LMFDB - Embedded newform 841.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(1,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("841.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.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:	$6.71541880999$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.2841328125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 3x^{6} + 23x^{5} - 43x^{3} + 2x^{2} + 24x + 1 $ x^8 - 4x^7 - 3x^6 + 23x^5 - 43x^3 + 2x^2 + 24x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.39427$ of defining polynomial
Character	$\chi$	$=$	841.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.39427 q^{2} -2.10137 q^{3} +3.73253 q^{4} -2.42792 q^{5} +5.03125 q^{6} -0.378370 q^{7} -4.14815 q^{8} +1.41576 q^{9} +5.81310 q^{10} -3.84036 q^{11} -7.84343 q^{12} +5.59342 q^{13} +0.905920 q^{14} +5.10197 q^{15} +2.46673 q^{16} +1.82620 q^{17} -3.38971 q^{18} +0.558292 q^{19} -9.06230 q^{20} +0.795095 q^{21} +9.19486 q^{22} -0.335234 q^{23} +8.71680 q^{24} +0.894809 q^{25} -13.3922 q^{26} +3.32908 q^{27} -1.41228 q^{28} -12.2155 q^{30} +4.06809 q^{31} +2.39028 q^{32} +8.07002 q^{33} -4.37242 q^{34} +0.918653 q^{35} +5.28437 q^{36} +8.06015 q^{37} -1.33670 q^{38} -11.7538 q^{39} +10.0714 q^{40} +7.79555 q^{41} -1.90367 q^{42} -0.0226949 q^{43} -14.3343 q^{44} -3.43735 q^{45} +0.802642 q^{46} -7.62892 q^{47} -5.18351 q^{48} -6.85684 q^{49} -2.14242 q^{50} -3.83753 q^{51} +20.8776 q^{52} +6.37259 q^{53} -7.97071 q^{54} +9.32410 q^{55} +1.56953 q^{56} -1.17318 q^{57} +8.13681 q^{59} +19.0433 q^{60} -10.3717 q^{61} -9.74010 q^{62} -0.535681 q^{63} -10.6564 q^{64} -13.5804 q^{65} -19.3218 q^{66} -12.0355 q^{67} +6.81636 q^{68} +0.704452 q^{69} -2.19950 q^{70} -15.5590 q^{71} -5.87278 q^{72} +6.93458 q^{73} -19.2982 q^{74} -1.88033 q^{75} +2.08384 q^{76} +1.45308 q^{77} +28.1419 q^{78} -7.46125 q^{79} -5.98903 q^{80} -11.2429 q^{81} -18.6647 q^{82} +0.671301 q^{83} +2.96772 q^{84} -4.43388 q^{85} +0.0543377 q^{86} +15.9304 q^{88} -9.63916 q^{89} +8.22996 q^{90} -2.11638 q^{91} -1.25127 q^{92} -8.54856 q^{93} +18.2657 q^{94} -1.35549 q^{95} -5.02287 q^{96} -4.45519 q^{97} +16.4171 q^{98} -5.43702 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 3 q^{6} - 15 q^{8} + 8 q^{9} + 3 q^{10} - 5 q^{11} - 12 q^{12} - 4 q^{13} - 15 q^{14} + 7 q^{15} - 2 q^{16} - 9 q^{17} + 11 q^{18} - 17 q^{19} - 7 q^{20} - 25 q^{21}+ \cdots - 15 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 - 6 * q^3 + 6 * q^4 - q^5 + 3 * q^6 - 15 * q^8 + 8 * q^9 + 3 * q^10 - 5 * q^11 - 12 * q^12 - 4 * q^13 - 15 * q^14 + 7 * q^15 - 2 * q^16 - 9 * q^17 + 11 * q^18 - 17 * q^19 - 7 * q^20 - 25 * q^21 + 25 * q^22 - 7 * q^23 + 25 * q^24 - 13 * q^25 - 3 * q^26 - 18 * q^27 + 25 * q^28 - 26 * q^30 - q^31 + 6 * q^32 + 12 * q^34 - 5 * q^35 - 19 * q^36 - 6 * q^37 - 14 * q^38 - 27 * q^39 + 25 * q^40 - 6 * q^41 + 15 * q^42 + 5 * q^43 - 20 * q^44 - q^45 - 14 * q^46 - 31 * q^47 - 11 * q^48 + 4 * q^49 - 11 * q^50 + 23 * q^51 + 2 * q^52 + 19 * q^53 - 46 * q^54 - 20 * q^55 - 20 * q^56 - 6 * q^57 - q^59 + 9 * q^60 - 21 * q^61 - 12 * q^62 - 10 * q^63 + q^64 - 17 * q^65 - 5 * q^66 + 2 * q^67 - 8 * q^68 + 14 * q^69 + 5 * q^70 - 12 * q^71 + 34 * q^73 - 27 * q^74 - 4 * q^75 + 16 * q^76 + 10 * q^77 + 6 * q^78 - 29 * q^79 - 41 * q^80 + 12 * q^81 - 22 * q^82 - 3 * q^83 + 10 * q^84 - 2 * q^85 - 5 * q^86 - 15 * q^88 - 36 * q^89 + 38 * q^90 - 4 * q^92 + 2 * q^93 + 13 * q^94 - 11 * q^95 - 2 * q^96 + 18 * q^97 - 27 * q^98 - 15 * q^99

Coefficient data

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

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

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

$2$	−2.39427	−1.69300	−0.846502	−	0.532385i	$-0.821296\pi$
$2$	−2.39427	−1.69300	−0.846502	+	0.532385i	$0.821296\pi$
$3$	−2.10137	−1.21323	−0.606614	−	0.794997i	$-0.707472\pi$
$3$	−2.10137	−1.21323	−0.606614	+	0.794997i	$0.707472\pi$
$4$	3.73253	1.86627
$5$	−2.42792	−1.08580	−0.542900	−	0.839797i	$-0.682674\pi$
$5$	−2.42792	−1.08580	−0.542900	+	0.839797i	$0.682674\pi$
$6$	5.03125	2.05400
$7$	−0.378370	−0.143010	−0.0715052	−	0.997440i	$-0.522780\pi$
$7$	−0.378370	−0.143010	−0.0715052	+	0.997440i	$0.522780\pi$
$8$	−4.14815	−1.46659
$9$	1.41576	0.471920
$10$	5.81310	1.83827
$11$	−3.84036	−1.15791	−0.578956	−	0.815359i	$-0.696540\pi$
$11$	−3.84036	−1.15791	−0.578956	+	0.815359i	$0.696540\pi$
$12$	−7.84343	−2.26420
$13$	5.59342	1.55133	0.775667	−	0.631142i	$-0.217414\pi$
$13$	5.59342	1.55133	0.775667	+	0.631142i	$0.217414\pi$
$14$	0.905920	0.242117
$15$	5.10197	1.31732
$16$	2.46673	0.616683
$17$	1.82620	0.442919	0.221460	−	0.975170i	$-0.428918\pi$
$17$	1.82620	0.442919	0.221460	+	0.975170i	$0.428918\pi$
$18$	−3.38971	−0.798963
$19$	0.558292	0.128081	0.0640405	−	0.997947i	$-0.479601\pi$
$19$	0.558292	0.128081	0.0640405	+	0.997947i	$0.479601\pi$
$20$	−9.06230	−2.02639
$21$	0.795095	0.173504
$22$	9.19486	1.96035
$23$	−0.335234	−0.0699012	−0.0349506	−	0.999389i	$-0.511127\pi$
$23$	−0.335234	−0.0699012	−0.0349506	+	0.999389i	$0.511127\pi$
$24$	8.71680	1.77931
$25$	0.894809	0.178962
$26$	−13.3922	−2.62642
$27$	3.32908	0.640681
$28$	−1.41228	−0.266895
$29$	0	0
$29$	0	0
$30$	−12.2155	−2.23023
$31$	4.06809	0.730650	0.365325	−	0.930880i	$-0.380958\pi$
$31$	4.06809	0.730650	0.365325	+	0.930880i	$0.380958\pi$
$32$	2.39028	0.422546
$33$	8.07002	1.40481
$34$	−4.37242	−0.749864
$35$	0.918653	0.155281
$36$	5.28437	0.880728
$37$	8.06015	1.32508	0.662540	−	0.749027i	$-0.269478\pi$
$37$	8.06015	1.32508	0.662540	+	0.749027i	$0.269478\pi$
$38$	−1.33670	−0.216842
$39$	−11.7538	−1.88212
$40$	10.0714	1.59243
$41$	7.79555	1.21746	0.608730	−	0.793377i	$-0.291679\pi$
$41$	7.79555	1.21746	0.608730	+	0.793377i	$0.291679\pi$
$42$	−1.90367	−0.293743
$43$	−0.0226949	−0.00346094	−0.00173047	−	0.999999i	$-0.500551\pi$
$43$	−0.0226949	−0.00346094	−0.00173047	+	0.999999i	$0.500551\pi$
$44$	−14.3343	−2.16097
$45$	−3.43735	−0.512411
$46$	0.802642	0.118343
$47$	−7.62892	−1.11279	−0.556396	−	0.830917i	$-0.687816\pi$
$47$	−7.62892	−1.11279	−0.556396	+	0.830917i	$0.687816\pi$
$48$	−5.18351	−0.748176
$49$	−6.85684	−0.979548
$50$	−2.14242	−0.302983
$51$	−3.83753	−0.537361
$52$	20.8776	2.89520
$53$	6.37259	0.875343	0.437672	−	0.899135i	$-0.355803\pi$
$53$	6.37259	0.875343	0.437672	+	0.899135i	$0.355803\pi$
$54$	−7.97071	−1.08468
$55$	9.32410	1.25726
$56$	1.56953	0.209738
$57$	−1.17318	−0.155391
$58$	0	0
$59$	8.13681	1.05932	0.529661	−	0.848209i	$-0.322319\pi$
$59$	8.13681	1.05932	0.529661	+	0.848209i	$0.322319\pi$
$60$	19.0433	2.45847
$61$	−10.3717	−1.32796	−0.663981	−	0.747749i	$-0.731135\pi$
$61$	−10.3717	−1.32796	−0.663981	+	0.747749i	$0.731135\pi$
$62$	−9.74010	−1.23699
$63$	−0.535681	−0.0674894
$64$	−10.6564	−1.33205
$65$	−13.5804	−1.68444
$66$	−19.3218	−2.37835
$67$	−12.0355	−1.47038	−0.735188	−	0.677863i	$-0.762906\pi$
$67$	−12.0355	−1.47038	−0.735188	+	0.677863i	$0.762906\pi$
$68$	6.81636	0.826605
$69$	0.704452	0.0848060
$70$	−2.19950	−0.262891
$71$	−15.5590	−1.84652	−0.923259	−	0.384178i	$-0.874485\pi$
$71$	−15.5590	−1.84652	−0.923259	+	0.384178i	$0.874485\pi$
$72$	−5.87278	−0.692114
$73$	6.93458	0.811631	0.405815	−	0.913955i	$-0.366988\pi$
$73$	6.93458	0.811631	0.405815	+	0.913955i	$0.366988\pi$
$74$	−19.2982	−2.24337
$75$	−1.88033	−0.217121
$76$	2.08384	0.239033
$77$	1.45308	0.165593
$78$	28.1419	3.18644
$79$	−7.46125	−0.839457	−0.419728	−	0.907650i	$-0.637875\pi$
$79$	−7.46125	−0.839457	−0.419728	+	0.907650i	$0.637875\pi$
$80$	−5.98903	−0.669594
$81$	−11.2429	−1.24921
$82$	−18.6647	−2.06117
$83$	0.671301	0.0736848	0.0368424	−	0.999321i	$-0.488270\pi$
$83$	0.671301	0.0736848	0.0368424	+	0.999321i	$0.488270\pi$
$84$	2.96772	0.323805
$85$	−4.43388	−0.480922
$86$	0.0543377	0.00585939
$87$	0	0
$88$	15.9304	1.69818
$89$	−9.63916	−1.02175	−0.510875	−	0.859655i	$-0.670678\pi$
$89$	−9.63916	−1.02175	−0.510875	+	0.859655i	$0.670678\pi$
$90$	8.22996	0.867514
$91$	−2.11638	−0.221857
$92$	−1.25127	−0.130454
$93$	−8.54856	−0.886444
$94$	18.2657	1.88396
$95$	−1.35549	−0.139070
$96$	−5.02287	−0.512644
$97$	−4.45519	−0.452356	−0.226178	−	0.974086i	$-0.572623\pi$
$97$	−4.45519	−0.452356	−0.226178	+	0.974086i	$0.572623\pi$
$98$	16.4171	1.65838
$99$	−5.43702	−0.546441
$100$	3.33990	0.333990
$101$	−6.03902	−0.600905	−0.300453	−	0.953797i	$-0.597138\pi$
$101$	−6.03902	−0.600905	−0.300453	+	0.953797i	$0.597138\pi$
$102$	9.18808	0.909755
$103$	6.19845	0.610752	0.305376	−	0.952232i	$-0.401218\pi$
$103$	6.19845	0.610752	0.305376	+	0.952232i	$0.401218\pi$
$104$	−23.2023	−2.27518
$105$	−1.93043	−0.188391
$106$	−15.2577	−1.48196
$107$	5.30153	0.512518	0.256259	−	0.966608i	$-0.417510\pi$
$107$	5.30153	0.512518	0.256259	+	0.966608i	$0.417510\pi$
$108$	12.4259	1.19568
$109$	6.29930	0.603364	0.301682	−	0.953409i	$-0.402452\pi$
$109$	6.29930	0.603364	0.301682	+	0.953409i	$0.402452\pi$
$110$	−22.3244	−2.12855
$111$	−16.9374	−1.60762
$112$	−0.933336	−0.0881920
$113$	−13.4101	−1.26151	−0.630757	−	0.775980i	$-0.717256\pi$
$113$	−13.4101	−1.26151	−0.630757	+	0.775980i	$0.717256\pi$
$114$	2.80891	0.263078
$115$	0.813923	0.0758987
$116$	0	0
$117$	7.91893	0.732105
$118$	−19.4817	−1.79344
$119$	−0.690980	−0.0633420
$120$	−21.1637	−1.93197
$121$	3.74836	0.340760
$122$	24.8327	2.24825
$123$	−16.3813	−1.47706
$124$	15.1843	1.36359
$125$	9.96709	0.891483
$126$	1.28256	0.114260
$127$	5.83032	0.517358	0.258679	−	0.965963i	$-0.416713\pi$
$127$	5.83032	0.517358	0.258679	+	0.965963i	$0.416713\pi$
$128$	20.7338	1.83263
$129$	0.0476904	0.00419891
$130$	32.5151	2.85176
$131$	0.528206	0.0461496	0.0230748	−	0.999734i	$-0.492654\pi$
$131$	0.528206	0.0461496	0.0230748	+	0.999734i	$0.492654\pi$
$132$	30.1216	2.62175
$133$	−0.211241	−0.0183169
$134$	28.8164	2.48935
$135$	−8.08274	−0.695652
$136$	−7.57536	−0.649582
$137$	−7.86200	−0.671696	−0.335848	−	0.941916i	$-0.609023\pi$
$137$	−7.86200	−0.671696	−0.335848	+	0.941916i	$0.609023\pi$
$138$	−1.68665	−0.143577
$139$	−9.24826	−0.784427	−0.392213	−	0.919874i	$-0.628291\pi$
$139$	−9.24826	−0.784427	−0.392213	+	0.919874i	$0.628291\pi$
$140$	3.42890	0.289795
$141$	16.0312	1.35007
$142$	37.2525	3.12616
$143$	−21.4807	−1.79631
$144$	3.49230	0.291025
$145$	0	0
$146$	−16.6033	−1.37410
$147$	14.4088	1.18841
$148$	30.0848	2.47295
$149$	−0.514276	−0.0421311	−0.0210656	−	0.999778i	$-0.506706\pi$
$149$	−0.514276	−0.0421311	−0.0210656	+	0.999778i	$0.506706\pi$
$150$	4.50201	0.367588
$151$	−6.55234	−0.533222	−0.266611	−	0.963804i	$-0.585904\pi$
$151$	−6.55234	−0.533222	−0.266611	+	0.963804i	$0.585904\pi$
$152$	−2.31588	−0.187843
$153$	2.58546	0.209022
$154$	−3.47906	−0.280350
$155$	−9.87700	−0.793340
$156$	−43.8716	−3.51254
$157$	−11.4559	−0.914282	−0.457141	−	0.889394i	$-0.651126\pi$
$157$	−11.4559	−0.914282	−0.457141	+	0.889394i	$0.651126\pi$
$158$	17.8643	1.42120
$159$	−13.3912	−1.06199
$160$	−5.80342	−0.458800
$161$	0.126843	0.00999660
$162$	26.9186	2.11492
$163$	−18.2330	−1.42812	−0.714060	−	0.700085i	$-0.753146\pi$
$163$	−18.2330	−1.42812	−0.714060	+	0.700085i	$0.753146\pi$
$164$	29.0972	2.27211
$165$	−19.5934	−1.52534
$166$	−1.60728	−0.124749
$167$	2.53384	0.196074	0.0980372	−	0.995183i	$-0.468744\pi$
$167$	2.53384	0.196074	0.0980372	+	0.995183i	$0.468744\pi$
$168$	−3.29817	−0.254460
$169$	18.2863	1.40664
$170$	10.6159	0.814203
$171$	0.790407	0.0604439
$172$	−0.0847094	−0.00645903
$173$	8.48473	0.645082	0.322541	−	0.946556i	$-0.395463\pi$
$173$	8.48473	0.645082	0.322541	+	0.946556i	$0.395463\pi$
$174$	0	0
$175$	−0.338569	−0.0255934
$176$	−9.47313	−0.714064
$177$	−17.0985	−1.28520
$178$	23.0788	1.72983
$179$	12.9529	0.968146	0.484073	−	0.875028i	$-0.339157\pi$
$179$	12.9529	0.968146	0.484073	+	0.875028i	$0.339157\pi$
$180$	−12.8300	−0.956294
$181$	16.4170	1.22026	0.610132	−	0.792300i	$-0.291116\pi$
$181$	16.4170	1.22026	0.610132	+	0.792300i	$0.291116\pi$
$182$	5.06719	0.375605
$183$	21.7948	1.61112
$184$	1.39060	0.102517
$185$	−19.5694	−1.43877
$186$	20.4676	1.50075
$187$	−7.01327	−0.512861
$188$	−28.4752	−2.07677
$189$	−1.25962	−0.0916240
$190$	3.24541	0.235447
$191$	−7.42138	−0.536992	−0.268496	−	0.963281i	$-0.586527\pi$
$191$	−7.42138	−0.536992	−0.268496	+	0.963281i	$0.586527\pi$
$192$	22.3931	1.61608
$193$	21.9696	1.58141	0.790704	−	0.612198i	$-0.209715\pi$
$193$	21.9696	1.58141	0.790704	+	0.612198i	$0.209715\pi$
$194$	10.6669	0.765841
$195$	28.5374	2.04361
$196$	−25.5934	−1.82810
$197$	19.7043	1.40387	0.701936	−	0.712240i	$-0.252319\pi$
$197$	19.7043	1.40387	0.701936	+	0.712240i	$0.252319\pi$
$198$	13.0177	0.925128
$199$	−16.2382	−1.15110	−0.575549	−	0.817767i	$-0.695212\pi$
$199$	−16.2382	−1.15110	−0.575549	+	0.817767i	$0.695212\pi$
$200$	−3.71180	−0.262464
$201$	25.2911	1.78390
$202$	14.4591	1.01734
$203$	0	0
$204$	−14.3237	−1.00286
$205$	−18.9270	−1.32192
$206$	−14.8408	−1.03401
$207$	−0.474611	−0.0329878
$208$	13.7974	0.956681
$209$	−2.14404	−0.148306
$210$	4.62197	0.318946
$211$	23.5359	1.62028	0.810138	−	0.586239i	$-0.199392\pi$
$211$	23.5359	1.62028	0.810138	+	0.586239i	$0.199392\pi$
$212$	23.7859	1.63362
$213$	32.6953	2.24025
$214$	−12.6933	−0.867696
$215$	0.0551015	0.00375789
$216$	−13.8095	−0.939618
$217$	−1.53924	−0.104490
$218$	−15.0822	−1.02150
$219$	−14.5721	−0.984693
$220$	34.8025	2.34638
$221$	10.2147	0.687116
$222$	40.5526	2.72171
$223$	−0.701931	−0.0470048	−0.0235024	−	0.999724i	$-0.507482\pi$
$223$	−0.701931	−0.0470048	−0.0235024	+	0.999724i	$0.507482\pi$
$224$	−0.904410	−0.0604284
$225$	1.26683	0.0844557
$226$	32.1074	2.13575
$227$	−7.40249	−0.491321	−0.245660	−	0.969356i	$-0.579005\pi$
$227$	−7.40249	−0.491321	−0.245660	+	0.969356i	$0.579005\pi$
$228$	−4.37893	−0.290002
$229$	−10.7457	−0.710094	−0.355047	−	0.934848i	$-0.615535\pi$
$229$	−10.7457	−0.710094	−0.355047	+	0.934848i	$0.615535\pi$
$230$	−1.94875	−0.128497
$231$	−3.05345	−0.200902
$232$	0	0
$233$	−15.5888	−1.02126	−0.510630	−	0.859801i	$-0.670588\pi$
$233$	−15.5888	−1.02126	−0.510630	+	0.859801i	$0.670588\pi$
$234$	−18.9601	−1.23946
$235$	18.5224	1.20827
$236$	30.3709	1.97698
$237$	15.6789	1.01845
$238$	1.65439	0.107238
$239$	7.44400	0.481512	0.240756	−	0.970586i	$-0.422605\pi$
$239$	7.44400	0.481512	0.240756	+	0.970586i	$0.422605\pi$
$240$	12.5852	0.812370
$241$	−5.26688	−0.339270	−0.169635	−	0.985507i	$-0.554259\pi$
$241$	−5.26688	−0.339270	−0.169635	+	0.985507i	$0.554259\pi$
$242$	−8.97458	−0.576908
$243$	13.6383	0.874896
$244$	−38.7128	−2.47833
$245$	16.6479	1.06359
$246$	39.2214	2.50066
$247$	3.12276	0.198696
$248$	−16.8750	−1.07157
$249$	−1.41065	−0.0893964
$250$	−23.8639	−1.50929
$251$	−27.9636	−1.76505	−0.882523	−	0.470269i	$-0.844157\pi$
$251$	−27.9636	−1.76505	−0.882523	+	0.470269i	$0.844157\pi$
$252$	−1.99944	−0.125953
$253$	1.28742	0.0809394
$254$	−13.9594	−0.875889
$255$	9.31722	0.583467
$256$	−28.3295	−1.77060
$257$	−19.0686	−1.18947	−0.594733	−	0.803923i	$-0.702742\pi$
$257$	−19.0686	−1.18947	−0.594733	+	0.803923i	$0.702742\pi$
$258$	−0.114184	−0.00710877
$259$	−3.04972	−0.189500
$260$	−50.6892	−3.14361
$261$	0	0
$262$	−1.26467	−0.0781315
$263$	7.11487	0.438721	0.219361	−	0.975644i	$-0.429603\pi$
$263$	7.11487	0.438721	0.219361	+	0.975644i	$0.429603\pi$
$264$	−33.4756	−2.06028
$265$	−15.4722	−0.950448
$266$	0.505768	0.0310106
$267$	20.2555	1.23961
$268$	−44.9231	−2.74411
$269$	−5.95607	−0.363148	−0.181574	−	0.983377i	$-0.558119\pi$
$269$	−5.95607	−0.363148	−0.181574	+	0.983377i	$0.558119\pi$
$270$	19.3523	1.17774
$271$	5.13351	0.311839	0.155919	−	0.987770i	$-0.450166\pi$
$271$	5.13351	0.311839	0.155919	+	0.987770i	$0.450166\pi$
$272$	4.50475	0.273140
$273$	4.44730	0.269163
$274$	18.8238	1.13719
$275$	−3.43639	−0.207222
$276$	2.62939	0.158271
$277$	20.0056	1.20202	0.601010	−	0.799241i	$-0.294765\pi$
$277$	20.0056	1.20202	0.601010	+	0.799241i	$0.294765\pi$
$278$	22.1428	1.32804
$279$	5.75943	0.344808
$280$	−3.81071	−0.227733
$281$	−11.7312	−0.699822	−0.349911	−	0.936783i	$-0.613788\pi$
$281$	−11.7312	−0.699822	−0.349911	+	0.936783i	$0.613788\pi$
$282$	−38.3830	−2.28568
$283$	−29.6224	−1.76087	−0.880435	−	0.474167i	$-0.842749\pi$
$283$	−29.6224	−1.76087	−0.880435	+	0.474167i	$0.842749\pi$
$284$	−58.0746	−3.44609
$285$	2.84839	0.168724
$286$	51.4307	3.04116
$287$	−2.94960	−0.174110
$288$	3.38406	0.199408
$289$	−13.6650	−0.803823
$290$	0	0
$291$	9.36201	0.548811
$292$	25.8835	1.51472
$293$	7.86184	0.459294	0.229647	−	0.973274i	$-0.426243\pi$
$293$	7.86184	0.459294	0.229647	+	0.973274i	$0.426243\pi$
$294$	−34.4985	−2.01199
$295$	−19.7556	−1.15021
$296$	−33.4347	−1.94335
$297$	−12.7849	−0.741852
$298$	1.23132	0.0713282
$299$	−1.87511	−0.108440
$300$	−7.01838	−0.405206
$301$	0.00858707	0.000494950 0
$302$	15.6881	0.902747
$303$	12.6902	0.729034
$304$	1.37716	0.0789853
$305$	25.1817	1.44190
$306$	−6.19030	−0.353876
$307$	−28.3811	−1.61979	−0.809897	−	0.586572i	$-0.800477\pi$
$307$	−28.3811	−1.61979	−0.809897	+	0.586572i	$0.800477\pi$
$308$	5.42365	0.309041
$309$	−13.0252	−0.740980
$310$	23.6482	1.34313
$311$	−8.57178	−0.486061	−0.243031	−	0.970019i	$-0.578142\pi$
$311$	−8.57178	−0.486061	−0.243031	+	0.970019i	$0.578142\pi$
$312$	48.7567	2.76030
$313$	0.879004	0.0496843	0.0248421	−	0.999691i	$-0.492092\pi$
$313$	0.879004	0.0496843	0.0248421	+	0.999691i	$0.492092\pi$
$314$	27.4286	1.54788
$315$	1.30059	0.0732800
$316$	−27.8494	−1.56665
$317$	26.6256	1.49544	0.747722	−	0.664012i	$-0.231148\pi$
$317$	26.6256	1.49544	0.747722	+	0.664012i	$0.231148\pi$
$318$	32.0621	1.79795
$319$	0	0
$320$	25.8730	1.44635
$321$	−11.1405	−0.621801
$322$	−0.303695	−0.0169243
$323$	1.01955	0.0567295
$324$	−41.9645	−2.33136
$325$	5.00504	0.277630
$326$	43.6548	2.41781
$327$	−13.2372	−0.732017
$328$	−32.3371	−1.78552
$329$	2.88655	0.159141
$330$	46.9119	2.58241
$331$	−18.6487	−1.02502	−0.512512	−	0.858680i	$-0.671285\pi$
$331$	−18.6487	−1.02502	−0.512512	+	0.858680i	$0.671285\pi$
$332$	2.50565	0.137515
$333$	11.4112	0.625331
$334$	−6.06670	−0.331955
$335$	29.2214	1.59653
$336$	1.96129	0.106997
$337$	6.04773	0.329441	0.164720	−	0.986340i	$-0.447328\pi$
$337$	6.04773	0.329441	0.164720	+	0.986340i	$0.447328\pi$
$338$	−43.7824	−2.38145
$339$	28.1795	1.53050
$340$	−16.5496	−0.897527
$341$	−15.6229	−0.846028
$342$	−1.89245	−0.102332
$343$	5.24301	0.283096
$344$	0.0941419	0.00507579
$345$	−1.71035	−0.0920824
$346$	−20.3147	−1.09213
$347$	−14.4469	−0.775548	−0.387774	−	0.921754i	$-0.626756\pi$
$347$	−14.4469	−0.775548	−0.387774	+	0.921754i	$0.626756\pi$
$348$	0	0
$349$	11.5232	0.616825	0.308413	−	0.951253i	$-0.400202\pi$
$349$	11.5232	0.616825	0.308413	+	0.951253i	$0.400202\pi$
$350$	0.810626	0.0433298
$351$	18.6209	0.993911
$352$	−9.17953	−0.489271
$353$	−8.47324	−0.450985	−0.225493	−	0.974245i	$-0.572399\pi$
$353$	−8.47324	−0.450985	−0.225493	+	0.974245i	$0.572399\pi$
$354$	40.9383	2.17585
$355$	37.7761	2.00495
$356$	−35.9785	−1.90686
$357$	1.45200	0.0768482
$358$	−31.0128	−1.63908
$359$	4.93526	0.260473	0.130237	−	0.991483i	$-0.458426\pi$
$359$	4.93526	0.260473	0.130237	+	0.991483i	$0.458426\pi$
$360$	14.2587	0.751497
$361$	−18.6883	−0.983595
$362$	−39.3067	−2.06591
$363$	−7.87669	−0.413419
$364$	−7.89945	−0.414044
$365$	−16.8366	−0.881269
$366$	−52.1827	−2.72763
$367$	−2.90702	−0.151745	−0.0758727	−	0.997118i	$-0.524174\pi$
$367$	−2.90702	−0.151745	−0.0758727	+	0.997118i	$0.524174\pi$
$368$	−0.826933	−0.0431069
$369$	11.0366	0.574544
$370$	46.8545	2.43585
$371$	−2.41120	−0.125183
$372$	−31.9078	−1.65434
$373$	−31.1460	−1.61268	−0.806340	−	0.591452i	$-0.798555\pi$
$373$	−31.1460	−1.61268	−0.806340	+	0.591452i	$0.798555\pi$
$374$	16.7917	0.868277
$375$	−20.9445	−1.08157
$376$	31.6459	1.63201
$377$	0	0
$378$	3.01588	0.155120
$379$	24.2758	1.24696	0.623482	−	0.781838i	$-0.285718\pi$
$379$	24.2758	1.24696	0.623482	+	0.781838i	$0.285718\pi$
$380$	−5.05941	−0.259542
$381$	−12.2517	−0.627672
$382$	17.7688	0.909130
$383$	−29.7679	−1.52107	−0.760533	−	0.649299i	$-0.775062\pi$
$383$	−29.7679	−1.52107	−0.760533	+	0.649299i	$0.775062\pi$
$384$	−43.5695	−2.22340
$385$	−3.52796	−0.179801
$386$	−52.6012	−2.67733
$387$	−0.0321305	−0.00163329
$388$	−16.6291	−0.844217
$389$	10.7861	0.546879	0.273440	−	0.961889i	$-0.411839\pi$
$389$	10.7861	0.546879	0.273440	+	0.961889i	$0.411839\pi$
$390$	−68.3263	−3.45984
$391$	−0.612206	−0.0309606
$392$	28.4432	1.43660
$393$	−1.10996	−0.0559899
$394$	−47.1774	−2.37676
$395$	18.1153	0.911482
$396$	−20.2939	−1.01981
$397$	1.14253	0.0573418	0.0286709	−	0.999589i	$-0.490873\pi$
$397$	1.14253	0.0573418	0.0286709	+	0.999589i	$0.490873\pi$
$398$	38.8787	1.94882
$399$	0.443895	0.0222226
$400$	2.20725	0.110363
$401$	−3.85161	−0.192340	−0.0961702	−	0.995365i	$-0.530659\pi$
$401$	−3.85161	−0.192340	−0.0961702	+	0.995365i	$0.530659\pi$
$402$	−60.5538	−3.02015
$403$	22.7545	1.13348
$404$	−22.5408	−1.12145
$405$	27.2969	1.35639
$406$	0	0
$407$	−30.9539	−1.53433
$408$	15.9186	0.788090
$409$	−19.2504	−0.951872	−0.475936	−	0.879480i	$-0.657891\pi$
$409$	−19.2504	−0.951872	−0.475936	+	0.879480i	$0.657891\pi$
$410$	45.3164	2.23802
$411$	16.5210	0.814920
$412$	23.1359	1.13983
$413$	−3.07872	−0.151494
$414$	1.13635	0.0558485
$415$	−1.62987	−0.0800070
$416$	13.3698	0.655510
$417$	19.4340	0.951688
$418$	5.13342	0.251084
$419$	−2.00471	−0.0979365	−0.0489682	−	0.998800i	$-0.515593\pi$
$419$	−2.00471	−0.0979365	−0.0489682	+	0.998800i	$0.515593\pi$
$420$	−7.20539	−0.351587
$421$	−39.8563	−1.94248	−0.971239	−	0.238106i	$-0.923474\pi$
$421$	−39.8563	−1.94248	−0.971239	+	0.238106i	$0.923474\pi$
$422$	−56.3512	−2.74314
$423$	−10.8007	−0.525149
$424$	−26.4345	−1.28377
$425$	1.63410	0.0792656
$426$	−78.2814	−3.79275
$427$	3.92435	0.189912
$428$	19.7881	0.956495
$429$	45.1390	2.17933
$430$	−0.131928	−0.00636213
$431$	14.3148	0.689520	0.344760	−	0.938691i	$-0.387960\pi$
$431$	14.3148	0.689520	0.344760	+	0.938691i	$0.387960\pi$
$432$	8.21194	0.395097
$433$	−8.83404	−0.424537	−0.212268	−	0.977211i	$-0.568085\pi$
$433$	−8.83404	−0.424537	−0.212268	+	0.977211i	$0.568085\pi$
$434$	3.68536	0.176903
$435$	0	0
$436$	23.5123	1.12604
$437$	−0.187159	−0.00895302
$438$	34.8896	1.66709
$439$	−11.5826	−0.552807	−0.276403	−	0.961042i	$-0.589143\pi$
$439$	−11.5826	−0.552807	−0.276403	+	0.961042i	$0.589143\pi$
$440$	−38.6777	−1.84389
$441$	−9.70763	−0.462268
$442$	−24.4568	−1.16329
$443$	−35.4891	−1.68614	−0.843068	−	0.537807i	$-0.819253\pi$
$443$	−35.4891	−1.68614	−0.843068	+	0.537807i	$0.819253\pi$
$444$	−63.2192	−3.00025
$445$	23.4031	1.10942
$446$	1.68061	0.0795793
$447$	1.08068	0.0511146
$448$	4.03207	0.190498
$449$	20.7485	0.979183	0.489592	−	0.871952i	$-0.337146\pi$
$449$	20.7485	0.979183	0.489592	+	0.871952i	$0.337146\pi$
$450$	−3.03315	−0.142984
$451$	−29.9377	−1.40971
$452$	−50.0535	−2.35432
$453$	13.7689	0.646919
$454$	17.7236	0.831808
$455$	5.13841	0.240892
$456$	4.86652	0.227896
$457$	38.1037	1.78242	0.891208	−	0.453596i	$-0.149859\pi$
$457$	38.1037	1.78242	0.891208	+	0.453596i	$0.149859\pi$
$458$	25.7281	1.20219
$459$	6.07957	0.283770
$460$	3.03799	0.141647
$461$	17.7244	0.825506	0.412753	−	0.910843i	$-0.364567\pi$
$461$	17.7244	0.825506	0.412753	+	0.910843i	$0.364567\pi$
$462$	7.31079	0.340129
$463$	9.21914	0.428450	0.214225	−	0.976784i	$-0.431277\pi$
$463$	9.21914	0.428450	0.214225	+	0.976784i	$0.431277\pi$
$464$	0	0
$465$	20.7552	0.962501
$466$	37.3239	1.72900
$467$	7.80216	0.361041	0.180521	−	0.983571i	$-0.442222\pi$
$467$	7.80216	0.361041	0.180521	+	0.983571i	$0.442222\pi$
$468$	29.5577	1.36630
$469$	4.55389	0.210279
$470$	−44.3477	−2.04561
$471$	24.0731	1.10923
$472$	−33.7527	−1.55359
$473$	0.0871566	0.00400746
$474$	−37.5394	−1.72424
$475$	0.499565	0.0229216
$476$	−2.57910	−0.118213
$477$	9.02206	0.413092
$478$	−17.8229	−0.815203
$479$	−17.7227	−0.809770	−0.404885	−	0.914368i	$-0.632688\pi$
$479$	−17.7227	−0.809770	−0.404885	+	0.914368i	$0.632688\pi$
$480$	12.1951	0.556629
$481$	45.0838	2.05564
$482$	12.6103	0.574385
$483$	−0.266543	−0.0121281
$484$	13.9909	0.635948
$485$	10.8169	0.491168
$486$	−32.6537	−1.48120
$487$	−21.1590	−0.958805	−0.479403	−	0.877595i	$-0.659147\pi$
$487$	−21.1590	−0.958805	−0.479403	+	0.877595i	$0.659147\pi$
$488$	43.0234	1.94758
$489$	38.3143	1.73263
$490$	−39.8595	−1.80067
$491$	17.7063	0.799076	0.399538	−	0.916717i	$-0.369171\pi$
$491$	17.7063	0.799076	0.399538	+	0.916717i	$0.369171\pi$
$492$	−61.1439	−2.75658
$493$	0	0
$494$	−7.47673	−0.336394
$495$	13.2007	0.593326
$496$	10.0349	0.450579
$497$	5.88707	0.264071
$498$	3.37748	0.151349
$499$	29.2051	1.30740	0.653700	−	0.756754i	$-0.273216\pi$
$499$	29.2051	1.30740	0.653700	+	0.756754i	$0.273216\pi$
$500$	37.2025	1.66374
$501$	−5.32453	−0.237883
$502$	66.9524	2.98823
$503$	−43.0842	−1.92103	−0.960516	−	0.278225i	$-0.910254\pi$
$503$	−43.0842	−1.92103	−0.960516	+	0.278225i	$0.910254\pi$
$504$	2.22208	0.0989795
$505$	14.6623	0.652463
$506$	−3.08243	−0.137031
$507$	−38.4263	−1.70657
$508$	21.7619	0.965527
$509$	32.4926	1.44021	0.720104	−	0.693866i	$-0.244094\pi$
$509$	32.4926	1.44021	0.720104	+	0.693866i	$0.244094\pi$
$510$	−22.3079	−0.987813
$511$	−2.62383	−0.116072
$512$	26.3609	1.16500
$513$	1.85860	0.0820591
$514$	45.6554	2.01377
$515$	−15.0494	−0.663154
$516$	0.178006	0.00783628
$517$	29.2978	1.28852
$518$	7.30185	0.320825
$519$	−17.8296	−0.782631
$520$	56.3335	2.47039
$521$	28.2658	1.23835	0.619173	−	0.785254i	$-0.287468\pi$
$521$	28.2658	1.23835	0.619173	+	0.785254i	$0.287468\pi$
$522$	0	0
$523$	−8.63499	−0.377582	−0.188791	−	0.982017i	$-0.560457\pi$
$523$	−8.63499	−0.377582	−0.188791	+	0.982017i	$0.560457\pi$
$524$	1.97155	0.0861274
$525$	0.711459	0.0310506
$526$	−17.0349	−0.742758
$527$	7.42915	0.323619
$528$	19.9066	0.866322
$529$	−22.8876	−0.995114
$530$	37.0446	1.60911
$531$	11.5198	0.499915
$532$	−0.788463	−0.0341842
$533$	43.6038	1.88869
$534$	−48.4970	−2.09867
$535$	−12.8717	−0.556492
$536$	49.9253	2.15644
$537$	−27.2189	−1.17458
$538$	14.2605	0.614812
$539$	26.3327	1.13423
$540$	−30.1691	−1.29827
$541$	−10.5447	−0.453352	−0.226676	−	0.973970i	$-0.572786\pi$
$541$	−10.5447	−0.453352	−0.226676	+	0.973970i	$0.572786\pi$
$542$	−12.2910	−0.527945
$543$	−34.4982	−1.48046
$544$	4.36513	0.187154
$545$	−15.2942	−0.655132
$546$	−10.6480	−0.455694
$547$	38.7197	1.65554	0.827768	−	0.561070i	$-0.189610\pi$
$547$	38.7197	1.65554	0.827768	+	0.561070i	$0.189610\pi$
$548$	−29.3452	−1.25356
$549$	−14.6839	−0.626692
$550$	8.22765	0.350828
$551$	0	0
$552$	−2.92217	−0.124376
$553$	2.82311	0.120051
$554$	−47.8989	−2.03503
$555$	41.1226	1.74556
$556$	−34.5194	−1.46395
$557$	8.38724	0.355379	0.177689	−	0.984087i	$-0.443138\pi$
$557$	8.38724	0.355379	0.177689	+	0.984087i	$0.443138\pi$
$558$	−13.7896	−0.583762
$559$	−0.126942	−0.00536908
$560$	2.26607	0.0957589
$561$	14.7375	0.622217
$562$	28.0876	1.18480
$563$	−27.6892	−1.16696	−0.583480	−	0.812127i	$-0.698309\pi$
$563$	−27.6892	−1.16696	−0.583480	+	0.812127i	$0.698309\pi$
$564$	59.8369	2.51959
$565$	32.5586	1.36975
$566$	70.9241	2.98116
$567$	4.25398	0.178650
$568$	64.5412	2.70809
$569$	−9.79842	−0.410771	−0.205386	−	0.978681i	$-0.565845\pi$
$569$	−9.79842	−0.410771	−0.205386	+	0.978681i	$0.565845\pi$
$570$	−6.81981	−0.285650
$571$	−14.4018	−0.602696	−0.301348	−	0.953514i	$-0.597437\pi$
$571$	−14.4018	−0.602696	−0.301348	+	0.953514i	$0.597437\pi$
$572$	−80.1775	−3.35239
$573$	15.5951	0.651493
$574$	7.06215	0.294768
$575$	−0.299971	−0.0125097
$576$	−15.0870	−0.628623
$577$	10.3904	0.432557	0.216279	−	0.976332i	$-0.430608\pi$
$577$	10.3904	0.432557	0.216279	+	0.976332i	$0.430608\pi$
$578$	32.7177	1.36088
$579$	−46.1663	−1.91861
$580$	0	0
$581$	−0.254000	−0.0105377
$582$	−22.4152	−0.929139
$583$	−24.4730	−1.01357
$584$	−28.7657	−1.19033
$585$	−19.2266	−0.794920
$586$	−18.8234	−0.777587
$587$	1.62703	0.0671548	0.0335774	−	0.999436i	$-0.489310\pi$
$587$	1.62703	0.0671548	0.0335774	+	0.999436i	$0.489310\pi$
$588$	53.7811	2.21790
$589$	2.27118	0.0935823
$590$	47.3001	1.94732
$591$	−41.4060	−1.70321
$592$	19.8822	0.817154
$593$	31.0584	1.27541	0.637707	−	0.770279i	$-0.279883\pi$
$593$	31.0584	1.27541	0.637707	+	0.770279i	$0.279883\pi$
$594$	30.6104	1.25596
$595$	1.67765	0.0687768
$596$	−1.91955	−0.0786278
$597$	34.1226	1.39654
$598$	4.48951	0.183590
$599$	16.3366	0.667495	0.333748	−	0.942662i	$-0.391687\pi$
$599$	16.3366	0.667495	0.333748	+	0.942662i	$0.391687\pi$
$600$	7.79988	0.318429
$601$	9.95353	0.406013	0.203007	−	0.979177i	$-0.434929\pi$
$601$	9.95353	0.406013	0.203007	+	0.979177i	$0.434929\pi$
$602$	−0.0205598	−0.000837953 0
$603$	−17.0394	−0.693900
$604$	−24.4568	−0.995133
$605$	−9.10072	−0.369997
$606$	−30.3838	−1.23426
$607$	−26.2102	−1.06384	−0.531920	−	0.846794i	$-0.678529\pi$
$607$	−26.2102	−1.06384	−0.531920	+	0.846794i	$0.678529\pi$
$608$	1.33447	0.0541201
$609$	0	0
$610$	−60.2919	−2.44115
$611$	−42.6717	−1.72631
$612$	9.65032	0.390091
$613$	−46.4433	−1.87583	−0.937913	−	0.346871i	$-0.887244\pi$
$613$	−46.4433	−1.87583	−0.937913	+	0.346871i	$0.887244\pi$
$614$	67.9520	2.74232
$615$	39.7727	1.60379
$616$	−6.02758	−0.242858
$617$	9.57571	0.385504	0.192752	−	0.981248i	$-0.438259\pi$
$617$	9.57571	0.385504	0.192752	+	0.981248i	$0.438259\pi$
$618$	31.1860	1.25448
$619$	20.0731	0.806808	0.403404	−	0.915022i	$-0.367827\pi$
$619$	20.0731	0.806808	0.403404	+	0.915022i	$0.367827\pi$
$620$	−36.8662	−1.48058
$621$	−1.11602	−0.0447844
$622$	20.5232	0.822904
$623$	3.64717	0.146121
$624$	−28.9936	−1.16067
$625$	−28.6734	−1.14693
$626$	−2.10457	−0.0841157
$627$	4.50543	0.179929
$628$	−42.7596	−1.70629
$629$	14.7195	0.586903
$630$	−3.11397	−0.124063
$631$	−28.5654	−1.13717	−0.568585	−	0.822625i	$-0.692509\pi$
$631$	−28.5654	−1.13717	−0.568585	+	0.822625i	$0.692509\pi$
$632$	30.9504	1.23114
$633$	−49.4576	−1.96576
$634$	−63.7489	−2.53179
$635$	−14.1556	−0.561747
$636$	−49.9830	−1.98196
$637$	−38.3531	−1.51961
$638$	0	0
$639$	−22.0278	−0.871408
$640$	−50.3402	−1.98987
$641$	7.47195	0.295124	0.147562	−	0.989053i	$-0.452857\pi$
$641$	7.47195	0.295124	0.147562	+	0.989053i	$0.452857\pi$
$642$	26.6733	1.05271
$643$	−20.9078	−0.824524	−0.412262	−	0.911065i	$-0.635261\pi$
$643$	−20.9078	−0.824524	−0.412262	+	0.911065i	$0.635261\pi$
$644$	0.473444	0.0186563
$645$	−0.115789	−0.00455917
$646$	−2.44109	−0.0960433
$647$	−9.13842	−0.359268	−0.179634	−	0.983733i	$-0.557491\pi$
$647$	−9.13842	−0.359268	−0.179634	+	0.983733i	$0.557491\pi$
$648$	46.6373	1.83208
$649$	−31.2483	−1.22660
$650$	−11.9834	−0.470029
$651$	3.23452	0.126771
$652$	−68.0553	−2.66525
$653$	−14.6392	−0.572876	−0.286438	−	0.958099i	$-0.592471\pi$
$653$	−14.6392	−0.572876	−0.286438	+	0.958099i	$0.592471\pi$
$654$	31.6934	1.23931
$655$	−1.28244	−0.0501092
$656$	19.2295	0.750787
$657$	9.81769	0.383025
$658$	−6.91119	−0.269426
$659$	−38.3439	−1.49367	−0.746833	−	0.665011i	$-0.768427\pi$
$659$	−38.3439	−1.49367	−0.746833	+	0.665011i	$0.768427\pi$
$660$	−73.1329	−2.84670
$661$	−24.0889	−0.936951	−0.468475	−	0.883477i	$-0.655197\pi$
$661$	−24.0889	−0.936951	−0.468475	+	0.883477i	$0.655197\pi$
$662$	44.6500	1.73537
$663$	−21.4649	−0.833627
$664$	−2.78466	−0.108066
$665$	0.512877	0.0198885
$666$	−27.3216	−1.05869
$667$	0	0
$668$	9.45763	0.365927
$669$	1.47502	0.0570275
$670$	−69.9639	−2.70294
$671$	39.8311	1.53766
$672$	1.90050	0.0733134
$673$	−2.01034	−0.0774928	−0.0387464	−	0.999249i	$-0.512336\pi$
$673$	−2.01034	−0.0774928	−0.0387464	+	0.999249i	$0.512336\pi$
$674$	−14.4799	−0.557745
$675$	2.97889	0.114658
$676$	68.2542	2.62516
$677$	16.4646	0.632788	0.316394	−	0.948628i	$-0.397528\pi$
$677$	16.4646	0.632788	0.316394	+	0.948628i	$0.397528\pi$
$678$	−67.4695	−2.59115
$679$	1.68571	0.0646916
$680$	18.3924	0.705316
$681$	15.5554	0.596084
$682$	37.4055	1.43233
$683$	−4.43283	−0.169617	−0.0848087	−	0.996397i	$-0.527028\pi$
$683$	−4.43283	−0.169617	−0.0848087	+	0.996397i	$0.527028\pi$
$684$	2.95022	0.112804
$685$	19.0883	0.729328
$686$	−12.5532	−0.479283
$687$	22.5807	0.861506
$688$	−0.0559822	−0.00213430
$689$	35.6446	1.35795
$690$	4.09505	0.155896
$691$	20.1730	0.767416	0.383708	−	0.923454i	$-0.374647\pi$
$691$	20.1730	0.767416	0.383708	+	0.923454i	$0.374647\pi$
$692$	31.6695	1.20389
$693$	2.05721	0.0781468
$694$	34.5897	1.31301
$695$	22.4541	0.851731
$696$	0	0
$697$	14.2363	0.539237
$698$	−27.5898	−1.04429
$699$	32.7579	1.23902
$700$	−1.26372	−0.0477641
$701$	−31.8660	−1.20356	−0.601781	−	0.798661i	$-0.705542\pi$
$701$	−31.8660	−1.20356	−0.601781	+	0.798661i	$0.705542\pi$
$702$	−44.5835	−1.68270
$703$	4.49992	0.169718
$704$	40.9246	1.54240
$705$	−38.9225	−1.46591
$706$	20.2872	0.763520
$707$	2.28498	0.0859356
$708$	−63.8205	−2.39852
$709$	−46.7660	−1.75633	−0.878167	−	0.478354i	$-0.841234\pi$
$709$	−46.7660	−1.75633	−0.878167	+	0.478354i	$0.841234\pi$
$710$	−90.4463	−3.39439
$711$	−10.5633	−0.396156
$712$	39.9847	1.49849
$713$	−1.36376	−0.0510733
$714$	−3.47649	−0.130104
$715$	52.1535	1.95043
$716$	48.3472	1.80682
$717$	−15.6426	−0.584184
$718$	−11.8164	−0.440983
$719$	34.7679	1.29662	0.648312	−	0.761375i	$-0.275475\pi$
$719$	34.7679	1.29662	0.648312	+	0.761375i	$0.275475\pi$
$720$	−8.47903	−0.315995
$721$	−2.34531	−0.0873438
$722$	44.7449	1.66523
$723$	11.0677	0.411611
$724$	61.2769	2.27734
$725$	0	0
$726$	18.8589	0.699920
$727$	−23.0602	−0.855257	−0.427628	−	0.903955i	$-0.640651\pi$
$727$	−23.0602	−0.855257	−0.427628	+	0.903955i	$0.640651\pi$
$728$	8.77906	0.325374
$729$	5.06963	0.187764
$730$	40.3114	1.49199
$731$	−0.0414455	−0.00153292
$732$	81.3499	3.00678
$733$	−38.4511	−1.42022	−0.710112	−	0.704088i	$-0.751356\pi$
$733$	−38.4511	−1.42022	−0.710112	+	0.704088i	$0.751356\pi$
$734$	6.96020	0.256906
$735$	−34.9833	−1.29038
$736$	−0.801304	−0.0295365
$737$	46.2208	1.70257
$738$	−26.4247	−0.972706
$739$	29.0800	1.06972	0.534862	−	0.844940i	$-0.320364\pi$
$739$	29.0800	1.06972	0.534862	+	0.844940i	$0.320364\pi$
$740$	−73.0435	−2.68513
$741$	−6.56208	−0.241064
$742$	5.77306	0.211936
$743$	−32.6399	−1.19744	−0.598721	−	0.800958i	$-0.704324\pi$
$743$	−32.6399	−1.19744	−0.598721	+	0.800958i	$0.704324\pi$
$744$	35.4607	1.30005
$745$	1.24862	0.0457460
$746$	74.5720	2.73028
$747$	0.950400	0.0347733
$748$	−26.1773	−0.957135
$749$	−2.00594	−0.0732954
$750$	50.1469	1.83111
$751$	0.139181	0.00507879	0.00253940	−	0.999997i	$-0.499192\pi$
$751$	0.139181	0.00507879	0.00253940	+	0.999997i	$0.499192\pi$
$752$	−18.8185	−0.686240
$753$	58.7619	2.14140
$754$	0	0
$755$	15.9086	0.578972
$756$	−4.70158	−0.170995
$757$	−1.71328	−0.0622703	−0.0311352	−	0.999515i	$-0.509912\pi$
$757$	−1.71328	−0.0622703	−0.0311352	+	0.999515i	$0.509912\pi$
$758$	−58.1228	−2.11112
$759$	−2.70535	−0.0981979
$760$	5.62278	0.203960
$761$	−38.7063	−1.40310	−0.701551	−	0.712620i	$-0.747509\pi$
$761$	−38.7063	−1.40310	−0.701551	+	0.712620i	$0.747509\pi$
$762$	29.3338	1.06265
$763$	−2.38347	−0.0862873
$764$	−27.7005	−1.00217
$765$	−6.27730	−0.226956
$766$	71.2723	2.57517
$767$	45.5126	1.64336
$768$	59.5309	2.14814
$769$	45.4266	1.63813	0.819063	−	0.573703i	$-0.194494\pi$
$769$	45.4266	1.63813	0.819063	+	0.573703i	$0.194494\pi$
$770$	8.44688	0.304404
$771$	40.0702	1.44309
$772$	82.0023	2.95133
$773$	50.6609	1.82215	0.911074	−	0.412243i	$-0.135255\pi$
$773$	50.6609	1.82215	0.911074	+	0.412243i	$0.135255\pi$
$774$	0.0769292	0.00276516
$775$	3.64016	0.130758
$776$	18.4808	0.663422
$777$	6.40858	0.229907
$778$	−25.8249	−0.925869
$779$	4.35220	0.155934
$780$	106.517	3.81391
$781$	59.7523	2.13810
$782$	1.46579	0.0524164
$783$	0	0
$784$	−16.9140	−0.604070
$785$	27.8141	0.992727
$786$	2.65754	0.0947912
$787$	40.0683	1.42828	0.714140	−	0.700003i	$-0.246818\pi$
$787$	40.0683	1.42828	0.714140	+	0.700003i	$0.246818\pi$
$788$	73.5468	2.62000
$789$	−14.9510	−0.532269
$790$	−43.3730	−1.54314
$791$	5.07397	0.180410
$792$	22.5536	0.801407
$793$	−58.0133	−2.06011
$794$	−2.73552	−0.0970800
$795$	32.5128	1.15311
$796$	−60.6098	−2.14826
$797$	9.11767	0.322965	0.161482	−	0.986876i	$-0.448373\pi$
$797$	9.11767	0.322965	0.161482	+	0.986876i	$0.448373\pi$
$798$	−1.06281	−0.0376229
$799$	−13.9320	−0.492877
$800$	2.13885	0.0756196
$801$	−13.6467	−0.482184
$802$	9.22181	0.325633
$803$	−26.6313	−0.939797
$804$	94.4000	3.32923
$805$	−0.307964	−0.0108543
$806$	−54.4804	−1.91899
$807$	12.5159	0.440581
$808$	25.0508	0.881283
$809$	−4.31479	−0.151700	−0.0758499	−	0.997119i	$-0.524167\pi$
$809$	−4.31479	−0.151700	−0.0758499	+	0.997119i	$0.524167\pi$
$810$	−65.3562	−2.29638
$811$	−46.7124	−1.64029	−0.820147	−	0.572153i	$-0.806108\pi$
$811$	−46.7124	−1.64029	−0.820147	+	0.572153i	$0.806108\pi$
$812$	0	0
$813$	−10.7874	−0.378331
$814$	74.1119	2.59762
$815$	44.2684	1.55065
$816$	−9.46615	−0.331381
$817$	−0.0126704	−0.000443281 0
$818$	46.0907	1.61152
$819$	−2.99628	−0.104699
$820$	−70.6456	−2.46705
$821$	−2.27936	−0.0795504	−0.0397752	−	0.999209i	$-0.512664\pi$
$821$	−2.27936	−0.0795504	−0.0397752	+	0.999209i	$0.512664\pi$
$822$	−39.5557	−1.37966
$823$	−26.1115	−0.910191	−0.455096	−	0.890443i	$-0.650395\pi$
$823$	−26.1115	−0.910191	−0.455096	+	0.890443i	$0.650395\pi$
$824$	−25.7121	−0.895724
$825$	7.22113	0.251407
$826$	7.37130	0.256480
$827$	25.6037	0.890327	0.445164	−	0.895449i	$-0.353146\pi$
$827$	25.6037	0.890327	0.445164	+	0.895449i	$0.353146\pi$
$828$	−1.77150	−0.0615639
$829$	−43.4804	−1.51014	−0.755069	−	0.655645i	$-0.772397\pi$
$829$	−43.4804	−1.51014	−0.755069	+	0.655645i	$0.772397\pi$
$830$	3.90234	0.135452
$831$	−42.0392	−1.45832
$832$	−59.6059	−2.06646
$833$	−12.5220	−0.433860
$834$	−46.5303	−1.61121
$835$	−6.15196	−0.212898
$836$	−8.00270	−0.276779
$837$	13.5430	0.468114
$838$	4.79982	0.165807
$839$	−46.9937	−1.62240	−0.811202	−	0.584766i	$-0.801186\pi$
$839$	−46.9937	−1.62240	−0.811202	+	0.584766i	$0.801186\pi$
$840$	8.00771	0.276292
$841$	0	0
$842$	95.4268	3.28863
$843$	24.6515	0.849043
$844$	87.8484	3.02387
$845$	−44.3977	−1.52733
$846$	25.8598	0.889079
$847$	−1.41827	−0.0487322
$848$	15.7195	0.539809
$849$	62.2477	2.13633
$850$	−3.91248	−0.134197
$851$	−2.70204	−0.0926247
$852$	122.036	4.18089
$853$	25.7853	0.882872	0.441436	−	0.897293i	$-0.354469\pi$
$853$	25.7853	0.882872	0.441436	+	0.897293i	$0.354469\pi$
$854$	−9.39594	−0.321523
$855$	−1.91905	−0.0656300
$856$	−21.9915	−0.751655
$857$	−51.3809	−1.75514	−0.877569	−	0.479450i	$-0.840836\pi$
$857$	−51.3809	−1.75514	−0.877569	+	0.479450i	$0.840836\pi$
$858$	−108.075	−3.68962
$859$	18.1709	0.619982	0.309991	−	0.950740i	$-0.399674\pi$
$859$	18.1709	0.619982	0.309991	+	0.950740i	$0.399674\pi$
$860$	0.205668	0.00701322
$861$	6.19821	0.211234
$862$	−34.2735	−1.16736
$863$	−14.6394	−0.498330	−0.249165	−	0.968461i	$-0.580156\pi$
$863$	−14.6394	−0.498330	−0.249165	+	0.968461i	$0.580156\pi$
$864$	7.95743	0.270717
$865$	−20.6003	−0.700430
$866$	21.1511	0.718743
$867$	28.7152	0.975219
$868$	−5.74527	−0.195007
$869$	28.6539	0.972017
$870$	0	0
$871$	−67.3198	−2.28105
$872$	−26.1305	−0.884889
$873$	−6.30748	−0.213476
$874$	0.448109	0.0151575
$875$	−3.77124	−0.127491
$876$	−54.3909	−1.83770
$877$	20.1811	0.681467	0.340734	−	0.940160i	$-0.389325\pi$
$877$	20.1811	0.681467	0.340734	+	0.940160i	$0.389325\pi$
$878$	27.7318	0.935904
$879$	−16.5206	−0.557228
$880$	23.0000	0.775331
$881$	17.5588	0.591572	0.295786	−	0.955254i	$-0.404419\pi$
$881$	17.5588	0.591572	0.295786	+	0.955254i	$0.404419\pi$
$882$	23.2427	0.782622
$883$	−8.49192	−0.285776	−0.142888	−	0.989739i	$-0.545639\pi$
$883$	−8.49192	−0.285776	−0.142888	+	0.989739i	$0.545639\pi$
$884$	38.1267	1.28234
$885$	41.5137	1.39547
$886$	84.9704	2.85464
$887$	−9.27160	−0.311310	−0.155655	−	0.987812i	$-0.549749\pi$
$887$	−9.27160	−0.311310	−0.155655	+	0.987812i	$0.549749\pi$
$888$	70.2587	2.35773
$889$	−2.20602	−0.0739875
$890$	−56.0335	−1.87825
$891$	43.1768	1.44648
$892$	−2.61998	−0.0877234
$893$	−4.25917	−0.142528
$894$	−2.58745	−0.0865373
$895$	−31.4487	−1.05121
$896$	−7.84506	−0.262085
$897$	3.94029	0.131563
$898$	−49.6776	−1.65776
$899$	0	0
$900$	4.72850	0.157617
$901$	11.6376	0.387706
$902$	71.6790	2.38665
$903$	−0.0180446	−0.000600487 0
$904$	55.6270	1.85013
$905$	−39.8592	−1.32496
$906$	−32.9664	−1.09524
$907$	19.9856	0.663610	0.331805	−	0.943348i	$-0.392342\pi$
$907$	19.9856	0.663610	0.331805	+	0.943348i	$0.392342\pi$
$908$	−27.6300	−0.916935
$909$	−8.54980	−0.283579
$910$	−12.3027	−0.407832
$911$	−14.2969	−0.473676	−0.236838	−	0.971549i	$-0.576111\pi$
$911$	−14.2969	−0.473676	−0.236838	+	0.971549i	$0.576111\pi$
$912$	−2.89392	−0.0958271
$913$	−2.57804	−0.0853205
$914$	−91.2305	−3.01764
$915$	−52.9162	−1.74935
$916$	−40.1086	−1.32523
$917$	−0.199857	−0.00659987
$918$	−14.5561	−0.480424
$919$	57.8964	1.90983	0.954913	−	0.296885i	$-0.0959476\pi$
$919$	57.8964	1.90983	0.954913	+	0.296885i	$0.0959476\pi$
$920$	−3.37628	−0.111313
$921$	59.6392	1.96518
$922$	−42.4369	−1.39759
$923$	−87.0282	−2.86457
$924$	−11.3971	−0.374937
$925$	7.21230	0.237139
$926$	−22.0731	−0.725367
$927$	8.77552	0.288226
$928$	0	0
$929$	28.1340	0.923048	0.461524	−	0.887128i	$-0.347303\pi$
$929$	28.1340	0.923048	0.461524	+	0.887128i	$0.347303\pi$
$930$	−49.6937	−1.62952
$931$	−3.82812	−0.125461
$932$	−58.1859	−1.90594
$933$	18.0125	0.589702
$934$	−18.6805	−0.611244
$935$	17.0277	0.556865
$936$	−32.8489	−1.07370
$937$	−47.1882	−1.54157	−0.770786	−	0.637094i	$-0.780136\pi$
$937$	−47.1882	−1.54157	−0.770786	+	0.637094i	$0.780136\pi$
$938$	−10.9032	−0.356003
$939$	−1.84711	−0.0602783
$940$	69.1356	2.25495
$941$	−22.1832	−0.723153	−0.361576	−	0.932342i	$-0.617761\pi$
$941$	−22.1832	−0.723153	−0.361576	+	0.932342i	$0.617761\pi$
$942$	−57.6376	−1.87793
$943$	−2.61334	−0.0851020
$944$	20.0713	0.653266
$945$	3.05827	0.0994854
$946$	−0.208676	−0.00678466
$947$	31.0675	1.00956	0.504780	−	0.863248i	$-0.331574\pi$
$947$	31.0675	1.00956	0.504780	+	0.863248i	$0.331574\pi$
$948$	58.5218	1.90070
$949$	38.7880	1.25911
$950$	−1.19609	−0.0388064
$951$	−55.9503	−1.81431
$952$	2.86629	0.0928969
$953$	20.2566	0.656175	0.328088	−	0.944647i	$-0.393596\pi$
$953$	20.2566	0.656175	0.328088	+	0.944647i	$0.393596\pi$
$954$	−21.6013	−0.699366
$955$	18.0185	0.583066
$956$	27.7850	0.898630
$957$	0	0
$958$	42.4329	1.37094
$959$	2.97475	0.0960595
$960$	−54.3688	−1.75475
$961$	−14.4507	−0.466151
$962$	−107.943	−3.48021
$963$	7.50569	0.241867
$964$	−19.6588	−0.633167
$965$	−53.3406	−1.71709
$966$	0.638177	0.0205330
$967$	26.0892	0.838972	0.419486	−	0.907762i	$-0.362210\pi$
$967$	26.0892	0.838972	0.419486	+	0.907762i	$0.362210\pi$
$968$	−15.5487	−0.499756
$969$	−2.14246	−0.0688258
$970$	−25.8985	−0.831551
$971$	−15.3621	−0.492993	−0.246496	−	0.969144i	$-0.579279\pi$
$971$	−15.3621	−0.492993	−0.246496	+	0.969144i	$0.579279\pi$
$972$	50.9053	1.63279
$973$	3.49926	0.112181
$974$	50.6603	1.62326
$975$	−10.5174	−0.336828
$976$	−25.5842	−0.818931
$977$	42.4438	1.35790	0.678949	−	0.734186i	$-0.262436\pi$
$977$	42.4438	1.35790	0.678949	+	0.734186i	$0.262436\pi$
$978$	−91.7349	−2.93336
$979$	37.0178	1.18310
$980$	62.1387	1.98495
$981$	8.91830	0.284739
$982$	−42.3938	−1.35284
$983$	−29.9384	−0.954886	−0.477443	−	0.878663i	$-0.658436\pi$
$983$	−29.9384	−0.954886	−0.477443	+	0.878663i	$0.658436\pi$
$984$	67.9523	2.16624
$985$	−47.8405	−1.52432
$986$	0	0
$987$	−6.06572	−0.193074
$988$	11.6558	0.370820
$989$	0.00760811	0.000241924 0
$990$	−31.6060	−1.00450
$991$	−50.7629	−1.61254	−0.806269	−	0.591550i	$-0.798516\pi$
$991$	−50.7629	−1.61254	−0.806269	+	0.591550i	$0.798516\pi$
$992$	9.72387	0.308733
$993$	39.1878	1.24359
$994$	−14.0952	−0.447074
$995$	39.4252	1.24986
$996$	−5.26530	−0.166838
$997$	8.21297	0.260107	0.130054	−	0.991507i	$-0.458485\pi$
$997$	8.21297	0.260107	0.130054	+	0.991507i	$0.458485\pi$
$998$	−69.9250	−2.21344
$999$	26.8328	0.848954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.2.a.i.1.2	✓	8
3.2	odd	2		7569.2.a.bi.1.7		8
29.2	odd	28		841.2.e.m.236.2		96
29.3	odd	28		841.2.e.m.270.2		96
29.4	even	14		841.2.d.p.190.2		48
29.5	even	14		841.2.d.p.605.2		48
29.6	even	14		841.2.d.p.645.2		48
29.7	even	7		841.2.d.q.571.7		48
29.8	odd	28		841.2.e.m.267.2		96
29.9	even	14		841.2.d.p.574.7		48
29.10	odd	28		841.2.e.m.651.2		96
29.11	odd	28		841.2.e.m.63.2		96
29.12	odd	4		841.2.b.f.840.2		16
29.13	even	14		841.2.d.p.778.7		48
29.14	odd	28		841.2.e.m.196.15		96
29.15	odd	28		841.2.e.m.196.2		96
29.16	even	7		841.2.d.q.778.2		48
29.17	odd	4		841.2.b.f.840.15		16
29.18	odd	28		841.2.e.m.63.15		96
29.19	odd	28		841.2.e.m.651.15		96
29.20	even	7		841.2.d.q.574.2		48
29.21	odd	28		841.2.e.m.267.15		96
29.22	even	14		841.2.d.p.571.2		48
29.23	even	7		841.2.d.q.645.7		48
29.24	even	7		841.2.d.q.605.7		48
29.25	even	7		841.2.d.q.190.7		48
29.26	odd	28		841.2.e.m.270.15		96
29.27	odd	28		841.2.e.m.236.15		96
29.28	even	2		841.2.a.j.1.7	yes	8
87.86	odd	2		7569.2.a.bd.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
841.2.a.i.1.2	✓	8	1.1	even	1	trivial
841.2.a.j.1.7	yes	8	29.28	even	2
841.2.b.f.840.2		16	29.12	odd	4
841.2.b.f.840.15		16	29.17	odd	4
841.2.d.p.190.2		48	29.4	even	14
841.2.d.p.571.2		48	29.22	even	14
841.2.d.p.574.7		48	29.9	even	14
841.2.d.p.605.2		48	29.5	even	14
841.2.d.p.645.2		48	29.6	even	14
841.2.d.p.778.7		48	29.13	even	14
841.2.d.q.190.7		48	29.25	even	7
841.2.d.q.571.7		48	29.7	even	7
841.2.d.q.574.2		48	29.20	even	7
841.2.d.q.605.7		48	29.24	even	7
841.2.d.q.645.7		48	29.23	even	7
841.2.d.q.778.2		48	29.16	even	7
841.2.e.m.63.2		96	29.11	odd	28
841.2.e.m.63.15		96	29.18	odd	28
841.2.e.m.196.2		96	29.15	odd	28
841.2.e.m.196.15		96	29.14	odd	28
841.2.e.m.236.2		96	29.2	odd	28
841.2.e.m.236.15		96	29.27	odd	28
841.2.e.m.267.2		96	29.8	odd	28
841.2.e.m.267.15		96	29.21	odd	28
841.2.e.m.270.2		96	29.3	odd	28
841.2.e.m.270.15		96	29.26	odd	28
841.2.e.m.651.2		96	29.10	odd	28
841.2.e.m.651.15		96	29.19	odd	28
7569.2.a.bd.1.2		8	87.86	odd	2
7569.2.a.bi.1.7		8	3.2	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:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39427 q^{2} -2.10137 q^{3} +3.73253 q^{4} -2.42792 q^{5} +5.03125 q^{6} -0.378370 q^{7} -4.14815 q^{8} +1.41576 q^{9} +5.81310 q^{10} -3.84036 q^{11} -7.84343 q^{12} +5.59342 q^{13} +0.905920 q^{14} +5.10197 q^{15} +2.46673 q^{16} +1.82620 q^{17} -3.38971 q^{18} +0.558292 q^{19} -9.06230 q^{20} +0.795095 q^{21} +9.19486 q^{22} -0.335234 q^{23} +8.71680 q^{24} +0.894809 q^{25} -13.3922 q^{26} +3.32908 q^{27} -1.41228 q^{28} -12.2155 q^{30} +4.06809 q^{31} +2.39028 q^{32} +8.07002 q^{33} -4.37242 q^{34} +0.918653 q^{35} +5.28437 q^{36} +8.06015 q^{37} -1.33670 q^{38} -11.7538 q^{39} +10.0714 q^{40} +7.79555 q^{41} -1.90367 q^{42} -0.0226949 q^{43} -14.3343 q^{44} -3.43735 q^{45} +0.802642 q^{46} -7.62892 q^{47} -5.18351 q^{48} -6.85684 q^{49} -2.14242 q^{50} -3.83753 q^{51} +20.8776 q^{52} +6.37259 q^{53} -7.97071 q^{54} +9.32410 q^{55} +1.56953 q^{56} -1.17318 q^{57} +8.13681 q^{59} +19.0433 q^{60} -10.3717 q^{61} -9.74010 q^{62} -0.535681 q^{63} -10.6564 q^{64} -13.5804 q^{65} -19.3218 q^{66} -12.0355 q^{67} +6.81636 q^{68} +0.704452 q^{69} -2.19950 q^{70} -15.5590 q^{71} -5.87278 q^{72} +6.93458 q^{73} -19.2982 q^{74} -1.88033 q^{75} +2.08384 q^{76} +1.45308 q^{77} +28.1419 q^{78} -7.46125 q^{79} -5.98903 q^{80} -11.2429 q^{81} -18.6647 q^{82} +0.671301 q^{83} +2.96772 q^{84} -4.43388 q^{85} +0.0543377 q^{86} +15.9304 q^{88} -9.63916 q^{89} +8.22996 q^{90} -2.11638 q^{91} -1.25127 q^{92} -8.54856 q^{93} +18.2657 q^{94} -1.35549 q^{95} -5.02287 q^{96} -4.45519 q^{97} +16.4171 q^{98} -5.43702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 3 q^{6} - 15 q^{8} + 8 q^{9} + 3 q^{10} - 5 q^{11} - 12 q^{12} - 4 q^{13} - 15 q^{14} + 7 q^{15} - 2 q^{16} - 9 q^{17} + 11 q^{18} - 17 q^{19} - 7 q^{20} - 25 q^{21}+ \cdots - 15 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 6 * q^3 + 6 * q^4 - q^5 + 3 * q^6 - 15 * q^8 + 8 * q^9 + 3 * q^10 - 5 * q^11 - 12 * q^12 - 4 * q^13 - 15 * q^14 + 7 * q^15 - 2 * q^16 - 9 * q^17 + 11 * q^18 - 17 * q^19 - 7 * q^20 - 25 * q^21 + 25 * q^22 - 7 * q^23 + 25 * q^24 - 13 * q^25 - 3 * q^26 - 18 * q^27 + 25 * q^28 - 26 * q^30 - q^31 + 6 * q^32 + 12 * q^34 - 5 * q^35 - 19 * q^36 - 6 * q^37 - 14 * q^38 - 27 * q^39 + 25 * q^40 - 6 * q^41 + 15 * q^42 + 5 * q^43 - 20 * q^44 - q^45 - 14 * q^46 - 31 * q^47 - 11 * q^48 + 4 * q^49 - 11 * q^50 + 23 * q^51 + 2 * q^52 + 19 * q^53 - 46 * q^54 - 20 * q^55 - 20 * q^56 - 6 * q^57 - q^59 + 9 * q^60 - 21 * q^61 - 12 * q^62 - 10 * q^63 + q^64 - 17 * q^65 - 5 * q^66 + 2 * q^67 - 8 * q^68 + 14 * q^69 + 5 * q^70 - 12 * q^71 + 34 * q^73 - 27 * q^74 - 4 * q^75 + 16 * q^76 + 10 * q^77 + 6 * q^78 - 29 * q^79 - 41 * q^80 + 12 * q^81 - 22 * q^82 - 3 * q^83 + 10 * q^84 - 2 * q^85 - 5 * q^86 - 15 * q^88 - 36 * q^89 + 38 * q^90 - 4 * q^92 + 2 * q^93 + 13 * q^94 - 11 * q^95 - 2 * q^96 + 18 * q^97 - 27 * q^98 - 15 * q^99

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