} -1.00000 q^{8} +1.88651 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.21054 q^{3} +1.00000 q^{4} -2.95515 q^{5} +2.21054 q^{6} -3.36619 q^{7} -1.00000 q^{8} +1.88651 q^{9} +2.95515 q^{10} -2.21054 q^{12} -3.45114 q^{13} +3.36619 q^{14} +6.53249 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.88651 q^{18} +0.923274 q^{19} -2.95515 q^{20} +7.44112 q^{21} +1.70547 q^{23} +2.21054 q^{24} +3.73291 q^{25} +3.45114 q^{26} +2.46142 q^{27} -3.36619 q^{28} -8.92413 q^{29} -6.53249 q^{30} -9.67978 q^{31} -1.00000 q^{32} -1.00000 q^{34} +9.94760 q^{35} +1.88651 q^{36} -2.34213 q^{37} -0.923274 q^{38} +7.62891 q^{39} +2.95515 q^{40} +5.29728 q^{41} -7.44112 q^{42} +11.7968 q^{43} -5.57492 q^{45} -1.70547 q^{46} +6.50633 q^{47} -2.21054 q^{48} +4.33125 q^{49} -3.73291 q^{50} -2.21054 q^{51} -3.45114 q^{52} +11.0366 q^{53} -2.46142 q^{54} +3.36619 q^{56} -2.04094 q^{57} +8.92413 q^{58} -1.87553 q^{59} +6.53249 q^{60} +8.16280 q^{61} +9.67978 q^{62} -6.35035 q^{63} +1.00000 q^{64} +10.1987 q^{65} +12.2524 q^{67} +1.00000 q^{68} -3.77002 q^{69} -9.94760 q^{70} -0.707032 q^{71} -1.88651 q^{72} +10.4546 q^{73} +2.34213 q^{74} -8.25177 q^{75} +0.923274 q^{76} -7.62891 q^{78} -13.3003 q^{79} -2.95515 q^{80} -11.1006 q^{81} -5.29728 q^{82} -7.09197 q^{83} +7.44112 q^{84} -2.95515 q^{85} -11.7968 q^{86} +19.7272 q^{87} +8.42835 q^{89} +5.57492 q^{90} +11.6172 q^{91} +1.70547 q^{92} +21.3976 q^{93} -6.50633 q^{94} -2.72841 q^{95} +2.21054 q^{96} +2.47057 q^{97} -4.33125 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Introduction

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