LMFDB - Embedded newform 4114.2.a.bg.1.7

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			1.7
Root			$2.04360$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.04360 q^{3} +1.00000 q^{4} +3.29527 q^{5} -1.04360 q^{6} -3.68858 q^{7} -1.00000 q^{8} -1.91090 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.04360 q^{3} +1.00000 q^{4} +3.29527 q^{5} -1.04360 q^{6} -3.68858 q^{7} -1.00000 q^{8} -1.91090 q^{9} -3.29527 q^{10} +1.04360 q^{12} +2.48474 q^{13} +3.68858 q^{14} +3.43894 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.91090 q^{18} -6.75683 q^{19} +3.29527 q^{20} -3.84940 q^{21} +3.00369 q^{23} -1.04360 q^{24} +5.85881 q^{25} -2.48474 q^{26} -5.12501 q^{27} -3.68858 q^{28} -0.327467 q^{29} -3.43894 q^{30} -6.65894 q^{31} -1.00000 q^{32} -1.00000 q^{34} -12.1549 q^{35} -1.91090 q^{36} -9.05457 q^{37} +6.75683 q^{38} +2.59307 q^{39} -3.29527 q^{40} +5.75930 q^{41} +3.84940 q^{42} -8.95131 q^{43} -6.29693 q^{45} -3.00369 q^{46} +12.5280 q^{47} +1.04360 q^{48} +6.60562 q^{49} -5.85881 q^{50} +1.04360 q^{51} +2.48474 q^{52} +1.61685 q^{53} +5.12501 q^{54} +3.68858 q^{56} -7.05142 q^{57} +0.327467 q^{58} -1.99129 q^{59} +3.43894 q^{60} +7.83580 q^{61} +6.65894 q^{62} +7.04851 q^{63} +1.00000 q^{64} +8.18788 q^{65} +6.46403 q^{67} +1.00000 q^{68} +3.13465 q^{69} +12.1549 q^{70} -11.1537 q^{71} +1.91090 q^{72} -15.2465 q^{73} +9.05457 q^{74} +6.11425 q^{75} -6.75683 q^{76} -2.59307 q^{78} -5.93084 q^{79} +3.29527 q^{80} +0.384242 q^{81} -5.75930 q^{82} -12.4167 q^{83} -3.84940 q^{84} +3.29527 q^{85} +8.95131 q^{86} -0.341744 q^{87} -11.0042 q^{89} +6.29693 q^{90} -9.16514 q^{91} +3.00369 q^{92} -6.94926 q^{93} -12.5280 q^{94} -22.2656 q^{95} -1.04360 q^{96} -9.82512 q^{97} -6.60562 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.04360	0.602522	0.301261	−	0.953542i	$-0.402592\pi$
$3$	1.04360	0.602522	0.301261	+	0.953542i	$0.402592\pi$
$4$	1.00000	0.500000
$5$	3.29527	1.47369	0.736845	−	0.676062i	$-0.236315\pi$
$5$	3.29527	1.47369	0.736845	+	0.676062i	$0.236315\pi$
$6$	−1.04360	−0.426048
$7$	−3.68858	−1.39415	−0.697076	−	0.716997i	$-0.745516\pi$
$7$	−3.68858	−1.39415	−0.697076	+	0.716997i	$0.745516\pi$
$8$	−1.00000	−0.353553
$9$	−1.91090	−0.636967
$10$	−3.29527	−1.04206
$11$	0	0
$11$	0	0
$12$	1.04360	0.301261
$13$	2.48474	0.689142	0.344571	−	0.938760i	$-0.388024\pi$
$13$	2.48474	0.689142	0.344571	+	0.938760i	$0.388024\pi$
$14$	3.68858	0.985814
$15$	3.43894	0.887931
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.91090	0.450404
$19$	−6.75683	−1.55012	−0.775061	−	0.631886i	$-0.782281\pi$
$19$	−6.75683	−1.55012	−0.775061	+	0.631886i	$0.782281\pi$
$20$	3.29527	0.736845
$21$	−3.84940	−0.840008
$22$	0	0
$23$	3.00369	0.626312	0.313156	−	0.949702i	$-0.398614\pi$
$23$	3.00369	0.626312	0.313156	+	0.949702i	$0.398614\pi$
$24$	−1.04360	−0.213024
$25$	5.85881	1.17176
$26$	−2.48474	−0.487297
$27$	−5.12501	−0.986309
$28$	−3.68858	−0.697076
$29$	−0.327467	−0.0608091	−0.0304045	−	0.999538i	$-0.509680\pi$
$29$	−0.327467	−0.0608091	−0.0304045	+	0.999538i	$0.509680\pi$
$30$	−3.43894	−0.627862
$31$	−6.65894	−1.19598	−0.597990	−	0.801503i	$-0.704034\pi$
$31$	−6.65894	−1.19598	−0.597990	+	0.801503i	$0.704034\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−12.1549	−2.05455
$36$	−1.91090	−0.318483
$37$	−9.05457	−1.48856	−0.744281	−	0.667866i	$-0.767208\pi$
$37$	−9.05457	−1.48856	−0.744281	+	0.667866i	$0.767208\pi$
$38$	6.75683	1.09610
$39$	2.59307	0.415223
$40$	−3.29527	−0.521028
$41$	5.75930	0.899452	0.449726	−	0.893167i	$-0.351522\pi$
$41$	5.75930	0.899452	0.449726	+	0.893167i	$0.351522\pi$
$42$	3.84940	0.593975
$43$	−8.95131	−1.36506	−0.682531	−	0.730856i	$-0.739121\pi$
$43$	−8.95131	−1.36506	−0.682531	+	0.730856i	$0.739121\pi$
$44$	0	0
$45$	−6.29693	−0.938691
$46$	−3.00369	−0.442870
$47$	12.5280	1.82740	0.913699	−	0.406391i	$-0.133213\pi$
$47$	12.5280	1.82740	0.913699	+	0.406391i	$0.133213\pi$
$48$	1.04360	0.150631
$49$	6.60562	0.943660
$50$	−5.85881	−0.828560
$51$	1.04360	0.146133
$52$	2.48474	0.344571
$53$	1.61685	0.222092	0.111046	−	0.993815i	$-0.464580\pi$
$53$	1.61685	0.222092	0.111046	+	0.993815i	$0.464580\pi$
$54$	5.12501	0.697426
$55$	0	0
$56$	3.68858	0.492907
$57$	−7.05142	−0.933983
$58$	0.327467	0.0429985
$59$	−1.99129	−0.259244	−0.129622	−	0.991563i	$-0.541376\pi$
$59$	−1.99129	−0.259244	−0.129622	+	0.991563i	$0.541376\pi$
$60$	3.43894	0.443965
$61$	7.83580	1.00327	0.501636	−	0.865079i	$-0.332732\pi$
$61$	7.83580	1.00327	0.501636	+	0.865079i	$0.332732\pi$
$62$	6.65894	0.845686
$63$	7.04851	0.888028
$64$	1.00000	0.125000
$65$	8.18788	1.01558
$66$	0	0
$67$	6.46403	0.789707	0.394854	−	0.918744i	$-0.370795\pi$
$67$	6.46403	0.789707	0.394854	+	0.918744i	$0.370795\pi$
$68$	1.00000	0.121268
$69$	3.13465	0.377367
$70$	12.1549	1.45278
$71$	−11.1537	−1.32370	−0.661849	−	0.749637i	$-0.730228\pi$
$71$	−11.1537	−1.32370	−0.661849	+	0.749637i	$0.730228\pi$
$72$	1.91090	0.225202
$73$	−15.2465	−1.78447	−0.892235	−	0.451571i	$-0.850864\pi$
$73$	−15.2465	−1.78447	−0.892235	+	0.451571i	$0.850864\pi$
$74$	9.05457	1.05257
$75$	6.11425	0.706012
$76$	−6.75683	−0.775061
$77$	0	0
$78$	−2.59307	−0.293607
$79$	−5.93084	−0.667271	−0.333636	−	0.942702i	$-0.608275\pi$
$79$	−5.93084	−0.667271	−0.333636	+	0.942702i	$0.608275\pi$
$80$	3.29527	0.368422
$81$	0.384242	0.0426935
$82$	−5.75930	−0.636008
$83$	−12.4167	−1.36290	−0.681452	−	0.731862i	$-0.738651\pi$
$83$	−12.4167	−1.36290	−0.681452	+	0.731862i	$0.738651\pi$
$84$	−3.84940	−0.420004
$85$	3.29527	0.357422
$86$	8.95131	0.965245
$87$	−0.341744	−0.0366388
$88$	0	0
$89$	−11.0042	−1.16644	−0.583219	−	0.812315i	$-0.698207\pi$
$89$	−11.0042	−1.16644	−0.583219	+	0.812315i	$0.698207\pi$
$90$	6.29693	0.663755
$91$	−9.16514	−0.960768
$92$	3.00369	0.313156
$93$	−6.94926	−0.720605
$94$	−12.5280	−1.29217
$95$	−22.2656	−2.28440
$96$	−1.04360	−0.106512
$97$	−9.82512	−0.997590	−0.498795	−	0.866720i	$-0.666224\pi$
$97$	−9.82512	−0.997590	−0.498795	+	0.866720i	$0.666224\pi$
$98$	−6.60562	−0.667268
$99$	0	0
$100$	5.85881	0.585881
$101$	14.1963	1.41258	0.706291	−	0.707922i	$-0.250367\pi$
$101$	14.1963	1.41258	0.706291	+	0.707922i	$0.250367\pi$
$102$	−1.04360	−0.103332
$103$	−15.4968	−1.52694	−0.763472	−	0.645841i	$-0.776507\pi$
$103$	−15.4968	−1.52694	−0.763472	+	0.645841i	$0.776507\pi$
$104$	−2.48474	−0.243648
$105$	−12.6848	−1.23791
$106$	−1.61685	−0.157043
$107$	−0.167926	−0.0162341	−0.00811703	−	0.999967i	$-0.502584\pi$
$107$	−0.167926	−0.0162341	−0.00811703	+	0.999967i	$0.502584\pi$
$108$	−5.12501	−0.493155
$109$	−3.91929	−0.375399	−0.187700	−	0.982226i	$-0.560103\pi$
$109$	−3.91929	−0.375399	−0.187700	+	0.982226i	$0.560103\pi$
$110$	0	0
$111$	−9.44935	−0.896892
$112$	−3.68858	−0.348538
$113$	1.57878	0.148519	0.0742595	−	0.997239i	$-0.476341\pi$
$113$	1.57878	0.148519	0.0742595	+	0.997239i	$0.476341\pi$
$114$	7.05142	0.660426
$115$	9.89796	0.922990
$116$	−0.327467	−0.0304045
$117$	−4.74808	−0.438960
$118$	1.99129	0.183313
$119$	−3.68858	−0.338132
$120$	−3.43894	−0.313931
$121$	0	0
$122$	−7.83580	−0.709420
$123$	6.01040	0.541940
$124$	−6.65894	−0.597990
$125$	2.82999	0.253122
$126$	−7.04851	−0.627931
$127$	−6.89631	−0.611949	−0.305974	−	0.952040i	$-0.598982\pi$
$127$	−6.89631	−0.611949	−0.305974	+	0.952040i	$0.598982\pi$
$128$	−1.00000	−0.0883883
$129$	−9.34159	−0.822481
$130$	−8.18788	−0.718124
$131$	−3.08556	−0.269587	−0.134793	−	0.990874i	$-0.543037\pi$
$131$	−3.08556	−0.269587	−0.134793	+	0.990874i	$0.543037\pi$
$132$	0	0
$133$	24.9231	2.16111
$134$	−6.46403	−0.558407
$135$	−16.8883	−1.45351
$136$	−1.00000	−0.0857493
$137$	4.19193	0.358141	0.179071	−	0.983836i	$-0.442691\pi$
$137$	4.19193	0.358141	0.179071	+	0.983836i	$0.442691\pi$
$138$	−3.13465	−0.266839
$139$	−1.77810	−0.150816	−0.0754081	−	0.997153i	$-0.524026\pi$
$139$	−1.77810	−0.150816	−0.0754081	+	0.997153i	$0.524026\pi$
$140$	−12.1549	−1.02727
$141$	13.0742	1.10105
$142$	11.1537	0.935996
$143$	0	0
$144$	−1.91090	−0.159242
$145$	−1.07909	−0.0896137
$146$	15.2465	1.26181
$147$	6.89362	0.568576
$148$	−9.05457	−0.744281
$149$	5.10075	0.417870	0.208935	−	0.977930i	$-0.433000\pi$
$149$	5.10075	0.417870	0.208935	+	0.977930i	$0.433000\pi$
$150$	−6.11425	−0.499226
$151$	13.0795	1.06439	0.532197	−	0.846621i	$-0.321367\pi$
$151$	13.0795	1.06439	0.532197	+	0.846621i	$0.321367\pi$
$152$	6.75683	0.548051
$153$	−1.91090	−0.154487
$154$	0	0
$155$	−21.9430	−1.76250
$156$	2.59307	0.207612
$157$	0.469645	0.0374817	0.0187409	−	0.999824i	$-0.494034\pi$
$157$	0.469645	0.0374817	0.0187409	+	0.999824i	$0.494034\pi$
$158$	5.93084	0.471832
$159$	1.68735	0.133815
$160$	−3.29527	−0.260514
$161$	−11.0793	−0.873175
$162$	−0.384242	−0.0301889
$163$	−20.9869	−1.64382	−0.821911	−	0.569616i	$-0.807092\pi$
$163$	−20.9869	−1.64382	−0.821911	+	0.569616i	$0.807092\pi$
$164$	5.75930	0.449726
$165$	0	0
$166$	12.4167	0.963719
$167$	9.07097	0.701933	0.350967	−	0.936388i	$-0.385853\pi$
$167$	9.07097	0.701933	0.350967	+	0.936388i	$0.385853\pi$
$168$	3.84940	0.296988
$169$	−6.82609	−0.525084
$170$	−3.29527	−0.252736
$171$	12.9116	0.987376
$172$	−8.95131	−0.682531
$173$	11.9688	0.909968	0.454984	−	0.890500i	$-0.349645\pi$
$173$	11.9688	0.909968	0.454984	+	0.890500i	$0.349645\pi$
$174$	0.341744	0.0259076
$175$	−21.6107	−1.63361
$176$	0	0
$177$	−2.07811	−0.156200
$178$	11.0042	0.824797
$179$	1.13869	0.0851094	0.0425547	−	0.999094i	$-0.486450\pi$
$179$	1.13869	0.0851094	0.0425547	+	0.999094i	$0.486450\pi$
$180$	−6.29693	−0.469346
$181$	−14.4963	−1.07751	−0.538753	−	0.842464i	$-0.681104\pi$
$181$	−14.4963	−1.07751	−0.538753	+	0.842464i	$0.681104\pi$
$182$	9.16514	0.679366
$183$	8.17744	0.604494
$184$	−3.00369	−0.221435
$185$	−29.8373	−2.19368
$186$	6.94926	0.509545
$187$	0	0
$188$	12.5280	0.913699
$189$	18.9040	1.37506
$190$	22.2656	1.61531
$191$	5.21506	0.377348	0.188674	−	0.982040i	$-0.439581\pi$
$191$	5.21506	0.377348	0.188674	+	0.982040i	$0.439581\pi$
$192$	1.04360	0.0753153
$193$	−15.1516	−1.09063	−0.545316	−	0.838230i	$-0.683590\pi$
$193$	−15.1516	−1.09063	−0.545316	+	0.838230i	$0.683590\pi$
$194$	9.82512	0.705403
$195$	8.54486	0.611910
$196$	6.60562	0.471830
$197$	6.17108	0.439671	0.219835	−	0.975537i	$-0.429448\pi$
$197$	6.17108	0.439671	0.219835	+	0.975537i	$0.429448\pi$
$198$	0	0
$199$	−3.07165	−0.217743	−0.108872	−	0.994056i	$-0.534724\pi$
$199$	−3.07165	−0.217743	−0.108872	+	0.994056i	$0.534724\pi$
$200$	−5.85881	−0.414280
$201$	6.74586	0.475816
$202$	−14.1963	−0.998846
$203$	1.20789	0.0847771
$204$	1.04360	0.0730666
$205$	18.9785	1.32551
$206$	15.4968	1.07971
$207$	−5.73975	−0.398940
$208$	2.48474	0.172285
$209$	0	0
$210$	12.6848	0.875335
$211$	20.7123	1.42590	0.712948	−	0.701217i	$-0.247360\pi$
$211$	20.7123	1.42590	0.712948	+	0.701217i	$0.247360\pi$
$212$	1.61685	0.111046
$213$	−11.6400	−0.797558
$214$	0.167926	0.0114792
$215$	−29.4970	−2.01168
$216$	5.12501	0.348713
$217$	24.5620	1.66738
$218$	3.91929	0.265447
$219$	−15.9113	−1.07518
$220$	0	0
$221$	2.48474	0.167141
$222$	9.44935	0.634199
$223$	3.03788	0.203432	0.101716	−	0.994813i	$-0.467567\pi$
$223$	3.03788	0.203432	0.101716	+	0.994813i	$0.467567\pi$
$224$	3.68858	0.246454
$225$	−11.1956	−0.746373
$226$	−1.57878	−0.105019
$227$	−4.57303	−0.303522	−0.151761	−	0.988417i	$-0.548494\pi$
$227$	−4.57303	−0.303522	−0.151761	+	0.988417i	$0.548494\pi$
$228$	−7.05142	−0.466992
$229$	−19.8539	−1.31198	−0.655992	−	0.754768i	$-0.727749\pi$
$229$	−19.8539	−1.31198	−0.655992	+	0.754768i	$0.727749\pi$
$230$	−9.89796	−0.652652
$231$	0	0
$232$	0.327467	0.0214993
$233$	5.40261	0.353937	0.176969	−	0.984217i	$-0.443371\pi$
$233$	5.40261	0.353937	0.176969	+	0.984217i	$0.443371\pi$
$234$	4.74808	0.310392
$235$	41.2832	2.69302
$236$	−1.99129	−0.129622
$237$	−6.18942	−0.402046
$238$	3.68858	0.239095
$239$	−26.0696	−1.68630	−0.843152	−	0.537676i	$-0.819302\pi$
$239$	−26.0696	−1.68630	−0.843152	+	0.537676i	$0.819302\pi$
$240$	3.43894	0.221983
$241$	19.0942	1.22996	0.614982	−	0.788541i	$-0.289163\pi$
$241$	19.0942	1.22996	0.614982	+	0.788541i	$0.289163\pi$
$242$	0	0
$243$	15.7760	1.01203
$244$	7.83580	0.501636
$245$	21.7673	1.39066
$246$	−6.01040	−0.383209
$247$	−16.7889	−1.06825
$248$	6.65894	0.422843
$249$	−12.9580	−0.821181
$250$	−2.82999	−0.178985
$251$	21.5238	1.35857	0.679284	−	0.733876i	$-0.262291\pi$
$251$	21.5238	1.35857	0.679284	+	0.733876i	$0.262291\pi$
$252$	7.04851	0.444014
$253$	0	0
$254$	6.89631	0.432713
$255$	3.43894	0.215355
$256$	1.00000	0.0625000
$257$	19.6228	1.22404	0.612018	−	0.790843i	$-0.290358\pi$
$257$	19.6228	1.22404	0.612018	+	0.790843i	$0.290358\pi$
$258$	9.34159	0.581582
$259$	33.3985	2.07528
$260$	8.18788	0.507790
$261$	0.625757	0.0387334
$262$	3.08556	0.190627
$263$	18.0824	1.11501	0.557504	−	0.830174i	$-0.311759\pi$
$263$	18.0824	1.11501	0.557504	+	0.830174i	$0.311759\pi$
$264$	0	0
$265$	5.32797	0.327295
$266$	−24.9231	−1.52813
$267$	−11.4839	−0.702805
$268$	6.46403	0.394854
$269$	−5.63740	−0.343719	−0.171859	−	0.985122i	$-0.554977\pi$
$269$	−5.63740	−0.343719	−0.171859	+	0.985122i	$0.554977\pi$
$270$	16.8883	1.02779
$271$	2.06414	0.125388	0.0626939	−	0.998033i	$-0.480031\pi$
$271$	2.06414	0.125388	0.0626939	+	0.998033i	$0.480031\pi$
$272$	1.00000	0.0606339
$273$	−9.56474	−0.578884
$274$	−4.19193	−0.253244
$275$	0	0
$276$	3.13465	0.188684
$277$	−28.7168	−1.72542	−0.862711	−	0.505697i	$-0.831235\pi$
$277$	−28.7168	−1.72542	−0.862711	+	0.505697i	$0.831235\pi$
$278$	1.77810	0.106643
$279$	12.7246	0.761800
$280$	12.1549	0.726392
$281$	−24.0286	−1.43342	−0.716712	−	0.697369i	$-0.754354\pi$
$281$	−24.0286	−1.43342	−0.716712	+	0.697369i	$0.754354\pi$
$282$	−13.0742	−0.778559
$283$	−27.6478	−1.64349	−0.821746	−	0.569854i	$-0.807000\pi$
$283$	−27.6478	−1.64349	−0.821746	+	0.569854i	$0.807000\pi$
$284$	−11.1537	−0.661849
$285$	−23.2363	−1.37640
$286$	0	0
$287$	−21.2436	−1.25397
$288$	1.91090	0.112601
$289$	1.00000	0.0588235
$290$	1.07909	0.0633665
$291$	−10.2535	−0.601070
$292$	−15.2465	−0.892235
$293$	15.1863	0.887195	0.443597	−	0.896226i	$-0.353702\pi$
$293$	15.1863	0.887195	0.443597	+	0.896226i	$0.353702\pi$
$294$	−6.89362	−0.402044
$295$	−6.56184	−0.382045
$296$	9.05457	0.526286
$297$	0	0
$298$	−5.10075	−0.295479
$299$	7.46337	0.431618
$300$	6.11425	0.353006
$301$	33.0176	1.90310
$302$	−13.0795	−0.752640
$303$	14.8152	0.851112
$304$	−6.75683	−0.387530
$305$	25.8211	1.47851
$306$	1.91090	0.109239
$307$	−22.4154	−1.27931	−0.639657	−	0.768660i	$-0.720924\pi$
$307$	−22.4154	−1.27931	−0.639657	+	0.768660i	$0.720924\pi$
$308$	0	0
$309$	−16.1724	−0.920018
$310$	21.9430	1.24628
$311$	2.30637	0.130782	0.0653912	−	0.997860i	$-0.479170\pi$
$311$	2.30637	0.130782	0.0653912	+	0.997860i	$0.479170\pi$
$312$	−2.59307	−0.146804
$313$	0.694135	0.0392348	0.0196174	−	0.999808i	$-0.493755\pi$
$313$	0.694135	0.0392348	0.0196174	+	0.999808i	$0.493755\pi$
$314$	−0.469645	−0.0265036
$315$	23.2267	1.30868
$316$	−5.93084	−0.333636
$317$	31.7598	1.78381	0.891904	−	0.452225i	$-0.149370\pi$
$317$	31.7598	1.78381	0.891904	+	0.452225i	$0.149370\pi$
$318$	−1.68735	−0.0946218
$319$	0	0
$320$	3.29527	0.184211
$321$	−0.175248	−0.00978138
$322$	11.0793	0.617428
$323$	−6.75683	−0.375960
$324$	0.384242	0.0213468
$325$	14.5576	0.807509
$326$	20.9869	1.16236
$327$	−4.09016	−0.226187
$328$	−5.75930	−0.318004
$329$	−46.2106	−2.54767
$330$	0	0
$331$	−8.38505	−0.460884	−0.230442	−	0.973086i	$-0.574017\pi$
$331$	−8.38505	−0.460884	−0.230442	+	0.973086i	$0.574017\pi$
$332$	−12.4167	−0.681452
$333$	17.3024	0.948165
$334$	−9.07097	−0.496342
$335$	21.3007	1.16378
$336$	−3.84940	−0.210002
$337$	11.5064	0.626792	0.313396	−	0.949623i	$-0.398533\pi$
$337$	11.5064	0.626792	0.313396	+	0.949623i	$0.398533\pi$
$338$	6.82609	0.371290
$339$	1.64761	0.0894861
$340$	3.29527	0.178711
$341$	0	0
$342$	−12.9116	−0.698180
$343$	1.45471	0.0785472
$344$	8.95131	0.482623
$345$	10.3295	0.556122
$346$	−11.9688	−0.643445
$347$	14.3643	0.771118	0.385559	−	0.922683i	$-0.374009\pi$
$347$	14.3643	0.771118	0.385559	+	0.922683i	$0.374009\pi$
$348$	−0.341744	−0.0183194
$349$	18.2852	0.978784	0.489392	−	0.872064i	$-0.337219\pi$
$349$	18.2852	0.978784	0.489392	+	0.872064i	$0.337219\pi$
$350$	21.6107	1.15514
$351$	−12.7343	−0.679707
$352$	0	0
$353$	−35.3333	−1.88060	−0.940300	−	0.340348i	$-0.889455\pi$
$353$	−35.3333	−1.88060	−0.940300	+	0.340348i	$0.889455\pi$
$354$	2.07811	0.110450
$355$	−36.7544	−1.95072
$356$	−11.0042	−0.583219
$357$	−3.84940	−0.203732
$358$	−1.13869	−0.0601815
$359$	−7.41124	−0.391150	−0.195575	−	0.980689i	$-0.562657\pi$
$359$	−7.41124	−0.391150	−0.195575	+	0.980689i	$0.562657\pi$
$360$	6.29693	0.331878
$361$	26.6547	1.40288
$362$	14.4963	0.761911
$363$	0	0
$364$	−9.16514	−0.480384
$365$	−50.2414	−2.62976
$366$	−8.17744	−0.427442
$367$	−29.5311	−1.54151	−0.770755	−	0.637132i	$-0.780121\pi$
$367$	−29.5311	−1.54151	−0.770755	+	0.637132i	$0.780121\pi$
$368$	3.00369	0.156578
$369$	−11.0055	−0.572921
$370$	29.8373	1.55117
$371$	−5.96389	−0.309630
$372$	−6.94926	−0.360302
$373$	−4.61889	−0.239157	−0.119578	−	0.992825i	$-0.538154\pi$
$373$	−4.61889	−0.239157	−0.119578	+	0.992825i	$0.538154\pi$
$374$	0	0
$375$	2.95338	0.152512
$376$	−12.5280	−0.646083
$377$	−0.813669	−0.0419061
$378$	−18.9040	−0.972318
$379$	16.1373	0.828915	0.414458	−	0.910069i	$-0.363971\pi$
$379$	16.1373	0.828915	0.414458	+	0.910069i	$0.363971\pi$
$380$	−22.2656	−1.14220
$381$	−7.19699	−0.368713
$382$	−5.21506	−0.266826
$383$	29.2797	1.49612	0.748061	−	0.663629i	$-0.230985\pi$
$383$	29.2797	1.49612	0.748061	+	0.663629i	$0.230985\pi$
$384$	−1.04360	−0.0532560
$385$	0	0
$386$	15.1516	0.771194
$387$	17.1051	0.869500
$388$	−9.82512	−0.498795
$389$	−7.99115	−0.405167	−0.202584	−	0.979265i	$-0.564934\pi$
$389$	−7.99115	−0.405167	−0.202584	+	0.979265i	$0.564934\pi$
$390$	−8.54486	−0.432686
$391$	3.00369	0.151903
$392$	−6.60562	−0.333634
$393$	−3.22009	−0.162432
$394$	−6.17108	−0.310894
$395$	−19.5437	−0.983351
$396$	0	0
$397$	31.6253	1.58723	0.793614	−	0.608422i	$-0.208197\pi$
$397$	31.6253	1.58723	0.793614	+	0.608422i	$0.208197\pi$
$398$	3.07165	0.153968
$399$	26.0097	1.30211
$400$	5.85881	0.292940
$401$	10.6813	0.533398	0.266699	−	0.963780i	$-0.414067\pi$
$401$	10.6813	0.533398	0.266699	+	0.963780i	$0.414067\pi$
$402$	−6.74586	−0.336453
$403$	−16.5457	−0.824200
$404$	14.1963	0.706291
$405$	1.26618	0.0629170
$406$	−1.20789	−0.0599465
$407$	0	0
$408$	−1.04360	−0.0516659
$409$	−0.418758	−0.0207062	−0.0103531	−	0.999946i	$-0.503296\pi$
$409$	−0.418758	−0.0207062	−0.0103531	+	0.999946i	$0.503296\pi$
$410$	−18.9785	−0.937279
$411$	4.37470	0.215788
$412$	−15.4968	−0.763472
$413$	7.34503	0.361425
$414$	5.73975	0.282093
$415$	−40.9162	−2.00850
$416$	−2.48474	−0.121824
$417$	−1.85562	−0.0908701
$418$	0	0
$419$	−9.82979	−0.480217	−0.240108	−	0.970746i	$-0.577183\pi$
$419$	−9.82979	−0.480217	−0.240108	+	0.970746i	$0.577183\pi$
$420$	−12.6848	−0.618955
$421$	14.0440	0.684465	0.342233	−	0.939615i	$-0.388817\pi$
$421$	14.0440	0.684465	0.342233	+	0.939615i	$0.388817\pi$
$422$	−20.7123	−1.00826
$423$	−23.9398	−1.16399
$424$	−1.61685	−0.0785214
$425$	5.85881	0.284194
$426$	11.6400	0.563959
$427$	−28.9030	−1.39871
$428$	−0.167926	−0.00811703
$429$	0	0
$430$	29.4970	1.42247
$431$	−13.3100	−0.641121	−0.320561	−	0.947228i	$-0.603871\pi$
$431$	−13.3100	−0.641121	−0.320561	+	0.947228i	$0.603871\pi$
$432$	−5.12501	−0.246577
$433$	22.7492	1.09326	0.546628	−	0.837376i	$-0.315911\pi$
$433$	22.7492	1.09326	0.546628	+	0.837376i	$0.315911\pi$
$434$	−24.5620	−1.17901
$435$	−1.12614	−0.0539943
$436$	−3.91929	−0.187700
$437$	−20.2954	−0.970861
$438$	15.9113	0.760269
$439$	−11.0402	−0.526922	−0.263461	−	0.964670i	$-0.584864\pi$
$439$	−11.0402	−0.526922	−0.263461	+	0.964670i	$0.584864\pi$
$440$	0	0
$441$	−12.6227	−0.601080
$442$	−2.48474	−0.118187
$443$	−21.4275	−1.01805	−0.509026	−	0.860751i	$-0.669994\pi$
$443$	−21.4275	−1.01805	−0.509026	+	0.860751i	$0.669994\pi$
$444$	−9.44935	−0.448446
$445$	−36.2617	−1.71897
$446$	−3.03788	−0.143848
$447$	5.32314	0.251776
$448$	−3.68858	−0.174269
$449$	−30.9179	−1.45910	−0.729552	−	0.683925i	$-0.760271\pi$
$449$	−30.9179	−1.45910	−0.729552	+	0.683925i	$0.760271\pi$
$450$	11.1956	0.527765
$451$	0	0
$452$	1.57878	0.0742595
$453$	13.6497	0.641321
$454$	4.57303	0.214623
$455$	−30.2016	−1.41587
$456$	7.05142	0.330213
$457$	6.97463	0.326259	0.163130	−	0.986605i	$-0.447841\pi$
$457$	6.97463	0.326259	0.163130	+	0.986605i	$0.447841\pi$
$458$	19.8539	0.927712
$459$	−5.12501	−0.239215
$460$	9.89796	0.461495
$461$	6.53587	0.304406	0.152203	−	0.988349i	$-0.451363\pi$
$461$	6.53587	0.304406	0.152203	+	0.988349i	$0.451363\pi$
$462$	0	0
$463$	27.3097	1.26919	0.634595	−	0.772845i	$-0.281167\pi$
$463$	27.3097	1.26919	0.634595	+	0.772845i	$0.281167\pi$
$464$	−0.327467	−0.0152023
$465$	−22.8997	−1.06195
$466$	−5.40261	−0.250271
$467$	−3.84984	−0.178149	−0.0890746	−	0.996025i	$-0.528391\pi$
$467$	−3.84984	−0.178149	−0.0890746	+	0.996025i	$0.528391\pi$
$468$	−4.74808	−0.219480
$469$	−23.8431	−1.10097
$470$	−41.2832	−1.90425
$471$	0.490121	0.0225836
$472$	1.99129	0.0916565
$473$	0	0
$474$	6.18942	0.284289
$475$	−39.5869	−1.81637
$476$	−3.68858	−0.169066
$477$	−3.08965	−0.141465
$478$	26.0696	1.19240
$479$	19.7747	0.903530	0.451765	−	0.892137i	$-0.350795\pi$
$479$	19.7747	0.903530	0.451765	+	0.892137i	$0.350795\pi$
$480$	−3.43894	−0.156965
$481$	−22.4982	−1.02583
$482$	−19.0942	−0.869716
$483$	−11.5624	−0.526107
$484$	0	0
$485$	−32.3764	−1.47014
$486$	−15.7760	−0.715615
$487$	−30.9964	−1.40458	−0.702290	−	0.711891i	$-0.747839\pi$
$487$	−30.9964	−1.40458	−0.702290	+	0.711891i	$0.747839\pi$
$488$	−7.83580	−0.354710
$489$	−21.9019	−0.990439
$490$	−21.7673	−0.983346
$491$	−17.9495	−0.810049	−0.405024	−	0.914306i	$-0.632737\pi$
$491$	−17.9495	−0.810049	−0.405024	+	0.914306i	$0.632737\pi$
$492$	6.01040	0.270970
$493$	−0.327467	−0.0147484
$494$	16.7889	0.755369
$495$	0	0
$496$	−6.65894	−0.298995
$497$	41.1413	1.84544
$498$	12.9580	0.580662
$499$	−8.52099	−0.381452	−0.190726	−	0.981643i	$-0.561084\pi$
$499$	−8.52099	−0.381452	−0.190726	+	0.981643i	$0.561084\pi$
$500$	2.82999	0.126561
$501$	9.46646	0.422930
$502$	−21.5238	−0.960652
$503$	3.27625	0.146081	0.0730403	−	0.997329i	$-0.476730\pi$
$503$	3.27625	0.146081	0.0730403	+	0.997329i	$0.476730\pi$
$504$	−7.04851	−0.313965
$505$	46.7805	2.08171
$506$	0	0
$507$	−7.12370	−0.316375
$508$	−6.89631	−0.305974
$509$	−10.4178	−0.461759	−0.230879	−	0.972982i	$-0.574160\pi$
$509$	−10.4178	−0.461759	−0.230879	+	0.972982i	$0.574160\pi$
$510$	−3.43894	−0.152279
$511$	56.2380	2.48782
$512$	−1.00000	−0.0441942
$513$	34.6288	1.52890
$514$	−19.6228	−0.865525
$515$	−51.0661	−2.25024
$516$	−9.34159	−0.411240
$517$	0	0
$518$	−33.3985	−1.46745
$519$	12.4906	0.548276
$520$	−8.18788	−0.359062
$521$	−2.49042	−0.109107	−0.0545537	−	0.998511i	$-0.517374\pi$
$521$	−2.49042	−0.109107	−0.0545537	+	0.998511i	$0.517374\pi$
$522$	−0.625757	−0.0273886
$523$	−32.0837	−1.40292	−0.701461	−	0.712708i	$-0.747469\pi$
$523$	−32.0837	−1.40292	−0.701461	+	0.712708i	$0.747469\pi$
$524$	−3.08556	−0.134793
$525$	−22.5529	−0.984288
$526$	−18.0824	−0.788430
$527$	−6.65894	−0.290068
$528$	0	0
$529$	−13.9779	−0.607733
$530$	−5.32797	−0.231432
$531$	3.80516	0.165130
$532$	24.9231	1.08055
$533$	14.3103	0.619850
$534$	11.4839	0.496958
$535$	−0.553363	−0.0239240
$536$	−6.46403	−0.279204
$537$	1.18833	0.0512803
$538$	5.63740	0.243046
$539$	0	0
$540$	−16.8883	−0.726757
$541$	−10.0017	−0.430007	−0.215004	−	0.976613i	$-0.568976\pi$
$541$	−10.0017	−0.430007	−0.215004	+	0.976613i	$0.568976\pi$
$542$	−2.06414	−0.0886625
$543$	−15.1284	−0.649221
$544$	−1.00000	−0.0428746
$545$	−12.9151	−0.553222
$546$	9.56474	0.409333
$547$	30.7063	1.31291	0.656453	−	0.754367i	$-0.272056\pi$
$547$	30.7063	1.31291	0.656453	+	0.754367i	$0.272056\pi$
$548$	4.19193	0.179071
$549$	−14.9734	−0.639051
$550$	0	0
$551$	2.21264	0.0942615
$552$	−3.13465	−0.133419
$553$	21.8764	0.930277
$554$	28.7168	1.22006
$555$	−31.1381	−1.32174
$556$	−1.77810	−0.0754081
$557$	−12.7730	−0.541210	−0.270605	−	0.962690i	$-0.587224\pi$
$557$	−12.7730	−0.541210	−0.270605	+	0.962690i	$0.587224\pi$
$558$	−12.7246	−0.538674
$559$	−22.2416	−0.940722
$560$	−12.1549	−0.513637
$561$	0	0
$562$	24.0286	1.01358
$563$	44.1499	1.86070	0.930348	−	0.366679i	$-0.119505\pi$
$563$	44.1499	1.86070	0.930348	+	0.366679i	$0.119505\pi$
$564$	13.0742	0.550524
$565$	5.20251	0.218871
$566$	27.6478	1.16212
$567$	−1.41731	−0.0595212
$568$	11.1537	0.467998
$569$	31.5442	1.32240	0.661201	−	0.750209i	$-0.270047\pi$
$569$	31.5442	1.32240	0.661201	+	0.750209i	$0.270047\pi$
$570$	23.2363	0.973263
$571$	18.5934	0.778110	0.389055	−	0.921214i	$-0.372802\pi$
$571$	18.5934	0.778110	0.389055	+	0.921214i	$0.372802\pi$
$572$	0	0
$573$	5.44243	0.227361
$574$	21.2436	0.886692
$575$	17.5980	0.733888
$576$	−1.91090	−0.0796208
$577$	46.1733	1.92222	0.961110	−	0.276166i	$-0.0890639\pi$
$577$	46.1733	1.92222	0.961110	+	0.276166i	$0.0890639\pi$
$578$	−1.00000	−0.0415945
$579$	−15.8121	−0.657130
$580$	−1.07909	−0.0448069
$581$	45.7998	1.90010
$582$	10.2535	0.425021
$583$	0	0
$584$	15.2465	0.630906
$585$	−15.6462	−0.646891
$586$	−15.1863	−0.627341
$587$	−11.8915	−0.490817	−0.245408	−	0.969420i	$-0.578922\pi$
$587$	−11.8915	−0.490817	−0.245408	+	0.969420i	$0.578922\pi$
$588$	6.89362	0.284288
$589$	44.9933	1.85392
$590$	6.56184	0.270147
$591$	6.44013	0.264912
$592$	−9.05457	−0.372141
$593$	−36.8708	−1.51410	−0.757051	−	0.653356i	$-0.773360\pi$
$593$	−36.8708	−1.51410	−0.757051	+	0.653356i	$0.773360\pi$
$594$	0	0
$595$	−12.1549	−0.498301
$596$	5.10075	0.208935
$597$	−3.20557	−0.131195
$598$	−7.46337	−0.305200
$599$	0.0907547	0.00370814	0.00185407	−	0.999998i	$-0.499410\pi$
$599$	0.0907547	0.00370814	0.00185407	+	0.999998i	$0.499410\pi$
$600$	−6.11425	−0.249613
$601$	−7.59600	−0.309847	−0.154924	−	0.987926i	$-0.549513\pi$
$601$	−7.59600	−0.309847	−0.154924	+	0.987926i	$0.549513\pi$
$602$	−33.0176	−1.34570
$603$	−12.3521	−0.503017
$604$	13.0795	0.532197
$605$	0	0
$606$	−14.8152	−0.601827
$607$	21.9436	0.890661	0.445331	−	0.895366i	$-0.353086\pi$
$607$	21.9436	0.890661	0.445331	+	0.895366i	$0.353086\pi$
$608$	6.75683	0.274025
$609$	1.26055	0.0510801
$610$	−25.8211	−1.04547
$611$	31.1288	1.25934
$612$	−1.91090	−0.0772436
$613$	−2.52090	−0.101818	−0.0509092	−	0.998703i	$-0.516212\pi$
$613$	−2.52090	−0.101818	−0.0509092	+	0.998703i	$0.516212\pi$
$614$	22.4154	0.904612
$615$	19.8059	0.798651
$616$	0	0
$617$	29.2656	1.17819	0.589095	−	0.808064i	$-0.299484\pi$
$617$	29.2656	1.17819	0.589095	+	0.808064i	$0.299484\pi$
$618$	16.1724	0.650551
$619$	−40.3346	−1.62118	−0.810592	−	0.585611i	$-0.800855\pi$
$619$	−40.3346	−1.62118	−0.810592	+	0.585611i	$0.800855\pi$
$620$	−21.9430	−0.881252
$621$	−15.3939	−0.617738
$622$	−2.30637	−0.0924771
$623$	40.5897	1.62619
$624$	2.59307	0.103806
$625$	−19.9684	−0.798737
$626$	−0.694135	−0.0277432
$627$	0	0
$628$	0.469645	0.0187409
$629$	−9.05457	−0.361029
$630$	−23.2267	−0.925375
$631$	10.9897	0.437495	0.218747	−	0.975782i	$-0.429803\pi$
$631$	10.9897	0.437495	0.218747	+	0.975782i	$0.429803\pi$
$632$	5.93084	0.235916
$633$	21.6154	0.859134
$634$	−31.7598	−1.26134
$635$	−22.7252	−0.901822
$636$	1.68735	0.0669077
$637$	16.4132	0.650315
$638$	0	0
$639$	21.3136	0.843152
$640$	−3.29527	−0.130257
$641$	−7.96289	−0.314515	−0.157258	−	0.987558i	$-0.550265\pi$
$641$	−7.96289	−0.314515	−0.157258	+	0.987558i	$0.550265\pi$
$642$	0.175248	0.00691648
$643$	27.2178	1.07336	0.536682	−	0.843784i	$-0.319677\pi$
$643$	27.2178	1.07336	0.536682	+	0.843784i	$0.319677\pi$
$644$	−11.0793	−0.436587
$645$	−30.7830	−1.21208
$646$	6.75683	0.265844
$647$	20.0212	0.787114	0.393557	−	0.919300i	$-0.371244\pi$
$647$	20.0212	0.787114	0.393557	+	0.919300i	$0.371244\pi$
$648$	−0.384242	−0.0150944
$649$	0	0
$650$	−14.5576	−0.570995
$651$	25.6329	1.00463
$652$	−20.9869	−0.821911
$653$	7.02316	0.274837	0.137419	−	0.990513i	$-0.456119\pi$
$653$	7.02316	0.274837	0.137419	+	0.990513i	$0.456119\pi$
$654$	4.09016	0.159938
$655$	−10.1678	−0.397287
$656$	5.75930	0.224863
$657$	29.1346	1.13665
$658$	46.2106	1.80148
$659$	9.99934	0.389519	0.194760	−	0.980851i	$-0.437607\pi$
$659$	9.99934	0.389519	0.194760	+	0.980851i	$0.437607\pi$
$660$	0	0
$661$	3.43391	0.133564	0.0667818	−	0.997768i	$-0.478727\pi$
$661$	3.43391	0.133564	0.0667818	+	0.997768i	$0.478727\pi$
$662$	8.38505	0.325894
$663$	2.59307	0.100706
$664$	12.4167	0.481860
$665$	82.1283	3.18480
$666$	−17.3024	−0.670454
$667$	−0.983609	−0.0380855
$668$	9.07097	0.350967
$669$	3.17033	0.122572
$670$	−21.3007	−0.822919
$671$	0	0
$672$	3.84940	0.148494
$673$	27.3649	1.05484	0.527419	−	0.849605i	$-0.323160\pi$
$673$	27.3649	1.05484	0.527419	+	0.849605i	$0.323160\pi$
$674$	−11.5064	−0.443209
$675$	−30.0264	−1.15572
$676$	−6.82609	−0.262542
$677$	21.5308	0.827496	0.413748	−	0.910392i	$-0.364220\pi$
$677$	21.5308	0.827496	0.413748	+	0.910392i	$0.364220\pi$
$678$	−1.64761	−0.0632762
$679$	36.2407	1.39079
$680$	−3.29527	−0.126368
$681$	−4.77241	−0.182879
$682$	0	0
$683$	−30.3867	−1.16271	−0.581357	−	0.813649i	$-0.697478\pi$
$683$	−30.3867	−1.16271	−0.581357	+	0.813649i	$0.697478\pi$
$684$	12.9116	0.493688
$685$	13.8136	0.527789
$686$	−1.45471	−0.0555412
$687$	−20.7195	−0.790499
$688$	−8.95131	−0.341266
$689$	4.01746	0.153053
$690$	−10.3295	−0.393238
$691$	−12.7255	−0.484102	−0.242051	−	0.970263i	$-0.577820\pi$
$691$	−12.7255	−0.484102	−0.242051	+	0.970263i	$0.577820\pi$
$692$	11.9688	0.454984
$693$	0	0
$694$	−14.3643	−0.545263
$695$	−5.85931	−0.222256
$696$	0.341744	0.0129538
$697$	5.75930	0.218149
$698$	−18.2852	−0.692104
$699$	5.63816	0.213255
$700$	−21.6107	−0.816806
$701$	−37.6729	−1.42289	−0.711443	−	0.702744i	$-0.751958\pi$
$701$	−37.6729	−1.42289	−0.711443	+	0.702744i	$0.751958\pi$
$702$	12.7343	0.480625
$703$	61.1802	2.30745
$704$	0	0
$705$	43.0831	1.62260
$706$	35.3333	1.32978
$707$	−52.3641	−1.96935
$708$	−2.07811	−0.0781001
$709$	−21.0709	−0.791335	−0.395668	−	0.918394i	$-0.629487\pi$
$709$	−21.0709	−0.791335	−0.395668	+	0.918394i	$0.629487\pi$
$710$	36.7544	1.37937
$711$	11.3332	0.425030
$712$	11.0042	0.412398
$713$	−20.0014	−0.749057
$714$	3.84940	0.144060
$715$	0	0
$716$	1.13869	0.0425547
$717$	−27.2062	−1.01604
$718$	7.41124	0.276585
$719$	−2.89831	−0.108089	−0.0540444	−	0.998539i	$-0.517211\pi$
$719$	−2.89831	−0.108089	−0.0540444	+	0.998539i	$0.517211\pi$
$720$	−6.29693	−0.234673
$721$	57.1611	2.12879
$722$	−26.6547	−0.991985
$723$	19.9267	0.741081
$724$	−14.4963	−0.538753
$725$	−1.91857	−0.0712537
$726$	0	0
$727$	45.2146	1.67692	0.838459	−	0.544965i	$-0.183457\pi$
$727$	45.2146	1.67692	0.838459	+	0.544965i	$0.183457\pi$
$728$	9.16514	0.339683
$729$	15.3111	0.567079
$730$	50.2414	1.85952
$731$	−8.95131	−0.331076
$732$	8.17744	0.302247
$733$	25.8788	0.955855	0.477927	−	0.878399i	$-0.341388\pi$
$733$	25.8788	0.955855	0.477927	+	0.878399i	$0.341388\pi$
$734$	29.5311	1.09001
$735$	22.7163	0.837904
$736$	−3.00369	−0.110717
$737$	0	0
$738$	11.0055	0.405116
$739$	−22.7867	−0.838224	−0.419112	−	0.907935i	$-0.637658\pi$
$739$	−22.7867	−0.838224	−0.419112	+	0.907935i	$0.637658\pi$
$740$	−29.8373	−1.09684
$741$	−17.5209	−0.643647
$742$	5.96389	0.218941
$743$	−16.3326	−0.599185	−0.299593	−	0.954067i	$-0.596851\pi$
$743$	−16.3326	−0.599185	−0.299593	+	0.954067i	$0.596851\pi$
$744$	6.94926	0.254772
$745$	16.8084	0.615810
$746$	4.61889	0.169110
$747$	23.7270	0.868125
$748$	0	0
$749$	0.619410	0.0226327
$750$	−2.95338	−0.107842
$751$	−22.0624	−0.805068	−0.402534	−	0.915405i	$-0.631871\pi$
$751$	−22.0624	−0.805068	−0.402534	+	0.915405i	$0.631871\pi$
$752$	12.5280	0.456850
$753$	22.4622	0.818567
$754$	0.813669	0.0296321
$755$	43.1004	1.56859
$756$	18.9040	0.687532
$757$	−3.52255	−0.128029	−0.0640146	−	0.997949i	$-0.520390\pi$
$757$	−3.52255	−0.128029	−0.0640146	+	0.997949i	$0.520390\pi$
$758$	−16.1373	−0.586131
$759$	0	0
$760$	22.2656	0.807657
$761$	−32.0828	−1.16300	−0.581501	−	0.813546i	$-0.697534\pi$
$761$	−32.0828	−1.16300	−0.581501	+	0.813546i	$0.697534\pi$
$762$	7.19699	0.260719
$763$	14.4566	0.523364
$764$	5.21506	0.188674
$765$	−6.29693	−0.227666
$766$	−29.2797	−1.05792
$767$	−4.94783	−0.178656
$768$	1.04360	0.0376576
$769$	17.8831	0.644882	0.322441	−	0.946590i	$-0.395497\pi$
$769$	17.8831	0.644882	0.322441	+	0.946590i	$0.395497\pi$
$770$	0	0
$771$	20.4783	0.737510
$772$	−15.1516	−0.545316
$773$	−25.8259	−0.928892	−0.464446	−	0.885601i	$-0.653746\pi$
$773$	−25.8259	−0.928892	−0.464446	+	0.885601i	$0.653746\pi$
$774$	−17.1051	−0.614829
$775$	−39.0134	−1.40140
$776$	9.82512	0.352701
$777$	34.8547	1.25040
$778$	7.99115	0.286497
$779$	−38.9146	−1.39426
$780$	8.54486	0.305955
$781$	0	0
$782$	−3.00369	−0.107412
$783$	1.67827	0.0599766
$784$	6.60562	0.235915
$785$	1.54761	0.0552364
$786$	3.22009	0.114857
$787$	37.2946	1.32941	0.664705	−	0.747106i	$-0.268557\pi$
$787$	37.2946	1.32941	0.664705	+	0.747106i	$0.268557\pi$
$788$	6.17108	0.219835
$789$	18.8708	0.671818
$790$	19.5437	0.695334
$791$	−5.82345	−0.207058
$792$	0	0
$793$	19.4699	0.691396
$794$	−31.6253	−1.12234
$795$	5.56027	0.197202
$796$	−3.07165	−0.108872
$797$	−16.2563	−0.575829	−0.287915	−	0.957656i	$-0.592962\pi$
$797$	−16.2563	−0.575829	−0.287915	+	0.957656i	$0.592962\pi$
$798$	−26.0097	−0.920734
$799$	12.5280	0.443209
$800$	−5.85881	−0.207140
$801$	21.0279	0.742983
$802$	−10.6813	−0.377169
$803$	0	0
$804$	6.74586	0.237908
$805$	−36.5094	−1.28679
$806$	16.5457	0.582797
$807$	−5.88319	−0.207098
$808$	−14.1963	−0.499423
$809$	−22.6417	−0.796039	−0.398019	−	0.917377i	$-0.630302\pi$
$809$	−22.6417	−0.796039	−0.398019	+	0.917377i	$0.630302\pi$
$810$	−1.26618	−0.0444890
$811$	−13.4769	−0.473237	−0.236618	−	0.971603i	$-0.576039\pi$
$811$	−13.4769	−0.473237	−0.236618	+	0.971603i	$0.576039\pi$
$812$	1.20789	0.0423886
$813$	2.15414	0.0755489
$814$	0	0
$815$	−69.1575	−2.42248
$816$	1.04360	0.0365333
$817$	60.4825	2.11601
$818$	0.418758	0.0146415
$819$	17.5137	0.611977
$820$	18.9785	0.662756
$821$	−32.2309	−1.12487	−0.562433	−	0.826843i	$-0.690134\pi$
$821$	−32.2309	−1.12487	−0.562433	+	0.826843i	$0.690134\pi$
$822$	−4.37470	−0.152585
$823$	−5.99896	−0.209110	−0.104555	−	0.994519i	$-0.533342\pi$
$823$	−5.99896	−0.209110	−0.104555	+	0.994519i	$0.533342\pi$
$824$	15.4968	0.539856
$825$	0	0
$826$	−7.34503	−0.255566
$827$	16.1017	0.559912	0.279956	−	0.960013i	$-0.409680\pi$
$827$	16.1017	0.559912	0.279956	+	0.960013i	$0.409680\pi$
$828$	−5.73975	−0.199470
$829$	−30.7355	−1.06749	−0.533744	−	0.845646i	$-0.679215\pi$
$829$	−30.7355	−1.06749	−0.533744	+	0.845646i	$0.679215\pi$
$830$	40.9162	1.42022
$831$	−29.9688	−1.03961
$832$	2.48474	0.0861427
$833$	6.60562	0.228871
$834$	1.85562	0.0642549
$835$	29.8913	1.03443
$836$	0	0
$837$	34.1271	1.17961
$838$	9.82979	0.339565
$839$	18.4478	0.636890	0.318445	−	0.947941i	$-0.396839\pi$
$839$	18.4478	0.636890	0.318445	+	0.947941i	$0.396839\pi$
$840$	12.6848	0.437667
$841$	−28.8928	−0.996302
$842$	−14.0440	−0.483990
$843$	−25.0762	−0.863670
$844$	20.7123	0.712948
$845$	−22.4938	−0.773810
$846$	23.9398	0.823067
$847$	0	0
$848$	1.61685	0.0555230
$849$	−28.8532	−0.990241
$850$	−5.85881	−0.200955
$851$	−27.1971	−0.932305
$852$	−11.6400	−0.398779
$853$	−37.4464	−1.28214	−0.641071	−	0.767482i	$-0.721509\pi$
$853$	−37.4464	−1.28214	−0.641071	+	0.767482i	$0.721509\pi$
$854$	28.9030	0.989039
$855$	42.5473	1.45509
$856$	0.167926	0.00573960
$857$	−10.0117	−0.341993	−0.170996	−	0.985272i	$-0.554699\pi$
$857$	−10.0117	−0.341993	−0.170996	+	0.985272i	$0.554699\pi$
$858$	0	0
$859$	−15.9211	−0.543220	−0.271610	−	0.962407i	$-0.587556\pi$
$859$	−15.9211	−0.543220	−0.271610	+	0.962407i	$0.587556\pi$
$860$	−29.4970	−1.00584
$861$	−22.1698	−0.755546
$862$	13.3100	0.453341
$863$	−0.291809	−0.00993329	−0.00496665	−	0.999988i	$-0.501581\pi$
$863$	−0.291809	−0.00993329	−0.00496665	+	0.999988i	$0.501581\pi$
$864$	5.12501	0.174356
$865$	39.4403	1.34101
$866$	−22.7492	−0.773048
$867$	1.04360	0.0354425
$868$	24.5620	0.833689
$869$	0	0
$870$	1.12614	0.0381797
$871$	16.0614	0.544220
$872$	3.91929	0.132724
$873$	18.7748	0.635432
$874$	20.2954	0.686502
$875$	−10.4387	−0.352891
$876$	−15.9113	−0.537592
$877$	17.2508	0.582517	0.291259	−	0.956644i	$-0.405926\pi$
$877$	17.2508	0.582517	0.291259	+	0.956644i	$0.405926\pi$
$878$	11.0402	0.372590
$879$	15.8484	0.534555
$880$	0	0
$881$	27.6463	0.931426	0.465713	−	0.884936i	$-0.345798\pi$
$881$	27.6463	0.931426	0.465713	+	0.884936i	$0.345798\pi$
$882$	12.6227	0.425028
$883$	−9.90909	−0.333467	−0.166734	−	0.986002i	$-0.553322\pi$
$883$	−9.90909	−0.333467	−0.166734	+	0.986002i	$0.553322\pi$
$884$	2.48474	0.0835707
$885$	−6.84793	−0.230191
$886$	21.4275	0.719871
$887$	−35.2534	−1.18369	−0.591846	−	0.806051i	$-0.701601\pi$
$887$	−35.2534	−1.18369	−0.591846	+	0.806051i	$0.701601\pi$
$888$	9.44935	0.317099
$889$	25.4376	0.853149
$890$	36.2617	1.21549
$891$	0	0
$892$	3.03788	0.101716
$893$	−84.6496	−2.83269
$894$	−5.32314	−0.178032
$895$	3.75228	0.125425
$896$	3.68858	0.123227
$897$	7.78877	0.260059
$898$	30.9179	1.03174
$899$	2.18058	0.0727265
$900$	−11.1956	−0.373186
$901$	1.61685	0.0538652
$902$	0	0
$903$	34.4572	1.14666
$904$	−1.57878	−0.0525094
$905$	−47.7694	−1.58791
$906$	−13.6497	−0.453482
$907$	20.2579	0.672652	0.336326	−	0.941746i	$-0.390816\pi$
$907$	20.2579	0.672652	0.336326	+	0.941746i	$0.390816\pi$
$908$	−4.57303	−0.151761
$909$	−27.1277	−0.899768
$910$	30.2016	1.00117
$911$	−4.49474	−0.148917	−0.0744586	−	0.997224i	$-0.523723\pi$
$911$	−4.49474	−0.148917	−0.0744586	+	0.997224i	$0.523723\pi$
$912$	−7.05142	−0.233496
$913$	0	0
$914$	−6.97463	−0.230700
$915$	26.9469	0.890836
$916$	−19.8539	−0.655992
$917$	11.3813	0.375845
$918$	5.12501	0.169151
$919$	38.8416	1.28127	0.640633	−	0.767847i	$-0.278672\pi$
$919$	38.8416	1.28127	0.640633	+	0.767847i	$0.278672\pi$
$920$	−9.89796	−0.326326
$921$	−23.3927	−0.770815
$922$	−6.53587	−0.215248
$923$	−27.7140	−0.912216
$924$	0	0
$925$	−53.0490	−1.74424
$926$	−27.3097	−0.897453
$927$	29.6128	0.972613
$928$	0.327467	0.0107496
$929$	−53.1852	−1.74495	−0.872474	−	0.488660i	$-0.837486\pi$
$929$	−53.1852	−1.74495	−0.872474	+	0.488660i	$0.837486\pi$
$930$	22.8997	0.750910
$931$	−44.6330	−1.46279
$932$	5.40261	0.176969
$933$	2.40693	0.0787993
$934$	3.84984	0.125971
$935$	0	0
$936$	4.74808	0.155196
$937$	−31.7967	−1.03875	−0.519377	−	0.854545i	$-0.673836\pi$
$937$	−31.7967	−1.03875	−0.519377	+	0.854545i	$0.673836\pi$
$938$	23.8431	0.778505
$939$	0.724399	0.0236399
$940$	41.2832	1.34651
$941$	11.9501	0.389562	0.194781	−	0.980847i	$-0.437600\pi$
$941$	11.9501	0.389562	0.194781	+	0.980847i	$0.437600\pi$
$942$	−0.490121	−0.0159690
$943$	17.2991	0.563338
$944$	−1.99129	−0.0648110
$945$	62.2938	2.02642
$946$	0	0
$947$	26.7616	0.869637	0.434818	−	0.900518i	$-0.356813\pi$
$947$	26.7616	0.869637	0.434818	+	0.900518i	$0.356813\pi$
$948$	−6.18942	−0.201023
$949$	−37.8836	−1.22975
$950$	39.5869	1.28437
$951$	33.1445	1.07478
$952$	3.68858	0.119548
$953$	10.5684	0.342345	0.171173	−	0.985241i	$-0.445244\pi$
$953$	10.5684	0.342345	0.171173	+	0.985241i	$0.445244\pi$
$954$	3.08965	0.100031
$955$	17.1850	0.556094
$956$	−26.0696	−0.843152
$957$	0	0
$958$	−19.7747	−0.638892
$959$	−15.4623	−0.499303
$960$	3.43894	0.110991
$961$	13.3414	0.430369
$962$	22.4982	0.725372
$963$	0.320890	0.0103406
$964$	19.0942	0.614982
$965$	−49.9285	−1.60725
$966$	11.5624	0.372014
$967$	6.80892	0.218960	0.109480	−	0.993989i	$-0.465081\pi$
$967$	6.80892	0.218960	0.109480	+	0.993989i	$0.465081\pi$
$968$	0	0
$969$	−7.05142	−0.226524
$970$	32.3764	1.03954
$971$	53.3329	1.71154	0.855768	−	0.517360i	$-0.173085\pi$
$971$	53.3329	1.71154	0.855768	+	0.517360i	$0.173085\pi$
$972$	15.7760	0.506016
$973$	6.55865	0.210261
$974$	30.9964	0.993189
$975$	15.1923	0.486542
$976$	7.83580	0.250818
$977$	26.0095	0.832119	0.416059	−	0.909337i	$-0.363411\pi$
$977$	26.0095	0.832119	0.416059	+	0.909337i	$0.363411\pi$
$978$	21.9019	0.700346
$979$	0	0
$980$	21.7673	0.695331
$981$	7.48936	0.239117
$982$	17.9495	0.572791
$983$	47.3544	1.51037	0.755185	−	0.655511i	$-0.227547\pi$
$983$	47.3544	1.51037	0.755185	+	0.655511i	$0.227547\pi$
$984$	−6.01040	−0.191605
$985$	20.3354	0.647938
$986$	0.327467	0.0104287
$987$	−48.2253	−1.53503
$988$	−16.7889	−0.534127
$989$	−26.8870	−0.854956
$990$	0	0
$991$	−6.11783	−0.194339	−0.0971697	−	0.995268i	$-0.530979\pi$
$991$	−6.11783	−0.194339	−0.0971697	+	0.995268i	$0.530979\pi$
$992$	6.65894	0.211421
$993$	−8.75063	−0.277693
$994$	−41.1413	−1.30492
$995$	−10.1219	−0.320886
$996$	−12.9580	−0.410590
$997$	17.9427	0.568250	0.284125	−	0.958787i	$-0.408297\pi$
$997$	17.9427	0.568250	0.284125	+	0.958787i	$0.408297\pi$
$998$	8.52099	0.269727
$999$	46.4048	1.46818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bg.1.7		8
11.7	odd	10		374.2.g.f.137.1	✓	16
11.8	odd	10		374.2.g.f.273.1	yes	16
11.10	odd	2		4114.2.a.bi.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.g.f.137.1	✓	16	11.7	odd	10
374.2.g.f.273.1	yes	16	11.8	odd	10
4114.2.a.bg.1.7		8	1.1	even	1	trivial
4114.2.a.bi.1.7		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.04360 q^{3} +1.00000 q^{4} +3.29527 q^{5} -1.04360 q^{6} -3.68858 q^{7} -1.00000 q^{8} -1.91090 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.04360 q^{3} +1.00000 q^{4} +3.29527 q^{5} -1.04360 q^{6} -3.68858 q^{7} -1.00000 q^{8} -1.91090 q^{9} -3.29527 q^{10} +1.04360 q^{12} +2.48474 q^{13} +3.68858 q^{14} +3.43894 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.91090 q^{18} -6.75683 q^{19} +3.29527 q^{20} -3.84940 q^{21} +3.00369 q^{23} -1.04360 q^{24} +5.85881 q^{25} -2.48474 q^{26} -5.12501 q^{27} -3.68858 q^{28} -0.327467 q^{29} -3.43894 q^{30} -6.65894 q^{31} -1.00000 q^{32} -1.00000 q^{34} -12.1549 q^{35} -1.91090 q^{36} -9.05457 q^{37} +6.75683 q^{38} +2.59307 q^{39} -3.29527 q^{40} +5.75930 q^{41} +3.84940 q^{42} -8.95131 q^{43} -6.29693 q^{45} -3.00369 q^{46} +12.5280 q^{47} +1.04360 q^{48} +6.60562 q^{49} -5.85881 q^{50} +1.04360 q^{51} +2.48474 q^{52} +1.61685 q^{53} +5.12501 q^{54} +3.68858 q^{56} -7.05142 q^{57} +0.327467 q^{58} -1.99129 q^{59} +3.43894 q^{60} +7.83580 q^{61} +6.65894 q^{62} +7.04851 q^{63} +1.00000 q^{64} +8.18788 q^{65} +6.46403 q^{67} +1.00000 q^{68} +3.13465 q^{69} +12.1549 q^{70} -11.1537 q^{71} +1.91090 q^{72} -15.2465 q^{73} +9.05457 q^{74} +6.11425 q^{75} -6.75683 q^{76} -2.59307 q^{78} -5.93084 q^{79} +3.29527 q^{80} +0.384242 q^{81} -5.75930 q^{82} -12.4167 q^{83} -3.84940 q^{84} +3.29527 q^{85} +8.95131 q^{86} -0.341744 q^{87} -11.0042 q^{89} +6.29693 q^{90} -9.16514 q^{91} +3.00369 q^{92} -6.94926 q^{93} -12.5280 q^{94} -22.2656 q^{95} -1.04360 q^{96} -9.82512 q^{97} -6.60562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) 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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