LMFDB - Embedded newform 841.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,4,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, 4, names="a")

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

chi := DirichletCharacter("841.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$49.6206063148$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} +0.656854 q^{5} -3.82843 q^{6} +6.14214 q^{7} +6.55635 q^{8} +58.4264 q^{9} +O(q^{10})$	\(q-0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} +0.656854 q^{5} -3.82843 q^{6} +6.14214 q^{7} +6.55635 q^{8} +58.4264 q^{9} -0.272078 q^{10} +65.3259 q^{11} -72.3553 q^{12} -49.7696 q^{13} -2.54416 q^{14} +6.07107 q^{15} +59.9117 q^{16} -55.4558 q^{17} -24.2010 q^{18} +64.7452 q^{19} -5.14214 q^{20} +56.7696 q^{21} -27.0589 q^{22} +93.8823 q^{23} +60.5980 q^{24} -124.569 q^{25} +20.6152 q^{26} +290.463 q^{27} -48.0833 q^{28} -2.51472 q^{30} +236.095 q^{31} -77.2670 q^{32} +603.784 q^{33} +22.9706 q^{34} +4.03449 q^{35} -457.387 q^{36} -76.8040 q^{37} -26.8183 q^{38} -460.002 q^{39} +4.30657 q^{40} -215.161 q^{41} -23.5147 q^{42} -80.8305 q^{43} -511.399 q^{44} +38.3776 q^{45} -38.8873 q^{46} +357.742 q^{47} +553.742 q^{48} -305.274 q^{49} +51.5980 q^{50} -512.558 q^{51} +389.617 q^{52} +328.466 q^{53} -120.314 q^{54} +42.9096 q^{55} +40.2700 q^{56} +598.416 q^{57} -99.2750 q^{59} -47.5269 q^{60} +725.730 q^{61} -97.7939 q^{62} +358.863 q^{63} -447.288 q^{64} -32.6913 q^{65} -250.095 q^{66} +844.479 q^{67} +434.132 q^{68} +867.720 q^{69} -1.67114 q^{70} -378.083 q^{71} +383.064 q^{72} +581.097 q^{73} +31.8133 q^{74} -1151.34 q^{75} -506.853 q^{76} +401.241 q^{77} +190.539 q^{78} +353.247 q^{79} +39.3532 q^{80} +1107.13 q^{81} +89.1228 q^{82} +696.510 q^{83} -444.416 q^{84} -36.4264 q^{85} +33.4811 q^{86} +428.299 q^{88} -1118.22 q^{89} -15.8965 q^{90} -305.691 q^{91} -734.950 q^{92} +2182.15 q^{93} -148.182 q^{94} +42.5281 q^{95} -714.151 q^{96} +805.415 q^{97} +126.449 q^{98} +3816.76 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{7} - 18 q^{8} + 32 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 10 * q^3 - 10 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^7 - 18 * q^8 + 32 * q^9	\( 2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{7} - 18 q^{8} + 32 q^{9} - 26 q^{10} + 26 q^{11} - 74 q^{12} - 26 q^{13} - 56 q^{14} - 2 q^{15} + 18 q^{16} - 60 q^{17} - 88 q^{18} + 220 q^{19} + 18 q^{20} + 40 q^{21} - 122 q^{22} + 52 q^{23} + 42 q^{24} - 136 q^{25} + 78 q^{26} + 250 q^{27} - 22 q^{30} + 294 q^{31} + 18 q^{32} + 574 q^{33} + 12 q^{34} + 240 q^{35} - 400 q^{36} - 312 q^{37} + 348 q^{38} - 442 q^{39} + 266 q^{40} - 40 q^{41} - 64 q^{42} + 322 q^{43} - 426 q^{44} + 320 q^{45} - 140 q^{46} + 130 q^{47} + 522 q^{48} - 158 q^{49} + 24 q^{50} - 516 q^{51} + 338 q^{52} + 1002 q^{53} - 218 q^{54} + 462 q^{55} + 584 q^{56} + 716 q^{57} - 900 q^{59} - 30 q^{60} + 948 q^{61} + 42 q^{62} + 944 q^{63} + 118 q^{64} - 286 q^{65} - 322 q^{66} + 320 q^{67} + 444 q^{68} + 836 q^{69} + 568 q^{70} - 660 q^{71} + 1032 q^{72} - 648 q^{73} - 536 q^{74} - 1160 q^{75} - 844 q^{76} + 1272 q^{77} + 234 q^{78} - 258 q^{79} + 486 q^{80} + 1790 q^{81} + 512 q^{82} + 1212 q^{83} - 408 q^{84} + 12 q^{85} + 1006 q^{86} + 1394 q^{88} - 760 q^{89} + 664 q^{90} - 832 q^{91} - 644 q^{92} + 2226 q^{93} - 698 q^{94} - 1612 q^{95} - 642 q^{96} - 24 q^{97} + 482 q^{98} + 4856 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 10 * q^3 - 10 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^7 - 18 * q^8 + 32 * q^9 - 26 * q^10 + 26 * q^11 - 74 * q^12 - 26 * q^13 - 56 * q^14 - 2 * q^15 + 18 * q^16 - 60 * q^17 - 88 * q^18 + 220 * q^19 + 18 * q^20 + 40 * q^21 - 122 * q^22 + 52 * q^23 + 42 * q^24 - 136 * q^25 + 78 * q^26 + 250 * q^27 - 22 * q^30 + 294 * q^31 + 18 * q^32 + 574 * q^33 + 12 * q^34 + 240 * q^35 - 400 * q^36 - 312 * q^37 + 348 * q^38 - 442 * q^39 + 266 * q^40 - 40 * q^41 - 64 * q^42 + 322 * q^43 - 426 * q^44 + 320 * q^45 - 140 * q^46 + 130 * q^47 + 522 * q^48 - 158 * q^49 + 24 * q^50 - 516 * q^51 + 338 * q^52 + 1002 * q^53 - 218 * q^54 + 462 * q^55 + 584 * q^56 + 716 * q^57 - 900 * q^59 - 30 * q^60 + 948 * q^61 + 42 * q^62 + 944 * q^63 + 118 * q^64 - 286 * q^65 - 322 * q^66 + 320 * q^67 + 444 * q^68 + 836 * q^69 + 568 * q^70 - 660 * q^71 + 1032 * q^72 - 648 * q^73 - 536 * q^74 - 1160 * q^75 - 844 * q^76 + 1272 * q^77 + 234 * q^78 - 258 * q^79 + 486 * q^80 + 1790 * q^81 + 512 * q^82 + 1212 * q^83 - 408 * q^84 + 12 * q^85 + 1006 * q^86 + 1394 * q^88 - 760 * q^89 + 664 * q^90 - 832 * q^91 - 644 * q^92 + 2226 * q^93 - 698 * q^94 - 1612 * q^95 - 642 * q^96 - 24 * q^97 + 482 * q^98 + 4856 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.414214	−0.146447	−0.0732233	−	0.997316i	$-0.523329\pi$
$2$	−0.414214	−0.146447	−0.0732233	+	0.997316i	$0.523329\pi$
$3$	9.24264	1.77875	0.889374	−	0.457181i	$-0.151141\pi$
$3$	9.24264	1.77875	0.889374	+	0.457181i	$0.151141\pi$
$4$	−7.82843	−0.978553
$5$	0.656854	0.0587508	0.0293754	−	0.999568i	$-0.490648\pi$
$5$	0.656854	0.0587508	0.0293754	+	0.999568i	$0.490648\pi$
$6$	−3.82843	−0.260491
$7$	6.14214	0.331644	0.165822	−	0.986156i	$-0.446972\pi$
$7$	6.14214	0.331644	0.165822	+	0.986156i	$0.446972\pi$
$8$	6.55635	0.289752
$9$	58.4264	2.16394
$10$	−0.272078	−0.00860386
$11$	65.3259	1.79059	0.895295	−	0.445473i	$-0.146964\pi$
$11$	65.3259	1.79059	0.895295	+	0.445473i	$0.146964\pi$
$12$	−72.3553	−1.74060
$13$	−49.7696	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−49.7696	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	−2.54416	−0.0485682
$15$	6.07107	0.104503
$16$	59.9117	0.936120
$17$	−55.4558	−0.791178	−0.395589	−	0.918428i	$-0.629459\pi$
$17$	−55.4558	−0.791178	−0.395589	+	0.918428i	$0.629459\pi$
$18$	−24.2010	−0.316902
$19$	64.7452	0.781766	0.390883	−	0.920440i	$-0.372170\pi$
$19$	64.7452	0.781766	0.390883	+	0.920440i	$0.372170\pi$
$20$	−5.14214	−0.0574908
$21$	56.7696	0.589911
$22$	−27.0589	−0.262226
$23$	93.8823	0.851122	0.425561	−	0.904930i	$-0.360077\pi$
$23$	93.8823	0.851122	0.425561	+	0.904930i	$0.360077\pi$
$24$	60.5980	0.515396
$25$	−124.569	−0.996548
$26$	20.6152	0.155499
$27$	290.463	2.07036
$28$	−48.0833	−0.324532
$29$	0	0
$29$	0	0
$30$	−2.51472	−0.0153041
$31$	236.095	1.36787	0.683935	−	0.729543i	$-0.260267\pi$
$31$	236.095	1.36787	0.683935	+	0.729543i	$0.260267\pi$
$32$	−77.2670	−0.426844
$33$	603.784	3.18501
$34$	22.9706	0.115865
$35$	4.03449	0.0194844
$36$	−457.387	−2.11753
$37$	−76.8040	−0.341257	−0.170628	−	0.985335i	$-0.554580\pi$
$37$	−76.8040	−0.341257	−0.170628	+	0.985335i	$0.554580\pi$
$38$	−26.8183	−0.114487
$39$	−460.002	−1.88870
$40$	4.30657	0.0170232
$41$	−215.161	−0.819575	−0.409788	−	0.912181i	$-0.634397\pi$
$41$	−215.161	−0.819575	−0.409788	+	0.912181i	$0.634397\pi$
$42$	−23.5147	−0.0863905
$43$	−80.8305	−0.286664	−0.143332	−	0.989675i	$-0.545782\pi$
$43$	−80.8305	−0.286664	−0.143332	+	0.989675i	$0.545782\pi$
$44$	−511.399	−1.75219
$45$	38.3776	0.127133
$46$	−38.8873	−0.124644
$47$	357.742	1.11026	0.555128	−	0.831765i	$-0.312669\pi$
$47$	357.742	1.11026	0.555128	+	0.831765i	$0.312669\pi$
$48$	553.742	1.66512
$49$	−305.274	−0.890012
$50$	51.5980	0.145941
$51$	−512.558	−1.40730
$52$	389.617	1.03904
$53$	328.466	0.851288	0.425644	−	0.904891i	$-0.360048\pi$
$53$	328.466	0.851288	0.425644	+	0.904891i	$0.360048\pi$
$54$	−120.314	−0.303197
$55$	42.9096	0.105199
$56$	40.2700	0.0960947
$57$	598.416	1.39056
$58$	0	0
$59$	−99.2750	−0.219059	−0.109530	−	0.993984i	$-0.534934\pi$
$59$	−99.2750	−0.219059	−0.109530	+	0.993984i	$0.534934\pi$
$60$	−47.5269	−0.102262
$61$	725.730	1.52328	0.761641	−	0.647999i	$-0.224394\pi$
$61$	725.730	1.52328	0.761641	+	0.647999i	$0.224394\pi$
$62$	−97.7939	−0.200320
$63$	358.863	0.717658
$64$	−447.288	−0.873610
$65$	−32.6913	−0.0623825
$66$	−250.095	−0.466434
$67$	844.479	1.53984	0.769922	−	0.638138i	$-0.220295\pi$
$67$	844.479	1.53984	0.769922	+	0.638138i	$0.220295\pi$
$68$	434.132	0.774209
$69$	867.720	1.51393
$70$	−1.67114	−0.00285342
$71$	−378.083	−0.631975	−0.315988	−	0.948763i	$-0.602336\pi$
$71$	−378.083	−0.631975	−0.315988	+	0.948763i	$0.602336\pi$
$72$	383.064	0.627007
$73$	581.097	0.931674	0.465837	−	0.884870i	$-0.345753\pi$
$73$	581.097	0.931674	0.465837	+	0.884870i	$0.345753\pi$
$74$	31.8133	0.0499759
$75$	−1151.34	−1.77261
$76$	−506.853	−0.765000
$77$	401.241	0.593839
$78$	190.539	0.276594
$79$	353.247	0.503081	0.251540	−	0.967847i	$-0.419063\pi$
$79$	353.247	0.503081	0.251540	+	0.967847i	$0.419063\pi$
$80$	39.3532	0.0549978
$81$	1107.13	1.51870
$82$	89.1228	0.120024
$83$	696.510	0.921107	0.460553	−	0.887632i	$-0.347651\pi$
$83$	696.510	0.921107	0.460553	+	0.887632i	$0.347651\pi$
$84$	−444.416	−0.577259
$85$	−36.4264	−0.0464823
$86$	33.4811	0.0419809
$87$	0	0
$88$	428.299	0.518828
$89$	−1118.22	−1.33181	−0.665905	−	0.746037i	$-0.731954\pi$
$89$	−1118.22	−1.33181	−0.665905	+	0.746037i	$0.731954\pi$
$90$	−15.8965	−0.0186182
$91$	−305.691	−0.352145
$92$	−734.950	−0.832868
$93$	2182.15	2.43310
$94$	−148.182	−0.162593
$95$	42.5281	0.0459294
$96$	−714.151	−0.759248
$97$	805.415	0.843068	0.421534	−	0.906813i	$-0.361492\pi$
$97$	805.415	0.843068	0.421534	+	0.906813i	$0.361492\pi$
$98$	126.449	0.130339
$99$	3816.76	3.87473
$100$	975.176	0.975176
$101$	1373.99	1.35363	0.676817	−	0.736151i	$-0.263359\pi$
$101$	1373.99	1.35363	0.676817	+	0.736151i	$0.263359\pi$
$102$	212.309	0.206095
$103$	−634.672	−0.607147	−0.303573	−	0.952808i	$-0.598180\pi$
$103$	−634.672	−0.607147	−0.303573	+	0.952808i	$0.598180\pi$
$104$	−326.307	−0.307663
$105$	37.2893	0.0346578
$106$	−136.055	−0.124668
$107$	−180.956	−0.163493	−0.0817463	−	0.996653i	$-0.526050\pi$
$107$	−180.956	−0.163493	−0.0817463	+	0.996653i	$0.526050\pi$
$108$	−2273.87	−2.02595
$109$	1038.14	0.912251	0.456125	−	0.889916i	$-0.349237\pi$
$109$	1038.14	0.912251	0.456125	+	0.889916i	$0.349237\pi$
$110$	−17.7737	−0.0154060
$111$	−709.872	−0.607010
$112$	367.986	0.310459
$113$	184.765	0.153816	0.0769082	−	0.997038i	$-0.475495\pi$
$113$	184.765	0.153816	0.0769082	+	0.997038i	$0.475495\pi$
$114$	−247.872	−0.203643
$115$	61.6670	0.0500041
$116$	0	0
$117$	−2907.86	−2.29770
$118$	41.1211	0.0320805
$119$	−340.617	−0.262389
$120$	39.8040	0.0302800
$121$	2936.47	2.20622
$122$	−300.607	−0.223079
$123$	−1988.66	−1.45782
$124$	−1848.26	−1.33853
$125$	−163.930	−0.117299
$126$	−148.646	−0.105099
$127$	−1999.58	−1.39712	−0.698558	−	0.715554i	$-0.746175\pi$
$127$	−1999.58	−1.39712	−0.698558	+	0.715554i	$0.746175\pi$
$128$	803.409	0.554781
$129$	−747.087	−0.509902
$130$	13.5412	0.00913570
$131$	−561.468	−0.374471	−0.187236	−	0.982315i	$-0.559953\pi$
$131$	−561.468	−0.374471	−0.187236	+	0.982315i	$0.559953\pi$
$132$	−4726.68	−3.11670
$133$	397.674	0.259268
$134$	−349.795	−0.225505
$135$	190.792	0.121635
$136$	−363.588	−0.229246
$137$	250.489	0.156210	0.0781050	−	0.996945i	$-0.475113\pi$
$137$	250.489	0.156210	0.0781050	+	0.996945i	$0.475113\pi$
$138$	−359.421	−0.221710
$139$	−242.244	−0.147819	−0.0739096	−	0.997265i	$-0.523548\pi$
$139$	−242.244	−0.147819	−0.0739096	+	0.997265i	$0.523548\pi$
$140$	−31.5837	−0.0190665
$141$	3306.48	1.97487
$142$	156.607	0.0925506
$143$	−3251.24	−1.90128
$144$	3500.42	2.02571
$145$	0	0
$146$	−240.698	−0.136441
$147$	−2821.54	−1.58311
$148$	601.255	0.333938
$149$	−1632.63	−0.897652	−0.448826	−	0.893619i	$-0.648158\pi$
$149$	−1632.63	−0.897652	−0.448826	+	0.893619i	$0.648158\pi$
$150$	476.902	0.259592
$151$	−121.582	−0.0655245	−0.0327623	−	0.999463i	$-0.510430\pi$
$151$	−121.582	−0.0655245	−0.0327623	+	0.999463i	$0.510430\pi$
$152$	424.492	0.226519
$153$	−3240.09	−1.71206
$154$	−166.199	−0.0869657
$155$	155.080	0.0803635
$156$	3601.09	1.84819
$157$	753.163	0.382860	0.191430	−	0.981506i	$-0.438688\pi$
$157$	753.163	0.382860	0.191430	+	0.981506i	$0.438688\pi$
$158$	−146.320	−0.0736745
$159$	3035.89	1.51423
$160$	−50.7532	−0.0250774
$161$	576.638	0.282270
$162$	−458.589	−0.222408
$163$	−537.917	−0.258484	−0.129242	−	0.991613i	$-0.541254\pi$
$163$	−537.917	−0.258484	−0.129242	+	0.991613i	$0.541254\pi$
$164$	1684.38	0.801998
$165$	396.598	0.187122
$166$	−288.504	−0.134893
$167$	484.613	0.224554	0.112277	−	0.993677i	$-0.464186\pi$
$167$	484.613	0.224554	0.112277	+	0.993677i	$0.464186\pi$
$168$	372.201	0.170928
$169$	280.008	0.127450
$170$	15.0883	0.00680718
$171$	3782.83	1.69170
$172$	632.776	0.280516
$173$	−3269.70	−1.43694	−0.718469	−	0.695559i	$-0.755157\pi$
$173$	−3269.70	−1.43694	−0.718469	+	0.695559i	$0.755157\pi$
$174$	0	0
$175$	−765.117	−0.330499
$176$	3913.79	1.67621
$177$	−917.563	−0.389651
$178$	463.182	0.195039
$179$	−562.267	−0.234781	−0.117390	−	0.993086i	$-0.537453\pi$
$179$	−562.267	−0.234781	−0.117390	+	0.993086i	$0.537453\pi$
$180$	−300.437	−0.124407
$181$	−1507.32	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$181$	−1507.32	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$182$	126.621	0.0515704
$183$	6707.66	2.70953
$184$	615.525	0.246615
$185$	−50.4491	−0.0200491
$186$	−903.874	−0.356319
$187$	−3622.70	−1.41668
$188$	−2800.56	−1.08645
$189$	1784.06	0.686622
$190$	−17.6157	−0.00672621
$191$	−4532.00	−1.71688	−0.858439	−	0.512915i	$-0.828566\pi$
$191$	−4532.00	−1.71688	−0.858439	+	0.512915i	$0.828566\pi$
$192$	−4134.13	−1.55393
$193$	2935.17	1.09471	0.547353	−	0.836902i	$-0.315636\pi$
$193$	2935.17	1.09471	0.547353	+	0.836902i	$0.315636\pi$
$194$	−333.614	−0.123464
$195$	−302.154	−0.110963
$196$	2389.82	0.870924
$197$	2682.20	0.970043	0.485022	−	0.874502i	$-0.338812\pi$
$197$	2682.20	0.970043	0.485022	+	0.874502i	$0.338812\pi$
$198$	−1580.95	−0.567442
$199$	−648.376	−0.230966	−0.115483	−	0.993309i	$-0.536842\pi$
$199$	−648.376	−0.230966	−0.115483	+	0.993309i	$0.536842\pi$
$200$	−816.715	−0.288752
$201$	7805.22	2.73899
$202$	−569.125	−0.198235
$203$	0	0
$204$	4012.53	1.37712
$205$	−141.330	−0.0481507
$206$	262.890	0.0889145
$207$	5485.20	1.84178
$208$	−2981.78	−0.993986
$209$	4229.54	1.39982
$210$	−15.4457	−0.00507551
$211$	4949.57	1.61489	0.807446	−	0.589941i	$-0.200849\pi$
$211$	4949.57	1.61489	0.807446	+	0.589941i	$0.200849\pi$
$212$	−2571.37	−0.833031
$213$	−3494.49	−1.12412
$214$	74.9545	0.0239429
$215$	−53.0939	−0.0168417
$216$	1904.38	0.599891
$217$	1450.13	0.453646
$218$	−430.010	−0.133596
$219$	5370.87	1.65721
$220$	−335.915	−0.102943
$221$	2760.01	0.840084
$222$	294.039	0.0888945
$223$	2216.94	0.665729	0.332864	−	0.942975i	$-0.391985\pi$
$223$	2216.94	0.665729	0.332864	+	0.942975i	$0.391985\pi$
$224$	−474.585	−0.141560
$225$	−7278.09	−2.15647
$226$	−76.5323	−0.0225259
$227$	4546.09	1.32923	0.664613	−	0.747187i	$-0.268596\pi$
$227$	4546.09	1.32923	0.664613	+	0.747187i	$0.268596\pi$
$228$	−4684.66	−1.36074
$229$	−3339.05	−0.963539	−0.481770	−	0.876298i	$-0.660006\pi$
$229$	−3339.05	−0.963539	−0.481770	+	0.876298i	$0.660006\pi$
$230$	−25.5433	−0.00732293
$231$	3708.52	1.05629
$232$	0	0
$233$	−3995.35	−1.12336	−0.561682	−	0.827353i	$-0.689846\pi$
$233$	−3995.35	−1.12336	−0.561682	+	0.827353i	$0.689846\pi$
$234$	1204.47	0.336491
$235$	234.984	0.0652285
$236$	777.167	0.214361
$237$	3264.93	0.894853
$238$	141.088	0.0384260
$239$	−1400.04	−0.378915	−0.189458	−	0.981889i	$-0.560673\pi$
$239$	−1400.04	−0.378915	−0.189458	+	0.981889i	$0.560673\pi$
$240$	363.728	0.0978272
$241$	−2040.94	−0.545513	−0.272756	−	0.962083i	$-0.587935\pi$
$241$	−2040.94	−0.545513	−0.272756	+	0.962083i	$0.587935\pi$
$242$	−1216.33	−0.323093
$243$	2390.32	0.631026
$244$	−5681.32	−1.49061
$245$	−200.521	−0.0522890
$246$	823.730	0.213492
$247$	−3222.34	−0.830091
$248$	1547.92	0.396344
$249$	6437.59	1.63842
$250$	67.9021	0.0171780
$251$	−802.648	−0.201843	−0.100922	−	0.994894i	$-0.532179\pi$
$251$	−802.648	−0.201843	−0.100922	+	0.994894i	$0.532179\pi$
$252$	−2809.33	−0.702267
$253$	6132.94	1.52401
$254$	828.252	0.204603
$255$	−336.676	−0.0826803
$256$	3245.52	0.792364
$257$	−4464.10	−1.08351	−0.541756	−	0.840536i	$-0.682240\pi$
$257$	−4464.10	−1.08351	−0.541756	+	0.840536i	$0.682240\pi$
$258$	309.454	0.0746734
$259$	−471.741	−0.113176
$260$	255.922	0.0610446
$261$	0	0
$262$	232.568	0.0548400
$263$	3815.21	0.894509	0.447255	−	0.894407i	$-0.352402\pi$
$263$	3815.21	0.894509	0.447255	+	0.894407i	$0.352402\pi$
$264$	3958.62	0.922864
$265$	215.754	0.0500139
$266$	−164.722	−0.0379690
$267$	−10335.3	−2.36895
$268$	−6610.95	−1.50682
$269$	−4523.98	−1.02540	−0.512699	−	0.858569i	$-0.671354\pi$
$269$	−4523.98	−1.02540	−0.512699	+	0.858569i	$0.671354\pi$
$270$	−79.0286	−0.0178131
$271$	−3962.65	−0.888242	−0.444121	−	0.895967i	$-0.646484\pi$
$271$	−3962.65	−0.888242	−0.444121	+	0.895967i	$0.646484\pi$
$272$	−3322.45	−0.740637
$273$	−2825.40	−0.626376
$274$	−103.756	−0.0228764
$275$	−8137.55	−1.78441
$276$	−6792.88	−1.48146
$277$	2217.59	0.481019	0.240509	−	0.970647i	$-0.422685\pi$
$277$	2217.59	0.481019	0.240509	+	0.970647i	$0.422685\pi$
$278$	100.341	0.0216476
$279$	13794.2	2.95999
$280$	26.4515	0.00564564
$281$	−2562.96	−0.544105	−0.272053	−	0.962282i	$-0.587702\pi$
$281$	−2562.96	−0.544105	−0.272053	+	0.962282i	$0.587702\pi$
$282$	−1369.59	−0.289212
$283$	−3869.29	−0.812741	−0.406370	−	0.913708i	$-0.633206\pi$
$283$	−3869.29	−0.812741	−0.406370	+	0.913708i	$0.633206\pi$
$284$	2959.80	0.618421
$285$	393.072	0.0816968
$286$	1346.71	0.278435
$287$	−1321.55	−0.271807
$288$	−4514.43	−0.923665
$289$	−1837.65	−0.374038
$290$	0	0
$291$	7444.17	1.49960
$292$	−4549.07	−0.911693
$293$	3883.83	0.774388	0.387194	−	0.921998i	$-0.373444\pi$
$293$	3883.83	0.774388	0.387194	+	0.921998i	$0.373444\pi$
$294$	1168.72	0.231841
$295$	−65.2092	−0.0128699
$296$	−503.554	−0.0988800
$297$	18974.8	3.70716
$298$	676.257	0.131458
$299$	−4672.48	−0.903734
$300$	9013.20	1.73459
$301$	−496.472	−0.0950703
$302$	50.3609	0.00959584
$303$	12699.3	2.40777
$304$	3878.99	0.731827
$305$	476.699	0.0894941
$306$	1342.09	0.250726
$307$	403.210	0.0749590	0.0374795	−	0.999297i	$-0.488067\pi$
$307$	403.210	0.0749590	0.0374795	+	0.999297i	$0.488067\pi$
$308$	−3141.08	−0.581103
$309$	−5866.05	−1.07996
$310$	−64.2364	−0.0117690
$311$	4838.71	0.882244	0.441122	−	0.897447i	$-0.354581\pi$
$311$	4838.71	0.882244	0.441122	+	0.897447i	$0.354581\pi$
$312$	−3015.93	−0.547255
$313$	−8544.28	−1.54298	−0.771488	−	0.636244i	$-0.780487\pi$
$313$	−8544.28	−1.54298	−0.771488	+	0.636244i	$0.780487\pi$
$314$	−311.970	−0.0560685
$315$	235.721	0.0421630
$316$	−2765.37	−0.492291
$317$	1773.06	0.314148	0.157074	−	0.987587i	$-0.449794\pi$
$317$	1773.06	0.314148	0.157074	+	0.987587i	$0.449794\pi$
$318$	−1257.51	−0.221753
$319$	0	0
$320$	−293.803	−0.0513253
$321$	−1672.51	−0.290812
$322$	−238.851	−0.0413374
$323$	−3590.50	−0.618516
$324$	−8667.10	−1.48613
$325$	6199.72	1.05815
$326$	222.813	0.0378541
$327$	9595.11	1.62266
$328$	−1410.67	−0.237474
$329$	2197.30	0.368210
$330$	−164.276	−0.0274034
$331$	−801.875	−0.133157	−0.0665786	−	0.997781i	$-0.521208\pi$
$331$	−801.875	−0.133157	−0.0665786	+	0.997781i	$0.521208\pi$
$332$	−5452.58	−0.901352
$333$	−4487.38	−0.738460
$334$	−200.733	−0.0328851
$335$	554.700	0.0904671
$336$	3401.16	0.552228
$337$	−8193.23	−1.32437	−0.662186	−	0.749339i	$-0.730371\pi$
$337$	−8193.23	−1.32437	−0.662186	+	0.749339i	$0.730371\pi$
$338$	−115.983	−0.0186647
$339$	1707.72	0.273601
$340$	285.161	0.0454854
$341$	15423.1	2.44930
$342$	−1566.90	−0.247743
$343$	−3981.79	−0.626811
$344$	−529.953	−0.0830615
$345$	569.966	0.0889447
$346$	1354.35	0.210435
$347$	−10914.9	−1.68860	−0.844301	−	0.535869i	$-0.819984\pi$
$347$	−10914.9	−1.68860	−0.844301	+	0.535869i	$0.819984\pi$
$348$	0	0
$349$	6697.83	1.02730	0.513649	−	0.858001i	$-0.328294\pi$
$349$	6697.83	1.02730	0.513649	+	0.858001i	$0.328294\pi$
$350$	316.922	0.0484005
$351$	−14456.2	−2.19833
$352$	−5047.54	−0.764303
$353$	−3764.83	−0.567654	−0.283827	−	0.958875i	$-0.591604\pi$
$353$	−3764.83	−0.567654	−0.283827	+	0.958875i	$0.591604\pi$
$354$	380.067	0.0570631
$355$	−248.346	−0.0371291
$356$	8753.90	1.30325
$357$	−3148.20	−0.466724
$358$	232.898	0.0343829
$359$	−6577.13	−0.966930	−0.483465	−	0.875364i	$-0.660622\pi$
$359$	−6577.13	−0.966930	−0.483465	+	0.875364i	$0.660622\pi$
$360$	251.617	0.0368372
$361$	−2667.06	−0.388841
$362$	624.354	0.0906501
$363$	27140.8	3.92430
$364$	2393.08	0.344592
$365$	381.696	0.0547366
$366$	−2778.40	−0.396802
$367$	−2274.27	−0.323477	−0.161738	−	0.986834i	$-0.551710\pi$
$367$	−2274.27	−0.323477	−0.161738	+	0.986834i	$0.551710\pi$
$368$	5624.64	0.796752
$369$	−12571.1	−1.77351
$370$	20.8967	0.00293613
$371$	2017.48	0.282325
$372$	−17082.8	−2.38091
$373$	1284.94	0.178369	0.0891844	−	0.996015i	$-0.471574\pi$
$373$	1284.94	0.178369	0.0891844	+	0.996015i	$0.471574\pi$
$374$	1500.57	0.207467
$375$	−1515.15	−0.208645
$376$	2345.48	0.321700
$377$	0	0
$378$	−738.983	−0.100553
$379$	−174.785	−0.0236890	−0.0118445	−	0.999930i	$-0.503770\pi$
$379$	−174.785	−0.0236890	−0.0118445	+	0.999930i	$0.503770\pi$
$380$	−332.928	−0.0449444
$381$	−18481.4	−2.48512
$382$	1877.22	0.251431
$383$	−5558.62	−0.741599	−0.370799	−	0.928713i	$-0.620916\pi$
$383$	−5558.62	−0.741599	−0.370799	+	0.928713i	$0.620916\pi$
$384$	7425.62	0.986816
$385$	263.557	0.0348885
$386$	−1215.79	−0.160316
$387$	−4722.64	−0.620323
$388$	−6305.14	−0.824987
$389$	2556.05	0.333154	0.166577	−	0.986028i	$-0.446729\pi$
$389$	2556.05	0.333154	0.166577	+	0.986028i	$0.446729\pi$
$390$	125.156	0.0162501
$391$	−5206.32	−0.673388
$392$	−2001.48	−0.257883
$393$	−5189.45	−0.666089
$394$	−1111.00	−0.142060
$395$	232.032	0.0295564
$396$	−29879.2	−3.79163
$397$	5927.27	0.749323	0.374662	−	0.927162i	$-0.377759\pi$
$397$	5927.27	0.749323	0.374662	+	0.927162i	$0.377759\pi$
$398$	268.566	0.0338241
$399$	3675.55	0.461173
$400$	−7463.11	−0.932889
$401$	4747.99	0.591280	0.295640	−	0.955299i	$-0.404467\pi$
$401$	4747.99	0.591280	0.295640	+	0.955299i	$0.404467\pi$
$402$	−3233.03	−0.401116
$403$	−11750.4	−1.45243
$404$	−10756.2	−1.32460
$405$	727.224	0.0892249
$406$	0	0
$407$	−5017.29	−0.611052
$408$	−3360.51	−0.407770
$409$	−5200.19	−0.628686	−0.314343	−	0.949309i	$-0.601784\pi$
$409$	−5200.19	−0.628686	−0.314343	+	0.949309i	$0.601784\pi$
$410$	58.5407	0.00705151
$411$	2315.18	0.277858
$412$	4968.48	0.594125
$413$	−609.761	−0.0726498
$414$	−2272.05	−0.269722
$415$	457.505	0.0541158
$416$	3845.55	0.453229
$417$	−2238.97	−0.262933
$418$	−1751.93	−0.204999
$419$	−6425.59	−0.749189	−0.374595	−	0.927189i	$-0.622218\pi$
$419$	−6425.59	−0.749189	−0.374595	+	0.927189i	$0.622218\pi$
$420$	−291.917	−0.0339145
$421$	−10037.6	−1.16201	−0.581003	−	0.813902i	$-0.697339\pi$
$421$	−10037.6	−1.16201	−0.581003	+	0.813902i	$0.697339\pi$
$422$	−2050.18	−0.236496
$423$	20901.6	2.40253
$424$	2153.54	0.246663
$425$	6908.05	0.788447
$426$	1447.46	0.164624
$427$	4457.53	0.505188
$428$	1416.60	0.159986
$429$	−30050.1	−3.38189
$430$	21.9922	0.00246641
$431$	16646.8	1.86044	0.930218	−	0.367006i	$-0.119617\pi$
$431$	16646.8	1.86044	0.930218	+	0.367006i	$0.119617\pi$
$432$	17402.1	1.93810
$433$	−15089.1	−1.67468	−0.837340	−	0.546682i	$-0.815891\pi$
$433$	−15089.1	−1.67468	−0.837340	+	0.546682i	$0.815891\pi$
$434$	−600.664	−0.0664350
$435$	0	0
$436$	−8126.97	−0.892686
$437$	6078.42	0.665378
$438$	−2224.69	−0.242693
$439$	−3777.24	−0.410656	−0.205328	−	0.978693i	$-0.565826\pi$
$439$	−3777.24	−0.410656	−0.205328	+	0.978693i	$0.565826\pi$
$440$	281.330	0.0304816
$441$	−17836.1	−1.92593
$442$	−1143.23	−0.123027
$443$	−7992.65	−0.857206	−0.428603	−	0.903493i	$-0.640994\pi$
$443$	−7992.65	−0.857206	−0.428603	+	0.903493i	$0.640994\pi$
$444$	5557.18	0.593991
$445$	−734.507	−0.0782449
$446$	−918.288	−0.0974937
$447$	−15089.8	−1.59670
$448$	−2747.31	−0.289728
$449$	−6433.54	−0.676209	−0.338104	−	0.941109i	$-0.609786\pi$
$449$	−6433.54	−0.676209	−0.338104	+	0.941109i	$0.609786\pi$
$450$	3014.68	0.315808
$451$	−14055.6	−1.46752
$452$	−1446.42	−0.150518
$453$	−1123.74	−0.116552
$454$	−1883.05	−0.194661
$455$	−200.795	−0.0206888
$456$	3923.43	0.402919
$457$	6975.18	0.713972	0.356986	−	0.934110i	$-0.383804\pi$
$457$	6975.18	0.713972	0.356986	+	0.934110i	$0.383804\pi$
$458$	1383.08	0.141107
$459$	−16107.9	−1.63802
$460$	−482.755	−0.0489317
$461$	14758.9	1.49109	0.745543	−	0.666458i	$-0.232190\pi$
$461$	14758.9	1.49109	0.745543	+	0.666458i	$0.232190\pi$
$462$	−1536.12	−0.154690
$463$	−18951.2	−1.90224	−0.951121	−	0.308818i	$-0.900067\pi$
$463$	−18951.2	−1.90224	−0.951121	+	0.308818i	$0.900067\pi$
$464$	0	0
$465$	1433.35	0.142946
$466$	1654.93	0.164513
$467$	−12442.4	−1.23290	−0.616449	−	0.787395i	$-0.711429\pi$
$467$	−12442.4	−1.23290	−0.616449	+	0.787395i	$0.711429\pi$
$468$	22763.9	2.24843
$469$	5186.91	0.510680
$470$	−97.3338	−0.00955249
$471$	6961.22	0.681010
$472$	−650.882	−0.0634730
$473$	−5280.33	−0.513297
$474$	−1352.38	−0.131048
$475$	−8065.21	−0.779068
$476$	2666.50	0.256762
$477$	19191.1	1.84214
$478$	579.914	0.0554909
$479$	−12947.5	−1.23504	−0.617522	−	0.786554i	$-0.711863\pi$
$479$	−12947.5	−1.23504	−0.617522	+	0.786554i	$0.711863\pi$
$480$	−469.093	−0.0446064
$481$	3822.50	0.362352
$482$	845.385	0.0798885
$483$	5329.65	0.502086
$484$	−22988.0	−2.15890
$485$	529.041	0.0495309
$486$	−990.104	−0.0924116
$487$	9844.72	0.916030	0.458015	−	0.888944i	$-0.348561\pi$
$487$	9844.72	0.916030	0.458015	+	0.888944i	$0.348561\pi$
$488$	4758.14	0.441375
$489$	−4971.77	−0.459778
$490$	83.0584	0.00765754
$491$	6809.50	0.625883	0.312941	−	0.949772i	$-0.398686\pi$
$491$	6809.50	0.625883	0.312941	+	0.949772i	$0.398686\pi$
$492$	15568.1	1.42655
$493$	0	0
$494$	1334.74	0.121564
$495$	2507.05	0.227644
$496$	14144.9	1.28049
$497$	−2322.24	−0.209591
$498$	−2666.54	−0.239940
$499$	−15953.8	−1.43124	−0.715622	−	0.698488i	$-0.753857\pi$
$499$	−15953.8	−1.43124	−0.715622	+	0.698488i	$0.753857\pi$
$500$	1283.32	0.114783
$501$	4479.11	0.399424
$502$	332.468	0.0295593
$503$	14582.7	1.29267	0.646334	−	0.763054i	$-0.276301\pi$
$503$	14582.7	1.29267	0.646334	+	0.763054i	$0.276301\pi$
$504$	2352.83	0.207943
$505$	902.511	0.0795272
$506$	−2540.35	−0.223186
$507$	2588.02	0.226702
$508$	15653.5	1.36715
$509$	20906.4	1.82055	0.910273	−	0.414008i	$-0.135871\pi$
$509$	20906.4	1.82055	0.910273	+	0.414008i	$0.135871\pi$
$510$	139.456	0.0121083
$511$	3569.17	0.308984
$512$	−7771.61	−0.670820
$513$	18806.1	1.61854
$514$	1849.09	0.158677
$515$	−416.887	−0.0356704
$516$	5848.52	0.498967
$517$	23369.8	1.98802
$518$	195.401	0.0165742
$519$	−30220.6	−2.55595
$520$	−214.336	−0.0180755
$521$	−15131.7	−1.27242	−0.636212	−	0.771515i	$-0.719500\pi$
$521$	−15131.7	−1.27242	−0.636212	+	0.771515i	$0.719500\pi$
$522$	0	0
$523$	12146.9	1.01558	0.507790	−	0.861481i	$-0.330463\pi$
$523$	12146.9	1.01558	0.507790	+	0.861481i	$0.330463\pi$
$524$	4395.41	0.366440
$525$	−7071.70	−0.587875
$526$	−1580.31	−0.130998
$527$	−13092.9	−1.08223
$528$	36173.7	2.98155
$529$	−3353.12	−0.275592
$530$	−89.3683	−0.00732436
$531$	−5800.28	−0.474032
$532$	−3113.16	−0.253708
$533$	10708.5	0.870237
$534$	4281.02	0.346925
$535$	−118.862	−0.00960532
$536$	5536.70	0.446174
$537$	−5196.83	−0.417616
$538$	1873.89	0.150166
$539$	−19942.3	−1.59365
$540$	−1493.60	−0.119027
$541$	22291.8	1.77153	0.885767	−	0.464130i	$-0.153633\pi$
$541$	22291.8	1.77153	0.885767	+	0.464130i	$0.153633\pi$
$542$	1641.38	0.130080
$543$	−13931.7	−1.10104
$544$	4284.91	0.337709
$545$	681.904	0.0535955
$546$	1170.32	0.0917307
$547$	−15439.4	−1.20684	−0.603421	−	0.797423i	$-0.706196\pi$
$547$	−15439.4	−1.20684	−0.603421	+	0.797423i	$0.706196\pi$
$548$	−1960.94	−0.152860
$549$	42401.8	3.29629
$550$	3370.68	0.261321
$551$	0	0
$552$	5689.07	0.438665
$553$	2169.69	0.166844
$554$	−918.557	−0.0704436
$555$	−466.283	−0.0356623
$556$	1896.39	0.144649
$557$	2336.99	0.177776	0.0888881	−	0.996042i	$-0.471669\pi$
$557$	2336.99	0.177776	0.0888881	+	0.996042i	$0.471669\pi$
$558$	−5713.75	−0.433481
$559$	4022.90	0.304384
$560$	241.713	0.0182397
$561$	−33483.3	−2.51991
$562$	1061.61	0.0796824
$563$	−19833.3	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$563$	−19833.3	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$564$	−25884.6	−1.93251
$565$	121.364	0.00903685
$566$	1602.71	0.119023
$567$	6800.16	0.503668
$568$	−2478.85	−0.183116
$569$	−11063.7	−0.815141	−0.407571	−	0.913174i	$-0.633624\pi$
$569$	−11063.7	−0.815141	−0.407571	+	0.913174i	$0.633624\pi$
$570$	−162.816	−0.0119642
$571$	−665.827	−0.0487986	−0.0243993	−	0.999702i	$-0.507767\pi$
$571$	−665.827	−0.0487986	−0.0243993	+	0.999702i	$0.507767\pi$
$572$	25452.1	1.86050
$573$	−41887.6	−3.05389
$574$	547.404	0.0398053
$575$	−11694.8	−0.848184
$576$	−26133.5	−1.89044
$577$	7165.21	0.516970	0.258485	−	0.966015i	$-0.416777\pi$
$577$	7165.21	0.516970	0.258485	+	0.966015i	$0.416777\pi$
$578$	761.179	0.0547766
$579$	27128.7	1.94720
$580$	0	0
$581$	4278.06	0.305480
$582$	−3083.47	−0.219612
$583$	21457.3	1.52431
$584$	3809.87	0.269955
$585$	−1910.04	−0.134992
$586$	−1608.73	−0.113406
$587$	−10375.2	−0.729525	−0.364763	−	0.931101i	$-0.618850\pi$
$587$	−10375.2	−0.729525	−0.364763	+	0.931101i	$0.618850\pi$
$588$	22088.2	1.54915
$589$	15286.0	1.06936
$590$	27.0105	0.00188476
$591$	24790.6	1.72546
$592$	−4601.46	−0.319457
$593$	18931.5	1.31100	0.655501	−	0.755194i	$-0.272458\pi$
$593$	18931.5	1.31100	0.655501	+	0.755194i	$0.272458\pi$
$594$	−7859.60	−0.542901
$595$	−223.736	−0.0154156
$596$	12780.9	0.878401
$597$	−5992.71	−0.410829
$598$	1935.40	0.132349
$599$	12244.2	0.835199	0.417600	−	0.908631i	$-0.362871\pi$
$599$	12244.2	0.835199	0.417600	+	0.908631i	$0.362871\pi$
$600$	−7548.60	−0.513617
$601$	−15596.9	−1.05859	−0.529293	−	0.848439i	$-0.677543\pi$
$601$	−15596.9	−1.05859	−0.529293	+	0.848439i	$0.677543\pi$
$602$	205.645	0.0139227
$603$	49339.9	3.33213
$604$	951.796	0.0641192
$605$	1928.84	0.129617
$606$	−5260.22	−0.352610
$607$	10155.5	0.679076	0.339538	−	0.940592i	$-0.389729\pi$
$607$	10155.5	0.679076	0.339538	+	0.940592i	$0.389729\pi$
$608$	−5002.67	−0.333692
$609$	0	0
$610$	−197.455	−0.0131061
$611$	−17804.7	−1.17889
$612$	25364.8	1.67534
$613$	−6227.67	−0.410331	−0.205166	−	0.978727i	$-0.565773\pi$
$613$	−6227.67	−0.410331	−0.205166	+	0.978727i	$0.565773\pi$
$614$	−167.015	−0.0109775
$615$	−1306.26	−0.0856479
$616$	2630.67	0.172066
$617$	−14357.5	−0.936808	−0.468404	−	0.883514i	$-0.655171\pi$
$617$	−14357.5	−0.936808	−0.468404	+	0.883514i	$0.655171\pi$
$618$	2429.80	0.158156
$619$	13220.6	0.858453	0.429227	−	0.903197i	$-0.358786\pi$
$619$	13220.6	0.858453	0.429227	+	0.903197i	$0.358786\pi$
$620$	−1214.03	−0.0786400
$621$	27269.3	1.76213
$622$	−2004.26	−0.129202
$623$	−6868.26	−0.441687
$624$	−27559.5	−1.76805
$625$	15463.4	0.989657
$626$	3539.16	0.225963
$627$	39092.1	2.48993
$628$	−5896.08	−0.374649
$629$	4259.23	0.269995
$630$	−97.6387	−0.00617463
$631$	1828.97	0.115389	0.0576943	−	0.998334i	$-0.481625\pi$
$631$	1828.97	0.115389	0.0576943	+	0.998334i	$0.481625\pi$
$632$	2316.01	0.145769
$633$	45747.1	2.87249
$634$	−734.426	−0.0460059
$635$	−1313.43	−0.0820817
$636$	−23766.3	−1.48175
$637$	15193.4	0.945028
$638$	0	0
$639$	−22090.0	−1.36756
$640$	527.723	0.0325939
$641$	22644.1	1.39530	0.697651	−	0.716437i	$-0.254228\pi$
$641$	22644.1	1.39530	0.697651	+	0.716437i	$0.254228\pi$
$642$	692.778	0.0425884
$643$	22728.4	1.39397	0.696983	−	0.717088i	$-0.254525\pi$
$643$	22728.4	1.39397	0.696983	+	0.717088i	$0.254525\pi$
$644$	−4514.16	−0.276216
$645$	−490.728	−0.0299572
$646$	1487.23	0.0905796
$647$	−5844.85	−0.355154	−0.177577	−	0.984107i	$-0.556826\pi$
$647$	−5844.85	−0.355154	−0.177577	+	0.984107i	$0.556826\pi$
$648$	7258.74	0.440047
$649$	−6485.23	−0.392246
$650$	−2568.01	−0.154962
$651$	13403.0	0.806922
$652$	4211.04	0.252941
$653$	−15174.1	−0.909355	−0.454677	−	0.890656i	$-0.650245\pi$
$653$	−15174.1	−0.909355	−0.454677	+	0.890656i	$0.650245\pi$
$654$	−3974.43	−0.237634
$655$	−368.803	−0.0220005
$656$	−12890.7	−0.767221
$657$	33951.4	2.01609
$658$	−910.152	−0.0539231
$659$	27857.0	1.64667	0.823333	−	0.567558i	$-0.192112\pi$
$659$	27857.0	1.64667	0.823333	+	0.567558i	$0.192112\pi$
$660$	−3104.74	−0.183109
$661$	−4966.64	−0.292254	−0.146127	−	0.989266i	$-0.546681\pi$
$661$	−4966.64	−0.292254	−0.146127	+	0.989266i	$0.546681\pi$
$662$	332.148	0.0195004
$663$	25509.8	1.49430
$664$	4566.56	0.266893
$665$	261.214	0.0152322
$666$	1858.74	0.108145
$667$	0	0
$668$	−3793.76	−0.219738
$669$	20490.4	1.18416
$670$	−229.764	−0.0132486
$671$	47409.0	2.72758
$672$	−4386.41	−0.251800
$673$	−2338.02	−0.133914	−0.0669569	−	0.997756i	$-0.521329\pi$
$673$	−2338.02	−0.133914	−0.0669569	+	0.997756i	$0.521329\pi$
$674$	3393.75	0.193950
$675$	−36182.6	−2.06321
$676$	−2192.03	−0.124717
$677$	−6342.30	−0.360051	−0.180025	−	0.983662i	$-0.557618\pi$
$677$	−6342.30	−0.360051	−0.180025	+	0.983662i	$0.557618\pi$
$678$	−707.361	−0.0400679
$679$	4946.97	0.279598
$680$	−238.824	−0.0134684
$681$	42017.9	2.36436
$682$	−6388.48	−0.358691
$683$	−30366.6	−1.70124	−0.850620	−	0.525780i	$-0.823773\pi$
$683$	−30366.6	−1.70124	−0.850620	+	0.525780i	$0.823773\pi$
$684$	−29613.6	−1.65542
$685$	164.535	0.00917746
$686$	1649.31	0.0917944
$687$	−30861.6	−1.71389
$688$	−4842.69	−0.268352
$689$	−16347.6	−0.903910
$690$	−236.087	−0.0130256
$691$	−11826.5	−0.651089	−0.325545	−	0.945527i	$-0.605548\pi$
$691$	−11826.5	−0.651089	−0.325545	+	0.945527i	$0.605548\pi$
$692$	25596.6	1.40612
$693$	23443.0	1.28503
$694$	4521.12	0.247290
$695$	−159.119	−0.00868450
$696$	0	0
$697$	11932.0	0.648429
$698$	−2774.33	−0.150444
$699$	−36927.6	−1.99818
$700$	5989.66	0.323411
$701$	−2776.33	−0.149587	−0.0747936	−	0.997199i	$-0.523830\pi$
$701$	−2776.33	−0.149587	−0.0747936	+	0.997199i	$0.523830\pi$
$702$	5987.96	0.321939
$703$	−4972.69	−0.266783
$704$	−29219.5	−1.56428
$705$	2171.88	0.116025
$706$	1559.45	0.0831310
$707$	8439.23	0.448925
$708$	7183.08	0.381295
$709$	−15962.7	−0.845543	−0.422772	−	0.906236i	$-0.638943\pi$
$709$	−15962.7	−0.845543	−0.422772	+	0.906236i	$0.638943\pi$
$710$	102.868	0.00543742
$711$	20638.9	1.08864
$712$	−7331.44	−0.385895
$713$	22165.2	1.16422
$714$	1304.03	0.0683502
$715$	−2135.59	−0.111702
$716$	4401.66	0.229746
$717$	−12940.0	−0.673994
$718$	2724.34	0.141604
$719$	−20832.9	−1.08058	−0.540289	−	0.841480i	$-0.681685\pi$
$719$	−20832.9	−1.08058	−0.540289	+	0.841480i	$0.681685\pi$
$720$	2299.27	0.119012
$721$	−3898.24	−0.201357
$722$	1104.73	0.0569445
$723$	−18863.7	−0.970329
$724$	11800.0	0.605722
$725$	0	0
$726$	−11242.1	−0.574700
$727$	−4452.04	−0.227121	−0.113561	−	0.993531i	$-0.536226\pi$
$727$	−4452.04	−0.227121	−0.113561	+	0.993531i	$0.536226\pi$
$728$	−2004.22	−0.102035
$729$	−7799.67	−0.396264
$730$	−158.104	−0.00801599
$731$	4482.52	0.226802
$732$	−52510.4	−2.65142
$733$	12107.2	0.610082	0.305041	−	0.952339i	$-0.401330\pi$
$733$	12107.2	0.610082	0.305041	+	0.952339i	$0.401330\pi$
$734$	942.034	0.0473721
$735$	−1853.34	−0.0930088
$736$	−7254.00	−0.363296
$737$	55166.4	2.75723
$738$	5207.12	0.259725
$739$	4506.27	0.224311	0.112156	−	0.993691i	$-0.464225\pi$
$739$	4506.27	0.224311	0.112156	+	0.993691i	$0.464225\pi$
$740$	394.937	0.0196191
$741$	−29782.9	−1.47652
$742$	−835.669	−0.0413455
$743$	−1177.19	−0.0581253	−0.0290626	−	0.999578i	$-0.509252\pi$
$743$	−1177.19	−0.0581253	−0.0290626	+	0.999578i	$0.509252\pi$
$744$	14306.9	0.704996
$745$	−1072.40	−0.0527378
$746$	−532.239	−0.0261215
$747$	40694.6	1.99322
$748$	28360.1	1.38629
$749$	−1111.46	−0.0542214
$750$	627.595	0.0305554
$751$	−27631.6	−1.34260	−0.671300	−	0.741186i	$-0.734264\pi$
$751$	−27631.6	−1.34260	−0.671300	+	0.741186i	$0.734264\pi$
$752$	21432.9	1.03933
$753$	−7418.59	−0.359028
$754$	0	0
$755$	−79.8616	−0.00384962
$756$	−13966.4	−0.671896
$757$	−11336.6	−0.544300	−0.272150	−	0.962255i	$-0.587735\pi$
$757$	−11336.6	−0.544300	−0.272150	+	0.962255i	$0.587735\pi$
$758$	72.3984	0.00346917
$759$	56684.6	2.71083
$760$	278.829	0.0133082
$761$	4356.58	0.207524	0.103762	−	0.994602i	$-0.466912\pi$
$761$	4356.58	0.207524	0.103762	+	0.994602i	$0.466912\pi$
$762$	7655.23	0.363937
$763$	6376.37	0.302543
$764$	35478.4	1.68006
$765$	−2128.26	−0.100585
$766$	2302.46	0.108605
$767$	4940.87	0.232601
$768$	29997.2	1.40942
$769$	21718.1	1.01843	0.509217	−	0.860638i	$-0.329935\pi$
$769$	21718.1	1.01843	0.509217	+	0.860638i	$0.329935\pi$
$770$	−109.169	−0.00510931
$771$	−41260.0	−1.92729
$772$	−22977.8	−1.07123
$773$	−22688.4	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$773$	−22688.4	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$774$	1956.18	0.0908442
$775$	−29410.1	−1.36315
$776$	5280.58	0.244281
$777$	−4360.13	−0.201311
$778$	−1058.75	−0.0487893
$779$	−13930.7	−0.640716
$780$	2365.39	0.108583
$781$	−24698.6	−1.13161
$782$	2156.53	0.0986155
$783$	0	0
$784$	−18289.5	−0.833158
$785$	494.718	0.0224933
$786$	2149.54	0.0975465
$787$	−32890.9	−1.48975	−0.744875	−	0.667204i	$-0.767491\pi$
$787$	−32890.9	−1.48975	−0.744875	+	0.667204i	$0.767491\pi$
$788$	−20997.4	−0.949239
$789$	35262.6	1.59111
$790$	−96.1107	−0.00432844
$791$	1134.85	0.0510123
$792$	25024.0	1.12271
$793$	−36119.3	−1.61744
$794$	−2455.16	−0.109736
$795$	1994.14	0.0889620
$796$	5075.76	0.226012
$797$	−30404.1	−1.35128	−0.675638	−	0.737233i	$-0.736132\pi$
$797$	−30404.1	−1.35128	−0.675638	+	0.737233i	$0.736132\pi$
$798$	−1522.46	−0.0675372
$799$	−19838.9	−0.878410
$800$	9625.04	0.425371
$801$	−65333.5	−2.88196
$802$	−1966.68	−0.0865909
$803$	37960.7	1.66825
$804$	−61102.6	−2.68025
$805$	378.767	0.0165836
$806$	4867.16	0.212703
$807$	−41813.5	−1.82392
$808$	9008.36	0.392219
$809$	37889.3	1.64662	0.823311	−	0.567591i	$-0.192124\pi$
$809$	37889.3	1.64662	0.823311	+	0.567591i	$0.192124\pi$
$810$	−301.226	−0.0130667
$811$	−8123.23	−0.351720	−0.175860	−	0.984415i	$-0.556271\pi$
$811$	−8123.23	−0.351720	−0.175860	+	0.984415i	$0.556271\pi$
$812$	0	0
$813$	−36625.3	−1.57996
$814$	2078.23	0.0894864
$815$	−353.333	−0.0151862
$816$	−30708.2	−1.31741
$817$	−5233.39	−0.224104
$818$	2153.99	0.0920690
$819$	−17860.4	−0.762020
$820$	1106.39	0.0471180
$821$	13226.8	0.562264	0.281132	−	0.959669i	$-0.409290\pi$
$821$	13226.8	0.562264	0.281132	+	0.959669i	$0.409290\pi$
$822$	−958.981	−0.0406914
$823$	29575.2	1.25265	0.626323	−	0.779563i	$-0.284559\pi$
$823$	29575.2	1.25265	0.626323	+	0.779563i	$0.284559\pi$
$824$	−4161.13	−0.175922
$825$	−75212.5	−3.17401
$826$	252.571	0.0106393
$827$	36661.2	1.54152	0.770758	−	0.637128i	$-0.219878\pi$
$827$	36661.2	1.54152	0.770758	+	0.637128i	$0.219878\pi$
$828$	−42940.5	−1.80228
$829$	−11277.0	−0.472455	−0.236228	−	0.971698i	$-0.575911\pi$
$829$	−11277.0	−0.472455	−0.236228	+	0.971698i	$0.575911\pi$
$830$	−189.505	−0.00792507
$831$	20496.4	0.855611
$832$	22261.3	0.927612
$833$	16929.2	0.704158
$834$	927.413	0.0385056
$835$	318.320	0.0131927
$836$	−33110.6	−1.36980
$837$	68577.0	2.83198
$838$	2661.56	0.109716
$839$	18965.7	0.780417	0.390208	−	0.920727i	$-0.372403\pi$
$839$	18965.7	0.780417	0.390208	+	0.920727i	$0.372403\pi$
$840$	244.482	0.0100422
$841$	0	0
$842$	4157.72	0.170172
$843$	−23688.5	−0.967825
$844$	−38747.3	−1.58026
$845$	183.925	0.00748781
$846$	−8657.72	−0.351842
$847$	18036.2	0.731679
$848$	19678.9	0.796908
$849$	−35762.5	−1.44566
$850$	−2861.41	−0.115465
$851$	−7210.54	−0.290451
$852$	27356.3	1.10002
$853$	−8067.23	−0.323818	−0.161909	−	0.986806i	$-0.551765\pi$
$853$	−8067.23	−0.323818	−0.161909	+	0.986806i	$0.551765\pi$
$854$	−1846.37	−0.0739830
$855$	2484.77	0.0993886
$856$	−1186.41	−0.0473724
$857$	15281.7	0.609118	0.304559	−	0.952493i	$-0.401491\pi$
$857$	15281.7	0.609118	0.304559	+	0.952493i	$0.401491\pi$
$858$	12447.1	0.495266
$859$	36789.1	1.46127	0.730634	−	0.682770i	$-0.239225\pi$
$859$	36789.1	1.46127	0.730634	+	0.682770i	$0.239225\pi$
$860$	415.641	0.0164805
$861$	−12214.6	−0.483476
$862$	−6895.33	−0.272455
$863$	40907.8	1.61358	0.806788	−	0.590840i	$-0.201204\pi$
$863$	40907.8	1.61358	0.806788	+	0.590840i	$0.201204\pi$
$864$	−22443.2	−0.883719
$865$	−2147.71	−0.0844213
$866$	6250.12	0.245251
$867$	−16984.7	−0.665319
$868$	−11352.2	−0.443917
$869$	23076.2	0.900812
$870$	0	0
$871$	−42029.4	−1.63503
$872$	6806.38	0.264327
$873$	47057.5	1.82435
$874$	−2517.76	−0.0974424
$875$	−1006.88	−0.0389015
$876$	−42045.4	−1.62167
$877$	2391.60	0.0920852	0.0460426	−	0.998939i	$-0.485339\pi$
$877$	2391.60	0.0920852	0.0460426	+	0.998939i	$0.485339\pi$
$878$	1564.59	0.0601392
$879$	35896.8	1.37744
$880$	2570.79	0.0984786
$881$	−5487.72	−0.209859	−0.104930	−	0.994480i	$-0.533462\pi$
$881$	−5487.72	−0.209859	−0.104930	+	0.994480i	$0.533462\pi$
$882$	7387.94	0.282046
$883$	170.008	0.00647931	0.00323966	−	0.999995i	$-0.498969\pi$
$883$	170.008	0.00647931	0.00323966	+	0.999995i	$0.498969\pi$
$884$	−21606.6	−0.822067
$885$	−602.705	−0.0228923
$886$	3310.66	0.125535
$887$	−25867.3	−0.979188	−0.489594	−	0.871950i	$-0.662855\pi$
$887$	−25867.3	−0.979188	−0.489594	+	0.871950i	$0.662855\pi$
$888$	−4654.17	−0.175883
$889$	−12281.7	−0.463345
$890$	304.243	0.0114587
$891$	72324.4	2.71937
$892$	−17355.2	−0.651451
$893$	23162.1	0.867961
$894$	6250.40	0.233831
$895$	−369.327	−0.0137936
$896$	4934.65	0.183990
$897$	−43186.0	−1.60751
$898$	2664.86	0.0990285
$899$	0	0
$900$	56976.0	2.11022
$901$	−18215.4	−0.673520
$902$	5822.03	0.214914
$903$	−4588.71	−0.169106
$904$	1211.39	0.0445687
$905$	−990.093	−0.0363666
$906$	465.468	0.0170686
$907$	−11411.5	−0.417765	−0.208882	−	0.977941i	$-0.566983\pi$
$907$	−11411.5	−0.417765	−0.208882	+	0.977941i	$0.566983\pi$
$908$	−35588.7	−1.30072
$909$	80277.3	2.92919
$910$	83.1719	0.00302980
$911$	38718.5	1.40813	0.704063	−	0.710138i	$-0.251367\pi$
$911$	38718.5	1.40813	0.704063	+	0.710138i	$0.251367\pi$
$912$	35852.1	1.30174
$913$	45500.1	1.64933
$914$	−2889.21	−0.104559
$915$	4405.96	0.159187
$916$	26139.5	0.942875
$917$	−3448.62	−0.124191
$918$	6672.10	0.239882
$919$	−48465.3	−1.73963	−0.869817	−	0.493374i	$-0.835763\pi$
$919$	−48465.3	−1.73963	−0.869817	+	0.493374i	$0.835763\pi$
$920$	404.310	0.0144888
$921$	3726.72	0.133333
$922$	−6113.34	−0.218364
$923$	18817.0	0.671040
$924$	−29031.9	−1.03364
$925$	9567.37	0.340079
$926$	7849.85	0.278577
$927$	−37081.6	−1.31383
$928$	0	0
$929$	10560.1	0.372946	0.186473	−	0.982460i	$-0.440294\pi$
$929$	10560.1	0.372946	0.186473	+	0.982460i	$0.440294\pi$
$930$	−593.714	−0.0209340
$931$	−19765.0	−0.695782
$932$	31277.3	1.09927
$933$	44722.4	1.56929
$934$	5153.79	0.180554
$935$	−2379.59	−0.0832309
$936$	−19064.9	−0.665765
$937$	−23025.0	−0.802769	−0.401384	−	0.915910i	$-0.631471\pi$
$937$	−23025.0	−0.802769	−0.401384	+	0.915910i	$0.631471\pi$
$938$	−2148.49	−0.0747874
$939$	−78971.7	−2.74456
$940$	−1839.56	−0.0638296
$941$	40778.1	1.41268	0.706338	−	0.707874i	$-0.250346\pi$
$941$	40778.1	1.41268	0.706338	+	0.707874i	$0.250346\pi$
$942$	−2883.43	−0.0997317
$943$	−20199.8	−0.697558
$944$	−5947.74	−0.205066
$945$	1171.87	0.0403396
$946$	2187.18	0.0751707
$947$	−36129.8	−1.23977	−0.619884	−	0.784693i	$-0.712821\pi$
$947$	−36129.8	−1.23977	−0.619884	+	0.784693i	$0.712821\pi$
$948$	−25559.3	−0.875662
$949$	−28920.9	−0.989265
$950$	3340.72	0.114092
$951$	16387.8	0.558790
$952$	−2233.21	−0.0760280
$953$	20831.4	0.708075	0.354037	−	0.935231i	$-0.384809\pi$
$953$	20831.4	0.708075	0.354037	+	0.935231i	$0.384809\pi$
$954$	−7949.21	−0.269775
$955$	−2976.86	−0.100868
$956$	10960.1	0.370789
$957$	0	0
$958$	5363.02	0.180868
$959$	1538.54	0.0518061
$960$	−2715.52	−0.0912948
$961$	25950.1	0.871071
$962$	−1583.33	−0.0530652
$963$	−10572.6	−0.353788
$964$	15977.4	0.533813
$965$	1927.98	0.0643149
$966$	−2207.61	−0.0735288
$967$	−49242.5	−1.63757	−0.818785	−	0.574100i	$-0.805352\pi$
$967$	−49242.5	−1.63757	−0.818785	+	0.574100i	$0.805352\pi$
$968$	19252.5	0.639256
$969$	−33185.7	−1.10018
$970$	−219.136	−0.00725363
$971$	2352.05	0.0777351	0.0388675	−	0.999244i	$-0.487625\pi$
$971$	2352.05	0.0777351	0.0388675	+	0.999244i	$0.487625\pi$
$972$	−18712.5	−0.617493
$973$	−1487.89	−0.0490233
$974$	−4077.82	−0.134150
$975$	57301.8	1.88218
$976$	43479.7	1.42598
$977$	18768.3	0.614588	0.307294	−	0.951615i	$-0.400577\pi$
$977$	18768.3	0.614588	0.307294	+	0.951615i	$0.400577\pi$
$978$	2059.38	0.0673329
$979$	−73048.7	−2.38473
$980$	1569.76	0.0511675
$981$	60654.5	1.97406
$982$	−2820.59	−0.0916584
$983$	49014.5	1.59036	0.795179	−	0.606375i	$-0.207377\pi$
$983$	49014.5	1.59036	0.795179	+	0.606375i	$0.207377\pi$
$984$	−13038.4	−0.422406
$985$	1761.81	0.0569908
$986$	0	0
$987$	20308.9	0.654953
$988$	25225.8	0.812288
$989$	−7588.55	−0.243986
$990$	−1038.46	−0.0333377
$991$	−48860.6	−1.56620	−0.783102	−	0.621893i	$-0.786364\pi$
$991$	−48860.6	−1.56620	−0.783102	+	0.621893i	$0.786364\pi$
$992$	−18242.4	−0.583868
$993$	−7411.44	−0.236853
$994$	961.903	0.0306939
$995$	−425.888	−0.0135694
$996$	−50396.2	−1.60328
$997$	−2934.57	−0.0932184	−0.0466092	−	0.998913i	$-0.514842\pi$
$997$	−2934.57	−0.0932184	−0.0466092	+	0.998913i	$0.514842\pi$
$998$	6608.29	0.209601
$999$	−22308.7	−0.706524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.4.a.a.1.1		2
29.28	even	2		29.4.a.a.1.2	✓	2
87.86	odd	2		261.4.a.b.1.1		2
116.115	odd	2		464.4.a.f.1.2		2
145.144	even	2		725.4.a.b.1.1		2
203.202	odd	2		1421.4.a.c.1.2		2
232.115	odd	2		1856.4.a.h.1.1		2
232.173	even	2		1856.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.a.1.2	✓	2	29.28	even	2
261.4.a.b.1.1		2	87.86	odd	2
464.4.a.f.1.2		2	116.115	odd	2
725.4.a.b.1.1		2	145.144	even	2
841.4.a.a.1.1		2	1.1	even	1	trivial
1421.4.a.c.1.2		2	203.202	odd	2
1856.4.a.h.1.1		2	232.115	odd	2
1856.4.a.n.1.2		2	232.173	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	841.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} +0.656854 q^{5} -3.82843 q^{6} +6.14214 q^{7} +6.55635 q^{8} +58.4264 q^{9} +O(q^{10})\)	\(q-0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} +0.656854 q^{5} -3.82843 q^{6} +6.14214 q^{7} +6.55635 q^{8} +58.4264 q^{9} -0.272078 q^{10} +65.3259 q^{11} -72.3553 q^{12} -49.7696 q^{13} -2.54416 q^{14} +6.07107 q^{15} +59.9117 q^{16} -55.4558 q^{17} -24.2010 q^{18} +64.7452 q^{19} -5.14214 q^{20} +56.7696 q^{21} -27.0589 q^{22} +93.8823 q^{23} +60.5980 q^{24} -124.569 q^{25} +20.6152 q^{26} +290.463 q^{27} -48.0833 q^{28} -2.51472 q^{30} +236.095 q^{31} -77.2670 q^{32} +603.784 q^{33} +22.9706 q^{34} +4.03449 q^{35} -457.387 q^{36} -76.8040 q^{37} -26.8183 q^{38} -460.002 q^{39} +4.30657 q^{40} -215.161 q^{41} -23.5147 q^{42} -80.8305 q^{43} -511.399 q^{44} +38.3776 q^{45} -38.8873 q^{46} +357.742 q^{47} +553.742 q^{48} -305.274 q^{49} +51.5980 q^{50} -512.558 q^{51} +389.617 q^{52} +328.466 q^{53} -120.314 q^{54} +42.9096 q^{55} +40.2700 q^{56} +598.416 q^{57} -99.2750 q^{59} -47.5269 q^{60} +725.730 q^{61} -97.7939 q^{62} +358.863 q^{63} -447.288 q^{64} -32.6913 q^{65} -250.095 q^{66} +844.479 q^{67} +434.132 q^{68} +867.720 q^{69} -1.67114 q^{70} -378.083 q^{71} +383.064 q^{72} +581.097 q^{73} +31.8133 q^{74} -1151.34 q^{75} -506.853 q^{76} +401.241 q^{77} +190.539 q^{78} +353.247 q^{79} +39.3532 q^{80} +1107.13 q^{81} +89.1228 q^{82} +696.510 q^{83} -444.416 q^{84} -36.4264 q^{85} +33.4811 q^{86} +428.299 q^{88} -1118.22 q^{89} -15.8965 q^{90} -305.691 q^{91} -734.950 q^{92} +2182.15 q^{93} -148.182 q^{94} +42.5281 q^{95} -714.151 q^{96} +805.415 q^{97} +126.449 q^{98} +3816.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{7} - 18 q^{8} + 32 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 10 * q^3 - 10 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^7 - 18 * q^8 + 32 * q^9	\( 2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{7} - 18 q^{8} + 32 q^{9} - 26 q^{10} + 26 q^{11} - 74 q^{12} - 26 q^{13} - 56 q^{14} - 2 q^{15} + 18 q^{16} - 60 q^{17} - 88 q^{18} + 220 q^{19} + 18 q^{20} + 40 q^{21} - 122 q^{22} + 52 q^{23} + 42 q^{24} - 136 q^{25} + 78 q^{26} + 250 q^{27} - 22 q^{30} + 294 q^{31} + 18 q^{32} + 574 q^{33} + 12 q^{34} + 240 q^{35} - 400 q^{36} - 312 q^{37} + 348 q^{38} - 442 q^{39} + 266 q^{40} - 40 q^{41} - 64 q^{42} + 322 q^{43} - 426 q^{44} + 320 q^{45} - 140 q^{46} + 130 q^{47} + 522 q^{48} - 158 q^{49} + 24 q^{50} - 516 q^{51} + 338 q^{52} + 1002 q^{53} - 218 q^{54} + 462 q^{55} + 584 q^{56} + 716 q^{57} - 900 q^{59} - 30 q^{60} + 948 q^{61} + 42 q^{62} + 944 q^{63} + 118 q^{64} - 286 q^{65} - 322 q^{66} + 320 q^{67} + 444 q^{68} + 836 q^{69} + 568 q^{70} - 660 q^{71} + 1032 q^{72} - 648 q^{73} - 536 q^{74} - 1160 q^{75} - 844 q^{76} + 1272 q^{77} + 234 q^{78} - 258 q^{79} + 486 q^{80} + 1790 q^{81} + 512 q^{82} + 1212 q^{83} - 408 q^{84} + 12 q^{85} + 1006 q^{86} + 1394 q^{88} - 760 q^{89} + 664 q^{90} - 832 q^{91} - 644 q^{92} + 2226 q^{93} - 698 q^{94} - 1612 q^{95} - 642 q^{96} - 24 q^{97} + 482 q^{98} + 4856 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 10 * q^3 - 10 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^7 - 18 * q^8 + 32 * q^9 - 26 * q^10 + 26 * q^11 - 74 * q^12 - 26 * q^13 - 56 * q^14 - 2 * q^15 + 18 * q^16 - 60 * q^17 - 88 * q^18 + 220 * q^19 + 18 * q^20 + 40 * q^21 - 122 * q^22 + 52 * q^23 + 42 * q^24 - 136 * q^25 + 78 * q^26 + 250 * q^27 - 22 * q^30 + 294 * q^31 + 18 * q^32 + 574 * q^33 + 12 * q^34 + 240 * q^35 - 400 * q^36 - 312 * q^37 + 348 * q^38 - 442 * q^39 + 266 * q^40 - 40 * q^41 - 64 * q^42 + 322 * q^43 - 426 * q^44 + 320 * q^45 - 140 * q^46 + 130 * q^47 + 522 * q^48 - 158 * q^49 + 24 * q^50 - 516 * q^51 + 338 * q^52 + 1002 * q^53 - 218 * q^54 + 462 * q^55 + 584 * q^56 + 716 * q^57 - 900 * q^59 - 30 * q^60 + 948 * q^61 + 42 * q^62 + 944 * q^63 + 118 * q^64 - 286 * q^65 - 322 * q^66 + 320 * q^67 + 444 * q^68 + 836 * q^69 + 568 * q^70 - 660 * q^71 + 1032 * q^72 - 648 * q^73 - 536 * q^74 - 1160 * q^75 - 844 * q^76 + 1272 * q^77 + 234 * q^78 - 258 * q^79 + 486 * q^80 + 1790 * q^81 + 512 * q^82 + 1212 * q^83 - 408 * q^84 + 12 * q^85 + 1006 * q^86 + 1394 * q^88 - 760 * q^89 + 664 * q^90 - 832 * q^91 - 644 * q^92 + 2226 * q^93 - 698 * q^94 - 1612 * q^95 - 642 * q^96 - 24 * q^97 + 482 * q^98 + 4856 * q^99

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